Properties

Label 177.8.a.b.1.16
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 1.16
Root \(18.2619\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.2619 q^{2} +27.0000 q^{3} +136.450 q^{4} -30.6671 q^{5} +439.072 q^{6} -1356.62 q^{7} +137.421 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+16.2619 q^{2} +27.0000 q^{3} +136.450 q^{4} -30.6671 q^{5} +439.072 q^{6} -1356.62 q^{7} +137.421 q^{8} +729.000 q^{9} -498.706 q^{10} +1825.79 q^{11} +3684.16 q^{12} -1173.55 q^{13} -22061.2 q^{14} -828.012 q^{15} -15230.9 q^{16} +13185.9 q^{17} +11854.9 q^{18} -18595.5 q^{19} -4184.54 q^{20} -36628.7 q^{21} +29690.8 q^{22} -14566.3 q^{23} +3710.36 q^{24} -77184.5 q^{25} -19084.3 q^{26} +19683.0 q^{27} -185111. q^{28} -183281. q^{29} -13465.1 q^{30} -265230. q^{31} -265274. q^{32} +49296.2 q^{33} +214428. q^{34} +41603.6 q^{35} +99472.4 q^{36} -260898. q^{37} -302399. q^{38} -31686.0 q^{39} -4214.29 q^{40} +719268. q^{41} -595654. q^{42} -707870. q^{43} +249129. q^{44} -22356.3 q^{45} -236876. q^{46} +1.30218e6 q^{47} -411235. q^{48} +1.01687e6 q^{49} -1.25517e6 q^{50} +356019. q^{51} -160132. q^{52} -1.80784e6 q^{53} +320084. q^{54} -55991.6 q^{55} -186427. q^{56} -502078. q^{57} -2.98051e6 q^{58} -205379. q^{59} -112983. q^{60} +1.38182e6 q^{61} -4.31315e6 q^{62} -988975. q^{63} -2.36431e6 q^{64} +35989.5 q^{65} +801652. q^{66} +2.41963e6 q^{67} +1.79922e6 q^{68} -393290. q^{69} +676555. q^{70} +2.72507e6 q^{71} +100180. q^{72} +3.97598e6 q^{73} -4.24271e6 q^{74} -2.08398e6 q^{75} -2.53736e6 q^{76} -2.47690e6 q^{77} -515275. q^{78} -6.91787e6 q^{79} +467089. q^{80} +531441. q^{81} +1.16967e7 q^{82} +8.32910e6 q^{83} -4.99800e6 q^{84} -404374. q^{85} -1.15113e7 q^{86} -4.94860e6 q^{87} +250901. q^{88} -6.71890e6 q^{89} -363557. q^{90} +1.59207e6 q^{91} -1.98758e6 q^{92} -7.16121e6 q^{93} +2.11760e7 q^{94} +570270. q^{95} -7.16240e6 q^{96} -4.77557e6 q^{97} +1.65363e7 q^{98} +1.33100e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.2619 1.43737 0.718683 0.695338i \(-0.244745\pi\)
0.718683 + 0.695338i \(0.244745\pi\)
\(3\) 27.0000 0.577350
\(4\) 136.450 1.06602
\(5\) −30.6671 −0.109718 −0.0548590 0.998494i \(-0.517471\pi\)
−0.0548590 + 0.998494i \(0.517471\pi\)
\(6\) 439.072 0.829863
\(7\) −1356.62 −1.49491 −0.747455 0.664313i \(-0.768724\pi\)
−0.747455 + 0.664313i \(0.768724\pi\)
\(8\) 137.421 0.0948936
\(9\) 729.000 0.333333
\(10\) −498.706 −0.157705
\(11\) 1825.79 0.413595 0.206798 0.978384i \(-0.433696\pi\)
0.206798 + 0.978384i \(0.433696\pi\)
\(12\) 3684.16 0.615466
\(13\) −1173.55 −0.148150 −0.0740750 0.997253i \(-0.523600\pi\)
−0.0740750 + 0.997253i \(0.523600\pi\)
\(14\) −22061.2 −2.14873
\(15\) −828.012 −0.0633457
\(16\) −15230.9 −0.929622
\(17\) 13185.9 0.650937 0.325468 0.945553i \(-0.394478\pi\)
0.325468 + 0.945553i \(0.394478\pi\)
\(18\) 11854.9 0.479122
\(19\) −18595.5 −0.621971 −0.310985 0.950415i \(-0.600659\pi\)
−0.310985 + 0.950415i \(0.600659\pi\)
\(20\) −4184.54 −0.116961
\(21\) −36628.7 −0.863086
\(22\) 29690.8 0.594488
\(23\) −14566.3 −0.249633 −0.124817 0.992180i \(-0.539834\pi\)
−0.124817 + 0.992180i \(0.539834\pi\)
\(24\) 3710.36 0.0547869
\(25\) −77184.5 −0.987962
\(26\) −19084.3 −0.212946
\(27\) 19683.0 0.192450
\(28\) −185111. −1.59360
\(29\) −183281. −1.39549 −0.697743 0.716348i \(-0.745812\pi\)
−0.697743 + 0.716348i \(0.745812\pi\)
\(30\) −13465.1 −0.0910509
\(31\) −265230. −1.59903 −0.799515 0.600646i \(-0.794910\pi\)
−0.799515 + 0.600646i \(0.794910\pi\)
\(32\) −265274. −1.43110
\(33\) 49296.2 0.238789
\(34\) 214428. 0.935634
\(35\) 41603.6 0.164018
\(36\) 99472.4 0.355340
\(37\) −260898. −0.846769 −0.423385 0.905950i \(-0.639158\pi\)
−0.423385 + 0.905950i \(0.639158\pi\)
\(38\) −302399. −0.893999
\(39\) −31686.0 −0.0855344
\(40\) −4214.29 −0.0104115
\(41\) 719268. 1.62985 0.814924 0.579568i \(-0.196779\pi\)
0.814924 + 0.579568i \(0.196779\pi\)
\(42\) −595654. −1.24057
\(43\) −707870. −1.35773 −0.678866 0.734262i \(-0.737528\pi\)
−0.678866 + 0.734262i \(0.737528\pi\)
\(44\) 249129. 0.440900
\(45\) −22356.3 −0.0365727
\(46\) −236876. −0.358814
\(47\) 1.30218e6 1.82948 0.914742 0.404038i \(-0.132394\pi\)
0.914742 + 0.404038i \(0.132394\pi\)
\(48\) −411235. −0.536718
\(49\) 1.01687e6 1.23475
\(50\) −1.25517e6 −1.42006
\(51\) 356019. 0.375818
\(52\) −160132. −0.157931
\(53\) −1.80784e6 −1.66799 −0.833996 0.551770i \(-0.813953\pi\)
−0.833996 + 0.551770i \(0.813953\pi\)
\(54\) 320084. 0.276621
\(55\) −55991.6 −0.0453788
\(56\) −186427. −0.141857
\(57\) −502078. −0.359095
\(58\) −2.98051e6 −2.00582
\(59\) −205379. −0.130189
\(60\) −112983. −0.0675277
\(61\) 1.38182e6 0.779468 0.389734 0.920928i \(-0.372567\pi\)
0.389734 + 0.920928i \(0.372567\pi\)
\(62\) −4.31315e6 −2.29839
\(63\) −988975. −0.498303
\(64\) −2.36431e6 −1.12739
\(65\) 35989.5 0.0162547
\(66\) 801652. 0.343228
\(67\) 2.41963e6 0.982849 0.491424 0.870920i \(-0.336476\pi\)
0.491424 + 0.870920i \(0.336476\pi\)
\(68\) 1.79922e6 0.693911
\(69\) −393290. −0.144126
\(70\) 676555. 0.235754
\(71\) 2.72507e6 0.903595 0.451797 0.892121i \(-0.350783\pi\)
0.451797 + 0.892121i \(0.350783\pi\)
\(72\) 100180. 0.0316312
\(73\) 3.97598e6 1.19623 0.598115 0.801411i \(-0.295917\pi\)
0.598115 + 0.801411i \(0.295917\pi\)
\(74\) −4.24271e6 −1.21712
\(75\) −2.08398e6 −0.570400
\(76\) −2.53736e6 −0.663033
