Properties

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

$q$-expansion

\(f(q)\) \(=\) \(q-4.55366 q^{2} -4.29077 q^{3} +12.7358 q^{4} -2.70051 q^{5} +19.5387 q^{6} +9.48123 q^{7} -21.5655 q^{8} -8.58929 q^{9} +O(q^{10})\) \(q-4.55366 q^{2} -4.29077 q^{3} +12.7358 q^{4} -2.70051 q^{5} +19.5387 q^{6} +9.48123 q^{7} -21.5655 q^{8} -8.58929 q^{9} +12.2972 q^{10} -16.2615 q^{11} -54.6466 q^{12} -58.6297 q^{13} -43.1743 q^{14} +11.5873 q^{15} -3.68497 q^{16} +67.1818 q^{17} +39.1127 q^{18} +41.5179 q^{19} -34.3933 q^{20} -40.6818 q^{21} +74.0492 q^{22} -85.8598 q^{23} +92.5324 q^{24} -117.707 q^{25} +266.980 q^{26} +152.705 q^{27} +120.752 q^{28} +162.947 q^{29} -52.7645 q^{30} +270.662 q^{31} +189.304 q^{32} +69.7742 q^{33} -305.923 q^{34} -25.6041 q^{35} -109.392 q^{36} -226.823 q^{37} -189.058 q^{38} +251.566 q^{39} +58.2377 q^{40} -360.349 q^{41} +185.251 q^{42} -207.103 q^{44} +23.1954 q^{45} +390.977 q^{46} +437.762 q^{47} +15.8114 q^{48} -253.106 q^{49} +535.999 q^{50} -288.262 q^{51} -746.698 q^{52} -377.821 q^{53} -695.369 q^{54} +43.9142 q^{55} -204.467 q^{56} -178.144 q^{57} -742.008 q^{58} -279.572 q^{59} +147.574 q^{60} +30.6533 q^{61} -1232.50 q^{62} -81.4370 q^{63} -832.546 q^{64} +158.330 q^{65} -317.728 q^{66} +904.150 q^{67} +855.617 q^{68} +368.405 q^{69} +116.593 q^{70} -121.791 q^{71} +185.232 q^{72} -372.695 q^{73} +1032.87 q^{74} +505.055 q^{75} +528.765 q^{76} -154.179 q^{77} -1145.55 q^{78} +730.616 q^{79} +9.95129 q^{80} -423.313 q^{81} +1640.91 q^{82} -73.2890 q^{83} -518.117 q^{84} -181.425 q^{85} -699.170 q^{87} +350.686 q^{88} +1156.75 q^{89} -105.624 q^{90} -555.881 q^{91} -1093.50 q^{92} -1161.35 q^{93} -1993.42 q^{94} -112.119 q^{95} -812.259 q^{96} -446.639 q^{97} +1152.56 q^{98} +139.674 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.55366 −1.60996 −0.804982 0.593300i \(-0.797825\pi\)
−0.804982 + 0.593300i \(0.797825\pi\)
\(3\) −4.29077 −0.825759 −0.412880 0.910786i \(-0.635477\pi\)
−0.412880 + 0.910786i \(0.635477\pi\)
\(4\) 12.7358 1.59198
\(5\) −2.70051 −0.241541 −0.120770 0.992680i \(-0.538536\pi\)
−0.120770 + 0.992680i \(0.538536\pi\)
\(6\) 19.5387 1.32944
\(7\) 9.48123 0.511938 0.255969 0.966685i \(-0.417605\pi\)
0.255969 + 0.966685i \(0.417605\pi\)
\(8\) −21.5655 −0.953067
\(9\) −8.58929 −0.318122
\(10\) 12.2972 0.388872
\(11\) −16.2615 −0.445729 −0.222864 0.974849i \(-0.571541\pi\)
−0.222864 + 0.974849i \(0.571541\pi\)
\(12\) −54.6466 −1.31459
\(13\) −58.6297 −1.25084 −0.625421 0.780288i \(-0.715073\pi\)
−0.625421 + 0.780288i \(0.715073\pi\)
\(14\) −43.1743 −0.824202
\(15\) 11.5873 0.199455
\(16\) −3.68497 −0.0575776
\(17\) 67.1818 0.958469 0.479235 0.877687i \(-0.340914\pi\)
0.479235 + 0.877687i \(0.340914\pi\)
\(18\) 39.1127 0.512164
\(19\) 41.5179 0.501308 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(20\) −34.3933 −0.384528
\(21\) −40.6818 −0.422738
\(22\) 74.0492 0.717607
\(23\) −85.8598 −0.778392 −0.389196 0.921155i \(-0.627247\pi\)
−0.389196 + 0.921155i \(0.627247\pi\)
\(24\) 92.5324 0.787004
\(25\) −117.707 −0.941658
\(26\) 266.980 2.01381
\(27\) 152.705 1.08845
\(28\) 120.752 0.814996
\(29\) 162.947 1.04340 0.521700 0.853129i \(-0.325298\pi\)
0.521700 + 0.853129i \(0.325298\pi\)
\(30\) −52.7645 −0.321114
\(31\) 270.662 1.56814 0.784069 0.620673i \(-0.213141\pi\)
0.784069 + 0.620673i \(0.213141\pi\)
\(32\) 189.304 1.04577
\(33\) 69.7742 0.368065
\(34\) −305.923 −1.54310
\(35\) −25.6041 −0.123654
\(36\) −109.392 −0.506444
\(37\) −226.823 −1.00782 −0.503911 0.863755i \(-0.668106\pi\)
−0.503911 + 0.863755i \(0.668106\pi\)
\(38\) −189.058 −0.807088
\(39\) 251.566 1.03289
\(40\) 58.2377 0.230205
\(41\) −360.349 −1.37261 −0.686305 0.727314i \(-0.740768\pi\)
−0.686305 + 0.727314i \(0.740768\pi\)
\(42\) 185.251 0.680592
\(43\) 0 0
\(44\) −207.103 −0.709592
\(45\) 23.1954 0.0768394
\(46\) 390.977 1.25318
\(47\) 437.762 1.35860 0.679299 0.733862i \(-0.262284\pi\)
0.679299 + 0.733862i \(0.262284\pi\)
\(48\) 15.8114 0.0475453
\(49\) −253.106 −0.737919
\(50\) 535.999 1.51603
\(51\) −288.262 −0.791465
\(52\) −746.698 −1.99132
\(53\) −377.821 −0.979203 −0.489601 0.871946i \(-0.662858\pi\)
−0.489601 + 0.871946i \(0.662858\pi\)
\(54\) −695.369 −1.75237
\(55\) 43.9142 0.107662
\(56\) −204.467 −0.487912
\(57\) −178.144 −0.413960
\(58\) −742.008 −1.67983
\(59\) −279.572 −0.616901 −0.308450 0.951240i \(-0.599810\pi\)
−0.308450 + 0.951240i \(0.599810\pi\)
\(60\) 147.574 0.317528
\(61\) 30.6533 0.0643403 0.0321701 0.999482i \(-0.489758\pi\)
0.0321701 + 0.999482i \(0.489758\pi\)
\(62\) −1232.50 −2.52465
\(63\) −81.4370 −0.162859
\(64\) −832.546 −1.62607
\(65\) 158.330 0.302129
\(66\) −317.728 −0.592570
\(67\) 904.150 1.64865 0.824325 0.566117i \(-0.191555\pi\)
0.824325 + 0.566117i \(0.191555\pi\)
\(68\) 855.617 1.52587
\(69\) 368.405 0.642764
\(70\) 116.593 0.199078
\(71\) −121.791 −0.203577 −0.101788 0.994806i \(-0.532456\pi\)
−0.101788 + 0.994806i \(0.532456\pi\)
\(72\) 185.232 0.303191
\(73\) −372.695 −0.597543 −0.298771 0.954325i \(-0.596577\pi\)
