Properties

Label 1849.4.a.i.1.42
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 1.42
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.08033 q^{2} +6.02226 q^{3} +8.64912 q^{4} -2.18258 q^{5} +24.5728 q^{6} -8.90728 q^{7} +2.64861 q^{8} +9.26767 q^{9} +O(q^{10})\) \(q+4.08033 q^{2} +6.02226 q^{3} +8.64912 q^{4} -2.18258 q^{5} +24.5728 q^{6} -8.90728 q^{7} +2.64861 q^{8} +9.26767 q^{9} -8.90564 q^{10} +2.96222 q^{11} +52.0873 q^{12} +15.9248 q^{13} -36.3447 q^{14} -13.1441 q^{15} -58.3857 q^{16} -54.4270 q^{17} +37.8152 q^{18} -136.251 q^{19} -18.8774 q^{20} -53.6420 q^{21} +12.0868 q^{22} +153.274 q^{23} +15.9506 q^{24} -120.236 q^{25} +64.9786 q^{26} -106.789 q^{27} -77.0401 q^{28} -69.4344 q^{29} -53.6321 q^{30} -13.1165 q^{31} -259.422 q^{32} +17.8393 q^{33} -222.080 q^{34} +19.4408 q^{35} +80.1572 q^{36} +306.611 q^{37} -555.948 q^{38} +95.9035 q^{39} -5.78080 q^{40} -27.1862 q^{41} -218.877 q^{42} +25.6206 q^{44} -20.2274 q^{45} +625.409 q^{46} -621.994 q^{47} -351.614 q^{48} -263.660 q^{49} -490.604 q^{50} -327.774 q^{51} +137.736 q^{52} -97.9357 q^{53} -435.734 q^{54} -6.46527 q^{55} -23.5919 q^{56} -820.538 q^{57} -283.315 q^{58} -417.172 q^{59} -113.684 q^{60} +608.416 q^{61} -53.5198 q^{62} -82.5498 q^{63} -591.443 q^{64} -34.7572 q^{65} +72.7902 q^{66} +772.902 q^{67} -470.745 q^{68} +923.057 q^{69} +79.3251 q^{70} -858.225 q^{71} +24.5464 q^{72} -14.7922 q^{73} +1251.07 q^{74} -724.095 q^{75} -1178.45 q^{76} -26.3853 q^{77} +391.318 q^{78} +872.351 q^{79} +127.431 q^{80} -893.337 q^{81} -110.929 q^{82} -1057.72 q^{83} -463.956 q^{84} +118.791 q^{85} -418.152 q^{87} +7.84576 q^{88} +444.661 q^{89} -82.5346 q^{90} -141.847 q^{91} +1325.69 q^{92} -78.9912 q^{93} -2537.94 q^{94} +297.378 q^{95} -1562.31 q^{96} +150.389 q^{97} -1075.82 q^{98} +27.4529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.08033 1.44262 0.721308 0.692615i \(-0.243541\pi\)
0.721308 + 0.692615i \(0.243541\pi\)
\(3\) 6.02226 1.15899 0.579493 0.814977i \(-0.303251\pi\)
0.579493 + 0.814977i \(0.303251\pi\)
\(4\) 8.64912 1.08114
\(5\) −2.18258 −0.195216 −0.0976078 0.995225i \(-0.531119\pi\)
−0.0976078 + 0.995225i \(0.531119\pi\)
\(6\) 24.5728 1.67197
\(7\) −8.90728 −0.480948 −0.240474 0.970656i \(-0.577303\pi\)
−0.240474 + 0.970656i \(0.577303\pi\)
\(8\) 2.64861 0.117053
\(9\) 9.26767 0.343247
\(10\) −8.90564 −0.281621
\(11\) 2.96222 0.0811948 0.0405974 0.999176i \(-0.487074\pi\)
0.0405974 + 0.999176i \(0.487074\pi\)
\(12\) 52.0873 1.25302
\(13\) 15.9248 0.339750 0.169875 0.985466i \(-0.445664\pi\)
0.169875 + 0.985466i \(0.445664\pi\)
\(14\) −36.3447 −0.693823
\(15\) −13.1441 −0.226252
\(16\) −58.3857 −0.912277
\(17\) −54.4270 −0.776499 −0.388250 0.921554i \(-0.626920\pi\)
−0.388250 + 0.921554i \(0.626920\pi\)
\(18\) 37.8152 0.495174
\(19\) −136.251 −1.64516 −0.822580 0.568649i \(-0.807466\pi\)
−0.822580 + 0.568649i \(0.807466\pi\)
\(20\) −18.8774 −0.211055
\(21\) −53.6420 −0.557412
\(22\) 12.0868 0.117133
\(23\) 153.274 1.38956 0.694780 0.719223i \(-0.255502\pi\)
0.694780 + 0.719223i \(0.255502\pi\)
\(24\) 15.9506 0.135663
\(25\) −120.236 −0.961891
\(26\) 64.9786 0.490129
\(27\) −106.789 −0.761167
\(28\) −77.0401 −0.519972
\(29\) −69.4344 −0.444608 −0.222304 0.974977i \(-0.571358\pi\)
−0.222304 + 0.974977i \(0.571358\pi\)
\(30\) −53.6321 −0.326395
\(31\) −13.1165 −0.0759935 −0.0379968 0.999278i \(-0.512098\pi\)
−0.0379968 + 0.999278i \(0.512098\pi\)
\(32\) −259.422 −1.43312
\(33\) 17.8393 0.0941036
\(34\) −222.080 −1.12019
\(35\) 19.4408 0.0938886
\(36\) 80.1572 0.371098
\(37\) 306.611 1.36234 0.681168 0.732127i \(-0.261472\pi\)
0.681168 + 0.732127i \(0.261472\pi\)
\(38\) −555.948 −2.37333
\(39\) 95.9035 0.393765
\(40\) −5.78080 −0.0228506
\(41\) −27.1862 −0.103555 −0.0517776 0.998659i \(-0.516489\pi\)
−0.0517776 + 0.998659i \(0.516489\pi\)
\(42\) −218.877 −0.804131
\(43\) 0 0
\(44\) 25.6206 0.0877829
\(45\) −20.2274 −0.0670072
\(46\) 625.409 2.00460
\(47\) −621.994 −1.93037 −0.965183 0.261576i \(-0.915758\pi\)
−0.965183 + 0.261576i \(0.915758\pi\)
\(48\) −351.614 −1.05732
\(49\) −263.660 −0.768689
\(50\) −490.604 −1.38764
\(51\) −327.774 −0.899951
\(52\) 137.736 0.367317
\(53\) −97.9357 −0.253821 −0.126910 0.991914i \(-0.540506\pi\)
−0.126910 + 0.991914i \(0.540506\pi\)
\(54\) −435.734 −1.09807
\(55\) −6.46527 −0.0158505
\(56\) −23.5919 −0.0562965
\(57\) −820.538 −1.90672
\(58\) −283.315 −0.641399
\(59\) −417.172 −0.920529 −0.460264 0.887782i \(-0.652245\pi\)
−0.460264 + 0.887782i \(0.652245\pi\)
\(60\) −113.684 −0.244610
\(61\) 608.416 1.27704 0.638522 0.769604i \(-0.279546\pi\)
0.638522 + 0.769604i \(0.279546\pi\)
\(62\) −53.5198 −0.109629
\(63\) −82.5498 −0.165084
\(64\) −591.443 −1.15516
\(65\) −34.7572 −0.0663245
\(66\) 72.7902 0.135755
\(67\) 772.902 1.40933 0.704664 0.709541i \(-0.251098\pi\)
0.704664 + 0.709541i \(0.251098\pi\)
\(68\) −470.745 −0.839504
\(69\) 923.057 1.61048
\(70\) 79.3251 0.135445
\(71\) −858.225 −1.43454 −0.717272 0.696793i \(-0.754609\pi\)
−0.717272 + 0.696793i \(0.754609\pi\)
