Properties

Label 8024.2.a.x.1.2
Level $8024$
Weight $2$
Character 8024.1
Self dual yes
Analytic conductor $64.072$
Analytic rank $1$
Dimension $22$
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] = [8024,2,Mod(1,8024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8024.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: \(64.0719625819\)
Analytic rank: \(1\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 8024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56466 q^{3} -3.26922 q^{5} +2.64610 q^{7} +3.57750 q^{9} +O(q^{10})\) \(q-2.56466 q^{3} -3.26922 q^{5} +2.64610 q^{7} +3.57750 q^{9} +4.86957 q^{11} +1.70896 q^{13} +8.38445 q^{15} -1.00000 q^{17} -0.816317 q^{19} -6.78637 q^{21} -7.69055 q^{23} +5.68779 q^{25} -1.48110 q^{27} -0.446626 q^{29} -5.86589 q^{31} -12.4888 q^{33} -8.65069 q^{35} +8.42139 q^{37} -4.38291 q^{39} +1.93260 q^{41} -9.18105 q^{43} -11.6956 q^{45} +0.424956 q^{47} +0.00185929 q^{49} +2.56466 q^{51} +3.07461 q^{53} -15.9197 q^{55} +2.09358 q^{57} +1.00000 q^{59} -8.31581 q^{61} +9.46644 q^{63} -5.58697 q^{65} +5.42814 q^{67} +19.7237 q^{69} +14.9065 q^{71} -9.04698 q^{73} -14.5873 q^{75} +12.8854 q^{77} -8.52263 q^{79} -6.93398 q^{81} +5.03121 q^{83} +3.26922 q^{85} +1.14545 q^{87} -0.630394 q^{89} +4.52209 q^{91} +15.0440 q^{93} +2.66872 q^{95} +12.4738 q^{97} +17.4209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 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
\(3\) −2.56466 −1.48071 −0.740355 0.672216i \(-0.765342\pi\)
−0.740355 + 0.672216i \(0.765342\pi\)
\(4\) 0 0
\(5\) −3.26922 −1.46204 −0.731020 0.682356i \(-0.760955\pi\)
−0.731020 + 0.682356i \(0.760955\pi\)
\(6\) 0 0
\(7\) 2.64610 1.00013 0.500066 0.865987i \(-0.333309\pi\)
0.500066 + 0.865987i \(0.333309\pi\)
\(8\) 0 0
\(9\) 3.57750 1.19250
\(10\) 0 0
\(11\) 4.86957 1.46823 0.734115 0.679025i \(-0.237597\pi\)
0.734115 + 0.679025i \(0.237597\pi\)
\(12\) 0 0
\(13\) 1.70896 0.473981 0.236990 0.971512i \(-0.423839\pi\)
0.236990 + 0.971512i \(0.423839\pi\)
\(14\) 0 0
\(15\) 8.38445 2.16486
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −0.816317 −0.187276 −0.0936380 0.995606i \(-0.529850\pi\)
−0.0936380 + 0.995606i \(0.529850\pi\)
\(20\) 0 0
\(21\) −6.78637 −1.48091
\(22\) 0 0
\(23\) −7.69055 −1.60359 −0.801795 0.597599i \(-0.796122\pi\)
−0.801795 + 0.597599i \(0.796122\pi\)
\(24\) 0 0
\(25\) 5.68779 1.13756
\(26\) 0 0
\(27\) −1.48110 −0.285038
\(28\) 0 0
\(29\) −0.446626 −0.0829364 −0.0414682 0.999140i \(-0.513204\pi\)
−0.0414682 + 0.999140i \(0.513204\pi\)
\(30\) 0 0
\(31\) −5.86589 −1.05354 −0.526772 0.850006i \(-0.676598\pi\)
−0.526772 + 0.850006i \(0.676598\pi\)
\(32\) 0 0
\(33\) −12.4888 −2.17402
\(34\) 0 0
\(35\) −8.65069 −1.46223
\(36\) 0 0
\(37\) 8.42139 1.38447 0.692234 0.721673i \(-0.256627\pi\)
0.692234 + 0.721673i \(0.256627\pi\)
\(38\) 0 0
\(39\) −4.38291 −0.701828
\(40\) 0 0
\(41\) 1.93260 0.301822 0.150911 0.988547i \(-0.451779\pi\)
0.150911 + 0.988547i \(0.451779\pi\)
\(42\) 0 0
\(43\) −9.18105 −1.40010 −0.700049 0.714095i \(-0.746838\pi\)
−0.700049 + 0.714095i \(0.746838\pi\)
\(44\) 0 0
\(45\) −11.6956 −1.74348
\(46\) 0 0
\(47\) 0.424956 0.0619862 0.0309931 0.999520i \(-0.490133\pi\)
0.0309931 + 0.999520i \(0.490133\pi\)
\(48\) 0 0
\(49\) 0.00185929 0.000265612 0
\(50\) 0 0
\(51\) 2.56466 0.359125
\(52\) 0 0
\(53\) 3.07461 0.422330 0.211165 0.977450i \(-0.432274\pi\)
0.211165 + 0.977450i \(0.432274\pi\)
\(54\) 0 0
\(55\) −15.9197 −2.14661
\(56\) 0 0
\(57\) 2.09358 0.277301
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) −8.31581 −1.06473 −0.532365 0.846515i \(-0.678697\pi\)
−0.532365 + 0.846515i \(0.678697\pi\)
\(62\) 0 0
\(63\) 9.46644 1.19266
\(64\) 0 0
\(65\) −5.58697 −0.692978
\(66\) 0 0
\(67\) 5.42814 0.663153 0.331576 0.943428i \(-0.392420\pi\)
0.331576 + 0.943428i \(0.392420\pi\)
\(68\) 0 0
\(69\) 19.7237 2.37445
\(70\) 0 0
\(71\) 14.9065 1.76908 0.884539 0.466466i \(-0.154473\pi\)
0.884539 + 0.466466i \(0.154473\pi\)
\(72\) 0 0
\(73\) −9.04698 −1.05887 −0.529434 0.848351i \(-0.677596\pi\)
−0.529434 + 0.848351i \(0.677596\pi\)
\(74\) 0 0
\(75\) −14.5873 −1.68439
