Properties

Label 208.10.a.h.1.3
Level $208$
Weight $10$
Character 208.1
Self dual yes
Analytic conductor $107.127$
Analytic rank $1$
Dimension $5$
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] = [208,10,Mod(1,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("208.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 208.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: \(107.127453922\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(35.1685\) of defining polynomial
Character \(\chi\) \(=\) 208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +O(q^{10})\) \(q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +25205.7 q^{11} +28561.0 q^{13} +5255.24 q^{15} +109318. q^{17} +904609. q^{19} +284754. q^{21} +435749. q^{23} -1.94108e6 q^{25} +1.77503e6 q^{27} +6.44791e6 q^{29} -6.62308e6 q^{31} -1.20681e6 q^{33} +652804. q^{35} +4.14357e6 q^{37} -1.36745e6 q^{39} +1.49568e7 q^{41} -4.01789e7 q^{43} +1.90884e6 q^{45} -6.30151e6 q^{47} -4.98153e6 q^{49} -5.23397e6 q^{51} +1.53111e7 q^{53} -2.76663e6 q^{55} -4.33112e7 q^{57} +1.52760e8 q^{59} +8.66321e7 q^{61} +1.03430e8 q^{63} -3.13492e6 q^{65} +1.01034e8 q^{67} -2.08630e7 q^{69} -4.13122e8 q^{71} -3.14453e8 q^{73} +9.29357e7 q^{75} -1.49909e8 q^{77} +2.00580e8 q^{79} +2.57315e8 q^{81} -6.34578e7 q^{83} -1.19990e7 q^{85} -3.08715e8 q^{87} +3.47074e7 q^{89} -1.69865e8 q^{91} +3.17102e8 q^{93} -9.92918e7 q^{95} -1.25403e9 q^{97} -4.38343e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −47.8784 −0.341267 −0.170633 0.985335i \(-0.554581\pi\)
−0.170633 + 0.985335i \(0.554581\pi\)
\(4\) 0 0
\(5\) −109.762 −0.0785394 −0.0392697 0.999229i \(-0.512503\pi\)
−0.0392697 + 0.999229i \(0.512503\pi\)
\(6\) 0 0
\(7\) −5947.44 −0.936244 −0.468122 0.883664i \(-0.655069\pi\)
−0.468122 + 0.883664i \(0.655069\pi\)
\(8\) 0 0
\(9\) −17390.7 −0.883537
\(10\) 0 0
\(11\) 25205.7 0.519076 0.259538 0.965733i \(-0.416430\pi\)
0.259538 + 0.965733i \(0.416430\pi\)
\(12\) 0 0
\(13\) 28561.0 0.277350
\(14\) 0 0
\(15\) 5255.24 0.0268029
\(16\) 0 0
\(17\) 109318. 0.317447 0.158724 0.987323i \(-0.449262\pi\)
0.158724 + 0.987323i \(0.449262\pi\)
\(18\) 0 0
\(19\) 904609. 1.59246 0.796232 0.604992i \(-0.206824\pi\)
0.796232 + 0.604992i \(0.206824\pi\)
\(20\) 0 0
\(21\) 284754. 0.319509
\(22\) 0 0
\(23\) 435749. 0.324684 0.162342 0.986735i \(-0.448095\pi\)
0.162342 + 0.986735i \(0.448095\pi\)
\(24\) 0 0
\(25\) −1.94108e6 −0.993832
\(26\) 0 0
\(27\) 1.77503e6 0.642789
\(28\) 0 0
\(29\) 6.44791e6 1.69289 0.846443 0.532479i \(-0.178740\pi\)
0.846443 + 0.532479i \(0.178740\pi\)
\(30\) 0 0
\(31\) −6.62308e6 −1.28805 −0.644024 0.765005i \(-0.722736\pi\)
−0.644024 + 0.765005i \(0.722736\pi\)
\(32\) 0 0
\(33\) −1.20681e6 −0.177143
\(34\) 0 0
\(35\) 652804. 0.0735320
\(36\) 0 0
\(37\) 4.14357e6 0.363469 0.181734 0.983348i \(-0.441829\pi\)
0.181734 + 0.983348i \(0.441829\pi\)
\(38\) 0 0
\(39\) −1.36745e6 −0.0946504
\(40\) 0 0
\(41\) 1.49568e7 0.826632 0.413316 0.910588i \(-0.364370\pi\)
0.413316 + 0.910588i \(0.364370\pi\)
\(42\) 0 0
\(43\) −4.01789e7 −1.79221 −0.896107 0.443838i \(-0.853616\pi\)
−0.896107 + 0.443838i \(0.853616\pi\)
\(44\) 0 0
\(45\) 1.90884e6 0.0693925
\(46\) 0 0
\(47\) −6.30151e6 −0.188367 −0.0941834 0.995555i \(-0.530024\pi\)
−0.0941834 + 0.995555i \(0.530024\pi\)
\(48\) 0 0
\(49\) −4.98153e6 −0.123447
\(50\) 0 0
\(51\) −5.23397e6 −0.108334
\(52\) 0 0
\(53\) 1.53111e7 0.266542 0.133271 0.991080i \(-0.457452\pi\)
0.133271 + 0.991080i \(0.457452\pi\)
\(54\) 0 0
\(55\) −2.76663e6 −0.0407679
\(56\) 0 0
\(57\) −4.33112e7 −0.543455
\(58\) 0 0
\(59\) 1.52760e8 1.64126 0.820629 0.571462i \(-0.193624\pi\)
0.820629 + 0.571462i \(0.193624\pi\)
\(60\) 0 0
\(61\) 8.66321e7 0.801114 0.400557 0.916272i \(-0.368817\pi\)
0.400557 + 0.916272i \(0.368817\pi\)
\(62\) 0 0
\(63\) 1.03430e8 0.827206
\(64\) 0 0
\(65\) −3.13492e6 −0.0217829
\(66\) 0 0
\(67\) 1.01034e8 0.612537 0.306268 0.951945i \(-0.400920\pi\)
0.306268 + 0.951945i \(0.400920\pi\)
\(68\) 0 0
\(69\) −2.08630e7 −0.110804
\(70\) 0 0
\(71\) −4.13122e8 −1.92937 −0.964685 0.263406i \(-0.915154\pi\)
−0.964685 + 0.263406i \(0.915154\pi\)
\(72\) 0 0
\(73\) −3.14453e8 −1.29599 −0.647997 0.761643i \(-0.724393\pi\)
−0.647997 + 0.761643i \(0.724393\pi\)
\(74\) 0 0
\(75\) 9.29357e7 0.339162
\(76\) 0 0
\(77\) −1.49909e8 −0.485982
\(78\) 0 0
