Properties

Label 2760.3.g.a.2161.13
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 2161.13
Character \(\chi\) \(=\) 2760.2161
Dual form 2760.3.g.a.2161.36

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} -2.23607i q^{5} +1.24234i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} -2.23607i q^{5} +1.24234i q^{7} +3.00000 q^{9} -6.29825i q^{11} +15.7139 q^{13} +3.87298i q^{15} +15.4592i q^{17} +32.9662i q^{19} -2.15179i q^{21} +(22.9031 - 2.10926i) q^{23} -5.00000 q^{25} -5.19615 q^{27} -35.4880 q^{29} -21.7015 q^{31} +10.9089i q^{33} +2.77795 q^{35} -15.1116i q^{37} -27.2172 q^{39} -29.1562 q^{41} -0.263675i q^{43} -6.70820i q^{45} -51.7458 q^{47} +47.4566 q^{49} -26.7761i q^{51} -15.7810i q^{53} -14.0833 q^{55} -57.0992i q^{57} +47.2802 q^{59} -79.4521i q^{61} +3.72701i q^{63} -35.1373i q^{65} +100.871i q^{67} +(-39.6693 + 3.65335i) q^{69} -70.2614 q^{71} +24.4214 q^{73} +8.66025 q^{75} +7.82453 q^{77} +27.0556i q^{79} +9.00000 q^{81} -40.8969i q^{83} +34.5678 q^{85} +61.4671 q^{87} +42.0337i q^{89} +19.5219i q^{91} +37.5881 q^{93} +73.7147 q^{95} -66.6484i q^{97} -18.8947i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 288 q^{9}+O(q^{10}) \) 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
\(7\) 1.24234i 0.177476i 0.996055 + 0.0887382i \(0.0282835\pi\)
−0.996055 + 0.0887382i \(0.971717\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 6.29825i 0.572568i −0.958145 0.286284i \(-0.907580\pi\)
0.958145 0.286284i \(-0.0924200\pi\)
\(12\) 0 0
\(13\) 15.7139 1.20876 0.604380 0.796697i \(-0.293421\pi\)
0.604380 + 0.796697i \(0.293421\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) 15.4592i 0.909364i 0.890654 + 0.454682i \(0.150247\pi\)
−0.890654 + 0.454682i \(0.849753\pi\)
\(18\) 0 0
\(19\) 32.9662i 1.73506i 0.497381 + 0.867532i \(0.334295\pi\)
−0.497381 + 0.867532i \(0.665705\pi\)
\(20\) 0 0
\(21\) 2.15179i 0.102466i
\(22\) 0 0
\(23\) 22.9031 2.10926i 0.995786 0.0917071i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) −35.4880 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(30\) 0 0
\(31\) −21.7015 −0.700048 −0.350024 0.936741i \(-0.613827\pi\)
−0.350024 + 0.936741i \(0.613827\pi\)
\(32\) 0 0
\(33\) 10.9089i 0.330572i
\(34\) 0 0
\(35\) 2.77795 0.0793699
\(36\) 0 0
\(37\) 15.1116i 0.408421i −0.978927 0.204211i \(-0.934537\pi\)
0.978927 0.204211i \(-0.0654627\pi\)
\(38\) 0 0
\(39\) −27.2172 −0.697877
\(40\) 0 0
\(41\) −29.1562 −0.711128 −0.355564 0.934652i \(-0.615711\pi\)
−0.355564 + 0.934652i \(0.615711\pi\)
\(42\) 0 0
\(43\) 0.263675i 0.00613198i −0.999995 0.00306599i \(-0.999024\pi\)
0.999995 0.00306599i \(-0.000975936\pi\)
\(44\) 0 0
\(45\) 6.70820i 0.149071i
\(46\) 0 0
\(47\) −51.7458 −1.10097 −0.550487 0.834844i \(-0.685558\pi\)
−0.550487 + 0.834844i \(0.685558\pi\)
\(48\) 0 0
\(49\) 47.4566 0.968502
\(50\) 0 0
\(51\) 26.7761i 0.525022i
\(52\) 0 0
\(53\) 15.7810i 0.297755i −0.988856 0.148877i \(-0.952434\pi\)
0.988856 0.148877i \(-0.0475660\pi\)
\(54\) 0 0
\(55\) −14.0833 −0.256060
\(56\) 0 0
\(57\) 57.0992i 1.00174i
\(58\) 0 0
\(59\) 47.2802 0.801360 0.400680 0.916218i \(-0.368774\pi\)
0.400680 + 0.916218i \(0.368774\pi\)
\(60\) 0 0
\(61\) 79.4521i 1.30249i −0.758866 0.651247i \(-0.774246\pi\)
0.758866 0.651247i \(-0.225754\pi\)
\(62\) 0 0
\(63\) 3.72701i 0.0591588i
\(64\) 0 0
\(65\) 35.1373i 0.540573i
\(66\) 0 0
