Properties

Label 177.8.a.d.1.18
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 1.18
Root \(20.8794\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +7358.16 q^{10} +2824.81 q^{11} +9469.15 q^{12} -5709.07 q^{13} -14302.0 q^{14} +9080.24 q^{15} +61722.2 q^{16} +27447.6 q^{17} +15950.1 q^{18} -52668.8 q^{19} +117945. q^{20} -17649.2 q^{21} +61805.2 q^{22} -112683. q^{23} +131564. q^{24} +34976.1 q^{25} -124911. q^{26} +19683.0 q^{27} -229250. q^{28} +129138. q^{29} +198670. q^{30} +264379. q^{31} +726735. q^{32} +76269.8 q^{33} +600538. q^{34} -219834. q^{35} +255667. q^{36} -162719. q^{37} -1.15236e6 q^{38} -154145. q^{39} +1.63873e6 q^{40} -570791. q^{41} -386154. q^{42} +468651. q^{43} +990686. q^{44} +245166. q^{45} -2.46544e6 q^{46} -451069. q^{47} +1.66650e6 q^{48} -396253. q^{49} +765257. q^{50} +741086. q^{51} -2.00222e6 q^{52} -1.09397e6 q^{53} +430653. q^{54} +949997. q^{55} -3.18519e6 q^{56} -1.42206e6 q^{57} +2.82547e6 q^{58} +205379. q^{59} +3.18452e6 q^{60} -1.17187e6 q^{61} +5.78445e6 q^{62} -476529. q^{63} +8.00009e6 q^{64} -1.91999e6 q^{65} +1.66874e6 q^{66} -1.24840e6 q^{67} +9.62614e6 q^{68} -3.04245e6 q^{69} -4.80984e6 q^{70} +2.90922e6 q^{71} +3.55224e6 q^{72} +1.95003e6 q^{73} -3.56019e6 q^{74} +944354. q^{75} -1.84714e7 q^{76} -1.84650e6 q^{77} -3.37260e6 q^{78} +529861. q^{79} +2.07575e7 q^{80} +531441. q^{81} -1.24886e7 q^{82} +7.53928e6 q^{83} -6.18974e6 q^{84} +9.23078e6 q^{85} +1.02538e7 q^{86} +3.48673e6 q^{87} +1.37646e7 q^{88} +6.07893e6 q^{89} +5.36410e6 q^{90} +3.73187e6 q^{91} -3.95190e7 q^{92} +7.13822e6 q^{93} -9.86913e6 q^{94} -1.77128e7 q^{95} +1.96218e7 q^{96} -2.74881e6 q^{97} -8.66979e6 q^{98} +2.05928e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8794 1.93389 0.966943 0.254992i \(-0.0820728\pi\)
0.966943 + 0.254992i \(0.0820728\pi\)
\(3\) 27.0000 0.577350
\(4\) 350.709 2.73992
\(5\) 336.305 1.20320 0.601601 0.798797i \(-0.294530\pi\)
0.601601 + 0.798797i \(0.294530\pi\)
\(6\) 590.745 1.11653
\(7\) −653.674 −0.720308 −0.360154 0.932893i \(-0.617276\pi\)
−0.360154 + 0.932893i \(0.617276\pi\)
\(8\) 4872.75 3.36480
\(9\) 729.000 0.333333
\(10\) 7358.16 2.32685
\(11\) 2824.81 0.639903 0.319952 0.947434i \(-0.396333\pi\)
0.319952 + 0.947434i \(0.396333\pi\)
\(12\) 9469.15 1.58189
\(13\) −5709.07 −0.720715 −0.360358 0.932814i \(-0.617345\pi\)
−0.360358 + 0.932814i \(0.617345\pi\)