−0.298771 + 0.954325i \(0.596577\pi\)
\(74\) 1032.87 1.62256
\(75\) 505.055 0.777583
\(76\) 528.765 0.798073
\(77\) −154.179 −0.228186
\(78\) −1145.55 −1.66292
\(79\) 730.616 1.04052 0.520258 0.854009i \(-0.325836\pi\)
0.520258 + 0.854009i \(0.325836\pi\)
\(80\) 9.95129 0.0139073
\(81\) −423.313 −0.580677
\(82\) 1640.91 2.20985
\(83\) −73.2890 −0.0969218 −0.0484609 0.998825i \(-0.515432\pi\)
−0.0484609 + 0.998825i \(0.515432\pi\)
\(84\) −518.117 −0.672991
\(85\) −181.425 −0.231509
\(86\) 0 0
\(87\) −699.170 −0.861596
\(88\) 350.686 0.424809
\(89\) 1156.75 1.37769 0.688847 0.724906i \(-0.258117\pi\)
0.688847 + 0.724906i \(0.258117\pi\)
\(90\) −105.624 −0.123709
\(91\) −555.881 −0.640354
\(92\) −1093.50 −1.23919
\(93\) −1161.35 −1.29490
\(94\) −1993.42 −2.18729
\(95\) −112.119 −0.121086
\(96\) −812.259 −0.863550
\(97\) −446.639 −0.467519 −0.233760 0.972294i \(-0.575103\pi\)
−0.233760 + 0.972294i \(0.575103\pi\)
\(98\) 1152.56 1.18802
\(99\) 139.674 0.141796
\(100\) −1499.10 −1.49910
\(101\) −1256.53 −1.23791 −0.618955 0.785426i \(-0.712444\pi\)
−0.618955 + 0.785426i \(0.712444\pi\)
\(102\) 1312.65 1.27423
\(103\) 1196.91 1.14500 0.572499 0.819905i \(-0.305974\pi\)
0.572499 + 0.819905i \(0.305974\pi\)
\(104\) 1264.38 1.19214
\(105\) 109.862 0.102108
\(106\) 1720.47 1.57648
\(107\) −1724.07 −1.55768 −0.778842 0.627220i \(-0.784193\pi\)
−0.778842 + 0.627220i \(0.784193\pi\)
\(108\) 1944.83 1.73279
\(109\) 1367.97 1.20209 0.601044 0.799216i \(-0.294752\pi\)
0.601044 + 0.799216i \(0.294752\pi\)
\(110\) −199.971 −0.173331
\(111\) 973.244 0.832219
\(112\) −34.9380 −0.0294762
\(113\) 1850.28 1.54035 0.770175 0.637833i \(-0.220169\pi\)
0.770175 + 0.637833i \(0.220169\pi\)
\(114\) 811.206 0.666460
\(115\) 231.865 0.188013
\(116\) 2075.27 1.66107
\(117\) 503.587 0.397920
\(118\) 1273.08 0.993188
\(119\) 636.966 0.490677
\(120\) −249.885 −0.190094
\(121\) −1066.56 −0.801326
\(122\) −139.585 −0.103585
\(123\) 1546.17 1.13344
\(124\) 3447.11 2.49645
\(125\) 655.433 0.468990
\(126\) 370.837 0.262197
\(127\) 1723.13 1.20396 0.601981 0.798510i \(-0.294378\pi\)
0.601981 + 0.798510i \(0.294378\pi\)
\(128\) 2276.70 1.57214
\(129\) 0 0
\(130\) −720.981 −0.486417
\(131\) −333.186 −0.222219 −0.111109 0.993808i \(-0.535440\pi\)
−0.111109 + 0.993808i \(0.535440\pi\)
\(132\) 888.634 0.585952
\(133\) 393.641 0.256639
\(134\) −4117.20 −2.65427
\(135\) −412.382 −0.262905
\(136\) −1448.81 −0.913486
\(137\) −2522.39 −1.57301 −0.786505 0.617584i \(-0.788112\pi\)
−0.786505 + 0.617584i \(0.788112\pi\)
\(138\) −1677.59 −1.03483
\(139\) 2241.41 1.36772 0.683862 0.729612i \(-0.260299\pi\)
0.683862 + 0.729612i \(0.260299\pi\)
\(140\) −326.091 −0.196855
\(141\) −1878.33 −1.12187
\(142\) 554.596 0.327751
\(143\) 953.404 0.557536
\(144\) 31.6512 0.0183167
\(145\) −440.041 −0.252023
\(146\) 1697.13 0.962022
\(147\) 1086.02 0.609343
\(148\) −2888.78 −1.60443
\(149\) 816.248 0.448789 0.224395 0.974498i \(-0.427960\pi\)
0.224395 + 0.974498i \(0.427960\pi\)
\(150\) −2299.85 −1.25188
\(151\) 33.6537 0.0181371 0.00906853 0.999959i \(-0.497113\pi\)
0.00906853 + 0.999959i \(0.497113\pi\)
\(152\) −895.352 −0.477781
\(153\) −577.044 −0.304910
\(154\) 702.078 0.367370
\(155\) −730.924 −0.378769
\(156\) 3203.91 1.64435
\(157\) 1660.28 0.843979 0.421990 0.906601i \(-0.361332\pi\)
0.421990 + 0.906601i \(0.361332\pi\)
\(158\) −3326.98 −1.67519
\(159\) 1621.14 0.808586
\(160\) −511.216 −0.252595
\(161\) −814.057 −0.398489
\(162\) 1927.63 0.934868
\(163\) 3488.21 1.67618 0.838091 0.545531i \(-0.183672\pi\)
0.838091 + 0.545531i \(0.183672\pi\)
\(164\) −4589.34 −2.18517
\(165\) −188.426 −0.0889026
\(166\) 333.733 0.156041
\(167\) 1841.55 0.853314 0.426657 0.904414i \(-0.359691\pi\)
0.426657 + 0.904414i \(0.359691\pi\)
\(168\) 877.321 0.402898
\(169\) 1240.44 0.564605
\(170\) 826.148 0.372722
\(171\) −356.609 −0.159477
\(172\) 0 0
\(173\) 2820.94 1.23972 0.619862 0.784711i \(-0.287189\pi\)
0.619862 + 0.784711i \(0.287189\pi\)
\(174\) 3183.78 1.38714
\(175\) −1116.01 −0.482071
\(176\) 59.9230 0.0256640
\(177\) 1199.58 0.509412
\(178\) −5267.43 −2.21804
\(179\) 1287.81 0.537738 0.268869 0.963177i \(-0.413350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(180\) 295.414 0.122327
\(181\) 2583.98 1.06114 0.530569 0.847642i \(-0.321978\pi\)
0.530569 + 0.847642i \(0.321978\pi\)
\(182\) 2531.30 1.03095
\(183\) −131.526 −0.0531296
\(184\) 1851.61 0.741860
\(185\) 612.537 0.243430
\(186\) 5288.39 2.08475
\(187\) −1092.47 −0.427217
\(188\) 5575.26 2.16286
\(189\) 1447.84 0.557220
\(190\) 510.554 0.194945
\(191\) −1810.98 −0.686064 −0.343032 0.939324i \(-0.611454\pi\)
−0.343032 + 0.939324i \(0.611454\pi\)
\(192\) 3572.26 1.34274
\(193\) −3979.51 −1.48420 −0.742101 0.670288i \(-0.766171\pi\)
−0.742101 + 0.670288i \(0.766171\pi\)
\(194\) 2033.85 0.752688
\(195\) −679.357 −0.249486
\(196\) −3223.52 −1.17475
\(197\) −2135.93 −0.772480 −0.386240 0.922398i \(-0.626226\pi\)
−0.386240 + 0.922398i \(0.626226\pi\)
\(198\) −636.030 −0.228286