\(67\) 100.871i 1.50553i 0.658287 + 0.752767i \(0.271281\pi\)
−0.658287 + 0.752767i \(0.728719\pi\)
\(68\) 0 0
\(69\) −39.6693 + 3.65335i −0.574917 + 0.0529471i
\(70\) 0 0
\(71\) −70.2614 −0.989598 −0.494799 0.869008i \(-0.664758\pi\)
−0.494799 + 0.869008i \(0.664758\pi\)
\(72\) 0 0
\(73\) 24.4214 0.334540 0.167270 0.985911i \(-0.446505\pi\)
0.167270 + 0.985911i \(0.446505\pi\)
\(74\) 0 0
\(75\) 8.66025 0.115470
\(76\) 0 0
\(77\) 7.82453 0.101617
\(78\) 0 0
\(79\) 27.0556i 0.342476i 0.985230 + 0.171238i \(0.0547768\pi\)
−0.985230 + 0.171238i \(0.945223\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 40.8969i 0.492734i −0.969177 0.246367i \(-0.920763\pi\)
0.969177 0.246367i \(-0.0792368\pi\)
\(84\) 0 0
\(85\) 34.5678 0.406680
\(86\) 0 0
\(87\) 61.4671 0.706518
\(88\) 0 0
\(89\) 42.0337i 0.472289i 0.971718 + 0.236144i \(0.0758839\pi\)
−0.971718 + 0.236144i \(0.924116\pi\)
\(90\) 0 0
\(91\) 19.5219i 0.214526i
\(92\) 0 0
\(93\) 37.5881 0.404173
\(94\) 0 0
\(95\) 73.7147 0.775945
\(96\) 0 0
\(97\) 66.6484i 0.687096i −0.939135 0.343548i \(-0.888371\pi\)
0.939135 0.343548i \(-0.111629\pi\)
\(98\) 0 0
\(99\) 18.8947i 0.190856i
\(100\) 0 0
\(101\) −74.6392 −0.739002 −0.369501 0.929230i \(-0.620471\pi\)
−0.369501 + 0.929230i \(0.620471\pi\)
\(102\) 0 0
\(103\) 157.054i 1.52480i 0.647106 + 0.762400i \(0.275979\pi\)
−0.647106 + 0.762400i \(0.724021\pi\)
\(104\) 0 0
\(105\) −4.81154 −0.0458242
\(106\) 0 0
\(107\) 69.3463i 0.648097i 0.946041 + 0.324048i \(0.105044\pi\)
−0.946041 + 0.324048i \(0.894956\pi\)
\(108\) 0 0
\(109\) 145.541i 1.33524i 0.744503 + 0.667619i \(0.232687\pi\)
−0.744503 + 0.667619i \(0.767313\pi\)
\(110\) 0 0
\(111\) 26.1740i 0.235802i
\(112\) 0 0
\(113\) 8.43652i 0.0746595i 0.999303 + 0.0373297i \(0.0118852\pi\)
−0.999303 + 0.0373297i \(0.988115\pi\)
\(114\) 0 0
\(115\) −4.71646 51.2128i −0.0410127 0.445329i
\(116\) 0 0
\(117\) 47.1416 0.402920
\(118\) 0 0
\(119\) −19.2055 −0.161391
\(120\) 0 0
\(121\) 81.3321 0.672166
\(122\) 0 0
\(123\) 50.5001 0.410570
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −24.3760 −0.191937 −0.0959685 0.995384i \(-0.530595\pi\)
−0.0959685 + 0.995384i \(0.530595\pi\)
\(128\) 0 0
\(129\) 0.456698i 0.00354030i
\(130\) 0 0
\(131\) 164.326 1.25440 0.627200 0.778858i \(-0.284201\pi\)
0.627200 + 0.778858i \(0.284201\pi\)
\(132\) 0 0
\(133\) −40.9551 −0.307933
\(134\) 0 0
\(135\) 11.6190i 0.0860663i
\(136\) 0 0
\(137\) 176.412i 1.28768i −0.765161 0.643840i \(-0.777340\pi\)
0.765161 0.643840i \(-0.222660\pi\)
\(138\) 0 0
\(139\) −21.3718 −0.153754 −0.0768769 0.997041i \(-0.524495\pi\)
−0.0768769 + 0.997041i \(0.524495\pi\)
\(140\) 0 0
\(141\) 89.6263 0.635647
\(142\) 0 0
\(143\) 98.9698i 0.692096i
\(144\) 0 0
\(145\) 79.3537i 0.547267i
\(146\) 0 0
\(147\) −82.1972 −0.559165
\(148\) 0 0
\(149\) 196.324i 1.31761i 0.752313 + 0.658806i \(0.228938\pi\)
−0.752313 + 0.658806i \(0.771062\pi\)
\(150\) 0 0
\(151\) −41.0892 −0.272114 −0.136057 0.990701i \(-0.543443\pi\)
−0.136057 + 0.990701i \(0.543443\pi\)
\(152\) 0 0
\(153\) 46.3776i 0.303121i
\(154\) 0 0
\(155\) 48.5260i 0.313071i
\(156\) 0 0
\(157\) 157.617i 1.00393i 0.864887 + 0.501966i \(0.167390\pi\)
−0.864887 + 0.501966i \(0.832610\pi\)
\(158\) 0 0
\(159\) 27.3335i 0.171909i
\(160\) 0 0
\(161\) 2.62041 + 28.4533i 0.0162759 + 0.176729i
\(162\) 0 0
\(163\) −312.849 −1.91932 −0.959658 0.281170i \(-0.909278\pi\)