\(14\) −14302.0 −1.39299
\(15\) 9080.24 0.694669
\(16\) 61722.2 3.76723
\(17\) 27447.6 1.35498 0.677491 0.735531i \(-0.263067\pi\)
0.677491 + 0.735531i \(0.263067\pi\)
\(18\) 15950.1 0.644629
\(19\) −52668.8 −1.76163 −0.880817 0.473458i \(-0.843006\pi\)
−0.880817 + 0.473458i \(0.843006\pi\)
\(20\) 117945. 3.29667
\(21\) −17649.2 −0.415870
\(22\) 61805.2 1.23750
\(23\) −112683. −1.93113 −0.965565 0.260161i \(-0.916224\pi\)
−0.965565 + 0.260161i \(0.916224\pi\)
\(24\) 131564. 1.94267
\(25\) 34976.1 0.447694
\(26\) −124911. −1.39378
\(27\) 19683.0 0.192450
\(28\) −229250. −1.97358
\(29\) 129138. 0.983246 0.491623 0.870808i \(-0.336404\pi\)
0.491623 + 0.870808i \(0.336404\pi\)
\(30\) 198670. 1.34341
\(31\) 264379. 1.59390 0.796949 0.604047i \(-0.206446\pi\)
0.796949 + 0.604047i \(0.206446\pi\)
\(32\) 726735. 3.92059
\(33\) 76269.8 0.369448
\(34\) 600538. 2.62038
\(35\) −219834. −0.866676
\(36\) 255667. 0.913305
\(37\) −162719. −0.528118 −0.264059 0.964506i \(-0.585061\pi\)
−0.264059 + 0.964506i \(0.585061\pi\)
\(38\) −1.15236e6 −3.40680
\(39\) −154145. −0.416105
\(40\) 1.63873e6 4.04853
\(41\) −570791. −1.29340 −0.646701 0.762743i \(-0.723852\pi\)
−0.646701 + 0.762743i \(0.723852\pi\)
\(42\) −386154. −0.804245
\(43\) 468651. 0.898898 0.449449 0.893306i \(-0.351620\pi\)
0.449449 + 0.893306i \(0.351620\pi\)
\(44\) 990686. 1.75328
\(45\) 245166. 0.401067
\(46\) −2.46544e6 −3.73459
\(47\) −451069. −0.633724 −0.316862 0.948472i \(-0.602629\pi\)
−0.316862 + 0.948472i \(0.602629\pi\)
\(48\) 1.66650e6 2.17501
\(49\) −396253. −0.481156
\(50\) 765257. 0.865789
\(51\) 741086. 0.782299
\(52\) −2.00222e6 −1.97470
\(53\) −1.09397e6 −1.00935 −0.504674 0.863310i \(-0.668387\pi\)
−0.504674 + 0.863310i \(0.668387\pi\)
\(54\) 430653. 0.372177
\(55\) 949997. 0.769933
\(56\) −3.18519e6 −2.42369
\(57\) −1.42206e6 −1.01708
\(58\) 2.82547e6 1.90149
\(59\) 205379. 0.130189
\(60\) 3.18452e6 1.90333
\(61\) −1.17187e6 −0.661038 −0.330519 0.943799i \(-0.607224\pi\)
−0.330519 + 0.943799i \(0.607224\pi\)
\(62\) 5.78445e6 3.08242
\(63\) −476529. −0.240103
\(64\) 8.00009e6 3.81474
\(65\) −1.91999e6 −0.867166
\(66\) 1.66874e6 0.714471
\(67\) −1.24840e6 −0.507096 −0.253548 0.967323i \(-0.581598\pi\)
−0.253548 + 0.967323i \(0.581598\pi\)
\(68\) 9.62614e6 3.71254
\(69\) −3.04245e6 −1.11494
\(70\) −4.80984e6 −1.67605
\(71\) 2.90922e6 0.964655 0.482327 0.875991i \(-0.339792\pi\)
