Properties

Label 177.8.a.b.1.17
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 1.17
Root \(24.0278\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+22.0278 q^{2} +27.0000 q^{3} +357.226 q^{4} -492.460 q^{5} +594.752 q^{6} -1209.81 q^{7} +5049.35 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+22.0278 q^{2} +27.0000 q^{3} +357.226 q^{4} -492.460 q^{5} +594.752 q^{6} -1209.81 q^{7} +5049.35 q^{8} +729.000 q^{9} -10847.8 q^{10} -7759.91 q^{11} +9645.10 q^{12} +1927.14 q^{13} -26649.5 q^{14} -13296.4 q^{15} +65501.4 q^{16} -21871.8 q^{17} +16058.3 q^{18} -31682.0 q^{19} -175919. q^{20} -32664.9 q^{21} -170934. q^{22} +76017.8 q^{23} +136332. q^{24} +164392. q^{25} +42450.8 q^{26} +19683.0 q^{27} -432176. q^{28} -107474. q^{29} -292891. q^{30} -56872.6 q^{31} +796537. q^{32} -209518. q^{33} -481789. q^{34} +595783. q^{35} +260418. q^{36} -72532.0 q^{37} -697885. q^{38} +52032.8 q^{39} -2.48660e6 q^{40} -598173. q^{41} -719537. q^{42} -127161. q^{43} -2.77204e6 q^{44} -359003. q^{45} +1.67451e6 q^{46} +20753.9 q^{47} +1.76854e6 q^{48} +640100. q^{49} +3.62120e6 q^{50} -590540. q^{51} +688425. q^{52} +1.52906e6 q^{53} +433574. q^{54} +3.82145e6 q^{55} -6.10876e6 q^{56} -855413. q^{57} -2.36741e6 q^{58} -205379. q^{59} -4.74982e6 q^{60} +2.40366e6 q^{61} -1.25278e6 q^{62} -881952. q^{63} +9.16182e6 q^{64} -949040. q^{65} -4.61522e6 q^{66} -606916. q^{67} -7.81319e6 q^{68} +2.05248e6 q^{69} +1.31238e7 q^{70} -3.52312e6 q^{71} +3.68098e6 q^{72} +1.27495e6 q^{73} -1.59772e6 q^{74} +4.43858e6 q^{75} -1.13176e7 q^{76} +9.38803e6 q^{77} +1.14617e6 q^{78} +3.89830e6 q^{79} -3.22568e7 q^{80} +531441. q^{81} -1.31765e7 q^{82} -4.03800e6 q^{83} -1.16687e7 q^{84} +1.07710e7 q^{85} -2.80108e6 q^{86} -2.90179e6 q^{87} -3.91825e7 q^{88} +5.97046e6 q^{89} -7.90807e6 q^{90} -2.33148e6 q^{91} +2.71555e7 q^{92} -1.53556e6 q^{93} +457163. q^{94} +1.56021e7 q^{95} +2.15065e7 q^{96} -1.98127e6 q^{97} +1.41000e7 q^{98} -5.65698e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.0278 1.94700 0.973502 0.228677i \(-0.0734401\pi\)
0.973502 + 0.228677i \(0.0734401\pi\)
\(3\) 27.0000 0.577350
\(4\) 357.226 2.79083
\(5\) −492.460 −1.76188 −0.880939 0.473230i \(-0.843088\pi\)
−0.880939 + 0.473230i \(0.843088\pi\)
\(6\) 594.752 1.12410
\(7\) −1209.81 −1.33314 −0.666568 0.745444i \(-0.732237\pi\)
−0.666568 + 0.745444i \(0.732237\pi\)
\(8\) 5049.35 3.48675
\(9\) 729.000 0.333333
\(10\) −10847.8 −3.43038
\(11\) −7759.91 −1.75785 −0.878926 0.476958i \(-0.841739\pi\)
−0.878926 + 0.476958i \(0.841739\pi\)
\(12\) 9645.10 1.61128
\(13\) 1927.14 0.243283 0.121642 0.992574i \(-0.461184\pi\)
0.121642 + 0.992574i \(0.461184\pi\)
\(14\) −26649.5 −2.59562
\(15\) −13296.4 −1.01722
\(16\) 65501.4 3.99789
\(17\) −21871.8 −1.07973 −0.539864 0.841752i \(-0.681524\pi\)
−0.539864 + 0.841752i \(0.681524\pi\)
\(18\) 16058.3 0.649001
\(19\) −31682.0 −1.05968 −0.529839 0.848098i \(-0.677748\pi\)
−0.529839 + 0.848098i \(0.677748\pi\)
\(20\) −175919. −4.91710
\(21\) −32664.9 −0.769686
\(22\) −170934. −3.42255
\(23\) 76017.8 1.30277 0.651385 0.758747i \(-0.274188\pi\)
0.651385 + 0.758747i \(0.274188\pi\)
\(24\) 136332. 2.01307
\(25\) 164392. 2.10421
\(26\) 42450.8 0.473673
\(27\) 19683.0 0.192450
\(28\) −432176. −3.72055
\(29\) −107474. −0.818293 −0.409146 0.912469i \(-0.634173\pi\)
−0.409146 + 0.912469i \(0.634173\pi\)
\(30\) −292891. −1.98053
\(31\) −56872.6 −0.342876 −0.171438 0.985195i \(-0.554841\pi\)
−0.171438 + 0.985195i \(0.554841\pi\)
\(32\) 796537. 4.29715
\(33\) −209518. −1.01490
\(34\) −481789. −2.10223
\(35\) 595783. 2.34882
\(36\) 260418. 0.930276
\(37\) −72532.0 −0.235409 −0.117705 0.993049i \(-0.537554\pi\)
−0.117705 + 0.993049i \(0.537554\pi\)
\(38\) −697885. −2.06320
\(39\) 52032.8 0.140460
\(40\) −2.48660e6 −6.14322
\(41\) −598173. −1.35545 −0.677725 0.735316i \(-0.737034\pi\)
−0.677725 + 0.735316i \(0.737034\pi\)
\(42\) −719537. −1.49858
\(43\) −127161. −0.243902 −0.121951 0.992536i \(-0.538915\pi\)
−0.121951 + 0.992536i \(0.538915\pi\)
\(44\) −2.77204e6 −4.90586
\(45\) −359003. −0.587293
\(46\) 1.67451e6 2.53650
\(47\) 20753.9 0.0291579 0.0145790 0.999894i \(-0.495359\pi\)
0.0145790 + 0.999894i \(0.495359\pi\)
\(48\) 1.76854e6 2.30818
\(49\) 640100. 0.777252
\(50\) 3.62120e6 4.09691
\(51\) −590540. −0.623381
\(52\) 688425. 0.678961
\(53\) 1.52906e6 1.41078 0.705389 0.708821i \(-0.250772\pi\)
0.705389 + 0.708821i \(0.250772\pi\)
\(54\) 433574. 0.374701
\(55\) 3.82145e6 3.09712
\(56\) −6.10876e6 −4.64831
\(57\) −855413. −0.611806
\(58\) −2.36741e6 −1.59322
\(59\) −205379. −0.130189
\(60\) −4.74982e6 −2.83889
\(61\) 2.40366e6 1.35587 0.677935 0.735122i \(-0.262875\pi\)
0.677935 + 0.735122i \(0.262875\pi\)
\(62\) −1.25278e6 −0.667582
\(63\) −881952. −0.444379
\(64\) 9.16182e6 4.36869
\(65\) −949040. −0.428635
\(66\) −4.61522e6 −1.97601
\(67\) −606916. −0.246528 −0.123264 0.992374i \(-0.539336\pi\)
−0.123264 + 0.992374i \(0.539336\pi\)
\(68\) −7.81319e6 −3.01333
\(69\) 2.05248e6 0.752155
\(70\) 1.31238e7 4.57317
\(71\) −3.52312e6 −1.16822 −0.584108 0.811676i \(-0.698556\pi\)
−0.584108 + 0.811676i \(0.698556\pi\)
\(72\) 3.68098e6 1.16225
\(73\) 1.27495e6 0.383585 0.191793 0.981435i \(-0.438570\pi\)
0.191793 + 0.981435i \(0.438570\pi\)
\(74\) −1.59772e6 −0.458343
\(75\) 4.43858e6 1.21487
