Properties

Label 177.8.a.c.1.3
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-14.3045\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.3045 q^{2} -27.0000 q^{3} +76.6177 q^{4} +348.885 q^{5} +386.221 q^{6} -1316.66 q^{7} +734.996 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-14.3045 q^{2} -27.0000 q^{3} +76.6177 q^{4} +348.885 q^{5} +386.221 q^{6} -1316.66 q^{7} +734.996 q^{8} +729.000 q^{9} -4990.62 q^{10} +4994.84 q^{11} -2068.68 q^{12} +3397.75 q^{13} +18834.2 q^{14} -9419.91 q^{15} -20320.8 q^{16} +13080.3 q^{17} -10428.0 q^{18} +8796.50 q^{19} +26730.8 q^{20} +35549.9 q^{21} -71448.5 q^{22} +107491. q^{23} -19844.9 q^{24} +43596.1 q^{25} -48603.0 q^{26} -19683.0 q^{27} -100880. q^{28} +31901.3 q^{29} +134747. q^{30} -154292. q^{31} +196599. q^{32} -134861. q^{33} -187107. q^{34} -459365. q^{35} +55854.3 q^{36} -528856. q^{37} -125829. q^{38} -91739.2 q^{39} +256430. q^{40} -480637. q^{41} -508523. q^{42} -438138. q^{43} +382693. q^{44} +254338. q^{45} -1.53760e6 q^{46} +1.23501e6 q^{47} +548661. q^{48} +910062. q^{49} -623619. q^{50} -353169. q^{51} +260328. q^{52} -1.18221e6 q^{53} +281555. q^{54} +1.74263e6 q^{55} -967743. q^{56} -237505. q^{57} -456331. q^{58} -205379. q^{59} -721732. q^{60} +873606. q^{61} +2.20706e6 q^{62} -959848. q^{63} -211175. q^{64} +1.18542e6 q^{65} +1.92911e6 q^{66} -832263. q^{67} +1.00219e6 q^{68} -2.90226e6 q^{69} +6.57097e6 q^{70} +3.15545e6 q^{71} +535812. q^{72} +2.69697e6 q^{73} +7.56500e6 q^{74} -1.17709e6 q^{75} +673968. q^{76} -6.57652e6 q^{77} +1.31228e6 q^{78} +6.93335e6 q^{79} -7.08963e6 q^{80} +531441. q^{81} +6.87525e6 q^{82} -3.14360e6 q^{83} +2.72375e6 q^{84} +4.56354e6 q^{85} +6.26733e6 q^{86} -861335. q^{87} +3.67119e6 q^{88} -470189. q^{89} -3.63816e6 q^{90} -4.47369e6 q^{91} +8.23572e6 q^{92} +4.16587e6 q^{93} -1.76661e7 q^{94} +3.06897e6 q^{95} -5.30816e6 q^{96} +1.03272e7 q^{97} -1.30179e7 q^{98} +3.64124e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.3045 −1.26435 −0.632174 0.774826i \(-0.717837\pi\)
−0.632174 + 0.774826i \(0.717837\pi\)
\(3\) −27.0000 −0.577350
\(4\) 76.6177 0.598576
\(5\) 348.885 1.24821 0.624105 0.781340i \(-0.285464\pi\)
0.624105 + 0.781340i \(0.285464\pi\)
\(6\) 386.221 0.729972
\(7\) −1316.66 −1.45088 −0.725441 0.688285i \(-0.758364\pi\)
−0.725441 + 0.688285i \(0.758364\pi\)
\(8\) 734.996 0.507540
\(9\) 729.000 0.333333
\(10\) −4990.62 −1.57817
\(11\) 4994.84 1.13148 0.565740 0.824584i \(-0.308591\pi\)
0.565740 + 0.824584i \(0.308591\pi\)
\(12\) −2068.68 −0.345588
\(13\) 3397.75 0.428933 0.214466 0.976731i \(-0.431199\pi\)
0.214466 + 0.976731i \(0.431199\pi\)
\(14\) 18834.2 1.83442
\(15\) −9419.91 −0.720655
\(16\) −20320.8 −1.24028
\(17\) 13080.3 0.645725 0.322863 0.946446i \(-0.395355\pi\)
0.322863 + 0.946446i \(0.395355\pi\)
\(18\) −10428.0 −0.421449
\(19\) 8796.50 0.294220 0.147110 0.989120i \(-0.453003\pi\)
0.147110 + 0.989120i \(0.453003\pi\)
\(20\) 26730.8 0.747149
\(21\) 35549.9 0.837667
\(22\) −71448.5 −1.43058
\(23\) 107491. 1.84215 0.921075 0.389385i \(-0.127313\pi\)
0.921075 + 0.389385i \(0.127313\pi\)
\(24\) −19844.9 −0.293028
\(25\) 43596.1 0.558030
\(26\) −48603.0 −0.542320
\(27\) −19683.0 −0.192450
\(28\) −100880. −0.868462
\(29\) 31901.3 0.242893 0.121447 0.992598i \(-0.461247\pi\)
0.121447 + 0.992598i \(0.461247\pi\)
\(30\) 134747. 0.911158
\(31\) −154292. −0.930200 −0.465100 0.885258i \(-0.653982\pi\)
−0.465100 + 0.885258i \(0.653982\pi\)
\(32\) 196599. 1.06061
\(33\) −134861. −0.653260
\(34\) −187107. −0.816422
\(35\) −459365. −1.81101
\(36\) 55854.3 0.199525
\(37\) −528856. −1.71645 −0.858225 0.513274i \(-0.828432\pi\)
−0.858225 + 0.513274i \(0.828432\pi\)
\(38\) −125829. −0.371996
\(39\) −91739.2 −0.247645
\(40\) 256430. 0.633517
\(41\) −480637. −1.08911 −0.544557 0.838724i \(-0.683302\pi\)
−0.544557 + 0.838724i \(0.683302\pi\)
\(42\) −508523. −1.05910
\(43\) −438138. −0.840371 −0.420186 0.907438i \(-0.638035\pi\)
−0.420186 + 0.907438i \(0.638035\pi\)
\(44\) 382693. 0.677277
\(45\) 254338. 0.416070
\(46\) −1.53760e6 −2.32912
\(47\) 1.23501e6 1.73511 0.867555 0.497341i \(-0.165690\pi\)
0.867555 + 0.497341i \(0.165690\pi\)
\(48\) 548661. 0.716078
\(49\) 910062. 1.10506
\(50\) −623619. −0.705544
\(51\) −353169. −0.372810
\(52\) 260328. 0.256749
\(53\) −1.18221e6 −1.09076 −0.545379 0.838190i \(-0.683614\pi\)
−0.545379 + 0.838190i \(0.683614\pi\)
\(54\) 281555. 0.243324
\(55\) 1.74263e6 1.41233
\(56\) −967743. −0.736380
\(57\) −237505. −0.169868
\(58\) −456331. −0.307102
\(59\) −205379. −0.130189
\(60\) −721732. −0.431367
\(61\) 873606. 0.492789 0.246395 0.969170i \(-0.420754\pi\)
0.246395 + 0.969170i \(0.420754\pi\)
\(62\) 2.20706e6 1.17610
\(63\) −959848. −0.483627
\(64\) −211175. −0.100696
\(65\) 1.18542e6 0.535399
\(66\) 1.92911e6 0.825948
\(67\) −832263. −0.338064 −0.169032 0.985611i \(-0.554064\pi\)
−0.169032 + 0.985611i \(0.554064\pi\)
\(68\) 1.00219e6 0.386516
\(69\) −2.90226e6 −1.06357
\(70\) 6.57097e6 2.28974
\(71\) 3.15545e6 1.04630 0.523152 0.852240i \(-0.324756\pi\)
0.523152 + 0.852240i \(0.324756\pi\)
\(72\) 535812. 0.169180
\(73\) 2.69697e6 0.811420 0.405710 0.914002i \(-0.367024\pi\)
0.405710 + 0.914002i \(0.367024\pi\)
\(74\) 7.56500e6 2.17019
\(75\) −1.17709e6 −0.322179
\(76\) 673968. 0.176113
\(77\) −6.57652e6 −1.64164
