Properties

Label 1764.4.k.l.1549.1
Level $1764$
Weight $4$
Character 1764.1549
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), 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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1549
Dual form 1764.4.k.l.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.00000 - 5.19615i) q^{5} +(18.0000 + 31.1769i) q^{11} +62.0000 q^{13} +(57.0000 + 98.7269i) q^{17} +(38.0000 - 65.8179i) q^{19} +(-12.0000 + 20.7846i) q^{23} +(44.5000 + 77.0763i) q^{25} -54.0000 q^{29} +(56.0000 + 96.9948i) q^{31} +(89.0000 - 154.153i) q^{37} -378.000 q^{41} -172.000 q^{43} +(-96.0000 + 166.277i) q^{47} +(-201.000 - 348.142i) q^{53} +216.000 q^{55} +(198.000 + 342.946i) q^{59} +(-127.000 + 219.970i) q^{61} +(186.000 - 322.161i) q^{65} +(506.000 + 876.418i) q^{67} -840.000 q^{71} +(-445.000 - 770.763i) q^{73} +(-40.0000 + 69.2820i) q^{79} +108.000 q^{83} +684.000 q^{85} +(-819.000 + 1418.55i) q^{89} +(-228.000 - 394.908i) q^{95} +1010.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5} + 36 q^{11} + 124 q^{13} + 114 q^{17} + 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} + 112 q^{31} + 178 q^{37} - 756 q^{41} - 344 q^{43} - 192 q^{47} - 402 q^{53} + 432 q^{55} + 396 q^{59}+ \cdots + 2020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 3.00000 5.19615i 0.268328 0.464758i −0.700102 0.714043i \(-0.746862\pi\)
0.968430 + 0.249285i \(0.0801955\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.0000 + 31.1769i 0.493382 + 0.854563i 0.999971 0.00762479i \(-0.00242707\pi\)
−0.506589 + 0.862188i \(0.669094\pi\)
\(12\) 0 0
\(13\) 62.0000 1.32275 0.661373 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 57.0000 + 98.7269i 0.813208 + 1.40852i 0.910608 + 0.413272i \(0.135614\pi\)
−0.0974001 + 0.995245i \(0.531053\pi\)
\(18\) 0 0
\(19\) 38.0000 65.8179i 0.458831 0.794719i −0.540068 0.841621i \(-0.681602\pi\)
0.998899 + 0.0469020i \(0.0149348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12.0000 + 20.7846i −0.108790 + 0.188430i −0.915280 0.402817i \(-0.868031\pi\)
0.806490 + 0.591247i \(0.201364\pi\)
\(24\) 0 0
\(25\) 44.5000 + 77.0763i 0.356000 + 0.616610i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −54.0000 −0.345778 −0.172889 0.984941i \(-0.555310\pi\)
−0.172889 + 0.984941i \(0.555310\pi\)
\(30\) 0 0
\(31\) 56.0000 + 96.9948i 0.324448 + 0.561961i 0.981401 0.191971i \(-0.0614880\pi\)
−0.656952 + 0.753932i \(0.728155\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 89.0000 154.153i 0.395446 0.684933i −0.597712 0.801711i \(-0.703923\pi\)
0.993158 + 0.116778i \(0.0372566\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −378.000 −1.43985 −0.719923 0.694054i \(-0.755823\pi\)
−0.719923 + 0.694054i \(0.755823\pi\)
\(42\) 0 0
\(43\) −172.000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −96.0000 + 166.277i −0.297937 + 0.516042i −0.975664 0.219272i \(-0.929632\pi\)
0.677727 + 0.735314i \(0.262965\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −201.000 348.142i −0.520933 0.902283i −0.999704 0.0243430i \(-0.992251\pi\)
0.478770 0.877940i \(-0.341083\pi\)
\(54\) 0 0
\(55\) 216.000 0.529553
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 198.000 + 342.946i 0.436905 + 0.756742i 0.997449 0.0713828i \(-0.0227412\pi\)
−0.560544 + 0.828125i \(0.689408\pi\)
\(60\) 0 0
\(61\) −127.000 + 219.970i −0.266569 + 0.461710i −0.967973 0.251053i \(-0.919223\pi\)
