Properties

Label 80.16.c.a.49.1
Level $80$
Weight $16$
Character 80.49
Analytic conductor $114.155$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,16,Mod(49,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.49");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 80.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: \(114.154804080\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 29397x^{4} + 153469728x^{2} + 65015354624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(21.5470i\) of defining polynomial
Character \(\chi\) \(=\) 80.49
Dual form 80.16.c.a.49.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6725.99i q^{3} +(-160365. - 69286.1i) q^{5} -1.29717e6i q^{7} -3.08900e7 q^{9} +O(q^{10})\) \(q-6725.99i q^{3} +(-160365. - 69286.1i) q^{5} -1.29717e6i q^{7} -3.08900e7 q^{9} +2.48560e7 q^{11} -1.64842e8i q^{13} +(-4.66018e8 + 1.07862e9i) q^{15} +2.49651e9i q^{17} -3.47974e9 q^{19} -8.72472e9 q^{21} -1.39929e10i q^{23} +(2.09165e10 + 2.22222e10i) q^{25} +1.11256e11i q^{27} +7.33587e10 q^{29} -6.63708e9 q^{31} -1.67181e11i q^{33} +(-8.98755e10 + 2.08020e11i) q^{35} +8.47925e11i q^{37} -1.10873e12 q^{39} -7.77655e11 q^{41} +2.03959e12i q^{43} +(4.95369e12 + 2.14025e12i) q^{45} +3.83108e12i q^{47} +3.06492e12 q^{49} +1.67915e13 q^{51} -2.49472e12i q^{53} +(-3.98604e12 - 1.72217e12i) q^{55} +2.34047e13i q^{57} -1.33880e12 q^{59} +7.16455e12 q^{61} +4.00695e13i q^{63} +(-1.14213e13 + 2.64349e13i) q^{65} -6.61388e12i q^{67} -9.41158e13 q^{69} -1.42046e13 q^{71} -5.85350e13i q^{73} +(1.49466e14 - 1.40684e14i) q^{75} -3.22423e13i q^{77} -2.48505e14 q^{79} +3.05065e14 q^{81} +6.66100e13i q^{83} +(1.72973e14 - 4.00353e14i) q^{85} -4.93410e14i q^{87} -5.07426e14 q^{89} -2.13827e14 q^{91} +4.46409e13i q^{93} +(5.58029e14 + 2.41097e14i) q^{95} -1.18734e15i q^{97} -7.67802e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 238350 q^{5} - 11416662 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 238350 q^{5} - 11416662 q^{9} + 108590088 q^{11} - 297743400 q^{15} + 3630995640 q^{19} - 8917537608 q^{21} + 18250878750 q^{25} + 286168468740 q^{29} + 276236748288 q^{31} - 1171274911800 q^{35} - 2186980965936 q^{39} - 6153278882388 q^{41} + 10442765857950 q^{45} + 11613390856242 q^{49} + 43487373385728 q^{51} - 33977390365800 q^{55} - 14903258326680 q^{59} - 11352061428588 q^{61} - 50675287275600 q^{65} - 150489671962824 q^{69} - 131693145807312 q^{71} + 360965926890000 q^{75} + 26081853939360 q^{79} + 11\!\cdots\!46 q^{81}+ \cdots - 506999099666376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 6725.99i 1.77561i −0.460223 0.887803i \(-0.652230\pi\)
0.460223 0.887803i \(-0.347770\pi\)
\(4\) 0 0
\(5\) −160365. 69286.1i −0.917984 0.396617i
\(6\) 0 0
\(7\) 1.29717e6i 0.595333i −0.954670 0.297667i \(-0.903792\pi\)
0.954670 0.297667i \(-0.0962084\pi\)
\(8\) 0 0
\(9\) −3.08900e7 −2.15278
\(10\) 0 0
\(11\) 2.48560e7 0.384579 0.192290 0.981338i \(-0.438409\pi\)
0.192290 + 0.981338i \(0.438409\pi\)
\(12\) 0 0
\(13\) 1.64842e8i 0.728606i −0.931281 0.364303i \(-0.881307\pi\)
0.931281 0.364303i \(-0.118693\pi\)
\(14\) 0 0
\(15\) −4.66018e8 + 1.07862e9i −0.704235 + 1.62998i
\(16\) 0 0
\(17\) 2.49651e9i 1.47559i 0.675024 + 0.737795i \(0.264133\pi\)
−0.675024 + 0.737795i \(0.735867\pi\)
\(18\) 0 0
\(19\) −3.47974e9 −0.893088 −0.446544 0.894762i \(-0.647345\pi\)
−0.446544 + 0.894762i \(0.647345\pi\)
\(20\) 0 0
\(21\) −8.72472e9 −1.05708
\(22\) 0 0
\(23\) 1.39929e10i 0.856934i −0.903557 0.428467i \(-0.859054\pi\)
0.903557 0.428467i \(-0.140946\pi\)
\(24\) 0 0
\(25\) 2.09165e10 + 2.22222e10i 0.685390 + 0.728176i
\(26\) 0 0
\(27\) 1.11256e11i 2.04688i
\(28\) 0 0
\(29\) 7.33587e10 0.789709 0.394854 0.918744i \(-0.370795\pi\)
0.394854 + 0.918744i \(0.370795\pi\)
\(30\) 0 0
\(31\) −6.63708e9 −0.0433276 −0.0216638 0.999765i \(-0.506896\pi\)
−0.0216638 + 0.999765i \(0.506896\pi\)
\(32\) 0 0
\(33\) 1.67181e11i 0.682862i
\(34\) 0 0
\(35\) −8.98755e10 + 2.08020e11i −0.236119 + 0.546506i
\(36\) 0 0
\(37\) 8.47925e11i 1.46840i 0.678933 + 0.734201i \(0.262443\pi\)
−0.678933 + 0.734201i \(0.737557\pi\)
\(38\) 0 0
\(39\) −1.10873e12 −1.29372
\(40\) 0 0
\(41\) −7.77655e11 −0.623603 −0.311801 0.950147i \(-0.600932\pi\)
−0.311801 + 0.950147i \(0.600932\pi\)
\(42\) 0 0
\(43\) 2.03959e12i 1.14427i 0.820158 + 0.572137i \(0.193885\pi\)
−0.820158 + 0.572137i \(0.806115\pi\)
\(44\) 0 0
\(45\) 4.95369e12 + 2.14025e12i 1.97622 + 0.853828i
\(46\) 0 0
\(47\) 3.83108e12i 1.10303i 0.834165 + 0.551514i \(0.185950\pi\)
−0.834165 + 0.551514i \(0.814050\pi\)
\(48\) 0 0
\(49\) 3.06492e12 0.645578
\(50\) 0 0
\(51\) 1.67915e13 2.62007
\(52\) 0 0
\(53\) 2.49472e12i 0.291711i −0.989306 0.145856i \(-0.953407\pi\)
0.989306 0.145856i \(-0.0465935\pi\)
\(54\) 0 0
\(55\) −3.98604e12 1.72217e12i −0.353038 0.152531i
\(56\) 0 0
