Properties

Label 177.8.a.b.1.9
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
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: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.9
Root \(2.39686\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.396855 q^{2} +27.0000 q^{3} -127.843 q^{4} +247.233 q^{5} +10.7151 q^{6} -652.929 q^{7} -101.532 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+0.396855 q^{2} +27.0000 q^{3} -127.843 q^{4} +247.233 q^{5} +10.7151 q^{6} -652.929 q^{7} -101.532 q^{8} +729.000 q^{9} +98.1157 q^{10} -2477.11 q^{11} -3451.75 q^{12} +9417.61 q^{13} -259.118 q^{14} +6675.29 q^{15} +16323.5 q^{16} +12577.2 q^{17} +289.307 q^{18} -41390.0 q^{19} -31606.9 q^{20} -17629.1 q^{21} -983.055 q^{22} -59781.4 q^{23} -2741.38 q^{24} -17000.9 q^{25} +3737.43 q^{26} +19683.0 q^{27} +83472.1 q^{28} +212559. q^{29} +2649.12 q^{30} -7793.74 q^{31} +19474.2 q^{32} -66882.0 q^{33} +4991.34 q^{34} -161426. q^{35} -93197.2 q^{36} -447343. q^{37} -16425.8 q^{38} +254276. q^{39} -25102.2 q^{40} +14016.1 q^{41} -6996.20 q^{42} -896369. q^{43} +316680. q^{44} +180233. q^{45} -23724.6 q^{46} +21326.5 q^{47} +440736. q^{48} -397227. q^{49} -6746.89 q^{50} +339585. q^{51} -1.20397e6 q^{52} +910896. q^{53} +7811.30 q^{54} -612424. q^{55} +66293.5 q^{56} -1.11753e6 q^{57} +84355.0 q^{58} -205379. q^{59} -853386. q^{60} +810282. q^{61} -3092.99 q^{62} -475985. q^{63} -2.08169e6 q^{64} +2.32834e6 q^{65} -26542.5 q^{66} -3.83166e6 q^{67} -1.60791e6 q^{68} -1.61410e6 q^{69} -64062.6 q^{70} -3.35674e6 q^{71} -74017.2 q^{72} +461542. q^{73} -177530. q^{74} -459024. q^{75} +5.29140e6 q^{76} +1.61738e6 q^{77} +100911. q^{78} -1.37060e6 q^{79} +4.03572e6 q^{80} +531441. q^{81} +5562.36 q^{82} -7.63422e6 q^{83} +2.25375e6 q^{84} +3.10951e6 q^{85} -355729. q^{86} +5.73908e6 q^{87} +251507. q^{88} -5.26057e6 q^{89} +71526.3 q^{90} -6.14903e6 q^{91} +7.64260e6 q^{92} -210431. q^{93} +8463.51 q^{94} -1.02330e7 q^{95} +525804. q^{96} -9.23173e6 q^{97} -157641. q^{98} -1.80582e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.396855 0.0350774 0.0175387 0.999846i \(-0.494417\pi\)
0.0175387 + 0.999846i \(0.494417\pi\)
\(3\) 27.0000 0.577350
\(4\) −127.843 −0.998770
\(5\) 247.233 0.884527 0.442264 0.896885i \(-0.354176\pi\)
0.442264 + 0.896885i \(0.354176\pi\)
\(6\) 10.7151 0.0202519
\(7\) −652.929 −0.719487 −0.359743 0.933051i \(-0.617136\pi\)
−0.359743 + 0.933051i \(0.617136\pi\)
\(8\) −101.532 −0.0701116
\(9\) 729.000 0.333333
\(10\) 98.1157 0.0310269
\(11\) −2477.11 −0.561140 −0.280570 0.959834i \(-0.590524\pi\)
−0.280570 + 0.959834i \(0.590524\pi\)
\(12\) −3451.75 −0.576640
\(13\) 9417.61 1.18888 0.594442 0.804139i \(-0.297373\pi\)
0.594442 + 0.804139i \(0.297373\pi\)
\(14\) −259.118 −0.0252377
\(15\) 6675.29 0.510682
\(16\) 16323.5 0.996310
\(17\) 12577.2 0.620889 0.310444 0.950592i \(-0.399522\pi\)
0.310444 + 0.950592i \(0.399522\pi\)
\(18\) 289.307 0.0116925
\(19\) −41390.0 −1.38439 −0.692194 0.721711i \(-0.743356\pi\)
−0.692194 + 0.721711i \(0.743356\pi\)
\(20\) −31606.9 −0.883439
\(21\) −17629.1 −0.415396
\(22\) −983.055 −0.0196833
\(23\) −59781.4 −1.02452 −0.512258 0.858832i \(-0.671191\pi\)
−0.512258 + 0.858832i \(0.671191\pi\)
\(24\) −2741.38 −0.0404790
\(25\) −17000.9 −0.217611
\(26\) 3737.43 0.0417029
\(27\) 19683.0 0.192450
\(28\) 83472.1 0.718602
\(29\) 212559. 1.61840 0.809199 0.587534i \(-0.199901\pi\)
0.809199 + 0.587534i \(0.199901\pi\)
\(30\) 2649.12 0.0179134
\(31\) −7793.74 −0.0469872 −0.0234936 0.999724i \(-0.507479\pi\)
−0.0234936 + 0.999724i \(0.507479\pi\)
\(32\) 19474.2 0.105060
\(33\) −66882.0 −0.323974
\(34\) 4991.34 0.0217792
\(35\) −161426. −0.636406
\(36\) −93197.2 −0.332923
\(37\) −447343. −1.45189 −0.725946 0.687751i \(-0.758598\pi\)
−0.725946 + 0.687751i \(0.758598\pi\)
\(38\) −16425.8 −0.0485607
\(39\) 254276. 0.686402
\(40\) −25102.2 −0.0620156
\(41\) 14016.1 0.0317602 0.0158801 0.999874i \(-0.494945\pi\)
0.0158801 + 0.999874i \(0.494945\pi\)
\(42\) −6996.20 −0.0145710
\(43\) −896369. −1.71928 −0.859641 0.510899i \(-0.829313\pi\)
−0.859641 + 0.510899i \(0.829313\pi\)
\(44\) 316680. 0.560450
\(45\) 180233. 0.294842
\(46\) −23724.6 −0.0359373
\(47\) 21326.5 0.0299624 0.0149812 0.999888i \(-0.495231\pi\)
0.0149812 + 0.999888i \(0.495231\pi\)
\(48\) 440736. 0.575220
\(49\) −397227. −0.482339
\(50\) −6746.89 −0.00763324
\(51\) 339585. 0.358470
\(52\) −1.20397e6 −1.18742
\(53\) 910896. 0.840433 0.420217 0.907424i \(-0.361954\pi\)
0.420217 + 0.907424i \(0.361954\pi\)
\(54\) 7811.30 0.00675065
\(55\) −612424. −0.496344
\(56\) 66293.5 0.0504444
\(57\) −1.11753e6 −0.799277
\(58\) 84355.0 0.0567692
\(59\) −205379. −0.130189
\(60\) −853386. −0.510054
\(61\) 810282. 0.457069 0.228535 0.973536i \(-0.426607\pi\)
0.228535 + 0.973536i \(0.426607\pi\)
\(62\) −3092.99 −0.00164819
\(63\) −475985. −0.239829
\(64\) −2.08169e6 −0.992625
\(65\) 2.32834e6 1.05160
\(66\) −26542.5 −0.0113642
\(67\) −3.83166e6 −1.55641 −0.778207 0.628007i \(-0.783871\pi\)
−0.778207 + 0.628007i \(0.783871\pi\)
\(68\) −1.60791e6 −0.620125
\(69\) −1.61410e6 −0.591504
\(70\) −64062.6 −0.0223234
\(71\) −3.35674e6 −1.11305 −0.556524 0.830832i \(-0.687865\pi\)
−0.556524 + 0.830832i \(0.687865\pi\)
\(72\) −74017.2 −0.0233705
\(73\) 461542. 0.138861 0.0694306 0.997587i \(-0.477882\pi\)
0.0694306 + 0.997587i \(0.477882\pi\)
