Properties

Label 252.9.z.e.145.7
Level $252$
Weight $9$
Character 252.145
Analytic conductor $102.659$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} + \cdots + 51\!\cdots\!63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{26}\cdot 7^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.7
Root \(-285.330 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 252.145
Dual form 252.9.z.e.73.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(428.745 + 247.536i) q^{5} +(1933.79 + 1423.12i) q^{7} +O(q^{10})\) \(q+(428.745 + 247.536i) q^{5} +(1933.79 + 1423.12i) q^{7} +(-1062.17 - 1839.73i) q^{11} -15639.3i q^{13} +(-48991.8 + 28285.4i) q^{17} +(-140021. - 80841.3i) q^{19} +(223626. - 387331. i) q^{23} +(-72764.3 - 126031. i) q^{25} -1.33434e6 q^{29} +(-467329. + 269813. i) q^{31} +(476829. + 1.08884e6i) q^{35} +(531934. - 921336. i) q^{37} -4.94716e6i q^{41} +2.05532e6 q^{43} +(-2.47316e6 - 1.42788e6i) q^{47} +(1.71427e6 + 5.50402e6i) q^{49} +(6.97472e6 + 1.20806e7i) q^{53} -1.05170e6i q^{55} +(-5.87945e6 + 3.39450e6i) q^{59} +(480161. + 277221. i) q^{61} +(3.87130e6 - 6.70529e6i) q^{65} +(-1.71524e7 - 2.97088e7i) q^{67} +2.21608e7 q^{71} +(-1.19338e7 + 6.89001e6i) q^{73} +(564145. - 5.06924e6i) q^{77} +(1.94917e7 - 3.37606e7i) q^{79} +6.23163e7i q^{83} -2.80066e7 q^{85} +(-4.64932e7 - 2.68429e7i) q^{89} +(2.22566e7 - 3.02432e7i) q^{91} +(-4.00223e7 - 6.93206e7i) q^{95} -2.92340e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 7238 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 7238 q^{7} - 107256 q^{19} + 1348832 q^{25} - 4734426 q^{31} + 6802748 q^{37} - 596128 q^{43} - 19249258 q^{49} + 32573772 q^{61} - 25999304 q^{67} - 52899612 q^{73} - 129945530 q^{79} - 83755728 q^{85} + 42306852 q^{91}+O(q^{100}) \) 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
\(5\) 428.745 + 247.536i 0.685992 + 0.396058i 0.802109 0.597178i \(-0.203711\pi\)
−0.116117 + 0.993236i \(0.537045\pi\)
\(6\) 0 0
\(7\) 1933.79 + 1423.12i 0.805409 + 0.592719i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1062.17 1839.73i −0.0725475 0.125656i 0.827470 0.561510i \(-0.189780\pi\)
−0.900017 + 0.435854i \(0.856446\pi\)
\(12\) 0 0
\(13\) 15639.3i 0.547577i −0.961790 0.273788i \(-0.911723\pi\)
0.961790 0.273788i \(-0.0882768\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −48991.8 + 28285.4i −0.586580 + 0.338662i −0.763744 0.645519i \(-0.776641\pi\)
0.177164 + 0.984181i \(0.443308\pi\)
\(18\) 0 0
\(19\) −140021. 80841.3i −1.07443 0.620325i −0.145044 0.989425i \(-0.546332\pi\)
−0.929390 + 0.369100i \(0.879666\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 223626. 387331.i 0.799117 1.38411i −0.121075 0.992643i \(-0.538634\pi\)
0.920192 0.391467i \(-0.128032\pi\)
\(24\) 0 0
\(25\) −72764.3 126031.i −0.186277 0.322641i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.33434e6 −1.88658 −0.943291 0.331966i \(-0.892288\pi\)
−0.943291 + 0.331966i \(0.892288\pi\)
\(30\) 0 0
\(31\) −467329. + 269813.i −0.506030 + 0.292156i −0.731200 0.682163i \(-0.761039\pi\)
0.225170 + 0.974319i \(0.427706\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 476829. + 1.08884e6i 0.317753 + 0.725589i
\(36\) 0 0
\(37\) 531934. 921336.i 0.283825 0.491599i −0.688499 0.725238i \(-0.741730\pi\)
0.972324 + 0.233638i \(0.0750632\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.94716e6i 1.75073i −0.483459 0.875367i \(-0.660620\pi\)
0.483459 0.875367i \(-0.339380\pi\)
\(42\) 0 0
\(43\) 2.05532e6 0.601180 0.300590 0.953753i \(-0.402816\pi\)
0.300590 + 0.953753i \(0.402816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.47316e6 1.42788e6i −0.506828 0.292617i 0.224701 0.974428i \(-0.427859\pi\)
−0.731529 + 0.681811i \(0.761193\pi\)
\(48\) 0 0
\(49\) 1.71427e6 + 5.50402e6i 0.297368 + 0.954763i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.97472e6 + 1.20806e7i 0.883941 + 1.53103i 0.846923 + 0.531715i \(0.178452\pi\)
0.0370175 + 0.999315i \(0.488214\pi\)
\(54\) 0 0
\(55\) 1.05170e6i 0.114932i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.87945e6 + 3.39450e6i −0.485209 + 0.280136i −0.722585 0.691282i \(-0.757046\pi\)
0.237376 + 0.971418i \(0.423713\pi\)
\(60\) 0 0
\(61\) 480161. + 277221.i 0.0346791 + 0.0200220i 0.517239 0.855841i \(-0.326960\pi\)
−0.482560 + 0.875863i \(0.660293\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.87130e6 6.70529e6i 0.216872 0.375633i
\(66\) 0 0
\(67\) −1.71524e7 2.97088e7i −0.851188 1.47430i −0.880137 0.474720i \(-0.842549\pi\)
0.0289486 0.999581i \(-0.490784\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.21608e7 0.872071 0.436036 0.899929i \(-0.356382\pi\)
0.436036 + 0.899929i \(0.356382\pi\)
\(72\) 0 0
\(73\) −1.19338e7 + 6.89001e6i −0.420232 + 0.242621i −0.695177 0.718839i \(-0.744674\pi\)