\(57\) 2.34047e13i 1.58577i
\(58\) 0 0
\(59\) −1.33880e12 −0.0700365 −0.0350183 0.999387i \(-0.511149\pi\)
−0.0350183 + 0.999387i \(0.511149\pi\)
\(60\) 0 0
\(61\) 7.16455e12 0.291887 0.145944 0.989293i \(-0.453378\pi\)
0.145944 + 0.989293i \(0.453378\pi\)
\(62\) 0 0
\(63\) 4.00695e13i 1.28162i
\(64\) 0 0
\(65\) −1.14213e13 + 2.64349e13i −0.288977 + 0.668849i
\(66\) 0 0
\(67\) 6.61388e12i 0.133320i −0.997776 0.0666600i \(-0.978766\pi\)
0.997776 0.0666600i \(-0.0212343\pi\)
\(68\) 0 0
\(69\) −9.41158e13 −1.52158
\(70\) 0 0
\(71\) −1.42046e13 −0.185349 −0.0926746 0.995696i \(-0.529542\pi\)
−0.0926746 + 0.995696i \(0.529542\pi\)
\(72\) 0 0
\(73\) 5.85350e13i 0.620147i −0.950713 0.310073i \(-0.899646\pi\)
0.950713 0.310073i \(-0.100354\pi\)
\(74\) 0 0
\(75\) 1.49466e14 1.40684e14i 1.29295 1.21698i
\(76\) 0 0
\(77\) 3.22423e13i 0.228953i
\(78\) 0 0
\(79\) −2.48505e14 −1.45590 −0.727950 0.685630i \(-0.759527\pi\)
−0.727950 + 0.685630i \(0.759527\pi\)
\(80\) 0 0
\(81\) 3.05065e14 1.48168
\(82\) 0 0
\(83\) 6.66100e13i 0.269435i 0.990884 + 0.134717i \(0.0430127\pi\)
−0.990884 + 0.134717i \(0.956987\pi\)
\(84\) 0 0
\(85\) 1.72973e14 4.00353e14i 0.585244 1.35457i
\(86\) 0 0
\(87\) 4.93410e14i 1.40221i
\(88\) 0 0
\(89\) −5.07426e14 −1.21604 −0.608020 0.793922i \(-0.708036\pi\)
−0.608020 + 0.793922i \(0.708036\pi\)
\(90\) 0 0
\(91\) −2.13827e14 −0.433763
\(92\) 0 0
\(93\) 4.46409e13i 0.0769327i
\(94\) 0 0
\(95\) 5.58029e14 + 2.41097e14i 0.819841 + 0.354214i
\(96\) 0 0
\(97\) 1.18734e15i 1.49206i −0.665912 0.746030i \(-0.731957\pi\)
0.665912 0.746030i \(-0.268043\pi\)
\(98\) 0 0
\(99\) −7.67802e14 −0.827915
\(100\) 0 0
\(101\) 1.97970e14 0.183734 0.0918671 0.995771i \(-0.470717\pi\)
0.0918671 + 0.995771i \(0.470717\pi\)
\(102\) 0 0
\(103\) 1.50015e15i 1.20186i −0.799301 0.600931i \(-0.794797\pi\)
0.799301 0.600931i \(-0.205203\pi\)
\(104\) 0 0
\(105\) 1.39914e15 + 6.04502e14i 0.970381 + 0.419255i
\(106\) 0 0
\(107\) 6.89575e14i 0.415148i 0.978219 + 0.207574i \(0.0665568\pi\)
−0.978219 + 0.207574i \(0.933443\pi\)
\(108\) 0 0
\(109\) 1.35369e15 0.709287 0.354644 0.935002i \(-0.384602\pi\)
0.354644 + 0.935002i \(0.384602\pi\)
\(110\) 0 0
\(111\) 5.70314e15 2.60730
\(112\) 0 0
\(113\) 2.28009e15i 0.911726i 0.890050 + 0.455863i \(0.150669\pi\)
−0.890050 + 0.455863i \(0.849331\pi\)
\(114\) 0 0
\(115\) −9.69510e14 + 2.24397e15i −0.339874 + 0.786652i
\(116\) 0 0
\(117\) 5.09197e15i 1.56853i
\(118\) 0 0
\(119\) 3.23838e15 0.878468
\(120\) 0 0
\(121\) −3.55943e15 −0.852099
\(122\) 0 0
\(123\) 5.23050e15i 1.10727i
\(124\) 0 0
\(125\) −1.81459e15 5.01288e15i −0.340371 0.940291i
\(126\) 0 0
\(127\) 2.82207e15i 0.469938i 0.972003 + 0.234969i \(0.0754988\pi\)
−0.972003 + 0.234969i \(0.924501\pi\)
\(128\) 0 0
\(129\) 1.37183e16 2.03178
\(130\) 0 0
\(131\) −6.54539e15 −0.863775 −0.431888 0.901927i \(-0.642152\pi\)
−0.431888 + 0.901927i \(0.642152\pi\)
\(132\) 0 0
\(133\) 4.51379e15i 0.531685i
\(134\) 0 0
\(135\) 7.70846e15 1.78415e16i 0.811828 1.87901i
\(136\) 0 0
\(137\) 1.17625e16i 1.10941i −0.832046 0.554707i \(-0.812830\pi\)
0.832046 0.554707i \(-0.187170\pi\)
\(138\) 0 0
\(139\) 2.03768e16 1.72395 0.861976 0.506949i \(-0.169227\pi\)
0.861976 + 0.506949i \(0.169227\pi\)
\(140\) 0 0
\(141\) 2.57678e16 1.95855
\(142\) 0 0
\(143\) 4.09731e15i 0.280207i
\(144\) 0 0
\(145\) −1.17642e16 5.08274e15i −0.724940 0.313212i
\(146\) 0 0
\(147\) 2.06146e16i 1.14629i
\(148\) 0 0
\(149\) 2.89572e16 1.45499 0.727493 0.686115i \(-0.240685\pi\)
0.727493 + 0.686115i \(0.240685\pi\)
\(150\) 0 0
\(151\) −1.77226e13 −0.000805748 −0.000402874 1.00000i \(-0.500128\pi\)
−0.000402874 1.00000i \(0.500128\pi\)
\(152\) 0 0
\(153\) 7.71172e16i 3.17662i
\(154\) 0 0
\(155\) 1.06436e15 + 4.59857e14i 0.0397740 + 0.0171844i
\(156\) 0 0
\(157\) 1.04137e16i 0.353475i 0.984258 + 0.176737i \(0.0565543\pi\)
−0.984258 + 0.176737i \(0.943446\pi\)
\(158\) 0 0
\(159\) −1.67795e16 −0.517964
\(160\) 0 0
\(161\) −1.81510e16 −0.510161
\(162\) 0 0
\(163\) 4.70622e16i 1.20577i −0.797827 0.602886i \(-0.794017\pi\)
0.797827 0.602886i \(-0.205983\pi\)
\(164\) 0 0
\(165\) −1.15833e16 + 2.68100e16i −0.270834 + 0.626856i
\(166\) 0 0
\(167\) 3.97751e16i 0.849646i 0.905276 + 0.424823i \(0.139664\pi\)
−0.905276 + 0.424823i \(0.860336\pi\)
\(168\) 0 0
\(169\) 2.40130e16 0.469134
\(170\) 0 0
\(171\) 1.07489e17 1.92262
\(172\) 0 0
\(173\) 6.30223e16i 1.03311i 0.856253 + 0.516557i \(0.172787\pi\)
−0.856253 + 0.516557i \(0.827213\pi\)
\(174\) 0 0
\(175\) 2.88258e16 2.71321e16i 0.433507 0.408036i
\(176\) 0 0
\(177\) 9.00474e15i 0.124357i
\(178\) 0 0
\(179\) −2.10684e16 −0.267445 −0.133722 0.991019i \(-0.542693\pi\)
−0.133722 + 0.991019i \(0.542693\pi\)
\(180\) 0 0
\(181\) 1.74536e16 0.203843 0.101922 0.994792i \(-0.467501\pi\)
0.101922 + 0.994792i \(0.467501\pi\)
\(182\) 0 0
\(183\) 4.81887e16i 0.518277i
