Properties

Label 1045.4.a.f.1.19
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $23$
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: \(23\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
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)
 
\(f(q)\) \(=\) \(q+3.55626 q^{2} -9.55461 q^{3} +4.64698 q^{4} -5.00000 q^{5} -33.9787 q^{6} -15.7816 q^{7} -11.9242 q^{8} +64.2906 q^{9} +O(q^{10})\) \(q+3.55626 q^{2} -9.55461 q^{3} +4.64698 q^{4} -5.00000 q^{5} -33.9787 q^{6} -15.7816 q^{7} -11.9242 q^{8} +64.2906 q^{9} -17.7813 q^{10} +11.0000 q^{11} -44.4001 q^{12} +63.5848 q^{13} -56.1236 q^{14} +47.7731 q^{15} -79.5814 q^{16} +82.1291 q^{17} +228.634 q^{18} -19.0000 q^{19} -23.2349 q^{20} +150.787 q^{21} +39.1188 q^{22} -51.7522 q^{23} +113.931 q^{24} +25.0000 q^{25} +226.124 q^{26} -356.298 q^{27} -73.3369 q^{28} -140.105 q^{29} +169.893 q^{30} +341.164 q^{31} -187.618 q^{32} -105.101 q^{33} +292.072 q^{34} +78.9082 q^{35} +298.757 q^{36} +298.575 q^{37} -67.5689 q^{38} -607.529 q^{39} +59.6211 q^{40} -380.297 q^{41} +536.239 q^{42} +544.593 q^{43} +51.1168 q^{44} -321.453 q^{45} -184.044 q^{46} -327.869 q^{47} +760.370 q^{48} -93.9398 q^{49} +88.9065 q^{50} -784.712 q^{51} +295.477 q^{52} -570.774 q^{53} -1267.09 q^{54} -55.0000 q^{55} +188.184 q^{56} +181.538 q^{57} -498.249 q^{58} -766.062 q^{59} +222.000 q^{60} +703.612 q^{61} +1213.27 q^{62} -1014.61 q^{63} -30.5684 q^{64} -317.924 q^{65} -373.766 q^{66} -183.134 q^{67} +381.652 q^{68} +494.472 q^{69} +280.618 q^{70} -361.012 q^{71} -766.615 q^{72} +112.254 q^{73} +1061.81 q^{74} -238.865 q^{75} -88.2926 q^{76} -173.598 q^{77} -2160.53 q^{78} -431.104 q^{79} +397.907 q^{80} +1668.44 q^{81} -1352.43 q^{82} -752.698 q^{83} +700.706 q^{84} -410.646 q^{85} +1936.71 q^{86} +1338.65 q^{87} -131.166 q^{88} -276.201 q^{89} -1143.17 q^{90} -1003.47 q^{91} -240.491 q^{92} -3259.69 q^{93} -1165.99 q^{94} +95.0000 q^{95} +1792.62 q^{96} +273.694 q^{97} -334.074 q^{98} +707.197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.55626 1.25733 0.628664 0.777677i \(-0.283602\pi\)
0.628664 + 0.777677i \(0.283602\pi\)
\(3\) −9.55461 −1.83879 −0.919393 0.393340i \(-0.871320\pi\)
−0.919393 + 0.393340i \(0.871320\pi\)
\(4\) 4.64698 0.580872
\(5\) −5.00000 −0.447214
\(6\) −33.9787 −2.31196
\(7\) −15.7816 −0.852128 −0.426064 0.904693i \(-0.640100\pi\)
−0.426064 + 0.904693i \(0.640100\pi\)
\(8\) −11.9242 −0.526981
\(9\) 64.2906 2.38114
\(10\) −17.7813 −0.562294
\(11\) 11.0000 0.301511
\(12\) −44.4001 −1.06810
\(13\) 63.5848 1.35656 0.678279 0.734804i \(-0.262726\pi\)
0.678279 + 0.734804i \(0.262726\pi\)
\(14\) −56.1236 −1.07140
\(15\) 47.7731 0.822330
\(16\) −79.5814 −1.24346
\(17\) 82.1291 1.17172 0.585860 0.810412i \(-0.300757\pi\)
0.585860 + 0.810412i \(0.300757\pi\)
\(18\) 228.634 2.99387
\(19\) −19.0000 −0.229416
\(20\) −23.2349 −0.259774
\(21\) 150.787 1.56688
\(22\) 39.1188 0.379098
\(23\) −51.7522 −0.469177 −0.234589 0.972095i \(-0.575374\pi\)
−0.234589 + 0.972095i \(0.575374\pi\)
\(24\) 113.931 0.969005
\(25\) 25.0000 0.200000
\(26\) 226.124 1.70564
\(27\) −356.298 −2.53961
\(28\) −73.3369 −0.494978
\(29\) −140.105 −0.897132 −0.448566 0.893750i \(-0.648065\pi\)
−0.448566 + 0.893750i \(0.648065\pi\)
\(30\) 169.893 1.03394
\(31\) 341.164 1.97661 0.988305 0.152489i \(-0.0487288\pi\)
0.988305 + 0.152489i \(0.0487288\pi\)
\(32\) −187.618 −1.03646
\(33\) −105.101 −0.554415
\(34\) 292.072 1.47324
\(35\) 78.9082 0.381083
\(36\) 298.757 1.38314
\(37\) 298.575 1.32663 0.663316 0.748339i \(-0.269148\pi\)
0.663316 + 0.748339i \(0.269148\pi\)
\(38\) −67.5689 −0.288451
\(39\) −607.529 −2.49442
\(40\) 59.6211 0.235673
\(41\) −380.297 −1.44859 −0.724297 0.689488i \(-0.757836\pi\)
−0.724297 + 0.689488i \(0.757836\pi\)
\(42\) 536.239 1.97008
\(43\) 544.593 1.93139 0.965693 0.259685i \(-0.0836186\pi\)
0.965693 + 0.259685i \(0.0836186\pi\)
\(44\) 51.1168 0.175140
\(45\) −321.453 −1.06488
\(46\) −184.044 −0.589909
\(47\) −327.869 −1.01755 −0.508773 0.860901i \(-0.669901\pi\)
−0.508773 + 0.860901i \(0.669901\pi\)
\(48\) 760.370 2.28646
\(49\) −93.9398 −0.273877
\(50\) 88.9065 0.251465
\(51\) −784.712 −2.15454
\(52\) 295.477 0.787987
\(53\) −570.774 −1.47928 −0.739640 0.673003i \(-0.765004\pi\)
−0.739640 + 0.673003i \(0.765004\pi\)
\(54\) −1267.09 −3.19312
\(55\) −55.0000 −0.134840
\(56\) 188.184 0.449055
\(57\) 181.538 0.421847
\(58\) −498.249 −1.12799
\(59\) −766.062 −1.69039 −0.845193 0.534461i \(-0.820514\pi\)
−0.845193 + 0.534461i \(0.820514\pi\)
\(60\) 222.000 0.477669
\(61\) 703.612 1.47686 0.738428 0.674332i \(-0.235568\pi\)
0.738428 + 0.674332i \(0.235568\pi\)
\(62\) 1213.27 2.48525
\(63\) −1014.61 −2.02903
\(64\) −30.5684 −0.0597039
\(65\) −317.924 −0.606671
\(66\) −373.766 −0.697081
\(67\) −183.134 −0.333930 −0.166965 0.985963i \(-0.553397\pi\)
−0.166965 + 0.985963i \(0.553397\pi\)
\(68\) 381.652 0.680620
\(69\) 494.472 0.862716
\(70\) 280.618 0.479147
\(71\) −361.012 −0.603439 −0.301720 0.953397i \(-0.597561\pi\)
−0.301720 + 0.953397i \(0.597561\pi\)