\(76\) −1.13176e7 −2.95738
\(77\) 9.38803e6 2.34346
\(78\) 1.14617e6 0.273475
\(79\) 3.89830e6 0.889571 0.444786 0.895637i \(-0.353280\pi\)
0.444786 + 0.895637i \(0.353280\pi\)
\(80\) −3.22568e7 −7.04379
\(81\) 531441. 0.111111
\(82\) −1.31765e7 −2.63907
\(83\) −4.03800e6 −0.775164 −0.387582 0.921835i \(-0.626690\pi\)
−0.387582 + 0.921835i \(0.626690\pi\)
\(84\) −1.16687e7 −2.14806
\(85\) 1.07710e7 1.90235
\(86\) −2.80108e6 −0.474877
\(87\) −2.90179e6 −0.472442
\(88\) −3.91825e7 −6.12919
\(89\) 5.97046e6 0.897724 0.448862 0.893601i \(-0.351830\pi\)
0.448862 + 0.893601i \(0.351830\pi\)
\(90\) −7.90807e6 −1.14346
\(91\) −2.33148e6 −0.324330
\(92\) 2.71555e7 3.63581
\(93\) −1.53556e6 −0.197960
\(94\) 457163. 0.0567706
\(95\) 1.56021e7 1.86703
\(96\) 2.15065e7 2.48096
\(97\) −1.98127e6 −0.220415 −0.110208 0.993909i \(-0.535152\pi\)
−0.110208 + 0.993909i \(0.535152\pi\)
\(98\) 1.41000e7 1.51331
\(99\) −5.65698e6 −0.585951
\(100\) 5.87250e7 5.87250
\(101\) −1.54871e7 −1.49570 −0.747851 0.663867i \(-0.768914\pi\)
−0.747851 + 0.663867i \(0.768914\pi\)
\(102\) −1.30083e7 −1.21373
\(103\) −1.20275e7 −1.08454 −0.542269 0.840205i \(-0.682435\pi\)
−0.542269 + 0.840205i \(0.682435\pi\)
\(104\) 9.73081e6 0.848267
\(105\) 1.60862e7 1.35609
\(106\) 3.36819e7 2.74679
\(107\) −7.69022e6 −0.606870 −0.303435 0.952852i \(-0.598134\pi\)
−0.303435 + 0.952852i \(0.598134\pi\)
\(108\) 7.03128e6 0.537095
\(109\) −2.11383e6 −0.156342 −0.0781711 0.996940i \(-0.524908\pi\)
−0.0781711 + 0.996940i \(0.524908\pi\)
\(110\) 8.41782e7 6.03011
\(111\) −1.95836e6 −0.135914
\(112\) −7.92443e7 −5.32973
\(113\) 2.03050e6 0.132382 0.0661909 0.997807i \(-0.478915\pi\)
0.0661909 + 0.997807i \(0.478915\pi\)
\(114\) −1.88429e7 −1.19119
\(115\) −3.74357e7 −2.29532
\(116\) −3.83923e7 −2.28371
\(117\) 1.40489e6 0.0810944
\(118\) −4.52406e6 −0.253478
\(119\) 2.64608e7 1.43942
\(120\) −6.71383e7 −3.54679
\(121\) 4.07291e7 2.09005
\(122\) 5.29474e7 2.63989
\(123\) −1.61507e7 −0.782569
\(124\) −2.03164e7 −0.956908
\(125\) −4.24829e7 −1.94549
\(126\) −1.94275e7 −0.865207
\(127\) 1.37874e7 0.597269 0.298635 0.954367i \(-0.403469\pi\)
0.298635 + 0.954367i \(0.403469\pi\)
\(128\) 9.98583e7 4.20871
\(129\) −3.43335e6 −0.140817
\(130\) −2.09053e7 −0.834555
\(131\) −845013. −0.0328408 −0.0164204 0.999865i \(-0.505227\pi\)
−0.0164204 + 0.999865i \(0.505227\pi\)
\(132\) −7.48451e7 −2.83240
\(133\) 3.83292e7 1.41270
\(134\) −1.33690e7 −0.479992
\(135\) −9.69309e6 −0.339074
\(136\) −1.10439e8 −3.76474
\(137\) −4.84595e7 −1.61012 −0.805058 0.593196i \(-0.797866\pi\)
−0.805058 + 0.593196i \(0.797866\pi\)
\(138\) 4.52117e7 1.46445
\(139\) −2.05210e6 −0.0648106 −0.0324053 0.999475i \(-0.510317\pi\)
−0.0324053 + 0.999475i \(0.510317\pi\)
\(140\) 2.12829e8 6.55516
\(141\) 560354. 0.0168343
\(142\) −7.76067e7 −2.27452
\(143\) −1.49545e7 −0.427656
\(144\) 4.77505e7 1.33263
\(145\) 5.29264e7 1.44173
\(146\) 2.80843e7 0.746842
\(147\) 1.72827e7 0.448746
\(148\) −2.59103e7 −0.656987
\(149\) −2.53433e6 −0.0627641 −0.0313821 0.999507i \(-0.509991\pi\)
−0.0313821 + 0.999507i \(0.509991\pi\)
\(150\) 9.77723e7 2.36535
\(151\) −5.49477e7 −1.29876 −0.649382 0.760462i \(-0.724972\pi\)
−0.649382 + 0.760462i \(0.724972\pi\)
\(152\) −1.59973e8 −3.69483
\(153\) −1.59446e7 −0.359909
\(154\) 2.06798e8 4.56272
\(155\) 2.80075e7 0.604106
\(156\) 1.85875e7 0.391998
\(157\) 7.27529e7 1.50038 0.750191 0.661221i \(-0.229962\pi\)
0.750191 + 0.661221i \(0.229962\pi\)
\(158\) 8.58712e7 1.73200
\(159\) 4.12846e7 0.814513
\(160\) −3.92263e8 −7.57106
\(161\) −9.19672e7 −1.73677
\(162\) 1.17065e7 0.216334
\(163\) 1.99055e6 0.0360011 0.0180006 0.999838i \(-0.494270\pi\)
0.0180006 + 0.999838i \(0.494270\pi\)
\(164\) −2.13683e8 −3.78283
\(165\) 1.03179e8 1.78812
\(166\) −8.89485e7 −1.50925
\(167\) −3.49476e7 −0.580643 −0.290322 0.956929i \(-0.593762\pi\)
−0.290322 + 0.956929i \(0.593762\pi\)
\(168\) −1.64937e8 −2.68370
\(169\) −5.90346e7 −0.940813
\(170\) 2.37262e8 3.70388
\(171\) −2.30961e7 −0.353226
\(172\) −4.54252e7 −0.680687
\(173\) −2.91994e7 −0.428759 −0.214380 0.976750i \(-0.568773\pi\)
−0.214380 + 0.976750i \(0.568773\pi\)
\(174\) −6.39201e7 −0.919846
\(175\) −1.98883e8 −2.80520
\(176\) −5.08285e8 −7.02769
\(177\) −5.54523e6 −0.0751646
\(178\) 1.31516e8 1.74787
\(179\) 1.31626e8 1.71536 0.857680 0.514184i \(-0.171905\pi\)
0.857680 + 0.514184i \(0.171905\pi\)
\(180\) −1.28245e8 −1.63903
\(181\) −1.51957e8 −1.90479 −0.952393 0.304872i \(-0.901386\pi\)
−0.952393 + 0.304872i \(0.901386\pi\)
\(182\) −5.13574e7 −0.631471
\(183\) 6.48987e7 0.782812
\(184\) 3.83840e8 4.54243
\(185\) 3.57191e7 0.414763
\(186\) −3.38251e7 −0.385428
\(187\) 1.69724e8 1.89800
\(188\) 7.41382e6 0.0813747
\(189\) −2.38127e7 −0.256562
\(190\) 3.43680e8 3.63511
\(191\) −1.21713e8 −1.26392 −0.631962 0.775000i \(-0.717750\pi\)
−0.631962 + 0.775000i \(0.717750\pi\)
\(192\) 2.47369e8 2.52227
\(193\) 1.32612e8 1.32780 0.663900 0.747821i \(-0.268900\pi\)
0.663900 + 0.747821i \(0.268900\pi\)
\(194\) −4.36430e7 −0.429149
\(195\) −2.56241e7 −0.247473
\(196\) 2.28660e8 2.16917
\(197\) 1.11991e8 1.04364 0.521819 0.853056i \(-0.325254\pi\)
0.521819 + 0.853056i \(0.325254\pi\)
\(198\) −1.24611e8 −1.14085
\(199\) −1.81688e8 −1.63433 −0.817165 0.576403i \(-0.804456\pi\)
−0.817165 + 0.576403i \(0.804456\pi\)