\(78\) 1.31228e6 0.313109
\(79\) 6.93335e6 1.58215 0.791076 0.611718i \(-0.209521\pi\)
0.791076 + 0.611718i \(0.209521\pi\)
\(80\) −7.08963e6 −1.54813
\(81\) 531441. 0.111111
\(82\) 6.87525e6 1.37702
\(83\) −3.14360e6 −0.603468 −0.301734 0.953392i \(-0.597565\pi\)
−0.301734 + 0.953392i \(0.597565\pi\)
\(84\) 2.72375e6 0.501407
\(85\) 4.56354e6 0.806001
\(86\) 6.26733e6 1.06252
\(87\) −861335. −0.140235
\(88\) 3.67119e6 0.574271
\(89\) −470189. −0.0706980 −0.0353490 0.999375i \(-0.511254\pi\)
−0.0353490 + 0.999375i \(0.511254\pi\)
\(90\) −3.63816e6 −0.526058
\(91\) −4.47369e6 −0.622331
\(92\) 8.23572e6 1.10267
\(93\) 4.16587e6 0.537051
\(94\) −1.76661e7 −2.19378
\(95\) 3.06897e6 0.367249
\(96\) −5.30816e6 −0.612343
\(97\) 1.03272e7 1.14890 0.574448 0.818541i \(-0.305217\pi\)
0.574448 + 0.818541i \(0.305217\pi\)
\(98\) −1.30179e7 −1.39718
\(99\) 3.64124e6 0.377160
\(100\) 3.34023e6 0.334023
\(101\) −1.36297e7 −1.31632 −0.658159 0.752879i \(-0.728664\pi\)
−0.658159 + 0.752879i \(0.728664\pi\)
\(102\) 5.05190e6 0.471361
\(103\) −1.58162e6 −0.142618 −0.0713088 0.997454i \(-0.522718\pi\)
−0.0713088 + 0.997454i \(0.522718\pi\)
\(104\) 2.49733e6 0.217701
\(105\) 1.24029e7 1.04558
\(106\) 1.69109e7 1.37910
\(107\) 1.70106e7 1.34238 0.671191 0.741284i \(-0.265783\pi\)
0.671191 + 0.741284i \(0.265783\pi\)
\(108\) −1.50807e6 −0.115196
\(109\) −5.78972e6 −0.428218 −0.214109 0.976810i \(-0.568685\pi\)
−0.214109 + 0.976810i \(0.568685\pi\)
\(110\) −2.49273e7 −1.78567
\(111\) 1.42791e7 0.990993
\(112\) 2.67557e7 1.79950
\(113\) 2.12637e7 1.38632 0.693161 0.720783i \(-0.256217\pi\)
0.693161 + 0.720783i \(0.256217\pi\)
\(114\) 3.39739e6 0.214772
\(115\) 3.75021e7 2.29939
\(116\) 2.44421e6 0.145390
\(117\) 2.47696e6 0.142978
\(118\) 2.93784e6 0.164604
\(119\) −1.72224e7 −0.936871
\(120\) −6.92360e6 −0.365761
\(121\) 5.46122e6 0.280247
\(122\) −1.24965e7 −0.623057
\(123\) 1.29772e7 0.628801
\(124\) −1.18215e7 −0.556795
\(125\) −1.20466e7 −0.551672
\(126\) 1.37301e7 0.611473
\(127\) 1.24454e7 0.539132 0.269566 0.962982i \(-0.413120\pi\)
0.269566 + 0.962982i \(0.413120\pi\)
\(128\) −2.21439e7 −0.933294
\(129\) 1.18297e7 0.485189
\(130\) −1.69569e7 −0.676930
\(131\) 8.16356e6 0.317271 0.158635 0.987337i \(-0.449291\pi\)
0.158635 + 0.987337i \(0.449291\pi\)
\(132\) −1.03327e7 −0.391026
\(133\) −1.15820e7 −0.426878
\(134\) 1.19051e7 0.427430
\(135\) −6.86711e6 −0.240218
\(136\) 9.61400e6 0.327731
\(137\) 4.91255e7 1.63224 0.816122 0.577880i \(-0.196120\pi\)
0.816122 + 0.577880i \(0.196120\pi\)
\(138\) 4.15153e7 1.34472
\(139\) −2.62822e6 −0.0830060 −0.0415030 0.999138i \(-0.513215\pi\)
−0.0415030 + 0.999138i \(0.513215\pi\)
\(140\) −3.51955e7 −1.08402
\(141\) −3.33452e7 −1.00177
\(142\) −4.51371e7 −1.32289
\(143\) 1.69712e7 0.485329
\(144\) −1.48139e7 −0.413428
\(145\) 1.11299e7 0.303182
\(146\) −3.85787e7 −1.02592
\(147\) −2.45717e7 −0.638005
\(148\) −4.05197e7 −1.02743
\(149\) −1.97981e7 −0.490311 −0.245156 0.969484i \(-0.578839\pi\)
−0.245156 + 0.969484i \(0.578839\pi\)
\(150\) 1.68377e7 0.407346
\(151\) 1.33779e7 0.316205 0.158103 0.987423i \(-0.449462\pi\)
0.158103 + 0.987423i \(0.449462\pi\)
\(152\) 6.46539e6 0.149328
\(153\) 9.53557e6 0.215242
\(154\) 9.40736e7 2.07561
\(155\) −5.38301e7 −1.16109
\(156\) −7.02885e6 −0.148234
\(157\) −2.96493e7 −0.611457 −0.305728 0.952119i \(-0.598900\pi\)
−0.305728 + 0.952119i \(0.598900\pi\)
\(158\) −9.91778e7 −2.00039
\(159\) 3.19196e7 0.629749
\(160\) 6.85904e7 1.32386
\(161\) −1.41530e8 −2.67274
\(162\) −7.60198e6 −0.140483
\(163\) −5.91942e7 −1.07059 −0.535294 0.844666i \(-0.679799\pi\)
−0.535294 + 0.844666i \(0.679799\pi\)
\(164\) −3.68253e7 −0.651918
\(165\) −4.70509e7 −0.815406
\(166\) 4.49675e7 0.762993
\(167\) 1.03566e8 1.72071 0.860357 0.509692i \(-0.170241\pi\)
0.860357 + 0.509692i \(0.170241\pi\)
\(168\) 2.61291e7 0.425149
\(169\) −5.12038e7 −0.816017
\(170\) −6.52790e7 −1.01907
\(171\) 6.41265e6 0.0980733
\(172\) −3.35691e7 −0.503026
\(173\) 7.49114e7 1.09998 0.549992 0.835170i \(-0.314631\pi\)
0.549992 + 0.835170i \(0.314631\pi\)
\(174\) 1.23209e7 0.177305
\(175\) −5.74014e7 −0.809635
\(176\) −1.01499e8 −1.40336
\(177\) 5.54523e6 0.0751646
\(178\) 6.72580e6 0.0893869
\(179\) 1.39539e8 1.81849 0.909244 0.416263i \(-0.136661\pi\)
0.909244 + 0.416263i \(0.136661\pi\)
\(180\) 1.94868e7 0.249050
\(181\) 1.39686e7 0.175096 0.0875480 0.996160i \(-0.472097\pi\)
0.0875480 + 0.996160i \(0.472097\pi\)
\(182\) 6.39938e7 0.786843
\(183\) −2.35874e7 −0.284512
\(184\) 7.90055e7 0.934965
\(185\) −1.84510e8 −2.14249
\(186\) −5.95906e7 −0.679019
\(187\) 6.53342e7 0.730625
\(188\) 9.46235e7 1.03860
\(189\) 2.59159e7 0.279222
\(190\) −4.39000e7 −0.464330
\(191\) −3.81975e7 −0.396660 −0.198330 0.980135i \(-0.563552\pi\)
−0.198330 + 0.980135i \(0.563552\pi\)
\(192\) 5.70174e6 0.0581370
\(193\) −3.69244e7 −0.369711 −0.184856 0.982766i \(-0.559182\pi\)
−0.184856 + 0.982766i \(0.559182\pi\)
\(194\) −1.47725e8 −1.45260
\(195\) −3.20065e7 −0.309113
\(196\) 6.97268e7 0.661460
\(197\) 1.04700e8 0.975695 0.487847 0.872929i \(-0.337782\pi\)
0.487847 + 0.872929i \(0.337782\pi\)
\(198\) −5.20859e7 −0.476861
\(199\) −1.24096e8 −1.11628 −0.558138 0.829748i \(-0.688484\pi\)
−0.558138 + 0.829748i \(0.688484\pi\)
\(200\) 3.20430e7 0.283222
\(201\) 2.24711e7 0.195181