0.701405 + 0.712763i \(0.252557\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 186.000 322.161i 0.354930 0.614757i
\(66\) 0 0
\(67\) 506.000 + 876.418i 0.922653 + 1.59808i 0.795293 + 0.606225i \(0.207317\pi\)
0.127360 + 0.991857i \(0.459350\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −840.000 −1.40408 −0.702040 0.712138i \(-0.747727\pi\)
−0.702040 + 0.712138i \(0.747727\pi\)
\(72\) 0 0
\(73\) −445.000 770.763i −0.713470 1.23577i −0.963547 0.267540i \(-0.913789\pi\)
0.250077 0.968226i \(-0.419544\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −40.0000 + 69.2820i −0.0569665 + 0.0986688i −0.893102 0.449854i \(-0.851476\pi\)
0.836136 + 0.548522i \(0.184809\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 108.000 0.142826 0.0714129 0.997447i \(-0.477249\pi\)
0.0714129 + 0.997447i \(0.477249\pi\)
\(84\) 0 0
\(85\) 684.000 0.872826
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −819.000 + 1418.55i −0.975436 + 1.68951i −0.296948 + 0.954894i \(0.595969\pi\)
−0.678488 + 0.734612i \(0.737364\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −228.000 394.908i −0.246235 0.426491i
\(96\) 0 0
\(97\) 1010.00 1.05722 0.528608 0.848866i \(-0.322714\pi\)
0.528608 + 0.848866i \(0.322714\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 + 5.19615i 0.00295556 + 0.00511917i 0.867499 0.497438i \(-0.165726\pi\)
−0.864544 + 0.502557i \(0.832393\pi\)
\(102\) 0 0
\(103\) 236.000 408.764i 0.225765 0.391036i −0.730784 0.682609i \(-0.760845\pi\)
0.956549 + 0.291573i \(0.0941786\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −486.000 + 841.777i −0.439097 + 0.760539i −0.997620 0.0689505i \(-0.978035\pi\)
0.558523 + 0.829489i \(0.311368\pi\)
\(108\) 0 0
\(109\) 893.000 + 1546.72i 0.784715 + 1.35917i 0.929169 + 0.369654i \(0.120524\pi\)
−0.144455 + 0.989511i \(0.546143\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2286.00 1.90309 0.951543 0.307515i \(-0.0994973\pi\)
0.951543 + 0.307515i \(0.0994973\pi\)
\(114\) 0 0
\(115\) 72.0000 + 124.708i 0.0583829 + 0.101122i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 17.5000 30.3109i 0.0131480 0.0227730i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) 1328.00 0.927881 0.463941 0.885866i \(-0.346435\pi\)
0.463941 + 0.885866i \(0.346435\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −606.000 + 1049.62i −0.404171 + 0.700046i −0.994225 0.107319i \(-0.965773\pi\)
0.590053 + 0.807364i \(0.299107\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −627.000 1086.00i −0.391009 0.677247i 0.601574 0.798817i \(-0.294541\pi\)
−0.992583 + 0.121570i \(0.961207\pi\)
\(138\) 0 0
\(139\) −340.000 −0.207471 −0.103735 0.994605i \(-0.533079\pi\)
−0.103735 + 0.994605i \(0.533079\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1116.00 + 1932.97i 0.652620 + 1.13037i
\(144\) 0 0
\(145\) −162.000 + 280.592i −0.0927818 + 0.160703i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 519.000 898.934i 0.285357 0.494252i −0.687339 0.726337i \(-0.741221\pi\)
0.972696 + 0.232085i \(0.0745546\pi\)
\(150\) 0 0
\(151\) −1468.00 2542.65i −0.791153 1.37032i −0.925253 0.379350i \(-0.876148\pi\)
0.134100 0.990968i \(-0.457186\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 672.000 0.348234
\(156\) 0 0
\(157\) 665.000 + 1151.81i 0.338043 + 0.585508i 0.984065 0.177811i \(-0.0569017\pi\)
