Properties

Label 144.9.g.i.127.4
Level $144$
Weight $9$
Character 144.127
Analytic conductor $58.663$
Analytic rank $0$
Dimension $4$
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] = [144,9,Mod(127,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, 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("144.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1801})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 451x^{2} + 450x + 202500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.4
Root \(-10.3595 - 17.9433i\) of defining polynomial
Character \(\chi\) \(=\) 144.127
Dual form 144.9.g.i.127.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1084.52 q^{5} +426.979i q^{7} +O(q^{10})\) \(q+1084.52 q^{5} +426.979i q^{7} +14232.3i q^{11} +34213.5 q^{13} -20078.0 q^{17} +196118. i q^{19} +347985. i q^{23} +785551. q^{25} -1.00247e6 q^{29} -1.63986e6i q^{31} +463066. i q^{35} -791050. q^{37} -1.36054e6 q^{41} -1.50116e6i q^{43} +1.49258e6i q^{47} +5.58249e6 q^{49} +8.94860e6 q^{53} +1.54352e7i q^{55} +8.50216e6i q^{59} -1.78549e7 q^{61} +3.71051e7 q^{65} +3.49438e7i q^{67} +3.84799e7i q^{71} +2.11205e7 q^{73} -6.07690e6 q^{77} -3.67300e7i q^{79} -2.94243e7i q^{83} -2.17749e7 q^{85} +3.70040e7 q^{89} +1.46085e7i q^{91} +2.12694e8i q^{95} +1.26423e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 264 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 264 q^{5} + 14632 q^{13} - 332904 q^{17} + 2604428 q^{25} - 2343576 q^{29} + 4315784 q^{37} - 9035496 q^{41} + 3458884 q^{49} + 42186600 q^{53} - 48148408 q^{61} + 125450832 q^{65} - 21215480 q^{73} + 32354496 q^{77} + 235297584 q^{85} + 12675576 q^{89} + 263153800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1084.52 1.73523 0.867613 0.497240i \(-0.165653\pi\)
0.867613 + 0.497240i \(0.165653\pi\)
\(6\) 0 0
\(7\) 426.979i 0.177834i 0.996039 + 0.0889170i \(0.0283406\pi\)
−0.996039 + 0.0889170i \(0.971659\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14232.3i 0.972087i 0.873935 + 0.486043i \(0.161560\pi\)
−0.873935 + 0.486043i \(0.838440\pi\)
\(12\) 0 0
\(13\) 34213.5 1.19791 0.598955 0.800783i \(-0.295583\pi\)
0.598955 + 0.800783i \(0.295583\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −20078.0 −0.240394 −0.120197 0.992750i \(-0.538353\pi\)
−0.120197 + 0.992750i \(0.538353\pi\)
\(18\) 0 0
\(19\) 196118.i 1.50489i 0.658657 + 0.752443i \(0.271125\pi\)
−0.658657 + 0.752443i \(0.728875\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 347985.i 1.24351i 0.783212 + 0.621754i \(0.213580\pi\)
−0.783212 + 0.621754i \(0.786420\pi\)
\(24\) 0 0
\(25\) 785551. 2.01101
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00247e6 −1.41735 −0.708677 0.705533i \(-0.750708\pi\)
−0.708677 + 0.705533i \(0.750708\pi\)
\(30\) 0 0
\(31\) − 1.63986e6i − 1.77566i −0.460175 0.887828i \(-0.652213\pi\)
0.460175 0.887828i \(-0.347787\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 463066.i 0.308582i
\(36\) 0 0
\(37\) −791050. −0.422082 −0.211041 0.977477i \(-0.567685\pi\)
−0.211041 + 0.977477i \(0.567685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.36054e6 −0.481478 −0.240739 0.970590i \(-0.577390\pi\)
−0.240739 + 0.970590i \(0.577390\pi\)
\(42\) 0 0
\(43\) − 1.50116e6i − 0.439091i −0.975602 0.219545i \(-0.929543\pi\)
0.975602 0.219545i \(-0.0704574\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.49258e6i 0.305878i 0.988236 + 0.152939i \(0.0488737\pi\)
−0.988236 + 0.152939i \(0.951126\pi\)
\(48\) 0 0
\(49\) 5.58249e6 0.968375
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.94860e6 1.13410 0.567050 0.823683i \(-0.308085\pi\)
0.567050 + 0.823683i \(0.308085\pi\)
\(54\) 0 0
\(55\) 1.54352e7i 1.68679i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.50216e6i 0.701651i 0.936441 + 0.350826i \(0.114099\pi\)
−0.936441 + 0.350826i \(0.885901\pi\)
\(60\) 0 0
\(61\) −1.78549e7 −1.28955 −0.644774 0.764374i \(-0.723048\pi\)
−0.644774 + 0.764374i \(0.723048\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.71051e7 2.07864
\(66\) 0 0
\(67\) 3.49438e7i 1.73409i 0.498232 + 0.867044i \(0.333983\pi\)
−0.498232 + 0.867044i \(0.666017\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.84799e7i 1.51426i 0.653263 + 0.757131i \(0.273400\pi\)
