Properties

Label 588.4.k.c.521.2
Level $588$
Weight $4$
Character 588.521
Analytic conductor $34.693$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} - 23328 x^{3} - 91854 x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 521.2
Root \(2.88784 + 0.812653i\) of defining polynomial
Character \(\chi\) \(=\) 588.521
Dual form 588.4.k.c.509.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.62798 + 3.71992i) q^{3} +(-9.68891 - 16.7817i) q^{5} +(-0.675594 - 26.9915i) q^{9} +O(q^{10})\) \(q+(-3.62798 + 3.71992i) q^{3} +(-9.68891 - 16.7817i) q^{5} +(-0.675594 - 26.9915i) q^{9} +(36.3221 + 20.9706i) q^{11} +77.1820i q^{13} +(97.5776 + 24.8416i) q^{15} +(-2.98415 + 5.16871i) q^{17} +(59.0330 - 34.0827i) q^{19} +(30.8090 - 17.7876i) q^{23} +(-125.250 + 216.939i) q^{25} +(102.857 + 95.4115i) q^{27} -228.525i q^{29} +(-62.3989 - 36.0260i) q^{31} +(-209.785 + 59.0346i) q^{33} +(-34.6696 - 60.0496i) q^{37} +(-287.111 - 280.014i) q^{39} +132.789 q^{41} -366.304 q^{43} +(-446.418 + 272.856i) q^{45} +(-90.9542 - 157.537i) q^{47} +(-8.40073 - 29.8527i) q^{51} +(15.3092 + 8.83877i) q^{53} -812.728i q^{55} +(-87.3853 + 343.249i) q^{57} +(247.185 - 428.137i) q^{59} +(-498.419 + 287.763i) q^{61} +(1295.24 - 747.810i) q^{65} +(202.162 - 350.154i) q^{67} +(-45.6060 + 179.140i) q^{69} -489.797i q^{71} +(336.670 + 194.377i) q^{73} +(-352.593 - 1252.97i) q^{75} +(-126.706 - 219.461i) q^{79} +(-728.087 + 36.4707i) q^{81} -356.881 q^{83} +115.653 q^{85} +(850.096 + 829.084i) q^{87} +(-711.578 - 1232.49i) q^{89} +(360.395 - 101.417i) q^{93} +(-1143.93 - 660.449i) q^{95} +694.168i q^{97} +(541.489 - 994.557i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{9} + 132 q^{15} - 204 q^{19} - 444 q^{25} - 1458 q^{31} + 108 q^{33} + 240 q^{37} - 432 q^{39} + 342 q^{45} - 300 q^{51} + 180 q^{57} - 2148 q^{61} + 1980 q^{67} + 3084 q^{73} + 3384 q^{75} - 438 q^{79} + 1008 q^{81} - 6144 q^{85} + 2898 q^{87} + 882 q^{93} + 9216 q^{99}+O(q^{100}) \) 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
\(3\) −3.62798 + 3.71992i −0.698204 + 0.715899i
\(4\) 0 0
\(5\) −9.68891 16.7817i −0.866602 1.50100i −0.865447 0.501000i \(-0.832966\pi\)
−0.00115523 0.999999i \(-0.500368\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.675594 26.9915i −0.0250220 0.999687i
\(10\) 0 0
\(11\) 36.3221 + 20.9706i 0.995593 + 0.574806i 0.906942 0.421257i \(-0.138411\pi\)
0.0886519 + 0.996063i \(0.471744\pi\)
\(12\) 0 0
\(13\) 77.1820i 1.64665i 0.567571 + 0.823325i \(0.307883\pi\)
−0.567571 + 0.823325i \(0.692117\pi\)
\(14\) 0 0
\(15\) 97.5776 + 24.8416i 1.67963 + 0.427604i
\(16\) 0 0
\(17\) −2.98415 + 5.16871i −0.0425743 + 0.0737409i −0.886527 0.462676i \(-0.846889\pi\)
0.843953 + 0.536417i \(0.180223\pi\)
\(18\) 0 0
\(19\) 59.0330 34.0827i 0.712794 0.411532i −0.0993004 0.995057i \(-0.531660\pi\)
0.812095 + 0.583525i \(0.198327\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 30.8090 17.7876i 0.279310 0.161260i −0.353801 0.935321i \(-0.615111\pi\)
0.633111 + 0.774061i \(0.281778\pi\)
\(24\) 0 0
\(25\) −125.250 + 216.939i −1.00200 + 1.73551i
\(26\) 0 0
\(27\) 102.857 + 95.4115i 0.733145 + 0.680072i
\(28\) 0 0
\(29\) 228.525i 1.46331i −0.681673 0.731657i \(-0.738747\pi\)
0.681673 0.731657i \(-0.261253\pi\)
\(30\) 0 0
\(31\) −62.3989 36.0260i −0.361522 0.208725i 0.308226 0.951313i \(-0.400265\pi\)
−0.669748 + 0.742588i \(0.733598\pi\)
\(32\) 0 0
\(33\) −209.785 + 59.0346i −1.10663 + 0.311412i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −34.6696 60.0496i −0.154045 0.266813i 0.778666 0.627439i \(-0.215897\pi\)
−0.932711 + 0.360625i \(0.882563\pi\)
\(38\) 0 0
\(39\) −287.111 280.014i −1.17883 1.14970i
\(40\) 0 0
\(41\) 132.789 0.505810 0.252905 0.967491i \(-0.418614\pi\)
0.252905 + 0.967491i \(0.418614\pi\)
\(42\) 0 0
\(43\) −366.304 −1.29909 −0.649544 0.760324i \(-0.725040\pi\)
−0.649544 + 0.760324i \(0.725040\pi\)
\(44\) 0 0
\(45\) −446.418 + 272.856i −1.47885 + 0.903889i
\(46\) 0 0
\(47\) −90.9542 157.537i −0.282277 0.488919i 0.689668 0.724126i \(-0.257756\pi\)
−0.971945 + 0.235207i \(0.924423\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.40073 29.8527i −0.0230654 0.0819651i
\(52\) 0 0
\(53\) 15.3092 + 8.83877i 0.0396770 + 0.0229075i 0.519707 0.854344i \(-0.326041\pi\)
−0.480030 + 0.877252i \(0.659374\pi\)
\(54\) 0 0
\(55\) 812.728i 1.99251i
\(56\) 0 0
\(57\) −87.3853 + 343.249i −0.203061 + 0.797622i
\(58\) 0 0
\(59\) 247.185 428.137i 0.545437 0.944724i −0.453142 0.891438i \(-0.649697\pi\)
0.998579 0.0532862i \(-0.0169696\pi\)
\(60\) 0 0
\(61\) −498.419 + 287.763i −1.04617 + 0.604004i −0.921573 0.388204i \(-0.873095\pi\)
−0.124592 + 0.992208i \(0.539762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1295.24 747.810i 2.47162 1.42699i
\(66\) 0 0
\(67\) 202.162 350.154i 0.368627 0.638480i −0.620725 0.784029i \(-0.713161\pi\)
0.989351 + 0.145549i \(0.0464948\pi\)
\(68\) 0 0
\(69\) −45.6060 + 179.140i −0.0795697 + 0.312550i
\(70\) 0 0
\(71\) 489.797i 0.818707i −0.912376 0.409353i \(-0.865754\pi\)
0.912376 0.409353i \(-0.134246\pi\)
\(72\) 0 0
\(73\) 336.670 + 194.377i 0.539785 + 0.311645i 0.744992 0.667074i \(-0.232453\pi\)