\(72\) 24.5464 0.0401781
\(73\) −14.7922 −0.0237164 −0.0118582 0.999930i \(-0.503775\pi\)
−0.0118582 + 0.999930i \(0.503775\pi\)
\(74\) 1251.07 1.96533
\(75\) −724.095 −1.11482
\(76\) −1178.45 −1.77865
\(77\) −26.3853 −0.0390505
\(78\) 391.318 0.568052
\(79\) 872.351 1.24237 0.621185 0.783664i \(-0.286652\pi\)
0.621185 + 0.783664i \(0.286652\pi\)
\(80\) 127.431 0.178091
\(81\) −893.337 −1.22543
\(82\) −110.929 −0.149390
\(83\) −1057.72 −1.39880 −0.699399 0.714732i \(-0.746549\pi\)
−0.699399 + 0.714732i \(0.746549\pi\)
\(84\) −463.956 −0.602640
\(85\) 118.791 0.151585
\(86\) 0 0
\(87\) −418.152 −0.515295
\(88\) 7.84576 0.00950411
\(89\) 444.661 0.529596 0.264798 0.964304i \(-0.414695\pi\)
0.264798 + 0.964304i \(0.414695\pi\)
\(90\) −82.5346 −0.0966656
\(91\) −141.847 −0.163402
\(92\) 1325.69 1.50231
\(93\) −78.9912 −0.0880754
\(94\) −2537.94 −2.78478
\(95\) 297.378 0.321161
\(96\) −1562.31 −1.66096
\(97\) 150.389 0.157420 0.0787098 0.996898i \(-0.474920\pi\)
0.0787098 + 0.996898i \(0.474920\pi\)
\(98\) −1075.82 −1.10892
\(99\) 27.4529 0.0278699
\(100\) −1039.94 −1.03994
\(101\) 400.658 0.394722 0.197361 0.980331i \(-0.436763\pi\)
0.197361 + 0.980331i \(0.436763\pi\)
\(102\) −1337.43 −1.29828
\(103\) 956.042 0.914579 0.457290 0.889318i \(-0.348820\pi\)
0.457290 + 0.889318i \(0.348820\pi\)
\(104\) 42.1786 0.0397688
\(105\) 117.078 0.108816
\(106\) −399.610 −0.366166
\(107\) 589.174 0.532314 0.266157 0.963930i \(-0.414246\pi\)
0.266157 + 0.963930i \(0.414246\pi\)
\(108\) −923.629 −0.822928
\(109\) 845.145 0.742662 0.371331 0.928500i \(-0.378901\pi\)
0.371331 + 0.928500i \(0.378901\pi\)
\(110\) −26.3805 −0.0228662
\(111\) 1846.49 1.57893
\(112\) 520.058 0.438758
\(113\) 216.642 0.180353 0.0901767 0.995926i \(-0.471257\pi\)
0.0901767 + 0.995926i \(0.471257\pi\)
\(114\) −3348.07 −2.75066
\(115\) −334.533 −0.271264
\(116\) −600.546 −0.480684
\(117\) 147.586 0.116618
\(118\) −1702.20 −1.32797
\(119\) 484.797 0.373456
\(120\) −34.8135 −0.0264835
\(121\) −1322.23 −0.993407
\(122\) 2482.54 1.84228
\(123\) −163.722 −0.120019
\(124\) −113.446 −0.0821596
\(125\) 535.247 0.382992
\(126\) −336.831 −0.238153
\(127\) 448.568 0.313417 0.156708 0.987645i \(-0.449912\pi\)
0.156708 + 0.987645i \(0.449912\pi\)
\(128\) −337.906 −0.233336
\(129\) 0 0
\(130\) −141.821 −0.0956808
\(131\) −1293.46 −0.862675 −0.431338 0.902191i \(-0.641958\pi\)
−0.431338 + 0.902191i \(0.641958\pi\)
\(132\) 154.294 0.101739
\(133\) 1213.62 0.791237
\(134\) 3153.70 2.03312
\(135\) 233.075 0.148592
\(136\) −144.156 −0.0908917
\(137\) −2103.91 −1.31204 −0.656019 0.754744i \(-0.727761\pi\)
−0.656019 + 0.754744i \(0.727761\pi\)
\(138\) 3766.38 2.32330
\(139\) −2683.27 −1.63735 −0.818676 0.574255i \(-0.805292\pi\)
−0.818676 + 0.574255i \(0.805292\pi\)
\(140\) 168.146 0.101507
\(141\) −3745.81 −2.23727
\(142\) −3501.85 −2.06950
\(143\) 47.1728 0.0275859
\(144\) −541.100 −0.313136
\(145\) 151.546 0.0867945
\(146\) −60.3571 −0.0342136
\(147\) −1587.83 −0.890899
\(148\) 2651.91 1.47288
\(149\) 2733.30 1.50283 0.751413 0.659832i \(-0.229373\pi\)
0.751413 + 0.659832i \(0.229373\pi\)
\(150\) −2954.55 −1.60825
\(151\) −2187.16 −1.17873 −0.589367 0.807865i \(-0.700623\pi\)
−0.589367 + 0.807865i \(0.700623\pi\)
\(152\) −360.875 −0.192571
\(153\) −504.411 −0.266531
\(154\) −107.661 −0.0563348
\(155\) 28.6278 0.0148351
\(156\) 829.480 0.425715
\(157\) −2618.77 −1.33122 −0.665608 0.746302i \(-0.731828\pi\)
−0.665608 + 0.746302i \(0.731828\pi\)
\(158\) 3559.48 1.79226
\(159\) −589.795 −0.294175
\(160\) 566.209 0.279767
\(161\) −1365.26 −0.668306
\(162\) −3645.11 −1.76782
\(163\) −556.769 −0.267543 −0.133772 0.991012i \(-0.542709\pi\)
−0.133772 + 0.991012i \(0.542709\pi\)
\(164\) −235.136 −0.111958
\(165\) −38.9356 −0.0183705
\(166\) −4315.86 −2.01793
\(167\) −84.4752 −0.0391430 −0.0195715 0.999808i \(-0.506230\pi\)
−0.0195715 + 0.999808i \(0.506230\pi\)
\(168\) −142.077 −0.0652468
\(169\) −1943.40 −0.884570
\(170\) 484.707 0.218679
\(171\) −1262.73 −0.564697
\(172\) 0 0
\(173\) −1870.04 −0.821829 −0.410914 0.911674i \(-0.634790\pi\)
−0.410914 + 0.911674i \(0.634790\pi\)
\(174\) −1706.20 −0.743372
\(175\) 1070.98 0.462620
\(176\) −172.951 −0.0740721
\(177\) −2512.32 −1.06688
\(178\) 1814.37 0.764003
\(179\) 590.661 0.246637 0.123319 0.992367i \(-0.460646\pi\)
0.123319 + 0.992367i \(0.460646\pi\)
\(180\) −174.949 −0.0724441
\(181\) 1963.05 0.806147 0.403073 0.915168i \(-0.367942\pi\)
0.403073 + 0.915168i \(0.367942\pi\)
\(182\) −578.783 −0.235726
\(183\) 3664.04 1.48007
\(184\) 405.963 0.162652
\(185\) −669.201 −0.265949
\(186\) −322.311 −0.127059
\(187\) −161.225 −0.0630477
\(188\) −5379.70 −2.08699
\(189\) 951.198 0.366082
\(190\) 1213.40 0.463312
\(191\) 3347.99 1.26833 0.634167 0.773196i \(-0.281343\pi\)
0.634167 + 0.773196i \(0.281343\pi\)
\(192\) −3561.82 −1.33881
\(193\) 3819.93 1.42469 0.712344 0.701831i \(-0.247634\pi\)
0.712344 + 0.701831i \(0.247634\pi\)
\(194\) 613.637 0.227096
\(195\) −209.317 −0.0768692
\(196\) −2280.43 −0.831060
