Properties

Label 1620.4.a.g.1.3
Level $1620$
Weight $4$
Character 1620.1
Self dual yes
Analytic conductor $95.583$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,4,Mod(1,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

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: yes
Analytic conductor: \(95.5830942093\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.438516.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.58654\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +7.61161 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} +7.61161 q^{7} +12.5749 q^{11} -20.3581 q^{13} -22.6483 q^{17} -18.8592 q^{19} -6.29244 q^{23} +25.0000 q^{25} +206.960 q^{29} +33.7288 q^{31} -38.0580 q^{35} +13.4727 q^{37} -290.559 q^{41} -338.477 q^{43} +263.145 q^{47} -285.063 q^{49} +240.338 q^{53} -62.8746 q^{55} -373.694 q^{59} -454.553 q^{61} +101.791 q^{65} +442.622 q^{67} +371.467 q^{71} +901.209 q^{73} +95.7154 q^{77} -1034.81 q^{79} +34.6531 q^{83} +113.241 q^{85} +1070.24 q^{89} -154.958 q^{91} +94.2962 q^{95} -1382.50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} - 13 q^{7} + 57 q^{11} + 14 q^{13} + 3 q^{17} - 31 q^{19} - 69 q^{23} + 100 q^{25} - 69 q^{29} - 58 q^{31} + 65 q^{35} - 388 q^{37} + 396 q^{41} + 371 q^{43} + 129 q^{47} + 111 q^{49} + 1356 q^{53} - 285 q^{55} + 15 q^{59} - 1441 q^{61} - 70 q^{65} + 368 q^{67} + 168 q^{71} - 955 q^{73} + 342 q^{77} - 1408 q^{79} - 789 q^{83} - 15 q^{85} + 1617 q^{89} - 1406 q^{91} + 155 q^{95} - 1495 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 7.61161 0.410988 0.205494 0.978658i \(-0.434120\pi\)
0.205494 + 0.978658i \(0.434120\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 12.5749 0.344680 0.172340 0.985038i \(-0.444867\pi\)
0.172340 + 0.985038i \(0.444867\pi\)
\(12\) 0 0
\(13\) −20.3581 −0.434333 −0.217166 0.976135i \(-0.569681\pi\)
−0.217166 + 0.976135i \(0.569681\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −22.6483 −0.323119 −0.161559 0.986863i \(-0.551652\pi\)
−0.161559 + 0.986863i \(0.551652\pi\)
\(18\) 0 0
\(19\) −18.8592 −0.227716 −0.113858 0.993497i \(-0.536321\pi\)
−0.113858 + 0.993497i \(0.536321\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.29244 −0.0570463 −0.0285232 0.999593i \(-0.509080\pi\)
−0.0285232 + 0.999593i \(0.509080\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 206.960 1.32522 0.662612 0.748963i \(-0.269448\pi\)
0.662612 + 0.748963i \(0.269448\pi\)
\(30\) 0 0
\(31\) 33.7288 0.195415 0.0977077 0.995215i \(-0.468849\pi\)
0.0977077 + 0.995215i \(0.468849\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −38.0580 −0.183800
\(36\) 0 0
\(37\) 13.4727 0.0598623 0.0299312 0.999552i \(-0.490471\pi\)
0.0299312 + 0.999552i \(0.490471\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −290.559 −1.10677 −0.553387 0.832924i \(-0.686665\pi\)
−0.553387 + 0.832924i \(0.686665\pi\)
\(42\) 0 0
\(43\) −338.477 −1.20040 −0.600200 0.799850i \(-0.704912\pi\)
−0.600200 + 0.799850i \(0.704912\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 263.145 0.816674 0.408337 0.912831i \(-0.366109\pi\)
0.408337 + 0.912831i \(0.366109\pi\)
\(48\) 0 0
\(49\) −285.063 −0.831089
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 240.338 0.622887 0.311444 0.950265i \(-0.399188\pi\)
0.311444 + 0.950265i \(0.399188\pi\)
\(54\) 0 0
\(55\) −62.8746 −0.154146
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −373.694 −0.824590 −0.412295 0.911050i \(-0.635273\pi\)
−0.412295 + 0.911050i \(0.635273\pi\)
\(60\) 0 0
\(61\) −454.553 −0.954091 −0.477045 0.878879i \(-0.658292\pi\)
−0.477045 + 0.878879i \(0.658292\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 101.791 0.194240
\(66\) 0 0
\(67\) 442.622 0.807088 0.403544 0.914960i \(-0.367778\pi\)
0.403544 + 0.914960i \(0.367778\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 371.467 0.620917 0.310458 0.950587i \(-0.399517\pi\)
0.310458 + 0.950587i \(0.399517\pi\)
\(72\) 0 0
\(73\) 901.209 1.44491 0.722456 0.691417i \(-0.243013\pi\)
