Properties

Label 1849.4.a.g.1.11
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.94581 q^{2} -1.92433 q^{3} -4.21381 q^{4} +20.3652 q^{5} +3.74438 q^{6} -13.1013 q^{7} +23.7658 q^{8} -23.2970 q^{9} +O(q^{10})\) \(q-1.94581 q^{2} -1.92433 q^{3} -4.21381 q^{4} +20.3652 q^{5} +3.74438 q^{6} -13.1013 q^{7} +23.7658 q^{8} -23.2970 q^{9} -39.6269 q^{10} -10.4869 q^{11} +8.10875 q^{12} -22.0421 q^{13} +25.4926 q^{14} -39.1893 q^{15} -12.5333 q^{16} -32.5210 q^{17} +45.3315 q^{18} +81.0332 q^{19} -85.8152 q^{20} +25.2111 q^{21} +20.4056 q^{22} -43.2090 q^{23} -45.7331 q^{24} +289.742 q^{25} +42.8897 q^{26} +96.7878 q^{27} +55.2063 q^{28} +263.358 q^{29} +76.2551 q^{30} -269.983 q^{31} -165.739 q^{32} +20.1803 q^{33} +63.2797 q^{34} -266.810 q^{35} +98.1691 q^{36} +228.341 q^{37} -157.675 q^{38} +42.4161 q^{39} +483.995 q^{40} +23.1758 q^{41} -49.0560 q^{42} +44.1899 q^{44} -474.448 q^{45} +84.0766 q^{46} +143.455 q^{47} +24.1181 q^{48} -171.357 q^{49} -563.784 q^{50} +62.5810 q^{51} +92.8812 q^{52} -438.723 q^{53} -188.331 q^{54} -213.568 q^{55} -311.362 q^{56} -155.934 q^{57} -512.446 q^{58} -446.768 q^{59} +165.137 q^{60} +680.512 q^{61} +525.336 q^{62} +305.220 q^{63} +422.763 q^{64} -448.892 q^{65} -39.2670 q^{66} +91.4540 q^{67} +137.037 q^{68} +83.1482 q^{69} +519.162 q^{70} -1024.93 q^{71} -553.671 q^{72} -342.419 q^{73} -444.310 q^{74} -557.558 q^{75} -341.459 q^{76} +137.392 q^{77} -82.5339 q^{78} +816.013 q^{79} -255.242 q^{80} +442.767 q^{81} -45.0957 q^{82} +1111.95 q^{83} -106.235 q^{84} -662.297 q^{85} -506.788 q^{87} -249.230 q^{88} -1266.79 q^{89} +923.186 q^{90} +288.779 q^{91} +182.075 q^{92} +519.535 q^{93} -279.136 q^{94} +1650.26 q^{95} +318.936 q^{96} -632.249 q^{97} +333.429 q^{98} +244.313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.94581 −0.687949 −0.343974 0.938979i \(-0.611773\pi\)
−0.343974 + 0.938979i \(0.611773\pi\)
\(3\) −1.92433 −0.370337 −0.185168 0.982707i \(-0.559283\pi\)
−0.185168 + 0.982707i \(0.559283\pi\)
\(4\) −4.21381 −0.526727
\(5\) 20.3652 1.82152 0.910760 0.412936i \(-0.135497\pi\)
0.910760 + 0.412936i \(0.135497\pi\)
\(6\) 3.74438 0.254773
\(7\) −13.1013 −0.707401 −0.353701 0.935359i \(-0.615077\pi\)
−0.353701 + 0.935359i \(0.615077\pi\)
\(8\) 23.7658 1.05031
\(9\) −23.2970 −0.862851
\(10\) −39.6269 −1.25311
\(11\) −10.4869 −0.287448 −0.143724 0.989618i \(-0.545908\pi\)
−0.143724 + 0.989618i \(0.545908\pi\)
\(12\) 8.10875 0.195066
\(13\) −22.0421 −0.470259 −0.235130 0.971964i \(-0.575551\pi\)
−0.235130 + 0.971964i \(0.575551\pi\)
\(14\) 25.4926 0.486656
\(15\) −39.1893 −0.674576
\(16\) −12.5333 −0.195832
\(17\) −32.5210 −0.463970 −0.231985 0.972719i \(-0.574522\pi\)
−0.231985 + 0.972719i \(0.574522\pi\)
\(18\) 45.3315 0.593597
\(19\) 81.0332 0.978436 0.489218 0.872162i \(-0.337282\pi\)
0.489218 + 0.872162i \(0.337282\pi\)
\(20\) −85.8152 −0.959443
\(21\) 25.2111 0.261977
\(22\) 20.4056 0.197749
\(23\) −43.2090 −0.391726 −0.195863 0.980631i \(-0.562751\pi\)
−0.195863 + 0.980631i \(0.562751\pi\)
\(24\) −45.7331 −0.388968
\(25\) 289.742 2.31794
\(26\) 42.8897 0.323514
\(27\) 96.7878 0.689882
\(28\) 55.2063 0.372607
\(29\) 263.358 1.68636 0.843180 0.537631i \(-0.180681\pi\)
0.843180 + 0.537631i \(0.180681\pi\)
\(30\) 76.2551 0.464074
\(31\) −269.983 −1.56421 −0.782103 0.623149i \(-0.785853\pi\)
−0.782103 + 0.623149i \(0.785853\pi\)
\(32\) −165.739 −0.915587
\(33\) 20.1803 0.106452
\(34\) 63.2797 0.319188
\(35\) −266.810 −1.28855
\(36\) 98.1691 0.454487
\(37\) 228.341 1.01457 0.507285 0.861778i \(-0.330649\pi\)
0.507285 + 0.861778i \(0.330649\pi\)
\(38\) −157.675 −0.673114
\(39\) 42.4161 0.174154
\(40\) 483.995 1.91316
\(41\) 23.1758 0.0882792 0.0441396 0.999025i \(-0.485945\pi\)
0.0441396 + 0.999025i \(0.485945\pi\)
\(42\) −49.0560 −0.180226
\(43\) 0 0
\(44\) 44.1899 0.151406
\(45\) −474.448 −1.57170
\(46\) 84.0766 0.269487
\(47\) 143.455 0.445213 0.222606 0.974908i \(-0.428544\pi\)
0.222606 + 0.974908i \(0.428544\pi\)
\(48\) 24.1181 0.0725238
\(49\) −171.357 −0.499583
\(50\) −563.784 −1.59462
\(51\) 62.5810 0.171825
\(52\) 92.8812 0.247698
\(53\) −438.723 −1.13704 −0.568521 0.822669i \(-0.692484\pi\)
−0.568521 + 0.822669i \(0.692484\pi\)
\(54\) −188.331 −0.474603
\(55\) −213.568 −0.523592
\(56\) −311.362 −0.742990
\(57\) −155.934 −0.362351
\(58\) −512.446 −1.16013
\(59\) −446.768 −0.985835 −0.492917 0.870076i \(-0.664069\pi\)
−0.492917 + 0.870076i \(0.664069\pi\)
\(60\) 165.137 0.355317
\(61\) 680.512 1.42837 0.714185 0.699957i \(-0.246797\pi\)
0.714185 + 0.699957i \(0.246797\pi\)
\(62\) 525.336 1.07609
\(63\) 305.220 0.610382
\(64\) 422.763 0.825709
\(65\) −448.892 −0.856587
\(66\) −39.2670 −0.0732338
\(67\) 91.4540 0.166759 0.0833797 0.996518i \(-0.473429\pi\)
0.0833797 + 0.996518i \(0.473429\pi\)
\(68\) 137.037 0.244386
\(69\) 83.1482 0.145071
\(70\) 519.162 0.886453
\(71\) −1024.93 −1.71319 −0.856596 0.515987i \(-0.827425\pi\)
−0.856596 + 0.515987i \(0.827425\pi\)
\(72\) −553.671 −0.906260
\(73\) −342.419 −0.549002 −0.274501 0.961587i \(-0.588513\pi\)