\(197\) 3154.91 1.14100 0.570502 0.821296i \(-0.306749\pi\)
0.570502 + 0.821296i \(0.306749\pi\)
\(198\) 112.017 0.0402055
\(199\) −5307.46 −1.89063 −0.945317 0.326154i \(-0.894247\pi\)
−0.945317 + 0.326154i \(0.894247\pi\)
\(200\) −318.459 −0.112592
\(201\) 4654.62 1.63339
\(202\) 1634.82 0.569432
\(203\) 618.472 0.213834
\(204\) −2834.95 −0.972973
\(205\) 59.3359 0.0202156
\(206\) 3900.97 1.31939
\(207\) 1420.49 0.476962
\(208\) −929.782 −0.309946
\(209\) −403.604 −0.133578
\(210\) 477.717 0.156979
\(211\) 2869.87 0.936352 0.468176 0.883635i \(-0.344911\pi\)
0.468176 + 0.883635i \(0.344911\pi\)
\(212\) −847.058 −0.274416
\(213\) −5168.46 −1.66262
\(214\) 2404.02 0.767924
\(215\) 0 0
\(216\) −282.842 −0.0890970
\(217\) 116.833 0.0365489
\(218\) 3448.47 1.07138
\(219\) −89.0825 −0.0274869
\(220\) −55.9189 −0.0171366
\(221\) −866.740 −0.263816
\(222\) 7534.29 2.27779
\(223\) 6510.38 1.95501 0.977506 0.210908i \(-0.0676422\pi\)
0.977506 + 0.210908i \(0.0676422\pi\)
\(224\) 2310.75 0.689255
\(225\) −1114.31 −0.330166
\(226\) 883.970 0.260181
\(227\) −1210.57 −0.353958 −0.176979 0.984215i \(-0.556632\pi\)
−0.176979 + 0.984215i \(0.556632\pi\)
\(228\) −7096.93 −2.06143
\(229\) −1712.57 −0.494192 −0.247096 0.968991i \(-0.579476\pi\)
−0.247096 + 0.968991i \(0.579476\pi\)
\(230\) −1365.00 −0.391329
\(231\) −158.899 −0.0452589
\(232\) −183.905 −0.0520428
\(233\) 4497.51 1.26456 0.632278 0.774741i \(-0.282120\pi\)
0.632278 + 0.774741i \(0.282120\pi\)
\(234\) 602.200 0.168235
\(235\) 1357.55 0.376838
\(236\) −3608.17 −0.995220
\(237\) 5253.53 1.43989
\(238\) 1978.13 0.538753
\(239\) 6253.68 1.69254 0.846270 0.532754i \(-0.178843\pi\)
0.846270 + 0.532754i \(0.178843\pi\)
\(240\) 767.425 0.206405
\(241\) −4314.12 −1.15310 −0.576550 0.817062i \(-0.695601\pi\)
−0.576550 + 0.817062i \(0.695601\pi\)
\(242\) −5395.12 −1.43310
\(243\) −2496.62 −0.659087
\(244\) 5262.26 1.38066
\(245\) 575.459 0.150060
\(246\) −668.041 −0.173141
\(247\) −2169.77 −0.558943
\(248\) −34.7406 −0.00889528
\(249\) −6369.89 −1.62119
\(250\) 2183.99 0.552510
\(251\) −6821.84 −1.71550 −0.857750 0.514067i \(-0.828138\pi\)
−0.857750 + 0.514067i \(0.828138\pi\)
\(252\) −713.983 −0.178479
\(253\) 454.032 0.112825
\(254\) 1830.31 0.452140
\(255\) 715.392 0.175685
\(256\) 3352.77 0.818548
\(257\) −2048.98 −0.497322 −0.248661 0.968591i \(-0.579991\pi\)
−0.248661 + 0.968591i \(0.579991\pi\)
\(258\) 0 0
\(259\) −2731.07 −0.655213
\(260\) −300.619 −0.0717061
\(261\) −643.495 −0.152611
\(262\) −5277.76 −1.24451
\(263\) 2225.75 0.521847 0.260923 0.965360i \(-0.415973\pi\)
0.260923 + 0.965360i \(0.415973\pi\)
\(264\) 47.2493 0.0110151
\(265\) 213.752 0.0495498
\(266\) 4951.99 1.14145
\(267\) 2677.87 0.613794
\(268\) 6684.92 1.52368
\(269\) −639.777 −0.145011 −0.0725054 0.997368i \(-0.523099\pi\)
−0.0725054 + 0.997368i \(0.523099\pi\)
\(270\) 951.023 0.214361
\(271\) 2033.22 0.455754 0.227877 0.973690i \(-0.426822\pi\)
0.227877 + 0.973690i \(0.426822\pi\)
\(272\) 3177.76 0.708382
\(273\) −854.240 −0.189381
\(274\) −8584.66 −1.89277
\(275\) −356.166 −0.0781005
\(276\) 7983.63 1.74115
\(277\) −855.321 −0.185528 −0.0927640 0.995688i \(-0.529570\pi\)
−0.0927640 + 0.995688i \(0.529570\pi\)
\(278\) −10948.6 −2.36207
\(279\) −121.560 −0.0260845
\(280\) 51.4912 0.0109900
\(281\) 6456.77 1.37074 0.685372 0.728194i \(-0.259640\pi\)
0.685372 + 0.728194i \(0.259640\pi\)
\(282\) −15284.2 −3.22751
\(283\) 2550.24 0.535676 0.267838 0.963464i \(-0.413691\pi\)
0.267838 + 0.963464i \(0.413691\pi\)
\(284\) −7422.89 −1.55094
\(285\) 1790.89 0.372221
\(286\) 192.481 0.0397959
\(287\) 242.155 0.0498047
\(288\) −2404.24 −0.491913
\(289\) −1950.70 −0.397049
\(290\) 618.358 0.125211
\(291\) 905.683 0.182447
\(292\) −127.939 −0.0256407
\(293\) −5438.66 −1.08440 −0.542201 0.840249i \(-0.682409\pi\)
−0.542201 + 0.840249i \(0.682409\pi\)
\(294\) −6478.88 −1.28523
\(295\) 910.511 0.179702
\(296\) 812.092 0.159466
\(297\) −316.332 −0.0618028
\(298\) 11152.8 2.16800
\(299\) 2440.86 0.472103
\(300\) −6262.78 −1.20527
\(301\) 0 0
\(302\) −8924.36 −1.70046
\(303\) 2412.87 0.457477
\(304\) 7955.09 1.50084
\(305\) −1327.91 −0.249299
\(306\) −2058.17 −0.384502
\(307\) −5098.12 −0.947769 −0.473884 0.880587i \(-0.657149\pi\)
−0.473884 + 0.880587i \(0.657149\pi\)
\(308\) −228.210 −0.0422190
\(309\) 5757.54 1.05998
\(310\) 116.811 0.0214014
\(311\) 608.278 0.110908 0.0554539 0.998461i \(-0.482339\pi\)
0.0554539 + 0.998461i \(0.482339\pi\)
\(312\) 254.011 0.0460915
\(313\) 8591.81 1.55156 0.775779 0.631005i \(-0.217357\pi\)
0.775779 + 0.631005i \(0.217357\pi\)
\(314\) −10685.5 −1.92043
\(315\) 180.171 0.0322270
\(316\) 7545.07 1.34317
\(317\) −3275.74 −0.580391 −0.290196 0.956967i \(-0.593720\pi\)
−0.290196 + 0.956967i \(0.593720\pi\)
\(318\) −2406.56 −0.424381
\(319\) −205.680 −0.0360999
\(320\) 1290.87 0.225506
\(321\) 3548.16 0.616944
\(322\) −5570.70 −0.964108
\(323\) 7415.72 1.27747
\(324\) −7726.58 −1.32486
\(325\) −1914.74 −0.326802
\(326\) −2271.80 −0.385962
\(327\) 5089.69 0.860735
\(328\) −72.0055 −0.0121215
