Properties

Label 2760.3.g.a.2161.16
Level $2760$
Weight $3$
Character 2760.2161
Analytic conductor $75.205$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,3,Mod(2161,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2760.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(75.2045529634\)
Analytic rank: \(0\)
Dimension: \(96\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.16
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.33

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +4.35205i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +4.35205i q^{7} +3.00000 q^{9} +20.3567i q^{11} -4.08582 q^{13} +3.87298i q^{15} +20.5409i q^{17} -18.5285i q^{19} -7.53798i q^{21} +(-20.6057 - 10.2178i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -52.0085 q^{29} -16.7033 q^{31} -35.2589i q^{33} +9.73148 q^{35} +63.5835i q^{37} +7.07685 q^{39} -34.2493 q^{41} -79.2781i q^{43} -6.70820i q^{45} +36.2240 q^{47} +30.0596 q^{49} -35.5778i q^{51} +56.2658i q^{53} +45.5190 q^{55} +32.0922i q^{57} +41.7313 q^{59} -38.2679i q^{61} +13.0562i q^{63} +9.13618i q^{65} +41.9915i q^{67} +(35.6902 + 17.6977i) q^{69} +39.6662 q^{71} +39.7633 q^{73} +8.66025 q^{75} -88.5935 q^{77} +58.3897i q^{79} +9.00000 q^{81} -162.289i q^{83} +45.9308 q^{85} +90.0813 q^{87} -5.48194i q^{89} -17.7817i q^{91} +28.9310 q^{93} -41.4309 q^{95} -72.2362i q^{97} +61.0701i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 96 q + 288 q^{9} - 16 q^{23} - 480 q^{25} - 80 q^{31} + 80 q^{35} + 48 q^{39} + 112 q^{41} + 32 q^{47} - 688 q^{49} - 80 q^{55} - 496 q^{59} - 96 q^{69} - 416 q^{71} - 320 q^{73} + 864 q^{81} + 192 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 4.35205i 0.621722i 0.950455 + 0.310861i \(0.100617\pi\)
−0.950455 + 0.310861i \(0.899383\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 20.3567i 1.85061i 0.379223 + 0.925305i \(0.376191\pi\)
−0.379223 + 0.925305i \(0.623809\pi\)
\(12\) 0 0
\(13\) −4.08582 −0.314294 −0.157147 0.987575i \(-0.550230\pi\)
−0.157147 + 0.987575i \(0.550230\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 20.5409i 1.20829i 0.796876 + 0.604143i \(0.206484\pi\)
−0.796876 + 0.604143i \(0.793516\pi\)
\(18\) 0 0
\(19\) 18.5285i 0.975182i −0.873072 0.487591i \(-0.837876\pi\)
0.873072 0.487591i \(-0.162124\pi\)
\(20\) 0 0
\(21\) 7.53798i 0.358951i
\(22\) 0 0
\(23\) −20.6057 10.2178i −0.895902 0.444252i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −52.0085 −1.79340 −0.896698 0.442644i \(-0.854041\pi\)
−0.896698 + 0.442644i \(0.854041\pi\)
\(30\) 0 0
\(31\) −16.7033 −0.538818 −0.269409 0.963026i \(-0.586828\pi\)
−0.269409 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) 35.2589i 1.06845i
\(34\) 0 0
\(35\) 9.73148 0.278042
\(36\) 0 0
\(37\) 63.5835i 1.71847i 0.511579 + 0.859236i \(0.329061\pi\)
−0.511579 + 0.859236i \(0.670939\pi\)
\(38\) 0 0
\(39\) 7.07685 0.181458
\(40\) 0 0
\(41\) −34.2493 −0.835349 −0.417674 0.908597i \(-0.637155\pi\)
−0.417674 + 0.908597i \(0.637155\pi\)
\(42\) 0 0
\(43\) 79.2781i 1.84368i −0.387574 0.921839i \(-0.626687\pi\)
0.387574 0.921839i \(-0.373313\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) 36.2240 0.770724 0.385362 0.922765i \(-0.374077\pi\)
0.385362 + 0.922765i \(0.374077\pi\)
\(48\) 0 0
\(49\) 30.0596 0.613462
\(50\) 0 0
\(51\) 35.5778i 0.697604i
\(52\) 0 0
\(53\) 56.2658i 1.06162i 0.847491 + 0.530810i \(0.178112\pi\)
−0.847491 + 0.530810i \(0.821888\pi\)
