Properties

Label 160.6.f.a.49.2
Level $160$
Weight $6$
Character 160.49
Analytic conductor $25.661$
Analytic rank $0$
Dimension $28$
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] = [160,6,Mod(49,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.f (of order \(2\), degree \(1\), 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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Character \(\chi\) \(=\) 160.49
Dual form 160.6.f.a.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-28.9041 q^{3} +(-40.5064 + 38.5257i) q^{5} -128.100i q^{7} +592.448 q^{9} +O(q^{10})\) \(q-28.9041 q^{3} +(-40.5064 + 38.5257i) q^{5} -128.100i q^{7} +592.448 q^{9} -433.553i q^{11} +78.5787 q^{13} +(1170.80 - 1113.55i) q^{15} -1487.27i q^{17} +98.3252i q^{19} +3702.61i q^{21} +2363.01i q^{23} +(156.542 - 3121.08i) q^{25} -10100.5 q^{27} -1024.44i q^{29} -4308.96 q^{31} +12531.5i q^{33} +(4935.13 + 5188.86i) q^{35} +6647.54 q^{37} -2271.25 q^{39} -4441.16 q^{41} -7066.93 q^{43} +(-23998.0 + 22824.5i) q^{45} +15560.5i q^{47} +397.456 q^{49} +42988.3i q^{51} -26793.6 q^{53} +(16702.9 + 17561.7i) q^{55} -2842.00i q^{57} -32902.0i q^{59} +19242.7i q^{61} -75892.5i q^{63} +(-3182.94 + 3027.30i) q^{65} +1066.77 q^{67} -68300.6i q^{69} -12397.2 q^{71} -37248.9i q^{73} +(-4524.72 + 90212.0i) q^{75} -55538.0 q^{77} +50784.5 q^{79} +147981. q^{81} -38676.4 q^{83} +(57298.2 + 60244.1i) q^{85} +29610.5i q^{87} -132425. q^{89} -10065.9i q^{91} +124547. q^{93} +(-3788.05 - 3982.80i) q^{95} +95071.6i q^{97} -256858. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 1940 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 1940 q^{9} + 488 q^{15} + 1556 q^{25} - 4368 q^{31} - 23360 q^{39} - 2480 q^{41} - 38420 q^{49} + 48776 q^{55} + 37200 q^{65} + 69232 q^{71} + 35984 q^{79} + 122596 q^{81} - 178744 q^{89} - 89416 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −28.9041 −1.85420 −0.927100 0.374814i \(-0.877707\pi\)
−0.927100 + 0.374814i \(0.877707\pi\)
\(4\) 0 0
\(5\) −40.5064 + 38.5257i −0.724601 + 0.689169i
\(6\) 0 0
\(7\) 128.100i 0.988105i −0.869432 0.494053i \(-0.835515\pi\)
0.869432 0.494053i \(-0.164485\pi\)
\(8\) 0 0
\(9\) 592.448 2.43806
\(10\) 0 0
\(11\) 433.553i 1.08034i −0.841556 0.540170i \(-0.818360\pi\)
0.841556 0.540170i \(-0.181640\pi\)
\(12\) 0 0
\(13\) 78.5787 0.128957 0.0644787 0.997919i \(-0.479462\pi\)
0.0644787 + 0.997919i \(0.479462\pi\)
\(14\) 0 0
\(15\) 1170.80 1113.55i 1.34356 1.27786i
\(16\) 0 0
\(17\) 1487.27i 1.24815i −0.781363 0.624077i \(-0.785475\pi\)
0.781363 0.624077i \(-0.214525\pi\)
\(18\) 0 0
\(19\) 98.3252i 0.0624857i 0.999512 + 0.0312429i \(0.00994653\pi\)
−0.999512 + 0.0312429i \(0.990053\pi\)
\(20\) 0 0
\(21\) 3702.61i 1.83214i
\(22\) 0 0
\(23\) 2363.01i 0.931419i 0.884938 + 0.465710i \(0.154201\pi\)
−0.884938 + 0.465710i \(0.845799\pi\)
\(24\) 0 0
\(25\) 156.542 3121.08i 0.0500935 0.998745i
\(26\) 0 0
\(27\) −10100.5 −2.66645
\(28\) 0 0
\(29\) 1024.44i 0.226199i −0.993584 0.113100i \(-0.963922\pi\)
0.993584 0.113100i \(-0.0360779\pi\)
\(30\) 0 0
\(31\) −4308.96 −0.805320 −0.402660 0.915350i \(-0.631914\pi\)
−0.402660 + 0.915350i \(0.631914\pi\)
\(32\) 0 0
\(33\) 12531.5i 2.00317i
\(34\) 0 0
\(35\) 4935.13 + 5188.86i 0.680971 + 0.715982i
\(36\) 0 0
\(37\) 6647.54 0.798283 0.399141 0.916889i \(-0.369308\pi\)
0.399141 + 0.916889i \(0.369308\pi\)
\(38\) 0 0
\(39\) −2271.25 −0.239113
\(40\) 0 0
\(41\) −4441.16 −0.412608 −0.206304 0.978488i \(-0.566144\pi\)
−0.206304 + 0.978488i \(0.566144\pi\)
\(42\) 0 0
\(43\) −7066.93 −0.582854 −0.291427 0.956593i \(-0.594130\pi\)
−0.291427 + 0.956593i \(0.594130\pi\)
\(44\) 0 0
\(45\) −23998.0 + 22824.5i −1.76662 + 1.68023i
\(46\) 0 0
\(47\) 15560.5i 1.02750i 0.857941 + 0.513748i \(0.171743\pi\)
−0.857941 + 0.513748i \(0.828257\pi\)
\(48\) 0 0
\(49\) 397.456 0.0236482
\(50\) 0 0
\(51\) 42988.3i 2.31433i
\(52\) 0 0
\(53\) −26793.6 −1.31021 −0.655105 0.755538i \(-0.727376\pi\)
−0.655105 + 0.755538i \(0.727376\pi\)
\(54\) 0 0
\(55\) 16702.9 + 17561.7i 0.744536 + 0.782816i
\(56\) 0 0
\(57\) 2842.00i 0.115861i
\(58\) 0 0
\(59\) 32902.0i 1.23053i −0.788320 0.615266i \(-0.789049\pi\)
0.788320 0.615266i \(-0.210951\pi\)
\(60\) 0 0
\(61\) 19242.7i 0.662126i 0.943609 + 0.331063i \(0.107407\pi\)
−0.943609 + 0.331063i \(0.892593\pi\)
\(62\) 0 0
\(63\) 75892.5i 2.40906i
\(64\) 0 0
\(65\) −3182.94 + 3027.30i −0.0934427 + 0.0888734i
\(66\) 0 0
\(67\) 1066.77 0.0290326 0.0145163 0.999895i \(-0.495379\pi\)
0.0145163 + 0.999895i \(0.495379\pi\)
\(68\) 0 0
\(69\) 68300.6i 1.72704i
\(70\) 0 0
\(71\) −12397.2 −0.291862 −0.145931 0.989295i \(-0.546618\pi\)
−0.145931 + 0.989295i \(0.546618\pi\)
\(72\) 0 0