0.722456 + 0.691417i \(0.243013\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 95.7154 0.141660
\(78\) 0 0
\(79\) −1034.81 −1.47373 −0.736866 0.676039i \(-0.763695\pi\)
−0.736866 + 0.676039i \(0.763695\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 34.6531 0.0458273 0.0229137 0.999737i \(-0.492706\pi\)
0.0229137 + 0.999737i \(0.492706\pi\)
\(84\) 0 0
\(85\) 113.241 0.144503
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1070.24 1.27467 0.637333 0.770588i \(-0.280037\pi\)
0.637333 + 0.770588i \(0.280037\pi\)
\(90\) 0 0
\(91\) −154.958 −0.178506
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 94.2962 0.101838
\(96\) 0 0
\(97\) −1382.50 −1.44713 −0.723567 0.690254i \(-0.757499\pi\)
−0.723567 + 0.690254i \(0.757499\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 138.480 0.136429 0.0682144 0.997671i \(-0.478270\pi\)
0.0682144 + 0.997671i \(0.478270\pi\)
\(102\) 0 0
\(103\) −51.1086 −0.0488920 −0.0244460 0.999701i \(-0.507782\pi\)
−0.0244460 + 0.999701i \(0.507782\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1519.23 −1.37261 −0.686304 0.727315i \(-0.740768\pi\)
−0.686304 + 0.727315i \(0.740768\pi\)
\(108\) 0 0
\(109\) 1692.75 1.48748 0.743742 0.668467i \(-0.233049\pi\)
0.743742 + 0.668467i \(0.233049\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1308.17 −1.08905 −0.544523 0.838746i \(-0.683289\pi\)
−0.544523 + 0.838746i \(0.683289\pi\)
\(114\) 0 0
\(115\) 31.4622 0.0255119
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −172.390 −0.132798
\(120\) 0 0
\(121\) −1172.87 −0.881196
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1977.69 −1.38182 −0.690912 0.722939i \(-0.742791\pi\)
−0.690912 + 0.722939i \(0.742791\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1947.63 −1.29897 −0.649486 0.760374i \(-0.725016\pi\)
−0.649486 + 0.760374i \(0.725016\pi\)
\(132\) 0 0
\(133\) −143.549 −0.0935886
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1260.91 0.786325 0.393163 0.919469i \(-0.371381\pi\)
0.393163 + 0.919469i \(0.371381\pi\)
\(138\) 0 0
\(139\) −661.071 −0.403391 −0.201695 0.979448i \(-0.564645\pi\)
−0.201695 + 0.979448i \(0.564645\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −256.002 −0.149706
\(144\) 0 0
\(145\) −1034.80 −0.592658
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −86.9994 −0.0478340 −0.0239170 0.999714i \(-0.507614\pi\)
−0.0239170 + 0.999714i \(0.507614\pi\)
\(150\) 0 0
\(151\) 1348.25 0.726618 0.363309 0.931669i \(-0.381647\pi\)
0.363309 + 0.931669i \(0.381647\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −168.644 −0.0873924
\(156\) 0 0
\(157\) −516.367 −0.262488 −0.131244 0.991350i \(-0.541897\pi\)
−0.131244 + 0.991350i \(0.541897\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −47.8956 −0.0234454
\(162\) 0 0
\(163\) −2464.89 −1.18445 −0.592224 0.805774i \(-0.701750\pi\)
−0.592224 + 0.805774i \(0.701750\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2981.02 1.38131 0.690653 0.723187i \(-0.257323\pi\)
0.690653 + 0.723187i \(0.257323\pi\)
\(168\) 0 0
\(169\) −1782.55 −0.811355
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1026.45 0.451096 0.225548 0.974232i \(-0.427583\pi\)
0.225548 + 0.974232i \(0.427583\pi\)
\(174\) 0 0
\(175\) 190.290 0.0821977
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 492.431 0.205620 0.102810 0.994701i \(-0.467217\pi\)
0.102810 + 0.994701i \(0.467217\pi\)
\(180\) 0 0
\(181\) −3523.74 −1.44706 −0.723529 0.690294i \(-0.757481\pi\)
−0.723529 + 0.690294i \(0.757481\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −67.3637 −0.0267712
\(186\) 0 0
\(187\) −284.801 −0.111373
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2981.86 −1.12963 −0.564816 0.825217i \(-0.691053\pi\)
−0.564816 + 0.825217i \(0.691053\pi\)
\(192\) 0 0
\(193\) −3252.42 −1.21303 −0.606514 0.795073i \(-0.707432\pi\)
−0.606514 + 0.795073i \(0.707432\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4300.46 −1.55531 −0.777653 0.628694i \(-0.783590\pi\)
