Properties

Label 192.9.g.d.127.5
Level $192$
Weight $9$
Character 192.127
Analytic conductor $78.217$
Analytic rank $0$
Dimension $8$
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] = [192,9,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3468738816.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.5
Root \(-1.27467 + 1.27467i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.9.g.d.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654i q^{3} -1079.08 q^{5} -3980.70i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q+46.7654i q^{3} -1079.08 q^{5} -3980.70i q^{7} -2187.00 q^{9} +20108.0i q^{11} +36104.9 q^{13} -50463.5i q^{15} +59944.2 q^{17} -126309. i q^{19} +186159. q^{21} -197109. i q^{23} +773786. q^{25} -102276. i q^{27} -485230. q^{29} +671904. i q^{31} -940357. q^{33} +4.29549e6i q^{35} +3.40344e6 q^{37} +1.68846e6i q^{39} -3.55723e6 q^{41} -736659. i q^{43} +2.35995e6 q^{45} -883461. i q^{47} -1.00811e7 q^{49} +2.80331e6i q^{51} -1.44913e7 q^{53} -2.16981e7i q^{55} +5.90690e6 q^{57} -1.08326e6i q^{59} +3.45104e6 q^{61} +8.70578e6i q^{63} -3.89601e7 q^{65} +1.43441e7i q^{67} +9.21785e6 q^{69} -3.97522e6i q^{71} +3.92004e6 q^{73} +3.61864e7i q^{75} +8.00437e7 q^{77} -5.94312e7i q^{79} +4.78297e6 q^{81} +5.42483e7i q^{83} -6.46846e7 q^{85} -2.26920e7i q^{87} -2.43253e7 q^{89} -1.43723e8i q^{91} -3.14218e7 q^{93} +1.36298e8i q^{95} -3.83933e7 q^{97} -4.39761e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 560 q^{5} - 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 560 q^{5} - 17496 q^{9} + 54256 q^{13} + 436176 q^{17} - 54432 q^{21} + 2456472 q^{25} - 195952 q^{29} - 1065312 q^{33} + 7023408 q^{37} - 12035120 q^{41} + 1224720 q^{45} - 9602040 q^{49} - 27342192 q^{53} + 2744928 q^{57} - 50803280 q^{61} + 34049888 q^{65} + 26687232 q^{69} + 59541648 q^{73} + 178489472 q^{77} + 38263752 q^{81} - 29428704 q^{85} - 170794992 q^{89} + 121951008 q^{93} + 272647184 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 46.7654i 0.577350i
\(4\) 0 0
\(5\) −1079.08 −1.72653 −0.863263 0.504754i \(-0.831583\pi\)
−0.863263 + 0.504754i \(0.831583\pi\)
\(6\) 0 0
\(7\) − 3980.70i − 1.65793i −0.559298 0.828966i \(-0.688929\pi\)
0.559298 0.828966i \(-0.311071\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) 20108.0i 1.37340i 0.726940 + 0.686701i \(0.240942\pi\)
−0.726940 + 0.686701i \(0.759058\pi\)
\(12\) 0 0
\(13\) 36104.9 1.26413 0.632067 0.774914i \(-0.282207\pi\)
0.632067 + 0.774914i \(0.282207\pi\)
\(14\) 0 0
\(15\) − 50463.5i − 0.996810i
\(16\) 0 0
\(17\) 59944.2 0.717715 0.358857 0.933392i \(-0.383166\pi\)
0.358857 + 0.933392i \(0.383166\pi\)
\(18\) 0 0
\(19\) − 126309.i − 0.969217i −0.874731 0.484608i \(-0.838962\pi\)
0.874731 0.484608i \(-0.161038\pi\)
\(20\) 0 0
\(21\) 186159. 0.957208
\(22\) 0 0
\(23\) − 197109.i − 0.704359i −0.935932 0.352179i \(-0.885441\pi\)
0.935932 0.352179i \(-0.114559\pi\)
\(24\) 0 0
\(25\) 773786. 1.98089
\(26\) 0 0
\(27\) − 102276.i − 0.192450i
\(28\) 0 0
\(29\) −485230. −0.686050 −0.343025 0.939326i \(-0.611452\pi\)
−0.343025 + 0.939326i \(0.611452\pi\)
\(30\) 0 0
\(31\) 671904.i 0.727546i 0.931488 + 0.363773i \(0.118512\pi\)
−0.931488 + 0.363773i \(0.881488\pi\)
\(32\) 0 0
\(33\) −940357. −0.792934
\(34\) 0 0
\(35\) 4.29549e6i 2.86246i
\(36\) 0 0
\(37\) 3.40344e6 1.81598 0.907991 0.418990i \(-0.137616\pi\)
0.907991 + 0.418990i \(0.137616\pi\)
\(38\) 0 0
\(39\) 1.68846e6i 0.729848i
\(40\) 0 0
\(41\) −3.55723e6 −1.25886 −0.629429 0.777058i \(-0.716711\pi\)
−0.629429 + 0.777058i \(0.716711\pi\)
\(42\) 0 0
\(43\) − 736659.i − 0.215473i −0.994179 0.107736i \(-0.965640\pi\)
0.994179 0.107736i \(-0.0343603\pi\)
\(44\) 0 0
\(45\) 2.35995e6 0.575509
\(46\) 0 0
\(47\) − 883461.i − 0.181049i −0.995894 0.0905245i \(-0.971146\pi\)
0.995894 0.0905245i \(-0.0288543\pi\)
\(48\) 0 0
\(49\) −1.00811e7 −1.74874
\(50\) 0 0
\(51\) 2.80331e6i 0.414373i
\(52\) 0 0
\(53\) −1.44913e7 −1.83655 −0.918275 0.395943i \(-0.870418\pi\)
−0.918275 + 0.395943i \(0.870418\pi\)
\(54\) 0 0
\(55\) − 2.16981e7i − 2.37121i
\(56\) 0 0
\(57\) 5.90690e6 0.559578
\(58\) 0 0
\(59\) − 1.08326e6i − 0.0893969i −0.999001 0.0446985i \(-0.985767\pi\)
0.999001 0.0446985i \(-0.0142327\pi\)
\(60\) 0 0
\(61\) 3.45104e6 0.249248 0.124624 0.992204i \(-0.460228\pi\)
0.124624 + 0.992204i \(0.460228\pi\)
\(62\) 0 0
\(63\) 8.70578e6i 0.552644i
\(64\) 0 0
\(65\) −3.89601e7 −2.18256
\(66\) 0 0
\(67\) 1.43441e7i 0.711828i 0.934519 + 0.355914i \(0.115830\pi\)
