Properties

Label 1045.4.a.d.1.8
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $22$
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] = [1045,4,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52988 q^{2} +7.31543 q^{3} -1.59971 q^{4} +5.00000 q^{5} -18.5072 q^{6} +2.79284 q^{7} +24.2861 q^{8} +26.5155 q^{9} +O(q^{10})\) \(q-2.52988 q^{2} +7.31543 q^{3} -1.59971 q^{4} +5.00000 q^{5} -18.5072 q^{6} +2.79284 q^{7} +24.2861 q^{8} +26.5155 q^{9} -12.6494 q^{10} -11.0000 q^{11} -11.7026 q^{12} +30.0745 q^{13} -7.06555 q^{14} +36.5772 q^{15} -48.6433 q^{16} -93.2244 q^{17} -67.0811 q^{18} -19.0000 q^{19} -7.99854 q^{20} +20.4308 q^{21} +27.8287 q^{22} -157.354 q^{23} +177.663 q^{24} +25.0000 q^{25} -76.0848 q^{26} -3.54404 q^{27} -4.46773 q^{28} -209.195 q^{29} -92.5358 q^{30} +151.518 q^{31} -71.2273 q^{32} -80.4697 q^{33} +235.847 q^{34} +13.9642 q^{35} -42.4171 q^{36} -440.562 q^{37} +48.0677 q^{38} +220.008 q^{39} +121.431 q^{40} -296.286 q^{41} -51.6875 q^{42} +514.211 q^{43} +17.5968 q^{44} +132.578 q^{45} +398.087 q^{46} +455.543 q^{47} -355.846 q^{48} -335.200 q^{49} -63.2470 q^{50} -681.977 q^{51} -48.1104 q^{52} +37.2991 q^{53} +8.96600 q^{54} -55.0000 q^{55} +67.8272 q^{56} -138.993 q^{57} +529.238 q^{58} +39.5060 q^{59} -58.5128 q^{60} -411.791 q^{61} -383.321 q^{62} +74.0537 q^{63} +569.343 q^{64} +150.372 q^{65} +203.579 q^{66} +227.684 q^{67} +149.132 q^{68} -1151.11 q^{69} -35.3278 q^{70} -364.300 q^{71} +643.959 q^{72} -293.949 q^{73} +1114.57 q^{74} +182.886 q^{75} +30.3945 q^{76} -30.7212 q^{77} -556.593 q^{78} +598.450 q^{79} -243.216 q^{80} -741.846 q^{81} +749.567 q^{82} -68.9269 q^{83} -32.6834 q^{84} -466.122 q^{85} -1300.89 q^{86} -1530.35 q^{87} -267.147 q^{88} -896.631 q^{89} -335.406 q^{90} +83.9933 q^{91} +251.721 q^{92} +1108.42 q^{93} -1152.47 q^{94} -95.0000 q^{95} -521.058 q^{96} -1202.45 q^{97} +848.016 q^{98} -291.671 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52988 −0.894448 −0.447224 0.894422i \(-0.647587\pi\)
−0.447224 + 0.894422i \(0.647587\pi\)
\(3\) 7.31543 1.40786 0.703928 0.710272i \(-0.251428\pi\)
0.703928 + 0.710272i \(0.251428\pi\)
\(4\) −1.59971 −0.199964
\(5\) 5.00000 0.447214
\(6\) −18.5072 −1.25925
\(7\) 2.79284 0.150799 0.0753996 0.997153i \(-0.475977\pi\)
0.0753996 + 0.997153i \(0.475977\pi\)
\(8\) 24.2861 1.07330
\(9\) 26.5155 0.982057
\(10\) −12.6494 −0.400009
\(11\) −11.0000 −0.301511
\(12\) −11.7026 −0.281520
\(13\) 30.0745 0.641628 0.320814 0.947142i \(-0.396044\pi\)
0.320814 + 0.947142i \(0.396044\pi\)
\(14\) −7.06555 −0.134882
\(15\) 36.5772 0.629612
\(16\) −48.6433 −0.760051
\(17\) −93.2244 −1.33001 −0.665007 0.746837i \(-0.731571\pi\)
−0.665007 + 0.746837i \(0.731571\pi\)
\(18\) −67.0811 −0.878398
\(19\) −19.0000 −0.229416
\(20\) −7.99854 −0.0894264
\(21\) 20.4308 0.212303
\(22\) 27.8287 0.269686
\(23\) −157.354 −1.42655 −0.713274 0.700885i \(-0.752789\pi\)
−0.713274 + 0.700885i \(0.752789\pi\)
\(24\) 177.663 1.51106
\(25\) 25.0000 0.200000
\(26\) −76.0848 −0.573903
\(27\) −3.54404 −0.0252611
\(28\) −4.46773 −0.0301543
\(29\) −209.195 −1.33954 −0.669768 0.742570i \(-0.733606\pi\)
−0.669768 + 0.742570i \(0.733606\pi\)
\(30\) −92.5358 −0.563155
\(31\) 151.518 0.877850 0.438925 0.898524i \(-0.355359\pi\)
0.438925 + 0.898524i \(0.355359\pi\)
\(32\) −71.2273 −0.393479
\(33\) −80.4697 −0.424484
\(34\) 235.847 1.18963
\(35\) 13.9642 0.0674395
\(36\) −42.4171 −0.196376
\(37\) −440.562 −1.95751 −0.978755 0.205032i \(-0.934270\pi\)
−0.978755 + 0.205032i \(0.934270\pi\)
\(38\) 48.0677 0.205200
\(39\) 220.008 0.903319
\(40\) 121.431 0.479996
\(41\) −296.286 −1.12859 −0.564294 0.825574i \(-0.690851\pi\)
−0.564294 + 0.825574i \(0.690851\pi\)
\(42\) −51.6875 −0.189894
\(43\) 514.211 1.82364 0.911818 0.410594i \(-0.134678\pi\)
0.911818 + 0.410594i \(0.134678\pi\)
\(44\) 17.5968 0.0602913
\(45\) 132.578 0.439189
\(46\) 398.087 1.27597
\(47\) 455.543 1.41378 0.706891 0.707323i \(-0.250097\pi\)
0.706891 + 0.707323i \(0.250097\pi\)
\(48\) −355.846 −1.07004
\(49\) −335.200 −0.977260
\(50\) −63.2470 −0.178890
\(51\) −681.977 −1.87247
\(52\) −48.1104 −0.128302
\(53\) 37.2991 0.0966684 0.0483342 0.998831i \(-0.484609\pi\)
0.0483342 + 0.998831i \(0.484609\pi\)
\(54\) 8.96600 0.0225948
\(55\) −55.0000 −0.134840
\(56\) 67.8272 0.161853
\(57\) −138.993 −0.322984
\(58\) 529.238 1.19814
\(59\) 39.5060 0.0871735 0.0435868 0.999050i \(-0.486122\pi\)
0.0435868 + 0.999050i \(0.486122\pi\)
\(60\) −58.5128 −0.125899
\(61\) −411.791 −0.864335 −0.432168 0.901793i \(-0.642251\pi\)
−0.432168 + 0.901793i \(0.642251\pi\)
\(62\) −383.321 −0.785191
\(63\) 74.0537 0.148093
\(64\) 569.343 1.11200
\(65\) 150.372 0.286945
\(66\) 203.579 0.379679
\(67\) 227.684 0.415164 0.207582 0.978218i \(-0.433441\pi\)
0.207582 + 0.978218i \(0.433441\pi\)
\(68\) 149.132 0.265954
\(69\) −1151.11 −2.00837
\(70\) −35.3278 −0.0603211
\(71\) −364.300 −0.608936 −0.304468 0.952523i \(-0.598479\pi\)
−0.304468 + 0.952523i \(0.598479\pi\)
\(72\) 643.959 1.05405
\(73\) −293.949 −0.471289 −0.235644 0.971839i \(-0.575720\pi\)