−0.205207 + 0.978719i \(0.565787\pi\)
\(74\) 0 0
\(75\) −352.593 1252.97i −0.542852 1.92907i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −126.706 219.461i −0.180449 0.312547i 0.761584 0.648066i \(-0.224422\pi\)
−0.942034 + 0.335518i \(0.891089\pi\)
\(80\) 0 0
\(81\) −728.087 + 36.4707i −0.998748 + 0.0500284i
\(82\) 0 0
\(83\) −356.881 −0.471961 −0.235981 0.971758i \(-0.575830\pi\)
−0.235981 + 0.971758i \(0.575830\pi\)
\(84\) 0 0
\(85\) 115.653 0.147580
\(86\) 0 0
\(87\) 850.096 + 829.084i 1.04758 + 1.02169i
\(88\) 0 0
\(89\) −711.578 1232.49i −0.847496 1.46791i −0.883436 0.468552i \(-0.844776\pi\)
0.0359398 0.999354i \(-0.488558\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 360.395 101.417i 0.401842 0.113080i
\(94\) 0 0
\(95\) −1143.93 660.449i −1.23542 0.713269i
\(96\) 0 0
\(97\) 694.168i 0.726619i 0.931668 + 0.363310i \(0.118353\pi\)
−0.931668 + 0.363310i \(0.881647\pi\)
\(98\) 0 0
\(99\) 541.489 994.557i 0.549714 1.00966i
\(100\) 0 0
\(101\) 7.67034 13.2854i 0.00755671 0.0130886i −0.862222 0.506530i \(-0.830928\pi\)
0.869779 + 0.493441i \(0.164261\pi\)
\(102\) 0 0
\(103\) 305.835 176.574i 0.292571 0.168916i −0.346530 0.938039i \(-0.612640\pi\)
0.639101 + 0.769123i \(0.279307\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −44.6169 + 25.7596i −0.0403110 + 0.0232736i −0.520020 0.854154i \(-0.674076\pi\)
0.479709 + 0.877428i \(0.340742\pi\)
\(108\) 0 0
\(109\) 335.314 580.781i 0.294654 0.510355i −0.680250 0.732980i \(-0.738129\pi\)
0.974904 + 0.222624i \(0.0714623\pi\)
\(110\) 0 0
\(111\) 349.160 + 88.8901i 0.298566 + 0.0760097i
\(112\) 0 0
\(113\) 461.944i 0.384567i −0.981339 0.192283i \(-0.938411\pi\)
0.981339 0.192283i \(-0.0615893\pi\)
\(114\) 0 0
\(115\) −597.012 344.685i −0.484101 0.279496i
\(116\) 0 0
\(117\) 2083.26 52.1437i 1.64613 0.0412025i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 214.030 + 370.711i 0.160804 + 0.278521i
\(122\) 0 0
\(123\) −481.757 + 493.966i −0.353159 + 0.362109i
\(124\) 0 0
\(125\) 2431.91 1.74014
\(126\) 0 0
\(127\) −388.699 −0.271587 −0.135793 0.990737i \(-0.543358\pi\)
−0.135793 + 0.990737i \(0.543358\pi\)
\(128\) 0 0
\(129\) 1328.94 1362.62i 0.907028 0.930015i
\(130\) 0 0
\(131\) −313.837 543.582i −0.209314 0.362542i 0.742185 0.670195i \(-0.233790\pi\)
−0.951499 + 0.307653i \(0.900456\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 604.590 2650.55i 0.385443 1.68980i
\(136\) 0 0
\(137\) −1822.30 1052.10i −1.13642 0.656111i −0.190877 0.981614i \(-0.561133\pi\)
−0.945541 + 0.325502i \(0.894467\pi\)
\(138\) 0 0
\(139\) 1030.03i 0.628533i −0.949335 0.314266i \(-0.898241\pi\)
0.949335 0.314266i \(-0.101759\pi\)
\(140\) 0 0
\(141\) 916.005 + 233.199i 0.547103 + 0.139283i
\(142\) 0 0
\(143\) −1618.55 + 2803.41i −0.946504 + 1.63939i
\(144\) 0 0
\(145\) −3835.04 + 2214.16i −2.19643 + 1.26811i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 174.214 100.583i 0.0957866 0.0553024i −0.451342 0.892351i \(-0.649054\pi\)
0.547128 + 0.837049i \(0.315721\pi\)
\(150\) 0 0
\(151\) −596.238 + 1032.71i −0.321332 + 0.556564i −0.980763 0.195202i \(-0.937464\pi\)
0.659431 + 0.751765i \(0.270797\pi\)
\(152\) 0 0
\(153\) 141.527 + 77.0550i 0.0747831 + 0.0407158i
\(154\) 0 0
\(155\) 1396.21i 0.723525i
\(156\) 0 0
\(157\) −945.892 546.111i −0.480830 0.277608i 0.239932 0.970790i \(-0.422875\pi\)
−0.720762 + 0.693182i \(0.756208\pi\)
\(158\) 0 0
\(159\) −88.4208 + 24.8821i −0.0441021 + 0.0124106i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 289.802 + 501.952i 0.139258 + 0.241202i 0.927216 0.374527i \(-0.122195\pi\)
−0.787958 + 0.615729i \(0.788862\pi\)
\(164\) 0 0
\(165\) 3023.28 + 2948.56i 1.42644 + 1.39118i
\(166\) 0 0
\(167\) −3715.71 −1.72174 −0.860869 0.508827i \(-0.830079\pi\)
−0.860869 + 0.508827i \(0.830079\pi\)
\(168\) 0 0
\(169\) −3760.06 −1.71145
\(170\) 0 0
\(171\) −959.827 1570.37i −0.429239 0.702274i
\(172\) 0 0
\(173\) −669.204 1159.10i −0.294096 0.509390i 0.680678 0.732583i \(-0.261685\pi\)
−0.974774 + 0.223193i \(0.928352\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 695.854 + 2472.78i 0.295501 + 1.05009i
\(178\) 0 0
\(179\) 2441.07 + 1409.35i 1.01930 + 0.588492i 0.913900 0.405938i \(-0.133055\pi\)
0.105397 + 0.994430i \(0.466389\pi\)
\(180\) 0 0
\(181\) 3688.28i 1.51463i −0.653050 0.757315i \(-0.726511\pi\)
0.653050 0.757315i \(-0.273489\pi\)
\(182\) 0 0
\(183\) 737.800 2898.08i 0.298031 1.17067i
\(184\) 0 0
\(185\) −671.822 + 1163.63i −0.266991 + 0.462442i
\(186\) 0 0
\(187\) −216.781 + 125.159i −0.0847734 + 0.0489440i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3905.03 + 2254.57i −1.47936 + 0.854111i −0.999727 0.0233572i \(-0.992564\pi\)
−0.479636 + 0.877468i \(0.659231\pi\)
\(192\) 0 0
\(193\) 1105.75 1915.21i 0.412401 0.714300i −0.582751 0.812651i \(-0.698024\pi\)
0.995152 + 0.0983513i \(0.0313569\pi\)
\(194\) 0 0
\(195\) −1917.32 + 7531.24i −0.704114 + 2.76576i
\(196\) 0 0
\(197\) 4177.21i 1.51073i 0.655304 + 0.755365i \(0.272540\pi\)
−0.655304 + 0.755365i \(0.727460\pi\)
\(198\) 0 0
\(199\) −3823.65 2207.59i −1.36207 0.786391i −0.372169 0.928165i \(-0.621386\pi\)
−0.989899 + 0.141774i \(0.954719\pi\)
