Properties

Label 2151.4.a.f.1.5
Level $2151$
Weight $4$
Character 2151.1
Self dual yes
Analytic conductor $126.913$
Analytic rank $0$
Dimension $37$
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] = [2151,4,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2151.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(126.913108422\)
Analytic rank: \(0\)
Dimension: \(37\)
Twist minimal: no (minimal twist has level 239)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 2151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.79841 q^{2} +15.0248 q^{4} +14.4400 q^{5} +1.92684 q^{7} -33.7078 q^{8} +O(q^{10})\) \(q-4.79841 q^{2} +15.0248 q^{4} +14.4400 q^{5} +1.92684 q^{7} -33.7078 q^{8} -69.2892 q^{10} -24.5238 q^{11} -11.6663 q^{13} -9.24579 q^{14} +41.5456 q^{16} +49.4720 q^{17} +106.669 q^{19} +216.958 q^{20} +117.675 q^{22} +102.672 q^{23} +83.5141 q^{25} +55.9795 q^{26} +28.9504 q^{28} -64.1149 q^{29} -280.276 q^{31} +70.3092 q^{32} -237.387 q^{34} +27.8237 q^{35} -224.640 q^{37} -511.844 q^{38} -486.741 q^{40} +65.1232 q^{41} -11.6265 q^{43} -368.465 q^{44} -492.663 q^{46} +171.648 q^{47} -339.287 q^{49} -400.735 q^{50} -175.283 q^{52} +347.764 q^{53} -354.124 q^{55} -64.9496 q^{56} +307.650 q^{58} -753.785 q^{59} +865.111 q^{61} +1344.88 q^{62} -669.737 q^{64} -168.461 q^{65} +664.993 q^{67} +743.305 q^{68} -133.509 q^{70} +624.120 q^{71} +750.992 q^{73} +1077.92 q^{74} +1602.68 q^{76} -47.2536 q^{77} +1171.54 q^{79} +599.919 q^{80} -312.488 q^{82} -1237.79 q^{83} +714.376 q^{85} +55.7888 q^{86} +826.643 q^{88} +558.663 q^{89} -22.4791 q^{91} +1542.63 q^{92} -823.637 q^{94} +1540.31 q^{95} -698.747 q^{97} +1628.04 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.79841 −1.69650 −0.848248 0.529600i \(-0.822342\pi\)
−0.848248 + 0.529600i \(0.822342\pi\)
\(3\) 0 0
\(4\) 15.0248 1.87810
\(5\) 14.4400 1.29155 0.645777 0.763526i \(-0.276533\pi\)
0.645777 + 0.763526i \(0.276533\pi\)
\(6\) 0 0
\(7\) 1.92684 0.104040 0.0520199 0.998646i \(-0.483434\pi\)
0.0520199 + 0.998646i \(0.483434\pi\)
\(8\) −33.7078 −1.48969
\(9\) 0 0
\(10\) −69.2892 −2.19112
\(11\) −24.5238 −0.672201 −0.336100 0.941826i \(-0.609108\pi\)
−0.336100 + 0.941826i \(0.609108\pi\)
\(12\) 0 0
\(13\) −11.6663 −0.248895 −0.124448 0.992226i \(-0.539716\pi\)
−0.124448 + 0.992226i \(0.539716\pi\)
\(14\) −9.24579 −0.176503
\(15\) 0 0
\(16\) 41.5456 0.649149
\(17\) 49.4720 0.705807 0.352903 0.935660i \(-0.385194\pi\)
0.352903 + 0.935660i \(0.385194\pi\)
\(18\) 0 0
\(19\) 106.669 1.28798 0.643990 0.765034i \(-0.277278\pi\)
0.643990 + 0.765034i \(0.277278\pi\)
\(20\) 216.958 2.42566
\(21\) 0 0
\(22\) 117.675 1.14039
\(23\) 102.672 0.930810 0.465405 0.885098i \(-0.345909\pi\)
0.465405 + 0.885098i \(0.345909\pi\)
\(24\) 0 0
\(25\) 83.5141 0.668113
\(26\) 55.9795 0.422250
\(27\) 0 0
\(28\) 28.9504 0.195397
\(29\) −64.1149 −0.410546 −0.205273 0.978705i \(-0.565808\pi\)
−0.205273 + 0.978705i \(0.565808\pi\)
\(30\) 0 0
\(31\) −280.276 −1.62384 −0.811920 0.583769i \(-0.801577\pi\)
−0.811920 + 0.583769i \(0.801577\pi\)
\(32\) 70.3092 0.388407
\(33\) 0 0
\(34\) −237.387 −1.19740
\(35\) 27.8237 0.134373
\(36\) 0 0
\(37\) −224.640 −0.998124 −0.499062 0.866566i \(-0.666322\pi\)
−0.499062 + 0.866566i \(0.666322\pi\)
\(38\) −511.844 −2.18505
\(39\) 0 0
\(40\) −486.741 −1.92401
\(41\) 65.1232 0.248062 0.124031 0.992278i \(-0.460418\pi\)
0.124031 + 0.992278i \(0.460418\pi\)
\(42\) 0 0
\(43\) −11.6265 −0.0412331 −0.0206166 0.999787i \(-0.506563\pi\)
−0.0206166 + 0.999787i \(0.506563\pi\)
\(44\) −368.465 −1.26246
\(45\) 0 0
\(46\) −492.663 −1.57911
\(47\) 171.648 0.532711 0.266355 0.963875i \(-0.414181\pi\)
0.266355 + 0.963875i \(0.414181\pi\)
\(48\) 0 0
\(49\) −339.287 −0.989176
\(50\) −400.735 −1.13345
\(51\) 0 0
\(52\) −175.283 −0.467449
\(53\) 347.764 0.901302 0.450651 0.892700i \(-0.351192\pi\)
0.450651 + 0.892700i \(0.351192\pi\)
\(54\) 0 0
\(55\) −354.124 −0.868184
\(56\) −64.9496 −0.154987
\(57\) 0 0
\(58\) 307.650 0.696490
\(59\) −753.785 −1.66330 −0.831648 0.555303i \(-0.812602\pi\)
−0.831648 + 0.555303i \(0.812602\pi\)
\(60\) 0 0
\(61\) 865.111 1.81584 0.907919 0.419145i \(-0.137670\pi\)
0.907919 + 0.419145i \(0.137670\pi\)
\(62\) 1344.88 2.75484
\(63\) 0 0
\(64\) −669.737 −1.30808
\(65\) −168.461 −0.321462
\(66\) 0 0
\(67\) 664.993 1.21257 0.606283 0.795249i \(-0.292660\pi\)
0.606283 + 0.795249i \(0.292660\pi\)
\(68\) 743.305 1.32557
\(69\) 0 0
