Properties

Label 384.6.c.d.383.16
Level $384$
Weight $6$
Character 384.383
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
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] = [384,6,Mod(383,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.16
Root \(0.406991i\) of defining polynomial
Character \(\chi\) \(=\) 384.383
Dual form 384.6.c.d.383.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(10.1204 + 11.8566i) q^{3} -19.3214i q^{5} +82.7816i q^{7} +(-38.1561 + 239.986i) q^{9} +O(q^{10})\) \(q+(10.1204 + 11.8566i) q^{3} -19.3214i q^{5} +82.7816i q^{7} +(-38.1561 + 239.986i) q^{9} +332.396 q^{11} +1171.14 q^{13} +(229.086 - 195.540i) q^{15} -1984.58i q^{17} -1346.21i q^{19} +(-981.506 + 837.781i) q^{21} -1833.50 q^{23} +2751.68 q^{25} +(-3231.56 + 1976.34i) q^{27} -1923.64i q^{29} +5801.19i q^{31} +(3363.97 + 3941.07i) q^{33} +1599.46 q^{35} +6805.63 q^{37} +(11852.4 + 13885.7i) q^{39} -4150.50i q^{41} -8370.22i q^{43} +(4636.87 + 737.231i) q^{45} +1805.52 q^{47} +9954.20 q^{49} +(23530.3 - 20084.7i) q^{51} +22703.3i q^{53} -6422.36i q^{55} +(15961.5 - 13624.2i) q^{57} -18653.9 q^{59} -19502.5 q^{61} +(-19866.4 - 3158.63i) q^{63} -22628.1i q^{65} +50953.6i q^{67} +(-18555.7 - 21739.0i) q^{69} +69899.0 q^{71} +37035.5 q^{73} +(27848.0 + 32625.5i) q^{75} +27516.3i q^{77} +50543.7i q^{79} +(-56137.2 - 18313.8i) q^{81} +96377.3 q^{83} -38345.0 q^{85} +(22807.8 - 19468.0i) q^{87} -17995.1i q^{89} +96949.0i q^{91} +(-68782.2 + 58710.2i) q^{93} -26010.8 q^{95} +48381.9 q^{97} +(-12682.9 + 79770.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 948 q^{11} + 852 q^{15} - 1640 q^{21} + 328 q^{23} - 12500 q^{25} + 2030 q^{27} + 2836 q^{33} - 7184 q^{35} - 15056 q^{37} - 12980 q^{39} - 11800 q^{45} + 36640 q^{47} - 33388 q^{49} + 1936 q^{51} + 15404 q^{57} + 62908 q^{59} - 73264 q^{61} + 23608 q^{63} + 84024 q^{69} + 34888 q^{71} + 52568 q^{73} + 115698 q^{75} + 55444 q^{81} - 225172 q^{83} + 30112 q^{85} - 225700 q^{87} + 148016 q^{93} + 418616 q^{95} + 7600 q^{97} + 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 10.1204 + 11.8566i 0.649222 + 0.760599i
\(4\) 0 0
\(5\) 19.3214i 0.345632i −0.984954 0.172816i \(-0.944713\pi\)
0.984954 0.172816i \(-0.0552867\pi\)
\(6\) 0 0
\(7\) 82.7816i 0.638541i 0.947664 + 0.319271i \(0.103438\pi\)
−0.947664 + 0.319271i \(0.896562\pi\)
\(8\) 0 0
\(9\) −38.1561 + 239.986i −0.157021 + 0.987595i
\(10\) 0 0
\(11\) 332.396 0.828273 0.414137 0.910215i \(-0.364084\pi\)
0.414137 + 0.910215i \(0.364084\pi\)
\(12\) 0 0
\(13\) 1171.14 1.92199 0.960994 0.276568i \(-0.0891971\pi\)
0.960994 + 0.276568i \(0.0891971\pi\)
\(14\) 0 0
\(15\) 229.086 195.540i 0.262888 0.224392i
\(16\) 0 0
\(17\) 1984.58i 1.66551i −0.553644 0.832754i \(-0.686763\pi\)
0.553644 0.832754i \(-0.313237\pi\)
\(18\) 0 0
\(19\) 1346.21i 0.855520i −0.903892 0.427760i \(-0.859303\pi\)
0.903892 0.427760i \(-0.140697\pi\)
\(20\) 0 0
\(21\) −981.506 + 837.781i −0.485674 + 0.414555i
\(22\) 0 0
\(23\) −1833.50 −0.722704 −0.361352 0.932429i \(-0.617685\pi\)
−0.361352 + 0.932429i \(0.617685\pi\)
\(24\) 0 0
\(25\) 2751.68 0.880538
\(26\) 0 0
\(27\) −3231.56 + 1976.34i −0.853105 + 0.521739i
\(28\) 0 0
\(29\) 1923.64i 0.424747i −0.977189 0.212373i \(-0.931881\pi\)
0.977189 0.212373i \(-0.0681193\pi\)
\(30\) 0 0
\(31\) 5801.19i 1.08421i 0.840311 + 0.542104i \(0.182372\pi\)
−0.840311 + 0.542104i \(0.817628\pi\)
\(32\) 0 0
\(33\) 3363.97 + 3941.07i 0.537733 + 0.629984i
\(34\) 0 0
\(35\) 1599.46 0.220701
\(36\) 0 0
\(37\) 6805.63 0.817267 0.408634 0.912698i \(-0.366005\pi\)
0.408634 + 0.912698i \(0.366005\pi\)
\(38\) 0 0
\(39\) 11852.4 + 13885.7i 1.24780 + 1.46186i
\(40\) 0 0
\(41\) 4150.50i 0.385603i −0.981238 0.192801i \(-0.938243\pi\)
0.981238 0.192801i \(-0.0617573\pi\)
\(42\) 0 0
\(43\) 8370.22i 0.690344i −0.938539 0.345172i \(-0.887821\pi\)
0.938539 0.345172i \(-0.112179\pi\)
\(44\) 0 0
\(45\) 4636.87 + 737.231i 0.341345 + 0.0542716i
\(46\) 0 0
\(47\) 1805.52 0.119222 0.0596111 0.998222i \(-0.481014\pi\)
0.0596111 + 0.998222i \(0.481014\pi\)
\(48\) 0 0
\(49\) 9954.20 0.592265
\(50\) 0 0
\(51\) 23530.3 20084.7i 1.26678 1.08128i
\(52\) 0 0
\(53\) 22703.3i 1.11019i 0.831786 + 0.555097i \(0.187319\pi\)
−0.831786 + 0.555097i \(0.812681\pi\)
\(54\) 0 0
\(55\) 6422.36i 0.286278i
\(56\) 0 0
\(57\) 15961.5 13624.2i 0.650708 0.555423i
\(58\) 0 0
\(59\) −18653.9 −0.697654 −0.348827 0.937187i \(-0.613420\pi\)
−0.348827 + 0.937187i \(0.613420\pi\)
\(60\) 0 0
\(61\) −19502.5 −0.671066 −0.335533 0.942028i \(-0.608916\pi\)
−0.335533 + 0.942028i \(0.608916\pi\)
\(62\) 0 0
\(63\) −19866.4 3158.63i −0.630620 0.100264i
\(64\) 0 0
\(65\) 22628.1i 0.664302i
\(66\) 0 0
\(67\) 50953.6i 1.38672i 0.720592 + 0.693359i \(0.243870\pi\)
−0.720592 + 0.693359i \(0.756130\pi\)
\(68\) 0 0
\(69\) −18555.7 21739.0i −0.469196 0.549688i
\(70\) 0 0