\(77\) −2.47690e6 −0.618287
\(78\) −515275. −0.122944
\(79\) −6.91787e6 −1.57862 −0.789310 0.613995i \(-0.789562\pi\)
−0.789310 + 0.613995i \(0.789562\pi\)
\(80\) 467089. 0.101996
\(81\) 531441. 0.111111
\(82\) 1.16967e7 2.34269
\(83\) 8.32910e6 1.59891 0.799456 0.600725i \(-0.205121\pi\)
0.799456 + 0.600725i \(0.205121\pi\)
\(84\) −4.99800e6 −0.920066
\(85\) −404374. −0.0714195
\(86\) −1.15113e7 −1.95156
\(87\) −4.94860e6 −0.805684
\(88\) 250901. 0.0392476
\(89\) −6.71890e6 −1.01026 −0.505130 0.863043i \(-0.668556\pi\)
−0.505130 + 0.863043i \(0.668556\pi\)
\(90\) −363557. −0.0525683
\(91\) 1.59207e6 0.221471
\(92\) −1.98758e6 −0.266114
\(93\) −7.16121e6 −0.923200
\(94\) 2.11760e7 2.62964
\(95\) 570270. 0.0682414
\(96\) −7.16240e6 −0.826246
\(97\) −4.77557e6 −0.531281 −0.265640 0.964072i \(-0.585583\pi\)
−0.265640 + 0.964072i \(0.585583\pi\)
\(98\) 1.65363e7 1.77479
\(99\) 1.33100e6 0.137865
\(100\) −1.05319e7 −1.05319
\(101\) 1.79352e7 1.73213 0.866066 0.499930i \(-0.166641\pi\)
0.866066 + 0.499930i \(0.166641\pi\)
\(102\) 5.78956e6 0.540188
\(103\) −1.83344e7 −1.65324 −0.826619 0.562762i \(-0.809739\pi\)
−0.826619 + 0.562762i \(0.809739\pi\)
\(104\) −161271. −0.0140585
\(105\) 1.12330e6 0.0946961
\(106\) −2.93990e7 −2.39752
\(107\) 1.58268e7 1.24896 0.624480 0.781041i \(-0.285311\pi\)
0.624480 + 0.781041i \(0.285311\pi\)
\(108\) 2.68575e6 0.205155
\(109\) 4.43687e6 0.328159 0.164079 0.986447i \(-0.447535\pi\)
0.164079 + 0.986447i \(0.447535\pi\)
\(110\) −910531. −0.0652260
\(111\) −7.04425e6 −0.488882
\(112\) 2.06626e7 1.38970
\(113\) 2.03544e7 1.32704 0.663520 0.748159i \(-0.269062\pi\)
0.663520 + 0.748159i \(0.269062\pi\)
\(114\) −8.16476e6 −0.516151
\(115\) 446707. 0.0273892
\(116\) −2.50088e7 −1.48761
\(117\) −855521. −0.0493833
\(118\) −3.33986e6 −0.187129
\(119\) −1.78883e7 −0.973091
\(120\) −113786. −0.00601110
\(121\) −1.61537e7 −0.828939
\(122\) 2.24711e7 1.12038
\(123\) 1.94202e7 0.940993
\(124\) −3.61907e7 −1.70460
\(125\) 4.76289e6 0.218115
\(126\) −1.60827e7 −0.716244
\(127\) −9.80807e6 −0.424884 −0.212442 0.977174i \(-0.568142\pi\)
−0.212442 + 0.977174i \(0.568142\pi\)
\(128\) −4.49319e6 −0.189374
\(129\) −1.91125e7 −0.783887
\(130\) 585259. 0.0233640
\(131\) −2.55344e7 −0.992377 −0.496189 0.868215i \(-0.665268\pi\)
−0.496189 + 0.868215i \(0.665268\pi\)
\(132\) 6.72649e6 0.254554
\(133\) 2.52270e7 0.929790
\(134\) 3.93478e7 1.41271
\(135\) −603621. −0.0211152
\(136\) 1.81202e6 0.0617698
\(137\) −5.60036e6 −0.186078 −0.0930388 0.995662i \(-0.529658\pi\)
−0.0930388 + 0.995662i \(0.529658\pi\)
\(138\) −6.39566e6 −0.207161
\(139\) −994215. −0.0313999 −0.0157000 0.999877i \(-0.504998\pi\)
−0.0157000 + 0.999877i \(0.504998\pi\)
\(140\) 5.67683e6 0.174847
\(141\) 3.51589e7 1.05625
\(142\) 4.43149e7 1.29880
\(143\) −2.14266e6 −0.0612741
\(144\) −1.11033e7 −0.309874
\(145\) 5.62071e6 0.153110
\(146\) 6.46572e7 1.71942
\(147\) 2.74556e7 0.712885
\(148\) −3.55997e7 −0.902672
\(149\) 2.61359e7 0.647269 0.323635 0.946182i \(-0.395095\pi\)
0.323635 + 0.946182i \(0.395095\pi\)
\(150\) −3.38896e7 −0.819873
\(151\) −7.02885e7 −1.66136 −0.830682 0.556747i \(-0.812049\pi\)
−0.830682 + 0.556747i \(0.812049\pi\)
\(152\) −2.55540e6 −0.0590211
\(153\) 9.61253e6 0.216979
\(154\) −4.02791e7 −0.888705
\(155\) 8.13383e6 0.175442
\(156\) −4.32357e6 −0.0911813
\(157\) −2.28654e7 −0.471553 −0.235776 0.971807i \(-0.575763\pi\)
−0.235776 + 0.971807i \(0.575763\pi\)
\(158\) −1.12498e8 −2.26905
\(159\) −4.88117e7 −0.963016
\(160\) 8.13519e6 0.157017
\(161\) 1.97609e7 0.373179
\(162\) 8.64226e6 0.159707
\(163\) −6.26397e7 −1.13290 −0.566452 0.824095i \(-0.691684\pi\)
−0.566452 + 0.824095i \(0.691684\pi\)
\(164\) 9.81444e7 1.73745
\(165\) −1.51177e6 −0.0261995
\(166\) 1.35447e8 2.29822
\(167\) −2.02935e7 −0.337171 −0.168586 0.985687i \(-0.553920\pi\)
−0.168586 + 0.985687i \(0.553920\pi\)
\(168\) −5.03354e6 −0.0819014
\(169\) −6.13713e7 −0.978052
\(170\) −6.57590e6 −0.102656
\(171\) −1.35561e7 −0.207324
\(172\) −9.65892e7 −1.44737
\(173\) −4.43065e7 −0.650588 −0.325294 0.945613i \(-0.605463\pi\)
−0.325294 + 0.945613i \(0.605463\pi\)
\(174\) −8.04737e7 −1.15806
\(175\) 1.04710e8 1.47691
\(176\) −2.78084e7 −0.384487
\(177\) −5.54523e6 −0.0751646
\(178\) −1.09262e8 −1.45211
\(179\) 1.30339e8 1.69858 0.849292 0.527923i \(-0.177029\pi\)
0.849292 + 0.527923i \(0.177029\pi\)
\(180\) −3.05053e6 −0.0389872
\(181\) −8.67394e7 −1.08728 −0.543640 0.839319i \(-0.682954\pi\)
−0.543640 + 0.839319i \(0.682954\pi\)
\(182\) 2.58901e7 0.318334
\(183\) 3.73092e7 0.450026
\(184\) −2.00171e6 −0.0236886
\(185\) 8.00099e6 0.0929058
\(186\) −1.16455e8 −1.32698
\(187\) 2.40746e7 0.269224
\(188\) 1.77683e8 1.95027
\(189\) −2.67023e7 −0.287695
\(190\) 9.27369e6 0.0980878
\(191\) −1.00406e8 −1.04266 −0.521330 0.853355i \(-0.674564\pi\)
−0.521330 + 0.853355i \(0.674564\pi\)
\(192\) −6.38364e7 −0.650900
\(193\) 1.63066e8 1.63272 0.816361 0.577542i \(-0.195988\pi\)
0.816361 + 0.577542i \(0.195988\pi\)
\(194\) −7.76600e7 −0.763645
\(195\) 971717. 0.00938466
\(196\) 1.38753e8 1.31627
\(197\) 1.22832e8 1.14467 0.572335 0.820020i \(-0.306038\pi\)
0.572335 + 0.820020i \(0.306038\pi\)
\(198\) 2.16446e7 0.198163
\(199\) 1.85902e8 1.67224 0.836119 0.548549i \(-0.184820\pi\)
0.836119 + 0.548549i \(0.184820\pi\)
\(200\) −1.06067e7 −0.0937513
\(201\) 6.53299e7 0.567448