\(200\) 8.30071e8 7.33686
\(201\) −1.63867e7 −0.142333
\(202\) −3.41147e8 −2.91214
\(203\) 1.30023e8 1.09090
\(204\) −2.10956e8 −1.73975
\(205\) 2.94576e8 2.38814
\(206\) −2.64940e8 −2.11160
\(207\) 5.54170e7 0.434257
\(208\) 1.26230e8 0.972618
\(209\) 2.45849e8 1.86276
\(210\) 3.54343e8 2.64032
\(211\) −3.34639e6 −0.0245238 −0.0122619 0.999925i \(-0.503903\pi\)
−0.0122619 + 0.999925i \(0.503903\pi\)
\(212\) 5.46219e8 3.93724
\(213\) −9.51243e7 −0.674470
\(214\) −1.69399e8 −1.18158
\(215\) 6.26217e7 0.429725
\(216\) 9.93863e7 0.671025
\(217\) 6.88051e7 0.457101
\(218\) −4.65630e7 −0.304399
\(219\) 3.44236e7 0.221463
\(220\) 1.36512e9 8.64353
\(221\) −4.21501e7 −0.262679
\(222\) −4.31385e7 −0.264624
\(223\) 3.23603e7 0.195410 0.0977048 0.995215i \(-0.468850\pi\)
0.0977048 + 0.995215i \(0.468850\pi\)
\(224\) −9.63659e8 −5.72869
\(225\) 1.19842e8 0.701405
\(226\) 4.47275e7 0.257748
\(227\) 1.60575e8 0.911145 0.455573 0.890199i \(-0.349435\pi\)
0.455573 + 0.890199i \(0.349435\pi\)
\(228\) −3.05575e8 −1.70744
\(229\) 4.34106e7 0.238876 0.119438 0.992842i \(-0.461891\pi\)
0.119438 + 0.992842i \(0.461891\pi\)
\(230\) −8.24628e8 −4.46900
\(231\) 2.53477e8 1.35300
\(232\) −5.42672e8 −2.85318
\(233\) −2.86270e7 −0.148262 −0.0741310 0.997249i \(-0.523618\pi\)
−0.0741310 + 0.997249i \(0.523618\pi\)
\(234\) 3.09466e7 0.157891
\(235\) −1.02204e7 −0.0513727
\(236\) −7.33667e7 −0.363335
\(237\) 1.05254e8 0.513594
\(238\) 5.82874e8 2.80256
\(239\) −6.06584e7 −0.287408 −0.143704 0.989621i \(-0.545901\pi\)
−0.143704 + 0.989621i \(0.545901\pi\)
\(240\) −8.70933e8 −4.06673
\(241\) 5.03046e7 0.231498 0.115749 0.993278i \(-0.463073\pi\)
0.115749 + 0.993278i \(0.463073\pi\)
\(242\) 8.97173e8 4.06933
\(243\) 1.43489e7 0.0641500
\(244\) 8.58648e8 3.78400
\(245\) −3.15224e8 −1.36942
\(246\) −3.55765e8 −1.52367
\(247\) −6.10556e7 −0.257802
\(248\) −2.87170e8 −1.19552
\(249\) −1.09026e8 −0.447541
\(250\) −9.35807e8 −3.78788
\(251\) 1.79753e8 0.717494 0.358747 0.933435i \(-0.383204\pi\)
0.358747 + 0.933435i \(0.383204\pi\)
\(252\) −3.15056e8 −1.24018
\(253\) −5.89891e8 −2.29008
\(254\) 3.03707e8 1.16289
\(255\) 2.90817e8 1.09832
\(256\) 1.02695e9 3.82569
\(257\) 8.52960e7 0.313446 0.156723 0.987643i \(-0.449907\pi\)
0.156723 + 0.987643i \(0.449907\pi\)
\(258\) −7.56293e7 −0.274171
\(259\) 8.77500e7 0.313833
\(260\) −3.39022e8 −1.19625
\(261\) −7.83483e7 −0.272764
\(262\) −1.86138e7 −0.0639412
\(263\) 4.14430e8 1.40477 0.702386 0.711796i \(-0.252118\pi\)
0.702386 + 0.711796i \(0.252118\pi\)
\(264\) −1.05793e9 −3.53869
\(265\) −7.53000e8 −2.48562
\(266\) 8.44309e8 2.75053
\(267\) 1.61202e8 0.518301
\(268\) −2.16806e8 −0.688018
\(269\) −3.07821e8 −0.964196 −0.482098 0.876117i \(-0.660125\pi\)
−0.482098 + 0.876117i \(0.660125\pi\)
\(270\) −2.13518e8 −0.660178
\(271\) −2.35783e8 −0.719646 −0.359823 0.933021i \(-0.617163\pi\)
−0.359823 + 0.933021i \(0.617163\pi\)
\(272\) −1.43264e9 −4.31663
\(273\) −6.29499e7 −0.187252
\(274\) −1.06746e9 −3.13490
\(275\) −1.27567e9 −3.69890
\(276\) 7.33199e8 2.09913
\(277\) −2.15483e8 −0.609165 −0.304582 0.952486i \(-0.598517\pi\)
−0.304582 + 0.952486i \(0.598517\pi\)
\(278\) −4.52032e7 −0.126186
\(279\) −4.14601e7 −0.114292
\(280\) 3.00832e9 8.18975
\(281\) −2.06356e8 −0.554810 −0.277405 0.960753i \(-0.589474\pi\)
−0.277405 + 0.960753i \(0.589474\pi\)
\(282\) 1.23434e7 0.0327765
\(283\) 6.73678e8 1.76685 0.883426 0.468571i \(-0.155231\pi\)
0.883426 + 0.468571i \(0.155231\pi\)
\(284\) −1.25855e9 −3.26029
\(285\) 4.21256e8 1.07793
\(286\) −3.29414e8 −0.832648
\(287\) 7.23677e8 1.80700
\(288\) 5.80675e8 1.43238
\(289\) 6.80388e7 0.165811
\(290\) 1.16586e9 2.80706
\(291\) −5.34942e7 −0.127257
\(292\) 4.55444e8 1.07052
\(293\) −2.68265e8 −0.623056 −0.311528 0.950237i \(-0.600841\pi\)
−0.311528 + 0.950237i \(0.600841\pi\)
\(294\) 3.80701e8 0.873711
\(295\) 1.01141e8 0.229377
\(296\) −3.66239e8 −0.820813
\(297\) −1.52738e8 −0.338299
\(298\) −5.58259e7 −0.122202
\(299\) 1.46497e8 0.316942
\(300\) 1.58557e9 3.39049
\(301\) 1.53841e8 0.325154
\(302\) −1.21038e9 −2.52870
\(303\) −4.18152e8 −0.863544
\(304\) −2.07521e9 −4.23648
\(305\) −1.18370e9 −2.38888
\(306\) −3.51225e8 −0.700745
\(307\) 6.59253e8 1.30037 0.650186 0.759775i \(-0.274691\pi\)
0.650186 + 0.759775i \(0.274691\pi\)
\(308\) 3.35365e9 6.54018
\(309\) −3.24742e8 −0.626158
\(310\) 6.16945e8 1.17620
\(311\) −8.73613e8 −1.64686 −0.823432 0.567415i \(-0.807944\pi\)
−0.823432 + 0.567415i \(0.807944\pi\)
\(312\) 2.62732e8 0.489747
\(313\) 2.84022e8 0.523537 0.261768 0.965131i \(-0.415694\pi\)
0.261768 + 0.965131i \(0.415694\pi\)
\(314\) 1.60259e9 2.92125
\(315\) 4.34326e8 0.782941
\(316\) 1.39257e9 2.48264
\(317\) −2.16935e8 −0.382491 −0.191246 0.981542i \(-0.561253\pi\)
−0.191246 + 0.981542i \(0.561253\pi\)
\(318\) 9.09410e8 1.58586
\(319\) 8.33986e8 1.43844
\(320\) −4.51183e9 −7.69711
\(321\) −2.07636e8 −0.350377
\(322\) −2.02584e9 −3.38150
\(323\) 6.92943e8 1.14416
\(324\) 1.89844e8 0.310092
\(325\) 3.16806e8 0.511920
\(326\) 4.38475e7 0.0700943
\(327\) −5.70733e7 −0.0902643
\(328\) −3.02039e9 −4.72611
\(329\) −2.51083e7 −0.0388715
\(330\) 2.27281e9 3.48149
\(331\) −9.98498e8 −1.51338 −0.756692 0.653772i \(-0.773186\pi\)
−0.756692 + 0.653772i \(0.773186\pi\)
\(332\) −1.44248e9 −2.16335
\(333\) −5.28758e7 −0.0784698
