Properties

Label 1620.3.t.f.269.20
Level $1620$
Weight $3$
Character 1620.269
Analytic conductor $44.142$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,3,Mod(269,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.269");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), degree \(2\), not 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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.20
Character \(\chi\) \(=\) 1620.269
Dual form 1620.3.t.f.1349.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.45514 + 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +O(q^{10})\) \(q+(4.45514 + 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +(11.1039 + 6.41086i) q^{11} +(1.55354 - 0.896938i) q^{13} +30.9523 q^{17} -19.2201 q^{19} +(1.19201 + 2.06461i) q^{23} +(14.6966 + 20.2240i) q^{25} +(-31.0770 - 17.9423i) q^{29} +(20.4538 + 35.4270i) q^{31} +(-9.36459 - 14.4187i) q^{35} -53.6302i q^{37} +(2.14983 - 1.24120i) q^{41} +(47.7611 + 27.5749i) q^{43} +(-28.5250 + 49.4067i) q^{47} +(-18.5881 - 32.1956i) q^{49} +19.4030 q^{53} +(34.9187 + 53.7643i) q^{55} +(59.8830 - 34.5735i) q^{59} +(-8.80471 + 15.2502i) q^{61} +(8.95707 - 0.469861i) q^{65} +(44.3133 - 25.5843i) q^{67} +53.9194i q^{71} +42.7832i q^{73} +(-22.0442 - 38.1816i) q^{77} +(-44.4228 + 76.9425i) q^{79} +(-14.0472 + 24.3305i) q^{83} +(137.897 + 70.2536i) q^{85} +68.3114i q^{89} -6.16836 q^{91} +(-85.6282 - 43.6245i) q^{95} +(135.186 + 78.0498i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 4.45514 + 2.26973i 0.891029 + 0.453947i
\(6\) 0 0
\(7\) −2.97789 1.71928i −0.425412 0.245612i 0.271978 0.962303i \(-0.412322\pi\)
−0.697390 + 0.716692i \(0.745656\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 11.1039 + 6.41086i 1.00945 + 0.582806i 0.911030 0.412339i \(-0.135288\pi\)
0.0984188 + 0.995145i \(0.468622\pi\)
\(12\) 0 0
\(13\) 1.55354 0.896938i 0.119503 0.0689952i −0.439057 0.898459i \(-0.644687\pi\)
0.558560 + 0.829464i \(0.311354\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 30.9523 1.82073 0.910363 0.413811i \(-0.135803\pi\)
0.910363 + 0.413811i \(0.135803\pi\)
\(18\) 0 0
\(19\) −19.2201 −1.01158 −0.505792 0.862656i \(-0.668800\pi\)
−0.505792 + 0.862656i \(0.668800\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.19201 + 2.06461i 0.0518263 + 0.0897659i 0.890775 0.454445i \(-0.150162\pi\)
−0.838948 + 0.544211i \(0.816829\pi\)
\(24\) 0 0
\(25\) 14.6966 + 20.2240i 0.587865 + 0.808959i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −31.0770 17.9423i −1.07162 0.618701i −0.142998 0.989723i \(-0.545674\pi\)
−0.928624 + 0.371022i \(0.879007\pi\)
\(30\) 0 0
\(31\) 20.4538 + 35.4270i 0.659800 + 1.14281i 0.980667 + 0.195683i \(0.0626924\pi\)
−0.320867 + 0.947124i \(0.603974\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.36459 14.4187i −0.267560 0.411962i
\(36\) 0 0
\(37\) 53.6302i 1.44947i −0.689030 0.724733i \(-0.741963\pi\)
0.689030 0.724733i \(-0.258037\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.14983 1.24120i 0.0524348 0.0302732i −0.473553 0.880765i \(-0.657029\pi\)
0.525988 + 0.850492i \(0.323696\pi\)
\(42\) 0 0
\(43\) 47.7611 + 27.5749i 1.11072 + 0.641276i 0.939016 0.343873i \(-0.111739\pi\)
0.171706 + 0.985148i \(0.445072\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −28.5250 + 49.4067i −0.606914 + 1.05121i 0.384831 + 0.922987i \(0.374260\pi\)
−0.991746 + 0.128220i \(0.959074\pi\)
\(48\) 0 0
\(49\) −18.5881 32.1956i −0.379350 0.657053i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 19.4030 0.366094 0.183047 0.983104i \(-0.441404\pi\)
0.183047 + 0.983104i \(0.441404\pi\)
\(54\) 0 0
\(55\) 34.9187 + 53.7643i 0.634886 + 0.977533i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 59.8830 34.5735i 1.01497 0.585991i 0.102324 0.994751i \(-0.467372\pi\)
0.912642 + 0.408760i \(0.134039\pi\)
\(60\) 0 0
\(61\) −8.80471 + 15.2502i −0.144340 + 0.250003i −0.929126 0.369762i \(-0.879439\pi\)
0.784787 + 0.619766i \(0.212772\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.95707 0.469861i 0.137801 0.00722864i
\(66\) 0 0
\(67\) 44.3133 25.5843i 0.661393 0.381855i −0.131415 0.991327i \(-0.541952\pi\)
0.792807 + 0.609472i \(0.208619\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 53.9194i 0.759428i 0.925104 + 0.379714i \(0.123978\pi\)
−0.925104 + 0.379714i \(0.876022\pi\)
\(72\) 0 0
\(73\) 42.7832i 0.586072i 0.956101 + 0.293036i \(0.0946655\pi\)
−0.956101 + 0.293036i \(0.905334\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22.0442 38.1816i −0.286288 0.495865i
\(78\) 0 0
\(79\) −44.4228 + 76.9425i −0.562314 + 0.973956i 0.434980 + 0.900440i \(0.356755\pi\)
−0.997294 + 0.0735161i \(0.976578\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −14.0472 + 24.3305i −0.169243 + 0.293138i −0.938154 0.346218i \(-0.887466\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(84\) 0 0
