Properties

Label 8001.2.a.y.1.15
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.0958522 q^{2} -1.99081 q^{4} -1.26637 q^{5} -1.00000 q^{7} -0.382528 q^{8} +O(q^{10})\) \(q+0.0958522 q^{2} -1.99081 q^{4} -1.26637 q^{5} -1.00000 q^{7} -0.382528 q^{8} -0.121384 q^{10} +3.81636 q^{11} +4.33409 q^{13} -0.0958522 q^{14} +3.94496 q^{16} +6.65918 q^{17} +0.635612 q^{19} +2.52110 q^{20} +0.365807 q^{22} +3.34860 q^{23} -3.39631 q^{25} +0.415432 q^{26} +1.99081 q^{28} +5.89589 q^{29} -3.22259 q^{31} +1.14319 q^{32} +0.638297 q^{34} +1.26637 q^{35} +1.02231 q^{37} +0.0609248 q^{38} +0.484422 q^{40} -7.10867 q^{41} +4.51017 q^{43} -7.59765 q^{44} +0.320970 q^{46} +5.62563 q^{47} +1.00000 q^{49} -0.325544 q^{50} -8.62836 q^{52} -1.17123 q^{53} -4.83292 q^{55} +0.382528 q^{56} +0.565134 q^{58} -6.02243 q^{59} +5.62912 q^{61} -0.308893 q^{62} -7.78034 q^{64} -5.48855 q^{65} +5.74306 q^{67} -13.2572 q^{68} +0.121384 q^{70} -11.1586 q^{71} +7.30292 q^{73} +0.0979904 q^{74} -1.26538 q^{76} -3.81636 q^{77} -6.82806 q^{79} -4.99577 q^{80} -0.681382 q^{82} +7.82200 q^{83} -8.43298 q^{85} +0.432310 q^{86} -1.45987 q^{88} -1.47470 q^{89} -4.33409 q^{91} -6.66643 q^{92} +0.539229 q^{94} -0.804918 q^{95} -14.1081 q^{97} +0.0958522 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0958522 0.0677778 0.0338889 0.999426i \(-0.489211\pi\)
0.0338889 + 0.999426i \(0.489211\pi\)
\(3\) 0 0
\(4\) −1.99081 −0.995406
\(5\) −1.26637 −0.566337 −0.283168 0.959070i \(-0.591386\pi\)
−0.283168 + 0.959070i \(0.591386\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.382528 −0.135244
\(9\) 0 0
\(10\) −0.121384 −0.0383851
\(11\) 3.81636 1.15068 0.575338 0.817916i \(-0.304871\pi\)
0.575338 + 0.817916i \(0.304871\pi\)
\(12\) 0 0
\(13\) 4.33409 1.20206 0.601030 0.799226i \(-0.294757\pi\)
0.601030 + 0.799226i \(0.294757\pi\)
\(14\) −0.0958522 −0.0256176
\(15\) 0 0
\(16\) 3.94496 0.986240
\(17\) 6.65918 1.61509 0.807544 0.589807i \(-0.200796\pi\)
0.807544 + 0.589807i \(0.200796\pi\)
\(18\) 0 0
\(19\) 0.635612 0.145819 0.0729097 0.997339i \(-0.476772\pi\)
0.0729097 + 0.997339i \(0.476772\pi\)
\(20\) 2.52110 0.563735
\(21\) 0 0
\(22\) 0.365807 0.0779902
\(23\) 3.34860 0.698230 0.349115 0.937080i \(-0.386482\pi\)
0.349115 + 0.937080i \(0.386482\pi\)
\(24\) 0 0
\(25\) −3.39631 −0.679262
\(26\) 0.415432 0.0814730
\(27\) 0 0
\(28\) 1.99081 0.376228
\(29\) 5.89589 1.09484 0.547420 0.836858i \(-0.315610\pi\)
0.547420 + 0.836858i \(0.315610\pi\)
\(30\) 0 0
\(31\) −3.22259 −0.578795 −0.289397 0.957209i \(-0.593455\pi\)
−0.289397 + 0.957209i \(0.593455\pi\)
\(32\) 1.14319 0.202089
\(33\) 0 0
\(34\) 0.638297 0.109467
\(35\) 1.26637 0.214055
\(36\) 0 0
\(37\) 1.02231 0.168066 0.0840332 0.996463i \(-0.473220\pi\)
0.0840332 + 0.996463i \(0.473220\pi\)
\(38\) 0.0609248 0.00988331
\(39\) 0 0
\(40\) 0.484422 0.0765938
\(41\) −7.10867 −1.11019 −0.555094 0.831788i \(-0.687318\pi\)
−0.555094 + 0.831788i \(0.687318\pi\)
\(42\) 0 0
\(43\) 4.51017 0.687794 0.343897 0.939007i \(-0.388253\pi\)
0.343897 + 0.939007i \(0.388253\pi\)
\(44\) −7.59765 −1.14539
\(45\) 0 0
\(46\) 0.320970 0.0473245
\(47\) 5.62563 0.820582 0.410291 0.911955i \(-0.365427\pi\)
0.410291 + 0.911955i \(0.365427\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.325544 −0.0460389
\(51\) 0 0
\(52\) −8.62836 −1.19654
\(53\) −1.17123 −0.160881 −0.0804407 0.996759i \(-0.525633\pi\)
−0.0804407 + 0.996759i \(0.525633\pi\)
\(54\) 0 0
\(55\) −4.83292 −0.651670
\(56\) 0.382528 0.0511175
\(57\) 0 0
\(58\) 0.565134 0.0742058
\(59\) −6.02243 −0.784054 −0.392027 0.919954i \(-0.628226\pi\)
−0.392027 + 0.919954i \(0.628226\pi\)
\(60\) 0 0
\(61\) 5.62912 0.720735 0.360367 0.932810i \(-0.382651\pi\)
0.360367 + 0.932810i \(0.382651\pi\)
\(62\) −0.308893 −0.0392294
\(63\) 0 0
\(64\) −7.78034 −0.972542
\(65\) −5.48855 −0.680771
\(66\) 0 0
\(67\) 5.74306 0.701626 0.350813 0.936446i \(-0.385905\pi\)
0.350813 + 0.936446i \(0.385905\pi\)
\(68\) −13.2572 −1.60767
\(69\) 0 0
\(70\) 0.121384 0.0145082
\(71\) −11.1586 −1.32428 −0.662140 0.749380i \(-0.730352\pi\)
−0.662140 + 0.749380i \(0.730352\pi\)
\(72\) 0 0
\(73\) 7.30292 0.854742 0.427371 0.904076i \(-0.359440\pi\)
0.427371 + 0.904076i \(0.359440\pi\)
\(74\) 0.0979904 0.0113912
\(75\) 0 0
\(76\) −1.26538 −0.145149
\(77\) −3.81636 −0.434914
\(78\) 0 0
\(79\) −6.82806 −0.768217 −0.384109 0.923288i \(-0.625491\pi\)
−0.384109 + 0.923288i \(0.625491\pi\)
\(80\) −4.99577 −0.558544