−0.653263 + 0.757131i \(0.726600\pi\)
\(72\) 0 0
\(73\) 2.11205e7 0.743726 0.371863 0.928288i \(-0.378719\pi\)
0.371863 + 0.928288i \(0.378719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.07690e6 −0.172870
\(78\) 0 0
\(79\) − 3.67300e7i − 0.943002i −0.881865 0.471501i \(-0.843712\pi\)
0.881865 0.471501i \(-0.156288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 2.94243e7i − 0.620003i −0.950736 0.310002i \(-0.899670\pi\)
0.950736 0.310002i \(-0.100330\pi\)
\(84\) 0 0
\(85\) −2.17749e7 −0.417139
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.70040e7 0.589778 0.294889 0.955531i \(-0.404717\pi\)
0.294889 + 0.955531i \(0.404717\pi\)
\(90\) 0 0
\(91\) 1.46085e7i 0.213029i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.12694e8i 2.61132i
\(96\) 0 0
\(97\) 1.26423e8 1.42803 0.714017 0.700129i \(-0.246874\pi\)
0.714017 + 0.700129i \(0.246874\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.91121e7 −0.664154 −0.332077 0.943252i \(-0.607749\pi\)
−0.332077 + 0.943252i \(0.607749\pi\)
\(102\) 0 0
\(103\) − 1.20199e7i − 0.106795i −0.998573 0.0533975i \(-0.982995\pi\)
0.998573 0.0533975i \(-0.0170050\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.48941e8i 1.89916i 0.313524 + 0.949580i \(0.398490\pi\)
−0.313524 + 0.949580i \(0.601510\pi\)
\(108\) 0 0
\(109\) 1.61675e8 1.14534 0.572672 0.819784i \(-0.305907\pi\)
0.572672 + 0.819784i \(0.305907\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.05559e7 0.432733 0.216366 0.976312i \(-0.430579\pi\)
0.216366 + 0.976312i \(0.430579\pi\)
\(114\) 0 0
\(115\) 3.77395e8i 2.15777i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 8.57288e6i − 0.0427503i
\(120\) 0 0
\(121\) 1.18000e7 0.0550478
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.28304e8 1.75433
\(126\) 0 0
\(127\) − 9.17381e7i − 0.352643i −0.984333 0.176321i \(-0.943580\pi\)
0.984333 0.176321i \(-0.0564198\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.58852e8i − 0.878953i −0.898254 0.439477i \(-0.855164\pi\)
0.898254 0.439477i \(-0.144836\pi\)
\(132\) 0 0
\(133\) −8.37384e7 −0.267620
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.73710e8 −1.06085 −0.530423 0.847733i \(-0.677967\pi\)
−0.530423 + 0.847733i \(0.677967\pi\)
\(138\) 0 0
\(139\) 1.61105e7i 0.0431569i 0.999767 + 0.0215785i \(0.00686917\pi\)
−0.999767 + 0.0215785i \(0.993131\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.86937e8i 1.16447i
\(144\) 0 0
\(145\) −1.08719e9 −2.45943
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.78564e8 1.17383 0.586917 0.809647i \(-0.300342\pi\)
0.586917 + 0.809647i \(0.300342\pi\)
\(150\) 0 0
\(151\) − 6.01470e8i − 1.15693i −0.815708 0.578463i \(-0.803653\pi\)
0.815708 0.578463i \(-0.196347\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.77845e9i − 3.08117i
\(156\) 0 0
\(157\) 1.28860e8 0.212090 0.106045 0.994361i \(-0.466181\pi\)
0.106045 + 0.994361i \(0.466181\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.48582e8 −0.221138
\(162\) 0 0
\(163\) 2.84647e7i 0.0403233i 0.999797 + 0.0201617i \(0.00641809\pi\)
−0.999797 + 0.0201617i \(0.993582\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 178227.i 0 0.000229144i 1.00000 0.000114572i \(3.64693e-5\pi\)
−1.00000 0.000114572i \(0.999964\pi\)
\(168\) 0 0
\(169\) 3.54833e8 0.434987
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.60561e9 −1.79248 −0.896240 0.443569i \(-0.853712\pi\)
−0.896240 + 0.443569i \(0.853712\pi\)
\(174\) 0 0
\(175\) 3.35414e8i 0.357626i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.44579e9i 1.40829i 0.710054 + 0.704147i \(0.248671\pi\)
−0.710054 + 0.704147i \(0.751329\pi\)
\(180\) 0 0
\(181\) 1.78411e9 1.66229 0.831146 0.556054i \(-0.187685\pi\)
0.831146 + 0.556054i \(0.187685\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.57907e8 −0.732409
\(186\) 0 0
\(187\) − 2.85756e8i − 0.233684i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 2.30763e9i − 1.73394i −0.498364 0.866968i \(-0.666066\pi\)
0.498364 0.866968i \(-0.333934\pi\)
\(192\) 0 0
\(193\) −5.47360e8 −0.394497 −0.197248 0.980354i \(-0.563201\pi\)
