Properties

Label 177.8.a.d.1.12
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(8.74396\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.74396 q^{2} +27.0000 q^{3} -33.0553 q^{4} +321.882 q^{5} +263.087 q^{6} -939.921 q^{7} -1569.32 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+9.74396 q^{2} +27.0000 q^{3} -33.0553 q^{4} +321.882 q^{5} +263.087 q^{6} -939.921 q^{7} -1569.32 q^{8} +729.000 q^{9} +3136.41 q^{10} +4998.14 q^{11} -892.494 q^{12} -1233.64 q^{13} -9158.55 q^{14} +8690.82 q^{15} -11060.3 q^{16} +9783.69 q^{17} +7103.34 q^{18} +19322.1 q^{19} -10639.9 q^{20} -25377.9 q^{21} +48701.6 q^{22} +79805.1 q^{23} -42371.5 q^{24} +25483.2 q^{25} -12020.5 q^{26} +19683.0 q^{27} +31069.4 q^{28} +144604. q^{29} +84683.0 q^{30} +148320. q^{31} +93101.7 q^{32} +134950. q^{33} +95331.8 q^{34} -302544. q^{35} -24097.3 q^{36} +203959. q^{37} +188274. q^{38} -33308.2 q^{39} -505135. q^{40} -393071. q^{41} -247281. q^{42} +24364.6 q^{43} -165215. q^{44} +234652. q^{45} +777617. q^{46} -18634.3 q^{47} -298627. q^{48} +59908.4 q^{49} +248307. q^{50} +264160. q^{51} +40778.3 q^{52} -228261. q^{53} +191790. q^{54} +1.60881e6 q^{55} +1.47503e6 q^{56} +521697. q^{57} +1.40901e6 q^{58} +205379. q^{59} -287278. q^{60} +2.54932e6 q^{61} +1.44523e6 q^{62} -685202. q^{63} +2.32289e6 q^{64} -397086. q^{65} +1.31494e6 q^{66} +2.19172e6 q^{67} -323403. q^{68} +2.15474e6 q^{69} -2.94797e6 q^{70} -3.19514e6 q^{71} -1.14403e6 q^{72} +2.62084e6 q^{73} +1.98737e6 q^{74} +688045. q^{75} -638699. q^{76} -4.69785e6 q^{77} -324554. q^{78} -2.93367e6 q^{79} -3.56010e6 q^{80} +531441. q^{81} -3.83007e6 q^{82} -5.63931e6 q^{83} +838873. q^{84} +3.14920e6 q^{85} +237407. q^{86} +3.90429e6 q^{87} -7.84365e6 q^{88} +8.11136e6 q^{89} +2.28644e6 q^{90} +1.15952e6 q^{91} -2.63798e6 q^{92} +4.00465e6 q^{93} -181571. q^{94} +6.21944e6 q^{95} +2.51375e6 q^{96} +6.86976e6 q^{97} +583745. q^{98} +3.64364e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.74396 0.861252 0.430626 0.902530i \(-0.358293\pi\)
0.430626 + 0.902530i \(0.358293\pi\)
\(3\) 27.0000 0.577350
\(4\) −33.0553 −0.258245
\(5\) 321.882 1.15160 0.575800 0.817590i \(-0.304691\pi\)
0.575800 + 0.817590i \(0.304691\pi\)
\(6\) 263.087 0.497244
\(7\) −939.921 −1.03573 −0.517867 0.855461i \(-0.673274\pi\)
−0.517867 + 0.855461i \(0.673274\pi\)
\(8\) −1569.32 −1.08367
\(9\) 729.000 0.333333
\(10\) 3136.41 0.991819
\(11\) 4998.14 1.13223 0.566114 0.824327i \(-0.308446\pi\)
0.566114 + 0.824327i \(0.308446\pi\)
\(12\) −892.494 −0.149098
\(13\) −1233.64 −0.155735 −0.0778674 0.996964i \(-0.524811\pi\)
−0.0778674 + 0.996964i \(0.524811\pi\)
\(14\) −9158.55 −0.892028
\(15\) 8690.82 0.664877
\(16\) −11060.3 −0.675065
\(17\) 9783.69 0.482983 0.241491 0.970403i \(-0.422363\pi\)
0.241491 + 0.970403i \(0.422363\pi\)
\(18\) 7103.34 0.287084
\(19\) 19322.1 0.646275 0.323137 0.946352i \(-0.395262\pi\)
0.323137 + 0.946352i \(0.395262\pi\)
\(20\) −10639.9 −0.297395
\(21\) −25377.9 −0.597981
\(22\) 48701.6 0.975133
\(23\) 79805.1 1.36768 0.683838 0.729634i \(-0.260310\pi\)
0.683838 + 0.729634i \(0.260310\pi\)
\(24\) −42371.5 −0.625655
\(25\) 25483.2 0.326184
\(26\) −12020.5 −0.134127
\(27\) 19683.0 0.192450
\(28\) 31069.4 0.267473
\(29\) 144604. 1.10100 0.550498 0.834836i \(-0.314438\pi\)
0.550498 + 0.834836i \(0.314438\pi\)
\(30\) 84683.0 0.572627
\(31\) 148320. 0.894200 0.447100 0.894484i \(-0.352457\pi\)
0.447100 + 0.894484i \(0.352457\pi\)
\(32\) 93101.7 0.502265
\(33\) 134950. 0.653692
\(34\) 95331.8 0.415970
\(35\) −302544. −1.19275
\(36\) −24097.3 −0.0860816
\(37\) 203959. 0.661969 0.330985 0.943636i \(-0.392619\pi\)
0.330985 + 0.943636i \(0.392619\pi\)
\(38\) 188274. 0.556605
\(39\) −33308.2 −0.0899136
\(40\) −505135. −1.24795
\(41\) −393071. −0.890693 −0.445346 0.895358i \(-0.646919\pi\)
−0.445346 + 0.895358i \(0.646919\pi\)
\(42\) −247281. −0.515013
\(43\) 24364.6 0.0467325 0.0233663 0.999727i \(-0.492562\pi\)
0.0233663 + 0.999727i \(0.492562\pi\)
\(44\) −165215. −0.292392
\(45\) 234652. 0.383867
\(46\) 777617. 1.17791
\(47\) −18634.3 −0.0261800 −0.0130900 0.999914i \(-0.504167\pi\)
−0.0130900 + 0.999914i \(0.504167\pi\)
\(48\) −298627. −0.389749
\(49\) 59908.4 0.0727448
\(50\) 248307. 0.280927
\(51\) 264160. 0.278850
\(52\) 40778.3 0.0402177
\(53\) −228261. −0.210603 −0.105302 0.994440i \(-0.533581\pi\)
−0.105302 + 0.994440i \(0.533581\pi\)
\(54\) 191790. 0.165748
\(55\) 1.60881e6 1.30387
\(56\) 1.47503e6 1.12239
\(57\) 521697. 0.373127
\(58\) 1.40901e6 0.948235
\(59\) 205379. 0.130189
\(60\) −287278. −0.171701
\(61\) 2.54932e6 1.43804 0.719019 0.694991i \(-0.244592\pi\)
0.719019 + 0.694991i \(0.244592\pi\)
\(62\) 1.44523e6 0.770132
\(63\) −685202. −0.345245
\(64\) 2.32289e6 1.10764
\(65\) −397086. −0.179344
\(66\) 1.31494e6 0.562993
\(67\) 2.19172e6 0.890271 0.445136 0.895463i \(-0.353155\pi\)
0.445136 + 0.895463i \(0.353155\pi\)
\(68\) −323403. −0.124728
\(69\) 2.15474e6 0.789628
\(70\) −2.94797e6 −1.02726
\(71\) −3.19514e6 −1.05946 −0.529732 0.848165i \(-0.677707\pi\)
−0.529732 + 0.848165i \(0.677707\pi\)
\(72\) −1.14403e6 −0.361222
\(73\) 2.62084e6 0.788516 0.394258 0.919000i \(-0.371002\pi\)
0.394258 + 0.919000i \(0.371002\pi\)
\(74\) 1.98737e6 0.570122
\(75\) 688045. 0.188323
\(76\) −638699. −0.166897
\(77\) −4.69785e6 −1.17269
\(78\) −324554. −0.0774382
