Properties

Label 8001.2.a.y.1.11
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 1.11
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)
 
\(f(q)\) \(=\) \(q-0.662421 q^{2} -1.56120 q^{4} -3.26980 q^{5} -1.00000 q^{7} +2.35901 q^{8} +O(q^{10})\) \(q-0.662421 q^{2} -1.56120 q^{4} -3.26980 q^{5} -1.00000 q^{7} +2.35901 q^{8} +2.16599 q^{10} +1.50096 q^{11} -4.27180 q^{13} +0.662421 q^{14} +1.55974 q^{16} -2.42284 q^{17} -3.39105 q^{19} +5.10481 q^{20} -0.994269 q^{22} -4.67878 q^{23} +5.69162 q^{25} +2.82973 q^{26} +1.56120 q^{28} -2.49283 q^{29} +5.16438 q^{31} -5.75123 q^{32} +1.60494 q^{34} +3.26980 q^{35} -9.93232 q^{37} +2.24631 q^{38} -7.71351 q^{40} -1.42938 q^{41} +0.721457 q^{43} -2.34330 q^{44} +3.09932 q^{46} +2.25922 q^{47} +1.00000 q^{49} -3.77025 q^{50} +6.66913 q^{52} -1.84702 q^{53} -4.90785 q^{55} -2.35901 q^{56} +1.65131 q^{58} -3.44025 q^{59} -1.15426 q^{61} -3.42100 q^{62} +0.690263 q^{64} +13.9679 q^{65} -14.4075 q^{67} +3.78254 q^{68} -2.16599 q^{70} -11.7156 q^{71} +3.97876 q^{73} +6.57938 q^{74} +5.29411 q^{76} -1.50096 q^{77} -3.64679 q^{79} -5.10003 q^{80} +0.946851 q^{82} -12.5424 q^{83} +7.92222 q^{85} -0.477908 q^{86} +3.54079 q^{88} -13.9694 q^{89} +4.27180 q^{91} +7.30451 q^{92} -1.49655 q^{94} +11.0881 q^{95} -9.82688 q^{97} -0.662421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.662421 −0.468403 −0.234201 0.972188i \(-0.575247\pi\)
−0.234201 + 0.972188i \(0.575247\pi\)
\(3\) 0 0
\(4\) −1.56120 −0.780599
\(5\) −3.26980 −1.46230 −0.731150 0.682216i \(-0.761016\pi\)
−0.731150 + 0.682216i \(0.761016\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.35901 0.834037
\(9\) 0 0
\(10\) 2.16599 0.684945
\(11\) 1.50096 0.452557 0.226279 0.974063i \(-0.427344\pi\)
0.226279 + 0.974063i \(0.427344\pi\)
\(12\) 0 0
\(13\) −4.27180 −1.18478 −0.592392 0.805650i \(-0.701816\pi\)
−0.592392 + 0.805650i \(0.701816\pi\)
\(14\) 0.662421 0.177040
\(15\) 0 0
\(16\) 1.55974 0.389934
\(17\) −2.42284 −0.587625 −0.293813 0.955863i \(-0.594924\pi\)
−0.293813 + 0.955863i \(0.594924\pi\)
\(18\) 0 0
\(19\) −3.39105 −0.777961 −0.388981 0.921246i \(-0.627173\pi\)
−0.388981 + 0.921246i \(0.627173\pi\)
\(20\) 5.10481 1.14147
\(21\) 0 0
\(22\) −0.994269 −0.211979
\(23\) −4.67878 −0.975594 −0.487797 0.872957i \(-0.662199\pi\)
−0.487797 + 0.872957i \(0.662199\pi\)
\(24\) 0 0
\(25\) 5.69162 1.13832
\(26\) 2.82973 0.554956
\(27\) 0 0
\(28\) 1.56120 0.295039
\(29\) −2.49283 −0.462907 −0.231454 0.972846i \(-0.574348\pi\)
−0.231454 + 0.972846i \(0.574348\pi\)
\(30\) 0 0
\(31\) 5.16438 0.927551 0.463775 0.885953i \(-0.346494\pi\)
0.463775 + 0.885953i \(0.346494\pi\)
\(32\) −5.75123 −1.01668
\(33\) 0 0
\(34\) 1.60494 0.275245
\(35\) 3.26980 0.552698
\(36\) 0 0
\(37\) −9.93232 −1.63286 −0.816432 0.577442i \(-0.804051\pi\)
−0.816432 + 0.577442i \(0.804051\pi\)
\(38\) 2.24631 0.364399
\(39\) 0 0
\(40\) −7.71351 −1.21961
\(41\) −1.42938 −0.223232 −0.111616 0.993751i \(-0.535603\pi\)
−0.111616 + 0.993751i \(0.535603\pi\)
\(42\) 0 0
\(43\) 0.721457 0.110021 0.0550106 0.998486i \(-0.482481\pi\)
0.0550106 + 0.998486i \(0.482481\pi\)
\(44\) −2.34330 −0.353266
\(45\) 0 0
\(46\) 3.09932 0.456971
\(47\) 2.25922 0.329541 0.164770 0.986332i \(-0.447312\pi\)
0.164770 + 0.986332i \(0.447312\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.77025 −0.533194
\(51\) 0 0
\(52\) 6.66913 0.924841
\(53\) −1.84702 −0.253707 −0.126854 0.991921i \(-0.540488\pi\)
−0.126854 + 0.991921i \(0.540488\pi\)
\(54\) 0 0
\(55\) −4.90785 −0.661775
\(56\) −2.35901 −0.315236
\(57\) 0 0
\(58\) 1.65131 0.216827
\(59\) −3.44025 −0.447883 −0.223941 0.974603i \(-0.571892\pi\)
−0.223941 + 0.974603i \(0.571892\pi\)
\(60\) 0 0
\(61\) −1.15426 −0.147788 −0.0738942 0.997266i \(-0.523543\pi\)
−0.0738942 + 0.997266i \(0.523543\pi\)
\(62\) −3.42100 −0.434467
\(63\) 0 0
\(64\) 0.690263 0.0862829
\(65\) 13.9679 1.73251
\(66\) 0 0
\(67\) −14.4075 −1.76016 −0.880079 0.474827i \(-0.842511\pi\)
−0.880079 + 0.474827i \(0.842511\pi\)
\(68\) 3.78254 0.458700
\(69\) 0 0
\(70\) −2.16599 −0.258885
\(71\) −11.7156 −1.39038 −0.695192 0.718824i \(-0.744681\pi\)
−0.695192 + 0.718824i \(0.744681\pi\)
\(72\) 0 0
\(73\) 3.97876 0.465679 0.232839 0.972515i \(-0.425198\pi\)
0.232839 + 0.972515i \(0.425198\pi\)
\(74\) 6.57938 0.764837
\(75\) 0 0
\(76\) 5.29411 0.607276
\(77\) −1.50096 −0.171050
\(78\) 0 0
\(79\) −3.64679 −0.410295 −0.205148 0.978731i \(-0.565767\pi\)
