Properties

Label 177.8.a.c.1.10
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 1.10
Root \(3.56813\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+3.56813 q^{2} -27.0000 q^{3} -115.268 q^{4} -210.232 q^{5} -96.3395 q^{6} +1544.27 q^{7} -868.013 q^{8} +729.000 q^{9} -750.136 q^{10} -1820.60 q^{11} +3112.25 q^{12} -12376.9 q^{13} +5510.17 q^{14} +5676.27 q^{15} +11657.2 q^{16} -36030.6 q^{17} +2601.17 q^{18} +18266.5 q^{19} +24233.1 q^{20} -41695.4 q^{21} -6496.13 q^{22} -99780.9 q^{23} +23436.4 q^{24} -33927.4 q^{25} -44162.4 q^{26} -19683.0 q^{27} -178006. q^{28} +14104.4 q^{29} +20253.7 q^{30} +55263.0 q^{31} +152700. q^{32} +49156.2 q^{33} -128562. q^{34} -324656. q^{35} -84030.7 q^{36} +532023. q^{37} +65177.1 q^{38} +334176. q^{39} +182484. q^{40} -27816.2 q^{41} -148774. q^{42} +784533. q^{43} +209858. q^{44} -153259. q^{45} -356031. q^{46} +115560. q^{47} -314744. q^{48} +1.56124e6 q^{49} -121058. q^{50} +972825. q^{51} +1.42667e6 q^{52} +566369. q^{53} -70231.5 q^{54} +382748. q^{55} -1.34045e6 q^{56} -493195. q^{57} +50326.3 q^{58} -205379. q^{59} -654295. q^{60} -126809. q^{61} +197186. q^{62} +1.12577e6 q^{63} -947265. q^{64} +2.60202e6 q^{65} +175396. q^{66} -4.67077e6 q^{67} +4.15319e6 q^{68} +2.69408e6 q^{69} -1.15841e6 q^{70} +1.27912e6 q^{71} -632782. q^{72} +55292.5 q^{73} +1.89833e6 q^{74} +916041. q^{75} -2.10555e6 q^{76} -2.81150e6 q^{77} +1.19238e6 q^{78} -1.32950e6 q^{79} -2.45071e6 q^{80} +531441. q^{81} -99251.9 q^{82} +2.54431e6 q^{83} +4.80616e6 q^{84} +7.57478e6 q^{85} +2.79932e6 q^{86} -380819. q^{87} +1.58030e6 q^{88} +8.04237e6 q^{89} -546849. q^{90} -1.91133e7 q^{91} +1.15016e7 q^{92} -1.49210e6 q^{93} +412332. q^{94} -3.84020e6 q^{95} -4.12290e6 q^{96} +1.17167e7 q^{97} +5.57069e6 q^{98} -1.32722e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.56813 0.315381 0.157691 0.987489i \(-0.449595\pi\)
0.157691 + 0.987489i \(0.449595\pi\)
\(3\) −27.0000 −0.577350
\(4\) −115.268 −0.900535
\(5\) −210.232 −0.752149 −0.376075 0.926589i \(-0.622726\pi\)
−0.376075 + 0.926589i \(0.622726\pi\)
\(6\) −96.3395 −0.182085
\(7\) 1544.27 1.70169 0.850846 0.525415i \(-0.176090\pi\)
0.850846 + 0.525415i \(0.176090\pi\)
\(8\) −868.013 −0.599393
\(9\) 729.000 0.333333
\(10\) −750.136 −0.237214
\(11\) −1820.60 −0.412420 −0.206210 0.978508i \(-0.566113\pi\)
−0.206210 + 0.978508i \(0.566113\pi\)
\(12\) 3112.25 0.519924
\(13\) −12376.9 −1.56246 −0.781232 0.624241i \(-0.785408\pi\)
−0.781232 + 0.624241i \(0.785408\pi\)
\(14\) 5510.17 0.536681
\(15\) 5676.27 0.434254
\(16\) 11657.2 0.711498
\(17\) −36030.6 −1.77869 −0.889344 0.457238i \(-0.848839\pi\)
−0.889344 + 0.457238i \(0.848839\pi\)
\(18\) 2601.17 0.105127
\(19\) 18266.5 0.610966 0.305483 0.952198i \(-0.401182\pi\)
0.305483 + 0.952198i \(0.401182\pi\)
\(20\) 24233.1 0.677337
\(21\) −41695.4 −0.982472
\(22\) −6496.13 −0.130069
\(23\) −99780.9 −1.71001 −0.855007 0.518616i \(-0.826447\pi\)
−0.855007 + 0.518616i \(0.826447\pi\)
\(24\) 23436.4 0.346060
\(25\) −33927.4 −0.434271
\(26\) −44162.4 −0.492771
\(27\) −19683.0 −0.192450
\(28\) −178006. −1.53243
\(29\) 14104.4 0.107389 0.0536947 0.998557i \(-0.482900\pi\)
0.0536947 + 0.998557i \(0.482900\pi\)
\(30\) 20253.7 0.136955
\(31\) 55263.0 0.333172 0.166586 0.986027i \(-0.446726\pi\)
0.166586 + 0.986027i \(0.446726\pi\)
\(32\) 152700. 0.823786
\(33\) 49156.2 0.238111
\(34\) −128562. −0.560965
\(35\) −324656. −1.27993
\(36\) −84030.7 −0.300178
\(37\) 532023. 1.72673 0.863364 0.504581i \(-0.168353\pi\)
0.863364 + 0.504581i \(0.168353\pi\)
\(38\) 65177.1 0.192687
\(39\) 334176. 0.902089
\(40\) 182484. 0.450833
\(41\) −27816.2 −0.0630311 −0.0315155 0.999503i \(-0.510033\pi\)
−0.0315155 + 0.999503i \(0.510033\pi\)
\(42\) −148774. −0.309853
\(43\) 784533. 1.50478 0.752388 0.658720i \(-0.228902\pi\)
0.752388 + 0.658720i \(0.228902\pi\)
\(44\) 209858. 0.371399
\(45\) −153259. −0.250716
\(46\) −356031. −0.539306
\(47\) 115560. 0.162355 0.0811773 0.996700i \(-0.474132\pi\)
0.0811773 + 0.996700i \(0.474132\pi\)
\(48\) −314744. −0.410783
\(49\) 1.56124e6 1.89575
\(50\) −121058. −0.136961
\(51\) 972825. 1.02693
\(52\) 1.42667e6 1.40705
\(53\) 566369. 0.522558 0.261279 0.965263i \(-0.415856\pi\)
0.261279 + 0.965263i \(0.415856\pi\)
\(54\) −70231.5 −0.0606951
\(55\) 382748. 0.310202
\(56\) −1.34045e6 −1.01998
\(57\) −493195. −0.352741
\(58\) 50326.3 0.0338686
\(59\) −205379. −0.130189
\(60\) −654295. −0.391061
\(61\) −126809. −0.0715313 −0.0357656 0.999360i \(-0.511387\pi\)
−0.0357656 + 0.999360i \(0.511387\pi\)
\(62\) 197186. 0.105076
\(63\) 1.12577e6 0.567231
\(64\) −947265. −0.451691
\(65\) 2.60202e6 1.17521
\(66\) 175396. 0.0750957
\(67\) −4.67077e6 −1.89726 −0.948629 0.316391i \(-0.897529\pi\)
−0.948629 + 0.316391i \(0.897529\pi\)
\(68\) 4.15319e6 1.60177
\(69\) 2.69408e6 0.987277
\(70\) −1.15841e6 −0.403665
\(71\) 1.27912e6 0.424137 0.212068 0.977255i \(-0.431980\pi\)
0.212068 + 0.977255i \(0.431980\pi\)
\(72\) −632782. −0.199798
\(73\) 55292.5 0.0166355 0.00831775 0.999965i \(-0.497352\pi\)
0.00831775 + 0.999965i \(0.497352\pi\)
\(74\) 1.89833e6 0.544578
\(75\) 916041. 0.250727
\(76\) −2.10555e6 −0.550196
\(77\) −2.81150e6 −0.701812
\(78\) 1.19238e6 0.284502
\(79\) −1.32950e6 −0.303385 −0.151693 0.988428i \(-0.548472\pi\)
−0.151693 + 0.988428i \(0.548472\pi\)
