Properties

Label 320.6.c.d.129.1
Level $320$
Weight $6$
Character 320.129
Analytic conductor $51.323$
Analytic rank $0$
Dimension $2$
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] = [320,6,Mod(129,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.129");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.c (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: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(0.500000 + 2.78388i\) of defining polynomial
Character \(\chi\) \(=\) 320.129
Dual form 320.6.c.d.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-11.1355i q^{3} +(5.00000 + 55.6776i) q^{5} +122.491i q^{7} +119.000 q^{9} +O(q^{10})\) \(q-11.1355i q^{3} +(5.00000 + 55.6776i) q^{5} +122.491i q^{7} +119.000 q^{9} -100.000 q^{11} -734.945i q^{13} +(620.000 - 55.6776i) q^{15} +979.927i q^{17} +2244.00 q^{19} +1364.00 q^{21} +3418.61i q^{23} +(-3075.00 + 556.776i) q^{25} -4031.06i q^{27} -7854.00 q^{29} +2144.00 q^{31} +1113.55i q^{33} +(-6820.00 + 612.454i) q^{35} -10400.6i q^{37} -8184.00 q^{39} -7414.00 q^{41} +17761.2i q^{43} +(595.000 + 6625.64i) q^{45} +9431.79i q^{47} +1803.00 q^{49} +10912.0 q^{51} +24253.2i q^{53} +(-500.000 - 5567.76i) q^{55} -24988.1i q^{57} -25972.0 q^{59} +3058.00 q^{61} +14576.4i q^{63} +(40920.0 - 3674.72i) q^{65} +58784.5i q^{67} +38068.0 q^{69} -37608.0 q^{71} +24008.2i q^{73} +(6200.00 + 34241.8i) q^{75} -12249.1i q^{77} +79728.0 q^{79} -15971.0 q^{81} +16291.3i q^{83} +(-54560.0 + 4899.63i) q^{85} +87458.4i q^{87} +826.000 q^{89} +90024.0 q^{91} -23874.6i q^{93} +(11220.0 + 124941. i) q^{95} -37593.5i q^{97} -11900.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 238 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} + 238 q^{9} - 200 q^{11} + 1240 q^{15} + 4488 q^{19} + 2728 q^{21} - 6150 q^{25} - 15708 q^{29} + 4288 q^{31} - 13640 q^{35} - 16368 q^{39} - 14828 q^{41} + 1190 q^{45} + 3606 q^{49} + 21824 q^{51} - 1000 q^{55} - 51944 q^{59} + 6116 q^{61} + 81840 q^{65} + 76136 q^{69} - 75216 q^{71} + 12400 q^{75} + 159456 q^{79} - 31942 q^{81} - 109120 q^{85} + 1652 q^{89} + 180048 q^{91} + 22440 q^{95} - 23800 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 11.1355i 0.714345i −0.934039 0.357172i \(-0.883741\pi\)
0.934039 0.357172i \(-0.116259\pi\)
\(4\) 0 0
\(5\) 5.00000 + 55.6776i 0.0894427 + 0.995992i
\(6\) 0 0
\(7\) 122.491i 0.944840i 0.881373 + 0.472420i \(0.156620\pi\)
−0.881373 + 0.472420i \(0.843380\pi\)
\(8\) 0 0
\(9\) 119.000 0.489712
\(10\) 0 0
\(11\) −100.000 −0.249183 −0.124591 0.992208i \(-0.539762\pi\)
−0.124591 + 0.992208i \(0.539762\pi\)
\(12\) 0 0
\(13\) 734.945i 1.20614i −0.797690 0.603068i \(-0.793945\pi\)
0.797690 0.603068i \(-0.206055\pi\)
\(14\) 0 0
\(15\) 620.000 55.6776i 0.711481 0.0638929i
\(16\) 0 0
\(17\) 979.927i 0.822377i 0.911550 + 0.411189i \(0.134886\pi\)
−0.911550 + 0.411189i \(0.865114\pi\)
\(18\) 0 0
\(19\) 2244.00 1.42606 0.713032 0.701132i \(-0.247322\pi\)
0.713032 + 0.701132i \(0.247322\pi\)
\(20\) 0 0
\(21\) 1364.00 0.674942
\(22\) 0 0
\(23\) 3418.61i 1.34750i 0.738958 + 0.673751i \(0.235318\pi\)
−0.738958 + 0.673751i \(0.764682\pi\)
\(24\) 0 0
\(25\) −3075.00 + 556.776i −0.984000 + 0.178168i
\(26\) 0 0
\(27\) 4031.06i 1.06417i
\(28\) 0 0
\(29\) −7854.00 −1.73419 −0.867093 0.498146i \(-0.834015\pi\)
−0.867093 + 0.498146i \(0.834015\pi\)
\(30\) 0 0
\(31\) 2144.00 0.400701 0.200351 0.979724i \(-0.435792\pi\)
0.200351 + 0.979724i \(0.435792\pi\)
\(32\) 0 0
\(33\) 1113.55i 0.178002i
\(34\) 0 0
\(35\) −6820.00 + 612.454i −0.941053 + 0.0845091i
\(36\) 0 0
\(37\) 10400.6i 1.24897i −0.781035 0.624487i \(-0.785308\pi\)
0.781035 0.624487i \(-0.214692\pi\)
\(38\) 0 0
\(39\) −8184.00 −0.861597
\(40\) 0 0
\(41\) −7414.00 −0.688800 −0.344400 0.938823i \(-0.611918\pi\)
−0.344400 + 0.938823i \(0.611918\pi\)
\(42\) 0 0
\(43\) 17761.2i 1.46487i 0.680835 + 0.732437i \(0.261617\pi\)
−0.680835 + 0.732437i \(0.738383\pi\)
\(44\) 0 0
\(45\) 595.000 + 6625.64i 0.0438012 + 0.487749i
\(46\) 0 0
\(47\) 9431.79i 0.622801i 0.950279 + 0.311401i \(0.100798\pi\)
−0.950279 + 0.311401i \(0.899202\pi\)
\(48\) 0 0
\(49\) 1803.00 0.107277
\(50\) 0 0
\(51\) 10912.0 0.587461
\(52\) 0 0
\(53\) 24253.2i 1.18598i 0.805208 + 0.592992i \(0.202054\pi\)
−0.805208 + 0.592992i \(0.797946\pi\)
\(54\) 0 0
\(55\) −500.000 5567.76i −0.0222876 0.248184i
\(56\) 0 0
\(57\) 24988.1i 1.01870i
\(58\) 0 0
\(59\) −25972.0 −0.971349 −0.485675 0.874140i \(-0.661426\pi\)
−0.485675 + 0.874140i \(0.661426\pi\)
\(60\) 0 0
\(61\) 3058.00 0.105224 0.0526118 0.998615i \(-0.483245\pi\)
0.0526118 + 0.998615i \(0.483245\pi\)
\(62\) 0 0
\(63\) 14576.4i 0.462700i
\(64\) 0 0
\(65\) 40920.0 3674.72i 1.20130 0.107880i
\(66\) 0 0
\(67\) 58784.5i 1.59984i 0.600109 + 0.799918i \(0.295124\pi\)
