Properties

Label 531.5.b.a.296.16
Level $531$
Weight $5$
Character 531.296
Analytic conductor $54.889$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,5,Mod(296,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.296");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.b (of order \(2\), degree \(1\), 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: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 296.16
Character \(\chi\) \(=\) 531.296
Dual form 531.5.b.a.296.61

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.40748i q^{2} -13.2409 q^{4} +5.33509i q^{5} -40.2707 q^{7} -14.9200i q^{8} +O(q^{10})\) \(q-5.40748i q^{2} -13.2409 q^{4} +5.33509i q^{5} -40.2707 q^{7} -14.9200i q^{8} +28.8494 q^{10} +236.242i q^{11} +228.643 q^{13} +217.763i q^{14} -292.533 q^{16} -143.085i q^{17} -422.626 q^{19} -70.6411i q^{20} +1277.48 q^{22} -486.101i q^{23} +596.537 q^{25} -1236.38i q^{26} +533.219 q^{28} -464.257i q^{29} +1107.45 q^{31} +1343.15i q^{32} -773.731 q^{34} -214.848i q^{35} -60.3545 q^{37} +2285.34i q^{38} +79.5994 q^{40} -404.305i q^{41} +1540.56 q^{43} -3128.05i q^{44} -2628.58 q^{46} -1125.85i q^{47} -779.269 q^{49} -3225.76i q^{50} -3027.44 q^{52} -5199.94i q^{53} -1260.37 q^{55} +600.838i q^{56} -2510.46 q^{58} +453.188i q^{59} -126.227 q^{61} -5988.51i q^{62} +2582.52 q^{64} +1219.83i q^{65} +5064.97 q^{67} +1894.57i q^{68} -1161.79 q^{70} -3201.77i q^{71} +4567.64 q^{73} +326.366i q^{74} +5595.94 q^{76} -9513.65i q^{77} +71.8464 q^{79} -1560.69i q^{80} -2186.27 q^{82} -4147.02i q^{83} +763.372 q^{85} -8330.56i q^{86} +3524.73 q^{88} -7286.37i q^{89} -9207.63 q^{91} +6436.40i q^{92} -6088.03 q^{94} -2254.75i q^{95} -11003.2 q^{97} +4213.88i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.40748i − 1.35187i −0.736961 0.675935i \(-0.763740\pi\)
0.736961 0.675935i \(-0.236260\pi\)
\(3\) 0 0
\(4\) −13.2409 −0.827554
\(5\) 5.33509i 0.213403i 0.994291 + 0.106702i \(0.0340290\pi\)
−0.994291 + 0.106702i \(0.965971\pi\)
\(6\) 0 0
\(7\) −40.2707 −0.821851 −0.410926 0.911669i \(-0.634794\pi\)
−0.410926 + 0.911669i \(0.634794\pi\)
\(8\) − 14.9200i − 0.233125i
\(9\) 0 0
\(10\) 28.8494 0.288494
\(11\) 236.242i 1.95242i 0.216836 + 0.976208i \(0.430426\pi\)
−0.216836 + 0.976208i \(0.569574\pi\)
\(12\) 0 0
\(13\) 228.643 1.35292 0.676460 0.736480i \(-0.263513\pi\)
0.676460 + 0.736480i \(0.263513\pi\)
\(14\) 217.763i 1.11104i
\(15\) 0 0
\(16\) −292.533 −1.14271
\(17\) − 143.085i − 0.495105i −0.968874 0.247552i \(-0.920374\pi\)
0.968874 0.247552i \(-0.0796262\pi\)
\(18\) 0 0
\(19\) −422.626 −1.17071 −0.585355 0.810777i \(-0.699045\pi\)
−0.585355 + 0.810777i \(0.699045\pi\)
\(20\) − 70.6411i − 0.176603i
\(21\) 0 0
\(22\) 1277.48 2.63941
\(23\) − 486.101i − 0.918906i −0.888202 0.459453i \(-0.848045\pi\)
0.888202 0.459453i \(-0.151955\pi\)
\(24\) 0 0
\(25\) 596.537 0.954459
\(26\) − 1236.38i − 1.82897i
\(27\) 0 0
\(28\) 533.219 0.680126
\(29\) − 464.257i − 0.552030i −0.961153 0.276015i \(-0.910986\pi\)
0.961153 0.276015i \(-0.0890139\pi\)
\(30\) 0 0
\(31\) 1107.45 1.15239 0.576196 0.817312i \(-0.304537\pi\)
0.576196 + 0.817312i \(0.304537\pi\)
\(32\) 1343.15i 1.31167i
\(33\) 0 0
\(34\) −773.731 −0.669318
\(35\) − 214.848i − 0.175386i
\(36\) 0 0
\(37\) −60.3545 −0.0440865 −0.0220433 0.999757i \(-0.507017\pi\)
−0.0220433 + 0.999757i \(0.507017\pi\)
\(38\) 2285.34i 1.58265i
\(39\) 0 0
\(40\) 79.5994 0.0497496
\(41\) − 404.305i − 0.240514i −0.992743 0.120257i \(-0.961628\pi\)
0.992743 0.120257i \(-0.0383719\pi\)
\(42\) 0 0
\(43\) 1540.56 0.833186 0.416593 0.909093i \(-0.363224\pi\)
0.416593 + 0.909093i \(0.363224\pi\)
\(44\) − 3128.05i − 1.61573i
\(45\) 0 0
\(46\) −2628.58 −1.24224
\(47\) − 1125.85i − 0.509666i −0.966985 0.254833i \(-0.917979\pi\)
0.966985 0.254833i \(-0.0820205\pi\)
\(48\) 0 0
\(49\) −779.269 −0.324560
\(50\) − 3225.76i − 1.29030i
\(51\) 0 0
\(52\) −3027.44 −1.11961
\(53\) − 5199.94i − 1.85117i −0.378537 0.925586i \(-0.623573\pi\)
0.378537 0.925586i \(-0.376427\pi\)
\(54\) 0 0
\(55\) −1260.37 −0.416652
\(56\) 600.838i 0.191594i
\(57\) 0 0
\(58\) −2510.46 −0.746273
\(59\) 453.188i 0.130189i
\(60\) 0 0
\(61\) −126.227 −0.0339229 −0.0169614 0.999856i \(-0.505399\pi\)
−0.0169614 + 0.999856i \(0.505399\pi\)
\(62\) − 5988.51i − 1.55788i
\(63\) 0 0
\(64\) 2582.52 0.630499
\(65\) 1219.83i 0.288718i
\(66\) 0 0
\(67\) 5064.97 1.12831 0.564153 0.825670i \(-0.309203\pi\)
0.564153 + 0.825670i \(0.309203\pi\)
\(68\) 1894.57i 0.409726i
\(69\) 0 0
\(70\) −1161.79 −0.237099
\(71\) − 3201.77i − 0.635147i −0.948234 0.317573i \(-0.897132\pi\)
0.948234 0.317573i \(-0.102868\pi\)
