Properties

Label 225.8.a.c.1.1
Level $225$
Weight $8$
Character 225.1
Self dual yes
Analytic conductor $70.287$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,8,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

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: yes
Analytic conductor: \(70.2866307339\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-13.0000 q^{2} +41.0000 q^{4} -1380.00 q^{7} +1131.00 q^{8} +O(q^{10})\) \(q-13.0000 q^{2} +41.0000 q^{4} -1380.00 q^{7} +1131.00 q^{8} +3304.00 q^{11} -8506.00 q^{13} +17940.0 q^{14} -19951.0 q^{16} -9994.00 q^{17} +41236.0 q^{19} -42952.0 q^{22} +84120.0 q^{23} +110578. q^{26} -56580.0 q^{28} -132802. q^{29} -55800.0 q^{31} +114595. q^{32} +129922. q^{34} -228170. q^{37} -536068. q^{38} +139670. q^{41} +755492. q^{43} +135464. q^{44} -1.09356e6 q^{46} +836984. q^{47} +1.08086e6 q^{49} -348746. q^{52} +1.64165e6 q^{53} -1.56078e6 q^{56} +1.72643e6 q^{58} +989656. q^{59} -1.65816e6 q^{61} +725400. q^{62} +1.06399e6 q^{64} +4.52384e6 q^{67} -409754. q^{68} +389408. q^{71} -5.61733e6 q^{73} +2.96621e6 q^{74} +1.69068e6 q^{76} -4.55952e6 q^{77} +3.90108e6 q^{79} -1.81571e6 q^{82} -9.39412e6 q^{83} -9.82140e6 q^{86} +3.73682e6 q^{88} -2.80375e6 q^{89} +1.17383e7 q^{91} +3.44892e6 q^{92} -1.08808e7 q^{94} -5.09943e6 q^{97} -1.40511e7 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.0000 −1.14905 −0.574524 0.818488i \(-0.694813\pi\)
−0.574524 + 0.818488i \(0.694813\pi\)
\(3\) 0 0
\(4\) 41.0000 0.320312
\(5\) 0 0
\(6\) 0 0
\(7\) −1380.00 −1.52067 −0.760337 0.649529i \(-0.774966\pi\)
−0.760337 + 0.649529i \(0.774966\pi\)
\(8\) 1131.00 0.780994
\(9\) 0 0
\(10\) 0 0
\(11\) 3304.00 0.748455 0.374227 0.927337i \(-0.377908\pi\)
0.374227 + 0.927337i \(0.377908\pi\)
\(12\) 0 0
\(13\) −8506.00 −1.07380 −0.536900 0.843646i \(-0.680405\pi\)
−0.536900 + 0.843646i \(0.680405\pi\)
\(14\) 17940.0 1.74733
\(15\) 0 0
\(16\) −19951.0 −1.21771
\(17\) −9994.00 −0.493365 −0.246682 0.969096i \(-0.579340\pi\)
−0.246682 + 0.969096i \(0.579340\pi\)
\(18\) 0 0
\(19\) 41236.0 1.37924 0.689619 0.724173i \(-0.257778\pi\)
0.689619 + 0.724173i \(0.257778\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −42952.0 −0.860011
\(23\) 84120.0 1.44162 0.720812 0.693131i \(-0.243769\pi\)
0.720812 + 0.693131i \(0.243769\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 110578. 1.23385
\(27\) 0 0
\(28\) −56580.0 −0.487091
\(29\) −132802. −1.01114 −0.505570 0.862785i \(-0.668718\pi\)
−0.505570 + 0.862785i \(0.668718\pi\)
\(30\) 0 0
\(31\) −55800.0 −0.336410 −0.168205 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(32\) 114595. 0.618217
\(33\) 0 0
\(34\) 129922. 0.566900
\(35\) 0 0
\(36\) 0 0
\(37\) −228170. −0.740547 −0.370273 0.928923i \(-0.620736\pi\)
−0.370273 + 0.928923i \(0.620736\pi\)
\(38\) −536068. −1.58481
\(39\) 0 0
\(40\) 0 0
\(41\) 139670. 0.316490 0.158245 0.987400i \(-0.449416\pi\)
0.158245 + 0.987400i \(0.449416\pi\)
\(42\) 0 0
\(43\) 755492. 1.44907 0.724537 0.689236i \(-0.242054\pi\)
0.724537 + 0.689236i \(0.242054\pi\)
\(44\) 135464. 0.239739
\(45\) 0 0
\(46\) −1.09356e6 −1.65650
\(47\) 836984. 1.17591 0.587956 0.808893i \(-0.299933\pi\)
0.587956 + 0.808893i \(0.299933\pi\)
\(48\) 0 0
\(49\) 1.08086e6 1.31245
\(50\) 0 0
\(51\) 0 0
\(52\) −348746. −0.343952
\(53\) 1.64165e6 1.51466 0.757330 0.653033i \(-0.226504\pi\)
0.757330 + 0.653033i \(0.226504\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.56078e6 −1.18764
\(57\) 0 0
\(58\) 1.72643e6 1.16185
\(59\) 989656. 0.627339 0.313669 0.949532i \(-0.398442\pi\)
0.313669 + 0.949532i \(0.398442\pi\)
\(60\) 0 0
\(61\) −1.65816e6 −0.935347 −0.467673 0.883901i \(-0.654908\pi\)
−0.467673 + 0.883901i \(0.654908\pi\)
\(62\) 725400. 0.386551
\(63\) 0 0
\(64\) 1.06399e6 0.507351
\(65\) 0 0
\(66\) 0 0
\(67\) 4.52384e6 1.83758 0.918789 0.394749i \(-0.129168\pi\)
0.918789 + 0.394749i \(0.129168\pi\)
\(68\) −409754. −0.158031
\(69\) 0 0
\(70\) 0 0
\(71\) 389408. 0.129122 0.0645611 0.997914i \(-0.479435\pi\)
0.0645611 + 0.997914i \(0.479435\pi\)
\(72\) 0 0
\(73\) −5.61733e6 −1.69005 −0.845026 0.534726i \(-0.820415\pi\)
−0.845026 + 0.534726i \(0.820415\pi\)
\(74\) 2.96621e6 0.850924
\(75\) 0 0
\(76\) 1.69068e6 0.441787
\(77\) −4.55952e6 −1.13816
\(78\) 0 0
\(79\) 3.90108e6 0.890205 0.445103 0.895480i \(-0.353167\pi\)
0.445103 + 0.895480i \(0.353167\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.81571e6 −0.363662
