Properties

Label 588.4.k.e.521.3
Level $588$
Weight $4$
Character 588.521
Analytic conductor $34.693$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,4,Mod(509,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.509");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 521.3
Character \(\chi\) \(=\) 588.521
Dual form 588.4.k.e.509.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.14452 + 0.730696i) q^{3} +(-3.83273 - 6.63849i) q^{5} +(25.9322 - 7.51816i) q^{9} +O(q^{10})\) \(q+(-5.14452 + 0.730696i) q^{3} +(-3.83273 - 6.63849i) q^{5} +(25.9322 - 7.51816i) q^{9} +(1.13857 + 0.657354i) q^{11} +23.9754i q^{13} +(24.5683 + 31.3513i) q^{15} +(-29.5490 + 51.1804i) q^{17} +(-24.4639 + 14.1243i) q^{19} +(65.7234 - 37.9454i) q^{23} +(33.1203 - 57.3661i) q^{25} +(-127.915 + 57.6258i) q^{27} +302.001i q^{29} +(-80.9549 - 46.7393i) q^{31} +(-6.33773 - 2.54982i) q^{33} +(-133.433 - 231.113i) q^{37} +(-17.5187 - 123.342i) q^{39} -142.471 q^{41} +284.654 q^{43} +(-149.300 - 143.335i) q^{45} +(-104.876 - 181.651i) q^{47} +(114.618 - 284.890i) q^{51} +(545.190 + 314.765i) q^{53} -10.0779i q^{55} +(115.535 - 90.5383i) q^{57} +(365.186 - 632.521i) q^{59} +(471.964 - 272.488i) q^{61} +(159.160 - 91.8912i) q^{65} +(240.556 - 416.655i) q^{67} +(-310.389 + 243.235i) q^{69} -46.5477i q^{71} +(-834.095 - 481.565i) q^{73} +(-128.471 + 319.322i) q^{75} +(-630.818 - 1092.61i) q^{79} +(615.955 - 389.924i) q^{81} +841.130 q^{83} +453.014 q^{85} +(-220.671 - 1553.65i) q^{87} +(641.169 + 1110.54i) q^{89} +(450.626 + 181.298i) q^{93} +(187.528 + 108.269i) q^{95} -60.2806i q^{97} +(34.4677 + 8.48666i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 64 q^{9} - 192 q^{15} - 456 q^{25} + 432 q^{37} - 688 q^{39} + 1248 q^{43} + 1536 q^{51} - 2720 q^{57} + 528 q^{67} - 3744 q^{79} - 3408 q^{81} + 13824 q^{85} + 5088 q^{93} - 15472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −5.14452 + 0.730696i −0.990063 + 0.140622i
\(4\) 0 0
\(5\) −3.83273 6.63849i −0.342810 0.593764i 0.642143 0.766585i \(-0.278045\pi\)
−0.984953 + 0.172820i \(0.944712\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 25.9322 7.51816i 0.960451 0.278450i
\(10\) 0 0
\(11\) 1.13857 + 0.657354i 0.0312084 + 0.0180182i 0.515523 0.856876i \(-0.327598\pi\)
−0.484315 + 0.874894i \(0.660931\pi\)
\(12\) 0 0
\(13\) 23.9754i 0.511505i 0.966742 + 0.255753i \(0.0823233\pi\)
−0.966742 + 0.255753i \(0.917677\pi\)
\(14\) 0 0
\(15\) 24.5683 + 31.3513i 0.422900 + 0.539658i
\(16\) 0 0
\(17\) −29.5490 + 51.1804i −0.421570 + 0.730181i −0.996093 0.0883076i \(-0.971854\pi\)
0.574523 + 0.818488i \(0.305187\pi\)
\(18\) 0 0
\(19\) −24.4639 + 14.1243i −0.295390 + 0.170544i −0.640370 0.768066i \(-0.721219\pi\)
0.344980 + 0.938610i \(0.387886\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 65.7234 37.9454i 0.595838 0.344008i −0.171564 0.985173i \(-0.554882\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(24\) 0 0
\(25\) 33.1203 57.3661i 0.264963 0.458929i
\(26\) 0 0
\(27\) −127.915 + 57.6258i −0.911751 + 0.410744i
\(28\) 0 0
\(29\) 302.001i 1.93380i 0.255151 + 0.966901i \(0.417875\pi\)
−0.255151 + 0.966901i \(0.582125\pi\)
\(30\) 0 0
\(31\) −80.9549 46.7393i −0.469030 0.270795i 0.246804 0.969066i \(-0.420620\pi\)
−0.715834 + 0.698271i \(0.753953\pi\)
\(32\) 0 0
\(33\) −6.33773 2.54982i −0.0334320 0.0134505i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −133.433 231.113i −0.592871 1.02688i −0.993843 0.110793i \(-0.964661\pi\)
0.400972 0.916090i \(-0.368672\pi\)
\(38\) 0 0
\(39\) −17.5187 123.342i −0.0719292 0.506423i
\(40\) 0 0
\(41\) −142.471 −0.542688 −0.271344 0.962482i \(-0.587468\pi\)
−0.271344 + 0.962482i \(0.587468\pi\)
\(42\) 0 0
\(43\) 284.654 1.00952 0.504760 0.863260i \(-0.331581\pi\)
0.504760 + 0.863260i \(0.331581\pi\)
\(44\) 0 0
\(45\) −149.300 143.335i −0.494586 0.474826i
\(46\) 0 0
\(47\) −104.876 181.651i −0.325484 0.563755i 0.656126 0.754651i \(-0.272194\pi\)
−0.981610 + 0.190896i \(0.938861\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 114.618 284.890i 0.314701 0.782207i
\(52\) 0 0
\(53\) 545.190 + 314.765i 1.41297 + 0.815780i 0.995667 0.0929859i \(-0.0296412\pi\)
0.417306 + 0.908766i \(0.362974\pi\)
\(54\) 0 0
\(55\) 10.0779i 0.0247072i
\(56\) 0 0
\(57\) 115.535 90.5383i 0.268473 0.210387i
\(58\) 0 0
\(59\) 365.186 632.521i 0.805817 1.39572i −0.109922 0.993940i \(-0.535060\pi\)
0.915738 0.401775i \(-0.131607\pi\)
\(60\) 0 0
\(61\) 471.964 272.488i 0.990635 0.571943i 0.0851710 0.996366i \(-0.472856\pi\)
0.905464 + 0.424423i \(0.139523\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 159.160 91.8912i 0.303714 0.175349i
\(66\) 0 0
\(67\) 240.556 416.655i 0.438635 0.759739i −0.558949 0.829202i \(-0.688795\pi\)
0.997585 + 0.0694632i \(0.0221286\pi\)
\(68\) 0 0
\(69\) −310.389 + 243.235i −0.541543 + 0.424377i
\(70\) 0 0
\(71\) 46.5477i 0.0778056i −0.999243 0.0389028i \(-0.987614\pi\)
0.999243 0.0389028i \(-0.0123863\pi\)
\(72\) 0 0
\(73\) −834.095 481.565i −1.33731 0.772095i −0.350900 0.936413i \(-0.614124\pi\)
−0.986407 + 0.164318i \(0.947458\pi\)
\(74\) 0 0
