Properties

Label 531.8.a.e.1.8
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,8,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(6.43791\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.43791 q^{2} -72.6775 q^{4} -171.827 q^{5} +1529.87 q^{7} +1492.62 q^{8} +O(q^{10})\) \(q-7.43791 q^{2} -72.6775 q^{4} -171.827 q^{5} +1529.87 q^{7} +1492.62 q^{8} +1278.03 q^{10} +1638.92 q^{11} -7315.98 q^{13} -11379.0 q^{14} -1799.27 q^{16} -33214.7 q^{17} +55570.3 q^{19} +12488.0 q^{20} -12190.1 q^{22} +51886.9 q^{23} -48600.5 q^{25} +54415.6 q^{26} -111187. q^{28} -10383.5 q^{29} +92770.0 q^{31} -177673. q^{32} +247048. q^{34} -262873. q^{35} -504232. q^{37} -413327. q^{38} -256473. q^{40} -635489. q^{41} +101770. q^{43} -119112. q^{44} -385930. q^{46} +72413.1 q^{47} +1.51695e6 q^{49} +361486. q^{50} +531707. q^{52} -1.67234e6 q^{53} -281610. q^{55} +2.28351e6 q^{56} +77231.8 q^{58} -205379. q^{59} +727353. q^{61} -690015. q^{62} +1.55182e6 q^{64} +1.25708e6 q^{65} +2.81505e6 q^{67} +2.41396e6 q^{68} +1.95522e6 q^{70} +850222. q^{71} +5.84245e6 q^{73} +3.75043e6 q^{74} -4.03871e6 q^{76} +2.50732e6 q^{77} +2.74755e6 q^{79} +309162. q^{80} +4.72671e6 q^{82} -2.78632e6 q^{83} +5.70719e6 q^{85} -756957. q^{86} +2.44628e6 q^{88} +6.16092e6 q^{89} -1.11925e7 q^{91} -3.77101e6 q^{92} -538602. q^{94} -9.54849e6 q^{95} -1.67686e7 q^{97} -1.12830e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.43791 −0.657425 −0.328712 0.944430i \(-0.606615\pi\)
−0.328712 + 0.944430i \(0.606615\pi\)
\(3\) 0 0
\(4\) −72.6775 −0.567793
\(5\) −171.827 −0.614747 −0.307374 0.951589i \(-0.599450\pi\)
−0.307374 + 0.951589i \(0.599450\pi\)
\(6\) 0 0
\(7\) 1529.87 1.68582 0.842909 0.538056i \(-0.180841\pi\)
0.842909 + 0.538056i \(0.180841\pi\)
\(8\) 1492.62 1.03071
\(9\) 0 0
\(10\) 1278.03 0.404150
\(11\) 1638.92 0.371263 0.185632 0.982619i \(-0.440567\pi\)
0.185632 + 0.982619i \(0.440567\pi\)
\(12\) 0 0
\(13\) −7315.98 −0.923572 −0.461786 0.886991i \(-0.652791\pi\)
−0.461786 + 0.886991i \(0.652791\pi\)
\(14\) −11379.0 −1.10830
\(15\) 0 0
\(16\) −1799.27 −0.109818
\(17\) −33214.7 −1.63968 −0.819841 0.572592i \(-0.805938\pi\)
−0.819841 + 0.572592i \(0.805938\pi\)
\(18\) 0 0
\(19\) 55570.3 1.85868 0.929342 0.369221i \(-0.120375\pi\)
0.929342 + 0.369221i \(0.120375\pi\)
\(20\) 12488.0 0.349049
\(21\) 0 0
\(22\) −12190.1 −0.244078
\(23\) 51886.9 0.889222 0.444611 0.895724i \(-0.353342\pi\)
0.444611 + 0.895724i \(0.353342\pi\)
\(24\) 0 0
\(25\) −48600.5 −0.622086
\(26\) 54415.6 0.607179
\(27\) 0 0
\(28\) −111187. −0.957196
\(29\) −10383.5 −0.0790591 −0.0395296 0.999218i \(-0.512586\pi\)
−0.0395296 + 0.999218i \(0.512586\pi\)
\(30\) 0 0
\(31\) 92770.0 0.559296 0.279648 0.960103i \(-0.409782\pi\)
0.279648 + 0.960103i \(0.409782\pi\)
\(32\) −177673. −0.958508
\(33\) 0 0
\(34\) 247048. 1.07797
\(35\) −262873. −1.03635
\(36\) 0 0
\(37\) −504232. −1.63653 −0.818266 0.574840i \(-0.805064\pi\)
−0.818266 + 0.574840i \(0.805064\pi\)
\(38\) −413327. −1.22194
\(39\) 0 0
\(40\) −256473. −0.633623
\(41\) −635489. −1.44001 −0.720003 0.693971i \(-0.755860\pi\)
−0.720003 + 0.693971i \(0.755860\pi\)
\(42\) 0 0
\(43\) 101770. 0.195200 0.0976002 0.995226i \(-0.468883\pi\)
0.0976002 + 0.995226i \(0.468883\pi\)
\(44\) −119112. −0.210801
\(45\) 0 0
\(46\) −385930. −0.584597
\(47\) 72413.1 0.101736 0.0508680 0.998705i \(-0.483801\pi\)
0.0508680 + 0.998705i \(0.483801\pi\)
\(48\) 0 0
\(49\) 1.51695e6 1.84198
\(50\) 361486. 0.408975
\(51\) 0 0
\(52\) 531707. 0.524397
\(53\) −1.67234e6 −1.54297 −0.771487 0.636245i \(-0.780487\pi\)
−0.771487 + 0.636245i \(0.780487\pi\)
\(54\) 0 0
\(55\) −281610. −0.228233
\(56\) 2.28351e6 1.73758
\(57\) 0 0
\(58\) 77231.8 0.0519754
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 727353. 0.410290 0.205145 0.978732i \(-0.434233\pi\)
0.205145 + 0.978732i \(0.434233\pi\)
\(62\) −690015. −0.367695
\(63\) 0 0
\(64\) 1.55182e6 0.739965
\(65\) 1.25708e6 0.567763
\(66\) 0 0
\(67\) 2.81505e6 1.14347 0.571735 0.820439i \(-0.306271\pi\)
0.571735 + 0.820439i \(0.306271\pi\)
\(68\) 2.41396e6 0.930999
\(69\) 0 0
\(70\) 1.95522e6 0.681323
\(71\) 850222. 0.281921 0.140961 0.990015i \(-0.454981\pi\)
0.140961 + 0.990015i \(0.454981\pi\)
\(72\) 0 0
\(73\) 5.84245e6 1.75778 0.878891 0.477023i \(-0.158284\pi\)
0.878891 + 0.477023i \(0.158284\pi\)
\(74\) 3.75043e6 1.07590
\(75\) 0 0
\(76\) −4.03871e6 −1.05535
\(77\) 2.50732e6 0.625883
\(78\) 0 0
\(79\) 2.74755e6 0.626977 0.313488 0.949592i \(-0.398502\pi\)
0.313488 + 0.949592i \(0.398502\pi\)
\(80\) 309162. 0.0675106
\(81\) 0 0
\(82\) 4.72671e6 0.946695