\(85\) 137.897 + 70.2536i 1.62232 + 0.826512i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 68.3114i 0.767544i 0.923428 + 0.383772i \(0.125375\pi\)
−0.923428 + 0.383772i \(0.874625\pi\)
\(90\) 0 0
\(91\) −6.16836 −0.0677842
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −85.6282 43.6245i −0.901350 0.459205i
\(96\) 0 0
\(97\) 135.186 + 78.0498i 1.39367 + 0.804637i 0.993720 0.111899i \(-0.0356933\pi\)
0.399952 + 0.916536i \(0.369027\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −60.5918 34.9827i −0.599919 0.346364i 0.169091 0.985601i \(-0.445917\pi\)
−0.769010 + 0.639237i \(0.779250\pi\)
\(102\) 0 0
\(103\) 166.405 96.0740i 1.61558 0.932757i 0.627539 0.778585i \(-0.284062\pi\)
0.988044 0.154172i \(-0.0492710\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 78.5826 0.734417 0.367209 0.930139i \(-0.380314\pi\)
0.367209 + 0.930139i \(0.380314\pi\)
\(108\) 0 0
\(109\) 3.54619 0.0325339 0.0162669 0.999868i \(-0.494822\pi\)
0.0162669 + 0.999868i \(0.494822\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −106.243 184.018i −0.940201 1.62848i −0.765086 0.643928i \(-0.777304\pi\)
−0.175115 0.984548i \(-0.556030\pi\)
\(114\) 0 0
\(115\) 0.624433 + 11.9037i 0.00542985 + 0.103510i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −92.1725 53.2158i −0.774559 0.447192i
\(120\) 0 0
\(121\) 21.6983 + 37.5826i 0.179325 + 0.310600i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 19.5725 + 123.458i 0.156580 + 0.987665i
\(126\) 0 0
\(127\) 183.486i 1.44477i 0.691489 + 0.722387i \(0.256955\pi\)
−0.691489 + 0.722387i \(0.743045\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 52.5451 30.3369i 0.401107 0.231579i −0.285854 0.958273i \(-0.592277\pi\)
0.686962 + 0.726694i \(0.258944\pi\)
\(132\) 0 0
\(133\) 57.2352 + 33.0448i 0.430340 + 0.248457i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.23685 10.8025i 0.0455244 0.0788506i −0.842365 0.538907i \(-0.818837\pi\)
0.887890 + 0.460056i \(0.152171\pi\)
\(138\) 0 0
\(139\) 36.8400 + 63.8088i 0.265036 + 0.459056i 0.967573 0.252591i \(-0.0812828\pi\)
−0.702537 + 0.711647i \(0.747949\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 23.0006 0.160843
\(144\) 0 0
\(145\) −97.7283 150.472i −0.673989 1.03774i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.02350 + 2.32297i −0.0270033 + 0.0155904i −0.513441 0.858125i \(-0.671629\pi\)
0.486438 + 0.873715i \(0.338296\pi\)
\(150\) 0 0
\(151\) −118.510 + 205.265i −0.784835 + 1.35937i 0.144263 + 0.989539i \(0.453919\pi\)
−0.929098 + 0.369835i \(0.879414\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.7147 + 204.257i 0.0691273 + 1.31779i
\(156\) 0 0
\(157\) 136.484 78.7991i 0.869325 0.501905i 0.00220122 0.999998i \(-0.499299\pi\)
0.867124 + 0.498092i \(0.165966\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.19758i 0.0509166i
\(162\) 0 0
\(163\) 82.9389i 0.508828i −0.967095 0.254414i \(-0.918117\pi\)
0.967095 0.254414i \(-0.0818826\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 45.2172 + 78.3185i 0.270762 + 0.468973i 0.969057 0.246837i \(-0.0793911\pi\)
−0.698295 + 0.715810i \(0.746058\pi\)
\(168\) 0 0
\(169\) −82.8910 + 143.571i −0.490479 + 0.849535i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −129.905 + 225.003i −0.750898 + 1.30059i 0.196490 + 0.980506i \(0.437046\pi\)
−0.947388 + 0.320088i \(0.896288\pi\)
\(174\) 0 0
\(175\) −8.99411 85.4923i −0.0513949 0.488528i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 228.604i 1.27712i −0.769573 0.638559i \(-0.779531\pi\)
0.769573 0.638559i \(-0.220469\pi\)
\(180\) 0 0
\(181\) 30.8901 0.170664 0.0853318 0.996353i \(-0.472805\pi\)
0.0853318 + 0.996353i \(0.472805\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 121.726 238.930i 0.657980 1.29152i
\(186\) 0 0
\(187\) 343.693 + 198.431i 1.83793 + 1.06113i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.27714 1.31471i −0.0119222 0.00688330i 0.494027 0.869447i \(-0.335524\pi\)
−0.505949 + 0.862563i \(0.668858\pi\)
\(192\) 0 0
\(193\) 18.9076 10.9163i 0.0979666 0.0565610i −0.450216 0.892920i \(-0.648653\pi\)
0.548183 + 0.836359i \(0.315320\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 68.8358 0.349421 0.174710 0.984620i \(-0.444101\pi\)
0.174710 + 0.984620i \(0.444101\pi\)
\(198\) 0 0
\(199\) 116.185 0.583846 0.291923 0.956442i \(-0.405705\pi\)
0.291923 + 0.956442i \(0.405705\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 61.6959 + 106.860i 0.303921 + 0.526406i
\(204\) 0 0
\(205\) 12.3950 0.650205i 0.0604633 0.00317173i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −213.419 123.217i −1.02114 0.589557i
\(210\) 0 0
