Properties

Label 1176.4.a.q.1.1
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1176,4,Mod(1,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1176.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.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: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.81507\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -13.6301 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -13.6301 q^{5} +9.00000 q^{9} +32.8904 q^{11} -42.5206 q^{13} +40.8904 q^{15} -25.6301 q^{17} +15.2603 q^{19} +134.671 q^{23} +60.7809 q^{25} -27.0000 q^{27} -163.343 q^{29} +40.3015 q^{31} -98.6713 q^{33} +351.343 q^{37} +127.562 q^{39} +256.754 q^{41} +188.000 q^{43} -122.671 q^{45} +290.082 q^{47} +76.8904 q^{51} -354.904 q^{53} -448.301 q^{55} -45.7809 q^{57} -616.165 q^{59} +394.822 q^{61} +579.562 q^{65} +176.466 q^{67} -404.014 q^{69} +716.014 q^{71} -258.657 q^{73} -182.343 q^{75} -1011.59 q^{79} +81.0000 q^{81} -271.835 q^{83} +349.343 q^{85} +490.028 q^{87} +1061.79 q^{89} -120.904 q^{93} -208.000 q^{95} -1174.55 q^{97} +296.014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9} + 2 q^{11} + 18 q^{15} - 30 q^{17} - 12 q^{19} + 78 q^{23} - 6 q^{25} - 54 q^{27} + 56 q^{29} - 132 q^{31} - 6 q^{33} + 320 q^{37} - 18 q^{41} + 376 q^{43} - 54 q^{45} + 240 q^{47} + 90 q^{51} - 72 q^{53} - 684 q^{55} + 36 q^{57} - 552 q^{59} + 492 q^{61} + 904 q^{65} - 540 q^{67} - 234 q^{69} + 858 q^{71} - 900 q^{73} + 18 q^{75} - 620 q^{79} + 162 q^{81} - 1224 q^{83} + 316 q^{85} - 168 q^{87} + 1422 q^{89} + 396 q^{93} - 416 q^{95} - 1116 q^{97} + 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −13.6301 −1.21912 −0.609559 0.792741i \(-0.708653\pi\)
−0.609559 + 0.792741i \(0.708653\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 32.8904 0.901531 0.450765 0.892642i \(-0.351151\pi\)
0.450765 + 0.892642i \(0.351151\pi\)
\(12\) 0 0
\(13\) −42.5206 −0.907161 −0.453580 0.891215i \(-0.649853\pi\)
−0.453580 + 0.891215i \(0.649853\pi\)
\(14\) 0 0
\(15\) 40.8904 0.703858
\(16\) 0 0
\(17\) −25.6301 −0.365660 −0.182830 0.983145i \(-0.558526\pi\)
−0.182830 + 0.983145i \(0.558526\pi\)
\(18\) 0 0
\(19\) 15.2603 0.184261 0.0921303 0.995747i \(-0.470632\pi\)
0.0921303 + 0.995747i \(0.470632\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 134.671 1.22091 0.610455 0.792051i \(-0.290987\pi\)
0.610455 + 0.792051i \(0.290987\pi\)
\(24\) 0 0
\(25\) 60.7809 0.486247
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −163.343 −1.04593 −0.522965 0.852354i \(-0.675174\pi\)
−0.522965 + 0.852354i \(0.675174\pi\)
\(30\) 0 0
\(31\) 40.3015 0.233495 0.116748 0.993162i \(-0.462753\pi\)
0.116748 + 0.993162i \(0.462753\pi\)
\(32\) 0 0
\(33\) −98.6713 −0.520499
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 351.343 1.56109 0.780546 0.625099i \(-0.214941\pi\)
0.780546 + 0.625099i \(0.214941\pi\)
\(38\) 0 0
\(39\) 127.562 0.523749
\(40\) 0 0
\(41\) 256.754 0.978004 0.489002 0.872283i \(-0.337361\pi\)
0.489002 + 0.872283i \(0.337361\pi\)
\(42\) 0 0
\(43\) 188.000 0.666738 0.333369 0.942796i \(-0.391815\pi\)
0.333369 + 0.942796i \(0.391815\pi\)
\(44\) 0 0
\(45\) −122.671 −0.406372
\(46\) 0 0
\(47\) 290.082 0.900274 0.450137 0.892960i \(-0.351375\pi\)
0.450137 + 0.892960i \(0.351375\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 76.8904 0.211114
\(52\) 0 0
\(53\) −354.904 −0.919809 −0.459904 0.887968i \(-0.652116\pi\)
−0.459904 + 0.887968i \(0.652116\pi\)
\(54\) 0 0
\(55\) −448.301 −1.09907
\(56\) 0 0
\(57\) −45.7809 −0.106383
\(58\) 0 0
\(59\) −616.165 −1.35962 −0.679812 0.733386i \(-0.737939\pi\)
−0.679812 + 0.733386i \(0.737939\pi\)
\(60\) 0 0
\(61\) 394.822 0.828718 0.414359 0.910114i \(-0.364006\pi\)
0.414359 + 0.910114i \(0.364006\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 579.562 1.10594
\(66\) 0 0
\(67\) 176.466 0.321773 0.160886 0.986973i \(-0.448565\pi\)
0.160886 + 0.986973i \(0.448565\pi\)
\(68\) 0 0
\(69\) −404.014 −0.704892
\(70\) 0 0
\(71\) 716.014 1.19683 0.598417 0.801185i \(-0.295797\pi\)
0.598417 + 0.801185i \(0.295797\pi\)