\(75\) −128.471 + 319.322i −0.197794 + 0.491628i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −630.818 1092.61i −0.898386 1.55605i −0.829557 0.558422i \(-0.811407\pi\)
−0.0688294 0.997628i \(-0.521926\pi\)
\(80\) 0 0
\(81\) 615.955 389.924i 0.844931 0.534876i
\(82\) 0 0
\(83\) 841.130 1.11236 0.556181 0.831061i \(-0.312266\pi\)
0.556181 + 0.831061i \(0.312266\pi\)
\(84\) 0 0
\(85\) 453.014 0.578074
\(86\) 0 0
\(87\) −220.671 1553.65i −0.271936 1.91459i
\(88\) 0 0
\(89\) 641.169 + 1110.54i 0.763638 + 1.32266i 0.940964 + 0.338507i \(0.109922\pi\)
−0.177326 + 0.984152i \(0.556745\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 450.626 + 181.298i 0.502449 + 0.202148i
\(94\) 0 0
\(95\) 187.528 + 108.269i 0.202525 + 0.116928i
\(96\) 0 0
\(97\) 60.2806i 0.0630986i −0.999502 0.0315493i \(-0.989956\pi\)
0.999502 0.0315493i \(-0.0100441\pi\)
\(98\) 0 0
\(99\) 34.4677 + 8.48666i 0.0349913 + 0.00861557i
\(100\) 0 0
\(101\) 584.358 1012.14i 0.575701 0.997144i −0.420264 0.907402i \(-0.638062\pi\)
0.995965 0.0897419i \(-0.0286042\pi\)
\(102\) 0 0
\(103\) −15.8686 + 9.16173i −0.0151804 + 0.00876439i −0.507571 0.861610i \(-0.669456\pi\)
0.492391 + 0.870374i \(0.336123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 824.968 476.295i 0.745352 0.430329i −0.0786602 0.996901i \(-0.525064\pi\)
0.824012 + 0.566572i \(0.191731\pi\)
\(108\) 0 0
\(109\) 353.940 613.042i 0.311021 0.538704i −0.667563 0.744554i \(-0.732662\pi\)
0.978584 + 0.205849i \(0.0659957\pi\)
\(110\) 0 0
\(111\) 855.322 + 1091.46i 0.731383 + 0.933309i
\(112\) 0 0
\(113\) 9.01287i 0.00750318i 0.999993 + 0.00375159i \(0.00119417\pi\)
−0.999993 + 0.00375159i \(0.998806\pi\)
\(114\) 0 0
\(115\) −503.801 290.870i −0.408519 0.235858i
\(116\) 0 0
\(117\) 180.251 + 621.733i 0.142429 + 0.491276i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −664.636 1151.18i −0.499351 0.864901i
\(122\) 0 0
\(123\) 732.944 104.103i 0.537296 0.0763141i
\(124\) 0 0
\(125\) −1465.95 −1.04895
\(126\) 0 0
\(127\) 387.174 0.270521 0.135260 0.990810i \(-0.456813\pi\)
0.135260 + 0.990810i \(0.456813\pi\)
\(128\) 0 0
\(129\) −1464.41 + 207.996i −0.999488 + 0.141961i
\(130\) 0 0
\(131\) −705.778 1222.44i −0.470718 0.815308i 0.528721 0.848796i \(-0.322672\pi\)
−0.999439 + 0.0334876i \(0.989339\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 872.813 + 628.298i 0.556443 + 0.400558i
\(136\) 0 0
\(137\) 631.197 + 364.422i 0.393626 + 0.227260i 0.683730 0.729735i \(-0.260357\pi\)
−0.290104 + 0.956995i \(0.593690\pi\)
\(138\) 0 0
\(139\) 1262.40i 0.770324i −0.922849 0.385162i \(-0.874146\pi\)
0.922849 0.385162i \(-0.125854\pi\)
\(140\) 0 0
\(141\) 672.269 + 857.874i 0.401527 + 0.512383i
\(142\) 0 0
\(143\) −15.7603 + 27.2977i −0.00921639 + 0.0159632i
\(144\) 0 0
\(145\) 2004.83 1157.49i 1.14822 0.662927i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 817.509 471.989i 0.449483 0.259509i −0.258129 0.966111i \(-0.583106\pi\)
0.707612 + 0.706601i \(0.249773\pi\)
\(150\) 0 0
\(151\) −153.771 + 266.339i −0.0828722 + 0.143539i −0.904483 0.426511i \(-0.859743\pi\)
0.821610 + 0.570050i \(0.193076\pi\)
\(152\) 0 0
\(153\) −381.488 + 1549.37i −0.201578 + 0.818689i
\(154\) 0 0
\(155\) 716.557i 0.371324i
\(156\) 0 0
\(157\) 1614.46 + 932.111i 0.820690 + 0.473825i 0.850654 0.525726i \(-0.176206\pi\)
−0.0299645 + 0.999551i \(0.509539\pi\)
\(158\) 0 0
\(159\) −3034.74 1220.95i −1.51365 0.608978i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1620.61 2806.98i −0.778749 1.34883i −0.932663 0.360749i \(-0.882521\pi\)
0.153913 0.988084i \(-0.450812\pi\)
\(164\) 0 0
\(165\) 7.36384 + 51.8457i 0.00347439 + 0.0244617i
\(166\) 0 0
\(167\) −2021.13 −0.936523 −0.468262 0.883590i \(-0.655119\pi\)
−0.468262 + 0.883590i \(0.655119\pi\)
\(168\) 0 0
\(169\) 1622.18 0.738362
\(170\) 0 0
\(171\) −528.215 + 550.197i −0.236220 + 0.246050i
\(172\) 0 0
\(173\) 350.294 + 606.727i 0.153944 + 0.266639i 0.932674 0.360720i \(-0.117469\pi\)
−0.778730 + 0.627359i \(0.784136\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1416.53 + 3520.86i −0.601541 + 1.49516i
\(178\) 0 0
\(179\) −2484.84 1434.62i −1.03757 0.599043i −0.118428 0.992963i \(-0.537785\pi\)
−0.919145 + 0.393920i \(0.871119\pi\)
\(180\) 0 0
\(181\) 1974.33i 0.810778i −0.914144 0.405389i \(-0.867136\pi\)
0.914144 0.405389i \(-0.132864\pi\)
\(182\) 0 0
\(183\) −2228.92 + 1746.68i −0.900363 + 0.705566i
\(184\) 0 0
\(185\) −1022.83 + 1771.59i −0.406485 + 0.704052i
\(186\) 0 0
\(187\) −67.2873 + 38.8483i −0.0263130 + 0.0151918i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −941.978 + 543.851i −0.356854 + 0.206030i −0.667700 0.744431i \(-0.732721\pi\)
0.310846 + 0.950460i \(0.399388\pi\)
\(192\) 0 0
\(193\) −1917.52 + 3321.24i −0.715161 + 1.23870i 0.247736 + 0.968828i \(0.420314\pi\)
−0.962897 + 0.269868i \(0.913020\pi\)
\(194\) 0 0
\(195\) −751.658 + 589.034i −0.276038 + 0.216316i
\(196\) 0 0
\(197\) 1635.24i 0.591400i 0.955281 + 0.295700i \(0.0955528\pi\)
−0.955281 + 0.295700i \(0.904447\pi\)
\(198\) 0 0
\(199\) 3734.89 + 2156.34i 1.33045 + 0.768134i 0.985368 0.170439i \(-0.0545187\pi\)
0.345079 + 0.938574i \(0.387852\pi\)
\(200\) 0 0