−0.205148 + 0.978731i \(0.565767\pi\)
\(80\) −5.10003 −0.570201
\(81\) 0 0
\(82\) 0.946851 0.104562
\(83\) −12.5424 −1.37671 −0.688353 0.725375i \(-0.741666\pi\)
−0.688353 + 0.725375i \(0.741666\pi\)
\(84\) 0 0
\(85\) 7.92222 0.859285
\(86\) −0.477908 −0.0515342
\(87\) 0 0
\(88\) 3.54079 0.377449
\(89\) −13.9694 −1.48075 −0.740376 0.672193i \(-0.765352\pi\)
−0.740376 + 0.672193i \(0.765352\pi\)
\(90\) 0 0
\(91\) 4.27180 0.447806
\(92\) 7.30451 0.761548
\(93\) 0 0
\(94\) −1.49655 −0.154358
\(95\) 11.0881 1.13761
\(96\) 0 0
\(97\) −9.82688 −0.997769 −0.498884 0.866669i \(-0.666257\pi\)
−0.498884 + 0.866669i \(0.666257\pi\)
\(98\) −0.662421 −0.0669146
\(99\) 0 0
\(100\) −8.88575 −0.888575
\(101\) −2.11397 −0.210348 −0.105174 0.994454i \(-0.533540\pi\)
−0.105174 + 0.994454i \(0.533540\pi\)
\(102\) 0 0
\(103\) 7.93085 0.781450 0.390725 0.920507i \(-0.372224\pi\)
0.390725 + 0.920507i \(0.372224\pi\)
\(104\) −10.0772 −0.988154
\(105\) 0 0
\(106\) 1.22350 0.118837
\(107\) −9.27389 −0.896541 −0.448271 0.893898i \(-0.647960\pi\)
−0.448271 + 0.893898i \(0.647960\pi\)
\(108\) 0 0
\(109\) 8.91701 0.854095 0.427048 0.904229i \(-0.359554\pi\)
0.427048 + 0.904229i \(0.359554\pi\)
\(110\) 3.25106 0.309977
\(111\) 0 0
\(112\) −1.55974 −0.147381
\(113\) −14.8106 −1.39326 −0.696632 0.717428i \(-0.745319\pi\)
−0.696632 + 0.717428i \(0.745319\pi\)
\(114\) 0 0
\(115\) 15.2987 1.42661
\(116\) 3.89181 0.361345
\(117\) 0 0
\(118\) 2.27890 0.209789
\(119\) 2.42284 0.222102
\(120\) 0 0
\(121\) −8.74711 −0.795192
\(122\) 0.764609 0.0692244
\(123\) 0 0
\(124\) −8.06263 −0.724045
\(125\) −2.26146 −0.202271
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 11.0452 0.976268
\(129\) 0 0
\(130\) −9.25266 −0.811512
\(131\) −7.10395 −0.620675 −0.310337 0.950626i \(-0.600442\pi\)
−0.310337 + 0.950626i \(0.600442\pi\)
\(132\) 0 0
\(133\) 3.39105 0.294042
\(134\) 9.54384 0.824462
\(135\) 0 0
\(136\) −5.71551 −0.490101
\(137\) −8.96915 −0.766286 −0.383143 0.923689i \(-0.625158\pi\)
−0.383143 + 0.923689i \(0.625158\pi\)
\(138\) 0 0
\(139\) 3.30357 0.280205 0.140102 0.990137i \(-0.455257\pi\)
0.140102 + 0.990137i \(0.455257\pi\)
\(140\) −5.10481 −0.431435
\(141\) 0 0
\(142\) 7.76065 0.651259
\(143\) −6.41181 −0.536182
\(144\) 0 0
\(145\) 8.15108 0.676910
\(146\) −2.63562 −0.218125
\(147\) 0 0
\(148\) 15.5063 1.27461
\(149\) −6.69098 −0.548147 −0.274073 0.961709i \(-0.588371\pi\)
−0.274073 + 0.961709i \(0.588371\pi\)
\(150\) 0 0
\(151\) 15.4206 1.25491 0.627455 0.778653i \(-0.284097\pi\)
0.627455 + 0.778653i \(0.284097\pi\)
\(152\) −7.99954 −0.648848
\(153\) 0 0
\(154\) 0.994269 0.0801205
\(155\) −16.8865 −1.35636
\(156\) 0 0
\(157\) −8.27724 −0.660596 −0.330298 0.943877i \(-0.607149\pi\)
−0.330298 + 0.943877i \(0.607149\pi\)
\(158\) 2.41571 0.192183
\(159\) 0 0
\(160\) 18.8054 1.48670
\(161\) 4.67878 0.368740
\(162\) 0 0
\(163\) 4.67491 0.366167 0.183084 0.983097i \(-0.441392\pi\)
0.183084 + 0.983097i \(0.441392\pi\)
\(164\) 2.23154 0.174254
\(165\) 0 0
\(166\) 8.30835 0.644853
\(167\) −10.9392 −0.846498 −0.423249 0.906014i \(-0.639110\pi\)
−0.423249 + 0.906014i \(0.639110\pi\)
\(168\) 0 0
\(169\) 5.24827 0.403713
\(170\) −5.24784 −0.402491
\(171\) 0 0
\(172\) −1.12634 −0.0858824
\(173\) 6.16889 0.469012 0.234506 0.972115i \(-0.424653\pi\)
0.234506 + 0.972115i \(0.424653\pi\)
\(174\) 0 0
\(175\) −5.69162 −0.430246
\(176\) 2.34110 0.176467
\(177\) 0 0
\(178\) 9.25362 0.693588
\(179\) −8.30227 −0.620541 −0.310270 0.950648i \(-0.600420\pi\)
−0.310270 + 0.950648i \(0.600420\pi\)
\(180\) 0 0
\(181\) 1.49944 0.111453 0.0557263 0.998446i \(-0.482253\pi\)
0.0557263 + 0.998446i \(0.482253\pi\)
\(182\) −2.82973 −0.209754
\(183\) 0 0
\(184\) −11.0373 −0.813681
\(185\) 32.4768 2.38774
\(186\) 0 0
\(187\) −3.63659 −0.265934
\(188\) −3.52709 −0.257239
\(189\) 0 0
\(190\) −7.34498 −0.532861
\(191\) 0.767209 0.0555133 0.0277566 0.999615i \(-0.491164\pi\)
0.0277566 + 0.999615i \(0.491164\pi\)
\(192\) 0 0
\(193\) −11.3034 −0.813636 −0.406818 0.913509i \(-0.633362\pi\)
−0.406818 + 0.913509i \(0.633362\pi\)
\(194\) 6.50953 0.467357
\(195\) 0 0
\(196\) −1.56120 −0.111514
\(197\) −14.1865 −1.01074 −0.505371 0.862902i \(-0.668645\pi\)
−0.505371 + 0.862902i \(0.668645\pi\)
\(198\) 0 0
\(199\) 4.78068 0.338893 0.169447 0.985539i \(-0.445802\pi\)
0.169447 + 0.985539i \(0.445802\pi\)
\(200\) 13.4266 0.949404
\(201\) 0 0
\(202\) 1.40034 0.0985276