\(54\) 0 0
\(55\) 45.5190 0.827618
\(56\) 0 0
\(57\) 32.0922i 0.563022i
\(58\) 0 0
\(59\) 41.7313 0.707311 0.353655 0.935376i \(-0.384939\pi\)
0.353655 + 0.935376i \(0.384939\pi\)
\(60\) 0 0
\(61\) 38.2679i 0.627343i −0.949532 0.313671i \(-0.898441\pi\)
0.949532 0.313671i \(-0.101559\pi\)
\(62\) 0 0
\(63\) 13.0562i 0.207241i
\(64\) 0 0
\(65\) 9.13618i 0.140557i
\(66\) 0 0
\(67\) 41.9915i 0.626738i 0.949631 + 0.313369i \(0.101458\pi\)
−0.949631 + 0.313369i \(0.898542\pi\)
\(68\) 0 0
\(69\) 35.6902 + 17.6977i 0.517249 + 0.256489i
\(70\) 0 0
\(71\) 39.6662 0.558678 0.279339 0.960192i \(-0.409885\pi\)
0.279339 + 0.960192i \(0.409885\pi\)
\(72\) 0 0
\(73\) 39.7633 0.544703 0.272352 0.962198i \(-0.412199\pi\)
0.272352 + 0.962198i \(0.412199\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) −88.5935 −1.15056
\(78\) 0 0
\(79\) 58.3897i 0.739110i 0.929209 + 0.369555i \(0.120490\pi\)
−0.929209 + 0.369555i \(0.879510\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 162.289i 1.95529i −0.210261 0.977645i \(-0.567431\pi\)
0.210261 0.977645i \(-0.432569\pi\)
\(84\) 0 0
\(85\) 45.9308 0.540362
\(86\) 0 0
\(87\) 90.0813 1.03542
\(88\) 0 0
\(89\) 5.48194i 0.0615949i −0.999526 0.0307974i \(-0.990195\pi\)
0.999526 0.0307974i \(-0.00980468\pi\)
\(90\) 0 0
\(91\) 17.7817i 0.195403i
\(92\) 0 0
\(93\) 28.9310 0.311087
\(94\) 0 0
\(95\) −41.4309 −0.436115
\(96\) 0 0
\(97\) 72.2362i 0.744703i −0.928092 0.372352i \(-0.878552\pi\)
0.928092 0.372352i \(-0.121448\pi\)
\(98\) 0 0
\(99\) 61.0701i 0.616870i
\(100\) 0 0
\(101\) 130.537 1.29244 0.646221 0.763150i \(-0.276348\pi\)
0.646221 + 0.763150i \(0.276348\pi\)
\(102\) 0 0
\(103\) 136.286i 1.32316i −0.749874 0.661581i \(-0.769886\pi\)
0.749874 0.661581i \(-0.230114\pi\)
\(104\) 0 0
\(105\) −16.8554 −0.160528
\(106\) 0 0
\(107\) 60.3219i 0.563756i −0.959450 0.281878i \(-0.909043\pi\)
0.959450 0.281878i \(-0.0909574\pi\)
\(108\) 0 0
\(109\) 38.9480i 0.357321i −0.983911 0.178661i \(-0.942824\pi\)
0.983911 0.178661i \(-0.0571764\pi\)
\(110\) 0 0
\(111\) 110.130i 0.992160i
\(112\) 0 0
\(113\) 41.2146i 0.364731i 0.983231 + 0.182366i \(0.0583754\pi\)
−0.983231 + 0.182366i \(0.941625\pi\)
\(114\) 0 0
\(115\) −22.8477 + 46.0758i −0.198675 + 0.400660i
\(116\) 0 0
\(117\) −12.2575 −0.104765
\(118\) 0 0
\(119\) −89.3949 −0.751218
\(120\) 0 0
\(121\) −293.396 −2.42476
\(122\) 0 0
\(123\) 59.3215 0.482289
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 196.756 1.54926 0.774629 0.632416i \(-0.217937\pi\)
0.774629 + 0.632416i \(0.217937\pi\)
\(128\) 0 0
\(129\) 137.314i 1.06445i
\(130\) 0 0
\(131\) −128.977 −0.984555 −0.492277 0.870438i \(-0.663835\pi\)
−0.492277 + 0.870438i \(0.663835\pi\)
\(132\) 0 0
\(133\) 80.6368 0.606292
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 83.4965i 0.609464i −0.952438 0.304732i \(-0.901433\pi\)
0.952438 0.304732i \(-0.0985669\pi\)
\(138\) 0 0
\(139\) −129.260 −0.929925 −0.464963 0.885330i \(-0.653932\pi\)
−0.464963 + 0.885330i \(0.653932\pi\)
\(140\) 0 0
\(141\) −62.7419 −0.444978
\(142\) 0 0
\(143\) 83.1739i 0.581636i
\(144\) 0 0
\(145\) 116.294i 0.802031i
\(146\) 0 0
\(147\) −52.0648 −0.354183
\(148\) 0 0
\(149\) 14.8648i 0.0997639i 0.998755 + 0.0498820i \(0.0158845\pi\)
−0.998755 + 0.0498820i \(0.984115\pi\)
\(150\) 0 0
\(151\) −152.751 −1.01159 −0.505797 0.862653i \(-0.668801\pi\)
−0.505797 + 0.862653i \(0.668801\pi\)
\(152\) 0 0
