Properties

Label 1617.4.a.be.1.12
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.95461\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.95461 q^{2} -3.00000 q^{3} +0.729713 q^{4} -19.4605 q^{5} -8.86383 q^{6} -21.4809 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.95461 q^{2} -3.00000 q^{3} +0.729713 q^{4} -19.4605 q^{5} -8.86383 q^{6} -21.4809 q^{8} +9.00000 q^{9} -57.4982 q^{10} +11.0000 q^{11} -2.18914 q^{12} +10.0839 q^{13} +58.3816 q^{15} -69.3052 q^{16} +58.8446 q^{17} +26.5915 q^{18} +80.1194 q^{19} -14.2006 q^{20} +32.5007 q^{22} +175.686 q^{23} +64.4426 q^{24} +253.712 q^{25} +29.7941 q^{26} -27.0000 q^{27} +106.905 q^{29} +172.495 q^{30} -292.488 q^{31} -32.9230 q^{32} -33.0000 q^{33} +173.863 q^{34} +6.56742 q^{36} -108.160 q^{37} +236.722 q^{38} -30.2518 q^{39} +418.029 q^{40} -509.446 q^{41} +526.547 q^{43} +8.02684 q^{44} -175.145 q^{45} +519.085 q^{46} +19.3449 q^{47} +207.916 q^{48} +749.620 q^{50} -176.534 q^{51} +7.35838 q^{52} +94.1301 q^{53} -79.7744 q^{54} -214.066 q^{55} -240.358 q^{57} +315.861 q^{58} -497.679 q^{59} +42.6018 q^{60} -470.144 q^{61} -864.188 q^{62} +457.167 q^{64} -196.239 q^{65} -97.5021 q^{66} -555.872 q^{67} +42.9397 q^{68} -527.059 q^{69} -485.502 q^{71} -193.328 q^{72} +159.914 q^{73} -319.571 q^{74} -761.137 q^{75} +58.4642 q^{76} -89.3823 q^{78} -678.814 q^{79} +1348.72 q^{80} +81.0000 q^{81} -1505.21 q^{82} -133.458 q^{83} -1145.15 q^{85} +1555.74 q^{86} -320.714 q^{87} -236.289 q^{88} +1545.32 q^{89} -517.484 q^{90} +128.201 q^{92} +877.465 q^{93} +57.1565 q^{94} -1559.17 q^{95} +98.7690 q^{96} +254.234 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.95461 1.04461 0.522306 0.852758i \(-0.325072\pi\)
0.522306 + 0.852758i \(0.325072\pi\)
\(3\) −3.00000 −0.577350
\(4\) 0.729713 0.0912141
\(5\) −19.4605 −1.74060 −0.870301 0.492520i \(-0.836076\pi\)
−0.870301 + 0.492520i \(0.836076\pi\)
\(6\) −8.86383 −0.603107
\(7\) 0 0
\(8\) −21.4809 −0.949329
\(9\) 9.00000 0.333333
\(10\) −57.4982 −1.81825
\(11\) 11.0000 0.301511
\(12\) −2.18914 −0.0526625
\(13\) 10.0839 0.215137 0.107569 0.994198i \(-0.465693\pi\)
0.107569 + 0.994198i \(0.465693\pi\)
\(14\) 0 0
\(15\) 58.3816 1.00494
\(16\) −69.3052 −1.08289
\(17\) 58.8446 0.839524 0.419762 0.907634i \(-0.362114\pi\)
0.419762 + 0.907634i \(0.362114\pi\)
\(18\) 26.5915 0.348204
\(19\) 80.1194 0.967403 0.483702 0.875233i \(-0.339292\pi\)
0.483702 + 0.875233i \(0.339292\pi\)
\(20\) −14.2006 −0.158768
\(21\) 0 0
\(22\) 32.5007 0.314962
\(23\) 175.686 1.59275 0.796373 0.604806i \(-0.206749\pi\)
0.796373 + 0.604806i \(0.206749\pi\)
\(24\) 64.4426 0.548095
\(25\) 253.712 2.02970
\(26\) 29.7941 0.224735
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 106.905 0.684541 0.342270 0.939602i \(-0.388804\pi\)
0.342270 + 0.939602i \(0.388804\pi\)
\(30\) 172.495 1.04977
\(31\) −292.488 −1.69459 −0.847297 0.531119i \(-0.821772\pi\)
−0.847297 + 0.531119i \(0.821772\pi\)
\(32\) −32.9230 −0.181876
\(33\) −33.0000 −0.174078
\(34\) 173.863 0.876977
\(35\) 0 0
\(36\) 6.56742 0.0304047
\(37\) −108.160 −0.480578 −0.240289 0.970701i \(-0.577242\pi\)
−0.240289 + 0.970701i \(0.577242\pi\)
\(38\) 236.722 1.01056
\(39\) −30.2518 −0.124209
\(40\) 418.029 1.65240
\(41\) −509.446 −1.94054 −0.970270 0.242024i \(-0.922189\pi\)
−0.970270 + 0.242024i \(0.922189\pi\)
\(42\) 0 0
\(43\) 526.547 1.86739 0.933693 0.358074i \(-0.116566\pi\)
0.933693 + 0.358074i \(0.116566\pi\)
\(44\) 8.02684 0.0275021
\(45\) −175.145 −0.580201
\(46\) 519.085 1.66380
\(47\) 19.3449 0.0600370 0.0300185 0.999549i \(-0.490443\pi\)
0.0300185 + 0.999549i \(0.490443\pi\)
\(48\) 207.916 0.625209
\(49\) 0 0
\(50\) 749.620 2.12025
\(51\) −176.534 −0.484700
\(52\) 7.35838 0.0196235
\(53\) 94.1301 0.243958 0.121979 0.992533i \(-0.461076\pi\)
0.121979 + 0.992533i \(0.461076\pi\)
\(54\) −79.7744 −0.201036
\(55\) −214.066 −0.524811
\(56\) 0 0
\(57\) −240.358 −0.558530
\(58\) 315.861 0.715079
\(59\) −497.679 −1.09818 −0.549088 0.835765i \(-0.685025\pi\)
−0.549088 + 0.835765i \(0.685025\pi\)
\(60\) 42.6018 0.0916645
\(61\) −470.144 −0.986816 −0.493408 0.869798i \(-0.664249\pi\)
−0.493408 + 0.869798i \(0.664249\pi\)
\(62\) −864.188 −1.77019
\(63\) 0 0
\(64\) 457.167 0.892905
\(65\) −196.239 −0.374468
\(66\) −97.5021 −0.181844
\(67\) −555.872 −1.01359 −0.506795 0.862067i \(-0.669170\pi\)
−0.506795 + 0.862067i \(0.669170\pi\)
\(68\) 42.9397 0.0765765
\(69\) −527.059 −0.919572
\(70\) 0 0
\(71\) −485.502 −0.811527 −0.405764 0.913978i \(-0.632994\pi\)
−0.405764 + 0.913978i \(0.632994\pi\)
\(72\) −193.328 −0.316443
\(73\) 159.914 0.256391 0.128196 0.991749i \(-0.459081\pi\)
0.128196 + 0.991749i \(0.459081\pi\)
\(74\) −319.571 −0.502018
\(75\) −761.137 −1.17185
\(76\) 58.4642 0.0882408
\(77\) 0 0
\(78\) −89.3823 −0.129751
