Properties

Label 160.6.o.a.47.17
Level $160$
Weight $6$
Character 160.47
Analytic conductor $25.661$
Analytic rank $0$
Dimension $56$
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(47,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.47");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.o (of order \(4\), 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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 47.17
Character \(\chi\) \(=\) 160.47
Dual form 160.6.o.a.143.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.85746 - 5.85746i) q^{3} +(13.4330 - 54.2637i) q^{5} +(100.487 - 100.487i) q^{7} +174.380i q^{9} +O(q^{10})\) \(q+(5.85746 - 5.85746i) q^{3} +(13.4330 - 54.2637i) q^{5} +(100.487 - 100.487i) q^{7} +174.380i q^{9} +546.323 q^{11} +(619.525 + 619.525i) q^{13} +(-239.164 - 396.531i) q^{15} +(-1509.84 - 1509.84i) q^{17} -1145.03i q^{19} -1177.20i q^{21} +(2136.97 + 2136.97i) q^{23} +(-2764.11 - 1457.85i) q^{25} +(2444.79 + 2444.79i) q^{27} -1309.79 q^{29} -4695.04i q^{31} +(3200.07 - 3200.07i) q^{33} +(-4102.97 - 6802.67i) q^{35} +(7598.37 - 7598.37i) q^{37} +7257.68 q^{39} -8811.44 q^{41} +(745.885 - 745.885i) q^{43} +(9462.53 + 2342.46i) q^{45} +(26.5684 - 26.5684i) q^{47} -3388.42i q^{49} -17687.6 q^{51} +(-189.847 - 189.847i) q^{53} +(7338.77 - 29645.5i) q^{55} +(-6706.96 - 6706.96i) q^{57} -39197.3i q^{59} +1750.39i q^{61} +(17523.0 + 17523.0i) q^{63} +(41939.8 - 25295.7i) q^{65} +(-20889.0 - 20889.0i) q^{67} +25034.4 q^{69} +15059.6i q^{71} +(-2616.06 + 2616.06i) q^{73} +(-24730.0 + 7651.34i) q^{75} +(54898.6 - 54898.6i) q^{77} +17424.9 q^{79} -13734.0 q^{81} +(-31132.3 + 31132.3i) q^{83} +(-102211. + 61647.7i) q^{85} +(-7672.05 + 7672.05i) q^{87} +33401.4i q^{89} +124509. q^{91} +(-27501.0 - 27501.0i) q^{93} +(-62133.6 - 15381.2i) q^{95} +(3265.58 + 3265.58i) q^{97} +95268.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 4 q^{3} + 8 q^{11} - 408 q^{17} - 3120 q^{25} - 968 q^{27} - 976 q^{33} + 4780 q^{35} - 8 q^{41} - 1308 q^{43} - 20872 q^{51} + 968 q^{57} + 17680 q^{65} - 89252 q^{67} - 25184 q^{73} + 127740 q^{75} - 67792 q^{81} + 126444 q^{83} - 329432 q^{91} + 212576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 5.85746 5.85746i 0.375756 0.375756i −0.493812 0.869568i \(-0.664397\pi\)
0.869568 + 0.493812i \(0.164397\pi\)
\(4\) 0 0
\(5\) 13.4330 54.2637i 0.240297 0.970699i
\(6\) 0 0
\(7\) 100.487 100.487i 0.775115 0.775115i −0.203880 0.978996i \(-0.565355\pi\)
0.978996 + 0.203880i \(0.0653554\pi\)
\(8\) 0 0
\(9\) 174.380i 0.717615i
\(10\) 0 0
\(11\) 546.323 1.36134 0.680672 0.732588i \(-0.261688\pi\)
0.680672 + 0.732588i \(0.261688\pi\)
\(12\) 0 0
\(13\) 619.525 + 619.525i 1.01672 + 1.01672i 0.999858 + 0.0168600i \(0.00536695\pi\)
0.0168600 + 0.999858i \(0.494633\pi\)
\(14\) 0 0
\(15\) −239.164 396.531i −0.274453 0.455039i
\(16\) 0 0
\(17\) −1509.84 1509.84i −1.26709 1.26709i −0.947585 0.319505i \(-0.896483\pi\)
−0.319505 0.947585i \(-0.603517\pi\)
\(18\) 0 0
\(19\) 1145.03i 0.727667i −0.931464 0.363833i \(-0.881468\pi\)
0.931464 0.363833i \(-0.118532\pi\)
\(20\) 0 0
\(21\) 1177.20i 0.582509i
\(22\) 0 0
\(23\) 2136.97 + 2136.97i 0.842321 + 0.842321i 0.989160 0.146839i \(-0.0469099\pi\)
−0.146839 + 0.989160i \(0.546910\pi\)
\(24\) 0 0
\(25\) −2764.11 1457.85i −0.884514 0.466513i
\(26\) 0 0
\(27\) 2444.79 + 2444.79i 0.645404 + 0.645404i
\(28\) 0 0
\(29\) −1309.79 −0.289206 −0.144603 0.989490i \(-0.546190\pi\)
−0.144603 + 0.989490i \(0.546190\pi\)
\(30\) 0 0
\(31\) 4695.04i 0.877476i −0.898615 0.438738i \(-0.855426\pi\)
0.898615 0.438738i \(-0.144574\pi\)
\(32\) 0 0
\(33\) 3200.07 3200.07i 0.511533 0.511533i
\(34\) 0 0
\(35\) −4102.97 6802.67i −0.566146 0.938662i
\(36\) 0 0
\(37\) 7598.37 7598.37i 0.912464 0.912464i −0.0840014 0.996466i \(-0.526770\pi\)
0.996466 + 0.0840014i \(0.0267700\pi\)
\(38\) 0 0
\(39\) 7257.68 0.764076
\(40\) 0 0
\(41\) −8811.44 −0.818629 −0.409314 0.912393i \(-0.634232\pi\)
−0.409314 + 0.912393i \(0.634232\pi\)
\(42\) 0 0
\(43\) 745.885 745.885i 0.0615178 0.0615178i −0.675679 0.737196i \(-0.736149\pi\)
0.737196 + 0.675679i \(0.236149\pi\)
\(44\) 0 0
\(45\) 9462.53 + 2342.46i 0.696588 + 0.172441i
\(46\) 0 0
\(47\) 26.5684 26.5684i 0.00175437 0.00175437i −0.706229 0.707983i \(-0.749605\pi\)
0.707983 + 0.706229i \(0.249605\pi\)
\(48\) 0 0
\(49\) 3388.42i 0.201608i
\(50\) 0 0
\(51\) −17687.6 −0.952233
\(52\) 0 0
\(53\) −189.847 189.847i −0.00928357 0.00928357i 0.702450 0.711733i \(-0.252090\pi\)
−0.711733 + 0.702450i \(0.752090\pi\)
\(54\) 0 0
\(55\) 7338.77 29645.5i 0.327127 1.32146i
\(56\) 0 0
\(57\) −6706.96 6706.96i −0.273425 0.273425i
\(58\) 0 0
\(59\) 39197.3i 1.46597i −0.680242 0.732987i \(-0.738125\pi\)
0.680242 0.732987i \(-0.261875\pi\)
\(60\) 0 0
\(61\) 1750.39i 0.0602296i 0.999546 + 0.0301148i \(0.00958729\pi\)
−0.999546 + 0.0301148i \(0.990413\pi\)
\(62\) 0 0
\(63\) 17523.0 + 17523.0i 0.556234 + 0.556234i
\(64\) 0 0
\(65\) 41939.8 25295.7i 1.23124 0.742613i
\(66\) 0 0
\(67\) −20889.0 20889.0i −0.568501 0.568501i 0.363207 0.931708i \(-0.381682\pi\)
−0.931708 + 0.363207i \(0.881682\pi\)
\(68\) 0 0
\(69\) 25034.4 0.633015
\(70\) 0 0
\(71\) 15059.6i 0.354541i 0.984162 + 0.177271i \(0.0567268\pi\)
