Properties

Label 32.18.b.a.17.14
Level $32$
Weight $18$
Character 32.17
Analytic conductor $58.631$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,18,Mod(17,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.17");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 32.b (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: \(58.6310679503\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 83403052 x^{14} - 583821224 x^{13} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{240}\cdot 3^{14}\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.14
Root \(0.500000 - 3466.80i\) of defining polynomial
Character \(\chi\) \(=\) 32.17
Dual form 32.18.b.a.17.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13867.2i q^{3} +665059. i q^{5} +7.57536e6 q^{7} -6.31593e7 q^{9} +O(q^{10})\) \(q+13867.2i q^{3} +665059. i q^{5} +7.57536e6 q^{7} -6.31593e7 q^{9} -1.97863e8i q^{11} +4.74981e9i q^{13} -9.22251e9 q^{15} +3.37447e10 q^{17} +1.00829e11i q^{19} +1.05049e11i q^{21} -3.16250e11 q^{23} +3.20636e11 q^{25} +9.14971e11i q^{27} +3.54291e12i q^{29} -2.76965e11 q^{31} +2.74380e12 q^{33} +5.03806e12i q^{35} -2.12889e13i q^{37} -6.58666e13 q^{39} +8.47798e13 q^{41} -1.38239e14i q^{43} -4.20047e13i q^{45} -8.21469e13 q^{47} -1.75244e14 q^{49} +4.67945e14i q^{51} +3.16386e14i q^{53} +1.31590e14 q^{55} -1.39822e15 q^{57} +3.29635e14i q^{59} +6.60732e14i q^{61} -4.78454e14 q^{63} -3.15891e15 q^{65} -3.64311e15i q^{67} -4.38551e15i q^{69} -9.19058e15 q^{71} -5.44264e15 q^{73} +4.44633e15i q^{75} -1.49888e15i q^{77} -1.06400e16 q^{79} -2.08445e16 q^{81} -9.38898e15i q^{83} +2.24422e16i q^{85} -4.91303e16 q^{87} +2.24169e16 q^{89} +3.59815e16i q^{91} -3.84074e15i q^{93} -6.70573e16 q^{95} -4.00962e16 q^{97} +1.24969e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 11529600 q^{7} - 602654096 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 11529600 q^{7} - 602654096 q^{9} + 9993282176 q^{15} - 7489125600 q^{17} - 746845345920 q^{23} - 1809682431664 q^{25} + 318979758592 q^{31} + 5633526177600 q^{33} + 18457706051456 q^{39} + 7482251536032 q^{41} + 376698804821760 q^{47} + 127691292101520 q^{49} - 22\!\cdots\!52 q^{55}+ \cdots + 95\!\cdots\!40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 13867.2i 1.22028i 0.792295 + 0.610138i \(0.208886\pi\)
−0.792295 + 0.610138i \(0.791114\pi\)
\(4\) 0 0
\(5\) 665059.i 0.761404i 0.924698 + 0.380702i \(0.124318\pi\)
−0.924698 + 0.380702i \(0.875682\pi\)
\(6\) 0 0
\(7\) 7.57536e6 0.496672 0.248336 0.968674i \(-0.420116\pi\)
0.248336 + 0.968674i \(0.420116\pi\)
\(8\) 0 0
\(9\) −6.31593e7 −0.489075
\(10\) 0 0
\(11\) − 1.97863e8i − 0.278308i −0.990271 0.139154i \(-0.955562\pi\)
0.990271 0.139154i \(-0.0444384\pi\)
\(12\) 0 0
\(13\) 4.74981e9i 1.61495i 0.589905 + 0.807473i \(0.299165\pi\)
−0.589905 + 0.807473i \(0.700835\pi\)
\(14\) 0 0
\(15\) −9.22251e9 −0.929124
\(16\) 0 0
\(17\) 3.37447e10 1.17325 0.586624 0.809859i \(-0.300457\pi\)
0.586624 + 0.809859i \(0.300457\pi\)
\(18\) 0 0
\(19\) 1.00829e11i 1.36201i 0.732278 + 0.681005i \(0.238457\pi\)
−0.732278 + 0.681005i \(0.761543\pi\)
\(20\) 0 0
\(21\) 1.05049e11i 0.606078i
\(22\) 0 0
\(23\) −3.16250e11 −0.842062 −0.421031 0.907046i \(-0.638332\pi\)
−0.421031 + 0.907046i \(0.638332\pi\)
\(24\) 0 0
\(25\) 3.20636e11 0.420264
\(26\) 0 0
\(27\) 9.14971e11i 0.623469i
\(28\) 0 0
\(29\) 3.54291e12i 1.31516i 0.753386 + 0.657578i \(0.228419\pi\)
−0.753386 + 0.657578i \(0.771581\pi\)
\(30\) 0 0
\(31\) −2.76965e11 −0.0583245 −0.0291623 0.999575i \(-0.509284\pi\)
−0.0291623 + 0.999575i \(0.509284\pi\)
\(32\) 0 0
\(33\) 2.74380e12 0.339613
\(34\) 0 0
\(35\) 5.03806e12i 0.378168i
\(36\) 0 0
\(37\) − 2.12889e13i − 0.996412i −0.867059 0.498206i \(-0.833992\pi\)
0.867059 0.498206i \(-0.166008\pi\)
\(38\) 0 0
\(39\) −6.58666e13 −1.97068
\(40\) 0 0
\(41\) 8.47798e13 1.65817 0.829086 0.559121i \(-0.188861\pi\)
0.829086 + 0.559121i \(0.188861\pi\)
\(42\) 0 0
\(43\) − 1.38239e14i − 1.80363i −0.432122 0.901815i \(-0.642235\pi\)
0.432122 0.901815i \(-0.357765\pi\)
\(44\) 0 0
\(45\) − 4.20047e13i − 0.372384i
\(46\) 0 0
\(47\) −8.21469e13 −0.503221 −0.251611 0.967829i \(-0.580960\pi\)
−0.251611 + 0.967829i \(0.580960\pi\)
\(48\) 0 0
\(49\) −1.75244e14 −0.753317
\(50\) 0 0
\(51\) 4.67945e14i 1.43169i
\(52\) 0 0
\(53\) 3.16386e14i 0.698028i 0.937118 + 0.349014i \(0.113483\pi\)
−0.937118 + 0.349014i \(0.886517\pi\)
\(54\) 0 0
\(55\) 1.31590e14 0.211905
\(56\) 0 0
\(57\) −1.39822e15 −1.66203
\(58\) 0 0
\(59\) 3.29635e14i 0.292275i 0.989264 + 0.146138i \(0.0466842\pi\)