\(79\) −678.814 −0.966741 −0.483370 0.875416i \(-0.660588\pi\)
−0.483370 + 0.875416i \(0.660588\pi\)
\(80\) 1348.72 1.88489
\(81\) 81.0000 0.111111
\(82\) −1505.21 −2.02711
\(83\) −133.458 −0.176493 −0.0882467 0.996099i \(-0.528126\pi\)
−0.0882467 + 0.996099i \(0.528126\pi\)
\(84\) 0 0
\(85\) −1145.15 −1.46128
\(86\) 1555.74 1.95069
\(87\) −320.714 −0.395220
\(88\) −236.289 −0.286233
\(89\) 1545.32 1.84049 0.920244 0.391346i \(-0.127990\pi\)
0.920244 + 0.391346i \(0.127990\pi\)
\(90\) −517.484 −0.606085
\(91\) 0 0
\(92\) 128.201 0.145281
\(93\) 877.465 0.978375
\(94\) 57.1565 0.0627154
\(95\) −1559.17 −1.68386
\(96\) 98.7690 0.105006
\(97\) 254.234 0.266119 0.133059 0.991108i \(-0.457520\pi\)
0.133059 + 0.991108i \(0.457520\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 185.137 0.185137
\(101\) −1013.25 −0.998237 −0.499119 0.866534i \(-0.666343\pi\)
−0.499119 + 0.866534i \(0.666343\pi\)
\(102\) −521.588 −0.506323
\(103\) 1164.77 1.11426 0.557129 0.830426i \(-0.311903\pi\)
0.557129 + 0.830426i \(0.311903\pi\)
\(104\) −216.612 −0.204236
\(105\) 0 0
\(106\) 278.118 0.254841
\(107\) 669.727 0.605093 0.302546 0.953135i \(-0.402163\pi\)
0.302546 + 0.953135i \(0.402163\pi\)
\(108\) −19.7023 −0.0175542
\(109\) 281.310 0.247198 0.123599 0.992332i \(-0.460556\pi\)
0.123599 + 0.992332i \(0.460556\pi\)
\(110\) −632.481 −0.548224
\(111\) 324.480 0.277462
\(112\) 0 0
\(113\) 655.577 0.545766 0.272883 0.962047i \(-0.412023\pi\)
0.272883 + 0.962047i \(0.412023\pi\)
\(114\) −710.165 −0.583448
\(115\) −3418.95 −2.77234
\(116\) 78.0096 0.0624398
\(117\) 90.7555 0.0717123
\(118\) −1470.45 −1.14717
\(119\) 0 0
\(120\) −1254.09 −0.954016
\(121\) 121.000 0.0909091
\(122\) −1389.09 −1.03084
\(123\) 1528.34 1.12037
\(124\) −213.432 −0.154571
\(125\) −2504.81 −1.79229
\(126\) 0 0
\(127\) 982.234 0.686293 0.343146 0.939282i \(-0.388507\pi\)
0.343146 + 0.939282i \(0.388507\pi\)
\(128\) 1614.13 1.11461
\(129\) −1579.64 −1.07814
\(130\) −579.809 −0.391174
\(131\) 639.961 0.426822 0.213411 0.976963i \(-0.431543\pi\)
0.213411 + 0.976963i \(0.431543\pi\)
\(132\) −24.0805 −0.0158783
\(133\) 0 0
\(134\) −1642.38 −1.05881
\(135\) 525.434 0.334979
\(136\) −1264.03 −0.796984
\(137\) −1624.90 −1.01332 −0.506659 0.862146i \(-0.669120\pi\)
−0.506659 + 0.862146i \(0.669120\pi\)
\(138\) −1557.25 −0.960596
\(139\) 2111.17 1.28825 0.644127 0.764918i \(-0.277221\pi\)
0.644127 + 0.764918i \(0.277221\pi\)
\(140\) 0 0
\(141\) −58.0346 −0.0346624
\(142\) −1434.47 −0.847731
\(143\) 110.923 0.0648663
\(144\) −623.747 −0.360965
\(145\) −2080.42 −1.19151
\(146\) 472.485 0.267829
\(147\) 0 0
\(148\) −78.9258 −0.0438355
\(149\) −772.509 −0.424741 −0.212371 0.977189i \(-0.568118\pi\)
−0.212371 + 0.977189i \(0.568118\pi\)
\(150\) −2248.86 −1.22412
\(151\) −2706.87 −1.45882 −0.729410 0.684077i \(-0.760205\pi\)
−0.729410 + 0.684077i \(0.760205\pi\)
\(152\) −1721.03 −0.918383
\(153\) 529.601 0.279841
\(154\) 0 0
\(155\) 5691.98 2.94962
\(156\) −22.0751 −0.0113297
\(157\) −3272.93 −1.66374 −0.831872 0.554967i \(-0.812731\pi\)
−0.831872 + 0.554967i \(0.812731\pi\)
\(158\) −2005.63 −1.00987
\(159\) −282.390 −0.140849
\(160\) 640.699 0.316573
\(161\) 0 0
\(162\) 239.323 0.116068
\(163\) 2630.65 1.26410 0.632051 0.774927i \(-0.282213\pi\)
0.632051 + 0.774927i \(0.282213\pi\)
\(164\) −371.750 −0.177005
\(165\) 642.197 0.303000
\(166\) −394.317 −0.184367
\(167\) 3370.14 1.56161 0.780805 0.624775i \(-0.214809\pi\)
0.780805 + 0.624775i \(0.214809\pi\)
\(168\) 0 0
\(169\) −2095.31 −0.953716
\(170\) −3383.46 −1.52647
\(171\) 721.075 0.322468
\(172\) 384.228 0.170332
\(173\) 2708.22 1.19018 0.595092 0.803657i \(-0.297116\pi\)
0.595092 + 0.803657i \(0.297116\pi\)
\(174\) −947.583 −0.412851
\(175\) 0 0
\(176\) −762.357 −0.326505
\(177\) 1493.04 0.634032
\(178\) 4565.81 1.92259
\(179\) −2793.98 −1.16666 −0.583329 0.812236i \(-0.698250\pi\)
−0.583329 + 0.812236i \(0.698250\pi\)
\(180\) −127.805 −0.0529225
\(181\) −3762.00 −1.54490 −0.772452 0.635074i \(-0.780970\pi\)
−0.772452 + 0.635074i \(0.780970\pi\)
\(182\) 0 0
\(183\) 1410.43 0.569739
\(184\) −3773.89 −1.51204
\(185\) 2104.85 0.836496
\(186\) 2592.57 1.02202
\(187\) 647.291 0.253126
\(188\) 14.1162 0.00547622
\(189\) 0 0
\(190\) −4606.73 −1.75898
\(191\) 474.111 0.179610 0.0898048 0.995959i \(-0.471376\pi\)
0.0898048 + 0.995959i \(0.471376\pi\)
\(192\) −1371.50 −0.515519
\(193\) −2840.87 −1.05954 −0.529768 0.848143i \(-0.677721\pi\)
−0.529768 + 0.848143i \(0.677721\pi\)
\(194\) 751.161 0.277991
\(195\) 588.716 0.216199
\(196\) 0 0
\(197\) −768.127 −0.277801 −0.138900 0.990306i \(-0.544357\pi\)
−0.138900 + 0.990306i \(0.544357\pi\)
\(198\) 292.506 0.104987
\(199\) −4292.00 −1.52890 −0.764452 0.644681i \(-0.776990\pi\)
−0.764452 + 0.644681i \(0.776990\pi\)
\(200\) −5449.95 −1.92685
