Properties

Label 225.10.a.f.1.1
Level $225$
Weight $10$
Character 225.1
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
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] = [225,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(115.883063137\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \(q+22.0000 q^{2} -28.0000 q^{4} +5988.00 q^{7} -11880.0 q^{8} +O(q^{10})\) \(q+22.0000 q^{2} -28.0000 q^{4} +5988.00 q^{7} -11880.0 q^{8} +14648.0 q^{11} -37906.0 q^{13} +131736. q^{14} -247024. q^{16} -441098. q^{17} +441820. q^{19} +322256. q^{22} +2.26414e6 q^{23} -833932. q^{26} -167664. q^{28} +1.04935e6 q^{29} -7.91057e6 q^{31} +648032. q^{32} -9.70416e6 q^{34} +2.09926e7 q^{37} +9.72004e6 q^{38} -1.32856e7 q^{41} +2.31308e7 q^{43} -410144. q^{44} +4.98110e7 q^{46} -1.38737e7 q^{47} -4.49746e6 q^{49} +1.06137e6 q^{52} -5.76352e7 q^{53} -7.11374e7 q^{56} +2.30857e7 q^{58} +3.20421e7 q^{59} +1.10664e8 q^{61} -1.74032e8 q^{62} +1.40733e8 q^{64} +1.18568e8 q^{67} +1.23507e7 q^{68} -2.76680e8 q^{71} +2.64023e8 q^{73} +4.61836e8 q^{74} -1.23710e7 q^{76} +8.77122e7 q^{77} +4.48203e8 q^{79} -2.92282e8 q^{82} +8.51016e8 q^{83} +5.08877e8 q^{86} -1.74018e8 q^{88} -1.89895e8 q^{89} -2.26981e8 q^{91} -6.33958e7 q^{92} -3.05221e8 q^{94} +1.01415e9 q^{97} -9.89442e7 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.0000 0.972272 0.486136 0.873883i \(-0.338406\pi\)
0.486136 + 0.873883i \(0.338406\pi\)
\(3\) 0 0
\(4\) −28.0000 −0.0546875
\(5\) 0 0
\(6\) 0 0
\(7\) 5988.00 0.942629 0.471314 0.881965i \(-0.343780\pi\)
0.471314 + 0.881965i \(0.343780\pi\)
\(8\) −11880.0 −1.02544
\(9\) 0 0
\(10\) 0 0
\(11\) 14648.0 0.301655 0.150828 0.988560i \(-0.451806\pi\)
0.150828 + 0.988560i \(0.451806\pi\)
\(12\) 0 0
\(13\) −37906.0 −0.368098 −0.184049 0.982917i \(-0.558920\pi\)
−0.184049 + 0.982917i \(0.558920\pi\)
\(14\) 131736. 0.916491
\(15\) 0 0
\(16\) −247024. −0.942322
\(17\) −441098. −1.28090 −0.640450 0.768000i \(-0.721252\pi\)
−0.640450 + 0.768000i \(0.721252\pi\)
\(18\) 0 0
\(19\) 441820. 0.777775 0.388888 0.921285i \(-0.372859\pi\)
0.388888 + 0.921285i \(0.372859\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 322256. 0.293291
\(23\) 2.26414e6 1.68705 0.843524 0.537092i \(-0.180477\pi\)
0.843524 + 0.537092i \(0.180477\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −833932. −0.357891
\(27\) 0 0
\(28\) −167664. −0.0515500
\(29\) 1.04935e6 0.275505 0.137752 0.990467i \(-0.456012\pi\)
0.137752 + 0.990467i \(0.456012\pi\)
\(30\) 0 0
\(31\) −7.91057e6 −1.53844 −0.769219 0.638985i \(-0.779355\pi\)
−0.769219 + 0.638985i \(0.779355\pi\)
\(32\) 648032. 0.109250
\(33\) 0 0
\(34\) −9.70416e6 −1.24538
\(35\) 0 0
\(36\) 0 0
\(37\) 2.09926e7 1.84144 0.920720 0.390224i \(-0.127602\pi\)
0.920720 + 0.390224i \(0.127602\pi\)
\(38\) 9.72004e6 0.756209
\(39\) 0 0
\(40\) 0 0
\(41\) −1.32856e7 −0.734265 −0.367132 0.930169i \(-0.619660\pi\)
−0.367132 + 0.930169i \(0.619660\pi\)
\(42\) 0 0
\(43\) 2.31308e7 1.03177 0.515884 0.856659i \(-0.327464\pi\)
0.515884 + 0.856659i \(0.327464\pi\)
\(44\) −410144. −0.0164968
\(45\) 0 0
\(46\) 4.98110e7 1.64027
\(47\) −1.38737e7 −0.414717 −0.207358 0.978265i \(-0.566487\pi\)
−0.207358 + 0.978265i \(0.566487\pi\)
\(48\) 0 0
\(49\) −4.49746e6 −0.111451
\(50\) 0 0
\(51\) 0 0
\(52\) 1.06137e6 0.0201303
\(53\) −5.76352e7 −1.00334 −0.501668 0.865060i \(-0.667280\pi\)
−0.501668 + 0.865060i \(0.667280\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.11374e7 −0.966612
\(57\) 0 0
\(58\) 2.30857e7 0.267866
\(59\) 3.20421e7 0.344260 0.172130 0.985074i \(-0.444935\pi\)
0.172130 + 0.985074i \(0.444935\pi\)
\(60\) 0 0
\(61\) 1.10664e8 1.02335 0.511673 0.859180i \(-0.329026\pi\)
0.511673 + 0.859180i \(0.329026\pi\)
\(62\) −1.74032e8 −1.49578
\(63\) 0 0
\(64\) 1.40733e8 1.04854
\(65\) 0 0
\(66\) 0 0
\(67\) 1.18568e8 0.718839 0.359420 0.933176i \(-0.382975\pi\)
0.359420 + 0.933176i \(0.382975\pi\)
\(68\) 1.23507e7 0.0700492
\(69\) 0 0
\(70\) 0 0
\(71\) −2.76680e8 −1.29216 −0.646078 0.763272i \(-0.723592\pi\)
−0.646078 + 0.763272i \(0.723592\pi\)
\(72\) 0 0
\(73\) 2.64023e8 1.08815 0.544076 0.839036i \(-0.316880\pi\)
0.544076 + 0.839036i \(0.316880\pi\)
\(74\) 4.61836e8 1.79038
\(75\) 0 0
\(76\) −1.23710e7 −0.0425346
\(77\) 8.77122e7 0.284349
\(78\) 0 0
\(79\) 4.48203e8 1.29465 0.647325 0.762214i \(-0.275888\pi\)
0.647325 + 0.762214i \(0.275888\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.92282e8 −0.713905
\(83\) 8.51016e8 1.96828 0.984138 0.177402i \(-0.0567693\pi\)