−0.274501 + 0.961587i \(0.588513\pi\)
\(74\) −444.310 −0.697972
\(75\) −557.558 −0.858417
\(76\) −341.459 −0.515368
\(77\) 137.392 0.203341
\(78\) −82.5339 −0.119809
\(79\) 816.013 1.16213 0.581067 0.813855i \(-0.302635\pi\)
0.581067 + 0.813855i \(0.302635\pi\)
\(80\) −255.242 −0.356712
\(81\) 442.767 0.607362
\(82\) −45.0957 −0.0607316
\(83\) 1111.95 1.47051 0.735255 0.677791i \(-0.237062\pi\)
0.735255 + 0.677791i \(0.237062\pi\)
\(84\) −106.235 −0.137990
\(85\) −662.297 −0.845132
\(86\) 0 0
\(87\) −506.788 −0.624521
\(88\) −249.230 −0.301909
\(89\) −1266.79 −1.50876 −0.754380 0.656438i \(-0.772062\pi\)
−0.754380 + 0.656438i \(0.772062\pi\)
\(90\) 923.186 1.08125
\(91\) 288.779 0.332662
\(92\) 182.075 0.206333
\(93\) 519.535 0.579283
\(94\) −279.136 −0.306283
\(95\) 1650.26 1.78224
\(96\) 318.936 0.339076
\(97\) −632.249 −0.661806 −0.330903 0.943665i \(-0.607353\pi\)
−0.330903 + 0.943665i \(0.607353\pi\)
\(98\) 333.429 0.343688
\(99\) 244.313 0.248024
\(100\) −1220.92 −1.22092
\(101\) 1812.13 1.78529 0.892644 0.450763i \(-0.148848\pi\)
0.892644 + 0.450763i \(0.148848\pi\)
\(102\) −121.771 −0.118207
\(103\) 231.655 0.221608 0.110804 0.993842i \(-0.464657\pi\)
0.110804 + 0.993842i \(0.464657\pi\)
\(104\) −523.847 −0.493918
\(105\) 513.429 0.477196
\(106\) 853.672 0.782226
\(107\) −1562.15 −1.41139 −0.705697 0.708514i \(-0.749366\pi\)
−0.705697 + 0.708514i \(0.749366\pi\)
\(108\) −407.846 −0.363379
\(109\) 5.29886 0.00465632 0.00232816 0.999997i \(-0.499259\pi\)
0.00232816 + 0.999997i \(0.499259\pi\)
\(110\) 415.564 0.360204
\(111\) −439.403 −0.375733
\(112\) 164.201 0.138532
\(113\) 509.549 0.424198 0.212099 0.977248i \(-0.431970\pi\)
0.212099 + 0.977248i \(0.431970\pi\)
\(114\) 303.419 0.249279
\(115\) −879.961 −0.713537
\(116\) −1109.74 −0.888251
\(117\) 513.514 0.405764
\(118\) 869.327 0.678204
\(119\) 426.066 0.328213
\(120\) −931.365 −0.708513
\(121\) −1221.02 −0.917374
\(122\) −1324.15 −0.982645
\(123\) −44.5978 −0.0326930
\(124\) 1137.66 0.823909
\(125\) 3355.01 2.40065
\(126\) −593.900 −0.419911
\(127\) 898.096 0.627505 0.313752 0.949505i \(-0.398414\pi\)
0.313752 + 0.949505i \(0.398414\pi\)
\(128\) 503.294 0.347542
\(129\) 0 0
\(130\) 873.459 0.589288
\(131\) 1158.48 0.772648 0.386324 0.922363i \(-0.373745\pi\)
0.386324 + 0.922363i \(0.373745\pi\)
\(132\) −85.0358 −0.0560714
\(133\) −1061.64 −0.692147
\(134\) −177.952 −0.114722
\(135\) 1971.10 1.25663
\(136\) −772.887 −0.487313
\(137\) 553.843 0.345387 0.172694 0.984976i \(-0.444753\pi\)
0.172694 + 0.984976i \(0.444753\pi\)
\(138\) −161.791 −0.0998011
\(139\) −2176.41 −1.32806 −0.664030 0.747706i \(-0.731155\pi\)
−0.664030 + 0.747706i \(0.731155\pi\)
\(140\) 1124.29 0.678712
\(141\) −276.053 −0.164879
\(142\) 1994.32 1.17859
\(143\) 231.153 0.135175
\(144\) 291.987 0.168974
\(145\) 5363.35 3.07174
\(146\) 666.284 0.377685
\(147\) 329.747 0.185014
\(148\) −962.188 −0.534401
\(149\) 496.910 0.273211 0.136606 0.990626i \(-0.456381\pi\)
0.136606 + 0.990626i \(0.456381\pi\)
\(150\) 1084.90 0.590547
\(151\) −1109.84 −0.598131 −0.299065 0.954233i \(-0.596675\pi\)
−0.299065 + 0.954233i \(0.596675\pi\)
\(152\) 1925.82 1.02766
\(153\) 757.640 0.400337
\(154\) −267.339 −0.139888
\(155\) −5498.26 −2.84923
\(156\) −178.734 −0.0917318
\(157\) 1707.18 0.867821 0.433910 0.900956i \(-0.357133\pi\)
0.433910 + 0.900956i \(0.357133\pi\)
\(158\) −1587.81 −0.799489
\(159\) 844.245 0.421088
\(160\) −3375.31 −1.66776
\(161\) 566.092 0.277108
\(162\) −861.541 −0.417834
\(163\) −3329.39 −1.59987 −0.799933 0.600089i \(-0.795132\pi\)
−0.799933 + 0.600089i \(0.795132\pi\)
\(164\) −97.6584 −0.0464990
\(165\) 410.975 0.193905
\(166\) −2163.65 −1.01164
\(167\) −271.350 −0.125735 −0.0628673 0.998022i \(-0.520024\pi\)
−0.0628673 + 0.998022i \(0.520024\pi\)
\(168\) 599.161 0.275157
\(169\) −1711.15 −0.778856
\(170\) 1288.71 0.581407
\(171\) −1887.83 −0.844244
\(172\) 0 0
\(173\) 968.753 0.425739 0.212870 0.977081i \(-0.431719\pi\)
0.212870 + 0.977081i \(0.431719\pi\)
\(174\) 986.114 0.429638
\(175\) −3795.98 −1.63971
\(176\) 131.435 0.0562915
\(177\) 859.727 0.365091
\(178\) 2464.94 1.03795
\(179\) 3974.82 1.65973 0.829867 0.557962i \(-0.188417\pi\)
0.829867 + 0.557962i \(0.188417\pi\)
\(180\) 1999.23 0.827856
\(181\) −612.811 −0.251657 −0.125828 0.992052i \(-0.540159\pi\)
−0.125828 + 0.992052i \(0.540159\pi\)
\(182\) −561.909 −0.228854
\(183\) −1309.53 −0.528978
\(184\) −1026.90 −0.411434
\(185\) 4650.22 1.84806
\(186\) −1010.92 −0.398517
\(187\) 341.045 0.133367
\(188\) −604.491 −0.234505
\(189\) −1268.04 −0.488023
\(190\) −3211.09 −1.22609
\(191\) −1007.84 −0.381806 −0.190903 0.981609i \(-0.561142\pi\)
−0.190903 + 0.981609i \(0.561142\pi\)
\(192\) −813.534 −0.305790
\(193\) −1728.70 −0.644737 −0.322369 0.946614i \(-0.604479\pi\)
−0.322369 + 0.946614i \(0.604479\pi\)
\(194\) 1230.24 0.455288
\(195\) 863.814 0.317226
\(196\) 722.067 0.263144
\(197\) 3184.78 1.15181 0.575903 0.817518i \(-0.304650\pi\)
0.575903 + 0.817518i \(0.304650\pi\)
\(198\) −475.388 −0.170628
