Properties

Label 495.4.a.j.1.1
Level $495$
Weight $4$
Character 495.1
Self dual yes
Analytic conductor $29.206$
Analytic rank $1$
Dimension $3$
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] = [495,4,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(29.2059454528\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.32803\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.32803 q^{2} +3.07579 q^{4} +5.00000 q^{5} +12.1102 q^{7} +16.3879 q^{8} +O(q^{10})\) \(q-3.32803 q^{2} +3.07579 q^{4} +5.00000 q^{5} +12.1102 q^{7} +16.3879 q^{8} -16.6402 q^{10} -11.0000 q^{11} -57.2210 q^{13} -40.3032 q^{14} -79.1458 q^{16} -22.7802 q^{17} -46.2204 q^{19} +15.3790 q^{20} +36.6083 q^{22} +128.447 q^{23} +25.0000 q^{25} +190.433 q^{26} +37.2485 q^{28} +71.6674 q^{29} -88.7110 q^{31} +132.296 q^{32} +75.8131 q^{34} +60.5511 q^{35} -30.5848 q^{37} +153.823 q^{38} +81.9396 q^{40} +223.209 q^{41} -170.565 q^{43} -33.8337 q^{44} -427.477 q^{46} -247.523 q^{47} -196.343 q^{49} -83.2008 q^{50} -176.000 q^{52} +76.4692 q^{53} -55.0000 q^{55} +198.461 q^{56} -238.511 q^{58} +258.248 q^{59} -97.3761 q^{61} +295.233 q^{62} +192.880 q^{64} -286.105 q^{65} -278.018 q^{67} -70.0671 q^{68} -201.516 q^{70} -292.087 q^{71} -482.377 q^{73} +101.787 q^{74} -142.164 q^{76} -133.212 q^{77} -93.3473 q^{79} -395.729 q^{80} -742.847 q^{82} -1011.13 q^{83} -113.901 q^{85} +567.646 q^{86} -180.267 q^{88} +60.3336 q^{89} -692.959 q^{91} +395.077 q^{92} +823.766 q^{94} -231.102 q^{95} +662.927 q^{97} +653.435 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} + 15 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{4} + 15 q^{5} + 2 q^{7} - 3 q^{8} + 5 q^{10} - 33 q^{11} - 66 q^{13} - 88 q^{14} - 103 q^{16} - 100 q^{17} - 70 q^{19} + 5 q^{20} - 11 q^{22} - 10 q^{23} + 75 q^{25} + 88 q^{26} - 64 q^{28} + 34 q^{29} - 104 q^{31} + 37 q^{32} - 278 q^{34} + 10 q^{35} - 270 q^{37} + 154 q^{38} - 15 q^{40} + 82 q^{41} - 428 q^{43} - 11 q^{44} - 706 q^{46} - 246 q^{47} - 685 q^{49} + 25 q^{50} - 528 q^{52} - 298 q^{53} - 165 q^{55} + 288 q^{56} + 114 q^{58} + 204 q^{59} - 492 q^{61} + 584 q^{62} - 319 q^{64} - 330 q^{65} - 812 q^{67} - 798 q^{68} - 440 q^{70} + 442 q^{71} - 1044 q^{73} + 42 q^{74} + 106 q^{76} - 22 q^{77} - 1436 q^{79} - 515 q^{80} - 1246 q^{82} - 734 q^{83} - 500 q^{85} - 82 q^{86} + 33 q^{88} + 556 q^{89} - 176 q^{91} + 630 q^{92} + 38 q^{94} - 350 q^{95} - 1038 q^{97} - 143 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.32803 −1.17664 −0.588318 0.808629i \(-0.700210\pi\)
−0.588318 + 0.808629i \(0.700210\pi\)
\(3\) 0 0
\(4\) 3.07579 0.384474
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 12.1102 0.653890 0.326945 0.945043i \(-0.393981\pi\)
0.326945 + 0.945043i \(0.393981\pi\)
\(8\) 16.3879 0.724250
\(9\) 0 0
\(10\) −16.6402 −0.526208
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −57.2210 −1.22079 −0.610395 0.792098i \(-0.708989\pi\)
−0.610395 + 0.792098i \(0.708989\pi\)
\(14\) −40.3032 −0.769391
\(15\) 0 0
\(16\) −79.1458 −1.23665
\(17\) −22.7802 −0.325000 −0.162500 0.986709i \(-0.551956\pi\)
−0.162500 + 0.986709i \(0.551956\pi\)
\(18\) 0 0
\(19\) −46.2204 −0.558089 −0.279045 0.960278i \(-0.590018\pi\)
−0.279045 + 0.960278i \(0.590018\pi\)
\(20\) 15.3790 0.171942
\(21\) 0 0
\(22\) 36.6083 0.354769
\(23\) 128.447 1.16448 0.582241 0.813016i \(-0.302176\pi\)
0.582241 + 0.813016i \(0.302176\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 190.433 1.43643
\(27\) 0 0
\(28\) 37.2485 0.251404
\(29\) 71.6674 0.458907 0.229454 0.973320i \(-0.426306\pi\)
0.229454 + 0.973320i \(0.426306\pi\)
\(30\) 0 0
\(31\) −88.7110 −0.513967 −0.256983 0.966416i \(-0.582729\pi\)
−0.256983 + 0.966416i \(0.582729\pi\)
\(32\) 132.296 0.730842
\(33\) 0 0
\(34\) 75.8131 0.382407
\(35\) 60.5511 0.292429
\(36\) 0 0
\(37\) −30.5848 −0.135895 −0.0679475 0.997689i \(-0.521645\pi\)
−0.0679475 + 0.997689i \(0.521645\pi\)
\(38\) 153.823 0.656668
\(39\) 0 0
\(40\) 81.9396 0.323895
\(41\) 223.209 0.850230 0.425115 0.905139i \(-0.360234\pi\)
0.425115 + 0.905139i \(0.360234\pi\)
\(42\) 0 0
\(43\) −170.565 −0.604905 −0.302452 0.953164i \(-0.597805\pi\)
−0.302452 + 0.953164i \(0.597805\pi\)
\(44\) −33.8337 −0.115923
\(45\) 0 0
\(46\) −427.477 −1.37017
\(47\) −247.523 −0.768192 −0.384096 0.923293i \(-0.625487\pi\)
−0.384096 + 0.923293i \(0.625487\pi\)
\(48\) 0 0
\(49\) −196.343 −0.572428
\(50\) −83.2008 −0.235327
\(51\) 0 0
\(52\) −176.000 −0.469362
\(53\) 76.4692 0.198186 0.0990929 0.995078i \(-0.468406\pi\)
0.0990929 + 0.995078i \(0.468406\pi\)
\(54\) 0 0
\(55\) −55.0000 −0.134840
