Properties

Label 48.9.g.c.31.3
Level $48$
Weight $9$
Character 48.31
Analytic conductor $19.554$
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] = [48,9,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.31");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(19.5541732829\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.3
Root \(10.8595 - 18.8093i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.9.g.c.31.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+46.7654i q^{3} -1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q+46.7654i q^{3} -1084.52 q^{5} +426.979i q^{7} -2187.00 q^{9} -14232.3i q^{11} +34213.5 q^{13} -50717.8i q^{15} +20078.0 q^{17} +196118. i q^{19} -19967.8 q^{21} -347985. i q^{23} +785551. q^{25} -102276. i q^{27} +1.00247e6 q^{29} -1.63986e6i q^{31} +665580. q^{33} -463066. i q^{35} -791050. q^{37} +1.60001e6i q^{39} +1.36054e6 q^{41} -1.50116e6i q^{43} +2.37184e6 q^{45} -1.49258e6i q^{47} +5.58249e6 q^{49} +938954. i q^{51} -8.94860e6 q^{53} +1.54352e7i q^{55} -9.17155e6 q^{57} -8.50216e6i q^{59} -1.78549e7 q^{61} -933804. i q^{63} -3.71051e7 q^{65} +3.49438e7i q^{67} +1.62736e7 q^{69} -3.84799e7i q^{71} +2.11205e7 q^{73} +3.67366e7i q^{75} +6.07690e6 q^{77} -3.67300e7i q^{79} +4.78297e6 q^{81} +2.94243e7i q^{83} -2.17749e7 q^{85} +4.68808e7i q^{87} -3.70040e7 q^{89} +1.46085e7i q^{91} +7.66885e7 q^{93} -2.12694e8i q^{95} +1.26423e8 q^{97} +3.11261e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 264 q^{5} - 8748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 264 q^{5} - 8748 q^{9} + 14632 q^{13} + 332904 q^{17} + 250128 q^{21} + 2604428 q^{25} + 2343576 q^{29} + 2002320 q^{33} + 4315784 q^{37} + 9035496 q^{41} + 577368 q^{45} + 3458884 q^{49} - 42186600 q^{53} + 2253744 q^{57} - 48148408 q^{61} - 125450832 q^{65} + 23514624 q^{69} - 21215480 q^{73} - 32354496 q^{77} + 19131876 q^{81} + 235297584 q^{85} - 12675576 q^{89} + 193564080 q^{93} + 263153800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 46.7654i 0.577350i
\(4\) 0 0
\(5\) −1084.52 −1.73523 −0.867613 0.497240i \(-0.834347\pi\)
−0.867613 + 0.497240i \(0.834347\pi\)
\(6\) 0 0
\(7\) 426.979i 0.177834i 0.996039 + 0.0889170i \(0.0283406\pi\)
−0.996039 + 0.0889170i \(0.971659\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) − 14232.3i − 0.972087i −0.873935 0.486043i \(-0.838440\pi\)
0.873935 0.486043i \(-0.161560\pi\)
\(12\) 0 0
\(13\) 34213.5 1.19791 0.598955 0.800783i \(-0.295583\pi\)
0.598955 + 0.800783i \(0.295583\pi\)
\(14\) 0 0
\(15\) − 50717.8i − 1.00183i
\(16\) 0 0
\(17\) 20078.0 0.240394 0.120197 0.992750i \(-0.461647\pi\)
0.120197 + 0.992750i \(0.461647\pi\)
\(18\) 0 0
\(19\) 196118.i 1.50489i 0.658657 + 0.752443i \(0.271125\pi\)
−0.658657 + 0.752443i \(0.728875\pi\)
\(20\) 0 0
\(21\) −19967.8 −0.102672
\(22\) 0 0
\(23\) − 347985.i − 1.24351i −0.783212 0.621754i \(-0.786420\pi\)
0.783212 0.621754i \(-0.213580\pi\)
\(24\) 0 0
\(25\) 785551. 2.01101
\(26\) 0 0
\(27\) − 102276.i − 0.192450i
\(28\) 0 0
\(29\) 1.00247e6 1.41735 0.708677 0.705533i \(-0.249292\pi\)
0.708677 + 0.705533i \(0.249292\pi\)
\(30\) 0 0
\(31\) − 1.63986e6i − 1.77566i −0.460175 0.887828i \(-0.652213\pi\)
0.460175 0.887828i \(-0.347787\pi\)
\(32\) 0 0
\(33\) 665580. 0.561234
\(34\) 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
\(39\) 1.60001e6i 0.691613i
\(40\) 0 0
\(41\) 1.36054e6 0.481478 0.240739 0.970590i \(-0.422610\pi\)
0.240739 + 0.970590i \(0.422610\pi\)
\(42\) 0 0
\(43\) − 1.50116e6i − 0.439091i −0.975602 0.219545i \(-0.929543\pi\)
0.975602 0.219545i \(-0.0704574\pi\)
\(44\) 0 0
\(45\) 2.37184e6 0.578409
\(46\) 0 0
\(47\) − 1.49258e6i − 0.305878i −0.988236 0.152939i \(-0.951126\pi\)
0.988236 0.152939i \(-0.0488737\pi\)
\(48\) 0 0
\(49\) 5.58249e6 0.968375
\(50\) 0 0
\(51\) 938954.i 0.138792i
\(52\) 0 0
\(53\) −8.94860e6 −1.13410 −0.567050 0.823683i \(-0.691915\pi\)