−0.984162 + 0.177271i \(0.943273\pi\)
\(72\) 0 0
\(73\) −2616.06 + 2616.06i −0.0574566 + 0.0574566i −0.735251 0.677795i \(-0.762936\pi\)
0.677795 + 0.735251i \(0.262936\pi\)
\(74\) 0 0
\(75\) −24730.0 + 7651.34i −0.507657 + 0.157067i
\(76\) 0 0
\(77\) 54898.6 54898.6i 1.05520 1.05520i
\(78\) 0 0
\(79\) 17424.9 0.314125 0.157062 0.987589i \(-0.449798\pi\)
0.157062 + 0.987589i \(0.449798\pi\)
\(80\) 0 0
\(81\) −13734.0 −0.232586
\(82\) 0 0
\(83\) −31132.3 + 31132.3i −0.496039 + 0.496039i −0.910203 0.414163i \(-0.864074\pi\)
0.414163 + 0.910203i \(0.364074\pi\)
\(84\) 0 0
\(85\) −102211. + 61647.7i −1.53444 + 0.925485i
\(86\) 0 0
\(87\) −7672.05 + 7672.05i −0.108671 + 0.108671i
\(88\) 0 0
\(89\) 33401.4i 0.446982i 0.974706 + 0.223491i \(0.0717454\pi\)
−0.974706 + 0.223491i \(0.928255\pi\)
\(90\) 0 0
\(91\) 124509. 1.57615
\(92\) 0 0
\(93\) −27501.0 27501.0i −0.329717 0.329717i
\(94\) 0 0
\(95\) −62133.6 15381.2i −0.706346 0.174856i
\(96\) 0 0
\(97\) 3265.58 + 3265.58i 0.0352396 + 0.0352396i 0.724507 0.689267i \(-0.242067\pi\)
−0.689267 + 0.724507i \(0.742067\pi\)
\(98\) 0 0
\(99\) 95268.1i 0.976921i
\(100\) 0 0
\(101\) 17293.5i 0.168686i 0.996437 + 0.0843431i \(0.0268792\pi\)
−0.996437 + 0.0843431i \(0.973121\pi\)
\(102\) 0 0
\(103\) 41987.6 + 41987.6i 0.389967 + 0.389967i 0.874676 0.484709i \(-0.161074\pi\)
−0.484709 + 0.874676i \(0.661074\pi\)
\(104\) 0 0
\(105\) −63879.3 15813.4i −0.565441 0.139975i
\(106\) 0 0
\(107\) 107361. + 107361.i 0.906538 + 0.906538i 0.995991 0.0894528i \(-0.0285118\pi\)
−0.0894528 + 0.995991i \(0.528512\pi\)
\(108\) 0 0
\(109\) 51260.4 0.413253 0.206626 0.978420i \(-0.433752\pi\)
0.206626 + 0.978420i \(0.433752\pi\)
\(110\) 0 0
\(111\) 89014.2i 0.685728i
\(112\) 0 0
\(113\) −150086. + 150086.i −1.10571 + 1.10571i −0.112006 + 0.993707i \(0.535728\pi\)
−0.993707 + 0.112006i \(0.964272\pi\)
\(114\) 0 0
\(115\) 144666. 87253.8i 1.02005 0.615233i
\(116\) 0 0
\(117\) −108033. + 108033.i −0.729612 + 0.729612i
\(118\) 0 0
\(119\) −303439. −1.96428
\(120\) 0 0
\(121\) 137418. 0.853259
\(122\) 0 0
\(123\) −51612.6 + 51612.6i −0.307605 + 0.307605i
\(124\) 0 0
\(125\) −116239. + 130407.i −0.665390 + 0.746496i
\(126\) 0 0
\(127\) −196504. + 196504.i −1.08109 + 1.08109i −0.0846831 + 0.996408i \(0.526988\pi\)
−0.996408 + 0.0846831i \(0.973012\pi\)
\(128\) 0 0
\(129\) 8737.98i 0.0462314i
\(130\) 0 0
\(131\) −146773. −0.747254 −0.373627 0.927579i \(-0.621886\pi\)
−0.373627 + 0.927579i \(0.621886\pi\)
\(132\) 0 0
\(133\) −115061. 115061.i −0.564026 0.564026i
\(134\) 0 0
\(135\) 165504. 99822.4i 0.781582 0.471405i
\(136\) 0 0
\(137\) 127282. + 127282.i 0.579381 + 0.579381i 0.934733 0.355351i \(-0.115639\pi\)
−0.355351 + 0.934733i \(0.615639\pi\)
\(138\) 0 0
\(139\) 127272.i 0.558722i 0.960186 + 0.279361i \(0.0901226\pi\)
−0.960186 + 0.279361i \(0.909877\pi\)
\(140\) 0 0
\(141\) 311.247i 0.00131843i
\(142\) 0 0
\(143\) 338461. + 338461.i 1.38410 + 1.38410i
\(144\) 0 0
\(145\) −17594.5 + 71074.2i −0.0694954 + 0.280732i
\(146\) 0 0
\(147\) −19847.5 19847.5i −0.0757553 0.0757553i
\(148\) 0 0
\(149\) 206774. 0.763011 0.381506 0.924367i \(-0.375406\pi\)
0.381506 + 0.924367i \(0.375406\pi\)
\(150\) 0 0
\(151\) 251221.i 0.896632i −0.893875 0.448316i \(-0.852024\pi\)
0.893875 0.448316i \(-0.147976\pi\)
\(152\) 0 0
\(153\) 263286. 263286.i 0.909282 0.909282i
\(154\) 0 0
\(155\) −254771. 63068.6i −0.851765 0.210855i
\(156\) 0 0
\(157\) 74099.0 74099.0i 0.239918 0.239918i −0.576898 0.816816i \(-0.695737\pi\)
0.816816 + 0.576898i \(0.195737\pi\)
\(158\) 0 0
\(159\) −2224.05 −0.00697672
\(160\) 0 0
\(161\) 429476. 1.30579
\(162\) 0 0
\(163\) 80376.6 80376.6i 0.236952 0.236952i −0.578635 0.815587i \(-0.696414\pi\)
0.815587 + 0.578635i \(0.196414\pi\)
\(164\) 0 0
\(165\) −130661. 216634.i −0.373625 0.619465i
\(166\) 0 0
\(167\) 67918.4 67918.4i 0.188450 0.188450i −0.606576 0.795026i \(-0.707457\pi\)
0.795026 + 0.606576i \(0.207457\pi\)
\(168\) 0 0
\(169\) 396329.i 1.06743i
\(170\) 0 0
\(171\) 199671. 0.522185
\(172\) 0 0
\(173\) 68660.9 + 68660.9i 0.174419 + 0.174419i 0.788918 0.614499i \(-0.210642\pi\)
−0.614499 + 0.788918i \(0.710642\pi\)
\(174\) 0 0
\(175\) −424254. + 131262.i −1.04720 + 0.324000i
\(176\) 0 0
\(177\) −229597. 229597.i −0.550849 0.550849i
\(178\) 0 0
\(179\) 346919.i 0.809274i 0.914477 + 0.404637i \(0.132602\pi\)
−0.914477 + 0.404637i \(0.867398\pi\)
\(180\) 0 0
\(181\) 79034.8i 0.179317i 0.995973 + 0.0896586i \(0.0285776\pi\)
−0.995973 + 0.0896586i \(0.971422\pi\)
\(182\) 0 0
\(183\) 10252.8 + 10252.8i 0.0226316 + 0.0226316i
\(184\) 0 0
\(185\) −310247. 514385.i −0.666466 1.10499i
\(186\) 0 0
\(187\) −824858. 824858.i −1.72495 1.72495i
\(188\) 0 0
\(189\) 491341. 1.00053
\(190\) 0 0
\(191\) 724824.i 1.43764i −0.695198 0.718819i \(-0.744683\pi\)
0.695198 0.718819i \(-0.255317\pi\)
\(192\) 0 0
\(193\) −508619. + 508619.i −0.982877 + 0.982877i −0.999856 0.0169786i \(-0.994595\pi\)
0.0169786 + 0.999856i \(0.494595\pi\)
\(194\) 0 0
\(195\) 97492.6 393829.i 0.183605 0.741688i
\(196\) 0 0
\(197\) −190061. + 190061.i −0.348922 + 0.348922i −0.859708 0.510786i \(-0.829354\pi\)
0.510786 + 0.859708i \(0.329354\pi\)
\(198\) 0 0
\(199\) 945435. 1.69238 0.846192 0.532878i \(-0.178889\pi\)