\(334\) −7.69820e8 −1.13052
\(335\) 2.98882e8 0.434353
\(336\) −2.13960e9 −3.07712
\(337\) −1.17882e9 −1.67780 −0.838902 0.544282i \(-0.816802\pi\)
−0.838902 + 0.544282i \(0.816802\pi\)
\(338\) −1.30041e9 −1.83177
\(339\) 5.48234e7 0.0764306
\(340\) 3.84768e9 5.30912
\(341\) 4.41327e8 0.602726
\(342\) −5.08758e8 −0.687733
\(343\) 2.21931e8 0.296954
\(344\) −6.42081e8 −0.850423
\(345\) −1.01076e9 −1.32521
\(346\) −6.43201e8 −0.834796
\(347\) −8.46545e8 −1.08767 −0.543835 0.839192i \(-0.683028\pi\)
−0.543835 + 0.839192i \(0.683028\pi\)
\(348\) −1.03659e9 −1.31850
\(349\) 6.62378e8 0.834098 0.417049 0.908884i \(-0.363064\pi\)
0.417049 + 0.908884i \(0.363064\pi\)
\(350\) −4.38096e9 −5.46174
\(351\) 3.79319e7 0.0468199
\(352\) −6.18106e9 −7.55376
\(353\) 8.83004e8 1.06844 0.534221 0.845345i \(-0.320605\pi\)
0.534221 + 0.845345i \(0.320605\pi\)
\(354\) −1.22150e8 −0.146346
\(355\) 1.73500e9 2.05825
\(356\) 2.13280e9 2.50539
\(357\) 7.14442e8 0.831052
\(358\) 2.89943e9 3.33981
\(359\) 7.02849e7 0.0801736 0.0400868 0.999196i \(-0.487237\pi\)
0.0400868 + 0.999196i \(0.487237\pi\)
\(360\) −1.81273e9 −2.04774
\(361\) 1.09874e8 0.122919
\(362\) −3.34729e9 −3.70863
\(363\) 1.09968e9 1.20669
\(364\) −8.32864e8 −0.905147
\(365\) −6.27860e8 −0.675830
\(366\) 1.42958e9 1.52414
\(367\) −1.46843e9 −1.55068 −0.775340 0.631544i \(-0.782421\pi\)
−0.775340 + 0.631544i \(0.782421\pi\)
\(368\) 4.97927e9 5.20833
\(369\) −4.36068e8 −0.451817
\(370\) 7.86815e8 0.807544
\(371\) −1.84987e9 −1.88076
\(372\) −5.48542e8 −0.552471
\(373\) 1.19545e9 1.19275 0.596376 0.802705i \(-0.296607\pi\)
0.596376 + 0.802705i \(0.296607\pi\)
\(374\) 3.73864e9 3.69542
\(375\) −1.14704e9 −1.12323
\(376\) 1.04794e8 0.101666
\(377\) −2.07117e8 −0.199077
\(378\) −5.24543e8 −0.499528
\(379\) 1.21004e9 1.14173 0.570863 0.821045i \(-0.306609\pi\)
0.570863 + 0.821045i \(0.306609\pi\)
\(380\) 5.57347e9 5.21054
\(381\) 3.72261e8 0.344834
\(382\) −2.68108e9 −2.46086
\(383\) −1.02900e9 −0.935876 −0.467938 0.883761i \(-0.655003\pi\)
−0.467938 + 0.883761i \(0.655003\pi\)
\(384\) 2.69617e9 2.42990
\(385\) −4.62323e9 −4.12888
\(386\) 2.92116e9 2.58523
\(387\) −9.27004e7 −0.0813005
\(388\) −7.07759e8 −0.615141
\(389\) −2.22572e9 −1.91711 −0.958556 0.284904i \(-0.908038\pi\)
−0.958556 + 0.284904i \(0.908038\pi\)
\(390\) −5.64443e8 −0.481830
\(391\) −1.66265e9 −1.40664
\(392\) 3.23209e9 2.71008
\(393\) −2.28153e7 −0.0189606
\(394\) 2.46691e9 2.03197
\(395\) −1.91976e9 −1.56732
\(396\) −2.02082e9 −1.63529
\(397\) 9.68593e8 0.776917 0.388458 0.921466i \(-0.373008\pi\)
0.388458 + 0.921466i \(0.373008\pi\)
\(398\) −4.00219e9 −3.18205
\(399\) 1.03489e9 0.815620
\(400\) 1.07679e10 8.41241
\(401\) 9.25332e6 0.00716626 0.00358313 0.999994i \(-0.498859\pi\)
0.00358313 + 0.999994i \(0.498859\pi\)
\(402\) −3.60964e8 −0.277123
\(403\) −1.09602e8 −0.0834160
\(404\) −5.53239e9 −4.17425
\(405\) −2.61713e8 −0.195764
\(406\) 2.86412e9 2.12398
\(407\) 5.62842e8 0.413815
\(408\) −2.98184e9 −2.17357
\(409\) 4.22397e8 0.305274 0.152637 0.988282i \(-0.451224\pi\)
0.152637 + 0.988282i \(0.451224\pi\)
\(410\) 6.48888e9 4.64971
\(411\) −1.30841e9 −0.929601
\(412\) −4.29653e9 −3.02676
\(413\) 2.48470e8 0.173560
\(414\) 1.22072e9 0.845500
\(415\) 1.98856e9 1.36574
\(416\) 1.53504e9 1.04543
\(417\) −5.54066e7 −0.0374184
\(418\) 5.41553e9 3.62680
\(419\) −1.30988e9 −0.869925 −0.434962 0.900449i \(-0.643238\pi\)
−0.434962 + 0.900449i \(0.643238\pi\)
\(420\) 5.74639e9 3.78462
\(421\) 2.15584e9 1.40809 0.704043 0.710158i \(-0.251376\pi\)
0.704043 + 0.710158i \(0.251376\pi\)
\(422\) −7.37137e7 −0.0477479
\(423\) 1.51296e7 0.00971931
\(424\) 7.72075e9 4.91902
\(425\) −3.59555e9 −2.27198
\(426\) −2.09538e9 −1.31320
\(427\) −2.90797e9 −1.80756
\(428\) −2.74715e9 −1.69367
\(429\) −4.03770e8 −0.246907
\(430\) 1.37942e9 0.836676
\(431\) 8.50792e8 0.511862 0.255931 0.966695i \(-0.417618\pi\)
0.255931 + 0.966695i \(0.417618\pi\)
\(432\) 1.28926e9 0.769394
\(433\) −1.01714e9 −0.602109 −0.301054 0.953607i \(-0.597339\pi\)
−0.301054 + 0.953607i \(0.597339\pi\)
\(434\) 1.51563e9 0.889977
\(435\) 1.42901e9 0.832385
\(436\) −7.55113e8 −0.436324
\(437\) −2.40839e9 −1.38052
\(438\) 7.58277e8 0.431190
\(439\) −1.42920e9 −0.806245 −0.403123 0.915146i \(-0.632075\pi\)
−0.403123 + 0.915146i \(0.632075\pi\)
\(440\) 1.92958e10 10.7989
\(441\) 4.66633e8 0.259084
\(442\) −9.28477e8 −0.511438
\(443\) 2.71763e8 0.148517 0.0742586 0.997239i \(-0.476341\pi\)
0.0742586 + 0.997239i \(0.476341\pi\)
\(444\) −6.99578e8 −0.379311
\(445\) −2.94021e9 −1.58168
\(446\) 7.12828e8 0.380463
\(447\) −6.84270e7 −0.0362369
\(448\) −1.10841e10 −5.82406
\(449\) −1.24125e9 −0.647138 −0.323569 0.946205i \(-0.604883\pi\)
−0.323569 + 0.946205i \(0.604883\pi\)
\(450\) 2.63985e9 1.36564
\(451\) 4.64177e9 2.38268
\(452\) 7.25346e8 0.369454
\(453\) −1.48359e9 −0.749842
\(454\) 3.53712e9 1.77400
\(455\) 1.14816e9 0.571429
\(456\) −4.31928e9 −2.13321
\(457\) −3.37198e9 −1.65264 −0.826320 0.563201i \(-0.809570\pi\)
−0.826320 + 0.563201i \(0.809570\pi\)
\(458\) 9.56242e8 0.465092
\(459\) −4.30503e8 −0.207794
\(460\) −1.33730e10 −6.40585
\(461\) −2.86356e9 −1.36130 −0.680650 0.732609i \(-0.738302\pi\)
−0.680650 + 0.732609i \(0.738302\pi\)
\(462\) 5.58355e9 2.63429
\(463\) 4.00910e8 0.187721 0.0938607 0.995585i \(-0.470079\pi\)