0.482327 + 0.875991i \(0.339792\pi\)
\(72\) 3.55224e6 1.12160
\(73\) 1.95003e6 0.586694 0.293347 0.956006i \(-0.405231\pi\)
0.293347 + 0.956006i \(0.405231\pi\)
\(74\) −3.56019e6 −1.02132
\(75\) 944354. 0.258476
\(76\) −1.84714e7 −4.82673
\(77\) −1.84650e6 −0.460928
\(78\) −3.37260e6 −0.804700
\(79\) 529861. 0.120911 0.0604557 0.998171i \(-0.480745\pi\)
0.0604557 + 0.998171i \(0.480745\pi\)
\(80\) 2.07575e7 4.53273
\(81\) 531441. 0.111111
\(82\) −1.24886e7 −2.50129
\(83\) 7.53928e6 1.44729 0.723646 0.690171i \(-0.242465\pi\)
0.723646 + 0.690171i \(0.242465\pi\)
\(84\) −6.18974e6 −1.13945
\(85\) 9.23078e6 1.63032
\(86\) 1.02538e7 1.73837
\(87\) 3.48673e6 0.567677
\(88\) 1.37646e7 2.15315
\(89\) 6.07893e6 0.914034 0.457017 0.889458i \(-0.348918\pi\)
0.457017 + 0.889458i \(0.348918\pi\)
\(90\) 5.36410e6 0.775618
\(91\) 3.73187e6 0.519137
\(92\) −3.95190e7 −5.29114
\(93\) 7.13822e6 0.920237
\(94\) −9.86913e6 −1.22555
\(95\) −1.77128e7 −2.11960
\(96\) 1.96218e7 2.26355
\(97\) −2.74881e6 −0.305804 −0.152902 0.988241i \(-0.548862\pi\)
−0.152902 + 0.988241i \(0.548862\pi\)
\(98\) −8.66979e6 −0.930502
\(99\) 2.05928e6 0.213301
\(100\) 1.22664e7 1.22664
\(101\) −1.90082e7 −1.83576 −0.917880 0.396858i \(-0.870100\pi\)
−0.917880 + 0.396858i \(0.870100\pi\)
\(102\) 1.62145e7 1.51288
\(103\) 1.79174e7 1.61564 0.807818 0.589432i \(-0.200648\pi\)
0.807818 + 0.589432i \(0.200648\pi\)
\(104\) −2.78189e7 −2.42506
\(105\) −5.93552e6 −0.500375
\(106\) −2.39355e7 −1.95196
\(107\) 85176.8 0.00672168 0.00336084 0.999994i \(-0.498930\pi\)
0.00336084 + 0.999994i \(0.498930\pi\)
\(108\) 6.90301e6 0.527297
\(109\) 1.58698e7 1.17376 0.586881 0.809673i \(-0.300356\pi\)
0.586881 + 0.809673i \(0.300356\pi\)
\(110\) 2.07854e7 1.48896
\(111\) −4.39340e6 −0.304909
\(112\) −4.03462e7 −2.71356
\(113\) −2.00702e7 −1.30851 −0.654257 0.756273i \(-0.727018\pi\)
−0.654257 + 0.756273i \(0.727018\pi\)
\(114\) −3.11138e7 −1.96692
\(115\) −3.78959e7 −2.32354
\(116\) 4.52900e7 2.69401
\(117\) −4.16191e6 −0.240238
\(118\) 4.49357e6 0.251771
\(119\) −1.79418e7 −0.976005
\(120\) 4.42457e7 2.33742
\(121\) −1.15076e7 −0.590524
\(122\) −2.56399e7 −1.27837
\(123\) −1.54114e7 −0.746747
\(124\) 9.27200e7 4.36715
\(125\) −1.45112e7 −0.664536
\(126\) −1.04262e7 −0.464331
\(127\) 1.19818e7 0.519048 0.259524 0.965737i \(-0.416434\pi\)
0.259524 + 0.965737i \(0.416434\pi\)
\(128\) 8.20154e7 3.45669
\(129\) 1.26536e7 0.518979
