Properties

Label 177.8.a.c.1.5
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-12.3679\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.3679 q^{2} -27.0000 q^{3} +24.9655 q^{4} -477.980 q^{5} +333.934 q^{6} +1344.57 q^{7} +1274.32 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-12.3679 q^{2} -27.0000 q^{3} +24.9655 q^{4} -477.980 q^{5} +333.934 q^{6} +1344.57 q^{7} +1274.32 q^{8} +729.000 q^{9} +5911.62 q^{10} +6221.77 q^{11} -674.068 q^{12} +10930.7 q^{13} -16629.5 q^{14} +12905.5 q^{15} -18956.3 q^{16} -31807.5 q^{17} -9016.21 q^{18} -36490.6 q^{19} -11933.0 q^{20} -36303.3 q^{21} -76950.3 q^{22} +38312.8 q^{23} -34406.7 q^{24} +150340. q^{25} -135190. q^{26} -19683.0 q^{27} +33567.7 q^{28} -138973. q^{29} -159614. q^{30} +256427. q^{31} +71336.8 q^{32} -167988. q^{33} +393392. q^{34} -642676. q^{35} +18199.8 q^{36} +22109.8 q^{37} +451313. q^{38} -295128. q^{39} -609101. q^{40} +316134. q^{41} +448996. q^{42} +148255. q^{43} +155329. q^{44} -348448. q^{45} -473850. q^{46} +748870. q^{47} +511820. q^{48} +984312. q^{49} -1.85940e6 q^{50} +858802. q^{51} +272890. q^{52} -1.94662e6 q^{53} +243438. q^{54} -2.97388e6 q^{55} +1.71341e6 q^{56} +985246. q^{57} +1.71880e6 q^{58} -205379. q^{59} +322191. q^{60} +1.76648e6 q^{61} -3.17147e6 q^{62} +980188. q^{63} +1.54412e6 q^{64} -5.22465e6 q^{65} +2.07766e6 q^{66} -425592. q^{67} -794089. q^{68} -1.03445e6 q^{69} +7.94856e6 q^{70} -3.19791e6 q^{71} +928981. q^{72} -4.35846e6 q^{73} -273452. q^{74} -4.05918e6 q^{75} -911006. q^{76} +8.36557e6 q^{77} +3.65013e6 q^{78} -2.91855e6 q^{79} +9.06074e6 q^{80} +531441. q^{81} -3.90993e6 q^{82} -1.97344e6 q^{83} -906328. q^{84} +1.52033e7 q^{85} -1.83361e6 q^{86} +3.75226e6 q^{87} +7.92854e6 q^{88} +7.72572e6 q^{89} +4.30957e6 q^{90} +1.46970e7 q^{91} +956499. q^{92} -6.92352e6 q^{93} -9.26197e6 q^{94} +1.74418e7 q^{95} -1.92609e6 q^{96} +9.26843e6 q^{97} -1.21739e7 q^{98} +4.53567e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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 \(-0.684074\pi\)
−0.546590 + 0.837400i \(0.684074\pi\)
\(3\) −27.0000 −0.577350
\(4\) 24.9655 0.195043
\(5\) −477.980 −1.71007 −0.855037 0.518567i \(-0.826466\pi\)
−0.855037 + 0.518567i \(0.826466\pi\)
\(6\) 333.934 0.631148
\(7\) 1344.57 1.48163 0.740813 0.671711i \(-0.234440\pi\)
0.740813 + 0.671711i \(0.234440\pi\)
\(8\) 1274.32 0.879963
\(9\) 729.000 0.333333
\(10\) 5911.62 1.86942
\(11\) 6221.77 1.40942 0.704708 0.709497i \(-0.251078\pi\)
0.704708 + 0.709497i \(0.251078\pi\)
\(12\) −674.068 −0.112608
\(13\) 10930.7 1.37989 0.689947 0.723860i \(-0.257634\pi\)
0.689947 + 0.723860i \(0.257634\pi\)
\(14\) −16629.5 −1.61968
\(15\) 12905.5 0.987312
\(16\) −18956.3 −1.15700
\(17\) −31807.5 −1.57021 −0.785105 0.619362i \(-0.787391\pi\)
−0.785105 + 0.619362i \(0.787391\pi\)
\(18\) −9016.21 −0.364393
\(19\) −36490.6 −1.22052 −0.610258 0.792203i \(-0.708934\pi\)
−0.610258 + 0.792203i \(0.708934\pi\)
\(20\) −11933.0 −0.333538
\(21\) −36303.3 −0.855417
\(22\) −76950.3 −1.54075
\(23\) 38312.8 0.656594 0.328297 0.944575i \(-0.393525\pi\)
0.328297 + 0.944575i \(0.393525\pi\)
\(24\) −34406.7 −0.508047
\(25\) 150340. 1.92435
\(26\) −135190. −1.50847
\(27\) −19683.0 −0.192450
\(28\) 33567.7 0.288981
\(29\) −138973. −1.05812 −0.529062 0.848583i \(-0.677456\pi\)
−0.529062 + 0.848583i \(0.677456\pi\)
\(30\) −159614. −1.07931
\(31\) 256427. 1.54596 0.772979 0.634432i \(-0.218766\pi\)
0.772979 + 0.634432i \(0.218766\pi\)
\(32\) 71336.8 0.384847
\(33\) −167988. −0.813727
\(34\) 393392. 1.71652
\(35\) −642676. −2.53369
\(36\) 18199.8 0.0650143
\(37\) 22109.8 0.0717594 0.0358797 0.999356i \(-0.488577\pi\)
0.0358797 + 0.999356i \(0.488577\pi\)
\(38\) 451313. 1.33424
\(39\) −295128. −0.796682
\(40\) −609101. −1.50480
\(41\) 316134. 0.716355 0.358178 0.933653i \(-0.383398\pi\)
0.358178 + 0.933653i \(0.383398\pi\)
\(42\) 448996. 0.935125
\(43\) 148255. 0.284361 0.142181 0.989841i \(-0.454589\pi\)
0.142181 + 0.989841i \(0.454589\pi\)
\(44\) 155329. 0.274897
\(45\) −348448. −0.570025
\(46\) −473850. −0.717776
\(47\) 748870. 1.05212 0.526058 0.850448i \(-0.323669\pi\)
0.526058 + 0.850448i \(0.323669\pi\)
\(48\) 511820. 0.667995
\(49\) 984312. 1.19522
\(50\) −1.85940e6 −2.10367
\(51\) 858802. 0.906562
\(52\) 272890. 0.269138
\(53\) −1.94662e6 −1.79604 −0.898021 0.439953i \(-0.854995\pi\)
−0.898021 + 0.439953i \(0.854995\pi\)
\(54\) 243438. 0.210383
\(55\) −2.97388e6 −2.41021
\(56\) 1.71341e6 1.30378
\(57\) 985246. 0.704665
\(58\) 1.71880e6 1.15672
\(59\) −205379. −0.130189
\(60\) 322191. 0.192568
\(61\) 1.76648e6 0.996445 0.498222 0.867049i \(-0.333986\pi\)
0.498222 + 0.867049i \(0.333986\pi\)
\(62\) −3.17147e6 −1.69001
\(63\) 980188. 0.493875
\(64\) 1.54412e6 0.736294
\(65\) −5.22465e6 −2.35972
\(66\) 2.07766e6 0.889550
\(67\) −425592. −0.172875 −0.0864374 0.996257i \(-0.527548\pi\)
−0.0864374 + 0.996257i \(0.527548\pi\)
\(68\) −794089. −0.306258
\(69\) −1.03445e6 −0.379085
\(70\) 7.94856e6 2.76978
\(71\) −3.19791e6 −1.06038 −0.530191 0.847878i \(-0.677880\pi\)
−0.530191 + 0.847878i \(0.677880\pi\)
\(72\) 928981. 0.293321
\(73\) −4.35846e6 −1.31130 −0.655651 0.755064i \(-0.727606\pi\)
−0.655651 + 0.755064i \(0.727606\pi\)
\(74\) −273452. −0.0784459
\(75\) −4.05918e6 −1.11103
\(76\) −911006. −0.238053