\(72\) 0 0
\(73\) 4567.64 0.857128 0.428564 0.903511i \(-0.359020\pi\)
0.428564 + 0.903511i \(0.359020\pi\)
\(74\) 326.366i 0.0595993i
\(75\) 0 0
\(76\) 5595.94 0.968825
\(77\) − 9513.65i − 1.60460i
\(78\) 0 0
\(79\) 71.8464 0.0115120 0.00575600 0.999983i \(-0.498168\pi\)
0.00575600 + 0.999983i \(0.498168\pi\)
\(80\) − 1560.69i − 0.243858i
\(81\) 0 0
\(82\) −2186.27 −0.325144
\(83\) − 4147.02i − 0.601977i −0.953628 0.300989i \(-0.902683\pi\)
0.953628 0.300989i \(-0.0973166\pi\)
\(84\) 0 0
\(85\) 763.372 0.105657
\(86\) − 8330.56i − 1.12636i
\(87\) 0 0
\(88\) 3524.73 0.455156
\(89\) − 7286.37i − 0.919880i −0.887950 0.459940i \(-0.847871\pi\)
0.887950 0.459940i \(-0.152129\pi\)
\(90\) 0 0
\(91\) −9207.63 −1.11190
\(92\) 6436.40i 0.760444i
\(93\) 0 0
\(94\) −6088.03 −0.689003
\(95\) − 2254.75i − 0.249833i
\(96\) 0 0
\(97\) −11003.2 −1.16944 −0.584718 0.811237i \(-0.698795\pi\)
−0.584718 + 0.811237i \(0.698795\pi\)
\(98\) 4213.88i 0.438763i
\(99\) 0 0
\(100\) −7898.66 −0.789866
\(101\) − 13790.2i − 1.35185i −0.736971 0.675924i \(-0.763745\pi\)
0.736971 0.675924i \(-0.236255\pi\)
\(102\) 0 0
\(103\) 845.921 0.0797361 0.0398681 0.999205i \(-0.487306\pi\)
0.0398681 + 0.999205i \(0.487306\pi\)
\(104\) − 3411.35i − 0.315399i
\(105\) 0 0
\(106\) −28118.6 −2.50255
\(107\) 9260.27i 0.808828i 0.914576 + 0.404414i \(0.132525\pi\)
−0.914576 + 0.404414i \(0.867475\pi\)
\(108\) 0 0
\(109\) −16263.0 −1.36883 −0.684413 0.729094i \(-0.739941\pi\)
−0.684413 + 0.729094i \(0.739941\pi\)
\(110\) 6815.45i 0.563260i
\(111\) 0 0
\(112\) 11780.5 0.939136
\(113\) − 7683.57i − 0.601737i −0.953666 0.300868i \(-0.902724\pi\)
0.953666 0.300868i \(-0.0972765\pi\)
\(114\) 0 0
\(115\) 2593.39 0.196098
\(116\) 6147.17i 0.456835i
\(117\) 0 0
\(118\) 2450.60 0.175999
\(119\) 5762.15i 0.406903i
\(120\) 0 0
\(121\) −41169.4 −2.81193
\(122\) 682.570i 0.0458593i
\(123\) 0 0
\(124\) −14663.6 −0.953666
\(125\) 6517.00i 0.417088i
\(126\) 0 0
\(127\) 27830.1 1.72547 0.862734 0.505658i \(-0.168750\pi\)
0.862734 + 0.505658i \(0.168750\pi\)
\(128\) 7525.45i 0.459317i
\(129\) 0 0
\(130\) 6596.22 0.390309
\(131\) 21420.2i 1.24819i 0.781348 + 0.624095i \(0.214532\pi\)
−0.781348 + 0.624095i \(0.785468\pi\)
\(132\) 0 0
\(133\) 17019.5 0.962149
\(134\) − 27388.7i − 1.52532i
\(135\) 0 0
\(136\) −2134.83 −0.115421
\(137\) − 5082.92i − 0.270814i −0.990790 0.135407i \(-0.956766\pi\)
0.990790 0.135407i \(-0.0432343\pi\)
\(138\) 0 0
\(139\) 6781.40 0.350986 0.175493 0.984481i \(-0.443848\pi\)
0.175493 + 0.984481i \(0.443848\pi\)
\(140\) 2844.77i 0.145141i
\(141\) 0 0
\(142\) −17313.5 −0.858636
\(143\) 54015.2i 2.64146i
\(144\) 0 0
\(145\) 2476.85 0.117805
\(146\) − 24699.4i − 1.15873i
\(147\) 0 0
\(148\) 799.146 0.0364840
\(149\) − 36064.7i − 1.62446i −0.583335 0.812232i \(-0.698252\pi\)
0.583335 0.812232i \(-0.301748\pi\)
\(150\) 0 0
\(151\) 7344.83 0.322127 0.161064 0.986944i \(-0.448508\pi\)
0.161064 + 0.986944i \(0.448508\pi\)
\(152\) 6305.57i 0.272921i
\(153\) 0 0
\(154\) −51444.9 −2.16921
\(155\) 5908.33i 0.245924i
\(156\) 0 0
\(157\) 9876.14 0.400671 0.200336 0.979727i \(-0.435797\pi\)
0.200336 + 0.979727i \(0.435797\pi\)
\(158\) − 388.508i − 0.0155627i
\(159\) 0 0
\(160\) −7165.82 −0.279915
\(161\) 19575.6i 0.755204i
\(162\) 0 0
\(163\) −36136.4 −1.36010 −0.680049 0.733167i \(-0.738041\pi\)
−0.680049 + 0.733167i \(0.738041\pi\)
\(164\) 5353.35i 0.199039i
\(165\) 0 0
\(166\) −22424.9 −0.813795
\(167\) − 22251.4i − 0.797856i −0.916982 0.398928i \(-0.869382\pi\)
0.916982 0.398928i \(-0.130618\pi\)
\(168\) 0 0
\(169\) 23716.8 0.830390
\(170\) − 4127.92i − 0.142835i
\(171\) 0 0
\(172\) −20398.4 −0.689507
\(173\) 24165.7i 0.807436i 0.914884 + 0.403718i \(0.132282\pi\)
−0.914884 + 0.403718i \(0.867718\pi\)
\(174\) 0 0
\(175\) −24023.0 −0.784423
\(176\) − 69108.8i − 2.23104i
\(177\) 0 0
\(178\) −39400.9 −1.24356
\(179\) 47777.0i 1.49112i 0.666437 + 0.745561i \(0.267818\pi\)
−0.666437 + 0.745561i \(0.732182\pi\)
\(180\) 0 0
\(181\) 55467.3 1.69309 0.846545 0.532317i \(-0.178678\pi\)
0.846545 + 0.532317i \(0.178678\pi\)
\(182\) 49790.1i 1.50314i
\(183\) 0 0
\(184\) −7252.62 −0.214220
\(185\) − 321.996i − 0.00940822i
\(186\) 0 0
\(187\) 33802.8 0.966651
\(188\) 14907.3i 0.421776i
\(189\) 0 0
\(190\) −12192.5 −0.337743
\(191\) − 26889.8i − 0.737089i −0.929610 0.368545i \(-0.879856\pi\)
0.929610 0.368545i \(-0.120144\pi\)
\(192\) 0 0
\(193\) 43456.7 1.16665 0.583327 0.812238i \(-0.301751\pi\)
0.583327 + 0.812238i \(0.301751\pi\)
\(194\) 59499.7i 1.58093i
\(195\) 0 0
\(196\) 10318.2 0.268591
\(197\) − 8804.71i − 0.226873i −0.993545 0.113436i \(-0.963814\pi\)