0.846192 + 0.532878i \(0.178889\pi\)
\(200\) 0 0
\(201\) −244713. −0.427236
\(202\) 0 0
\(203\) −131617. + 131617.i −0.224168 + 0.224168i
\(204\) 0 0
\(205\) −118364. + 478141.i −0.196714 + 0.794642i
\(206\) 0 0
\(207\) −372645. + 372645.i −0.604462 + 0.604462i
\(208\) 0 0
\(209\) 625556.i 0.990605i
\(210\) 0 0
\(211\) 1.01059e6 1.56267 0.781337 0.624109i \(-0.214538\pi\)
0.781337 + 0.624109i \(0.214538\pi\)
\(212\) 0 0
\(213\) 88210.8 + 88210.8i 0.133221 + 0.133221i
\(214\) 0 0
\(215\) −30455.0 50494.0i −0.0449327 0.0744978i
\(216\) 0 0
\(217\) −471792. 471792.i −0.680145 0.680145i
\(218\) 0 0
\(219\) 30646.9i 0.0431794i
\(220\) 0 0
\(221\) 1.87076e6i 2.57655i
\(222\) 0 0
\(223\) 320886. + 320886.i 0.432104 + 0.432104i 0.889344 0.457239i \(-0.151162\pi\)
−0.457239 + 0.889344i \(0.651162\pi\)
\(224\) 0 0
\(225\) 254221. 482006.i 0.334776 0.634741i
\(226\) 0 0
\(227\) 307496. + 307496.i 0.396073 + 0.396073i 0.876845 0.480772i \(-0.159644\pi\)
−0.480772 + 0.876845i \(0.659644\pi\)
\(228\) 0 0
\(229\) 821991. 1.03581 0.517903 0.855439i \(-0.326713\pi\)
0.517903 + 0.855439i \(0.326713\pi\)
\(230\) 0 0
\(231\) 643132.i 0.792995i
\(232\) 0 0
\(233\) −444211. + 444211.i −0.536043 + 0.536043i −0.922364 0.386321i \(-0.873746\pi\)
0.386321 + 0.922364i \(0.373746\pi\)
\(234\) 0 0
\(235\) −1084.81 1798.60i −0.00128140 0.00212454i
\(236\) 0 0
\(237\) 102066. 102066.i 0.118034 0.118034i
\(238\) 0 0
\(239\) 668811. 0.757371 0.378685 0.925526i \(-0.376376\pi\)
0.378685 + 0.925526i \(0.376376\pi\)
\(240\) 0 0
\(241\) 923504. 1.02423 0.512114 0.858918i \(-0.328863\pi\)
0.512114 + 0.858918i \(0.328863\pi\)
\(242\) 0 0
\(243\) −674529. + 674529.i −0.732800 + 0.732800i
\(244\) 0 0
\(245\) −183868. 45516.7i −0.195700 0.0484457i
\(246\) 0 0
\(247\) 709374. 709374.i 0.739832 0.739832i
\(248\) 0 0
\(249\) 364712.i 0.372780i
\(250\) 0 0
\(251\) 428698. 0.429504 0.214752 0.976669i \(-0.431106\pi\)
0.214752 + 0.976669i \(0.431106\pi\)
\(252\) 0 0
\(253\) 1.16747e6 + 1.16747e6i 1.14669 + 1.14669i
\(254\) 0 0
\(255\) −237598. + 959795.i −0.228819 + 0.924332i
\(256\) 0 0
\(257\) −935603. 935603.i −0.883606 0.883606i 0.110293 0.993899i \(-0.464821\pi\)
−0.993899 + 0.110293i \(0.964821\pi\)
\(258\) 0 0
\(259\) 1.52708e6i 1.41453i
\(260\) 0 0
\(261\) 228402.i 0.207538i
\(262\) 0 0
\(263\) −1.11658e6 1.11658e6i −0.995411 0.995411i 0.00457866 0.999990i \(-0.498543\pi\)
−0.999990 + 0.00457866i \(0.998543\pi\)
\(264\) 0 0
\(265\) −12852.1 + 7751.61i −0.0112424 + 0.00678074i
\(266\) 0 0
\(267\) 195648. + 195648.i 0.167956 + 0.167956i
\(268\) 0 0
\(269\) −2.11682e6 −1.78362 −0.891810 0.452410i \(-0.850564\pi\)
−0.891810 + 0.452410i \(0.850564\pi\)
\(270\) 0 0
\(271\) 2.01962e6i 1.67050i 0.549868 + 0.835252i \(0.314678\pi\)
−0.549868 + 0.835252i \(0.685322\pi\)
\(272\) 0 0
\(273\) 729305. 729305.i 0.592247 0.592247i
\(274\) 0 0
\(275\) −1.51010e6 796459.i −1.20413 0.635084i
\(276\) 0 0
\(277\) −418515. + 418515.i −0.327727 + 0.327727i −0.851721 0.523995i \(-0.824441\pi\)
0.523995 + 0.851721i \(0.324441\pi\)
\(278\) 0 0
\(279\) 818723. 0.629690
\(280\) 0 0
\(281\) 293805. 0.221969 0.110985 0.993822i \(-0.464600\pi\)
0.110985 + 0.993822i \(0.464600\pi\)
\(282\) 0 0
\(283\) 133050. 133050.i 0.0987527 0.0987527i −0.656004 0.754757i \(-0.727755\pi\)
0.754757 + 0.656004i \(0.227755\pi\)
\(284\) 0 0
\(285\) −454040. + 273850.i −0.331117 + 0.199710i
\(286\) 0 0
\(287\) −885438. + 885438.i −0.634532 + 0.634532i
\(288\) 0 0
\(289\) 3.13935e6i 2.21103i
\(290\) 0 0
\(291\) 38256.0 0.0264830
\(292\) 0 0
\(293\) 686968. + 686968.i 0.467484 + 0.467484i 0.901099 0.433614i \(-0.142762\pi\)
−0.433614 + 0.901099i \(0.642762\pi\)
\(294\) 0 0
\(295\) −2.12699e6 526539.i −1.42302 0.352270i
\(296\) 0 0
\(297\) 1.33564e6 + 1.33564e6i 0.878617 + 0.878617i
\(298\) 0 0
\(299\) 2.64781e6i 1.71281i
\(300\) 0 0
\(301\) 149904.i 0.0953667i
\(302\) 0 0
\(303\) 101296. + 101296.i 0.0633849 + 0.0633849i
\(304\) 0 0
\(305\) 94982.7 + 23513.0i 0.0584649 + 0.0144730i
\(306\) 0 0
\(307\) 397372. + 397372.i 0.240631 + 0.240631i 0.817111 0.576480i \(-0.195574\pi\)
−0.576480 + 0.817111i \(0.695574\pi\)
\(308\) 0 0
\(309\) 491881. 0.293065
\(310\) 0 0
\(311\) 2.55376e6i 1.49720i −0.663023 0.748599i \(-0.730727\pi\)
0.663023 0.748599i \(-0.269273\pi\)
\(312\) 0 0
\(313\) 93894.6 93894.6i 0.0541726 0.0541726i −0.679501 0.733674i \(-0.737804\pi\)
0.733674 + 0.679501i \(0.237804\pi\)
\(314\) 0 0
\(315\) 1.18625e6 715478.i 0.673598 0.406275i
\(316\) 0 0
\(317\) −1.55567e6 + 1.55567e6i −0.869497 + 0.869497i −0.992417 0.122920i \(-0.960774\pi\)
0.122920 + 0.992417i \(0.460774\pi\)
\(318\) 0 0
\(319\) −715569. −0.393709
\(320\) 0 0
\(321\) 1.25772e6 0.681274
\(322\) 0 0
\(323\) −1.72881e6 + 1.72881e6i −0.922019 + 0.922019i
\(324\) 0 0
\(325\) −809258. 2.61561e6i −0.424990 1.37361i
\(326\) 0 0
\(327\) 300256. 300256.i 0.155282 0.155282i
\(328\) 0 0
\(329\) 5339.58i 0.00271968i
\(330\) 0 0
\(331\) 976781. 0.490035 0.245018 0.969519i \(-0.421206\pi\)
0.245018 + 0.969519i \(0.421206\pi\)
\(332\) 0 0
\(333\) 1.32501e6 + 1.32501e6i 0.654798 + 0.654798i
\(334\) 0 0
\(335\) −1.41412e6 + 852915.i −0.688453 + 0.415235i
\(336\) 0 0