−0.235644 + 0.971839i \(0.575720\pi\)
\(74\) 1114.57 1.75089
\(75\) 182.886 0.281571
\(76\) 30.3945 0.0458748
\(77\) −30.7212 −0.0454677
\(78\) −556.593 −0.807972
\(79\) 598.450 0.852290 0.426145 0.904655i \(-0.359871\pi\)
0.426145 + 0.904655i \(0.359871\pi\)
\(80\) −243.216 −0.339905
\(81\) −741.846 −1.01762
\(82\) 749.567 1.00946
\(83\) −68.9269 −0.0911531 −0.0455765 0.998961i \(-0.514512\pi\)
−0.0455765 + 0.998961i \(0.514512\pi\)
\(84\) −32.6834 −0.0424530
\(85\) −466.122 −0.594800
\(86\) −1300.89 −1.63115
\(87\) −1530.35 −1.88587
\(88\) −267.147 −0.323613
\(89\) −896.631 −1.06790 −0.533948 0.845517i \(-0.679292\pi\)
−0.533948 + 0.845517i \(0.679292\pi\)
\(90\) −335.406 −0.392832
\(91\) 83.9933 0.0967570
\(92\) 251.721 0.285258
\(93\) 1108.42 1.23589
\(94\) −1152.47 −1.26455
\(95\) −95.0000 −0.102598
\(96\) −521.058 −0.553961
\(97\) −1202.45 −1.25867 −0.629334 0.777135i \(-0.716672\pi\)
−0.629334 + 0.777135i \(0.716672\pi\)
\(98\) 848.016 0.874107
\(99\) −291.671 −0.296101
\(100\) −39.9927 −0.0399927
\(101\) −1837.97 −1.81074 −0.905372 0.424619i \(-0.860408\pi\)
−0.905372 + 0.424619i \(0.860408\pi\)
\(102\) 1725.32 1.67482
\(103\) 625.355 0.598234 0.299117 0.954216i \(-0.403308\pi\)
0.299117 + 0.954216i \(0.403308\pi\)
\(104\) 730.392 0.688662
\(105\) 102.154 0.0949450
\(106\) −94.3623 −0.0864648
\(107\) 1213.17 1.09609 0.548046 0.836448i \(-0.315372\pi\)
0.548046 + 0.836448i \(0.315372\pi\)
\(108\) 5.66943 0.00505131
\(109\) 1586.52 1.39414 0.697068 0.717005i \(-0.254488\pi\)
0.697068 + 0.717005i \(0.254488\pi\)
\(110\) 139.143 0.120607
\(111\) −3222.90 −2.75589
\(112\) −135.853 −0.114615
\(113\) 690.468 0.574812 0.287406 0.957809i \(-0.407207\pi\)
0.287406 + 0.957809i \(0.407207\pi\)
\(114\) 351.636 0.288892
\(115\) −786.771 −0.637972
\(116\) 334.651 0.267858
\(117\) 797.441 0.630115
\(118\) −99.9453 −0.0779722
\(119\) −260.361 −0.200565
\(120\) 888.317 0.675765
\(121\) 121.000 0.0909091
\(122\) 1041.78 0.773103
\(123\) −2167.46 −1.58889
\(124\) −242.384 −0.175538
\(125\) 125.000 0.0894427
\(126\) −187.347 −0.132462
\(127\) −638.173 −0.445895 −0.222948 0.974830i \(-0.571568\pi\)
−0.222948 + 0.974830i \(0.571568\pi\)
\(128\) −870.550 −0.601144
\(129\) 3761.67 2.56742
\(130\) −380.424 −0.256657
\(131\) 162.200 0.108179 0.0540895 0.998536i \(-0.482774\pi\)
0.0540895 + 0.998536i \(0.482774\pi\)
\(132\) 128.728 0.0848814
\(133\) −53.0640 −0.0345957
\(134\) −576.012 −0.371342
\(135\) −17.7202 −0.0112971
\(136\) −2264.06 −1.42751
\(137\) 1192.01 0.743362 0.371681 0.928361i \(-0.378782\pi\)
0.371681 + 0.928361i \(0.378782\pi\)
\(138\) 2912.18 1.79639
\(139\) 483.185 0.294843 0.147422 0.989074i \(-0.452903\pi\)
0.147422 + 0.989074i \(0.452903\pi\)
\(140\) −22.3387 −0.0134854
\(141\) 3332.49 1.99040
\(142\) 921.635 0.544661
\(143\) −330.819 −0.193458
\(144\) −1289.80 −0.746413
\(145\) −1045.98 −0.599059
\(146\) 743.655 0.421543
\(147\) −2452.13 −1.37584
\(148\) 704.770 0.391431
\(149\) −359.318 −0.197560 −0.0987801 0.995109i \(-0.531494\pi\)
−0.0987801 + 0.995109i \(0.531494\pi\)
\(150\) −462.679 −0.251851
\(151\) 330.088 0.177895 0.0889475 0.996036i \(-0.471650\pi\)
0.0889475 + 0.996036i \(0.471650\pi\)
\(152\) −461.436 −0.246233
\(153\) −2471.90 −1.30615
\(154\) 77.7211 0.0406684
\(155\) 757.588 0.392587
\(156\) −351.948 −0.180631
\(157\) −3721.78 −1.89191 −0.945957 0.324293i \(-0.894874\pi\)
−0.945957 + 0.324293i \(0.894874\pi\)
\(158\) −1514.01 −0.762329
\(159\) 272.859 0.136095
\(160\) −356.136 −0.175969
\(161\) −439.465 −0.215122
\(162\) 1876.78 0.910209
\(163\) −631.831 −0.303612 −0.151806 0.988410i \(-0.548509\pi\)
−0.151806 + 0.988410i \(0.548509\pi\)
\(164\) 473.971 0.225676
\(165\) −402.349 −0.189835
\(166\) 174.377 0.0815316
\(167\) 2185.74 1.01280 0.506401 0.862298i \(-0.330975\pi\)
0.506401 + 0.862298i \(0.330975\pi\)
\(168\) 496.185 0.227866
\(169\) −1292.52 −0.588314
\(170\) 1179.23 0.532018
\(171\) −503.795 −0.225299
\(172\) −822.587 −0.364661
\(173\) 4065.62 1.78672 0.893362 0.449338i \(-0.148340\pi\)
0.893362 + 0.449338i \(0.148340\pi\)
\(174\) 3871.61 1.68681
\(175\) 69.8210 0.0301598
\(176\) 535.076 0.229164
\(177\) 289.003 0.122728
\(178\) 2268.37 0.955177
\(179\) −1911.11 −0.798004 −0.399002 0.916950i \(-0.630643\pi\)
−0.399002 + 0.916950i \(0.630643\pi\)
\(180\) −212.086 −0.0878218
\(181\) −1127.61 −0.463062 −0.231531 0.972828i \(-0.574373\pi\)
−0.231531 + 0.972828i \(0.574373\pi\)
\(182\) −212.493 −0.0865440
\(183\) −3012.43 −1.21686
\(184\) −3821.52 −1.53112
\(185\) −2202.81 −0.875425
\(186\) −2804.16 −1.10544
\(187\) 1025.47 0.401014
\(188\) −728.735 −0.282705
\(189\) −9.89794 −0.00380936
\(190\) 240.339 0.0917684
\(191\) 966.220 0.366038 0.183019 0.983109i \(-0.441413\pi\)
0.183019 + 0.983109i \(0.441413\pi\)
\(192\) 4164.99 1.56553
\(193\) 915.165 0.341321 0.170661 0.985330i \(-0.445410\pi\)
0.170661 + 0.985330i \(0.445410\pi\)
\(194\) 3042.07 1.12581
\(195\) 1100.04 0.403977
\(196\) 536.222 0.195416
\(197\) −4781.39 −1.72924 −0.864619 0.502429i \(-0.832440\pi\)
−0.864619 + 0.502429i \(0.832440\pi\)