−0.934519 + 0.355914i \(0.884170\pi\)
\(68\) 0 0
\(69\) 9.21785e6 0.406662
\(70\) 0 0
\(71\) − 3.97522e6i − 0.156433i −0.996936 0.0782163i \(-0.975078\pi\)
0.996936 0.0782163i \(-0.0249225\pi\)
\(72\) 0 0
\(73\) 3.92004e6 0.138038 0.0690191 0.997615i \(-0.478013\pi\)
0.0690191 + 0.997615i \(0.478013\pi\)
\(74\) 0 0
\(75\) 3.61864e7i 1.14367i
\(76\) 0 0
\(77\) 8.00437e7 2.27701
\(78\) 0 0
\(79\) − 5.94312e7i − 1.52583i −0.646500 0.762914i \(-0.723768\pi\)
0.646500 0.762914i \(-0.276232\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 5.42483e7i 1.14307i 0.820577 + 0.571536i \(0.193652\pi\)
−0.820577 + 0.571536i \(0.806348\pi\)
\(84\) 0 0
\(85\) −6.46846e7 −1.23915
\(86\) 0 0
\(87\) − 2.26920e7i − 0.396091i
\(88\) 0 0
\(89\) −2.43253e7 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(90\) 0 0
\(91\) − 1.43723e8i − 2.09585i
\(92\) 0 0
\(93\) −3.14218e7 −0.420049
\(94\) 0 0
\(95\) 1.36298e8i 1.67338i
\(96\) 0 0
\(97\) −3.83933e7 −0.433679 −0.216839 0.976207i \(-0.569575\pi\)
−0.216839 + 0.976207i \(0.569575\pi\)
\(98\) 0 0
\(99\) − 4.39761e7i − 0.457800i
\(100\) 0 0
\(101\) −1.49683e8 −1.43842 −0.719212 0.694791i \(-0.755497\pi\)
−0.719212 + 0.694791i \(0.755497\pi\)
\(102\) 0 0
\(103\) − 7.61086e7i − 0.676215i −0.941107 0.338108i \(-0.890213\pi\)
0.941107 0.338108i \(-0.109787\pi\)
\(104\) 0 0
\(105\) −2.00880e8 −1.65264
\(106\) 0 0
\(107\) − 1.33636e8i − 1.01950i −0.860322 0.509751i \(-0.829738\pi\)
0.860322 0.509751i \(-0.170262\pi\)
\(108\) 0 0
\(109\) −1.88815e8 −1.33762 −0.668808 0.743436i \(-0.733195\pi\)
−0.668808 + 0.743436i \(0.733195\pi\)
\(110\) 0 0
\(111\) 1.59163e8i 1.04846i
\(112\) 0 0
\(113\) 1.75442e7 0.107602 0.0538009 0.998552i \(-0.482866\pi\)
0.0538009 + 0.998552i \(0.482866\pi\)
\(114\) 0 0
\(115\) 2.12696e8i 1.21609i
\(116\) 0 0
\(117\) −7.89615e7 −0.421378
\(118\) 0 0
\(119\) − 2.38620e8i − 1.18992i
\(120\) 0 0
\(121\) −1.89971e8 −0.886231
\(122\) 0 0
\(123\) − 1.66355e8i − 0.726802i
\(124\) 0 0
\(125\) −4.13461e8 −1.69354
\(126\) 0 0
\(127\) 1.91097e8i 0.734579i 0.930107 + 0.367289i \(0.119714\pi\)
−0.930107 + 0.367289i \(0.880286\pi\)
\(128\) 0 0
\(129\) 3.44501e7 0.124403
\(130\) 0 0
\(131\) 4.29123e7i 0.145712i 0.997342 + 0.0728562i \(0.0232114\pi\)
−0.997342 + 0.0728562i \(0.976789\pi\)
\(132\) 0 0
\(133\) −5.02799e8 −1.60690
\(134\) 0 0
\(135\) 1.10364e8i 0.332270i
\(136\) 0 0
\(137\) −2.12185e7 −0.0602328 −0.0301164 0.999546i \(-0.509588\pi\)
−0.0301164 + 0.999546i \(0.509588\pi\)
\(138\) 0 0
\(139\) 4.01495e8i 1.07552i 0.843096 + 0.537762i \(0.180730\pi\)
−0.843096 + 0.537762i \(0.819270\pi\)
\(140\) 0 0
\(141\) 4.13154e7 0.104529
\(142\) 0 0
\(143\) 7.25997e8i 1.73616i
\(144\) 0 0
\(145\) 5.23601e8 1.18448
\(146\) 0 0
\(147\) − 4.71449e8i − 1.00964i
\(148\) 0 0
\(149\) 2.84042e8 0.576285 0.288143 0.957588i \(-0.406962\pi\)
0.288143 + 0.957588i \(0.406962\pi\)
\(150\) 0 0
\(151\) 8.35682e8i 1.60744i 0.595011 + 0.803718i \(0.297148\pi\)
−0.595011 + 0.803718i \(0.702852\pi\)
\(152\) 0 0
\(153\) −1.31098e8 −0.239238
\(154\) 0 0
\(155\) − 7.25037e8i − 1.25613i
\(156\) 0 0
\(157\) −7.20894e8 −1.18651 −0.593257 0.805013i \(-0.702158\pi\)
−0.593257 + 0.805013i \(0.702158\pi\)
\(158\) 0 0
\(159\) − 6.77689e8i − 1.06033i
\(160\) 0 0
\(161\) −7.84629e8 −1.16778
\(162\) 0 0
\(163\) 5.61685e8i 0.795687i 0.917453 + 0.397843i \(0.130241\pi\)
−0.917453 + 0.397843i \(0.869759\pi\)
\(164\) 0 0
\(165\) 1.01472e9 1.36902
\(166\) 0 0
\(167\) − 7.29640e7i − 0.0938086i −0.998899 0.0469043i \(-0.985064\pi\)
0.998899 0.0469043i \(-0.0149356\pi\)
\(168\) 0 0
\(169\) 4.87835e8 0.598034
\(170\) 0 0
\(171\) 2.76238e8i 0.323072i
\(172\) 0 0
\(173\) 6.80934e8 0.760187 0.380093 0.924948i \(-0.375892\pi\)
0.380093 + 0.924948i \(0.375892\pi\)
\(174\) 0 0
\(175\) − 3.08021e9i − 3.28419i
\(176\) 0 0
\(177\) 5.06588e7 0.0516134
\(178\) 0 0
\(179\) − 4.03178e8i − 0.392721i −0.980532 0.196361i \(-0.937088\pi\)
0.980532 0.196361i \(-0.0629123\pi\)
\(180\) 0 0
\(181\) −1.19998e9 −1.11805 −0.559023 0.829152i \(-0.688823\pi\)
−0.559023 + 0.829152i \(0.688823\pi\)
\(182\) 0 0
\(183\) 1.61389e8i 0.143903i
\(184\) 0 0
\(185\) −3.67258e9 −3.13534
\(186\) 0 0
\(187\) 1.20536e9i 0.985710i
\(188\) 0 0
\(189\) −4.07129e8 −0.319069
\(190\) 0 0
\(191\) 4.09810e8i 0.307928i 0.988076 + 0.153964i \(0.0492040\pi\)
−0.988076 + 0.153964i \(0.950796\pi\)
\(192\) 0 0
\(193\) 8.43979e7 0.0608278 0.0304139 0.999537i \(-0.490317\pi\)
0.0304139 + 0.999537i \(0.490317\pi\)