\(201\) −933.096 + 2319.26i −0.327440 + 0.813871i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 546.053 + 945.791i 0.186039 + 0.322229i
\(206\) 0 0
\(207\) 1419.07 1478.13i 0.476484 0.496314i
\(208\) 0 0
\(209\) −37.1386 −0.0122915
\(210\) 0 0
\(211\) −2882.49 −0.940468 −0.470234 0.882542i \(-0.655831\pi\)
−0.470234 + 0.882542i \(0.655831\pi\)
\(212\) 0 0
\(213\) 34.0122 + 239.466i 0.0109412 + 0.0770325i
\(214\) 0 0
\(215\) −1091.00 1889.67i −0.346073 0.599417i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4642.90 + 1867.95i 1.43259 + 0.576367i
\(220\) 0 0
\(221\) −1227.07 708.449i −0.373491 0.215635i
\(222\) 0 0
\(223\) 3378.49i 1.01453i −0.861790 0.507265i \(-0.830656\pi\)
0.861790 0.507265i \(-0.169344\pi\)
\(224\) 0 0
\(225\) 427.594 1736.63i 0.126695 0.514557i
\(226\) 0 0
\(227\) −2538.84 + 4397.39i −0.742328 + 1.28575i 0.209105 + 0.977893i \(0.432945\pi\)
−0.951433 + 0.307857i \(0.900388\pi\)
\(228\) 0 0
\(229\) 3784.94 2185.24i 1.09221 0.630587i 0.158046 0.987432i \(-0.449481\pi\)
0.934164 + 0.356844i \(0.116147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3335.56 + 1925.79i −0.937854 + 0.541470i −0.889287 0.457350i \(-0.848799\pi\)
−0.0485671 + 0.998820i \(0.515465\pi\)
\(234\) 0 0
\(235\) −803.925 + 1392.44i −0.223159 + 0.386522i
\(236\) 0 0
\(237\) 4043.62 + 5160.01i 1.10827 + 1.41426i
\(238\) 0 0
\(239\) 7067.79i 1.91288i −0.291937 0.956438i \(-0.594300\pi\)
0.291937 0.956438i \(-0.405700\pi\)
\(240\) 0 0
\(241\) 2636.64 + 1522.26i 0.704733 + 0.406878i 0.809108 0.587660i \(-0.199951\pi\)
−0.104375 + 0.994538i \(0.533284\pi\)
\(242\) 0 0
\(243\) −2883.87 + 2456.05i −0.761319 + 0.648377i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −338.634 586.532i −0.0872340 0.151094i
\(248\) 0 0
\(249\) −4327.21 + 614.610i −1.10131 + 0.156423i
\(250\) 0 0
\(251\) 5439.29 1.36783 0.683914 0.729562i \(-0.260276\pi\)
0.683914 + 0.729562i \(0.260276\pi\)
\(252\) 0 0
\(253\) 99.7744 0.0247935
\(254\) 0 0
\(255\) −2330.54 + 331.015i −0.572330 + 0.0812902i
\(256\) 0 0
\(257\) 1139.37 + 1973.45i 0.276545 + 0.478990i 0.970524 0.241005i \(-0.0774771\pi\)
−0.693979 + 0.719996i \(0.744144\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2270.50 + 7831.55i 0.538468 + 1.85732i
\(262\) 0 0
\(263\) 1003.97 + 579.643i 0.235390 + 0.135902i 0.613056 0.790039i \(-0.289940\pi\)
−0.377666 + 0.925942i \(0.623273\pi\)
\(264\) 0 0
\(265\) 4825.65i 1.11863i
\(266\) 0 0
\(267\) −4109.97 5244.68i −0.942045 1.20213i
\(268\) 0 0
\(269\) 1873.94 3245.77i 0.424745 0.735680i −0.571651 0.820497i \(-0.693697\pi\)
0.996397 + 0.0848164i \(0.0270304\pi\)
\(270\) 0 0
\(271\) −1771.74 + 1022.91i −0.397142 + 0.229290i −0.685250 0.728308i \(-0.740307\pi\)
0.288108 + 0.957598i \(0.406974\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 75.4197 43.5436i 0.0165381 0.00954827i
\(276\) 0 0
\(277\) −2421.58 + 4194.29i −0.525265 + 0.909785i 0.474302 + 0.880362i \(0.342700\pi\)
−0.999567 + 0.0294232i \(0.990633\pi\)
\(278\) 0 0
\(279\) −2450.73 603.420i −0.525883 0.129483i
\(280\) 0 0
\(281\) 827.961i 0.175772i −0.996131 0.0878862i \(-0.971989\pi\)
0.996131 0.0878862i \(-0.0280112\pi\)
\(282\) 0 0
\(283\) 7402.44 + 4273.80i 1.55487 + 0.897707i 0.997734 + 0.0672878i \(0.0214346\pi\)
0.557140 + 0.830419i \(0.311899\pi\)
\(284\) 0 0
\(285\) −1043.85 419.967i −0.216956 0.0872866i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 710.211 + 1230.12i 0.144557 + 0.250381i
\(290\) 0 0
\(291\) 44.0468 + 310.115i 0.00887308 + 0.0624716i
\(292\) 0 0
\(293\) 7266.94 1.44894 0.724469 0.689307i \(-0.242085\pi\)
0.724469 + 0.689307i \(0.242085\pi\)
\(294\) 0 0
\(295\) −5598.64 −1.10497
\(296\) 0 0
\(297\) −183.521 18.4744i −0.0358551 0.00360940i
\(298\) 0 0
\(299\) 909.756 + 1575.74i 0.175962 + 0.304775i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2266.68 + 5633.95i −0.429760 + 1.06819i
\(304\) 0 0
\(305\) −3617.82 2088.75i −0.679199 0.392136i
\(306\) 0 0
\(307\) 4124.73i 0.766811i −0.923580 0.383406i \(-0.874751\pi\)
0.923580 0.383406i \(-0.125249\pi\)
\(308\) 0 0
\(309\) 74.9418 58.7278i 0.0137971 0.0108120i
\(310\) 0 0
\(311\) −1649.15 + 2856.41i −0.300691 + 0.520812i −0.976293 0.216455i \(-0.930551\pi\)
0.675602 + 0.737267i \(0.263884\pi\)
\(312\) 0 0
\(313\) 5188.30 2995.47i 0.936934 0.540939i 0.0479358 0.998850i \(-0.484736\pi\)
0.888998 + 0.457912i \(0.151402\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6144.35 + 3547.44i −1.08865 + 0.628531i −0.933216 0.359316i \(-0.883010\pi\)
−0.155432 + 0.987847i \(0.549677\pi\)
\(318\) 0 0
\(319\) −198.522 + 343.850i −0.0348436 + 0.0603508i
\(320\) 0 0
\(321\) −3896.04 + 3053.11i −0.677431 + 0.530866i
\(322\) 0 0
\(323\) 1669.43i 0.287584i
\(324\) 0 0
\(325\) 1375.37 + 794.072i 0.234744 + 0.135530i
\(326\) 0 0
\(327\) −1372.90 + 3412.43i −0.232177 + 0.577088i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4581.91 7936.10i −0.760860 1.31785i −0.942408 0.334466i \(-0.891444\pi\)
0.181548 0.983382i \(-0.441889\pi\)
\(332\) 0 0
\(333\) −5197.75 4990.08i −0.855360 0.821186i
\(334\) 0 0
\(335\) −3687.94 −0.601474
\(336\) 0 0
\(337\) 1546.78 0.250025 0.125012 0.992155i \(-0.460103\pi\)