0.274945 + 0.961460i \(0.411340\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 564145. 5.06924e6i 0.0160483 0.144205i
\(78\) 0 0
\(79\) 1.94917e7 3.37606e7i 0.500428 0.866767i −0.499572 0.866272i \(-0.666509\pi\)
1.00000 0.000494391i \(-0.000157370\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.23163e7i 1.31307i 0.754294 + 0.656537i \(0.227979\pi\)
−0.754294 + 0.656537i \(0.772021\pi\)
\(84\) 0 0
\(85\) −2.80066e7 −0.536519
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.64932e7 2.68429e7i −0.741019 0.427827i 0.0814208 0.996680i \(-0.474054\pi\)
−0.822440 + 0.568852i \(0.807388\pi\)
\(90\) 0 0
\(91\) 2.22566e7 3.02432e7i 0.324559 0.441023i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00223e7 6.93206e7i −0.491369 0.851076i
\(96\) 0 0
\(97\) 2.92340e7i 0.330218i −0.986275 0.165109i \(-0.947202\pi\)
0.986275 0.165109i \(-0.0527977\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.15074e8 6.64378e7i 1.10583 0.638454i 0.168087 0.985772i \(-0.446241\pi\)
0.937747 + 0.347318i \(0.112908\pi\)
\(102\) 0 0
\(103\) −8.24124e7 4.75808e7i −0.732224 0.422750i 0.0870114 0.996207i \(-0.472268\pi\)
−0.819235 + 0.573458i \(0.805602\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.88716e7 + 6.73275e7i −0.296549 + 0.513638i −0.975344 0.220690i \(-0.929169\pi\)
0.678795 + 0.734328i \(0.262503\pi\)
\(108\) 0 0
\(109\) −1.53205e7 2.65360e7i −0.108535 0.187987i 0.806642 0.591040i \(-0.201282\pi\)
−0.915177 + 0.403053i \(0.867949\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.30284e8 −0.799056 −0.399528 0.916721i \(-0.630826\pi\)
−0.399528 + 0.916721i \(0.630826\pi\)
\(114\) 0 0
\(115\) 1.91757e8 1.10711e8i 1.09638 0.632993i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.34993e8 1.50231e7i −0.673169 0.0749156i
\(120\) 0 0
\(121\) 1.04923e8 1.81732e8i 0.489474 0.847793i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.65435e8i 1.08722i
\(126\) 0 0
\(127\) −2.78124e8 −1.06911 −0.534557 0.845132i \(-0.679522\pi\)
−0.534557 + 0.845132i \(0.679522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.16821e8 + 1.82917e8i 1.07579 + 0.621110i 0.929759 0.368169i \(-0.120015\pi\)
0.146036 + 0.989279i \(0.453349\pi\)
\(132\) 0 0
\(133\) −1.55725e8 3.55597e8i −0.497681 1.13645i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.91348e8 + 3.31425e8i 0.543178 + 0.940812i 0.998719 + 0.0505971i \(0.0161124\pi\)
−0.455541 + 0.890215i \(0.650554\pi\)
\(138\) 0 0
\(139\) 5.95007e8i 1.59391i −0.604040 0.796954i \(-0.706443\pi\)
0.604040 0.796954i \(-0.293557\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.87722e7 + 1.66116e7i −0.0688063 + 0.0397253i
\(144\) 0 0
\(145\) −5.72093e8 3.30298e8i −1.29418 0.747196i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.12693e8 + 1.95190e8i −0.228639 + 0.396015i −0.957405 0.288748i \(-0.906761\pi\)
0.728766 + 0.684763i \(0.240094\pi\)
\(150\) 0 0
\(151\) −4.57375e8 7.92196e8i −0.879760 1.52379i −0.851604 0.524185i \(-0.824370\pi\)
−0.0281557 0.999604i \(-0.508963\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.67153e8 −0.462843
\(156\) 0 0
\(157\) 1.82590e8 1.05418e8i 0.300523 0.173507i −0.342155 0.939644i \(-0.611157\pi\)
0.642678 + 0.766136i \(0.277823\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.83662e8 4.30770e8i 1.46400 0.641124i
\(162\) 0 0
\(163\) 1.37742e8 2.38576e8i 0.195126 0.337969i −0.751816 0.659373i \(-0.770822\pi\)
0.946942 + 0.321405i \(0.104155\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.66593e8i 0.728460i 0.931309 + 0.364230i \(0.118668\pi\)
−0.931309 + 0.364230i \(0.881332\pi\)
\(168\) 0 0
\(169\) 5.71142e8 0.700160
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.27950e9 7.38720e8i −1.42842 0.824699i −0.431425 0.902149i \(-0.641989\pi\)
−0.996996 + 0.0774499i \(0.975322\pi\)
\(174\) 0 0
\(175\) 3.86470e7 3.47270e8i 0.0412063 0.370267i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.83010e8 + 6.63393e8i 0.373077 + 0.646188i 0.990037 0.140806i \(-0.0449694\pi\)
−0.616960 + 0.786994i \(0.711636\pi\)
\(180\) 0 0
\(181\) 1.84765e9i 1.72150i −0.509030 0.860749i \(-0.669996\pi\)
0.509030 0.860749i \(-0.330004\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.56128e8 2.63345e8i 0.389403 0.224822i
\(186\) 0 0
\(187\) 1.04075e8 + 6.00878e7i 0.0851099 + 0.0491382i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.14741e8 1.98737e8i 0.0862152 0.149329i −0.819693 0.572803i \(-0.805856\pi\)
0.905908 + 0.423474i \(0.139189\pi\)
\(192\) 0 0
\(193\) −7.44633e8 1.28974e9i −0.536677 0.929552i −0.999080 0.0428820i \(-0.986346\pi\)
0.462403 0.886670i \(-0.346987\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.44196e8 −0.294924 −0.147462 0.989068i \(-0.547110\pi\)
−0.147462 + 0.989068i \(0.547110\pi\)
\(198\) 0 0
\(199\) −1.25444e9 + 7.24251e8i −0.799904 + 0.461825i −0.843437 0.537227i \(-0.819472\pi\)