0.0938607 + 0.995585i \(0.470079\pi\)
\(464\) −7.03967e9 −3.27144
\(465\) 7.56202e8 0.348781
\(466\) −6.30591e8 −0.288667
\(467\) −8.05487e8 −0.365973 −0.182987 0.983115i \(-0.558577\pi\)
−0.182987 + 0.983115i \(0.558577\pi\)
\(468\) 5.01862e8 0.226320
\(469\) 7.34254e8 0.328656
\(470\) −2.25134e8 −0.100023
\(471\) 1.96433e9 0.866246
\(472\) −1.03703e9 −0.453936
\(473\) 9.86759e8 0.428743
\(474\) 2.31852e9 0.999970
\(475\) −5.20825e9 −2.22979
\(476\) 9.45248e9 4.01718
\(477\) 1.11468e9 0.470259
\(478\) −1.33617e9 −0.559584
\(479\) −1.45641e9 −0.605495 −0.302747 0.953071i \(-0.597904\pi\)
−0.302747 + 0.953071i \(0.597904\pi\)
\(480\) −1.05911e10 −4.37116
\(481\) −1.39779e8 −0.0572711
\(482\) 1.10810e9 0.450728
\(483\) −2.48311e9 −1.00272
\(484\) 1.45495e10 5.83295
\(485\) 9.75694e8 0.388345
\(486\) 3.16075e8 0.124900
\(487\) 1.91160e9 0.749973 0.374986 0.927030i \(-0.377647\pi\)
0.374986 + 0.927030i \(0.377647\pi\)
\(488\) 1.21369e10 4.72758
\(489\) 5.37448e7 0.0207853
\(490\) −6.94370e9 −2.66627
\(491\) 3.11951e8 0.118933 0.0594663 0.998230i \(-0.481060\pi\)
0.0594663 + 0.998230i \(0.481060\pi\)
\(492\) −5.76944e9 −2.18402
\(493\) 2.35065e9 0.883533
\(494\) −1.34492e9 −0.501942
\(495\) 2.78583e9 1.03237
\(496\) −3.72524e9 −1.37078
\(497\) 4.26231e9 1.55739
\(498\) −2.40161e9 −0.871364
\(499\) −1.10424e9 −0.397844 −0.198922 0.980015i \(-0.563744\pi\)
−0.198922 + 0.980015i \(0.563744\pi\)
\(500\) −1.51760e10 −5.42953
\(501\) −9.43585e8 −0.335235
\(502\) 3.95957e9 1.39696
\(503\) −3.17067e9 −1.11087 −0.555434 0.831560i \(-0.687448\pi\)
−0.555434 + 0.831560i \(0.687448\pi\)
\(504\) −4.45329e9 −1.54944
\(505\) 7.62677e9 2.63524
\(506\) −1.29940e10 −4.45879
\(507\) −1.59394e9 −0.543179
\(508\) 4.92523e9 1.66688
\(509\) 1.80020e9 0.605074 0.302537 0.953138i \(-0.402166\pi\)
0.302537 + 0.953138i \(0.402166\pi\)
\(510\) 6.40607e9 2.13844
\(511\) −1.54244e9 −0.511371
\(512\) 9.83963e9 3.23992
\(513\) −6.23596e8 −0.203935
\(514\) 1.87889e9 0.610281
\(515\) 5.92306e9 1.91082
\(516\) −1.22648e9 −0.392995
\(517\) −1.61048e8 −0.0512553
\(518\) 1.93294e9 0.611034
\(519\) −7.88385e8 −0.247544
\(520\) −4.79204e9 −1.49454
\(521\) −1.90764e9 −0.590968 −0.295484 0.955348i \(-0.595481\pi\)
−0.295484 + 0.955348i \(0.595481\pi\)
\(522\) −1.72584e9 −0.531073
\(523\) 1.01205e9 0.309347 0.154673 0.987966i \(-0.450567\pi\)
0.154673 + 0.987966i \(0.450567\pi\)
\(524\) −3.01860e8 −0.0916530
\(525\) −5.36984e9 −1.61959
\(526\) 9.12900e9 2.73510
\(527\) 1.24391e9 0.370213
\(528\) −1.37237e10 −4.05744
\(529\) 2.37388e9 0.697211
\(530\) −1.65870e10 −4.83951
\(531\) −1.49721e8 −0.0433963
\(532\) 1.36922e10 3.94259
\(533\) −1.15276e9 −0.329758
\(534\) 3.55094e9 1.00913
\(535\) 3.78713e9 1.06923
\(536\) −3.06453e9 −0.859582
\(537\) 3.55389e9 0.990363
\(538\) −6.78063e9 −1.87729
\(539\) −4.96712e9 −1.36629
\(540\) −3.46262e9 −0.946296
\(541\) −5.50100e9 −1.49366 −0.746829 0.665016i \(-0.768425\pi\)
−0.746829 + 0.665016i \(0.768425\pi\)
\(542\) −5.19378e9 −1.40115
\(543\) −4.10285e9 −1.09973
\(544\) −1.74217e10 −4.63976
\(545\) 1.04097e9 0.275456
\(546\) −1.38665e9 −0.364580
\(547\) 4.40039e9 1.14957 0.574786 0.818304i \(-0.305085\pi\)
0.574786 + 0.818304i \(0.305085\pi\)
\(548\) −1.73110e10 −4.49355
\(549\) 1.75227e9 0.451957
\(550\) −2.81002e10 −7.20177
\(551\) 3.40497e9 0.867128
\(552\) 1.03637e10 2.62257
\(553\) −4.71621e9 −1.18592
\(554\) −4.74664e9 −1.18605
\(555\) 9.64416e8 0.239463
\(556\) −7.33062e8 −0.180875
\(557\) 5.20574e9 1.27641 0.638203 0.769868i \(-0.279678\pi\)
0.638203 + 0.769868i \(0.279678\pi\)
\(558\) −9.13278e8 −0.222527
\(559\) −2.45057e8 −0.0593371
\(560\) 3.90246e10 9.39033
\(561\) 4.58254e9 1.09581
\(562\) −4.54557e9 −1.08022
\(563\) 6.20253e9 1.46484 0.732419 0.680854i \(-0.238391\pi\)
0.732419 + 0.680854i \(0.238391\pi\)
\(564\) 2.00173e8 0.0469817
\(565\) −9.99939e8 −0.233240
\(566\) 1.48397e10 3.44007
\(567\) −6.42943e8 −0.148126
\(568\) −1.77895e10 −4.07328
\(569\) 6.72094e9 1.52946 0.764728 0.644353i \(-0.222873\pi\)
0.764728 + 0.644353i \(0.222873\pi\)
\(570\) 9.27937e9 2.09873
\(571\) 3.74441e8 0.0841700 0.0420850 0.999114i \(-0.486600\pi\)
0.0420850 + 0.999114i \(0.486600\pi\)
\(572\) −5.34212e9 −1.19351
\(573\) −3.28626e9 −0.729727
\(574\) 1.59410e10 3.51824
\(575\) 1.24967e10 2.74131
\(576\) 6.67896e9 1.45623
\(577\) 1.91890e9 0.415851 0.207925 0.978145i \(-0.433329\pi\)
0.207925 + 0.978145i \(0.433329\pi\)
\(578\) 1.49875e9 0.322835
\(579\) 3.58053e9 0.766606
\(580\) 1.89067e10 4.02362
\(581\) 4.88522e9 1.03340
\(582\) −1.17836e9 −0.247770
\(583\) −1.18654e10 −2.47994
\(584\) 6.43765e9 1.33746
\(585\) −6.91850e8 −0.142878
\(586\) −5.90929e9 −1.21309
\(587\) −9.08556e9 −1.85404 −0.927018 0.375016i \(-0.877637\pi\)
−0.927018 + 0.375016i \(0.877637\pi\)
\(588\) 6.17383e9 1.25237
\(589\) 1.80184e9 0.363339
\(590\) 2.22792e9 0.446598
\(591\) 3.02375e9 0.602545
\(592\) −4.75095e9 −0.941140
\(593\) −1.90930e9 −0.375997 −0.187998 0.982169i \(-0.560200\pi\)
−0.187998 + 0.982169i \(0.560200\pi\)
\(594\) −3.36450e9 −0.658669
\(595\) −1.30309e10 −2.53609
\(596\) −9.05329e8 −0.175164
\(597\) −4.90557e9 −0.943581
\(598\) 3.22701e9 0.617088
\(599\) −5.47139e9 −1.04017 −0.520084 0.854115i \(-0.674099\pi\)
−0.520084 + 0.854115i \(0.674099\pi\)
\(600\) 2.24119e10 4.23594