\(71\) 69899.0 1.64560 0.822801 0.568329i \(-0.192410\pi\)
0.822801 + 0.568329i \(0.192410\pi\)
\(72\) 0 0
\(73\) 37035.5 0.813414 0.406707 0.913559i \(-0.366677\pi\)
0.406707 + 0.913559i \(0.366677\pi\)
\(74\) 0 0
\(75\) 27848.0 + 32625.5i 0.571665 + 0.669736i
\(76\) 0 0
\(77\) 27516.3i 0.528887i
\(78\) 0 0
\(79\) 50543.7i 0.911170i 0.890192 + 0.455585i \(0.150570\pi\)
−0.890192 + 0.455585i \(0.849430\pi\)
\(80\) 0 0
\(81\) −56137.2 18313.8i −0.950689 0.310147i
\(82\) 0 0
\(83\) 96377.3 1.53560 0.767802 0.640687i \(-0.221350\pi\)
0.767802 + 0.640687i \(0.221350\pi\)
\(84\) 0 0
\(85\) −38345.0 −0.575653
\(86\) 0 0
\(87\) 22807.8 19468.0i 0.323062 0.275755i
\(88\) 0 0
\(89\) 17995.1i 0.240812i −0.992725 0.120406i \(-0.961580\pi\)
0.992725 0.120406i \(-0.0384196\pi\)
\(90\) 0 0
\(91\) 96949.0i 1.22727i
\(92\) 0 0
\(93\) −68782.2 + 58710.2i −0.824648 + 0.703892i
\(94\) 0 0
\(95\) −26010.8 −0.295695
\(96\) 0 0
\(97\) 48381.9 0.522100 0.261050 0.965325i \(-0.415931\pi\)
0.261050 + 0.965325i \(0.415931\pi\)
\(98\) 0 0
\(99\) −12682.9 + 79770.2i −0.130056 + 0.817999i
\(100\) 0 0
\(101\) 101239.i 0.987512i 0.869600 + 0.493756i \(0.164376\pi\)
−0.869600 + 0.493756i \(0.835624\pi\)
\(102\) 0 0
\(103\) 26652.9i 0.247544i 0.992311 + 0.123772i \(0.0394991\pi\)
−0.992311 + 0.123772i \(0.960501\pi\)
\(104\) 0 0
\(105\) 16187.1 + 18964.1i 0.143284 + 0.167865i
\(106\) 0 0
\(107\) 12661.8 0.106914 0.0534571 0.998570i \(-0.482976\pi\)
0.0534571 + 0.998570i \(0.482976\pi\)
\(108\) 0 0
\(109\) −198283. −1.59852 −0.799262 0.600983i \(-0.794776\pi\)
−0.799262 + 0.600983i \(0.794776\pi\)
\(110\) 0 0
\(111\) 68875.5 + 80691.4i 0.530588 + 0.621613i
\(112\) 0 0
\(113\) 225671.i 1.66257i −0.555849 0.831284i \(-0.687607\pi\)
0.555849 0.831284i \(-0.312393\pi\)
\(114\) 0 0
\(115\) 35425.8i 0.249790i
\(116\) 0 0
\(117\) −44686.2 + 281057.i −0.301793 + 1.89815i
\(118\) 0 0
\(119\) 164287. 1.06350
\(120\) 0 0
\(121\) −50564.1 −0.313963
\(122\) 0 0
\(123\) 49210.6 42004.6i 0.293289 0.250342i
\(124\) 0 0
\(125\) 113546.i 0.649975i
\(126\) 0 0
\(127\) 99291.9i 0.546266i −0.961976 0.273133i \(-0.911940\pi\)
0.961976 0.273133i \(-0.0880600\pi\)
\(128\) 0 0
\(129\) 99242.0 84709.7i 0.525075 0.448187i
\(130\) 0 0
\(131\) 253584. 1.29105 0.645527 0.763737i \(-0.276638\pi\)
0.645527 + 0.763737i \(0.276638\pi\)
\(132\) 0 0
\(133\) 111442. 0.546285
\(134\) 0 0
\(135\) 38185.8 + 62438.4i 0.180330 + 0.294861i
\(136\) 0 0
\(137\) 47885.6i 0.217973i −0.994043 0.108987i \(-0.965239\pi\)
0.994043 0.108987i \(-0.0347606\pi\)
\(138\) 0 0
\(139\) 199921.i 0.877652i 0.898572 + 0.438826i \(0.144606\pi\)
−0.898572 + 0.438826i \(0.855394\pi\)
\(140\) 0 0
\(141\) 18272.5 + 21407.2i 0.0774017 + 0.0906802i
\(142\) 0 0
\(143\) 389282. 1.59193
\(144\) 0 0
\(145\) −37167.6 −0.146806
\(146\) 0 0
\(147\) 100740. + 118023.i 0.384512 + 0.450476i
\(148\) 0 0
\(149\) 63727.7i 0.235159i −0.993063 0.117580i \(-0.962486\pi\)
0.993063 0.117580i \(-0.0375136\pi\)
\(150\) 0 0
\(151\) 446578.i 1.59388i 0.604060 + 0.796939i \(0.293549\pi\)
−0.604060 + 0.796939i \(0.706451\pi\)
\(152\) 0 0
\(153\) 476271. + 75723.9i 1.64485 + 0.261520i
\(154\) 0 0
\(155\) 112087. 0.374738
\(156\) 0 0
\(157\) −266348. −0.862384 −0.431192 0.902260i \(-0.641907\pi\)
−0.431192 + 0.902260i \(0.641907\pi\)
\(158\) 0 0
\(159\) −269183. + 229766.i −0.844412 + 0.720762i
\(160\) 0 0
\(161\) 151780.i 0.461476i
\(162\) 0 0
\(163\) 244693.i 0.721361i −0.932689 0.360680i \(-0.882545\pi\)
0.932689 0.360680i \(-0.117455\pi\)
\(164\) 0 0
\(165\) 76147.1 64996.7i 0.217743 0.185858i
\(166\) 0 0
\(167\) −199361. −0.553157 −0.276579 0.960991i \(-0.589201\pi\)
−0.276579 + 0.960991i \(0.589201\pi\)
\(168\) 0 0
\(169\) 1.00028e6 2.69404
\(170\) 0 0
\(171\) 323072. + 51366.3i 0.844908 + 0.134335i
\(172\) 0 0
\(173\) 322299.i 0.818736i 0.912369 + 0.409368i \(0.134251\pi\)
−0.912369 + 0.409368i \(0.865749\pi\)
\(174\) 0 0
\(175\) 227789.i 0.562260i
\(176\) 0 0
\(177\) −188785. 221171.i −0.452933 0.530635i
\(178\) 0 0
\(179\) −853012. −1.98986 −0.994930 0.100570i \(-0.967933\pi\)
−0.994930 + 0.100570i \(0.967933\pi\)
\(180\) 0 0
\(181\) 434775. 0.986434 0.493217 0.869906i \(-0.335821\pi\)
0.493217 + 0.869906i \(0.335821\pi\)
\(182\) 0 0
\(183\) −197372. 231232.i −0.435671 0.510412i
\(184\) 0 0
\(185\) 131495.i 0.282474i
\(186\) 0 0
\(187\) 659666.i 1.37950i
\(188\) 0 0
\(189\) −163605. 267514.i −0.333152 0.544743i
\(190\) 0 0
\(191\) 946078. 1.87648 0.938239 0.345987i \(-0.112456\pi\)
0.938239 + 0.345987i \(0.112456\pi\)
\(192\) 0 0
\(193\) −512156. −0.989712 −0.494856 0.868975i \(-0.664779\pi\)
−0.494856 + 0.868975i \(0.664779\pi\)
\(194\) 0 0
\(195\) 268292. 229005.i 0.505267 0.431279i
\(196\) 0 0
\(197\) 589256.i 1.08178i 0.841094 + 0.540890i \(0.181912\pi\)
−0.841094 + 0.540890i \(0.818088\pi\)
\(198\) 0 0