\(79\) −2.93367e6 −0.669448 −0.334724 0.942316i \(-0.608643\pi\)
−0.334724 + 0.942316i \(0.608643\pi\)
\(80\) −3.56010e6 −0.777405
\(81\) 531441. 0.111111
\(82\) −3.83007e6 −0.767111
\(83\) −5.63931e6 −1.08256 −0.541280 0.840842i \(-0.682060\pi\)
−0.541280 + 0.840842i \(0.682060\pi\)
\(84\) 838873. 0.154425
\(85\) 3.14920e6 0.556203
\(86\) 237407. 0.0402485
\(87\) 3.90429e6 0.635661
\(88\) −7.84365e6 −1.22696
\(89\) 8.11136e6 1.21963 0.609816 0.792543i \(-0.291243\pi\)
0.609816 + 0.792543i \(0.291243\pi\)
\(90\) 2.28644e6 0.330606
\(91\) 1.15952e6 0.161300
\(92\) −2.63798e6 −0.353195
\(93\) 4.00465e6 0.516267
\(94\) −181571. −0.0225476
\(95\) 6.21944e6 0.744250
\(96\) 2.51375e6 0.289983
\(97\) 6.86976e6 0.764259 0.382130 0.924109i \(-0.375191\pi\)
0.382130 + 0.924109i \(0.375191\pi\)
\(98\) 583745. 0.0626516
\(99\) 3.64364e6 0.377409
\(100\) −842354. −0.0842354
\(101\) 2.38664e6 0.230495 0.115248 0.993337i \(-0.463234\pi\)
0.115248 + 0.993337i \(0.463234\pi\)
\(102\) 2.57396e6 0.240160
\(103\) −1.07956e7 −0.973456 −0.486728 0.873553i \(-0.661810\pi\)
−0.486728 + 0.873553i \(0.661810\pi\)
\(104\) 1.93597e6 0.168765
\(105\) −8.16868e6 −0.688636
\(106\) −2.22416e6 −0.181383
\(107\) −7.34597e6 −0.579703 −0.289852 0.957072i \(-0.593606\pi\)
−0.289852 + 0.957072i \(0.593606\pi\)
\(108\) −650628. −0.0496992
\(109\) 9.08395e6 0.671865 0.335933 0.941886i \(-0.390949\pi\)
0.335933 + 0.941886i \(0.390949\pi\)
\(110\) 1.56762e7 1.12296
\(111\) 5.50690e6 0.382188
\(112\) 1.03958e7 0.699188
\(113\) 2.23282e7 1.45573 0.727863 0.685722i \(-0.240513\pi\)
0.727863 + 0.685722i \(0.240513\pi\)
\(114\) 5.08339e6 0.321356
\(115\) 2.56878e7 1.57502
\(116\) −4.77992e6 −0.284326
\(117\) −899322. −0.0519116
\(118\) 2.00120e6 0.112125
\(119\) −9.19589e6 −0.500241
\(120\) −1.36386e7 −0.720505
\(121\) 5.49419e6 0.281939
\(122\) 2.48405e7 1.23851
\(123\) −1.06129e7 −0.514242
\(124\) −4.90277e6 −0.230922
\(125\) −1.69445e7 −0.775967
\(126\) −6.67658e6 −0.297343
\(127\) −6.47924e6 −0.280680 −0.140340 0.990103i \(-0.544820\pi\)
−0.140340 + 0.990103i \(0.544820\pi\)
\(128\) 1.07171e7 0.451694
\(129\) 657843. 0.0269810
\(130\) −3.86919e6 −0.154461
\(131\) 2.30891e7 0.897339 0.448670 0.893698i \(-0.351898\pi\)
0.448670 + 0.893698i \(0.351898\pi\)
\(132\) −4.46080e6 −0.168812
\(133\) −1.81613e7 −0.669368
\(134\) 2.13560e7 0.766748
\(135\) 6.33561e6 0.221626
\(136\) −1.53537e7 −0.523392
\(137\) −4.31217e7 −1.43276 −0.716381 0.697709i \(-0.754203\pi\)
−0.716381 + 0.697709i \(0.754203\pi\)
\(138\) 2.09957e7 0.680069
\(139\) 1.97420e7 0.623504 0.311752 0.950163i \(-0.399084\pi\)
0.311752 + 0.950163i \(0.399084\pi\)
\(140\) 1.00007e7 0.308022
\(141\) −503125. −0.0151150
\(142\) −3.11333e7 −0.912465
\(143\) −6.16589e6 −0.176327
\(144\) −8.06293e6 −0.225022
\(145\) 4.65453e7 1.26791
\(146\) 2.55374e7 0.679111
\(147\) 1.61753e6 0.0419992
\(148\) −6.74194e6 −0.170950
\(149\) −5.76420e7 −1.42754 −0.713768 0.700382i \(-0.753013\pi\)
−0.713768 + 0.700382i \(0.753013\pi\)
\(150\) 6.70428e6 0.162193
\(151\) 3.07490e7 0.726794 0.363397 0.931634i \(-0.381617\pi\)
0.363397 + 0.931634i \(0.381617\pi\)
\(152\) −3.03225e7 −0.700346
\(153\) 7.13231e6 0.160994
\(154\) −4.57757e7 −1.00998
\(155\) 4.77417e7 1.02976
\(156\) 1.10101e6 0.0232197
\(157\) 5.96166e6 0.122947 0.0614736 0.998109i \(-0.480420\pi\)
0.0614736 + 0.998109i \(0.480420\pi\)
\(158\) −2.85856e7 −0.576564
\(159\) −6.16304e6 −0.121592
\(160\) 2.99678e7 0.578409
\(161\) −7.50105e7 −1.41655
\(162\) 5.17834e6 0.0956947
\(163\) 2.08332e7 0.376790 0.188395 0.982093i \(-0.439671\pi\)
0.188395 + 0.982093i \(0.439671\pi\)
\(164\) 1.29931e7 0.230017
\(165\) 4.34379e7 0.752792
\(166\) −5.49491e7 −0.932358
\(167\) −4.41080e7 −0.732840 −0.366420 0.930449i \(-0.619417\pi\)
−0.366420 + 0.930449i \(0.619417\pi\)
\(168\) 3.98259e7 0.648012
\(169\) −6.12267e7 −0.975747
\(170\) 3.06856e7 0.479031
\(171\) 1.40858e7 0.215425
\(172\) −805379. −0.0120684
\(173\) −6.63421e7 −0.974154 −0.487077 0.873359i \(-0.661937\pi\)
−0.487077 + 0.873359i \(0.661937\pi\)
\(174\) 3.80433e7 0.547464
\(175\) −2.39522e7 −0.337840
\(176\) −5.52807e7 −0.764327
\(177\) 5.54523e6 0.0751646
\(178\) 7.90367e7 1.05041
\(179\) −1.18508e8 −1.54441 −0.772206 0.635372i \(-0.780847\pi\)
−0.772206 + 0.635372i \(0.780847\pi\)
\(180\) −7.75650e6 −0.0991316
\(181\) −5.40992e7 −0.678134 −0.339067 0.940762i \(-0.610111\pi\)
−0.339067 + 0.940762i \(0.610111\pi\)
\(182\) 1.12983e7 0.138920
\(183\) 6.88317e7 0.830251
\(184\) −1.25239e8 −1.48210
\(185\) 6.56509e7 0.762324
\(186\) 3.90211e7 0.444636
\(187\) 4.89002e7 0.546846
\(188\) 615961. 0.00676084
\(189\) −1.85005e7 −0.199327
\(190\) 6.06020e7 0.640987
\(191\) 8.73557e7 0.907140 0.453570 0.891221i \(-0.350150\pi\)
0.453570 + 0.891221i \(0.350150\pi\)
\(192\) 6.27181e7 0.639497
\(193\) −206564. −0.00206826 −0.00103413 0.999999i \(-0.500329\pi\)
−0.00103413 + 0.999999i \(0.500329\pi\)
\(194\) 6.69387e7 0.658220
\(195\) −1.07213e7 −0.103545
\(196\) −1.98029e6 −0.0187859
\(197\) −1.06894e8 −0.996142 −0.498071 0.867136i \(-0.665958\pi\)
−0.498071 + 0.867136i \(0.665958\pi\)
\(198\) 3.55035e7 0.325044
\(199\) −7.19278e7 −0.647010 −0.323505 0.946226i \(-0.604861\pi\)
−0.323505 + 0.946226i \(0.604861\pi\)
\(200\) −3.99911e7 −0.353475
\(201\) 5.91763e7 0.513998
\(202\) 2.32553e7 0.198514
\(203\) −1.35916e8 −1.14034