\(198\) 737.892 0.264847
\(199\) −3438.46 −1.22485 −0.612427 0.790527i \(-0.709807\pi\)
−0.612427 + 0.790527i \(0.709807\pi\)
\(200\) 607.153 0.214661
\(201\) 1665.60 0.584491
\(202\) 4649.85 1.61962
\(203\) −584.248 −0.202001
\(204\) 1090.96 0.374425
\(205\) −1481.43 −0.504719
\(206\) −1582.07 −0.535089
\(207\) −4172.33 −1.40095
\(208\) −1462.92 −0.487670
\(209\) 209.000 0.0691714
\(210\) −258.438 −0.0849233
\(211\) −2149.64 −0.701362 −0.350681 0.936495i \(-0.614050\pi\)
−0.350681 + 0.936495i \(0.614050\pi\)
\(212\) −59.6677 −0.0193302
\(213\) −2665.01 −0.857294
\(214\) −3069.18 −0.980397
\(215\) 2571.05 0.815555
\(216\) −86.0710 −0.0271129
\(217\) 423.164 0.132379
\(218\) −4013.70 −1.24698
\(219\) −2150.36 −0.663507
\(220\) 87.9840 0.0269631
\(221\) −2803.68 −0.853374
\(222\) 8153.55 2.46500
\(223\) −1036.94 −0.311383 −0.155691 0.987806i \(-0.549761\pi\)
−0.155691 + 0.987806i \(0.549761\pi\)
\(224\) −198.926 −0.0593363
\(225\) 662.888 0.196411
\(226\) −1746.80 −0.514139
\(227\) −676.719 −0.197865 −0.0989326 0.995094i \(-0.531543\pi\)
−0.0989326 + 0.995094i \(0.531543\pi\)
\(228\) 222.349 0.0645851
\(229\) 3166.82 0.913840 0.456920 0.889508i \(-0.348953\pi\)
0.456920 + 0.889508i \(0.348953\pi\)
\(230\) 1990.44 0.570632
\(231\) −224.739 −0.0640119
\(232\) −5080.53 −1.43773
\(233\) 826.349 0.232343 0.116171 0.993229i \(-0.462938\pi\)
0.116171 + 0.993229i \(0.462938\pi\)
\(234\) −2017.43 −0.563605
\(235\) 2277.71 0.632262
\(236\) −63.1980 −0.0174315
\(237\) 4377.92 1.19990
\(238\) 658.682 0.179395
\(239\) 2581.31 0.698623 0.349311 0.937007i \(-0.386416\pi\)
0.349311 + 0.937007i \(0.386416\pi\)
\(240\) −1779.23 −0.478537
\(241\) 3108.18 0.830770 0.415385 0.909646i \(-0.363647\pi\)
0.415385 + 0.909646i \(0.363647\pi\)
\(242\) −306.115 −0.0813134
\(243\) −5331.23 −1.40740
\(244\) 658.746 0.172836
\(245\) −1676.00 −0.437044
\(246\) 5483.41 1.42118
\(247\) −571.415 −0.147200
\(248\) 3679.77 0.942201
\(249\) −504.230 −0.128330
\(250\) −316.235 −0.0800018
\(251\) −1220.33 −0.306878 −0.153439 0.988158i \(-0.549035\pi\)
−0.153439 + 0.988158i \(0.549035\pi\)
\(252\) −118.464 −0.0296133
\(253\) 1730.90 0.430121
\(254\) 1614.50 0.398830
\(255\) −3409.88 −0.837393
\(256\) −2352.35 −0.574305
\(257\) −6329.14 −1.53619 −0.768096 0.640335i \(-0.778795\pi\)
−0.768096 + 0.640335i \(0.778795\pi\)
\(258\) −9516.58 −2.29642
\(259\) −1230.42 −0.295191
\(260\) −240.552 −0.0573785
\(261\) −5546.92 −1.31550
\(262\) −410.345 −0.0967604
\(263\) −2870.36 −0.672980 −0.336490 0.941687i \(-0.609240\pi\)
−0.336490 + 0.941687i \(0.609240\pi\)
\(264\) −1954.30 −0.455601
\(265\) 186.496 0.0432314
\(266\) 134.245 0.0309440
\(267\) −6559.25 −1.50344
\(268\) −364.227 −0.0830176
\(269\) −8202.73 −1.85922 −0.929609 0.368548i \(-0.879855\pi\)
−0.929609 + 0.368548i \(0.879855\pi\)
\(270\) 44.8300 0.0101047
\(271\) −2243.52 −0.502894 −0.251447 0.967871i \(-0.580906\pi\)
−0.251447 + 0.967871i \(0.580906\pi\)
\(272\) 4534.74 1.01088
\(273\) 614.447 0.136220
\(274\) −3015.65 −0.664898
\(275\) −275.000 −0.0603023
\(276\) 1841.45 0.401602
\(277\) −2622.08 −0.568757 −0.284378 0.958712i \(-0.591787\pi\)
−0.284378 + 0.958712i \(0.591787\pi\)
\(278\) −1222.40 −0.263722
\(279\) 4017.57 0.862099
\(280\) 339.136 0.0723831
\(281\) −3802.38 −0.807228 −0.403614 0.914929i \(-0.632246\pi\)
−0.403614 + 0.914929i \(0.632246\pi\)
\(282\) −8430.80 −1.78031
\(283\) −5545.13 −1.16475 −0.582374 0.812921i \(-0.697876\pi\)
−0.582374 + 0.812921i \(0.697876\pi\)
\(284\) 582.774 0.121765
\(285\) −694.966 −0.144443
\(286\) 836.933 0.173038
\(287\) −827.479 −0.170190
\(288\) −1888.63 −0.386419
\(289\) 3777.79 0.768938
\(290\) 2646.19 0.535827
\(291\) −8796.48 −1.77202
\(292\) 470.232 0.0942406
\(293\) −3593.74 −0.716549 −0.358274 0.933616i \(-0.616635\pi\)
−0.358274 + 0.933616i \(0.616635\pi\)
\(294\) 6203.60 1.23062
\(295\) 197.530 0.0389852
\(296\) −10699.5 −2.10100
\(297\) 38.9845 0.00761652
\(298\) 909.031 0.176707
\(299\) −4732.35 −0.915313
\(300\) −292.564 −0.0563040
\(301\) 1436.11 0.275003
\(302\) −835.082 −0.159118
\(303\) −13445.6 −2.54927
\(304\) 924.222 0.174368
\(305\) −2058.96 −0.386543
\(306\) 6253.60 1.16828
\(307\) 7243.62 1.34663 0.673315 0.739356i \(-0.264870\pi\)
0.673315 + 0.739356i \(0.264870\pi\)
\(308\) 49.1450 0.00909188
\(309\) 4574.74 0.842227
\(310\) −1916.61 −0.351148
\(311\) −7399.74 −1.34920 −0.674599 0.738184i \(-0.735684\pi\)
−0.674599 + 0.738184i \(0.735684\pi\)
\(312\) 5343.14 0.969537
\(313\) 1196.71 0.216109 0.108055 0.994145i \(-0.465538\pi\)
0.108055 + 0.994145i \(0.465538\pi\)
\(314\) 9415.66 1.69222
\(315\) 370.268 0.0662294
\(316\) −957.346 −0.170427
\(317\) 6918.56 1.22582 0.612910 0.790153i \(-0.289999\pi\)
0.612910 + 0.790153i \(0.289999\pi\)
\(318\) −690.301 −0.121730
\(319\) 2301.15 0.403885
\(320\) 2846.71 0.497300
\(321\) 8874.89 1.54314
\(322\) 1111.79 0.192416
\(323\) 1771.26 0.305126
\(324\) 1186.74 0.203487
\(325\) 751.862 0.128326
\(326\) 1598.46 0.271565
\(327\) 11606.1 1.96274
\(328\) −7195.63 −1.21132
\(329\) 1272.26 0.213197