\(184\) 0 0
\(185\) 5.87494e16 1.35978e17i 0.582392 1.34797i
\(186\) 0 0
\(187\) 6.20531e16i 0.567482i
\(188\) 0 0
\(189\) 1.44317e17 1.21858
\(190\) 0 0
\(191\) 1.82919e17 1.42728 0.713639 0.700514i \(-0.247046\pi\)
0.713639 + 0.700514i \(0.247046\pi\)
\(192\) 0 0
\(193\) 4.54299e16i 0.327840i −0.986474 0.163920i \(-0.947586\pi\)
0.986474 0.163920i \(-0.0524139\pi\)
\(194\) 0 0
\(195\) 1.77801e17 + 7.68192e16i 1.18761 + 0.513110i
\(196\) 0 0
\(197\) 2.54430e17i 1.57425i 0.616797 + 0.787123i \(0.288430\pi\)
−0.616797 + 0.787123i \(0.711570\pi\)
\(198\) 0 0
\(199\) −1.75148e17 −1.00463 −0.502316 0.864684i \(-0.667518\pi\)
−0.502316 + 0.864684i \(0.667518\pi\)
\(200\) 0 0
\(201\) −4.44849e16 −0.236724
\(202\) 0 0
\(203\) 9.51584e16i 0.470140i
\(204\) 0 0
\(205\) 1.24709e17 + 5.38807e16i 0.572457 + 0.247331i
\(206\) 0 0
\(207\) 4.32240e17i 1.84479i
\(208\) 0 0
\(209\) −8.64923e16 −0.343463
\(210\) 0 0
\(211\) −3.58937e17 −1.32709 −0.663545 0.748137i \(-0.730949\pi\)
−0.663545 + 0.748137i \(0.730949\pi\)
\(212\) 0 0
\(213\) 9.55399e16i 0.329107i
\(214\) 0 0
\(215\) 1.41315e17 3.27080e17i 0.453838 1.05043i
\(216\) 0 0
\(217\) 8.60939e15i 0.0257943i
\(218\) 0 0
\(219\) −3.93706e17 −1.10114
\(220\) 0 0
\(221\) 4.11529e17 1.07512
\(222\) 0 0
\(223\) 8.51111e16i 0.207826i 0.994586 + 0.103913i \(0.0331364\pi\)
−0.994586 + 0.103913i \(0.966864\pi\)
\(224\) 0 0
\(225\) −6.46110e17 6.86443e17i −1.47549 1.56760i
\(226\) 0 0
\(227\) 8.22273e16i 0.175720i 0.996133 + 0.0878602i \(0.0280029\pi\)
−0.996133 + 0.0878602i \(0.971997\pi\)
\(228\) 0 0
\(229\) −3.91258e17 −0.782884 −0.391442 0.920203i \(-0.628024\pi\)
−0.391442 + 0.920203i \(0.628024\pi\)
\(230\) 0 0
\(231\) −2.16862e17 −0.406530
\(232\) 0 0
\(233\) 9.50055e17i 1.66947i −0.550650 0.834736i \(-0.685620\pi\)
0.550650 0.834736i \(-0.314380\pi\)
\(234\) 0 0
\(235\) 2.65440e17 6.14371e17i 0.437480 1.01256i
\(236\) 0 0
\(237\) 1.67144e18i 2.58511i
\(238\) 0 0
\(239\) −3.05534e17 −0.443685 −0.221843 0.975082i \(-0.571207\pi\)
−0.221843 + 0.975082i \(0.571207\pi\)
\(240\) 0 0
\(241\) −1.24585e18 −1.69956 −0.849780 0.527137i \(-0.823266\pi\)
−0.849780 + 0.527137i \(0.823266\pi\)
\(242\) 0 0
\(243\) 4.55471e17i 0.584001i
\(244\) 0 0
\(245\) −4.91507e17 2.12357e17i −0.592631 0.256047i
\(246\) 0 0
\(247\) 5.73606e17i 0.650709i
\(248\) 0 0
\(249\) 4.48018e17 0.478411
\(250\) 0 0
\(251\) −3.78704e17 −0.380844 −0.190422 0.981702i \(-0.560986\pi\)
−0.190422 + 0.981702i \(0.560986\pi\)
\(252\) 0 0
\(253\) 3.47806e17i 0.329559i
\(254\) 0 0
\(255\) −2.69277e18 1.16342e18i −2.40518 1.03916i
\(256\) 0 0
\(257\) 6.98680e16i 0.0588545i 0.999567 + 0.0294273i \(0.00936834\pi\)
−0.999567 + 0.0294273i \(0.990632\pi\)
\(258\) 0 0
\(259\) 1.09990e18 0.874188
\(260\) 0 0
\(261\) −2.26605e18 −1.70007
\(262\) 0 0
\(263\) 6.94838e17i 0.492284i 0.969234 + 0.246142i \(0.0791629\pi\)
−0.969234 + 0.246142i \(0.920837\pi\)
\(264\) 0 0
\(265\) −1.72849e17 + 4.00067e17i −0.115698 + 0.267786i
\(266\) 0 0
\(267\) 3.41294e18i 2.15921i
\(268\) 0 0
\(269\) 1.75703e18 1.05108 0.525542 0.850768i \(-0.323863\pi\)
0.525542 + 0.850768i \(0.323863\pi\)
\(270\) 0 0
\(271\) 1.86384e18 1.05472 0.527362 0.849641i \(-0.323181\pi\)
0.527362 + 0.849641i \(0.323181\pi\)
\(272\) 0 0
\(273\) 1.43820e18i 0.770193i
\(274\) 0 0
\(275\) 5.19899e17 + 5.52354e17i 0.263587 + 0.280041i
\(276\) 0 0
\(277\) 3.28806e18i 1.57885i 0.613846 + 0.789425i \(0.289621\pi\)
−0.613846 + 0.789425i \(0.710379\pi\)
\(278\) 0 0
\(279\) 2.05020e17 0.0932747
\(280\) 0 0
\(281\) 2.41481e18 1.04132 0.520662 0.853763i \(-0.325685\pi\)
0.520662 + 0.853763i \(0.325685\pi\)
\(282\) 0 0
\(283\) 1.11571e17i 0.0456199i −0.999740 0.0228099i \(-0.992739\pi\)
0.999740 0.0228099i \(-0.00726126\pi\)
\(284\) 0 0
\(285\) 1.62162e18 3.75330e18i 0.628944 1.45571i
\(286\) 0 0
\(287\) 1.00875e18i 0.371251i
\(288\) 0 0
\(289\) −3.37012e18 −1.17737
\(290\) 0 0
\(291\) −7.98602e18 −2.64931
\(292\) 0 0
\(293\) 2.59868e18i 0.818927i 0.912327 + 0.409463i \(0.134284\pi\)
−0.912327 + 0.409463i \(0.865716\pi\)
\(294\) 0 0
\(295\) 2.14697e17 + 9.27600e16i 0.0642924 + 0.0277777i
\(296\) 0 0
\(297\) 2.76537e18i 0.787189i
\(298\) 0 0
\(299\) −2.30661e18 −0.624367
\(300\) 0 0
\(301\) 2.64569e18 0.681224
\(302\) 0 0
\(303\) 1.33155e18i 0.326240i
\(304\) 0 0
\(305\) −1.14895e18 4.96404e17i −0.267948 0.115767i
\(306\) 0 0
\(307\) 7.96608e18i 1.76892i 0.466621 + 0.884458i \(0.345471\pi\)
−0.466621 + 0.884458i \(0.654529\pi\)
\(308\) 0 0
\(309\) −1.00900e19 −2.13403
\(310\) 0 0
\(311\) 8.72923e18 1.75903 0.879515 0.475872i \(-0.157867\pi\)
0.879515 + 0.475872i \(0.157867\pi\)
\(312\) 0 0
\(313\) 3.04567e17i 0.0584926i −0.999572 0.0292463i \(-0.990689\pi\)
0.999572 0.0292463i \(-0.00931071\pi\)
\(314\) 0 0
\(315\) 2.77626e18 6.42575e18i 0.508312 1.17651i
\(316\) 0 0