\(199\) 339.933 0.121091 0.0605457 0.998165i \(-0.480716\pi\)
0.0605457 + 0.998165i \(0.480716\pi\)
\(200\) 6885.95 2.43455
\(201\) −175.987 −0.0617572
\(202\) −3526.07 −1.22819
\(203\) −3450.33 −1.19293
\(204\) −263.705 −0.0905050
\(205\) 471.980 0.160802
\(206\) −450.757 −0.152455
\(207\) 1006.64 0.338001
\(208\) 276.259 0.0920919
\(209\) −849.788 −0.281249
\(210\) −999.037 −0.328286
\(211\) 978.950 0.319401 0.159701 0.987165i \(-0.448947\pi\)
0.159701 + 0.987165i \(0.448947\pi\)
\(212\) 1848.70 0.598910
\(213\) 1972.30 0.634458
\(214\) 3039.66 0.970967
\(215\) 0 0
\(216\) 2300.24 0.724590
\(217\) 3537.12 1.10652
\(218\) −10.3106 −0.00320331
\(219\) 658.927 0.203316
\(220\) 899.937 0.275790
\(221\) 716.830 0.218186
\(222\) 854.997 0.258485
\(223\) −3952.31 −1.18684 −0.593422 0.804892i \(-0.702223\pi\)
−0.593422 + 0.804892i \(0.702223\pi\)
\(224\) 2171.39 0.647687
\(225\) −6750.11 −2.00003
\(226\) −991.487 −0.291826
\(227\) −4291.75 −1.25486 −0.627431 0.778672i \(-0.715893\pi\)
−0.627431 + 0.778672i \(0.715893\pi\)
\(228\) 657.078 0.190860
\(229\) −2771.21 −0.799679 −0.399839 0.916585i \(-0.630934\pi\)
−0.399839 + 0.916585i \(0.630934\pi\)
\(230\) 1712.24 0.490877
\(231\) −264.387 −0.0753046
\(232\) 6258.92 1.77120
\(233\) 879.619 0.247321 0.123660 0.992325i \(-0.460537\pi\)
0.123660 + 0.992325i \(0.460537\pi\)
\(234\) −999.201 −0.279144
\(235\) 2921.48 0.810964
\(236\) 1882.60 0.519266
\(237\) −1570.28 −0.430381
\(238\) −829.044 −0.225794
\(239\) −3245.14 −0.878286 −0.439143 0.898417i \(-0.644718\pi\)
−0.439143 + 0.898417i \(0.644718\pi\)
\(240\) 491.170 0.132104
\(241\) 229.371 0.0613074 0.0306537 0.999530i \(-0.490241\pi\)
0.0306537 + 0.999530i \(0.490241\pi\)
\(242\) 2375.88 0.631106
\(243\) −3465.30 −0.914811
\(244\) −2867.55 −0.752361
\(245\) −3489.72 −0.910001
\(246\) 86.7789 0.0224911
\(247\) −1786.14 −0.460119
\(248\) −6416.36 −1.64290
\(249\) −2139.75 −0.544584
\(250\) −6528.21 −1.65152
\(251\) −578.374 −0.145445 −0.0727224 0.997352i \(-0.523169\pi\)
−0.0727224 + 0.997352i \(0.523169\pi\)
\(252\) −1286.14 −0.321504
\(253\) 453.129 0.112601
\(254\) −1747.53 −0.431691
\(255\) 1274.48 0.312983
\(256\) −4361.42 −1.06480
\(257\) −6846.67 −1.66180 −0.830902 0.556419i \(-0.812175\pi\)
−0.830902 + 0.556419i \(0.812175\pi\)
\(258\) 0 0
\(259\) −2991.56 −0.717708
\(260\) 1891.55 0.451187
\(261\) −6135.45 −1.45508
\(262\) −2254.18 −0.531542
\(263\) −388.631 −0.0911179 −0.0455589 0.998962i \(-0.514507\pi\)
−0.0455589 + 0.998962i \(0.514507\pi\)
\(264\) 479.600 0.111808
\(265\) −8934.68 −2.07114
\(266\) 2065.74 0.476161
\(267\) 2437.72 0.558749
\(268\) −385.370 −0.0878367
\(269\) −5908.90 −1.33930 −0.669651 0.742676i \(-0.733556\pi\)
−0.669651 + 0.742676i \(0.733556\pi\)
\(270\) −3835.40 −0.864500
\(271\) 2427.11 0.544045 0.272023 0.962291i \(-0.412307\pi\)
0.272023 + 0.962291i \(0.412307\pi\)
\(272\) 407.594 0.0908603
\(273\) −555.705 −0.123197
\(274\) −1077.68 −0.237609
\(275\) −3038.50 −0.666285
\(276\) −350.371 −0.0764125
\(277\) 6618.68 1.43566 0.717830 0.696218i \(-0.245135\pi\)
0.717830 + 0.696218i \(0.245135\pi\)
\(278\) 4234.88 0.913638
\(279\) 6289.79 1.34968
\(280\) −6340.95 −1.35337
\(281\) −6248.79 −1.32659 −0.663294 0.748358i \(-0.730842\pi\)
−0.663294 + 0.748358i \(0.730842\pi\)
\(282\) 537.148 0.113428
\(283\) −6339.90 −1.33169 −0.665845 0.746090i \(-0.731929\pi\)
−0.665845 + 0.746090i \(0.731929\pi\)
\(284\) 4318.86 0.902385
\(285\) −3175.63 −0.660029
\(286\) −449.781 −0.0929934
\(287\) −303.632 −0.0624488
\(288\) 3861.22 0.790015
\(289\) −3855.39 −0.784731
\(290\) −10436.1 −2.11320
\(291\) 1216.65 0.245091
\(292\) 1442.89 0.289174
\(293\) −4848.30 −0.966691 −0.483346 0.875430i \(-0.660579\pi\)
−0.483346 + 0.875430i \(0.660579\pi\)
\(294\) −641.626 −0.127280
\(295\) −9098.53 −1.79572
\(296\) 5426.71 1.06561
\(297\) −1015.01 −0.198305
\(298\) −966.894 −0.187955
\(299\) 952.416 0.184213
\(300\) 2349.45 0.452151
\(301\) 0 0
\(302\) 2159.55 0.411483
\(303\) −3487.14 −0.661158
\(304\) −1015.61 −0.191609
\(305\) 13858.8 2.60181
\(306\) −1474.23 −0.275411
\(307\) −9380.78 −1.74394 −0.871970 0.489560i \(-0.837157\pi\)
−0.871970 + 0.489560i \(0.837157\pi\)
\(308\) −578.944 −0.107105
\(309\) −445.779 −0.0820696
\(310\) 10698.6 1.96013
\(311\) −2556.27 −0.466087 −0.233043 0.972466i \(-0.574868\pi\)
−0.233043 + 0.972466i \(0.574868\pi\)
\(312\) 1008.05 0.182916
\(313\) 670.669 0.121113 0.0605566 0.998165i \(-0.480712\pi\)
0.0605566 + 0.998165i \(0.480712\pi\)
\(314\) −3321.85 −0.597016
\(315\) 6215.86 1.11182
\(316\) −3438.53 −0.612127
\(317\) −1897.82 −0.336253 −0.168126 0.985765i \(-0.553772\pi\)
−0.168126 + 0.985765i \(0.553772\pi\)
\(318\) −1642.74 −0.289687
\(319\) −2761.82 −0.484740
\(320\) 8609.66 1.50405
\(321\) 3006.10 0.522691
\(322\) −1101.51 −0.190636
\(323\) −2635.28 −0.453965
\(324\) −1865.74 −0.319914
\(325\) −6386.51 −1.09003
\(326\) 6478.37 1.10063
\(327\) −10.1967 −0.00172441
\(328\) 550.791 0.0927205
\(329\) −1879.43 −0.314944
\(330\) −799.681 −0.133397