−0.646021 + 0.763319i \(0.723568\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1682.00 2913.31i 0.808248 1.39993i −0.105828 0.994384i \(-0.533749\pi\)
0.914076 0.405542i \(-0.132917\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3048.00 −1.41234 −0.706172 0.708041i \(-0.749579\pi\)
−0.706172 + 0.708041i \(0.749579\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1353.00 + 2343.46i −0.594605 + 1.02989i 0.398997 + 0.916952i \(0.369358\pi\)
−0.993602 + 0.112934i \(0.963975\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2358.00 + 4084.18i 0.984610 + 1.70539i 0.643657 + 0.765314i \(0.277416\pi\)
0.340953 + 0.940080i \(0.389250\pi\)
\(180\) 0 0
\(181\) 1910.00 0.784360 0.392180 0.919888i \(-0.371721\pi\)
0.392180 + 0.919888i \(0.371721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −534.000 924.915i −0.212219 0.367574i
\(186\) 0 0
\(187\) −2052.00 + 3554.17i −0.802444 + 1.38987i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2040.00 3533.38i 0.772823 1.33857i −0.163187 0.986595i \(-0.552177\pi\)
0.936010 0.351974i \(-0.114489\pi\)
\(192\) 0 0
\(193\) 1343.00 + 2326.14i 0.500887 + 0.867562i 0.999999 + 0.00102491i \(0.000326238\pi\)
−0.499112 + 0.866537i \(0.666340\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −510.000 −0.184447 −0.0922233 0.995738i \(-0.529397\pi\)
−0.0922233 + 0.995738i \(0.529397\pi\)
\(198\) 0 0
\(199\) −676.000 1170.87i −0.240806 0.417088i 0.720138 0.693831i \(-0.244078\pi\)
−0.960944 + 0.276743i \(0.910745\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1134.00 + 1964.15i −0.386351 + 0.669180i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2736.00 0.905517
\(210\) 0 0
\(211\) −3364.00 −1.09757 −0.548785 0.835963i \(-0.684909\pi\)
−0.548785 + 0.835963i \(0.684909\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −516.000 + 893.738i −0.163679 + 0.283500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3534.00 + 6121.07i 1.07567 + 1.86311i
\(222\) 0 0
\(223\) −4768.00 −1.43179 −0.715894 0.698209i \(-0.753981\pi\)
−0.715894 + 0.698209i \(0.753981\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 210.000 + 363.731i 0.0614017 + 0.106351i 0.895092 0.445881i \(-0.147110\pi\)
−0.833690 + 0.552232i \(0.813776\pi\)
\(228\) 0 0
\(229\) 941.000 1629.86i 0.271542 0.470324i −0.697715 0.716375i \(-0.745800\pi\)
0.969257 + 0.246051i \(0.0791332\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2541.00 4401.14i 0.714448 1.23746i −0.248724 0.968574i \(-0.580011\pi\)
0.963172 0.268886i \(-0.0866556\pi\)
\(234\) 0 0
\(235\) 576.000 + 997.661i 0.159890 + 0.276937i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5424.00 1.46799 0.733995 0.679155i \(-0.237654\pi\)
0.733995 + 0.679155i \(0.237654\pi\)
\(240\) 0 0
\(241\) 1295.00 + 2243.01i 0.346134 + 0.599522i 0.985559 0.169332i \(-0.0541609\pi\)
−0.639425 + 0.768853i \(0.720828\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2356.00 4080.71i 0.606918 1.05121i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4932.00 1.24026 0.620130 0.784499i \(-0.287080\pi\)
0.620130 + 0.784499i \(0.287080\pi\)
\(252\) 0 0
\(253\) −864.000 −0.214700
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1719.00 + 2977.40i −0.417231 + 0.722665i −0.995660 0.0930680i \(-0.970333\pi\)
0.578429 + 0.815733i \(0.303666\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3060.00 + 5300.08i 0.717444 + 1.24265i 0.962009 + 0.273016i \(0.0880213\pi\)
