Properties

Label 531.8.a.c.1.13
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-12.3679\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.3679 q^{2} +24.9655 q^{4} +477.980 q^{5} +1344.57 q^{7} -1274.32 q^{8} +O(q^{10})\) \(q+12.3679 q^{2} +24.9655 q^{4} +477.980 q^{5} +1344.57 q^{7} -1274.32 q^{8} +5911.62 q^{10} -6221.77 q^{11} +10930.7 q^{13} +16629.5 q^{14} -18956.3 q^{16} +31807.5 q^{17} -36490.6 q^{19} +11933.0 q^{20} -76950.3 q^{22} -38312.8 q^{23} +150340. q^{25} +135190. q^{26} +33567.7 q^{28} +138973. q^{29} +256427. q^{31} -71336.8 q^{32} +393392. q^{34} +642676. q^{35} +22109.8 q^{37} -451313. q^{38} -609101. q^{40} -316134. q^{41} +148255. q^{43} -155329. q^{44} -473850. q^{46} -748870. q^{47} +984312. q^{49} +1.85940e6 q^{50} +272890. q^{52} +1.94662e6 q^{53} -2.97388e6 q^{55} -1.71341e6 q^{56} +1.71880e6 q^{58} +205379. q^{59} +1.76648e6 q^{61} +3.17147e6 q^{62} +1.54412e6 q^{64} +5.22465e6 q^{65} -425592. q^{67} +794089. q^{68} +7.94856e6 q^{70} +3.19791e6 q^{71} -4.35846e6 q^{73} +273452. q^{74} -911006. q^{76} -8.36557e6 q^{77} -2.91855e6 q^{79} -9.06074e6 q^{80} -3.90993e6 q^{82} +1.97344e6 q^{83} +1.52033e7 q^{85} +1.83361e6 q^{86} +7.92854e6 q^{88} -7.72572e6 q^{89} +1.46970e7 q^{91} -956499. q^{92} -9.26197e6 q^{94} -1.74418e7 q^{95} +9.26843e6 q^{97} +1.21739e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.3679 1.09318 0.546590 0.837400i \(-0.315926\pi\)
0.546590 + 0.837400i \(0.315926\pi\)
\(3\) 0 0
\(4\) 24.9655 0.195043
\(5\) 477.980 1.71007 0.855037 0.518567i \(-0.173534\pi\)
0.855037 + 0.518567i \(0.173534\pi\)
\(6\) 0 0
\(7\) 1344.57 1.48163 0.740813 0.671711i \(-0.234440\pi\)
0.740813 + 0.671711i \(0.234440\pi\)
\(8\) −1274.32 −0.879963
\(9\) 0 0
\(10\) 5911.62 1.86942
\(11\) −6221.77 −1.40942 −0.704708 0.709497i \(-0.748922\pi\)
−0.704708 + 0.709497i \(0.748922\pi\)
\(12\) 0 0
\(13\) 10930.7 1.37989 0.689947 0.723860i \(-0.257634\pi\)
0.689947 + 0.723860i \(0.257634\pi\)
\(14\) 16629.5 1.61968
\(15\) 0 0
\(16\) −18956.3 −1.15700
\(17\) 31807.5 1.57021 0.785105 0.619362i \(-0.212609\pi\)
0.785105 + 0.619362i \(0.212609\pi\)
\(18\) 0 0
\(19\) −36490.6 −1.22052 −0.610258 0.792203i \(-0.708934\pi\)
−0.610258 + 0.792203i \(0.708934\pi\)
\(20\) 11933.0 0.333538
\(21\) 0 0
\(22\) −76950.3 −1.54075
\(23\) −38312.8 −0.656594 −0.328297 0.944575i \(-0.606475\pi\)
−0.328297 + 0.944575i \(0.606475\pi\)
\(24\) 0 0
\(25\) 150340. 1.92435
\(26\) 135190. 1.50847
\(27\) 0 0
\(28\) 33567.7 0.288981
\(29\) 138973. 1.05812 0.529062 0.848583i \(-0.322544\pi\)
0.529062 + 0.848583i \(0.322544\pi\)
\(30\) 0 0
\(31\) 256427. 1.54596 0.772979 0.634432i \(-0.218766\pi\)
0.772979 + 0.634432i \(0.218766\pi\)
\(32\) −71336.8 −0.384847
\(33\) 0 0
\(34\) 393392. 1.71652
\(35\) 642676. 2.53369
\(36\) 0 0
\(37\) 22109.8 0.0717594 0.0358797 0.999356i \(-0.488577\pi\)
0.0358797 + 0.999356i \(0.488577\pi\)
\(38\) −451313. −1.33424
\(39\) 0 0
\(40\) −609101. −1.50480
\(41\) −316134. −0.716355 −0.358178 0.933653i \(-0.616602\pi\)
−0.358178 + 0.933653i \(0.616602\pi\)
\(42\) 0 0
\(43\) 148255. 0.284361 0.142181 0.989841i \(-0.454589\pi\)
0.142181 + 0.989841i \(0.454589\pi\)
\(44\) −155329. −0.274897
\(45\) 0 0
\(46\) −473850. −0.717776
\(47\) −748870. −1.05212 −0.526058 0.850448i \(-0.676331\pi\)
−0.526058 + 0.850448i \(0.676331\pi\)
\(48\) 0 0
\(49\) 984312. 1.19522
\(50\) 1.85940e6 2.10367
\(51\) 0 0
\(52\) 272890. 0.269138
\(53\) 1.94662e6 1.79604 0.898021 0.439953i \(-0.145005\pi\)
0.898021 + 0.439953i \(0.145005\pi\)
\(54\) 0 0
\(55\) −2.97388e6 −2.41021
\(56\) −1.71341e6 −1.30378
\(57\) 0 0
\(58\) 1.71880e6 1.15672
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 1.76648e6 0.996445 0.498222 0.867049i \(-0.333986\pi\)
0.498222 + 0.867049i \(0.333986\pi\)
\(62\) 3.17147e6 1.69001
\(63\) 0 0
\(64\) 1.54412e6 0.736294
\(65\) 5.22465e6 2.35972
\(66\) 0 0
\(67\) −425592. −0.172875 −0.0864374 0.996257i \(-0.527548\pi\)
−0.0864374 + 0.996257i \(0.527548\pi\)
\(68\) 794089. 0.306258
\(69\) 0 0
\(70\) 7.94856e6 2.76978
\(71\) 3.19791e6 1.06038 0.530191 0.847878i \(-0.322120\pi\)
0.530191 + 0.847878i \(0.322120\pi\)
\(72\) 0 0
\(73\) −4.35846e6 −1.31130 −0.655651 0.755064i \(-0.727606\pi\)
−0.655651 + 0.755064i \(0.727606\pi\)
\(74\) 273452. 0.0784459
\(75\) 0 0
\(76\) −911006. −0.238053
\(77\) −8.36557e6 −2.08823
\(78\) 0 0
\(79\) −2.91855e6 −0.665996 −0.332998 0.942928i \(-0.608060\pi\)
−0.332998 + 0.942928i \(0.608060\pi\)
\(80\) −9.06074e6 −1.97856
\(81\) 0 0
\(82\) −3.90993e6 −0.783105