\(204\) −8.73188e6 −0.0720116
\(205\) −1.26523e8 −1.02572
\(206\) −1.05192e8 −0.838391
\(207\) 5.81779e7 0.455892
\(208\) 1.36444e7 0.105131
\(209\) 9.65745e7 0.731730
\(210\) −7.95953e7 −0.593089
\(211\) −1.64503e8 −1.20555 −0.602776 0.797910i \(-0.705939\pi\)
−0.602776 + 0.797910i \(0.705939\pi\)
\(212\) 7.54523e6 0.0543872
\(213\) −8.62688e7 −0.611681
\(214\) −7.15788e7 −0.499271
\(215\) 7.84252e6 0.0538172
\(216\) −3.08888e7 −0.208552
\(217\) −1.39409e8 −0.926153
\(218\) 8.85137e7 0.578645
\(219\) 7.07627e7 0.455250
\(220\) −5.31798e7 −0.336718
\(221\) −1.20695e7 −0.0752172
\(222\) 5.36590e7 0.329160
\(223\) −6.38980e7 −0.385852 −0.192926 0.981213i \(-0.561798\pi\)
−0.192926 + 0.981213i \(0.561798\pi\)
\(224\) −8.75082e7 −0.520213
\(225\) 1.85772e7 0.108728
\(226\) 2.17565e8 1.25375
\(227\) 2.31309e8 1.31251 0.656256 0.754539i \(-0.272139\pi\)
0.656256 + 0.754539i \(0.272139\pi\)
\(228\) −1.72449e7 −0.0963580
\(229\) −2.79302e8 −1.53692 −0.768458 0.639900i \(-0.778976\pi\)
−0.768458 + 0.639900i \(0.778976\pi\)
\(230\) 2.50301e8 1.35649
\(231\) −1.26842e8 −0.677051
\(232\) −2.26929e8 −1.19311
\(233\) 2.23510e8 1.15758 0.578790 0.815477i \(-0.303525\pi\)
0.578790 + 0.815477i \(0.303525\pi\)
\(234\) −8.76295e6 −0.0447090
\(235\) −5.99803e6 −0.0301489
\(236\) −6.78887e6 −0.0336206
\(237\) −7.92092e7 −0.386506
\(238\) −8.96044e7 −0.430834
\(239\) 7.42988e7 0.352038 0.176019 0.984387i \(-0.443678\pi\)
0.176019 + 0.984387i \(0.443678\pi\)
\(240\) −9.61228e7 −0.448835
\(241\) −1.59415e8 −0.733615 −0.366808 0.930297i \(-0.619549\pi\)
−0.366808 + 0.930297i \(0.619549\pi\)
\(242\) 5.35351e7 0.242820
\(243\) 1.43489e7 0.0641500
\(244\) −8.42686e7 −0.371365
\(245\) 1.92835e7 0.0837729
\(246\) −1.03412e8 −0.442892
\(247\) −2.38365e7 −0.100647
\(248\) −2.32761e8 −0.969014
\(249\) −1.52261e8 −0.625017
\(250\) −1.65106e8 −0.668303
\(251\) −1.11214e8 −0.443918 −0.221959 0.975056i \(-0.571245\pi\)
−0.221959 + 0.975056i \(0.571245\pi\)
\(252\) 2.26496e7 0.0891576
\(253\) 3.98877e8 1.54852
\(254\) −6.31334e7 −0.241736
\(255\) 8.50283e7 0.321124
\(256\) −1.92903e8 −0.718619
\(257\) −1.27820e8 −0.469712 −0.234856 0.972030i \(-0.575462\pi\)
−0.234856 + 0.972030i \(0.575462\pi\)
\(258\) 6.41000e6 0.0232375
\(259\) −1.91706e8 −0.685624
\(260\) 1.31258e7 0.0463147
\(261\) 1.05416e8 0.366999
\(262\) 2.24979e8 0.772836
\(263\) 3.37653e8 1.14453 0.572263 0.820070i \(-0.306066\pi\)
0.572263 + 0.820070i \(0.306066\pi\)
\(264\) −2.11779e8 −0.708383
\(265\) −7.34730e7 −0.242531
\(266\) −1.76963e8 −0.576495
\(267\) 2.19007e8 0.704154
\(268\) −7.24479e7 −0.229908
\(269\) −3.70415e8 −1.16026 −0.580130 0.814524i \(-0.696998\pi\)
−0.580130 + 0.814524i \(0.696998\pi\)
\(270\) 6.17339e7 0.190876
\(271\) 2.34791e8 0.716619 0.358309 0.933603i \(-0.383353\pi\)
0.358309 + 0.933603i \(0.383353\pi\)
\(272\) −1.08210e8 −0.326045
\(273\) 3.13071e7 0.0931265
\(274\) −4.20176e8 −1.23397
\(275\) 1.27368e8 0.369315
\(276\) −7.12255e7 −0.203917
\(277\) −1.55266e8 −0.438932 −0.219466 0.975620i \(-0.570432\pi\)
−0.219466 + 0.975620i \(0.570432\pi\)
\(278\) 1.92365e8 0.536994
\(279\) 1.08125e8 0.298067
\(280\) 4.74787e8 1.29254
\(281\) −1.15239e8 −0.309834 −0.154917 0.987927i \(-0.549511\pi\)
−0.154917 + 0.987927i \(0.549511\pi\)
\(282\) −4.90243e6 −0.0130178
\(283\) 4.45082e8 1.16731 0.583657 0.812000i \(-0.301621\pi\)
0.583657 + 0.812000i \(0.301621\pi\)
\(284\) 1.05616e8 0.273601
\(285\) 1.67925e8 0.429693
\(286\) −6.00801e7 −0.151862
\(287\) 3.69456e8 0.922521
\(288\) 6.78711e7 0.167422
\(289\) −3.14618e8 −0.766728
\(290\) 4.53535e8 1.09199
\(291\) 1.85484e8 0.441245
\(292\) −8.66327e7 −0.203630
\(293\) 7.32530e8 1.70133 0.850666 0.525707i \(-0.176199\pi\)
0.850666 + 0.525707i \(0.176199\pi\)
\(294\) 1.57611e7 0.0361719
\(295\) 6.61078e7 0.149926
\(296\) −3.20077e8 −0.717353
\(297\) 9.83783e7 0.217897
\(298\) −5.61661e8 −1.22947
\(299\) −9.84505e7 −0.212995
\(300\) −2.27436e7 −0.0486333
\(301\) −2.29008e7 −0.0484024
\(302\) 2.99617e8 0.625953
\(303\) 6.44392e7 0.133076
\(304\) −2.13708e8 −0.436277
\(305\) 8.20581e8 1.65604
\(306\) 6.94969e7 0.138657
\(307\) −2.59871e7 −0.0512594 −0.0256297 0.999672i \(-0.508159\pi\)
−0.0256297 + 0.999672i \(0.508159\pi\)
\(308\) 1.55289e8 0.302840
\(309\) −2.91481e8 −0.562025
\(310\) 4.65193e8 0.886884
\(311\) 7.71976e8 1.45527 0.727633 0.685966i \(-0.240620\pi\)
0.727633 + 0.685966i \(0.240620\pi\)
\(312\) 5.22711e7 0.0974363
\(313\) −5.17834e8 −0.954521 −0.477261 0.878762i \(-0.658370\pi\)
−0.477261 + 0.878762i \(0.658370\pi\)
\(314\) 5.80901e7 0.105888
\(315\) −2.20554e8 −0.397584
\(316\) 9.69735e7 0.172881
\(317\) 5.63749e8 0.993982 0.496991 0.867756i \(-0.334438\pi\)
0.496991 + 0.867756i \(0.334438\pi\)
\(318\) −6.00524e7 −0.104721
\(319\) 7.22748e8 1.24658
\(320\) 7.47698e8 1.27556
\(321\) −1.98341e8 −0.334692
\(322\) −7.30899e8 −1.22000
\(323\) 1.89042e8 0.312139
\(324\) −1.75670e7 −0.0286939
\(325\) −3.14370e7 −0.0507983
\(326\) 2.02998e8 0.324511
\(327\) 2.45267e8 0.387902
\(328\) 6.16853e8 0.965213
\(329\) 1.75147e7 0.0271155
\(330\) 4.23257e8 0.648344
\(331\) −4.79482e8 −0.726733 −0.363366 0.931646i \(-0.618373\pi\)
−0.363366 + 0.931646i \(0.618373\pi\)
\(332\) 1.86409e8 0.279566
\(333\) 1.48686e8 0.220656
\(334\) −4.29786e8 −0.631160
\(335\) 7.05474e8 1.02524
\(336\) 2.80686e8 0.403676