\(199\) −2804.85 −0.999149 −0.499575 0.866271i \(-0.666510\pi\)
−0.499575 + 0.866271i \(0.666510\pi\)
\(200\) 2538.41 0.897464
\(201\) −3879.50 −1.36139
\(202\) 5721.79 1.99299
\(203\) 1544.94 0.534156
\(204\) −3671.26 −1.26000
\(205\) 973.124 0.331541
\(206\) −5450.31 −1.84340
\(207\) 737.475 0.247623
\(208\) 216.048 0.0720205
\(209\) −675.141 −0.223447
\(210\) −500.272 −0.164391
\(211\) 2756.80 0.899459 0.449730 0.893165i \(-0.351520\pi\)
0.449730 + 0.893165i \(0.351520\pi\)
\(212\) −4811.88 −1.55887
\(213\) 522.578 0.168105
\(214\) 7850.84 2.50781
\(215\) 0 0
\(216\) −3293.16 −1.03737
\(217\) 2566.21 0.802790
\(218\) −6229.27 −1.93532
\(219\) 1599.15 0.493427
\(220\) 559.285 0.171395
\(221\) −3938.85 −1.19889
\(222\) −4431.83 −1.33984
\(223\) 3779.12 1.13484 0.567418 0.823430i \(-0.307942\pi\)
0.567418 + 0.823430i \(0.307942\pi\)
\(224\) 1794.83 0.535367
\(225\) 1011.02 0.299562
\(226\) −8425.54 −2.47991
\(227\) −3727.76 −1.08996 −0.544979 0.838450i \(-0.683462\pi\)
−0.544979 + 0.838450i \(0.683462\pi\)
\(228\) −2268.81 −0.659016
\(229\) 3616.46 1.04359 0.521796 0.853070i \(-0.325262\pi\)
0.521796 + 0.853070i \(0.325262\pi\)
\(230\) −1055.84 −0.302695
\(231\) 661.545 0.188426
\(232\) −3514.03 −0.994430
\(233\) −52.8357 −0.0148557 −0.00742786 0.999972i \(-0.502364\pi\)
−0.00742786 + 0.999972i \(0.502364\pi\)
\(234\) −2293.17 −0.640636
\(235\) −1182.18 −0.328157
\(236\) −3560.58 −0.982095
\(237\) −3134.91 −0.859215
\(238\) −2900.53 −0.789972
\(239\) −1707.71 −0.462187 −0.231094 0.972932i \(-0.574230\pi\)
−0.231094 + 0.972932i \(0.574230\pi\)
\(240\) −42.6987 −0.0114841
\(241\) −604.111 −0.161470 −0.0807348 0.996736i \(-0.525727\pi\)
−0.0807348 + 0.996736i \(0.525727\pi\)
\(242\) 4856.78 1.29011
\(243\) −2306.71 −0.608952
\(244\) 390.396 0.102429
\(245\) 683.516 0.178238
\(246\) −7040.75 −1.82480
\(247\) −2434.18 −0.627057
\(248\) −5836.94 −1.49454
\(249\) 314.466 0.0800341
\(250\) −2984.62 −0.755056
\(251\) −1637.24 −0.411720 −0.205860 0.978581i \(-0.565999\pi\)
−0.205860 + 0.978581i \(0.565999\pi\)
\(252\) −1037.17 −0.259268
\(253\) 1396.21 0.346952
\(254\) −7846.56 −1.93833
\(255\) 778.453 0.191171
\(256\) −3706.97 −0.905022
\(257\) −7248.10 −1.75924 −0.879619 0.475679i \(-0.842202\pi\)
−0.879619 + 0.475679i \(0.842202\pi\)
\(258\) 0 0
\(259\) −2150.56 −0.515943
\(260\) 2016.47 0.480984
\(261\) −1399.60 −0.331928
\(262\) 1517.22 0.357764
\(263\) 4668.09 1.09447 0.547237 0.836977i \(-0.315679\pi\)
0.547237 + 0.836977i \(0.315679\pi\)
\(264\) −1504.71 −0.350790
\(265\) 1020.31 0.236517
\(266\) −1792.51 −0.413179
\(267\) −4963.33 −1.13764
\(268\) 11515.1 2.62462
\(269\) 2919.35 0.661696 0.330848 0.943684i \(-0.392665\pi\)
0.330848 + 0.943684i \(0.392665\pi\)
\(270\) 1877.85 0.423268
\(271\) −4747.60 −1.06419 −0.532096 0.846684i \(-0.678595\pi\)
−0.532096 + 0.846684i \(0.678595\pi\)
\(272\) −247.563 −0.0551864
\(273\) 2385.16 0.528778
\(274\) 11486.1 2.53249
\(275\) 1914.09 0.419724
\(276\) 4691.95 1.02327
\(277\) 429.145 0.0930861 0.0465431 0.998916i \(-0.485180\pi\)
0.0465431 + 0.998916i \(0.485180\pi\)
\(278\) −10206.6 −2.20198
\(279\) −2324.79 −0.498859
\(280\) 552.165 0.117851
\(281\) 59.2317 0.0125746 0.00628731 0.999980i \(-0.497999\pi\)
0.00628731 + 0.999980i \(0.497999\pi\)
\(282\) 8553.30 1.80618
\(283\) −5470.70 −1.14911 −0.574557 0.818464i \(-0.694826\pi\)
−0.574557 + 0.818464i \(0.694826\pi\)
\(284\) −1551.11 −0.324090
\(285\) 481.079 0.0999882
\(286\) −4341.48 −0.897612
\(287\) −3416.55 −0.702691
\(288\) −1625.98 −0.332681
\(289\) −399.606 −0.0813364
\(290\) 2003.80 0.405748
\(291\) 1916.43 0.386058
\(292\) −4746.58 −0.951277
\(293\) 2179.76 0.434617 0.217308 0.976103i \(-0.430272\pi\)
0.217308 + 0.976103i \(0.430272\pi\)
\(294\) −4945.37 −0.981020
\(295\) 754.986 0.149007
\(296\) 4891.54 0.960523
\(297\) −2483.21 −0.485154
\(298\) −3716.92 −0.722534
\(299\) 5033.93 0.973645
\(300\) 6432.30 1.23790
\(301\) 0 0
\(302\) −153.247 −0.0292000
\(303\) 5391.46 1.02222
\(304\) −152.992 −0.0288641
\(305\) −82.7796 −0.0155408
\(306\) 2627.66 0.490894
\(307\) 7118.86 1.32344 0.661718 0.749753i \(-0.269827\pi\)
0.661718 + 0.749753i \(0.269827\pi\)
\(308\) −1963.60 −0.363267
\(309\) −5135.66 −0.945493
\(310\) 3328.38 0.609805
\(311\) 3986.21 0.726808 0.363404 0.931632i \(-0.381614\pi\)
0.363404 + 0.931632i \(0.381614\pi\)
\(312\) −5425.14 −0.984418
\(313\) 9318.69 1.68282 0.841411 0.540395i \(-0.181725\pi\)
0.841411 + 0.540395i \(0.181725\pi\)
\(314\) −7560.35 −1.35878
\(315\) 219.921 0.0393370
\(316\) 9305.01 1.65648
\(317\) −8476.89 −1.50192 −0.750961 0.660346i \(-0.770409\pi\)
−0.750961 + 0.660346i \(0.770409\pi\)
\(318\) −7382.15 −1.30179
\(319\) −2649.76 −0.465073
\(320\) 2248.30 0.392761
\(321\) 7397.59 1.28627
\(322\) 3706.94 0.641552
\(323\) 2789.25 0.480489
\(324\) −5391.25 −0.924426
\(325\) 6901.14 1.17787
\(326\) −15884.1 −2.69859
\(327\) −5869.64 −0.992635
\(328\) 7771.08 1.30819
\(329\) 4150.52 0.695518
\(330\) 858.028 0.143130