\(130\) −4.20083e7 −1.67700
\(131\) 8.95447e6 0.348009 0.174004 0.984745i \(-0.444329\pi\)
0.174004 + 0.984745i \(0.444329\pi\)
\(132\) 2.67485e7 1.01226
\(133\) 3.44282e7 1.26892
\(134\) −2.73142e7 −0.980666
\(135\) 6.61949e6 0.231556
\(136\) 1.33745e8 4.55924
\(137\) −1.70175e7 −0.565424 −0.282712 0.959205i \(-0.591234\pi\)
−0.282712 + 0.959205i \(0.591234\pi\)
\(138\) −6.65670e7 −2.15616
\(139\) −1.49569e7 −0.472379 −0.236189 0.971707i \(-0.575899\pi\)
−0.236189 + 0.971707i \(0.575899\pi\)
\(140\) −7.70978e7 −2.37462
\(141\) −1.21789e7 −0.365881
\(142\) 6.36520e7 1.86553
\(143\) −1.61270e7 −0.461188
\(144\) 4.49955e7 1.25574
\(145\) 4.34299e7 1.18304
\(146\) 4.26655e7 1.13460
\(147\) −1.06988e7 −0.277796
\(148\) −5.70669e7 −1.44700
\(149\) −5.19387e7 −1.28629 −0.643145 0.765744i \(-0.722371\pi\)
−0.643145 + 0.765744i \(0.722371\pi\)
\(150\) 2.06619e7 0.499863
\(151\) −4.79328e6 −0.113296 −0.0566478 0.998394i \(-0.518041\pi\)
−0.0566478 + 0.998394i \(0.518041\pi\)
\(152\) −2.56642e8 −5.92754
\(153\) 2.00093e7 0.451661
\(154\) −4.04004e7 −0.891381
\(155\) 8.89119e7 1.91778
\(156\) −5.40601e7 −1.14009
\(157\) 5.35304e7 1.10396 0.551978 0.833859i \(-0.313873\pi\)
0.551978 + 0.833859i \(0.313873\pi\)
\(158\) 1.15931e7 0.233829
\(159\) −2.95372e7 −0.582747
\(160\) 2.44404e8 4.71725
\(161\) 7.36581e7 1.39101
\(162\) 1.16276e7 0.214876
\(163\) −8.41640e7 −1.52219 −0.761096 0.648639i \(-0.775339\pi\)
−0.761096 + 0.648639i \(0.775339\pi\)
\(164\) −2.00182e8 −3.54382
\(165\) 2.56499e7 0.444521
\(166\) 1.64955e8 2.79890
\(167\) 6.28714e6 0.104459 0.0522295 0.998635i \(-0.483367\pi\)
0.0522295 + 0.998635i \(0.483367\pi\)
\(168\) −8.60002e7 −1.39932
\(169\) −3.01550e7 −0.480569
\(170\) 2.01964e8 3.15285
\(171\) −3.83955e7 −0.587211
\(172\) 1.64360e8 2.46290
\(173\) −4.64860e7 −0.682591 −0.341296 0.939956i \(-0.610866\pi\)
−0.341296 + 0.939956i \(0.610866\pi\)
\(174\) 7.62877e7 1.09782
\(175\) −2.28630e7 −0.322477
\(176\) 1.74353e8 2.41066
\(177\) 5.54523e6 0.0751646
\(178\) 1.33004e8 1.76764
\(179\) 1.12901e8 1.47134 0.735671 0.677339i \(-0.236867\pi\)
0.735671 + 0.677339i \(0.236867\pi\)
\(180\) 8.59821e7 1.09889
\(181\) −1.04425e8 −1.30897 −0.654487 0.756073i \(-0.727115\pi\)
−0.654487 + 0.756073i \(0.727115\pi\)
\(182\) 8.16513e7 1.00395
\(183\) −3.16406e7 −0.381651
\(184\) −5.49077e8 −6.49787
\(185\) −5.47231e7 −0.635433
\(186\) 1.56180e8 1.77963