−0.777653 + 0.628694i \(0.783590\pi\)
\(198\) 0 0
\(199\) −1549.59 −0.551997 −0.275998 0.961158i \(-0.589008\pi\)
−0.275998 + 0.961158i \(0.589008\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1575.30 0.544652
\(204\) 0 0
\(205\) 1452.80 0.494964
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −237.153 −0.0784892
\(210\) 0 0
\(211\) −2564.99 −0.836877 −0.418439 0.908245i \(-0.637422\pi\)
−0.418439 + 0.908245i \(0.637422\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1692.38 0.536835
\(216\) 0 0
\(217\) 256.731 0.0803134
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 461.077 0.140341
\(222\) 0 0
\(223\) −2499.47 −0.750570 −0.375285 0.926909i \(-0.622455\pi\)
−0.375285 + 0.926909i \(0.622455\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2917.54 −0.853057 −0.426528 0.904474i \(-0.640264\pi\)
−0.426528 + 0.904474i \(0.640264\pi\)
\(228\) 0 0
\(229\) −3904.29 −1.12665 −0.563324 0.826236i \(-0.690478\pi\)
−0.563324 + 0.826236i \(0.690478\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 529.427 0.148858 0.0744290 0.997226i \(-0.476287\pi\)
0.0744290 + 0.997226i \(0.476287\pi\)
\(234\) 0 0
\(235\) −1315.73 −0.365228
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3854.43 1.04319 0.521595 0.853193i \(-0.325337\pi\)
0.521595 + 0.853193i \(0.325337\pi\)
\(240\) 0 0
\(241\) 1713.23 0.457921 0.228960 0.973436i \(-0.426467\pi\)
0.228960 + 0.973436i \(0.426467\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1425.32 0.371674
\(246\) 0 0
\(247\) 383.939 0.0989046
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7789.93 1.95895 0.979474 0.201573i \(-0.0646053\pi\)
0.979474 + 0.201573i \(0.0646053\pi\)
\(252\) 0 0
\(253\) −79.1270 −0.0196627
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4350.88 1.05603 0.528016 0.849234i \(-0.322936\pi\)
0.528016 + 0.849234i \(0.322936\pi\)
\(258\) 0 0
\(259\) 102.549 0.0246027
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6189.73 −1.45124 −0.725618 0.688098i \(-0.758446\pi\)
−0.725618 + 0.688098i \(0.758446\pi\)
\(264\) 0 0
\(265\) −1201.69 −0.278564
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5497.65 −1.24609 −0.623044 0.782187i \(-0.714104\pi\)
−0.623044 + 0.782187i \(0.714104\pi\)
\(270\) 0 0
\(271\) −6473.63 −1.45109 −0.725545 0.688175i \(-0.758412\pi\)
−0.725545 + 0.688175i \(0.758412\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 314.373 0.0689360
\(276\) 0 0
\(277\) −6639.69 −1.44022 −0.720109 0.693861i \(-0.755908\pi\)
−0.720109 + 0.693861i \(0.755908\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2423.44 −0.514486 −0.257243 0.966347i \(-0.582814\pi\)
−0.257243 + 0.966347i \(0.582814\pi\)
\(282\) 0 0
\(283\) 2182.30 0.458390 0.229195 0.973380i \(-0.426391\pi\)
0.229195 + 0.973380i \(0.426391\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2211.62 −0.454871
\(288\) 0 0
\(289\) −4400.05 −0.895594
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5783.64 1.15319 0.576594 0.817031i \(-0.304382\pi\)
0.576594 + 0.817031i \(0.304382\pi\)
\(294\) 0 0
\(295\) 1868.47 0.368768
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 128.102 0.0247771
\(300\) 0 0
\(301\) −2576.35 −0.493351
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2272.76 0.426682
\(306\) 0 0
\(307\) 3781.94 0.703084 0.351542 0.936172i \(-0.385657\pi\)
0.351542 + 0.936172i \(0.385657\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5358.79 −0.977072 −0.488536 0.872544i \(-0.662469\pi\)
−0.488536 + 0.872544i \(0.662469\pi\)
\(312\) 0 0
\(313\) −1146.63 −0.207066 −0.103533 0.994626i \(-0.533015\pi\)
−0.103533 + 0.994626i \(0.533015\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1328.96 0.235464 0.117732 0.993045i \(-0.462438\pi\)
0.117732 + 0.993045i \(0.462438\pi\)
\(318\) 0 0
\(319\) 2602.51 0.456779
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 427.129 0.0735793
\(324\) 0 0
\(325\) −508.953 −0.0868666
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2002.96 0.335643
\(330\) 0 0