\(203\) 2.49283 0.174963
\(204\) 0 0
\(205\) 4.67379 0.326432
\(206\) −5.25357 −0.366033
\(207\) 0 0
\(208\) −6.66288 −0.461988
\(209\) −5.08984 −0.352072
\(210\) 0 0
\(211\) −1.56528 −0.107759 −0.0538793 0.998547i \(-0.517159\pi\)
−0.0538793 + 0.998547i \(0.517159\pi\)
\(212\) 2.88356 0.198044
\(213\) 0 0
\(214\) 6.14322 0.419942
\(215\) −2.35902 −0.160884
\(216\) 0 0
\(217\) −5.16438 −0.350581
\(218\) −5.90682 −0.400060
\(219\) 0 0
\(220\) 7.66213 0.516581
\(221\) 10.3499 0.696209
\(222\) 0 0
\(223\) −5.42472 −0.363266 −0.181633 0.983366i \(-0.558138\pi\)
−0.181633 + 0.983366i \(0.558138\pi\)
\(224\) 5.75123 0.384270
\(225\) 0 0
\(226\) 9.81086 0.652609
\(227\) 13.4870 0.895167 0.447583 0.894242i \(-0.352285\pi\)
0.447583 + 0.894242i \(0.352285\pi\)
\(228\) 0 0
\(229\) 26.6212 1.75918 0.879590 0.475732i \(-0.157817\pi\)
0.879590 + 0.475732i \(0.157817\pi\)
\(230\) −10.1342 −0.668228
\(231\) 0 0
\(232\) −5.88062 −0.386082
\(233\) −12.2959 −0.805528 −0.402764 0.915304i \(-0.631951\pi\)
−0.402764 + 0.915304i \(0.631951\pi\)
\(234\) 0 0
\(235\) −7.38720 −0.481888
\(236\) 5.37092 0.349617
\(237\) 0 0
\(238\) −1.60494 −0.104033
\(239\) −21.7254 −1.40530 −0.702651 0.711535i \(-0.748000\pi\)
−0.702651 + 0.711535i \(0.748000\pi\)
\(240\) 0 0
\(241\) 24.2273 1.56062 0.780308 0.625396i \(-0.215062\pi\)
0.780308 + 0.625396i \(0.215062\pi\)
\(242\) 5.79427 0.372470
\(243\) 0 0
\(244\) 1.80203 0.115363
\(245\) −3.26980 −0.208900
\(246\) 0 0
\(247\) 14.4859 0.921716
\(248\) 12.1829 0.773612
\(249\) 0 0
\(250\) 1.49804 0.0947444
\(251\) −12.6868 −0.800784 −0.400392 0.916344i \(-0.631126\pi\)
−0.400392 + 0.916344i \(0.631126\pi\)
\(252\) 0 0
\(253\) −7.02268 −0.441512
\(254\) −0.662421 −0.0415640
\(255\) 0 0
\(256\) −8.69711 −0.543569
\(257\) 9.00660 0.561816 0.280908 0.959735i \(-0.409364\pi\)
0.280908 + 0.959735i \(0.409364\pi\)
\(258\) 0 0
\(259\) 9.93232 0.617164
\(260\) −21.8067 −1.35240
\(261\) 0 0
\(262\) 4.70581 0.290726
\(263\) −0.569700 −0.0351292 −0.0175646 0.999846i \(-0.505591\pi\)
−0.0175646 + 0.999846i \(0.505591\pi\)
\(264\) 0 0
\(265\) 6.03938 0.370996
\(266\) −2.24631 −0.137730
\(267\) 0 0
\(268\) 22.4930 1.37398
\(269\) 8.24294 0.502581 0.251290 0.967912i \(-0.419145\pi\)
0.251290 + 0.967912i \(0.419145\pi\)
\(270\) 0 0
\(271\) −13.0643 −0.793597 −0.396799 0.917906i \(-0.629879\pi\)
−0.396799 + 0.917906i \(0.629879\pi\)
\(272\) −3.77899 −0.229135
\(273\) 0 0
\(274\) 5.94136 0.358930
\(275\) 8.54291 0.515157
\(276\) 0 0
\(277\) −26.0311 −1.56406 −0.782029 0.623242i \(-0.785815\pi\)
−0.782029 + 0.623242i \(0.785815\pi\)
\(278\) −2.18835 −0.131249
\(279\) 0 0
\(280\) 7.71351 0.460970
\(281\) 9.62664 0.574277 0.287139 0.957889i \(-0.407296\pi\)
0.287139 + 0.957889i \(0.407296\pi\)
\(282\) 0 0
\(283\) −6.55517 −0.389664 −0.194832 0.980837i \(-0.562416\pi\)
−0.194832 + 0.980837i \(0.562416\pi\)
\(284\) 18.2903 1.08533
\(285\) 0 0
\(286\) 4.24732 0.251149
\(287\) 1.42938 0.0843736
\(288\) 0 0
\(289\) −11.1298 −0.654696
\(290\) −5.39944 −0.317066
\(291\) 0 0
\(292\) −6.21164 −0.363508
\(293\) 18.1387 1.05968 0.529838 0.848099i \(-0.322253\pi\)
0.529838 + 0.848099i \(0.322253\pi\)
\(294\) 0 0
\(295\) 11.2490 0.654940
\(296\) −23.4305 −1.36187
\(297\) 0 0
\(298\) 4.43225 0.256753
\(299\) 19.9868 1.15587
\(300\) 0 0
\(301\) −0.721457 −0.0415841
\(302\) −10.2149 −0.587803
\(303\) 0 0
\(304\) −5.28915 −0.303353
\(305\) 3.77422 0.216111
\(306\) 0 0
\(307\) −19.1900 −1.09523 −0.547615 0.836731i \(-0.684464\pi\)
−0.547615 + 0.836731i \(0.684464\pi\)
\(308\) 2.34330 0.133522
\(309\) 0 0
\(310\) 11.1860 0.635322
\(311\) 7.90235 0.448101 0.224051 0.974577i \(-0.428072\pi\)
0.224051 + 0.974577i \(0.428072\pi\)
\(312\) 0 0
\(313\) −0.379749 −0.0214647 −0.0107324 0.999942i \(-0.503416\pi\)
−0.0107324 + 0.999942i \(0.503416\pi\)
\(314\) 5.48302 0.309425
\(315\) 0 0
\(316\) 5.69336 0.320276
\(317\) −4.64692 −0.260997 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(318\) 0 0
\(319\) −3.74165 −0.209492
\(320\) −2.25703 −0.126172
\(321\) 0 0
\(322\) −3.09932 −0.172719
\(323\) 8.21598 0.457150
\(324\) 0 0
\(325\) −24.3135 −1.34867
\(326\) −3.09676 −0.171514
\(327\) 0 0
\(328\) −3.37192 −0.186183
\(329\) −2.25922 −0.124555
\(330\) 0 0
\(331\) −2.44366 −0.134316 −0.0671578 0.997742i \(-0.521393\pi\)
−0.0671578 + 0.997742i \(0.521393\pi\)
\(332\) 19.5812 1.07466
\(333\) 0 0
\(334\) 7.24633 0.396502
\(335\) 47.1098 2.57388
