Properties

Label 2760.3.g.a.2161.17
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 2161.17
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.32

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +5.23512i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +5.23512i q^{7} +3.00000 q^{9} -0.200771i q^{11} -23.8661 q^{13} +3.87298i q^{15} -1.11718i q^{17} -19.7042i q^{19} -9.06750i q^{21} +(-18.8058 - 13.2416i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +2.39590 q^{29} +38.2118 q^{31} +0.347746i q^{33} +11.7061 q^{35} -38.0912i q^{37} +41.3372 q^{39} +26.9899 q^{41} +35.3606i q^{43} -6.70820i q^{45} -9.80621 q^{47} +21.5935 q^{49} +1.93501i q^{51} +78.2168i q^{53} -0.448939 q^{55} +34.1286i q^{57} -63.9063 q^{59} -0.253327i q^{61} +15.7054i q^{63} +53.3661i q^{65} -17.8439i q^{67} +(32.5727 + 22.9352i) q^{69} +76.3329 q^{71} -77.6479 q^{73} +8.66025 q^{75} +1.05106 q^{77} -25.9565i q^{79} +9.00000 q^{81} -18.6692i q^{83} -2.49809 q^{85} -4.14982 q^{87} +28.6300i q^{89} -124.942i q^{91} -66.1847 q^{93} -44.0598 q^{95} -47.1436i q^{97} -0.602314i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 5.23512i 0.747875i 0.927454 + 0.373937i \(0.121992\pi\)
−0.927454 + 0.373937i \(0.878008\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 0.200771i 0.0182519i −0.999958 0.00912597i \(-0.997095\pi\)
0.999958 0.00912597i \(-0.00290493\pi\)
\(12\) 0 0
\(13\) −23.8661 −1.83585 −0.917925 0.396754i \(-0.870137\pi\)
−0.917925 + 0.396754i \(0.870137\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 1.11718i 0.0657164i −0.999460 0.0328582i \(-0.989539\pi\)
0.999460 0.0328582i \(-0.0104610\pi\)
\(18\) 0 0
\(19\) 19.7042i 1.03706i −0.855059 0.518531i \(-0.826479\pi\)
0.855059 0.518531i \(-0.173521\pi\)
\(20\) 0 0
\(21\) 9.06750i 0.431786i
\(22\) 0 0
\(23\) −18.8058 13.2416i −0.817645 0.575723i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 2.39590 0.0826173 0.0413086 0.999146i \(-0.486847\pi\)
0.0413086 + 0.999146i \(0.486847\pi\)
\(30\) 0 0
\(31\) 38.2118 1.23264 0.616319 0.787497i \(-0.288623\pi\)
0.616319 + 0.787497i \(0.288623\pi\)
\(32\) 0 0
\(33\) 0.347746i 0.0105378i
\(34\) 0 0
\(35\) 11.7061 0.334460
\(36\) 0 0
\(37\) 38.0912i 1.02949i −0.857343 0.514745i \(-0.827886\pi\)
0.857343 0.514745i \(-0.172114\pi\)
\(38\) 0 0
\(39\) 41.3372 1.05993
\(40\) 0 0
\(41\) 26.9899 0.658291 0.329145 0.944279i \(-0.393239\pi\)
0.329145 + 0.944279i \(0.393239\pi\)
\(42\) 0 0
\(43\) 35.3606i 0.822338i 0.911559 + 0.411169i \(0.134879\pi\)
−0.911559 + 0.411169i \(0.865121\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −9.80621 −0.208643 −0.104321 0.994544i \(-0.533267\pi\)
−0.104321 + 0.994544i \(0.533267\pi\)
\(48\) 0 0
\(49\) 21.5935 0.440684
\(50\) 0 0
\(51\) 1.93501i 0.0379414i
\(52\) 0 0
\(53\) 78.2168i 1.47579i 0.674916 + 0.737895i \(0.264180\pi\)
−0.674916 + 0.737895i \(0.735820\pi\)
\(54\) 0 0
\(55\) −0.448939 −0.00816252
\(56\) 0 0
\(57\) 34.1286i 0.598748i
\(58\) 0 0
\(59\) −63.9063 −1.08316 −0.541579 0.840650i \(-0.682173\pi\)
−0.541579 + 0.840650i \(0.682173\pi\)
\(60\) 0 0
\(61\) 0.253327i 0.00415290i −0.999998 0.00207645i \(-0.999339\pi\)
0.999998 0.00207645i \(-0.000660955\pi\)
\(62\) 0 0
\(63\) 15.7054i 0.249292i
\(64\) 0 0
\(65\) 53.3661i 0.821017i
\(66\) 0 0
\(67\) 17.8439i 0.266326i −0.991094 0.133163i \(-0.957487\pi\)
0.991094 0.133163i \(-0.0425134\pi\)