\(200\) 0 0
\(201\) 569.108 + 2022.38i 0.199710 + 0.709689i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1286.58 2228.43i −0.438336 0.759221i
\(206\) 0 0
\(207\) −500.929 819.566i −0.168198 0.275187i
\(208\) 0 0
\(209\) 2858.94 0.946205
\(210\) 0 0
\(211\) −2002.19 −0.653252 −0.326626 0.945154i \(-0.605912\pi\)
−0.326626 + 0.945154i \(0.605912\pi\)
\(212\) 0 0
\(213\) 1822.00 + 1776.97i 0.586111 + 0.571624i
\(214\) 0 0
\(215\) 3549.08 + 6147.19i 1.12579 + 1.94993i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1944.50 + 547.193i −0.599986 + 0.168839i
\(220\) 0 0
\(221\) −398.931 230.323i −0.121425 0.0701050i
\(222\) 0 0
\(223\) 2122.04i 0.637230i −0.947884 0.318615i \(-0.896782\pi\)
0.947884 0.318615i \(-0.103218\pi\)
\(224\) 0 0
\(225\) 5940.14 + 3234.13i 1.76004 + 0.958260i
\(226\) 0 0
\(227\) 2431.41 4211.33i 0.710918 1.23135i −0.253594 0.967311i \(-0.581613\pi\)
0.964513 0.264036i \(-0.0850538\pi\)
\(228\) 0 0
\(229\) 4903.46 2831.01i 1.41498 0.816937i 0.419125 0.907928i \(-0.362337\pi\)
0.995852 + 0.0909910i \(0.0290035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2817.82 1626.87i 0.792280 0.457423i −0.0484845 0.998824i \(-0.515439\pi\)
0.840765 + 0.541401i \(0.182106\pi\)
\(234\) 0 0
\(235\) −1762.49 + 3052.73i −0.489244 + 0.847396i
\(236\) 0 0
\(237\) 1276.06 + 324.863i 0.349743 + 0.0890385i
\(238\) 0 0
\(239\) 5823.69i 1.57616i −0.615571 0.788082i \(-0.711074\pi\)
0.615571 0.788082i \(-0.288926\pi\)
\(240\) 0 0
\(241\) 5349.06 + 3088.28i 1.42972 + 0.825451i 0.997098 0.0761237i \(-0.0242544\pi\)
0.432624 + 0.901574i \(0.357588\pi\)
\(242\) 0 0
\(243\) 2505.81 2840.74i 0.661515 0.749932i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2630.57 + 4556.28i 0.677649 + 1.17372i
\(248\) 0 0
\(249\) 1294.76 1327.57i 0.329525 0.337877i
\(250\) 0 0
\(251\) 1648.38 0.414522 0.207261 0.978286i \(-0.433545\pi\)
0.207261 + 0.978286i \(0.433545\pi\)
\(252\) 0 0
\(253\) 1492.07 0.370772
\(254\) 0 0
\(255\) −419.585 + 430.219i −0.103041 + 0.105652i
\(256\) 0 0
\(257\) 1602.27 + 2775.22i 0.388899 + 0.673592i 0.992302 0.123844i \(-0.0395223\pi\)
−0.603403 + 0.797436i \(0.706189\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6168.25 + 154.390i −1.46286 + 0.0366151i
\(262\) 0 0
\(263\) 2995.95 + 1729.71i 0.702426 + 0.405546i 0.808250 0.588839i \(-0.200415\pi\)
−0.105824 + 0.994385i \(0.533748\pi\)
\(264\) 0 0
\(265\) 342.552i 0.0794068i
\(266\) 0 0
\(267\) 7166.35 + 1824.43i 1.64260 + 0.418177i
\(268\) 0 0
\(269\) 665.298 1152.33i 0.150795 0.261185i −0.780725 0.624875i \(-0.785150\pi\)
0.931520 + 0.363690i \(0.118483\pi\)
\(270\) 0 0
\(271\) −2894.39 + 1671.08i −0.648789 + 0.374578i −0.787992 0.615686i \(-0.788879\pi\)
0.139203 + 0.990264i \(0.455546\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9098.68 + 5253.13i −1.99517 + 1.15191i
\(276\) 0 0
\(277\) 2400.57 4157.91i 0.520709 0.901894i −0.479001 0.877814i \(-0.659001\pi\)
0.999710 0.0240796i \(-0.00766553\pi\)
\(278\) 0 0
\(279\) −930.241 + 1708.58i −0.199613 + 0.366631i
\(280\) 0 0
\(281\) 4548.87i 0.965704i 0.875702 + 0.482852i \(0.160399\pi\)
−0.875702 + 0.482852i \(0.839601\pi\)
\(282\) 0 0
\(283\) 3922.91 + 2264.89i 0.824003 + 0.475739i 0.851795 0.523875i \(-0.175514\pi\)
−0.0277917 + 0.999614i \(0.508848\pi\)
\(284\) 0 0
\(285\) 6606.97 1859.24i 1.37320 0.386427i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2438.69 + 4223.93i 0.496375 + 0.859746i
\(290\) 0 0
\(291\) −2582.25 2518.42i −0.520186 0.507328i
\(292\) 0 0
\(293\) −9525.19 −1.89921 −0.949603 0.313454i \(-0.898514\pi\)
−0.949603 + 0.313454i \(0.898514\pi\)
\(294\) 0 0
\(295\) −9579.82 −1.89071
\(296\) 0 0
\(297\) 1735.16 + 5622.53i 0.339005 + 1.09849i
\(298\) 0 0
\(299\) 1372.88 + 2377.90i 0.265538 + 0.459925i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 21.5929 + 76.7322i 0.00409399 + 0.0145484i
\(304\) 0 0
\(305\) 9658.28 + 5576.21i 1.81322 + 1.04686i
\(306\) 0 0
\(307\) 3743.83i 0.696000i −0.937495 0.348000i \(-0.886861\pi\)
0.937495 0.348000i \(-0.113139\pi\)
\(308\) 0 0
\(309\) −452.720 + 1778.28i −0.0833475 + 0.327389i
\(310\) 0 0
\(311\) 5201.50 9009.27i 0.948393 1.64266i 0.199582 0.979881i \(-0.436042\pi\)
0.748811 0.662783i \(-0.230625\pi\)
\(312\) 0 0
\(313\) −2850.12 + 1645.52i −0.514690 + 0.297157i −0.734760 0.678328i \(-0.762705\pi\)
0.220069 + 0.975484i \(0.429372\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5177.79 2989.40i 0.917393 0.529657i 0.0345906 0.999402i \(-0.488987\pi\)
0.882802 + 0.469744i \(0.155654\pi\)
\(318\) 0 0
\(319\) 4792.31 8300.52i 0.841122 1.45687i
\(320\) 0 0
\(321\) 66.0454 259.426i 0.0114838 0.0451083i
\(322\) 0 0
\(323\) 406.832i 0.0700828i
\(324\) 0 0
\(325\) −16743.8 9667.04i −2.85778 1.64994i
\(326\) 0 0
\(327\) 943.947 + 3354.40i 0.159634 + 0.567275i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.9169 36.2291i −0.00347340 0.00601611i 0.864283 0.503005i \(-0.167772\pi\)
−0.867757 + 0.496989i \(0.834439\pi\)
\(332\) 0 0
\(333\) −1597.41 + 976.356i −0.262875 + 0.160673i
\(334\) 0 0
\(335\) −7834.90 −1.27781
\(336\) 0 0
\(337\) −5477.82 −0.885447 −0.442723 0.896658i \(-0.645988\pi\)
