Properties

Label 252.9.z.d.145.4
Level $252$
Weight $9$
Character 252.145
Analytic conductor $102.659$
Analytic rank $0$
Dimension $12$
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] = [252,9,Mod(73,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.73");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.z (of order \(6\), degree \(2\), 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: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{10}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.4
Root \(-28.8366 - 49.9465i\) of defining polynomial
Character \(\chi\) \(=\) 252.145
Dual form 252.9.z.d.73.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(203.366 + 117.413i) q^{5} +(1130.34 - 2118.29i) q^{7} +O(q^{10})\) \(q+(203.366 + 117.413i) q^{5} +(1130.34 - 2118.29i) q^{7} +(7474.44 + 12946.1i) q^{11} +39673.5i q^{13} +(-84679.4 + 48889.7i) q^{17} +(-177843. - 102678. i) q^{19} +(185312. - 320970. i) q^{23} +(-167741. - 290535. i) q^{25} -117048. q^{29} +(437915. - 252830. i) q^{31} +(478588. - 298070. i) q^{35} +(-986447. + 1.70858e6i) q^{37} +3.25966e6i q^{41} -4.13332e6 q^{43} +(4.79532e6 + 2.76858e6i) q^{47} +(-3.20947e6 - 4.78876e6i) q^{49} +(3.39093e6 + 5.87326e6i) q^{53} +3.51040e6i q^{55} +(1.91904e7 - 1.10796e7i) q^{59} +(-3.97144e6 - 2.29291e6i) q^{61} +(-4.65820e6 + 8.06823e6i) q^{65} +(-1.86385e7 - 3.22828e7i) q^{67} -4.19870e7 q^{71} +(1.08607e7 - 6.27044e6i) q^{73} +(3.58722e7 - 1.19951e6i) q^{77} +(-2.01064e7 + 3.48253e7i) q^{79} +5.31441e6i q^{83} -2.29612e7 q^{85} +(-6.45415e7 - 3.72630e7i) q^{89} +(8.40397e7 + 4.48445e7i) q^{91} +(-2.41115e7 - 4.17623e7i) q^{95} +3.76415e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 285 q^{5} + 198 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 285 q^{5} + 198 q^{7} + 17919 q^{11} + 205782 q^{17} + 74313 q^{19} + 62832 q^{23} + 878679 q^{25} + 575454 q^{29} + 1442952 q^{31} + 3989514 q^{35} - 2058621 q^{37} + 7721322 q^{43} - 12088194 q^{47} - 16964694 q^{49} + 5506743 q^{53} - 7511901 q^{59} - 37215576 q^{61} - 5047122 q^{65} - 36824553 q^{67} + 30011556 q^{71} + 95080185 q^{73} + 38333727 q^{77} + 8514456 q^{79} + 20121540 q^{85} - 83038554 q^{89} - 198538635 q^{91} + 221605224 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 203.366 + 117.413i 0.325386 + 0.187862i 0.653791 0.756676i \(-0.273178\pi\)
−0.328405 + 0.944537i \(0.606511\pi\)
\(6\) 0 0
\(7\) 1130.34 2118.29i 0.470778 0.882251i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7474.44 + 12946.1i 0.510514 + 0.884237i 0.999926 + 0.0121839i \(0.00387834\pi\)
−0.489411 + 0.872053i \(0.662788\pi\)
\(12\) 0 0
\(13\) 39673.5i 1.38908i 0.719455 + 0.694539i \(0.244392\pi\)
−0.719455 + 0.694539i \(0.755608\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −84679.4 + 48889.7i −1.01387 + 0.585358i −0.912322 0.409473i \(-0.865713\pi\)
−0.101547 + 0.994831i \(0.532379\pi\)
\(18\) 0 0
\(19\) −177843. 102678.i −1.36465 0.787882i −0.374413 0.927262i \(-0.622156\pi\)
−0.990239 + 0.139380i \(0.955489\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 185312. 320970.i 0.662205 1.14697i −0.317830 0.948148i \(-0.602954\pi\)
0.980035 0.198825i \(-0.0637124\pi\)
\(24\) 0 0
\(25\) −167741. 290535.i −0.429416 0.743771i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −117048. −0.165490 −0.0827448 0.996571i \(-0.526369\pi\)
−0.0827448 + 0.996571i \(0.526369\pi\)
\(30\) 0 0
\(31\) 437915. 252830.i 0.474180 0.273768i −0.243808 0.969823i \(-0.578397\pi\)
0.717988 + 0.696056i \(0.245063\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 478588. 298070.i 0.318926 0.198631i
\(36\) 0 0
\(37\) −986447. + 1.70858e6i −0.526341 + 0.911649i 0.473188 + 0.880961i \(0.343103\pi\)
−0.999529 + 0.0306877i \(0.990230\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.25966e6i 1.15355i 0.816903 + 0.576776i \(0.195689\pi\)
−0.816903 + 0.576776i \(0.804311\pi\)
\(42\) 0 0
\(43\) −4.13332e6 −1.20900 −0.604498 0.796607i \(-0.706626\pi\)
−0.604498 + 0.796607i \(0.706626\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.79532e6 + 2.76858e6i 0.982711 + 0.567369i 0.903088 0.429456i \(-0.141295\pi\)
0.0796238 + 0.996825i \(0.474628\pi\)
\(48\) 0 0
\(49\) −3.20947e6 4.78876e6i −0.556735 0.830690i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.39093e6 + 5.87326e6i 0.429749 + 0.744347i 0.996851 0.0793006i \(-0.0252687\pi\)
−0.567102 + 0.823648i \(0.691935\pi\)
\(54\) 0 0
\(55\) 3.51040e6i 0.383624i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.91904e7 1.10796e7i 1.58371 0.914355i 0.589397 0.807843i \(-0.299365\pi\)
0.994312 0.106511i \(-0.0339680\pi\)
\(60\) 0 0
\(61\) −3.97144e6 2.29291e6i −0.286833 0.165603i 0.349680 0.936869i \(-0.386290\pi\)
−0.636513 + 0.771266i \(0.719624\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.65820e6 + 8.06823e6i −0.260954 + 0.451986i
\(66\) 0 0