\(331\) −6332.67 −1.05159 −0.525793 0.850612i \(-0.676231\pi\)
−0.525793 + 0.850612i \(0.676231\pi\)
\(332\) −4685.55 −0.774557
\(333\) −5319.66 −0.875423
\(334\) 527.996 0.0864990
\(335\) 1862.48 0.303756
\(336\) −315.977 −0.0513034
\(337\) 1125.14 0.181870 0.0909349 0.995857i \(-0.471014\pi\)
0.0909349 + 0.995857i \(0.471014\pi\)
\(338\) 3329.57 0.535813
\(339\) −980.539 −0.157096
\(340\) 2790.80 0.445153
\(341\) 2831.29 0.449627
\(342\) 3673.36 0.580796
\(343\) 6738.72 1.06081
\(344\) 0 0
\(345\) 1693.33 0.264249
\(346\) −1885.01 −0.292887
\(347\) 7157.63 1.10732 0.553662 0.832741i \(-0.313230\pi\)
0.553662 + 0.832741i \(0.313230\pi\)
\(348\) 2135.51 0.328952
\(349\) −3179.32 −0.487637 −0.243818 0.969821i \(-0.578400\pi\)
−0.243818 + 0.969821i \(0.578400\pi\)
\(350\) 7386.27 1.12804
\(351\) −2133.40 −0.324424
\(352\) 1738.09 0.263183
\(353\) 4097.11 0.617755 0.308877 0.951102i \(-0.400047\pi\)
0.308877 + 0.951102i \(0.400047\pi\)
\(354\) −1672.87 −0.251164
\(355\) −20872.9 −3.12062
\(356\) 5338.02 0.794704
\(357\) −819.889 −0.121549
\(358\) −7734.26 −1.14181
\(359\) 5437.68 0.799414 0.399707 0.916643i \(-0.369112\pi\)
0.399707 + 0.916643i \(0.369112\pi\)
\(360\) −11275.6 −1.65077
\(361\) −292.627 −0.0426632
\(362\) 1192.41 0.173127
\(363\) 2349.65 0.339737
\(364\) −1216.86 −0.175222
\(365\) −6973.44 −1.00002
\(366\) 2548.09 0.363910
\(367\) −10507.7 −1.49455 −0.747274 0.664516i \(-0.768638\pi\)
−0.747274 + 0.664516i \(0.768638\pi\)
\(368\) 541.549 0.0767125
\(369\) −539.925 −0.0761718
\(370\) −9048.46 −1.27137
\(371\) 5747.82 0.804344
\(372\) −2189.23 −0.305124
\(373\) −2314.05 −0.321225 −0.160613 0.987018i \(-0.551347\pi\)
−0.160613 + 0.987018i \(0.551347\pi\)
\(374\) −663.610 −0.0917498
\(375\) −6456.13 −0.889048
\(376\) 3409.31 0.467611
\(377\) −5804.97 −0.793027
\(378\) 2467.37 0.335735
\(379\) −5322.63 −0.721386 −0.360693 0.932685i \(-0.617460\pi\)
−0.360693 + 0.932685i \(0.617460\pi\)
\(380\) −6953.88 −0.938754
\(381\) −1728.23 −0.232388
\(382\) 1961.07 0.262663
\(383\) 974.765 0.130047 0.0650237 0.997884i \(-0.479288\pi\)
0.0650237 + 0.997884i \(0.479288\pi\)
\(384\) −968.502 −0.128708
\(385\) 2798.01 0.370390
\(386\) 3363.72 0.443546
\(387\) 0 0
\(388\) 2664.18 0.348591
\(389\) −1161.73 −0.151419 −0.0757096 0.997130i \(-0.524122\pi\)
−0.0757096 + 0.997130i \(0.524122\pi\)
\(390\) −1680.82 −0.218235
\(391\) 1405.20 0.181749
\(392\) −4072.44 −0.524717
\(393\) −2229.29 −0.286140
\(394\) −6196.98 −0.792384
\(395\) 16618.3 2.11685
\(396\) −1029.49 −0.130641
\(397\) 5403.67 0.683129 0.341564 0.939858i \(-0.389043\pi\)
0.341564 + 0.939858i \(0.389043\pi\)
\(398\) −661.445 −0.0833046
\(399\) 2042.93 0.256327
\(400\) −3631.41 −0.453926
\(401\) 6586.98 0.820294 0.410147 0.912019i \(-0.365477\pi\)
0.410147 + 0.912019i \(0.365477\pi\)
\(402\) 342.438 0.0424858
\(403\) 5950.99 0.735582
\(404\) −7635.99 −0.940359
\(405\) 9017.04 1.10632
\(406\) 6713.69 0.820677
\(407\) −2394.60 −0.291636
\(408\) 1487.29 0.180470
\(409\) 3664.02 0.442969 0.221484 0.975164i \(-0.428910\pi\)
0.221484 + 0.975164i \(0.428910\pi\)
\(410\) −918.384 −0.110624
\(411\) −1065.78 −0.127910
\(412\) −976.150 −0.116727
\(413\) 5853.22 0.697381
\(414\) −1958.73 −0.232527
\(415\) 22645.1 2.67856
\(416\) 3653.23 0.430563
\(417\) 4188.12 0.491830
\(418\) 1653.53 0.193485
\(419\) −5113.21 −0.596173 −0.298087 0.954539i \(-0.596348\pi\)
−0.298087 + 0.954539i \(0.596348\pi\)
\(420\) −2163.50 −0.251352
\(421\) −10270.3 −1.18894 −0.594468 0.804119i \(-0.702637\pi\)
−0.594468 + 0.804119i \(0.702637\pi\)
\(422\) −1904.85 −0.219732
\(423\) −3342.06 −0.384152
\(424\) −10426.6 −1.19424
\(425\) −9422.69 −1.07545
\(426\) −3837.72 −0.436475
\(427\) −8915.56 −1.01043
\(428\) 6582.63 0.743419
\(429\) −444.815 −0.0500603
\(430\) 0 0
\(431\) 1086.40 0.121415 0.0607077 0.998156i \(-0.480664\pi\)
0.0607077 + 0.998156i \(0.480664\pi\)
\(432\) −1213.07 −0.135101
\(433\) −5392.96 −0.598543 −0.299272 0.954168i \(-0.596744\pi\)
−0.299272 + 0.954168i \(0.596744\pi\)
\(434\) −6882.56 −0.761230
\(435\) −10320.8 −1.13758
\(436\) −22.3284 −0.00245261
\(437\) −3501.36 −0.383279
\(438\) −1282.15 −0.139871
\(439\) −7125.71 −0.774696 −0.387348 0.921934i \(-0.626609\pi\)
−0.387348 + 0.921934i \(0.626609\pi\)
\(440\) −5075.62 −0.549934
\(441\) 3992.10 0.431066
\(442\) −1394.82 −0.150101
\(443\) 8187.50 0.878103 0.439052 0.898462i \(-0.355315\pi\)
0.439052 + 0.898462i \(0.355315\pi\)
\(444\) 1851.56 0.197908
\(445\) −25798.5 −2.74824
\(446\) 7690.45 0.816487
\(447\) −956.217 −0.101180
\(448\) −5538.73 −0.584108
\(449\) −7136.80 −0.750125 −0.375063 0.926999i \(-0.622379\pi\)
−0.375063 + 0.926999i \(0.622379\pi\)
\(450\) 13134.4 1.37592
\(451\) −243.043 −0.0253757
\(452\) −2147.15 −0.223436
\(453\) 2135.70 0.221510
\(454\) 8350.94 0.863280
\(455\) 5881.04 0.605951
\(456\) −3705.90 −0.380580
\(457\) −5491.92 −0.562147 −0.281073 0.959686i \(-0.590690\pi\)
−0.281073 + 0.959686i \(0.590690\pi\)
\(458\) 5392.25 0.550138
\(459\) −3147.63 −0.320085
\(460\) 3707.99 0.375839
\(461\) 6867.45 0.693815 0.346908 0.937899i \(-0.387232\pi\)