\(83\) −9.39412e6 −1.80336 −0.901680 0.432403i \(-0.857666\pi\)
−0.901680 + 0.432403i \(0.857666\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −9.82140e6 −1.66506
\(87\) 0 0
\(88\) 3.73682e6 0.584539
\(89\) −2.80375e6 −0.421574 −0.210787 0.977532i \(-0.567603\pi\)
−0.210787 + 0.977532i \(0.567603\pi\)
\(90\) 0 0
\(91\) 1.17383e7 1.63290
\(92\) 3.44892e6 0.461770
\(93\) 0 0
\(94\) −1.08808e7 −1.35118
\(95\) 0 0
\(96\) 0 0
\(97\) −5.09943e6 −0.567310 −0.283655 0.958926i \(-0.591547\pi\)
−0.283655 + 0.958926i \(0.591547\pi\)
\(98\) −1.40511e7 −1.50807
\(99\) 0 0
\(100\) 0 0
\(101\) −1.51723e7 −1.46530 −0.732648 0.680607i \(-0.761716\pi\)
−0.732648 + 0.680607i \(0.761716\pi\)
\(102\) 0 0
\(103\) −4.70527e6 −0.424281 −0.212141 0.977239i \(-0.568044\pi\)
−0.212141 + 0.977239i \(0.568044\pi\)
\(104\) −9.62029e6 −0.838632
\(105\) 0 0
\(106\) −2.13414e7 −1.74042
\(107\) 2.63120e6 0.207640 0.103820 0.994596i \(-0.466893\pi\)
0.103820 + 0.994596i \(0.466893\pi\)
\(108\) 0 0
\(109\) −4.30059e6 −0.318080 −0.159040 0.987272i \(-0.550840\pi\)
−0.159040 + 0.987272i \(0.550840\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.75324e7 1.85174
\(113\) 3.98233e6 0.259635 0.129817 0.991538i \(-0.458561\pi\)
0.129817 + 0.991538i \(0.458561\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.44488e6 −0.323881
\(117\) 0 0
\(118\) −1.28655e7 −0.720843
\(119\) 1.37917e7 0.750247
\(120\) 0 0
\(121\) −8.57076e6 −0.439815
\(122\) 2.15561e7 1.07476
\(123\) 0 0
\(124\) −2.28780e6 −0.107756
\(125\) 0 0
\(126\) 0 0
\(127\) −2.80177e7 −1.21372 −0.606861 0.794808i \(-0.707571\pi\)
−0.606861 + 0.794808i \(0.707571\pi\)
\(128\) −2.85001e7 −1.20119
\(129\) 0 0
\(130\) 0 0
\(131\) 8.19919e6 0.318656 0.159328 0.987226i \(-0.449067\pi\)
0.159328 + 0.987226i \(0.449067\pi\)
\(132\) 0 0
\(133\) −5.69057e7 −2.09737
\(134\) −5.88100e7 −2.11147
\(135\) 0 0
\(136\) −1.13032e7 −0.385315
\(137\) −1.66646e6 −0.0553697 −0.0276849 0.999617i \(-0.508813\pi\)
−0.0276849 + 0.999617i \(0.508813\pi\)
\(138\) 0 0
\(139\) −5.87456e7 −1.85534 −0.927670 0.373401i \(-0.878192\pi\)
−0.927670 + 0.373401i \(0.878192\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.06230e6 −0.148368
\(143\) −2.81038e7 −0.803691
\(144\) 0 0
\(145\) 0 0
\(146\) 7.30253e7 1.94195
\(147\) 0 0
\(148\) −9.35497e6 −0.237206
\(149\) 1.93697e7 0.479702 0.239851 0.970810i \(-0.422901\pi\)
0.239851 + 0.970810i \(0.422901\pi\)
\(150\) 0 0
\(151\) −5.33952e7 −1.26207 −0.631034 0.775755i \(-0.717369\pi\)
−0.631034 + 0.775755i \(0.717369\pi\)
\(152\) 4.66379e7 1.07718
\(153\) 0 0
\(154\) 5.92738e7 1.30780
\(155\) 0 0
\(156\) 0 0
\(157\) −2.04529e7 −0.421800 −0.210900 0.977508i \(-0.567639\pi\)
−0.210900 + 0.977508i \(0.567639\pi\)
\(158\) −5.07140e7 −1.02289
\(159\) 0 0
\(160\) 0 0
\(161\) −1.16086e8 −2.19224
\(162\) 0 0
\(163\) 733588. 0.0132677 0.00663385 0.999978i \(-0.497888\pi\)
0.00663385 + 0.999978i \(0.497888\pi\)
\(164\) 5.72647e6 0.101376
\(165\) 0 0
\(166\) 1.22124e8 2.07215
\(167\) 1.68925e7 0.280664 0.140332 0.990105i \(-0.455183\pi\)
0.140332 + 0.990105i \(0.455183\pi\)
\(168\) 0 0
\(169\) 9.60352e6 0.153048
\(170\) 0 0
\(171\) 0 0
\(172\) 3.09752e7 0.464156
\(173\) 1.18186e7 0.173541 0.0867707 0.996228i \(-0.472345\pi\)
0.0867707 + 0.996228i \(0.472345\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.59181e7 −0.911403
\(177\) 0 0
\(178\) 3.64487e7 0.484409
\(179\) 3.13746e7 0.408877 0.204439 0.978879i \(-0.434463\pi\)
0.204439 + 0.978879i \(0.434463\pi\)
\(180\) 0 0
\(181\) −5.83555e7 −0.731488 −0.365744 0.930716i \(-0.619185\pi\)
−0.365744 + 0.930716i \(0.619185\pi\)
\(182\) −1.52598e8 −1.87628
\(183\) 0 0
\(184\) 9.51397e7 1.12590
\(185\) 0 0
\(186\) 0 0
\(187\) −3.30202e7 −0.369261
\(188\) 3.43163e7 0.376659
\(189\) 0 0
\(190\) 0 0
\(191\) −4.06166e6 −0.0421780 −0.0210890 0.999778i \(-0.506713\pi\)
−0.0210890 + 0.999778i \(0.506713\pi\)
\(192\) 0 0
\(193\) 1.33221e8 1.33389 0.666946 0.745106i \(-0.267601\pi\)
0.666946 + 0.745106i \(0.267601\pi\)
\(194\) 6.62925e7 0.651866
\(195\) 0 0
\(196\) 4.43151e7 0.420393
\(197\) 1.30771e7 0.121866 0.0609328 0.998142i \(-0.480592\pi\)
0.0609328 + 0.998142i \(0.480592\pi\)
\(198\) 0 0
\(199\) −6.98502e7 −0.628322 −0.314161 0.949370i \(-0.601723\pi\)
−0.314161 + 0.949370i \(0.601723\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.97239e8 1.68370
\(203\) 1.83267e8 1.53761
\(204\) 0 0
\(205\) 0 0
\(206\) 6.11685e7 0.487520