\(74\) −177530. −0.0509286
\(75\) −459024. −0.125638
\(76\) 5.29140e6 1.38269
\(77\) 1.61738e6 0.403733
\(78\) 100911. 0.0240772
\(79\) −1.37060e6 −0.312762 −0.156381 0.987697i \(-0.549983\pi\)
−0.156381 + 0.987697i \(0.549983\pi\)
\(80\) 4.03572e6 0.881264
\(81\) 531441. 0.111111
\(82\) 5562.36 0.00111407
\(83\) −7.63422e6 −1.46552 −0.732759 0.680488i \(-0.761768\pi\)
−0.732759 + 0.680488i \(0.761768\pi\)
\(84\) 2.25375e6 0.414885
\(85\) 3.10951e6 0.549193
\(86\) −355729. −0.0603079
\(87\) 5.73908e6 0.934383
\(88\) 251507. 0.0393424
\(89\) −5.26057e6 −0.790984 −0.395492 0.918469i \(-0.629426\pi\)
−0.395492 + 0.918469i \(0.629426\pi\)
\(90\) 71526.3 0.0103423
\(91\) −6.14903e6 −0.855386
\(92\) 7.64260e6 1.02326
\(93\) −210431. −0.0271281
\(94\) 8463.51 0.00105100
\(95\) −1.02330e7 −1.22453
\(96\) 525804. 0.0606562
\(97\) −9.23173e6 −1.02703 −0.513514 0.858081i \(-0.671657\pi\)
−0.513514 + 0.858081i \(0.671657\pi\)
\(98\) −157641. −0.0169192
\(99\) −1.80582e6 −0.187047
\(100\) 2.17344e6 0.217344
\(101\) 4.85554e6 0.468935 0.234468 0.972124i \(-0.424665\pi\)
0.234468 + 0.972124i \(0.424665\pi\)
\(102\) 134766. 0.0125742
\(103\) −4.34436e6 −0.391738 −0.195869 0.980630i \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(104\) −956193. −0.0833545
\(105\) −4.35849e6 −0.367429
\(106\) 361494. 0.0294802
\(107\) −6.60510e6 −0.521238 −0.260619 0.965442i \(-0.583927\pi\)
−0.260619 + 0.965442i \(0.583927\pi\)
\(108\) −2.51632e6 −0.192213
\(109\) −2.74784e6 −0.203235 −0.101617 0.994824i \(-0.532402\pi\)
−0.101617 + 0.994824i \(0.532402\pi\)
\(110\) −243044. −0.0174104
\(111\) −1.20783e7 −0.838251
\(112\) −1.06581e7 −0.716832
\(113\) −3.08697e6 −0.201260 −0.100630 0.994924i \(-0.532086\pi\)
−0.100630 + 0.994924i \(0.532086\pi\)
\(114\) −443498. −0.0280365
\(115\) −1.47799e7 −0.906212
\(116\) −2.71740e7 −1.61641
\(117\) 6.86544e6 0.396294
\(118\) −81505.7 −0.00456669
\(119\) −8.21204e6 −0.446721
\(120\) −677758. −0.0358047
\(121\) −1.33511e7 −0.685122
\(122\) 321565. 0.0160328
\(123\) 378435. 0.0183368
\(124\) 996371. 0.0469294
\(125\) −2.35182e7 −1.07701
\(126\) −188897. −0.00841257
\(127\) 1.02662e7 0.444731 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(128\) −3.31883e6 −0.139878
\(129\) −2.42020e7 −0.992628
\(130\) 924016. 0.0368874
\(131\) −3.04995e7 −1.18534 −0.592671 0.805445i \(-0.701927\pi\)
−0.592671 + 0.805445i \(0.701927\pi\)
\(132\) 8.55037e6 0.323576
\(133\) 2.70247e7 0.996049
\(134\) −1.52062e6 −0.0545950
\(135\) 4.86629e6 0.170227
\(136\) −1.27700e6 −0.0435315
\(137\) 2.18105e7 0.724676 0.362338 0.932047i \(-0.381979\pi\)
0.362338 + 0.932047i \(0.381979\pi\)
\(138\) −640563. −0.0207484
\(139\) 5.59222e7 1.76617 0.883085 0.469213i \(-0.155462\pi\)
0.883085 + 0.469213i \(0.155462\pi\)
\(140\) 2.06370e7 0.635623
\(141\) 575814. 0.0172988
\(142\) −1.33214e6 −0.0390428
\(143\) −2.33285e7 −0.667130
\(144\) 1.18999e7 0.332103
\(145\) 5.25515e7 1.43152
\(146\) 183165. 0.00487089
\(147\) −1.07251e7 −0.278478
\(148\) 5.71894e7 1.45011
\(149\) 4.25096e7 1.05277 0.526387 0.850245i \(-0.323546\pi\)
0.526387 + 0.850245i \(0.323546\pi\)
\(150\) −182166. −0.00440705
\(151\) −2.69136e7 −0.636140 −0.318070 0.948067i \(-0.603035\pi\)
−0.318070 + 0.948067i \(0.603035\pi\)
\(152\) 4.20243e6 0.0970617
\(153\) 9.16880e6 0.206963
\(154\) 641865. 0.0141619
\(155\) −1.92687e6 −0.0415615
\(156\) −3.25072e7 −0.685557
\(157\) −6.09453e7 −1.25687 −0.628436 0.777861i \(-0.716305\pi\)
−0.628436 + 0.777861i \(0.716305\pi\)
\(158\) −543928. −0.0109709
\(159\) 2.45942e7 0.485224
\(160\) 4.81467e6 0.0929281
\(161\) 3.90330e7 0.737126
\(162\) 210905. 0.00389749
\(163\) 2.43192e7 0.439839 0.219919 0.975518i \(-0.429421\pi\)
0.219919 + 0.975518i \(0.429421\pi\)
\(164\) −1.79185e6 −0.0317212
\(165\) −1.65354e7 −0.286564
\(166\) −3.02968e6 −0.0514066
\(167\) 2.18099e7 0.362366 0.181183 0.983449i \(-0.442007\pi\)
0.181183 + 0.983449i \(0.442007\pi\)
\(168\) 1.78992e6 0.0291241
\(169\) 2.59429e7 0.413443
\(170\) 1.23402e6 0.0192643
\(171\) −3.01733e7 −0.461463
\(172\) 1.14594e8 1.71717
\(173\) 7.57905e7 1.11289 0.556446 0.830884i \(-0.312164\pi\)
0.556446 + 0.830884i \(0.312164\pi\)
\(174\) 2.27758e6 0.0327757
\(175\) 1.11004e7 0.156568
\(176\) −4.04353e7 −0.559070
\(177\) −5.54523e6 −0.0751646
\(178\) −2.08769e6 −0.0277457
\(179\) −1.14237e6 −0.0148874 −0.00744372 0.999972i \(-0.502369\pi\)
−0.00744372 + 0.999972i \(0.502369\pi\)
\(180\) −2.30414e7 −0.294480
\(181\) 1.35346e8 1.69656 0.848280 0.529548i \(-0.177638\pi\)
0.848280 + 0.529548i \(0.177638\pi\)
\(182\) −2.44028e6 −0.0300047
\(183\) 2.18776e7 0.263889
\(184\) 6.06975e6 0.0718304
\(185\) −1.10598e8 −1.28424
\(186\) −83510.6 −0.000951582 0
\(187\) −3.11552e7 −0.348406
\(188\) −2.72643e6 −0.0299255
\(189\) −1.28516e7 −0.138465
\(190\) −4.06101e6 −0.0429533
\(191\) 1.11443e8 1.15727 0.578637 0.815585i \(-0.303585\pi\)
0.578637 + 0.815585i \(0.303585\pi\)
\(192\) −5.62055e7 −0.573092
\(193\) −4.58821e7 −0.459402 −0.229701 0.973261i \(-0.573775\pi\)
−0.229701 + 0.973261i \(0.573775\pi\)
\(194\) −3.66366e6 −0.0360254
\(195\) 6.28653e7 0.607141
\(196\) 5.07824e7 0.481745
\(197\) 1.22654e8 1.14301 0.571504 0.820600i \(-0.306360\pi\)
0.571504 + 0.820600i \(0.306360\pi\)
\(198\) −716647. −0.00656111
\(199\) −1.02506e7 −0.0922072 −0.0461036 0.998937i \(-0.514680\pi\)
−0.0461036 + 0.998937i \(0.514680\pi\)