−0.959658 + 0.281170i \(0.909278\pi\)
\(164\) 0 0
\(165\) 24.3930 0.147836
\(166\) 0 0
\(167\) −328.097 −1.96465 −0.982327 0.187173i \(-0.940068\pi\)
−0.982327 + 0.187173i \(0.940068\pi\)
\(168\) 0 0
\(169\) 77.9256 0.461098
\(170\) 0 0
\(171\) 98.8987i 0.578355i
\(172\) 0 0
\(173\) −68.1794 −0.394101 −0.197050 0.980393i \(-0.563136\pi\)
−0.197050 + 0.980393i \(0.563136\pi\)
\(174\) 0 0
\(175\) 6.21168i 0.0354953i
\(176\) 0 0
\(177\) −81.8917 −0.462665
\(178\) 0 0
\(179\) −17.7636 −0.0992377 −0.0496189 0.998768i \(-0.515801\pi\)
−0.0496189 + 0.998768i \(0.515801\pi\)
\(180\) 0 0
\(181\) 1.93866i 0.0107108i −0.999986 0.00535542i \(-0.998295\pi\)
0.999986 0.00535542i \(-0.00170469\pi\)
\(182\) 0 0
\(183\) 137.615i 0.751995i
\(184\) 0 0
\(185\) −33.7905 −0.182651
\(186\) 0 0
\(187\) 97.3658 0.520673
\(188\) 0 0
\(189\) 6.45536i 0.0341554i
\(190\) 0 0
\(191\) 299.790i 1.56958i 0.619761 + 0.784790i \(0.287229\pi\)
−0.619761 + 0.784790i \(0.712771\pi\)
\(192\) 0 0
\(193\) −186.357 −0.965580 −0.482790 0.875736i \(-0.660377\pi\)
−0.482790 + 0.875736i \(0.660377\pi\)
\(194\) 0 0
\(195\) 60.8595i 0.312100i
\(196\) 0 0
\(197\) 14.3099 0.0726392 0.0363196 0.999340i \(-0.488437\pi\)
0.0363196 + 0.999340i \(0.488437\pi\)
\(198\) 0 0
\(199\) 80.9853i 0.406961i 0.979079 + 0.203481i \(0.0652254\pi\)
−0.979079 + 0.203481i \(0.934775\pi\)
\(200\) 0 0
\(201\) 174.713i 0.869221i
\(202\) 0 0
\(203\) 44.0881i 0.217183i
\(204\) 0 0
\(205\) 65.1954i 0.318026i
\(206\) 0 0
\(207\) 68.7092 6.32779i 0.331929 0.0305690i
\(208\) 0 0
\(209\) 207.629 0.993442
\(210\) 0 0
\(211\) 207.667 0.984202 0.492101 0.870538i \(-0.336229\pi\)
0.492101 + 0.870538i \(0.336229\pi\)
\(212\) 0 0
\(213\) 121.696 0.571344
\(214\) 0 0
\(215\) −0.589595 −0.00274230
\(216\) 0 0
\(217\) 26.9605i 0.124242i
\(218\) 0 0
\(219\) −42.2992 −0.193147
\(220\) 0 0
\(221\) 242.924i 1.09920i
\(222\) 0 0
\(223\) 101.350 0.454485 0.227243 0.973838i \(-0.427029\pi\)
0.227243 + 0.973838i \(0.427029\pi\)
\(224\) 0 0
\(225\) −15.0000 −0.0666667
\(226\) 0 0
\(227\) 185.415i 0.816807i 0.912801 + 0.408404i \(0.133914\pi\)
−0.912801 + 0.408404i \(0.866086\pi\)
\(228\) 0 0
\(229\) 262.115i 1.14460i −0.820043 0.572302i \(-0.806050\pi\)
0.820043 0.572302i \(-0.193950\pi\)
\(230\) 0 0
\(231\) −13.5525 −0.0586688
\(232\) 0 0
\(233\) −332.521 −1.42713 −0.713564 0.700590i \(-0.752920\pi\)
−0.713564 + 0.700590i \(0.752920\pi\)
\(234\) 0 0
\(235\) 115.707i 0.492370i
\(236\) 0 0
\(237\) 46.8617i 0.197729i
\(238\) 0 0
\(239\) −93.9530 −0.393109 −0.196554 0.980493i \(-0.562975\pi\)
−0.196554 + 0.980493i \(0.562975\pi\)
\(240\) 0 0
\(241\) 211.201i 0.876355i 0.898889 + 0.438177i \(0.144376\pi\)
−0.898889 + 0.438177i \(0.855624\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 106.116i 0.433127i
\(246\) 0 0
\(247\) 518.027i 2.09728i
\(248\) 0 0
\(249\) 70.8355i 0.284480i
\(250\) 0 0
\(251\) 451.452i 1.79861i 0.437319 + 0.899306i \(0.355928\pi\)
−0.437319 + 0.899306i \(0.644072\pi\)
\(252\) 0 0
\(253\) −13.2847 144.249i −0.0525086 0.570155i
\(254\) 0 0
\(255\) −59.8732 −0.234797
\(256\) 0 0
\(257\) 433.504 1.68679 0.843393 0.537297i \(-0.180554\pi\)
0.843393 + 0.537297i \(0.180554\pi\)
\(258\) 0 0
\(259\) 18.7736 0.0724851
\(260\) 0 0
\(261\) −106.464 −0.407909
\(262\) 0 0
\(263\) 82.8288i 0.314939i 0.987524 + 0.157469i \(0.0503335\pi\)
−0.987524 + 0.157469i \(0.949666\pi\)
\(264\) 0 0