\(72\) 0 0
\(73\) −258.657 −0.414706 −0.207353 0.978266i \(-0.566485\pi\)
−0.207353 + 0.978266i \(0.566485\pi\)
\(74\) 0 0
\(75\) −182.343 −0.280735
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1011.59 −1.44067 −0.720334 0.693628i \(-0.756011\pi\)
−0.720334 + 0.693628i \(0.756011\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −271.835 −0.359492 −0.179746 0.983713i \(-0.557528\pi\)
−0.179746 + 0.983713i \(0.557528\pi\)
\(84\) 0 0
\(85\) 349.343 0.445783
\(86\) 0 0
\(87\) 490.028 0.603868
\(88\) 0 0
\(89\) 1061.79 1.26461 0.632304 0.774721i \(-0.282110\pi\)
0.632304 + 0.774721i \(0.282110\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −120.904 −0.134809
\(94\) 0 0
\(95\) −208.000 −0.224635
\(96\) 0 0
\(97\) −1174.55 −1.22946 −0.614728 0.788739i \(-0.710734\pi\)
−0.614728 + 0.788739i \(0.710734\pi\)
\(98\) 0 0
\(99\) 296.014 0.300510
\(100\) 0 0
\(101\) −1485.33 −1.46332 −0.731662 0.681668i \(-0.761255\pi\)
−0.731662 + 0.681668i \(0.761255\pi\)
\(102\) 0 0
\(103\) −1688.96 −1.61571 −0.807855 0.589381i \(-0.799372\pi\)
−0.807855 + 0.589381i \(0.799372\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1160.92 −1.04888 −0.524440 0.851447i \(-0.675725\pi\)
−0.524440 + 0.851447i \(0.675725\pi\)
\(108\) 0 0
\(109\) −1812.30 −1.59254 −0.796271 0.604940i \(-0.793197\pi\)
−0.796271 + 0.604940i \(0.793197\pi\)
\(110\) 0 0
\(111\) −1054.03 −0.901296
\(112\) 0 0
\(113\) −526.000 −0.437893 −0.218947 0.975737i \(-0.570262\pi\)
−0.218947 + 0.975737i \(0.570262\pi\)
\(114\) 0 0
\(115\) −1835.59 −1.48843
\(116\) 0 0
\(117\) −382.685 −0.302387
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −249.219 −0.187242
\(122\) 0 0
\(123\) −770.261 −0.564651
\(124\) 0 0
\(125\) 875.316 0.626325
\(126\) 0 0
\(127\) 98.6574 0.0689325 0.0344662 0.999406i \(-0.489027\pi\)
0.0344662 + 0.999406i \(0.489027\pi\)
\(128\) 0 0
\(129\) −564.000 −0.384941
\(130\) 0 0
\(131\) 2286.47 1.52496 0.762479 0.647013i \(-0.223982\pi\)
0.762479 + 0.647013i \(0.223982\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 368.014 0.234619
\(136\) 0 0
\(137\) 1858.49 1.15899 0.579496 0.814975i \(-0.303250\pi\)
0.579496 + 0.814975i \(0.303250\pi\)
\(138\) 0 0
\(139\) −1485.15 −0.906250 −0.453125 0.891447i \(-0.649691\pi\)
−0.453125 + 0.891447i \(0.649691\pi\)
\(140\) 0 0
\(141\) −870.247 −0.519773
\(142\) 0 0
\(143\) −1398.52 −0.817833
\(144\) 0 0
\(145\) 2226.38 1.27511
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1023.40 −0.562685 −0.281342 0.959607i \(-0.590780\pi\)
−0.281342 + 0.959607i \(0.590780\pi\)
\(150\) 0 0
\(151\) −1670.25 −0.900151 −0.450075 0.892991i \(-0.648603\pi\)
−0.450075 + 0.892991i \(0.648603\pi\)
\(152\) 0 0
\(153\) −230.671 −0.121887
\(154\) 0 0
\(155\) −549.315 −0.284658
\(156\) 0 0
\(157\) −1889.89 −0.960698 −0.480349 0.877077i \(-0.659490\pi\)
−0.480349 + 0.877077i \(0.659490\pi\)
\(158\) 0 0
\(159\) 1064.71 0.531052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −647.478 −0.311131 −0.155566 0.987826i \(-0.549720\pi\)
−0.155566 + 0.987826i \(0.549720\pi\)
\(164\) 0 0
\(165\) 1344.90 0.634549
\(166\) 0 0
\(167\) −2919.40 −1.35275 −0.676376 0.736556i \(-0.736451\pi\)
−0.676376 + 0.736556i \(0.736451\pi\)
\(168\) 0 0
\(169\) −389.000 −0.177060
\(170\) 0 0
\(171\) 137.343 0.0614202
\(172\) 0 0
\(173\) 1432.26 0.629437 0.314719 0.949185i \(-0.398090\pi\)
0.314719 + 0.949185i \(0.398090\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1848.49 0.784979
\(178\) 0 0
\(179\) 1566.23 0.653999 0.326999 0.945025i \(-0.393963\pi\)
0.326999 + 0.945025i \(0.393963\pi\)
\(180\) 0 0
\(181\) −1564.93 −0.642655 −0.321327 0.946968i \(-0.604129\pi\)
−0.321327 + 0.946968i \(0.604129\pi\)
\(182\) 0 0
\(183\) −1184.47 −0.478460
\(184\) 0 0
\(185\) −4788.85 −1.90315
\(186\) 0 0
\(187\) −842.987 −0.329654
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2251.19 0.852830 0.426415 0.904528i \(-0.359776\pi\)
0.426415 + 0.904528i \(0.359776\pi\)
\(192\) 0 0
\(193\) 4972.00 1.85437 0.927183 0.374609i \(-0.122223\pi\)
0.927183 + 0.374609i \(0.122223\pi\)
\(194\) 0 0
\(195\) −1738.69 −0.638512