0.984138 + 0.177402i \(0.0567693\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.08877e8 1.00316
\(87\) 0 0
\(88\) −1.74018e8 −0.309331
\(89\) −1.89895e8 −0.320818 −0.160409 0.987051i \(-0.551281\pi\)
−0.160409 + 0.987051i \(0.551281\pi\)
\(90\) 0 0
\(91\) −2.26981e8 −0.346979
\(92\) −6.33958e7 −0.0922604
\(93\) 0 0
\(94\) −3.05221e8 −0.403217
\(95\) 0 0
\(96\) 0 0
\(97\) 1.01415e9 1.16313 0.581566 0.813499i \(-0.302440\pi\)
0.581566 + 0.813499i \(0.302440\pi\)
\(98\) −9.89442e7 −0.108361
\(99\) 0 0
\(100\) 0 0
\(101\) 1.31537e9 1.25777 0.628885 0.777498i \(-0.283512\pi\)
0.628885 + 0.777498i \(0.283512\pi\)
\(102\) 0 0
\(103\) 1.82797e9 1.60030 0.800151 0.599798i \(-0.204752\pi\)
0.800151 + 0.599798i \(0.204752\pi\)
\(104\) 4.50323e8 0.377463
\(105\) 0 0
\(106\) −1.26797e9 −0.975515
\(107\) 1.85367e8 0.136712 0.0683559 0.997661i \(-0.478225\pi\)
0.0683559 + 0.997661i \(0.478225\pi\)
\(108\) 0 0
\(109\) 1.97869e9 1.34264 0.671318 0.741169i \(-0.265728\pi\)
0.671318 + 0.741169i \(0.265728\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.47918e9 −0.888259
\(113\) 1.57875e9 0.910876 0.455438 0.890267i \(-0.349483\pi\)
0.455438 + 0.890267i \(0.349483\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.93818e7 −0.0150667
\(117\) 0 0
\(118\) 7.04927e8 0.334715
\(119\) −2.64129e9 −1.20741
\(120\) 0 0
\(121\) −2.14338e9 −0.909004
\(122\) 2.43461e9 0.994970
\(123\) 0 0
\(124\) 2.21496e8 0.0841333
\(125\) 0 0
\(126\) 0 0
\(127\) −2.40001e9 −0.818645 −0.409322 0.912390i \(-0.634235\pi\)
−0.409322 + 0.912390i \(0.634235\pi\)
\(128\) 2.76433e9 0.910218
\(129\) 0 0
\(130\) 0 0
\(131\) 1.96840e9 0.583971 0.291986 0.956423i \(-0.405684\pi\)
0.291986 + 0.956423i \(0.405684\pi\)
\(132\) 0 0
\(133\) 2.64562e9 0.733153
\(134\) 2.60850e9 0.698907
\(135\) 0 0
\(136\) 5.24024e9 1.31349
\(137\) −2.02909e9 −0.492107 −0.246054 0.969256i \(-0.579134\pi\)
−0.246054 + 0.969256i \(0.579134\pi\)
\(138\) 0 0
\(139\) −1.13673e9 −0.258280 −0.129140 0.991626i \(-0.541222\pi\)
−0.129140 + 0.991626i \(0.541222\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.08695e9 −1.25633
\(143\) −5.55247e8 −0.111039
\(144\) 0 0
\(145\) 0 0
\(146\) 5.80851e9 1.05798
\(147\) 0 0
\(148\) −5.87792e8 −0.100704
\(149\) 4.73854e9 0.787601 0.393800 0.919196i \(-0.371160\pi\)
0.393800 + 0.919196i \(0.371160\pi\)
\(150\) 0 0
\(151\) 2.26216e9 0.354101 0.177051 0.984202i \(-0.443344\pi\)
0.177051 + 0.984202i \(0.443344\pi\)
\(152\) −5.24882e9 −0.797564
\(153\) 0 0
\(154\) 1.92967e9 0.276465
\(155\) 0 0
\(156\) 0 0
\(157\) 1.17889e10 1.54854 0.774272 0.632853i \(-0.218116\pi\)
0.774272 + 0.632853i \(0.218116\pi\)
\(158\) 9.86046e9 1.25875
\(159\) 0 0
\(160\) 0 0
\(161\) 1.35576e10 1.59026
\(162\) 0 0
\(163\) −1.14608e10 −1.27166 −0.635829 0.771830i \(-0.719342\pi\)
−0.635829 + 0.771830i \(0.719342\pi\)
\(164\) 3.71996e8 0.0401551
\(165\) 0 0
\(166\) 1.87223e10 1.91370
\(167\) −1.33707e10 −1.33024 −0.665118 0.746738i \(-0.731619\pi\)
−0.665118 + 0.746738i \(0.731619\pi\)
\(168\) 0 0
\(169\) −9.16763e9 −0.864504
\(170\) 0 0
\(171\) 0 0
\(172\) −6.47661e8 −0.0564248
\(173\) 1.06264e10 0.901939 0.450970 0.892539i \(-0.351078\pi\)
0.450970 + 0.892539i \(0.351078\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.61841e9 −0.284257
\(177\) 0 0
\(178\) −4.17769e9 −0.311922
\(179\) −2.61254e10 −1.90206 −0.951031 0.309094i \(-0.899974\pi\)
−0.951031 + 0.309094i \(0.899974\pi\)
\(180\) 0 0
\(181\) 2.34689e9 0.162532 0.0812660 0.996692i \(-0.474104\pi\)
0.0812660 + 0.996692i \(0.474104\pi\)
\(182\) −4.99358e9 −0.337358
\(183\) 0 0
\(184\) −2.68979e10 −1.72997
\(185\) 0 0
\(186\) 0 0
\(187\) −6.46120e9 −0.386390
\(188\) 3.88463e8 0.0226798
\(189\) 0 0
\(190\) 0 0
\(191\) −2.24064e10 −1.21821 −0.609105 0.793089i \(-0.708471\pi\)
−0.609105 + 0.793089i \(0.708471\pi\)
\(192\) 0 0
\(193\) 3.65959e10 1.89856 0.949282 0.314427i \(-0.101812\pi\)
0.949282 + 0.314427i \(0.101812\pi\)
\(194\) 2.23113e10 1.13088
\(195\) 0 0
\(196\) 1.25929e8 0.00609499
\(197\) −5.41546e9 −0.256175 −0.128088 0.991763i \(-0.540884\pi\)
−0.128088 + 0.991763i \(0.540884\pi\)
\(198\) 0 0
\(199\) 2.62714e8 0.0118753 0.00593764 0.999982i \(-0.498110\pi\)
0.00593764 + 0.999982i \(0.498110\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.89381e10 1.22289
\(203\) 6.28351e9 0.259699
\(204\) 0 0
\(205\) 0 0
\(206\) 4.02154e10 1.55593
\(207\) 0 0
\(208\) 9.36369e9 0.346866
\(209\) 6.47178e9 0.234620
\(210\) 0 0