−0.567050 + 0.823683i \(0.691915\pi\)
\(54\) 0 0
\(55\) 1.54352e7i 1.68679i
\(56\) 0 0
\(57\) −9.17155e6 −0.868847
\(58\) 0 0
\(59\) − 8.50216e6i − 0.701651i −0.936441 0.350826i \(-0.885901\pi\)
0.936441 0.350826i \(-0.114099\pi\)
\(60\) 0 0
\(61\) −1.78549e7 −1.28955 −0.644774 0.764374i \(-0.723048\pi\)
−0.644774 + 0.764374i \(0.723048\pi\)
\(62\) 0 0
\(63\) − 933804.i − 0.0592780i
\(64\) 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
\(69\) 1.62736e7 0.717940
\(70\) 0 0
\(71\) − 3.84799e7i − 1.51426i −0.653263 0.757131i \(-0.726600\pi\)
0.653263 0.757131i \(-0.273400\pi\)
\(72\) 0 0
\(73\) 2.11205e7 0.743726 0.371863 0.928288i \(-0.378719\pi\)
0.371863 + 0.928288i \(0.378719\pi\)
\(74\) 0 0
\(75\) 3.67366e7i 1.16106i
\(76\) 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
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 2.94243e7i 0.620003i 0.950736 + 0.310002i \(0.100330\pi\)
−0.950736 + 0.310002i \(0.899670\pi\)
\(84\) 0 0
\(85\) −2.17749e7 −0.417139
\(86\) 0 0
\(87\) 4.68808e7i 0.818309i
\(88\) 0 0
\(89\) −3.70040e7 −0.589778 −0.294889 0.955531i \(-0.595283\pi\)
−0.294889 + 0.955531i \(0.595283\pi\)
\(90\) 0 0
\(91\) 1.46085e7i 0.213029i
\(92\) 0 0
\(93\) 7.66885e7 1.02518
\(94\) 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
\(99\) 3.11261e7i 0.324029i
\(100\) 0 0
\(101\) 6.91121e7 0.664154 0.332077 0.943252i \(-0.392251\pi\)
0.332077 + 0.943252i \(0.392251\pi\)
\(102\) 0 0
\(103\) − 1.20199e7i − 0.106795i −0.998573 0.0533975i \(-0.982995\pi\)
0.998573 0.0533975i \(-0.0170050\pi\)
\(104\) 0 0
\(105\) 2.16555e7 0.178160
\(106\) 0 0
\(107\) − 2.48941e8i − 1.89916i −0.313524 0.949580i \(-0.601510\pi\)
0.313524 0.949580i \(-0.398490\pi\)
\(108\) 0 0
\(109\) 1.61675e8 1.14534 0.572672 0.819784i \(-0.305907\pi\)
0.572672 + 0.819784i \(0.305907\pi\)
\(110\) 0 0
\(111\) − 3.69938e7i − 0.243689i
\(112\) 0 0
\(113\) −7.05559e7 −0.432733 −0.216366 0.976312i \(-0.569421\pi\)
−0.216366 + 0.976312i \(0.569421\pi\)
\(114\) 0 0
\(115\) 3.77395e8i 2.15777i
\(116\) 0 0
\(117\) −7.48249e7 −0.399303
\(118\) 0 0
\(119\) 8.57288e6i 0.0427503i
\(120\) 0 0
\(121\) 1.18000e7 0.0550478
\(122\) 0 0
\(123\) 6.36263e7i 0.277982i
\(124\) 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
\(129\) 7.02025e7 0.253509
\(130\) 0 0
\(131\) 2.58852e8i 0.878953i 0.898254 + 0.439477i \(0.144836\pi\)
−0.898254 + 0.439477i \(0.855164\pi\)
\(132\) 0 0
\(133\) −8.37384e7 −0.267620
\(134\) 0 0
\(135\) 1.10920e8i 0.333944i
\(136\) 0 0
\(137\) 3.73710e8 1.06085 0.530423 0.847733i \(-0.322033\pi\)
0.530423 + 0.847733i \(0.322033\pi\)
\(138\) 0 0
\(139\) 1.61105e7i 0.0431569i 0.999767 + 0.0215785i \(0.00686917\pi\)
−0.999767 + 0.0215785i \(0.993131\pi\)
\(140\) 0 0
\(141\) 6.98013e7 0.176598
\(142\) 0 0
\(143\) − 4.86937e8i − 1.16447i
\(144\) 0 0
\(145\) −1.08719e9 −2.45943
\(146\) 0 0
\(147\) 2.61067e8i 0.559092i
\(148\) 0 0
\(149\) −5.78564e8 −1.17383 −0.586917 0.809647i \(-0.699658\pi\)
−0.586917 + 0.809647i \(0.699658\pi\)
\(150\) 0 0
\(151\) − 6.01470e8i − 1.15693i −0.815708 0.578463i \(-0.803653\pi\)
0.815708 0.578463i \(-0.196347\pi\)
\(152\) 0 0
\(153\) −4.39105e7 −0.0801314
\(154\) 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
\(159\) − 4.18484e8i − 0.654773i
\(160\) 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
\(165\) −7.21832e8 −0.973869
\(166\) 0 0
\(167\) − 178227.i 0 0.000229144i −1.00000 0.000114572i \(-0.999964\pi\)
1.00000 0.000114572i \(-3.64693e-5\pi\)
\(168\) 0 0
\(169\) 3.54833e8 0.434987
\(170\) 0 0
\(171\) − 4.28911e8i − 0.501629i
\(172\) 0 0
\(173\) 1.60561e9 1.79248 0.896240 0.443569i \(-0.146288\pi\)
0.896240 + 0.443569i \(0.146288\pi\)
\(174\) 0 0
\(175\) 3.35414e8i 0.357626i
\(176\) 0 0
\(177\) 3.97607e8 0.405098
\(178\) 0 0
\(179\) − 1.44579e9i − 1.40829i −0.710054 0.704147i \(-0.751329\pi\)