\(199\) 2967.84i 0.00531260i −0.999996 0.00265630i \(-0.999154\pi\)
0.999996 0.00265630i \(-0.000845528\pi\)
\(200\) 0 0
\(201\) −604135. + 515670.i −1.05474 + 0.900288i
\(202\) 0 0
\(203\) 159242. 0.271218
\(204\) 0 0
\(205\) −80193.6 −0.133277
\(206\) 0 0
\(207\) 69959.1 440013.i 0.113480 0.713739i
\(208\) 0 0
\(209\) 447476.i 0.708604i
\(210\) 0 0
\(211\) 768116.i 1.18774i 0.804562 + 0.593869i \(0.202400\pi\)
−0.804562 + 0.593869i \(0.797600\pi\)
\(212\) 0 0
\(213\) 707404. + 828762.i 1.06836 + 1.25164i
\(214\) 0 0
\(215\) −161725. −0.238605
\(216\) 0 0
\(217\) −480232. −0.692312
\(218\) 0 0
\(219\) 374813. + 439114.i 0.528086 + 0.618682i
\(220\) 0 0
\(221\) 2.32422e6i 3.20109i
\(222\) 0 0
\(223\) 288136.i 0.388003i 0.981001 + 0.194001i \(0.0621466\pi\)
−0.981001 + 0.194001i \(0.937853\pi\)
\(224\) 0 0
\(225\) −104994. + 660364.i −0.138263 + 0.869615i
\(226\) 0 0
\(227\) −805116. −1.03704 −0.518518 0.855067i \(-0.673516\pi\)
−0.518518 + 0.855067i \(0.673516\pi\)
\(228\) 0 0
\(229\) −164519. −0.207313 −0.103657 0.994613i \(-0.533054\pi\)
−0.103657 + 0.994613i \(0.533054\pi\)
\(230\) 0 0
\(231\) −326248. + 278475.i −0.402270 + 0.343365i
\(232\) 0 0
\(233\) 470691.i 0.567997i 0.958825 + 0.283998i \(0.0916610\pi\)
−0.958825 + 0.283998i \(0.908339\pi\)
\(234\) 0 0
\(235\) 34885.2i 0.0412070i
\(236\) 0 0
\(237\) −599274. + 511521.i −0.693035 + 0.591552i
\(238\) 0 0
\(239\) −1.53286e6 −1.73584 −0.867919 0.496706i \(-0.834543\pi\)
−0.867919 + 0.496706i \(0.834543\pi\)
\(240\) 0 0
\(241\) −924322. −1.02513 −0.512567 0.858647i \(-0.671305\pi\)
−0.512567 + 0.858647i \(0.671305\pi\)
\(242\) 0 0
\(243\) −350990. 850937.i −0.381311 0.924447i
\(244\) 0 0
\(245\) 192329.i 0.204706i
\(246\) 0 0
\(247\) 1.57661e6i 1.64430i
\(248\) 0 0
\(249\) 975374. + 1.14270e6i 0.996949 + 1.16798i
\(250\) 0 0
\(251\) −1.28007e6 −1.28248 −0.641238 0.767342i \(-0.721579\pi\)
−0.641238 + 0.767342i \(0.721579\pi\)
\(252\) 0 0
\(253\) −609446. −0.598596
\(254\) 0 0
\(255\) −388065. 454639.i −0.373727 0.437841i
\(256\) 0 0
\(257\) 1.72426e6i 1.62843i −0.580561 0.814217i \(-0.697167\pi\)
0.580561 0.814217i \(-0.302833\pi\)
\(258\) 0 0
\(259\) 563381.i 0.521859i
\(260\) 0 0
\(261\) 461647. + 73398.8i 0.419478 + 0.0666942i
\(262\) 0 0
\(263\) −39731.5 −0.0354197 −0.0177099 0.999843i \(-0.505638\pi\)
−0.0177099 + 0.999843i \(0.505638\pi\)
\(264\) 0 0
\(265\) 438660. 0.383719
\(266\) 0 0
\(267\) 213359. 182117.i 0.183161 0.156341i
\(268\) 0 0
\(269\) 280025.i 0.235948i −0.993017 0.117974i \(-0.962360\pi\)
0.993017 0.117974i \(-0.0376399\pi\)
\(270\) 0 0
\(271\) 2.25288e6i 1.86344i −0.363178 0.931720i \(-0.618308\pi\)
0.363178 0.931720i \(-0.381692\pi\)
\(272\) 0 0
\(273\) −1.14948e6 + 981160.i −0.933459 + 0.796770i
\(274\) 0 0
\(275\) 914647. 0.729326
\(276\) 0 0
\(277\) 1.14847e6 0.899334 0.449667 0.893196i \(-0.351543\pi\)
0.449667 + 0.893196i \(0.351543\pi\)
\(278\) 0 0
\(279\) −1.39220e6 221351.i −1.07076 0.170244i
\(280\) 0 0
\(281\) 1.52077e6i 1.14894i 0.818524 + 0.574472i \(0.194793\pi\)
−0.818524 + 0.574472i \(0.805207\pi\)
\(282\) 0 0
\(283\) 610831.i 0.453372i −0.973968 0.226686i \(-0.927211\pi\)
0.973968 0.226686i \(-0.0727891\pi\)
\(284\) 0 0
\(285\) −263239. 308399.i −0.191972 0.224906i
\(286\) 0 0
\(287\) 343585. 0.246223
\(288\) 0 0
\(289\) −2.51871e6 −1.77392
\(290\) 0 0
\(291\) 489643. + 573643.i 0.338959 + 0.397108i
\(292\) 0 0
\(293\) 2.29419e6i 1.56121i −0.625025 0.780605i \(-0.714911\pi\)
0.625025 0.780605i \(-0.285089\pi\)
\(294\) 0 0
\(295\) 360421.i 0.241132i
\(296\) 0 0
\(297\) −1.07416e6 + 656928.i −0.706604 + 0.432142i
\(298\) 0 0
\(299\) −2.14728e6 −1.38903
\(300\) 0 0
\(301\) 692901. 0.440813
\(302\) 0 0
\(303\) −1.20034e6 + 1.02457e6i −0.751101 + 0.641115i
\(304\) 0 0
\(305\) 376816.i 0.231942i
\(306\) 0 0
\(307\) 2.25996e6i 1.36853i −0.729233 0.684265i \(-0.760123\pi\)
0.729233 0.684265i \(-0.239877\pi\)
\(308\) 0 0
\(309\) −316012. + 269737.i −0.188281 + 0.160711i
\(310\) 0 0
\(311\) 2.99587e6 1.75639 0.878196 0.478301i \(-0.158747\pi\)
0.878196 + 0.478301i \(0.158747\pi\)
\(312\) 0 0
\(313\) 686360. 0.395996 0.197998 0.980202i \(-0.436556\pi\)
0.197998 + 0.980202i \(0.436556\pi\)
\(314\) 0 0
\(315\) −61029.2 + 383848.i −0.0346546 + 0.217963i
\(316\) 0 0
\(317\) 2.99429e6i 1.67358i −0.547525 0.836789i \(-0.684430\pi\)
0.547525 0.836789i \(-0.315570\pi\)
\(318\) 0 0
\(319\) 639411.i 0.351806i
\(320\) 0 0
\(321\) 128142. + 150125.i 0.0694111 + 0.0813188i
\(322\) 0 0
\(323\) −2.67167e6 −1.42488
\(324\) 0 0
\(325\) 3.22261e6 1.69238
\(326\) 0 0
\(327\) −2.00670e6 2.35095e6i −1.03780 1.21584i
\(328\) 0 0
\(329\) 149464.i 0.0761282i
\(330\) 0 0
\(331\) 1.25988e6i 0.632064i −0.948749 0.316032i \(-0.897649\pi\)
0.948749 0.316032i \(-0.102351\pi\)
\(332\) 0 0
\(333\) −259677. + 1.63325e6i −0.128328 + 0.807129i
\(334\) 0 0