\(336\) 0 0
\(337\) 27.6343 1.50533 0.752667 0.658402i \(-0.228767\pi\)
0.752667 + 0.658402i \(0.228767\pi\)
\(338\) −3.47657 −0.189100
\(339\) 0 0
\(340\) −12.3682 −0.670757
\(341\) 7.75154 0.419770
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.70193 0.0917617
\(345\) 0 0
\(346\) −4.08640 −0.219686
\(347\) −33.0149 −1.77233 −0.886167 0.463366i \(-0.846642\pi\)
−0.886167 + 0.463366i \(0.846642\pi\)
\(348\) 0 0
\(349\) 14.2869 0.764763 0.382381 0.924005i \(-0.375104\pi\)
0.382381 + 0.924005i \(0.375104\pi\)
\(350\) 3.77025 0.201528
\(351\) 0 0
\(352\) −8.63237 −0.460107
\(353\) 27.1604 1.44560 0.722800 0.691057i \(-0.242855\pi\)
0.722800 + 0.691057i \(0.242855\pi\)
\(354\) 0 0
\(355\) 38.3077 2.03316
\(356\) 21.8090 1.15587
\(357\) 0 0
\(358\) 5.49960 0.290663
\(359\) −27.3202 −1.44190 −0.720952 0.692984i \(-0.756295\pi\)
−0.720952 + 0.692984i \(0.756295\pi\)
\(360\) 0 0
\(361\) −7.50076 −0.394777
\(362\) −0.993262 −0.0522047
\(363\) 0 0
\(364\) −6.66913 −0.349557
\(365\) −13.0098 −0.680963
\(366\) 0 0
\(367\) 24.1868 1.26254 0.631271 0.775562i \(-0.282534\pi\)
0.631271 + 0.775562i \(0.282534\pi\)
\(368\) −7.29767 −0.380417
\(369\) 0 0
\(370\) −21.5133 −1.11842
\(371\) 1.84702 0.0958923
\(372\) 0 0
\(373\) 16.7437 0.866958 0.433479 0.901164i \(-0.357286\pi\)
0.433479 + 0.901164i \(0.357286\pi\)
\(374\) 2.40896 0.124564
\(375\) 0 0
\(376\) 5.32953 0.274849
\(377\) 10.6489 0.548445
\(378\) 0 0
\(379\) 33.7897 1.73566 0.867830 0.496861i \(-0.165514\pi\)
0.867830 + 0.496861i \(0.165514\pi\)
\(380\) −17.3107 −0.888020
\(381\) 0 0
\(382\) −0.508215 −0.0260026
\(383\) 12.8406 0.656125 0.328063 0.944656i \(-0.393604\pi\)
0.328063 + 0.944656i \(0.393604\pi\)
\(384\) 0 0
\(385\) 4.90785 0.250127
\(386\) 7.48761 0.381109
\(387\) 0 0
\(388\) 15.3417 0.778857
\(389\) 1.62314 0.0822967 0.0411484 0.999153i \(-0.486898\pi\)
0.0411484 + 0.999153i \(0.486898\pi\)
\(390\) 0 0
\(391\) 11.3359 0.573284
\(392\) 2.35901 0.119148
\(393\) 0 0
\(394\) 9.39741 0.473435
\(395\) 11.9243 0.599975
\(396\) 0 0
\(397\) 0.0450194 0.00225946 0.00112973 0.999999i \(-0.499640\pi\)
0.00112973 + 0.999999i \(0.499640\pi\)
\(398\) −3.16682 −0.158738
\(399\) 0 0
\(400\) 8.87743 0.443871
\(401\) −11.0217 −0.550398 −0.275199 0.961387i \(-0.588744\pi\)
−0.275199 + 0.961387i \(0.588744\pi\)
\(402\) 0 0
\(403\) −22.0612 −1.09895
\(404\) 3.30033 0.164197
\(405\) 0 0
\(406\) −1.65131 −0.0819529
\(407\) −14.9080 −0.738964
\(408\) 0 0
\(409\) 28.1749 1.39316 0.696580 0.717479i \(-0.254704\pi\)
0.696580 + 0.717479i \(0.254704\pi\)
\(410\) −3.09602 −0.152901
\(411\) 0 0
\(412\) −12.3816 −0.609999
\(413\) 3.44025 0.169284
\(414\) 0 0
\(415\) 41.0112 2.01316
\(416\) 24.5681 1.20455
\(417\) 0 0
\(418\) 3.37162 0.164911
\(419\) −7.08817 −0.346280 −0.173140 0.984897i \(-0.555391\pi\)
−0.173140 + 0.984897i \(0.555391\pi\)
\(420\) 0 0
\(421\) −3.66854 −0.178794 −0.0893969 0.995996i \(-0.528494\pi\)
−0.0893969 + 0.995996i \(0.528494\pi\)
\(422\) 1.03688 0.0504744
\(423\) 0 0
\(424\) −4.35714 −0.211601
\(425\) −13.7899 −0.668908
\(426\) 0 0
\(427\) 1.15426 0.0558587
\(428\) 14.4784 0.699839
\(429\) 0 0
\(430\) 1.56267 0.0753585
\(431\) 40.0756 1.93037 0.965187 0.261561i \(-0.0842371\pi\)
0.965187 + 0.261561i \(0.0842371\pi\)
\(432\) 0 0
\(433\) 5.97880 0.287323 0.143661 0.989627i \(-0.454112\pi\)
0.143661 + 0.989627i \(0.454112\pi\)
\(434\) 3.42100 0.164213
\(435\) 0 0
\(436\) −13.9212 −0.666706
\(437\) 15.8660 0.758974
\(438\) 0 0
\(439\) 16.8939 0.806302 0.403151 0.915134i \(-0.367915\pi\)
0.403151 + 0.915134i \(0.367915\pi\)
\(440\) −11.5777 −0.551945
\(441\) 0 0
\(442\) −6.85599 −0.326106
\(443\) −18.8237 −0.894341 −0.447171 0.894449i \(-0.647568\pi\)
−0.447171 + 0.894449i \(0.647568\pi\)
\(444\) 0 0
\(445\) 45.6772 2.16530
\(446\) 3.59345 0.170155
\(447\) 0 0
\(448\) −0.690263 −0.0326119
\(449\) 12.7387 0.601177 0.300588 0.953754i \(-0.402817\pi\)
0.300588 + 0.953754i \(0.402817\pi\)
\(450\) 0 0
\(451\) −2.14544 −0.101025
\(452\) 23.1223 1.08758
\(453\) 0 0
\(454\) −8.93411 −0.419298
\(455\) −13.9679 −0.654828
\(456\) 0 0
\(457\) 38.4582 1.79900 0.899500 0.436920i \(-0.143931\pi\)
0.899500 + 0.436920i \(0.143931\pi\)
\(458\) −17.6345 −0.824005
\(459\) 0 0
\(460\) −23.8843 −1.11361
\(461\) −12.2755 −0.571728 −0.285864 0.958270i \(-0.592281\pi\)
−0.285864 + 0.958270i \(0.592281\pi\)
\(462\) 0 0
\(463\) 23.8693 1.10930 0.554649 0.832084i \(-0.312853\pi\)