\(76\) 0 0
\(77\) 12.8854 1.46843
\(78\) 0 0
\(79\) −8.52263 −0.958871 −0.479436 0.877577i \(-0.659159\pi\)
−0.479436 + 0.877577i \(0.659159\pi\)
\(80\) 0 0
\(81\) −6.93398 −0.770442
\(82\) 0 0
\(83\) 5.03121 0.552247 0.276123 0.961122i \(-0.410950\pi\)
0.276123 + 0.961122i \(0.410950\pi\)
\(84\) 0 0
\(85\) 3.26922 0.354597
\(86\) 0 0
\(87\) 1.14545 0.122805
\(88\) 0 0
\(89\) −0.630394 −0.0668216 −0.0334108 0.999442i \(-0.510637\pi\)
−0.0334108 + 0.999442i \(0.510637\pi\)
\(90\) 0 0
\(91\) 4.52209 0.474043
\(92\) 0 0
\(93\) 15.0440 1.55999
\(94\) 0 0
\(95\) 2.66872 0.273805
\(96\) 0 0
\(97\) 12.4738 1.26652 0.633259 0.773940i \(-0.281717\pi\)
0.633259 + 0.773940i \(0.281717\pi\)
\(98\) 0 0
\(99\) 17.4209 1.75087
\(100\) 0 0
\(101\) −6.53453 −0.650210 −0.325105 0.945678i \(-0.605400\pi\)
−0.325105 + 0.945678i \(0.605400\pi\)
\(102\) 0 0
\(103\) −6.01515 −0.592690 −0.296345 0.955081i \(-0.595768\pi\)
−0.296345 + 0.955081i \(0.595768\pi\)
\(104\) 0 0
\(105\) 22.1861 2.16514
\(106\) 0 0
\(107\) 7.70657 0.745022 0.372511 0.928028i \(-0.378497\pi\)
0.372511 + 0.928028i \(0.378497\pi\)
\(108\) 0 0
\(109\) 1.37323 0.131531 0.0657656 0.997835i \(-0.479051\pi\)
0.0657656 + 0.997835i \(0.479051\pi\)
\(110\) 0 0
\(111\) −21.5980 −2.04999
\(112\) 0 0
\(113\) −6.23707 −0.586734 −0.293367 0.956000i \(-0.594776\pi\)
−0.293367 + 0.956000i \(0.594776\pi\)
\(114\) 0 0
\(115\) 25.1421 2.34451
\(116\) 0 0
\(117\) 6.11381 0.565222
\(118\) 0 0
\(119\) −2.64610 −0.242568
\(120\) 0 0
\(121\) 12.7127 1.15570
\(122\) 0 0
\(123\) −4.95648 −0.446911
\(124\) 0 0
\(125\) −2.24855 −0.201116
\(126\) 0 0
\(127\) 2.55441 0.226667 0.113334 0.993557i \(-0.463847\pi\)
0.113334 + 0.993557i \(0.463847\pi\)
\(128\) 0 0
\(129\) 23.5463 2.07314
\(130\) 0 0
\(131\) 17.3680 1.51745 0.758723 0.651414i \(-0.225824\pi\)
0.758723 + 0.651414i \(0.225824\pi\)
\(132\) 0 0
\(133\) −2.16006 −0.187301
\(134\) 0 0
\(135\) 4.84205 0.416737
\(136\) 0 0
\(137\) −17.6166 −1.50509 −0.752543 0.658544i \(-0.771173\pi\)
−0.752543 + 0.658544i \(0.771173\pi\)
\(138\) 0 0
\(139\) −6.85306 −0.581269 −0.290634 0.956834i \(-0.593866\pi\)
−0.290634 + 0.956834i \(0.593866\pi\)
\(140\) 0 0
\(141\) −1.08987 −0.0917836
\(142\) 0 0
\(143\) 8.32191 0.695913
\(144\) 0 0
\(145\) 1.46012 0.121256
\(146\) 0 0
\(147\) −0.00476845 −0.000393295 0
\(148\) 0 0
\(149\) 17.6242 1.44383 0.721915 0.691982i \(-0.243262\pi\)
0.721915 + 0.691982i \(0.243262\pi\)
\(150\) 0 0
\(151\) 5.19339 0.422632 0.211316 0.977418i \(-0.432225\pi\)
0.211316 + 0.977418i \(0.432225\pi\)
\(152\) 0 0
\(153\) −3.57750 −0.289224
\(154\) 0 0
\(155\) 19.1769 1.54032
\(156\) 0 0
\(157\) −6.15908 −0.491548 −0.245774 0.969327i \(-0.579042\pi\)
−0.245774 + 0.969327i \(0.579042\pi\)
\(158\) 0 0
\(159\) −7.88534 −0.625348
\(160\) 0 0
\(161\) −20.3500 −1.60380
\(162\) 0 0
\(163\) −20.7190 −1.62284 −0.811420 0.584464i \(-0.801305\pi\)
−0.811420 + 0.584464i \(0.801305\pi\)
\(164\) 0 0
\(165\) 40.8287 3.17851
\(166\) 0 0
\(167\) 25.2322 1.95253 0.976263 0.216589i \(-0.0694931\pi\)
0.976263 + 0.216589i \(0.0694931\pi\)
\(168\) 0 0
\(169\) −10.0795 −0.775342
\(170\) 0 0
\(171\) −2.92038 −0.223327
\(172\) 0 0
\(173\) 15.9528 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(174\) 0 0
\(175\) 15.0505 1.13771
\(176\) 0 0
\(177\) −2.56466 −0.192772
\(178\) 0 0
\(179\) 5.47667 0.409346 0.204673 0.978830i \(-0.434387\pi\)
0.204673 + 0.978830i \(0.434387\pi\)
\(180\) 0 0
\(181\) −9.68134 −0.719608 −0.359804 0.933028i \(-0.617157\pi\)
−0.359804 + 0.933028i \(0.617157\pi\)
\(182\) 0 0
\(183\) 21.3273 1.57656
\(184\) 0 0
\(185\) −27.5314 −2.02415
\(186\) 0 0
\(187\) −4.86957 −0.356098
\(188\) 0 0
\(189\) −3.91915 −0.285076
\(190\) 0 0
\(191\) −12.1195 −0.876938 −0.438469 0.898746i \(-0.644479\pi\)
−0.438469 + 0.898746i \(0.644479\pi\)
\(192\) 0 0
\(193\) 5.25546 0.378296 0.189148 0.981949i \(-0.439427\pi\)
0.189148 + 0.981949i \(0.439427\pi\)
\(194\) 0 0
\(195\) 14.3287 1.02610
\(196\) 0 0
\(197\) 23.8164 1.69685 0.848423 0.529320i \(-0.177553\pi\)
0.848423 + 0.529320i \(0.177553\pi\)
\(198\) 0 0
\(199\) −5.01170 −0.355270 −0.177635 0.984096i \(-0.556845\pi\)