\(196\) 0 0
\(197\) −212.602 −0.0768895 −0.0384448 0.999261i \(-0.512240\pi\)
−0.0384448 + 0.999261i \(0.512240\pi\)
\(198\) 0 0
\(199\) −2358.25 −0.840059 −0.420029 0.907511i \(-0.637980\pi\)
−0.420029 + 0.907511i \(0.637980\pi\)
\(200\) 0 0
\(201\) −529.398 −0.185776
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3499.59 −1.19230
\(206\) 0 0
\(207\) 1212.04 0.406970
\(208\) 0 0
\(209\) 501.918 0.166117
\(210\) 0 0
\(211\) −3081.43 −1.00538 −0.502688 0.864468i \(-0.667655\pi\)
−0.502688 + 0.864468i \(0.667655\pi\)
\(212\) 0 0
\(213\) −2148.04 −0.690992
\(214\) 0 0
\(215\) −2562.47 −0.812832
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 775.972 0.239431
\(220\) 0 0
\(221\) 1089.81 0.331713
\(222\) 0 0
\(223\) −5299.01 −1.59125 −0.795624 0.605791i \(-0.792857\pi\)
−0.795624 + 0.605791i \(0.792857\pi\)
\(224\) 0 0
\(225\) 547.028 0.162082
\(226\) 0 0
\(227\) 3624.66 1.05981 0.529906 0.848057i \(-0.322227\pi\)
0.529906 + 0.848057i \(0.322227\pi\)
\(228\) 0 0
\(229\) 6154.36 1.77595 0.887973 0.459896i \(-0.152113\pi\)
0.887973 + 0.459896i \(0.152113\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5860.41 1.64776 0.823881 0.566763i \(-0.191804\pi\)
0.823881 + 0.566763i \(0.191804\pi\)
\(234\) 0 0
\(235\) −3953.86 −1.09754
\(236\) 0 0
\(237\) 3034.77 0.831770
\(238\) 0 0
\(239\) −2390.81 −0.647066 −0.323533 0.946217i \(-0.604871\pi\)
−0.323533 + 0.946217i \(0.604871\pi\)
\(240\) 0 0
\(241\) 1652.91 0.441797 0.220899 0.975297i \(-0.429101\pi\)
0.220899 + 0.975297i \(0.429101\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −648.877 −0.167154
\(248\) 0 0
\(249\) 815.506 0.207553
\(250\) 0 0
\(251\) −7586.03 −1.90767 −0.953836 0.300326i \(-0.902904\pi\)
−0.953836 + 0.300326i \(0.902904\pi\)
\(252\) 0 0
\(253\) 4429.40 1.10069
\(254\) 0 0
\(255\) −1048.03 −0.257373
\(256\) 0 0
\(257\) 153.245 0.0371952 0.0185976 0.999827i \(-0.494080\pi\)
0.0185976 + 0.999827i \(0.494080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1470.08 −0.348643
\(262\) 0 0
\(263\) 601.245 0.140967 0.0704836 0.997513i \(-0.477546\pi\)
0.0704836 + 0.997513i \(0.477546\pi\)
\(264\) 0 0
\(265\) 4837.40 1.12135
\(266\) 0 0
\(267\) −3185.38 −0.730121
\(268\) 0 0
\(269\) −4974.56 −1.12753 −0.563763 0.825937i \(-0.690647\pi\)
−0.563763 + 0.825937i \(0.690647\pi\)
\(270\) 0 0
\(271\) −1830.16 −0.410238 −0.205119 0.978737i \(-0.565758\pi\)
−0.205119 + 0.978737i \(0.565758\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1999.11 0.438367
\(276\) 0 0
\(277\) −6568.33 −1.42474 −0.712370 0.701805i \(-0.752378\pi\)
−0.712370 + 0.701805i \(0.752378\pi\)
\(278\) 0 0
\(279\) 362.713 0.0778318
\(280\) 0 0
\(281\) −2562.99 −0.544110 −0.272055 0.962282i \(-0.587703\pi\)
−0.272055 + 0.962282i \(0.587703\pi\)
\(282\) 0 0
\(283\) −5547.65 −1.16528 −0.582639 0.812731i \(-0.697980\pi\)
−0.582639 + 0.812731i \(0.697980\pi\)
\(284\) 0 0
\(285\) 624.000 0.129693
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4256.10 −0.866293
\(290\) 0 0
\(291\) 3523.65 0.709827
\(292\) 0 0
\(293\) −3783.69 −0.754422 −0.377211 0.926127i \(-0.623117\pi\)
−0.377211 + 0.926127i \(0.623117\pi\)
\(294\) 0 0
\(295\) 8398.41 1.65754
\(296\) 0 0
\(297\) −888.042 −0.173500
\(298\) 0 0
\(299\) −5726.30 −1.10756
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4455.99 0.844851
\(304\) 0 0
\(305\) −5381.48 −1.01030
\(306\) 0 0
\(307\) 2388.25 0.443989 0.221995 0.975048i \(-0.428743\pi\)
0.221995 + 0.975048i \(0.428743\pi\)
\(308\) 0 0
\(309\) 5066.88 0.932831
\(310\) 0 0
\(311\) −1014.69 −0.185009 −0.0925043 0.995712i \(-0.529487\pi\)
−0.0925043 + 0.995712i \(0.529487\pi\)
\(312\) 0 0
\(313\) 7801.98 1.40893 0.704463 0.709741i \(-0.251188\pi\)
0.704463 + 0.709741i \(0.251188\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4418.31 −0.782829 −0.391415 0.920214i \(-0.628014\pi\)
−0.391415 + 0.920214i \(0.628014\pi\)
\(318\) 0 0
\(319\) −5372.41 −0.942938
\(320\) 0 0
\(321\) 3482.75 0.605571
\(322\) 0 0
\(323\) −391.123 −0.0673768
\(324\) 0 0
\(325\) −2584.44 −0.441104
\(326\) 0 0
\(327\) 5436.91 0.919455