0.993545 0.113436i \(-0.0361858\pi\)
\(198\) 0 0
\(199\) 62780.3 1.58532 0.792661 0.609663i \(-0.208695\pi\)
0.792661 + 0.609663i \(0.208695\pi\)
\(200\) − 8900.32i − 0.222508i
\(201\) 0 0
\(202\) −74570.3 −1.82752
\(203\) 18696.0i 0.453687i
\(204\) 0 0
\(205\) 2157.00 0.0513266
\(206\) − 4574.30i − 0.107793i
\(207\) 0 0
\(208\) −66885.8 −1.54599
\(209\) − 99842.2i − 2.28571i
\(210\) 0 0
\(211\) 13469.2 0.302536 0.151268 0.988493i \(-0.451664\pi\)
0.151268 + 0.988493i \(0.451664\pi\)
\(212\) 68851.7i 1.53194i
\(213\) 0 0
\(214\) 50074.8 1.09343
\(215\) 8219.03i 0.177805i
\(216\) 0 0
\(217\) −44597.7 −0.947095
\(218\) 87942.0i 1.85048i
\(219\) 0 0
\(220\) 16688.4 0.344802
\(221\) − 32715.5i − 0.669837i
\(222\) 0 0
\(223\) 20730.4 0.416868 0.208434 0.978036i \(-0.433163\pi\)
0.208434 + 0.978036i \(0.433163\pi\)
\(224\) − 54089.6i − 1.07800i
\(225\) 0 0
\(226\) −41548.8 −0.813470
\(227\) − 26973.0i − 0.523453i −0.965142 0.261727i \(-0.915708\pi\)
0.965142 0.261727i \(-0.0842919\pi\)
\(228\) 0 0
\(229\) −97728.9 −1.86360 −0.931798 0.362977i \(-0.881760\pi\)
−0.931798 + 0.362977i \(0.881760\pi\)
\(230\) − 14023.7i − 0.265099i
\(231\) 0 0
\(232\) −6926.71 −0.128692
\(233\) − 46974.8i − 0.865273i −0.901568 0.432637i \(-0.857583\pi\)
0.901568 0.432637i \(-0.142417\pi\)
\(234\) 0 0
\(235\) 6006.52 0.108764
\(236\) − 6000.60i − 0.107738i
\(237\) 0 0
\(238\) 31158.7 0.550080
\(239\) − 36154.4i − 0.632944i −0.948602 0.316472i \(-0.897502\pi\)
0.948602 0.316472i \(-0.102498\pi\)
\(240\) 0 0
\(241\) 41569.2 0.715711 0.357856 0.933777i \(-0.383508\pi\)
0.357856 + 0.933777i \(0.383508\pi\)
\(242\) 222623.i 3.80136i
\(243\) 0 0
\(244\) 1671.35 0.0280730
\(245\) − 4157.47i − 0.0692623i
\(246\) 0 0
\(247\) −96630.7 −1.58388
\(248\) − 16523.1i − 0.268651i
\(249\) 0 0
\(250\) 35240.6 0.563849
\(251\) − 37383.6i − 0.593381i −0.954974 0.296690i \(-0.904117\pi\)
0.954974 0.296690i \(-0.0958829\pi\)
\(252\) 0 0
\(253\) 114838. 1.79409
\(254\) − 150491.i − 2.33261i
\(255\) 0 0
\(256\) 82014.1 1.25144
\(257\) 13617.3i 0.206170i 0.994673 + 0.103085i \(0.0328714\pi\)
−0.994673 + 0.103085i \(0.967129\pi\)
\(258\) 0 0
\(259\) 2430.52 0.0362326
\(260\) − 16151.6i − 0.238929i
\(261\) 0 0
\(262\) 115829. 1.68739
\(263\) − 109111.i − 1.57746i −0.614738 0.788731i \(-0.710738\pi\)
0.614738 0.788731i \(-0.289262\pi\)
\(264\) 0 0
\(265\) 27742.1 0.395047
\(266\) − 92032.4i − 1.30070i
\(267\) 0 0
\(268\) −67064.6 −0.933735
\(269\) − 114932.i − 1.58832i −0.607709 0.794160i \(-0.707911\pi\)
0.607709 0.794160i \(-0.292089\pi\)
\(270\) 0 0
\(271\) 6066.71 0.0826065 0.0413033 0.999147i \(-0.486849\pi\)
0.0413033 + 0.999147i \(0.486849\pi\)
\(272\) 41857.2i 0.565760i
\(273\) 0 0
\(274\) −27485.8 −0.366106
\(275\) 140927.i 1.86350i
\(276\) 0 0
\(277\) 510.178 0.00664908 0.00332454 0.999994i \(-0.498942\pi\)
0.00332454 + 0.999994i \(0.498942\pi\)
\(278\) − 36670.3i − 0.474487i
\(279\) 0 0
\(280\) −3205.52 −0.0408868
\(281\) 119073.i 1.50799i 0.656878 + 0.753997i \(0.271877\pi\)
−0.656878 + 0.753997i \(0.728123\pi\)
\(282\) 0 0
\(283\) 39843.2 0.497487 0.248743 0.968569i \(-0.419982\pi\)
0.248743 + 0.968569i \(0.419982\pi\)
\(284\) 42394.3i 0.525618i
\(285\) 0 0
\(286\) 292086. 3.57091
\(287\) 16281.6i 0.197667i
\(288\) 0 0
\(289\) 63047.6 0.754871
\(290\) − 13393.5i − 0.159257i
\(291\) 0 0
\(292\) −60479.4 −0.709320
\(293\) 34537.5i 0.402306i 0.979560 + 0.201153i \(0.0644688\pi\)
−0.979560 + 0.201153i \(0.935531\pi\)
\(294\) 0 0
\(295\) −2417.79 −0.0277828
\(296\) 900.488i 0.0102777i
\(297\) 0 0
\(298\) −195019. −2.19606
\(299\) − 111144.i − 1.24321i
\(300\) 0 0
\(301\) −62039.5 −0.684755
\(302\) − 39717.0i − 0.435474i
\(303\) 0 0
\(304\) 123632. 1.33778
\(305\) − 673.432i − 0.00723926i
\(306\) 0 0
\(307\) 101616. 1.07816 0.539080 0.842255i \(-0.318772\pi\)
0.539080 + 0.842255i \(0.318772\pi\)
\(308\) 125969.i 1.32789i
\(309\) 0 0
\(310\) 31949.2 0.332458
\(311\) − 160917.i − 1.66372i −0.554985 0.831860i \(-0.687276\pi\)
0.554985 0.831860i \(-0.312724\pi\)
\(312\) 0 0
\(313\) 177997. 1.81687 0.908437 0.418022i \(-0.137276\pi\)
0.908437 + 0.418022i \(0.137276\pi\)
\(314\) − 53405.1i − 0.541656i
\(315\) 0 0
\(316\) −951.309 −0.00952681
\(317\) 31885.4i 0.317303i 0.987335 + 0.158651i \(0.0507146\pi\)
−0.987335 + 0.158651i \(0.949285\pi\)
\(318\) 0 0
\(319\) 109677. 1.07779
\(320\) 13778.0i 0.134551i
\(321\) 0 0
\(322\) 105855. 1.02094
\(323\) 60471.6i 0.579624i
\(324\) 0 0
\(325\) 136394. 1.29131
\(326\) 195407.i 1.83868i
\(327\) 0 0
\(328\) −6032.22 −0.0560699
\(329\) 45338.9i 0.418870i
\(330\) 0 0