\(68\) 0 0
\(69\) 32.5727 + 22.9352i 0.472067 + 0.332394i
\(70\) 0 0
\(71\) 76.3329 1.07511 0.537555 0.843228i \(-0.319348\pi\)
0.537555 + 0.843228i \(0.319348\pi\)
\(72\) 0 0
\(73\) −77.6479 −1.06367 −0.531835 0.846848i \(-0.678497\pi\)
−0.531835 + 0.846848i \(0.678497\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 1.05106 0.0136502
\(78\) 0 0
\(79\) 25.9565i 0.328563i −0.986413 0.164282i \(-0.947469\pi\)
0.986413 0.164282i \(-0.0525306\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 18.6692i 0.224930i −0.993656 0.112465i \(-0.964125\pi\)
0.993656 0.112465i \(-0.0358746\pi\)
\(84\) 0 0
\(85\) −2.49809 −0.0293893
\(86\) 0 0
\(87\) −4.14982 −0.0476991
\(88\) 0 0
\(89\) 28.6300i 0.321686i 0.986980 + 0.160843i \(0.0514212\pi\)
−0.986980 + 0.160843i \(0.948579\pi\)
\(90\) 0 0
\(91\) 124.942i 1.37299i
\(92\) 0 0
\(93\) −66.1847 −0.711663
\(94\) 0 0
\(95\) −44.0598 −0.463788
\(96\) 0 0
\(97\) 47.1436i 0.486017i −0.970024 0.243008i \(-0.921866\pi\)
0.970024 0.243008i \(-0.0781342\pi\)
\(98\) 0 0
\(99\) 0.602314i 0.00608398i
\(100\) 0 0
\(101\) −69.8248 −0.691334 −0.345667 0.938357i \(-0.612347\pi\)
−0.345667 + 0.938357i \(0.612347\pi\)
\(102\) 0 0
\(103\) 50.4967i 0.490260i 0.969490 + 0.245130i \(0.0788306\pi\)
−0.969490 + 0.245130i \(0.921169\pi\)
\(104\) 0 0
\(105\) −20.2755 −0.193100
\(106\) 0 0
\(107\) 119.697i 1.11866i −0.828945 0.559330i \(-0.811059\pi\)
0.828945 0.559330i \(-0.188941\pi\)
\(108\) 0 0
\(109\) 137.224i 1.25894i 0.777026 + 0.629469i \(0.216728\pi\)
−0.777026 + 0.629469i \(0.783272\pi\)
\(110\) 0 0
\(111\) 65.9758i 0.594377i
\(112\) 0 0
\(113\) 197.504i 1.74783i 0.486083 + 0.873913i \(0.338425\pi\)
−0.486083 + 0.873913i \(0.661575\pi\)
\(114\) 0 0
\(115\) −29.6092 + 42.0511i −0.257471 + 0.365662i
\(116\) 0 0
\(117\) −71.5982 −0.611950
\(118\) 0 0
\(119\) 5.84856 0.0491476
\(120\) 0 0
\(121\) 120.960 0.999667
\(122\) 0 0
\(123\) −46.7479 −0.380064
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) 126.780 0.998267 0.499133 0.866525i \(-0.333652\pi\)
0.499133 + 0.866525i \(0.333652\pi\)
\(128\) 0 0
\(129\) 61.2463i 0.474777i
\(130\) 0 0
\(131\) −8.63807 −0.0659395 −0.0329697 0.999456i \(-0.510496\pi\)
−0.0329697 + 0.999456i \(0.510496\pi\)
\(132\) 0 0
\(133\) 103.154 0.775592
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 77.8979i 0.568598i −0.958736 0.284299i \(-0.908239\pi\)
0.958736 0.284299i \(-0.0917608\pi\)
\(138\) 0 0
\(139\) −0.316859 −0.00227956 −0.00113978 0.999999i \(-0.500363\pi\)
−0.00113978 + 0.999999i \(0.500363\pi\)
\(140\) 0 0
\(141\) 16.9849 0.120460
\(142\) 0 0
\(143\) 4.79162i 0.0335078i
\(144\) 0 0
\(145\) 5.35740i 0.0369476i
\(146\) 0 0
\(147\) −37.4010 −0.254429
\(148\) 0 0
\(149\) 253.992i 1.70465i 0.523015 + 0.852323i \(0.324807\pi\)
−0.523015 + 0.852323i \(0.675193\pi\)
\(150\) 0 0
\(151\) 106.476 0.705140 0.352570 0.935785i \(-0.385308\pi\)
0.352570 + 0.935785i \(0.385308\pi\)
\(152\) 0 0
\(153\) 3.35153i 0.0219055i
\(154\) 0 0
\(155\) 85.4441i 0.551252i
\(156\) 0 0
\(157\) 54.7747i 0.348883i 0.984668 + 0.174442i \(0.0558120\pi\)
−0.984668 + 0.174442i \(0.944188\pi\)
\(158\) 0 0
\(159\) 135.476i 0.852047i
\(160\) 0 0
\(161\) 69.3215 98.4508i 0.430569 0.611496i
\(162\) 0 0
\(163\) −217.561 −1.33473 −0.667365 0.744731i \(-0.732578\pi\)
−0.667365 + 0.744731i \(0.732578\pi\)
\(164\) 0 0