\(80\) −2.45071e6 −0.535153
\(81\) 531441. 0.111111
\(82\) −99251.9 −0.0198788
\(83\) 2.54431e6 0.488423 0.244211 0.969722i \(-0.421471\pi\)
0.244211 + 0.969722i \(0.421471\pi\)
\(84\) 4.80616e6 0.884750
\(85\) 7.57478e6 1.33784
\(86\) 2.79932e6 0.474578
\(87\) −380819. −0.0620013
\(88\) 1.58030e6 0.247202
\(89\) 8.04237e6 1.20926 0.604629 0.796507i \(-0.293321\pi\)
0.604629 + 0.796507i \(0.293321\pi\)
\(90\) −546849. −0.0790712
\(91\) −1.91133e7 −2.65883
\(92\) 1.15016e7 1.53993
\(93\) −1.49210e6 −0.192357
\(94\) 412332. 0.0512035
\(95\) −3.84020e6 −0.459538
\(96\) −4.12290e6 −0.475613
\(97\) 1.17167e7 1.30348 0.651739 0.758443i \(-0.274040\pi\)
0.651739 + 0.758443i \(0.274040\pi\)
\(98\) 5.57069e6 0.597885
\(99\) −1.32722e6 −0.137473
\(100\) 3.91076e6 0.391076
\(101\) −826632. −0.0798339 −0.0399169 0.999203i \(-0.512709\pi\)
−0.0399169 + 0.999203i \(0.512709\pi\)
\(102\) 3.47117e6 0.323873
\(103\) 9.34032e6 0.842232 0.421116 0.907007i \(-0.361639\pi\)
0.421116 + 0.907007i \(0.361639\pi\)
\(104\) 1.07433e7 0.936529
\(105\) 8.76571e6 0.738966
\(106\) 2.02088e6 0.164805
\(107\) −1.21889e7 −0.961884 −0.480942 0.876752i \(-0.659705\pi\)
−0.480942 + 0.876752i \(0.659705\pi\)
\(108\) 2.26883e6 0.173308
\(109\) −1.32306e7 −0.978556 −0.489278 0.872128i \(-0.662740\pi\)
−0.489278 + 0.872128i \(0.662740\pi\)
\(110\) 1.36570e6 0.0978317
\(111\) −1.43646e7 −0.996927
\(112\) 1.80019e7 1.21075
\(113\) 1.51047e7 0.984777 0.492389 0.870375i \(-0.336124\pi\)
0.492389 + 0.870375i \(0.336124\pi\)
\(114\) −1.75978e6 −0.111248
\(115\) 2.09771e7 1.28619
\(116\) −1.62579e6 −0.0967079
\(117\) −9.02275e6 −0.520821
\(118\) −732819. −0.0410591
\(119\) −5.56410e7 −3.02678
\(120\) −4.92708e6 −0.260288
\(121\) −1.61726e7 −0.829910
\(122\) −452471. −0.0225596
\(123\) 751038. 0.0363910
\(124\) −6.37008e6 −0.300033
\(125\) 2.35570e7 1.07879
\(126\) 4.01691e6 0.178894
\(127\) 567728. 0.0245939 0.0122969 0.999924i \(-0.496086\pi\)
0.0122969 + 0.999924i \(0.496086\pi\)
\(128\) −2.29256e7 −0.966240
\(129\) −2.11824e7 −0.868783
\(130\) 9.28435e6 0.370638
\(131\) −4.38980e6 −0.170606 −0.0853032 0.996355i \(-0.527186\pi\)
−0.0853032 + 0.996355i \(0.527186\pi\)
\(132\) −5.66615e6 −0.214427
\(133\) 2.82084e7 1.03968
\(134\) −1.66659e7 −0.598359
\(135\) 4.13800e6 0.144751
\(136\) 3.12750e7 1.06613
\(137\) 5.41462e7 1.79906 0.899530 0.436858i \(-0.143909\pi\)
0.899530 + 0.436858i \(0.143909\pi\)
\(138\) 9.61284e6 0.311369
\(139\) 5.04738e7 1.59410 0.797048 0.603917i \(-0.206394\pi\)
0.797048 + 0.603917i \(0.206394\pi\)
\(140\) 3.74226e7 1.15262
\(141\) −3.12012e6 −0.0937354
\(142\) 4.56405e6 0.133765
\(143\) 2.25334e7 0.644391
\(144\) 8.49808e6 0.237166
\(145\) −2.96520e6 −0.0807729
\(146\) 197291. 0.00524652
\(147\) −4.21533e7 −1.09451
\(148\) −6.13254e7 −1.55498
\(149\) 5.85576e7 1.45021 0.725106 0.688638i \(-0.241791\pi\)
0.725106 + 0.688638i \(0.241791\pi\)
\(150\) 3.26855e6 0.0790744
\(151\) −1.16831e6 −0.0276147 −0.0138074 0.999905i \(-0.504395\pi\)
−0.0138074 + 0.999905i \(0.504395\pi\)
\(152\) −1.58555e7 −0.366209
\(153\) −2.62663e7 −0.592896
\(154\) −1.00318e7 −0.221338
\(155\) −1.16181e7 −0.250595
\(156\) −3.85200e7 −0.812362
\(157\) 2.26283e7 0.466663 0.233332 0.972397i \(-0.425037\pi\)
0.233332 + 0.972397i \(0.425037\pi\)
\(158\) −4.74384e6 −0.0956820
\(159\) −1.52920e7 −0.301699
\(160\) −3.21025e7 −0.619610
\(161\) −1.54089e8 −2.90992
\(162\) 1.89625e6 0.0350423
\(163\) 8.98984e7 1.62591 0.812953 0.582329i \(-0.197858\pi\)
0.812953 + 0.582329i \(0.197858\pi\)
\(164\) 3.20633e6 0.0567617
\(165\) −1.03342e7 −0.179095
\(166\) 9.07841e6 0.154039
\(167\) 8.00457e7 1.32994 0.664968 0.746872i \(-0.268445\pi\)
0.664968 + 0.746872i \(0.268445\pi\)
\(168\) 3.61921e7 0.588887
\(169\) 9.04390e7 1.44129
\(170\) 2.70278e7 0.421929
\(171\) 1.33163e7 0.203655
\(172\) −9.04320e7 −1.35510
\(173\) −2.61935e7 −0.384620 −0.192310 0.981334i \(-0.561598\pi\)
−0.192310 + 0.981334i \(0.561598\pi\)
\(174\) −1.35881e6 −0.0195540
\(175\) −5.23932e7 −0.738996
\(176\) −2.12230e7 −0.293436
\(177\) 5.54523e6 0.0751646
\(178\) 2.86962e7 0.381377
\(179\) −1.04938e8 −1.36756 −0.683781 0.729687i \(-0.739666\pi\)
−0.683781 + 0.729687i \(0.739666\pi\)
\(180\) 1.76660e7 0.225779
\(181\) 320272. 0.00401461 0.00200731 0.999998i \(-0.499361\pi\)
0.00200731 + 0.999998i \(0.499361\pi\)
\(182\) −6.81987e7 −0.838545
\(183\) 3.42385e6 0.0412986
\(184\) 8.66111e7 1.02497
\(185\) −1.11848e8 −1.29876
\(186\) −5.32401e6 −0.0606658
\(187\) 6.55972e7 0.733567
\(188\) −1.33204e7 −0.146206
\(189\) −3.03959e7 −0.327491
\(190\) −1.37023e7 −0.144929
\(191\) −4.56002e7 −0.473533 −0.236767 0.971567i \(-0.576088\pi\)
−0.236767 + 0.971567i \(0.576088\pi\)
\(192\) 2.55762e7 0.260784
\(193\) −4.97679e6 −0.0498309 −0.0249154 0.999690i \(-0.507932\pi\)
−0.0249154 + 0.999690i \(0.507932\pi\)
\(194\) 4.18066e7 0.411092
\(195\) −7.02546e7 −0.678506
\(196\) −1.79961e8 −1.70719
\(197\) −1.34591e8 −1.25425 −0.627126 0.778917i \(-0.715769\pi\)
−0.627126 + 0.778917i \(0.715769\pi\)
\(198\) −4.73568e6 −0.0433565
\(199\) −696482. −0.00626504 −0.00313252 0.999995i \(-0.500997\pi\)
−0.00313252 + 0.999995i \(0.500997\pi\)
\(200\) 2.94495e7 0.260299
\(201\) 1.26111e8 1.09538
\(202\) −2.94953e6 −0.0251781
\(203\) 2.17810e7 0.182744
\(204\) −1.12136e8 −0.924783
\(205\) 5.84786e6 0.0474088