0.0435338 + 0.999052i \(0.486138\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.58034e9 1.89893e9i −1.51947 1.11821i
\(204\) 0 0
\(205\) 1.22460e9 2.12107e9i 0.693392 1.20099i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.43469e8i 0.180012i
\(210\) 0 0
\(211\) −1.24503e8 −0.0628129 −0.0314065 0.999507i \(-0.509999\pi\)
−0.0314065 + 0.999507i \(0.509999\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.81206e8 + 5.08765e8i 0.412405 + 0.238102i
\(216\) 0 0
\(217\) −1.28769e9 1.43304e8i −0.580728 0.0646280i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.42365e8 + 7.66199e8i 0.185444 + 0.321198i
\(222\) 0 0
\(223\) 1.07401e9i 0.434300i −0.976138 0.217150i \(-0.930324\pi\)
0.976138 0.217150i \(-0.0696761\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.61590e9 2.08764e9i 1.36180 0.786234i 0.371935 0.928259i \(-0.378695\pi\)
0.989863 + 0.142025i \(0.0453612\pi\)
\(228\) 0 0
\(229\) −2.27312e9 1.31238e9i −0.826571 0.477221i 0.0261064 0.999659i \(-0.491689\pi\)
−0.852677 + 0.522438i \(0.825022\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.84691e8 + 1.18592e9i −0.232312 + 0.402376i −0.958488 0.285133i \(-0.907962\pi\)
0.726176 + 0.687509i \(0.241296\pi\)
\(234\) 0 0
\(235\) −7.06903e8 1.22439e9i −0.231786 0.401466i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.37306e9 −0.727307 −0.363653 0.931534i \(-0.618471\pi\)
−0.363653 + 0.931534i \(0.618471\pi\)
\(240\) 0 0
\(241\) 2.85917e9 1.65074e9i 0.847564 0.489341i −0.0122645 0.999925i \(-0.503904\pi\)
0.859828 + 0.510584i \(0.170571\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.27458e8 + 2.78416e9i −0.174149 + 0.772735i
\(246\) 0 0
\(247\) −1.26430e9 + 2.18984e9i −0.339675 + 0.588335i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.12960e7i 0.00788487i 0.999992 + 0.00394244i \(0.00125492\pi\)
−0.999992 + 0.00394244i \(0.998745\pi\)
\(252\) 0 0
\(253\) −9.50112e8 −0.231896
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.40586e8 + 3.69842e8i 0.146840 + 0.0847782i 0.571620 0.820518i \(-0.306315\pi\)
−0.424780 + 0.905297i \(0.639648\pi\)
\(258\) 0 0
\(259\) 2.33982e9 1.02466e9i 0.519975 0.227710i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.59579e9 + 4.49604e9i 0.542558 + 0.939738i 0.998756 + 0.0498601i \(0.0158775\pi\)
−0.456198 + 0.889878i \(0.650789\pi\)
\(264\) 0 0
\(265\) 6.90598e9i 1.40037i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.76648e9 2.17458e9i 0.719328 0.415304i −0.0951772 0.995460i \(-0.530342\pi\)
0.814505 + 0.580156i \(0.197008\pi\)
\(270\) 0 0
\(271\) 3.75508e9 + 2.16799e9i 0.696212 + 0.401958i 0.805935 0.592004i \(-0.201663\pi\)
−0.109723 + 0.993962i \(0.534996\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.54576e8 + 2.67733e8i −0.0270278 + 0.0468136i
\(276\) 0 0
\(277\) 5.19364e9 + 8.99565e9i 0.882171 + 1.52797i 0.848922 + 0.528518i \(0.177252\pi\)
0.0332491 + 0.999447i \(0.489415\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.25066e9 −0.842149 −0.421074 0.907026i \(-0.638347\pi\)
−0.421074 + 0.907026i \(0.638347\pi\)
\(282\) 0 0
\(283\) −4.61734e8 + 2.66582e8i −0.0719857 + 0.0415610i −0.535561 0.844497i \(-0.679900\pi\)
0.463575 + 0.886058i \(0.346566\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.04039e9 9.56675e9i 1.03769 1.41006i
\(288\) 0 0
\(289\) −1.88775e9 + 3.26968e9i −0.270616 + 0.468720i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.02293e9i 0.545849i 0.962036 + 0.272924i \(0.0879908\pi\)
−0.962036 + 0.272924i \(0.912009\pi\)
\(294\) 0 0
\(295\) −3.36105e9 −0.443799
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.05760e9 3.49736e9i −0.757907 0.437578i
\(300\) 0 0
\(301\) 3.97454e9 + 2.92496e9i 0.484196 + 0.356331i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.37245e8 + 2.37714e8i 0.0158597 + 0.0274699i
\(306\) 0 0
\(307\) 7.66029e9i 0.862366i 0.902265 + 0.431183i \(0.141904\pi\)
−0.902265 + 0.431183i \(0.858096\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.30940e10 + 7.55982e9i −1.39969 + 0.808109i −0.994359 0.106063i \(-0.966175\pi\)
−0.405326 + 0.914172i \(0.632842\pi\)
\(312\) 0 0
\(313\) 2.40149e9 + 1.38650e9i 0.250209 + 0.144458i 0.619860 0.784712i \(-0.287189\pi\)
−0.369651 + 0.929171i \(0.620523\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.98768e9 + 3.44277e9i −0.196839 + 0.340934i −0.947502 0.319751i \(-0.896401\pi\)
0.750663 + 0.660685i \(0.229734\pi\)
\(318\) 0 0
\(319\) 1.41730e9 + 2.45483e9i 0.136867 + 0.237061i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.14652e9 0.840322
\(324\) 0 0
\(325\) −1.97105e9 + 1.13799e9i −0.176670 + 0.102001i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.75052e9 6.28081e9i −0.234764 0.536083i
\(330\) 0 0
\(331\) −4.95902e9 + 8.58927e9i −0.413127 + 0.715557i −0.995230 0.0975583i \(-0.968897\pi\)
0.582103 + 0.813115i \(0.302230\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.69833e10i 1.34848i