\(153\) 61.6226i 0.402762i
\(154\) 0 0
\(155\) 37.3498i 0.240967i
\(156\) 0 0
\(157\) 46.7368i 0.297686i −0.988861 0.148843i \(-0.952445\pi\)
0.988861 0.148843i \(-0.0475550\pi\)
\(158\) 0 0
\(159\) 97.4553i 0.612926i
\(160\) 0 0
\(161\) 44.4684 89.6773i 0.276201 0.557002i
\(162\) 0 0
\(163\) −290.611 −1.78289 −0.891446 0.453127i \(-0.850308\pi\)
−0.891446 + 0.453127i \(0.850308\pi\)
\(164\) 0 0
\(165\) −78.8412 −0.477826
\(166\) 0 0
\(167\) 28.1745 0.168709 0.0843547 0.996436i \(-0.473117\pi\)
0.0843547 + 0.996436i \(0.473117\pi\)
\(168\) 0 0
\(169\) −152.306 −0.901219
\(170\) 0 0
\(171\) 55.5854i 0.325061i
\(172\) 0 0
\(173\) 76.4129 0.441693 0.220846 0.975309i \(-0.429118\pi\)
0.220846 + 0.975309i \(0.429118\pi\)
\(174\) 0 0
\(175\) 21.7603i 0.124344i
\(176\) 0 0
\(177\) −72.2808 −0.408366
\(178\) 0 0
\(179\) 162.475 0.907679 0.453840 0.891083i \(-0.350054\pi\)
0.453840 + 0.891083i \(0.350054\pi\)
\(180\) 0 0
\(181\) 68.9428i 0.380899i 0.981697 + 0.190450i \(0.0609946\pi\)
−0.981697 + 0.190450i \(0.939005\pi\)
\(182\) 0 0
\(183\) 66.2820i 0.362197i
\(184\) 0 0
\(185\) 142.177 0.768524
\(186\) 0 0
\(187\) −418.144 −2.23607
\(188\) 0 0
\(189\) 22.6139i 0.119650i
\(190\) 0 0
\(191\) 155.603i 0.814678i 0.913277 + 0.407339i \(0.133543\pi\)
−0.913277 + 0.407339i \(0.866457\pi\)
\(192\) 0 0
\(193\) 144.410 0.748241 0.374120 0.927380i \(-0.377945\pi\)
0.374120 + 0.927380i \(0.377945\pi\)
\(194\) 0 0
\(195\) 15.8243i 0.0811504i
\(196\) 0 0
\(197\) 267.825 1.35952 0.679759 0.733436i \(-0.262084\pi\)
0.679759 + 0.733436i \(0.262084\pi\)
\(198\) 0 0
\(199\) 294.477i 1.47978i −0.672726 0.739891i \(-0.734877\pi\)
0.672726 0.739891i \(-0.265123\pi\)
\(200\) 0 0
\(201\) 72.7314i 0.361848i
\(202\) 0 0
\(203\) 226.344i 1.11499i
\(204\) 0 0
\(205\) 76.5838i 0.373579i
\(206\) 0 0
\(207\) −61.8172 30.6534i −0.298634 0.148084i
\(208\) 0 0
\(209\) 377.179 1.80468
\(210\) 0 0
\(211\) −278.029 −1.31767 −0.658837 0.752286i \(-0.728951\pi\)
−0.658837 + 0.752286i \(0.728951\pi\)
\(212\) 0 0
\(213\) −68.7038 −0.322553
\(214\) 0 0
\(215\) −177.271 −0.824518
\(216\) 0 0
\(217\) 72.6938i 0.334995i
\(218\) 0 0
\(219\) −68.8721 −0.314485
\(220\) 0 0
\(221\) 83.9263i 0.379757i
\(222\) 0 0
\(223\) −136.509 −0.612146 −0.306073 0.952008i \(-0.599015\pi\)
−0.306073 + 0.952008i \(0.599015\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 132.271i 0.582694i −0.956618 0.291347i \(-0.905897\pi\)
0.956618 0.291347i \(-0.0941033\pi\)
\(228\) 0 0
\(229\) 290.872i 1.27018i −0.772437 0.635091i \(-0.780963\pi\)
0.772437 0.635091i \(-0.219037\pi\)
\(230\) 0 0
\(231\) 153.448 0.664279
\(232\) 0 0
\(233\) −146.851 −0.630260 −0.315130 0.949049i \(-0.602048\pi\)
−0.315130 + 0.949049i \(0.602048\pi\)
\(234\) 0 0
\(235\) 80.9994i 0.344678i
\(236\) 0 0
\(237\) 101.134i 0.426725i
\(238\) 0 0
\(239\) −106.834 −0.447003 −0.223501 0.974704i \(-0.571749\pi\)
−0.223501 + 0.974704i \(0.571749\pi\)
\(240\) 0 0
\(241\) 138.206i 0.573470i −0.958010 0.286735i \(-0.907430\pi\)
0.958010 0.286735i \(-0.0925699\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 67.2154i 0.274349i
\(246\) 0 0
\(247\) 75.7040i 0.306494i
\(248\) 0 0
\(249\) 281.093i 1.12889i
\(250\) 0 0
\(251\) 27.3495i 0.108962i 0.998515 + 0.0544811i \(0.0173505\pi\)
−0.998515 + 0.0544811i \(0.982650\pi\)
\(252\) 0 0
\(253\) 208.001 419.465i 0.822137 1.65797i
\(254\) 0 0
\(255\) −79.5544 −0.311978
\(256\) 0 0