\(202\) 1.94965e8 1.66428
\(203\) −4.20033e7 −0.352409
\(204\) −2.70590e7 −0.223155
\(205\) −1.67687e8 −1.35944
\(206\) 2.26243e7 0.180318
\(207\) 7.83610e7 0.614050
\(208\) −6.90449e7 −0.531998
\(209\) 4.39371e7 0.332904
\(210\) −1.77416e8 −1.32198
\(211\) −8.39750e7 −0.615406 −0.307703 0.951483i \(-0.599560\pi\)
−0.307703 + 0.951483i \(0.599560\pi\)
\(212\) −9.05781e7 −0.652901
\(213\) −8.51973e7 −0.604084
\(214\) −2.43327e8 −1.69724
\(215\) −1.52860e8 −1.04896
\(216\) −1.44669e7 −0.0976761
\(217\) 2.03150e8 1.34961
\(218\) 8.28188e7 0.541416
\(219\) −7.28181e7 −0.468474
\(220\) 1.33516e8 0.845384
\(221\) 4.44437e7 0.276973
\(222\) −2.04255e8 −1.25296
\(223\) 2.55405e8 1.54227 0.771137 0.636669i \(-0.219688\pi\)
0.771137 + 0.636669i \(0.219688\pi\)
\(224\) −2.58854e8 −1.53882
\(225\) 3.17815e7 0.186010
\(226\) −3.04166e8 −1.75279
\(227\) 2.94977e8 1.67377 0.836887 0.547375i \(-0.184373\pi\)
0.836887 + 0.547375i \(0.184373\pi\)
\(228\) −1.81971e7 −0.101679
\(229\) 3.95841e7 0.217819 0.108910 0.994052i \(-0.465264\pi\)
0.108910 + 0.994052i \(0.465264\pi\)
\(230\) −5.36447e8 −2.90723
\(231\) 1.77566e8 0.947803
\(232\) 2.34473e7 0.123278
\(233\) −1.72426e7 −0.0893009 −0.0446505 0.999003i \(-0.514217\pi\)
−0.0446505 + 0.999003i \(0.514217\pi\)
\(234\) −3.54316e7 −0.180773
\(235\) 4.30876e8 2.16578
\(236\) −1.57357e7 −0.0779279
\(237\) −1.87200e8 −0.913456
\(238\) 2.46357e8 1.18453
\(239\) 3.47868e6 0.0164824 0.00824121 0.999966i \(-0.497377\pi\)
0.00824121 + 0.999966i \(0.497377\pi\)
\(240\) 1.91420e8 0.893816
\(241\) 2.24809e8 1.03456 0.517279 0.855817i \(-0.326945\pi\)
0.517279 + 0.855817i \(0.326945\pi\)
\(242\) −7.81198e7 −0.354330
\(243\) −1.43489e7 −0.0641500
\(244\) 6.69337e7 0.294972
\(245\) 3.17507e8 1.37934
\(246\) −1.85632e8 −0.795023
\(247\) 2.98883e7 0.126201
\(248\) −1.13404e8 −0.472113
\(249\) 8.48772e7 0.348412
\(250\) 1.72321e8 0.697505
\(251\) 4.20064e8 1.67671 0.838354 0.545127i \(-0.183519\pi\)
0.838354 + 0.545127i \(0.183519\pi\)
\(252\) −7.35414e7 −0.289487
\(253\) 5.36900e8 2.08436
\(254\) −1.78024e8 −0.681650
\(255\) −1.23216e8 −0.465345
\(256\) 3.43787e8 1.28070
\(257\) 1.99799e8 0.734221 0.367111 0.930177i \(-0.380347\pi\)
0.367111 + 0.930177i \(0.380347\pi\)
\(258\) −1.69218e8 −0.613447
\(259\) 6.96325e8 2.49036
\(260\) 9.08245e7 0.320477
\(261\) 2.32561e7 0.0809644
\(262\) −1.16775e8 −0.401141
\(263\) 5.10505e8 1.73043 0.865216 0.501399i \(-0.167181\pi\)
0.865216 + 0.501399i \(0.167181\pi\)
\(264\) −9.91220e7 −0.331556
\(265\) −4.12455e8 −1.36150
\(266\) 1.65675e8 0.539723
\(267\) 1.26951e7 0.0408175
\(268\) −6.37661e7 −0.202357
\(269\) −6.12082e8 −1.91724 −0.958621 0.284686i \(-0.908111\pi\)
−0.958621 + 0.284686i \(0.908111\pi\)
\(270\) 9.82304e7 0.303719
\(271\) 1.95076e8 0.595404 0.297702 0.954659i \(-0.403780\pi\)
0.297702 + 0.954659i \(0.403780\pi\)
\(272\) −2.65803e8 −0.800882
\(273\) 1.20790e8 0.359303
\(274\) −7.02714e8 −2.06372
\(275\) 2.17755e8 0.631400
\(276\) −2.22364e8 −0.636625
\(277\) −4.11429e8 −1.16310 −0.581548 0.813512i \(-0.697553\pi\)
−0.581548 + 0.813512i \(0.697553\pi\)
\(278\) 3.75953e7 0.104949
\(279\) −1.12479e8 −0.310067
\(280\) −3.37632e8 −0.919157
\(281\) −1.45356e8 −0.390805 −0.195403 0.980723i \(-0.562601\pi\)
−0.195403 + 0.980723i \(0.562601\pi\)
\(282\) 4.76985e8 1.26658
\(283\) −7.33769e8 −1.92445 −0.962226 0.272252i \(-0.912232\pi\)
−0.962226 + 0.272252i \(0.912232\pi\)
\(284\) 2.41764e8 0.626292
\(285\) −8.28622e7 −0.212031
\(286\) −2.42764e8 −0.613625
\(287\) 6.32837e8 1.58018
\(288\) 1.43320e8 0.353536
\(289\) −2.39243e8 −0.583039
\(290\) −1.59207e8 −0.383328
\(291\) −2.78834e8 −0.663315
\(292\) 2.06635e8 0.485696
\(293\) −3.59570e8 −0.835117 −0.417558 0.908650i \(-0.637114\pi\)
−0.417558 + 0.908650i \(0.637114\pi\)
\(294\) 3.51484e8 0.806660
\(295\) −7.16538e7 −0.162503
\(296\) −3.88707e8 −0.871167
\(297\) −9.83134e7 −0.217753
\(298\) 2.83202e8 0.619924
\(299\) 3.65228e8 0.790159
\(300\) −9.01862e7 −0.192848
\(301\) 5.76880e8 1.21928
\(302\) −1.91364e8 −0.399794
\(303\) 3.68001e8 0.759976
\(304\) −1.78752e8 −0.364916
\(305\) 3.04788e8 0.615105
\(306\) −1.36401e8 −0.272141
\(307\) −2.59711e8 −0.512278 −0.256139 0.966640i \(-0.582450\pi\)
−0.256139 + 0.966640i \(0.582450\pi\)
\(308\) −5.03878e8 −0.982648
\(309\) 4.27039e7 0.0823403
\(310\) 7.70010e8 1.46802
\(311\) 4.42793e8 0.834717 0.417359 0.908742i \(-0.362956\pi\)
0.417359 + 0.908742i \(0.362956\pi\)
\(312\) −6.74280e7 −0.125689
\(313\) 4.37731e7 0.0806867 0.0403433 0.999186i \(-0.487155\pi\)
0.0403433 + 0.999186i \(0.487155\pi\)
\(314\) 4.24117e8 0.773094
\(315\) −3.34877e8 −0.603668
\(316\) 5.31217e8 0.947038
\(317\) 1.74353e8 0.307413 0.153706 0.988117i \(-0.450879\pi\)
0.153706 + 0.988117i \(0.450879\pi\)
\(318\) −4.56593e8 −0.796222
\(319\) 1.59342e8 0.274829
\(320\) −7.36761e7 −0.125690
\(321\) −4.59286e8 −0.775025
\(322\) 2.02451e9 3.37927
\(323\) 1.15061e8 0.189985
\(324\) 4.07178e7 0.0665084
\(325\) 1.48128e8 0.239357
\(326\) 8.46741e8 1.35360
\(327\) 1.56322e8 0.247232
\(328\) −3.53266e8 −0.552769
\(329\) −1.62609e9 −2.51744
\(330\) 6.73038e8 1.03096
\(331\) 4.19111e8 0.635231 0.317615 0.948220i \(-0.397118\pi\)
0.317615 + 0.948220i \(0.397118\pi\)
\(332\) −2.40856e8 −0.361221
\(333\) −3.85536e8 −0.572150
\(334\) −1.48145e9 −2.17558