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7709.04 −1.28014 −0.640072 0.768315i \(-0.721095\pi\)
−0.640072 + 0.768315i \(0.721095\pi\)
\(332\) 0 0
\(333\) 3162.08 0.520364
\(334\) 0 0
\(335\) −2405.26 −0.392279
\(336\) 0 0
\(337\) 1980.02 0.320056 0.160028 0.987112i \(-0.448842\pi\)
0.160028 + 0.987112i \(0.448842\pi\)
\(338\) 0 0
\(339\) 1578.00 0.252818
\(340\) 0 0
\(341\) 1325.53 0.210503
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5506.77 0.859346
\(346\) 0 0
\(347\) 5671.05 0.877343 0.438672 0.898647i \(-0.355449\pi\)
0.438672 + 0.898647i \(0.355449\pi\)
\(348\) 0 0
\(349\) 5680.52 0.871265 0.435632 0.900125i \(-0.356525\pi\)
0.435632 + 0.900125i \(0.356525\pi\)
\(350\) 0 0
\(351\) 1148.06 0.174583
\(352\) 0 0
\(353\) 8078.65 1.21808 0.609041 0.793139i \(-0.291554\pi\)
0.609041 + 0.793139i \(0.291554\pi\)
\(354\) 0 0
\(355\) −9759.37 −1.45908
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 222.365 0.0326907 0.0163453 0.999866i \(-0.494797\pi\)
0.0163453 + 0.999866i \(0.494797\pi\)
\(360\) 0 0
\(361\) −6626.12 −0.966048
\(362\) 0 0
\(363\) 747.657 0.108104
\(364\) 0 0
\(365\) 3525.54 0.505576
\(366\) 0 0
\(367\) 10963.7 1.55941 0.779703 0.626150i \(-0.215370\pi\)
0.779703 + 0.626150i \(0.215370\pi\)
\(368\) 0 0
\(369\) 2310.78 0.326001
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13881.3 −1.92694 −0.963470 0.267815i \(-0.913698\pi\)
−0.963470 + 0.267815i \(0.913698\pi\)
\(374\) 0 0
\(375\) −2625.95 −0.361609
\(376\) 0 0
\(377\) 6945.42 0.948826
\(378\) 0 0
\(379\) 7909.27 1.07196 0.535979 0.844232i \(-0.319943\pi\)
0.535979 + 0.844232i \(0.319943\pi\)
\(380\) 0 0
\(381\) −295.972 −0.0397982
\(382\) 0 0
\(383\) −4329.15 −0.577570 −0.288785 0.957394i \(-0.593251\pi\)
−0.288785 + 0.957394i \(0.593251\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1692.00 0.222246
\(388\) 0 0
\(389\) −6914.58 −0.901242 −0.450621 0.892715i \(-0.648797\pi\)
−0.450621 + 0.892715i \(0.648797\pi\)
\(390\) 0 0
\(391\) −3451.65 −0.446438
\(392\) 0 0
\(393\) −6859.40 −0.880435
\(394\) 0 0
\(395\) 13788.1 1.75634
\(396\) 0 0
\(397\) 142.434 0.0180065 0.00900325 0.999959i \(-0.497134\pi\)
0.00900325 + 0.999959i \(0.497134\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10229.3 1.27389 0.636944 0.770910i \(-0.280198\pi\)
0.636944 + 0.770910i \(0.280198\pi\)
\(402\) 0 0
\(403\) −1713.64 −0.211818
\(404\) 0 0
\(405\) −1104.04 −0.135457
\(406\) 0 0
\(407\) 11555.8 1.40737
\(408\) 0 0
\(409\) −5552.47 −0.671277 −0.335638 0.941991i \(-0.608952\pi\)
−0.335638 + 0.941991i \(0.608952\pi\)
\(410\) 0 0
\(411\) −5575.48 −0.669144
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3705.16 0.438262
\(416\) 0 0
\(417\) 4455.45 0.523224
\(418\) 0 0
\(419\) −11571.1 −1.34913 −0.674563 0.738217i \(-0.735668\pi\)
−0.674563 + 0.738217i \(0.735668\pi\)
\(420\) 0 0
\(421\) −2150.80 −0.248987 −0.124494 0.992220i \(-0.539731\pi\)
−0.124494 + 0.992220i \(0.539731\pi\)
\(422\) 0 0
\(423\) 2610.74 0.300091
\(424\) 0 0
\(425\) −1557.82 −0.177801
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4195.56 0.472176
\(430\) 0 0
\(431\) 16771.5 1.87438 0.937188 0.348825i \(-0.113419\pi\)
0.937188 + 0.348825i \(0.113419\pi\)
\(432\) 0 0
\(433\) 11983.6 1.33001 0.665007 0.746837i \(-0.268429\pi\)
0.665007 + 0.746837i \(0.268429\pi\)
\(434\) 0 0
\(435\) −6679.15 −0.736186
\(436\) 0 0
\(437\) 2055.12 0.224965
\(438\) 0 0
\(439\) −3542.96 −0.385185 −0.192592 0.981279i \(-0.561689\pi\)
−0.192592 + 0.981279i \(0.561689\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6522.07 0.699488 0.349744 0.936845i \(-0.386269\pi\)
0.349744 + 0.936845i \(0.386269\pi\)
\(444\) 0 0
\(445\) −14472.4 −1.54170
\(446\) 0 0
\(447\) 3070.20 0.324866
\(448\) 0 0
\(449\) −15612.5 −1.64098 −0.820491 0.571660i \(-0.806300\pi\)
−0.820491 + 0.571660i \(0.806300\pi\)
\(450\) 0 0
\(451\) 8444.74 0.881701
\(452\) 0 0
\(453\) 5010.74 0.519702
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4059.04 −0.415479 −0.207739 0.978184i \(-0.566611\pi\)
−0.207739 + 0.978184i \(0.566611\pi\)
\(458\) 0 0
\(459\) 692.014 0.0703713
\(460\) 0 0