−0.600109 + 0.799918i \(0.704876\pi\)
\(68\) 0 0
\(69\) 38068.0 0.962581
\(70\) 0 0
\(71\) −37608.0 −0.885389 −0.442695 0.896672i \(-0.645977\pi\)
−0.442695 + 0.896672i \(0.645977\pi\)
\(72\) 0 0
\(73\) 24008.2i 0.527294i 0.964619 + 0.263647i \(0.0849253\pi\)
−0.964619 + 0.263647i \(0.915075\pi\)
\(74\) 0 0
\(75\) 6200.00 + 34241.8i 0.127274 + 0.702915i
\(76\) 0 0
\(77\) 12249.1i 0.235438i
\(78\) 0 0
\(79\) 79728.0 1.43729 0.718643 0.695379i \(-0.244764\pi\)
0.718643 + 0.695379i \(0.244764\pi\)
\(80\) 0 0
\(81\) −15971.0 −0.270470
\(82\) 0 0
\(83\) 16291.3i 0.259573i 0.991542 + 0.129787i \(0.0414292\pi\)
−0.991542 + 0.129787i \(0.958571\pi\)
\(84\) 0 0
\(85\) −54560.0 + 4899.63i −0.819081 + 0.0735557i
\(86\) 0 0
\(87\) 87458.4i 1.23881i
\(88\) 0 0
\(89\) 826.000 0.0110536 0.00552682 0.999985i \(-0.498241\pi\)
0.00552682 + 0.999985i \(0.498241\pi\)
\(90\) 0 0
\(91\) 90024.0 1.13961
\(92\) 0 0
\(93\) 23874.6i 0.286239i
\(94\) 0 0
\(95\) 11220.0 + 124941.i 0.127551 + 1.42035i
\(96\) 0 0
\(97\) 37593.5i 0.405680i −0.979212 0.202840i \(-0.934983\pi\)
0.979212 0.202840i \(-0.0650172\pi\)
\(98\) 0 0
\(99\) −11900.0 −0.122028
\(100\) 0 0
\(101\) 143594. 1.40066 0.700330 0.713819i \(-0.253036\pi\)
0.700330 + 0.713819i \(0.253036\pi\)
\(102\) 0 0
\(103\) 111834.i 1.03868i 0.854568 + 0.519339i \(0.173822\pi\)
−0.854568 + 0.519339i \(0.826178\pi\)
\(104\) 0 0
\(105\) 6820.00 + 75944.3i 0.0603686 + 0.672236i
\(106\) 0 0
\(107\) 92235.6i 0.778824i 0.921064 + 0.389412i \(0.127322\pi\)
−0.921064 + 0.389412i \(0.872678\pi\)
\(108\) 0 0
\(109\) −106238. −0.856473 −0.428236 0.903667i \(-0.640865\pi\)
−0.428236 + 0.903667i \(0.640865\pi\)
\(110\) 0 0
\(111\) −115816. −0.892198
\(112\) 0 0
\(113\) 113048.i 0.832849i 0.909170 + 0.416425i \(0.136717\pi\)
−0.909170 + 0.416425i \(0.863283\pi\)
\(114\) 0 0
\(115\) −190340. + 17093.0i −1.34210 + 0.120524i
\(116\) 0 0
\(117\) 87458.4i 0.590659i
\(118\) 0 0
\(119\) −120032. −0.777015
\(120\) 0 0
\(121\) −151051. −0.937908
\(122\) 0 0
\(123\) 82558.8i 0.492040i
\(124\) 0 0
\(125\) −46375.0 168425.i −0.265466 0.964120i
\(126\) 0 0
\(127\) 51568.6i 0.283711i 0.989887 + 0.141856i \(0.0453069\pi\)
−0.989887 + 0.141856i \(0.954693\pi\)
\(128\) 0 0
\(129\) 197780. 1.04642
\(130\) 0 0
\(131\) −89100.0 −0.453628 −0.226814 0.973938i \(-0.572831\pi\)
−0.226814 + 0.973938i \(0.572831\pi\)
\(132\) 0 0
\(133\) 274869.i 1.34740i
\(134\) 0 0
\(135\) 224440. 20155.3i 1.05990 0.0951820i
\(136\) 0 0
\(137\) 38350.8i 0.174571i −0.996183 0.0872856i \(-0.972181\pi\)
0.996183 0.0872856i \(-0.0278193\pi\)
\(138\) 0 0
\(139\) 134684. 0.591261 0.295630 0.955302i \(-0.404470\pi\)
0.295630 + 0.955302i \(0.404470\pi\)
\(140\) 0 0
\(141\) 105028. 0.444895
\(142\) 0 0
\(143\) 73494.5i 0.300549i
\(144\) 0 0
\(145\) −39270.0 437292.i −0.155110 1.72724i
\(146\) 0 0
\(147\) 20077.4i 0.0766325i
\(148\) 0 0
\(149\) −248006. −0.915159 −0.457579 0.889169i \(-0.651283\pi\)
−0.457579 + 0.889169i \(0.651283\pi\)
\(150\) 0 0
\(151\) −313720. −1.11970 −0.559848 0.828596i \(-0.689140\pi\)
−0.559848 + 0.828596i \(0.689140\pi\)
\(152\) 0 0
\(153\) 116611.i 0.402728i
\(154\) 0 0
\(155\) 10720.0 + 119373.i 0.0358398 + 0.399095i
\(156\) 0 0
\(157\) 245583.i 0.795150i 0.917570 + 0.397575i \(0.130148\pi\)
−0.917570 + 0.397575i \(0.869852\pi\)
\(158\) 0 0
\(159\) 270072. 0.847202
\(160\) 0 0
\(161\) −418748. −1.27317
\(162\) 0 0
\(163\) 397483.i 1.17179i 0.810388 + 0.585894i \(0.199257\pi\)
−0.810388 + 0.585894i \(0.800743\pi\)
\(164\) 0 0
\(165\) −62000.0 + 5567.76i −0.177289 + 0.0159210i
\(166\) 0 0
\(167\) 189983.i 0.527138i −0.964641 0.263569i \(-0.915100\pi\)
0.964641 0.263569i \(-0.0848996\pi\)
\(168\) 0 0
\(169\) −168851. −0.454765
\(170\) 0 0
\(171\) 267036. 0.698360
\(172\) 0 0
\(173\) 81088.9i 0.205990i −0.994682 0.102995i \(-0.967157\pi\)
0.994682 0.102995i \(-0.0328426\pi\)
\(174\) 0 0
\(175\) −68200.0 376659.i −0.168341 0.929723i
\(176\) 0 0
\(177\) 289212.i 0.693878i
\(178\) 0 0
\(179\) −142108. −0.331502 −0.165751 0.986168i \(-0.553005\pi\)
−0.165751 + 0.986168i \(0.553005\pi\)
\(180\) 0 0
\(181\) −250790. −0.569002 −0.284501 0.958676i \(-0.591828\pi\)
−0.284501 + 0.958676i \(0.591828\pi\)
\(182\) 0 0
\(183\) 34052.4i 0.0751659i
\(184\) 0 0
\(185\) 579080. 52002.9i 1.24397 0.111712i
\(186\) 0 0
\(187\) 97992.7i 0.204922i
\(188\) 0 0
\(189\) 493768. 1.00547
\(190\) 0 0
\(191\) 209472. 0.415473 0.207736 0.978185i \(-0.433390\pi\)
0.207736 + 0.978185i \(0.433390\pi\)
\(192\) 0 0
\(193\) 356693.i 0.689289i −0.938733 0.344645i \(-0.887999\pi\)
0.938733 0.344645i \(-0.112001\pi\)
\(194\) 0 0
\(195\) −40920.0 455666.i −0.0770636 0.858144i
\(196\) 0 0
\(197\) 86478.5i 0.158761i −0.996844 0.0793803i \(-0.974706\pi\)