\(335\) −2.90364e8 −0.421975
\(336\) −7.22403e8 −1.03894
\(337\) −5.81583e8 −0.827765 −0.413882 0.910330i \(-0.635827\pi\)
−0.413882 + 0.910330i \(0.635827\pi\)
\(338\) 7.32443e8 1.03173
\(339\) −5.74120e8 −0.800393
\(340\) 3.49648e8 0.482453
\(341\) −7.70661e8 −1.05250
\(342\) −9.17295e7 −0.123999
\(343\) −1.13916e8 −0.152425
\(344\) −3.22030e8 −0.426522
\(345\) −1.01256e9 −1.32755
\(346\) −1.07157e9 −1.39076
\(347\) 6.12801e8 0.787347 0.393673 0.919250i \(-0.371204\pi\)
0.393673 + 0.919250i \(0.371204\pi\)
\(348\) −6.59935e7 −0.0839410
\(349\) 5.26220e8 0.662642 0.331321 0.943518i \(-0.392506\pi\)
0.331321 + 0.943518i \(0.392506\pi\)
\(350\) 8.21096e8 1.02366
\(351\) −6.68779e7 −0.0825482
\(352\) 9.81978e8 1.20006
\(353\) 7.42142e8 0.897999 0.448999 0.893532i \(-0.351781\pi\)
0.448999 + 0.893532i \(0.351781\pi\)
\(354\) −7.93216e7 −0.0950342
\(355\) 1.10089e9 1.30601
\(356\) −3.60248e7 −0.0423181
\(357\) 4.65005e8 0.540903
\(358\) −1.99603e9 −2.29920
\(359\) 4.68258e8 0.534139 0.267070 0.963677i \(-0.413945\pi\)
0.267070 + 0.963677i \(0.413945\pi\)
\(360\) 1.86937e8 0.211172
\(361\) −8.16493e8 −0.913435
\(362\) −1.99813e8 −0.221382
\(363\) −1.47453e8 −0.161801
\(364\) −3.42764e8 −0.372512
\(365\) 9.40933e8 1.01282
\(366\) 3.37405e8 0.359722
\(367\) 4.07678e8 0.430513 0.215257 0.976558i \(-0.430941\pi\)
0.215257 + 0.976558i \(0.430941\pi\)
\(368\) −2.18430e9 −2.28479
\(369\) −3.50384e8 −0.363038
\(370\) 2.63932e9 2.70885
\(371\) 1.55657e9 1.58256
\(372\) 3.19180e8 0.321466
\(373\) 1.76997e9 1.76598 0.882989 0.469393i \(-0.155527\pi\)
0.882989 + 0.469393i \(0.155527\pi\)
\(374\) −9.34570e8 −0.923765
\(375\) 3.25259e8 0.318508
\(376\) 9.07726e8 0.880638
\(377\) 1.08393e8 0.104185
\(378\) −3.70713e8 −0.353034
\(379\) −2.03436e8 −0.191951 −0.0959757 0.995384i \(-0.530597\pi\)
−0.0959757 + 0.995384i \(0.530597\pi\)
\(380\) 2.35137e8 0.219826
\(381\) −3.36025e8 −0.311268
\(382\) 5.46395e8 0.501516
\(383\) −1.58998e9 −1.44609 −0.723044 0.690802i \(-0.757258\pi\)
−0.723044 + 0.690802i \(0.757258\pi\)
\(384\) 5.97884e8 0.538838
\(385\) −2.29445e9 −2.04912
\(386\) 5.28184e8 0.467444
\(387\) −3.19402e8 −0.280124
\(388\) 7.91244e8 0.687701
\(389\) −6.10709e8 −0.526030 −0.263015 0.964792i \(-0.584717\pi\)
−0.263015 + 0.964792i \(0.584717\pi\)
\(390\) 4.57835e8 0.390826
\(391\) 1.40602e9 1.18952
\(392\) 6.68892e8 0.560860
\(393\) −2.20416e8 −0.183176
\(394\) −1.49767e9 −1.23362
\(395\) 2.41894e9 1.97486
\(396\) 2.78983e8 0.225759
\(397\) 1.32027e8 0.105900 0.0529501 0.998597i \(-0.483138\pi\)
0.0529501 + 0.998597i \(0.483138\pi\)
\(398\) 1.77513e9 1.41136
\(399\) 3.12715e8 0.246458
\(400\) −8.85907e8 −0.692115
\(401\) 6.34337e8 0.491263 0.245632 0.969363i \(-0.421005\pi\)
0.245632 + 0.969363i \(0.421005\pi\)
\(402\) −3.21437e8 −0.246777
\(403\) −5.24244e8 −0.398993
\(404\) −1.04427e9 −0.787916
\(405\) 1.85412e8 0.138690
\(406\) 6.00835e8 0.445568
\(407\) −2.64155e9 −1.94213
\(408\) −2.59578e8 −0.189216
\(409\) 9.45727e8 0.683493 0.341746 0.939792i \(-0.388982\pi\)
0.341746 + 0.939792i \(0.388982\pi\)
\(410\) 2.39868e9 1.71881
\(411\) −1.32639e9 −0.942376
\(412\) −1.21180e8 −0.0853675
\(413\) 2.70415e8 0.188889
\(414\) −1.12091e9 −0.776373
\(415\) −1.09676e9 −0.753255
\(416\) 6.67992e8 0.454930
\(417\) 7.09619e7 0.0479236
\(418\) −6.28496e8 −0.420907
\(419\) −1.99014e9 −1.32170 −0.660852 0.750517i \(-0.729805\pi\)
−0.660852 + 0.750517i \(0.729805\pi\)
\(420\) 9.50278e8 0.625862
\(421\) 2.39417e8 0.156375 0.0781875 0.996939i \(-0.475087\pi\)
0.0781875 + 0.996939i \(0.475087\pi\)
\(422\) 1.20122e9 0.778087
\(423\) 9.00321e8 0.578370
\(424\) −8.68918e8 −0.553603
\(425\) 5.70252e8 0.360334
\(426\) 1.21870e9 0.763772
\(427\) −1.15025e9 −0.714979
\(428\) 1.30331e9 0.803518
\(429\) −4.58222e8 −0.280205
\(430\) 2.18658e9 1.32625
\(431\) −1.41644e9 −0.852171 −0.426086 0.904683i \(-0.640108\pi\)
−0.426086 + 0.904683i \(0.640108\pi\)
\(432\) 3.99974e8 0.238693
\(433\) 2.49096e9 1.47455 0.737273 0.675595i \(-0.236113\pi\)
0.737273 + 0.675595i \(0.236113\pi\)
\(434\) −2.90595e9 −1.70638
\(435\) −3.00507e8 −0.175042
\(436\) −4.43595e8 −0.256321
\(437\) 9.45545e8 0.541997
\(438\) 1.04162e9 0.592314
\(439\) 1.29699e9 0.731661 0.365831 0.930681i \(-0.380785\pi\)
0.365831 + 0.930681i \(0.380785\pi\)
\(440\) 1.28082e9 0.716811
\(441\) 6.63435e8 0.368352
\(442\) −6.35743e8 −0.350190
\(443\) −4.26015e8 −0.232815 −0.116408 0.993202i \(-0.537138\pi\)
−0.116408 + 0.993202i \(0.537138\pi\)
\(444\) 1.09403e9 0.593184
\(445\) −1.64042e8 −0.0882460
\(446\) −3.65343e9 −1.94997
\(447\) 5.34549e8 0.283081
\(448\) 2.78047e8 0.146098
\(449\) 3.29234e9 1.71649 0.858247 0.513236i \(-0.171554\pi\)
0.858247 + 0.513236i \(0.171554\pi\)
\(450\) −4.54618e8 −0.235181
\(451\) −2.40070e9 −1.23231
\(452\) 1.62918e9 0.829819
\(453\) −3.61204e8 −0.182561
\(454\) −4.21948e9 −2.11623
\(455\) −1.56081e9 −0.776800
\(456\) −1.74566e8 −0.0862148
\(457\) −4.28524e8 −0.210024 −0.105012 0.994471i \(-0.533488\pi\)
−0.105012 + 0.994471i \(0.533488\pi\)
\(458\) −5.66229e8 −0.275399
\(459\) −2.57460e8 −0.124270
\(460\) 2.87332e9 1.37636
\(461\) −2.18183e9 −1.03721 −0.518605 0.855014i \(-0.673548\pi\)
−0.518605 + 0.855014i \(0.673548\pi\)
\(462\) −2.53999e9 −1.19835
\(463\) −3.49857e9 −1.63816 −0.819082 0.573677i \(-0.805517\pi\)