\(165\) 0.777584 0.00471263
\(166\) 0 0
\(167\) −153.009 −0.916222 −0.458111 0.888895i \(-0.651474\pi\)
−0.458111 + 0.888895i \(0.651474\pi\)
\(168\) 0 0
\(169\) 400.589 2.37035
\(170\) 0 0
\(171\) 59.1125i 0.345687i
\(172\) 0 0
\(173\) 105.144 0.607769 0.303884 0.952709i \(-0.401716\pi\)
0.303884 + 0.952709i \(0.401716\pi\)
\(174\) 0 0
\(175\) 26.1756i 0.149575i
\(176\) 0 0
\(177\) 110.689 0.625362
\(178\) 0 0
\(179\) −109.111 −0.609558 −0.304779 0.952423i \(-0.598583\pi\)
−0.304779 + 0.952423i \(0.598583\pi\)
\(180\) 0 0
\(181\) 283.290i 1.56514i 0.622562 + 0.782570i \(0.286092\pi\)
−0.622562 + 0.782570i \(0.713908\pi\)
\(182\) 0 0
\(183\) 0.438775i 0.00239768i
\(184\) 0 0
\(185\) −85.1744 −0.460402
\(186\) 0 0
\(187\) −0.224297 −0.00119945
\(188\) 0 0
\(189\) 27.2025i 0.143929i
\(190\) 0 0
\(191\) 66.8202i 0.349844i 0.984582 + 0.174922i \(0.0559673\pi\)
−0.984582 + 0.174922i \(0.944033\pi\)
\(192\) 0 0
\(193\) 179.398 0.929525 0.464763 0.885435i \(-0.346140\pi\)
0.464763 + 0.885435i \(0.346140\pi\)
\(194\) 0 0
\(195\) 92.4328i 0.474015i
\(196\) 0 0
\(197\) 294.073 1.49276 0.746378 0.665522i \(-0.231791\pi\)
0.746378 + 0.665522i \(0.231791\pi\)
\(198\) 0 0
\(199\) 106.727i 0.536316i 0.963375 + 0.268158i \(0.0864149\pi\)
−0.963375 + 0.268158i \(0.913585\pi\)
\(200\) 0 0
\(201\) 30.9065i 0.153764i
\(202\) 0 0
\(203\) 12.5428i 0.0617874i
\(204\) 0 0
\(205\) 60.3513i 0.294397i
\(206\) 0 0
\(207\) −56.4175 39.7249i −0.272548 0.191908i
\(208\) 0 0
\(209\) −3.95603 −0.0189284
\(210\) 0 0
\(211\) 87.6103 0.415215 0.207607 0.978212i \(-0.433432\pi\)
0.207607 + 0.978212i \(0.433432\pi\)
\(212\) 0 0
\(213\) −132.212 −0.620716
\(214\) 0 0
\(215\) 79.0686 0.367761
\(216\) 0 0
\(217\) 200.043i 0.921858i
\(218\) 0 0
\(219\) 134.490 0.614110
\(220\) 0 0
\(221\) 26.6626i 0.120645i
\(222\) 0 0
\(223\) 226.958 1.01775 0.508874 0.860841i \(-0.330062\pi\)
0.508874 + 0.860841i \(0.330062\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 269.830i 1.18868i 0.804214 + 0.594340i \(0.202587\pi\)
−0.804214 + 0.594340i \(0.797413\pi\)
\(228\) 0 0
\(229\) 193.430i 0.844673i 0.906439 + 0.422337i \(0.138790\pi\)
−0.906439 + 0.422337i \(0.861210\pi\)
\(230\) 0 0
\(231\) −1.82049 −0.00788093
\(232\) 0 0
\(233\) −27.9090 −0.119781 −0.0598906 0.998205i \(-0.519075\pi\)
−0.0598906 + 0.998205i \(0.519075\pi\)
\(234\) 0 0
\(235\) 21.9274i 0.0933079i
\(236\) 0 0
\(237\) 44.9580i 0.189696i
\(238\) 0 0
\(239\) −203.637 −0.852039 −0.426020 0.904714i \(-0.640085\pi\)
−0.426020 + 0.904714i \(0.640085\pi\)
\(240\) 0 0
\(241\) 466.175i 1.93434i 0.254134 + 0.967169i \(0.418210\pi\)
−0.254134 + 0.967169i \(0.581790\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 48.2845i 0.197080i
\(246\) 0 0
\(247\) 470.261i 1.90389i
\(248\) 0 0
\(249\) 32.3360i 0.129863i
\(250\) 0 0
\(251\) 167.843i 0.668696i −0.942450 0.334348i \(-0.891484\pi\)
0.942450 0.334348i \(-0.108516\pi\)
\(252\) 0 0
\(253\) −2.65854 + 3.77567i −0.0105081 + 0.0149236i
\(254\) 0 0
\(255\) 4.32681 0.0169679
\(256\) 0 0
\(257\) 82.9455 0.322745 0.161373 0.986894i \(-0.448408\pi\)
0.161373 + 0.986894i \(0.448408\pi\)
\(258\) 0 0
\(259\) 199.412 0.769930
\(260\) 0 0
\(261\) 7.18770 0.0275391
\(262\) 0 0
\(263\) 231.115i 0.878765i 0.898300 + 0.439383i \(0.144803\pi\)
−0.898300 + 0.439383i \(0.855197\pi\)
\(264\) 0 0
\(265\) 174.898 0.659993
\(266\) 0 0