\(83\) −2.78632e6 −0.534882 −0.267441 0.963574i \(-0.586178\pi\)
−0.267441 + 0.963574i \(0.586178\pi\)
\(84\) 0 0
\(85\) 5.70719e6 1.00799
\(86\) −756957. −0.128330
\(87\) 0 0
\(88\) 2.44628e6 0.382663
\(89\) 6.16092e6 0.926361 0.463181 0.886264i \(-0.346708\pi\)
0.463181 + 0.886264i \(0.346708\pi\)
\(90\) 0 0
\(91\) −1.11925e7 −1.55697
\(92\) −3.77101e6 −0.504894
\(93\) 0 0
\(94\) −538602. −0.0668838
\(95\) −9.54849e6 −1.14262
\(96\) 0 0
\(97\) −1.67686e7 −1.86550 −0.932751 0.360521i \(-0.882599\pi\)
−0.932751 + 0.360521i \(0.882599\pi\)
\(98\) −1.12830e7 −1.21097
\(99\) 0 0
\(100\) 3.53216e6 0.353216
\(101\) −7.93944e6 −0.766770 −0.383385 0.923589i \(-0.625242\pi\)
−0.383385 + 0.923589i \(0.625242\pi\)
\(102\) 0 0
\(103\) 1.76750e7 1.59378 0.796889 0.604125i \(-0.206477\pi\)
0.796889 + 0.604125i \(0.206477\pi\)
\(104\) −1.09200e7 −0.951931
\(105\) 0 0
\(106\) 1.24387e7 1.01439
\(107\) 9.16951e6 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(108\) 0 0
\(109\) 1.32228e7 0.977982 0.488991 0.872289i \(-0.337365\pi\)
0.488991 + 0.872289i \(0.337365\pi\)
\(110\) 2.09459e6 0.150046
\(111\) 0 0
\(112\) −2.75264e6 −0.185134
\(113\) 2.66031e7 1.73443 0.867216 0.497932i \(-0.165907\pi\)
0.867216 + 0.497932i \(0.165907\pi\)
\(114\) 0 0
\(115\) −8.91557e6 −0.546647
\(116\) 754649. 0.0448892
\(117\) 0 0
\(118\) 1.52759e6 0.0855894
\(119\) −5.08142e7 −2.76421
\(120\) 0 0
\(121\) −1.68011e7 −0.862163
\(122\) −5.40999e6 −0.269735
\(123\) 0 0
\(124\) −6.74229e6 −0.317564
\(125\) 2.17749e7 0.997173
\(126\) 0 0
\(127\) −2.04342e7 −0.885209 −0.442604 0.896717i \(-0.645945\pi\)
−0.442604 + 0.896717i \(0.645945\pi\)
\(128\) 1.11998e7 0.472037
\(129\) 0 0
\(130\) −9.35007e6 −0.373261
\(131\) 4.18739e7 1.62740 0.813699 0.581286i \(-0.197450\pi\)
0.813699 + 0.581286i \(0.197450\pi\)
\(132\) 0 0
\(133\) 8.50153e7 3.13340
\(134\) −2.09381e7 −0.751745
\(135\) 0 0
\(136\) −4.95770e7 −1.69003
\(137\) −5.60709e7 −1.86301 −0.931506 0.363725i \(-0.881505\pi\)
−0.931506 + 0.363725i \(0.881505\pi\)
\(138\) 0 0
\(139\) −3.90420e7 −1.23305 −0.616524 0.787336i \(-0.711460\pi\)
−0.616524 + 0.787336i \(0.711460\pi\)
\(140\) 1.91049e7 0.588433
\(141\) 0 0
\(142\) −6.32387e6 −0.185342
\(143\) −1.19903e7 −0.342888
\(144\) 0 0
\(145\) 1.78417e6 0.0486014
\(146\) −4.34556e7 −1.15561
\(147\) 0 0
\(148\) 3.66463e7 0.929211
\(149\) −5.58520e7 −1.38320 −0.691602 0.722278i \(-0.743095\pi\)
−0.691602 + 0.722278i \(0.743095\pi\)
\(150\) 0 0
\(151\) −6.78120e7 −1.60283 −0.801414 0.598110i \(-0.795919\pi\)
−0.801414 + 0.598110i \(0.795919\pi\)
\(152\) 8.29455e7 1.91576
\(153\) 0 0
\(154\) −1.86493e7 −0.411471
\(155\) −1.59404e7 −0.343826
\(156\) 0 0
\(157\) 4.45770e7 0.919310 0.459655 0.888098i \(-0.347973\pi\)
0.459655 + 0.888098i \(0.347973\pi\)
\(158\) −2.04361e7 −0.412190
\(159\) 0 0
\(160\) 3.05290e7 0.589240
\(161\) 7.93801e7 1.49907
\(162\) 0 0
\(163\) −9.62679e7 −1.74111 −0.870553 0.492075i \(-0.836238\pi\)
−0.870553 + 0.492075i \(0.836238\pi\)
\(164\) 4.61857e7 0.817625
\(165\) 0 0
\(166\) 2.07244e7 0.351645
\(167\) 3.06157e7 0.508671 0.254335 0.967116i \(-0.418143\pi\)
0.254335 + 0.967116i \(0.418143\pi\)
\(168\) 0 0
\(169\) −9.22500e6 −0.147015
\(170\) −4.24496e7 −0.662677
\(171\) 0 0
\(172\) −7.39640e6 −0.110833
\(173\) 1.60370e7 0.235484 0.117742 0.993044i \(-0.462434\pi\)
0.117742 + 0.993044i \(0.462434\pi\)
\(174\) 0 0
\(175\) −7.43523e7 −1.04872
\(176\) −2.94884e6 −0.0407716
\(177\) 0 0
\(178\) −4.58244e7 −0.609013
\(179\) 2.33939e7 0.304872 0.152436 0.988313i \(-0.451288\pi\)
0.152436 + 0.988313i \(0.451288\pi\)
\(180\) 0 0
\(181\) −9.84290e7 −1.23381 −0.616904 0.787038i \(-0.711614\pi\)
−0.616904 + 0.787038i \(0.711614\pi\)
\(182\) 8.32487e7 1.02359
\(183\) 0 0
\(184\) 7.74475e7 0.916527
\(185\) 8.66407e7 1.00605
\(186\) 0 0
\(187\) −5.44361e7 −0.608754
\(188\) −5.26280e6 −0.0577650
\(189\) 0 0
\(190\) 7.10208e7 0.751187
\(191\) −3.21552e7 −0.333914 −0.166957 0.985964i \(-0.553394\pi\)
−0.166957 + 0.985964i \(0.553394\pi\)
\(192\) 0 0
\(193\) 2.59381e7 0.259709 0.129854 0.991533i \(-0.458549\pi\)
0.129854 + 0.991533i \(0.458549\pi\)
\(194\) 1.24723e8 1.22643
\(195\) 0 0
\(196\) −1.10248e8 −1.04587
\(197\) 1.60676e7 0.149733 0.0748666 0.997194i \(-0.476147\pi\)
0.0748666 + 0.997194i \(0.476147\pi\)
\(198\) 0 0
\(199\) 6.47444e7 0.582394 0.291197 0.956663i \(-0.405947\pi\)
0.291197 + 0.956663i \(0.405947\pi\)
\(200\) −7.25421e7 −0.641187
\(201\) 0 0
\(202\) 5.90529e7 0.504094
\(203\) −1.58854e7 −0.133279
\(204\) 0 0
\(205\) 1.09194e8 0.885240
\(206\) −1.31465e8 −1.04779
\(207\) 0 0