−0.989264 + 0.146138i \(0.953316\pi\)
\(60\) 0 0
\(61\) 6.60732e14i 0.441288i 0.975354 + 0.220644i \(0.0708159\pi\)
−0.975354 + 0.220644i \(0.929184\pi\)
\(62\) 0 0
\(63\) −4.78454e14 −0.242910
\(64\) 0 0
\(65\) −3.15891e15 −1.22963
\(66\) 0 0
\(67\) − 3.64311e15i − 1.09607i −0.836457 0.548033i \(-0.815377\pi\)
0.836457 0.548033i \(-0.184623\pi\)
\(68\) 0 0
\(69\) − 4.38551e15i − 1.02755i
\(70\) 0 0
\(71\) −9.19058e15 −1.68907 −0.844534 0.535502i \(-0.820122\pi\)
−0.844534 + 0.535502i \(0.820122\pi\)
\(72\) 0 0
\(73\) −5.44264e15 −0.789888 −0.394944 0.918705i \(-0.629236\pi\)
−0.394944 + 0.918705i \(0.629236\pi\)
\(74\) 0 0
\(75\) 4.44633e15i 0.512838i
\(76\) 0 0
\(77\) − 1.49888e15i − 0.138228i
\(78\) 0 0
\(79\) −1.06400e16 −0.789059 −0.394529 0.918883i \(-0.629092\pi\)
−0.394529 + 0.918883i \(0.629092\pi\)
\(80\) 0 0
\(81\) −2.08445e16 −1.24988
\(82\) 0 0
\(83\) − 9.38898e15i − 0.457567i −0.973477 0.228783i \(-0.926525\pi\)
0.973477 0.228783i \(-0.0734748\pi\)
\(84\) 0 0
\(85\) 2.24422e16i 0.893316i
\(86\) 0 0
\(87\) −4.91303e16 −1.60485
\(88\) 0 0
\(89\) 2.24169e16 0.603616 0.301808 0.953369i \(-0.402410\pi\)
0.301808 + 0.953369i \(0.402410\pi\)
\(90\) 0 0
\(91\) 3.59815e16i 0.802099i
\(92\) 0 0
\(93\) − 3.84074e15i − 0.0711721i
\(94\) 0 0
\(95\) −6.70573e16 −1.03704
\(96\) 0 0
\(97\) −4.00962e16 −0.519450 −0.259725 0.965683i \(-0.583632\pi\)
−0.259725 + 0.965683i \(0.583632\pi\)
\(98\) 0 0
\(99\) 1.24969e16i 0.136114i
\(100\) 0 0
\(101\) − 3.80297e16i − 0.349455i −0.984617 0.174728i \(-0.944096\pi\)
0.984617 0.174728i \(-0.0559045\pi\)
\(102\) 0 0
\(103\) 2.07629e17 1.61500 0.807498 0.589870i \(-0.200821\pi\)
0.807498 + 0.589870i \(0.200821\pi\)
\(104\) 0 0
\(105\) −6.98638e16 −0.461470
\(106\) 0 0
\(107\) − 1.68191e17i − 0.946325i −0.880975 0.473162i \(-0.843112\pi\)
0.880975 0.473162i \(-0.156888\pi\)
\(108\) 0 0
\(109\) 4.08973e15i 0.0196593i 0.999952 + 0.00982967i \(0.00312893\pi\)
−0.999952 + 0.00982967i \(0.996871\pi\)
\(110\) 0 0
\(111\) 2.95218e17 1.21590
\(112\) 0 0
\(113\) 2.83148e16 0.100195 0.0500976 0.998744i \(-0.484047\pi\)
0.0500976 + 0.998744i \(0.484047\pi\)
\(114\) 0 0
\(115\) − 2.10325e17i − 0.641149i
\(116\) 0 0
\(117\) − 2.99995e17i − 0.789830i
\(118\) 0 0
\(119\) 2.55628e17 0.582720
\(120\) 0 0
\(121\) 4.66297e17 0.922544
\(122\) 0 0
\(123\) 1.17566e18i 2.02343i
\(124\) 0 0
\(125\) 7.20642e17i 1.08139i
\(126\) 0 0
\(127\) 1.04849e17 0.137478 0.0687388 0.997635i \(-0.478102\pi\)
0.0687388 + 0.997635i \(0.478102\pi\)
\(128\) 0 0
\(129\) 1.91698e18 2.20093
\(130\) 0 0
\(131\) − 7.82237e17i − 0.788011i −0.919108 0.394005i \(-0.871089\pi\)
0.919108 0.394005i \(-0.128911\pi\)
\(132\) 0 0
\(133\) 7.63817e17i 0.676473i
\(134\) 0 0
\(135\) −6.08509e17 −0.474712
\(136\) 0 0
\(137\) 6.27442e17 0.431965 0.215983 0.976397i \(-0.430705\pi\)
0.215983 + 0.976397i \(0.430705\pi\)
\(138\) 0 0
\(139\) 2.57262e17i 0.156585i 0.996930 + 0.0782924i \(0.0249468\pi\)
−0.996930 + 0.0782924i \(0.975053\pi\)
\(140\) 0 0
\(141\) − 1.13915e18i − 0.614069i
\(142\) 0 0
\(143\) 9.39811e17 0.449453
\(144\) 0 0
\(145\) −2.35625e18 −1.00137
\(146\) 0 0
\(147\) − 2.43015e18i − 0.919255i
\(148\) 0 0
\(149\) − 5.12586e18i − 1.72855i −0.503016 0.864277i \(-0.667776\pi\)
0.503016 0.864277i \(-0.332224\pi\)
\(150\) 0 0
\(151\) 1.51399e18 0.455848 0.227924 0.973679i \(-0.426806\pi\)
0.227924 + 0.973679i \(0.426806\pi\)
\(152\) 0 0
\(153\) −2.13129e18 −0.573807
\(154\) 0 0
\(155\) − 1.84198e17i − 0.0444085i
\(156\) 0 0
\(157\) 6.31795e18i 1.36593i 0.730450 + 0.682966i \(0.239310\pi\)
−0.730450 + 0.682966i \(0.760690\pi\)
\(158\) 0 0
\(159\) −4.38739e18 −0.851787
\(160\) 0 0
\(161\) −2.39571e18 −0.418229
\(162\) 0 0
\(163\) 7.62696e18i 1.19883i 0.800439 + 0.599414i \(0.204600\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(164\) 0 0
\(165\) 1.82479e18i 0.258583i
\(166\) 0 0
\(167\) −7.60518e18 −0.972791 −0.486396 0.873739i \(-0.661689\pi\)
−0.486396 + 0.873739i \(0.661689\pi\)
\(168\) 0 0
\(169\) −1.39103e19 −1.60805
\(170\) 0 0
\(171\) − 6.36830e18i − 0.666126i
\(172\) 0 0
\(173\) 9.85927e18i 0.934228i 0.884197 + 0.467114i \(0.154706\pi\)
−0.884197 + 0.467114i \(0.845294\pi\)
\(174\) 0 0
\(175\) 2.42893e18 0.208734
\(176\) 0 0
\(177\) −4.57112e18 −0.356657
\(178\) 0 0
\(179\) 1.05371e19i 0.747257i 0.927578 + 0.373628i \(0.121886\pi\)
−0.927578 + 0.373628i \(0.878114\pi\)
\(180\) 0 0
\(181\) − 2.64819e19i − 1.70876i −0.519649 0.854380i \(-0.673937\pi\)
0.519649 0.854380i \(-0.326063\pi\)
\(182\) 0 0
\(183\) −9.16251e18 −0.538493
\(184\) 0 0