\(77\) 8.36557e6 2.08823
\(78\) 3.65013e6 0.870917
\(79\) −2.91855e6 −0.665996 −0.332998 0.942928i \(-0.608060\pi\)
−0.332998 + 0.942928i \(0.608060\pi\)
\(80\) 9.06074e6 1.97856
\(81\) 531441. 0.111111
\(82\) −3.90993e6 −0.783105
\(83\) −1.97344e6 −0.378835 −0.189418 0.981897i \(-0.560660\pi\)
−0.189418 + 0.981897i \(0.560660\pi\)
\(84\) −906328. −0.166843
\(85\) 1.52033e7 2.68518
\(86\) −1.83361e6 −0.310858
\(87\) 3.75226e6 0.610908
\(88\) 7.92854e6 1.24023
\(89\) 7.72572e6 1.16165 0.580823 0.814030i \(-0.302731\pi\)
0.580823 + 0.814030i \(0.302731\pi\)
\(90\) 4.30957e6 0.623140
\(91\) 1.46970e7 2.04449
\(92\) 956499. 0.128064
\(93\) −6.92352e6 −0.892559
\(94\) −9.26197e6 −1.15015
\(95\) 1.74418e7 2.08717
\(96\) −1.92609e6 −0.222192
\(97\) 9.26843e6 1.03111 0.515555 0.856857i \(-0.327586\pi\)
0.515555 + 0.856857i \(0.327586\pi\)
\(98\) −1.21739e7 −1.30659
\(99\) 4.53567e6 0.469805
\(100\) 3.75331e6 0.375331
\(101\) 5.77631e6 0.557861 0.278930 0.960311i \(-0.410020\pi\)
0.278930 + 0.960311i \(0.410020\pi\)
\(102\) −1.06216e7 −0.991035
\(103\) −1.72461e7 −1.55511 −0.777555 0.628815i \(-0.783540\pi\)
−0.777555 + 0.628815i \(0.783540\pi\)
\(104\) 1.39292e7 1.21426
\(105\) 1.73522e7 1.46283
\(106\) 2.40757e7 1.96340
\(107\) 5.41952e6 0.427679 0.213839 0.976869i \(-0.431403\pi\)
0.213839 + 0.976869i \(0.431403\pi\)
\(108\) −491396. −0.0375360
\(109\) 4.21169e6 0.311504 0.155752 0.987796i \(-0.450220\pi\)
0.155752 + 0.987796i \(0.450220\pi\)
\(110\) 3.67807e7 2.63479
\(111\) −596965. −0.0414303
\(112\) −2.54880e7 −1.71424
\(113\) −1.72308e7 −1.12339 −0.561695 0.827345i \(-0.689850\pi\)
−0.561695 + 0.827345i \(0.689850\pi\)
\(114\) −1.21855e7 −0.770326
\(115\) −1.83128e7 −1.12282
\(116\) −3.46952e6 −0.206379
\(117\) 7.96847e6 0.459965
\(118\) 2.54011e6 0.142320
\(119\) −4.27672e7 −2.32647
\(120\) 1.64457e7 0.868798
\(121\) 1.92232e7 0.986455
\(122\) −2.18476e7 −1.08929
\(123\) −8.53563e6 −0.413588
\(124\) 6.40182e6 0.301528
\(125\) −3.45174e7 −1.58071
\(126\) −1.21229e7 −0.539895
\(127\) 1.99947e7 0.866169 0.433085 0.901353i \(-0.357425\pi\)
0.433085 + 0.901353i \(0.357425\pi\)
\(128\) −2.82287e7 −1.18975
\(129\) −4.00289e6 −0.164176
\(130\) 6.46181e7 2.57960
\(131\) 1.04080e6 0.0404501 0.0202251 0.999795i \(-0.493562\pi\)
0.0202251 + 0.999795i \(0.493562\pi\)
\(132\) −4.19389e6 −0.158712
\(133\) −4.90640e7 −1.80835
\(134\) 5.26369e6 0.188983
\(135\) 9.40809e6 0.329104
\(136\) −4.05330e7 −1.38173
\(137\) 4.65153e7 1.54552 0.772759 0.634699i \(-0.218876\pi\)
0.772759 + 0.634699i \(0.218876\pi\)
\(138\) 1.27940e7 0.414408
\(139\) −1.91042e7 −0.603360 −0.301680 0.953409i \(-0.597547\pi\)
−0.301680 + 0.953409i \(0.597547\pi\)
\(140\) −1.60447e7 −0.494178
\(141\) −2.02195e7 −0.607440
\(142\) 3.95515e7 1.15919
\(143\) 6.80082e7 1.94484
\(144\) −1.38191e7 −0.385667
\(145\) 6.64262e7 1.80947
\(146\) 5.39050e7 1.43349
\(147\) −2.65764e7 −0.690059
\(148\) 551982. 0.0139962
\(149\) −5.31787e7 −1.31700 −0.658500 0.752580i \(-0.728809\pi\)
−0.658500 + 0.752580i \(0.728809\pi\)
\(150\) 5.02037e7 1.21455
\(151\) 4.97786e7 1.17658 0.588292 0.808648i \(-0.299800\pi\)
0.588292 + 0.808648i \(0.299800\pi\)
\(152\) −4.65008e7 −1.07401
\(153\) −2.31876e7 −0.523404
\(154\) −1.03465e8 −2.28281
\(155\) −1.22567e8 −2.64370
\(156\) −7.36802e6 −0.155387
\(157\) 7.17256e7 1.47920 0.739598 0.673049i \(-0.235016\pi\)
0.739598 + 0.673049i \(0.235016\pi\)
\(158\) 3.60963e7 0.728054
\(159\) 5.25588e7 1.03694
\(160\) −3.40976e7 −0.658118
\(161\) 5.15141e7 0.972827
\(162\) −6.57282e6 −0.121464
\(163\) −6.67080e7 −1.20648 −0.603241 0.797559i \(-0.706124\pi\)
−0.603241 + 0.797559i \(0.706124\pi\)
\(164\) 7.89245e6 0.139720
\(165\) 8.02948e7 1.39153
\(166\) 2.44073e7 0.414135
\(167\) −3.70110e7 −0.614927 −0.307463 0.951560i \(-0.599480\pi\)
−0.307463 + 0.951560i \(0.599480\pi\)
\(168\) −4.62621e7 −0.752736
\(169\) 5.67313e7 0.904106
\(170\) −1.88034e8 −2.93538
\(171\) −2.66017e7 −0.406839
\(172\) 3.70127e6 0.0554627
\(173\) 3.73146e7 0.547920 0.273960 0.961741i \(-0.411666\pi\)
0.273960 + 0.961741i \(0.411666\pi\)
\(174\) −4.64077e7 −0.667832
\(175\) 2.02142e8 2.85117
\(176\) −1.17942e8 −1.63070
\(177\) 5.54523e6 0.0751646
\(178\) −9.55511e7 −1.26989
\(179\) 7.93461e7 1.03405 0.517023 0.855972i \(-0.327040\pi\)
0.517023 + 0.855972i \(0.327040\pi\)
\(180\) −8.69916e6 −0.111179
\(181\) 8.73656e7 1.09513 0.547565 0.836763i \(-0.315555\pi\)
0.547565 + 0.836763i \(0.315555\pi\)
\(182\) −1.81772e8 −2.23499
\(183\) −4.76948e7 −0.575298
\(184\) 4.88229e7 0.577779
\(185\) −1.05680e7 −0.122714
\(186\) 8.56296e7 0.975727
\(187\) −1.97899e8 −2.21308
\(188\) 1.86959e7 0.205208
\(189\) −2.64651e7 −0.285139
\(190\) −2.15719e8 −2.28166
\(191\) −1.36478e8 −1.41725 −0.708625 0.705586i \(-0.750684\pi\)
−0.708625 + 0.705586i \(0.750684\pi\)
\(192\) −4.16912e7 −0.425099
\(193\) 1.59359e8 1.59561 0.797804 0.602917i \(-0.205995\pi\)
0.797804 + 0.602917i \(0.205995\pi\)
\(194\) −1.14631e8 −1.12719
\(195\) 1.41066e8 1.36239
\(196\) 2.45738e7 0.233118
\(197\) −2.20118e7 −0.205127 −0.102564 0.994726i \(-0.532705\pi\)
−0.102564 + 0.994726i \(0.532705\pi\)
\(198\) −5.60968e7 −0.513582
\(199\) 7.23177e7 0.650517 0.325259 0.945625i \(-0.394549\pi\)
0.325259 + 0.945625i \(0.394549\pi\)
\(200\) 1.91582e8 1.69336
\(201\) 1.14910e7 0.0998093