\(331\) −1225.47 −0.203499 −0.101749 0.994810i \(-0.532444\pi\)
−0.101749 + 0.994810i \(0.532444\pi\)
\(332\) −933.398 −0.154298
\(333\) 1948.25 0.320610
\(334\) −8385.80 −1.37380
\(335\) −2441.67 −0.398216
\(336\) 149.911 0.0243402
\(337\) 1486.27 0.240244 0.120122 0.992759i \(-0.461671\pi\)
0.120122 + 0.992759i \(0.461671\pi\)
\(338\) −5648.53 −0.908993
\(339\) −7939.12 −1.27196
\(340\) −2310.60 −0.368559
\(341\) −4401.36 −0.698964
\(342\) 1623.88 0.256752
\(343\) −5651.82 −0.889708
\(344\) 0 0
\(345\) −994.881 −0.155254
\(346\) −12845.6 −1.99591
\(347\) 8460.85 1.30894 0.654470 0.756088i \(-0.272892\pi\)
0.654470 + 0.756088i \(0.272892\pi\)
\(348\) −8904.52 −1.37165
\(349\) 11788.1 1.80803 0.904015 0.427501i \(-0.140606\pi\)
0.904015 + 0.427501i \(0.140606\pi\)
\(350\) 5081.93 0.776116
\(351\) −8953.07 −1.36148
\(352\) −3078.36 −0.466128
\(353\) −4732.50 −0.713557 −0.356778 0.934189i \(-0.616125\pi\)
−0.356778 + 0.934189i \(0.616125\pi\)
\(354\) −5462.48 −0.820134
\(355\) 328.898 0.0491721
\(356\) 14732.1 2.19326
\(357\) −2733.08 −0.405181
\(358\) −5864.23 −0.865739
\(359\) −6028.18 −0.886226 −0.443113 0.896466i \(-0.646126\pi\)
−0.443113 + 0.896466i \(0.646126\pi\)
\(360\) −500.220 −0.0732331
\(361\) −5135.27 −0.748690
\(362\) −11766.6 −1.70839
\(363\) 4576.39 0.661702
\(364\) −7079.62 −1.01943
\(365\) 1006.47 0.144331
\(366\) 598.927 0.0855367
\(367\) −2429.27 −0.345523 −0.172761 0.984964i \(-0.555269\pi\)
−0.172761 + 0.984964i \(0.555269\pi\)
\(368\) 316.391 0.0448180
\(369\) 3095.14 0.436657
\(370\) −2789.29 −0.391914
\(371\) −3582.21 −0.501292
\(372\) −14790.7 −2.06146
\(373\) 8992.11 1.24824 0.624120 0.781328i \(-0.285458\pi\)
0.624120 + 0.781328i \(0.285458\pi\)
\(374\) 4974.76 0.687804
\(375\) −2812.31 −0.387273
\(376\) −9440.53 −1.29484
\(377\) −9553.55 −1.30513
\(378\) −6592.96 −0.897104
\(379\) −5750.17 −0.779331 −0.389666 0.920956i \(-0.627409\pi\)
−0.389666 + 0.920956i \(0.627409\pi\)
\(380\) −1427.94 −0.192767
\(381\) −7393.56 −0.994183
\(382\) 8246.61 1.10454
\(383\) 282.965 0.0377516 0.0188758 0.999822i \(-0.493991\pi\)
0.0188758 + 0.999822i \(0.493991\pi\)
\(384\) −9768.81 −1.29821
\(385\) 416.361 0.0551161
\(386\) 18121.3 2.38951
\(387\) 0 0
\(388\) −5688.33 −0.744282
\(389\) −11295.8 −1.47229 −0.736145 0.676824i \(-0.763356\pi\)
−0.736145 + 0.676824i \(0.763356\pi\)
\(390\) 3093.56 0.401663
\(391\) −5768.22 −0.746065
\(392\) 5458.35 0.703287
\(393\) 1429.63 0.183499
\(394\) 9726.30 1.24366
\(395\) −1973.03 −0.251327
\(396\) 1778.87 0.225737
\(397\) 3635.34 0.459579 0.229789 0.973240i \(-0.426196\pi\)
0.229789 + 0.973240i \(0.426196\pi\)
\(398\) 12772.3 1.60859
\(399\) −1689.02 −0.211922
\(400\) 433.747 0.0542184
\(401\) −11126.5 −1.38562 −0.692809 0.721121i \(-0.743627\pi\)
−0.692809 + 0.721121i \(0.743627\pi\)
\(402\) 17665.9 2.19178
\(403\) −15868.8 −1.96149
\(404\) −16002.9 −1.97073
\(405\) 1143.16 0.140257
\(406\) −7035.15 −0.859972
\(407\) 3688.47 0.449215
\(408\) 6216.49 0.754319
\(409\) −9047.71 −1.09384 −0.546920 0.837185i \(-0.684200\pi\)
−0.546920 + 0.837185i \(0.684200\pi\)
\(410\) −4431.28 −0.533769
\(411\) 10823.0 1.29893
\(412\) 15243.6 1.82282
\(413\) −2650.69 −0.315815
\(414\) −3358.21 −0.398665
\(415\) 197.918 0.0234106
\(416\) −11098.8 −1.30809
\(417\) −9617.36 −1.12941
\(418\) 3074.37 0.359742
\(419\) −12094.5 −1.41015 −0.705077 0.709130i \(-0.749088\pi\)
−0.705077 + 0.709130i \(0.749088\pi\)
\(420\) 1399.18 0.162555
\(421\) −4362.59 −0.505035 −0.252517 0.967592i \(-0.581258\pi\)
−0.252517 + 0.967592i \(0.581258\pi\)
\(422\) −12553.5 −1.44810
\(423\) −3760.06 −0.432199
\(424\) 8147.89 0.933246
\(425\) −7907.79 −0.902550
\(426\) −2379.64 −0.270643
\(427\) 290.631 0.0329383
\(428\) −21957.5 −2.47980
\(429\) −4090.84 −0.460390
\(430\) 0 0
\(431\) 12667.9 1.41576 0.707880 0.706333i \(-0.249652\pi\)
0.707880 + 0.706333i \(0.249652\pi\)
\(432\) −562.715 −0.0626704
\(433\) −15698.9 −1.74236 −0.871180 0.490963i \(-0.836645\pi\)
−0.871180 + 0.490963i \(0.836645\pi\)
\(434\) −11685.6 −1.29246
\(435\) 1888.11 0.208111
\(436\) 17422.2 1.91370
\(437\) −3564.72 −0.390214
\(438\) −7281.98 −0.794398
\(439\) −2524.91 −0.274505 −0.137252 0.990536i \(-0.543827\pi\)
−0.137252 + 0.990536i \(0.543827\pi\)
\(440\) −947.030 −0.102609
\(441\) 2174.00 0.234748
\(442\) 17936.2 1.93017
\(443\) −8347.92 −0.895308 −0.447654 0.894207i \(-0.647740\pi\)
−0.447654 + 0.894207i \(0.647740\pi\)
\(444\) 12395.1 1.32488
\(445\) −3123.80 −0.332770
\(446\) −17208.8 −1.82705
\(447\) −3502.33 −0.370592
\(448\) −7893.56 −0.832446
\(449\) −3832.87 −0.402861 −0.201430 0.979503i \(-0.564559\pi\)
−0.201430 + 0.979503i \(0.564559\pi\)
\(450\) −4603.85 −0.482284
\(451\) 5859.79 0.611811
\(452\) 23564.9 2.45221
\(453\) −144.400 −0.0149768
\(454\) 16975.0 1.75479
\(455\) 1501.16 0.154672
\(456\) 3841.75 0.394532
\(457\) −7651.81 −0.783231 −0.391616 0.920129i \(-0.628084\pi\)
−0.391616 + 0.920129i \(0.628084\pi\)
\(458\) −16468.2 −1.68014
\(459\) 10259.0 1.04325
\(460\) 2953.00 0.299314