\(56\) 198.461 0.473580
\(57\) 0 0
\(58\) −238.511 −0.539967
\(59\) 258.248 0.569849 0.284924 0.958550i \(-0.408032\pi\)
0.284924 + 0.958550i \(0.408032\pi\)
\(60\) 0 0
\(61\) −97.3761 −0.204389 −0.102195 0.994764i \(-0.532586\pi\)
−0.102195 + 0.994764i \(0.532586\pi\)
\(62\) 295.233 0.604752
\(63\) 0 0
\(64\) 192.880 0.376718
\(65\) −286.105 −0.545953
\(66\) 0 0
\(67\) −278.018 −0.506945 −0.253472 0.967343i \(-0.581573\pi\)
−0.253472 + 0.967343i \(0.581573\pi\)
\(68\) −70.0671 −0.124954
\(69\) 0 0
\(70\) −201.516 −0.344082
\(71\) −292.087 −0.488231 −0.244116 0.969746i \(-0.578498\pi\)
−0.244116 + 0.969746i \(0.578498\pi\)
\(72\) 0 0
\(73\) −482.377 −0.773397 −0.386699 0.922206i \(-0.626385\pi\)
−0.386699 + 0.922206i \(0.626385\pi\)
\(74\) 101.787 0.159899
\(75\) 0 0
\(76\) −142.164 −0.214571
\(77\) −133.212 −0.197155
\(78\) 0 0
\(79\) −93.3473 −0.132942 −0.0664708 0.997788i \(-0.521174\pi\)
−0.0664708 + 0.997788i \(0.521174\pi\)
\(80\) −395.729 −0.553048
\(81\) 0 0
\(82\) −742.847 −1.00041
\(83\) −1011.13 −1.33718 −0.668588 0.743633i \(-0.733101\pi\)
−0.668588 + 0.743633i \(0.733101\pi\)
\(84\) 0 0
\(85\) −113.901 −0.145345
\(86\) 567.646 0.711753
\(87\) 0 0
\(88\) −180.267 −0.218370
\(89\) 60.3336 0.0718579 0.0359290 0.999354i \(-0.488561\pi\)
0.0359290 + 0.999354i \(0.488561\pi\)
\(90\) 0 0
\(91\) −692.959 −0.798262
\(92\) 395.077 0.447713
\(93\) 0 0
\(94\) 823.766 0.903883
\(95\) −231.102 −0.249585
\(96\) 0 0
\(97\) 662.927 0.693918 0.346959 0.937880i \(-0.387214\pi\)
0.346959 + 0.937880i \(0.387214\pi\)
\(98\) 653.435 0.673540
\(99\) 0 0
\(100\) 76.8948 0.0768948
\(101\) −1585.81 −1.56231 −0.781156 0.624336i \(-0.785370\pi\)
−0.781156 + 0.624336i \(0.785370\pi\)
\(102\) 0 0
\(103\) −2024.01 −1.93623 −0.968115 0.250508i \(-0.919402\pi\)
−0.968115 + 0.250508i \(0.919402\pi\)
\(104\) −937.734 −0.884157
\(105\) 0 0
\(106\) −254.492 −0.233193
\(107\) −611.358 −0.552357 −0.276179 0.961106i \(-0.589068\pi\)
−0.276179 + 0.961106i \(0.589068\pi\)
\(108\) 0 0
\(109\) 1180.33 1.03721 0.518604 0.855015i \(-0.326452\pi\)
0.518604 + 0.855015i \(0.326452\pi\)
\(110\) 183.042 0.158658
\(111\) 0 0
\(112\) −958.473 −0.808636
\(113\) −1098.57 −0.914557 −0.457278 0.889324i \(-0.651176\pi\)
−0.457278 + 0.889324i \(0.651176\pi\)
\(114\) 0 0
\(115\) 642.236 0.520773
\(116\) 220.434 0.176438
\(117\) 0 0
\(118\) −859.459 −0.670505
\(119\) −275.873 −0.212514
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 324.071 0.240492
\(123\) 0 0
\(124\) −272.857 −0.197607
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −794.434 −0.555076 −0.277538 0.960715i \(-0.589518\pi\)
−0.277538 + 0.960715i \(0.589518\pi\)
\(128\) −1700.28 −1.17410
\(129\) 0 0
\(130\) 952.167 0.642389
\(131\) −364.955 −0.243406 −0.121703 0.992567i \(-0.538836\pi\)
−0.121703 + 0.992567i \(0.538836\pi\)
\(132\) 0 0
\(133\) −559.739 −0.364929
\(134\) 925.252 0.596490
\(135\) 0 0
\(136\) −373.320 −0.235382
\(137\) −2057.56 −1.28313 −0.641567 0.767067i \(-0.721715\pi\)
−0.641567 + 0.767067i \(0.721715\pi\)
\(138\) 0 0
\(139\) 471.536 0.287735 0.143867 0.989597i \(-0.454046\pi\)
0.143867 + 0.989597i \(0.454046\pi\)
\(140\) 186.243 0.112431
\(141\) 0 0
\(142\) 972.076 0.574471
\(143\) 629.431 0.368082
\(144\) 0 0
\(145\) 358.337 0.205230
\(146\) 1605.37 0.910008
\(147\) 0 0
\(148\) −94.0725 −0.0522481
\(149\) 135.727 0.0746255 0.0373128 0.999304i \(-0.488120\pi\)
0.0373128 + 0.999304i \(0.488120\pi\)
\(150\) 0 0
\(151\) −1908.95 −1.02879 −0.514397 0.857552i \(-0.671984\pi\)
−0.514397 + 0.857552i \(0.671984\pi\)
\(152\) −757.457 −0.404196
\(153\) 0 0
\(154\) 443.335 0.231980
\(155\) −443.555 −0.229853
\(156\) 0 0
\(157\) −1028.53 −0.522840 −0.261420 0.965225i \(-0.584191\pi\)
−0.261420 + 0.965225i \(0.584191\pi\)
\(158\) 310.663 0.156424
\(159\) 0 0
\(160\) 661.482 0.326842
\(161\) 1555.52 0.761444
\(162\) 0 0
\(163\) 684.628 0.328983 0.164492 0.986378i \(-0.447402\pi\)
0.164492 + 0.986378i \(0.447402\pi\)
\(164\) 686.545 0.326891
\(165\) 0 0
\(166\) 3365.06 1.57337
\(167\) 799.231 0.370337 0.185169 0.982707i \(-0.440717\pi\)
0.185169 + 0.982707i \(0.440717\pi\)
\(168\) 0 0
\(169\) 1077.25 0.490326
\(170\) 379.066 0.171018
\(171\) 0 0
\(172\) −524.622 −0.232570
\(173\) −3670.62 −1.61313 −0.806567 0.591143i \(-0.798677\pi\)
−0.806567 + 0.591143i \(0.798677\pi\)
\(174\) 0 0
\(175\) 302.755 0.130778
\(176\) 870.604 0.372865
\(177\) 0 0
\(178\) −200.792 −0.0845506
\(179\) 3971.27 1.65825 0.829125 0.559064i \(-0.188839\pi\)
0.829125 + 0.559064i \(0.188839\pi\)
\(180\) 0 0
\(181\) 2720.15 1.11706 0.558529 0.829485i \(-0.311366\pi\)