\(202\) −7.14410e7 −0.609842
\(203\) −1.86858e8 −1.56774
\(204\) 2.14404e7 0.176818
\(205\) −1.51106e8 −1.22502
\(206\) 2.13299e8 1.70001
\(207\) 2.79301e7 0.218865
\(208\) −2.07205e8 −1.59654
\(209\) −2.27036e8 −1.72022
\(210\) −2.14611e8 −1.59913
\(211\) 1.37262e8 1.00592 0.502959 0.864310i \(-0.332245\pi\)
0.502959 + 0.864310i \(0.332245\pi\)
\(212\) −4.85984e7 −0.350305
\(213\) 8.63437e7 0.612212
\(214\) −6.70282e7 −0.467530
\(215\) −7.08631e7 −0.486279
\(216\) −2.50825e7 −0.169349
\(217\) 3.44782e8 2.29053
\(218\) −5.20899e7 −0.340530
\(219\) 1.17678e8 0.757080
\(220\) −7.42444e7 −0.470094
\(221\) −3.47677e8 −2.16672
\(222\) 7.38321e6 0.0452908
\(223\) 5.13329e7 0.309977 0.154988 0.987916i \(-0.450466\pi\)
0.154988 + 0.987916i \(0.450466\pi\)
\(224\) 9.59170e7 0.570200
\(225\) 1.09598e8 0.641451
\(226\) 2.13109e8 1.22807
\(227\) 2.24268e8 1.27256 0.636278 0.771459i \(-0.280473\pi\)
0.636278 + 0.771459i \(0.280473\pi\)
\(228\) 2.45972e7 0.137440
\(229\) 1.09514e8 0.602625 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(230\) 2.26491e8 1.22745
\(231\) −2.25870e8 −1.20564
\(232\) −1.77096e8 −0.931110
\(233\) 1.49211e8 0.772779 0.386390 0.922336i \(-0.373722\pi\)
0.386390 + 0.922336i \(0.373722\pi\)
\(234\) −9.85534e7 −0.502824
\(235\) −3.57945e8 −1.79920
\(236\) −5.12739e6 −0.0253924
\(237\) 7.88007e7 0.384513
\(238\) 5.28942e8 2.54325
\(239\) −1.38832e8 −0.657804 −0.328902 0.944364i \(-0.606679\pi\)
−0.328902 + 0.944364i \(0.606679\pi\)
\(240\) −2.44640e8 −1.14232
\(241\) 2.34611e8 1.07966 0.539831 0.841773i \(-0.318488\pi\)
0.539831 + 0.841773i \(0.318488\pi\)
\(242\) −2.37751e8 −1.07837
\(243\) −1.43489e7 −0.0641500
\(244\) 4.41009e7 0.194349
\(245\) −4.70482e8 −2.04391
\(246\) 1.05568e8 0.452126
\(247\) −3.98867e8 −1.68418
\(248\) 3.26770e8 1.36039
\(249\) 5.32828e7 0.218721
\(250\) 4.26909e8 1.72800
\(251\) 2.52396e8 1.00745 0.503725 0.863864i \(-0.331962\pi\)
0.503725 + 0.863864i \(0.331962\pi\)
\(252\) 2.44709e7 0.0963269
\(253\) 2.38374e8 0.925414
\(254\) −2.47293e8 −0.946879
\(255\) −4.10490e8 −1.55029
\(256\) 1.51483e8 0.564316
\(257\) 4.59969e8 1.69029 0.845147 0.534534i \(-0.179513\pi\)
0.845147 + 0.534534i \(0.179513\pi\)
\(258\) 4.95075e7 0.179474
\(259\) 2.97281e7 0.106321
\(260\) −1.30436e8 −0.460247
\(261\) −1.01311e8 −0.352708
\(262\) −1.28726e7 −0.0442193
\(263\) 3.87682e8 1.31411 0.657053 0.753844i \(-0.271803\pi\)
0.657053 + 0.753844i \(0.271803\pi\)
\(264\) −2.14071e8 −0.716050
\(265\) 9.30447e8 3.07136
\(266\) 6.06820e8 1.97685
\(267\) −2.08594e8 −0.670677
\(268\) −1.06251e7 −0.0337180
\(269\) 2.41785e7 0.0757351 0.0378675 0.999283i \(-0.487944\pi\)
0.0378675 + 0.999283i \(0.487944\pi\)
\(270\) −1.16358e8 −0.359770
\(271\) −2.33181e7 −0.0711707 −0.0355853 0.999367i \(-0.511330\pi\)
−0.0355853 + 0.999367i \(0.511330\pi\)
\(272\) 6.02952e8 1.81674
\(273\) −3.96819e8 −1.18038
\(274\) −5.75298e8 −1.68953
\(275\) 9.35381e8 2.71222
\(276\) −2.58255e7 −0.0739378
\(277\) 3.40111e7 0.0961482 0.0480741 0.998844i \(-0.484692\pi\)
0.0480741 + 0.998844i \(0.484692\pi\)
\(278\) 2.36279e8 0.659581
\(279\) 1.86935e8 0.515319
\(280\) −8.18976e8 −2.22955
\(281\) −9.09433e7 −0.244511 −0.122256 0.992499i \(-0.539013\pi\)
−0.122256 + 0.992499i \(0.539013\pi\)
\(282\) 2.50073e8 0.664041
\(283\) −1.54868e8 −0.406171 −0.203085 0.979161i \(-0.565097\pi\)
−0.203085 + 0.979161i \(0.565097\pi\)
\(284\) −7.98374e7 −0.206820
\(285\) −4.70928e8 −1.20503
\(286\) −8.41120e8 −2.12607
\(287\) 4.25063e8 1.06137
\(288\) 5.20045e7 0.128282
\(289\) 6.01377e8 1.46556
\(290\) −8.21554e8 −1.97808
\(291\) −2.50247e8 −0.595311
\(292\) −1.08811e8 −0.255760
\(293\) −2.68346e8 −0.623245 −0.311623 0.950206i \(-0.600872\pi\)
−0.311623 + 0.950206i \(0.600872\pi\)
\(294\) 3.28695e8 0.754358
\(295\) 9.81671e7 0.222633
\(296\) 2.81750e7 0.0631456
\(297\) −1.22463e8 −0.271242
\(298\) 6.57710e8 1.43972
\(299\) 4.18786e8 0.906030
\(300\) −1.01339e8 −0.216698
\(301\) 1.99339e8 0.421317
\(302\) −6.15658e8 −1.28622
\(303\) −1.55960e8 −0.322081
\(304\) 6.91727e8 1.41214
\(305\) −8.44340e8 −1.70399
\(306\) 2.86783e8 0.572174
\(307\) −8.22676e8 −1.62272 −0.811362 0.584544i \(-0.801273\pi\)
−0.811362 + 0.584544i \(0.801273\pi\)
\(308\) 2.08850e8 0.407294
\(309\) 4.65645e8 0.897843
\(310\) 1.51590e9 2.89004
\(311\) −8.75600e8 −1.65061 −0.825305 0.564687i \(-0.808997\pi\)
−0.825305 + 0.564687i \(0.808997\pi\)
\(312\) −3.76089e8 −0.701051
\(313\) −4.06490e8 −0.749280 −0.374640 0.927170i \(-0.622234\pi\)
−0.374640 + 0.927170i \(0.622234\pi\)
\(314\) −8.87097e8 −1.61703
\(315\) −4.68510e8 −0.844564
\(316\) −7.28629e7 −0.129898
\(317\) −6.46224e8 −1.13940 −0.569699 0.821853i \(-0.692940\pi\)
−0.569699 + 0.821853i \(0.692940\pi\)
\(318\) −6.50043e8 −1.13357
\(319\) −8.64655e8 −1.49134
\(320\) −7.38059e8 −1.25912
\(321\) −1.46327e8 −0.246921
\(322\) −6.37122e8 −1.06348
\(323\) 1.16067e9 1.91647
\(324\) 1.32677e7 0.0216714
\(325\) 1.64332e9 2.65540
\(326\) 8.25039e8 1.31890
\(327\) −1.13716e8 −0.179847
\(328\) 4.02857e8 0.630366
\(329\) 1.00690e9 1.55884
\(330\) −9.93080e8 −1.52120
\(331\) −7.67395e8 −1.16311 −0.581555 0.813507i \(-0.697556\pi\)
−0.581555 + 0.813507i \(0.697556\pi\)
\(332\) −4.92678e7 −0.0738891
\(333\) 1.61180e7 0.0239198
\(334\) 4.57749e8 0.672226
\(335\) 2.03425e8 0.295629