0.996844 0.0793803i \(-0.0252941\pi\)
\(198\) 0 0
\(199\) 749208. 1.34113 0.670563 0.741852i \(-0.266053\pi\)
0.670563 + 0.741852i \(0.266053\pi\)
\(200\) 0 0
\(201\) 654596. 1.14283
\(202\) 0 0
\(203\) 962043.i 1.63853i
\(204\) 0 0
\(205\) −37070.0 412794.i −0.0616081 0.686039i
\(206\) 0 0
\(207\) 406814.i 0.659888i
\(208\) 0 0
\(209\) −224400. −0.355351
\(210\) 0 0
\(211\) 287364. 0.444351 0.222176 0.975007i \(-0.428684\pi\)
0.222176 + 0.975007i \(0.428684\pi\)
\(212\) 0 0
\(213\) 418785.i 0.632473i
\(214\) 0 0
\(215\) −988900. + 88805.8i −1.45900 + 0.131022i
\(216\) 0 0
\(217\) 262620.i 0.378599i
\(218\) 0 0
\(219\) 267344. 0.376669
\(220\) 0 0
\(221\) 720192. 0.991899
\(222\) 0 0
\(223\) 1.18866e6i 1.60065i −0.599567 0.800325i \(-0.704660\pi\)
0.599567 0.800325i \(-0.295340\pi\)
\(224\) 0 0
\(225\) −365925. + 66256.4i −0.481877 + 0.0872512i
\(226\) 0 0
\(227\) 978334.i 1.26015i −0.776534 0.630075i \(-0.783024\pi\)
0.776534 0.630075i \(-0.216976\pi\)
\(228\) 0 0
\(229\) 506474. 0.638217 0.319109 0.947718i \(-0.396617\pi\)
0.319109 + 0.947718i \(0.396617\pi\)
\(230\) 0 0
\(231\) −136400. −0.168184
\(232\) 0 0
\(233\) 1.55465e6i 1.87605i 0.346571 + 0.938024i \(0.387346\pi\)
−0.346571 + 0.938024i \(0.612654\pi\)
\(234\) 0 0
\(235\) −525140. + 47159.0i −0.620305 + 0.0557051i
\(236\) 0 0
\(237\) 887813.i 1.02672i
\(238\) 0 0
\(239\) −374704. −0.424320 −0.212160 0.977235i \(-0.568050\pi\)
−0.212160 + 0.977235i \(0.568050\pi\)
\(240\) 0 0
\(241\) 843634. 0.935646 0.467823 0.883822i \(-0.345039\pi\)
0.467823 + 0.883822i \(0.345039\pi\)
\(242\) 0 0
\(243\) 801702.i 0.870959i
\(244\) 0 0
\(245\) 9015.00 + 100387.i 0.00959512 + 0.106847i
\(246\) 0 0
\(247\) 1.64922e6i 1.72003i
\(248\) 0 0
\(249\) 181412. 0.185425
\(250\) 0 0
\(251\) −1.72050e6 −1.72373 −0.861867 0.507134i \(-0.830705\pi\)
−0.861867 + 0.507134i \(0.830705\pi\)
\(252\) 0 0
\(253\) 341861.i 0.335775i
\(254\) 0 0
\(255\) 54560.0 + 607554.i 0.0525441 + 0.585106i
\(256\) 0 0
\(257\) 1.55220e6i 1.46594i −0.680262 0.732969i \(-0.738134\pi\)
0.680262 0.732969i \(-0.261866\pi\)
\(258\) 0 0
\(259\) 1.27398e6 1.18008
\(260\) 0 0
\(261\) −934626. −0.849252
\(262\) 0 0
\(263\) 407772.i 0.363520i 0.983343 + 0.181760i \(0.0581793\pi\)
−0.983343 + 0.181760i \(0.941821\pi\)
\(264\) 0 0
\(265\) −1.35036e6 + 121266.i −1.18123 + 0.106078i
\(266\) 0 0
\(267\) 9197.95i 0.00789610i
\(268\) 0 0
\(269\) −1.82710e6 −1.53951 −0.769754 0.638340i \(-0.779621\pi\)
−0.769754 + 0.638340i \(0.779621\pi\)
\(270\) 0 0
\(271\) 616880. 0.510243 0.255122 0.966909i \(-0.417884\pi\)
0.255122 + 0.966909i \(0.417884\pi\)
\(272\) 0 0
\(273\) 1.00246e6i 0.814071i
\(274\) 0 0
\(275\) 307500. 55677.6i 0.245196 0.0443965i
\(276\) 0 0
\(277\) 1.83712e6i 1.43859i 0.694704 + 0.719296i \(0.255535\pi\)
−0.694704 + 0.719296i \(0.744465\pi\)
\(278\) 0 0
\(279\) 255136. 0.196228
\(280\) 0 0
\(281\) −1.22093e6 −0.922415 −0.461208 0.887292i \(-0.652584\pi\)
−0.461208 + 0.887292i \(0.652584\pi\)
\(282\) 0 0
\(283\) 688766.i 0.511217i −0.966780 0.255609i \(-0.917724\pi\)
0.966780 0.255609i \(-0.0822758\pi\)
\(284\) 0 0
\(285\) 1.39128e6 124941.i 1.01462 0.0911154i
\(286\) 0 0
\(287\) 908147.i 0.650806i
\(288\) 0 0
\(289\) 459601. 0.323695
\(290\) 0 0
\(291\) −418624. −0.289796
\(292\) 0 0
\(293\) 856211.i 0.582655i 0.956623 + 0.291328i \(0.0940970\pi\)
−0.956623 + 0.291328i \(0.905903\pi\)
\(294\) 0 0
\(295\) −129860. 1.44606e6i −0.0868801 0.967456i
\(296\) 0 0
\(297\) 403106.i 0.265172i
\(298\) 0 0
\(299\) 2.51249e6 1.62527
\(300\) 0 0
\(301\) −2.17558e6 −1.38407
\(302\) 0 0
\(303\) 1.59900e6i 1.00055i
\(304\) 0 0
\(305\) 15290.0 + 170262.i 0.00941148 + 0.104802i
\(306\) 0 0
\(307\) 1.09617e6i 0.663792i 0.943316 + 0.331896i \(0.107688\pi\)
−0.943316 + 0.331896i \(0.892312\pi\)
\(308\) 0 0
\(309\) 1.24533e6 0.741974
\(310\) 0 0
\(311\) 2.12465e6 1.24562 0.622811 0.782373i \(-0.285991\pi\)
0.622811 + 0.782373i \(0.285991\pi\)
\(312\) 0 0
\(313\) 294824.i 0.170099i −0.996377 0.0850496i \(-0.972895\pi\)
0.996377 0.0850496i \(-0.0271049\pi\)
\(314\) 0 0
\(315\) −811580. + 72882.0i −0.460845 + 0.0413851i
\(316\) 0 0
\(317\) 2.53153e6i 1.41493i −0.706749 0.707465i \(-0.749839\pi\)
0.706749 0.707465i \(-0.250161\pi\)
\(318\) 0 0
\(319\) 785400. 0.432130
\(320\) 0 0
\(321\) 1.02709e6 0.556348
\(322\) 0 0
\(323\) 2.19896e6i 1.17276i
\(324\) 0 0
\(325\) 409200. + 2.25996e6i 0.214895 + 1.18684i
\(326\) 0 0
\(327\) 1.18302e6i 0.611817i
\(328\) 0 0
\(329\) −1.15531e6 −0.588448
\(330\) 0 0
\(331\) −1.17021e6 −0.587076 −0.293538 0.955947i \(-0.594833\pi\)
−0.293538 + 0.955947i \(0.594833\pi\)
\(332\) 0 0
\(333\) 1.23767e6i 0.611637i
\(334\) 0 0
\(335\) −3.27298e6 + 293922.i −1.59342 + 0.143094i