0.710054 0.704147i \(-0.248671\pi\)
\(180\) 0 0
\(181\) 1.78411e9 1.66229 0.831146 0.556054i \(-0.187685\pi\)
0.831146 + 0.556054i \(0.187685\pi\)
\(182\) 0 0
\(183\) − 8.34990e8i − 0.744521i
\(184\) 0 0
\(185\) 8.57907e8 0.732409
\(186\) 0 0
\(187\) − 2.85756e8i − 0.233684i
\(188\) 0 0
\(189\) 4.36697e7 0.0342242
\(190\) 0 0
\(191\) 2.30763e9i 1.73394i 0.498364 + 0.866968i \(0.333934\pi\)
−0.498364 + 0.866968i \(0.666066\pi\)
\(192\) 0 0
\(193\) −5.47360e8 −0.394497 −0.197248 0.980354i \(-0.563201\pi\)
−0.197248 + 0.980354i \(0.563201\pi\)
\(194\) 0 0
\(195\) − 1.73523e9i − 1.20011i
\(196\) 0 0
\(197\) −3.43899e8 −0.228332 −0.114166 0.993462i \(-0.536420\pi\)
−0.114166 + 0.993462i \(0.536420\pi\)
\(198\) 0 0
\(199\) − 1.12814e9i − 0.719369i −0.933074 0.359684i \(-0.882884\pi\)
0.933074 0.359684i \(-0.117116\pi\)
\(200\) 0 0
\(201\) −1.63416e9 −1.00118
\(202\) 0 0
\(203\) 4.28033e8i 0.252054i
\(204\) 0 0
\(205\) −1.47553e9 −0.835474
\(206\) 0 0
\(207\) 7.61043e8i 0.414503i
\(208\) 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
\(213\) 1.79953e9 0.874259
\(214\) 0 0
\(215\) 1.62804e9i 0.761922i
\(216\) 0 0
\(217\) 7.00184e8 0.315772
\(218\) 0 0
\(219\) 9.87709e8i 0.429391i
\(220\) 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
\(225\) −1.71800e9 −0.670337
\(226\) 0 0
\(227\) 2.32327e9i 0.874977i 0.899224 + 0.437488i \(0.144132\pi\)
−0.899224 + 0.437488i \(0.855868\pi\)
\(228\) 0 0
\(229\) −2.17161e9 −0.789659 −0.394830 0.918754i \(-0.629196\pi\)
−0.394830 + 0.918754i \(0.629196\pi\)
\(230\) 0 0
\(231\) 2.84189e8i 0.0998065i
\(232\) 0 0
\(233\) −1.99024e9 −0.675275 −0.337638 0.941276i \(-0.609628\pi\)
−0.337638 + 0.941276i \(0.609628\pi\)
\(234\) 0 0
\(235\) 1.61873e9i 0.530767i
\(236\) 0 0
\(237\) 1.71769e9 0.544443
\(238\) 0 0
\(239\) 1.47132e9i 0.450937i 0.974250 + 0.225469i \(0.0723914\pi\)
−0.974250 + 0.225469i \(0.927609\pi\)
\(240\) 0 0
\(241\) −2.94516e9 −0.873052 −0.436526 0.899692i \(-0.643791\pi\)
−0.436526 + 0.899692i \(0.643791\pi\)
\(242\) 0 0
\(243\) 2.23677e8i 0.0641500i
\(244\) 0 0
\(245\) −6.05430e9 −1.68035
\(246\) 0 0
\(247\) 6.70989e9i 1.80272i
\(248\) 0 0
\(249\) −1.37604e9 −0.357959
\(250\) 0 0
\(251\) − 2.41458e9i − 0.608341i −0.952618 0.304170i \(-0.901621\pi\)
0.952618 0.304170i \(-0.0983792\pi\)
\(252\) 0 0
\(253\) −4.95263e9 −1.20880
\(254\) 0 0
\(255\) − 1.01831e9i − 0.240835i
\(256\) 0 0
\(257\) 5.48578e8 0.125749 0.0628747 0.998021i \(-0.479973\pi\)
0.0628747 + 0.998021i \(0.479973\pi\)
\(258\) 0 0
\(259\) − 3.37762e8i − 0.0750606i
\(260\) 0 0
\(261\) −2.19240e9 −0.472451
\(262\) 0 0
\(263\) − 2.76986e9i − 0.578942i −0.957187 0.289471i \(-0.906521\pi\)
0.957187 0.289471i \(-0.0934794\pi\)
\(264\) 0 0
\(265\) 9.70490e9 1.96792
\(266\) 0 0
\(267\) − 1.73051e9i − 0.340509i
\(268\) 0 0
\(269\) −2.47401e9 −0.472490 −0.236245 0.971694i \(-0.575917\pi\)
−0.236245 + 0.971694i \(0.575917\pi\)
\(270\) 0 0
\(271\) 2.51223e9i 0.465781i 0.972503 + 0.232890i \(0.0748183\pi\)
−0.972503 + 0.232890i \(0.925182\pi\)
\(272\) 0 0
\(273\) −6.83170e8 −0.122992
\(274\) 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
\(279\) 3.58636e9i 0.591885i
\(280\) 0 0
\(281\) −2.60700e9 −0.418134 −0.209067 0.977901i \(-0.567043\pi\)
−0.209067 + 0.977901i \(0.567043\pi\)
\(282\) 0 0
\(283\) − 1.75548e8i − 0.0273684i −0.999906 0.0136842i \(-0.995644\pi\)
0.999906 0.0136842i \(-0.00435595\pi\)
\(284\) 0 0
\(285\) 9.94669e9 1.50765
\(286\) 0 0
\(287\) 5.80923e8i 0.0856232i
\(288\) 0 0
\(289\) −6.57263e9 −0.942211
\(290\) 0 0
\(291\) 5.91221e9i 0.824476i
\(292\) 0 0
\(293\) −1.83363e9 −0.248795 −0.124397 0.992232i \(-0.539700\pi\)
−0.124397 + 0.992232i \(0.539700\pi\)
\(294\) 0 0
\(295\) 9.22073e9i 1.21752i
\(296\) 0 0
\(297\) −1.45562e9 −0.187078
\(298\) 0 0
\(299\) − 1.19058e10i − 1.48961i
\(300\) 0 0
\(301\) 6.40966e8 0.0780852
\(302\) 0 0
\(303\) 3.23205e9i 0.383450i
