Properties

Label 588.4.k.e.521.15
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 521.15
Character \(\chi\) \(=\) 588.521
Dual form 588.4.k.e.509.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.93946 + 4.82063i) q^{3} +(-3.83273 - 6.63849i) q^{5} +(-19.4770 + 18.6988i) q^{9} +O(q^{10})\) \(q+(1.93946 + 4.82063i) q^{3} +(-3.83273 - 6.63849i) q^{5} +(-19.4770 + 18.6988i) q^{9} +(-1.13857 - 0.657354i) q^{11} -23.9754i q^{13} +(24.5683 - 31.3513i) q^{15} +(-29.5490 + 51.1804i) q^{17} +(24.4639 - 14.1243i) q^{19} +(-65.7234 + 37.9454i) q^{23} +(33.1203 - 57.3661i) q^{25} +(-127.915 - 57.6258i) q^{27} -302.001i q^{29} +(80.9549 + 46.7393i) q^{31} +(0.960652 - 6.76354i) q^{33} +(-133.433 - 231.113i) q^{37} +(115.576 - 46.4992i) q^{39} -142.471 q^{41} +284.654 q^{43} +(198.782 + 57.6302i) q^{45} +(-104.876 - 181.651i) q^{47} +(-304.031 - 43.1827i) q^{51} +(-545.190 - 314.765i) q^{53} +10.0779i q^{55} +(115.535 + 90.5383i) q^{57} +(365.186 - 632.521i) q^{59} +(-471.964 + 272.488i) q^{61} +(-159.160 + 91.8912i) q^{65} +(240.556 - 416.655i) q^{67} +(-310.389 - 243.235i) q^{69} +46.5477i q^{71} +(834.095 + 481.565i) q^{73} +(340.776 + 48.4018i) q^{75} +(-630.818 - 1092.61i) q^{79} +(29.7071 - 728.394i) q^{81} +841.130 q^{83} +453.014 q^{85} +(1455.84 - 585.719i) q^{87} +(641.169 + 1110.54i) q^{89} +(-68.3045 + 480.903i) q^{93} +(-187.528 - 108.269i) q^{95} +60.2806i q^{97} +(34.4677 - 8.48666i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 64 q^{9}+O(q^{10}) \) 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
\(3\) 1.93946 + 4.82063i 0.373249 + 0.927731i
\(4\) 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
\(9\) −19.4770 + 18.6988i −0.721370 + 0.692549i
\(10\) 0 0
\(11\) −1.13857 0.657354i −0.0312084 0.0180182i 0.484315 0.874894i \(-0.339069\pi\)
−0.515523 + 0.856876i \(0.672402\pi\)
\(12\) 0 0
\(13\) 23.9754i 0.511505i −0.966742 0.255753i \(-0.917677\pi\)
0.966742 0.255753i \(-0.0823233\pi\)
\(14\) 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 −0.344980 0.938610i \(-0.612114\pi\)
0.640370 + 0.768066i \(0.278781\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −65.7234 + 37.9454i −0.595838 + 0.344008i −0.767403 0.641165i \(-0.778451\pi\)
0.171564 + 0.985173i \(0.445118\pi\)
\(24\) 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 \(-0.582125\pi\)
0.255151 0.966901i \(-0.417875\pi\)
\(30\) 0 0
\(31\) 80.9549 + 46.7393i 0.469030 + 0.270795i 0.715834 0.698271i \(-0.246047\pi\)
−0.246804 + 0.969066i \(0.579380\pi\)
\(32\) 0 0
\(33\) 0.960652 6.76354i 0.00506752 0.0356782i
\(34\) 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
\(39\) 115.576 46.4992i 0.474540 0.190919i
\(40\) 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
\(45\) 198.782 + 57.6302i 0.658504 + 0.190911i
\(46\) 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
\(51\) −304.031 43.1827i −0.834762 0.118564i
\(52\) 0 0
\(53\) −545.190 314.765i −1.41297 0.815780i −0.417306 0.908766i \(-0.637026\pi\)
−0.995667 + 0.0929859i \(0.970359\pi\)
\(54\) 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 −0.905464 0.424423i \(-0.860477\pi\)
−0.0851710 + 0.996366i \(0.527144\pi\)
\(62\) 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 \(0.0123863\pi\)
−0.999243 + 0.0389028i \(0.987614\pi\)
\(72\) 0 0
\(73\) 834.095 + 481.565i 1.33731 + 0.772095i 0.986407 0.164318i \(-0.0525423\pi\)
0.350900 + 0.936413i \(0.385876\pi\)
\(74\) 0 0
\(75\) 340.776 + 48.4018i 0.524659 + 0.0745194i
\(76\) 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
