Properties

Label 8012.2.a.b.1.7
Level $8012$
Weight $2$
Character 8012.1
Self dual yes
Analytic conductor $63.976$
Analytic rank $0$
Dimension $88$
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] = [8012,2,Mod(1,8012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8012 = 2^{2} \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8012.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: \(63.9761420994\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 8012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.80739 q^{3} +2.13337 q^{5} -1.30534 q^{7} +4.88143 q^{9} +O(q^{10})\) \(q-2.80739 q^{3} +2.13337 q^{5} -1.30534 q^{7} +4.88143 q^{9} +2.64263 q^{11} -1.23341 q^{13} -5.98920 q^{15} -1.89582 q^{17} +4.77330 q^{19} +3.66458 q^{21} +3.45037 q^{23} -0.448727 q^{25} -5.28191 q^{27} +2.68254 q^{29} -10.2146 q^{31} -7.41889 q^{33} -2.78477 q^{35} -0.381771 q^{37} +3.46267 q^{39} +2.80784 q^{41} +10.5225 q^{43} +10.4139 q^{45} +11.0749 q^{47} -5.29610 q^{49} +5.32230 q^{51} -4.75066 q^{53} +5.63771 q^{55} -13.4005 q^{57} +14.4162 q^{59} +0.262609 q^{61} -6.37190 q^{63} -2.63133 q^{65} +2.07270 q^{67} -9.68653 q^{69} +3.11912 q^{71} -9.33681 q^{73} +1.25975 q^{75} -3.44952 q^{77} -15.5084 q^{79} +0.184073 q^{81} +13.7805 q^{83} -4.04449 q^{85} -7.53092 q^{87} -16.3267 q^{89} +1.61002 q^{91} +28.6764 q^{93} +10.1832 q^{95} +4.59892 q^{97} +12.8998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 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
\(3\) −2.80739 −1.62085 −0.810423 0.585845i \(-0.800763\pi\)
−0.810423 + 0.585845i \(0.800763\pi\)
\(4\) 0 0
\(5\) 2.13337 0.954073 0.477036 0.878884i \(-0.341711\pi\)
0.477036 + 0.878884i \(0.341711\pi\)
\(6\) 0 0
\(7\) −1.30534 −0.493370 −0.246685 0.969096i \(-0.579341\pi\)
−0.246685 + 0.969096i \(0.579341\pi\)
\(8\) 0 0
\(9\) 4.88143 1.62714
\(10\) 0 0
\(11\) 2.64263 0.796783 0.398392 0.917215i \(-0.369569\pi\)
0.398392 + 0.917215i \(0.369569\pi\)
\(12\) 0 0
\(13\) −1.23341 −0.342087 −0.171044 0.985263i \(-0.554714\pi\)
−0.171044 + 0.985263i \(0.554714\pi\)
\(14\) 0 0
\(15\) −5.98920 −1.54641
\(16\) 0 0
\(17\) −1.89582 −0.459804 −0.229902 0.973214i \(-0.573841\pi\)
−0.229902 + 0.973214i \(0.573841\pi\)
\(18\) 0 0
\(19\) 4.77330 1.09507 0.547535 0.836783i \(-0.315566\pi\)
0.547535 + 0.836783i \(0.315566\pi\)
\(20\) 0 0
\(21\) 3.66458 0.799678
\(22\) 0 0
\(23\) 3.45037 0.719452 0.359726 0.933058i \(-0.382870\pi\)
0.359726 + 0.933058i \(0.382870\pi\)
\(24\) 0 0
\(25\) −0.448727 −0.0897454
\(26\) 0 0
\(27\) −5.28191 −1.01650
\(28\) 0 0
\(29\) 2.68254 0.498134 0.249067 0.968486i \(-0.419876\pi\)
0.249067 + 0.968486i \(0.419876\pi\)
\(30\) 0 0
\(31\) −10.2146 −1.83460 −0.917301 0.398194i \(-0.869637\pi\)
−0.917301 + 0.398194i \(0.869637\pi\)
\(32\) 0 0
\(33\) −7.41889 −1.29146
\(34\) 0 0
\(35\) −2.78477 −0.470711
\(36\) 0 0
\(37\) −0.381771 −0.0627627 −0.0313814 0.999507i \(-0.509991\pi\)
−0.0313814 + 0.999507i \(0.509991\pi\)
\(38\) 0 0
\(39\) 3.46267 0.554471
\(40\) 0 0
\(41\) 2.80784 0.438511 0.219255 0.975668i \(-0.429637\pi\)
0.219255 + 0.975668i \(0.429637\pi\)
\(42\) 0 0
\(43\) 10.5225 1.60467 0.802337 0.596872i \(-0.203590\pi\)
0.802337 + 0.596872i \(0.203590\pi\)
\(44\) 0 0
\(45\) 10.4139 1.55241
\(46\) 0 0
\(47\) 11.0749 1.61545 0.807723 0.589563i \(-0.200700\pi\)
0.807723 + 0.589563i \(0.200700\pi\)
\(48\) 0 0
\(49\) −5.29610 −0.756586
\(50\) 0 0
\(51\) 5.32230 0.745272
\(52\) 0 0
\(53\) −4.75066 −0.652554 −0.326277 0.945274i \(-0.605794\pi\)
−0.326277 + 0.945274i \(0.605794\pi\)
\(54\) 0 0
\(55\) 5.63771 0.760189
\(56\) 0 0
\(57\) −13.4005 −1.77494
\(58\) 0 0
\(59\) 14.4162 1.87683 0.938413 0.345515i \(-0.112296\pi\)
0.938413 + 0.345515i \(0.112296\pi\)
\(60\) 0 0
\(61\) 0.262609 0.0336236 0.0168118 0.999859i \(-0.494648\pi\)
0.0168118 + 0.999859i \(0.494648\pi\)
\(62\) 0 0
\(63\) −6.37190 −0.802785
\(64\) 0 0
\(65\) −2.63133 −0.326376
\(66\) 0 0
\(67\) 2.07270 0.253220 0.126610 0.991953i \(-0.459590\pi\)
0.126610 + 0.991953i \(0.459590\pi\)
\(68\) 0 0
\(69\) −9.68653 −1.16612
\(70\) 0 0
\(71\) 3.11912 0.370172 0.185086 0.982722i \(-0.440744\pi\)