\(337\) 3.39194e8 0.482773 0.241387 0.970429i \(-0.422398\pi\)
0.241387 + 0.970429i \(0.422398\pi\)
\(338\) −5.96590e8 −0.840364
\(339\) 6.02862e8 0.840464
\(340\) −1.04098e8 −0.143636
\(341\) 7.41325e8 1.01244
\(342\) 1.37252e8 0.185535
\(343\) 7.17756e8 0.960390
\(344\) −3.82357e7 −0.0506424
\(345\) 6.93571e8 0.909336
\(346\) −6.46434e8 −0.838992
\(347\) 9.78350e8 1.25702 0.628508 0.777803i \(-0.283666\pi\)
0.628508 + 0.777803i \(0.283666\pi\)
\(348\) −1.29058e8 −0.164156
\(349\) −2.36450e8 −0.297749 −0.148875 0.988856i \(-0.547565\pi\)
−0.148875 + 0.988856i \(0.547565\pi\)
\(350\) −2.33389e8 −0.290966
\(351\) −2.42817e7 −0.0299712
\(352\) 4.65335e8 0.568678
\(353\) −1.61081e8 −0.194910 −0.0974548 0.995240i \(-0.531070\pi\)
−0.0974548 + 0.995240i \(0.531070\pi\)
\(354\) 5.40325e7 0.0647357
\(355\) −1.02846e9 −1.22008
\(356\) −2.68124e8 −0.314963
\(357\) −2.48289e8 −0.288814
\(358\) −1.15474e9 −1.33013
\(359\) −1.10346e9 −1.25871 −0.629355 0.777118i \(-0.716681\pi\)
−0.629355 + 0.777118i \(0.716681\pi\)
\(360\) −3.68243e8 −0.415984
\(361\) −5.20528e8 −0.582329
\(362\) −5.27140e8 −0.584044
\(363\) 1.48343e8 0.162777
\(364\) −3.83284e7 −0.0416548
\(365\) 8.43602e8 0.908056
\(366\) 6.70693e8 0.715056
\(367\) −1.44038e8 −0.152105 −0.0760527 0.997104i \(-0.524232\pi\)
−0.0760527 + 0.997104i \(0.524232\pi\)
\(368\) −8.82665e8 −0.923270
\(369\) −2.86549e8 −0.296898
\(370\) 6.39700e8 0.656553
\(371\) 2.14547e8 0.218129
\(372\) −1.32375e8 −0.133323
\(373\) 1.06553e9 1.06312 0.531561 0.847020i \(-0.321606\pi\)
0.531561 + 0.847020i \(0.321606\pi\)
\(374\) 4.76481e8 0.470972
\(375\) −4.57501e8 −0.448004
\(376\) 2.92430e7 0.0283704
\(377\) −1.78388e8 −0.171463
\(378\) −1.80268e8 −0.171671
\(379\) 1.49139e9 1.40719 0.703597 0.710600i \(-0.251576\pi\)
0.703597 + 0.710600i \(0.251576\pi\)
\(380\) −2.05586e8 −0.192199
\(381\) −1.74939e8 −0.162051
\(382\) 8.51190e8 0.781276
\(383\) −1.49307e9 −1.35795 −0.678976 0.734160i \(-0.737576\pi\)
−0.678976 + 0.734160i \(0.737576\pi\)
\(384\) 2.89363e8 0.260786
\(385\) −1.51216e9 −1.35047
\(386\) −2.01275e6 −0.00178129
\(387\) 1.77618e7 0.0155775
\(388\) −2.27082e8 −0.197366
\(389\) 7.33205e8 0.631541 0.315771 0.948836i \(-0.397737\pi\)
0.315771 + 0.948836i \(0.397737\pi\)
\(390\) −1.04468e8 −0.0891779
\(391\) 7.80788e8 0.660563
\(392\) −9.40153e7 −0.0788310
\(393\) 6.23405e8 0.518079
\(394\) −1.04157e9 −0.857930
\(395\) −9.44297e8 −0.770937
\(396\) −1.20442e8 −0.0974639
\(397\) −2.09263e9 −1.67851 −0.839257 0.543735i \(-0.817010\pi\)
−0.839257 + 0.543735i \(0.817010\pi\)
\(398\) −7.00861e8 −0.557239
\(399\) −4.90354e8 −0.386460
\(400\) −2.81851e8 −0.220196
\(401\) −1.30799e9 −1.01298 −0.506489 0.862247i \(-0.669057\pi\)
−0.506489 + 0.862247i \(0.669057\pi\)
\(402\) 5.76612e8 0.442682
\(403\) −1.82973e8 −0.139258
\(404\) −7.88910e7 −0.0595241
\(405\) 1.71061e8 0.127956
\(406\) −1.32436e9 −0.982120
\(407\) 1.01942e9 0.749499
\(408\) −4.14550e8 −0.302180
\(409\) 2.74486e9 1.98376 0.991878 0.127193i \(-0.0405967\pi\)
0.991878 + 0.127193i \(0.0405967\pi\)
\(410\) −1.23283e9 −0.883406
\(411\) −1.16429e9 −0.827206
\(412\) 3.56852e8 0.251390
\(413\) −1.93040e8 −0.134841
\(414\) 5.66883e8 0.392638
\(415\) −1.81519e9 −1.24668
\(416\) −1.14854e8 −0.0782201
\(417\) 5.33034e8 0.359980
\(418\) 9.41018e8 0.630204
\(419\) −6.51049e7 −0.0432379 −0.0216189 0.999766i \(-0.506882\pi\)
−0.0216189 + 0.999766i \(0.506882\pi\)
\(420\) 2.70018e8 0.177837
\(421\) 1.48843e9 0.972170 0.486085 0.873912i \(-0.338425\pi\)
0.486085 + 0.873912i \(0.338425\pi\)
\(422\) −1.60291e9 −1.03828
\(423\) −1.35844e7 −0.00872666
\(424\) 3.58213e8 0.228224
\(425\) 2.49319e8 0.157541
\(426\) −8.40600e8 −0.526812
\(427\) −2.39616e9 −1.48942
\(428\) 2.42823e8 0.149705
\(429\) −1.66479e8 −0.101803
\(430\) 7.64172e7 0.0463502
\(431\) 1.85011e9 1.11308 0.556540 0.830821i \(-0.312129\pi\)
0.556540 + 0.830821i \(0.312129\pi\)
\(432\) −2.17699e8 −0.129916
\(433\) 6.17068e8 0.365280 0.182640 0.983180i \(-0.441536\pi\)
0.182640 + 0.983180i \(0.441536\pi\)
\(434\) −1.35840e9 −0.797651
\(435\) 1.25672e9 0.732027
\(436\) −3.00273e8 −0.173506
\(437\) 1.54200e9 0.883894
\(438\) 6.89509e8 0.392085
\(439\) −7.44414e8 −0.419941 −0.209971 0.977708i \(-0.567337\pi\)
−0.209971 + 0.977708i \(0.567337\pi\)
\(440\) −2.52473e9 −1.41296
\(441\) 4.36733e7 0.0242483
\(442\) −1.17605e8 −0.0647810
\(443\) −1.54073e9 −0.842004 −0.421002 0.907060i \(-0.638321\pi\)
−0.421002 + 0.907060i \(0.638321\pi\)
\(444\) −1.82032e8 −0.0986980
\(445\) 2.61090e9 1.40453
\(446\) −6.22619e8 −0.332316
\(447\) −1.55633e9 −0.824188
\(448\) −2.18334e9 −1.14722
\(449\) −3.21711e8 −0.167727 −0.0838636 0.996477i \(-0.526726\pi\)
−0.0838636 + 0.996477i \(0.526726\pi\)
\(450\) 1.81016e8 0.0936424
\(451\) −1.96462e9 −1.00847
\(452\) −7.38067e8 −0.375934
\(453\) 8.30222e8 0.419615
\(454\) 2.25387e9 1.13040
\(455\) 3.73229e8 0.185753
\(456\) −8.18708e8 −0.404345
\(457\) −1.64590e9 −0.806673 −0.403336 0.915052i \(-0.632150\pi\)
−0.403336 + 0.915052i \(0.632150\pi\)
\(458\) −2.72151e9 −1.32367
\(459\) 1.92572e8 0.0929500
\(460\) −8.49119e8 −0.406740
\(461\) 1.86228e9 0.885301 0.442650 0.896694i \(-0.354038\pi\)
0.442650 + 0.896694i \(0.354038\pi\)
\(462\) −1.23594e9 −0.583111
\(463\) 1.30026e9 0.608833 0.304417 0.952539i \(-0.401538\pi\)
0.304417 + 0.952539i \(0.401538\pi\)