\(211\) 112.307 + 194.521i 0.532258 + 0.921898i 0.999291 + 0.0376582i \(0.0119898\pi\)
−0.467032 + 0.884240i \(0.654677\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 150.195 + 231.255i 0.698580 + 1.07560i
\(216\) 0 0
\(217\) 140.664i 0.648219i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 48.0858 27.7623i 0.217583 0.125621i
\(222\) 0 0
\(223\) 219.753 + 126.874i 0.985439 + 0.568944i 0.903908 0.427727i \(-0.140686\pi\)
0.0815313 + 0.996671i \(0.474019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −27.7996 + 48.1503i −0.122465 + 0.212116i −0.920739 0.390179i \(-0.872413\pi\)
0.798274 + 0.602294i \(0.205747\pi\)
\(228\) 0 0
\(229\) 111.391 + 192.935i 0.486423 + 0.842509i 0.999878 0.0156074i \(-0.00496819\pi\)
−0.513456 + 0.858116i \(0.671635\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 399.458 1.71441 0.857207 0.514972i \(-0.172198\pi\)
0.857207 + 0.514972i \(0.172198\pi\)
\(234\) 0 0
\(235\) −239.223 + 155.370i −1.01797 + 0.661149i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.4959 + 10.1013i −0.0732046 + 0.0422647i −0.536156 0.844119i \(-0.680124\pi\)
0.462951 + 0.886384i \(0.346791\pi\)
\(240\) 0 0
\(241\) 168.299 291.502i 0.698336 1.20955i −0.270707 0.962662i \(-0.587257\pi\)
0.969043 0.246892i \(-0.0794092\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.73740 185.626i −0.0397445 0.757658i
\(246\) 0 0
\(247\) −29.8592 + 17.2392i −0.120887 + 0.0697944i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 280.569i 1.11781i 0.829233 + 0.558903i \(0.188778\pi\)
−0.829233 + 0.558903i \(0.811222\pi\)
\(252\) 0 0
\(253\) 30.5671i 0.120819i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −82.1464 142.282i −0.319636 0.553625i 0.660776 0.750583i \(-0.270227\pi\)
−0.980412 + 0.196958i \(0.936894\pi\)
\(258\) 0 0
\(259\) −92.2056 + 159.705i −0.356006 + 0.616621i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 147.024 254.654i 0.559028 0.968266i −0.438550 0.898707i \(-0.644508\pi\)
0.997578 0.0695585i \(-0.0221590\pi\)
\(264\) 0 0
\(265\) 86.4430 + 44.0396i 0.326200 + 0.166187i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 373.214i 1.38741i −0.720259 0.693706i \(-0.755977\pi\)
0.720259 0.693706i \(-0.244023\pi\)
\(270\) 0 0
\(271\) −388.909 −1.43509 −0.717543 0.696514i \(-0.754733\pi\)
−0.717543 + 0.696514i \(0.754733\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 33.5372 + 318.784i 0.121954 + 1.15921i
\(276\) 0 0
\(277\) −321.698 185.732i −1.16136 0.670514i −0.209733 0.977759i \(-0.567260\pi\)
−0.951630 + 0.307245i \(0.900593\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 234.725 + 135.519i 0.835321 + 0.482273i 0.855671 0.517520i \(-0.173145\pi\)
−0.0203504 + 0.999793i \(0.506478\pi\)
\(282\) 0 0
\(283\) 28.8742 16.6705i 0.102029 0.0589065i −0.448117 0.893975i \(-0.647905\pi\)
0.550146 + 0.835068i \(0.314572\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.53591 −0.0297419
\(288\) 0 0
\(289\) 669.047 2.31504
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 59.5109 + 103.076i 0.203109 + 0.351795i 0.949529 0.313681i \(-0.101562\pi\)
−0.746420 + 0.665476i \(0.768229\pi\)
\(294\) 0 0
\(295\) 345.260 18.1113i 1.17037 0.0613943i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.70366 + 2.13831i 0.0123868 + 0.00715154i
\(300\) 0 0
\(301\) −94.8180 164.230i −0.315010 0.545613i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −73.8402 + 47.9575i −0.242099 + 0.157238i
\(306\) 0 0
\(307\) 63.9942i 0.208450i −0.994554 0.104225i \(-0.966764\pi\)
0.994554 0.104225i \(-0.0332362\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 92.3321 53.3079i 0.296888 0.171408i −0.344156 0.938912i \(-0.611835\pi\)
0.641044 + 0.767504i \(0.278502\pi\)
\(312\) 0 0
\(313\) −189.421 109.362i −0.605179 0.349401i 0.165897 0.986143i \(-0.446948\pi\)
−0.771076 + 0.636743i \(0.780281\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −176.345 + 305.438i −0.556292 + 0.963526i 0.441510 + 0.897256i \(0.354443\pi\)
−0.997802 + 0.0662695i \(0.978890\pi\)
\(318\) 0 0
\(319\) −230.052 398.461i −0.721165 1.24909i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −594.906 −1.84182
\(324\) 0 0
\(325\) 40.9715 + 18.2369i 0.126066 + 0.0561134i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 169.888 98.0850i 0.516378 0.298131i
\(330\) 0 0
\(331\) −30.3510 + 52.5695i −0.0916949 + 0.158820i −0.908224 0.418483i \(-0.862562\pi\)
0.816529 + 0.577304i \(0.195895\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 255.492 13.4024i 0.762662 0.0400070i
\(336\) 0 0
\(337\) 19.1598 11.0619i 0.0568539 0.0328246i −0.471304 0.881971i \(-0.656216\pi\)
0.528158 + 0.849146i \(0.322883\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 524.506i 1.53814i
\(342\) 0 0