\(330\) 1017.89 0.169798
\(331\) 807.068 0.134020 0.0670098 0.997752i \(-0.478654\pi\)
0.0670098 + 0.997752i \(0.478654\pi\)
\(332\) 110.263 0.0182273
\(333\) −11681.7 −1.92239
\(334\) −5529.67 −0.905899
\(335\) 1138.42 0.185667
\(336\) −993.822 −0.161361
\(337\) 7218.46 1.16681 0.583404 0.812182i \(-0.301720\pi\)
0.583404 + 0.812182i \(0.301720\pi\)
\(338\) 3269.93 0.526216
\(339\) 5051.07 0.809253
\(340\) 745.659 0.118938
\(341\) −1666.69 −0.264682
\(342\) 1274.54 0.201518
\(343\) −1894.10 −0.298169
\(344\) 12488.2 1.95732
\(345\) −5755.57 −0.898172
\(346\) −10285.5 −1.59813
\(347\) 6598.36 1.02080 0.510401 0.859936i \(-0.329497\pi\)
0.510401 + 0.859936i \(0.329497\pi\)
\(348\) 2448.12 0.377106
\(349\) 10762.2 1.65068 0.825342 0.564633i \(-0.190982\pi\)
0.825342 + 0.564633i \(0.190982\pi\)
\(350\) −176.639 −0.0269764
\(351\) −106.585 −0.0162083
\(352\) 783.500 0.118638
\(353\) −5983.40 −0.902165 −0.451082 0.892482i \(-0.648962\pi\)
−0.451082 + 0.892482i \(0.648962\pi\)
\(354\) −731.143 −0.109774
\(355\) −1821.50 −0.272324
\(356\) 1434.35 0.213540
\(357\) −1904.65 −0.282367
\(358\) 4834.87 0.713773
\(359\) 4034.06 0.593062 0.296531 0.955023i \(-0.404170\pi\)
0.296531 + 0.955023i \(0.404170\pi\)
\(360\) 3219.80 0.471384
\(361\) 361.000 0.0526316
\(362\) 2852.71 0.414185
\(363\) 885.167 0.127987
\(364\) −134.365 −0.0193479
\(365\) −1469.74 −0.210767
\(366\) 7621.09 1.08842
\(367\) 5455.32 0.775928 0.387964 0.921675i \(-0.373178\pi\)
0.387964 + 0.921675i \(0.373178\pi\)
\(368\) 7654.22 1.08425
\(369\) −7856.18 −1.10834
\(370\) 5572.84 0.783022
\(371\) 104.170 0.0145775
\(372\) −1773.14 −0.247132
\(373\) −5023.06 −0.697277 −0.348639 0.937257i \(-0.613356\pi\)
−0.348639 + 0.937257i \(0.613356\pi\)
\(374\) −2594.31 −0.358686
\(375\) 914.429 0.125922
\(376\) 11063.4 1.51742
\(377\) −6291.44 −0.859484
\(378\) 25.0406 0.00340727
\(379\) 8955.56 1.21376 0.606882 0.794792i \(-0.292420\pi\)
0.606882 + 0.794792i \(0.292420\pi\)
\(380\) 151.972 0.0205158
\(381\) −4668.51 −0.627756
\(382\) −2444.42 −0.327402
\(383\) −7565.40 −1.00933 −0.504666 0.863315i \(-0.668384\pi\)
−0.504666 + 0.863315i \(0.668384\pi\)
\(384\) −6368.45 −0.846324
\(385\) −153.606 −0.0203338
\(386\) −2315.26 −0.305294
\(387\) 13634.6 1.79092
\(388\) 1923.58 0.251688
\(389\) −10239.4 −1.33460 −0.667301 0.744788i \(-0.732551\pi\)
−0.667301 + 0.744788i \(0.732551\pi\)
\(390\) −2782.97 −0.361336
\(391\) 14669.3 1.89733
\(392\) −8140.70 −1.04890
\(393\) 1186.56 0.152300
\(394\) 12096.3 1.54671
\(395\) 2992.25 0.381156
\(396\) 466.588 0.0592095
\(397\) 6185.73 0.781997 0.390998 0.920391i \(-0.372130\pi\)
0.390998 + 0.920391i \(0.372130\pi\)
\(398\) 8698.89 1.09557
\(399\) −388.186 −0.0487058
\(400\) −1216.08 −0.152010
\(401\) 14441.7 1.79847 0.899233 0.437469i \(-0.144125\pi\)
0.899233 + 0.437469i \(0.144125\pi\)
\(402\) −4213.78 −0.522796
\(403\) 4556.81 0.563253
\(404\) 2940.22 0.362083
\(405\) −3709.23 −0.455094
\(406\) 1478.08 0.180679
\(407\) 4846.18 0.590212
\(408\) −16562.6 −2.00973
\(409\) −4076.86 −0.492879 −0.246440 0.969158i \(-0.579261\pi\)
−0.246440 + 0.969158i \(0.579261\pi\)
\(410\) 3747.84 0.451445
\(411\) 8720.09 1.04655
\(412\) −1000.39 −0.119625
\(413\) 110.334 0.0131457
\(414\) 10555.5 1.25308
\(415\) −344.634 −0.0407649
\(416\) −2142.12 −0.252467
\(417\) 3534.71 0.415097
\(418\) −528.745 −0.0618702
\(419\) −7212.76 −0.840970 −0.420485 0.907299i \(-0.638140\pi\)
−0.420485 + 0.907299i \(0.638140\pi\)
\(420\) −163.417 −0.0189855
\(421\) −1114.24 −0.128990 −0.0644949 0.997918i \(-0.520544\pi\)
−0.0644949 + 0.997918i \(0.520544\pi\)
\(422\) 5438.34 0.627332
\(423\) 12079.0 1.38841
\(424\) 905.850 0.103755
\(425\) −2330.61 −0.266003
\(426\) 6742.16 0.766804
\(427\) −1150.07 −0.130341
\(428\) −1940.72 −0.219179
\(429\) −2420.09 −0.272361
\(430\) −6504.46 −0.729471
\(431\) 11702.2 1.30783 0.653913 0.756570i \(-0.273126\pi\)
0.653913 + 0.756570i \(0.273126\pi\)
\(432\) 172.394 0.0191998
\(433\) 3023.18 0.335530 0.167765 0.985827i \(-0.446345\pi\)
0.167765 + 0.985827i \(0.446345\pi\)
\(434\) −1070.56 −0.118406
\(435\) −7651.76 −0.843388
\(436\) −2537.97 −0.278776
\(437\) 2989.73 0.327273
\(438\) 5440.16 0.593472
\(439\) 7581.15 0.824211 0.412105 0.911136i \(-0.364794\pi\)
0.412105 + 0.911136i \(0.364794\pi\)
\(440\) −1335.74 −0.144724
\(441\) −8888.01 −0.959725
\(442\) 7092.96 0.763298
\(443\) 9398.82 1.00802 0.504008 0.863699i \(-0.331858\pi\)
0.504008 + 0.863699i \(0.331858\pi\)
\(444\) 5155.70 0.551078
\(445\) −4483.16 −0.477578
\(446\) 2623.32 0.278516
\(447\) −2628.57 −0.278136
\(448\) 1590.08 0.167688
\(449\) −8511.21 −0.894586 −0.447293 0.894388i \(-0.647612\pi\)
−0.447293 + 0.894388i \(0.647612\pi\)
\(450\) −1677.03 −0.175680
\(451\) 3259.14 0.340282
\(452\) −1104.55 −0.114942
\(453\) 2414.73 0.250451
\(454\) 1712.02 0.176980
\(455\) 419.966 0.0432710
\(456\) −3375.60 −0.346660
\(457\) 15373.8 1.57364 0.786820 0.617182i \(-0.211726\pi\)
0.786820 + 0.617182i \(0.211726\pi\)
\(458\) −8011.68 −0.817382
\(459\) 330.391 0.0335977
\(460\) 1258.60 0.127571