\(329\) 5540.28 0.928406
\(330\) −158.870 −0.0265016
\(331\) −7793.65 −1.29419 −0.647096 0.762409i \(-0.724017\pi\)
−0.647096 + 0.762409i \(0.724017\pi\)
\(332\) −9148.37 −1.51230
\(333\) 2841.57 0.467618
\(334\) −344.687 −0.0564684
\(335\) −1686.92 −0.275123
\(336\) 3131.93 0.508514
\(337\) −1219.14 −0.197065 −0.0985324 0.995134i \(-0.531415\pi\)
−0.0985324 + 0.995134i \(0.531415\pi\)
\(338\) −7929.72 −1.27609
\(339\) 1304.67 0.209027
\(340\) 1027.44 0.163884
\(341\) −38.8540 −0.00617028
\(342\) −5152.34 −0.814640
\(343\) 5403.70 0.850648
\(344\) 0 0
\(345\) −2014.64 −0.314391
\(346\) −7630.38 −1.18558
\(347\) −8942.44 −1.38344 −0.691722 0.722164i \(-0.743148\pi\)
−0.691722 + 0.722164i \(0.743148\pi\)
\(348\) −3616.65 −0.557105
\(349\) 11578.0 1.77581 0.887904 0.460030i \(-0.152161\pi\)
0.887904 + 0.460030i \(0.152161\pi\)
\(350\) 4369.95 0.667382
\(351\) −1700.59 −0.258607
\(352\) −768.465 −0.116362
\(353\) −7587.38 −1.14401 −0.572005 0.820250i \(-0.693834\pi\)
−0.572005 + 0.820250i \(0.693834\pi\)
\(354\) −10251.1 −1.53910
\(355\) 1873.14 0.280045
\(356\) 3845.93 0.572567
\(357\) 2919.57 0.432830
\(358\) 2410.09 0.355803
\(359\) 8406.81 1.23592 0.617959 0.786211i \(-0.287960\pi\)
0.617959 + 0.786211i \(0.287960\pi\)
\(360\) −53.5745 −0.00784340
\(361\) 11705.3 1.70655
\(362\) 8009.91 1.16296
\(363\) −7962.79 −1.15134
\(364\) −1226.85 −0.176661
\(365\) 32.2851 0.00462981
\(366\) 14950.5 2.13518
\(367\) −5733.44 −0.815486 −0.407743 0.913097i \(-0.633684\pi\)
−0.407743 + 0.913097i \(0.633684\pi\)
\(368\) −8949.02 −1.26766
\(369\) −251.952 −0.0355450
\(370\) −2730.56 −0.383663
\(371\) 872.341 0.122075
\(372\) −683.204 −0.0952218
\(373\) −3701.87 −0.513875 −0.256938 0.966428i \(-0.582714\pi\)
−0.256938 + 0.966428i \(0.582714\pi\)
\(374\) −657.850 −0.0909536
\(375\) 3223.40 0.443882
\(376\) −1647.42 −0.225955
\(377\) −1105.73 −0.151056
\(378\) 3881.20 0.528115
\(379\) −1981.11 −0.268504 −0.134252 0.990947i \(-0.542863\pi\)
−0.134252 + 0.990947i \(0.542863\pi\)
\(380\) 2572.05 0.347220
\(381\) 2701.39 0.363246
\(382\) 13660.9 1.82972
\(383\) 4324.44 0.576942 0.288471 0.957489i \(-0.406853\pi\)
0.288471 + 0.957489i \(0.406853\pi\)
\(384\) −2034.96 −0.270433
\(385\) 57.5880 0.00762327
\(386\) 15586.6 2.05528
\(387\) 0 0
\(388\) 1300.73 0.170192
\(389\) 13257.2 1.72793 0.863967 0.503548i \(-0.167972\pi\)
0.863967 + 0.503548i \(0.167972\pi\)
\(390\) −854.082 −0.110893
\(391\) −8342.25 −1.07899
\(392\) −698.333 −0.0899774
\(393\) −7789.58 −0.999828
\(394\) 12873.1 1.64603
\(395\) −1903.97 −0.242530
\(396\) 237.443 0.0301312
\(397\) 2355.52 0.297784 0.148892 0.988853i \(-0.452429\pi\)
0.148892 + 0.988853i \(0.452429\pi\)
\(398\) −21656.2 −2.72746
\(399\) 7308.76 0.917032
\(400\) 7020.09 0.877511
\(401\) −3462.47 −0.431191 −0.215595 0.976483i \(-0.569169\pi\)
−0.215595 + 0.976483i \(0.569169\pi\)
\(402\) 18992.4 2.35636
\(403\) −208.878 −0.0258188
\(404\) 3465.33 0.426750
\(405\) 1949.78 0.239223
\(406\) 2523.57 0.308480
\(407\) 908.248 0.110615
\(408\) −868.145 −0.105342
\(409\) −7101.00 −0.858489 −0.429244 0.903188i \(-0.641220\pi\)
−0.429244 + 0.903188i \(0.641220\pi\)
\(410\) 242.110 0.0291633
\(411\) −12670.3 −1.52063
\(412\) 8268.92 0.988788
\(413\) 3715.87 0.442727
\(414\) 5796.09 0.688073
\(415\) 2308.56 0.273067
\(416\) −4131.25 −0.486902
\(417\) −16159.4 −1.89767
\(418\) −1646.84 −0.192702
\(419\) 7575.88 0.883308 0.441654 0.897185i \(-0.354392\pi\)
0.441654 + 0.897185i \(0.354392\pi\)
\(420\) 1012.62 0.117645
\(421\) −2054.30 −0.237816 −0.118908 0.992905i \(-0.537939\pi\)
−0.118908 + 0.992905i \(0.537939\pi\)
\(422\) 11710.0 1.35080
\(423\) −5764.44 −0.662592
\(424\) −259.394 −0.0297105
\(425\) 6544.10 0.746907
\(426\) −21089.0 −2.39851
\(427\) −5419.33 −0.614192
\(428\) 5095.83 0.575505
\(429\) 284.087 0.0319717
\(430\) 0 0
\(431\) −17175.1 −1.91948 −0.959742 0.280884i \(-0.909372\pi\)
−0.959742 + 0.280884i \(0.909372\pi\)
\(432\) 6234.94 0.694395
\(433\) −7628.21 −0.846624 −0.423312 0.905984i \(-0.639133\pi\)
−0.423312 + 0.905984i \(0.639133\pi\)
\(434\) 476.716 0.0527261
\(435\) 912.650 0.100594
\(436\) 7309.76 0.802922
\(437\) −20883.7 −2.28605
\(438\) −363.486 −0.0396531
\(439\) −6478.18 −0.704298 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(440\) −17.1240 −0.00185535
\(441\) −2443.52 −0.263850
\(442\) −3536.59 −0.380584
\(443\) 3350.48 0.359336 0.179668 0.983727i \(-0.442498\pi\)
0.179668 + 0.983727i \(0.442498\pi\)
\(444\) 15970.5 1.70704
\(445\) −970.508 −0.103385
\(446\) 26564.5 2.82033
\(447\) 16460.7 1.74175
\(448\) 5268.15 0.555573
\(449\) 14358.1 1.50913 0.754567 0.656223i \(-0.227847\pi\)
0.754567 + 0.656223i \(0.227847\pi\)
\(450\) −4546.76 −0.476303
\(451\) −80.5313 −0.00840814
\(452\) 1873.76 0.194987
\(453\) −13171.7 −1.36614
\(454\) −4939.53 −0.510625
\(455\) 309.592 0.0318987
\(456\) −2173.28 −0.223187
\(457\) 3151.07 0.322540 0.161270 0.986910i \(-0.448441\pi\)
0.161270 + 0.986910i \(0.448441\pi\)
\(458\) −6987.86 −0.712929
\(459\) 5812.19 0.591046
\(460\) −2893.41 −0.293274