0.554649 + 0.832084i \(0.312853\pi\)
\(464\) −3.88816 −0.180503
\(465\) 0 0
\(466\) 8.14504 0.377311
\(467\) 11.5335 0.533708 0.266854 0.963737i \(-0.414016\pi\)
0.266854 + 0.963737i \(0.414016\pi\)
\(468\) 0 0
\(469\) 14.4075 0.665277
\(470\) 4.89344 0.225718
\(471\) 0 0
\(472\) −8.11560 −0.373551
\(473\) 1.08288 0.0497908
\(474\) 0 0
\(475\) −19.3006 −0.885572
\(476\) −3.78254 −0.173372
\(477\) 0 0
\(478\) 14.3914 0.658247
\(479\) −41.2351 −1.88408 −0.942041 0.335498i \(-0.891095\pi\)
−0.942041 + 0.335498i \(0.891095\pi\)
\(480\) 0 0
\(481\) 42.4289 1.93459
\(482\) −16.0487 −0.730996
\(483\) 0 0
\(484\) 13.6560 0.620726
\(485\) 32.1320 1.45904
\(486\) 0 0
\(487\) −25.0962 −1.13722 −0.568608 0.822608i \(-0.692518\pi\)
−0.568608 + 0.822608i \(0.692518\pi\)
\(488\) −2.72292 −0.123261
\(489\) 0 0
\(490\) 2.16599 0.0978493
\(491\) 30.7343 1.38702 0.693509 0.720448i \(-0.256064\pi\)
0.693509 + 0.720448i \(0.256064\pi\)
\(492\) 0 0
\(493\) 6.03974 0.272016
\(494\) −9.59577 −0.431734
\(495\) 0 0
\(496\) 8.05508 0.361684
\(497\) 11.7156 0.525516
\(498\) 0 0
\(499\) 37.8270 1.69337 0.846684 0.532097i \(-0.178596\pi\)
0.846684 + 0.532097i \(0.178596\pi\)
\(500\) 3.53059 0.157893
\(501\) 0 0
\(502\) 8.40400 0.375089
\(503\) −1.91290 −0.0852921 −0.0426461 0.999090i \(-0.513579\pi\)
−0.0426461 + 0.999090i \(0.513579\pi\)
\(504\) 0 0
\(505\) 6.91227 0.307592
\(506\) 4.65197 0.206805
\(507\) 0 0
\(508\) −1.56120 −0.0692670
\(509\) 31.9035 1.41410 0.707049 0.707164i \(-0.250026\pi\)
0.707049 + 0.707164i \(0.250026\pi\)
\(510\) 0 0
\(511\) −3.97876 −0.176010
\(512\) −16.3293 −0.721659
\(513\) 0 0
\(514\) −5.96616 −0.263156
\(515\) −25.9323 −1.14272
\(516\) 0 0
\(517\) 3.39100 0.149136
\(518\) −6.57938 −0.289081
\(519\) 0 0
\(520\) 32.9506 1.44498
\(521\) 40.7782 1.78652 0.893262 0.449536i \(-0.148411\pi\)
0.893262 + 0.449536i \(0.148411\pi\)
\(522\) 0 0
\(523\) −6.54698 −0.286280 −0.143140 0.989702i \(-0.545720\pi\)
−0.143140 + 0.989702i \(0.545720\pi\)
\(524\) 11.0907 0.484498
\(525\) 0 0
\(526\) 0.377382 0.0164546
\(527\) −12.5125 −0.545052
\(528\) 0 0
\(529\) −1.10899 −0.0482169
\(530\) −4.00061 −0.173776
\(531\) 0 0
\(532\) −5.29411 −0.229529
\(533\) 6.10602 0.264481
\(534\) 0 0
\(535\) 30.3238 1.31101
\(536\) −33.9875 −1.46804
\(537\) 0 0
\(538\) −5.46030 −0.235410
\(539\) 1.50096 0.0646510
\(540\) 0 0
\(541\) −14.3882 −0.618595 −0.309297 0.950965i \(-0.600094\pi\)
−0.309297 + 0.950965i \(0.600094\pi\)
\(542\) 8.65404 0.371723
\(543\) 0 0
\(544\) 13.9343 0.597429
\(545\) −29.1569 −1.24894
\(546\) 0 0
\(547\) −38.8696 −1.66194 −0.830972 0.556314i \(-0.812215\pi\)
−0.830972 + 0.556314i \(0.812215\pi\)
\(548\) 14.0026 0.598163
\(549\) 0 0
\(550\) −5.65900 −0.241301
\(551\) 8.45333 0.360124
\(552\) 0 0
\(553\) 3.64679 0.155077
\(554\) 17.2436 0.732609
\(555\) 0 0
\(556\) −5.15752 −0.218728
\(557\) −4.06639 −0.172299 −0.0861493 0.996282i \(-0.527456\pi\)
−0.0861493 + 0.996282i \(0.527456\pi\)
\(558\) 0 0
\(559\) −3.08192 −0.130351
\(560\) 5.10003 0.215516
\(561\) 0 0
\(562\) −6.37689 −0.268993
\(563\) −4.72486 −0.199129 −0.0995646 0.995031i \(-0.531745\pi\)
−0.0995646 + 0.995031i \(0.531745\pi\)
\(564\) 0 0
\(565\) 48.4278 2.03737
\(566\) 4.34228 0.182520
\(567\) 0 0
\(568\) −27.6372 −1.15963
\(569\) −6.56790 −0.275341 −0.137670 0.990478i \(-0.543961\pi\)
−0.137670 + 0.990478i \(0.543961\pi\)
\(570\) 0 0
\(571\) 9.73369 0.407342 0.203671 0.979039i \(-0.434713\pi\)
0.203671 + 0.979039i \(0.434713\pi\)
\(572\) 10.0101 0.418543
\(573\) 0 0
\(574\) −0.946851 −0.0395208
\(575\) −26.6299 −1.11054
\(576\) 0 0
\(577\) 8.68524 0.361571 0.180786 0.983523i \(-0.442136\pi\)
0.180786 + 0.983523i \(0.442136\pi\)
\(578\) 7.37264 0.306661
\(579\) 0 0
\(580\) −12.7254 −0.528395
\(581\) 12.5424 0.520346
\(582\) 0 0
\(583\) −2.77230 −0.114817
\(584\) 9.38595 0.388393
\(585\) 0 0
\(586\) −12.0155 −0.496355
\(587\) −28.0699 −1.15857 −0.579285 0.815125i \(-0.696668\pi\)
−0.579285 + 0.815125i \(0.696668\pi\)
\(588\) 0 0
\(589\) −17.5127 −0.721598
\(590\) −7.45155 −0.306775
\(591\) 0 0
\(592\) −15.4918 −0.636709
\(593\) 17.6643 0.725385 0.362693 0.931909i \(-0.381858\pi\)
0.362693 + 0.931909i \(0.381858\pi\)
\(594\) 0 0
\(595\) −7.92222 −0.324779
\(596\) 10.4459 0.427883
\(597\) 0 0
\(598\) −13.2397 −0.541411
\(599\) 19.1441 0.782205 0.391102 0.920347i \(-0.372094\pi\)
0.391102 + 0.920347i \(0.372094\pi\)