\(72\) −766.615 −1.25481
\(73\) 112.254 0.179977 0.0899884 0.995943i \(-0.471317\pi\)
0.0899884 + 0.995943i \(0.471317\pi\)
\(74\) 1061.81 1.66801
\(75\) −238.865 −0.367757
\(76\) −88.2926 −0.133261
\(77\) −173.598 −0.256926
\(78\) −2160.53 −3.13630
\(79\) −431.104 −0.613962 −0.306981 0.951716i \(-0.599319\pi\)
−0.306981 + 0.951716i \(0.599319\pi\)
\(80\) 397.907 0.556092
\(81\) 1668.44 2.28867
\(82\) −1352.43 −1.82136
\(83\) −752.698 −0.995414 −0.497707 0.867345i \(-0.665824\pi\)
−0.497707 + 0.867345i \(0.665824\pi\)
\(84\) 700.706 0.910158
\(85\) −410.646 −0.524009
\(86\) 1936.71 2.42839
\(87\) 1338.65 1.64963
\(88\) −131.166 −0.158891
\(89\) −276.201 −0.328958 −0.164479 0.986381i \(-0.552594\pi\)
−0.164479 + 0.986381i \(0.552594\pi\)
\(90\) −1143.17 −1.33890
\(91\) −1003.47 −1.15596
\(92\) −240.491 −0.272532
\(93\) −3259.69 −3.63456
\(94\) −1165.99 −1.27939
\(95\) 95.0000 0.102598
\(96\) 1792.62 1.90582
\(97\) 273.694 0.286489 0.143245 0.989687i \(-0.454246\pi\)
0.143245 + 0.989687i \(0.454246\pi\)
\(98\) −334.074 −0.344353
\(99\) 707.197 0.717939
\(100\) 116.174 0.116174
\(101\) 310.068 0.305475 0.152737 0.988267i \(-0.451191\pi\)
0.152737 + 0.988267i \(0.451191\pi\)
\(102\) −2790.64 −2.70897
\(103\) −1400.13 −1.33940 −0.669701 0.742631i \(-0.733578\pi\)
−0.669701 + 0.742631i \(0.733578\pi\)
\(104\) −758.199 −0.714880
\(105\) −753.937 −0.700731
\(106\) −2029.82 −1.85994
\(107\) 743.667 0.671897 0.335948 0.941880i \(-0.390943\pi\)
0.335948 + 0.941880i \(0.390943\pi\)
\(108\) −1655.71 −1.47519
\(109\) 230.370 0.202435 0.101218 0.994864i \(-0.467726\pi\)
0.101218 + 0.994864i \(0.467726\pi\)
\(110\) −195.594 −0.169538
\(111\) −2852.77 −2.43939
\(112\) 1255.93 1.05959
\(113\) −248.240 −0.206659 −0.103329 0.994647i \(-0.532950\pi\)
−0.103329 + 0.994647i \(0.532950\pi\)
\(114\) 645.595 0.530399
\(115\) 258.761 0.209822
\(116\) −651.065 −0.521119
\(117\) 4087.91 3.23015
\(118\) −2724.31 −2.12537
\(119\) −1296.13 −0.998456
\(120\) −569.656 −0.433352
\(121\) 121.000 0.0909091
\(122\) 2502.23 1.85689
\(123\) 3633.59 2.66365
\(124\) 1585.38 1.14816
\(125\) −125.000 −0.0894427
\(126\) −3608.22 −2.55116
\(127\) 1531.01 1.06973 0.534863 0.844939i \(-0.320363\pi\)
0.534863 + 0.844939i \(0.320363\pi\)
\(128\) 1392.24 0.961388
\(129\) −5203.38 −3.55141
\(130\) −1130.62 −0.762785
\(131\) −2608.54 −1.73977 −0.869883 0.493258i \(-0.835806\pi\)
−0.869883 + 0.493258i \(0.835806\pi\)
\(132\) −488.401 −0.322044
\(133\) 299.851 0.195492
\(134\) −651.271 −0.419860
\(135\) 1781.49 1.13575
\(136\) −979.325 −0.617474
\(137\) 480.774 0.299820 0.149910 0.988700i \(-0.452102\pi\)
0.149910 + 0.988700i \(0.452102\pi\)
\(138\) 1758.47 1.08472
\(139\) −392.957 −0.239786 −0.119893 0.992787i \(-0.538255\pi\)
−0.119893 + 0.992787i \(0.538255\pi\)
\(140\) 366.685 0.221361
\(141\) 3132.66 1.87105
\(142\) −1283.85 −0.758721
\(143\) 699.433 0.409018
\(144\) −5116.34 −2.96085
\(145\) 700.525 0.401210
\(146\) 399.203 0.226290
\(147\) 897.559 0.503601
\(148\) 1387.47 0.770604
\(149\) 2142.22 1.17784 0.588919 0.808192i \(-0.299554\pi\)
0.588919 + 0.808192i \(0.299554\pi\)
\(150\) −849.467 −0.462391
\(151\) −675.336 −0.363961 −0.181980 0.983302i \(-0.558251\pi\)
−0.181980 + 0.983302i \(0.558251\pi\)
\(152\) 226.560 0.120898
\(153\) 5280.14 2.79002
\(154\) −617.360 −0.323041
\(155\) −1705.82 −0.883967
\(156\) −2823.17 −1.44894
\(157\) −600.936 −0.305477 −0.152739 0.988267i \(-0.548809\pi\)
−0.152739 + 0.988267i \(0.548809\pi\)
\(158\) −1533.12 −0.771952
\(159\) 5453.52 2.72008
\(160\) 938.092 0.463517
\(161\) 816.734 0.399799
\(162\) 5933.40 2.87761
\(163\) −184.371 −0.0885956 −0.0442978 0.999018i \(-0.514105\pi\)
−0.0442978 + 0.999018i \(0.514105\pi\)
\(164\) −1767.23 −0.841448
\(165\) 525.504 0.247942
\(166\) −2676.79 −1.25156
\(167\) 3600.99 1.66858 0.834290 0.551326i \(-0.185878\pi\)
0.834290 + 0.551326i \(0.185878\pi\)
\(168\) −1798.02 −0.825717
\(169\) 1846.03 0.840251
\(170\) −1460.36 −0.658851
\(171\) −1221.52 −0.546270
\(172\) 2530.71 1.12189
\(173\) −1881.61 −0.826912 −0.413456 0.910524i \(-0.635678\pi\)
−0.413456 + 0.910524i \(0.635678\pi\)
\(174\) 4760.58 2.07413
\(175\) −394.541 −0.170426
\(176\) −875.396 −0.374917
\(177\) 7319.43 3.10826
\(178\) −982.242 −0.413608
\(179\) −3120.71 −1.30309 −0.651543 0.758611i \(-0.725878\pi\)
−0.651543 + 0.758611i \(0.725878\pi\)
\(180\) −1493.79 −0.618557
\(181\) 1735.82 0.712829 0.356415 0.934328i \(-0.383999\pi\)
0.356415 + 0.934328i \(0.383999\pi\)
\(182\) −3568.61 −1.45342
\(183\) −6722.74 −2.71562
\(184\) 617.104 0.247247
\(185\) −1492.87 −0.593288
\(186\) −11592.3 −4.56984
\(187\) 903.420 0.353287
\(188\) −1523.60 −0.591064
\(189\) 5622.96 2.16408
\(190\) 337.845 0.128999
\(191\) −3171.49 −1.20147 −0.600736 0.799448i \(-0.705126\pi\)
−0.600736 + 0.799448i \(0.705126\pi\)
\(192\) 292.069 0.109783
\(193\) 2839.28 1.05894 0.529471 0.848328i \(-0.322391\pi\)
0.529471 + 0.848328i \(0.322391\pi\)
\(194\) 973.327 0.360210
\(195\) 3037.64 1.11554
\(196\) −436.536 −0.159088
\(197\) −3002.12 −1.08575 −0.542873 0.839815i \(-0.682664\pi\)