\(343\) 296.323i 0.863915i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −174.972 303.060i −0.504241 0.873371i −0.999988 0.00490371i \(-0.998439\pi\)
0.495747 0.868467i \(-0.334894\pi\)
\(348\) 0 0
\(349\) −140.703 + 243.705i −0.403161 + 0.698296i −0.994106 0.108416i \(-0.965422\pi\)
0.590944 + 0.806712i \(0.298755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −79.9931 + 138.552i −0.226609 + 0.392499i −0.956801 0.290743i \(-0.906097\pi\)
0.730192 + 0.683242i \(0.239431\pi\)
\(354\) 0 0
\(355\) −122.383 + 240.219i −0.344740 + 0.676673i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 643.967i 1.79378i −0.442254 0.896890i \(-0.645821\pi\)
0.442254 0.896890i \(-0.354179\pi\)
\(360\) 0 0
\(361\) 8.41153 0.0233006
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −97.1065 + 190.605i −0.266045 + 0.522207i
\(366\) 0 0
\(367\) −536.920 309.991i −1.46300 0.844663i −0.463850 0.885914i \(-0.653532\pi\)
−0.999149 + 0.0412512i \(0.986866\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −57.7798 33.3592i −0.155741 0.0899169i
\(372\) 0 0
\(373\) −123.864 + 71.5129i −0.332075 + 0.191724i −0.656762 0.754098i \(-0.728074\pi\)
0.324687 + 0.945822i \(0.394741\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −64.3726 −0.170750
\(378\) 0 0
\(379\) −57.1486 −0.150788 −0.0753940 0.997154i \(-0.524021\pi\)
−0.0753940 + 0.997154i \(0.524021\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −145.454 251.933i −0.379775 0.657789i 0.611254 0.791434i \(-0.290665\pi\)
−0.991029 + 0.133645i \(0.957332\pi\)
\(384\) 0 0
\(385\) −11.5479 220.139i −0.0299944 0.571790i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −566.472 327.053i −1.45623 0.840753i −0.457404 0.889259i \(-0.651221\pi\)
−0.998823 + 0.0485060i \(0.984554\pi\)
\(390\) 0 0
\(391\) 36.8954 + 63.9046i 0.0943615 + 0.163439i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −372.549 + 241.962i −0.943162 + 0.612563i
\(396\) 0 0
\(397\) 229.991i 0.579323i −0.957129 0.289661i \(-0.906457\pi\)
0.957129 0.289661i \(-0.0935427\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −442.493 + 255.474i −1.10347 + 0.637091i −0.937131 0.348977i \(-0.886529\pi\)
−0.166343 + 0.986068i \(0.553196\pi\)
\(402\) 0 0
\(403\) 63.5517 + 36.6916i 0.157697 + 0.0910461i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 343.816 595.507i 0.844757 1.46316i
\(408\) 0 0
\(409\) −299.632 518.979i −0.732598 1.26890i −0.955769 0.294117i \(-0.904974\pi\)
0.223172 0.974779i \(-0.428359\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −237.766 −0.575705
\(414\) 0 0
\(415\) −117.806 + 76.5123i −0.283870 + 0.184367i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −525.674 + 303.498i −1.25459 + 0.724339i −0.972018 0.234906i \(-0.924522\pi\)
−0.282574 + 0.959245i \(0.591188\pi\)
\(420\) 0 0
\(421\) 226.077 391.578i 0.537001 0.930113i −0.462062 0.886847i \(-0.652890\pi\)
0.999064 0.0432659i \(-0.0137763\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 454.895 + 625.979i 1.07034 + 1.47289i
\(426\) 0 0
\(427\) 52.4389 30.2756i 0.122808 0.0709030i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 684.701i 1.58863i −0.607505 0.794316i \(-0.707829\pi\)
0.607505 0.794316i \(-0.292171\pi\)
\(432\) 0 0
\(433\) 458.058i 1.05787i −0.848662 0.528936i \(-0.822591\pi\)
0.848662 0.528936i \(-0.177409\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22.9104 39.6821i −0.0524267 0.0908056i
\(438\) 0 0
\(439\) −6.07259 + 10.5180i −0.0138328 + 0.0239591i −0.872859 0.487973i \(-0.837737\pi\)
0.859026 + 0.511932i \(0.171070\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 237.224 410.885i 0.535495 0.927505i −0.463644 0.886022i \(-0.653458\pi\)
0.999139 0.0414836i \(-0.0132084\pi\)
\(444\) 0 0
\(445\) −155.049 + 304.337i −0.348424 + 0.683904i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 269.662i 0.600583i −0.953847 0.300291i \(-0.902916\pi\)
0.953847 0.300291i \(-0.0970840\pi\)
\(450\) 0 0
\(451\) 31.8287 0.0705737
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −27.4809 14.0005i −0.0603977 0.0307704i
\(456\) 0 0
\(457\) −483.695 279.261i −1.05841 0.611075i −0.133420 0.991060i \(-0.542596\pi\)
−0.924993 + 0.379985i \(0.875929\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 606.316 + 350.056i 1.31522 + 0.759341i 0.982955 0.183845i \(-0.0588546\pi\)
0.332263 + 0.943187i \(0.392188\pi\)
\(462\) 0 0
\(463\) −694.612 + 401.035i −1.50024 + 0.866165i −0.500242 + 0.865885i \(0.666756\pi\)
−1.00000 0.000279830i \(0.999911\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −868.960 −1.86073 −0.930364 0.366636i \(-0.880509\pi\)
−0.930364 + 0.366636i \(0.880509\pi\)
\(468\) 0 0
\(469\) −175.947 −0.375153
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 353.557 + 612.379i 0.747478 + 1.29467i