\(601\) 3.55169e9 0.667381 0.333691 0.942683i \(-0.391706\pi\)
0.333691 + 0.942683i \(0.391706\pi\)
\(602\) 3.38878e9 0.633076
\(603\) −4.42442e8 −0.0821761
\(604\) −1.96287e10 −3.62463
\(605\) −2.00574e10 −3.68240
\(606\) −9.21098e9 −1.68132
\(607\) 5.38370e9 0.977057 0.488529 0.872548i \(-0.337534\pi\)
0.488529 + 0.872548i \(0.337534\pi\)
\(608\) −2.52358e10 −4.55360
\(609\) 3.51061e9 0.629829
\(610\) −2.60745e10 −4.65116
\(611\) 3.99956e7 0.00709363
\(612\) −5.69581e9 −1.00444
\(613\) 5.24608e9 0.919863 0.459932 0.887954i \(-0.347874\pi\)
0.459932 + 0.887954i \(0.347874\pi\)
\(614\) 1.45219e10 2.53183
\(615\) 7.95356e9 1.37879
\(616\) 4.74034e10 8.17104
\(617\) 6.88172e9 1.17950 0.589751 0.807585i \(-0.299226\pi\)
0.589751 + 0.807585i \(0.299226\pi\)
\(618\) −7.15337e9 −1.21913
\(619\) 5.67281e9 0.961350 0.480675 0.876899i \(-0.340392\pi\)
0.480675 + 0.876899i \(0.340392\pi\)
\(620\) 1.00050e10 1.68596
\(621\) 1.49626e9 0.250718
\(622\) −1.92438e10 −3.20645
\(623\) −7.22313e9 −1.19679
\(624\) 3.40822e9 0.561541
\(625\) 8.07802e9 1.32350
\(626\) 6.25640e9 1.01933
\(627\) 6.63793e9 1.07546
\(628\) 2.59892e10 4.18730
\(629\) 1.58641e9 0.254178
\(630\) 9.56727e9 1.52439
\(631\) 1.88338e9 0.298425 0.149212 0.988805i \(-0.452326\pi\)
0.149212 + 0.988805i \(0.452326\pi\)
\(632\) 1.96839e10 3.10171
\(633\) −9.03524e7 −0.0141588
\(634\) −4.77860e9 −0.744713
\(635\) −6.78976e9 −1.05232
\(636\) 1.47479e10 2.27316
\(637\) 1.23356e9 0.189092
\(638\) 1.83709e10 2.80065
\(639\) −2.56836e9 −0.389405
\(640\) −4.91762e10 −7.41524
\(641\) 5.23503e9 0.785084 0.392542 0.919734i \(-0.371596\pi\)
0.392542 + 0.919734i \(0.371596\pi\)
\(642\) −4.57377e9 −0.682185
\(643\) −7.39762e8 −0.109737 −0.0548686 0.998494i \(-0.517474\pi\)
−0.0548686 + 0.998494i \(0.517474\pi\)
\(644\) −3.28530e10 −4.84702
\(645\) 1.69079e9 0.248102
\(646\) 1.52640e10 2.22769
\(647\) −7.49507e9 −1.08796 −0.543978 0.839100i \(-0.683082\pi\)
−0.543978 + 0.839100i \(0.683082\pi\)
\(648\) 2.68343e9 0.387416
\(649\) 1.59372e9 0.228853
\(650\) 6.97856e9 0.996710
\(651\) 1.85774e9 0.263907
\(652\) 7.11075e8 0.100473
\(653\) 9.12022e9 1.28177 0.640884 0.767638i \(-0.278568\pi\)
0.640884 + 0.767638i \(0.278568\pi\)
\(654\) −1.25720e9 −0.175745
\(655\) 4.16135e8 0.0578615
\(656\) −3.91812e10 −5.41893
\(657\) 9.29436e8 0.127862
\(658\) −5.53081e8 −0.0756829
\(659\) 8.11531e9 1.10460 0.552301 0.833645i \(-0.313750\pi\)
0.552301 + 0.833645i \(0.313750\pi\)
\(660\) 3.68582e10 4.99034
\(661\) 6.72245e9 0.905362 0.452681 0.891672i \(-0.350468\pi\)
0.452681 + 0.891672i \(0.350468\pi\)
\(662\) −2.19947e10 −2.94656
\(663\) −1.13805e9 −0.151658
\(664\) −2.03893e10 −2.70280
\(665\) −1.88756e10 −2.48900
\(666\) −1.16474e9 −0.152781
\(667\) −8.16991e9 −1.06605
\(668\) −1.24842e10 −1.62047
\(669\) 8.73729e8 0.112820
\(670\) 6.58372e9 0.845687
\(671\) −1.86522e10 −2.38342
\(672\) −2.60188e10 −3.30746
\(673\) 1.14461e10 1.44745 0.723725 0.690089i \(-0.242429\pi\)
0.723725 + 0.690089i \(0.242429\pi\)
\(674\) −2.59668e10 −3.26669
\(675\) 3.23572e9 0.404956
\(676\) −2.10887e10 −2.62565
\(677\) −4.26750e9 −0.528583 −0.264292 0.964443i \(-0.585138\pi\)
−0.264292 + 0.964443i \(0.585138\pi\)
\(678\) 1.20764e9 0.148811
\(679\) 2.39696e9 0.293843
\(680\) 5.43866e10 6.63301
\(681\) 4.33553e9 0.526050
\(682\) 9.72147e9 1.17351
\(683\) 6.74829e9 0.810441 0.405220 0.914219i \(-0.367195\pi\)
0.405220 + 0.914219i \(0.367195\pi\)
\(684\) −8.25054e9 −0.985793
\(685\) 2.38644e10 2.83683
\(686\) 4.88867e9 0.578171
\(687\) 1.17209e9 0.137915
\(688\) −8.32922e9 −0.975091
\(689\) 2.94671e9 0.343218
\(690\) −2.22650e10 −2.58018
\(691\) −6.41648e9 −0.739816 −0.369908 0.929068i \(-0.620611\pi\)
−0.369908 + 0.929068i \(0.620611\pi\)
\(692\) −1.04308e10 −1.19659
\(693\) 6.84387e9 0.781152
\(694\) −1.86476e10 −2.11770
\(695\) 1.01057e9 0.114188
\(696\) −1.46521e10 −1.64728
\(697\) 1.30832e10 1.46352
\(698\) 1.45908e10 1.62399
\(699\) −7.72929e8 −0.0855991
\(700\) −7.10461e10 −7.82884
\(701\) −1.49165e10 −1.63552 −0.817758 0.575562i \(-0.804784\pi\)
−0.817758 + 0.575562i \(0.804784\pi\)
\(702\) 8.35559e8 0.0911585
\(703\) 2.29796e9 0.249458
\(704\) −7.10949e10 −7.67952
\(705\) −2.75952e8 −0.0296600
\(706\) 1.94507e10 2.08026
\(707\) 1.87365e10 1.99397
\(708\) −1.98090e9 −0.209771
\(709\) 5.36963e9 0.565826 0.282913 0.959146i \(-0.408699\pi\)
0.282913 + 0.959146i \(0.408699\pi\)
\(710\) 3.82182e10 4.00743
\(711\) 2.84186e9 0.296524
\(712\) 3.01469e10 3.13014
\(713\) −4.32333e9 −0.446689
\(714\) 1.57376e10 1.61806
\(715\) 7.36447e9 0.753477
\(716\) 4.70201e10 4.78727
\(717\) −1.63778e9 −0.165935
\(718\) 1.54822e9 0.156098
\(719\) −2.42754e9 −0.243565 −0.121783 0.992557i \(-0.538861\pi\)
−0.121783 + 0.992557i \(0.538861\pi\)
\(720\) −2.35152e10 −2.34793
\(721\) 1.45510e10 1.44584
\(722\) 2.42029e9 0.239325
\(723\) 1.35822e9 0.133656
\(724\) −5.42830e10 −5.31593
\(725\) −1.76678e10 −1.72186
\(726\) 2.42237e10 2.34943
\(727\) 1.20971e10 1.16764 0.583822 0.811882i \(-0.301557\pi\)
0.583822 + 0.811882i \(0.301557\pi\)
\(728\) −1.17724e10 −1.13086
\(729\) 3.87420e8 0.0370370
\(730\) −1.38304e10 −1.31584
\(731\) 2.78125e9 0.263347
\(732\) 2.31835e10 2.18469
\(733\) 3.52480e7 0.00330575 0.00165288 0.999999i \(-0.499474\pi\)
0.00165288 + 0.999999i \(0.499474\pi\)