\(461\) 14452.5 1.46013 0.730067 0.683375i \(-0.239489\pi\)
0.730067 + 0.683375i \(0.239489\pi\)
\(462\) 0 0
\(463\) −7970.34 −0.800028 −0.400014 0.916509i \(-0.630995\pi\)
−0.400014 + 0.916509i \(0.630995\pi\)
\(464\) 0 0
\(465\) 1647.94 0.164347
\(466\) 0 0
\(467\) −6695.51 −0.663450 −0.331725 0.943376i \(-0.607631\pi\)
−0.331725 + 0.943376i \(0.607631\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5669.67 0.554659
\(472\) 0 0
\(473\) 6183.40 0.601085
\(474\) 0 0
\(475\) 927.534 0.0895962
\(476\) 0 0
\(477\) −3194.14 −0.306603
\(478\) 0 0
\(479\) 1434.68 0.136852 0.0684261 0.997656i \(-0.478202\pi\)
0.0684261 + 0.997656i \(0.478202\pi\)
\(480\) 0 0
\(481\) −14939.3 −1.41616
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16009.3 1.49885
\(486\) 0 0
\(487\) −5522.46 −0.513853 −0.256927 0.966431i \(-0.582710\pi\)
−0.256927 + 0.966431i \(0.582710\pi\)
\(488\) 0 0
\(489\) 1942.43 0.179632
\(490\) 0 0
\(491\) −2340.34 −0.215108 −0.107554 0.994199i \(-0.534302\pi\)
−0.107554 + 0.994199i \(0.534302\pi\)
\(492\) 0 0
\(493\) 4186.50 0.382455
\(494\) 0 0
\(495\) −4034.71 −0.366357
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17693.0 −1.58727 −0.793637 0.608392i \(-0.791815\pi\)
−0.793637 + 0.608392i \(0.791815\pi\)
\(500\) 0 0
\(501\) 8758.19 0.781012
\(502\) 0 0
\(503\) 17042.9 1.51074 0.755371 0.655297i \(-0.227457\pi\)
0.755371 + 0.655297i \(0.227457\pi\)
\(504\) 0 0
\(505\) 20245.2 1.78396
\(506\) 0 0
\(507\) 1167.00 0.102225
\(508\) 0 0
\(509\) 6252.35 0.544460 0.272230 0.962232i \(-0.412239\pi\)
0.272230 + 0.962232i \(0.412239\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −412.028 −0.0354610
\(514\) 0 0
\(515\) 23020.8 1.96974
\(516\) 0 0
\(517\) 9540.93 0.811624
\(518\) 0 0
\(519\) −4296.78 −0.363406
\(520\) 0 0
\(521\) 9375.13 0.788353 0.394177 0.919035i \(-0.371030\pi\)
0.394177 + 0.919035i \(0.371030\pi\)
\(522\) 0 0
\(523\) 13852.2 1.15816 0.579078 0.815272i \(-0.303413\pi\)
0.579078 + 0.815272i \(0.303413\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1032.93 −0.0853800
\(528\) 0 0
\(529\) 5969.36 0.490619
\(530\) 0 0
\(531\) −5545.48 −0.453208
\(532\) 0 0
\(533\) −10917.3 −0.887207
\(534\) 0 0
\(535\) 15823.5 1.27871
\(536\) 0 0
\(537\) −4698.70 −0.377586
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4303.77 −0.342022 −0.171011 0.985269i \(-0.554703\pi\)
−0.171011 + 0.985269i \(0.554703\pi\)
\(542\) 0 0
\(543\) 4694.80 0.371037
\(544\) 0 0
\(545\) 24702.0 1.94150
\(546\) 0 0
\(547\) −18406.3 −1.43875 −0.719377 0.694620i \(-0.755573\pi\)
−0.719377 + 0.694620i \(0.755573\pi\)
\(548\) 0 0
\(549\) 3553.40 0.276239
\(550\) 0 0
\(551\) −2492.66 −0.192724
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 14366.6 1.09879
\(556\) 0 0
\(557\) 5473.81 0.416397 0.208198 0.978087i \(-0.433240\pi\)
0.208198 + 0.978087i \(0.433240\pi\)
\(558\) 0 0
\(559\) −7993.87 −0.604838
\(560\) 0 0
\(561\) 2528.96 0.190326
\(562\) 0 0
\(563\) −7361.93 −0.551098 −0.275549 0.961287i \(-0.588860\pi\)
−0.275549 + 0.961287i \(0.588860\pi\)
\(564\) 0 0
\(565\) 7169.46 0.533843
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5628.42 −0.414685 −0.207342 0.978268i \(-0.566481\pi\)
−0.207342 + 0.978268i \(0.566481\pi\)
\(570\) 0 0
\(571\) −15398.8 −1.12858 −0.564291 0.825576i \(-0.690851\pi\)
−0.564291 + 0.825576i \(0.690851\pi\)
\(572\) 0 0
\(573\) −6753.58 −0.492382
\(574\) 0 0
\(575\) 8185.44 0.593663
\(576\) 0 0
\(577\) 23587.9 1.70187 0.850933 0.525274i \(-0.176037\pi\)
0.850933 + 0.525274i \(0.176037\pi\)
\(578\) 0 0
\(579\) −14916.0 −1.07062
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −11673.0 −0.829236
\(584\) 0 0
\(585\) 5216.06 0.368645
\(586\) 0 0
\(587\) 10358.1 0.728324 0.364162 0.931336i \(-0.381355\pi\)
0.364162 + 0.931336i \(0.381355\pi\)
\(588\) 0 0
\(589\) 615.012 0.0430240
\(590\) 0 0
\(591\) 637.805 0.0443922
\(592\) 0 0
\(593\) 4159.31 0.288031 0.144015 0.989575i \(-0.453999\pi\)
0.144015 + 0.989575i \(0.453999\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7074.74 0.485008
\(598\) 0 0
\(599\) −12888.3 −0.879137 −0.439568 0.898209i \(-0.644869\pi\)