\(265\) −35.2874 −0.133160
\(266\) 0 0
\(267\) 72.8045i 0.272676i
\(268\) 0 0
\(269\) 240.724 0.894886 0.447443 0.894312i \(-0.352335\pi\)
0.447443 + 0.894312i \(0.352335\pi\)
\(270\) 0 0
\(271\) 419.383 1.54754 0.773769 0.633468i \(-0.218369\pi\)
0.773769 + 0.633468i \(0.218369\pi\)
\(272\) 0 0
\(273\) 33.8129i 0.123857i
\(274\) 0 0
\(275\) 31.4912i 0.114514i
\(276\) 0 0
\(277\) −45.1007 −0.162818 −0.0814092 0.996681i \(-0.525942\pi\)
−0.0814092 + 0.996681i \(0.525942\pi\)
\(278\) 0 0
\(279\) −65.1044 −0.233349
\(280\) 0 0
\(281\) 102.013i 0.363035i −0.983388 0.181518i \(-0.941899\pi\)
0.983388 0.181518i \(-0.0581010\pi\)
\(282\) 0 0
\(283\) 406.507i 1.43642i 0.695826 + 0.718210i \(0.255038\pi\)
−0.695826 + 0.718210i \(0.744962\pi\)
\(284\) 0 0
\(285\) −127.678 −0.447992
\(286\) 0 0
\(287\) 36.2218i 0.126208i
\(288\) 0 0
\(289\) 50.0135 0.173057
\(290\) 0 0
\(291\) 115.438i 0.396695i
\(292\) 0 0
\(293\) 20.0698i 0.0684975i −0.999413 0.0342488i \(-0.989096\pi\)
0.999413 0.0342488i \(-0.0109039\pi\)
\(294\) 0 0
\(295\) 105.722i 0.358379i
\(296\) 0 0
\(297\) 32.7266i 0.110191i
\(298\) 0 0
\(299\) 359.896 33.1447i 1.20367 0.110852i
\(300\) 0 0
\(301\) 0.327573 0.00108828
\(302\) 0 0
\(303\) 129.279 0.426663
\(304\) 0 0
\(305\) −177.660 −0.582493
\(306\) 0 0
\(307\) 150.284 0.489524 0.244762 0.969583i \(-0.421290\pi\)
0.244762 + 0.969583i \(0.421290\pi\)
\(308\) 0 0
\(309\) 272.026i 0.880344i
\(310\) 0 0
\(311\) −200.804 −0.645672 −0.322836 0.946455i \(-0.604636\pi\)
−0.322836 + 0.946455i \(0.604636\pi\)
\(312\) 0 0
\(313\) 261.619i 0.835845i 0.908483 + 0.417922i \(0.137242\pi\)
−0.908483 + 0.417922i \(0.862758\pi\)
\(314\) 0 0
\(315\) 8.33384 0.0264566
\(316\) 0 0
\(317\) 536.131 1.69126 0.845632 0.533766i \(-0.179224\pi\)
0.845632 + 0.533766i \(0.179224\pi\)
\(318\) 0 0
\(319\) 223.512i 0.700666i
\(320\) 0 0
\(321\) 120.111i 0.374179i
\(322\) 0 0
\(323\) −509.631 −1.57781
\(324\) 0 0
\(325\) −78.5693 −0.241752
\(326\) 0 0
\(327\) 252.084i 0.770901i
\(328\) 0 0
\(329\) 64.2856i 0.195397i
\(330\) 0 0
\(331\) 21.2317 0.0641440 0.0320720 0.999486i \(-0.489789\pi\)
0.0320720 + 0.999486i \(0.489789\pi\)
\(332\) 0 0
\(333\) 45.3347i 0.136140i
\(334\) 0 0
\(335\) 225.554 0.673296
\(336\) 0 0
\(337\) 90.4432i 0.268377i 0.990956 + 0.134189i \(0.0428428\pi\)
−0.990956 + 0.134189i \(0.957157\pi\)
\(338\) 0 0
\(339\) 14.6125i 0.0431047i
\(340\) 0 0
\(341\) 136.681i 0.400825i
\(342\) 0 0
\(343\) 119.831i 0.349363i
\(344\) 0 0
\(345\) 8.16915 + 88.7032i 0.0236787 + 0.257111i
\(346\) 0 0
\(347\) 294.422 0.848478 0.424239 0.905550i \(-0.360542\pi\)
0.424239 + 0.905550i \(0.360542\pi\)
\(348\) 0 0
\(349\) −0.168469 −0.000482718 −0.000241359 1.00000i \(-0.500077\pi\)
−0.000241359 1.00000i \(0.500077\pi\)
\(350\) 0 0
\(351\) −81.6517 −0.232626
\(352\) 0 0
\(353\) −419.988 −1.18977 −0.594884 0.803812i \(-0.702802\pi\)
−0.594884 + 0.803812i \(0.702802\pi\)
\(354\) 0 0
\(355\) 157.109i 0.442562i
\(356\) 0 0
\(357\) 33.2649 0.0931790
\(358\) 0 0
\(359\) 211.488i 0.589104i 0.955635 + 0.294552i \(0.0951705\pi\)
−0.955635 + 0.294552i \(0.904830\pi\)
\(360\) 0 0
\(361\) −725.772 −2.01045
\(362\) 0 0
\(363\) −140.871 −0.388075
\(364\) 0 0
\(365\) 54.6080i 0.149611i
\(366\) 0 0
\(367\) 543.125i 1.47990i 0.672659 + 0.739952i \(0.265152\pi\)
−0.672659 + 0.739952i \(0.734848\pi\)
\(368\) 0 0
\(369\) −87.4687 −0.237043