\(202\) 2.91661e8 2.48971
\(203\) 2.48643e8 2.08612
\(204\) 4.85790e7 0.400630
\(205\) −2.20579e7 −0.178824
\(206\) −2.98152e8 −2.37631
\(207\) −1.06188e7 −0.0832110
\(208\) 1.78743e7 0.137724
\(209\) −3.39514e7 −0.257244
\(210\) 1.82670e7 0.136113
\(211\) 4.25639e6 0.0311927 0.0155963 0.999878i \(-0.495035\pi\)
0.0155963 + 0.999878i \(0.495035\pi\)
\(212\) −2.46680e8 −1.77811
\(213\) 7.35769e7 0.521691
\(214\) 2.57374e8 1.79521
\(215\) 2.17083e7 0.148968
\(216\) 2.70485e6 0.0182623
\(217\) 3.59816e8 2.39040
\(218\) 7.21521e7 0.471684
\(219\) 1.07352e8 0.690643
\(220\) −7.64008e6 −0.0483747
\(221\) −1.54744e7 −0.0964362
\(222\) −1.14553e8 −0.702703
\(223\) −1.01090e8 −0.610437 −0.305219 0.952282i \(-0.598730\pi\)
−0.305219 + 0.952282i \(0.598730\pi\)
\(224\) 3.59876e8 2.13937
\(225\) −5.62675e7 −0.329321
\(226\) 3.31002e8 1.90744
\(227\) −1.56725e8 −0.889299 −0.444650 0.895705i \(-0.646672\pi\)
−0.444650 + 0.895705i \(0.646672\pi\)
\(228\) −6.85088e7 −0.382802
\(229\) 2.30104e8 1.26619 0.633097 0.774072i \(-0.281783\pi\)
0.633097 + 0.774072i \(0.281783\pi\)
\(230\) 7.26431e6 0.0393683
\(231\) −6.68762e7 −0.356968
\(232\) −2.51866e7 −0.132423
\(233\) 9.45434e7 0.489649 0.244825 0.969567i \(-0.421270\pi\)
0.244825 + 0.969567i \(0.421270\pi\)
\(234\) −1.39124e7 −0.0709819
\(235\) −3.99341e7 −0.200727
\(236\) −2.80241e7 −0.138784
\(237\) −1.86783e8 −0.911417
\(238\) −2.90898e8 −1.39869
\(239\) −2.18199e8 −1.03385 −0.516927 0.856029i \(-0.672924\pi\)
−0.516927 + 0.856029i \(0.672924\pi\)
\(240\) 1.26114e7 0.0588876
\(241\) −7.52814e7 −0.346440 −0.173220 0.984883i \(-0.555417\pi\)
−0.173220 + 0.984883i \(0.555417\pi\)
\(242\) −2.62690e8 −1.19149
\(243\) 1.43489e7 0.0641500
\(244\) 1.88550e8 0.830927
\(245\) −3.11845e7 −0.135475
\(246\) 3.15811e8 1.35255
\(247\) 2.18228e7 0.0921450
\(248\) −3.64481e7 −0.151738
\(249\) 2.24886e8 0.923132
\(250\) 7.74539e7 0.313511
\(251\) −2.48536e8 −0.992046 −0.496023 0.868309i \(-0.665207\pi\)
−0.496023 + 0.868309i \(0.665207\pi\)
\(252\) −1.34946e8 −0.531201
\(253\) −2.65950e7 −0.103247
\(254\) −1.59498e8 −0.610714
\(255\) −1.09181e7 −0.0412340
\(256\) 2.29564e8 0.855193
\(257\) 9.48986e7 0.348734 0.174367 0.984681i \(-0.444212\pi\)
0.174367 + 0.984681i \(0.444212\pi\)
\(258\) −3.10806e8 −1.12673
\(259\) 3.53939e8 1.26584
\(260\) 4.91079e6 0.0173278
\(261\) −1.33612e8 −0.465162
\(262\) −4.15239e8 −1.42641
\(263\) 7.74187e7 0.262422 0.131211 0.991354i \(-0.458113\pi\)
0.131211 + 0.991354i \(0.458113\pi\)
\(264\) 6.77432e6 0.0226596
\(265\) 5.54412e7 0.183009
\(266\) 4.10240e8 1.33645
\(267\) −1.81410e8 −0.583274
\(268\) 3.30159e8 1.04774
\(269\) −1.48368e8 −0.464738 −0.232369 0.972628i \(-0.574648\pi\)
−0.232369 + 0.972628i \(0.574648\pi\)
\(270\) −9.81604e6 −0.0303503
\(271\) 2.14037e8 0.653275 0.326637 0.945150i \(-0.394084\pi\)
0.326637 + 0.945150i \(0.394084\pi\)
\(272\) −2.00834e8 −0.605125
\(273\) 4.29858e7 0.127866
\(274\) −9.10727e7 −0.267462
\(275\) −1.40922e8 −0.408616
\(276\) −5.36647e7 −0.153641
\(277\) −4.49384e8 −1.27039 −0.635196 0.772351i \(-0.719081\pi\)
−0.635196 + 0.772351i \(0.719081\pi\)
\(278\) −1.61679e7 −0.0451331
\(279\) −1.93353e8 −0.533010
\(280\) 5.71719e6 0.0155643
\(281\) 5.47292e7 0.147145 0.0735727 0.997290i \(-0.476560\pi\)
0.0735727 + 0.997290i \(0.476560\pi\)
\(282\) 5.71751e8 1.51822
\(283\) −6.96310e8 −1.82621 −0.913104 0.407727i \(-0.866322\pi\)
−0.913104 + 0.407727i \(0.866322\pi\)
\(284\) 3.71837e8 0.963249
\(285\) 1.53973e7 0.0393992
\(286\) −3.48438e7 −0.0880733
\(287\) −9.75773e8 −2.43648
\(288\) −1.93385e8 −0.477034
\(289\) −2.36471e8 −0.576281
\(290\) 9.14036e7 0.220075
\(291\) −1.28940e8 −0.306735
\(292\) 5.42525e8 1.27520
\(293\) −1.58823e8 −0.368874 −0.184437 0.982844i \(-0.559046\pi\)
−0.184437 + 0.982844i \(0.559046\pi\)
\(294\) 4.46480e8 1.02468
\(295\) 6.29838e6 0.0142841
\(296\) −3.58528e7 −0.0803530
\(297\) 3.59370e7 0.0795964
\(298\) 4.25020e8 0.930363
\(299\) 1.70944e7 0.0369831
\(300\) −2.84360e8 −0.608057
\(301\) 9.60310e8 2.02969
\(302\) −1.14303e9 −2.38799
\(303\) 4.84250e8 1.00005
\(304\) 2.83227e8 0.578198
\(305\) −4.23765e7 −0.0855216
\(306\) 1.56318e8 0.311878
\(307\) 2.01551e8 0.397558 0.198779 0.980044i \(-0.436302\pi\)
0.198779 + 0.980044i \(0.436302\pi\)
\(308\) −3.37974e8 −0.659106
\(309\) −4.95028e8 −0.954497
\(310\) 1.32272e8 0.252175
\(311\) 3.80395e8 0.717090 0.358545 0.933513i \(-0.383273\pi\)
0.358545 + 0.933513i \(0.383273\pi\)
\(312\) −4.35431e6 −0.00811667
\(313\) 1.43971e8 0.265382 0.132691 0.991157i \(-0.457638\pi\)
0.132691 + 0.991157i \(0.457638\pi\)
\(314\) −3.71836e8 −0.677794
\(315\) 3.03290e7 0.0546728
\(316\) −9.43947e8 −1.68284
\(317\) −3.85345e8 −0.679426 −0.339713 0.940529i \(-0.610330\pi\)
−0.339713 + 0.940529i \(0.610330\pi\)
\(318\) −7.93772e8 −1.38421
\(319\) −3.34633e8 −0.577166
\(320\) 7.25066e7 0.123695
\(321\) 4.27322e8 0.721087
\(322\) 3.21351e8 0.536394
\(323\) −2.45198e8 −0.404864
\(324\) 7.25154e7 0.118447
\(325\) 9.05803e7 0.146367
\(326\) −1.01864e9 −1.62840
\(327\) 1.19796e8 0.189463
\(328\) 9.88423e7 0.154662
\(329\) −1.76656e9 −2.73491
\(330\) −2.45843e7 −0.0376582
\(331\) −5.09634e8 −0.772432 −0.386216 0.922408i \(-0.626218\pi\)
−0.386216 + 0.922408i \(0.626218\pi\)
\(332\) 1.13651e9 1.70447
\(333\) −1.90195e8 −0.282256
\(334\) −3.30012e8 −0.484638