\(461\) 11658.4 1.17785 0.588923 0.808189i \(-0.299552\pi\)
0.588923 + 0.808189i \(0.299552\pi\)
\(462\) 568.563 0.0572553
\(463\) −10435.9 −1.04751 −0.523757 0.851868i \(-0.675470\pi\)
−0.523757 + 0.851868i \(0.675470\pi\)
\(464\) 10175.9 1.01812
\(465\) 5542.08 0.552705
\(466\) −2090.56 −0.207819
\(467\) −17050.1 −1.68948 −0.844739 0.535179i \(-0.820244\pi\)
−0.844739 + 0.535179i \(0.820244\pi\)
\(468\) −1275.67 −0.126000
\(469\) 635.884 0.0626064
\(470\) −5762.34 −0.565525
\(471\) −27226.4 −2.66354
\(472\) 959.446 0.0935638
\(473\) −5656.32 −0.549847
\(474\) −11075.6 −1.07325
\(475\) −475.000 −0.0458831
\(476\) 416.502 0.0401057
\(477\) 989.006 0.0949339
\(478\) −6530.39 −0.624881
\(479\) 973.555 0.0928661 0.0464331 0.998921i \(-0.485215\pi\)
0.0464331 + 0.998921i \(0.485215\pi\)
\(480\) −2605.29 −0.247739
\(481\) −13249.7 −1.25599
\(482\) −7863.32 −0.743080
\(483\) −3214.88 −0.302861
\(484\) −193.565 −0.0181785
\(485\) −6012.27 −0.562893
\(486\) 13487.4 1.25885
\(487\) 6959.17 0.647536 0.323768 0.946137i \(-0.395050\pi\)
0.323768 + 0.946137i \(0.395050\pi\)
\(488\) −10000.8 −0.927695
\(489\) −4622.11 −0.427442
\(490\) 4240.08 0.390913
\(491\) 21221.7 1.95055 0.975276 0.220989i \(-0.0709285\pi\)
0.975276 + 0.220989i \(0.0709285\pi\)
\(492\) 3467.30 0.317720
\(493\) 19502.1 1.78160
\(494\) 1445.61 0.131662
\(495\) −1458.35 −0.132421
\(496\) −7370.31 −0.667211
\(497\) −1017.43 −0.0918271
\(498\) 1275.64 0.114785
\(499\) 19202.6 1.72269 0.861347 0.508017i \(-0.169621\pi\)
0.861347 + 0.508017i \(0.169621\pi\)
\(500\) −199.964 −0.0178853
\(501\) 15989.7 1.42588
\(502\) 3087.28 0.274486
\(503\) −18828.7 −1.66905 −0.834524 0.550971i \(-0.814257\pi\)
−0.834524 + 0.550971i \(0.814257\pi\)
\(504\) 1798.48 0.158949
\(505\) −9189.86 −0.809789
\(506\) −4378.96 −0.384720
\(507\) −9455.38 −0.828261
\(508\) 1020.89 0.0891628
\(509\) 8686.23 0.756406 0.378203 0.925723i \(-0.376542\pi\)
0.378203 + 0.925723i \(0.376542\pi\)
\(510\) 8626.60 0.749004
\(511\) −820.952 −0.0710700
\(512\) 12915.6 1.11483
\(513\) 67.3368 0.00579531
\(514\) 16012.0 1.37404
\(515\) 3126.78 0.267538
\(516\) −6017.58 −0.513390
\(517\) −5010.97 −0.426271
\(518\) 3112.81 0.264033
\(519\) 29741.8 2.51545
\(520\) 3651.96 0.307979
\(521\) 20241.2 1.70208 0.851040 0.525101i \(-0.175973\pi\)
0.851040 + 0.525101i \(0.175973\pi\)
\(522\) 14033.0 1.17665
\(523\) −20045.4 −1.67596 −0.837978 0.545705i \(-0.816262\pi\)
−0.837978 + 0.545705i \(0.816262\pi\)
\(524\) −259.472 −0.0216318
\(525\) 510.771 0.0424607
\(526\) 7261.65 0.601945
\(527\) −14125.1 −1.16755
\(528\) 3914.31 0.322630
\(529\) 12593.3 1.03504
\(530\) −471.811 −0.0386683
\(531\) 1047.52 0.0856094
\(532\) 84.8869 0.00691788
\(533\) −8910.64 −0.724133
\(534\) 16594.1 1.34475
\(535\) 6065.87 0.490188
\(536\) 5529.55 0.445597
\(537\) −13980.6 −1.12347
\(538\) 20751.9 1.66297
\(539\) 3687.20 0.294655
\(540\) 28.3472 0.00225901
\(541\) −4019.89 −0.319462 −0.159731 0.987161i \(-0.551063\pi\)
−0.159731 + 0.987161i \(0.551063\pi\)
\(542\) 5675.85 0.449813
\(543\) −8248.92 −0.651925
\(544\) 6640.12 0.523332
\(545\) 7932.59 0.623477
\(546\) −1554.48 −0.121842
\(547\) −10463.0 −0.817852 −0.408926 0.912568i \(-0.634097\pi\)
−0.408926 + 0.912568i \(0.634097\pi\)
\(548\) −1906.87 −0.148645
\(549\) −10918.9 −0.848827
\(550\) 695.717 0.0539372
\(551\) 3974.71 0.307311
\(552\) −27956.1 −2.15560
\(553\) 1671.38 0.128525
\(554\) 6633.55 0.508723
\(555\) −16114.5 −1.23247
\(556\) −772.955 −0.0589579
\(557\) 2301.85 0.175103 0.0875516 0.996160i \(-0.472096\pi\)
0.0875516 + 0.996160i \(0.472096\pi\)
\(558\) −10164.0 −0.771102
\(559\) 15464.6 1.17010
\(560\) −679.264 −0.0512574
\(561\) 7501.74 0.564570
\(562\) 9619.57 0.722023
\(563\) 23635.2 1.76928 0.884641 0.466273i \(-0.154403\pi\)
0.884641 + 0.466273i \(0.154403\pi\)
\(564\) −5331.01 −0.398007
\(565\) 3452.34 0.257064
\(566\) 14028.5 1.04181
\(567\) −2071.86 −0.153456
\(568\) −8847.43 −0.653574
\(569\) 2213.32 0.163071 0.0815355 0.996670i \(-0.474018\pi\)
0.0815355 + 0.996670i \(0.474018\pi\)
\(570\) 1758.18 0.129197
\(571\) −3401.12 −0.249269 −0.124634 0.992203i \(-0.539776\pi\)
−0.124634 + 0.992203i \(0.539776\pi\)
\(572\) 529.215 0.0386846
\(573\) 7068.32 0.515328
\(574\) 2093.42 0.152226
\(575\) −3933.85 −0.285310
\(576\) 15096.4 1.09204
\(577\) 18870.2 1.36149 0.680743 0.732523i \(-0.261657\pi\)
0.680743 + 0.732523i \(0.261657\pi\)
\(578\) −9557.36 −0.687774
\(579\) 6694.83 0.480531
\(580\) 1673.26 0.119790
\(581\) −192.502 −0.0137458
\(582\) 22254.0 1.58498
\(583\) −410.290 −0.0291466
\(584\) −7138.87 −0.505836
\(585\) 3987.21 0.281796
\(586\) 9091.74 0.640915
\(587\) −15453.5 −1.08660 −0.543300 0.839538i \(-0.682825\pi\)
−0.543300 + 0.839538i \(0.682825\pi\)
\(588\) 3922.70 0.275118
\(589\) −2878.83 −0.201393
\(590\) −499.727 −0.0348702
\(591\) −34977.9 −2.43452
\(592\) 21430.4 1.48781
\(593\) −4246.51 −0.294069 −0.147035 0.989131i \(-0.546973\pi\)
−0.147035 + 0.989131i \(0.546973\pi\)
\(594\) −98.6260 −0.00681258
\(595\) −1301.80 −0.0896954
\(596\) 574.804 0.0395049
\(597\) −25153.8 −1.72442