\(67\) −1.86385e7 3.22828e7i −0.924934 1.60203i −0.791667 0.610952i \(-0.790787\pi\)
−0.133267 0.991080i \(-0.542547\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.19870e7 −1.65227 −0.826135 0.563472i \(-0.809465\pi\)
−0.826135 + 0.563472i \(0.809465\pi\)
\(72\) 0 0
\(73\) 1.08607e7 6.27044e6i 0.382443 0.220804i −0.296438 0.955052i \(-0.595799\pi\)
0.678881 + 0.734249i \(0.262465\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.58722e7 1.19951e6i 1.02046 0.0341224i
\(78\) 0 0
\(79\) −2.01064e7 + 3.48253e7i −0.516209 + 0.894100i 0.483614 + 0.875281i \(0.339324\pi\)
−0.999823 + 0.0188184i \(0.994010\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.31441e6i 0.111981i 0.998431 + 0.0559903i \(0.0178316\pi\)
−0.998431 + 0.0559903i \(0.982168\pi\)
\(84\) 0 0
\(85\) −2.29612e7 −0.439865
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.45415e7 3.72630e7i −1.02868 0.593907i −0.112072 0.993700i \(-0.535749\pi\)
−0.916605 + 0.399793i \(0.869082\pi\)
\(90\) 0 0
\(91\) 8.40397e7 + 4.48445e7i 1.22552 + 0.653948i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.41115e7 4.17623e7i −0.296026 0.512731i
\(96\) 0 0
\(97\) 3.76415e7i 0.425187i 0.977141 + 0.212593i \(0.0681909\pi\)
−0.977141 + 0.212593i \(0.931809\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.57943e8 + 9.11884e7i −1.51780 + 0.876303i −0.518020 + 0.855368i \(0.673331\pi\)
−0.999781 + 0.0209346i \(0.993336\pi\)
\(102\) 0 0
\(103\) −9.37340e7 5.41173e7i −0.832814 0.480826i 0.0220010 0.999758i \(-0.492996\pi\)
−0.854815 + 0.518932i \(0.826330\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.48762e6 1.64330e7i 0.0723806 0.125367i −0.827564 0.561372i \(-0.810274\pi\)
0.899944 + 0.436005i \(0.143607\pi\)
\(108\) 0 0
\(109\) 3.98197e7 + 6.89697e7i 0.282093 + 0.488599i 0.971900 0.235394i \(-0.0756381\pi\)
−0.689807 + 0.723993i \(0.742305\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.14140e8 −1.31336 −0.656680 0.754169i \(-0.728040\pi\)
−0.656680 + 0.754169i \(0.728040\pi\)
\(114\) 0 0
\(115\) 7.53723e7 4.35162e7i 0.430944 0.248806i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 7.84586e6 + 2.34637e8i 0.0391249 + 1.17006i
\(120\) 0 0
\(121\) −4.55511e6 + 7.88969e6i −0.0212499 + 0.0368060i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.70509e8i 0.698406i
\(126\) 0 0
\(127\) 2.22002e8 0.853381 0.426690 0.904398i \(-0.359679\pi\)
0.426690 + 0.904398i \(0.359679\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.38205e8 2.52998e8i −1.48796 0.859075i −0.488056 0.872812i \(-0.662294\pi\)
−0.999906 + 0.0137373i \(0.995627\pi\)
\(132\) 0 0
\(133\) −4.18523e8 + 2.60661e8i −1.33756 + 0.833048i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.73086e8 2.99795e8i −0.491339 0.851023i 0.508612 0.860996i \(-0.330159\pi\)
−0.999950 + 0.00997260i \(0.996826\pi\)
\(138\) 0 0
\(139\) 1.04014e8i 0.278632i 0.990248 + 0.139316i \(0.0444904\pi\)
−0.990248 + 0.139316i \(0.955510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.13617e8 + 2.96537e8i −1.22827 + 0.709144i
\(144\) 0 0
\(145\) −2.38035e7 1.37430e7i −0.0538480 0.0310891i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.71762e8 + 2.97501e8i −0.348483 + 0.603591i −0.985980 0.166862i \(-0.946637\pi\)
0.637497 + 0.770453i \(0.279970\pi\)
\(150\) 0 0
\(151\) −4.21266e8 7.29654e8i −0.810305 1.40349i −0.912651 0.408740i \(-0.865968\pi\)
0.102346 0.994749i \(-0.467365\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.18743e8 0.205722
\(156\) 0 0
\(157\) −3.00370e8 + 1.73419e8i −0.494377 + 0.285429i −0.726388 0.687284i \(-0.758803\pi\)
0.232011 + 0.972713i \(0.425469\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.70440e8 7.55349e8i −0.700166 1.12420i
\(162\) 0 0
\(163\) 6.38287e7 1.10555e8i 0.0904203 0.156612i −0.817268 0.576258i \(-0.804512\pi\)
0.907688 + 0.419646i \(0.137846\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.84232e8i 0.622569i 0.950317 + 0.311284i \(0.100759\pi\)
−0.950317 + 0.311284i \(0.899241\pi\)
\(168\) 0 0
\(169\) −7.58252e8 −0.929537
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.57674e8 + 4.37444e8i 0.845860 + 0.488357i 0.859252 0.511553i \(-0.170930\pi\)
−0.0133921 + 0.999910i \(0.504263\pi\)
\(174\) 0 0
\(175\) −8.05041e8 + 2.69192e7i −0.858352 + 0.0287019i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.06371e8 + 1.05027e9i 0.590645 + 1.02303i 0.994146 + 0.108048i \(0.0344599\pi\)
−0.403501 + 0.914979i \(0.632207\pi\)
\(180\) 0 0
\(181\) 2.75613e8i 0.256794i −0.991723 0.128397i \(-0.959017\pi\)
0.991723 0.128397i \(-0.0409832\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.01220e8 + 2.31644e8i −0.342528 + 0.197758i
\(186\) 0 0
\(187\) −1.26586e9 7.30846e8i −1.03519 0.597667i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.56485e8 + 1.65668e9i −0.718695 + 1.24482i 0.242821 + 0.970071i \(0.421927\pi\)
−0.961517 + 0.274746i \(0.911406\pi\)
\(192\) 0 0