\(336\) 0 0
\(337\) 1.86872e6i 0.896333i −0.893950 0.448167i \(-0.852077\pi\)
0.893950 0.448167i \(-0.147923\pi\)
\(338\) 0 0
\(339\) 1.25885e6 0.594941
\(340\) 0 0
\(341\) −214400. −0.0998479
\(342\) 0 0
\(343\) 2.27955e6i 1.04620i
\(344\) 0 0
\(345\) 190340. + 2.11954e6i 0.0860959 + 0.958723i
\(346\) 0 0
\(347\) 1.63342e6i 0.728237i −0.931353 0.364119i \(-0.881370\pi\)
0.931353 0.364119i \(-0.118630\pi\)
\(348\) 0 0
\(349\) 2.00629e6 0.881719 0.440859 0.897576i \(-0.354674\pi\)
0.440859 + 0.897576i \(0.354674\pi\)
\(350\) 0 0
\(351\) −2.96261e6 −1.28353
\(352\) 0 0
\(353\) 1.80859e6i 0.772508i −0.922392 0.386254i \(-0.873769\pi\)
0.922392 0.386254i \(-0.126231\pi\)
\(354\) 0 0
\(355\) −188040. 2.09392e6i −0.0791916 0.881841i
\(356\) 0 0
\(357\) 1.33662e6i 0.555057i
\(358\) 0 0
\(359\) 4.50674e6 1.84555 0.922777 0.385334i \(-0.125914\pi\)
0.922777 + 0.385334i \(0.125914\pi\)
\(360\) 0 0
\(361\) 2.55944e6 1.03366
\(362\) 0 0
\(363\) 1.68203e6i 0.669989i
\(364\) 0 0
\(365\) −1.33672e6 + 120041.i −0.525180 + 0.0471626i
\(366\) 0 0
\(367\) 3.02796e6i 1.17351i 0.809766 + 0.586753i \(0.199594\pi\)
−0.809766 + 0.586753i \(0.800406\pi\)
\(368\) 0 0
\(369\) −882266. −0.337313
\(370\) 0 0
\(371\) −2.97079e6 −1.12057
\(372\) 0 0
\(373\) 1.16342e6i 0.432976i −0.976285 0.216488i \(-0.930540\pi\)
0.976285 0.216488i \(-0.0694602\pi\)
\(374\) 0 0
\(375\) −1.87550e6 + 516410.i −0.688714 + 0.189634i
\(376\) 0 0
\(377\) 5.77226e6i 2.09167i
\(378\) 0 0
\(379\) −832052. −0.297545 −0.148772 0.988871i \(-0.547532\pi\)
−0.148772 + 0.988871i \(0.547532\pi\)
\(380\) 0 0
\(381\) 574244. 0.202667
\(382\) 0 0
\(383\) 2.86948e6i 0.999554i 0.866154 + 0.499777i \(0.166585\pi\)
−0.866154 + 0.499777i \(0.833415\pi\)
\(384\) 0 0
\(385\) 682000. 61245.4i 0.234494 0.0210582i
\(386\) 0 0
\(387\) 2.11358e6i 0.717366i
\(388\) 0 0
\(389\) −311926. −0.104515 −0.0522574 0.998634i \(-0.516642\pi\)
−0.0522574 + 0.998634i \(0.516642\pi\)
\(390\) 0 0
\(391\) −3.34998e6 −1.10816
\(392\) 0 0
\(393\) 992176.i 0.324046i
\(394\) 0 0
\(395\) 398640. + 4.43907e6i 0.128555 + 1.43153i
\(396\) 0 0
\(397\) 2.95619e6i 0.941362i −0.882304 0.470681i \(-0.844008\pi\)
0.882304 0.470681i \(-0.155992\pi\)
\(398\) 0 0
\(399\) 3.06082e6 0.962509
\(400\) 0 0
\(401\) 2770.00 0.000860238 0.000430119 1.00000i \(-0.499863\pi\)
0.000430119 1.00000i \(0.499863\pi\)
\(402\) 0 0
\(403\) 1.57572e6i 0.483300i
\(404\) 0 0
\(405\) −79855.0 889228.i −0.0241916 0.269386i
\(406\) 0 0
\(407\) 1.04006e6i 0.311223i
\(408\) 0 0
\(409\) −1.97985e6 −0.585225 −0.292613 0.956231i \(-0.594525\pi\)
−0.292613 + 0.956231i \(0.594525\pi\)
\(410\) 0 0
\(411\) −427056. −0.124704
\(412\) 0 0
\(413\) 3.18133e6i 0.917770i
\(414\) 0 0
\(415\) −907060. + 81456.4i −0.258533 + 0.0232169i
\(416\) 0 0
\(417\) 1.49978e6i 0.422364i
\(418\) 0 0
\(419\) 5.10120e6 1.41951 0.709754 0.704450i \(-0.248806\pi\)
0.709754 + 0.704450i \(0.248806\pi\)
\(420\) 0 0
\(421\) 2.43223e6 0.668806 0.334403 0.942430i \(-0.391465\pi\)
0.334403 + 0.942430i \(0.391465\pi\)
\(422\) 0 0
\(423\) 1.12238e6i 0.304993i
\(424\) 0 0
\(425\) −545600. 3.01327e6i −0.146522 0.809219i
\(426\) 0 0
\(427\) 374577.i 0.0994194i
\(428\) 0 0
\(429\) 818400. 0.214695
\(430\) 0 0
\(431\) −918896. −0.238272 −0.119136 0.992878i \(-0.538012\pi\)
−0.119136 + 0.992878i \(0.538012\pi\)
\(432\) 0 0
\(433\) 2.05455e6i 0.526619i −0.964711 0.263310i \(-0.915186\pi\)
0.964711 0.263310i \(-0.0848141\pi\)
\(434\) 0 0
\(435\) −4.86948e6 + 437292.i −1.23384 + 0.110802i
\(436\) 0 0
\(437\) 7.67135e6i 1.92162i
\(438\) 0 0
\(439\) −676632. −0.167568 −0.0837840 0.996484i \(-0.526701\pi\)
−0.0837840 + 0.996484i \(0.526701\pi\)
\(440\) 0 0
\(441\) 214557. 0.0525347
\(442\) 0 0
\(443\) 2.53092e6i 0.612729i 0.951914 + 0.306365i \(0.0991126\pi\)
−0.951914 + 0.306365i \(0.900887\pi\)
\(444\) 0 0
\(445\) 4130.00 + 45989.7i 0.000988667 + 0.0110093i
\(446\) 0 0
\(447\) 2.76168e6i 0.653739i
\(448\) 0 0
\(449\) 5.17619e6 1.21170 0.605849 0.795579i \(-0.292833\pi\)
0.605849 + 0.795579i \(0.292833\pi\)
\(450\) 0 0
\(451\) 741400. 0.171637
\(452\) 0 0
\(453\) 3.49344e6i 0.799848i
\(454\) 0 0
\(455\) 450120. + 5.01232e6i 0.101929 + 1.13504i
\(456\) 0 0
\(457\) 3.11274e6i 0.697191i 0.937273 + 0.348596i \(0.113341\pi\)
−0.937273 + 0.348596i \(0.886659\pi\)
\(458\) 0 0
\(459\) 3.95014e6 0.875147
\(460\) 0 0
\(461\) 2.64957e6 0.580662 0.290331 0.956926i \(-0.406235\pi\)
0.290331 + 0.956926i \(0.406235\pi\)
\(462\) 0 0
\(463\) 2.59165e6i 0.561854i −0.959729 0.280927i \(-0.909358\pi\)
0.959729 0.280927i \(-0.0906420\pi\)
\(464\) 0 0
\(465\) 1.32928e6 119373.i 0.285091 0.0256020i
\(466\) 0 0
\(467\) 6.62135e6i 1.40493i 0.711719 + 0.702465i \(0.247917\pi\)
−0.711719 + 0.702465i \(0.752083\pi\)