\(81\) 0 0
\(82\) −0.681382 −0.0752461
\(83\) 7.82200 0.858576 0.429288 0.903168i \(-0.358764\pi\)
0.429288 + 0.903168i \(0.358764\pi\)
\(84\) 0 0
\(85\) −8.43298 −0.914685
\(86\) 0.432310 0.0466172
\(87\) 0 0
\(88\) −1.45987 −0.155622
\(89\) −1.47470 −0.156318 −0.0781590 0.996941i \(-0.524904\pi\)
−0.0781590 + 0.996941i \(0.524904\pi\)
\(90\) 0 0
\(91\) −4.33409 −0.454336
\(92\) −6.66643 −0.695023
\(93\) 0 0
\(94\) 0.539229 0.0556172
\(95\) −0.804918 −0.0825829
\(96\) 0 0
\(97\) −14.1081 −1.43246 −0.716231 0.697863i \(-0.754134\pi\)
−0.716231 + 0.697863i \(0.754134\pi\)
\(98\) 0.0958522 0.00968254
\(99\) 0 0
\(100\) 6.76142 0.676142
\(101\) 6.15691 0.612635 0.306318 0.951929i \(-0.400903\pi\)
0.306318 + 0.951929i \(0.400903\pi\)
\(102\) 0 0
\(103\) 17.7177 1.74578 0.872888 0.487920i \(-0.162244\pi\)
0.872888 + 0.487920i \(0.162244\pi\)
\(104\) −1.65791 −0.162572
\(105\) 0 0
\(106\) −0.112265 −0.0109042
\(107\) −0.266921 −0.0258043 −0.0129021 0.999917i \(-0.504107\pi\)
−0.0129021 + 0.999917i \(0.504107\pi\)
\(108\) 0 0
\(109\) −5.58786 −0.535220 −0.267610 0.963527i \(-0.586234\pi\)
−0.267610 + 0.963527i \(0.586234\pi\)
\(110\) −0.463246 −0.0441687
\(111\) 0 0
\(112\) −3.94496 −0.372764
\(113\) 7.43321 0.699257 0.349629 0.936888i \(-0.386308\pi\)
0.349629 + 0.936888i \(0.386308\pi\)
\(114\) 0 0
\(115\) −4.24055 −0.395434
\(116\) −11.7376 −1.08981
\(117\) 0 0
\(118\) −0.577264 −0.0531414
\(119\) −6.65918 −0.610446
\(120\) 0 0
\(121\) 3.56460 0.324054
\(122\) 0.539564 0.0488498
\(123\) 0 0
\(124\) 6.41558 0.576136
\(125\) 10.6328 0.951028
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) −3.03214 −0.268006
\(129\) 0 0
\(130\) −0.526090 −0.0461411
\(131\) −10.0704 −0.879852 −0.439926 0.898034i \(-0.644995\pi\)
−0.439926 + 0.898034i \(0.644995\pi\)
\(132\) 0 0
\(133\) −0.635612 −0.0551145
\(134\) 0.550485 0.0475546
\(135\) 0 0
\(136\) −2.54733 −0.218431
\(137\) 1.08742 0.0929046 0.0464523 0.998921i \(-0.485208\pi\)
0.0464523 + 0.998921i \(0.485208\pi\)
\(138\) 0 0
\(139\) −16.8187 −1.42655 −0.713274 0.700886i \(-0.752788\pi\)
−0.713274 + 0.700886i \(0.752788\pi\)
\(140\) −2.52110 −0.213072
\(141\) 0 0
\(142\) −1.06958 −0.0897568
\(143\) 16.5404 1.38318
\(144\) 0 0
\(145\) −7.46637 −0.620048
\(146\) 0.700001 0.0579325
\(147\) 0 0
\(148\) −2.03522 −0.167294
\(149\) −7.42345 −0.608153 −0.304076 0.952648i \(-0.598348\pi\)
−0.304076 + 0.952648i \(0.598348\pi\)
\(150\) 0 0
\(151\) 14.2503 1.15968 0.579838 0.814732i \(-0.303116\pi\)
0.579838 + 0.814732i \(0.303116\pi\)
\(152\) −0.243139 −0.0197212
\(153\) 0 0
\(154\) −0.365807 −0.0294775
\(155\) 4.08099 0.327793
\(156\) 0 0
\(157\) −3.75230 −0.299466 −0.149733 0.988726i \(-0.547841\pi\)
−0.149733 + 0.988726i \(0.547841\pi\)
\(158\) −0.654485 −0.0520680
\(159\) 0 0
\(160\) −1.44770 −0.114451
\(161\) −3.34860 −0.263906
\(162\) 0 0
\(163\) 16.2486 1.27269 0.636344 0.771406i \(-0.280446\pi\)
0.636344 + 0.771406i \(0.280446\pi\)
\(164\) 14.1520 1.10509
\(165\) 0 0
\(166\) 0.749756 0.0581924
\(167\) −9.85761 −0.762804 −0.381402 0.924409i \(-0.624559\pi\)
−0.381402 + 0.924409i \(0.624559\pi\)
\(168\) 0 0
\(169\) 5.78434 0.444949
\(170\) −0.808319 −0.0619953
\(171\) 0 0
\(172\) −8.97890 −0.684635
\(173\) 5.01326 0.381151 0.190575 0.981673i \(-0.438965\pi\)
0.190575 + 0.981673i \(0.438965\pi\)
\(174\) 0 0
\(175\) 3.39631 0.256737
\(176\) 15.0554 1.13484
\(177\) 0 0
\(178\) −0.141353 −0.0105949
\(179\) 6.91717 0.517014 0.258507 0.966009i \(-0.416770\pi\)
0.258507 + 0.966009i \(0.416770\pi\)
\(180\) 0 0
\(181\) −6.55394 −0.487151 −0.243575 0.969882i \(-0.578320\pi\)
−0.243575 + 0.969882i \(0.578320\pi\)
\(182\) −0.415432 −0.0307939
\(183\) 0 0
\(184\) −1.28093 −0.0944316
\(185\) −1.29462 −0.0951822
\(186\) 0 0
\(187\) 25.4138 1.85844
\(188\) −11.1996 −0.816813
\(189\) 0 0
\(190\) −0.0771532 −0.00559728
\(191\) −17.3623 −1.25629 −0.628147 0.778095i \(-0.716186\pi\)
−0.628147 + 0.778095i \(0.716186\pi\)
\(192\) 0 0
\(193\) −13.1367 −0.945601 −0.472801 0.881169i \(-0.656757\pi\)
−0.472801 + 0.881169i \(0.656757\pi\)
\(194\) −1.35229 −0.0970891
\(195\) 0 0
\(196\) −1.99081 −0.142201
\(197\) 0.622155 0.0443267 0.0221634 0.999754i \(-0.492945\pi\)
0.0221634 + 0.999754i \(0.492945\pi\)
\(198\) 0 0
\(199\) −1.27384 −0.0903001 −0.0451501 0.998980i \(-0.514377\pi\)
−0.0451501 + 0.998980i \(0.514377\pi\)
\(200\) 1.29919 0.0918663
\(201\) 0 0
\(202\) 0.590153 0.0415231
\(203\) −5.89589 −0.413810