\(73\) 37248.9i 0.818100i −0.912512 0.409050i \(-0.865860\pi\)
0.912512 0.409050i \(-0.134140\pi\)
\(74\) 0 0
\(75\) −4524.72 + 90212.0i −0.0928834 + 1.85187i
\(76\) 0 0
\(77\) −55538.0 −1.06749
\(78\) 0 0
\(79\) 50784.5 0.915510 0.457755 0.889078i \(-0.348654\pi\)
0.457755 + 0.889078i \(0.348654\pi\)
\(80\) 0 0
\(81\) 147981. 2.50607
\(82\) 0 0
\(83\) −38676.4 −0.616242 −0.308121 0.951347i \(-0.599700\pi\)
−0.308121 + 0.951347i \(0.599700\pi\)
\(84\) 0 0
\(85\) 57298.2 + 60244.1i 0.860188 + 0.904413i
\(86\) 0 0
\(87\) 29610.5i 0.419419i
\(88\) 0 0
\(89\) −132425. −1.77213 −0.886067 0.463558i \(-0.846573\pi\)
−0.886067 + 0.463558i \(0.846573\pi\)
\(90\) 0 0
\(91\) 10065.9i 0.127424i
\(92\) 0 0
\(93\) 124547. 1.49322
\(94\) 0 0
\(95\) −3788.05 3982.80i −0.0430632 0.0452772i
\(96\) 0 0
\(97\) 95071.6i 1.02594i 0.858407 + 0.512969i \(0.171455\pi\)
−0.858407 + 0.512969i \(0.828545\pi\)
\(98\) 0 0
\(99\) 256858.i 2.63393i
\(100\) 0 0
\(101\) 49284.2i 0.480734i 0.970682 + 0.240367i \(0.0772677\pi\)
−0.970682 + 0.240367i \(0.922732\pi\)
\(102\) 0 0
\(103\) 143185.i 1.32986i 0.746906 + 0.664930i \(0.231539\pi\)
−0.746906 + 0.664930i \(0.768461\pi\)
\(104\) 0 0
\(105\) −142646. 149980.i −1.26266 1.32757i
\(106\) 0 0
\(107\) 168772. 1.42508 0.712541 0.701630i \(-0.247544\pi\)
0.712541 + 0.701630i \(0.247544\pi\)
\(108\) 0 0
\(109\) 169878.i 1.36953i −0.728763 0.684766i \(-0.759905\pi\)
0.728763 0.684766i \(-0.240095\pi\)
\(110\) 0 0
\(111\) −192141. −1.48018
\(112\) 0 0
\(113\) 176203.i 1.29813i 0.760733 + 0.649064i \(0.224839\pi\)
−0.760733 + 0.649064i \(0.775161\pi\)
\(114\) 0 0
\(115\) −91036.5 95717.0i −0.641905 0.674908i
\(116\) 0 0
\(117\) 46553.8 0.314406
\(118\) 0 0
\(119\) −190519. −1.23331
\(120\) 0 0
\(121\) −26917.2 −0.167135
\(122\) 0 0
\(123\) 128368. 0.765057
\(124\) 0 0
\(125\) 113901. + 132455.i 0.652005 + 0.758214i
\(126\) 0 0
\(127\) 28671.3i 0.157739i 0.996885 + 0.0788694i \(0.0251310\pi\)
−0.996885 + 0.0788694i \(0.974869\pi\)
\(128\) 0 0
\(129\) 204263. 1.08073
\(130\) 0 0
\(131\) 42028.2i 0.213975i 0.994260 + 0.106987i \(0.0341204\pi\)
−0.994260 + 0.106987i \(0.965880\pi\)
\(132\) 0 0
\(133\) 12595.4 0.0617425
\(134\) 0 0
\(135\) 409135. 389129.i 1.93211 1.83763i
\(136\) 0 0
\(137\) 49062.2i 0.223329i 0.993746 + 0.111665i \(0.0356182\pi\)
−0.993746 + 0.111665i \(0.964382\pi\)
\(138\) 0 0
\(139\) 85502.0i 0.375352i 0.982231 + 0.187676i \(0.0600956\pi\)
−0.982231 + 0.187676i \(0.939904\pi\)
\(140\) 0 0
\(141\) 449764.i 1.90518i
\(142\) 0 0
\(143\) 34068.0i 0.139318i
\(144\) 0 0
\(145\) 39467.2 + 41496.4i 0.155889 + 0.163904i
\(146\) 0 0
\(147\) −11488.1 −0.0438486
\(148\) 0 0
\(149\) 29442.1i 0.108643i 0.998523 + 0.0543216i \(0.0172996\pi\)
−0.998523 + 0.0543216i \(0.982700\pi\)
\(150\) 0 0
\(151\) −131978. −0.471041 −0.235520 0.971869i \(-0.575679\pi\)
−0.235520 + 0.971869i \(0.575679\pi\)
\(152\) 0 0
\(153\) 881132.i 3.04307i
\(154\) 0 0
\(155\) 174541. 166006.i 0.583536 0.555001i
\(156\) 0 0
\(157\) 360997. 1.16884 0.584419 0.811452i \(-0.301322\pi\)
0.584419 + 0.811452i \(0.301322\pi\)
\(158\) 0 0
\(159\) 774444. 2.42939
\(160\) 0 0
\(161\) 302701. 0.920340
\(162\) 0 0
\(163\) 139344. 0.410788 0.205394 0.978679i \(-0.434152\pi\)
0.205394 + 0.978679i \(0.434152\pi\)
\(164\) 0 0
\(165\) −482784. 507605.i −1.38052 1.45150i
\(166\) 0 0
\(167\) 440004.i 1.22086i 0.792071 + 0.610429i \(0.209003\pi\)
−0.792071 + 0.610429i \(0.790997\pi\)
\(168\) 0 0
\(169\) −365118. −0.983370
\(170\) 0 0
\(171\) 58252.6i 0.152344i
\(172\) 0 0
\(173\) −711657. −1.80782 −0.903910 0.427722i \(-0.859316\pi\)
−0.903910 + 0.427722i \(0.859316\pi\)
\(174\) 0 0
\(175\) −399809. 20053.0i −0.986865 0.0494977i
\(176\) 0 0
\(177\) 951005.i 2.28165i
\(178\) 0 0
\(179\) 813579.i 1.89787i 0.315468 + 0.948936i \(0.397839\pi\)
−0.315468 + 0.948936i \(0.602161\pi\)
\(180\) 0 0
\(181\) 415019.i 0.941612i 0.882237 + 0.470806i \(0.156037\pi\)
−0.882237 + 0.470806i \(0.843963\pi\)
\(182\) 0 0
\(183\) 556192.i 1.22771i
\(184\) 0 0
\(185\) −269268. + 256101.i −0.578437 + 0.550151i
\(186\) 0 0
\(187\) −644811. −1.34843
\(188\) 0 0
\(189\) 1.29387e6i 2.63473i
\(190\) 0 0
\(191\) −598242. −1.18657 −0.593285 0.804992i \(-0.702169\pi\)
−0.593285 + 0.804992i \(0.702169\pi\)
\(192\) 0 0
\(193\) 150045.i 0.289954i 0.989435 + 0.144977i \(0.0463107\pi\)
−0.989435 + 0.144977i \(0.953689\pi\)
\(194\) 0 0
\(195\) 92000.2 87501.4i 0.173262 0.164789i
\(196\) 0 0
\(197\) 120180. 0.220630 0.110315 0.993897i \(-0.464814\pi\)
0.110315 + 0.993897i \(0.464814\pi\)
\(198\) 0 0
\(199\) −994244. −1.77976 −0.889878 0.456199i \(-0.849211\pi\)
−0.889878 + 0.456199i \(0.849211\pi\)
\(200\) 0 0
\(201\) −30834.2 −0.0538322