\(464\) −1.59935e9 −0.743244
\(465\) 1.28902e9 0.594533
\(466\) 2.17787e9 0.996969
\(467\) 2.19765e9 0.998501 0.499251 0.866458i \(-0.333609\pi\)
0.499251 + 0.866458i \(0.333609\pi\)
\(468\) 2.97274e7 0.0134059
\(469\) −2.06004e9 −0.922084
\(470\) −5.84446e7 −0.0259658
\(471\) 1.60965e8 0.0709836
\(472\) −3.22305e8 −0.141081
\(473\) 1.21777e8 0.0529118
\(474\) −7.71811e8 −0.332879
\(475\) 4.92389e8 0.210805
\(476\) 3.03973e8 0.129185
\(477\) −1.66402e8 −0.0702011
\(478\) 7.23964e8 0.303193
\(479\) −4.13730e9 −1.72006 −0.860029 0.510245i \(-0.829555\pi\)
−0.860029 + 0.510245i \(0.829555\pi\)
\(480\) 8.09130e8 0.333944
\(481\) −2.51612e8 −0.103092
\(482\) −1.55333e9 −0.631828
\(483\) −2.02528e9 −0.817844
\(484\) −1.81612e8 −0.0728092
\(485\) 2.21125e9 0.880121
\(486\) 1.39815e8 0.0552494
\(487\) 5.06252e9 1.98617 0.993083 0.117413i \(-0.0374601\pi\)
0.993083 + 0.117413i \(0.0374601\pi\)
\(488\) −4.00069e9 −1.55835
\(489\) 5.62496e8 0.217540
\(490\) 1.87897e8 0.0721496
\(491\) 5.29752e8 0.201970 0.100985 0.994888i \(-0.467801\pi\)
0.100985 + 0.994888i \(0.467801\pi\)
\(492\) 3.50814e8 0.132800
\(493\) 1.41476e9 0.531762
\(494\) −2.32262e8 −0.0866828
\(495\) 1.17282e9 0.434625
\(496\) −1.64046e9 −0.603643
\(497\) 3.00318e9 1.09732
\(498\) −1.48363e9 −0.538297
\(499\) 1.14827e9 0.413705 0.206852 0.978372i \(-0.433678\pi\)
0.206852 + 0.978372i \(0.433678\pi\)
\(500\) 5.60105e8 0.200389
\(501\) −1.19092e9 −0.423106
\(502\) −1.08367e9 −0.382326
\(503\) 2.33999e9 0.819835 0.409918 0.912123i \(-0.365557\pi\)
0.409918 + 0.912123i \(0.365557\pi\)
\(504\) 1.07530e9 0.374130
\(505\) 7.68216e8 0.265438
\(506\) 3.88664e9 1.33367
\(507\) −1.65312e9 −0.563348
\(508\) 2.14173e8 0.0724840
\(509\) 3.44383e9 1.15752 0.578762 0.815497i \(-0.303536\pi\)
0.578762 + 0.815497i \(0.303536\pi\)
\(510\) 8.28512e8 0.276569
\(511\) −2.46338e9 −0.816693
\(512\) −3.25143e9 −1.07061
\(513\) 3.80317e8 0.124376
\(514\) −1.24547e9 −0.404540
\(515\) −3.47491e9 −1.12103
\(516\) −2.17452e7 −0.00696771
\(517\) −9.31365e7 −0.0296417
\(518\) −1.86797e9 −0.590495
\(519\) −1.79124e9 −0.562428
\(520\) 6.23153e8 0.194349
\(521\) −5.23947e9 −1.62314 −0.811568 0.584258i \(-0.801386\pi\)
−0.811568 + 0.584258i \(0.801386\pi\)
\(522\) 1.02717e9 0.316078
\(523\) 1.85224e9 0.566164 0.283082 0.959096i \(-0.408643\pi\)
0.283082 + 0.959096i \(0.408643\pi\)
\(524\) −7.63216e8 −0.231733
\(525\) −6.46708e8 −0.195052
\(526\) 3.29008e9 0.985725
\(527\) 1.45112e9 0.431883
\(528\) −1.49258e9 −0.441284
\(529\) 2.96402e9 0.870536
\(530\) −7.15918e8 −0.208880
\(531\) 1.49721e8 0.0433963
\(532\) 6.00326e8 0.172861
\(533\) 4.84908e8 0.138712
\(534\) 2.13399e9 0.606455
\(535\) −2.36454e9 −0.667587
\(536\) −3.43949e9 −0.964757
\(537\) −3.19972e9 −0.891667
\(538\) −3.60931e9 −0.999277
\(539\) 2.99431e8 0.0823636
\(540\) −2.09426e8 −0.0572337
\(541\) 7.04253e8 0.191222 0.0956111 0.995419i \(-0.469519\pi\)
0.0956111 + 0.995419i \(0.469519\pi\)
\(542\) 2.28779e9 0.617190
\(543\) −1.46068e9 −0.391521
\(544\) 9.10878e8 0.242585
\(545\) 2.92396e9 0.773721
\(546\) 3.05055e8 0.0802054
\(547\) 3.55125e9 0.927738 0.463869 0.885904i \(-0.346461\pi\)
0.463869 + 0.885904i \(0.346461\pi\)
\(548\) 1.42540e9 0.370003
\(549\) 1.85846e9 0.479346
\(550\) 1.24107e9 0.318073
\(551\) 2.79405e9 0.711546
\(552\) −3.38146e9 −0.855693
\(553\) 2.75742e9 0.693370
\(554\) −1.51291e9 −0.378032
\(555\) 1.77257e9 0.440128
\(556\) −6.52578e8 −0.161017
\(557\) −1.10300e9 −0.270447 −0.135223 0.990815i \(-0.543175\pi\)
−0.135223 + 0.990815i \(0.543175\pi\)
\(558\) 1.05357e9 0.256711
\(559\) −3.00570e7 −0.00727788
\(560\) 3.34622e9 0.805185
\(561\) 1.32031e9 0.315722
\(562\) −1.12289e9 −0.266845
\(563\) 5.07113e8 0.119764 0.0598818 0.998205i \(-0.480928\pi\)
0.0598818 + 0.998205i \(0.480928\pi\)
\(564\) 1.66309e7 0.00390337
\(565\) 7.18706e9 1.67642
\(566\) 4.33686e9 1.00535
\(567\) −4.99513e8 −0.115082
\(568\) 5.01419e9 1.14810
\(569\) −6.60666e9 −1.50345 −0.751726 0.659476i \(-0.770778\pi\)
−0.751726 + 0.659476i \(0.770778\pi\)
\(570\) 1.63625e9 0.370074
\(571\) 3.31524e9 0.745227 0.372613 0.927987i \(-0.378462\pi\)
0.372613 + 0.927987i \(0.378462\pi\)
\(572\) 2.03815e8 0.0455356
\(573\) 2.35860e9 0.523737
\(574\) 3.59996e9 0.794523
\(575\) 2.03369e9 0.446114
\(576\) 1.69339e9 0.369214
\(577\) −6.56723e9 −1.42320 −0.711601 0.702584i \(-0.752030\pi\)
−0.711601 + 0.702584i \(0.752030\pi\)
\(578\) −3.06562e9 −0.660346
\(579\) −5.57723e6 −0.00119411
\(580\) −1.53857e9 −0.327431
\(581\) 5.30050e9 1.12125
\(582\) 1.80734e9 0.380023
\(583\) −1.14088e9 −0.238451
\(584\) −4.11293e9 −0.854488
\(585\) −2.89476e8 −0.0597815
\(586\) 7.13774e9 1.46528
\(587\) −2.12865e9 −0.434380 −0.217190 0.976129i \(-0.569689\pi\)
−0.217190 + 0.976129i \(0.569689\pi\)
\(588\) −5.34679e7 −0.0108461
\(589\) 2.86586e9 0.577899
\(590\) 6.44152e8 0.129124
\(591\) −2.88614e9 −0.575123
\(592\) −2.25585e9 −0.446872
\(593\) 6.79461e7 0.0133805 0.00669026 0.999978i \(-0.497870\pi\)
0.00669026 + 0.999978i \(0.497870\pi\)
\(594\) 9.58594e8 0.187664
\(595\) −2.95999e9 −0.576078
\(596\) 1.90537e9 0.368653
\(597\) −1.94205e9 −0.373551
\(598\) −9.59297e8 −0.183442
\(599\) −3.92845e9 −0.746840 −0.373420 0.927662i \(-0.621815\pi\)
−0.373420 + 0.927662i \(0.621815\pi\)
\(600\) −1.07976e9 −0.204079
\(601\) −5.25955e9 −0.988298 −0.494149 0.869377i \(-0.664520\pi\)