\(267\) 49.5887i 0.185725i
\(268\) 0 0
\(269\) 10.6727 0.0396756 0.0198378 0.999803i \(-0.493685\pi\)
0.0198378 + 0.999803i \(0.493685\pi\)
\(270\) 0 0
\(271\) 323.563 1.19396 0.596980 0.802256i \(-0.296367\pi\)
0.596980 + 0.802256i \(0.296367\pi\)
\(272\) 0 0
\(273\) 216.405i 0.792694i
\(274\) 0 0
\(275\) 1.00386i 0.00365039i
\(276\) 0 0
\(277\) 71.4145 0.257814 0.128907 0.991657i \(-0.458853\pi\)
0.128907 + 0.991657i \(0.458853\pi\)
\(278\) 0 0
\(279\) 114.635 0.410879
\(280\) 0 0
\(281\) 393.137i 1.39906i −0.714601 0.699532i \(-0.753392\pi\)
0.714601 0.699532i \(-0.246608\pi\)
\(282\) 0 0
\(283\) 96.7722i 0.341951i 0.985275 + 0.170976i \(0.0546919\pi\)
−0.985275 + 0.170976i \(0.945308\pi\)
\(284\) 0 0
\(285\) 76.3139 0.267768
\(286\) 0 0
\(287\) 141.296i 0.492319i
\(288\) 0 0
\(289\) 287.752 0.995681
\(290\) 0 0
\(291\) 81.6551i 0.280602i
\(292\) 0 0
\(293\) 43.7569i 0.149341i −0.997208 0.0746704i \(-0.976210\pi\)
0.997208 0.0746704i \(-0.0237905\pi\)
\(294\) 0 0
\(295\) 142.899i 0.484403i
\(296\) 0 0
\(297\) 1.04324i 0.00351259i
\(298\) 0 0
\(299\) 448.821 + 316.025i 1.50107 + 1.05694i
\(300\) 0 0
\(301\) −185.117 −0.615006
\(302\) 0 0
\(303\) 120.940 0.399142
\(304\) 0 0
\(305\) −0.566456 −0.00185723
\(306\) 0 0
\(307\) −338.800 −1.10358 −0.551791 0.833982i \(-0.686056\pi\)
−0.551791 + 0.833982i \(0.686056\pi\)
\(308\) 0 0
\(309\) 87.4629i 0.283052i
\(310\) 0 0
\(311\) 15.6524 0.0503293 0.0251646 0.999683i \(-0.491989\pi\)
0.0251646 + 0.999683i \(0.491989\pi\)
\(312\) 0 0
\(313\) 495.024i 1.58155i 0.612110 + 0.790773i \(0.290321\pi\)
−0.612110 + 0.790773i \(0.709679\pi\)
\(314\) 0 0
\(315\) 35.1183 0.111487
\(316\) 0 0
\(317\) 209.614 0.661244 0.330622 0.943763i \(-0.392741\pi\)
0.330622 + 0.943763i \(0.392741\pi\)
\(318\) 0 0
\(319\) 0.481028i 0.00150793i
\(320\) 0 0
\(321\) 207.320i 0.645858i
\(322\) 0 0
\(323\) −22.0131 −0.0681519
\(324\) 0 0
\(325\) 119.330 0.367170
\(326\) 0 0
\(327\) 237.679i 0.726848i
\(328\) 0 0
\(329\) 51.3367i 0.156039i
\(330\) 0 0
\(331\) −341.668 −1.03223 −0.516115 0.856520i \(-0.672622\pi\)
−0.516115 + 0.856520i \(0.672622\pi\)
\(332\) 0 0
\(333\) 114.273i 0.343164i
\(334\) 0 0
\(335\) −39.9001 −0.119105
\(336\) 0 0
\(337\) 67.5483i 0.200440i −0.994965 0.100220i \(-0.968045\pi\)
0.994965 0.100220i \(-0.0319547\pi\)
\(338\) 0 0
\(339\) 342.087i 1.00911i
\(340\) 0 0
\(341\) 7.67183i 0.0224980i
\(342\) 0 0
\(343\) 369.566i 1.07745i
\(344\) 0 0
\(345\) 51.2846 72.8347i 0.148651 0.211115i
\(346\) 0 0
\(347\) 156.348 0.450572 0.225286 0.974293i \(-0.427668\pi\)
0.225286 + 0.974293i \(0.427668\pi\)
\(348\) 0 0
\(349\) 231.064 0.662074 0.331037 0.943618i \(-0.392602\pi\)
0.331037 + 0.943618i \(0.392602\pi\)
\(350\) 0 0
\(351\) 124.012 0.353310
\(352\) 0 0
\(353\) 396.420 1.12300 0.561501 0.827476i \(-0.310224\pi\)
0.561501 + 0.827476i \(0.310224\pi\)
\(354\) 0 0
\(355\) 170.686i 0.480804i
\(356\) 0 0
\(357\) −10.1300 −0.0283754
\(358\) 0 0
\(359\) 302.034i 0.841320i −0.907218 0.420660i \(-0.861799\pi\)
0.907218 0.420660i \(-0.138201\pi\)
\(360\) 0 0
\(361\) −27.2540 −0.0754957
\(362\) 0 0
\(363\) −209.508 −0.577158
\(364\) 0 0
\(365\) 173.626i 0.475688i
\(366\) 0 0
\(367\) 402.123i 1.09570i −0.836576 0.547851i \(-0.815446\pi\)
0.836576 0.547851i \(-0.184554\pi\)
\(368\) 0 0
\(369\) 80.9698 0.219430
\(370\) 0 0
\(371\) −409.475 −1.10371
\(372\) 0 0