\(204\) 0 0
\(205\) 9.00220 0.628741
\(206\) 1.69828 0.118325
\(207\) 0 0
\(208\) 17.0978 1.18552
\(209\) 2.42572 0.167791
\(210\) 0 0
\(211\) 15.3752 1.05847 0.529237 0.848474i \(-0.322478\pi\)
0.529237 + 0.848474i \(0.322478\pi\)
\(212\) 2.33171 0.160142
\(213\) 0 0
\(214\) −0.0255850 −0.00174895
\(215\) −5.71153 −0.389523
\(216\) 0 0
\(217\) 3.22259 0.218764
\(218\) −0.535609 −0.0362760
\(219\) 0 0
\(220\) 9.62143 0.648676
\(221\) 28.8615 1.94143
\(222\) 0 0
\(223\) 11.7223 0.784983 0.392492 0.919756i \(-0.371613\pi\)
0.392492 + 0.919756i \(0.371613\pi\)
\(224\) −1.14319 −0.0763826
\(225\) 0 0
\(226\) 0.712489 0.0473941
\(227\) 15.0463 0.998655 0.499327 0.866413i \(-0.333580\pi\)
0.499327 + 0.866413i \(0.333580\pi\)
\(228\) 0 0
\(229\) 10.5521 0.697301 0.348651 0.937253i \(-0.386640\pi\)
0.348651 + 0.937253i \(0.386640\pi\)
\(230\) −0.406467 −0.0268016
\(231\) 0 0
\(232\) −2.25535 −0.148071
\(233\) 2.55680 0.167501 0.0837507 0.996487i \(-0.473310\pi\)
0.0837507 + 0.996487i \(0.473310\pi\)
\(234\) 0 0
\(235\) −7.12412 −0.464726
\(236\) 11.9895 0.780452
\(237\) 0 0
\(238\) −0.638297 −0.0413747
\(239\) 11.8858 0.768830 0.384415 0.923160i \(-0.374403\pi\)
0.384415 + 0.923160i \(0.374403\pi\)
\(240\) 0 0
\(241\) 12.0407 0.775607 0.387804 0.921742i \(-0.373234\pi\)
0.387804 + 0.921742i \(0.373234\pi\)
\(242\) 0.341675 0.0219637
\(243\) 0 0
\(244\) −11.2065 −0.717424
\(245\) −1.26637 −0.0809053
\(246\) 0 0
\(247\) 2.75480 0.175284
\(248\) 1.23273 0.0782786
\(249\) 0 0
\(250\) 1.01918 0.0644586
\(251\) −14.0925 −0.889513 −0.444756 0.895652i \(-0.646710\pi\)
−0.444756 + 0.895652i \(0.646710\pi\)
\(252\) 0 0
\(253\) 12.7794 0.803437
\(254\) 0.0958522 0.00601430
\(255\) 0 0
\(256\) 15.2700 0.954378
\(257\) 21.1606 1.31996 0.659981 0.751283i \(-0.270564\pi\)
0.659981 + 0.751283i \(0.270564\pi\)
\(258\) 0 0
\(259\) −1.02231 −0.0635231
\(260\) 10.9267 0.677644
\(261\) 0 0
\(262\) −0.965267 −0.0596344
\(263\) −5.22017 −0.321890 −0.160945 0.986963i \(-0.551454\pi\)
−0.160945 + 0.986963i \(0.551454\pi\)
\(264\) 0 0
\(265\) 1.48321 0.0911131
\(266\) −0.0609248 −0.00373554
\(267\) 0 0
\(268\) −11.4333 −0.698403
\(269\) −18.2060 −1.11004 −0.555019 0.831838i \(-0.687289\pi\)
−0.555019 + 0.831838i \(0.687289\pi\)
\(270\) 0 0
\(271\) 8.95597 0.544036 0.272018 0.962292i \(-0.412309\pi\)
0.272018 + 0.962292i \(0.412309\pi\)
\(272\) 26.2702 1.59286
\(273\) 0 0
\(274\) 0.104232 0.00629687
\(275\) −12.9615 −0.781611
\(276\) 0 0
\(277\) 28.7235 1.72583 0.862915 0.505349i \(-0.168636\pi\)
0.862915 + 0.505349i \(0.168636\pi\)
\(278\) −1.61211 −0.0966882
\(279\) 0 0
\(280\) −0.484422 −0.0289497
\(281\) −20.5362 −1.22509 −0.612545 0.790436i \(-0.709854\pi\)
−0.612545 + 0.790436i \(0.709854\pi\)
\(282\) 0 0
\(283\) 25.9163 1.54056 0.770281 0.637704i \(-0.220116\pi\)
0.770281 + 0.637704i \(0.220116\pi\)
\(284\) 22.2147 1.31820
\(285\) 0 0
\(286\) 1.58544 0.0937489
\(287\) 7.10867 0.419612
\(288\) 0 0
\(289\) 27.3447 1.60851
\(290\) −0.715668 −0.0420255
\(291\) 0 0
\(292\) −14.5387 −0.850815
\(293\) −6.15259 −0.359438 −0.179719 0.983718i \(-0.557519\pi\)
−0.179719 + 0.983718i \(0.557519\pi\)
\(294\) 0 0
\(295\) 7.62662 0.444039
\(296\) −0.391061 −0.0227300
\(297\) 0 0
\(298\) −0.711554 −0.0412192
\(299\) 14.5131 0.839315
\(300\) 0 0
\(301\) −4.51017 −0.259962
\(302\) 1.36593 0.0786003
\(303\) 0 0
\(304\) 2.50746 0.143813
\(305\) −7.12854 −0.408179
\(306\) 0 0
\(307\) 11.4262 0.652126 0.326063 0.945348i \(-0.394278\pi\)
0.326063 + 0.945348i \(0.394278\pi\)
\(308\) 7.59765 0.432917
\(309\) 0 0
\(310\) 0.391172 0.0222171
\(311\) −28.7019 −1.62753 −0.813767 0.581191i \(-0.802587\pi\)
−0.813767 + 0.581191i \(0.802587\pi\)
\(312\) 0 0
\(313\) 2.05815 0.116334 0.0581668 0.998307i \(-0.481474\pi\)
0.0581668 + 0.998307i \(0.481474\pi\)
\(314\) −0.359666 −0.0202971
\(315\) 0 0
\(316\) 13.5934 0.764688
\(317\) −15.6084 −0.876654 −0.438327 0.898816i \(-0.644429\pi\)
−0.438327 + 0.898816i \(0.644429\pi\)
\(318\) 0 0
\(319\) 22.5008 1.25981
\(320\) 9.85277 0.550787
\(321\) 0 0
\(322\) −0.320970 −0.0178870
\(323\) 4.23265 0.235511
\(324\) 0 0
\(325\) −14.7199 −0.816514
\(326\) 1.55746 0.0862599
\(327\) 0 0
\(328\) 2.71927 0.150146
\(329\) −5.62563 −0.310151
\(330\) 0 0
\(331\) 15.7809 0.867394 0.433697 0.901059i \(-0.357209\pi\)
0.433697 + 0.901059i \(0.357209\pi\)
\(332\) −15.5721 −0.854632
\(333\) 0 0
\(334\) −0.944874 −0.0517012
\(335\) −7.27282 −0.397357