\(201\) 1667.62 0.585197
\(202\) −2993.75 −1.04277
\(203\) 0 0
\(204\) −128.819 −0.0442114
\(205\) 9914.10 3.37771
\(206\) 3441.45 1.16397
\(207\) 1581.18 0.530915
\(208\) −698.870 −0.232971
\(209\) 881.314 0.291683
\(210\) 0 0
\(211\) −2240.91 −0.731140 −0.365570 0.930784i \(-0.619126\pi\)
−0.365570 + 0.930784i \(0.619126\pi\)
\(212\) 68.6880 0.0222524
\(213\) 1456.50 0.468535
\(214\) 1978.78 0.632087
\(215\) −10246.9 −3.25038
\(216\) 579.983 0.182698
\(217\) 0 0
\(218\) 831.160 0.258226
\(219\) −479.743 −0.148028
\(220\) −156.207 −0.0478702
\(221\) 593.385 0.180613
\(222\) 958.712 0.289840
\(223\) −1897.49 −0.569799 −0.284900 0.958557i \(-0.591960\pi\)
−0.284900 + 0.958557i \(0.591960\pi\)
\(224\) 0 0
\(225\) 2283.41 0.676566
\(226\) 1936.97 0.570113
\(227\) 6783.09 1.98330 0.991651 0.128953i \(-0.0411615\pi\)
0.991651 + 0.128953i \(0.0411615\pi\)
\(228\) −175.393 −0.0509459
\(229\) −4021.82 −1.16057 −0.580283 0.814415i \(-0.697058\pi\)
−0.580283 + 0.814415i \(0.697058\pi\)
\(230\) −10101.7 −2.89602
\(231\) 0 0
\(232\) −2296.40 −0.649854
\(233\) −881.553 −0.247865 −0.123932 0.992291i \(-0.539551\pi\)
−0.123932 + 0.992291i \(0.539551\pi\)
\(234\) 268.147 0.0749116
\(235\) −376.461 −0.104501
\(236\) −363.163 −0.100169
\(237\) 2036.44 0.558148
\(238\) 0 0
\(239\) −4509.47 −1.22048 −0.610238 0.792218i \(-0.708926\pi\)
−0.610238 + 0.792218i \(0.708926\pi\)
\(240\) −4046.15 −1.08824
\(241\) −952.815 −0.254673 −0.127336 0.991860i \(-0.540643\pi\)
−0.127336 + 0.991860i \(0.540643\pi\)
\(242\) 357.508 0.0949647
\(243\) −243.000 −0.0641500
\(244\) −343.070 −0.0900116
\(245\) 0 0
\(246\) 4515.64 1.17035
\(247\) 807.919 0.208124
\(248\) 6282.90 1.60873
\(249\) 400.375 0.101898
\(250\) −7400.72 −1.87225
\(251\) −2509.30 −0.631018 −0.315509 0.948923i \(-0.602175\pi\)
−0.315509 + 0.948923i \(0.602175\pi\)
\(252\) 0 0
\(253\) 1932.55 0.480231
\(254\) 2902.12 0.716910
\(255\) 3435.44 0.843669
\(256\) 1111.80 0.271435
\(257\) 620.282 0.150553 0.0752766 0.997163i \(-0.476016\pi\)
0.0752766 + 0.997163i \(0.476016\pi\)
\(258\) −4667.22 −1.12623
\(259\) 0 0
\(260\) −143.198 −0.0341568
\(261\) 962.141 0.228180
\(262\) 1890.83 0.445863
\(263\) −3556.97 −0.833963 −0.416981 0.908915i \(-0.636912\pi\)
−0.416981 + 0.908915i \(0.636912\pi\)
\(264\) 708.868 0.165257
\(265\) −1831.82 −0.424634
\(266\) 0 0
\(267\) −4635.95 −1.06261
\(268\) −405.627 −0.0924537
\(269\) 4603.67 1.04346 0.521730 0.853111i \(-0.325287\pi\)
0.521730 + 0.853111i \(0.325287\pi\)
\(270\) 1552.45 0.349923
\(271\) −2439.76 −0.546882 −0.273441 0.961889i \(-0.588162\pi\)
−0.273441 + 0.961889i \(0.588162\pi\)
\(272\) −4078.24 −0.909116
\(273\) 0 0
\(274\) −4800.94 −1.05852
\(275\) 2790.83 0.611977
\(276\) −384.602 −0.0838780
\(277\) 2407.35 0.522180 0.261090 0.965314i \(-0.415918\pi\)
0.261090 + 0.965314i \(0.415918\pi\)
\(278\) 6237.69 1.34573
\(279\) −2632.39 −0.564865
\(280\) 0 0
\(281\) −32.7749 −0.00695795 −0.00347898 0.999994i \(-0.501107\pi\)
−0.00347898 + 0.999994i \(0.501107\pi\)
\(282\) −171.470 −0.0362087
\(283\) −1681.58 −0.353214 −0.176607 0.984281i \(-0.556512\pi\)
−0.176607 + 0.984281i \(0.556512\pi\)
\(284\) −354.277 −0.0740228
\(285\) 4677.50 0.972179
\(286\) 327.735 0.0677601
\(287\) 0 0
\(288\) −296.307 −0.0606252
\(289\) −1450.31 −0.295199
\(290\) −6146.82 −1.24467
\(291\) −762.701 −0.153644
\(292\) 116.692 0.0233865
\(293\) −636.150 −0.126841 −0.0634203 0.997987i \(-0.520201\pi\)
−0.0634203 + 0.997987i \(0.520201\pi\)
\(294\) 0 0
\(295\) 9685.10 1.91149
\(296\) 2323.37 0.456227
\(297\) −297.000 −0.0580259
\(298\) −2282.46 −0.443690
\(299\) 1771.61 0.342659
\(300\) −555.411 −0.106889
\(301\) 0 0
\(302\) −7997.73 −1.52390
\(303\) 3039.74 0.576332
\(304\) −5552.69 −1.04760
\(305\) 9149.26 1.71766
\(306\) 1564.76 0.292326
\(307\) −6592.58 −1.22560 −0.612799 0.790239i \(-0.709956\pi\)
−0.612799 + 0.790239i \(0.709956\pi\)
\(308\) 0 0
\(309\) −3494.32 −0.643317
\(310\) 16817.6 3.08120
\(311\) 1462.01 0.266569 0.133284 0.991078i \(-0.457448\pi\)
0.133284 + 0.991078i \(0.457448\pi\)
\(312\) 649.835 0.117916
\(313\) 2498.36 0.451168 0.225584 0.974224i \(-0.427571\pi\)
0.225584 + 0.974224i \(0.427571\pi\)
\(314\) −9670.22 −1.73797
\(315\) 0 0
\(316\) −495.339 −0.0881804
\(317\) 658.803 0.116726 0.0583628 0.998295i \(-0.481412\pi\)
0.0583628 + 0.998295i \(0.481412\pi\)
\(318\) −834.353 −0.147133
\(319\) 1175.95 0.206397
\(320\) −8896.72 −1.55419
\(321\) −2009.18 −0.349350
\(322\) 0 0
\(323\) 4714.60 0.812158
\(324\) 59.1068 0.0101349
\(325\) 2558.42 0.436663
\(326\) 7772.55 1.32050
\(327\) −843.929 −0.142720
\(328\) 10943.3 1.84221
\(329\) 0 0
\(330\) 1897.44 0.316517
\(331\) 6590.47 1.09440 0.547198 0.837003i \(-0.315695\pi\)
0.547198 + 0.837003i \(0.315695\pi\)
\(332\) −97.3862 −0.0160987
\(333\) −973.440 −0.160193