\(187\) 7.75343e7 0.867058
\(188\) −1.58194e8 −1.73635
\(189\) −1.28663e7 −0.138623
\(190\) −3.87545e8 −4.09906
\(191\) −1.38829e8 −1.44167 −0.720833 0.693108i \(-0.756241\pi\)
−0.720833 + 0.693108i \(0.756241\pi\)
\(192\) 2.16002e8 2.20244
\(193\) 3.06880e7 0.307268 0.153634 0.988128i \(-0.450902\pi\)
0.153634 + 0.988128i \(0.450902\pi\)
\(194\) −6.01424e7 −0.591391
\(195\) −5.18397e7 −0.500658
\(196\) −1.38970e8 −1.31833
\(197\) −2.78823e7 −0.259834 −0.129917 0.991525i \(-0.541471\pi\)
−0.129917 + 0.991525i \(0.541471\pi\)
\(198\) 4.50560e7 0.412500
\(199\) 1.31062e8 1.17894 0.589469 0.807791i \(-0.299337\pi\)
0.589469 + 0.807791i \(0.299337\pi\)
\(200\) 1.70430e8 1.50640
\(201\) −3.37067e7 −0.292772
\(202\) −4.15888e8 −3.55015
\(203\) −8.44144e7 −0.708240
\(204\) 2.59906e8 2.14343
\(205\) −1.91960e8 −1.55622
\(206\) 3.92021e8 3.12446
\(207\) −8.21460e7 −0.643710
\(208\) −3.52377e8 −2.71510
\(209\) −1.48779e8 −1.12727
\(210\) −1.29866e8 −0.967669
\(211\) 7.00592e7 0.513424 0.256712 0.966488i \(-0.417361\pi\)
0.256712 + 0.966488i \(0.417361\pi\)
\(212\) −3.83666e8 −2.76553
\(213\) 7.85489e7 0.556944
\(214\) 1.86362e6 0.0129990
\(215\) 1.57610e8 1.08155
\(216\) 9.59104e7 0.647556
\(217\) −1.72817e8 −1.14810
\(218\) 3.47223e8 2.26992
\(219\) 5.26508e7 0.338728
\(220\) 3.33173e8 2.10955
\(221\) −1.56700e8 −0.976557
\(222\) −9.61252e7 −0.589660
\(223\) 1.42923e8 0.863049 0.431525 0.902101i \(-0.357976\pi\)
0.431525 + 0.902101i \(0.357976\pi\)
\(224\) −4.75048e8 −2.82403
\(225\) 2.54976e7 0.149231
\(226\) −4.39126e8 −2.53052
\(227\) −1.16361e8 −0.660262 −0.330131 0.943935i \(-0.607093\pi\)
−0.330131 + 0.943935i \(0.607093\pi\)
\(228\) −4.98728e8 −2.78671
\(229\) 2.48550e8 1.36770 0.683849 0.729624i \(-0.260305\pi\)
0.683849 + 0.729624i \(0.260305\pi\)
\(230\) −8.29141e8 −4.49346
\(231\) −4.98556e7 −0.266117
\(232\) 6.29259e8 3.30842
\(233\) −1.28332e8 −0.664645 −0.332323 0.943166i \(-0.607832\pi\)
−0.332323 + 0.943166i \(0.607832\pi\)
\(234\) −9.10603e7 −0.464594
\(235\) −1.51697e8 −0.762498
\(236\) 7.20283e7 0.356707
\(237\) 1.43063e7 0.0698082
\(238\) −3.92556e8 −1.88748
\(239\) 1.52392e8 0.722056 0.361028 0.932555i \(-0.382426\pi\)
0.361028 + 0.932555i \(0.382426\pi\)
\(240\) 5.60452e8 2.61697
\(241\) −2.46545e8 −1.13458 −0.567292 0.823517i \(-0.692009\pi\)
−0.567292 + 0.823517i \(0.692009\pi\)
\(242\) −2.51780e8 −1.14201
\(243\) 1.43489e7 0.0641500