\(317\) 5.64109e18i 0.984961i −0.870324 0.492480i \(-0.836090\pi\)
0.870324 0.492480i \(-0.163910\pi\)
\(318\) 0 0
\(319\) 1.82340e18 0.303706
\(320\) 0 0
\(321\) 4.63807e18 0.737140
\(322\) 0 0
\(323\) 8.68718e18i 1.31783i
\(324\) 0 0
\(325\) 3.66314e18 3.44791e18i 0.530553 0.499379i
\(326\) 0 0
\(327\) 9.10494e18i 1.25941i
\(328\) 0 0
\(329\) 4.96954e18 0.656670
\(330\) 0 0
\(331\) −6.05049e18 −0.763978 −0.381989 0.924167i \(-0.624761\pi\)
−0.381989 + 0.924167i \(0.624761\pi\)
\(332\) 0 0
\(333\) 2.61925e19i 3.16114i
\(334\) 0 0
\(335\) −4.58250e17 + 1.06064e18i −0.0528770 + 0.122386i
\(336\) 0 0
\(337\) 1.58672e19i 1.75096i 0.483254 + 0.875480i \(0.339455\pi\)
−0.483254 + 0.875480i \(0.660545\pi\)
\(338\) 0 0
\(339\) 1.53359e19 1.61887
\(340\) 0 0
\(341\) −1.64971e17 −0.0166629
\(342\) 0 0
\(343\) 1.01341e19i 0.979667i
\(344\) 0 0
\(345\) 1.50929e19 + 6.52092e18i 1.39679 + 0.603484i
\(346\) 0 0
\(347\) 5.47026e18i 0.484771i −0.970180 0.242386i \(-0.922070\pi\)
0.970180 0.242386i \(-0.0779299\pi\)
\(348\) 0 0
\(349\) −7.51980e18 −0.638287 −0.319143 0.947706i \(-0.603395\pi\)
−0.319143 + 0.947706i \(0.603395\pi\)
\(350\) 0 0
\(351\) 1.83396e19 1.49137
\(352\) 0 0
\(353\) 3.96219e18i 0.308763i 0.988011 + 0.154382i \(0.0493385\pi\)
−0.988011 + 0.154382i \(0.950662\pi\)
\(354\) 0 0
\(355\) 2.27792e18 + 9.84179e17i 0.170148 + 0.0735126i
\(356\) 0 0
\(357\) 2.17813e19i 1.55981i
\(358\) 0 0
\(359\) −1.88540e19 −1.29478 −0.647390 0.762159i \(-0.724140\pi\)
−0.647390 + 0.762159i \(0.724140\pi\)
\(360\) 0 0
\(361\) −3.07257e18 −0.202394
\(362\) 0 0
\(363\) 2.39407e19i 1.51299i
\(364\) 0 0
\(365\) −4.05566e18 + 9.38699e18i −0.245961 + 0.569285i
\(366\) 0 0
\(367\) 1.02228e19i 0.595077i 0.954710 + 0.297538i \(0.0961656\pi\)
−0.954710 + 0.297538i \(0.903834\pi\)
\(368\) 0 0
\(369\) 2.40218e19 1.34248
\(370\) 0 0
\(371\) −3.23607e18 −0.173665
\(372\) 0 0
\(373\) 1.86547e19i 0.961550i 0.876844 + 0.480775i \(0.159645\pi\)
−0.876844 + 0.480775i \(0.840355\pi\)
\(374\) 0 0
\(375\) −3.37166e19 + 1.22049e19i −1.66959 + 0.604365i
\(376\) 0 0
\(377\) 1.20926e19i 0.575386i
\(378\) 0 0
\(379\) 3.62923e19 1.65967 0.829833 0.558012i \(-0.188436\pi\)
0.829833 + 0.558012i \(0.188436\pi\)
\(380\) 0 0
\(381\) 1.89812e19 0.834425
\(382\) 0 0
\(383\) 3.39832e19i 1.43640i 0.695839 + 0.718198i \(0.255033\pi\)
−0.695839 + 0.718198i \(0.744967\pi\)
\(384\) 0 0
\(385\) −2.23394e18 + 5.17055e18i −0.0908065 + 0.210175i
\(386\) 0 0
\(387\) 6.30031e19i 2.46337i
\(388\) 0 0
\(389\) 4.43693e19 1.66902 0.834508 0.550996i \(-0.185752\pi\)
0.834508 + 0.550996i \(0.185752\pi\)
\(390\) 0 0
\(391\) 3.49333e19 1.26448
\(392\) 0 0
\(393\) 4.40242e19i 1.53373i
\(394\) 0 0
\(395\) 3.98516e19 + 1.72179e19i 1.33649 + 0.577434i
\(396\) 0 0
\(397\) 1.44302e19i 0.465953i 0.972482 + 0.232977i \(0.0748466\pi\)
−0.972482 + 0.232977i \(0.925153\pi\)
\(398\) 0 0
\(399\) 3.03597e19 0.944063
\(400\) 0 0
\(401\) 2.34827e19 0.703339 0.351669 0.936124i \(-0.385614\pi\)
0.351669 + 0.936124i \(0.385614\pi\)
\(402\) 0 0
\(403\) 1.09407e18i 0.0315687i
\(404\) 0 0
\(405\) −4.89219e19 2.11368e19i −1.36016 0.587660i
\(406\) 0 0
\(407\) 2.10760e19i 0.564717i
\(408\) 0 0
\(409\) −3.99167e19 −1.03093 −0.515466 0.856910i \(-0.672381\pi\)
−0.515466 + 0.856910i \(0.672381\pi\)
\(410\) 0 0
\(411\) −7.91142e19 −1.96988
\(412\) 0 0
\(413\) 1.73664e18i 0.0416951i
\(414\) 0 0
\(415\) 4.61515e18 1.06819e19i 0.106862 0.247337i
\(416\) 0 0
\(417\) 1.37054e20i 3.06106i
\(418\) 0 0
\(419\) −7.16449e19 −1.54376 −0.771881 0.635767i \(-0.780684\pi\)
−0.771881 + 0.635767i \(0.780684\pi\)
\(420\) 0 0
\(421\) −6.22202e18 −0.129365 −0.0646823 0.997906i \(-0.520603\pi\)
−0.0646823 + 0.997906i \(0.520603\pi\)
\(422\) 0 0
\(423\) 1.18342e20i 2.37458i
\(424\) 0 0
\(425\) −5.54778e19 + 5.22181e19i −1.07449 + 1.01136i
\(426\) 0 0
\(427\) 9.29361e18i 0.173770i
\(428\) 0 0
\(429\) −2.75585e19 −0.497537
\(430\) 0 0
\(431\) 4.93032e19 0.859598 0.429799 0.902925i \(-0.358584\pi\)
0.429799 + 0.902925i \(0.358584\pi\)
\(432\) 0 0
\(433\) 1.59414e19i 0.268453i −0.990951 0.134226i \(-0.957145\pi\)
0.990951 0.134226i \(-0.0428549\pi\)
\(434\) 0 0
\(435\) −3.41865e19 + 7.91259e19i −0.556141 + 1.28721i
\(436\) 0 0
\(437\) 4.86914e19i 0.765318i
\(438\) 0 0
\(439\) 1.26174e19 0.191640 0.0958201 0.995399i \(-0.469453\pi\)
0.0958201 + 0.995399i \(0.469453\pi\)
\(440\) 0 0
\(441\) −9.46756e19 −1.38979
\(442\) 0 0
\(443\) 7.89961e19i 1.12093i 0.828179 + 0.560464i \(0.189377\pi\)
−0.828179 + 0.560464i \(0.810623\pi\)
\(444\) 0 0
\(445\) 8.13735e19 + 3.51576e19i 1.11630 + 0.482301i
\(446\) 0 0
\(447\) 1.94766e20i 2.58348i
\(448\) 0 0
\(449\) 4.36438e19 0.559854 0.279927 0.960021i \(-0.409690\pi\)
0.279927 + 0.960021i \(0.409690\pi\)
\(450\) 0 0
\(451\) −1.93294e19 −0.239825
\(452\) 0 0
\(453\) 1.19202e17i 0.00143069i