\(474\) 0 0
\(475\) −282.470 388.707i −0.594674 0.818330i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −610.797 352.644i −1.27515 0.736208i −0.299197 0.954191i \(-0.596719\pi\)
−0.975953 + 0.217983i \(0.930052\pi\)
\(480\) 0 0
\(481\) −48.1030 83.3169i −0.100006 0.173216i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 425.122 + 654.560i 0.876540 + 1.34961i
\(486\) 0 0
\(487\) 715.753i 1.46972i 0.678220 + 0.734859i \(0.262752\pi\)
−0.678220 + 0.734859i \(0.737248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −87.8066 + 50.6952i −0.178832 + 0.103249i −0.586744 0.809773i \(-0.699590\pi\)
0.407912 + 0.913021i \(0.366257\pi\)
\(492\) 0 0
\(493\) −961.907 555.357i −1.95113 1.12648i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 92.7027 160.566i 0.186525 0.323070i
\(498\) 0 0
\(499\) −468.963 812.267i −0.939805 1.62779i −0.765833 0.643040i \(-0.777673\pi\)
−0.173973 0.984751i \(-0.555660\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 766.791 1.52444 0.762218 0.647321i \(-0.224111\pi\)
0.762218 + 0.647321i \(0.224111\pi\)
\(504\) 0 0
\(505\) −190.544 293.380i −0.377315 0.580951i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 647.269 373.701i 1.27165 0.734186i 0.296350 0.955079i \(-0.404231\pi\)
0.975298 + 0.220893i \(0.0708972\pi\)
\(510\) 0 0
\(511\) 73.5565 127.404i 0.143946 0.249322i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 959.421 50.3284i 1.86295 0.0977251i
\(516\) 0 0
\(517\) −633.479 + 365.739i −1.22530 + 0.707426i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 39.7011i 0.0762018i 0.999274 + 0.0381009i \(0.0121308\pi\)
−0.999274 + 0.0381009i \(0.987869\pi\)
\(522\) 0 0
\(523\) 135.722i 0.259506i 0.991546 + 0.129753i \(0.0414184\pi\)
−0.991546 + 0.129753i \(0.958582\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 633.093 + 1096.55i 1.20132 + 2.08074i
\(528\) 0 0
\(529\) 261.658 453.205i 0.494628 0.856721i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.22656 3.85652i 0.00417742 0.00723550i
\(534\) 0 0
\(535\) 350.097 + 178.362i 0.654387 + 0.333386i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 476.664i 0.884349i
\(540\) 0 0
\(541\) 840.659 1.55390 0.776949 0.629563i \(-0.216766\pi\)
0.776949 + 0.629563i \(0.216766\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.7988 + 8.04892i 0.0289886 + 0.0147687i
\(546\) 0 0
\(547\) 819.259 + 472.999i 1.49773 + 0.864715i 0.999997 0.00261458i \(-0.000832248\pi\)
0.497734 + 0.867330i \(0.334166\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 597.303 + 344.853i 1.08403 + 0.625868i
\(552\) 0 0
\(553\) 264.572 152.751i 0.478430 0.276222i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 78.6518 0.141206 0.0706031 0.997504i \(-0.477508\pi\)
0.0706031 + 0.997504i \(0.477508\pi\)
\(558\) 0 0
\(559\) 98.9317 0.176980
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 139.427 + 241.495i 0.247651 + 0.428944i 0.962874 0.269953i \(-0.0870080\pi\)
−0.715223 + 0.698897i \(0.753675\pi\)
\(564\) 0 0
\(565\) −55.6553 1060.97i −0.0985049 1.87782i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −504.075 291.028i −0.885895 0.511472i −0.0132978 0.999912i \(-0.504233\pi\)
−0.872598 + 0.488440i \(0.837566\pi\)
\(570\) 0 0
\(571\) 153.625 + 266.087i 0.269046 + 0.466001i 0.968616 0.248563i \(-0.0799583\pi\)
−0.699570 + 0.714564i \(0.746625\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.2363 + 54.4500i −0.0421500 + 0.0946956i
\(576\) 0 0
\(577\) 24.6333i 0.0426921i 0.999772 + 0.0213460i \(0.00679517\pi\)
−0.999772 + 0.0213460i \(0.993205\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 83.6619 48.3022i 0.143996 0.0831363i
\(582\) 0 0
\(583\) 215.449 + 124.390i 0.369553 + 0.213362i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.2262 17.7123i 0.0174211 0.0301743i −0.857183 0.515011i \(-0.827788\pi\)
0.874605 + 0.484837i \(0.161121\pi\)
\(588\) 0 0
\(589\) −393.124 680.910i −0.667443 1.15604i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −676.813 −1.14134 −0.570669 0.821180i \(-0.693316\pi\)
−0.570669 + 0.821180i \(0.693316\pi\)
\(594\) 0 0
\(595\) −289.856 446.291i −0.487153 0.750069i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −47.0075 + 27.1398i −0.0784766 + 0.0453085i −0.538725 0.842482i \(-0.681094\pi\)
0.460248 + 0.887790i \(0.347760\pi\)
\(600\) 0 0
\(601\) 142.651 247.078i 0.237355 0.411111i −0.722599 0.691267i \(-0.757053\pi\)
0.959955 + 0.280156i \(0.0903861\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.3667 + 216.685i 0.0187879 + 0.358158i
\(606\) 0 0
\(607\) −850.741 + 491.175i −1.40155 + 0.809185i −0.994552 0.104244i \(-0.966758\pi\)
−0.406998 + 0.913429i \(0.633424\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 102.341i 0.167497i
\(612\) 0 0