\(331\) 10544.9 1.75106 0.875528 0.483167i \(-0.160514\pi\)
0.875528 + 0.483167i \(0.160514\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2213.11 −0.360941
\(336\) 0 0
\(337\) −3318.42 −0.536398 −0.268199 0.963364i \(-0.586428\pi\)
−0.268199 + 0.963364i \(0.586428\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 424.138 0.0673558
\(342\) 0 0
\(343\) −4780.57 −0.752556
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5230.96 −0.809258 −0.404629 0.914481i \(-0.632599\pi\)
−0.404629 + 0.914481i \(0.632599\pi\)
\(348\) 0 0
\(349\) −2859.66 −0.438608 −0.219304 0.975657i \(-0.570379\pi\)
−0.219304 + 0.975657i \(0.570379\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10993.4 1.65757 0.828784 0.559569i \(-0.189033\pi\)
0.828784 + 0.559569i \(0.189033\pi\)
\(354\) 0 0
\(355\) −1857.34 −0.277682
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3419.15 −0.502662 −0.251331 0.967901i \(-0.580868\pi\)
−0.251331 + 0.967901i \(0.580868\pi\)
\(360\) 0 0
\(361\) −6503.33 −0.948145
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4506.05 −0.646184
\(366\) 0 0
\(367\) −7785.47 −1.10735 −0.553676 0.832732i \(-0.686775\pi\)
−0.553676 + 0.832732i \(0.686775\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1829.36 0.255999
\(372\) 0 0
\(373\) −9123.79 −1.26652 −0.633260 0.773939i \(-0.718284\pi\)
−0.633260 + 0.773939i \(0.718284\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4213.32 −0.575589
\(378\) 0 0
\(379\) 2117.60 0.287003 0.143501 0.989650i \(-0.454164\pi\)
0.143501 + 0.989650i \(0.454164\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3239.27 0.432165 0.216082 0.976375i \(-0.430672\pi\)
0.216082 + 0.976375i \(0.430672\pi\)
\(384\) 0 0
\(385\) −478.577 −0.0633521
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6103.04 0.795466 0.397733 0.917501i \(-0.369797\pi\)
0.397733 + 0.917501i \(0.369797\pi\)
\(390\) 0 0
\(391\) 142.513 0.0184327
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5174.03 0.659073
\(396\) 0 0
\(397\) 9935.69 1.25607 0.628033 0.778187i \(-0.283860\pi\)
0.628033 + 0.778187i \(0.283860\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11715.9 −1.45902 −0.729508 0.683973i \(-0.760251\pi\)
−0.729508 + 0.683973i \(0.760251\pi\)
\(402\) 0 0
\(403\) −686.656 −0.0848753
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 169.419 0.0206334
\(408\) 0 0
\(409\) 7690.95 0.929812 0.464906 0.885360i \(-0.346088\pi\)
0.464906 + 0.885360i \(0.346088\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2844.41 −0.338897
\(414\) 0 0
\(415\) −173.265 −0.0204946
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4633.54 0.540246 0.270123 0.962826i \(-0.412936\pi\)
0.270123 + 0.962826i \(0.412936\pi\)
\(420\) 0 0
\(421\) 3867.10 0.447675 0.223837 0.974626i \(-0.428142\pi\)
0.223837 + 0.974626i \(0.428142\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −566.207 −0.0646237
\(426\) 0 0
\(427\) −3459.88 −0.392120
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5735.22 0.640965 0.320482 0.947254i \(-0.396155\pi\)
0.320482 + 0.947254i \(0.396155\pi\)
\(432\) 0 0
\(433\) 2696.51 0.299274 0.149637 0.988741i \(-0.452189\pi\)
0.149637 + 0.988741i \(0.452189\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 118.671 0.0129904
\(438\) 0 0
\(439\) −9039.61 −0.982773 −0.491386 0.870942i \(-0.663510\pi\)
−0.491386 + 0.870942i \(0.663510\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8959.37 0.960886 0.480443 0.877026i \(-0.340476\pi\)
0.480443 + 0.877026i \(0.340476\pi\)
\(444\) 0 0
\(445\) −5351.21 −0.570048
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10716.1 −1.12634 −0.563169 0.826342i \(-0.690418\pi\)
−0.563169 + 0.826342i \(0.690418\pi\)
\(450\) 0 0
\(451\) −3653.76 −0.381483
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 774.790 0.0798302
\(456\) 0 0
\(457\) −15370.6 −1.57331 −0.786657 0.617390i \(-0.788190\pi\)
−0.786657 + 0.617390i \(0.788190\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7155.35 −0.722902 −0.361451 0.932391i \(-0.617719\pi\)