\(79\) 2.00580e8 0.579383 0.289692 0.957120i \(-0.406447\pi\)
0.289692 + 0.957120i \(0.406447\pi\)
\(80\) 0 0
\(81\) 2.57315e8 0.664175
\(82\) 0 0
\(83\) −6.34578e7 −0.146769 −0.0733843 0.997304i \(-0.523380\pi\)
−0.0733843 + 0.997304i \(0.523380\pi\)
\(84\) 0 0
\(85\) −1.19990e7 −0.0249321
\(86\) 0 0
\(87\) −3.08715e8 −0.577726
\(88\) 0 0
\(89\) 3.47074e7 0.0586364 0.0293182 0.999570i \(-0.490666\pi\)
0.0293182 + 0.999570i \(0.490666\pi\)
\(90\) 0 0
\(91\) −1.69865e8 −0.259667
\(92\) 0 0
\(93\) 3.17102e8 0.439568
\(94\) 0 0
\(95\) −9.92918e7 −0.125071
\(96\) 0 0
\(97\) −1.25403e9 −1.43825 −0.719127 0.694879i \(-0.755458\pi\)
−0.719127 + 0.694879i \(0.755458\pi\)
\(98\) 0 0
\(99\) −4.38343e8 −0.458623
\(100\) 0 0
\(101\) −9.06459e8 −0.866766 −0.433383 0.901210i \(-0.642680\pi\)
−0.433383 + 0.901210i \(0.642680\pi\)
\(102\) 0 0
\(103\) 4.17013e8 0.365075 0.182537 0.983199i \(-0.441569\pi\)
0.182537 + 0.983199i \(0.441569\pi\)
\(104\) 0 0
\(105\) −3.12552e7 −0.0250940
\(106\) 0 0
\(107\) −6.71636e8 −0.495344 −0.247672 0.968844i \(-0.579666\pi\)
−0.247672 + 0.968844i \(0.579666\pi\)
\(108\) 0 0
\(109\) 1.66748e9 1.13147 0.565734 0.824588i \(-0.308593\pi\)
0.565734 + 0.824588i \(0.308593\pi\)
\(110\) 0 0
\(111\) −1.98387e8 −0.124040
\(112\) 0 0
\(113\) −1.84580e9 −1.06496 −0.532478 0.846444i \(-0.678739\pi\)
−0.532478 + 0.846444i \(0.678739\pi\)
\(114\) 0 0
\(115\) −4.78287e7 −0.0255005
\(116\) 0 0
\(117\) −4.96695e8 −0.245049
\(118\) 0 0
\(119\) −6.50162e8 −0.297208
\(120\) 0 0
\(121\) −1.72262e9 −0.730560
\(122\) 0 0
\(123\) −7.16109e8 −0.282102
\(124\) 0 0
\(125\) 4.27436e8 0.156594
\(126\) 0 0
\(127\) −1.87922e9 −0.641006 −0.320503 0.947248i \(-0.603852\pi\)
−0.320503 + 0.947248i \(0.603852\pi\)
\(128\) 0 0
\(129\) 1.92370e9 0.611623
\(130\) 0 0
\(131\) −3.46045e9 −1.02663 −0.513313 0.858201i \(-0.671582\pi\)
−0.513313 + 0.858201i \(0.671582\pi\)
\(132\) 0 0
\(133\) −5.38011e9 −1.49093
\(134\) 0 0
\(135\) −1.94831e8 −0.0504842
\(136\) 0 0
\(137\) 5.04786e9 1.22424 0.612118 0.790766i \(-0.290318\pi\)
0.612118 + 0.790766i \(0.290318\pi\)
\(138\) 0 0
\(139\) −3.59716e9 −0.817322 −0.408661 0.912686i \(-0.634004\pi\)
−0.408661 + 0.912686i \(0.634004\pi\)
\(140\) 0 0
\(141\) 3.01706e8 0.0642833
\(142\) 0 0
\(143\) 7.19899e8 0.143966
\(144\) 0 0
\(145\) −7.07736e8 −0.132958
\(146\) 0 0
\(147\) 2.38508e8 0.0421283
\(148\) 0 0
\(149\) 1.27561e9 0.212022 0.106011 0.994365i \(-0.466192\pi\)
0.106011 + 0.994365i \(0.466192\pi\)
\(150\) 0 0
\(151\) −4.35273e9 −0.681342 −0.340671 0.940183i \(-0.610654\pi\)
−0.340671 + 0.940183i \(0.610654\pi\)
\(152\) 0 0
\(153\) −1.90111e9 −0.280476
\(154\) 0 0
\(155\) 7.26963e8 0.101163
\(156\) 0 0
\(157\) −1.41002e10 −1.85215 −0.926075 0.377339i \(-0.876839\pi\)
−0.926075 + 0.377339i \(0.876839\pi\)
\(158\) 0 0
\(159\) −7.33072e8 −0.0909619
\(160\) 0 0
\(161\) −2.59159e9 −0.303983
\(162\) 0 0
\(163\) −7.24812e8 −0.0804231 −0.0402116 0.999191i \(-0.512803\pi\)
−0.0402116 + 0.999191i \(0.512803\pi\)
\(164\) 0 0
\(165\) 1.32462e8 0.0139127
\(166\) 0 0
\(167\) 8.33031e8 0.0828776 0.0414388 0.999141i \(-0.486806\pi\)
0.0414388 + 0.999141i \(0.486806\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) 0 0
\(171\) −1.57317e10 −1.40700
\(172\) 0 0
\(173\) 5.77955e8 0.0490553 0.0245277 0.999699i \(-0.492192\pi\)
0.0245277 + 0.999699i \(0.492192\pi\)
\(174\) 0 0
\(175\) 1.15444e10 0.930469
\(176\) 0 0
\(177\) −7.31392e9 −0.560106
\(178\) 0 0
\(179\) 1.56569e10 1.13990 0.569952 0.821678i \(-0.306962\pi\)
0.569952 + 0.821678i \(0.306962\pi\)
\(180\) 0 0
\(181\) −2.18552e10 −1.51356 −0.756781 0.653668i \(-0.773229\pi\)
−0.756781 + 0.653668i \(0.773229\pi\)
\(182\) 0 0
\(183\) −4.14780e9 −0.273394
\(184\) 0 0
\(185\) −4.54807e8 −0.0285466
\(186\) 0 0
\(187\) 2.75543e9 0.164779
\(188\) 0 0
\(189\) −1.05569e10 −0.601807
\(190\) 0 0
\(191\) −1.67784e10 −0.912222 −0.456111 0.889923i \(-0.650758\pi\)
−0.456111 + 0.889923i \(0.650758\pi\)
\(192\) 0 0
\(193\) −2.70036e10 −1.40092 −0.700462 0.713690i \(-0.747023\pi\)
−0.700462 + 0.713690i \(0.747023\pi\)
\(194\) 0 0
\(195\) 1.50095e8 0.00743378
\(196\) 0 0
\(197\) 1.58277e10 0.748720 0.374360 0.927283i \(-0.377862\pi\)
0.374360 + 0.927283i \(0.377862\pi\)
\(198\) 0 0
\(199\) −8.80397e9 −0.397960 −0.198980 0.980004i \(-0.563763\pi\)
−0.198980 + 0.980004i \(0.563763\pi\)