\(202\) 0 0
\(203\) −131230. −0.223509
\(204\) 0 0
\(205\) 179896. 171099.i 0.298976 0.284356i
\(206\) 0 0
\(207\) 1.39996e6i 2.27086i
\(208\) 0 0
\(209\) 42629.2 0.0675059
\(210\) 0 0
\(211\) 854741.i 1.32169i −0.750524 0.660843i \(-0.770199\pi\)
0.750524 0.660843i \(-0.229801\pi\)
\(212\) 0 0
\(213\) 358330. 0.541171
\(214\) 0 0
\(215\) 286256. 272258.i 0.422336 0.401684i
\(216\) 0 0
\(217\) 551977.i 0.795741i
\(218\) 0 0
\(219\) 1.07665e6i 1.51692i
\(220\) 0 0
\(221\) 116868.i 0.160959i
\(222\) 0 0
\(223\) 135422.i 0.182359i 0.995834 + 0.0911797i \(0.0290638\pi\)
−0.995834 + 0.0911797i \(0.970936\pi\)
\(224\) 0 0
\(225\) 92743.2 1.84908e6i 0.122131 2.43500i
\(226\) 0 0
\(227\) −538983. −0.694241 −0.347120 0.937821i \(-0.612840\pi\)
−0.347120 + 0.937821i \(0.612840\pi\)
\(228\) 0 0
\(229\) 1.04360e6i 1.31507i 0.753426 + 0.657533i \(0.228400\pi\)
−0.753426 + 0.657533i \(0.771600\pi\)
\(230\) 0 0
\(231\) 1.60528e6 1.97934
\(232\) 0 0
\(233\) 466145.i 0.562512i 0.959633 + 0.281256i \(0.0907510\pi\)
−0.959633 + 0.281256i \(0.909249\pi\)
\(234\) 0 0
\(235\) −599481. 630302.i −0.708118 0.744525i
\(236\) 0 0
\(237\) −1.46788e6 −1.69754
\(238\) 0 0
\(239\) 1.49073e6 1.68812 0.844061 0.536248i \(-0.180159\pi\)
0.844061 + 0.536248i \(0.180159\pi\)
\(240\) 0 0
\(241\) −248170. −0.275237 −0.137619 0.990485i \(-0.543945\pi\)
−0.137619 + 0.990485i \(0.543945\pi\)
\(242\) 0 0
\(243\) −1.82284e6 −1.98031
\(244\) 0 0
\(245\) −16099.5 + 15312.3i −0.0171355 + 0.0162976i
\(246\) 0 0
\(247\) 7726.27i 0.00805800i
\(248\) 0 0
\(249\) 1.11791e6 1.14264
\(250\) 0 0
\(251\) 1.70931e6i 1.71252i 0.516545 + 0.856260i \(0.327218\pi\)
−0.516545 + 0.856260i \(0.672782\pi\)
\(252\) 0 0
\(253\) 1.02449e6 1.00625
\(254\) 0 0
\(255\) −1.65615e6 1.74130e6i −1.59496 1.67696i
\(256\) 0 0
\(257\) 445935.i 0.421152i −0.977578 0.210576i \(-0.932466\pi\)
0.977578 0.210576i \(-0.0675339\pi\)
\(258\) 0 0
\(259\) 851549.i 0.788787i
\(260\) 0 0
\(261\) 606928.i 0.551487i
\(262\) 0 0
\(263\) 616116.i 0.549254i −0.961551 0.274627i \(-0.911446\pi\)
0.961551 0.274627i \(-0.0885544\pi\)
\(264\) 0 0
\(265\) 1.08531e6 1.03224e6i 0.949379 0.902955i
\(266\) 0 0
\(267\) 3.82764e6 3.28589
\(268\) 0 0
\(269\) 945945.i 0.797049i −0.917158 0.398524i \(-0.869522\pi\)
0.917158 0.398524i \(-0.130478\pi\)
\(270\) 0 0
\(271\) 859288. 0.710748 0.355374 0.934724i \(-0.384354\pi\)
0.355374 + 0.934724i \(0.384354\pi\)
\(272\) 0 0
\(273\) 290946.i 0.236269i
\(274\) 0 0
\(275\) −1.35315e6 67869.4i −1.07898 0.0541180i
\(276\) 0 0
\(277\) 1.32686e6 1.03903 0.519514 0.854462i \(-0.326113\pi\)
0.519514 + 0.854462i \(0.326113\pi\)
\(278\) 0 0
\(279\) −2.55284e6 −1.96342
\(280\) 0 0
\(281\) −651484. −0.492196 −0.246098 0.969245i \(-0.579148\pi\)
−0.246098 + 0.969245i \(0.579148\pi\)
\(282\) 0 0
\(283\) 1.94978e6 1.44717 0.723585 0.690235i \(-0.242493\pi\)
0.723585 + 0.690235i \(0.242493\pi\)
\(284\) 0 0
\(285\) 109490. + 115119.i 0.0798478 + 0.0839531i
\(286\) 0 0
\(287\) 568912.i 0.407700i
\(288\) 0 0
\(289\) −792120. −0.557887
\(290\) 0 0
\(291\) 2.74796e6i 1.90230i
\(292\) 0 0
\(293\) −1.80561e6 −1.22872 −0.614362 0.789024i \(-0.710587\pi\)
−0.614362 + 0.789024i \(0.710587\pi\)
\(294\) 0 0
\(295\) 1.26757e6 + 1.33274e6i 0.848044 + 0.891645i
\(296\) 0 0
\(297\) 4.37910e6i 2.88067i
\(298\) 0 0
\(299\) 185682.i 0.120113i
\(300\) 0 0
\(301\) 905272.i 0.575921i
\(302\) 0 0
\(303\) 1.42452e6i 0.891377i
\(304\) 0 0
\(305\) −741337. 779451.i −0.456316 0.479777i
\(306\) 0 0
\(307\) −333416. −0.201902 −0.100951 0.994891i \(-0.532188\pi\)
−0.100951 + 0.994891i \(0.532188\pi\)
\(308\) 0 0
\(309\) 4.13865e6i 2.46583i
\(310\) 0 0
\(311\) −126148. −0.0739573 −0.0369787 0.999316i \(-0.511773\pi\)
−0.0369787 + 0.999316i \(0.511773\pi\)
\(312\) 0 0
\(313\) 1.61358e6i 0.930955i −0.885060 0.465477i \(-0.845883\pi\)
0.885060 0.465477i \(-0.154117\pi\)
\(314\) 0 0
\(315\) 2.92381e6 + 3.07413e6i 1.66025 + 1.74561i
\(316\) 0 0
\(317\) −1.62558e6 −0.908571 −0.454286 0.890856i \(-0.650105\pi\)
−0.454286 + 0.890856i \(0.650105\pi\)
\(318\) 0 0
\(319\) −444149. −0.244372
\(320\) 0 0
\(321\) −4.87820e6 −2.64239
\(322\) 0 0
\(323\) 146236. 0.0779918
\(324\) 0 0
\(325\) 12300.9 245250.i 0.00645993 0.128796i
\(326\) 0 0
\(327\) 4.91019e6i 2.53939i
\(328\) 0 0
\(329\) 1.99330e6 1.01527
\(330\) 0 0
\(331\) 325105.i 0.163100i 0.996669 + 0.0815499i \(0.0259870\pi\)
−0.996669 + 0.0815499i \(0.974013\pi\)
\(332\) 0 0
\(333\) 3.93833e6 1.94626
\(334\) 0 0
\(335\) −43211.2 + 41098.2i −0.0210370 + 0.0200083i
\(336\) 0 0
\(337\) 805090.i 0.386162i −0.981183 0.193081i \(-0.938152\pi\)
0.981183 0.193081i \(-0.0618480\pi\)
\(338\) 0 0
\(339\) 5.09300e6i 2.40699i