\(334\) 9957.43 1.63128
\(335\) 10817.6 1.76426
\(336\) 0 0
\(337\) 3934.01 0.635902 0.317951 0.948107i \(-0.397005\pi\)
0.317951 + 0.948107i \(0.397005\pi\)
\(338\) −6190.83 −0.996263
\(339\) −1966.73 −0.315098
\(340\) −835.629 −0.133289
\(341\) −3217.37 −0.510940
\(342\) 2130.49 0.336854
\(343\) 0 0
\(344\) −11310.7 −1.77276
\(345\) 10256.9 1.60061
\(346\) 8001.72 1.24328
\(347\) −2745.98 −0.424819 −0.212409 0.977181i \(-0.568131\pi\)
−0.212409 + 0.977181i \(0.568131\pi\)
\(348\) −234.029 −0.0360496
\(349\) 1552.59 0.238133 0.119066 0.992886i \(-0.462010\pi\)
0.119066 + 0.992886i \(0.462010\pi\)
\(350\) 0 0
\(351\) −272.266 −0.0414031
\(352\) −362.153 −0.0548375
\(353\) −12079.4 −1.82131 −0.910655 0.413167i \(-0.864423\pi\)
−0.910655 + 0.413167i \(0.864423\pi\)
\(354\) 4411.34 0.662317
\(355\) 9448.12 1.41255
\(356\) 1127.64 0.167878
\(357\) 0 0
\(358\) −8255.12 −1.21871
\(359\) 51.5378 0.00757677 0.00378838 0.999993i \(-0.498794\pi\)
0.00378838 + 0.999993i \(0.498794\pi\)
\(360\) 3762.26 0.550801
\(361\) −439.877 −0.0641314
\(362\) −11115.2 −1.61382
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −3112.02 −0.446275
\(366\) 4167.28 0.595156
\(367\) 8662.75 1.23213 0.616065 0.787695i \(-0.288726\pi\)
0.616065 + 0.787695i \(0.288726\pi\)
\(368\) −12176.0 −1.72477
\(369\) −4585.02 −0.646847
\(370\) 6219.01 0.873814
\(371\) 0 0
\(372\) 640.297 0.0892416
\(373\) −11710.6 −1.62561 −0.812804 0.582537i \(-0.802060\pi\)
−0.812804 + 0.582537i \(0.802060\pi\)
\(374\) 1912.49 0.264419
\(375\) 7514.42 1.03478
\(376\) −415.544 −0.0569949
\(377\) 1078.02 0.147270
\(378\) 0 0
\(379\) −1556.07 −0.210897 −0.105448 0.994425i \(-0.533628\pi\)
−0.105448 + 0.994425i \(0.533628\pi\)
\(380\) −1137.74 −0.153592
\(381\) −2946.70 −0.396231
\(382\) 1400.81 0.187622
\(383\) −8840.51 −1.17945 −0.589725 0.807604i \(-0.700764\pi\)
−0.589725 + 0.807604i \(0.700764\pi\)
\(384\) −4842.40 −0.643523
\(385\) 0 0
\(386\) −8393.66 −1.10680
\(387\) 4738.92 0.622462
\(388\) 185.518 0.0242738
\(389\) −2865.47 −0.373484 −0.186742 0.982409i \(-0.559793\pi\)
−0.186742 + 0.982409i \(0.559793\pi\)
\(390\) 1739.43 0.225844
\(391\) 10338.2 1.33715
\(392\) 0 0
\(393\) −1919.88 −0.246426
\(394\) −2269.51 −0.290194
\(395\) 13210.1 1.68271
\(396\) 72.2416 0.00916736
\(397\) 7962.49 1.00661 0.503307 0.864108i \(-0.332116\pi\)
0.503307 + 0.864108i \(0.332116\pi\)
\(398\) −12681.2 −1.59711
\(399\) 0 0
\(400\) −17583.6 −2.19795
\(401\) −9038.91 −1.12564 −0.562820 0.826579i \(-0.690284\pi\)
−0.562820 + 0.826579i \(0.690284\pi\)
\(402\) 4927.15 0.611303
\(403\) −2949.43 −0.364570
\(404\) −739.380 −0.0910533
\(405\) −1576.30 −0.193400
\(406\) 0 0
\(407\) −1189.76 −0.144900
\(408\) 3792.10 0.460139
\(409\) −4996.92 −0.604112 −0.302056 0.953290i \(-0.597673\pi\)
−0.302056 + 0.953290i \(0.597673\pi\)
\(410\) 29292.3 3.52840
\(411\) 4874.70 0.585040
\(412\) 849.951 0.101636
\(413\) 0 0
\(414\) 4671.76 0.554600
\(415\) 2597.17 0.307205
\(416\) −331.993 −0.0391282
\(417\) −6333.52 −0.743774
\(418\) 2603.94 0.304696
\(419\) −7088.86 −0.826524 −0.413262 0.910612i \(-0.635611\pi\)
−0.413262 + 0.910612i \(0.635611\pi\)
\(420\) 0 0
\(421\) −9865.96 −1.14213 −0.571066 0.820904i \(-0.693470\pi\)
−0.571066 + 0.820904i \(0.693470\pi\)
\(422\) −6621.01 −0.763757
\(423\) 174.104 0.0200123
\(424\) −2022.00 −0.231596
\(425\) 14929.6 1.70398
\(426\) 4303.40 0.489438
\(427\) 0 0
\(428\) 488.708 0.0551930
\(429\) −332.770 −0.0374506
\(430\) −30275.5 −3.39538
\(431\) 1580.42 0.176626 0.0883132 0.996093i \(-0.471852\pi\)
0.0883132 + 0.996093i \(0.471852\pi\)
\(432\) 1871.24 0.208403
\(433\) −3564.55 −0.395615 −0.197808 0.980241i \(-0.563382\pi\)
−0.197808 + 0.980241i \(0.563382\pi\)
\(434\) 0 0
\(435\) 6241.26 0.687920
\(436\) 205.275 0.0225479
\(437\) 14075.9 1.54083
\(438\) −1417.45 −0.154631
\(439\) −10676.4 −1.16073 −0.580363 0.814358i \(-0.697089\pi\)
−0.580363 + 0.814358i \(0.697089\pi\)
\(440\) 4598.32 0.498218
\(441\) 0 0
\(442\) 1753.22 0.188670
\(443\) 8169.88 0.876214 0.438107 0.898923i \(-0.355649\pi\)
0.438107 + 0.898923i \(0.355649\pi\)
\(444\) 236.777 0.0253085
\(445\) −30072.7 −3.20356
\(446\) −5606.34 −0.595219
\(447\) 2317.53 0.245224
\(448\) 0 0
\(449\) 15082.1 1.58523 0.792615 0.609723i \(-0.208719\pi\)
0.792615 + 0.609723i \(0.208719\pi\)
\(450\) 6746.58 0.706749
\(451\) −5603.91 −0.585095
\(452\) 478.383 0.0497816
\(453\) 8120.60 0.842250
\(454\) 20041.4 2.07178
\(455\) 0 0
\(456\) 5163.10 0.530229
\(457\) −5615.54 −0.574800 −0.287400 0.957811i \(-0.592791\pi\)
−0.287400 + 0.957811i \(0.592791\pi\)
\(458\) −11882.9 −1.21234
\(459\) −1588.80 −0.161567
\(460\) −2494.85 −0.252876
\(461\) 6677.03 0.674578 0.337289 0.941401i \(-0.390490\pi\)
0.337289 + 0.941401i \(0.390490\pi\)
\(462\) 0 0