\(336\) 6.88176e8 0.989719
\(337\) 6.41971e8 0.913715 0.456858 0.889540i \(-0.348975\pi\)
0.456858 + 0.889540i \(0.348975\pi\)
\(338\) −7.01649e8 −0.988351
\(339\) 4.65231e8 0.648589
\(340\) 3.79559e8 0.523725
\(341\) 1.59543e9 2.17890
\(342\) 3.29007e8 0.444748
\(343\) 2.16165e8 0.289238
\(344\) 1.88925e8 0.250228
\(345\) 4.94445e8 0.648263
\(346\) −4.61504e8 −0.598976
\(347\) 6.85227e8 0.880403 0.440201 0.897899i \(-0.354907\pi\)
0.440201 + 0.897899i \(0.354907\pi\)
\(348\) 9.36770e7 0.119153
\(349\) −3.31439e8 −0.417363 −0.208682 0.977984i \(-0.566917\pi\)
−0.208682 + 0.977984i \(0.566917\pi\)
\(350\) −2.50008e9 −3.11685
\(351\) −2.15149e8 −0.265561
\(352\) 4.43841e8 0.542410
\(353\) 1.17646e9 1.42353 0.711766 0.702416i \(-0.247896\pi\)
0.711766 + 0.702416i \(0.247896\pi\)
\(354\) −6.85830e7 −0.0821685
\(355\) 1.52854e9 1.81333
\(356\) 1.92876e8 0.226571
\(357\) 1.15471e9 1.34319
\(358\) −9.81346e8 −1.13040
\(359\) −5.67306e8 −0.647123 −0.323562 0.946207i \(-0.604880\pi\)
−0.323562 + 0.946207i \(0.604880\pi\)
\(360\) −4.44035e8 −0.501601
\(361\) 4.37693e8 0.489660
\(362\) −1.08053e9 −1.19717
\(363\) −5.19027e8 −0.569530
\(364\) 3.66918e8 0.398762
\(365\) 2.08326e9 2.24242
\(366\) 5.89886e8 0.628904
\(367\) 1.09692e8 0.115836 0.0579182 0.998321i \(-0.481554\pi\)
0.0579182 + 0.998321i \(0.481554\pi\)
\(368\) −7.26270e8 −0.759680
\(369\) 2.30462e8 0.238785
\(370\) 1.30705e8 0.134148
\(371\) −2.61736e9 −2.66106
\(372\) −1.72849e8 −0.174087
\(373\) −1.29535e9 −1.29243 −0.646214 0.763156i \(-0.723649\pi\)
−0.646214 + 0.763156i \(0.723649\pi\)
\(374\) 2.44760e9 2.41930
\(375\) 9.31970e8 0.912625
\(376\) 9.54302e8 0.925824
\(377\) −1.51907e9 −1.46010
\(378\) 3.27318e8 0.311708
\(379\) −3.39868e8 −0.320681 −0.160341 0.987062i \(-0.551259\pi\)
−0.160341 + 0.987062i \(0.551259\pi\)
\(380\) 4.35443e8 0.407088
\(381\) −5.39858e8 −0.500083
\(382\) 1.68795e9 1.54931
\(383\) 1.07778e9 0.980246 0.490123 0.871653i \(-0.336952\pi\)
0.490123 + 0.871653i \(0.336952\pi\)
\(384\) 7.62174e8 0.686902
\(385\) −3.99858e9 −3.57103
\(386\) −1.97094e9 −1.74429
\(387\) 1.08078e8 0.0947872
\(388\) 2.31391e8 0.201111
\(389\) 7.02547e8 0.605134 0.302567 0.953128i \(-0.402156\pi\)
0.302567 + 0.953128i \(0.402156\pi\)
\(390\) −1.74469e9 −1.48933
\(391\) −1.21863e9 −1.03099
\(392\) 1.25433e9 1.05175
\(393\) −2.81017e7 −0.0233539
\(394\) 2.72240e8 0.224241
\(395\) 1.39501e9 1.13890
\(396\) 1.13235e8 0.0916322
\(397\) −4.23518e7 −0.0339708 −0.0169854 0.999856i \(-0.505407\pi\)
−0.0169854 + 0.999856i \(0.505407\pi\)
\(398\) −8.94419e8 −0.711132
\(399\) 1.32473e9 1.04405
\(400\) −2.84989e9 −2.22648
\(401\) 1.82357e9 1.41227 0.706133 0.708079i \(-0.250438\pi\)
0.706133 + 0.708079i \(0.250438\pi\)
\(402\) −1.42120e8 −0.109110
\(403\) 2.80292e9 2.13326
\(404\) 1.44208e8 0.108807
\(405\) −2.54018e8 −0.190008
\(406\) 2.31104e9 1.71383
\(407\) 1.37562e8 0.101139
\(408\) 1.09439e9 0.797741
\(409\) −3.94853e8 −0.285367 −0.142684 0.989768i \(-0.545573\pi\)
−0.142684 + 0.989768i \(0.545573\pi\)
\(410\) 1.86887e9 1.33917
\(411\) −1.25591e9 −0.892306
\(412\) −4.30558e8 −0.303313
\(413\) −2.76145e8 −0.192891
\(414\) −3.45437e8 −0.239259
\(415\) 9.43265e8 0.647836
\(416\) 7.79760e8 0.531049
\(417\) 5.15813e8 0.348350
\(418\) 2.80796e9 1.88051
\(419\) 1.27926e9 0.849591 0.424795 0.905289i \(-0.360346\pi\)
0.424795 + 0.905289i \(0.360346\pi\)
\(420\) 4.33207e8 0.285314
\(421\) −4.37766e8 −0.285926 −0.142963 0.989728i \(-0.545663\pi\)
−0.142963 + 0.989728i \(0.545663\pi\)
\(422\) −1.69765e9 −1.09965
\(423\) 5.45926e8 0.350706
\(424\) −2.48063e9 −1.58045
\(425\) −4.78194e9 −3.02164
\(426\) −1.06789e9 −0.669258
\(427\) 2.37514e9 1.47636
\(428\) 1.35301e8 0.0834157
\(429\) −1.83622e9 −1.12286
\(430\) 8.76430e8 0.531591
\(431\) 2.13470e9 1.28430 0.642150 0.766579i \(-0.278043\pi\)
0.642150 + 0.766579i \(0.278043\pi\)
\(432\) 3.73117e8 0.222665
\(433\) −2.81072e9 −1.66383 −0.831917 0.554900i \(-0.812757\pi\)
−0.831917 + 0.554900i \(0.812757\pi\)
\(434\) −4.26424e9 −2.50396
\(435\) −1.79351e9 −1.04470
\(436\) 1.05147e8 0.0607567
\(437\) −1.39806e9 −0.801384
\(438\) −1.45544e9 −0.827625
\(439\) 2.30629e9 1.30103 0.650516 0.759493i \(-0.274553\pi\)
0.650516 + 0.759493i \(0.274553\pi\)
\(440\) −3.78969e9 −2.12089
\(441\) 7.17564e8 0.398405
\(442\) 4.30005e9 2.36862
\(443\) 9.88032e8 0.539956 0.269978 0.962867i \(-0.412984\pi\)
0.269978 + 0.962867i \(0.412984\pi\)
\(444\) −1.49035e7 −0.00808068
\(445\) −3.69274e9 −1.98650
\(446\) −6.34882e8 −0.338860
\(447\) 1.43583e9 0.760371
\(448\) 2.07617e9 1.09091
\(449\) −3.41363e8 −0.177973 −0.0889866 0.996033i \(-0.528363\pi\)
−0.0889866 + 0.996033i \(0.528363\pi\)
\(450\) −1.35550e9 −0.701222
\(451\) 1.96692e9 1.00964
\(452\) −4.30175e8 −0.219109
\(453\) −1.34402e9 −0.679302
\(454\) −2.77373e9 −1.39113
\(455\) −7.02488e9 −3.49622
\(456\) 1.25552e9 0.620080
\(457\) −2.86701e6 −0.00140515 −0.000702575 1.00000i \(-0.500224\pi\)
−0.000702575 1.00000i \(0.500224\pi\)
\(458\) −1.35447e9 −0.658778
\(459\) 6.26067e8 0.302187
\(460\) −4.57187e8 −0.218999
\(461\) 2.81859e9 1.33992 0.669959 0.742398i \(-0.266312\pi\)
0.669959 + 0.742398i \(0.266312\pi\)
\(462\) 2.79355e9 1.31798
\(463\) 1.30688e9 0.611932 0.305966 0.952042i \(-0.401021\pi\)
0.305966 + 0.952042i \(0.401021\pi\)