\(200\) 1.72614e6 0.0152571
\(201\) −1.03455e8 −0.898596
\(202\) 1.92695e6 0.0164490
\(203\) −1.38786e8 −1.16442
\(204\) −4.34134e7 −0.358029
\(205\) 3.46524e6 0.0280928
\(206\) −1.72408e6 −0.0137411
\(207\) −4.35806e7 −0.341505
\(208\) 1.53729e8 1.18450
\(209\) 1.02528e8 0.776836
\(210\) −1.72969e6 −0.0128884
\(211\) −1.63127e8 −1.19547 −0.597734 0.801694i \(-0.703932\pi\)
−0.597734 + 0.801694i \(0.703932\pi\)
\(212\) −1.16451e8 −0.839399
\(213\) −9.06320e7 −0.642618
\(214\) −2.62127e6 −0.0182837
\(215\) −2.21612e8 −1.52075
\(216\) −1.99846e6 −0.0134930
\(217\) 5.08876e6 0.0338067
\(218\) −1.09049e6 −0.00712895
\(219\) 1.24616e7 0.0801716
\(220\) 7.82938e7 0.495733
\(221\) 1.18448e8 0.738164
\(222\) −4.79332e6 −0.0294036
\(223\) 2.16056e8 1.30466 0.652332 0.757933i \(-0.273791\pi\)
0.652332 + 0.757933i \(0.273791\pi\)
\(224\) −1.27153e7 −0.0755890
\(225\) −1.23936e7 −0.0725371
\(226\) −1.22508e6 −0.00705968
\(227\) −9.46539e7 −0.537091 −0.268546 0.963267i \(-0.586543\pi\)
−0.268546 + 0.963267i \(0.586543\pi\)
\(228\) 1.42868e8 0.798294
\(229\) 5.62090e7 0.309301 0.154651 0.987969i \(-0.450575\pi\)
0.154651 + 0.987969i \(0.450575\pi\)
\(230\) −5.86549e6 −0.0317876
\(231\) 4.36692e7 0.233095
\(232\) −2.15816e7 −0.113469
\(233\) 3.12602e8 1.61900 0.809500 0.587120i \(-0.199739\pi\)
0.809500 + 0.587120i \(0.199739\pi\)
\(234\) 2.72459e6 0.0139010
\(235\) 5.27260e6 0.0265025
\(236\) 2.62562e7 0.130029
\(237\) −3.70061e7 −0.180573
\(238\) −3.25899e6 −0.0156698
\(239\) 3.24955e8 1.53968 0.769840 0.638237i \(-0.220336\pi\)
0.769840 + 0.638237i \(0.220336\pi\)
\(240\) 1.08964e8 0.508798
\(241\) 2.55841e8 1.17737 0.588683 0.808364i \(-0.299647\pi\)
0.588683 + 0.808364i \(0.299647\pi\)
\(242\) −5.29845e6 −0.0240323
\(243\) 1.43489e7 0.0641500
\(244\) −1.03589e8 −0.456507
\(245\) −9.82075e7 −0.426642
\(246\) 150184. 0.000643206 0
\(247\) −3.89795e8 −1.64588
\(248\) 791317. 0.00329435
\(249\) −2.06124e8 −0.846118
\(250\) −9.33334e6 −0.0377787
\(251\) 1.47199e8 0.587552 0.293776 0.955874i \(-0.405088\pi\)
0.293776 + 0.955874i \(0.405088\pi\)
\(252\) 6.08512e7 0.239534
\(253\) 1.48085e8 0.574897
\(254\) 4.07420e6 0.0156000
\(255\) 8.39567e7 0.317077
\(256\) 2.65139e8 0.987718
\(257\) −3.26518e8 −1.19989 −0.599944 0.800042i \(-0.704810\pi\)
−0.599944 + 0.800042i \(0.704810\pi\)
\(258\) −9.60467e6 −0.0348188
\(259\) 2.92083e8 1.04462
\(260\) −2.97661e8 −1.05031
\(261\) 1.54955e8 0.539466
\(262\) −1.21039e7 −0.0415787
\(263\) −1.12281e8 −0.380592 −0.190296 0.981727i \(-0.560945\pi\)
−0.190296 + 0.981727i \(0.560945\pi\)
\(264\) 6.79070e6 0.0227144
\(265\) 2.25203e8 0.743386
\(266\) 1.07249e7 0.0349388
\(267\) −1.42035e8 −0.456675
\(268\) 4.89849e8 1.55450
\(269\) −5.12589e7 −0.160560 −0.0802798 0.996772i \(-0.525581\pi\)
−0.0802798 + 0.996772i \(0.525581\pi\)
\(270\) 1.93121e6 0.00597113
\(271\) −5.28326e8 −1.61254 −0.806269 0.591549i \(-0.798516\pi\)
−0.806269 + 0.591549i \(0.798516\pi\)
\(272\) 2.05305e8 0.618598
\(273\) −1.66024e8 −0.493857
\(274\) 8.65561e6 0.0254197
\(275\) 4.21131e7 0.122110
\(276\) 2.06350e8 0.590777
\(277\) −4.48700e7 −0.126846 −0.0634230 0.997987i \(-0.520202\pi\)
−0.0634230 + 0.997987i \(0.520202\pi\)
\(278\) 2.21930e7 0.0619526
\(279\) −5.68163e6 −0.0156624
\(280\) 1.63899e7 0.0446194
\(281\) −5.27134e8 −1.41726 −0.708629 0.705581i \(-0.750686\pi\)
−0.708629 + 0.705581i \(0.750686\pi\)
\(282\) 228515. 0.000606796 0
\(283\) 3.08524e8 0.809163 0.404582 0.914502i \(-0.367417\pi\)
0.404582 + 0.914502i \(0.367417\pi\)
\(284\) 4.29134e8 1.11168
\(285\) −2.76290e8 −0.706982
\(286\) −9.25804e6 −0.0234012
\(287\) −9.15152e6 −0.0228511
\(288\) 1.41967e7 0.0350199
\(289\) −2.52152e8 −0.614497
\(290\) 2.08553e7 0.0502139
\(291\) −2.49257e8 −0.592955
\(292\) −5.90047e7 −0.138690
\(293\) −8.22037e8 −1.90921 −0.954607 0.297869i \(-0.903724\pi\)
−0.954607 + 0.297869i \(0.903724\pi\)
\(294\) −4.25632e6 −0.00976829
\(295\) −5.07764e7 −0.115156
\(296\) 4.54198e7 0.101795
\(297\) −4.87570e7 −0.107991
\(298\) 1.68702e7 0.0369285
\(299\) −5.62998e8 −1.21803
\(300\) 5.86828e7 0.125483
\(301\) 5.85265e8 1.23700
\(302\) −1.06808e7 −0.0223141
\(303\) 1.31100e8 0.270740
\(304\) −6.75632e8 −1.37928
\(305\) 2.00328e8 0.404290
\(306\) 3.63869e6 0.00725972
\(307\) 1.21225e8 0.239116 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(308\) −2.06770e8 −0.403236
\(309\) −1.17298e8 −0.226170
\(310\) −764688. −0.00145787
\(311\) −4.99117e8 −0.940894 −0.470447 0.882428i \(-0.655907\pi\)
−0.470447 + 0.882428i \(0.655907\pi\)
\(312\) −2.58172e7 −0.0481247
\(313\) 6.71603e8 1.23796 0.618981 0.785406i \(-0.287546\pi\)
0.618981 + 0.785406i \(0.287546\pi\)
\(314\) −2.41865e7 −0.0440878
\(315\) −1.17679e8 −0.212135
\(316\) 1.75220e8 0.312377
\(317\) −7.55861e8 −1.33271 −0.666353 0.745636i \(-0.732146\pi\)
−0.666353 + 0.745636i \(0.732146\pi\)
\(318\) 9.76033e6 0.0170204
\(319\) −5.26531e8 −0.908149
\(320\) −5.14661e8 −0.878004
\(321\) −1.78338e8 −0.300937
\(322\) 1.54905e7 0.0258564
\(323\) −5.20572e8 −0.859552
\(324\) −6.79407e7 −0.110974
\(325\) −1.60108e8 −0.258714
\(326\) 9.65122e6 0.0154284
\(327\) −7.41916e7 −0.117338
\(328\) −1.42309e6 −0.00222676
\(329\) −1.39247e7 −0.0215575
\(330\) −6.56218e6 −0.0100519
\(331\) 7.06940e7 0.107148 0.0535740 0.998564i \(-0.482939\pi\)
0.0535740 + 0.998564i \(0.482939\pi\)
\(332\) 9.75978e8 1.46372
\(333\) −3.26113e8 −0.483964