0.346908 + 0.937899i \(0.387232\pi\)
\(462\) 514.447 0.0518057
\(463\) 11786.2 1.18305 0.591524 0.806288i \(-0.298527\pi\)
0.591524 + 0.806288i \(0.298527\pi\)
\(464\) −3300.74 −0.330243
\(465\) 10580.5 1.05518
\(466\) −1711.57 −0.170144
\(467\) −4927.74 −0.488284 −0.244142 0.969739i \(-0.578506\pi\)
−0.244142 + 0.969739i \(0.578506\pi\)
\(468\) −2163.85 −0.213727
\(469\) −1198.16 −0.117966
\(470\) −5684.66 −0.557902
\(471\) −3285.17 −0.321386
\(472\) −10617.8 −1.03543
\(473\) 0 0
\(474\) 3055.46 0.296080
\(475\) 23478.7 2.26795
\(476\) −1795.36 −0.172879
\(477\) 10220.9 0.981097
\(478\) 6314.42 0.604216
\(479\) −2859.41 −0.272755 −0.136378 0.990657i \(-0.543546\pi\)
−0.136378 + 0.990657i \(0.543546\pi\)
\(480\) 6495.20 0.617633
\(481\) −5033.12 −0.477111
\(482\) −446.313 −0.0421763
\(483\) −1089.35 −0.102623
\(484\) 5145.17 0.483205
\(485\) −12875.9 −1.20549
\(486\) 6742.82 0.629343
\(487\) −20667.3 −1.92305 −0.961525 0.274716i \(-0.911416\pi\)
−0.961525 + 0.274716i \(0.911416\pi\)
\(488\) 16172.9 1.50023
\(489\) 6406.84 0.592489
\(490\) 6790.35 0.626034
\(491\) 754.889 0.0693842 0.0346921 0.999398i \(-0.488955\pi\)
0.0346921 + 0.999398i \(0.488955\pi\)
\(492\) 187.927 0.0172203
\(493\) −8564.68 −0.782421
\(494\) 3475.49 0.316538
\(495\) 4975.50 0.451782
\(496\) 3383.77 0.306322
\(497\) 13427.9 1.21191
\(498\) 4163.56 0.374646
\(499\) 15119.3 1.35638 0.678190 0.734887i \(-0.262765\pi\)
0.678190 + 0.734887i \(0.262765\pi\)
\(500\) −14137.4 −1.26448
\(501\) 522.166 0.0465642
\(502\) 1125.41 0.100059
\(503\) −5614.73 −0.497711 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(504\) 7253.78 0.641090
\(505\) 36904.5 3.25194
\(506\) −881.705 −0.0774635
\(507\) 3292.80 0.288439
\(508\) −3784.41 −0.330524
\(509\) −3114.06 −0.271175 −0.135588 0.990765i \(-0.543292\pi\)
−0.135588 + 0.990765i \(0.543292\pi\)
\(510\) −2479.89 −0.215316
\(511\) 4486.12 0.388365
\(512\) 4460.15 0.384986
\(513\) 7843.02 0.675005
\(514\) 13322.3 1.14324
\(515\) 4717.70 0.403663
\(516\) 0 0
\(517\) −1504.40 −0.127975
\(518\) 5821.01 0.493746
\(519\) −1864.20 −0.157667
\(520\) −10668.3 −0.899681
\(521\) −8397.20 −0.706119 −0.353060 0.935601i \(-0.614859\pi\)
−0.353060 + 0.935601i \(0.614859\pi\)
\(522\) 11938.4 1.00102
\(523\) 8152.10 0.681580 0.340790 0.940139i \(-0.389306\pi\)
0.340790 + 0.940139i \(0.389306\pi\)
\(524\) −4881.62 −0.406974
\(525\) 7304.71 0.607245
\(526\) 756.202 0.0626844
\(527\) 8780.11 0.725745
\(528\) −252.924 −0.0208468
\(529\) −10300.0 −0.846551
\(530\) 17385.2 1.42484
\(531\) 10408.3 0.850628
\(532\) 4473.54 0.364572
\(533\) −510.842 −0.0415141
\(534\) −4743.34 −0.384391
\(535\) −31813.6 −2.57088
\(536\) 2173.48 0.175149
\(537\) −7648.86 −0.614660
\(538\) 11497.6 0.921370
\(539\) 1797.01 0.143604
\(540\) −8305.87 −0.661903
\(541\) −15296.5 −1.21561 −0.607807 0.794085i \(-0.707950\pi\)
−0.607807 + 0.794085i \(0.707950\pi\)
\(542\) −4722.69 −0.374275
\(543\) 1179.25 0.0931977
\(544\) 5389.99 0.424805
\(545\) 107.913 0.00848159
\(546\) 1081.30 0.0847532
\(547\) −3667.39 −0.286666 −0.143333 0.989675i \(-0.545782\pi\)
−0.143333 + 0.989675i \(0.545782\pi\)
\(548\) −2333.79 −0.181925
\(549\) −15853.9 −1.23247
\(550\) 5912.35 0.458370
\(551\) 21340.8 1.64999
\(552\) 1976.08 0.152369
\(553\) −10690.8 −0.822096
\(554\) −12878.7 −0.987661
\(555\) −8948.55 −0.684405
\(556\) 9170.98 0.699525
\(557\) 19084.9 1.45180 0.725900 0.687800i \(-0.241424\pi\)
0.725900 + 0.687800i \(0.241424\pi\)
\(558\) −12238.7 −0.928508
\(559\) 0 0
\(560\) 3344.00 0.252339
\(561\) −656.282 −0.0493908
\(562\) 12159.0 0.912625
\(563\) −21727.6 −1.62648 −0.813241 0.581926i \(-0.802299\pi\)
−0.813241 + 0.581926i \(0.802299\pi\)
\(564\) 1163.24 0.0868460
\(565\) 10377.1 0.772685
\(566\) 12336.3 0.916134
\(567\) −5800.80 −0.429649
\(568\) −24358.2 −1.79938
\(569\) 15887.8 1.17057 0.585283 0.810829i \(-0.300984\pi\)
0.585283 + 0.810829i \(0.300984\pi\)
\(570\) 6179.19 0.454066
\(571\) −9090.46 −0.666242 −0.333121 0.942884i \(-0.608102\pi\)
−0.333121 + 0.942884i \(0.608102\pi\)
\(572\) −974.038 −0.0712003
\(573\) 1939.42 0.141397
\(574\) 590.810 0.0429616
\(575\) −12519.5 −0.907996
\(576\) −9849.10 −0.712463
\(577\) −19120.0 −1.37951 −0.689755 0.724043i \(-0.742282\pi\)
−0.689755 + 0.724043i \(0.742282\pi\)
\(578\) 7501.86 0.539855
\(579\) 3326.57 0.238770
\(580\) −22600.2 −1.61797
\(581\) −14567.9 −1.04024
\(582\) −2367.38 −0.168610
\(583\) 4600.85 0.326840
\(584\) −8137.87 −0.576622
\(585\) 10457.8 0.739107
\(586\) 9433.88 0.665034
\(587\) −15313.7 −1.07677 −0.538383 0.842700i \(-0.680965\pi\)
−0.538383 + 0.842700i \(0.680965\pi\)
\(588\) −1389.49 −0.0974519
\(589\) −21877.6 −1.53048
\(590\) 17704.0 1.23536
\(591\) −6128.55 −0.426556
\(592\) −2861.86 −0.198685
\(593\) −4684.47 −0.324399 −0.162199 0.986758i \(-0.551859\pi\)
−0.162199 + 0.986758i \(0.551859\pi\)
\(594\) 1975.01 0.136424
\(595\) 8676.92 0.597847
\(596\) −2093.89 −0.143908
\(597\) −654.141 −0.0448446
\(598\) −1853.22 −0.126729
\(599\) −22451.6 −1.53147 −0.765734 0.643157i \(-0.777624\pi\)