\(194\) 0 0
\(195\) − 1.82198e9i − 1.26010i
\(196\) 0 0
\(197\) −1.18786e9 −0.788677 −0.394339 0.918965i \(-0.629026\pi\)
−0.394339 + 0.918965i \(0.629026\pi\)
\(198\) 0 0
\(199\) − 4.47256e8i − 0.285196i −0.989781 0.142598i \(-0.954454\pi\)
0.989781 0.142598i \(-0.0455457\pi\)
\(200\) 0 0
\(201\) −6.70809e8 −0.410974
\(202\) 0 0
\(203\) 1.93155e9i 1.13742i
\(204\) 0 0
\(205\) 3.83853e9 2.17345
\(206\) 0 0
\(207\) 4.31076e8i 0.234786i
\(208\) 0 0
\(209\) 2.53982e9 1.33112
\(210\) 0 0
\(211\) 6.71991e8i 0.339026i 0.985528 + 0.169513i \(0.0542195\pi\)
−0.985528 + 0.169513i \(0.945780\pi\)
\(212\) 0 0
\(213\) 1.85903e8 0.0903164
\(214\) 0 0
\(215\) 7.94913e8i 0.372019i
\(216\) 0 0
\(217\) 2.67465e9 1.20622
\(218\) 0 0
\(219\) 1.83322e8i 0.0796963i
\(220\) 0 0
\(221\) 2.16428e9 0.907287
\(222\) 0 0
\(223\) − 3.40869e9i − 1.37838i −0.724582 0.689188i \(-0.757967\pi\)
0.724582 0.689188i \(-0.242033\pi\)
\(224\) 0 0
\(225\) −1.69227e9 −0.660297
\(226\) 0 0
\(227\) 4.05729e9i 1.52803i 0.645198 + 0.764016i \(0.276775\pi\)
−0.645198 + 0.764016i \(0.723225\pi\)
\(228\) 0 0
\(229\) −3.28401e9 −1.19416 −0.597080 0.802182i \(-0.703673\pi\)
−0.597080 + 0.802182i \(0.703673\pi\)
\(230\) 0 0
\(231\) 3.74327e9i 1.31463i
\(232\) 0 0
\(233\) 4.91934e9 1.66910 0.834552 0.550929i \(-0.185727\pi\)
0.834552 + 0.550929i \(0.185727\pi\)
\(234\) 0 0
\(235\) 9.53325e8i 0.312586i
\(236\) 0 0
\(237\) 2.77932e9 0.880938
\(238\) 0 0
\(239\) 1.11979e9i 0.343199i 0.985167 + 0.171599i \(0.0548935\pi\)
−0.985167 + 0.171599i \(0.945107\pi\)
\(240\) 0 0
\(241\) −3.54686e9 −1.05142 −0.525710 0.850664i \(-0.676200\pi\)
−0.525710 + 0.850664i \(0.676200\pi\)
\(242\) 0 0
\(243\) 2.23677e8i 0.0641500i
\(244\) 0 0
\(245\) 1.08784e10 3.01925
\(246\) 0 0
\(247\) − 4.56039e9i − 1.22522i
\(248\) 0 0
\(249\) −2.53694e9 −0.659953
\(250\) 0 0
\(251\) − 5.42783e9i − 1.36751i −0.729710 0.683756i \(-0.760345\pi\)
0.729710 0.683756i \(-0.239655\pi\)
\(252\) 0 0
\(253\) 3.96345e9 0.967367
\(254\) 0 0
\(255\) − 3.02500e9i − 0.715425i
\(256\) 0 0
\(257\) 5.11925e9 1.17348 0.586738 0.809777i \(-0.300412\pi\)
0.586738 + 0.809777i \(0.300412\pi\)
\(258\) 0 0
\(259\) − 1.35481e10i − 3.01078i
\(260\) 0 0
\(261\) 1.06120e9 0.228683
\(262\) 0 0
\(263\) 7.63972e9i 1.59681i 0.602118 + 0.798407i \(0.294324\pi\)
−0.602118 + 0.798407i \(0.705676\pi\)
\(264\) 0 0
\(265\) 1.56372e10 3.17085
\(266\) 0 0
\(267\) − 1.13758e9i − 0.223840i
\(268\) 0 0
\(269\) −5.74735e9 −1.09764 −0.548818 0.835942i \(-0.684922\pi\)
−0.548818 + 0.835942i \(0.684922\pi\)
\(270\) 0 0
\(271\) 4.27034e9i 0.791746i 0.918305 + 0.395873i \(0.129558\pi\)
−0.918305 + 0.395873i \(0.870442\pi\)
\(272\) 0 0
\(273\) 6.72125e9 1.21004
\(274\) 0 0
\(275\) 1.55593e10i 2.72056i
\(276\) 0 0
\(277\) −9.64494e9 −1.63825 −0.819125 0.573614i \(-0.805541\pi\)
−0.819125 + 0.573614i \(0.805541\pi\)
\(278\) 0 0
\(279\) − 1.46945e9i − 0.242515i
\(280\) 0 0
\(281\) −2.63511e9 −0.422643 −0.211322 0.977417i \(-0.567777\pi\)
−0.211322 + 0.977417i \(0.567777\pi\)
\(282\) 0 0
\(283\) − 3.18282e8i − 0.0496212i −0.999692 0.0248106i \(-0.992102\pi\)
0.999692 0.0248106i \(-0.00789826\pi\)
\(284\) 0 0
\(285\) −6.37401e9 −0.966125
\(286\) 0 0
\(287\) 1.41603e10i 2.08710i
\(288\) 0 0
\(289\) −3.38245e9 −0.484886
\(290\) 0 0
\(291\) − 1.79547e9i − 0.250384i
\(292\) 0 0
\(293\) 1.15368e9 0.156536 0.0782682 0.996932i \(-0.475061\pi\)
0.0782682 + 0.996932i \(0.475061\pi\)
\(294\) 0 0
\(295\) 1.16892e9i 0.154346i
\(296\) 0 0
\(297\) 2.05656e9 0.264311
\(298\) 0 0
\(299\) − 7.11659e9i − 0.890404i
\(300\) 0 0
\(301\) −2.93242e9 −0.357239
\(302\) 0 0
\(303\) − 6.99998e9i − 0.830474i
\(304\) 0 0
\(305\) −3.72395e9 −0.430332
\(306\) 0 0
\(307\) 1.55195e10i 1.74712i 0.486715 + 0.873561i \(0.338195\pi\)
−0.486715 + 0.873561i \(0.661805\pi\)
\(308\) 0 0
\(309\) 3.55925e9 0.390413
\(310\) 0 0
\(311\) 3.12251e9i 0.333782i 0.985975 + 0.166891i \(0.0533728\pi\)
−0.985975 + 0.166891i \(0.946627\pi\)
\(312\) 0 0
\(313\) −9.18882e9 −0.957375 −0.478688 0.877985i \(-0.658887\pi\)
−0.478688 + 0.877985i \(0.658887\pi\)
\(314\) 0 0
\(315\) − 9.39423e9i − 0.954155i
\(316\) 0 0
\(317\) −1.03286e10 −1.02283 −0.511417 0.859333i \(-0.670879\pi\)
−0.511417 + 0.859333i \(0.670879\pi\)
\(318\) 0 0
\(319\) − 9.75699e9i − 0.942222i
\(320\) 0 0
\(321\) 6.24953e9 0.588609
\(322\) 0 0
\(323\) − 7.57152e9i − 0.695621i
\(324\) 0 0
\(325\) 2.79375e10 2.50411
\(326\) 0 0
\(327\) − 8.83002e9i − 0.772272i
\(328\) 0 0
\(329\) −3.51679e9 −0.300167