\(370\) 0 0
\(371\) 19.6053 0.0528445
\(372\) 0 0
\(373\) 91.4621i 0.245207i 0.992456 + 0.122603i \(0.0391243\pi\)
−0.992456 + 0.122603i \(0.960876\pi\)
\(374\) 0 0
\(375\) 19.3649i 0.0516398i
\(376\) 0 0
\(377\) −557.654 −1.47919
\(378\) 0 0
\(379\) 630.233i 1.66288i 0.555612 + 0.831441i \(0.312484\pi\)
−0.555612 + 0.831441i \(0.687516\pi\)
\(380\) 0 0
\(381\) 42.2205 0.110815
\(382\) 0 0
\(383\) 189.667i 0.495215i −0.968860 0.247607i \(-0.920356\pi\)
0.968860 0.247607i \(-0.0796443\pi\)
\(384\) 0 0
\(385\) 17.4962i 0.0454446i
\(386\) 0 0
\(387\) 0.791025i 0.00204399i
\(388\) 0 0
\(389\) 312.003i 0.802064i −0.916064 0.401032i \(-0.868652\pi\)
0.916064 0.401032i \(-0.131348\pi\)
\(390\) 0 0
\(391\) 32.6075 + 354.063i 0.0833952 + 0.905532i
\(392\) 0 0
\(393\) −284.622 −0.724228
\(394\) 0 0
\(395\) 60.4982 0.153160
\(396\) 0 0
\(397\) −520.015 −1.30986 −0.654931 0.755688i \(-0.727302\pi\)
−0.654931 + 0.755688i \(0.727302\pi\)
\(398\) 0 0
\(399\) 70.9363 0.177785
\(400\) 0 0
\(401\) 513.574i 1.28073i −0.768070 0.640366i \(-0.778783\pi\)
0.768070 0.640366i \(-0.221217\pi\)
\(402\) 0 0
\(403\) −341.014 −0.846189
\(404\) 0 0
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −95.1764 −0.233849
\(408\) 0 0
\(409\) 188.162 0.460053 0.230027 0.973184i \(-0.426119\pi\)
0.230027 + 0.973184i \(0.426119\pi\)
\(410\) 0 0
\(411\) 305.555i 0.743442i
\(412\) 0 0
\(413\) 58.7379i 0.142223i
\(414\) 0 0
\(415\) −91.4483 −0.220357
\(416\) 0 0
\(417\) 37.0170 0.0887698
\(418\) 0 0
\(419\) 535.223i 1.27738i 0.769464 + 0.638691i \(0.220524\pi\)
−0.769464 + 0.638691i \(0.779476\pi\)
\(420\) 0 0
\(421\) 383.423i 0.910744i 0.890301 + 0.455372i \(0.150494\pi\)
−0.890301 + 0.455372i \(0.849506\pi\)
\(422\) 0 0
\(423\) −155.237 −0.366991
\(424\) 0 0
\(425\) 77.2959i 0.181873i
\(426\) 0 0
\(427\) 98.7061 0.231162
\(428\) 0 0
\(429\) 171.421i 0.399582i
\(430\) 0 0
\(431\) 449.014i 1.04180i −0.853619 0.520898i \(-0.825597\pi\)
0.853619 0.520898i \(-0.174403\pi\)
\(432\) 0 0
\(433\) 424.231i 0.979748i 0.871793 + 0.489874i \(0.162957\pi\)
−0.871793 + 0.489874i \(0.837043\pi\)
\(434\) 0 0
\(435\) 137.445i 0.315965i
\(436\) 0 0
\(437\) 69.5345 + 755.028i 0.159118 + 1.72775i
\(438\) 0 0
\(439\) −291.872 −0.664857 −0.332428 0.943128i \(-0.607868\pi\)
−0.332428 + 0.943128i \(0.607868\pi\)
\(440\) 0 0
\(441\) 142.370 0.322834
\(442\) 0 0
\(443\) −489.250 −1.10440 −0.552201 0.833711i \(-0.686212\pi\)
−0.552201 + 0.833711i \(0.686212\pi\)
\(444\) 0 0
\(445\) 93.9902 0.211214
\(446\) 0 0
\(447\) 340.043i 0.760723i
\(448\) 0 0
\(449\) 357.166 0.795469 0.397735 0.917500i \(-0.369796\pi\)
0.397735 + 0.917500i \(0.369796\pi\)
\(450\) 0 0
\(451\) 183.633i 0.407169i
\(452\) 0 0
\(453\) 71.1686 0.157105
\(454\) 0 0
\(455\) 43.6523 0.0959391
\(456\) 0 0
\(457\) 103.016i 0.225417i −0.993628 0.112709i \(-0.964047\pi\)
0.993628 0.112709i \(-0.0359527\pi\)
\(458\) 0 0
\(459\) 80.3283i 0.175007i
\(460\) 0 0
\(461\) 438.875 0.952007 0.476004 0.879443i \(-0.342085\pi\)
0.476004 + 0.879443i \(0.342085\pi\)
\(462\) 0 0
\(463\) 41.5278 0.0896928 0.0448464 0.998994i \(-0.485720\pi\)
0.0448464 + 0.998994i \(0.485720\pi\)
\(464\) 0 0
\(465\) 84.0495i 0.180752i
\(466\) 0 0
\(467\) 47.8079i 0.102372i −0.998689 0.0511862i \(-0.983700\pi\)
0.998689 0.0511862i \(-0.0163002\pi\)
\(468\) 0 0
\(469\) −125.315 −0.267197
\(470\) 0 0
\(471\) 273.001i 0.579621i