\(734\) −3.23463e10 −3.01918
\(735\) −8.51104e9 −0.790636
\(736\) 6.05510e10 5.59821
\(737\) 4.70962e9 0.433360
\(738\) −9.60564e9 −0.879689
\(739\) 9.27198e9 0.845118 0.422559 0.906336i \(-0.361132\pi\)
0.422559 + 0.906336i \(0.361132\pi\)
\(740\) 1.27598e10 1.15753
\(741\) −1.64850e9 −0.148842
\(742\) −4.07487e10 −3.66184
\(743\) −1.63793e10 −1.46499 −0.732494 0.680773i \(-0.761644\pi\)
−0.732494 + 0.680773i \(0.761644\pi\)
\(744\) −7.75358e9 −0.690235
\(745\) 1.24806e9 0.110583
\(746\) 2.63331e10 2.32229
\(747\) −2.94371e9 −0.258388
\(748\) 6.06296e10 5.29699
\(749\) 9.30372e9 0.809040
\(750\) −2.52668e10 −2.18693
\(751\) 2.46248e9 0.212145 0.106073 0.994358i \(-0.466172\pi\)
0.106073 + 0.994358i \(0.466172\pi\)
\(752\) 1.35941e9 0.116570
\(753\) 4.85334e9 0.414246
\(754\) −4.56234e9 −0.387604
\(755\) 2.70596e10 2.28826
\(756\) −8.50652e9 −0.716020
\(757\) 7.88850e9 0.660935 0.330468 0.943817i \(-0.392794\pi\)
0.330468 + 0.943817i \(0.392794\pi\)
\(758\) 2.66545e10 2.22295
\(759\) −1.59271e10 −1.32218
\(760\) 7.87804e10 6.50984
\(761\) −2.28543e10 −1.87984 −0.939922 0.341390i \(-0.889102\pi\)
−0.939922 + 0.341390i \(0.889102\pi\)
\(762\) 8.20010e9 0.671393
\(763\) 2.55733e9 0.208426
\(764\) −4.34791e10 −3.52739
\(765\) 7.85206e9 0.634116
\(766\) −2.26666e10 −1.82216
\(767\) −3.95795e8 −0.0316728
\(768\) 2.77276e10 2.20876
\(769\) 3.75836e9 0.298028 0.149014 0.988835i \(-0.452390\pi\)
0.149014 + 0.988835i \(0.452390\pi\)
\(770\) −1.01840e11 −8.03896
\(771\) 2.30299e9 0.180968
\(772\) 4.73725e10 3.70566
\(773\) −2.31574e10 −1.80327 −0.901637 0.432494i \(-0.857634\pi\)
−0.901637 + 0.432494i \(0.857634\pi\)
\(774\) −2.04199e9 −0.158292
\(775\) −9.34939e9 −0.721485
\(776\) −1.00041e10 −0.768532
\(777\) 2.36925e9 0.181191
\(778\) −4.90279e10 −3.73263
\(779\) 1.89513e10 1.43634
\(780\) −9.15358e9 −0.690653
\(781\) 2.73391e10 2.05355
\(782\) −3.66246e10 −2.73873
\(783\) −2.11540e9 −0.157481
\(784\) 4.19274e10 3.10736
\(785\) −3.58279e10 −2.64349
\(786\) −5.02573e8 −0.0369165
\(787\) 7.24985e8 0.0530173 0.0265087 0.999649i \(-0.491561\pi\)
0.0265087 + 0.999649i \(0.491561\pi\)
\(788\) 4.00059e10 2.91261
\(789\) 1.11896e10 0.811046
\(790\) −4.22881e10 −3.05157
\(791\) −2.45652e9 −0.176483
\(792\) −2.85640e10 −2.04306
\(793\) 4.63219e9 0.329860
\(794\) 2.13360e10 1.51266
\(795\) −2.03310e10 −1.43507
\(796\) −6.49036e10 −4.56113
\(797\) 1.66646e10 1.16598 0.582989 0.812480i \(-0.301883\pi\)
0.582989 + 0.812480i \(0.301883\pi\)
\(798\) 2.27963e10 1.58802
\(799\) −4.53925e8 −0.0314826
\(800\) 1.30944e11 9.04213
\(801\) 4.35247e9 0.299241
\(802\) 2.03831e8 0.0139527
\(803\) −9.89347e9 −0.674286
\(804\) −5.85376e9 −0.397227
\(805\) 4.52901e10 3.05998
\(806\) −2.41429e9 −0.162411
\(807\) −8.31117e9 −0.556679
\(808\) −7.81997e10 −5.21514
\(809\) −1.50299e10 −0.998010 −0.499005 0.866599i \(-0.666301\pi\)
−0.499005 + 0.866599i \(0.666301\pi\)
\(810\) −5.76498e9 −0.381154
\(811\) −1.62482e10 −1.06963 −0.534813 0.844971i \(-0.679618\pi\)
−0.534813 + 0.844971i \(0.679618\pi\)
\(812\) 4.64475e10 3.04450
\(813\) −6.36613e9 −0.415488
\(814\) 1.23982e10 0.805699
\(815\) −9.80265e8 −0.0634296
\(816\) −3.86812e10 −2.49221
\(817\) 4.02871e9 0.258457
\(818\) 9.30449e9 0.594369
\(819\) −1.69965e9 −0.108110
\(820\) 1.05230e11 6.66488
\(821\) −2.31309e10 −1.45879 −0.729393 0.684095i \(-0.760197\pi\)
−0.729393 + 0.684095i \(0.760197\pi\)
\(822\) −2.88214e10 −1.80994
\(823\) −2.00237e10 −1.25211 −0.626057 0.779777i \(-0.715332\pi\)
−0.626057 + 0.779777i \(0.715332\pi\)
\(824\) −6.07310e10 −3.78151
\(825\) −3.44430e10 −2.13556
\(826\) 5.47325e9 0.337921
\(827\) −1.75153e10 −1.07683 −0.538417 0.842679i \(-0.680977\pi\)
−0.538417 + 0.842679i \(0.680977\pi\)
\(828\) 1.97964e10 1.21194
\(829\) 1.27473e10 0.777099 0.388550 0.921428i \(-0.372976\pi\)
0.388550 + 0.921428i \(0.372976\pi\)
\(830\) 4.38036e10 2.65911
\(831\) −5.81805e9 −0.351701
\(832\) 1.76561e10 1.06283
\(833\) −1.40002e10 −0.839220
\(834\) −1.22049e9 −0.0728538
\(835\) 1.72103e10 1.02302
\(836\) 8.78237e10 5.19864
\(837\) −1.11942e9 −0.0659866
\(838\) −2.88538e10 −1.69375
\(839\) 1.43905e10 0.841217 0.420609 0.907242i \(-0.361817\pi\)
0.420609 + 0.907242i \(0.361817\pi\)
\(840\) 8.12246e10 4.72836
\(841\) −5.69930e9 −0.330397
\(842\) 4.74885e10 2.74155
\(843\) −5.57160e9 −0.320320
\(844\) −1.19542e9 −0.0684416
\(845\) 2.90722e10 1.65760
\(846\) 3.33272e8 0.0189235
\(847\) −4.92745e10 −2.78631
\(848\) 1.00155e11 5.64013
\(849\) 1.81893e10 1.02009
\(850\) −7.92022e10 −4.42355
\(851\) −5.51372e9 −0.306684
\(852\) −3.39808e10 −1.88233
\(853\) −2.47811e10 −1.36710 −0.683549 0.729905i \(-0.739564\pi\)
−0.683549 + 0.729905i \(0.739564\pi\)
\(854\) −6.40563e10 −3.51933
\(855\) 1.13739e10 0.622342
\(856\) −3.88306e10 −2.11600
\(857\) 1.34333e10 0.729036 0.364518 0.931196i \(-0.381234\pi\)
0.364518 + 0.931196i \(0.381234\pi\)
\(858\) −8.89419e9 −0.480729
\(859\) 1.54319e10 0.830701 0.415350 0.909662i \(-0.363659\pi\)
0.415350 + 0.909662i \(0.363659\pi\)
\(860\) 2.23701e10 1.19929
\(861\) 1.95393e10 1.04327
\(862\) 1.87411e10 0.996598
\(863\) −1.32034e10 −0.699273 −0.349636 0.936885i \(-0.613695\pi\)
−0.349636 + 0.936885i \(0.613695\pi\)
\(864\) 1.56782e10 0.826988
\(865\) 1.43796e10 0.755421
\(866\) −2.24055e10 −1.17231
\(867\) 1.83705e9 0.0957312