\(600\) 0 0
\(601\) −39.2226 −1.59992 −0.799961 0.600052i \(-0.795147\pi\)
−0.799961 + 0.600052i \(0.795147\pi\)
\(602\) 0.477908 0.0194781
\(603\) 0 0
\(604\) −24.0746 −0.979581
\(605\) 28.6013 1.16281
\(606\) 0 0
\(607\) 0.762927 0.0309662 0.0154831 0.999880i \(-0.495071\pi\)
0.0154831 + 0.999880i \(0.495071\pi\)
\(608\) 19.5027 0.790940
\(609\) 0 0
\(610\) −2.50012 −0.101227
\(611\) −9.65093 −0.390435
\(612\) 0 0
\(613\) −21.3445 −0.862097 −0.431048 0.902329i \(-0.641856\pi\)
−0.431048 + 0.902329i \(0.641856\pi\)
\(614\) 12.7118 0.513008
\(615\) 0 0
\(616\) −3.54079 −0.142662
\(617\) −16.2462 −0.654046 −0.327023 0.945016i \(-0.606046\pi\)
−0.327023 + 0.945016i \(0.606046\pi\)
\(618\) 0 0
\(619\) 25.6478 1.03087 0.515436 0.856928i \(-0.327630\pi\)
0.515436 + 0.856928i \(0.327630\pi\)
\(620\) 26.3632 1.05877
\(621\) 0 0
\(622\) −5.23469 −0.209892
\(623\) 13.9694 0.559672
\(624\) 0 0
\(625\) −21.0636 −0.842542
\(626\) 0.251554 0.0100541
\(627\) 0 0
\(628\) 12.9224 0.515660
\(629\) 24.0644 0.959512
\(630\) 0 0
\(631\) −4.71692 −0.187777 −0.0938887 0.995583i \(-0.529930\pi\)
−0.0938887 + 0.995583i \(0.529930\pi\)
\(632\) −8.60282 −0.342202
\(633\) 0 0
\(634\) 3.07822 0.122252
\(635\) −3.26980 −0.129758
\(636\) 0 0
\(637\) −4.27180 −0.169255
\(638\) 2.47855 0.0981266
\(639\) 0 0
\(640\) −36.1157 −1.42760
\(641\) 1.83451 0.0724588 0.0362294 0.999344i \(-0.488465\pi\)
0.0362294 + 0.999344i \(0.488465\pi\)
\(642\) 0 0
\(643\) −29.1448 −1.14936 −0.574679 0.818379i \(-0.694873\pi\)
−0.574679 + 0.818379i \(0.694873\pi\)
\(644\) −7.30451 −0.287838
\(645\) 0 0
\(646\) −5.44244 −0.214130
\(647\) −20.8372 −0.819197 −0.409598 0.912266i \(-0.634331\pi\)
−0.409598 + 0.912266i \(0.634331\pi\)
\(648\) 0 0
\(649\) −5.16369 −0.202693
\(650\) 16.1058 0.631720
\(651\) 0 0
\(652\) −7.29846 −0.285830
\(653\) 13.0640 0.511235 0.255617 0.966778i \(-0.417721\pi\)
0.255617 + 0.966778i \(0.417721\pi\)
\(654\) 0 0
\(655\) 23.2285 0.907613
\(656\) −2.22945 −0.0870456
\(657\) 0 0
\(658\) 1.49655 0.0583418
\(659\) 9.10837 0.354812 0.177406 0.984138i \(-0.443230\pi\)
0.177406 + 0.984138i \(0.443230\pi\)
\(660\) 0 0
\(661\) 11.0180 0.428549 0.214274 0.976774i \(-0.431261\pi\)
0.214274 + 0.976774i \(0.431261\pi\)
\(662\) 1.61873 0.0629137
\(663\) 0 0
\(664\) −29.5877 −1.14822
\(665\) −11.0881 −0.429977
\(666\) 0 0
\(667\) 11.6634 0.451610
\(668\) 17.0782 0.660775
\(669\) 0 0
\(670\) −31.2065 −1.20561
\(671\) −1.73251 −0.0668827
\(672\) 0 0
\(673\) 20.0887 0.774362 0.387181 0.922004i \(-0.373449\pi\)
0.387181 + 0.922004i \(0.373449\pi\)
\(674\) −18.3055 −0.705102
\(675\) 0 0
\(676\) −8.19359 −0.315138
\(677\) −40.4365 −1.55410 −0.777051 0.629438i \(-0.783285\pi\)
−0.777051 + 0.629438i \(0.783285\pi\)
\(678\) 0 0
\(679\) 9.82688 0.377121
\(680\) 18.6886 0.716676
\(681\) 0 0
\(682\) −5.13479 −0.196621
\(683\) −29.3606 −1.12345 −0.561727 0.827323i \(-0.689863\pi\)
−0.561727 + 0.827323i \(0.689863\pi\)
\(684\) 0 0
\(685\) 29.3274 1.12054
\(686\) 0.662421 0.0252914
\(687\) 0 0
\(688\) 1.12528 0.0429010
\(689\) 7.89008 0.300588
\(690\) 0 0
\(691\) −46.0332 −1.75118 −0.875592 0.483051i \(-0.839528\pi\)
−0.875592 + 0.483051i \(0.839528\pi\)
\(692\) −9.63086 −0.366110
\(693\) 0 0
\(694\) 21.8698 0.830166
\(695\) −10.8020 −0.409744
\(696\) 0 0
\(697\) 3.46316 0.131177
\(698\) −9.46397 −0.358217
\(699\) 0 0
\(700\) 8.88575 0.335850
\(701\) −17.3356 −0.654756 −0.327378 0.944893i \(-0.606165\pi\)
−0.327378 + 0.944893i \(0.606165\pi\)
\(702\) 0 0
\(703\) 33.6810 1.27030
\(704\) 1.03606 0.0390479
\(705\) 0 0
\(706\) −17.9916 −0.677123
\(707\) 2.11397 0.0795041
\(708\) 0 0
\(709\) 11.4015 0.428194 0.214097 0.976812i \(-0.431319\pi\)
0.214097 + 0.976812i \(0.431319\pi\)
\(710\) −25.3758 −0.952337
\(711\) 0 0
\(712\) −32.9540 −1.23500
\(713\) −24.1630 −0.904913
\(714\) 0 0
\(715\) 20.9654 0.784060
\(716\) 12.9615 0.484393
\(717\) 0 0
\(718\) 18.0975 0.675392
\(719\) −10.3304 −0.385258 −0.192629 0.981272i \(-0.561701\pi\)
−0.192629 + 0.981272i \(0.561701\pi\)
\(720\) 0 0
\(721\) −7.93085 −0.295360
\(722\) 4.96866 0.184914
\(723\) 0 0
\(724\) −2.34092 −0.0869998
\(725\) −14.1883 −0.526939
\(726\) 0 0
\(727\) 39.8392 1.47755 0.738777 0.673950i \(-0.235404\pi\)
0.738777 + 0.673950i \(0.235404\pi\)
\(728\) 10.0772 0.373487
\(729\) 0 0
\(730\) 8.61795 0.318965
\(731\) −1.74798 −0.0646512
\(732\) 0 0
\(733\) 6.61936 0.244492 0.122246 0.992500i \(-0.460990\pi\)