−0.197248 + 0.980354i \(0.563201\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.43899e8 0.228332 0.114166 0.993462i \(-0.463580\pi\)
0.114166 + 0.993462i \(0.463580\pi\)
\(198\) 0 0
\(199\) − 1.12814e9i − 0.719369i −0.933074 0.359684i \(-0.882884\pi\)
0.933074 0.359684i \(-0.117116\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 4.28033e8i − 0.252054i
\(204\) 0 0
\(205\) −1.47553e9 −0.835474
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.79122e9 −1.46288
\(210\) 0 0
\(211\) − 3.25742e9i − 1.64340i −0.569919 0.821701i \(-0.693025\pi\)
0.569919 0.821701i \(-0.306975\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 1.62804e9i − 0.761922i
\(216\) 0 0
\(217\) 7.00184e8 0.315772
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.86938e8 −0.287971
\(222\) 0 0
\(223\) − 7.96939e8i − 0.322259i −0.986933 0.161130i \(-0.948486\pi\)
0.986933 0.161130i \(-0.0515137\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.32327e9i − 0.874977i −0.899224 0.437488i \(-0.855868\pi\)
0.899224 0.437488i \(-0.144132\pi\)
\(228\) 0 0
\(229\) −2.17161e9 −0.789659 −0.394830 0.918754i \(-0.629196\pi\)
−0.394830 + 0.918754i \(0.629196\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.99024e9 0.675275 0.337638 0.941276i \(-0.390372\pi\)
0.337638 + 0.941276i \(0.390372\pi\)
\(234\) 0 0
\(235\) 1.61873e9i 0.530767i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 1.47132e9i − 0.450937i −0.974250 0.225469i \(-0.927609\pi\)
0.974250 0.225469i \(-0.0723914\pi\)
\(240\) 0 0
\(241\) −2.94516e9 −0.873052 −0.436526 0.899692i \(-0.643791\pi\)
−0.436526 + 0.899692i \(0.643791\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.05430e9 1.68035
\(246\) 0 0
\(247\) 6.70989e9i 1.80272i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.41458e9i 0.608341i 0.952618 + 0.304170i \(0.0983792\pi\)
−0.952618 + 0.304170i \(0.901621\pi\)
\(252\) 0 0
\(253\) −4.95263e9 −1.20880
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.48578e8 −0.125749 −0.0628747 0.998021i \(-0.520027\pi\)
−0.0628747 + 0.998021i \(0.520027\pi\)
\(258\) 0 0
\(259\) − 3.37762e8i − 0.0750606i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.76986e9i 0.578942i 0.957187 + 0.289471i \(0.0934794\pi\)
−0.957187 + 0.289471i \(0.906521\pi\)
\(264\) 0 0
\(265\) 9.70490e9 1.96792
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.47401e9 0.472490 0.236245 0.971694i \(-0.424083\pi\)
0.236245 + 0.971694i \(0.424083\pi\)
\(270\) 0 0
\(271\) 2.51223e9i 0.465781i 0.972503 + 0.232890i \(0.0748183\pi\)
−0.972503 + 0.232890i \(0.925182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.11802e10i 1.95488i
\(276\) 0 0
\(277\) 1.06906e10 1.81585 0.907927 0.419127i \(-0.137664\pi\)
0.907927 + 0.419127i \(0.137664\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.60700e9 0.418134 0.209067 0.977901i \(-0.432957\pi\)
0.209067 + 0.977901i \(0.432957\pi\)
\(282\) 0 0
\(283\) − 1.75548e8i − 0.0273684i −0.999906 0.0136842i \(-0.995644\pi\)
0.999906 0.0136842i \(-0.00435595\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 5.80923e8i − 0.0856232i
\(288\) 0 0
\(289\) −6.57263e9 −0.942211
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.83363e9 0.248795 0.124397 0.992232i \(-0.460300\pi\)
0.124397 + 0.992232i \(0.460300\pi\)
\(294\) 0 0
\(295\) 9.22073e9i 1.21752i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.19058e10i 1.48961i
\(300\) 0 0
\(301\) 6.40966e8 0.0780852
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.93639e10 −2.23766
\(306\) 0 0
\(307\) − 1.39249e10i − 1.56761i −0.621007 0.783805i \(-0.713276\pi\)
0.621007 0.783805i \(-0.286724\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 3.88787e9i − 0.415595i −0.978172 0.207798i \(-0.933370\pi\)
0.978172 0.207798i \(-0.0666295\pi\)
\(312\) 0 0
\(313\) −3.59882e9 −0.374958 −0.187479 0.982269i \(-0.560032\pi\)
−0.187479 + 0.982269i \(0.560032\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.64268e9 0.855878 0.427939 0.903808i \(-0.359240\pi\)
0.427939 + 0.903808i \(0.359240\pi\)
\(318\) 0 0
\(319\) − 1.42674e10i − 1.37779i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 3.93766e9i − 0.361766i
\(324\) 0 0
\(325\) 2.68765e10 2.40901
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.37303e8 −0.0543954