\(206\) 3.33275e7 0.265624
\(207\) −7.27402e7 −0.570005
\(208\) −1.44280e8 −1.11169
\(209\) −3.32559e7 −0.251975
\(210\) 3.12772e7 0.233056
\(211\) −7.56254e7 −0.554216 −0.277108 0.960839i \(-0.589376\pi\)
−0.277108 + 0.960839i \(0.589376\pi\)
\(212\) −6.52845e7 −0.470581
\(213\) −3.45361e7 −0.244875
\(214\) −4.34917e7 −0.303360
\(215\) −1.64934e8 −1.13182
\(216\) 1.70851e7 0.115353
\(217\) 8.53412e7 0.566956
\(218\) −4.72084e7 −0.308618
\(219\) −1.49290e6 −0.00960451
\(220\) −4.41188e7 −0.279347
\(221\) 4.45946e8 2.77914
\(222\) −5.12548e7 −0.314412
\(223\) 2.78187e8 1.67984 0.839922 0.542706i \(-0.182600\pi\)
0.839922 + 0.542706i \(0.182600\pi\)
\(224\) 2.35810e8 1.40183
\(225\) −2.47331e7 −0.144757
\(226\) 5.38956e7 0.310580
\(227\) −2.26384e8 −1.28457 −0.642283 0.766468i \(-0.722012\pi\)
−0.642283 + 0.766468i \(0.722012\pi\)
\(228\) 5.68498e7 0.317656
\(229\) −1.60387e8 −0.882559 −0.441279 0.897370i \(-0.645475\pi\)
−0.441279 + 0.897370i \(0.645475\pi\)
\(230\) 7.48492e7 0.405639
\(231\) 7.59105e7 0.405191
\(232\) −1.22428e7 −0.0643684
\(233\) 7.39393e7 0.382939 0.191469 0.981499i \(-0.438675\pi\)
0.191469 + 0.981499i \(0.438675\pi\)
\(234\) −3.21944e7 −0.164257
\(235\) −2.42944e7 −0.122115
\(236\) 2.36737e7 0.117240
\(237\) 3.58966e7 0.175160
\(238\) −1.98534e8 −0.954589
\(239\) 9.87669e7 0.467971 0.233985 0.972240i \(-0.424823\pi\)
0.233985 + 0.972240i \(0.424823\pi\)
\(240\) 6.61693e7 0.308970
\(241\) −1.16325e8 −0.535321 −0.267661 0.963513i \(-0.586251\pi\)
−0.267661 + 0.963513i \(0.586251\pi\)
\(242\) −5.77059e7 −0.261738
\(243\) −1.43489e7 −0.0641500
\(244\) 1.46171e7 0.0644164
\(245\) −3.28222e8 −1.42589
\(246\) 2.67980e6 0.0114770
\(247\) −2.26082e8 −0.954612
\(248\) −4.79690e7 −0.199701
\(249\) −6.86962e7 −0.281991
\(250\) 8.40545e7 0.340229
\(251\) −3.82452e8 −1.52658 −0.763290 0.646056i \(-0.776417\pi\)
−0.763290 + 0.646056i \(0.776417\pi\)
\(252\) −1.29766e8 −0.510811
\(253\) 1.81661e8 0.705244
\(254\) 2.02573e6 0.00775645
\(255\) −2.04519e8 −0.772402
\(256\) 3.94485e7 0.146957
\(257\) −1.87540e8 −0.689174 −0.344587 0.938754i \(-0.611981\pi\)
−0.344587 + 0.938754i \(0.611981\pi\)
\(258\) −7.55816e7 −0.273998
\(259\) 8.21588e8 2.93836
\(260\) −2.99931e8 −1.05831
\(261\) 1.02821e7 0.0357965
\(262\) −1.56634e7 −0.0538061
\(263\) −3.36862e8 −1.14185 −0.570923 0.821004i \(-0.693414\pi\)
−0.570923 + 0.821004i \(0.693414\pi\)
\(264\) −4.26682e7 −0.142722
\(265\) −1.19069e8 −0.393041
\(266\) 1.00651e8 0.327894
\(267\) −2.17144e8 −0.698165
\(268\) 5.38392e8 1.70855
\(269\) −6.01783e7 −0.188498 −0.0942490 0.995549i \(-0.530045\pi\)
−0.0942490 + 0.995549i \(0.530045\pi\)
\(270\) 1.47649e7 0.0456518
\(271\) −1.24433e8 −0.379788 −0.189894 0.981805i \(-0.560814\pi\)
−0.189894 + 0.981805i \(0.560814\pi\)
\(272\) −4.20015e8 −1.26553
\(273\) 5.16059e8 1.53508
\(274\) 1.93201e8 0.567390
\(275\) 6.17683e7 0.179102
\(276\) −3.10543e8 −0.889078
\(277\) −4.48164e8 −1.26695 −0.633473 0.773765i \(-0.718371\pi\)
−0.633473 + 0.773765i \(0.718371\pi\)
\(278\) 1.80097e8 0.502747
\(279\) 4.02867e7 0.111057
\(280\) 2.81806e8 0.767179
\(281\) 5.57459e8 1.49879 0.749395 0.662124i \(-0.230345\pi\)
0.749395 + 0.662124i \(0.230345\pi\)
\(282\) −1.11330e7 −0.0295624
\(283\) 6.19049e8 1.62358 0.811788 0.583952i \(-0.198495\pi\)
0.811788 + 0.583952i \(0.198495\pi\)
\(284\) −1.47442e8 −0.381950
\(285\) 1.03685e8 0.265314
\(286\) 8.04019e7 0.203229
\(287\) −4.29558e7 −0.107259
\(288\) 1.11318e8 0.274595
\(289\) 8.87863e8 2.16373
\(290\) −1.05802e7 −0.0254742
\(291\) −3.16350e8 −0.752563
\(292\) −6.37348e6 −0.0149809
\(293\) 5.08839e8 1.18180 0.590899 0.806745i \(-0.298773\pi\)
0.590899 + 0.806745i \(0.298773\pi\)
\(294\) −1.50409e8 −0.345189
\(295\) 4.31773e7 0.0979215
\(296\) −4.61803e8 −1.03499
\(297\) 3.58348e7 0.0793703
\(298\) 2.08941e8 0.457369
\(299\) 1.23498e9 2.67184
\(300\) −1.05591e8 −0.225788
\(301\) 1.21153e9 2.56066
\(302\) −4.16870e6 −0.00870915
\(303\) 2.23191e7 0.0460921
\(304\) 2.12935e8 0.434701
\(305\) 2.66593e7 0.0538022
\(306\) −9.37215e7 −0.186988
\(307\) −2.00181e8 −0.394855 −0.197428 0.980317i \(-0.563259\pi\)
−0.197428 + 0.980317i \(0.563259\pi\)
\(308\) 3.24077e8 0.632006
\(309\) −2.52189e8 −0.486263
\(310\) −4.14548e7 −0.0790330
\(311\) −9.52557e7 −0.179568 −0.0897841 0.995961i \(-0.528618\pi\)
−0.0897841 + 0.995961i \(0.528618\pi\)
\(312\) −2.90069e8 −0.540705
\(313\) −3.67649e8 −0.677686 −0.338843 0.940843i \(-0.610035\pi\)
−0.338843 + 0.940843i \(0.610035\pi\)
\(314\) 8.07408e7 0.147177
\(315\) −2.36674e8 −0.426642
\(316\) 1.53250e8 0.273209
\(317\) 3.80959e8 0.671693 0.335847 0.941917i \(-0.390978\pi\)
0.335847 + 0.941917i \(0.390978\pi\)
\(318\) −5.45637e7 −0.0951501
\(319\) −2.56784e7 −0.0442895
\(320\) 1.99146e8 0.339739
\(321\) 3.29101e8 0.555344
\(322\) −5.49809e8 −0.917733
\(323\) −6.58151e8 −1.08672
\(324\) −6.12584e7 −0.100059
\(325\) 4.19916e8 0.678533
\(326\) 3.20769e8 0.512780
\(327\) 3.57225e8 0.564970
\(328\) 2.41449e7 0.0377804
\(329\) 1.78456e8 0.276277
\(330\) −3.68738e7 −0.0564832
\(331\) −8.19344e8 −1.24185 −0.620924 0.783871i \(-0.713242\pi\)
−0.620924 + 0.783871i \(0.713242\pi\)
\(332\) −2.93278e8 −0.439842
\(333\) 3.87845e8 0.575576
\(334\) 2.85614e8 0.419436
\(335\) 9.81945e8 1.42702
\(336\) −4.86050e8 −0.699027