\(185\) 1.41584e19 0.758672
\(186\) 0 0
\(187\) − 6.67682e18i − 0.326525i
\(188\) 0 0
\(189\) 6.93123e18i 0.309660i
\(190\) 0 0
\(191\) −2.32908e19 −0.951484 −0.475742 0.879585i \(-0.657820\pi\)
−0.475742 + 0.879585i \(0.657820\pi\)
\(192\) 0 0
\(193\) −2.61228e17 −0.00976748 −0.00488374 0.999988i \(-0.501555\pi\)
−0.00488374 + 0.999988i \(0.501555\pi\)
\(194\) 0 0
\(195\) − 4.38052e19i − 1.50048i
\(196\) 0 0
\(197\) − 6.46253e18i − 0.202974i −0.994837 0.101487i \(-0.967640\pi\)
0.994837 0.101487i \(-0.0323600\pi\)
\(198\) 0 0
\(199\) 1.82250e19 0.525311 0.262655 0.964890i \(-0.415402\pi\)
0.262655 + 0.964890i \(0.415402\pi\)
\(200\) 0 0
\(201\) 5.05198e19 1.33750
\(202\) 0 0
\(203\) 2.68388e19i 0.653202i
\(204\) 0 0
\(205\) 5.63836e19i 1.26254i
\(206\) 0 0
\(207\) 1.99741e19 0.411832
\(208\) 0 0
\(209\) 1.99503e19 0.379059
\(210\) 0 0
\(211\) 7.73859e19i 1.35600i 0.735060 + 0.678002i \(0.237154\pi\)
−0.735060 + 0.678002i \(0.762846\pi\)
\(212\) 0 0
\(213\) − 1.27448e20i − 2.06113i
\(214\) 0 0
\(215\) 9.19369e19 1.37329
\(216\) 0 0
\(217\) −2.09811e18 −0.0289682
\(218\) 0 0
\(219\) − 7.54743e19i − 0.963882i
\(220\) 0 0
\(221\) 1.60281e20i 1.89473i
\(222\) 0 0
\(223\) 1.34556e20 1.47337 0.736685 0.676236i \(-0.236390\pi\)
0.736685 + 0.676236i \(0.236390\pi\)
\(224\) 0 0
\(225\) −2.02511e19 −0.205541
\(226\) 0 0
\(227\) 7.95558e19i 0.748949i 0.927237 + 0.374474i \(0.122177\pi\)
−0.927237 + 0.374474i \(0.877823\pi\)
\(228\) 0 0
\(229\) 3.46397e19i 0.302673i 0.988482 + 0.151336i \(0.0483576\pi\)
−0.988482 + 0.151336i \(0.951642\pi\)
\(230\) 0 0
\(231\) 2.07853e19 0.168676
\(232\) 0 0
\(233\) −1.21528e20 −0.916538 −0.458269 0.888814i \(-0.651530\pi\)
−0.458269 + 0.888814i \(0.651530\pi\)
\(234\) 0 0
\(235\) − 5.46325e19i − 0.383155i
\(236\) 0 0
\(237\) − 1.47546e20i − 0.962870i
\(238\) 0 0
\(239\) 2.25849e19 0.137226 0.0686129 0.997643i \(-0.478143\pi\)
0.0686129 + 0.997643i \(0.478143\pi\)
\(240\) 0 0
\(241\) 2.85622e20 1.61677 0.808383 0.588657i \(-0.200343\pi\)
0.808383 + 0.588657i \(0.200343\pi\)
\(242\) 0 0
\(243\) − 1.70895e20i − 0.901731i
\(244\) 0 0
\(245\) − 1.16548e20i − 0.573578i
\(246\) 0 0
\(247\) −4.78920e20 −2.19957
\(248\) 0 0
\(249\) 1.30199e20 0.558358
\(250\) 0 0
\(251\) 1.47186e20i 0.589713i 0.955541 + 0.294857i \(0.0952719\pi\)
−0.955541 + 0.294857i \(0.904728\pi\)
\(252\) 0 0
\(253\) 6.25741e19i 0.234353i
\(254\) 0 0
\(255\) −3.11211e20 −1.09009
\(256\) 0 0
\(257\) −3.49632e20 −1.14599 −0.572993 0.819560i \(-0.694218\pi\)
−0.572993 + 0.819560i \(0.694218\pi\)
\(258\) 0 0
\(259\) − 1.61271e20i − 0.494890i
\(260\) 0 0
\(261\) − 2.23768e20i − 0.643211i
\(262\) 0 0
\(263\) 2.56287e20 0.690402 0.345201 0.938529i \(-0.387811\pi\)
0.345201 + 0.938529i \(0.387811\pi\)
\(264\) 0 0
\(265\) −2.10415e20 −0.531481
\(266\) 0 0
\(267\) 3.10860e20i 0.736579i
\(268\) 0 0
\(269\) 6.85596e20i 1.52466i 0.647187 + 0.762331i \(0.275945\pi\)
−0.647187 + 0.762331i \(0.724055\pi\)
\(270\) 0 0
\(271\) −5.06814e20 −1.05830 −0.529151 0.848528i \(-0.677489\pi\)
−0.529151 + 0.848528i \(0.677489\pi\)
\(272\) 0 0
\(273\) −4.98964e20 −0.978783
\(274\) 0 0
\(275\) − 6.34419e19i − 0.116963i
\(276\) 0 0
\(277\) 5.87639e20i 1.01867i 0.860569 + 0.509334i \(0.170108\pi\)
−0.860569 + 0.509334i \(0.829892\pi\)
\(278\) 0 0
\(279\) 1.74929e19 0.0285251
\(280\) 0 0
\(281\) −5.24336e20 −0.804647 −0.402324 0.915498i \(-0.631797\pi\)
−0.402324 + 0.915498i \(0.631797\pi\)
\(282\) 0 0
\(283\) 7.51546e20i 1.08585i 0.839780 + 0.542927i \(0.182684\pi\)
−0.839780 + 0.542927i \(0.817316\pi\)
\(284\) 0 0
\(285\) − 9.29898e20i − 1.26548i
\(286\) 0 0
\(287\) 6.42237e20 0.823568
\(288\) 0 0
\(289\) 3.11465e20 0.376511
\(290\) 0 0
\(291\) − 5.56023e20i − 0.633873i
\(292\) 0 0
\(293\) − 7.51033e20i − 0.807764i −0.914811 0.403882i \(-0.867661\pi\)
0.914811 0.403882i \(-0.132339\pi\)
\(294\) 0 0
\(295\) −2.19227e20 −0.222539
\(296\) 0 0
\(297\) 1.81039e20 0.173517
\(298\) 0 0
\(299\) − 1.50213e21i − 1.35988i
\(300\) 0 0
\(301\) − 1.04721e21i − 0.895813i
\(302\) 0 0
\(303\) 5.27366e20 0.426432
\(304\) 0 0
\(305\) −4.39426e20 −0.335998
\(306\) 0 0
\(307\) 5.89249e19i 0.0426209i 0.999773 + 0.0213105i \(0.00678384\pi\)
−0.999773 + 0.0213105i \(0.993216\pi\)
\(308\) 0 0
\(309\) 2.87923e21i 1.97074i
\(310\) 0 0
\(311\) 9.05717e20 0.586853 0.293427 0.955982i \(-0.405204\pi\)
0.293427 + 0.955982i \(0.405204\pi\)
\(312\) 0 0
\(313\) 2.51738e21 1.54462 0.772311 0.635245i \(-0.219101\pi\)
0.772311 + 0.635245i \(0.219101\pi\)
\(314\) 0 0
\(315\) − 3.18200e20i − 0.184953i
\(316\) 0 0