\(208\) 1.31634e7 0.101425
\(209\) 9.10751e7 0.690061
\(210\) 0 0
\(211\) −4.05485e7 −0.297157 −0.148578 0.988901i \(-0.547470\pi\)
−0.148578 + 0.988901i \(0.547470\pi\)
\(212\) 1.21541e8 0.876090
\(213\) 0 0
\(214\) −6.82020e7 −0.475717
\(215\) −1.74869e7 −0.119999
\(216\) 0 0
\(217\) 1.41926e8 0.942872
\(218\) −9.83501e7 −0.642950
\(219\) 0 0
\(220\) 2.04667e7 0.129589
\(221\) 2.42998e8 1.51436
\(222\) 0 0
\(223\) −6.68937e7 −0.403941 −0.201971 0.979392i \(-0.564735\pi\)
−0.201971 + 0.979392i \(0.564735\pi\)
\(224\) −2.71816e8 −1.61587
\(225\) 0 0
\(226\) −1.97871e8 −1.14026
\(227\) −1.22298e8 −0.693949 −0.346975 0.937875i \(-0.612791\pi\)
−0.346975 + 0.937875i \(0.612791\pi\)
\(228\) 0 0
\(229\) −2.32389e8 −1.27876 −0.639382 0.768889i \(-0.720810\pi\)
−0.639382 + 0.768889i \(0.720810\pi\)
\(230\) 6.63132e7 0.359379
\(231\) 0 0
\(232\) −1.54987e7 −0.0814867
\(233\) −7.69697e7 −0.398634 −0.199317 0.979935i \(-0.563872\pi\)
−0.199317 + 0.979935i \(0.563872\pi\)
\(234\) 0 0
\(235\) −1.24425e7 −0.0625419
\(236\) 1.49264e7 0.0739203
\(237\) 0 0
\(238\) 3.77951e8 1.81726
\(239\) 1.06849e7 0.0506264 0.0253132 0.999680i \(-0.491942\pi\)
0.0253132 + 0.999680i \(0.491942\pi\)
\(240\) 0 0
\(241\) −3.74132e8 −1.72173 −0.860866 0.508831i \(-0.830078\pi\)
−0.860866 + 0.508831i \(0.830078\pi\)
\(242\) 1.24965e8 0.566808
\(243\) 0 0
\(244\) −5.28622e7 −0.232960
\(245\) −2.60654e8 −1.13235
\(246\) 0 0
\(247\) −4.06551e8 −1.71663
\(248\) 1.38470e8 0.576470
\(249\) 0 0
\(250\) −1.61959e8 −0.655566
\(251\) −2.63434e8 −1.05151 −0.525755 0.850636i \(-0.676217\pi\)
−0.525755 + 0.850636i \(0.676217\pi\)
\(252\) 0 0
\(253\) 8.50383e7 0.330136
\(254\) 1.51988e8 0.581958
\(255\) 0 0
\(256\) −2.81936e8 −1.05029
\(257\) 3.92808e8 1.44349 0.721746 0.692158i \(-0.243340\pi\)
0.721746 + 0.692158i \(0.243340\pi\)
\(258\) 0 0
\(259\) −7.71409e8 −2.75890
\(260\) −9.13616e7 −0.322372
\(261\) 0 0
\(262\) −3.11454e8 −1.06989
\(263\) 2.41410e8 0.818297 0.409148 0.912468i \(-0.365826\pi\)
0.409148 + 0.912468i \(0.365826\pi\)
\(264\) 0 0
\(265\) 2.87353e8 0.948539
\(266\) −6.32336e8 −2.05998
\(267\) 0 0
\(268\) −2.04591e8 −0.649254
\(269\) −2.35755e8 −0.738463 −0.369231 0.929337i \(-0.620379\pi\)
−0.369231 + 0.929337i \(0.620379\pi\)
\(270\) 0 0
\(271\) −1.49957e8 −0.457692 −0.228846 0.973463i \(-0.573495\pi\)
−0.228846 + 0.973463i \(0.573495\pi\)
\(272\) 5.97621e7 0.180067
\(273\) 0 0
\(274\) 4.17051e8 1.22479
\(275\) −7.96521e7 −0.230958
\(276\) 0 0
\(277\) −9.52984e7 −0.269406 −0.134703 0.990886i \(-0.543008\pi\)
−0.134703 + 0.990886i \(0.543008\pi\)
\(278\) 2.90391e8 0.810637
\(279\) 0 0
\(280\) −3.92369e8 −1.06817
\(281\) 7.30263e7 0.196339 0.0981696 0.995170i \(-0.468701\pi\)
0.0981696 + 0.995170i \(0.468701\pi\)
\(282\) 0 0
\(283\) 1.79740e8 0.471404 0.235702 0.971825i \(-0.424261\pi\)
0.235702 + 0.971825i \(0.424261\pi\)
\(284\) −6.17920e7 −0.160073
\(285\) 0 0
\(286\) 8.91825e7 0.225423
\(287\) −9.72214e8 −2.42759
\(288\) 0 0
\(289\) 6.92880e8 1.68856
\(290\) −1.32705e7 −0.0319517
\(291\) 0 0
\(292\) −4.24614e8 −0.998056
\(293\) −8.12673e8 −1.88747 −0.943733 0.330709i \(-0.892712\pi\)
−0.943733 + 0.330709i \(0.892712\pi\)
\(294\) 0 0
\(295\) 3.52897e7 0.0800333
\(296\) −7.52628e8 −1.68678
\(297\) 0 0
\(298\) 4.15422e8 0.909353
\(299\) −3.79603e8 −0.821261
\(300\) 0 0
\(301\) 1.55695e8 0.329073
\(302\) 5.04379e8 1.05374
\(303\) 0 0
\(304\) −9.99858e7 −0.204118
\(305\) −1.24979e8 −0.252225
\(306\) 0 0
\(307\) −1.93647e8 −0.381968 −0.190984 0.981593i \(-0.561168\pi\)
−0.190984 + 0.981593i \(0.561168\pi\)
\(308\) −1.82226e8 −0.355372
\(309\) 0 0
\(310\) 1.18563e8 0.226039
\(311\) −1.76764e8 −0.333220 −0.166610 0.986023i \(-0.553282\pi\)
−0.166610 + 0.986023i \(0.553282\pi\)
\(312\) 0 0
\(313\) 1.36535e8 0.251675 0.125837 0.992051i \(-0.459838\pi\)
0.125837 + 0.992051i \(0.459838\pi\)
\(314\) −3.31560e8 −0.604377
\(315\) 0 0
\(316\) −1.99685e8 −0.355993
\(317\) 1.67078e8 0.294586 0.147293 0.989093i \(-0.452944\pi\)
0.147293 + 0.989093i \(0.452944\pi\)
\(318\) 0 0
\(319\) −1.70177e7 −0.0293518
\(320\) −2.66645e8 −0.454892
\(321\) 0 0
\(322\) −5.90422e8 −0.985524
\(323\) −1.84575e9 −3.04765
\(324\) 0 0
\(325\) 3.55560e8 0.574541
\(326\) 7.16032e8 1.14465
\(327\) 0 0
\(328\) −9.48544e8 −1.48422
\(329\) 1.10783e8 0.171508
\(330\) 0 0
\(331\) 3.82887e8 0.580327 0.290163 0.956977i \(-0.406290\pi\)
0.290163 + 0.956977i \(0.406290\pi\)
\(332\) 2.02503e8 0.303702
\(333\) 0 0
\(334\) −2.27717e8 −0.334413
\(335\) −4.83702e8 −0.702944
\(336\) 0 0
\(337\) 7.71340e8 1.09785 0.548923 0.835873i \(-0.315038\pi\)