−0.244566 + 0.969633i \(0.578645\pi\)
\(264\) 0 0
\(265\) −2412.00 −0.559124
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.00000 15.5885i −0.00203992 0.00353325i 0.865004 0.501766i \(-0.167316\pi\)
−0.867044 + 0.498232i \(0.833983\pi\)
\(270\) 0 0
\(271\) −3448.00 + 5972.11i −0.772882 + 1.33867i 0.163095 + 0.986610i \(0.447852\pi\)
−0.935977 + 0.352061i \(0.885481\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1602.00 + 2774.75i −0.351288 + 0.608449i
\(276\) 0 0
\(277\) −3127.00 5416.12i −0.678279 1.17481i −0.975499 0.220004i \(-0.929393\pi\)
0.297220 0.954809i \(-0.403940\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1194.00 −0.253481 −0.126740 0.991936i \(-0.540451\pi\)
−0.126740 + 0.991936i \(0.540451\pi\)
\(282\) 0 0
\(283\) 3578.00 + 6197.28i 0.751555 + 1.30173i 0.947069 + 0.321030i \(0.104029\pi\)
−0.195514 + 0.980701i \(0.562638\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4041.50 + 7000.08i −0.822613 + 1.42481i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3738.00 0.745312 0.372656 0.927970i \(-0.378447\pi\)
0.372656 + 0.927970i \(0.378447\pi\)
\(294\) 0 0
\(295\) 2376.00 0.468936
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −744.000 + 1288.65i −0.143902 + 0.249245i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 762.000 + 1319.82i 0.143056 + 0.247780i
\(306\) 0 0
\(307\) −844.000 −0.156904 −0.0784522 0.996918i \(-0.524998\pi\)
−0.0784522 + 0.996918i \(0.524998\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3156.00 + 5466.35i 0.575435 + 0.996683i 0.995994 + 0.0894178i \(0.0285006\pi\)
−0.420559 + 0.907265i \(0.638166\pi\)
\(312\) 0 0
\(313\) −4141.00 + 7172.42i −0.747806 + 1.29524i 0.201067 + 0.979578i \(0.435559\pi\)
−0.948872 + 0.315660i \(0.897774\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4659.00 8069.62i 0.825475 1.42976i −0.0760811 0.997102i \(-0.524241\pi\)
0.901556 0.432663i \(-0.142426\pi\)
\(318\) 0 0
\(319\) −972.000 1683.55i −0.170600 0.295489i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8664.00 1.49250
\(324\) 0 0
\(325\) 2759.00 + 4778.73i 0.470898 + 0.815619i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −826.000 + 1430.67i −0.137163 + 0.237574i −0.926422 0.376487i \(-0.877132\pi\)
0.789258 + 0.614061i \(0.210465\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6072.00 0.990295
\(336\) 0 0
\(337\) −1294.00 −0.209165 −0.104583 0.994516i \(-0.533351\pi\)
−0.104583 + 0.994516i \(0.533351\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2016.00 + 3491.81i −0.320154 + 0.554523i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1818.00 + 3148.87i 0.281255 + 0.487147i 0.971694 0.236243i \(-0.0759161\pi\)
−0.690439 + 0.723390i \(0.742583\pi\)
\(348\) 0 0
\(349\) 10478.0 1.60709 0.803545 0.595244i \(-0.202945\pi\)
0.803545 + 0.595244i \(0.202945\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3783.00 6552.35i −0.570393 0.987950i −0.996525 0.0832890i \(-0.973458\pi\)
0.426132 0.904661i \(-0.359876\pi\)
\(354\) 0 0
\(355\) −2520.00 + 4364.77i −0.376754 + 0.652557i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4020.00 + 6962.84i −0.590996 + 1.02363i 0.403103 + 0.915155i \(0.367932\pi\)
−0.994099 + 0.108480i \(0.965402\pi\)
\(360\) 0 0
\(361\) 541.500 + 937.906i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5340.00 −0.765776
\(366\) 0 0