\(335\) 984497. 0.479295
\(336\) 0 0
\(337\) 1.52288e6 0.730452 0.365226 0.930919i \(-0.380992\pi\)
0.365226 + 0.930919i \(0.380992\pi\)
\(338\) 0 0
\(339\) 2.67568e6 2.28387e6i 1.26455 1.07938i
\(340\) 0 0
\(341\) 1.92829e6i 0.898021i
\(342\) 0 0
\(343\) 2.21534e6i 1.01673i
\(344\) 0 0
\(345\) −420028. + 358522.i −0.189990 + 0.162169i
\(346\) 0 0
\(347\) −2.32360e6 −1.03595 −0.517974 0.855396i \(-0.673314\pi\)
−0.517974 + 0.855396i \(0.673314\pi\)
\(348\) 0 0
\(349\) −481189. −0.211472 −0.105736 0.994394i \(-0.533720\pi\)
−0.105736 + 0.994394i \(0.533720\pi\)
\(350\) 0 0
\(351\) −3.78461e6 + 2.31458e6i −1.63966 + 1.00278i
\(352\) 0 0
\(353\) 168846.i 0.0721197i 0.999350 + 0.0360598i \(0.0114807\pi\)
−0.999350 + 0.0360598i \(0.988519\pi\)
\(354\) 0 0
\(355\) 1.35055e6i 0.568774i
\(356\) 0 0
\(357\) 1.66264e6 + 1.94788e6i 0.690445 + 0.808893i
\(358\) 0 0
\(359\) −272248. −0.111488 −0.0557441 0.998445i \(-0.517753\pi\)
−0.0557441 + 0.998445i \(0.517753\pi\)
\(360\) 0 0
\(361\) 663806. 0.268085
\(362\) 0 0
\(363\) −511728. 599517.i −0.203832 0.238800i
\(364\) 0 0
\(365\) 715580.i 0.281142i
\(366\) 0 0
\(367\) 118260.i 0.0458325i 0.999737 + 0.0229163i \(0.00729511\pi\)
−0.999737 + 0.0229163i \(0.992705\pi\)
\(368\) 0 0
\(369\) 996059. + 158367.i 0.380820 + 0.0605478i
\(370\) 0 0
\(371\) −1.87941e6 −0.708904
\(372\) 0 0
\(373\) 525577. 0.195598 0.0977989 0.995206i \(-0.468820\pi\)
0.0977989 + 0.995206i \(0.468820\pi\)
\(374\) 0 0
\(375\) 1.34626e6 1.14913e6i 0.494370 0.421978i
\(376\) 0 0
\(377\) 2.25286e6i 0.816358i
\(378\) 0 0
\(379\) 806611.i 0.288447i 0.989545 + 0.144224i \(0.0460684\pi\)
−0.989545 + 0.144224i \(0.953932\pi\)
\(380\) 0 0
\(381\) 1.17726e6 1.00487e6i 0.415490 0.354648i
\(382\) 0 0
\(383\) −1.37292e6 −0.478241 −0.239121 0.970990i \(-0.576859\pi\)
−0.239121 + 0.970990i \(0.576859\pi\)
\(384\) 0 0
\(385\) 531654. 0.182800
\(386\) 0 0
\(387\) 2.00873e6 + 319375.i 0.681781 + 0.108399i
\(388\) 0 0
\(389\) 5.48298e6i 1.83714i −0.395258 0.918570i \(-0.629345\pi\)
0.395258 0.918570i \(-0.370655\pi\)
\(390\) 0 0
\(391\) 3.63872e6i 1.20367i
\(392\) 0 0
\(393\) 2.56637e6 + 3.00664e6i 0.838181 + 0.981974i
\(394\) 0 0
\(395\) 976577. 0.314930
\(396\) 0 0
\(397\) 65596.8 0.0208885 0.0104442 0.999945i \(-0.496675\pi\)
0.0104442 + 0.999945i \(0.496675\pi\)
\(398\) 0 0
\(399\) 1.12783e6 + 1.32132e6i 0.354660 + 0.415504i
\(400\) 0 0
\(401\) 2.35838e6i 0.732409i −0.930534 0.366205i \(-0.880657\pi\)
0.930534 0.366205i \(-0.119343\pi\)
\(402\) 0 0
\(403\) 6.79401e6i 2.08384i
\(404\) 0 0
\(405\) −353850. + 1.08465e6i −0.107197 + 0.328589i
\(406\) 0 0
\(407\) 2.26216e6 0.676921
\(408\) 0 0
\(409\) −40774.2 −0.0120525 −0.00602625 0.999982i \(-0.501918\pi\)
−0.00602625 + 0.999982i \(0.501918\pi\)
\(410\) 0 0
\(411\) 567758. 484620.i 0.165790 0.141513i
\(412\) 0 0
\(413\) 1.54420e6i 0.445481i
\(414\) 0 0
\(415\) 1.86215e6i 0.530755i
\(416\) 0 0
\(417\) −2.37038e6 + 2.02328e6i −0.667541 + 0.569791i
\(418\) 0 0
\(419\) 2.09087e6 0.581824 0.290912 0.956750i \(-0.406041\pi\)
0.290912 + 0.956750i \(0.406041\pi\)
\(420\) 0 0
\(421\) −5.59228e6 −1.53774 −0.768871 0.639403i \(-0.779181\pi\)
−0.768871 + 0.639403i \(0.779181\pi\)
\(422\) 0 0
\(423\) −68891.5 + 433298.i −0.0187204 + 0.117743i
\(424\) 0 0
\(425\) 5.46094e6i 1.46654i
\(426\) 0 0
\(427\) 1.61445e6i 0.428503i
\(428\) 0 0
\(429\) 3.93968e6 + 4.61555e6i 1.03352 + 1.21082i
\(430\) 0 0
\(431\) 3.47562e6 0.901236 0.450618 0.892717i \(-0.351204\pi\)
0.450618 + 0.892717i \(0.351204\pi\)
\(432\) 0 0
\(433\) 262773. 0.0673537 0.0336768 0.999433i \(-0.489278\pi\)
0.0336768 + 0.999433i \(0.489278\pi\)
\(434\) 0 0
\(435\) −376150. 440680.i −0.0953098 0.111661i
\(436\) 0 0
\(437\) 2.46828e6i 0.618288i
\(438\) 0 0
\(439\) 5.40059e6i 1.33746i −0.743506 0.668729i \(-0.766839\pi\)
0.743506 0.668729i \(-0.233161\pi\)
\(440\) 0 0
\(441\) −379814. + 2.38887e6i −0.0929981 + 0.584918i
\(442\) 0 0
\(443\) 4.14902e6 1.00447 0.502234 0.864732i \(-0.332512\pi\)
0.502234 + 0.864732i \(0.332512\pi\)
\(444\) 0 0
\(445\) −347690. −0.0832324
\(446\) 0 0
\(447\) 755591. 644948.i 0.178862 0.152671i
\(448\) 0 0
\(449\) 5.27971e6i 1.23593i −0.786205 0.617966i \(-0.787957\pi\)
0.786205 0.617966i \(-0.212043\pi\)
\(450\) 0 0
\(451\) 1.37961e6i 0.319385i
\(452\) 0 0
\(453\) −5.29488e6 + 4.51954e6i −1.21230 + 1.03478i
\(454\) 0 0
\(455\) 1.87319e6 0.424184
\(456\) 0 0
\(457\) −6.04997e6 −1.35507 −0.677537 0.735489i \(-0.736953\pi\)
−0.677537 + 0.735489i \(0.736953\pi\)
\(458\) 0 0
\(459\) 3.92221e6 + 6.41329e6i 0.868960 + 1.42085i
\(460\) 0 0
\(461\) 5.14860e6i 1.12833i 0.825661 + 0.564166i \(0.190802\pi\)
−0.825661 + 0.564166i \(0.809198\pi\)
\(462\) 0 0
\(463\) 311898.i 0.0676176i 0.999428 + 0.0338088i \(0.0107637\pi\)
−0.999428 + 0.0338088i \(0.989236\pi\)
\(464\) 0 0
\(465\) 1.13437e6 + 1.32897e6i 0.243288 + 0.285025i