\(461\) 1289.67 0.130295 0.0651475 0.997876i \(-0.479248\pi\)
0.0651475 + 0.997876i \(0.479248\pi\)
\(462\) −648.363 −0.0652912
\(463\) −9807.31 −0.984416 −0.492208 0.870478i \(-0.663810\pi\)
−0.492208 + 0.870478i \(0.663810\pi\)
\(464\) 4053.98 0.405606
\(465\) 172.404 0.0171937
\(466\) 18351.3 1.82427
\(467\) 2759.08 0.273394 0.136697 0.990613i \(-0.456351\pi\)
0.136697 + 0.990613i \(0.456351\pi\)
\(468\) 1276.49 0.126081
\(469\) −6884.46 −0.677814
\(470\) 5539.26 0.543632
\(471\) −15770.9 −1.54286
\(472\) −1104.93 −0.107751
\(473\) 0 0
\(474\) 21436.1 2.07720
\(475\) 16382.3 1.58246
\(476\) 4193.06 0.403758
\(477\) −907.636 −0.0871233
\(478\) 25517.1 2.44168
\(479\) 6246.31 0.595828 0.297914 0.954593i \(-0.403709\pi\)
0.297914 + 0.954593i \(0.403709\pi\)
\(480\) 3409.86 0.324246
\(481\) 4882.72 0.462854
\(482\) −17603.0 −1.66348
\(483\) −8221.93 −0.774557
\(484\) −11436.1 −1.07401
\(485\) −328.236 −0.0307308
\(486\) −10187.0 −0.950809
\(487\) −7223.74 −0.672154 −0.336077 0.941835i \(-0.609100\pi\)
−0.336077 + 0.941835i \(0.609100\pi\)
\(488\) 1611.46 0.149482
\(489\) −3353.01 −0.310079
\(490\) 2348.06 0.216479
\(491\) −9289.33 −0.853812 −0.426906 0.904296i \(-0.640397\pi\)
−0.426906 + 0.904296i \(0.640397\pi\)
\(492\) −1416.05 −0.129757
\(493\) 3779.11 0.345238
\(494\) −8853.37 −0.806340
\(495\) −59.9180 −0.00544064
\(496\) 765.818 0.0693271
\(497\) 7644.46 0.689941
\(498\) −25991.3 −2.33875
\(499\) −3220.27 −0.288896 −0.144448 0.989512i \(-0.546141\pi\)
−0.144448 + 0.989512i \(0.546141\pi\)
\(500\) 4629.42 0.414068
\(501\) −508.732 −0.0453662
\(502\) −27835.4 −2.47481
\(503\) 164.938 0.0146207 0.00731035 0.999973i \(-0.497673\pi\)
0.00731035 + 0.999973i \(0.497673\pi\)
\(504\) −218.642 −0.0193236
\(505\) −874.466 −0.0770559
\(506\) 1852.60 0.162763
\(507\) −11703.7 −1.02520
\(508\) 3879.71 0.338847
\(509\) −11574.1 −1.00788 −0.503940 0.863738i \(-0.668117\pi\)
−0.503940 + 0.863738i \(0.668117\pi\)
\(510\) 2919.04 0.253445
\(511\) 131.758 0.0114063
\(512\) 16383.7 1.41419
\(513\) 14550.0 1.25224
\(514\) −8360.52 −0.717445
\(515\) −2086.64 −0.178540
\(516\) 0 0
\(517\) −1842.48 −0.156736
\(518\) −11143.7 −0.945221
\(519\) −11261.9 −0.952487
\(520\) −92.0581 −0.00776349
\(521\) 16300.3 1.37069 0.685343 0.728220i \(-0.259652\pi\)
0.685343 + 0.728220i \(0.259652\pi\)
\(522\) −2625.67 −0.220158
\(523\) −12912.3 −1.07957 −0.539785 0.841803i \(-0.681494\pi\)
−0.539785 + 0.841803i \(0.681494\pi\)
\(524\) −11187.3 −0.932672
\(525\) 6449.72 0.536169
\(526\) 9081.81 0.752824
\(527\) 713.893 0.0590089
\(528\) −1041.56 −0.0858485
\(529\) 11326.0 0.930875
\(530\) 872.181 0.0714813
\(531\) −3866.22 −0.315969
\(532\) 10496.8 0.855437
\(533\) −432.935 −0.0351829
\(534\) 10926.6 0.885468
\(535\) −1285.92 −0.103916
\(536\) 2047.12 0.164966
\(537\) 3557.12 0.285849
\(538\) −2610.50 −0.209195
\(539\) −781.020 −0.0624135
\(540\) 2015.89 0.160648
\(541\) −2148.68 −0.170756 −0.0853780 0.996349i \(-0.527210\pi\)
−0.0853780 + 0.996349i \(0.527210\pi\)
\(542\) 8296.22 0.657478
\(543\) 11822.0 0.934312
\(544\) 14119.6 1.11281
\(545\) −1844.59 −0.144979
\(546\) −3485.58 −0.273204
\(547\) −7515.40 −0.587450 −0.293725 0.955890i \(-0.594895\pi\)
−0.293725 + 0.955890i \(0.594895\pi\)
\(548\) −18197.0 −1.41850
\(549\) 5638.60 0.438341
\(550\) −1453.28 −0.112669
\(551\) 9460.49 0.731452
\(552\) 2444.82 0.188512
\(553\) −7770.28 −0.597515
\(554\) −3489.99 −0.267646
\(555\) −4030.11 −0.308232
\(556\) −23207.9 −1.77021
\(557\) 13442.7 1.02259 0.511297 0.859404i \(-0.329165\pi\)
0.511297 + 0.859404i \(0.329165\pi\)
\(558\) −496.004 −0.0376300
\(559\) 0 0
\(560\) −1135.07 −0.0856524
\(561\) −970.938 −0.0730713
\(562\) 26345.8 1.97746
\(563\) 14488.6 1.08458 0.542291 0.840191i \(-0.317557\pi\)
0.542291 + 0.840191i \(0.317557\pi\)
\(564\) −32398.0 −2.41880
\(565\) −472.837 −0.0352078
\(566\) 10405.8 0.772774
\(567\) 7957.21 0.589368
\(568\) −2273.10 −0.167918
\(569\) −7940.71 −0.585047 −0.292524 0.956258i \(-0.594495\pi\)
−0.292524 + 0.956258i \(0.594495\pi\)
\(570\) 7307.41 0.536972
\(571\) 16716.9 1.22518 0.612592 0.790400i \(-0.290127\pi\)
0.612592 + 0.790400i \(0.290127\pi\)
\(572\) 408.003 0.0298242
\(573\) 20162.5 1.46998
\(574\) 988.072 0.0718490
\(575\) −18429.1 −1.33660
\(576\) −5481.29 −0.396506
\(577\) −22899.6 −1.65221 −0.826104 0.563518i \(-0.809448\pi\)
−0.826104 + 0.563518i \(0.809448\pi\)
\(578\) −7959.51 −0.572789
\(579\) 23004.6 1.65119
\(580\) 1310.74 0.0938370
\(581\) 9421.44 0.672749
\(582\) 3695.49 0.263201
\(583\) −290.107 −0.0206089
\(584\) −39.1788 −0.00277608
\(585\) −322.118 −0.0227657
\(586\) −22191.6 −1.56438
\(587\) 21550.7 1.51532 0.757661 0.652648i \(-0.226342\pi\)
0.757661 + 0.652648i \(0.226342\pi\)
\(588\) −13733.3 −0.963186
\(589\) 1787.14 0.125022
\(590\) 3715.19 0.259240
\(591\) 18999.7 1.32241
\(592\) −17901.7 −1.24283
\(593\) −26777.0 −1.85430 −0.927149 0.374694i \(-0.877748\pi\)
−0.927149 + 0.374694i \(0.877748\pi\)
\(594\) −1290.74 −0.0891577
\(595\) −1058.11 −0.0729044