\(340\) 0 0
\(341\) 1.86816e6i 0.870020i
\(342\) 0 0
\(343\) 2.20389e6i 1.01147i
\(344\) 0 0
\(345\) 2.63133e6 + 2.76661e6i 1.19022 + 1.25141i
\(346\) 0 0
\(347\) −2.17010e6 −0.967512 −0.483756 0.875203i \(-0.660728\pi\)
−0.483756 + 0.875203i \(0.660728\pi\)
\(348\) 0 0
\(349\) 3.51347e6i 1.54409i −0.635569 0.772044i \(-0.719234\pi\)
0.635569 0.772044i \(-0.280766\pi\)
\(350\) 0 0
\(351\) −793684. −0.343859
\(352\) 0 0
\(353\) 2.21866e6i 0.947665i −0.880615 0.473833i \(-0.842870\pi\)
0.880615 0.473833i \(-0.157130\pi\)
\(354\) 0 0
\(355\) 502166. 477611.i 0.211484 0.201142i
\(356\) 0 0
\(357\) 5.50679e6 2.28680
\(358\) 0 0
\(359\) 1.82259e6 0.746366 0.373183 0.927758i \(-0.378266\pi\)
0.373183 + 0.927758i \(0.378266\pi\)
\(360\) 0 0
\(361\) 2.46643e6 0.996096
\(362\) 0 0
\(363\) 778019. 0.309901
\(364\) 0 0
\(365\) 1.43504e6 + 1.50882e6i 0.563809 + 0.592796i
\(366\) 0 0
\(367\) 2.02561e6i 0.785036i −0.919744 0.392518i \(-0.871604\pi\)
0.919744 0.392518i \(-0.128396\pi\)
\(368\) 0 0
\(369\) −2.63116e6 −1.00596
\(370\) 0 0
\(371\) 3.43225e6i 1.29462i
\(372\) 0 0
\(373\) −3.39891e6 −1.26493 −0.632466 0.774588i \(-0.717957\pi\)
−0.632466 + 0.774588i \(0.717957\pi\)
\(374\) 0 0
\(375\) −3.29220e6 3.82848e6i −1.20895 1.40588i
\(376\) 0 0
\(377\) 80499.2i 0.0291701i
\(378\) 0 0
\(379\) 3.13155e6i 1.11986i −0.828542 0.559928i \(-0.810829\pi\)
0.828542 0.559928i \(-0.189171\pi\)
\(380\) 0 0
\(381\) 828719.i 0.292479i
\(382\) 0 0
\(383\) 1.68971e6i 0.588592i 0.955714 + 0.294296i \(0.0950852\pi\)
−0.955714 + 0.294296i \(0.904915\pi\)
\(384\) 0 0
\(385\) 2.24965e6 2.13964e6i 0.773504 0.735680i
\(386\) 0 0
\(387\) −4.18679e6 −1.42103
\(388\) 0 0
\(389\) 3.88788e6i 1.30268i 0.758784 + 0.651342i \(0.225794\pi\)
−0.758784 + 0.651342i \(0.774206\pi\)
\(390\) 0 0
\(391\) 3.51443e6 1.16255
\(392\) 0 0
\(393\) 1.21479e6i 0.396752i
\(394\) 0 0
\(395\) −2.05710e6 + 1.95651e6i −0.663380 + 0.630941i
\(396\) 0 0
\(397\) 213582. 0.0680123 0.0340061 0.999422i \(-0.489173\pi\)
0.0340061 + 0.999422i \(0.489173\pi\)
\(398\) 0 0
\(399\) −364060. −0.114483
\(400\) 0 0
\(401\) 3.71761e6 1.15452 0.577261 0.816560i \(-0.304121\pi\)
0.577261 + 0.816560i \(0.304121\pi\)
\(402\) 0 0
\(403\) −338593. −0.103852
\(404\) 0 0
\(405\) −5.99419e6 + 5.70107e6i −1.81590 + 1.72711i
\(406\) 0 0
\(407\) 2.88206e6i 0.862417i
\(408\) 0 0
\(409\) −1.38621e6 −0.409752 −0.204876 0.978788i \(-0.565679\pi\)
−0.204876 + 0.978788i \(0.565679\pi\)
\(410\) 0 0
\(411\) 1.41810e6i 0.414097i
\(412\) 0 0
\(413\) −4.21474e6 −1.21589
\(414\) 0 0
\(415\) 1.56664e6 1.49004e6i 0.446530 0.424695i
\(416\) 0 0
\(417\) 2.47136e6i 0.695979i
\(418\) 0 0
\(419\) 721117.i 0.200665i −0.994954 0.100332i \(-0.968009\pi\)
0.994954 0.100332i \(-0.0319906\pi\)
\(420\) 0 0
\(421\) 1.36768e6i 0.376079i −0.982161 0.188040i \(-0.939787\pi\)
0.982161 0.188040i \(-0.0602133\pi\)
\(422\) 0 0
\(423\) 9.21882e6i 2.50510i
\(424\) 0 0
\(425\) −4.64189e6 232821.i −1.24659 0.0625244i
\(426\) 0 0
\(427\) 2.46498e6 0.654250
\(428\) 0 0
\(429\) 984707.i 0.258323i
\(430\) 0 0
\(431\) 2.51906e6 0.653198 0.326599 0.945163i \(-0.394097\pi\)
0.326599 + 0.945163i \(0.394097\pi\)
\(432\) 0 0
\(433\) 4.29790e6i 1.10163i 0.834626 + 0.550817i \(0.185684\pi\)
−0.834626 + 0.550817i \(0.814316\pi\)
\(434\) 0 0
\(435\) −1.14077e6 1.19942e6i −0.289050 0.303911i
\(436\) 0 0
\(437\) −232343. −0.0582004
\(438\) 0 0
\(439\) 386299. 0.0956670 0.0478335 0.998855i \(-0.484768\pi\)
0.0478335 + 0.998855i \(0.484768\pi\)
\(440\) 0 0
\(441\) 235472. 0.0576558
\(442\) 0 0
\(443\) 1.13129e6 0.273884 0.136942 0.990579i \(-0.456273\pi\)
0.136942 + 0.990579i \(0.456273\pi\)
\(444\) 0 0
\(445\) 5.36408e6 5.10178e6i 1.28409 1.22130i
\(446\) 0 0
\(447\) 850998.i 0.201446i
\(448\) 0 0
\(449\) 4.41752e6 1.03410 0.517051 0.855955i \(-0.327030\pi\)
0.517051 + 0.855955i \(0.327030\pi\)
\(450\) 0 0
\(451\) 1.92548e6i 0.445757i
\(452\) 0 0
\(453\) 3.81470e6 0.873404
\(454\) 0 0
\(455\) 387796. + 407734.i 0.0878163 + 0.0923312i
\(456\) 0 0
\(457\) 2.26658e6i 0.507668i 0.967248 + 0.253834i \(0.0816917\pi\)
−0.967248 + 0.253834i \(0.918308\pi\)
\(458\) 0 0
\(459\) 1.50222e7i 3.32814i
\(460\) 0 0
\(461\) 8.97491e6i 1.96688i 0.181233 + 0.983440i \(0.441991\pi\)
−0.181233 + 0.983440i \(0.558009\pi\)
\(462\) 0 0
\(463\) 2.13135e6i 0.462063i 0.972946 + 0.231032i \(0.0742101\pi\)
−0.972946 + 0.231032i \(0.925790\pi\)
\(464\) 0 0
\(465\) −5.04495e6 + 4.79825e6i −1.08199 + 1.02908i
\(466\) 0 0
\(467\) −1.21641e6 −0.258099 −0.129050 0.991638i \(-0.541193\pi\)
−0.129050 + 0.991638i \(0.541193\pi\)
\(468\) 0 0
\(469\) 136653.i 0.0286872i
\(470\) 0 0
\(471\) −1.04343e7 −2.16726
\(472\) 0 0
\(473\) 3.06389e6i 0.629680i