\(200\) 0 0
\(201\) −4.83736e9 −0.209038
\(202\) 0 0
\(203\) −3.83485e10 −1.58495
\(204\) 0 0
\(205\) −1.64169e9 −0.0649232
\(206\) 0 0
\(207\) −7.57796e9 −0.286870
\(208\) 0 0
\(209\) 2.28013e10 0.826610
\(210\) 0 0
\(211\) 1.82054e10 0.632308 0.316154 0.948708i \(-0.397608\pi\)
0.316154 + 0.948708i \(0.397608\pi\)
\(212\) 0 0
\(213\) 1.97796e10 0.658430
\(214\) 0 0
\(215\) 4.41012e9 0.140759
\(216\) 0 0
\(217\) 3.93904e10 1.20593
\(218\) 0 0
\(219\) 1.50555e10 0.442280
\(220\) 0 0
\(221\) 3.12223e9 0.0880440
\(222\) 0 0
\(223\) −1.53511e10 −0.415688 −0.207844 0.978162i \(-0.566645\pi\)
−0.207844 + 0.978162i \(0.566645\pi\)
\(224\) 0 0
\(225\) 3.37566e10 0.878087
\(226\) 0 0
\(227\) −4.20620e10 −1.05141 −0.525707 0.850666i \(-0.676199\pi\)
−0.525707 + 0.850666i \(0.676199\pi\)
\(228\) 0 0
\(229\) −6.68760e10 −1.60698 −0.803490 0.595318i \(-0.797026\pi\)
−0.803490 + 0.595318i \(0.797026\pi\)
\(230\) 0 0
\(231\) 7.17741e9 0.165849
\(232\) 0 0
\(233\) −5.19268e10 −1.15422 −0.577112 0.816665i \(-0.695820\pi\)
−0.577112 + 0.816665i \(0.695820\pi\)
\(234\) 0 0
\(235\) 6.91668e8 0.0147942
\(236\) 0 0
\(237\) −9.60345e9 −0.197724
\(238\) 0 0
\(239\) −7.45881e10 −1.47870 −0.739348 0.673323i \(-0.764866\pi\)
−0.739348 + 0.673323i \(0.764866\pi\)
\(240\) 0 0
\(241\) 5.74852e10 1.09769 0.548845 0.835924i \(-0.315068\pi\)
0.548845 + 0.835924i \(0.315068\pi\)
\(242\) 0 0
\(243\) −4.72577e10 −0.869449
\(244\) 0 0
\(245\) 5.46784e8 0.00969545
\(246\) 0 0
\(247\) 2.58365e10 0.441670
\(248\) 0 0
\(249\) 3.03826e9 0.0500873
\(250\) 0 0
\(251\) 1.07873e11 1.71546 0.857732 0.514097i \(-0.171873\pi\)
0.857732 + 0.514097i \(0.171873\pi\)
\(252\) 0 0
\(253\) 1.09833e10 0.168536
\(254\) 0 0
\(255\) 5.74492e8 0.00850850
\(256\) 0 0
\(257\) −6.64074e10 −0.949550 −0.474775 0.880107i \(-0.657470\pi\)
−0.474775 + 0.880107i \(0.657470\pi\)
\(258\) 0 0
\(259\) −2.46436e10 −0.340295
\(260\) 0 0
\(261\) −1.12133e11 −1.49573
\(262\) 0 0
\(263\) 8.15356e10 1.05086 0.525432 0.850836i \(-0.323904\pi\)
0.525432 + 0.850836i \(0.323904\pi\)
\(264\) 0 0
\(265\) −1.68058e9 −0.0209340
\(266\) 0 0
\(267\) −1.66174e9 −0.0200107
\(268\) 0 0
\(269\) 1.00568e11 1.17105 0.585523 0.810656i \(-0.300889\pi\)
0.585523 + 0.810656i \(0.300889\pi\)
\(270\) 0 0
\(271\) 2.98757e10 0.336477 0.168239 0.985746i \(-0.446192\pi\)
0.168239 + 0.985746i \(0.446192\pi\)
\(272\) 0 0
\(273\) 8.13286e9 0.0886158
\(274\) 0 0
\(275\) −4.89261e10 −0.515874
\(276\) 0 0
\(277\) −3.17195e10 −0.323718 −0.161859 0.986814i \(-0.551749\pi\)
−0.161859 + 0.986814i \(0.551749\pi\)
\(278\) 0 0
\(279\) 1.15180e11 1.13804
\(280\) 0 0
\(281\) 9.86953e10 0.944318 0.472159 0.881513i \(-0.343475\pi\)
0.472159 + 0.881513i \(0.343475\pi\)
\(282\) 0 0
\(283\) 1.06164e11 0.983871 0.491935 0.870632i \(-0.336290\pi\)
0.491935 + 0.870632i \(0.336290\pi\)
\(284\) 0 0
\(285\) 4.75393e9 0.0426826
\(286\) 0 0
\(287\) −8.89549e10 −0.773929
\(288\) 0 0
\(289\) −1.06637e11 −0.899227
\(290\) 0 0
\(291\) 6.00410e10 0.490828
\(292\) 0 0
\(293\) 2.26355e11 1.79426 0.897132 0.441763i \(-0.145647\pi\)
0.897132 + 0.441763i \(0.145647\pi\)
\(294\) 0 0
\(295\) −1.67673e10 −0.128903
\(296\) 0 0
\(297\) 4.47407e10 0.333656
\(298\) 0 0
\(299\) 1.24454e10 0.0900511
\(300\) 0 0
\(301\) 2.38962e11 1.67795
\(302\) 0 0
\(303\) 4.33998e10 0.295798
\(304\) 0 0
\(305\) −9.50892e9 −0.0629190
\(306\) 0 0
\(307\) 2.23009e10 0.143285 0.0716424 0.997430i \(-0.477176\pi\)
0.0716424 + 0.997430i \(0.477176\pi\)
\(308\) 0 0
\(309\) −1.99659e10 −0.124588
\(310\) 0 0
\(311\) 2.71805e11 1.64754 0.823770 0.566925i \(-0.191867\pi\)
0.823770 + 0.566925i \(0.191867\pi\)
\(312\) 0 0
\(313\) −7.90775e10 −0.465697 −0.232849 0.972513i \(-0.574805\pi\)
−0.232849 + 0.972513i \(0.574805\pi\)
\(314\) 0 0
\(315\) −1.13527e10 −0.0649683
\(316\) 0 0
\(317\) 2.03032e11 1.12927 0.564636 0.825340i \(-0.309017\pi\)
0.564636 + 0.825340i \(0.309017\pi\)
\(318\) 0 0
\(319\) 1.62524e11 0.878736
\(320\) 0 0
\(321\) 3.21568e10 0.169044
\(322\) 0 0
\(323\) 9.88899e10 0.505523
\(324\) 0 0
\(325\) −5.54391e10 −0.275639
\(326\) 0 0
\(327\) −7.98365e10 −0.386132
\(328\) 0 0
\(329\) 3.74779e10 0.176357
\(330\) 0 0
\(331\) −3.24137e11 −1.48424 −0.742118 0.670270i \(-0.766178\pi\)
−0.742118 + 0.670270i \(0.766178\pi\)
\(332\) 0 0
\(333\) −7.20594e10 −0.321138
\(334\) 0 0