−0.542873 + 0.839815i \(0.682664\pi\)
\(198\) 2514.98 0.902685
\(199\) −3715.16 −1.32342 −0.661711 0.749759i \(-0.730169\pi\)
−0.661711 + 0.749759i \(0.730169\pi\)
\(200\) −298.105 −0.105396
\(201\) 1749.77 0.614027
\(202\) 1102.68 0.384082
\(203\) 2211.09 0.764472
\(204\) −3646.54 −1.25151
\(205\) 1901.48 0.647831
\(206\) −4979.21 −1.68407
\(207\) −3327.18 −1.11717
\(208\) −5060.17 −1.68683
\(209\) −209.000 −0.0691714
\(210\) −2681.20 −0.881048
\(211\) −1142.70 −0.372827 −0.186413 0.982471i \(-0.559686\pi\)
−0.186413 + 0.982471i \(0.559686\pi\)
\(212\) −2652.37 −0.859272
\(213\) 3449.33 1.10960
\(214\) 2644.67 0.844794
\(215\) −2722.96 −0.863743
\(216\) 4248.57 1.33833
\(217\) −5384.13 −1.68433
\(218\) 819.255 0.254527
\(219\) −1072.54 −0.330939
\(220\) −255.584 −0.0783248
\(221\) 5222.17 1.58951
\(222\) −10145.2 −3.06712
\(223\) 2792.07 0.838435 0.419218 0.907886i \(-0.362304\pi\)
0.419218 + 0.907886i \(0.362304\pi\)
\(224\) 2960.93 0.883193
\(225\) 1607.27 0.476227
\(226\) −882.806 −0.259838
\(227\) −1023.26 −0.299190 −0.149595 0.988747i \(-0.547797\pi\)
−0.149595 + 0.988747i \(0.547797\pi\)
\(228\) 843.602 0.245039
\(229\) −4438.45 −1.28079 −0.640395 0.768046i \(-0.721229\pi\)
−0.640395 + 0.768046i \(0.721229\pi\)
\(230\) 920.221 0.263815
\(231\) 1658.66 0.472433
\(232\) 1670.64 0.472771
\(233\) 566.920 0.159400 0.0796999 0.996819i \(-0.474604\pi\)
0.0796999 + 0.996819i \(0.474604\pi\)
\(234\) 14537.7 4.06136
\(235\) 1639.35 0.455060
\(236\) −3559.87 −0.981898
\(237\) 4119.04 1.12895
\(238\) −4609.38 −1.25539
\(239\) −5127.28 −1.38768 −0.693841 0.720128i \(-0.744083\pi\)
−0.693841 + 0.720128i \(0.744083\pi\)
\(240\) −3801.85 −1.02253
\(241\) −2785.96 −0.744645 −0.372323 0.928103i \(-0.621438\pi\)
−0.372323 + 0.928103i \(0.621438\pi\)
\(242\) 430.307 0.114302
\(243\) −6321.26 −1.66876
\(244\) 3269.67 0.857865
\(245\) 469.699 0.122482
\(246\) 12922.0 3.34909
\(247\) −1208.11 −0.311216
\(248\) −4068.12 −1.04164
\(249\) 7191.74 1.83035
\(250\) −444.532 −0.112459
\(251\) −4187.44 −1.05302 −0.526512 0.850168i \(-0.676500\pi\)
−0.526512 + 0.850168i \(0.676500\pi\)
\(252\) −4714.88 −1.17861
\(253\) −569.274 −0.141462
\(254\) 5444.67 1.34499
\(255\) 3923.56 0.963541
\(256\) 5195.71 1.26848
\(257\) 5754.34 1.39668 0.698338 0.715768i \(-0.253923\pi\)
0.698338 + 0.715768i \(0.253923\pi\)
\(258\) −18504.5 −4.46528
\(259\) −4712.00 −1.13046
\(260\) −1477.39 −0.352399
\(261\) −9007.44 −2.13619
\(262\) −9276.65 −2.18746
\(263\) −6386.37 −1.49734 −0.748670 0.662943i \(-0.769307\pi\)
−0.748670 + 0.662943i \(0.769307\pi\)
\(264\) 1253.24 0.292166
\(265\) 2853.87 0.661554
\(266\) 1066.35 0.245797
\(267\) 2638.99 0.604883
\(268\) −851.018 −0.193971
\(269\) −8596.44 −1.94846 −0.974228 0.225567i \(-0.927577\pi\)
−0.974228 + 0.225567i \(0.927577\pi\)
\(270\) 6335.44 1.42801
\(271\) −5116.34 −1.14685 −0.573424 0.819259i \(-0.694385\pi\)
−0.573424 + 0.819259i \(0.694385\pi\)
\(272\) −6535.95 −1.45699
\(273\) 9587.80 2.12557
\(274\) 1709.76 0.376972
\(275\) 275.000 0.0603023
\(276\) 2297.80 0.501128
\(277\) −2057.55 −0.446304 −0.223152 0.974784i \(-0.571635\pi\)
−0.223152 + 0.974784i \(0.571635\pi\)
\(278\) −1397.46 −0.301489
\(279\) 21933.7 4.70658
\(280\) −940.918 −0.200824
\(281\) 5042.93 1.07059 0.535295 0.844665i \(-0.320201\pi\)
0.535295 + 0.844665i \(0.320201\pi\)
\(282\) 11140.6 2.35252
\(283\) −2156.19 −0.452904 −0.226452 0.974022i \(-0.572713\pi\)
−0.226452 + 0.974022i \(0.572713\pi\)
\(284\) −1677.61 −0.350521
\(285\) −907.688 −0.188655
\(286\) 2487.37 0.514269
\(287\) 6001.70 1.23439
\(288\) −12062.1 −2.46794
\(289\) 1832.19 0.372928
\(290\) 2491.25 0.504452
\(291\) −2615.04 −0.526792
\(292\) 521.641 0.104544
\(293\) 4712.80 0.939674 0.469837 0.882753i \(-0.344313\pi\)
0.469837 + 0.882753i \(0.344313\pi\)
\(294\) 3191.95 0.633192
\(295\) 3830.31 0.755964
\(296\) −3560.27 −0.699110
\(297\) −3919.28 −0.765722
\(298\) 7618.30 1.48093
\(299\) −3290.65 −0.636466
\(300\) −1110.00 −0.213620
\(301\) −8594.57 −1.64579
\(302\) −2401.67 −0.457618
\(303\) −2962.58 −0.561703
\(304\) 1512.05 0.285269
\(305\) −3518.06 −0.660470
\(306\) 18777.5 3.50797
\(307\) −3284.79 −0.610661 −0.305331 0.952246i \(-0.598767\pi\)
−0.305331 + 0.952246i \(0.598767\pi\)
\(308\) −806.706 −0.149241
\(309\) 13377.7 2.46287
\(310\) −6066.34 −1.11144
\(311\) 5682.11 1.03602 0.518011 0.855374i \(-0.326672\pi\)
0.518011 + 0.855374i \(0.326672\pi\)
\(312\) 7244.30 1.31451
\(313\) 9225.31 1.66596 0.832980 0.553304i \(-0.186633\pi\)
0.832980 + 0.553304i \(0.186633\pi\)
\(314\) −2137.09 −0.384085
\(315\) 5073.06 0.907411
\(316\) −2003.33 −0.356634
\(317\) −2168.94 −0.384290 −0.192145 0.981367i \(-0.561544\pi\)
−0.192145 + 0.981367i \(0.561544\pi\)
\(318\) 19394.1 3.42003
\(319\) −1541.15 −0.270496
\(320\) 152.842 0.0267004
\(321\) −7105.45 −1.23547
\(322\) 2904.52 0.502678
\(323\) −1560.45 −0.268811
\(324\) 7753.20 1.32942
\(325\) 1589.62 0.271312
\(326\) −655.672 −0.111394
\(327\) −2201.10 −0.372235
\(328\) 4534.74 0.763381