\(373\) 203.213i 0.544806i −0.962183 0.272403i \(-0.912182\pi\)
0.962183 0.272403i \(-0.0878185\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −57.1807 −0.151673
\(378\) 0 0
\(379\) 223.194i 0.588903i −0.955666 0.294451i \(-0.904863\pi\)
0.955666 0.294451i \(-0.0951369\pi\)
\(380\) 0 0
\(381\) −219.589 −0.576350
\(382\) 0 0
\(383\) 154.120i 0.402403i 0.979550 + 0.201202i \(0.0644847\pi\)
−0.979550 + 0.201202i \(0.935515\pi\)
\(384\) 0 0
\(385\) 2.35025i 0.00610454i
\(386\) 0 0
\(387\) 106.082i 0.274113i
\(388\) 0 0
\(389\) 344.101i 0.884578i 0.896873 + 0.442289i \(0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(390\) 0 0
\(391\) −14.7933 + 21.0095i −0.0378344 + 0.0537327i
\(392\) 0 0
\(393\) 14.9616 0.0380702
\(394\) 0 0
\(395\) −58.0405 −0.146938
\(396\) 0 0
\(397\) 62.7088 0.157957 0.0789783 0.996876i \(-0.474834\pi\)
0.0789783 + 0.996876i \(0.474834\pi\)
\(398\) 0 0
\(399\) −178.667 −0.447788
\(400\) 0 0
\(401\) 151.037i 0.376650i 0.982107 + 0.188325i \(0.0603058\pi\)
−0.982107 + 0.188325i \(0.939694\pi\)
\(402\) 0 0
\(403\) −911.964 −2.26294
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −7.64762 −0.0187902
\(408\) 0 0
\(409\) −173.909 −0.425206 −0.212603 0.977139i \(-0.568194\pi\)
−0.212603 + 0.977139i \(0.568194\pi\)
\(410\) 0 0
\(411\) 134.923i 0.328280i
\(412\) 0 0
\(413\) 334.557i 0.810066i
\(414\) 0 0
\(415\) −41.7456 −0.100592
\(416\) 0 0
\(417\) 0.548816 0.00131611
\(418\) 0 0
\(419\) 182.183i 0.434805i −0.976082 0.217402i \(-0.930242\pi\)
0.976082 0.217402i \(-0.0697584\pi\)
\(420\) 0 0
\(421\) 365.856i 0.869017i 0.900668 + 0.434508i \(0.143078\pi\)
−0.900668 + 0.434508i \(0.856922\pi\)
\(422\) 0 0
\(423\) −29.4186 −0.0695476
\(424\) 0 0
\(425\) 5.58589i 0.0131433i
\(426\) 0 0
\(427\) 1.32620 0.00310585
\(428\) 0 0
\(429\) 8.29933i 0.0193458i
\(430\) 0 0
\(431\) 114.799i 0.266355i 0.991092 + 0.133177i \(0.0425180\pi\)
−0.991092 + 0.133177i \(0.957482\pi\)
\(432\) 0 0
\(433\) 268.115i 0.619202i 0.950867 + 0.309601i \(0.100196\pi\)
−0.950867 + 0.309601i \(0.899804\pi\)
\(434\) 0 0
\(435\) 9.27928i 0.0213317i
\(436\) 0 0
\(437\) −260.915 + 370.553i −0.597060 + 0.847948i
\(438\) 0 0
\(439\) −44.3726 −0.101077 −0.0505383 0.998722i \(-0.516094\pi\)
−0.0505383 + 0.998722i \(0.516094\pi\)
\(440\) 0 0
\(441\) 64.7805 0.146895
\(442\) 0 0
\(443\) 630.841 1.42402 0.712010 0.702169i \(-0.247785\pi\)
0.712010 + 0.702169i \(0.247785\pi\)
\(444\) 0 0
\(445\) 64.0187 0.143862
\(446\) 0 0
\(447\) 439.928i 0.984178i
\(448\) 0 0
\(449\) −67.6896 −0.150756 −0.0753781 0.997155i \(-0.524016\pi\)
−0.0753781 + 0.997155i \(0.524016\pi\)
\(450\) 0 0
\(451\) 5.41881i 0.0120151i
\(452\) 0 0
\(453\) −184.422 −0.407113
\(454\) 0 0
\(455\) −279.378 −0.614018
\(456\) 0 0
\(457\) 379.024i 0.829373i 0.909964 + 0.414687i \(0.136109\pi\)
−0.909964 + 0.414687i \(0.863891\pi\)
\(458\) 0 0
\(459\) 5.80503i 0.0126471i
\(460\) 0 0
\(461\) 58.6697 0.127266 0.0636330 0.997973i \(-0.479731\pi\)
0.0636330 + 0.997973i \(0.479731\pi\)
\(462\) 0 0
\(463\) −422.962 −0.913525 −0.456762 0.889589i \(-0.650991\pi\)
−0.456762 + 0.889589i \(0.650991\pi\)
\(464\) 0 0
\(465\) 147.993i 0.318266i
\(466\) 0 0
\(467\) 600.100i 1.28501i −0.766281 0.642506i \(-0.777895\pi\)
0.766281 0.642506i \(-0.222105\pi\)
\(468\) 0 0
\(469\) 93.4148 0.199179
\(470\) 0 0
\(471\) 94.8725i 0.201428i
\(472\) 0 0