\(330\) 0 0
\(331\) 9.59670e9i 0.799484i 0.916628 + 0.399742i \(0.130900\pi\)
−0.916628 + 0.399742i \(0.869100\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.78971e10i 3.00904i
\(336\) 0 0
\(337\) −1.75447e10 −1.36027 −0.680137 0.733085i \(-0.738080\pi\)
−0.680137 + 0.733085i \(0.738080\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.33390e10 1.72609
\(342\) 0 0
\(343\) 4.84506e9i 0.350044i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.71291e10i − 1.18146i −0.806871 0.590728i \(-0.798841\pi\)
0.806871 0.590728i \(-0.201159\pi\)
\(348\) 0 0
\(349\) 1.99032e9 0.134160 0.0670799 0.997748i \(-0.478632\pi\)
0.0670799 + 0.997748i \(0.478632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.53774e8 −0.0163436 −0.00817181 0.999967i \(-0.502601\pi\)
−0.00817181 + 0.999967i \(0.502601\pi\)
\(354\) 0 0
\(355\) 4.17321e10i 2.62759i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 2.26677e10i − 1.36468i −0.731036 0.682339i \(-0.760963\pi\)
0.731036 0.682339i \(-0.239037\pi\)
\(360\) 0 0
\(361\) −2.14788e10 −1.26468
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.29056e10 1.29053
\(366\) 0 0
\(367\) − 7.30278e9i − 0.402554i −0.979534 0.201277i \(-0.935491\pi\)
0.979534 0.201277i \(-0.0645091\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.82087e9i 0.201682i
\(372\) 0 0
\(373\) 1.11279e10 0.574879 0.287439 0.957799i \(-0.407196\pi\)
0.287439 + 0.957799i \(0.407196\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.42979e10 −1.69786
\(378\) 0 0
\(379\) 5.68377e9i 0.275473i 0.990469 + 0.137737i \(0.0439827\pi\)
−0.990469 + 0.137737i \(0.956017\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 1.92155e9i − 0.0893009i −0.999003 0.0446504i \(-0.985783\pi\)
0.999003 0.0446504i \(-0.0142174\pi\)
\(384\) 0 0
\(385\) −6.59050e9 −0.299969
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.05878e10 0.462389 0.231194 0.972908i \(-0.425737\pi\)
0.231194 + 0.972908i \(0.425737\pi\)
\(390\) 0 0
\(391\) − 6.98683e9i − 0.298932i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 3.98343e10i − 1.63632i
\(396\) 0 0
\(397\) −2.38291e10 −0.959282 −0.479641 0.877465i \(-0.659233\pi\)
−0.479641 + 0.877465i \(0.659233\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.59390e10 −0.616428 −0.308214 0.951317i \(-0.599731\pi\)
−0.308214 + 0.951317i \(0.599731\pi\)
\(402\) 0 0
\(403\) − 5.61052e10i − 2.12708i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.12585e10i − 0.410301i
\(408\) 0 0
\(409\) 8.96615e9 0.320415 0.160208 0.987083i \(-0.448784\pi\)
0.160208 + 0.987083i \(0.448784\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.63025e9 −0.124777
\(414\) 0 0
\(415\) − 3.19112e10i − 1.07585i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 1.06743e10i − 0.346326i −0.984893 0.173163i \(-0.944601\pi\)
0.984893 0.173163i \(-0.0553987\pi\)
\(420\) 0 0
\(421\) −2.60816e10 −0.830245 −0.415123 0.909765i \(-0.636261\pi\)
−0.415123 + 0.909765i \(0.636261\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.57723e10 −0.483436
\(426\) 0 0
\(427\) − 7.62366e9i − 0.229325i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 1.20312e9i − 0.0348658i −0.999848 0.0174329i \(-0.994451\pi\)
0.999848 0.0174329i \(-0.00554935\pi\)
\(432\) 0 0
\(433\) 1.13142e10 0.321865 0.160932 0.986965i \(-0.448550\pi\)
0.160932 + 0.986965i \(0.448550\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.82462e10 −1.87134
\(438\) 0 0
\(439\) − 1.76972e10i − 0.476482i −0.971206 0.238241i \(-0.923429\pi\)
0.971206 0.238241i \(-0.0765708\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 6.47192e10i − 1.68042i −0.542260 0.840211i \(-0.682431\pi\)
0.542260 0.840211i \(-0.317569\pi\)
\(444\) 0 0
\(445\) 4.01315e10 1.02340
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.07525e10 −1.49478 −0.747392 0.664383i \(-0.768694\pi\)
−0.747392 + 0.664383i \(0.768694\pi\)
\(450\) 0 0
\(451\) − 1.93637e10i − 0.468038i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.58431e10i 0.369653i
\(456\) 0 0
\(457\) 4.77740e10 1.09528 0.547642 0.836713i \(-0.315526\pi\)
0.547642 + 0.836713i \(0.315526\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.17546e9 0.158872 0.0794358 0.996840i \(-0.474688\pi\)