\(336\) 0 0
\(337\) 24.8367 1.35294 0.676472 0.736468i \(-0.263508\pi\)
0.676472 + 0.736468i \(0.263508\pi\)
\(338\) 0.554442 0.0301576
\(339\) 0 0
\(340\) 16.7885 0.910483
\(341\) −12.2986 −0.666005
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −1.72527 −0.0930202
\(345\) 0 0
\(346\) 0.480532 0.0258336
\(347\) 5.35114 0.287265 0.143632 0.989631i \(-0.454122\pi\)
0.143632 + 0.989631i \(0.454122\pi\)
\(348\) 0 0
\(349\) −4.47824 −0.239714 −0.119857 0.992791i \(-0.538244\pi\)
−0.119857 + 0.992791i \(0.538244\pi\)
\(350\) 0.325544 0.0174011
\(351\) 0 0
\(352\) 4.36282 0.232539
\(353\) 11.9927 0.638306 0.319153 0.947703i \(-0.396602\pi\)
0.319153 + 0.947703i \(0.396602\pi\)
\(354\) 0 0
\(355\) 14.1309 0.749989
\(356\) 2.93585 0.155600
\(357\) 0 0
\(358\) 0.663027 0.0350420
\(359\) 1.85710 0.0980142 0.0490071 0.998798i \(-0.484394\pi\)
0.0490071 + 0.998798i \(0.484394\pi\)
\(360\) 0 0
\(361\) −18.5960 −0.978737
\(362\) −0.628210 −0.0330180
\(363\) 0 0
\(364\) 8.62836 0.452249
\(365\) −9.24818 −0.484072
\(366\) 0 0
\(367\) 2.96824 0.154941 0.0774705 0.996995i \(-0.475316\pi\)
0.0774705 + 0.996995i \(0.475316\pi\)
\(368\) 13.2101 0.688623
\(369\) 0 0
\(370\) −0.124092 −0.00645123
\(371\) 1.17123 0.0608075
\(372\) 0 0
\(373\) 24.5776 1.27258 0.636290 0.771450i \(-0.280468\pi\)
0.636290 + 0.771450i \(0.280468\pi\)
\(374\) 2.43597 0.125961
\(375\) 0 0
\(376\) −2.15196 −0.110979
\(377\) 25.5533 1.31606
\(378\) 0 0
\(379\) 33.4482 1.71812 0.859058 0.511878i \(-0.171050\pi\)
0.859058 + 0.511878i \(0.171050\pi\)
\(380\) 1.60244 0.0822035
\(381\) 0 0
\(382\) −1.66422 −0.0851488
\(383\) 17.8344 0.911296 0.455648 0.890160i \(-0.349408\pi\)
0.455648 + 0.890160i \(0.349408\pi\)
\(384\) 0 0
\(385\) 4.83292 0.246308
\(386\) −1.25918 −0.0640907
\(387\) 0 0
\(388\) 28.0866 1.42588
\(389\) 20.0265 1.01539 0.507693 0.861538i \(-0.330498\pi\)
0.507693 + 0.861538i \(0.330498\pi\)
\(390\) 0 0
\(391\) 22.2989 1.12770
\(392\) −0.382528 −0.0193206
\(393\) 0 0
\(394\) 0.0596350 0.00300437
\(395\) 8.64684 0.435070
\(396\) 0 0
\(397\) −12.3422 −0.619439 −0.309719 0.950828i \(-0.600235\pi\)
−0.309719 + 0.950828i \(0.600235\pi\)
\(398\) −0.122100 −0.00612034
\(399\) 0 0
\(400\) −13.3983 −0.669916
\(401\) −31.8047 −1.58825 −0.794126 0.607754i \(-0.792071\pi\)
−0.794126 + 0.607754i \(0.792071\pi\)
\(402\) 0 0
\(403\) −13.9670 −0.695746
\(404\) −12.2573 −0.609821
\(405\) 0 0
\(406\) −0.565134 −0.0280471
\(407\) 3.90149 0.193390
\(408\) 0 0
\(409\) −3.64012 −0.179992 −0.0899961 0.995942i \(-0.528685\pi\)
−0.0899961 + 0.995942i \(0.528685\pi\)
\(410\) 0.862881 0.0426146
\(411\) 0 0
\(412\) −35.2726 −1.73776
\(413\) 6.02243 0.296345
\(414\) 0 0
\(415\) −9.90553 −0.486243
\(416\) 4.95469 0.242924
\(417\) 0 0
\(418\) 0.232511 0.0113725
\(419\) 9.23283 0.451053 0.225527 0.974237i \(-0.427590\pi\)
0.225527 + 0.974237i \(0.427590\pi\)
\(420\) 0 0
\(421\) 5.42277 0.264289 0.132145 0.991230i \(-0.457814\pi\)
0.132145 + 0.991230i \(0.457814\pi\)
\(422\) 1.47375 0.0717410
\(423\) 0 0
\(424\) 0.448030 0.0217583
\(425\) −22.6167 −1.09707
\(426\) 0 0
\(427\) −5.62912 −0.272412
\(428\) 0.531390 0.0256857
\(429\) 0 0
\(430\) −0.547463 −0.0264010
\(431\) 13.8921 0.669160 0.334580 0.942367i \(-0.391406\pi\)
0.334580 + 0.942367i \(0.391406\pi\)
\(432\) 0 0
\(433\) −20.8389 −1.00145 −0.500727 0.865605i \(-0.666934\pi\)
−0.500727 + 0.865605i \(0.666934\pi\)
\(434\) 0.308893 0.0148273
\(435\) 0 0
\(436\) 11.1244 0.532762
\(437\) 2.12841 0.101815
\(438\) 0 0
\(439\) −13.5145 −0.645011 −0.322505 0.946568i \(-0.604525\pi\)
−0.322505 + 0.946568i \(0.604525\pi\)
\(440\) 1.84873 0.0881346
\(441\) 0 0
\(442\) 2.76644 0.131586
\(443\) −29.3679 −1.39531 −0.697656 0.716433i \(-0.745773\pi\)
−0.697656 + 0.716433i \(0.745773\pi\)
\(444\) 0 0
\(445\) 1.86751 0.0885287
\(446\) 1.12361 0.0532044
\(447\) 0 0
\(448\) 7.78034 0.367587
\(449\) −3.66521 −0.172972 −0.0864859 0.996253i \(-0.527564\pi\)
−0.0864859 + 0.996253i \(0.527564\pi\)
\(450\) 0 0
\(451\) −27.1292 −1.27747
\(452\) −14.7981 −0.696045
\(453\) 0 0
\(454\) 1.44222 0.0676866
\(455\) 5.48855 0.257307
\(456\) 0 0
\(457\) 4.44008 0.207698 0.103849 0.994593i \(-0.466884\pi\)
0.103849 + 0.994593i \(0.466884\pi\)
\(458\) 1.01144 0.0472615
\(459\) 0 0
\(460\) 8.44215 0.393617
\(461\) 25.7832 1.20084 0.600421 0.799684i \(-0.295000\pi\)
0.600421 + 0.799684i \(0.295000\pi\)
\(462\) 0 0
\(463\) −37.5798 −1.74648 −0.873240 0.487291i \(-0.837985\pi\)