\(336\) 0 0
\(337\) 7.98626e9 0.619189 0.309595 0.950869i \(-0.399807\pi\)
0.309595 + 0.950869i \(0.399807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.92765e8 + 5.73173e8i 0.0734224 + 0.0423905i
\(342\) 0 0
\(343\) −4.51784e9 + 1.30832e10i −0.326403 + 0.945231i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.76637e9 3.05945e9i −0.121833 0.211021i 0.798658 0.601786i \(-0.205544\pi\)
−0.920491 + 0.390765i \(0.872211\pi\)
\(348\) 0 0
\(349\) 1.17888e10i 0.794638i 0.917680 + 0.397319i \(0.130059\pi\)
−0.917680 + 0.397319i \(0.869941\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.39483e10 8.05305e9i 0.898301 0.518634i 0.0216525 0.999766i \(-0.493107\pi\)
0.876649 + 0.481131i \(0.159774\pi\)
\(354\) 0 0
\(355\) 9.50133e9 + 5.48560e9i 0.598234 + 0.345391i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.78100e9 + 8.28093e9i −0.287833 + 0.498541i −0.973292 0.229570i \(-0.926268\pi\)
0.685459 + 0.728111i \(0.259601\pi\)
\(360\) 0 0
\(361\) 4.57886e9 + 7.93082e9i 0.269605 + 0.466970i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.82210e9 −0.384368
\(366\) 0 0
\(367\) −2.48860e10 + 1.43680e10i −1.37180 + 0.792010i −0.991155 0.132710i \(-0.957632\pi\)
−0.380647 + 0.924720i \(0.624299\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.70445e9 + 3.32871e10i −0.195537 + 1.75703i
\(372\) 0 0
\(373\) 2.48457e9 4.30341e9i 0.128356 0.222319i −0.794684 0.607024i \(-0.792363\pi\)
0.923040 + 0.384704i \(0.125697\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.08683e10i 1.03305i
\(378\) 0 0
\(379\) 3.03220e10 1.46961 0.734804 0.678280i \(-0.237274\pi\)
0.734804 + 0.678280i \(0.237274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.45211e10 + 8.38379e9i 0.674848 + 0.389624i 0.797911 0.602775i \(-0.205938\pi\)
−0.123063 + 0.992399i \(0.539272\pi\)
\(384\) 0 0
\(385\) 1.49669e9 2.03376e9i 0.0681224 0.0925674i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.83553e9 6.64334e9i −0.167505 0.290127i 0.770037 0.637999i \(-0.220238\pi\)
−0.937542 + 0.347872i \(0.886904\pi\)
\(390\) 0 0
\(391\) 2.53014e10i 1.08252i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.67140e10 9.64980e9i 0.686579 0.396397i
\(396\) 0 0
\(397\) 1.70191e10 + 9.82597e9i 0.685132 + 0.395561i 0.801786 0.597612i \(-0.203884\pi\)
−0.116654 + 0.993173i \(0.537217\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.16330e10 3.74694e10i 0.836639 1.44910i −0.0560491 0.998428i \(-0.517850\pi\)
0.892689 0.450674i \(-0.148816\pi\)
\(402\) 0 0
\(403\) 4.21969e9 + 7.30872e9i 0.159978 + 0.277090i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.26001e9 −0.0823632
\(408\) 0 0
\(409\) 1.76795e10 1.02073e10i 0.631796 0.364768i −0.149651 0.988739i \(-0.547815\pi\)
0.781447 + 0.623971i \(0.214482\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.62004e10 1.80291e9i −0.556834 0.0619688i
\(414\) 0 0
\(415\) −1.54255e10 + 2.67178e10i −0.520053 + 0.900758i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.55195e10i 0.503524i 0.967789 + 0.251762i \(0.0810101\pi\)
−0.967789 + 0.251762i \(0.918990\pi\)
\(420\) 0 0
\(421\) −2.58425e9 −0.0822633 −0.0411317 0.999154i \(-0.513096\pi\)
−0.0411317 + 0.999154i \(0.513096\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.12971e9 + 4.11634e9i 0.218532 + 0.126170i
\(426\) 0 0
\(427\) 5.34011e8 + 1.21941e9i 0.0160635 + 0.0366809i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.10297e10 1.91041e10i −0.319637 0.553627i 0.660776 0.750583i \(-0.270227\pi\)
−0.980412 + 0.196957i \(0.936894\pi\)
\(432\) 0 0
\(433\) 6.09726e10i 1.73453i −0.497843 0.867267i \(-0.665874\pi\)
0.497843 0.867267i \(-0.334126\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.26247e10 + 3.61564e10i −1.71720 + 0.991424i
\(438\) 0 0
\(439\) −5.10681e10 2.94842e10i −1.37497 0.793836i −0.383417 0.923575i \(-0.625253\pi\)
−0.991548 + 0.129739i \(0.958586\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.02703e10 + 1.77888e10i −0.266668 + 0.461882i −0.967999 0.250953i \(-0.919256\pi\)
0.701332 + 0.712835i \(0.252589\pi\)
\(444\) 0 0
\(445\) −1.32891e10 2.30175e10i −0.338889 0.586972i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.11616e10 1.50485 0.752425 0.658678i \(-0.228884\pi\)
0.752425 + 0.658678i \(0.228884\pi\)
\(450\) 0 0
\(451\) −9.10143e9 + 5.25471e9i −0.219990 + 0.127011i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.70287e10 7.45729e9i 0.397316 0.173994i
\(456\) 0 0
\(457\) 6.93723e9 1.20156e10i 0.159045 0.275475i −0.775479 0.631373i \(-0.782492\pi\)
0.934525 + 0.355898i \(0.115825\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.22295e10i 1.59923i −0.600513 0.799615i \(-0.705037\pi\)
0.600513 0.799615i \(-0.294963\pi\)
\(462\) 0 0
\(463\) −5.19540e10 −1.13056 −0.565281 0.824898i \(-0.691232\pi\)