\(207\) 0 0
\(208\) 1.69703e8 1.30758
\(209\) 1.36244e8 1.03230
\(210\) 0 0
\(211\) 3.28535e7 0.240765 0.120382 0.992728i \(-0.461588\pi\)
0.120382 + 0.992728i \(0.461588\pi\)
\(212\) 6.73076e7 0.485164
\(213\) 0 0
\(214\) −3.42057e7 −0.238589
\(215\) 0 0
\(216\) 0 0
\(217\) 7.70040e7 0.511569
\(218\) 5.59077e7 0.365489
\(219\) 0 0
\(220\) 0 0
\(221\) 8.50090e7 0.529775
\(222\) 0 0
\(223\) −6.95194e7 −0.419796 −0.209898 0.977723i \(-0.567313\pi\)
−0.209898 + 0.977723i \(0.567313\pi\)
\(224\) −1.58141e8 −0.940106
\(225\) 0 0
\(226\) −5.17703e7 −0.298333
\(227\) −2.30779e8 −1.30950 −0.654750 0.755845i \(-0.727226\pi\)
−0.654750 + 0.755845i \(0.727226\pi\)
\(228\) 0 0
\(229\) 1.46157e8 0.804258 0.402129 0.915583i \(-0.368270\pi\)
0.402129 + 0.915583i \(0.368270\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.50199e8 −0.789695
\(233\) 3.11907e8 1.61540 0.807700 0.589594i \(-0.200712\pi\)
0.807700 + 0.589594i \(0.200712\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.05759e7 0.200944
\(237\) 0 0
\(238\) −1.79292e8 −0.862070
\(239\) −2.27310e8 −1.07703 −0.538513 0.842617i \(-0.681014\pi\)
−0.538513 + 0.842617i \(0.681014\pi\)
\(240\) 0 0
\(241\) −1.98483e8 −0.913404 −0.456702 0.889620i \(-0.650969\pi\)
−0.456702 + 0.889620i \(0.650969\pi\)
\(242\) 1.11420e8 0.505369
\(243\) 0 0
\(244\) −6.79846e7 −0.299603
\(245\) 0 0
\(246\) 0 0
\(247\) −3.50753e8 −1.48103
\(248\) −6.31098e7 −0.262734
\(249\) 0 0
\(250\) 0 0
\(251\) 1.32536e8 0.529024 0.264512 0.964382i \(-0.414789\pi\)
0.264512 + 0.964382i \(0.414789\pi\)
\(252\) 0 0
\(253\) 2.77932e8 1.07899
\(254\) 3.64230e8 1.39463
\(255\) 0 0
\(256\) 2.34310e8 0.872872
\(257\) −3.58642e7 −0.131794 −0.0658970 0.997826i \(-0.520991\pi\)
−0.0658970 + 0.997826i \(0.520991\pi\)
\(258\) 0 0
\(259\) 3.14875e8 1.12613
\(260\) 0 0
\(261\) 0 0
\(262\) −1.06589e8 −0.366151
\(263\) −4.79640e8 −1.62581 −0.812907 0.582394i \(-0.802116\pi\)
−0.812907 + 0.582394i \(0.802116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.39774e8 2.40998
\(267\) 0 0
\(268\) 1.85478e8 0.588599
\(269\) 7.08764e7 0.222008 0.111004 0.993820i \(-0.464593\pi\)
0.111004 + 0.993820i \(0.464593\pi\)
\(270\) 0 0
\(271\) 4.07490e8 1.24373 0.621863 0.783126i \(-0.286376\pi\)
0.621863 + 0.783126i \(0.286376\pi\)
\(272\) 1.99390e8 0.600776
\(273\) 0 0
\(274\) 2.16640e7 0.0636225
\(275\) 0 0
\(276\) 0 0
\(277\) 7.93477e7 0.224313 0.112157 0.993691i \(-0.464224\pi\)
0.112157 + 0.993691i \(0.464224\pi\)
\(278\) 7.63693e8 2.13188
\(279\) 0 0
\(280\) 0 0
\(281\) 8.70068e7 0.233928 0.116964 0.993136i \(-0.462684\pi\)
0.116964 + 0.993136i \(0.462684\pi\)
\(282\) 0 0
\(283\) −2.77612e8 −0.728091 −0.364045 0.931381i \(-0.618605\pi\)
−0.364045 + 0.931381i \(0.618605\pi\)
\(284\) 1.59657e7 0.0413594
\(285\) 0 0
\(286\) 3.65350e8 0.923480
\(287\) −1.92745e8 −0.481278
\(288\) 0 0
\(289\) −3.10459e8 −0.756591
\(290\) 0 0
\(291\) 0 0
\(292\) −2.30311e8 −0.541345
\(293\) 2.46490e8 0.572484 0.286242 0.958157i \(-0.407594\pi\)
0.286242 + 0.958157i \(0.407594\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.58060e8 −0.578363
\(297\) 0 0
\(298\) −2.51807e8 −0.551201
\(299\) −7.15525e8 −1.54802
\(300\) 0 0
\(301\) −1.04258e9 −2.20357
\(302\) 6.94138e8 1.45018
\(303\) 0 0
\(304\) −8.22699e8 −1.67951
\(305\) 0 0
\(306\) 0 0
\(307\) −3.84965e8 −0.759340 −0.379670 0.925122i \(-0.623963\pi\)
−0.379670 + 0.925122i \(0.623963\pi\)
\(308\) −1.86940e8 −0.364565
\(309\) 0 0
\(310\) 0 0
\(311\) −4.64435e7 −0.0875514 −0.0437757 0.999041i \(-0.513939\pi\)
−0.0437757 + 0.999041i \(0.513939\pi\)
\(312\) 0 0
\(313\) −2.10558e8 −0.388120 −0.194060 0.980990i \(-0.562166\pi\)
−0.194060 + 0.980990i \(0.562166\pi\)
\(314\) 2.65888e8 0.484669
\(315\) 0 0
\(316\) 1.59944e8 0.285144
\(317\) −9.60971e8 −1.69435 −0.847175 0.531314i \(-0.821698\pi\)
−0.847175 + 0.531314i \(0.821698\pi\)
\(318\) 0 0
\(319\) −4.38778e8 −0.756793
\(320\) 0 0
\(321\) 0 0
\(322\) 1.50911e9 2.51899
\(323\) −4.12113e8 −0.680467
\(324\) 0 0
\(325\) 0 0
\(326\) −9.53664e6 −0.0152452
\(327\) 0 0
\(328\) 1.57967e8 0.247177
\(329\) −1.15504e9 −1.78818
\(330\) 0 0
\(331\) 3.99923e8 0.606147 0.303074 0.952967i \(-0.401987\pi\)
0.303074 + 0.952967i \(0.401987\pi\)
\(332\) −3.85159e8 −0.577639
\(333\) 0 0
\(334\) −2.19603e8 −0.322497
\(335\) 0 0
\(336\) 0 0
\(337\) −2.69185e8 −0.383129 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(338\) −1.24846e8 −0.175859
\(339\) 0 0
\(340\) 0 0