\(334\) 8.65539e6 0.0127108
\(335\) −9.47313e8 −1.37669
\(336\) −2.87769e8 −0.413863
\(337\) −6.96987e8 −0.992020 −0.496010 0.868317i \(-0.665202\pi\)
−0.496010 + 0.868317i \(0.665202\pi\)
\(338\) 1.02956e7 0.0145025
\(339\) −8.33481e7 −0.116198
\(340\) −3.97527e8 −0.548518
\(341\) 1.93060e7 0.0263664
\(342\) −1.19744e7 −0.0161869
\(343\) 7.97076e8 1.06652
\(344\) 9.10105e7 0.120542
\(345\) −3.99058e8 −0.523202
\(346\) 3.00778e7 0.0390374
\(347\) −3.13443e8 −0.402721 −0.201361 0.979517i \(-0.564536\pi\)
−0.201361 + 0.979517i \(0.564536\pi\)
\(348\) −7.33698e8 −0.933233
\(349\) 6.69924e8 0.843599 0.421800 0.906689i \(-0.361399\pi\)
0.421800 + 0.906689i \(0.361399\pi\)
\(350\) 4.40524e6 0.00549201
\(351\) 1.85367e8 0.228801
\(352\) −4.82399e7 −0.0589531
\(353\) −1.16673e9 −1.41175 −0.705875 0.708336i \(-0.749446\pi\)
−0.705875 + 0.708336i \(0.749446\pi\)
\(354\) −2.20065e6 −0.00263658
\(355\) −8.29897e8 −0.984521
\(356\) 6.72525e8 0.790011
\(357\) −2.21725e8 −0.257915
\(358\) −453354. −0.000522212 0
\(359\) 1.70302e9 1.94263 0.971316 0.237794i \(-0.0764242\pi\)
0.971316 + 0.237794i \(0.0764242\pi\)
\(360\) −1.82995e7 −0.0206719
\(361\) 8.19262e8 0.916532
\(362\) 5.37126e7 0.0595109
\(363\) −3.60479e8 −0.395555
\(364\) 7.86108e8 0.854333
\(365\) 1.14108e8 0.122827
\(366\) 8.68225e6 0.00925653
\(367\) 9.54527e8 1.00799 0.503996 0.863706i \(-0.331863\pi\)
0.503996 + 0.863706i \(0.331863\pi\)
\(368\) −9.75845e8 −1.02074
\(369\) 1.02177e7 0.0105867
\(370\) −4.38914e7 −0.0450477
\(371\) −5.94751e8 −0.604681
\(372\) 2.69020e7 0.0270947
\(373\) 1.42139e9 1.41819 0.709093 0.705115i \(-0.249105\pi\)
0.709093 + 0.705115i \(0.249105\pi\)
\(374\) −1.23641e7 −0.0122212
\(375\) −6.34993e8 −0.621812
\(376\) −2.16533e6 −0.00210071
\(377\) 2.00179e9 1.92409
\(378\) −5.10023e6 −0.00485700
\(379\) 2.98909e8 0.282035 0.141017 0.990007i \(-0.454963\pi\)
0.141017 + 0.990007i \(0.454963\pi\)
\(380\) 1.30821e9 1.22302
\(381\) 2.77188e8 0.256765
\(382\) 4.42268e7 0.0405941
\(383\) −2.63974e8 −0.240086 −0.120043 0.992769i \(-0.538303\pi\)
−0.120043 + 0.992769i \(0.538303\pi\)
\(384\) −8.96084e7 −0.0807587
\(385\) 3.99869e8 0.357113
\(386\) −1.82086e7 −0.0161146
\(387\) −6.53453e8 −0.573094
\(388\) 1.18021e9 1.02576
\(389\) −1.71619e9 −1.47823 −0.739116 0.673578i \(-0.764757\pi\)
−0.739116 + 0.673578i \(0.764757\pi\)
\(390\) 2.49484e7 0.0212969
\(391\) −7.51885e8 −0.636111
\(392\) 4.03314e7 0.0338175
\(393\) −8.23487e8 −0.684357
\(394\) 4.86758e7 0.0400937
\(395\) −3.38856e8 −0.276647
\(396\) 2.30860e8 0.186817
\(397\) 5.74825e8 0.461072 0.230536 0.973064i \(-0.425952\pi\)
0.230536 + 0.973064i \(0.425952\pi\)
\(398\) −4.06802e6 −0.00323439
\(399\) 7.29668e8 0.575069
\(400\) −2.77515e8 −0.216808
\(401\) −1.14785e9 −0.888953 −0.444476 0.895791i \(-0.646610\pi\)
−0.444476 + 0.895791i \(0.646610\pi\)
\(402\) −4.10566e7 −0.0315204
\(403\) −7.33984e7 −0.0558623
\(404\) −6.20745e8 −0.468358
\(405\) 1.31390e8 0.0982808
\(406\) −5.50778e7 −0.0408447
\(407\) 1.10812e9 0.814715
\(408\) −3.44789e7 −0.0251329
\(409\) −4.45907e6 −0.00322265 −0.00161132 0.999999i \(-0.500513\pi\)
−0.00161132 + 0.999999i \(0.500513\pi\)
\(410\) 1.37520e6 0.000985422 0
\(411\) 5.88884e8 0.418392
\(412\) 5.55394e8 0.391256
\(413\) 1.34098e8 0.0936692
\(414\) −1.72952e7 −0.0119791
\(415\) −1.88743e9 −1.29629
\(416\) 1.83401e8 0.124904
\(417\) 1.50990e9 1.01970
\(418\) 4.06887e7 0.0272494
\(419\) 2.72506e9 1.80978 0.904892 0.425642i \(-0.139952\pi\)
0.904892 + 0.425642i \(0.139952\pi\)
\(420\) 5.57200e8 0.366977
\(421\) −1.98552e9 −1.29684 −0.648420 0.761283i \(-0.724570\pi\)
−0.648420 + 0.761283i \(0.724570\pi\)
\(422\) −6.47379e7 −0.0419339
\(423\) 1.55470e7 0.00998746
\(424\) −9.24855e7 −0.0589241
\(425\) −2.13824e8 −0.135112
\(426\) −3.59678e7 −0.0225414
\(427\) −5.29057e8 −0.328855
\(428\) 8.44413e8 0.520597
\(429\) −6.29869e8 −0.385168
\(430\) −8.79478e7 −0.0533440
\(431\) −8.94337e8 −0.538060 −0.269030 0.963132i \(-0.586703\pi\)
−0.269030 + 0.963132i \(0.586703\pi\)
\(432\) 3.21296e8 0.191740
\(433\) −2.81913e9 −1.66881 −0.834405 0.551152i \(-0.814188\pi\)
−0.834405 + 0.551152i \(0.814188\pi\)
\(434\) 2.01950e6 0.00118585
\(435\) 1.41889e9 0.826487
\(436\) 3.51290e8 0.202985
\(437\) 2.47435e9 1.41833
\(438\) 4.94546e6 0.00281221
\(439\) 1.50706e9 0.850166 0.425083 0.905154i \(-0.360245\pi\)
0.425083 + 0.905154i \(0.360245\pi\)
\(440\) 6.21809e7 0.0347995
\(441\) −2.89578e8 −0.160780
\(442\) 4.70065e7 0.0258929
\(443\) 1.30737e9 0.714471 0.357235 0.934014i \(-0.383719\pi\)
0.357235 + 0.934014i \(0.383719\pi\)
\(444\) 1.54411e9 0.837219
\(445\) −1.30059e9 −0.699647
\(446\) 8.57428e7 0.0457642
\(447\) 1.14776e9 0.607819
\(448\) 1.35919e9 0.714181
\(449\) −2.59024e9 −1.35045 −0.675223 0.737613i \(-0.735953\pi\)
−0.675223 + 0.737613i \(0.735953\pi\)
\(450\) −4.91848e6 −0.00254441
\(451\) −3.47195e7 −0.0178219
\(452\) 3.94646e8 0.201012
\(453\) −7.26668e8 −0.367276
\(454\) −3.75639e7 −0.0188397
\(455\) −1.52024e9 −0.756612
\(456\) 1.13466e8 0.0560386
\(457\) −3.00834e9 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(458\) 2.23068e7 0.0108495
\(459\) 2.47558e8 0.119490
\(460\) 1.88950e9 0.905097
\(461\) 1.29563e9 0.615926 0.307963 0.951398i \(-0.400353\pi\)
0.307963 + 0.951398i \(0.400353\pi\)
\(462\) 1.73304e7 0.00817637
\(463\) 7.89028e7 0.0369453 0.0184726 0.999829i \(-0.494120\pi\)