0.548923 + 0.835873i \(0.315038\pi\)
\(338\) 6.86147e7 0.0966515
\(339\) 0 0
\(340\) −4.14784e8 −0.572329
\(341\) 1.52042e8 0.207646
\(342\) 0 0
\(343\) 1.06083e9 1.41943
\(344\) 1.51904e8 0.201194
\(345\) 0 0
\(346\) −1.19281e8 −0.154813
\(347\) 1.31263e9 1.68651 0.843257 0.537511i \(-0.180635\pi\)
0.843257 + 0.537511i \(0.180635\pi\)
\(348\) 0 0
\(349\) −5.67291e8 −0.714360 −0.357180 0.934036i \(-0.616262\pi\)
−0.357180 + 0.934036i \(0.616262\pi\)
\(350\) 5.53026e8 0.689457
\(351\) 0 0
\(352\) −2.91191e8 −0.355859
\(353\) −1.79304e8 −0.216959 −0.108480 0.994099i \(-0.534598\pi\)
−0.108480 + 0.994099i \(0.534598\pi\)
\(354\) 0 0
\(355\) −1.46091e8 −0.173310
\(356\) −4.47760e8 −0.525981
\(357\) 0 0
\(358\) −1.74002e8 −0.200430
\(359\) 1.28294e9 1.46344 0.731722 0.681603i \(-0.238717\pi\)
0.731722 + 0.681603i \(0.238717\pi\)
\(360\) 0 0
\(361\) 2.19419e9 2.45470
\(362\) 7.32106e8 0.811136
\(363\) 0 0
\(364\) 8.13441e8 0.884039
\(365\) −1.00389e9 −1.08059
\(366\) 0 0
\(367\) 1.71884e7 0.0181512 0.00907560 0.999959i \(-0.497111\pi\)
0.00907560 + 0.999959i \(0.497111\pi\)
\(368\) −9.33583e7 −0.0976530
\(369\) 0 0
\(370\) −6.44426e8 −0.661404
\(371\) −2.55846e9 −2.60118
\(372\) 0 0
\(373\) −8.02127e8 −0.800318 −0.400159 0.916446i \(-0.631045\pi\)
−0.400159 + 0.916446i \(0.631045\pi\)
\(374\) 4.04891e8 0.400210
\(375\) 0 0
\(376\) 1.08085e8 0.104860
\(377\) 7.59657e7 0.0730168
\(378\) 0 0
\(379\) 9.63664e6 0.00909261 0.00454630 0.999990i \(-0.498553\pi\)
0.00454630 + 0.999990i \(0.498553\pi\)
\(380\) 6.93960e8 0.648772
\(381\) 0 0
\(382\) 2.39168e8 0.219523
\(383\) −8.26495e8 −0.751700 −0.375850 0.926681i \(-0.622649\pi\)
−0.375850 + 0.926681i \(0.622649\pi\)
\(384\) 0 0
\(385\) −4.30826e8 −0.384760
\(386\) −1.92925e8 −0.170739
\(387\) 0 0
\(388\) 1.21870e9 1.05922
\(389\) −7.27469e8 −0.626601 −0.313300 0.949654i \(-0.601435\pi\)
−0.313300 + 0.949654i \(0.601435\pi\)
\(390\) 0 0
\(391\) −1.72341e9 −1.45804
\(392\) 2.26424e9 1.89854
\(393\) 0 0
\(394\) −1.19509e8 −0.0984383
\(395\) −4.72104e8 −0.385432
\(396\) 0 0
\(397\) −1.15679e9 −0.927871 −0.463936 0.885869i \(-0.653563\pi\)
−0.463936 + 0.885869i \(0.653563\pi\)
\(398\) −4.81563e8 −0.382880
\(399\) 0 0
\(400\) 8.74451e7 0.0683165
\(401\) −9.57807e7 −0.0741776 −0.0370888 0.999312i \(-0.511808\pi\)
−0.0370888 + 0.999312i \(0.511808\pi\)
\(402\) 0 0
\(403\) −6.78703e8 −0.516550
\(404\) 5.77019e8 0.435367
\(405\) 0 0
\(406\) 1.18154e8 0.0876211
\(407\) −8.26394e8 −0.607585
\(408\) 0 0
\(409\) −1.12641e9 −0.814079 −0.407040 0.913410i \(-0.633439\pi\)
−0.407040 + 0.913410i \(0.633439\pi\)
\(410\) −8.12176e8 −0.581978
\(411\) 0 0
\(412\) −1.28457e9 −0.904936
\(413\) −3.14203e8 −0.219475
\(414\) 0 0
\(415\) 4.78766e8 0.328817
\(416\) 1.29985e9 0.885251
\(417\) 0 0
\(418\) −6.77408e8 −0.453663
\(419\) 2.46333e9 1.63596 0.817981 0.575245i \(-0.195093\pi\)
0.817981 + 0.575245i \(0.195093\pi\)
\(420\) 0 0
\(421\) 6.02698e8 0.393652 0.196826 0.980438i \(-0.436937\pi\)
0.196826 + 0.980438i \(0.436937\pi\)
\(422\) 3.01596e8 0.195358
\(423\) 0 0
\(424\) −2.49617e9 −1.59035
\(425\) 1.61425e9 1.02002
\(426\) 0 0
\(427\) 1.11275e9 0.691674
\(428\) −6.66417e8 −0.410859
\(429\) 0 0
\(430\) 1.30066e8 0.0788902
\(431\) −1.69548e9 −1.02005 −0.510027 0.860159i \(-0.670364\pi\)
−0.510027 + 0.860159i \(0.670364\pi\)
\(432\) 0 0
\(433\) −5.31537e8 −0.314649 −0.157324 0.987547i \(-0.550287\pi\)
−0.157324 + 0.987547i \(0.550287\pi\)
\(434\) −1.05563e9 −0.619867
\(435\) 0 0
\(436\) −9.61000e8 −0.555291
\(437\) 2.88337e9 1.65278
\(438\) 0 0
\(439\) −2.39663e9 −1.35200 −0.675998 0.736904i \(-0.736287\pi\)
−0.675998 + 0.736904i \(0.736287\pi\)
\(440\) −4.20337e8 −0.235241
\(441\) 0 0
\(442\) −1.80740e9 −0.995580
\(443\) 2.03310e9 1.11108 0.555542 0.831489i \(-0.312511\pi\)
0.555542 + 0.831489i \(0.312511\pi\)
\(444\) 0 0
\(445\) −1.05861e9 −0.569478
\(446\) 4.97550e8 0.265561
\(447\) 0 0
\(448\) 2.37408e9 1.24745
\(449\) −3.46417e9 −1.80608 −0.903040 0.429557i \(-0.858670\pi\)
−0.903040 + 0.429557i \(0.858670\pi\)
\(450\) 0 0
\(451\) −1.04151e9 −0.534622
\(452\) −1.93344e9 −0.984798
\(453\) 0 0
\(454\) 9.09639e8 0.456219
\(455\) 1.92317e9 0.957146
\(456\) 0 0
\(457\) −2.70035e8 −0.132347 −0.0661735 0.997808i \(-0.521079\pi\)
−0.0661735 + 0.997808i \(0.521079\pi\)
\(458\) 1.72849e9 0.840692
\(459\) 0 0
\(460\) 6.47962e8 0.310382
\(461\) 1.94305e9 0.923698 0.461849 0.886959i \(-0.347186\pi\)
0.461849 + 0.886959i \(0.347186\pi\)
\(462\) 0 0
\(463\) −1.57266e8 −0.0736378 −0.0368189 0.999322i \(-0.511722\pi\)
−0.0368189 + 0.999322i \(0.511722\pi\)