\(367\) −3784.00 6554.08i −0.538210 0.932208i −0.999001 0.0446985i \(-0.985767\pi\)
0.460790 0.887509i \(-0.347566\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6761.00 11710.4i 0.938529 1.62558i 0.170312 0.985390i \(-0.445522\pi\)
0.768217 0.640190i \(-0.221144\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3348.00 −0.457376
\(378\) 0 0
\(379\) 2468.00 0.334492 0.167246 0.985915i \(-0.446513\pi\)
0.167246 + 0.985915i \(0.446513\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6168.00 + 10683.3i −0.822898 + 1.42530i 0.0806166 + 0.996745i \(0.474311\pi\)
−0.903515 + 0.428557i \(0.859022\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1881.00 3257.99i −0.245168 0.424644i 0.717011 0.697062i \(-0.245510\pi\)
−0.962179 + 0.272418i \(0.912177\pi\)
\(390\) 0 0
\(391\) −2736.00 −0.353876
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 240.000 + 415.692i 0.0305714 + 0.0529513i
\(396\) 0 0
\(397\) 4385.00 7595.04i 0.554350 0.960162i −0.443604 0.896223i \(-0.646300\pi\)
0.997954 0.0639390i \(-0.0203663\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3321.00 5752.14i 0.413573 0.716330i −0.581704 0.813400i \(-0.697614\pi\)
0.995277 + 0.0970706i \(0.0309473\pi\)
\(402\) 0 0
\(403\) 3472.00 + 6013.68i 0.429163 + 0.743332i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6408.00 0.780424
\(408\) 0 0
\(409\) 755.000 + 1307.70i 0.0912771 + 0.158097i 0.908049 0.418864i \(-0.137572\pi\)
−0.816772 + 0.576961i \(0.804238\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 324.000 561.184i 0.0383242 0.0663794i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1260.00 0.146909 0.0734547 0.997299i \(-0.476598\pi\)
0.0734547 + 0.997299i \(0.476598\pi\)
\(420\) 0 0
\(421\) 3998.00 0.462828 0.231414 0.972855i \(-0.425665\pi\)
0.231414 + 0.972855i \(0.425665\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5073.00 + 8786.69i −0.579004 + 1.00286i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1368.00 2369.45i −0.152887 0.264808i 0.779401 0.626526i \(-0.215524\pi\)
−0.932288 + 0.361718i \(0.882190\pi\)
\(432\) 0 0
\(433\) 2690.00 0.298552 0.149276 0.988796i \(-0.452306\pi\)
0.149276 + 0.988796i \(0.452306\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 912.000 + 1579.63i 0.0998327 + 0.172915i
\(438\) 0 0
\(439\) 620.000 1073.87i 0.0674054 0.116750i −0.830353 0.557238i \(-0.811861\pi\)
0.897758 + 0.440488i \(0.145195\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1950.00 + 3377.50i −0.209136 + 0.362234i −0.951443 0.307826i \(-0.900399\pi\)
0.742307 + 0.670060i \(0.233732\pi\)
\(444\) 0 0
\(445\) 4914.00 + 8511.30i 0.523474 + 0.906684i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10878.0 1.14335 0.571675 0.820480i \(-0.306294\pi\)
0.571675 + 0.820480i \(0.306294\pi\)
\(450\) 0 0
\(451\) −6804.00 11784.9i −0.710394 1.23044i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1165.00 + 2017.84i −0.119248 + 0.206544i −0.919470 0.393160i \(-0.871382\pi\)
0.800222 + 0.599704i \(0.204715\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15150.0 −1.53060 −0.765299 0.643675i \(-0.777409\pi\)
−0.765299 + 0.643675i \(0.777409\pi\)
\(462\) 0 0
\(463\) −2992.00 −0.300324 −0.150162 0.988661i \(-0.547980\pi\)
−0.150162 + 0.988661i \(0.547980\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4362.00 7555.21i 0.432225 0.748636i −0.564839 0.825201i \(-0.691062\pi\)