\(83\) 1.97344e6 0.378835 0.189418 0.981897i \(-0.439340\pi\)
0.189418 + 0.981897i \(0.439340\pi\)
\(84\) 0 0
\(85\) 1.52033e7 2.68518
\(86\) 1.83361e6 0.310858
\(87\) 0 0
\(88\) 7.92854e6 1.24023
\(89\) −7.72572e6 −1.16165 −0.580823 0.814030i \(-0.697269\pi\)
−0.580823 + 0.814030i \(0.697269\pi\)
\(90\) 0 0
\(91\) 1.46970e7 2.04449
\(92\) −956499. −0.128064
\(93\) 0 0
\(94\) −9.26197e6 −1.15015
\(95\) −1.74418e7 −2.08717
\(96\) 0 0
\(97\) 9.26843e6 1.03111 0.515555 0.856857i \(-0.327586\pi\)
0.515555 + 0.856857i \(0.327586\pi\)
\(98\) 1.21739e7 1.30659
\(99\) 0 0
\(100\) 3.75331e6 0.375331
\(101\) −5.77631e6 −0.557861 −0.278930 0.960311i \(-0.589980\pi\)
−0.278930 + 0.960311i \(0.589980\pi\)
\(102\) 0 0
\(103\) −1.72461e7 −1.55511 −0.777555 0.628815i \(-0.783540\pi\)
−0.777555 + 0.628815i \(0.783540\pi\)
\(104\) −1.39292e7 −1.21426
\(105\) 0 0
\(106\) 2.40757e7 1.96340
\(107\) −5.41952e6 −0.427679 −0.213839 0.976869i \(-0.568597\pi\)
−0.213839 + 0.976869i \(0.568597\pi\)
\(108\) 0 0
\(109\) 4.21169e6 0.311504 0.155752 0.987796i \(-0.450220\pi\)
0.155752 + 0.987796i \(0.450220\pi\)
\(110\) −3.67807e7 −2.63479
\(111\) 0 0
\(112\) −2.54880e7 −1.71424
\(113\) 1.72308e7 1.12339 0.561695 0.827345i \(-0.310150\pi\)
0.561695 + 0.827345i \(0.310150\pi\)
\(114\) 0 0
\(115\) −1.83128e7 −1.12282
\(116\) 3.46952e6 0.206379
\(117\) 0 0
\(118\) 2.54011e6 0.142320
\(119\) 4.27672e7 2.32647
\(120\) 0 0
\(121\) 1.92232e7 0.986455
\(122\) 2.18476e7 1.08929
\(123\) 0 0
\(124\) 6.40182e6 0.301528
\(125\) 3.45174e7 1.58071
\(126\) 0 0
\(127\) 1.99947e7 0.866169 0.433085 0.901353i \(-0.357425\pi\)
0.433085 + 0.901353i \(0.357425\pi\)
\(128\) 2.82287e7 1.18975
\(129\) 0 0
\(130\) 6.46181e7 2.57960
\(131\) −1.04080e6 −0.0404501 −0.0202251 0.999795i \(-0.506438\pi\)
−0.0202251 + 0.999795i \(0.506438\pi\)
\(132\) 0 0
\(133\) −4.90640e7 −1.80835
\(134\) −5.26369e6 −0.188983
\(135\) 0 0
\(136\) −4.05330e7 −1.38173
\(137\) −4.65153e7 −1.54552 −0.772759 0.634699i \(-0.781124\pi\)
−0.772759 + 0.634699i \(0.781124\pi\)
\(138\) 0 0
\(139\) −1.91042e7 −0.603360 −0.301680 0.953409i \(-0.597547\pi\)
−0.301680 + 0.953409i \(0.597547\pi\)
\(140\) 1.60447e7 0.494178
\(141\) 0 0
\(142\) 3.95515e7 1.15919
\(143\) −6.80082e7 −1.94484
\(144\) 0 0
\(145\) 6.64262e7 1.80947
\(146\) −5.39050e7 −1.43349
\(147\) 0 0
\(148\) 551982. 0.0139962
\(149\) 5.31787e7 1.31700 0.658500 0.752580i \(-0.271191\pi\)
0.658500 + 0.752580i \(0.271191\pi\)
\(150\) 0 0
\(151\) 4.97786e7 1.17658 0.588292 0.808648i \(-0.299800\pi\)
0.588292 + 0.808648i \(0.299800\pi\)
\(152\) 4.65008e7 1.07401
\(153\) 0 0
\(154\) −1.03465e8 −2.28281
\(155\) 1.22567e8 2.64370
\(156\) 0 0
\(157\) 7.17256e7 1.47920 0.739598 0.673049i \(-0.235016\pi\)
0.739598 + 0.673049i \(0.235016\pi\)
\(158\) −3.60963e7 −0.728054
\(159\) 0 0
\(160\) −3.40976e7 −0.658118
\(161\) −5.15141e7 −0.972827
\(162\) 0 0
\(163\) −6.67080e7 −1.20648 −0.603241 0.797559i \(-0.706124\pi\)
−0.603241 + 0.797559i \(0.706124\pi\)
\(164\) −7.89245e6 −0.139720
\(165\) 0 0
\(166\) 2.44073e7 0.414135
\(167\) 3.70110e7 0.614927 0.307463 0.951560i \(-0.400520\pi\)
0.307463 + 0.951560i \(0.400520\pi\)
\(168\) 0 0
\(169\) 5.67313e7 0.904106
\(170\) 1.88034e8 2.93538
\(171\) 0 0
\(172\) 3.70127e6 0.0554627
\(173\) −3.73146e7 −0.547920 −0.273960 0.961741i \(-0.588334\pi\)
−0.273960 + 0.961741i \(0.588334\pi\)
\(174\) 0 0
\(175\) 2.02142e8 2.85117
\(176\) 1.17942e8 1.63070
\(177\) 0 0
\(178\) −9.55511e7 −1.26989
\(179\) −7.93461e7 −1.03405 −0.517023 0.855972i \(-0.672960\pi\)
−0.517023 + 0.855972i \(0.672960\pi\)
\(180\) 0 0
\(181\) 8.73656e7 1.09513 0.547565 0.836763i \(-0.315555\pi\)
0.547565 + 0.836763i \(0.315555\pi\)
\(182\) 1.81772e8 2.23499
\(183\) 0 0
\(184\) 4.88229e7 0.577779
\(185\) 1.05680e7 0.122714
\(186\) 0 0
\(187\) −1.97899e8 −2.21308
\(188\) −1.86959e7 −0.205208
\(189\) 0 0
\(190\) −2.15719e8 −2.28166
\(191\) 1.36478e8 1.41725 0.708625 0.705586i \(-0.249316\pi\)
0.708625 + 0.705586i \(0.249316\pi\)
\(192\) 0 0
\(193\) 1.59359e8 1.59561 0.797804 0.602917i \(-0.205995\pi\)
0.797804 + 0.602917i \(0.205995\pi\)
\(194\) 1.14631e8 1.12719
\(195\) 0 0
\(196\) 2.45738e7 0.233118
\(197\) 2.20118e7 0.205127 0.102564 0.994726i \(-0.467295\pi\)
0.102564 + 0.994726i \(0.467295\pi\)
\(198\) 0 0
\(199\) 7.23177e7 0.650517 0.325259 0.945625i \(-0.394549\pi\)
0.325259 + 0.945625i \(0.394549\pi\)
\(200\) −1.91582e8 −1.69336
\(201\) 0 0
\(202\) −7.14410e7 −0.609842
\(203\) 1.86858e8 1.56774
\(204\) 0 0
\(205\) −1.51106e8 −1.22502
\(206\) −2.13299e8 −1.70001
\(207\) 0 0