−0.442723 + 0.896658i \(0.645988\pi\)
\(338\) 0 0
\(339\) 1718.40 + 1675.92i 0.275311 + 0.268506i
\(340\) 0 0
\(341\) −1510.97 2617.08i −0.239952 0.415610i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3448.14 970.327i 0.538092 0.151422i
\(346\) 0 0
\(347\) −958.002 553.103i −0.148208 0.0855681i 0.424062 0.905633i \(-0.360604\pi\)
−0.572270 + 0.820065i \(0.693937\pi\)
\(348\) 0 0
\(349\) 7172.24i 1.10006i −0.835145 0.550030i \(-0.814616\pi\)
0.835145 0.550030i \(-0.185384\pi\)
\(350\) 0 0
\(351\) −7364.05 + 7938.74i −1.11984 + 1.20723i
\(352\) 0 0
\(353\) −3686.86 + 6385.84i −0.555898 + 0.962843i 0.441935 + 0.897047i \(0.354292\pi\)
−0.997833 + 0.0657965i \(0.979041\pi\)
\(354\) 0 0
\(355\) −8219.61 + 4745.60i −1.22888 + 0.709493i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −827.565 + 477.795i −0.121664 + 0.0702425i −0.559597 0.828765i \(-0.689044\pi\)
0.437933 + 0.899008i \(0.355711\pi\)
\(360\) 0 0
\(361\) −1106.24 + 1916.06i −0.161283 + 0.279350i
\(362\) 0 0
\(363\) −2155.51 548.756i −0.311667 0.0793450i
\(364\) 0 0
\(365\) 7533.19i 1.08029i
\(366\) 0 0
\(367\) 4935.71 + 2849.63i 0.702021 + 0.405312i 0.808100 0.589046i \(-0.200496\pi\)
−0.106079 + 0.994358i \(0.533830\pi\)
\(368\) 0 0
\(369\) −89.7118 3584.19i −0.0126564 0.505652i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4957.03 8585.83i −0.688111 1.19184i −0.972448 0.233119i \(-0.925107\pi\)
0.284337 0.958724i \(-0.408226\pi\)
\(374\) 0 0
\(375\) −8822.92 + 9046.52i −1.21497 + 1.24576i
\(376\) 0 0
\(377\) 17638.1 2.40956
\(378\) 0 0
\(379\) 8051.92 1.09129 0.545646 0.838016i \(-0.316284\pi\)
0.545646 + 0.838016i \(0.316284\pi\)
\(380\) 0 0
\(381\) 1410.19 1445.93i 0.189623 0.194428i
\(382\) 0 0
\(383\) 2184.21 + 3783.17i 0.291405 + 0.504728i 0.974142 0.225936i \(-0.0725440\pi\)
−0.682737 + 0.730664i \(0.739211\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 247.473 + 9887.10i 0.0325058 + 1.29868i
\(388\) 0 0
\(389\) 2123.46 + 1225.98i 0.276770 + 0.159793i 0.631960 0.775001i \(-0.282251\pi\)
−0.355190 + 0.934794i \(0.615584\pi\)
\(390\) 0 0
\(391\) 212.324i 0.0274621i
\(392\) 0 0
\(393\) 3160.68 + 804.654i 0.405687 + 0.103281i
\(394\) 0 0
\(395\) −2455.28 + 4252.67i −0.312756 + 0.541709i
\(396\) 0 0
\(397\) −7282.29 + 4204.43i −0.920624 + 0.531523i −0.883834 0.467801i \(-0.845047\pi\)
−0.0367899 + 0.999323i \(0.511713\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6960.80 + 4018.82i −0.866847 + 0.500474i −0.866299 0.499526i \(-0.833508\pi\)
−0.000547684 1.00000i \(0.500174\pi\)
\(402\) 0 0
\(403\) 2780.56 4816.07i 0.343696 0.595299i
\(404\) 0 0
\(405\) 7666.41 + 11865.2i 0.940610 + 1.45577i
\(406\) 0 0
\(407\) 2908.17i 0.354183i
\(408\) 0 0
\(409\) −4257.86 2458.28i −0.514762 0.297198i 0.220027 0.975494i \(-0.429386\pi\)
−0.734789 + 0.678296i \(0.762719\pi\)
\(410\) 0 0
\(411\) 10525.0 2961.79i 1.26316 0.355461i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3457.79 + 5989.07i 0.409003 + 0.708414i
\(416\) 0 0
\(417\) 3831.63 + 3736.93i 0.449966 + 0.438844i
\(418\) 0 0
\(419\) −4713.84 −0.549608 −0.274804 0.961500i \(-0.588613\pi\)
−0.274804 + 0.961500i \(0.588613\pi\)
\(420\) 0 0
\(421\) 9921.92 1.14861 0.574305 0.818641i \(-0.305272\pi\)
0.574305 + 0.818641i \(0.305272\pi\)
\(422\) 0 0
\(423\) −4190.73 + 2561.43i −0.481702 + 0.294423i
\(424\) 0 0
\(425\) −747.530 1294.76i −0.0853189 0.147777i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4556.41 16191.6i −0.512786 1.82223i
\(430\) 0 0
\(431\) −10901.0 6293.70i −1.21829 0.703380i −0.253739 0.967273i \(-0.581660\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(432\) 0 0
\(433\) 3715.19i 0.412334i 0.978517 + 0.206167i \(0.0660989\pi\)
−0.978517 + 0.206167i \(0.933901\pi\)
\(434\) 0 0
\(435\) 5676.93 22299.0i 0.625719 2.45782i
\(436\) 0 0
\(437\) 1212.50 2100.11i 0.132727 0.229890i
\(438\) 0 0
\(439\) 5079.47 2932.63i 0.552232 0.318831i −0.197790 0.980244i \(-0.563376\pi\)
0.750022 + 0.661413i \(0.230043\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13372.1 7720.38i 1.43415 0.828005i 0.436713 0.899601i \(-0.356142\pi\)
0.997434 + 0.0715954i \(0.0228090\pi\)
\(444\) 0 0
\(445\) −13788.8 + 23883.0i −1.46888 + 2.54418i
\(446\) 0 0
\(447\) −257.886 + 1012.98i −0.0272877 + 0.107186i
\(448\) 0 0
\(449\) 16373.4i 1.72095i 0.509491 + 0.860476i \(0.329834\pi\)
−0.509491 + 0.860476i \(0.670166\pi\)
\(450\) 0 0
\(451\) 4823.19 + 2784.67i 0.503581 + 0.290743i
\(452\) 0 0
\(453\) −1678.48 5964.62i −0.174088 0.618636i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1377.21 2385.40i −0.140970 0.244167i 0.786892 0.617090i \(-0.211689\pi\)
−0.927862 + 0.372923i \(0.878355\pi\)
\(458\) 0 0
\(459\) −800.096 + 246.917i −0.0813623 + 0.0251092i
\(460\) 0 0
\(461\) 15142.3 1.52982 0.764909 0.644139i \(-0.222784\pi\)
0.764909 + 0.644139i \(0.222784\pi\)
\(462\) 0 0
\(463\) −10138.9 −1.01770 −0.508851 0.860855i \(-0.669930\pi\)
−0.508851 + 0.860855i \(0.669930\pi\)
\(464\) 0 0
\(465\) −5193.79 5065.42i −0.517971 0.505168i
\(466\) 0 0
\(467\) −8074.64 13985.7i −0.800106 1.38583i −0.919545 0.392984i \(-0.871443\pi\)