−0.819082 + 0.573677i \(0.805517\pi\)
\(464\) −6.48260e8 −0.301256
\(465\) 1.45341e9 0.670353
\(466\) 2.46646e8 0.112907
\(467\) 5.99769e8 0.272505 0.136253 0.990674i \(-0.456494\pi\)
0.136253 + 0.990674i \(0.456494\pi\)
\(468\) 1.89779e8 0.0855830
\(469\) 1.09581e9 0.490490
\(470\) −6.16345e9 −2.73830
\(471\) 8.00531e8 0.353025
\(472\) −1.50953e8 −0.0660761
\(473\) −2.18843e9 −0.950863
\(474\) 2.67780e9 1.15493
\(475\) 3.83493e8 0.164184
\(476\) −1.31954e9 −0.560788
\(477\) −8.61830e8 −0.363586
\(478\) −4.97606e7 −0.0208395
\(479\) 1.04242e9 0.433381 0.216691 0.976240i \(-0.430474\pi\)
0.216691 + 0.976240i \(0.430474\pi\)
\(480\) −1.85194e9 −0.764333
\(481\) −1.79692e9 −0.736242
\(482\) −3.21578e9 −1.30804
\(483\) 3.82130e9 1.54311
\(484\) 4.18426e8 0.167749
\(485\) 3.60300e9 1.43406
\(486\) 2.05253e8 0.0811080
\(487\) 4.49609e9 1.76394 0.881970 0.471305i \(-0.156217\pi\)
0.881970 + 0.471305i \(0.156217\pi\)
\(488\) 6.42097e8 0.250110
\(489\) 1.59824e9 0.618104
\(490\) −4.54177e9 −1.74397
\(491\) −3.34941e9 −1.27698 −0.638489 0.769631i \(-0.720440\pi\)
−0.638489 + 0.769631i \(0.720440\pi\)
\(492\) 9.94283e8 0.376385
\(493\) 4.17280e8 0.156842
\(494\) −4.27536e8 −0.159562
\(495\) 1.27037e9 0.470775
\(496\) 3.13533e9 1.15371
\(497\) −4.15467e9 −1.51806
\(498\) −1.21412e9 −0.440514
\(499\) −2.53248e9 −0.912419 −0.456209 0.889872i \(-0.650793\pi\)
−0.456209 + 0.889872i \(0.650793\pi\)
\(500\) −9.22986e8 −0.330217
\(501\) −2.79628e9 −0.993455
\(502\) −6.00879e9 −2.11994
\(503\) −2.14316e9 −0.750874 −0.375437 0.926848i \(-0.622507\pi\)
−0.375437 + 0.926848i \(0.622507\pi\)
\(504\) −7.05485e8 −0.245460
\(505\) −4.75520e9 −1.64304
\(506\) −7.68007e9 −2.63535
\(507\) 1.38250e9 0.471127
\(508\) 9.53536e8 0.322711
\(509\) 3.85176e9 1.29463 0.647316 0.762222i \(-0.275891\pi\)
0.647316 + 0.762222i \(0.275891\pi\)
\(510\) 1.76253e9 0.588358
\(511\) −3.55100e9 −1.17727
\(512\) −2.08327e9 −0.685962
\(513\) −1.73141e8 −0.0566227
\(514\) −2.85801e9 −0.928311
\(515\) −5.51806e8 −0.178017
\(516\) 9.06366e8 0.290422
\(517\) 6.16866e9 1.96324
\(518\) −9.96056e9 −3.14869
\(519\) −2.02261e9 −0.635076
\(520\) 8.71283e8 0.271736
\(521\) −4.67938e9 −1.44963 −0.724814 0.688945i \(-0.758074\pi\)
−0.724814 + 0.688945i \(0.758074\pi\)
\(522\) −3.32665e8 −0.102367
\(523\) 3.37502e9 1.03162 0.515810 0.856703i \(-0.327491\pi\)
0.515810 + 0.856703i \(0.327491\pi\)
\(524\) 6.25474e8 0.189911
\(525\) 1.54984e9 0.467443
\(526\) −7.30250e9 −2.18787
\(527\) −2.01819e9 −0.600654
\(528\) 2.74047e9 0.810228
\(529\) 8.14951e9 2.39352
\(530\) 5.89995e9 1.72140
\(531\) −1.49721e8 −0.0433963
\(532\) −8.87389e8 −0.255519
\(533\) −1.63308e9 −0.467157
\(534\) −1.81596e8 −0.0516075
\(535\) 5.93475e9 1.67558
\(536\) −6.11710e8 −0.171581
\(537\) −3.76756e9 −1.04990
\(538\) 8.75551e9 2.42406
\(539\) 4.54561e9 1.25035
\(540\) −5.26142e8 −0.143789
\(541\) 3.20625e9 0.870576 0.435288 0.900291i \(-0.356647\pi\)
0.435288 + 0.900291i \(0.356647\pi\)
\(542\) −2.79046e9 −0.752798
\(543\) −3.77151e8 −0.101092
\(544\) 2.57158e9 0.684862
\(545\) −2.01995e9 −0.534506
\(546\) −1.72783e9 −0.454284
\(547\) −4.82907e8 −0.126156 −0.0630779 0.998009i \(-0.520092\pi\)
−0.0630779 + 0.998009i \(0.520092\pi\)
\(548\) 3.76388e9 0.977022
\(549\) 6.36859e8 0.164263
\(550\) −3.11487e9 −0.798309
\(551\) 2.80620e8 0.0714641
\(552\) −2.13315e9 −0.539802
\(553\) −9.12889e9 −2.29551
\(554\) 5.88527e9 1.47056
\(555\) 4.98177e9 1.23697
\(556\) −2.01368e8 −0.0496854
\(557\) −3.29285e9 −0.807380 −0.403690 0.914896i \(-0.632273\pi\)
−0.403690 + 0.914896i \(0.632273\pi\)
\(558\) 1.60895e9 0.392032
\(559\) −1.48868e9 −0.360463
\(560\) 9.33466e9 2.24616
\(561\) −1.76402e9 −0.421827
\(562\) 2.07924e9 0.494114
\(563\) 1.67347e9 0.395220 0.197610 0.980281i \(-0.436682\pi\)
0.197610 + 0.980281i \(0.436682\pi\)
\(564\) −2.55483e9 −0.599633
\(565\) 7.41859e9 1.73042
\(566\) 1.04962e10 2.43318
\(567\) −6.99729e8 −0.161209
\(568\) 2.31925e9 0.531041
\(569\) 5.59667e9 1.27361 0.636805 0.771025i \(-0.280255\pi\)
0.636805 + 0.771025i \(0.280255\pi\)
\(570\) 1.18530e9 0.268081
\(571\) −6.50267e9 −1.46172 −0.730862 0.682525i \(-0.760882\pi\)
−0.730862 + 0.682525i \(0.760882\pi\)
\(572\) 1.30029e9 0.290506
\(573\) 1.03133e9 0.229012
\(574\) −9.05240e9 −1.99789
\(575\) 4.68619e9 1.02797
\(576\) −1.53947e8 −0.0335654
\(577\) −4.50802e9 −0.976945 −0.488473 0.872579i \(-0.662446\pi\)
−0.488473 + 0.872579i \(0.662446\pi\)
\(578\) 3.42225e9 0.737164
\(579\) 9.96959e8 0.213453
\(580\) 8.52748e8 0.181477
\(581\) 4.13907e9 0.875560
\(582\) 3.98857e9 0.838661
\(583\) −5.90494e9 −1.23417
\(584\) 1.98226e9 0.411828
\(585\) 8.64175e8 0.178466
\(586\) 5.14346e9 1.05588
\(587\) 6.96930e9 1.42218 0.711092 0.703099i \(-0.248201\pi\)
0.711092 + 0.703099i \(0.248201\pi\)
\(588\) −1.88262e9 −0.381894
\(589\) −1.35723e9 −0.273683
\(590\) 1.02497e9 0.205461
\(591\) −2.82689e9 −0.563318
\(592\) 1.07468e10 2.12888
\(593\) 1.72102e9 0.338917 0.169459 0.985537i \(-0.445798\pi\)
0.169459 + 0.985537i \(0.445798\pi\)
\(594\) 1.40632e9 0.275316
\(595\) −6.00865e9 −1.16941
\(596\) −1.51689e9 −0.293489
\(597\) 3.35059e9 0.644483
\(598\) −5.22438e9 −0.999036
\(599\) 4.56819e9 0.868460 0.434230 0.900802i \(-0.357021\pi\)
0.434230 + 0.900802i \(0.357021\pi\)
\(600\) −8.65160e8 −0.163518