0.0794358 + 0.996840i \(0.474688\pi\)
\(462\) 0 0
\(463\) 5.86354e9i 0.127596i 0.997963 + 0.0637978i \(0.0203213\pi\)
−0.997963 + 0.0637978i \(0.979679\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.53384e10i 1.58398i 0.610537 + 0.791988i \(0.290954\pi\)
−0.610537 + 0.791988i \(0.709046\pi\)
\(468\) 0 0
\(469\) −1.49203e10 −0.308380
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.13650e10 0.426834
\(474\) 0 0
\(475\) 1.54061e11i 3.02634i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 1.97647e10i − 0.375446i −0.982222 0.187723i \(-0.939889\pi\)
0.982222 0.187723i \(-0.0601107\pi\)
\(480\) 0 0
\(481\) −2.70646e10 −0.505617
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.37108e11 2.47796
\(486\) 0 0
\(487\) − 5.47310e10i − 0.973011i −0.873677 0.486506i \(-0.838271\pi\)
0.873677 0.486506i \(-0.161729\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.44118e10i − 0.420024i −0.977699 0.210012i \(-0.932650\pi\)
0.977699 0.210012i \(-0.0673502\pi\)
\(492\) 0 0
\(493\) 2.01275e10 0.340724
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.64301e10 −0.269287
\(498\) 0 0
\(499\) − 3.29856e10i − 0.532014i −0.963971 0.266007i \(-0.914296\pi\)
0.963971 0.266007i \(-0.0857044\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 7.46160e9i − 0.116563i −0.998300 0.0582814i \(-0.981438\pi\)
0.998300 0.0582814i \(-0.0185621\pi\)
\(504\) 0 0
\(505\) −7.49533e10 −1.15246
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.46374e10 −0.367049 −0.183524 0.983015i \(-0.558751\pi\)
−0.183524 + 0.983015i \(0.558751\pi\)
\(510\) 0 0
\(511\) 9.01802e9i 0.132260i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 1.30357e10i − 0.185313i
\(516\) 0 0
\(517\) −2.12429e10 −0.297339
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.99951e10 0.407099 0.203549 0.979065i \(-0.434752\pi\)
0.203549 + 0.979065i \(0.434752\pi\)
\(522\) 0 0
\(523\) − 1.04737e11i − 1.39988i −0.714199 0.699942i \(-0.753209\pi\)
0.714199 0.699942i \(-0.246791\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.29250e10i 0.426858i
\(528\) 0 0
\(529\) −4.27824e10 −0.546314
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.65489e10 −0.576767
\(534\) 0 0
\(535\) 2.69981e11i 3.29547i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.94518e10i 0.941344i
\(540\) 0 0
\(541\) −5.69348e10 −0.664644 −0.332322 0.943166i \(-0.607832\pi\)
−0.332322 + 0.943166i \(0.607832\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.75339e11 1.98743
\(546\) 0 0
\(547\) − 1.13814e11i − 1.27130i −0.771979 0.635648i \(-0.780733\pi\)
0.771979 0.635648i \(-0.219267\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1.96602e11i − 2.13296i
\(552\) 0 0
\(553\) 1.56830e10 0.167698
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.32489e11 1.37645 0.688225 0.725497i \(-0.258390\pi\)
0.688225 + 0.725497i \(0.258390\pi\)
\(558\) 0 0
\(559\) − 5.13600e10i − 0.525991i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.68664e11i 1.67876i 0.543545 + 0.839380i \(0.317082\pi\)
−0.543545 + 0.839380i \(0.682918\pi\)
\(564\) 0 0
\(565\) 7.65191e10 0.750889
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.36419e11 −1.30145 −0.650725 0.759314i \(-0.725535\pi\)
−0.650725 + 0.759314i \(0.725535\pi\)
\(570\) 0 0
\(571\) 5.62677e10i 0.529315i 0.964342 + 0.264658i \(0.0852590\pi\)
−0.964342 + 0.264658i \(0.914741\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.73360e11i 2.50071i
\(576\) 0 0
\(577\) 1.35733e11 1.22457 0.612285 0.790637i \(-0.290250\pi\)
0.612285 + 0.790637i \(0.290250\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.25636e10 0.110258
\(582\) 0 0
\(583\) 1.27359e11i 1.10244i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.54607e11i 1.30220i 0.758993 + 0.651098i \(0.225691\pi\)
−0.758993 + 0.651098i \(0.774309\pi\)
\(588\) 0 0
\(589\) 3.21606e11 2.67216
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.44789e11 −1.17089 −0.585447 0.810711i \(-0.699081\pi\)
−0.585447 + 0.810711i \(0.699081\pi\)
\(594\) 0 0
\(595\) − 9.29743e9i − 0.0741814i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.20988e10i 0.171657i 0.996310 + 0.0858284i \(0.0273537\pi\)