\(337\) 1.23100e8 0.175208 0.0876039 0.996155i \(-0.472079\pi\)
0.0876039 + 0.996155i \(0.472079\pi\)
\(338\) 3.22698e8 0.454556
\(339\) −4.07827e8 −0.568561
\(340\) −8.73134e8 −1.20477
\(341\) −1.00612e8 −0.137407
\(342\) 4.75141e7 0.0642290
\(343\) 1.13920e9 1.52430
\(344\) −6.80986e8 −0.901952
\(345\) −5.66383e8 −0.742580
\(346\) −9.34617e7 −0.121302
\(347\) −5.04884e8 −0.648692 −0.324346 0.945939i \(-0.605144\pi\)
−0.324346 + 0.945939i \(0.605144\pi\)
\(348\) 4.38964e7 0.0558343
\(349\) 8.90989e8 1.12197 0.560987 0.827824i \(-0.310422\pi\)
0.560987 + 0.827824i \(0.310422\pi\)
\(350\) −1.86946e8 −0.233065
\(351\) 2.43614e8 0.300696
\(352\) −2.78005e8 −0.339746
\(353\) −7.69217e8 −0.930759 −0.465380 0.885111i \(-0.654082\pi\)
−0.465380 + 0.885111i \(0.654082\pi\)
\(354\) 1.97861e7 0.0237055
\(355\) −2.68911e8 −0.319014
\(356\) −9.27031e8 −1.08898
\(357\) 1.50231e9 1.74751
\(358\) −3.74432e8 −0.431303
\(359\) −8.50524e8 −0.970188 −0.485094 0.874462i \(-0.661215\pi\)
−0.485094 + 0.874462i \(0.661215\pi\)
\(360\) 1.33031e8 0.150278
\(361\) −5.60208e8 −0.626721
\(362\) 1.14277e6 0.00126613
\(363\) 4.36660e8 0.479149
\(364\) 2.20316e9 2.39437
\(365\) −1.16243e7 −0.0125124
\(366\) 1.22167e7 0.0130248
\(367\) 8.45450e8 0.892805 0.446402 0.894832i \(-0.352705\pi\)
0.446402 + 0.894832i \(0.352705\pi\)
\(368\) −1.16316e9 −1.21667
\(369\) −2.02780e7 −0.0210104
\(370\) −3.99089e8 −0.409604
\(371\) 8.74629e8 0.889232
\(372\) 1.71992e8 0.173224
\(373\) −7.54053e7 −0.0752352 −0.0376176 0.999292i \(-0.511977\pi\)
−0.0376176 + 0.999292i \(0.511977\pi\)
\(374\) 2.34059e8 0.231353
\(375\) −6.36040e8 −0.622838
\(376\) −1.00307e8 −0.0973141
\(377\) −1.74569e8 −0.167792
\(378\) −1.08457e8 −0.103284
\(379\) 2.52216e8 0.237977 0.118988 0.992896i \(-0.462035\pi\)
0.118988 + 0.992896i \(0.462035\pi\)
\(380\) 4.42654e8 0.413830
\(381\) −1.53287e7 −0.0141993
\(382\) −1.62708e8 −0.149343
\(383\) 5.53340e8 0.503265 0.251632 0.967823i \(-0.419033\pi\)
0.251632 + 0.967823i \(0.419033\pi\)
\(384\) 6.18990e8 0.557859
\(385\) 5.91068e8 0.527867
\(386\) −1.77578e7 −0.0157157
\(387\) 5.71925e8 0.501592
\(388\) −1.35056e9 −1.17383
\(389\) −1.67614e9 −1.44373 −0.721865 0.692034i \(-0.756715\pi\)
−0.721865 + 0.692034i \(0.756715\pi\)
\(390\) −2.50677e8 −0.213988
\(391\) 3.59516e9 3.04158
\(392\) −1.35517e9 −1.13630
\(393\) 1.18525e8 0.0984997
\(394\) −4.80239e8 −0.395568
\(395\) 2.79504e8 0.228191
\(396\) 1.52986e8 0.123800
\(397\) 6.56873e7 0.0526883 0.0263442 0.999653i \(-0.491613\pi\)
0.0263442 + 0.999653i \(0.491613\pi\)
\(398\) −2.48514e6 −0.00197588
\(399\) −7.61627e8 −0.600257
\(400\) −3.95498e8 −0.308983
\(401\) 2.32525e9 1.80080 0.900398 0.435067i \(-0.143275\pi\)
0.900398 + 0.435067i \(0.143275\pi\)
\(402\) 4.49979e8 0.345463
\(403\) −6.83985e8 −0.520569
\(404\) 9.52846e7 0.0718932
\(405\) −1.11726e8 −0.0835722
\(406\) 7.77175e7 0.0576339
\(407\) −9.68600e8 −0.712138
\(408\) −8.44425e8 −0.615532
\(409\) 1.53360e9 1.10836 0.554178 0.832398i \(-0.313033\pi\)
0.554178 + 0.832398i \(0.313033\pi\)
\(410\) 2.08659e7 0.0149518
\(411\) −1.46195e9 −1.03869
\(412\) −1.07664e9 −0.758459
\(413\) −3.17161e8 −0.221541
\(414\) −2.59547e8 −0.179769
\(415\) −5.34895e8 −0.367367
\(416\) −1.88995e9 −1.28714
\(417\) −1.36279e9 −0.920351
\(418\) −1.18661e8 −0.0794680
\(419\) 7.21485e8 0.479158 0.239579 0.970877i \(-0.422991\pi\)
0.239579 + 0.970877i \(0.422991\pi\)
\(420\) −1.01041e9 −0.665464
\(421\) −4.02560e8 −0.262932 −0.131466 0.991321i \(-0.541968\pi\)
−0.131466 + 0.991321i \(0.541968\pi\)
\(422\) −2.69841e8 −0.174789
\(423\) 8.42431e7 0.0541182
\(424\) −4.91616e8 −0.313217
\(425\) 1.22243e9 0.772433
\(426\) −1.23229e8 −0.0772291
\(427\) −1.95828e8 −0.121724
\(428\) 1.40500e9 0.866210
\(429\) −6.08401e8 −0.372040
\(430\) −5.88506e8 −0.356953
\(431\) −7.13192e8 −0.429078 −0.214539 0.976715i \(-0.568825\pi\)
−0.214539 + 0.976715i \(0.568825\pi\)
\(432\) −2.29448e8 −0.136928
\(433\) −1.48657e9 −0.879989 −0.439995 0.898000i \(-0.645020\pi\)
−0.439995 + 0.898000i \(0.645020\pi\)
\(434\) 3.04508e8 0.178807
\(435\) 8.00603e7 0.0466342
\(436\) 1.52507e9 0.881224
\(437\) −1.82264e9 −1.04476
\(438\) −5.32685e6 −0.00302908
\(439\) 2.80769e9 1.58388 0.791942 0.610597i \(-0.209070\pi\)
0.791942 + 0.610597i \(0.209070\pi\)
\(440\) −3.32231e8 −0.185933
\(441\) 1.13814e9 0.631918
\(442\) 1.59119e9 0.876487
\(443\) −2.23536e9 −1.22161 −0.610807 0.791780i \(-0.709155\pi\)
−0.610807 + 0.791780i \(0.709155\pi\)
\(444\) 1.65579e9 0.897768
\(445\) −1.69076e9 −0.909542
\(446\) 9.92606e8 0.529791
\(447\) −1.58105e9 −0.837280
\(448\) −1.46284e9 −0.768639
\(449\) −2.11217e9 −1.10120 −0.550600 0.834769i \(-0.685601\pi\)
−0.550600 + 0.834769i \(0.685601\pi\)
\(450\) −8.82509e7 −0.0456536
\(451\) 5.06422e7 0.0259953
\(452\) −1.74110e9 −0.886826
\(453\) 3.15445e7 0.0159434
\(454\) −8.07769e8 −0.405128
\(455\) 4.01823e9 1.99984
\(456\) 4.28100e8 0.211431
\(457\) −2.25521e9 −1.10530 −0.552650 0.833413i \(-0.686383\pi\)
−0.552650 + 0.833413i \(0.686383\pi\)
\(458\) −5.72280e8 −0.278342
\(459\) 7.09190e8 0.342309
\(460\) −2.41800e9 −1.15826
\(461\) 1.54873e9 0.736244 0.368122 0.929778i \(-0.380001\pi\)
0.368122 + 0.929778i \(0.380001\pi\)
\(462\) 2.70859e8 0.127790
\(463\) 2.79861e9 1.31041 0.655207 0.755449i \(-0.272581\pi\)
0.655207 + 0.755449i \(0.272581\pi\)