\(473\) 7.09939 0.0150093
\(474\) 0 0
\(475\) 98.5208i 0.207412i
\(476\) 0 0
\(477\) 234.650i 0.491930i
\(478\) 0 0
\(479\) 491.467i 1.02603i 0.858381 + 0.513013i \(0.171471\pi\)
−0.858381 + 0.513013i \(0.828529\pi\)
\(480\) 0 0
\(481\) 909.086i 1.88999i
\(482\) 0 0
\(483\) −120.068 + 170.522i −0.248589 + 0.353047i
\(484\) 0 0
\(485\) −105.416 −0.217353
\(486\) 0 0
\(487\) 409.006 0.839849 0.419925 0.907559i \(-0.362057\pi\)
0.419925 + 0.907559i \(0.362057\pi\)
\(488\) 0 0
\(489\) 376.827 0.770607
\(490\) 0 0
\(491\) −109.571 −0.223159 −0.111580 0.993756i \(-0.535591\pi\)
−0.111580 + 0.993756i \(0.535591\pi\)
\(492\) 0 0
\(493\) 2.67665i 0.00542931i
\(494\) 0 0
\(495\) −1.34682 −0.00272084
\(496\) 0 0
\(497\) 399.612i 0.804048i
\(498\) 0 0
\(499\) 185.334 0.371412 0.185706 0.982605i \(-0.440543\pi\)
0.185706 + 0.982605i \(0.440543\pi\)
\(500\) 0 0
\(501\) 265.019 0.528981
\(502\) 0 0
\(503\) 582.863i 1.15877i −0.815053 0.579387i \(-0.803292\pi\)
0.815053 0.579387i \(-0.196708\pi\)
\(504\) 0 0
\(505\) 156.133i 0.309174i
\(506\) 0 0
\(507\) −693.840 −1.36852
\(508\) 0 0
\(509\) 148.389 0.291530 0.145765 0.989319i \(-0.453436\pi\)
0.145765 + 0.989319i \(0.453436\pi\)
\(510\) 0 0
\(511\) 406.496i 0.795492i
\(512\) 0 0
\(513\) 102.386i 0.199583i
\(514\) 0 0
\(515\) 112.914 0.219251
\(516\) 0 0
\(517\) 1.96881i 0.00380814i
\(518\) 0 0
\(519\) −182.115 −0.350895
\(520\) 0 0
\(521\) 862.617i 1.65569i −0.560954 0.827847i \(-0.689565\pi\)
0.560954 0.827847i \(-0.310435\pi\)
\(522\) 0 0
\(523\) 874.917i 1.67288i 0.548057 + 0.836441i \(0.315368\pi\)
−0.548057 + 0.836441i \(0.684632\pi\)
\(524\) 0 0
\(525\) 45.3375i 0.0863571i
\(526\) 0 0
\(527\) 42.6893i 0.0810044i
\(528\) 0 0
\(529\) 178.319 + 498.040i 0.337086 + 0.941474i
\(530\) 0 0
\(531\) −191.719 −0.361053
\(532\) 0 0
\(533\) −644.143 −1.20852
\(534\) 0 0
\(535\) −267.650 −0.500280
\(536\) 0 0
\(537\) 188.986 0.351928
\(538\) 0 0
\(539\) 4.33536i 0.00804333i
\(540\) 0 0
\(541\) 528.756 0.977368 0.488684 0.872461i \(-0.337477\pi\)
0.488684 + 0.872461i \(0.337477\pi\)
\(542\) 0 0
\(543\) 490.673i 0.903634i
\(544\) 0 0
\(545\) 306.843 0.563014
\(546\) 0 0
\(547\) 746.212 1.36419 0.682095 0.731264i \(-0.261069\pi\)
0.682095 + 0.731264i \(0.261069\pi\)
\(548\) 0 0
\(549\) 0.759980i 0.00138430i
\(550\) 0 0
\(551\) 47.2092i 0.0856792i
\(552\) 0 0
\(553\) 135.885 0.245724
\(554\) 0 0
\(555\) 147.526 0.265813
\(556\) 0 0
\(557\) 225.174i 0.404262i −0.979358 0.202131i \(-0.935213\pi\)
0.979358 0.202131i \(-0.0647867\pi\)
\(558\) 0 0
\(559\) 843.917i 1.50969i
\(560\) 0 0
\(561\) 0.388495 0.000692504
\(562\) 0 0
\(563\) 274.306i 0.487222i 0.969873 + 0.243611i \(0.0783320\pi\)
−0.969873 + 0.243611i \(0.921668\pi\)
\(564\) 0 0
\(565\) 441.633 0.781651
\(566\) 0 0
\(567\) 47.1161i 0.0830972i
\(568\) 0 0
\(569\) 100.058i 0.175848i −0.996127 0.0879241i \(-0.971977\pi\)
0.996127 0.0879241i \(-0.0280233\pi\)
\(570\) 0 0
\(571\) 951.356i 1.66612i 0.553180 + 0.833061i \(0.313414\pi\)
−0.553180 + 0.833061i \(0.686586\pi\)
\(572\) 0 0
\(573\) 115.736i 0.201983i
\(574\) 0 0
\(575\) 94.0292 + 66.2081i 0.163529 + 0.115145i
\(576\) 0 0
\(577\) −585.207 −1.01422 −0.507112 0.861880i \(-0.669287\pi\)
−0.507112 + 0.861880i \(0.669287\pi\)
\(578\) 0 0
\(579\) −310.727 −0.536662
\(580\) 0 0
\(581\) 97.7355 0.168219
\(582\) 0 0
\(583\) 15.7037 0.0269360
\(584\) 0 0