\(601\) −2.34532e9 −0.440698 −0.220349 0.975421i \(-0.570720\pi\)
−0.220349 + 0.975421i \(0.570720\pi\)
\(602\) −8.25197e9 −1.54159
\(603\) −6.06720e8 −0.112688
\(604\) 1.02499e9 0.189273
\(605\) 1.90534e9 0.349807
\(606\) −5.26406e9 −0.960875
\(607\) 3.67032e9 0.666107 0.333053 0.942908i \(-0.391921\pi\)
0.333053 + 0.942908i \(0.391921\pi\)
\(608\) 1.72938e9 0.312052
\(609\) 1.13409e9 0.203464
\(610\) −4.35984e9 −0.777706
\(611\) 4.19624e9 0.744246
\(612\) 7.30593e8 0.128839
\(613\) 3.62973e9 0.636447 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(614\) 3.71502e9 0.647697
\(615\) 4.52756e9 0.784876
\(616\) −4.83372e9 −0.833199
\(617\) 4.89208e9 0.838486 0.419243 0.907874i \(-0.362296\pi\)
0.419243 + 0.907874i \(0.362296\pi\)
\(618\) −6.10856e8 −0.104107
\(619\) −8.57865e9 −1.45379 −0.726895 0.686749i \(-0.759037\pi\)
−0.726895 + 0.686749i \(0.759037\pi\)
\(620\) −4.12434e9 −0.694997
\(621\) −2.11575e9 −0.354522
\(622\) −6.33392e9 −1.05537
\(623\) 6.19080e8 0.102574
\(624\) 1.86421e9 0.307149
\(625\) −7.60884e9 −1.24663
\(626\) −6.26150e8 −0.102016
\(627\) −1.18630e9 −0.192202
\(628\) −2.27166e9 −0.366003
\(629\) −6.91761e9 −1.10836
\(630\) 4.79024e9 0.763247
\(631\) 5.81549e9 0.921475 0.460737 0.887537i \(-0.347585\pi\)
0.460737 + 0.887537i \(0.347585\pi\)
\(632\) 5.09598e9 0.803005
\(633\) 2.26733e9 0.355305
\(634\) −2.49403e9 −0.388677
\(635\) 4.34201e9 0.672950
\(636\) 2.44561e9 0.376953
\(637\) 3.09216e9 0.473995
\(638\) −2.27930e9 −0.347479
\(639\) 2.30033e9 0.348768
\(640\) −7.72567e9 −1.16495
\(641\) −1.97168e6 −0.000295688 0 −0.000147844 1.00000i \(-0.500047\pi\)
−0.000147844 1.00000i \(0.500047\pi\)
\(642\) 6.56984e9 0.979901
\(643\) −5.68675e9 −0.843580 −0.421790 0.906694i \(-0.638598\pi\)
−0.421790 + 0.906694i \(0.638598\pi\)
\(644\) −1.08437e10 −1.59984
\(645\) 4.12722e9 0.605618
\(646\) −1.64589e9 −0.240208
\(647\) 8.00900e9 1.16256 0.581278 0.813705i \(-0.302553\pi\)
0.581278 + 0.813705i \(0.302553\pi\)
\(648\) 3.90607e8 0.0563933
\(649\) −1.02583e9 −0.147306
\(650\) −2.11890e9 −0.302631
\(651\) −5.48505e9 −0.779197
\(652\) −4.53532e9 −0.640828
\(653\) 4.57134e9 0.642462 0.321231 0.947001i \(-0.395903\pi\)
0.321231 + 0.947001i \(0.395903\pi\)
\(654\) −2.23611e9 −0.312587
\(655\) 2.84815e9 0.396021
\(656\) 9.76692e9 1.35081
\(657\) 1.96609e9 0.270473
\(658\) 2.32604e10 3.18292
\(659\) 1.31699e10 1.79260 0.896298 0.443452i \(-0.146246\pi\)
0.896298 + 0.443452i \(0.146246\pi\)
\(660\) −3.60493e9 −0.488083
\(661\) −7.04155e9 −0.948338 −0.474169 0.880434i \(-0.657251\pi\)
−0.474169 + 0.880434i \(0.657251\pi\)
\(662\) −5.99517e9 −0.803153
\(663\) −1.19998e9 −0.159910
\(664\) −2.31053e9 −0.306284
\(665\) −4.04080e9 −0.532834
\(666\) 5.51488e9 0.723397
\(667\) 3.42911e9 0.447446
\(668\) 7.93497e9 1.02998
\(669\) −6.89592e9 −0.890433
\(670\) 4.15351e9 0.533523
\(671\) 4.36352e9 0.557581
\(672\) 6.98907e9 0.888437
\(673\) 8.33165e8 0.105361 0.0526803 0.998611i \(-0.483224\pi\)
0.0526803 + 0.998611i \(0.483224\pi\)
\(674\) 8.31923e9 1.04658
\(675\) −8.58102e8 −0.107393
\(676\) −3.92312e9 −0.488448
\(677\) 1.49936e10 1.85715 0.928573 0.371150i \(-0.121036\pi\)
0.928573 + 0.371150i \(0.121036\pi\)
\(678\) 8.21247e9 1.01198
\(679\) −1.35974e10 −1.66691
\(680\) 3.35419e9 0.409078
\(681\) −7.96437e9 −0.966354
\(682\) 1.10239e10 1.33073
\(683\) 6.61866e9 0.794873 0.397436 0.917630i \(-0.369900\pi\)
0.397436 + 0.917630i \(0.369900\pi\)
\(684\) 4.91322e8 0.0587043
\(685\) 1.71392e10 2.03738
\(686\) 1.62951e9 0.192718
\(687\) −1.06877e9 −0.125758
\(688\) 8.90331e9 1.04230
\(689\) −4.01684e9 −0.467862
\(690\) 1.44841e10 1.67849
\(691\) 7.24311e9 0.835125 0.417563 0.908648i \(-0.362884\pi\)
0.417563 + 0.908648i \(0.362884\pi\)
\(692\) 5.73954e9 0.658424
\(693\) −4.79428e9 −0.547214
\(694\) −8.76579e9 −0.995480
\(695\) −9.16947e8 −0.103609
\(696\) −6.33078e8 −0.0711746
\(697\) −6.28689e9 −0.703269
\(698\) −7.52730e9 −0.837810
\(699\) 4.65549e8 0.0515579
\(700\) −4.39796e9 −0.484628
\(701\) −2.33354e9 −0.255860 −0.127930 0.991783i \(-0.540833\pi\)
−0.127930 + 0.991783i \(0.540833\pi\)
\(702\) 9.56652e8 0.104370
\(703\) −4.65208e9 −0.505014
\(704\) −1.05479e9 −0.113936
\(705\) −1.16337e10 −1.25042
\(706\) −1.06160e10 −1.13538
\(707\) 1.79457e10 1.90982
\(708\) 4.24863e8 0.0449917
\(709\) 4.23075e9 0.445816 0.222908 0.974839i \(-0.428445\pi\)
0.222908 + 0.974839i \(0.428445\pi\)
\(710\) −1.57477e10 −1.65125
\(711\) 5.05441e9 0.527384
\(712\) −3.45587e8 −0.0358820
\(713\) −1.65850e10 −1.71357
\(714\) −6.65165e9 −0.683889
\(715\) 5.92100e9 0.605793
\(716\) 1.06912e10 1.08850
\(717\) −9.39242e7 −0.00951614
\(718\) −6.69818e9 −0.675338
\(719\) −1.25609e10 −1.26029 −0.630143 0.776479i \(-0.717004\pi\)
−0.630143 + 0.776479i \(0.717004\pi\)
\(720\) −5.16834e9 −0.516045
\(721\) 2.08247e9 0.206921
\(722\) 1.16795e10 1.15490
\(723\) −6.06985e9 −0.597302
\(724\) 1.07024e9 0.104808
\(725\) 1.39077e9 0.135542
\(726\) 2.10924e9 0.204572
\(727\) 9.77361e9 0.943375 0.471688 0.881766i \(-0.343645\pi\)
0.471688 + 0.881766i \(0.343645\pi\)
\(728\) −3.28815e9 −0.315858
\(729\) 3.87420e8 0.0370370
\(730\) −1.34595e10 −1.28056
\(731\) −5.73099e9 −0.542649
\(732\) −1.80721e9 −0.170302
\(733\) −4.89324e9 −0.458915 −0.229458 0.973319i \(-0.573695\pi\)
−0.229458 + 0.973319i \(0.573695\pi\)