\(463\) 2242.67 0.225109 0.112555 0.993646i \(-0.464097\pi\)
0.112555 + 0.993646i \(0.464097\pi\)
\(464\) −7409.04 −0.741285
\(465\) −17075.9 −1.70296
\(466\) −2604.64 −0.258922
\(467\) −9036.35 −0.895401 −0.447700 0.894184i \(-0.647757\pi\)
−0.447700 + 0.894184i \(0.647757\pi\)
\(468\) 66.2254 0.00654118
\(469\) 0 0
\(470\) −1112.30 −0.109163
\(471\) 9818.78 0.960563
\(472\) 10690.6 1.04253
\(473\) 5792.01 0.563038
\(474\) 6016.89 0.583048
\(475\) 20327.3 1.96354
\(476\) 0 0
\(477\) 847.171 0.0813193
\(478\) −13323.7 −1.27492
\(479\) 20084.5 1.91584 0.957918 0.287043i \(-0.0926723\pi\)
0.957918 + 0.287043i \(0.0926723\pi\)
\(480\) −1922.10 −0.182774
\(481\) −1090.68 −0.103390
\(482\) −2815.19 −0.266034
\(483\) 0 0
\(484\) 88.2953 0.00829219
\(485\) −4947.52 −0.463207
\(486\) −717.970 −0.0670119
\(487\) −13607.8 −1.26617 −0.633087 0.774081i \(-0.718212\pi\)
−0.633087 + 0.774081i \(0.718212\pi\)
\(488\) 10099.1 0.936813
\(489\) −7891.96 −0.729829
\(490\) 0 0
\(491\) −18422.9 −1.69331 −0.846654 0.532143i \(-0.821387\pi\)
−0.846654 + 0.532143i \(0.821387\pi\)
\(492\) 1115.25 0.102194
\(493\) 6290.76 0.574688
\(494\) 2387.09 0.217409
\(495\) −1926.59 −0.174937
\(496\) 20271.0 1.83507
\(497\) 0 0
\(498\) 1182.95 0.106444
\(499\) 20971.8 1.88142 0.940710 0.339212i \(-0.110160\pi\)
0.940710 + 0.339212i \(0.110160\pi\)
\(500\) −1827.79 −0.163483
\(501\) −10110.4 −0.901596
\(502\) −7413.99 −0.659169
\(503\) −12613.8 −1.11813 −0.559067 0.829122i \(-0.688841\pi\)
−0.559067 + 0.829122i \(0.688841\pi\)
\(504\) 0 0
\(505\) 19718.3 1.73753
\(506\) 5709.93 0.501655
\(507\) 6285.94 0.550628
\(508\) 716.749 0.0625996
\(509\) −15094.2 −1.31442 −0.657211 0.753707i \(-0.728264\pi\)
−0.657211 + 0.753707i \(0.728264\pi\)
\(510\) 10150.4 0.881307
\(511\) 0 0
\(512\) −9628.15 −0.831070
\(513\) −2163.22 −0.186177
\(514\) 1832.69 0.157270
\(515\) −22667.1 −1.93948
\(516\) −1152.68 −0.0983412
\(517\) 212.794 0.0181018
\(518\) 0 0
\(519\) −8124.65 −0.687153
\(520\) 4215.38 0.355493
\(521\) 10229.4 0.860185 0.430093 0.902785i \(-0.358481\pi\)
0.430093 + 0.902785i \(0.358481\pi\)
\(522\) 2842.75 0.238360
\(523\) 6047.72 0.505637 0.252819 0.967514i \(-0.418642\pi\)
0.252819 + 0.967514i \(0.418642\pi\)
\(524\) 466.988 0.0389322
\(525\) 0 0
\(526\) −10509.5 −0.871167
\(527\) −17211.4 −1.42265
\(528\) 2287.07 0.188508
\(529\) 18698.7 1.53684
\(530\) −5412.32 −0.443577
\(531\) −4479.11 −0.366058
\(532\) 0 0
\(533\) −5137.23 −0.417482
\(534\) −13697.4 −1.11001
\(535\) −13033.2 −1.05323
\(536\) 11940.6 0.962230
\(537\) 8381.94 0.673571
\(538\) 13602.0 1.09001
\(539\) 0 0
\(540\) 383.416 0.0305548
\(541\) −22904.9 −1.82026 −0.910128 0.414328i \(-0.864017\pi\)
−0.910128 + 0.414328i \(0.864017\pi\)
\(542\) −7208.55 −0.571280
\(543\) 11286.0 0.891950
\(544\) −1937.34 −0.152689
\(545\) −5474.43 −0.430273
\(546\) 0 0
\(547\) −1927.97 −0.150702 −0.0753511 0.997157i \(-0.524008\pi\)
−0.0753511 + 0.997157i \(0.524008\pi\)
\(548\) −1185.71 −0.0924289
\(549\) −4231.30 −0.328939
\(550\) 8245.82 0.639278
\(551\) 8565.13 0.662227
\(552\) 11321.7 0.872976
\(553\) 0 0
\(554\) 7112.79 0.545475
\(555\) −6314.55 −0.482951
\(556\) 1540.55 0.117507
\(557\) 19140.5 1.45603 0.728016 0.685561i \(-0.240443\pi\)
0.728016 + 0.685561i \(0.240443\pi\)
\(558\) −7777.70 −0.590065
\(559\) 5309.66 0.401744
\(560\) 0 0
\(561\) −1941.87 −0.146142
\(562\) −96.8369 −0.00726836
\(563\) −19312.6 −1.44570 −0.722850 0.691005i \(-0.757168\pi\)
−0.722850 + 0.691005i \(0.757168\pi\)
\(564\) −42.3486 −0.00316170
\(565\) −12757.9 −0.949961
\(566\) −4968.41 −0.368971
\(567\) 0 0
\(568\) 10429.0 0.770406
\(569\) −17541.6 −1.29241 −0.646206 0.763163i \(-0.723646\pi\)
−0.646206 + 0.763163i \(0.723646\pi\)
\(570\) 13820.2 1.01555
\(571\) −2208.44 −0.161857 −0.0809285 0.996720i \(-0.525789\pi\)
−0.0809285 + 0.996720i \(0.525789\pi\)
\(572\) 80.9422 0.00591672
\(573\) −1422.33 −0.103698
\(574\) 0 0
\(575\) 44573.8 3.23279
\(576\) 4114.51 0.297635
\(577\) −15015.3 −1.08335 −0.541677 0.840586i \(-0.682211\pi\)
−0.541677 + 0.840586i \(0.682211\pi\)
\(578\) −4285.11 −0.308369
\(579\) 8522.61 0.611723
\(580\) −1518.11 −0.108683
\(581\) 0 0
\(582\) −2253.48 −0.160498
\(583\) 1035.43 0.0735561
\(584\) −3435.10 −0.243400
\(585\) −1766.15 −0.124823
\(586\) −1879.57 −0.132499
\(587\) −466.061 −0.0327707 −0.0163854 0.999866i \(-0.505216\pi\)
−0.0163854 + 0.999866i \(0.505216\pi\)
\(588\) 0 0
\(589\) −23434.0 −1.63936
\(590\) 28615.7 1.99676
\(591\) 2304.38 0.160388
\(592\) 7496.05 0.520415
\(593\) 23767.5 1.64589 0.822945 0.568120i \(-0.192329\pi\)
0.822945 + 0.568120i \(0.192329\pi\)
\(594\) −877.519 −0.0606145
\(595\) 0 0
\(596\) −563.710 −0.0387424
\(597\) 12876.0 0.882713
\(598\) 5234.42 0.357945
\(599\) 19513.0 1.33102 0.665510 0.746389i \(-0.268214\pi\)