\(304\) 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
\(309\) 5.62113e8 0.0616581
\(310\) 0 0
\(311\) 3.88787e9i 0.415595i 0.978172 + 0.207798i \(0.0666295\pi\)
−0.978172 + 0.207798i \(0.933370\pi\)
\(312\) 0 0
\(313\) −3.59882e9 −0.374958 −0.187479 0.982269i \(-0.560032\pi\)
−0.187479 + 0.982269i \(0.560032\pi\)
\(314\) 0 0
\(315\) 1.01273e9i 0.102861i
\(316\) 0 0
\(317\) −8.64268e9 −0.855878 −0.427939 0.903808i \(-0.640760\pi\)
−0.427939 + 0.903808i \(0.640760\pi\)
\(318\) 0 0
\(319\) − 1.42674e10i − 1.37779i
\(320\) 0 0
\(321\) 1.16418e10 1.09648
\(322\) 0 0
\(323\) 3.93766e9i 0.361766i
\(324\) 0 0
\(325\) 2.68765e10 2.40901
\(326\) 0 0
\(327\) 7.56078e9i 0.661265i
\(328\) 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
\(333\) 1.73003e9 0.140694
\(334\) 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
\(339\) − 3.29957e9i − 0.249838i
\(340\) 0 0
\(341\) −2.33390e10 −1.72609
\(342\) 0 0
\(343\) 4.84506e9i 0.350044i
\(344\) 0 0
\(345\) −1.76490e10 −1.24579
\(346\) 0 0
\(347\) 1.71291e10i 1.18146i 0.806871 + 0.590728i \(0.201159\pi\)
−0.806871 + 0.590728i \(0.798841\pi\)
\(348\) 0 0
\(349\) 1.99032e9 0.134160 0.0670799 0.997748i \(-0.478632\pi\)
0.0670799 + 0.997748i \(0.478632\pi\)
\(350\) 0 0
\(351\) − 3.49922e9i − 0.230538i
\(352\) 0 0
\(353\) 2.53774e8 0.0163436 0.00817181 0.999967i \(-0.497399\pi\)
0.00817181 + 0.999967i \(0.497399\pi\)
\(354\) 0 0
\(355\) 4.17321e10i 2.62759i
\(356\) 0 0
\(357\) −4.00914e8 −0.0246819
\(358\) 0 0
\(359\) 2.26677e10i 1.36468i 0.731036 + 0.682339i \(0.239037\pi\)
−0.731036 + 0.682339i \(0.760963\pi\)
\(360\) 0 0
\(361\) −2.14788e10 −1.26468
\(362\) 0 0
\(363\) 5.51830e8i 0.0317818i
\(364\) 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
\(369\) −2.97551e9 −0.160493
\(370\) 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
\(375\) − 2.00298e10i − 1.01286i
\(376\) 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
\(381\) 4.29017e9 0.203598
\(382\) 0 0
\(383\) 1.92155e9i 0.0893009i 0.999003 + 0.0446504i \(0.0142174\pi\)
−0.999003 + 0.0446504i \(0.985783\pi\)
\(384\) 0 0
\(385\) −6.59050e9 −0.299969
\(386\) 0 0
\(387\) 3.28304e9i 0.146364i
\(388\) 0 0
\(389\) −1.05878e10 −0.462389 −0.231194 0.972908i \(-0.574263\pi\)
−0.231194 + 0.972908i \(0.574263\pi\)
\(390\) 0 0
\(391\) − 6.98683e9i − 0.298932i
\(392\) 0 0
\(393\) −1.21053e10 −0.507464
\(394\) 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
\(399\) − 3.91606e9i − 0.154510i
\(400\) 0 0
\(401\) 1.59390e10 0.616428 0.308214 0.951317i \(-0.400269\pi\)
0.308214 + 0.951317i \(0.400269\pi\)
\(402\) 0 0
\(403\) − 5.61052e10i − 2.12708i
\(404\) 0 0
\(405\) −5.18721e9 −0.192803
\(406\) 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
\(411\) 1.74767e10i 0.612480i
\(412\) 0 0
\(413\) 3.63025e9 0.124777
\(414\) 0 0
\(415\) − 3.19112e10i − 1.07585i
\(416\) 0 0
\(417\) −7.53415e8 −0.0249167
\(418\) 0 0
\(419\) 1.06743e10i 0.346326i 0.984893 + 0.173163i \(0.0553987\pi\)
−0.984893 + 0.173163i \(0.944601\pi\)
\(420\) 0 0
\(421\) −2.60816e10 −0.830245 −0.415123 0.909765i \(-0.636261\pi\)
−0.415123 + 0.909765i \(0.636261\pi\)
\(422\) 0 0
\(423\) 3.26428e9i 0.101959i
\(424\) 0 0
\(425\) 1.57723e10 0.483436
\(426\) 0 0
\(427\) − 7.62366e9i − 0.229325i
\(428\) 0 0
\(429\) 2.27718e10 0.672308
\(430\) 0 0
\(431\) 1.20312e9i 0.0348658i 0.999848 + 0.0174329i \(0.00554935\pi\)
−0.999848 + 0.0174329i \(0.994451\pi\)
\(432\) 0 0
\(433\) 1.13142e10 0.321865 0.160932 0.986965i \(-0.448550\pi\)
0.160932 + 0.986965i \(0.448550\pi\)
\(434\) 0 0
\(435\) − 5.08430e10i − 1.41995i
\(436\) 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
\(441\) −1.22089e10 −0.322792
\(442\) 0 0
\(443\) 6.47192e10i 1.68042i 0.542260 + 0.840211i \(0.317569\pi\)
−0.542260 + 0.840211i \(0.682431\pi\)
\(444\) 0 0
\(445\) 4.01315e10 1.02340
\(446\) 0 0
\(447\) − 2.70568e10i − 0.677713i