\(81\) 29.7071 728.394i 0.0407504 0.999169i
\(82\) 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
\(87\) 1455.84 585.719i 1.79405 0.721790i
\(88\) 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
\(93\) −68.3045 + 480.903i −0.0761596 + 0.536207i
\(94\) 0 0
\(95\) −187.528 108.269i −0.202525 0.116928i
\(96\) 0 0
\(97\) 60.2806i 0.0630986i 0.999502 + 0.0315493i \(0.0100441\pi\)
−0.999502 + 0.0315493i \(0.989956\pi\)
\(98\) 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 −0.492391 0.870374i \(-0.663877\pi\)
0.507571 + 0.861610i \(0.330544\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −824.968 + 476.295i −0.745352 + 0.430329i −0.824012 0.566572i \(-0.808269\pi\)
0.0786602 + 0.996901i \(0.474936\pi\)
\(108\) 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 \(-0.998806\pi\)
0.999993 0.00375159i \(-0.00119417\pi\)
\(114\) 0 0
\(115\) 503.801 + 290.870i 0.408519 + 0.235858i
\(116\) 0 0
\(117\) 448.311 + 466.968i 0.354243 + 0.368985i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −664.636 1151.18i −0.499351 0.864901i
\(122\) 0 0
\(123\) −276.316 686.800i −0.202558 0.503469i
\(124\) 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
\(129\) 552.075 + 1372.21i 0.376802 + 0.936563i
\(130\) 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
\(135\) 107.716 + 1070.03i 0.0686718 + 0.682172i
\(136\) 0 0
\(137\) −631.197 364.422i −0.393626 0.227260i 0.290104 0.956995i \(-0.406310\pi\)
−0.683730 + 0.729735i \(0.739643\pi\)
\(138\) 0 0
\(139\) 1262.40i 0.770324i 0.922849 + 0.385162i \(0.125854\pi\)
−0.922849 + 0.385162i \(0.874146\pi\)
\(140\) 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 −0.707612 0.706601i \(-0.750227\pi\)
0.258129 + 0.966111i \(0.416894\pi\)
\(150\) 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 0.0299645 0.999551i \(-0.490461\pi\)
−0.850654 + 0.525726i \(0.823794\pi\)
\(158\) 0 0
\(159\) 459.996 3238.63i 0.229434 1.61535i
\(160\) 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
\(165\) −48.5816 + 19.5456i −0.0229217 + 0.00922195i
\(166\) 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
\(171\) −212.377 + 732.546i −0.0949758 + 0.327597i
\(172\) 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
\(177\) 3757.41 + 533.680i 1.59562 + 0.226632i
\(178\) 0 0
\(179\) 2484.84 + 1434.62i 1.03757 + 0.599043i 0.919145 0.393920i \(-0.128881\pi\)
0.118428 + 0.992963i \(0.462215\pi\)
\(180\) 0 0
\(181\) 1974.33i 0.810778i 0.914144 + 0.405389i \(0.132864\pi\)
−0.914144 + 0.405389i \(0.867136\pi\)
\(182\) 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 −0.310846 0.950460i \(-0.600612\pi\)
0.667700 + 0.744431i \(0.267279\pi\)
\(192\) 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 \(-0.904447\pi\)
0.955281 0.295700i \(-0.0955528\pi\)
\(198\) 0 0
\(199\) −3734.89 2156.34i −1.33045 0.768134i −0.345079 0.938574i \(-0.612148\pi\)
−0.985368 + 0.170439i \(0.945481\pi\)
\(200\) 0 0
\(201\) 2475.09 + 351.546i 0.868554 + 0.123364i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 546.053 + 945.791i 0.186039 + 0.322229i
\(206\) 0 0
\(207\) 570.560 1968.02i 0.191578 0.660804i
\(208\) 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
\(213\) −224.390 + 90.2774i −0.0721827 + 0.0290409i
\(214\) 0 0
\(215\) −1091.00 1889.67i −0.346073 0.599417i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −703.755 + 4954.84i −0.217148 + 1.52885i
\(220\) 0 0
\(221\) 1227.07 + 708.449i 0.373491 + 0.215635i
\(222\) 0 0
\(223\) 3378.49i 1.01453i 0.861790 + 0.507265i \(0.169344\pi\)