\(193\) 4.90016e8 + 8.48733e8i 0.353168 + 0.611705i 0.986803 0.161928i \(-0.0517711\pi\)
−0.633635 + 0.773632i \(0.718438\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.49081e9 −0.989824 −0.494912 0.868943i \(-0.664800\pi\)
−0.494912 + 0.868943i \(0.664800\pi\)
\(198\) 0 0
\(199\) −8.60781e8 + 4.96972e8i −0.548884 + 0.316898i −0.748672 0.662941i \(-0.769308\pi\)
0.199788 + 0.979839i \(0.435975\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.32304e8 + 2.47940e8i −0.0779090 + 0.146004i
\(204\) 0 0
\(205\) −3.82728e8 + 6.62904e8i −0.216708 + 0.375349i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.06983e9i 1.60890i
\(210\) 0 0
\(211\) −4.64537e8 −0.234364 −0.117182 0.993110i \(-0.537386\pi\)
−0.117182 + 0.993110i \(0.537386\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.40576e8 4.85307e8i −0.393390 0.227124i
\(216\) 0 0
\(217\) −4.05745e7 1.21341e9i −0.0182984 0.547229i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.93962e9 3.35952e9i −0.813107 1.40834i
\(222\) 0 0
\(223\) 6.58116e7i 0.0266123i 0.999911 + 0.0133062i \(0.00423561\pi\)
−0.999911 + 0.0133062i \(0.995764\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.98442e9 + 1.14570e9i −0.747360 + 0.431489i −0.824739 0.565513i \(-0.808678\pi\)
0.0773791 + 0.997002i \(0.475345\pi\)
\(228\) 0 0
\(229\) 2.77398e9 + 1.60156e9i 1.00870 + 0.582373i 0.910811 0.412825i \(-0.135458\pi\)
0.0978887 + 0.995197i \(0.468791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.43830e8 + 1.28835e9i −0.252377 + 0.437130i −0.964180 0.265249i \(-0.914546\pi\)
0.711803 + 0.702380i \(0.247879\pi\)
\(234\) 0 0
\(235\) 6.50137e8 + 1.12607e9i 0.213173 + 0.369227i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.50233e8 −0.291232 −0.145616 0.989341i \(-0.546516\pi\)
−0.145616 + 0.989341i \(0.546516\pi\)
\(240\) 0 0
\(241\) −3.55088e9 + 2.05010e9i −1.05261 + 0.607725i −0.923379 0.383889i \(-0.874584\pi\)
−0.129232 + 0.991614i \(0.541251\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.04318e7 1.35071e9i −0.0250990 0.374884i
\(246\) 0 0
\(247\) 4.07358e9 7.05564e9i 1.09443 1.89561i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.77515e9i 1.20307i 0.798845 + 0.601537i \(0.205445\pi\)
−0.798845 + 0.601537i \(0.794555\pi\)
\(252\) 0 0
\(253\) 5.54042e9 1.35226
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.28158e9 + 2.47197e9i 0.981457 + 0.566645i 0.902710 0.430250i \(-0.141575\pi\)
0.0787473 + 0.996895i \(0.474908\pi\)
\(258\) 0 0
\(259\) 2.50423e9 + 4.02085e9i 0.556514 + 0.893550i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.41497e9 5.91490e9i −0.713779 1.23630i −0.963429 0.267965i \(-0.913649\pi\)
0.249650 0.968336i \(-0.419685\pi\)
\(264\) 0 0
\(265\) 1.59256e9i 0.322933i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.01169e9 1.73880e9i 0.575176 0.332078i −0.184038 0.982919i \(-0.558917\pi\)
0.759214 + 0.650841i \(0.225584\pi\)
\(270\) 0 0
\(271\) −5.59984e9 3.23307e9i −1.03824 0.599429i −0.118907 0.992905i \(-0.537939\pi\)
−0.919335 + 0.393477i \(0.871272\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.50754e9 4.34318e9i 0.438446 0.759411i
\(276\) 0 0
\(277\) 2.15914e9 + 3.73973e9i 0.366742 + 0.635216i 0.989054 0.147553i \(-0.0471397\pi\)
−0.622312 + 0.782769i \(0.713806\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.64214e9 −0.904938 −0.452469 0.891780i \(-0.649457\pi\)
−0.452469 + 0.891780i \(0.649457\pi\)
\(282\) 0 0
\(283\) 3.51708e9 2.03059e9i 0.548323 0.316574i −0.200122 0.979771i \(-0.564134\pi\)
0.748445 + 0.663197i \(0.230801\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.90489e9 + 3.68452e9i 1.01772 + 0.543067i
\(288\) 0 0
\(289\) 1.29252e9 2.23871e9i 0.185287 0.320927i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.43223e7i 0.0114412i 0.999984 + 0.00572061i \(0.00182094\pi\)
−0.999984 + 0.00572061i \(0.998179\pi\)
\(294\) 0 0
\(295\) 5.20356e9 0.687088
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.27340e10 + 7.35197e9i 1.59323 + 0.919854i
\(300\) 0 0
\(301\) −4.67205e9 + 8.75554e9i −0.569169 + 1.06664i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.38438e8 9.32601e8i −0.0622209 0.107770i
\(306\) 0 0
\(307\) 6.10423e9i 0.687191i 0.939118 + 0.343596i \(0.111645\pi\)
−0.939118 + 0.343596i \(0.888355\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.74096e9 1.00514e9i 0.186100 0.107445i −0.404055 0.914735i \(-0.632400\pi\)
0.590156 + 0.807289i \(0.299066\pi\)
\(312\) 0 0
\(313\) −1.10102e10 6.35672e9i −1.14714 0.662302i −0.198951 0.980009i \(-0.563754\pi\)
−0.948189 + 0.317708i \(0.897087\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.98773e9 + 5.17489e9i −0.295872 + 0.512465i −0.975187 0.221381i \(-0.928943\pi\)
0.679315 + 0.733846i \(0.262277\pi\)
\(318\) 0 0
\(319\) −8.74866e8 1.51531e9i −0.0844849 0.146332i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00795e10 1.84477
\(324\) 0 0