\(211\) 1.34493e10 0.467121 0.233560 0.972342i \(-0.424962\pi\)
0.233560 + 0.972342i \(0.424962\pi\)
\(212\) 1.61378e9 0.0548699
\(213\) 0 0
\(214\) 4.07808e9 0.132921
\(215\) 0 0
\(216\) 0 0
\(217\) −4.73685e10 −1.45018
\(218\) 4.35312e10 1.30541
\(219\) 0 0
\(220\) 0 0
\(221\) 1.67203e10 0.471496
\(222\) 0 0
\(223\) 2.66463e10 0.721547 0.360773 0.932654i \(-0.382513\pi\)
0.360773 + 0.932654i \(0.382513\pi\)
\(224\) 3.88042e9 0.102982
\(225\) 0 0
\(226\) 3.47324e10 0.885619
\(227\) 3.36318e10 0.840686 0.420343 0.907365i \(-0.361910\pi\)
0.420343 + 0.907365i \(0.361910\pi\)
\(228\) 0 0
\(229\) 5.00453e10 1.20255 0.601276 0.799042i \(-0.294659\pi\)
0.601276 + 0.799042i \(0.294659\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.24663e10 −0.282515
\(233\) −3.29626e10 −0.732688 −0.366344 0.930479i \(-0.619391\pi\)
−0.366344 + 0.930479i \(0.619391\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.97179e8 −0.0188267
\(237\) 0 0
\(238\) −5.81085e10 −1.17393
\(239\) 7.95422e9 0.157691 0.0788455 0.996887i \(-0.474877\pi\)
0.0788455 + 0.996887i \(0.474877\pi\)
\(240\) 0 0
\(241\) −7.52477e10 −1.43687 −0.718434 0.695595i \(-0.755141\pi\)
−0.718434 + 0.695595i \(0.755141\pi\)
\(242\) −4.71544e10 −0.883799
\(243\) 0 0
\(244\) −3.09859e9 −0.0559642
\(245\) 0 0
\(246\) 0 0
\(247\) −1.67476e10 −0.286297
\(248\) 9.39775e10 1.57758
\(249\) 0 0
\(250\) 0 0
\(251\) 9.84631e10 1.56582 0.782910 0.622135i \(-0.213734\pi\)
0.782910 + 0.622135i \(0.213734\pi\)
\(252\) 0 0
\(253\) 3.31651e10 0.508907
\(254\) −5.28001e10 −0.795945
\(255\) 0 0
\(256\) −1.12400e10 −0.163563
\(257\) 8.52399e9 0.121883 0.0609416 0.998141i \(-0.480590\pi\)
0.0609416 + 0.998141i \(0.480590\pi\)
\(258\) 0 0
\(259\) 1.25703e11 1.73579
\(260\) 0 0
\(261\) 0 0
\(262\) 4.33047e10 0.567779
\(263\) 5.90654e9 0.0761259 0.0380630 0.999275i \(-0.487881\pi\)
0.0380630 + 0.999275i \(0.487881\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.82036e10 0.712824
\(267\) 0 0
\(268\) −3.31991e9 −0.0393115
\(269\) 7.15961e10 0.833689 0.416845 0.908978i \(-0.363136\pi\)
0.416845 + 0.908978i \(0.363136\pi\)
\(270\) 0 0
\(271\) −1.24755e11 −1.40506 −0.702530 0.711654i \(-0.747946\pi\)
−0.702530 + 0.711654i \(0.747946\pi\)
\(272\) 1.08962e11 1.20702
\(273\) 0 0
\(274\) −4.46401e10 −0.478462
\(275\) 0 0
\(276\) 0 0
\(277\) −1.12824e11 −1.15145 −0.575723 0.817645i \(-0.695279\pi\)
−0.575723 + 0.817645i \(0.695279\pi\)
\(278\) −2.50080e10 −0.251118
\(279\) 0 0
\(280\) 0 0
\(281\) −8.70208e10 −0.832616 −0.416308 0.909224i \(-0.636676\pi\)
−0.416308 + 0.909224i \(0.636676\pi\)
\(282\) 0 0
\(283\) −3.27696e9 −0.0303692 −0.0151846 0.999885i \(-0.504834\pi\)
−0.0151846 + 0.999885i \(0.504834\pi\)
\(284\) 7.74703e9 0.0706647
\(285\) 0 0
\(286\) −1.22154e10 −0.107960
\(287\) −7.95539e10 −0.692139
\(288\) 0 0
\(289\) 7.59796e10 0.640703
\(290\) 0 0
\(291\) 0 0
\(292\) −7.39265e9 −0.0595083
\(293\) −1.49860e11 −1.18791 −0.593953 0.804500i \(-0.702433\pi\)
−0.593953 + 0.804500i \(0.702433\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.49392e11 −1.88829
\(297\) 0 0
\(298\) 1.04248e11 0.765762
\(299\) −8.58243e10 −0.620998
\(300\) 0 0
\(301\) 1.38507e11 0.972574
\(302\) 4.97676e10 0.344283
\(303\) 0 0
\(304\) −1.09140e11 −0.732915
\(305\) 0 0
\(306\) 0 0
\(307\) 1.84570e11 1.18587 0.592937 0.805249i \(-0.297968\pi\)
0.592937 + 0.805249i \(0.297968\pi\)
\(308\) −2.45594e9 −0.0155503
\(309\) 0 0
\(310\) 0 0
\(311\) −9.04650e10 −0.548351 −0.274176 0.961680i \(-0.588405\pi\)
−0.274176 + 0.961680i \(0.588405\pi\)
\(312\) 0 0
\(313\) −1.07930e10 −0.0635615 −0.0317808 0.999495i \(-0.510118\pi\)
−0.0317808 + 0.999495i \(0.510118\pi\)
\(314\) 2.59355e11 1.50560
\(315\) 0 0
\(316\) −1.25497e10 −0.0708012
\(317\) 4.18319e10 0.232670 0.116335 0.993210i \(-0.462885\pi\)
0.116335 + 0.993210i \(0.462885\pi\)
\(318\) 0 0
\(319\) 1.53709e10 0.0831076
\(320\) 0 0
\(321\) 0 0
\(322\) 2.98268e11 1.54616
\(323\) −1.94886e11 −0.996252
\(324\) 0 0
\(325\) 0 0
\(326\) −2.52137e11 −1.23640
\(327\) 0 0
\(328\) 1.57832e11 0.752946
\(329\) −8.30756e10 −0.390924
\(330\) 0 0
\(331\) −1.23310e10 −0.0564640 −0.0282320 0.999601i \(-0.508988\pi\)
−0.0282320 + 0.999601i \(0.508988\pi\)
\(332\) −2.38284e10 −0.107640
\(333\) 0 0
\(334\) −2.94154e11 −1.29335
\(335\) 0 0
\(336\) 0 0
\(337\) 2.17976e10 0.0920606 0.0460303 0.998940i \(-0.485343\pi\)
0.0460303 + 0.998940i \(0.485343\pi\)
\(338\) −2.01688e11 −0.840533
\(339\) 0 0
\(340\) 0 0
\(341\) −1.15874e11 −0.464078
\(342\) 0 0
\(343\) −2.68568e11 −1.04769