\(454\) 0 0
\(455\) 3.42905e19 + 1.48153e19i 0.398188 + 0.172038i
\(456\) 0 0
\(457\) 1.00520e20i 1.12949i −0.825266 0.564744i \(-0.808975\pi\)
0.825266 0.564744i \(-0.191025\pi\)
\(458\) 0 0
\(459\) −2.77750e20 −3.02036
\(460\) 0 0
\(461\) −1.09258e20 −1.15000 −0.574999 0.818154i \(-0.694998\pi\)
−0.574999 + 0.818154i \(0.694998\pi\)
\(462\) 0 0
\(463\) 1.81520e20i 1.84955i 0.380510 + 0.924777i \(0.375748\pi\)
−0.380510 + 0.924777i \(0.624252\pi\)
\(464\) 0 0
\(465\) 3.09299e18 7.15885e18i 0.0305128 0.0706230i
\(466\) 0 0
\(467\) 1.14048e20i 1.08946i 0.838611 + 0.544731i \(0.183368\pi\)
−0.838611 + 0.544731i \(0.816632\pi\)
\(468\) 0 0
\(469\) −8.57930e18 −0.0793699
\(470\) 0 0
\(471\) 7.00425e19 0.627632
\(472\) 0 0
\(473\) 5.06961e19i 0.440064i
\(474\) 0 0
\(475\) −7.27837e19 7.73272e19i −0.612114 0.650325i
\(476\) 0 0
\(477\) 7.70621e19i 0.627990i
\(478\) 0 0
\(479\) −1.21110e20 −0.956451 −0.478226 0.878237i \(-0.658720\pi\)
−0.478226 + 0.878237i \(0.658720\pi\)
\(480\) 0 0
\(481\) 1.39774e20 1.06989
\(482\) 0 0
\(483\) 1.22084e20i 0.905846i
\(484\) 0 0
\(485\) −8.22660e19 + 1.90408e20i −0.591776 + 1.36969i
\(486\) 0 0
\(487\) 1.00069e20i 0.697963i 0.937130 + 0.348982i \(0.113472\pi\)
−0.937130 + 0.348982i \(0.886528\pi\)
\(488\) 0 0
\(489\) −3.16540e20 −2.14098
\(490\) 0 0
\(491\) 1.32724e20 0.870641 0.435321 0.900276i \(-0.356635\pi\)
0.435321 + 0.900276i \(0.356635\pi\)
\(492\) 0 0
\(493\) 1.83141e20i 1.16529i
\(494\) 0 0
\(495\) 1.23129e20 + 5.31980e19i 0.760013 + 0.328365i
\(496\) 0 0
\(497\) 1.84257e19i 0.110345i
\(498\) 0 0
\(499\) 9.34938e18 0.0543286 0.0271643 0.999631i \(-0.491352\pi\)
0.0271643 + 0.999631i \(0.491352\pi\)
\(500\) 0 0
\(501\) 2.67527e20 1.50864
\(502\) 0 0
\(503\) 1.56622e20i 0.857223i −0.903489 0.428612i \(-0.859003\pi\)
0.903489 0.428612i \(-0.140997\pi\)
\(504\) 0 0
\(505\) −3.17476e19 1.37166e19i −0.168665 0.0728720i
\(506\) 0 0
\(507\) 1.61511e20i 0.832997i
\(508\) 0 0
\(509\) 1.94050e20 0.971698 0.485849 0.874043i \(-0.338511\pi\)
0.485849 + 0.874043i \(0.338511\pi\)
\(510\) 0 0
\(511\) −7.59296e19 −0.369194
\(512\) 0 0
\(513\) 3.87140e20i 1.82805i
\(514\) 0 0
\(515\) −1.03939e20 + 2.40571e20i −0.476678 + 1.10329i
\(516\) 0 0
\(517\) 9.52252e19i 0.424202i
\(518\) 0 0
\(519\) 4.23887e20 1.83441
\(520\) 0 0
\(521\) −6.43719e19 −0.270653 −0.135327 0.990801i \(-0.543208\pi\)
−0.135327 + 0.990801i \(0.543208\pi\)
\(522\) 0 0
\(523\) 1.94571e20i 0.794904i −0.917623 0.397452i \(-0.869895\pi\)
0.917623 0.397452i \(-0.130105\pi\)
\(524\) 0 0
\(525\) −1.82490e20 1.93882e20i −0.724511 0.769738i
\(526\) 0 0
\(527\) 1.65695e19i 0.0639337i
\(528\) 0 0
\(529\) 7.08352e19 0.265663
\(530\) 0 0
\(531\) 4.13555e19 0.150773
\(532\) 0 0
\(533\) 1.28190e20i 0.454360i
\(534\) 0 0
\(535\) 4.77779e19 1.10584e20i 0.164655 0.381099i
\(536\) 0 0
\(537\) 1.41706e20i 0.474877i
\(538\) 0 0
\(539\) 7.61817e19 0.248276
\(540\) 0 0
\(541\) 1.81403e19 0.0574996 0.0287498 0.999587i \(-0.490847\pi\)
0.0287498 + 0.999587i \(0.490847\pi\)
\(542\) 0 0
\(543\) 1.17393e20i 0.361945i
\(544\) 0 0
\(545\) −2.17086e20 9.37922e19i −0.651114 0.281315i
\(546\) 0 0
\(547\) 6.00592e20i 1.75257i 0.481796 + 0.876283i \(0.339985\pi\)
−0.481796 + 0.876283i \(0.660015\pi\)
\(548\) 0 0
\(549\) −2.21313e20 −0.628369
\(550\) 0 0
\(551\) −2.55269e20 −0.705279
\(552\) 0 0
\(553\) 3.22352e20i 0.866746i
\(554\) 0 0
\(555\) −9.14585e20 3.95148e20i −2.39346 1.03410i
\(556\) 0 0
\(557\) 2.29611e20i 0.584895i −0.956282 0.292447i \(-0.905530\pi\)
0.956282 0.292447i \(-0.0944697\pi\)
\(558\) 0 0
\(559\) 3.36210e20 0.833725
\(560\) 0 0
\(561\) 4.17369e20 1.00762
\(562\) 0 0
\(563\) 5.26651e20i 1.23797i −0.785403 0.618985i \(-0.787544\pi\)
0.785403 0.618985i \(-0.212456\pi\)
\(564\) 0 0
\(565\) 1.57979e20 3.65648e20i 0.361606 0.836950i
\(566\) 0 0
\(567\) 3.95720e20i 0.882095i
\(568\) 0 0
\(569\) 5.86825e20 1.27399 0.636996 0.770868i \(-0.280177\pi\)
0.636996 + 0.770868i \(0.280177\pi\)
\(570\) 0 0
\(571\) 2.76080e20 0.583800 0.291900 0.956449i \(-0.405713\pi\)
0.291900 + 0.956449i \(0.405713\pi\)
\(572\) 0 0
\(573\) 1.23031e21i 2.53428i
\(574\) 0 0
\(575\) 3.10952e20 2.92681e20i 0.623999 0.587335i
\(576\) 0 0
\(577\) 1.42599e20i 0.278804i 0.990236 + 0.139402i \(0.0445180\pi\)
−0.990236 + 0.139402i \(0.955482\pi\)
\(578\) 0 0
\(579\) −3.05561e20 −0.582115
\(580\) 0 0
\(581\) 8.64042e19 0.160404
\(582\) 0 0
\(583\) 6.20088e19i 0.112186i
\(584\) 0 0
\(585\) 3.52803e20 8.16576e20i 0.622104 1.43988i
\(586\) 0 0
\(587\) 7.48323e20i 1.28618i 0.765789 + 0.643092i \(0.222349\pi\)
−0.765789 + 0.643092i \(0.777651\pi\)
\(588\) 0 0
\(589\) 2.30953e19 0.0386953
\(590\) 0 0
\(591\) 1.71130e21 2.79524
\(592\) 0 0
\(593\) 3.21784e20i 0.512454i 0.966617 + 0.256227i \(0.0824794\pi\)
−0.966617 + 0.256227i \(0.917521\pi\)
\(594\) 0 0
