Properties

Label 1764.4.k.f.1549.1
Level $1764$
Weight $4$
Character 1764.1549
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
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] = [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.f.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.00000 + 5.19615i) q^{5} +O(q^{10})\) \(q+(-3.00000 + 5.19615i) q^{5} +(18.0000 + 31.1769i) q^{11} -62.0000 q^{13} +(-57.0000 - 98.7269i) q^{17} +(-38.0000 + 65.8179i) q^{19} +(-12.0000 + 20.7846i) q^{23} +(44.5000 + 77.0763i) q^{25} -54.0000 q^{29} +(-56.0000 - 96.9948i) q^{31} +(89.0000 - 154.153i) q^{37} +378.000 q^{41} -172.000 q^{43} +(96.0000 - 166.277i) q^{47} +(-201.000 - 348.142i) q^{53} -216.000 q^{55} +(-198.000 - 342.946i) q^{59} +(127.000 - 219.970i) q^{61} +(186.000 - 322.161i) q^{65} +(506.000 + 876.418i) q^{67} -840.000 q^{71} +(445.000 + 770.763i) q^{73} +(-40.0000 + 69.2820i) q^{79} -108.000 q^{83} +684.000 q^{85} +(819.000 - 1418.55i) q^{89} +(-228.000 - 394.908i) q^{95} -1010.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{5} + 36 q^{11} - 124 q^{13} - 114 q^{17} - 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} - 112 q^{31} + 178 q^{37} + 756 q^{41} - 344 q^{43} + 192 q^{47} - 402 q^{53} - 432 q^{55} - 396 q^{59} + 254 q^{61} + 372 q^{65} + 1012 q^{67} - 1680 q^{71} + 890 q^{73} - 80 q^{79} - 216 q^{83} + 1368 q^{85} + 1638 q^{89} - 456 q^{95} - 2020 q^{97}+O(q^{100}) \) 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 −0.968430 0.249285i \(-0.919804\pi\)
0.700102 + 0.714043i \(0.253138\pi\)
\(6\) 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 \(-0.730026\pi\)
−0.661373 + 0.750057i \(0.730026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −57.0000 98.7269i −0.813208 1.40852i −0.910608 0.413272i \(-0.864386\pi\)
0.0974001 0.995245i \(-0.468947\pi\)
\(18\) 0 0
\(19\) −38.0000 + 65.8179i −0.458831 + 0.794719i −0.998899 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) 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 0.656952 0.753932i \(-0.271845\pi\)
−0.981401 + 0.191971i \(0.938512\pi\)
\(32\) 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 \(-0.244177\pi\)
0.719923 + 0.694054i \(0.244177\pi\)
\(42\) 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 −0.677727 0.735314i \(-0.737035\pi\)
0.975664 + 0.219272i \(0.0703681\pi\)
\(48\) 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 0.560544 0.828125i \(-0.310592\pi\)
−0.997449 + 0.0713828i \(0.977259\pi\)
\(60\) 0 0
\(61\) 127.000 219.970i 0.266569 0.461710i −0.701405 0.712763i \(-0.747443\pi\)
0.967973 + 0.251053i \(0.0807768\pi\)
\(62\) 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 \(0.0862108\pi\)
−0.250077 + 0.968226i \(0.580456\pi\)
\(74\) 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 \(-0.522751\pi\)
−0.0714129 + 0.997447i \(0.522751\pi\)
\(84\) 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 \(-0.404031\pi\)
0.678488 0.734612i \(-0.262636\pi\)
\(90\) 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 \(-0.677286\pi\)
−0.528608 + 0.848866i \(0.677286\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.00295556 0.00511917i 0.864544 0.502557i \(-0.167607\pi\)
−0.867499 + 0.497438i \(0.834274\pi\)
\(102\) 0 0
\(103\) −236.000 + 408.764i −0.225765 + 0.391036i −0.956549 0.291573i \(-0.905821\pi\)
0.730784 + 0.682609i \(0.239155\pi\)
\(104\) 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 −0.590053 0.807364i \(-0.700893\pi\)
0.994225 + 0.107319i \(0.0342266\pi\)
\(132\) 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 \(-0.466921\pi\)
0.103735 + 0.994605i \(0.466921\pi\)