\(257\) −77.1480 −0.300187 −0.150093 0.988672i \(-0.547957\pi\)
−0.150093 + 0.988672i \(0.547957\pi\)
\(258\) 0 0
\(259\) −276.719 −1.06841
\(260\) 0 0
\(261\) −156.025 −0.597798
\(262\) 0 0
\(263\) 310.689i 1.18133i −0.806918 0.590663i \(-0.798866\pi\)
0.806918 0.590663i \(-0.201134\pi\)
\(264\) 0 0
\(265\) 125.814 0.474771
\(266\) 0 0
\(267\) 9.49500i 0.0355618i
\(268\) 0 0
\(269\) −469.313 −1.74466 −0.872329 0.488920i \(-0.837391\pi\)
−0.872329 + 0.488920i \(0.837391\pi\)
\(270\) 0 0
\(271\) 412.914 1.52367 0.761833 0.647773i \(-0.224300\pi\)
0.761833 + 0.647773i \(0.224300\pi\)
\(272\) 0 0
\(273\) 30.7988i 0.112816i
\(274\) 0 0
\(275\) 101.784i 0.370122i
\(276\) 0 0
\(277\) −54.6176 −0.197175 −0.0985877 0.995128i \(-0.531432\pi\)
−0.0985877 + 0.995128i \(0.531432\pi\)
\(278\) 0 0
\(279\) −50.1100 −0.179606
\(280\) 0 0
\(281\) 102.404i 0.364429i 0.983259 + 0.182214i \(0.0583265\pi\)
−0.983259 + 0.182214i \(0.941674\pi\)
\(282\) 0 0
\(283\) 486.390i 1.71869i −0.511395 0.859346i \(-0.670871\pi\)
0.511395 0.859346i \(-0.329129\pi\)
\(284\) 0 0
\(285\) 71.7604 0.251791
\(286\) 0 0
\(287\) 149.055i 0.519354i
\(288\) 0 0
\(289\) −132.927 −0.459955
\(290\) 0 0
\(291\) 125.117i 0.429955i
\(292\) 0 0
\(293\) 32.9181i 0.112349i −0.998421 0.0561743i \(-0.982110\pi\)
0.998421 0.0561743i \(-0.0178902\pi\)
\(294\) 0 0
\(295\) 93.3141i 0.316319i
\(296\) 0 0
\(297\) 105.777i 0.356150i
\(298\) 0 0
\(299\) 84.1914 + 41.7481i 0.281577 + 0.139626i
\(300\) 0 0
\(301\) 345.023 1.14625
\(302\) 0 0
\(303\) −226.096 −0.746192
\(304\) 0 0
\(305\) −85.5697 −0.280556
\(306\) 0 0
\(307\) −180.336 −0.587413 −0.293706 0.955896i \(-0.594889\pi\)
−0.293706 + 0.955896i \(0.594889\pi\)
\(308\) 0 0
\(309\) 236.054i 0.763928i
\(310\) 0 0
\(311\) 62.6272 0.201374 0.100687 0.994918i \(-0.467896\pi\)
0.100687 + 0.994918i \(0.467896\pi\)
\(312\) 0 0
\(313\) 46.8523i 0.149688i 0.997195 + 0.0748439i \(0.0238459\pi\)
−0.997195 + 0.0748439i \(0.976154\pi\)
\(314\) 0 0
\(315\) 29.1945 0.0926808
\(316\) 0 0
\(317\) −426.928 −1.34678 −0.673388 0.739290i \(-0.735161\pi\)
−0.673388 + 0.739290i \(0.735161\pi\)
\(318\) 0 0
\(319\) 1058.72i 3.31888i
\(320\) 0 0
\(321\) 104.481i 0.325485i
\(322\) 0 0
\(323\) 380.590 1.17830
\(324\) 0 0
\(325\) 20.4291 0.0628588
\(326\) 0 0
\(327\) 67.4599i 0.206300i
\(328\) 0 0
\(329\) 157.649i 0.479176i
\(330\) 0 0
\(331\) 68.7366 0.207663 0.103832 0.994595i \(-0.466890\pi\)
0.103832 + 0.994595i \(0.466890\pi\)
\(332\) 0 0
\(333\) 190.750i 0.572824i
\(334\) 0 0
\(335\) 93.8958 0.280286
\(336\) 0 0
\(337\) 54.1771i 0.160763i 0.996764 + 0.0803814i \(0.0256138\pi\)
−0.996764 + 0.0803814i \(0.974386\pi\)
\(338\) 0 0
\(339\) 71.3858i 0.210578i
\(340\) 0 0
\(341\) 340.025i 0.997142i
\(342\) 0 0
\(343\) 344.072i 1.00312i
\(344\) 0 0
\(345\) 39.5733 79.8057i 0.114705 0.231321i
\(346\) 0 0
\(347\) 230.753 0.664994 0.332497 0.943104i \(-0.392109\pi\)
0.332497 + 0.943104i \(0.392109\pi\)
\(348\) 0 0
\(349\) −286.486 −0.820877 −0.410438 0.911888i \(-0.634624\pi\)
−0.410438 + 0.911888i \(0.634624\pi\)
\(350\) 0 0
\(351\) 21.2306 0.0604859
\(352\) 0 0
\(353\) −83.4342 −0.236358 −0.118179 0.992992i \(-0.537706\pi\)
−0.118179 + 0.992992i \(0.537706\pi\)
\(354\) 0 0
\(355\) 88.6962i 0.249849i
\(356\) 0 0
\(357\) 154.836 0.433716
\(358\) 0 0
\(359\) 654.224i 1.82235i 0.412019 + 0.911175i \(0.364824\pi\)
−0.412019 + 0.911175i \(0.635176\pi\)
\(360\) 0 0
\(361\) 17.6962 0.0490199