\(598\) 11972.3 0.818700
\(599\) 3506.61 0.239192 0.119596 0.992823i \(-0.461840\pi\)
0.119596 + 0.992823i \(0.461840\pi\)
\(600\) 4441.58 0.302212
\(601\) 4288.64 0.291077 0.145539 0.989353i \(-0.453509\pi\)
0.145539 + 0.989353i \(0.453509\pi\)
\(602\) −3633.18 −0.245976
\(603\) 6037.15 0.407715
\(604\) −528.044 −0.0355725
\(605\) 605.000 0.0406558
\(606\) 34015.7 2.28018
\(607\) 3585.23 0.239736 0.119868 0.992790i \(-0.461753\pi\)
0.119868 + 0.992790i \(0.461753\pi\)
\(608\) 1353.32 0.0902702
\(609\) −4274.03 −0.284388
\(610\) 5208.91 0.345742
\(611\) 13700.2 0.907122
\(612\) 3954.31 0.261182
\(613\) 22008.4 1.45010 0.725049 0.688697i \(-0.241817\pi\)
0.725049 + 0.688697i \(0.241817\pi\)
\(614\) −18325.5 −1.20449
\(615\) −10837.3 −0.710572
\(616\) −746.099 −0.0488007
\(617\) −29571.3 −1.92949 −0.964747 0.263181i \(-0.915228\pi\)
−0.964747 + 0.263181i \(0.915228\pi\)
\(618\) −11573.5 −0.753327
\(619\) 8655.78 0.562044 0.281022 0.959701i \(-0.409327\pi\)
0.281022 + 0.959701i \(0.409327\pi\)
\(620\) −1211.92 −0.0785030
\(621\) 557.670 0.0360363
\(622\) 18720.5 1.20679
\(623\) −2504.15 −0.161038
\(624\) −10701.9 −0.686569
\(625\) 625.000 0.0400000
\(626\) −3027.54 −0.193298
\(627\) 1528.93 0.0973834
\(628\) 5953.76 0.378314
\(629\) 41071.1 2.60352
\(630\) −936.734 −0.0592387
\(631\) 15795.8 0.996544 0.498272 0.867021i \(-0.333968\pi\)
0.498272 + 0.867021i \(0.333968\pi\)
\(632\) 14534.0 0.914767
\(633\) −15725.6 −0.987417
\(634\) −17503.1 −1.09643
\(635\) −3190.86 −0.199410
\(636\) −436.495 −0.0272141
\(637\) −10081.0 −0.627037
\(638\) −5821.62 −0.361254
\(639\) −9659.61 −0.598010
\(640\) −4352.75 −0.268840
\(641\) −11949.2 −0.736295 −0.368147 0.929767i \(-0.620008\pi\)
−0.368147 + 0.929767i \(0.620008\pi\)
\(642\) −22452.4 −1.38026
\(643\) −9936.99 −0.609451 −0.304725 0.952440i \(-0.598565\pi\)
−0.304725 + 0.952440i \(0.598565\pi\)
\(644\) 703.016 0.0430166
\(645\) 18808.4 1.14818
\(646\) −4481.08 −0.272919
\(647\) −384.498 −0.0233635 −0.0116818 0.999932i \(-0.503719\pi\)
−0.0116818 + 0.999932i \(0.503719\pi\)
\(648\) −18016.5 −1.09222
\(649\) −434.566 −0.0262838
\(650\) −1902.12 −0.114781
\(651\) 3095.63 0.186371
\(652\) 1010.75 0.0607114
\(653\) −4589.11 −0.275016 −0.137508 0.990501i \(-0.543909\pi\)
−0.137508 + 0.990501i \(0.543909\pi\)
\(654\) −29361.9 −1.75557
\(655\) 810.998 0.0483791
\(656\) 14412.3 0.857784
\(657\) −7794.21 −0.462833
\(658\) −3218.66 −0.190694
\(659\) −27923.7 −1.65061 −0.825305 0.564687i \(-0.808997\pi\)
−0.825305 + 0.564687i \(0.808997\pi\)
\(660\) 643.641 0.0379601
\(661\) 1983.81 0.116734 0.0583671 0.998295i \(-0.481411\pi\)
0.0583671 + 0.998295i \(0.481411\pi\)
\(662\) −2041.79 −0.119873
\(663\) −20510.1 −1.20143
\(664\) −1673.97 −0.0978350
\(665\) −265.320 −0.0154717
\(666\) 29553.4 1.71947
\(667\) 32917.7 1.91091
\(668\) −3496.55 −0.202524
\(669\) −7585.63 −0.438382
\(670\) −2880.06 −0.166069
\(671\) 4529.70 0.260607
\(672\) −1455.23 −0.0835369
\(673\) −18166.4 −1.04051 −0.520254 0.854011i \(-0.674163\pi\)
−0.520254 + 0.854011i \(0.674163\pi\)
\(674\) −18261.8 −1.04365
\(675\) −88.6010 −0.00505223
\(676\) 2067.66 0.117641
\(677\) 13351.3 0.757948 0.378974 0.925407i \(-0.376277\pi\)
0.378974 + 0.925407i \(0.376277\pi\)
\(678\) −12778.6 −0.723834
\(679\) −3358.26 −0.189806
\(680\) −11320.3 −0.638402
\(681\) −4950.49 −0.278566
\(682\) 4216.53 0.236744
\(683\) 13189.1 0.738896 0.369448 0.929251i \(-0.379547\pi\)
0.369448 + 0.929251i \(0.379547\pi\)
\(684\) 805.926 0.0450517
\(685\) 5960.07 0.332442
\(686\) 4791.86 0.266697
\(687\) 23166.7 1.28655
\(688\) −25012.9 −1.38606
\(689\) 1121.75 0.0620252
\(690\) 14560.9 0.803368
\(691\) −23878.4 −1.31458 −0.657292 0.753636i \(-0.728298\pi\)
−0.657292 + 0.753636i \(0.728298\pi\)
\(692\) −6503.80 −0.357280
\(693\) −814.590 −0.0446518
\(694\) −16693.1 −0.913055
\(695\) 2415.93 0.131858
\(696\) −37166.3 −2.02412
\(697\) 27621.1 1.50104
\(698\) −27227.1 −1.47645
\(699\) 6045.10 0.327105
\(700\) −111.693 −0.00603087
\(701\) −2769.78 −0.149234 −0.0746172 0.997212i \(-0.523773\pi\)
−0.0746172 + 0.997212i \(0.523773\pi\)
\(702\) 269.648 0.0144974
\(703\) 8370.67 0.449084
\(704\) −6262.77 −0.335280
\(705\) 16662.5 0.890134
\(706\) 15137.3 0.806939
\(707\) −5133.16 −0.273059
\(708\) −462.321 −0.0245411
\(709\) −23939.3 −1.26807 −0.634034 0.773305i \(-0.718602\pi\)
−0.634034 + 0.773305i \(0.718602\pi\)
\(710\) 4608.18 0.243580
\(711\) 15868.2 0.836997
\(712\) −21775.7 −1.14618
\(713\) −23841.9 −1.25230
\(714\) 4818.54 0.252562
\(715\) −1654.10 −0.0865171
\(716\) 3057.21 0.159572
\(717\) 18883.4 0.983560
\(718\) −10205.7 −0.530463
\(719\) −18218.6 −0.944976 −0.472488 0.881337i \(-0.656644\pi\)
−0.472488 + 0.881337i \(0.656644\pi\)
\(720\) −6449.01 −0.333806
\(721\) 1746.52 0.0902132
\(722\) −913.287 −0.0470762
\(723\) 22737.7 1.16960
\(724\) 1803.84 0.0925955
\(725\) −5229.88 −0.267907
\(726\) −2239.37 −0.114478
\(727\) −21155.8 −1.07926 −0.539632 0.841901i \(-0.681437\pi\)
−0.539632 + 0.841901i \(0.681437\pi\)
\(728\) 2039.87 0.103850
\(729\) −18970.4 −0.963798