−0.861790 + 0.507265i \(0.830656\pi\)
\(224\) 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 −0.934164 0.356844i \(-0.883853\pi\)
−0.158046 + 0.987432i \(0.550519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3335.56 1925.79i 0.937854 0.541470i 0.0485671 0.998820i \(-0.484535\pi\)
0.889287 + 0.457350i \(0.151201\pi\)
\(234\) 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 \(0.405700\pi\)
−0.291937 + 0.956438i \(0.594300\pi\)
\(240\) 0 0
\(241\) −2636.64 1522.26i −0.704733 0.406878i 0.104375 0.994538i \(-0.466716\pi\)
−0.809108 + 0.587660i \(0.800049\pi\)
\(242\) 0 0
\(243\) 3568.94 1269.48i 0.942171 0.335134i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −338.634 586.532i −0.0872340 0.151094i
\(248\) 0 0
\(249\) 1631.34 + 4054.78i 0.415188 + 1.03197i
\(250\) 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
\(255\) 878.602 + 2183.81i 0.215765 + 0.536297i
\(256\) 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
\(261\) 5647.08 + 5882.08i 1.33925 + 1.39499i
\(262\) 0 0
\(263\) −1003.97 579.643i −0.235390 0.135902i 0.377666 0.925942i \(-0.376727\pi\)
−0.613056 + 0.790039i \(0.710060\pi\)
\(264\) 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 −0.288108 0.957598i \(-0.593026\pi\)
0.685250 + 0.728308i \(0.259693\pi\)
\(272\) 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 \(0.0280112\pi\)
−0.996131 + 0.0878862i \(0.971989\pi\)
\(282\) 0 0
\(283\) −7402.44 4273.80i −1.55487 0.897707i −0.997734 0.0672878i \(-0.978565\pi\)
−0.557140 0.830419i \(-0.688101\pi\)
\(284\) 0 0
\(285\) 158.224 1113.98i 0.0328854 0.231532i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 710.211 + 1230.12i 0.144557 + 0.250381i
\(290\) 0 0
\(291\) −290.590 + 116.912i −0.0585385 + 0.0235515i
\(292\) 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
\(297\) 107.760 + 149.697i 0.0210534 + 0.0292467i
\(298\) 0 0
\(299\) 909.756 + 1575.74i 0.175962 + 0.304775i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6012.49 + 853.977i 1.13996 + 0.161913i
\(304\) 0 0
\(305\) 3617.82 + 2088.75i 0.679199 + 0.392136i
\(306\) 0 0
\(307\) 4124.73i 0.766811i 0.923580 + 0.383406i \(0.125249\pi\)
−0.923580 + 0.383406i \(0.874751\pi\)
\(308\) 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 −0.888998 0.457912i \(-0.848598\pi\)
−0.0479358 + 0.998850i \(0.515264\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6144.35 3547.44i 1.08865 0.628531i 0.155432 0.987847i \(-0.450323\pi\)
0.933216 + 0.359316i \(0.116990\pi\)
\(318\) 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
\(327\) 3641.70 + 517.245i 0.615861 + 0.0874731i
\(328\) 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
\(333\) 6920.41 + 2006.34i 1.13885 + 0.330170i
\(334\) 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
\(339\) 43.4477 17.4801i 0.00696094 0.00280055i
\(340\) 0 0
\(341\) −61.4486 106.432i −0.00975844 0.0169021i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −425.074 + 2992.77i −0.0663340 + 0.467030i
\(346\) 0 0
\(347\) −7966.18 4599.27i −1.23241 0.711533i −0.264879 0.964282i \(-0.585332\pi\)
−0.967532 + 0.252749i \(0.918665\pi\)
\(348\) 0 0
\(349\) 6916.38i 1.06082i 0.847742 + 0.530409i \(0.177962\pi\)
−0.847742 + 0.530409i \(0.822038\pi\)
\(350\) 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 −0.256224 0.966618i \(-0.582478\pi\)
0.709004 + 0.705205i \(0.249145\pi\)
\(360\) 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 0.119150 0.992876i \(-0.461983\pi\)
−0.800281 + 0.599625i \(0.795316\pi\)
\(368\) 0 0