\(461\) −11080.0 −1.11941 −0.559704 0.828692i \(-0.689085\pi\)
−0.559704 + 0.828692i \(0.689085\pi\)
\(462\) −3012.45 −0.303359
\(463\) 12167.8 1.22135 0.610674 0.791882i \(-0.290899\pi\)
0.610674 + 0.791882i \(0.290899\pi\)
\(464\) −600.456 −0.0600764
\(465\) 3136.23 0.312772
\(466\) 240.596 0.0239172
\(467\) −9164.29 −0.908079 −0.454039 0.890982i \(-0.650017\pi\)
−0.454039 + 0.890982i \(0.650017\pi\)
\(468\) 6413.61 0.633481
\(469\) 8572.46 0.844007
\(470\) 5383.24 0.528320
\(471\) −7123.88 −0.696924
\(472\) 6029.09 0.587948
\(473\) 0 0
\(474\) 14275.3 1.38330
\(475\) −4886.96 −0.472061
\(476\) 8112.30 0.781149
\(477\) 3245.22 0.311506
\(478\) 7776.34 0.744104
\(479\) −2661.40 −0.253867 −0.126934 0.991911i \(-0.540514\pi\)
−0.126934 + 0.991911i \(0.540514\pi\)
\(480\) 2193.51 0.208583
\(481\) 13298.5 1.26063
\(482\) 2750.92 0.259960
\(483\) 3492.93 0.329056
\(484\) −13583.6 −1.27570
\(485\) 1206.15 0.112925
\(486\) 10504.0 0.980390
\(487\) −15963.7 −1.48539 −0.742696 0.669629i \(-0.766453\pi\)
−0.742696 + 0.669629i \(0.766453\pi\)
\(488\) −661.053 −0.0613206
\(489\) −14967.1 −1.38412
\(490\) −3112.50 −0.286956
\(491\) −13980.6 −1.28500 −0.642501 0.766285i \(-0.722103\pi\)
−0.642501 + 0.766285i \(0.722103\pi\)
\(492\) 19691.8 1.80442
\(493\) 10947.1 1.00007
\(494\) 11084.4 1.00954
\(495\) −377.192 −0.0342495
\(496\) −997.380 −0.0902897
\(497\) −1154.73 −0.104219
\(498\) −1431.97 −0.128852
\(499\) −11789.0 −1.05761 −0.528804 0.848744i \(-0.677359\pi\)
−0.528804 + 0.848744i \(0.677359\pi\)
\(500\) 8347.49 0.746623
\(501\) −7901.67 −0.704632
\(502\) 7455.45 0.662855
\(503\) 10616.1 0.941050 0.470525 0.882387i \(-0.344065\pi\)
0.470525 + 0.882387i \(0.344065\pi\)
\(504\) 1756.23 0.155215
\(505\) 3393.26 0.299006
\(506\) −6357.85 −0.558579
\(507\) −5322.43 −0.466228
\(508\) 21945.5 1.91668
\(509\) −16222.9 −1.41270 −0.706352 0.707861i \(-0.749660\pi\)
−0.706352 + 0.707861i \(0.749660\pi\)
\(510\) −3544.81 −0.307778
\(511\) −3533.61 −0.305905
\(512\) −1333.32 −0.115088
\(513\) 6340.01 0.545649
\(514\) 33005.4 2.83231
\(515\) −3232.26 −0.276564
\(516\) 0 0
\(517\) −7118.64 −0.605566
\(518\) 9792.92 0.830649
\(519\) −12104.0 −1.02371
\(520\) −3414.46 −0.287950
\(521\) 2799.09 0.235375 0.117688 0.993051i \(-0.462452\pi\)
0.117688 + 0.993051i \(0.462452\pi\)
\(522\) 6373.32 0.534392
\(523\) −11873.6 −0.992726 −0.496363 0.868115i \(-0.665331\pi\)
−0.496363 + 0.868115i \(0.665331\pi\)
\(524\) −4243.41 −0.353768
\(525\) 4788.54 0.398074
\(526\) −21256.9 −1.76206
\(527\) 18183.5 1.50301
\(528\) −257.116 −0.0211923
\(529\) −4795.09 −0.394106
\(530\) −4646.15 −0.380784
\(531\) 2401.32 0.196250
\(532\) 5013.35 0.408564
\(533\) 21127.1 1.71692
\(534\) 22601.3 1.83157
\(535\) 4655.87 0.376244
\(536\) −19498.4 −1.57127
\(537\) −5525.68 −0.444042
\(538\) −13293.8 −1.06531
\(539\) 4115.88 0.328912
\(540\) −5252.04 −0.418540
\(541\) −15732.1 −1.25023 −0.625116 0.780532i \(-0.714948\pi\)
−0.625116 + 0.780532i \(0.714948\pi\)
\(542\) 21619.0 1.71331
\(543\) −11087.3 −0.876244
\(544\) 12717.8 1.00233
\(545\) −3694.21 −0.290353
\(546\) −10861.2 −0.851313
\(547\) 11050.4 0.863764 0.431882 0.901930i \(-0.357850\pi\)
0.431882 + 0.901930i \(0.357850\pi\)
\(548\) −32124.8 −2.50420
\(549\) −263.290 −0.0204680
\(550\) −8716.13 −0.675740
\(551\) 6765.23 0.523065
\(552\) −7944.82 −0.612598
\(553\) 6927.14 0.532680
\(554\) −1954.18 −0.149865
\(555\) −2628.25 −0.201015
\(556\) 28546.2 2.17739
\(557\) −9079.45 −0.690680 −0.345340 0.938478i \(-0.612236\pi\)
−0.345340 + 0.938478i \(0.612236\pi\)
\(558\) 10586.3 0.803145
\(559\) 0 0
\(560\) 94.3505 0.00711970
\(561\) 4687.56 0.352779
\(562\) −269.721 −0.0202447
\(563\) 3751.33 0.280817 0.140408 0.990094i \(-0.455158\pi\)
0.140408 + 0.990094i \(0.455158\pi\)
\(564\) −23922.2 −1.78600
\(565\) −4996.69 −0.372057
\(566\) 24911.7 1.85003
\(567\) −4013.53 −0.297271
\(568\) 2626.48 0.194022
\(569\) 19349.0 1.42557 0.712787 0.701380i \(-0.247432\pi\)
0.712787 + 0.701380i \(0.247432\pi\)
\(570\) −2190.67 −0.160977
\(571\) −4876.77 −0.357419 −0.178710 0.983902i \(-0.557192\pi\)
−0.178710 + 0.983902i \(0.557192\pi\)
\(572\) 12142.4 0.887587
\(573\) 7770.51 0.566523
\(574\) 15557.8 1.13131
\(575\) 10106.3 0.732979
\(576\) 7150.97 0.517287
\(577\) 11927.2 0.860546 0.430273 0.902699i \(-0.358417\pi\)
0.430273 + 0.902699i \(0.358417\pi\)
\(578\) 1819.67 0.130949
\(579\) 17075.1 1.22559
\(580\) −5604.29 −0.401217
\(581\) −694.870 −0.0496180
\(582\) −8726.76 −0.621539
\(583\) 6143.93 0.436459
\(584\) 8037.33 0.569499
\(585\) −1359.94 −0.0961139
\(586\) −9925.87 −0.699717
\(587\) 20820.2 1.46395 0.731977 0.681330i \(-0.238598\pi\)
0.731977 + 0.681330i \(0.238598\pi\)
\(588\) 13831.4 0.970063
\(589\) 11237.3 0.786121
\(590\) −3437.95 −0.239895
\(591\) 9164.78 0.637883
\(592\) 835.835 0.0580280
\(593\) 15515.4 1.07444 0.537218 0.843443i \(-0.319475\pi\)
0.537218 + 0.843443i \(0.319475\pi\)
\(594\) 11307.7 0.781080
\(595\) −1720.13 −0.118519
\(596\) 10395.6 0.714464
\(597\) 12035.0 0.825056
\(598\) −22922.8 −1.56753