−0.765734 + 0.643157i \(0.777624\pi\)
\(600\) −13250.8 −0.901603
\(601\) 14772.9 1.00266 0.501332 0.865255i \(-0.332844\pi\)
0.501332 + 0.865255i \(0.332844\pi\)
\(602\) 0 0
\(603\) −2130.60 −0.143889
\(604\) 4676.67 0.315051
\(605\) −24866.4 −1.67101
\(606\) 6785.31 0.454842
\(607\) 13455.8 0.899760 0.449880 0.893089i \(-0.351467\pi\)
0.449880 + 0.893089i \(0.351467\pi\)
\(608\) −13430.4 −0.895843
\(609\) 6639.55 0.441787
\(610\) −26966.6 −1.78991
\(611\) −3162.04 −0.209365
\(612\) −3192.56 −0.210868
\(613\) 17123.4 1.12823 0.564116 0.825695i \(-0.309217\pi\)
0.564116 + 0.825695i \(0.309217\pi\)
\(614\) 18253.2 1.19974
\(615\) −908.243 −0.0595510
\(616\) 3265.22 0.213571
\(617\) −15914.5 −1.03840 −0.519201 0.854652i \(-0.673770\pi\)
−0.519201 + 0.854652i \(0.673770\pi\)
\(618\) 867.403 0.0564597
\(619\) 8669.99 0.562967 0.281483 0.959566i \(-0.409174\pi\)
0.281483 + 0.959566i \(0.409174\pi\)
\(620\) 23168.7 1.50077
\(621\) −4182.10 −0.270245
\(622\) 4974.03 0.320644
\(623\) 16596.6 1.06730
\(624\) −531.612 −0.0341050
\(625\) 32107.7 2.05489
\(626\) −1305.00 −0.0833197
\(627\) 1635.27 0.104157
\(628\) −7193.74 −0.457105
\(629\) −7425.89 −0.470731
\(630\) −12094.9 −0.764877
\(631\) −3419.60 −0.215740 −0.107870 0.994165i \(-0.534403\pi\)
−0.107870 + 0.994165i \(0.534403\pi\)
\(632\) 19393.2 1.22060
\(633\) −1883.82 −0.118286
\(634\) 3692.80 0.231325
\(635\) 18289.9 1.14301
\(636\) −3557.49 −0.221798
\(637\) 3777.07 0.234934
\(638\) 5373.98 0.333476
\(639\) 23877.7 1.47823
\(640\) 10249.7 0.633054
\(641\) 9707.88 0.598188 0.299094 0.954224i \(-0.403316\pi\)
0.299094 + 0.954224i \(0.403316\pi\)
\(642\) −5849.30 −0.359585
\(643\) −28804.8 −1.76664 −0.883321 0.468768i \(-0.844698\pi\)
−0.883321 + 0.468768i \(0.844698\pi\)
\(644\) −2385.41 −0.145960
\(645\) 0 0
\(646\) 5127.76 0.312305
\(647\) −30097.3 −1.82882 −0.914410 0.404790i \(-0.867345\pi\)
−0.914410 + 0.404790i \(0.867345\pi\)
\(648\) 10522.7 0.637918
\(649\) 4685.22 0.283376
\(650\) 12427.0 0.749885
\(651\) −6806.57 −0.409785
\(652\) 14029.4 0.842692
\(653\) 26942.7 1.61463 0.807313 0.590124i \(-0.200921\pi\)
0.807313 + 0.590124i \(0.200921\pi\)
\(654\) 19.8410 0.00118630
\(655\) 23592.7 1.40739
\(656\) −290.468 −0.0172879
\(657\) 7977.33 0.473707
\(658\) 3657.03 0.216665
\(659\) 5904.52 0.349025 0.174512 0.984655i \(-0.444165\pi\)
0.174512 + 0.984655i \(0.444165\pi\)
\(660\) −1731.77 −0.102135
\(661\) 18772.0 1.10461 0.552303 0.833643i \(-0.313749\pi\)
0.552303 + 0.833643i \(0.313749\pi\)
\(662\) 12322.2 0.723438
\(663\) −1379.41 −0.0808025
\(664\) 26426.4 1.54449
\(665\) −21620.4 −1.26076
\(666\) 10351.1 0.602246
\(667\) −11379.5 −0.660591
\(668\) 1143.42 0.0662278
\(669\) 7605.53 0.439532
\(670\) −3624.04 −0.208968
\(671\) −7136.47 −0.410582
\(672\) −4178.46 −0.239862
\(673\) −15231.2 −0.872390 −0.436195 0.899852i \(-0.643674\pi\)
−0.436195 + 0.899852i \(0.643674\pi\)
\(674\) −2189.31 −0.125117
\(675\) 28043.5 1.59910
\(676\) 7210.46 0.410244
\(677\) −18405.3 −1.04486 −0.522432 0.852681i \(-0.674975\pi\)
−0.522432 + 0.852681i \(0.674975\pi\)
\(678\) 1907.95 0.108074
\(679\) 8283.25 0.468162
\(680\) −15740.0 −0.887650
\(681\) 8258.73 0.464721
\(682\) −5509.16 −0.309321
\(683\) 16819.0 0.942256 0.471128 0.882065i \(-0.343847\pi\)
0.471128 + 0.882065i \(0.343847\pi\)
\(684\) 7954.95 0.444686
\(685\) 11279.1 0.629129
\(686\) −13112.3 −0.729781
\(687\) 5332.71 0.296150
\(688\) 0 0
\(689\) 9670.36 0.534704
\(690\) −3294.90 −0.181790
\(691\) 24645.7 1.35682 0.678412 0.734681i \(-0.262668\pi\)
0.678412 + 0.734681i \(0.262668\pi\)
\(692\) −4082.14 −0.224248
\(693\) −3200.81 −0.175453
\(694\) −13927.4 −0.761783
\(695\) −44323.0 −2.41909
\(696\) −12044.2 −0.655940
\(697\) −753.699 −0.0409590
\(698\) 6186.36 0.335469
\(699\) −1692.67 −0.0915920
\(700\) 15995.6 0.863680
\(701\) −23686.8 −1.27623 −0.638115 0.769941i \(-0.720285\pi\)
−0.638115 + 0.769941i \(0.720285\pi\)
\(702\) 4151.20 0.223187
\(703\) 18503.2 0.992692
\(704\) −4433.48 −0.237348
\(705\) −5621.89 −0.300330
\(706\) −7972.22 −0.424983
\(707\) −23741.2 −1.26291
\(708\) −3622.73 −0.192303
\(709\) 481.648 0.0255129 0.0127565 0.999919i \(-0.495939\pi\)
0.0127565 + 0.999919i \(0.495939\pi\)
\(710\) 40614.7 2.14682
\(711\) −19010.6 −1.00275
\(712\) −30106.3 −1.58466
\(713\) 11665.7 0.612740
\(714\) 1595.35 0.0836198
\(715\) 4707.49 0.246224
\(716\) −16749.2 −0.874226
\(717\) 6244.70 0.325262
\(718\) −10580.7 −0.549956
\(719\) −3202.21 −0.166095 −0.0830475 0.996546i \(-0.526465\pi\)
−0.0830475 + 0.996546i \(0.526465\pi\)
\(720\) 5946.37 0.307789
\(721\) −3034.97 −0.156766
\(722\) 569.397 0.0293501
\(723\) −441.385 −0.0227044
\(724\) 2582.27 0.132554
\(725\) 76306.0 3.90887
\(726\) −4571.98 −0.233722
\(727\) 9608.33 0.490170 0.245085 0.969502i \(-0.421184\pi\)
0.245085 + 0.969502i \(0.421184\pi\)
\(728\) 6863.06 0.349398
\(729\) −5286.34 −0.268574
\(730\) 13569.0 0.687961
\(731\) 0 0
\(732\) 5518.10 0.278627
\(733\) −29472.6 −1.48512 −0.742562 0.669777i \(-0.766390\pi\)