\(70\) −133.509 −0.227963
\(71\) 624.120 1.04323 0.521616 0.853180i \(-0.325329\pi\)
0.521616 + 0.853180i \(0.325329\pi\)
\(72\) 0 0
\(73\) 750.992 1.20407 0.602034 0.798470i \(-0.294357\pi\)
0.602034 + 0.798470i \(0.294357\pi\)
\(74\) 1077.92 1.69331
\(75\) 0 0
\(76\) 1602.68 2.41895
\(77\) −47.2536 −0.0699356
\(78\) 0 0
\(79\) 1171.54 1.66846 0.834231 0.551415i \(-0.185912\pi\)
0.834231 + 0.551415i \(0.185912\pi\)
\(80\) 599.919 0.838412
\(81\) 0 0
\(82\) −312.488 −0.420836
\(83\) −1237.79 −1.63693 −0.818466 0.574554i \(-0.805175\pi\)
−0.818466 + 0.574554i \(0.805175\pi\)
\(84\) 0 0
\(85\) 714.376 0.911588
\(86\) 55.7888 0.0699518
\(87\) 0 0
\(88\) 826.643 1.00137
\(89\) 558.663 0.665372 0.332686 0.943038i \(-0.392045\pi\)
0.332686 + 0.943038i \(0.392045\pi\)
\(90\) 0 0
\(91\) −22.4791 −0.0258950
\(92\) 1542.63 1.74815
\(93\) 0 0
\(94\) −823.637 −0.903741
\(95\) 1540.31 1.66350
\(96\) 0 0
\(97\) −698.747 −0.731412 −0.365706 0.930730i \(-0.619172\pi\)
−0.365706 + 0.930730i \(0.619172\pi\)
\(98\) 1628.04 1.67813
\(99\) 0 0
\(100\) 1254.78 1.25478
\(101\) −1492.55 −1.47044 −0.735219 0.677829i \(-0.762921\pi\)
−0.735219 + 0.677829i \(0.762921\pi\)
\(102\) 0 0
\(103\) 42.1689 0.0403401 0.0201700 0.999797i \(-0.493579\pi\)
0.0201700 + 0.999797i \(0.493579\pi\)
\(104\) 393.243 0.370776
\(105\) 0 0
\(106\) −1668.71 −1.52906
\(107\) −680.282 −0.614629 −0.307315 0.951608i \(-0.599430\pi\)
−0.307315 + 0.951608i \(0.599430\pi\)
\(108\) 0 0
\(109\) 1890.61 1.66135 0.830675 0.556757i \(-0.187955\pi\)
0.830675 + 0.556757i \(0.187955\pi\)
\(110\) 1699.23 1.47287
\(111\) 0 0
\(112\) 80.0518 0.0675374
\(113\) 1819.51 1.51473 0.757367 0.652989i \(-0.226485\pi\)
0.757367 + 0.652989i \(0.226485\pi\)
\(114\) 0 0
\(115\) 1482.59 1.20219
\(116\) −963.312 −0.771045
\(117\) 0 0
\(118\) 3616.97 2.82178
\(119\) 95.3248 0.0734320
\(120\) 0 0
\(121\) −729.582 −0.548146
\(122\) −4151.16 −3.08056
\(123\) 0 0
\(124\) −4211.08 −3.04973
\(125\) −599.057 −0.428651
\(126\) 0 0
\(127\) 1034.80 0.723019 0.361509 0.932369i \(-0.382262\pi\)
0.361509 + 0.932369i \(0.382262\pi\)
\(128\) 2651.20 1.83075
\(129\) 0 0
\(130\) 808.346 0.545358
\(131\) −1271.02 −0.847704 −0.423852 0.905732i \(-0.639322\pi\)
−0.423852 + 0.905732i \(0.639322\pi\)
\(132\) 0 0
\(133\) 205.535 0.134001
\(134\) −3190.91 −2.05711
\(135\) 0 0
\(136\) −1667.59 −1.05143
\(137\) 1067.27 0.665571 0.332786 0.943003i \(-0.392011\pi\)
0.332786 + 0.943003i \(0.392011\pi\)
\(138\) 0 0
\(139\) 1148.09 0.700574 0.350287 0.936642i \(-0.386084\pi\)
0.350287 + 0.936642i \(0.386084\pi\)
\(140\) 418.044 0.252366
\(141\) 0 0
\(142\) −2994.79 −1.76984
\(143\) 286.101 0.167308
\(144\) 0 0
\(145\) −925.821 −0.530243
\(146\) −3603.57 −2.04270
\(147\) 0 0
\(148\) −3375.16 −1.87457
\(149\) 2430.70 1.33645 0.668223 0.743961i \(-0.267055\pi\)
0.668223 + 0.743961i \(0.267055\pi\)
\(150\) 0 0
\(151\) 1720.37 0.927165 0.463582 0.886054i \(-0.346564\pi\)
0.463582 + 0.886054i \(0.346564\pi\)
\(152\) −3595.59 −1.91869
\(153\) 0 0
\(154\) 226.742 0.118645
\(155\) −4047.19 −2.09728
\(156\) 0 0
\(157\) −346.753 −0.176267 −0.0881335 0.996109i \(-0.528090\pi\)
−0.0881335 + 0.996109i \(0.528090\pi\)
\(158\) −5621.53 −2.83054
\(159\) 0 0
\(160\) 1015.27 0.501649
\(161\) 197.833 0.0968413
\(162\) 0 0
\(163\) −1373.55 −0.660027 −0.330014 0.943976i \(-0.607053\pi\)
−0.330014 + 0.943976i \(0.607053\pi\)
\(164\) 978.461 0.465884
\(165\) 0 0
\(166\) 5939.44 2.77705
\(167\) 1252.43 0.580336 0.290168 0.956976i \(-0.406289\pi\)
0.290168 + 0.956976i \(0.406289\pi\)
\(168\) 0 0
\(169\) −2060.90 −0.938051
\(170\) −3427.87 −1.54650
\(171\) 0 0
\(172\) −174.686 −0.0774398
\(173\) 3957.84 1.73936 0.869679 0.493619i \(-0.164326\pi\)
0.869679 + 0.493619i \(0.164326\pi\)
\(174\) 0 0
\(175\) 160.919 0.0695103
\(176\) −1018.86 −0.436359
\(177\) 0 0
\(178\) −2680.69 −1.12880
\(179\) −1886.60 −0.787774 −0.393887 0.919159i \(-0.628870\pi\)
−0.393887 + 0.919159i \(0.628870\pi\)
\(180\) 0 0
\(181\) −1706.92 −0.700964 −0.350482 0.936569i \(-0.613982\pi\)
−0.350482 + 0.936569i \(0.613982\pi\)
\(182\) 107.864 0.0439308
\(183\) 0 0
\(184\) −3460.85 −1.38661
\(185\) −3243.80 −1.28913
\(186\) 0 0
\(187\) −1213.24 −0.474444
\(188\) 2578.97 1.00048
\(189\) 0 0
\(190\) −7391.03 −2.82212
\(191\) −1407.60 −0.533247 −0.266623 0.963801i \(-0.585908\pi\)
−0.266623 + 0.963801i \(0.585908\pi\)
\(192\) 0 0
\(193\) −4498.42 −1.67774 −0.838868 0.544334i \(-0.816782\pi\)