\(474\) 0 0
\(475\) 306881. + 15392.1i 0.0624073 + 0.00313013i
\(476\) 0 0
\(477\) −1.58738e7 −3.19437
\(478\) 0 0
\(479\) −9.14993e6 −1.82213 −0.911065 0.412264i \(-0.864738\pi\)
−0.911065 + 0.412264i \(0.864738\pi\)
\(480\) 0 0
\(481\) 522355. 0.102945
\(482\) 0 0
\(483\) −8.74929e6 −1.70650
\(484\) 0 0
\(485\) −3.66270e6 3.85101e6i −0.707045 0.743396i
\(486\) 0 0
\(487\) 5.29419e6i 1.01153i −0.862672 0.505763i \(-0.831211\pi\)
0.862672 0.505763i \(-0.168789\pi\)
\(488\) 0 0
\(489\) −4.02760e6 −0.761684
\(490\) 0 0
\(491\) 5.58951e6i 1.04633i 0.852231 + 0.523166i \(0.175249\pi\)
−0.852231 + 0.523166i \(0.824751\pi\)
\(492\) 0 0
\(493\) −1.52362e6 −0.282332
\(494\) 0 0
\(495\) 9.89562e6 + 1.04044e7i 1.81522 + 1.90855i
\(496\) 0 0
\(497\) 1.58808e6i 0.288390i
\(498\) 0 0
\(499\) 3.40479e6i 0.612123i 0.952012 + 0.306062i \(0.0990114\pi\)
−0.952012 + 0.306062i \(0.900989\pi\)
\(500\) 0 0
\(501\) 1.27179e7i 2.26372i
\(502\) 0 0
\(503\) 1.05370e7i 1.85693i −0.371421 0.928465i \(-0.621129\pi\)
0.371421 0.928465i \(-0.378871\pi\)
\(504\) 0 0
\(505\) −1.89871e6 1.99633e6i −0.331307 0.348340i
\(506\) 0 0
\(507\) 1.05534e7 1.82336
\(508\) 0 0
\(509\) 7.81588e6i 1.33716i 0.743640 + 0.668580i \(0.233098\pi\)
−0.743640 + 0.668580i \(0.766902\pi\)
\(510\) 0 0
\(511\) −4.77158e6 −0.808369
\(512\) 0 0
\(513\) 993134.i 0.166615i
\(514\) 0 0
\(515\) −5.51632e6 5.79993e6i −0.916498 0.963618i
\(516\) 0 0
\(517\) 6.74632e6 1.11005
\(518\) 0 0
\(519\) 2.05698e7 3.35206
\(520\) 0 0
\(521\) −977378. −0.157750 −0.0788748 0.996885i \(-0.525133\pi\)
−0.0788748 + 0.996885i \(0.525133\pi\)
\(522\) 0 0
\(523\) 6.82809e6 1.09155 0.545777 0.837931i \(-0.316235\pi\)
0.545777 + 0.837931i \(0.316235\pi\)
\(524\) 0 0
\(525\) 1.15561e7 + 579615.i 1.82984 + 0.0917786i
\(526\) 0 0
\(527\) 6.40860e6i 1.00516i
\(528\) 0 0
\(529\) 852544. 0.132458
\(530\) 0 0
\(531\) 1.94928e7i 3.00011i
\(532\) 0 0
\(533\) −348981. −0.0532088
\(534\) 0 0
\(535\) −6.83634e6 + 6.50204e6i −1.03262 + 0.982122i
\(536\) 0 0
\(537\) 2.35158e7i 3.51904i
\(538\) 0 0
\(539\) 172318.i 0.0255481i
\(540\) 0 0
\(541\) 4.14464e6i 0.608827i −0.952540 0.304414i \(-0.901540\pi\)
0.952540 0.304414i \(-0.0984605\pi\)
\(542\) 0 0
\(543\) 1.19958e7i 1.74594i
\(544\) 0 0
\(545\) 6.54468e6 + 6.88117e6i 0.943838 + 0.992364i
\(546\) 0 0
\(547\) −2.25850e6 −0.322739 −0.161369 0.986894i \(-0.551591\pi\)
−0.161369 + 0.986894i \(0.551591\pi\)
\(548\) 0 0
\(549\) 1.14003e7i 1.61430i
\(550\) 0 0
\(551\) 100728. 0.0141342
\(552\) 0 0
\(553\) 6.50548e6i 0.904620i
\(554\) 0 0
\(555\) 7.78296e6 7.40238e6i 1.07254 1.02009i
\(556\) 0 0
\(557\) −1.08783e7 −1.48567 −0.742835 0.669474i \(-0.766519\pi\)
−0.742835 + 0.669474i \(0.766519\pi\)
\(558\) 0 0
\(559\) −555310. −0.0751633
\(560\) 0 0
\(561\) 1.86377e7 2.50026
\(562\) 0 0
\(563\) −3.08080e6 −0.409630 −0.204815 0.978801i \(-0.565659\pi\)
−0.204815 + 0.978801i \(0.565659\pi\)
\(564\) 0 0
\(565\) −6.78835e6 7.13737e6i −0.894630 0.940626i
\(566\) 0 0
\(567\) 1.89563e7i 2.47626i
\(568\) 0 0
\(569\) −828106. −0.107227 −0.0536136 0.998562i \(-0.517074\pi\)
−0.0536136 + 0.998562i \(0.517074\pi\)
\(570\) 0 0
\(571\) 1.09946e7i 1.41120i 0.708609 + 0.705601i \(0.249323\pi\)
−0.708609 + 0.705601i \(0.750677\pi\)
\(572\) 0 0
\(573\) 1.72917e7 2.20014
\(574\) 0 0
\(575\) 7.37512e6 + 369910.i 0.930250 + 0.0466581i
\(576\) 0 0
\(577\) 1.05176e7i 1.31516i 0.753384 + 0.657580i \(0.228420\pi\)
−0.753384 + 0.657580i \(0.771580\pi\)
\(578\) 0 0
\(579\) 4.33692e6i 0.537632i
\(580\) 0 0
\(581\) 4.95444e6i 0.608912i
\(582\) 0 0
\(583\) 1.16164e7i 1.41547i
\(584\) 0 0
\(585\) −1.88573e6 + 1.79352e6i −0.227819 + 0.216679i
\(586\) 0 0
\(587\) −1.07272e7 −1.28497 −0.642484 0.766299i \(-0.722096\pi\)
−0.642484 + 0.766299i \(0.722096\pi\)
\(588\) 0 0
\(589\) 423680.i 0.0503210i
\(590\) 0 0
\(591\) −3.47368e6 −0.409093
\(592\) 0 0
\(593\) 8.76839e6i 1.02396i 0.858997 + 0.511980i \(0.171088\pi\)
−0.858997 + 0.511980i \(0.828912\pi\)
\(594\) 0 0
\(595\) 7.71725e6 7.33988e6i 0.893655 0.849956i
\(596\) 0 0
\(597\) 2.87378e7 3.30002
\(598\) 0 0
\(599\) −6.28188e6 −0.715356 −0.357678 0.933845i \(-0.616432\pi\)
−0.357678 + 0.933845i \(0.616432\pi\)
\(600\) 0 0
\(601\) −1.18519e7 −1.33844 −0.669222 0.743063i \(-0.733372\pi\)
−0.669222 + 0.743063i \(0.733372\pi\)
\(602\) 0 0
\(603\) 632008. 0.0707831
\(604\) 0 0
\(605\) 1.09032e6 1.03700e6i 0.121106 0.115184i
\(606\) 0 0
\(607\) 7.80939e6i 0.860291i −0.902760 0.430145i \(-0.858462\pi\)
0.902760 0.430145i \(-0.141538\pi\)
\(608\) 0 0
\(609\) 3.79310e6 0.414430
\(610\) 0 0
\(611\) 1.22273e6i 0.132503i
\(612\) 0 0
\(613\) −9.77560e6 −1.05073 −0.525366 0.850876i \(-0.676072\pi\)