\(337\) 1.02826e6 + 1.02826e6i 0.493207 + 0.493207i 0.909315 0.416108i \(-0.136606\pi\)
−0.416108 + 0.909315i \(0.636606\pi\)
\(338\) 0 0
\(339\) 1.75824e6i 0.830957i
\(340\) 0 0
\(341\) 2.56501e6i 1.19455i
\(342\) 0 0
\(343\) 1.34840e6 + 1.34840e6i 0.618846 + 0.618846i
\(344\) 0 0
\(345\) 336287. 1.35846e6i 0.152112 0.614467i
\(346\) 0 0
\(347\) −404120. 404120.i −0.180172 0.180172i 0.611259 0.791431i \(-0.290663\pi\)
−0.791431 + 0.611259i \(0.790663\pi\)
\(348\) 0 0
\(349\) −2.80363e6 −1.23213 −0.616065 0.787695i \(-0.711274\pi\)
−0.616065 + 0.787695i \(0.711274\pi\)
\(350\) 0 0
\(351\) 3.02921e6i 1.31239i
\(352\) 0 0
\(353\) −2.76762e6 + 2.76762e6i −1.18214 + 1.18214i −0.202956 + 0.979188i \(0.565055\pi\)
−0.979188 + 0.202956i \(0.934945\pi\)
\(354\) 0 0
\(355\) 817189. + 202296.i 0.344153 + 0.0851953i
\(356\) 0 0
\(357\) −1.77738e6 + 1.77738e6i −0.738090 + 0.738090i
\(358\) 0 0
\(359\) 1.57792e6 0.646175 0.323088 0.946369i \(-0.395279\pi\)
0.323088 + 0.946369i \(0.395279\pi\)
\(360\) 0 0
\(361\) 1.16501e6 0.470501
\(362\) 0 0
\(363\) 804921. 804921.i 0.320617 0.320617i
\(364\) 0 0
\(365\) 106816. + 177099.i 0.0419665 + 0.0695798i
\(366\) 0 0
\(367\) −1.27870e6 + 1.27870e6i −0.495567 + 0.495567i −0.910055 0.414488i \(-0.863961\pi\)
0.414488 + 0.910055i \(0.363961\pi\)
\(368\) 0 0
\(369\) 1.53654e6i 0.587460i
\(370\) 0 0
\(371\) −38154.5 −0.0143917
\(372\) 0 0
\(373\) 954700. + 954700.i 0.355300 + 0.355300i 0.862077 0.506777i \(-0.169163\pi\)
−0.506777 + 0.862077i \(0.669163\pi\)
\(374\) 0 0
\(375\) 82992.2 + 1.44472e6i 0.0304761 + 0.530525i
\(376\) 0 0
\(377\) −811448. 811448.i −0.294041 0.294041i
\(378\) 0 0
\(379\) 732358.i 0.261894i −0.991389 0.130947i \(-0.958198\pi\)
0.991389 0.130947i \(-0.0418017\pi\)
\(380\) 0 0
\(381\) 2.30203e6i 0.812453i
\(382\) 0 0
\(383\) −1.21103e6 1.21103e6i −0.421851 0.421851i 0.463989 0.885841i \(-0.346418\pi\)
−0.885841 + 0.463989i \(0.846418\pi\)
\(384\) 0 0
\(385\) −2.24155e6 3.71646e6i −0.770720 1.27784i
\(386\) 0 0
\(387\) 130068. + 130068.i 0.0441461 + 0.0441461i
\(388\) 0 0
\(389\) 3.87126e6 1.29711 0.648557 0.761166i \(-0.275373\pi\)
0.648557 + 0.761166i \(0.275373\pi\)
\(390\) 0 0
\(391\) 6.45293e6i 2.13459i
\(392\) 0 0
\(393\) −859717. + 859717.i −0.280785 + 0.280785i
\(394\) 0 0
\(395\) 234069. 945540.i 0.0754834 0.304921i
\(396\) 0 0
\(397\) 3.28430e6 3.28430e6i 1.04584 1.04584i 0.0469472 0.998897i \(-0.485051\pi\)
0.998897 0.0469472i \(-0.0149492\pi\)
\(398\) 0 0
\(399\) −1.34793e6 −0.423872
\(400\) 0 0
\(401\) −4.96788e6 −1.54280 −0.771401 0.636349i \(-0.780444\pi\)
−0.771401 + 0.636349i \(0.780444\pi\)
\(402\) 0 0
\(403\) 2.90870e6 2.90870e6i 0.892145 0.892145i
\(404\) 0 0
\(405\) −184489. + 745256.i −0.0558897 + 0.225771i
\(406\) 0 0
\(407\) 4.15116e6 4.15116e6i 1.24218 1.24218i
\(408\) 0 0
\(409\) 4.77068e6i 1.41017i −0.709122 0.705085i \(-0.750909\pi\)
0.709122 0.705085i \(-0.249091\pi\)
\(410\) 0 0
\(411\) 1.49109e6 0.435412
\(412\) 0 0
\(413\) −3.93884e6 3.93884e6i −1.13630 1.13630i
\(414\) 0 0
\(415\) 1.27115e6 + 2.10756e6i 0.362308 + 0.600702i
\(416\) 0 0
\(417\) 745490. + 745490.i 0.209943 + 0.209943i
\(418\) 0 0
\(419\) 144488.i 0.0402065i −0.999798 0.0201032i \(-0.993601\pi\)
0.999798 0.0201032i \(-0.00639949\pi\)
\(420\) 0 0
\(421\) 4.80073e6i 1.32008i 0.751229 + 0.660042i \(0.229462\pi\)
−0.751229 + 0.660042i \(0.770538\pi\)
\(422\) 0 0
\(423\) 4633.01 + 4633.01i 0.00125896 + 0.00125896i
\(424\) 0 0
\(425\) 1.97223e6 + 6.37446e6i 0.529646 + 1.71187i
\(426\) 0 0
\(427\) 175892. + 175892.i 0.0466849 + 0.0466849i
\(428\) 0 0
\(429\) 3.96504e6 1.04017
\(430\) 0 0
\(431\) 5.19074e6i 1.34597i 0.739655 + 0.672986i \(0.234989\pi\)
−0.739655 + 0.672986i \(0.765011\pi\)
\(432\) 0 0
\(433\) 3.29626e6 3.29626e6i 0.844894 0.844894i −0.144597 0.989491i \(-0.546188\pi\)
0.989491 + 0.144597i \(0.0461885\pi\)
\(434\) 0 0
\(435\) 313255. + 519373.i 0.0793734 + 0.131600i
\(436\) 0 0
\(437\) 2.44689e6 2.44689e6i 0.612929 0.612929i
\(438\) 0 0
\(439\) 906765. 0.224561 0.112280 0.993677i \(-0.464185\pi\)
0.112280 + 0.993677i \(0.464185\pi\)
\(440\) 0 0
\(441\) 590874. 0.144677
\(442\) 0 0
\(443\) −4.04046e6 + 4.04046e6i −0.978186 + 0.978186i −0.999767 0.0215808i \(-0.993130\pi\)
0.0215808 + 0.999767i \(0.493130\pi\)
\(444\) 0 0
\(445\) 1.81249e6 + 448682.i 0.433885 + 0.107409i
\(446\) 0 0
\(447\) 1.21117e6 1.21117e6i 0.286706 0.286706i
\(448\) 0 0
\(449\) 567297.i 0.132799i 0.997793 + 0.0663995i \(0.0211512\pi\)
−0.997793 + 0.0663995i \(0.978849\pi\)
\(450\) 0 0
\(451\) −4.81389e6 −1.11444
\(452\) 0 0
\(453\) −1.47152e6 1.47152e6i −0.336915 0.336915i
\(454\) 0 0
\(455\) 1.67253e6 6.75632e6i 0.378744 1.52997i
\(456\) 0 0
\(457\) −3.73989e6 3.73989e6i −0.837662 0.837662i 0.150889 0.988551i \(-0.451787\pi\)
−0.988551 + 0.150889i \(0.951787\pi\)
\(458\) 0 0
\(459\) 7.38245e6i 1.63557i
\(460\) 0 0
\(461\) 3.97764e6i 0.871711i −0.900017 0.435856i \(-0.856446\pi\)
0.900017 0.435856i \(-0.143554\pi\)
\(462\) 0 0
\(463\) 4.06996e6 + 4.06996e6i 0.882343 + 0.882343i 0.993772 0.111429i \(-0.0355427\pi\)
−0.111429 + 0.993772i \(0.535543\pi\)
\(464\) 0 0
\(465\) −1.86173e6 + 1.12289e6i −0.399286 + 0.240826i
\(466\) 0 0