0.119439 0.992842i \(-0.461890\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5463.16 1537.36i 0.534457 0.150399i
\(472\) 0 0
\(473\) −13304.9 7681.60i −1.29336 0.746724i
\(474\) 0 0
\(475\) 17075.4i 1.64942i
\(476\) 0 0
\(477\) 228.229 419.190i 0.0219075 0.0402377i
\(478\) 0 0
\(479\) −4209.90 + 7291.77i −0.401577 + 0.695552i −0.993916 0.110137i \(-0.964871\pi\)
0.592339 + 0.805689i \(0.298204\pi\)
\(480\) 0 0
\(481\) 4634.75 2675.87i 0.439348 0.253657i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11649.3 6725.73i 1.09065 0.629690i
\(486\) 0 0
\(487\) 7039.12 12192.1i 0.654975 1.13445i −0.326925 0.945050i \(-0.606013\pi\)
0.981900 0.189400i \(-0.0606541\pi\)
\(488\) 0 0
\(489\) −2918.61 743.029i −0.269907 0.0687135i
\(490\) 0 0
\(491\) 17853.2i 1.64094i −0.571688 0.820471i \(-0.693711\pi\)
0.571688 0.820471i \(-0.306289\pi\)
\(492\) 0 0
\(493\) 1181.18 + 681.955i 0.107906 + 0.0622996i
\(494\) 0 0
\(495\) −21936.8 + 549.075i −1.99189 + 0.0498567i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −6802.88 11782.9i −0.610298 1.05707i −0.991190 0.132447i \(-0.957716\pi\)
0.380892 0.924619i \(-0.375617\pi\)
\(500\) 0 0
\(501\) 13480.5 13822.1i 1.20212 1.23259i
\(502\) 0 0
\(503\) −7503.79 −0.665164 −0.332582 0.943074i \(-0.607920\pi\)
−0.332582 + 0.943074i \(0.607920\pi\)
\(504\) 0 0
\(505\) −297.269 −0.0261946
\(506\) 0 0
\(507\) 13641.4 13987.1i 1.19494 1.22523i
\(508\) 0 0
\(509\) −5239.61 9075.27i −0.456270 0.790283i 0.542490 0.840062i \(-0.317482\pi\)
−0.998760 + 0.0497789i \(0.984148\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 9323.86 + 2126.77i 0.802453 + 0.183039i
\(514\) 0 0
\(515\) −5926.41 3421.61i −0.507085 0.292766i
\(516\) 0 0
\(517\) 7629.45i 0.649019i
\(518\) 0 0
\(519\) 6739.60 + 1715.79i 0.570011 + 0.145115i
\(520\) 0 0
\(521\) −3647.88 + 6318.31i −0.306749 + 0.531305i −0.977649 0.210243i \(-0.932575\pi\)
0.670900 + 0.741548i \(0.265908\pi\)
\(522\) 0 0
\(523\) 13159.3 7597.52i 1.10022 0.635213i 0.163942 0.986470i \(-0.447579\pi\)
0.936279 + 0.351257i \(0.114246\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 372.416 215.014i 0.0307831 0.0177726i
\(528\) 0 0
\(529\) −5450.70 + 9440.89i −0.447991 + 0.775943i
\(530\) 0 0
\(531\) −11723.1 6382.66i −0.958076 0.521627i
\(532\) 0 0
\(533\) 10249.0i 0.832892i
\(534\) 0 0
\(535\) 864.578 + 499.164i 0.0698672 + 0.0403378i
\(536\) 0 0
\(537\) −14098.8 + 3967.49i −1.13298 + 0.318827i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9186.61 + 15911.7i 0.730061 + 1.26450i 0.956857 + 0.290561i \(0.0938418\pi\)
−0.226795 + 0.973942i \(0.572825\pi\)
\(542\) 0 0
\(543\) 13720.1 + 13381.0i 1.08432 + 1.05752i
\(544\) 0 0
\(545\) −12995.3 −1.02139
\(546\) 0 0
\(547\) −20764.3 −1.62307 −0.811534 0.584305i \(-0.801367\pi\)
−0.811534 + 0.584305i \(0.801367\pi\)
\(548\) 0 0
\(549\) 8103.89 + 13258.7i 0.629992 + 1.03072i
\(550\) 0 0
\(551\) −7788.76 13490.5i −0.602201 1.04304i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1891.25 6720.74i −0.144647 0.514017i
\(556\) 0 0
\(557\) −4512.21 2605.13i −0.343247 0.198174i 0.318460 0.947936i \(-0.396834\pi\)
−0.661707 + 0.749763i \(0.730168\pi\)
\(558\) 0 0
\(559\) 28272.1i 2.13914i
\(560\) 0 0
\(561\) 320.897 1260.48i 0.0241502 0.0948621i
\(562\) 0 0
\(563\) −5094.34 + 8823.65i −0.381351 + 0.660520i −0.991256 0.131955i \(-0.957874\pi\)
0.609904 + 0.792475i \(0.291208\pi\)
\(564\) 0 0
\(565\) −7752.20 + 4475.74i −0.577235 + 0.333267i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1617.54 + 933.886i −0.119175 + 0.0688059i −0.558403 0.829570i \(-0.688586\pi\)
0.439227 + 0.898376i \(0.355252\pi\)
\(570\) 0 0
\(571\) 454.422 787.082i 0.0333047 0.0576854i −0.848893 0.528565i \(-0.822730\pi\)
0.882197 + 0.470880i \(0.156063\pi\)
\(572\) 0 0
\(573\) 5780.54 22705.9i 0.421441 1.65542i
\(574\) 0 0
\(575\) 8911.59i 0.646328i
\(576\) 0 0
\(577\) 13028.7 + 7522.10i 0.940018 + 0.542720i 0.889966 0.456027i \(-0.150728\pi\)
0.0500521 + 0.998747i \(0.484061\pi\)
\(578\) 0 0
\(579\) 3112.80 + 11061.6i 0.223426 + 0.793965i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 370.708 + 642.085i 0.0263347 + 0.0456131i
\(584\) 0 0
\(585\) −21059.6 34455.4i −1.48839 2.43514i
\(586\) 0 0
\(587\) 22979.9 1.61581 0.807906 0.589312i \(-0.200601\pi\)
0.807906 + 0.589312i \(0.200601\pi\)
\(588\) 0 0
\(589\) −4911.46 −0.343587
\(590\) 0 0
\(591\) −15538.9 15154.8i −1.08153 1.05480i
\(592\) 0 0
\(593\) −2110.33 3655.20i −0.146140 0.253121i 0.783658 0.621193i \(-0.213352\pi\)
−0.929798 + 0.368071i \(0.880018\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22084.2 6214.61i 1.51398 0.426042i
\(598\) 0 0
\(599\) −19538.6 11280.6i −1.33276 0.769471i −0.347041 0.937850i \(-0.612813\pi\)
−0.985722 + 0.168379i \(0.946147\pi\)
\(600\) 0 0
\(601\) 2004.06i 0.136019i 0.997685 + 0.0680095i \(0.0216648\pi\)
−0.997685 + 0.0680095i \(0.978335\pi\)
\(602\) 0 0
\(603\) −9587.78 5220.09i −0.647504 0.352535i
\(604\) 0 0
\(605\) 4147.44 7183.58i 0.278707 0.482734i
\(606\) 0 0
\(607\) −8503.70 + 4909.61i −0.568624 + 0.328295i −0.756599 0.653879i \(-0.773141\pi\)
0.187976 + 0.982174i \(0.439807\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12159.0 7020.03i 0.805077 0.464812i