\(335\) −7.42030e7 −0.107836
\(336\) 5.57889e8 0.802344
\(337\) 3.29994e8 0.469679 0.234840 0.972034i \(-0.424543\pi\)
0.234840 + 0.972034i \(0.424543\pi\)
\(338\) −9.98016e8 −1.40582
\(339\) 5.49569e8 0.766167
\(340\) −5.51770e7 −0.0761345
\(341\) −4.84253e8 −0.661351
\(342\) −2.20449e8 −0.298000
\(343\) −2.62274e8 −0.350935
\(344\) −9.72760e7 −0.128840
\(345\) 1.20611e7 0.0158132
\(346\) −7.20510e8 −0.935133
\(347\) −1.54205e9 −1.98128 −0.990642 0.136488i \(-0.956418\pi\)
−0.990642 + 0.136488i \(0.956418\pi\)
\(348\) −6.75238e8 −0.858874
\(349\) −3.37260e8 −0.424694 −0.212347 0.977194i \(-0.568111\pi\)
−0.212347 + 0.977194i \(0.568111\pi\)
\(350\) 1.70279e9 2.12286
\(351\) −2.30991e7 −0.0285115
\(352\) −4.84334e8 −0.591896
\(353\) −8.38966e8 −1.01516 −0.507578 0.861606i \(-0.669459\pi\)
−0.507578 + 0.861606i \(0.669459\pi\)
\(354\) −9.01762e7 −0.108039
\(355\) −8.35701e7 −0.0991406
\(356\) −9.16797e8 −1.07696
\(357\) −4.82983e8 −0.561814
\(358\) 2.11956e9 2.44149
\(359\) −5.75290e8 −0.656231 −0.328115 0.944638i \(-0.606413\pi\)
−0.328115 + 0.944638i \(0.606413\pi\)
\(360\) −3.07222e6 −0.00347051
\(361\) −5.48079e8 −0.613152
\(362\) −1.41055e9 −1.56282
\(363\) −4.36149e8 −0.478588
\(364\) 2.17238e8 0.236092
\(365\) −1.21932e8 −0.131248
\(366\) 6.06720e8 0.646852
\(367\) −1.63254e8 −0.172398 −0.0861990 0.996278i \(-0.527472\pi\)
−0.0861990 + 0.996278i \(0.527472\pi\)
\(368\) 2.21859e8 0.232065
\(369\) 5.24346e8 0.543283
\(370\) 1.30112e8 0.133540
\(371\) 2.45255e9 2.49350
\(372\) −9.77150e8 −0.984149
\(373\) −5.02935e8 −0.501801 −0.250900 0.968013i \(-0.580727\pi\)
−0.250900 + 0.968013i \(0.580727\pi\)
\(374\) 3.91500e8 0.386974
\(375\) 1.28598e8 0.125929
\(376\) 1.78947e8 0.173606
\(377\) 2.15091e8 0.206741
\(378\) −4.34232e8 −0.413523
\(379\) 1.84658e9 1.74234 0.871168 0.490986i \(-0.163363\pi\)
0.871168 + 0.490986i \(0.163363\pi\)
\(380\) 7.78136e7 0.0727466
\(381\) −2.64818e8 −0.245307
\(382\) −1.63279e9 −1.49868
\(383\) 78579.0 7.14679e−5 0 3.57339e−5 1.00000i \(-0.499989\pi\)
3.57339e−5 1.00000i \(0.499989\pi\)
\(384\) −1.21316e8 −0.109335
\(385\) 7.59593e7 0.0678372
\(386\) 2.65176e9 2.34682
\(387\) −5.16037e8 −0.452577
\(388\) −6.51629e8 −0.566355
\(389\) 1.01883e9 0.877561 0.438780 0.898594i \(-0.355411\pi\)
0.438780 + 0.898594i \(0.355411\pi\)
\(390\) 1.58020e7 0.0134892
\(391\) −1.92070e8 −0.162495
\(392\) 1.39739e8 0.117170
\(393\) −6.89430e8 −0.572949
\(394\) 1.99749e9 1.64531
\(395\) 2.12151e8 0.173203
\(396\) 1.81615e8 0.146967
\(397\) 3.45551e8 0.277170 0.138585 0.990351i \(-0.455745\pi\)
0.138585 + 0.990351i \(0.455745\pi\)
\(398\) 3.02312e9 2.40362
\(399\) 6.81129e8 0.536814
\(400\) 1.17559e9 0.918431
\(401\) −2.15091e9 −1.66578 −0.832890 0.553439i \(-0.813315\pi\)
−0.832890 + 0.553439i \(0.813315\pi\)
\(402\) 1.06239e9 0.815630
\(403\) 3.11262e8 0.236896
\(404\) 2.44726e9 1.84649
\(405\) −1.62978e7 −0.0121909
\(406\) 4.04342e9 2.99852
\(407\) −4.76344e8 −0.350220
\(408\) 4.89244e7 0.0356628
\(409\) −2.23244e8 −0.161342 −0.0806710 0.996741i \(-0.525706\pi\)
−0.0806710 + 0.996741i \(0.525706\pi\)
\(410\) −3.58704e8 −0.257035
\(411\) −1.51210e8 −0.107432
\(412\) −2.50173e9 −1.76238
\(413\) 2.78621e8 0.194621
\(414\) −1.72683e8 −0.119605
\(415\) −2.55429e8 −0.175429
\(416\) 3.11314e8 0.212017
\(417\) −2.68438e7 −0.0181287
\(418\) −5.52115e8 −0.369754
\(419\) −1.16588e9 −0.774291 −0.387146 0.922019i \(-0.626539\pi\)
−0.387146 + 0.922019i \(0.626539\pi\)
\(420\) 1.53274e8 0.100948
\(421\) −9.44097e8 −0.616637 −0.308318 0.951283i \(-0.599766\pi\)
−0.308318 + 0.951283i \(0.599766\pi\)
\(422\) 6.92171e7 0.0448353
\(423\) 9.49290e8 0.609828
\(424\) −2.48434e8 −0.158282
\(425\) −1.01775e9 −0.643101
\(426\) 1.19650e9 0.749860
\(427\) −1.87461e9 −1.16523
\(428\) 2.15957e9 1.33142
\(429\) −5.78518e7 −0.0353766
\(430\) 3.53019e8 0.214121
\(431\) −1.01990e9 −0.613600 −0.306800 0.951774i \(-0.599258\pi\)
−0.306800 + 0.951774i \(0.599258\pi\)
\(432\) −2.99790e8 −0.178906
\(433\) −4.96072e8 −0.293655 −0.146827 0.989162i \(-0.546906\pi\)
−0.146827 + 0.989162i \(0.546906\pi\)
\(434\) 5.85130e9 3.43588
\(435\) 1.51759e8 0.0883980
\(436\) 6.05413e8 0.349824
\(437\) 2.70868e8 0.155265
\(438\) 1.74574e9 0.992707
\(439\) 1.84581e9 1.04126 0.520632 0.853781i \(-0.325697\pi\)
0.520632 + 0.853781i \(0.325697\pi\)
\(440\) −7.69440e6 −0.00430616
\(441\) 7.41300e8 0.411584
\(442\) −2.51643e8 −0.138614
\(443\) 2.12406e8 0.116079 0.0580394 0.998314i \(-0.481515\pi\)
0.0580394 + 0.998314i \(0.481515\pi\)
\(444\) −9.61191e8 −0.521158
\(445\) 2.06049e8 0.110844
\(446\) −1.64392e9 −0.877421
\(447\) 7.05669e8 0.373701
\(448\) 3.20747e9 1.68535
\(449\) −3.53221e9 −1.84155 −0.920777 0.390090i \(-0.872444\pi\)
−0.920777 + 0.390090i \(0.872444\pi\)
\(450\) −9.15019e8 −0.473354
\(451\) 1.31323e9 0.674098
\(452\) 2.77737e9 1.41465
\(453\) −1.89779e9 −0.959189
\(454\) −2.54865e9 −1.27825
\(455\) −4.88241e7 −0.0242993
\(456\) −6.89959e7 −0.0340758
\(457\) 2.05595e9 1.00764 0.503821 0.863808i \(-0.331927\pi\)
0.503821 + 0.863808i \(0.331927\pi\)
\(458\) 3.74194e9 1.81998
\(459\) 2.59538e8 0.125273
\(460\) 6.09533e7 0.0291975
\(461\) −1.34533e9 −0.639551 −0.319776 0.947493i \(-0.603608\pi\)
−0.319776 + 0.947493i \(0.603608\pi\)
\(462\) −1.08754e9 −0.513094
\(463\) 1.16277e9 0.544452 0.272226 0.962233i \(-0.412240\pi\)