\(335\) −1.10897e10 −0.0481083
\(336\) 0 0
\(337\) 2.68510e11 1.13403 0.567017 0.823706i \(-0.308097\pi\)
0.567017 + 0.823706i \(0.308097\pi\)
\(338\) 0 0
\(339\) 8.83739e10 0.363434
\(340\) 0 0
\(341\) −1.66939e11 −0.668595
\(342\) 0 0
\(343\) 2.69628e11 1.05182
\(344\) 0 0
\(345\) 2.28996e9 0.00870247
\(346\) 0 0
\(347\) −2.74715e11 −1.01719 −0.508593 0.861007i \(-0.669834\pi\)
−0.508593 + 0.861007i \(0.669834\pi\)
\(348\) 0 0
\(349\) −3.54006e11 −1.27731 −0.638655 0.769494i \(-0.720509\pi\)
−0.638655 + 0.769494i \(0.720509\pi\)
\(350\) 0 0
\(351\) 5.06966e10 0.178277
\(352\) 0 0
\(353\) −2.25650e11 −0.773481 −0.386741 0.922189i \(-0.626399\pi\)
−0.386741 + 0.922189i \(0.626399\pi\)
\(354\) 0 0
\(355\) 4.53451e10 0.151532
\(356\) 0 0
\(357\) 3.11287e10 0.101427
\(358\) 0 0
\(359\) −3.39022e10 −0.107722 −0.0538608 0.998548i \(-0.517153\pi\)
−0.0538608 + 0.998548i \(0.517153\pi\)
\(360\) 0 0
\(361\) 4.95629e11 1.53594
\(362\) 0 0
\(363\) 8.24764e10 0.249316
\(364\) 0 0
\(365\) 3.45150e10 0.101787
\(366\) 0 0
\(367\) 7.74387e10 0.222823 0.111412 0.993774i \(-0.464463\pi\)
0.111412 + 0.993774i \(0.464463\pi\)
\(368\) 0 0
\(369\) −2.60109e11 −0.730360
\(370\) 0 0
\(371\) −9.10620e10 −0.249548
\(372\) 0 0
\(373\) 6.50821e11 1.74089 0.870446 0.492264i \(-0.163831\pi\)
0.870446 + 0.492264i \(0.163831\pi\)
\(374\) 0 0
\(375\) −2.04649e10 −0.0534404
\(376\) 0 0
\(377\) 1.84159e11 0.469522
\(378\) 0 0
\(379\) −1.93989e11 −0.482948 −0.241474 0.970407i \(-0.577631\pi\)
−0.241474 + 0.970407i \(0.577631\pi\)
\(380\) 0 0
\(381\) 8.99743e10 0.218754
\(382\) 0 0
\(383\) −4.44645e11 −1.05589 −0.527946 0.849278i \(-0.677037\pi\)
−0.527946 + 0.849278i \(0.677037\pi\)
\(384\) 0 0
\(385\) 1.64544e10 0.0381687
\(386\) 0 0
\(387\) 6.98737e11 1.58349
\(388\) 0 0
\(389\) −1.69623e11 −0.375589 −0.187794 0.982208i \(-0.560134\pi\)
−0.187794 + 0.982208i \(0.560134\pi\)
\(390\) 0 0
\(391\) 4.76352e10 0.103070
\(392\) 0 0
\(393\) 1.65681e11 0.350353
\(394\) 0 0
\(395\) −2.20161e10 −0.0455044
\(396\) 0 0
\(397\) 2.94468e11 0.594951 0.297476 0.954729i \(-0.403855\pi\)
0.297476 + 0.954729i \(0.403855\pi\)
\(398\) 0 0
\(399\) 2.57591e11 0.508806
\(400\) 0 0
\(401\) 4.43362e11 0.856265 0.428133 0.903716i \(-0.359172\pi\)
0.428133 + 0.903716i \(0.359172\pi\)
\(402\) 0 0
\(403\) −1.89162e11 −0.357240
\(404\) 0 0
\(405\) −2.82434e10 −0.0521639
\(406\) 0 0
\(407\) 1.04441e11 0.188668
\(408\) 0 0
\(409\) −1.01253e11 −0.178918 −0.0894592 0.995990i \(-0.528514\pi\)
−0.0894592 + 0.995990i \(0.528514\pi\)
\(410\) 0 0
\(411\) −2.41684e11 −0.417791
\(412\) 0 0
\(413\) −9.08533e11 −1.53662
\(414\) 0 0
\(415\) 6.96526e9 0.0115271
\(416\) 0 0
\(417\) 1.72226e11 0.278925
\(418\) 0 0
\(419\) −7.93982e10 −0.125848 −0.0629242 0.998018i \(-0.520043\pi\)
−0.0629242 + 0.998018i \(0.520043\pi\)
\(420\) 0 0
\(421\) −6.06765e11 −0.941350 −0.470675 0.882307i \(-0.655990\pi\)
−0.470675 + 0.882307i \(0.655990\pi\)
\(422\) 0 0
\(423\) 1.09587e11 0.166429
\(424\) 0 0
\(425\) −2.12195e11 −0.315489
\(426\) 0 0
\(427\) −5.15239e11 −0.750038
\(428\) 0 0
\(429\) −3.44676e10 −0.0491307
\(430\) 0 0
\(431\) 2.72544e11 0.380442 0.190221 0.981741i \(-0.439080\pi\)
0.190221 + 0.981741i \(0.439080\pi\)
\(432\) 0 0
\(433\) −1.15522e12 −1.57931 −0.789655 0.613551i \(-0.789740\pi\)
−0.789655 + 0.613551i \(0.789740\pi\)
\(434\) 0 0
\(435\) 3.38853e10 0.0453742
\(436\) 0 0
\(437\) 3.94182e11 0.517047
\(438\) 0 0
\(439\) −1.78053e11 −0.228801 −0.114401 0.993435i \(-0.536495\pi\)
−0.114401 + 0.993435i \(0.536495\pi\)
\(440\) 0 0
\(441\) 8.66321e10 0.109070
\(442\) 0 0
\(443\) 6.55260e10 0.0808345 0.0404173 0.999183i \(-0.487131\pi\)
0.0404173 + 0.999183i \(0.487131\pi\)
\(444\) 0 0
\(445\) −3.80956e9 −0.00460527
\(446\) 0 0
\(447\) −6.10742e10 −0.0723559
\(448\) 0 0
\(449\) −7.98150e11 −0.926779 −0.463390 0.886155i \(-0.653367\pi\)
−0.463390 + 0.886155i \(0.653367\pi\)
\(450\) 0 0
\(451\) 3.76997e11 0.429085
\(452\) 0 0
\(453\) 2.08402e11 0.232519
\(454\) 0 0
\(455\) 1.86447e10 0.0203941
\(456\) 0 0
\(457\) −8.92736e11 −0.957415 −0.478708 0.877974i \(-0.658895\pi\)
−0.478708 + 0.877974i \(0.658895\pi\)
\(458\) 0 0
\(459\) 1.94042e11 0.204051
\(460\) 0 0
\(461\) 2.32490e11 0.239745 0.119872 0.992789i \(-0.461751\pi\)
0.119872 + 0.992789i \(0.461751\pi\)
\(462\) 0 0
\(463\) 1.54421e12 1.56168 0.780841 0.624730i \(-0.214791\pi\)