−0.838868 + 0.544334i \(0.816782\pi\)
\(194\) 3352.87 1.24084
\(195\) 0 0
\(196\) −5097.71 −1.85777
\(197\) 2768.03 1.00109 0.500543 0.865712i \(-0.333134\pi\)
0.500543 + 0.865712i \(0.333134\pi\)
\(198\) 0 0
\(199\) 5558.62 1.98010 0.990051 0.140706i \(-0.0449373\pi\)
0.990051 + 0.140706i \(0.0449373\pi\)
\(200\) −2815.07 −0.995278
\(201\) 0 0
\(202\) 7161.87 2.49459
\(203\) −123.539 −0.0427132
\(204\) 0 0
\(205\) 940.380 0.320385
\(206\) −202.344 −0.0684368
\(207\) 0 0
\(208\) −484.681 −0.161570
\(209\) −2615.94 −0.865782
\(210\) 0 0
\(211\) −960.310 −0.313320 −0.156660 0.987653i \(-0.550073\pi\)
−0.156660 + 0.987653i \(0.550073\pi\)
\(212\) 5225.07 1.69273
\(213\) 0 0
\(214\) 3264.27 1.04272
\(215\) −167.887 −0.0532548
\(216\) 0 0
\(217\) −540.048 −0.168944
\(218\) −9071.91 −2.81847
\(219\) 0 0
\(220\) −5320.64 −1.63053
\(221\) −577.153 −0.175672
\(222\) 0 0
\(223\) 614.790 0.184616 0.0923080 0.995731i \(-0.470576\pi\)
0.0923080 + 0.995731i \(0.470576\pi\)
\(224\) 135.475 0.0404098
\(225\) 0 0
\(226\) −8730.75 −2.56974
\(227\) −893.396 −0.261219 −0.130610 0.991434i \(-0.541693\pi\)
−0.130610 + 0.991434i \(0.541693\pi\)
\(228\) 0 0
\(229\) 4864.05 1.40360 0.701802 0.712372i \(-0.252379\pi\)
0.701802 + 0.712372i \(0.252379\pi\)
\(230\) −7114.07 −2.03951
\(231\) 0 0
\(232\) 2161.17 0.611585
\(233\) 3134.42 0.881298 0.440649 0.897679i \(-0.354748\pi\)
0.440649 + 0.897679i \(0.354748\pi\)
\(234\) 0 0
\(235\) 2478.60 0.688025
\(236\) −11325.5 −3.12383
\(237\) 0 0
\(238\) −457.408 −0.124577
\(239\) 239.000 0.0646846
\(240\) 0 0
\(241\) −567.016 −0.151555 −0.0757774 0.997125i \(-0.524144\pi\)
−0.0757774 + 0.997125i \(0.524144\pi\)
\(242\) 3500.84 0.929927
\(243\) 0 0
\(244\) 12998.1 3.41032
\(245\) −4899.31 −1.27757
\(246\) 0 0
\(247\) −1244.43 −0.320572
\(248\) 9447.47 2.41901
\(249\) 0 0
\(250\) 2874.53 0.727204
\(251\) 2548.35 0.640839 0.320419 0.947276i \(-0.396176\pi\)
0.320419 + 0.947276i \(0.396176\pi\)
\(252\) 0 0
\(253\) −2517.91 −0.625691
\(254\) −4965.38 −1.22660
\(255\) 0 0
\(256\) −7363.67 −1.79777
\(257\) 2265.46 0.549866 0.274933 0.961463i \(-0.411344\pi\)
0.274933 + 0.961463i \(0.411344\pi\)
\(258\) 0 0
\(259\) −432.846 −0.103845
\(260\) −2531.09 −0.603736
\(261\) 0 0
\(262\) 6098.86 1.43813
\(263\) −4385.52 −1.02822 −0.514111 0.857723i \(-0.671878\pi\)
−0.514111 + 0.857723i \(0.671878\pi\)
\(264\) 0 0
\(265\) 5021.71 1.16408
\(266\) −986.243 −0.227333
\(267\) 0 0
\(268\) 9991.37 2.27731
\(269\) −132.642 −0.0300645 −0.0150322 0.999887i \(-0.504785\pi\)
−0.0150322 + 0.999887i \(0.504785\pi\)
\(270\) 0 0
\(271\) 4955.89 1.11088 0.555441 0.831556i \(-0.312549\pi\)
0.555441 + 0.831556i \(0.312549\pi\)
\(272\) 2055.34 0.458174
\(273\) 0 0
\(274\) −5121.21 −1.12914
\(275\) −2048.08 −0.449106
\(276\) 0 0
\(277\) −6105.47 −1.32434 −0.662169 0.749354i \(-0.730364\pi\)
−0.662169 + 0.749354i \(0.730364\pi\)
\(278\) −5509.02 −1.18852
\(279\) 0 0
\(280\) −937.873 −0.200174
\(281\) −5141.47 −1.09151 −0.545755 0.837945i \(-0.683757\pi\)
−0.545755 + 0.837945i \(0.683757\pi\)
\(282\) 0 0
\(283\) −2162.85 −0.454304 −0.227152 0.973859i \(-0.572941\pi\)
−0.227152 + 0.973859i \(0.572941\pi\)
\(284\) 9377.26 1.95929
\(285\) 0 0
\(286\) −1372.83 −0.283837
\(287\) 125.482 0.0258083
\(288\) 0 0
\(289\) −2465.52 −0.501837
\(290\) 4442.47 0.899555
\(291\) 0 0
\(292\) 11283.5 2.26136
\(293\) −1525.62 −0.304189 −0.152095 0.988366i \(-0.548602\pi\)
−0.152095 + 0.988366i \(0.548602\pi\)
\(294\) 0 0
\(295\) −10884.7 −2.14824
\(296\) 7572.11 1.48689
\(297\) 0 0
\(298\) −11663.5 −2.26728
\(299\) −1197.80 −0.231674
\(300\) 0 0
\(301\) −22.4025 −0.00428989
\(302\) −8255.06 −1.57293
\(303\) 0 0
\(304\) 4431.64 0.836092
\(305\) 12492.2 2.34525
\(306\) 0 0
\(307\) −1612.54 −0.299781 −0.149890 0.988703i \(-0.547892\pi\)
−0.149890 + 0.988703i \(0.547892\pi\)
\(308\) −709.974 −0.131346
\(309\) 0 0
\(310\) 19420.1 3.55802
\(311\) 6087.96 1.11002 0.555011 0.831843i \(-0.312714\pi\)
0.555011 + 0.831843i \(0.312714\pi\)
\(312\) 0 0
\(313\) 6272.21 1.13267 0.566336 0.824175i \(-0.308361\pi\)
0.566336 + 0.824175i \(0.308361\pi\)
\(314\) 1663.86 0.299036
\(315\) 0 0
\(316\) 17602.1 3.13353
\(317\) −8710.44 −1.54330 −0.771652 0.636045i \(-0.780569\pi\)
−0.771652 + 0.636045i \(0.780569\pi\)
\(318\) 0 0
\(319\) 1572.34 0.275970
\(320\) −9671.02 −1.68946
\(321\) 0 0
\(322\) −949.286 −0.164291
\(323\) 5277.15 0.909066
\(324\) 0 0
\(325\) −974.297 −0.166290