\(317\) − 1.88675e21i − 1.03923i −0.854401 0.519614i \(-0.826076\pi\)
0.854401 0.519614i \(-0.173924\pi\)
\(318\) 0 0
\(319\) 7.01010e20 0.366019
\(320\) 0 0
\(321\) 2.33234e21 1.15478
\(322\) 0 0
\(323\) 3.40245e21i 1.59798i
\(324\) 0 0
\(325\) 1.52296e21i 0.678704i
\(326\) 0 0
\(327\) −5.67131e19 −0.0239898
\(328\) 0 0
\(329\) −6.22292e20 −0.249936
\(330\) 0 0
\(331\) 1.03084e21i 0.393236i 0.980480 + 0.196618i \(0.0629959\pi\)
−0.980480 + 0.196618i \(0.937004\pi\)
\(332\) 0 0
\(333\) 1.34459e21i 0.487321i
\(334\) 0 0
\(335\) 2.42288e21 0.834549
\(336\) 0 0
\(337\) 2.14822e21 0.703435 0.351717 0.936106i \(-0.385598\pi\)
0.351717 + 0.936106i \(0.385598\pi\)
\(338\) 0 0
\(339\) 3.92648e20i 0.122266i
\(340\) 0 0
\(341\) 5.48012e19i 0.0162322i
\(342\) 0 0
\(343\) −3.08980e21 −0.870824
\(344\) 0 0
\(345\) 2.91662e21 0.782380
\(346\) 0 0
\(347\) − 4.67482e21i − 1.19389i −0.802282 0.596945i \(-0.796381\pi\)
0.802282 0.596945i \(-0.203619\pi\)
\(348\) 0 0
\(349\) − 4.72313e21i − 1.14872i −0.818603 0.574360i \(-0.805251\pi\)
0.818603 0.574360i \(-0.194749\pi\)
\(350\) 0 0
\(351\) −4.34594e21 −1.00687
\(352\) 0 0
\(353\) −4.29922e21 −0.949084 −0.474542 0.880233i \(-0.657386\pi\)
−0.474542 + 0.880233i \(0.657386\pi\)
\(354\) 0 0
\(355\) − 6.11228e21i − 1.28606i
\(356\) 0 0
\(357\) 3.54485e21i 0.711079i
\(358\) 0 0
\(359\) 5.26993e21 1.00810 0.504048 0.863675i \(-0.331843\pi\)
0.504048 + 0.863675i \(0.331843\pi\)
\(360\) 0 0
\(361\) −4.68613e21 −0.855073
\(362\) 0 0
\(363\) 6.46624e21i 1.12576i
\(364\) 0 0
\(365\) − 3.61968e21i − 0.601424i
\(366\) 0 0
\(367\) 7.39416e21 1.17281 0.586405 0.810018i \(-0.300543\pi\)
0.586405 + 0.810018i \(0.300543\pi\)
\(368\) 0 0
\(369\) −5.35463e21 −0.810971
\(370\) 0 0
\(371\) 2.39674e21i 0.346691i
\(372\) 0 0
\(373\) − 3.45714e21i − 0.477740i −0.971052 0.238870i \(-0.923223\pi\)
0.971052 0.238870i \(-0.0767769\pi\)
\(374\) 0 0
\(375\) −9.99329e21 −1.31960
\(376\) 0 0
\(377\) −1.68282e22 −2.12391
\(378\) 0 0
\(379\) − 3.17249e21i − 0.382795i −0.981513 0.191398i \(-0.938698\pi\)
0.981513 0.191398i \(-0.0613020\pi\)
\(380\) 0 0
\(381\) 1.45396e21i 0.167761i
\(382\) 0 0
\(383\) −3.27971e21 −0.361947 −0.180974 0.983488i \(-0.557925\pi\)
−0.180974 + 0.983488i \(0.557925\pi\)
\(384\) 0 0
\(385\) 9.96845e20 0.105247
\(386\) 0 0
\(387\) 8.73105e21i 0.882112i
\(388\) 0 0
\(389\) 1.09936e22i 1.06308i 0.847032 + 0.531542i \(0.178387\pi\)
−0.847032 + 0.531542i \(0.821613\pi\)
\(390\) 0 0
\(391\) −1.06718e22 −0.987947
\(392\) 0 0
\(393\) 1.08474e22 0.961591
\(394\) 0 0
\(395\) − 7.07620e21i − 0.600792i
\(396\) 0 0
\(397\) 9.01100e21i 0.732915i 0.930435 + 0.366458i \(0.119430\pi\)
−0.930435 + 0.366458i \(0.880570\pi\)
\(398\) 0 0
\(399\) −1.05920e22 −0.825484
\(400\) 0 0
\(401\) 1.05004e22 0.784296 0.392148 0.919902i \(-0.371732\pi\)
0.392148 + 0.919902i \(0.371732\pi\)
\(402\) 0 0
\(403\) − 1.31553e21i − 0.0941910i
\(404\) 0 0
\(405\) − 1.38628e22i − 0.951664i
\(406\) 0 0
\(407\) −4.21228e21 −0.277310
\(408\) 0 0
\(409\) 5.86541e21 0.370382 0.185191 0.982703i \(-0.440710\pi\)
0.185191 + 0.982703i \(0.440710\pi\)
\(410\) 0 0
\(411\) 8.70087e21i 0.527117i
\(412\) 0 0
\(413\) 2.49711e21i 0.145165i
\(414\) 0 0
\(415\) 6.24423e21 0.348393
\(416\) 0 0
\(417\) −3.56750e21 −0.191077
\(418\) 0 0
\(419\) − 5.02281e21i − 0.258302i −0.991625 0.129151i \(-0.958775\pi\)
0.991625 0.129151i \(-0.0412252\pi\)
\(420\) 0 0
\(421\) − 1.55885e22i − 0.769848i −0.922948 0.384924i \(-0.874228\pi\)
0.922948 0.384924i \(-0.125772\pi\)
\(422\) 0 0
\(423\) 5.18834e21 0.246113
\(424\) 0 0
\(425\) 1.08198e22 0.493074
\(426\) 0 0
\(427\) 5.00528e21i 0.219175i
\(428\) 0 0
\(429\) 1.30326e22i 0.548457i
\(430\) 0 0
\(431\) 2.78561e22 1.12684 0.563421 0.826170i \(-0.309485\pi\)
0.563421 + 0.826170i \(0.309485\pi\)
\(432\) 0 0
\(433\) −1.24470e21 −0.0484079 −0.0242040 0.999707i \(-0.507705\pi\)
−0.0242040 + 0.999707i \(0.507705\pi\)
\(434\) 0 0
\(435\) − 3.26745e22i − 1.22194i
\(436\) 0 0
\(437\) − 3.18872e22i − 1.14690i
\(438\) 0 0
\(439\) 2.09614e22 0.725224 0.362612 0.931940i \(-0.381885\pi\)
0.362612 + 0.931940i \(0.381885\pi\)
\(440\) 0 0
\(441\) 1.10683e22 0.368429
\(442\) 0 0
\(443\) 1.11978e22i 0.358674i 0.983788 + 0.179337i \(0.0573952\pi\)
−0.983788 + 0.179337i \(0.942605\pi\)
\(444\) 0 0
\(445\) 1.49086e22i 0.459596i
\(446\) 0 0
\(447\) 7.10813e22 2.10931
\(448\) 0 0
\(449\) −1.44200e22 −0.411975 −0.205987 0.978555i \(-0.566041\pi\)
−0.205987 + 0.978555i \(0.566041\pi\)
\(450\) 0 0
\(451\) − 1.67748e22i − 0.461483i
\(452\) 0 0