\(734\) −5.83162e9 −0.544318
\(735\) −8.57270e9 −0.796364
\(736\) 2.11326e10 1.95380
\(737\) −4.15702e9 −0.382512
\(738\) 5.01206e9 0.459007
\(739\) −1.16838e10 −1.06495 −0.532473 0.846447i \(-0.678737\pi\)
−0.532473 + 0.846447i \(0.678737\pi\)
\(740\) −1.41367e10 −1.28244
\(741\) −8.06984e8 −0.0728620
\(742\) −2.22659e10 −2.00091
\(743\) −2.02352e9 −0.180987 −0.0904934 0.995897i \(-0.528844\pi\)
−0.0904934 + 0.995897i \(0.528844\pi\)
\(744\) 3.06190e9 0.272575
\(745\) −6.90728e9 −0.612012
\(746\) −2.53185e10 −2.23281
\(747\) −2.29169e9 −0.201156
\(748\) 5.00575e9 0.437335
\(749\) −2.23972e10 −1.94764
\(750\) −4.65266e9 −0.402705
\(751\) −5.88364e8 −0.0506881 −0.0253441 0.999679i \(-0.508068\pi\)
−0.0253441 + 0.999679i \(0.508068\pi\)
\(752\) −2.50963e10 −2.15203
\(753\) −1.13417e10 −0.968047
\(754\) −1.55050e9 −0.131726
\(755\) 4.66736e9 0.394691
\(756\) 1.98562e9 0.167136
\(757\) −1.84565e10 −1.54637 −0.773185 0.634181i \(-0.781337\pi\)
−0.773185 + 0.634181i \(0.781337\pi\)
\(758\) 2.91005e9 0.242693
\(759\) −1.44963e10 −1.20340
\(760\) 2.25568e9 0.186393
\(761\) 1.56129e10 1.28421 0.642105 0.766617i \(-0.278061\pi\)
0.642105 + 0.766617i \(0.278061\pi\)
\(762\) 4.80666e9 0.393551
\(763\) 7.62311e9 0.621293
\(764\) −2.92660e9 −0.237431
\(765\) 3.32682e9 0.268667
\(766\) 2.27438e10 1.82836
\(767\) −6.97826e8 −0.0558423
\(768\) −9.28224e9 −0.739415
\(769\) 2.12983e10 1.68889 0.844446 0.535641i \(-0.179930\pi\)
0.844446 + 0.535641i \(0.179930\pi\)
\(770\) 3.28209e10 2.59080
\(771\) −5.39457e9 −0.423903
\(772\) −2.82906e9 −0.221300
\(773\) 1.58105e10 1.23116 0.615582 0.788073i \(-0.288921\pi\)
0.615582 + 0.788073i \(0.288921\pi\)
\(774\) 4.56888e9 0.354174
\(775\) −6.72651e9 −0.519079
\(776\) 7.59043e9 0.583110
\(777\) −1.88008e10 −1.43781
\(778\) 8.73587e9 0.665086
\(779\) −4.22792e9 −0.320439
\(780\) −2.45226e9 −0.185027
\(781\) 1.57610e10 1.18387
\(782\) −2.01124e10 −1.50397
\(783\) −6.27913e8 −0.0467448
\(784\) −1.84932e10 −1.37058
\(785\) −1.03442e10 −0.763227
\(786\) 3.15294e9 0.231599
\(787\) 2.69394e9 0.197005 0.0985024 0.995137i \(-0.468595\pi\)
0.0985024 + 0.995137i \(0.468595\pi\)
\(788\) 8.02186e9 0.584027
\(789\) −1.37836e10 −0.999066
\(790\) −3.46017e10 −2.49691
\(791\) −2.79971e10 −2.01139
\(792\) 2.67629e9 0.191424
\(793\) 2.96829e9 0.211374
\(794\) −1.88858e9 −0.133895
\(795\) 1.11363e10 0.786060
\(796\) −9.50795e9 −0.668176
\(797\) 5.54182e9 0.387747 0.193873 0.981027i \(-0.437895\pi\)
0.193873 + 0.981027i \(0.437895\pi\)
\(798\) −4.47322e9 −0.311609
\(799\) 1.61543e10 1.12040
\(800\) 8.57093e9 0.591852
\(801\) −3.42767e8 −0.0235660
\(802\) −9.07385e9 −0.621128
\(803\) 1.34709e10 0.918106
\(804\) 1.72168e9 0.116831
\(805\) −4.93776e10 −3.33614
\(806\) 7.49903e9 0.504466
\(807\) 1.65262e10 1.10692
\(808\) −1.00178e10 −0.668084
\(809\) 1.56002e10 1.03589 0.517943 0.855415i \(-0.326698\pi\)
0.517943 + 0.855415i \(0.326698\pi\)
\(810\) −2.65222e9 −0.175353
\(811\) −7.79846e9 −0.513377 −0.256688 0.966494i \(-0.582631\pi\)
−0.256688 + 0.966494i \(0.582631\pi\)
\(812\) −3.21820e9 −0.210944
\(813\) −5.26706e9 −0.343757
\(814\) 3.77859e10 2.45553
\(815\) −2.06520e10 −1.33632
\(816\) 7.17668e9 0.462390
\(817\) −3.85408e9 −0.247254
\(818\) −1.35281e10 −0.864173
\(819\) −3.26132e9 −0.207444
\(820\) −1.28478e10 −0.813731
\(821\) 5.49348e9 0.346455 0.173227 0.984882i \(-0.444580\pi\)
0.173227 + 0.984882i \(0.444580\pi\)
\(822\) 1.89733e10 1.19149
\(823\) 2.49423e10 1.55969 0.779843 0.625976i \(-0.215299\pi\)
0.779843 + 0.625976i \(0.215299\pi\)
\(824\) −1.16249e9 −0.0723841
\(825\) −5.87939e9 −0.364539
\(826\) −3.86814e9 −0.238821
\(827\) −3.09149e10 −1.90063 −0.950317 0.311285i \(-0.899241\pi\)
−0.950317 + 0.311285i \(0.899241\pi\)
\(828\) 6.00384e9 0.367555
\(829\) −7.17903e8 −0.0437648 −0.0218824 0.999761i \(-0.506966\pi\)
−0.0218824 + 0.999761i \(0.506966\pi\)
\(830\) 1.56885e10 0.952376
\(831\) 1.11086e10 0.671514
\(832\) −7.17521e8 −0.0431920
\(833\) 1.19039e10 0.713563
\(834\) −1.01507e9 −0.0605921
\(835\) 3.61326e10 2.14781
\(836\) 3.36636e9 0.199268
\(837\) 3.03692e9 0.179017
\(838\) 2.84678e10 1.67109
\(839\) −1.00495e10 −0.587460 −0.293730 0.955888i \(-0.594897\pi\)
−0.293730 + 0.955888i \(0.594897\pi\)
\(840\) 9.11605e9 0.530676
\(841\) −1.62322e10 −0.941003
\(842\) −3.42473e9 −0.197713
\(843\) 3.92461e9 0.225631
\(844\) −6.43397e9 −0.368367
\(845\) −1.78643e10 −1.01856
\(846\) −1.28786e10 −0.731261
\(847\) −7.19059e9 −0.406605
\(848\) 2.40234e10 1.35285
\(849\) 1.98118e10 1.11108
\(850\) −8.15714e9 −0.455588
\(851\) −5.68473e10 −3.16196
\(852\) −6.52762e9 −0.361590
\(853\) 7.41016e9 0.408795 0.204398 0.978888i \(-0.434476\pi\)
0.204398 + 0.978888i \(0.434476\pi\)
\(854\) 1.64536e10 0.903982
\(855\) 2.23728e9 0.122416
\(856\) 1.25027e10 0.681313
\(857\) 1.81207e10 0.983429 0.491714 0.870757i \(-0.336370\pi\)
0.491714 + 0.870757i \(0.336370\pi\)
\(858\) 6.55462e9 0.354276
\(859\) 1.04208e10 0.560953 0.280477 0.959861i \(-0.409508\pi\)
0.280477 + 0.959861i \(0.409508\pi\)
\(860\) −1.17118e10 −0.627882
\(861\) −1.70866e10 −0.912315
\(862\) 2.02614e10 1.07744
\(863\) −2.49924e10 −1.32364 −0.661821 0.749662i \(-0.730216\pi\)
−0.661821 + 0.749662i \(0.730216\pi\)
\(864\) −3.86965e9 −0.204114
\(865\) 2.61355e10 1.37301
\(866\) −3.56318e10 −1.86434