0.780841 + 0.624730i \(0.214791\pi\)
\(464\) 0 0
\(465\) −3.48058e10 −0.0345234
\(466\) 0 0
\(467\) −1.63317e12 −1.58893 −0.794465 0.607310i \(-0.792249\pi\)
−0.794465 + 0.607310i \(0.792249\pi\)
\(468\) 0 0
\(469\) −6.00895e11 −0.573484
\(470\) 0 0
\(471\) 6.75094e11 0.632077
\(472\) 0 0
\(473\) −1.01274e12 −0.930295
\(474\) 0 0
\(475\) −1.75592e12 −1.58264
\(476\) 0 0
\(477\) −2.66271e11 −0.235500
\(478\) 0 0
\(479\) −1.54414e12 −1.34022 −0.670112 0.742260i \(-0.733754\pi\)
−0.670112 + 0.742260i \(0.733754\pi\)
\(480\) 0 0
\(481\) 1.18344e11 0.100808
\(482\) 0 0
\(483\) 1.24081e11 0.103739
\(484\) 0 0
\(485\) 1.37645e11 0.112960
\(486\) 0 0
\(487\) −2.55439e11 −0.205782 −0.102891 0.994693i \(-0.532809\pi\)
−0.102891 + 0.994693i \(0.532809\pi\)
\(488\) 0 0
\(489\) 3.47028e10 0.0274457
\(490\) 0 0
\(491\) −1.58407e12 −1.23000 −0.615002 0.788526i \(-0.710845\pi\)
−0.615002 + 0.788526i \(0.710845\pi\)
\(492\) 0 0
\(493\) 7.04872e11 0.537402
\(494\) 0 0
\(495\) 4.81135e10 0.0360200
\(496\) 0 0
\(497\) 2.45702e12 1.80636
\(498\) 0 0
\(499\) 6.29948e11 0.454833 0.227417 0.973798i \(-0.426972\pi\)
0.227417 + 0.973798i \(0.426972\pi\)
\(500\) 0 0
\(501\) −3.98842e10 −0.0282834
\(502\) 0 0
\(503\) −5.49263e11 −0.382582 −0.191291 0.981533i \(-0.561267\pi\)
−0.191291 + 0.981533i \(0.561267\pi\)
\(504\) 0 0
\(505\) 9.94949e10 0.0680753
\(506\) 0 0
\(507\) −3.90559e10 −0.0262513
\(508\) 0 0
\(509\) 1.36923e11 0.0904165 0.0452083 0.998978i \(-0.485605\pi\)
0.0452083 + 0.998978i \(0.485605\pi\)
\(510\) 0 0
\(511\) 1.87019e12 1.21337
\(512\) 0 0
\(513\) 1.60570e12 1.02362
\(514\) 0 0
\(515\) −4.57722e10 −0.0286727
\(516\) 0 0
\(517\) −1.58834e11 −0.0977767
\(518\) 0 0
\(519\) −2.76715e10 −0.0167410
\(520\) 0 0
\(521\) −1.66627e12 −0.990779 −0.495389 0.868671i \(-0.664975\pi\)
−0.495389 + 0.868671i \(0.664975\pi\)
\(522\) 0 0
\(523\) −1.97187e12 −1.15244 −0.576222 0.817293i \(-0.695474\pi\)
−0.576222 + 0.817293i \(0.695474\pi\)
\(524\) 0 0
\(525\) −5.52730e11 −0.317538
\(526\) 0 0
\(527\) −7.24021e11 −0.408887
\(528\) 0 0
\(529\) −1.61128e12 −0.894580
\(530\) 0 0
\(531\) −2.65660e12 −1.45011
\(532\) 0 0
\(533\) 4.27182e11 0.229266
\(534\) 0 0
\(535\) 7.37202e10 0.0389040
\(536\) 0 0
\(537\) −7.49629e11 −0.389011
\(538\) 0 0
\(539\) −1.25563e11 −0.0640784
\(540\) 0 0
\(541\) 1.54916e12 0.777513 0.388756 0.921341i \(-0.372905\pi\)
0.388756 + 0.921341i \(0.372905\pi\)
\(542\) 0 0
\(543\) 1.04639e12 0.516529
\(544\) 0 0
\(545\) −1.83027e11 −0.0888648
\(546\) 0 0
\(547\) −1.98052e12 −0.945879 −0.472939 0.881095i \(-0.656807\pi\)
−0.472939 + 0.881095i \(0.656807\pi\)
\(548\) 0 0
\(549\) −1.50659e12 −0.707814
\(550\) 0 0
\(551\) 5.83283e12 2.69586
\(552\) 0 0
\(553\) −1.19294e12 −0.542444
\(554\) 0 0
\(555\) 2.17754e10 0.00974200
\(556\) 0 0
\(557\) 8.86577e11 0.390273 0.195136 0.980776i \(-0.437485\pi\)
0.195136 + 0.980776i \(0.437485\pi\)
\(558\) 0 0
\(559\) −1.14755e12 −0.497071
\(560\) 0 0
\(561\) −1.31926e11 −0.0562336
\(562\) 0 0
\(563\) −3.89786e12 −1.63508 −0.817539 0.575873i \(-0.804662\pi\)
−0.817539 + 0.575873i \(0.804662\pi\)
\(564\) 0 0
\(565\) 2.02599e11 0.0836410
\(566\) 0 0
\(567\) −1.53037e12 −0.621830
\(568\) 0 0
\(569\) −2.26590e12 −0.906223 −0.453112 0.891454i \(-0.649686\pi\)
−0.453112 + 0.891454i \(0.649686\pi\)
\(570\) 0 0
\(571\) 3.09546e12 1.21860 0.609302 0.792938i \(-0.291450\pi\)
0.609302 + 0.792938i \(0.291450\pi\)
\(572\) 0 0
\(573\) 8.03323e11 0.311311
\(574\) 0 0
\(575\) −8.45822e11 −0.322681
\(576\) 0 0
\(577\) −3.23415e12 −1.21470 −0.607350 0.794434i \(-0.707767\pi\)
−0.607350 + 0.794434i \(0.707767\pi\)
\(578\) 0 0
\(579\) 1.29289e12 0.478089
\(580\) 0 0
\(581\) 3.77411e11 0.137411
\(582\) 0 0
\(583\) 3.85927e11 0.138356
\(584\) 0 0
\(585\) 5.45183e10 0.0192460
\(586\) 0 0
\(587\) 8.78491e11 0.305398 0.152699 0.988273i \(-0.451204\pi\)
0.152699 + 0.988273i \(0.451204\pi\)
\(588\) 0 0
\(589\) −5.99129e12 −2.05117
\(590\) 0 0
\(591\) −7.57805e11 −0.255513
\(592\) 0 0
\(593\) −1.91730e12 −0.636712 −0.318356 0.947971i \(-0.603131\pi\)
−0.318356 + 0.947971i \(0.603131\pi\)
\(594\) 0 0
\(595\) 7.13632e10 0.0233425
\(596\) 0 0
\(597\) 4.21520e11 0.135811
\(598\) 0 0
\(599\) 5.10464e12 1.62011 0.810054 0.586355i \(-0.199438\pi\)
0.810054 + 0.586355i \(0.199438\pi\)
\(600\) 0 0