0.185086 + 0.982722i \(0.440744\pi\)
\(72\) 0 0
\(73\) −9.33681 −1.09279 −0.546396 0.837527i \(-0.684000\pi\)
−0.546396 + 0.837527i \(0.684000\pi\)
\(74\) 0 0
\(75\) 1.25975 0.145464
\(76\) 0 0
\(77\) −3.44952 −0.393109
\(78\) 0 0
\(79\) −15.5084 −1.74483 −0.872416 0.488764i \(-0.837448\pi\)
−0.872416 + 0.488764i \(0.837448\pi\)
\(80\) 0 0
\(81\) 0.184073 0.0204526
\(82\) 0 0
\(83\) 13.7805 1.51260 0.756302 0.654223i \(-0.227004\pi\)
0.756302 + 0.654223i \(0.227004\pi\)
\(84\) 0 0
\(85\) −4.04449 −0.438686
\(86\) 0 0
\(87\) −7.53092 −0.807399
\(88\) 0 0
\(89\) −16.3267 −1.73063 −0.865314 0.501230i \(-0.832881\pi\)
−0.865314 + 0.501230i \(0.832881\pi\)
\(90\) 0 0
\(91\) 1.61002 0.168776
\(92\) 0 0
\(93\) 28.6764 2.97361
\(94\) 0 0
\(95\) 10.1832 1.04478
\(96\) 0 0
\(97\) 4.59892 0.466949 0.233475 0.972363i \(-0.424990\pi\)
0.233475 + 0.972363i \(0.424990\pi\)
\(98\) 0 0
\(99\) 12.8998 1.29648
\(100\) 0 0
\(101\) 8.95495 0.891051 0.445525 0.895269i \(-0.353017\pi\)
0.445525 + 0.895269i \(0.353017\pi\)
\(102\) 0 0
\(103\) −0.197783 −0.0194882 −0.00974408 0.999953i \(-0.503102\pi\)
−0.00974408 + 0.999953i \(0.503102\pi\)
\(104\) 0 0
\(105\) 7.81792 0.762951
\(106\) 0 0
\(107\) 8.32578 0.804883 0.402442 0.915446i \(-0.368162\pi\)
0.402442 + 0.915446i \(0.368162\pi\)
\(108\) 0 0
\(109\) −3.67112 −0.351630 −0.175815 0.984423i \(-0.556256\pi\)
−0.175815 + 0.984423i \(0.556256\pi\)
\(110\) 0 0
\(111\) 1.07178 0.101729
\(112\) 0 0
\(113\) 6.86361 0.645674 0.322837 0.946455i \(-0.395363\pi\)
0.322837 + 0.946455i \(0.395363\pi\)
\(114\) 0 0
\(115\) 7.36092 0.686409
\(116\) 0 0
\(117\) −6.02082 −0.556625
\(118\) 0 0
\(119\) 2.47468 0.226854
\(120\) 0 0
\(121\) −4.01650 −0.365137
\(122\) 0 0
\(123\) −7.88269 −0.710758
\(124\) 0 0
\(125\) −11.6242 −1.03970
\(126\) 0 0
\(127\) 13.1033 1.16273 0.581364 0.813643i \(-0.302519\pi\)
0.581364 + 0.813643i \(0.302519\pi\)
\(128\) 0 0
\(129\) −29.5409 −2.60093
\(130\) 0 0
\(131\) −8.30900 −0.725960 −0.362980 0.931797i \(-0.618241\pi\)
−0.362980 + 0.931797i \(0.618241\pi\)
\(132\) 0 0
\(133\) −6.23075 −0.540275
\(134\) 0 0
\(135\) −11.2683 −0.969818
\(136\) 0 0
\(137\) 19.4619 1.66274 0.831371 0.555717i \(-0.187556\pi\)
0.831371 + 0.555717i \(0.187556\pi\)
\(138\) 0 0
\(139\) 1.90790 0.161826 0.0809131 0.996721i \(-0.474216\pi\)
0.0809131 + 0.996721i \(0.474216\pi\)
\(140\) 0 0
\(141\) −31.0916 −2.61839
\(142\) 0 0
\(143\) −3.25946 −0.272569
\(144\) 0 0
\(145\) 5.72284 0.475256
\(146\) 0 0
\(147\) 14.8682 1.22631
\(148\) 0 0
\(149\) −7.48829 −0.613464 −0.306732 0.951796i \(-0.599236\pi\)
−0.306732 + 0.951796i \(0.599236\pi\)
\(150\) 0 0
\(151\) −2.72876 −0.222064 −0.111032 0.993817i \(-0.535416\pi\)
−0.111032 + 0.993817i \(0.535416\pi\)
\(152\) 0 0
\(153\) −9.25432 −0.748167
\(154\) 0 0
\(155\) −21.7916 −1.75034
\(156\) 0 0
\(157\) 0.526050 0.0419834 0.0209917 0.999780i \(-0.493318\pi\)
0.0209917 + 0.999780i \(0.493318\pi\)
\(158\) 0 0
\(159\) 13.3370 1.05769
\(160\) 0 0
\(161\) −4.50389 −0.354956
\(162\) 0 0
\(163\) 5.98689 0.468929 0.234465 0.972125i \(-0.424666\pi\)
0.234465 + 0.972125i \(0.424666\pi\)
\(164\) 0 0
\(165\) −15.8272 −1.23215
\(166\) 0 0
\(167\) 9.09559 0.703838 0.351919 0.936030i \(-0.385529\pi\)
0.351919 + 0.936030i \(0.385529\pi\)
\(168\) 0 0
\(169\) −11.4787 −0.882976
\(170\) 0 0
\(171\) 23.3005 1.78183
\(172\) 0 0
\(173\) 0.303013 0.0230377 0.0115188 0.999934i \(-0.496333\pi\)
0.0115188 + 0.999934i \(0.496333\pi\)
\(174\) 0 0
\(175\) 0.585739 0.0442777
\(176\) 0 0
\(177\) −40.4718 −3.04205
\(178\) 0 0
\(179\) −16.2797 −1.21680 −0.608402 0.793629i \(-0.708189\pi\)
−0.608402 + 0.793629i \(0.708189\pi\)
\(180\) 0 0
\(181\) −13.5830 −1.00961 −0.504807 0.863232i \(-0.668436\pi\)
−0.504807 + 0.863232i \(0.668436\pi\)
\(182\) 0 0
\(183\) −0.737246 −0.0544988
\(184\) 0 0
\(185\) −0.814459 −0.0598802
\(186\) 0 0
\(187\) −5.00995 −0.366364
\(188\) 0 0
\(189\) 6.89466 0.501513
\(190\) 0 0
\(191\) −13.7009 −0.991361 −0.495681 0.868505i \(-0.665081\pi\)
−0.495681 + 0.868505i \(0.665081\pi\)
\(192\) 0 0
\(193\) −22.9353 −1.65092 −0.825458 0.564463i \(-0.809083\pi\)
−0.825458 + 0.564463i \(0.809083\pi\)