\(325\) 1.15265e10 6.65485e9i 1.03316 0.596492i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.12850e10 7.02842e9i 0.963201 0.599894i
\(330\) 0 0
\(331\) 6.42439e9 1.11274e10i 0.535205 0.927002i −0.463948 0.885862i \(-0.653568\pi\)
0.999153 0.0411400i \(-0.0130990\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.75362e9i 0.695038i
\(336\) 0 0
\(337\) −9.70137e9 −0.752166 −0.376083 0.926586i \(-0.622729\pi\)
−0.376083 + 0.926586i \(0.622729\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.54634e9 + 3.77953e9i 0.484151 + 0.279525i
\(342\) 0 0
\(343\) −1.37718e10 + 1.38565e9i −0.994976 + 0.100110i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.48892e8 + 1.12391e9i 0.0447563 + 0.0775203i 0.887536 0.460739i \(-0.152416\pi\)
−0.842779 + 0.538259i \(0.819082\pi\)
\(348\) 0 0
\(349\) 2.01643e10i 1.35919i −0.733587 0.679596i \(-0.762155\pi\)
0.733587 0.679596i \(-0.237845\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.27487e10 1.31340e10i 1.46507 0.845857i 0.465828 0.884875i \(-0.345756\pi\)
0.999238 + 0.0390186i \(0.0124232\pi\)
\(354\) 0 0
\(355\) −8.53872e9 4.92983e9i −0.537625 0.310398i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.13441e10 + 1.96486e10i −0.682958 + 1.18292i 0.291116 + 0.956688i \(0.405973\pi\)
−0.974074 + 0.226230i \(0.927360\pi\)
\(360\) 0 0
\(361\) 1.25936e10 + 2.18128e10i 0.741517 + 1.28435i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.94493e9 0.165922
\(366\) 0 0
\(367\) 9.38693e8 5.41955e8i 0.0517439 0.0298744i −0.473905 0.880576i \(-0.657156\pi\)
0.525649 + 0.850702i \(0.323823\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.62741e10 5.44179e8i 0.859018 0.0287241i
\(372\) 0 0
\(373\) 1.91674e9 3.31989e9i 0.0990211 0.171510i −0.812259 0.583297i \(-0.801762\pi\)
0.911280 + 0.411788i \(0.135096\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.64369e9i 0.229878i
\(378\) 0 0
\(379\) 3.26734e10 1.58357 0.791785 0.610800i \(-0.209152\pi\)
0.791785 + 0.610800i \(0.209152\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.54251e10 + 8.90569e9i 0.716858 + 0.413878i 0.813595 0.581432i \(-0.197507\pi\)
−0.0967371 + 0.995310i \(0.530841\pi\)
\(384\) 0 0
\(385\) 7.43603e9 + 3.96794e9i 0.338453 + 0.180602i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.92749e9 + 1.19988e10i 0.302536 + 0.524008i 0.976710 0.214565i \(-0.0688333\pi\)
−0.674173 + 0.738573i \(0.735500\pi\)
\(390\) 0 0
\(391\) 3.62394e10i 1.55051i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.17791e9 + 4.72152e9i −0.335934 + 0.193951i
\(396\) 0 0
\(397\) −1.66863e10 9.63385e9i −0.671735 0.387827i 0.124998 0.992157i \(-0.460107\pi\)
−0.796734 + 0.604330i \(0.793441\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.58138e9 + 1.65954e10i −0.370553 + 0.641817i −0.989651 0.143497i \(-0.954165\pi\)
0.619097 + 0.785314i \(0.287499\pi\)
\(402\) 0 0
\(403\) 1.00306e10 + 1.73736e10i 0.380285 + 0.658672i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.94926e10 −1.07482
\(408\) 0 0
\(409\) 3.45544e10 1.99500e10i 1.23484 0.712935i 0.266805 0.963751i \(-0.414032\pi\)
0.968035 + 0.250815i \(0.0806987\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.77806e9 5.31744e10i −0.0611148 1.82769i
\(414\) 0 0
\(415\) −6.23984e8 + 1.08077e9i −0.0210369 + 0.0364369i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.17883e10i 1.68026i −0.542387 0.840129i \(-0.682479\pi\)
0.542387 0.840129i \(-0.317521\pi\)
\(420\) 0 0
\(421\) 5.78196e9 0.184055 0.0920273 0.995756i \(-0.470665\pi\)
0.0920273 + 0.995756i \(0.470665\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.84083e10 + 1.64016e10i 0.870743 + 0.502724i
\(426\) 0 0
\(427\) −9.34612e9 + 5.82088e9i −0.281138 + 0.175096i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.29999e10 2.25165e10i −0.376730 0.652516i 0.613854 0.789420i \(-0.289618\pi\)
−0.990584 + 0.136903i \(0.956285\pi\)
\(432\) 0 0
\(433\) 3.85244e10i 1.09593i −0.836500 0.547967i \(-0.815402\pi\)
0.836500 0.547967i \(-0.184598\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.59128e10 + 3.80548e10i −1.80736 + 1.04348i
\(438\) 0 0
\(439\) 5.36255e10 + 3.09607e10i 1.44382 + 0.833591i 0.998102 0.0615814i \(-0.0196144\pi\)
0.445720 + 0.895172i \(0.352948\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.82777e9 + 1.18260e10i −0.177282 + 0.307061i −0.940949 0.338550i \(-0.890064\pi\)
0.763667 + 0.645611i \(0.223397\pi\)
\(444\) 0 0
\(445\) −8.75037e9 1.51561e10i −0.223144 0.386498i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.45335e10 −1.58782 −0.793908 0.608038i \(-0.791957\pi\)
−0.793908 + 0.608038i \(0.791957\pi\)
\(450\) 0 0
\(451\) −4.21999e10 + 2.43641e10i −1.02001 + 0.588904i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.18255e10 + 1.89872e10i 0.275914 + 0.443013i
\(456\) 0 0
\(457\) 2.13781e10 3.70279e10i 0.490122 0.848916i −0.509814 0.860285i \(-0.670286\pi\)