\(330\) 0 0
\(331\) − 2.10055e10i − 1.74993i −0.484184 0.874966i \(-0.660883\pi\)
0.484184 0.874966i \(-0.339117\pi\)
\(332\) 0 0
\(333\) −7.44333e9 −0.605327
\(334\) 0 0
\(335\) − 1.54784e10i − 1.22899i
\(336\) 0 0
\(337\) 6.33904e9 0.491478 0.245739 0.969336i \(-0.420969\pi\)
0.245739 + 0.969336i \(0.420969\pi\)
\(338\) 0 0
\(339\) 8.20461e8i 0.0621239i
\(340\) 0 0
\(341\) −1.35106e10 −0.999212
\(342\) 0 0
\(343\) 1.71821e10i 1.24136i
\(344\) 0 0
\(345\) −9.94679e9 −0.702112
\(346\) 0 0
\(347\) 9.29606e9i 0.641181i 0.947218 + 0.320591i \(0.103881\pi\)
−0.947218 + 0.320591i \(0.896119\pi\)
\(348\) 0 0
\(349\) 2.11040e10 1.42254 0.711269 0.702920i \(-0.248121\pi\)
0.711269 + 0.702920i \(0.248121\pi\)
\(350\) 0 0
\(351\) − 3.69266e9i − 0.243283i
\(352\) 0 0
\(353\) 1.30618e10 0.841209 0.420604 0.907244i \(-0.361818\pi\)
0.420604 + 0.907244i \(0.361818\pi\)
\(354\) 0 0
\(355\) 4.28957e9i 0.270085i
\(356\) 0 0
\(357\) 1.11591e10 0.687002
\(358\) 0 0
\(359\) − 1.18146e10i − 0.711279i −0.934623 0.355639i \(-0.884263\pi\)
0.934623 0.355639i \(-0.115737\pi\)
\(360\) 0 0
\(361\) 1.02952e9 0.0606185
\(362\) 0 0
\(363\) − 8.88409e9i − 0.511666i
\(364\) 0 0
\(365\) −4.23003e9 −0.238326
\(366\) 0 0
\(367\) 2.69814e10i 1.48730i 0.668567 + 0.743652i \(0.266908\pi\)
−0.668567 + 0.743652i \(0.733092\pi\)
\(368\) 0 0
\(369\) 7.77967e9 0.419620
\(370\) 0 0
\(371\) 5.76853e10i 3.04488i
\(372\) 0 0
\(373\) −2.07520e10 −1.07208 −0.536038 0.844194i \(-0.680080\pi\)
−0.536038 + 0.844194i \(0.680080\pi\)
\(374\) 0 0
\(375\) − 1.93356e10i − 0.977763i
\(376\) 0 0
\(377\) −1.75192e10 −0.867259
\(378\) 0 0
\(379\) − 3.91651e10i − 1.89820i −0.314973 0.949101i \(-0.601995\pi\)
0.314973 0.949101i \(-0.398005\pi\)
\(380\) 0 0
\(381\) −8.93671e9 −0.424109
\(382\) 0 0
\(383\) 1.04984e8i 0.00487897i 0.999997 + 0.00243949i \(0.000776513\pi\)
−0.999997 + 0.00243949i \(0.999223\pi\)
\(384\) 0 0
\(385\) −8.63735e10 −3.93131
\(386\) 0 0
\(387\) 1.61107e9i 0.0718243i
\(388\) 0 0
\(389\) 9.73662e9 0.425216 0.212608 0.977138i \(-0.431804\pi\)
0.212608 + 0.977138i \(0.431804\pi\)
\(390\) 0 0
\(391\) − 1.18155e10i − 0.505529i
\(392\) 0 0
\(393\) −2.00681e9 −0.0841271
\(394\) 0 0
\(395\) 6.41309e10i 2.63438i
\(396\) 0 0
\(397\) −2.07590e9 −0.0835687 −0.0417843 0.999127i \(-0.513304\pi\)
−0.0417843 + 0.999127i \(0.513304\pi\)
\(398\) 0 0
\(399\) − 2.35136e10i − 0.927742i
\(400\) 0 0
\(401\) −2.62141e10 −1.01381 −0.506905 0.862002i \(-0.669211\pi\)
−0.506905 + 0.862002i \(0.669211\pi\)
\(402\) 0 0
\(403\) 2.42590e10i 0.919715i
\(404\) 0 0
\(405\) −5.16120e9 −0.191836
\(406\) 0 0
\(407\) 6.84363e10i 2.49407i
\(408\) 0 0
\(409\) −1.51824e10 −0.542559 −0.271280 0.962501i \(-0.587447\pi\)
−0.271280 + 0.962501i \(0.587447\pi\)
\(410\) 0 0
\(411\) − 9.92293e8i − 0.0347754i
\(412\) 0 0
\(413\) −4.31211e9 −0.148214
\(414\) 0 0
\(415\) − 5.85382e10i − 1.97354i
\(416\) 0 0
\(417\) −1.87760e10 −0.620955
\(418\) 0 0
\(419\) − 2.04890e10i − 0.664760i −0.943145 0.332380i \(-0.892148\pi\)
0.943145 0.332380i \(-0.107852\pi\)
\(420\) 0 0
\(421\) −5.40434e10 −1.72034 −0.860170 0.510007i \(-0.829643\pi\)
−0.860170 + 0.510007i \(0.829643\pi\)
\(422\) 0 0
\(423\) 1.93213e9i 0.0603497i
\(424\) 0 0
\(425\) 4.63840e10 1.42172
\(426\) 0 0
\(427\) − 1.37376e10i − 0.413236i
\(428\) 0 0
\(429\) −3.39515e10 −1.00237
\(430\) 0 0
\(431\) − 4.04198e10i − 1.17134i −0.810548 0.585672i \(-0.800831\pi\)
0.810548 0.585672i \(-0.199169\pi\)
\(432\) 0 0
\(433\) −4.32070e9 −0.122914 −0.0614572 0.998110i \(-0.519575\pi\)
−0.0614572 + 0.998110i \(0.519575\pi\)
\(434\) 0 0
\(435\) 2.44864e10i 0.683861i
\(436\) 0 0
\(437\) −2.48966e10 −0.682677
\(438\) 0 0
\(439\) 3.97062e10i 1.06905i 0.845151 + 0.534527i \(0.179510\pi\)
−0.845151 + 0.534527i \(0.820490\pi\)
\(440\) 0 0
\(441\) 2.20475e10 0.582914
\(442\) 0 0
\(443\) − 1.97115e10i − 0.511804i −0.966703 0.255902i \(-0.917628\pi\)
0.966703 0.255902i \(-0.0823725\pi\)
\(444\) 0 0
\(445\) 2.62489e10 0.669378
\(446\) 0 0
\(447\) 1.32833e10i 0.332718i
\(448\) 0 0
\(449\) −1.89134e10 −0.465355 −0.232677 0.972554i \(-0.574749\pi\)
−0.232677 + 0.972554i \(0.574749\pi\)
\(450\) 0 0
\(451\) − 7.15287e10i − 1.72892i
\(452\) 0 0
\(453\) −3.90810e10 −0.928053
\(454\) 0 0
\(455\) 1.55088e11i 3.61854i
\(456\) 0 0
\(457\) −5.24800e10 −1.20318 −0.601588 0.798806i \(-0.705465\pi\)
−0.601588 + 0.798806i \(0.705465\pi\)
\(458\) 0 0
\(459\) − 6.13085e9i − 0.138124i
\(460\) 0 0