\(464\) 2.63441e9 1.22425
\(465\) 3.30931e9 1.52634
\(466\) −1.84543e9 −0.844787
\(467\) −1.72823e9 −0.785221 −0.392611 0.919705i \(-0.628428\pi\)
−0.392611 + 0.919705i \(0.628428\pi\)
\(468\) 1.98937e8 0.0897128
\(469\) −5.72236e8 −0.256136
\(470\) 4.42704e9 1.96685
\(471\) −1.93659e9 −0.854014
\(472\) −2.61719e8 −0.114561
\(473\) 9.22410e8 0.400784
\(474\) −9.74601e8 −0.420342
\(475\) −5.48600e9 −2.34870
\(476\) −1.06770e9 −0.453760
\(477\) −1.41909e9 −0.598680
\(478\) 1.71706e9 0.719099
\(479\) 2.64444e9 1.09941 0.549705 0.835359i \(-0.314740\pi\)
0.549705 + 0.835359i \(0.314740\pi\)
\(480\) 9.20635e8 0.379964
\(481\) 2.41675e8 0.0990203
\(482\) −2.90165e9 −1.18027
\(483\) −1.39088e9 −0.561662
\(484\) 4.79917e8 0.192401
\(485\) −4.43012e9 −1.76327
\(486\) 1.77466e8 0.0701275
\(487\) 2.49943e9 0.980594 0.490297 0.871556i \(-0.336888\pi\)
0.490297 + 0.871556i \(0.336888\pi\)
\(488\) 2.25106e9 0.876835
\(489\) 1.80112e9 0.696563
\(490\) 5.81888e9 2.23436
\(491\) −1.41128e9 −0.538057 −0.269029 0.963132i \(-0.586703\pi\)
−0.269029 + 0.963132i \(0.586703\pi\)
\(492\) −2.13096e8 −0.0806673
\(493\) 4.42037e9 1.66148
\(494\) 4.93316e9 1.84111
\(495\) −2.16796e9 −0.803402
\(496\) −4.86090e9 −1.78867
\(497\) −4.29980e9 −1.57109
\(498\) −6.58998e8 −0.239101
\(499\) 2.12697e8 0.0766319 0.0383159 0.999266i \(-0.487801\pi\)
0.0383159 + 0.999266i \(0.487801\pi\)
\(500\) −8.61744e8 −0.308307
\(501\) 9.99298e8 0.355028
\(502\) −3.12161e9 −1.10133
\(503\) −2.85799e7 −0.0100132 −0.00500660 0.999987i \(-0.501594\pi\)
−0.00500660 + 0.999987i \(0.501594\pi\)
\(504\) 1.24908e9 0.434592
\(505\) −2.76096e9 −0.953984
\(506\) −2.94819e9 −1.01164
\(507\) −1.53175e9 −0.521986
\(508\) 4.99178e8 0.168940
\(509\) −1.91356e9 −0.643178 −0.321589 0.946879i \(-0.604217\pi\)
−0.321589 + 0.946879i \(0.604217\pi\)
\(510\) 5.07691e9 1.69474
\(511\) −5.86023e9 −1.94286
\(512\) 1.73974e9 0.572849
\(513\) 7.18245e8 0.234888
\(514\) −5.68885e9 −1.84780
\(515\) 8.24330e9 2.65935
\(516\) −9.99342e7 −0.0320214
\(517\) 4.65929e9 1.48287
\(518\) −3.67674e8 −0.116228
\(519\) −1.00749e9 −0.316342
\(520\) −6.65789e9 −2.07647
\(521\) 1.49555e9 0.463306 0.231653 0.972798i \(-0.425587\pi\)
0.231653 + 0.972798i \(0.425587\pi\)
\(522\) 1.25301e9 0.385573
\(523\) 2.48258e9 0.758835 0.379417 0.925226i \(-0.376124\pi\)
0.379417 + 0.925226i \(0.376124\pi\)
\(524\) 2.59842e7 0.00788951
\(525\) −5.45784e9 −1.64613
\(526\) −4.79482e9 −1.43656
\(527\) −8.15629e9 −2.42748
\(528\) 3.18443e9 0.941483
\(529\) −1.93695e9 −0.568884
\(530\) −1.15077e10 −3.35755
\(531\) −1.49721e8 −0.0433963
\(532\) −1.22491e9 −0.352705
\(533\) 3.45557e9 0.988494
\(534\) 2.57988e9 0.733171
\(535\) −2.59043e9 −0.731363
\(536\) −5.42342e8 −0.152124
\(537\) −2.14234e9 −0.597007
\(538\) −2.99038e8 −0.0827921
\(539\) 6.12416e9 1.68456
\(540\) 2.34877e8 0.0641894
\(541\) −6.44103e9 −1.74890 −0.874450 0.485115i \(-0.838778\pi\)
−0.874450 + 0.485115i \(0.838778\pi\)
\(542\) 2.88397e8 0.0778024
\(543\) −2.35887e9 −0.632273
\(544\) −2.26904e9 −0.604292
\(545\) −2.01311e9 −0.532695
\(546\) 4.90783e9 1.29037
\(547\) −5.40495e8 −0.141200 −0.0706002 0.997505i \(-0.522491\pi\)
−0.0706002 + 0.997505i \(0.522491\pi\)
\(548\) 1.16128e9 0.301442
\(549\) 1.28776e9 0.332148
\(550\) −1.15687e10 −2.96494
\(551\) 5.07120e9 1.29146
\(552\) −1.31822e9 −0.333581
\(553\) −3.92417e9 −0.986757
\(554\) −4.20646e8 −0.105107
\(555\) 2.85337e8 0.0708489
\(556\) −4.76945e8 −0.117681
\(557\) 2.83692e9 0.695591 0.347795 0.937570i \(-0.386930\pi\)
0.347795 + 0.937570i \(0.386930\pi\)
\(558\) −2.31200e9 −0.563337
\(559\) 1.62053e9 0.392389
\(560\) 1.21828e10 2.93148
\(561\) 5.34326e9 1.27772
\(562\) 1.12478e9 0.267295
\(563\) 7.86504e9 1.85747 0.928734 0.370747i \(-0.120898\pi\)
0.928734 + 0.370747i \(0.120898\pi\)
\(564\) −5.04789e8 −0.118477
\(565\) 8.23597e9 1.92108
\(566\) 1.91539e9 0.444018
\(567\) 7.14557e8 0.164625
\(568\) −4.07517e9 −0.933097
\(569\) 6.12820e9 1.39457 0.697285 0.716794i \(-0.254391\pi\)
0.697285 + 0.716794i \(0.254391\pi\)
\(570\) 5.82441e9 1.31731
\(571\) 7.88550e9 1.77257 0.886284 0.463142i \(-0.153278\pi\)
0.886284 + 0.463142i \(0.153278\pi\)
\(572\) 1.69786e9 0.379328
\(573\) 3.68491e9 0.818249
\(574\) −5.25715e9 −1.16027
\(575\) 5.75996e9 1.26352
\(576\) 1.12566e9 0.245431
\(577\) 4.87423e9 1.05631 0.528154 0.849149i \(-0.322884\pi\)
0.528154 + 0.849149i \(0.322884\pi\)
\(578\) −7.43778e9 −1.60212
\(579\) −4.30269e9 −0.921224
\(580\) 1.65836e9 0.352924
\(581\) −2.65342e9 −0.561292
\(582\) 3.09504e9 0.650783
\(583\) −1.21114e10 −2.53137
\(584\) −5.55408e9 −1.15390
\(585\) −3.80877e9 −0.786573
\(586\) 3.31889e9 0.681319
\(587\) 1.45833e9 0.297593 0.148796 0.988868i \(-0.452460\pi\)
0.148796 + 0.988868i \(0.452460\pi\)
\(588\) −6.63493e8 −0.134591
\(589\) −9.35717e9 −1.88687
\(590\) −1.21412e9 −0.243378
\(591\) 5.94318e8 0.118430
\(592\) −4.19120e8 −0.0830257
\(593\) 2.71973e9 0.535592 0.267796 0.963476i \(-0.413705\pi\)
0.267796 + 0.963476i \(0.413705\pi\)
\(594\) 1.51461e9 0.296517
\(595\) 2.04419e10 3.97843
\(596\) −1.32763e9 −0.256872
\(597\) −1.95258e9 −0.375576
\(598\) −5.17951e9 −0.990454
\(599\) −8.04358e9 −1.52917 −0.764584 0.644524i \(-0.777056\pi\)
−0.764584 + 0.644524i \(0.777056\pi\)
\(600\) −5.17271e9 −0.977662
\(601\) 3.08370e9 0.579444 0.289722 0.957111i \(-0.406437\pi\)