\(867\) 6.45957e9 0.336618
\(868\) 1.55649e10 0.807843
\(869\) 3.46309e10 1.79017
\(870\) 4.29860e9 0.221314
\(871\) −2.82782e9 −0.145007
\(872\) −4.25542e9 −0.217337
\(873\) 7.52851e9 0.382965
\(874\) −1.35255e10 −0.685273
\(875\) 1.58614e10 0.800410
\(876\) −5.57916e9 −0.280417
\(877\) −1.00470e10 −0.502964 −0.251482 0.967862i \(-0.580918\pi\)
−0.251482 + 0.967862i \(0.580918\pi\)
\(878\) −1.85527e10 −0.925075
\(879\) 9.70840e9 0.482155
\(880\) −3.54115e10 −1.75168
\(881\) −3.06184e10 −1.50858 −0.754288 0.656544i \(-0.772018\pi\)
−0.754288 + 0.656544i \(0.772018\pi\)
\(882\) −9.49008e9 −0.465725
\(883\) 5.86608e9 0.286738 0.143369 0.989669i \(-0.454206\pi\)
0.143369 + 0.989669i \(0.454206\pi\)
\(884\) 3.40517e9 0.165789
\(885\) 1.93465e9 0.0938213
\(886\) 6.09391e9 0.294359
\(887\) 1.17203e9 0.0563905 0.0281953 0.999602i \(-0.491024\pi\)
0.0281953 + 0.999602i \(0.491024\pi\)
\(888\) 1.04951e10 0.502968
\(889\) −1.63864e10 −0.782217
\(890\) 2.34653e9 0.111574
\(891\) 2.65446e9 0.125720
\(892\) 1.95685e10 0.923168
\(893\) 1.08637e10 0.510504
\(894\) −7.64644e9 −0.357913
\(895\) 4.86832e10 2.26986
\(896\) 2.91560e10 1.35410
\(897\) −9.86114e9 −0.456198
\(898\) −4.70952e10 −2.17025
\(899\) −4.92210e9 −0.225939
\(900\) 2.43503e9 0.111341
\(901\) −1.54637e10 −0.704330
\(902\) 3.43408e10 1.55807
\(903\) −1.55758e10 −0.703951
\(904\) 1.56287e10 0.703614
\(905\) 4.87342e9 0.218557
\(906\) 5.16683e9 0.230821
\(907\) 1.61777e10 0.719932 0.359966 0.932965i \(-0.382788\pi\)
0.359966 + 0.932965i \(0.382788\pi\)
\(908\) 2.26004e10 1.00188
\(909\) −9.93603e9 −0.438773
\(910\) 2.23265e10 0.982145
\(911\) −1.77685e10 −0.778637 −0.389319 0.921103i \(-0.627290\pi\)
−0.389319 + 0.921103i \(0.627290\pi\)
\(912\) 4.82630e9 0.210684
\(913\) −1.57018e10 −0.682812
\(914\) 6.12981e9 0.265543
\(915\) −8.22929e9 −0.355131
\(916\) 3.03284e9 0.130381
\(917\) −1.07487e10 −0.460322
\(918\) 3.68283e9 0.157120
\(919\) 2.24473e10 0.954024 0.477012 0.878897i \(-0.341720\pi\)
0.477012 + 0.878897i \(0.341720\pi\)
\(920\) 2.75639e10 1.16703
\(921\) 7.01219e9 0.295764
\(922\) 3.12098e10 1.31139
\(923\) 1.07214e10 0.448794
\(924\) 1.36047e10 0.567332
\(925\) −2.30560e10 −0.957830
\(926\) 5.00452e10 2.07121
\(927\) −1.15300e9 −0.0475392
\(928\) 6.27175e9 0.257615
\(929\) −2.97565e10 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(930\) −2.07903e10 −0.847559
\(931\) 8.00535e9 0.325130
\(932\) −1.32109e9 −0.0534534
\(933\) −1.19554e10 −0.481924
\(934\) −8.57937e9 −0.344541
\(935\) 2.27941e10 0.911974
\(936\) 1.82055e9 0.0725669
\(937\) 2.60896e9 0.103605 0.0518023 0.998657i \(-0.483503\pi\)
0.0518023 + 0.998657i \(0.483503\pi\)
\(938\) −1.56750e10 −0.620151
\(939\) −1.18187e9 −0.0465845
\(940\) 3.30127e10 1.29639
\(941\) 3.46382e10 1.35516 0.677582 0.735447i \(-0.263028\pi\)
0.677582 + 0.735447i \(0.263028\pi\)
\(942\) −1.14512e10 −0.446346
\(943\) −5.16642e10 −2.00631
\(944\) 4.17346e9 0.161471
\(945\) 9.04168e9 0.348528
\(946\) 3.13043e10 1.20222
\(947\) −4.81487e10 −1.84230 −0.921149 0.389209i \(-0.872748\pi\)
−0.921149 + 0.389209i \(0.872748\pi\)
\(948\) −1.43429e10 −0.546772
\(949\) 9.16362e9 0.348045
\(950\) −5.48566e9 −0.207585
\(951\) −4.70753e9 −0.177485
\(952\) −1.26584e10 −0.475499
\(953\) −1.99375e9 −0.0746182 −0.0373091 0.999304i \(-0.511879\pi\)
−0.0373091 + 0.999304i \(0.511879\pi\)
\(954\) 1.23280e10 0.459699
\(955\) −1.33266e10 −0.495115
\(956\) 2.66528e8 0.00986598
\(957\) −4.30223e9 −0.158673
\(958\) −1.49113e10 −0.547945
\(959\) −6.46818e10 −2.36819
\(960\) 1.98925e9 0.0725673
\(961\) −3.70674e9 −0.134729
\(962\) 2.57039e10 0.930866
\(963\) 1.24007e10 0.447461
\(964\) 1.72244e10 0.619261
\(965\) −1.28824e10 −0.461478
\(966\) −5.46617e10 −1.95103
\(967\) −7.83920e9 −0.278791 −0.139396 0.990237i \(-0.544516\pi\)
−0.139396 + 0.990237i \(0.544516\pi\)
\(968\) 4.01398e9 0.142237
\(969\) −3.10665e9 −0.109688
\(970\) −5.15390e10 −1.81315
\(971\) −4.89013e10 −1.71417 −0.857084 0.515177i \(-0.827726\pi\)
−0.857084 + 0.515177i \(0.827726\pi\)
\(972\) −1.09938e9 −0.0383987
\(973\) 3.46048e9 0.120432
\(974\) −6.43142e10 −2.23023
\(975\) −3.99947e9 −0.138193
\(976\) −1.77524e10 −0.611198
\(977\) −2.82603e10 −0.969494 −0.484747 0.874654i \(-0.661088\pi\)
−0.484747 + 0.874654i \(0.661088\pi\)
\(978\) −2.28620e10 −0.781499
\(979\) −2.34852e9 −0.0799934
\(980\) 2.43267e10 0.825642
\(981\) −4.22070e9 −0.142739
\(982\) 4.79115e10 1.61454
\(983\) 4.62408e10 1.55270 0.776351 0.630301i \(-0.217068\pi\)
0.776351 + 0.630301i \(0.217068\pi\)
\(984\) 9.53819e9 0.319141
\(985\) 3.65282e10 1.21787
\(986\) −5.96897e9 −0.198303
\(987\) 4.39044e10 1.45344
\(988\) 2.28997e9 0.0755407
\(989\) −4.70959e10 −1.54809
\(990\) −1.81720e10 −0.595224
\(991\) 1.15771e10 0.377870 0.188935 0.981990i \(-0.439497\pi\)
0.188935 + 0.981990i \(0.439497\pi\)
\(992\) −3.03335e10 −0.986578
\(993\) −1.13160e10 −0.366751
\(994\) 5.94304e10 1.91936
\(995\) −4.32953e10 −1.39335
\(996\) 6.50310e9 0.208551
\(997\) 4.71011e9 0.150521 0.0752607 0.997164i \(-0.476021\pi\)
0.0752607 + 0.997164i \(0.476021\pi\)
\(998\) 3.62258e10 1.15361
\(999\) 1.04095e10 0.330331
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.c.1.3 17
3.2 odd 2 531.8.a.c.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.3 17 1.1 even 1 trivial
531.8.a.c.1.15 17 3.2 odd 2