\(612\) 0 0
\(613\) 9496.72 16448.8i 0.625724 1.08379i −0.362676 0.931915i \(-0.618137\pi\)
0.988400 0.151871i \(-0.0485298\pi\)
\(614\) 0 0
\(615\) 12957.3 + 3298.70i 0.849573 + 0.216287i
\(616\) 0 0
\(617\) 13306.4i 0.868223i −0.900859 0.434112i \(-0.857062\pi\)
0.900859 0.434112i \(-0.142938\pi\)
\(618\) 0 0
\(619\) 1611.02 + 930.120i 0.104608 + 0.0603953i 0.551391 0.834247i \(-0.314097\pi\)
−0.446783 + 0.894642i \(0.647431\pi\)
\(620\) 0 0
\(621\) 4866.08 + 1109.95i 0.314443 + 0.0717242i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7906.36 13694.2i −0.506007 0.876429i
\(626\) 0 0
\(627\) −10372.1 + 10635.0i −0.660644 + 0.677387i
\(628\) 0 0
\(629\) 413.838 0.0262334
\(630\) 0 0
\(631\) 16908.2 1.06673 0.533364 0.845886i \(-0.320928\pi\)
0.533364 + 0.845886i \(0.320928\pi\)
\(632\) 0 0
\(633\) 7263.88 7447.97i 0.456103 0.467662i
\(634\) 0 0
\(635\) 3766.07 + 6523.03i 0.235358 + 0.407651i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −13220.4 + 330.904i −0.818450 + 0.0204857i
\(640\) 0 0
\(641\) 4361.32 + 2518.01i 0.268739 + 0.155157i 0.628315 0.777959i \(-0.283745\pi\)
−0.359575 + 0.933116i \(0.617078\pi\)
\(642\) 0 0
\(643\) 7842.19i 0.480973i 0.970652 + 0.240487i \(0.0773070\pi\)
−0.970652 + 0.240487i \(0.922693\pi\)
\(644\) 0 0
\(645\) −35743.0 9099.56i −2.18199 0.555496i
\(646\) 0 0
\(647\) 5238.32 9073.04i 0.318299 0.551311i −0.661834 0.749651i \(-0.730222\pi\)
0.980133 + 0.198340i \(0.0635549\pi\)
\(648\) 0 0
\(649\) 17956.6 10367.2i 1.08607 0.627041i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10925.7 + 6307.97i −0.654758 + 0.378025i −0.790277 0.612750i \(-0.790063\pi\)
0.135519 + 0.990775i \(0.456730\pi\)
\(654\) 0 0
\(655\) −6081.49 + 10533.4i −0.362784 + 0.628360i
\(656\) 0 0
\(657\) 5019.08 9218.57i 0.298041 0.547414i
\(658\) 0 0
\(659\) 1347.17i 0.0796334i −0.999207 0.0398167i \(-0.987323\pi\)
0.999207 0.0398167i \(-0.0126774\pi\)
\(660\) 0 0
\(661\) 3269.64 + 1887.73i 0.192397 + 0.111080i 0.593104 0.805126i \(-0.297902\pi\)
−0.400707 + 0.916206i \(0.631236\pi\)
\(662\) 0 0
\(663\) 2304.09 648.385i 0.134968 0.0379807i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4064.92 7040.65i −0.235973 0.408718i
\(668\) 0 0
\(669\) 7893.81 + 7698.70i 0.456192 + 0.444916i
\(670\) 0 0
\(671\) −24138.2 −1.38874
\(672\) 0 0
\(673\) 27877.7 1.59674 0.798370 0.602167i \(-0.205696\pi\)
0.798370 + 0.602167i \(0.205696\pi\)
\(674\) 0 0
\(675\) −33581.4 + 10363.5i −1.91489 + 0.590952i
\(676\) 0 0
\(677\) 10172.5 + 17619.3i 0.577492 + 1.00024i 0.995766 + 0.0919241i \(0.0293017\pi\)
−0.418274 + 0.908321i \(0.637365\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6844.70 + 24323.3i 0.385153 + 1.36868i
\(682\) 0 0
\(683\) 10909.0 + 6298.30i 0.611157 + 0.352852i 0.773418 0.633896i \(-0.218545\pi\)
−0.162261 + 0.986748i \(0.551879\pi\)
\(684\) 0 0
\(685\) 40774.9i 2.27435i
\(686\) 0 0
\(687\) −7258.49 + 28511.3i −0.403098 + 1.58337i
\(688\) 0 0
\(689\) −682.194 + 1181.59i −0.0377206 + 0.0653340i
\(690\) 0 0
\(691\) 8495.72 4905.01i 0.467717 0.270037i −0.247567 0.968871i \(-0.579631\pi\)
0.715284 + 0.698834i \(0.246298\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17285.6 + 9979.87i −0.943427 + 0.544688i
\(696\) 0 0
\(697\) −396.264 + 686.349i −0.0215345 + 0.0372989i
\(698\) 0 0
\(699\) −4171.15 + 16384.3i −0.225705 + 0.886567i
\(700\) 0 0
\(701\) 21928.4i 1.18149i −0.806858 0.590746i \(-0.798834\pi\)
0.806858 0.590746i \(-0.201166\pi\)
\(702\) 0 0
\(703\) −4093.30 2363.27i −0.219604 0.126789i
\(704\) 0 0
\(705\) −4961.62 17631.6i −0.265057 0.941905i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10010.9 + 17339.4i 0.530279 + 0.918471i 0.999376 + 0.0353240i \(0.0112463\pi\)
−0.469096 + 0.883147i \(0.655420\pi\)
\(710\) 0 0
\(711\) −5837.98 + 3568.25i −0.307934 + 0.188213i
\(712\) 0 0
\(713\) −2563.27 −0.134635
\(714\) 0 0
\(715\) 62728.0 3.28097
\(716\) 0 0
\(717\) 21663.6 + 21128.2i 1.12837 + 1.10048i
\(718\) 0 0
\(719\) −6260.61 10843.7i −0.324731 0.562450i 0.656727 0.754128i \(-0.271940\pi\)
−0.981458 + 0.191678i \(0.938607\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −30894.4 + 8693.86i −1.58918 + 0.447203i
\(724\) 0 0
\(725\) 49576.1 + 28622.8i 2.53960 + 1.46624i
\(726\) 0 0
\(727\) 27158.2i 1.38547i 0.721190 + 0.692737i \(0.243595\pi\)
−0.721190 + 0.692737i \(0.756405\pi\)
\(728\) 0 0
\(729\) 1476.29 + 19627.6i 0.0750034 + 0.997183i
\(730\) 0 0
\(731\) 1093.11 1893.32i 0.0553078 0.0957959i
\(732\) 0 0
\(733\) −4714.05 + 2721.66i −0.237541 + 0.137144i −0.614046 0.789270i \(-0.710459\pi\)
0.376505 + 0.926415i \(0.377126\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14685.9 8478.89i 0.734004 0.423778i
\(738\) 0 0
\(739\) −13386.6 + 23186.2i −0.666351 + 1.15415i 0.312566 + 0.949896i \(0.398811\pi\)
−0.978917 + 0.204257i \(0.934522\pi\)
\(740\) 0 0
\(741\) −26492.7 6744.57i −1.31340 0.334370i
\(742\) 0 0
\(743\) 6409.29i 0.316466i −0.987402 0.158233i \(-0.949420\pi\)
0.987402 0.158233i \(-0.0505797\pi\)
\(744\) 0 0
\(745\) −3375.90 1949.07i −0.166018 0.0958504i
\(746\) 0 0