0.558529 + 0.829485i \(0.311366\pi\)
\(182\) 2306.19 0.939264
\(183\) 0 0
\(184\) 2104.98 0.843377
\(185\) −152.924 −0.0607741
\(186\) 0 0
\(187\) 250.582 0.0979913
\(188\) −761.331 −0.295350
\(189\) 0 0
\(190\) 769.115 0.293671
\(191\) −23.4786 −0.00889450 −0.00444725 0.999990i \(-0.501416\pi\)
−0.00444725 + 0.999990i \(0.501416\pi\)
\(192\) 0 0
\(193\) −4974.19 −1.85518 −0.927590 0.373599i \(-0.878124\pi\)
−0.927590 + 0.373599i \(0.878124\pi\)
\(194\) −2206.24 −0.816489
\(195\) 0 0
\(196\) −603.909 −0.220084
\(197\) 175.739 0.0635576 0.0317788 0.999495i \(-0.489883\pi\)
0.0317788 + 0.999495i \(0.489883\pi\)
\(198\) 0 0
\(199\) 5328.65 1.89818 0.949091 0.315002i \(-0.102005\pi\)
0.949091 + 0.315002i \(0.102005\pi\)
\(200\) 409.698 0.144850
\(201\) 0 0
\(202\) 5277.61 1.83827
\(203\) 867.908 0.300075
\(204\) 0 0
\(205\) 1116.05 0.380234
\(206\) 6735.97 2.27824
\(207\) 0 0
\(208\) 4528.81 1.50969
\(209\) 508.425 0.168270
\(210\) 0 0
\(211\) −4931.62 −1.60904 −0.804518 0.593928i \(-0.797576\pi\)
−0.804518 + 0.593928i \(0.797576\pi\)
\(212\) 235.203 0.0761973
\(213\) 0 0
\(214\) 2034.62 0.649924
\(215\) −852.825 −0.270522
\(216\) 0 0
\(217\) −1074.31 −0.336078
\(218\) −3928.19 −1.22042
\(219\) 0 0
\(220\) −169.169 −0.0518425
\(221\) 1303.51 0.396757
\(222\) 0 0
\(223\) −2070.50 −0.621754 −0.310877 0.950450i \(-0.600623\pi\)
−0.310877 + 0.950450i \(0.600623\pi\)
\(224\) 1602.14 0.477890
\(225\) 0 0
\(226\) 3656.08 1.07610
\(227\) 5474.76 1.60076 0.800380 0.599494i \(-0.204631\pi\)
0.800380 + 0.599494i \(0.204631\pi\)
\(228\) 0 0
\(229\) 5061.11 1.46047 0.730234 0.683197i \(-0.239411\pi\)
0.730234 + 0.683197i \(0.239411\pi\)
\(230\) −2137.38 −0.612760
\(231\) 0 0
\(232\) 1174.48 0.332364
\(233\) −1740.43 −0.489354 −0.244677 0.969605i \(-0.578682\pi\)
−0.244677 + 0.969605i \(0.578682\pi\)
\(234\) 0 0
\(235\) −1237.62 −0.343546
\(236\) 794.318 0.219092
\(237\) 0 0
\(238\) 918.113 0.250052
\(239\) −3557.50 −0.962827 −0.481413 0.876494i \(-0.659876\pi\)
−0.481413 + 0.876494i \(0.659876\pi\)
\(240\) 0 0
\(241\) −5027.91 −1.34388 −0.671942 0.740603i \(-0.734540\pi\)
−0.671942 + 0.740603i \(0.734540\pi\)
\(242\) −402.692 −0.106967
\(243\) 0 0
\(244\) −299.509 −0.0785823
\(245\) −981.714 −0.255997
\(246\) 0 0
\(247\) 2644.78 0.681309
\(248\) −1453.79 −0.372241
\(249\) 0 0
\(250\) −416.004 −0.105242
\(251\) 1319.91 0.331919 0.165960 0.986133i \(-0.446928\pi\)
0.165960 + 0.986133i \(0.446928\pi\)
\(252\) 0 0
\(253\) −1412.92 −0.351105
\(254\) 2643.90 0.653123
\(255\) 0 0
\(256\) 4115.55 1.00477
\(257\) 5827.46 1.41443 0.707213 0.707001i \(-0.249952\pi\)
0.707213 + 0.707001i \(0.249952\pi\)
\(258\) 0 0
\(259\) −370.389 −0.0888603
\(260\) −880.000 −0.209905
\(261\) 0 0
\(262\) 1214.58 0.286401
\(263\) −4452.54 −1.04394 −0.521968 0.852965i \(-0.674802\pi\)
−0.521968 + 0.852965i \(0.674802\pi\)
\(264\) 0 0
\(265\) 382.346 0.0886314
\(266\) 1862.83 0.429389
\(267\) 0 0
\(268\) −855.125 −0.194907
\(269\) 1558.71 0.353295 0.176647 0.984274i \(-0.443475\pi\)
0.176647 + 0.984274i \(0.443475\pi\)
\(270\) 0 0
\(271\) −5194.68 −1.16441 −0.582203 0.813043i \(-0.697809\pi\)
−0.582203 + 0.813043i \(0.697809\pi\)
\(272\) 1802.96 0.401913
\(273\) 0 0
\(274\) 6847.63 1.50978
\(275\) −275.000 −0.0603023
\(276\) 0 0
\(277\) −7425.82 −1.61074 −0.805369 0.592774i \(-0.798033\pi\)
−0.805369 + 0.592774i \(0.798033\pi\)
\(278\) −1569.29 −0.338559
\(279\) 0 0
\(280\) 992.306 0.211791
\(281\) 2209.95 0.469163 0.234581 0.972096i \(-0.424628\pi\)
0.234581 + 0.972096i \(0.424628\pi\)
\(282\) 0 0
\(283\) 3668.07 0.770475 0.385237 0.922817i \(-0.374120\pi\)
0.385237 + 0.922817i \(0.374120\pi\)
\(284\) −898.400 −0.187712
\(285\) 0 0
\(286\) −2094.77 −0.433099
\(287\) 2703.11 0.555957
\(288\) 0 0
\(289\) −4394.06 −0.894375
\(290\) −1192.56 −0.241481
\(291\) 0 0
\(292\) −1483.69 −0.297351
\(293\) 1455.91 0.290291 0.145145 0.989410i \(-0.453635\pi\)
0.145145 + 0.989410i \(0.453635\pi\)
\(294\) 0 0
\(295\) 1291.24 0.254844
\(296\) −501.221 −0.0984220
\(297\) 0 0
\(298\) −451.704 −0.0878071
\(299\) −7349.89 −1.42159
\(300\) 0 0
\(301\) −2065.58 −0.395541
\(302\) 6353.04 1.21052
\(303\) 0 0
\(304\) 3658.15 0.690163
\(305\) −486.881 −0.0914056
\(306\) 0 0
\(307\) 8129.61 1.51134 0.755670 0.654953i \(-0.227311\pi\)
0.755670 + 0.654953i \(0.227311\pi\)
\(308\) −409.734 −0.0758011
\(309\) 0 0
\(310\) 1476.16 0.270453
\(311\) −2030.06 −0.370142 −0.185071 0.982725i \(-0.559251\pi\)
−0.185071 + 0.982725i \(0.559251\pi\)
\(312\) 0 0
\(313\) −513.357 −0.0927050 −0.0463525 0.998925i \(-0.514760\pi\)