0.289722 + 0.957111i \(0.406437\pi\)
\(602\) −2.46541e9 −0.460576
\(603\) −3.10257e8 −0.0576250
\(604\) 1.24275e9 0.229484
\(605\) −9.18832e9 −1.68691
\(606\) 1.92891e9 0.352093
\(607\) 5.53440e9 1.00441 0.502204 0.864749i \(-0.332523\pi\)
0.502204 + 0.864749i \(0.332523\pi\)
\(608\) −2.60312e9 −0.469713
\(609\) 5.04516e9 0.905137
\(610\) 1.04427e10 1.86277
\(611\) 8.18566e9 1.45181
\(612\) −5.78891e8 −0.102086
\(613\) −1.57182e9 −0.275608 −0.137804 0.990460i \(-0.544004\pi\)
−0.137804 + 0.990460i \(0.544004\pi\)
\(614\) 1.01748e10 1.77393
\(615\) 4.07986e9 0.707266
\(616\) 1.06604e10 1.83756
\(617\) −3.83210e9 −0.656809 −0.328404 0.944537i \(-0.606511\pi\)
−0.328404 + 0.944537i \(0.606511\pi\)
\(618\) −5.75906e9 −0.981504
\(619\) −5.03030e9 −0.852465 −0.426233 0.904614i \(-0.640159\pi\)
−0.426233 + 0.904614i \(0.640159\pi\)
\(620\) −3.05994e9 −0.515635
\(621\) −7.54112e8 −0.126362
\(622\) 1.08294e10 1.80441
\(623\) 1.03877e10 1.72113
\(624\) 5.59455e9 0.921762
\(625\) 4.75332e9 0.778784
\(626\) 5.02743e9 0.819098
\(627\) 6.12997e9 0.993167
\(628\) 1.79066e9 0.288506
\(629\) −7.03257e8 −0.112677
\(630\) 5.79450e9 0.923260
\(631\) 5.76799e9 0.913949 0.456974 0.889480i \(-0.348933\pi\)
0.456974 + 0.889480i \(0.348933\pi\)
\(632\) −3.71917e9 −0.586052
\(633\) −3.70608e9 −0.580767
\(634\) 7.99244e9 1.24557
\(635\) −9.55709e9 −1.48121
\(636\) 1.31216e9 0.202249
\(637\) 1.07592e10 1.64927
\(638\) 1.06940e10 1.63030
\(639\) −2.33128e9 −0.353461
\(640\) 1.34927e10 2.03456
\(641\) −1.37617e9 −0.206381 −0.103191 0.994662i \(-0.532905\pi\)
−0.103191 + 0.994662i \(0.532905\pi\)
\(642\) 1.80976e9 0.269929
\(643\) 8.74856e9 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(644\) 1.28607e9 0.189743
\(645\) 1.91330e9 0.280753
\(646\) −1.43551e10 −2.09504
\(647\) 8.82702e9 1.28130 0.640648 0.767835i \(-0.278666\pi\)
0.640648 + 0.767835i \(0.278666\pi\)
\(648\) 6.77227e8 0.0977737
\(649\) −1.27782e9 −0.183490
\(650\) −2.03245e10 −2.90283
\(651\) −9.30913e9 −1.32244
\(652\) −1.66540e9 −0.235316
\(653\) −2.39631e9 −0.336780 −0.168390 0.985720i \(-0.553857\pi\)
−0.168390 + 0.985720i \(0.553857\pi\)
\(654\) 1.40643e9 0.196605
\(655\) −4.97484e8 −0.0691727
\(656\) −5.99274e9 −0.828824
\(657\) −3.17731e9 −0.437101
\(658\) −1.24533e10 −1.70410
\(659\) −3.92961e9 −0.534873 −0.267437 0.963575i \(-0.586177\pi\)
−0.267437 + 0.963575i \(0.586177\pi\)
\(660\) 2.00460e9 0.271409
\(661\) 6.16515e9 0.830307 0.415153 0.909751i \(-0.363728\pi\)
0.415153 + 0.909751i \(0.363728\pi\)
\(662\) 9.49108e9 1.27149
\(663\) 9.38729e9 1.25096
\(664\) −2.51480e9 −0.333361
\(665\) 2.34516e10 3.09241
\(666\) −1.99347e8 −0.0261486
\(667\) −5.32444e9 −0.694757
\(668\) −9.23998e8 −0.119937
\(669\) −1.38599e9 −0.178965
\(670\) −2.51594e9 −0.323176
\(671\) 1.09906e10 1.40441
\(672\) −2.58976e9 −0.329205
\(673\) −8.84954e9 −1.11910 −0.559549 0.828798i \(-0.689025\pi\)
−0.559549 + 0.828798i \(0.689025\pi\)
\(674\) −7.93985e9 −0.998856
\(675\) −2.95914e9 −0.370342
\(676\) 1.41633e9 0.176339
\(677\) 5.42902e9 0.672452 0.336226 0.941781i \(-0.390849\pi\)
0.336226 + 0.941781i \(0.390849\pi\)
\(678\) −5.75394e9 −0.709025
\(679\) 1.24620e10 1.52772
\(680\) 1.93740e10 2.36286
\(681\) −6.05524e9 −0.734711
\(682\) −1.97321e10 −2.38193
\(683\) 5.65530e9 0.679178 0.339589 0.940574i \(-0.389712\pi\)
0.339589 + 0.940574i \(0.389712\pi\)
\(684\) −6.64123e8 −0.0793510
\(685\) −2.22334e10 −2.64295
\(686\) −2.67351e9 −0.316189
\(687\) −2.95689e9 −0.347926
\(688\) −2.81037e9 −0.329007
\(689\) −2.12779e10 −2.47835
\(690\) −6.11526e9 −0.708668
\(691\) −6.11832e9 −0.705439 −0.352719 0.935729i \(-0.614743\pi\)
−0.352719 + 0.935729i \(0.614743\pi\)
\(692\) 9.31577e8 0.106868
\(693\) 6.09850e9 0.696076
\(694\) −8.47484e9 −0.962439
\(695\) 9.13142e9 1.03179
\(696\) 4.78159e9 0.537576
\(697\) −1.00554e10 −1.12483
\(698\) 4.09921e9 0.456253
\(699\) −4.02870e9 −0.446164
\(700\) 5.04657e9 0.556101
\(701\) −1.75688e9 −0.192632 −0.0963160 0.995351i \(-0.530706\pi\)
−0.0963160 + 0.995351i \(0.530706\pi\)
\(702\) 2.66094e9 0.290306
\(703\) −8.06800e8 −0.0875835
\(704\) 9.60715e9 1.03774
\(705\) 9.66452e9 1.03877
\(706\) −1.45504e10 −1.55618
\(707\) 7.76663e9 0.826541
\(708\) 1.38439e8 0.0146603
\(709\) −9.00661e9 −0.949072 −0.474536 0.880236i \(-0.657384\pi\)
−0.474536 + 0.880236i \(0.657384\pi\)
\(710\) −1.89049e10 −1.98230
\(711\) −2.12762e9 −0.221999
\(712\) 9.84506e9 1.02221
\(713\) 9.82444e9 1.01507
\(714\) −1.42814e10 −1.46834
\(715\) −3.25066e10 −3.32583
\(716\) 1.98091e9 0.201683
\(717\) 3.74846e9 0.379784
\(718\) 7.01640e9 0.707422
\(719\) 3.19207e8 0.0320274 0.0160137 0.999872i \(-0.494902\pi\)
0.0160137 + 0.999872i \(0.494902\pi\)
\(720\) 6.60528e9 0.659519
\(721\) −2.31885e10 −2.30409
\(722\) −5.41335e9 −0.535286
\(723\) −6.33449e9 −0.623344
\(724\) 2.18112e9 0.213597
\(725\) −2.08932e10 −2.03620
\(726\) 6.41928e9 0.622599
\(727\) 7.47713e9 0.721713 0.360856 0.932621i \(-0.382484\pi\)
0.360856 + 0.932621i \(0.382484\pi\)
\(728\) 1.87287e10 1.79907
\(729\) 3.87420e8 0.0370370
\(730\) −2.57655e10 −2.45137
\(731\) −4.71563e9 −0.446507
\(732\) −1.19072e9 −0.112208
\(733\) −7.27192e9 −0.682002 −0.341001 0.940063i \(-0.610766\pi\)
−0.341001 + 0.940063i \(0.610766\pi\)
\(734\) −1.35667e9 −0.126630
\(735\) 1.27030e10 1.18005