−0.565281 + 0.824898i \(0.691232\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.77578e10 3.33465e10i −1.21435 0.701104i −0.250644 0.968079i \(-0.580642\pi\)
−0.963703 + 0.266975i \(0.913976\pi\)
\(468\) 0 0
\(469\) 9.11008e9 8.18604e10i 0.188291 1.69193i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.18309e9 3.78122e9i −0.0436141 0.0755419i
\(474\) 0 0
\(475\) 2.35295e10i 0.462208i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.78886e10 + 2.18750e10i −0.719724 + 0.415533i −0.814651 0.579951i \(-0.803072\pi\)
0.0949269 + 0.995484i \(0.469738\pi\)
\(480\) 0 0
\(481\) −1.44091e10 8.31909e9i −0.269188 0.155416i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.23647e9 1.25339e10i 0.130786 0.226527i
\(486\) 0 0
\(487\) −2.08822e9 3.61691e9i −0.0371245 0.0643016i 0.846866 0.531806i \(-0.178486\pi\)
−0.883991 + 0.467505i \(0.845153\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.01240e10 0.518307 0.259153 0.965836i \(-0.416557\pi\)
0.259153 + 0.965836i \(0.416557\pi\)
\(492\) 0 0
\(493\) 6.53719e10 3.77425e10i 1.10663 0.638915i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.28543e10 + 3.15374e10i 0.702374 + 0.516893i
\(498\) 0 0
\(499\) 3.25369e10 5.63555e10i 0.524776 0.908938i −0.474808 0.880089i \(-0.657483\pi\)
0.999584 0.0288486i \(-0.00918408\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.63371e10i 0.723864i 0.932204 + 0.361932i \(0.117883\pi\)
−0.932204 + 0.361932i \(0.882117\pi\)
\(504\) 0 0
\(505\) 6.57830e10 1.01146
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.46633e10 + 2.00129e10i 0.516415 + 0.298152i 0.735466 0.677561i \(-0.236963\pi\)
−0.219052 + 0.975713i \(0.570296\pi\)
\(510\) 0 0
\(511\) −3.28828e10 3.65946e9i −0.482265 0.0536702i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.35559e10 4.08001e10i −0.334866 0.580006i
\(516\) 0 0
\(517\) 6.06659e9i 0.0849146i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.05814e10 + 2.34297e10i −0.550777 + 0.317992i −0.749436 0.662077i \(-0.769675\pi\)
0.198658 + 0.980069i \(0.436342\pi\)
\(522\) 0 0
\(523\) −7.50830e10 4.33492e10i −1.00354 0.579395i −0.0942468 0.995549i \(-0.530044\pi\)
−0.909294 + 0.416154i \(0.863378\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.52635e10 2.64372e10i 0.197885 0.342746i
\(528\) 0 0
\(529\) −6.08613e10 1.05415e11i −0.777175 1.34611i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.73702e10 −0.958661
\(534\) 0 0
\(535\) −3.33320e10 + 1.92442e10i −0.406861 + 0.234901i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.30506e9 8.99999e9i 0.0983984 0.106632i
\(540\) 0 0
\(541\) 3.33406e10 5.77477e10i 0.389211 0.674133i −0.603133 0.797641i \(-0.706081\pi\)
0.992344 + 0.123508i \(0.0394145\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.51695e10i 0.171944i
\(546\) 0 0
\(547\) −3.37378e10 −0.376849 −0.188425 0.982088i \(-0.560338\pi\)
−0.188425 + 0.982088i \(0.560338\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.86837e11 + 1.07870e11i 2.02701 + 1.17029i
\(552\) 0 0
\(553\) 8.57382e10 3.75469e10i 0.916799 0.401489i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.38794e10 + 4.13604e10i 0.248087 + 0.429699i 0.962995 0.269520i \(-0.0868649\pi\)
−0.714908 + 0.699218i \(0.753532\pi\)
\(558\) 0 0
\(559\) 3.21438e10i 0.329192i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.30170e10 5.37034e10i 0.925825 0.534525i 0.0403360 0.999186i \(-0.487157\pi\)
0.885489 + 0.464661i \(0.153824\pi\)
\(564\) 0 0
\(565\) −5.58586e10 3.22500e10i −0.548146 0.316472i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.54714e10 + 2.67973e10i −0.147598 + 0.255648i −0.930339 0.366700i \(-0.880488\pi\)
0.782741 + 0.622348i \(0.213821\pi\)
\(570\) 0 0
\(571\) −5.37493e10 9.30965e10i −0.505625 0.875768i −0.999979 0.00650703i \(-0.997929\pi\)
0.494354 0.869261i \(-0.335405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.50878e10 −0.595427
\(576\) 0 0
\(577\) −5.22110e10 + 3.01440e10i −0.471041 + 0.271955i −0.716675 0.697407i \(-0.754337\pi\)
0.245635 + 0.969363i \(0.421004\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8.86834e10 + 1.20506e11i −0.778284 + 1.05756i
\(582\) 0 0
\(583\) 1.48167e10 2.56632e10i 0.128255 0.222145i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.16561e10i 0.182401i −0.995833 0.0912005i \(-0.970930\pi\)
0.995833 0.0912005i \(-0.0290704\pi\)
\(588\) 0 0
\(589\) 8.72480e10 0.724927
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.44121e11 8.32084e10i −1.16549 0.672897i −0.212878 0.977079i \(-0.568284\pi\)
−0.952614 + 0.304182i \(0.901617\pi\)
\(594\) 0 0
\(595\) −5.41589e10 3.98568e10i −0.432118 0.318005i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.40903e10 1.45649e11i −0.653189 1.13136i −0.982345 0.187080i \(-0.940098\pi\)
0.329156 0.944276i \(-0.393236\pi\)
\(600\) 0 0
\(601\) 7.87638e10i 0.603710i −0.953354 0.301855i \(-0.902394\pi\)