\(331\) 183181. 1.67195 0.835975 0.548767i \(-0.184903\pi\)
0.835975 + 0.548767i \(0.184903\pi\)
\(332\) 54910.2i 0.498169i
\(333\) 0 0
\(334\) −120324. −1.07860
\(335\) 27022.0i 0.240785i
\(336\) 0 0
\(337\) −48086.1 −0.423408 −0.211704 0.977334i \(-0.567901\pi\)
−0.211704 + 0.977334i \(0.567901\pi\)
\(338\) − 128248.i − 1.12258i
\(339\) 0 0
\(340\) −10107.7 −0.0874369
\(341\) 261626.i 2.24995i
\(342\) 0 0
\(343\) 128072. 1.08859
\(344\) − 22985.1i − 0.194236i
\(345\) 0 0
\(346\) 130676. 1.09155
\(347\) 89342.6i 0.741993i 0.928634 + 0.370996i \(0.120984\pi\)
−0.928634 + 0.370996i \(0.879016\pi\)
\(348\) 0 0
\(349\) −152276. −1.25020 −0.625102 0.780543i \(-0.714943\pi\)
−0.625102 + 0.780543i \(0.714943\pi\)
\(350\) 129904.i 1.06044i
\(351\) 0 0
\(352\) −317309. −2.56092
\(353\) 4377.51i 0.0351300i 0.999846 + 0.0175650i \(0.00559140\pi\)
−0.999846 + 0.0175650i \(0.994409\pi\)
\(354\) 0 0
\(355\) 17081.7 0.135542
\(356\) 96477.9i 0.761251i
\(357\) 0 0
\(358\) 258354. 2.01580
\(359\) 84682.0i 0.657056i 0.944494 + 0.328528i \(0.106553\pi\)
−0.944494 + 0.328528i \(0.893447\pi\)
\(360\) 0 0
\(361\) 48291.9 0.370561
\(362\) − 299939.i − 2.28884i
\(363\) 0 0
\(364\) 121917. 0.920156
\(365\) 24368.7i 0.182914i
\(366\) 0 0
\(367\) −62433.7 −0.463540 −0.231770 0.972771i \(-0.574452\pi\)
−0.231770 + 0.972771i \(0.574452\pi\)
\(368\) 142201.i 1.05004i
\(369\) 0 0
\(370\) −1741.19 −0.0127187
\(371\) 209405.i 1.52139i
\(372\) 0 0
\(373\) 93463.9 0.671778 0.335889 0.941901i \(-0.390963\pi\)
0.335889 + 0.941901i \(0.390963\pi\)
\(374\) − 182788.i − 1.30679i
\(375\) 0 0
\(376\) −16797.7 −0.118816
\(377\) − 106149.i − 0.746852i
\(378\) 0 0
\(379\) −194054. −1.35097 −0.675483 0.737376i \(-0.736065\pi\)
−0.675483 + 0.737376i \(0.736065\pi\)
\(380\) 29854.8i 0.206751i
\(381\) 0 0
\(382\) −145406. −0.996449
\(383\) − 234963.i − 1.60177i −0.598816 0.800887i \(-0.704362\pi\)
0.598816 0.800887i \(-0.295638\pi\)
\(384\) 0 0
\(385\) 50756.1 0.342426
\(386\) − 234991.i − 1.57716i
\(387\) 0 0
\(388\) 145692. 0.967771
\(389\) 13392.4i 0.0885032i 0.999020 + 0.0442516i \(0.0140903\pi\)
−0.999020 + 0.0442516i \(0.985910\pi\)
\(390\) 0 0
\(391\) −69553.9 −0.454955
\(392\) 11626.7i 0.0756630i
\(393\) 0 0
\(394\) −47611.3 −0.306703
\(395\) 383.307i 0.00245670i
\(396\) 0 0
\(397\) −244261. −1.54979 −0.774894 0.632091i \(-0.782197\pi\)
−0.774894 + 0.632091i \(0.782197\pi\)
\(398\) − 339483.i − 2.14315i
\(399\) 0 0
\(400\) −174507. −1.09067
\(401\) − 47912.6i − 0.297962i −0.988840 0.148981i \(-0.952401\pi\)
0.988840 0.148981i \(-0.0475993\pi\)
\(402\) 0 0
\(403\) 253211. 1.55909
\(404\) 182594.i 1.11873i
\(405\) 0 0
\(406\) 101098. 0.613325
\(407\) − 14258.3i − 0.0860753i
\(408\) 0 0
\(409\) −50042.1 −0.299150 −0.149575 0.988750i \(-0.547791\pi\)
−0.149575 + 0.988750i \(0.547791\pi\)
\(410\) − 11663.9i − 0.0693869i
\(411\) 0 0
\(412\) −11200.7 −0.0659860
\(413\) − 18250.2i − 0.106996i
\(414\) 0 0
\(415\) 22124.7 0.128464
\(416\) 307102.i 1.77458i
\(417\) 0 0
\(418\) −539895. −3.08999
\(419\) 236589.i 1.34762i 0.738906 + 0.673809i \(0.235343\pi\)
−0.738906 + 0.673809i \(0.764657\pi\)
\(420\) 0 0
\(421\) −100454. −0.566767 −0.283383 0.959007i \(-0.591457\pi\)
−0.283383 + 0.959007i \(0.591457\pi\)
\(422\) − 72834.5i − 0.408989i
\(423\) 0 0
\(424\) −77583.0 −0.431554
\(425\) − 85355.7i − 0.472557i
\(426\) 0 0
\(427\) 5083.25 0.0278796
\(428\) − 122614.i − 0.669349i
\(429\) 0 0
\(430\) 44444.3 0.240369
\(431\) − 45004.6i − 0.242271i −0.992636 0.121136i \(-0.961346\pi\)
0.992636 0.121136i \(-0.0386536\pi\)
\(432\) 0 0
\(433\) 81854.8 0.436584 0.218292 0.975883i \(-0.429951\pi\)
0.218292 + 0.975883i \(0.429951\pi\)
\(434\) 241161.i 1.28035i
\(435\) 0 0
\(436\) 215336. 1.13278
\(437\) 205439.i 1.07577i
\(438\) 0 0
\(439\) 203970. 1.05837 0.529185 0.848506i \(-0.322498\pi\)
0.529185 + 0.848506i \(0.322498\pi\)
\(440\) 18804.7i 0.0971319i
\(441\) 0 0
\(442\) −176908. −0.905533
\(443\) − 13039.6i − 0.0664443i −0.999448 0.0332222i \(-0.989423\pi\)
0.999448 0.0332222i \(-0.0105769\pi\)
\(444\) 0 0
\(445\) 38873.4 0.196306
\(446\) − 112099.i − 0.563552i
\(447\) 0 0
\(448\) −104000. −0.518176
\(449\) 132025.i 0.654884i 0.944871 + 0.327442i \(0.106187\pi\)
−0.944871 + 0.327442i \(0.893813\pi\)
\(450\) 0 0
\(451\) 95513.9 0.469584
\(452\) 101737.i 0.497970i
\(453\) 0 0
\(454\) −145856. −0.707641
\(455\) − 49123.5i − 0.237283i
\(456\) 0 0
\(457\) −416478. −1.99416 −0.997078 0.0763845i \(-0.975662\pi\)
−0.997078 + 0.0763845i \(0.975662\pi\)
\(458\) 528467.i 2.51934i
\(459\) 0 0
\(460\) −34338.8 −0.162281