\(344\) −2.74793e11 −1.05802
\(345\) 0 0
\(346\) 2.33780e11 0.876930
\(347\) −8.43613e10 −0.312364 −0.156182 0.987728i \(-0.549919\pi\)
−0.156182 + 0.987728i \(0.549919\pi\)
\(348\) 0 0
\(349\) −1.23295e9 −0.00444867 −0.00222434 0.999998i \(-0.500708\pi\)
−0.00222434 + 0.999998i \(0.500708\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 9.49237e9 0.0329559
\(353\) 1.73388e11 0.594336 0.297168 0.954825i \(-0.403958\pi\)
0.297168 + 0.954825i \(0.403958\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5.31706e9 0.0175447
\(357\) 0 0
\(358\) −5.74760e11 −1.84932
\(359\) −6.38159e10 −0.202770 −0.101385 0.994847i \(-0.532327\pi\)
−0.101385 + 0.994847i \(0.532327\pi\)
\(360\) 0 0
\(361\) −1.27483e11 −0.395066
\(362\) 5.16316e10 0.158025
\(363\) 0 0
\(364\) 6.35547e9 0.0189754
\(365\) 0 0
\(366\) 0 0
\(367\) −1.12242e11 −0.322966 −0.161483 0.986876i \(-0.551628\pi\)
−0.161483 + 0.986876i \(0.551628\pi\)
\(368\) −5.59296e11 −1.58974
\(369\) 0 0
\(370\) 0 0
\(371\) −3.45119e11 −0.945773
\(372\) 0 0
\(373\) −7.21004e11 −1.92863 −0.964313 0.264766i \(-0.914705\pi\)
−0.964313 + 0.264766i \(0.914705\pi\)
\(374\) −1.42146e11 −0.375676
\(375\) 0 0
\(376\) 1.64819e11 0.425268
\(377\) −3.97767e10 −0.101413
\(378\) 0 0
\(379\) −2.04331e11 −0.508695 −0.254348 0.967113i \(-0.581861\pi\)
−0.254348 + 0.967113i \(0.581861\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.92941e11 −1.18443
\(383\) −5.47180e11 −1.29938 −0.649689 0.760200i \(-0.725101\pi\)
−0.649689 + 0.760200i \(0.725101\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.05111e11 1.84592
\(387\) 0 0
\(388\) −2.83962e10 −0.0636088
\(389\) 3.46262e11 0.766711 0.383356 0.923601i \(-0.374768\pi\)
0.383356 + 0.923601i \(0.374768\pi\)
\(390\) 0 0
\(391\) −9.98706e11 −2.16094
\(392\) 5.34299e10 0.114287
\(393\) 0 0
\(394\) −1.19140e11 −0.249072
\(395\) 0 0
\(396\) 0 0
\(397\) −2.56758e11 −0.518760 −0.259380 0.965775i \(-0.583518\pi\)
−0.259380 + 0.965775i \(0.583518\pi\)
\(398\) 5.77971e9 0.0115460
\(399\) 0 0
\(400\) 0 0
\(401\) −2.01679e10 −0.0389504 −0.0194752 0.999810i \(-0.506200\pi\)
−0.0194752 + 0.999810i \(0.506200\pi\)
\(402\) 0 0
\(403\) 2.99858e11 0.566295
\(404\) −3.68303e10 −0.0687843
\(405\) 0 0
\(406\) 1.38237e11 0.252498
\(407\) 3.07499e11 0.555481
\(408\) 0 0
\(409\) −4.33405e10 −0.0765842 −0.0382921 0.999267i \(-0.512192\pi\)
−0.0382921 + 0.999267i \(0.512192\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −5.11832e10 −0.0875166
\(413\) 1.91868e11 0.324510
\(414\) 0 0
\(415\) 0 0
\(416\) −2.45643e10 −0.0402147
\(417\) 0 0
\(418\) 1.42379e11 0.228115
\(419\) −5.10680e11 −0.809443 −0.404721 0.914440i \(-0.632631\pi\)
−0.404721 + 0.914440i \(0.632631\pi\)
\(420\) 0 0
\(421\) 3.21228e11 0.498361 0.249181 0.968457i \(-0.419839\pi\)
0.249181 + 0.968457i \(0.419839\pi\)
\(422\) 2.95885e11 0.454168
\(423\) 0 0
\(424\) 6.84706e11 1.02886
\(425\) 0 0
\(426\) 0 0
\(427\) 6.62656e11 0.964635
\(428\) −5.19029e9 −0.00747643
\(429\) 0 0
\(430\) 0 0
\(431\) −4.47617e11 −0.624826 −0.312413 0.949946i \(-0.601137\pi\)
−0.312413 + 0.949946i \(0.601137\pi\)
\(432\) 0 0
\(433\) 1.41186e11 0.193017 0.0965085 0.995332i \(-0.469233\pi\)
0.0965085 + 0.995332i \(0.469233\pi\)
\(434\) −1.04211e12 −1.40997
\(435\) 0 0
\(436\) −5.54033e10 −0.0734254
\(437\) 1.00034e12 1.31214
\(438\) 0 0
\(439\) −8.50498e11 −1.09291 −0.546453 0.837490i \(-0.684022\pi\)
−0.546453 + 0.837490i \(0.684022\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.67846e11 0.458422
\(443\) 3.03188e11 0.374020 0.187010 0.982358i \(-0.440120\pi\)
0.187010 + 0.982358i \(0.440120\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.86218e11 0.701540
\(447\) 0 0
\(448\) 8.42709e11 0.988386
\(449\) 1.40328e12 1.62943 0.814714 0.579863i \(-0.196894\pi\)
0.814714 + 0.579863i \(0.196894\pi\)
\(450\) 0 0
\(451\) −1.94607e11 −0.221495
\(452\) −4.42049e10 −0.0498136
\(453\) 0 0
\(454\) 7.39899e11 0.817375
\(455\) 0 0
\(456\) 0 0
\(457\) −3.35354e11 −0.359650 −0.179825 0.983699i \(-0.557553\pi\)
−0.179825 + 0.983699i \(0.557553\pi\)
\(458\) 1.10100e12 1.16921
\(459\) 0 0
\(460\) 0 0
\(461\) −1.04275e12 −1.07529 −0.537645 0.843171i \(-0.680686\pi\)
−0.537645 + 0.843171i \(0.680686\pi\)
\(462\) 0 0
\(463\) 9.74084e11 0.985103 0.492552 0.870283i \(-0.336064\pi\)
0.492552 + 0.870283i \(0.336064\pi\)
\(464\) −2.59215e11 −0.259614
\(465\) 0 0
\(466\) −7.25176e11 −0.712372
\(467\) −1.97851e11 −0.192492 −0.0962458 0.995358i \(-0.530683\pi\)
