Properties

Label 8048.2.a.y.1.5
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $0$
Dimension $33$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, 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("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(0\)
Dimension: \(33\)
Twist minimal: no (minimal twist has level 4024)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.60209 q^{3} +1.12945 q^{5} +1.15113 q^{7} +3.77086 q^{9} +O(q^{10})\) \(q-2.60209 q^{3} +1.12945 q^{5} +1.15113 q^{7} +3.77086 q^{9} -4.62351 q^{11} +6.29696 q^{13} -2.93893 q^{15} -4.65086 q^{17} -4.55120 q^{19} -2.99535 q^{21} -8.09579 q^{23} -3.72434 q^{25} -2.00585 q^{27} -4.35654 q^{29} +3.23462 q^{31} +12.0308 q^{33} +1.30015 q^{35} -1.34814 q^{37} -16.3852 q^{39} -0.554523 q^{41} -2.45396 q^{43} +4.25900 q^{45} +4.98010 q^{47} -5.67489 q^{49} +12.1019 q^{51} +11.5645 q^{53} -5.22202 q^{55} +11.8426 q^{57} +8.43649 q^{59} -0.328087 q^{61} +4.34076 q^{63} +7.11209 q^{65} -2.55790 q^{67} +21.0659 q^{69} +9.67099 q^{71} -1.53114 q^{73} +9.69107 q^{75} -5.32227 q^{77} +12.1977 q^{79} -6.09319 q^{81} -4.49665 q^{83} -5.25291 q^{85} +11.3361 q^{87} +13.9749 q^{89} +7.24863 q^{91} -8.41677 q^{93} -5.14035 q^{95} +14.5290 q^{97} -17.4346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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
\(3\) −2.60209 −1.50232 −0.751158 0.660122i \(-0.770504\pi\)
−0.751158 + 0.660122i \(0.770504\pi\)
\(4\) 0 0
\(5\) 1.12945 0.505105 0.252553 0.967583i \(-0.418730\pi\)
0.252553 + 0.967583i \(0.418730\pi\)
\(6\) 0 0
\(7\) 1.15113 0.435087 0.217544 0.976051i \(-0.430196\pi\)
0.217544 + 0.976051i \(0.430196\pi\)
\(8\) 0 0
\(9\) 3.77086 1.25695
\(10\) 0 0
\(11\) −4.62351 −1.39404 −0.697020 0.717052i \(-0.745491\pi\)
−0.697020 + 0.717052i \(0.745491\pi\)
\(12\) 0 0
\(13\) 6.29696 1.74646 0.873231 0.487307i \(-0.162021\pi\)
0.873231 + 0.487307i \(0.162021\pi\)
\(14\) 0 0
\(15\) −2.93893 −0.758828
\(16\) 0 0
\(17\) −4.65086 −1.12800 −0.564000 0.825775i \(-0.690738\pi\)
−0.564000 + 0.825775i \(0.690738\pi\)
\(18\) 0 0
\(19\) −4.55120 −1.04412 −0.522058 0.852910i \(-0.674836\pi\)
−0.522058 + 0.852910i \(0.674836\pi\)
\(20\) 0 0
\(21\) −2.99535 −0.653638
\(22\) 0 0
\(23\) −8.09579 −1.68809 −0.844044 0.536274i \(-0.819831\pi\)
−0.844044 + 0.536274i \(0.819831\pi\)
\(24\) 0 0
\(25\) −3.72434 −0.744869
\(26\) 0 0
\(27\) −2.00585 −0.386025
\(28\) 0 0
\(29\) −4.35654 −0.808989 −0.404494 0.914541i \(-0.632552\pi\)
−0.404494 + 0.914541i \(0.632552\pi\)
\(30\) 0 0
\(31\) 3.23462 0.580955 0.290477 0.956882i \(-0.406186\pi\)
0.290477 + 0.956882i \(0.406186\pi\)
\(32\) 0 0
\(33\) 12.0308 2.09429
\(34\) 0 0
\(35\) 1.30015 0.219765
\(36\) 0 0
\(37\) −1.34814 −0.221632 −0.110816 0.993841i \(-0.535346\pi\)
−0.110816 + 0.993841i \(0.535346\pi\)
\(38\) 0 0
\(39\) −16.3852 −2.62374
\(40\) 0 0
\(41\) −0.554523 −0.0866019 −0.0433010 0.999062i \(-0.513787\pi\)
−0.0433010 + 0.999062i \(0.513787\pi\)
\(42\) 0 0
\(43\) −2.45396 −0.374225 −0.187113 0.982338i \(-0.559913\pi\)
−0.187113 + 0.982338i \(0.559913\pi\)
\(44\) 0 0
\(45\) 4.25900 0.634894
\(46\) 0 0
\(47\) 4.98010 0.726423 0.363211 0.931707i \(-0.381680\pi\)
0.363211 + 0.931707i \(0.381680\pi\)
\(48\) 0 0
\(49\) −5.67489 −0.810699
\(50\) 0 0
\(51\) 12.1019 1.69461
\(52\) 0 0
\(53\) 11.5645 1.58851 0.794253 0.607587i \(-0.207862\pi\)
0.794253 + 0.607587i \(0.207862\pi\)
\(54\) 0 0
\(55\) −5.22202 −0.704137
\(56\) 0 0
\(57\) 11.8426 1.56859
\(58\) 0 0
\(59\) 8.43649 1.09834 0.549169 0.835712i \(-0.314944\pi\)
0.549169 + 0.835712i \(0.314944\pi\)
\(60\) 0 0
\(61\) −0.328087 −0.0420073 −0.0210036 0.999779i \(-0.506686\pi\)
−0.0210036 + 0.999779i \(0.506686\pi\)
\(62\) 0 0
\(63\) 4.34076 0.546884
\(64\) 0 0
\(65\) 7.11209 0.882147
\(66\) 0 0
\(67\) −2.55790 −0.312498 −0.156249 0.987718i \(-0.549940\pi\)
−0.156249 + 0.987718i \(0.549940\pi\)
\(68\) 0 0
\(69\) 21.0659 2.53604
\(70\) 0 0
\(71\) 9.67099 1.14774 0.573868 0.818948i \(-0.305442\pi\)
0.573868 + 0.818948i \(0.305442\pi\)
\(72\) 0 0
\(73\) −1.53114 −0.179206 −0.0896031 0.995978i \(-0.528560\pi\)