\(329\) 5174.31 0.867080
\(330\) 1868.83 0.311744
\(331\) 141.422 0.0234842 0.0117421 0.999931i \(-0.496262\pi\)
0.0117421 + 0.999931i \(0.496262\pi\)
\(332\) −3497.77 −0.578208
\(333\) 19195.6 3.15889
\(334\) 12806.0 2.09795
\(335\) 915.669 0.149338
\(336\) −11999.9 −1.94835
\(337\) −5247.39 −0.848200 −0.424100 0.905615i \(-0.639409\pi\)
−0.424100 + 0.905615i \(0.639409\pi\)
\(338\) 6564.97 1.05647
\(339\) 2371.84 0.380001
\(340\) −1908.26 −0.304382
\(341\) 3752.81 0.595970
\(342\) −4344.05 −0.686840
\(343\) 6895.63 1.08551
\(344\) −6493.84 −1.01780
\(345\) −2472.36 −0.385819
\(346\) −6691.48 −1.03970
\(347\) 9012.15 1.39423 0.697115 0.716959i \(-0.254467\pi\)
0.697115 + 0.716959i \(0.254467\pi\)
\(348\) 6220.67 0.958227
\(349\) 5256.23 0.806188 0.403094 0.915159i \(-0.367935\pi\)
0.403094 + 0.915159i \(0.367935\pi\)
\(350\) −1403.09 −0.214281
\(351\) −22655.1 −3.44513
\(352\) −2063.80 −0.312503
\(353\) −2126.07 −0.320565 −0.160282 0.987071i \(-0.551240\pi\)
−0.160282 + 0.987071i \(0.551240\pi\)
\(354\) 26029.8 3.90810
\(355\) 1805.06 0.269866
\(356\) −1283.50 −0.191082
\(357\) 12384.0 1.83595
\(358\) −11098.0 −1.63841
\(359\) 12471.5 1.83349 0.916744 0.399474i \(-0.130807\pi\)
0.916744 + 0.399474i \(0.130807\pi\)
\(360\) 3833.08 0.561169
\(361\) 361.000 0.0526316
\(362\) 6173.01 0.896260
\(363\) −1156.11 −0.167162
\(364\) −4663.12 −0.671466
\(365\) −561.269 −0.0804880
\(366\) −23907.8 −3.41443
\(367\) −3574.01 −0.508342 −0.254171 0.967159i \(-0.581803\pi\)
−0.254171 + 0.967159i \(0.581803\pi\)
\(368\) 4118.51 0.583403
\(369\) −24449.5 −3.44930
\(370\) −5309.05 −0.745957
\(371\) 9007.75 1.26054
\(372\) −15147.7 −2.11122
\(373\) −4030.01 −0.559427 −0.279713 0.960084i \(-0.590239\pi\)
−0.279713 + 0.960084i \(0.590239\pi\)
\(374\) 3212.80 0.444197
\(375\) 1194.33 0.164466
\(376\) 3909.58 0.536227
\(377\) −8908.55 −1.21701
\(378\) 19996.7 2.72095
\(379\) 6530.98 0.885155 0.442578 0.896730i \(-0.354064\pi\)
0.442578 + 0.896730i \(0.354064\pi\)
\(380\) 441.463 0.0595962
\(381\) −14628.2 −1.96700
\(382\) −11278.6 −1.51064
\(383\) −10114.6 −1.34944 −0.674718 0.738076i \(-0.735735\pi\)
−0.674718 + 0.738076i \(0.735735\pi\)
\(384\) −13302.3 −1.76779
\(385\) 867.990 0.114901
\(386\) 10097.2 1.33144
\(387\) 35012.2 4.59889
\(388\) 1271.85 0.166414
\(389\) 8183.23 1.06660 0.533299 0.845927i \(-0.320952\pi\)
0.533299 + 0.845927i \(0.320952\pi\)
\(390\) 10802.6 1.40260
\(391\) −4250.36 −0.549744
\(392\) 1120.16 0.144328
\(393\) 24923.6 3.19906
\(394\) −10676.3 −1.36514
\(395\) 2155.52 0.274572
\(396\) 3286.33 0.417031
\(397\) −4617.19 −0.583704 −0.291852 0.956464i \(-0.594271\pi\)
−0.291852 + 0.956464i \(0.594271\pi\)
\(398\) −13212.1 −1.66397
\(399\) −2864.96 −0.359467
\(400\) −1989.54 −0.248692
\(401\) 9493.37 1.18223 0.591117 0.806586i \(-0.298687\pi\)
0.591117 + 0.806586i \(0.298687\pi\)
\(402\) 6222.64 0.772033
\(403\) 21692.9 2.68139
\(404\) 1440.88 0.177442
\(405\) −8342.20 −1.02352
\(406\) 7863.19 0.961191
\(407\) 3284.32 0.399995
\(408\) 9357.08 1.13540
\(409\) −823.193 −0.0995214 −0.0497607 0.998761i \(-0.515846\pi\)
−0.0497607 + 0.998761i \(0.515846\pi\)
\(410\) 6762.17 0.814536
\(411\) −4593.61 −0.551305
\(412\) −6506.35 −0.778022
\(413\) 12089.7 1.44043
\(414\) −11832.3 −1.40465
\(415\) 3763.49 0.445163
\(416\) −11929.7 −1.40601
\(417\) 3754.55 0.440914
\(418\) −743.258 −0.0869712
\(419\) −4611.76 −0.537707 −0.268854 0.963181i \(-0.586645\pi\)
−0.268854 + 0.963181i \(0.586645\pi\)
\(420\) −3503.53 −0.407035
\(421\) −13857.9 −1.60425 −0.802127 0.597154i \(-0.796298\pi\)
−0.802127 + 0.597154i \(0.796298\pi\)
\(422\) −4063.72 −0.468765
\(423\) −21078.9 −2.42291
\(424\) 6806.03 0.779552
\(425\) 2053.23 0.234344
\(426\) 12266.7 1.39513
\(427\) −11104.1 −1.25847
\(428\) 3455.80 0.390286
\(429\) −6682.81 −0.752096
\(430\) −9683.57 −1.08601
\(431\) −9836.15 −1.09928 −0.549641 0.835401i \(-0.685236\pi\)
−0.549641 + 0.835401i \(0.685236\pi\)
\(432\) 28354.7 3.15791
\(433\) −15250.8 −1.69262 −0.846311 0.532688i \(-0.821182\pi\)
−0.846311 + 0.532688i \(0.821182\pi\)
\(434\) −19147.4 −2.11775
\(435\) −6693.24 −0.737739
\(436\) 1070.52 0.117589
\(437\) 983.291 0.107637
\(438\) −3814.23 −0.416098
\(439\) −1098.00 −0.119372 −0.0596862 0.998217i \(-0.519010\pi\)
−0.0596862 + 0.998217i \(0.519010\pi\)
\(440\) 655.832 0.0710581
\(441\) −6039.45 −0.652138
\(442\) 18571.4 1.99853
\(443\) 13504.2 1.44831 0.724157 0.689635i \(-0.242229\pi\)
0.724157 + 0.689635i \(0.242229\pi\)
\(444\) −13256.7 −1.41698
\(445\) 1381.00 0.147114
\(446\) 9929.33 1.05419
\(447\) −20468.1 −2.16579
\(448\) 482.419 0.0508754
\(449\) −14196.1 −1.49210 −0.746051 0.665888i \(-0.768053\pi\)
−0.746051 + 0.665888i \(0.768053\pi\)
\(450\) 5715.86 0.598773
\(451\) −4183.26 −0.436767
\(452\) −1153.57 −0.120042
\(453\) 6452.57 0.669246
\(454\) −3638.98 −0.376180
\(455\) 5017.37 0.516962
\(456\) −2164.69 −0.222305
\(457\) 8889.24 0.909893 0.454947 0.890519i \(-0.349658\pi\)
0.454947 + 0.890519i \(0.349658\pi\)
\(458\) −15784.3 −1.61037
\(459\) −29262.4 −2.97571