0.0184726 + 0.999829i \(0.494120\pi\)
\(464\) 3.46971e9 1.61243
\(465\) −5.20254e7 −0.0239955
\(466\) 1.24058e8 0.0567903
\(467\) −1.97263e9 −0.896264 −0.448132 0.893967i \(-0.647911\pi\)
−0.448132 + 0.893967i \(0.647911\pi\)
\(468\) −8.77695e8 −0.395807
\(469\) 2.50180e9 1.11982
\(470\) 2.09246e6 0.000929639 0
\(471\) −1.64552e9 −0.725656
\(472\) 2.08526e7 0.00912775
\(473\) 2.22041e9 0.964758
\(474\) −1.46861e7 −0.00633404
\(475\) 7.03667e8 0.301259
\(476\) 1.04985e9 0.446172
\(477\) 6.64043e8 0.280144
\(478\) 1.28960e8 0.0540080
\(479\) 8.20556e7 0.0341141 0.0170571 0.999855i \(-0.494570\pi\)
0.0170571 + 0.999855i \(0.494570\pi\)
\(480\) 1.29996e8 0.0536520
\(481\) −4.21290e9 −1.72613
\(482\) 1.01532e8 0.0412989
\(483\) 1.05389e9 0.425580
\(484\) 1.70684e9 0.684279
\(485\) −2.28239e9 −0.908434
\(486\) 5.69444e6 0.00225022
\(487\) 2.81731e9 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(488\) −8.22699e7 −0.0320458
\(489\) 6.56619e8 0.253941
\(490\) −3.89742e7 −0.0149655
\(491\) −4.34251e9 −1.65560 −0.827800 0.561023i \(-0.810408\pi\)
−0.827800 + 0.561023i \(0.810408\pi\)
\(492\) −4.83800e7 −0.0183142
\(493\) 2.67340e9 1.00485
\(494\) −1.54692e8 −0.0577330
\(495\) −4.46457e8 −0.165448
\(496\) −1.27221e8 −0.0468138
\(497\) 2.19171e9 0.800823
\(498\) −8.18014e7 −0.0296796
\(499\) −2.81881e9 −1.01558 −0.507789 0.861481i \(-0.669537\pi\)
−0.507789 + 0.861481i \(0.669537\pi\)
\(500\) 3.00663e9 1.07569
\(501\) 5.88869e8 0.209212
\(502\) 5.84167e7 0.0206098
\(503\) 3.80145e9 1.33187 0.665933 0.746011i \(-0.268033\pi\)
0.665933 + 0.746011i \(0.268033\pi\)
\(504\) 4.83280e7 0.0168148
\(505\) 1.20045e9 0.414786
\(506\) 5.87684e7 0.0201659
\(507\) 7.00460e8 0.238702
\(508\) −1.31246e9 −0.444184
\(509\) 1.38927e9 0.466956 0.233478 0.972362i \(-0.424989\pi\)
0.233478 + 0.972362i \(0.424989\pi\)
\(510\) 3.33186e7 0.0111222
\(511\) −3.01354e8 −0.0999089
\(512\) 5.30032e8 0.174525
\(513\) −8.14680e8 −0.266426
\(514\) −1.29580e8 −0.0420889
\(515\) −1.07407e9 −0.346503
\(516\) 3.09404e9 0.991407
\(517\) −5.28280e7 −0.0168131
\(518\) 1.15915e8 0.0366425
\(519\) 2.04634e9 0.642529
\(520\) −2.36402e8 −0.0737293
\(521\) −2.67044e9 −0.827277 −0.413639 0.910441i \(-0.635742\pi\)
−0.413639 + 0.910441i \(0.635742\pi\)
\(522\) 6.14948e7 0.0189231
\(523\) 2.60473e8 0.0796173 0.0398086 0.999207i \(-0.487325\pi\)
0.0398086 + 0.999207i \(0.487325\pi\)
\(524\) 3.89914e9 1.18388
\(525\) 2.99710e8 0.0903949
\(526\) −4.45592e7 −0.0133502
\(527\) −9.80236e7 −0.0291738
\(528\) −1.09175e9 −0.322779
\(529\) 1.68991e8 0.0496327
\(530\) 8.93732e7 0.0260760
\(531\) −1.49721e8 −0.0433963
\(532\) −3.45491e9 −0.994824
\(533\) 1.31998e8 0.0377592
\(534\) −5.63675e7 −0.0160190
\(535\) −1.63300e9 −0.461049
\(536\) 3.89038e8 0.109123
\(537\) −3.08439e7 −0.00859527
\(538\) −2.03424e7 −0.00563201
\(539\) 9.83975e8 0.270660
\(540\) −6.22118e8 −0.170018
\(541\) −9.32970e7 −0.0253324 −0.0126662 0.999920i \(-0.504032\pi\)
−0.0126662 + 0.999920i \(0.504032\pi\)
\(542\) −2.09669e8 −0.0565636
\(543\) 3.65433e9 0.979509
\(544\) 2.44932e8 0.0652303
\(545\) −6.79356e8 −0.179767
\(546\) −6.58875e7 −0.0173232
\(547\) 6.23723e9 1.62943 0.814716 0.579861i \(-0.196893\pi\)
0.814716 + 0.579861i \(0.196893\pi\)
\(548\) −2.78831e9 −0.723784
\(549\) 5.90696e8 0.152356
\(550\) 1.67128e7 0.00428332
\(551\) −8.79780e9 −2.24049
\(552\) 1.63883e8 0.0414713
\(553\) 8.94902e8 0.225028
\(554\) −1.78069e7 −0.00444943
\(555\) −2.98614e9 −0.741456
\(556\) −7.14923e9 −1.76400
\(557\) 1.77646e9 0.435574 0.217787 0.975996i \(-0.430116\pi\)
0.217787 + 0.975996i \(0.430116\pi\)
\(558\) −2.25479e6 −0.000549396 0
\(559\) −8.44166e9 −2.04403
\(560\) −2.63504e9 −0.634058
\(561\) −8.41191e8 −0.201152
\(562\) −2.09196e8 −0.0497137
\(563\) 2.36725e9 0.559067 0.279534 0.960136i \(-0.409820\pi\)
0.279534 + 0.960136i \(0.409820\pi\)
\(564\) −7.36135e7 −0.0172775
\(565\) −7.63200e8 −0.178020
\(566\) 1.22439e8 0.0283833
\(567\) −3.46993e8 −0.0799430
\(568\) 3.40818e8 0.0780375
\(569\) −4.20570e9 −0.957073 −0.478536 0.878068i \(-0.658833\pi\)
−0.478536 + 0.878068i \(0.658833\pi\)
\(570\) −1.09647e8 −0.0247991
\(571\) 5.59664e9 1.25806 0.629029 0.777382i \(-0.283453\pi\)
0.629029 + 0.777382i \(0.283453\pi\)
\(572\) 2.98237e9 0.666309
\(573\) 3.00896e9 0.668153
\(574\) −3.63183e6 −0.000801556 0
\(575\) 1.01634e9 0.222946
\(576\) −1.51755e9 −0.330875
\(577\) −3.81070e9 −0.825828 −0.412914 0.910770i \(-0.635489\pi\)
−0.412914 + 0.910770i \(0.635489\pi\)
\(578\) −1.00068e8 −0.0215549
\(579\) −1.23882e9 −0.265236
\(580\) −6.71831e9 −1.42976
\(581\) 4.98460e9 1.05442
\(582\) −9.89189e7 −0.0207993
\(583\) −2.25639e9 −0.471601
\(584\) −4.68615e7 −0.00973579
\(585\) 1.69736e9 0.350533
\(586\) −3.26230e8 −0.0669702
\(587\) 1.29560e9 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(588\) 1.37113e9 0.278136
\(589\) 3.22583e8 0.0650486
\(590\) −2.01509e7 −0.00403936
\(591\) 3.31165e9 0.659915
\(592\) −7.30222e9 −1.44654
\(593\) 3.30125e9 0.650111 0.325055 0.945695i \(-0.394617\pi\)
0.325055 + 0.945695i \(0.394617\pi\)
\(594\) −1.93495e7 −0.00378806
\(595\) −2.03029e9 −0.395137
\(596\) −5.43453e9 −1.05148
\(597\) −2.76767e8 −0.0532358
\(598\) −2.23429e8 −0.0427253
\(599\) 3.03218e9 0.576448 0.288224 0.957563i \(-0.406935\pi\)
0.288224 + 0.957563i \(0.406935\pi\)
\(600\) 4.66058e7 0.00880868