−0.494149 + 0.869377i \(0.664520\pi\)
\(602\) −2.23144e8 −0.0416867
\(603\) 1.59776e9 0.296757
\(604\) −1.01642e9 −0.187691
\(605\) 1.76848e9 0.324681
\(606\) 6.27893e8 0.114612
\(607\) −7.11758e9 −1.29173 −0.645865 0.763452i \(-0.723503\pi\)
−0.645865 + 0.763452i \(0.723503\pi\)
\(608\) 1.79892e9 0.324601
\(609\) −3.66973e9 −0.658375
\(610\) 7.99571e9 1.42627
\(611\) 2.29879e7 0.00407714
\(612\) −2.35761e8 −0.0415759
\(613\) −8.31463e9 −1.45791 −0.728956 0.684561i \(-0.759994\pi\)
−0.728956 + 0.684561i \(0.759994\pi\)
\(614\) −2.53217e8 −0.0441473
\(615\) −3.41611e9 −0.592201
\(616\) 7.37241e9 1.27080
\(617\) −8.10336e9 −1.38889 −0.694444 0.719547i \(-0.744349\pi\)
−0.694444 + 0.719547i \(0.744349\pi\)
\(618\) −2.84018e9 −0.484046
\(619\) 6.25611e8 0.106020 0.0530099 0.998594i \(-0.483119\pi\)
0.0530099 + 0.998594i \(0.483119\pi\)
\(620\) −1.57812e9 −0.265930
\(621\) 1.57080e9 0.263209
\(622\) 7.52210e9 1.25335
\(623\) −7.62404e9 −1.26321
\(624\) 3.68398e8 0.0606975
\(625\) −7.44500e9 −1.21979
\(626\) −5.04575e9 −0.822083
\(627\) 2.60751e9 0.422464
\(628\) −1.97065e8 −0.0317504
\(629\) 1.99548e9 0.319719
\(630\) −2.14907e9 −0.342420
\(631\) −5.93874e9 −0.941004 −0.470502 0.882399i \(-0.655927\pi\)
−0.470502 + 0.882399i \(0.655927\pi\)
\(632\) 4.60386e9 0.725458
\(633\) −4.44159e9 −0.696026
\(634\) 5.49315e9 0.856070
\(635\) −2.08555e9 −0.323231
\(636\) 2.03721e8 0.0314005
\(637\) −7.39053e7 −0.0113289
\(638\) 7.04242e9 1.07362
\(639\) −2.32926e9 −0.353154
\(640\) 3.44966e9 0.520171
\(641\) 7.93154e9 1.18947 0.594736 0.803921i \(-0.297257\pi\)
0.594736 + 0.803921i \(0.297257\pi\)
\(642\) −1.93263e9 −0.288254
\(643\) 1.20200e9 0.178306 0.0891530 0.996018i \(-0.471584\pi\)
0.0891530 + 0.996018i \(0.471584\pi\)
\(644\) 2.47949e9 0.365816
\(645\) 2.11748e8 0.0310714
\(646\) 1.84201e9 0.268831
\(647\) 1.87158e9 0.271671 0.135835 0.990731i \(-0.456628\pi\)
0.135835 + 0.990731i \(0.456628\pi\)
\(648\) −8.33999e8 −0.120407
\(649\) 1.02651e9 0.147403
\(650\) −3.06321e8 −0.0437501
\(651\) −3.76405e9 −0.534715
\(652\) −6.88648e8 −0.0973040
\(653\) −9.58524e9 −1.34712 −0.673561 0.739132i \(-0.735236\pi\)
−0.673561 + 0.739132i \(0.735236\pi\)
\(654\) 2.38987e9 0.334081
\(655\) 7.43196e9 1.03338
\(656\) 4.34747e9 0.601276
\(657\) 1.91059e9 0.262839
\(658\) 1.70663e8 0.0233533
\(659\) −1.22998e10 −1.67417 −0.837087 0.547070i \(-0.815743\pi\)
−0.837087 + 0.547070i \(0.815743\pi\)
\(660\) −1.43585e9 −0.194405
\(661\) 1.28794e9 0.173456 0.0867280 0.996232i \(-0.472359\pi\)
0.0867280 + 0.996232i \(0.472359\pi\)
\(662\) −4.67206e9 −0.625900
\(663\) −3.25877e8 −0.0434267
\(664\) 8.84985e9 1.17313
\(665\) −5.84579e9 −0.770845
\(666\) 1.44879e9 0.190041
\(667\) 1.15401e10 1.50581
\(668\) 1.45800e9 0.189252
\(669\) −1.72525e9 −0.222772
\(670\) 6.87411e9 0.882988
\(671\) 1.27419e10 1.62818
\(672\) −2.36272e9 −0.300345
\(673\) 9.85741e8 0.124655 0.0623275 0.998056i \(-0.480148\pi\)
0.0623275 + 0.998056i \(0.480148\pi\)
\(674\) 3.30509e9 0.415789
\(675\) 5.01585e8 0.0627742
\(676\) 2.02387e9 0.251981
\(677\) −1.00272e10 −1.24200 −0.620999 0.783812i \(-0.713273\pi\)
−0.620999 + 0.783812i \(0.713273\pi\)
\(678\) 5.87426e9 0.723852
\(679\) −6.45703e9 −0.791569
\(680\) −4.94208e9 −0.602738
\(681\) 6.24536e9 0.757779
\(682\) 7.22344e9 0.871964
\(683\) −2.51763e8 −0.0302356 −0.0151178 0.999886i \(-0.504812\pi\)
−0.0151178 + 0.999886i \(0.504812\pi\)
\(684\) −4.65611e8 −0.0556323
\(685\) −1.38801e10 −1.64997
\(686\) 6.99378e9 0.827138
\(687\) −7.54116e9 −0.887339
\(688\) −2.69479e8 −0.0315475
\(689\) 2.81591e8 0.0327983
\(690\) 6.75813e9 0.783168
\(691\) 1.20798e10 1.39279 0.696396 0.717657i \(-0.254786\pi\)
0.696396 + 0.717657i \(0.254786\pi\)
\(692\) 2.19296e9 0.251570
\(693\) −3.42473e9 −0.390895
\(694\) 9.53300e9 1.08261
\(695\) 6.35460e9 0.718028
\(696\) −6.12707e9 −0.688844
\(697\) −3.84569e9 −0.430189
\(698\) −2.30396e9 −0.256437
\(699\) 6.03477e9 0.668329
\(700\) 7.91746e8 0.0872455
\(701\) 1.78453e10 1.95664 0.978321 0.207096i \(-0.0664013\pi\)
0.978321 + 0.207096i \(0.0664013\pi\)
\(702\) −2.36600e8 −0.0258127
\(703\) 3.94093e9 0.427814
\(704\) 1.16101e10 1.25410
\(705\) −1.61947e8 −0.0174065
\(706\) −1.56957e9 −0.167866
\(707\) −2.24325e9 −0.238732
\(708\) −1.83299e8 −0.0194109
\(709\) −6.60811e9 −0.696331 −0.348165 0.937433i \(-0.613195\pi\)
−0.348165 + 0.937433i \(0.613195\pi\)
\(710\) −1.00213e10 −1.05080
\(711\) −2.13865e9 −0.223149
\(712\) −1.27293e10 −1.32167
\(713\) 1.18367e10 1.22298
\(714\) −2.41932e9 −0.248742
\(715\) −1.98469e9 −0.203059
\(716\) 3.91733e9 0.398836
\(717\) 2.00607e9 0.203249
\(718\) −1.07521e10 −1.08407
\(719\) 1.04223e10 1.04572 0.522859 0.852419i \(-0.324866\pi\)
0.522859 + 0.852419i \(0.324866\pi\)
\(720\) −2.59531e9 −0.259135
\(721\) 1.01470e10 1.00824
\(722\) −5.07200e9 −0.501532
\(723\) −4.30419e9 −0.423553
\(724\) 1.78826e9 0.175124
\(725\) 3.68495e9 0.359128
\(726\) 1.44545e9 0.140192
\(727\) −1.07207e10 −1.03479 −0.517394 0.855747i \(-0.673098\pi\)
−0.517394 + 0.855747i \(0.673098\pi\)
\(728\) −1.81966e9 −0.174795
\(729\) 3.87420e8 0.0370370
\(730\) 8.22002e9 0.782065
\(731\) 2.38375e8 0.0225710
\(732\) −2.27525e9 −0.214408
\(733\) 2.61994e9 0.245712 0.122856 0.992424i \(-0.460795\pi\)
0.122856 + 0.992424i \(0.460795\pi\)
\(734\) −1.40350e9 −0.131001