0.953354 0.301855i \(-0.0976059\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 8.99705e10 5.19445e10i 0.671550 0.387720i
\(606\) 0 0
\(607\) −2.54048e10 1.46674e10i −0.187137 0.108044i 0.403504 0.914978i \(-0.367792\pi\)
−0.590642 + 0.806934i \(0.701125\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.23311e10 + 3.86785e10i −0.160230 + 0.277527i
\(612\) 0 0
\(613\) −5.10346e10 8.83945e10i −0.361429 0.626013i 0.626767 0.779206i \(-0.284378\pi\)
−0.988196 + 0.153193i \(0.951044\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.49461e11 1.72132 0.860661 0.509179i \(-0.170051\pi\)
0.860661 + 0.509179i \(0.170051\pi\)
\(618\) 0 0
\(619\) −1.46061e11 + 8.43281e10i −0.994879 + 0.574394i −0.906729 0.421713i \(-0.861429\pi\)
−0.0881501 + 0.996107i \(0.528096\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.17074e10 1.18074e11i −0.343242 0.783792i
\(624\) 0 0
\(625\) 3.72811e10 6.45728e10i 0.244325 0.423184i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.01839e10i 0.384483i
\(630\) 0 0
\(631\) 8.03011e10 0.506529 0.253265 0.967397i \(-0.418496\pi\)
0.253265 + 0.967397i \(0.418496\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.19244e11 6.88458e10i −0.733404 0.423431i
\(636\) 0 0
\(637\) 8.60792e10 2.68100e10i 0.522806 0.162832i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.51587e10 + 7.82172e10i 0.267491 + 0.463308i 0.968213 0.250126i \(-0.0804721\pi\)
−0.700722 + 0.713434i \(0.747139\pi\)
\(642\) 0 0
\(643\) 2.94542e11i 1.72307i 0.507696 + 0.861536i \(0.330497\pi\)
−0.507696 + 0.861536i \(0.669503\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.46833e11 + 8.47741e10i −0.837928 + 0.483778i −0.856559 0.516048i \(-0.827402\pi\)
0.0186314 + 0.999826i \(0.494069\pi\)
\(648\) 0 0
\(649\) 1.24899e10 + 7.21107e9i 0.0704015 + 0.0406463i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.31238e10 + 7.46926e10i −0.237172 + 0.410795i −0.959902 0.280337i \(-0.909554\pi\)
0.722729 + 0.691131i \(0.242887\pi\)
\(654\) 0 0
\(655\) 9.05571e10 + 1.56849e11i 0.491991 + 0.852153i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.84405e10 −0.415909 −0.207955 0.978138i \(-0.566681\pi\)
−0.207955 + 0.978138i \(0.566681\pi\)
\(660\) 0 0
\(661\) 1.79270e11 1.03502e11i 0.939080 0.542178i 0.0494080 0.998779i \(-0.484267\pi\)
0.889672 + 0.456601i \(0.150933\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.12569e10 1.91008e11i 0.108696 0.976708i
\(666\) 0 0
\(667\) −2.98394e11 + 5.16833e11i −1.50760 + 2.61124i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.17782e9i 0.00581018i
\(672\) 0 0
\(673\) 3.74989e11 1.82792 0.913961 0.405801i \(-0.133008\pi\)
0.913961 + 0.405801i \(0.133008\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.82953e11 1.05628e11i −0.870932 0.502833i −0.00327401 0.999995i \(-0.501042\pi\)
−0.867658 + 0.497162i \(0.834375\pi\)
\(678\) 0 0
\(679\) 4.16034e10 5.65323e10i 0.195727 0.265961i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.80288e11 + 3.12268e11i 0.828484 + 1.43498i 0.899227 + 0.437482i \(0.144130\pi\)
−0.0707428 + 0.997495i \(0.522537\pi\)
\(684\) 0 0
\(685\) 1.89462e11i 0.860519i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.88932e11 1.09080e11i 0.838356 0.484025i
\(690\) 0 0
\(691\) −1.19035e11 6.87249e10i −0.522110 0.301441i 0.215687 0.976462i \(-0.430801\pi\)
−0.737798 + 0.675022i \(0.764134\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.47286e11 2.55106e11i 0.631279 1.09341i
\(696\) 0 0
\(697\) 1.39932e11 + 2.42370e11i 0.592908 + 1.02695i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.01153e11 −1.24714 −0.623570 0.781767i \(-0.714318\pi\)
−0.623570 + 0.781767i \(0.714318\pi\)
\(702\) 0 0
\(703\) −1.48964e11 + 8.60044e10i −0.609902 + 0.352127i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.17077e11 + 3.52868e10i 1.26907 + 0.141232i
\(708\) 0 0
\(709\) 1.24726e11 2.16031e11i 0.493595 0.854931i −0.506378 0.862312i \(-0.669016\pi\)
0.999973 + 0.00738024i \(0.00234922\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.41348e11i 0.933868i
\(714\) 0 0
\(715\) −1.64479e10 −0.0629341
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.58304e11 + 1.49132e11i 0.966531 + 0.558027i 0.898177 0.439634i \(-0.144892\pi\)
0.0683543 + 0.997661i \(0.478225\pi\)
\(720\) 0 0
\(721\) −9.16550e10 2.09294e11i −0.339168 0.774489i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.70926e10 + 1.68169e11i 0.351426 + 0.608688i
\(726\) 0 0
\(727\) 5.07469e11i 1.81665i 0.418263 + 0.908326i \(0.362639\pi\)
−0.418263 + 0.908326i \(0.637361\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.00694e11 + 5.81354e10i −0.352640 + 0.203597i
\(732\) 0 0
\(733\) 7.01877e9 + 4.05229e9i 0.0243134 + 0.0140373i 0.512107 0.858921i \(-0.328865\pi\)
−0.487794 + 0.872959i \(0.662198\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.64375e10 + 6.31116e10i −0.123503 + 0.213914i