\(341\) −1.84363e8 −0.251787
\(342\) 0 0
\(343\) −3.55093e8 −0.475131
\(344\) 8.54461e8 1.13172
\(345\) 0 0
\(346\) −1.53641e8 −0.199407
\(347\) −8.21868e8 −1.05596 −0.527982 0.849256i \(-0.677051\pi\)
−0.527982 + 0.849256i \(0.677051\pi\)
\(348\) 0 0
\(349\) 6.48354e8 0.816438 0.408219 0.912884i \(-0.366150\pi\)
0.408219 + 0.912884i \(0.366150\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.78622e8 0.462707
\(353\) 6.52666e8 0.789732 0.394866 0.918739i \(-0.370791\pi\)
0.394866 + 0.918739i \(0.370791\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.14954e8 −0.135035
\(357\) 0 0
\(358\) −4.07870e8 −0.469820
\(359\) 8.77431e8 1.00088 0.500440 0.865771i \(-0.333171\pi\)
0.500440 + 0.865771i \(0.333171\pi\)
\(360\) 0 0
\(361\) 8.06536e8 0.902295
\(362\) 7.58622e8 0.840515
\(363\) 0 0
\(364\) 4.81269e8 0.523038
\(365\) 0 0
\(366\) 0 0
\(367\) −2.86989e8 −0.303064 −0.151532 0.988452i \(-0.548421\pi\)
−0.151532 + 0.988452i \(0.548421\pi\)
\(368\) −1.67828e9 −1.75548
\(369\) 0 0
\(370\) 0 0
\(371\) −2.26548e9 −2.30330
\(372\) 0 0
\(373\) 1.77013e9 1.76614 0.883069 0.469242i \(-0.155473\pi\)
0.883069 + 0.469242i \(0.155473\pi\)
\(374\) 4.29262e8 0.424299
\(375\) 0 0
\(376\) 9.46629e8 0.918380
\(377\) 1.12961e9 1.08576
\(378\) 0 0
\(379\) 1.46311e9 1.38051 0.690257 0.723565i \(-0.257498\pi\)
0.690257 + 0.723565i \(0.257498\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 5.28015e7 0.0484646
\(383\) −9.90456e8 −0.900823 −0.450412 0.892821i \(-0.648723\pi\)
−0.450412 + 0.892821i \(0.648723\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.73187e9 −1.53271
\(387\) 0 0
\(388\) −2.09076e8 −0.181716
\(389\) 7.96901e8 0.686406 0.343203 0.939261i \(-0.388488\pi\)
0.343203 + 0.939261i \(0.388488\pi\)
\(390\) 0 0
\(391\) −8.40695e8 −0.711246
\(392\) 1.22245e9 1.02501
\(393\) 0 0
\(394\) −1.70003e8 −0.140029
\(395\) 0 0
\(396\) 0 0
\(397\) −7.95584e8 −0.638145 −0.319073 0.947730i \(-0.603371\pi\)
−0.319073 + 0.947730i \(0.603371\pi\)
\(398\) 9.08053e8 0.721972
\(399\) 0 0
\(400\) 0 0
\(401\) 2.01632e9 1.56154 0.780771 0.624817i \(-0.214826\pi\)
0.780771 + 0.624817i \(0.214826\pi\)
\(402\) 0 0
\(403\) 4.74635e8 0.361237
\(404\) −6.22063e8 −0.469353
\(405\) 0 0
\(406\) −2.38247e9 −1.76679
\(407\) −7.53874e8 −0.554266
\(408\) 0 0
\(409\) −5.48030e7 −0.0396070 −0.0198035 0.999804i \(-0.506304\pi\)
−0.0198035 + 0.999804i \(0.506304\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.92916e8 −0.135903
\(413\) −1.36573e9 −0.953978
\(414\) 0 0
\(415\) 0 0
\(416\) −9.74745e8 −0.663841
\(417\) 0 0
\(418\) −1.77117e9 −1.18616
\(419\) −1.10925e8 −0.0736679 −0.0368340 0.999321i \(-0.511727\pi\)
−0.0368340 + 0.999321i \(0.511727\pi\)
\(420\) 0 0
\(421\) −2.12064e9 −1.38509 −0.692547 0.721373i \(-0.743511\pi\)
−0.692547 + 0.721373i \(0.743511\pi\)
\(422\) −4.27096e8 −0.276650
\(423\) 0 0
\(424\) 1.85671e9 1.18294
\(425\) 0 0
\(426\) 0 0
\(427\) 2.28826e9 1.42236
\(428\) 1.07879e8 0.0665097
\(429\) 0 0
\(430\) 0 0
\(431\) 2.48533e9 1.49525 0.747624 0.664123i \(-0.231195\pi\)
0.747624 + 0.664123i \(0.231195\pi\)
\(432\) 0 0
\(433\) −2.99956e7 −0.0177562 −0.00887811 0.999961i \(-0.502826\pi\)
−0.00887811 + 0.999961i \(0.502826\pi\)
\(434\) −1.00105e9 −0.587818
\(435\) 0 0
\(436\) −1.76324e8 −0.101885
\(437\) 3.46877e9 1.98834
\(438\) 0 0
\(439\) −1.04873e9 −0.591616 −0.295808 0.955247i \(-0.595589\pi\)
−0.295808 + 0.955247i \(0.595589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.10512e9 −0.608738
\(443\) 2.04679e8 0.111856 0.0559282 0.998435i \(-0.482188\pi\)
0.0559282 + 0.998435i \(0.482188\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9.03752e8 0.482366
\(447\) 0 0
\(448\) −1.46831e9 −0.771516
\(449\) −2.63962e9 −1.37619 −0.688097 0.725619i \(-0.741554\pi\)
−0.688097 + 0.725619i \(0.741554\pi\)
\(450\) 0 0
\(451\) 4.61470e8 0.236878
\(452\) 1.63276e8 0.0831643
\(453\) 0 0
\(454\) 3.00013e9 1.50468
\(455\) 0 0
\(456\) 0 0
\(457\) −1.83738e9 −0.900519 −0.450260 0.892898i \(-0.648669\pi\)
−0.450260 + 0.892898i \(0.648669\pi\)
\(458\) −1.90004e9 −0.924132
\(459\) 0 0
\(460\) 0 0
\(461\) −3.57678e9 −1.70035 −0.850176 0.526499i \(-0.823505\pi\)
−0.850176 + 0.526499i \(0.823505\pi\)
\(462\) 0 0
\(463\) −1.28068e9 −0.599662 −0.299831 0.953992i \(-0.596930\pi\)
−0.299831 + 0.953992i \(0.596930\pi\)
\(464\) 2.64953e9 1.23128
\(465\) 0 0
\(466\) −4.05480e9 −1.85617
\(467\) −4.19984e9 −1.90820 −0.954100 0.299490i \(-0.903184\pi\)