\(194\) 0 0
\(195\) 7.38716 0.529006
\(196\) 0 0
\(197\) −1.35603 −0.0966128 −0.0483064 0.998833i \(-0.515382\pi\)
−0.0483064 + 0.998833i \(0.515382\pi\)
\(198\) 0 0
\(199\) 25.8043 1.82922 0.914608 0.404341i \(-0.132499\pi\)
0.914608 + 0.404341i \(0.132499\pi\)
\(200\) 0 0
\(201\) −5.81887 −0.410431
\(202\) 0 0
\(203\) −3.50161 −0.245765
\(204\) 0 0
\(205\) 5.99016 0.418371
\(206\) 0 0
\(207\) 16.8427 1.17065
\(208\) 0 0
\(209\) 12.6141 0.872533
\(210\) 0 0
\(211\) −4.96593 −0.341869 −0.170934 0.985282i \(-0.554679\pi\)
−0.170934 + 0.985282i \(0.554679\pi\)
\(212\) 0 0
\(213\) −8.75660 −0.599992
\(214\) 0 0
\(215\) 22.4485 1.53097
\(216\) 0 0
\(217\) 13.3335 0.905139
\(218\) 0 0
\(219\) 26.2121 1.77125
\(220\) 0 0
\(221\) 2.33833 0.157293
\(222\) 0 0
\(223\) −20.6187 −1.38073 −0.690365 0.723461i \(-0.742550\pi\)
−0.690365 + 0.723461i \(0.742550\pi\)
\(224\) 0 0
\(225\) −2.19043 −0.146029
\(226\) 0 0
\(227\) −17.2988 −1.14816 −0.574081 0.818798i \(-0.694641\pi\)
−0.574081 + 0.818798i \(0.694641\pi\)
\(228\) 0 0
\(229\) 14.8279 0.979856 0.489928 0.871763i \(-0.337023\pi\)
0.489928 + 0.871763i \(0.337023\pi\)
\(230\) 0 0
\(231\) 9.68414 0.637170
\(232\) 0 0
\(233\) 18.6404 1.22117 0.610587 0.791950i \(-0.290934\pi\)
0.610587 + 0.791950i \(0.290934\pi\)
\(234\) 0 0
\(235\) 23.6269 1.54125
\(236\) 0 0
\(237\) 43.5381 2.82810
\(238\) 0 0
\(239\) −20.5288 −1.32789 −0.663947 0.747779i \(-0.731120\pi\)
−0.663947 + 0.747779i \(0.731120\pi\)
\(240\) 0 0
\(241\) 22.1279 1.42538 0.712690 0.701479i \(-0.247477\pi\)
0.712690 + 0.701479i \(0.247477\pi\)
\(242\) 0 0
\(243\) 15.3290 0.983353
\(244\) 0 0
\(245\) −11.2985 −0.721838
\(246\) 0 0
\(247\) −5.88745 −0.374609
\(248\) 0 0
\(249\) −38.6871 −2.45170
\(250\) 0 0
\(251\) 5.20213 0.328356 0.164178 0.986431i \(-0.447503\pi\)
0.164178 + 0.986431i \(0.447503\pi\)
\(252\) 0 0
\(253\) 9.11805 0.573247
\(254\) 0 0
\(255\) 11.3545 0.711043
\(256\) 0 0
\(257\) 25.5121 1.59140 0.795702 0.605688i \(-0.207102\pi\)
0.795702 + 0.605688i \(0.207102\pi\)
\(258\) 0 0
\(259\) 0.498339 0.0309653
\(260\) 0 0
\(261\) 13.0946 0.810536
\(262\) 0 0
\(263\) −10.1137 −0.623639 −0.311819 0.950141i \(-0.600938\pi\)
−0.311819 + 0.950141i \(0.600938\pi\)
\(264\) 0 0
\(265\) −10.1349 −0.622584
\(266\) 0 0
\(267\) 45.8354 2.80508
\(268\) 0 0
\(269\) 26.1915 1.59692 0.798462 0.602045i \(-0.205647\pi\)
0.798462 + 0.602045i \(0.205647\pi\)
\(270\) 0 0
\(271\) −10.6976 −0.649831 −0.324916 0.945743i \(-0.605336\pi\)
−0.324916 + 0.945743i \(0.605336\pi\)
\(272\) 0 0
\(273\) −4.51995 −0.273560
\(274\) 0 0
\(275\) −1.18582 −0.0715076
\(276\) 0 0
\(277\) 19.3522 1.16276 0.581380 0.813632i \(-0.302513\pi\)
0.581380 + 0.813632i \(0.302513\pi\)
\(278\) 0 0
\(279\) −49.8620 −2.98516
\(280\) 0 0
\(281\) 30.9173 1.84437 0.922186 0.386748i \(-0.126402\pi\)
0.922186 + 0.386748i \(0.126402\pi\)
\(282\) 0 0
\(283\) 13.8357 0.822445 0.411222 0.911535i \(-0.365102\pi\)
0.411222 + 0.911535i \(0.365102\pi\)
\(284\) 0 0
\(285\) −28.5882 −1.69342
\(286\) 0 0
\(287\) −3.66517 −0.216348
\(288\) 0 0
\(289\) −13.4059 −0.788580
\(290\) 0 0
\(291\) −12.9110 −0.756853
\(292\) 0 0
\(293\) −3.06755 −0.179208 −0.0896042 0.995977i \(-0.528560\pi\)
−0.0896042 + 0.995977i \(0.528560\pi\)
\(294\) 0 0
\(295\) 30.7551 1.79063
\(296\) 0 0
\(297\) −13.9581 −0.809933
\(298\) 0 0
\(299\) −4.25573 −0.246115
\(300\) 0 0
\(301\) −13.7355 −0.791698
\(302\) 0 0
\(303\) −25.1400 −1.44426
\(304\) 0 0
\(305\) 0.560243 0.0320794
\(306\) 0 0
\(307\) −33.5692 −1.91590 −0.957949 0.286940i \(-0.907362\pi\)
−0.957949 + 0.286940i \(0.907362\pi\)
\(308\) 0 0
\(309\) 0.555255 0.0315873
\(310\) 0 0
\(311\) 0.493489 0.0279832 0.0139916 0.999902i \(-0.495546\pi\)
0.0139916 + 0.999902i \(0.495546\pi\)
\(312\) 0 0
\(313\) 31.2159 1.76443 0.882215 0.470847i \(-0.156052\pi\)
0.882215 + 0.470847i \(0.156052\pi\)
\(314\) 0 0
\(315\) −13.5936 −0.765915
\(316\) 0 0
\(317\) 6.75608 0.379459 0.189730 0.981836i \(-0.439239\pi\)
0.189730 + 0.981836i \(0.439239\pi\)
\(318\) 0 0
\(319\) 7.08895 0.396905
\(320\) 0 0
\(321\) −23.3737 −1.30459
\(322\) 0 0
\(323\) −9.04931 −0.503517