\(464\) 1.64417e8 0.0764073
\(465\) 3.13688e8 0.144681
\(466\) 2.63825e8 0.120772
\(467\) 3.32921e9 1.51263 0.756314 0.654209i \(-0.226998\pi\)
0.756314 + 0.654209i \(0.226998\pi\)
\(468\) 1.04004e9 0.469018
\(469\) −7.21294e9 −3.22855
\(470\) −8.66855e7 −0.0385127
\(471\) −6.10965e8 −0.269428
\(472\) 1.78272e8 0.0780343
\(473\) −1.42832e9 −0.620600
\(474\) 1.28084e8 0.0552420
\(475\) −6.19735e8 −0.265325
\(476\) 6.41365e9 2.72572
\(477\) 4.12883e8 0.174186
\(478\) 3.52413e8 0.147589
\(479\) −2.33364e9 −0.970196 −0.485098 0.874460i \(-0.661216\pi\)
−0.485098 + 0.874460i \(0.661216\pi\)
\(480\) 8.66766e8 0.357732
\(481\) −6.58479e9 −2.69795
\(482\) −4.15064e8 −0.168830
\(483\) 4.16040e9 1.68004
\(484\) 1.86419e9 0.747363
\(485\) −2.46322e9 −0.980410
\(486\) −5.11988e7 −0.0202317
\(487\) 9.35274e8 0.366934 0.183467 0.983026i \(-0.441268\pi\)
0.183467 + 0.983026i \(0.441268\pi\)
\(488\) 1.10072e8 0.0428753
\(489\) −2.42726e9 −0.938718
\(490\) −1.17114e9 −0.449699
\(491\) 2.77010e9 1.05611 0.528056 0.849210i \(-0.322921\pi\)
0.528056 + 0.849210i \(0.322921\pi\)
\(492\) −8.65710e7 −0.0327714
\(493\) −5.08189e8 −0.191012
\(494\) −8.06690e8 −0.301067
\(495\) 2.79024e8 0.103401
\(496\) 6.44211e8 0.237051
\(497\) 1.97530e9 0.721750
\(498\) −2.45117e8 −0.0889346
\(499\) −1.78899e9 −0.644547 −0.322274 0.946647i \(-0.604447\pi\)
−0.322274 + 0.946647i \(0.604447\pi\)
\(500\) −2.71538e9 −0.971485
\(501\) −2.16123e9 −0.767839
\(502\) −1.36464e9 −0.481454
\(503\) 2.69835e9 0.945388 0.472694 0.881227i \(-0.343282\pi\)
0.472694 + 0.881227i \(0.343282\pi\)
\(504\) −9.77188e8 −0.339994
\(505\) 1.73785e8 0.0600470
\(506\) 6.48190e8 0.222421
\(507\) −2.44185e9 −0.832131
\(508\) −6.54411e7 −0.0221477
\(509\) 2.29043e8 0.0769848 0.0384924 0.999259i \(-0.487744\pi\)
0.0384924 + 0.999259i \(0.487744\pi\)
\(510\) −7.29751e8 −0.243601
\(511\) 8.53867e7 0.0283085
\(512\) 3.07523e9 1.01259
\(513\) −3.59539e8 −0.117580
\(514\) −6.69169e8 −0.217353
\(515\) −1.96364e9 −0.633484
\(516\) 2.44166e9 0.782369
\(517\) −2.10388e8 −0.0669583
\(518\) 2.93153e9 0.926703
\(519\) 7.07223e8 0.222060
\(520\) −2.25859e9 −0.704410
\(521\) −1.11723e9 −0.346107 −0.173054 0.984912i \(-0.555363\pi\)
−0.173054 + 0.984912i \(0.555363\pi\)
\(522\) 3.66879e7 0.0112895
\(523\) −3.64533e8 −0.111425 −0.0557123 0.998447i \(-0.517743\pi\)
−0.0557123 + 0.998447i \(0.517743\pi\)
\(524\) 5.06006e8 0.153637
\(525\) 1.41462e9 0.426659
\(526\) −1.20197e9 −0.360116
\(527\) −1.99116e9 −0.592609
\(528\) 5.73022e8 0.169415
\(529\) 6.55140e9 1.92415
\(530\) −4.24854e8 −0.123958
\(531\) −1.49721e8 −0.0433963
\(532\) −3.25154e9 −0.936264
\(533\) 3.44278e8 0.0984837
\(534\) −7.74798e8 −0.220188
\(535\) 2.56251e9 0.723481
\(536\) 4.05429e9 1.13720
\(537\) 2.83333e9 0.789562
\(538\) −2.14724e8 −0.0594487
\(539\) −2.84238e9 −0.781847
\(540\) −4.76981e8 −0.130354
\(541\) −3.21998e9 −0.874305 −0.437153 0.899387i \(-0.644013\pi\)
−0.437153 + 0.899387i \(0.644013\pi\)
\(542\) −4.43991e8 −0.119778
\(543\) −8.64734e6 −0.00231784
\(544\) −5.50187e9 −1.46526
\(545\) 2.78149e9 0.736020
\(546\) 1.84137e9 0.484134
\(547\) 3.19865e9 0.835625 0.417812 0.908533i \(-0.362797\pi\)
0.417812 + 0.908533i \(0.362797\pi\)
\(548\) −6.24135e9 −1.62012
\(549\) −9.24438e7 −0.0238438
\(550\) 2.20397e8 0.0564854
\(551\) 2.57637e8 0.0656113
\(552\) −2.33850e9 −0.591767
\(553\) −2.05312e9 −0.516268
\(554\) −1.59911e9 −0.399571
\(555\) 3.01990e9 0.749838
\(556\) −5.81804e9 −1.43554
\(557\) −4.79806e9 −1.17645 −0.588223 0.808699i \(-0.700172\pi\)
−0.588223 + 0.808699i \(0.700172\pi\)
\(558\) 1.43748e8 0.0350254
\(559\) −9.71009e9 −2.35116
\(560\) −3.78457e9 −0.910665
\(561\) −1.77112e9 −0.423525
\(562\) 1.98909e9 0.472690
\(563\) 2.61139e9 0.616726 0.308363 0.951269i \(-0.400219\pi\)
0.308363 + 0.951269i \(0.400219\pi\)
\(564\) 3.59651e8 0.0844120
\(565\) −3.17550e9 −0.740700
\(566\) 2.20885e9 0.512045
\(567\) 8.20690e8 0.189077
\(568\) −1.11029e9 −0.254224
\(569\) 8.32085e9 1.89354 0.946771 0.321909i \(-0.104324\pi\)
0.946771 + 0.321909i \(0.104324\pi\)
\(570\) 3.69963e8 0.0836751
\(571\) −3.91195e8 −0.0879360 −0.0439680 0.999033i \(-0.514000\pi\)
−0.0439680 + 0.999033i \(0.514000\pi\)
\(572\) −2.59738e9 −0.580297
\(573\) 1.23121e9 0.273395
\(574\) −1.53272e8 −0.0338276
\(575\) 3.38531e9 0.742610
\(576\) −6.90556e8 −0.150564
\(577\) 6.50587e9 1.40991 0.704953 0.709254i \(-0.250968\pi\)
0.704953 + 0.709254i \(0.250968\pi\)
\(578\) 3.16801e9 0.682400
\(579\) 1.34373e8 0.0287699
\(580\) 3.41794e8 0.0727388
\(581\) 3.92910e9 0.831145
\(582\) −1.12878e9 −0.237344
\(583\) −1.03113e9 −0.215513
\(584\) −4.79946e7 −0.00997120
\(585\) 1.89687e9 0.391735
\(586\) 1.81560e9 0.372717
\(587\) −7.65590e9 −1.56230 −0.781148 0.624346i \(-0.785365\pi\)
−0.781148 + 0.624346i \(0.785365\pi\)
\(588\) 4.85895e9 0.985648
\(589\) 1.00946e9 0.203557
\(590\) 1.54062e8 0.0308826
\(591\) 3.63396e9 0.724143
\(592\) 6.20188e9 1.22856
\(593\) 2.64479e9 0.520835 0.260418 0.965496i \(-0.416140\pi\)
0.260418 + 0.965496i \(0.416140\pi\)
\(594\) 1.27863e8 0.0250319
\(595\) 1.16975e10 2.27659
\(596\) −6.74984e9 −1.30597
\(597\) 1.88050e7 0.00361712
\(598\) 4.40656e9 0.842646
\(599\) −3.92980e8 −0.0747096 −0.0373548 0.999302i \(-0.511893\pi\)
−0.0373548 + 0.999302i \(0.511893\pi\)
\(600\) −7.95136e8 −0.150284