0.122246 + 0.992500i \(0.460990\pi\)
\(734\) −16.0219 −0.591378
\(735\) 0 0
\(736\) 26.9087 0.991870
\(737\) −21.6251 −0.796572
\(738\) 0 0
\(739\) 2.68575 0.0987971 0.0493985 0.998779i \(-0.484270\pi\)
0.0493985 + 0.998779i \(0.484270\pi\)
\(740\) −50.7026 −1.86387
\(741\) 0 0
\(742\) −1.22350 −0.0449162
\(743\) 0.915186 0.0335749 0.0167875 0.999859i \(-0.494656\pi\)
0.0167875 + 0.999859i \(0.494656\pi\)
\(744\) 0 0
\(745\) 21.8782 0.801555
\(746\) −11.0914 −0.406085
\(747\) 0 0
\(748\) 5.67744 0.207588
\(749\) 9.27389 0.338861
\(750\) 0 0
\(751\) 24.3489 0.888505 0.444252 0.895902i \(-0.353469\pi\)
0.444252 + 0.895902i \(0.353469\pi\)
\(752\) 3.52379 0.128499
\(753\) 0 0
\(754\) −7.05404 −0.256893
\(755\) −50.4223 −1.83505
\(756\) 0 0
\(757\) −42.8153 −1.55615 −0.778074 0.628173i \(-0.783803\pi\)
−0.778074 + 0.628173i \(0.783803\pi\)
\(758\) −22.3830 −0.812987
\(759\) 0 0
\(760\) 26.1569 0.948812
\(761\) −21.3773 −0.774927 −0.387463 0.921885i \(-0.626649\pi\)
−0.387463 + 0.921885i \(0.626649\pi\)
\(762\) 0 0
\(763\) −8.91701 −0.322818
\(764\) −1.19776 −0.0433336
\(765\) 0 0
\(766\) −8.50590 −0.307331
\(767\) 14.6961 0.530645
\(768\) 0 0
\(769\) −8.30249 −0.299395 −0.149698 0.988732i \(-0.547830\pi\)
−0.149698 + 0.988732i \(0.547830\pi\)
\(770\) −3.25106 −0.117160
\(771\) 0 0
\(772\) 17.6468 0.635124
\(773\) 24.0880 0.866384 0.433192 0.901302i \(-0.357387\pi\)
0.433192 + 0.901302i \(0.357387\pi\)
\(774\) 0 0
\(775\) 29.3937 1.05585
\(776\) −23.1817 −0.832176
\(777\) 0 0
\(778\) −1.07521 −0.0385480
\(779\) 4.84710 0.173665
\(780\) 0 0
\(781\) −17.5846 −0.629228
\(782\) −7.50917 −0.268527
\(783\) 0 0
\(784\) 1.55974 0.0557049
\(785\) 27.0650 0.965990
\(786\) 0 0
\(787\) 29.2925 1.04416 0.522082 0.852895i \(-0.325156\pi\)
0.522082 + 0.852895i \(0.325156\pi\)
\(788\) 22.1479 0.788985
\(789\) 0 0
\(790\) −7.89889 −0.281030
\(791\) 14.8106 0.526605
\(792\) 0 0
\(793\) 4.93078 0.175097
\(794\) −0.0298218 −0.00105834
\(795\) 0 0
\(796\) −7.46358 −0.264540
\(797\) −14.0851 −0.498919 −0.249460 0.968385i \(-0.580253\pi\)
−0.249460 + 0.968385i \(0.580253\pi\)
\(798\) 0 0
\(799\) −5.47373 −0.193647
\(800\) −32.7338 −1.15731
\(801\) 0 0
\(802\) 7.30102 0.257808
\(803\) 5.97197 0.210746
\(804\) 0 0
\(805\) −15.2987 −0.539208
\(806\) 14.6138 0.514750
\(807\) 0 0
\(808\) −4.98689 −0.175438
\(809\) −15.5617 −0.547121 −0.273561 0.961855i \(-0.588201\pi\)
−0.273561 + 0.961855i \(0.588201\pi\)
\(810\) 0 0
\(811\) −34.6939 −1.21827 −0.609133 0.793068i \(-0.708483\pi\)
−0.609133 + 0.793068i \(0.708483\pi\)
\(812\) −3.89181 −0.136576
\(813\) 0 0
\(814\) 9.87540 0.346133
\(815\) −15.2860 −0.535447
\(816\) 0 0
\(817\) −2.44650 −0.0855922
\(818\) −18.6637 −0.652560
\(819\) 0 0
\(820\) −7.29671 −0.254812
\(821\) 53.7305 1.87521 0.937604 0.347705i \(-0.113039\pi\)
0.937604 + 0.347705i \(0.113039\pi\)
\(822\) 0 0
\(823\) 31.8700 1.11092 0.555458 0.831544i \(-0.312543\pi\)
0.555458 + 0.831544i \(0.312543\pi\)
\(824\) 18.7090 0.651759
\(825\) 0 0
\(826\) −2.27890 −0.0792930
\(827\) −47.6852 −1.65818 −0.829089 0.559117i \(-0.811140\pi\)
−0.829089 + 0.559117i \(0.811140\pi\)
\(828\) 0 0
\(829\) −21.6153 −0.750731 −0.375365 0.926877i \(-0.622483\pi\)
−0.375365 + 0.926877i \(0.622483\pi\)
\(830\) −27.1667 −0.942969
\(831\) 0 0
\(832\) −2.94867 −0.102227
\(833\) −2.42284 −0.0839465
\(834\) 0 0
\(835\) 35.7689 1.23783
\(836\) 7.94625 0.274827
\(837\) 0 0
\(838\) 4.69536 0.162198
\(839\) −27.1939 −0.938838 −0.469419 0.882976i \(-0.655537\pi\)
−0.469419 + 0.882976i \(0.655537\pi\)
\(840\) 0 0
\(841\) −22.7858 −0.785717
\(842\) 2.43012 0.0837475
\(843\) 0 0
\(844\) 2.44372 0.0841162
\(845\) −17.1608 −0.590350
\(846\) 0 0
\(847\) 8.74711 0.300554
\(848\) −2.88086 −0.0989291
\(849\) 0 0
\(850\) 9.13472 0.313318
\(851\) 46.4712 1.59301
\(852\) 0 0
\(853\) 3.03848 0.104035 0.0520177 0.998646i \(-0.483435\pi\)
0.0520177 + 0.998646i \(0.483435\pi\)
\(854\) −0.764609 −0.0261644
\(855\) 0 0
\(856\) −21.8772 −0.747748
\(857\) 29.6832 1.01396 0.506979 0.861959i \(-0.330762\pi\)
0.506979 + 0.861959i \(0.330762\pi\)
\(858\) 0 0
\(859\) −44.7450 −1.52668 −0.763339 0.645998i \(-0.776442\pi\)
−0.763339 + 0.645998i \(0.776442\pi\)
\(860\) 3.68290 0.125586
\(861\) 0 0
\(862\) −26.5469 −0.904192
\(863\) 32.0831 1.09212 0.546060 0.837746i \(-0.316127\pi\)
0.546060 + 0.837746i \(0.316127\pi\)
\(864\) 0 0
\(865\) −20.1711 −0.685837
\(866\) −3.96048 −0.134583