−0.525366 + 0.850876i \(0.676072\pi\)
\(614\) 0 0
\(615\) −5.19973e6 + 4.94547e6i −0.554361 + 0.527253i
\(616\) 0 0
\(617\) 1.46302e7i 1.54717i −0.633691 0.773586i \(-0.718461\pi\)
0.633691 0.773586i \(-0.281539\pi\)
\(618\) 0 0
\(619\) 7.48407e6i 0.785076i −0.919736 0.392538i \(-0.871597\pi\)
0.919736 0.392538i \(-0.128403\pi\)
\(620\) 0 0
\(621\) 2.38675e7i 2.48358i
\(622\) 0 0
\(623\) 1.69637e7i 1.75105i
\(624\) 0 0
\(625\) −9.71661e6 977161.i −0.994981 0.100061i
\(626\) 0 0
\(627\) −1.23216e6 −0.125169
\(628\) 0 0
\(629\) 9.88670e6i 0.996380i
\(630\) 0 0
\(631\) −9.89503e6 −0.989335 −0.494668 0.869082i \(-0.664710\pi\)
−0.494668 + 0.869082i \(0.664710\pi\)
\(632\) 0 0
\(633\) 2.47055e7i 2.45067i
\(634\) 0 0
\(635\) −1.10458e6 1.16137e6i −0.108709 0.114298i
\(636\) 0 0
\(637\) 31231.6 0.00304962
\(638\) 0 0
\(639\) −7.34470e6 −0.711577
\(640\) 0 0
\(641\) −1.34899e7 −1.29677 −0.648386 0.761312i \(-0.724556\pi\)
−0.648386 + 0.761312i \(0.724556\pi\)
\(642\) 0 0
\(643\) 1.56805e7 1.49566 0.747828 0.663893i \(-0.231097\pi\)
0.747828 + 0.663893i \(0.231097\pi\)
\(644\) 0 0
\(645\) −8.27398e6 + 7.86939e6i −0.783096 + 0.744803i
\(646\) 0 0
\(647\) 1.70592e7i 1.60213i 0.598579 + 0.801064i \(0.295732\pi\)
−0.598579 + 0.801064i \(0.704268\pi\)
\(648\) 0 0
\(649\) −1.42648e7 −1.32939
\(650\) 0 0
\(651\) 1.59544e7i 1.47546i
\(652\) 0 0
\(653\) 8.71978e6 0.800245 0.400122 0.916462i \(-0.368968\pi\)
0.400122 + 0.916462i \(0.368968\pi\)
\(654\) 0 0
\(655\) −1.61917e6 1.70241e6i −0.147465 0.155046i
\(656\) 0 0
\(657\) 2.20681e7i 1.99458i
\(658\) 0 0
\(659\) 9.58270e6i 0.859556i 0.902935 + 0.429778i \(0.141408\pi\)
−0.902935 + 0.429778i \(0.858592\pi\)
\(660\) 0 0
\(661\) 1.66657e7i 1.48361i 0.670616 + 0.741804i \(0.266030\pi\)
−0.670616 + 0.741804i \(0.733970\pi\)
\(662\) 0 0
\(663\) 3.37796e6i 0.298450i
\(664\) 0 0
\(665\) −510196. + 485248.i −0.0447387 + 0.0425510i
\(666\) 0 0
\(667\) 2.42076e6 0.210686
\(668\) 0 0
\(669\) 3.91427e6i 0.338131i
\(670\) 0 0
\(671\) 8.34271e6 0.715321
\(672\) 0 0
\(673\) 4.89257e6i 0.416389i 0.978087 + 0.208194i \(0.0667587\pi\)
−0.978087 + 0.208194i \(0.933241\pi\)
\(674\) 0 0
\(675\) −1.58115e6 + 3.15244e7i −0.133572 + 2.66310i
\(676\) 0 0
\(677\) 1.32965e7 1.11498 0.557490 0.830184i \(-0.311765\pi\)
0.557490 + 0.830184i \(0.311765\pi\)
\(678\) 0 0
\(679\) 1.21786e7 1.01374
\(680\) 0 0
\(681\) 1.55788e7 1.28726
\(682\) 0 0
\(683\) −4.92824e6 −0.404241 −0.202120 0.979361i \(-0.564783\pi\)
−0.202120 + 0.979361i \(0.564783\pi\)
\(684\) 0 0
\(685\) −1.89015e6 1.98733e6i −0.153911 0.161824i
\(686\) 0 0
\(687\) 3.01645e7i 2.43839i
\(688\) 0 0
\(689\) −2.10540e6 −0.168961
\(690\) 0 0
\(691\) 1.24185e7i 0.989403i 0.869063 + 0.494702i \(0.164723\pi\)
−0.869063 + 0.494702i \(0.835277\pi\)
\(692\) 0 0
\(693\) −3.29034e7 −2.60260
\(694\) 0 0
\(695\) −3.29402e6 3.46338e6i −0.258681 0.271981i
\(696\) 0 0
\(697\) 6.60522e6i 0.514998i
\(698\) 0 0
\(699\) 1.34735e7i 1.04301i
\(700\) 0 0
\(701\) 2.45892e7i 1.88994i 0.327153 + 0.944971i \(0.393911\pi\)
−0.327153 + 0.944971i \(0.606089\pi\)
\(702\) 0 0
\(703\) 653621.i 0.0498813i
\(704\) 0 0
\(705\) 1.73275e7 + 1.82183e7i 1.31299 + 1.38050i
\(706\) 0 0
\(707\) 6.31330e6 0.475015
\(708\) 0 0
\(709\) 2.05841e7i 1.53786i −0.639334 0.768930i \(-0.720790\pi\)
0.639334 0.768930i \(-0.279210\pi\)
\(710\) 0 0
\(711\) 3.00872e7 2.23207
\(712\) 0 0
\(713\) 1.01821e7i 0.750091i
\(714\) 0 0
\(715\) 1.31249e6 + 1.37997e6i 0.0960135 + 0.100950i
\(716\) 0 0
\(717\) −4.30882e7 −3.13011
\(718\) 0 0
\(719\) −5.88556e6 −0.424586 −0.212293 0.977206i \(-0.568093\pi\)
−0.212293 + 0.977206i \(0.568093\pi\)
\(720\) 0 0
\(721\) 1.83420e7 1.31404
\(722\) 0 0
\(723\) 7.17315e6 0.510345
\(724\) 0 0
\(725\) −3.19735e6 160368.i −0.225915 0.0113311i
\(726\) 0 0
\(727\) 7.65798e6i 0.537376i −0.963227 0.268688i \(-0.913410\pi\)
0.963227 0.268688i \(-0.0865900\pi\)
\(728\) 0 0
\(729\) 1.67283e7 1.16582
\(730\) 0 0
\(731\) 1.05104e7i 0.727491i
\(732\) 0 0
\(733\) 1.64697e7 1.13220 0.566102 0.824335i \(-0.308451\pi\)
0.566102 + 0.824335i \(0.308451\pi\)
\(734\) 0 0
\(735\) 465343. 442588.i 0.0317727 0.0302191i
\(736\) 0 0
\(737\) 462503.i 0.0313650i
\(738\) 0 0
\(739\) 1.03197e7i 0.695116i −0.937658 0.347558i \(-0.887011\pi\)
0.937658 0.347558i \(-0.112989\pi\)
\(740\) 0 0
\(741\) 223321.i 0.0149412i
\(742\) 0 0
\(743\) 1.43842e7i 0.955903i −0.878386 0.477952i \(-0.841379\pi\)
0.878386 0.477952i \(-0.158621\pi\)
\(744\) 0 0
\(745\) −1.13428e6 1.19259e6i −0.0748735 0.0787230i
\(746\) 0 0
\(747\) −2.29138e7 −1.50243
\(748\) 0 0
\(749\) 2.16196e7i 1.40813i
\(750\) 0 0
\(751\) 3.39860e6 0.219887 0.109944 0.993938i \(-0.464933\pi\)