\(140\) 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 0.646021 0.763319i \(-0.276432\pi\)
−0.984065 + 0.177811i \(0.943098\pi\)
\(158\) 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 \(-0.250421\pi\)
0.706172 + 0.708041i \(0.250421\pi\)
\(168\) 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 \(-0.630642\pi\)
0.993602 0.112934i \(-0.0360249\pi\)
\(174\) 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 \(-0.628279\pi\)
−0.392180 + 0.919888i \(0.628279\pi\)
\(182\) 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 0.960944 0.276743i \(-0.0892550\pi\)
−0.720138 + 0.693831i \(0.755922\pi\)
\(200\) 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 \(-0.246019\pi\)
0.715894 + 0.698209i \(0.246019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −210.000 363.731i −0.0614017 0.106351i 0.833690 0.552232i \(-0.186224\pi\)
−0.895092 + 0.445881i \(0.852890\pi\)
\(228\) 0 0
\(229\) −941.000 + 1629.86i −0.271542 + 0.470324i −0.969257 0.246051i \(-0.920867\pi\)
0.697715 + 0.716375i \(0.254200\pi\)
\(230\) 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 0.639425 0.768853i \(-0.279172\pi\)
−0.985559 + 0.169332i \(0.945839\pi\)
\(242\) 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 \(-0.712920\pi\)
−0.620130 + 0.784499i \(0.712920\pi\)
\(252\) 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 −0.578429 0.815733i \(-0.696334\pi\)
0.995660 + 0.0930680i \(0.0296674\pi\)
\(258\) 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 0.867044 0.498232i \(-0.166017\pi\)
−0.865004 + 0.501766i \(0.832684\pi\)
\(270\) 0 0
\(271\) 3448.00 5972.11i 0.772882 1.33867i −0.163095 0.986610i \(-0.552148\pi\)
0.935977 0.352061i \(-0.114519\pi\)
\(272\) 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 \(-0.895971\pi\)
0.195514 0.980701i \(-0.437362\pi\)
\(284\) 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 \(-0.621553\pi\)
−0.372656 + 0.927970i \(0.621553\pi\)
\(294\) 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 \(-0.475002\pi\)
0.0784522 + 0.996918i \(0.475002\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3156.00 5466.35i −0.575435 0.996683i −0.995994 0.0894178i \(-0.971499\pi\)
0.420559 0.907265i \(-0.361834\pi\)
\(312\) 0 0
\(313\) 4141.00 7172.42i 0.747806 1.29524i −0.201067 0.979578i \(-0.564441\pi\)
0.948872 0.315660i \(-0.102226\pi\)
\(314\) 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 \(-0.797055\pi\)
−0.803545 + 0.595244i \(0.797055\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3783.00 + 6552.35i 0.570393 + 0.987950i 0.996525 + 0.0832890i \(0.0265425\pi\)
−0.426132 + 0.904661i \(0.640124\pi\)
\(354\) 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 \(0.0142327\pi\)
−0.460790 + 0.887509i \(0.652434\pi\)
\(368\) 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 \(-0.525689\pi\)
0.903515 0.428557i \(-0.140978\pi\)
\(384\) 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 \(0.353700\pi\)
−0.997954 + 0.0639390i \(0.979634\pi\)
\(398\) 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 0.816772 0.576961i \(-0.195762\pi\)
−0.908049 + 0.418864i \(0.862428\pi\)
\(410\) 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 \(-0.523402\pi\)
−0.0734547 + 0.997299i \(0.523402\pi\)
\(420\) 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 \(-0.547694\pi\)
−0.149276 + 0.988796i \(0.547694\pi\)
\(434\) 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 −0.897758 0.440488i \(-0.854805\pi\)
0.830353 + 0.557238i \(0.188139\pi\)
\(440\) 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 \(-0.222591\pi\)
0.765299 + 0.643675i \(0.222591\pi\)