\(208\) −2.07205e8 −1.59654
\(209\) 2.27036e8 1.72022
\(210\) 0 0
\(211\) 1.37262e8 1.00592 0.502959 0.864310i \(-0.332245\pi\)
0.502959 + 0.864310i \(0.332245\pi\)
\(212\) 4.85984e7 0.350305
\(213\) 0 0
\(214\) −6.70282e7 −0.467530
\(215\) 7.08631e7 0.486279
\(216\) 0 0
\(217\) 3.44782e8 2.29053
\(218\) 5.20899e7 0.340530
\(219\) 0 0
\(220\) −7.42444e7 −0.470094
\(221\) 3.47677e8 2.16672
\(222\) 0 0
\(223\) 5.13329e7 0.309977 0.154988 0.987916i \(-0.450466\pi\)
0.154988 + 0.987916i \(0.450466\pi\)
\(224\) −9.59170e7 −0.570200
\(225\) 0 0
\(226\) 2.13109e8 1.22807
\(227\) −2.24268e8 −1.27256 −0.636278 0.771459i \(-0.719527\pi\)
−0.636278 + 0.771459i \(0.719527\pi\)
\(228\) 0 0
\(229\) 1.09514e8 0.602625 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(230\) −2.26491e8 −1.22745
\(231\) 0 0
\(232\) −1.77096e8 −0.931110
\(233\) −1.49211e8 −0.772779 −0.386390 0.922336i \(-0.626278\pi\)
−0.386390 + 0.922336i \(0.626278\pi\)
\(234\) 0 0
\(235\) −3.57945e8 −1.79920
\(236\) 5.12739e6 0.0253924
\(237\) 0 0
\(238\) 5.28942e8 2.54325
\(239\) 1.38832e8 0.657804 0.328902 0.944364i \(-0.393321\pi\)
0.328902 + 0.944364i \(0.393321\pi\)
\(240\) 0 0
\(241\) 2.34611e8 1.07966 0.539831 0.841773i \(-0.318488\pi\)
0.539831 + 0.841773i \(0.318488\pi\)
\(242\) 2.37751e8 1.07837
\(243\) 0 0
\(244\) 4.41009e7 0.194349
\(245\) 4.70482e8 2.04391
\(246\) 0 0
\(247\) −3.98867e8 −1.68418
\(248\) −3.26770e8 −1.36039
\(249\) 0 0
\(250\) 4.26909e8 1.72800
\(251\) −2.52396e8 −1.00745 −0.503725 0.863864i \(-0.668038\pi\)
−0.503725 + 0.863864i \(0.668038\pi\)
\(252\) 0 0
\(253\) 2.38374e8 0.925414
\(254\) 2.47293e8 0.946879
\(255\) 0 0
\(256\) 1.51483e8 0.564316
\(257\) −4.59969e8 −1.69029 −0.845147 0.534534i \(-0.820487\pi\)
−0.845147 + 0.534534i \(0.820487\pi\)
\(258\) 0 0
\(259\) 2.97281e7 0.106321
\(260\) 1.30436e8 0.460247
\(261\) 0 0
\(262\) −1.28726e7 −0.0442193
\(263\) −3.87682e8 −1.31411 −0.657053 0.753844i \(-0.728197\pi\)
−0.657053 + 0.753844i \(0.728197\pi\)
\(264\) 0 0
\(265\) 9.30447e8 3.07136
\(266\) −6.06820e8 −1.97685
\(267\) 0 0
\(268\) −1.06251e7 −0.0337180
\(269\) −2.41785e7 −0.0757351 −0.0378675 0.999283i \(-0.512056\pi\)
−0.0378675 + 0.999283i \(0.512056\pi\)
\(270\) 0 0
\(271\) −2.33181e7 −0.0711707 −0.0355853 0.999367i \(-0.511330\pi\)
−0.0355853 + 0.999367i \(0.511330\pi\)
\(272\) −6.02952e8 −1.81674
\(273\) 0 0
\(274\) −5.75298e8 −1.68953
\(275\) −9.35381e8 −2.71222
\(276\) 0 0
\(277\) 3.40111e7 0.0961482 0.0480741 0.998844i \(-0.484692\pi\)
0.0480741 + 0.998844i \(0.484692\pi\)
\(278\) −2.36279e8 −0.659581
\(279\) 0 0
\(280\) −8.18976e8 −2.22955
\(281\) 9.09433e7 0.244511 0.122256 0.992499i \(-0.460987\pi\)
0.122256 + 0.992499i \(0.460987\pi\)
\(282\) 0 0
\(283\) −1.54868e8 −0.406171 −0.203085 0.979161i \(-0.565097\pi\)
−0.203085 + 0.979161i \(0.565097\pi\)
\(284\) 7.98374e7 0.206820
\(285\) 0 0
\(286\) −8.41120e8 −2.12607
\(287\) −4.25063e8 −1.06137
\(288\) 0 0
\(289\) 6.01377e8 1.46556
\(290\) 8.21554e8 1.97808
\(291\) 0 0
\(292\) −1.08811e8 −0.255760
\(293\) 2.68346e8 0.623245 0.311623 0.950206i \(-0.399128\pi\)
0.311623 + 0.950206i \(0.399128\pi\)
\(294\) 0 0
\(295\) 9.81671e7 0.222633
\(296\) −2.81750e7 −0.0631456
\(297\) 0 0
\(298\) 6.57710e8 1.43972
\(299\) −4.18786e8 −0.906030
\(300\) 0 0
\(301\) 1.99339e8 0.421317
\(302\) 6.15658e8 1.28622
\(303\) 0 0
\(304\) 6.91727e8 1.41214
\(305\) 8.44340e8 1.70399
\(306\) 0 0
\(307\) −8.22676e8 −1.62272 −0.811362 0.584544i \(-0.801273\pi\)
−0.811362 + 0.584544i \(0.801273\pi\)
\(308\) −2.08850e8 −0.407294
\(309\) 0 0
\(310\) 1.51590e9 2.89004
\(311\) 8.75600e8 1.65061 0.825305 0.564687i \(-0.191003\pi\)
0.825305 + 0.564687i \(0.191003\pi\)
\(312\) 0 0
\(313\) −4.06490e8 −0.749280 −0.374640 0.927170i \(-0.622234\pi\)
−0.374640 + 0.927170i \(0.622234\pi\)
\(314\) 8.87097e8 1.61703
\(315\) 0 0
\(316\) −7.28629e7 −0.129898
\(317\) 6.46224e8 1.13940 0.569699 0.821853i \(-0.307060\pi\)
0.569699 + 0.821853i \(0.307060\pi\)
\(318\) 0 0
\(319\) −8.64655e8 −1.49134
\(320\) 7.38059e8 1.25912
\(321\) 0 0
\(322\) −6.37122e8 −1.06348
\(323\) −1.16067e9 −1.91647
\(324\) 0 0
\(325\) 1.64332e9 2.65540
\(326\) −8.25039e8 −1.31890
\(327\) 0 0
\(328\) 4.02857e8 0.630366
\(329\) −1.00690e9 −1.55884
\(330\) 0 0
\(331\) −7.67395e8 −1.16311 −0.581555 0.813507i \(-0.697556\pi\)
−0.581555 + 0.813507i \(0.697556\pi\)
\(332\) 4.92678e7 0.0738891
\(333\) 0 0
\(334\) 4.57749e8 0.672226
\(335\) −2.03425e8 −0.295629
\(336\) 0 0
\(337\) 6.41971e8 0.913715 0.456858 0.889540i \(-0.348975\pi\)