0.272226 + 0.962233i \(0.412240\pi\)
\(464\) 2.79155e9 1.29727
\(465\) 2.19613e8 0.101292
\(466\) 1.53746e9 0.703805
\(467\) 3.37053e9 1.53140 0.765700 0.643198i \(-0.222393\pi\)
0.765700 + 0.643198i \(0.222393\pi\)
\(468\) −1.16736e8 −0.0526436
\(469\) −3.28251e9 −1.46927
\(470\) −6.49406e8 −0.288519
\(471\) −6.17366e8 −0.272251
\(472\) −2.82233e7 −0.0123541
\(473\) −1.29242e9 −0.561552
\(474\) −3.03744e9 −1.31004
\(475\) 1.43528e9 0.614484
\(476\) −2.44086e9 −1.03733
\(477\) −1.31791e9 −0.555998
\(478\) −3.54833e9 −1.48603
\(479\) −1.66586e9 −0.692571 −0.346286 0.938129i \(-0.612557\pi\)
−0.346286 + 0.938129i \(0.612557\pi\)
\(480\) 2.19650e8 0.0906541
\(481\) 3.06178e8 0.125449
\(482\) −1.22422e9 −0.497961
\(483\) 5.33545e8 0.215455
\(484\) −2.20418e9 −0.883665
\(485\) 1.46453e8 0.0582910
\(486\) 2.33341e8 0.0922070
\(487\) −1.02449e9 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(488\) 1.89891e8 0.0739665
\(489\) −1.69127e9 −0.654082
\(490\) −5.07121e8 −0.194727
\(491\) 2.27469e9 0.867235 0.433618 0.901097i \(-0.357237\pi\)
0.433618 + 0.901097i \(0.357237\pi\)
\(492\) 2.64990e9 1.00312
\(493\) −2.41673e9 −0.908373
\(494\) 3.54881e8 0.132446
\(495\) −4.08179e7 −0.0151263
\(496\) 4.03970e9 1.48649
\(497\) −3.69688e9 −1.35079
\(498\) 3.65707e9 1.32688
\(499\) 2.02609e9 0.729971 0.364986 0.931013i \(-0.381074\pi\)
0.364986 + 0.931013i \(0.381074\pi\)
\(500\) 6.49899e8 0.232515
\(501\) −5.47926e8 −0.194666
\(502\) −4.04168e9 −1.42593
\(503\) 1.54149e9 0.540073 0.270037 0.962850i \(-0.412964\pi\)
0.270037 + 0.962850i \(0.412964\pi\)
\(504\) −1.35906e8 −0.0472858
\(505\) −5.50020e8 −0.190046
\(506\) −4.32486e8 −0.148404
\(507\) −1.65702e9 −0.564678
\(508\) −1.33832e9 −0.452935
\(509\) 3.87817e9 1.30351 0.651755 0.758430i \(-0.274033\pi\)
0.651755 + 0.758430i \(0.274033\pi\)
\(510\) −1.77549e8 −0.0592684
\(511\) −5.39389e9 −1.78825
\(512\) 4.30828e9 1.41860
\(513\) −3.66015e8 −0.119698
\(514\) 1.54323e9 0.501258
\(515\) 5.62262e8 0.181390
\(516\) −2.60791e9 −0.835638
\(517\) 2.37750e9 0.756666
\(518\) 5.75574e9 1.81948
\(519\) −1.19628e9 −0.375617
\(520\) 4.94570e6 0.00154247
\(521\) −4.39564e8 −0.136173 −0.0680864 0.997679i \(-0.521689\pi\)
−0.0680864 + 0.997679i \(0.521689\pi\)
\(522\) −2.17279e9 −0.668607
\(523\) −6.36830e9 −1.94656 −0.973280 0.229623i \(-0.926251\pi\)
−0.973280 + 0.229623i \(0.926251\pi\)
\(524\) −3.48419e9 −1.05789
\(525\) 2.82717e9 0.852696
\(526\) 1.25898e9 0.377197
\(527\) −3.49730e9 −1.04087
\(528\) −7.50828e8 −0.221984
\(529\) −3.19265e9 −0.937683
\(530\) 9.01581e8 0.263051
\(531\) −1.49721e8 −0.0433963
\(532\) 3.44224e9 0.991174
\(533\) −8.44100e8 −0.241462
\(534\) −2.95008e9 −0.838378
\(535\) −4.85361e8 −0.137033
\(536\) 3.32507e8 0.0932661
\(537\) 3.51914e9 0.980678
\(538\) −2.41276e9 −0.667999
\(539\) 1.85659e9 0.510688
\(540\) −8.23643e7 −0.0225092
\(541\) −1.34859e9 −0.366175 −0.183088 0.983097i \(-0.558609\pi\)
−0.183088 + 0.983097i \(0.558609\pi\)
\(542\) 3.48065e9 0.938995
\(543\) −2.34196e9 −0.627741
\(544\) −3.49788e9 −0.931556
\(545\) −1.36066e8 −0.0360049
\(546\) 6.99032e8 0.183790
\(547\) −2.38331e9 −0.622622 −0.311311 0.950308i \(-0.600768\pi\)
−0.311311 + 0.950308i \(0.600768\pi\)
\(548\) −7.64172e8 −0.198362
\(549\) 1.00735e9 0.259823
\(550\) −2.29167e9 −0.587331
\(551\) 3.40821e9 0.867951
\(552\) −5.40462e7 −0.0136766
\(553\) 9.38492e9 2.35989
\(554\) −7.30785e9 −1.82602
\(555\) 2.16027e8 0.0536392
\(556\) −1.35661e8 −0.0334729
\(557\) −2.39992e8 −0.0588442 −0.0294221 0.999567i \(-0.509367\pi\)
−0.0294221 + 0.999567i \(0.509367\pi\)
\(558\) −3.14429e9 −0.766130
\(559\) 8.30724e8 0.201148
\(560\) −6.33661e8 −0.152475
\(561\) 6.50016e8 0.155437
\(562\) 8.90002e8 0.211502
\(563\) −1.44852e9 −0.342093 −0.171047 0.985263i \(-0.554715\pi\)
−0.171047 + 0.985263i \(0.554715\pi\)
\(564\) 4.79745e9 1.12599
\(565\) −6.24211e8 −0.145600
\(566\) −1.13234e10 −2.62493
\(567\) −7.20963e8 −0.166101
\(568\) 3.74481e8 0.0857454
\(569\) 7.46267e9 1.69825 0.849124 0.528193i \(-0.177130\pi\)
0.849124 + 0.528193i \(0.177130\pi\)
\(570\) 2.50390e8 0.0566310
\(571\) 1.56608e9 0.352036 0.176018 0.984387i \(-0.443678\pi\)
0.176018 + 0.984387i \(0.443678\pi\)
\(572\) −2.92367e8 −0.0653194
\(573\) −2.71096e9 −0.601980
\(574\) −1.58679e10 −3.50211
\(575\) 1.12429e9 0.246628
\(576\) −1.72358e9 −0.375797
\(577\) −6.11728e7 −0.0132569 −0.00662847 0.999978i \(-0.502110\pi\)
−0.00662847 + 0.999978i \(0.502110\pi\)
\(578\) −3.84547e9 −0.828327
\(579\) 4.40278e9 0.942653
\(580\) 7.66948e8 0.163218
\(581\) −1.12994e10 −2.39023
\(582\) −2.09682e9 −0.440890
\(583\) −3.30073e9 −0.689874
\(584\) 5.46382e8 0.113515
\(585\) 2.62364e7 0.00541824
\(586\) −2.58277e9 −0.530206
\(587\) 2.95616e9 0.603245 0.301623 0.953427i \(-0.402472\pi\)
0.301623 + 0.953427i \(0.402472\pi\)
\(588\) 3.74632e9 0.759949
\(589\) 4.93208e9 0.994550
\(590\) 1.02424e8 0.0205314
\(591\) 3.31647e9 0.660875
\(592\) 3.97372e9 0.787175
\(593\) 1.97406e8 0.0388748 0.0194374 0.999811i \(-0.493812\pi\)
0.0194374 + 0.999811i \(0.493812\pi\)
\(594\) 5.84404e8 0.114409
\(595\) 5.48581e8 0.106766
\(596\) 3.56625e9 0.690002
\(597\) 5.01935e9 0.965467
\(598\) 2.77987e8 0.0531583
\(599\) 8.20815e8 0.156046 0.0780228 0.996952i \(-0.475139\pi\)
0.0780228 + 0.996952i \(0.475139\pi\)
\(600\) −2.86382e8 −0.0541273