\(613\) 850.701i 1.38777i −0.720087 0.693883i \(-0.755898\pi\)
0.720087 0.693883i \(-0.244102\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 64.4265 + 111.590i 0.104419 + 0.180859i 0.913501 0.406837i \(-0.133368\pi\)
−0.809082 + 0.587696i \(0.800035\pi\)
\(618\) 0 0
\(619\) −452.830 + 784.325i −0.731552 + 1.26708i 0.224668 + 0.974435i \(0.427870\pi\)
−0.956220 + 0.292649i \(0.905463\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 117.447 203.424i 0.188518 0.326523i
\(624\) 0 0
\(625\) −193.019 + 594.448i −0.308830 + 0.951117i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1659.98i 2.63908i
\(630\) 0 0
\(631\) −1083.23 −1.71669 −0.858345 0.513073i \(-0.828507\pi\)
−0.858345 + 0.513073i \(0.828507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −416.465 + 817.458i −0.655850 + 1.28734i
\(636\) 0 0
\(637\) −57.7549 33.3448i −0.0906670 0.0523466i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −875.815 505.652i −1.36633 0.788848i −0.375869 0.926673i \(-0.622656\pi\)
−0.990457 + 0.137825i \(0.955989\pi\)
\(642\) 0 0
\(643\) −984.875 + 568.618i −1.53169 + 0.884320i −0.532403 + 0.846491i \(0.678711\pi\)
−0.999284 + 0.0378292i \(0.987956\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −273.788 −0.423166 −0.211583 0.977360i \(-0.567862\pi\)
−0.211583 + 0.977360i \(0.567862\pi\)
\(648\) 0 0
\(649\) 886.583 1.36608
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −200.616 347.478i −0.307223 0.532125i 0.670531 0.741882i \(-0.266066\pi\)
−0.977754 + 0.209756i \(0.932733\pi\)
\(654\) 0 0
\(655\) 302.953 15.8920i 0.462523 0.0242626i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −323.745 186.914i −0.491267 0.283633i 0.233833 0.972277i \(-0.424873\pi\)
−0.725100 + 0.688644i \(0.758206\pi\)
\(660\) 0 0
\(661\) −65.4788 113.413i −0.0990602 0.171577i 0.812236 0.583329i \(-0.198250\pi\)
−0.911296 + 0.411752i \(0.864917\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 179.988 + 277.128i 0.270659 + 0.416734i
\(666\) 0 0
\(667\) 85.5494i 0.128260i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −195.534 + 112.892i −0.291407 + 0.168244i
\(672\) 0 0
\(673\) −603.505 348.434i −0.896738 0.517732i −0.0205979 0.999788i \(-0.506557\pi\)
−0.876141 + 0.482056i \(0.839890\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 246.437 426.841i 0.364013 0.630489i −0.624604 0.780941i \(-0.714740\pi\)
0.988617 + 0.150453i \(0.0480731\pi\)
\(678\) 0 0
\(679\) −268.379 464.847i −0.395257 0.684605i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −628.686 −0.920477 −0.460238 0.887795i \(-0.652236\pi\)
−0.460238 + 0.887795i \(0.652236\pi\)
\(684\) 0 0
\(685\) 52.3049 33.9709i 0.0763576 0.0495925i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 30.1433 17.4033i 0.0437494 0.0252587i
\(690\) 0 0
\(691\) 617.313 1069.22i 0.893362 1.54735i 0.0575441 0.998343i \(-0.481673\pi\)
0.835818 0.549006i \(-0.184994\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.2987 + 367.894i 0.0277679 + 0.529344i
\(696\) 0 0
\(697\) 66.5421 38.4181i 0.0954694 0.0551193i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 834.094i 1.18986i 0.803777 + 0.594931i \(0.202821\pi\)
−0.803777 + 0.594931i \(0.797179\pi\)
\(702\) 0 0
\(703\) 1030.78i 1.46626i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 120.290 + 208.349i 0.170142 + 0.294695i
\(708\) 0 0
\(709\) −51.1022 + 88.5117i −0.0720765 + 0.124840i −0.899811 0.436279i \(-0.856296\pi\)
0.827735 + 0.561120i \(0.189629\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −48.7621 + 84.4584i −0.0683901 + 0.118455i
\(714\) 0 0
\(715\) 102.471 + 52.2052i 0.143316 + 0.0730143i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 593.832i 0.825913i −0.910751 0.412957i \(-0.864496\pi\)
0.910751 0.412957i \(-0.135504\pi\)
\(720\) 0 0
\(721\) −660.714 −0.916385
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −93.8619 892.193i −0.129465 1.23061i
\(726\) 0 0
\(727\) −580.916 335.392i −0.799059 0.461337i 0.0440827 0.999028i \(-0.485963\pi\)
−0.843142 + 0.537691i \(0.819297\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1478.32 + 853.506i 2.02232 + 1.16759i
\(732\) 0 0
\(733\) 729.510 421.183i 0.995239 0.574602i 0.0884030 0.996085i \(-0.471824\pi\)
0.906836 + 0.421483i \(0.138490\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 656.070 0.890190
\(738\) 0 0
\(739\) 1423.99 1.92692 0.963460 0.267853i \(-0.0863141\pi\)
0.963460 + 0.267853i \(0.0863141\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 275.646 + 477.432i 0.370990 + 0.642574i 0.989718 0.143032i \(-0.0456850\pi\)
−0.618728 + 0.785605i \(0.712352\pi\)
\(744\) 0 0
\(745\) −23.1978 + 1.21689i −0.0311380 + 0.00163341i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −234.010 135.106i −0.312430 0.180382i