\(453\) 2.09949e22i 0.556261i
\(454\) 0 0
\(455\) −2.39298e22 −0.610721
\(456\) 0 0
\(457\) 2.99851e22 0.737254 0.368627 0.929577i \(-0.379828\pi\)
0.368627 + 0.929577i \(0.379828\pi\)
\(458\) 0 0
\(459\) 3.08754e22i 0.731484i
\(460\) 0 0
\(461\) 4.32706e22i 0.987950i 0.869476 + 0.493975i \(0.164456\pi\)
−0.869476 + 0.493975i \(0.835544\pi\)
\(462\) 0 0
\(463\) 1.79925e22 0.395962 0.197981 0.980206i \(-0.436562\pi\)
0.197981 + 0.980206i \(0.436562\pi\)
\(464\) 0 0
\(465\) 2.55432e21 0.0541907
\(466\) 0 0
\(467\) 5.53566e22i 1.13234i 0.824289 + 0.566169i \(0.191575\pi\)
−0.824289 + 0.566169i \(0.808425\pi\)
\(468\) 0 0
\(469\) − 2.75979e22i − 0.544386i
\(470\) 0 0
\(471\) −8.76123e22 −1.66682
\(472\) 0 0
\(473\) −2.73523e22 −0.501965
\(474\) 0 0
\(475\) 3.23295e22i 0.572404i
\(476\) 0 0
\(477\) − 1.99827e22i − 0.341388i
\(478\) 0 0
\(479\) 1.18108e22 0.194727 0.0973637 0.995249i \(-0.468959\pi\)
0.0973637 + 0.995249i \(0.468959\pi\)
\(480\) 0 0
\(481\) 1.01118e23 1.60915
\(482\) 0 0
\(483\) − 3.32218e22i − 0.510355i
\(484\) 0 0
\(485\) − 2.66664e22i − 0.395511i
\(486\) 0 0
\(487\) −2.16692e22 −0.310347 −0.155173 0.987887i \(-0.549594\pi\)
−0.155173 + 0.987887i \(0.549594\pi\)
\(488\) 0 0
\(489\) −1.05765e23 −1.46290
\(490\) 0 0
\(491\) − 1.30265e23i − 1.74034i −0.492749 0.870172i \(-0.664008\pi\)
0.492749 0.870172i \(-0.335992\pi\)
\(492\) 0 0
\(493\) 1.19555e23i 1.54300i
\(494\) 0 0
\(495\) −8.31116e21 −0.103638
\(496\) 0 0
\(497\) −6.96220e22 −0.838913
\(498\) 0 0
\(499\) 5.17276e22i 0.602376i 0.953565 + 0.301188i \(0.0973832\pi\)
−0.953565 + 0.301188i \(0.902617\pi\)
\(500\) 0 0
\(501\) − 1.05463e23i − 1.18707i
\(502\) 0 0
\(503\) −1.07219e23 −1.16665 −0.583327 0.812237i \(-0.698249\pi\)
−0.583327 + 0.812237i \(0.698249\pi\)
\(504\) 0 0
\(505\) 2.52920e22 0.266076
\(506\) 0 0
\(507\) − 1.92897e23i − 1.96227i
\(508\) 0 0
\(509\) − 1.48328e23i − 1.45922i −0.683861 0.729612i \(-0.739701\pi\)
0.683861 0.729612i \(-0.260299\pi\)
\(510\) 0 0
\(511\) −4.12300e22 −0.392316
\(512\) 0 0
\(513\) −9.22557e22 −0.849172
\(514\) 0 0
\(515\) 1.38086e23i 1.22966i
\(516\) 0 0
\(517\) 1.62538e22i 0.140051i
\(518\) 0 0
\(519\) −1.36721e23 −1.14002
\(520\) 0 0
\(521\) 7.01242e22 0.565910 0.282955 0.959133i \(-0.408685\pi\)
0.282955 + 0.959133i \(0.408685\pi\)
\(522\) 0 0
\(523\) 2.14598e23i 1.67634i 0.545410 + 0.838169i \(0.316374\pi\)
−0.545410 + 0.838169i \(0.683626\pi\)
\(524\) 0 0
\(525\) 3.36825e22i 0.254713i
\(526\) 0 0
\(527\) −9.34612e21 −0.0684291
\(528\) 0 0
\(529\) −4.10359e22 −0.290932
\(530\) 0 0
\(531\) − 2.08195e22i − 0.142945i
\(532\) 0 0
\(533\) 4.02688e23i 2.67786i
\(534\) 0 0
\(535\) 1.11857e23 0.720536
\(536\) 0 0
\(537\) −1.46120e23 −0.911860
\(538\) 0 0
\(539\) 3.46743e22i 0.209654i
\(540\) 0 0
\(541\) − 1.34761e23i − 0.789565i −0.918775 0.394783i \(-0.870820\pi\)
0.918775 0.394783i \(-0.129180\pi\)
\(542\) 0 0
\(543\) 3.67230e23 2.08516
\(544\) 0 0
\(545\) −2.71991e21 −0.0149687
\(546\) 0 0
\(547\) − 2.17960e23i − 1.16274i −0.813638 0.581372i \(-0.802516\pi\)
0.813638 0.581372i \(-0.197484\pi\)
\(548\) 0 0
\(549\) − 4.17314e22i − 0.215823i
\(550\) 0 0
\(551\) −3.57229e23 −1.79126
\(552\) 0 0
\(553\) −8.06015e22 −0.391904
\(554\) 0 0
\(555\) 1.96337e23i 0.925790i
\(556\) 0 0
\(557\) − 7.39588e22i − 0.338237i −0.985596 0.169118i \(-0.945908\pi\)
0.985596 0.169118i \(-0.0540920\pi\)
\(558\) 0 0
\(559\) 6.56608e23 2.91277
\(560\) 0 0
\(561\) 9.25888e22 0.398450
\(562\) 0 0
\(563\) 6.46778e21i 0.0270044i 0.999909 + 0.0135022i \(0.00429801\pi\)
−0.999909 + 0.0135022i \(0.995702\pi\)
\(564\) 0 0
\(565\) 1.88310e22i 0.0762891i
\(566\) 0 0
\(567\) −1.57904e23 −0.620781
\(568\) 0 0
\(569\) 3.40211e23 1.29806 0.649029 0.760764i \(-0.275176\pi\)
0.649029 + 0.760764i \(0.275176\pi\)
\(570\) 0 0
\(571\) − 4.01054e23i − 1.48524i −0.669715 0.742619i \(-0.733584\pi\)
0.669715 0.742619i \(-0.266416\pi\)
\(572\) 0 0
\(573\) − 3.22979e23i − 1.16107i
\(574\) 0 0
\(575\) −1.01401e23 −0.353888
\(576\) 0 0
\(577\) 1.92304e23 0.651619 0.325810 0.945435i \(-0.394363\pi\)
0.325810 + 0.945435i \(0.394363\pi\)
\(578\) 0 0
\(579\) − 3.62250e21i − 0.0119190i
\(580\) 0 0
\(581\) − 7.11249e22i − 0.227261i
\(582\) 0 0
\(583\) 6.26010e22 0.194267
\(584\) 0 0
\(585\) 1.99514e23 0.601380
\(586\) 0 0
\(587\) 1.13316e23i 0.331793i 0.986143 + 0.165896i \(0.0530517\pi\)
−0.986143 + 0.165896i \(0.946948\pi\)
\(588\) 0 0
\(589\) − 2.79262e22i − 0.0794386i
\(590\) 0 0
\(591\) 8.96173e22 0.247684
\(592\) 0 0
\(593\) 4.06278e23 1.09108 0.545542 0.838084i \(-0.316324\pi\)