\(735\) 5.20653e8 0.0483663
\(736\) 7.42999e9 0.686935
\(737\) 1.09545e10 1.00799
\(738\) −2.79212e9 −0.255704
\(739\) 1.57747e10 1.43783 0.718913 0.695100i \(-0.244640\pi\)
0.718913 + 0.695100i \(0.244640\pi\)
\(740\) −2.17011e9 −0.196866
\(741\) −6.43585e8 −0.0581088
\(742\) 2.09054e9 0.187864
\(743\) −942965. −8.43402e−5 0 −4.21701e−5 1.00000i \(-0.500013\pi\)
−4.21701e−5 1.00000i \(0.500013\pi\)
\(744\) −6.28456e9 −0.559460
\(745\) −1.85539e10 −1.64395
\(746\) 1.03824e10 0.915617
\(747\) −4.11105e9 −0.360854
\(748\) −1.61641e9 −0.141220
\(749\) 6.90463e9 0.600418
\(750\) −4.45787e9 −0.385845
\(751\) −1.30768e10 −1.12658 −0.563288 0.826261i \(-0.690464\pi\)
−0.563288 + 0.826261i \(0.690464\pi\)
\(752\) 2.06100e8 0.0176732
\(753\) −3.00279e9 −0.256296
\(754\) −1.73821e9 −0.147673
\(755\) 9.89755e9 0.836976
\(756\) 6.11539e8 0.0514752
\(757\) 2.15132e10 1.80247 0.901237 0.433326i \(-0.142660\pi\)
0.901237 + 0.433326i \(0.142660\pi\)
\(758\) 1.45320e10 1.21195
\(759\) 1.07697e10 0.894038
\(760\) −9.76027e9 −0.806519
\(761\) −1.50838e10 −1.24069 −0.620347 0.784328i \(-0.713008\pi\)
−0.620347 + 0.784328i \(0.713008\pi\)
\(762\) −1.70460e9 −0.139566
\(763\) −8.53820e9 −0.695874
\(764\) −2.88757e9 −0.234264
\(765\) 2.29576e9 0.185401
\(766\) −1.45484e10 −1.16954
\(767\) −2.53363e8 −0.0202749
\(768\) −5.20838e9 −0.414895
\(769\) −2.21876e9 −0.175941 −0.0879707 0.996123i \(-0.528038\pi\)
−0.0879707 + 0.996123i \(0.528038\pi\)
\(770\) −1.47344e10 −1.16309
\(771\) −3.45113e9 −0.271188
\(772\) 6.82804e6 0.000534116 0
\(773\) −8.81943e9 −0.686771 −0.343386 0.939194i \(-0.611574\pi\)
−0.343386 + 0.939194i \(0.611574\pi\)
\(774\) 1.73070e8 0.0134162
\(775\) 3.77967e9 0.291674
\(776\) −1.07808e10 −0.828202
\(777\) −5.17605e9 −0.395845
\(778\) 7.14431e9 0.543916
\(779\) −7.59497e9 −0.575632
\(780\) 3.54397e8 0.0267398
\(781\) −1.59697e10 −1.19955
\(782\) 7.60796e9 0.568912
\(783\) 2.84623e9 0.211887
\(784\) −6.62603e8 −0.0491074
\(785\) 1.91895e9 0.141586
\(786\) 6.07443e9 0.446197
\(787\) −1.26134e10 −0.922406 −0.461203 0.887295i \(-0.652582\pi\)
−0.461203 + 0.887295i \(0.652582\pi\)
\(788\) 3.53341e9 0.257248
\(789\) 9.11663e9 0.660792
\(790\) −9.20119e9 −0.663971
\(791\) −2.09868e10 −1.50775
\(792\) −5.71802e9 −0.408985
\(793\) −3.14494e9 −0.223952
\(794\) −2.03905e10 −1.44562
\(795\) −1.98377e9 −0.140025
\(796\) 2.37760e9 0.167087
\(797\) 2.61316e9 0.182836 0.0914179 0.995813i \(-0.470860\pi\)
0.0914179 + 0.995813i \(0.470860\pi\)
\(798\) −4.77799e9 −0.332840
\(799\) −1.82312e8 −0.0126445
\(800\) 2.37253e9 0.163831
\(801\) 5.91318e9 0.406544
\(802\) −1.27450e10 −0.872429
\(803\) 1.30993e10 0.892779
\(804\) −1.95609e9 −0.132737
\(805\) −2.41445e10 −1.63130
\(806\) −1.78289e9 −0.119936
\(807\) −1.00012e10 −0.669876
\(808\) −3.74539e9 −0.249780
\(809\) −2.83771e10 −1.88429 −0.942147 0.335199i \(-0.891196\pi\)
−0.942147 + 0.335199i \(0.891196\pi\)
\(810\) 1.66681e9 0.110202
\(811\) −2.32326e10 −1.52941 −0.764705 0.644380i \(-0.777115\pi\)
−0.764705 + 0.644380i \(0.777115\pi\)
\(812\) 4.49274e9 0.294487
\(813\) 6.33935e9 0.413740
\(814\) 9.93315e9 0.645508
\(815\) 6.70584e9 0.433912
\(816\) −2.92168e9 −0.188242
\(817\) 4.70775e8 0.0302020
\(818\) 2.67458e10 1.70851
\(819\) 8.45291e8 0.0537666
\(820\) 4.18225e9 0.264887
\(821\) 1.23630e10 0.779691 0.389845 0.920880i \(-0.372528\pi\)
0.389845 + 0.920880i \(0.372528\pi\)
\(822\) −1.13448e10 −0.712433
\(823\) 1.47567e10 0.922763 0.461382 0.887202i \(-0.347354\pi\)
0.461382 + 0.887202i \(0.347354\pi\)
\(824\) 1.69417e10 1.05490
\(825\) 3.43894e9 0.213224
\(826\) −1.88097e9 −0.116132
\(827\) 7.08701e9 0.435706 0.217853 0.975982i \(-0.430095\pi\)
0.217853 + 0.975982i \(0.430095\pi\)
\(828\) −1.92309e9 −0.117732
\(829\) −5.40146e7 −0.00329284 −0.00164642 0.999999i \(-0.500524\pi\)
−0.00164642 + 0.999999i \(0.500524\pi\)
\(830\) −1.76872e10 −1.07370
\(831\) −4.19219e9 −0.253418
\(832\) −2.86561e9 −0.172498
\(833\) 5.86126e8 0.0351345
\(834\) 5.19386e9 0.310034
\(835\) −1.41976e10 −0.843940
\(836\) −3.19230e9 −0.188965
\(837\) 2.91939e9 0.172089
\(838\) −6.34379e8 −0.0372387
\(839\) −4.08313e9 −0.238686 −0.119343 0.992853i \(-0.538079\pi\)
−0.119343 + 0.992853i \(0.538079\pi\)
\(840\) 1.28192e10 0.746251
\(841\) 3.66030e9 0.212193
\(842\) 1.45032e10 0.837283
\(843\) −3.11146e9 −0.178883
\(844\) 5.43771e9 0.311328
\(845\) −1.97078e10 −1.12367
\(846\) −1.32365e8 −0.00751586
\(847\) −5.16410e9 −0.292013
\(848\) 2.52462e9 0.142171
\(849\) 1.20172e10 0.673949
\(850\) 2.42936e9 0.135683
\(851\) 1.62770e10 0.905359
\(852\) 2.85164e9 0.157963
\(853\) 4.28167e8 0.0236207 0.0118103 0.999930i \(-0.496241\pi\)
0.0118103 + 0.999930i \(0.496241\pi\)
\(854\) −2.33481e10 −1.28277
\(855\) 4.53398e9 0.248083
\(856\) 1.15281e10 0.628205
\(857\) −7.39852e9 −0.401524 −0.200762 0.979640i \(-0.564342\pi\)
−0.200762 + 0.979640i \(0.564342\pi\)
\(858\) −1.62216e9 −0.0876777
\(859\) −2.72977e9 −0.146943 −0.0734717 0.997297i \(-0.523408\pi\)
−0.0734717 + 0.997297i \(0.523408\pi\)
\(860\) −2.59237e8 −0.0138980
\(861\) 9.97531e9 0.532618
\(862\) 1.80274e10 0.958642
\(863\) 1.38149e10 0.731663 0.365832 0.930681i \(-0.380785\pi\)
0.365832 + 0.930681i \(0.380785\pi\)
\(864\) 1.83252e9 0.0966609
\(865\) −2.13543e10 −1.12184
\(866\) 6.01268e9 0.314598
\(867\) −8.49469e9 −0.442671
\(868\) 4.60822e9 0.239174