\(324\) 0 0
\(325\) 0.553466 0.0307008
\(326\) 0 0
\(327\) 10.3063 0.569938
\(328\) 0 0
\(329\) −14.4565 −0.797013
\(330\) 0 0
\(331\) −22.0730 −1.21324 −0.606620 0.794992i \(-0.707475\pi\)
−0.606620 + 0.794992i \(0.707475\pi\)
\(332\) 0 0
\(333\) −1.86359 −0.102124
\(334\) 0 0
\(335\) 4.42184 0.241591
\(336\) 0 0
\(337\) 0.308315 0.0167950 0.00839749 0.999965i \(-0.497327\pi\)
0.00839749 + 0.999965i \(0.497327\pi\)
\(338\) 0 0
\(339\) −19.2688 −1.04654
\(340\) 0 0
\(341\) −26.9935 −1.46178
\(342\) 0 0
\(343\) 16.0505 0.866647
\(344\) 0 0
\(345\) −20.6650 −1.11256
\(346\) 0 0
\(347\) 9.33516 0.501138 0.250569 0.968099i \(-0.419382\pi\)
0.250569 + 0.968099i \(0.419382\pi\)
\(348\) 0 0
\(349\) −30.0277 −1.60735 −0.803673 0.595072i \(-0.797124\pi\)
−0.803673 + 0.595072i \(0.797124\pi\)
\(350\) 0 0
\(351\) 6.51478 0.347733
\(352\) 0 0
\(353\) 24.8659 1.32348 0.661739 0.749734i \(-0.269819\pi\)
0.661739 + 0.749734i \(0.269819\pi\)
\(354\) 0 0
\(355\) 6.65425 0.353171
\(356\) 0 0
\(357\) −6.94739 −0.367695
\(358\) 0 0
\(359\) 32.2427 1.70170 0.850852 0.525406i \(-0.176087\pi\)
0.850852 + 0.525406i \(0.176087\pi\)
\(360\) 0 0
\(361\) 3.78435 0.199177
\(362\) 0 0
\(363\) 11.2759 0.591830
\(364\) 0 0
\(365\) −19.9189 −1.04260
\(366\) 0 0
\(367\) −28.3223 −1.47841 −0.739206 0.673479i \(-0.764799\pi\)
−0.739206 + 0.673479i \(0.764799\pi\)
\(368\) 0 0
\(369\) 13.7063 0.713520
\(370\) 0 0
\(371\) 6.20121 0.321951
\(372\) 0 0
\(373\) −12.0391 −0.623360 −0.311680 0.950187i \(-0.600892\pi\)
−0.311680 + 0.950187i \(0.600892\pi\)
\(374\) 0 0
\(375\) 32.6335 1.68519
\(376\) 0 0
\(377\) −3.30868 −0.170405
\(378\) 0 0
\(379\) 27.6192 1.41870 0.709352 0.704854i \(-0.248988\pi\)
0.709352 + 0.704854i \(0.248988\pi\)
\(380\) 0 0
\(381\) −36.7860 −1.88461
\(382\) 0 0
\(383\) 12.9082 0.659576 0.329788 0.944055i \(-0.393023\pi\)
0.329788 + 0.944055i \(0.393023\pi\)
\(384\) 0 0
\(385\) −7.35911 −0.375055
\(386\) 0 0
\(387\) 51.3651 2.61103
\(388\) 0 0
\(389\) 34.2037 1.73420 0.867098 0.498138i \(-0.165983\pi\)
0.867098 + 0.498138i \(0.165983\pi\)
\(390\) 0 0
\(391\) −6.54128 −0.330807
\(392\) 0 0
\(393\) 23.3266 1.17667
\(394\) 0 0
\(395\) −33.0852 −1.66470
\(396\) 0 0
\(397\) 11.2872 0.566486 0.283243 0.959048i \(-0.408590\pi\)
0.283243 + 0.959048i \(0.408590\pi\)
\(398\) 0 0
\(399\) 17.4921 0.875703
\(400\) 0 0
\(401\) 12.9681 0.647594 0.323797 0.946127i \(-0.395041\pi\)
0.323797 + 0.946127i \(0.395041\pi\)
\(402\) 0 0
\(403\) 12.5989 0.627594
\(404\) 0 0
\(405\) 0.392697 0.0195133
\(406\) 0 0
\(407\) −1.00888 −0.0500083
\(408\) 0 0
\(409\) 9.07234 0.448598 0.224299 0.974520i \(-0.427991\pi\)
0.224299 + 0.974520i \(0.427991\pi\)
\(410\) 0 0
\(411\) −54.6371 −2.69505
\(412\) 0 0
\(413\) −18.8179 −0.925971
\(414\) 0 0
\(415\) 29.3989 1.44313
\(416\) 0 0
\(417\) −5.35623 −0.262296
\(418\) 0 0
\(419\) 20.3301 0.993188 0.496594 0.867983i \(-0.334584\pi\)
0.496594 + 0.867983i \(0.334584\pi\)
\(420\) 0 0
\(421\) −4.08127 −0.198909 −0.0994544 0.995042i \(-0.531710\pi\)
−0.0994544 + 0.995042i \(0.531710\pi\)
\(422\) 0 0
\(423\) 54.0615 2.62856
\(424\) 0 0
\(425\) 0.850706 0.0412653
\(426\) 0 0
\(427\) −0.342793 −0.0165889
\(428\) 0 0
\(429\) 9.15056 0.441793
\(430\) 0 0
\(431\) −31.5483 −1.51963 −0.759815 0.650139i \(-0.774711\pi\)
−0.759815 + 0.650139i \(0.774711\pi\)
\(432\) 0 0
\(433\) 0.606781 0.0291600 0.0145800 0.999894i \(-0.495359\pi\)
0.0145800 + 0.999894i \(0.495359\pi\)
\(434\) 0 0
\(435\) −16.0662 −0.770318
\(436\) 0 0
\(437\) 16.4696 0.787849
\(438\) 0 0
\(439\) −7.57029 −0.361310 −0.180655 0.983547i \(-0.557822\pi\)
−0.180655 + 0.983547i \(0.557822\pi\)
\(440\) 0 0
\(441\) −25.8525 −1.23107
\(442\) 0 0
\(443\) 11.3363 0.538602 0.269301 0.963056i \(-0.413207\pi\)
0.269301 + 0.963056i \(0.413207\pi\)
\(444\) 0 0
\(445\) −34.8309 −1.65114
\(446\) 0 0
\(447\) 21.0225 0.994332
\(448\) 0 0
\(449\) −28.6132 −1.35034 −0.675171 0.737662i \(-0.735930\pi\)
−0.675171 + 0.737662i \(0.735930\pi\)
\(450\) 0 0
\(451\) 7.42008 0.349398
\(452\) 0 0
\(453\) 7.66070 0.359931
\(454\) 0 0
\(455\) 3.43477 0.161024
\(456\) 0 0