\(369\) 2774.90 2664.04i 0.391479 0.375838i
\(370\) 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
\(375\) −2843.15 7066.80i −0.391519 0.973141i
\(376\) 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
\(381\) 750.908 + 1866.42i 0.100972 + 0.250970i
\(382\) 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
\(387\) −5544.21 + 5322.70i −0.728237 + 0.699142i
\(388\) 0 0
\(389\) 9257.16 + 5344.63i 1.20657 + 0.696615i 0.962009 0.273018i \(-0.0880219\pi\)
0.244564 + 0.969633i \(0.421355\pi\)
\(390\) 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 −0.938785 0.344504i \(-0.888047\pi\)
−0.171043 + 0.985263i \(0.554714\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7292.21 + 4210.16i −0.908119 + 0.524303i −0.879825 0.475297i \(-0.842341\pi\)
−0.0282934 + 0.999600i \(0.509007\pi\)
\(402\) 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 0.899173 0.437594i \(-0.144169\pi\)
0.0706189 + 0.997503i \(0.477503\pi\)
\(410\) 0 0
\(411\) 532.563 3749.55i 0.0639158 0.450004i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3223.83 5583.83i −0.381329 0.660480i
\(416\) 0 0
\(417\) −6085.55 + 2448.37i −0.714654 + 0.287523i
\(418\) 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
\(423\) 5439.33 + 1576.95i 0.625223 + 0.181262i
\(424\) 0 0
\(425\) 1957.35 + 3390.22i 0.223401 + 0.386941i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −162.158 23.0320i −0.0182496 0.00259206i
\(430\) 0 0
\(431\) −6291.22 3632.24i −0.703103 0.405937i 0.105399 0.994430i \(-0.466388\pi\)
−0.808502 + 0.588493i \(0.799721\pi\)
\(432\) 0 0
\(433\) 4242.72i 0.470883i −0.971889 0.235441i \(-0.924346\pi\)
0.971889 0.235441i \(-0.0756536\pi\)
\(434\) 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 0.109648 0.993971i \(-0.465028\pi\)
0.915628 + 0.402028i \(0.131694\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13925.6 + 8039.92i −1.49351 + 0.862276i −0.999972 0.00745048i \(-0.997628\pi\)
−0.493534 + 0.869727i \(0.664295\pi\)
\(444\) 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 \(-0.745634\pi\)
0.697341 0.716740i \(-0.254366\pi\)
\(450\) 0 0
\(451\) 162.213 + 93.6538i 0.0169364 + 0.00977824i
\(452\) 0 0
\(453\) −1582.16 224.720i −0.164098 0.0233074i
\(454\) 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
\(459\) 6729.08 4843.96i 0.684284 0.492585i
\(460\) 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
\(465\) 3454.26 1389.73i 0.344489 0.138596i
\(466\) 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
\(471\) 1362.18 9590.53i 0.133261 0.938234i
\(472\) 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 \(0.0988747\pi\)
−0.952143 + 0.305653i \(0.901125\pi\)
\(492\) 0 0
\(493\) 15456.6 + 8923.85i 1.41203 + 0.815233i
\(494\) 0 0
\(495\) −188.444 196.286i −0.0171110 0.0178231i
\(496\) 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
\(501\) −3919.89 9743.10i −0.349556 0.868842i
\(502\) 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
\(507\) 3146.15 + 7819.94i 0.275593 + 0.685002i
\(508\) 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
\(513\) −3943.23 + 396.951i −0.339372 + 0.0341634i
\(514\) 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 −0.293395 0.955991i \(-0.594785\pi\)
0.681215 + 0.732083i \(0.261452\pi\)
\(524\) 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
\(537\) −2096.54 + 14760.9i −0.168478 + 1.18618i
\(538\) 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
\(543\) −9517.52 + 3829.13i −0.752184 + 0.302622i
\(544\) 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