−0.873240 + 0.487291i \(0.837985\pi\)
\(464\) 23.2590 1.07977
\(465\) 0 0
\(466\) 0.245075 0.0113529
\(467\) 19.4322 0.899216 0.449608 0.893226i \(-0.351564\pi\)
0.449608 + 0.893226i \(0.351564\pi\)
\(468\) 0 0
\(469\) −5.74306 −0.265190
\(470\) −0.682862 −0.0314981
\(471\) 0 0
\(472\) 2.30375 0.106039
\(473\) 17.2124 0.791428
\(474\) 0 0
\(475\) −2.15874 −0.0990496
\(476\) 13.2572 0.607642
\(477\) 0 0
\(478\) 1.13928 0.0521096
\(479\) 26.0260 1.18916 0.594579 0.804038i \(-0.297319\pi\)
0.594579 + 0.804038i \(0.297319\pi\)
\(480\) 0 0
\(481\) 4.43077 0.202026
\(482\) 1.15412 0.0525689
\(483\) 0 0
\(484\) −7.09644 −0.322566
\(485\) 17.8661 0.811256
\(486\) 0 0
\(487\) 10.2081 0.462573 0.231287 0.972886i \(-0.425706\pi\)
0.231287 + 0.972886i \(0.425706\pi\)
\(488\) −2.15330 −0.0974752
\(489\) 0 0
\(490\) −0.121384 −0.00548358
\(491\) −23.4197 −1.05692 −0.528458 0.848959i \(-0.677230\pi\)
−0.528458 + 0.848959i \(0.677230\pi\)
\(492\) 0 0
\(493\) 39.2618 1.76826
\(494\) 0.264054 0.0118803
\(495\) 0 0
\(496\) −12.7130 −0.570830
\(497\) 11.1586 0.500531
\(498\) 0 0
\(499\) 10.8342 0.485006 0.242503 0.970151i \(-0.422032\pi\)
0.242503 + 0.970151i \(0.422032\pi\)
\(500\) −21.1680 −0.946660
\(501\) 0 0
\(502\) −1.35080 −0.0602892
\(503\) −31.1992 −1.39110 −0.695552 0.718475i \(-0.744840\pi\)
−0.695552 + 0.718475i \(0.744840\pi\)
\(504\) 0 0
\(505\) −7.79691 −0.346958
\(506\) 1.22494 0.0544551
\(507\) 0 0
\(508\) −1.99081 −0.0883280
\(509\) −5.38386 −0.238635 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(510\) 0 0
\(511\) −7.30292 −0.323062
\(512\) 7.52795 0.332692
\(513\) 0 0
\(514\) 2.02829 0.0894640
\(515\) −22.4371 −0.988698
\(516\) 0 0
\(517\) 21.4694 0.944224
\(518\) −0.0979904 −0.00430545
\(519\) 0 0
\(520\) 2.09953 0.0920703
\(521\) −20.8338 −0.912746 −0.456373 0.889788i \(-0.650852\pi\)
−0.456373 + 0.889788i \(0.650852\pi\)
\(522\) 0 0
\(523\) 32.6149 1.42615 0.713075 0.701088i \(-0.247302\pi\)
0.713075 + 0.701088i \(0.247302\pi\)
\(524\) 20.0482 0.875810
\(525\) 0 0
\(526\) −0.500365 −0.0218170
\(527\) −21.4598 −0.934805
\(528\) 0 0
\(529\) −11.7869 −0.512474
\(530\) 0.142169 0.00617544
\(531\) 0 0
\(532\) 1.26538 0.0548613
\(533\) −30.8096 −1.33451
\(534\) 0 0
\(535\) 0.338021 0.0146139
\(536\) −2.19688 −0.0948908
\(537\) 0 0
\(538\) −1.74508 −0.0752359
\(539\) 3.81636 0.164382
\(540\) 0 0
\(541\) 0.0614040 0.00263996 0.00131998 0.999999i \(-0.499580\pi\)
0.00131998 + 0.999999i \(0.499580\pi\)
\(542\) 0.858449 0.0368736
\(543\) 0 0
\(544\) 7.61271 0.326392
\(545\) 7.07629 0.303115
\(546\) 0 0
\(547\) −3.50340 −0.149795 −0.0748973 0.997191i \(-0.523863\pi\)
−0.0748973 + 0.997191i \(0.523863\pi\)
\(548\) −2.16485 −0.0924778
\(549\) 0 0
\(550\) −1.24239 −0.0529758
\(551\) 3.74750 0.159649
\(552\) 0 0
\(553\) 6.82806 0.290359
\(554\) 2.75322 0.116973
\(555\) 0 0
\(556\) 33.4830 1.41999
\(557\) −32.3766 −1.37184 −0.685920 0.727677i \(-0.740600\pi\)
−0.685920 + 0.727677i \(0.740600\pi\)
\(558\) 0 0
\(559\) 19.5475 0.826770
\(560\) 4.99577 0.211110
\(561\) 0 0
\(562\) −1.96844 −0.0830338
\(563\) 7.69977 0.324507 0.162253 0.986749i \(-0.448124\pi\)
0.162253 + 0.986749i \(0.448124\pi\)
\(564\) 0 0
\(565\) −9.41318 −0.396015
\(566\) 2.48413 0.104416
\(567\) 0 0
\(568\) 4.26847 0.179101
\(569\) 24.3873 1.02237 0.511183 0.859472i \(-0.329207\pi\)
0.511183 + 0.859472i \(0.329207\pi\)
\(570\) 0 0
\(571\) −23.5247 −0.984479 −0.492240 0.870460i \(-0.663822\pi\)
−0.492240 + 0.870460i \(0.663822\pi\)
\(572\) −32.9289 −1.37683
\(573\) 0 0
\(574\) 0.681382 0.0284403
\(575\) −11.3729 −0.474282
\(576\) 0 0
\(577\) −6.11550 −0.254591 −0.127296 0.991865i \(-0.540630\pi\)
−0.127296 + 0.991865i \(0.540630\pi\)
\(578\) 2.62105 0.109021
\(579\) 0 0
\(580\) 14.8641 0.617200
\(581\) −7.82200 −0.324511
\(582\) 0 0
\(583\) −4.46985 −0.185122
\(584\) −2.79357 −0.115599
\(585\) 0 0
\(586\) −0.589739 −0.0243619
\(587\) −41.9754 −1.73251 −0.866256 0.499600i \(-0.833480\pi\)
−0.866256 + 0.499600i \(0.833480\pi\)
\(588\) 0 0
\(589\) −2.04832 −0.0843995
\(590\) 0.731028 0.0300960
\(591\) 0 0
\(592\) 4.03296 0.165754
\(593\) −12.0895 −0.496456 −0.248228 0.968702i \(-0.579848\pi\)
−0.248228 + 0.968702i \(0.579848\pi\)
\(594\) 0 0
\(595\) 8.43298 0.345718
\(596\) 14.7787 0.605359
\(597\) 0 0
\(598\) 1.39111 0.0568869
\(599\) −25.1578 −1.02792 −0.513959 0.857815i \(-0.671822\pi\)
−0.513959 + 0.857815i \(0.671822\pi\)
\(600\) 0 0