\(601\) 3.38890e9 0.636792 0.318396 0.947958i \(-0.396856\pi\)
0.318396 + 0.947958i \(0.396856\pi\)
\(602\) 4.32291e9 0.807585
\(603\) −3.40499e9 −0.632419
\(604\) 1.34670e8 0.0248680
\(605\) 3.40000e9 0.624216
\(606\) 7.96373e7 0.0145366
\(607\) 6.03144e9 1.09461 0.547307 0.836932i \(-0.315653\pi\)
0.547307 + 0.836932i \(0.315653\pi\)
\(608\) 2.78929e9 0.503305
\(609\) −5.88088e8 −0.105507
\(610\) 9.51240e7 0.0169682
\(611\) −1.43027e9 −0.253673
\(612\) 3.02767e9 0.533924
\(613\) 4.12690e9 0.723623 0.361812 0.932251i \(-0.382158\pi\)
0.361812 + 0.932251i \(0.382158\pi\)
\(614\) −7.14271e8 −0.124530
\(615\) −1.57892e8 −0.0273715
\(616\) 2.44042e9 0.420661
\(617\) 1.09713e10 1.88045 0.940224 0.340557i \(-0.110615\pi\)
0.940224 + 0.340557i \(0.110615\pi\)
\(618\) −8.99842e8 −0.153358
\(619\) 6.38450e9 1.08196 0.540978 0.841037i \(-0.318054\pi\)
0.540978 + 0.841037i \(0.318054\pi\)
\(620\) 1.33920e9 0.225670
\(621\) 1.96399e9 0.329092
\(622\) −3.39885e8 −0.0566324
\(623\) 1.24196e10 2.05778
\(624\) 3.89555e9 0.641834
\(625\) −2.30186e9 −0.377137
\(626\) −1.31182e9 −0.213729
\(627\) 8.97910e8 0.145478
\(628\) −2.60833e9 −0.420246
\(629\) −1.91691e10 −3.07131
\(630\) −8.44484e8 −0.134555
\(631\) 4.25053e9 0.673504 0.336752 0.941593i \(-0.390672\pi\)
0.336752 + 0.941593i \(0.390672\pi\)
\(632\) 1.15403e9 0.181847
\(633\) 2.04189e9 0.319977
\(634\) 1.35931e9 0.211839
\(635\) −1.19355e8 −0.0184983
\(636\) 1.76268e9 0.271690
\(637\) −1.93232e10 −2.96205
\(638\) −9.16240e7 −0.0139681
\(639\) 9.32476e8 0.141379
\(640\) 4.81969e9 0.726757
\(641\) 7.36203e9 1.10406 0.552032 0.833823i \(-0.313853\pi\)
0.552032 + 0.833823i \(0.313853\pi\)
\(642\) 1.17428e9 0.175145
\(643\) 4.96388e8 0.0736348 0.0368174 0.999322i \(-0.488278\pi\)
0.0368174 + 0.999322i \(0.488278\pi\)
\(644\) 1.77616e10 2.62048
\(645\) 4.45322e9 0.653455
\(646\) −2.34837e9 −0.342730
\(647\) 1.80994e8 0.0262724 0.0131362 0.999914i \(-0.495819\pi\)
0.0131362 + 0.999914i \(0.495819\pi\)
\(648\) −4.61298e8 −0.0665992
\(649\) 3.73913e8 0.0536925
\(650\) 1.49832e9 0.213996
\(651\) −2.30421e9 −0.327332
\(652\) −1.03625e10 −1.46419
\(653\) −9.58608e9 −1.34724 −0.673620 0.739078i \(-0.735262\pi\)
−0.673620 + 0.739078i \(0.735262\pi\)
\(654\) 1.27463e9 0.178181
\(655\) 9.22878e8 0.128322
\(656\) −3.24259e8 −0.0448465
\(657\) 4.03082e7 0.00554517
\(658\) 6.36754e8 0.0871326
\(659\) −6.06150e9 −0.825051 −0.412526 0.910946i \(-0.635353\pi\)
−0.412526 + 0.910946i \(0.635353\pi\)
\(660\) 1.19121e9 0.161281
\(661\) −9.42282e9 −1.26904 −0.634520 0.772906i \(-0.718802\pi\)
−0.634520 + 0.772906i \(0.718802\pi\)
\(662\) −2.92352e9 −0.391655
\(663\) −1.20406e10 −1.60453
\(664\) −2.20849e9 −0.292757
\(665\) −5.93031e9 −0.781991
\(666\) 1.38388e9 0.181526
\(667\) −1.40735e9 −0.183637
\(668\) −9.22675e9 −1.19765
\(669\) −7.51104e9 −0.969859
\(670\) 3.50371e9 0.450056
\(671\) 2.30868e8 0.0295009
\(672\) −6.36688e9 −0.809346
\(673\) 5.79599e9 0.732951 0.366475 0.930428i \(-0.380564\pi\)
0.366475 + 0.930428i \(0.380564\pi\)
\(674\) 4.39237e8 0.0552572
\(675\) 6.67794e8 0.0835755
\(676\) −1.04248e10 −1.29793
\(677\) −1.36618e10 −1.69219 −0.846093 0.533036i \(-0.821051\pi\)
−0.846093 + 0.533036i \(0.821051\pi\)
\(678\) −1.45518e9 −0.179314
\(679\) 1.80938e10 2.21812
\(680\) −6.57501e9 −0.801891
\(681\) 6.11238e9 0.741644
\(682\) −3.58996e8 −0.0433355
\(683\) 1.32763e10 1.59443 0.797214 0.603697i \(-0.206306\pi\)
0.797214 + 0.603697i \(0.206306\pi\)
\(684\) −1.53494e9 −0.183399
\(685\) −1.13833e10 −1.35316
\(686\) 4.06481e9 0.480735
\(687\) 4.33044e9 0.509546
\(688\) 9.14545e9 1.07064
\(689\) −7.00989e9 −0.816477
\(690\) −2.02093e9 −0.234196
\(691\) −1.61989e10 −1.86773 −0.933863 0.357630i \(-0.883585\pi\)
−0.933863 + 0.357630i \(0.883585\pi\)
\(692\) 3.01928e9 0.346363
\(693\) −2.04958e9 −0.233937
\(694\) −1.80149e9 −0.204585
\(695\) −1.06112e10 −1.19900
\(696\) 3.30556e8 0.0371631
\(697\) 1.00223e9 0.112113
\(698\) 3.17916e9 0.353850
\(699\) −1.99636e9 −0.221090
\(700\) 6.03929e9 0.665491
\(701\) 1.52630e10 1.67351 0.836754 0.547579i \(-0.184451\pi\)
0.836754 + 0.547579i \(0.184451\pi\)
\(702\) 8.69248e8 0.0948339
\(703\) 9.71818e9 1.05497
\(704\) 1.72459e9 0.186287
\(705\) 6.55949e8 0.0705030
\(706\) −2.74467e9 −0.293544
\(707\) −1.27654e9 −0.135853
\(708\) −6.39190e8 −0.0676883
\(709\) 5.05134e9 0.532286 0.266143 0.963934i \(-0.414251\pi\)
0.266143 + 0.963934i \(0.414251\pi\)
\(710\) −9.59511e8 −0.100611
\(711\) −9.69208e8 −0.101128
\(712\) −6.98088e9 −0.724820
\(713\) −5.51419e9 −0.569729
\(714\) 5.36043e9 0.551132
\(715\) −4.73724e9 −0.484679
\(716\) 1.20960e10 1.23154
\(717\) −2.66671e9 −0.270183
\(718\) −3.03478e9 −0.305979
\(719\) 1.20715e10 1.21118 0.605592 0.795775i \(-0.292936\pi\)
0.605592 + 0.795775i \(0.292936\pi\)
\(720\) −1.78657e9 −0.178384
\(721\) 1.44240e10 1.43322
\(722\) −1.99889e9 −0.197656
\(723\) 3.14078e9 0.309068
\(724\) −3.69172e7 −0.00361530
\(725\) −4.78526e8 −0.0466361
\(726\) 1.55806e9 0.151114
\(727\) −1.02032e10 −0.984845 −0.492423 0.870356i \(-0.663889\pi\)
−0.492423 + 0.870356i \(0.663889\pi\)
\(728\) 1.65906e10 1.59368
\(729\) 3.87420e8 0.0370370
\(730\) −4.14768e7 −0.00394617
\(731\) −2.82672e10 −2.67653
\(732\) −3.94661e8 −0.0371908
\(733\) 1.27539e10 1.19613 0.598067 0.801446i \(-0.295936\pi\)
0.598067 + 0.801446i \(0.295936\pi\)