0.456858 + 0.889540i \(0.348975\pi\)
\(338\) 7.01649e8 0.988351
\(339\) 0 0
\(340\) 3.79559e8 0.523725
\(341\) −1.59543e9 −2.17890
\(342\) 0 0
\(343\) 2.16165e8 0.289238
\(344\) −1.88925e8 −0.250228
\(345\) 0 0
\(346\) −4.61504e8 −0.598976
\(347\) −6.85227e8 −0.880403 −0.440201 0.897899i \(-0.645093\pi\)
−0.440201 + 0.897899i \(0.645093\pi\)
\(348\) 0 0
\(349\) −3.31439e8 −0.417363 −0.208682 0.977984i \(-0.566917\pi\)
−0.208682 + 0.977984i \(0.566917\pi\)
\(350\) 2.50008e9 3.11685
\(351\) 0 0
\(352\) 4.43841e8 0.542410
\(353\) −1.17646e9 −1.42353 −0.711766 0.702416i \(-0.752104\pi\)
−0.711766 + 0.702416i \(0.752104\pi\)
\(354\) 0 0
\(355\) 1.52854e9 1.81333
\(356\) −1.92876e8 −0.226571
\(357\) 0 0
\(358\) −9.81346e8 −1.13040
\(359\) 5.67306e8 0.647123 0.323562 0.946207i \(-0.395120\pi\)
0.323562 + 0.946207i \(0.395120\pi\)
\(360\) 0 0
\(361\) 4.37693e8 0.489660
\(362\) 1.08053e9 1.19717
\(363\) 0 0
\(364\) 3.66918e8 0.398762
\(365\) −2.08326e9 −2.24242
\(366\) 0 0
\(367\) 1.09692e8 0.115836 0.0579182 0.998321i \(-0.481554\pi\)
0.0579182 + 0.998321i \(0.481554\pi\)
\(368\) 7.26270e8 0.759680
\(369\) 0 0
\(370\) 1.30705e8 0.134148
\(371\) 2.61736e9 2.66106
\(372\) 0 0
\(373\) −1.29535e9 −1.29243 −0.646214 0.763156i \(-0.723649\pi\)
−0.646214 + 0.763156i \(0.723649\pi\)
\(374\) −2.44760e9 −2.41930
\(375\) 0 0
\(376\) 9.54302e8 0.925824
\(377\) 1.51907e9 1.46010
\(378\) 0 0
\(379\) −3.39868e8 −0.320681 −0.160341 0.987062i \(-0.551259\pi\)
−0.160341 + 0.987062i \(0.551259\pi\)
\(380\) −4.35443e8 −0.407088
\(381\) 0 0
\(382\) 1.68795e9 1.54931
\(383\) −1.07778e9 −0.980246 −0.490123 0.871653i \(-0.663048\pi\)
−0.490123 + 0.871653i \(0.663048\pi\)
\(384\) 0 0
\(385\) −3.99858e9 −3.57103
\(386\) 1.97094e9 1.74429
\(387\) 0 0
\(388\) 2.31391e8 0.201111
\(389\) −7.02547e8 −0.605134 −0.302567 0.953128i \(-0.597844\pi\)
−0.302567 + 0.953128i \(0.597844\pi\)
\(390\) 0 0
\(391\) −1.21863e9 −1.03099
\(392\) −1.25433e9 −1.05175
\(393\) 0 0
\(394\) 2.72240e8 0.224241
\(395\) −1.39501e9 −1.13890
\(396\) 0 0
\(397\) −4.23518e7 −0.0339708 −0.0169854 0.999856i \(-0.505407\pi\)
−0.0169854 + 0.999856i \(0.505407\pi\)
\(398\) 8.94419e8 0.711132
\(399\) 0 0
\(400\) −2.84989e9 −2.22648
\(401\) −1.82357e9 −1.41227 −0.706133 0.708079i \(-0.749562\pi\)
−0.706133 + 0.708079i \(0.749562\pi\)
\(402\) 0 0
\(403\) 2.80292e9 2.13326
\(404\) −1.44208e8 −0.108807
\(405\) 0 0
\(406\) 2.31104e9 1.71383
\(407\) −1.37562e8 −0.101139
\(408\) 0 0
\(409\) −3.94853e8 −0.285367 −0.142684 0.989768i \(-0.545573\pi\)
−0.142684 + 0.989768i \(0.545573\pi\)
\(410\) −1.86887e9 −1.33917
\(411\) 0 0
\(412\) −4.30558e8 −0.303313
\(413\) 2.76145e8 0.192891
\(414\) 0 0
\(415\) 9.43265e8 0.647836
\(416\) −7.79760e8 −0.531049
\(417\) 0 0
\(418\) 2.80796e9 1.88051
\(419\) −1.27926e9 −0.849591 −0.424795 0.905289i \(-0.639654\pi\)
−0.424795 + 0.905289i \(0.639654\pi\)
\(420\) 0 0
\(421\) −4.37766e8 −0.285926 −0.142963 0.989728i \(-0.545663\pi\)
−0.142963 + 0.989728i \(0.545663\pi\)
\(422\) 1.69765e9 1.09965
\(423\) 0 0
\(424\) −2.48063e9 −1.58045
\(425\) 4.78194e9 3.02164
\(426\) 0 0
\(427\) 2.37514e9 1.47636
\(428\) −1.35301e8 −0.0834157
\(429\) 0 0
\(430\) 8.76430e8 0.531591
\(431\) −2.13470e9 −1.28430 −0.642150 0.766579i \(-0.721957\pi\)
−0.642150 + 0.766579i \(0.721957\pi\)
\(432\) 0 0
\(433\) −2.81072e9 −1.66383 −0.831917 0.554900i \(-0.812757\pi\)
−0.831917 + 0.554900i \(0.812757\pi\)
\(434\) 4.26424e9 2.50396
\(435\) 0 0
\(436\) 1.05147e8 0.0607567
\(437\) 1.39806e9 0.801384
\(438\) 0 0
\(439\) 2.30629e9 1.30103 0.650516 0.759493i \(-0.274553\pi\)
0.650516 + 0.759493i \(0.274553\pi\)
\(440\) 3.78969e9 2.12089
\(441\) 0 0
\(442\) 4.30005e9 2.36862
\(443\) −9.88032e8 −0.539956 −0.269978 0.962867i \(-0.587016\pi\)
−0.269978 + 0.962867i \(0.587016\pi\)
\(444\) 0 0
\(445\) −3.69274e9 −1.98650
\(446\) 6.34882e8 0.338860
\(447\) 0 0
\(448\) 2.07617e9 1.09091
\(449\) 3.41363e8 0.177973 0.0889866 0.996033i \(-0.471637\pi\)
0.0889866 + 0.996033i \(0.471637\pi\)
\(450\) 0 0
\(451\) 1.96692e9 1.00964
\(452\) 4.30175e8 0.219109
\(453\) 0 0
\(454\) −2.77373e9 −1.39113
\(455\) 7.02488e9 3.49622
\(456\) 0 0
\(457\) −2.86701e6 −0.00140515 −0.000702575 1.00000i \(-0.500224\pi\)
−0.000702575 1.00000i \(0.500224\pi\)
\(458\) 1.35447e9 0.658778
\(459\) 0 0
\(460\) −4.57187e8 −0.218999
\(461\) −2.81859e9 −1.33992 −0.669959 0.742398i \(-0.733688\pi\)
−0.669959 + 0.742398i \(0.733688\pi\)
\(462\) 0 0
\(463\) 1.30688e9 0.611932 0.305966 0.952042i \(-0.401021\pi\)
0.305966 + 0.952042i \(0.401021\pi\)
\(464\) −2.63441e9 −1.22425