−0.954100 + 0.299490i \(0.903184\pi\)
\(468\) 0 0
\(469\) −6.24290e9 −2.79436
\(470\) 0 0
\(471\) 0 0
\(472\) 1.11930e9 0.489948
\(473\) 2.49615e9 1.08457
\(474\) 0 0
\(475\) 0 0
\(476\) 5.65461e8 0.240313
\(477\) 0 0
\(478\) 2.95503e9 1.23756
\(479\) −2.48202e9 −1.03188 −0.515942 0.856623i \(-0.672558\pi\)
−0.515942 + 0.856623i \(0.672558\pi\)
\(480\) 0 0
\(481\) 1.94081e9 0.795200
\(482\) 2.58027e9 1.04955
\(483\) 0 0
\(484\) −3.51401e8 −0.140878
\(485\) 0 0
\(486\) 0 0
\(487\) 1.08065e9 0.423970 0.211985 0.977273i \(-0.432007\pi\)
0.211985 + 0.977273i \(0.432007\pi\)
\(488\) −1.87538e9 −0.730500
\(489\) 0 0
\(490\) 0 0
\(491\) 9.47787e8 0.361348 0.180674 0.983543i \(-0.442172\pi\)
0.180674 + 0.983543i \(0.442172\pi\)
\(492\) 0 0
\(493\) 1.32722e9 0.498861
\(494\) 4.55979e9 1.70177
\(495\) 0 0
\(496\) 1.11327e9 0.409650
\(497\) −5.37383e8 −0.196353
\(498\) 0 0
\(499\) −2.07692e9 −0.748286 −0.374143 0.927371i \(-0.622063\pi\)
−0.374143 + 0.927371i \(0.622063\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.72297e9 −0.607875
\(503\) 2.73265e9 0.957408 0.478704 0.877976i \(-0.341107\pi\)
0.478704 + 0.877976i \(0.341107\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.61312e9 −1.23981
\(507\) 0 0
\(508\) −1.14872e9 −0.388770
\(509\) −4.52470e9 −1.52082 −0.760409 0.649444i \(-0.775002\pi\)
−0.760409 + 0.649444i \(0.775002\pi\)
\(510\) 0 0
\(511\) 7.75192e9 2.57002
\(512\) 6.01982e8 0.198216
\(513\) 0 0
\(514\) 4.66235e8 0.151438
\(515\) 0 0
\(516\) 0 0
\(517\) 2.76540e9 0.880117
\(518\) −4.09337e9 −1.29398
\(519\) 0 0
\(520\) 0 0
\(521\) 2.79533e8 0.0865965 0.0432983 0.999062i \(-0.486213\pi\)
0.0432983 + 0.999062i \(0.486213\pi\)
\(522\) 0 0
\(523\) −1.46376e9 −0.447419 −0.223710 0.974656i \(-0.571817\pi\)
−0.223710 + 0.974656i \(0.571817\pi\)
\(524\) 3.36167e8 0.102069
\(525\) 0 0
\(526\) 6.23533e9 1.86814
\(527\) 5.57665e8 0.165973
\(528\) 0 0
\(529\) 3.67135e9 1.07828
\(530\) 0 0
\(531\) 0 0
\(532\) −2.33313e9 −0.671814
\(533\) −1.18803e9 −0.339847
\(534\) 0 0
\(535\) 0 0
\(536\) 5.11647e9 1.43514
\(537\) 0 0
\(538\) −9.21393e8 −0.255098
\(539\) 3.57115e9 0.982308
\(540\) 0 0
\(541\) −2.72194e9 −0.739073 −0.369537 0.929216i \(-0.620484\pi\)
−0.369537 + 0.929216i \(0.620484\pi\)
\(542\) −5.29737e9 −1.42910
\(543\) 0 0
\(544\) −1.14526e9 −0.305006
\(545\) 0 0
\(546\) 0 0
\(547\) 6.73048e8 0.175829 0.0879145 0.996128i \(-0.471980\pi\)
0.0879145 + 0.996128i \(0.471980\pi\)
\(548\) −6.83248e7 −0.0177356
\(549\) 0 0
\(550\) 0 0
\(551\) −5.47622e9 −1.39460
\(552\) 0 0
\(553\) −5.38349e9 −1.35371
\(554\) −1.03152e9 −0.257747
\(555\) 0 0
\(556\) −2.40857e9 −0.594289
\(557\) −7.19994e9 −1.76537 −0.882685 0.469964i \(-0.844267\pi\)
−0.882685 + 0.469964i \(0.844267\pi\)
\(558\) 0 0
\(559\) −6.42621e9 −1.55602
\(560\) 0 0
\(561\) 0 0
\(562\) −1.13109e9 −0.268794
\(563\) 5.80114e9 1.37004 0.685021 0.728524i \(-0.259793\pi\)
0.685021 + 0.728524i \(0.259793\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3.60895e9 0.836612
\(567\) 0 0
\(568\) 4.40420e8 0.100844
\(569\) 1.67069e9 0.380192 0.190096 0.981766i \(-0.439120\pi\)
0.190096 + 0.981766i \(0.439120\pi\)
\(570\) 0 0
\(571\) 5.70139e9 1.28160 0.640802 0.767706i \(-0.278602\pi\)
0.640802 + 0.767706i \(0.278602\pi\)
\(572\) −1.15226e9 −0.257432
\(573\) 0 0
\(574\) 2.50568e9 0.553011
\(575\) 0 0
\(576\) 0 0
\(577\) −2.43063e9 −0.526750 −0.263375 0.964694i \(-0.584836\pi\)
−0.263375 + 0.964694i \(0.584836\pi\)
\(578\) 4.03596e9 0.869360
\(579\) 0 0
\(580\) 0 0
\(581\) 1.29639e10 2.74232
\(582\) 0 0
\(583\) 5.42401e9 1.13365
\(584\) −6.35320e9 −1.31992
\(585\) 0 0
\(586\) −3.20438e9 −0.657812
\(587\) −3.33378e9 −0.680305 −0.340153 0.940370i \(-0.610479\pi\)
−0.340153 + 0.940370i \(0.610479\pi\)
\(588\) 0 0
\(589\) −2.30097e9 −0.463988
\(590\) 0 0
\(591\) 0 0
\(592\) 4.55222e9 0.901773
\(593\) 3.42280e9 0.674048 0.337024 0.941496i \(-0.390580\pi\)
0.337024 + 0.941496i \(0.390580\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.94159e8 0.153655
\(597\) 0 0
\(598\) 9.30182e9 1.77875
\(599\) 9.28661e8 0.176548 0.0882741 0.996096i \(-0.471865\pi\)
0.0882741 + 0.996096i \(0.471865\pi\)
\(600\) 0 0
\(601\) −6.56974e9 −1.23449 −0.617245 0.786771i \(-0.711751\pi\)
−0.617245 + 0.786771i \(0.711751\pi\)
\(602\) 1.35535e10 2.53201
\(603\) 0 0
\(604\) −2.18920e9 −0.404256
\(605\) 0 0
\(606\) 0 0
\(607\) 3.50790e9 0.636629 0.318315 0.947985i \(-0.396883\pi\)