−0.0896031 + 0.995978i \(0.528560\pi\)
\(74\) 0 0
\(75\) 9.69107 1.11903
\(76\) 0 0
\(77\) −5.32227 −0.606529
\(78\) 0 0
\(79\) 12.1977 1.37235 0.686176 0.727436i \(-0.259288\pi\)
0.686176 + 0.727436i \(0.259288\pi\)
\(80\) 0 0
\(81\) −6.09319 −0.677021
\(82\) 0 0
\(83\) −4.49665 −0.493571 −0.246786 0.969070i \(-0.579374\pi\)
−0.246786 + 0.969070i \(0.579374\pi\)
\(84\) 0 0
\(85\) −5.25291 −0.569758
\(86\) 0 0
\(87\) 11.3361 1.21536
\(88\) 0 0
\(89\) 13.9749 1.48133 0.740667 0.671872i \(-0.234510\pi\)
0.740667 + 0.671872i \(0.234510\pi\)
\(90\) 0 0
\(91\) 7.24863 0.759863
\(92\) 0 0
\(93\) −8.41677 −0.872778
\(94\) 0 0
\(95\) −5.14035 −0.527389
\(96\) 0 0
\(97\) 14.5290 1.47519 0.737596 0.675242i \(-0.235961\pi\)
0.737596 + 0.675242i \(0.235961\pi\)
\(98\) 0 0
\(99\) −17.4346 −1.75224
\(100\) 0 0
\(101\) −12.3268 −1.22657 −0.613283 0.789863i \(-0.710151\pi\)
−0.613283 + 0.789863i \(0.710151\pi\)
\(102\) 0 0
\(103\) −11.2957 −1.11300 −0.556501 0.830847i \(-0.687856\pi\)
−0.556501 + 0.830847i \(0.687856\pi\)
\(104\) 0 0
\(105\) −3.38309 −0.330156
\(106\) 0 0
\(107\) −18.0989 −1.74969 −0.874845 0.484402i \(-0.839037\pi\)
−0.874845 + 0.484402i \(0.839037\pi\)
\(108\) 0 0
\(109\) −7.42098 −0.710801 −0.355400 0.934714i \(-0.615655\pi\)
−0.355400 + 0.934714i \(0.615655\pi\)
\(110\) 0 0
\(111\) 3.50797 0.332962
\(112\) 0 0
\(113\) −15.9717 −1.50249 −0.751246 0.660022i \(-0.770547\pi\)
−0.751246 + 0.660022i \(0.770547\pi\)
\(114\) 0 0
\(115\) −9.14378 −0.852662
\(116\) 0 0
\(117\) 23.7449 2.19522
\(118\) 0 0
\(119\) −5.35376 −0.490778
\(120\) 0 0
\(121\) 10.3768 0.943347
\(122\) 0 0
\(123\) 1.44292 0.130103
\(124\) 0 0
\(125\) −9.85371 −0.881342
\(126\) 0 0
\(127\) −13.6115 −1.20783 −0.603914 0.797050i \(-0.706393\pi\)
−0.603914 + 0.797050i \(0.706393\pi\)
\(128\) 0 0
\(129\) 6.38542 0.562205
\(130\) 0 0
\(131\) 16.1024 1.40688 0.703438 0.710757i \(-0.251647\pi\)
0.703438 + 0.710757i \(0.251647\pi\)
\(132\) 0 0
\(133\) −5.23903 −0.454282
\(134\) 0 0
\(135\) −2.26550 −0.194983
\(136\) 0 0
\(137\) 4.35245 0.371854 0.185927 0.982564i \(-0.440471\pi\)
0.185927 + 0.982564i \(0.440471\pi\)
\(138\) 0 0
\(139\) 14.7434 1.25052 0.625260 0.780416i \(-0.284993\pi\)
0.625260 + 0.780416i \(0.284993\pi\)
\(140\) 0 0
\(141\) −12.9587 −1.09132
\(142\) 0 0
\(143\) −29.1140 −2.43464
\(144\) 0 0
\(145\) −4.92049 −0.408624
\(146\) 0 0
\(147\) 14.7666 1.21793
\(148\) 0 0
\(149\) 7.64454 0.626265 0.313133 0.949709i \(-0.398622\pi\)
0.313133 + 0.949709i \(0.398622\pi\)
\(150\) 0 0
\(151\) −9.45004 −0.769033 −0.384517 0.923118i \(-0.625632\pi\)
−0.384517 + 0.923118i \(0.625632\pi\)
\(152\) 0 0
\(153\) −17.5377 −1.41784
\(154\) 0 0
\(155\) 3.65334 0.293443
\(156\) 0 0
\(157\) 18.8117 1.50133 0.750667 0.660681i \(-0.229732\pi\)
0.750667 + 0.660681i \(0.229732\pi\)
\(158\) 0 0
\(159\) −30.0919 −2.38644
\(160\) 0 0
\(161\) −9.31932 −0.734465
\(162\) 0 0
\(163\) 23.3994 1.83278 0.916390 0.400287i \(-0.131090\pi\)
0.916390 + 0.400287i \(0.131090\pi\)
\(164\) 0 0
\(165\) 13.5881 1.05784
\(166\) 0 0
\(167\) −10.5886 −0.819374 −0.409687 0.912226i \(-0.634362\pi\)
−0.409687 + 0.912226i \(0.634362\pi\)
\(168\) 0 0
\(169\) 26.6517 2.05013
\(170\) 0 0
\(171\) −17.1619 −1.31241
\(172\) 0 0
\(173\) 24.8869 1.89212 0.946058 0.323998i \(-0.105027\pi\)
0.946058 + 0.323998i \(0.105027\pi\)
\(174\) 0 0
\(175\) −4.28721 −0.324083
\(176\) 0 0
\(177\) −21.9525 −1.65005
\(178\) 0 0
\(179\) −11.8944 −0.889028 −0.444514 0.895772i \(-0.646624\pi\)
−0.444514 + 0.895772i \(0.646624\pi\)
\(180\) 0 0
\(181\) 8.52664 0.633781 0.316890 0.948462i \(-0.397361\pi\)
0.316890 + 0.948462i \(0.397361\pi\)
\(182\) 0 0
\(183\) 0.853712 0.0631082
\(184\) 0 0
\(185\) −1.52265 −0.111948
\(186\) 0 0
\(187\) 21.5033 1.57248
\(188\) 0 0
\(189\) −2.30900 −0.167955
\(190\) 0 0
\(191\) −3.12540 −0.226146 −0.113073 0.993587i \(-0.536069\pi\)
−0.113073 + 0.993587i \(0.536069\pi\)
\(192\) 0 0
\(193\) −8.18560 −0.589212 −0.294606 0.955619i \(-0.595188\pi\)
−0.294606 + 0.955619i \(0.595188\pi\)
\(194\) 0 0
\(195\) −18.5063 −1.32526
\(196\) 0 0
\(197\) 11.7451 0.836802 0.418401 0.908262i \(-0.362591\pi\)