\(326\) 6590.84 1.11973
\(327\) 0 0
\(328\) −2195.16 −0.369534
\(329\) 330.738 0.0554231
\(330\) 0 0
\(331\) 3094.97 0.513943 0.256972 0.966419i \(-0.417275\pi\)
0.256972 + 0.966419i \(0.417275\pi\)
\(332\) −18597.6 −3.07432
\(333\) 0 0
\(334\) −6009.69 −0.984537
\(335\) 9602.52 1.56609
\(336\) 0 0
\(337\) −907.417 −0.146677 −0.0733385 0.997307i \(-0.523365\pi\)
−0.0733385 + 0.997307i \(0.523365\pi\)
\(338\) 9889.04 1.59140
\(339\) 0 0
\(340\) 10733.3 1.71205
\(341\) 6873.43 1.09155
\(342\) 0 0
\(343\) −1314.66 −0.206953
\(344\) 391.903 0.0614244
\(345\) 0 0
\(346\) −18991.3 −2.95081
\(347\) −1470.59 −0.227509 −0.113754 0.993509i \(-0.536288\pi\)
−0.113754 + 0.993509i \(0.536288\pi\)
\(348\) 0 0
\(349\) −12338.1 −1.89239 −0.946194 0.323601i \(-0.895107\pi\)
−0.946194 + 0.323601i \(0.895107\pi\)
\(350\) −772.154 −0.117924
\(351\) 0 0
\(352\) −1724.25 −0.261088
\(353\) −204.115 −0.0307761 −0.0153881 0.999882i \(-0.504898\pi\)
−0.0153881 + 0.999882i \(0.504898\pi\)
\(354\) 0 0
\(355\) 9012.31 1.34739
\(356\) 8393.78 1.24963
\(357\) 0 0
\(358\) 9052.71 1.33645
\(359\) 4940.39 0.726306 0.363153 0.931729i \(-0.381700\pi\)
0.363153 + 0.931729i \(0.381700\pi\)
\(360\) 0 0
\(361\) 4519.36 0.658895
\(362\) 8190.52 1.18918
\(363\) 0 0
\(364\) −337.743 −0.0486333
\(365\) 10844.3 1.55512
\(366\) 0 0
\(367\) −11947.9 −1.69939 −0.849697 0.527271i \(-0.823215\pi\)
−0.849697 + 0.527271i \(0.823215\pi\)
\(368\) 4265.57 0.604235
\(369\) 0 0
\(370\) 15565.1 2.18701
\(371\) 670.087 0.0937713
\(372\) 0 0
\(373\) 4902.14 0.680491 0.340246 0.940337i \(-0.389490\pi\)
0.340246 + 0.940337i \(0.389490\pi\)
\(374\) 5821.63 0.804892
\(375\) 0 0
\(376\) −5785.86 −0.793572
\(377\) 747.981 0.102183
\(378\) 0 0
\(379\) 1510.16 0.204675 0.102338 0.994750i \(-0.467368\pi\)
0.102338 + 0.994750i \(0.467368\pi\)
\(380\) 23142.8 3.12421
\(381\) 0 0
\(382\) 6754.23 0.904650
\(383\) 7967.82 1.06302 0.531510 0.847052i \(-0.321625\pi\)
0.531510 + 0.847052i \(0.321625\pi\)
\(384\) 0 0
\(385\) −682.342 −0.0903257
\(386\) 21585.3 2.84627
\(387\) 0 0
\(388\) −10498.5 −1.37366
\(389\) 1896.84 0.247233 0.123616 0.992330i \(-0.460551\pi\)
0.123616 + 0.992330i \(0.460551\pi\)
\(390\) 0 0
\(391\) 5079.39 0.656972
\(392\) 11436.6 1.47356
\(393\) 0 0
\(394\) −13282.2 −1.69834
\(395\) 16917.1 2.15491
\(396\) 0 0
\(397\) 6439.34 0.814059 0.407029 0.913415i \(-0.366565\pi\)
0.407029 + 0.913415i \(0.366565\pi\)
\(398\) −26672.6 −3.35924
\(399\) 0 0
\(400\) 3469.64 0.433705
\(401\) 1714.30 0.213487 0.106743 0.994287i \(-0.465958\pi\)
0.106743 + 0.994287i \(0.465958\pi\)
\(402\) 0 0
\(403\) 3269.77 0.404166
\(404\) −22425.2 −2.76162
\(405\) 0 0
\(406\) 592.793 0.0724627
\(407\) 5509.03 0.670940
\(408\) 0 0
\(409\) 6901.15 0.834327 0.417163 0.908831i \(-0.363024\pi\)
0.417163 + 0.908831i \(0.363024\pi\)
\(410\) −4512.33 −0.543532
\(411\) 0 0
\(412\) 633.579 0.0757626
\(413\) −1452.43 −0.173049
\(414\) 0 0
\(415\) −17873.8 −2.11419
\(416\) −820.246 −0.0966727
\(417\) 0 0
\(418\) 12552.4 1.46879
\(419\) 6255.85 0.729399 0.364699 0.931125i \(-0.381172\pi\)
0.364699 + 0.931125i \(0.381172\pi\)
\(420\) 0 0
\(421\) 4921.75 0.569766 0.284883 0.958562i \(-0.408045\pi\)
0.284883 + 0.958562i \(0.408045\pi\)
\(422\) 4607.96 0.531545
\(423\) 0 0
\(424\) −11722.3 −1.34266
\(425\) 4131.61 0.471558
\(426\) 0 0
\(427\) 1666.93 0.188920
\(428\) −10221.1 −1.15433
\(429\) 0 0
\(430\) 805.591 0.0903466
\(431\) −841.948 −0.0940957 −0.0470478 0.998893i \(-0.514981\pi\)
−0.0470478 + 0.998893i \(0.514981\pi\)
\(432\) 0 0
\(433\) 14074.4 1.56206 0.781031 0.624492i \(-0.214694\pi\)
0.781031 + 0.624492i \(0.214694\pi\)
\(434\) 2591.37 0.286613
\(435\) 0 0
\(436\) 28405.9 3.12018
\(437\) 10952.0 1.19887
\(438\) 0 0
\(439\) −3415.28 −0.371304 −0.185652 0.982616i \(-0.559440\pi\)
−0.185652 + 0.982616i \(0.559440\pi\)
\(440\) 11936.7 1.29332
\(441\) 0 0
\(442\) 2769.42 0.298027
\(443\) −9565.85 −1.02593 −0.512965 0.858409i \(-0.671453\pi\)
−0.512965 + 0.858409i \(0.671453\pi\)
\(444\) 0 0
\(445\) 8067.10 0.859365
\(446\) −2950.01 −0.313200
\(447\) 0 0
\(448\) −1290.48 −0.136092
\(449\) 11487.5 1.20741 0.603707 0.797207i \(-0.293690\pi\)
0.603707 + 0.797207i \(0.293690\pi\)
\(450\) 0 0
\(451\) −1597.07 −0.166747
\(452\) 27337.7 2.84482
\(453\) 0 0
\(454\) 4286.88 0.443157
\(455\) −324.598 −0.0334448
\(456\) 0 0
\(457\) 14153.0 1.44869 0.724344 0.689439i \(-0.242143\pi\)
0.724344 + 0.689439i \(0.242143\pi\)