0.545542 + 0.838084i \(0.316324\pi\)
\(594\) 0 0
\(595\) 1.70008e23i 0.443685i
\(596\) 0 0
\(597\) 2.52730e23i 0.641025i
\(598\) 0 0
\(599\) −5.15444e23 −1.27073 −0.635365 0.772212i \(-0.719150\pi\)
−0.635365 + 0.772212i \(0.719150\pi\)
\(600\) 0 0
\(601\) 3.97639e23 0.952918 0.476459 0.879197i \(-0.341920\pi\)
0.476459 + 0.879197i \(0.341920\pi\)
\(602\) 0 0
\(603\) 2.30096e23i 0.536059i
\(604\) 0 0
\(605\) 3.10115e23i 0.702429i
\(606\) 0 0
\(607\) 6.76629e23 1.49021 0.745104 0.666948i \(-0.232400\pi\)
0.745104 + 0.666948i \(0.232400\pi\)
\(608\) 0 0
\(609\) −3.72180e23 −0.797087
\(610\) 0 0
\(611\) − 3.90182e23i − 0.812675i
\(612\) 0 0
\(613\) 6.69956e23i 1.35716i 0.734525 + 0.678582i \(0.237405\pi\)
−0.734525 + 0.678582i \(0.762595\pi\)
\(614\) 0 0
\(615\) −7.81882e23 −1.54065
\(616\) 0 0
\(617\) −2.38236e23 −0.456649 −0.228325 0.973585i \(-0.573325\pi\)
−0.228325 + 0.973585i \(0.573325\pi\)
\(618\) 0 0
\(619\) 1.75944e23i 0.328099i 0.986452 + 0.164049i \(0.0524556\pi\)
−0.986452 + 0.164049i \(0.947544\pi\)
\(620\) 0 0
\(621\) − 2.89360e23i − 0.525000i
\(622\) 0 0
\(623\) 1.69816e23 0.299799
\(624\) 0 0
\(625\) −2.34643e23 −0.403114
\(626\) 0 0
\(627\) 2.76655e23i 0.462557i
\(628\) 0 0
\(629\) − 7.18388e23i − 1.16904i
\(630\) 0 0
\(631\) 5.34532e23 0.846689 0.423344 0.905969i \(-0.360856\pi\)
0.423344 + 0.905969i \(0.360856\pi\)
\(632\) 0 0
\(633\) −1.07313e24 −1.65470
\(634\) 0 0
\(635\) 6.97306e22i 0.104676i
\(636\) 0 0
\(637\) − 8.32378e23i − 1.21657i
\(638\) 0 0
\(639\) 5.80471e23 0.826082
\(640\) 0 0
\(641\) −1.21663e24 −1.68603 −0.843017 0.537887i \(-0.819223\pi\)
−0.843017 + 0.537887i \(0.819223\pi\)
\(642\) 0 0
\(643\) 2.18465e23i 0.294842i 0.989074 + 0.147421i \(0.0470972\pi\)
−0.989074 + 0.147421i \(0.952903\pi\)
\(644\) 0 0
\(645\) 1.27491e24i 1.67580i
\(646\) 0 0
\(647\) −1.14703e24 −1.46855 −0.734275 0.678852i \(-0.762478\pi\)
−0.734275 + 0.678852i \(0.762478\pi\)
\(648\) 0 0
\(649\) 6.52226e22 0.0813426
\(650\) 0 0
\(651\) − 2.90950e22i − 0.0353492i
\(652\) 0 0
\(653\) 1.50867e23i 0.178580i 0.996006 + 0.0892898i \(0.0284597\pi\)
−0.996006 + 0.0892898i \(0.971540\pi\)
\(654\) 0 0
\(655\) 5.20234e23 0.599995
\(656\) 0 0
\(657\) 3.43754e23 0.386315
\(658\) 0 0
\(659\) − 1.33218e24i − 1.45894i −0.684013 0.729470i \(-0.739767\pi\)
0.684013 0.729470i \(-0.260233\pi\)
\(660\) 0 0
\(661\) − 1.50165e24i − 1.60272i −0.598183 0.801359i \(-0.704111\pi\)
0.598183 0.801359i \(-0.295889\pi\)
\(662\) 0 0
\(663\) −2.22265e24 −2.31210
\(664\) 0 0
\(665\) −5.07983e23 −0.515069
\(666\) 0 0
\(667\) − 1.12045e24i − 1.10744i
\(668\) 0 0
\(669\) 1.86592e24i 1.79792i
\(670\) 0 0
\(671\) 1.30734e23 0.122814
\(672\) 0 0
\(673\) 1.95354e24 1.78935 0.894673 0.446722i \(-0.147409\pi\)
0.894673 + 0.446722i \(0.147409\pi\)
\(674\) 0 0
\(675\) 2.93373e23i 0.262022i
\(676\) 0 0
\(677\) − 1.79219e24i − 1.56092i −0.625208 0.780458i \(-0.714986\pi\)
0.625208 0.780458i \(-0.285014\pi\)
\(678\) 0 0
\(679\) −3.03743e23 −0.257997
\(680\) 0 0
\(681\) −1.10322e24 −0.913925
\(682\) 0 0
\(683\) 2.23745e24i 1.80791i 0.427628 + 0.903955i \(0.359349\pi\)
−0.427628 + 0.903955i \(0.640651\pi\)
\(684\) 0 0
\(685\) 4.17286e23i 0.328900i
\(686\) 0 0
\(687\) −4.80356e23 −0.369344
\(688\) 0 0
\(689\) −1.50278e24 −1.12728
\(690\) 0 0
\(691\) 1.07548e24i 0.787114i 0.919300 + 0.393557i \(0.128756\pi\)
−0.919300 + 0.393557i \(0.871244\pi\)
\(692\) 0 0
\(693\) 9.46683e22i 0.0676040i
\(694\) 0 0
\(695\) −1.71094e23 −0.119224
\(696\) 0 0
\(697\) 2.86087e24 1.94545
\(698\) 0 0
\(699\) − 1.68525e24i − 1.11843i
\(700\) 0 0
\(701\) 4.83049e23i 0.312887i 0.987687 + 0.156444i \(0.0500030\pi\)
−0.987687 + 0.156444i \(0.949997\pi\)
\(702\) 0 0
\(703\) 2.14654e24 1.35712
\(704\) 0 0
\(705\) 7.57600e23 0.467555
\(706\) 0 0
\(707\) − 2.88089e23i − 0.173565i
\(708\) 0 0
\(709\) 6.46817e23i 0.380442i 0.981741 + 0.190221i \(0.0609204\pi\)
−0.981741 + 0.190221i \(0.939080\pi\)
\(710\) 0 0
\(711\) 6.72012e23 0.385909
\(712\) 0 0
\(713\) 8.75904e22 0.0491129
\(714\) 0 0
\(715\) 6.25030e23i 0.342215i
\(716\) 0 0
\(717\) 3.13189e23i 0.167454i
\(718\) 0 0
\(719\) 2.08660e24 1.08954 0.544770 0.838586i \(-0.316617\pi\)
0.544770 + 0.838586i \(0.316617\pi\)
\(720\) 0 0
\(721\) 1.57286e24 0.802124
\(722\) 0 0
\(723\) 3.96078e24i 1.97290i
\(724\) 0 0
\(725\) 1.13599e24i 0.552713i
\(726\) 0 0
\(727\) −2.50710e24 −1.19160 −0.595799 0.803134i \(-0.703164\pi\)
−0.595799 + 0.803134i \(0.703164\pi\)
\(728\) 0 0
\(729\) −3.22019e23 −0.149519
\(730\) 0 0
\(731\) − 4.66482e24i − 2.11611i
\(732\) 0 0