\(462\) 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 −0.997065 0.0765646i \(-0.975605\pi\)
0.564839 + 0.825201i \(0.308938\pi\)
\(468\) 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 0.534456 0.845196i \(-0.320517\pi\)
−0.999189 + 0.0402542i \(0.987183\pi\)
\(480\) 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 \(-0.302326\pi\)
0.581857 + 0.813291i \(0.302326\pi\)
\(504\) 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 \(0.354803\pi\)
−0.997726 + 0.0673977i \(0.978530\pi\)
\(510\) 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 0.667936 0.744218i \(-0.267178\pi\)
−0.978480 + 0.206341i \(0.933844\pi\)
\(522\) 0 0
\(523\) 2590.00 4486.01i 0.216545 0.375066i −0.737205 0.675669i \(-0.763855\pi\)
0.953749 + 0.300603i \(0.0971880\pi\)
\(524\) 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 0.714049 0.700096i \(-0.246860\pi\)
−0.963325 + 0.268336i \(0.913526\pi\)
\(564\) 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 0.740902 0.671613i \(-0.234398\pi\)
−0.952085 + 0.305833i \(0.901065\pi\)
\(578\) 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 \(-0.287918\pi\)
0.618062 + 0.786130i \(0.287918\pi\)
\(588\) 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 \(0.369101\pi\)
−0.993693 + 0.112131i \(0.964232\pi\)
\(594\) 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 \(-0.703021\pi\)
−0.595438 + 0.803401i \(0.703021\pi\)
\(602\) 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 −0.999885 0.0151784i \(-0.995168\pi\)
0.513087 + 0.858336i \(0.328502\pi\)
\(608\) 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 0.947302 0.320342i \(-0.103798\pi\)
−0.751075 + 0.660217i \(0.770464\pi\)
\(620\) 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 \(-0.536823\pi\)
−0.115426 + 0.993316i \(0.536823\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8844.00 + 15318.3i 0.537393 + 0.930793i 0.999043 + 0.0437305i \(0.0139243\pi\)
−0.461650 + 0.887062i \(0.652742\pi\)
\(648\) 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 0.846173 0.532908i \(-0.178901\pi\)
−0.884598 + 0.466354i \(0.845567\pi\)
\(662\) 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 −0.928711 0.370804i \(-0.879082\pi\)
0.785481 + 0.618886i \(0.212416\pi\)
\(678\) 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 \(0.387297\pi\)
−0.985664 + 0.168720i \(0.946037\pi\)
\(692\) 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 −0.798559 0.601917i \(-0.794404\pi\)
0.920555 + 0.390614i \(0.127737\pi\)
\(720\) 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 \(-0.412851\pi\)
0.270380 + 0.962754i \(0.412851\pi\)
\(728\) 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 −0.663653 0.748040i \(-0.730995\pi\)
0.979649 + 0.200720i \(0.0643282\pi\)
\(734\) 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 \(0.359851\pi\)
−0.996532 + 0.0832121i \(0.973482\pi\)
\(762\) 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 \(-0.645055\pi\)
−0.440093 + 0.897952i \(0.645055\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15171.0 26276.9i −0.705903 1.22266i −0.966365 0.257176i \(-0.917208\pi\)
0.260462 0.965484i \(-0.416125\pi\)
\(774\) 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 \(-0.964580\pi\)
0.400739 0.916192i \(-0.368753\pi\)
\(788\) 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 −1.54745 −0.773724 0.633522i \(-0.781609\pi\)
−0.773724 + 0.633522i \(0.781609\pi\)
\(798\) 0 0
\(799\) −21888.0 −0.969139
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −16020.0 + 27747.5i −0.704027 + 1.21941i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10851.0 18794.5i −0.471571 0.816785i 0.527900 0.849306i \(-0.322980\pi\)