\(468\) 0 0
\(469\) −7.20056e6 −1.51159
\(470\) 0 0
\(471\) 2.73470e6 0.568011
\(472\) 0 0
\(473\) 1.77612e6i 0.365022i
\(474\) 0 0
\(475\) −6.90030e6 + 1.24941e6i −1.40325 + 0.254080i
\(476\) 0 0
\(477\) 2.88613e6i 0.580791i
\(478\) 0 0
\(479\) 6.89322e6 1.37272 0.686362 0.727260i \(-0.259207\pi\)
0.686362 + 0.727260i \(0.259207\pi\)
\(480\) 0 0
\(481\) −7.64386e6 −1.50643
\(482\) 0 0
\(483\) 4.66298e6i 0.909485i
\(484\) 0 0
\(485\) 2.09312e6 187968.i 0.404054 0.0362852i
\(486\) 0 0
\(487\) 5.65370e6i 1.08021i −0.841596 0.540107i \(-0.818384\pi\)
0.841596 0.540107i \(-0.181616\pi\)
\(488\) 0 0
\(489\) 4.42618e6 0.837061
\(490\) 0 0
\(491\) 5.88390e6 1.10144 0.550721 0.834689i \(-0.314353\pi\)
0.550721 + 0.834689i \(0.314353\pi\)
\(492\) 0 0
\(493\) 7.69634e6i 1.42616i
\(494\) 0 0
\(495\) −59500.0 662564.i −0.0109145 0.121539i
\(496\) 0 0
\(497\) 4.60663e6i 0.836552i
\(498\) 0 0
\(499\) 6.72080e6 1.20829 0.604143 0.796876i \(-0.293515\pi\)
0.604143 + 0.796876i \(0.293515\pi\)
\(500\) 0 0
\(501\) −2.11556e6 −0.376558
\(502\) 0 0
\(503\) 469262.i 0.0826981i −0.999145 0.0413491i \(-0.986834\pi\)
0.999145 0.0413491i \(-0.0131656\pi\)
\(504\) 0 0
\(505\) 717970. + 7.99498e6i 0.125279 + 1.39505i
\(506\) 0 0
\(507\) 1.88025e6i 0.324859i
\(508\) 0 0
\(509\) −294414. −0.0503691 −0.0251845 0.999683i \(-0.508017\pi\)
−0.0251845 + 0.999683i \(0.508017\pi\)
\(510\) 0 0
\(511\) −2.94078e6 −0.498208
\(512\) 0 0
\(513\) 9.04570e6i 1.51757i
\(514\) 0 0
\(515\) −6.22666e6 + 559171.i −1.03452 + 0.0929023i
\(516\) 0 0
\(517\) 943179.i 0.155191i
\(518\) 0 0
\(519\) −902968. −0.147148
\(520\) 0 0
\(521\) −7.10025e6 −1.14599 −0.572993 0.819560i \(-0.694218\pi\)
−0.572993 + 0.819560i \(0.694218\pi\)
\(522\) 0 0
\(523\) 5.96567e6i 0.953685i −0.878989 0.476843i \(-0.841781\pi\)
0.878989 0.476843i \(-0.158219\pi\)
\(524\) 0 0
\(525\) −4.19430e6 + 759443.i −0.664142 + 0.120253i
\(526\) 0 0
\(527\) 2.10096e6i 0.329528i
\(528\) 0 0
\(529\) −5.25053e6 −0.815763
\(530\) 0 0
\(531\) −3.09067e6 −0.475681
\(532\) 0 0
\(533\) 5.44888e6i 0.830786i
\(534\) 0 0
\(535\) −5.13546e6 + 461178.i −0.775702 + 0.0696601i
\(536\) 0 0
\(537\) 1.58245e6i 0.236807i
\(538\) 0 0
\(539\) −180300. −0.0267315
\(540\) 0 0
\(541\) 2.72367e6 0.400093 0.200046 0.979786i \(-0.435891\pi\)
0.200046 + 0.979786i \(0.435891\pi\)
\(542\) 0 0
\(543\) 2.79268e6i 0.406463i
\(544\) 0 0
\(545\) −531190. 5.91508e6i −0.0766053 0.853040i
\(546\) 0 0
\(547\) 9.22148e6i 1.31775i −0.752254 0.658874i \(-0.771033\pi\)
0.752254 0.658874i \(-0.228967\pi\)
\(548\) 0 0
\(549\) 363902. 0.0515292
\(550\) 0 0
\(551\) −1.76244e7 −2.47306
\(552\) 0 0
\(553\) 9.76595e6i 1.35801i
\(554\) 0 0
\(555\) −579080. 6.44836e6i −0.0798006 0.888622i
\(556\) 0 0
\(557\) 3.42852e6i 0.468240i −0.972208 0.234120i \(-0.924779\pi\)
0.972208 0.234120i \(-0.0752209\pi\)
\(558\) 0 0
\(559\) 1.30535e7 1.76684
\(560\) 0 0
\(561\) −1.09120e6 −0.146385
\(562\) 0 0
\(563\) 5.77899e6i 0.768389i −0.923252 0.384195i \(-0.874479\pi\)
0.923252 0.384195i \(-0.125521\pi\)
\(564\) 0 0
\(565\) −6.29424e6 + 565239.i −0.829511 + 0.0744923i
\(566\) 0 0
\(567\) 1.95630e6i 0.255551i
\(568\) 0 0
\(569\) 3.89257e6 0.504029 0.252015 0.967723i \(-0.418907\pi\)
0.252015 + 0.967723i \(0.418907\pi\)
\(570\) 0 0
\(571\) −5.06277e6 −0.649828 −0.324914 0.945744i \(-0.605335\pi\)
−0.324914 + 0.945744i \(0.605335\pi\)
\(572\) 0 0
\(573\) 2.33258e6i 0.296791i
\(574\) 0 0
\(575\) −1.90340e6 1.05122e7i −0.240082 1.32594i
\(576\) 0 0
\(577\) 3.30075e6i 0.412737i 0.978474 + 0.206368i \(0.0661645\pi\)
−0.978474 + 0.206368i \(0.933836\pi\)
\(578\) 0 0
\(579\) −3.97197e6 −0.492390
\(580\) 0 0
\(581\) −1.99553e6 −0.245255
\(582\) 0 0
\(583\) 2.42532e6i 0.295527i
\(584\) 0 0
\(585\) 4.86948e6 437292.i 0.588292 0.0528302i
\(586\) 0 0
\(587\) 5.16997e6i 0.619288i −0.950853 0.309644i \(-0.899790\pi\)
0.950853 0.309644i \(-0.100210\pi\)
\(588\) 0 0
\(589\) 4.81114e6 0.571425
\(590\) 0 0
\(591\) −962984. −0.113410
\(592\) 0 0
\(593\) 1.58484e7i 1.85075i 0.379055 + 0.925374i \(0.376249\pi\)
−0.379055 + 0.925374i \(0.623751\pi\)
\(594\) 0 0
\(595\) −600160. 6.68310e6i −0.0694984 0.773901i
\(596\) 0 0
\(597\) 8.34283e6i 0.958026i
\(598\) 0 0
\(599\) 1.66146e7 1.89201 0.946004 0.324156i \(-0.105080\pi\)
0.946004 + 0.324156i \(0.105080\pi\)
\(600\) 0 0
\(601\) 7.88249e6 0.890179 0.445089 0.895486i \(-0.353172\pi\)
0.445089 + 0.895486i \(0.353172\pi\)
\(602\) 0 0
\(603\) 6.99535e6i 0.783459i
\(604\) 0 0
\(605\) −755255. 8.41016e6i −0.0838890 0.934149i
\(606\) 0 0
\(607\) 782594.i 0.0862114i 0.999071 + 0.0431057i \(0.0137252\pi\)
−0.999071 + 0.0431057i \(0.986275\pi\)
\(608\) 0 0
\(609\) −1.07129e7 −1.17047
\(610\) 0 0
\(611\) 6.93185e6 0.751183
\(612\) 0 0