\(733\) 2.50202e24i 1.10894i 0.832205 + 0.554469i \(0.187079\pi\)
−0.832205 + 0.554469i \(0.812921\pi\)
\(734\) 0 0
\(735\) 1.61619e24 0.699924
\(736\) 0 0
\(737\) −7.20836e23 −0.305044
\(738\) 0 0
\(739\) − 1.98150e24i − 0.819440i −0.912211 0.409720i \(-0.865626\pi\)
0.912211 0.409720i \(-0.134374\pi\)
\(740\) 0 0
\(741\) − 6.64128e24i − 2.68409i
\(742\) 0 0
\(743\) 1.48113e24 0.585043 0.292521 0.956259i \(-0.405506\pi\)
0.292521 + 0.956259i \(0.405506\pi\)
\(744\) 0 0
\(745\) 3.40900e24 1.31613
\(746\) 0 0
\(747\) 5.93001e23i 0.223785i
\(748\) 0 0
\(749\) − 1.27411e24i − 0.470013i
\(750\) 0 0
\(751\) 5.06055e24 1.82498 0.912490 0.409100i \(-0.134157\pi\)
0.912490 + 0.409100i \(0.134157\pi\)
\(752\) 0 0
\(753\) −2.04106e24 −0.719614
\(754\) 0 0
\(755\) 1.00690e24i 0.347085i
\(756\) 0 0
\(757\) − 1.15475e23i − 0.0389200i −0.999811 0.0194600i \(-0.993805\pi\)
0.999811 0.0194600i \(-0.00619470\pi\)
\(758\) 0 0
\(759\) −8.67728e23 −0.285975
\(760\) 0 0
\(761\) −3.07347e24 −0.990510 −0.495255 0.868748i \(-0.664925\pi\)
−0.495255 + 0.868748i \(0.664925\pi\)
\(762\) 0 0
\(763\) 3.09811e22i 0.00976425i
\(764\) 0 0
\(765\) − 1.41743e24i − 0.436899i
\(766\) 0 0
\(767\) −1.56571e24 −0.472008
\(768\) 0 0
\(769\) 1.56828e24 0.462434 0.231217 0.972902i \(-0.425729\pi\)
0.231217 + 0.972902i \(0.425729\pi\)
\(770\) 0 0
\(771\) − 4.84842e24i − 1.39842i
\(772\) 0 0
\(773\) 3.06812e24i 0.865659i 0.901476 + 0.432830i \(0.142485\pi\)
−0.901476 + 0.432830i \(0.857515\pi\)
\(774\) 0 0
\(775\) −8.88051e22 −0.0245117
\(776\) 0 0
\(777\) 2.23638e24 0.603903
\(778\) 0 0
\(779\) 8.54827e24i 2.25845i
\(780\) 0 0
\(781\) 1.81847e24i 0.470082i
\(782\) 0 0
\(783\) −3.24166e24 −0.819960
\(784\) 0 0
\(785\) −4.20181e24 −1.04003
\(786\) 0 0
\(787\) − 7.03196e23i − 0.170330i −0.996367 0.0851650i \(-0.972858\pi\)
0.996367 0.0851650i \(-0.0271417\pi\)
\(788\) 0 0
\(789\) 3.55398e24i 0.842482i
\(790\) 0 0
\(791\) 2.14495e23 0.0497642
\(792\) 0 0
\(793\) −3.13835e24 −0.712656
\(794\) 0 0
\(795\) − 2.91788e24i − 0.648554i
\(796\) 0 0
\(797\) 4.69599e24i 1.02172i 0.859664 + 0.510859i \(0.170673\pi\)
−0.859664 + 0.510859i \(0.829327\pi\)
\(798\) 0 0
\(799\) −2.77202e24 −0.590403
\(800\) 0 0
\(801\) −1.41584e24 −0.295214
\(802\) 0 0
\(803\) 1.07690e24i 0.219832i
\(804\) 0 0
\(805\) − 1.59329e24i − 0.318441i
\(806\) 0 0
\(807\) −9.50730e24 −1.86051
\(808\) 0 0
\(809\) −4.37783e24 −0.838873 −0.419436 0.907785i \(-0.637772\pi\)
−0.419436 + 0.907785i \(0.637772\pi\)
\(810\) 0 0
\(811\) 8.61180e24i 1.61591i 0.589245 + 0.807954i \(0.299425\pi\)
−0.589245 + 0.807954i \(0.700575\pi\)
\(812\) 0 0
\(813\) − 7.02809e24i − 1.29142i
\(814\) 0 0
\(815\) −5.07238e24 −0.912793
\(816\) 0 0
\(817\) 1.39385e25 2.45656
\(818\) 0 0
\(819\) − 2.27257e24i − 0.392287i
\(820\) 0 0
\(821\) 4.09648e24i 0.692619i 0.938120 + 0.346309i \(0.112565\pi\)
−0.938120 + 0.346309i \(0.887435\pi\)
\(822\) 0 0
\(823\) 4.02308e24 0.666284 0.333142 0.942877i \(-0.391891\pi\)
0.333142 + 0.942877i \(0.391891\pi\)
\(824\) 0 0
\(825\) 8.79762e23 0.142727
\(826\) 0 0
\(827\) − 6.80277e24i − 1.08116i −0.841294 0.540579i \(-0.818205\pi\)
0.841294 0.540579i \(-0.181795\pi\)
\(828\) 0 0
\(829\) − 9.06827e24i − 1.41192i −0.708251 0.705961i \(-0.750515\pi\)
0.708251 0.705961i \(-0.249485\pi\)
\(830\) 0 0
\(831\) −8.14891e24 −1.24306
\(832\) 0 0
\(833\) −5.91357e24 −0.883827
\(834\) 0 0
\(835\) − 5.05790e24i − 0.740687i
\(836\) 0 0
\(837\) − 2.53415e23i − 0.0363636i
\(838\) 0 0
\(839\) 2.26281e24 0.318179 0.159090 0.987264i \(-0.449144\pi\)
0.159090 + 0.987264i \(0.449144\pi\)
\(840\) 0 0
\(841\) −5.29508e24 −0.729637
\(842\) 0 0
\(843\) − 7.27107e24i − 0.981892i
\(844\) 0 0
\(845\) − 9.25117e24i − 1.22438i
\(846\) 0 0
\(847\) 3.53237e24 0.458202
\(848\) 0 0
\(849\) −1.04219e25 −1.32504
\(850\) 0 0
\(851\) 6.73262e24i 0.839040i
\(852\) 0 0
\(853\) 9.22743e24i 1.12723i 0.826036 + 0.563617i \(0.190591\pi\)
−0.826036 + 0.563617i \(0.809409\pi\)
\(854\) 0 0
\(855\) 4.23529e24 0.507191
\(856\) 0 0
\(857\) 4.23968e24 0.497733 0.248867 0.968538i \(-0.419942\pi\)
0.248867 + 0.968538i \(0.419942\pi\)
\(858\) 0 0
\(859\) − 2.28165e24i − 0.262608i −0.991342 0.131304i \(-0.958084\pi\)
0.991342 0.131304i \(-0.0419164\pi\)
\(860\) 0 0
\(861\) 8.90604e24i 1.00498i
\(862\) 0 0
\(863\) 1.03258e25 1.14244 0.571220 0.820797i \(-0.306470\pi\)
0.571220 + 0.820797i \(0.306470\pi\)
\(864\) 0 0
\(865\) −6.55700e24 −0.711325
\(866\) 0 0
\(867\) 4.31915e24i 0.459447i
\(868\) 0 0
\(869\) 2.10525e24i 0.219602i
\(870\) 0 0
\(871\) 1.73041e25 1.77009
\(872\) 0 0