\(464\) 1.86827e7 0.00868215
\(465\) 0 0
\(466\) 5.72494e8 0.262072
\(467\) 1.00777e9 0.457881 0.228941 0.973440i \(-0.426474\pi\)
0.228941 + 0.973440i \(0.426474\pi\)
\(468\) 0 0
\(469\) 4.30666e9 1.92768
\(470\) 9.25465e7 0.0411166
\(471\) 0 0
\(472\) −3.06553e8 −0.134186
\(473\) 1.66793e8 0.0724708
\(474\) 0 0
\(475\) −2.70074e9 −1.15626
\(476\) 3.69305e9 1.56950
\(477\) 0 0
\(478\) −7.94732e7 −0.0332830
\(479\) −1.14824e9 −0.477372 −0.238686 0.971097i \(-0.576717\pi\)
−0.238686 + 0.971097i \(0.576717\pi\)
\(480\) 0 0
\(481\) 3.68895e9 1.51145
\(482\) 2.78276e9 1.13191
\(483\) 0 0
\(484\) 1.22106e9 0.489530
\(485\) 2.88130e9 1.14681
\(486\) 0 0
\(487\) 2.65456e9 1.04146 0.520729 0.853722i \(-0.325660\pi\)
0.520729 + 0.853722i \(0.325660\pi\)
\(488\) 1.08566e9 0.422888
\(489\) 0 0
\(490\) 1.93872e9 0.744438
\(491\) 2.25039e9 0.857971 0.428986 0.903311i \(-0.358871\pi\)
0.428986 + 0.903311i \(0.358871\pi\)
\(492\) 0 0
\(493\) 3.44886e8 0.129632
\(494\) 3.02389e9 1.12855
\(495\) 0 0
\(496\) −1.66918e8 −0.0614210
\(497\) 1.30073e9 0.475268
\(498\) 0 0
\(499\) 7.47962e8 0.269481 0.134740 0.990881i \(-0.456980\pi\)
0.134740 + 0.990881i \(0.456980\pi\)
\(500\) −1.58254e9 −0.566188
\(501\) 0 0
\(502\) 1.95940e9 0.691288
\(503\) −5.29706e9 −1.85587 −0.927933 0.372746i \(-0.878416\pi\)
−0.927933 + 0.372746i \(0.878416\pi\)
\(504\) 0 0
\(505\) 1.36421e9 0.471370
\(506\) −6.32507e8 −0.217039
\(507\) 0 0
\(508\) 1.48511e9 0.502615
\(509\) −2.52094e9 −0.847327 −0.423663 0.905820i \(-0.639256\pi\)
−0.423663 + 0.905820i \(0.639256\pi\)
\(510\) 0 0
\(511\) 8.93818e9 2.96330
\(512\) 6.63440e8 0.218452
\(513\) 0 0
\(514\) −2.92167e9 −0.948988
\(515\) −3.03703e9 −0.979771
\(516\) 0 0
\(517\) 1.18679e8 0.0377709
\(518\) 5.73767e9 1.81377
\(519\) 0 0
\(520\) 1.87635e9 0.585197
\(521\) −6.27028e9 −1.94247 −0.971236 0.238118i \(-0.923470\pi\)
−0.971236 + 0.238118i \(0.923470\pi\)
\(522\) 0 0
\(523\) −6.45997e8 −0.197458 −0.0987289 0.995114i \(-0.531478\pi\)
−0.0987289 + 0.995114i \(0.531478\pi\)
\(524\) −3.04329e9 −0.924025
\(525\) 0 0
\(526\) −1.79559e9 −0.537969
\(527\) −3.08133e9 −0.917067
\(528\) 0 0
\(529\) −7.12574e8 −0.209284
\(530\) −2.13731e9 −0.623593
\(531\) 0 0
\(532\) −6.17870e9 −1.77912
\(533\) 4.64922e9 1.32995
\(534\) 0 0
\(535\) −1.57557e9 −0.444835
\(536\) 4.20181e9 1.17858
\(537\) 0 0
\(538\) 1.75353e9 0.485484
\(539\) 2.48616e9 0.683861
\(540\) 0 0
\(541\) −3.10398e9 −0.842807 −0.421404 0.906873i \(-0.638462\pi\)
−0.421404 + 0.906873i \(0.638462\pi\)
\(542\) 1.11536e9 0.300898
\(543\) 0 0
\(544\) 5.90135e9 1.57165
\(545\) −2.27204e9 −0.601212
\(546\) 0 0
\(547\) −2.73990e9 −0.715779 −0.357890 0.933764i \(-0.616504\pi\)
−0.357890 + 0.933764i \(0.616504\pi\)
\(548\) 4.07509e9 1.05781
\(549\) 0 0
\(550\) 5.92445e8 0.151837
\(551\) −5.77016e8 −0.146946
\(552\) 0 0
\(553\) 4.20339e9 1.05697
\(554\) 7.08821e8 0.177114
\(555\) 0 0
\(556\) 2.83747e9 0.700116
\(557\) −2.13952e9 −0.524595 −0.262297 0.964987i \(-0.584480\pi\)
−0.262297 + 0.964987i \(0.584480\pi\)
\(558\) 0 0
\(559\) −7.44548e8 −0.180282
\(560\) 4.72978e8 0.113811
\(561\) 0 0
\(562\) −5.43163e8 −0.129078
\(563\) 2.85723e9 0.674786 0.337393 0.941364i \(-0.390455\pi\)
0.337393 + 0.941364i \(0.390455\pi\)
\(564\) 0 0
\(565\) −4.57113e9 −1.06624
\(566\) −1.33689e9 −0.309912
\(567\) 0 0
\(568\) 1.26906e9 0.290578
\(569\) 3.13031e9 0.712352 0.356176 0.934419i \(-0.384080\pi\)
0.356176 + 0.934419i \(0.384080\pi\)
\(570\) 0 0
\(571\) −4.57416e9 −1.02822 −0.514109 0.857725i \(-0.671877\pi\)
−0.514109 + 0.857725i \(0.671877\pi\)
\(572\) 8.71422e8 0.194690
\(573\) 0 0
\(574\) 7.23124e9 1.59596
\(575\) −2.52173e9 −0.553173
\(576\) 0 0
\(577\) −5.19496e9 −1.12581 −0.562907 0.826520i \(-0.690317\pi\)
−0.562907 + 0.826520i \(0.690317\pi\)
\(578\) −5.15358e9 −1.11010
\(579\) 0 0
\(580\) −1.29669e8 −0.0275955
\(581\) −4.26271e9 −0.901714
\(582\) 0 0
\(583\) −2.74082e9 −0.572850
\(584\) 8.72056e9 1.81176
\(585\) 0 0
\(586\) 6.04459e9 1.24087
\(587\) −2.60032e9 −0.530632 −0.265316 0.964162i \(-0.585476\pi\)
−0.265316 + 0.964162i \(0.585476\pi\)
\(588\) 0 0
\(589\) 5.15526e9 1.03955
\(590\) −2.62481e8 −0.0526158
\(591\) 0 0
\(592\) 9.07248e8 0.179721
\(593\) −7.30947e9 −1.43944 −0.719721 0.694263i \(-0.755731\pi\)
−0.719721 + 0.694263i \(0.755731\pi\)
\(594\) 0 0
\(595\) 8.73125e9 1.69929
\(596\) 4.05918e9 0.785374
\(597\) 0 0
\(598\) 2.82346e9 0.539917
\(599\) −1.02859e9 −0.195545 −0.0977725 0.995209i \(-0.531172\pi\)
−0.0977725 + 0.995209i \(0.531172\pi\)
\(600\) 0 0
\(601\) −2.84013e9 −0.533676 −0.266838 0.963741i \(-0.585979\pi\)