0.418401 + 0.908262i \(0.362591\pi\)
\(198\) 0 0
\(199\) 3.24878 0.230300 0.115150 0.993348i \(-0.463265\pi\)
0.115150 + 0.993348i \(0.463265\pi\)
\(200\) 0 0
\(201\) 6.65589 0.469470
\(202\) 0 0
\(203\) −5.01495 −0.351981
\(204\) 0 0
\(205\) −0.626306 −0.0437431
\(206\) 0 0
\(207\) −30.5281 −2.12185
\(208\) 0 0
\(209\) 21.0425 1.45554
\(210\) 0 0
\(211\) 8.14046 0.560412 0.280206 0.959940i \(-0.409597\pi\)
0.280206 + 0.959940i \(0.409597\pi\)
\(212\) 0 0
\(213\) −25.1648 −1.72426
\(214\) 0 0
\(215\) −2.77162 −0.189023
\(216\) 0 0
\(217\) 3.72348 0.252766
\(218\) 0 0
\(219\) 3.98416 0.269224
\(220\) 0 0
\(221\) −29.2863 −1.97001
\(222\) 0 0
\(223\) 0.323283 0.0216486 0.0108243 0.999941i \(-0.496554\pi\)
0.0108243 + 0.999941i \(0.496554\pi\)
\(224\) 0 0
\(225\) −14.0440 −0.936265
\(226\) 0 0
\(227\) 14.7413 0.978412 0.489206 0.872168i \(-0.337287\pi\)
0.489206 + 0.872168i \(0.337287\pi\)
\(228\) 0 0
\(229\) −4.49931 −0.297323 −0.148661 0.988888i \(-0.547496\pi\)
−0.148661 + 0.988888i \(0.547496\pi\)
\(230\) 0 0
\(231\) 13.8490 0.911198
\(232\) 0 0
\(233\) −3.11271 −0.203921 −0.101960 0.994788i \(-0.532511\pi\)
−0.101960 + 0.994788i \(0.532511\pi\)
\(234\) 0 0
\(235\) 5.62478 0.366920
\(236\) 0 0
\(237\) −31.7396 −2.06171
\(238\) 0 0
\(239\) −23.8753 −1.54436 −0.772181 0.635403i \(-0.780834\pi\)
−0.772181 + 0.635403i \(0.780834\pi\)
\(240\) 0 0
\(241\) −7.35938 −0.474059 −0.237030 0.971502i \(-0.576174\pi\)
−0.237030 + 0.971502i \(0.576174\pi\)
\(242\) 0 0
\(243\) 21.8726 1.40313
\(244\) 0 0
\(245\) −6.40951 −0.409488
\(246\) 0 0
\(247\) −28.6587 −1.82351
\(248\) 0 0
\(249\) 11.7007 0.741500
\(250\) 0 0
\(251\) 27.5069 1.73622 0.868109 0.496374i \(-0.165335\pi\)
0.868109 + 0.496374i \(0.165335\pi\)
\(252\) 0 0
\(253\) 37.4309 2.35326
\(254\) 0 0
\(255\) 13.6685 0.855957
\(256\) 0 0
\(257\) −11.1452 −0.695219 −0.347609 0.937639i \(-0.613006\pi\)
−0.347609 + 0.937639i \(0.613006\pi\)
\(258\) 0 0
\(259\) −1.55188 −0.0964294
\(260\) 0 0
\(261\) −16.4279 −1.01686
\(262\) 0 0
\(263\) −7.56516 −0.466488 −0.233244 0.972418i \(-0.574934\pi\)
−0.233244 + 0.972418i \(0.574934\pi\)
\(264\) 0 0
\(265\) 13.0615 0.802363
\(266\) 0 0
\(267\) −36.3638 −2.22543
\(268\) 0 0
\(269\) −11.6010 −0.707326 −0.353663 0.935373i \(-0.615064\pi\)
−0.353663 + 0.935373i \(0.615064\pi\)
\(270\) 0 0
\(271\) 10.4477 0.634650 0.317325 0.948317i \(-0.397215\pi\)
0.317325 + 0.948317i \(0.397215\pi\)
\(272\) 0 0
\(273\) −18.8616 −1.14155
\(274\) 0 0
\(275\) 17.2195 1.03838
\(276\) 0 0
\(277\) 3.67812 0.220997 0.110499 0.993876i \(-0.464755\pi\)
0.110499 + 0.993876i \(0.464755\pi\)
\(278\) 0 0
\(279\) 12.1973 0.730233
\(280\) 0 0
\(281\) −32.8358 −1.95882 −0.979409 0.201887i \(-0.935293\pi\)
−0.979409 + 0.201887i \(0.935293\pi\)
\(282\) 0 0
\(283\) 29.9506 1.78038 0.890190 0.455590i \(-0.150572\pi\)
0.890190 + 0.455590i \(0.150572\pi\)
\(284\) 0 0
\(285\) 13.3756 0.792305
\(286\) 0 0
\(287\) −0.638329 −0.0376794
\(288\) 0 0
\(289\) 4.63050 0.272382
\(290\) 0 0
\(291\) −37.8056 −2.21620
\(292\) 0 0
\(293\) 11.2661 0.658174 0.329087 0.944300i \(-0.393259\pi\)
0.329087 + 0.944300i \(0.393259\pi\)
\(294\) 0 0
\(295\) 9.52859 0.554776
\(296\) 0 0
\(297\) 9.27405 0.538135
\(298\) 0 0
\(299\) −50.9788 −2.94818
\(300\) 0 0
\(301\) −2.82483 −0.162821
\(302\) 0 0
\(303\) 32.0755 1.84269
\(304\) 0 0
\(305\) −0.370558 −0.0212181
\(306\) 0 0
\(307\) 15.2757 0.871833 0.435916 0.899987i \(-0.356424\pi\)
0.435916 + 0.899987i \(0.356424\pi\)
\(308\) 0 0
\(309\) 29.3925 1.67208
\(310\) 0 0
\(311\) −0.451128 −0.0255811 −0.0127906 0.999918i \(-0.504071\pi\)
−0.0127906 + 0.999918i \(0.504071\pi\)
\(312\) 0 0
\(313\) 13.6107 0.769321 0.384660 0.923058i \(-0.374319\pi\)
0.384660 + 0.923058i \(0.374319\pi\)
\(314\) 0 0
\(315\) 4.90267 0.276234
\(316\) 0 0
\(317\) −29.1587 −1.63772 −0.818858 0.573996i \(-0.805393\pi\)
−0.818858 + 0.573996i \(0.805393\pi\)
\(318\) 0 0
\(319\) 20.1425 1.12776
\(320\) 0 0
\(321\) 47.0950 2.62859
\(322\) 0 0
\(323\) 21.1670 1.17776
\(324\) 0 0
\(325\) −23.4520 −1.30088
\(326\) 0 0
\(327\) 19.3100 1.06785