−0.0463525 + 0.998925i \(0.514760\pi\)
\(314\) 3422.99 0.615192
\(315\) 0 0
\(316\) −287.117 −0.0511126
\(317\) −5230.37 −0.926710 −0.463355 0.886173i \(-0.653354\pi\)
−0.463355 + 0.886173i \(0.653354\pi\)
\(318\) 0 0
\(319\) −788.342 −0.138366
\(320\) 964.399 0.168474
\(321\) 0 0
\(322\) −5176.83 −0.895943
\(323\) 1052.91 0.181379
\(324\) 0 0
\(325\) −1430.53 −0.244158
\(326\) −2278.46 −0.387094
\(327\) 0 0
\(328\) 3657.93 0.615779
\(329\) −2997.56 −0.502313
\(330\) 0 0
\(331\) 7261.31 1.20579 0.602897 0.797819i \(-0.294013\pi\)
0.602897 + 0.797819i \(0.294013\pi\)
\(332\) −3110.02 −0.514110
\(333\) 0 0
\(334\) −2659.87 −0.435753
\(335\) −1390.09 −0.226713
\(336\) 0 0
\(337\) 1707.70 0.276037 0.138019 0.990430i \(-0.455927\pi\)
0.138019 + 0.990430i \(0.455927\pi\)
\(338\) −3585.11 −0.576936
\(339\) 0 0
\(340\) −350.335 −0.0558812
\(341\) 975.821 0.154967
\(342\) 0 0
\(343\) −6531.56 −1.02819
\(344\) −2795.20 −0.438103
\(345\) 0 0
\(346\) 12215.9 1.89807
\(347\) 3948.37 0.610834 0.305417 0.952219i \(-0.401204\pi\)
0.305417 + 0.952219i \(0.401204\pi\)
\(348\) 0 0
\(349\) 1623.51 0.249010 0.124505 0.992219i \(-0.460266\pi\)
0.124505 + 0.992219i \(0.460266\pi\)
\(350\) −1007.58 −0.153878
\(351\) 0 0
\(352\) −1455.26 −0.220357
\(353\) −5926.17 −0.893535 −0.446768 0.894650i \(-0.647425\pi\)
−0.446768 + 0.894650i \(0.647425\pi\)
\(354\) 0 0
\(355\) −1460.44 −0.218344
\(356\) 185.574 0.0276275
\(357\) 0 0
\(358\) −13216.5 −1.95116
\(359\) −3545.16 −0.521187 −0.260594 0.965449i \(-0.583918\pi\)
−0.260594 + 0.965449i \(0.583918\pi\)
\(360\) 0 0
\(361\) −4722.67 −0.688537
\(362\) −9052.76 −1.31437
\(363\) 0 0
\(364\) −2131.40 −0.306911
\(365\) −2411.89 −0.345874
\(366\) 0 0
\(367\) 3429.59 0.487802 0.243901 0.969800i \(-0.421573\pi\)
0.243901 + 0.969800i \(0.421573\pi\)
\(368\) −10166.1 −1.44006
\(369\) 0 0
\(370\) 508.936 0.0715090
\(371\) 926.058 0.129592
\(372\) 0 0
\(373\) 2462.08 0.341773 0.170887 0.985291i \(-0.445337\pi\)
0.170887 + 0.985291i \(0.445337\pi\)
\(374\) −833.945 −0.115300
\(375\) 0 0
\(376\) −4056.39 −0.556363
\(377\) −4100.88 −0.560229
\(378\) 0 0
\(379\) −6713.31 −0.909867 −0.454933 0.890525i \(-0.650337\pi\)
−0.454933 + 0.890525i \(0.650337\pi\)
\(380\) −710.822 −0.0959590
\(381\) 0 0
\(382\) 78.1374 0.0104656
\(383\) −3230.45 −0.430988 −0.215494 0.976505i \(-0.569136\pi\)
−0.215494 + 0.976505i \(0.569136\pi\)
\(384\) 0 0
\(385\) −666.062 −0.0881705
\(386\) 16554.3 2.18287
\(387\) 0 0
\(388\) 2039.03 0.266793
\(389\) −3075.88 −0.400908 −0.200454 0.979703i \(-0.564242\pi\)
−0.200454 + 0.979703i \(0.564242\pi\)
\(390\) 0 0
\(391\) −2926.05 −0.378457
\(392\) −3217.65 −0.414581
\(393\) 0 0
\(394\) −584.863 −0.0747843
\(395\) −466.736 −0.0594533
\(396\) 0 0
\(397\) 4864.47 0.614964 0.307482 0.951554i \(-0.400514\pi\)
0.307482 + 0.951554i \(0.400514\pi\)
\(398\) −17733.9 −2.23347
\(399\) 0 0
\(400\) −1978.65 −0.247331
\(401\) 11676.4 1.45409 0.727044 0.686591i \(-0.240893\pi\)
0.727044 + 0.686591i \(0.240893\pi\)
\(402\) 0 0
\(403\) 5076.14 0.627445
\(404\) −4877.61 −0.600669
\(405\) 0 0
\(406\) −2888.42 −0.353079
\(407\) 336.433 0.0409739
\(408\) 0 0
\(409\) −16394.0 −1.98199 −0.990993 0.133917i \(-0.957244\pi\)
−0.990993 + 0.133917i \(0.957244\pi\)
\(410\) −3714.24 −0.447398
\(411\) 0 0
\(412\) −6225.43 −0.744430
\(413\) 3127.44 0.372618
\(414\) 0 0
\(415\) −5055.64 −0.598004
\(416\) −7570.14 −0.892204
\(417\) 0 0
\(418\) −1692.05 −0.197993
\(419\) 8451.89 0.985446 0.492723 0.870186i \(-0.336002\pi\)
0.492723 + 0.870186i \(0.336002\pi\)
\(420\) 0 0
\(421\) −10486.2 −1.21393 −0.606967 0.794727i \(-0.707614\pi\)
−0.606967 + 0.794727i \(0.707614\pi\)
\(422\) 16412.6 1.89325
\(423\) 0 0
\(424\) 1253.17 0.143536
\(425\) −569.504 −0.0650000
\(426\) 0 0
\(427\) −1179.25 −0.133648
\(428\) −1880.41 −0.212367
\(429\) 0 0
\(430\) 2838.23 0.318306
\(431\) 7443.30 0.831859 0.415929 0.909397i \(-0.363456\pi\)
0.415929 + 0.909397i \(0.363456\pi\)
\(432\) 0 0
\(433\) −3778.56 −0.419367 −0.209683 0.977769i \(-0.567243\pi\)
−0.209683 + 0.977769i \(0.567243\pi\)
\(434\) 3575.33 0.395441
\(435\) 0 0
\(436\) 3630.46 0.398779
\(437\) −5936.89 −0.649885
\(438\) 0 0
\(439\) −5405.03 −0.587627 −0.293813 0.955863i \(-0.594924\pi\)
−0.293813 + 0.955863i \(0.594924\pi\)
\(440\) −901.335 −0.0976579
\(441\) 0 0
\(442\) −4338.11 −0.466839
\(443\) 2183.94 0.234226 0.117113 0.993119i \(-0.462636\pi\)
0.117113 + 0.993119i \(0.462636\pi\)
\(444\) 0 0
\(445\) 301.668 0.0321358
\(446\) 6890.70 0.731579
\(447\) 0 0
\(448\) 2335.82 0.246332