\(458\) −23339.7 −2.38121
\(459\) 0 0
\(460\) 22275.5 2.25783
\(461\) −4852.98 −0.490295 −0.245147 0.969486i \(-0.578836\pi\)
−0.245147 + 0.969486i \(0.578836\pi\)
\(462\) 0 0
\(463\) 4232.92 0.424882 0.212441 0.977174i \(-0.431859\pi\)
0.212441 + 0.977174i \(0.431859\pi\)
\(464\) −2663.69 −0.266506
\(465\) 0 0
\(466\) −15040.2 −1.49512
\(467\) 8525.46 0.844778 0.422389 0.906415i \(-0.361192\pi\)
0.422389 + 0.906415i \(0.361192\pi\)
\(468\) 0 0
\(469\) 1281.34 0.126155
\(470\) −11893.3 −1.16723
\(471\) 0 0
\(472\) 25408.4 2.47779
\(473\) 285.126 0.0277169
\(474\) 0 0
\(475\) 8908.40 0.860516
\(476\) 1432.23 0.137912
\(477\) 0 0
\(478\) −1146.82 −0.109737
\(479\) 1731.93 0.165207 0.0826033 0.996583i \(-0.473677\pi\)
0.0826033 + 0.996583i \(0.473677\pi\)
\(480\) 0 0
\(481\) 2620.71 0.248428
\(482\) 2720.78 0.257112
\(483\) 0 0
\(484\) −10961.8 −1.02947
\(485\) −10089.9 −0.944658
\(486\) 0 0
\(487\) 12107.2 1.12655 0.563276 0.826269i \(-0.309541\pi\)
0.563276 + 0.826269i \(0.309541\pi\)
\(488\) −29161.0 −2.70503
\(489\) 0 0
\(490\) 23508.9 2.16740
\(491\) −17318.6 −1.59180 −0.795902 0.605425i \(-0.793003\pi\)
−0.795902 + 0.605425i \(0.793003\pi\)
\(492\) 0 0
\(493\) −3171.89 −0.289766
\(494\) 5971.30 0.543849
\(495\) 0 0
\(496\) −11644.2 −1.05411
\(497\) 1202.58 0.108538
\(498\) 0 0
\(499\) 10591.8 0.950205 0.475103 0.879930i \(-0.342411\pi\)
0.475103 + 0.879930i \(0.342411\pi\)
\(500\) −9000.70 −0.805047
\(501\) 0 0
\(502\) −12228.0 −1.08718
\(503\) −12622.8 −1.11894 −0.559468 0.828852i \(-0.688994\pi\)
−0.559468 + 0.828852i \(0.688994\pi\)
\(504\) 0 0
\(505\) −21552.4 −1.89915
\(506\) 12082.0 1.06148
\(507\) 0 0
\(508\) 15547.6 1.35790
\(509\) 21821.7 1.90025 0.950127 0.311862i \(-0.100953\pi\)
0.950127 + 0.311862i \(0.100953\pi\)
\(510\) 0 0
\(511\) 1447.04 0.125271
\(512\) 14124.3 1.21916
\(513\) 0 0
\(514\) −10870.6 −0.932846
\(515\) 608.920 0.0521014
\(516\) 0 0
\(517\) −4209.46 −0.358088
\(518\) 2076.97 0.176172
\(519\) 0 0
\(520\) 5678.44 0.478877
\(521\) 2616.35 0.220008 0.110004 0.993931i \(-0.464914\pi\)
0.110004 + 0.993931i \(0.464914\pi\)
\(522\) 0 0
\(523\) −15427.1 −1.28983 −0.644915 0.764254i \(-0.723107\pi\)
−0.644915 + 0.764254i \(0.723107\pi\)
\(524\) −19096.7 −1.59207
\(525\) 0 0
\(526\) 21043.5 1.74438
\(527\) −13865.8 −1.14612
\(528\) 0 0
\(529\) −1625.43 −0.133593
\(530\) −24096.3 −1.97486
\(531\) 0 0
\(532\) 3088.12 0.251667
\(533\) −759.744 −0.0617414
\(534\) 0 0
\(535\) −9823.28 −0.793827
\(536\) −22415.4 −1.80634
\(537\) 0 0
\(538\) 636.472 0.0510042
\(539\) 8320.62 0.664925
\(540\) 0 0
\(541\) 12474.7 0.991369 0.495685 0.868503i \(-0.334917\pi\)
0.495685 + 0.868503i \(0.334917\pi\)
\(542\) −23780.4 −1.88461
\(543\) 0 0
\(544\) 3478.34 0.274140
\(545\) 27300.4 2.14572
\(546\) 0 0
\(547\) −6216.78 −0.485942 −0.242971 0.970034i \(-0.578122\pi\)
−0.242971 + 0.970034i \(0.578122\pi\)
\(548\) 16035.5 1.25001
\(549\) 0 0
\(550\) 9827.55 0.761906
\(551\) −6839.10 −0.528776
\(552\) 0 0
\(553\) 2257.37 0.173586
\(554\) 29296.5 2.24673
\(555\) 0 0
\(556\) 17249.8 1.31575
\(557\) 17629.7 1.34110 0.670551 0.741864i \(-0.266058\pi\)
0.670551 + 0.741864i \(0.266058\pi\)
\(558\) 0 0
\(559\) 135.638 0.0102627
\(560\) 1155.95 0.0872282
\(561\) 0 0
\(562\) 24670.9 1.85174
\(563\) 8727.05 0.653288 0.326644 0.945147i \(-0.394082\pi\)
0.326644 + 0.945147i \(0.394082\pi\)
\(564\) 0 0
\(565\) 26273.7 1.95636
\(566\) 10378.2 0.770725
\(567\) 0 0
\(568\) −21037.7 −1.55409
\(569\) 917.170 0.0675742 0.0337871 0.999429i \(-0.489243\pi\)
0.0337871 + 0.999429i \(0.489243\pi\)
\(570\) 0 0
\(571\) −10983.3 −0.804966 −0.402483 0.915428i \(-0.631853\pi\)
−0.402483 + 0.915428i \(0.631853\pi\)
\(572\) 4298.61 0.314220
\(573\) 0 0
\(574\) −602.116 −0.0437837
\(575\) 8574.57 0.621886
\(576\) 0 0
\(577\) −11383.1 −0.821292 −0.410646 0.911795i \(-0.634697\pi\)
−0.410646 + 0.911795i \(0.634697\pi\)
\(578\) 11830.6 0.851364
\(579\) 0 0
\(580\) −13910.2 −0.995847
\(581\) −2385.03 −0.170306
\(582\) 0 0
\(583\) −8528.49 −0.605856
\(584\) −25314.2 −1.79368
\(585\) 0 0
\(586\) 7320.53 0.516055
\(587\) −18677.0 −1.31326 −0.656628 0.754215i \(-0.728018\pi\)
−0.656628 + 0.754215i \(0.728018\pi\)
\(588\) 0 0
\(589\) −29896.8 −2.09147
\(590\) 52229.2 3.64448
\(591\) 0 0
\(592\) −9332.79 −0.647932
\(593\) −14315.0 −0.991307 −0.495653 0.868520i \(-0.665071\pi\)
−0.495653 + 0.868520i \(0.665071\pi\)
\(594\) 0 0
\(595\) 1376.49 0.0948414
\(596\) 36520.7 2.50998