\(461\) 240078.i 1.12967i 0.825205 + 0.564833i \(0.191060\pi\)
−0.825205 + 0.564833i \(0.808940\pi\)
\(462\) 0 0
\(463\) 256699. 1.19746 0.598731 0.800950i \(-0.295672\pi\)
0.598731 + 0.800950i \(0.295672\pi\)
\(464\) 135811.i 0.630809i
\(465\) 0 0
\(466\) −254016. −1.16974
\(467\) − 117445.i − 0.538520i −0.963068 0.269260i \(-0.913221\pi\)
0.963068 0.269260i \(-0.0867791\pi\)
\(468\) 0 0
\(469\) −203970. −0.927301
\(470\) − 32480.1i − 0.147036i
\(471\) 0 0
\(472\) 6761.55 0.0303502
\(473\) 363946.i 1.62673i
\(474\) 0 0
\(475\) −252112. −1.11739
\(476\) − 76295.8i − 0.336734i
\(477\) 0 0
\(478\) −195504. −0.855659
\(479\) − 374316.i − 1.63143i −0.578456 0.815714i \(-0.696344\pi\)
0.578456 0.815714i \(-0.303656\pi\)
\(480\) 0 0
\(481\) −13799.7 −0.0596455
\(482\) − 224785.i − 0.967549i
\(483\) 0 0
\(484\) 545119. 2.32702
\(485\) − 58703.1i − 0.249562i
\(486\) 0 0
\(487\) 109835. 0.463107 0.231553 0.972822i \(-0.425619\pi\)
0.231553 + 0.972822i \(0.425619\pi\)
\(488\) 1883.30i 0.00790826i
\(489\) 0 0
\(490\) −22481.4 −0.0936336
\(491\) 222390.i 0.922471i 0.887278 + 0.461236i \(0.152594\pi\)
−0.887278 + 0.461236i \(0.847406\pi\)
\(492\) 0 0
\(493\) −66428.4 −0.273313
\(494\) 522529.i 2.14119i
\(495\) 0 0
\(496\) −323966. −1.31685
\(497\) 128938.i 0.521996i
\(498\) 0 0
\(499\) −111625. −0.448292 −0.224146 0.974556i \(-0.571959\pi\)
−0.224146 + 0.974556i \(0.571959\pi\)
\(500\) − 86290.8i − 0.345163i
\(501\) 0 0
\(502\) −202151. −0.802174
\(503\) 139370.i 0.550850i 0.961323 + 0.275425i \(0.0888186\pi\)
−0.961323 + 0.275425i \(0.911181\pi\)
\(504\) 0 0
\(505\) 73571.9 0.288489
\(506\) − 620983.i − 2.42537i
\(507\) 0 0
\(508\) −368494. −1.42792
\(509\) − 12592.1i − 0.0486030i −0.999705 0.0243015i \(-0.992264\pi\)
0.999705 0.0243015i \(-0.00773616\pi\)
\(510\) 0 0
\(511\) −183942. −0.704432
\(512\) − 323082.i − 1.23246i
\(513\) 0 0
\(514\) 73635.5 0.278715
\(515\) 4513.06i 0.0170160i
\(516\) 0 0
\(517\) 265974. 0.995080
\(518\) − 13143.0i − 0.0489818i
\(519\) 0 0
\(520\) 18199.9 0.0673072
\(521\) − 44155.3i − 0.162670i −0.996687 0.0813350i \(-0.974082\pi\)
0.996687 0.0813350i \(-0.0259184\pi\)
\(522\) 0 0
\(523\) 140348. 0.513102 0.256551 0.966531i \(-0.417414\pi\)
0.256551 + 0.966531i \(0.417414\pi\)
\(524\) − 283622.i − 1.03295i
\(525\) 0 0
\(526\) −590018. −2.13252
\(527\) − 158460.i − 0.570555i
\(528\) 0 0
\(529\) 43546.6 0.155612
\(530\) − 150015.i − 0.534052i
\(531\) 0 0
\(532\) −225352. −0.796231
\(533\) − 92441.6i − 0.325397i
\(534\) 0 0
\(535\) −49404.3 −0.172607
\(536\) − 75569.2i − 0.263036i
\(537\) 0 0
\(538\) −621495. −2.14720
\(539\) − 184096.i − 0.633677i
\(540\) 0 0
\(541\) 335690. 1.14695 0.573474 0.819224i \(-0.305595\pi\)
0.573474 + 0.819224i \(0.305595\pi\)
\(542\) − 32805.6i − 0.111673i
\(543\) 0 0
\(544\) 192185. 0.649414
\(545\) − 86764.6i − 0.292112i
\(546\) 0 0
\(547\) 25995.2 0.0868798 0.0434399 0.999056i \(-0.486168\pi\)
0.0434399 + 0.999056i \(0.486168\pi\)
\(548\) 67302.2i 0.224114i
\(549\) 0 0
\(550\) 762062. 2.51921
\(551\) 196207.i 0.646267i
\(552\) 0 0
\(553\) −2893.31 −0.00946116
\(554\) − 2758.78i − 0.00898870i
\(555\) 0 0
\(556\) −89791.6 −0.290460
\(557\) − 375672.i − 1.21087i −0.795894 0.605437i \(-0.792999\pi\)
0.795894 0.605437i \(-0.207001\pi\)
\(558\) 0 0
\(559\) 352239. 1.12723
\(560\) 62850.1i 0.200415i
\(561\) 0 0
\(562\) 643884. 2.03861
\(563\) − 5956.29i − 0.0187914i −0.999956 0.00939570i \(-0.997009\pi\)
0.999956 0.00939570i \(-0.00299079\pi\)
\(564\) 0 0
\(565\) 40992.5 0.128413
\(566\) − 215452.i − 0.672538i
\(567\) 0 0
\(568\) −47770.4 −0.148068
\(569\) 598655.i 1.84907i 0.381103 + 0.924533i \(0.375544\pi\)
−0.381103 + 0.924533i \(0.624456\pi\)
\(570\) 0 0
\(571\) −291238. −0.893257 −0.446629 0.894719i \(-0.647375\pi\)
−0.446629 + 0.894719i \(0.647375\pi\)
\(572\) − 715208.i − 2.18595i
\(573\) 0 0
\(574\) 88042.7 0.267220
\(575\) − 289977.i − 0.877058i
\(576\) 0 0
\(577\) −97573.3 −0.293075 −0.146538 0.989205i \(-0.546813\pi\)
−0.146538 + 0.989205i \(0.546813\pi\)
\(578\) − 340929.i − 1.02049i
\(579\) 0 0
\(580\) −32795.7 −0.0974901
\(581\) 167004.i 0.494736i
\(582\) 0 0
\(583\) 1.22845e6 3.61426
\(584\) − 68149.0i − 0.199818i
\(585\) 0 0
\(586\) 186761. 0.543865
\(587\) 610947.i 1.77308i 0.462656 + 0.886538i \(0.346896\pi\)
−0.462656 + 0.886538i \(0.653104\pi\)
\(588\) 0 0
\(589\) −468037. −1.34912
\(590\) 13074.2i 0.0375587i
\(591\) 0 0
\(592\) 17655.7 0.0503781
\(593\) 289519.i 0.823319i 0.911338 + 0.411660i \(0.135051\pi\)
−0.911338 + 0.411660i \(0.864949\pi\)
\(594\) 0 0
\(595\) −30741.6 −0.0868344
\(596\) 477528.i 1.34433i
\(597\) 0 0
\(598\) −601008. −1.68065