\(461\) 4.67940e9 0.103606 0.0518032 0.998657i \(-0.483503\pi\)
0.0518032 + 0.998657i \(0.483503\pi\)
\(462\) 0 0
\(463\) − 8.06631e10i − 1.75530i −0.479304 0.877649i \(-0.659111\pi\)
0.479304 0.877649i \(-0.340889\pi\)
\(464\) 0 0
\(465\) 3.39066e10 0.725225
\(466\) 0 0
\(467\) − 2.84821e10i − 0.598832i −0.954123 0.299416i \(-0.903208\pi\)
0.954123 0.299416i \(-0.0967918\pi\)
\(468\) 0 0
\(469\) 5.70996e10 1.18016
\(470\) 0 0
\(471\) − 3.37129e10i − 0.685034i
\(472\) 0 0
\(473\) 1.48127e10 0.295931
\(474\) 0 0
\(475\) − 9.77364e10i − 1.91991i
\(476\) 0 0
\(477\) 3.16924e10 0.612183
\(478\) 0 0
\(479\) − 3.38937e10i − 0.643838i −0.946767 0.321919i \(-0.895672\pi\)
0.946767 0.321919i \(-0.104328\pi\)
\(480\) 0 0
\(481\) 1.22881e11 2.29564
\(482\) 0 0
\(483\) − 3.66935e10i − 0.674218i
\(484\) 0 0
\(485\) 4.14293e10 0.748757
\(486\) 0 0
\(487\) 1.38440e10i 0.246120i 0.992399 + 0.123060i \(0.0392708\pi\)
−0.992399 + 0.123060i \(0.960729\pi\)
\(488\) 0 0
\(489\) −2.62674e10 −0.459390
\(490\) 0 0
\(491\) 8.71026e10i 1.49867i 0.662193 + 0.749333i \(0.269626\pi\)
−0.662193 + 0.749333i \(0.730374\pi\)
\(492\) 0 0
\(493\) −2.90867e10 −0.492388
\(494\) 0 0
\(495\) 4.74537e10i 0.790404i
\(496\) 0 0
\(497\) −1.58241e10 −0.259355
\(498\) 0 0
\(499\) 5.51402e10i 0.889337i 0.895695 + 0.444668i \(0.146678\pi\)
−0.895695 + 0.444668i \(0.853322\pi\)
\(500\) 0 0
\(501\) 3.41219e9 0.0541604
\(502\) 0 0
\(503\) − 5.65908e10i − 0.884045i −0.897004 0.442022i \(-0.854261\pi\)
0.897004 0.442022i \(-0.145739\pi\)
\(504\) 0 0
\(505\) 1.61520e11 2.48348
\(506\) 0 0
\(507\) 2.28138e10i 0.345275i
\(508\) 0 0
\(509\) −8.63292e10 −1.28614 −0.643068 0.765809i \(-0.722339\pi\)
−0.643068 + 0.765809i \(0.722339\pi\)
\(510\) 0 0
\(511\) − 1.56045e10i − 0.228858i
\(512\) 0 0
\(513\) −1.29184e10 −0.186526
\(514\) 0 0
\(515\) 8.21272e10i 1.16750i
\(516\) 0 0
\(517\) 1.77646e10 0.248653
\(518\) 0 0
\(519\) 3.18441e10i 0.438894i
\(520\) 0 0
\(521\) −3.77681e10 −0.512595 −0.256298 0.966598i \(-0.582503\pi\)
−0.256298 + 0.966598i \(0.582503\pi\)
\(522\) 0 0
\(523\) − 5.86646e10i − 0.784096i −0.919945 0.392048i \(-0.871767\pi\)
0.919945 0.392048i \(-0.128233\pi\)
\(524\) 0 0
\(525\) 1.44047e11 1.89613
\(526\) 0 0
\(527\) 4.02768e10i 0.522170i
\(528\) 0 0
\(529\) 3.94592e10 0.503878
\(530\) 0 0
\(531\) 2.36908e9i 0.0297990i
\(532\) 0 0
\(533\) −1.28434e11 −1.59137
\(534\) 0 0
\(535\) 1.44204e11i 1.76020i
\(536\) 0 0
\(537\) 1.88548e10 0.226738
\(538\) 0 0
\(539\) − 2.02711e11i − 2.40172i
\(540\) 0 0
\(541\) −7.10071e10 −0.828920 −0.414460 0.910067i \(-0.636030\pi\)
−0.414460 + 0.910067i \(0.636030\pi\)
\(542\) 0 0
\(543\) − 5.61175e10i − 0.645504i
\(544\) 0 0
\(545\) 2.03747e11 2.30943
\(546\) 0 0
\(547\) 1.23143e11i 1.37550i 0.725947 + 0.687750i \(0.241402\pi\)
−0.725947 + 0.687750i \(0.758598\pi\)
\(548\) 0 0
\(549\) −7.54743e9 −0.0830825
\(550\) 0 0
\(551\) 6.12891e10i 0.664931i
\(552\) 0 0
\(553\) −2.36577e11 −2.52972
\(554\) 0 0
\(555\) − 1.71750e11i − 1.81019i
\(556\) 0 0
\(557\) 1.47407e11 1.53143 0.765717 0.643177i \(-0.222384\pi\)
0.765717 + 0.643177i \(0.222384\pi\)
\(558\) 0 0
\(559\) − 2.65970e10i − 0.272386i
\(560\) 0 0
\(561\) −5.63690e10 −0.569100
\(562\) 0 0
\(563\) 1.33296e11i 1.32673i 0.748296 + 0.663365i \(0.230872\pi\)
−0.748296 + 0.663365i \(0.769128\pi\)
\(564\) 0 0
\(565\) −1.89316e10 −0.185777
\(566\) 0 0
\(567\) − 1.90395e10i − 0.184215i
\(568\) 0 0
\(569\) −1.29497e11 −1.23541 −0.617703 0.786412i \(-0.711937\pi\)
−0.617703 + 0.786412i \(0.711937\pi\)
\(570\) 0 0
\(571\) 1.08901e11i 1.02444i 0.858854 + 0.512221i \(0.171177\pi\)
−0.858854 + 0.512221i \(0.828823\pi\)
\(572\) 0 0
\(573\) −1.91649e10 −0.177782
\(574\) 0 0
\(575\) − 1.52520e11i − 1.39526i
\(576\) 0 0
\(577\) 1.23772e11 1.11666 0.558330 0.829619i \(-0.311442\pi\)
0.558330 + 0.829619i \(0.311442\pi\)
\(578\) 0 0
\(579\) 3.94690e9i 0.0351190i
\(580\) 0 0
\(581\) 2.15946e11 1.89514
\(582\) 0 0
\(583\) − 2.91390e11i − 2.52232i
\(584\) 0 0
\(585\) 8.52056e10 0.727520
\(586\) 0 0
\(587\) − 9.78754e10i − 0.824368i −0.911101 0.412184i \(-0.864766\pi\)
0.911101 0.412184i \(-0.135234\pi\)
\(588\) 0 0
\(589\) 8.48677e10 0.705150
\(590\) 0 0
\(591\) − 5.55506e10i − 0.455343i
\(592\) 0 0
\(593\) −6.13187e10 −0.495877 −0.247938 0.968776i \(-0.579753\pi\)
−0.247938 + 0.968776i \(0.579753\pi\)
\(594\) 0 0
\(595\) 2.57490e11i 2.05443i
\(596\) 0 0
\(597\) 2.09161e10 0.164658
\(598\) 0 0
\(599\) 4.28422e10i 0.332786i 0.986060 + 0.166393i \(0.0532120\pi\)