\(448\) 0 0
\(449\) 6.07525e10 1.49478 0.747392 0.664383i \(-0.231306\pi\)
0.747392 + 0.664383i \(0.231306\pi\)
\(450\) 0 0
\(451\) − 1.93637e10i − 0.468038i
\(452\) 0 0
\(453\) 2.81279e10 0.667952
\(454\) 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
\(459\) − 2.05349e9i − 0.0462639i
\(460\) 0 0
\(461\) −7.17546e9 −0.158872 −0.0794358 0.996840i \(-0.525312\pi\)
−0.0794358 + 0.996840i \(0.525312\pi\)
\(462\) 0 0
\(463\) 5.86354e9i 0.127596i 0.997963 + 0.0637978i \(0.0203213\pi\)
−0.997963 + 0.0637978i \(0.979679\pi\)
\(464\) 0 0
\(465\) −8.31699e10 −1.77891
\(466\) 0 0
\(467\) − 7.53384e10i − 1.58398i −0.610537 0.791988i \(-0.709046\pi\)
0.610537 0.791988i \(-0.290954\pi\)
\(468\) 0 0
\(469\) −1.49203e10 −0.308380
\(470\) 0 0
\(471\) 6.02620e9i 0.122450i
\(472\) 0 0
\(473\) −2.13650e10 −0.426834
\(474\) 0 0
\(475\) 1.54061e11i 3.02634i
\(476\) 0 0
\(477\) 1.95706e10 0.378033
\(478\) 0 0
\(479\) 1.97647e10i 0.375446i 0.982222 + 0.187723i \(0.0601107\pi\)
−0.982222 + 0.187723i \(0.939889\pi\)
\(480\) 0 0
\(481\) −2.70646e10 −0.505617
\(482\) 0 0
\(483\) 6.94850e9i 0.127674i
\(484\) 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
\(489\) −1.33116e9 −0.0232807
\(490\) 0 0
\(491\) 2.44118e10i 0.420024i 0.977699 + 0.210012i \(0.0673502\pi\)
−0.977699 + 0.210012i \(0.932650\pi\)
\(492\) 0 0
\(493\) 2.01275e10 0.340724
\(494\) 0 0
\(495\) − 3.37568e10i − 0.562263i
\(496\) 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
\(501\) 8.33485e6 0.000132296 0
\(502\) 0 0
\(503\) 7.46160e9i 0.116563i 0.998300 + 0.0582814i \(0.0185621\pi\)
−0.998300 + 0.0582814i \(0.981438\pi\)
\(504\) 0 0
\(505\) −7.49533e10 −1.15246
\(506\) 0 0
\(507\) 1.65939e10i 0.251140i
\(508\) 0 0
\(509\) 2.46374e10 0.367049 0.183524 0.983015i \(-0.441249\pi\)
0.183524 + 0.983015i \(0.441249\pi\)
\(510\) 0 0
\(511\) 9.01802e9i 0.132260i
\(512\) 0 0
\(513\) 2.00582e10 0.289616
\(514\) 0 0
\(515\) 1.30357e10i 0.185313i
\(516\) 0 0
\(517\) −2.12429e10 −0.297339
\(518\) 0 0
\(519\) 7.50867e10i 1.03489i
\(520\) 0 0
\(521\) −2.99951e10 −0.407099 −0.203549 0.979065i \(-0.565248\pi\)
−0.203549 + 0.979065i \(0.565248\pi\)
\(522\) 0 0
\(523\) − 1.04737e11i − 1.39988i −0.714199 0.699942i \(-0.753209\pi\)
0.714199 0.699942i \(-0.246791\pi\)
\(524\) 0 0
\(525\) −1.56858e10 −0.206475
\(526\) 0 0
\(527\) − 3.29250e10i − 0.426858i
\(528\) 0 0
\(529\) −4.27824e10 −0.546314
\(530\) 0 0
\(531\) 1.85942e10i 0.233884i
\(532\) 0 0
\(533\) 4.65489e10 0.576767
\(534\) 0 0
\(535\) 2.69981e11i 3.29547i
\(536\) 0 0
\(537\) 6.76130e10 0.813079
\(538\) 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
\(543\) 8.34346e10i 0.959725i
\(544\) 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
\(549\) 3.90486e10 0.429849
\(550\) 0 0
\(551\) 1.96602e11i 2.13296i
\(552\) 0 0
\(553\) 1.56830e10 0.167698
\(554\) 0 0
\(555\) 4.01204e10i 0.422856i
\(556\) 0 0
\(557\) −1.32489e11 −1.37645 −0.688225 0.725497i \(-0.741610\pi\)
−0.688225 + 0.725497i \(0.741610\pi\)
\(558\) 0 0
\(559\) − 5.13600e10i − 0.525991i
\(560\) 0 0
\(561\) 1.33635e10 0.134918
\(562\) 0 0
\(563\) − 1.68664e11i − 1.67876i −0.543545 0.839380i \(-0.682918\pi\)
0.543545 0.839380i \(-0.317082\pi\)
\(564\) 0 0
\(565\) 7.65191e10 0.750889
\(566\) 0 0
\(567\) 2.04223e9i 0.0197593i
\(568\) 0 0
\(569\) 1.36419e11 1.30145 0.650725 0.759314i \(-0.274465\pi\)
0.650725 + 0.759314i \(0.274465\pi\)
\(570\) 0 0
\(571\) 5.62677e10i 0.529315i 0.964342 + 0.264658i \(0.0852590\pi\)
−0.964342 + 0.264658i \(0.914741\pi\)
\(572\) 0 0
\(573\) −1.07917e11 −1.00109
\(574\) 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
\(579\) − 2.55975e10i − 0.227763i
\(580\) 0 0
\(581\) −1.25636e10 −0.110258
\(582\) 0 0
\(583\) 1.27359e11i 1.10244i
\(584\) 0 0
\(585\) 8.11489e10 0.692882
\(586\) 0 0
\(587\) − 1.54607e11i − 1.30220i −0.758993 0.651098i \(-0.774309\pi\)