−0.439568 + 0.898209i \(0.644869\pi\)
\(600\) 0 0
\(601\) 2768.82 0.187924 0.0939620 0.995576i \(-0.470047\pi\)
0.0939620 + 0.995576i \(0.470047\pi\)
\(602\) 0 0
\(603\) 1588.20 0.107258
\(604\) 0 0
\(605\) 3396.89 0.228270
\(606\) 0 0
\(607\) −26084.0 −1.74418 −0.872090 0.489345i \(-0.837236\pi\)
−0.872090 + 0.489345i \(0.837236\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12334.5 −0.816693
\(612\) 0 0
\(613\) 7507.98 0.494689 0.247344 0.968928i \(-0.420442\pi\)
0.247344 + 0.968928i \(0.420442\pi\)
\(614\) 0 0
\(615\) 10498.8 0.688376
\(616\) 0 0
\(617\) −13414.4 −0.875275 −0.437638 0.899151i \(-0.644185\pi\)
−0.437638 + 0.899151i \(0.644185\pi\)
\(618\) 0 0
\(619\) 19775.5 1.28408 0.642038 0.766673i \(-0.278089\pi\)
0.642038 + 0.766673i \(0.278089\pi\)
\(620\) 0 0
\(621\) −3636.13 −0.234964
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19528.3 −1.24981
\(626\) 0 0
\(627\) −1505.75 −0.0959075
\(628\) 0 0
\(629\) −9004.96 −0.570829
\(630\) 0 0
\(631\) 12474.1 0.786984 0.393492 0.919328i \(-0.371267\pi\)
0.393492 + 0.919328i \(0.371267\pi\)
\(632\) 0 0
\(633\) 9244.28 0.580454
\(634\) 0 0
\(635\) −1344.71 −0.0840368
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6444.13 0.398945
\(640\) 0 0
\(641\) −8075.38 −0.497595 −0.248797 0.968556i \(-0.580035\pi\)
−0.248797 + 0.968556i \(0.580035\pi\)
\(642\) 0 0
\(643\) 17525.2 1.07485 0.537423 0.843313i \(-0.319398\pi\)
0.537423 + 0.843313i \(0.319398\pi\)
\(644\) 0 0
\(645\) 7687.40 0.469289
\(646\) 0 0
\(647\) 3190.19 0.193848 0.0969238 0.995292i \(-0.469100\pi\)
0.0969238 + 0.995292i \(0.469100\pi\)
\(648\) 0 0
\(649\) −20265.9 −1.22574
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6581.37 0.394409 0.197204 0.980362i \(-0.436814\pi\)
0.197204 + 0.980362i \(0.436814\pi\)
\(654\) 0 0
\(655\) −31164.9 −1.85910
\(656\) 0 0
\(657\) −2327.92 −0.138235
\(658\) 0 0
\(659\) −14239.0 −0.841686 −0.420843 0.907134i \(-0.638266\pi\)
−0.420843 + 0.907134i \(0.638266\pi\)
\(660\) 0 0
\(661\) −24996.2 −1.47086 −0.735431 0.677600i \(-0.763020\pi\)
−0.735431 + 0.677600i \(0.763020\pi\)
\(662\) 0 0
\(663\) −3269.43 −0.191514
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21997.6 −1.27699
\(668\) 0 0
\(669\) 15897.0 0.918707
\(670\) 0 0
\(671\) 12985.9 0.747115
\(672\) 0 0
\(673\) 23298.9 1.33448 0.667242 0.744841i \(-0.267475\pi\)
0.667242 + 0.744841i \(0.267475\pi\)
\(674\) 0 0
\(675\) −1641.08 −0.0935783
\(676\) 0 0
\(677\) 16825.4 0.955172 0.477586 0.878585i \(-0.341512\pi\)
0.477586 + 0.878585i \(0.341512\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −10874.0 −0.611882
\(682\) 0 0
\(683\) −10039.3 −0.562436 −0.281218 0.959644i \(-0.590738\pi\)
−0.281218 + 0.959644i \(0.590738\pi\)
\(684\) 0 0
\(685\) −25331.5 −1.41295
\(686\) 0 0
\(687\) −18463.1 −1.02534
\(688\) 0 0
\(689\) 15090.7 0.834414
\(690\) 0 0
\(691\) 35180.7 1.93681 0.968405 0.249382i \(-0.0802273\pi\)
0.968405 + 0.249382i \(0.0802273\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20242.8 1.10483
\(696\) 0 0
\(697\) −6580.63 −0.357617
\(698\) 0 0
\(699\) −17581.2 −0.951336
\(700\) 0 0
\(701\) −14725.4 −0.793395 −0.396697 0.917949i \(-0.629844\pi\)
−0.396697 + 0.917949i \(0.629844\pi\)
\(702\) 0 0
\(703\) 5361.59 0.287648
\(704\) 0 0
\(705\) 11861.6 0.633664
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6264.80 −0.331847 −0.165923 0.986139i \(-0.553060\pi\)
−0.165923 + 0.986139i \(0.553060\pi\)
\(710\) 0 0
\(711\) −9104.31 −0.480222
\(712\) 0 0
\(713\) 5427.45 0.285077
\(714\) 0 0
\(715\) 19062.0 0.997035
\(716\) 0 0
\(717\) 7172.43 0.373584
\(718\) 0 0
\(719\) 6764.99 0.350892 0.175446 0.984489i \(-0.443863\pi\)
0.175446 + 0.984489i \(0.443863\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −4958.72 −0.255072
\(724\) 0 0
\(725\) −9928.11 −0.508580
\(726\) 0 0
\(727\) −30930.1 −1.57790 −0.788950 0.614458i \(-0.789375\pi\)
−0.788950 + 0.614458i \(0.789375\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4818.47 −0.243800
\(732\) 0 0
\(733\) −31046.2 −1.56441 −0.782207 0.623018i \(-0.785906\pi\)