\(549\) 4097.22 14132.4i 0.318516 1.09865i
\(550\) 0 0
\(551\) −4265.55 7388.15i −0.329798 0.571226i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10523.9 1494.75i −0.804891 0.114322i
\(556\) 0 0
\(557\) −13316.1 7688.05i −1.01296 0.584835i −0.100906 0.994896i \(-0.532174\pi\)
−0.912058 + 0.410061i \(0.865507\pi\)
\(558\) 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 −0.384086 0.923297i \(-0.625483\pi\)
0.607556 + 0.794277i \(0.292150\pi\)
\(570\) 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 0.769910 0.638153i \(-0.220301\pi\)
−0.167702 + 0.985838i \(0.553635\pi\)
\(578\) 0 0
\(579\) −19729.4 2802.25i −1.41611 0.201136i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 413.825 + 716.765i 0.0293977 + 0.0509183i
\(584\) 0 0
\(585\) 1381.70 4765.87i 0.0976521 0.336828i
\(586\) 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
\(591\) 7882.87 3171.47i 0.548660 0.220739i
\(592\) 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
\(597\) 3151.25 22186.6i 0.216034 1.52100i
\(598\) 0 0
\(599\) −700.491 404.428i −0.0477818 0.0275868i 0.475919 0.879489i \(-0.342116\pi\)
−0.523701 + 0.851902i \(0.675449\pi\)
\(600\) 0 0
\(601\) 6813.76i 0.462461i −0.972899 0.231231i \(-0.925725\pi\)
0.972899 0.231231i \(-0.0742752\pi\)
\(602\) 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 −0.590655 0.806924i \(-0.701130\pi\)
0.403490 + 0.914984i \(0.367797\pi\)
\(608\) 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 \(0.276681\pi\)
−0.645422 + 0.763826i \(0.723319\pi\)
\(618\) 0 0
\(619\) 9042.06 + 5220.44i 0.587126 + 0.338977i 0.763960 0.645263i \(-0.223252\pi\)
−0.176834 + 0.984241i \(0.556586\pi\)
\(620\) 0 0
\(621\) 10593.7 1066.43i 0.684555 0.0689117i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1478.55 + 2560.92i 0.0946271 + 0.163899i
\(626\) 0 0
\(627\) −72.0287 179.031i −0.00458780 0.0114032i
\(628\) 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
\(633\) −5590.47 13895.4i −0.351029 0.872502i
\(634\) 0 0
\(635\) −1483.93 2570.25i −0.0927372 0.160626i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −870.388 906.610i −0.0538842 0.0561267i
\(640\) 0 0
\(641\) −1197.94 691.630i −0.0738155 0.0426174i 0.462638 0.886547i \(-0.346903\pi\)
−0.536453 + 0.843930i \(0.680236\pi\)
\(642\) 0 0
\(643\) 13841.5i 0.848918i −0.905447 0.424459i \(-0.860464\pi\)
0.905447 0.424459i \(-0.139536\pi\)
\(644\) 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 −0.348814 0.937192i \(-0.613416\pi\)
0.637225 + 0.770678i \(0.280082\pi\)
\(654\) 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 \(-0.600996\pi\)
0.311991 0.950085i \(-0.399004\pi\)
\(660\) 0 0
\(661\) −602.813 348.034i −0.0354715 0.0204795i 0.482159 0.876084i \(-0.339853\pi\)
−0.517631 + 0.855604i \(0.673186\pi\)
\(662\) 0 0
\(663\) −1035.32 + 7289.26i −0.0606464 + 0.426985i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11459.6 + 19848.6i 0.665243 + 1.15223i
\(668\) 0 0
\(669\) −16286.5 + 6552.44i −0.941212 + 0.378673i
\(670\) 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
\(675\) −7542.36 + 5429.40i −0.430082 + 0.309597i
\(676\) 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
\(681\) −26122.2 3710.23i −1.46990 0.208776i
\(682\) 0 0
\(683\) 25452.0 + 14694.7i 1.42591 + 0.823247i 0.996795 0.0800044i \(-0.0254934\pi\)
0.429111 + 0.903252i \(0.358827\pi\)
\(684\) 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 \(-0.329856\pi\)
0.999940 0.0109252i \(-0.00347768\pi\)
\(692\) 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 \(-0.953765\pi\)