\(738\) 0 0
\(739\) 1.21836e11 + 2.11026e11i 0.408506 + 0.707553i 0.994723 0.102601i \(-0.0327166\pi\)
−0.586217 + 0.810154i \(0.699383\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.89064e11 −0.620376 −0.310188 0.950675i \(-0.600392\pi\)
−0.310188 + 0.950675i \(0.600392\pi\)
\(744\) 0 0
\(745\) −9.66329e10 + 5.57910e10i −0.313689 + 0.181109i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.70984e11 + 7.48783e10i −0.543287 + 0.237919i
\(750\) 0 0
\(751\) −2.32130e11 + 4.02061e11i −0.729746 + 1.26396i 0.227245 + 0.973838i \(0.427028\pi\)
−0.956991 + 0.290119i \(0.906305\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.52867e11i 1.39374i
\(756\) 0 0
\(757\) 2.81923e11 0.858513 0.429257 0.903183i \(-0.358776\pi\)
0.429257 + 0.903183i \(0.358776\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.54973e9 + 1.47209e9i 0.00760248 + 0.00438929i 0.503796 0.863822i \(-0.331936\pi\)
−0.496194 + 0.868212i \(0.665270\pi\)
\(762\) 0 0
\(763\) 8.13713e9 7.31178e10i 0.0240089 0.215737i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.30878e10 + 9.19508e10i 0.153396 + 0.265689i
\(768\) 0 0
\(769\) 3.03649e11i 0.868293i 0.900842 + 0.434146i \(0.142950\pi\)
−0.900842 + 0.434146i \(0.857050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.73971e11 2.73647e11i 1.32750 0.766431i 0.342585 0.939487i \(-0.388698\pi\)
0.984912 + 0.173056i \(0.0553642\pi\)
\(774\) 0 0
\(775\) 6.80098e10 + 3.92654e10i 0.188523 + 0.108844i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.99935e11 + 6.92707e11i −1.08602 + 1.88105i
\(780\) 0 0
\(781\) −2.35385e10 4.07699e10i −0.0632666 0.109581i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.04379e11 0.274875
\(786\) 0 0
\(787\) −3.54149e11 + 2.04468e11i −0.923182 + 0.532999i −0.884649 0.466258i \(-0.845602\pi\)
−0.0385331 + 0.999257i \(0.512268\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.51942e11 1.85410e11i −0.643567 0.473616i
\(792\) 0 0
\(793\) 4.33556e9 7.50940e9i 0.0109636 0.0189895i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.43730e11i 0.604055i −0.953299 0.302027i \(-0.902337\pi\)
0.953299 0.302027i \(-0.0976634\pi\)
\(798\) 0 0
\(799\) 1.61553e11 0.396394
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.53515e10 + 1.46367e10i 0.0609736 + 0.0352031i
\(804\) 0 0
\(805\) 5.28371e11 + 5.88013e10i 1.25822 + 0.140024i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.88172e11 3.25924e11i −0.439300 0.760890i 0.558335 0.829615i \(-0.311440\pi\)
−0.997636 + 0.0687249i \(0.978107\pi\)
\(810\) 0 0
\(811\) 6.15386e11i 1.42254i 0.702920 + 0.711269i \(0.251879\pi\)
−0.702920 + 0.711269i \(0.748121\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.18112e11 6.81922e10i 0.267710 0.154563i
\(816\) 0 0
\(817\) −2.87788e11 1.66154e11i −0.645928 0.372927i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.08388e11 5.34144e11i 0.678774 1.17567i −0.296576 0.955009i \(-0.595845\pi\)
0.975350 0.220662i \(-0.0708218\pi\)
\(822\) 0 0
\(823\) −3.77630e10 6.54074e10i −0.0823127 0.142570i 0.821930 0.569588i \(-0.192897\pi\)
−0.904243 + 0.427018i \(0.859564\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.93362e11 −0.413380 −0.206690 0.978406i \(-0.566269\pi\)
−0.206690 + 0.978406i \(0.566269\pi\)
\(828\) 0 0
\(829\) −1.02147e11 + 5.89748e10i −0.216276 + 0.124867i −0.604225 0.796814i \(-0.706517\pi\)
0.387949 + 0.921681i \(0.373184\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.39669e11 2.21163e11i −0.497773 0.459338i
\(834\) 0 0
\(835\) −1.40252e11 + 2.42924e11i −0.288512 + 0.499718i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.17469e11i 1.44796i 0.689823 + 0.723978i \(0.257688\pi\)
−0.689823 + 0.723978i \(0.742312\pi\)
\(840\) 0 0
\(841\) 1.28023e12 2.55920
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.44874e11 + 1.41378e11i 0.480304 + 0.277304i
\(846\) 0 0
\(847\) 4.61525e11 2.02113e11i 0.896730 0.392700i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.37908e11 4.12069e11i −0.453618 0.785690i
\(852\) 0 0
\(853\) 6.84954e11i 1.29379i 0.762577 + 0.646897i \(0.223934\pi\)
−0.762577 + 0.646897i \(0.776066\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −4.95605e11 + 2.86137e11i −0.918781 + 0.530458i −0.883246 0.468910i \(-0.844647\pi\)
−0.0355351 + 0.999368i \(0.511314\pi\)
\(858\) 0 0
\(859\) −1.20619e11 6.96395e10i −0.221536 0.127904i 0.385126 0.922864i \(-0.374158\pi\)
−0.606661 + 0.794961i \(0.707491\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.25312e11 + 3.90251e11i −0.406200 + 0.703560i −0.994460 0.105112i \(-0.966480\pi\)
0.588260 + 0.808672i \(0.299813\pi\)
\(864\) 0 0
\(865\) −3.65720e11 6.33445e11i −0.653257 1.13147i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.28140e10 −0.145219
\(870\) 0 0