\(868\) 2.45790e10 1.27569
\(869\) −3.02505e10 −1.56373
\(870\) 3.14781e10 1.62066
\(871\) −1.16961e9 −0.0599762
\(872\) −1.06734e10 −0.545126
\(873\) −1.44434e9 −0.0734717
\(874\) −5.30517e10 −2.68788
\(875\) 5.13963e10 2.59360
\(876\) 1.22970e10 0.618065
\(877\) −1.39419e10 −0.697949 −0.348975 0.937132i \(-0.613470\pi\)
−0.348975 + 0.937132i \(0.613470\pi\)
\(878\) −3.14822e10 −1.56976
\(879\) −7.24315e9 −0.359721
\(880\) 2.50310e11 12.3819
\(881\) 1.37479e10 0.677360 0.338680 0.940902i \(-0.390020\pi\)
0.338680 + 0.940902i \(0.390020\pi\)
\(882\) 1.02789e10 0.504437
\(883\) 8.50641e9 0.415799 0.207900 0.978150i \(-0.433337\pi\)
0.207900 + 0.978150i \(0.433337\pi\)
\(884\) −1.50571e10 −0.733093
\(885\) 2.73080e9 0.132431
\(886\) 5.98635e9 0.289164
\(887\) −1.27701e10 −0.614417 −0.307208 0.951642i \(-0.599395\pi\)
−0.307208 + 0.951642i \(0.599395\pi\)
\(888\) −9.88846e9 −0.473896
\(889\) −1.66802e10 −0.796241
\(890\) −6.47665e10 −3.07954
\(891\) −4.12394e9 −0.195317
\(892\) 1.15599e10 0.545354
\(893\) −6.57523e8 −0.0308980
\(894\) −1.50730e9 −0.0705534
\(895\) −6.48204e10 −3.02225
\(896\) −1.20810e11 −5.61079
\(897\) 3.95542e9 0.182987
\(898\) −2.73420e10 −1.25998
\(899\) 6.11231e9 0.280573
\(900\) 4.28105e10 1.95750
\(901\) −3.34433e10 −1.52326
\(902\) 1.02248e11 4.63909
\(903\) 4.15370e9 0.187728
\(904\) 1.02527e10 0.461582
\(905\) 7.48328e10 3.35600
\(906\) −3.26803e10 −1.45995
\(907\) 3.70884e10 1.65049 0.825244 0.564776i \(-0.191037\pi\)
0.825244 + 0.564776i \(0.191037\pi\)
\(908\) 5.73616e10 2.54285
\(909\) −1.12901e10 −0.498567
\(910\) 2.52915e10 1.11257
\(911\) 1.58470e10 0.694436 0.347218 0.937784i \(-0.387126\pi\)
0.347218 + 0.937784i \(0.387126\pi\)
\(912\) −5.60307e10 −2.44593
\(913\) 3.13346e10 1.36262
\(914\) −7.42774e10 −3.21770
\(915\) −3.19600e10 −1.37922
\(916\) 1.55074e10 0.666660
\(917\) 1.02231e9 0.0437813
\(918\) −9.48306e9 −0.404575
\(919\) 1.09730e10 0.466359 0.233180 0.972434i \(-0.425087\pi\)
0.233180 + 0.972434i \(0.425087\pi\)
\(920\) −1.89026e11 −8.00321
\(921\) 1.77998e10 0.750770
\(922\) −6.30781e10 −2.65046
\(923\) −6.78955e9 −0.284207
\(924\) 9.05484e10 3.77597
\(925\) −1.19237e10 −0.495352
\(926\) 8.83119e9 0.365494
\(927\) −8.76804e9 −0.361513
\(928\) −8.56067e10 −3.51633
\(929\) 3.30298e10 1.35161 0.675804 0.737081i \(-0.263797\pi\)
0.675804 + 0.737081i \(0.263797\pi\)
\(930\) 1.66575e10 0.679078
\(931\) −2.02796e10 −0.823637
\(932\) −1.02263e10 −0.413774
\(933\) −2.35876e10 −0.950818
\(934\) −1.77431e10 −0.712552
\(935\) −8.35821e10 −3.34405
\(936\) 7.09376e9 0.282756
\(937\) −1.81069e10 −0.719045 −0.359522 0.933136i \(-0.617060\pi\)
−0.359522 + 0.933136i \(0.617060\pi\)
\(938\) 1.61740e10 0.639894
\(939\) 7.66860e9 0.302264
\(940\) −3.65101e9 −0.143372
\(941\) 2.38460e10 0.932938 0.466469 0.884538i \(-0.345526\pi\)
0.466469 + 0.884538i \(0.345526\pi\)
\(942\) 4.32699e10 1.68658
\(943\) −4.54718e10 −1.76584
\(944\) −1.34526e10 −0.520480
\(945\) 1.17268e10 0.452031
\(946\) 2.17362e10 0.834764
\(947\) 2.50480e10 0.958404 0.479202 0.877705i \(-0.340926\pi\)
0.479202 + 0.877705i \(0.340926\pi\)
\(948\) 3.75995e10 1.43335
\(949\) 2.45700e9 0.0933198
\(950\) −1.14727e11 −4.34141
\(951\) −5.85724e9 −0.220832
\(952\) 1.33610e11 5.01891
\(953\) −2.27700e10 −0.852194 −0.426097 0.904677i \(-0.640112\pi\)
−0.426097 + 0.904677i \(0.640112\pi\)
\(954\) 2.45541e10 0.915597
\(955\) 5.99389e10 2.22688
\(956\) −2.16688e10 −0.802105
\(957\) 2.25176e10 0.830483
\(958\) −3.20816e10 −1.17890
\(959\) 5.86269e10 2.14650
\(960\) −1.21819e11 −4.44393
\(961\) −2.42781e10 −0.882436
\(962\) −3.07904e9 −0.111507
\(963\) −5.60617e9 −0.202290
\(964\) 1.79701e10 0.646072
\(965\) −6.53062e10 −2.33942
\(966\) −5.46976e10 −1.95231
\(967\) 1.53182e10 0.544772 0.272386 0.962188i \(-0.412187\pi\)
0.272386 + 0.962188i \(0.412187\pi\)
\(968\) 2.05655e11 7.28746
\(969\) 1.87095e10 0.660584
\(970\) 2.14924e10 0.756109
\(971\) 1.35596e10 0.475314 0.237657 0.971349i \(-0.423621\pi\)
0.237657 + 0.971349i \(0.423621\pi\)
\(972\) 5.12580e9 0.179032
\(973\) 2.48265e9 0.0864013
\(974\) 4.21084e10 1.46020
\(975\) 8.55377e9 0.295557
\(976\) 1.57443e11 5.42061
\(977\) 5.16415e10 1.77161 0.885804 0.464060i \(-0.153608\pi\)
0.885804 + 0.464060i \(0.153608\pi\)
\(978\) 1.18388e9 0.0404690
\(979\) −4.63302e10 −1.57807
\(980\) −1.12606e11 −3.82182
\(981\) −1.54098e9 −0.0521141
\(982\) 6.87160e9 0.231562
\(983\) 2.68451e10 0.901421 0.450711 0.892670i \(-0.351171\pi\)
0.450711 + 0.892670i \(0.351171\pi\)
\(984\) −8.15504e10 −2.72862
\(985\) −5.51509e10 −1.83876
\(986\) 5.17796e10 1.72024
\(987\) −6.77923e8 −0.0224425
\(988\) −2.18106e10 −0.719481
\(989\) −9.66650e9 −0.317748
\(990\) 6.13659e10 2.01004
\(991\) 3.98369e10 1.30025 0.650126 0.759826i \(-0.274716\pi\)
0.650126 + 0.759826i \(0.274716\pi\)
\(992\) −4.53012e10 −1.47339
\(993\) −2.69594e10 −0.873753
\(994\) 9.38895e10 3.03225
\(995\) 8.94740e10 2.87949
\(996\) −3.89469e10 −1.24901
\(997\) −5.65997e9 −0.180876 −0.0904380 0.995902i \(-0.528827\pi\)
−0.0904380 + 0.995902i \(0.528827\pi\)
\(998\) −2.43241e10 −0.774605
\(999\) −1.42765e9 −0.0453045
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.1.17 17
3.2 odd 2 531.8.a.d.1.1 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.17 17 1.1 even 1 trivial
531.8.a.d.1.1 17 3.2 odd 2