\(466\) 0 0
\(467\) −3.10534e6 −0.658895 −0.329448 0.944174i \(-0.606863\pi\)
−0.329448 + 0.944174i \(0.606863\pi\)
\(468\) 0 0
\(469\) −4.21803e6 −0.885477
\(470\) 0 0
\(471\) −2.69554e6 3.15797e6i −0.559879 0.655928i
\(472\) 0 0
\(473\) 2.78222e6i 0.571794i
\(474\) 0 0
\(475\) 3.70435e6i 0.753318i
\(476\) 0 0
\(477\) −5.44846e6 866269.i −1.09642 0.174324i
\(478\) 0 0
\(479\) 3.51933e6 0.700845 0.350422 0.936592i \(-0.386038\pi\)
0.350422 + 0.936592i \(0.386038\pi\)
\(480\) 0 0
\(481\) 7.97036e6 1.57078
\(482\) 0 0
\(483\) 1.79959e6 1.53607e6i 0.350998 0.299601i
\(484\) 0 0
\(485\) 934807.i 0.180455i
\(486\) 0 0
\(487\) 1.00090e7i 1.91235i 0.292791 + 0.956177i \(0.405416\pi\)
−0.292791 + 0.956177i \(0.594584\pi\)
\(488\) 0 0
\(489\) 2.90122e6 2.47638e6i 0.548666 0.468323i
\(490\) 0 0
\(491\) −4.74286e6 −0.887844 −0.443922 0.896066i \(-0.646413\pi\)
−0.443922 + 0.896066i \(0.646413\pi\)
\(492\) 0 0
\(493\) −3.81763e6 −0.707419
\(494\) 0 0
\(495\) 1.54127e6 + 245052.i 0.282727 + 0.0449517i
\(496\) 0 0
\(497\) 5.78635e6i 1.05079i
\(498\) 0 0
\(499\) 1.00050e7i 1.79872i 0.437209 + 0.899360i \(0.355967\pi\)
−0.437209 + 0.899360i \(0.644033\pi\)
\(500\) 0 0
\(501\) −2.01761e6 2.36373e6i −0.359122 0.420731i
\(502\) 0 0
\(503\) −118388. −0.0208635 −0.0104317 0.999946i \(-0.503321\pi\)
−0.0104317 + 0.999946i \(0.503321\pi\)
\(504\) 0 0
\(505\) 1.95607e6 0.341316
\(506\) 0 0
\(507\) 1.01232e7 + 1.18599e7i 1.74903 + 2.04908i
\(508\) 0 0
\(509\) 6.60168e6i 1.12943i 0.825285 + 0.564716i \(0.191014\pi\)
−0.825285 + 0.564716i \(0.808986\pi\)
\(510\) 0 0
\(511\) 3.06586e6i 0.519398i
\(512\) 0 0
\(513\) 2.66058e6 + 4.35037e6i 0.446358 + 0.729849i
\(514\) 0 0
\(515\) 514973. 0.0855591
\(516\) 0 0
\(517\) 600146. 0.0987485
\(518\) 0 0
\(519\) −3.82136e6 + 3.26179e6i −0.622729 + 0.531541i
\(520\) 0 0
\(521\) 6.04770e6i 0.976103i −0.872815 0.488052i \(-0.837708\pi\)
0.872815 0.488052i \(-0.162292\pi\)
\(522\) 0 0
\(523\) 1.86066e6i 0.297449i 0.988879 + 0.148724i \(0.0475167\pi\)
−0.988879 + 0.148724i \(0.952483\pi\)
\(524\) 0 0
\(525\) −2.70079e6 + 2.30531e6i −0.427654 + 0.365032i
\(526\) 0 0
\(527\) 1.15129e7 1.80576
\(528\) 0 0
\(529\) −3.07463e6 −0.477699
\(530\) 0 0
\(531\) 711762. 4.47668e6i 0.109546 0.689000i
\(532\) 0 0
\(533\) 4.86082e6i 0.741125i
\(534\) 0 0
\(535\) 244644.i 0.0369530i
\(536\) 0 0
\(537\) −8.63280e6 1.01138e7i −1.29186 1.51349i
\(538\) 0 0
\(539\) 3.30873e6 0.490557
\(540\) 0 0
\(541\) 3.48260e6 0.511577 0.255788 0.966733i \(-0.417665\pi\)
0.255788 + 0.966733i \(0.417665\pi\)
\(542\) 0 0
\(543\) 4.40009e6 + 5.15494e6i 0.640415 + 0.750281i
\(544\) 0 0
\(545\) 3.83111e6i 0.552502i
\(546\) 0 0
\(547\) 7.39011e6i 1.05605i 0.849230 + 0.528023i \(0.177066\pi\)
−0.849230 + 0.528023i \(0.822934\pi\)
\(548\) 0 0
\(549\) 744139. 4.68032e6i 0.105372 0.662742i
\(550\) 0 0
\(551\) −2.58964e6 −0.363379
\(552\) 0 0
\(553\) −4.18409e6 −0.581819
\(554\) 0 0
\(555\) 1.55907e6 1.33077e6i 0.214849 0.183388i
\(556\) 0 0
\(557\) 1.55684e6i 0.212620i 0.994333 + 0.106310i \(0.0339037\pi\)
−0.994333 + 0.106310i \(0.966096\pi\)
\(558\) 0 0
\(559\) 9.80271e6i 1.32683i
\(560\) 0 0
\(561\) 7.82137e6 6.67607e6i 1.04924 0.895599i
\(562\) 0 0
\(563\) 1.43554e6 0.190873 0.0954364 0.995436i \(-0.469575\pi\)
0.0954364 + 0.995436i \(0.469575\pi\)
\(564\) 0 0
\(565\) −4.36028e6 −0.574637
\(566\) 0 0
\(567\) 1.51605e6 4.64713e6i 0.198041 0.607054i
\(568\) 0 0
\(569\) 4.69381e6i 0.607778i −0.952707 0.303889i \(-0.901715\pi\)
0.952707 0.303889i \(-0.0982852\pi\)
\(570\) 0 0
\(571\) 6.38793e6i 0.819917i −0.912104 0.409959i \(-0.865543\pi\)
0.912104 0.409959i \(-0.134457\pi\)
\(572\) 0 0
\(573\) 9.57466e6 + 1.12172e7i 1.21825 + 1.42725i
\(574\) 0 0
\(575\) −5.04520e6 −0.636369
\(576\) 0 0
\(577\) 9.73246e6 1.21698 0.608490 0.793562i \(-0.291776\pi\)
0.608490 + 0.793562i \(0.291776\pi\)
\(578\) 0 0
\(579\) −5.18321e6 6.07241e6i −0.642543 0.752774i
\(580\) 0 0
\(581\) 7.97827e6i 0.980547i
\(582\) 0 0
\(583\) 7.54647e6i 0.919544i
\(584\) 0 0
\(585\) 5.43043e6 + 863402.i 0.656061 + 0.104309i
\(586\) 0 0
\(587\) 5.95499e6 0.713323 0.356661 0.934234i \(-0.383915\pi\)
0.356661 + 0.934234i \(0.383915\pi\)
\(588\) 0 0
\(589\) 7.80964e6 0.927562
\(590\) 0 0
\(591\) −6.98655e6 + 5.96349e6i −0.822800 + 0.702315i
\(592\) 0 0
\(593\) 1.08446e7i 1.26642i −0.773979 0.633211i \(-0.781736\pi\)
0.773979 0.633211i \(-0.218264\pi\)
\(594\) 0 0
\(595\) 3.17426e6i 0.367578i
\(596\) 0 0
\(597\) 35188.4 30035.6i 0.00404076 0.00344906i
\(598\) 0 0
\(599\) 1.05961e7 1.20664 0.603321 0.797499i \(-0.293844\pi\)
0.603321 + 0.797499i \(0.293844\pi\)
\(600\) 0 0
\(601\) 7.04222e6 0.795286 0.397643 0.917540i \(-0.369828\pi\)
0.397643 + 0.917540i \(0.369828\pi\)
\(602\) 0 0
\(603\) −1.22281e7 1.94419e6i −1.36952 0.217744i
\(604\) 0 0