0.989470 0.144741i \(-0.0462349\pi\)
\(702\) 0 0
\(703\) −6528.59 3769.28i −0.350257 0.202221i
\(704\) 0 0
\(705\) −8271.61 1174.85i −0.441882 0.0627622i
\(706\) 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
\(711\) 32716.9 + 9485.17i 1.72571 + 0.500312i
\(712\) 0 0
\(713\) −7094.18 −0.372621
\(714\) 0 0
\(715\) 241.620 0.0126379
\(716\) 0 0
\(717\) −34071.2 + 13707.7i −1.77463 + 0.713979i
\(718\) 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
\(723\) 2224.62 15662.6i 0.114432 0.805670i
\(724\) 0 0
\(725\) −17324.6 10002.4i −0.887477 0.512385i
\(726\) 0 0
\(727\) 16801.8i 0.857143i −0.903508 0.428572i \(-0.859017\pi\)
0.903508 0.428572i \(-0.140983\pi\)
\(728\) 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 −0.0777364 0.996974i \(-0.524769\pi\)
0.824537 + 0.565809i \(0.191436\pi\)
\(734\) 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 \(-0.858287\pi\)
0.902523 0.430642i \(-0.141713\pi\)
\(744\) 0 0
\(745\) 6266.59 + 3618.02i 0.308175 + 0.177925i
\(746\) 0 0
\(747\) −16382.7 + 15728.1i −0.802424 + 0.770365i
\(748\) 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
\(753\) 10549.3 + 26220.8i 0.510541 + 1.26898i
\(754\) 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
\(759\) 193.508 + 480.976i 0.00925416 + 0.0230017i
\(760\) 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
\(765\) −8823.35 + 8470.83i −0.417005 + 0.400345i
\(766\) 0 0
\(767\) −15164.9 8755.47i −0.713916 0.412180i
\(768\) 0 0
\(769\) 22223.5i 1.04213i −0.853517 0.521065i \(-0.825535\pi\)
0.853517 0.521065i \(-0.174465\pi\)
\(770\) 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 −0.0343191 0.999411i \(-0.510926\pi\)
−0.882675 + 0.469984i \(0.844260\pi\)
\(788\) 0 0
\(789\) 847.085 5963.97i 0.0382218 0.269104i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6533.01 + 11315.5i 0.292552 + 0.506715i
\(794\) 0 0
\(795\) −23262.7 + 9359.14i −1.03779 + 0.417528i
\(796\) 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
\(801\) −33253.8 9640.82i −1.46687 0.425270i
\(802\) 0 0
\(803\) −633.118 1096.59i −0.0278235 0.0481916i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19281.1 + 2738.57i 0.841049 + 0.119457i
\(808\) 0 0
\(809\) 28161.4 + 16259.0i 1.22386 + 0.706597i 0.965739 0.259516i \(-0.0835629\pi\)
0.258122 + 0.966112i \(0.416896\pi\)
\(810\) 0 0
\(811\) 16408.0i 0.710437i −0.934783 0.355218i \(-0.884406\pi\)
0.934783 0.355218i \(-0.115594\pi\)
\(812\) 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 −0.258707 0.965956i \(-0.583297\pi\)
0.707188 + 0.707025i \(0.249963\pi\)
\(822\) 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 \(-0.983647\pi\)
0.998681 0.0513513i \(-0.0163528\pi\)
\(828\) 0 0
\(829\) −8327.05 4807.62i −0.348866 0.201418i 0.315319 0.948986i \(-0.397888\pi\)
−0.664186 + 0.747567i \(0.731222\pi\)
\(830\) 0 0
\(831\) −24915.7 3538.87i −1.04009 0.147728i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7746.43 + 13417.2i 0.321050 + 0.556074i
\(836\) 0 0
\(837\) −7661.95 10643.8i −0.316411 0.439548i
\(838\) 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
\(843\) −3991.30 + 1605.80i −0.163069 + 0.0656068i
\(844\) 0 0
\(845\) −6217.39 10768.8i −0.253118 0.438413i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 6245.69 43973.3i 0.252475 1.77757i
\(850\) 0 0
\(851\) 17539.3 + 10126.3i 0.706511 + 0.407904i
\(852\) 0 0
\(853\) 49160.6i 1.97330i 0.162848 + 0.986651i \(0.447932\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(854\) 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 −0.616604 0.787273i \(-0.711492\pi\)