\(328\) 0 0
\(329\) 5.73276 0.316057
\(330\) 0 0
\(331\) 8.87857 0.488010 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(332\) 0 0
\(333\) −5.08364 −0.278582
\(334\) 0 0
\(335\) −2.88902 −0.157844
\(336\) 0 0
\(337\) 25.2390 1.37486 0.687428 0.726253i \(-0.258740\pi\)
0.687428 + 0.726253i \(0.258740\pi\)
\(338\) 0 0
\(339\) 41.5598 2.25722
\(340\) 0 0
\(341\) −14.9553 −0.809874
\(342\) 0 0
\(343\) −14.5905 −0.787812
\(344\) 0 0
\(345\) 23.7929 1.28097
\(346\) 0 0
\(347\) −23.3324 −1.25255 −0.626275 0.779603i \(-0.715421\pi\)
−0.626275 + 0.779603i \(0.715421\pi\)
\(348\) 0 0
\(349\) 25.0473 1.34075 0.670376 0.742022i \(-0.266133\pi\)
0.670376 + 0.742022i \(0.266133\pi\)
\(350\) 0 0
\(351\) −12.6307 −0.674179
\(352\) 0 0
\(353\) −27.4167 −1.45924 −0.729621 0.683851i \(-0.760304\pi\)
−0.729621 + 0.683851i \(0.760304\pi\)
\(354\) 0 0
\(355\) 10.9229 0.579727
\(356\) 0 0
\(357\) 13.9309 0.737304
\(358\) 0 0
\(359\) 29.4902 1.55643 0.778216 0.627997i \(-0.216125\pi\)
0.778216 + 0.627997i \(0.216125\pi\)
\(360\) 0 0
\(361\) 1.71342 0.0901798
\(362\) 0 0
\(363\) −27.0014 −1.41720
\(364\) 0 0
\(365\) −1.72934 −0.0905180
\(366\) 0 0
\(367\) 2.98251 0.155686 0.0778430 0.996966i \(-0.475197\pi\)
0.0778430 + 0.996966i \(0.475197\pi\)
\(368\) 0 0
\(369\) −2.09103 −0.108855
\(370\) 0 0
\(371\) 13.3123 0.691139
\(372\) 0 0
\(373\) −20.2692 −1.04950 −0.524749 0.851257i \(-0.675841\pi\)
−0.524749 + 0.851257i \(0.675841\pi\)
\(374\) 0 0
\(375\) 25.6402 1.32405
\(376\) 0 0
\(377\) −27.4329 −1.41287
\(378\) 0 0
\(379\) 31.7494 1.63086 0.815430 0.578856i \(-0.196501\pi\)
0.815430 + 0.578856i \(0.196501\pi\)
\(380\) 0 0
\(381\) 35.4184 1.81454
\(382\) 0 0
\(383\) 18.7728 0.959244 0.479622 0.877475i \(-0.340774\pi\)
0.479622 + 0.877475i \(0.340774\pi\)
\(384\) 0 0
\(385\) −6.01123 −0.306361
\(386\) 0 0
\(387\) −9.25354 −0.470384
\(388\) 0 0
\(389\) 4.83208 0.244996 0.122498 0.992469i \(-0.460909\pi\)
0.122498 + 0.992469i \(0.460909\pi\)
\(390\) 0 0
\(391\) 37.6524 1.90416
\(392\) 0 0
\(393\) −41.8999 −2.11357
\(394\) 0 0
\(395\) 13.7767 0.693182
\(396\) 0 0
\(397\) −11.4136 −0.572833 −0.286416 0.958105i \(-0.592464\pi\)
−0.286416 + 0.958105i \(0.592464\pi\)
\(398\) 0 0
\(399\) 13.6324 0.682475
\(400\) 0 0
\(401\) 17.6191 0.879856 0.439928 0.898033i \(-0.355004\pi\)
0.439928 + 0.898033i \(0.355004\pi\)
\(402\) 0 0
\(403\) 20.3683 1.01462
\(404\) 0 0
\(405\) −6.88195 −0.341967
\(406\) 0 0
\(407\) 6.23312 0.308964
\(408\) 0 0
\(409\) −20.1583 −0.996766 −0.498383 0.866957i \(-0.666073\pi\)
−0.498383 + 0.866957i \(0.666073\pi\)
\(410\) 0 0
\(411\) −11.3254 −0.558643
\(412\) 0 0
\(413\) 9.71151 0.477872
\(414\) 0 0
\(415\) −5.07874 −0.249305
\(416\) 0 0
\(417\) −38.3637 −1.87868
\(418\) 0 0
\(419\) −24.1359 −1.17912 −0.589558 0.807726i \(-0.700698\pi\)
−0.589558 + 0.807726i \(0.700698\pi\)
\(420\) 0 0
\(421\) 12.4653 0.607521 0.303761 0.952748i \(-0.401758\pi\)
0.303761 + 0.952748i \(0.401758\pi\)
\(422\) 0 0
\(423\) 18.7793 0.913080
\(424\) 0 0
\(425\) 17.3214 0.840211
\(426\) 0 0
\(427\) −0.377672 −0.0182768
\(428\) 0 0
\(429\) 75.7572 3.65759
\(430\) 0 0
\(431\) −27.5636 −1.32769 −0.663846 0.747869i \(-0.731077\pi\)
−0.663846 + 0.747869i \(0.731077\pi\)
\(432\) 0 0
\(433\) 32.3610 1.55517 0.777585 0.628777i \(-0.216444\pi\)
0.777585 + 0.628777i \(0.216444\pi\)
\(434\) 0 0
\(435\) 12.8035 0.613883
\(436\) 0 0
\(437\) 36.8455 1.76256
\(438\) 0 0
\(439\) −9.07883 −0.433309 −0.216655 0.976248i \(-0.569515\pi\)
−0.216655 + 0.976248i \(0.569515\pi\)
\(440\) 0 0
\(441\) −21.3992 −1.01901
\(442\) 0 0
\(443\) 26.5951 1.26357 0.631785 0.775144i \(-0.282323\pi\)
0.631785 + 0.775144i \(0.282323\pi\)
\(444\) 0 0
\(445\) 15.7839 0.748229
\(446\) 0 0
\(447\) −19.8918 −0.940849
\(448\) 0 0
\(449\) 25.1248 1.18571 0.592856 0.805309i \(-0.298000\pi\)
0.592856 + 0.805309i \(0.298000\pi\)
\(450\) 0 0
\(451\) 2.56384 0.120727
\(452\) 0 0
\(453\) 24.5898 1.15533
\(454\) 0 0
\(455\) 8.18696 0.383811
\(456\) 0 0
\(457\) −15.1200 −0.707286 −0.353643 0.935381i \(-0.615057\pi\)
−0.353643 + 0.935381i \(0.615057\pi\)
\(458\) 0 0