−0.177635 + 0.984096i \(0.556845\pi\)
\(200\) 0 0
\(201\) −13.9214 −0.981937
\(202\) 0 0
\(203\) −1.18182 −0.0829474
\(204\) 0 0
\(205\) −6.31810 −0.441276
\(206\) 0 0
\(207\) −27.5130 −1.91228
\(208\) 0 0
\(209\) −3.97511 −0.274964
\(210\) 0 0
\(211\) 4.88156 0.336060 0.168030 0.985782i \(-0.446259\pi\)
0.168030 + 0.985782i \(0.446259\pi\)
\(212\) 0 0
\(213\) −38.2302 −2.61949
\(214\) 0 0
\(215\) 30.0149 2.04700
\(216\) 0 0
\(217\) −15.5217 −1.05368
\(218\) 0 0
\(219\) 23.2025 1.56788
\(220\) 0 0
\(221\) −1.70896 −0.114957
\(222\) 0 0
\(223\) 10.4142 0.697388 0.348694 0.937237i \(-0.386625\pi\)
0.348694 + 0.937237i \(0.386625\pi\)
\(224\) 0 0
\(225\) 20.3481 1.35654
\(226\) 0 0
\(227\) 4.49672 0.298458 0.149229 0.988803i \(-0.452321\pi\)
0.149229 + 0.988803i \(0.452321\pi\)
\(228\) 0 0
\(229\) −19.2079 −1.26930 −0.634648 0.772801i \(-0.718855\pi\)
−0.634648 + 0.772801i \(0.718855\pi\)
\(230\) 0 0
\(231\) −33.0467 −2.17431
\(232\) 0 0
\(233\) 7.65638 0.501586 0.250793 0.968041i \(-0.419309\pi\)
0.250793 + 0.968041i \(0.419309\pi\)
\(234\) 0 0
\(235\) −1.38927 −0.0906263
\(236\) 0 0
\(237\) 21.8577 1.41981
\(238\) 0 0
\(239\) −3.18581 −0.206073 −0.103036 0.994678i \(-0.532856\pi\)
−0.103036 + 0.994678i \(0.532856\pi\)
\(240\) 0 0
\(241\) −24.4039 −1.57200 −0.785998 0.618229i \(-0.787850\pi\)
−0.785998 + 0.618229i \(0.787850\pi\)
\(242\) 0 0
\(243\) 22.2266 1.42584
\(244\) 0 0
\(245\) −0.00607842 −0.000388336 0
\(246\) 0 0
\(247\) −1.39505 −0.0887652
\(248\) 0 0
\(249\) −12.9034 −0.817717
\(250\) 0 0
\(251\) −6.03155 −0.380708 −0.190354 0.981715i \(-0.560964\pi\)
−0.190354 + 0.981715i \(0.560964\pi\)
\(252\) 0 0
\(253\) −37.4497 −2.35444
\(254\) 0 0
\(255\) −8.38445 −0.525055
\(256\) 0 0
\(257\) −27.6479 −1.72463 −0.862315 0.506373i \(-0.830986\pi\)
−0.862315 + 0.506373i \(0.830986\pi\)
\(258\) 0 0
\(259\) 22.2839 1.38465
\(260\) 0 0
\(261\) −1.59781 −0.0989017
\(262\) 0 0
\(263\) 10.6355 0.655810 0.327905 0.944711i \(-0.393657\pi\)
0.327905 + 0.944711i \(0.393657\pi\)
\(264\) 0 0
\(265\) −10.0516 −0.617463
\(266\) 0 0
\(267\) 1.61675 0.0989434
\(268\) 0 0
\(269\) 10.6905 0.651814 0.325907 0.945402i \(-0.394330\pi\)
0.325907 + 0.945402i \(0.394330\pi\)
\(270\) 0 0
\(271\) −7.14955 −0.434304 −0.217152 0.976138i \(-0.569677\pi\)
−0.217152 + 0.976138i \(0.569677\pi\)
\(272\) 0 0
\(273\) −11.5976 −0.701921
\(274\) 0 0
\(275\) 27.6971 1.67020
\(276\) 0 0
\(277\) 3.88059 0.233162 0.116581 0.993181i \(-0.462807\pi\)
0.116581 + 0.993181i \(0.462807\pi\)
\(278\) 0 0
\(279\) −20.9852 −1.25635
\(280\) 0 0
\(281\) −16.6086 −0.990787 −0.495393 0.868669i \(-0.664976\pi\)
−0.495393 + 0.868669i \(0.664976\pi\)
\(282\) 0 0
\(283\) −4.16530 −0.247601 −0.123801 0.992307i \(-0.539508\pi\)
−0.123801 + 0.992307i \(0.539508\pi\)
\(284\) 0 0
\(285\) −6.84437 −0.405426
\(286\) 0 0
\(287\) 5.11387 0.301862
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −31.9910 −1.87534
\(292\) 0 0
\(293\) −12.6479 −0.738900 −0.369450 0.929251i \(-0.620454\pi\)
−0.369450 + 0.929251i \(0.620454\pi\)
\(294\) 0 0
\(295\) −3.26922 −0.190341
\(296\) 0 0
\(297\) −7.21233 −0.418502
\(298\) 0 0
\(299\) −13.1429 −0.760071
\(300\) 0 0
\(301\) −24.2940 −1.40028
\(302\) 0 0
\(303\) 16.7589 0.962772
\(304\) 0 0
\(305\) 27.1862 1.55668
\(306\) 0 0
\(307\) 19.1944 1.09548 0.547740 0.836648i \(-0.315488\pi\)
0.547740 + 0.836648i \(0.315488\pi\)
\(308\) 0 0
\(309\) 15.4268 0.877602
\(310\) 0 0
\(311\) 14.7635 0.837159 0.418579 0.908180i \(-0.362528\pi\)
0.418579 + 0.908180i \(0.362528\pi\)
\(312\) 0 0
\(313\) −30.0953 −1.70108 −0.850542 0.525907i \(-0.823726\pi\)
−0.850542 + 0.525907i \(0.823726\pi\)
\(314\) 0 0
\(315\) −30.9479 −1.74371
\(316\) 0 0
\(317\) −8.74801 −0.491337 −0.245669 0.969354i \(-0.579008\pi\)
−0.245669 + 0.969354i \(0.579008\pi\)
\(318\) 0 0
\(319\) −2.17488 −0.121770
\(320\) 0 0
\(321\) −19.7648 −1.10316
\(322\) 0 0
\(323\) 0.816317 0.0454211
\(324\) 0 0
\(325\) 9.72022 0.539181
\(326\) 0 0
\(327\) −3.52187 −0.194760
\(328\) 0 0
\(329\) 1.12448 0.0619944
\(330\) 0 0
\(331\) 14.0942 0.774689 0.387345 0.921935i \(-0.373392\pi\)