\(595\) −5.19324e20 2.24375e20i −0.806420 0.348415i
\(596\) 0 0
\(597\) 1.17804e21i 1.78383i
\(598\) 0 0
\(599\) 1.08146e21 1.59702 0.798509 0.601983i \(-0.205623\pi\)
0.798509 + 0.601983i \(0.205623\pi\)
\(600\) 0 0
\(601\) 5.36803e20 0.773136 0.386568 0.922261i \(-0.373660\pi\)
0.386568 + 0.922261i \(0.373660\pi\)
\(602\) 0 0
\(603\) 2.04303e20i 0.287009i
\(604\) 0 0
\(605\) 5.70809e20 + 2.46619e20i 0.782213 + 0.337957i
\(606\) 0 0
\(607\) 4.11434e20i 0.550029i 0.961440 + 0.275014i \(0.0886826\pi\)
−0.961440 + 0.275014i \(0.911317\pi\)
\(608\) 0 0
\(609\) −6.40035e20 −0.834783
\(610\) 0 0
\(611\) 6.31522e20 0.803673
\(612\) 0 0
\(613\) 5.44101e20i 0.675656i −0.941208 0.337828i \(-0.890308\pi\)
0.941208 0.337828i \(-0.109692\pi\)
\(614\) 0 0
\(615\) 3.62401e20 8.38790e20i 0.439163 1.01646i
\(616\) 0 0
\(617\) 4.65828e20i 0.550918i 0.961313 + 0.275459i \(0.0888298\pi\)
−0.961313 + 0.275459i \(0.911170\pi\)
\(618\) 0 0
\(619\) −1.51736e21 −1.75150 −0.875749 0.482767i \(-0.839632\pi\)
−0.875749 + 0.482767i \(0.839632\pi\)
\(620\) 0 0
\(621\) 1.55678e21 1.75405
\(622\) 0 0
\(623\) 6.58215e20i 0.723948i
\(624\) 0 0
\(625\) −5.63262e19 + 9.29618e20i −0.0604798 + 0.998169i
\(626\) 0 0
\(627\) 5.81746e20i 0.609855i
\(628\) 0 0
\(629\) −2.11685e21 −2.16676
\(630\) 0 0
\(631\) −5.94310e20 −0.594009 −0.297004 0.954876i \(-0.595988\pi\)
−0.297004 + 0.954876i \(0.595988\pi\)
\(632\) 0 0
\(633\) 2.41421e21i 2.35639i
\(634\) 0 0
\(635\) 1.95530e20 4.52563e20i 0.186385 0.431396i
\(636\) 0 0
\(637\) 5.05228e20i 0.470372i
\(638\) 0 0
\(639\) 4.38780e20 0.399016
\(640\) 0 0
\(641\) 9.48383e20 0.842458 0.421229 0.906954i \(-0.361599\pi\)
0.421229 + 0.906954i \(0.361599\pi\)
\(642\) 0 0
\(643\) 2.15909e21i 1.87365i 0.349800 + 0.936824i \(0.386250\pi\)
−0.349800 + 0.936824i \(0.613750\pi\)
\(644\) 0 0
\(645\) −2.19994e21 9.50486e20i −1.86514 0.805838i
\(646\) 0 0
\(647\) 2.26989e20i 0.188028i −0.995571 0.0940139i \(-0.970030\pi\)
0.995571 0.0940139i \(-0.0299698\pi\)
\(648\) 0 0
\(649\) −3.32771e19 −0.0269346
\(650\) 0 0
\(651\) 5.79067e19 0.0458006
\(652\) 0 0
\(653\) 2.12810e21i 1.64491i −0.568830 0.822455i \(-0.692604\pi\)
0.568830 0.822455i \(-0.307396\pi\)
\(654\) 0 0
\(655\) 1.04965e21 + 4.53504e20i 0.792932 + 0.342588i
\(656\) 0 0
\(657\) 1.80815e21i 1.33504i
\(658\) 0 0
\(659\) −3.97715e20 −0.287032 −0.143516 0.989648i \(-0.545841\pi\)
−0.143516 + 0.989648i \(0.545841\pi\)
\(660\) 0 0
\(661\) 1.97089e20 0.139044 0.0695218 0.997580i \(-0.477853\pi\)
0.0695218 + 0.997580i \(0.477853\pi\)
\(662\) 0 0
\(663\) 2.76794e21i 1.90900i
\(664\) 0 0
\(665\) 3.12743e20 7.23855e20i 0.210875 0.488078i
\(666\) 0 0
\(667\) 1.02650e21i 0.676729i
\(668\) 0 0
\(669\) 5.72457e20 0.369017
\(670\) 0 0
\(671\) 1.78082e20 0.112254
\(672\) 0 0
\(673\) 1.12563e21i 0.693876i −0.937888 0.346938i \(-0.887221\pi\)
0.937888 0.346938i \(-0.112779\pi\)
\(674\) 0 0
\(675\) −2.47234e21 + 2.32707e21i −1.49049 + 1.40292i
\(676\) 0 0
\(677\) 2.01414e21i 1.18761i −0.804608 0.593806i \(-0.797625\pi\)
0.804608 0.593806i \(-0.202375\pi\)
\(678\) 0 0
\(679\) −1.54017e21 −0.888273
\(680\) 0 0
\(681\) 5.53060e20 0.312010
\(682\) 0 0
\(683\) 2.93860e21i 1.62176i −0.585215 0.810878i \(-0.698990\pi\)
0.585215 0.810878i \(-0.301010\pi\)
\(684\) 0 0
\(685\) −8.14974e20 + 1.88629e21i −0.440012 + 1.01842i
\(686\) 0 0
\(687\) 2.63160e21i 1.39009i
\(688\) 0 0
\(689\) −4.11235e20 −0.212542
\(690\) 0 0
\(691\) 1.85211e20 0.0936661 0.0468330 0.998903i \(-0.485087\pi\)
0.0468330 + 0.998903i \(0.485087\pi\)
\(692\) 0 0
\(693\) 9.95967e20i 0.492885i
\(694\) 0 0
\(695\) −3.26773e21 1.41183e21i −1.58256 0.683748i
\(696\) 0 0
\(697\) 1.94142e21i 0.920182i
\(698\) 0 0
\(699\) −6.39006e21 −2.96433
\(700\) 0 0
\(701\) −1.53115e21 −0.695236 −0.347618 0.937636i \(-0.613009\pi\)
−0.347618 + 0.937636i \(0.613009\pi\)
\(702\) 0 0
\(703\) 2.95056e21i 1.31141i
\(704\) 0 0
\(705\) −4.13226e21 1.78535e21i −1.79791 0.776792i
\(706\) 0 0
\(707\) 2.56800e20i 0.109383i
\(708\) 0 0
\(709\) −1.28361e19 −0.00535287 −0.00267643 0.999996i \(-0.500852\pi\)
−0.00267643 + 0.999996i \(0.500852\pi\)
\(710\) 0 0
\(711\) 7.67633e21 3.13423
\(712\) 0 0
\(713\) 9.28717e19i 0.0371289i
\(714\) 0 0
\(715\) −2.83886e20 + 6.57066e20i −0.111135 + 0.257225i
\(716\) 0 0
\(717\) 2.05502e21i 0.787811i
\(718\) 0 0
\(719\) 3.14657e21 1.18133 0.590665 0.806917i \(-0.298866\pi\)
0.590665 + 0.806917i \(0.298866\pi\)
\(720\) 0 0
\(721\) −1.94594e21 −0.715508
\(722\) 0 0
\(723\) 8.37955e21i 3.01775i
\(724\) 0 0
\(725\) 1.53440e21 + 1.63019e21i 0.541259 + 0.575047i
\(726\) 0 0
\(727\) 4.52609e21i 1.56392i −0.623327 0.781962i \(-0.714219\pi\)
0.623327 0.781962i \(-0.285781\pi\)
\(728\) 0 0
\(729\) 1.31386e21 0.444726
\(730\) 0 0
\(731\) −5.09186e21 −1.68848
\(732\) 0 0
\(733\) 2.68116e21i 0.871048i 0.900177 + 0.435524i \(0.143437\pi\)