\(750\) 0 0
\(751\) −142.586 246.967i −0.189862 0.328851i 0.755342 0.655331i \(-0.227471\pi\)
−0.945204 + 0.326480i \(0.894137\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −993.877 + 645.501i −1.31639 + 0.854968i
\(756\) 0 0
\(757\) 787.273i 1.03999i −0.854169 0.519995i \(-0.825934\pi\)
0.854169 0.519995i \(-0.174066\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −87.6204 + 50.5876i −0.115138 + 0.0664752i −0.556463 0.830872i \(-0.687842\pi\)
0.441325 + 0.897347i \(0.354509\pi\)
\(762\) 0 0
\(763\) −10.5602 6.09691i −0.0138403 0.00799071i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 62.0205 107.423i 0.0808612 0.140056i
\(768\) 0 0
\(769\) 1.84498 + 3.19561i 0.00239920 + 0.00415554i 0.867223 0.497921i \(-0.165903\pi\)
−0.864823 + 0.502076i \(0.832570\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 504.260 0.652341 0.326171 0.945311i \(-0.394242\pi\)
0.326171 + 0.945311i \(0.394242\pi\)
\(774\) 0 0
\(775\) −415.874 + 934.315i −0.536611 + 1.20557i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −41.3198 + 23.8560i −0.0530422 + 0.0306239i
\(780\) 0 0
\(781\) −345.670 + 598.718i −0.442599 + 0.766604i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 786.909 41.2789i 1.00243 0.0525846i
\(786\) 0 0
\(787\) 1223.57 706.429i 1.55473 0.897622i 0.556981 0.830525i \(-0.311960\pi\)
0.997746 0.0670969i \(-0.0213737\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 730.645i 0.923698i
\(792\) 0 0
\(793\) 31.5891i 0.0398350i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −241.383 418.088i −0.302865 0.524577i 0.673919 0.738805i \(-0.264610\pi\)
−0.976784 + 0.214228i \(0.931276\pi\)
\(798\) 0 0
\(799\) −882.915 + 1529.25i −1.10502 + 1.91396i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −274.277 + 475.063i −0.341566 + 0.591610i
\(804\) 0 0
\(805\) 18.6063 36.5214i 0.0231134 0.0453682i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 874.774i 1.08130i −0.841247 0.540652i \(-0.818178\pi\)
0.841247 0.540652i \(-0.181822\pi\)
\(810\) 0 0
\(811\) −568.734 −0.701275 −0.350638 0.936511i \(-0.614035\pi\)
−0.350638 + 0.936511i \(0.614035\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 188.249 369.505i 0.230981 0.453380i
\(816\) 0 0
\(817\) −917.971 529.991i −1.12359 0.648704i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 128.787 + 74.3550i 0.156866 + 0.0905664i 0.576378 0.817183i \(-0.304466\pi\)
−0.419512 + 0.907750i \(0.637799\pi\)
\(822\) 0 0
\(823\) 437.829 252.780i 0.531991 0.307145i −0.209836 0.977737i \(-0.567293\pi\)
0.741827 + 0.670591i \(0.233960\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 980.267 1.18533 0.592664 0.805450i \(-0.298076\pi\)
0.592664 + 0.805450i \(0.298076\pi\)
\(828\) 0 0
\(829\) 190.269 0.229516 0.114758 0.993393i \(-0.463391\pi\)
0.114758 + 0.993393i \(0.463391\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −575.346 996.529i −0.690692 1.19631i
\(834\) 0 0
\(835\) 23.6871 + 451.551i 0.0283677 + 0.540780i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1021.86 589.969i −1.21795 0.703181i −0.253467 0.967344i \(-0.581571\pi\)
−0.964478 + 0.264163i \(0.914904\pi\)
\(840\) 0 0
\(841\) 223.354 + 386.861i 0.265582 + 0.460002i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −695.160 + 451.491i −0.822675 + 0.534309i
\(846\) 0 0
\(847\) 149.222i 0.176177i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 110.726 63.9276i 0.130113 0.0751205i
\(852\) 0 0
\(853\) 970.251 + 560.175i 1.13746 + 0.656711i 0.945800 0.324751i \(-0.105280\pi\)
0.191657 + 0.981462i \(0.438614\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −650.362 + 1126.46i −0.758882 + 1.31442i 0.184540 + 0.982825i \(0.440921\pi\)
−0.943421 + 0.331597i \(0.892413\pi\)
\(858\) 0 0
\(859\) 433.951 + 751.625i 0.505182 + 0.875000i 0.999982 + 0.00599363i \(0.00190784\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −258.845 −0.299937 −0.149968 0.988691i \(-0.547917\pi\)
−0.149968 + 0.988691i \(0.547917\pi\)
\(864\) 0 0
\(865\) −1089.44 + 707.569i −1.25947 + 0.817999i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −986.536 + 569.577i −1.13525 + 0.655439i
\(870\) 0 0
\(871\) 45.8951 79.4926i 0.0526924 0.0912659i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 153.975 401.295i 0.175971 0.458623i
\(876\) 0 0
\(877\) −16.4844 + 9.51730i −0.0187964 + 0.0108521i −0.509369 0.860548i \(-0.670121\pi\)
0.490572 + 0.871400i \(0.336788\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1688.86i 1.91699i 0.285117 + 0.958493i \(0.407968\pi\)
−0.285117 + 0.958493i \(0.592032\pi\)
\(882\) 0 0
\(883\) 373.276i 0.422736i 0.977407 + 0.211368i \(0.0677918\pi\)
−0.977407 + 0.211368i \(0.932208\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −598.296 1036.28i −0.674516 1.16830i −0.976610 0.215018i \(-0.931019\pi\)