\(585\) 160.098i 0.273672i
\(586\) 0 0
\(587\) 555.851 0.946936 0.473468 0.880811i \(-0.343002\pi\)
0.473468 + 0.880811i \(0.343002\pi\)
\(588\) 0 0
\(589\) 752.930i 1.27832i
\(590\) 0 0
\(591\) −509.349 −0.861843
\(592\) 0 0
\(593\) −995.087 −1.67806 −0.839028 0.544089i \(-0.816875\pi\)
−0.839028 + 0.544089i \(0.816875\pi\)
\(594\) 0 0
\(595\) 13.0778i 0.0219795i
\(596\) 0 0
\(597\) 184.856i 0.309642i
\(598\) 0 0
\(599\) 249.063 0.415797 0.207899 0.978150i \(-0.433338\pi\)
0.207899 + 0.978150i \(0.433338\pi\)
\(600\) 0 0
\(601\) 678.462 1.12889 0.564445 0.825471i \(-0.309090\pi\)
0.564445 + 0.825471i \(0.309090\pi\)
\(602\) 0 0
\(603\) 53.5316i 0.0887754i
\(604\) 0 0
\(605\) 270.474i 0.447065i
\(606\) 0 0
\(607\) 484.630 0.798403 0.399201 0.916863i \(-0.369287\pi\)
0.399201 + 0.916863i \(0.369287\pi\)
\(608\) 0 0
\(609\) 21.7248i 0.0356729i
\(610\) 0 0
\(611\) 234.036 0.383037
\(612\) 0 0
\(613\) 56.3353i 0.0919009i 0.998944 + 0.0459505i \(0.0146316\pi\)
−0.998944 + 0.0459505i \(0.985368\pi\)
\(614\) 0 0
\(615\) 104.532i 0.169970i
\(616\) 0 0
\(617\) 259.103i 0.419940i −0.977708 0.209970i \(-0.932663\pi\)
0.977708 0.209970i \(-0.0673367\pi\)
\(618\) 0 0
\(619\) 1067.70i 1.72488i −0.506163 0.862438i \(-0.668937\pi\)
0.506163 0.862438i \(-0.331063\pi\)
\(620\) 0 0
\(621\) 97.7180 + 68.8055i 0.157356 + 0.110798i
\(622\) 0 0
\(623\) −149.882 −0.240581
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 6.85205 0.0109283
\(628\) 0 0
\(629\) −42.5546 −0.0676544
\(630\) 0 0
\(631\) 999.923i 1.58466i −0.610091 0.792332i \(-0.708867\pi\)
0.610091 0.792332i \(-0.291133\pi\)
\(632\) 0 0
\(633\) −151.745 −0.239724
\(634\) 0 0
\(635\) 283.488i 0.446438i
\(636\) 0 0
\(637\) −515.352 −0.809029
\(638\) 0 0
\(639\) 228.999 0.358370
\(640\) 0 0
\(641\) 568.784i 0.887338i 0.896191 + 0.443669i \(0.146323\pi\)
−0.896191 + 0.443669i \(0.853677\pi\)
\(642\) 0 0
\(643\) 748.876i 1.16466i 0.812953 + 0.582330i \(0.197859\pi\)
−0.812953 + 0.582330i \(0.802141\pi\)
\(644\) 0 0
\(645\) −136.951 −0.212327
\(646\) 0 0
\(647\) −67.0944 −0.103701 −0.0518504 0.998655i \(-0.516512\pi\)
−0.0518504 + 0.998655i \(0.516512\pi\)
\(648\) 0 0
\(649\) 12.8306i 0.0197697i
\(650\) 0 0
\(651\) 346.485i 0.532235i
\(652\) 0 0
\(653\) −1250.62 −1.91519 −0.957593 0.288125i \(-0.906968\pi\)
−0.957593 + 0.288125i \(0.906968\pi\)
\(654\) 0 0
\(655\) 19.3153i 0.0294890i
\(656\) 0 0
\(657\) −232.944 −0.354557
\(658\) 0 0
\(659\) 1297.90i 1.96950i −0.173980 0.984749i \(-0.555663\pi\)
0.173980 0.984749i \(-0.444337\pi\)
\(660\) 0 0
\(661\) 939.253i 1.42096i −0.703719 0.710479i \(-0.748478\pi\)
0.703719 0.710479i \(-0.251522\pi\)
\(662\) 0 0
\(663\) 46.1810i 0.0696547i
\(664\) 0 0
\(665\) 230.659i 0.346855i
\(666\) 0 0
\(667\) −45.0569 31.7256i −0.0675516 0.0475647i
\(668\) 0 0
\(669\) −393.103 −0.587597
\(670\) 0 0
\(671\) −0.0508608 −7.57985e−5
\(672\) 0 0
\(673\) 772.694 1.14813 0.574067 0.818808i \(-0.305365\pi\)
0.574067 + 0.818808i \(0.305365\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 766.136i 1.13166i 0.824521 + 0.565832i \(0.191445\pi\)
−0.824521 + 0.565832i \(0.808555\pi\)
\(678\) 0 0
\(679\) 246.803 0.363479
\(680\) 0 0
\(681\) 467.360i 0.686285i
\(682\) 0 0
\(683\) −1010.44 −1.47942 −0.739709 0.672926i \(-0.765037\pi\)
−0.739709 + 0.672926i \(0.765037\pi\)
\(684\) 0 0
\(685\) −174.185 −0.254285