0.373497 + 0.927631i \(0.378159\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23350.3 13481.3i 0.921035 0.531760i 0.0370701 0.999313i \(-0.488198\pi\)
0.883965 + 0.467553i \(0.154864\pi\)
\(864\) 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
\(873\) −1127.18 1174.08i −0.0436989 0.0455175i
\(874\) 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
\(879\) 14093.9 + 35031.2i 0.540815 + 1.34423i
\(880\) 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
\(885\) −10858.3 26989.0i −0.412428 1.02511i
\(886\) 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
\(891\) −512.637 + 809.801i −0.0192749 + 0.0304482i
\(892\) 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 \(0.312510\pi\)
−0.555545 + 0.831487i \(0.687490\pi\)
\(912\) 0 0
\(913\) −957.686 552.920i −0.0347150 0.0200427i
\(914\) 0 0
\(915\) −3052.48 + 21491.2i −0.110286 + 0.776479i
\(916\) 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
\(921\) −19883.8 + 7999.75i −0.711395 + 0.286211i
\(922\) 0 0
\(923\) 1116.00 0.0397980
\(924\) 0 0
\(925\) −17677.4 −0.628355
\(926\) 0 0
\(927\) −137.759 + 475.167i −0.00488089 + 0.0168355i
\(928\) 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
\(933\) −16968.2 2410.06i −0.595406 0.0845678i
\(934\) 0 0
\(935\) −515.788 297.791i −0.0180407 0.0104158i
\(936\) 0 0
\(937\) 25708.8i 0.896339i −0.893949 0.448170i \(-0.852076\pi\)
0.893949 0.448170i \(-0.147924\pi\)
\(938\) 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 0.323000 0.946399i \(-0.395309\pi\)
0.981105 + 0.193474i \(0.0619754\pi\)
\(948\) 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 \(0.167928\pi\)
−0.864037 + 0.503429i \(0.832072\pi\)
\(954\) 0 0
\(955\) −7220.70 4168.87i −0.244666 0.141258i
\(956\) 0 0
\(957\) −2042.60 290.118i −0.0689947 0.00979958i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −10526.4 18232.2i −0.353341 0.612004i
\(962\) 0 0
\(963\) 7161.73 24702.7i 0.239650 0.826620i
\(964\) 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
\(969\) −8047.72 + 3237.80i −0.266801 + 0.107341i
\(970\) 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
\(975\) 1160.45 8170.24i 0.0381171 0.268366i
\(976\) 0 0
\(977\) 23589.9 + 13619.6i 0.772474 + 0.445988i 0.833756 0.552132i \(-0.186186\pi\)
−0.0612826 + 0.998120i \(0.519519\pi\)
\(978\) 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 0.999670 0.0256990i \(-0.00818116\pi\)
0.477579 + 0.878589i \(0.341514\pi\)
\(998\) 0 0
\(999\) 3750.02 + 37252.0i 0.118764 + 1.17978i
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.521.15 48
3.2 odd 2 inner 588.4.k.e.521.22 48
7.2 even 3 inner 588.4.k.e.509.3 48
7.3 odd 6 588.4.f.d.293.6 yes 24
7.4 even 3 588.4.f.d.293.19 yes 24
7.5 odd 6 inner 588.4.k.e.509.22 48
7.6 odd 2 inner 588.4.k.e.521.10 48
21.2 odd 6 inner 588.4.k.e.509.10 48
21.5 even 6 inner 588.4.k.e.509.15 48
21.11 odd 6 588.4.f.d.293.5 24
21.17 even 6 588.4.f.d.293.20 yes 24
21.20 even 2 inner 588.4.k.e.521.3 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.f.d.293.5 24 21.11 odd 6
588.4.f.d.293.6 yes 24 7.3 odd 6
588.4.f.d.293.19 yes 24 7.4 even 3
588.4.f.d.293.20 yes 24 21.17 even 6
588.4.k.e.509.3 48 7.2 even 3 inner
588.4.k.e.509.10 48 21.2 odd 6 inner
588.4.k.e.509.15 48 21.5 even 6 inner
588.4.k.e.509.22 48 7.5 odd 6 inner
588.4.k.e.521.3 48 21.20 even 2 inner
588.4.k.e.521.10 48 7.6 odd 2 inner
588.4.k.e.521.15 48 1.1 even 1 trivial
588.4.k.e.521.22 48 3.2 odd 2 inner