0.758993 0.651098i \(-0.225691\pi\)
\(588\) 0 0
\(589\) 3.21606e11 2.67216
\(590\) 0 0
\(591\) − 1.60826e10i − 0.131827i
\(592\) 0 0
\(593\) 1.44789e11 1.17089 0.585447 0.810711i \(-0.300919\pi\)
0.585447 + 0.810711i \(0.300919\pi\)
\(594\) 0 0
\(595\) − 9.29743e9i − 0.0741814i
\(596\) 0 0
\(597\) 5.27580e10 0.415328
\(598\) 0 0
\(599\) − 2.20988e10i − 0.171657i −0.996310 0.0858284i \(-0.972646\pi\)
0.996310 0.0858284i \(-0.0273537\pi\)
\(600\) 0 0
\(601\) −1.29762e11 −0.994602 −0.497301 0.867578i \(-0.665676\pi\)
−0.497301 + 0.867578i \(0.665676\pi\)
\(602\) 0 0
\(603\) − 7.64221e10i − 0.578029i
\(604\) 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
\(609\) −2.00171e10 −0.145523
\(610\) 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
\(615\) − 6.90037e10i − 0.482361i
\(616\) 0 0
\(617\) −3.94877e10 −0.272471 −0.136236 0.990676i \(-0.543500\pi\)
−0.136236 + 0.990676i \(0.543500\pi\)
\(618\) 0 0
\(619\) 1.02727e11i 0.699718i 0.936802 + 0.349859i \(0.113770\pi\)
−0.936802 + 0.349859i \(0.886230\pi\)
\(620\) 0 0
\(621\) −3.55904e10 −0.239313
\(622\) 0 0
\(623\) − 1.57999e10i − 0.104883i
\(624\) 0 0
\(625\) 1.57647e11 1.03315
\(626\) 0 0
\(627\) 1.30532e11i 0.844594i
\(628\) 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
\(633\) 1.52334e11 0.948819
\(634\) 0 0
\(635\) 9.94915e10i 0.611915i
\(636\) 0 0
\(637\) 1.90996e11 1.16003
\(638\) 0 0
\(639\) 8.41556e10i 0.504754i
\(640\) 0 0
\(641\) −2.17056e11 −1.28570 −0.642851 0.765992i \(-0.722248\pi\)
−0.642851 + 0.765992i \(0.722248\pi\)
\(642\) 0 0
\(643\) 1.02152e11i 0.597590i 0.954317 + 0.298795i \(0.0965848\pi\)
−0.954317 + 0.298795i \(0.903415\pi\)
\(644\) 0 0
\(645\) −7.61357e10 −0.439896
\(646\) 0 0
\(647\) 9.09290e10i 0.518902i 0.965756 + 0.259451i \(0.0835416\pi\)
−0.965756 + 0.259451i \(0.916458\pi\)
\(648\) 0 0
\(649\) −1.21005e11 −0.682065
\(650\) 0 0
\(651\) 3.27444e10i 0.182311i
\(652\) 0 0
\(653\) 1.71409e11 0.942717 0.471359 0.881942i \(-0.343764\pi\)
0.471359 + 0.881942i \(0.343764\pi\)
\(654\) 0 0
\(655\) − 2.80729e11i − 1.52518i
\(656\) 0 0
\(657\) −4.61906e10 −0.247909
\(658\) 0 0
\(659\) − 2.27365e9i − 0.0120554i −0.999982 0.00602769i \(-0.998081\pi\)
0.999982 0.00602769i \(-0.00191869\pi\)
\(660\) 0 0
\(661\) −2.92200e11 −1.53065 −0.765323 0.643647i \(-0.777421\pi\)
−0.765323 + 0.643647i \(0.777421\pi\)
\(662\) 0 0
\(663\) 3.21249e10i 0.166260i
\(664\) 0 0
\(665\) 9.08157e10 0.464381
\(666\) 0 0
\(667\) − 3.48843e11i − 1.76249i
\(668\) 0 0
\(669\) 3.72691e10 0.186056
\(670\) 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
\(675\) − 8.03429e10i − 0.387019i
\(676\) 0 0
\(677\) 3.80775e11 1.81265 0.906324 0.422584i \(-0.138877\pi\)
0.906324 + 0.422584i \(0.138877\pi\)
\(678\) 0 0
\(679\) 5.39799e10i 0.253953i
\(680\) 0 0
\(681\) −1.08649e11 −0.505168
\(682\) 0 0
\(683\) 4.18285e11i 1.92216i 0.276269 + 0.961080i \(0.410902\pi\)
−0.276269 + 0.961080i \(0.589098\pi\)
\(684\) 0 0
\(685\) −4.05295e11 −1.84081
\(686\) 0 0
\(687\) − 1.01556e11i − 0.455910i
\(688\) 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
\(693\) −1.32902e10 −0.0576233
\(694\) 0 0
\(695\) − 1.74721e10i − 0.0748870i
\(696\) 0 0
\(697\) 2.73169e10 0.115745
\(698\) 0 0
\(699\) − 9.30741e10i − 0.389870i
\(700\) 0 0
\(701\) −3.22597e11 −1.33594 −0.667971 0.744187i \(-0.732837\pi\)
−0.667971 + 0.744187i \(0.732837\pi\)
\(702\) 0 0
\(703\) − 1.55139e11i − 0.635186i
\(704\) 0 0
\(705\) −7.57006e10 −0.306438
\(706\) 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
\(711\) 8.03286e10i 0.314334i
\(712\) 0 0
\(713\) −5.70645e11 −2.20804
\(714\) 0 0
\(715\) 5.28092e11i 2.02062i
\(716\) 0 0
\(717\) −6.88070e10 −0.260349
\(718\) 0 0
\(719\) 1.62338e11i 0.607442i 0.952761 + 0.303721i \(0.0982290\pi\)
−0.952761 + 0.303721i \(0.901771\pi\)
\(720\) 0 0
\(721\) 5.13223e9 0.0189918
\(722\) 0 0
\(723\) − 1.37731e11i − 0.504057i