\(613\) 2.41233e6i 0.259290i 0.991560 + 0.129645i \(0.0413838\pi\)
−0.991560 + 0.129645i \(0.958616\pi\)
\(614\) 0 0
\(615\) −4.59668e6 + 412794.i −0.490068 + 0.0440094i
\(616\) 0 0
\(617\) 9.66355e6i 1.02194i −0.859600 0.510968i \(-0.829287\pi\)
0.859600 0.510968i \(-0.170713\pi\)
\(618\) 0 0
\(619\) 1.80036e7 1.88857 0.944283 0.329134i \(-0.106757\pi\)
0.944283 + 0.329134i \(0.106757\pi\)
\(620\) 0 0
\(621\) 1.37806e7 1.43397
\(622\) 0 0
\(623\) 101177.i 0.0104439i
\(624\) 0 0
\(625\) 9.14562e6 3.42418e6i 0.936512 0.350636i
\(626\) 0 0
\(627\) 2.49881e6i 0.253843i
\(628\) 0 0
\(629\) 1.01918e7 1.02713
\(630\) 0 0
\(631\) 4.80081e6 0.480000 0.240000 0.970773i \(-0.422853\pi\)
0.240000 + 0.970773i \(0.422853\pi\)
\(632\) 0 0
\(633\) 3.19995e6i 0.317420i
\(634\) 0 0
\(635\) −2.87122e6 + 257843.i −0.282574 + 0.0253759i
\(636\) 0 0
\(637\) 1.32511e6i 0.129390i
\(638\) 0 0
\(639\) −4.47535e6 −0.433586
\(640\) 0 0
\(641\) 1.44950e7 1.39340 0.696698 0.717365i \(-0.254652\pi\)
0.696698 + 0.717365i \(0.254652\pi\)
\(642\) 0 0
\(643\) 1.82430e7i 1.74008i −0.492979 0.870041i \(-0.664092\pi\)
0.492979 0.870041i \(-0.335908\pi\)
\(644\) 0 0
\(645\) 988900. + 1.10119e7i 0.0935951 + 1.04223i
\(646\) 0 0
\(647\) 9.64592e6i 0.905905i −0.891535 0.452953i \(-0.850371\pi\)
0.891535 0.452953i \(-0.149629\pi\)
\(648\) 0 0
\(649\) 2.59720e6 0.242044
\(650\) 0 0
\(651\) 2.92442e6 0.270450
\(652\) 0 0
\(653\) 1.92807e7i 1.76945i 0.466111 + 0.884726i \(0.345655\pi\)
−0.466111 + 0.884726i \(0.654345\pi\)
\(654\) 0 0
\(655\) −445500. 4.96088e6i −0.0405737 0.451809i
\(656\) 0 0
\(657\) 2.85698e6i 0.258222i
\(658\) 0 0
\(659\) −9.70592e6 −0.870609 −0.435304 0.900283i \(-0.643359\pi\)
−0.435304 + 0.900283i \(0.643359\pi\)
\(660\) 0 0
\(661\) 4.28396e6 0.381366 0.190683 0.981652i \(-0.438930\pi\)
0.190683 + 0.981652i \(0.438930\pi\)
\(662\) 0 0
\(663\) 8.01972e6i 0.708558i
\(664\) 0 0
\(665\) −1.53041e7 + 1.37435e6i −1.34200 + 0.120515i
\(666\) 0 0
\(667\) 2.68497e7i 2.33682i
\(668\) 0 0
\(669\) −1.32364e7 −1.14342
\(670\) 0 0
\(671\) −305800. −0.0262199
\(672\) 0 0
\(673\) 1.30585e7i 1.11136i −0.831395 0.555681i \(-0.812458\pi\)
0.831395 0.555681i \(-0.187542\pi\)
\(674\) 0 0
\(675\) 2.24440e6 + 1.23955e7i 0.189601 + 1.04714i
\(676\) 0 0
\(677\) 8.42565e6i 0.706532i −0.935523 0.353266i \(-0.885071\pi\)
0.935523 0.353266i \(-0.114929\pi\)
\(678\) 0 0
\(679\) 4.60486e6 0.383303
\(680\) 0 0
\(681\) −1.08943e7 −0.900182
\(682\) 0 0
\(683\) 1.64100e7i 1.34603i 0.739627 + 0.673017i \(0.235002\pi\)
−0.739627 + 0.673017i \(0.764998\pi\)
\(684\) 0 0
\(685\) 2.13528e6 191754.i 0.173872 0.0156141i
\(686\) 0 0
\(687\) 5.63986e6i 0.455907i
\(688\) 0 0
\(689\) 1.78248e7 1.43046
\(690\) 0 0
\(691\) −1.12139e7 −0.893428 −0.446714 0.894677i \(-0.647406\pi\)
−0.446714 + 0.894677i \(0.647406\pi\)
\(692\) 0 0
\(693\) 1.45764e6i 0.115297i
\(694\) 0 0
\(695\) 673420. + 7.49889e6i 0.0528840 + 0.588891i
\(696\) 0 0
\(697\) 7.26518e6i 0.566453i
\(698\) 0 0
\(699\) 1.73119e7 1.34014
\(700\) 0 0
\(701\) −2.04707e7 −1.57339 −0.786696 0.617340i \(-0.788210\pi\)
−0.786696 + 0.617340i \(0.788210\pi\)
\(702\) 0 0
\(703\) 2.33389e7i 1.78112i
\(704\) 0 0
\(705\) 525140. + 5.84771e6i 0.0397926 + 0.443112i
\(706\) 0 0
\(707\) 1.75889e7i 1.32340i
\(708\) 0 0
\(709\) −81654.0 −0.00610045 −0.00305023 0.999995i \(-0.500971\pi\)
−0.00305023 + 0.999995i \(0.500971\pi\)
\(710\) 0 0
\(711\) 9.48763e6 0.703856
\(712\) 0 0
\(713\) 7.32949e6i 0.539946i
\(714\) 0 0
\(715\) −4.09200e6 + 367472.i −0.299344 + 0.0268819i
\(716\) 0 0
\(717\) 4.17253e6i 0.303111i
\(718\) 0 0
\(719\) −8.61006e6 −0.621132 −0.310566 0.950552i \(-0.600519\pi\)
−0.310566 + 0.950552i \(0.600519\pi\)
\(720\) 0 0
\(721\) −1.36987e7 −0.981386
\(722\) 0 0
\(723\) 9.39431e6i 0.668373i
\(724\) 0 0
\(725\) 2.41510e7 4.37292e6i 1.70644 0.308977i
\(726\) 0 0
\(727\) 1.17682e7i 0.825796i −0.910777 0.412898i \(-0.864517\pi\)
0.910777 0.412898i \(-0.135483\pi\)
\(728\) 0 0
\(729\) −1.28083e7 −0.892635
\(730\) 0 0
\(731\) −1.74046e7 −1.20468
\(732\) 0 0
\(733\) 3.93759e6i 0.270689i 0.990799 + 0.135344i \(0.0432141\pi\)
−0.990799 + 0.135344i \(0.956786\pi\)
\(734\) 0 0
\(735\) 1.11786e6 100387.i 0.0763254 0.00685422i
\(736\) 0 0
\(737\) 5.87845e6i 0.398652i
\(738\) 0 0
\(739\) 2.30602e7 1.55329 0.776643 0.629941i \(-0.216921\pi\)
0.776643 + 0.629941i \(0.216921\pi\)
\(740\) 0 0
\(741\) −1.83649e7 −1.22869
\(742\) 0 0
\(743\) 1.72675e6i 0.114751i 0.998353 + 0.0573757i \(0.0182733\pi\)
−0.998353 + 0.0573757i \(0.981727\pi\)
\(744\) 0 0
\(745\) −1.24003e6 1.38084e7i −0.0818543 0.911491i
\(746\) 0 0
\(747\) 1.93866e6i 0.127116i
\(748\) 0 0
\(749\) −1.12980e7 −0.735864
\(750\) 0 0
\(751\) 2.58030e6 0.166944 0.0834720 0.996510i \(-0.473399\pi\)