\(460\) 1202.46 0.121880
\(461\) −16936.9 −1.71113 −0.855566 0.517695i \(-0.826790\pi\)
−0.855566 + 0.517695i \(0.826790\pi\)
\(462\) 5898.63 0.594003
\(463\) −1071.27 −0.107530 −0.0537649 0.998554i \(-0.517122\pi\)
−0.0537649 + 0.998554i \(0.517122\pi\)
\(464\) 11149.7 1.11555
\(465\) 16298.5 1.62543
\(466\) 2016.11 0.200418
\(467\) −10354.0 −1.02597 −0.512984 0.858398i \(-0.671460\pi\)
−0.512984 + 0.858398i \(0.671460\pi\)
\(468\) 18996.4 1.87630
\(469\) 2890.15 0.284552
\(470\) 5829.94 0.572160
\(471\) 5741.71 0.561707
\(472\) 9134.69 0.890801
\(473\) 5990.52 0.582335
\(474\) 14648.4 1.41945
\(475\) −475.000 −0.0458831
\(476\) −6023.10 −0.579975
\(477\) −36695.4 −3.52236
\(478\) −18233.9 −1.74477
\(479\) 1957.37 0.186711 0.0933554 0.995633i \(-0.470241\pi\)
0.0933554 + 0.995633i \(0.470241\pi\)
\(480\) −8963.11 −0.852308
\(481\) 18984.8 1.79965
\(482\) −9907.60 −0.936263
\(483\) −7803.58 −0.735145
\(484\) 562.284 0.0528066
\(485\) −1368.47 −0.128122
\(486\) −22480.0 −2.09818
\(487\) −9116.55 −0.848275 −0.424138 0.905598i \(-0.639423\pi\)
−0.424138 + 0.905598i \(0.639423\pi\)
\(488\) −8390.02 −0.778275
\(489\) 1761.60 0.162908
\(490\) 1670.37 0.153999
\(491\) −184.420 −0.0169506 −0.00847532 0.999964i \(-0.502698\pi\)
−0.00847532 + 0.999964i \(0.502698\pi\)
\(492\) 16885.2 1.54724
\(493\) −11506.7 −1.05119
\(494\) −4296.36 −0.391300
\(495\) −3535.99 −0.321072
\(496\) −27150.3 −2.45784
\(497\) 5697.35 0.514208
\(498\) 25575.7 2.30135
\(499\) 14600.0 1.30979 0.654896 0.755719i \(-0.272713\pi\)
0.654896 + 0.755719i \(0.272713\pi\)
\(500\) −580.872 −0.0519548
\(501\) −34406.0 −3.06816
\(502\) −14891.6 −1.32400
\(503\) −22433.4 −1.98858 −0.994288 0.106731i \(-0.965962\pi\)
−0.994288 + 0.106731i \(0.965962\pi\)
\(504\) 12098.4 1.06926
\(505\) −1550.34 −0.136612
\(506\) −2024.49 −0.177864
\(507\) −17638.1 −1.54504
\(508\) 7114.57 0.621374
\(509\) −15778.4 −1.37400 −0.687001 0.726657i \(-0.741073\pi\)
−0.687001 + 0.726657i \(0.741073\pi\)
\(510\) 13953.2 1.21149
\(511\) −1771.55 −0.153363
\(512\) 7339.37 0.633511
\(513\) 6769.66 0.582627
\(514\) 20463.9 1.75608
\(515\) 7000.63 0.598999
\(516\) −24180.0 −2.06291
\(517\) −3606.56 −0.306802
\(518\) −16757.1 −1.42136
\(519\) 17978.0 1.52051
\(520\) 3791.00 0.319704
\(521\) 7388.88 0.621329 0.310665 0.950520i \(-0.399448\pi\)
0.310665 + 0.950520i \(0.399448\pi\)
\(522\) −32032.8 −2.68589
\(523\) −21072.4 −1.76182 −0.880909 0.473286i \(-0.843068\pi\)
−0.880909 + 0.473286i \(0.843068\pi\)
\(524\) −12121.8 −1.01058
\(525\) 3769.69 0.313376
\(526\) −22711.6 −1.88265
\(527\) 28019.5 2.31603
\(528\) 8364.07 0.689393
\(529\) −9488.71 −0.779873
\(530\) 10149.1 0.831790
\(531\) −49250.6 −4.02504
\(532\) 1393.40 0.113556
\(533\) −24181.1 −1.96510
\(534\) 9384.95 0.760536
\(535\) −3718.33 −0.300481
\(536\) 2183.73 0.175975
\(537\) 29817.1 2.39610
\(538\) −30571.2 −2.44985
\(539\) −1033.34 −0.0825770
\(540\) 8278.54 0.659725
\(541\) 7955.28 0.632207 0.316104 0.948725i \(-0.397625\pi\)
0.316104 + 0.948725i \(0.397625\pi\)
\(542\) −18195.0 −1.44196
\(543\) −16585.0 −1.31074
\(544\) −15408.9 −1.21444
\(545\) −1151.85 −0.0905317
\(546\) 34096.7 2.67253
\(547\) −1204.66 −0.0941636 −0.0470818 0.998891i \(-0.514992\pi\)
−0.0470818 + 0.998891i \(0.514992\pi\)
\(548\) 2234.15 0.174157
\(549\) 45235.7 3.51659
\(550\) 977.971 0.0758197
\(551\) 2661.99 0.205816
\(552\) −5896.19 −0.454635
\(553\) 6803.53 0.523175
\(554\) −7317.18 −0.561150
\(555\) 14263.8 1.09093
\(556\) −1826.06 −0.139285
\(557\) 13218.3 1.00553 0.502763 0.864424i \(-0.332317\pi\)
0.502763 + 0.864424i \(0.332317\pi\)
\(558\) 78001.8 5.91771
\(559\) 34627.9 2.62004
\(560\) −6279.63 −0.473862
\(561\) −8631.83 −0.649619
\(562\) 17933.9 1.34608
\(563\) −4353.73 −0.325911 −0.162956 0.986633i \(-0.552103\pi\)
−0.162956 + 0.986633i \(0.552103\pi\)
\(564\) 14557.4 1.08684
\(565\) 1241.20 0.0924207
\(566\) −7667.95 −0.569449
\(567\) −26330.7 −1.95024
\(568\) 4304.78 0.318001
\(569\) −10725.2 −0.790202 −0.395101 0.918638i \(-0.629290\pi\)
−0.395101 + 0.918638i \(0.629290\pi\)
\(570\) −3227.97 −0.237202
\(571\) −17478.1 −1.28098 −0.640488 0.767968i \(-0.721268\pi\)
−0.640488 + 0.767968i \(0.721268\pi\)
\(572\) 3250.25 0.237587
\(573\) 30302.4 2.20925
\(574\) 21343.6 1.55203
\(575\) −1293.80 −0.0938354
\(576\) −1965.26 −0.142163
\(577\) 25176.5 1.81649 0.908244 0.418441i \(-0.137423\pi\)
0.908244 + 0.418441i \(0.137423\pi\)
\(578\) 6515.76 0.468892
\(579\) −27128.2 −1.94717
\(580\) 3255.32 0.233052
\(581\) 11878.8 0.848221
\(582\) −9299.77 −0.662350
\(583\) −6278.51 −0.446020
\(584\) −1338.54 −0.0948443
\(585\) −20439.6 −1.44457
\(586\) 16759.9 1.18148
\(587\) −10522.1 −0.739854 −0.369927 0.929061i \(-0.620617\pi\)
−0.369927 + 0.929061i \(0.620617\pi\)
\(588\) 4170.94 0.292528
\(589\) −6482.12 −0.453466
\(590\) 13621.6 0.950494
\(591\) 28684.1 1.99645
\(592\) −23761.0 −1.64961
\(593\) −12806.4 −0.886837 −0.443418 0.896315i \(-0.646234\pi\)
−0.443418 + 0.896315i \(0.646234\pi\)
\(594\) −13938.0 −0.962763
\(595\) 6480.66 0.446523