\(459\) 9.32891 0.435436
\(460\) 0 0
\(461\) 6.88979 0.320890 0.160445 0.987045i \(-0.448707\pi\)
0.160445 + 0.987045i \(0.448707\pi\)
\(462\) 0 0
\(463\) 5.67494 0.263737 0.131869 0.991267i \(-0.457902\pi\)
0.131869 + 0.991267i \(0.457902\pi\)
\(464\) 0 0
\(465\) −9.50631 −0.440845
\(466\) 0 0
\(467\) 34.1968 1.58244 0.791220 0.611532i \(-0.209447\pi\)
0.791220 + 0.611532i \(0.209447\pi\)
\(468\) 0 0
\(469\) −2.94449 −0.135964
\(470\) 0 0
\(471\) −48.9496 −2.25548
\(472\) 0 0
\(473\) 11.3459 0.521685
\(474\) 0 0
\(475\) 16.9502 0.777730
\(476\) 0 0
\(477\) 43.6081 1.99668
\(478\) 0 0
\(479\) 13.1549 0.601062 0.300531 0.953772i \(-0.402836\pi\)
0.300531 + 0.953772i \(0.402836\pi\)
\(480\) 0 0
\(481\) −8.48916 −0.387072
\(482\) 0 0
\(483\) 24.2497 1.10340
\(484\) 0 0
\(485\) 16.4097 0.745127
\(486\) 0 0
\(487\) 24.0148 1.08822 0.544108 0.839015i \(-0.316868\pi\)
0.544108 + 0.839015i \(0.316868\pi\)
\(488\) 0 0
\(489\) −60.8872 −2.75341
\(490\) 0 0
\(491\) 7.55445 0.340928 0.170464 0.985364i \(-0.445473\pi\)
0.170464 + 0.985364i \(0.445473\pi\)
\(492\) 0 0
\(493\) 20.2616 0.912539
\(494\) 0 0
\(495\) −19.6915 −0.885067
\(496\) 0 0
\(497\) 11.1326 0.499365
\(498\) 0 0
\(499\) 8.71408 0.390096 0.195048 0.980794i \(-0.437514\pi\)
0.195048 + 0.980794i \(0.437514\pi\)
\(500\) 0 0
\(501\) 27.5526 1.23096
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −13.9225 −0.619545
\(506\) 0 0
\(507\) −69.3500 −3.07994
\(508\) 0 0
\(509\) −10.3551 −0.458982 −0.229491 0.973311i \(-0.573706\pi\)
−0.229491 + 0.973311i \(0.573706\pi\)
\(510\) 0 0
\(511\) −1.76254 −0.0779703
\(512\) 0 0
\(513\) 9.12901 0.403056
\(514\) 0 0
\(515\) −12.7580 −0.562183
\(516\) 0 0
\(517\) −23.0255 −1.01266
\(518\) 0 0
\(519\) −64.7579 −2.84256
\(520\) 0 0
\(521\) 1.38974 0.0608855 0.0304428 0.999537i \(-0.490308\pi\)
0.0304428 + 0.999537i \(0.490308\pi\)
\(522\) 0 0
\(523\) 1.05713 0.0462252 0.0231126 0.999733i \(-0.492642\pi\)
0.0231126 + 0.999733i \(0.492642\pi\)
\(524\) 0 0
\(525\) 11.1557 0.486875
\(526\) 0 0
\(527\) −15.0438 −0.655317
\(528\) 0 0
\(529\) 42.5417 1.84964
\(530\) 0 0
\(531\) 31.8128 1.38056
\(532\) 0 0
\(533\) −3.49181 −0.151247
\(534\) 0 0
\(535\) −20.4418 −0.883778
\(536\) 0 0
\(537\) 30.9502 1.33560
\(538\) 0 0
\(539\) 26.2379 1.13015
\(540\) 0 0
\(541\) 5.96612 0.256503 0.128252 0.991742i \(-0.459063\pi\)
0.128252 + 0.991742i \(0.459063\pi\)
\(542\) 0 0
\(543\) −22.1871 −0.952139
\(544\) 0 0
\(545\) −8.38162 −0.359029
\(546\) 0 0
\(547\) −22.9973 −0.983292 −0.491646 0.870795i \(-0.663605\pi\)
−0.491646 + 0.870795i \(0.663605\pi\)
\(548\) 0 0
\(549\) −1.23717 −0.0528012
\(550\) 0 0
\(551\) 19.8275 0.844679
\(552\) 0 0
\(553\) 14.0412 0.597092
\(554\) 0 0
\(555\) 3.96208 0.168181
\(556\) 0 0
\(557\) 4.09009 0.173303 0.0866514 0.996239i \(-0.472383\pi\)
0.0866514 + 0.996239i \(0.472383\pi\)
\(558\) 0 0
\(559\) −15.4525 −0.653570
\(560\) 0 0
\(561\) −55.9534 −2.36236
\(562\) 0 0
\(563\) −44.2568 −1.86520 −0.932600 0.360911i \(-0.882466\pi\)
−0.932600 + 0.360911i \(0.882466\pi\)
\(564\) 0 0
\(565\) −18.0392 −0.758917
\(566\) 0 0
\(567\) −7.01407 −0.294563
\(568\) 0 0
\(569\) −7.59686 −0.318477 −0.159239 0.987240i \(-0.550904\pi\)
−0.159239 + 0.987240i \(0.550904\pi\)
\(570\) 0 0
\(571\) 12.0736 0.505265 0.252632 0.967562i \(-0.418704\pi\)
0.252632 + 0.967562i \(0.418704\pi\)
\(572\) 0 0
\(573\) 8.13257 0.339743
\(574\) 0 0
\(575\) 30.1515 1.25740
\(576\) 0 0
\(577\) −11.2700 −0.469174 −0.234587 0.972095i \(-0.575374\pi\)
−0.234587 + 0.972095i \(0.575374\pi\)
\(578\) 0 0
\(579\) 21.2997 0.885183
\(580\) 0 0
\(581\) −5.17624 −0.214747
\(582\) 0 0
\(583\) −53.4686 −2.21444
\(584\) 0 0
\(585\) 26.8187 1.10882
\(586\) 0 0
\(587\) 30.1680 1.24517 0.622583 0.782553i \(-0.286083\pi\)
0.622583 + 0.782553i \(0.286083\pi\)
\(588\) 0 0
\(589\) −14.7214 −0.606585
\(590\) 0 0
\(591\) −30.5617 −1.25714
\(592\) 0 0
\(593\) 6.24886 0.256610 0.128305 0.991735i \(-0.459046\pi\)
0.128305 + 0.991735i \(0.459046\pi\)
\(594\) 0 0
\(595\) −6.04680 −0.247895
\(596\) 0 0
\(597\) −8.45360 −0.345983
\(598\) 0 0