−0.986060 + 0.166393i \(0.946788\pi\)
\(600\) 0 0
\(601\) 9.04002e10 0.692901 0.346451 0.938068i \(-0.387387\pi\)
0.346451 + 0.938068i \(0.387387\pi\)
\(602\) 0 0
\(603\) − 3.13706e10i − 0.237276i
\(604\) 0 0
\(605\) 2.04994e11 1.53010
\(606\) 0 0
\(607\) 1.05084e11i 0.774069i 0.922065 + 0.387035i \(0.126501\pi\)
−0.922065 + 0.387035i \(0.873499\pi\)
\(608\) 0 0
\(609\) −9.03298e10 −0.656692
\(610\) 0 0
\(611\) − 3.18973e10i − 0.228870i
\(612\) 0 0
\(613\) 2.50049e11 1.77085 0.885427 0.464778i \(-0.153866\pi\)
0.885427 + 0.464778i \(0.153866\pi\)
\(614\) 0 0
\(615\) 1.79511e11i 1.25484i
\(616\) 0 0
\(617\) 5.63479e10 0.388810 0.194405 0.980921i \(-0.437722\pi\)
0.194405 + 0.980921i \(0.437722\pi\)
\(618\) 0 0
\(619\) 2.37696e11i 1.61904i 0.587090 + 0.809522i \(0.300274\pi\)
−0.587090 + 0.809522i \(0.699726\pi\)
\(620\) 0 0
\(621\) −2.01594e10 −0.135554
\(622\) 0 0
\(623\) 9.68316e10i 0.642784i
\(624\) 0 0
\(625\) 1.43897e11 0.943041
\(626\) 0 0
\(627\) 1.18776e11i 0.768525i
\(628\) 0 0
\(629\) 2.04017e11 1.30336
\(630\) 0 0
\(631\) 3.96989e10i 0.250416i 0.992131 + 0.125208i \(0.0399598\pi\)
−0.992131 + 0.125208i \(0.960040\pi\)
\(632\) 0 0
\(633\) −3.14259e10 −0.195737
\(634\) 0 0
\(635\) − 2.06208e11i − 1.26827i
\(636\) 0 0
\(637\) −3.63979e11 −2.21064
\(638\) 0 0
\(639\) 8.69380e9i 0.0521442i
\(640\) 0 0
\(641\) −2.16918e11 −1.28488 −0.642442 0.766334i \(-0.722079\pi\)
−0.642442 + 0.766334i \(0.722079\pi\)
\(642\) 0 0
\(643\) 3.00637e10i 0.175873i 0.996126 + 0.0879365i \(0.0280273\pi\)
−0.996126 + 0.0879365i \(0.971973\pi\)
\(644\) 0 0
\(645\) −3.71744e10 −0.214786
\(646\) 0 0
\(647\) − 2.17396e11i − 1.24061i −0.784361 0.620305i \(-0.787009\pi\)
0.784361 0.620305i \(-0.212991\pi\)
\(648\) 0 0
\(649\) 2.17821e10 0.122778
\(650\) 0 0
\(651\) 1.25081e11i 0.696413i
\(652\) 0 0
\(653\) 9.82145e10 0.540161 0.270080 0.962838i \(-0.412950\pi\)
0.270080 + 0.962838i \(0.412950\pi\)
\(654\) 0 0
\(655\) − 4.63058e10i − 0.251576i
\(656\) 0 0
\(657\) −8.57313e9 −0.0460127
\(658\) 0 0
\(659\) − 8.83128e10i − 0.468255i −0.972206 0.234127i \(-0.924777\pi\)
0.972206 0.234127i \(-0.0752233\pi\)
\(660\) 0 0
\(661\) −9.04522e10 −0.473820 −0.236910 0.971532i \(-0.576135\pi\)
−0.236910 + 0.971532i \(0.576135\pi\)
\(662\) 0 0
\(663\) 1.01213e11i 0.523823i
\(664\) 0 0
\(665\) 5.42560e11 2.77435
\(666\) 0 0
\(667\) 9.56429e10i 0.483225i
\(668\) 0 0
\(669\) 1.59409e11 0.795806
\(670\) 0 0
\(671\) 6.93934e10i 0.342317i
\(672\) 0 0
\(673\) 2.22996e11 1.08702 0.543510 0.839403i \(-0.317095\pi\)
0.543510 + 0.839403i \(0.317095\pi\)
\(674\) 0 0
\(675\) − 7.91396e10i − 0.381223i
\(676\) 0 0
\(677\) −3.19359e11 −1.52028 −0.760142 0.649757i \(-0.774871\pi\)
−0.760142 + 0.649757i \(0.774871\pi\)
\(678\) 0 0
\(679\) 1.52832e11i 0.719010i
\(680\) 0 0
\(681\) −1.89741e11 −0.882209
\(682\) 0 0
\(683\) − 3.69050e11i − 1.69591i −0.530070 0.847954i \(-0.677834\pi\)
0.530070 0.847954i \(-0.322166\pi\)
\(684\) 0 0
\(685\) 2.28965e10 0.103994
\(686\) 0 0
\(687\) − 1.53578e11i − 0.689449i
\(688\) 0 0
\(689\) −5.23206e11 −2.32164
\(690\) 0 0
\(691\) − 3.14143e11i − 1.37789i −0.724813 0.688946i \(-0.758074\pi\)
0.724813 0.688946i \(-0.241926\pi\)
\(692\) 0 0
\(693\) −1.75056e11 −0.759002
\(694\) 0 0
\(695\) − 4.33244e11i − 1.85692i
\(696\) 0 0
\(697\) −2.13236e11 −0.903501
\(698\) 0 0
\(699\) 2.30055e11i 0.963657i
\(700\) 0 0
\(701\) −2.36655e11 −0.980040 −0.490020 0.871711i \(-0.663011\pi\)
−0.490020 + 0.871711i \(0.663011\pi\)
\(702\) 0 0
\(703\) − 4.29887e11i − 1.76008i
\(704\) 0 0
\(705\) −4.45826e10 −0.180472
\(706\) 0 0
\(707\) 5.95843e11i 2.38481i
\(708\) 0 0
\(709\) −1.91093e11 −0.756240 −0.378120 0.925757i \(-0.623429\pi\)
−0.378120 + 0.925757i \(0.623429\pi\)
\(710\) 0 0
\(711\) 1.29976e11i 0.508610i
\(712\) 0 0
\(713\) 1.32438e11 0.512453
\(714\) 0 0
\(715\) − 7.83408e11i − 2.99753i
\(716\) 0 0
\(717\) −5.23675e10 −0.198146
\(718\) 0 0
\(719\) − 1.08030e11i − 0.404230i −0.979362 0.202115i \(-0.935219\pi\)
0.979362 0.202115i \(-0.0647814\pi\)
\(720\) 0 0
\(721\) −3.02965e11 −1.12112
\(722\) 0 0
\(723\) − 1.65870e11i − 0.607038i
\(724\) 0 0
\(725\) −3.75464e11 −1.35899
\(726\) 0 0
\(727\) 2.11129e11i 0.755804i 0.925846 + 0.377902i \(0.123354\pi\)
−0.925846 + 0.377902i \(0.876646\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) − 4.41584e10i − 0.154648i
\(732\) 0 0
\(733\) 1.48237e10 0.0513500 0.0256750 0.999670i \(-0.491826\pi\)
0.0256750 + 0.999670i \(0.491826\pi\)
\(734\) 0 0
\(735\) 5.08730e11i 1.74316i