\(596\) 9954.87 0.684173
\(597\) 35496.9 2.43349
\(598\) −11702.4 −0.800246
\(599\) −13801.7 −0.941440 −0.470720 0.882283i \(-0.656006\pi\)
−0.470720 + 0.882283i \(0.656006\pi\)
\(600\) 2848.28 0.193801
\(601\) −10114.3 −0.686472 −0.343236 0.939249i \(-0.611523\pi\)
−0.343236 + 0.939249i \(0.611523\pi\)
\(602\) −30564.5 −2.06930
\(603\) −11773.8 −0.795134
\(604\) −3138.27 −0.211415
\(605\) −605.000 −0.0406558
\(606\) −10535.7 −0.706244
\(607\) 14070.1 0.940835 0.470417 0.882444i \(-0.344103\pi\)
0.470417 + 0.882444i \(0.344103\pi\)
\(608\) 3564.75 0.237779
\(609\) −21126.1 −1.40570
\(610\) −12511.1 −0.830427
\(611\) −20847.5 −1.38036
\(612\) 24536.7 1.62065
\(613\) 12314.7 0.811396 0.405698 0.914007i \(-0.367028\pi\)
0.405698 + 0.914007i \(0.367028\pi\)
\(614\) −11681.6 −0.767801
\(615\) −18167.9 −1.19122
\(616\) 2070.02 0.135395
\(617\) −104.090 −0.00679176 −0.00339588 0.999994i \(-0.501081\pi\)
−0.00339588 + 0.999994i \(0.501081\pi\)
\(618\) 47574.4 3.09664
\(619\) 7499.08 0.486936 0.243468 0.969909i \(-0.421715\pi\)
0.243468 + 0.969909i \(0.421715\pi\)
\(620\) −7926.92 −0.513472
\(621\) 18439.2 1.19153
\(622\) 20207.1 1.30262
\(623\) 4358.90 0.280314
\(624\) 48348.0 3.10171
\(625\) 625.000 0.0400000
\(626\) 32807.6 2.09466
\(627\) 1996.91 0.127192
\(628\) −2792.54 −0.177443
\(629\) 24521.7 1.55444
\(630\) 18041.1 1.14091
\(631\) 7178.78 0.452904 0.226452 0.974022i \(-0.427287\pi\)
0.226452 + 0.974022i \(0.427287\pi\)
\(632\) 5140.58 0.323546
\(633\) 10918.0 0.685549
\(634\) −7713.32 −0.483178
\(635\) −7655.05 −0.478396
\(636\) 25342.4 1.58002
\(637\) −5973.15 −0.371530
\(638\) −5480.74 −0.340101
\(639\) −23209.7 −1.43687
\(640\) −6961.19 −0.429946
\(641\) 14085.2 0.867915 0.433958 0.900933i \(-0.357117\pi\)
0.433958 + 0.900933i \(0.357117\pi\)
\(642\) −25268.8 −1.55340
\(643\) −13814.3 −0.847249 −0.423624 0.905838i \(-0.639242\pi\)
−0.423624 + 0.905838i \(0.639242\pi\)
\(644\) 3795.35 0.232232
\(645\) 26016.9 1.58824
\(646\) −5549.38 −0.337983
\(647\) −6237.19 −0.378994 −0.189497 0.981881i \(-0.560686\pi\)
−0.189497 + 0.981881i \(0.560686\pi\)
\(648\) −19894.8 −1.20608
\(649\) −8426.68 −0.509670
\(650\) 5653.10 0.341128
\(651\) 51443.3 3.09712
\(652\) −856.770 −0.0514627
\(653\) −28608.0 −1.71442 −0.857212 0.514964i \(-0.827805\pi\)
−0.857212 + 0.514964i \(0.827805\pi\)
\(654\) −7827.66 −0.468021
\(655\) 13042.7 0.778047
\(656\) 30264.5 1.80127
\(657\) 7216.86 0.428549
\(658\) 18401.2 1.09020
\(659\) 2255.66 0.133335 0.0666677 0.997775i \(-0.478763\pi\)
0.0666677 + 0.997775i \(0.478763\pi\)
\(660\) 2442.00 0.144023
\(661\) −12873.4 −0.757517 −0.378759 0.925496i \(-0.623649\pi\)
−0.378759 + 0.925496i \(0.623649\pi\)
\(662\) 502.935 0.0295274
\(663\) −49895.8 −2.92276
\(664\) 8975.33 0.524564
\(665\) −1499.26 −0.0874265
\(666\) 68264.4 3.97176
\(667\) 7250.73 0.420914
\(668\) 16733.7 0.969232
\(669\) −26677.2 −1.54170
\(670\) 3256.35 0.187767
\(671\) 7739.73 0.445289
\(672\) −28290.5 −1.62400
\(673\) −10154.3 −0.581606 −0.290803 0.956783i \(-0.593922\pi\)
−0.290803 + 0.956783i \(0.593922\pi\)
\(674\) −18661.1 −1.06647
\(675\) −8907.44 −0.507922
\(676\) 8578.47 0.488079
\(677\) −19891.8 −1.12925 −0.564626 0.825347i \(-0.690980\pi\)
−0.564626 + 0.825347i \(0.690980\pi\)
\(678\) 8434.87 0.477786
\(679\) −4319.34 −0.244125
\(680\) 4896.63 0.276143
\(681\) 9776.85 0.550147
\(682\) 13346.0 0.749330
\(683\) −498.789 −0.0279438 −0.0139719 0.999902i \(-0.504448\pi\)
−0.0139719 + 0.999902i \(0.504448\pi\)
\(684\) −5676.39 −0.317313
\(685\) −2403.87 −0.134083
\(686\) 24522.6 1.36484
\(687\) 42407.7 2.35510
\(688\) −43339.5 −2.40160
\(689\) −36292.6 −2.00673
\(690\) −8792.35 −0.485100
\(691\) −1697.97 −0.0934787 −0.0467394 0.998907i \(-0.514883\pi\)
−0.0467394 + 0.998907i \(0.514883\pi\)
\(692\) −8743.78 −0.480330
\(693\) −11160.7 −0.611776
\(694\) 32049.6 1.75300
\(695\) 1964.79 0.107235
\(696\) −15962.3 −0.869325
\(697\) −31233.4 −1.69735
\(698\) 18692.5 1.01364
\(699\) −5416.70 −0.293102
\(700\) −1833.42 −0.0989956
\(701\) −5049.07 −0.272041 −0.136020 0.990706i \(-0.543431\pi\)
−0.136020 + 0.990706i \(0.543431\pi\)
\(702\) −80567.5 −4.33166
\(703\) −5672.92 −0.304350
\(704\) −336.252 −0.0180014
\(705\) −15663.3 −0.836759
\(706\) −7560.86 −0.403055
\(707\) −4893.38 −0.260304
\(708\) 34013.2 1.80550
\(709\) −28098.4 −1.48837 −0.744186 0.667972i \(-0.767163\pi\)
−0.744186 + 0.667972i \(0.767163\pi\)
\(710\) 6419.25 0.339310
\(711\) −27716.0 −1.46193
\(712\) 3293.48 0.173354
\(713\) −17656.0 −0.927380
\(714\) 44040.9 2.30839
\(715\) −3497.17 −0.182918
\(716\) −14501.9 −0.756927
\(717\) 48989.2 2.55165
\(718\) 44352.0 2.30530
\(719\) 23561.7 1.22212 0.611059 0.791585i \(-0.290744\pi\)
0.611059 + 0.791585i \(0.290744\pi\)
\(720\) 25581.7 1.32413
\(721\) 22096.3 1.14134
\(722\) 1283.81 0.0661751
\(723\) 26618.8 1.36924
\(724\) 8066.30 0.414063
\(725\) −3502.62 −0.179426
\(726\) −4111.42 −0.210178
\(727\) −5103.57 −0.260359 −0.130179 0.991490i \(-0.541555\pi\)
−0.130179 + 0.991490i \(0.541555\pi\)
\(728\) 11965.6 0.609170