\(599\) 13.6174 0.556390 0.278195 0.960525i \(-0.410264\pi\)
0.278195 + 0.960525i \(0.410264\pi\)
\(600\) 0 0
\(601\) 37.7493 1.53982 0.769912 0.638150i \(-0.220300\pi\)
0.769912 + 0.638150i \(0.220300\pi\)
\(602\) 0 0
\(603\) −9.64550 −0.392795
\(604\) 0 0
\(605\) 11.7201 0.476489
\(606\) 0 0
\(607\) 17.9252 0.727562 0.363781 0.931485i \(-0.381486\pi\)
0.363781 + 0.931485i \(0.381486\pi\)
\(608\) 0 0
\(609\) 13.0493 0.528786
\(610\) 0 0
\(611\) 31.3595 1.26867
\(612\) 0 0
\(613\) −13.0328 −0.526391 −0.263195 0.964743i \(-0.584776\pi\)
−0.263195 + 0.964743i \(0.584776\pi\)
\(614\) 0 0
\(615\) 1.62970 0.0657159
\(616\) 0 0
\(617\) −34.1504 −1.37484 −0.687421 0.726259i \(-0.741257\pi\)
−0.687421 + 0.726259i \(0.741257\pi\)
\(618\) 0 0
\(619\) −26.0115 −1.04549 −0.522746 0.852489i \(-0.675092\pi\)
−0.522746 + 0.852489i \(0.675092\pi\)
\(620\) 0 0
\(621\) 16.2389 0.651645
\(622\) 0 0
\(623\) 16.0869 0.644509
\(624\) 0 0
\(625\) 7.49246 0.299698
\(626\) 0 0
\(627\) −54.7544 −2.18668
\(628\) 0 0
\(629\) 6.27000 0.250001
\(630\) 0 0
\(631\) −26.9082 −1.07120 −0.535599 0.844472i \(-0.679914\pi\)
−0.535599 + 0.844472i \(0.679914\pi\)
\(632\) 0 0
\(633\) −21.1822 −0.841916
\(634\) 0 0
\(635\) −15.3735 −0.610080
\(636\) 0 0
\(637\) −35.7346 −1.41585
\(638\) 0 0
\(639\) 36.4680 1.44265
\(640\) 0 0
\(641\) 16.9550 0.669684 0.334842 0.942274i \(-0.391317\pi\)
0.334842 + 0.942274i \(0.391317\pi\)
\(642\) 0 0
\(643\) 9.26548 0.365395 0.182697 0.983169i \(-0.441517\pi\)
0.182697 + 0.983169i \(0.441517\pi\)
\(644\) 0 0
\(645\) 7.21201 0.283973
\(646\) 0 0
\(647\) 36.2625 1.42563 0.712814 0.701353i \(-0.247421\pi\)
0.712814 + 0.701353i \(0.247421\pi\)
\(648\) 0 0
\(649\) −39.0062 −1.53113
\(650\) 0 0
\(651\) −9.68881 −0.379734
\(652\) 0 0
\(653\) 26.2455 1.02706 0.513532 0.858070i \(-0.328337\pi\)
0.513532 + 0.858070i \(0.328337\pi\)
\(654\) 0 0
\(655\) 18.1869 0.710620
\(656\) 0 0
\(657\) −5.77371 −0.225254
\(658\) 0 0
\(659\) 3.87094 0.150790 0.0753952 0.997154i \(-0.475978\pi\)
0.0753952 + 0.997154i \(0.475978\pi\)
\(660\) 0 0
\(661\) 43.8920 1.70720 0.853601 0.520928i \(-0.174414\pi\)
0.853601 + 0.520928i \(0.174414\pi\)
\(662\) 0 0
\(663\) 76.2054 2.95957
\(664\) 0 0
\(665\) −5.91722 −0.229460
\(666\) 0 0
\(667\) 35.2696 1.36564
\(668\) 0 0
\(669\) −0.841211 −0.0325231
\(670\) 0 0
\(671\) 1.51691 0.0585598
\(672\) 0 0
\(673\) 12.8052 0.493603 0.246802 0.969066i \(-0.420620\pi\)
0.246802 + 0.969066i \(0.420620\pi\)
\(674\) 0 0
\(675\) 7.47046 0.287538
\(676\) 0 0
\(677\) −19.9187 −0.765538 −0.382769 0.923844i \(-0.625030\pi\)
−0.382769 + 0.923844i \(0.625030\pi\)
\(678\) 0 0
\(679\) 16.7248 0.641837
\(680\) 0 0
\(681\) −38.3581 −1.46988
\(682\) 0 0
\(683\) 3.48171 0.133224 0.0666120 0.997779i \(-0.478781\pi\)
0.0666120 + 0.997779i \(0.478781\pi\)
\(684\) 0 0
\(685\) 4.91587 0.187826
\(686\) 0 0
\(687\) 11.7076 0.446672
\(688\) 0 0
\(689\) 72.8212 2.77427
\(690\) 0 0
\(691\) 35.8255 1.36287 0.681434 0.731880i \(-0.261357\pi\)
0.681434 + 0.731880i \(0.261357\pi\)
\(692\) 0 0
\(693\) −20.0695 −0.762378
\(694\) 0 0
\(695\) 16.6519 0.631644
\(696\) 0 0
\(697\) 2.57901 0.0976869
\(698\) 0 0
\(699\) 8.09956 0.306353
\(700\) 0 0
\(701\) −16.4520 −0.621382 −0.310691 0.950511i \(-0.600560\pi\)
−0.310691 + 0.950511i \(0.600560\pi\)
\(702\) 0 0
\(703\) 6.13564 0.231410
\(704\) 0 0
\(705\) −14.6362 −0.551230
\(706\) 0 0
\(707\) −14.1898 −0.533663
\(708\) 0 0
\(709\) 53.0877 1.99375 0.996874 0.0790033i \(-0.0251738\pi\)
0.996874 + 0.0790033i \(0.0251738\pi\)
\(710\) 0 0
\(711\) 45.9959 1.72498
\(712\) 0 0
\(713\) −26.1868 −0.980703
\(714\) 0 0
\(715\) −32.8828 −1.22975
\(716\) 0 0
\(717\) 62.1255 2.32012
\(718\) 0 0
\(719\) −12.1024 −0.451343 −0.225672 0.974203i \(-0.572458\pi\)
−0.225672 + 0.974203i \(0.572458\pi\)
\(720\) 0 0
\(721\) −13.0029 −0.484253
\(722\) 0 0
\(723\) 19.1498 0.712187
\(724\) 0 0
\(725\) 16.2252 0.602590
\(726\) 0 0
\(727\) 30.8602 1.14454 0.572271 0.820064i \(-0.306062\pi\)
0.572271 + 0.820064i \(0.306062\pi\)
\(728\) 0 0
\(729\) −38.6347 −1.43092
\(730\) 0 0
\(731\) 11.4130 0.422126
\(732\) 0 0