\(601\) 3.00419e12 0.939273 0.469636 0.882860i \(-0.344385\pi\)
0.469636 + 0.882860i \(0.344385\pi\)
\(602\) 0 0
\(603\) −1.75705e12 −0.541199
\(604\) 0 0
\(605\) 1.89079e11 0.0573778
\(606\) 0 0
\(607\) −1.18434e12 −0.354101 −0.177051 0.984202i \(-0.556656\pi\)
−0.177051 + 0.984202i \(0.556656\pi\)
\(608\) 0 0
\(609\) 1.83607e12 0.540892
\(610\) 0 0
\(611\) −1.79978e11 −0.0522436
\(612\) 0 0
\(613\) 6.42597e11 0.183809 0.0919045 0.995768i \(-0.470705\pi\)
0.0919045 + 0.995768i \(0.470705\pi\)
\(614\) 0 0
\(615\) 7.86017e10 0.0221561
\(616\) 0 0
\(617\) 5.05474e12 1.40416 0.702078 0.712100i \(-0.252256\pi\)
0.702078 + 0.712100i \(0.252256\pi\)
\(618\) 0 0
\(619\) 3.09492e11 0.0847308 0.0423654 0.999102i \(-0.486511\pi\)
0.0423654 + 0.999102i \(0.486511\pi\)
\(620\) 0 0
\(621\) 7.73466e11 0.208703
\(622\) 0 0
\(623\) −2.06421e11 −0.0548980
\(624\) 0 0
\(625\) 3.74425e12 0.981533
\(626\) 0 0
\(627\) −1.09169e12 −0.282094
\(628\) 0 0
\(629\) 4.52966e11 0.115382
\(630\) 0 0
\(631\) 7.94237e11 0.199443 0.0997214 0.995015i \(-0.468205\pi\)
0.0997214 + 0.995015i \(0.468205\pi\)
\(632\) 0 0
\(633\) −8.71644e11 −0.215786
\(634\) 0 0
\(635\) 2.06268e11 0.0503442
\(636\) 0 0
\(637\) −1.42278e11 −0.0342380
\(638\) 0 0
\(639\) 7.18446e12 1.70467
\(640\) 0 0
\(641\) 2.30298e12 0.538802 0.269401 0.963028i \(-0.413174\pi\)
0.269401 + 0.963028i \(0.413174\pi\)
\(642\) 0 0
\(643\) 2.54593e12 0.587350 0.293675 0.955905i \(-0.405122\pi\)
0.293675 + 0.955905i \(0.405122\pi\)
\(644\) 0 0
\(645\) −2.11149e11 −0.0480365
\(646\) 0 0
\(647\) 2.74568e11 0.0616001 0.0308000 0.999526i \(-0.490194\pi\)
0.0308000 + 0.999526i \(0.490194\pi\)
\(648\) 0 0
\(649\) 3.85043e12 0.851937
\(650\) 0 0
\(651\) −1.88595e12 −0.411543
\(652\) 0 0
\(653\) −6.05719e12 −1.30365 −0.651826 0.758369i \(-0.725997\pi\)
−0.651826 + 0.758369i \(0.725997\pi\)
\(654\) 0 0
\(655\) 3.79827e11 0.0806306
\(656\) 0 0
\(657\) 5.46855e12 1.14506
\(658\) 0 0
\(659\) −4.97718e12 −1.02801 −0.514007 0.857786i \(-0.671840\pi\)
−0.514007 + 0.857786i \(0.671840\pi\)
\(660\) 0 0
\(661\) 1.79834e12 0.366409 0.183204 0.983075i \(-0.441353\pi\)
0.183204 + 0.983075i \(0.441353\pi\)
\(662\) 0 0
\(663\) −1.49487e11 −0.0300465
\(664\) 0 0
\(665\) 5.90532e11 0.117097
\(666\) 0 0
\(667\) 2.80967e12 0.549653
\(668\) 0 0
\(669\) 7.34986e11 0.141860
\(670\) 0 0
\(671\) 2.18362e12 0.415839
\(672\) 0 0
\(673\) 1.37693e12 0.258728 0.129364 0.991597i \(-0.458706\pi\)
0.129364 + 0.991597i \(0.458706\pi\)
\(674\) 0 0
\(675\) −3.44547e12 −0.638824
\(676\) 0 0
\(677\) 6.49642e11 0.118857 0.0594285 0.998233i \(-0.481072\pi\)
0.0594285 + 0.998233i \(0.481072\pi\)
\(678\) 0 0
\(679\) 7.45828e12 1.34656
\(680\) 0 0
\(681\) 2.01386e12 0.358813
\(682\) 0 0
\(683\) 1.10011e12 0.193438 0.0967190 0.995312i \(-0.469165\pi\)
0.0967190 + 0.995312i \(0.469165\pi\)
\(684\) 0 0
\(685\) −5.54064e11 −0.0961507
\(686\) 0 0
\(687\) 3.20192e12 0.548409
\(688\) 0 0
\(689\) 4.37301e11 0.0739254
\(690\) 0 0
\(691\) −7.72289e12 −1.28863 −0.644315 0.764760i \(-0.722858\pi\)
−0.644315 + 0.764760i \(0.722858\pi\)
\(692\) 0 0
\(693\) 2.60702e12 0.429383
\(694\) 0 0
\(695\) 3.94832e11 0.0641920
\(696\) 0 0
\(697\) 1.63505e12 0.262412
\(698\) 0 0
\(699\) 2.48617e12 0.393898
\(700\) 0 0
\(701\) 1.40436e12 0.219658 0.109829 0.993950i \(-0.464970\pi\)
0.109829 + 0.993950i \(0.464970\pi\)
\(702\) 0 0
\(703\) 3.74831e12 0.578810
\(704\) 0 0
\(705\) −3.31159e10 −0.00504877
\(706\) 0 0
\(707\) 5.39111e12 0.811505
\(708\) 0 0
\(709\) 5.19764e12 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(710\) 0 0
\(711\) −3.48822e12 −0.511907
\(712\) 0 0
\(713\) −2.88600e12 −0.418209
\(714\) 0 0
\(715\) −7.90176e10 −0.0113070
\(716\) 0 0
\(717\) 3.57116e12 0.504630
\(718\) 0 0
\(719\) 1.27042e13 1.77284 0.886418 0.462886i \(-0.153186\pi\)
0.886418 + 0.462886i \(0.153186\pi\)
\(720\) 0 0
\(721\) −2.48016e12 −0.341799
\(722\) 0 0
\(723\) −2.75230e12 −0.374605
\(724\) 0 0
\(725\) −1.25159e13 −1.68244
\(726\) 0 0
\(727\) −9.91998e11 −0.131706 −0.0658531 0.997829i \(-0.520977\pi\)
−0.0658531 + 0.997829i \(0.520977\pi\)
\(728\) 0 0
\(729\) −2.80211e12 −0.367461
\(730\) 0 0
\(731\) −4.39227e12 −0.568933
\(732\) 0 0
\(733\) 5.54849e12 0.709916 0.354958 0.934882i \(-0.384495\pi\)
0.354958 + 0.934882i \(0.384495\pi\)
\(734\) 0 0
\(735\) −2.61791e10 −0.00330873
\(736\) 0 0
\(737\) 2.54663e12 0.317953