0.665510 + 0.746389i \(0.268214\pi\)
\(600\) 16349.9 1.11247
\(601\) −2710.53 −0.183968 −0.0919840 0.995760i \(-0.529321\pi\)
−0.0919840 + 0.995760i \(0.529321\pi\)
\(602\) 0 0
\(603\) −5002.85 −0.337863
\(604\) −1975.24 −0.133065
\(605\) −2354.72 −0.158237
\(606\) 8981.25 0.602044
\(607\) −251.382 −0.0168094 −0.00840468 0.999965i \(-0.502675\pi\)
−0.00840468 + 0.999965i \(0.502675\pi\)
\(608\) −2637.77 −0.175947
\(609\) 0 0
\(610\) 27032.5 1.79428
\(611\) 195.073 0.0129162
\(612\) 386.457 0.0255255
\(613\) −16798.6 −1.10683 −0.553416 0.832905i \(-0.686676\pi\)
−0.553416 + 0.832905i \(0.686676\pi\)
\(614\) −19478.5 −1.28027
\(615\) −29742.3 −1.95012
\(616\) 0 0
\(617\) 6179.81 0.403225 0.201613 0.979465i \(-0.435382\pi\)
0.201613 + 0.979465i \(0.435382\pi\)
\(618\) −10324.4 −0.672017
\(619\) 12197.9 0.792045 0.396023 0.918241i \(-0.370390\pi\)
0.396023 + 0.918241i \(0.370390\pi\)
\(620\) 4153.51 0.269047
\(621\) −4743.53 −0.306524
\(622\) 4319.67 0.278461
\(623\) 0 0
\(624\) 2096.61 0.134506
\(625\) 17030.8 1.08997
\(626\) 7381.68 0.471296
\(627\) −2643.94 −0.168403
\(628\) −2388.30 −0.151757
\(629\) −6364.63 −0.403457
\(630\) 0 0
\(631\) 13589.7 0.857365 0.428682 0.903455i \(-0.358978\pi\)
0.428682 + 0.903455i \(0.358978\pi\)
\(632\) 14581.5 0.917755
\(633\) 6722.72 0.422124
\(634\) 1946.50 0.121933
\(635\) −19114.8 −1.19456
\(636\) −206.064 −0.0128474
\(637\) 0 0
\(638\) 3474.47 0.215605
\(639\) −4369.51 −0.270509
\(640\) −31411.9 −1.94010
\(641\) −30329.7 −1.86888 −0.934439 0.356124i \(-0.884098\pi\)
−0.934439 + 0.356124i \(0.884098\pi\)
\(642\) −5936.34 −0.364936
\(643\) −12832.3 −0.787021 −0.393511 0.919320i \(-0.628740\pi\)
−0.393511 + 0.919320i \(0.628740\pi\)
\(644\) 0 0
\(645\) 30740.6 1.87661
\(646\) 13929.8 0.848390
\(647\) −5954.78 −0.361834 −0.180917 0.983498i \(-0.557907\pi\)
−0.180917 + 0.983498i \(0.557907\pi\)
\(648\) −1739.95 −0.105481
\(649\) −5474.47 −0.331112
\(650\) 7559.12 0.456143
\(651\) 0 0
\(652\) 1919.62 0.115304
\(653\) −10555.8 −0.632588 −0.316294 0.948661i \(-0.602439\pi\)
−0.316294 + 0.948661i \(0.602439\pi\)
\(654\) −2493.48 −0.149087
\(655\) −12454.0 −0.742927
\(656\) 35307.3 2.10140
\(657\) 1439.23 0.0854638
\(658\) 0 0
\(659\) 16990.8 1.00435 0.502176 0.864766i \(-0.332533\pi\)
0.502176 + 0.864766i \(0.332533\pi\)
\(660\) 468.620 0.0276379
\(661\) −2320.16 −0.136526 −0.0682629 0.997667i \(-0.521746\pi\)
−0.0682629 + 0.997667i \(0.521746\pi\)
\(662\) 19472.3 1.14322
\(663\) −1780.16 −0.104277
\(664\) 2866.80 0.167550
\(665\) 0 0
\(666\) −2876.13 −0.167339
\(667\) 18781.7 1.09030
\(668\) 2459.23 0.142441
\(669\) 5692.47 0.328974
\(670\) 31961.7 1.84296
\(671\) −5171.59 −0.297536
\(672\) 0 0
\(673\) −3681.46 −0.210862 −0.105431 0.994427i \(-0.533622\pi\)
−0.105431 + 0.994427i \(0.533622\pi\)
\(674\) 11623.5 0.664271
\(675\) −6850.23 −0.390615
\(676\) −1528.98 −0.0869924
\(677\) 28591.4 1.62313 0.811565 0.584263i \(-0.198616\pi\)
0.811565 + 0.584263i \(0.198616\pi\)
\(678\) −5810.92 −0.329155
\(679\) 0 0
\(680\) 24598.7 1.38723
\(681\) −20349.3 −1.14506
\(682\) −9506.07 −0.533734
\(683\) 21871.4 1.22531 0.612654 0.790351i \(-0.290102\pi\)
0.612654 + 0.790351i \(0.290102\pi\)
\(684\) 526.178 0.0294136
\(685\) 31621.4 1.76378
\(686\) 0 0
\(687\) 12065.5 0.670053
\(688\) −36492.4 −2.02218
\(689\) 949.203 0.0524844
\(690\) 30305.0 1.67202
\(691\) 20972.1 1.15458 0.577291 0.816538i \(-0.304110\pi\)
0.577291 + 0.816538i \(0.304110\pi\)
\(692\) 1976.22 0.108562
\(693\) 0 0
\(694\) −8113.31 −0.443771
\(695\) −41084.6 −2.24234
\(696\) 6889.20 0.375193
\(697\) −29978.2 −1.62913
\(698\) 4587.30 0.248756
\(699\) 2644.66 0.143105
\(700\) 0 0
\(701\) 21899.5 1.17993 0.589967 0.807427i \(-0.299141\pi\)
0.589967 + 0.807427i \(0.299141\pi\)
\(702\) −804.441 −0.0432502
\(703\) −8665.72 −0.464913
\(704\) 5028.84 0.269221
\(705\) 1129.38 0.0603334
\(706\) −35690.0 −1.90256
\(707\) 0 0
\(708\) 1089.49 0.0578327
\(709\) −22181.7 −1.17497 −0.587484 0.809236i \(-0.699881\pi\)
−0.587484 + 0.809236i \(0.699881\pi\)
\(710\) 27915.5 1.47556
\(711\) −6109.33 −0.322247
\(712\) −33194.7 −1.74723
\(713\) −51386.2 −2.69906
\(714\) 0 0
\(715\) −2158.63 −0.112906
\(716\) −2038.80 −0.106416
\(717\) 13528.4 0.704642
\(718\) 152.274 0.00791478
\(719\) 5912.69 0.306684 0.153342 0.988173i \(-0.450996\pi\)
0.153342 + 0.988173i \(0.450996\pi\)
\(720\) 12138.4 0.628296
\(721\) 0 0
\(722\) −1299.66 −0.0669924
\(723\) 2858.44 0.147036
\(724\) −2745.18 −0.140917
\(725\) 27123.0 1.38941
\(726\) −1072.52 −0.0548279
\(727\) −5922.24 −0.302123 −0.151062 0.988524i \(-0.548269\pi\)
−0.151062 + 0.988524i \(0.548269\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −9194.80 −0.466185
\(731\) 30984.4 1.56772
\(732\) 1029.21 0.0519682
\(733\) 14726.7 0.742075 0.371038 0.928618i \(-0.379002\pi\)