0.318315 + 0.947985i \(0.396883\pi\)
\(608\) 4.72544e9 0.852667
\(609\) 0 0
\(610\) 0 0
\(611\) −7.11939e9 −1.26269
\(612\) 0 0
\(613\) −3.63970e9 −0.638196 −0.319098 0.947722i \(-0.603380\pi\)
−0.319098 + 0.947722i \(0.603380\pi\)
\(614\) 5.00454e9 0.872519
\(615\) 0 0
\(616\) −5.15682e9 −0.888892
\(617\) 7.82559e9 1.34128 0.670639 0.741784i \(-0.266020\pi\)
0.670639 + 0.741784i \(0.266020\pi\)
\(618\) 0 0
\(619\) 8.35173e9 1.41533 0.707667 0.706546i \(-0.249748\pi\)
0.707667 + 0.706546i \(0.249748\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.03765e8 0.100601
\(623\) 3.86917e9 0.641076
\(624\) 0 0
\(625\) 0 0
\(626\) 2.73725e9 0.445969
\(627\) 0 0
\(628\) −8.38570e8 −0.135108
\(629\) 2.28033e9 0.365360
\(630\) 0 0
\(631\) −2.16820e7 −0.00343555 −0.00171777 0.999999i \(-0.500547\pi\)
−0.00171777 + 0.999999i \(0.500547\pi\)
\(632\) 4.41212e9 0.695245
\(633\) 0 0
\(634\) 1.24926e10 1.94689
\(635\) 0 0
\(636\) 0 0
\(637\) −9.19377e9 −1.40931
\(638\) 5.70411e9 0.869592
\(639\) 0 0
\(640\) 0 0
\(641\) 7.85361e9 1.17779 0.588893 0.808211i \(-0.299564\pi\)
0.588893 + 0.808211i \(0.299564\pi\)
\(642\) 0 0
\(643\) 8.81820e9 1.30810 0.654051 0.756451i \(-0.273068\pi\)
0.654051 + 0.756451i \(0.273068\pi\)
\(644\) −4.75951e9 −0.702201
\(645\) 0 0
\(646\) 5.35746e9 0.781890
\(647\) −8.71997e9 −1.26576 −0.632878 0.774252i \(-0.718127\pi\)
−0.632878 + 0.774252i \(0.718127\pi\)
\(648\) 0 0
\(649\) 3.26982e9 0.469535
\(650\) 0 0
\(651\) 0 0
\(652\) 3.00771e7 0.00424981
\(653\) −6.65755e8 −0.0935661 −0.0467830 0.998905i \(-0.514897\pi\)
−0.0467830 + 0.998905i \(0.514897\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.78656e9 −0.385393
\(657\) 0 0
\(658\) 1.50155e10 2.05470
\(659\) 9.99513e9 1.36047 0.680236 0.732994i \(-0.261877\pi\)
0.680236 + 0.732994i \(0.261877\pi\)
\(660\) 0 0
\(661\) −2.89160e9 −0.389434 −0.194717 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(662\) −5.19900e9 −0.696493
\(663\) 0 0
\(664\) −1.06247e10 −1.40841
\(665\) 0 0
\(666\) 0 0
\(667\) −1.11713e10 −1.45768
\(668\) 6.92593e8 0.0899002
\(669\) 0 0
\(670\) 0 0
\(671\) −5.47857e9 −0.700065
\(672\) 0 0
\(673\) 5.83236e9 0.737550 0.368775 0.929519i \(-0.379777\pi\)
0.368775 + 0.929519i \(0.379777\pi\)
\(674\) 3.49940e9 0.440234
\(675\) 0 0
\(676\) 3.93744e8 0.0490231
\(677\) −1.22524e10 −1.51762 −0.758808 0.651315i \(-0.774218\pi\)
−0.758808 + 0.651315i \(0.774218\pi\)
\(678\) 0 0
\(679\) 7.03721e9 0.862693
\(680\) 0 0
\(681\) 0 0
\(682\) 2.39672e9 0.289316
\(683\) −9.69293e9 −1.16408 −0.582040 0.813160i \(-0.697745\pi\)
−0.582040 + 0.813160i \(0.697745\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 4.61621e9 0.545948
\(687\) 0 0
\(688\) −1.50728e10 −1.76455
\(689\) −1.39639e10 −1.62644
\(690\) 0 0
\(691\) 3.33285e9 0.384276 0.192138 0.981368i \(-0.438458\pi\)
0.192138 + 0.981368i \(0.438458\pi\)
\(692\) 4.84561e8 0.0555875
\(693\) 0 0
\(694\) 1.06843e10 1.21335
\(695\) 0 0
\(696\) 0 0
\(697\) −1.39586e9 −0.156145
\(698\) −8.42860e9 −0.938127
\(699\) 0 0
\(700\) 0 0
\(701\) 2.73199e9 0.299548 0.149774 0.988720i \(-0.452145\pi\)
0.149774 + 0.988720i \(0.452145\pi\)
\(702\) 0 0
\(703\) −9.40882e9 −1.02139
\(704\) 3.51543e9 0.379730
\(705\) 0 0
\(706\) −8.48466e9 −0.907440
\(707\) 2.09377e10 2.22824
\(708\) 0 0
\(709\) 3.44620e8 0.0363144 0.0181572 0.999835i \(-0.494220\pi\)
0.0181572 + 0.999835i \(0.494220\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.17104e9 −0.329247
\(713\) −4.69390e9 −0.484976
\(714\) 0 0
\(715\) 0 0
\(716\) 1.28636e9 0.130969
\(717\) 0 0
\(718\) −1.14066e10 −1.15006
\(719\) −3.35749e9 −0.336871 −0.168436 0.985713i \(-0.553872\pi\)
−0.168436 + 0.985713i \(0.553872\pi\)
\(720\) 0 0
\(721\) 6.49327e9 0.645194
\(722\) −1.04850e10 −1.03678
\(723\) 0 0
\(724\) −2.39258e9 −0.234305
\(725\) 0 0
\(726\) 0 0
\(727\) 1.32017e10 1.27426 0.637132 0.770755i \(-0.280121\pi\)
0.637132 + 0.770755i \(0.280121\pi\)
\(728\) 1.32760e10 1.27528
\(729\) 0 0
\(730\) 0 0
\(731\) −7.55039e9 −0.714922
\(732\) 0 0
\(733\) −4.30179e9 −0.403446 −0.201723 0.979443i \(-0.564654\pi\)
−0.201723 + 0.979443i \(0.564654\pi\)
\(734\) 3.73086e9 0.348235
\(735\) 0 0
\(736\) 9.63973e9 0.891236
\(737\) 1.49468e10 1.37534
\(738\) 0 0
\(739\) −1.82414e10 −1.66266 −0.831330 0.555780i \(-0.812420\pi\)
−0.831330 + 0.555780i \(0.812420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.94512e10 2.64661