\(601\) 3.17671e9 0.596922 0.298461 0.954422i \(-0.403527\pi\)
0.298461 + 0.954422i \(0.403527\pi\)
\(602\) 2.32266e8 0.0433908
\(603\) −2.79328e9 −0.518805
\(604\) 3.44071e9 0.635358
\(605\) −3.30083e9 −0.606009
\(606\) 5.20276e7 0.00949685
\(607\) 2.80510e9 0.509083 0.254541 0.967062i \(-0.418075\pi\)
0.254541 + 0.967062i \(0.418075\pi\)
\(608\) −8.06039e8 −0.145443
\(609\) −3.74721e9 −0.672276
\(610\) 7.95014e7 0.0141814
\(611\) 2.00844e8 0.0356218
\(612\) −1.17216e9 −0.206708
\(613\) −9.45694e9 −1.65821 −0.829104 0.559094i \(-0.811149\pi\)
−0.829104 + 0.559094i \(0.811149\pi\)
\(614\) 4.81088e7 0.00838755
\(615\) 9.35615e7 0.0162194
\(616\) −1.64216e8 −0.0283064
\(617\) 5.14249e9 0.881405 0.440702 0.897653i \(-0.354729\pi\)
0.440702 + 0.897653i \(0.354729\pi\)
\(618\) −4.65502e7 −0.00793345
\(619\) 7.42513e9 1.25831 0.629153 0.777281i \(-0.283402\pi\)
0.629153 + 0.777281i \(0.283402\pi\)
\(620\) 2.46336e8 0.0415103
\(621\) −1.17668e9 −0.197168
\(622\) −1.98077e8 −0.0330041
\(623\) 3.43478e9 0.569103
\(624\) 4.15068e9 0.683869
\(625\) −4.48629e9 −0.735034
\(626\) 2.66529e8 0.0434245
\(627\) 2.76825e9 0.448507
\(628\) 7.79140e9 1.25533
\(629\) −5.62634e9 −0.901464
\(630\) −4.67016e7 −0.00744115
\(631\) 6.01569e8 0.0953197 0.0476598 0.998864i \(-0.484824\pi\)
0.0476598 + 0.998864i \(0.484824\pi\)
\(632\) 1.39160e8 0.0219283
\(633\) −4.40444e9 −0.690204
\(634\) −2.99967e8 −0.0467479
\(635\) 2.53815e9 0.393377
\(636\) −3.14418e9 −0.484627
\(637\) −3.74093e9 −0.573444
\(638\) −2.08957e8 −0.0318555
\(639\) −2.44706e9 −0.371016
\(640\) −8.20524e8 −0.123726
\(641\) 3.94333e9 0.591371 0.295685 0.955285i \(-0.404452\pi\)
0.295685 + 0.955285i \(0.404452\pi\)
\(642\) −7.07743e7 −0.0105561
\(643\) 3.67937e9 0.545802 0.272901 0.962042i \(-0.412017\pi\)
0.272901 + 0.962042i \(0.412017\pi\)
\(644\) −4.99008e9 −0.736219
\(645\) −5.98352e9 −0.878007
\(646\) −2.06592e8 −0.0301508
\(647\) 9.80484e9 1.42323 0.711616 0.702569i \(-0.247964\pi\)
0.711616 + 0.702569i \(0.247964\pi\)
\(648\) −5.39585e7 −0.00779018
\(649\) 5.08747e8 0.0730542
\(650\) −6.35396e7 −0.00907503
\(651\) 1.37396e8 0.0195183
\(652\) −3.10903e9 −0.439297
\(653\) −4.40696e9 −0.619360 −0.309680 0.950841i \(-0.600222\pi\)
−0.309680 + 0.950841i \(0.600222\pi\)
\(654\) −2.94433e7 −0.00411590
\(655\) −7.54049e9 −1.04847
\(656\) 2.28792e8 0.0316430
\(657\) 3.36464e8 0.0462871
\(658\) −5.52607e6 −0.000756182 0
\(659\) −1.35935e10 −1.85025 −0.925127 0.379657i \(-0.876042\pi\)
−0.925127 + 0.379657i \(0.876042\pi\)
\(660\) 2.11393e9 0.286212
\(661\) −5.30552e9 −0.714534 −0.357267 0.934002i \(-0.616291\pi\)
−0.357267 + 0.934002i \(0.616291\pi\)
\(662\) 2.80553e7 0.00375847
\(663\) 3.19808e9 0.426179
\(664\) 7.75121e8 0.102750
\(665\) 6.68141e9 0.881033
\(666\) −1.29420e8 −0.0169762
\(667\) −1.27070e10 −1.65807
\(668\) −2.78824e9 −0.361920
\(669\) 5.83350e9 0.753248
\(670\) −3.75946e8 −0.0482907
\(671\) −2.00716e9 −0.256480
\(672\) −3.43313e8 −0.0436413
\(673\) 3.77543e9 0.477434 0.238717 0.971089i \(-0.423273\pi\)
0.238717 + 0.971089i \(0.423273\pi\)
\(674\) −2.76603e8 −0.0347975
\(675\) −3.34628e8 −0.0418793
\(676\) −3.31661e9 −0.412934
\(677\) −7.29031e9 −0.902995 −0.451498 0.892272i \(-0.649110\pi\)
−0.451498 + 0.892272i \(0.649110\pi\)
\(678\) −3.30771e7 −0.00407591
\(679\) 6.02767e9 0.738933
\(680\) −3.15716e8 −0.0385048
\(681\) −2.55565e9 −0.310090
\(682\) 7.66167e6 0.000924865 0
\(683\) 7.10884e9 0.853742 0.426871 0.904313i \(-0.359616\pi\)
0.426871 + 0.904313i \(0.359616\pi\)
\(684\) 3.85743e9 0.460895
\(685\) 5.39227e9 0.640996
\(686\) 3.16324e8 0.0374108
\(687\) 1.51764e9 0.178575
\(688\) −1.46319e10 −1.71294
\(689\) 8.57847e9 0.999177
\(690\) −1.58368e8 −0.0183526
\(691\) −1.26683e10 −1.46065 −0.730324 0.683100i \(-0.760631\pi\)
−0.730324 + 0.683100i \(0.760631\pi\)
\(692\) −9.68924e9 −1.11152
\(693\) 1.17907e9 0.134578
\(694\) −1.24391e8 −0.0141264
\(695\) 1.38258e10 1.56223
\(696\) −5.82703e8 −0.0655111
\(697\) 1.76284e8 0.0197196
\(698\) 2.65863e8 0.0295913
\(699\) 8.44027e9 0.934730
\(700\) −1.41910e9 −0.156376
\(701\) −1.78862e10 −1.96113 −0.980564 0.196201i \(-0.937139\pi\)
−0.980564 + 0.196201i \(0.937139\pi\)
\(702\) 7.35638e7 0.00802573
\(703\) 1.85155e10 2.00998
\(704\) 5.15657e9 0.557002
\(705\) 1.42360e8 0.0153012
\(706\) −4.63022e8 −0.0495205
\(707\) −3.17033e9 −0.337393
\(708\) 7.08916e8 0.0750721
\(709\) −1.24346e10 −1.31030 −0.655150 0.755499i \(-0.727395\pi\)
−0.655150 + 0.755499i \(0.727395\pi\)
\(710\) −3.29349e8 −0.0345344
\(711\) −9.99164e8 −0.104254
\(712\) 5.34119e8 0.0554572
\(713\) 4.65920e8 0.0481392
\(714\) −8.79928e7 −0.00904697
\(715\) −5.76757e9 −0.590095
\(716\) 1.46043e8 0.0148691
\(717\) 8.77379e9 0.888935
\(718\) 6.75854e8 0.0681424
\(719\) 2.69952e9 0.270854 0.135427 0.990787i \(-0.456759\pi\)
0.135427 + 0.990787i \(0.456759\pi\)
\(720\) 2.94204e9 0.293755
\(721\) 2.83656e9 0.281850
\(722\) 3.25128e8 0.0321495
\(723\) 6.90772e9 0.679752
\(724\) −1.73029e10 −1.69447
\(725\) −3.61368e9 −0.352182
\(726\) −1.43058e8 −0.0138750
\(727\) −4.43778e9 −0.428346 −0.214173 0.976796i \(-0.568706\pi\)
−0.214173 + 0.976796i \(0.568706\pi\)
\(728\) 6.24327e8 0.0599725
\(729\) 3.87420e8 0.0370370
\(730\) 4.52845e7 0.00430844
\(731\) −1.12738e10 −1.06748
\(732\) −2.79689e9 −0.263564
\(733\) 8.21884e9 0.770808 0.385404 0.922748i \(-0.374062\pi\)
0.385404 + 0.922748i \(0.374062\pi\)