\(747\) 241.107 + 9632.77i 0.0118094 + 0.471814i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −9007.68 15601.8i −0.437676 0.758078i 0.559834 0.828605i \(-0.310865\pi\)
−0.997510 + 0.0705276i \(0.977532\pi\)
\(752\) 0 0
\(753\) −5980.30 + 6131.85i −0.289421 + 0.296756i
\(754\) 0 0
\(755\) 23107.6 1.11387
\(756\) 0 0
\(757\) 4064.93 0.195168 0.0975840 0.995227i \(-0.468889\pi\)
0.0975840 + 0.995227i \(0.468889\pi\)
\(758\) 0 0
\(759\) −5413.18 + 5550.36i −0.258875 + 0.265435i
\(760\) 0 0
\(761\) 10719.5 + 18566.8i 0.510621 + 0.884421i 0.999924 + 0.0123076i \(0.00391774\pi\)
−0.489303 + 0.872114i \(0.662749\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −78.1344 3121.65i −0.00369275 0.147534i
\(766\) 0 0
\(767\) 33044.5 + 19078.3i 1.55563 + 0.898143i
\(768\) 0 0
\(769\) 18622.9i 0.873286i 0.899635 + 0.436643i \(0.143833\pi\)
−0.899635 + 0.436643i \(0.856167\pi\)
\(770\) 0 0
\(771\) −16136.6 4108.09i −0.753754 0.191893i
\(772\) 0 0
\(773\) 20786.3 36002.9i 0.967180 1.67520i 0.263538 0.964649i \(-0.415110\pi\)
0.703641 0.710555i \(-0.251556\pi\)
\(774\) 0 0
\(775\) 15630.9 9024.51i 0.724489 0.418284i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7838.95 4525.82i 0.360539 0.208157i
\(780\) 0 0
\(781\) 10271.3 17790.4i 0.470598 0.815099i
\(782\) 0 0
\(783\) 21804.0 23505.5i 0.995159 1.07282i
\(784\) 0 0
\(785\) 21164.9i 0.962301i
\(786\) 0 0
\(787\) −2973.05 1716.49i −0.134661 0.0777463i 0.431156 0.902277i \(-0.358106\pi\)
−0.565817 + 0.824531i \(0.691439\pi\)
\(788\) 0 0
\(789\) −17303.6 + 4869.33i −0.780766 + 0.219712i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −22210.1 38469.0i −0.994582 1.72267i
\(794\) 0 0
\(795\) 1274.27 + 1242.77i 0.0568472 + 0.0554422i
\(796\) 0 0
\(797\) −9577.83 −0.425676 −0.212838 0.977087i \(-0.568271\pi\)
−0.212838 + 0.977087i \(0.568271\pi\)
\(798\) 0 0
\(799\) 1085.69 0.0480711
\(800\) 0 0
\(801\) −32786.1 + 20039.3i −1.44624 + 0.883961i
\(802\) 0 0
\(803\) 8152.38 + 14120.3i 0.358271 + 0.620543i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1872.89 + 6655.48i 0.0816962 + 0.290315i
\(808\) 0 0
\(809\) −12196.7 7041.78i −0.530054 0.306027i 0.210984 0.977489i \(-0.432333\pi\)
−0.741039 + 0.671463i \(0.765666\pi\)
\(810\) 0 0
\(811\) 8029.38i 0.347657i 0.984776 + 0.173828i \(0.0556138\pi\)
−0.984776 + 0.173828i \(0.944386\pi\)
\(812\) 0 0
\(813\) 4284.50 16829.5i 0.184827 0.725999i
\(814\) 0 0
\(815\) 5615.73 9726.73i 0.241363 0.418052i
\(816\) 0 0
\(817\) −21624.0 + 12484.6i −0.925983 + 0.534616i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15403.4 + 8893.18i −0.654791 + 0.378044i −0.790290 0.612734i \(-0.790070\pi\)
0.135498 + 0.990778i \(0.456737\pi\)
\(822\) 0 0
\(823\) 4759.27 8243.29i 0.201577 0.349141i −0.747460 0.664307i \(-0.768727\pi\)
0.949037 + 0.315166i \(0.102060\pi\)
\(824\) 0 0
\(825\) 13468.6 52904.6i 0.568383 2.23261i
\(826\) 0 0
\(827\) 37498.5i 1.57672i 0.615212 + 0.788362i \(0.289071\pi\)
−0.615212 + 0.788362i \(0.710929\pi\)
\(828\) 0 0
\(829\) −23820.3 13752.7i −0.997967 0.576176i −0.0903208 0.995913i \(-0.528789\pi\)
−0.907646 + 0.419736i \(0.862123\pi\)
\(830\) 0 0
\(831\) 6757.88 + 24014.7i 0.282104 + 1.00248i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 36001.2 + 62355.8i 1.49206 + 2.58433i
\(836\) 0 0
\(837\) −2980.89 9659.11i −0.123100 0.398886i
\(838\) 0 0
\(839\) −2040.23 −0.0839528 −0.0419764 0.999119i \(-0.513365\pi\)
−0.0419764 + 0.999119i \(0.513365\pi\)
\(840\) 0 0
\(841\) −27834.9 −1.14129
\(842\) 0 0
\(843\) −16921.4 16503.2i −0.691346 0.674259i
\(844\) 0 0
\(845\) 36430.9 + 63100.2i 1.48315 + 2.56889i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −22657.5 + 6375.94i −0.915903 + 0.257740i
\(850\) 0 0
\(851\) −2136.28 1233.38i −0.0860524 0.0496824i
\(852\) 0 0
\(853\) 22267.7i 0.893825i 0.894578 + 0.446913i \(0.147476\pi\)
−0.894578 + 0.446913i \(0.852524\pi\)
\(854\) 0 0
\(855\) −17053.7 + 31322.6i −0.682133 + 1.25288i
\(856\) 0 0
\(857\) −13711.2 + 23748.5i −0.546517 + 0.946595i 0.451993 + 0.892022i \(0.350713\pi\)
−0.998510 + 0.0545736i \(0.982620\pi\)
\(858\) 0 0
\(859\) 37954.1 21912.8i 1.50754 0.870380i 0.507580 0.861605i \(-0.330540\pi\)
0.999961 0.00877488i \(-0.00279317\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23255.2 + 13426.4i −0.917286 + 0.529595i −0.882768 0.469809i \(-0.844323\pi\)
−0.0345178 + 0.999404i \(0.510990\pi\)
\(864\) 0 0
\(865\) −12967.7 + 22460.8i −0.509729 + 0.882877i
\(866\) 0 0
\(867\) −24560.2 6252.60i −0.962062 0.244924i
\(868\) 0 0
\(869\) 10628.4i 0.414894i
\(870\) 0 0
\(871\) 27025.6 + 15603.2i 1.05135 + 0.606999i
\(872\) 0 0
\(873\) 18736.7 468.976i 0.726392 0.0181815i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19557.7 + 33874.9i 0.753039 + 1.30430i 0.946343 + 0.323163i \(0.104746\pi\)
−0.193304 + 0.981139i \(0.561920\pi\)
\(878\) 0 0
\(879\) 34557.2 35432.9i 1.32603 1.35964i
\(880\) 0 0
\(881\) −39685.7 −1.51764 −0.758822 0.651299i \(-0.774225\pi\)
−0.758822 + 0.651299i \(0.774225\pi\)
\(882\) 0 0
\(883\) −37785.0 −1.44005 −0.720026 0.693947i \(-0.755870\pi\)
−0.720026 + 0.693947i \(0.755870\pi\)
\(884\) 0 0
\(885\) 34755.3 35636.2i 1.32010 1.35356i
\(886\) 0 0