0.0834720 + 0.996510i \(0.473399\pi\)
\(752\) 0 0
\(753\) 1.91587e7i 1.23134i
\(754\) 0 0
\(755\) −1.56860e6 1.74672e7i −0.100149 1.11521i
\(756\) 0 0
\(757\) 5.75878e6i 0.365251i 0.983183 + 0.182625i \(0.0584595\pi\)
−0.983183 + 0.182625i \(0.941540\pi\)
\(758\) 0 0
\(759\) −3.80680e6 −0.239859
\(760\) 0 0
\(761\) 1.40499e7 0.879450 0.439725 0.898133i \(-0.355076\pi\)
0.439725 + 0.898133i \(0.355076\pi\)
\(762\) 0 0
\(763\) 1.30132e7i 0.809230i
\(764\) 0 0
\(765\) −6.49264e6 + 583056.i −0.401114 + 0.0360211i
\(766\) 0 0
\(767\) 1.90880e7i 1.17158i
\(768\) 0 0
\(769\) 5.59898e6 0.341423 0.170712 0.985321i \(-0.445393\pi\)
0.170712 + 0.985321i \(0.445393\pi\)
\(770\) 0 0
\(771\) −1.72846e7 −1.04719
\(772\) 0 0
\(773\) 6.34625e6i 0.382004i −0.981590 0.191002i \(-0.938826\pi\)
0.981590 0.191002i \(-0.0611738\pi\)
\(774\) 0 0
\(775\) −6.59280e6 + 1.19373e6i −0.394290 + 0.0713923i
\(776\) 0 0
\(777\) 1.41864e7i 0.842984i
\(778\) 0 0
\(779\) −1.66370e7 −0.982272
\(780\) 0 0
\(781\) 3.76080e6 0.220624
\(782\) 0 0
\(783\) 3.16600e7i 1.84547i
\(784\) 0 0
\(785\) −1.36735e7 + 1.22791e6i −0.791963 + 0.0711204i
\(786\) 0 0
\(787\) 1.73688e7i 0.999617i 0.866136 + 0.499809i \(0.166596\pi\)
−0.866136 + 0.499809i \(0.833404\pi\)
\(788\) 0 0
\(789\) 4.54076e6 0.259678
\(790\) 0 0
\(791\) −1.38473e7 −0.786909
\(792\) 0 0
\(793\) 2.24746e6i 0.126914i
\(794\) 0 0
\(795\) 1.35036e6 + 1.50370e7i 0.0757760 + 0.843806i
\(796\) 0 0
\(797\) 8.06932e6i 0.449978i 0.974361 + 0.224989i \(0.0722346\pi\)
−0.974361 + 0.224989i \(0.927765\pi\)
\(798\) 0 0
\(799\) −9.24246e6 −0.512178
\(800\) 0 0
\(801\) 98294.0 0.00541310
\(802\) 0 0
\(803\) 2.40082e6i 0.131393i
\(804\) 0 0
\(805\) −2.09374e6 2.33149e7i −0.113876 1.26807i
\(806\) 0 0
\(807\) 2.03457e7i 1.09974i
\(808\) 0 0
\(809\) −1.94554e7 −1.04513 −0.522564 0.852600i \(-0.675024\pi\)
−0.522564 + 0.852600i \(0.675024\pi\)
\(810\) 0 0
\(811\) −2.85204e6 −0.152266 −0.0761330 0.997098i \(-0.524257\pi\)
−0.0761330 + 0.997098i \(0.524257\pi\)
\(812\) 0 0
\(813\) 6.86928e6i 0.364490i
\(814\) 0 0
\(815\) −2.21309e7 + 1.98741e6i −1.16709 + 0.104808i
\(816\) 0 0
\(817\) 3.98561e7i 2.08900i
\(818\) 0 0
\(819\) 1.07129e7 0.558079
\(820\) 0 0
\(821\) 5.48431e6 0.283965 0.141982 0.989869i \(-0.454652\pi\)
0.141982 + 0.989869i \(0.454652\pi\)
\(822\) 0 0
\(823\) 6.57652e6i 0.338452i −0.985577 0.169226i \(-0.945873\pi\)
0.985577 0.169226i \(-0.0541267\pi\)
\(824\) 0 0
\(825\) −620000. 3.42418e6i −0.0317144 0.175154i
\(826\) 0 0
\(827\) 4.19715e6i 0.213398i 0.994291 + 0.106699i \(0.0340281\pi\)
−0.994291 + 0.106699i \(0.965972\pi\)
\(828\) 0 0
\(829\) 2.78889e7 1.40943 0.704717 0.709488i \(-0.251074\pi\)
0.704717 + 0.709488i \(0.251074\pi\)
\(830\) 0 0
\(831\) 2.04573e7 1.02765
\(832\) 0 0
\(833\) 1.76681e6i 0.0882220i
\(834\) 0 0
\(835\) 1.05778e7 949916.i 0.525025 0.0471486i
\(836\) 0 0
\(837\) 8.64260e6i 0.426413i
\(838\) 0 0
\(839\) 1.27752e7 0.626560 0.313280 0.949661i \(-0.398572\pi\)
0.313280 + 0.949661i \(0.398572\pi\)
\(840\) 0 0
\(841\) 4.11742e7 2.00740
\(842\) 0 0
\(843\) 1.35957e7i 0.658922i
\(844\) 0 0
\(845\) −844255. 9.40123e6i −0.0406754 0.452942i
\(846\) 0 0
\(847\) 1.85024e7i 0.886173i
\(848\) 0 0
\(849\) −7.66977e6 −0.365185
\(850\) 0 0
\(851\) 3.55555e7 1.68300
\(852\) 0 0
\(853\) 5.37122e6i 0.252755i −0.991982 0.126378i \(-0.959665\pi\)
0.991982 0.126378i \(-0.0403351\pi\)
\(854\) 0 0
\(855\) 1.33518e6 + 1.48679e7i 0.0624632 + 0.695561i
\(856\) 0 0
\(857\) 1.61585e7i 0.751535i 0.926714 + 0.375767i \(0.122621\pi\)
−0.926714 + 0.375767i \(0.877379\pi\)
\(858\) 0 0
\(859\) −3.34643e7 −1.54739 −0.773693 0.633561i \(-0.781593\pi\)
−0.773693 + 0.633561i \(0.781593\pi\)
\(860\) 0 0
\(861\) −1.01127e7 −0.464899
\(862\) 0 0
\(863\) 2.51788e7i 1.15082i 0.817864 + 0.575412i \(0.195158\pi\)
−0.817864 + 0.575412i \(0.804842\pi\)
\(864\) 0 0
\(865\) 4.51484e6 405445.i 0.205164 0.0184243i
\(866\) 0 0
\(867\) 5.11790e6i 0.231230i
\(868\) 0 0
\(869\) −7.97280e6 −0.358147
\(870\) 0 0
\(871\) 4.32033e7 1.92962
\(872\) 0 0
\(873\) 4.47363e6i 0.198666i
\(874\) 0 0
\(875\) 2.06305e7 5.68051e6i 0.910940 0.250823i
\(876\) 0 0
\(877\) 2.46327e7i 1.08146i 0.841195 + 0.540732i \(0.181853\pi\)
−0.841195 + 0.540732i \(0.818147\pi\)
\(878\) 0 0
\(879\) 9.53436e6 0.416217
\(880\) 0 0
\(881\) −1.10564e7 −0.479926 −0.239963 0.970782i \(-0.577135\pi\)
−0.239963 + 0.970782i \(0.577135\pi\)
\(882\) 0 0
\(883\) 3.74671e7i 1.61714i 0.588398 + 0.808572i \(0.299759\pi\)
−0.588398 + 0.808572i \(0.700241\pi\)
\(884\) 0 0
\(885\) −1.61026e7 + 1.44606e6i −0.691097 + 0.0620623i
\(886\) 0 0
\(887\) 2.96341e7i 1.26469i 0.774689 + 0.632343i \(0.217906\pi\)
−0.774689 + 0.632343i \(0.782094\pi\)