\(729\) 15349.3 0.779827
\(730\) −1996.02 −0.101200
\(731\) 44726.9 2.26304
\(732\) −31240.4 −1.57743
\(733\) −16522.7 −0.832580 −0.416290 0.909232i \(-0.636670\pi\)
−0.416290 + 0.909232i \(0.636670\pi\)
\(734\) −12710.1 −0.639153
\(735\) −4487.79 −0.225217
\(736\) 9709.66 0.486281
\(737\) −2014.47 −0.100684
\(738\) −86948.8 −4.33690
\(739\) 4168.34 0.207490 0.103745 0.994604i \(-0.466917\pi\)
0.103745 + 0.994604i \(0.466917\pi\)
\(740\) −6937.35 −0.344625
\(741\) 11543.0 0.572259
\(742\) 32033.9 1.58491
\(743\) 32527.1 1.60606 0.803032 0.595937i \(-0.203219\pi\)
0.803032 + 0.595937i \(0.203219\pi\)
\(744\) 38869.3 1.91535
\(745\) −10711.1 −0.526745
\(746\) −14331.8 −0.703382
\(747\) −48391.5 −2.37022
\(748\) 4198.18 0.205215
\(749\) −11736.3 −0.572542
\(750\) 4247.34 0.206788
\(751\) −16121.8 −0.783346 −0.391673 0.920104i \(-0.628104\pi\)
−0.391673 + 0.920104i \(0.628104\pi\)
\(752\) 26092.3 1.26528
\(753\) 40009.4 1.93629
\(754\) −31681.1 −1.53018
\(755\) 3376.68 0.162768
\(756\) 26129.8 1.25705
\(757\) −1959.32 −0.0940722 −0.0470361 0.998893i \(-0.514978\pi\)
−0.0470361 + 0.998893i \(0.514978\pi\)
\(758\) 23225.9 1.11293
\(759\) 5439.19 0.260119
\(760\) −1132.80 −0.0540671
\(761\) 4037.12 0.192307 0.0961534 0.995367i \(-0.469346\pi\)
0.0961534 + 0.995367i \(0.469346\pi\)
\(762\) −52021.7 −2.47316
\(763\) −3635.61 −0.172501
\(764\) −14737.8 −0.697901
\(765\) −26400.7 −1.24774
\(766\) −35970.3 −1.69668
\(767\) −48709.9 −2.29311
\(768\) −49643.0 −2.33247
\(769\) −20471.7 −0.959985 −0.479993 0.877272i \(-0.659361\pi\)
−0.479993 + 0.877272i \(0.659361\pi\)
\(770\) 3086.80 0.144468
\(771\) −54980.5 −2.56819
\(772\) 13194.1 0.615110
\(773\) 11032.7 0.513347 0.256673 0.966498i \(-0.417374\pi\)
0.256673 + 0.966498i \(0.417374\pi\)
\(774\) 124513. 5.78231
\(775\) 8529.11 0.395322
\(776\) −3263.59 −0.150974
\(777\) 45021.3 2.07868
\(778\) 29101.7 1.34106
\(779\) 7225.64 0.332330
\(780\) 14115.9 0.647986
\(781\) −3971.13 −0.181944
\(782\) −15115.4 −0.691208
\(783\) 49919.1 2.27837
\(784\) 7475.87 0.340555
\(785\) 3004.68 0.136614
\(786\) 88634.8 4.02226
\(787\) −11394.8 −0.516111 −0.258056 0.966130i \(-0.583082\pi\)
−0.258056 + 0.966130i \(0.583082\pi\)
\(788\) −13950.8 −0.630680
\(789\) 61019.3 2.75329
\(790\) 7665.59 0.345227
\(791\) 3917.63 0.176100
\(792\) −8432.77 −0.378340
\(793\) 44739.0 2.00344
\(794\) −16419.9 −0.733906
\(795\) −27267.6 −1.21646
\(796\) −17264.3 −0.768739
\(797\) −19693.0 −0.875236 −0.437618 0.899161i \(-0.644178\pi\)
−0.437618 + 0.899161i \(0.644178\pi\)
\(798\) −10188.5 −0.451968
\(799\) −26927.6 −1.19228
\(800\) −4690.46 −0.207291
\(801\) −17757.1 −0.783293
\(802\) 33760.9 1.48646
\(803\) 1234.79 0.0542650
\(804\) 8131.15 0.356671
\(805\) −4083.67 −0.178796
\(806\) 77145.5 3.37138
\(807\) 82135.7 3.58279
\(808\) −3697.32 −0.160979
\(809\) 24939.6 1.08384 0.541921 0.840429i \(-0.317697\pi\)
0.541921 + 0.840429i \(0.317697\pi\)
\(810\) −29667.0 −1.28690
\(811\) 9789.67 0.423874 0.211937 0.977283i \(-0.432023\pi\)
0.211937 + 0.977283i \(0.432023\pi\)
\(812\) 10274.9 0.444060
\(813\) 48884.7 2.10881
\(814\) 11679.9 0.502924
\(815\) 921.857 0.0396211
\(816\) 62448.5 2.67909
\(817\) −10347.3 −0.443091
\(818\) −2927.49 −0.125131
\(819\) −64513.9 −2.75250
\(820\) 8836.15 0.376307
\(821\) 6187.91 0.263045 0.131522 0.991313i \(-0.458014\pi\)
0.131522 + 0.991313i \(0.458014\pi\)
\(822\) −16336.1 −0.693170
\(823\) 34354.7 1.45508 0.727538 0.686067i \(-0.240664\pi\)
0.727538 + 0.686067i \(0.240664\pi\)
\(824\) 16695.4 0.705839
\(825\) −2627.52 −0.110883
\(826\) 42994.2 1.81109
\(827\) 20640.7 0.867894 0.433947 0.900938i \(-0.357121\pi\)
0.433947 + 0.900938i \(0.357121\pi\)
\(828\) −15461.3 −0.648935
\(829\) −35471.4 −1.48610 −0.743048 0.669238i \(-0.766621\pi\)
−0.743048 + 0.669238i \(0.766621\pi\)
\(830\) 13383.9 0.559715
\(831\) 19659.1 0.820657
\(832\) −1943.69 −0.0809918
\(833\) −7715.20 −0.320907
\(834\) 13352.2 0.554374
\(835\) −18004.9 −0.746211
\(836\) −971.218 −0.0401798
\(837\) −121556. −5.01982
\(838\) −16400.6 −0.676074
\(839\) −12387.4 −0.509727 −0.254864 0.966977i \(-0.582031\pi\)
−0.254864 + 0.966977i \(0.582031\pi\)
\(840\) 8990.11 0.369272
\(841\) −4759.61 −0.195154
\(842\) −49282.1 −2.01707
\(843\) −48183.2 −1.96859
\(844\) −5310.08 −0.216565
\(845\) −9230.16 −0.375772
\(846\) −74962.1 −3.04640
\(847\) −1909.58 −0.0774662
\(848\) 45423.0 1.83942
\(849\) 20601.5 0.832794
\(850\) 7301.81 0.294647
\(851\) −15451.9 −0.622426
\(852\) 16028.9 0.644534
\(853\) 13430.3 0.539093 0.269546 0.962987i \(-0.413126\pi\)
0.269546 + 0.962987i \(0.413126\pi\)
\(854\) −39489.2 −1.58231
\(855\) 6107.61 0.244299
\(856\) −8867.64 −0.354077
\(857\) 33098.6 1.31929 0.659643 0.751579i \(-0.270708\pi\)
0.659643 + 0.751579i \(0.270708\pi\)
\(858\) −23765.8 −0.945631
\(859\) −34099.5 −1.35443 −0.677217 0.735783i \(-0.736814\pi\)
−0.677217 + 0.735783i \(0.736814\pi\)
\(860\) −12653.6 −0.501724
\(861\) −57344.0 −2.26978
\(862\) −34979.9 −1.38216
\(863\) −5836.94 −0.230234 −0.115117 0.993352i \(-0.536724\pi\)