0.387345 + 0.921935i \(0.373392\pi\)
\(332\) 0 0
\(333\) 30.1275 1.65098
\(334\) 0 0
\(335\) −17.7458 −0.969555
\(336\) 0 0
\(337\) −0.944659 −0.0514589 −0.0257294 0.999669i \(-0.508191\pi\)
−0.0257294 + 0.999669i \(0.508191\pi\)
\(338\) 0 0
\(339\) 15.9960 0.868783
\(340\) 0 0
\(341\) −28.5644 −1.54685
\(342\) 0 0
\(343\) −18.5178 −0.999867
\(344\) 0 0
\(345\) −64.4810 −3.47154
\(346\) 0 0
\(347\) −25.5103 −1.36947 −0.684733 0.728794i \(-0.740081\pi\)
−0.684733 + 0.728794i \(0.740081\pi\)
\(348\) 0 0
\(349\) −29.5143 −1.57986 −0.789932 0.613195i \(-0.789884\pi\)
−0.789932 + 0.613195i \(0.789884\pi\)
\(350\) 0 0
\(351\) −2.53115 −0.135103
\(352\) 0 0
\(353\) −20.4536 −1.08863 −0.544317 0.838880i \(-0.683211\pi\)
−0.544317 + 0.838880i \(0.683211\pi\)
\(354\) 0 0
\(355\) −48.7327 −2.58646
\(356\) 0 0
\(357\) 6.78637 0.359173
\(358\) 0 0
\(359\) 12.6994 0.670247 0.335124 0.942174i \(-0.391222\pi\)
0.335124 + 0.942174i \(0.391222\pi\)
\(360\) 0 0
\(361\) −18.3336 −0.964928
\(362\) 0 0
\(363\) −32.6038 −1.71126
\(364\) 0 0
\(365\) 29.5765 1.54811
\(366\) 0 0
\(367\) −1.72214 −0.0898952 −0.0449476 0.998989i \(-0.514312\pi\)
−0.0449476 + 0.998989i \(0.514312\pi\)
\(368\) 0 0
\(369\) 6.91390 0.359923
\(370\) 0 0
\(371\) 8.13573 0.422386
\(372\) 0 0
\(373\) 25.3648 1.31334 0.656670 0.754178i \(-0.271964\pi\)
0.656670 + 0.754178i \(0.271964\pi\)
\(374\) 0 0
\(375\) 5.76678 0.297795
\(376\) 0 0
\(377\) −0.763266 −0.0393102
\(378\) 0 0
\(379\) −11.8219 −0.607251 −0.303625 0.952791i \(-0.598197\pi\)
−0.303625 + 0.952791i \(0.598197\pi\)
\(380\) 0 0
\(381\) −6.55120 −0.335628
\(382\) 0 0
\(383\) −10.8150 −0.552623 −0.276311 0.961068i \(-0.589112\pi\)
−0.276311 + 0.961068i \(0.589112\pi\)
\(384\) 0 0
\(385\) −42.1251 −2.14690
\(386\) 0 0
\(387\) −32.8452 −1.66962
\(388\) 0 0
\(389\) −27.9524 −1.41724 −0.708621 0.705589i \(-0.750682\pi\)
−0.708621 + 0.705589i \(0.750682\pi\)
\(390\) 0 0
\(391\) 7.69055 0.388928
\(392\) 0 0
\(393\) −44.5430 −2.24690
\(394\) 0 0
\(395\) 27.8624 1.40191
\(396\) 0 0
\(397\) −16.4024 −0.823215 −0.411608 0.911361i \(-0.635033\pi\)
−0.411608 + 0.911361i \(0.635033\pi\)
\(398\) 0 0
\(399\) 5.53983 0.277338
\(400\) 0 0
\(401\) 1.29379 0.0646087 0.0323044 0.999478i \(-0.489715\pi\)
0.0323044 + 0.999478i \(0.489715\pi\)
\(402\) 0 0
\(403\) −10.0246 −0.499360
\(404\) 0 0
\(405\) 22.6687 1.12642
\(406\) 0 0
\(407\) 41.0085 2.03272
\(408\) 0 0
\(409\) −35.1342 −1.73727 −0.868637 0.495449i \(-0.835003\pi\)
−0.868637 + 0.495449i \(0.835003\pi\)
\(410\) 0 0
\(411\) 45.1806 2.22859
\(412\) 0 0
\(413\) 2.64610 0.130206
\(414\) 0 0
\(415\) −16.4481 −0.807407
\(416\) 0 0
\(417\) 17.5758 0.860690
\(418\) 0 0
\(419\) −12.6658 −0.618765 −0.309382 0.950938i \(-0.600122\pi\)
−0.309382 + 0.950938i \(0.600122\pi\)
\(420\) 0 0
\(421\) 9.87488 0.481272 0.240636 0.970615i \(-0.422644\pi\)
0.240636 + 0.970615i \(0.422644\pi\)
\(422\) 0 0
\(423\) 1.52028 0.0739186
\(424\) 0 0
\(425\) −5.68779 −0.275899
\(426\) 0 0
\(427\) −22.0045 −1.06487
\(428\) 0 0
\(429\) −21.3429 −1.03044
\(430\) 0 0
\(431\) −20.0785 −0.967149 −0.483575 0.875303i \(-0.660662\pi\)
−0.483575 + 0.875303i \(0.660662\pi\)
\(432\) 0 0
\(433\) 39.0793 1.87803 0.939015 0.343876i \(-0.111740\pi\)
0.939015 + 0.343876i \(0.111740\pi\)
\(434\) 0 0
\(435\) −3.74471 −0.179545
\(436\) 0 0
\(437\) 6.27793 0.300314
\(438\) 0 0
\(439\) −32.7106 −1.56119 −0.780595 0.625038i \(-0.785084\pi\)
−0.780595 + 0.625038i \(0.785084\pi\)
\(440\) 0 0
\(441\) 0.00665161 0.000316743 0
\(442\) 0 0
\(443\) −31.0887 −1.47707 −0.738534 0.674216i \(-0.764482\pi\)
−0.738534 + 0.674216i \(0.764482\pi\)
\(444\) 0 0
\(445\) 2.06089 0.0976958
\(446\) 0 0
\(447\) −45.2001 −2.13789
\(448\) 0 0
\(449\) 14.1453 0.667558 0.333779 0.942651i \(-0.391676\pi\)
0.333779 + 0.942651i \(0.391676\pi\)
\(450\) 0 0
\(451\) 9.41095 0.443144
\(452\) 0 0
\(453\) −13.3193 −0.625796
\(454\) 0 0
\(455\) −14.7837 −0.693070
\(456\) 0 0
\(457\) 7.09246 0.331771 0.165886 0.986145i \(-0.446952\pi\)
0.165886 + 0.986145i \(0.446952\pi\)
\(458\) 0 0
\(459\) 1.48110 0.0691319
\(460\) 0 0