\(597\) 0 0
\(598\) 5747.54 0.393034
\(599\) 21968.9 1.49854 0.749269 0.662266i \(-0.230405\pi\)
0.749269 + 0.662266i \(0.230405\pi\)
\(600\) 0 0
\(601\) 17018.9 1.15510 0.577549 0.816356i \(-0.304009\pi\)
0.577549 + 0.816356i \(0.304009\pi\)
\(602\) 107.496 0.00727777
\(603\) 0 0
\(604\) 25848.2 1.74131
\(605\) −10535.2 −0.707960
\(606\) 0 0
\(607\) 19695.6 1.31700 0.658500 0.752581i \(-0.271191\pi\)
0.658500 + 0.752581i \(0.271191\pi\)
\(608\) 7499.84 0.500261
\(609\) 0 0
\(610\) −59942.8 −3.97871
\(611\) −2002.49 −0.132589
\(612\) 0 0
\(613\) −21790.3 −1.43573 −0.717866 0.696181i \(-0.754881\pi\)
−0.717866 + 0.696181i \(0.754881\pi\)
\(614\) 7737.65 0.508577
\(615\) 0 0
\(616\) 1592.81 0.104182
\(617\) 1280.93 0.0835792 0.0417896 0.999126i \(-0.486694\pi\)
0.0417896 + 0.999126i \(0.486694\pi\)
\(618\) 0 0
\(619\) 22357.2 1.45171 0.725857 0.687846i \(-0.241444\pi\)
0.725857 + 0.687846i \(0.241444\pi\)
\(620\) −60808.0 −3.93889
\(621\) 0 0
\(622\) −29212.5 −1.88315
\(623\) 1076.46 0.0692252
\(624\) 0 0
\(625\) −19089.7 −1.22174
\(626\) −30096.6 −1.92157
\(627\) 0 0
\(628\) −5209.88 −0.331046
\(629\) −11113.4 −0.704483
\(630\) 0 0
\(631\) 362.824 0.0228903 0.0114452 0.999935i \(-0.496357\pi\)
0.0114452 + 0.999935i \(0.496357\pi\)
\(632\) −39490.0 −2.48548
\(633\) 0 0
\(634\) 41796.3 2.61821
\(635\) 14942.5 0.933818
\(636\) 0 0
\(637\) 3958.21 0.246201
\(638\) −7544.75 −0.468181
\(639\) 0 0
\(640\) 38283.4 2.36451
\(641\) 10762.7 0.663187 0.331594 0.943422i \(-0.392414\pi\)
0.331594 + 0.943422i \(0.392414\pi\)
\(642\) 0 0
\(643\) −19982.4 −1.22555 −0.612774 0.790258i \(-0.709946\pi\)
−0.612774 + 0.790258i \(0.709946\pi\)
\(644\) 2972.40 0.181877
\(645\) 0 0
\(646\) −25321.9 −1.54223
\(647\) −20270.5 −1.23171 −0.615854 0.787860i \(-0.711189\pi\)
−0.615854 + 0.787860i \(0.711189\pi\)
\(648\) 0 0
\(649\) 18485.7 1.11807
\(650\) 4675.08 0.282110
\(651\) 0 0
\(652\) −20637.2 −1.23959
\(653\) 14667.0 0.878967 0.439483 0.898251i \(-0.355162\pi\)
0.439483 + 0.898251i \(0.355162\pi\)
\(654\) 0 0
\(655\) −18353.5 −1.09486
\(656\) 2705.58 0.161029
\(657\) 0 0
\(658\) −1587.02 −0.0940250
\(659\) 30650.2 1.81178 0.905890 0.423513i \(-0.139203\pi\)
0.905890 + 0.423513i \(0.139203\pi\)
\(660\) 0 0
\(661\) 22478.6 1.32272 0.661360 0.750069i \(-0.269980\pi\)
0.661360 + 0.750069i \(0.269980\pi\)
\(662\) −14851.0 −0.871902
\(663\) 0 0
\(664\) 41723.2 2.43852
\(665\) 2967.93 0.173070
\(666\) 0 0
\(667\) −6582.82 −0.382140
\(668\) 18817.5 1.08993
\(669\) 0 0
\(670\) −46076.8 −2.65687
\(671\) −21215.8 −1.22061
\(672\) 0 0
\(673\) −29582.8 −1.69440 −0.847202 0.531271i \(-0.821714\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(674\) 4354.16 0.248837
\(675\) 0 0
\(676\) −30964.5 −1.76175
\(677\) 26863.8 1.52505 0.762526 0.646957i \(-0.223959\pi\)
0.762526 + 0.646957i \(0.223959\pi\)
\(678\) 0 0
\(679\) −1346.38 −0.0760960
\(680\) −24080.0 −1.35798
\(681\) 0 0
\(682\) −32981.6 −1.85180
\(683\) −10052.2 −0.563159 −0.281579 0.959538i \(-0.590858\pi\)
−0.281579 + 0.959538i \(0.590858\pi\)
\(684\) 0 0
\(685\) 15411.4 0.859621
\(686\) 6308.29 0.351096
\(687\) 0 0
\(688\) −483.030 −0.0267665
\(689\) −4057.10 −0.224330
\(690\) 0 0
\(691\) 28224.7 1.55386 0.776930 0.629587i \(-0.216776\pi\)
0.776930 + 0.629587i \(0.216776\pi\)
\(692\) 59465.6 3.26668
\(693\) 0 0
\(694\) 7056.51 0.385968
\(695\) 16578.5 0.904830
\(696\) 0 0
\(697\) 3221.77 0.175084
\(698\) 59203.3 3.21043
\(699\) 0 0
\(700\) 2417.77 0.130547
\(701\) 6567.19 0.353837 0.176918 0.984226i \(-0.443387\pi\)
0.176918 + 0.984226i \(0.443387\pi\)
\(702\) 0 0
\(703\) −23962.2 −1.28556
\(704\) 16424.5 0.879293
\(705\) 0 0
\(706\) 979.430 0.0522115
\(707\) −2875.91 −0.152984
\(708\) 0 0
\(709\) 6459.04 0.342136 0.171068 0.985259i \(-0.445278\pi\)
0.171068 + 0.985259i \(0.445278\pi\)
\(710\) −43244.8 −2.28584
\(711\) 0 0
\(712\) −18831.3 −0.991196
\(713\) −28776.5 −1.51149
\(714\) 0 0
\(715\) 4131.31 0.216087
\(716\) −28345.8 −1.47951
\(717\) 0 0
\(718\) −23706.0 −1.23217
\(719\) 22411.0 1.16243 0.581217 0.813749i \(-0.302577\pi\)
0.581217 + 0.813749i \(0.302577\pi\)
\(720\) 0 0
\(721\) 81.2530 0.00419698
\(722\) −21685.8 −1.11781
\(723\) 0 0
\(724\) −25646.1 −1.31648
\(725\) −5354.50 −0.274291
\(726\) 0 0
\(727\) 10632.3 0.542407 0.271203 0.962522i \(-0.412578\pi\)
0.271203 + 0.962522i \(0.412578\pi\)
\(728\) 757.719 0.0385754
\(729\) 0 0
\(730\) −52035.6 −2.63825
\(731\) −575.186 −0.0291026