−0.0962458 + 0.995358i \(0.530683\pi\)
\(468\) 0 0
\(469\) 7.09987e11 0.677599
\(470\) 0 0
\(471\) 0 0
\(472\) −3.80660e11 −0.353019
\(473\) 3.38819e11 0.311238
\(474\) 0 0
\(475\) 0 0
\(476\) 7.39563e10 0.0660304
\(477\) 0 0
\(478\) 1.74993e11 0.153319
\(479\) 9.08731e11 0.788725 0.394362 0.918955i \(-0.370965\pi\)
0.394362 + 0.918955i \(0.370965\pi\)
\(480\) 0 0
\(481\) −7.95744e11 −0.677830
\(482\) −1.65545e12 −1.39703
\(483\) 0 0
\(484\) 6.00147e10 0.0497112
\(485\) 0 0
\(486\) 0 0
\(487\) −1.16963e12 −0.942254 −0.471127 0.882065i \(-0.656153\pi\)
−0.471127 + 0.882065i \(0.656153\pi\)
\(488\) −1.31469e12 −1.04938
\(489\) 0 0
\(490\) 0 0
\(491\) 2.49149e12 1.93460 0.967301 0.253630i \(-0.0816246\pi\)
0.967301 + 0.253630i \(0.0816246\pi\)
\(492\) 0 0
\(493\) −4.62866e11 −0.352894
\(494\) −3.68448e11 −0.278359
\(495\) 0 0
\(496\) 1.95410e12 1.44970
\(497\) −1.65676e12 −1.21802
\(498\) 0 0
\(499\) −5.57571e11 −0.402576 −0.201288 0.979532i \(-0.564513\pi\)
−0.201288 + 0.979532i \(0.564513\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.16619e12 1.52240
\(503\) 1.80137e12 1.25472 0.627359 0.778730i \(-0.284136\pi\)
0.627359 + 0.778730i \(0.284136\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 7.29631e11 0.494796
\(507\) 0 0
\(508\) 6.72001e10 0.0447696
\(509\) 2.40110e12 1.58555 0.792774 0.609515i \(-0.208636\pi\)
0.792774 + 0.609515i \(0.208636\pi\)
\(510\) 0 0
\(511\) 1.58097e12 1.02572
\(512\) −1.66262e12 −1.06925
\(513\) 0 0
\(514\) 1.87528e11 0.118504
\(515\) 0 0
\(516\) 0 0
\(517\) −2.03222e11 −0.125102
\(518\) 2.76548e12 1.68766
\(519\) 0 0
\(520\) 0 0
\(521\) −1.03008e12 −0.612491 −0.306246 0.951953i \(-0.599073\pi\)
−0.306246 + 0.951953i \(0.599073\pi\)
\(522\) 0 0
\(523\) −1.06637e12 −0.623232 −0.311616 0.950208i \(-0.600870\pi\)
−0.311616 + 0.950208i \(0.600870\pi\)
\(524\) −5.51151e10 −0.0319359
\(525\) 0 0
\(526\) 1.29944e11 0.0740151
\(527\) 3.48934e12 1.97058
\(528\) 0 0
\(529\) 3.32516e12 1.84613
\(530\) 0 0
\(531\) 0 0
\(532\) −7.40773e10 −0.0400943
\(533\) 5.03603e11 0.270281
\(534\) 0 0
\(535\) 0 0
\(536\) −1.40859e12 −0.737129
\(537\) 0 0
\(538\) 1.57511e12 0.810572
\(539\) −6.58788e10 −0.0336199
\(540\) 0 0
\(541\) 2.51749e12 1.26351 0.631756 0.775167i \(-0.282334\pi\)
0.631756 + 0.775167i \(0.282334\pi\)
\(542\) −2.74460e12 −1.36610
\(543\) 0 0
\(544\) −2.85846e11 −0.139938
\(545\) 0 0
\(546\) 0 0
\(547\) 1.10353e12 0.527037 0.263518 0.964654i \(-0.415117\pi\)
0.263518 + 0.964654i \(0.415117\pi\)
\(548\) 5.68146e10 0.0269121
\(549\) 0 0
\(550\) 0 0
\(551\) 4.63624e11 0.214281
\(552\) 0 0
\(553\) 2.68384e12 1.22037
\(554\) −2.48213e12 −1.11952
\(555\) 0 0
\(556\) 3.18284e10 0.0141247
\(557\) −1.09072e12 −0.480137 −0.240068 0.970756i \(-0.577170\pi\)
−0.240068 + 0.970756i \(0.577170\pi\)
\(558\) 0 0
\(559\) −8.76795e11 −0.379791
\(560\) 0 0
\(561\) 0 0
\(562\) −1.91446e12 −0.809529
\(563\) 1.03081e12 0.432403 0.216202 0.976349i \(-0.430633\pi\)
0.216202 + 0.976349i \(0.430633\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7.20932e10 −0.0295271
\(567\) 0 0
\(568\) 3.28695e12 1.32503
\(569\) 4.06618e12 1.62623 0.813113 0.582106i \(-0.197771\pi\)
0.813113 + 0.582106i \(0.197771\pi\)
\(570\) 0 0
\(571\) 9.86144e11 0.388220 0.194110 0.980980i \(-0.437818\pi\)
0.194110 + 0.980980i \(0.437818\pi\)
\(572\) 1.55469e10 0.00607243
\(573\) 0 0
\(574\) −1.75019e12 −0.672947
\(575\) 0 0
\(576\) 0 0
\(577\) 2.86947e12 1.07773 0.538866 0.842391i \(-0.318853\pi\)
0.538866 + 0.842391i \(0.318853\pi\)
\(578\) 1.67155e12 0.622937
\(579\) 0 0
\(580\) 0 0
\(581\) 5.09588e12 1.85535
\(582\) 0 0
\(583\) −8.44240e11 −0.302662
\(584\) −3.13660e12 −1.11584
\(585\) 0 0
\(586\) −3.29692e12 −1.15497
\(587\) 1.14169e12 0.396898 0.198449 0.980111i \(-0.436410\pi\)
0.198449 + 0.980111i \(0.436410\pi\)
\(588\) 0 0
\(589\) −3.49505e12 −1.19656
\(590\) 0 0
\(591\) 0 0
\(592\) −5.18567e12 −1.73523
\(593\) −2.97176e12 −0.986888 −0.493444 0.869777i \(-0.664262\pi\)
−0.493444 + 0.869777i \(0.664262\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.32679e11 −0.0430719
\(597\) 0 0
\(598\) −1.88814e12 −0.603779
\(599\) −1.77262e12 −0.562593 −0.281297 0.959621i \(-0.590764\pi\)
−0.281297 + 0.959621i \(0.590764\pi\)
\(600\) 0 0
\(601\) 1.25838e12 0.393439 0.196719 0.980460i \(-0.436971\pi\)
0.196719 + 0.980460i \(0.436971\pi\)
\(602\) 3.04715e12 0.945606
\(603\) 0 0
\(604\) −6.33405e10 −0.0193649
\(605\) 0 0
\(606\) 0 0
\(607\) −1.74535e11 −0.0521834 −0.0260917 0.999660i \(-0.508306\pi\)