\(596\) 23640.7 1.62476
\(597\) −31962.9 −2.19122
\(598\) 9959.53 0.681063
\(599\) −8296.25 −0.565903 −0.282951 0.959134i \(-0.591313\pi\)
−0.282951 + 0.959134i \(0.591313\pi\)
\(600\) −1917.85 −0.130493
\(601\) 1874.08 0.127197 0.0635984 0.997976i \(-0.479742\pi\)
0.0635984 + 0.997976i \(0.479742\pi\)
\(602\) 0 0
\(603\) 7163.00 0.483748
\(604\) −18917.0 −1.27438
\(605\) 2885.86 0.193929
\(606\) 9845.30 0.659963
\(607\) −2574.28 −0.172136 −0.0860681 0.996289i \(-0.527430\pi\)
−0.0860681 + 0.996289i \(0.527430\pi\)
\(608\) 35346.4 2.35771
\(609\) 3724.60 0.247830
\(610\) −5418.33 −0.359642
\(611\) −9905.15 −0.655842
\(612\) −4362.71 −0.288157
\(613\) 8303.26 0.547089 0.273545 0.961859i \(-0.411804\pi\)
0.273545 + 0.961859i \(0.411804\pi\)
\(614\) −20802.0 −1.36727
\(615\) 357.336 0.0234296
\(616\) −69.8844 −0.00457098
\(617\) −28299.9 −1.84653 −0.923267 0.384159i \(-0.874491\pi\)
−0.923267 + 0.384159i \(0.874491\pi\)
\(618\) 23492.7 1.52915
\(619\) −24822.2 −1.61178 −0.805888 0.592069i \(-0.798311\pi\)
−0.805888 + 0.592069i \(0.798311\pi\)
\(620\) 247.606 0.0160388
\(621\) −16368.0 −1.05769
\(622\) 2481.98 0.159997
\(623\) −3960.73 −0.254708
\(624\) −5599.39 −0.359223
\(625\) 13861.3 0.887125
\(626\) 35057.4 2.23830
\(627\) −2430.61 −0.154816
\(628\) −22650.1 −1.43923
\(629\) −16687.9 −1.05785
\(630\) 735.159 0.0464911
\(631\) 15805.6 0.997162 0.498581 0.866843i \(-0.333855\pi\)
0.498581 + 0.866843i \(0.333855\pi\)
\(632\) 2310.52 0.145423
\(633\) 17283.1 1.08522
\(634\) −13366.1 −0.837281
\(635\) −979.034 −0.0611839
\(636\) −5101.20 −0.318044
\(637\) −4198.74 −0.261162
\(638\) −839.243 −0.0520783
\(639\) −7953.75 −0.492403
\(640\) 737.506 0.0455508
\(641\) 4409.29 0.271695 0.135847 0.990730i \(-0.456624\pi\)
0.135847 + 0.990730i \(0.456624\pi\)
\(642\) 14477.7 0.890013
\(643\) −22435.9 −1.37603 −0.688014 0.725698i \(-0.741517\pi\)
−0.688014 + 0.725698i \(0.741517\pi\)
\(644\) −11808.3 −0.722532
\(645\) 0 0
\(646\) 30258.6 1.84289
\(647\) 7104.96 0.431723 0.215862 0.976424i \(-0.430744\pi\)
0.215862 + 0.976424i \(0.430744\pi\)
\(648\) −2366.10 −0.143440
\(649\) −1235.76 −0.0747422
\(650\) −7812.79 −0.471450
\(651\) 703.597 0.0423597
\(652\) −4815.56 −0.289252
\(653\) 6201.10 0.371620 0.185810 0.982586i \(-0.440509\pi\)
0.185810 + 0.982586i \(0.440509\pi\)
\(654\) 20767.6 1.24171
\(655\) 2823.08 0.168408
\(656\) 1587.28 0.0944710
\(657\) −137.089 −0.00814058
\(658\) 22606.2 1.33933
\(659\) −24325.9 −1.43794 −0.718971 0.695041i \(-0.755386\pi\)
−0.718971 + 0.695041i \(0.755386\pi\)
\(660\) −336.758 −0.0198611
\(661\) 4195.11 0.246855 0.123427 0.992354i \(-0.460611\pi\)
0.123427 + 0.992354i \(0.460611\pi\)
\(662\) −31800.7 −1.86702
\(663\) −5219.74 −0.305758
\(664\) −2801.50 −0.163734
\(665\) −2648.83 −0.154462
\(666\) 11594.5 0.674593
\(667\) −10642.5 −0.617810
\(668\) −730.636 −0.0423191
\(669\) 39207.3 2.26583
\(670\) −6883.19 −0.396897
\(671\) 1802.26 0.103689
\(672\) 13915.9 0.798837
\(673\) −30476.3 −1.74558 −0.872789 0.488097i \(-0.837691\pi\)
−0.872789 + 0.488097i \(0.837691\pi\)
\(674\) −4974.50 −0.284289
\(675\) 12839.9 0.732160
\(676\) −16808.7 −0.956343
\(677\) 13889.1 0.788482 0.394241 0.919007i \(-0.371008\pi\)
0.394241 + 0.919007i \(0.371008\pi\)
\(678\) 5323.50 0.301545
\(679\) −1339.56 −0.0757106
\(680\) 314.631 0.0177435
\(681\) −7290.38 −0.410232
\(682\) −158.537 −0.00890134
\(683\) −10181.3 −0.570388 −0.285194 0.958470i \(-0.592058\pi\)
−0.285194 + 0.958470i \(0.592058\pi\)
\(684\) −10921.5 −0.610516
\(685\) 4591.95 0.256130
\(686\) 22048.9 1.22716
\(687\) −10313.6 −0.572761
\(688\) 0 0
\(689\) −1559.61 −0.0862357
\(690\) −8220.42 −0.453545
\(691\) 16877.9 0.929182 0.464591 0.885525i \(-0.346201\pi\)
0.464591 + 0.885525i \(0.346201\pi\)
\(692\) −16174.2 −0.888511
\(693\) −244.531 −0.0134040
\(694\) −36488.1 −1.99578
\(695\) 5856.44 0.319637
\(696\) −1107.52 −0.0603169
\(697\) 1479.66 0.0804105
\(698\) 47242.1 2.56181
\(699\) 27085.2 1.46560
\(700\) 9263.02 0.500156
\(701\) −22820.0 −1.22953 −0.614765 0.788710i \(-0.710749\pi\)
−0.614765 + 0.788710i \(0.710749\pi\)
\(702\) −6938.98 −0.373070
\(703\) −41775.9 −2.24126
\(704\) −1751.98 −0.0937931
\(705\) 8175.53 0.436749
\(706\) −30959.1 −1.65037
\(707\) −3568.77 −0.189841
\(708\) −21729.4 −1.15345
\(709\) 1810.88 0.0959226 0.0479613 0.998849i \(-0.484728\pi\)
0.0479613 + 0.998849i \(0.484728\pi\)
\(710\) 7643.05 0.403998
\(711\) 8084.66 0.426440
\(712\) 1177.73 0.0619908
\(713\) −2010.43 −0.105597
\(714\) 11912.8 0.624407
\(715\) −102.958 −0.00538521
\(716\) 5108.69 0.266649
\(717\) 37661.3 1.96163
\(718\) 34302.6 1.78295
\(719\) 16290.1 0.844951 0.422476 0.906374i \(-0.361161\pi\)
0.422476 + 0.906374i \(0.361161\pi\)
\(720\) 1180.99 0.0611291
\(721\) −8515.74 −0.439865
\(722\) 47761.3 2.46190
\(723\) −25980.8 −1.33643
\(724\) 16978.7 0.871557
\(725\) 8348.54 0.427665
\(726\) −32490.8 −1.66095
\(727\) 9094.37 0.463950 0.231975 0.972722i \(-0.425481\pi\)
0.231975 + 0.972722i \(0.425481\pi\)
\(728\) −375.697 −0.0191267
\(729\) 9084.82 0.461557