\(599\) 666953.i 1.85884i 0.369025 + 0.929420i \(0.379692\pi\)
−0.369025 + 0.929420i \(0.620308\pi\)
\(600\) 0 0
\(601\) 267999. 0.741965 0.370983 0.928640i \(-0.379021\pi\)
0.370983 + 0.928640i \(0.379021\pi\)
\(602\) 335478.i 0.925701i
\(603\) 0 0
\(604\) −97251.8 −0.266578
\(605\) − 219642.i − 0.600075i
\(606\) 0 0
\(607\) −651200. −1.76741 −0.883704 0.468046i \(-0.844958\pi\)
−0.883704 + 0.468046i \(0.844958\pi\)
\(608\) − 567650.i − 1.53558i
\(609\) 0 0
\(610\) −3641.57 −0.00978654
\(611\) − 257419.i − 0.689537i
\(612\) 0 0
\(613\) −123203. −0.327868 −0.163934 0.986471i \(-0.552418\pi\)
−0.163934 + 0.986471i \(0.552418\pi\)
\(614\) − 549484.i − 1.45753i
\(615\) 0 0
\(616\) −141943. −0.374071
\(617\) − 427945.i − 1.12413i −0.827092 0.562067i \(-0.810006\pi\)
0.827092 0.562067i \(-0.189994\pi\)
\(618\) 0 0
\(619\) −67763.5 −0.176854 −0.0884269 0.996083i \(-0.528184\pi\)
−0.0884269 + 0.996083i \(0.528184\pi\)
\(620\) − 78231.4i − 0.203516i
\(621\) 0 0
\(622\) −870154. −2.24913
\(623\) 293427.i 0.756005i
\(624\) 0 0
\(625\) 338067. 0.865451
\(626\) − 962517.i − 2.45618i
\(627\) 0 0
\(628\) −130769. −0.331577
\(629\) 8635.84i 0.0218275i
\(630\) 0 0
\(631\) −479812. −1.20507 −0.602536 0.798092i \(-0.705843\pi\)
−0.602536 + 0.798092i \(0.705843\pi\)
\(632\) − 1071.95i − 0.00268373i
\(633\) 0 0
\(634\) 172420. 0.428952
\(635\) 148476.i 0.368221i
\(636\) 0 0
\(637\) −178175. −0.439104
\(638\) − 593077.i − 1.45704i
\(639\) 0 0
\(640\) −40148.9 −0.0980198
\(641\) 499820.i 1.21646i 0.793761 + 0.608230i \(0.208120\pi\)
−0.793761 + 0.608230i \(0.791880\pi\)
\(642\) 0 0
\(643\) −512244. −1.23895 −0.619476 0.785015i \(-0.712655\pi\)
−0.619476 + 0.785015i \(0.712655\pi\)
\(644\) − 259198.i − 0.624972i
\(645\) 0 0
\(646\) 326999. 0.783577
\(647\) − 246169.i − 0.588063i −0.955796 0.294032i \(-0.905003\pi\)
0.955796 0.294032i \(-0.0949971\pi\)
\(648\) 0 0
\(649\) −107062. −0.254183
\(650\) − 737549.i − 1.74568i
\(651\) 0 0
\(652\) 478477. 1.12555
\(653\) 232200.i 0.544549i 0.962220 + 0.272274i \(0.0877758\pi\)
−0.962220 + 0.272274i \(0.912224\pi\)
\(654\) 0 0
\(655\) −114279. −0.266368
\(656\) 118273.i 0.274838i
\(657\) 0 0
\(658\) 245169. 0.566258
\(659\) − 254880.i − 0.586902i −0.955974 0.293451i \(-0.905196\pi\)
0.955974 0.293451i \(-0.0948037\pi\)
\(660\) 0 0
\(661\) −735421. −1.68319 −0.841595 0.540109i \(-0.818383\pi\)
−0.841595 + 0.540109i \(0.818383\pi\)
\(662\) − 990546.i − 2.26026i
\(663\) 0 0
\(664\) −61873.5 −0.140336
\(665\) 90800.3i 0.205326i
\(666\) 0 0
\(667\) −225676. −0.507264
\(668\) 294628.i 0.660269i
\(669\) 0 0
\(670\) 146121. 0.325510
\(671\) − 29820.2i − 0.0662315i
\(672\) 0 0
\(673\) 678968. 1.49906 0.749530 0.661971i \(-0.230280\pi\)
0.749530 + 0.661971i \(0.230280\pi\)
\(674\) 260025.i 0.572393i
\(675\) 0 0
\(676\) −314030. −0.687193
\(677\) 574807.i 1.25414i 0.778965 + 0.627068i \(0.215745\pi\)
−0.778965 + 0.627068i \(0.784255\pi\)
\(678\) 0 0
\(679\) 443108. 0.961102
\(680\) − 11389.5i − 0.0246313i
\(681\) 0 0
\(682\) 1.41474e6 3.04164
\(683\) 415608.i 0.890928i 0.895300 + 0.445464i \(0.146961\pi\)
−0.895300 + 0.445464i \(0.853039\pi\)
\(684\) 0 0
\(685\) 27117.8 0.0577927
\(686\) − 692546.i − 1.47164i
\(687\) 0 0
\(688\) −450666. −0.952089
\(689\) − 1.18893e6i − 2.50449i
\(690\) 0 0
\(691\) −77705.9 −0.162741 −0.0813706 0.996684i \(-0.525930\pi\)
−0.0813706 + 0.996684i \(0.525930\pi\)
\(692\) − 319975.i − 0.668197i
\(693\) 0 0
\(694\) 483118. 1.00308
\(695\) 36179.3i 0.0749016i
\(696\) 0 0
\(697\) −57850.1 −0.119080
\(698\) 823430.i 1.69011i
\(699\) 0 0
\(700\) 318085. 0.649153
\(701\) − 754544.i − 1.53549i −0.640753 0.767747i \(-0.721378\pi\)
0.640753 0.767747i \(-0.278622\pi\)
\(702\) 0 0
\(703\) 25507.4 0.0516125
\(704\) 610101.i 1.23100i
\(705\) 0 0
\(706\) 23671.3 0.0474912
\(707\) 555342.i 1.11102i
\(708\) 0 0
\(709\) −312848. −0.622360 −0.311180 0.950351i \(-0.600724\pi\)
−0.311180 + 0.950351i \(0.600724\pi\)
\(710\) − 92369.2i − 0.183236i
\(711\) 0 0
\(712\) −108713. −0.214447
\(713\) − 538332.i − 1.05894i
\(714\) 0 0
\(715\) −288176. −0.563697
\(716\) − 632609.i − 1.23398i
\(717\) 0 0
\(718\) 457917. 0.888255
\(719\) 518853.i 1.00366i 0.864966 + 0.501830i \(0.167340\pi\)
−0.864966 + 0.501830i \(0.832660\pi\)
\(720\) 0 0
\(721\) −34065.8 −0.0655313
\(722\) − 261137.i − 0.500951i
\(723\) 0 0
\(724\) −734436. −1.40112
\(725\) − 276946.i − 0.526890i
\(726\) 0 0
\(727\) 352533. 0.667007 0.333504 0.942749i \(-0.391769\pi\)
0.333504 + 0.942749i \(0.391769\pi\)
\(728\) 137378.i 0.259211i
\(729\) 0 0
\(730\) 131773. 0.247276
\(731\) − 220432.i − 0.412515i