\(733\) −40.5216 −1.49670 −0.748351 0.663303i \(-0.769154\pi\)
−0.748351 + 0.663303i \(0.769154\pi\)
\(734\) 0 0
\(735\) 16.6781 0.615181
\(736\) 0 0
\(737\) 11.8265 0.435634
\(738\) 0 0
\(739\) 20.6325 0.758979 0.379489 0.925196i \(-0.376100\pi\)
0.379489 + 0.925196i \(0.376100\pi\)
\(740\) 0 0
\(741\) 74.5725 2.73949
\(742\) 0 0
\(743\) −49.2312 −1.80612 −0.903059 0.429517i \(-0.858684\pi\)
−0.903059 + 0.429517i \(0.858684\pi\)
\(744\) 0 0
\(745\) 8.63413 0.316330
\(746\) 0 0
\(747\) −16.9562 −0.620396
\(748\) 0 0
\(749\) −20.8343 −0.761268
\(750\) 0 0
\(751\) 21.0160 0.766883 0.383442 0.923565i \(-0.374739\pi\)
0.383442 + 0.923565i \(0.374739\pi\)
\(752\) 0 0
\(753\) −71.5753 −2.60835
\(754\) 0 0
\(755\) −10.6733 −0.388443
\(756\) 0 0
\(757\) −3.41383 −0.124078 −0.0620389 0.998074i \(-0.519760\pi\)
−0.0620389 + 0.998074i \(0.519760\pi\)
\(758\) 0 0
\(759\) −97.3985 −3.53534
\(760\) 0 0
\(761\) 44.6666 1.61916 0.809582 0.587007i \(-0.199694\pi\)
0.809582 + 0.587007i \(0.199694\pi\)
\(762\) 0 0
\(763\) −8.54253 −0.309260
\(764\) 0 0
\(765\) −19.8080 −0.716160
\(766\) 0 0
\(767\) 53.1242 1.91820
\(768\) 0 0
\(769\) 32.5210 1.17274 0.586369 0.810044i \(-0.300557\pi\)
0.586369 + 0.810044i \(0.300557\pi\)
\(770\) 0 0
\(771\) 29.0008 1.04444
\(772\) 0 0
\(773\) −9.27718 −0.333677 −0.166838 0.985984i \(-0.553356\pi\)
−0.166838 + 0.985984i \(0.553356\pi\)
\(774\) 0 0
\(775\) −12.0468 −0.432735
\(776\) 0 0
\(777\) 4.03814 0.144867
\(778\) 0 0
\(779\) 2.52374 0.0904225
\(780\) 0 0
\(781\) −44.7139 −1.59999
\(782\) 0 0
\(783\) 8.73855 0.312290
\(784\) 0 0
\(785\) 21.2468 0.758332
\(786\) 0 0
\(787\) 23.5268 0.838640 0.419320 0.907839i \(-0.362269\pi\)
0.419320 + 0.907839i \(0.362269\pi\)
\(788\) 0 0
\(789\) 19.6852 0.700812
\(790\) 0 0
\(791\) −18.3856 −0.653715
\(792\) 0 0
\(793\) −2.06595 −0.0733641
\(794\) 0 0
\(795\) −33.9872 −1.20540
\(796\) 0 0
\(797\) 42.8975 1.51951 0.759754 0.650210i \(-0.225319\pi\)
0.759754 + 0.650210i \(0.225319\pi\)
\(798\) 0 0
\(799\) −23.1618 −0.819404
\(800\) 0 0
\(801\) 52.6973 1.86197
\(802\) 0 0
\(803\) 7.07923 0.249821
\(804\) 0 0
\(805\) −10.5257 −0.370982
\(806\) 0 0
\(807\) 30.1869 1.06263
\(808\) 0 0
\(809\) −2.29970 −0.0808533 −0.0404266 0.999183i \(-0.512872\pi\)
−0.0404266 + 0.999183i \(0.512872\pi\)
\(810\) 0 0
\(811\) −17.5348 −0.615732 −0.307866 0.951430i \(-0.599615\pi\)
−0.307866 + 0.951430i \(0.599615\pi\)
\(812\) 0 0
\(813\) −27.1857 −0.953444
\(814\) 0 0
\(815\) 26.4284 0.925747
\(816\) 0 0
\(817\) 11.1685 0.390735
\(818\) 0 0
\(819\) 27.3336 0.955112
\(820\) 0 0
\(821\) 5.91753 0.206523 0.103262 0.994654i \(-0.467072\pi\)
0.103262 + 0.994654i \(0.467072\pi\)
\(822\) 0 0
\(823\) 17.4796 0.609300 0.304650 0.952464i \(-0.401460\pi\)
0.304650 + 0.952464i \(0.401460\pi\)
\(824\) 0 0
\(825\) −44.8067 −1.55997
\(826\) 0 0
\(827\) 42.1717 1.46645 0.733226 0.679985i \(-0.238014\pi\)
0.733226 + 0.679985i \(0.238014\pi\)
\(828\) 0 0
\(829\) −45.7023 −1.58731 −0.793653 0.608371i \(-0.791823\pi\)
−0.793653 + 0.608371i \(0.791823\pi\)
\(830\) 0 0
\(831\) −9.57080 −0.332007
\(832\) 0 0
\(833\) 26.3931 0.914468
\(834\) 0 0
\(835\) −11.9593 −0.413870
\(836\) 0 0
\(837\) −6.48815 −0.224263
\(838\) 0 0
\(839\) −47.6299 −1.64437 −0.822184 0.569222i \(-0.807245\pi\)
−0.822184 + 0.569222i \(0.807245\pi\)
\(840\) 0 0
\(841\) −10.0206 −0.345537
\(842\) 0 0
\(843\) 85.4415 2.94276
\(844\) 0 0
\(845\) 30.1017 1.03553
\(846\) 0 0
\(847\) 11.9451 0.410438
\(848\) 0 0
\(849\) −77.9341 −2.67469
\(850\) 0 0
\(851\) 10.9142 0.374135
\(852\) 0 0
\(853\) −15.0088 −0.513890 −0.256945 0.966426i \(-0.582716\pi\)
−0.256945 + 0.966426i \(0.582716\pi\)
\(854\) 0 0
\(855\) −19.3835 −0.662903
\(856\) 0 0
\(857\) 38.2556 1.30679 0.653393 0.757019i \(-0.273345\pi\)
0.653393 + 0.757019i \(0.273345\pi\)
\(858\) 0 0
\(859\) 26.0573 0.889062 0.444531 0.895763i \(-0.353370\pi\)
0.444531 + 0.895763i \(0.353370\pi\)
\(860\) 0 0
\(861\) 1.66099 0.0566064
\(862\) 0 0
\(863\) 29.3364 0.998624 0.499312 0.866422i \(-0.333586\pi\)
0.499312 + 0.866422i \(0.333586\pi\)
\(864\) 0 0
\(865\) 28.1085 0.955717
\(866\) 0 0