0.999935 + 0.0113692i \(0.00361901\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.19885e10i 0.708256i −0.935197 0.354128i \(-0.884778\pi\)
0.935197 0.354128i \(-0.115222\pi\)
\(462\) 0 0
\(463\) −6.84419e9 −0.148935 −0.0744677 0.997223i \(-0.523726\pi\)
−0.0744677 + 0.997223i \(0.523726\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.83350e10 + 1.05857e10i 0.385490 + 0.222563i 0.680204 0.733023i \(-0.261891\pi\)
−0.294714 + 0.955585i \(0.595224\pi\)
\(468\) 0 0
\(469\) −8.94519e10 + 2.99112e9i −1.84883 + 0.0618219i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.08942e10 5.35104e10i −0.617210 1.06904i
\(474\) 0 0
\(475\) 6.88928e10i 1.35332i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.99972e9 + 5.77334e9i −0.189953 + 0.109669i −0.591960 0.805967i \(-0.701646\pi\)
0.402008 + 0.915636i \(0.368313\pi\)
\(480\) 0 0
\(481\) −6.77852e10 3.91358e10i −1.26635 0.731128i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.41961e9 + 7.65500e9i −0.0798762 + 0.138350i
\(486\) 0 0
\(487\) 3.40949e10 + 5.90541e10i 0.606141 + 1.04987i 0.991870 + 0.127255i \(0.0406165\pi\)
−0.385729 + 0.922612i \(0.626050\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.71885e10 0.639857 0.319929 0.947442i \(-0.396341\pi\)
0.319929 + 0.947442i \(0.396341\pi\)
\(492\) 0 0
\(493\) 9.91152e9 5.72242e9i 0.167785 0.0968706i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.74595e10 + 8.89404e10i −0.777853 + 1.45772i
\(498\) 0 0
\(499\) 2.66255e10 4.61166e10i 0.429432 0.743799i −0.567390 0.823449i \(-0.692047\pi\)
0.996823 + 0.0796501i \(0.0253803\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.10881e10i 1.42295i 0.702711 + 0.711475i \(0.251973\pi\)
−0.702711 + 0.711475i \(0.748027\pi\)
\(504\) 0 0
\(505\) −4.28270e10 −0.658494
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.02031e9 2.89848e9i −0.0747927 0.0431816i 0.462137 0.886808i \(-0.347083\pi\)
−0.536930 + 0.843627i \(0.680416\pi\)
\(510\) 0 0
\(511\) −1.00629e9 3.00938e10i −0.0147584 0.441361i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.27082e10 2.20113e10i −0.180657 0.312907i
\(516\) 0 0
\(517\) 8.27743e10i 1.15860i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.38767e10 3.68792e10i 0.866945 0.500531i 0.000613137 1.00000i \(-0.499805\pi\)
0.866332 + 0.499469i \(0.166471\pi\)
\(522\) 0 0
\(523\) −8.49467e10 4.90440e10i −1.13538 0.655510i −0.190095 0.981766i \(-0.560880\pi\)
−0.945281 + 0.326256i \(0.894213\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.47216e10 + 4.28190e10i −0.320504 + 0.555129i
\(528\) 0 0
\(529\) −2.95256e10 5.11398e10i −0.377030 0.653035i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.29322e11 −1.60237
\(534\) 0 0
\(535\) 3.85892e9 2.22795e9i 0.0471032 0.0271950i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.80069e10 7.73435e10i 0.450305 0.916365i
\(540\) 0 0
\(541\) 1.61177e10 2.79168e10i 0.188155 0.325894i −0.756480 0.654017i \(-0.773083\pi\)
0.944635 + 0.328123i \(0.106416\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.87015e10i 0.211978i
\(546\) 0 0
\(547\) −9.97958e10 −1.11471 −0.557356 0.830273i \(-0.688184\pi\)
−0.557356 + 0.830273i \(0.688184\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.08161e10 + 1.20182e10i 0.225836 + 0.130386i
\(552\) 0 0
\(553\) 5.10428e10 + 8.19554e10i 0.545801 + 0.876349i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.01561e10 + 5.22319e10i 0.313295 + 0.542644i 0.979074 0.203506i \(-0.0652337\pi\)
−0.665778 + 0.746150i \(0.731900\pi\)
\(558\) 0 0
\(559\) 1.63983e11i 1.67939i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.10263e8 6.36604e7i 0.00109748 0.000633630i −0.499451 0.866342i \(-0.666465\pi\)
0.500549 + 0.865708i \(0.333132\pi\)
\(564\) 0 0
\(565\) −4.35488e10 2.51429e10i −0.427349 0.246730i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.12713e10 8.88046e10i 0.489132 0.847201i −0.510790 0.859705i \(-0.670647\pi\)
0.999922 + 0.0125047i \(0.00398046\pi\)
\(570\) 0 0
\(571\) 3.04428e10 + 5.27285e10i 0.286378 + 0.496022i 0.972943 0.231047i \(-0.0742152\pi\)
−0.686564 + 0.727069i \(0.740882\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.24337e11 −1.13745
\(576\) 0 0
\(577\) −2.41312e10 + 1.39322e10i −0.217709 + 0.125694i −0.604889 0.796310i \(-0.706783\pi\)
0.387180 + 0.922004i \(0.373449\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.12574e10 + 6.00709e9i 0.0987951 + 0.0527181i
\(582\) 0 0
\(583\) −5.06906e10 + 8.77986e10i −0.438786 + 0.760000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.01355e9i 0.0338047i 0.999857 + 0.0169023i \(0.00538043\pi\)
−0.999857 + 0.0169023i \(0.994620\pi\)
\(588\) 0 0
\(589\) −1.03840e11 −0.862787
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.32222e11 7.63384e10i −1.06926 0.617340i −0.141283 0.989969i \(-0.545123\pi\)
−0.927980 + 0.372630i \(0.878456\pi\)
\(594\) 0 0
\(595\) −2.59540e10 + 4.86384e10i −0.207079 + 0.388071i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.45784e10 + 9.45325e10i 0.423949 + 0.734301i 0.996322 0.0856928i \(-0.0273104\pi\)