\(732\) 0 0
\(733\) 9413.78 0.474360 0.237180 0.971466i \(-0.423777\pi\)
0.237180 + 0.971466i \(0.423777\pi\)
\(734\) 57331.2 2.88301
\(735\) 0 0
\(736\) 7218.80 0.361533
\(737\) −16308.2 −0.815087
\(738\) 0 0
\(739\) −13114.7 −0.652819 −0.326409 0.945228i \(-0.605839\pi\)
−0.326409 + 0.945228i \(0.605839\pi\)
\(740\) −48737.4 −2.42111
\(741\) 0 0
\(742\) −3215.35 −0.159083
\(743\) −14499.7 −0.715938 −0.357969 0.933733i \(-0.616531\pi\)
−0.357969 + 0.933733i \(0.616531\pi\)
\(744\) 0 0
\(745\) 35099.3 1.72609
\(746\) −23522.5 −1.15445
\(747\) 0 0
\(748\) −18228.7 −0.891051
\(749\) −1310.80 −0.0639459
\(750\) 0 0
\(751\) −13932.1 −0.676950 −0.338475 0.940975i \(-0.609911\pi\)
−0.338475 + 0.940975i \(0.609911\pi\)
\(752\) 7131.20 0.345809
\(753\) 0 0
\(754\) −3589.12 −0.173353
\(755\) 24842.2 1.19748
\(756\) 0 0
\(757\) 15664.4 0.752088 0.376044 0.926602i \(-0.377284\pi\)
0.376044 + 0.926602i \(0.377284\pi\)
\(758\) −7246.39 −0.347231
\(759\) 0 0
\(760\) −51920.3 −2.47809
\(761\) 18040.5 0.859353 0.429677 0.902983i \(-0.358628\pi\)
0.429677 + 0.902983i \(0.358628\pi\)
\(762\) 0 0
\(763\) 3642.90 0.172847
\(764\) −21148.8 −1.00149
\(765\) 0 0
\(766\) −38232.9 −1.80341
\(767\) 8793.86 0.413987
\(768\) 0 0
\(769\) −1523.11 −0.0714238 −0.0357119 0.999362i \(-0.511370\pi\)
−0.0357119 + 0.999362i \(0.511370\pi\)
\(770\) 3274.16 0.153237
\(771\) 0 0
\(772\) −67587.7 −3.15095
\(773\) 40569.1 1.88767 0.943836 0.330414i \(-0.107189\pi\)
0.943836 + 0.330414i \(0.107189\pi\)
\(774\) 0 0
\(775\) −23407.0 −1.08491
\(776\) 23553.2 1.08957
\(777\) 0 0
\(778\) −9101.81 −0.419429
\(779\) 6946.65 0.319499
\(780\) 0 0
\(781\) −15305.8 −0.701261
\(782\) −24373.0 −1.11455
\(783\) 0 0
\(784\) −14095.9 −0.642123
\(785\) −5007.12 −0.227658
\(786\) 0 0
\(787\) 19586.0 0.887124 0.443562 0.896244i \(-0.353715\pi\)
0.443562 + 0.896244i \(0.353715\pi\)
\(788\) 41589.0 1.88014
\(789\) 0 0
\(790\) −81175.0 −3.65579
\(791\) 3505.91 0.157593
\(792\) 0 0
\(793\) −10092.6 −0.451954
\(794\) −30898.6 −1.38105
\(795\) 0 0
\(796\) 83517.1 3.71882
\(797\) 9014.22 0.400628 0.200314 0.979732i \(-0.435804\pi\)
0.200314 + 0.979732i \(0.435804\pi\)
\(798\) 0 0
\(799\) 8491.75 0.375991
\(800\) 5871.81 0.259500
\(801\) 0 0
\(802\) −8225.94 −0.362180
\(803\) −18417.2 −0.809375
\(804\) 0 0
\(805\) 2856.72 0.125076
\(806\) −15689.7 −0.685666
\(807\) 0 0
\(808\) 50310.5 2.19049
\(809\) 37058.4 1.61051 0.805255 0.592929i \(-0.202028\pi\)
0.805255 + 0.592929i \(0.202028\pi\)
\(810\) 0 0
\(811\) −12988.4 −0.562375 −0.281187 0.959653i \(-0.590728\pi\)
−0.281187 + 0.959653i \(0.590728\pi\)
\(812\) −1856.15 −0.0802194
\(813\) 0 0
\(814\) −26434.6 −1.13825
\(815\) −19834.0 −0.852461
\(816\) 0 0
\(817\) −1240.19 −0.0531075
\(818\) −33114.6 −1.41543
\(819\) 0 0
\(820\) 14129.0 0.601715
\(821\) −37857.7 −1.60931 −0.804654 0.593744i \(-0.797649\pi\)
−0.804654 + 0.593744i \(0.797649\pi\)
\(822\) 0 0
\(823\) −24099.3 −1.02072 −0.510358 0.859962i \(-0.670487\pi\)
−0.510358 + 0.859962i \(0.670487\pi\)
\(824\) −1421.42 −0.0600941
\(825\) 0 0
\(826\) 6969.34 0.293577
\(827\) −17199.6 −0.723201 −0.361601 0.932333i \(-0.617770\pi\)
−0.361601 + 0.932333i \(0.617770\pi\)
\(828\) 0 0
\(829\) 5547.26 0.232406 0.116203 0.993226i \(-0.462928\pi\)
0.116203 + 0.993226i \(0.462928\pi\)
\(830\) 85765.6 3.58671
\(831\) 0 0
\(832\) 7813.33 0.325575
\(833\) −16785.2 −0.698167
\(834\) 0 0
\(835\) 18085.1 0.749536
\(836\) −39303.9 −1.62602
\(837\) 0 0
\(838\) −30018.1 −1.23742
\(839\) −11780.4 −0.484748 −0.242374 0.970183i \(-0.577926\pi\)
−0.242374 + 0.970183i \(0.577926\pi\)
\(840\) 0 0
\(841\) −20278.3 −0.831452
\(842\) −23616.6 −0.966605
\(843\) 0 0
\(844\) −14428.4 −0.588445
\(845\) −29759.4 −1.21154
\(846\) 0 0
\(847\) −1405.79 −0.0570290
\(848\) 14448.0 0.585080
\(849\) 0 0
\(850\) −19825.2 −0.799997
\(851\) −23064.3 −0.929063
\(852\) 0 0
\(853\) −27508.0 −1.10417 −0.552084 0.833789i \(-0.686167\pi\)
−0.552084 + 0.833789i \(0.686167\pi\)
\(854\) −7998.64 −0.320501
\(855\) 0 0
\(856\) 22930.8 0.915605
\(857\) 8744.22 0.348538 0.174269 0.984698i \(-0.444244\pi\)
0.174269 + 0.984698i \(0.444244\pi\)
\(858\) 0 0
\(859\) −4332.09 −0.172071 −0.0860355 0.996292i \(-0.527420\pi\)
−0.0860355 + 0.996292i \(0.527420\pi\)
\(860\) −2522.46 −0.100018
\(861\) 0 0
\(862\) 4040.02 0.159633
\(863\) 47669.0 1.88027 0.940135 0.340802i \(-0.110698\pi\)
0.940135 + 0.340802i \(0.110698\pi\)
\(864\) 0 0
\(865\) 57151.2 2.24647