\(724\) 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
\(729\) −1.04604e10 −0.0370370
\(730\) 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
\(735\) − 2.83132e11i − 0.970151i
\(736\) 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
\(741\) −3.13791e11 −1.04080
\(742\) 0 0
\(743\) − 1.89167e11i − 0.620713i −0.950620 0.310357i \(-0.899552\pi\)
0.950620 0.310357i \(-0.100448\pi\)
\(744\) 0 0
\(745\) 6.27462e11 2.03687
\(746\) 0 0
\(747\) − 6.43510e10i − 0.206668i
\(748\) 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
\(753\) 1.12919e11 0.351226
\(754\) 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
\(759\) − 2.31612e11i − 0.697900i
\(760\) 0 0
\(761\) −3.92845e11 −1.17134 −0.585670 0.810550i \(-0.699168\pi\)
−0.585670 + 0.810550i \(0.699168\pi\)
\(762\) 0 0
\(763\) 6.90318e10i 0.203681i
\(764\) 0 0
\(765\) 4.76217e10 0.139046
\(766\) 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
\(771\) 2.56545e10i 0.0726015i
\(772\) 0 0
\(773\) 3.88295e11 1.08754 0.543769 0.839235i \(-0.316997\pi\)
0.543769 + 0.839235i \(0.316997\pi\)
\(774\) 0 0
\(775\) − 1.28819e12i − 3.57086i
\(776\) 0 0
\(777\) 1.57956e10 0.0433362
\(778\) 0 0
\(779\) 2.66827e11i 0.724570i
\(780\) 0 0
\(781\) −5.47659e11 −1.47199
\(782\) 0 0
\(783\) − 1.02528e11i − 0.272770i
\(784\) 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
\(789\) 1.29534e11 0.334253
\(790\) 0 0
\(791\) − 3.01259e10i − 0.0769546i
\(792\) 0 0
\(793\) −6.10877e11 −1.54476
\(794\) 0 0
\(795\) 4.53853e11i 1.13618i
\(796\) 0 0
\(797\) 5.81694e11 1.44165 0.720827 0.693115i \(-0.243762\pi\)
0.720827 + 0.693115i \(0.243762\pi\)
\(798\) 0 0
\(799\) − 2.99681e10i − 0.0735312i
\(800\) 0 0
\(801\) 8.09278e10 0.196593
\(802\) 0 0
\(803\) − 3.00594e11i − 0.722966i
\(804\) 0 0
\(805\) −1.61140e11 −0.383725
\(806\) 0 0
\(807\) − 1.15698e11i − 0.272792i
\(808\) 0 0
\(809\) 5.29788e11 1.23682 0.618412 0.785854i \(-0.287776\pi\)
0.618412 + 0.785854i \(0.287776\pi\)
\(810\) 0 0
\(811\) − 1.27857e11i − 0.295556i −0.989021 0.147778i \(-0.952788\pi\)
0.989021 0.147778i \(-0.0472121\pi\)
\(812\) 0 0
\(813\) −1.17485e11 −0.268919
\(814\) 0 0
\(815\) − 3.08705e10i − 0.0699701i
\(816\) 0 0
\(817\) 2.94406e11 0.660781
\(818\) 0 0
\(819\) − 3.19487e10i − 0.0710097i
\(820\) 0 0
\(821\) −5.23718e11 −1.15272 −0.576361 0.817195i \(-0.695528\pi\)
−0.576361 + 0.817195i \(0.695528\pi\)
\(822\) 0 0
\(823\) − 4.16147e11i − 0.907083i −0.891235 0.453541i \(-0.850160\pi\)
0.891235 0.453541i \(-0.149840\pi\)
\(824\) 0 0
\(825\) 5.22847e11 1.12865
\(826\) 0 0
\(827\) − 1.32442e11i − 0.283141i −0.989928 0.141570i \(-0.954785\pi\)
0.989928 0.141570i \(-0.0452152\pi\)
\(828\) 0 0
\(829\) 1.50571e11 0.318804 0.159402 0.987214i \(-0.449043\pi\)
0.159402 + 0.987214i \(0.449043\pi\)
\(830\) 0 0
\(831\) 4.99948e11i 1.04838i
\(832\) 0 0
\(833\) 1.12085e11 0.232792
\(834\) 0 0
\(835\) 1.93290e8i 0 0.000397616i
\(836\) 0 0
\(837\) −1.67718e11 −0.341725
\(838\) 0 0
\(839\) − 5.30040e11i − 1.06970i −0.844948 0.534848i \(-0.820369\pi\)
0.844948 0.534848i \(-0.179631\pi\)
\(840\) 0 0
\(841\) 5.04694e11 1.00889
\(842\) 0 0
\(843\) − 1.21917e11i − 0.241410i
\(844\) 0 0
\(845\) −3.84822e11 −0.754802
\(846\) 0 0
\(847\) 5.03835e9i 0.00978936i
\(848\) 0 0
\(849\) 8.20955e9 0.0158011
\(850\) 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
\(855\) 4.65161e11i 0.870440i
\(856\) 0 0
\(857\) −2.91969e11 −0.541270 −0.270635 0.962682i \(-0.587234\pi\)
−0.270635 + 0.962682i \(0.587234\pi\)
\(858\) 0 0
\(859\) 3.42628e11i 0.629289i 0.949210 + 0.314645i \(0.101885\pi\)
−0.949210 + 0.314645i \(0.898115\pi\)
\(860\) 0 0
\(861\) −2.71671e10 −0.0494346
\(862\) 0 0
\(863\) 6.00307e11i 1.08226i 0.840940 + 0.541128i \(0.182003\pi\)
−0.840940 + 0.541128i \(0.817997\pi\)
\(864\) 0 0
\(865\) −1.74131e12 −3.11036
\(866\) 0 0
\(867\) − 3.07372e11i − 0.543986i