\(736\) 0 0
\(737\) −2.88431e11 −0.977625
\(738\) 0 0
\(739\) 2.87737e10i 0.0964757i 0.998836 + 0.0482378i \(0.0153605\pi\)
−0.998836 + 0.0482378i \(0.984639\pi\)
\(740\) 0 0
\(741\) 2.13268e11 0.707381
\(742\) 0 0
\(743\) 4.75481e11i 1.56019i 0.625660 + 0.780096i \(0.284830\pi\)
−0.625660 + 0.780096i \(0.715170\pi\)
\(744\) 0 0
\(745\) −3.06504e11 −0.994971
\(746\) 0 0
\(747\) − 1.18641e11i − 0.381024i
\(748\) 0 0
\(749\) −5.31964e11 −1.69026
\(750\) 0 0
\(751\) 2.15430e11i 0.677245i 0.940922 + 0.338623i \(0.109961\pi\)
−0.940922 + 0.338623i \(0.890039\pi\)
\(752\) 0 0
\(753\) 2.53835e11 0.789534
\(754\) 0 0
\(755\) − 9.01767e11i − 2.77528i
\(756\) 0 0
\(757\) −3.87925e11 −1.18131 −0.590656 0.806924i \(-0.701131\pi\)
−0.590656 + 0.806924i \(0.701131\pi\)
\(758\) 0 0
\(759\) 1.85352e11i 0.558510i
\(760\) 0 0
\(761\) −1.81201e11 −0.540284 −0.270142 0.962821i \(-0.587071\pi\)
−0.270142 + 0.962821i \(0.587071\pi\)
\(762\) 0 0
\(763\) 7.51616e11i 2.21768i
\(764\) 0 0
\(765\) 1.41465e11 0.413051
\(766\) 0 0
\(767\) − 3.91108e10i − 0.113010i
\(768\) 0 0
\(769\) 1.65080e10 0.0472052 0.0236026 0.999721i \(-0.492486\pi\)
0.0236026 + 0.999721i \(0.492486\pi\)
\(770\) 0 0
\(771\) 2.39404e11i 0.677506i
\(772\) 0 0
\(773\) −1.35215e11 −0.378710 −0.189355 0.981909i \(-0.560640\pi\)
−0.189355 + 0.981909i \(0.560640\pi\)
\(774\) 0 0
\(775\) 5.19910e11i 1.44119i
\(776\) 0 0
\(777\) 6.33581e11 1.73827
\(778\) 0 0
\(779\) 4.49312e11i 1.22011i
\(780\) 0 0
\(781\) 7.99335e10 0.214845
\(782\) 0 0
\(783\) 4.96273e10i 0.132030i
\(784\) 0 0
\(785\) 7.77901e11 2.04855
\(786\) 0 0
\(787\) − 3.83876e11i − 1.00067i −0.865831 0.500337i \(-0.833209\pi\)
0.865831 0.500337i \(-0.166791\pi\)
\(788\) 0 0
\(789\) −3.57274e11 −0.921921
\(790\) 0 0
\(791\) − 6.98381e10i − 0.178397i
\(792\) 0 0
\(793\) 1.24600e11 0.315082
\(794\) 0 0
\(795\) 7.31280e11i 1.83069i
\(796\) 0 0
\(797\) −2.54150e10 −0.0629878 −0.0314939 0.999504i \(-0.510026\pi\)
−0.0314939 + 0.999504i \(0.510026\pi\)
\(798\) 0 0
\(799\) − 5.29584e10i − 0.129942i
\(800\) 0 0
\(801\) 5.31994e10 0.129234
\(802\) 0 0
\(803\) 7.88240e10i 0.189582i
\(804\) 0 0
\(805\) 8.46677e11 2.01620
\(806\) 0 0
\(807\) − 2.68777e11i − 0.633720i
\(808\) 0 0
\(809\) −1.93064e11 −0.450721 −0.225361 0.974275i \(-0.572356\pi\)
−0.225361 + 0.974275i \(0.572356\pi\)
\(810\) 0 0
\(811\) 1.36589e11i 0.315741i 0.987460 + 0.157871i \(0.0504629\pi\)
−0.987460 + 0.157871i \(0.949537\pi\)
\(812\) 0 0
\(813\) −1.99704e11 −0.457115
\(814\) 0 0
\(815\) − 6.06102e11i − 1.37377i
\(816\) 0 0
\(817\) −9.30469e10 −0.208840
\(818\) 0 0
\(819\) 3.14322e11i 0.698616i
\(820\) 0 0
\(821\) 6.34575e11 1.39672 0.698362 0.715745i \(-0.253913\pi\)
0.698362 + 0.715745i \(0.253913\pi\)
\(822\) 0 0
\(823\) − 6.86712e10i − 0.149684i −0.997195 0.0748419i \(-0.976155\pi\)
0.997195 0.0748419i \(-0.0238452\pi\)
\(824\) 0 0
\(825\) −7.27635e11 −1.57072
\(826\) 0 0
\(827\) − 9.21952e11i − 1.97100i −0.169679 0.985499i \(-0.554273\pi\)
0.169679 0.985499i \(-0.445727\pi\)
\(828\) 0 0
\(829\) −7.03277e10 −0.148905 −0.0744523 0.997225i \(-0.523721\pi\)
−0.0744523 + 0.997225i \(0.523721\pi\)
\(830\) 0 0
\(831\) − 4.51049e11i − 0.945845i
\(832\) 0 0
\(833\) −6.04307e11 −1.25510
\(834\) 0 0
\(835\) 7.87339e10i 0.161963i
\(836\) 0 0
\(837\) 6.87195e10 0.140016
\(838\) 0 0
\(839\) − 8.07774e11i − 1.63020i −0.579317 0.815102i \(-0.696681\pi\)
0.579317 0.815102i \(-0.303319\pi\)
\(840\) 0 0
\(841\) −2.64798e11 −0.529336
\(842\) 0 0
\(843\) − 1.23232e11i − 0.244013i
\(844\) 0 0
\(845\) −5.26412e11 −1.03252
\(846\) 0 0
\(847\) 7.56219e11i 1.46931i
\(848\) 0 0
\(849\) 1.48846e10 0.0286488
\(850\) 0 0
\(851\) − 6.70847e11i − 1.27910i
\(852\) 0 0
\(853\) 1.50102e11 0.283525 0.141762 0.989901i \(-0.454723\pi\)
0.141762 + 0.989901i \(0.454723\pi\)
\(854\) 0 0
\(855\) − 2.98083e11i − 0.557793i
\(856\) 0 0
\(857\) −7.35461e11 −1.36344 −0.681720 0.731613i \(-0.738768\pi\)
−0.681720 + 0.731613i \(0.738768\pi\)
\(858\) 0 0
\(859\) 5.72116e11i 1.05078i 0.850862 + 0.525389i \(0.176080\pi\)
−0.850862 + 0.525389i \(0.823920\pi\)
\(860\) 0 0
\(861\) −6.62210e11 −1.20499
\(862\) 0 0
\(863\) − 3.82637e11i − 0.689832i −0.938634 0.344916i \(-0.887907\pi\)
0.938634 0.344916i \(-0.112093\pi\)
\(864\) 0 0
\(865\) −7.34781e11 −1.31248
\(866\) 0 0
\(867\) − 1.58181e11i − 0.279949i
\(868\) 0 0
\(869\) 1.19504e12 2.09558
\(870\) 0 0
\(871\) 5.17894e11i 0.899846i
\(872\) 0 0
\(873\) 8.39660e10 0.144560
\(874\) 0 0