\(467\) −5.80864e6 5.80864e6i −1.23249 1.23249i −0.963006 0.269481i \(-0.913148\pi\)
−0.269481 0.963006i \(-0.586852\pi\)
\(468\) 0 0
\(469\) −4.19817e6 −0.881308
\(470\) 0 0
\(471\) 868064.i 0.180302i
\(472\) 0 0
\(473\) 407494. 407494.i 0.0837469 0.0837469i
\(474\) 0 0
\(475\) −1.66928e6 + 3.16498e6i −0.339466 + 0.643632i
\(476\) 0 0
\(477\) 33105.7 33105.7i 0.00666203 0.00666203i
\(478\) 0 0
\(479\) −5.53752e6 −1.10275 −0.551374 0.834258i \(-0.685896\pi\)
−0.551374 + 0.834258i \(0.685896\pi\)
\(480\) 0 0
\(481\) 9.41475e6 1.85544
\(482\) 0 0
\(483\) 2.51564e6 2.51564e6i 0.490659 0.490659i
\(484\) 0 0
\(485\) 221069. 133336.i 0.0426750 0.0257391i
\(486\) 0 0
\(487\) −1.55138e6 + 1.55138e6i −0.296412 + 0.296412i −0.839607 0.543195i \(-0.817214\pi\)
0.543195 + 0.839607i \(0.317214\pi\)
\(488\) 0 0
\(489\) 941605.i 0.178072i
\(490\) 0 0
\(491\) −8.21995e6 −1.53874 −0.769370 0.638803i \(-0.779430\pi\)
−0.769370 + 0.638803i \(0.779430\pi\)
\(492\) 0 0
\(493\) 1.97757e6 + 1.97757e6i 0.366450 + 0.366450i
\(494\) 0 0
\(495\) 5.16960e6 + 1.27974e6i 0.948296 + 0.234751i
\(496\) 0 0
\(497\) 1.51330e6 + 1.51330e6i 0.274810 + 0.274810i
\(498\) 0 0
\(499\) 3.17479e6i 0.570774i 0.958412 + 0.285387i \(0.0921221\pi\)
−0.958412 + 0.285387i \(0.907878\pi\)
\(500\) 0 0
\(501\) 795658.i 0.141622i
\(502\) 0 0
\(503\) −3.83730e6 3.83730e6i −0.676247 0.676247i 0.282902 0.959149i \(-0.408703\pi\)
−0.959149 + 0.282902i \(0.908703\pi\)
\(504\) 0 0
\(505\) 938410. + 232304.i 0.163744 + 0.0405348i
\(506\) 0 0
\(507\) 2.32148e6 + 2.32148e6i 0.401093 + 0.401093i
\(508\) 0 0
\(509\) 4.82930e6 0.826208 0.413104 0.910684i \(-0.364445\pi\)
0.413104 + 0.910684i \(0.364445\pi\)
\(510\) 0 0
\(511\) 525762.i 0.0890711i
\(512\) 0 0
\(513\) 2.79935e6 2.79935e6i 0.469639 0.469639i
\(514\) 0 0
\(515\) 2.84242e6 1.71438e6i 0.472249 0.284833i
\(516\) 0 0
\(517\) 14514.9 14514.9i 0.00238830 0.00238830i
\(518\) 0 0
\(519\) 804356. 0.131078
\(520\) 0 0
\(521\) −1.06985e7 −1.72674 −0.863372 0.504569i \(-0.831652\pi\)
−0.863372 + 0.504569i \(0.831652\pi\)
\(522\) 0 0
\(523\) 367714. 367714.i 0.0587836 0.0587836i −0.677104 0.735887i \(-0.736765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(524\) 0 0
\(525\) −1.71618e6 + 3.25391e6i −0.271748 + 0.515237i
\(526\) 0 0
\(527\) −7.08874e6 + 7.08874e6i −1.11184 + 1.11184i
\(528\) 0 0
\(529\) 2.69690e6i 0.419011i
\(530\) 0 0
\(531\) 6.83525e6 1.05201
\(532\) 0 0
\(533\) −5.45890e6 5.45890e6i −0.832315 0.832315i
\(534\) 0 0
\(535\) 7.26798e6 4.38362e6i 1.09781 0.662137i
\(536\) 0 0
\(537\) 2.03206e6 + 2.03206e6i 0.304090 + 0.304090i
\(538\) 0 0
\(539\) 1.85117e6i 0.274457i
\(540\) 0 0
\(541\) 4.61678e6i 0.678182i 0.940754 + 0.339091i \(0.110119\pi\)
−0.940754 + 0.339091i \(0.889881\pi\)
\(542\) 0 0
\(543\) 462943. + 462943.i 0.0673795 + 0.0673795i
\(544\) 0 0
\(545\) 688582. 2.78158e6i 0.0993035 0.401144i
\(546\) 0 0
\(547\) 8.25355e6 + 8.25355e6i 1.17943 + 1.17943i 0.979889 + 0.199542i \(0.0639453\pi\)
0.199542 + 0.979889i \(0.436055\pi\)
\(548\) 0 0
\(549\) −305234. −0.0432217
\(550\) 0 0
\(551\) 1.49975e6i 0.210446i
\(552\) 0 0
\(553\) 1.75098e6 1.75098e6i 0.243483 0.243483i
\(554\) 0 0
\(555\) −4.83024e6 1.19573e6i −0.665636 0.164779i
\(556\) 0 0
\(557\) −1.93310e6 + 1.93310e6i −0.264008 + 0.264008i −0.826680 0.562672i \(-0.809773\pi\)
0.562672 + 0.826680i \(0.309773\pi\)
\(558\) 0 0
\(559\) 924189. 0.125092
\(560\) 0 0
\(561\) −9.66314e6 −1.29632
\(562\) 0 0
\(563\) 6.10842e6 6.10842e6i 0.812190 0.812190i −0.172772 0.984962i \(-0.555272\pi\)
0.984962 + 0.172772i \(0.0552723\pi\)
\(564\) 0 0
\(565\) 6.12810e6 + 1.01603e7i 0.807616 + 1.33902i
\(566\) 0 0
\(567\) −1.38009e6 + 1.38009e6i −0.180281 + 0.180281i
\(568\) 0 0
\(569\) 6.95880e6i 0.901060i 0.892761 + 0.450530i \(0.148765\pi\)
−0.892761 + 0.450530i \(0.851235\pi\)
\(570\) 0 0
\(571\) 3.46936e6 0.445306 0.222653 0.974898i \(-0.428528\pi\)
0.222653 + 0.974898i \(0.428528\pi\)
\(572\) 0 0
\(573\) −4.24563e6 4.24563e6i −0.540201 0.540201i
\(574\) 0 0
\(575\) −2.79142e6 9.02218e6i −0.352092 1.13800i
\(576\) 0 0
\(577\) −2.25638e6 2.25638e6i −0.282145 0.282145i 0.551819 0.833964i \(-0.313934\pi\)
−0.833964 + 0.551819i \(0.813934\pi\)
\(578\) 0 0
\(579\) 5.95843e6i 0.738644i
\(580\) 0 0
\(581\) 6.25681e6i 0.768975i
\(582\) 0 0
\(583\) −103718. 103718.i −0.0126381 0.0126381i
\(584\) 0 0
\(585\) 4.41107e6 + 7.31349e6i 0.532910 + 0.883557i
\(586\) 0 0
\(587\) 4.93652e6 + 4.93652e6i 0.591324 + 0.591324i 0.937989 0.346665i \(-0.112686\pi\)
−0.346665 + 0.937989i \(0.612686\pi\)
\(588\) 0 0
\(589\) −5.37596e6 −0.638510
\(590\) 0 0
\(591\) 2.22655e6i 0.262219i
\(592\) 0 0
\(593\) −227180. + 227180.i −0.0265298 + 0.0265298i −0.720247 0.693717i \(-0.755972\pi\)
0.693717 + 0.720247i \(0.255972\pi\)
\(594\) 0 0
\(595\) −4.07610e6 + 1.64657e7i −0.472011 + 1.90673i
\(596\) 0 0
\(597\) 5.53785e6 5.53785e6i 0.635924 0.635924i
\(598\) 0 0
\(599\) −1.51009e7 −1.71963 −0.859815 0.510605i \(-0.829421\pi\)
−0.859815 + 0.510605i \(0.829421\pi\)
\(600\) 0 0
\(601\) −4.90619e6 −0.554061 −0.277031 0.960861i \(-0.589350\pi\)
−0.277031 + 0.960861i \(0.589350\pi\)
\(602\) 0 0
\(603\) 3.64264e6 3.64264e6i 0.407965 0.407965i
\(604\) 0 0
\(605\) 1.84594e6 7.45682e6i 0.205036 0.828258i