\(866\) −67534.9 −2.65003
\(867\) 0 0
\(868\) −8114.09 −0.317293
\(869\) −28730.6 −1.12154
\(870\) 0 0
\(871\) −7757.99 −0.301802
\(872\) −63728.1 −2.47489
\(873\) 0 0
\(874\) −52552.1 −2.03387
\(875\) −1154.29 −0.0445967
\(876\) 0 0
\(877\) 8876.79 0.341788 0.170894 0.985289i \(-0.445334\pi\)
0.170894 + 0.985289i \(0.445334\pi\)
\(878\) 16387.9 0.629916
\(879\) 0 0
\(880\) −14712.3 −0.563581
\(881\) −17796.2 −0.680554 −0.340277 0.940325i \(-0.610521\pi\)
−0.340277 + 0.940325i \(0.610521\pi\)
\(882\) 0 0
\(883\) 36680.9 1.39797 0.698987 0.715134i \(-0.253634\pi\)
0.698987 + 0.715134i \(0.253634\pi\)
\(884\) −8671.59 −0.329929
\(885\) 0 0
\(886\) 45900.9 1.74049
\(887\) 21245.9 0.804248 0.402124 0.915585i \(-0.368272\pi\)
0.402124 + 0.915585i \(0.368272\pi\)
\(888\) 0 0
\(889\) 1993.89 0.0752227
\(890\) −38709.3 −1.45791
\(891\) 0 0
\(892\) 9237.07 0.346727
\(893\) 18309.6 0.686121
\(894\) 0 0
\(895\) −27242.6 −1.01745
\(896\) 5108.45 0.190470
\(897\) 0 0
\(898\) −55121.8 −2.04837
\(899\) 17969.9 0.666661
\(900\) 0 0
\(901\) 17204.6 0.636145
\(902\) 7663.40 0.282886
\(903\) 0 0
\(904\) −61331.5 −2.25648
\(905\) −24648.0 −0.905333
\(906\) 0 0
\(907\) 22616.2 0.827960 0.413980 0.910286i \(-0.364138\pi\)
0.413980 + 0.910286i \(0.364138\pi\)
\(908\) −13423.1 −0.490595
\(909\) 0 0
\(910\) 1557.56 0.0567390
\(911\) 22225.8 0.808315 0.404157 0.914689i \(-0.367565\pi\)
0.404157 + 0.914689i \(0.367565\pi\)
\(912\) 0 0
\(913\) 30355.4 1.10035
\(914\) −67912.0 −2.45769
\(915\) 0 0
\(916\) 73081.2 2.63611
\(917\) −2449.05 −0.0881949
\(918\) 0 0
\(919\) 15412.2 0.553212 0.276606 0.960983i \(-0.410790\pi\)
0.276606 + 0.960983i \(0.410790\pi\)
\(920\) −49974.7 −1.79089
\(921\) 0 0
\(922\) 23286.6 0.831783
\(923\) −7281.15 −0.259655
\(924\) 0 0
\(925\) −18760.6 −0.666859
\(926\) −20311.3 −0.720810
\(927\) 0 0
\(928\) −4507.87 −0.159459
\(929\) 53009.2 1.87209 0.936047 0.351875i \(-0.114456\pi\)
0.936047 + 0.351875i \(0.114456\pi\)
\(930\) 0 0
\(931\) −36191.6 −1.27404
\(932\) 47093.9 1.65516
\(933\) 0 0
\(934\) −40908.7 −1.43316
\(935\) −17519.2 −0.612770
\(936\) 0 0
\(937\) 43533.3 1.51779 0.758896 0.651212i \(-0.225739\pi\)
0.758896 + 0.651212i \(0.225739\pi\)
\(938\) −6148.39 −0.214021
\(939\) 0 0
\(940\) 37240.3 1.29218
\(941\) −15767.1 −0.546221 −0.273110 0.961983i \(-0.588052\pi\)
−0.273110 + 0.961983i \(0.588052\pi\)
\(942\) 0 0
\(943\) 6686.34 0.230898
\(944\) −31316.4 −1.07973
\(945\) 0 0
\(946\) −1368.15 −0.0470217
\(947\) −10349.7 −0.355144 −0.177572 0.984108i \(-0.556824\pi\)
−0.177572 + 0.984108i \(0.556824\pi\)
\(948\) 0 0
\(949\) −8761.27 −0.299687
\(950\) −42746.2 −1.45986
\(951\) 0 0
\(952\) −3213.18 −0.109391
\(953\) 31061.4 1.05580 0.527899 0.849307i \(-0.322980\pi\)
0.527899 + 0.849307i \(0.322980\pi\)
\(954\) 0 0
\(955\) −20325.7 −0.688717
\(956\) 3590.92 0.121484
\(957\) 0 0
\(958\) −8310.53 −0.280272
\(959\) 2056.47 0.0692459
\(960\) 0 0
\(961\) 48763.5 1.63685
\(962\) −12575.2 −0.421457
\(963\) 0 0
\(964\) −8519.29 −0.284635
\(965\) −64957.2 −2.16689
\(966\) 0 0
\(967\) −16381.8 −0.544781 −0.272390 0.962187i \(-0.587814\pi\)
−0.272390 + 0.962187i \(0.587814\pi\)
\(968\) 24592.6 0.816566
\(969\) 0 0
\(970\) 48415.6 1.60261
\(971\) −13611.5 −0.449861 −0.224930 0.974375i \(-0.572215\pi\)
−0.224930 + 0.974375i \(0.572215\pi\)
\(972\) 0 0
\(973\) 2212.19 0.0728876
\(974\) −58095.5 −1.91119
\(975\) 0 0
\(976\) 35941.5 1.17875
\(977\) 24497.0 0.802178 0.401089 0.916039i \(-0.368632\pi\)
0.401089 + 0.916039i \(0.368632\pi\)
\(978\) 0 0
\(979\) −13700.5 −0.447264
\(980\) −73611.1 −2.39941
\(981\) 0 0
\(982\) 83101.6 2.70049
\(983\) −16984.8 −0.551101 −0.275550 0.961287i \(-0.588860\pi\)
−0.275550 + 0.961287i \(0.588860\pi\)
\(984\) 0 0
\(985\) 39970.4 1.29296
\(986\) 15220.0 0.491587
\(987\) 0 0
\(988\) −18697.3 −0.602066
\(989\) −1193.72 −0.0383802
\(990\) 0 0
\(991\) −47834.1 −1.53330 −0.766650 0.642065i \(-0.778078\pi\)
−0.766650 + 0.642065i \(0.778078\pi\)
\(992\) −19706.0 −0.630711
\(993\) 0 0
\(994\) −5770.49 −0.184134
\(995\) 80266.6 2.55741
\(996\) 0 0
\(997\) 2869.40 0.0911483 0.0455742 0.998961i \(-0.485488\pi\)
0.0455742 + 0.998961i \(0.485488\pi\)
\(998\) −50823.7 −1.61202
\(999\) 0 0
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 2151.4.a.f.1.5 37
3.2 odd 2 239.4.a.b.1.33 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
239.4.a.b.1.33 37 3.2 odd 2
2151.4.a.f.1.5 37 1.1 even 1 trivial