\(601\) −5.79385 −0.236336 −0.118168 0.992994i \(-0.537702\pi\)
−0.118168 + 0.992994i \(0.537702\pi\)
\(602\) −0.432310 −0.0176196
\(603\) 0 0
\(604\) −28.3698 −1.15435
\(605\) −4.51409 −0.183524
\(606\) 0 0
\(607\) 3.32817 0.135086 0.0675431 0.997716i \(-0.478484\pi\)
0.0675431 + 0.997716i \(0.478484\pi\)
\(608\) 0.726625 0.0294685
\(609\) 0 0
\(610\) −0.683286 −0.0276654
\(611\) 24.3820 0.986389
\(612\) 0 0
\(613\) 8.89568 0.359293 0.179647 0.983731i \(-0.442505\pi\)
0.179647 + 0.983731i \(0.442505\pi\)
\(614\) 1.09522 0.0441996
\(615\) 0 0
\(616\) 1.45987 0.0588196
\(617\) −3.16076 −0.127248 −0.0636238 0.997974i \(-0.520266\pi\)
−0.0636238 + 0.997974i \(0.520266\pi\)
\(618\) 0 0
\(619\) 36.8371 1.48061 0.740304 0.672273i \(-0.234682\pi\)
0.740304 + 0.672273i \(0.234682\pi\)
\(620\) −8.12448 −0.326287
\(621\) 0 0
\(622\) −2.75114 −0.110311
\(623\) 1.47470 0.0590827
\(624\) 0 0
\(625\) 3.51650 0.140660
\(626\) 0.197278 0.00788483
\(627\) 0 0
\(628\) 7.47012 0.298090
\(629\) 6.80773 0.271442
\(630\) 0 0
\(631\) −17.0398 −0.678345 −0.339172 0.940724i \(-0.610147\pi\)
−0.339172 + 0.940724i \(0.610147\pi\)
\(632\) 2.61193 0.103897
\(633\) 0 0
\(634\) −1.49610 −0.0594176
\(635\) −1.26637 −0.0502543
\(636\) 0 0
\(637\) 4.33409 0.171723
\(638\) 2.15676 0.0853868
\(639\) 0 0
\(640\) 3.83981 0.151782
\(641\) 32.9956 1.30325 0.651623 0.758543i \(-0.274088\pi\)
0.651623 + 0.758543i \(0.274088\pi\)
\(642\) 0 0
\(643\) 1.55797 0.0614405 0.0307203 0.999528i \(-0.490220\pi\)
0.0307203 + 0.999528i \(0.490220\pi\)
\(644\) 6.66643 0.262694
\(645\) 0 0
\(646\) 0.405709 0.0159624
\(647\) −14.7373 −0.579381 −0.289691 0.957120i \(-0.593552\pi\)
−0.289691 + 0.957120i \(0.593552\pi\)
\(648\) 0 0
\(649\) −22.9838 −0.902192
\(650\) −1.41094 −0.0553415
\(651\) 0 0
\(652\) −32.3479 −1.26684
\(653\) −14.5714 −0.570223 −0.285111 0.958494i \(-0.592031\pi\)
−0.285111 + 0.958494i \(0.592031\pi\)
\(654\) 0 0
\(655\) 12.7528 0.498293
\(656\) −28.0434 −1.09491
\(657\) 0 0
\(658\) −0.539229 −0.0210213
\(659\) −16.4172 −0.639523 −0.319762 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(660\) 0 0
\(661\) 13.8785 0.539809 0.269905 0.962887i \(-0.413008\pi\)
0.269905 + 0.962887i \(0.413008\pi\)
\(662\) 1.51263 0.0587900
\(663\) 0 0
\(664\) −2.99214 −0.116117
\(665\) 0.804918 0.0312134
\(666\) 0 0
\(667\) 19.7430 0.764450
\(668\) 19.6246 0.759300
\(669\) 0 0
\(670\) −0.697116 −0.0269319
\(671\) 21.4827 0.829332
\(672\) 0 0
\(673\) 16.9690 0.654106 0.327053 0.945006i \(-0.393945\pi\)
0.327053 + 0.945006i \(0.393945\pi\)
\(674\) 2.38066 0.0916995
\(675\) 0 0
\(676\) −11.5155 −0.442905
\(677\) 41.6406 1.60038 0.800190 0.599747i \(-0.204732\pi\)
0.800190 + 0.599747i \(0.204732\pi\)
\(678\) 0 0
\(679\) 14.1081 0.541420
\(680\) 3.22585 0.123706
\(681\) 0 0
\(682\) −1.17885 −0.0451403
\(683\) 12.8820 0.492916 0.246458 0.969153i \(-0.420733\pi\)
0.246458 + 0.969153i \(0.420733\pi\)
\(684\) 0 0
\(685\) −1.37707 −0.0526153
\(686\) −0.0958522 −0.00365966
\(687\) 0 0
\(688\) 17.7924 0.678330
\(689\) −5.07624 −0.193389
\(690\) 0 0
\(691\) 23.6851 0.901023 0.450512 0.892771i \(-0.351242\pi\)
0.450512 + 0.892771i \(0.351242\pi\)
\(692\) −9.98046 −0.379400
\(693\) 0 0
\(694\) 0.512919 0.0194701
\(695\) 21.2987 0.807906
\(696\) 0 0
\(697\) −47.3379 −1.79305
\(698\) −0.429249 −0.0162473
\(699\) 0 0
\(700\) −6.76142 −0.255558
\(701\) −51.4264 −1.94235 −0.971175 0.238369i \(-0.923387\pi\)
−0.971175 + 0.238369i \(0.923387\pi\)
\(702\) 0 0
\(703\) 0.649791 0.0245073
\(704\) −29.6926 −1.11908
\(705\) 0 0
\(706\) 1.14952 0.0432629
\(707\) −6.15691 −0.231554
\(708\) 0 0
\(709\) 33.3449 1.25229 0.626147 0.779705i \(-0.284631\pi\)
0.626147 + 0.779705i \(0.284631\pi\)
\(710\) 1.35448 0.0508326
\(711\) 0 0
\(712\) 0.564115 0.0211411
\(713\) −10.7912 −0.404132
\(714\) 0 0
\(715\) −20.9463 −0.783347
\(716\) −13.7708 −0.514639
\(717\) 0 0
\(718\) 0.178007 0.00664318
\(719\) 25.4167 0.947883 0.473941 0.880556i \(-0.342831\pi\)
0.473941 + 0.880556i \(0.342831\pi\)
\(720\) 0 0
\(721\) −17.7177 −0.659842
\(722\) −1.78247 −0.0663366
\(723\) 0 0
\(724\) 13.0477 0.484913
\(725\) −20.0243 −0.743683
\(726\) 0 0
\(727\) −33.7669 −1.25235 −0.626173 0.779684i \(-0.715380\pi\)
−0.626173 + 0.779684i \(0.715380\pi\)
\(728\) 1.65791 0.0614463
\(729\) 0 0
\(730\) −0.886459 −0.0328093
\(731\) 30.0340 1.11085
\(732\) 0 0
\(733\) −25.1390 −0.928531 −0.464266 0.885696i \(-0.653682\pi\)