\(457\) −9.22205 −0.431389 −0.215695 0.976461i \(-0.569202\pi\)
−0.215695 + 0.976461i \(0.569202\pi\)
\(458\) 0 0
\(459\) 10.0135 0.467392
\(460\) 0 0
\(461\) 21.8277 1.01662 0.508309 0.861175i \(-0.330271\pi\)
0.508309 + 0.861175i \(0.330271\pi\)
\(462\) 0 0
\(463\) 0.383705 0.0178323 0.00891614 0.999960i \(-0.497162\pi\)
0.00891614 + 0.999960i \(0.497162\pi\)
\(464\) 0 0
\(465\) 61.1775 2.83704
\(466\) 0 0
\(467\) 33.1149 1.53237 0.766186 0.642618i \(-0.222152\pi\)
0.766186 + 0.642618i \(0.222152\pi\)
\(468\) 0 0
\(469\) −2.70557 −0.124931
\(470\) 0 0
\(471\) −1.47683 −0.0680486
\(472\) 0 0
\(473\) 27.8072 1.27858
\(474\) 0 0
\(475\) −2.14191 −0.0982775
\(476\) 0 0
\(477\) −23.1900 −1.06180
\(478\) 0 0
\(479\) 24.7935 1.13284 0.566422 0.824116i \(-0.308327\pi\)
0.566422 + 0.824116i \(0.308327\pi\)
\(480\) 0 0
\(481\) 0.470881 0.0214703
\(482\) 0 0
\(483\) 12.6442 0.575330
\(484\) 0 0
\(485\) 9.81120 0.445504
\(486\) 0 0
\(487\) 16.8922 0.765458 0.382729 0.923861i \(-0.374984\pi\)
0.382729 + 0.923861i \(0.374984\pi\)
\(488\) 0 0
\(489\) −16.8075 −0.760062
\(490\) 0 0
\(491\) −16.5978 −0.749048 −0.374524 0.927217i \(-0.622194\pi\)
−0.374524 + 0.927217i \(0.622194\pi\)
\(492\) 0 0
\(493\) −5.08560 −0.229044
\(494\) 0 0
\(495\) 27.5201 1.23694
\(496\) 0 0
\(497\) −4.07150 −0.182632
\(498\) 0 0
\(499\) −26.8499 −1.20196 −0.600982 0.799262i \(-0.705224\pi\)
−0.600982 + 0.799262i \(0.705224\pi\)
\(500\) 0 0
\(501\) −25.5349 −1.14081
\(502\) 0 0
\(503\) 41.1035 1.83272 0.916358 0.400359i \(-0.131115\pi\)
0.916358 + 0.400359i \(0.131115\pi\)
\(504\) 0 0
\(505\) 19.1042 0.850127
\(506\) 0 0
\(507\) 32.2251 1.43117
\(508\) 0 0
\(509\) −11.6386 −0.515870 −0.257935 0.966162i \(-0.583042\pi\)
−0.257935 + 0.966162i \(0.583042\pi\)
\(510\) 0 0
\(511\) 12.1877 0.539151
\(512\) 0 0
\(513\) −25.2121 −1.11314
\(514\) 0 0
\(515\) −0.421945 −0.0185931
\(516\) 0 0
\(517\) 29.2670 1.28716
\(518\) 0 0
\(519\) −0.850676 −0.0373405
\(520\) 0 0
\(521\) 10.4871 0.459448 0.229724 0.973256i \(-0.426218\pi\)
0.229724 + 0.973256i \(0.426218\pi\)
\(522\) 0 0
\(523\) 36.6885 1.60427 0.802137 0.597139i \(-0.203696\pi\)
0.802137 + 0.597139i \(0.203696\pi\)
\(524\) 0 0
\(525\) −1.64440 −0.0717674
\(526\) 0 0
\(527\) 19.3651 0.843558
\(528\) 0 0
\(529\) −11.0950 −0.482389
\(530\) 0 0
\(531\) 70.3716 3.05387
\(532\) 0 0
\(533\) −3.46323 −0.150009
\(534\) 0 0
\(535\) 17.7620 0.767917
\(536\) 0 0
\(537\) 45.7036 1.97225
\(538\) 0 0
\(539\) −13.9956 −0.602835
\(540\) 0 0
\(541\) 24.6895 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(542\) 0 0
\(543\) 38.1327 1.63643
\(544\) 0 0
\(545\) −7.83186 −0.335480
\(546\) 0 0
\(547\) 13.1350 0.561612 0.280806 0.959765i \(-0.409398\pi\)
0.280806 + 0.959765i \(0.409398\pi\)
\(548\) 0 0
\(549\) 1.28191 0.0547105
\(550\) 0 0
\(551\) 12.8045 0.545492
\(552\) 0 0
\(553\) 20.2437 0.860848
\(554\) 0 0
\(555\) 2.28650 0.0970566
\(556\) 0 0
\(557\) 11.1909 0.474175 0.237088 0.971488i \(-0.423807\pi\)
0.237088 + 0.971488i \(0.423807\pi\)
\(558\) 0 0
\(559\) −12.9787 −0.548938
\(560\) 0 0
\(561\) 14.0649 0.593820
\(562\) 0 0
\(563\) −1.66122 −0.0700123 −0.0350061 0.999387i \(-0.511145\pi\)
−0.0350061 + 0.999387i \(0.511145\pi\)
\(564\) 0 0
\(565\) 14.6426 0.616020
\(566\) 0 0
\(567\) −0.240278 −0.0100907
\(568\) 0 0
\(569\) 27.7642 1.16394 0.581969 0.813211i \(-0.302283\pi\)
0.581969 + 0.813211i \(0.302283\pi\)
\(570\) 0 0
\(571\) 0.321419 0.0134510 0.00672548 0.999977i \(-0.497859\pi\)
0.00672548 + 0.999977i \(0.497859\pi\)
\(572\) 0 0
\(573\) 38.4637 1.60684
\(574\) 0 0
\(575\) −1.54827 −0.0645675
\(576\) 0 0
\(577\) 26.0897 1.08613 0.543065 0.839691i \(-0.317264\pi\)
0.543065 + 0.839691i \(0.317264\pi\)
\(578\) 0 0
\(579\) 64.3882 2.67588
\(580\) 0 0
\(581\) −17.9881 −0.746274
\(582\) 0 0
\(583\) −12.5542 −0.519944
\(584\) 0 0
\(585\) −12.8446 −0.531061
\(586\) 0 0
\(587\) 29.9747 1.23719 0.618594 0.785711i \(-0.287703\pi\)
0.618594 + 0.785711i \(0.287703\pi\)
\(588\) 0 0
\(589\) −48.7575 −2.00902
\(590\) 0 0
\(591\) 3.80689 0.156595
\(592\) 0 0
\(593\) 3.89995 0.160152 0.0800759 0.996789i \(-0.474484\pi\)
0.0800759 + 0.996789i \(0.474484\pi\)