0.997065 + 0.0765646i \(0.0243951\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3096.00 5362.43i −0.300960 0.521279i
\(474\) 0 0
\(475\) 6764.00 0.653376
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4872.00 + 8438.55i 0.464734 + 0.804942i 0.999189 0.0402542i \(-0.0128168\pi\)
−0.534456 + 0.845196i \(0.679483\pi\)
\(480\) 0 0
\(481\) 5518.00 9557.46i 0.523075 0.905993i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3030.00 5248.11i 0.283681 0.491350i
\(486\) 0 0
\(487\) −2068.00 3581.88i −0.192423 0.333286i 0.753630 0.657299i \(-0.228301\pi\)
−0.946053 + 0.324013i \(0.894968\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −16212.0 −1.49010 −0.745048 0.667011i \(-0.767574\pi\)
−0.745048 + 0.667011i \(0.767574\pi\)
\(492\) 0 0
\(493\) −3078.00 5331.25i −0.281189 0.487034i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1198.00 + 2075.00i −0.107475 + 0.186152i −0.914747 0.404028i \(-0.867610\pi\)
0.807272 + 0.590180i \(0.200943\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13128.0 −1.16371 −0.581857 0.813291i \(-0.697674\pi\)
−0.581857 + 0.813291i \(0.697674\pi\)
\(504\) 0 0
\(505\) 36.0000 0.00317224
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6399.00 11083.4i 0.557231 0.965153i −0.440495 0.897755i \(-0.645197\pi\)
0.997726 0.0673977i \(-0.0214696\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1416.00 2452.58i −0.121158 0.209852i
\(516\) 0 0
\(517\) −6912.00 −0.587987
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3693.00 + 6396.46i 0.310544 + 0.537877i 0.978480 0.206341i \(-0.0661556\pi\)
−0.667936 + 0.744218i \(0.732822\pi\)
\(522\) 0 0
\(523\) −2590.00 + 4486.01i −0.216545 + 0.375066i −0.953749 0.300603i \(-0.902812\pi\)
0.737205 + 0.675669i \(0.236145\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6384.00 + 11057.4i −0.527688 + 0.913982i
\(528\) 0 0
\(529\) 5795.50 + 10038.1i 0.476329 + 0.825027i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −23436.0 −1.90455
\(534\) 0 0
\(535\) 2916.00 + 5050.66i 0.235644 + 0.408148i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2035.00 + 3524.72i −0.161722 + 0.280110i −0.935486 0.353363i \(-0.885038\pi\)
0.773764 + 0.633473i \(0.218371\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10716.0 0.842244
\(546\) 0 0
\(547\) 14780.0 1.15530 0.577648 0.816286i \(-0.303971\pi\)
0.577648 + 0.816286i \(0.303971\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2052.00 + 3554.17i −0.158654 + 0.274796i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3429.00 5939.20i −0.260846 0.451799i 0.705621 0.708590i \(-0.250668\pi\)
−0.966467 + 0.256791i \(0.917335\pi\)
\(558\) 0 0
\(559\) −10664.0 −0.806868
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3330.00 + 5767.73i 0.249277 + 0.431760i 0.963325 0.268336i \(-0.0864738\pi\)
−0.714049 + 0.700096i \(0.753140\pi\)
\(564\) 0 0
\(565\) 6858.00 11878.4i 0.510652 0.884475i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −75.0000 + 129.904i −0.00552577 + 0.00957092i −0.868775 0.495207i \(-0.835092\pi\)
0.863249 + 0.504778i \(0.168426\pi\)
\(570\) 0 0
\(571\) 4094.00 + 7091.02i 0.300050 + 0.519702i 0.976147 0.217111i \(-0.0696633\pi\)
−0.676097 + 0.736813i \(0.736330\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2136.00 −0.154917
\(576\) 0 0
\(577\) 2927.00 + 5069.71i 0.211183 + 0.365780i 0.952085 0.305833i \(-0.0989351\pi\)