−0.782207 + 0.623018i \(0.785906\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5804.05 0.290088
\(738\) 0 0
\(739\) −26261.8 −1.30725 −0.653625 0.756819i \(-0.726752\pi\)
−0.653625 + 0.756819i \(0.726752\pi\)
\(740\) 0 0
\(741\) 1946.63 0.0965064
\(742\) 0 0
\(743\) 21911.5 1.08191 0.540953 0.841053i \(-0.318064\pi\)
0.540953 + 0.841053i \(0.318064\pi\)
\(744\) 0 0
\(745\) 13949.1 0.685979
\(746\) 0 0
\(747\) −2446.52 −0.119831
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22265.8 −1.08188 −0.540938 0.841062i \(-0.681931\pi\)
−0.540938 + 0.841062i \(0.681931\pi\)
\(752\) 0 0
\(753\) 22758.1 1.10140
\(754\) 0 0
\(755\) 22765.7 1.09739
\(756\) 0 0
\(757\) 246.869 0.0118528 0.00592642 0.999982i \(-0.498114\pi\)
0.00592642 + 0.999982i \(0.498114\pi\)
\(758\) 0 0
\(759\) −13288.2 −0.635482
\(760\) 0 0
\(761\) −6297.54 −0.299981 −0.149991 0.988687i \(-0.547924\pi\)
−0.149991 + 0.988687i \(0.547924\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3144.08 0.148594
\(766\) 0 0
\(767\) 26199.7 1.23340
\(768\) 0 0
\(769\) 38562.6 1.80833 0.904163 0.427187i \(-0.140495\pi\)
0.904163 + 0.427187i \(0.140495\pi\)
\(770\) 0 0
\(771\) −459.735 −0.0214747
\(772\) 0 0
\(773\) −33338.8 −1.55125 −0.775623 0.631196i \(-0.782564\pi\)
−0.775623 + 0.631196i \(0.782564\pi\)
\(774\) 0 0
\(775\) 2449.56 0.113536
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3918.14 0.180208
\(780\) 0 0
\(781\) 23550.0 1.07898
\(782\) 0 0
\(783\) 4410.25 0.201289
\(784\) 0 0
\(785\) 25759.5 1.17120
\(786\) 0 0
\(787\) −29492.7 −1.33583 −0.667917 0.744235i \(-0.732814\pi\)
−0.667917 + 0.744235i \(0.732814\pi\)
\(788\) 0 0
\(789\) −1803.74 −0.0813874
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16788.1 −0.751780
\(794\) 0 0
\(795\) −14512.2 −0.647414
\(796\) 0 0
\(797\) −17821.6 −0.792062 −0.396031 0.918237i \(-0.629613\pi\)
−0.396031 + 0.918237i \(0.629613\pi\)
\(798\) 0 0
\(799\) −7434.85 −0.329194
\(800\) 0 0
\(801\) 9556.15 0.421536
\(802\) 0 0
\(803\) −8507.35 −0.373870
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14923.7 0.650977
\(808\) 0 0
\(809\) −23280.4 −1.01174 −0.505869 0.862611i \(-0.668828\pi\)
−0.505869 + 0.862611i \(0.668828\pi\)
\(810\) 0 0
\(811\) 29822.4 1.29125 0.645627 0.763653i \(-0.276596\pi\)
0.645627 + 0.763653i \(0.276596\pi\)
\(812\) 0 0
\(813\) 5490.49 0.236851
\(814\) 0 0
\(815\) 8825.22 0.379306
\(816\) 0 0
\(817\) 2868.93 0.122854
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 40215.0 1.70952 0.854758 0.519027i \(-0.173706\pi\)
0.854758 + 0.519027i \(0.173706\pi\)
\(822\) 0 0
\(823\) 2077.68 0.0879991 0.0439996 0.999032i \(-0.485990\pi\)
0.0439996 + 0.999032i \(0.485990\pi\)
\(824\) 0 0
\(825\) −5997.33 −0.253091
\(826\) 0 0
\(827\) 25858.4 1.08728 0.543642 0.839317i \(-0.317045\pi\)
0.543642 + 0.839317i \(0.317045\pi\)
\(828\) 0 0
\(829\) −27781.4 −1.16392 −0.581958 0.813219i \(-0.697713\pi\)
−0.581958 + 0.813219i \(0.697713\pi\)
\(830\) 0 0
\(831\) 19705.0 0.822574
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 39791.8 1.64916
\(836\) 0 0
\(837\) −1088.14 −0.0449362
\(838\) 0 0
\(839\) −5199.25 −0.213943 −0.106971 0.994262i \(-0.534115\pi\)
−0.106971 + 0.994262i \(0.534115\pi\)
\(840\) 0 0
\(841\) 2291.81 0.0939691
\(842\) 0 0
\(843\) 7688.96 0.314142
\(844\) 0 0
\(845\) 5302.13 0.215856
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 16642.9 0.672773
\(850\) 0 0
\(851\) 47315.8 1.90595
\(852\) 0 0
\(853\) −23538.6 −0.944839 −0.472420 0.881374i \(-0.656619\pi\)
−0.472420 + 0.881374i \(0.656619\pi\)
\(854\) 0 0
\(855\) −1872.00 −0.0748784
\(856\) 0 0
\(857\) 12563.3 0.500762 0.250381 0.968147i \(-0.419444\pi\)
0.250381 + 0.968147i \(0.419444\pi\)
\(858\) 0 0
\(859\) 3807.37 0.151229 0.0756145 0.997137i \(-0.475908\pi\)
0.0756145 + 0.997137i \(0.475908\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30845.9 −1.21669 −0.608347 0.793671i \(-0.708167\pi\)
−0.608347 + 0.793671i \(0.708167\pi\)
\(864\) 0 0
\(865\) −19521.9 −0.767358
\(866\) 0 0
\(867\) 12768.3 0.500154
\(868\) 0 0
\(869\) −33271.6 −1.29881
\(870\) 0 0