−0.361451 + 0.932391i \(0.617719\pi\)
\(462\) 0 0
\(463\) 18448.9 1.85182 0.925911 0.377743i \(-0.123300\pi\)
0.925911 + 0.377743i \(0.123300\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3276.70 0.324684 0.162342 0.986735i \(-0.448095\pi\)
0.162342 + 0.986735i \(0.448095\pi\)
\(468\) 0 0
\(469\) 3369.07 0.331704
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4256.32 −0.413754
\(474\) 0 0
\(475\) −471.481 −0.0455432
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7742.87 −0.738582 −0.369291 0.929314i \(-0.620399\pi\)
−0.369291 + 0.929314i \(0.620399\pi\)
\(480\) 0 0
\(481\) −274.280 −0.0260002
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6912.52 0.647178
\(486\) 0 0
\(487\) 18292.5 1.70208 0.851041 0.525099i \(-0.175972\pi\)
0.851041 + 0.525099i \(0.175972\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19322.1 −1.77596 −0.887979 0.459883i \(-0.847891\pi\)
−0.887979 + 0.459883i \(0.847891\pi\)
\(492\) 0 0
\(493\) −4687.29 −0.428205
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2827.47 0.255189
\(498\) 0 0
\(499\) 2480.64 0.222543 0.111271 0.993790i \(-0.464508\pi\)
0.111271 + 0.993790i \(0.464508\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5598.09 0.496236 0.248118 0.968730i \(-0.420188\pi\)
0.248118 + 0.968730i \(0.420188\pi\)
\(504\) 0 0
\(505\) −692.402 −0.0610128
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8550.84 −0.744616 −0.372308 0.928109i \(-0.621433\pi\)
−0.372308 + 0.928109i \(0.621433\pi\)
\(510\) 0 0
\(511\) 6859.65 0.593842
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 255.543 0.0218652
\(516\) 0 0
\(517\) 3309.03 0.281491
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9207.53 0.774260 0.387130 0.922025i \(-0.373467\pi\)
0.387130 + 0.922025i \(0.373467\pi\)
\(522\) 0 0
\(523\) 4862.02 0.406503 0.203252 0.979127i \(-0.434849\pi\)
0.203252 + 0.979127i \(0.434849\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −763.901 −0.0631424
\(528\) 0 0
\(529\) −12127.4 −0.996746
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5915.24 0.480708
\(534\) 0 0
\(535\) 7596.13 0.613849
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3584.65 −0.286460
\(540\) 0 0
\(541\) 5654.94 0.449399 0.224700 0.974428i \(-0.427860\pi\)
0.224700 + 0.974428i \(0.427860\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8463.74 −0.665223
\(546\) 0 0
\(547\) −3574.52 −0.279407 −0.139703 0.990193i \(-0.544615\pi\)
−0.139703 + 0.990193i \(0.544615\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3903.11 −0.301775
\(552\) 0 0
\(553\) −7876.55 −0.605687
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10361.8 −0.788230 −0.394115 0.919061i \(-0.628949\pi\)
−0.394115 + 0.919061i \(0.628949\pi\)
\(558\) 0 0
\(559\) 6890.75 0.521373
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12751.9 0.954580 0.477290 0.878746i \(-0.341619\pi\)
0.477290 + 0.878746i \(0.341619\pi\)
\(564\) 0 0
\(565\) 6540.84 0.487036
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5557.24 0.409441 0.204720 0.978821i \(-0.434372\pi\)
0.204720 + 0.978821i \(0.434372\pi\)
\(570\) 0 0
\(571\) 1543.35 0.113112 0.0565562 0.998399i \(-0.481988\pi\)
0.0565562 + 0.998399i \(0.481988\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −157.311 −0.0114093
\(576\) 0 0
\(577\) 4507.17 0.325192 0.162596 0.986693i \(-0.448013\pi\)
0.162596 + 0.986693i \(0.448013\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 263.766 0.0188345
\(582\) 0 0
\(583\) 3022.24 0.214697
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1968.90 −0.138442 −0.0692208 0.997601i \(-0.522051\pi\)
−0.0692208 + 0.997601i \(0.522051\pi\)
\(588\) 0 0
\(589\) −636.100 −0.0444992
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18751.9 1.29857 0.649283 0.760547i \(-0.275069\pi\)
0.649283 + 0.760547i \(0.275069\pi\)
\(594\) 0 0
\(595\) 861.950 0.0593891
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2736.59 0.186668 0.0933340 0.995635i \(-0.470248\pi\)