\(449\) 13367.2 1.40498 0.702492 0.711692i \(-0.252071\pi\)
0.702492 + 0.711692i \(0.252071\pi\)
\(450\) 0 0
\(451\) −2455.30 −0.256354
\(452\) −3378.98 −0.351623
\(453\) 0 0
\(454\) −18220.2 −1.88351
\(455\) −3464.79 −0.356994
\(456\) 0 0
\(457\) −13897.3 −1.42251 −0.711256 0.702933i \(-0.751873\pi\)
−0.711256 + 0.702933i \(0.751873\pi\)
\(458\) −16843.5 −1.71844
\(459\) 0 0
\(460\) 1975.39 0.200224
\(461\) 12543.9 1.26730 0.633652 0.773618i \(-0.281555\pi\)
0.633652 + 0.773618i \(0.281555\pi\)
\(462\) 0 0
\(463\) 16288.3 1.63495 0.817475 0.575965i \(-0.195373\pi\)
0.817475 + 0.575965i \(0.195373\pi\)
\(464\) −5672.18 −0.567509
\(465\) 0 0
\(466\) 5792.21 0.575792
\(467\) 4689.45 0.464672 0.232336 0.972636i \(-0.425363\pi\)
0.232336 + 0.972636i \(0.425363\pi\)
\(468\) 0 0
\(469\) −3366.86 −0.331486
\(470\) 4118.83 0.404229
\(471\) 0 0
\(472\) 4232.15 0.412713
\(473\) 1876.21 0.182386
\(474\) 0 0
\(475\) −1155.51 −0.111618
\(476\) −848.527 −0.0817063
\(477\) 0 0
\(478\) 11839.5 1.13290
\(479\) 15228.0 1.45258 0.726291 0.687387i \(-0.241242\pi\)
0.726291 + 0.687387i \(0.241242\pi\)
\(480\) 0 0
\(481\) 1750.09 0.165899
\(482\) 16733.0 1.58126
\(483\) 0 0
\(484\) 372.171 0.0349522
\(485\) 3314.63 0.310330
\(486\) 0 0
\(487\) −15805.7 −1.47068 −0.735342 0.677697i \(-0.762978\pi\)
−0.735342 + 0.677697i \(0.762978\pi\)
\(488\) −1595.79 −0.148029
\(489\) 0 0
\(490\) 3267.17 0.301216
\(491\) −9239.13 −0.849198 −0.424599 0.905382i \(-0.639585\pi\)
−0.424599 + 0.905382i \(0.639585\pi\)
\(492\) 0 0
\(493\) −1632.60 −0.149145
\(494\) −8801.91 −0.801653
\(495\) 0 0
\(496\) 7021.11 0.635599
\(497\) −3537.24 −0.319249
\(498\) 0 0
\(499\) 346.738 0.0311064 0.0155532 0.999879i \(-0.495049\pi\)
0.0155532 + 0.999879i \(0.495049\pi\)
\(500\) 384.474 0.0343884
\(501\) 0 0
\(502\) −4392.69 −0.390548
\(503\) 12519.4 1.10977 0.554884 0.831928i \(-0.312763\pi\)
0.554884 + 0.831928i \(0.312763\pi\)
\(504\) 0 0
\(505\) −7929.03 −0.698687
\(506\) 4702.24 0.413123
\(507\) 0 0
\(508\) −2443.51 −0.213412
\(509\) 9405.03 0.818999 0.409500 0.912310i \(-0.365703\pi\)
0.409500 + 0.912310i \(0.365703\pi\)
\(510\) 0 0
\(511\) −5841.69 −0.505717
\(512\) −94.4331 −0.00815116
\(513\) 0 0
\(514\) −19394.0 −1.66426
\(515\) −10120.0 −0.865908
\(516\) 0 0
\(517\) 2722.76 0.231619
\(518\) 1232.66 0.104556
\(519\) 0 0
\(520\) −4688.67 −0.395407
\(521\) 17884.4 1.50390 0.751949 0.659222i \(-0.229114\pi\)
0.751949 + 0.659222i \(0.229114\pi\)
\(522\) 0 0
\(523\) 7617.35 0.636871 0.318435 0.947945i \(-0.396843\pi\)
0.318435 + 0.947945i \(0.396843\pi\)
\(524\) −1122.52 −0.0935834
\(525\) 0 0
\(526\) 14818.2 1.22833
\(527\) 2020.85 0.167039
\(528\) 0 0
\(529\) 4331.70 0.356021
\(530\) −1272.46 −0.104287
\(531\) 0 0
\(532\) −1721.64 −0.140306
\(533\) −12772.3 −1.03795
\(534\) 0 0
\(535\) −3056.79 −0.247022
\(536\) −4556.13 −0.367155
\(537\) 0 0
\(538\) −5187.44 −0.415700
\(539\) 2159.77 0.172593
\(540\) 0 0
\(541\) −6309.12 −0.501387 −0.250694 0.968066i \(-0.580659\pi\)
−0.250694 + 0.968066i \(0.580659\pi\)
\(542\) 17288.0 1.37008
\(543\) 0 0
\(544\) −3013.74 −0.237524
\(545\) 5901.67 0.463853
\(546\) 0 0
\(547\) −957.117 −0.0748142 −0.0374071 0.999300i \(-0.511910\pi\)
−0.0374071 + 0.999300i \(0.511910\pi\)
\(548\) −6328.63 −0.493332
\(549\) 0 0
\(550\) 915.209 0.0709539
\(551\) −3312.50 −0.256111
\(552\) 0 0
\(553\) −1130.46 −0.0869292
\(554\) 24713.4 1.89525
\(555\) 0 0
\(556\) 1450.35 0.110627
\(557\) −9284.17 −0.706253 −0.353127 0.935576i \(-0.614881\pi\)
−0.353127 + 0.935576i \(0.614881\pi\)
\(558\) 0 0
\(559\) 9759.90 0.738461
\(560\) −4792.37 −0.361633
\(561\) 0 0
\(562\) −7354.79 −0.552034
\(563\) −5842.70 −0.437372 −0.218686 0.975795i \(-0.570177\pi\)
−0.218686 + 0.975795i \(0.570177\pi\)
\(564\) 0 0
\(565\) −5492.86 −0.409002
\(566\) −12207.5 −0.906569
\(567\) 0 0
\(568\) −4786.71 −0.353602
\(569\) 6805.46 0.501405 0.250703 0.968064i \(-0.419338\pi\)
0.250703 + 0.968064i \(0.419338\pi\)
\(570\) 0 0
\(571\) 21649.6 1.58671 0.793353 0.608762i \(-0.208334\pi\)
0.793353 + 0.608762i \(0.208334\pi\)
\(572\) 1936.00 0.141518
\(573\) 0 0
\(574\) −8996.04 −0.654159
\(575\) 3211.18 0.232897
\(576\) 0 0
\(577\) 23967.4 1.72925 0.864624 0.502420i \(-0.167557\pi\)
0.864624 + 0.502420i \(0.167557\pi\)
\(578\) 14623.6 1.05235
\(579\) 0 0
\(580\) 1102.17 0.0789054
\(581\) −12245.0 −0.874367
\(582\) 0 0
\(583\) −841.161 −0.0597553
\(584\) −7905.16 −0.560133
\(585\) 0 0
\(586\) −4845.31 −0.341566
\(587\) 7580.37 0.533007 0.266504 0.963834i \(-0.414132\pi\)
0.266504 + 0.963834i \(0.414132\pi\)
\(588\) 0 0
\(589\) 4100.26 0.286839