−0.996310 + 0.0858284i \(0.972646\pi\)
\(600\) 0 0
\(601\) −1.29762e11 −0.994602 −0.497301 0.867578i \(-0.665676\pi\)
−0.497301 + 0.867578i \(0.665676\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.27973e10 0.0955204
\(606\) 0 0
\(607\) − 1.15676e10i − 0.0852099i −0.999092 0.0426050i \(-0.986434\pi\)
0.999092 0.0426050i \(-0.0135657\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.10665e10i 0.366414i
\(612\) 0 0
\(613\) −1.57261e11 −1.11373 −0.556865 0.830603i \(-0.687996\pi\)
−0.556865 + 0.830603i \(0.687996\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.94877e10 0.272471 0.136236 0.990676i \(-0.456500\pi\)
0.136236 + 0.990676i \(0.456500\pi\)
\(618\) 0 0
\(619\) 1.02727e11i 0.699718i 0.936802 + 0.349859i \(0.113770\pi\)
−0.936802 + 0.349859i \(0.886230\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.57999e10i 0.104883i
\(624\) 0 0
\(625\) 1.57647e11 1.03315
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.58827e10 0.101466
\(630\) 0 0
\(631\) − 2.04152e10i − 0.128777i −0.997925 0.0643883i \(-0.979490\pi\)
0.997925 0.0643883i \(-0.0205096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 9.94915e10i − 0.611915i
\(636\) 0 0
\(637\) 1.90996e11 1.16003
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.17056e11 1.28570 0.642851 0.765992i \(-0.277752\pi\)
0.642851 + 0.765992i \(0.277752\pi\)
\(642\) 0 0
\(643\) 1.02152e11i 0.597590i 0.954317 + 0.298795i \(0.0965848\pi\)
−0.954317 + 0.298795i \(0.903415\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 9.09290e10i − 0.518902i −0.965756 0.259451i \(-0.916458\pi\)
0.965756 0.259451i \(-0.0835416\pi\)
\(648\) 0 0
\(649\) −1.21005e11 −0.682065
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.71409e11 −0.942717 −0.471359 0.881942i \(-0.656236\pi\)
−0.471359 + 0.881942i \(0.656236\pi\)
\(654\) 0 0
\(655\) − 2.80729e11i − 1.52518i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.27365e9i 0.0120554i 0.999982 + 0.00602769i \(0.00191869\pi\)
−0.999982 + 0.00602769i \(0.998081\pi\)
\(660\) 0 0
\(661\) −2.92200e11 −1.53065 −0.765323 0.643647i \(-0.777421\pi\)
−0.765323 + 0.643647i \(0.777421\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9.08157e10 −0.464381
\(666\) 0 0
\(667\) − 3.48843e11i − 1.76249i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 2.54116e11i − 1.25355i
\(672\) 0 0
\(673\) −1.51294e10 −0.0737501 −0.0368751 0.999320i \(-0.511740\pi\)
−0.0368751 + 0.999320i \(0.511740\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.80775e11 −1.81265 −0.906324 0.422584i \(-0.861123\pi\)
−0.906324 + 0.422584i \(0.861123\pi\)
\(678\) 0 0
\(679\) 5.39799e10i 0.253953i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 4.18285e11i − 1.92216i −0.276269 0.961080i \(-0.589098\pi\)
0.276269 0.961080i \(-0.410902\pi\)
\(684\) 0 0
\(685\) −4.05295e11 −1.84081
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.06163e11 1.35855
\(690\) 0 0
\(691\) 8.10077e10i 0.355315i 0.984092 + 0.177658i \(0.0568520\pi\)
−0.984092 + 0.177658i \(0.943148\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.74721e10i 0.0748870i
\(696\) 0 0
\(697\) 2.73169e10 0.115745
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.22597e11 1.33594 0.667971 0.744187i \(-0.267163\pi\)
0.667971 + 0.744187i \(0.267163\pi\)
\(702\) 0 0
\(703\) − 1.55139e11i − 0.635186i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.95094e10i − 0.118109i
\(708\) 0 0
\(709\) −2.99903e11 −1.18685 −0.593424 0.804890i \(-0.702224\pi\)
−0.593424 + 0.804890i \(0.702224\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.70645e11 2.20804
\(714\) 0 0
\(715\) 5.28092e11i 2.02062i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1.62338e11i − 0.607442i −0.952761 0.303721i \(-0.901771\pi\)
0.952761 0.303721i \(-0.0982290\pi\)
\(720\) 0 0
\(721\) 5.13223e9 0.0189918
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.87489e11 −2.85031
\(726\) 0 0
\(727\) 2.20900e11i 0.790786i 0.918512 + 0.395393i \(0.129392\pi\)
−0.918512 + 0.395393i \(0.870608\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.01403e10i 0.105555i
\(732\) 0 0
\(733\) 1.15228e11 0.399154 0.199577 0.979882i \(-0.436043\pi\)