\(472\) 0 0
\(473\) −1.66069 −0.00351097
\(474\) 0 0
\(475\) 164.831i 0.347013i
\(476\) 0 0
\(477\) 47.3430i 0.0992516i
\(478\) 0 0
\(479\) 249.701i 0.521297i −0.965434 0.260649i \(-0.916064\pi\)
0.965434 0.260649i \(-0.0839364\pi\)
\(480\) 0 0
\(481\) 237.461i 0.493683i
\(482\) 0 0
\(483\) −4.53869 49.2826i −0.00939687 0.102034i
\(484\) 0 0
\(485\) −149.030 −0.307279
\(486\) 0 0
\(487\) −301.120 −0.618316 −0.309158 0.951011i \(-0.600047\pi\)
−0.309158 + 0.951011i \(0.600047\pi\)
\(488\) 0 0
\(489\) 541.870 1.10812
\(490\) 0 0
\(491\) 221.360 0.450835 0.225418 0.974262i \(-0.427625\pi\)
0.225418 + 0.974262i \(0.427625\pi\)
\(492\) 0 0
\(493\) 548.616i 1.11281i
\(494\) 0 0
\(495\) −42.2499 −0.0853534
\(496\) 0 0
\(497\) 87.2883i 0.175630i
\(498\) 0 0
\(499\) 230.086 0.461095 0.230548 0.973061i \(-0.425948\pi\)
0.230548 + 0.973061i \(0.425948\pi\)
\(500\) 0 0
\(501\) 568.281 1.13429
\(502\) 0 0
\(503\) 171.230i 0.340417i 0.985408 + 0.170208i \(0.0544441\pi\)
−0.985408 + 0.170208i \(0.945556\pi\)
\(504\) 0 0
\(505\) 166.898i 0.330492i
\(506\) 0 0
\(507\) −134.971 −0.266215
\(508\) 0 0
\(509\) 154.438 0.303415 0.151708 0.988425i \(-0.451523\pi\)
0.151708 + 0.988425i \(0.451523\pi\)
\(510\) 0 0
\(511\) 30.3396i 0.0593730i
\(512\) 0 0
\(513\) 171.298i 0.333913i
\(514\) 0 0
\(515\) 351.184 0.681911
\(516\) 0 0
\(517\) 325.907i 0.630382i
\(518\) 0 0
\(519\) 118.090 0.227534
\(520\) 0 0
\(521\) 974.687i 1.87080i −0.353591 0.935400i \(-0.615040\pi\)
0.353591 0.935400i \(-0.384960\pi\)
\(522\) 0 0
\(523\) 73.8469i 0.141199i 0.997505 + 0.0705994i \(0.0224912\pi\)
−0.997505 + 0.0705994i \(0.977509\pi\)
\(524\) 0 0
\(525\) 10.7589i 0.0204932i
\(526\) 0 0
\(527\) 335.487i 0.636598i
\(528\) 0 0
\(529\) 520.102 96.6173i 0.983180 0.182641i
\(530\) 0 0
\(531\) 141.841 0.267120
\(532\) 0 0
\(533\) −458.157 −0.859582
\(534\) 0 0
\(535\) 155.063 0.289838
\(536\) 0 0
\(537\) 30.7674 0.0572949
\(538\) 0 0
\(539\) 298.893i 0.554533i
\(540\) 0 0
\(541\) −814.783 −1.50607 −0.753034 0.657981i \(-0.771411\pi\)
−0.753034 + 0.657981i \(0.771411\pi\)
\(542\) 0 0
\(543\) 3.35786i 0.00618391i
\(544\) 0 0
\(545\) 325.440 0.597137
\(546\) 0 0
\(547\) −771.313 −1.41008 −0.705039 0.709168i \(-0.749071\pi\)
−0.705039 + 0.709168i \(0.749071\pi\)
\(548\) 0 0
\(549\) 238.356i 0.434164i
\(550\) 0 0
\(551\) 1169.91i 2.12324i
\(552\) 0 0
\(553\) −33.6122 −0.0607815
\(554\) 0 0
\(555\) 58.5269 0.105454
\(556\) 0 0
\(557\) 13.6129i 0.0244396i −0.999925 0.0122198i \(-0.996110\pi\)
0.999925 0.0122198i \(-0.00388978\pi\)
\(558\) 0 0
\(559\) 4.14335i 0.00741208i
\(560\) 0 0
\(561\) −168.642 −0.300610
\(562\) 0 0
\(563\) 76.2610i 0.135455i 0.997704 + 0.0677273i \(0.0215748\pi\)
−0.997704 + 0.0677273i \(0.978425\pi\)
\(564\) 0 0
\(565\) 18.8646 0.0333887
\(566\) 0 0
\(567\) 11.1810i 0.0197196i
\(568\) 0 0
\(569\) 76.5199i 0.134481i −0.997737 0.0672407i \(-0.978580\pi\)
0.997737 0.0672407i \(-0.0214195\pi\)
\(570\) 0 0
\(571\) 384.219i 0.672889i 0.941703 + 0.336444i \(0.109224\pi\)
−0.941703 + 0.336444i \(0.890776\pi\)
\(572\) 0 0
\(573\) 519.251i 0.906198i
\(574\) 0 0
\(575\) −114.515 + 10.5463i −0.199157 + 0.0183414i
\(576\) 0 0
\(577\) 471.594 0.817321 0.408660 0.912687i \(-0.365996\pi\)
0.408660 + 0.912687i \(0.365996\pi\)
\(578\) 0 0
\(579\) 322.780 0.557478
\(580\) 0 0
\(581\) 50.8077 0.0874487
\(582\) 0 0
\(583\) −99.3926 −0.170485