0.125012 + 0.992155i \(0.460103\pi\)
\(338\) 0 0
\(339\) −6.58567 46.3669i −0.00105512 0.00742862i
\(340\) 0 0
\(341\) −61.4486 106.432i −0.00975844 0.0169021i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2804.35 + 1128.26i 0.437627 + 0.176068i
\(346\) 0 0
\(347\) 7966.18 + 4599.27i 1.23241 + 0.711533i 0.967532 0.252749i \(-0.0813346\pi\)
0.264879 + 0.964282i \(0.414668\pi\)
\(348\) 0 0
\(349\) 6916.38i 1.06082i −0.847742 0.530409i \(-0.822038\pi\)
0.847742 0.530409i \(-0.177962\pi\)
\(350\) 0 0
\(351\) −1381.60 3066.81i −0.210098 0.466365i
\(352\) 0 0
\(353\) 1376.71 2384.52i 0.207577 0.359534i −0.743374 0.668876i \(-0.766776\pi\)
0.950951 + 0.309343i \(0.100109\pi\)
\(354\) 0 0
\(355\) −309.007 + 178.405i −0.0461982 + 0.0266725i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3079.84 + 1778.15i −0.452780 + 0.261413i −0.709004 0.705205i \(-0.750855\pi\)
0.256224 + 0.966618i \(0.417522\pi\)
\(360\) 0 0
\(361\) −3030.51 + 5249.00i −0.441830 + 0.765272i
\(362\) 0 0
\(363\) 4260.40 + 5436.64i 0.616013 + 0.786087i
\(364\) 0 0
\(365\) 7382.84i 1.05873i
\(366\) 0 0
\(367\) 4788.83 + 2764.83i 0.681131 + 0.393251i 0.800281 0.599625i \(-0.204684\pi\)
−0.119150 + 0.992876i \(0.538017\pi\)
\(368\) 0 0
\(369\) −3694.58 + 1071.12i −0.521225 + 0.151112i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 566.970 + 982.020i 0.0787039 + 0.136319i 0.902691 0.430289i \(-0.141588\pi\)
−0.823987 + 0.566609i \(0.808255\pi\)
\(374\) 0 0
\(375\) 7541.60 1071.16i 1.03852 0.147506i
\(376\) 0 0
\(377\) −7240.60 −0.989150
\(378\) 0 0
\(379\) 4071.36 0.551799 0.275900 0.961186i \(-0.411024\pi\)
0.275900 + 0.961186i \(0.411024\pi\)
\(380\) 0 0
\(381\) −1991.82 + 282.906i −0.267833 + 0.0380413i
\(382\) 0 0
\(383\) −7069.27 12244.3i −0.943140 1.63357i −0.759433 0.650586i \(-0.774523\pi\)
−0.183707 0.982981i \(-0.558810\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7381.70 2140.07i 0.969594 0.281101i
\(388\) 0 0
\(389\) −9257.16 5344.63i −1.20657 0.696615i −0.244564 0.969633i \(-0.578645\pi\)
−0.962009 + 0.273018i \(0.911978\pi\)
\(390\) 0 0
\(391\) 4485.00i 0.580093i
\(392\) 0 0
\(393\) 4524.12 + 5773.18i 0.580692 + 0.741013i
\(394\) 0 0
\(395\) −4835.51 + 8375.35i −0.615952 + 1.06686i
\(396\) 0 0
\(397\) 8778.93 5068.52i 1.10983 0.640760i 0.171043 0.985263i \(-0.445286\pi\)
0.938785 + 0.344504i \(0.111953\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7292.21 4210.16i 0.908119 0.524303i 0.0282934 0.999600i \(-0.490993\pi\)
0.879825 + 0.475297i \(0.157659\pi\)
\(402\) 0 0
\(403\) 1120.59 1940.92i 0.138513 0.239911i
\(404\) 0 0
\(405\) −4949.30 2594.53i −0.607241 0.318329i
\(406\) 0 0
\(407\) 350.851i 0.0427298i
\(408\) 0 0
\(409\) −8021.64 4631.30i −0.969792 0.559909i −0.0706189 0.997503i \(-0.522497\pi\)
−0.899173 + 0.437594i \(0.855831\pi\)
\(410\) 0 0
\(411\) −3513.49 1413.56i −0.421673 0.169649i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3223.83 5583.83i −0.381329 0.660480i
\(416\) 0 0
\(417\) 922.428 + 6494.42i 0.108325 + 0.762670i
\(418\) 0 0
\(419\) 883.038 0.102958 0.0514788 0.998674i \(-0.483607\pi\)
0.0514788 + 0.998674i \(0.483607\pi\)
\(420\) 0 0
\(421\) 6313.53 0.730886 0.365443 0.930834i \(-0.380918\pi\)
0.365443 + 0.930834i \(0.380918\pi\)
\(422\) 0 0
\(423\) −4085.35 3922.12i −0.469589 0.450828i
\(424\) 0 0
\(425\) 1957.35 + 3390.22i 0.223401 + 0.386941i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 61.1329 151.949i 0.00688001 0.0171007i
\(430\) 0 0
\(431\) 6291.22 + 3632.24i 0.703103 + 0.405937i 0.808502 0.588493i \(-0.200279\pi\)
−0.105399 + 0.994430i \(0.533612\pi\)
\(432\) 0 0
\(433\) 4242.72i 0.470883i 0.971889 + 0.235441i \(0.0756536\pi\)
−0.971889 + 0.235441i \(0.924346\pi\)
\(434\) 0 0
\(435\) −9468.13 + 7419.66i −1.04359 + 0.817806i
\(436\) 0 0
\(437\) −1071.90 + 1856.59i −0.117337 + 0.203233i
\(438\) 0 0
\(439\) −9430.55 + 5444.73i −1.02528 + 0.591943i −0.915628 0.402028i \(-0.868306\pi\)
−0.109648 + 0.993971i \(0.534972\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13925.6 8039.92i 1.49351 0.862276i 0.493534 0.869727i \(-0.335705\pi\)
0.999972 + 0.00745048i \(0.00237158\pi\)
\(444\) 0 0
\(445\) 4914.86 8512.78i 0.523565 0.906842i
\(446\) 0 0
\(447\) −3860.81 + 3025.51i −0.408524 + 0.320138i
\(448\) 0 0
\(449\) 13638.3i 1.43348i 0.697341 + 0.716740i \(0.254366\pi\)
−0.697341 + 0.716740i \(0.745634\pi\)
\(450\) 0 0
\(451\) −162.213 93.6538i −0.0169364 0.00977824i
\(452\) 0 0
\(453\) 596.465 1482.55i 0.0618640 0.153766i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −939.399 1627.09i −0.0961559 0.166547i 0.813935 0.580957i \(-0.197321\pi\)
−0.910090 + 0.414410i \(0.863988\pi\)
\(458\) 0 0
\(459\) 830.451 8249.53i 0.0844490 0.838900i
\(460\) 0 0
\(461\) −2579.48 −0.260604 −0.130302 0.991474i \(-0.541595\pi\)
−0.130302 + 0.991474i \(0.541595\pi\)
\(462\) 0 0
\(463\) 6099.88 0.612280 0.306140 0.951986i \(-0.400962\pi\)
0.306140 + 0.951986i \(0.400962\pi\)
\(464\) 0 0
\(465\) −523.585 3686.34i −0.0522165 0.367635i
\(466\) 0 0
\(467\) 9526.74 + 16500.8i 0.943993 + 1.63504i 0.757755 + 0.652540i \(0.226296\pi\)