0.0933340 + 0.995635i \(0.470248\pi\)
\(600\) 0 0
\(601\) −6214.47 −0.421786 −0.210893 0.977509i \(-0.567637\pi\)
−0.210893 + 0.977509i \(0.567637\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5864.36 0.394083
\(606\) 0 0
\(607\) 15237.7 1.01891 0.509457 0.860496i \(-0.329846\pi\)
0.509457 + 0.860496i \(0.329846\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5357.14 −0.354708
\(612\) 0 0
\(613\) 9390.77 0.618743 0.309371 0.950941i \(-0.399881\pi\)
0.309371 + 0.950941i \(0.399881\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9597.64 0.626234 0.313117 0.949715i \(-0.398627\pi\)
0.313117 + 0.949715i \(0.398627\pi\)
\(618\) 0 0
\(619\) 20136.4 1.30751 0.653756 0.756705i \(-0.273192\pi\)
0.653756 + 0.756705i \(0.273192\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8146.26 0.523873
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −305.135 −0.0193426
\(630\) 0 0
\(631\) −6466.60 −0.407974 −0.203987 0.978974i \(-0.565390\pi\)
−0.203987 + 0.978974i \(0.565390\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9888.45 0.617971
\(636\) 0 0
\(637\) 5803.36 0.360969
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22555.5 −1.38984 −0.694921 0.719086i \(-0.744560\pi\)
−0.694921 + 0.719086i \(0.744560\pi\)
\(642\) 0 0
\(643\) 18114.3 1.11098 0.555488 0.831525i \(-0.312532\pi\)
0.555488 + 0.831525i \(0.312532\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23680.5 −1.43891 −0.719456 0.694538i \(-0.755609\pi\)
−0.719456 + 0.694538i \(0.755609\pi\)
\(648\) 0 0
\(649\) −4699.17 −0.284220
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10761.9 0.644939 0.322469 0.946580i \(-0.395487\pi\)
0.322469 + 0.946580i \(0.395487\pi\)
\(654\) 0 0
\(655\) 9738.15 0.580918
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3650.17 0.215767 0.107883 0.994164i \(-0.465593\pi\)
0.107883 + 0.994164i \(0.465593\pi\)
\(660\) 0 0
\(661\) −3031.97 −0.178411 −0.0892056 0.996013i \(-0.528433\pi\)
−0.0892056 + 0.996013i \(0.528433\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 717.746 0.0418541
\(666\) 0 0
\(667\) −1302.28 −0.0755992
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5715.97 −0.328856
\(672\) 0 0
\(673\) 7852.53 0.449766 0.224883 0.974386i \(-0.427800\pi\)
0.224883 + 0.974386i \(0.427800\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1629.74 −0.0925202 −0.0462601 0.998929i \(-0.514730\pi\)
−0.0462601 + 0.998929i \(0.514730\pi\)
\(678\) 0 0
\(679\) −10523.1 −0.594755
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14384.1 0.805847 0.402923 0.915234i \(-0.367994\pi\)
0.402923 + 0.915234i \(0.367994\pi\)
\(684\) 0 0
\(685\) −6304.54 −0.351655
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4892.84 −0.270540
\(690\) 0 0
\(691\) 33385.6 1.83799 0.918993 0.394273i \(-0.129004\pi\)
0.918993 + 0.394273i \(0.129004\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3305.36 0.180402
\(696\) 0 0
\(697\) 6580.67 0.357619
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13436.1 −0.723929 −0.361965 0.932192i \(-0.617894\pi\)
−0.361965 + 0.932192i \(0.617894\pi\)
\(702\) 0 0
\(703\) −254.086 −0.0136316
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1054.06 0.0560707
\(708\) 0 0
\(709\) 25316.2 1.34100 0.670501 0.741908i \(-0.266079\pi\)
0.670501 + 0.741908i \(0.266079\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −212.237 −0.0111477
\(714\) 0 0
\(715\) 1280.01 0.0669505
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1384.99 −0.0718378 −0.0359189 0.999355i \(-0.511436\pi\)
−0.0359189 + 0.999355i \(0.511436\pi\)
\(720\) 0 0
\(721\) −389.019 −0.0200940
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5174.00 0.265045
\(726\) 0 0
\(727\) 34360.0 1.75288 0.876439 0.481513i \(-0.159912\pi\)
0.876439 + 0.481513i \(0.159912\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7665.92 0.387872
\(732\) 0 0
\(733\) −12812.9 −0.645642 −0.322821 0.946460i \(-0.604631\pi\)