\(590\) −4297.29 −0.299859
\(591\) 0 0
\(592\) 2420.66 0.168055
\(593\) −2996.14 −0.207481 −0.103741 0.994604i \(-0.533081\pi\)
−0.103741 + 0.994604i \(0.533081\pi\)
\(594\) 0 0
\(595\) −1379.36 −0.0950393
\(596\) 417.469 0.0286916
\(597\) 0 0
\(598\) 24460.6 1.67269
\(599\) 21317.9 1.45413 0.727066 0.686568i \(-0.240883\pi\)
0.727066 + 0.686568i \(0.240883\pi\)
\(600\) 0 0
\(601\) 5703.47 0.387104 0.193552 0.981090i \(-0.437999\pi\)
0.193552 + 0.981090i \(0.437999\pi\)
\(602\) 6874.31 0.465408
\(603\) 0 0
\(604\) −5871.53 −0.395545
\(605\) 605.000 0.0406558
\(606\) 0 0
\(607\) 23658.6 1.58200 0.790998 0.611819i \(-0.209562\pi\)
0.790998 + 0.611819i \(0.209562\pi\)
\(608\) −6114.80 −0.407875
\(609\) 0 0
\(610\) 1620.35 0.107551
\(611\) 14163.5 0.937800
\(612\) 0 0
\(613\) 8792.89 0.579350 0.289675 0.957125i \(-0.406453\pi\)
0.289675 + 0.957125i \(0.406453\pi\)
\(614\) −27055.6 −1.77830
\(615\) 0 0
\(616\) −2183.07 −0.142790
\(617\) −18080.8 −1.17975 −0.589876 0.807494i \(-0.700823\pi\)
−0.589876 + 0.807494i \(0.700823\pi\)
\(618\) 0 0
\(619\) −9077.15 −0.589405 −0.294702 0.955589i \(-0.595221\pi\)
−0.294702 + 0.955589i \(0.595221\pi\)
\(620\) −1364.28 −0.0883725
\(621\) 0 0
\(622\) 6756.10 0.435522
\(623\) 730.653 0.0469872
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 1708.47 0.109080
\(627\) 0 0
\(628\) −3163.55 −0.201018
\(629\) 696.727 0.0441659
\(630\) 0 0
\(631\) −10646.4 −0.671672 −0.335836 0.941920i \(-0.609019\pi\)
−0.335836 + 0.941920i \(0.609019\pi\)
\(632\) −1529.77 −0.0962830
\(633\) 0 0
\(634\) 17406.8 1.09040
\(635\) −3972.17 −0.248237
\(636\) 0 0
\(637\) 11234.9 0.698814
\(638\) 2623.63 0.162806
\(639\) 0 0
\(640\) −8501.41 −0.525075
\(641\) −14313.4 −0.881972 −0.440986 0.897514i \(-0.645371\pi\)
−0.440986 + 0.897514i \(0.645371\pi\)
\(642\) 0 0
\(643\) 20481.6 1.25617 0.628084 0.778146i \(-0.283839\pi\)
0.628084 + 0.778146i \(0.283839\pi\)
\(644\) 4784.47 0.292755
\(645\) 0 0
\(646\) −3504.12 −0.213417
\(647\) −13408.9 −0.814773 −0.407387 0.913256i \(-0.633560\pi\)
−0.407387 + 0.913256i \(0.633560\pi\)
\(648\) 0 0
\(649\) −2840.73 −0.171816
\(650\) 4760.83 0.287285
\(651\) 0 0
\(652\) 2105.77 0.126485
\(653\) 27525.1 1.64953 0.824764 0.565477i \(-0.191308\pi\)
0.824764 + 0.565477i \(0.191308\pi\)
\(654\) 0 0
\(655\) −1824.77 −0.108855
\(656\) −17666.1 −1.05144
\(657\) 0 0
\(658\) 9975.98 0.591040
\(659\) −19977.6 −1.18091 −0.590453 0.807072i \(-0.701051\pi\)
−0.590453 + 0.807072i \(0.701051\pi\)
\(660\) 0 0
\(661\) 1661.53 0.0977698 0.0488849 0.998804i \(-0.484433\pi\)
0.0488849 + 0.998804i \(0.484433\pi\)
\(662\) −24165.9 −1.41878
\(663\) 0 0
\(664\) −16570.3 −0.968451
\(665\) −2798.70 −0.163201
\(666\) 0 0
\(667\) 9205.49 0.534390
\(668\) 2458.27 0.142385
\(669\) 0 0
\(670\) 4626.26 0.266758
\(671\) 1071.14 0.0616256
\(672\) 0 0
\(673\) −13546.5 −0.775898 −0.387949 0.921681i \(-0.626816\pi\)
−0.387949 + 0.921681i \(0.626816\pi\)
\(674\) −5683.29 −0.324796
\(675\) 0 0
\(676\) 3313.39 0.188518
\(677\) 9859.39 0.559715 0.279858 0.960042i \(-0.409713\pi\)
0.279858 + 0.960042i \(0.409713\pi\)
\(678\) 0 0
\(679\) 8028.19 0.453746
\(680\) −1866.60 −0.105266
\(681\) 0 0
\(682\) −3247.56 −0.182340
\(683\) 1182.82 0.0662654 0.0331327 0.999451i \(-0.489452\pi\)
0.0331327 + 0.999451i \(0.489452\pi\)
\(684\) 0 0
\(685\) −10287.8 −0.573835
\(686\) 21737.2 1.20981
\(687\) 0 0
\(688\) 13499.5 0.748058
\(689\) −4375.65 −0.241943
\(690\) 0 0
\(691\) −4312.15 −0.237398 −0.118699 0.992930i \(-0.537872\pi\)
−0.118699 + 0.992930i \(0.537872\pi\)
\(692\) −11290.1 −0.620208
\(693\) 0 0
\(694\) −13140.3 −0.718730
\(695\) 2357.68 0.128679
\(696\) 0 0
\(697\) −5084.75 −0.276325
\(698\) −5403.08 −0.292994
\(699\) 0 0
\(700\) 931.213 0.0502807
\(701\) 23850.3 1.28504 0.642519 0.766270i \(-0.277889\pi\)
0.642519 + 0.766270i \(0.277889\pi\)
\(702\) 0 0
\(703\) 1413.64 0.0758415
\(704\) −2121.68 −0.113585
\(705\) 0 0
\(706\) 19722.5 1.05137
\(707\) −19204.4 −1.02158
\(708\) 0 0
\(709\) −28492.0 −1.50922 −0.754612 0.656172i \(-0.772175\pi\)
−0.754612 + 0.656172i \(0.772175\pi\)
\(710\) 4860.38 0.256911
\(711\) 0 0
\(712\) 988.743 0.0520431
\(713\) −11394.7 −0.598505
\(714\) 0 0
\(715\) 3147.16 0.164611
\(716\) 12214.8 0.637554
\(717\) 0 0
\(718\) 11798.4 0.613248
\(719\) −14182.8 −0.735647 −0.367824 0.929896i \(-0.619897\pi\)
−0.367824 + 0.929896i \(0.619897\pi\)
\(720\) 0 0
\(721\) −24511.2 −1.26608
\(722\) 15717.2 0.810157
\(723\) 0 0
\(724\) 8366.63 0.429480
\(725\) 1791.69 0.0917814
\(726\) 0 0
\(727\) 31836.8 1.62416 0.812078 0.583549i \(-0.198336\pi\)