\(730\) 3718.27 0.188520
\(731\) −47937.0 −2.42546
\(732\) 4819.01 0.243328
\(733\) 18400.8 0.927218 0.463609 0.886040i \(-0.346554\pi\)
0.463609 + 0.886040i \(0.346554\pi\)
\(734\) −13801.3 −0.694027
\(735\) −12260.7 −0.615294
\(736\) 11207.9 0.561316
\(737\) −2504.52 −0.125177
\(738\) 19875.2 0.991349
\(739\) 27770.3 1.38234 0.691169 0.722693i \(-0.257096\pi\)
0.691169 + 0.722693i \(0.257096\pi\)
\(740\) 3523.85 0.175053
\(741\) −4180.15 −0.207236
\(742\) −263.539 −0.0130388
\(743\) 37492.2 1.85122 0.925608 0.378483i \(-0.123554\pi\)
0.925608 + 0.378483i \(0.123554\pi\)
\(744\) 26919.1 1.32648
\(745\) −1796.59 −0.0883516
\(746\) 12707.7 0.623678
\(747\) −1827.63 −0.0895175
\(748\) −1640.45 −0.0801883
\(749\) 3388.20 0.165290
\(750\) −2313.40 −0.112631
\(751\) −9405.97 −0.457029 −0.228514 0.973541i \(-0.573387\pi\)
−0.228514 + 0.973541i \(0.573387\pi\)
\(752\) −22159.1 −1.07455
\(753\) −8927.21 −0.432040
\(754\) 15916.6 0.768763
\(755\) 1650.44 0.0795571
\(756\) 15.8338 0.000761733 0
\(757\) 3911.31 0.187793 0.0938963 0.995582i \(-0.470068\pi\)
0.0938963 + 0.995582i \(0.470068\pi\)
\(758\) −22656.5 −1.08565
\(759\) 12662.3 0.605548
\(760\) −2307.18 −0.110119
\(761\) −16994.1 −0.809508 −0.404754 0.914426i \(-0.632643\pi\)
−0.404754 + 0.914426i \(0.632643\pi\)
\(762\) 11810.8 0.561495
\(763\) 4430.89 0.210235
\(764\) −1545.67 −0.0731942
\(765\) −12359.5 −0.584128
\(766\) 19139.6 0.902794
\(767\) 1188.12 0.0559330
\(768\) −17208.5 −0.808538
\(769\) 3264.54 0.153085 0.0765424 0.997066i \(-0.475612\pi\)
0.0765424 + 0.997066i \(0.475612\pi\)
\(770\) 388.605 0.0181875
\(771\) −46300.4 −2.16273
\(772\) −1464.00 −0.0682518
\(773\) −28840.6 −1.34194 −0.670972 0.741483i \(-0.734123\pi\)
−0.670972 + 0.741483i \(0.734123\pi\)
\(774\) −34493.8 −1.60188
\(775\) 3787.94 0.175570
\(776\) −29202.9 −1.35093
\(777\) −9001.04 −0.415586
\(778\) 25904.6 1.19373
\(779\) 5629.43 0.258916
\(780\) −1759.74 −0.0807806
\(781\) 4007.30 0.183601
\(782\) −37111.4 −1.69706
\(783\) 741.396 0.0338382
\(784\) 16305.2 0.742767
\(785\) −18608.9 −0.846089
\(786\) −3001.85 −0.136225
\(787\) −11380.4 −0.515462 −0.257731 0.966217i \(-0.582975\pi\)
−0.257731 + 0.966217i \(0.582975\pi\)
\(788\) 7648.82 0.345784
\(789\) −20997.9 −0.947458
\(790\) −7570.03 −0.340924
\(791\) 1928.37 0.0866812
\(792\) −7083.55 −0.317807
\(793\) −12384.4 −0.554582
\(794\) −15649.1 −0.699455
\(795\) 1364.30 0.0608636
\(796\) 5500.54 0.244926
\(797\) −24474.2 −1.08773 −0.543866 0.839172i \(-0.683040\pi\)
−0.543866 + 0.839172i \(0.683040\pi\)
\(798\) 982.063 0.0435647
\(799\) −42467.7 −1.88035
\(800\) −1780.68 −0.0786957
\(801\) −23774.7 −1.04873
\(802\) −36535.8 −1.60863
\(803\) 3233.44 0.142099
\(804\) −2664.48 −0.116877
\(805\) −2197.33 −0.0962056
\(806\) −11528.2 −0.503801
\(807\) −60006.5 −2.61751
\(808\) −44637.2 −1.94348
\(809\) 19401.4 0.843162 0.421581 0.906791i \(-0.361475\pi\)
0.421581 + 0.906791i \(0.361475\pi\)
\(810\) 9383.90 0.407058
\(811\) −4251.69 −0.184090 −0.0920451 0.995755i \(-0.529340\pi\)
−0.0920451 + 0.995755i \(0.529340\pi\)
\(812\) 934.627 0.0403928
\(813\) −16412.3 −0.708003
\(814\) −12260.2 −0.527913
\(815\) −3159.15 −0.135780
\(816\) 33173.6 1.42317
\(817\) −9770.00 −0.418371
\(818\) 10314.0 0.440855
\(819\) 2227.13 0.0950209
\(820\) 2369.85 0.100925
\(821\) 13776.6 0.585636 0.292818 0.956168i \(-0.405407\pi\)
0.292818 + 0.956168i \(0.405407\pi\)
\(822\) −22060.8 −0.936081
\(823\) 14914.8 0.631710 0.315855 0.948807i \(-0.397709\pi\)
0.315855 + 0.948807i \(0.397709\pi\)
\(824\) 15187.4 0.642087
\(825\) −2011.74 −0.0848969
\(826\) −279.131 −0.0117581
\(827\) −20855.9 −0.876943 −0.438472 0.898745i \(-0.644480\pi\)
−0.438472 + 0.898745i \(0.644480\pi\)
\(828\) 6674.51 0.280139
\(829\) −25184.2 −1.05511 −0.527554 0.849522i \(-0.676891\pi\)
−0.527554 + 0.849522i \(0.676891\pi\)
\(830\) 871.883 0.0364621
\(831\) −19181.7 −0.800727
\(832\) 17122.7 0.713488
\(833\) 31248.8 1.29977
\(834\) −8942.39 −0.371282
\(835\) 10928.7 0.452939
\(836\) −334.339 −0.0138318
\(837\) −536.985 −0.0221755
\(838\) 18247.4 0.752204
\(839\) 15722.2 0.646948 0.323474 0.946237i \(-0.395149\pi\)
0.323474 + 0.946237i \(0.395149\pi\)
\(840\) 2480.93 0.101905
\(841\) 19373.6 0.794357
\(842\) 2818.89 0.115375
\(843\) −27816.1 −1.13646
\(844\) 3438.80 0.140247
\(845\) −6462.62 −0.263102
\(846\) −30558.3 −1.24186
\(847\) 337.934 0.0137090
\(848\) −1814.35 −0.0734729
\(849\) −40565.0 −1.63980
\(850\) 5896.16 0.237926
\(851\) 69324.2 2.79248
\(852\) 4263.24 0.171428
\(853\) 27788.1 1.11541 0.557706 0.830039i \(-0.311682\pi\)
0.557706 + 0.830039i \(0.311682\pi\)
\(854\) 2909.53 0.116583
\(855\) −2518.98 −0.100757
\(856\) 29463.3 1.17644
\(857\) −38883.1 −1.54985 −0.774924 0.632054i \(-0.782212\pi\)
−0.774924 + 0.632054i \(0.782212\pi\)
\(858\) 6122.53 0.243613
\(859\) 28685.6 1.13939 0.569697 0.821855i \(-0.307061\pi\)
0.569697 + 0.821855i \(0.307061\pi\)
\(860\) −4112.94 −0.163081
\(861\) −6053.37 −0.239603
\(862\) −29605.0 −1.16978
\(863\) −41287.5 −1.62856 −0.814278 0.580475i \(-0.802867\pi\)