\(362\) 0 0
\(363\) 508.176 1.39993
\(364\) 0 0
\(365\) 88.9135i 0.243599i
\(366\) 0 0
\(367\) 78.2379i 0.213182i −0.994303 0.106591i \(-0.966006\pi\)
0.994303 0.106591i \(-0.0339936\pi\)
\(368\) 0 0
\(369\) −102.748 −0.278450
\(370\) 0 0
\(371\) −244.872 −0.660032
\(372\) 0 0
\(373\) 121.606i 0.326020i 0.986624 + 0.163010i \(0.0521203\pi\)
−0.986624 + 0.163010i \(0.947880\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) 212.497 0.563654
\(378\) 0 0
\(379\) 365.190i 0.963562i −0.876292 0.481781i \(-0.839990\pi\)
0.876292 0.481781i \(-0.160010\pi\)
\(380\) 0 0
\(381\) −340.791 −0.894464
\(382\) 0 0
\(383\) 50.1255i 0.130876i 0.997857 + 0.0654379i \(0.0208444\pi\)
−0.997857 + 0.0654379i \(0.979156\pi\)
\(384\) 0 0
\(385\) 198.101i 0.514548i
\(386\) 0 0
\(387\) 237.834i 0.614559i
\(388\) 0 0
\(389\) 212.581i 0.546480i −0.961946 0.273240i \(-0.911905\pi\)
0.961946 0.273240i \(-0.0880953\pi\)
\(390\) 0 0
\(391\) 209.882 423.260i 0.536783 1.08251i
\(392\) 0 0
\(393\) 223.394 0.568433
\(394\) 0 0
\(395\) 130.563 0.330540
\(396\) 0 0
\(397\) 357.976 0.901703 0.450851 0.892599i \(-0.351120\pi\)
0.450851 + 0.892599i \(0.351120\pi\)
\(398\) 0 0
\(399\) −139.667 −0.350043
\(400\) 0 0
\(401\) 11.6301i 0.0290027i 0.999895 + 0.0145014i \(0.00461609\pi\)
−0.999895 + 0.0145014i \(0.995384\pi\)
\(402\) 0 0
\(403\) 68.2469 0.169347
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −1294.35 −3.18022
\(408\) 0 0
\(409\) −295.715 −0.723019 −0.361510 0.932368i \(-0.617739\pi\)
−0.361510 + 0.932368i \(0.617739\pi\)
\(410\) 0 0
\(411\) 144.620i 0.351874i
\(412\) 0 0
\(413\) 181.617i 0.439750i
\(414\) 0 0
\(415\) −362.889 −0.874433
\(416\) 0 0
\(417\) 223.884 0.536893
\(418\) 0 0
\(419\) 391.238i 0.933742i −0.884325 0.466871i \(-0.845381\pi\)
0.884325 0.466871i \(-0.154619\pi\)
\(420\) 0 0
\(421\) 230.016i 0.546357i 0.961963 + 0.273179i \(0.0880750\pi\)
−0.961963 + 0.273179i \(0.911925\pi\)
\(422\) 0 0
\(423\) 108.672 0.256908
\(424\) 0 0
\(425\) 102.704i 0.241657i
\(426\) 0 0
\(427\) 166.544 0.390033
\(428\) 0 0
\(429\) 144.061i 0.335808i
\(430\) 0 0
\(431\) 57.3445i 0.133050i −0.997785 0.0665250i \(-0.978809\pi\)
0.997785 0.0665250i \(-0.0211912\pi\)
\(432\) 0 0
\(433\) 556.989i 1.28635i 0.765719 + 0.643175i \(0.222383\pi\)
−0.765719 + 0.643175i \(0.777617\pi\)
\(434\) 0 0
\(435\) 201.428i 0.463053i
\(436\) 0 0
\(437\) −189.320 + 381.793i −0.433226 + 0.873668i
\(438\) 0 0
\(439\) −567.844 −1.29349 −0.646747 0.762705i \(-0.723871\pi\)
−0.646747 + 0.762705i \(0.723871\pi\)
\(440\) 0 0
\(441\) 90.1789 0.204487
\(442\) 0 0
\(443\) 122.228 0.275910 0.137955 0.990439i \(-0.455947\pi\)
0.137955 + 0.990439i \(0.455947\pi\)
\(444\) 0 0
\(445\) −12.2580 −0.0275461
\(446\) 0 0
\(447\) 25.7466i 0.0575987i
\(448\) 0 0
\(449\) −704.802 −1.56972 −0.784858 0.619676i \(-0.787264\pi\)
−0.784858 + 0.619676i \(0.787264\pi\)
\(450\) 0 0
\(451\) 697.203i 1.54590i
\(452\) 0 0
\(453\) 264.572 0.584044
\(454\) 0 0
\(455\) −39.7611 −0.0873871
\(456\) 0 0
\(457\) 391.760i 0.857244i −0.903484 0.428622i \(-0.858999\pi\)
0.903484 0.428622i \(-0.141001\pi\)
\(458\) 0 0
\(459\) 106.733i 0.232535i
\(460\) 0 0
\(461\) −139.599 −0.302818 −0.151409 0.988471i \(-0.548381\pi\)
−0.151409 + 0.988471i \(0.548381\pi\)
\(462\) 0 0
\(463\) 881.132 1.90309 0.951546 0.307505i \(-0.0994942\pi\)
0.951546 + 0.307505i \(0.0994942\pi\)
\(464\) 0 0
\(465\) 64.6918i 0.139122i