\(584\) 0 0
\(585\) 105.412i 0.180191i
\(586\) 0 0
\(587\) 665.132 1.13310 0.566552 0.824026i \(-0.308277\pi\)
0.566552 + 0.824026i \(0.308277\pi\)
\(588\) 0 0
\(589\) 715.416i 1.21463i
\(590\) 0 0
\(591\) −24.7855 −0.0419383
\(592\) 0 0
\(593\) −18.5726 −0.0313197 −0.0156599 0.999877i \(-0.504985\pi\)
−0.0156599 + 0.999877i \(0.504985\pi\)
\(594\) 0 0
\(595\) 42.9448i 0.0721761i
\(596\) 0 0
\(597\) 140.271i 0.234959i
\(598\) 0 0
\(599\) 562.550 0.939149 0.469574 0.882893i \(-0.344407\pi\)
0.469574 + 0.882893i \(0.344407\pi\)
\(600\) 0 0
\(601\) 829.929 1.38091 0.690456 0.723374i \(-0.257410\pi\)
0.690456 + 0.723374i \(0.257410\pi\)
\(602\) 0 0
\(603\) 302.612i 0.501845i
\(604\) 0 0
\(605\) 181.864i 0.300602i
\(606\) 0 0
\(607\) 751.250 1.23764 0.618822 0.785531i \(-0.287610\pi\)
0.618822 + 0.785531i \(0.287610\pi\)
\(608\) 0 0
\(609\) 76.3627i 0.125390i
\(610\) 0 0
\(611\) −813.126 −1.33081
\(612\) 0 0
\(613\) 581.611i 0.948795i 0.880311 + 0.474398i \(0.157334\pi\)
−0.880311 + 0.474398i \(0.842666\pi\)
\(614\) 0 0
\(615\) 112.922i 0.183612i
\(616\) 0 0
\(617\) 490.945i 0.795696i −0.917451 0.397848i \(-0.869757\pi\)
0.917451 0.397848i \(-0.130243\pi\)
\(618\) 0 0
\(619\) 1058.92i 1.71070i 0.518055 + 0.855348i \(0.326656\pi\)
−0.518055 + 0.855348i \(0.673344\pi\)
\(620\) 0 0
\(621\) −119.008 + 10.9601i −0.191639 + 0.0176490i
\(622\) 0 0
\(623\) −52.2200 −0.0838202
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −359.625 −0.573564
\(628\) 0 0
\(629\) 233.613 0.371403
\(630\) 0 0
\(631\) 14.8327i 0.0235067i 0.999931 + 0.0117533i \(0.00374129\pi\)
−0.999931 + 0.0117533i \(0.996259\pi\)
\(632\) 0 0
\(633\) −359.689 −0.568229
\(634\) 0 0
\(635\) 54.5064i 0.0858368i
\(636\) 0 0
\(637\) 745.727 1.17069
\(638\) 0 0
\(639\) −210.784 −0.329866
\(640\) 0 0
\(641\) 295.411i 0.460860i 0.973089 + 0.230430i \(0.0740132\pi\)
−0.973089 + 0.230430i \(0.925987\pi\)
\(642\) 0 0
\(643\) 135.632i 0.210936i 0.994423 + 0.105468i \(0.0336340\pi\)
−0.994423 + 0.105468i \(0.966366\pi\)
\(644\) 0 0
\(645\) 1.02121 0.00158327
\(646\) 0 0
\(647\) −372.540 −0.575796 −0.287898 0.957661i \(-0.592956\pi\)
−0.287898 + 0.957661i \(0.592956\pi\)
\(648\) 0 0
\(649\) 297.782i 0.458833i
\(650\) 0 0
\(651\) 46.6970i 0.0717312i
\(652\) 0 0
\(653\) −1042.07 −1.59581 −0.797906 0.602782i \(-0.794059\pi\)
−0.797906 + 0.602782i \(0.794059\pi\)
\(654\) 0 0
\(655\) 367.445i 0.560985i
\(656\) 0 0
\(657\) 73.2643 0.111513
\(658\) 0 0
\(659\) 112.996i 0.171465i 0.996318 + 0.0857326i \(0.0273231\pi\)
−0.996318 + 0.0857326i \(0.972677\pi\)
\(660\) 0 0
\(661\) 380.305i 0.575347i 0.957729 + 0.287674i \(0.0928818\pi\)
−0.957729 + 0.287674i \(0.907118\pi\)
\(662\) 0 0
\(663\) 420.756i 0.634625i
\(664\) 0 0
\(665\) 91.5784i 0.137712i
\(666\) 0 0
\(667\) −812.785 + 74.8537i −1.21857 + 0.112224i
\(668\) 0 0
\(669\) −175.544 −0.262397
\(670\) 0 0
\(671\) −500.409 −0.745766
\(672\) 0 0
\(673\) −36.6534 −0.0544628 −0.0272314 0.999629i \(-0.508669\pi\)
−0.0272314 + 0.999629i \(0.508669\pi\)
\(674\) 0 0
\(675\) 25.9808 0.0384900
\(676\) 0 0
\(677\) 863.582i 1.27560i −0.770201 0.637801i \(-0.779844\pi\)
0.770201 0.637801i \(-0.220156\pi\)
\(678\) 0 0
\(679\) 82.7996 0.121943
\(680\) 0 0
\(681\) 321.149i 0.471584i
\(682\) 0 0
\(683\) 1267.73 1.85612 0.928059 0.372433i \(-0.121476\pi\)
0.928059 + 0.372433i \(0.121476\pi\)
\(684\) 0 0