\(461\) −30.5736 −1.42395 −0.711977 0.702203i \(-0.752200\pi\)
−0.711977 + 0.702203i \(0.752200\pi\)
\(462\) 0 0
\(463\) 11.3222 0.526189 0.263094 0.964770i \(-0.415257\pi\)
0.263094 + 0.964770i \(0.415257\pi\)
\(464\) 0 0
\(465\) −49.1823 −2.28077
\(466\) 0 0
\(467\) −1.87967 −0.0869809 −0.0434904 0.999054i \(-0.513848\pi\)
−0.0434904 + 0.999054i \(0.513848\pi\)
\(468\) 0 0
\(469\) 14.3634 0.663241
\(470\) 0 0
\(471\) 15.7960 0.727840
\(472\) 0 0
\(473\) −44.7078 −2.05567
\(474\) 0 0
\(475\) −4.64304 −0.213037
\(476\) 0 0
\(477\) 10.9994 0.503629
\(478\) 0 0
\(479\) 12.8189 0.585708 0.292854 0.956157i \(-0.405395\pi\)
0.292854 + 0.956157i \(0.405395\pi\)
\(480\) 0 0
\(481\) 14.3918 0.656211
\(482\) 0 0
\(483\) 52.1909 2.37477
\(484\) 0 0
\(485\) −40.7794 −1.85170
\(486\) 0 0
\(487\) 10.6486 0.482534 0.241267 0.970459i \(-0.422437\pi\)
0.241267 + 0.970459i \(0.422437\pi\)
\(488\) 0 0
\(489\) 53.1374 2.40295
\(490\) 0 0
\(491\) 42.4725 1.91676 0.958378 0.285502i \(-0.0921602\pi\)
0.958378 + 0.285502i \(0.0921602\pi\)
\(492\) 0 0
\(493\) 0.446626 0.0201150
\(494\) 0 0
\(495\) −56.9527 −2.55984
\(496\) 0 0
\(497\) 39.4442 1.76931
\(498\) 0 0
\(499\) 27.9722 1.25221 0.626103 0.779740i \(-0.284649\pi\)
0.626103 + 0.779740i \(0.284649\pi\)
\(500\) 0 0
\(501\) −64.7121 −2.89112
\(502\) 0 0
\(503\) −18.1497 −0.809255 −0.404628 0.914482i \(-0.632599\pi\)
−0.404628 + 0.914482i \(0.632599\pi\)
\(504\) 0 0
\(505\) 21.3628 0.950632
\(506\) 0 0
\(507\) 25.8504 1.14806
\(508\) 0 0
\(509\) −6.43748 −0.285336 −0.142668 0.989771i \(-0.545568\pi\)
−0.142668 + 0.989771i \(0.545568\pi\)
\(510\) 0 0
\(511\) −23.9392 −1.05901
\(512\) 0 0
\(513\) 1.20905 0.0533808
\(514\) 0 0
\(515\) 19.6648 0.866537
\(516\) 0 0
\(517\) 2.06935 0.0910101
\(518\) 0 0
\(519\) −40.9136 −1.79591
\(520\) 0 0
\(521\) −37.9810 −1.66398 −0.831989 0.554792i \(-0.812798\pi\)
−0.831989 + 0.554792i \(0.812798\pi\)
\(522\) 0 0
\(523\) 1.39808 0.0611339 0.0305669 0.999533i \(-0.490269\pi\)
0.0305669 + 0.999533i \(0.490269\pi\)
\(524\) 0 0
\(525\) −38.5994 −1.68462
\(526\) 0 0
\(527\) 5.86589 0.255522
\(528\) 0 0
\(529\) 36.1446 1.57150
\(530\) 0 0
\(531\) 3.57750 0.155250
\(532\) 0 0
\(533\) 3.30274 0.143058
\(534\) 0 0
\(535\) −25.1945 −1.08925
\(536\) 0 0
\(537\) −14.0458 −0.606122
\(538\) 0 0
\(539\) 0.00905393 0.000389980 0
\(540\) 0 0
\(541\) 30.3633 1.30542 0.652710 0.757608i \(-0.273632\pi\)
0.652710 + 0.757608i \(0.273632\pi\)
\(542\) 0 0
\(543\) 24.8294 1.06553
\(544\) 0 0
\(545\) −4.48938 −0.192304
\(546\) 0 0
\(547\) 4.23866 0.181232 0.0906159 0.995886i \(-0.471116\pi\)
0.0906159 + 0.995886i \(0.471116\pi\)
\(548\) 0 0
\(549\) −29.7498 −1.26969
\(550\) 0 0
\(551\) 0.364588 0.0155320
\(552\) 0 0
\(553\) −22.5518 −0.958999
\(554\) 0 0
\(555\) 70.6087 2.99717
\(556\) 0 0
\(557\) −14.1581 −0.599898 −0.299949 0.953955i \(-0.596970\pi\)
−0.299949 + 0.953955i \(0.596970\pi\)
\(558\) 0 0
\(559\) −15.6901 −0.663619
\(560\) 0 0
\(561\) 12.4888 0.527278
\(562\) 0 0
\(563\) −14.7101 −0.619955 −0.309978 0.950744i \(-0.600322\pi\)
−0.309978 + 0.950744i \(0.600322\pi\)
\(564\) 0 0
\(565\) 20.3904 0.857829
\(566\) 0 0
\(567\) −18.3480 −0.770545
\(568\) 0 0
\(569\) 18.6477 0.781751 0.390875 0.920444i \(-0.372172\pi\)
0.390875 + 0.920444i \(0.372172\pi\)
\(570\) 0 0
\(571\) −3.25266 −0.136120 −0.0680598 0.997681i \(-0.521681\pi\)
−0.0680598 + 0.997681i \(0.521681\pi\)
\(572\) 0 0
\(573\) 31.0825 1.29849
\(574\) 0 0
\(575\) −43.7423 −1.82418
\(576\) 0 0
\(577\) 10.9268 0.454889 0.227444 0.973791i \(-0.426963\pi\)
0.227444 + 0.973791i \(0.426963\pi\)
\(578\) 0 0
\(579\) −13.4785 −0.560147
\(580\) 0 0
\(581\) 13.3131 0.552320
\(582\) 0 0
\(583\) 14.9720 0.620078
\(584\) 0 0
\(585\) −19.9874 −0.826377
\(586\) 0 0
\(587\) −4.06489 −0.167776 −0.0838880 0.996475i \(-0.526734\pi\)
−0.0838880 + 0.996475i \(0.526734\pi\)
\(588\) 0 0
\(589\) 4.78843 0.197304
\(590\) 0 0
\(591\) −61.0810 −2.51253
\(592\) 0 0
\(593\) −11.5320 −0.473562 −0.236781 0.971563i \(-0.576092\pi\)
−0.236781 + 0.971563i \(0.576092\pi\)
\(594\) 0 0
\(595\) 8.65069 0.354644
\(596\) 0 0
\(597\) 12.8533 0.526052
\(598\) 0 0