\(868\) 0 0
\(869\) −5.22753e11 −0.916680
\(870\) 0 0
\(871\) 1.19555e12i 2.07728i
\(872\) 0 0
\(873\) −2.76487e11 −0.476011
\(874\) 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
\(879\) − 8.57504e10i − 0.143642i
\(880\) 0 0
\(881\) −8.36651e11 −1.38880 −0.694402 0.719587i \(-0.744331\pi\)
−0.694402 + 0.719587i \(0.744331\pi\)
\(882\) 0 0
\(883\) 6.58705e11i 1.08355i 0.840524 + 0.541774i \(0.182247\pi\)
−0.840524 + 0.541774i \(0.817753\pi\)
\(884\) 0 0
\(885\) −4.31211e11 −0.702937
\(886\) 0 0
\(887\) − 1.84018e11i − 0.297281i −0.988891 0.148640i \(-0.952510\pi\)
0.988891 0.148640i \(-0.0474897\pi\)
\(888\) 0 0
\(889\) 3.91703e10 0.0627119
\(890\) 0 0
\(891\) − 6.80727e10i − 0.108010i
\(892\) 0 0
\(893\) 2.92723e11 0.460311
\(894\) 0 0
\(895\) 1.56799e12i 2.44371i
\(896\) 0 0
\(897\) 5.56778e11 0.860027
\(898\) 0 0
\(899\) − 1.64390e12i − 2.51673i
\(900\) 0 0
\(901\) −1.79670e11 −0.272631
\(902\) 0 0
\(903\) 2.99750e10i 0.0450825i
\(904\) 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
\(909\) −1.51148e11 −0.221385
\(910\) 0 0
\(911\) − 2.57326e11i − 0.373602i −0.982398 0.186801i \(-0.940188\pi\)
0.982398 0.186801i \(-0.0598120\pi\)
\(912\) 0 0
\(913\) 4.18776e11 0.602697
\(914\) 0 0
\(915\) 9.05560e11i 1.29191i
\(916\) 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
\(921\) 6.51202e11 0.905060
\(922\) 0 0
\(923\) − 1.31653e12i − 1.81395i
\(924\) 0 0
\(925\) −6.21411e11 −0.848812
\(926\) 0 0
\(927\) 2.62874e10i 0.0355983i
\(928\) 0 0
\(929\) 1.13798e12 1.52782 0.763910 0.645322i \(-0.223277\pi\)
0.763910 + 0.645322i \(0.223277\pi\)
\(930\) 0 0
\(931\) 1.09483e12i 1.45729i
\(932\) 0 0
\(933\) −1.81818e11 −0.239944
\(934\) 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
\(939\) − 1.68300e11i − 0.216482i
\(940\) 0 0
\(941\) −1.54350e10 −0.0196856 −0.00984279 0.999952i \(-0.503133\pi\)
−0.00984279 + 0.999952i \(0.503133\pi\)
\(942\) 0 0
\(943\) − 4.73448e11i − 0.598722i
\(944\) 0 0
\(945\) −4.73605e10 −0.0593867
\(946\) 0 0
\(947\) 1.10486e12i 1.37374i 0.726779 + 0.686872i \(0.241017\pi\)
−0.726779 + 0.686872i \(0.758983\pi\)
\(948\) 0 0
\(949\) 7.22607e11 0.890917
\(950\) 0 0
\(951\) − 4.04178e11i − 0.494141i
\(952\) 0 0
\(953\) −3.94710e11 −0.478527 −0.239264 0.970955i \(-0.576906\pi\)
−0.239264 + 0.970955i \(0.576906\pi\)
\(954\) 0 0
\(955\) − 2.50267e12i − 3.00877i
\(956\) 0 0
\(957\) 6.67222e11 0.795468
\(958\) 0 0
\(959\) 1.59566e11i 0.188654i
\(960\) 0 0
\(961\) −1.83624e12 −2.15295
\(962\) 0 0
\(963\) 5.44434e11i 0.633053i
\(964\) 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
\(969\) −1.84146e11 −0.208866
\(970\) 0 0
\(971\) − 9.15239e11i − 1.02957i −0.857318 0.514787i \(-0.827871\pi\)
0.857318 0.514787i \(-0.172129\pi\)
\(972\) 0 0
\(973\) −6.87886e9 −0.00767477
\(974\) 0 0
\(975\) 1.25689e12i 1.39084i
\(976\) 0 0
\(977\) 1.39678e12 1.53303 0.766515 0.642226i \(-0.221989\pi\)
0.766515 + 0.642226i \(0.221989\pi\)
\(978\) 0 0
\(979\) 5.26653e11i 0.573316i
\(980\) 0 0
\(981\) −3.53583e11 −0.381782
\(982\) 0 0
\(983\) 4.88577e11i 0.523261i 0.965168 + 0.261631i \(0.0842602\pi\)
−0.965168 + 0.261631i \(0.915740\pi\)
\(984\) 0 0
\(985\) 3.72964e11 0.396207
\(986\) 0 0
\(987\) 2.98037e10i 0.0314052i
\(988\) 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
\(993\) −4.48793e11 −0.461582
\(994\) 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
\(999\) 8.09054e10i 0.0812298i
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 48.9.g.c.31.3 yes 4
3.2 odd 2 144.9.g.i.127.4 4
4.3 odd 2 inner 48.9.g.c.31.1 4
8.3 odd 2 192.9.g.c.127.4 4
8.5 even 2 192.9.g.c.127.2 4
12.11 even 2 144.9.g.i.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.c.31.1 4 4.3 odd 2 inner
48.9.g.c.31.3 yes 4 1.1 even 1 trivial
144.9.g.i.127.3 4 12.11 even 2
144.9.g.i.127.4 4 3.2 odd 2
192.9.g.c.127.2 4 8.5 even 2
192.9.g.c.127.4 4 8.3 odd 2