\(465\) 0 0
\(466\) −1.84543e9 −0.844787
\(467\) 1.72823e9 0.785221 0.392611 0.919705i \(-0.371572\pi\)
0.392611 + 0.919705i \(0.371572\pi\)
\(468\) 0 0
\(469\) −5.72236e8 −0.256136
\(470\) −4.42704e9 −1.96685
\(471\) 0 0
\(472\) −2.61719e8 −0.114561
\(473\) −9.22410e8 −0.400784
\(474\) 0 0
\(475\) −5.48600e9 −2.34870
\(476\) 1.06770e9 0.453760
\(477\) 0 0
\(478\) 1.71706e9 0.719099
\(479\) −2.64444e9 −1.09941 −0.549705 0.835359i \(-0.685260\pi\)
−0.549705 + 0.835359i \(0.685260\pi\)
\(480\) 0 0
\(481\) 2.41675e8 0.0990203
\(482\) 2.90165e9 1.18027
\(483\) 0 0
\(484\) 4.79917e8 0.192401
\(485\) 4.43012e9 1.76327
\(486\) 0 0
\(487\) 2.49943e9 0.980594 0.490297 0.871556i \(-0.336888\pi\)
0.490297 + 0.871556i \(0.336888\pi\)
\(488\) −2.25106e9 −0.876835
\(489\) 0 0
\(490\) 5.81888e9 2.23436
\(491\) 1.41128e9 0.538057 0.269029 0.963132i \(-0.413297\pi\)
0.269029 + 0.963132i \(0.413297\pi\)
\(492\) 0 0
\(493\) 4.42037e9 1.66148
\(494\) −4.93316e9 −1.84111
\(495\) 0 0
\(496\) −4.86090e9 −1.78867
\(497\) 4.29980e9 1.57109
\(498\) 0 0
\(499\) 2.12697e8 0.0766319 0.0383159 0.999266i \(-0.487801\pi\)
0.0383159 + 0.999266i \(0.487801\pi\)
\(500\) 8.61744e8 0.308307
\(501\) 0 0
\(502\) −3.12161e9 −1.10133
\(503\) 2.85799e7 0.0100132 0.00500660 0.999987i \(-0.498406\pi\)
0.00500660 + 0.999987i \(0.498406\pi\)
\(504\) 0 0
\(505\) −2.76096e9 −0.953984
\(506\) 2.94819e9 1.01164
\(507\) 0 0
\(508\) 4.99178e8 0.168940
\(509\) 1.91356e9 0.643178 0.321589 0.946879i \(-0.395783\pi\)
0.321589 + 0.946879i \(0.395783\pi\)
\(510\) 0 0
\(511\) −5.86023e9 −1.94286
\(512\) −1.73974e9 −0.572849
\(513\) 0 0
\(514\) −5.68885e9 −1.84780
\(515\) −8.24330e9 −2.65935
\(516\) 0 0
\(517\) 4.65929e9 1.48287
\(518\) 3.67674e8 0.116228
\(519\) 0 0
\(520\) −6.65789e9 −2.07647
\(521\) −1.49555e9 −0.463306 −0.231653 0.972798i \(-0.574413\pi\)
−0.231653 + 0.972798i \(0.574413\pi\)
\(522\) 0 0
\(523\) 2.48258e9 0.758835 0.379417 0.925226i \(-0.376124\pi\)
0.379417 + 0.925226i \(0.376124\pi\)
\(524\) −2.59842e7 −0.00788951
\(525\) 0 0
\(526\) −4.79482e9 −1.43656
\(527\) 8.15629e9 2.42748
\(528\) 0 0
\(529\) −1.93695e9 −0.568884
\(530\) 1.15077e10 3.35755
\(531\) 0 0
\(532\) −1.22491e9 −0.352705
\(533\) −3.45557e9 −0.988494
\(534\) 0 0
\(535\) −2.59043e9 −0.731363
\(536\) 5.42342e8 0.152124
\(537\) 0 0
\(538\) −2.99038e8 −0.0827921
\(539\) −6.12416e9 −1.68456
\(540\) 0 0
\(541\) −6.44103e9 −1.74890 −0.874450 0.485115i \(-0.838778\pi\)
−0.874450 + 0.485115i \(0.838778\pi\)
\(542\) −2.88397e8 −0.0778024
\(543\) 0 0
\(544\) −2.26904e9 −0.604292
\(545\) 2.01311e9 0.532695
\(546\) 0 0
\(547\) −5.40495e8 −0.141200 −0.0706002 0.997505i \(-0.522491\pi\)
−0.0706002 + 0.997505i \(0.522491\pi\)
\(548\) −1.16128e9 −0.301442
\(549\) 0 0
\(550\) −1.15687e10 −2.96494
\(551\) −5.07120e9 −1.29146
\(552\) 0 0
\(553\) −3.92417e9 −0.986757
\(554\) 4.20646e8 0.105107
\(555\) 0 0
\(556\) −4.76945e8 −0.117681
\(557\) −2.83692e9 −0.695591 −0.347795 0.937570i \(-0.613070\pi\)
−0.347795 + 0.937570i \(0.613070\pi\)
\(558\) 0 0
\(559\) 1.62053e9 0.392389
\(560\) −1.21828e10 −2.93148
\(561\) 0 0
\(562\) 1.12478e9 0.267295
\(563\) −7.86504e9 −1.85747 −0.928734 0.370747i \(-0.879102\pi\)
−0.928734 + 0.370747i \(0.879102\pi\)
\(564\) 0 0
\(565\) 8.23597e9 1.92108
\(566\) −1.91539e9 −0.444018
\(567\) 0 0
\(568\) −4.07517e9 −0.933097
\(569\) −6.12820e9 −1.39457 −0.697285 0.716794i \(-0.745609\pi\)
−0.697285 + 0.716794i \(0.745609\pi\)
\(570\) 0 0
\(571\) 7.88550e9 1.77257 0.886284 0.463142i \(-0.153278\pi\)
0.886284 + 0.463142i \(0.153278\pi\)
\(572\) −1.69786e9 −0.379328
\(573\) 0 0
\(574\) −5.25715e9 −1.16027
\(575\) −5.75996e9 −1.26352
\(576\) 0 0
\(577\) 4.87423e9 1.05631 0.528154 0.849149i \(-0.322884\pi\)
0.528154 + 0.849149i \(0.322884\pi\)
\(578\) 7.43778e9 1.60212
\(579\) 0 0
\(580\) 1.65836e9 0.352924
\(581\) 2.65342e9 0.561292
\(582\) 0 0
\(583\) −1.21114e10 −2.53137
\(584\) 5.55408e9 1.15390
\(585\) 0 0
\(586\) 3.31889e9 0.681319
\(587\) −1.45833e9 −0.297593 −0.148796 0.988868i \(-0.547540\pi\)
−0.148796 + 0.988868i \(0.547540\pi\)
\(588\) 0 0
\(589\) −9.35717e9 −1.88687
\(590\) 1.21412e9 0.243378
\(591\) 0 0
\(592\) −4.19120e8 −0.0830257
\(593\) −2.71973e9 −0.535592 −0.267796 0.963476i \(-0.586295\pi\)
−0.267796 + 0.963476i \(0.586295\pi\)
\(594\) 0 0
\(595\) 2.04419e10 3.97843
\(596\) 1.32763e9 0.256872
\(597\) 0 0
\(598\) −5.17951e9 −0.990454
\(599\) 8.04358e9 1.52917 0.764584 0.644524i \(-0.222944\pi\)
0.764584 + 0.644524i \(0.222944\pi\)
\(600\) 0 0