\(594\) 0 0
\(595\) 5.27941 0.216435
\(596\) 0 0
\(597\) −72.4426 −2.96488
\(598\) 0 0
\(599\) 18.5532 0.758062 0.379031 0.925384i \(-0.376257\pi\)
0.379031 + 0.925384i \(0.376257\pi\)
\(600\) 0 0
\(601\) 26.5534 1.08313 0.541567 0.840658i \(-0.317831\pi\)
0.541567 + 0.840658i \(0.317831\pi\)
\(602\) 0 0
\(603\) 10.1177 0.412026
\(604\) 0 0
\(605\) −8.56869 −0.348367
\(606\) 0 0
\(607\) 30.2820 1.22911 0.614555 0.788874i \(-0.289336\pi\)
0.614555 + 0.788874i \(0.289336\pi\)
\(608\) 0 0
\(609\) 9.83038 0.398347
\(610\) 0 0
\(611\) −13.6600 −0.552623
\(612\) 0 0
\(613\) −6.62975 −0.267773 −0.133887 0.990997i \(-0.542746\pi\)
−0.133887 + 0.990997i \(0.542746\pi\)
\(614\) 0 0
\(615\) −16.8167 −0.678115
\(616\) 0 0
\(617\) 31.4413 1.26578 0.632889 0.774243i \(-0.281869\pi\)
0.632889 + 0.774243i \(0.281869\pi\)
\(618\) 0 0
\(619\) 9.71192 0.390355 0.195178 0.980768i \(-0.437472\pi\)
0.195178 + 0.980768i \(0.437472\pi\)
\(620\) 0 0
\(621\) −18.2245 −0.731325
\(622\) 0 0
\(623\) 21.3118 0.853841
\(624\) 0 0
\(625\) −22.5550 −0.902200
\(626\) 0 0
\(627\) −35.4126 −1.41424
\(628\) 0 0
\(629\) 0.723769 0.0288585
\(630\) 0 0
\(631\) −38.9579 −1.55089 −0.775445 0.631415i \(-0.782474\pi\)
−0.775445 + 0.631415i \(0.782474\pi\)
\(632\) 0 0
\(633\) 13.9413 0.554117
\(634\) 0 0
\(635\) 27.9542 1.10933
\(636\) 0 0
\(637\) 6.53228 0.258818
\(638\) 0 0
\(639\) 15.2258 0.602323
\(640\) 0 0
\(641\) 6.08952 0.240522 0.120261 0.992742i \(-0.461627\pi\)
0.120261 + 0.992742i \(0.461627\pi\)
\(642\) 0 0
\(643\) 12.3951 0.488814 0.244407 0.969673i \(-0.421407\pi\)
0.244407 + 0.969673i \(0.421407\pi\)
\(644\) 0 0
\(645\) −63.0217 −2.48148
\(646\) 0 0
\(647\) 10.4905 0.412425 0.206212 0.978507i \(-0.433886\pi\)
0.206212 + 0.978507i \(0.433886\pi\)
\(648\) 0 0
\(649\) 38.0966 1.49542
\(650\) 0 0
\(651\) −37.4324 −1.46709
\(652\) 0 0
\(653\) 14.4212 0.564344 0.282172 0.959364i \(-0.408945\pi\)
0.282172 + 0.959364i \(0.408945\pi\)
\(654\) 0 0
\(655\) −17.7262 −0.692619
\(656\) 0 0
\(657\) −45.5770 −1.77813
\(658\) 0 0
\(659\) 37.2443 1.45083 0.725416 0.688311i \(-0.241647\pi\)
0.725416 + 0.688311i \(0.241647\pi\)
\(660\) 0 0
\(661\) −23.2991 −0.906228 −0.453114 0.891453i \(-0.649687\pi\)
−0.453114 + 0.891453i \(0.649687\pi\)
\(662\) 0 0
\(663\) −6.56460 −0.254948
\(664\) 0 0
\(665\) −13.2925 −0.515461
\(666\) 0 0
\(667\) 9.25574 0.358384
\(668\) 0 0
\(669\) 57.8847 2.23795
\(670\) 0 0
\(671\) 0.693979 0.0267908
\(672\) 0 0
\(673\) 38.3854 1.47965 0.739825 0.672799i \(-0.234908\pi\)
0.739825 + 0.672799i \(0.234908\pi\)
\(674\) 0 0
\(675\) 2.37014 0.0912265
\(676\) 0 0
\(677\) −45.0194 −1.73023 −0.865117 0.501570i \(-0.832756\pi\)
−0.865117 + 0.501570i \(0.832756\pi\)
\(678\) 0 0
\(679\) −6.00313 −0.230379
\(680\) 0 0
\(681\) 48.5645 1.86100
\(682\) 0 0
\(683\) −0.840213 −0.0321498 −0.0160749 0.999871i \(-0.505117\pi\)
−0.0160749 + 0.999871i \(0.505117\pi\)
\(684\) 0 0
\(685\) 41.5195 1.58638
\(686\) 0 0
\(687\) −41.6277 −1.58820
\(688\) 0 0
\(689\) 5.85953 0.223230
\(690\) 0 0
\(691\) 32.9672 1.25413 0.627065 0.778967i \(-0.284256\pi\)
0.627065 + 0.778967i \(0.284256\pi\)
\(692\) 0 0
\(693\) −16.8386 −0.639645
\(694\) 0 0
\(695\) 4.07027 0.154394
\(696\) 0 0
\(697\) −5.32316 −0.201629
\(698\) 0 0
\(699\) −52.3308 −1.97933
\(700\) 0 0
\(701\) −36.9407 −1.39523 −0.697616 0.716472i \(-0.745756\pi\)
−0.697616 + 0.716472i \(0.745756\pi\)
\(702\) 0 0
\(703\) −1.82230 −0.0687295
\(704\) 0 0
\(705\) −66.3300 −2.49813
\(706\) 0 0
\(707\) −11.6892 −0.439618
\(708\) 0 0
\(709\) 4.32612 0.162471 0.0812353 0.996695i \(-0.474113\pi\)
0.0812353 + 0.996695i \(0.474113\pi\)
\(710\) 0 0
\(711\) −75.7032 −2.83909
\(712\) 0 0
\(713\) −35.2443 −1.31991
\(714\) 0 0
\(715\) −6.95363 −0.260051
\(716\) 0 0
\(717\) 57.6322 2.15231
\(718\) 0 0
\(719\) 28.7216 1.07113 0.535567 0.844493i \(-0.320098\pi\)
0.535567 + 0.844493i \(0.320098\pi\)
\(720\) 0 0
\(721\) 0.258174 0.00961489
\(722\) 0 0
\(723\) −62.1215 −2.31032
\(724\) 0 0
\(725\) −1.20373 −0.0447053
\(726\) 0 0
\(727\) −20.5287 −0.761367 −0.380684 0.924705i \(-0.624311\pi\)
−0.380684 + 0.924705i \(0.624311\pi\)
\(728\) 0 0