\(888\) 0 0
\(889\) −6.31668e6 −0.268062
\(890\) 0 0
\(891\) 1.59710e6 0.0673966
\(892\) 0 0
\(893\) 2.11649e7i 0.888154i
\(894\) 0 0
\(895\) −710540. 7.91224e6i −0.0296504 0.330173i
\(896\) 0 0
\(897\) 2.79779e7i 1.16100i
\(898\) 0 0
\(899\) −1.68390e7 −0.694891
\(900\) 0 0
\(901\) −2.37663e7 −0.975327
\(902\) 0 0
\(903\) 2.42262e7i 0.988705i
\(904\) 0 0
\(905\) −1.25395e6 1.39634e7i −0.0508931 0.566721i
\(906\) 0 0
\(907\) 1.94430e6i 0.0784773i 0.999230 + 0.0392387i \(0.0124933\pi\)
−0.999230 + 0.0392387i \(0.987507\pi\)
\(908\) 0 0
\(909\) 1.70877e7 0.685920
\(910\) 0 0
\(911\) −1.28064e7 −0.511245 −0.255623 0.966777i \(-0.582280\pi\)
−0.255623 + 0.966777i \(0.582280\pi\)
\(912\) 0 0
\(913\) 1.62913e6i 0.0646812i
\(914\) 0 0
\(915\) 1.89596e6 170262.i 0.0748646 0.00672304i
\(916\) 0 0
\(917\) 1.09139e7i 0.428606i
\(918\) 0 0
\(919\) −3.58404e7 −1.39986 −0.699929 0.714213i \(-0.746785\pi\)
−0.699929 + 0.714213i \(0.746785\pi\)
\(920\) 0 0
\(921\) 1.22064e7 0.474176
\(922\) 0 0
\(923\) 2.76398e7i 1.06790i
\(924\) 0 0
\(925\) 5.79080e6 + 3.19818e7i 0.222528 + 1.22899i
\(926\) 0 0
\(927\) 1.33083e7i 0.508653i
\(928\) 0 0
\(929\) 8.47792e6 0.322292 0.161146 0.986931i \(-0.448481\pi\)
0.161146 + 0.986931i \(0.448481\pi\)
\(930\) 0 0
\(931\) 4.04593e6 0.152983
\(932\) 0 0
\(933\) 2.36591e7i 0.889803i
\(934\) 0 0
\(935\) 5.45600e6 489963.i 0.204101 0.0183288i
\(936\) 0 0
\(937\) 1.60566e7i 0.597454i −0.954339 0.298727i \(-0.903438\pi\)
0.954339 0.298727i \(-0.0965619\pi\)
\(938\) 0 0
\(939\) −3.28302e6 −0.121509
\(940\) 0 0
\(941\) 4.24675e7 1.56344 0.781722 0.623627i \(-0.214342\pi\)
0.781722 + 0.623627i \(0.214342\pi\)
\(942\) 0 0
\(943\) 2.53456e7i 0.928159i
\(944\) 0 0
\(945\) 2.46884e6 + 2.74918e7i 0.0899318 + 1.00144i
\(946\) 0 0
\(947\) 2.78877e7i 1.01050i 0.862972 + 0.505252i \(0.168600\pi\)
−0.862972 + 0.505252i \(0.831400\pi\)
\(948\) 0 0
\(949\) 1.76447e7 0.635988
\(950\) 0 0
\(951\) −2.81899e7 −1.01075
\(952\) 0 0
\(953\) 4.77895e7i 1.70451i −0.523123 0.852257i \(-0.675233\pi\)
0.523123 0.852257i \(-0.324767\pi\)
\(954\) 0 0
\(955\) 1.04736e6 + 1.16629e7i 0.0371610 + 0.413808i
\(956\) 0 0
\(957\) 8.74584e6i 0.308690i
\(958\) 0 0
\(959\) 4.69762e6 0.164942
\(960\) 0 0
\(961\) −2.40324e7 −0.839439
\(962\) 0 0
\(963\) 1.09760e7i 0.381399i
\(964\) 0 0
\(965\) 1.98598e7 1.78347e6i 0.686527 0.0616519i
\(966\) 0 0
\(967\) 4.51588e7i 1.55302i −0.630107 0.776509i \(-0.716989\pi\)
0.630107 0.776509i \(-0.283011\pi\)
\(968\) 0 0
\(969\) 2.44865e7 0.837756
\(970\) 0 0
\(971\) −2.70901e7 −0.922067 −0.461033 0.887383i \(-0.652521\pi\)
−0.461033 + 0.887383i \(0.652521\pi\)
\(972\) 0 0
\(973\) 1.64976e7i 0.558647i
\(974\) 0 0
\(975\) 2.51658e7 4.55666e6i 0.847811 0.153509i
\(976\) 0 0
\(977\) 4.34630e7i 1.45675i 0.685181 + 0.728373i \(0.259723\pi\)
−0.685181 + 0.728373i \(0.740277\pi\)
\(978\) 0 0
\(979\) −82600.0 −0.00275438
\(980\) 0 0
\(981\) −1.26423e7 −0.419425
\(982\) 0 0
\(983\) 6.56037e6i 0.216543i 0.994121 + 0.108272i \(0.0345316\pi\)
−0.994121 + 0.108272i \(0.965468\pi\)
\(984\) 0 0
\(985\) 4.81492e6 432393.i 0.158124 0.0142000i
\(986\) 0 0
\(987\) 1.28650e7i 0.420355i
\(988\) 0 0
\(989\) −6.07185e7 −1.97392
\(990\) 0 0
\(991\) −1.72942e7 −0.559391 −0.279696 0.960089i \(-0.590234\pi\)
−0.279696 + 0.960089i \(0.590234\pi\)
\(992\) 0 0
\(993\) 1.30309e7i 0.419375i
\(994\) 0 0
\(995\) 3.74604e6 + 4.17141e7i 0.119954 + 1.33575i
\(996\) 0 0
\(997\) 2.24107e7i 0.714031i 0.934098 + 0.357015i \(0.116206\pi\)
−0.934098 + 0.357015i \(0.883794\pi\)
\(998\) 0 0
\(999\) −4.19254e7 −1.32912
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 320.6.c.d.129.1 2
4.3 odd 2 320.6.c.e.129.2 2
5.4 even 2 inner 320.6.c.d.129.2 2
8.3 odd 2 20.6.c.a.9.1 2
8.5 even 2 80.6.c.b.49.2 2
20.19 odd 2 320.6.c.e.129.1 2
24.5 odd 2 720.6.f.d.289.2 2
24.11 even 2 180.6.d.b.109.2 2
40.3 even 4 100.6.a.d.1.2 2
40.13 odd 4 400.6.a.s.1.1 2
40.19 odd 2 20.6.c.a.9.2 yes 2
40.27 even 4 100.6.a.d.1.1 2
40.29 even 2 80.6.c.b.49.1 2
40.37 odd 4 400.6.a.s.1.2 2
120.29 odd 2 720.6.f.d.289.1 2
120.59 even 2 180.6.d.b.109.1 2
120.83 odd 4 900.6.a.q.1.1 2
120.107 odd 4 900.6.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.6.c.a.9.1 2 8.3 odd 2
20.6.c.a.9.2 yes 2 40.19 odd 2
80.6.c.b.49.1 2 40.29 even 2
80.6.c.b.49.2 2 8.5 even 2
100.6.a.d.1.1 2 40.27 even 4
100.6.a.d.1.2 2 40.3 even 4
180.6.d.b.109.1 2 120.59 even 2
180.6.d.b.109.2 2 24.11 even 2
320.6.c.d.129.1 2 1.1 even 1 trivial
320.6.c.d.129.2 2 5.4 even 2 inner
320.6.c.e.129.1 2 20.19 odd 2
320.6.c.e.129.2 2 4.3 odd 2
400.6.a.s.1.1 2 40.13 odd 4
400.6.a.s.1.2 2 40.37 odd 4
720.6.f.d.289.1 2 120.29 odd 2
720.6.f.d.289.2 2 24.5 odd 2
900.6.a.q.1.1 2 120.83 odd 4
900.6.a.q.1.2 2 120.107 odd 4