0.186239 + 0.982505i \(0.440370\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8986.73 3615.58i −0.879165 0.353710i
\(472\) 0 0
\(473\) 324.099 + 187.119i 0.0315054 + 0.0181897i
\(474\) 0 0
\(475\) 1871.20i 0.180751i
\(476\) 0 0
\(477\) 16504.4 + 4063.73i 1.58425 + 0.390074i
\(478\) 0 0
\(479\) 8990.07 15571.3i 0.857551 1.48532i −0.0167077 0.999860i \(-0.505318\pi\)
0.874258 0.485461i \(-0.161348\pi\)
\(480\) 0 0
\(481\) 5541.01 3199.10i 0.525256 0.303257i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −400.172 + 231.039i −0.0374657 + 0.0216308i
\(486\) 0 0
\(487\) 6157.30 10664.8i 0.572924 0.992333i −0.423340 0.905971i \(-0.639142\pi\)
0.996264 0.0863623i \(-0.0275243\pi\)
\(488\) 0 0
\(489\) 10388.3 + 13256.4i 0.960688 + 1.22592i
\(490\) 0 0
\(491\) 6650.90i 0.611305i −0.952143 0.305653i \(-0.901125\pi\)
0.952143 0.305653i \(-0.0988747\pi\)
\(492\) 0 0
\(493\) −15456.6 8923.85i −1.41203 0.815233i
\(494\) 0 0
\(495\) −75.7669 261.341i −0.00687973 0.0237301i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6376.49 11044.4i −0.572046 0.990813i −0.996356 0.0852957i \(-0.972816\pi\)
0.424310 0.905517i \(-0.360517\pi\)
\(500\) 0 0
\(501\) 10397.7 1476.83i 0.927217 0.131696i
\(502\) 0 0
\(503\) −7044.47 −0.624448 −0.312224 0.950009i \(-0.601074\pi\)
−0.312224 + 0.950009i \(0.601074\pi\)
\(504\) 0 0
\(505\) −8958.76 −0.789425
\(506\) 0 0
\(507\) −8345.35 + 1185.32i −0.731025 + 0.103830i
\(508\) 0 0
\(509\) −2386.47 4133.49i −0.207816 0.359948i 0.743210 0.669058i \(-0.233302\pi\)
−0.951026 + 0.309110i \(0.899969\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2315.38 3216.46i 0.199272 0.276823i
\(514\) 0 0
\(515\) 121.640 + 70.2289i 0.0104080 + 0.00600904i
\(516\) 0 0
\(517\) 275.763i 0.0234585i
\(518\) 0 0
\(519\) −2245.43 2865.36i −0.189910 0.242342i
\(520\) 0 0
\(521\) 7491.53 12975.7i 0.629961 1.09113i −0.357597 0.933876i \(-0.616404\pi\)
0.987559 0.157249i \(-0.0502627\pi\)
\(522\) 0 0
\(523\) −4638.56 + 2678.07i −0.387820 + 0.223908i −0.681215 0.732083i \(-0.738548\pi\)
0.293395 + 0.955991i \(0.405215\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4784.27 2762.20i 0.395458 0.228318i
\(528\) 0 0
\(529\) −3203.79 + 5549.12i −0.263318 + 0.456080i
\(530\) 0 0
\(531\) 4714.68 19148.2i 0.385310 1.56490i
\(532\) 0 0
\(533\) 3415.79i 0.277588i
\(534\) 0 0
\(535\) −6323.76 3651.03i −0.511028 0.295042i
\(536\) 0 0
\(537\) 13831.6 + 5564.78i 1.11150 + 0.447184i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.18626 + 7.25082i 0.000332683 + 0.000576224i 0.866192 0.499712i \(-0.166561\pi\)
−0.865859 + 0.500288i \(0.833227\pi\)
\(542\) 0 0
\(543\) 1442.63 + 10157.0i 0.114014 + 0.802721i
\(544\) 0 0
\(545\) −5426.23 −0.426485
\(546\) 0 0
\(547\) 10274.7 0.803137 0.401568 0.915829i \(-0.368465\pi\)
0.401568 + 0.915829i \(0.368465\pi\)
\(548\) 0 0
\(549\) 10190.4 10614.5i 0.792198 0.825166i
\(550\) 0 0
\(551\) −4265.55 7388.15i −0.329798 0.571226i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3967.46 9861.33i 0.303440 0.754217i
\(556\) 0 0
\(557\) 13316.1 + 7688.05i 1.01296 + 0.584835i 0.912058 0.410061i \(-0.134493\pi\)
0.100906 + 0.994896i \(0.467826\pi\)
\(558\) 0 0
\(559\) 6824.68i 0.516375i
\(560\) 0 0
\(561\) 317.775 249.023i 0.0239152 0.0187411i
\(562\) 0 0
\(563\) 9229.97 15986.8i 0.690936 1.19674i −0.280596 0.959826i \(-0.590532\pi\)
0.971532 0.236910i \(-0.0761346\pi\)
\(564\) 0 0
\(565\) 59.8318 34.5439i 0.00445512 0.00257217i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3033.11 + 1751.17i −0.223471 + 0.129021i −0.607556 0.794277i \(-0.707850\pi\)
0.384086 + 0.923297i \(0.374517\pi\)
\(570\) 0 0
\(571\) 5593.74 9688.63i 0.409966 0.710082i −0.584920 0.811091i \(-0.698874\pi\)
0.994885 + 0.101009i \(0.0322072\pi\)
\(572\) 0 0
\(573\) 4448.64 3486.15i 0.324336 0.254164i
\(574\) 0 0
\(575\) 5027.06i 0.364596i
\(576\) 0 0
\(577\) −8346.61 4818.92i −0.602208 0.347685i 0.167702 0.985838i \(-0.446365\pi\)
−0.769910 + 0.638153i \(0.779699\pi\)
\(578\) 0 0
\(579\) 7437.90 18487.3i 0.533867 1.32696i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 413.825 + 716.765i 0.0293977 + 0.0509183i
\(584\) 0 0
\(585\) 3436.52 3579.53i 0.242876 0.252983i
\(586\) 0 0
\(587\) 3998.01 0.281117 0.140559 0.990072i \(-0.455110\pi\)
0.140559 + 0.990072i \(0.455110\pi\)
\(588\) 0 0
\(589\) 2640.63 0.184729
\(590\) 0 0
\(591\) −1194.86 8412.50i −0.0831641 0.585523i
\(592\) 0 0
\(593\) 5716.00 + 9900.41i 0.395832 + 0.685600i 0.993207 0.116361i \(-0.0371231\pi\)
−0.597375 + 0.801962i \(0.703790\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −20789.8 8364.25i −1.42524 0.573411i
\(598\) 0 0
\(599\) 700.491 + 404.428i 0.0477818 + 0.0275868i 0.523701 0.851902i \(-0.324551\pi\)
−0.475919 + 0.879489i \(0.657884\pi\)
\(600\) 0 0
\(601\) 6813.76i 0.462461i 0.972899 + 0.231231i \(0.0742752\pi\)
−0.972899 + 0.231231i \(0.925725\pi\)
\(602\) 0 0
\(603\) 3105.66 12613.3i 0.209738 0.851830i
\(604\) 0 0
\(605\) −5094.74 + 8824.35i −0.342365 + 0.592993i
\(606\) 0 0
\(607\) 2799.03 1616.02i 0.187165 0.108060i −0.403490 0.914984i \(-0.632203\pi\)