\(601\) 9.75118e9 1.83230 0.916150 0.400837i \(-0.131280\pi\)
0.916150 + 0.400837i \(0.131280\pi\)
\(602\) 1.56165e10 2.91740
\(603\) 1.76391e9 0.327616
\(604\) −9.59089e9 −1.77105
\(605\) 4.95386e8 0.0909495
\(606\) 7.87484e9 1.43743
\(607\) −5.21137e9 −0.945784 −0.472892 0.881120i \(-0.656790\pi\)
−0.472892 + 0.881120i \(0.656790\pi\)
\(608\) 4.93290e9 0.890103
\(609\) 6.71336e9 1.20442
\(610\) −6.89124e8 −0.122926
\(611\) −1.52818e9 −0.271038
\(612\) 1.31163e9 0.231304
\(613\) −1.02466e9 −0.179666 −0.0898330 0.995957i \(-0.528633\pi\)
−0.0898330 + 0.995957i \(0.528633\pi\)
\(614\) 3.27761e9 0.571437
\(615\) −5.95562e8 −0.103244
\(616\) −3.40377e8 −0.0586715
\(617\) −1.71355e9 −0.293696 −0.146848 0.989159i \(-0.546913\pi\)
−0.146848 + 0.989159i \(0.546913\pi\)
\(618\) −8.05010e9 −1.37196
\(619\) 3.38211e9 0.573153 0.286577 0.958057i \(-0.407483\pi\)
0.286577 + 0.958057i \(0.407483\pi\)
\(620\) 1.10987e9 0.187025
\(621\) −2.86709e8 −0.0480419
\(622\) 6.18596e9 1.03072
\(623\) 9.11499e9 1.51025
\(624\) 4.82607e8 0.0795147
\(625\) 5.88398e9 0.964031
\(626\) 2.34125e9 0.381451
\(627\) −9.16688e8 −0.148520
\(628\) −3.12000e9 −0.502684
\(629\) −3.44018e9 −0.551193
\(630\) 4.93208e8 0.0785848
\(631\) 2.96207e8 0.0469345 0.0234673 0.999725i \(-0.492529\pi\)
0.0234673 + 0.999725i \(0.492529\pi\)
\(632\) −9.50658e8 −0.149801
\(633\) 1.14922e8 0.0180091
\(634\) −6.26645e9 −0.976583
\(635\) 3.00785e8 0.0466174
\(636\) −6.66037e9 −1.02659
\(637\) −1.19336e9 −0.182929
\(638\) −5.44177e9 −0.829599
\(639\) 1.98658e9 0.301198
\(640\) 1.37793e8 0.0207777
\(641\) 5.82615e9 0.873732 0.436866 0.899526i \(-0.356088\pi\)
0.436866 + 0.899526i \(0.356088\pi\)
\(642\) 6.94909e9 1.03647
\(643\) −5.33573e9 −0.791508 −0.395754 0.918357i \(-0.629517\pi\)
−0.395754 + 0.918357i \(0.629517\pi\)
\(644\) 2.69639e9 0.397816
\(645\) 5.86125e8 0.0860065
\(646\) −3.98740e9 −0.581937
\(647\) 1.10418e10 1.60279 0.801393 0.598138i \(-0.204093\pi\)
0.801393 + 0.598138i \(0.204093\pi\)
\(648\) 7.30310e7 0.0105437
\(649\) −3.74978e8 −0.0538455
\(650\) 1.47301e9 0.210382
\(651\) 9.71503e9 1.38010
\(652\) −8.54722e9 −1.20770
\(653\) −7.56490e9 −1.06318 −0.531590 0.847002i \(-0.678405\pi\)
−0.531590 + 0.847002i \(0.678405\pi\)
\(654\) 1.94811e9 0.272327
\(655\) 7.83068e8 0.108882
\(656\) −1.09551e10 −1.51514
\(657\) 2.89849e9 0.398743
\(658\) −2.87277e10 −3.93107
\(659\) −5.59162e9 −0.761095 −0.380547 0.924761i \(-0.624264\pi\)
−0.380547 + 0.924761i \(0.624264\pi\)
\(660\) −2.06282e8 −0.0279292
\(661\) −1.88074e9 −0.253293 −0.126647 0.991948i \(-0.540421\pi\)
−0.126647 + 0.991948i \(0.540421\pi\)
\(662\) −8.28763e9 −1.11027
\(663\) −4.17808e8 −0.0556775
\(664\) 1.14459e9 0.151727
\(665\) −7.73639e8 −0.102015
\(666\) −3.09293e9 −0.405705
\(667\) 2.66973e9 0.348359
\(668\) −2.76906e9 −0.359431
\(669\) −2.72943e9 −0.352436
\(670\) −1.20668e9 −0.155000
\(671\) 2.52291e9 0.322384
\(672\) 9.71665e9 1.23516
\(673\) 8.51715e9 1.07706 0.538532 0.842605i \(-0.318979\pi\)
0.538532 + 0.842605i \(0.318979\pi\)
\(674\) 5.36634e9 0.675101
\(675\) −1.51922e9 −0.190133
\(676\) −8.37414e9 −1.04262
\(677\) −1.19888e10 −1.48497 −0.742484 0.669864i \(-0.766352\pi\)
−0.742484 + 0.669864i \(0.766352\pi\)
\(678\) 8.93705e9 1.10126
\(679\) 6.47863e9 0.794216
\(680\) −5.55693e7 −0.00677725
\(681\) −4.23158e9 −0.513437
\(682\) −7.87489e9 −0.950603
\(683\) −7.29599e9 −0.876217 −0.438109 0.898922i \(-0.644351\pi\)
−0.438109 + 0.898922i \(0.644351\pi\)
\(684\) −1.84974e9 −0.221011
\(685\) 1.71747e8 0.0204161
\(686\) −4.26509e9 −0.504422
\(687\) 6.21282e9 0.731038
\(688\) 1.07815e10 1.26218
\(689\) 2.12160e9 0.247113
\(690\) 1.96136e8 0.0227293
\(691\) 2.48922e9 0.287006 0.143503 0.989650i \(-0.454163\pi\)
0.143503 + 0.989650i \(0.454163\pi\)
\(692\) −6.04564e9 −0.693540
\(693\) −1.80566e9 −0.206096
\(694\) −2.50768e10 −2.84783
\(695\) 3.04897e7 0.00344513
\(696\) −6.80039e8 −0.0764543
\(697\) 9.48420e9 1.06093
\(698\) −5.48450e9 −0.610440
\(699\) 2.55267e9 0.282699
\(700\) 1.42877e10 1.57442
\(701\) 1.04516e10 1.14596 0.572980 0.819570i \(-0.305787\pi\)
0.572980 + 0.819570i \(0.305787\pi\)
\(702\) −3.75636e8 −0.0409814
\(703\) 4.85153e9 0.526666
\(704\) −4.31673e9 −0.466284
\(705\) −1.07822e9 −0.115890
\(706\) −1.36432e10 −1.45915
\(707\) −2.43312e10 −2.58938
\(708\) −7.56650e8 −0.0801269
\(709\) 4.68714e9 0.493908 0.246954 0.969027i \(-0.420570\pi\)
0.246954 + 0.969027i \(0.420570\pi\)
\(710\) −1.35901e9 −0.142501
\(711\) −5.04313e9 −0.526207
\(712\) −9.23316e8 −0.0958673
\(713\) 3.86342e9 0.399171
\(714\) −7.85423e9 −0.807533
\(715\) 6.57092e7 0.00672287
\(716\) 1.77847e10 1.81072
\(717\) −5.89136e9 −0.596896
\(718\) −9.35533e9 −0.943243
\(719\) −1.88048e10 −1.88676 −0.943381 0.331710i \(-0.892374\pi\)
−0.943381 + 0.331710i \(0.892374\pi\)
\(720\) 3.40508e8 0.0339988
\(721\) 2.48727e10 2.47144
\(722\) −8.91283e9 −0.881324
\(723\) −2.03260e9 −0.200017
\(724\) −1.18356e10 −1.15906
\(725\) 1.41465e10 1.37869
\(726\) −7.09263e9 −0.687906
\(727\) −1.41494e10 −1.36574 −0.682869 0.730541i \(-0.739268\pi\)
−0.682869 + 0.730541i \(0.739268\pi\)
\(728\) 2.18783e8 0.0210162
\(729\) 3.87420e8 0.0370370
\(730\) −1.98285e9 −0.188651
\(731\) −9.33391e9 −0.883798
\(732\) 5.09086e9 0.479736
\(733\) −1.16913e10 −1.09647 −0.548237 0.836323i \(-0.684701\pi\)
−0.548237 + 0.836323i \(0.684701\pi\)