\(730\) 131.734 0.00667903
\(731\) 0 0
\(732\) 31690.7 1.60017
\(733\) 12269.4 0.618256 0.309128 0.951020i \(-0.399963\pi\)
0.309128 + 0.951020i \(0.399963\pi\)
\(734\) −23394.4 −1.17643
\(735\) 3465.57 0.173917
\(736\) −39762.7 −1.99140
\(737\) 2289.51 0.114430
\(738\) −1028.05 −0.0512778
\(739\) −1437.99 −0.0715794 −0.0357897 0.999359i \(-0.511395\pi\)
−0.0357897 + 0.999359i \(0.511395\pi\)
\(740\) −5788.00 −0.287529
\(741\) −13066.9 −0.647807
\(742\) 3559.44 0.176107
\(743\) −28488.6 −1.40666 −0.703328 0.710865i \(-0.748304\pi\)
−0.703328 + 0.710865i \(0.748304\pi\)
\(744\) −209.217 −0.0103095
\(745\) −5965.65 −0.293375
\(746\) −15104.9 −0.741325
\(747\) −9802.63 −0.480133
\(748\) −1394.45 −0.0681633
\(749\) −5247.94 −0.256015
\(750\) 13152.5 0.640351
\(751\) 16276.2 0.790850 0.395425 0.918498i \(-0.370597\pi\)
0.395425 + 0.918498i \(0.370597\pi\)
\(752\) 36315.6 1.76103
\(753\) −41082.9 −1.98824
\(754\) −4511.75 −0.217915
\(755\) 4773.65 0.230107
\(756\) 8227.02 0.395786
\(757\) −12549.2 −0.602519 −0.301259 0.953542i \(-0.597407\pi\)
−0.301259 + 0.953542i \(0.597407\pi\)
\(758\) −8083.60 −0.387348
\(759\) 2734.30 0.130763
\(760\) 787.637 0.0375929
\(761\) 20472.0 0.975179 0.487589 0.873073i \(-0.337876\pi\)
0.487589 + 0.873073i \(0.337876\pi\)
\(762\) 11022.6 0.524024
\(763\) −7527.94 −0.357182
\(764\) 28957.1 1.37125
\(765\) 1100.92 0.0520310
\(766\) 17645.2 0.832305
\(767\) −6643.39 −0.312750
\(768\) 20191.3 0.948685
\(769\) 21892.6 1.02662 0.513309 0.858204i \(-0.328420\pi\)
0.513309 + 0.858204i \(0.328420\pi\)
\(770\) 234.978 0.0109974
\(771\) −12339.5 −0.576389
\(772\) 33039.0 1.54029
\(773\) −33744.5 −1.57012 −0.785061 0.619419i \(-0.787368\pi\)
−0.785061 + 0.619419i \(0.787368\pi\)
\(774\) 0 0
\(775\) 1577.08 0.0730975
\(776\) 398.322 0.0184264
\(777\) −16447.2 −0.759383
\(778\) 54093.8 2.49275
\(779\) 3704.13 0.170365
\(780\) −1810.41 −0.0831063
\(781\) −2542.25 −0.116477
\(782\) −34039.2 −1.55657
\(783\) 7414.82 0.338421
\(784\) 15394.0 0.701257
\(785\) 5715.67 0.259874
\(786\) −31784.1 −1.44237
\(787\) 40966.7 1.85553 0.927767 0.373159i \(-0.121725\pi\)
0.927767 + 0.373159i \(0.121725\pi\)
\(788\) 27287.2 1.23358
\(789\) 13404.1 0.604813
\(790\) −7768.85 −0.349877
\(791\) −1929.69 −0.0867406
\(792\) 72.7120 0.00326226
\(793\) 9688.91 0.433876
\(794\) 9611.30 0.429587
\(795\) 1287.27 0.0574275
\(796\) −45904.9 −2.04404
\(797\) −253.103 −0.0112489 −0.00562445 0.999984i \(-0.501790\pi\)
−0.00562445 + 0.999984i \(0.501790\pi\)
\(798\) 29822.2 1.32292
\(799\) 33853.3 1.49893
\(800\) 31192.0 1.37850
\(801\) 4120.98 0.181782
\(802\) −14128.0 −0.622042
\(803\) −43.8177 −0.00192565
\(804\) 40258.4 1.76592
\(805\) 2979.78 0.130464
\(806\) −852.294 −0.0372466
\(807\) −3852.91 −0.168065
\(808\) 1061.19 0.0462034
\(809\) −25849.5 −1.12339 −0.561694 0.827345i \(-0.689850\pi\)
−0.561694 + 0.827345i \(0.689850\pi\)
\(810\) 7955.74 0.345107
\(811\) −24298.3 −1.05207 −0.526036 0.850463i \(-0.676322\pi\)
−0.526036 + 0.850463i \(0.676322\pi\)
\(812\) 5349.24 0.231184
\(813\) 12244.6 0.528212
\(814\) 3705.95 0.159574
\(815\) 1215.19 0.0522286
\(816\) 19137.3 0.821005
\(817\) 0 0
\(818\) −28974.4 −1.23847
\(819\) −1314.59 −0.0560873
\(820\) 513.203 0.0218559
\(821\) −16567.0 −0.704253 −0.352127 0.935952i \(-0.614541\pi\)
−0.352127 + 0.935952i \(0.614541\pi\)
\(822\) −51699.1 −2.19369
\(823\) −19573.2 −0.829013 −0.414506 0.910046i \(-0.636046\pi\)
−0.414506 + 0.910046i \(0.636046\pi\)
\(824\) 2532.18 0.107054
\(825\) −2144.93 −0.0905174
\(826\) 15162.0 0.638684
\(827\) −34803.1 −1.46339 −0.731695 0.681632i \(-0.761270\pi\)
−0.731695 + 0.681632i \(0.761270\pi\)
\(828\) 12286.0 0.515663
\(829\) −8559.16 −0.358591 −0.179295 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(830\) 9419.70 0.393931
\(831\) −5150.97 −0.215024
\(832\) −9418.62 −0.392466
\(833\) 14350.2 0.596886
\(834\) −65935.6 −2.73760
\(835\) 184.374 0.00764134
\(836\) −3490.82 −0.144417
\(837\) 1400.70 0.0578438
\(838\) 30912.1 1.27427
\(839\) 11133.3 0.458124 0.229062 0.973412i \(-0.426434\pi\)
0.229062 + 0.973412i \(0.426434\pi\)
\(840\) 310.094 0.0127372
\(841\) −19567.9 −0.802323
\(842\) −8382.24 −0.343077
\(843\) 38884.4 1.58867
\(844\) 24821.9 1.01233
\(845\) 4241.62 0.172682
\(846\) −23520.8 −0.955866
\(847\) 11777.4 0.477777
\(848\) 5718.05 0.231555
\(849\) 15358.2 0.620841
\(850\) 26702.1 1.07750
\(851\) 46995.5 1.89305
\(852\) −44702.6 −1.79752
\(853\) 39508.0 1.58585 0.792923 0.609321i \(-0.208558\pi\)
0.792923 + 0.609321i \(0.208558\pi\)
\(854\) −22112.7 −0.886043
\(855\) 2756.00 0.110238
\(856\) 1560.49 0.0623090
\(857\) −25368.9 −1.01118 −0.505592 0.862773i \(-0.668726\pi\)
−0.505592 + 0.862773i \(0.668726\pi\)
\(858\) 1159.17 0.0461229
\(859\) 15928.9 0.632696 0.316348 0.948643i \(-0.397543\pi\)
0.316348 + 0.948643i \(0.397543\pi\)
\(860\) 0 0
\(861\) 1458.32 0.0577229
\(862\) −70080.3 −2.76908
\(863\) −18377.5 −0.724885 −0.362442 0.932006i \(-0.618057\pi\)
−0.362442 + 0.932006i \(0.618057\pi\)
\(864\) 27703.4 1.09084