\(466\) 0 0
\(467\) 401.486i 0.859714i −0.902897 0.429857i \(-0.858564\pi\)
0.902897 0.429857i \(-0.141436\pi\)
\(468\) 0 0
\(469\) −182.749 −0.389657
\(470\) 0 0
\(471\) 80.9505i 0.171869i
\(472\) 0 0
\(473\) 1613.84 3.41193
\(474\) 0 0
\(475\) 92.6423i 0.195036i
\(476\) 0 0
\(477\) 168.798i 0.353873i
\(478\) 0 0
\(479\) 45.0345i 0.0940177i −0.998894 0.0470089i \(-0.985031\pi\)
0.998894 0.0470089i \(-0.0149689\pi\)
\(480\) 0 0
\(481\) 259.791i 0.540106i
\(482\) 0 0
\(483\) −77.0215 + 155.326i −0.159465 + 0.321585i
\(484\) 0 0
\(485\) −161.525 −0.333041
\(486\) 0 0
\(487\) 533.284 1.09504 0.547519 0.836793i \(-0.315572\pi\)
0.547519 + 0.836793i \(0.315572\pi\)
\(488\) 0 0
\(489\) 503.354 1.02935
\(490\) 0 0
\(491\) −598.592 −1.21913 −0.609565 0.792736i \(-0.708656\pi\)
−0.609565 + 0.792736i \(0.708656\pi\)
\(492\) 0 0
\(493\) 1068.30i 2.16693i
\(494\) 0 0
\(495\) 136.557 0.275873
\(496\) 0 0
\(497\) 172.629i 0.347342i
\(498\) 0 0
\(499\) −375.742 −0.752991 −0.376495 0.926419i \(-0.622871\pi\)
−0.376495 + 0.926419i \(0.622871\pi\)
\(500\) 0 0
\(501\) −48.7996 −0.0974044
\(502\) 0 0
\(503\) 785.917i 1.56246i 0.624243 + 0.781230i \(0.285407\pi\)
−0.624243 + 0.781230i \(0.714593\pi\)
\(504\) 0 0
\(505\) 291.889i 0.577998i
\(506\) 0 0
\(507\) 263.802 0.520319
\(508\) 0 0
\(509\) 588.753 1.15668 0.578342 0.815794i \(-0.303700\pi\)
0.578342 + 0.815794i \(0.303700\pi\)
\(510\) 0 0
\(511\) 173.052i 0.338654i
\(512\) 0 0
\(513\) 96.2767i 0.187674i
\(514\) 0 0
\(515\) −304.744 −0.591736
\(516\) 0 0
\(517\) 737.402i 1.42631i
\(518\) 0 0
\(519\) −132.351 −0.255012
\(520\) 0 0
\(521\) 644.174i 1.23642i 0.786014 + 0.618209i \(0.212142\pi\)
−0.786014 + 0.618209i \(0.787858\pi\)
\(522\) 0 0
\(523\) 584.220i 1.11705i −0.829486 0.558527i \(-0.811367\pi\)
0.829486 0.558527i \(-0.188633\pi\)
\(524\) 0 0
\(525\) 37.6899i 0.0717902i
\(526\) 0 0
\(527\) 343.101i 0.651046i
\(528\) 0 0
\(529\) 320.193 + 421.090i 0.605281 + 0.796012i
\(530\) 0 0
\(531\) 125.194 0.235770
\(532\) 0 0
\(533\) 139.937 0.262545
\(534\) 0 0
\(535\) −134.884 −0.252119
\(536\) 0 0
\(537\) −281.414 −0.524049
\(538\) 0 0
\(539\) 611.916i 1.13528i
\(540\) 0 0
\(541\) −382.638 −0.707279 −0.353640 0.935382i \(-0.615056\pi\)
−0.353640 + 0.935382i \(0.615056\pi\)
\(542\) 0 0
\(543\) 119.412i 0.219912i
\(544\) 0 0
\(545\) −87.0904 −0.159799
\(546\) 0 0
\(547\) −179.304 −0.327795 −0.163897 0.986477i \(-0.552407\pi\)
−0.163897 + 0.986477i \(0.552407\pi\)
\(548\) 0 0
\(549\) 114.804i 0.209114i
\(550\) 0 0
\(551\) 963.637i 1.74889i
\(552\) 0 0
\(553\) −254.115 −0.459521
\(554\) 0 0
\(555\) −246.258 −0.443708
\(556\) 0 0
\(557\) 672.738i 1.20779i 0.797065 + 0.603894i \(0.206385\pi\)
−0.797065 + 0.603894i \(0.793615\pi\)
\(558\) 0 0
\(559\) 323.916i 0.579457i
\(560\) 0 0
\(561\) 724.247 1.29099
\(562\) 0 0
\(563\) 318.411i 0.565560i 0.959185 + 0.282780i \(0.0912567\pi\)
−0.959185 + 0.282780i \(0.908743\pi\)
\(564\) 0 0
\(565\) 92.1587 0.163113
\(566\) 0 0
\(567\) 39.1685i 0.0690802i
\(568\) 0 0
\(569\) 903.751i 1.58831i −0.607712 0.794157i \(-0.707913\pi\)
0.607712 0.794157i \(-0.292087\pi\)
\(570\) 0 0
\(571\) 155.592i 0.272490i −0.990675 0.136245i \(-0.956497\pi\)
0.990675 0.136245i \(-0.0435034\pi\)
\(572\) 0 0
\(573\) 269.513i 0.470355i
\(574\) 0 0
\(575\) 103.029 + 51.0890i 0.179180 + 0.0888504i
\(576\) 0 0
\(577\) −360.066 −0.624032 −0.312016 0.950077i \(-0.601004\pi\)