\(734\) −2.65482e9 −0.247799
\(735\) −8.41982e8 −0.0782163
\(736\) 3.86407e9 0.357250
\(737\) 4.41772e9 0.406502
\(738\) 8.52688e9 0.780896
\(739\) 8.53323e9 0.777782 0.388891 0.921284i \(-0.372858\pi\)
0.388891 + 0.921284i \(0.372858\pi\)
\(740\) 1.09174e9 0.0990393
\(741\) 5.89216e8 0.0531999
\(742\) 3.98832e10 3.58407
\(743\) −2.88722e9 −0.258237 −0.129119 0.991629i \(-0.541215\pi\)
−0.129119 + 0.991629i \(0.541215\pi\)
\(744\) −9.84098e8 −0.0876058
\(745\) −8.01512e8 −0.0710171
\(746\) −8.17870e9 −0.721271
\(747\) 6.07191e9 0.532971
\(748\) 3.28500e9 0.286998
\(749\) −2.14709e10 −1.86708
\(750\) 2.09125e9 0.181006
\(751\) 1.71155e10 1.47452 0.737258 0.675611i \(-0.236120\pi\)
0.737258 + 0.675611i \(0.236120\pi\)
\(752\) −1.98334e10 −1.70073
\(753\) −6.71048e9 −0.572758
\(754\) 3.49779e9 0.297162
\(755\) 2.15554e9 0.182281
\(756\) −3.64355e9 −0.306689
\(757\) 1.12521e10 0.942751 0.471375 0.881933i \(-0.343758\pi\)
0.471375 + 0.881933i \(0.343758\pi\)
\(758\) 3.00290e10 2.50437
\(759\) −7.18064e8 −0.0596097
\(760\) 7.83669e7 0.00647567
\(761\) 9.66355e9 0.794859 0.397430 0.917633i \(-0.369902\pi\)
0.397430 + 0.917633i \(0.369902\pi\)
\(762\) −4.30645e9 −0.352596
\(763\) −6.01915e9 −0.490568
\(764\) −1.37004e10 −1.11150
\(765\) −2.94788e8 −0.0238065
\(766\) 1.27785e6 0.000102725 0
\(767\) 2.41023e8 0.0192875
\(768\) 6.19823e9 0.493746
\(769\) 6.93012e8 0.0549539 0.0274769 0.999622i \(-0.491253\pi\)
0.0274769 + 0.999622i \(0.491253\pi\)
\(770\) 1.23524e9 0.0975069
\(771\) 2.56226e9 0.201342
\(772\) 2.22504e10 1.74051
\(773\) −6.71030e8 −0.0522533 −0.0261266 0.999659i \(-0.508317\pi\)
−0.0261266 + 0.999659i \(0.508317\pi\)
\(774\) −8.39176e9 −0.650519
\(775\) 2.04716e10 1.57978
\(776\) −6.56262e8 −0.0504152
\(777\) 9.55636e9 0.730835
\(778\) 1.65681e10 1.26138
\(779\) −1.33751e10 −1.01372
\(780\) 1.32591e8 0.0100042
\(781\) 4.97540e9 0.373723
\(782\) −3.12343e9 −0.233565
\(783\) −3.60753e9 −0.268561
\(784\) −1.54879e10 −1.14785
\(785\) 7.01216e8 0.0517378
\(786\) −1.12115e10 −0.823538
\(787\) −1.62943e10 −1.19158 −0.595790 0.803141i \(-0.703161\pi\)
−0.595790 + 0.803141i \(0.703161\pi\)
\(788\) 1.67605e10 1.22024
\(789\) 2.09031e9 0.151510
\(790\) 3.44999e9 0.248956
\(791\) −2.76132e10 −1.98380
\(792\) 1.82907e8 0.0130825
\(793\) −1.62165e9 −0.115478
\(794\) 5.61933e9 0.398394
\(795\) 1.49691e9 0.105660
\(796\) 2.53664e10 1.78264
\(797\) 6.31608e9 0.441920 0.220960 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(798\) 1.10765e10 0.771599
\(799\) 1.71704e10 1.19088
\(800\) 2.04751e10 1.41387
\(801\) −4.89808e9 −0.336753
\(802\) −3.49780e10 −2.39433
\(803\) 7.25929e9 0.494755
\(804\) 8.91430e9 0.604910
\(805\) −6.06011e8 −0.0409444
\(806\) 5.06172e9 0.340506
\(807\) −4.00595e9 −0.268317
\(808\) 2.46466e9 0.164368
\(809\) −5.93171e9 −0.393876 −0.196938 0.980416i \(-0.563100\pi\)
−0.196938 + 0.980416i \(0.563100\pi\)
\(810\) −2.65033e8 −0.0175228
\(811\) 3.74376e9 0.246454 0.123227 0.992379i \(-0.460676\pi\)
0.123227 + 0.992379i \(0.460676\pi\)
\(812\) 3.39274e10 2.22385
\(813\) 5.77899e9 0.377168
\(814\) −7.74628e9 −0.503394
\(815\) 1.92098e9 0.124300
\(816\) −5.42251e9 −0.349369
\(817\) 1.31632e10 0.844470
\(818\) −3.63037e9 −0.231907
\(819\) 1.16062e9 0.0738236
\(820\) −3.00981e9 −0.190629
\(821\) 2.22338e10 1.40221 0.701103 0.713060i \(-0.252691\pi\)
0.701103 + 0.713060i \(0.252691\pi\)
\(822\) −2.45896e9 −0.154419
\(823\) −2.57937e10 −1.61292 −0.806462 0.591286i \(-0.798620\pi\)
−0.806462 + 0.591286i \(0.798620\pi\)
\(824\) −2.51952e9 −0.156882
\(825\) −3.80491e9 −0.235915
\(826\) 4.53092e9 0.279741
\(827\) 2.36853e10 1.45616 0.728082 0.685490i \(-0.240412\pi\)
0.728082 + 0.685490i \(0.240412\pi\)
\(828\) −1.44895e9 −0.0887046
\(829\) −3.07672e10 −1.87563 −0.937815 0.347137i \(-0.887154\pi\)
−0.937815 + 0.347137i \(0.887154\pi\)
\(830\) −4.15377e9 −0.252156
\(831\) −1.21334e10 −0.733462
\(832\) 2.77465e9 0.167023
\(833\) 1.34084e10 0.803746
\(834\) −4.36532e8 −0.0260576
\(835\) 6.22344e8 0.0369937
\(836\) −4.63268e9 −0.274227
\(837\) −5.22052e9 −0.307733
\(838\) −1.89594e10 −1.11294
\(839\) −9.92051e9 −0.579919 −0.289959 0.957039i \(-0.593642\pi\)
−0.289959 + 0.957039i \(0.593642\pi\)
\(840\) 1.54364e8 0.00898605
\(841\) 1.63422e10 0.947379
\(842\) −1.53528e10 −0.886332
\(843\) 1.47769e9 0.0849544
\(844\) 5.80786e8 0.0332520
\(845\) 1.88208e9 0.107310
\(846\) 1.54373e10 0.876546
\(847\) 2.19144e10 1.23919
\(848\) 2.75351e10 1.55060
\(849\) −1.88004e10 −1.05436
\(850\) −1.65505e10 −0.924371
\(851\) 3.80032e9 0.211382
\(852\) 1.00396e10 0.556132
\(853\) 1.89035e10 1.04285 0.521424 0.853298i \(-0.325401\pi\)
0.521424 + 0.853298i \(0.325401\pi\)
\(854\) −3.04847e10 −1.67487
\(855\) 4.15727e8 0.0227471
\(856\) 2.17492e9 0.118518
\(857\) 1.56380e10 0.848688 0.424344 0.905501i \(-0.360505\pi\)
0.424344 + 0.905501i \(0.360505\pi\)
\(858\) −9.40782e8 −0.0508491
\(859\) 3.59002e10 1.93251 0.966254 0.257592i \(-0.0829291\pi\)
0.966254 + 0.257592i \(0.0829291\pi\)
\(860\) 2.96211e9 0.158802
\(861\) −2.63459e10 −1.40670
\(862\) −1.65855e10 −0.881968
\(863\) −3.31160e10 −1.75388 −0.876941 0.480599i \(-0.840419\pi\)
−0.876941 + 0.480599i \(0.840419\pi\)
\(864\) −5.22139e9 −0.275415
\(865\) 1.35875e9 0.0713812
\(866\) −8.06708e9 −0.422089
\(867\) −6.38470e9 −0.332716