\(734\) 3.01667e9 0.281574
\(735\) 8.86199e9 0.823238
\(736\) −1.52365e10 −1.40869
\(737\) 8.50359e9 0.782467
\(738\) −7.23546e7 −0.00662627
\(739\) 1.16465e10 1.06155 0.530774 0.847514i \(-0.321901\pi\)
0.530774 + 0.847514i \(0.321901\pi\)
\(740\) 1.28926e10 1.16958
\(741\) 6.10422e9 0.551146
\(742\) 3.12079e9 0.280447
\(743\) −1.66695e10 −1.49095 −0.745475 0.666534i \(-0.767777\pi\)
−0.745475 + 0.666534i \(0.767777\pi\)
\(744\) 1.29516e9 0.115297
\(745\) −1.23107e10 −1.09078
\(746\) −2.69056e8 −0.0237277
\(747\) 1.85480e9 0.162808
\(748\) −7.56129e9 −0.660602
\(749\) −1.88230e10 −1.63683
\(750\) −2.26947e9 −0.196431
\(751\) −1.52168e10 −1.31094 −0.655471 0.755220i \(-0.727530\pi\)
−0.655471 + 0.755220i \(0.727530\pi\)
\(752\) 1.34710e9 0.115515
\(753\) 1.03262e10 0.881371
\(754\) −6.22883e8 −0.0529184
\(755\) 2.45617e8 0.0207704
\(756\) 3.50369e9 0.294917
\(757\) −8.51162e9 −0.713143 −0.356572 0.934268i \(-0.616054\pi\)
−0.356572 + 0.934268i \(0.616054\pi\)
\(758\) 8.99938e8 0.0750534
\(759\) −4.90484e9 −0.407173
\(760\) 3.33334e9 0.275444
\(761\) 1.77193e10 1.45748 0.728738 0.684793i \(-0.240107\pi\)
0.728738 + 0.684793i \(0.240107\pi\)
\(762\) −5.46946e7 −0.00447819
\(763\) −2.04316e10 −1.66520
\(764\) 5.25627e9 0.426433
\(765\) 5.52202e9 0.445947
\(766\) 1.97439e9 0.158720
\(767\) 2.54195e9 0.203415
\(768\) −1.06511e9 −0.0848458
\(769\) −1.41872e10 −1.12500 −0.562502 0.826796i \(-0.690161\pi\)
−0.562502 + 0.826796i \(0.690161\pi\)
\(770\) 2.10901e9 0.166479
\(771\) 5.06359e9 0.397895
\(772\) 5.73667e8 0.0448744
\(773\) −5.86760e9 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(774\) 2.04070e9 0.158193
\(775\) −1.87493e9 −0.144687
\(776\) −1.01702e10 −0.781295
\(777\) −2.21829e10 −1.69646
\(778\) −5.98067e9 −0.455325
\(779\) −5.08104e8 −0.0385098
\(780\) 8.09813e9 0.611018
\(781\) −2.32876e9 −0.174923
\(782\) 1.28280e10 0.959258
\(783\) −2.77617e8 −0.0206671
\(784\) 1.81996e10 1.34882
\(785\) −4.75720e9 −0.351001
\(786\) 4.22911e8 0.0310649
\(787\) 6.68562e9 0.488911 0.244456 0.969660i \(-0.421391\pi\)
0.244456 + 0.969660i \(0.421391\pi\)
\(788\) 1.55141e10 1.12950
\(789\) 9.09528e9 0.659245
\(790\) 9.97308e8 0.0719672
\(791\) 2.33258e10 1.67579
\(792\) 1.15204e9 0.0824005
\(793\) 1.56950e9 0.111765
\(794\) 2.34381e8 0.0166169
\(795\) 3.21486e9 0.226923
\(796\) 8.02824e7 0.00564189
\(797\) 1.99892e10 1.39860 0.699298 0.714831i \(-0.253496\pi\)
0.699298 + 0.714831i \(0.253496\pi\)
\(798\) −2.71758e9 −0.189310
\(799\) −4.16369e9 −0.288778
\(800\) −5.18072e9 −0.357746
\(801\) 5.86289e9 0.403086
\(802\) 8.29680e9 0.567937
\(803\) −1.00665e8 −0.00686082
\(804\) −1.45366e10 −0.986430
\(805\) 3.23944e10 2.18869
\(806\) −2.44055e9 −0.164178
\(807\) 1.62481e9 0.108829
\(808\) 7.17527e8 0.0478519
\(809\) −7.22991e9 −0.480079 −0.240040 0.970763i \(-0.577160\pi\)
−0.240040 + 0.970763i \(0.577160\pi\)
\(810\) −3.98653e8 −0.0263571
\(811\) −1.55913e10 −1.02639 −0.513193 0.858273i \(-0.671537\pi\)
−0.513193 + 0.858273i \(0.671537\pi\)
\(812\) −2.51067e9 −0.164567
\(813\) 3.35968e9 0.219271
\(814\) −3.45609e9 −0.224595
\(815\) −1.88995e10 −1.22292
\(816\) 1.13404e10 0.730656
\(817\) 1.43307e10 0.919367
\(818\) 5.47207e9 0.349555
\(819\) −1.39336e10 −0.886277
\(820\) −6.74074e8 −0.0426933
\(821\) 3.10602e8 0.0195886 0.00979431 0.999952i \(-0.496882\pi\)
0.00979431 + 0.999952i \(0.496882\pi\)
\(822\) −5.21642e9 −0.327583
\(823\) −2.52102e10 −1.57644 −0.788218 0.615396i \(-0.788996\pi\)
−0.788218 + 0.615396i \(0.788996\pi\)
\(824\) −8.10752e9 −0.504828
\(825\) −1.66774e9 −0.103405
\(826\) −1.13167e9 −0.0698700
\(827\) 1.09598e8 0.00673804 0.00336902 0.999994i \(-0.498928\pi\)
0.00336902 + 0.999994i \(0.498928\pi\)
\(828\) 8.38466e9 0.513309
\(829\) −2.09237e10 −1.27555 −0.637776 0.770222i \(-0.720145\pi\)
−0.637776 + 0.770222i \(0.720145\pi\)
\(830\) −1.90857e9 −0.115861
\(831\) 1.21004e10 0.731472
\(832\) 1.17242e10 0.705751
\(833\) −5.62522e10 −3.37196
\(834\) −4.86262e9 −0.290261
\(835\) −1.68282e10 −1.00031
\(836\) 3.83336e9 0.226912
\(837\) −1.08774e9 −0.0641190
\(838\) 2.57435e9 0.151117
\(839\) 2.64073e9 0.154368 0.0771839 0.997017i \(-0.475407\pi\)
0.0771839 + 0.997017i \(0.475407\pi\)
\(840\) −7.60875e9 −0.442931
\(841\) −1.70509e10 −0.988468
\(842\) −1.43639e9 −0.0829237
\(843\) −1.50514e10 −0.865326
\(844\) 8.71722e9 0.499091
\(845\) −1.90132e10 −1.08407
\(846\) 3.00590e8 0.0170678
\(847\) −2.49749e10 −1.41225
\(848\) 6.60227e9 0.371798
\(849\) −1.67143e10 −0.937372
\(850\) 4.36177e9 0.243611
\(851\) −5.30857e10 −2.95273
\(852\) 3.98093e9 0.220519
\(853\) 1.11033e10 0.612534 0.306267 0.951946i \(-0.400920\pi\)
0.306267 + 0.951946i \(0.400920\pi\)
\(854\) −6.98739e8 −0.0383895
\(855\) −2.79951e9 −0.153179
\(856\) 1.05802e10 0.576546
\(857\) −2.50546e9 −0.135974 −0.0679868 0.997686i \(-0.521658\pi\)
−0.0679868 + 0.997686i \(0.521658\pi\)
\(858\) −2.17085e9 −0.117334
\(859\) −1.03144e10 −0.555225 −0.277612 0.960693i \(-0.589543\pi\)
−0.277612 + 0.960693i \(0.589543\pi\)
\(860\) 1.90117e10 1.01924
\(861\) 1.15981e9 0.0619263
\(862\) −2.54476e9 −0.135323
\(863\) −1.04059e10 −0.551112 −0.275556 0.961285i \(-0.588862\pi\)
−0.275556 + 0.961285i \(0.588862\pi\)
\(864\) −3.00559e9 −0.158538
\(865\) 5.50671e9 0.289291
\(866\) −5.30427e9 −0.277532