\(867\) 0 0
\(868\) 8.06263 0.273663
\(869\) −5.47369 −0.185682
\(870\) 0 0
\(871\) 61.5460 2.08541
\(872\) 21.0354 0.712347
\(873\) 0 0
\(874\) −10.5100 −0.355505
\(875\) 2.26146 0.0764514
\(876\) 0 0
\(877\) −42.1566 −1.42353 −0.711763 0.702420i \(-0.752103\pi\)
−0.711763 + 0.702420i \(0.752103\pi\)
\(878\) −11.1909 −0.377674
\(879\) 0 0
\(880\) −7.65495 −0.258048
\(881\) −3.01953 −0.101731 −0.0508653 0.998706i \(-0.516198\pi\)
−0.0508653 + 0.998706i \(0.516198\pi\)
\(882\) 0 0
\(883\) −48.2690 −1.62438 −0.812190 0.583393i \(-0.801725\pi\)
−0.812190 + 0.583393i \(0.801725\pi\)
\(884\) −16.1582 −0.543460
\(885\) 0 0
\(886\) 12.4692 0.418912
\(887\) 17.5918 0.590676 0.295338 0.955393i \(-0.404568\pi\)
0.295338 + 0.955393i \(0.404568\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −30.2575 −1.01423
\(891\) 0 0
\(892\) 8.46906 0.283565
\(893\) −7.66113 −0.256370
\(894\) 0 0
\(895\) 27.1468 0.907417
\(896\) −11.0452 −0.368995
\(897\) 0 0
\(898\) −8.43839 −0.281593
\(899\) −12.8739 −0.429370
\(900\) 0 0
\(901\) 4.47503 0.149085
\(902\) 1.42119 0.0473204
\(903\) 0 0
\(904\) −34.9384 −1.16203
\(905\) −4.90288 −0.162977
\(906\) 0 0
\(907\) −4.41456 −0.146583 −0.0732915 0.997311i \(-0.523350\pi\)
−0.0732915 + 0.997311i \(0.523350\pi\)
\(908\) −21.0560 −0.698766
\(909\) 0 0
\(910\) 9.25266 0.306723
\(911\) 21.8078 0.722525 0.361263 0.932464i \(-0.382346\pi\)
0.361263 + 0.932464i \(0.382346\pi\)
\(912\) 0 0
\(913\) −18.8257 −0.623038
\(914\) −25.4756 −0.842656
\(915\) 0 0
\(916\) −41.5610 −1.37321
\(917\) 7.10395 0.234593
\(918\) 0 0
\(919\) 44.9714 1.48347 0.741735 0.670693i \(-0.234003\pi\)
0.741735 + 0.670693i \(0.234003\pi\)
\(920\) 36.0898 1.18985
\(921\) 0 0
\(922\) 8.13157 0.267799
\(923\) 50.0466 1.64730
\(924\) 0 0
\(925\) −56.5310 −1.85873
\(926\) −15.8115 −0.519598
\(927\) 0 0
\(928\) 14.3369 0.470630
\(929\) −6.22681 −0.204295 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(930\) 0 0
\(931\) −3.39105 −0.111137
\(932\) 19.1963 0.628795
\(933\) 0 0
\(934\) −7.64005 −0.249990
\(935\) 11.8909 0.388876
\(936\) 0 0
\(937\) 29.3666 0.959365 0.479683 0.877442i \(-0.340752\pi\)
0.479683 + 0.877442i \(0.340752\pi\)
\(938\) −9.54384 −0.311618
\(939\) 0 0
\(940\) 11.5329 0.376161
\(941\) 20.1629 0.657290 0.328645 0.944453i \(-0.393408\pi\)
0.328645 + 0.944453i \(0.393408\pi\)
\(942\) 0 0
\(943\) 6.68776 0.217783
\(944\) −5.36589 −0.174645
\(945\) 0 0
\(946\) −0.717322 −0.0233222
\(947\) 5.62411 0.182759 0.0913796 0.995816i \(-0.470872\pi\)
0.0913796 + 0.995816i \(0.470872\pi\)
\(948\) 0 0
\(949\) −16.9965 −0.551729
\(950\) 12.7851 0.414804
\(951\) 0 0
\(952\) 5.71551 0.185241
\(953\) −42.2249 −1.36780 −0.683899 0.729577i \(-0.739717\pi\)
−0.683899 + 0.729577i \(0.739717\pi\)
\(954\) 0 0
\(955\) −2.50862 −0.0811771
\(956\) 33.9177 1.09698
\(957\) 0 0
\(958\) 27.3150 0.882509
\(959\) 8.96915 0.289629
\(960\) 0 0
\(961\) −4.32913 −0.139649
\(962\) −28.1058 −0.906167
\(963\) 0 0
\(964\) −37.8236 −1.21822
\(965\) 36.9599 1.18978
\(966\) 0 0
\(967\) 6.32562 0.203418 0.101709 0.994814i \(-0.467569\pi\)
0.101709 + 0.994814i \(0.467569\pi\)
\(968\) −20.6346 −0.663220
\(969\) 0 0
\(970\) −21.2849 −0.683417
\(971\) −55.0518 −1.76670 −0.883348 0.468717i \(-0.844716\pi\)
−0.883348 + 0.468717i \(0.844716\pi\)
\(972\) 0 0
\(973\) −3.30357 −0.105907
\(974\) 16.6242 0.532675
\(975\) 0 0
\(976\) −1.80035 −0.0576277
\(977\) 1.20715 0.0386200 0.0193100 0.999814i \(-0.493853\pi\)
0.0193100 + 0.999814i \(0.493853\pi\)
\(978\) 0 0
\(979\) −20.9675 −0.670125
\(980\) 5.10481 0.163067
\(981\) 0 0
\(982\) −20.3590 −0.649683
\(983\) 5.96804 0.190351 0.0951755 0.995461i \(-0.469659\pi\)
0.0951755 + 0.995461i \(0.469659\pi\)
\(984\) 0 0
\(985\) 46.3869 1.47801
\(986\) −4.00085 −0.127413
\(987\) 0 0
\(988\) −22.6154 −0.719491
\(989\) −3.37554 −0.107336
\(990\) 0 0
\(991\) 16.4137 0.521398 0.260699 0.965420i \(-0.416047\pi\)
0.260699 + 0.965420i \(0.416047\pi\)
\(992\) −29.7016 −0.943025
\(993\) 0 0
\(994\) −7.76065 −0.246153
\(995\) −15.6319 −0.495564
\(996\) 0 0
\(997\) −3.64536 −0.115450 −0.0577249 0.998333i \(-0.518385\pi\)
−0.0577249 + 0.998333i \(0.518385\pi\)
\(998\) −25.0574 −0.793177
\(999\) 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.1.11 28
3.2 odd 2 inner 8001.2.a.y.1.18 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.y.1.11 28 1.1 even 1 trivial
8001.2.a.y.1.18 yes 28 3.2 odd 2 inner