\(605\) 976972.i 0.108516i
\(606\) 0 0
\(607\) 638984.i 0.0703911i −0.999380 0.0351956i \(-0.988795\pi\)
0.999380 0.0351956i \(-0.0112054\pi\)
\(608\) 0 0
\(609\) 1.61159e6 + 1.88807e6i 0.176081 + 0.206288i
\(610\) 0 0
\(611\) 2.11451e6 0.229144
\(612\) 0 0
\(613\) −1.19943e7 −1.28921 −0.644603 0.764518i \(-0.722977\pi\)
−0.644603 + 0.764518i \(0.722977\pi\)
\(614\) 0 0
\(615\) −811589. 950820.i −0.0865263 0.101370i
\(616\) 0 0
\(617\) 1.01397e7i 1.07229i 0.844126 + 0.536144i \(0.180120\pi\)
−0.844126 + 0.536144i \(0.819880\pi\)
\(618\) 0 0
\(619\) 6.43700e6i 0.675239i −0.941283 0.337619i \(-0.890378\pi\)
0.941283 0.337619i \(-0.109622\pi\)
\(620\) 0 0
\(621\) 5.92505e6 3.62362e6i 0.616543 0.377063i
\(622\) 0 0
\(623\) 1.48966e6 0.153768
\(624\) 0 0
\(625\) 6.40513e6 0.655886
\(626\) 0 0
\(627\) 5.30552e6 4.52862e6i 0.538964 0.460042i
\(628\) 0 0
\(629\) 1.35063e7i 1.36116i
\(630\) 0 0
\(631\) 1.35312e7i 1.35289i 0.736493 + 0.676445i \(0.236480\pi\)
−0.736493 + 0.676445i \(0.763520\pi\)
\(632\) 0 0
\(633\) −9.10721e6 + 7.77362e6i −0.903392 + 0.771106i
\(634\) 0 0
\(635\) −1.91846e6 −0.188807
\(636\) 0 0
\(637\) 1.16578e7 1.13833
\(638\) 0 0
\(639\) −2.66707e6 + 1.67748e7i −0.258394 + 1.62519i
\(640\) 0 0
\(641\) 3.36979e6i 0.323935i 0.986796 + 0.161967i \(0.0517839\pi\)
−0.986796 + 0.161967i \(0.948216\pi\)
\(642\) 0 0
\(643\) 5.93188e6i 0.565802i 0.959149 + 0.282901i \(0.0912968\pi\)
−0.959149 + 0.282901i \(0.908703\pi\)
\(644\) 0 0
\(645\) −1.63671e6 1.91750e6i −0.154908 0.181483i
\(646\) 0 0
\(647\) −1.62942e7 −1.53029 −0.765143 0.643860i \(-0.777332\pi\)
−0.765143 + 0.643860i \(0.777332\pi\)
\(648\) 0 0
\(649\) −6.20048e6 −0.577848
\(650\) 0 0
\(651\) −4.86013e6 5.69390e6i −0.449464 0.526572i
\(652\) 0 0
\(653\) 8.96912e6i 0.823127i −0.911381 0.411564i \(-0.864983\pi\)
0.911381 0.411564i \(-0.135017\pi\)
\(654\) 0 0
\(655\) 4.89962e6i 0.446230i
\(656\) 0 0
\(657\) −1.41313e6 + 8.88800e6i −0.127723 + 0.803324i
\(658\) 0 0
\(659\) 8.28105e6 0.742800 0.371400 0.928473i \(-0.378878\pi\)
0.371400 + 0.928473i \(0.378878\pi\)
\(660\) 0 0
\(661\) 7.16993e6 0.638280 0.319140 0.947708i \(-0.396606\pi\)
0.319140 + 0.947708i \(0.396606\pi\)
\(662\) 0 0
\(663\) 2.75573e7 2.35220e7i 2.43474 2.07822i
\(664\) 0 0
\(665\) 2.15322e6i 0.188814i
\(666\) 0 0
\(667\) 3.52700e6i 0.306966i
\(668\) 0 0
\(669\) −3.41630e6 + 2.91604e6i −0.295114 + 0.251900i
\(670\) 0 0
\(671\) −6.48254e6 −0.555826
\(672\) 0 0
\(673\) 781505. 0.0665111 0.0332555 0.999447i \(-0.489412\pi\)
0.0332555 + 0.999447i \(0.489412\pi\)
\(674\) 0 0
\(675\) −8.89222e6 + 5.43827e6i −0.751192 + 0.459411i
\(676\) 0 0
\(677\) 976634.i 0.0818956i −0.999161 0.0409478i \(-0.986962\pi\)
0.999161 0.0409478i \(-0.0130377\pi\)
\(678\) 0 0
\(679\) 4.00513e6i 0.333382i
\(680\) 0 0
\(681\) −8.14808e6 9.54591e6i −0.673267 0.788769i
\(682\) 0 0
\(683\) 1.17475e6 0.0963590 0.0481795 0.998839i \(-0.484658\pi\)
0.0481795 + 0.998839i \(0.484658\pi\)
\(684\) 0 0
\(685\) −925218. −0.0753386
\(686\) 0 0
\(687\) −1.66499e6 1.95063e6i −0.134592 0.157682i
\(688\) 0 0
\(689\) 2.65887e7i 2.13378i
\(690\) 0 0
\(691\) 1.25020e7i 0.996058i −0.867161 0.498029i \(-0.834057\pi\)
0.867161 0.498029i \(-0.165943\pi\)
\(692\) 0 0
\(693\) −6.60351e6 1.04991e6i −0.522326 0.0830464i
\(694\) 0 0
\(695\) 3.86277e6 0.303345
\(696\) 0 0
\(697\) −8.23700e6 −0.642225
\(698\) 0 0
\(699\) −5.58077e6 + 4.76356e6i −0.432017 + 0.368756i
\(700\) 0 0
\(701\) 1.36812e7i 1.05155i 0.850624 + 0.525774i \(0.176224\pi\)
−0.850624 + 0.525774i \(0.823776\pi\)
\(702\) 0 0
\(703\) 9.16184e6i 0.699189i
\(704\) 0 0
\(705\) 413618. 353051.i 0.0313420 0.0267525i
\(706\) 0 0
\(707\) −8.38069e6 −0.630567
\(708\) 0 0
\(709\) 4.64829e6 0.347278 0.173639 0.984809i \(-0.444447\pi\)
0.173639 + 0.984809i \(0.444447\pi\)
\(710\) 0 0
\(711\) −1.21298e7 1.92855e6i −0.899867 0.143073i
\(712\) 0 0
\(713\) 1.06365e7i 0.783562i
\(714\) 0 0
\(715\) 7.52149e6i 0.550223i
\(716\) 0 0
\(717\) −1.55132e7 1.81745e7i −1.12694 1.32028i
\(718\) 0 0
\(719\) −1.82672e7 −1.31780 −0.658899 0.752231i \(-0.728978\pi\)
−0.658899 + 0.752231i \(0.728978\pi\)
\(720\) 0 0
\(721\) −2.20637e6 −0.158067
\(722\) 0 0
\(723\) −9.35449e6 1.09593e7i −0.665540 0.779716i
\(724\) 0 0
\(725\) 5.29326e6i 0.374006i
\(726\) 0 0
\(727\) 2.11523e7i 1.48430i −0.670235 0.742149i \(-0.733807\pi\)
0.670235 0.742149i \(-0.266193\pi\)
\(728\) 0 0
\(729\) 6.53704e6 1.27733e7i 0.455577 0.890196i
\(730\) 0 0
\(731\) −1.66114e7 −1.14977
\(732\) 0 0
\(733\) 2.38899e6 0.164231 0.0821153 0.996623i \(-0.473832\pi\)
0.0821153 + 0.996623i \(0.473832\pi\)
\(734\) 0 0
\(735\) 2.28037e6 1.94645e6i 0.155699 0.132900i
\(736\) 0 0
\(737\) 1.69368e7i 1.14858i
\(738\) 0 0
\(739\) 6.98369e6i 0.470407i 0.971946 + 0.235203i \(0.0755756\pi\)
−0.971946 + 0.235203i \(0.924424\pi\)
\(740\) 0 0