−0.322821 + 0.946460i \(0.604631\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5565.94 0.278187
\(738\) 0 0
\(739\) 32839.5 1.63467 0.817335 0.576163i \(-0.195450\pi\)
0.817335 + 0.576163i \(0.195450\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25350.6 1.25172 0.625858 0.779937i \(-0.284749\pi\)
0.625858 + 0.779937i \(0.284749\pi\)
\(744\) 0 0
\(745\) 434.997 0.0213920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11563.8 −0.564126
\(750\) 0 0
\(751\) 3471.46 0.168675 0.0843377 0.996437i \(-0.473123\pi\)
0.0843377 + 0.996437i \(0.473123\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6741.27 −0.324954
\(756\) 0 0
\(757\) −33545.9 −1.61063 −0.805316 0.592846i \(-0.798004\pi\)
−0.805316 + 0.592846i \(0.798004\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5998.42 −0.285733 −0.142866 0.989742i \(-0.545632\pi\)
−0.142866 + 0.989742i \(0.545632\pi\)
\(762\) 0 0
\(763\) 12884.5 0.611339
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7607.71 0.358146
\(768\) 0 0
\(769\) 24208.1 1.13520 0.567598 0.823306i \(-0.307873\pi\)
0.567598 + 0.823306i \(0.307873\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2134.55 −0.0993199 −0.0496600 0.998766i \(-0.515814\pi\)
−0.0496600 + 0.998766i \(0.515814\pi\)
\(774\) 0 0
\(775\) 843.221 0.0390831
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5479.72 0.252030
\(780\) 0 0
\(781\) 4671.18 0.214018
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2581.84 0.117388
\(786\) 0 0
\(787\) 15744.3 0.713117 0.356558 0.934273i \(-0.383950\pi\)
0.356558 + 0.934273i \(0.383950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9957.27 −0.447585
\(792\) 0 0
\(793\) 9253.84 0.414393
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30160.5 1.34045 0.670226 0.742157i \(-0.266197\pi\)
0.670226 + 0.742157i \(0.266197\pi\)
\(798\) 0 0
\(799\) −5959.79 −0.263883
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11332.6 0.498032
\(804\) 0 0
\(805\) 239.478 0.0104851
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31677.9 1.37668 0.688340 0.725388i \(-0.258340\pi\)
0.688340 + 0.725388i \(0.258340\pi\)
\(810\) 0 0
\(811\) 14294.2 0.618912 0.309456 0.950914i \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12324.4 0.529701
\(816\) 0 0
\(817\) 6383.41 0.273350
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25274.3 −1.07440 −0.537198 0.843456i \(-0.680517\pi\)
−0.537198 + 0.843456i \(0.680517\pi\)
\(822\) 0 0
\(823\) −1414.08 −0.0598928 −0.0299464 0.999552i \(-0.509534\pi\)
−0.0299464 + 0.999552i \(0.509534\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −19366.4 −0.814311 −0.407155 0.913359i \(-0.633479\pi\)
−0.407155 + 0.913359i \(0.633479\pi\)
\(828\) 0 0
\(829\) −14280.3 −0.598283 −0.299141 0.954209i \(-0.596700\pi\)
−0.299141 + 0.954209i \(0.596700\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6456.20 0.268540
\(834\) 0 0
\(835\) −14905.1 −0.617739
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26112.9 −1.07451 −0.537257 0.843418i \(-0.680540\pi\)
−0.537257 + 0.843418i \(0.680540\pi\)
\(840\) 0 0
\(841\) 18443.4 0.756220
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8912.73 0.362849
\(846\) 0 0
\(847\) −8927.44 −0.362161
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −84.7765 −0.00341493
\(852\) 0 0
\(853\) 21239.0 0.852534 0.426267 0.904597i \(-0.359828\pi\)
0.426267 + 0.904597i \(0.359828\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41563.4 −1.65669 −0.828343 0.560222i \(-0.810716\pi\)
−0.828343 + 0.560222i \(0.810716\pi\)
\(858\) 0 0
\(859\) −5885.90 −0.233788 −0.116894 0.993144i \(-0.537294\pi\)
−0.116894 + 0.993144i \(0.537294\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11237.9 0.443272 0.221636 0.975130i \(-0.428860\pi\)
0.221636 + 0.975130i \(0.428860\pi\)
\(864\) 0 0
\(865\) −5132.25 −0.201736
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −13012.6 −0.507966
\(870\) 0 0
\(871\) −9010.96 −0.350545
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −951.451 −0.0367599