0.109944 + 0.993938i \(0.464933\pi\)
\(752\) 0 0
\(753\) 4.94060e7i 3.17536i
\(754\) 0 0
\(755\) 5.34595e6 5.08454e6i 0.341317 0.324627i
\(756\) 0 0
\(757\) −8.89557e6 −0.564201 −0.282101 0.959385i \(-0.591031\pi\)
−0.282101 + 0.959385i \(0.591031\pi\)
\(758\) 0 0
\(759\) −2.96119e7 −1.86579
\(760\) 0 0
\(761\) −1.68168e7 −1.05264 −0.526321 0.850286i \(-0.676429\pi\)
−0.526321 + 0.850286i \(0.676429\pi\)
\(762\) 0 0
\(763\) −2.17614e7 −1.35324
\(764\) 0 0
\(765\) 3.39462e7 + 3.56915e7i 2.09719 + 2.20501i
\(766\) 0 0
\(767\) 2.58540e6i 0.158686i
\(768\) 0 0
\(769\) 1.14740e7 0.699681 0.349840 0.936809i \(-0.386236\pi\)
0.349840 + 0.936809i \(0.386236\pi\)
\(770\) 0 0
\(771\) 1.28894e7i 0.780900i
\(772\) 0 0
\(773\) 5.97879e6 0.359886 0.179943 0.983677i \(-0.442409\pi\)
0.179943 + 0.983677i \(0.442409\pi\)
\(774\) 0 0
\(775\) −674535. + 1.34486e7i −0.0403413 + 0.804309i
\(776\) 0 0
\(777\) 2.46133e7i 1.46257i
\(778\) 0 0
\(779\) 436679.i 0.0257821i
\(780\) 0 0
\(781\) 5.37484e6i 0.315310i
\(782\) 0 0
\(783\) 1.03474e7i 0.603149i
\(784\) 0 0
\(785\) −1.46227e7 + 1.39077e7i −0.846942 + 0.805527i
\(786\) 0 0
\(787\) −1.28293e6 −0.0738356 −0.0369178 0.999318i \(-0.511754\pi\)
−0.0369178 + 0.999318i \(0.511754\pi\)
\(788\) 0 0
\(789\) 1.78083e7i 1.01843i
\(790\) 0 0
\(791\) 2.25716e7 1.28269
\(792\) 0 0
\(793\) 1.51206e6i 0.0853860i
\(794\) 0 0
\(795\) −3.13700e7 + 2.98360e7i −1.76034 + 1.67426i
\(796\) 0 0
\(797\) −1.38090e7 −0.770047 −0.385024 0.922907i \(-0.625807\pi\)
−0.385024 + 0.922907i \(0.625807\pi\)
\(798\) 0 0
\(799\) 2.31428e7 1.28247
\(800\) 0 0
\(801\) −7.84552e7 −4.32057
\(802\) 0 0
\(803\) −1.61494e7 −0.883826
\(804\) 0 0
\(805\) −1.22613e7 + 1.16617e7i −0.666880 + 0.634270i
\(806\) 0 0
\(807\) 2.73417e7i 1.47789i
\(808\) 0 0
\(809\) 2.36073e7 1.26816 0.634082 0.773266i \(-0.281378\pi\)
0.634082 + 0.773266i \(0.281378\pi\)
\(810\) 0 0
\(811\) 1.59317e6i 0.0850569i 0.999095 + 0.0425284i \(0.0135413\pi\)
−0.999095 + 0.0425284i \(0.986459\pi\)
\(812\) 0 0
\(813\) −2.48370e7 −1.31787
\(814\) 0 0
\(815\) −5.64431e6 + 5.36831e6i −0.297658 + 0.283102i
\(816\) 0 0
\(817\) 694857.i 0.0364200i
\(818\) 0 0
\(819\) 5.96353e6i 0.310666i
\(820\) 0 0
\(821\) 9.05321e6i 0.468754i 0.972146 + 0.234377i \(0.0753050\pi\)
−0.972146 + 0.234377i \(0.924695\pi\)
\(822\) 0 0
\(823\) 1.59075e7i 0.818657i 0.912387 + 0.409329i \(0.134237\pi\)
−0.912387 + 0.409329i \(0.865763\pi\)
\(824\) 0 0
\(825\) 3.91117e7 + 1.96170e6i 2.00065 + 0.100346i
\(826\) 0 0
\(827\) 2.64781e7 1.34624 0.673122 0.739532i \(-0.264953\pi\)
0.673122 + 0.739532i \(0.264953\pi\)
\(828\) 0 0
\(829\) 1.54704e7i 0.781835i 0.920426 + 0.390918i \(0.127842\pi\)
−0.920426 + 0.390918i \(0.872158\pi\)
\(830\) 0 0
\(831\) −3.83518e7 −1.92657
\(832\) 0 0
\(833\) 591125.i 0.0295166i
\(834\) 0 0
\(835\) −1.69514e7 1.78230e7i −0.841377 0.884635i
\(836\) 0 0
\(837\) 4.35227e7 2.14735
\(838\) 0 0
\(839\) −3.82791e7 −1.87740 −0.938701 0.344733i \(-0.887970\pi\)
−0.938701 + 0.344733i \(0.887970\pi\)
\(840\) 0 0
\(841\) 1.94617e7 0.948834
\(842\) 0 0
\(843\) 1.88306e7 0.912630
\(844\) 0 0
\(845\) 1.47896e7 1.40664e7i 0.712551 0.677708i
\(846\) 0 0
\(847\) 3.44809e6i 0.165147i
\(848\) 0 0
\(849\) −5.63567e7 −2.68334
\(850\) 0 0
\(851\) 1.57082e7i 0.743536i
\(852\) 0 0
\(853\) 2.94396e7 1.38535 0.692676 0.721249i \(-0.256432\pi\)
0.692676 + 0.721249i \(0.256432\pi\)
\(854\) 0 0
\(855\) −2.24422e6 2.35961e6i −0.104991 0.110389i
\(856\) 0 0
\(857\) 1.36598e7i 0.635320i 0.948205 + 0.317660i \(0.102897\pi\)
−0.948205 + 0.317660i \(0.897103\pi\)
\(858\) 0 0
\(859\) 1.49700e7i 0.692212i −0.938195 0.346106i \(-0.887504\pi\)
0.938195 0.346106i \(-0.112496\pi\)
\(860\) 0 0
\(861\) 1.64439e7i 0.755957i
\(862\) 0 0
\(863\) 2.36162e6i 0.107940i −0.998543 0.0539701i \(-0.982812\pi\)
0.998543 0.0539701i \(-0.0171876\pi\)
\(864\) 0 0
\(865\) 2.88267e7 2.74171e7i 1.30995 1.24589i
\(866\) 0 0
\(867\) 2.28955e7 1.03443
\(868\) 0 0
\(869\) 2.20178e7i 0.989063i
\(870\) 0 0
\(871\) 83825.7 0.00374397
\(872\) 0 0
\(873\) 5.63250e7i 2.50130i
\(874\) 0 0
\(875\) 1.69674e7 1.45906e7i 0.749195 0.644250i
\(876\) 0 0
\(877\) −1.52212e7 −0.668268 −0.334134 0.942526i \(-0.608444\pi\)
−0.334134 + 0.942526i \(0.608444\pi\)
\(878\) 0 0
\(879\) 5.21895e7 2.27830
\(880\) 0 0
\(881\) 2.72576e7 1.18317 0.591586 0.806242i \(-0.298502\pi\)
0.591586 + 0.806242i \(0.298502\pi\)
\(882\) 0 0
\(883\) −1.90157e7 −0.820748 −0.410374 0.911917i \(-0.634602\pi\)
−0.410374 + 0.911917i \(0.634602\pi\)
\(884\) 0 0
\(885\) −3.66381e7 3.85218e7i −1.57244 1.65329i
\(886\) 0 0
\(887\) 2.99966e7i 1.28016i −0.768310 0.640078i \(-0.778902\pi\)
0.768310 0.640078i \(-0.221098\pi\)