\(736\) 2.73312e9 0.252689
\(737\) −2.64794e9 −0.243653
\(738\) −2.85034e9 −0.261035
\(739\) 6.36734e8 0.0580367 0.0290183 0.999579i \(-0.490762\pi\)
0.0290183 + 0.999579i \(0.490762\pi\)
\(740\) −2.63836e8 −0.0239345
\(741\) 1.07694e10 0.972363
\(742\) 3.23713e10 2.90902
\(743\) 1.53224e10 1.37045 0.685227 0.728329i \(-0.259703\pi\)
0.685227 + 0.728329i \(0.259703\pi\)
\(744\) −8.82280e9 −0.785419
\(745\) 2.54184e10 2.25217
\(746\) 1.60208e10 1.41286
\(747\) −1.43864e9 −0.126278
\(748\) −4.94064e9 −0.431646
\(749\) 7.28690e9 0.633660
\(750\) −1.15265e10 −0.997664
\(751\) −1.47426e10 −1.27009 −0.635045 0.772475i \(-0.719018\pi\)
−0.635045 + 0.772475i \(0.719018\pi\)
\(752\) −1.41958e10 −1.21730
\(753\) −6.81468e9 −0.581652
\(754\) 1.87877e10 1.59615
\(755\) −2.37932e10 −2.01205
\(756\) −6.60713e8 −0.0556143
\(757\) 1.03556e10 0.867643 0.433822 0.900999i \(-0.357165\pi\)
0.433822 + 0.900999i \(0.357165\pi\)
\(758\) 4.20347e9 0.350562
\(759\) −6.43609e9 −0.534288
\(760\) 2.22265e10 1.83664
\(761\) 4.05993e9 0.333943 0.166971 0.985962i \(-0.446601\pi\)
0.166971 + 0.985962i \(0.446601\pi\)
\(762\) 6.67692e9 0.546681
\(763\) 5.66289e9 0.461533
\(764\) −3.40724e9 −0.276424
\(765\) 1.10832e10 0.895059
\(766\) −1.33299e10 −1.07159
\(767\) −2.24493e9 −0.179647
\(768\) −4.09003e9 −0.325808
\(769\) −2.02272e10 −1.60396 −0.801981 0.597349i \(-0.796221\pi\)
−0.801981 + 0.597349i \(0.796221\pi\)
\(770\) 4.94541e10 3.90377
\(771\) −1.24192e10 −0.975892
\(772\) 3.97847e9 0.311212
\(773\) 1.26500e10 0.985062 0.492531 0.870295i \(-0.336072\pi\)
0.492531 + 0.870295i \(0.336072\pi\)
\(774\) −1.33670e9 −0.103619
\(775\) 3.85512e10 2.97497
\(776\) 1.18110e10 0.907339
\(777\) −8.02658e8 −0.0613842
\(778\) −8.68905e9 −0.661521
\(779\) −1.15359e10 −0.874323
\(780\) 3.52177e9 0.265723
\(781\) −1.98967e10 −1.49452
\(782\) 1.50720e10 1.12706
\(783\) 2.73540e9 0.203636
\(784\) −1.86589e10 −1.38287
\(785\) −3.42834e10 −2.52953
\(786\) 3.47560e8 0.0255300
\(787\) −2.58438e9 −0.188992 −0.0944962 0.995525i \(-0.530124\pi\)
−0.0944962 + 0.995525i \(0.530124\pi\)
\(788\) −5.49535e8 −0.0400086
\(789\) −1.04674e10 −0.758700
\(790\) −1.72533e10 −1.24503
\(791\) −2.31679e10 −1.66444
\(792\) 5.77991e9 0.413412
\(793\) 1.93088e10 1.37499
\(794\) 5.23804e8 0.0371362
\(795\) −2.51221e10 −1.77325
\(796\) 1.80545e9 0.126879
\(797\) 2.54750e10 1.78242 0.891209 0.453593i \(-0.149858\pi\)
0.891209 + 0.453593i \(0.149858\pi\)
\(798\) −1.63841e10 −1.14134
\(799\) −2.38197e10 −1.65205
\(800\) 1.07248e10 0.740583
\(801\) 5.63205e9 0.387215
\(802\) −2.25537e10 −1.54386
\(803\) −2.71173e10 −1.84817
\(804\) 2.86878e8 0.0194671
\(805\) −2.46227e10 −1.66361
\(806\) −3.46663e10 −2.33203
\(807\) −6.52820e8 −0.0437257
\(808\) 7.36089e9 0.490897
\(809\) 2.50645e10 1.66433 0.832163 0.554531i \(-0.187102\pi\)
0.832163 + 0.554531i \(0.187102\pi\)
\(810\) 3.14168e9 0.207713
\(811\) −9.63538e9 −0.634301 −0.317151 0.948375i \(-0.602726\pi\)
−0.317151 + 0.948375i \(0.602726\pi\)
\(812\) −4.66499e9 −0.305777
\(813\) 6.29589e8 0.0410904
\(814\) −1.70136e9 −0.110563
\(815\) 3.18851e10 2.06317
\(816\) −1.62797e10 −1.04889
\(817\) −5.40993e9 −0.347068
\(818\) 4.88351e9 0.311958
\(819\) 1.07141e10 0.681496
\(820\) −3.77244e9 −0.238931
\(821\) −2.27996e10 −1.43789 −0.718946 0.695066i \(-0.755375\pi\)
−0.718946 + 0.695066i \(0.755375\pi\)
\(822\) 1.55331e10 0.975451
\(823\) 7.33439e9 0.458632 0.229316 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(824\) −2.19771e10 −1.36844
\(825\) −2.52553e10 −1.56590
\(826\) 3.41534e9 0.210865
\(827\) 1.34883e9 0.0829252 0.0414626 0.999140i \(-0.486798\pi\)
0.0414626 + 0.999140i \(0.486798\pi\)
\(828\) 6.97287e8 0.0426880
\(829\) 1.61073e10 0.981932 0.490966 0.871179i \(-0.336644\pi\)
0.490966 + 0.871179i \(0.336644\pi\)
\(830\) −1.16662e10 −0.708202
\(831\) −9.18299e8 −0.0555112
\(832\) 1.68783e10 1.01601
\(833\) −3.13085e10 −1.87674
\(834\) −6.37953e9 −0.380810
\(835\) 1.76905e10 1.05157
\(836\) −5.66806e9 −0.335516
\(837\) −5.04725e9 −0.297520
\(838\) −1.58218e10 −0.928756
\(839\) −1.49944e10 −0.876523 −0.438261 0.898848i \(-0.644406\pi\)
−0.438261 + 0.898848i \(0.644406\pi\)
\(840\) 2.21124e10 1.28723
\(841\) 2.06352e9 0.119625
\(842\) 5.41425e9 0.312569
\(843\) 2.45547e9 0.141169
\(844\) 3.42682e9 0.196197
\(845\) −2.71165e10 −1.54609
\(846\) −6.75197e9 −0.383384
\(847\) 2.58469e10 1.46156
\(848\) 3.69008e10 2.07802
\(849\) 4.18143e9 0.234503
\(850\) 5.91427e10 3.30320
\(851\) 8.47089e8 0.0471168
\(852\) 2.15561e9 0.119408
\(853\) −2.48117e9 −0.136879 −0.0684393 0.997655i \(-0.521802\pi\)
−0.0684393 + 0.997655i \(0.521802\pi\)
\(854\) −2.93756e10 −1.61393
\(855\) 1.27151e10 0.695724
\(856\) 6.90622e9 0.376342
\(857\) 1.30344e10 0.707388 0.353694 0.935361i \(-0.384925\pi\)
0.353694 + 0.935361i \(0.384925\pi\)
\(858\) 2.27102e10 1.22748
\(859\) −3.01941e10 −1.62535 −0.812673 0.582720i \(-0.801989\pi\)
−0.812673 + 0.582720i \(0.801989\pi\)
\(860\) −1.76913e9 −0.0948453
\(861\) −1.14767e10 −0.612783
\(862\) −2.64018e10 −1.40397
\(863\) 4.40460e9 0.233275 0.116638 0.993175i \(-0.462788\pi\)
0.116638 + 0.993175i \(0.462788\pi\)
\(864\) −1.40412e9 −0.0740639
\(865\) −1.78356e10 −0.936984
\(866\) 3.47628e10 1.81887
\(867\) −1.62372e10 −0.846143
\(868\) 8.60766e9 0.446752
