Properties

Label 1849.4.a.i.1.46
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 1.46
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 \(-0.202175\pi\)
0.804982 + 0.593300i \(0.202175\pi\)
\(3\) 4.29077 0.825759 0.412880 0.910786i \(-0.364523\pi\)
0.412880 + 0.910786i \(0.364523\pi\)
\(4\) 12.7358 1.59198
\(5\) 2.70051 0.241541 0.120770 0.992680i \(-0.461464\pi\)
0.120770 + 0.992680i \(0.461464\pi\)
\(6\) 19.5387 1.32944
\(7\) −9.48123 −0.511938 −0.255969 0.966685i \(-0.582395\pi\)
−0.255969 + 0.966685i \(0.582395\pi\)
\(8\) 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 \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(20\) 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 \(-0.674702\pi\)
−0.521700 + 0.853129i \(0.674702\pi\)
\(30\) 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 \(-0.331894\pi\)
0.503911 + 0.863755i \(0.331894\pi\)
\(38\) −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 \(-0.510242\pi\)
−0.0321701 + 0.999482i \(0.510242\pi\)
\(62\) 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 \(-0.467544\pi\)
0.101788 + 0.994806i \(0.467544\pi\)
\(72\) −185.232 −0.303191
\(73\) 372.695 0.597543 0.298771 0.954325i \(-0.403423\pi\)
0.298771 + 0.954325i \(0.403423\pi\)
\(74\) 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 \(-0.741883\pi\)
−0.688847 + 0.724906i \(0.741883\pi\)
\(90\) −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 \(-0.779831\pi\)
−0.770175 + 0.637833i \(0.779831\pi\)
\(114\) −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 \(-0.464560\pi\)
0.111109 + 0.993808i \(0.464560\pi\)
\(132\) −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 \(-0.211888\pi\)
0.786505 + 0.617584i \(0.211888\pi\)
\(138\) −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 \(-0.572040\pi\)
−0.224395 + 0.974498i \(0.572040\pi\)
\(150\) −2299.85 −1.25188
\(151\) −33.6537 −0.0181371 −0.00906853 0.999959i \(-0.502887\pi\)
−0.00906853 + 0.999959i \(0.502887\pi\)
\(152\) −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 \(-0.638668\pi\)
−0.421990 + 0.906601i \(0.638668\pi\)
\(158\) 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 \(-0.816328\pi\)
−0.838091 + 0.545531i \(0.816328\pi\)
\(164\) −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 \(-0.586650\pi\)
−0.268869 + 0.963177i \(0.586650\pi\)
\(180\) −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 \(-0.388546\pi\)
0.343032 + 0.939324i \(0.388546\pi\)
\(192\) −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 \(-0.333490\pi\)
0.499575 + 0.866271i \(0.333490\pi\)
\(200\) −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 \(-0.648480\pi\)
−0.449730 + 0.893165i \(0.648480\pi\)
\(212\) −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 \(-0.692058\pi\)
−0.567418 + 0.823430i \(0.692058\pi\)
\(224\) 1794.83 0.535367
\(225\) 1011.02 0.299562
\(226\) −8425.54 −2.47991
\(227\) 3727.76 1.08996 0.544979 0.838450i \(-0.316538\pi\)
0.544979 + 0.838450i \(0.316538\pi\)
\(228\) −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 \(-0.497636\pi\)
0.00742786 + 0.999972i \(0.497636\pi\)
\(234\) 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 \(-0.474273\pi\)
0.0807348 + 0.996736i \(0.474273\pi\)
\(242\) −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 \(-0.157798\pi\)
0.879619 + 0.475679i \(0.157798\pi\)
\(258\) 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 \(-0.684321\pi\)
−0.547237 + 0.836977i \(0.684321\pi\)
\(264\) −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 \(-0.514820\pi\)
−0.0465431 + 0.998916i \(0.514820\pi\)
\(278\) 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 \(-0.818275\pi\)
−0.841411 + 0.540395i \(0.818275\pi\)
\(314\) −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 \(-0.467556\pi\)
0.101749 + 0.994810i \(0.467556\pi\)
\(332\) −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 \(-0.727108\pi\)
−0.654470 + 0.756088i \(0.727108\pi\)
\(348\) −8904.52 −1.37165
\(349\) −11788.1 −1.80803 −0.904015 0.427501i \(-0.859394\pi\)
−0.904015 + 0.427501i \(0.859394\pi\)
\(350\) 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 \(-0.714542\pi\)
−0.624120 + 0.781328i \(0.714542\pi\)
\(374\) −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 \(-0.506009\pi\)
−0.0188758 + 0.999822i \(0.506009\pi\)
\(384\) −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 \(-0.236644\pi\)
0.736145 + 0.676824i \(0.236644\pi\)
\(390\) −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 \(-0.315800\pi\)
0.546920 + 0.837185i \(0.315800\pi\)
\(410\) −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 \(-0.250912\pi\)
0.705077 + 0.709130i \(0.250912\pi\)
\(420\) −1399.18 −0.162555
\(421\) 4362.59 0.505035 0.252517 0.967592i \(-0.418742\pi\)
0.252517 + 0.967592i \(0.418742\pi\)
\(422\) −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 \(-0.163355\pi\)
0.871180 + 0.490963i \(0.163355\pi\)
\(434\) −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 \(-0.435441\pi\)
0.201430 + 0.979503i \(0.435441\pi\)
\(450\) 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 \(-0.371916\pi\)
0.391616 + 0.920129i \(0.371916\pi\)
\(458\) 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 \(-0.709101\pi\)
−0.610674 + 0.791882i \(0.709101\pi\)
\(464\) 600.456 0.0600764