\(686\) 0 0
\(687\) 335.031i 0.487672i
\(688\) 0 0
\(689\) 1866.73i 2.70933i
\(690\) 0 0
\(691\) −53.5977 −0.0775654 −0.0387827 0.999248i \(-0.512348\pi\)
−0.0387827 + 0.999248i \(0.512348\pi\)
\(692\) 0 0
\(693\) 3.15319 0.00455006
\(694\) 0 0
\(695\) 0.708519i 0.00101945i
\(696\) 0 0
\(697\) 30.1526i 0.0432605i
\(698\) 0 0
\(699\) 48.3398 0.0691557
\(700\) 0 0
\(701\) 943.325i 1.34568i −0.739786 0.672842i \(-0.765073\pi\)
0.739786 0.672842i \(-0.234927\pi\)
\(702\) 0 0
\(703\) −750.554 −1.06765
\(704\) 0 0
\(705\) 37.9793i 0.0538713i
\(706\) 0 0
\(707\) 365.541i 0.517031i
\(708\) 0 0
\(709\) 304.039i 0.428828i 0.976743 + 0.214414i \(0.0687842\pi\)
−0.976743 + 0.214414i \(0.931216\pi\)
\(710\) 0 0
\(711\) 77.8695i 0.109521i
\(712\) 0 0
\(713\) −718.604 505.986i −1.00786 0.709657i
\(714\) 0 0
\(715\) 10.7144 0.0149852
\(716\) 0 0
\(717\) 352.710 0.491925
\(718\) 0 0
\(719\) 89.2663 0.124153 0.0620767 0.998071i \(-0.480228\pi\)
0.0620767 + 0.998071i \(0.480228\pi\)
\(720\) 0 0
\(721\) −264.357 −0.366653
\(722\) 0 0
\(723\) 807.440i 1.11679i
\(724\) 0 0
\(725\) −11.9795 −0.0165235
\(726\) 0 0
\(727\) 172.629i 0.237454i 0.992927 + 0.118727i \(0.0378813\pi\)
−0.992927 + 0.118727i \(0.962119\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 39.5040 0.0540411
\(732\) 0 0
\(733\) 382.885i 0.522353i 0.965291 + 0.261176i \(0.0841104\pi\)
−0.965291 + 0.261176i \(0.915890\pi\)
\(734\) 0 0
\(735\) 83.6312i 0.113784i
\(736\) 0 0
\(737\) −3.58254 −0.00486097
\(738\) 0 0
\(739\) 1099.12 1.48730 0.743652 0.668567i \(-0.233092\pi\)
0.743652 + 0.668567i \(0.233092\pi\)
\(740\) 0 0
\(741\) 814.515i 1.09921i
\(742\) 0 0
\(743\) 647.782i 0.871846i −0.899984 0.435923i \(-0.856422\pi\)
0.899984 0.435923i \(-0.143578\pi\)
\(744\) 0 0
\(745\) 567.944 0.762341
\(746\) 0 0
\(747\) 56.0076i 0.0749767i
\(748\) 0 0
\(749\) 626.626 0.836617
\(750\) 0 0
\(751\) 848.139i 1.12935i 0.825315 + 0.564673i \(0.190998\pi\)
−0.825315 + 0.564673i \(0.809002\pi\)
\(752\) 0 0
\(753\) 290.712i 0.386072i
\(754\) 0 0
\(755\) 238.088i 0.315348i
\(756\) 0 0
\(757\) 1499.22i 1.98048i 0.139388 + 0.990238i \(0.455487\pi\)
−0.139388 + 0.990238i \(0.544513\pi\)
\(758\) 0 0
\(759\) 4.60473 6.53966i 0.00606683 0.00861615i
\(760\) 0 0
\(761\) −625.715 −0.822228 −0.411114 0.911584i \(-0.634860\pi\)
−0.411114 + 0.911584i \(0.634860\pi\)
\(762\) 0 0
\(763\) −718.386 −0.941528
\(764\) 0 0
\(765\) −7.49426 −0.00979642
\(766\) 0 0
\(767\) 1525.19 1.98852
\(768\) 0 0
\(769\) 10.9625i 0.0142555i −0.999975 0.00712774i \(-0.997731\pi\)
0.999975 0.00712774i \(-0.00226885\pi\)
\(770\) 0 0
\(771\) −143.666 −0.186337
\(772\) 0 0
\(773\) 948.455i 1.22698i 0.789703 + 0.613490i \(0.210235\pi\)
−0.789703 + 0.613490i \(0.789765\pi\)
\(774\) 0 0
\(775\) −191.059 −0.246527
\(776\) 0 0
\(777\) −345.392 −0.444519
\(778\) 0 0
\(779\) 531.814i 0.682688i
\(780\) 0 0
\(781\) 15.3255i 0.0196229i
\(782\) 0 0
\(783\) −12.4495 −0.0158997
\(784\) 0 0
\(785\) 122.480 0.156025
\(786\) 0 0
\(787\) 846.711i 1.07587i −0.842986 0.537936i \(-0.819204\pi\)
0.842986 0.537936i \(-0.180796\pi\)
\(788\) 0 0
\(789\) 400.303i 0.507355i
\(790\) 0 0
\(791\) −1033.96 −1.30715
\(792\) 0 0
\(793\) 6.04591i 0.00762410i
\(794\) 0 0
\(795\) −302.932 −0.381047
\(796\) 0 0
\(797\) 1107.68i 1.38981i −0.719102 0.694905i \(-0.755447\pi\)
0.719102 0.694905i \(-0.244553\pi\)
\(798\) 0 0