−0.572373 + 0.819993i \(0.693977\pi\)
\(600\) 0 0
\(601\) 1.79228e11i 1.37375i −0.726776 0.686875i \(-0.758982\pi\)
0.726776 0.686875i \(-0.241018\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.85271e9 + 1.06966e9i −0.0138289 + 0.00798409i
\(606\) 0 0
\(607\) 3.96127e10 + 2.28704e10i 0.291796 + 0.168469i 0.638752 0.769413i \(-0.279451\pi\)
−0.346955 + 0.937882i \(0.612784\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.09839e11 + 1.90247e11i −0.788119 + 1.36506i
\(612\) 0 0
\(613\) 6.70156e10 + 1.16074e11i 0.474607 + 0.822044i 0.999577 0.0290771i \(-0.00925685\pi\)
−0.524970 + 0.851121i \(0.675924\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.21594e11 −0.839015 −0.419508 0.907752i \(-0.637797\pi\)
−0.419508 + 0.907752i \(0.637797\pi\)
\(618\) 0 0
\(619\) −3.51231e10 + 2.02783e10i −0.239238 + 0.138124i −0.614826 0.788662i \(-0.710774\pi\)
0.375589 + 0.926787i \(0.377441\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.51888e11 + 9.45974e10i −1.00825 + 0.627953i
\(624\) 0 0
\(625\) −4.55036e10 + 7.88146e10i −0.298212 + 0.516519i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.92908e11i 1.23239i
\(630\) 0 0
\(631\) 2.35096e11 1.48296 0.741478 0.670978i \(-0.234125\pi\)
0.741478 + 0.670978i \(0.234125\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.51478e10 + 2.60661e10i 0.277678 + 0.160317i
\(636\) 0 0
\(637\) 1.89987e11 1.27331e11i 1.15389 0.773349i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.51414e11 2.62257e11i −0.896881 1.55344i −0.831460 0.555585i \(-0.812494\pi\)
−0.0654207 0.997858i \(-0.520839\pi\)
\(642\) 0 0
\(643\) 1.17949e11i 0.690001i 0.938603 + 0.345000i \(0.112121\pi\)
−0.938603 + 0.345000i \(0.887879\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.55386e11 8.97120e10i 0.886735 0.511957i 0.0138619 0.999904i \(-0.495587\pi\)
0.872873 + 0.487947i \(0.162254\pi\)
\(648\) 0 0
\(649\) 2.86875e11 + 1.65627e11i 1.61701 + 0.933582i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.14200e11 1.97801e11i 0.628079 1.08786i −0.359858 0.933007i \(-0.617175\pi\)
0.987937 0.154858i \(-0.0494918\pi\)
\(654\) 0 0
\(655\) −5.94106e10 1.02902e11i −0.322774 0.559061i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.80179e11 1.48557 0.742785 0.669530i \(-0.233504\pi\)
0.742785 + 0.669530i \(0.233504\pi\)
\(660\) 0 0
\(661\) −8.31508e9 + 4.80071e9i −0.0435573 + 0.0251478i −0.521621 0.853178i \(-0.674672\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.15719e11 + 3.86943e9i −0.591720 + 0.0197861i
\(666\) 0 0
\(667\) −2.16903e10 + 3.75688e10i −0.109588 + 0.189812i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.85530e10i 0.338171i
\(672\) 0 0
\(673\) −1.19348e11 −0.581776 −0.290888 0.956757i \(-0.593951\pi\)
−0.290888 + 0.956757i \(0.593951\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.44704e10 + 4.87690e10i 0.402115 + 0.232161i 0.687396 0.726283i \(-0.258754\pi\)
−0.285281 + 0.958444i \(0.592087\pi\)
\(678\) 0 0
\(679\) 7.97354e10 + 4.25476e10i 0.375121 + 0.200169i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.69347e11 + 2.93317e11i 0.778204 + 1.34789i 0.932976 + 0.359939i \(0.117203\pi\)
−0.154772 + 0.987950i \(0.549464\pi\)
\(684\) 0 0
\(685\) 8.12907e10i 0.369214i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.33012e11 + 1.34530e11i −1.03396 + 0.596955i
\(690\) 0 0
\(691\) 2.57158e11 + 1.48470e11i 1.12794 + 0.651219i 0.943417 0.331608i \(-0.107591\pi\)
0.184527 + 0.982827i \(0.440925\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.22126e10 + 2.11529e10i −0.0523443 + 0.0906630i
\(696\) 0 0
\(697\) −1.59364e11 2.76026e11i −0.675240 1.16955i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.94718e11 0.806370 0.403185 0.915118i \(-0.367903\pi\)
0.403185 + 0.915118i \(0.367903\pi\)
\(702\) 0 0
\(703\) 3.50865e11 2.02572e11i 1.43654 0.829389i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.46340e10 + 4.37642e11i 0.0585714 + 1.75163i
\(708\) 0 0
\(709\) −2.00882e11 + 3.47938e11i −0.794979 + 1.37694i 0.127873 + 0.991791i \(0.459185\pi\)
−0.922852 + 0.385154i \(0.874148\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.87410e11i 0.725161i
\(714\) 0 0
\(715\) −1.39270e11 −0.532884
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.54791e11 + 8.93684e10i 0.579201 + 0.334402i 0.760816 0.648968i \(-0.224799\pi\)
−0.181615 + 0.983370i \(0.558132\pi\)
\(720\) 0 0
\(721\) −2.20587e11 + 1.37384e11i −0.816280 + 0.508389i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.96337e10 + 3.40065e10i 0.0710639 + 0.123086i
\(726\) 0 0
\(727\) 1.97437e11i 0.706790i −0.935474 0.353395i \(-0.885027\pi\)
0.935474 0.353395i \(-0.114973\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.50006e11 2.02076e11i 1.22576 0.707695i
\(732\) 0 0
\(733\) 1.01144e11 + 5.83958e10i 0.350370 + 0.202286i 0.664848 0.746979i \(-0.268496\pi\)