\(888\) 0 0
\(889\) 3.67279e6 0.155862
\(890\) 0 0
\(891\) 6.41576e7i 2.70741i
\(892\) 0 0
\(893\) −1.52999e6 −0.0642038
\(894\) 0 0
\(895\) −3.13437e7 3.29552e7i −1.30795 1.37520i
\(896\) 0 0
\(897\) 5.36698e6i 0.222714i
\(898\) 0 0
\(899\) 4.41427e6i 0.182163i
\(900\) 0 0
\(901\) 3.98493e7i 1.63534i
\(902\) 0 0
\(903\) 2.61661e7i 1.06787i
\(904\) 0 0
\(905\) −1.59889e7 1.68110e7i −0.648929 0.682293i
\(906\) 0 0
\(907\) −1.81689e7 −0.733348 −0.366674 0.930349i \(-0.619504\pi\)
−0.366674 + 0.930349i \(0.619504\pi\)
\(908\) 0 0
\(909\) 2.91984e7i 1.17206i
\(910\) 0 0
\(911\) 2.14560e6 0.0856550 0.0428275 0.999082i \(-0.486363\pi\)
0.0428275 + 0.999082i \(0.486363\pi\)
\(912\) 0 0
\(913\) 1.67683e7i 0.665751i
\(914\) 0 0
\(915\) 2.14277e7 + 2.25294e7i 0.846101 + 0.889603i
\(916\) 0 0
\(917\) 5.38380e6 0.211430
\(918\) 0 0
\(919\) 4.88257e6 0.190704 0.0953519 0.995444i \(-0.469602\pi\)
0.0953519 + 0.995444i \(0.469602\pi\)
\(920\) 0 0
\(921\) 9.63709e6 0.374366
\(922\) 0 0
\(923\) −974156. −0.0376378
\(924\) 0 0
\(925\) 1.04062e6 2.07475e7i 0.0399888 0.797281i
\(926\) 0 0
\(927\) 8.48300e7i 3.24228i
\(928\) 0 0
\(929\) 3.34515e7 1.27168 0.635838 0.771823i \(-0.280655\pi\)
0.635838 + 0.771823i \(0.280655\pi\)
\(930\) 0 0
\(931\) 39080.0i 0.00147768i
\(932\) 0 0
\(933\) 3.64621e6 0.137132
\(934\) 0 0
\(935\) 2.61190e7 2.48418e7i 0.977074 0.929296i
\(936\) 0 0
\(937\) 3.63111e7i 1.35111i −0.737310 0.675554i \(-0.763904\pi\)
0.737310 0.675554i \(-0.236096\pi\)
\(938\) 0 0
\(939\) 4.66390e7i 1.72618i
\(940\) 0 0
\(941\) 3.70043e7i 1.36232i −0.732136 0.681158i \(-0.761477\pi\)
0.732136 0.681158i \(-0.238523\pi\)
\(942\) 0 0
\(943\) 1.04945e7i 0.384311i
\(944\) 0 0
\(945\) −4.98473e7 5.24101e7i −1.81577 1.90913i
\(946\) 0 0
\(947\) 1.25663e7 0.455335 0.227668 0.973739i \(-0.426890\pi\)
0.227668 + 0.973739i \(0.426890\pi\)
\(948\) 0 0
\(949\) 2.92697e6i 0.105500i
\(950\) 0 0
\(951\) 4.69858e7 1.68467
\(952\) 0 0
\(953\) 3.08701e7i 1.10105i −0.834819 0.550524i \(-0.814428\pi\)
0.834819 0.550524i \(-0.185572\pi\)
\(954\) 0 0
\(955\) 2.42327e7 2.30477e7i 0.859791 0.817747i
\(956\) 0 0
\(957\) 1.28377e7 0.453115
\(958\) 0 0
\(959\) 6.28485e6 0.220673
\(960\) 0 0
\(961\) −1.00620e7 −0.351460
\(962\) 0 0
\(963\) 9.99885e7 3.47443
\(964\) 0 0
\(965\) −5.78059e6 6.07779e6i −0.199827 0.210101i
\(966\) 0 0
\(967\) 2.55508e7i 0.878696i −0.898317 0.439348i \(-0.855209\pi\)
0.898317 0.439348i \(-0.144791\pi\)
\(968\) 0 0
\(969\) −4.22683e6 −0.144612
\(970\) 0 0
\(971\) 2.78585e7i 0.948221i −0.880465 0.474110i \(-0.842770\pi\)
0.880465 0.474110i \(-0.157230\pi\)
\(972\) 0 0
\(973\) 1.09528e7 0.370888
\(974\) 0 0
\(975\) −355546. + 7.08874e6i −0.0119780 + 0.238813i
\(976\) 0 0
\(977\) 1.73322e7i 0.580920i 0.956887 + 0.290460i \(0.0938084\pi\)
−0.956887 + 0.290460i \(0.906192\pi\)
\(978\) 0 0
\(979\) 5.74135e7i 1.91451i
\(980\) 0 0
\(981\) 1.00644e8i 3.33900i
\(982\) 0 0
\(983\) 1.02000e7i 0.336681i −0.985729 0.168340i \(-0.946159\pi\)
0.985729 0.168340i \(-0.0538408\pi\)
\(984\) 0 0
\(985\) −4.86805e6 + 4.63000e6i −0.159869 + 0.152051i
\(986\) 0 0
\(987\) −5.76146e7 −1.88252
\(988\) 0 0
\(989\) 1.66992e7i 0.542881i
\(990\) 0 0
\(991\) −1.05292e7 −0.340573 −0.170286 0.985395i \(-0.554469\pi\)
−0.170286 + 0.985395i \(0.554469\pi\)
\(992\) 0 0
\(993\) 9.39687e6i 0.302420i
\(994\) 0 0
\(995\) 4.02733e7 3.83040e7i 1.28961 1.22655i
\(996\) 0 0
\(997\) 4.69241e7 1.49506 0.747529 0.664229i \(-0.231240\pi\)
0.747529 + 0.664229i \(0.231240\pi\)
\(998\) 0 0
\(999\) −6.71435e7 −2.12858
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 160.6.f.a.49.2 28
4.3 odd 2 40.6.f.a.29.14 yes 28
5.2 odd 4 800.6.d.e.401.6 28
5.3 odd 4 800.6.d.e.401.23 28
5.4 even 2 inner 160.6.f.a.49.27 28
8.3 odd 2 40.6.f.a.29.16 yes 28
8.5 even 2 inner 160.6.f.a.49.28 28
20.3 even 4 200.6.d.e.101.28 28
20.7 even 4 200.6.d.e.101.1 28
20.19 odd 2 40.6.f.a.29.15 yes 28
40.3 even 4 200.6.d.e.101.27 28
40.13 odd 4 800.6.d.e.401.24 28
40.19 odd 2 40.6.f.a.29.13 28
40.27 even 4 200.6.d.e.101.2 28
40.29 even 2 inner 160.6.f.a.49.1 28
40.37 odd 4 800.6.d.e.401.5 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.f.a.29.13 28 40.19 odd 2
40.6.f.a.29.14 yes 28 4.3 odd 2
40.6.f.a.29.15 yes 28 20.19 odd 2
40.6.f.a.29.16 yes 28 8.3 odd 2
160.6.f.a.49.1 28 40.29 even 2 inner
160.6.f.a.49.2 28 1.1 even 1 trivial
160.6.f.a.49.27 28 5.4 even 2 inner
160.6.f.a.49.28 28 8.5 even 2 inner
200.6.d.e.101.1 28 20.7 even 4
200.6.d.e.101.2 28 40.27 even 4
200.6.d.e.101.27 28 40.3 even 4
200.6.d.e.101.28 28 20.3 even 4
800.6.d.e.401.5 28 40.37 odd 4
800.6.d.e.401.6 28 5.2 odd 4
800.6.d.e.401.23 28 5.3 odd 4
800.6.d.e.401.24 28 40.13 odd 4