−0.0260917 + 0.999660i \(0.508306\pi\)
\(608\) 2.86313e11 0.0849720
\(609\) 0 0
\(610\) 0 0
\(611\) 5.25896e11 0.152656
\(612\) 0 0
\(613\) −4.63037e12 −1.32447 −0.662237 0.749294i \(-0.730393\pi\)
−0.662237 + 0.749294i \(0.730393\pi\)
\(614\) 4.06054e12 1.15299
\(615\) 0 0
\(616\) −1.04202e12 −0.291584
\(617\) 2.50518e12 0.695914 0.347957 0.937511i \(-0.386876\pi\)
0.347957 + 0.937511i \(0.386876\pi\)
\(618\) 0 0
\(619\) 5.44228e11 0.148995 0.0744977 0.997221i \(-0.476265\pi\)
0.0744977 + 0.997221i \(0.476265\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −1.99023e12 −0.533146
\(623\) −1.13709e12 −0.302412
\(624\) 0 0
\(625\) 0 0
\(626\) −2.37447e11 −0.0617991
\(627\) 0 0
\(628\) −3.30088e11 −0.0846860
\(629\) −9.25978e12 −2.35870
\(630\) 0 0
\(631\) −6.51975e12 −1.63719 −0.818594 0.574372i \(-0.805246\pi\)
−0.818594 + 0.574372i \(0.805246\pi\)
\(632\) −5.32465e12 −1.32759
\(633\) 0 0
\(634\) 9.20303e11 0.226219
\(635\) 0 0
\(636\) 0 0
\(637\) 1.70481e11 0.0410250
\(638\) 3.38159e11 0.0808032
\(639\) 0 0
\(640\) 0 0
\(641\) 3.36297e12 0.786794 0.393397 0.919369i \(-0.371300\pi\)
0.393397 + 0.919369i \(0.371300\pi\)
\(642\) 0 0
\(643\) −1.18082e12 −0.272417 −0.136208 0.990680i \(-0.543492\pi\)
−0.136208 + 0.990680i \(0.543492\pi\)
\(644\) −3.79614e11 −0.0869673
\(645\) 0 0
\(646\) −4.28749e12 −0.968628
\(647\) 6.63176e12 1.48785 0.743926 0.668262i \(-0.232962\pi\)
0.743926 + 0.668262i \(0.232962\pi\)
\(648\) 0 0
\(649\) 4.69353e11 0.103848
\(650\) 0 0
\(651\) 0 0
\(652\) 3.20902e11 0.0695438
\(653\) 8.25189e11 0.177601 0.0888003 0.996049i \(-0.471697\pi\)
0.0888003 + 0.996049i \(0.471697\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.28185e12 0.691913
\(657\) 0 0
\(658\) −1.82766e12 −0.380084
\(659\) 4.40214e11 0.0909242 0.0454621 0.998966i \(-0.485524\pi\)
0.0454621 + 0.998966i \(0.485524\pi\)
\(660\) 0 0
\(661\) 9.77788e12 1.99222 0.996112 0.0880914i \(-0.0280767\pi\)
0.996112 + 0.0880914i \(0.0280767\pi\)
\(662\) −2.71281e11 −0.0548983
\(663\) 0 0
\(664\) −1.01101e13 −2.01836
\(665\) 0 0
\(666\) 0 0
\(667\) 2.37587e12 0.464790
\(668\) 3.74378e11 0.0727473
\(669\) 0 0
\(670\) 0 0
\(671\) 1.62101e12 0.308698
\(672\) 0 0
\(673\) −4.22591e12 −0.794058 −0.397029 0.917806i \(-0.629959\pi\)
−0.397029 + 0.917806i \(0.629959\pi\)
\(674\) 4.79547e11 0.0895079
\(675\) 0 0
\(676\) 2.56694e11 0.0472776
\(677\) −7.56977e12 −1.38495 −0.692474 0.721442i \(-0.743479\pi\)
−0.692474 + 0.721442i \(0.743479\pi\)
\(678\) 0 0
\(679\) 6.07273e12 1.09640
\(680\) 0 0
\(681\) 0 0
\(682\) −2.54923e12 −0.451210
\(683\) −1.92029e12 −0.337655 −0.168827 0.985646i \(-0.553998\pi\)
−0.168827 + 0.985646i \(0.553998\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −5.90850e12 −1.01864
\(687\) 0 0
\(688\) −5.71385e12 −0.972257
\(689\) 2.18472e12 0.369325
\(690\) 0 0
\(691\) −3.99839e12 −0.667166 −0.333583 0.942721i \(-0.608258\pi\)
−0.333583 + 0.942721i \(0.608258\pi\)
\(692\) −2.97538e11 −0.0493248
\(693\) 0 0
\(694\) −1.85595e12 −0.303702
\(695\) 0 0
\(696\) 0 0
\(697\) 5.86023e12 0.940519
\(698\) −2.71249e10 −0.00432532
\(699\) 0 0
\(700\) 0 0
\(701\) −1.12736e13 −1.76332 −0.881662 0.471881i \(-0.843575\pi\)
−0.881662 + 0.471881i \(0.843575\pi\)
\(702\) 0 0
\(703\) 9.27493e12 1.43223
\(704\) 2.06146e12 0.316299
\(705\) 0 0
\(706\) 3.81453e12 0.577856
\(707\) 7.87643e12 1.18561
\(708\) 0 0
\(709\) 1.12679e13 1.67469 0.837346 0.546673i \(-0.184106\pi\)
0.837346 + 0.546673i \(0.184106\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.25595e12 0.328980
\(713\) −1.79106e13 −2.59542
\(714\) 0 0
\(715\) 0 0
\(716\) 7.31512e11 0.104019
\(717\) 0 0
\(718\) −1.40395e12 −0.197148
\(719\) 9.05247e12 1.26324 0.631622 0.775277i \(-0.282390\pi\)
0.631622 + 0.775277i \(0.282390\pi\)
\(720\) 0 0
\(721\) 1.09459e13 1.50849
\(722\) −2.80462e12 −0.384111
\(723\) 0 0
\(724\) −6.57129e10 −0.00888847
\(725\) 0 0
\(726\) 0 0
\(727\) −1.34408e12 −0.178452 −0.0892260 0.996011i \(-0.528439\pi\)
−0.0892260 + 0.996011i \(0.528439\pi\)
\(728\) 2.69654e12 0.355807
\(729\) 0 0
\(730\) 0 0
\(731\) −1.02029e13 −1.32159
\(732\) 0 0
\(733\) −6.57401e11 −0.0841129 −0.0420564 0.999115i \(-0.513391\pi\)
−0.0420564 + 0.999115i \(0.513391\pi\)
\(734\) −2.46932e12 −0.314011
\(735\) 0 0
\(736\) 1.46723e12 0.184310
\(737\) 1.73679e12 0.216842
\(738\) 0 0
\(739\) 2.15587e12 0.265903 0.132951 0.991123i \(-0.457555\pi\)
0.132951 + 0.991123i \(0.457555\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −7.59263e12 −0.919548