0.590655 + 0.806924i \(0.298870\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4355.15 2514.44i 0.288364 0.166487i
\(612\) 0 0
\(613\) 6607.92 11445.2i 0.435385 0.754110i −0.561942 0.827177i \(-0.689945\pi\)
0.997327 + 0.0730673i \(0.0232788\pi\)
\(614\) 0 0
\(615\) −3500.26 4466.64i −0.229503 0.292866i
\(616\) 0 0
\(617\) 23412.7i 1.52765i −0.645422 0.763826i \(-0.723319\pi\)
0.645422 0.763826i \(-0.276681\pi\)
\(618\) 0 0
\(619\) −9042.06 5220.44i −0.587126 0.338977i 0.176834 0.984241i \(-0.443414\pi\)
−0.763960 + 0.645263i \(0.776748\pi\)
\(620\) 0 0
\(621\) −6220.38 + 8641.16i −0.401957 + 0.558386i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1478.55 + 2560.92i 0.0946271 + 0.163899i
\(626\) 0 0
\(627\) 191.060 27.1370i 0.0121694 0.00172846i
\(628\) 0 0
\(629\) 15771.3 0.999747
\(630\) 0 0
\(631\) −1116.76 −0.0704556 −0.0352278 0.999379i \(-0.511216\pi\)
−0.0352278 + 0.999379i \(0.511216\pi\)
\(632\) 0 0
\(633\) 14829.0 2106.22i 0.931123 0.132251i
\(634\) 0 0
\(635\) −1483.93 2570.25i −0.0927372 0.160626i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −349.953 1207.08i −0.0216650 0.0747285i
\(640\) 0 0
\(641\) 1197.94 + 691.630i 0.0738155 + 0.0426174i 0.536453 0.843930i \(-0.319764\pi\)
−0.462638 + 0.886547i \(0.653097\pi\)
\(642\) 0 0
\(643\) 13841.5i 0.848918i 0.905447 + 0.424459i \(0.139536\pi\)
−0.905447 + 0.424459i \(0.860464\pi\)
\(644\) 0 0
\(645\) 6993.46 + 8924.27i 0.426926 + 0.544795i
\(646\) 0 0
\(647\) 1188.54 2058.62i 0.0722201 0.125089i −0.827654 0.561239i \(-0.810325\pi\)
0.899874 + 0.436150i \(0.143658\pi\)
\(648\) 0 0
\(649\) 831.581 480.113i 0.0502964 0.0290387i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4812.62 + 2778.57i −0.288411 + 0.166514i −0.637225 0.770678i \(-0.719918\pi\)
0.348814 + 0.937192i \(0.386584\pi\)
\(654\) 0 0
\(655\) −5410.12 + 9370.60i −0.322734 + 0.558992i
\(656\) 0 0
\(657\) −25250.4 6217.17i −1.49941 0.369185i
\(658\) 0 0
\(659\) 32145.5i 1.90017i 0.311991 + 0.950085i \(0.399004\pi\)
−0.311991 + 0.950085i \(0.600996\pi\)
\(660\) 0 0
\(661\) 602.813 + 348.034i 0.0354715 + 0.0204795i 0.517631 0.855604i \(-0.326814\pi\)
−0.482159 + 0.876084i \(0.660147\pi\)
\(662\) 0 0
\(663\) 6830.34 + 2748.01i 0.400103 + 0.160971i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11459.6 + 19848.6i 0.665243 + 1.15223i
\(668\) 0 0
\(669\) 2468.65 + 17380.7i 0.142666 + 1.00445i
\(670\) 0 0
\(671\) 716.485 0.0412215
\(672\) 0 0
\(673\) −12403.6 −0.710436 −0.355218 0.934783i \(-0.615593\pi\)
−0.355218 + 0.934783i \(0.615593\pi\)
\(674\) 0 0
\(675\) −930.819 + 9246.57i −0.0530774 + 0.527260i
\(676\) 0 0
\(677\) 9752.83 + 16892.4i 0.553666 + 0.958977i 0.998006 + 0.0631191i \(0.0201048\pi\)
−0.444340 + 0.895858i \(0.646562\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9847.94 24477.6i 0.554146 1.37736i
\(682\) 0 0
\(683\) −25452.0 14694.7i −1.42591 0.823247i −0.429111 0.903252i \(-0.641173\pi\)
−0.996795 + 0.0800044i \(0.974507\pi\)
\(684\) 0 0
\(685\) 5586.92i 0.311628i
\(686\) 0 0
\(687\) −17875.0 + 14007.6i −0.992682 + 0.777911i
\(688\) 0 0
\(689\) −7546.62 + 13071.1i −0.417276 + 0.722743i
\(690\) 0 0
\(691\) −27416.6 + 15829.0i −1.50937 + 0.871436i −0.509432 + 0.860511i \(0.670144\pi\)
−0.999940 + 0.0109252i \(0.996522\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8380.40 + 4838.43i −0.457391 + 0.264075i
\(696\) 0 0
\(697\) 4209.87 7291.72i 0.228781 0.396260i
\(698\) 0 0
\(699\) 15752.7 12344.5i 0.852392 0.667973i
\(700\) 0 0
\(701\) 5372.77i 0.289482i 0.989470 + 0.144741i \(0.0462349\pi\)
−0.989470 + 0.144741i \(0.953765\pi\)
\(702\) 0 0
\(703\) 6528.59 + 3769.28i 0.350257 + 0.202221i
\(704\) 0 0
\(705\) 3118.36 7750.85i 0.166587 0.414062i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16713.7 + 28949.0i 0.885328 + 1.53343i 0.845337 + 0.534233i \(0.179400\pi\)
0.0399911 + 0.999200i \(0.487267\pi\)
\(710\) 0 0
\(711\) −24572.9 23591.1i −1.29614 1.24435i
\(712\) 0 0
\(713\) −7094.18 −0.372621
\(714\) 0 0
\(715\) 241.620 0.0126379
\(716\) 0 0
\(717\) 5164.40 + 36360.4i 0.268993 + 1.89387i
\(718\) 0 0
\(719\) −13795.7 23894.9i −0.715567 1.23940i −0.962740 0.270428i \(-0.912835\pi\)
0.247173 0.968971i \(-0.420498\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −14676.5 5904.73i −0.754947 0.303734i
\(724\) 0 0
\(725\) 17324.6 + 10002.4i 0.887477 + 0.512385i
\(726\) 0 0
\(727\) 16801.8i 0.857143i 0.903508 + 0.428572i \(0.140983\pi\)
−0.903508 + 0.428572i \(0.859017\pi\)
\(728\) 0 0
\(729\) 13041.5 14742.4i 0.662578 0.748993i
\(730\) 0 0
\(731\) −8411.25 + 14568.7i −0.425583 + 0.737131i
\(732\) 0 0
\(733\) −14820.4 + 8556.57i −0.746800 + 0.431165i −0.824537 0.565809i \(-0.808564\pi\)
0.0777364 + 0.996974i \(0.475231\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 547.780 316.261i 0.0273782 0.0158068i
\(738\) 0 0
\(739\) −9855.83 + 17070.8i −0.490599 + 0.849742i −0.999941 0.0108215i \(-0.996555\pi\)
0.509342 + 0.860564i \(0.329889\pi\)
\(740\) 0 0
\(741\) 2170.69 + 2769.99i 0.107614 + 0.137325i
\(742\) 0 0
\(743\) 17443.3i 0.861283i 0.902523 + 0.430642i \(0.141713\pi\)
−0.902523 + 0.430642i \(0.858287\pi\)
\(744\) 0 0
\(745\) −6266.59 3618.02i −0.308175 0.177925i