\(741\) 1.86931e7 1.59558e7i 1.25065 1.06752i
\(742\) 0 0
\(743\) −1.91909e7 −1.27533 −0.637665 0.770314i \(-0.720099\pi\)
−0.637665 + 0.770314i \(0.720099\pi\)
\(744\) 0 0
\(745\) −1.23131e6 −0.0812787
\(746\) 0 0
\(747\) −3.67738e6 + 2.31292e7i −0.241122 + 1.51656i
\(748\) 0 0
\(749\) 1.04816e6i 0.0682691i
\(750\) 0 0
\(751\) 3.04159e7i 1.96789i −0.178465 0.983946i \(-0.557113\pi\)
0.178465 0.983946i \(-0.442887\pi\)
\(752\) 0 0
\(753\) −1.29548e7 1.51772e7i −0.832612 0.975450i
\(754\) 0 0
\(755\) 8.62853e6 0.550896
\(756\) 0 0
\(757\) −6.16178e6 −0.390811 −0.195405 0.980723i \(-0.562602\pi\)
−0.195405 + 0.980723i \(0.562602\pi\)
\(758\) 0 0
\(759\) −6.16782e6 7.22594e6i −0.388622 0.455292i
\(760\) 0 0
\(761\) 2.27781e7i 1.42579i 0.701269 + 0.712897i \(0.252617\pi\)
−0.701269 + 0.712897i \(0.747383\pi\)
\(762\) 0 0
\(763\) 1.64142e7i 1.02072i
\(764\) 0 0
\(765\) 1.46310e6 9.20224e6i 0.0903897 0.568513i
\(766\) 0 0
\(767\) −2.18464e7 −1.34088
\(768\) 0 0
\(769\) −3.17621e7 −1.93684 −0.968419 0.249326i \(-0.919791\pi\)
−0.968419 + 0.249326i \(0.919791\pi\)
\(770\) 0 0
\(771\) 2.04438e7 1.74502e7i 1.23858 1.05721i
\(772\) 0 0
\(773\) 1.28739e7i 0.774930i 0.921884 + 0.387465i \(0.126649\pi\)
−0.921884 + 0.387465i \(0.873351\pi\)
\(774\) 0 0
\(775\) 1.59630e7i 0.954687i
\(776\) 0 0
\(777\) −6.67977e6 + 5.70163e6i −0.396925 + 0.338802i
\(778\) 0 0
\(779\) −5.58746e6 −0.329891
\(780\) 0 0
\(781\) 2.32341e7 1.36301
\(782\) 0 0
\(783\) 3.80178e6 + 6.21637e6i 0.221607 + 0.362354i
\(784\) 0 0
\(785\) 5.14623e6i 0.298068i
\(786\) 0 0
\(787\) 2.15917e7i 1.24265i 0.783552 + 0.621327i \(0.213406\pi\)
−0.783552 + 0.621327i \(0.786594\pi\)
\(788\) 0 0
\(789\) −402097. 471079.i −0.0229953 0.0269402i
\(790\) 0 0
\(791\) 1.86814e7 1.06162
\(792\) 0 0
\(793\) −2.28402e7 −1.28978
\(794\) 0 0
\(795\) 4.43940e6 + 5.20100e6i 0.249119 + 0.291856i
\(796\) 0 0
\(797\) 1.94436e7i 1.08425i 0.840297 + 0.542126i \(0.182380\pi\)
−0.840297 + 0.542126i \(0.817620\pi\)
\(798\) 0 0
\(799\) 3.58319e6i 0.198565i
\(800\) 0 0
\(801\) 4.31855e6 + 686622.i 0.237825 + 0.0378126i
\(802\) 0 0
\(803\) 1.23105e7 0.673729
\(804\) 0 0
\(805\) −2.93261e6 −0.159501
\(806\) 0 0
\(807\) 3.32013e6 2.83396e6i 0.179462 0.153183i
\(808\) 0 0
\(809\) 2.43124e7i 1.30604i 0.757341 + 0.653020i \(0.226498\pi\)
−0.757341 + 0.653020i \(0.773502\pi\)
\(810\) 0 0
\(811\) 1.09629e7i 0.585293i −0.956221 0.292647i \(-0.905464\pi\)
0.956221 0.292647i \(-0.0945360\pi\)
\(812\) 0 0
\(813\) 2.67114e7 2.28000e7i 1.41733 1.20979i
\(814\) 0 0
\(815\) −4.72782e6 −0.249326
\(816\) 0 0
\(817\) −1.12681e7 −0.590603
\(818\) 0 0
\(819\) −2.32664e7 3.69920e6i −1.21204 0.192707i
\(820\) 0 0
\(821\) 2.05702e6i 0.106507i −0.998581 0.0532537i \(-0.983041\pi\)
0.998581 0.0532537i \(-0.0169592\pi\)
\(822\) 0 0
\(823\) 2.72914e7i 1.40451i −0.711924 0.702256i \(-0.752176\pi\)
0.711924 0.702256i \(-0.247824\pi\)
\(824\) 0 0
\(825\) 9.25657e6 + 1.08446e7i 0.473495 + 0.554725i
\(826\) 0 0
\(827\) −1.02167e7 −0.519456 −0.259728 0.965682i \(-0.583633\pi\)
−0.259728 + 0.965682i \(0.583633\pi\)
\(828\) 0 0
\(829\) −2.19066e7 −1.10710 −0.553551 0.832815i \(-0.686728\pi\)
−0.553551 + 0.832815i \(0.686728\pi\)
\(830\) 0 0
\(831\) 1.16230e7 + 1.36169e7i 0.583868 + 0.684032i
\(832\) 0 0
\(833\) 1.97549e7i 0.986422i
\(834\) 0 0
\(835\) 3.85194e6i 0.191189i
\(836\) 0 0
\(837\) −1.14651e7 1.87469e7i −0.565674 0.924944i
\(838\) 0 0
\(839\) −1.65586e7 −0.812116 −0.406058 0.913847i \(-0.633097\pi\)
−0.406058 + 0.913847i \(0.633097\pi\)
\(840\) 0 0
\(841\) 1.68107e7 0.819590
\(842\) 0 0
\(843\) −1.80312e7 + 1.53908e7i −0.873886 + 0.745920i
\(844\) 0 0
\(845\) 1.93268e7i 0.931148i
\(846\) 0 0
\(847\) 4.18578e6i 0.200479i
\(848\) 0 0
\(849\) 7.24235e6 6.18183e6i 0.344834 0.294339i
\(850\) 0 0
\(851\) −1.24781e7 −0.590643
\(852\) 0 0
\(853\) 7.67539e6 0.361183 0.180592 0.983558i \(-0.442199\pi\)
0.180592 + 0.983558i \(0.442199\pi\)
\(854\) 0 0
\(855\) 992471. 6.24222e6i 0.0464304 0.292027i
\(856\) 0 0
\(857\) 3.53759e6i 0.164534i 0.996610 + 0.0822670i \(0.0262160\pi\)
−0.996610 + 0.0822670i \(0.973784\pi\)
\(858\) 0 0
\(859\) 1.36445e7i 0.630921i −0.948939 0.315461i \(-0.897841\pi\)
0.948939 0.315461i \(-0.102159\pi\)
\(860\) 0 0
\(861\) 3.47721e6 + 4.07374e6i 0.159854 + 0.187277i
\(862\) 0 0
\(863\) −2.32847e7 −1.06425 −0.532126 0.846665i \(-0.678607\pi\)
−0.532126 + 0.846665i \(0.678607\pi\)
\(864\) 0 0
\(865\) 6.22728e6 0.282982
\(866\) 0 0
\(867\) −2.54902e7 2.98632e7i −1.15167 1.34924i
\(868\) 0 0
\(869\) 1.68005e7i 0.754697i
\(870\) 0 0
\(871\) 5.96739e7i 2.66526i
\(872\) 0 0
\(873\) −1.84606e6 + 1.16110e7i −0.0819806 + 0.515623i
\(874\) 0 0
\(875\) 9.39952e6 0.415036
\(876\) 0 0
\(877\) −4.33930e7 −1.90511 −0.952556 0.304364i \(-0.901556\pi\)
−0.952556 + 0.304364i \(0.901556\pi\)
\(878\) 0 0
\(879\) 2.72013e7 2.32181e7i 1.18745 1.01357i