\(871\) −4.64626e11 + 2.68252e11i −0.807293 + 0.466091i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.77745e11 5.13294e11i 0.644416 0.875658i
\(876\) 0 0
\(877\) 1.71149e11 2.96439e11i 0.289319 0.501114i −0.684329 0.729174i \(-0.739905\pi\)
0.973647 + 0.228059i \(0.0732380\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.49186e11i 1.07762i 0.842427 + 0.538810i \(0.181126\pi\)
−0.842427 + 0.538810i \(0.818874\pi\)
\(882\) 0 0
\(883\) 1.09905e11 0.180790 0.0903951 0.995906i \(-0.471187\pi\)
0.0903951 + 0.995906i \(0.471187\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.02639e11 1.74729e11i −0.488912 0.282274i 0.235211 0.971944i \(-0.424422\pi\)
−0.724123 + 0.689671i \(0.757755\pi\)
\(888\) 0 0
\(889\) −5.37833e11 3.95804e11i −0.861074 0.633684i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.30863e11 + 3.99867e11i 0.363035 + 0.628795i
\(894\) 0 0
\(895\) 3.79235e11i 0.591040i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.23578e11 3.60023e11i 0.954667 0.551177i
\(900\) 0 0
\(901\) −6.83408e11 3.94566e11i −1.03700 0.598715i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.57361e11 7.92172e11i 0.681812 1.18093i
\(906\) 0 0
\(907\) −5.29895e11 9.17805e11i −0.782998 1.35619i −0.930188 0.367084i \(-0.880356\pi\)
0.147190 0.989108i \(-0.452977\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.66809e11 1.11330 0.556652 0.830746i \(-0.312086\pi\)
0.556652 + 0.830746i \(0.312086\pi\)
\(912\) 0 0
\(913\) 1.14645e11 6.61904e10i 0.164996 0.0952603i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.52353e11 + 8.04597e11i 0.498311 + 1.13789i
\(918\) 0 0
\(919\) 3.25499e11 5.63780e11i 0.456339 0.790402i −0.542425 0.840104i \(-0.682494\pi\)
0.998764 + 0.0497021i \(0.0158272\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.46580e11i 0.477526i
\(924\) 0 0
\(925\) −1.54823e11 −0.211480
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.11519e11 5.26266e11i −1.22378 0.706549i −0.258058 0.966130i \(-0.583082\pi\)
−0.965722 + 0.259580i \(0.916416\pi\)
\(930\) 0 0
\(931\) 2.04918e11 9.09264e11i 0.272760 1.21029i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.97478e10 + 5.15247e10i 0.0389232 + 0.0674169i
\(936\) 0 0
\(937\) 3.22936e11i 0.418946i −0.977814 0.209473i \(-0.932825\pi\)
0.977814 0.209473i \(-0.0671748\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −9.73086e11 + 5.61812e11i −1.24106 + 0.716526i −0.969310 0.245841i \(-0.920936\pi\)
−0.271750 + 0.962368i \(0.587602\pi\)
\(942\) 0 0
\(943\) −1.91619e12 1.10631e12i −2.42321 1.39904i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.59737e11 + 6.23083e11i −0.447286 + 0.774723i −0.998208 0.0598341i \(-0.980943\pi\)
0.550922 + 0.834557i \(0.314276\pi\)
\(948\) 0 0
\(949\) 1.07755e11 + 1.86637e11i 0.132854 + 0.230109i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.43394e11 0.537549 0.268775 0.963203i \(-0.413381\pi\)
0.268775 + 0.963203i \(0.413381\pi\)
\(954\) 0 0
\(955\) 9.83890e10 5.68049e10i 0.118286 0.0682924i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.01630e11 + 9.13216e11i −0.120156 + 1.07969i
\(960\) 0 0
\(961\) −2.80848e11 + 4.86443e11i −0.329289 + 0.570346i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7.37294e11i 0.850220i
\(966\) 0 0
\(967\) 1.62631e12 1.85993 0.929965 0.367647i \(-0.119837\pi\)
0.929965 + 0.367647i \(0.119837\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.05556e12 + 6.09425e11i 1.18742 + 0.685557i 0.957719 0.287704i \(-0.0928921\pi\)
0.229700 + 0.973261i \(0.426225\pi\)
\(972\) 0 0
\(973\) 8.46766e11 1.15062e12i 0.944739 1.28375i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.03817e11 + 1.04584e12i 0.662715 + 1.14786i 0.979899 + 0.199493i \(0.0639294\pi\)
−0.317184 + 0.948364i \(0.602737\pi\)
\(978\) 0 0
\(979\) 1.14047e11i 0.124151i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.14970e11 + 1.81848e11i −0.337330 + 0.194758i −0.659091 0.752063i \(-0.729059\pi\)
0.321760 + 0.946821i \(0.395725\pi\)
\(984\) 0 0
\(985\) −1.90447e11 1.09955e11i −0.202316 0.116807i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.59621e11 7.96087e11i 0.480413 0.832100i
\(990\) 0 0
\(991\) −2.17673e11 3.77021e11i −0.225689 0.390905i 0.730837 0.682552i \(-0.239130\pi\)
−0.956526 + 0.291647i \(0.905797\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.17113e11 −0.731637
\(996\) 0 0
\(997\) −1.88123e11 + 1.08613e11i −0.190397 + 0.109926i −0.592169 0.805814i \(-0.701728\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(998\) 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.e.145.7 yes 20
3.2 odd 2 inner 252.9.z.e.145.4 yes 20
7.3 odd 6 inner 252.9.z.e.73.7 yes 20
21.17 even 6 inner 252.9.z.e.73.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.z.e.73.4 20 21.17 even 6 inner
252.9.z.e.73.7 yes 20 7.3 odd 6 inner
252.9.z.e.145.4 yes 20 3.2 odd 2 inner
252.9.z.e.145.7 yes 20 1.1 even 1 trivial