−0.266838 + 0.963741i \(0.585979\pi\)
\(602\) −1.15804e9 −0.216340
\(603\) 0 0
\(604\) 4.92840e9 0.910074
\(605\) 2.88689e9 0.530013
\(606\) 0 0
\(607\) 8.39067e9 1.52278 0.761388 0.648296i \(-0.224518\pi\)
0.761388 + 0.648296i \(0.224518\pi\)
\(608\) −9.87333e9 −1.78156
\(609\) 0 0
\(610\) 9.29582e8 0.165819
\(611\) −5.29773e8 −0.0939605
\(612\) 0 0
\(613\) −2.69978e9 −0.473388 −0.236694 0.971584i \(-0.576064\pi\)
−0.236694 + 0.971584i \(0.576064\pi\)
\(614\) 1.44033e9 0.251115
\(615\) 0 0
\(616\) 3.74249e9 0.645101
\(617\) −3.54215e9 −0.607112 −0.303556 0.952814i \(-0.598174\pi\)
−0.303556 + 0.952814i \(0.598174\pi\)
\(618\) 0 0
\(619\) −7.53401e9 −1.27676 −0.638380 0.769721i \(-0.720395\pi\)
−0.638380 + 0.769721i \(0.720395\pi\)
\(620\) 1.15851e9 0.195222
\(621\) 0 0
\(622\) 1.31475e9 0.219067
\(623\) 9.42539e9 1.56168
\(624\) 0 0
\(625\) 5.54007e7 0.00907685
\(626\) −1.01554e9 −0.165457
\(627\) 0 0
\(628\) −3.23974e9 −0.521978
\(629\) 1.67479e10 2.68339
\(630\) 0 0
\(631\) 1.21541e10 1.92584 0.962920 0.269787i \(-0.0869533\pi\)
0.962920 + 0.269787i \(0.0869533\pi\)
\(632\) 4.10106e9 0.646228
\(633\) 0 0
\(634\) −1.24271e9 −0.193668
\(635\) 3.51116e9 0.544179
\(636\) 0 0
\(637\) −1.10980e10 −1.70120
\(638\) 1.26576e8 0.0192966
\(639\) 0 0
\(640\) −1.92443e9 −0.290183
\(641\) −6.74739e9 −1.01189 −0.505944 0.862566i \(-0.668856\pi\)
−0.505944 + 0.862566i \(0.668856\pi\)
\(642\) 0 0
\(643\) −8.85998e9 −1.31430 −0.657150 0.753760i \(-0.728238\pi\)
−0.657150 + 0.753760i \(0.728238\pi\)
\(644\) −5.76915e9 −0.851160
\(645\) 0 0
\(646\) 1.37286e10 2.00360
\(647\) 5.90409e9 0.857015 0.428507 0.903538i \(-0.359040\pi\)
0.428507 + 0.903538i \(0.359040\pi\)
\(648\) 0 0
\(649\) −3.36599e8 −0.0483344
\(650\) −2.64462e9 −0.377717
\(651\) 0 0
\(652\) 6.99651e9 0.988587
\(653\) 8.95278e9 1.25823 0.629117 0.777310i \(-0.283416\pi\)
0.629117 + 0.777310i \(0.283416\pi\)
\(654\) 0 0
\(655\) −7.19507e9 −1.00044
\(656\) 1.14341e9 0.158139
\(657\) 0 0
\(658\) −8.23991e8 −0.112754
\(659\) −1.19309e10 −1.62396 −0.811979 0.583687i \(-0.801609\pi\)
−0.811979 + 0.583687i \(0.801609\pi\)
\(660\) 0 0
\(661\) −3.52536e9 −0.474786 −0.237393 0.971414i \(-0.576293\pi\)
−0.237393 + 0.971414i \(0.576293\pi\)
\(662\) −2.84788e9 −0.381521
\(663\) 0 0
\(664\) −4.15892e9 −0.551306
\(665\) −1.46079e10 −1.92625
\(666\) 0 0
\(667\) −5.38769e8 −0.0703011
\(668\) −2.22507e9 −0.288820
\(669\) 0 0
\(670\) 3.59773e9 0.462133
\(671\) 1.19207e9 0.152326
\(672\) 0 0
\(673\) 2.13472e9 0.269953 0.134976 0.990849i \(-0.456904\pi\)
0.134976 + 0.990849i \(0.456904\pi\)
\(674\) −5.73716e9 −0.721751
\(675\) 0 0
\(676\) 6.70450e8 0.0834743
\(677\) −1.03034e10 −1.27621 −0.638103 0.769951i \(-0.720281\pi\)
−0.638103 + 0.769951i \(0.720281\pi\)
\(678\) 0 0
\(679\) −2.56537e10 −3.14490
\(680\) 8.51867e9 1.03894
\(681\) 0 0
\(682\) −1.13088e9 −0.136512
\(683\) −4.34566e9 −0.521895 −0.260947 0.965353i \(-0.584035\pi\)
−0.260947 + 0.965353i \(0.584035\pi\)
\(684\) 0 0
\(685\) 9.63450e9 1.14528
\(686\) −7.89033e9 −0.933170
\(687\) 0 0
\(688\) −1.83112e8 −0.0214366
\(689\) 1.22348e10 1.42505
\(690\) 0 0
\(691\) 1.26968e9 0.146393 0.0731965 0.997318i \(-0.476680\pi\)
0.0731965 + 0.997318i \(0.476680\pi\)
\(692\) −1.16553e9 −0.133706
\(693\) 0 0
\(694\) −9.76324e9 −1.10876
\(695\) 6.70847e9 0.758013
\(696\) 0 0
\(697\) 2.11076e10 2.36115
\(698\) 4.21946e9 0.469638
\(699\) 0 0
\(700\) 5.40374e9 0.595458
\(701\) 1.71178e9 0.187687 0.0938435 0.995587i \(-0.470085\pi\)
0.0938435 + 0.995587i \(0.470085\pi\)
\(702\) 0 0
\(703\) −2.80204e10 −3.04180
\(704\) 2.54330e9 0.274722
\(705\) 0 0
\(706\) 1.33365e9 0.142634
\(707\) −1.21463e10 −1.29264
\(708\) 0 0
\(709\) 1.22235e10 1.28805 0.644025 0.765004i \(-0.277263\pi\)
0.644025 + 0.765004i \(0.277263\pi\)
\(710\) 1.08661e9 0.113938
\(711\) 0 0
\(712\) 9.19592e9 0.954806
\(713\) 4.81355e9 0.497338
\(714\) 0 0
\(715\) 2.06025e9 0.210790
\(716\) −1.70021e9 −0.173104
\(717\) 0 0
\(718\) −9.54240e9 −0.962104
\(719\) 5.76282e8 0.0578207 0.0289104 0.999582i \(-0.490796\pi\)
0.0289104 + 0.999582i \(0.490796\pi\)
\(720\) 0 0
\(721\) 2.70403e10 2.68682
\(722\) −1.63202e10 −1.61378
\(723\) 0 0
\(724\) 7.15357e9 0.700548
\(725\) 5.04644e8 0.0491816
\(726\) 0 0
\(727\) 3.68604e9 0.355787 0.177894 0.984050i \(-0.443072\pi\)
0.177894 + 0.984050i \(0.443072\pi\)
\(728\) −1.67061e10 −1.60478
\(729\) 0 0
\(730\) 7.46685e9 0.710407
\(731\) −3.38027e9 −0.320067
\(732\) 0 0
\(733\) −5.75370e9 −0.539614 −0.269807 0.962914i \(-0.586960\pi\)
−0.269807 + 0.962914i \(0.586960\pi\)