−0.314478 + 0.949265i \(0.601830\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.78624e11 4.82591e11i 0.944384 1.63572i
\(738\) 0 0
\(739\) −8.45542e10 1.46452e11i −0.283503 0.491042i 0.688742 0.725006i \(-0.258163\pi\)
−0.972245 + 0.233965i \(0.924830\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.84466e11 0.605287 0.302644 0.953104i \(-0.402131\pi\)
0.302644 + 0.953104i \(0.402131\pi\)
\(744\) 0 0
\(745\) −6.98611e10 + 4.03343e10i −0.226783 + 0.130933i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.40856e10 3.86724e10i −0.0765298 0.122878i
\(750\) 0 0
\(751\) 1.50650e11 2.60933e11i 0.473596 0.820293i −0.525947 0.850518i \(-0.676289\pi\)
0.999543 + 0.0302244i \(0.00962218\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.97849e11i 0.608900i
\(756\) 0 0
\(757\) 4.94329e11 1.50533 0.752666 0.658403i \(-0.228768\pi\)
0.752666 + 0.658403i \(0.228768\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.78956e10 + 3.34260e10i 0.172626 + 0.0996658i 0.583824 0.811881i \(-0.301556\pi\)
−0.411197 + 0.911546i \(0.634889\pi\)
\(762\) 0 0
\(763\) 1.91107e11 6.39031e9i 0.563870 0.0188549i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.39565e11 + 7.61348e11i 1.27011 + 2.19989i
\(768\) 0 0
\(769\) 1.20153e11i 0.343581i −0.985134 0.171790i \(-0.945045\pi\)
0.985134 0.171790i \(-0.0549552\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.05856e11 1.76586e11i 0.856640 0.494582i −0.00624545 0.999980i \(-0.501988\pi\)
0.862886 + 0.505399i \(0.168655\pi\)
\(774\) 0 0
\(775\) −1.46912e11 8.48198e10i −0.407241 0.235120i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.34694e11 5.79707e11i 0.908862 1.57420i
\(780\) 0 0
\(781\) −3.13829e11 5.43568e11i −0.843508 1.46100i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.14468e10 −0.214484
\(786\) 0 0
\(787\) −4.79863e11 + 2.77049e11i −1.25089 + 0.722201i −0.971286 0.237915i \(-0.923536\pi\)
−0.279602 + 0.960116i \(0.590203\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.42051e11 + 4.53610e11i −0.618302 + 1.15871i
\(792\) 0 0
\(793\) 9.09678e10 1.57561e11i 0.230036 0.398433i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.84023e11i 1.19959i −0.800154 0.599795i \(-0.795249\pi\)
0.800154 0.599795i \(-0.204751\pi\)
\(798\) 0 0
\(799\) −5.41419e11 −1.32845
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.62356e11 + 9.37360e10i 0.390486 + 0.225447i
\(804\) 0 0
\(805\) −6.98353e9 2.08848e11i −0.0166300 0.497333i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.05691e11 + 1.83062e11i 0.246743 + 0.427371i 0.962620 0.270855i \(-0.0873064\pi\)
−0.715877 + 0.698226i \(0.753973\pi\)
\(810\) 0 0
\(811\) 1.05159e11i 0.243089i 0.992586 + 0.121544i \(0.0387847\pi\)
−0.992586 + 0.121544i \(0.961215\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.59612e10 1.49887e10i 0.0588429 0.0339730i
\(816\) 0 0
\(817\) 7.35080e11 + 4.24399e11i 1.64986 + 0.952546i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.78260e10 + 1.00158e11i −0.127277 + 0.220450i −0.922621 0.385708i \(-0.873957\pi\)
0.795344 + 0.606159i \(0.207290\pi\)
\(822\) 0 0
\(823\) 4.35537e11 + 7.54373e11i 0.949349 + 1.64432i 0.746799 + 0.665049i \(0.231590\pi\)
0.202550 + 0.979272i \(0.435077\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.49297e11 −1.17432 −0.587159 0.809472i \(-0.699754\pi\)
−0.587159 + 0.809472i \(0.699754\pi\)
\(828\) 0 0
\(829\) −3.02301e11 + 1.74533e11i −0.640060 + 0.369539i −0.784638 0.619954i \(-0.787151\pi\)
0.144577 + 0.989493i \(0.453818\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.05897e11 + 2.48600e11i 1.05071 + 0.516322i
\(834\) 0 0
\(835\) −5.68553e10 + 9.84763e10i −0.116957 + 0.202575i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.74760e11i 1.56358i 0.623543 + 0.781789i \(0.285693\pi\)
−0.623543 + 0.781789i \(0.714307\pi\)
\(840\) 0 0
\(841\) −4.86546e11 −0.972613
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.54203e11 8.90290e10i −0.302458 0.174624i
\(846\) 0 0
\(847\) 1.15638e10 + 1.85671e10i 0.0224681 + 0.0360752i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.65601e11 + 6.33240e11i 0.697091 + 1.20740i
\(852\) 0 0
\(853\) 4.31150e11i 0.814390i −0.913341 0.407195i \(-0.866507\pi\)
0.913341 0.407195i \(-0.133493\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.55977e11 1.47788e11i 0.474545 0.273979i −0.243596 0.969877i \(-0.578327\pi\)
0.718140 + 0.695898i \(0.244994\pi\)
\(858\) 0 0
\(859\) 8.08069e11 + 4.66539e11i 1.48414 + 0.856870i 0.999837 0.0180286i \(-0.00573900\pi\)
0.484305 + 0.874899i \(0.339072\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.21149e11 5.56246e11i 0.578980 1.00282i −0.416617 0.909082i \(-0.636784\pi\)
0.995597 0.0937400i \(-0.0298822\pi\)
\(864\) 0 0
\(865\) 1.02724e11 + 1.77922e11i 0.183487 + 0.317809i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.01136e11 −1.05413
\(870\) 0 0