0.371038 + 0.928618i \(0.379002\pi\)
\(734\) 25595.0 1.28710
\(735\) 0 0
\(736\) −5784.12 −0.289681
\(737\) −6114.59 −0.305609
\(738\) −13546.9 −0.675704
\(739\) −20820.7 −1.03640 −0.518201 0.855259i \(-0.673398\pi\)
−0.518201 + 0.855259i \(0.673398\pi\)
\(740\) 1535.94 0.0763002
\(741\) −2423.76 −0.120161
\(742\) 0 0
\(743\) −15989.7 −0.789509 −0.394754 0.918787i \(-0.629170\pi\)
−0.394754 + 0.918787i \(0.629170\pi\)
\(744\) −18848.7 −0.928799
\(745\) 15033.4 0.739306
\(746\) −34600.2 −1.69813
\(747\) −1201.12 −0.0588311
\(748\) 472.336 0.0230887
\(749\) 0 0
\(750\) 22202.2 1.08094
\(751\) −23683.2 −1.15075 −0.575375 0.817890i \(-0.695144\pi\)
−0.575375 + 0.817890i \(0.695144\pi\)
\(752\) −1340.70 −0.0650137
\(753\) 7527.89 0.364318
\(754\) 3185.12 0.153840
\(755\) 52677.1 2.53922
\(756\) 0 0
\(757\) −11576.7 −0.555827 −0.277914 0.960606i \(-0.589643\pi\)
−0.277914 + 0.960606i \(0.589643\pi\)
\(758\) −4597.57 −0.220305
\(759\) −5797.65 −0.277261
\(760\) 33492.2 1.59854
\(761\) 9835.54 0.468512 0.234256 0.972175i \(-0.424735\pi\)
0.234256 + 0.972175i \(0.424735\pi\)
\(762\) −8706.35 −0.413908
\(763\) 0 0
\(764\) 345.965 0.0163829
\(765\) −10306.3 −0.487093
\(766\) −26120.3 −1.23207
\(767\) −5018.57 −0.236258
\(768\) −3335.39 −0.156713
\(769\) −14281.1 −0.669689 −0.334845 0.942273i \(-0.608684\pi\)
−0.334845 + 0.942273i \(0.608684\pi\)
\(770\) 0 0
\(771\) −1860.85 −0.0869219
\(772\) −2073.02 −0.0966446
\(773\) 7509.32 0.349407 0.174703 0.984621i \(-0.444103\pi\)
0.174703 + 0.984621i \(0.444103\pi\)
\(774\) 14001.7 0.650231
\(775\) −74207.8 −3.43951
\(776\) −5461.16 −0.252634
\(777\) 0 0
\(778\) −8466.36 −0.390146
\(779\) −40816.6 −1.87728
\(780\) 429.594 0.0197204
\(781\) −5340.52 −0.244685
\(782\) 30545.3 1.39680
\(783\) −2886.42 −0.131740
\(784\) 0 0
\(785\) 63692.9 2.89592
\(786\) −5672.50 −0.257419
\(787\) −23481.2 −1.06355 −0.531775 0.846886i \(-0.678475\pi\)
−0.531775 + 0.846886i \(0.678475\pi\)
\(788\) −560.512 −0.0253394
\(789\) 10670.9 0.481489
\(790\) 39030.6 1.75778
\(791\) 0 0
\(792\) −2126.60 −0.0954111
\(793\) −4740.91 −0.212301
\(794\) 23526.0 1.05152
\(795\) 5495.47 0.245162
\(796\) −3131.93 −0.139458
\(797\) −2063.85 −0.0917258 −0.0458629 0.998948i \(-0.514604\pi\)
−0.0458629 + 0.998948i \(0.514604\pi\)
\(798\) 0 0
\(799\) 1138.34 0.0504025
\(800\) −8352.96 −0.369152
\(801\) 13907.9 0.613496
\(802\) −26706.5 −1.17586
\(803\) 1759.06 0.0773049
\(804\) 1216.88 0.0533782
\(805\) 0 0
\(806\) −8714.42 −0.380834
\(807\) −13811.0 −0.602441
\(808\) 21765.4 0.947655
\(809\) −40464.7 −1.75854 −0.879272 0.476320i \(-0.841970\pi\)
−0.879272 + 0.476320i \(0.841970\pi\)
\(810\) −4657.36 −0.202028
\(811\) −18422.2 −0.797645 −0.398823 0.917028i \(-0.630581\pi\)
−0.398823 + 0.917028i \(0.630581\pi\)
\(812\) 0 0
\(813\) 7319.29 0.315743
\(814\) −3515.28 −0.151364
\(815\) −51193.9 −2.20030
\(816\) 12234.7 0.524878
\(817\) 42186.6 1.80652
\(818\) −14764.0 −0.631063
\(819\) 0 0
\(820\) 7234.45 0.308095
\(821\) −23189.3 −0.985764 −0.492882 0.870096i \(-0.664056\pi\)
−0.492882 + 0.870096i \(0.664056\pi\)
\(822\) 14402.8 0.611139
\(823\) 27653.0 1.17123 0.585616 0.810589i \(-0.300853\pi\)
0.585616 + 0.810589i \(0.300853\pi\)
\(824\) −25020.3 −1.05780
\(825\) −8372.50 −0.353325
\(826\) 0 0
\(827\) 43713.5 1.83805 0.919025 0.394199i \(-0.128978\pi\)
0.919025 + 0.394199i \(0.128978\pi\)
\(828\) 1153.81 0.0484270
\(829\) −431.124 −0.0180622 −0.00903110 0.999959i \(-0.502875\pi\)
−0.00903110 + 0.999959i \(0.502875\pi\)
\(830\) 7673.61 0.320910
\(831\) −7222.06 −0.301481
\(832\) 4610.05 0.192097
\(833\) 0 0
\(834\) −18713.1 −0.776956
\(835\) −65584.6 −2.71814
\(836\) 643.106 0.0266056
\(837\) 7897.18 0.326125
\(838\) −20944.8 −0.863397
\(839\) 43764.1 1.80084 0.900420 0.435022i \(-0.143259\pi\)
0.900420 + 0.435022i \(0.143259\pi\)
\(840\) 0 0
\(841\) −12960.4 −0.531404
\(842\) −29150.1 −1.19308
\(843\) 98.3246 0.00401718
\(844\) −1635.22 −0.0666903
\(845\) 40775.9 1.66004
\(846\) 514.409 0.0209051
\(847\) 0 0
\(848\) −6523.71 −0.264181
\(849\) 5044.74 0.203928
\(850\) 44111.1 1.78000
\(851\) −19002.2 −0.765439
\(852\) 1062.83 0.0427371
\(853\) −24431.1 −0.980662 −0.490331 0.871536i \(-0.663124\pi\)
−0.490331 + 0.871536i \(0.663124\pi\)
\(854\) 0 0
\(855\) −14032.5 −0.561288
\(856\) −14386.3 −0.574432
\(857\) 14310.0 0.570385 0.285193 0.958470i \(-0.407942\pi\)
0.285193 + 0.958470i \(0.407942\pi\)
\(858\) −983.205 −0.0391213
\(859\) −19309.5 −0.766976 −0.383488 0.923546i \(-0.625277\pi\)
−0.383488 + 0.923546i \(0.625277\pi\)
\(860\) −7477.28 −0.296480
\(861\) 0 0
\(862\) 4669.51 0.184506
\(863\) 15892.6 0.626871 0.313436 0.949609i \(-0.398520\pi\)
0.313436 + 0.949609i \(0.398520\pi\)
\(864\) 888.921 0.0350020
\(865\) −52703.3 −2.07164
\(866\) −10531.9 −0.413264