\(734\) −1.27846e8 −0.0119331
\(735\) 0 0
\(736\) −9.21889e9 −0.852327
\(737\) 4.61363e9 0.424528
\(738\) 0 0
\(739\) −7.57514e9 −0.690455 −0.345227 0.938519i \(-0.612198\pi\)
−0.345227 + 0.938519i \(0.612198\pi\)
\(740\) −6.29683e9 −0.571230
\(741\) 0 0
\(742\) 1.90296e10 1.71008
\(743\) −9.30976e9 −0.832679 −0.416340 0.909209i \(-0.636687\pi\)
−0.416340 + 0.909209i \(0.636687\pi\)
\(744\) 0 0
\(745\) 9.59688e9 0.850321
\(746\) 5.96615e9 0.526149
\(747\) 0 0
\(748\) 3.95628e9 0.345646
\(749\) 1.40281e10 1.21987
\(750\) 0 0
\(751\) 1.04130e10 0.897091 0.448546 0.893760i \(-0.351942\pi\)
0.448546 + 0.893760i \(0.351942\pi\)
\(752\) −1.30290e8 −0.0111725
\(753\) 0 0
\(754\) −5.65026e8 −0.0480030
\(755\) 1.16519e10 0.985334
\(756\) 0 0
\(757\) −5.37115e9 −0.450020 −0.225010 0.974356i \(-0.572241\pi\)
−0.225010 + 0.974356i \(0.572241\pi\)
\(758\) −7.16765e7 −0.00597771
\(759\) 0 0
\(760\) −1.42523e10 −1.17771
\(761\) 2.00035e10 1.64535 0.822677 0.568509i \(-0.192479\pi\)
0.822677 + 0.568509i \(0.192479\pi\)
\(762\) 0 0
\(763\) 2.02292e10 1.64870
\(764\) 2.33696e9 0.189594
\(765\) 0 0
\(766\) 6.14740e9 0.494186
\(767\) 1.50255e9 0.120239
\(768\) 0 0
\(769\) −1.99672e10 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(770\) 3.20445e9 0.252950
\(771\) 0 0
\(772\) −1.88511e9 −0.147461
\(773\) 1.32942e10 1.03522 0.517611 0.855616i \(-0.326821\pi\)
0.517611 + 0.855616i \(0.326821\pi\)
\(774\) 0 0
\(775\) −4.50867e9 −0.347930
\(776\) −2.50292e10 −1.92278
\(777\) 0 0
\(778\) 5.41085e9 0.411943
\(779\) −3.53143e10 −2.67652
\(780\) 0 0
\(781\) 1.39344e9 0.104667
\(782\) 1.28186e10 0.958552
\(783\) 0 0
\(784\) −2.72940e9 −0.202284
\(785\) −7.65954e9 −0.565143
\(786\) 0 0
\(787\) −1.67731e10 −1.22659 −0.613297 0.789853i \(-0.710157\pi\)
−0.613297 + 0.789853i \(0.710157\pi\)
\(788\) −1.16775e9 −0.0850174
\(789\) 0 0
\(790\) 3.51147e9 0.253393
\(791\) 4.06992e10 2.92394
\(792\) 0 0
\(793\) −5.32130e9 −0.378932
\(794\) 8.60410e9 0.610005
\(795\) 0 0
\(796\) −4.70546e9 −0.330679
\(797\) 1.36721e10 0.956603 0.478301 0.878196i \(-0.341253\pi\)
0.478301 + 0.878196i \(0.341253\pi\)
\(798\) 0 0
\(799\) −2.40518e9 −0.166815
\(800\) 8.63498e9 0.596275
\(801\) 0 0
\(802\) 7.12408e8 0.0487662
\(803\) 9.57528e9 0.652600
\(804\) 0 0
\(805\) −1.36397e10 −0.921548
\(806\) 5.04813e9 0.339593
\(807\) 0 0
\(808\) −1.18506e10 −0.790314
\(809\) −2.47296e10 −1.64209 −0.821046 0.570862i \(-0.806609\pi\)
−0.821046 + 0.570862i \(0.806609\pi\)
\(810\) 0 0
\(811\) −5.97641e9 −0.393430 −0.196715 0.980461i \(-0.563027\pi\)
−0.196715 + 0.980461i \(0.563027\pi\)
\(812\) 1.15451e9 0.0756751
\(813\) 0 0
\(814\) 6.14664e9 0.399441
\(815\) 1.65414e10 1.07034
\(816\) 0 0
\(817\) 5.65540e9 0.362816
\(818\) 8.37817e9 0.535196
\(819\) 0 0
\(820\) −7.93596e9 −0.502633
\(821\) −6.86651e9 −0.433047 −0.216523 0.976277i \(-0.569472\pi\)
−0.216523 + 0.976277i \(0.569472\pi\)
\(822\) 0 0
\(823\) −1.06812e10 −0.667916 −0.333958 0.942588i \(-0.608384\pi\)
−0.333958 + 0.942588i \(0.608384\pi\)
\(824\) 2.63820e10 1.64272
\(825\) 0 0
\(826\) 2.33701e9 0.144288
\(827\) −1.28871e10 −0.792291 −0.396146 0.918188i \(-0.629652\pi\)
−0.396146 + 0.918188i \(0.629652\pi\)
\(828\) 0 0
\(829\) 2.54525e10 1.55164 0.775819 0.630956i \(-0.217337\pi\)
0.775819 + 0.630956i \(0.217337\pi\)
\(830\) −3.56102e9 −0.216173
\(831\) 0 0
\(832\) −1.13531e10 −0.683411
\(833\) −5.03852e10 −3.02027
\(834\) 0 0
\(835\) −5.26061e9 −0.312704
\(836\) −6.61911e9 −0.391812
\(837\) 0 0
\(838\) −1.83220e10 −1.07552
\(839\) −1.16543e10 −0.681269 −0.340635 0.940196i \(-0.610642\pi\)
−0.340635 + 0.940196i \(0.610642\pi\)
\(840\) 0 0
\(841\) −1.71421e10 −0.993750
\(842\) −4.48282e9 −0.258797
\(843\) 0 0
\(844\) 2.94696e9 0.168724
\(845\) 1.58510e9 0.0903773
\(846\) 0 0
\(847\) −2.57035e10 −1.45345
\(848\) 3.00898e9 0.169447
\(849\) 0 0
\(850\) −1.20067e10 −0.670588
\(851\) −2.61631e10 −1.45524
\(852\) 0 0
\(853\) −8.84397e9 −0.487894 −0.243947 0.969789i \(-0.578442\pi\)
−0.243947 + 0.969789i \(0.578442\pi\)
\(854\) −8.27657e9 −0.454724
\(855\) 0 0
\(856\) 1.36866e10 0.745826
\(857\) 9.53214e9 0.517318 0.258659 0.965969i \(-0.416719\pi\)
0.258659 + 0.965969i \(0.416719\pi\)
\(858\) 0 0
\(859\) −1.01239e10 −0.544971 −0.272485 0.962160i \(-0.587846\pi\)
−0.272485 + 0.962160i \(0.587846\pi\)
\(860\) 1.27090e9 0.0681345
\(861\) 0 0
\(862\) 1.26108e10 0.670608
\(863\) −3.34111e9 −0.176951 −0.0884755 0.996078i \(-0.528199\pi\)
−0.0884755 + 0.996078i \(0.528199\pi\)
\(864\) 0 0
\(865\) −2.75558e9 −0.144763
\(866\) 3.95353e9 0.206858
\(867\) 0 0
\(868\) −1.03148e10 −0.535356