\(880\) 0 0
\(881\) 2.09569e7i 0.909678i 0.890574 + 0.454839i \(0.150303\pi\)
−0.890574 + 0.454839i \(0.849697\pi\)
\(882\) 0 0
\(883\) 1.32913e7i 0.573674i 0.957979 + 0.286837i \(0.0926037\pi\)
−0.957979 + 0.286837i \(0.907396\pi\)
\(884\) 0 0
\(885\) −4.27335e6 + 3.64759e6i −0.183405 + 0.156548i
\(886\) 0 0
\(887\) −2.96276e7 −1.26441 −0.632205 0.774801i \(-0.717850\pi\)
−0.632205 + 0.774801i \(0.717850\pi\)
\(888\) 0 0
\(889\) 8.21955e6 0.348814
\(890\) 0 0
\(891\) −1.86598e7 6.08744e6i −0.787430 0.256886i
\(892\) 0 0
\(893\) 2.43061e6i 0.101997i
\(894\) 0 0
\(895\) 1.64814e7i 0.687760i
\(896\) 0 0
\(897\) −2.17313e7 2.54594e7i −0.901789 1.05649i
\(898\) 0 0
\(899\) 1.11594e7 0.460514
\(900\) 0 0
\(901\) 4.50565e7 1.84904
\(902\) 0 0
\(903\) 7.01241e6 + 8.21542e6i 0.286186 + 0.335282i
\(904\) 0 0
\(905\) 8.40048e6i 0.340944i
\(906\) 0 0
\(907\) 2.59307e7i 1.04664i −0.852137 0.523318i \(-0.824694\pi\)
0.852137 0.523318i \(-0.175306\pi\)
\(908\) 0 0
\(909\) −2.42958e7 3.86287e6i −0.975262 0.155060i
\(910\) 0 0
\(911\) 2.62650e7 1.04853 0.524265 0.851555i \(-0.324340\pi\)
0.524265 + 0.851555i \(0.324340\pi\)
\(912\) 0 0
\(913\) 3.20354e7 1.27190
\(914\) 0 0
\(915\) −4.46774e6 + 3.81352e6i −0.176415 + 0.150582i
\(916\) 0 0
\(917\) 2.09921e7i 0.824391i
\(918\) 0 0
\(919\) 2.40150e7i 0.937982i −0.883203 0.468991i \(-0.844618\pi\)
0.883203 0.468991i \(-0.155382\pi\)
\(920\) 0 0
\(921\) 2.67953e7 2.28716e7i 1.04090 0.888480i
\(922\) 0 0
\(923\) 8.18616e7 3.16283
\(924\) 0 0
\(925\) 1.87269e7 0.719635
\(926\) 0 0
\(927\) −6.39632e6 1.01697e6i −0.244473 0.0388696i
\(928\) 0 0
\(929\) 199297.i 0.00757638i 0.999993 + 0.00378819i \(0.00120582\pi\)
−0.999993 + 0.00378819i \(0.998794\pi\)
\(930\) 0 0
\(931\) 1.34005e7i 0.506695i
\(932\) 0 0
\(933\) 3.03193e7 + 3.55207e7i 1.14029 + 1.33591i
\(934\) 0 0
\(935\) −1.27457e7 −0.476798
\(936\) 0 0
\(937\) 2.11339e7 0.786375 0.393188 0.919458i \(-0.371372\pi\)
0.393188 + 0.919458i \(0.371372\pi\)
\(938\) 0 0
\(939\) 6.94622e6 + 8.13787e6i 0.257089 + 0.301194i
\(940\) 0 0
\(941\) 2.91277e7i 1.07234i −0.844110 0.536169i \(-0.819871\pi\)
0.844110 0.536169i \(-0.180129\pi\)
\(942\) 0 0
\(943\) 7.60992e6i 0.278677i
\(944\) 0 0
\(945\) −5.16875e6 + 3.16108e6i −0.188281 + 0.115148i
\(946\) 0 0
\(947\) −1.30882e7 −0.474248 −0.237124 0.971479i \(-0.576205\pi\)
−0.237124 + 0.971479i \(0.576205\pi\)
\(948\) 0 0
\(949\) 4.33738e7 1.56337
\(950\) 0 0
\(951\) 3.55020e7 3.03034e7i 1.27292 1.08652i
\(952\) 0 0
\(953\) 3.88256e7i 1.38480i 0.721515 + 0.692398i \(0.243446\pi\)
−0.721515 + 0.692398i \(0.756554\pi\)
\(954\) 0 0
\(955\) 1.82796e7i 0.648572i
\(956\) 0 0
\(957\) 7.58122e6 6.47108e6i 0.267583 0.228400i
\(958\) 0 0
\(959\) 3.96405e6 0.139185
\(960\) 0 0
\(961\) −5.02465e6 −0.175508
\(962\) 0 0
\(963\) −483125. + 3.03865e6i −0.0167878 + 0.105588i
\(964\) 0 0
\(965\) 9.89559e6i 0.342076i
\(966\) 0 0
\(967\) 3.38341e7i 1.16356i 0.813346 + 0.581780i \(0.197644\pi\)
−0.813346 + 0.581780i \(0.802356\pi\)
\(968\) 0 0
\(969\) −2.70383e7 3.16768e7i −0.925061 1.08376i
\(970\) 0 0
\(971\) 3.86651e7 1.31604 0.658022 0.752998i \(-0.271393\pi\)
0.658022 + 0.752998i \(0.271393\pi\)
\(972\) 0 0
\(973\) −1.65498e7 −0.560417
\(974\) 0 0
\(975\) 3.26140e7 + 3.82090e7i 1.09873 + 1.28723i
\(976\) 0 0
\(977\) 4.73321e7i 1.58643i −0.608944 0.793213i \(-0.708407\pi\)
0.608944 0.793213i \(-0.291593\pi\)
\(978\) 0 0
\(979\) 5.98148e6i 0.199458i
\(980\) 0 0
\(981\) 7.56571e6 4.75851e7i 0.251002 1.57869i
\(982\) 0 0
\(983\) 2.15996e7 0.712954 0.356477 0.934304i \(-0.383978\pi\)
0.356477 + 0.934304i \(0.383978\pi\)
\(984\) 0 0
\(985\) 1.13853e7 0.373898
\(986\) 0 0
\(987\) −1.77212e6 + 1.51263e6i −0.0579031 + 0.0494241i
\(988\) 0 0
\(989\) 1.53468e7i 0.498915i
\(990\) 0 0
\(991\) 2.42907e7i 0.785699i 0.919603 + 0.392849i \(0.128511\pi\)
−0.919603 + 0.392849i \(0.871489\pi\)
\(992\) 0 0
\(993\) 1.49379e7 1.27505e7i 0.480747 0.410350i
\(994\) 0 0
\(995\) −57342.9 −0.00183621
\(996\) 0 0
\(997\) −5.08871e7 −1.62132 −0.810662 0.585514i \(-0.800893\pi\)
−0.810662 + 0.585514i \(0.800893\pi\)
\(998\) 0 0
\(999\) −2.19928e7 + 1.34503e7i −0.697215 + 0.426400i
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 384.6.c.d.383.16 yes 20
3.2 odd 2 384.6.c.a.383.6 yes 20
4.3 odd 2 384.6.c.a.383.5 20
8.3 odd 2 384.6.c.c.383.16 yes 20
8.5 even 2 384.6.c.b.383.5 yes 20
12.11 even 2 inner 384.6.c.d.383.15 yes 20
24.5 odd 2 384.6.c.c.383.15 yes 20
24.11 even 2 384.6.c.b.383.6 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.c.a.383.5 20 4.3 odd 2
384.6.c.a.383.6 yes 20 3.2 odd 2
384.6.c.b.383.5 yes 20 8.5 even 2
384.6.c.b.383.6 yes 20 24.11 even 2
384.6.c.c.383.15 yes 20 24.5 odd 2
384.6.c.c.383.16 yes 20 8.3 odd 2
384.6.c.d.383.15 yes 20 12.11 even 2 inner
384.6.c.d.383.16 yes 20 1.1 even 1 trivial