−0.742562 + 0.669777i \(0.766390\pi\)
\(734\) 20446.1 1.02817
\(735\) 6715.37 0.337007
\(736\) 7161.41 0.358659
\(737\) −959.071 −0.0479346
\(738\) 1050.59 0.0524023
\(739\) 14639.6 0.728725 0.364362 0.931257i \(-0.381287\pi\)
0.364362 + 0.931257i \(0.381287\pi\)
\(740\) −19595.2 −0.973423
\(741\) 3437.11 0.170399
\(742\) −11184.2 −0.553347
\(743\) 2765.18 0.136534 0.0682669 0.997667i \(-0.478253\pi\)
0.0682669 + 0.997667i \(0.478253\pi\)
\(744\) 12347.2 0.608426
\(745\) 10119.7 0.497660
\(746\) 4502.71 0.220986
\(747\) −25905.1 −1.26883
\(748\) −1437.10 −0.0702481
\(749\) 20466.2 0.998422
\(750\) 12562.4 0.611619
\(751\) −35094.8 −1.70523 −0.852616 0.522538i \(-0.824985\pi\)
−0.852616 + 0.522538i \(0.824985\pi\)
\(752\) −1797.95 −0.0871870
\(753\) 1112.98 0.0538636
\(754\) 11295.4 0.545561
\(755\) −22602.2 −1.08951
\(756\) 5343.29 0.257055
\(757\) 31615.4 1.51794 0.758970 0.651126i \(-0.225703\pi\)
0.758970 + 0.651126i \(0.225703\pi\)
\(758\) 10356.8 0.496276
\(759\) −871.969 −0.0417002
\(760\) 39219.7 1.87190
\(761\) −7423.91 −0.353635 −0.176818 0.984244i \(-0.556580\pi\)
−0.176818 + 0.984244i \(0.556580\pi\)
\(762\) 3362.81 0.159871
\(763\) −69.4218 −0.00329389
\(764\) 4246.86 0.201107
\(765\) 15429.5 0.729222
\(766\) −1896.71 −0.0894659
\(767\) 9847.69 0.463598
\(768\) 8392.80 0.394335
\(769\) −11020.6 −0.516793 −0.258396 0.966039i \(-0.583194\pi\)
−0.258396 + 0.966039i \(0.583194\pi\)
\(770\) −5444.41 −0.254809
\(771\) 13175.2 0.615427
\(772\) 7284.40 0.339600
\(773\) −12846.3 −0.597735 −0.298868 0.954295i \(-0.596609\pi\)
−0.298868 + 0.954295i \(0.596609\pi\)
\(774\) 0 0
\(775\) −78225.4 −3.62573
\(776\) −15025.9 −0.695101
\(777\) 5756.74 0.265794
\(778\) 2260.51 0.104169
\(779\) 1878.01 0.0863756
\(780\) −3639.95 −0.167091
\(781\) 10748.3 0.492453
\(782\) −2734.25 −0.125034
\(783\) 25489.9 1.16339
\(784\) 2147.66 0.0978345
\(785\) 34767.1 1.58075
\(786\) 4337.79 0.196849
\(787\) −6209.95 −0.281272 −0.140636 0.990061i \(-0.544915\pi\)
−0.140636 + 0.990061i \(0.544915\pi\)
\(788\) −13420.1 −0.606687
\(789\) 747.852 0.0337443
\(790\) −32336.1 −1.45629
\(791\) −6675.74 −0.300078
\(792\) 5806.30 0.260502
\(793\) −14999.9 −0.671705
\(794\) −10514.5 −0.469958
\(795\) 17193.2 0.767020
\(796\) −1432.41 −0.0637821
\(797\) −4570.33 −0.203123 −0.101562 0.994829i \(-0.532384\pi\)
−0.101562 + 0.994829i \(0.532384\pi\)
\(798\) −3975.17 −0.176340
\(799\) −4665.28 −0.206566
\(800\) −48021.5 −2.12227
\(801\) 29512.4 1.30183
\(802\) −12817.0 −0.564320
\(803\) 3590.92 0.157809
\(804\) 741.578 0.0325292
\(805\) 11528.6 0.504757
\(806\) −11579.5 −0.506043
\(807\) 11370.7 0.495992
\(808\) 43066.8 1.87510
\(809\) −11509.3 −0.500179 −0.250090 0.968223i \(-0.580460\pi\)
−0.250090 + 0.968223i \(0.580460\pi\)
\(810\) −17545.5 −0.761093
\(811\) 27492.6 1.19038 0.595189 0.803585i \(-0.297077\pi\)
0.595189 + 0.803585i \(0.297077\pi\)
\(812\) 14539.0 0.628350
\(813\) −4670.55 −0.201480
\(814\) 4659.44 0.200631
\(815\) −67803.8 −2.91419
\(816\) −784.343 −0.0336489
\(817\) 0 0
\(818\) −7129.50 −0.304740
\(819\) −6727.67 −0.287038
\(820\) −1988.83 −0.0846989
\(821\) 20475.2 0.870390 0.435195 0.900336i \(-0.356679\pi\)
0.435195 + 0.900336i \(0.356679\pi\)
\(822\) 2073.80 0.0879952
\(823\) 8523.09 0.360992 0.180496 0.983576i \(-0.442230\pi\)
0.180496 + 0.983576i \(0.442230\pi\)
\(824\) 5505.46 0.232757
\(825\) 5847.07 0.246750
\(826\) −11389.3 −0.479762
\(827\) −44719.4 −1.88035 −0.940173 0.340697i \(-0.889337\pi\)
−0.940173 + 0.340697i \(0.889337\pi\)
\(828\) −4241.79 −0.178034
\(829\) 16064.5 0.673031 0.336515 0.941678i \(-0.390752\pi\)
0.336515 + 0.941678i \(0.390752\pi\)
\(830\) −44063.1 −1.84271
\(831\) −12736.5 −0.531678
\(832\) −9318.57 −0.388297
\(833\) 5572.70 0.231792
\(834\) −8149.29 −0.338354
\(835\) −5526.10 −0.229028
\(836\) 3580.85 0.148141
\(837\) −26131.1 −1.07912
\(838\) 9949.34 0.410136
\(839\) −24208.5 −0.996150 −0.498075 0.867134i \(-0.665960\pi\)
−0.498075 + 0.867134i \(0.665960\pi\)
\(840\) 12202.1 0.501203
\(841\) 44968.7 1.84381
\(842\) 19984.0 0.817927
\(843\) 12024.7 0.491285
\(844\) −4125.11 −0.168237
\(845\) −34847.9 −1.41870
\(846\) 6503.01 0.264277
\(847\) 15997.0 0.648951
\(848\) 5498.62 0.222669
\(849\) 12200.0 0.493174
\(850\) 18334.8 0.739857
\(851\) −9866.40 −0.397434
\(852\) −8310.90 −0.334186
\(853\) 27563.0 1.10638 0.553189 0.833056i \(-0.313411\pi\)
0.553189 + 0.833056i \(0.313411\pi\)
\(854\) 17348.0 0.695125
\(855\) −38446.0 −1.53781
\(856\) −37125.8 −1.48240
\(857\) −510.284 −0.0203395 −0.0101698 0.999948i \(-0.503237\pi\)
−0.0101698 + 0.999948i \(0.503237\pi\)
\(858\) 865.526 0.0344389
\(859\) −6837.12 −0.271571 −0.135786 0.990738i \(-0.543356\pi\)
−0.135786 + 0.990738i \(0.543356\pi\)
\(860\) 0 0
\(861\) 584.287 0.0231271
\(862\) −2113.93 −0.0835275
\(863\) 7846.08 0.309483 0.154741 0.987955i \(-0.450546\pi\)
0.154741 + 0.987955i \(0.450546\pi\)
\(864\) −16041.5 −0.631647
\(865\) 19728.9 0.775493
\(866\) 10493.7 0.411767
\(867\) 7419.02 0.290615
\(868\) −14904.8 −0.582834
\(869\) −8557.46 −0.334053