0.199577 + 0.979882i \(0.436043\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.97332e11 −1.68568
\(738\) 0 0
\(739\) − 5.64364e11i − 1.89226i −0.323781 0.946132i \(-0.604954\pi\)
0.323781 0.946132i \(-0.395046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.89167e11i 0.620713i 0.950620 + 0.310357i \(0.100448\pi\)
−0.950620 + 0.310357i \(0.899552\pi\)
\(744\) 0 0
\(745\) 6.27462e11 2.03687
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.06293e11 −0.337735
\(750\) 0 0
\(751\) 3.66355e11i 1.15171i 0.817553 + 0.575853i \(0.195330\pi\)
−0.817553 + 0.575853i \(0.804670\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.52304e11i − 2.00753i
\(756\) 0 0
\(757\) 1.30666e11 0.397904 0.198952 0.980009i \(-0.436246\pi\)
0.198952 + 0.980009i \(0.436246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.92845e11 1.17134 0.585670 0.810550i \(-0.300832\pi\)
0.585670 + 0.810550i \(0.300832\pi\)
\(762\) 0 0
\(763\) 6.90318e10i 0.203681i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.90889e11i 0.840515i
\(768\) 0 0
\(769\) 5.36177e11 1.53322 0.766608 0.642116i \(-0.221943\pi\)
0.766608 + 0.642116i \(0.221943\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.88295e11 −1.08754 −0.543769 0.839235i \(-0.683003\pi\)
−0.543769 + 0.839235i \(0.683003\pi\)
\(774\) 0 0
\(775\) − 1.28819e12i − 3.57086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2.66827e11i − 0.724570i
\(780\) 0 0
\(781\) −5.47659e11 −1.47199
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.39751e11 0.368025
\(786\) 0 0
\(787\) 3.56234e11i 0.928616i 0.885674 + 0.464308i \(0.153697\pi\)
−0.885674 + 0.464308i \(0.846303\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.01259e10i 0.0769546i
\(792\) 0 0
\(793\) −6.10877e11 −1.54476
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.81694e11 −1.44165 −0.720827 0.693115i \(-0.756238\pi\)
−0.720827 + 0.693115i \(0.756238\pi\)
\(798\) 0 0
\(799\) − 2.99681e10i − 0.0735312i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.00594e11i 0.722966i
\(804\) 0 0
\(805\) −1.61140e11 −0.383725
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.29788e11 −1.23682 −0.618412 0.785854i \(-0.712224\pi\)
−0.618412 + 0.785854i \(0.712224\pi\)
\(810\) 0 0
\(811\) − 1.27857e11i − 0.295556i −0.989021 0.147778i \(-0.952788\pi\)
0.989021 0.147778i \(-0.0472121\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.08705e10i 0.0699701i
\(816\) 0 0
\(817\) 2.94406e11 0.660781
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.23718e11 1.15272 0.576361 0.817195i \(-0.304472\pi\)
0.576361 + 0.817195i \(0.304472\pi\)
\(822\) 0 0
\(823\) − 4.16147e11i − 0.907083i −0.891235 0.453541i \(-0.850160\pi\)
0.891235 0.453541i \(-0.149840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.32442e11i 0.283141i 0.989928 + 0.141570i \(0.0452152\pi\)
−0.989928 + 0.141570i \(0.954785\pi\)
\(828\) 0 0
\(829\) 1.50571e11 0.318804 0.159402 0.987214i \(-0.449043\pi\)
0.159402 + 0.987214i \(0.449043\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.12085e11 −0.232792
\(834\) 0 0
\(835\) 1.93290e8i 0 0.000397616i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.30040e11i 1.06970i 0.844948 + 0.534848i \(0.179631\pi\)
−0.844948 + 0.534848i \(0.820369\pi\)
\(840\) 0 0
\(841\) 5.04694e11 1.00889
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.84822e11 0.754802
\(846\) 0 0
\(847\) 5.03835e9i 0.00978936i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.75273e11i − 0.524863i
\(852\) 0 0
\(853\) 7.37919e11 1.39384 0.696919 0.717150i \(-0.254554\pi\)
0.696919 + 0.717150i \(0.254554\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.91969e11 0.541270 0.270635 0.962682i \(-0.412766\pi\)
0.270635 + 0.962682i \(0.412766\pi\)
\(858\) 0 0
\(859\) 3.42628e11i 0.629289i 0.949210 + 0.314645i \(0.101885\pi\)
−0.949210 + 0.314645i \(0.898115\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 6.00307e11i − 1.08226i −0.840940 0.541128i \(-0.817997\pi\)
0.840940 0.541128i \(-0.182003\pi\)
\(864\) 0 0
\(865\) −1.74131e12 −3.11036
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.22753e11 0.916680
\(870\) 0 0
\(871\) 1.19555e12i 2.07728i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.82877e11i 0.311980i