\(873\) 2.53245e24 0.254050
\(874\) 0 0
\(875\) 5.45912e24i 0.537099i
\(876\) 0 0
\(877\) 1.49582e24i 0.144338i 0.997392 + 0.0721692i \(0.0229921\pi\)
−0.997392 + 0.0721692i \(0.977008\pi\)
\(878\) 0 0
\(879\) 1.04147e25 0.985695
\(880\) 0 0
\(881\) 9.02128e24 0.837477 0.418739 0.908107i \(-0.362472\pi\)
0.418739 + 0.908107i \(0.362472\pi\)
\(882\) 0 0
\(883\) 5.61859e24i 0.511636i 0.966725 + 0.255818i \(0.0823448\pi\)
−0.966725 + 0.255818i \(0.917655\pi\)
\(884\) 0 0
\(885\) − 3.04007e24i − 0.271560i
\(886\) 0 0
\(887\) −1.48729e25 −1.30330 −0.651649 0.758521i \(-0.725922\pi\)
−0.651649 + 0.758521i \(0.725922\pi\)
\(888\) 0 0
\(889\) 7.94267e23 0.0682813
\(890\) 0 0
\(891\) 4.12435e24i 0.347852i
\(892\) 0 0
\(893\) − 8.28280e24i − 0.685393i
\(894\) 0 0
\(895\) −7.00778e24 −0.568964
\(896\) 0 0
\(897\) 2.08303e25 1.65944
\(898\) 0 0
\(899\) − 9.81264e23i − 0.0767059i
\(900\) 0 0
\(901\) 1.06764e25i 0.818959i
\(902\) 0 0
\(903\) 1.45218e25 1.09314
\(904\) 0 0
\(905\) 1.76120e25 1.30106
\(906\) 0 0
\(907\) 1.26650e25i 0.918215i 0.888381 + 0.459107i \(0.151831\pi\)
−0.888381 + 0.459107i \(0.848169\pi\)
\(908\) 0 0
\(909\) 2.40193e24i 0.170910i
\(910\) 0 0
\(911\) −2.22052e25 −1.55077 −0.775387 0.631487i \(-0.782445\pi\)
−0.775387 + 0.631487i \(0.782445\pi\)
\(912\) 0 0
\(913\) −1.85773e24 −0.127345
\(914\) 0 0
\(915\) − 6.09361e24i − 0.410011i
\(916\) 0 0
\(917\) − 5.92572e24i − 0.391383i
\(918\) 0 0
\(919\) 2.34055e25 1.51753 0.758763 0.651367i \(-0.225804\pi\)
0.758763 + 0.651367i \(0.225804\pi\)
\(920\) 0 0
\(921\) −8.17124e23 −0.0520093
\(922\) 0 0
\(923\) − 4.36536e25i − 2.72775i
\(924\) 0 0
\(925\) − 6.82599e24i − 0.418756i
\(926\) 0 0
\(927\) −1.31137e25 −0.789855
\(928\) 0 0
\(929\) 6.30561e24 0.372901 0.186450 0.982464i \(-0.440302\pi\)
0.186450 + 0.982464i \(0.440302\pi\)
\(930\) 0 0
\(931\) − 1.76697e25i − 1.02603i
\(932\) 0 0
\(933\) 1.25598e25i 0.716123i
\(934\) 0 0
\(935\) 4.44048e24 0.248617
\(936\) 0 0
\(937\) −8.24230e23 −0.0453171 −0.0226586 0.999743i \(-0.507213\pi\)
−0.0226586 + 0.999743i \(0.507213\pi\)
\(938\) 0 0
\(939\) 3.49090e25i 1.88487i
\(940\) 0 0
\(941\) − 6.73452e24i − 0.357104i −0.983930 0.178552i \(-0.942859\pi\)
0.983930 0.178552i \(-0.0571413\pi\)
\(942\) 0 0
\(943\) −2.68116e25 −1.39628
\(944\) 0 0
\(945\) −4.60968e24 −0.235776
\(946\) 0 0
\(947\) 2.19407e25i 1.10224i 0.834426 + 0.551119i \(0.185799\pi\)
−0.834426 + 0.551119i \(0.814201\pi\)
\(948\) 0 0
\(949\) − 2.58515e25i − 1.27563i
\(950\) 0 0
\(951\) 2.61639e25 1.26814
\(952\) 0 0
\(953\) 1.04120e25 0.495727 0.247864 0.968795i \(-0.420272\pi\)
0.247864 + 0.968795i \(0.420272\pi\)
\(954\) 0 0
\(955\) − 1.54898e25i − 0.724464i
\(956\) 0 0
\(957\) 9.72106e24i 0.446645i
\(958\) 0 0
\(959\) 4.75310e24 0.214545
\(960\) 0 0
\(961\) −2.24734e25 −0.996598
\(962\) 0 0
\(963\) 1.06228e25i 0.462824i
\(964\) 0 0
\(965\) − 1.73732e23i − 0.00743700i
\(966\) 0 0
\(967\) −2.11036e25 −0.887630 −0.443815 0.896118i \(-0.646375\pi\)
−0.443815 + 0.896118i \(0.646375\pi\)
\(968\) 0 0
\(969\) −4.71825e25 −1.94997
\(970\) 0 0
\(971\) − 6.24704e24i − 0.253694i −0.991922 0.126847i \(-0.959514\pi\)
0.991922 0.126847i \(-0.0404858\pi\)
\(972\) 0 0
\(973\) 1.94885e24i 0.0777713i
\(974\) 0 0
\(975\) −2.11192e25 −0.828206
\(976\) 0 0
\(977\) −1.86564e25 −0.718992 −0.359496 0.933147i \(-0.617051\pi\)
−0.359496 + 0.933147i \(0.617051\pi\)
\(978\) 0 0
\(979\) − 4.43547e24i − 0.167991i
\(980\) 0 0
\(981\) − 2.58304e23i − 0.00961490i
\(982\) 0 0
\(983\) 2.39881e25 0.877589 0.438794 0.898588i \(-0.355406\pi\)
0.438794 + 0.898588i \(0.355406\pi\)
\(984\) 0 0
\(985\) 4.29797e24 0.154545
\(986\) 0 0
\(987\) − 8.62945e24i − 0.304991i
\(988\) 0 0
\(989\) 4.37180e25i 1.51877i
\(990\) 0 0
\(991\) −2.85777e25 −0.975889 −0.487945 0.872875i \(-0.662253\pi\)
−0.487945 + 0.872875i \(0.662253\pi\)
\(992\) 0 0
\(993\) −1.42949e25 −0.479857
\(994\) 0 0
\(995\) 1.21207e25i 0.399974i
\(996\) 0 0
\(997\) − 1.00050e25i − 0.324571i −0.986744 0.162285i \(-0.948113\pi\)
0.986744 0.162285i \(-0.0518865\pi\)
\(998\) 0 0
\(999\) 1.94787e25 0.621232
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 32.18.b.a.17.14 16
4.3 odd 2 8.18.b.a.5.2 yes 16
8.3 odd 2 8.18.b.a.5.1 16
8.5 even 2 inner 32.18.b.a.17.3 16
12.11 even 2 72.18.d.b.37.15 16
24.11 even 2 72.18.d.b.37.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.18.b.a.5.1 16 8.3 odd 2
8.18.b.a.5.2 yes 16 4.3 odd 2
32.18.b.a.17.3 16 8.5 even 2 inner
32.18.b.a.17.14 16 1.1 even 1 trivial
72.18.d.b.37.15 16 12.11 even 2
72.18.d.b.37.16 16 24.11 even 2