\(601\) 3.08370e9 0.579444 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(602\) 2.46541e9 0.460576
\(603\) 0 0
\(604\) 1.24275e9 0.229484
\(605\) 9.18832e9 1.68691
\(606\) 0 0
\(607\) 5.53440e9 1.00441 0.502204 0.864749i \(-0.332523\pi\)
0.502204 + 0.864749i \(0.332523\pi\)
\(608\) 2.60312e9 0.469713
\(609\) 0 0
\(610\) 1.04427e10 1.86277
\(611\) −8.18566e9 −1.45181
\(612\) 0 0
\(613\) −1.57182e9 −0.275608 −0.137804 0.990460i \(-0.544004\pi\)
−0.137804 + 0.990460i \(0.544004\pi\)
\(614\) −1.01748e10 −1.77393
\(615\) 0 0
\(616\) 1.06604e10 1.83756
\(617\) 3.83210e9 0.656809 0.328404 0.944537i \(-0.393489\pi\)
0.328404 + 0.944537i \(0.393489\pi\)
\(618\) 0 0
\(619\) −5.03030e9 −0.852465 −0.426233 0.904614i \(-0.640159\pi\)
−0.426233 + 0.904614i \(0.640159\pi\)
\(620\) 3.05994e9 0.515635
\(621\) 0 0
\(622\) 1.08294e10 1.80441
\(623\) −1.03877e10 −1.72113
\(624\) 0 0
\(625\) 4.75332e9 0.778784
\(626\) −5.02743e9 −0.819098
\(627\) 0 0
\(628\) 1.79066e9 0.288506
\(629\) 7.03257e8 0.112677
\(630\) 0 0
\(631\) 5.76799e9 0.913949 0.456974 0.889480i \(-0.348933\pi\)
0.456974 + 0.889480i \(0.348933\pi\)
\(632\) 3.71917e9 0.586052
\(633\) 0 0
\(634\) 7.99244e9 1.24557
\(635\) 9.55709e9 1.48121
\(636\) 0 0
\(637\) 1.07592e10 1.64927
\(638\) −1.06940e10 −1.63030
\(639\) 0 0
\(640\) 1.34927e10 2.03456
\(641\) 1.37617e9 0.206381 0.103191 0.994662i \(-0.467095\pi\)
0.103191 + 0.994662i \(0.467095\pi\)
\(642\) 0 0
\(643\) 8.74856e9 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(644\) −1.28607e9 −0.189743
\(645\) 0 0
\(646\) −1.43551e10 −2.09504
\(647\) −8.82702e9 −1.28130 −0.640648 0.767835i \(-0.721334\pi\)
−0.640648 + 0.767835i \(0.721334\pi\)
\(648\) 0 0
\(649\) −1.27782e9 −0.183490
\(650\) 2.03245e10 2.90283
\(651\) 0 0
\(652\) −1.66540e9 −0.235316
\(653\) 2.39631e9 0.336780 0.168390 0.985720i \(-0.446143\pi\)
0.168390 + 0.985720i \(0.446143\pi\)
\(654\) 0 0
\(655\) −4.97484e8 −0.0691727
\(656\) 5.99274e9 0.828824
\(657\) 0 0
\(658\) −1.24533e10 −1.70410
\(659\) 3.92961e9 0.534873 0.267437 0.963575i \(-0.413823\pi\)
0.267437 + 0.963575i \(0.413823\pi\)
\(660\) 0 0
\(661\) 6.16515e9 0.830307 0.415153 0.909751i \(-0.363728\pi\)
0.415153 + 0.909751i \(0.363728\pi\)
\(662\) −9.49108e9 −1.27149
\(663\) 0 0
\(664\) −2.51480e9 −0.333361
\(665\) −2.34516e10 −3.09241
\(666\) 0 0
\(667\) −5.32444e9 −0.694757
\(668\) 9.23998e8 0.119937
\(669\) 0 0
\(670\) −2.51594e9 −0.323176
\(671\) −1.09906e10 −1.40441
\(672\) 0 0
\(673\) −8.84954e9 −1.11910 −0.559549 0.828798i \(-0.689025\pi\)
−0.559549 + 0.828798i \(0.689025\pi\)
\(674\) 7.93985e9 0.998856
\(675\) 0 0
\(676\) 1.41633e9 0.176339
\(677\) −5.42902e9 −0.672452 −0.336226 0.941781i \(-0.609151\pi\)
−0.336226 + 0.941781i \(0.609151\pi\)
\(678\) 0 0
\(679\) 1.24620e10 1.52772
\(680\) −1.93740e10 −2.36286
\(681\) 0 0
\(682\) −1.97321e10 −2.38193
\(683\) −5.65530e9 −0.679178 −0.339589 0.940574i \(-0.610288\pi\)
−0.339589 + 0.940574i \(0.610288\pi\)
\(684\) 0 0
\(685\) −2.22334e10 −2.64295
\(686\) 2.67351e9 0.316189
\(687\) 0 0
\(688\) −2.81037e9 −0.329007
\(689\) 2.12779e10 2.47835
\(690\) 0 0
\(691\) −6.11832e9 −0.705439 −0.352719 0.935729i \(-0.614743\pi\)
−0.352719 + 0.935729i \(0.614743\pi\)
\(692\) −9.31577e8 −0.106868
\(693\) 0 0
\(694\) −8.47484e9 −0.962439
\(695\) −9.13142e9 −1.03179
\(696\) 0 0
\(697\) −1.00554e10 −1.12483
\(698\) −4.09921e9 −0.456253
\(699\) 0 0
\(700\) 5.04657e9 0.556101
\(701\) 1.75688e9 0.192632 0.0963160 0.995351i \(-0.469294\pi\)
0.0963160 + 0.995351i \(0.469294\pi\)
\(702\) 0 0
\(703\) −8.06800e8 −0.0875835
\(704\) −9.60715e9 −1.03774
\(705\) 0 0
\(706\) −1.45504e10 −1.55618
\(707\) −7.76663e9 −0.826541
\(708\) 0 0
\(709\) −9.00661e9 −0.949072 −0.474536 0.880236i \(-0.657384\pi\)
−0.474536 + 0.880236i \(0.657384\pi\)
\(710\) 1.89049e10 1.98230
\(711\) 0 0
\(712\) 9.84506e9 1.02221
\(713\) −9.82444e9 −1.01507
\(714\) 0 0
\(715\) −3.25066e10 −3.32583
\(716\) −1.98091e9 −0.201683
\(717\) 0 0
\(718\) 7.01640e9 0.707422
\(719\) −3.19207e8 −0.0320274 −0.0160137 0.999872i \(-0.505098\pi\)
−0.0160137 + 0.999872i \(0.505098\pi\)
\(720\) 0 0
\(721\) −2.31885e10 −2.30409
\(722\) 5.41335e9 0.535286
\(723\) 0 0
\(724\) 2.18112e9 0.213597
\(725\) 2.08932e10 2.03620
\(726\) 0 0
\(727\) 7.47713e9 0.721713 0.360856 0.932621i \(-0.382484\pi\)
0.360856 + 0.932621i \(0.382484\pi\)
\(728\) −1.87287e10 −1.79907
\(729\) 0 0
\(730\) −2.57655e10 −2.45137
\(731\) 4.71563e9 0.446507
\(732\) 0 0
\(733\) −7.27192e9 −0.682002 −0.341001 0.940063i \(-0.610766\pi\)
−0.341001 + 0.940063i \(0.610766\pi\)