\(738\) 0 0
\(739\) −8.78597e12 −1.08365 −0.541826 0.840491i \(-0.682267\pi\)
−0.541826 + 0.840491i \(0.682267\pi\)
\(740\) 0 0
\(741\) −1.23701e12 −0.150727
\(742\) 0 0
\(743\) −4.04326e12 −0.486724 −0.243362 0.969936i \(-0.578250\pi\)
−0.243362 + 0.969936i \(0.578250\pi\)
\(744\) 0 0
\(745\) −1.40014e11 −0.0166521
\(746\) 0 0
\(747\) 1.10357e12 0.129676
\(748\) 0 0
\(749\) 3.99451e12 0.463763
\(750\) 0 0
\(751\) 2.52936e11 0.0290156 0.0145078 0.999895i \(-0.495382\pi\)
0.0145078 + 0.999895i \(0.495382\pi\)
\(752\) 0 0
\(753\) −5.16479e12 −0.585431
\(754\) 0 0
\(755\) 4.77765e11 0.0535122
\(756\) 0 0
\(757\) 9.99638e12 1.10640 0.553199 0.833049i \(-0.313407\pi\)
0.553199 + 0.833049i \(0.313407\pi\)
\(758\) 0 0
\(759\) −5.25864e11 −0.0575156
\(760\) 0 0
\(761\) −2.98848e12 −0.323013 −0.161507 0.986872i \(-0.551635\pi\)
−0.161507 + 0.986872i \(0.551635\pi\)
\(762\) 0 0
\(763\) −9.91726e12 −1.05933
\(764\) 0 0
\(765\) 2.08670e11 0.0220284
\(766\) 0 0
\(767\) 4.36299e12 0.455203
\(768\) 0 0
\(769\) 1.54887e13 1.59715 0.798576 0.601894i \(-0.205587\pi\)
0.798576 + 0.601894i \(0.205587\pi\)
\(770\) 0 0
\(771\) 3.17948e12 0.324050
\(772\) 0 0
\(773\) 1.07589e13 1.08383 0.541915 0.840434i \(-0.317700\pi\)
0.541915 + 0.840434i \(0.317700\pi\)
\(774\) 0 0
\(775\) 1.28559e13 1.28010
\(776\) 0 0
\(777\) 1.17990e12 0.116131
\(778\) 0 0
\(779\) 1.35301e13 1.31638
\(780\) 0 0
\(781\) −1.04130e13 −1.00149
\(782\) 0 0
\(783\) 1.14452e13 1.08817
\(784\) 0 0
\(785\) 1.54767e12 0.145467
\(786\) 0 0
\(787\) −1.29304e13 −1.20151 −0.600755 0.799433i \(-0.705133\pi\)
−0.600755 + 0.799433i \(0.705133\pi\)
\(788\) 0 0
\(789\) −3.90379e12 −0.358625
\(790\) 0 0
\(791\) 1.09778e13 0.997059
\(792\) 0 0
\(793\) 2.47430e12 0.222189
\(794\) 0 0
\(795\) 8.04635e10 0.00714409
\(796\) 0 0
\(797\) 1.24713e13 1.09484 0.547418 0.836859i \(-0.315611\pi\)
0.547418 + 0.836859i \(0.315611\pi\)
\(798\) 0 0
\(799\) −6.88868e11 −0.0597965
\(800\) 0 0
\(801\) −6.03585e11 −0.0518075
\(802\) 0 0
\(803\) −7.92600e12 −0.672719
\(804\) 0 0
\(805\) 2.84459e11 0.0238747
\(806\) 0 0
\(807\) −4.81502e12 −0.399639
\(808\) 0 0
\(809\) −2.40763e13 −1.97616 −0.988079 0.153948i \(-0.950801\pi\)
−0.988079 + 0.153948i \(0.950801\pi\)
\(810\) 0 0
\(811\) −1.14686e13 −0.930928 −0.465464 0.885067i \(-0.654112\pi\)
−0.465464 + 0.885067i \(0.654112\pi\)
\(812\) 0 0
\(813\) −1.43040e12 −0.114829
\(814\) 0 0
\(815\) 7.95569e10 0.00631638
\(816\) 0 0
\(817\) −3.63462e13 −2.85403
\(818\) 0 0
\(819\) 2.95406e12 0.229426
\(820\) 0 0
\(821\) −2.50113e13 −1.92129 −0.960643 0.277786i \(-0.910399\pi\)
−0.960643 + 0.277786i \(0.910399\pi\)
\(822\) 0 0
\(823\) −1.29351e13 −0.982808 −0.491404 0.870932i \(-0.663516\pi\)
−0.491404 + 0.870932i \(0.663516\pi\)
\(824\) 0 0
\(825\) 2.34250e12 0.176051
\(826\) 0 0
\(827\) −1.00464e13 −0.746851 −0.373425 0.927660i \(-0.621817\pi\)
−0.373425 + 0.927660i \(0.621817\pi\)
\(828\) 0 0
\(829\) 8.77304e12 0.645141 0.322570 0.946545i \(-0.395453\pi\)
0.322570 + 0.946545i \(0.395453\pi\)
\(830\) 0 0
\(831\) 1.51868e12 0.110474
\(832\) 0 0
\(833\) −5.44571e11 −0.0391879
\(834\) 0 0
\(835\) −9.14353e10 −0.00650916
\(836\) 0 0
\(837\) −1.17561e13 −0.827943
\(838\) 0 0
\(839\) 1.54259e13 1.07478 0.537392 0.843332i \(-0.319410\pi\)
0.537392 + 0.843332i \(0.319410\pi\)
\(840\) 0 0
\(841\) 2.70683e13 1.86586
\(842\) 0 0
\(843\) −4.72537e12 −0.322264
\(844\) 0 0
\(845\) −8.95364e10 −0.00604149
\(846\) 0 0
\(847\) 1.02452e13 0.683983
\(848\) 0 0
\(849\) −5.08296e12 −0.335762
\(850\) 0 0
\(851\) 1.80556e12 0.118012
\(852\) 0 0
\(853\) −1.84928e12 −0.119600 −0.0598002 0.998210i \(-0.519046\pi\)
−0.0598002 + 0.998210i \(0.519046\pi\)
\(854\) 0 0
\(855\) 1.72675e12 0.110505
\(856\) 0 0
\(857\) −2.34715e13 −1.48637 −0.743185 0.669086i \(-0.766686\pi\)
−0.743185 + 0.669086i \(0.766686\pi\)
\(858\) 0 0
\(859\) −1.25121e12 −0.0784079 −0.0392039 0.999231i \(-0.512482\pi\)
−0.0392039 + 0.999231i \(0.512482\pi\)
\(860\) 0 0
\(861\) 4.25902e12 0.264116
\(862\) 0 0
\(863\) −2.38217e13 −1.46192 −0.730961 0.682419i \(-0.760928\pi\)
−0.730961 + 0.682419i \(0.760928\pi\)
\(864\) 0 0
\(865\) −6.34375e10 −0.00385278
\(866\) 0 0
\(867\) 5.10563e12 0.306876
\(868\) 0 0
\(869\) 5.05575e12 0.300744
\(870\) 0 0
\(871\) 2.88564e12 0.169887
\(872\) 0 0
\(873\) 2.18084e13 1.27075
\(874\) 0 0