−0.312016 + 0.950077i \(0.601004\pi\)
\(578\) 0 0
\(579\) −250.126 −0.431997
\(580\) 0 0
\(581\) 706.291 1.21565
\(582\) 0 0
\(583\) −1145.39 −1.96464
\(584\) 0 0
\(585\) 27.4085i 0.0468522i
\(586\) 0 0
\(587\) −146.158 −0.248992 −0.124496 0.992220i \(-0.539731\pi\)
−0.124496 + 0.992220i \(0.539731\pi\)
\(588\) 0 0
\(589\) 309.487i 0.525445i
\(590\) 0 0
\(591\) −463.886 −0.784918
\(592\) 0 0
\(593\) −857.655 −1.44630 −0.723149 0.690692i \(-0.757306\pi\)
−0.723149 + 0.690692i \(0.757306\pi\)
\(594\) 0 0
\(595\) 199.893i 0.335955i
\(596\) 0 0
\(597\) 510.049i 0.854353i
\(598\) 0 0
\(599\) 78.2060 0.130561 0.0652805 0.997867i \(-0.479206\pi\)
0.0652805 + 0.997867i \(0.479206\pi\)
\(600\) 0 0
\(601\) −252.086 −0.419445 −0.209722 0.977761i \(-0.567256\pi\)
−0.209722 + 0.977761i \(0.567256\pi\)
\(602\) 0 0
\(603\) 125.974i 0.208913i
\(604\) 0 0
\(605\) 656.053i 1.08438i
\(606\) 0 0
\(607\) 1010.86 1.66533 0.832666 0.553776i \(-0.186814\pi\)
0.832666 + 0.553776i \(0.186814\pi\)
\(608\) 0 0
\(609\) 392.038i 0.643741i
\(610\) 0 0
\(611\) −148.005 −0.242234
\(612\) 0 0
\(613\) 1184.06i 1.93158i −0.259318 0.965792i \(-0.583498\pi\)
0.259318 0.965792i \(-0.416502\pi\)
\(614\) 0 0
\(615\) 132.647i 0.215686i
\(616\) 0 0
\(617\) 271.156i 0.439475i −0.975559 0.219738i \(-0.929480\pi\)
0.975559 0.219738i \(-0.0705201\pi\)
\(618\) 0 0
\(619\) 618.264i 0.998811i −0.866368 0.499405i \(-0.833552\pi\)
0.866368 0.499405i \(-0.166448\pi\)
\(620\) 0 0
\(621\) 107.071 + 53.0932i 0.172416 + 0.0854963i
\(622\) 0 0
\(623\) 23.8577 0.0382949
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −653.292 −1.04193
\(628\) 0 0
\(629\) −1306.06 −2.07641
\(630\) 0 0
\(631\) 837.071i 1.32658i −0.748363 0.663289i \(-0.769160\pi\)
0.748363 0.663289i \(-0.230840\pi\)
\(632\) 0 0
\(633\) 481.561 0.760759
\(634\) 0 0
\(635\) 439.959i 0.692849i
\(636\) 0 0
\(637\) −122.818 −0.192808
\(638\) 0 0
\(639\) 118.998 0.186226
\(640\) 0 0
\(641\) 864.535i 1.34873i 0.738399 + 0.674365i \(0.235582\pi\)
−0.738399 + 0.674365i \(0.764418\pi\)
\(642\) 0 0
\(643\) 223.427i 0.347475i 0.984792 + 0.173738i \(0.0555845\pi\)
−0.984792 + 0.173738i \(0.944416\pi\)
\(644\) 0 0
\(645\) 307.043 0.476035
\(646\) 0 0
\(647\) 1005.78 1.55453 0.777266 0.629172i \(-0.216606\pi\)
0.777266 + 0.629172i \(0.216606\pi\)
\(648\) 0 0
\(649\) 849.512i 1.30896i
\(650\) 0 0
\(651\) 125.909i 0.193409i
\(652\) 0 0
\(653\) 711.174 1.08909 0.544543 0.838733i \(-0.316703\pi\)
0.544543 + 0.838733i \(0.316703\pi\)
\(654\) 0 0
\(655\) 288.401i 0.440306i
\(656\) 0 0
\(657\) 119.290 0.181568
\(658\) 0 0
\(659\) 609.922i 0.925526i 0.886482 + 0.462763i \(0.153142\pi\)
−0.886482 + 0.462763i \(0.846858\pi\)
\(660\) 0 0
\(661\) 445.554i 0.674060i −0.941494 0.337030i \(-0.890578\pi\)
0.941494 0.337030i \(-0.109422\pi\)
\(662\) 0 0
\(663\) 145.365i 0.219253i
\(664\) 0 0
\(665\) 180.309i 0.271142i
\(666\) 0 0
\(667\) 1071.67 + 531.412i 1.60671 + 0.796719i
\(668\) 0 0
\(669\) 236.440 0.353423
\(670\) 0 0
\(671\) 779.009 1.16097
\(672\) 0 0
\(673\) 542.076 0.805462 0.402731 0.915318i \(-0.368061\pi\)
0.402731 + 0.915318i \(0.368061\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 140.601i 0.207682i 0.994594 + 0.103841i \(0.0331134\pi\)
−0.994594 + 0.103841i \(0.966887\pi\)
\(678\) 0 0
\(679\) 314.376 0.462998
\(680\) 0 0
\(681\) 229.101i 0.336418i
\(682\) 0 0
\(683\) 283.775 0.415484 0.207742 0.978184i \(-0.433389\pi\)