\(729\) −43.5866 −1.61432
\(730\) 0 0
\(731\) −19.9489 −0.737835
\(732\) 0 0
\(733\) −25.3244 −0.935378 −0.467689 0.883893i \(-0.654913\pi\)
−0.467689 + 0.883893i \(0.654913\pi\)
\(734\) 0 0
\(735\) 31.7194 1.16999
\(736\) 0 0
\(737\) 5.47738 0.201762
\(738\) 0 0
\(739\) 40.9084 1.50484 0.752421 0.658683i \(-0.228886\pi\)
0.752421 + 0.658683i \(0.228886\pi\)
\(740\) 0 0
\(741\) 16.5284 0.607184
\(742\) 0 0
\(743\) −12.4513 −0.456796 −0.228398 0.973568i \(-0.573349\pi\)
−0.228398 + 0.973568i \(0.573349\pi\)
\(744\) 0 0
\(745\) −15.9753 −0.585290
\(746\) 0 0
\(747\) 67.2684 2.46122
\(748\) 0 0
\(749\) −10.8679 −0.397106
\(750\) 0 0
\(751\) −0.904509 −0.0330060 −0.0165030 0.999864i \(-0.505253\pi\)
−0.0165030 + 0.999864i \(0.505253\pi\)
\(752\) 0 0
\(753\) −14.6044 −0.532214
\(754\) 0 0
\(755\) −5.82147 −0.211865
\(756\) 0 0
\(757\) −25.4117 −0.923605 −0.461802 0.886983i \(-0.652797\pi\)
−0.461802 + 0.886983i \(0.652797\pi\)
\(758\) 0 0
\(759\) −25.5979 −0.929145
\(760\) 0 0
\(761\) −26.6769 −0.967037 −0.483518 0.875334i \(-0.660641\pi\)
−0.483518 + 0.875334i \(0.660641\pi\)
\(762\) 0 0
\(763\) 4.79204 0.173484
\(764\) 0 0
\(765\) −19.7429 −0.713806
\(766\) 0 0
\(767\) −17.7811 −0.642039
\(768\) 0 0
\(769\) −38.7904 −1.39882 −0.699409 0.714721i \(-0.746554\pi\)
−0.699409 + 0.714721i \(0.746554\pi\)
\(770\) 0 0
\(771\) −71.6225 −2.57942
\(772\) 0 0
\(773\) 18.8460 0.677844 0.338922 0.940814i \(-0.389938\pi\)
0.338922 + 0.940814i \(0.389938\pi\)
\(774\) 0 0
\(775\) 4.58358 0.164647
\(776\) 0 0
\(777\) −1.39903 −0.0501899
\(778\) 0 0
\(779\) 13.4026 0.480199
\(780\) 0 0
\(781\) 8.24270 0.294947
\(782\) 0 0
\(783\) −14.1689 −0.506355
\(784\) 0 0
\(785\) 1.12226 0.0400552
\(786\) 0 0
\(787\) 19.1304 0.681924 0.340962 0.940077i \(-0.389247\pi\)
0.340962 + 0.940077i \(0.389247\pi\)
\(788\) 0 0
\(789\) 28.3931 1.01082
\(790\) 0 0
\(791\) −8.95931 −0.318556
\(792\) 0 0
\(793\) −0.323906 −0.0115022
\(794\) 0 0
\(795\) 28.4527 1.00911
\(796\) 0 0
\(797\) −23.6477 −0.837644 −0.418822 0.908068i \(-0.637557\pi\)
−0.418822 + 0.908068i \(0.637557\pi\)
\(798\) 0 0
\(799\) −20.9961 −0.742788
\(800\) 0 0
\(801\) −79.6977 −2.81598
\(802\) 0 0
\(803\) −24.6737 −0.870717
\(804\) 0 0
\(805\) −9.60847 −0.338654
\(806\) 0 0
\(807\) −73.5298 −2.58837
\(808\) 0 0
\(809\) 50.2406 1.76637 0.883184 0.469027i \(-0.155395\pi\)
0.883184 + 0.469027i \(0.155395\pi\)
\(810\) 0 0
\(811\) 39.4904 1.38670 0.693348 0.720603i \(-0.256135\pi\)
0.693348 + 0.720603i \(0.256135\pi\)
\(812\) 0 0
\(813\) 30.0323 1.05328
\(814\) 0 0
\(815\) 12.7723 0.447392
\(816\) 0 0
\(817\) 50.2272 1.75723
\(818\) 0 0
\(819\) 7.85919 0.274622
\(820\) 0 0
\(821\) 56.1720 1.96042 0.980208 0.197970i \(-0.0634349\pi\)
0.980208 + 0.197970i \(0.0634349\pi\)
\(822\) 0 0
\(823\) −36.4527 −1.27066 −0.635331 0.772240i \(-0.719137\pi\)
−0.635331 + 0.772240i \(0.719137\pi\)
\(824\) 0 0
\(825\) 3.32906 0.115903
\(826\) 0 0
\(827\) 42.8105 1.48867 0.744334 0.667808i \(-0.232767\pi\)
0.744334 + 0.667808i \(0.232767\pi\)
\(828\) 0 0
\(829\) 29.2448 1.01571 0.507857 0.861442i \(-0.330438\pi\)
0.507857 + 0.861442i \(0.330438\pi\)
\(830\) 0 0
\(831\) −54.3291 −1.88465
\(832\) 0 0
\(833\) 10.0405 0.347881
\(834\) 0 0
\(835\) 19.4043 0.671513
\(836\) 0 0
\(837\) 53.9527 1.86488
\(838\) 0 0
\(839\) −22.0705 −0.761959 −0.380979 0.924584i \(-0.624413\pi\)
−0.380979 + 0.924584i \(0.624413\pi\)
\(840\) 0 0
\(841\) −21.8040 −0.751862
\(842\) 0 0
\(843\) −86.7969 −2.98944
\(844\) 0 0
\(845\) −24.4883 −0.842423
\(846\) 0 0
\(847\) 5.24288 0.180148
\(848\) 0 0
\(849\) −38.8421 −1.33306
\(850\) 0 0
\(851\) −1.31725 −0.0451547
\(852\) 0 0
\(853\) −12.9347 −0.442875 −0.221437 0.975175i \(-0.571075\pi\)
−0.221437 + 0.975175i \(0.571075\pi\)
\(854\) 0 0
\(855\) 49.7086 1.70000
\(856\) 0 0
\(857\) −16.6451 −0.568586 −0.284293 0.958738i \(-0.591759\pi\)
−0.284293 + 0.958738i \(0.591759\pi\)
\(858\) 0 0
\(859\) 14.3279 0.488860 0.244430 0.969667i \(-0.421399\pi\)
0.244430 + 0.969667i \(0.421399\pi\)
\(860\) 0 0
\(861\) 10.2896 0.350667
\(862\) 0 0
\(863\) −22.0785 −0.751562 −0.375781 0.926708i \(-0.622626\pi\)