\(746\) 0 0
\(747\) 21812.3 6323.75i 1.06837 0.309737i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 9793.78 + 16963.3i 0.475872 + 0.824235i 0.999618 0.0276399i \(-0.00879916\pi\)
−0.523746 + 0.851875i \(0.675466\pi\)
\(752\) 0 0
\(753\) −27982.5 + 3974.47i −1.35424 + 0.192348i
\(754\) 0 0
\(755\) 2357.45 0.113638
\(756\) 0 0
\(757\) −4635.88 −0.222581 −0.111291 0.993788i \(-0.535498\pi\)
−0.111291 + 0.993788i \(0.535498\pi\)
\(758\) 0 0
\(759\) −513.291 + 72.9047i −0.0245472 + 0.00348653i
\(760\) 0 0
\(761\) −9585.45 16602.5i −0.456600 0.790854i 0.542179 0.840263i \(-0.317600\pi\)
−0.998779 + 0.0494092i \(0.984266\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 11747.6 3405.83i 0.555211 0.160965i
\(766\) 0 0
\(767\) 15164.9 + 8755.47i 0.713916 + 0.412180i
\(768\) 0 0
\(769\) 22223.5i 1.04213i 0.853517 + 0.521065i \(0.174465\pi\)
−0.853517 + 0.521065i \(0.825535\pi\)
\(770\) 0 0
\(771\) −7303.52 9319.93i −0.341154 0.435342i
\(772\) 0 0
\(773\) −10497.8 + 18182.7i −0.488460 + 0.846037i −0.999912 0.0132746i \(-0.995774\pi\)
0.511452 + 0.859312i \(0.329108\pi\)
\(774\) 0 0
\(775\) −5362.50 + 3096.04i −0.248551 + 0.143501i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3485.40 2012.30i 0.160305 0.0925520i
\(780\) 0 0
\(781\) 30.5983 52.9979i 0.00140191 0.00242819i
\(782\) 0 0
\(783\) −17403.1 38630.5i −0.794299 1.76315i
\(784\) 0 0
\(785\) 14290.1i 0.649728i
\(786\) 0 0
\(787\) 20245.5 + 11688.7i 0.916994 + 0.529427i 0.882675 0.469984i \(-0.155740\pi\)
0.0343191 + 0.999411i \(0.489074\pi\)
\(788\) 0 0
\(789\) −5588.49 2248.39i −0.252162 0.101451i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6533.01 + 11315.5i 0.292552 + 0.506715i
\(794\) 0 0
\(795\) 3526.08 + 24825.6i 0.157305 + 1.10752i
\(796\) 0 0
\(797\) −5445.93 −0.242039 −0.121019 0.992650i \(-0.538616\pi\)
−0.121019 + 0.992650i \(0.538616\pi\)
\(798\) 0 0
\(799\) 12396.0 0.548858
\(800\) 0 0
\(801\) 24976.1 + 23978.2i 1.10173 + 1.05771i
\(802\) 0 0
\(803\) −633.118 1096.59i −0.0278235 0.0481916i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −7268.88 + 18067.2i −0.317071 + 0.788099i
\(808\) 0 0
\(809\) −28161.4 16259.0i −1.22386 0.706597i −0.258122 0.966112i \(-0.583104\pi\)
−0.965739 + 0.259516i \(0.916437\pi\)
\(810\) 0 0
\(811\) 16408.0i 0.710437i 0.934783 + 0.355218i \(0.115594\pi\)
−0.934783 + 0.355218i \(0.884406\pi\)
\(812\) 0 0
\(813\) 8367.31 6557.01i 0.360953 0.282859i
\(814\) 0 0
\(815\) −12422.7 + 21516.8i −0.533926 + 0.924787i
\(816\) 0 0
\(817\) −6963.76 + 4020.53i −0.298202 + 0.172167i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10550.2 + 6091.13i −0.448481 + 0.258931i −0.707188 0.707025i \(-0.750037\pi\)
0.258707 + 0.965956i \(0.416703\pi\)
\(822\) 0 0
\(823\) −2231.32 + 3864.75i −0.0945064 + 0.163690i −0.909402 0.415917i \(-0.863461\pi\)
0.814896 + 0.579607i \(0.196794\pi\)
\(824\) 0 0
\(825\) −356.181 + 279.120i −0.0150311 + 0.0117790i
\(826\) 0 0
\(827\) 2442.53i 0.102703i 0.998681 + 0.0513513i \(0.0163528\pi\)
−0.998681 + 0.0513513i \(0.983647\pi\)
\(828\) 0 0
\(829\) 8327.05 + 4807.62i 0.348866 + 0.201418i 0.664186 0.747567i \(-0.268778\pi\)
−0.315319 + 0.948986i \(0.602112\pi\)
\(830\) 0 0
\(831\) 9393.09 23347.0i 0.392109 0.974609i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7746.43 + 13417.2i 0.321050 + 0.556074i
\(836\) 0 0
\(837\) 13048.7 + 1313.57i 0.538866 + 0.0542457i
\(838\) 0 0
\(839\) 19611.2 0.806979 0.403489 0.914984i \(-0.367797\pi\)
0.403489 + 0.914984i \(0.367797\pi\)
\(840\) 0 0
\(841\) −66815.9 −2.73959
\(842\) 0 0
\(843\) 604.988 + 4259.46i 0.0247175 + 0.174026i
\(844\) 0 0
\(845\) −6217.39 10768.8i −0.253118 0.438413i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −41204.8 16577.7i −1.66566 0.670136i
\(850\) 0 0
\(851\) −17539.3 10126.3i −0.706511 0.407904i
\(852\) 0 0
\(853\) 49160.6i 1.97330i −0.162848 0.986651i \(-0.552068\pi\)
0.162848 0.986651i \(-0.447932\pi\)
\(854\) 0 0
\(855\) 5676.98 + 1397.79i 0.227074 + 0.0559104i
\(856\) 0 0
\(857\) −5684.31 + 9845.52i −0.226572 + 0.392435i −0.956790 0.290780i \(-0.906085\pi\)
0.730218 + 0.683215i \(0.239419\pi\)
\(858\) 0 0
\(859\) 6120.51 3533.68i 0.243107 0.140358i −0.373497 0.927631i \(-0.621841\pi\)
0.616604 + 0.787273i \(0.288508\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23350.3 + 13481.3i −0.921035 + 0.531760i −0.883965 0.467553i \(-0.845136\pi\)
−0.0370701 + 0.999313i \(0.511802\pi\)
\(864\) 0 0
\(865\) 2685.17 4650.85i 0.105547 0.182813i
\(866\) 0 0
\(867\) −4552.54 5809.44i −0.178330 0.227565i
\(868\) 0 0
\(869\) 1658.68i 0.0647491i
\(870\) 0 0
\(871\) 9989.45 + 5767.41i 0.388610 + 0.224364i
\(872\) 0 0
\(873\) −453.199 1563.21i −0.0175698 0.0606031i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19303.6 + 33434.9i 0.743258 + 1.28736i 0.951004 + 0.309177i \(0.100054\pi\)
−0.207747 + 0.978183i \(0.566613\pi\)
\(878\) 0 0
\(879\) −37384.9 + 5309.92i −1.43454 + 0.203753i
\(880\) 0 0
\(881\) −3460.09 −0.132320 −0.0661598 0.997809i \(-0.521075\pi\)
−0.0661598 + 0.997809i \(0.521075\pi\)
\(882\) 0 0
\(883\) 10380.8 0.395631 0.197815 0.980239i \(-0.436615\pi\)