\(734\) 3.78809e8 0.0353577
\(735\) −2.65160e9 −0.246322
\(736\) −1.16420e9 −0.107635
\(737\) 9.49146e9 0.873367
\(738\) 4.05496e6 0.000371355 0
\(739\) −8.71653e9 −0.794490 −0.397245 0.917713i \(-0.630034\pi\)
−0.397245 + 0.917713i \(0.630034\pi\)
\(740\) 1.41391e10 1.28266
\(741\) −1.05245e10 −0.950247
\(742\) −2.36030e8 −0.0212106
\(743\) −1.01228e9 −0.0905396 −0.0452698 0.998975i \(-0.514415\pi\)
−0.0452698 + 0.998975i \(0.514415\pi\)
\(744\) 2.13656e7 0.00190199
\(745\) 1.05098e10 0.931207
\(746\) 5.64087e8 0.0497462
\(747\) −5.56535e9 −0.488506
\(748\) 3.98296e9 0.347977
\(749\) 4.31266e9 0.375024
\(750\) −2.52000e8 −0.0218115
\(751\) 1.76039e10 1.51659 0.758295 0.651911i \(-0.226032\pi\)
0.758295 + 0.651911i \(0.226032\pi\)
\(752\) 3.48123e8 0.0298518
\(753\) 3.97437e9 0.339224
\(754\) 7.94422e8 0.0674919
\(755\) −6.65393e9 −0.562684
\(756\) 1.64298e9 0.138295
\(757\) 1.48620e10 1.24520 0.622602 0.782539i \(-0.286076\pi\)
0.622602 + 0.782539i \(0.286076\pi\)
\(758\) 1.18624e8 0.00989304
\(759\) 3.99830e9 0.331917
\(760\) 1.03898e9 0.0858537
\(761\) 1.01144e10 0.831939 0.415970 0.909378i \(-0.363442\pi\)
0.415970 + 0.909378i \(0.363442\pi\)
\(762\) 1.10003e8 0.00900666
\(763\) 1.79414e9 0.146225
\(764\) −1.42472e10 −1.15585
\(765\) 2.26683e9 0.183064
\(766\) −1.04760e8 −0.00842157
\(767\) −1.93418e9 −0.154779
\(768\) 7.15874e9 0.570260
\(769\) −1.01233e10 −0.802748 −0.401374 0.915914i \(-0.631467\pi\)
−0.401374 + 0.915914i \(0.631467\pi\)
\(770\) 1.58690e8 0.0125266
\(771\) −8.81597e9 −0.692756
\(772\) 5.86568e9 0.458836
\(773\) 1.87926e10 1.46339 0.731693 0.681635i \(-0.238731\pi\)
0.731693 + 0.681635i \(0.238731\pi\)
\(774\) −2.59326e8 −0.0201026
\(775\) 1.32500e8 0.0102250
\(776\) 9.37320e8 0.0720066
\(777\) 7.88625e9 0.603110
\(778\) −6.81081e8 −0.0518525
\(779\) −5.80127e8 −0.0439685
\(780\) −8.03686e9 −0.606394
\(781\) 8.31503e9 0.624576
\(782\) −2.98389e8 −0.0223131
\(783\) 4.18379e9 0.311461
\(784\) −6.48415e9 −0.480559
\(785\) −1.50677e10 −1.11174
\(786\) −3.26805e8 −0.0240055
\(787\) 1.04611e10 0.765009 0.382505 0.923954i \(-0.375062\pi\)
0.382505 + 0.923954i \(0.375062\pi\)
\(788\) −1.56804e10 −1.14160
\(789\) −3.03158e9 −0.219735
\(790\) −1.34477e8 −0.00970405
\(791\) 2.01557e9 0.144804
\(792\) 1.83349e8 0.0131141
\(793\) 7.63092e9 0.543402
\(794\) 2.28122e8 0.0161732
\(795\) 6.08049e9 0.429194
\(796\) 1.31047e9 0.0920937
\(797\) 2.15776e10 1.50973 0.754865 0.655880i \(-0.227702\pi\)
0.754865 + 0.655880i \(0.227702\pi\)
\(798\) 2.89573e8 0.0201719
\(799\) 2.68228e8 0.0186033
\(800\) −3.31079e8 −0.0228622
\(801\) −3.83496e9 −0.263661
\(802\) −4.55529e8 −0.0311821
\(803\) −1.14329e9 −0.0779206
\(804\) 1.32259e10 0.897491
\(805\) 9.65025e9 0.652008
\(806\) −2.91285e7 −0.00195950
\(807\) −1.38399e9 −0.0926992
\(808\) −4.92995e8 −0.0328778
\(809\) 1.89775e10 1.26014 0.630070 0.776538i \(-0.283026\pi\)
0.630070 + 0.776538i \(0.283026\pi\)
\(810\) 5.21427e7 0.00344743
\(811\) −4.70722e9 −0.309879 −0.154939 0.987924i \(-0.549518\pi\)
−0.154939 + 0.987924i \(0.549518\pi\)
\(812\) 1.77427e10 1.16298
\(813\) −1.42648e10 −0.930999
\(814\) 4.39763e8 0.0285781
\(815\) 6.01252e9 0.389049
\(816\) 5.54324e9 0.357148
\(817\) 3.71007e10 2.38015
\(818\) −1.76961e6 −0.000113042 0
\(819\) −4.48265e9 −0.285129
\(820\) −4.43005e8 −0.0280582
\(821\) −2.74939e10 −1.73395 −0.866973 0.498355i \(-0.833938\pi\)
−0.866973 + 0.498355i \(0.833938\pi\)
\(822\) 2.33702e8 0.0146761
\(823\) −1.47691e9 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(824\) 4.41093e8 0.0274654
\(825\) 1.13705e9 0.0705005
\(826\) 5.32175e7 0.00328567
\(827\) −3.11067e10 −1.91243 −0.956213 0.292671i \(-0.905456\pi\)
−0.956213 + 0.292671i \(0.905456\pi\)
\(828\) 5.57146e9 0.341085
\(829\) 2.06946e9 0.126158 0.0630792 0.998009i \(-0.479908\pi\)
0.0630792 + 0.998009i \(0.479908\pi\)
\(830\) −7.49037e8 −0.0454705
\(831\) −1.21149e9 −0.0732346
\(832\) −1.96045e10 −1.18012
\(833\) −4.99601e9 −0.299479
\(834\) 5.99212e8 0.0357684
\(835\) 5.39214e9 0.320522
\(836\) −1.31074e10 −0.775880
\(837\) −1.53404e8 −0.00904269
\(838\) 1.08145e9 0.0634825
\(839\) 2.02258e9 0.118233 0.0591166 0.998251i \(-0.481172\pi\)
0.0591166 + 0.998251i \(0.481172\pi\)
\(840\) 4.42528e8 0.0257610
\(841\) 2.79312e10 1.61921
\(842\) −7.87963e8 −0.0454898
\(843\) −1.42326e10 −0.818254
\(844\) 2.08546e10 1.19400
\(845\) 6.41395e9 0.365702
\(846\) 6.16990e6 0.000350334 0
\(847\) 8.71731e9 0.492936
\(848\) 1.48691e10 0.837332
\(849\) 8.33014e9 0.467171
\(850\) −8.48572e7 −0.00473939
\(851\) 2.67428e10 1.48749
\(852\) 1.15866e10 0.641828
\(853\) −3.22240e10 −1.77770 −0.888849 0.458201i \(-0.848494\pi\)
−0.888849 + 0.458201i \(0.848494\pi\)
\(854\) −2.09959e8 −0.0115354
\(855\) −7.45984e9 −0.408177
\(856\) 6.70632e8 0.0365448
\(857\) 6.94237e9 0.376769 0.188384 0.982095i \(-0.439675\pi\)
0.188384 + 0.982095i \(0.439675\pi\)
\(858\) −2.49967e8 −0.0135107
\(859\) −6.91744e9 −0.372365 −0.186183 0.982515i \(-0.559612\pi\)
−0.186183 + 0.982515i \(0.559612\pi\)
\(860\) 2.83314e10 1.51888
\(861\) −2.47091e8 −0.0131931
\(862\) −3.54922e8 −0.0188737
\(863\) −2.83705e10 −1.50255 −0.751275 0.659990i \(-0.770561\pi\)
−0.751275 + 0.659990i \(0.770561\pi\)
\(864\) 3.83311e8 0.0202187
\(865\) 1.87379e10 0.984384
\(866\) −1.11878e9 −0.0585375
\(867\) −6.80810e9 −0.354780
\(868\) −6.50559e8 −0.0337651