−0.740902 + 0.671613i \(0.765602\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7236.00 12533.1i 0.514039 0.890341i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17580.0 −1.23612 −0.618062 0.786130i \(-0.712082\pi\)
−0.618062 + 0.786130i \(0.712082\pi\)
\(588\) 0 0
\(589\) 8512.00 0.595468
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8577.00 14855.8i 0.593955 1.02876i −0.399739 0.916629i \(-0.630899\pi\)
0.993693 0.112131i \(-0.0357675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9060.00 15692.4i −0.617999 1.07041i −0.989850 0.142114i \(-0.954610\pi\)
0.371851 0.928292i \(-0.378723\pi\)
\(600\) 0 0
\(601\) 17546.0 1.19088 0.595438 0.803401i \(-0.296979\pi\)
0.595438 + 0.803401i \(0.296979\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −105.000 181.865i −0.00705596 0.0122213i
\(606\) 0 0
\(607\) 7280.00 12609.3i 0.486798 0.843158i −0.513087 0.858336i \(-0.671498\pi\)
0.999885 + 0.0151784i \(0.00483163\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5952.00 + 10309.2i −0.394095 + 0.682593i
\(612\) 0 0
\(613\) 2249.00 + 3895.38i 0.148183 + 0.256661i 0.930556 0.366150i \(-0.119324\pi\)
−0.782373 + 0.622810i \(0.785991\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5478.00 0.357433 0.178716 0.983901i \(-0.442806\pi\)
0.178716 + 0.983901i \(0.442806\pi\)
\(618\) 0 0
\(619\) −3022.00 5234.26i −0.196227 0.339875i 0.751075 0.660217i \(-0.229536\pi\)
−0.947302 + 0.320342i \(0.896202\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1710.50 + 2962.67i −0.109472 + 0.189611i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20292.0 1.28632
\(630\) 0 0
\(631\) −15352.0 −0.968547 −0.484274 0.874917i \(-0.660916\pi\)
−0.484274 + 0.874917i \(0.660916\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3984.00 6900.49i 0.248977 0.431240i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11199.0 19397.2i −0.690068 1.19523i −0.971815 0.235745i \(-0.924247\pi\)
0.281746 0.959489i \(-0.409086\pi\)
\(642\) 0 0
\(643\) 3764.00 0.230852 0.115426 0.993316i \(-0.463177\pi\)
0.115426 + 0.993316i \(0.463177\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8844.00 15318.3i −0.537393 0.930793i −0.999043 0.0437305i \(-0.986076\pi\)
0.461650 0.887062i \(-0.347258\pi\)
\(648\) 0 0
\(649\) −7128.00 + 12346.1i −0.431122 + 0.746726i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9939.00 17214.9i 0.595625 1.03165i −0.397833 0.917458i \(-0.630238\pi\)
0.993458 0.114195i \(-0.0364289\pi\)
\(654\) 0 0
\(655\) 3636.00 + 6297.74i 0.216901 + 0.375684i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20004.0 1.18247 0.591233 0.806501i \(-0.298641\pi\)
0.591233 + 0.806501i \(0.298641\pi\)
\(660\) 0 0
\(661\) 653.000 + 1131.03i 0.0384247 + 0.0665536i 0.884598 0.466354i \(-0.154433\pi\)
−0.846173 + 0.532908i \(0.821099\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 648.000 1122.37i 0.0376172 0.0651549i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9144.00 −0.526081
\(672\) 0 0
\(673\) −13054.0 −0.747689 −0.373845 0.927491i \(-0.621961\pi\)
−0.373845 + 0.927491i \(0.621961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2523.00 4369.96i 0.143230 0.248082i −0.785481 0.618886i \(-0.787584\pi\)
0.928711 + 0.370804i \(0.120918\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6234.00 + 10797.6i 0.349249 + 0.604918i 0.986116 0.166056i \(-0.0531032\pi\)