\(865\) 4081.50 0.160434
\(866\) −31125.6 −1.22135
\(867\) −11747.6 −0.460174
\(868\) 1010.50 0.0395145
\(869\) 2584.10 0.100874
\(870\) 3723.91 0.145118
\(871\) 12308.3 0.478820
\(872\) 2238.46 0.0869310
\(873\) 1393.76 0.0540338
\(874\) −85212.5 −3.29789
\(875\) −4767.60 −0.184199
\(876\) −770.485 −0.0297172
\(877\) 46868.1 1.80459 0.902293 0.431124i \(-0.141883\pi\)
0.902293 + 0.431124i \(0.141883\pi\)
\(878\) −26433.1 −1.01603
\(879\) −32753.1 −1.25681
\(880\) 377.480 0.0144600
\(881\) −24976.8 −0.955153 −0.477577 0.878590i \(-0.658485\pi\)
−0.477577 + 0.878590i \(0.658485\pi\)
\(882\) −9970.36 −0.380634
\(883\) 44574.5 1.69881 0.849406 0.527740i \(-0.176960\pi\)
0.849406 + 0.527740i \(0.176960\pi\)
\(884\) −7496.54 −0.285221
\(885\) 5483.34 0.208272
\(886\) 13671.1 0.518384
\(887\) 18914.5 0.715994 0.357997 0.933723i \(-0.383460\pi\)
0.357997 + 0.933723i \(0.383460\pi\)
\(888\) 4890.63 0.184819
\(889\) −3995.52 −0.150737
\(890\) −3960.00 −0.149145
\(891\) −2646.26 −0.0994984
\(892\) 56309.1 2.11364
\(893\) 84747.1 3.17576
\(894\) 67165.1 2.51268
\(895\) −1289.16 −0.0481474
\(896\) 3009.83 0.112222
\(897\) 14699.5 0.547160
\(898\) 58585.8 2.17710
\(899\) 910.739 0.0337874
\(900\) −9637.80 −0.356956
\(901\) 5330.35 0.197092
\(902\) −328.595 −0.0121297
\(903\) 0 0
\(904\) 573.799 0.0211109
\(905\) −4284.51 −0.157372
\(906\) −53744.8 −1.97081
\(907\) −9458.37 −0.346262 −0.173131 0.984899i \(-0.555388\pi\)
−0.173131 + 0.984899i \(0.555388\pi\)
\(908\) −10470.4 −0.382678
\(909\) 3713.16 0.135487
\(910\) 1263.24 0.0460175
\(911\) −33612.0 −1.22241 −0.611206 0.791472i \(-0.709315\pi\)
−0.611206 + 0.791472i \(0.709315\pi\)
\(912\) 47907.7 1.73945
\(913\) −3133.21 −0.113575
\(914\) 12857.4 0.465302
\(915\) −7997.05 −0.288934
\(916\) −14812.2 −0.534290
\(917\) 11521.2 0.414902
\(918\) 23715.7 0.852651
\(919\) −28693.2 −1.02993 −0.514963 0.857212i \(-0.672195\pi\)
−0.514963 + 0.857212i \(0.672195\pi\)
\(920\) −886.046 −0.0317523
\(921\) −30702.2 −1.09845
\(922\) 5262.29 0.187965
\(923\) −13667.1 −0.487386
\(924\) −1374.34 −0.0489312
\(925\) −36865.7 −1.31042
\(926\) −40017.1 −1.42013
\(927\) 8860.28 0.313927
\(928\) 18012.8 0.637176
\(929\) 18092.0 0.638946 0.319473 0.947595i \(-0.396494\pi\)
0.319473 + 0.947595i \(0.396494\pi\)
\(930\) 703.468 0.0248039
\(931\) 35923.9 1.26462
\(932\) 38899.5 1.36716
\(933\) 3663.21 0.128540
\(934\) 11258.0 0.394402
\(935\) 351.885 0.0123079
\(936\) 390.898 0.0136505
\(937\) −17209.8 −0.600021 −0.300010 0.953936i \(-0.596990\pi\)
−0.300010 + 0.953936i \(0.596990\pi\)
\(938\) −28090.9 −0.977825
\(939\) 51742.1 1.79823
\(940\) 11741.6 0.407414
\(941\) −48758.8 −1.68915 −0.844576 0.535436i \(-0.820147\pi\)
−0.844576 + 0.535436i \(0.820147\pi\)
\(942\) −64350.7 −2.22575
\(943\) −4166.93 −0.143896
\(944\) 24356.9 0.839777
\(945\) −2076.06 −0.0714649
\(946\) 0 0
\(947\) −32685.7 −1.12159 −0.560794 0.827956i \(-0.689504\pi\)
−0.560794 + 0.827956i \(0.689504\pi\)
\(948\) 45438.4 1.55672
\(949\) −235.563 −0.00805764
\(950\) 66845.2 2.28289
\(951\) −19727.4 −0.672665
\(952\) 1284.04 0.0437142
\(953\) −14271.7 −0.485105 −0.242552 0.970138i \(-0.577985\pi\)
−0.242552 + 0.970138i \(0.577985\pi\)
\(954\) −3703.46 −0.125685
\(955\) −7307.24 −0.247599
\(956\) 54088.8 1.82987
\(957\) −1238.66 −0.0418392
\(958\) 25487.0 0.859550
\(959\) 18740.1 0.631022
\(960\) 7773.95 0.261358
\(961\) −29619.0 −0.994225
\(962\) 19923.1 0.667720
\(963\) 5460.27 0.182715
\(964\) −37313.3 −1.24666
\(965\) −8337.30 −0.278121
\(966\) −33548.2 −1.11739
\(967\) 18223.7 0.606034 0.303017 0.952985i \(-0.402006\pi\)
0.303017 + 0.952985i \(0.402006\pi\)
\(968\) −3502.06 −0.116281
\(969\) 44659.4 1.48056
\(970\) −1339.31 −0.0443327
\(971\) 27591.5 0.911899 0.455949 0.890006i \(-0.349300\pi\)
0.455949 + 0.890006i \(0.349300\pi\)
\(972\) −21593.5 −0.712565
\(973\) 23900.6 0.787482
\(974\) −29475.3 −0.969660
\(975\) −11531.1 −0.378759
\(976\) −35522.8 −1.16502
\(977\) −26884.2 −0.880348 −0.440174 0.897912i \(-0.645083\pi\)
−0.440174 + 0.897912i \(0.645083\pi\)
\(978\) −13681.4 −0.447324
\(979\) 1317.18 0.0430004
\(980\) 4977.21 0.162236
\(981\) 7832.52 0.254917
\(982\) −37903.6 −1.23172
\(983\) 11678.2 0.378918 0.189459 0.981889i \(-0.439327\pi\)
0.189459 + 0.981889i \(0.439327\pi\)
\(984\) −433.636 −0.0140486
\(985\) −6885.83 −0.222742
\(986\) 15420.0 0.498046
\(987\) 33365.0 1.07601
\(988\) −18766.6 −0.604296
\(989\) 0 0
\(990\) −244.485 −0.00784875
\(991\) −37149.5 −1.19081 −0.595406 0.803425i \(-0.703009\pi\)
−0.595406 + 0.803425i \(0.703009\pi\)
\(992\) 3402.72 0.108908
\(993\) −46935.4 −1.49995
\(994\) 31191.9 0.995320
\(995\) 11583.9 0.369081
\(996\) −55093.9 −1.75273
\(997\) −59233.9 −1.88160 −0.940800 0.338962i \(-0.889924\pi\)
−0.940800 + 0.338962i \(0.889924\pi\)
\(998\) −13139.8 −0.416765
\(999\) −32742.6 −1.03697
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.1.42 50
43.42 odd 2 1849.4.a.j.1.9 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.42 50 1.1 even 1 trivial
1849.4.a.j.1.9 yes 50 43.42 odd 2