\(685\) −394.469 −0.575868
\(686\) 0 0
\(687\) 453.996i 0.660838i
\(688\) 0 0
\(689\) 247.981i 0.359914i
\(690\) 0 0
\(691\) −368.415 −0.533162 −0.266581 0.963813i \(-0.585894\pi\)
−0.266581 + 0.963813i \(0.585894\pi\)
\(692\) 0 0
\(693\) 23.4736 0.0338724
\(694\) 0 0
\(695\) 47.7887i 0.0687608i
\(696\) 0 0
\(697\) 450.732i 0.646674i
\(698\) 0 0
\(699\) 575.943 0.823953
\(700\) 0 0
\(701\) 1090.96i 1.55630i 0.628080 + 0.778149i \(0.283841\pi\)
−0.628080 + 0.778149i \(0.716159\pi\)
\(702\) 0 0
\(703\) 498.172 0.708637
\(704\) 0 0
\(705\) 200.410i 0.284270i
\(706\) 0 0
\(707\) 92.7270i 0.131156i
\(708\) 0 0
\(709\) 1145.85i 1.61614i 0.589084 + 0.808072i \(0.299489\pi\)
−0.589084 + 0.808072i \(0.700511\pi\)
\(710\) 0 0
\(711\) 81.1669i 0.114159i
\(712\) 0 0
\(713\) −497.031 + 45.7742i −0.697098 + 0.0641994i
\(714\) 0 0
\(715\) −221.303 −0.309515
\(716\) 0 0
\(717\) 162.731 0.226962
\(718\) 0 0
\(719\) −621.099 −0.863837 −0.431918 0.901913i \(-0.642163\pi\)
−0.431918 + 0.901913i \(0.642163\pi\)
\(720\) 0 0
\(721\) −195.114 −0.270616
\(722\) 0 0
\(723\) 365.812i 0.505964i
\(724\) 0 0
\(725\) 177.440 0.244745
\(726\) 0 0
\(727\) 679.718i 0.934962i −0.884003 0.467481i \(-0.845162\pi\)
0.884003 0.467481i \(-0.154838\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 4.07620 0.00557620
\(732\) 0 0
\(733\) 1184.71i 1.61625i −0.589012 0.808124i \(-0.700483\pi\)
0.589012 0.808124i \(-0.299517\pi\)
\(734\) 0 0
\(735\) 183.799i 0.250066i
\(736\) 0 0
\(737\) 635.309 0.862021
\(738\) 0 0
\(739\) 1143.08 1.54679 0.773394 0.633926i \(-0.218558\pi\)
0.773394 + 0.633926i \(0.218558\pi\)
\(740\) 0 0
\(741\) 897.249i 1.21086i
\(742\) 0 0
\(743\) 530.519i 0.714023i −0.934100 0.357012i \(-0.883796\pi\)
0.934100 0.357012i \(-0.116204\pi\)
\(744\) 0 0
\(745\) 438.994 0.589254
\(746\) 0 0
\(747\) 122.691i 0.164245i
\(748\) 0 0
\(749\) −86.1514 −0.115022
\(750\) 0 0
\(751\) 1369.14i 1.82309i −0.411197 0.911546i \(-0.634889\pi\)
0.411197 0.911546i \(-0.365111\pi\)
\(752\) 0 0
\(753\) 781.937i 1.03843i
\(754\) 0 0
\(755\) 91.8783i 0.121693i
\(756\) 0 0
\(757\) 428.467i 0.566007i 0.959119 + 0.283003i \(0.0913307\pi\)
−0.959119 + 0.283003i \(0.908669\pi\)
\(758\) 0 0
\(759\) 23.0097 + 249.847i 0.0303158 + 0.329179i
\(760\) 0 0
\(761\) 1121.11 1.47320 0.736602 0.676327i \(-0.236429\pi\)
0.736602 + 0.676327i \(0.236429\pi\)
\(762\) 0 0
\(763\) −180.811 −0.236974
\(764\) 0 0
\(765\) 103.703 0.135560
\(766\) 0 0
\(767\) 742.955 0.968651
\(768\) 0 0
\(769\) 849.172i 1.10425i −0.833760 0.552127i \(-0.813816\pi\)
0.833760 0.552127i \(-0.186184\pi\)
\(770\) 0 0
\(771\) −750.851 −0.973867
\(772\) 0 0
\(773\) 180.216i 0.233138i 0.993183 + 0.116569i \(0.0371896\pi\)
−0.993183 + 0.116569i \(0.962810\pi\)
\(774\) 0 0
\(775\) 108.507 0.140010
\(776\) 0 0
\(777\) −32.5169 −0.0418493
\(778\) 0 0
\(779\) 961.172i 1.23385i
\(780\) 0 0
\(781\) 442.524i 0.566612i
\(782\) 0 0
\(783\) 184.401 0.235506
\(784\) 0 0
\(785\) 352.443 0.448972
\(786\) 0 0
\(787\) 615.955i 0.782662i 0.920250 + 0.391331i \(0.127985\pi\)
−0.920250 + 0.391331i \(0.872015\pi\)
\(788\) 0 0
\(789\) 143.464i 0.181830i
\(790\) 0 0
\(791\) −10.4810 −0.0132503
\(792\) 0 0
\(793\) 1248.50i 1.57440i
\(794\) 0 0
\(795\) 61.1196 0.0768799
\(796\) 0 0
\(797\) 0.598340i 0.000750740i −1.00000 0.000375370i \(-0.999881\pi\)
1.00000 0.000375370i \(-0.000119484\pi\)
\(798\) 0 0
\(799\)