\(871\) −7503.44 −0.291899
\(872\) 0 0
\(873\) −10570.9 −0.409819
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18058.9 0.695331 0.347666 0.937619i \(-0.386974\pi\)
0.347666 + 0.937619i \(0.386974\pi\)
\(878\) 0 0
\(879\) 11351.1 0.435565
\(880\) 0 0
\(881\) −51388.8 −1.96519 −0.982596 0.185756i \(-0.940527\pi\)
−0.982596 + 0.185756i \(0.940527\pi\)
\(882\) 0 0
\(883\) −25852.4 −0.985281 −0.492640 0.870233i \(-0.663968\pi\)
−0.492640 + 0.870233i \(0.663968\pi\)
\(884\) 0 0
\(885\) −25195.2 −0.956982
\(886\) 0 0
\(887\) 33982.4 1.28638 0.643189 0.765707i \(-0.277611\pi\)
0.643189 + 0.765707i \(0.277611\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2664.13 0.100170
\(892\) 0 0
\(893\) 4426.74 0.165885
\(894\) 0 0
\(895\) −21348.0 −0.797301
\(896\) 0 0
\(897\) 17178.9 0.639450
\(898\) 0 0
\(899\) −6582.95 −0.244220
\(900\) 0 0
\(901\) 9096.25 0.336337
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21330.3 0.783471
\(906\) 0 0
\(907\) 1730.58 0.0633550 0.0316775 0.999498i \(-0.489915\pi\)
0.0316775 + 0.999498i \(0.489915\pi\)
\(908\) 0 0
\(909\) −13368.0 −0.487775
\(910\) 0 0
\(911\) 57.2017 0.00208033 0.00104016 0.999999i \(-0.499669\pi\)
0.00104016 + 0.999999i \(0.499669\pi\)
\(912\) 0 0
\(913\) −8940.78 −0.324093
\(914\) 0 0
\(915\) 16144.4 0.583299
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 44264.3 1.58884 0.794420 0.607369i \(-0.207775\pi\)
0.794420 + 0.607369i \(0.207775\pi\)
\(920\) 0 0
\(921\) −7164.75 −0.256337
\(922\) 0 0
\(923\) −30445.3 −1.08572
\(924\) 0 0
\(925\) 21354.9 0.759076
\(926\) 0 0
\(927\) −15200.6 −0.538570
\(928\) 0 0
\(929\) −37811.9 −1.33538 −0.667690 0.744440i \(-0.732717\pi\)
−0.667690 + 0.744440i \(0.732717\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 3044.06 0.106815
\(934\) 0 0
\(935\) 11490.0 0.401887
\(936\) 0 0
\(937\) −40391.3 −1.40824 −0.704122 0.710079i \(-0.748659\pi\)
−0.704122 + 0.710079i \(0.748659\pi\)
\(938\) 0 0
\(939\) −23405.9 −0.813444
\(940\) 0 0
\(941\) −4662.49 −0.161523 −0.0807613 0.996733i \(-0.525735\pi\)
−0.0807613 + 0.996733i \(0.525735\pi\)
\(942\) 0 0
\(943\) 34577.4 1.19405
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27594.1 −0.946874 −0.473437 0.880828i \(-0.656987\pi\)
−0.473437 + 0.880828i \(0.656987\pi\)
\(948\) 0 0
\(949\) 10998.3 0.376205
\(950\) 0 0
\(951\) 13254.9 0.451967
\(952\) 0 0
\(953\) −32317.6 −1.09850 −0.549249 0.835659i \(-0.685086\pi\)
−0.549249 + 0.835659i \(0.685086\pi\)
\(954\) 0 0
\(955\) −30684.1 −1.03970
\(956\) 0 0
\(957\) 16117.2 0.544406
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28166.8 −0.945480
\(962\) 0 0
\(963\) −10448.3 −0.349627
\(964\) 0 0
\(965\) −67769.1 −2.26069
\(966\) 0 0
\(967\) 10669.7 0.354825 0.177413 0.984137i \(-0.443227\pi\)
0.177413 + 0.984137i \(0.443227\pi\)
\(968\) 0 0
\(969\) 1173.37 0.0389000
\(970\) 0 0
\(971\) −11821.9 −0.390715 −0.195358 0.980732i \(-0.562587\pi\)
−0.195358 + 0.980732i \(0.562587\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7753.31 0.254672
\(976\) 0 0
\(977\) −32994.5 −1.08044 −0.540219 0.841525i \(-0.681659\pi\)
−0.540219 + 0.841525i \(0.681659\pi\)
\(978\) 0 0
\(979\) 34922.9 1.14008
\(980\) 0 0
\(981\) −16310.7 −0.530847
\(982\) 0 0
\(983\) −52785.2 −1.71270 −0.856351 0.516395i \(-0.827274\pi\)
−0.856351 + 0.516395i \(0.827274\pi\)
\(984\) 0 0
\(985\) 2897.79 0.0937374
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25318.2 0.814026
\(990\) 0 0
\(991\) 999.387 0.0320349 0.0160174 0.999872i \(-0.494901\pi\)
0.0160174 + 0.999872i \(0.494901\pi\)
\(992\) 0 0
\(993\) 23127.1 0.739091
\(994\) 0 0
\(995\) 32143.3 1.02413
\(996\) 0 0
\(997\) 12363.2 0.392724 0.196362 0.980531i \(-0.437087\pi\)
0.196362 + 0.980531i \(0.437087\pi\)
\(998\) 0 0
\(999\) −9486.25 −0.300432
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 1176.4.a.q.1.1 2
4.3 odd 2 2352.4.a.by.1.1 2
7.6 odd 2 1176.4.a.v.1.2 yes 2
28.27 even 2 2352.4.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.q.1.1 2 1.1 even 1 trivial
1176.4.a.v.1.2 yes 2 7.6 odd 2
2352.4.a.bs.1.2 2 28.27 even 2
2352.4.a.by.1.1 2 4.3 odd 2