\(867\) 4350.94 0.170433
\(868\) 0 0
\(869\) −7466.95 −0.291483
\(870\) 18440.5 0.718610
\(871\) −5605.38 −0.218061
\(872\) −6042.77 −0.234672
\(873\) 2288.10 0.0887062
\(874\) 41588.8 1.60957
\(875\) 0 0
\(876\) −350.075 −0.0135022
\(877\) −35783.0 −1.37777 −0.688886 0.724870i \(-0.741900\pi\)
−0.688886 + 0.724870i \(0.741900\pi\)
\(878\) −31544.7 −1.21251
\(879\) 1908.45 0.0732314
\(880\) 14835.9 0.568315
\(881\) 8642.65 0.330509 0.165254 0.986251i \(-0.447155\pi\)
0.165254 + 0.986251i \(0.447155\pi\)
\(882\) 0 0
\(883\) 42937.0 1.63640 0.818202 0.574931i \(-0.194971\pi\)
0.818202 + 0.574931i \(0.194971\pi\)
\(884\) 433.001 0.0164744
\(885\) −29055.3 −1.10360
\(886\) 24138.8 0.915303
\(887\) 3547.08 0.134272 0.0671360 0.997744i \(-0.478614\pi\)
0.0671360 + 0.997744i \(0.478614\pi\)
\(888\) −6970.11 −0.263403
\(889\) 0 0
\(890\) −88853.1 −3.34647
\(891\) 891.000 0.0335013
\(892\) −1384.62 −0.0519738
\(893\) 1549.90 0.0580800
\(894\) 6847.39 0.256164
\(895\) 54372.3 2.03069
\(896\) 0 0
\(897\) −5314.83 −0.197834
\(898\) 44561.7 1.65595
\(899\) −31268.3 −1.16002
\(900\) 1666.23 0.0617124
\(901\) 5539.05 0.204809
\(902\) −16557.4 −0.611197
\(903\) 0 0
\(904\) −14082.4 −0.518111
\(905\) 73210.6 2.68906
\(906\) 23993.2 0.879824
\(907\) −1930.62 −0.0706783 −0.0353392 0.999375i \(-0.511251\pi\)
−0.0353392 + 0.999375i \(0.511251\pi\)
\(908\) 4949.71 0.180905
\(909\) −9119.23 −0.332746
\(910\) 0 0
\(911\) 50242.6 1.82724 0.913618 0.406574i \(-0.133277\pi\)
0.913618 + 0.406574i \(0.133277\pi\)
\(912\) 16658.1 0.604829
\(913\) −1468.04 −0.0532147
\(914\) −16591.7 −0.600443
\(915\) −27447.8 −0.991689
\(916\) −2934.78 −0.105860
\(917\) 0 0
\(918\) −4694.29 −0.168774
\(919\) −2235.80 −0.0802526 −0.0401263 0.999195i \(-0.512776\pi\)
−0.0401263 + 0.999195i \(0.512776\pi\)
\(920\) 73442.0 2.63186
\(921\) 19777.7 0.707599
\(922\) 19728.0 0.704672
\(923\) −4895.77 −0.174590
\(924\) 0 0
\(925\) −27441.5 −0.975429
\(926\) 6626.20 0.235152
\(927\) 10483.0 0.371419
\(928\) −3519.62 −0.124501
\(929\) 6906.37 0.243908 0.121954 0.992536i \(-0.461084\pi\)
0.121954 + 0.992536i \(0.461084\pi\)
\(930\) −50452.7 −1.77893
\(931\) 0 0
\(932\) −643.281 −0.0226088
\(933\) −4386.03 −0.153904
\(934\) −26698.9 −0.935346
\(935\) −12596.6 −0.440592
\(936\) −1949.50 −0.0680786
\(937\) −23094.9 −0.805207 −0.402603 0.915375i \(-0.631895\pi\)
−0.402603 + 0.915375i \(0.631895\pi\)
\(938\) 0 0
\(939\) −7495.08 −0.260482
\(940\) −274.709 −0.00953193
\(941\) −18573.5 −0.643441 −0.321720 0.946835i \(-0.604261\pi\)
−0.321720 + 0.946835i \(0.604261\pi\)
\(942\) 29010.6 1.00342
\(943\) −89502.8 −3.09079
\(944\) 34491.8 1.18921
\(945\) 0 0
\(946\) 17113.1 0.588156
\(947\) 49733.9 1.70658 0.853292 0.521433i \(-0.174602\pi\)
0.853292 + 0.521433i \(0.174602\pi\)
\(948\) 1486.02 0.0509110
\(949\) 1612.57 0.0551593
\(950\) 60059.1 2.05113
\(951\) −1976.41 −0.0673916
\(952\) 0 0
\(953\) −46966.8 −1.59644 −0.798218 0.602369i \(-0.794223\pi\)
−0.798218 + 0.602369i \(0.794223\pi\)
\(954\) 2503.06 0.0849471
\(955\) −9226.44 −0.312629
\(956\) −3290.62 −0.111325
\(957\) −3527.85 −0.119163
\(958\) 59341.9 2.00130
\(959\) 0 0
\(960\) 26690.1 0.897313
\(961\) 55758.4 1.87165
\(962\) −3222.53 −0.108003
\(963\) 6027.54 0.201698
\(964\) −695.281 −0.0232298
\(965\) 55284.8 1.84423
\(966\) 0 0
\(967\) −42590.4 −1.41635 −0.708177 0.706035i \(-0.750482\pi\)
−0.708177 + 0.706035i \(0.750482\pi\)
\(968\) −2599.18 −0.0863026
\(969\) −14143.8 −0.468900
\(970\) −14618.0 −0.483871
\(971\) −1641.83 −0.0542626 −0.0271313 0.999632i \(-0.508637\pi\)
−0.0271313 + 0.999632i \(0.508637\pi\)
\(972\) −177.320 −0.00585139
\(973\) 0 0
\(974\) −40205.6 −1.32266
\(975\) −7675.25 −0.252108
\(976\) 32583.5 1.06862
\(977\) 15375.4 0.503484 0.251742 0.967794i \(-0.418997\pi\)
0.251742 + 0.967794i \(0.418997\pi\)
\(978\) −23317.6 −0.762388
\(979\) 16998.5 0.554928
\(980\) 0 0
\(981\) 2531.79 0.0823993
\(982\) −54432.5 −1.76885
\(983\) −35992.9 −1.16785 −0.583924 0.811809i \(-0.698483\pi\)
−0.583924 + 0.811809i \(0.698483\pi\)
\(984\) −32830.0 −1.06360
\(985\) 14948.2 0.483541
\(986\) 18586.7 0.600326
\(987\) 0 0
\(988\) 589.549 0.0189839
\(989\) 92507.1 2.97427
\(990\) −5692.33 −0.182741
\(991\) −23208.1 −0.743925 −0.371963 0.928248i \(-0.621315\pi\)
−0.371963 + 0.928248i \(0.621315\pi\)
\(992\) 9629.59 0.308205
\(993\) −19771.4 −0.631850
\(994\) 0 0
\(995\) 83524.6 2.66121
\(996\) 292.159 0.00929458
\(997\) −3692.36 −0.117290 −0.0586451 0.998279i \(-0.518678\pi\)
−0.0586451 + 0.998279i \(0.518678\pi\)
\(998\) 61963.6 1.96535
\(999\) 2920.32 0.0924873
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 1617.4.a.be.1.12 16
7.6 odd 2 1617.4.a.bf.1.12 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.12 16 1.1 even 1 trivial
1617.4.a.bf.1.12 yes 16 7.6 odd 2