\(869\) 4.50301e9 0.232773
\(870\) 0 0
\(871\) −2.05948e10 −1.05608
\(872\) 1.97366e10 1.00801
\(873\) 0 0
\(874\) −2.14463e10 −1.08658
\(875\) 3.33127e10 1.68105
\(876\) 0 0
\(877\) −1.70327e10 −0.852679 −0.426339 0.904563i \(-0.640197\pi\)
−0.426339 + 0.904563i \(0.640197\pi\)
\(878\) 1.78259e10 0.888835
\(879\) 0 0
\(880\) 5.06691e8 0.0250642
\(881\) 6.92335e9 0.341115 0.170558 0.985348i \(-0.445443\pi\)
0.170558 + 0.985348i \(0.445443\pi\)
\(882\) 0 0
\(883\) −2.45719e9 −0.120109 −0.0600547 0.998195i \(-0.519128\pi\)
−0.0600547 + 0.998195i \(0.519128\pi\)
\(884\) −1.76605e10 −0.859845
\(885\) 0 0
\(886\) −1.51220e10 −0.730453
\(887\) −1.99592e10 −0.960307 −0.480154 0.877184i \(-0.659419\pi\)
−0.480154 + 0.877184i \(0.659419\pi\)
\(888\) 0 0
\(889\) −3.12617e10 −1.49230
\(890\) 7.87387e9 0.374389
\(891\) 0 0
\(892\) 4.86167e9 0.229355
\(893\) 4.02402e9 0.189095
\(894\) 0 0
\(895\) −4.01971e9 −0.187419
\(896\) 1.71342e10 0.795768
\(897\) 0 0
\(898\) 2.57662e10 1.18736
\(899\) −9.63280e8 −0.0442174
\(900\) 0 0
\(901\) 5.55463e10 2.52999
\(902\) 7.74667e9 0.351473
\(903\) 0 0
\(904\) 3.97083e10 1.78769
\(905\) 1.69128e10 0.758481
\(906\) 0 0
\(907\) 1.22070e10 0.543227 0.271614 0.962406i \(-0.412443\pi\)
0.271614 + 0.962406i \(0.412443\pi\)
\(908\) 8.88829e9 0.394019
\(909\) 0 0
\(910\) −1.43044e10 −0.629251
\(911\) −1.78824e10 −0.783629 −0.391815 0.920044i \(-0.628153\pi\)
−0.391815 + 0.920044i \(0.628153\pi\)
\(912\) 0 0
\(913\) −4.56655e9 −0.198582
\(914\) 2.00850e9 0.0870081
\(915\) 0 0
\(916\) 1.68894e10 0.726074
\(917\) 6.40616e10 2.74350
\(918\) 0 0
\(919\) −1.01138e10 −0.429843 −0.214921 0.976631i \(-0.568949\pi\)
−0.214921 + 0.976631i \(0.568949\pi\)
\(920\) −1.33076e10 −0.563432
\(921\) 0 0
\(922\) −1.44522e10 −0.607262
\(923\) −6.22020e9 −0.260375
\(924\) 0 0
\(925\) 2.45059e10 1.01806
\(926\) 1.16973e9 0.0484113
\(927\) 0 0
\(928\) 1.84487e9 0.0757788
\(929\) −7.28094e9 −0.297942 −0.148971 0.988842i \(-0.547596\pi\)
−0.148971 + 0.988842i \(0.547596\pi\)
\(930\) 0 0
\(931\) 8.42976e10 3.42367
\(932\) 5.59397e9 0.226342
\(933\) 0 0
\(934\) −7.49571e9 −0.301022
\(935\) 9.35360e9 0.374230
\(936\) 0 0
\(937\) −2.64374e10 −1.04986 −0.524928 0.851147i \(-0.675908\pi\)
−0.524928 + 0.851147i \(0.675908\pi\)
\(938\) −3.20325e10 −1.26731
\(939\) 0 0
\(940\) 9.04292e8 0.0355109
\(941\) 7.30500e9 0.285796 0.142898 0.989737i \(-0.454358\pi\)
0.142898 + 0.989737i \(0.454358\pi\)
\(942\) 0 0
\(943\) −3.29735e10 −1.28049
\(944\) 3.69531e8 0.0142971
\(945\) 0 0
\(946\) −1.24059e9 −0.0476441
\(947\) −6.35717e9 −0.243242 −0.121621 0.992577i \(-0.538809\pi\)
−0.121621 + 0.992577i \(0.538809\pi\)
\(948\) 0 0
\(949\) −4.27432e10 −1.62344
\(950\) 2.00879e10 0.760154
\(951\) 0 0
\(952\) −7.58463e10 −2.84908
\(953\) 6.22176e9 0.232856 0.116428 0.993199i \(-0.462856\pi\)
0.116428 + 0.993199i \(0.462856\pi\)
\(954\) 0 0
\(955\) 5.52514e9 0.205273
\(956\) −7.76550e8 −0.0287453
\(957\) 0 0
\(958\) 8.54047e9 0.313836
\(959\) −8.57811e10 −3.14070
\(960\) 0 0
\(961\) −1.89063e10 −0.687188
\(962\) −2.74381e10 −0.993668
\(963\) 0 0
\(964\) 2.71910e10 0.977587
\(965\) −4.45686e9 −0.159655
\(966\) 0 0
\(967\) −3.37902e10 −1.20171 −0.600853 0.799360i \(-0.705172\pi\)
−0.600853 + 0.799360i \(0.705172\pi\)
\(968\) −2.50777e10 −0.888637
\(969\) 0 0
\(970\) −2.14308e10 −0.753943
\(971\) 4.96021e10 1.73873 0.869366 0.494169i \(-0.164528\pi\)
0.869366 + 0.494169i \(0.164528\pi\)
\(972\) 0 0
\(973\) −5.97291e10 −2.07870
\(974\) −1.97444e10 −0.684680
\(975\) 0 0
\(976\) −1.30870e9 −0.0450574
\(977\) 3.46137e10 1.18745 0.593727 0.804667i \(-0.297656\pi\)
0.593727 + 0.804667i \(0.297656\pi\)
\(978\) 0 0
\(979\) 1.00972e10 0.343924
\(980\) 1.89437e10 0.642943
\(981\) 0 0
\(982\) −1.67382e10 −0.564052
\(983\) 9.52456e9 0.319822 0.159911 0.987131i \(-0.448879\pi\)
0.159911 + 0.987131i \(0.448879\pi\)
\(984\) 0 0
\(985\) −2.76084e9 −0.0920480
\(986\) −2.56523e9 −0.0852231
\(987\) 0 0
\(988\) 2.95471e10 0.974689
\(989\) 5.28054e9 0.173577
\(990\) 0 0
\(991\) 4.31860e10 1.40956 0.704782 0.709424i \(-0.251045\pi\)
0.704782 + 0.709424i \(0.251045\pi\)
\(992\) −1.64827e10 −0.536090
\(993\) 0 0
\(994\) −9.67469e9 −0.312453
\(995\) −1.11248e10 −0.358025
\(996\) 0 0
\(997\) 5.74177e9 0.183490 0.0917451 0.995783i \(-0.470755\pi\)
0.0917451 + 0.995783i \(0.470755\pi\)
\(998\) −5.56327e9 −0.177163
\(999\) 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 531.8.a.e.1.8 18
3.2 odd 2 177.8.a.d.1.11 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.11 18 3.2 odd 2
531.8.a.e.1.8 18 1.1 even 1 trivial