0.207742 + 0.978184i \(0.433389\pi\)
\(684\) 0 0
\(685\) −186.704 −0.272560
\(686\) 0 0
\(687\) 503.805i 0.733340i
\(688\) 0 0
\(689\) 229.892i 0.333661i
\(690\) 0 0
\(691\) −1132.94 −1.63957 −0.819786 0.572669i \(-0.805908\pi\)
−0.819786 + 0.572669i \(0.805908\pi\)
\(692\) 0 0
\(693\) −265.780 −0.383522
\(694\) 0 0
\(695\) 289.033i 0.415875i
\(696\) 0 0
\(697\) 703.510i 1.00934i
\(698\) 0 0
\(699\) 254.353 0.363881
\(700\) 0 0
\(701\) 52.1383i 0.0743770i 0.999308 + 0.0371885i \(0.0118402\pi\)
−0.999308 + 0.0371885i \(0.988160\pi\)
\(702\) 0 0
\(703\) 1178.10 1.67582
\(704\) 0 0
\(705\) 140.295i 0.199000i
\(706\) 0 0
\(707\) 568.103i 0.803540i
\(708\) 0 0
\(709\) 1111.26i 1.56736i 0.621165 + 0.783680i \(0.286660\pi\)
−0.621165 + 0.783680i \(0.713340\pi\)
\(710\) 0 0
\(711\) 175.169i 0.246370i
\(712\) 0 0
\(713\) 344.185 + 170.671i 0.482728 + 0.239371i
\(714\) 0 0
\(715\) −185.983 −0.260115
\(716\) 0 0
\(717\) 185.041 0.258077
\(718\) 0 0
\(719\) 98.1047 0.136446 0.0682230 0.997670i \(-0.478267\pi\)
0.0682230 + 0.997670i \(0.478267\pi\)
\(720\) 0 0
\(721\) 593.122 0.822639
\(722\) 0 0
\(723\) 239.380i 0.331093i
\(724\) 0 0
\(725\) 260.042 0.358679
\(726\) 0 0
\(727\) 814.763i 1.12072i −0.828250 0.560359i \(-0.810663\pi\)
0.828250 0.560359i \(-0.189337\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 1628.44 2.22769
\(732\) 0 0
\(733\) 648.304i 0.884453i −0.896903 0.442227i \(-0.854189\pi\)
0.896903 0.442227i \(-0.145811\pi\)
\(734\) 0 0
\(735\) 116.421i 0.158395i
\(736\) 0 0
\(737\) −854.808 −1.15985
\(738\) 0 0
\(739\) −1295.09 −1.75249 −0.876245 0.481866i \(-0.839959\pi\)
−0.876245 + 0.481866i \(0.839959\pi\)
\(740\) 0 0
\(741\) 131.123i 0.176954i
\(742\) 0 0
\(743\) 516.058i 0.694560i −0.937762 0.347280i \(-0.887105\pi\)
0.937762 0.347280i \(-0.112895\pi\)
\(744\) 0 0
\(745\) 33.2388 0.0446158
\(746\) 0 0
\(747\) 486.867i 0.651764i
\(748\) 0 0
\(749\) 262.524 0.350500
\(750\) 0 0
\(751\) 388.093i 0.516769i −0.966042 0.258384i \(-0.916810\pi\)
0.966042 0.258384i \(-0.0831901\pi\)
\(752\) 0 0
\(753\) 47.3708i 0.0629094i
\(754\) 0 0
\(755\) 341.561i 0.452398i
\(756\) 0 0
\(757\) 502.322i 0.663569i 0.943355 + 0.331784i \(0.107651\pi\)
−0.943355 + 0.331784i \(0.892349\pi\)
\(758\) 0 0
\(759\) −360.268 + 726.535i −0.474661 + 0.957227i
\(760\) 0 0
\(761\) 241.763 0.317691 0.158846 0.987303i \(-0.449223\pi\)
0.158846 + 0.987303i \(0.449223\pi\)
\(762\) 0 0
\(763\) 169.504 0.222154
\(764\) 0 0
\(765\) 137.792 0.180121
\(766\) 0 0
\(767\) −170.507 −0.222304
\(768\) 0 0
\(769\) 873.218i 1.13552i 0.823193 + 0.567762i \(0.192191\pi\)
−0.823193 + 0.567762i \(0.807809\pi\)
\(770\) 0 0
\(771\) 133.624 0.173313
\(772\) 0 0
\(773\) 203.496i 0.263255i 0.991299 + 0.131628i \(0.0420203\pi\)
−0.991299 + 0.131628i \(0.957980\pi\)
\(774\) 0 0
\(775\) 83.5167 0.107764
\(776\) 0 0
\(777\) 479.291 0.616848
\(778\) 0 0
\(779\) 634.587i 0.814617i
\(780\) 0 0
\(781\) 807.473i 1.03390i
\(782\) 0 0
\(783\) 270.244 0.345139
\(784\) 0 0
\(785\) −104.507 −0.133129
\(786\) 0 0
\(787\) 1313.63i 1.66916i 0.550884 + 0.834582i \(0.314291\pi\)
−0.550884 + 0.834582i \(0.685709\pi\)
\(788\) 0 0
\(789\) 538.129i 0.682039i
\(790\) 0 0
\(791\) −179.368 −0.226761
\(792\) 0 0
\(793\) 156.356i 0.197170i
\(794\) 0 0
\(795\) −217.917 −0.274109
\(796\) 0 0
\(797\) 444.355i 0.557534i 0.960359 + 0.278767i \(0.0899257\pi\)
−0.960359 + 0.278767i \(0.910074\pi\)
\(798\) 0 0
\(799\)