\(599\) 2.70831 0.110658 0.0553292 0.998468i \(-0.482379\pi\)
0.0553292 + 0.998468i \(0.482379\pi\)
\(600\) 0 0
\(601\) −9.29209 −0.379032 −0.189516 0.981878i \(-0.560692\pi\)
−0.189516 + 0.981878i \(0.560692\pi\)
\(602\) 0 0
\(603\) 19.4192 0.790810
\(604\) 0 0
\(605\) −41.5606 −1.68968
\(606\) 0 0
\(607\) 19.4328 0.788754 0.394377 0.918949i \(-0.370960\pi\)
0.394377 + 0.918949i \(0.370960\pi\)
\(608\) 0 0
\(609\) 3.03097 0.122821
\(610\) 0 0
\(611\) 0.726233 0.0293803
\(612\) 0 0
\(613\) 11.6537 0.470687 0.235343 0.971912i \(-0.424379\pi\)
0.235343 + 0.971912i \(0.424379\pi\)
\(614\) 0 0
\(615\) 16.2038 0.653401
\(616\) 0 0
\(617\) −1.16618 −0.0469484 −0.0234742 0.999724i \(-0.507473\pi\)
−0.0234742 + 0.999724i \(0.507473\pi\)
\(618\) 0 0
\(619\) −24.7148 −0.993373 −0.496686 0.867930i \(-0.665450\pi\)
−0.496686 + 0.867930i \(0.665450\pi\)
\(620\) 0 0
\(621\) 11.3905 0.457085
\(622\) 0 0
\(623\) −1.66809 −0.0668305
\(624\) 0 0
\(625\) −21.0880 −0.843519
\(626\) 0 0
\(627\) 10.1948 0.407142
\(628\) 0 0
\(629\) −8.42139 −0.335783
\(630\) 0 0
\(631\) 43.3103 1.72416 0.862079 0.506774i \(-0.169162\pi\)
0.862079 + 0.506774i \(0.169162\pi\)
\(632\) 0 0
\(633\) −12.5196 −0.497608
\(634\) 0 0
\(635\) −8.35092 −0.331396
\(636\) 0 0
\(637\) 0.00317745 0.000125895 0
\(638\) 0 0
\(639\) 53.3281 2.10963
\(640\) 0 0
\(641\) −21.1745 −0.836341 −0.418170 0.908369i \(-0.637328\pi\)
−0.418170 + 0.908369i \(0.637328\pi\)
\(642\) 0 0
\(643\) −15.3045 −0.603549 −0.301774 0.953379i \(-0.597579\pi\)
−0.301774 + 0.953379i \(0.597579\pi\)
\(644\) 0 0
\(645\) −76.9781 −3.03101
\(646\) 0 0
\(647\) −21.2202 −0.834252 −0.417126 0.908849i \(-0.636963\pi\)
−0.417126 + 0.908849i \(0.636963\pi\)
\(648\) 0 0
\(649\) 4.86957 0.191147
\(650\) 0 0
\(651\) 39.8081 1.56020
\(652\) 0 0
\(653\) 22.8483 0.894124 0.447062 0.894503i \(-0.352470\pi\)
0.447062 + 0.894503i \(0.352470\pi\)
\(654\) 0 0
\(655\) −56.7797 −2.21856
\(656\) 0 0
\(657\) −32.3656 −1.26270
\(658\) 0 0
\(659\) 0.273160 0.0106408 0.00532041 0.999986i \(-0.498306\pi\)
0.00532041 + 0.999986i \(0.498306\pi\)
\(660\) 0 0
\(661\) −13.3951 −0.521011 −0.260505 0.965472i \(-0.583889\pi\)
−0.260505 + 0.965472i \(0.583889\pi\)
\(662\) 0 0
\(663\) 4.38291 0.170218
\(664\) 0 0
\(665\) 7.06171 0.273841
\(666\) 0 0
\(667\) 3.43480 0.132996
\(668\) 0 0
\(669\) −26.7090 −1.03263
\(670\) 0 0
\(671\) −40.4944 −1.56327
\(672\) 0 0
\(673\) 6.76182 0.260649 0.130324 0.991471i \(-0.458398\pi\)
0.130324 + 0.991471i \(0.458398\pi\)
\(674\) 0 0
\(675\) −8.42420 −0.324248
\(676\) 0 0
\(677\) 41.4982 1.59490 0.797452 0.603382i \(-0.206181\pi\)
0.797452 + 0.603382i \(0.206181\pi\)
\(678\) 0 0
\(679\) 33.0068 1.26669
\(680\) 0 0
\(681\) −11.5326 −0.441929
\(682\) 0 0
\(683\) −3.36872 −0.128901 −0.0644503 0.997921i \(-0.520529\pi\)
−0.0644503 + 0.997921i \(0.520529\pi\)
\(684\) 0 0
\(685\) 57.5924 2.20049
\(686\) 0 0
\(687\) 49.2619 1.87946
\(688\) 0 0
\(689\) 5.25439 0.200176
\(690\) 0 0
\(691\) 16.4479 0.625708 0.312854 0.949801i \(-0.398715\pi\)
0.312854 + 0.949801i \(0.398715\pi\)
\(692\) 0 0
\(693\) 46.0975 1.75110
\(694\) 0 0
\(695\) 22.4041 0.849838
\(696\) 0 0
\(697\) −1.93260 −0.0732026
\(698\) 0 0
\(699\) −19.6360 −0.742703
\(700\) 0 0
\(701\) 25.9004 0.978244 0.489122 0.872215i \(-0.337317\pi\)
0.489122 + 0.872215i \(0.337317\pi\)
\(702\) 0 0
\(703\) −6.87452 −0.259278
\(704\) 0 0
\(705\) 3.56302 0.134191
\(706\) 0 0
\(707\) −17.2910 −0.650296
\(708\) 0 0
\(709\) −10.5244 −0.395253 −0.197627 0.980277i \(-0.563323\pi\)
−0.197627 + 0.980277i \(0.563323\pi\)
\(710\) 0 0
\(711\) −30.4898 −1.14345
\(712\) 0 0
\(713\) 45.1119 1.68945
\(714\) 0 0
\(715\) −27.2061 −1.01745
\(716\) 0 0
\(717\) 8.17052 0.305134
\(718\) 0 0
\(719\) −3.54966 −0.132380 −0.0661899 0.997807i \(-0.521084\pi\)
−0.0661899 + 0.997807i \(0.521084\pi\)
\(720\) 0 0
\(721\) −15.9167 −0.592769
\(722\) 0 0
\(723\) 62.5879 2.32767
\(724\) 0 0
\(725\) −2.54032 −0.0943450
\(726\) 0 0
\(727\) 16.0818 0.596441 0.298221 0.954497i \(-0.403607\pi\)
0.298221 + 0.954497i \(0.403607\pi\)
\(728\) 0 0
\(729\) −36.2019 −1.34081
\(730\) 0 0
\(731\) 9.18105 0.339574