−0.999471 + 0.0325217i \(0.989646\pi\)
\(810\) 0 0
\(811\) 20356.0 0.881376 0.440688 0.897660i \(-0.354735\pi\)
0.440688 + 0.897660i \(0.354735\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10092.0 + 17479.9i 0.433751 + 0.751279i
\(816\) 0 0
\(817\) 6536.00 11320.7i 0.279885 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9945.00 + 17225.2i −0.422756 + 0.732235i −0.996208 0.0870040i \(-0.972271\pi\)
0.573452 + 0.819239i \(0.305604\pi\)
\(822\) 0 0
\(823\) −2116.00 3665.02i −0.0896223 0.155230i 0.817729 0.575603i \(-0.195233\pi\)
−0.907352 + 0.420373i \(0.861899\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9636.00 −0.405171 −0.202586 0.979265i \(-0.564934\pi\)
−0.202586 + 0.979265i \(0.564934\pi\)
\(828\) 0 0
\(829\) 17647.0 + 30565.5i 0.739331 + 1.28056i 0.952797 + 0.303609i \(0.0981916\pi\)
−0.213465 + 0.976951i \(0.568475\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9144.00 + 15837.9i −0.378971 + 0.656398i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3768.00 −0.155049 −0.0775243 0.996990i \(-0.524702\pi\)
−0.0775243 + 0.996990i \(0.524702\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4941.00 + 8558.06i −0.201155 + 0.348410i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2136.00 + 3699.66i 0.0860413 + 0.149028i
\(852\) 0 0
\(853\) 39466.0 1.58416 0.792081 0.610416i \(-0.208998\pi\)
0.792081 + 0.610416i \(0.208998\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17019.0 + 29477.8i 0.678364 + 1.17496i 0.975473 + 0.220117i \(0.0706441\pi\)
−0.297109 + 0.954843i \(0.596023\pi\)
\(858\) 0 0
\(859\) −1682.00 + 2913.31i −0.0668092 + 0.115717i −0.897495 0.441024i \(-0.854615\pi\)
0.830686 + 0.556741i \(0.187949\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6552.00 11348.4i 0.258439 0.447629i −0.707385 0.706828i \(-0.750125\pi\)
0.965824 + 0.259199i \(0.0834586\pi\)
\(864\) 0 0
\(865\) 8118.00 + 14060.8i 0.319099 + 0.552695i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2880.00 −0.112425
\(870\) 0 0
\(871\) −31372.0 54337.9i −1.22044 2.11386i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 20429.0 35384.1i 0.786589 1.36241i −0.141457 0.989944i \(-0.545179\pi\)
0.928045 0.372467i \(-0.121488\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −37374.0 −1.42924 −0.714621 0.699512i \(-0.753401\pi\)
−0.714621 + 0.699512i \(0.753401\pi\)
\(882\) 0 0
\(883\) 9788.00 0.373038 0.186519 0.982451i \(-0.440279\pi\)
0.186519 + 0.982451i \(0.440279\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −25212.0 + 43668.5i −0.954381 + 1.65304i −0.218603 + 0.975814i \(0.570150\pi\)
−0.735778 + 0.677223i \(0.763183\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7296.00 + 12637.0i 0.273406 + 0.473553i
\(894\) 0 0
\(895\) −28296.0 −1.05679
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3024.00 + 5237.72i 0.112187 + 0.194313i
\(900\) 0 0
\(901\) −22914.0 + 39688.2i −0.847254 + 1.46749i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5730.00 9924.65i 0.210466 0.364538i
\(906\) 0 0
\(907\) 6206.00 + 10749.1i 0.227196 + 0.393515i 0.956976 0.290167i \(-0.0937108\pi\)
−0.729780 + 0.683682i \(0.760378\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6576.00 −0.239158 −0.119579 0.992825i \(-0.538154\pi\)
−0.119579 + 0.992825i \(0.538154\pi\)
\(912\) 0 0
\(913\) −1944.00 3367.11i −0.0704677 0.122054i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4132.00 + 7156.83i −0.148316 + 0.256890i −0.930605 0.366025i \(-0.880718\pi\)
0.782289 + 0.622915i \(0.214052\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 52080.0 1.85724
\(924\) 0 0