0.197815 + 0.980239i \(0.436615\pi\)
\(884\) 0 0
\(885\) 28802.3 4090.91i 1.09399 0.155383i
\(886\) 0 0
\(887\) 22566.0 + 39085.4i 0.854217 + 1.47955i 0.877369 + 0.479815i \(0.159296\pi\)
−0.0231523 + 0.999732i \(0.507370\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 957.626 39.0561i 0.0360064 0.00146850i
\(892\) 0 0
\(893\) 5131.37 + 2962.60i 0.192290 + 0.111019i
\(894\) 0 0
\(895\) 21994.1i 0.821431i
\(896\) 0 0
\(897\) −5831.65 7441.69i −0.217071 0.277002i
\(898\) 0 0
\(899\) 14115.3 24448.5i 0.523663 0.907011i
\(900\) 0 0
\(901\) −32219.6 + 18602.0i −1.19133 + 0.687817i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −13106.6 + 7567.08i −0.481411 + 0.277943i
\(906\) 0 0
\(907\) −21491.9 + 37225.1i −0.786800 + 1.36278i 0.141118 + 0.989993i \(0.454930\pi\)
−0.927918 + 0.372785i \(0.878403\pi\)
\(908\) 0 0
\(909\) 7544.26 30640.2i 0.275278 1.11801i
\(910\) 0 0
\(911\) 45726.0i 1.66297i −0.555545 0.831487i \(-0.687490\pi\)
0.555545 0.831487i \(-0.312510\pi\)
\(912\) 0 0
\(913\) 957.686 + 552.920i 0.0347150 + 0.0200427i
\(914\) 0 0
\(915\) 20138.2 + 8102.09i 0.727593 + 0.292729i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 12581.6 + 21791.9i 0.451608 + 0.782208i 0.998486 0.0550040i \(-0.0175172\pi\)
−0.546878 + 0.837212i \(0.684184\pi\)
\(920\) 0 0
\(921\) 3013.93 + 21219.8i 0.107831 + 0.759191i
\(922\) 0 0
\(923\) 1116.00 0.0397980
\(924\) 0 0
\(925\) −17677.4 −0.628355
\(926\) 0 0
\(927\) −342.627 + 356.886i −0.0121395 + 0.0126447i
\(928\) 0 0
\(929\) −10145.7 17572.8i −0.358309 0.620609i 0.629370 0.777106i \(-0.283313\pi\)
−0.987678 + 0.156497i \(0.949980\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 6396.92 15899.9i 0.224465 0.557920i
\(934\) 0 0
\(935\) 515.788 + 297.791i 0.0180407 + 0.0104158i
\(936\) 0 0
\(937\) 25708.8i 0.896339i 0.893949 + 0.448170i \(0.147924\pi\)
−0.893949 + 0.448170i \(0.852076\pi\)
\(938\) 0 0
\(939\) −24502.5 + 19201.3i −0.851555 + 0.667318i
\(940\) 0 0
\(941\) 4571.93 7918.81i 0.158385 0.274332i −0.775901 0.630854i \(-0.782704\pi\)
0.934287 + 0.356523i \(0.116038\pi\)
\(942\) 0 0
\(943\) −9363.68 + 5406.12i −0.323354 + 0.186689i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38004.7 + 21942.0i −1.30411 + 0.752925i −0.981105 0.193474i \(-0.938025\pi\)
−0.323000 + 0.946399i \(0.604691\pi\)
\(948\) 0 0
\(949\) 11545.7 19997.7i 0.394931 0.684040i
\(950\) 0 0
\(951\) 29017.7 22739.6i 0.989445 0.775374i
\(952\) 0 0
\(953\) 29621.5i 1.00686i −0.864037 0.503429i \(-0.832072\pi\)
0.864037 0.503429i \(-0.167928\pi\)
\(954\) 0 0
\(955\) 7220.70 + 4168.87i 0.244666 + 0.141258i
\(956\) 0 0
\(957\) 770.050 1914.00i 0.0260106 0.0646509i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −10526.4 18232.2i −0.353341 0.612004i
\(962\) 0 0
\(963\) 17812.3 18553.6i 0.596048 0.620853i
\(964\) 0 0
\(965\) 29397.4 0.980658
\(966\) 0 0
\(967\) 39437.0 1.31149 0.655744 0.754983i \(-0.272355\pi\)
0.655744 + 0.754983i \(0.272355\pi\)
\(968\) 0 0
\(969\) 1219.85 + 8588.43i 0.0404408 + 0.284727i
\(970\) 0 0
\(971\) 11231.4 + 19453.3i 0.371197 + 0.642931i 0.989750 0.142812i \(-0.0456143\pi\)
−0.618553 + 0.785743i \(0.712281\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7655.86 3080.14i −0.251470 0.101173i
\(976\) 0 0
\(977\) −23589.9 13619.6i −0.772474 0.445988i 0.0612826 0.998120i \(-0.480481\pi\)
−0.833756 + 0.552132i \(0.813814\pi\)
\(978\) 0 0
\(979\) 1685.90i 0.0550374i
\(980\) 0 0
\(981\) 4569.48 18558.5i 0.148718 0.604003i
\(982\) 0 0
\(983\) 4358.40 7548.98i 0.141416 0.244939i −0.786614 0.617445i \(-0.788168\pi\)
0.928030 + 0.372506i \(0.121501\pi\)
\(984\) 0 0
\(985\) 10855.5 6267.42i 0.351152 0.202738i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18708.4 10801.3i 0.601510 0.347282i
\(990\) 0 0
\(991\) 14888.8 25788.1i 0.477252 0.826625i −0.522408 0.852696i \(-0.674966\pi\)
0.999660 + 0.0260705i \(0.00829944\pi\)
\(992\) 0 0
\(993\) 29370.6 + 37479.4i 0.938618 + 1.19776i
\(994\) 0 0
\(995\) 33058.7i 1.05330i
\(996\) 0 0
\(997\) −46504.6 26849.5i −1.47725 0.852890i −0.477579 0.878589i \(-0.658486\pi\)
−0.999670 + 0.0256990i \(0.991819\pi\)
\(998\) 0 0
\(999\) 30386.1 + 21873.6i 0.962337 + 0.692743i
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 588.4.k.e.521.3 48
3.2 odd 2 inner 588.4.k.e.521.10 48
7.2 even 3 inner 588.4.k.e.509.15 48
7.3 odd 6 588.4.f.d.293.5 24
7.4 even 3 588.4.f.d.293.20 yes 24
7.5 odd 6 inner 588.4.k.e.509.10 48
7.6 odd 2 inner 588.4.k.e.521.22 48
21.2 odd 6 inner 588.4.k.e.509.22 48
21.5 even 6 inner 588.4.k.e.509.3 48
21.11 odd 6 588.4.f.d.293.6 yes 24
21.17 even 6 588.4.f.d.293.19 yes 24
21.20 even 2 inner 588.4.k.e.521.15 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.f.d.293.5 24 7.3 odd 6
588.4.f.d.293.6 yes 24 21.11 odd 6
588.4.f.d.293.19 yes 24 21.17 even 6
588.4.f.d.293.20 yes 24 7.4 even 3
588.4.k.e.509.3 48 21.5 even 6 inner
588.4.k.e.509.10 48 7.5 odd 6 inner
588.4.k.e.509.15 48 7.2 even 3 inner
588.4.k.e.509.22 48 21.2 odd 6 inner
588.4.k.e.521.3 48 1.1 even 1 trivial
588.4.k.e.521.10 48 3.2 odd 2 inner
588.4.k.e.521.15 48 21.20 even 2 inner
588.4.k.e.521.22 48 7.6 odd 2 inner