\(743\) −2.36847e8 −0.0211840 −0.0105920 0.999944i \(-0.503372\pi\)
−0.0105920 + 0.999944i \(0.503372\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.30117e10 −2.02938
\(747\) 0 0
\(748\) −1.35383e9 −0.118279
\(749\) −3.63106e9 −0.315753
\(750\) 0 0
\(751\) −1.44326e10 −1.24338 −0.621690 0.783264i \(-0.713553\pi\)
−0.621690 + 0.783264i \(0.713553\pi\)
\(752\) −1.66987e10 −1.43192
\(753\) 0 0
\(754\) −1.46850e10 −1.24760
\(755\) 0 0
\(756\) 0 0
\(757\) −1.64934e10 −1.38189 −0.690945 0.722907i \(-0.742805\pi\)
−0.690945 + 0.722907i \(0.742805\pi\)
\(758\) −1.90205e10 −1.58628
\(759\) 0 0
\(760\) 0 0
\(761\) 2.96845e9 0.244165 0.122083 0.992520i \(-0.461043\pi\)
0.122083 + 0.992520i \(0.461043\pi\)
\(762\) 0 0
\(763\) 5.93482e9 0.483695
\(764\) −1.66528e8 −0.0135102
\(765\) 0 0
\(766\) 1.28759e10 1.03509
\(767\) −8.41801e9 −0.673637
\(768\) 0 0
\(769\) 8.15400e9 0.646590 0.323295 0.946298i \(-0.395209\pi\)
0.323295 + 0.946298i \(0.395209\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.46204e9 0.427262
\(773\) −2.78059e9 −0.216525 −0.108263 0.994122i \(-0.534529\pi\)
−0.108263 + 0.994122i \(0.534529\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −5.76745e9 −0.443065
\(777\) 0 0
\(778\) −1.03597e10 −0.788713
\(779\) 5.75943e9 0.436514
\(780\) 0 0
\(781\) 1.28660e9 0.0966421
\(782\) 1.09290e10 0.817256
\(783\) 0 0
\(784\) −2.15642e10 −1.59818
\(785\) 0 0
\(786\) 0 0
\(787\) 8.20982e9 0.600374 0.300187 0.953880i \(-0.402951\pi\)
0.300187 + 0.953880i \(0.402951\pi\)
\(788\) 5.36163e8 0.0390351
\(789\) 0 0
\(790\) 0 0
\(791\) −5.49562e9 −0.394820
\(792\) 0 0
\(793\) 1.41043e10 1.00438
\(794\) 1.03426e10 0.733260
\(795\) 0 0
\(796\) −2.86386e9 −0.201259
\(797\) −5.47916e7 −0.00383363 −0.00191681 0.999998i \(-0.500610\pi\)
−0.00191681 + 0.999998i \(0.500610\pi\)
\(798\) 0 0
\(799\) −8.36482e9 −0.580153
\(800\) 0 0
\(801\) 0 0
\(802\) −2.62121e10 −1.79429
\(803\) −1.85597e10 −1.26493
\(804\) 0 0
\(805\) 0 0
\(806\) −6.17025e9 −0.415079
\(807\) 0 0
\(808\) −1.71598e10 −1.14439
\(809\) −2.04973e9 −0.136106 −0.0680529 0.997682i \(-0.521679\pi\)
−0.0680529 + 0.997682i \(0.521679\pi\)
\(810\) 0 0
\(811\) −1.59471e10 −1.04981 −0.524903 0.851162i \(-0.675899\pi\)
−0.524903 + 0.851162i \(0.675899\pi\)
\(812\) 7.51394e9 0.492517
\(813\) 0 0
\(814\) 9.80036e9 0.636878
\(815\) 0 0
\(816\) 0 0
\(817\) 3.11535e10 1.99862
\(818\) 7.12439e8 0.0455104
\(819\) 0 0
\(820\) 0 0
\(821\) −9.65923e9 −0.609174 −0.304587 0.952484i \(-0.598518\pi\)
−0.304587 + 0.952484i \(0.598518\pi\)
\(822\) 0 0
\(823\) −3.90398e9 −0.244123 −0.122061 0.992523i \(-0.538950\pi\)
−0.122061 + 0.992523i \(0.538950\pi\)
\(824\) −5.32166e9 −0.331361
\(825\) 0 0
\(826\) 1.77544e10 1.09617
\(827\) −1.32224e10 −0.812910 −0.406455 0.913671i \(-0.633235\pi\)
−0.406455 + 0.913671i \(0.633235\pi\)
\(828\) 0 0
\(829\) 1.20482e10 0.734484 0.367242 0.930125i \(-0.380302\pi\)
0.367242 + 0.930125i \(0.380302\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −9.05032e9 −0.544794
\(833\) −1.08021e10 −0.647515
\(834\) 0 0
\(835\) 0 0
\(836\) 5.58599e9 0.330658
\(837\) 0 0
\(838\) 1.44202e9 0.0846480
\(839\) −3.10603e10 −1.81568 −0.907839 0.419319i \(-0.862269\pi\)
−0.907839 + 0.419319i \(0.862269\pi\)
\(840\) 0 0
\(841\) 3.86495e8 0.0224057
\(842\) 2.75683e10 1.59154
\(843\) 0 0
\(844\) 1.34699e9 0.0771200
\(845\) 0 0
\(846\) 0 0
\(847\) 1.18276e10 0.668815
\(848\) −3.27526e10 −1.84442
\(849\) 0 0
\(850\) 0 0
\(851\) −1.91937e10 −1.06759
\(852\) 0 0
\(853\) −2.52157e10 −1.39107 −0.695536 0.718491i \(-0.744833\pi\)
−0.695536 + 0.718491i \(0.744833\pi\)
\(854\) −2.97474e10 −1.63436
\(855\) 0 0
\(856\) 2.97589e9 0.162166
\(857\) 2.25314e10 1.22280 0.611401 0.791321i \(-0.290606\pi\)
0.611401 + 0.791321i \(0.290606\pi\)
\(858\) 0 0
\(859\) 4.07863e9 0.219552 0.109776 0.993956i \(-0.464987\pi\)
0.109776 + 0.993956i \(0.464987\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −3.23092e10 −1.71811
\(863\) −1.10643e10 −0.585986 −0.292993 0.956115i \(-0.594651\pi\)
−0.292993 + 0.956115i \(0.594651\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.89943e8 0.0204028
\(867\) 0 0
\(868\) 3.15716e9 0.163862
\(869\) 1.28892e10 0.666278
\(870\) 0 0
\(871\) −3.84798e10 −1.97319
\(872\) −4.86397e9 −0.248418
\(873\) 0 0
\(874\) −4.50940e10 −2.28470
\(875\) 0 0
\(876\) 0 0
\(877\) −3.60595e10 −1.80518 −0.902590 0.430501i \(-0.858337\pi\)
−0.902590 + 0.430501i \(0.858337\pi\)
\(878\) 1.36336e10 0.679795