\(867\) −12.0490 −0.409205
\(868\) 0 0
\(869\) −56.3963 −1.91311
\(870\) 0 0
\(871\) −16.1070 −0.545765
\(872\) 0 0
\(873\) 54.7867 1.85425
\(874\) 0 0
\(875\) −11.3429 −0.383461
\(876\) 0 0
\(877\) 20.9083 0.706022 0.353011 0.935619i \(-0.385158\pi\)
0.353011 + 0.935619i \(0.385158\pi\)
\(878\) 0 0
\(879\) −29.3155 −0.988786
\(880\) 0 0
\(881\) 3.68502 0.124151 0.0620757 0.998071i \(-0.480228\pi\)
0.0620757 + 0.998071i \(0.480228\pi\)
\(882\) 0 0
\(883\) 0.00202305 6.80812e−5 0 3.40406e−5 1.00000i \(-0.499989\pi\)
3.40406e−5 1.00000i \(0.499989\pi\)
\(884\) 0 0
\(885\) −24.7942 −0.833449
\(886\) 0 0
\(887\) 19.5580 0.656694 0.328347 0.944557i \(-0.393508\pi\)
0.328347 + 0.944557i \(0.393508\pi\)
\(888\) 0 0
\(889\) −15.6687 −0.525510
\(890\) 0 0
\(891\) 28.1719 0.943795
\(892\) 0 0
\(893\) −22.6654 −0.758470
\(894\) 0 0
\(895\) −13.4341 −0.449053
\(896\) 0 0
\(897\) 132.651 4.42910
\(898\) 0 0
\(899\) −14.0917 −0.469986
\(900\) 0 0
\(901\) −53.7849 −1.79183
\(902\) 0 0
\(903\) 7.35047 0.244608
\(904\) 0 0
\(905\) 9.63042 0.320126
\(906\) 0 0
\(907\) 7.64015 0.253687 0.126844 0.991923i \(-0.459515\pi\)
0.126844 + 0.991923i \(0.459515\pi\)
\(908\) 0 0
\(909\) −46.4828 −1.54174
\(910\) 0 0
\(911\) −36.3768 −1.20522 −0.602608 0.798037i \(-0.705872\pi\)
−0.602608 + 0.798037i \(0.705872\pi\)
\(912\) 0 0
\(913\) 20.7903 0.688058
\(914\) 0 0
\(915\) 0.964225 0.0318763
\(916\) 0 0
\(917\) 18.5360 0.612113
\(918\) 0 0
\(919\) −55.9427 −1.84538 −0.922690 0.385543i \(-0.874014\pi\)
−0.922690 + 0.385543i \(0.874014\pi\)
\(920\) 0 0
\(921\) −39.7488 −1.30977
\(922\) 0 0
\(923\) 60.8978 2.00448
\(924\) 0 0
\(925\) 5.02093 0.165087
\(926\) 0 0
\(927\) −42.5946 −1.39899
\(928\) 0 0
\(929\) −3.47640 −0.114057 −0.0570285 0.998373i \(-0.518163\pi\)
−0.0570285 + 0.998373i \(0.518163\pi\)
\(930\) 0 0
\(931\) 25.8276 0.846465
\(932\) 0 0
\(933\) 1.17387 0.0384309
\(934\) 0 0
\(935\) 24.2869 0.794266
\(936\) 0 0
\(937\) −56.1396 −1.83400 −0.917001 0.398886i \(-0.869397\pi\)
−0.917001 + 0.398886i \(0.869397\pi\)
\(938\) 0 0
\(939\) −35.4162 −1.15576
\(940\) 0 0
\(941\) 21.7192 0.708024 0.354012 0.935241i \(-0.384817\pi\)
0.354012 + 0.935241i \(0.384817\pi\)
\(942\) 0 0
\(943\) 4.48930 0.146192
\(944\) 0 0
\(945\) −2.60789 −0.0848348
\(946\) 0 0
\(947\) −33.7946 −1.09818 −0.549088 0.835764i \(-0.685025\pi\)
−0.549088 + 0.835764i \(0.685025\pi\)
\(948\) 0 0
\(949\) −9.64151 −0.312977
\(950\) 0 0
\(951\) 75.8735 2.46037
\(952\) 0 0
\(953\) 10.0311 0.324938 0.162469 0.986714i \(-0.448054\pi\)
0.162469 + 0.986714i \(0.448054\pi\)
\(954\) 0 0
\(955\) −3.52998 −0.114228
\(956\) 0 0
\(957\) −52.4125 −1.69426
\(958\) 0 0
\(959\) 5.01024 0.161789
\(960\) 0 0
\(961\) −20.5372 −0.662491
\(962\) 0 0
\(963\) −68.2486 −2.19928
\(964\) 0 0
\(965\) −9.24522 −0.297614
\(966\) 0 0
\(967\) 52.6091 1.69180 0.845898 0.533345i \(-0.179065\pi\)
0.845898 + 0.533345i \(0.179065\pi\)
\(968\) 0 0
\(969\) −55.0784 −1.76937
\(970\) 0 0
\(971\) −54.9841 −1.76452 −0.882262 0.470759i \(-0.843980\pi\)
−0.882262 + 0.470759i \(0.843980\pi\)
\(972\) 0 0
\(973\) 16.9716 0.544085
\(974\) 0 0
\(975\) 61.0242 1.95434
\(976\) 0 0
\(977\) 24.6271 0.787892 0.393946 0.919134i \(-0.371110\pi\)
0.393946 + 0.919134i \(0.371110\pi\)
\(978\) 0 0
\(979\) −64.6129 −2.06504
\(980\) 0 0
\(981\) −27.9835 −0.893444
\(982\) 0 0
\(983\) 35.9409 1.14634 0.573169 0.819437i \(-0.305714\pi\)
0.573169 + 0.819437i \(0.305714\pi\)
\(984\) 0 0
\(985\) 13.2655 0.422673
\(986\) 0 0
\(987\) −14.9171 −0.474818
\(988\) 0 0
\(989\) 19.8667 0.631726
\(990\) 0 0
\(991\) 30.5366 0.970028 0.485014 0.874506i \(-0.338815\pi\)
0.485014 + 0.874506i \(0.338815\pi\)
\(992\) 0 0
\(993\) −23.1028 −0.733146
\(994\) 0 0
\(995\) 3.66933 0.116326
\(996\) 0 0
\(997\) −45.1530 −1.43001 −0.715006 0.699119i \(-0.753576\pi\)
−0.715006 + 0.699119i \(0.753576\pi\)
\(998\) 0 0
\(999\) 2.70416 0.0855557
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 8048.2.a.y.1.5 33
4.3 odd 2 4024.2.a.f.1.29 33
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4024.2.a.f.1.29 33 4.3 odd 2
8048.2.a.y.1.5 33 1.1 even 1 trivial