\(870\) 20082.4 0.782595
\(871\) −2015.84 −0.0784202
\(872\) 125.932 0.00489058
\(873\) 14729.5 0.571040
\(874\) 6812.99 0.263676
\(875\) −43954.8 −1.69822
\(876\) −2776.59 −0.107092
\(877\) 20448.0 0.787320 0.393660 0.919256i \(-0.371209\pi\)
0.393660 + 0.919256i \(0.371209\pi\)
\(878\) 13865.3 0.532951
\(879\) 9329.70 0.358001
\(880\) 2676.71 0.102536
\(881\) −1249.19 −0.0477709 −0.0238854 0.999715i \(-0.507604\pi\)
−0.0238854 + 0.999715i \(0.507604\pi\)
\(882\) −7767.88 −0.296551
\(883\) 17831.9 0.679606 0.339803 0.940497i \(-0.389640\pi\)
0.339803 + 0.940497i \(0.389640\pi\)
\(884\) −3020.59 −0.114925
\(885\) 17508.5 0.665020
\(886\) −15931.3 −0.604090
\(887\) −3260.32 −0.123417 −0.0617086 0.998094i \(-0.519655\pi\)
−0.0617086 + 0.998094i \(0.519655\pi\)
\(888\) −10442.8 −0.394636
\(889\) −11766.2 −0.443898
\(890\) 50199.0 1.89064
\(891\) −4643.26 −0.174585
\(892\) 16654.3 0.625142
\(893\) 11624.6 0.435612
\(894\) 1860.62 0.0696067
\(895\) 80948.1 3.02324
\(896\) −6593.79 −0.245852
\(897\) −1832.76 −0.0682208
\(898\) 13886.9 0.516048
\(899\) −71102.3 −2.63781
\(900\) 28443.7 1.05347
\(901\) 14267.7 0.527553
\(902\) 472.915 0.0174572
\(903\) 0 0
\(904\) 12109.8 0.445539
\(905\) −12480.0 −0.458398
\(906\) −4155.67 −0.152387
\(907\) −52697.3 −1.92920 −0.964601 0.263713i \(-0.915053\pi\)
−0.964601 + 0.263713i \(0.915053\pi\)
\(908\) 18084.6 0.660969
\(909\) −42217.2 −1.54044
\(910\) −11443.4 −0.416863
\(911\) −23392.6 −0.850749 −0.425374 0.905017i \(-0.639858\pi\)
−0.425374 + 0.905017i \(0.639858\pi\)
\(912\) 1954.36 0.0709599
\(913\) −11660.9 −0.422695
\(914\) 10686.2 0.386728
\(915\) −26668.8 −0.963544
\(916\) 11677.3 0.421212
\(917\) −15177.5 −0.546572
\(918\) 6124.71 0.220202
\(919\) 6550.12 0.235113 0.117556 0.993066i \(-0.462494\pi\)
0.117556 + 0.993066i \(0.462494\pi\)
\(920\) −20913.0 −0.749435
\(921\) 18051.7 0.645845
\(922\) −13362.8 −0.477309
\(923\) 22591.6 0.805645
\(924\) 1114.08 0.0396650
\(925\) 66160.1 2.35171
\(926\) −22933.7 −0.813875
\(927\) −5396.85 −0.191215
\(928\) −43648.8 −1.54401
\(929\) 51999.4 1.83643 0.918217 0.396079i \(-0.129629\pi\)
0.918217 + 0.396079i \(0.129629\pi\)
\(930\) −20587.6 −0.725906
\(931\) −13885.6 −0.488810
\(932\) −3706.55 −0.130270
\(933\) 4919.10 0.172609
\(934\) 9588.46 0.335914
\(935\) 6945.45 0.242931
\(936\) 12204.1 0.426177
\(937\) −10652.6 −0.371402 −0.185701 0.982606i \(-0.559456\pi\)
−0.185701 + 0.982606i \(0.559456\pi\)
\(938\) 2331.40 0.0811544
\(939\) −1290.59 −0.0448527
\(940\) −12310.6 −0.427156
\(941\) 8065.38 0.279409 0.139705 0.990193i \(-0.455385\pi\)
0.139705 + 0.990193i \(0.455385\pi\)
\(942\) 6392.33 0.221097
\(943\) −1001.40 −0.0345813
\(944\) 5599.46 0.193058
\(945\) −25823.9 −0.888945
\(946\) 0 0
\(947\) −31142.2 −1.06862 −0.534312 0.845287i \(-0.679429\pi\)
−0.534312 + 0.845287i \(0.679429\pi\)
\(948\) 6616.85 0.226693
\(949\) 7547.63 0.258173
\(950\) −45685.2 −1.56023
\(951\) 3652.02 0.124527
\(952\) 10125.8 0.344726
\(953\) 20888.2 0.710005 0.355002 0.934865i \(-0.384480\pi\)
0.355002 + 0.934865i \(0.384480\pi\)
\(954\) −19888.0 −0.674944
\(955\) −20524.9 −0.695467
\(956\) 13674.4 0.462617
\(957\) 5314.64 0.179517
\(958\) 5563.88 0.187642
\(959\) −7256.04 −0.244327
\(960\) −16567.8 −0.557003
\(961\) 43099.8 1.44674
\(962\) 9793.51 0.328228
\(963\) 36393.5 1.21782
\(964\) −966.527 −0.0322922
\(965\) −35205.3 −1.17440
\(966\) 2119.66 0.0705994
\(967\) 24446.5 0.812975 0.406487 0.913656i \(-0.366754\pi\)
0.406487 + 0.913656i \(0.366754\pi\)
\(968\) −29018.6 −0.963526
\(969\) 5071.14 0.168120
\(970\) 25054.1 0.829317
\(971\) −22613.9 −0.747389 −0.373695 0.927552i \(-0.621909\pi\)
−0.373695 + 0.927552i \(0.621909\pi\)
\(972\) 14602.1 0.481855
\(973\) 28513.7 0.939472
\(974\) 40214.7 1.32296
\(975\) 12289.7 0.403679
\(976\) −8529.03 −0.279721
\(977\) 31715.2 1.03855 0.519273 0.854608i \(-0.326203\pi\)
0.519273 + 0.854608i \(0.326203\pi\)
\(978\) −12466.5 −0.407602
\(979\) 13284.7 0.433690
\(980\) 14705.1 0.479322
\(981\) −123.447 −0.00401771
\(982\) −1468.87 −0.0477328
\(983\) 33948.3 1.10151 0.550754 0.834668i \(-0.314340\pi\)
0.550754 + 0.834668i \(0.314340\pi\)
\(984\) −1059.90 −0.0343378
\(985\) 64858.7 2.09804
\(986\) 16665.3 0.538265
\(987\) 3616.65 0.116635
\(988\) 7526.46 0.242357
\(989\) 0 0
\(990\) −9681.38 −0.310803
\(991\) 33659.2 1.07893 0.539465 0.842008i \(-0.318627\pi\)
0.539465 + 0.842008i \(0.318627\pi\)
\(992\) 44746.7 1.43217
\(993\) 12186.1 0.389441
\(994\) −26128.1 −0.833735
\(995\) 6922.80 0.220570
\(996\) 9016.52 0.286847
\(997\) −3785.19 −0.120239 −0.0601195 0.998191i \(-0.519148\pi\)
−0.0601195 + 0.998191i \(0.519148\pi\)
\(998\) −29419.3 −0.933119
\(999\) 22100.7 0.699934
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.g.1.11 30
43.8 odd 14 43.4.e.a.21.4 60
43.27 odd 14 43.4.e.a.41.4 yes 60
43.42 odd 2 1849.4.a.h.1.20 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.21.4 60 43.8 odd 14
43.4.e.a.41.4 yes 60 43.27 odd 14
1849.4.a.g.1.11 30 1.1 even 1 trivial
1849.4.a.h.1.20 30 43.42 odd 2