0.302094 0.953278i \(-0.402314\pi\)
\(888\) 0 0
\(889\) 315.465 546.401i 0.354854 0.614624i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 548.252 949.601i 0.613944 1.06338i
\(894\) 0 0
\(895\) 518.870 1018.46i 0.579743 1.13795i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1467.96i 1.63288i
\(900\) 0 0
\(901\) 600.567 0.666556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 137.620 + 70.1123i 0.152066 + 0.0774722i
\(906\) 0 0
\(907\) 760.118 + 438.855i 0.838058 + 0.483853i 0.856604 0.515975i \(-0.172570\pi\)
−0.0185458 + 0.999828i \(0.505904\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −212.780 122.849i −0.233567 0.134850i 0.378649 0.925540i \(-0.376389\pi\)
−0.612217 + 0.790690i \(0.709722\pi\)
\(912\) 0 0
\(913\) −311.959 + 180.109i −0.341685 + 0.197272i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −208.631 −0.227515
\(918\) 0 0
\(919\) 66.4769 0.0723361 0.0361681 0.999346i \(-0.488485\pi\)
0.0361681 + 0.999346i \(0.488485\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 48.3624 + 83.7661i 0.0523969 + 0.0907542i
\(924\) 0 0
\(925\) 1084.62 788.183i 1.17256 0.852090i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1179.99 681.269i −1.27017 0.733336i −0.295154 0.955450i \(-0.595371\pi\)
−0.975021 + 0.222114i \(0.928704\pi\)
\(930\) 0 0
\(931\) 357.265 + 618.802i 0.383744 + 0.664664i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1080.82 + 1664.13i 1.15595 + 1.77982i
\(936\) 0 0
\(937\) 750.141i 0.800578i −0.916389 0.400289i \(-0.868910\pi\)
0.916389 0.400289i \(-0.131090\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 227.057 131.091i 0.241293 0.139311i −0.374478 0.927236i \(-0.622178\pi\)
0.615771 + 0.787925i \(0.288845\pi\)
\(942\) 0 0
\(943\) 5.12521 + 2.95904i 0.00543501 + 0.00313790i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −329.271 + 570.313i −0.347699 + 0.602232i −0.985840 0.167687i \(-0.946370\pi\)
0.638142 + 0.769919i \(0.279703\pi\)
\(948\) 0 0
\(949\) 38.3739 + 66.4656i 0.0404362 + 0.0700375i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 122.390 0.128426 0.0642131 0.997936i \(-0.479546\pi\)
0.0642131 + 0.997936i \(0.479546\pi\)
\(954\) 0 0
\(955\) −7.16096 11.0257i −0.00749839 0.0115453i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −37.1452 + 21.4458i −0.0387333 + 0.0223627i
\(960\) 0 0
\(961\) −356.216 + 616.985i −0.370672 + 0.642024i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 109.013 5.71850i 0.112967 0.00592591i
\(966\) 0 0
\(967\) −923.389 + 533.119i −0.954901 + 0.551312i −0.894600 0.446868i \(-0.852539\pi\)
−0.0603010 + 0.998180i \(0.519206\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1798.30i 1.85201i −0.377508 0.926006i \(-0.623219\pi\)
0.377508 0.926006i \(-0.376781\pi\)
\(972\) 0 0
\(973\) 253.354i 0.260384i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −287.036 497.161i −0.293794 0.508865i 0.680910 0.732367i \(-0.261584\pi\)
−0.974704 + 0.223502i \(0.928251\pi\)
\(978\) 0 0
\(979\) −437.935 + 758.526i −0.447329 + 0.774797i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −382.134 + 661.875i −0.388742 + 0.673321i −0.992281 0.124013i \(-0.960424\pi\)
0.603538 + 0.797334i \(0.293757\pi\)
\(984\) 0 0
\(985\) 306.674 + 156.239i 0.311344 + 0.158618i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 131.478i 0.132940i
\(990\) 0 0
\(991\) 398.586 0.402206 0.201103 0.979570i \(-0.435547\pi\)
0.201103 + 0.979570i \(0.435547\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 517.622 + 263.710i 0.520224 + 0.265035i
\(996\) 0 0
\(997\) 1278.30 + 738.025i 1.28214 + 0.740246i 0.977240 0.212138i \(-0.0680426\pi\)
0.304903 + 0.952383i \(0.401376\pi\)
\(998\) 0 0
\(999\) 0 0
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 1620.3.t.f.269.20 48
3.2 odd 2 inner 1620.3.t.f.269.5 48
5.4 even 2 inner 1620.3.t.f.269.13 48
9.2 odd 6 1620.3.b.a.809.21 yes 24
9.4 even 3 inner 1620.3.t.f.1349.12 48
9.5 odd 6 inner 1620.3.t.f.1349.13 48
9.7 even 3 1620.3.b.a.809.4 yes 24
15.14 odd 2 inner 1620.3.t.f.269.12 48
45.4 even 6 inner 1620.3.t.f.1349.5 48
45.14 odd 6 inner 1620.3.t.f.1349.20 48
45.29 odd 6 1620.3.b.a.809.3 24
45.34 even 6 1620.3.b.a.809.22 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.3.b.a.809.3 24 45.29 odd 6
1620.3.b.a.809.4 yes 24 9.7 even 3
1620.3.b.a.809.21 yes 24 9.2 odd 6
1620.3.b.a.809.22 yes 24 45.34 even 6
1620.3.t.f.269.5 48 3.2 odd 2 inner
1620.3.t.f.269.12 48 15.14 odd 2 inner
1620.3.t.f.269.13 48 5.4 even 2 inner
1620.3.t.f.269.20 48 1.1 even 1 trivial
1620.3.t.f.1349.5 48 45.4 even 6 inner
1620.3.t.f.1349.12 48 9.4 even 3 inner
1620.3.t.f.1349.13 48 9.5 odd 6 inner
1620.3.t.f.1349.20 48 45.14 odd 6 inner