\(875\) 1.64586e12i 2.80777i
\(876\) 0 0
\(877\) −1.84222e11 −0.311418 −0.155709 0.987803i \(-0.549766\pi\)
−0.155709 + 0.987803i \(0.549766\pi\)
\(878\) 0 0
\(879\) 5.39523e10i 0.0903763i
\(880\) 0 0
\(881\) 5.11578e11 0.849196 0.424598 0.905382i \(-0.360415\pi\)
0.424598 + 0.905382i \(0.360415\pi\)
\(882\) 0 0
\(883\) 1.58004e11i 0.259912i 0.991520 + 0.129956i \(0.0414836\pi\)
−0.991520 + 0.129956i \(0.958516\pi\)
\(884\) 0 0
\(885\) −5.46649e10 −0.0891118
\(886\) 0 0
\(887\) − 4.26843e11i − 0.689562i −0.938683 0.344781i \(-0.887953\pi\)
0.938683 0.344781i \(-0.112047\pi\)
\(888\) 0 0
\(889\) 7.60698e11 1.21788
\(890\) 0 0
\(891\) 9.61758e10i 0.152600i
\(892\) 0 0
\(893\) −1.11589e11 −0.175476
\(894\) 0 0
\(895\) 4.35061e11i 0.678043i
\(896\) 0 0
\(897\) 3.32810e11 0.514075
\(898\) 0 0
\(899\) − 3.26028e11i − 0.499133i
\(900\) 0 0
\(901\) −8.68668e11 −1.31812
\(902\) 0 0
\(903\) − 1.37135e11i − 0.206252i
\(904\) 0 0
\(905\) 1.29487e12 1.93033
\(906\) 0 0
\(907\) − 9.37783e11i − 1.38571i −0.721075 0.692857i \(-0.756352\pi\)
0.721075 0.692857i \(-0.243648\pi\)
\(908\) 0 0
\(909\) 3.27357e11 0.479475
\(910\) 0 0
\(911\) 3.95324e11i 0.573958i 0.957937 + 0.286979i \(0.0926509\pi\)
−0.957937 + 0.286979i \(0.907349\pi\)
\(912\) 0 0
\(913\) −1.09082e12 −1.56990
\(914\) 0 0
\(915\) − 1.74152e11i − 0.248453i
\(916\) 0 0
\(917\) 1.70821e11 0.241582
\(918\) 0 0
\(919\) − 1.86250e11i − 0.261116i −0.991441 0.130558i \(-0.958323\pi\)
0.991441 0.130558i \(-0.0416769\pi\)
\(920\) 0 0
\(921\) −7.25773e11 −1.00870
\(922\) 0 0
\(923\) − 1.43525e11i − 0.197752i
\(924\) 0 0
\(925\) 2.63354e12 3.59726
\(926\) 0 0
\(927\) 1.66450e11i 0.225405i
\(928\) 0 0
\(929\) 4.97016e11 0.667279 0.333639 0.942701i \(-0.391723\pi\)
0.333639 + 0.942701i \(0.391723\pi\)
\(930\) 0 0
\(931\) 1.27334e12i 1.69491i
\(932\) 0 0
\(933\) −1.46025e11 −0.192709
\(934\) 0 0
\(935\) − 1.30067e12i − 1.70185i
\(936\) 0 0
\(937\) −9.17550e11 −1.19034 −0.595171 0.803599i \(-0.702915\pi\)
−0.595171 + 0.803599i \(0.702915\pi\)
\(938\) 0 0
\(939\) − 4.29718e11i − 0.552741i
\(940\) 0 0
\(941\) −1.67021e11 −0.213016 −0.106508 0.994312i \(-0.533967\pi\)
−0.106508 + 0.994312i \(0.533967\pi\)
\(942\) 0 0
\(943\) 7.01161e11i 0.886688i
\(944\) 0 0
\(945\) 4.39324e11 0.550881
\(946\) 0 0
\(947\) − 6.85774e11i − 0.852670i −0.904565 0.426335i \(-0.859805\pi\)
0.904565 0.426335i \(-0.140195\pi\)
\(948\) 0 0
\(949\) 1.41533e11 0.174499
\(950\) 0 0
\(951\) − 4.83022e11i − 0.590534i
\(952\) 0 0
\(953\) −5.38436e11 −0.652774 −0.326387 0.945236i \(-0.605831\pi\)
−0.326387 + 0.945236i \(0.605831\pi\)
\(954\) 0 0
\(955\) − 4.42218e11i − 0.531646i
\(956\) 0 0
\(957\) 4.56289e11 0.543992
\(958\) 0 0
\(959\) 8.44645e10i 0.0998619i
\(960\) 0 0
\(961\) 4.01436e11 0.470677
\(962\) 0 0
\(963\) 2.92262e11i 0.339834i
\(964\) 0 0
\(965\) −9.10719e10 −0.105021
\(966\) 0 0
\(967\) − 1.61685e12i − 1.84912i −0.381038 0.924559i \(-0.624433\pi\)
0.381038 0.924559i \(-0.375567\pi\)
\(968\) 0 0
\(969\) 3.54085e11 0.401617
\(970\) 0 0
\(971\) 1.35680e12i 1.52630i 0.646223 + 0.763148i \(0.276347\pi\)
−0.646223 + 0.763148i \(0.723653\pi\)
\(972\) 0 0
\(973\) 1.59823e12 1.78315
\(974\) 0 0
\(975\) 1.30651e12i 1.44575i
\(976\) 0 0
\(977\) 1.55870e12 1.71074 0.855372 0.518014i \(-0.173328\pi\)
0.855372 + 0.518014i \(0.173328\pi\)
\(978\) 0 0
\(979\) − 4.89132e11i − 0.532470i
\(980\) 0 0
\(981\) 4.12939e11 0.445872
\(982\) 0 0
\(983\) 3.31667e11i 0.355213i 0.984102 + 0.177606i \(0.0568354\pi\)
−0.984102 + 0.177606i \(0.943165\pi\)
\(984\) 0 0
\(985\) 1.28179e12 1.36167
\(986\) 0 0
\(987\) − 1.64464e11i − 0.173302i
\(988\) 0 0
\(989\) −1.45202e11 −0.151770
\(990\) 0 0
\(991\) 9.22831e11i 0.956814i 0.878138 + 0.478407i \(0.158786\pi\)
−0.878138 + 0.478407i \(0.841214\pi\)
\(992\) 0 0
\(993\) 9.82331e11 1.01032
\(994\) 0 0
\(995\) 4.82625e11i 0.492399i
\(996\) 0 0
\(997\) −1.00763e12 −1.01981 −0.509905 0.860231i \(-0.670320\pi\)
−0.509905 + 0.860231i \(0.670320\pi\)
\(998\) 0 0
\(999\) − 3.48090e11i − 0.349486i
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 192.9.g.d.127.5 8
4.3 odd 2 inner 192.9.g.d.127.1 8
8.3 odd 2 96.9.g.b.31.8 yes 8
8.5 even 2 96.9.g.b.31.4 8
24.5 odd 2 288.9.g.c.127.1 8
24.11 even 2 288.9.g.c.127.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.9.g.b.31.4 8 8.5 even 2
96.9.g.b.31.8 yes 8 8.3 odd 2
192.9.g.d.127.1 8 4.3 odd 2 inner
192.9.g.d.127.5 8 1.1 even 1 trivial
288.9.g.c.127.1 8 24.5 odd 2
288.9.g.c.127.2 8 24.11 even 2