\(925\) 15842.0 0.563115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19713.0 + 34143.9i −0.696192 + 1.20584i 0.273585 + 0.961848i \(0.411791\pi\)
−0.969777 + 0.243992i \(0.921543\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12312.0 + 21325.0i 0.430637 + 0.745885i
\(936\) 0 0
\(937\) 4678.00 0.163099 0.0815494 0.996669i \(-0.474013\pi\)
0.0815494 + 0.996669i \(0.474013\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8673.00 + 15022.1i 0.300459 + 0.520410i 0.976240 0.216692i \(-0.0695269\pi\)
−0.675781 + 0.737102i \(0.736194\pi\)
\(942\) 0 0
\(943\) −4536.00 + 7856.58i −0.156641 + 0.271310i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9726.00 16845.9i 0.333741 0.578056i −0.649501 0.760360i \(-0.725022\pi\)
0.983242 + 0.182304i \(0.0583556\pi\)
\(948\) 0 0
\(949\) −27590.0 47787.3i −0.943740 1.63461i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4458.00 −0.151531 −0.0757654 0.997126i \(-0.524140\pi\)
−0.0757654 + 0.997126i \(0.524140\pi\)
\(954\) 0 0
\(955\) 12240.0 + 21200.3i 0.414740 + 0.718351i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8623.50 14936.3i 0.289467 0.501371i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −16116.0 −0.537609
\(966\) 0 0
\(967\) 52520.0 1.74657 0.873283 0.487213i \(-0.161987\pi\)
0.873283 + 0.487213i \(0.161987\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5202.00 9010.13i 0.171926 0.297785i −0.767167 0.641447i \(-0.778334\pi\)
0.939093 + 0.343663i \(0.111668\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3783.00 6552.35i −0.123878 0.214563i 0.797416 0.603430i \(-0.206200\pi\)
−0.921294 + 0.388867i \(0.872867\pi\)
\(978\) 0 0
\(979\) 58968.0 1.92505
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22188.0 38430.7i −0.719926 1.24695i −0.961029 0.276449i \(-0.910842\pi\)
0.241103 0.970500i \(-0.422491\pi\)
\(984\) 0 0
\(985\) 1530.00 2650.04i 0.0494922 0.0857231i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2064.00 3574.95i 0.0663614 0.114941i
\(990\) 0 0
\(991\) 13664.0 + 23666.7i 0.437993 + 0.758626i 0.997535 0.0701759i \(-0.0223561\pi\)
−0.559541 + 0.828802i \(0.689023\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8112.00 −0.258460
\(996\) 0 0
\(997\) 1387.00 + 2402.35i 0.0440589 + 0.0763123i 0.887214 0.461358i \(-0.152638\pi\)
−0.843155 + 0.537671i \(0.819304\pi\)
\(998\) 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 1764.4.k.f.1549.1 2
3.2 odd 2 588.4.i.c.373.1 2
7.2 even 3 1764.4.a.j.1.1 1
7.3 odd 6 1764.4.k.l.361.1 2
7.4 even 3 inner 1764.4.k.f.361.1 2
7.5 odd 6 252.4.a.b.1.1 1
7.6 odd 2 1764.4.k.l.1549.1 2
21.2 odd 6 588.4.a.d.1.1 1
21.5 even 6 84.4.a.a.1.1 1
21.11 odd 6 588.4.i.c.361.1 2
21.17 even 6 588.4.i.f.361.1 2
21.20 even 2 588.4.i.f.373.1 2
28.19 even 6 1008.4.a.h.1.1 1
84.23 even 6 2352.4.a.d.1.1 1
84.47 odd 6 336.4.a.k.1.1 1
105.47 odd 12 2100.4.k.j.1849.2 2
105.68 odd 12 2100.4.k.j.1849.1 2
105.89 even 6 2100.4.a.l.1.1 1
168.5 even 6 1344.4.a.q.1.1 1
168.131 odd 6 1344.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.a.1.1 1 21.5 even 6
252.4.a.b.1.1 1 7.5 odd 6
336.4.a.k.1.1 1 84.47 odd 6
588.4.a.d.1.1 1 21.2 odd 6
588.4.i.c.361.1 2 21.11 odd 6
588.4.i.c.373.1 2 3.2 odd 2
588.4.i.f.361.1 2 21.17 even 6
588.4.i.f.373.1 2 21.20 even 2
1008.4.a.h.1.1 1 28.19 even 6
1344.4.a.d.1.1 1 168.131 odd 6
1344.4.a.q.1.1 1 168.5 even 6
1764.4.a.j.1.1 1 7.2 even 3
1764.4.k.f.361.1 2 7.4 even 3 inner
1764.4.k.f.1549.1 2 1.1 even 1 trivial
1764.4.k.l.361.1 2 7.3 odd 6
1764.4.k.l.1549.1 2 7.6 odd 2
2100.4.a.l.1.1 1 105.89 even 6
2100.4.k.j.1849.1 2 105.68 odd 12
2100.4.k.j.1849.2 2 105.47 odd 12
2352.4.a.d.1.1 1 84.23 even 6