\(734\) 1.35667e9 0.126630
\(735\) 0 0
\(736\) 2.73312e9 0.252689
\(737\) 2.64794e9 0.243653
\(738\) 0 0
\(739\) 6.36734e8 0.0580367 0.0290183 0.999579i \(-0.490762\pi\)
0.0290183 + 0.999579i \(0.490762\pi\)
\(740\) 2.63836e8 0.0239345
\(741\) 0 0
\(742\) 3.23713e10 2.90902
\(743\) −1.53224e10 −1.37045 −0.685227 0.728329i \(-0.740297\pi\)
−0.685227 + 0.728329i \(0.740297\pi\)
\(744\) 0 0
\(745\) 2.54184e10 2.25217
\(746\) −1.60208e10 −1.41286
\(747\) 0 0
\(748\) −4.94064e9 −0.431646
\(749\) −7.28690e9 −0.633660
\(750\) 0 0
\(751\) −1.47426e10 −1.27009 −0.635045 0.772475i \(-0.719018\pi\)
−0.635045 + 0.772475i \(0.719018\pi\)
\(752\) 1.41958e10 1.21730
\(753\) 0 0
\(754\) 1.87877e10 1.59615
\(755\) 2.37932e10 2.01205
\(756\) 0 0
\(757\) 1.03556e10 0.867643 0.433822 0.900999i \(-0.357165\pi\)
0.433822 + 0.900999i \(0.357165\pi\)
\(758\) −4.20347e9 −0.350562
\(759\) 0 0
\(760\) 2.22265e10 1.83664
\(761\) −4.05993e9 −0.333943 −0.166971 0.985962i \(-0.553399\pi\)
−0.166971 + 0.985962i \(0.553399\pi\)
\(762\) 0 0
\(763\) 5.66289e9 0.461533
\(764\) 3.40724e9 0.276424
\(765\) 0 0
\(766\) −1.33299e10 −1.07159
\(767\) 2.24493e9 0.179647
\(768\) 0 0
\(769\) −2.02272e10 −1.60396 −0.801981 0.597349i \(-0.796221\pi\)
−0.801981 + 0.597349i \(0.796221\pi\)
\(770\) −4.94541e10 −3.90377
\(771\) 0 0
\(772\) 3.97847e9 0.311212
\(773\) −1.26500e10 −0.985062 −0.492531 0.870295i \(-0.663928\pi\)
−0.492531 + 0.870295i \(0.663928\pi\)
\(774\) 0 0
\(775\) 3.85512e10 2.97497
\(776\) −1.18110e10 −0.907339
\(777\) 0 0
\(778\) −8.68905e9 −0.661521
\(779\) 1.15359e10 0.874323
\(780\) 0 0
\(781\) −1.98967e10 −1.49452
\(782\) −1.50720e10 −1.12706
\(783\) 0 0
\(784\) −1.86589e10 −1.38287
\(785\) 3.42834e10 2.52953
\(786\) 0 0
\(787\) −2.58438e9 −0.188992 −0.0944962 0.995525i \(-0.530124\pi\)
−0.0944962 + 0.995525i \(0.530124\pi\)
\(788\) 5.49535e8 0.0400086
\(789\) 0 0
\(790\) −1.72533e10 −1.24503
\(791\) 2.31679e10 1.66444
\(792\) 0 0
\(793\) 1.93088e10 1.37499
\(794\) −5.23804e8 −0.0371362
\(795\) 0 0
\(796\) 1.80545e9 0.126879
\(797\) −2.54750e10 −1.78242 −0.891209 0.453593i \(-0.850142\pi\)
−0.891209 + 0.453593i \(0.850142\pi\)
\(798\) 0 0
\(799\) −2.38197e10 −1.65205
\(800\) −1.07248e10 −0.740583
\(801\) 0 0
\(802\) −2.25537e10 −1.54386
\(803\) 2.71173e10 1.84817
\(804\) 0 0
\(805\) −2.46227e10 −1.66361
\(806\) 3.46663e10 2.33203
\(807\) 0 0
\(808\) 7.36089e9 0.490897
\(809\) −2.50645e10 −1.66433 −0.832163 0.554531i \(-0.812898\pi\)
−0.832163 + 0.554531i \(0.812898\pi\)
\(810\) 0 0
\(811\) −9.63538e9 −0.634301 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(812\) 4.66499e9 0.305777
\(813\) 0 0
\(814\) −1.70136e9 −0.110563
\(815\) −3.18851e10 −2.06317
\(816\) 0 0
\(817\) −5.40993e9 −0.347068
\(818\) −4.88351e9 −0.311958
\(819\) 0 0
\(820\) −3.77244e9 −0.238931
\(821\) 2.27996e10 1.43789 0.718946 0.695066i \(-0.244625\pi\)
0.718946 + 0.695066i \(0.244625\pi\)
\(822\) 0 0
\(823\) 7.33439e9 0.458632 0.229316 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(824\) 2.19771e10 1.36844
\(825\) 0 0
\(826\) 3.41534e9 0.210865
\(827\) −1.34883e9 −0.0829252 −0.0414626 0.999140i \(-0.513202\pi\)
−0.0414626 + 0.999140i \(0.513202\pi\)
\(828\) 0 0
\(829\) 1.61073e10 0.981932 0.490966 0.871179i \(-0.336644\pi\)
0.490966 + 0.871179i \(0.336644\pi\)
\(830\) 1.16662e10 0.708202
\(831\) 0 0
\(832\) 1.68783e10 1.01601
\(833\) 3.13085e10 1.87674
\(834\) 0 0
\(835\) 1.76905e10 1.05157
\(836\) 5.66806e9 0.335516
\(837\) 0 0
\(838\) −1.58218e10 −0.928756
\(839\) 1.49944e10 0.876523 0.438261 0.898848i \(-0.355594\pi\)
0.438261 + 0.898848i \(0.355594\pi\)
\(840\) 0 0
\(841\) 2.06352e9 0.119625
\(842\) −5.41425e9 −0.312569
\(843\) 0 0
\(844\) 3.42682e9 0.196197
\(845\) 2.71165e10 1.54609
\(846\) 0 0
\(847\) 2.58469e10 1.46156
\(848\) −3.69008e10 −2.07802
\(849\) 0 0
\(850\) 5.91427e10 3.30320
\(851\) −8.47089e8 −0.0471168
\(852\) 0 0
\(853\) −2.48117e9 −0.136879 −0.0684393 0.997655i \(-0.521802\pi\)
−0.0684393 + 0.997655i \(0.521802\pi\)
\(854\) 2.93756e10 1.61393
\(855\) 0 0
\(856\) 6.90622e9 0.376342
\(857\) −1.30344e10 −0.707388 −0.353694 0.935361i \(-0.615075\pi\)
−0.353694 + 0.935361i \(0.615075\pi\)
\(858\) 0 0
\(859\) −3.01941e10 −1.62535 −0.812673 0.582720i \(-0.801989\pi\)
−0.812673 + 0.582720i \(0.801989\pi\)
\(860\) 1.76913e9 0.0948453
\(861\) 0 0
\(862\) −2.64018e10 −1.40397
\(863\) −4.40460e9 −0.233275 −0.116638 0.993175i \(-0.537212\pi\)
−0.116638 + 0.993175i \(0.537212\pi\)
\(864\) 0 0
\(865\) −1.78356e10 −0.936984
\(866\) −3.47628e10 −1.81887