\(743\) −7.73594e12 −0.931244 −0.465622 0.884984i \(-0.654169\pi\)
−0.465622 + 0.884984i \(0.654169\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.58621e13 −1.87515
\(747\) 0 0
\(748\) 1.80914e11 0.0211307
\(749\) 1.10998e12 0.128869
\(750\) 0 0
\(751\) −1.12448e13 −1.28995 −0.644974 0.764205i \(-0.723132\pi\)
−0.644974 + 0.764205i \(0.723132\pi\)
\(752\) 3.42713e12 0.390797
\(753\) 0 0
\(754\) −8.75087e11 −0.0986007
\(755\) 0 0
\(756\) 0 0
\(757\) 1.18544e13 1.31205 0.656024 0.754740i \(-0.272237\pi\)
0.656024 + 0.754740i \(0.272237\pi\)
\(758\) −4.49528e12 −0.494590
\(759\) 0 0
\(760\) 0 0
\(761\) 2.24765e12 0.242939 0.121470 0.992595i \(-0.461239\pi\)
0.121470 + 0.992595i \(0.461239\pi\)
\(762\) 0 0
\(763\) 1.18484e13 1.26561
\(764\) 6.27380e11 0.0666209
\(765\) 0 0
\(766\) −1.20380e13 −1.26335
\(767\) −1.21459e12 −0.126721
\(768\) 0 0
\(769\) −1.44505e13 −1.49010 −0.745049 0.667009i \(-0.767574\pi\)
−0.745049 + 0.667009i \(0.767574\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.02469e12 −0.103828
\(773\) 5.19022e12 0.522852 0.261426 0.965224i \(-0.415807\pi\)
0.261426 + 0.965224i \(0.415807\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.20481e13 −1.19273
\(777\) 0 0
\(778\) 7.61777e12 0.745452
\(779\) −5.86983e12 −0.571093
\(780\) 0 0
\(781\) −4.05280e12 −0.389786
\(782\) −2.19715e13 −2.10102
\(783\) 0 0
\(784\) 1.11098e12 0.105023
\(785\) 0 0
\(786\) 0 0
\(787\) 1.39938e13 1.30032 0.650161 0.759797i \(-0.274701\pi\)
0.650161 + 0.759797i \(0.274701\pi\)
\(788\) 1.51633e11 0.0140096
\(789\) 0 0
\(790\) 0 0
\(791\) 9.45354e12 0.858618
\(792\) 0 0
\(793\) −4.19483e12 −0.376691
\(794\) −5.64867e12 −0.504375
\(795\) 0 0
\(796\) −7.35599e9 −0.000649430 0
\(797\) 2.01269e13 1.76691 0.883454 0.468518i \(-0.155212\pi\)
0.883454 + 0.468518i \(0.155212\pi\)
\(798\) 0 0
\(799\) 6.11966e12 0.531210
\(800\) 0 0
\(801\) 0 0
\(802\) −4.43695e11 −0.0378704
\(803\) 3.86741e12 0.328247
\(804\) 0 0
\(805\) 0 0
\(806\) 6.59688e12 0.550593
\(807\) 0 0
\(808\) −1.56266e13 −1.28977
\(809\) 8.95775e12 0.735242 0.367621 0.929976i \(-0.380172\pi\)
0.367621 + 0.929976i \(0.380172\pi\)
\(810\) 0 0
\(811\) −1.76446e12 −0.143225 −0.0716123 0.997433i \(-0.522814\pi\)
−0.0716123 + 0.997433i \(0.522814\pi\)
\(812\) −1.75938e11 −0.0142023
\(813\) 0 0
\(814\) 6.76498e12 0.540078
\(815\) 0 0
\(816\) 0 0
\(817\) 1.02196e13 0.802483
\(818\) −9.53491e11 −0.0744607
\(819\) 0 0
\(820\) 0 0
\(821\) −1.48082e13 −1.13752 −0.568758 0.822505i \(-0.692576\pi\)
−0.568758 + 0.822505i \(0.692576\pi\)
\(822\) 0 0
\(823\) 3.39651e12 0.258067 0.129034 0.991640i \(-0.458812\pi\)
0.129034 + 0.991640i \(0.458812\pi\)
\(824\) −2.17163e13 −1.64102
\(825\) 0 0
\(826\) 4.22110e12 0.315512
\(827\) 1.88344e13 1.40016 0.700080 0.714065i \(-0.253148\pi\)
0.700080 + 0.714065i \(0.253148\pi\)
\(828\) 0 0
\(829\) 1.18947e13 0.874695 0.437348 0.899292i \(-0.355918\pi\)
0.437348 + 0.899292i \(0.355918\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.33462e12 −0.385966
\(833\) 1.98382e12 0.142758
\(834\) 0 0
\(835\) 0 0
\(836\) −1.81210e11 −0.0128308
\(837\) 0 0
\(838\) −1.12350e13 −0.786998
\(839\) 1.22881e13 0.856165 0.428083 0.903740i \(-0.359189\pi\)
0.428083 + 0.903740i \(0.359189\pi\)
\(840\) 0 0
\(841\) −1.34060e13 −0.924097
\(842\) 7.06702e12 0.484543
\(843\) 0 0
\(844\) −3.76581e11 −0.0255457
\(845\) 0 0
\(846\) 0 0
\(847\) −1.28346e13 −0.856853
\(848\) 1.42373e13 0.945465
\(849\) 0 0
\(850\) 0 0
\(851\) 4.75300e13 3.10660
\(852\) 0 0
\(853\) 4.61355e12 0.298376 0.149188 0.988809i \(-0.452334\pi\)
0.149188 + 0.988809i \(0.452334\pi\)
\(854\) 1.45784e13 0.937887
\(855\) 0 0
\(856\) −2.20216e12 −0.140190
\(857\) −4.29363e12 −0.271901 −0.135950 0.990716i \(-0.543409\pi\)
−0.135950 + 0.990716i \(0.543409\pi\)
\(858\) 0 0
\(859\) −6.40428e12 −0.401330 −0.200665 0.979660i \(-0.564310\pi\)
−0.200665 + 0.979660i \(0.564310\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9.84758e12 −0.607501
\(863\) −3.24421e12 −0.199095 −0.0995474 0.995033i \(-0.531740\pi\)
−0.0995474 + 0.995033i \(0.531740\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.10609e12 0.187665
\(867\) 0 0
\(868\) 1.32632e12 0.0793065
\(869\) 6.56527e12 0.390539
\(870\) 0 0
\(871\) −4.49445e12 −0.264603
\(872\) −2.35068e13 −1.37680
\(873\) 0 0
\(874\) 2.20075e13 1.27576
\(875\) 0 0
\(876\) 0 0
\(877\) −2.89711e13 −1.65374 −0.826871 0.562392i \(-0.809881\pi\)
−0.826871 + 0.562392i \(0.809881\pi\)
\(878\) −1.87109e13 −1.06260