−0.900177 + 0.435524i \(0.856563\pi\)
\(734\) 0 0
\(735\) −1.42831e21 + 3.30587e21i −0.454639 + 1.05228i
\(736\) 0 0
\(737\) 1.64395e20i 0.0512721i
\(738\) 0 0
\(739\) 1.15277e21 0.352298 0.176149 0.984364i \(-0.443636\pi\)
0.176149 + 0.984364i \(0.443636\pi\)
\(740\) 0 0
\(741\) 3.85807e21 1.15540
\(742\) 0 0
\(743\) 2.64683e21i 0.776801i 0.921491 + 0.388400i \(0.126972\pi\)
−0.921491 + 0.388400i \(0.873028\pi\)
\(744\) 0 0
\(745\) −4.64372e21 2.00633e21i −1.33565 0.577072i
\(746\) 0 0
\(747\) 2.05759e21i 0.580034i
\(748\) 0 0
\(749\) 8.94493e20 0.247151
\(750\) 0 0
\(751\) 4.36972e21 1.18346 0.591730 0.806136i \(-0.298445\pi\)
0.591730 + 0.806136i \(0.298445\pi\)
\(752\) 0 0
\(753\) 2.54716e21i 0.676229i
\(754\) 0 0
\(755\) 2.84208e18 + 1.22793e18i 0.000739664 + 0.000319573i
\(756\) 0 0
\(757\) 7.25274e20i 0.185047i −0.995710 0.0925237i \(-0.970507\pi\)
0.995710 0.0925237i \(-0.0294934\pi\)
\(758\) 0 0
\(759\) −2.33934e21 −0.585168
\(760\) 0 0
\(761\) 2.04957e21 0.502665 0.251333 0.967901i \(-0.419131\pi\)
0.251333 + 0.967901i \(0.419131\pi\)
\(762\) 0 0
\(763\) 1.75597e21i 0.422262i
\(764\) 0 0
\(765\) −5.34315e21 + 1.23669e22i −1.25990 + 2.91609i
\(766\) 0 0
\(767\) 2.20690e20i 0.0510290i
\(768\) 0 0
\(769\) −2.52209e21 −0.571891 −0.285945 0.958246i \(-0.592308\pi\)
−0.285945 + 0.958246i \(0.592308\pi\)
\(770\) 0 0
\(771\) 4.69931e20 0.104503
\(772\) 0 0
\(773\) 2.53203e21i 0.552233i 0.961124 + 0.276117i \(0.0890476\pi\)
−0.961124 + 0.276117i \(0.910952\pi\)
\(774\) 0 0
\(775\) −1.38824e20 1.47490e20i −0.0296963 0.0315501i
\(776\) 0 0
\(777\) 7.39791e21i 1.55221i
\(778\) 0 0
\(779\) 2.70603e21 0.556932
\(780\) 0 0
\(781\) −3.53069e20 −0.0712815
\(782\) 0 0
\(783\) 8.16157e21i 1.61644i
\(784\) 0 0
\(785\) 7.21525e20 1.67000e21i 0.140194 0.324484i
\(786\) 0 0
\(787\) 1.39343e21i 0.265629i 0.991141 + 0.132814i \(0.0424014\pi\)
−0.991141 + 0.132814i \(0.957599\pi\)
\(788\) 0 0
\(789\) 4.67347e21 0.874102
\(790\) 0 0
\(791\) 2.95766e21 0.542781
\(792\) 0 0
\(793\) 1.18102e21i 0.212671i
\(794\) 0 0
\(795\) 2.69085e21 + 1.16258e21i 0.475483 + 0.205433i
\(796\) 0 0
\(797\) 9.55241e20i 0.165644i −0.996564 0.0828220i \(-0.973607\pi\)
0.996564 0.0828220i \(-0.0263933\pi\)
\(798\) 0 0
\(799\) −9.56431e21 −1.62762
\(800\) 0 0
\(801\) 1.56744e22 2.61787
\(802\) 0 0
\(803\) 1.45495e21i 0.238496i
\(804\) 0 0
\(805\) 2.91080e21 + 1.25762e21i 0.468320 + 0.202339i
\(806\) 0 0
\(807\) 1.18178e22i 1.86631i
\(808\) 0 0
\(809\) −9.73892e21 −1.50972 −0.754861 0.655885i \(-0.772296\pi\)
−0.754861 + 0.655885i \(0.772296\pi\)
\(810\) 0 0
\(811\) −4.03949e20 −0.0614711 −0.0307355 0.999528i \(-0.509785\pi\)
−0.0307355 + 0.999528i \(0.509785\pi\)
\(812\) 0 0
\(813\) 1.25362e22i 1.87277i
\(814\) 0 0
\(815\) −3.26076e21 + 7.54715e21i −0.478230 + 1.10688i
\(816\) 0 0
\(817\) 7.09724e21i 1.02194i
\(818\) 0 0
\(819\) 6.60513e21 0.933797
\(820\) 0 0
\(821\) 1.42068e22 1.97207 0.986034 0.166541i \(-0.0532599\pi\)
0.986034 + 0.166541i \(0.0532599\pi\)
\(822\) 0 0
\(823\) 6.03911e21i 0.823141i −0.911378 0.411570i \(-0.864980\pi\)
0.911378 0.411570i \(-0.135020\pi\)
\(824\) 0 0
\(825\) 3.71513e21 3.49684e21i 0.497243 0.468027i
\(826\) 0 0
\(827\) 1.20995e22i 1.59029i −0.606421 0.795144i \(-0.707395\pi\)
0.606421 0.795144i \(-0.292605\pi\)
\(828\) 0 0
\(829\) −5.90195e21 −0.761792 −0.380896 0.924618i \(-0.624384\pi\)
−0.380896 + 0.924618i \(0.624384\pi\)
\(830\) 0 0
\(831\) 2.21154e22 2.80342
\(832\) 0 0
\(833\) 7.65160e21i 0.952610i
\(834\) 0 0
\(835\) 2.75586e21 6.37855e21i 0.336984 0.779962i
\(836\) 0 0
\(837\) 7.38412e20i 0.0886865i
\(838\) 0 0
\(839\) −1.29661e22 −1.52966 −0.764831 0.644231i \(-0.777178\pi\)
−0.764831 + 0.644231i \(0.777178\pi\)
\(840\) 0 0
\(841\) −3.24768e21 −0.376360
\(842\) 0 0
\(843\) 1.62420e22i 1.84898i
\(844\) 0 0
\(845\) −3.85086e21 1.66377e21i −0.430657 0.186066i
\(846\) 0 0
\(847\) 4.61717e21i 0.507283i
\(848\) 0 0
\(849\) −7.50427e20 −0.0810029
\(850\) 0 0
\(851\) 1.18649e22 1.25832
\(852\) 0 0
\(853\) 9.05799e21i 0.943874i −0.881632 0.471937i \(-0.843555\pi\)
0.881632 0.471937i \(-0.156445\pi\)
\(854\) 0 0
\(855\) −1.72375e22 7.44750e21i −1.76494 0.762544i
\(856\) 0 0
\(857\) 1.80435e22i 1.81536i −0.419660 0.907682i \(-0.637851\pi\)
0.419660 0.907682i \(-0.362149\pi\)
\(858\) 0 0
\(859\) 2.44689e21 0.241916 0.120958 0.992658i \(-0.461403\pi\)
0.120958 + 0.992658i \(0.461403\pi\)
\(860\) 0 0
\(861\) 6.78482e21 0.659196
\(862\) 0 0
\(863\) 9.95152e21i 0.950185i 0.879936 + 0.475093i \(0.157585\pi\)
−0.879936 + 0.475093i \(0.842415\pi\)
\(864\) 0 0
\(865\) 4.36657e21 1.01066e22i 0.409750 0.948383i
\(866\) 0 0
\(867\) 2.26674e22i 2.09054i
\(868\) 0 0
\(869\) −6.17683e21 −0.559909
\(870\) 0 0
\(871\) −1.09025e21 −0.0971378
\(872\) 0 0
\(873\) 3.66769e22i 3.21208i
\(874\) 0 0