\(465\) 3136.23 0.312772
\(466\) 240.596 0.0239172
\(467\) 9164.29 0.908079 0.454039 0.890982i \(-0.349983\pi\)
0.454039 + 0.890982i \(0.349983\pi\)
\(468\) 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 \(-0.277897\pi\)
0.642501 + 0.766285i \(0.277897\pi\)
\(492\) −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 \(-0.322641\pi\)
0.528804 + 0.848744i \(0.322641\pi\)
\(500\) −8347.49 −0.746623
\(501\) 7901.67 0.704632
\(502\) −7455.45 −0.662855
\(503\) −10616.1 −0.941050 −0.470525 0.882387i \(-0.655935\pi\)
−0.470525 + 0.882387i \(0.655935\pi\)
\(504\) 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 \(-0.537548\pi\)
−0.117688 + 0.993051i \(0.537548\pi\)
\(522\) 6373.32 0.534392
\(523\) 11873.6 0.992726 0.496363 0.868115i \(-0.334669\pi\)
0.496363 + 0.868115i \(0.334669\pi\)
\(524\) 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 \(-0.442808\pi\)
0.178710 + 0.983902i \(0.442808\pi\)
\(572\) 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 \(-0.641583\pi\)
−0.430273 + 0.902699i \(0.641583\pi\)
\(578\) −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 \(-0.761402\pi\)
−0.731977 + 0.681330i \(0.761402\pi\)
\(588\) −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 \(-0.680525\pi\)
−0.537218 + 0.843443i \(0.680525\pi\)
\(594\) 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 \(-0.296800\pi\)
0.595888 + 0.803068i \(0.296800\pi\)
\(602\) 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 \(-0.815416\pi\)
−0.836525 + 0.547929i \(0.815416\pi\)
\(608\) 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 \(-0.121880\pi\)
0.927586 + 0.373610i \(0.121880\pi\)
\(632\) 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 \(-0.470645\pi\)
0.0920921 + 0.995750i \(0.470645\pi\)
\(642\) −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 \(-0.875967\pi\)
−0.925037 + 0.379876i \(0.875967\pi\)
\(648\) −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 \(-0.595253\pi\)
−0.294801 + 0.955559i \(0.595253\pi\)
\(654\) 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 \(-0.516673\pi\)
−0.0523573 + 0.998628i \(0.516673\pi\)
\(674\) 6767.97 0.386784
\(675\) 17974.5 1.02495
\(676\) 15798.0 0.898840
\(677\) 21093.1 1.19745 0.598724 0.800955i \(-0.295675\pi\)
0.598724 + 0.800955i \(0.295675\pi\)
\(678\) −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 \(-0.403421\pi\)
0.298778 + 0.954323i \(0.403421\pi\)
\(692\) 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 \(-0.702227\pi\)
−0.593431 + 0.804885i \(0.702227\pi\)
\(728\) 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 \(-0.417930\pi\)
0.254982 + 0.966946i \(0.417930\pi\)
\(734\) −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 \(-0.638671\pi\)
−0.421999 + 0.906596i \(0.638671\pi\)
\(740\) 7801.18 0.387536
\(741\) 10444.5 0.517798
\(742\) 16312.2 0.807061
\(743\) 30362.0 1.49916 0.749579 0.661915i \(-0.230256\pi\)
0.749579 + 0.661915i \(0.230256\pi\)
\(744\) 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 \(-0.583975\pi\)
−0.260765 + 0.965402i \(0.583975\pi\)
\(752\) −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 \(-0.328377\pi\)
0.513422 + 0.858136i \(0.328377\pi\)
\(758\) −26184.4 −1.25469
\(759\) 5990.80 0.286498
\(760\) −2417.91 −0.115403
\(761\) −12272.6 −0.584601 −0.292300 0.956327i \(-0.594421\pi\)
−0.292300 + 0.956327i \(0.594421\pi\)
\(762\) 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 \(-0.352386\pi\)
0.447300 + 0.894384i \(0.352386\pi\)
\(774\) 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 \(-0.705123\pi\)
−0.600730 + 0.799452i \(0.705123\pi\)
\(812\) 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 \(-0.101364\pi\)
0.949724 + 0.313090i \(0.101364\pi\)
\(830\) −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 \(-0.610163\pi\)
−0.339219 + 0.940707i \(0.610163\pi\)
\(840\) −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 \(-0.453039\pi\)
0.146999 + 0.989137i \(0.453039\pi\)
\(860\) 0 0
\(861\) 14659.6 0.580254
\(862\) 57685.4 2.27932
\(863\) 5356.38 0.211278 0.105639 0.994405i \(-0.466311\pi\)
0.105639 + 0.994405i \(0.466311\pi\)
\(864\) 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 \(-0.593663\pi\)
−0.290025 + 0.957019i \(0.593663\pi\)
\(888\) 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 \(-0.127523\pi\)
0.920817 + 0.389995i \(0.127523\pi\)
\(912\) 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 \(-0.639317\pi\)
−0.423836 + 0.905739i \(0.639317\pi\)
\(930\) 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 \(-0.561388\pi\)
−0.191662 + 0.981461i \(0.561388\pi\)
\(938\) −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 \(-0.313455\pi\)
0.553074 + 0.833132i \(0.313455\pi\)
\(954\) 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 \(-0.443518\pi\)
0.176513 + 0.984298i \(0.443518\pi\)
\(984\) −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 \(-0.223630\pi\)
0.763194 + 0.646170i \(0.223630\pi\)
\(992\) −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 \(-0.476986\pi\)
0.0722381 + 0.997387i \(0.476986\pi\)
\(998\) 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.i.1.46 50
43.42 odd 2 1849.4.a.j.1.5 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.46 50 1.1 even 1 trivial
1849.4.a.j.1.5 yes 50 43.42 odd 2