−0.636867 + 0.770974i \(0.719770\pi\)
\(684\) 0 0
\(685\) −7524.00 −0.419675
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12462.0 21584.8i −0.689063 1.19349i
\(690\) 0 0
\(691\) 11606.0 20102.2i 0.638948 1.10669i −0.346716 0.937970i \(-0.612703\pi\)
0.985664 0.168720i \(-0.0539634\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1020.00 + 1766.69i −0.0556702 + 0.0964237i
\(696\) 0 0
\(697\) −21546.0 37318.8i −1.17089 2.02805i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −35958.0 −1.93740 −0.968698 0.248241i \(-0.920147\pi\)
−0.968698 + 0.248241i \(0.920147\pi\)
\(702\) 0 0
\(703\) −6764.00 11715.6i −0.362886 0.628538i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3223.00 + 5582.40i −0.170723 + 0.295700i −0.938673 0.344809i \(-0.887944\pi\)
0.767950 + 0.640510i \(0.221277\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2688.00 −0.141187
\(714\) 0 0
\(715\) 13392.0 0.700465
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2352.00 + 4073.78i −0.121996 + 0.211302i −0.920555 0.390614i \(-0.872263\pi\)
0.798559 + 0.601917i \(0.205596\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2403.00 4162.12i −0.123097 0.213210i
\(726\) 0 0
\(727\) −10600.0 −0.540760 −0.270380 0.962754i \(-0.587149\pi\)
−0.270380 + 0.962754i \(0.587149\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9804.00 16981.0i −0.496052 0.859187i
\(732\) 0 0
\(733\) −6271.00 + 10861.7i −0.315995 + 0.547320i −0.979649 0.200720i \(-0.935672\pi\)
0.663653 + 0.748040i \(0.269005\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18216.0 + 31551.0i −0.910441 + 1.57693i
\(738\) 0 0
\(739\) −11662.0 20199.2i −0.580506 1.00547i −0.995419 0.0956044i \(-0.969522\pi\)
0.414914 0.909861i \(-0.363812\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6312.00 0.311662 0.155831 0.987784i \(-0.450194\pi\)
0.155831 + 0.987784i \(0.450194\pi\)
\(744\) 0 0
\(745\) −3114.00 5393.61i −0.153138 0.265244i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −17920.0 + 31038.4i −0.870719 + 1.50813i −0.00946509 + 0.999955i \(0.503013\pi\)
−0.861254 + 0.508175i \(0.830320\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17616.0 −0.849155
\(756\) 0 0
\(757\) −34594.0 −1.66095 −0.830476 0.557055i \(-0.811931\pi\)
−0.830476 + 0.557055i \(0.811931\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11973.0 20737.8i 0.570330 0.987840i −0.426202 0.904628i \(-0.640149\pi\)
0.996532 0.0832121i \(-0.0265179\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12276.0 + 21262.7i 0.577915 + 1.00098i
\(768\) 0 0
\(769\) 18770.0 0.880187 0.440093 0.897952i \(-0.354945\pi\)
0.440093 + 0.897952i \(0.354945\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15171.0 + 26276.9i 0.705903 + 1.22266i 0.966365 + 0.257176i \(0.0827919\pi\)
−0.260462 + 0.965484i \(0.583875\pi\)
\(774\) 0 0
\(775\) −4984.00 + 8632.54i −0.231007 + 0.400116i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −14364.0 + 24879.2i −0.660647 + 1.14427i
\(780\) 0 0
\(781\) −15120.0 26188.6i −0.692748 1.19987i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7980.00 0.362826
\(786\) 0 0
\(787\) 13094.0 + 22679.5i 0.593076 + 1.02724i 0.993815 + 0.111045i \(0.0354199\pi\)
−0.400739 + 0.916192i \(0.631247\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7874.00 + 13638.2i −0.352603 + 0.610726i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34818.0