\(876\) 0 0
\(877\) 3.40566e11 0.575709 0.287854 0.957674i \(-0.407058\pi\)
0.287854 + 0.957674i \(0.407058\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.36651e11 1.38880 0.694402 0.719587i \(-0.255669\pi\)
0.694402 + 0.719587i \(0.255669\pi\)
\(882\) 0 0
\(883\) 6.58705e11i 1.08355i 0.840524 + 0.541774i \(0.182247\pi\)
−0.840524 + 0.541774i \(0.817753\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.84018e11i 0.297281i 0.988891 + 0.148640i \(0.0474897\pi\)
−0.988891 + 0.148640i \(0.952510\pi\)
\(888\) 0 0
\(889\) 3.91703e10 0.0627119
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.92723e11 −0.460311
\(894\) 0 0
\(895\) 1.56799e12i 2.44371i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.64390e12i 2.51673i
\(900\) 0 0
\(901\) −1.79670e11 −0.272631
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.93490e12 2.88445
\(906\) 0 0
\(907\) − 7.74263e11i − 1.14409i −0.820223 0.572044i \(-0.806151\pi\)
0.820223 0.572044i \(-0.193849\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.57326e11i 0.373602i 0.982398 + 0.186801i \(0.0598120\pi\)
−0.982398 + 0.186801i \(0.940188\pi\)
\(912\) 0 0
\(913\) 4.18776e11 0.602697
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.10524e11 0.156308
\(918\) 0 0
\(919\) − 3.23480e11i − 0.453508i −0.973952 0.226754i \(-0.927189\pi\)
0.973952 0.226754i \(-0.0728114\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.31653e12i 1.81395i
\(924\) 0 0
\(925\) −6.21411e11 −0.848812
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.13798e12 −1.52782 −0.763910 0.645322i \(-0.776723\pi\)
−0.763910 + 0.645322i \(0.776723\pi\)
\(930\) 0 0
\(931\) 1.09483e12i 1.45729i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 3.09907e11i − 0.405495i
\(936\) 0 0
\(937\) 1.34507e12 1.74497 0.872483 0.488645i \(-0.162509\pi\)
0.872483 + 0.488645i \(0.162509\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.54350e10 0.0196856 0.00984279 0.999952i \(-0.496867\pi\)
0.00984279 + 0.999952i \(0.496867\pi\)
\(942\) 0 0
\(943\) − 4.73448e11i − 0.598722i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.10486e12i − 1.37374i −0.726779 0.686872i \(-0.758983\pi\)
0.726779 0.686872i \(-0.241017\pi\)
\(948\) 0 0
\(949\) 7.22607e11 0.890917
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.94710e11 0.478527 0.239264 0.970955i \(-0.423094\pi\)
0.239264 + 0.970955i \(0.423094\pi\)
\(954\) 0 0
\(955\) − 2.50267e12i − 3.00877i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1.59566e11i − 0.188654i
\(960\) 0 0
\(961\) −1.83624e12 −2.15295
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5.93621e11 −0.684541
\(966\) 0 0
\(967\) 6.43683e11i 0.736150i 0.929796 + 0.368075i \(0.119983\pi\)
−0.929796 + 0.368075i \(0.880017\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.15239e11i 1.02957i 0.857318 + 0.514787i \(0.172129\pi\)
−0.857318 + 0.514787i \(0.827871\pi\)
\(972\) 0 0
\(973\) −6.87886e9 −0.00767477
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.39678e12 −1.53303 −0.766515 0.642226i \(-0.778011\pi\)
−0.766515 + 0.642226i \(0.778011\pi\)
\(978\) 0 0
\(979\) 5.26653e11i 0.573316i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 4.88577e11i − 0.523261i −0.965168 0.261631i \(-0.915740\pi\)
0.965168 0.261631i \(-0.0842602\pi\)
\(984\) 0 0
\(985\) 3.72964e11 0.396207
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.22382e11 0.546013
\(990\) 0 0
\(991\) − 7.71826e11i − 0.800248i −0.916461 0.400124i \(-0.868967\pi\)
0.916461 0.400124i \(-0.131033\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.22349e12i − 1.24827i
\(996\) 0 0
\(997\) 1.24478e12 1.25983 0.629916 0.776663i \(-0.283089\pi\)
0.629916 + 0.776663i \(0.283089\pi\)
\(998\) 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 144.9.g.i.127.4 4
3.2 odd 2 48.9.g.c.31.3 yes 4
4.3 odd 2 inner 144.9.g.i.127.3 4
12.11 even 2 48.9.g.c.31.1 4
24.5 odd 2 192.9.g.c.127.2 4
24.11 even 2 192.9.g.c.127.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.c.31.1 4 12.11 even 2
48.9.g.c.31.3 yes 4 3.2 odd 2
144.9.g.i.127.3 4 4.3 odd 2 inner
144.9.g.i.127.4 4 1.1 even 1 trivial
192.9.g.c.127.2 4 24.5 odd 2
192.9.g.c.127.4 4 24.11 even 2