\(871\) 1.28077e12 7.39452e11i 2.22535 1.28481i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.61187e11 1.92733e11i −0.616170 0.328795i
\(876\) 0 0
\(877\) −2.91337e11 + 5.04610e11i −0.492490 + 0.853017i −0.999963 0.00865063i \(-0.997246\pi\)
0.507473 + 0.861668i \(0.330580\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.79255e11i 0.297554i 0.988871 + 0.148777i \(0.0475337\pi\)
−0.988871 + 0.148777i \(0.952466\pi\)
\(882\) 0 0
\(883\) −1.09789e12 −1.80599 −0.902993 0.429655i \(-0.858635\pi\)
−0.902993 + 0.429655i \(0.858635\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.05091e11 2.91615e11i −0.815972 0.471102i 0.0330533 0.999454i \(-0.489477\pi\)
−0.849026 + 0.528352i \(0.822810\pi\)
\(888\) 0 0
\(889\) 2.50938e11 4.70265e11i 0.401753 0.752896i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5.68542e11 9.84744e11i −0.894040 1.54852i
\(894\) 0 0
\(895\) 2.84784e11i 0.443838i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.12569e10 + 2.95932e10i −0.0784718 + 0.0453057i
\(900\) 0 0
\(901\) −5.74283e11 3.31562e11i −0.871419 0.503114i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.23607e10 5.60503e10i 0.0482418 0.0835572i
\(906\) 0 0
\(907\) −4.66059e11 8.07237e11i −0.688671 1.19281i −0.972268 0.233869i \(-0.924861\pi\)
0.283598 0.958943i \(-0.408472\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.47091e11 −1.08468 −0.542338 0.840160i \(-0.682461\pi\)
−0.542338 + 0.840160i \(0.682461\pi\)
\(912\) 0 0
\(913\) −6.88010e10 + 3.97223e10i −0.0990174 + 0.0571677i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.03124e12 + 6.42270e11i −1.45842 + 0.908322i
\(918\) 0 0
\(919\) 3.60674e11 6.24705e11i 0.505653 0.875817i −0.494326 0.869277i \(-0.664585\pi\)
0.999979 0.00653993i \(-0.00208174\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.66577e12i 2.29513i
\(924\) 0 0
\(925\) 6.61869e11 0.904077
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −5.66284e10 3.26944e10i −0.0760276 0.0438945i 0.461504 0.887138i \(-0.347310\pi\)
−0.537532 + 0.843243i \(0.680643\pi\)
\(930\) 0 0
\(931\) 7.90822e10 + 1.18119e12i 0.105264 + 1.57224i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.71622e11 2.97258e11i −0.224557 0.388945i
\(936\) 0 0
\(937\) 1.46812e11i 0.190460i −0.995455 0.0952301i \(-0.969641\pi\)
0.995455 0.0952301i \(-0.0303587\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.76414e11 + 1.59588e11i −0.352535 + 0.203536i −0.665801 0.746129i \(-0.731910\pi\)
0.313266 + 0.949665i \(0.398577\pi\)
\(942\) 0 0
\(943\) 1.04625e12 + 6.04054e11i 1.32309 + 0.763887i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.63248e11 2.82754e11i 0.202977 0.351567i −0.746509 0.665375i \(-0.768272\pi\)
0.949486 + 0.313808i \(0.101605\pi\)
\(948\) 0 0
\(949\) 2.48770e11 + 4.30882e11i 0.306714 + 0.531243i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.37626e12 −1.66851 −0.834254 0.551380i \(-0.814102\pi\)
−0.834254 + 0.551380i \(0.814102\pi\)
\(954\) 0 0
\(955\) −3.89033e11 + 2.24608e11i −0.467706 + 0.270030i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.30697e11 + 2.77771e10i −0.982128 + 0.0328407i
\(960\) 0 0
\(961\) −2.98599e11 + 5.17189e11i −0.350103 + 0.606395i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.30138e11i 0.265387i
\(966\) 0 0
\(967\) −4.03059e11 −0.460959 −0.230480 0.973077i \(-0.574029\pi\)
−0.230480 + 0.973077i \(0.574029\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.41001e12 + 8.14067e11i 1.58615 + 0.915764i 0.993933 + 0.109985i \(0.0350802\pi\)
0.592216 + 0.805779i \(0.298253\pi\)
\(972\) 0 0
\(973\) 2.20331e11 + 1.17571e11i 0.245824 + 0.131174i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.36831e11 + 9.29819e11i 0.589196 + 1.02052i 0.994338 + 0.106263i \(0.0338886\pi\)
−0.405143 + 0.914254i \(0.632778\pi\)
\(978\) 0 0
\(979\) 1.11408e12i 1.21279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.63659e9 + 3.83164e9i −0.00710773 + 0.00410365i −0.503550 0.863966i \(-0.667973\pi\)
0.496442 + 0.868070i \(0.334640\pi\)
\(984\) 0 0
\(985\) −3.03180e11 1.75041e11i −0.322074 0.185950i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.65953e11 + 1.32667e12i −0.800602 + 1.38668i
\(990\) 0 0
\(991\) −1.26964e11 2.19908e11i −0.131639 0.228006i 0.792669 0.609652i \(-0.208691\pi\)
−0.924309 + 0.381646i \(0.875357\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.33405e11 −0.238132
\(996\) 0 0
\(997\) −3.46237e11 + 1.99900e11i −0.350423 + 0.202317i −0.664872 0.746958i \(-0.731514\pi\)
0.314449 + 0.949275i \(0.398180\pi\)
\(998\) 0 0
\(999\) 0 0
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 252.9.z.d.145.4 12
3.2 odd 2 84.9.m.b.61.3 12
7.3 odd 6 inner 252.9.z.d.73.4 12
21.17 even 6 84.9.m.b.73.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.9.m.b.61.3 12 3.2 odd 2
84.9.m.b.73.3 yes 12 21.17 even 6
252.9.z.d.73.4 12 7.3 odd 6 inner
252.9.z.d.145.4 12 1.1 even 1 trivial