\(732\) 0 0
\(733\) 30.5819 1.12957 0.564785 0.825238i \(-0.308959\pi\)
0.564785 + 0.825238i \(0.308959\pi\)
\(734\) 0 0
\(735\) 0.0155891 0.000575013 0
\(736\) 0 0
\(737\) 26.4327 0.973661
\(738\) 0 0
\(739\) 15.0584 0.553934 0.276967 0.960879i \(-0.410671\pi\)
0.276967 + 0.960879i \(0.410671\pi\)
\(740\) 0 0
\(741\) 3.57785 0.131435
\(742\) 0 0
\(743\) 28.8727 1.05924 0.529618 0.848236i \(-0.322335\pi\)
0.529618 + 0.848236i \(0.322335\pi\)
\(744\) 0 0
\(745\) −57.6173 −2.11094
\(746\) 0 0
\(747\) 17.9992 0.658555
\(748\) 0 0
\(749\) 20.3924 0.745121
\(750\) 0 0
\(751\) −34.3230 −1.25247 −0.626233 0.779636i \(-0.715404\pi\)
−0.626233 + 0.779636i \(0.715404\pi\)
\(752\) 0 0
\(753\) 15.4689 0.563719
\(754\) 0 0
\(755\) −16.9783 −0.617905
\(756\) 0 0
\(757\) −0.975245 −0.0354459 −0.0177229 0.999843i \(-0.505642\pi\)
−0.0177229 + 0.999843i \(0.505642\pi\)
\(758\) 0 0
\(759\) 96.0458 3.48624
\(760\) 0 0
\(761\) −24.0776 −0.872811 −0.436406 0.899750i \(-0.643749\pi\)
−0.436406 + 0.899750i \(0.643749\pi\)
\(762\) 0 0
\(763\) 3.63370 0.131549
\(764\) 0 0
\(765\) 11.6956 0.422857
\(766\) 0 0
\(767\) 1.70896 0.0617070
\(768\) 0 0
\(769\) −20.7698 −0.748977 −0.374488 0.927232i \(-0.622182\pi\)
−0.374488 + 0.927232i \(0.622182\pi\)
\(770\) 0 0
\(771\) 70.9076 2.55368
\(772\) 0 0
\(773\) −39.4978 −1.42064 −0.710319 0.703880i \(-0.751449\pi\)
−0.710319 + 0.703880i \(0.751449\pi\)
\(774\) 0 0
\(775\) −33.3640 −1.19847
\(776\) 0 0
\(777\) −57.1506 −2.05027
\(778\) 0 0
\(779\) −1.57762 −0.0565240
\(780\) 0 0
\(781\) 72.5883 2.59741
\(782\) 0 0
\(783\) 0.661499 0.0236400
\(784\) 0 0
\(785\) 20.1354 0.718662
\(786\) 0 0
\(787\) −14.4965 −0.516744 −0.258372 0.966046i \(-0.583186\pi\)
−0.258372 + 0.966046i \(0.583186\pi\)
\(788\) 0 0
\(789\) −27.2764 −0.971065
\(790\) 0 0
\(791\) −16.5039 −0.586812
\(792\) 0 0
\(793\) −14.2114 −0.504661
\(794\) 0 0
\(795\) 25.7789 0.914284
\(796\) 0 0
\(797\) −10.6341 −0.376679 −0.188340 0.982104i \(-0.560311\pi\)
−0.188340 + 0.982104i \(0.560311\pi\)
\(798\) 0 0
\(799\) −0.424956 −0.0150339
\(800\) 0 0
\(801\) −2.25523 −0.0796848
\(802\) 0 0
\(803\) −44.0549 −1.55466
\(804\) 0 0
\(805\) 66.5286 2.34482
\(806\) 0 0
\(807\) −27.4177 −0.965147
\(808\) 0 0
\(809\) −10.4744 −0.368259 −0.184130 0.982902i \(-0.558947\pi\)
−0.184130 + 0.982902i \(0.558947\pi\)
\(810\) 0 0
\(811\) −48.8786 −1.71636 −0.858179 0.513350i \(-0.828404\pi\)
−0.858179 + 0.513350i \(0.828404\pi\)
\(812\) 0 0
\(813\) 18.3362 0.643078
\(814\) 0 0
\(815\) 67.7351 2.37266
\(816\) 0 0
\(817\) 7.49465 0.262205
\(818\) 0 0
\(819\) 16.1778 0.565297
\(820\) 0 0
\(821\) −19.6347 −0.685256 −0.342628 0.939471i \(-0.611317\pi\)
−0.342628 + 0.939471i \(0.611317\pi\)
\(822\) 0 0
\(823\) −15.6968 −0.547157 −0.273579 0.961850i \(-0.588207\pi\)
−0.273579 + 0.961850i \(0.588207\pi\)
\(824\) 0 0
\(825\) −71.0338 −2.47308
\(826\) 0 0
\(827\) 34.5659 1.20197 0.600987 0.799259i \(-0.294774\pi\)
0.600987 + 0.799259i \(0.294774\pi\)
\(828\) 0 0
\(829\) 16.7573 0.582005 0.291002 0.956722i \(-0.406011\pi\)
0.291002 + 0.956722i \(0.406011\pi\)
\(830\) 0 0
\(831\) −9.95241 −0.345245
\(832\) 0 0
\(833\) −0.00185929 −6.44205e−5 0
\(834\) 0 0
\(835\) −82.4896 −2.85467
\(836\) 0 0
\(837\) 8.68798 0.300300
\(838\) 0 0
\(839\) −6.58638 −0.227387 −0.113694 0.993516i \(-0.536268\pi\)
−0.113694 + 0.993516i \(0.536268\pi\)
\(840\) 0 0
\(841\) −28.8005 −0.993122
\(842\) 0 0
\(843\) 42.5955 1.46707
\(844\) 0 0
\(845\) 32.9519 1.13358
\(846\) 0 0
\(847\) 33.6391 1.15585
\(848\) 0 0
\(849\) 10.6826 0.366625
\(850\) 0 0
\(851\) −64.7651 −2.22012
\(852\) 0 0
\(853\) −44.5678 −1.52597 −0.762987 0.646414i \(-0.776268\pi\)
−0.762987 + 0.646414i \(0.776268\pi\)
\(854\) 0 0
\(855\) 9.54735 0.326513
\(856\) 0 0
\(857\) 16.7487 0.572123 0.286062 0.958211i \(-0.407654\pi\)
0.286062 + 0.958211i \(0.407654\pi\)
\(858\) 0 0
\(859\) −37.9026 −1.29322 −0.646610 0.762820i \(-0.723814\pi\)
−0.646610 + 0.762820i \(0.723814\pi\)
\(860\) 0 0
\(861\) −13.1154 −0.446970
\(862\) 0 0
\(863\) −16.7099 −0.568811 −0.284405 0.958704i \(-0.591796\pi\)
−0.284405 + 0.958704i \(0.591796\pi\)