\(867\) 0 0
\(868\) 8.60766e9 0.446752
\(869\) 1.81585e10 0.938666
\(870\) 0 0
\(871\) −4.65201e9 −0.238549
\(872\) −5.36706e9 −0.274112
\(873\) 0 0
\(874\) 1.72911e10 0.876057
\(875\) 4.64109e10 2.34203
\(876\) 0 0
\(877\) 1.87680e10 0.939549 0.469775 0.882786i \(-0.344335\pi\)
0.469775 + 0.882786i \(0.344335\pi\)
\(878\) 2.85240e10 1.42226
\(879\) 0 0
\(880\) 5.63738e10 2.78861
\(881\) −2.99234e10 −1.47433 −0.737165 0.675713i \(-0.763836\pi\)
−0.737165 + 0.675713i \(0.763836\pi\)
\(882\) 0 0
\(883\) 3.52933e10 1.72516 0.862582 0.505918i \(-0.168846\pi\)
0.862582 + 0.505918i \(0.168846\pi\)
\(884\) 8.67994e9 0.422604
\(885\) 0 0
\(886\) −1.22199e10 −0.590269
\(887\) 1.25706e10 0.604818 0.302409 0.953178i \(-0.402209\pi\)
0.302409 + 0.953178i \(0.402209\pi\)
\(888\) 0 0
\(889\) 2.68842e10 1.28334
\(890\) −4.56715e10 −2.17160
\(891\) 0 0
\(892\) 1.28155e9 0.0604587
\(893\) 2.73267e10 1.28413
\(894\) 0 0
\(895\) −3.79258e10 −1.76829
\(896\) 3.79553e10 1.76276
\(897\) 0 0
\(898\) 4.22196e9 0.194557
\(899\) 3.56363e10 1.63581
\(900\) 0 0
\(901\) 6.19172e10 2.82016
\(902\) 2.43267e10 1.10372
\(903\) 0 0
\(904\) −2.19576e10 −0.988541
\(905\) 4.17590e10 1.87275
\(906\) 0 0
\(907\) −1.02215e10 −0.454874 −0.227437 0.973793i \(-0.573035\pi\)
−0.227437 + 0.973793i \(0.573035\pi\)
\(908\) −5.59896e9 −0.248203
\(909\) 0 0
\(910\) 8.68832e10 3.82200
\(911\) 3.10518e9 0.136073 0.0680366 0.997683i \(-0.478327\pi\)
0.0680366 + 0.997683i \(0.478327\pi\)
\(912\) 0 0
\(913\) −1.22783e10 −0.533936
\(914\) −3.54590e7 −0.00153608
\(915\) 0 0
\(916\) 2.73408e9 0.117538
\(917\) −1.39943e9 −0.0599320
\(918\) 0 0
\(919\) 2.97646e10 1.26501 0.632507 0.774554i \(-0.282026\pi\)
0.632507 + 0.774554i \(0.282026\pi\)
\(920\) 2.33364e10 0.988044
\(921\) 0 0
\(922\) −3.48601e10 −1.46477
\(923\) 3.49554e10 1.46321
\(924\) 0 0
\(925\) 3.32399e9 0.138090
\(926\) 1.61634e10 0.668952
\(927\) 0 0
\(928\) −9.91386e9 −0.407216
\(929\) −3.44435e10 −1.40946 −0.704729 0.709477i \(-0.748931\pi\)
−0.704729 + 0.709477i \(0.748931\pi\)
\(930\) 0 0
\(931\) −3.59182e10 −1.45878
\(932\) −3.72513e9 −0.150725
\(933\) 0 0
\(934\) 2.13746e10 0.858388
\(935\) −9.45917e10 −3.78453
\(936\) 0 0
\(937\) 2.41950e10 0.960807 0.480404 0.877048i \(-0.340490\pi\)
0.480404 + 0.877048i \(0.340490\pi\)
\(938\) −7.07737e9 −0.280003
\(939\) 0 0
\(940\) −8.93627e9 −0.350921
\(941\) 3.24324e10 1.26886 0.634432 0.772979i \(-0.281234\pi\)
0.634432 + 0.772979i \(0.281234\pi\)
\(942\) 0 0
\(943\) 1.21120e10 0.470355
\(944\) −3.89323e9 −0.150629
\(945\) 0 0
\(946\) −1.14083e10 −0.438129
\(947\) 4.68426e9 0.179232 0.0896162 0.995976i \(-0.471436\pi\)
0.0896162 + 0.995976i \(0.471436\pi\)
\(948\) 0 0
\(949\) −4.76409e10 −1.80946
\(950\) −6.78505e10 −2.56756
\(951\) 0 0
\(952\) −5.44992e10 −2.04720
\(953\) 3.34959e10 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(954\) 0 0
\(955\) 6.52338e10 2.42360
\(956\) 3.46601e9 0.128300
\(957\) 0 0
\(958\) −3.27063e10 −1.20185
\(959\) −6.25429e10 −2.28988
\(960\) 0 0
\(961\) 3.82421e10 1.38998
\(962\) 2.98902e9 0.108247
\(963\) 0 0
\(964\) 5.85717e9 0.210580
\(965\) 7.61705e10 2.72861
\(966\) 0 0
\(967\) 4.48458e10 1.59489 0.797443 0.603395i \(-0.206186\pi\)
0.797443 + 0.603395i \(0.206186\pi\)
\(968\) −2.44966e10 −0.868044
\(969\) 0 0
\(970\) 5.47914e10 1.92758
\(971\) 3.33761e10 1.16995 0.584977 0.811050i \(-0.301104\pi\)
0.584977 + 0.811050i \(0.301104\pi\)
\(972\) 0 0
\(973\) −2.56868e10 −0.893954
\(974\) 3.09127e10 1.07197
\(975\) 0 0
\(976\) −3.34858e10 −1.15289
\(977\) 2.05126e10 0.703705 0.351853 0.936055i \(-0.385552\pi\)
0.351853 + 0.936055i \(0.385552\pi\)
\(978\) 0 0
\(979\) 4.80676e10 1.63724
\(980\) 1.17458e10 0.398650
\(981\) 0 0
\(982\) 1.74546e10 0.588193
\(983\) 1.39469e9 0.0468317 0.0234158 0.999726i \(-0.492546\pi\)
0.0234158 + 0.999726i \(0.492546\pi\)
\(984\) 0 0
\(985\) 1.05212e10 0.350783
\(986\) 5.46708e10 1.81629
\(987\) 0 0
\(988\) −9.95792e9 −0.328488
\(989\) −5.68008e9 −0.186710
\(990\) 0 0
\(991\) −5.58431e10 −1.82268 −0.911342 0.411650i \(-0.864953\pi\)
−0.911342 + 0.411650i \(0.864953\pi\)
\(992\) −1.82927e10 −0.594958
\(993\) 0 0
\(994\) 5.31796e10 1.71748
\(995\) 3.45664e10 1.11243
\(996\) 0 0
\(997\) −3.23901e9 −0.103509 −0.0517546 0.998660i \(-0.516481\pi\)
−0.0517546 + 0.998660i \(0.516481\pi\)
\(998\) 2.63062e9 0.0837725
\(999\) 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.c.1.13 17
3.2 odd 2 177.8.a.c.1.5 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.5 17 3.2 odd 2
531.8.a.c.1.13 17 1.1 even 1 trivial