−0.814278 + 0.580475i \(0.802867\pi\)
\(864\) 252.432 0.00993972
\(865\) 20328.1 0.799047
\(866\) −7648.27 −0.300114
\(867\) 27636.2 1.08255
\(868\) −676.940 −0.0264710
\(869\) −6582.95 −0.256975
\(870\) 19358.0 0.754366
\(871\) 6847.47 0.266381
\(872\) 38530.3 1.49633
\(873\) −31883.7 −1.23608
\(874\) −7563.66 −0.292728
\(875\) 349.105 0.0134879
\(876\) 3439.95 0.132677
\(877\) 5933.53 0.228462 0.114231 0.993454i \(-0.463560\pi\)
0.114231 + 0.993454i \(0.463560\pi\)
\(878\) −19179.4 −0.737213
\(879\) −26289.8 −1.00880
\(880\) 2675.38 0.102485
\(881\) 37757.9 1.44392 0.721961 0.691934i \(-0.243241\pi\)
0.721961 + 0.691934i \(0.243241\pi\)
\(882\) 22485.6 0.858423
\(883\) 24552.1 0.935725 0.467862 0.883801i \(-0.345024\pi\)
0.467862 + 0.883801i \(0.345024\pi\)
\(884\) 4485.07 0.170644
\(885\) 1445.02 0.0548855
\(886\) −23777.9 −0.901618
\(887\) 17032.9 0.644767 0.322383 0.946609i \(-0.395516\pi\)
0.322383 + 0.946609i \(0.395516\pi\)
\(888\) −78271.7 −2.95791
\(889\) −1782.31 −0.0672406
\(890\) 11341.8 0.427168
\(891\) 8160.30 0.306824
\(892\) 1658.79 0.0622652
\(893\) −8655.31 −0.324344
\(894\) 6649.96 0.248778
\(895\) −9555.53 −0.356878
\(896\) −2431.31 −0.0906521
\(897\) −34619.2 −1.28863
\(898\) 21532.3 0.800160
\(899\) −31696.7 −1.17591
\(900\) −1060.43 −0.0392751
\(901\) −3477.19 −0.128570
\(902\) −8245.24 −0.304364
\(903\) 10505.8 0.387164
\(904\) 16768.8 0.616949
\(905\) −5638.03 −0.207088
\(906\) −6108.99 −0.224015
\(907\) 1450.32 0.0530947 0.0265474 0.999648i \(-0.491549\pi\)
0.0265474 + 0.999648i \(0.491549\pi\)
\(908\) 1082.55 0.0395658
\(909\) −48734.8 −1.77825
\(910\) −1062.46 −0.0387037
\(911\) 33935.1 1.23416 0.617081 0.786900i \(-0.288315\pi\)
0.617081 + 0.786900i \(0.288315\pi\)
\(912\) 6761.08 0.245484
\(913\) 758.195 0.0274837
\(914\) −38893.7 −1.40754
\(915\) −15062.2 −0.544196
\(916\) −5065.99 −0.182735
\(917\) 452.998 0.0163133
\(918\) −835.850 −0.0300514
\(919\) 19042.8 0.683528 0.341764 0.939786i \(-0.388976\pi\)
0.341764 + 0.939786i \(0.388976\pi\)
\(920\) −19107.6 −0.684738
\(921\) 52990.2 1.89586
\(922\) −29494.4 −1.05352
\(923\) −10956.1 −0.390710
\(924\) 359.517 0.0128000
\(925\) −11014.0 −0.391502
\(926\) 26401.6 0.936946
\(927\) 16581.6 0.587500
\(928\) 14900.4 0.527079
\(929\) −7512.08 −0.265300 −0.132650 0.991163i \(-0.542349\pi\)
−0.132650 + 0.991163i \(0.542349\pi\)
\(930\) −14020.8 −0.494366
\(931\) 6368.80 0.224199
\(932\) −1321.92 −0.0464601
\(933\) −54132.3 −1.89948
\(934\) 43134.8 1.51115
\(935\) 5127.34 0.179339
\(936\) 19366.7 0.676305
\(937\) −3521.93 −0.122792 −0.0613962 0.998113i \(-0.519555\pi\)
−0.0613962 + 0.998113i \(0.519555\pi\)
\(938\) −1608.71 −0.0559981
\(939\) 8754.47 0.304250
\(940\) −3643.68 −0.126429
\(941\) −46438.0 −1.60875 −0.804376 0.594121i \(-0.797500\pi\)
−0.804376 + 0.594121i \(0.797500\pi\)
\(942\) 68879.6 2.38240
\(943\) 46621.8 1.60998
\(944\) −1921.70 −0.0662563
\(945\) −49.4897 −0.00170360
\(946\) 14309.8 0.491810
\(947\) 49642.1 1.70343 0.851717 0.524002i \(-0.175561\pi\)
0.851717 + 0.524002i \(0.175561\pi\)
\(948\) −7003.40 −0.239936
\(949\) −8840.36 −0.302392
\(950\) 1201.69 0.0410401
\(951\) 50612.2 1.72578
\(952\) −6323.15 −0.215267
\(953\) 25629.0 0.871148 0.435574 0.900153i \(-0.356546\pi\)
0.435574 + 0.900153i \(0.356546\pi\)
\(954\) −2502.07 −0.0849134
\(955\) 4831.10 0.163697
\(956\) −4129.34 −0.139699
\(957\) 16833.9 0.568612
\(958\) −2462.98 −0.0830639
\(959\) 3329.10 0.112098
\(960\) 20824.9 0.700127
\(961\) −6833.42 −0.229379
\(962\) 33520.1 1.12342
\(963\) 32168.0 1.07643
\(964\) −4972.18 −0.166124
\(965\) 4575.82 0.152644
\(966\) 8133.25 0.270893
\(967\) −31670.8 −1.05322 −0.526610 0.850107i \(-0.676537\pi\)
−0.526610 + 0.850107i \(0.676537\pi\)
\(968\) 2938.62 0.0975731
\(969\) 12957.6 0.429574
\(970\) 15210.3 0.503479
\(971\) 36224.6 1.19722 0.598612 0.801039i \(-0.295719\pi\)
0.598612 + 0.801039i \(0.295719\pi\)
\(972\) 8528.42 0.281429
\(973\) 1349.46 0.0444622
\(974\) −17605.9 −0.579187
\(975\) 5500.20 0.180664
\(976\) 20030.9 0.656939
\(977\) −59720.8 −1.95562 −0.977809 0.209498i \(-0.932817\pi\)
−0.977809 + 0.209498i \(0.932817\pi\)
\(978\) 11693.4 0.382325
\(979\) 9862.94 0.321983
\(980\) 2681.11 0.0873928
\(981\) 42067.4 1.36912
\(982\) −53688.3 −1.74467
\(983\) 47487.6 1.54081 0.770407 0.637553i \(-0.220053\pi\)
0.770407 + 0.637553i \(0.220053\pi\)
\(984\) −52639.1 −1.70536
\(985\) −23906.9 −0.773338
\(986\) −49337.9 −1.59355
\(987\) 9307.12 0.300151
\(988\) 914.098 0.0294345
\(989\) −80913.2 −2.60151
\(990\) 3689.46 0.118443
\(991\) 55419.6 1.77645 0.888224 0.459410i \(-0.151939\pi\)
0.888224 + 0.459410i \(0.151939\pi\)
\(992\) −10792.2 −0.345415
\(993\) 5904.05 0.188680
\(994\) 2573.98 0.0821345
\(995\) −17192.3 −0.547772
\(996\) 806.621 0.0256614
\(997\) −30231.2 −0.960312 −0.480156 0.877183i \(-0.659420\pi\)
−0.480156 + 0.877183i \(0.659420\pi\)
\(998\) −48580.1 −1.54086
\(999\) 1561.37 0.0494490
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 1045.4.a.d.1.8 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.d.1.8 22 1.1 even 1 trivial