\(869\) 3.39512e9 0.175504
\(870\) 5.63094e8 0.0289910
\(871\) −3.60851e10 −1.85040
\(872\) 2.78995e8 0.0142491
\(873\) −6.72993e9 −0.342343
\(874\) 9.81960e8 0.0497512
\(875\) 1.53557e10 0.774895
\(876\) −1.59313e9 −0.0800729
\(877\) −3.27420e9 −0.163910 −0.0819552 0.996636i \(-0.526116\pi\)
−0.0819552 + 0.996636i \(0.526116\pi\)
\(878\) 5.98083e8 0.0298216
\(879\) −2.21950e10 −1.10228
\(880\) −9.99693e9 −0.494512
\(881\) 1.44579e10 0.712346 0.356173 0.934420i \(-0.384081\pi\)
0.356173 + 0.934420i \(0.384081\pi\)
\(882\) −1.14921e8 −0.00563973
\(883\) 8.75453e9 0.427928 0.213964 0.976842i \(-0.431363\pi\)
0.213964 + 0.976842i \(0.431363\pi\)
\(884\) −1.51426e10 −0.737256
\(885\) −1.37096e9 −0.0664851
\(886\) 5.18835e8 0.0250618
\(887\) 1.54047e10 0.741175 0.370588 0.928798i \(-0.379156\pi\)
0.370588 + 0.928798i \(0.379156\pi\)
\(888\) 1.22634e9 0.0587711
\(889\) −6.70311e9 −0.319978
\(890\) −5.16144e8 −0.0245418
\(891\) −1.31644e9 −0.0623489
\(892\) −2.76211e10 −1.30306
\(893\) −8.82702e8 −0.0414796
\(894\) 4.55494e8 0.0213207
\(895\) −2.82431e8 −0.0131684
\(896\) 2.16696e9 0.100641
\(897\) −1.52010e10 −0.703230
\(898\) −1.02795e9 −0.0473701
\(899\) −1.65662e9 −0.0760441
\(900\) 1.58443e9 0.0724479
\(901\) 1.14566e10 0.521816
\(902\) −1.37786e7 −0.000625147 0
\(903\) 1.58022e10 0.714183
\(904\) 3.13427e8 0.0141107
\(905\) 3.34619e10 1.50065
\(906\) −2.88382e8 −0.0128831
\(907\) 4.05126e10 1.80287 0.901435 0.432915i \(-0.142515\pi\)
0.901435 + 0.432915i \(0.142515\pi\)
\(908\) 1.21008e10 0.536430
\(909\) 3.53969e9 0.156312
\(910\) −6.03317e8 −0.0265400
\(911\) 9.95690e9 0.436325 0.218162 0.975912i \(-0.429994\pi\)
0.218162 + 0.975912i \(0.429994\pi\)
\(912\) −1.82421e10 −0.796328
\(913\) 1.89108e10 0.822361
\(914\) −1.19387e9 −0.0517186
\(915\) 5.40887e9 0.233417
\(916\) −7.18590e9 −0.308921
\(917\) 1.99140e10 0.852838
\(918\) 9.82446e7 0.00419140
\(919\) 1.67638e10 0.712472 0.356236 0.934396i \(-0.384060\pi\)
0.356236 + 0.934396i \(0.384060\pi\)
\(920\) 1.50064e9 0.0635360
\(921\) 3.27307e9 0.138053
\(922\) 5.14179e8 0.0216051
\(923\) −3.16125e10 −1.32328
\(924\) −5.58278e9 −0.232809
\(925\) 7.60523e9 0.315948
\(926\) 3.13130e7 0.00129594
\(927\) −3.16704e9 −0.130579
\(928\) 4.13941e9 0.170028
\(929\) −2.97001e10 −1.21536 −0.607678 0.794184i \(-0.707899\pi\)
−0.607678 + 0.794184i \(0.707899\pi\)
\(930\) −2.06466e7 −0.000841700 0
\(931\) 1.64412e10 0.667744
\(932\) −3.99639e10 −1.61701
\(933\) −1.34762e10 −0.543226
\(934\) −7.82847e8 −0.0314386
\(935\) −7.70260e9 −0.308174
\(936\) −6.97065e8 −0.0277848
\(937\) 3.54569e10 1.40803 0.704015 0.710185i \(-0.251389\pi\)
0.704015 + 0.710185i \(0.251389\pi\)
\(938\) 9.92854e8 0.0392804
\(939\) 1.81333e10 0.714738
\(940\) −6.74063e8 −0.0264699
\(941\) 1.78640e10 0.698900 0.349450 0.936955i \(-0.386369\pi\)
0.349450 + 0.936955i \(0.386369\pi\)
\(942\) −6.53034e8 −0.0254541
\(943\) −8.37902e8 −0.0325389
\(944\) −3.35251e9 −0.129709
\(945\) −3.17734e9 −0.122476
\(946\) 8.81180e8 0.0338412
\(947\) −4.19426e10 −1.60483 −0.802417 0.596764i \(-0.796453\pi\)
−0.802417 + 0.596764i \(0.796453\pi\)
\(948\) 4.73095e9 0.180351
\(949\) 4.34662e9 0.165090
\(950\) 2.79254e8 0.0105674
\(951\) −2.04082e10 −0.769439
\(952\) 8.33789e8 0.0313204
\(953\) −2.78513e10 −1.04236 −0.521182 0.853445i \(-0.674509\pi\)
−0.521182 + 0.853445i \(0.674509\pi\)
\(954\) 2.63529e8 0.00982673
\(955\) 2.75524e10 1.02364
\(956\) −4.15431e10 −1.53779
\(957\) −1.42163e10 −0.524320
\(958\) 3.25642e7 0.00119663
\(959\) −1.42407e10 −0.521395
\(960\) −1.38959e10 −0.506916
\(961\) −2.74519e10 −0.997792
\(962\) −1.67191e9 −0.0605482
\(963\) −4.81512e9 −0.173746
\(964\) −3.27074e10 −1.17592
\(965\) −1.13436e10 −0.406353
\(966\) 4.18242e8 0.0149282
\(967\) −1.00043e10 −0.355789 −0.177895 0.984050i \(-0.556929\pi\)
−0.177895 + 0.984050i \(0.556929\pi\)
\(968\) 1.35557e9 0.0480350
\(969\) −1.40554e10 −0.496262
\(970\) −9.05778e8 −0.0318655
\(971\) −2.19879e10 −0.770755 −0.385377 0.922759i \(-0.625929\pi\)
−0.385377 + 0.922759i \(0.625929\pi\)
\(972\) −1.83440e9 −0.0640711
\(973\) −3.65132e10 −1.27074
\(974\) 1.11807e9 0.0387714
\(975\) −4.32291e9 −0.149369
\(976\) 1.32267e10 0.455383
\(977\) −4.29793e10 −1.47444 −0.737222 0.675651i \(-0.763863\pi\)
−0.737222 + 0.675651i \(0.763863\pi\)
\(978\) 2.60583e8 0.00890758
\(979\) 1.30310e10 0.443853
\(980\) 1.25551e10 0.426117
\(981\) −2.00317e9 −0.0677450
\(982\) −1.72335e9 −0.0580741
\(983\) −2.32107e10 −0.779383 −0.389691 0.920946i \(-0.627418\pi\)
−0.389691 + 0.920946i \(0.627418\pi\)
\(984\) −3.84234e7 −0.00128562
\(985\) 3.03240e10 1.01102
\(986\) 1.06095e9 0.0352474
\(987\) −3.75966e8 −0.0124462
\(988\) 4.98324e10 1.64385
\(989\) 5.35862e10 1.76143
\(990\) −1.77179e8 −0.00580348
\(991\) −1.22560e10 −0.400030 −0.200015 0.979793i \(-0.564099\pi\)
−0.200015 + 0.979793i \(0.564099\pi\)
\(992\) −1.51777e8 −0.00493646
\(993\) 1.90874e9 0.0618620
\(994\) 8.69793e8 0.0280908
\(995\) −2.53429e9 −0.0815598
\(996\) 2.63514e10 0.845077
\(997\) −1.51059e10 −0.482740 −0.241370 0.970433i \(-0.577597\pi\)
−0.241370 + 0.970433i \(0.577597\pi\)
\(998\) −1.11866e9 −0.0356238
\(999\) −8.80505e9 −0.279417
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.b.1.9 17
3.2 odd 2 531.8.a.d.1.9 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.9 17 1.1 even 1 trivial
531.8.a.d.1.9 17 3.2 odd 2