−0.464266 + 0.885696i \(0.653682\pi\)
\(734\) 0.284513 0.0105016
\(735\) 0 0
\(736\) 3.82808 0.141105
\(737\) 21.9176 0.807344
\(738\) 0 0
\(739\) −39.8845 −1.46718 −0.733588 0.679594i \(-0.762156\pi\)
−0.733588 + 0.679594i \(0.762156\pi\)
\(740\) 2.57734 0.0947449
\(741\) 0 0
\(742\) 0.112265 0.00412139
\(743\) 34.5585 1.26783 0.633914 0.773404i \(-0.281447\pi\)
0.633914 + 0.773404i \(0.281447\pi\)
\(744\) 0 0
\(745\) 9.40082 0.344419
\(746\) 2.35582 0.0862527
\(747\) 0 0
\(748\) −50.5942 −1.84991
\(749\) 0.266921 0.00975309
\(750\) 0 0
\(751\) 39.4560 1.43977 0.719885 0.694094i \(-0.244195\pi\)
0.719885 + 0.694094i \(0.244195\pi\)
\(752\) 22.1929 0.809291
\(753\) 0 0
\(754\) 2.44934 0.0891998
\(755\) −18.0462 −0.656767
\(756\) 0 0
\(757\) −1.66834 −0.0606370 −0.0303185 0.999540i \(-0.509652\pi\)
−0.0303185 + 0.999540i \(0.509652\pi\)
\(758\) 3.20608 0.116450
\(759\) 0 0
\(760\) 0.307904 0.0111689
\(761\) −22.3197 −0.809087 −0.404543 0.914519i \(-0.632570\pi\)
−0.404543 + 0.914519i \(0.632570\pi\)
\(762\) 0 0
\(763\) 5.58786 0.202294
\(764\) 34.5651 1.25052
\(765\) 0 0
\(766\) 1.70947 0.0617656
\(767\) −26.1018 −0.942480
\(768\) 0 0
\(769\) −41.1244 −1.48298 −0.741492 0.670962i \(-0.765881\pi\)
−0.741492 + 0.670962i \(0.765881\pi\)
\(770\) 0.463246 0.0166942
\(771\) 0 0
\(772\) 26.1527 0.941258
\(773\) 20.9238 0.752578 0.376289 0.926502i \(-0.377200\pi\)
0.376289 + 0.926502i \(0.377200\pi\)
\(774\) 0 0
\(775\) 10.9449 0.393154
\(776\) 5.39675 0.193732
\(777\) 0 0
\(778\) 1.91959 0.0688206
\(779\) −4.51836 −0.161887
\(780\) 0 0
\(781\) −42.5852 −1.52382
\(782\) 2.13740 0.0764333
\(783\) 0 0
\(784\) 3.94496 0.140891
\(785\) 4.75179 0.169599
\(786\) 0 0
\(787\) 19.9400 0.710784 0.355392 0.934717i \(-0.384347\pi\)
0.355392 + 0.934717i \(0.384347\pi\)
\(788\) −1.23859 −0.0441231
\(789\) 0 0
\(790\) 0.828819 0.0294881
\(791\) −7.43321 −0.264294
\(792\) 0 0
\(793\) 24.3971 0.866367
\(794\) −1.18303 −0.0419842
\(795\) 0 0
\(796\) 2.53598 0.0898853
\(797\) 17.7408 0.628411 0.314205 0.949355i \(-0.398262\pi\)
0.314205 + 0.949355i \(0.398262\pi\)
\(798\) 0 0
\(799\) 37.4621 1.32531
\(800\) −3.88263 −0.137272
\(801\) 0 0
\(802\) −3.04855 −0.107648
\(803\) 27.8705 0.983530
\(804\) 0 0
\(805\) 4.24055 0.149460
\(806\) −1.33877 −0.0471561
\(807\) 0 0
\(808\) −2.35519 −0.0828554
\(809\) −41.6391 −1.46395 −0.731976 0.681331i \(-0.761402\pi\)
−0.731976 + 0.681331i \(0.761402\pi\)
\(810\) 0 0
\(811\) 44.0807 1.54788 0.773942 0.633256i \(-0.218282\pi\)
0.773942 + 0.633256i \(0.218282\pi\)
\(812\) 11.7376 0.411910
\(813\) 0 0
\(814\) 0.373967 0.0131075
\(815\) −20.5767 −0.720770
\(816\) 0 0
\(817\) 2.86672 0.100294
\(818\) −0.348913 −0.0121995
\(819\) 0 0
\(820\) −17.9217 −0.625852
\(821\) 35.8438 1.25096 0.625478 0.780242i \(-0.284904\pi\)
0.625478 + 0.780242i \(0.284904\pi\)
\(822\) 0 0
\(823\) 10.7907 0.376141 0.188070 0.982156i \(-0.439777\pi\)
0.188070 + 0.982156i \(0.439777\pi\)
\(824\) −6.77752 −0.236106
\(825\) 0 0
\(826\) 0.577264 0.0200856
\(827\) −13.7809 −0.479207 −0.239604 0.970871i \(-0.577018\pi\)
−0.239604 + 0.970871i \(0.577018\pi\)
\(828\) 0 0
\(829\) −25.9498 −0.901273 −0.450636 0.892708i \(-0.648803\pi\)
−0.450636 + 0.892708i \(0.648803\pi\)
\(830\) −0.949467 −0.0329565
\(831\) 0 0
\(832\) −33.7207 −1.16905
\(833\) 6.65918 0.230727
\(834\) 0 0
\(835\) 12.4834 0.432004
\(836\) −4.82916 −0.167020
\(837\) 0 0
\(838\) 0.884987 0.0305714
\(839\) −29.2765 −1.01074 −0.505368 0.862904i \(-0.668643\pi\)
−0.505368 + 0.862904i \(0.668643\pi\)
\(840\) 0 0
\(841\) 5.76154 0.198674
\(842\) 0.519784 0.0179129
\(843\) 0 0
\(844\) −30.6092 −1.05361
\(845\) −7.32510 −0.251991
\(846\) 0 0
\(847\) −3.56460 −0.122481
\(848\) −4.62047 −0.158668
\(849\) 0 0
\(850\) −2.16786 −0.0743569
\(851\) 3.42329 0.117349
\(852\) 0 0
\(853\) 49.8635 1.70729 0.853646 0.520853i \(-0.174386\pi\)
0.853646 + 0.520853i \(0.174386\pi\)
\(854\) −0.539564 −0.0184635
\(855\) 0 0
\(856\) 0.102105 0.00348988
\(857\) −48.9517 −1.67216 −0.836080 0.548608i \(-0.815158\pi\)
−0.836080 + 0.548608i \(0.815158\pi\)
\(858\) 0 0
\(859\) 52.6366 1.79594 0.897968 0.440060i \(-0.145043\pi\)
0.897968 + 0.440060i \(0.145043\pi\)
\(860\) 11.3706 0.387734
\(861\) 0 0
\(862\) 1.33159 0.0453541
\(863\) 20.4155 0.694952 0.347476 0.937689i \(-0.387039\pi\)
0.347476 + 0.937689i \(0.387039\pi\)
\(864\) 0 0
\(865\) −6.34863 −0.215860
\(866\) −1.99746 −0.0678764
\(867\) 0 0
\(868\) −6.41558 −0.217759