\(876\) 0 0
\(877\) −9433.27 −0.363214 −0.181607 0.983371i \(-0.558130\pi\)
−0.181607 + 0.983371i \(0.558130\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −29845.4 −1.14134 −0.570668 0.821181i \(-0.693315\pi\)
−0.570668 + 0.821181i \(0.693315\pi\)
\(882\) 0 0
\(883\) 10785.9 0.411070 0.205535 0.978650i \(-0.434107\pi\)
0.205535 + 0.978650i \(0.434107\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −30705.7 −1.16234 −0.581171 0.813782i \(-0.697405\pi\)
−0.581171 + 0.813782i \(0.697405\pi\)
\(888\) 0 0
\(889\) −15053.4 −0.567914
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4962.72 −0.185970
\(894\) 0 0
\(895\) −2462.15 −0.0919561
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6980.52 0.258969
\(900\) 0 0
\(901\) −5443.26 −0.201267
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17618.7 0.647144
\(906\) 0 0
\(907\) 7062.97 0.258569 0.129285 0.991608i \(-0.458732\pi\)
0.129285 + 0.991608i \(0.458732\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5057.09 −0.183918 −0.0919588 0.995763i \(-0.529313\pi\)
−0.0919588 + 0.995763i \(0.529313\pi\)
\(912\) 0 0
\(913\) 435.760 0.0157958
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14824.6 −0.533862
\(918\) 0 0
\(919\) 48440.1 1.73873 0.869364 0.494173i \(-0.164529\pi\)
0.869364 + 0.494173i \(0.164529\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7562.38 −0.269684
\(924\) 0 0
\(925\) 336.819 0.0119725
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10086.5 −0.356220 −0.178110 0.984011i \(-0.556998\pi\)
−0.178110 + 0.984011i \(0.556998\pi\)
\(930\) 0 0
\(931\) 5376.08 0.189252
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1424.00 0.0498074
\(936\) 0 0
\(937\) −22271.7 −0.776504 −0.388252 0.921553i \(-0.626921\pi\)
−0.388252 + 0.921553i \(0.626921\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 18339.1 0.635322 0.317661 0.948204i \(-0.397103\pi\)
0.317661 + 0.948204i \(0.397103\pi\)
\(942\) 0 0
\(943\) 1828.33 0.0631374
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10141.9 −0.348012 −0.174006 0.984745i \(-0.555671\pi\)
−0.174006 + 0.984745i \(0.555671\pi\)
\(948\) 0 0
\(949\) −18346.9 −0.627573
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 46973.4 1.59666 0.798331 0.602219i \(-0.205717\pi\)
0.798331 + 0.602219i \(0.205717\pi\)
\(954\) 0 0
\(955\) 14909.3 0.505187
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9597.53 0.323171
\(960\) 0 0
\(961\) −28653.4 −0.961813
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16262.1 0.542482
\(966\) 0 0
\(967\) −3021.79 −0.100490 −0.0502452 0.998737i \(-0.516000\pi\)
−0.0502452 + 0.998737i \(0.516000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −13426.4 −0.443743 −0.221871 0.975076i \(-0.571217\pi\)
−0.221871 + 0.975076i \(0.571217\pi\)
\(972\) 0 0
\(973\) −5031.82 −0.165789
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −56527.9 −1.85106 −0.925531 0.378672i \(-0.876381\pi\)
−0.925531 + 0.378672i \(0.876381\pi\)
\(978\) 0 0
\(979\) 13458.2 0.439352
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 45626.5 1.48043 0.740213 0.672372i \(-0.234725\pi\)
0.740213 + 0.672372i \(0.234725\pi\)
\(984\) 0 0
\(985\) 21502.3 0.695554
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2129.85 0.0684784
\(990\) 0 0
\(991\) 55864.3 1.79071 0.895353 0.445358i \(-0.146924\pi\)
0.895353 + 0.445358i \(0.146924\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7747.94 0.246860
\(996\) 0 0
\(997\) 19012.0 0.603927 0.301963 0.953320i \(-0.402358\pi\)
0.301963 + 0.953320i \(0.402358\pi\)
\(998\) 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 1620.4.a.g.1.3 4
3.2 odd 2 1620.4.a.h.1.3 4
9.2 odd 6 540.4.i.b.361.2 8
9.4 even 3 180.4.i.b.61.4 8
9.5 odd 6 540.4.i.b.181.2 8
9.7 even 3 180.4.i.b.121.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.4.i.b.61.4 8 9.4 even 3
180.4.i.b.121.4 yes 8 9.7 even 3
540.4.i.b.181.2 8 9.5 odd 6
540.4.i.b.361.2 8 9.2 odd 6
1620.4.a.g.1.3 4 1.1 even 1 trivial
1620.4.a.h.1.3 4 3.2 odd 2