\(864\) 0 0
\(865\) −52.1532 −1.77326
\(866\) 0 0
\(867\) −2.56466 −0.0871006
\(868\) 0 0
\(869\) −41.5016 −1.40784
\(870\) 0 0
\(871\) 9.27648 0.314322
\(872\) 0 0
\(873\) 44.6249 1.51032
\(874\) 0 0
\(875\) −5.94989 −0.201143
\(876\) 0 0
\(877\) −47.9147 −1.61796 −0.808981 0.587834i \(-0.799981\pi\)
−0.808981 + 0.587834i \(0.799981\pi\)
\(878\) 0 0
\(879\) 32.4377 1.09410
\(880\) 0 0
\(881\) −1.34644 −0.0453626 −0.0226813 0.999743i \(-0.507220\pi\)
−0.0226813 + 0.999743i \(0.507220\pi\)
\(882\) 0 0
\(883\) −31.5391 −1.06137 −0.530687 0.847568i \(-0.678066\pi\)
−0.530687 + 0.847568i \(0.678066\pi\)
\(884\) 0 0
\(885\) 8.38445 0.281840
\(886\) 0 0
\(887\) −6.95728 −0.233603 −0.116801 0.993155i \(-0.537264\pi\)
−0.116801 + 0.993155i \(0.537264\pi\)
\(888\) 0 0
\(889\) 6.75923 0.226697
\(890\) 0 0
\(891\) −33.7655 −1.13119
\(892\) 0 0
\(893\) −0.346899 −0.0116085
\(894\) 0 0
\(895\) −17.9044 −0.598480
\(896\) 0 0
\(897\) 33.7070 1.12544
\(898\) 0 0
\(899\) 2.61986 0.0873772
\(900\) 0 0
\(901\) −3.07461 −0.102430
\(902\) 0 0
\(903\) 62.3060 2.07341
\(904\) 0 0
\(905\) 31.6504 1.05210
\(906\) 0 0
\(907\) 58.9293 1.95672 0.978358 0.206922i \(-0.0663445\pi\)
0.978358 + 0.206922i \(0.0663445\pi\)
\(908\) 0 0
\(909\) −23.3773 −0.775376
\(910\) 0 0
\(911\) 54.9891 1.82187 0.910935 0.412549i \(-0.135361\pi\)
0.910935 + 0.412549i \(0.135361\pi\)
\(912\) 0 0
\(913\) 24.4998 0.810826
\(914\) 0 0
\(915\) −69.7235 −2.30499
\(916\) 0 0
\(917\) 45.9574 1.51765
\(918\) 0 0
\(919\) −42.0470 −1.38700 −0.693502 0.720455i \(-0.743933\pi\)
−0.693502 + 0.720455i \(0.743933\pi\)
\(920\) 0 0
\(921\) −49.2271 −1.62209
\(922\) 0 0
\(923\) 25.4747 0.838509
\(924\) 0 0
\(925\) 47.8991 1.57491
\(926\) 0 0
\(927\) −21.5192 −0.706784
\(928\) 0 0
\(929\) −37.0139 −1.21439 −0.607193 0.794554i \(-0.707705\pi\)
−0.607193 + 0.794554i \(0.707705\pi\)
\(930\) 0 0
\(931\) −0.00151777 −4.97428e−5 0
\(932\) 0 0
\(933\) −37.8633 −1.23959
\(934\) 0 0
\(935\) 15.9197 0.520630
\(936\) 0 0
\(937\) −1.31474 −0.0429508 −0.0214754 0.999769i \(-0.506836\pi\)
−0.0214754 + 0.999769i \(0.506836\pi\)
\(938\) 0 0
\(939\) 77.1842 2.51881
\(940\) 0 0
\(941\) 57.3093 1.86823 0.934116 0.356970i \(-0.116190\pi\)
0.934116 + 0.356970i \(0.116190\pi\)
\(942\) 0 0
\(943\) −14.8628 −0.483999
\(944\) 0 0
\(945\) 12.8126 0.416792
\(946\) 0 0
\(947\) 10.6406 0.345772 0.172886 0.984942i \(-0.444691\pi\)
0.172886 + 0.984942i \(0.444691\pi\)
\(948\) 0 0
\(949\) −15.4609 −0.501883
\(950\) 0 0
\(951\) 22.4357 0.727528
\(952\) 0 0
\(953\) −37.9657 −1.22983 −0.614915 0.788593i \(-0.710810\pi\)
−0.614915 + 0.788593i \(0.710810\pi\)
\(954\) 0 0
\(955\) 39.6214 1.28212
\(956\) 0 0
\(957\) 5.57783 0.180306
\(958\) 0 0
\(959\) −46.6153 −1.50528
\(960\) 0 0
\(961\) 3.40865 0.109956
\(962\) 0 0
\(963\) 27.5703 0.888439
\(964\) 0 0
\(965\) −17.1813 −0.553084
\(966\) 0 0
\(967\) −23.6933 −0.761924 −0.380962 0.924591i \(-0.624407\pi\)
−0.380962 + 0.924591i \(0.624407\pi\)
\(968\) 0 0
\(969\) −2.09358 −0.0672555
\(970\) 0 0
\(971\) −50.9751 −1.63587 −0.817934 0.575312i \(-0.804881\pi\)
−0.817934 + 0.575312i \(0.804881\pi\)
\(972\) 0 0
\(973\) −18.1339 −0.581346
\(974\) 0 0
\(975\) −24.9291 −0.798370
\(976\) 0 0
\(977\) −13.2571 −0.424132 −0.212066 0.977255i \(-0.568019\pi\)
−0.212066 + 0.977255i \(0.568019\pi\)
\(978\) 0 0
\(979\) −3.06975 −0.0981095
\(980\) 0 0
\(981\) 4.91272 0.156851
\(982\) 0 0
\(983\) 43.2246 1.37865 0.689325 0.724452i \(-0.257907\pi\)
0.689325 + 0.724452i \(0.257907\pi\)
\(984\) 0 0
\(985\) −77.8609 −2.48085
\(986\) 0 0
\(987\) −2.88391 −0.0917958
\(988\) 0 0
\(989\) 70.6074 2.24518
\(990\) 0 0
\(991\) 51.6680 1.64129 0.820644 0.571440i \(-0.193615\pi\)
0.820644 + 0.571440i \(0.193615\pi\)
\(992\) 0 0
\(993\) −36.1470 −1.14709
\(994\) 0 0
\(995\) 16.3844 0.519419
\(996\) 0 0
\(997\) 12.8260 0.406203 0.203102 0.979158i \(-0.434898\pi\)
0.203102 + 0.979158i \(0.434898\pi\)
\(998\) 0 0
\(999\) −12.4729 −0.394626
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 8024.2.a.x.1.2 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8024.2.a.x.1.2 22 1.1 even 1 trivial