\(732\) 0 0
\(733\) 242825. 0.451944 0.225972 0.974134i \(-0.427444\pi\)
0.225972 + 0.974134i \(0.427444\pi\)
\(734\) 337609.i 0.626645i
\(735\) 0 0
\(736\) 652907. 1.20530
\(737\) 1.19656e6i 2.20292i
\(738\) 0 0
\(739\) −919000. −1.68278 −0.841388 0.540431i \(-0.818261\pi\)
−0.841388 + 0.540431i \(0.818261\pi\)
\(740\) 4263.51i 0.00778581i
\(741\) 0 0
\(742\) 1.13236e6 2.05672
\(743\) − 563145.i − 1.02010i −0.860145 0.510050i \(-0.829627\pi\)
0.860145 0.510050i \(-0.170373\pi\)
\(744\) 0 0
\(745\) 192408. 0.346666
\(746\) − 505404.i − 0.908157i
\(747\) 0 0
\(748\) −447578. −0.799956
\(749\) − 372918.i − 0.664736i
\(750\) 0 0
\(751\) 105723. 0.187451 0.0937255 0.995598i \(-0.470122\pi\)
0.0937255 + 0.995598i \(0.470122\pi\)
\(752\) 329349.i 0.582400i
\(753\) 0 0
\(754\) −574000. −1.00965
\(755\) 39185.3i 0.0687431i
\(756\) 0 0
\(757\) −318147. −0.555184 −0.277592 0.960699i \(-0.589536\pi\)
−0.277592 + 0.960699i \(0.589536\pi\)
\(758\) 1.04934e6i 1.82633i
\(759\) 0 0
\(760\) −33640.8 −0.0582423
\(761\) − 864313.i − 1.49246i −0.665690 0.746228i \(-0.731863\pi\)
0.665690 0.746228i \(-0.268137\pi\)
\(762\) 0 0
\(763\) 654924. 1.12497
\(764\) 356044.i 0.609981i
\(765\) 0 0
\(766\) −1.27056e6 −2.16539
\(767\) 103618.i 0.176135i
\(768\) 0 0
\(769\) 1.13314e6 1.91615 0.958077 0.286512i \(-0.0924958\pi\)
0.958077 + 0.286512i \(0.0924958\pi\)
\(770\) − 274463.i − 0.462916i
\(771\) 0 0
\(772\) −575404. −0.965469
\(773\) − 995777.i − 1.66649i −0.552903 0.833246i \(-0.686480\pi\)
0.552903 0.833246i \(-0.313520\pi\)
\(774\) 0 0
\(775\) 660634. 1.09991
\(776\) 164168.i 0.272624i
\(777\) 0 0
\(778\) 72419.1 0.119645
\(779\) 170870.i 0.281573i
\(780\) 0 0
\(781\) 756395. 1.24007
\(782\) 376112.i 0.615040i
\(783\) 0 0
\(784\) 227962. 0.370878
\(785\) 52690.1i 0.0855046i
\(786\) 0 0
\(787\) −1.08294e6 −1.74845 −0.874225 0.485521i \(-0.838630\pi\)
−0.874225 + 0.485521i \(0.838630\pi\)
\(788\) 116582.i 0.187750i
\(789\) 0 0
\(790\) 2072.73 0.00332114
\(791\) 309423.i 0.494538i
\(792\) 0 0
\(793\) −28861.0 −0.0458949
\(794\) 1.32083e6i 2.09511i
\(795\) 0 0
\(796\) −831265. −1.31194
\(797\) 1.09994e6i 1.73161i 0.500378 + 0.865807i \(0.333194\pi\)
−0.500378 + 0.865807i \(0.666806\pi\)
\(798\) 0 0
\(799\) −161093. −0.252338
\(800\) 801238.i 1.25193i
\(801\) 0 0
\(802\) −259086. −0.402806
\(803\) 1.07907e6i 1.67347i
\(804\) 0 0
\(805\) −104438. −0.161163
\(806\) − 1.36923e6i − 2.10769i
\(807\) 0 0
\(808\) −205750. −0.315149
\(809\) 1.06036e6i 1.62015i 0.586328 + 0.810074i \(0.300573\pi\)
−0.586328 + 0.810074i \(0.699427\pi\)
\(810\) 0 0
\(811\) −32747.6 −0.0497895 −0.0248948 0.999690i \(-0.507925\pi\)
−0.0248948 + 0.999690i \(0.507925\pi\)
\(812\) − 247551.i − 0.375450i
\(813\) 0 0
\(814\) −77101.4 −0.116363
\(815\) − 192791.i − 0.290249i
\(816\) 0 0
\(817\) −651082. −0.975419
\(818\) 270602.i 0.404412i
\(819\) 0 0
\(820\) −28560.6 −0.0424755
\(821\) − 286009.i − 0.424319i −0.977235 0.212160i \(-0.931950\pi\)
0.977235 0.212160i \(-0.0680497\pi\)
\(822\) 0 0
\(823\) 68867.6 0.101675 0.0508376 0.998707i \(-0.483811\pi\)
0.0508376 + 0.998707i \(0.483811\pi\)
\(824\) − 12621.1i − 0.0185885i
\(825\) 0 0
\(826\) −98687.6 −0.144645
\(827\) 214757.i 0.314005i 0.987598 + 0.157002i \(0.0501830\pi\)
−0.987598 + 0.157002i \(0.949817\pi\)
\(828\) 0 0
\(829\) −891503. −1.29722 −0.648610 0.761121i \(-0.724649\pi\)
−0.648610 + 0.761121i \(0.724649\pi\)
\(830\) − 119639.i − 0.173667i
\(831\) 0 0
\(832\) 590476. 0.853013
\(833\) 111502.i 0.160691i
\(834\) 0 0
\(835\) 118713. 0.170265
\(836\) 1.32200e6i 1.89155i
\(837\) 0 0
\(838\) 1.27935e6 1.82180
\(839\) 716368.i 1.01768i 0.860860 + 0.508841i \(0.169926\pi\)
−0.860860 + 0.508841i \(0.830074\pi\)
\(840\) 0 0
\(841\) 491746. 0.695263
\(842\) 543205.i 0.766195i
\(843\) 0 0
\(844\) −178344. −0.250365
\(845\) 126531.i 0.177208i
\(846\) 0 0
\(847\) 1.65792e6 2.31099
\(848\) 1.52116e6i 2.11535i
\(849\) 0 0
\(850\) −461559. −0.638836
\(851\) 29338.4i 0.0405114i
\(852\) 0 0
\(853\) −206839. −0.284273 −0.142136 0.989847i \(-0.545397\pi\)
−0.142136 + 0.989847i \(0.545397\pi\)
\(854\) − 27487.6i − 0.0376895i
\(855\) 0 0
\(856\) 138163. 0.188558
\(857\) 141720.i 0.192960i 0.995335 + 0.0964802i \(0.0307585\pi\)
−0.995335 + 0.0964802i \(0.969242\pi\)
\(858\) 0 0
\(859\) −587558. −0.796277 −0.398139 0.917325i \(-0.630344\pi\)
−0.398139 + 0.917325i \(0.630344\pi\)
\(860\) − 108827.i − 0.147143i
\(861\) 0 0
\(862\) −243362. −0.327520
\(863\) − 1.10938e6i − 1.48957i −0.667307 0.744783i \(-0.732553\pi\)
0.667307 0.744783i \(-0.267447\pi\)
\(864\) 0 0
\(865\) −128926. −0.172310