\(887\) 7332.70 + 12700.6i 0.277574 + 0.480772i 0.970781 0.239966i \(-0.0771364\pi\)
−0.693207 + 0.720738i \(0.743803\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −27210.5 13943.7i −1.02310 0.524278i
\(892\) 0 0
\(893\) −10738.6 6199.93i −0.402411 0.232332i
\(894\) 0 0
\(895\) 54620.4i 2.03995i
\(896\) 0 0
\(897\) −13826.4 3519.96i −0.514660 0.131023i
\(898\) 0 0
\(899\) −8232.86 + 14259.7i −0.305430 + 0.529020i
\(900\) 0 0
\(901\) −91.3699 + 52.7525i −0.00337844 + 0.00195054i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −61895.6 + 35735.4i −2.27346 + 1.31258i
\(906\) 0 0
\(907\) 19099.9 33081.9i 0.699229 1.21110i −0.269505 0.962999i \(-0.586860\pi\)
0.968734 0.248101i \(-0.0798066\pi\)
\(908\) 0 0
\(909\) −363.776 198.059i −0.0132736 0.00722684i
\(910\) 0 0
\(911\) 41209.5i 1.49872i −0.662165 0.749358i \(-0.730362\pi\)
0.662165 0.749358i \(-0.269638\pi\)
\(912\) 0 0
\(913\) −12962.7 7484.00i −0.469882 0.271286i
\(914\) 0 0
\(915\) −55783.1 + 15697.7i −2.01544 + 0.567157i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17925.7 31048.1i −0.643431 1.11445i −0.984662 0.174475i \(-0.944177\pi\)
0.341231 0.939979i \(-0.389156\pi\)
\(920\) 0 0
\(921\) 13926.8 + 13582.5i 0.498265 + 0.485950i
\(922\) 0 0
\(923\) 37803.5 1.34812
\(924\) 0 0
\(925\) 17369.5 0.617411
\(926\) 0 0
\(927\) −4972.62 8135.65i −0.176184 0.288252i
\(928\) 0 0
\(929\) −19657.8 34048.3i −0.694243 1.20246i −0.970435 0.241361i \(-0.922406\pi\)
0.276193 0.961102i \(-0.410927\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 14642.8 + 52034.6i 0.513810 + 1.82587i
\(934\) 0 0
\(935\) 4200.75 + 2425.31i 0.146930 + 0.0848299i
\(936\) 0 0
\(937\) 25396.6i 0.885455i 0.896656 + 0.442728i \(0.145989\pi\)
−0.896656 + 0.442728i \(0.854011\pi\)
\(938\) 0 0
\(939\) 4218.97 16572.1i 0.146625 0.575942i
\(940\) 0 0
\(941\) 9088.06 15741.0i 0.314838 0.545315i −0.664565 0.747230i \(-0.731383\pi\)
0.979403 + 0.201915i \(0.0647165\pi\)
\(942\) 0 0
\(943\) 4091.11 2362.00i 0.141278 0.0815668i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36468.5 21055.1i 1.25139 0.722491i 0.280005 0.959998i \(-0.409664\pi\)
0.971386 + 0.237507i \(0.0763304\pi\)
\(948\) 0 0
\(949\) −15002.4 + 25984.9i −0.513170 + 0.888836i
\(950\) 0 0
\(951\) −7664.57 + 30106.4i −0.261347 + 1.02657i
\(952\) 0 0
\(953\) 18521.2i 0.629551i 0.949166 + 0.314775i \(0.101929\pi\)
−0.949166 + 0.314775i \(0.898071\pi\)
\(954\) 0 0
\(955\) 75671.0 + 43688.7i 2.56404 + 1.48035i
\(956\) 0 0
\(957\) 13490.9 + 47941.1i 0.455694 + 1.61935i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −12299.8 21303.8i −0.412868 0.715109i
\(962\) 0 0
\(963\) 725.433 + 1186.88i 0.0242749 + 0.0397160i
\(964\) 0 0
\(965\) −42853.9 −1.42955
\(966\) 0 0
\(967\) 48504.3 1.61302 0.806512 0.591218i \(-0.201353\pi\)
0.806512 + 0.591218i \(0.201353\pi\)
\(968\) 0 0
\(969\) −1513.38 1475.98i −0.0501722 0.0489321i
\(970\) 0 0
\(971\) −16579.2 28716.0i −0.547943 0.949065i −0.998415 0.0562744i \(-0.982078\pi\)
0.450473 0.892790i \(-0.351255\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 96706.7 27213.8i 3.17651 0.893887i
\(976\) 0 0
\(977\) −29562.2 17067.8i −0.968045 0.558901i −0.0694053 0.997589i \(-0.522110\pi\)
−0.898640 + 0.438688i \(0.855443\pi\)
\(978\) 0 0
\(979\) 59688.8i 1.94858i
\(980\) 0 0
\(981\) −15902.7 8658.27i −0.517568 0.281791i
\(982\) 0 0
\(983\) −169.731 + 293.983i −0.00550721 + 0.00953877i −0.868766 0.495223i \(-0.835086\pi\)
0.863259 + 0.504762i \(0.168420\pi\)
\(984\) 0 0
\(985\) 70100.6 40472.6i 2.26761 1.30920i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11285.5 + 6515.66i −0.362848 + 0.209490i
\(990\) 0 0
\(991\) 279.970 484.922i 0.00897431 0.0155440i −0.861503 0.507752i \(-0.830477\pi\)
0.870478 + 0.492208i \(0.163810\pi\)
\(992\) 0 0
\(993\) 210.655 + 53.6292i 0.00673207 + 0.00171387i
\(994\) 0 0
\(995\) 85556.5i 2.72595i
\(996\) 0 0
\(997\) 11120.1 + 6420.22i 0.353238 + 0.203942i 0.666111 0.745853i \(-0.267958\pi\)
−0.312872 + 0.949795i \(0.601291\pi\)
\(998\) 0 0
\(999\) 2163.39 9484.42i 0.0685151 0.300374i
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.c.521.2 12
3.2 odd 2 inner 588.4.k.c.521.5 12
7.2 even 3 84.4.k.c.5.2 12
7.3 odd 6 588.4.f.c.293.1 12
7.4 even 3 588.4.f.c.293.12 12
7.5 odd 6 inner 588.4.k.c.509.5 12
7.6 odd 2 84.4.k.c.17.5 yes 12
21.2 odd 6 84.4.k.c.5.5 yes 12
21.5 even 6 inner 588.4.k.c.509.2 12
21.11 odd 6 588.4.f.c.293.2 12
21.17 even 6 588.4.f.c.293.11 12
21.20 even 2 84.4.k.c.17.2 yes 12
28.23 odd 6 336.4.bc.c.257.5 12
28.27 even 2 336.4.bc.c.17.2 12
84.23 even 6 336.4.bc.c.257.2 12
84.83 odd 2 336.4.bc.c.17.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.k.c.5.2 12 7.2 even 3
84.4.k.c.5.5 yes 12 21.2 odd 6
84.4.k.c.17.2 yes 12 21.20 even 2
84.4.k.c.17.5 yes 12 7.6 odd 2
336.4.bc.c.17.2 12 28.27 even 2
336.4.bc.c.17.5 12 84.83 odd 2
336.4.bc.c.257.2 12 84.23 even 6
336.4.bc.c.257.5 12 28.23 odd 6
588.4.f.c.293.1 12 7.3 odd 6
588.4.f.c.293.2 12 21.11 odd 6
588.4.f.c.293.11 12 21.17 even 6
588.4.f.c.293.12 12 7.4 even 3
588.4.k.c.509.2 12 21.5 even 6 inner
588.4.k.c.509.5 12 7.5 odd 6 inner
588.4.k.c.521.2 12 1.1 even 1 trivial
588.4.k.c.521.5 12 3.2 odd 2 inner