\(879\) 0 0
\(880\) 0 0
\(881\) 2.62954e10 1.29558 0.647789 0.761820i \(-0.275694\pi\)
0.647789 + 0.761820i \(0.275694\pi\)
\(882\) 0 0
\(883\) −2.78873e9 −0.136315 −0.0681575 0.997675i \(-0.521712\pi\)
−0.0681575 + 0.997675i \(0.521712\pi\)
\(884\) 3.48537e9 0.169694
\(885\) 0 0
\(886\) −2.66083e9 −0.128528
\(887\) −1.82970e10 −0.880332 −0.440166 0.897916i \(-0.645080\pi\)
−0.440166 + 0.897916i \(0.645080\pi\)
\(888\) 0 0
\(889\) 3.86644e10 1.84567
\(890\) 0 0
\(891\) 0 0
\(892\) −2.85029e9 −0.134466
\(893\) 3.45139e10 1.62186
\(894\) 0 0
\(895\) 0 0
\(896\) 3.93301e10 1.82661
\(897\) 0 0
\(898\) 3.43151e10 1.58131
\(899\) 7.41035e9 0.340157
\(900\) 0 0
\(901\) −1.64067e10 −0.747280
\(902\) −5.99911e9 −0.272185
\(903\) 0 0
\(904\) 4.50402e9 0.202773
\(905\) 0 0
\(906\) 0 0
\(907\) 1.21245e9 0.0539559 0.0269780 0.999636i \(-0.491412\pi\)
0.0269780 + 0.999636i \(0.491412\pi\)
\(908\) −9.46193e9 −0.419449
\(909\) 0 0
\(910\) 0 0
\(911\) 2.09181e10 0.916659 0.458329 0.888782i \(-0.348448\pi\)
0.458329 + 0.888782i \(0.348448\pi\)
\(912\) 0 0
\(913\) −3.10382e10 −1.34973
\(914\) 2.38860e10 1.03474
\(915\) 0 0
\(916\) 5.99244e9 0.257614
\(917\) −1.13149e10 −0.484571
\(918\) 0 0
\(919\) −1.81545e10 −0.771577 −0.385789 0.922587i \(-0.626071\pi\)
−0.385789 + 0.922587i \(0.626071\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 4.64981e10 1.95379
\(923\) −3.31230e9 −0.138651
\(924\) 0 0
\(925\) 0 0
\(926\) 1.66488e10 0.689040
\(927\) 0 0
\(928\) −1.52184e10 −0.625104
\(929\) 3.39225e10 1.38814 0.694069 0.719908i \(-0.255816\pi\)
0.694069 + 0.719908i \(0.255816\pi\)
\(930\) 0 0
\(931\) 4.45702e10 1.81018
\(932\) 1.27882e10 0.517433
\(933\) 0 0
\(934\) 5.45979e10 2.19261
\(935\) 0 0
\(936\) 0 0
\(937\) −3.64023e10 −1.44557 −0.722786 0.691072i \(-0.757139\pi\)
−0.722786 + 0.691072i \(0.757139\pi\)
\(938\) 8.11578e10 3.21085
\(939\) 0 0
\(940\) 0 0
\(941\) −5.07117e10 −1.98401 −0.992007 0.126181i \(-0.959728\pi\)
−0.992007 + 0.126181i \(0.959728\pi\)
\(942\) 0 0
\(943\) 1.17490e10 0.456259
\(944\) −1.97446e10 −0.763918
\(945\) 0 0
\(946\) −3.24499e10 −1.24622
\(947\) 2.30994e10 0.883845 0.441922 0.897053i \(-0.354297\pi\)
0.441922 + 0.897053i \(0.354297\pi\)
\(948\) 0 0
\(949\) 4.77810e10 1.81478
\(950\) 0 0
\(951\) 0 0
\(952\) 1.55984e10 0.585938
\(953\) −4.16368e9 −0.155830 −0.0779151 0.996960i \(-0.524826\pi\)
−0.0779151 + 0.996960i \(0.524826\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −9.31972e9 −0.344985
\(957\) 0 0
\(958\) 3.22663e10 1.18569
\(959\) 2.29971e9 0.0841993
\(960\) 0 0
\(961\) −2.43990e10 −0.886829
\(962\) −2.52306e10 −0.913723
\(963\) 0 0
\(964\) −8.13779e9 −0.292575
\(965\) 0 0
\(966\) 0 0
\(967\) 5.20468e10 1.85098 0.925489 0.378775i \(-0.123655\pi\)
0.925489 + 0.378775i \(0.123655\pi\)
\(968\) −9.69352e9 −0.343493
\(969\) 0 0
\(970\) 0 0
\(971\) 3.64573e8 0.0127796 0.00638979 0.999980i \(-0.497966\pi\)
0.00638979 + 0.999980i \(0.497966\pi\)
\(972\) 0 0
\(973\) 8.10689e10 2.82137
\(974\) −1.40485e10 −0.487162
\(975\) 0 0
\(976\) 3.30820e10 1.13898
\(977\) −1.81785e10 −0.623631 −0.311816 0.950143i \(-0.600937\pi\)
−0.311816 + 0.950143i \(0.600937\pi\)
\(978\) 0 0
\(979\) −9.26358e9 −0.315529
\(980\) 0 0
\(981\) 0 0
\(982\) −1.23212e10 −0.415206
\(983\) −1.85778e10 −0.623816 −0.311908 0.950112i \(-0.600968\pi\)
−0.311908 + 0.950112i \(0.600968\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.72539e10 −0.573216
\(987\) 0 0
\(988\) −1.43809e10 −0.474391
\(989\) 6.35520e10 2.08902
\(990\) 0 0
\(991\) 1.82520e10 0.595735 0.297867 0.954607i \(-0.403725\pi\)
0.297867 + 0.954607i \(0.403725\pi\)
\(992\) −6.39440e9 −0.207974
\(993\) 0 0
\(994\) 6.98598e9 0.225619
\(995\) 0 0
\(996\) 0 0
\(997\) 1.24534e9 0.0397975 0.0198987 0.999802i \(-0.493666\pi\)
0.0198987 + 0.999802i \(0.493666\pi\)
\(998\) 2.70000e10 0.859817
\(999\) 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 225.8.a.c.1.1 1
3.2 odd 2 75.8.a.b.1.1 1
5.2 odd 4 225.8.b.c.199.1 2
5.3 odd 4 225.8.b.c.199.2 2
5.4 even 2 45.8.a.e.1.1 1
15.2 even 4 75.8.b.b.49.2 2
15.8 even 4 75.8.b.b.49.1 2
15.14 odd 2 15.8.a.b.1.1 1
60.59 even 2 240.8.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.8.a.b.1.1 1 15.14 odd 2
45.8.a.e.1.1 1 5.4 even 2
75.8.a.b.1.1 1 3.2 odd 2
75.8.b.b.49.1 2 15.8 even 4
75.8.b.b.49.2 2 15.2 even 4
225.8.a.c.1.1 1 1.1 even 1 trivial
225.8.b.c.199.1 2 5.2 odd 4
225.8.b.c.199.2 2 5.3 odd 4
240.8.a.h.1.1 1 60.59 even 2