0.812078 + 0.583549i \(0.198336\pi\)
\(728\) −11356.2 −0.578142
\(729\) 0 0
\(730\) 8026.83 0.406968
\(731\) 3885.50 0.196594
\(732\) 0 0
\(733\) 10002.6 0.504030 0.252015 0.967723i \(-0.418907\pi\)
0.252015 + 0.967723i \(0.418907\pi\)
\(734\) −11413.8 −0.573965
\(735\) 0 0
\(736\) 16993.1 0.851053
\(737\) 3058.20 0.152850
\(738\) 0 0
\(739\) 18784.8 0.935060 0.467530 0.883977i \(-0.345144\pi\)
0.467530 + 0.883977i \(0.345144\pi\)
\(740\) −470.363 −0.0233660
\(741\) 0 0
\(742\) −3081.95 −0.152482
\(743\) −19455.4 −0.960633 −0.480317 0.877095i \(-0.659478\pi\)
−0.480317 + 0.877095i \(0.659478\pi\)
\(744\) 0 0
\(745\) 678.636 0.0333735
\(746\) −8193.87 −0.402143
\(747\) 0 0
\(748\) 770.738 0.0376751
\(749\) −7403.68 −0.361181
\(750\) 0 0
\(751\) 19365.5 0.940953 0.470477 0.882413i \(-0.344082\pi\)
0.470477 + 0.882413i \(0.344082\pi\)
\(752\) 19590.5 0.949987
\(753\) 0 0
\(754\) 13647.9 0.659186
\(755\) −9544.74 −0.460091
\(756\) 0 0
\(757\) 19298.9 0.926595 0.463297 0.886203i \(-0.346666\pi\)
0.463297 + 0.886203i \(0.346666\pi\)
\(758\) 22342.1 1.07058
\(759\) 0 0
\(760\) −3787.28 −0.180762
\(761\) −30510.0 −1.45333 −0.726666 0.686991i \(-0.758931\pi\)
−0.726666 + 0.686991i \(0.758931\pi\)
\(762\) 0 0
\(763\) 14294.1 0.678219
\(764\) −72.2152 −0.00341970
\(765\) 0 0
\(766\) 10751.1 0.507117
\(767\) −14777.2 −0.695665
\(768\) 0 0
\(769\) 11569.6 0.542536 0.271268 0.962504i \(-0.412557\pi\)
0.271268 + 0.962504i \(0.412557\pi\)
\(770\) 2216.67 0.103745
\(771\) 0 0
\(772\) −15299.6 −0.713269
\(773\) −42711.1 −1.98734 −0.993668 0.112353i \(-0.964161\pi\)
−0.993668 + 0.112353i \(0.964161\pi\)
\(774\) 0 0
\(775\) −2217.78 −0.102793
\(776\) 10864.0 0.502570
\(777\) 0 0
\(778\) 10236.6 0.471723
\(779\) −10316.8 −0.474504
\(780\) 0 0
\(781\) 3212.96 0.147207
\(782\) 9737.99 0.445307
\(783\) 0 0
\(784\) 15539.7 0.707895
\(785\) −5142.66 −0.233821
\(786\) 0 0
\(787\) −403.118 −0.0182587 −0.00912936 0.999958i \(-0.502906\pi\)
−0.00912936 + 0.999958i \(0.502906\pi\)
\(788\) 540.535 0.0244363
\(789\) 0 0
\(790\) 1553.31 0.0699549
\(791\) −13303.9 −0.598019
\(792\) 0 0
\(793\) 5571.96 0.249516
\(794\) −16189.1 −0.723589
\(795\) 0 0
\(796\) 16389.8 0.729802
\(797\) −11413.0 −0.507237 −0.253619 0.967304i \(-0.581621\pi\)
−0.253619 + 0.967304i \(0.581621\pi\)
\(798\) 0 0
\(799\) 5638.63 0.249662
\(800\) 3307.41 0.146168
\(801\) 0 0
\(802\) −38859.3 −1.71093
\(803\) 5306.15 0.233188
\(804\) 0 0
\(805\) 7777.62 0.340528
\(806\) −16893.5 −0.738275
\(807\) 0 0
\(808\) −25988.1 −1.13151
\(809\) 21799.0 0.947357 0.473679 0.880698i \(-0.342926\pi\)
0.473679 + 0.880698i \(0.342926\pi\)
\(810\) 0 0
\(811\) 404.042 0.0174942 0.00874711 0.999962i \(-0.497216\pi\)
0.00874711 + 0.999962i \(0.497216\pi\)
\(812\) 2669.50 0.115371
\(813\) 0 0
\(814\) −1119.66 −0.0482113
\(815\) 3423.14 0.147126
\(816\) 0 0
\(817\) 7883.59 0.337591
\(818\) 54559.8 2.33208
\(819\) 0 0
\(820\) 3432.73 0.146190
\(821\) 428.289 0.0182063 0.00910316 0.999959i \(-0.497102\pi\)
0.00910316 + 0.999959i \(0.497102\pi\)
\(822\) 0 0
\(823\) −21469.0 −0.909309 −0.454655 0.890668i \(-0.650237\pi\)
−0.454655 + 0.890668i \(0.650237\pi\)
\(824\) −33169.3 −1.40231
\(825\) 0 0
\(826\) −10408.2 −0.438437
\(827\) −24062.7 −1.01178 −0.505889 0.862599i \(-0.668835\pi\)
−0.505889 + 0.862599i \(0.668835\pi\)
\(828\) 0 0
\(829\) −15567.7 −0.652217 −0.326108 0.945332i \(-0.605737\pi\)
−0.326108 + 0.945332i \(0.605737\pi\)
\(830\) 16825.3 0.703633
\(831\) 0 0
\(832\) −11036.8 −0.459894
\(833\) 4472.72 0.186039
\(834\) 0 0
\(835\) 3996.15 0.165620
\(836\) 1563.81 0.0646955
\(837\) 0 0
\(838\) −28128.1 −1.15951
\(839\) 11335.4 0.466437 0.233218 0.972424i \(-0.425074\pi\)
0.233218 + 0.972424i \(0.425074\pi\)
\(840\) 0 0
\(841\) −19252.8 −0.789404
\(842\) 34898.4 1.42836
\(843\) 0 0
\(844\) −15168.6 −0.618633
\(845\) 5386.23 0.219280
\(846\) 0 0
\(847\) 1465.34 0.0594446
\(848\) −6052.22 −0.245087
\(849\) 0 0
\(850\) 1895.33 0.0764814
\(851\) −3928.54 −0.158247
\(852\) 0 0
\(853\) −5152.34 −0.206814 −0.103407 0.994639i \(-0.532974\pi\)
−0.103407 + 0.994639i \(0.532974\pi\)
\(854\) 3924.57 0.157255
\(855\) 0 0
\(856\) −10018.9 −0.400045
\(857\) 21195.2 0.844825 0.422413 0.906404i \(-0.361183\pi\)
0.422413 + 0.906404i \(0.361183\pi\)
\(858\) 0 0
\(859\) 33887.6 1.34602 0.673010 0.739634i \(-0.265001\pi\)
0.673010 + 0.739634i \(0.265001\pi\)
\(860\) −2623.11 −0.104009
\(861\) 0 0
\(862\) −24771.5 −0.978795
\(863\) 26947.3 1.06292 0.531458 0.847085i \(-0.321644\pi\)
0.531458 + 0.847085i \(0.321644\pi\)
\(864\) 0 0