−0.115117 + 0.993352i \(0.536724\pi\)
\(864\) 66848.0 2.63219
\(865\) 9408.03 0.369806
\(866\) −54235.7 −2.12818
\(867\) −17505.9 −0.685735
\(868\) −25019.9 −0.978378
\(869\) −4742.15 −0.185117
\(870\) −23802.9 −0.927579
\(871\) −11644.5 −0.452996
\(872\) −2746.98 −0.106679
\(873\) 17596.0 0.682169
\(874\) 3496.84 0.135334
\(875\) 1972.71 0.0762167
\(876\) −4984.07 −0.192233
\(877\) 23603.5 0.908816 0.454408 0.890794i \(-0.349851\pi\)
0.454408 + 0.890794i \(0.349851\pi\)
\(878\) −3904.76 −0.150090
\(879\) −45029.0 −1.72786
\(880\) 4376.98 0.167668
\(881\) 2235.23 0.0854788 0.0427394 0.999086i \(-0.486391\pi\)
0.0427394 + 0.999086i \(0.486391\pi\)
\(882\) −21477.9 −0.819951
\(883\) 20886.0 0.796004 0.398002 0.917385i \(-0.369704\pi\)
0.398002 + 0.917385i \(0.369704\pi\)
\(884\) 24267.3 0.923300
\(885\) −36597.1 −1.39006
\(886\) 48024.4 1.82101
\(887\) 37757.7 1.42929 0.714644 0.699489i \(-0.246589\pi\)
0.714644 + 0.699489i \(0.246589\pi\)
\(888\) 34017.0 1.28551
\(889\) −24161.8 −0.911543
\(890\) 4911.21 0.184971
\(891\) 18352.8 0.690060
\(892\) 12974.7 0.487024
\(893\) 6229.52 0.233441
\(894\) −72790.0 −2.72311
\(895\) 15603.5 0.582758
\(896\) −21971.8 −0.819226
\(897\) 31440.9 1.17033
\(898\) −50484.9 −1.87606
\(899\) −47798.8 −1.77328
\(900\) 7468.93 0.276627
\(901\) −46877.2 −1.73330
\(902\) −14876.8 −0.549160
\(903\) 82117.8 3.02626
\(904\) 2960.07 0.108905
\(905\) −8679.08 −0.318787
\(906\) 22947.0 0.841461
\(907\) −36817.6 −1.34786 −0.673930 0.738795i \(-0.735395\pi\)
−0.673930 + 0.738795i \(0.735395\pi\)
\(908\) −4755.07 −0.173791
\(909\) 19934.5 0.727376
\(910\) 17843.0 0.649990
\(911\) 7864.91 0.286033 0.143017 0.989720i \(-0.454320\pi\)
0.143017 + 0.989720i \(0.454320\pi\)
\(912\) −14447.0 −0.524549
\(913\) −8279.68 −0.300129
\(914\) 31612.5 1.14403
\(915\) 33613.7 1.21446
\(916\) −20625.4 −0.743975
\(917\) 41167.1 1.48250
\(918\) −104065. −3.74145
\(919\) 37629.1 1.35067 0.675337 0.737509i \(-0.263998\pi\)
0.675337 + 0.737509i \(0.263998\pi\)
\(920\) −3085.52 −0.110572
\(921\) 31384.9 1.12288
\(922\) −60232.1 −2.15145
\(923\) −22954.9 −0.818601
\(924\) 7707.77 0.274423
\(925\) 7464.37 0.265326
\(926\) −3809.73 −0.135200
\(927\) −90015.0 −3.18930
\(928\) 26286.3 0.929837
\(929\) 34809.4 1.22934 0.614672 0.788783i \(-0.289289\pi\)
0.614672 + 0.788783i \(0.289289\pi\)
\(930\) 57961.6 2.04369
\(931\) 1784.86 0.0628317
\(932\) 2634.46 0.0925909
\(933\) −54290.4 −1.90502
\(934\) −36821.6 −1.28998
\(935\) −4517.10 −0.157995
\(936\) −48745.1 −1.70223
\(937\) −33947.7 −1.18359 −0.591795 0.806088i \(-0.701581\pi\)
−0.591795 + 0.806088i \(0.701581\pi\)
\(938\) 10278.1 0.357775
\(939\) −88144.3 −3.06334
\(940\) 7618.01 0.264332
\(941\) 13691.9 0.474328 0.237164 0.971470i \(-0.423782\pi\)
0.237164 + 0.971470i \(0.423782\pi\)
\(942\) 20419.0 0.706250
\(943\) 19681.2 0.679647
\(944\) 60964.3 2.10193
\(945\) −28114.8 −0.967804
\(946\) 21303.8 0.732186
\(947\) 12598.2 0.432298 0.216149 0.976360i \(-0.430650\pi\)
0.216149 + 0.976360i \(0.430650\pi\)
\(948\) 19141.1 0.655773
\(949\) 7137.63 0.244149
\(950\) −1689.22 −0.0576901
\(951\) 20723.4 0.706627
\(952\) 15455.4 0.526167
\(953\) −13647.4 −0.463884 −0.231942 0.972730i \(-0.574508\pi\)
−0.231942 + 0.972730i \(0.574508\pi\)
\(954\) −130498. −4.42877
\(955\) 15857.5 0.537314
\(956\) −23826.4 −0.806066
\(957\) 14725.1 0.497383
\(958\) 6960.91 0.234757
\(959\) −7587.41 −0.255485
\(960\) −1460.35 −0.0490963
\(961\) 86602.1 2.90699
\(962\) 67515.0 2.26276
\(963\) 47810.8 1.59988
\(964\) −12946.3 −0.432544
\(965\) −14196.4 −0.473573
\(966\) −27751.5 −0.924318
\(967\) 33723.7 1.12149 0.560745 0.827988i \(-0.310515\pi\)
0.560745 + 0.827988i \(0.310515\pi\)
\(968\) −1442.83 −0.0479073
\(969\) 14909.5 0.494286
\(970\) −4866.64 −0.161091
\(971\) 5738.59 0.189660 0.0948301 0.995493i \(-0.469769\pi\)
0.0948301 + 0.995493i \(0.469769\pi\)
\(972\) −29374.8 −0.969337
\(973\) 6201.51 0.204328
\(974\) −32420.8 −1.06656
\(975\) −15188.2 −0.498884
\(976\) −55994.4 −1.83641
\(977\) 4949.31 0.162070 0.0810351 0.996711i \(-0.474177\pi\)
0.0810351 + 0.996711i \(0.474177\pi\)
\(978\) 6264.70 0.204829
\(979\) −3038.21 −0.0991845
\(980\) 2182.68 0.0711461
\(981\) 14810.6 0.482025
\(982\) −655.846 −0.0213125
\(983\) 8112.37 0.263219 0.131610 0.991302i \(-0.457985\pi\)
0.131610 + 0.991302i \(0.457985\pi\)
\(984\) −43327.7 −1.40369
\(985\) 15010.6 0.485560
\(986\) −40920.8 −1.32169
\(987\) −49438.6 −1.59437
\(988\) −5614.07 −0.180777
\(989\) −28183.9 −0.906163
\(990\) −12574.9 −0.403693
\(991\) 5787.92 0.185529 0.0927646 0.995688i \(-0.470430\pi\)
0.0927646 + 0.995688i \(0.470430\pi\)
\(992\) −64008.7 −2.04867
\(993\) −1351.24 −0.0431825
\(994\) 20261.3 0.646528
\(995\) 18575.8 0.591852
\(996\) 33419.9 1.06320
\(997\) 37407.4 1.18827 0.594135 0.804365i \(-0.297494\pi\)
0.594135 + 0.804365i \(0.297494\pi\)
\(998\) 51921.4 1.64684
\(999\) −106382. −3.36913
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.f.1.19 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.f.1.19 23 1.1 even 1 trivial