\(606\) 0 0
\(607\) −5.77545e6 + 5.77545e6i −0.636230 + 0.636230i −0.949623 0.313393i \(-0.898534\pi\)
0.313393 + 0.949623i \(0.398534\pi\)
\(608\) 0 0
\(609\) 1.54189e6i 0.168465i
\(610\) 0 0
\(611\) 32919.6 0.00356740
\(612\) 0 0
\(613\) −5.92087e6 5.92087e6i −0.636406 0.636406i 0.313261 0.949667i \(-0.398579\pi\)
−0.949667 + 0.313261i \(0.898579\pi\)
\(614\) 0 0
\(615\) 2.10738e6 + 3.49401e6i 0.224675 + 0.372508i
\(616\) 0 0
\(617\) 6.30723e6 + 6.30723e6i 0.667000 + 0.667000i 0.957020 0.290021i \(-0.0936622\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(618\) 0 0
\(619\) 5.28850e6i 0.554761i −0.960760 0.277380i \(-0.910534\pi\)
0.960760 0.277380i \(-0.0894662\pi\)
\(620\) 0 0
\(621\) 1.04489e7i 1.08728i
\(622\) 0 0
\(623\) 3.35642e6 + 3.35642e6i 0.346463 + 0.346463i
\(624\) 0 0
\(625\) 5.51496e6 + 8.05932e6i 0.564732 + 0.825275i
\(626\) 0 0
\(627\) −3.66417e6 3.66417e6i −0.372226 0.372226i
\(628\) 0 0
\(629\) −2.29446e7 −2.31235
\(630\) 0 0
\(631\) 1.31906e7i 1.31884i 0.751777 + 0.659418i \(0.229197\pi\)
−0.751777 + 0.659418i \(0.770803\pi\)
\(632\) 0 0
\(633\) 5.91948e6 5.91948e6i 0.587184 0.587184i
\(634\) 0 0
\(635\) 8.02340e6 + 1.33027e7i 0.789631 + 1.30920i
\(636\) 0 0
\(637\) 2.09921e6 2.09921e6i 0.204978 0.204978i
\(638\) 0 0
\(639\) −2.62609e6 −0.254424
\(640\) 0 0
\(641\) 3.49015e6 0.335505 0.167752 0.985829i \(-0.446349\pi\)
0.167752 + 0.985829i \(0.446349\pi\)
\(642\) 0 0
\(643\) −4.40501e6 + 4.40501e6i −0.420165 + 0.420165i −0.885260 0.465096i \(-0.846020\pi\)
0.465096 + 0.885260i \(0.346020\pi\)
\(644\) 0 0
\(645\) −474155. 117377.i −0.0448767 0.0111093i
\(646\) 0 0
\(647\) 1.25788e7 1.25788e7i 1.18135 1.18135i 0.201960 0.979394i \(-0.435269\pi\)
0.979394 0.201960i \(-0.0647312\pi\)
\(648\) 0 0
\(649\) 2.14144e7i 1.99570i
\(650\) 0 0
\(651\) −5.52701e6 −0.511137
\(652\) 0 0
\(653\) −1.35390e6 1.35390e6i −0.124252 0.124252i 0.642246 0.766498i \(-0.278003\pi\)
−0.766498 + 0.642246i \(0.778003\pi\)
\(654\) 0 0
\(655\) −1.97161e6 + 7.96446e6i −0.179563 + 0.725359i
\(656\) 0 0
\(657\) −456189. 456189.i −0.0412317 0.0412317i
\(658\) 0 0
\(659\) 9.42856e6i 0.845730i −0.906193 0.422865i \(-0.861024\pi\)
0.906193 0.422865i \(-0.138976\pi\)
\(660\) 0 0
\(661\) 4.10961e6i 0.365845i 0.983127 + 0.182923i \(0.0585558\pi\)
−0.983127 + 0.182923i \(0.941444\pi\)
\(662\) 0 0
\(663\) −1.09579e7 1.09579e7i −0.968152 0.968152i
\(664\) 0 0
\(665\) −7.78926e6 + 4.69802e6i −0.683033 + 0.411966i
\(666\) 0 0
\(667\) −2.79898e6 2.79898e6i −0.243604 0.243604i
\(668\) 0 0
\(669\) 3.75915e6 0.324732
\(670\) 0 0
\(671\) 956279.i 0.0819933i
\(672\) 0 0
\(673\) −1.18829e7 + 1.18829e7i −1.01131 + 1.01131i −0.0113747 + 0.999935i \(0.503621\pi\)
−0.999935 + 0.0113747i \(0.996379\pi\)
\(674\) 0 0
\(675\) −3.19352e6 1.03218e7i −0.269780 0.871959i
\(676\) 0 0
\(677\) −2.69398e6 + 2.69398e6i −0.225904 + 0.225904i −0.810979 0.585075i \(-0.801065\pi\)
0.585075 + 0.810979i \(0.301065\pi\)
\(678\) 0 0
\(679\) 656299. 0.0546295
\(680\) 0 0
\(681\) 3.60229e6 0.297654
\(682\) 0 0
\(683\) −5.78117e6 + 5.78117e6i −0.474203 + 0.474203i −0.903272 0.429069i \(-0.858842\pi\)
0.429069 + 0.903272i \(0.358842\pi\)
\(684\) 0 0
\(685\) 8.61656e6 5.19700e6i 0.701629 0.423181i
\(686\) 0 0
\(687\) 4.81478e6 4.81478e6i 0.389210 0.389210i
\(688\) 0 0
\(689\) 235230.i 0.0188775i
\(690\) 0 0
\(691\) −2.16912e7 −1.72818 −0.864091 0.503336i \(-0.832106\pi\)
−0.864091 + 0.503336i \(0.832106\pi\)
\(692\) 0 0
\(693\) 9.57324e6 + 9.57324e6i 0.757226 + 0.757226i
\(694\) 0 0
\(695\) 6.90626e6 + 1.70965e6i 0.542351 + 0.134259i
\(696\) 0 0
\(697\) 1.33038e7 + 1.33038e7i 1.03728 + 1.03728i
\(698\) 0 0
\(699\) 5.20390e6i 0.402843i
\(700\) 0 0
\(701\) 324553.i 0.0249454i 0.999922 + 0.0124727i \(0.00397029\pi\)
−0.999922 + 0.0124727i \(0.996030\pi\)
\(702\) 0 0
\(703\) −8.70035e6 8.70035e6i −0.663970 0.663970i
\(704\) 0 0
\(705\) −16889.4 4180.98i −0.00127980 0.000316815i
\(706\) 0 0
\(707\) 1.73778e6 + 1.73778e6i 0.130751 + 0.130751i
\(708\) 0 0
\(709\) 1.29646e7 0.968596 0.484298 0.874903i \(-0.339075\pi\)
0.484298 + 0.874903i \(0.339075\pi\)
\(710\) 0 0
\(711\) 3.03856e6i 0.225421i
\(712\) 0 0
\(713\) 1.00331e7 1.00331e7i 0.739117 0.739117i
\(714\) 0 0
\(715\) 2.29127e7 1.38196e7i 1.67614 1.01095i
\(716\) 0 0
\(717\) 3.91753e6 3.91753e6i 0.284587 0.284587i
\(718\) 0 0
\(719\) 1.03596e7 0.747346 0.373673 0.927561i \(-0.378098\pi\)
0.373673 + 0.927561i \(0.378098\pi\)
\(720\) 0 0
\(721\) 8.43844e6 0.604539
\(722\) 0 0
\(723\) 5.40939e6 5.40939e6i 0.384860 0.384860i
\(724\) 0 0
\(725\) 3.62040e6 + 1.90948e6i 0.255807 + 0.134918i
\(726\) 0 0
\(727\) 724947. 724947.i 0.0508710 0.0508710i −0.681214 0.732085i \(-0.738548\pi\)
0.732085 + 0.681214i \(0.238548\pi\)
\(728\) 0 0
\(729\) 4.56470e6i 0.318122i
\(730\) 0 0
\(731\) −2.25233e6 −0.155897
\(732\) 0 0
\(733\) −8.37353e6 8.37353e6i −0.575637 0.575637i 0.358061 0.933698i \(-0.383438\pi\)
−0.933698 + 0.358061i \(0.883438\pi\)
\(734\) 0 0
\(735\) −1.34361e6 + 810388.i −0.0917394 + 0.0553318i
\(736\) 0 0
\(737\) −1.14122e7 1.14122e7i −0.773926 0.773926i
\(738\) 0 0
\(739\) 1.89309e7i 1.27515i −0.770390 0.637573i \(-0.779939\pi\)
0.770390 0.637573i \(-0.220061\pi\)
\(740\) 0 0