\(879\) 0 0
\(880\) 0 0
\(881\) −7.50447e12 −0.419690 −0.209845 0.977735i \(-0.567296\pi\)
−0.209845 + 0.977735i \(0.567296\pi\)
\(882\) 0 0
\(883\) −2.87141e13 −1.58954 −0.794772 0.606908i \(-0.792410\pi\)
−0.794772 + 0.606908i \(0.792410\pi\)
\(884\) −4.68167e11 −0.0257849
\(885\) 0 0
\(886\) 6.67013e12 0.363649
\(887\) 9.99825e12 0.542335 0.271168 0.962532i \(-0.412590\pi\)
0.271168 + 0.962532i \(0.412590\pi\)
\(888\) 0 0
\(889\) −1.43712e13 −0.771678
\(890\) 0 0
\(891\) 0 0
\(892\) −7.46095e11 −0.0394596
\(893\) −6.12967e12 −0.322556
\(894\) 0 0
\(895\) 0 0
\(896\) 1.65528e13 0.857998
\(897\) 0 0
\(898\) 3.08721e13 1.58425
\(899\) −8.30095e12 −0.423847
\(900\) 0 0
\(901\) 2.54228e13 1.28517
\(902\) −4.28135e12 −0.215353
\(903\) 0 0
\(904\) −1.87555e13 −0.934052
\(905\) 0 0
\(906\) 0 0
\(907\) −4.70846e12 −0.231018 −0.115509 0.993306i \(-0.536850\pi\)
−0.115509 + 0.993306i \(0.536850\pi\)
\(908\) −9.41690e11 −0.0459750
\(909\) 0 0
\(910\) 0 0
\(911\) −2.69663e13 −1.29714 −0.648572 0.761154i \(-0.724633\pi\)
−0.648572 + 0.761154i \(0.724633\pi\)
\(912\) 0 0
\(913\) 1.24657e13 0.593742
\(914\) −7.37778e12 −0.349678
\(915\) 0 0
\(916\) −1.40127e12 −0.0657645
\(917\) 1.17868e13 0.550468
\(918\) 0 0
\(919\) −3.96618e12 −0.183422 −0.0917112 0.995786i \(-0.529234\pi\)
−0.0917112 + 0.995786i \(0.529234\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.29405e13 −1.04547
\(923\) 1.04878e13 0.475639
\(924\) 0 0
\(925\) 0 0
\(926\) 2.14298e13 0.957788
\(927\) 0 0
\(928\) 6.80012e11 0.0300989
\(929\) −1.96912e13 −0.867365 −0.433683 0.901066i \(-0.642786\pi\)
−0.433683 + 0.901066i \(0.642786\pi\)
\(930\) 0 0
\(931\) −1.98707e12 −0.0866841
\(932\) 9.22952e11 0.0400689
\(933\) 0 0
\(934\) −4.35271e12 −0.187154
\(935\) 0 0
\(936\) 0 0
\(937\) 2.11305e13 0.895532 0.447766 0.894151i \(-0.352220\pi\)
0.447766 + 0.894151i \(0.352220\pi\)
\(938\) 1.56197e13 0.658810
\(939\) 0 0
\(940\) 0 0
\(941\) −5.41072e12 −0.224958 −0.112479 0.993654i \(-0.535879\pi\)
−0.112479 + 0.993654i \(0.535879\pi\)
\(942\) 0 0
\(943\) −3.00803e13 −1.23874
\(944\) −7.91517e12 −0.324404
\(945\) 0 0
\(946\) 7.45403e12 0.302608
\(947\) 3.44259e12 0.139095 0.0695473 0.997579i \(-0.477845\pi\)
0.0695473 + 0.997579i \(0.477845\pi\)
\(948\) 0 0
\(949\) −1.00081e13 −0.400546
\(950\) 0 0
\(951\) 0 0
\(952\) 3.13786e13 1.23813
\(953\) −4.18440e12 −0.164329 −0.0821647 0.996619i \(-0.526183\pi\)
−0.0821647 + 0.996619i \(0.526183\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.22718e11 −0.00862373
\(957\) 0 0
\(958\) 1.99921e13 0.766855
\(959\) −1.21502e13 −0.463874
\(960\) 0 0
\(961\) 3.61375e13 1.36679
\(962\) −1.75064e13 −0.659035
\(963\) 0 0
\(964\) 2.10694e12 0.0785787
\(965\) 0 0
\(966\) 0 0
\(967\) −1.49800e13 −0.550926 −0.275463 0.961312i \(-0.588831\pi\)
−0.275463 + 0.961312i \(0.588831\pi\)
\(968\) 2.54634e13 0.932132
\(969\) 0 0
\(970\) 0 0
\(971\) 2.82460e13 1.01970 0.509848 0.860264i \(-0.329701\pi\)
0.509848 + 0.860264i \(0.329701\pi\)
\(972\) 0 0
\(973\) −6.80673e12 −0.243462
\(974\) −2.57318e13 −0.916127
\(975\) 0 0
\(976\) −2.73367e13 −0.964321
\(977\) −4.62242e13 −1.62309 −0.811547 0.584287i \(-0.801374\pi\)
−0.811547 + 0.584287i \(0.801374\pi\)
\(978\) 0 0
\(979\) −2.78158e12 −0.0967764
\(980\) 0 0
\(981\) 0 0
\(982\) 5.48127e13 1.88096
\(983\) 4.80363e13 1.64089 0.820443 0.571728i \(-0.193727\pi\)
0.820443 + 0.571728i \(0.193727\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.01831e13 −0.343109
\(987\) 0 0
\(988\) 4.68934e11 0.0156569
\(989\) 5.23712e13 1.74064
\(990\) 0 0
\(991\) 2.31211e13 0.761512 0.380756 0.924675i \(-0.375664\pi\)
0.380756 + 0.924675i \(0.375664\pi\)
\(992\) −5.12630e12 −0.168074
\(993\) 0 0
\(994\) −3.64487e13 −1.18425
\(995\) 0 0
\(996\) 0 0
\(997\) 1.35362e13 0.433878 0.216939 0.976185i \(-0.430393\pi\)
0.216939 + 0.976185i \(0.430393\pi\)
\(998\) −1.22666e13 −0.391413
\(999\) 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 225.10.a.f.1.1 1
3.2 odd 2 75.10.a.a.1.1 1
5.2 odd 4 225.10.b.b.199.2 2
5.3 odd 4 225.10.b.b.199.1 2
5.4 even 2 45.10.a.a.1.1 1
15.2 even 4 75.10.b.b.49.1 2
15.8 even 4 75.10.b.b.49.2 2
15.14 odd 2 15.10.a.b.1.1 1
60.59 even 2 240.10.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.b.1.1 1 15.14 odd 2
45.10.a.a.1.1 1 5.4 even 2
75.10.a.a.1.1 1 3.2 odd 2
75.10.b.b.49.1 2 15.2 even 4
75.10.b.b.49.2 2 15.8 even 4
225.10.a.f.1.1 1 1.1 even 1 trivial
225.10.b.b.199.1 2 5.3 odd 4
225.10.b.b.199.2 2 5.2 odd 4
240.10.a.g.1.1 1 60.59 even 2