\(599\) −1698.64 −0.115868 −0.0579338 0.998320i \(-0.518451\pi\)
−0.0579338 + 0.998320i \(0.518451\pi\)
\(600\) −10891.7 −0.741089
\(601\) −17559.3 −1.19178 −0.595888 0.803068i \(-0.703200\pi\)
−0.595888 + 0.803068i \(0.703200\pi\)
\(602\) 0 0
\(603\) −7766.01 −0.524472
\(604\) 428.608 0.0288738
\(605\) 2880.27 0.193553
\(606\) −24550.9 −1.64573
\(607\) 25020.3 1.67305 0.836525 0.547929i \(-0.184584\pi\)
0.836525 + 0.547929i \(0.184584\pi\)
\(608\) 7859.49 0.524251
\(609\) −6628.99 −0.441084
\(610\) 376.950 0.0250201
\(611\) −25665.8 −1.69939
\(612\) −7349.14 −0.485411
\(613\) −2905.86 −0.191462 −0.0957312 0.995407i \(-0.530519\pi\)
−0.0957312 + 0.995407i \(0.530519\pi\)
\(614\) −32416.9 −2.13068
\(615\) −4175.45 −0.273773
\(616\) 3324.93 0.217476
\(617\) −24437.2 −1.59450 −0.797249 0.603651i \(-0.793712\pi\)
−0.797249 + 0.603651i \(0.793712\pi\)
\(618\) 23386.0 1.52221
\(619\) 13786.3 0.895182 0.447591 0.894238i \(-0.352282\pi\)
0.447591 + 0.894238i \(0.352282\pi\)
\(620\) −9308.94 −0.602994
\(621\) −13111.3 −0.847242
\(622\) −18151.9 −1.17013
\(623\) 10967.4 0.705295
\(624\) −927.014 −0.0594716
\(625\) 12943.4 0.828378
\(626\) −42434.2 −2.70928
\(627\) 2896.88 0.184514
\(628\) 21145.1 1.34360
\(629\) −15238.4 −0.965967
\(630\) −1001.45 −0.0633312
\(631\) −29405.5 −1.85517 −0.927586 0.373610i \(-0.878120\pi\)
−0.927586 + 0.373610i \(0.878120\pi\)
\(632\) −15756.1 −0.991681
\(633\) −11828.8 −0.742737
\(634\) 38600.9 2.41804
\(635\) −4653.33 −0.290806
\(636\) 20646.7 1.28725
\(637\) 14839.5 0.923020
\(638\) 12066.1 0.748750
\(639\) 1046.10 0.0647622
\(640\) −6148.25 −0.379736
\(641\) −2989.09 −0.184184 −0.0920921 0.995750i \(-0.529355\pi\)
−0.0920921 + 0.995750i \(0.529355\pi\)
\(642\) −33686.2 −2.07085
\(643\) 15849.7 0.972083 0.486041 0.873936i \(-0.338440\pi\)
0.486041 + 0.873936i \(0.338440\pi\)
\(644\) −10367.7 −0.634386
\(645\) 0 0
\(646\) −12701.3 −0.773569
\(647\) 30447.1 1.85007 0.925037 0.379876i \(-0.124033\pi\)
0.925037 + 0.379876i \(0.124033\pi\)
\(648\) 9128.94 0.553424
\(649\) 4546.25 0.274970
\(650\) −31425.5 −1.89632
\(651\) −11011.0 −0.662911
\(652\) 44425.3 2.66845
\(653\) 9838.49 0.589601 0.294801 0.955559i \(-0.404747\pi\)
0.294801 + 0.955559i \(0.404747\pi\)
\(654\) 26728.4 1.59811
\(655\) 899.773 0.0536749
\(656\) 1327.87 0.0790316
\(657\) 3201.18 0.190091
\(658\) −18900.1 −1.11976
\(659\) −8170.61 −0.482977 −0.241488 0.970404i \(-0.577636\pi\)
−0.241488 + 0.970404i \(0.577636\pi\)
\(660\) −2399.76 −0.141531
\(661\) 22065.7 1.29842 0.649212 0.760608i \(-0.275099\pi\)
0.649212 + 0.760608i \(0.275099\pi\)
\(662\) 5580.39 0.327626
\(663\) 16900.7 0.989997
\(664\) 1580.51 0.0923731
\(665\) −1063.03 −0.0619888
\(666\) −8871.66 −0.516171
\(667\) −13990.6 −0.812173
\(668\) 23453.7 1.35846
\(669\) −16215.3 −0.937102
\(670\) 11118.5 0.641114
\(671\) −498.468 −0.0286783
\(672\) −7701.22 −0.442085
\(673\) 1828.23 0.104715 0.0523573 0.998628i \(-0.483327\pi\)
0.0523573 + 0.998628i \(0.483327\pi\)
\(674\) −6767.97 −0.386784
\(675\) −17974.5 −1.02495
\(676\) 15798.0 0.898840
\(677\) −21093.1 −1.19745 −0.598724 0.800955i \(-0.704325\pi\)
−0.598724 + 0.800955i \(0.704325\pi\)
\(678\) 36152.1 2.04780
\(679\) −4234.69 −0.239341
\(680\) 3912.51 0.220644
\(681\) 15995.0 0.900042
\(682\) 20042.3 1.12531
\(683\) −11074.6 −0.620438 −0.310219 0.950665i \(-0.600402\pi\)
−0.310219 + 0.950665i \(0.600402\pi\)
\(684\) −4541.72 −0.253884
\(685\) 6811.74 0.379946
\(686\) 25736.5 1.43240
\(687\) −15517.4 −0.861756
\(688\) 0 0
\(689\) 22151.5 1.22483
\(690\) 4530.35 0.249953
\(691\) −10854.2 −0.597557 −0.298778 0.954323i \(-0.596579\pi\)
−0.298778 + 0.954323i \(0.596579\pi\)
\(692\) 35927.1 1.97362
\(693\) 1324.29 0.0725908
\(694\) −38527.9 −2.10735
\(695\) −6052.94 −0.330361
\(696\) 15077.9 0.821159
\(697\) −24208.9 −1.31560
\(698\) −53679.0 −2.91086
\(699\) 226.706 0.0122672
\(700\) −14213.3 −0.767448
\(701\) 32035.1 1.72603 0.863015 0.505177i \(-0.168573\pi\)
0.863015 + 0.505177i \(0.168573\pi\)
\(702\) 40769.3 2.19193
\(703\) −9417.20 −0.505230
\(704\) 13538.4 0.724784
\(705\) 5072.46 0.270978
\(706\) 21550.2 1.14880
\(707\) −11913.4 −0.633734
\(708\) 15277.7 0.810974
\(709\) −14009.1 −0.742064 −0.371032 0.928620i \(-0.620996\pi\)
−0.371032 + 0.928620i \(0.620996\pi\)
\(710\) −1497.69 −0.0791653
\(711\) −6275.47 −0.331011
\(712\) −24945.8 −1.31304
\(713\) −23239.0 −1.22063
\(714\) 12445.5 0.652327
\(715\) −2574.68 −0.134668
\(716\) 16401.3 0.856069
\(717\) 7327.40 0.381655
\(718\) 27450.3 1.42679
\(719\) −12270.7 −0.636467 −0.318234 0.948012i \(-0.603090\pi\)
−0.318234 + 0.948012i \(0.603090\pi\)
\(720\) −85.4745 −0.00442423
\(721\) 11348.2 0.586168
\(722\) 23384.3 1.20536
\(723\) 2592.10 0.133335
\(724\) 32909.2 1.68931
\(725\) −19180.1 −0.982525
\(726\) −20839.3 −1.06532
\(727\) 23264.9 1.18686 0.593431 0.804885i \(-0.297773\pi\)
0.593431 + 0.804885i \(0.297773\pi\)
\(728\) 11987.8 0.610300
\(729\) 21327.0 1.08352
\(730\) −4583.11 −0.232368
\(731\) 0 0
\(732\) −1675.10 −0.0845813