\(741\) 8.31026e6i 0.555993i
\(742\) 0 0
\(743\) 8.69837e6 + 8.69837e6i 0.578051 + 0.578051i 0.934366 0.356315i \(-0.115967\pi\)
−0.356315 + 0.934366i \(0.615967\pi\)
\(744\) 0 0
\(745\) 2.77760e6 1.12203e7i 0.183349 0.740655i
\(746\) 0 0
\(747\) −5.42886e6 5.42886e6i −0.355965 0.355965i
\(748\) 0 0
\(749\) 2.15768e7 1.40534
\(750\) 0 0
\(751\) 1.96100e7i 1.26875i −0.773025 0.634376i \(-0.781257\pi\)
0.773025 0.634376i \(-0.218743\pi\)
\(752\) 0 0
\(753\) 2.51108e6 2.51108e6i 0.161389 0.161389i
\(754\) 0 0
\(755\) −1.36322e7 3.37466e6i −0.870360 0.215458i
\(756\) 0 0
\(757\) 3.43229e6 3.43229e6i 0.217693 0.217693i −0.589833 0.807525i \(-0.700806\pi\)
0.807525 + 0.589833i \(0.200806\pi\)
\(758\) 0 0
\(759\) 1.36769e7 0.861751
\(760\) 0 0
\(761\) −1.87574e7 −1.17412 −0.587059 0.809544i \(-0.699714\pi\)
−0.587059 + 0.809544i \(0.699714\pi\)
\(762\) 0 0
\(763\) 5.15102e6 5.15102e6i 0.320319 0.320319i
\(764\) 0 0
\(765\) −1.07501e7 1.78236e7i −0.664142 1.10114i
\(766\) 0 0
\(767\) 2.42837e7 2.42837e7i 1.49048 1.49048i
\(768\) 0 0
\(769\) 1.37065e7i 0.835815i −0.908490 0.417908i \(-0.862764\pi\)
0.908490 0.417908i \(-0.137236\pi\)
\(770\) 0 0
\(771\) −1.09605e7 −0.664041
\(772\) 0 0
\(773\) 1.40462e7 + 1.40462e7i 0.845490 + 0.845490i 0.989567 0.144076i \(-0.0460211\pi\)
−0.144076 + 0.989567i \(0.546021\pi\)
\(774\) 0 0
\(775\) −6.84468e6 + 1.29776e7i −0.409354 + 0.776140i
\(776\) 0 0
\(777\) −8.94480e6 8.94480e6i −0.531518 0.531518i
\(778\) 0 0
\(779\) 1.00894e7i 0.595689i
\(780\) 0 0
\(781\) 8.22740e6i 0.482653i
\(782\) 0 0
\(783\) −3.20216e6 3.20216e6i −0.186655 0.186655i
\(784\) 0 0
\(785\) −3.02552e6 5.01627e6i −0.175237 0.290540i
\(786\) 0 0
\(787\) 7.19838e6 + 7.19838e6i 0.414284 + 0.414284i 0.883228 0.468944i \(-0.155365\pi\)
−0.468944 + 0.883228i \(0.655365\pi\)
\(788\) 0 0
\(789\) −1.30807e7 −0.748063
\(790\) 0 0
\(791\) 3.01634e7i 1.71411i
\(792\) 0 0
\(793\) −1.08441e6 + 1.08441e6i −0.0612365 + 0.0612365i
\(794\) 0 0
\(795\) −29875.7 + 120685.i −0.00167649 + 0.00677229i
\(796\) 0 0
\(797\) −2.25277e7 + 2.25277e7i −1.25624 + 1.25624i −0.303361 + 0.952876i \(0.598109\pi\)
−0.952876 + 0.303361i \(0.901891\pi\)
\(798\) 0 0
\(799\) −80227.9 −0.00444589
\(800\) 0 0
\(801\) −5.82456e6 −0.320761
\(802\) 0 0
\(803\) −1.42921e6 + 1.42921e6i −0.0782183 + 0.0782183i
\(804\) 0 0
\(805\) 5.76916e6 2.33050e7i 0.313778 1.26753i
\(806\) 0 0
\(807\) −1.23992e7 + 1.23992e7i −0.670206 + 0.670206i
\(808\) 0 0
\(809\) 7.84195e6i 0.421263i −0.977566 0.210631i \(-0.932448\pi\)
0.977566 0.210631i \(-0.0675519\pi\)
\(810\) 0 0
\(811\) 9.65765e6 0.515608 0.257804 0.966197i \(-0.417001\pi\)
0.257804 + 0.966197i \(0.417001\pi\)
\(812\) 0 0
\(813\) 1.18299e7 + 1.18299e7i 0.627702 + 0.627702i
\(814\) 0 0
\(815\) −3.28184e6 5.44124e6i −0.173070 0.286948i
\(816\) 0 0
\(817\) −854060. 854060.i −0.0447645 0.0447645i
\(818\) 0 0
\(819\) 2.17119e7i 1.13107i
\(820\) 0 0
\(821\) 3.13878e7i 1.62519i −0.582830 0.812594i \(-0.698055\pi\)
0.582830 0.812594i \(-0.301945\pi\)
\(822\) 0 0
\(823\) −1.52238e6 1.52238e6i −0.0783474 0.0783474i 0.666847 0.745195i \(-0.267643\pi\)
−0.745195 + 0.666847i \(0.767643\pi\)
\(824\) 0 0
\(825\) −1.35105e7 + 4.18010e6i −0.691096 + 0.213822i
\(826\) 0 0
\(827\) −1.45364e7 1.45364e7i −0.739082 0.739082i 0.233318 0.972400i \(-0.425042\pi\)
−0.972400 + 0.233318i \(0.925042\pi\)
\(828\) 0 0
\(829\) −3.42957e7 −1.73322 −0.866609 0.498989i \(-0.833705\pi\)
−0.866609 + 0.498989i \(0.833705\pi\)
\(830\) 0 0
\(831\) 4.90287e6i 0.246291i
\(832\) 0 0
\(833\) −5.11595e6 + 5.11595e6i −0.255455 + 0.255455i
\(834\) 0 0
\(835\) −2.77316e6 4.59785e6i −0.137644 0.228212i
\(836\) 0 0
\(837\) 1.14784e7 1.14784e7i 0.566327 0.566327i
\(838\) 0 0
\(839\) −1.82753e7 −0.896313 −0.448157 0.893955i \(-0.647919\pi\)
−0.448157 + 0.893955i \(0.647919\pi\)
\(840\) 0 0
\(841\) −1.87956e7 −0.916360
\(842\) 0 0
\(843\) 1.72095e6 1.72095e6i 0.0834063 0.0834063i
\(844\) 0 0
\(845\) 2.15063e7 + 5.32390e6i 1.03615 + 0.256501i
\(846\) 0 0
\(847\) 1.38088e7 1.38088e7i 0.661374 0.661374i
\(848\) 0 0
\(849\) 1.55867e6i 0.0742138i
\(850\) 0 0
\(851\) 3.24749e7 1.53718
\(852\) 0 0
\(853\) 1.42805e7 + 1.42805e7i 0.672002 + 0.672002i 0.958177 0.286175i \(-0.0923840\pi\)
−0.286175 + 0.958177i \(0.592384\pi\)
\(854\) 0 0
\(855\) 2.68218e6 1.08349e7i 0.125479 0.506884i
\(856\) 0 0
\(857\) 6.07126e6 + 6.07126e6i 0.282376 + 0.282376i 0.834056 0.551680i \(-0.186013\pi\)
−0.551680 + 0.834056i \(0.686013\pi\)
\(858\) 0 0
\(859\) 1.28490e7i 0.594135i −0.954856 0.297068i \(-0.903991\pi\)
0.954856 0.297068i \(-0.0960087\pi\)
\(860\) 0 0
\(861\) 1.03728e7i 0.476858i
\(862\) 0 0
\(863\) −4.40683e6 4.40683e6i −0.201418 0.201418i 0.599189 0.800608i \(-0.295490\pi\)
−0.800608 + 0.599189i \(0.795490\pi\)
\(864\) 0 0
\(865\) 4.64812e6 2.80347e6i 0.211221 0.127396i
\(866\) 0 0
\(867\) 1.83886e7 + 1.83886e7i 0.830809 + 0.830809i
\(868\) 0 0
\(869\) 9.51963e6 0.427632
\(870\) 0 0
\(871\) 2.58826e7i 1.15601i
\(872\) 0 0
\(873\) −569453. + 569453.i −0.0252885 + 0.0252885i
\(874\) 0 0
\(875\) 1.42377e6 + 2.47848e7i 0.0628665 + 1.09437i
\(876\) 0 0
\(877\) 1.44143e6 1.44143e6i 0.0632839 0.0632839i −0.674757 0.738040i \(-0.735751\pi\)
0.738040 + 0.674757i \(0.235751\pi\)