\(869\) −1.46629e10 −0.757968
\(870\) 1.22455e10 0.630460
\(871\) −2.70378e9 −0.138646
\(872\) −1.42556e10 −0.728078
\(873\) 5.00806e9 0.254753
\(874\) 1.50252e10 0.761255
\(875\) 1.59265e10 0.803695
\(876\) −2.33908e9 −0.117566
\(877\) 2.96899e10 1.48631 0.743157 0.669117i \(-0.233327\pi\)
0.743157 + 0.669117i \(0.233327\pi\)
\(878\) −7.25354e9 −0.361675
\(879\) 1.97783e10 0.982264
\(880\) −1.77939e10 −0.880200
\(881\) −3.95312e10 −1.94771 −0.973855 0.227169i \(-0.927053\pi\)
−0.973855 + 0.227169i \(0.927053\pi\)
\(882\) 4.25550e8 0.0208839
\(883\) −6.41631e9 −0.313634 −0.156817 0.987628i \(-0.550123\pi\)
−0.156817 + 0.987628i \(0.550123\pi\)
\(884\) 3.98962e8 0.0194244
\(885\) 1.78491e9 0.0865596
\(886\) −1.50128e10 −0.725177
\(887\) −3.07401e10 −1.47901 −0.739507 0.673149i \(-0.764941\pi\)
−0.739507 + 0.673149i \(0.764941\pi\)
\(888\) −8.64207e9 −0.414164
\(889\) 6.08997e9 0.290710
\(890\) 2.54405e10 1.20965
\(891\) 2.65621e9 0.125803
\(892\) 2.11217e9 0.0996441
\(893\) −3.60053e8 −0.0169195
\(894\) −1.51648e10 −0.709834
\(895\) −3.81457e10 −1.77855
\(896\) −1.00733e10 −0.467835
\(897\) −2.65816e9 −0.122973
\(898\) −3.13474e9 −0.144455
\(899\) 2.14476e10 0.984511
\(900\) −6.14076e8 −0.0280785
\(901\) −2.23323e9 −0.101718
\(902\) −1.91432e10 −0.868544
\(903\) −6.18321e8 −0.0279452
\(904\) −3.50401e10 −1.57752
\(905\) −1.74136e10 −0.780940
\(906\) 8.08965e9 0.361394
\(907\) 6.93998e9 0.308839 0.154420 0.988005i \(-0.450649\pi\)
0.154420 + 0.988005i \(0.450649\pi\)
\(908\) −7.64601e9 −0.338949
\(909\) 1.73986e9 0.0768317
\(910\) 3.63673e9 0.159980
\(911\) 1.43108e10 0.627118 0.313559 0.949569i \(-0.398479\pi\)
0.313559 + 0.949569i \(0.398479\pi\)
\(912\) −5.77011e9 −0.251885
\(913\) −2.81860e10 −1.22570
\(914\) −1.60376e10 −0.694749
\(915\) 2.21557e10 0.956118
\(916\) 9.23243e9 0.396901
\(917\) −2.17019e10 −0.929405
\(918\) 1.87642e9 0.0800534
\(919\) 4.66570e10 1.98295 0.991477 0.130280i \(-0.0415876\pi\)
0.991477 + 0.130280i \(0.0415876\pi\)
\(920\) −4.03123e10 −1.70679
\(921\) −7.01652e8 −0.0295947
\(922\) 1.81459e10 0.762467
\(923\) 3.94165e9 0.164995
\(924\) 4.19280e9 0.174845
\(925\) 5.19753e9 0.215924
\(926\) 1.26697e10 0.524359
\(927\) −7.86999e9 −0.324485
\(928\) 1.34628e10 0.552992
\(929\) −2.83056e10 −1.15829 −0.579146 0.815224i \(-0.696614\pi\)
−0.579146 + 0.815224i \(0.696614\pi\)
\(930\) 1.25602e10 0.512043
\(931\) 1.15756e9 0.0470131
\(932\) −7.38819e9 −0.298939
\(933\) 2.08434e10 0.840199
\(934\) 2.14138e10 0.859962
\(935\) 1.57401e10 0.629748
\(936\) 1.41132e9 0.0562548
\(937\) 7.53166e9 0.299090 0.149545 0.988755i \(-0.452219\pi\)
0.149545 + 0.988755i \(0.452219\pi\)
\(938\) −2.00729e10 −0.794147
\(939\) −1.39815e10 −0.551093
\(940\) 1.98267e8 0.00778579
\(941\) −2.80301e10 −1.09663 −0.548316 0.836271i \(-0.684731\pi\)
−0.548316 + 0.836271i \(0.684731\pi\)
\(942\) 1.56843e9 0.0611347
\(943\) −3.13691e10 −1.21818
\(944\) −2.27155e9 −0.0878860
\(945\) −5.95497e9 −0.229545
\(946\) 1.18659e9 0.0455704
\(947\) 5.03691e10 1.92725 0.963627 0.267250i \(-0.0861150\pi\)
0.963627 + 0.267250i \(0.0861150\pi\)
\(948\) 2.61828e9 0.0998131
\(949\) −3.23317e9 −0.122799
\(950\) 4.79781e9 0.181556
\(951\) 1.52212e10 0.573876
\(952\) 1.44313e10 0.542095
\(953\) −1.09520e10 −0.409890 −0.204945 0.978773i \(-0.565702\pi\)
−0.204945 + 0.978773i \(0.565702\pi\)
\(954\) −1.62141e9 −0.0604609
\(955\) 2.81182e10 1.04466
\(956\) −2.45597e9 −0.0909118
\(957\) 1.95142e10 0.719712
\(958\) −4.03137e10 −1.48140
\(959\) 4.05310e10 1.48396
\(960\) 2.01878e10 0.736445
\(961\) −5.51371e9 −0.200407
\(962\) −2.45170e9 −0.0887879
\(963\) −5.35521e9 −0.193234
\(964\) 5.26950e9 0.189452
\(965\) −6.64893e7 −0.00238181
\(966\) −1.97343e10 −0.704370
\(967\) −1.15786e10 −0.411778 −0.205889 0.978575i \(-0.566009\pi\)
−0.205889 + 0.978575i \(0.566009\pi\)
\(968\) −8.62211e9 −0.305527
\(969\) 5.10412e9 0.180214
\(970\) 2.15464e10 0.758007
\(971\) −2.02758e10 −0.710739 −0.355370 0.934726i \(-0.615645\pi\)
−0.355370 + 0.934726i \(0.615645\pi\)
\(972\) −4.74308e8 −0.0165664
\(973\) −1.85559e10 −0.645784
\(974\) 4.93290e10 1.71059
\(975\) −8.48798e8 −0.0293284
\(976\) −2.81962e10 −0.970769
\(977\) −1.76622e10 −0.605917 −0.302958 0.953004i \(-0.597974\pi\)
−0.302958 + 0.953004i \(0.597974\pi\)
\(978\) 5.48094e9 0.187357
\(979\) 4.05417e10 1.38090
\(980\) −6.37421e8 −0.0216339
\(981\) 6.62220e9 0.223955
\(982\) 5.16188e9 0.173947
\(983\) 5.25893e10 1.76588 0.882938 0.469489i \(-0.155562\pi\)
0.882938 + 0.469489i \(0.155562\pi\)
\(984\) 1.66550e10 0.557266
\(985\) −3.44073e10 −1.14716
\(986\) 1.37853e10 0.457981
\(987\) 4.72898e8 0.0156551
\(988\) 7.87923e8 0.0259917
\(989\) 1.94442e9 0.0639149
\(990\) 1.14279e10 0.374321
\(991\) −2.06461e10 −0.673876 −0.336938 0.941527i \(-0.609391\pi\)
−0.336938 + 0.941527i \(0.609391\pi\)
\(992\) 1.38089e10 0.449125
\(993\) −1.29460e10 −0.419579
\(994\) 2.92629e10 0.945071
\(995\) −2.31523e10 −0.745097
\(996\) 5.03304e9 0.161407
\(997\) −2.76691e9 −0.0884224 −0.0442112 0.999022i \(-0.514077\pi\)
−0.0442112 + 0.999022i \(0.514077\pi\)
\(998\) 1.11887e10 0.356304
\(999\) 4.01453e9 0.127396
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.d.1.12 18
3.2 odd 2 531.8.a.e.1.7 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.12 18 1.1 even 1 trivial
531.8.a.e.1.7 18 3.2 odd 2