\(733\) −10120.4 −0.509965 −0.254982 0.966946i \(-0.582070\pi\)
−0.254982 + 0.966946i \(0.582070\pi\)
\(734\) 11062.1 0.556278
\(735\) −2932.81 −0.147181
\(736\) −16253.6 −0.814015
\(737\) −14702.8 −0.734851
\(738\) −14094.2 −0.703002
\(739\) 16955.4 0.843998 0.421999 0.906596i \(-0.361329\pi\)
0.421999 + 0.906596i \(0.361329\pi\)
\(740\) 7801.18 0.387536
\(741\) 10444.5 0.517798
\(742\) 16312.2 0.807061
\(743\) −30362.0 −1.49916 −0.749579 0.661915i \(-0.769744\pi\)
−0.749579 + 0.661915i \(0.769744\pi\)
\(744\) 25045.0 1.23413
\(745\) −2204.28 −0.108401
\(746\) −40947.0 −2.00962
\(747\) 629.500 0.0308330
\(748\) −13913.6 −0.680122
\(749\) −16346.3 −0.797439
\(750\) 12806.3 0.623494
\(751\) 10733.4 0.521529 0.260765 0.965402i \(-0.416025\pi\)
0.260765 + 0.965402i \(0.416025\pi\)
\(752\) −1613.14 −0.0782248
\(753\) 7025.03 0.339982
\(754\) 43503.7 2.10121
\(755\) −90.8820 −0.00438084
\(756\) 18439.4 0.887084
\(757\) −21386.9 −1.02684 −0.513422 0.858136i \(-0.671623\pi\)
−0.513422 + 0.858136i \(0.671623\pi\)
\(758\) 26184.4 1.25469
\(759\) −5990.80 −0.286498
\(760\) 2417.91 0.115403
\(761\) 12272.6 0.584601 0.292300 0.956327i \(-0.405579\pi\)
0.292300 + 0.956327i \(0.405579\pi\)
\(762\) 33667.8 1.60060
\(763\) 12970.0 0.615395
\(764\) −23064.4 −1.09220
\(765\) 1558.31 0.0736482
\(766\) −1288.53 −0.0607787
\(767\) 16391.2 0.771645
\(768\) 15905.8 0.747331
\(769\) −6111.07 −0.286568 −0.143284 0.989682i \(-0.545766\pi\)
−0.143284 + 0.989682i \(0.545766\pi\)
\(770\) −1895.97 −0.0887349
\(771\) 31099.9 1.45271
\(772\) −50682.4 −2.36282
\(773\) −19226.4 −0.894600 −0.447300 0.894384i \(-0.647614\pi\)
−0.447300 + 0.894384i \(0.647614\pi\)
\(774\) 0 0
\(775\) −31858.9 −1.47665
\(776\) 9631.98 0.445577
\(777\) 9227.56 0.426045
\(778\) 51437.4 2.37033
\(779\) −14960.9 −0.688100
\(780\) −8652.19 −0.397177
\(781\) 1980.50 0.0907400
\(782\) 26266.5 1.20114
\(783\) 24883.0 1.13569
\(784\) 932.688 0.0424876
\(785\) −4483.60 −0.203855
\(786\) −6510.04 −0.295427
\(787\) 17479.0 0.791690 0.395845 0.918317i \(-0.370452\pi\)
0.395845 + 0.918317i \(0.370452\pi\)
\(788\) −27202.9 −1.22977
\(789\) −20029.7 −0.903773
\(790\) 8984.53 0.404627
\(791\) 17542.9 0.788564
\(792\) −3012.14 −0.135141
\(793\) −1797.19 −0.0804795
\(794\) −16554.1 −0.739905
\(795\) −4377.92 −0.195306
\(796\) −35722.2 −1.59063
\(797\) −4386.69 −0.194962 −0.0974809 0.995237i \(-0.531078\pi\)
−0.0974809 + 0.995237i \(0.531078\pi\)
\(798\) 7691.24 0.341186
\(799\) 29409.6 1.30217
\(800\) −22282.4 −0.984753
\(801\) −9935.63 −0.438275
\(802\) 50666.5 2.23079
\(803\) 6060.56 0.266342
\(804\) −49408.7 −2.16730
\(805\) 2198.37 0.0962513
\(806\) 72261.2 3.15793
\(807\) −12526.3 −0.546401
\(808\) 27097.5 1.17981
\(809\) −42338.7 −1.83999 −0.919994 0.391932i \(-0.871807\pi\)
−0.919994 + 0.391932i \(0.871807\pi\)
\(810\) −5205.57 −0.225809
\(811\) 27748.6 1.20146 0.600730 0.799452i \(-0.294877\pi\)
0.600730 + 0.799452i \(0.294877\pi\)
\(812\) 19676.1 0.850366
\(813\) 20370.9 0.878766
\(814\) −16796.0 −0.723220
\(815\) −9419.94 −0.404866
\(816\) 1062.24 0.0455707
\(817\) 0 0
\(818\) 41200.2 1.76104
\(819\) 4774.63 0.203711
\(820\) 12393.6 0.527807
\(821\) −490.567 −0.0208537 −0.0104269 0.999946i \(-0.503319\pi\)
−0.0104269 + 0.999946i \(0.503319\pi\)
\(822\) −49284.3 −2.09123
\(823\) −26286.3 −1.11334 −0.556672 0.830732i \(-0.687922\pi\)
−0.556672 + 0.830732i \(0.687922\pi\)
\(824\) −25811.9 −1.09126
\(825\) −8212.93 −0.346591
\(826\) 12070.3 0.508451
\(827\) 545.162 0.0229228 0.0114614 0.999934i \(-0.496352\pi\)
0.0114614 + 0.999934i \(0.496352\pi\)
\(828\) 9392.37 0.394212
\(829\) −45337.7 −1.89945 −0.949724 0.313090i \(-0.898636\pi\)
−0.949724 + 0.313090i \(0.898636\pi\)
\(830\) −901.250 −0.0376902
\(831\) −1841.36 −0.0768667
\(832\) 48811.9 2.03395
\(833\) −17004.1 −0.707273
\(834\) 43794.2 1.81831
\(835\) −4973.12 −0.206110
\(836\) −8598.50 −0.355724
\(837\) 41331.5 1.70684
\(838\) 55074.3 2.27030
\(839\) 16487.5 0.678439 0.339219 0.940707i \(-0.389837\pi\)
0.339219 + 0.940707i \(0.389837\pi\)
\(840\) −2369.21 −0.0973162
\(841\) 2162.86 0.0886818
\(842\) 19865.8 0.813087
\(843\) −254.150 −0.0103836
\(844\) 35110.2 1.43192
\(845\) −3349.81 −0.136375
\(846\) 17122.0 0.695825
\(847\) −10112.3 −0.410230
\(848\) 1392.26 0.0563802
\(849\) 23473.5 0.948892
\(850\) 36009.4 1.45307
\(851\) 19475.0 0.784481
\(852\) 6655.47 0.267621
\(853\) 10283.7 0.412786 0.206393 0.978469i \(-0.433827\pi\)
0.206393 + 0.978469i \(0.433827\pi\)
\(854\) −1323.44 −0.0530294
\(855\) 963.026 0.0385202
\(856\) 37180.4 1.48458
\(857\) 7061.07 0.281449 0.140724 0.990049i \(-0.455057\pi\)
0.140724 + 0.990049i \(0.455057\pi\)
\(858\) 18628.3 0.741212
\(859\) −7401.74 −0.293998 −0.146999 0.989137i \(-0.546961\pi\)
−0.146999 + 0.989137i \(0.546961\pi\)
\(860\) 0 0
\(861\) 14659.6 0.580254
\(862\) −57685.4 −2.27932
\(863\) −5356.38 −0.211278 −0.105639 0.994405i \(-0.533689\pi\)
−0.105639 + 0.994405i \(0.533689\pi\)
\(864\) 28907.7 1.13826
\(865\) −7617.98 −0.299444