\(878\) 0 0
\(879\) 8.04777e6 0.351320
\(880\) 0 0
\(881\) 3.59518e7 1.56056 0.780280 0.625430i \(-0.215077\pi\)
0.780280 + 0.625430i \(0.215077\pi\)
\(882\) 0 0
\(883\) 1.03739e7 1.03739e7i 0.447755 0.447755i −0.446853 0.894608i \(-0.647455\pi\)
0.894608 + 0.446853i \(0.147455\pi\)
\(884\) 0 0
\(885\) −1.55430e7 + 9.37460e6i −0.667076 + 0.402341i
\(886\) 0 0
\(887\) −1.39804e7 + 1.39804e7i −0.596637 + 0.596637i −0.939416 0.342779i \(-0.888632\pi\)
0.342779 + 0.939416i \(0.388632\pi\)
\(888\) 0 0
\(889\) 3.94924e7i 1.67594i
\(890\) 0 0
\(891\) −7.50318e6 −0.316629
\(892\) 0 0
\(893\) −30421.6 30421.6i −0.00127660 0.00127660i
\(894\) 0 0
\(895\) 1.88251e7 + 4.66017e6i 0.785562 + 0.194466i
\(896\) 0 0
\(897\) 1.55094e7 + 1.55094e7i 0.643597 + 0.643597i
\(898\) 0 0
\(899\) 6.14952e6i 0.253771i
\(900\) 0 0
\(901\) 573277.i 0.0235262i
\(902\) 0 0
\(903\) −878056. 878056.i −0.0358346 0.0358346i
\(904\) 0 0
\(905\) 4.28872e6 + 1.06168e6i 0.174063 + 0.0430894i
\(906\) 0 0
\(907\) −8.08547e6 8.08547e6i −0.326353 0.326353i 0.524845 0.851198i \(-0.324123\pi\)
−0.851198 + 0.524845i \(0.824123\pi\)
\(908\) 0 0
\(909\) −3.01565e6 −0.121052
\(910\) 0 0
\(911\) 345072.i 0.0137757i −0.999976 0.00688785i \(-0.997808\pi\)
0.999976 0.00688785i \(-0.00219249\pi\)
\(912\) 0 0
\(913\) −1.70083e7 + 1.70083e7i −0.675280 + 0.675280i
\(914\) 0 0
\(915\) 694084. 418631.i 0.0274068 0.0165302i
\(916\) 0 0
\(917\) −1.47488e7 + 1.47488e7i −0.579208 + 0.579208i
\(918\) 0 0
\(919\) 9.48331e6 0.370400 0.185200 0.982701i \(-0.440707\pi\)
0.185200 + 0.982701i \(0.440707\pi\)
\(920\) 0 0
\(921\) 4.65518e6 0.180837
\(922\) 0 0
\(923\) −9.32978e6 + 9.32978e6i −0.360468 + 0.360468i
\(924\) 0 0
\(925\) −3.20800e7 + 9.92541e6i −1.23276 + 0.381412i
\(926\) 0 0
\(927\) −7.32181e6 + 7.32181e6i −0.279846 + 0.279846i
\(928\) 0 0
\(929\) 4.39463e6i 0.167064i −0.996505 0.0835320i \(-0.973380\pi\)
0.996505 0.0835320i \(-0.0266201\pi\)
\(930\) 0 0
\(931\) −3.87984e6 −0.146703
\(932\) 0 0
\(933\) −1.49585e7 1.49585e7i −0.562581 0.562581i
\(934\) 0 0
\(935\) −5.58402e7 + 3.36796e7i −2.08890 + 1.25990i
\(936\) 0 0
\(937\) −3.07299e6 3.07299e6i −0.114344 0.114344i 0.647620 0.761964i \(-0.275765\pi\)
−0.761964 + 0.647620i \(0.775765\pi\)
\(938\) 0 0
\(939\) 1.09997e6i 0.0407114i
\(940\) 0 0
\(941\) 2.12225e7i 0.781309i −0.920537 0.390654i \(-0.872249\pi\)
0.920537 0.390654i \(-0.127751\pi\)
\(942\) 0 0
\(943\) −1.88297e7 1.88297e7i −0.689549 0.689549i
\(944\) 0 0
\(945\) 6.60019e6 2.66620e7i 0.240423 0.971209i
\(946\) 0 0
\(947\) −5.55954e6 5.55954e6i −0.201448 0.201448i 0.599172 0.800620i \(-0.295497\pi\)
−0.800620 + 0.599172i \(0.795497\pi\)
\(948\) 0 0
\(949\) −3.24143e6 −0.116834
\(950\) 0 0
\(951\) 1.82245e7i 0.653437i
\(952\) 0 0
\(953\) 2.16053e7 2.16053e7i 0.770599 0.770599i −0.207612 0.978211i \(-0.566569\pi\)
0.978211 + 0.207612i \(0.0665691\pi\)
\(954\) 0 0
\(955\) −3.93317e7 9.73658e6i −1.39551 0.345460i
\(956\) 0 0
\(957\) −4.19142e6 + 4.19142e6i −0.147938 + 0.147938i
\(958\) 0 0
\(959\) 2.55804e7 0.898175
\(960\) 0 0
\(961\) 6.58574e6 0.230036
\(962\) 0 0
\(963\) −1.87216e7 + 1.87216e7i −0.650545 + 0.650545i
\(964\) 0 0
\(965\) 2.07673e7 + 3.44319e7i 0.717896 + 1.19026i
\(966\) 0 0
\(967\) −3.96257e7 + 3.96257e7i −1.36273 + 1.36273i −0.492316 + 0.870417i \(0.663849\pi\)
−0.870417 + 0.492316i \(0.836151\pi\)
\(968\) 0 0
\(969\) 2.02528e7i 0.692909i
\(970\) 0 0
\(971\) 3.48187e7 1.18513 0.592563 0.805524i \(-0.298116\pi\)
0.592563 + 0.805524i \(0.298116\pi\)
\(972\) 0 0
\(973\) 1.27892e7 + 1.27892e7i 0.433074 + 0.433074i
\(974\) 0 0
\(975\) −2.00610e7 1.05806e7i −0.675836 0.356451i
\(976\) 0 0
\(977\) −2.78180e6 2.78180e6i −0.0932373 0.0932373i 0.658950 0.752187i \(-0.271001\pi\)
−0.752187 + 0.658950i \(0.771001\pi\)
\(978\) 0 0
\(979\) 1.82480e7i 0.608497i
\(980\) 0 0
\(981\) 8.93881e6i 0.296556i
\(982\) 0 0
\(983\) 4.07608e7 + 4.07608e7i 1.34542 + 1.34542i 0.890563 + 0.454861i \(0.150311\pi\)
0.454861 + 0.890563i \(0.349689\pi\)
\(984\) 0 0
\(985\) 7.76034e6 + 1.28665e7i 0.254853 + 0.422543i
\(986\) 0 0
\(987\) −31276.4 31276.4i −0.00102194 0.00102194i
\(988\) 0 0
\(989\) 3.18786e6 0.103635
\(990\) 0 0
\(991\) 4.18330e7i 1.35311i −0.736390 0.676557i \(-0.763471\pi\)
0.736390 0.676557i \(-0.236529\pi\)
\(992\) 0 0
\(993\) 5.72145e6 5.72145e6i 0.184134 0.184134i
\(994\) 0 0
\(995\) 1.27001e7 5.13028e7i 0.406675 1.64280i
\(996\) 0 0
\(997\) −1.02022e7 + 1.02022e7i −0.325056 + 0.325056i −0.850703 0.525647i \(-0.823823\pi\)
0.525647 + 0.850703i \(0.323823\pi\)
\(998\) 0 0
\(999\) 3.71528e7 1.17782
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.o.a.47.17 56
4.3 odd 2 40.6.k.a.27.15 yes 56
5.3 odd 4 inner 160.6.o.a.143.18 56
8.3 odd 2 inner 160.6.o.a.47.18 56
8.5 even 2 40.6.k.a.27.1 yes 56
20.3 even 4 40.6.k.a.3.1 56
40.3 even 4 inner 160.6.o.a.143.17 56
40.13 odd 4 40.6.k.a.3.15 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.k.a.3.1 56 20.3 even 4
40.6.k.a.3.15 yes 56 40.13 odd 4
40.6.k.a.27.1 yes 56 8.5 even 2
40.6.k.a.27.15 yes 56 4.3 odd 2
160.6.o.a.47.17 56 1.1 even 1 trivial
160.6.o.a.47.18 56 8.3 odd 2 inner
160.6.o.a.143.17 56 40.3 even 4 inner
160.6.o.a.143.18 56 5.3 odd 4 inner