\(867\) −2.39723e10 −1.24923
\(868\) −9.83715e9 −0.510564
\(869\) 2.42049e9 0.125122
\(870\) 2.85666e8 0.0147076
\(871\) 5.78096e10 2.96440
\(872\) 1.14843e10 0.586539
\(873\) 8.54146e9 0.434493
\(874\) −6.50343e9 −0.329498
\(875\) 3.63785e10 1.83576
\(876\) 1.72084e8 0.00864920
\(877\) 3.14032e10 1.57208 0.786041 0.618175i \(-0.212128\pi\)
0.786041 + 0.618175i \(0.212128\pi\)
\(878\) 1.00182e10 0.499527
\(879\) −1.37386e10 −0.682312
\(880\) 4.46177e9 0.220708
\(881\) −9.24452e9 −0.455479 −0.227740 0.973722i \(-0.573133\pi\)
−0.227740 + 0.973722i \(0.573133\pi\)
\(882\) 4.06103e9 0.199295
\(883\) 2.47182e10 1.20824 0.604121 0.796893i \(-0.293524\pi\)
0.604121 + 0.796893i \(0.293524\pi\)
\(884\) −5.14036e10 −2.50271
\(885\) −1.16579e9 −0.0565350
\(886\) −7.97605e9 −0.385274
\(887\) 1.21845e10 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(888\) 1.24687e10 0.597551
\(889\) 8.76727e8 0.0418512
\(890\) −6.03287e9 −0.286852
\(891\) −9.67541e8 −0.0458245
\(892\) −3.20661e10 −1.51276
\(893\) 2.11087e9 0.0991931
\(894\) −5.64141e9 −0.264062
\(895\) 2.20613e10 1.02861
\(896\) −3.54033e10 −1.64424
\(897\) −3.33444e10 −1.54259
\(898\) −7.53649e9 −0.347298
\(899\) 7.79451e8 0.0357792
\(900\) 2.85095e9 0.130359
\(901\) −2.04066e10 −0.929467
\(902\) 1.80698e8 0.00819842
\(903\) −3.27114e10 −1.47840
\(904\) −1.31111e10 −0.590268
\(905\) −6.73314e7 −0.00301959
\(906\) 1.12555e8 0.00502823
\(907\) 4.11727e10 1.83225 0.916124 0.400895i \(-0.131301\pi\)
0.916124 + 0.400895i \(0.131301\pi\)
\(908\) 2.60950e10 1.15680
\(909\) −6.02615e8 −0.0266113
\(910\) 1.43376e10 0.630711
\(911\) 3.78466e10 1.65849 0.829244 0.558887i \(-0.188771\pi\)
0.829244 + 0.558887i \(0.188771\pi\)
\(912\) −5.74926e9 −0.250975
\(913\) −4.63216e9 −0.201435
\(914\) −8.04688e9 −0.348591
\(915\) −7.19802e8 −0.0310627
\(916\) 1.84875e10 0.794775
\(917\) −6.77905e9 −0.290320
\(918\) 2.53048e9 0.107958
\(919\) −5.50991e9 −0.234175 −0.117087 0.993122i \(-0.537356\pi\)
−0.117087 + 0.993122i \(0.537356\pi\)
\(920\) −1.82084e10 −0.770931
\(921\) 5.40488e9 0.227970
\(922\) 5.52606e9 0.232197
\(923\) −1.58315e10 −0.662698
\(924\) −8.75009e9 −0.364889
\(925\) −1.80502e10 −0.749869
\(926\) 9.98580e9 0.413280
\(927\) 6.80909e9 0.280744
\(928\) 2.15374e9 0.0884658
\(929\) 3.71808e10 1.52147 0.760736 0.649062i \(-0.224838\pi\)
0.760736 + 0.649062i \(0.224838\pi\)
\(930\) 1.11928e9 0.0456297
\(931\) 2.85183e10 1.15824
\(932\) −8.52286e9 −0.344850
\(933\) 2.57190e9 0.103674
\(934\) 1.18791e10 0.477054
\(935\) −1.37906e10 −0.551752
\(936\) 7.83187e9 0.312176
\(937\) 4.61321e10 1.83195 0.915977 0.401230i \(-0.131417\pi\)
0.915977 + 0.401230i \(0.131417\pi\)
\(938\) −2.57367e10 −1.01822
\(939\) 9.92652e9 0.391262
\(940\) 2.80038e9 0.109969
\(941\) −4.30417e10 −1.68394 −0.841968 0.539528i \(-0.818603\pi\)
−0.841968 + 0.539528i \(0.818603\pi\)
\(942\) −2.18000e9 −0.0849725
\(943\) 2.77553e9 0.107784
\(944\) −2.39414e9 −0.0926291
\(945\) 6.39020e9 0.246322
\(946\) −5.09643e9 −0.195725
\(947\) −2.96342e10 −1.13388 −0.566942 0.823758i \(-0.691874\pi\)
−0.566942 + 0.823758i \(0.691874\pi\)
\(948\) −4.13775e9 −0.157737
\(949\) −6.84349e8 −0.0259924
\(950\) −2.21129e9 −0.0836785
\(951\) −1.02859e10 −0.387802
\(952\) 4.82972e10 1.81423
\(953\) −1.01449e10 −0.379683 −0.189841 0.981815i \(-0.560797\pi\)
−0.189841 + 0.981815i \(0.560797\pi\)
\(954\) 1.47322e9 0.0549349
\(955\) 9.58664e9 0.356168
\(956\) −1.13847e10 −0.421424
\(957\) 6.93318e8 0.0255706
\(958\) −8.32673e9 −0.305981
\(959\) 8.36165e10 3.06145
\(960\) −5.37693e9 −0.196149
\(961\) −2.44586e10 −0.888996
\(962\) −2.34954e10 −0.850883
\(963\) −8.88574e9 −0.320628
\(964\) 1.34086e10 0.482075
\(965\) 1.04628e9 0.0374803
\(966\) 1.48448e10 0.529853
\(967\) −4.46512e9 −0.158796 −0.0793982 0.996843i \(-0.525300\pi\)
−0.0793982 + 0.996843i \(0.525300\pi\)
\(968\) 1.40380e10 0.497442
\(969\) 1.77701e10 0.627417
\(970\) −8.78910e9 −0.309203
\(971\) −7.60821e9 −0.266695 −0.133348 0.991069i \(-0.542573\pi\)
−0.133348 + 0.991069i \(0.542573\pi\)
\(972\) 1.65398e9 0.0577693
\(973\) 7.79453e10 2.71266
\(974\) 3.33718e9 0.115724
\(975\) −1.13377e10 −0.391751
\(976\) −1.47824e9 −0.0508943
\(977\) −4.60674e9 −0.158038 −0.0790192 0.996873i \(-0.525179\pi\)
−0.0790192 + 0.996873i \(0.525179\pi\)
\(978\) −8.66077e9 −0.296054
\(979\) −1.46419e10 −0.498722
\(980\) 3.78336e10 1.28406
\(981\) −9.64508e9 −0.326185
\(982\) 9.88407e9 0.333078
\(983\) 3.68529e10 1.23747 0.618735 0.785600i \(-0.287645\pi\)
0.618735 + 0.785600i \(0.287645\pi\)
\(984\) −6.51911e8 −0.0218125
\(985\) 2.82954e10 0.943386
\(986\) −1.81329e9 −0.0602417
\(987\) −4.81831e9 −0.159509
\(988\) 2.60601e10 0.859661
\(989\) −7.82814e10 −2.57319
\(990\) 9.95592e8 0.0326106
\(991\) −8.00041e9 −0.261129 −0.130564 0.991440i \(-0.541679\pi\)
−0.130564 + 0.991440i \(0.541679\pi\)
\(992\) 8.43867e9 0.274462
\(993\) 2.21223e10 0.716981
\(994\) 7.04814e9 0.227626
\(995\) 1.46423e8 0.00471225
\(996\) 7.91851e9 0.253943
\(997\) 5.99071e9 0.191445 0.0957227 0.995408i \(-0.469484\pi\)
0.0957227 + 0.995408i \(0.469484\pi\)
\(998\) −6.38333e9 −0.203278
\(999\) −1.04718e10 −0.332309
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.1.10 17
3.2 odd 2 531.8.a.c.1.8 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.10 17 1.1 even 1 trivial
531.8.a.c.1.8 17 3.2 odd 2