\(875\) −6.50254e21 + 2.35382e21i −0.559786 + 0.202634i
\(876\) 0 0
\(877\) 1.75476e22i 1.48498i 0.669855 + 0.742492i \(0.266356\pi\)
−0.669855 + 0.742492i \(0.733644\pi\)
\(878\) 0 0
\(879\) 1.74787e22 1.45409
\(880\) 0 0
\(881\) −1.45628e22 −1.19104 −0.595520 0.803341i \(-0.703054\pi\)
−0.595520 + 0.803341i \(0.703054\pi\)
\(882\) 0 0
\(883\) 1.28619e22i 1.03419i 0.855928 + 0.517095i \(0.172987\pi\)
−0.855928 + 0.517095i \(0.827013\pi\)
\(884\) 0 0
\(885\) 6.23903e20 1.44405e21i 0.0493222 0.114158i
\(886\) 0 0
\(887\) 1.46337e22i 1.13743i −0.822534 0.568716i \(-0.807440\pi\)
0.822534 0.568716i \(-0.192560\pi\)
\(888\) 0 0
\(889\) 3.66070e21 0.279770
\(890\) 0 0
\(891\) 7.58270e21 0.569824
\(892\) 0 0
\(893\) 1.33311e22i 0.985102i
\(894\) 0 0
\(895\) 3.37864e21 + 1.45975e21i 0.245510 + 0.106073i
\(896\) 0 0
\(897\) 1.55142e22i 1.10863i
\(898\) 0 0
\(899\) −4.86888e20 −0.0342162
\(900\) 0 0
\(901\) 6.22809e21 0.430446
\(902\) 0 0
\(903\) 1.77949e22i 1.20959i
\(904\) 0 0
\(905\) −2.79895e21 1.20929e21i −0.187125 0.0808476i
\(906\) 0 0
\(907\) 4.06427e21i 0.267256i −0.991032 0.133628i \(-0.957337\pi\)
0.991032 0.133628i \(-0.0426627\pi\)
\(908\) 0 0
\(909\) −6.11531e21 −0.395539
\(910\) 0 0
\(911\) −6.36749e21 −0.405117 −0.202558 0.979270i \(-0.564926\pi\)
−0.202558 + 0.979270i \(0.564926\pi\)
\(912\) 0 0
\(913\) 1.65566e21i 0.103619i
\(914\) 0 0
\(915\) −3.33881e21 + 7.72780e21i −0.205557 + 0.475770i
\(916\) 0 0
\(917\) 8.49045e21i 0.514234i
\(918\) 0 0
\(919\) 2.93596e22 1.74938 0.874690 0.484682i \(-0.161065\pi\)
0.874690 + 0.484682i \(0.161065\pi\)
\(920\) 0 0
\(921\) 5.35798e22 3.14090
\(922\) 0 0
\(923\) 2.34151e21i 0.135046i
\(924\) 0 0
\(925\) −1.88427e22 + 1.77356e22i −1.06925 + 1.00643i
\(926\) 0 0
\(927\) 4.63396e22i 2.58734i
\(928\) 0 0
\(929\) 3.30652e22 1.81658 0.908288 0.418347i \(-0.137390\pi\)
0.908288 + 0.418347i \(0.137390\pi\)
\(930\) 0 0
\(931\) −1.06651e22 −0.576558
\(932\) 0 0
\(933\) 5.87127e22i 3.12334i
\(934\) 0 0
\(935\) 4.29942e21 9.95117e21i 0.225073 0.520939i
\(936\) 0 0
\(937\) 3.47429e22i 1.78986i 0.446205 + 0.894931i \(0.352775\pi\)
−0.446205 + 0.894931i \(0.647225\pi\)
\(938\) 0 0
\(939\) −2.04852e21 −0.103860
\(940\) 0 0
\(941\) −2.80935e21 −0.140179 −0.0700897 0.997541i \(-0.522329\pi\)
−0.0700897 + 0.997541i \(0.522329\pi\)
\(942\) 0 0
\(943\) 1.08816e22i 0.534387i
\(944\) 0 0
\(945\) −2.31434e22 9.99915e21i −1.11864 0.483308i
\(946\) 0 0
\(947\) 9.11666e21i 0.433722i 0.976203 + 0.216861i \(0.0695818\pi\)
−0.976203 + 0.216861i \(0.930418\pi\)
\(948\) 0 0
\(949\) −9.64903e21 −0.451842
\(950\) 0 0
\(951\) −3.79419e22 −1.74890
\(952\) 0 0
\(953\) 1.41863e22i 0.643682i 0.946794 + 0.321841i \(0.104302\pi\)
−0.946794 + 0.321841i \(0.895698\pi\)
\(954\) 0 0
\(955\) −2.93338e22 1.26737e22i −1.31022 0.566082i
\(956\) 0 0
\(957\) 1.22642e22i 0.539262i
\(958\) 0 0
\(959\) −1.52578e22 −0.660471
\(960\) 0 0
\(961\) −2.34212e22 −0.998123
\(962\) 0 0
\(963\) 2.13010e22i 0.893723i
\(964\) 0 0
\(965\) −3.14766e21 + 7.28538e21i −0.130027 + 0.300952i
\(966\) 0 0
\(967\) 1.40091e22i 0.569785i 0.958560 + 0.284892i \(0.0919578\pi\)
−0.958560 + 0.284892i \(0.908042\pi\)
\(968\) 0 0
\(969\) −5.84299e22 −2.33995
\(970\) 0 0
\(971\) 3.55233e22 1.40078 0.700388 0.713762i \(-0.253010\pi\)
0.700388 + 0.713762i \(0.253010\pi\)
\(972\) 0 0
\(973\) 2.64321e22i 1.02633i
\(974\) 0 0
\(975\) −2.31906e22 2.46383e22i −0.886702 0.942054i
\(976\) 0 0
\(977\) 2.18841e22i 0.823985i 0.911187 + 0.411993i \(0.135167\pi\)
−0.911187 + 0.411993i \(0.864833\pi\)
\(978\) 0 0
\(979\) −1.26126e22 −0.467663
\(980\) 0 0
\(981\) −4.18157e22 −1.52694
\(982\) 0 0
\(983\) 6.12627e21i 0.220315i 0.993914 + 0.110158i \(0.0351356\pi\)
−0.993914 + 0.110158i \(0.964864\pi\)
\(984\) 0 0
\(985\) 1.76285e22 4.08018e22i 0.624372 1.44513i
\(986\) 0 0
\(987\) 3.34251e22i 1.16599i
\(988\) 0 0
\(989\) 2.85397e22 0.980568
\(990\) 0 0
\(991\) −3.50331e22 −1.18557 −0.592784 0.805361i \(-0.701971\pi\)
−0.592784 + 0.805361i \(0.701971\pi\)
\(992\) 0 0
\(993\) 4.06956e22i 1.35652i
\(994\) 0 0
\(995\) 2.80876e22 + 1.21353e22i 0.922236 + 0.398453i
\(996\) 0 0
\(997\) 3.84796e22i 1.24456i 0.782793 + 0.622282i \(0.213794\pi\)
−0.782793 + 0.622282i \(0.786206\pi\)
\(998\) 0 0
\(999\) −9.43364e22 −3.00565
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 80.16.c.a.49.1 6
4.3 odd 2 5.16.b.a.4.4 yes 6
5.4 even 2 inner 80.16.c.a.49.6 6
12.11 even 2 45.16.b.b.19.3 6
20.3 even 4 25.16.a.f.1.4 6
20.7 even 4 25.16.a.f.1.3 6
20.19 odd 2 5.16.b.a.4.3 6
60.59 even 2 45.16.b.b.19.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.16.b.a.4.3 6 20.19 odd 2
5.16.b.a.4.4 yes 6 4.3 odd 2
25.16.a.f.1.3 6 20.7 even 4
25.16.a.f.1.4 6 20.3 even 4
45.16.b.b.19.3 6 12.11 even 2
45.16.b.b.19.4 6 60.59 even 2
80.16.c.a.49.1 6 1.1 even 1 trivial
80.16.c.a.49.6 6 5.4 even 2 inner