\(866\) 71487.6 2.80514
\(867\) 1714.62 0.0671643
\(868\) 32682.8 1.27803
\(869\) −11880.9 −0.463787
\(870\) −8597.84 −0.335051
\(871\) −53010.0 −2.06220
\(872\) −29500.9 −1.14567
\(873\) 3836.31 0.148728
\(874\) 16232.5 0.628231
\(875\) 6214.31 0.240094
\(876\) 20366.5 0.785526
\(877\) −24690.2 −0.950660 −0.475330 0.879808i \(-0.657671\pi\)
−0.475330 + 0.879808i \(0.657671\pi\)
\(878\) 11497.6 0.441942
\(879\) −9352.83 −0.358889
\(880\) −161.822 −0.00619890
\(881\) −14555.9 −0.556640 −0.278320 0.960488i \(-0.589778\pi\)
−0.278320 + 0.960488i \(0.589778\pi\)
\(882\) −9899.67 −0.377936
\(883\) 25326.8 0.965251 0.482625 0.875827i \(-0.339683\pi\)
0.482625 + 0.875827i \(0.339683\pi\)
\(884\) −50164.5 −1.90862
\(885\) −3239.47 −0.123044
\(886\) 38013.6 1.44141
\(887\) 15323.2 0.580049 0.290025 0.957019i \(-0.406337\pi\)
0.290025 + 0.957019i \(0.406337\pi\)
\(888\) −20988.5 −0.793161
\(889\) 16337.4 0.616354
\(890\) 14224.7 0.535747
\(891\) 6883.69 0.258824
\(892\) 48130.3 1.80664
\(893\) 18174.9 0.681076
\(894\) 15948.4 0.596639
\(895\) −3477.73 −0.129886
\(896\) 21585.9 0.804839
\(897\) −21599.5 −0.803996
\(898\) 17453.6 0.648591
\(899\) 44103.6 1.63619
\(900\) 12876.2 0.476897
\(901\) −25382.7 −0.938536
\(902\) −26683.5 −0.984994
\(903\) 0 0
\(904\) −39902.1 −1.46806
\(905\) −6978.06 −0.256308
\(906\) 657.549 0.0241122
\(907\) 8536.27 0.312505 0.156253 0.987717i \(-0.450059\pi\)
0.156253 + 0.987717i \(0.450059\pi\)
\(908\) −47476.2 −1.73519
\(909\) 10792.7 0.393806
\(910\) −6835.79 −0.249016
\(911\) −50638.5 −1.84163 −0.920817 0.389995i \(-0.872477\pi\)
−0.920817 + 0.389995i \(0.872477\pi\)
\(912\) 656.454 0.0238348
\(913\) 1191.79 0.0432008
\(914\) 34843.8 1.26097
\(915\) 355.188 0.0128330
\(916\) 46058.7 1.66138
\(917\) −3159.02 −0.113762
\(918\) −46716.2 −1.67959
\(919\) −47429.0 −1.70244 −0.851218 0.524812i \(-0.824136\pi\)
−0.851218 + 0.524812i \(0.824136\pi\)
\(920\) −5000.28 −0.179189
\(921\) −30545.4 −1.09284
\(922\) 50454.6 1.80221
\(923\) 7140.57 0.254642
\(924\) 8425.34 0.299971
\(925\) 26698.7 0.949024
\(926\) −55407.9 −1.96633
\(927\) −10280.6 −0.364249
\(928\) 30846.6 1.09115
\(929\) 24002.2 0.847671 0.423836 0.905739i \(-0.360683\pi\)
0.423836 + 0.905739i \(0.360683\pi\)
\(930\) −14281.3 −0.503552
\(931\) −10508.4 −0.369925
\(932\) −672.907 −0.0236500
\(933\) −17103.9 −0.600169
\(934\) 41731.1 1.46197
\(935\) 2950.24 0.103190
\(936\) −10860.1 −0.379245
\(937\) 10994.5 0.383325 0.191662 0.981461i \(-0.438612\pi\)
0.191662 + 0.981461i \(0.438612\pi\)
\(938\) −39036.1 −1.35882
\(939\) −39984.4 −1.38961
\(940\) −15056.0 −0.522419
\(941\) −17775.3 −0.615790 −0.307895 0.951420i \(-0.599625\pi\)
−0.307895 + 0.951420i \(0.599625\pi\)
\(942\) 32439.7 1.12202
\(943\) 30939.5 1.06843
\(944\) 1030.21 0.0355197
\(945\) −3909.89 −0.134591
\(946\) 0 0
\(947\) −19576.9 −0.671767 −0.335884 0.941904i \(-0.609035\pi\)
−0.335884 + 0.941904i \(0.609035\pi\)
\(948\) −39925.7 −1.36785
\(949\) 21851.0 0.747432
\(950\) 22253.6 0.760001
\(951\) 36372.4 1.24023
\(952\) −13736.5 −0.467649
\(953\) −32542.6 −1.10615 −0.553074 0.833132i \(-0.686545\pi\)
−0.553074 + 0.833132i \(0.686545\pi\)
\(954\) −14777.6 −0.501513
\(955\) 4890.58 0.165712
\(956\) −21749.2 −0.735793
\(957\) 11369.5 0.384038
\(958\) 12119.1 0.408717
\(959\) −23915.4 −0.805284
\(960\) −9646.93 −0.324326
\(961\) 43466.8 1.45906
\(962\) −60557.1 −2.02956
\(963\) 14808.5 0.495533
\(964\) −7693.86 −0.257057
\(965\) 10746.7 0.358496
\(966\) −15905.6 −0.529768
\(967\) −31808.6 −1.05780 −0.528901 0.848683i \(-0.677396\pi\)
−0.528901 + 0.848683i \(0.677396\pi\)
\(968\) 23001.0 0.763718
\(969\) −11968.0 −0.396768
\(970\) −5492.42 −0.181805
\(971\) −34826.2 −1.15100 −0.575502 0.817800i \(-0.695193\pi\)
−0.575502 + 0.817800i \(0.695193\pi\)
\(972\) −29377.9 −0.969440
\(973\) 21251.3 0.700190
\(974\) 72693.5 2.39143
\(975\) −29611.2 −0.972633
\(976\) −112.957 −0.00370456
\(977\) −3393.59 −0.111126 −0.0555632 0.998455i \(-0.517695\pi\)
−0.0555632 + 0.998455i \(0.517695\pi\)
\(978\) 68155.1 2.22839
\(979\) −18810.4 −0.614078
\(980\) 8705.15 0.283751
\(981\) −11749.9 −0.382410
\(982\) 63663.0 2.06881
\(983\) −10880.2 −0.353025 −0.176513 0.984298i \(-0.556482\pi\)
−0.176513 + 0.984298i \(0.556482\pi\)
\(984\) −33343.9 −1.08025
\(985\) 5768.09 0.186585
\(986\) −49849.4 −1.61007
\(987\) −17808.9 −0.574331
\(988\) −31001.3 −0.998263
\(989\) 0 0
\(990\) 1717.60 0.0551405
\(991\) −47618.4 −1.52639 −0.763194 0.646170i \(-0.776370\pi\)
−0.763194 + 0.646170i \(0.776370\pi\)
\(992\) 51237.3 1.63990
\(993\) 5258.23 0.168041
\(994\) 5258.25 0.167788
\(995\) 7574.53 0.241335
\(996\) 4004.99 0.127413
\(997\) −4548.20 −0.144476 −0.0722381 0.997387i \(-0.523014\pi\)
−0.0722381 + 0.997387i \(0.523014\pi\)
\(998\) 53682.9 1.70271
\(999\) −34637.1 −1.09697
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.1.5 yes 50
43.42 odd 2 1849.4.a.i.1.46 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.46 50 43.42 odd 2
1849.4.a.j.1.5 yes 50 1.1 even 1 trivial