\(868\) 4.90970e10 2.54822
\(869\) −1.26306e10 −0.652910
\(870\) 2.46790e9 0.127060
\(871\) −2.83956e9 −0.145609
\(872\) 6.09718e8 0.0311402
\(873\) −3.48139e9 −0.177094
\(874\) 4.40483e9 0.223172
\(875\) −6.46143e9 −0.326062
\(876\) 1.46482e10 0.736239
\(877\) 3.12644e10 1.56513 0.782567 0.622566i \(-0.213910\pi\)
0.782567 + 0.622566i \(0.213910\pi\)
\(878\) 3.00164e10 1.49668
\(879\) −4.28823e9 −0.212969
\(880\) 8.52804e8 0.0421852
\(881\) −1.15281e9 −0.0567991 −0.0283996 0.999597i \(-0.509041\pi\)
−0.0283996 + 0.999597i \(0.509041\pi\)
\(882\) 1.20550e10 0.591597
\(883\) −7.36328e9 −0.359922 −0.179961 0.983674i \(-0.557597\pi\)
−0.179961 + 0.983674i \(0.557597\pi\)
\(884\) −2.11149e9 −0.102803
\(885\) 1.70056e8 0.00824691
\(886\) 3.45413e9 0.166848
\(887\) 2.01350e10 0.968765 0.484383 0.874856i \(-0.339044\pi\)
0.484383 + 0.874856i \(0.339044\pi\)
\(888\) −9.68025e8 −0.0463918
\(889\) 1.33058e10 0.635163
\(890\) 3.35076e9 0.159323
\(891\) 9.70298e8 0.0459550
\(892\) −1.37938e10 −0.650738
\(893\) −2.42147e10 −1.13789
\(894\) 1.14755e10 0.537145
\(895\) −3.99710e9 −0.186365
\(896\) 6.09554e9 0.283096
\(897\) 4.61548e8 0.0213522
\(898\) −5.74406e10 −2.64698
\(899\) 4.86117e10 2.23142
\(900\) −7.67773e9 −0.351062
\(901\) −2.38380e10 −1.08576
\(902\) 2.13557e10 0.968925
\(903\) 2.59284e10 1.17184
\(904\) 2.79712e9 0.125928
\(905\) 2.66005e9 0.119294
\(906\) −3.08617e10 −1.37870
\(907\) −4.37196e10 −1.94559 −0.972793 0.231677i \(-0.925579\pi\)
−0.972793 + 0.231677i \(0.925579\pi\)
\(908\) −2.13852e10 −0.948010
\(909\) 1.30747e10 0.577377
\(910\) −7.93974e8 −0.0349270
\(911\) 2.71783e10 1.19099 0.595494 0.803360i \(-0.296956\pi\)
0.595494 + 0.803360i \(0.296956\pi\)
\(912\) 7.64712e9 0.333823
\(913\) 1.52071e10 0.661302
\(914\) 3.34337e10 1.44835
\(915\) −1.14417e9 −0.0493759
\(916\) 3.13978e10 1.34979
\(917\) 3.46405e10 1.48351
\(918\) 4.22059e9 0.180063
\(919\) −1.97736e10 −0.840393 −0.420196 0.907433i \(-0.638039\pi\)
−0.420196 + 0.907433i \(0.638039\pi\)
\(920\) 6.13867e7 0.00259906
\(921\) 5.44188e9 0.229530
\(922\) −2.18777e10 −0.919269
\(923\) −3.19802e9 −0.133868
\(924\) −9.12529e9 −0.380535
\(925\) 2.01373e10 0.836576
\(926\) 1.89088e10 0.782576
\(927\) −1.33657e10 −0.551079
\(928\) 4.86198e10 1.99708
\(929\) −3.68821e10 −1.50925 −0.754625 0.656156i \(-0.772181\pi\)
−0.754625 + 0.656156i \(0.772181\pi\)
\(930\) 3.57134e9 0.145593
\(931\) −1.89092e10 −0.767981
\(932\) 1.29005e10 0.521976
\(933\) 1.02707e10 0.414012
\(934\) 5.48113e10 2.20118
\(935\) −7.38300e8 −0.0295388
\(936\) −1.17566e8 −0.00468616
\(937\) −2.54302e10 −1.00986 −0.504930 0.863160i \(-0.668482\pi\)
−0.504930 + 0.863160i \(0.668482\pi\)
\(938\) −5.33800e10 −2.11188
\(939\) 3.88723e9 0.153218
\(940\) −5.44903e9 −0.213979
\(941\) −1.08449e10 −0.424288 −0.212144 0.977238i \(-0.568045\pi\)
−0.212144 + 0.977238i \(0.568045\pi\)
\(942\) −1.00396e10 −0.391324
\(943\) −1.04771e10 −0.406864
\(944\) 3.12811e9 0.121027
\(945\) 8.18883e8 0.0315654
\(946\) −2.10172e10 −0.807155
\(947\) −2.44245e9 −0.0934547 −0.0467273 0.998908i \(-0.514879\pi\)
−0.0467273 + 0.998908i \(0.514879\pi\)
\(948\) −2.54866e10 −0.971588
\(949\) −4.66603e9 −0.177221
\(950\) 2.33405e10 0.883237
\(951\) −1.04043e10 −0.392267
\(952\) −2.45822e9 −0.0923402
\(953\) −3.90381e10 −1.46104 −0.730522 0.682890i \(-0.760723\pi\)
−0.730522 + 0.682890i \(0.760723\pi\)
\(954\) −2.14318e10 −0.799172
\(955\) 3.07916e9 0.114398
\(956\) −2.97733e10 −1.10211
\(957\) −9.03508e9 −0.333227
\(958\) −2.70901e10 −0.995478
\(959\) 7.59756e9 0.278169
\(960\) 1.95768e9 0.0714154
\(961\) 4.28343e10 1.55690
\(962\) 4.97905e9 0.180316
\(963\) 1.15377e10 0.416320
\(964\) −1.02722e10 −0.369312
\(965\) −5.00076e9 −0.179139
\(966\) 8.67648e9 0.309687
\(967\) −2.52051e10 −0.896388 −0.448194 0.893936i \(-0.647933\pi\)
−0.448194 + 0.893936i \(0.647933\pi\)
\(968\) −2.21985e9 −0.0786610
\(969\) −6.62036e9 −0.233748
\(970\) 2.38161e9 0.0837855
\(971\) −6.22835e9 −0.218326 −0.109163 0.994024i \(-0.534817\pi\)
−0.109163 + 0.994024i \(0.534817\pi\)
\(972\) 1.95791e9 0.0683852
\(973\) 1.34877e9 0.0469400
\(974\) −1.66601e10 −0.577726
\(975\) 2.44567e9 0.0845048
\(976\) −2.10465e10 −0.724611
\(977\) 1.86878e10 0.641103 0.320551 0.947231i \(-0.396132\pi\)
0.320551 + 0.947231i \(0.396132\pi\)
\(978\) −2.75034e10 −0.940155
\(979\) −1.22673e10 −0.417839
\(980\) −4.25514e9 −0.144419
\(981\) 3.23448e9 0.109386
\(982\) 3.69908e10 1.24653
\(983\) 4.26187e10 1.43108 0.715538 0.698574i \(-0.246182\pi\)
0.715538 + 0.698574i \(0.246182\pi\)
\(984\) 2.66874e9 0.0892943
\(985\) −3.76691e9 −0.125591
\(986\) −3.93007e10 −1.30566
\(987\) −4.76972e10 −1.57900
\(988\) 2.97773e9 0.0982283
\(989\) 1.03111e10 0.338935
\(990\) −6.63777e8 −0.0217420
\(991\) −3.72023e9 −0.121426 −0.0607130 0.998155i \(-0.519337\pi\)
−0.0607130 + 0.998155i \(0.519337\pi\)
\(992\) 7.03586e10 2.28837
\(993\) −1.37601e10 −0.445964
\(994\) −6.01185e10 −1.94158
\(995\) −5.70107e9 −0.183474
\(996\) 3.06857e10 0.984077
\(997\) 3.00026e10 0.958794 0.479397 0.877598i \(-0.340855\pi\)
0.479397 + 0.877598i \(0.340855\pi\)
\(998\) 3.29481e10 1.04924
\(999\) −5.13526e9 −0.162961
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.1.16 17
3.2 odd 2 531.8.a.d.1.2 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.16 17 1.1 even 1 trivial
531.8.a.d.1.2 17 3.2 odd 2