\(866\) − 442628.i − 0.590206i
\(867\) 0 0
\(868\) 590513. 0.783772
\(869\) 16973.2i 0.0224762i
\(870\) 0 0
\(871\) 1.15807e6 1.52651
\(872\) 242644.i 0.319107i
\(873\) 0 0
\(874\) 1.11091e6 1.45430
\(875\) − 262444.i − 0.342785i
\(876\) 0 0
\(877\) 299446. 0.389332 0.194666 0.980870i \(-0.437638\pi\)
0.194666 + 0.980870i \(0.437638\pi\)
\(878\) − 1.10297e6i − 1.43078i
\(879\) 0 0
\(880\) 368701. 0.476112
\(881\) − 743364.i − 0.957745i −0.877884 0.478872i \(-0.841046\pi\)
0.877884 0.478872i \(-0.158954\pi\)
\(882\) 0 0
\(883\) 294544. 0.377771 0.188886 0.981999i \(-0.439512\pi\)
0.188886 + 0.981999i \(0.439512\pi\)
\(884\) 433181.i 0.554326i
\(885\) 0 0
\(886\) −70511.6 −0.0898241
\(887\) − 441410.i − 0.561041i −0.959848 0.280521i \(-0.909493\pi\)
0.959848 0.280521i \(-0.0905071\pi\)
\(888\) 0 0
\(889\) −1.12074e6 −1.41808
\(890\) − 210207.i − 0.265380i
\(891\) 0 0
\(892\) −274489. −0.344981
\(893\) 475815.i 0.596671i
\(894\) 0 0
\(895\) −254895. −0.318211
\(896\) − 303055.i − 0.377490i
\(897\) 0 0
\(898\) 713924. 0.885318
\(899\) − 514141.i − 0.636155i
\(900\) 0 0
\(901\) −744035. −0.916524
\(902\) − 516490.i − 0.634817i
\(903\) 0 0
\(904\) −114639. −0.140280
\(905\) 295923.i 0.361311i
\(906\) 0 0
\(907\) −887819. −1.07922 −0.539610 0.841915i \(-0.681428\pi\)
−0.539610 + 0.841915i \(0.681428\pi\)
\(908\) 357146.i 0.433186i
\(909\) 0 0
\(910\) −265634. −0.320776
\(911\) 1.25649e6i 1.51399i 0.653422 + 0.756994i \(0.273333\pi\)
−0.653422 + 0.756994i \(0.726667\pi\)
\(912\) 0 0
\(913\) 979702. 1.17531
\(914\) 2.25210e6i 2.69584i
\(915\) 0 0
\(916\) 1.29401e6 1.54223
\(917\) − 862607.i − 1.02583i
\(918\) 0 0
\(919\) −700226. −0.829101 −0.414550 0.910026i \(-0.636061\pi\)
−0.414550 + 0.910026i \(0.636061\pi\)
\(920\) − 38693.4i − 0.0457152i
\(921\) 0 0
\(922\) 1.29822e6 1.52716
\(923\) − 732064.i − 0.859302i
\(924\) 0 0
\(925\) −36003.7 −0.0420788
\(926\) − 1.38809e6i − 1.61881i
\(927\) 0 0
\(928\) 623567. 0.724081
\(929\) 1.28131e6i 1.48464i 0.670043 + 0.742322i \(0.266276\pi\)
−0.670043 + 0.742322i \(0.733724\pi\)
\(930\) 0 0
\(931\) 329340. 0.379966
\(932\) 621987.i 0.716060i
\(933\) 0 0
\(934\) −635083. −0.728009
\(935\) 180341.i 0.206287i
\(936\) 0 0
\(937\) −1.20550e6 −1.37305 −0.686525 0.727106i \(-0.740865\pi\)
−0.686525 + 0.727106i \(0.740865\pi\)
\(938\) 1.10296e6i 1.25359i
\(939\) 0 0
\(940\) −79531.5 −0.0900085
\(941\) 404760.i 0.457107i 0.973531 + 0.228554i \(0.0733996\pi\)
−0.973531 + 0.228554i \(0.926600\pi\)
\(942\) 0 0
\(943\) −196533. −0.221010
\(944\) − 132572.i − 0.148768i
\(945\) 0 0
\(946\) 1.96803e6 2.19912
\(947\) 111370.i 0.124185i 0.998070 + 0.0620925i \(0.0197774\pi\)
−0.998070 + 0.0620925i \(0.980223\pi\)
\(948\) 0 0
\(949\) 1.04436e6 1.15962
\(950\) 1.36329e6i 1.51057i
\(951\) 0 0
\(952\) 85971.1 0.0948590
\(953\) 848130.i 0.933849i 0.884297 + 0.466925i \(0.154638\pi\)
−0.884297 + 0.466925i \(0.845362\pi\)
\(954\) 0 0
\(955\) 143459. 0.157297
\(956\) 478716.i 0.523796i
\(957\) 0 0
\(958\) −2.02411e6 −2.20548
\(959\) 204693.i 0.222569i
\(960\) 0 0
\(961\) 302921. 0.328006
\(962\) 74621.4i 0.0806330i
\(963\) 0 0
\(964\) −550412. −0.592290
\(965\) 231845.i 0.248968i
\(966\) 0 0
\(967\) −1.22760e6 −1.31282 −0.656408 0.754406i \(-0.727925\pi\)
−0.656408 + 0.754406i \(0.727925\pi\)
\(968\) 614247.i 0.655530i
\(969\) 0 0
\(970\) −317436. −0.337375
\(971\) 863529.i 0.915880i 0.888983 + 0.457940i \(0.151413\pi\)
−0.888983 + 0.457940i \(0.848587\pi\)
\(972\) 0 0
\(973\) −273092. −0.288458
\(974\) − 593928.i − 0.626060i
\(975\) 0 0
\(976\) 36925.6 0.0387639
\(977\) − 1.55838e6i − 1.63262i −0.577613 0.816311i \(-0.696016\pi\)
0.577613 0.816311i \(-0.303984\pi\)
\(978\) 0 0
\(979\) 1.72135e6 1.79599
\(980\) 55048.5i 0.0573183i
\(981\) 0 0
\(982\) 1.20257e6 1.24706
\(983\) 1.35456e6i 1.40182i 0.713251 + 0.700909i \(0.247222\pi\)
−0.713251 + 0.700909i \(0.752778\pi\)
\(984\) 0 0
\(985\) 46973.9 0.0484155
\(986\) 359210.i 0.369483i
\(987\) 0 0
\(988\) 1.27947e6 1.31074
\(989\) − 748869.i − 0.765620i
\(990\) 0 0
\(991\) 79523.9 0.0809749 0.0404874 0.999180i \(-0.487109\pi\)
0.0404874 + 0.999180i \(0.487109\pi\)
\(992\) 1.48747e6i 1.51156i
\(993\) 0 0
\(994\) 697229. 0.705671
\(995\) 334938.i 0.338313i
\(996\) 0 0
\(997\) −1.29801e6 −1.30583 −0.652917 0.757430i \(-0.726455\pi\)
−0.652917 + 0.757430i \(0.726455\pi\)
\(998\) 603611.i 0.606033i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.5.b.a.296.16 76
3.2 odd 2 inner 531.5.b.a.296.61 yes 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
531.5.b.a.296.16 76 1.1 even 1 trivial
531.5.b.a.296.61 yes 76 3.2 odd 2 inner