\(869\) −1.81585e10 −0.938666
\(870\) 2.21820e10 1.14204
\(871\) −4.65201e9 −0.238549
\(872\) 5.36706e9 0.274112
\(873\) 6.75668e9 0.343703
\(874\) 1.72911e10 0.876057
\(875\) −4.64109e10 −2.34203
\(876\) 2.93790e9 0.147663
\(877\) 1.87680e10 0.939549 0.469775 0.882786i \(-0.344335\pi\)
0.469775 + 0.882786i \(0.344335\pi\)
\(878\) −2.85240e10 −1.42226
\(879\) 7.24535e9 0.359831
\(880\) 5.63738e10 2.78861
\(881\) 2.99234e10 1.47433 0.737165 0.675713i \(-0.236164\pi\)
0.737165 + 0.675713i \(0.236164\pi\)
\(882\) −8.87477e9 −0.435529
\(883\) 3.52933e10 1.72516 0.862582 0.505918i \(-0.168846\pi\)
0.862582 + 0.505918i \(0.168846\pi\)
\(884\) −8.67994e9 −0.422604
\(885\) −2.65051e9 −0.128537
\(886\) −1.22199e10 −0.590269
\(887\) −1.25706e10 −0.604818 −0.302409 0.953178i \(-0.597791\pi\)
−0.302409 + 0.953178i \(0.597791\pi\)
\(888\) −7.60726e8 −0.0364571
\(889\) 2.68842e10 1.28334
\(890\) 4.56715e10 2.17160
\(891\) 3.30650e9 0.156602
\(892\) 1.28155e9 0.0604587
\(893\) −2.73267e10 −1.28413
\(894\) −1.77582e10 −0.831222
\(895\) −3.79258e10 −1.76829
\(896\) −3.79553e10 −1.76276
\(897\) −1.13072e10 −0.523097
\(898\) 4.22196e9 0.194557
\(899\) −3.56363e10 −1.63581
\(900\) 2.73617e9 0.125110
\(901\) 6.19172e10 2.82016
\(902\) −2.43267e10 −1.10372
\(903\) −5.38215e9 −0.243248
\(904\) −2.19576e10 −0.988541
\(905\) −4.17590e10 −1.87275
\(906\) 1.66228e10 0.742599
\(907\) −1.02215e10 −0.454874 −0.227437 0.973793i \(-0.573035\pi\)
−0.227437 + 0.973793i \(0.573035\pi\)
\(908\) 5.59896e9 0.248203
\(909\) 4.21093e9 0.185954
\(910\) 8.68832e10 3.82200
\(911\) −3.10518e9 −0.136073 −0.0680366 0.997683i \(-0.521673\pi\)
−0.0680366 + 0.997683i \(0.521673\pi\)
\(912\) −1.86766e10 −0.815299
\(913\) −1.22783e10 −0.533936
\(914\) 3.54590e7 0.00153608
\(915\) 2.27972e10 0.983801
\(916\) 2.73408e9 0.117538
\(917\) 1.39943e9 0.0599320
\(918\) −7.74314e9 −0.330345
\(919\) 2.97646e10 1.26501 0.632507 0.774554i \(-0.282026\pi\)
0.632507 + 0.774554i \(0.282026\pi\)
\(920\) −2.33364e10 −0.988044
\(921\) 2.22123e10 0.936880
\(922\) −3.48601e10 −1.46477
\(923\) −3.49554e10 −1.46321
\(924\) −5.63896e9 −0.235151
\(925\) 3.32399e9 0.138090
\(926\) −1.61634e10 −0.668952
\(927\) −1.25724e10 −0.518370
\(928\) −9.91386e9 −0.407216
\(929\) 3.44435e10 1.40946 0.704729 0.709477i \(-0.251069\pi\)
0.704729 + 0.709477i \(0.251069\pi\)
\(930\) −4.09293e10 −1.66857
\(931\) −3.59182e10 −1.45878
\(932\) 3.72513e9 0.150725
\(933\) 2.36412e10 0.952980
\(934\) 2.13746e10 0.858388
\(935\) 9.45917e10 3.78453
\(936\) 1.01544e10 0.404752
\(937\) 2.41950e10 0.960807 0.480404 0.877048i \(-0.340490\pi\)
0.480404 + 0.877048i \(0.340490\pi\)
\(938\) 7.07737e9 0.280003
\(939\) 1.09752e10 0.432597
\(940\) −8.93627e9 −0.350921
\(941\) −3.24324e10 −1.26886 −0.634432 0.772979i \(-0.718766\pi\)
−0.634432 + 0.772979i \(0.718766\pi\)
\(942\) 2.39516e10 0.933591
\(943\) 1.21120e10 0.470355
\(944\) 3.89323e9 0.150629
\(945\) 1.26498e10 0.487609
\(946\) −1.14083e10 −0.438129
\(947\) −4.68426e9 −0.179232 −0.0896162 0.995976i \(-0.528564\pi\)
−0.0896162 + 0.995976i \(0.528564\pi\)
\(948\) 1.96730e9 0.0749965
\(949\) −4.76409e10 −1.80946
\(950\) 6.78505e10 2.56756
\(951\) 1.74480e10 0.657832
\(952\) −5.44992e10 −2.04720
\(953\) −3.34959e10 −1.25362 −0.626811 0.779171i \(-0.715640\pi\)
−0.626811 + 0.779171i \(0.715640\pi\)
\(954\) 1.75512e10 0.654466
\(955\) 6.52338e10 2.42360
\(956\) −3.46601e9 −0.128300
\(957\) 2.33457e10 0.861023
\(958\) −3.27063e10 −1.20185
\(959\) 6.25429e10 2.28988
\(960\) 1.99276e10 0.726951
\(961\) 3.82421e10 1.38998
\(962\) −2.98902e9 −0.108247
\(963\) 3.95083e9 0.142560
\(964\) 5.85717e9 0.210580
\(965\) −7.61705e10 −2.72861
\(966\) 1.72023e10 0.613998
\(967\) 4.48458e10 1.59489 0.797443 0.603395i \(-0.206186\pi\)
0.797443 + 0.603395i \(0.206186\pi\)
\(968\) 2.44966e10 0.868044
\(969\) −3.13382e10 −1.10647
\(970\) 5.47914e10 1.92758
\(971\) −3.33761e10 −1.16995 −0.584977 0.811050i \(-0.698896\pi\)
−0.584977 + 0.811050i \(0.698896\pi\)
\(972\) −3.58227e8 −0.0125120
\(973\) −2.56868e10 −0.893954
\(974\) −3.09127e10 −1.07197
\(975\) −4.43697e10 −1.53310
\(976\) −3.34858e10 −1.15289
\(977\) −2.05126e10 −0.703705 −0.351853 0.936055i \(-0.614448\pi\)
−0.351853 + 0.936055i \(0.614448\pi\)
\(978\) −2.22761e10 −0.761469
\(979\) 4.80676e10 1.63724
\(980\) −1.17458e10 −0.398650
\(981\) 3.07032e9 0.103835
\(982\) 1.74546e10 0.588193
\(983\) −1.39469e9 −0.0468317 −0.0234158 0.999726i \(-0.507454\pi\)
−0.0234158 + 0.999726i \(0.507454\pi\)
\(984\) −1.08771e10 −0.363942
\(985\) 1.05212e10 0.350783
\(986\) −5.46708e10 −1.81629
\(987\) −2.71864e10 −0.899999
\(988\) −9.95792e9 −0.328488
\(989\) 5.68008e9 0.186710
\(990\) 2.68132e10 0.878263
\(991\) −5.58431e10 −1.82268 −0.911342 0.411650i \(-0.864953\pi\)
−0.911342 + 0.411650i \(0.864953\pi\)
\(992\) 1.82927e10 0.594958
\(993\) 2.07197e10 0.671522
\(994\) 5.31796e10 1.71748
\(995\) −3.45664e10 −1.11243
\(996\) 1.33023e9 0.0426599
\(997\) −3.23901e9 −0.103509 −0.0517546 0.998660i \(-0.516481\pi\)
−0.0517546 + 0.998660i \(0.516481\pi\)
\(998\) −2.63062e9 −0.0837725
\(999\) −4.35187e8 −0.0138101
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 177.8.a.c.1.5 17
3.2 odd 2 531.8.a.c.1.13 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.5 17 1.1 even 1 trivial
531.8.a.c.1.13 17 3.2 odd 2