\(869\) −26.0583 −0.883969
\(870\) 0 0
\(871\) 24.8909 0.843397
\(872\) 2.13752 0.0723854
\(873\) 0 0
\(874\) 0.204013 0.00690083
\(875\) −10.6328 −0.359455
\(876\) 0 0
\(877\) 18.0957 0.611047 0.305524 0.952185i \(-0.401169\pi\)
0.305524 + 0.952185i \(0.401169\pi\)
\(878\) −1.29539 −0.0437174
\(879\) 0 0
\(880\) −19.0656 −0.642703
\(881\) 53.9666 1.81818 0.909090 0.416600i \(-0.136778\pi\)
0.909090 + 0.416600i \(0.136778\pi\)
\(882\) 0 0
\(883\) −2.01437 −0.0677889 −0.0338945 0.999425i \(-0.510791\pi\)
−0.0338945 + 0.999425i \(0.510791\pi\)
\(884\) −57.4578 −1.93252
\(885\) 0 0
\(886\) −2.81498 −0.0945711
\(887\) 12.7082 0.426700 0.213350 0.976976i \(-0.431563\pi\)
0.213350 + 0.976976i \(0.431563\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0.179005 0.00600028
\(891\) 0 0
\(892\) −23.3369 −0.781377
\(893\) 3.57572 0.119657
\(894\) 0 0
\(895\) −8.75969 −0.292804
\(896\) 3.03214 0.101297
\(897\) 0 0
\(898\) −0.351318 −0.0117236
\(899\) −19.0001 −0.633688
\(900\) 0 0
\(901\) −7.79946 −0.259838
\(902\) −2.60040 −0.0865838
\(903\) 0 0
\(904\) −2.84341 −0.0945705
\(905\) 8.29970 0.275891
\(906\) 0 0
\(907\) −13.7859 −0.457753 −0.228877 0.973455i \(-0.573505\pi\)
−0.228877 + 0.973455i \(0.573505\pi\)
\(908\) −29.9543 −0.994067
\(909\) 0 0
\(910\) 0.526090 0.0174397
\(911\) −47.4797 −1.57307 −0.786536 0.617544i \(-0.788128\pi\)
−0.786536 + 0.617544i \(0.788128\pi\)
\(912\) 0 0
\(913\) 29.8516 0.987943
\(914\) 0.425591 0.0140773
\(915\) 0 0
\(916\) −21.0072 −0.694098
\(917\) 10.0704 0.332553
\(918\) 0 0
\(919\) −0.951786 −0.0313965 −0.0156983 0.999877i \(-0.504997\pi\)
−0.0156983 + 0.999877i \(0.504997\pi\)
\(920\) 1.62213 0.0534801
\(921\) 0 0
\(922\) 2.47138 0.0813904
\(923\) −48.3623 −1.59187
\(924\) 0 0
\(925\) −3.47208 −0.114161
\(926\) −3.60210 −0.118372
\(927\) 0 0
\(928\) 6.74012 0.221255
\(929\) 20.0992 0.659435 0.329717 0.944080i \(-0.393047\pi\)
0.329717 + 0.944080i \(0.393047\pi\)
\(930\) 0 0
\(931\) 0.635612 0.0208313
\(932\) −5.09011 −0.166732
\(933\) 0 0
\(934\) 1.86262 0.0609468
\(935\) −32.1833 −1.05251
\(936\) 0 0
\(937\) −32.8606 −1.07351 −0.536755 0.843738i \(-0.680350\pi\)
−0.536755 + 0.843738i \(0.680350\pi\)
\(938\) −0.550485 −0.0179740
\(939\) 0 0
\(940\) 14.1828 0.462591
\(941\) 2.15918 0.0703871 0.0351936 0.999381i \(-0.488795\pi\)
0.0351936 + 0.999381i \(0.488795\pi\)
\(942\) 0 0
\(943\) −23.8041 −0.775167
\(944\) −23.7582 −0.773265
\(945\) 0 0
\(946\) 1.64985 0.0536412
\(947\) −7.88837 −0.256338 −0.128169 0.991752i \(-0.540910\pi\)
−0.128169 + 0.991752i \(0.540910\pi\)
\(948\) 0 0
\(949\) 31.6515 1.02745
\(950\) −0.206920 −0.00671336
\(951\) 0 0
\(952\) 2.54733 0.0825593
\(953\) 4.91726 0.159286 0.0796428 0.996823i \(-0.474622\pi\)
0.0796428 + 0.996823i \(0.474622\pi\)
\(954\) 0 0
\(955\) 21.9871 0.711486
\(956\) −23.6625 −0.765299
\(957\) 0 0
\(958\) 2.49465 0.0805984
\(959\) −1.08742 −0.0351146
\(960\) 0 0
\(961\) −20.6149 −0.664997
\(962\) 0.424699 0.0136929
\(963\) 0 0
\(964\) −23.9707 −0.772044
\(965\) 16.6359 0.535529
\(966\) 0 0
\(967\) 6.23416 0.200477 0.100238 0.994963i \(-0.468039\pi\)
0.100238 + 0.994963i \(0.468039\pi\)
\(968\) −1.36356 −0.0438265
\(969\) 0 0
\(970\) 1.71250 0.0549851
\(971\) 46.9401 1.50638 0.753190 0.657803i \(-0.228514\pi\)
0.753190 + 0.657803i \(0.228514\pi\)
\(972\) 0 0
\(973\) 16.8187 0.539184
\(974\) 0.978469 0.0313522
\(975\) 0 0
\(976\) 22.2066 0.710817
\(977\) −29.2796 −0.936737 −0.468368 0.883533i \(-0.655158\pi\)
−0.468368 + 0.883533i \(0.655158\pi\)
\(978\) 0 0
\(979\) −5.62799 −0.179871
\(980\) 2.52110 0.0805336
\(981\) 0 0
\(982\) −2.24483 −0.0716355
\(983\) 33.1814 1.05832 0.529160 0.848522i \(-0.322507\pi\)
0.529160 + 0.848522i \(0.322507\pi\)
\(984\) 0 0
\(985\) −0.787877 −0.0251039
\(986\) 3.76333 0.119849
\(987\) 0 0
\(988\) −5.48429 −0.174478
\(989\) 15.1027 0.480239
\(990\) 0 0
\(991\) −44.8275 −1.42399 −0.711996 0.702184i \(-0.752208\pi\)
−0.711996 + 0.702184i \(0.752208\pi\)
\(992\) −3.68404 −0.116968
\(993\) 0 0
\(994\) 1.06958 0.0339249
\(995\) 1.61315 0.0511403
\(996\) 0 0
\(997\) −4.58468 −0.145198 −0.0725991 0.997361i \(-0.523129\pi\)
−0.0725991 + 0.997361i \(0.523129\pi\)
\(998\) 1.03848 0.0328726
\(999\) 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 8001.2.a.y.1.15 yes 28
3.2 odd 2 inner 8001.2.a.y.1.14 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.y.1.14 28 3.2 odd 2 inner
8001.2.a.y.1.15 yes 28 1.1 even 1 trivial