\(875\) −2.54215e12 −0.146611
\(876\) 0 0
\(877\) −1.35717e13 −0.774707 −0.387353 0.921931i \(-0.626611\pi\)
−0.387353 + 0.921931i \(0.626611\pi\)
\(878\) 0 0
\(879\) −1.08375e13 −0.612323
\(880\) 0 0
\(881\) 2.38436e12 0.133346 0.0666730 0.997775i \(-0.478762\pi\)
0.0666730 + 0.997775i \(0.478762\pi\)
\(882\) 0 0
\(883\) 5.99779e11 0.0332023 0.0166011 0.999862i \(-0.494715\pi\)
0.0166011 + 0.999862i \(0.494715\pi\)
\(884\) 0 0
\(885\) 8.02792e11 0.0439904
\(886\) 0 0
\(887\) −1.02483e13 −0.555897 −0.277949 0.960596i \(-0.589654\pi\)
−0.277949 + 0.960596i \(0.589654\pi\)
\(888\) 0 0
\(889\) 1.11766e13 0.600138
\(890\) 0 0
\(891\) 6.48579e12 0.344757
\(892\) 0 0
\(893\) −5.70040e12 −0.299967
\(894\) 0 0
\(895\) −1.71854e12 −0.0895274
\(896\) 0 0
\(897\) −5.95867e11 −0.0307315
\(898\) 0 0
\(899\) −4.27050e13 −2.18052
\(900\) 0 0
\(901\) 1.67378e12 0.0846130
\(902\) 0 0
\(903\) −1.14411e13 −0.572628
\(904\) 0 0
\(905\) 2.39887e12 0.118874
\(906\) 0 0
\(907\) 2.43855e13 1.19646 0.598231 0.801323i \(-0.295870\pi\)
0.598231 + 0.801323i \(0.295870\pi\)
\(908\) 0 0
\(909\) 1.57639e13 0.765820
\(910\) 0 0
\(911\) −2.90386e13 −1.39683 −0.698415 0.715693i \(-0.746111\pi\)
−0.698415 + 0.715693i \(0.746111\pi\)
\(912\) 0 0
\(913\) −1.59949e12 −0.0761841
\(914\) 0 0
\(915\) 4.55272e11 0.0214722
\(916\) 0 0
\(917\) 2.05809e13 0.961173
\(918\) 0 0
\(919\) 1.41793e13 0.655746 0.327873 0.944722i \(-0.393668\pi\)
0.327873 + 0.944722i \(0.393668\pi\)
\(920\) 0 0
\(921\) −1.06773e12 −0.0488984
\(922\) 0 0
\(923\) −1.17992e13 −0.535111
\(924\) 0 0
\(925\) −8.04299e12 −0.361226
\(926\) 0 0
\(927\) −7.25212e12 −0.322557
\(928\) 0 0
\(929\) −3.56246e13 −1.56920 −0.784601 0.620001i \(-0.787132\pi\)
−0.784601 + 0.620001i \(0.787132\pi\)
\(930\) 0 0
\(931\) −4.50634e12 −0.196585
\(932\) 0 0
\(933\) −1.30136e13 −0.562250
\(934\) 0 0
\(935\) −3.02442e11 −0.0129417
\(936\) 0 0
\(937\) 9.35575e12 0.396506 0.198253 0.980151i \(-0.436473\pi\)
0.198253 + 0.980151i \(0.436473\pi\)
\(938\) 0 0
\(939\) 3.78610e12 0.158927
\(940\) 0 0
\(941\) 1.23413e13 0.513106 0.256553 0.966530i \(-0.417413\pi\)
0.256553 + 0.966530i \(0.417413\pi\)
\(942\) 0 0
\(943\) 6.51742e12 0.268394
\(944\) 0 0
\(945\) 1.15875e12 0.0472656
\(946\) 0 0
\(947\) −2.93722e12 −0.118676 −0.0593379 0.998238i \(-0.518899\pi\)
−0.0593379 + 0.998238i \(0.518899\pi\)
\(948\) 0 0
\(949\) −8.98110e12 −0.359444
\(950\) 0 0
\(951\) −9.72086e12 −0.385383
\(952\) 0 0
\(953\) −3.90544e12 −0.153374 −0.0766870 0.997055i \(-0.524434\pi\)
−0.0766870 + 0.997055i \(0.524434\pi\)
\(954\) 0 0
\(955\) 1.84163e12 0.0716454
\(956\) 0 0
\(957\) −7.78137e12 −0.299883
\(958\) 0 0
\(959\) −3.00219e13 −1.14618
\(960\) 0 0
\(961\) 1.74255e13 0.659069
\(962\) 0 0
\(963\) 1.16802e13 0.437655
\(964\) 0 0
\(965\) 2.96398e12 0.110028
\(966\) 0 0
\(967\) 1.06697e13 0.392405 0.196203 0.980563i \(-0.437139\pi\)
0.196203 + 0.980563i \(0.437139\pi\)
\(968\) 0 0
\(969\) −4.73469e12 −0.172518
\(970\) 0 0
\(971\) 3.10264e12 0.112007 0.0560034 0.998431i \(-0.482164\pi\)
0.0560034 + 0.998431i \(0.482164\pi\)
\(972\) 0 0
\(973\) 2.13939e13 0.765213
\(974\) 0 0
\(975\) 2.65434e12 0.0940665
\(976\) 0 0
\(977\) −4.66167e13 −1.63688 −0.818438 0.574594i \(-0.805160\pi\)
−0.818438 + 0.574594i \(0.805160\pi\)
\(978\) 0 0
\(979\) 8.74824e11 0.0304368
\(980\) 0 0
\(981\) −2.89986e13 −0.999694
\(982\) 0 0
\(983\) 3.22316e13 1.10101 0.550505 0.834832i \(-0.314435\pi\)
0.550505 + 0.834832i \(0.314435\pi\)
\(984\) 0 0
\(985\) −1.73728e12 −0.0588040
\(986\) 0 0
\(987\) −1.79438e12 −0.0601849
\(988\) 0 0
\(989\) −1.75079e13 −0.581903
\(990\) 0 0
\(991\) 2.25008e13 0.741081 0.370540 0.928816i \(-0.379172\pi\)
0.370540 + 0.928816i \(0.379172\pi\)
\(992\) 0 0
\(993\) 1.55192e13 0.506520
\(994\) 0 0
\(995\) 9.66342e11 0.0312556
\(996\) 0 0
\(997\) −3.13756e13 −1.00569 −0.502844 0.864377i \(-0.667713\pi\)
−0.502844 + 0.864377i \(0.667713\pi\)
\(998\) 0 0
\(999\) 7.35495e12 0.233633
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 208.10.a.h.1.3 5
4.3 odd 2 13.10.a.b.1.5 5
12.11 even 2 117.10.a.e.1.1 5
20.19 odd 2 325.10.a.b.1.1 5
52.51 odd 2 169.10.a.b.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.b.1.5 5 4.3 odd 2
117.10.a.e.1.1 5 12.11 even 2
169.10.a.b.1.1 5 52.51 odd 2
208.10.a.h.1.3 5 1.1 even 1 trivial
325.10.a.b.1.1 5 20.19 odd 2