\(865\) −18353.1 −0.721415
\(866\) 12575.2 0.493442
\(867\) 0 0
\(868\) −3304.35 −0.129213
\(869\) 1026.82 0.0400834
\(870\) 0 0
\(871\) 15908.5 0.618873
\(872\) 19343.2 0.751198
\(873\) 0 0
\(874\) 19758.1 0.764679
\(875\) 1513.78 0.0584857
\(876\) 0 0
\(877\) −15854.7 −0.610462 −0.305231 0.952278i \(-0.598734\pi\)
−0.305231 + 0.952278i \(0.598734\pi\)
\(878\) 17988.1 0.691423
\(879\) 0 0
\(880\) 4353.02 0.166750
\(881\) 2198.99 0.0840930 0.0420465 0.999116i \(-0.486612\pi\)
0.0420465 + 0.999116i \(0.486612\pi\)
\(882\) 0 0
\(883\) 17607.6 0.671058 0.335529 0.942030i \(-0.391085\pi\)
0.335529 + 0.942030i \(0.391085\pi\)
\(884\) 4009.31 0.152543
\(885\) 0 0
\(886\) −7268.21 −0.275599
\(887\) −50391.4 −1.90753 −0.953764 0.300558i \(-0.902827\pi\)
−0.953764 + 0.300558i \(0.902827\pi\)
\(888\) 0 0
\(889\) −9620.77 −0.362959
\(890\) −1003.96 −0.0378122
\(891\) 0 0
\(892\) −6368.44 −0.239048
\(893\) 11440.6 0.428719
\(894\) 0 0
\(895\) 19856.3 0.741592
\(896\) −20590.8 −0.767734
\(897\) 0 0
\(898\) −44486.5 −1.65316
\(899\) −6357.69 −0.235863
\(900\) 0 0
\(901\) −1741.98 −0.0644105
\(902\) 8171.32 0.301635
\(903\) 0 0
\(904\) −18003.3 −0.662368
\(905\) 13600.8 0.499563
\(906\) 0 0
\(907\) 20022.6 0.733009 0.366505 0.930416i \(-0.380554\pi\)
0.366505 + 0.930416i \(0.380554\pi\)
\(908\) 16839.2 0.615450
\(909\) 0 0
\(910\) 11530.9 0.420052
\(911\) −14270.0 −0.518974 −0.259487 0.965747i \(-0.583553\pi\)
−0.259487 + 0.965747i \(0.583553\pi\)
\(912\) 0 0
\(913\) 11122.4 0.403174
\(914\) 46250.6 1.67378
\(915\) 0 0
\(916\) 15566.9 0.561512
\(917\) −4419.68 −0.159161
\(918\) 0 0
\(919\) 36781.2 1.32024 0.660120 0.751160i \(-0.270505\pi\)
0.660120 + 0.751160i \(0.270505\pi\)
\(920\) 10524.9 0.377170
\(921\) 0 0
\(922\) −41746.4 −1.49116
\(923\) 16713.5 0.596027
\(924\) 0 0
\(925\) −764.620 −0.0271790
\(926\) −54208.0 −1.92374
\(927\) 0 0
\(928\) 9481.35 0.335389
\(929\) 35798.6 1.26428 0.632139 0.774855i \(-0.282177\pi\)
0.632139 + 0.774855i \(0.282177\pi\)
\(930\) 0 0
\(931\) 9075.04 0.319466
\(932\) −5353.21 −0.188144
\(933\) 0 0
\(934\) −15606.6 −0.546750
\(935\) 1252.91 0.0438230
\(936\) 0 0
\(937\) 49934.6 1.74097 0.870486 0.492193i \(-0.163805\pi\)
0.870486 + 0.492193i \(0.163805\pi\)
\(938\) 11205.0 0.390039
\(939\) 0 0
\(940\) −3806.65 −0.132084
\(941\) −28437.4 −0.985156 −0.492578 0.870268i \(-0.663945\pi\)
−0.492578 + 0.870268i \(0.663945\pi\)
\(942\) 0 0
\(943\) 28670.6 0.990078
\(944\) −20439.3 −0.704706
\(945\) 0 0
\(946\) −6244.10 −0.214602
\(947\) 45305.0 1.55461 0.777304 0.629125i \(-0.216587\pi\)
0.777304 + 0.629125i \(0.216587\pi\)
\(948\) 0 0
\(949\) 27602.1 0.944155
\(950\) 3845.58 0.131334
\(951\) 0 0
\(952\) −4520.98 −0.153914
\(953\) 6991.64 0.237651 0.118825 0.992915i \(-0.462087\pi\)
0.118825 + 0.992915i \(0.462087\pi\)
\(954\) 0 0
\(955\) −117.393 −0.00397774
\(956\) −10942.1 −0.370182
\(957\) 0 0
\(958\) −50679.4 −1.70916
\(959\) −24917.5 −0.839029
\(960\) 0 0
\(961\) −21921.4 −0.735838
\(962\) −5824.37 −0.195203
\(963\) 0 0
\(964\) −15464.8 −0.516689
\(965\) −24870.9 −0.829662
\(966\) 0 0
\(967\) −1119.96 −0.0372444 −0.0186222 0.999827i \(-0.505928\pi\)
−0.0186222 + 0.999827i \(0.505928\pi\)
\(968\) 1982.94 0.0658410
\(969\) 0 0
\(970\) −11031.2 −0.365145
\(971\) −1998.46 −0.0660490 −0.0330245 0.999455i \(-0.510514\pi\)
−0.0330245 + 0.999455i \(0.510514\pi\)
\(972\) 0 0
\(973\) 5710.40 0.188147
\(974\) 52601.7 1.73046
\(975\) 0 0
\(976\) 7706.92 0.252759
\(977\) −9456.06 −0.309648 −0.154824 0.987942i \(-0.549481\pi\)
−0.154824 + 0.987942i \(0.549481\pi\)
\(978\) 0 0
\(979\) −663.670 −0.0216660
\(980\) −3019.55 −0.0984244
\(981\) 0 0
\(982\) 30748.1 0.999197
\(983\) 56956.1 1.84803 0.924016 0.382353i \(-0.124886\pi\)
0.924016 + 0.382353i \(0.124886\pi\)
\(984\) 0 0
\(985\) 878.693 0.0284238
\(986\) 5433.33 0.175489
\(987\) 0 0
\(988\) 8134.80 0.261946
\(989\) −21908.6 −0.704402
\(990\) 0 0
\(991\) −45020.8 −1.44312 −0.721561 0.692351i \(-0.756575\pi\)
−0.721561 + 0.692351i \(0.756575\pi\)
\(992\) −11736.2 −0.375628
\(993\) 0 0
\(994\) 11772.1 0.375641
\(995\) 26643.3 0.848893
\(996\) 0 0
\(997\) −51620.5 −1.63976 −0.819878 0.572538i \(-0.805959\pi\)
−0.819878 + 0.572538i \(0.805959\pi\)
\(998\) −1153.95 −0.0366010
\(999\) 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 495.4.a.j.1.1 yes 3
3.2 odd 2 495.4.a.h.1.3 3
5.4 even 2 2475.4.a.u.1.3 3
15.14 odd 2 2475.4.a.x.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.4.a.h.1.3 3 3.2 odd 2
495.4.a.j.1.1 yes 3 1.1 even 1 trivial
2475.4.a.u.1.3 3 5.4 even 2
2475.4.a.x.1.1 3 15.14 odd 2