−0.375781 + 0.926708i \(0.622626\pi\)
\(864\) 0 0
\(865\) 0.646440 0.0219796
\(866\) 0 0
\(867\) 37.6355 1.27817
\(868\) 0 0
\(869\) −40.9830 −1.39025
\(870\) 0 0
\(871\) −2.55649 −0.0866235
\(872\) 0 0
\(873\) 22.4493 0.759794
\(874\) 0 0
\(875\) 15.1734 0.512955
\(876\) 0 0
\(877\) 47.7181 1.61133 0.805663 0.592374i \(-0.201809\pi\)
0.805663 + 0.592374i \(0.201809\pi\)
\(878\) 0 0
\(879\) 8.61181 0.290469
\(880\) 0 0
\(881\) 1.73988 0.0586179 0.0293090 0.999570i \(-0.490669\pi\)
0.0293090 + 0.999570i \(0.490669\pi\)
\(882\) 0 0
\(883\) 13.1934 0.443992 0.221996 0.975048i \(-0.428743\pi\)
0.221996 + 0.975048i \(0.428743\pi\)
\(884\) 0 0
\(885\) −86.3414 −2.90233
\(886\) 0 0
\(887\) 26.7994 0.899836 0.449918 0.893070i \(-0.351453\pi\)
0.449918 + 0.893070i \(0.351453\pi\)
\(888\) 0 0
\(889\) −17.1042 −0.573656
\(890\) 0 0
\(891\) 0.486438 0.0162963
\(892\) 0 0
\(893\) 52.8639 1.76902
\(894\) 0 0
\(895\) −34.7307 −1.16092
\(896\) 0 0
\(897\) 11.9475 0.398915
\(898\) 0 0
\(899\) −27.4011 −0.913878
\(900\) 0 0
\(901\) 9.00640 0.300047
\(902\) 0 0
\(903\) 38.5608 1.28322
\(904\) 0 0
\(905\) −28.9775 −0.963245
\(906\) 0 0
\(907\) −48.7741 −1.61952 −0.809759 0.586762i \(-0.800402\pi\)
−0.809759 + 0.586762i \(0.800402\pi\)
\(908\) 0 0
\(909\) 43.7130 1.44987
\(910\) 0 0
\(911\) 29.3100 0.971083 0.485542 0.874213i \(-0.338622\pi\)
0.485542 + 0.874213i \(0.338622\pi\)
\(912\) 0 0
\(913\) 36.4167 1.20522
\(914\) 0 0
\(915\) −1.57282 −0.0519958
\(916\) 0 0
\(917\) 10.8460 0.358167
\(918\) 0 0
\(919\) 27.3685 0.902802 0.451401 0.892321i \(-0.350924\pi\)
0.451401 + 0.892321i \(0.350924\pi\)
\(920\) 0 0
\(921\) 94.2419 3.10538
\(922\) 0 0
\(923\) −3.84717 −0.126631
\(924\) 0 0
\(925\) 0.171311 0.00563267
\(926\) 0 0
\(927\) −0.965466 −0.0317100
\(928\) 0 0
\(929\) 43.2898 1.42029 0.710146 0.704055i \(-0.248629\pi\)
0.710146 + 0.704055i \(0.248629\pi\)
\(930\) 0 0
\(931\) −25.2798 −0.828514
\(932\) 0 0
\(933\) −1.38541 −0.0453564
\(934\) 0 0
\(935\) −10.6881 −0.349538
\(936\) 0 0
\(937\) −45.3443 −1.48134 −0.740668 0.671872i \(-0.765491\pi\)
−0.740668 + 0.671872i \(0.765491\pi\)
\(938\) 0 0
\(939\) −87.6353 −2.85987
\(940\) 0 0
\(941\) 19.6190 0.639562 0.319781 0.947491i \(-0.396391\pi\)
0.319781 + 0.947491i \(0.396391\pi\)
\(942\) 0 0
\(943\) 9.68808 0.315487
\(944\) 0 0
\(945\) 14.7089 0.478480
\(946\) 0 0
\(947\) −0.189029 −0.00614263 −0.00307132 0.999995i \(-0.500978\pi\)
−0.00307132 + 0.999995i \(0.500978\pi\)
\(948\) 0 0
\(949\) 11.5161 0.373830
\(950\) 0 0
\(951\) −18.9669 −0.615045
\(952\) 0 0
\(953\) −59.1968 −1.91757 −0.958786 0.284129i \(-0.908296\pi\)
−0.958786 + 0.284129i \(0.908296\pi\)
\(954\) 0 0
\(955\) −29.2291 −0.945831
\(956\) 0 0
\(957\) −19.9014 −0.643322
\(958\) 0 0
\(959\) −25.4043 −0.820348
\(960\) 0 0
\(961\) 73.3388 2.36577
\(962\) 0 0
\(963\) 40.6417 1.30966
\(964\) 0 0
\(965\) −48.9294 −1.57509
\(966\) 0 0
\(967\) 6.37056 0.204864 0.102432 0.994740i \(-0.467338\pi\)
0.102432 + 0.994740i \(0.467338\pi\)
\(968\) 0 0
\(969\) 25.4049 0.816124
\(970\) 0 0
\(971\) −32.0634 −1.02896 −0.514482 0.857501i \(-0.672016\pi\)
−0.514482 + 0.857501i \(0.672016\pi\)
\(972\) 0 0
\(973\) −2.49045 −0.0798403
\(974\) 0 0
\(975\) −1.55379 −0.0497613
\(976\) 0 0
\(977\) −45.9782 −1.47097 −0.735486 0.677540i \(-0.763046\pi\)
−0.735486 + 0.677540i \(0.763046\pi\)
\(978\) 0 0
\(979\) −43.1455 −1.37894
\(980\) 0 0
\(981\) −17.9203 −0.572152
\(982\) 0 0
\(983\) 31.9226 1.01817 0.509086 0.860716i \(-0.329983\pi\)
0.509086 + 0.860716i \(0.329983\pi\)
\(984\) 0 0
\(985\) −2.89291 −0.0921756
\(986\) 0 0
\(987\) 40.5850 1.29184
\(988\) 0 0
\(989\) 36.3067 1.15449
\(990\) 0 0
\(991\) 49.1835 1.56237 0.781183 0.624302i \(-0.214617\pi\)
0.781183 + 0.624302i \(0.214617\pi\)
\(992\) 0 0
\(993\) 61.9674 1.96648
\(994\) 0 0
\(995\) 55.0501 1.74521
\(996\) 0 0
\(997\) −40.1515 −1.27161 −0.635806 0.771849i \(-0.719332\pi\)
−0.635806 + 0.771849i \(0.719332\pi\)
\(998\) 0 0
\(999\) 2.01648 0.0637985
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 8012.2.a.b.1.7 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8012.2.a.b.1.7 88 1.1 even 1 trivial