\(244\) −4.10987e8 −1.81119
\(245\) −1.33262e8 −0.578928
\(246\) −3.37192e8 −1.44412
\(247\) 3.00690e8 1.26964
\(248\) 1.28825e9 5.36315
\(249\) 2.03560e8 0.835595
\(250\) −3.17497e8 −1.28514
\(251\) −1.53947e7 −0.0614486 −0.0307243 0.999528i \(-0.509781\pi\)
−0.0307243 + 0.999528i \(0.509781\pi\)
\(252\) −1.67123e8 −0.657861
\(253\) −3.18308e8 −1.23574
\(254\) 2.62154e8 1.00378
\(255\) 2.49231e8 0.941264
\(256\) 7.70438e8 2.87010
\(257\) 1.45269e7 0.0533836 0.0266918 0.999644i \(-0.491503\pi\)
0.0266918 + 0.999644i \(0.491503\pi\)
\(258\) 2.76853e8 1.00365
\(259\) 1.06365e8 0.380408
\(260\) −6.73358e8 −2.37596
\(261\) 9.41418e7 0.327749
\(262\) 1.95919e8 0.673010
\(263\) 4.08752e8 1.38553 0.692763 0.721166i \(-0.256393\pi\)
0.692763 + 0.721166i \(0.256393\pi\)
\(264\) 3.71644e8 1.24312
\(265\) −3.67908e8 −1.21445
\(266\) 7.53270e8 2.45394
\(267\) 1.64131e8 0.527718
\(268\) −4.37824e8 −1.38940
\(269\) 2.03154e6 0.00636345 0.00318173 0.999995i \(-0.498987\pi\)
0.00318173 + 0.999995i \(0.498987\pi\)
\(270\) 1.44831e8 0.447803
\(271\) 1.02015e8 0.311366 0.155683 0.987807i \(-0.450242\pi\)
0.155683 + 0.987807i \(0.450242\pi\)
\(272\) 1.69413e9 5.10452
\(273\) 1.00761e8 0.299724
\(274\) −3.72333e8 −1.09347
\(275\) 9.88007e7 0.286481
\(276\) −1.06701e9 −3.05484
\(277\) −1.80373e8 −0.509907 −0.254954 0.966953i \(-0.582060\pi\)
−0.254954 + 0.966953i \(0.582060\pi\)
\(278\) −3.27249e8 −0.913526
\(279\) 1.92732e8 0.531299
\(280\) −1.07120e9 −2.91619
\(281\) 3.39131e8 0.911791 0.455896 0.890033i \(-0.349319\pi\)
0.455896 + 0.890033i \(0.349319\pi\)
\(282\) −2.66467e8 −0.707572
\(283\) −1.22871e8 −0.322252 −0.161126 0.986934i \(-0.551513\pi\)
−0.161126 + 0.986934i \(0.551513\pi\)
\(284\) 1.02029e9 2.64307
\(285\) −4.78245e8 −1.22375
\(286\) −3.52850e8 −0.891886
\(287\) 3.73112e8 0.931649
\(288\) 5.29789e8 1.30686
\(289\) 3.43034e8 0.835977
\(290\) 9.50220e8 2.28787
\(291\) −7.42179e7 −0.176556
\(292\) 6.83894e8 1.60749
\(293\) −2.73920e8 −0.636190 −0.318095 0.948059i \(-0.603043\pi\)
−0.318095 + 0.948059i \(0.603043\pi\)
\(294\) −2.34084e8 −0.537225
\(295\) 6.90700e7 0.156643
\(296\) −7.92888e8 −1.77701
\(297\) 5.56007e7 0.123149
\(298\) −1.13639e9 −2.48754
\(299\) 6.43316e8 1.39180
\(300\) 3.31194e8 0.708203
\(301\) −3.06345e8 −0.647483
\(302\) −1.04874e8 −0.219101
\(303\) −5.13221e8 −1.05988
\(304\) −3.25083e9 −6.63647
\(305\) −3.94107e8 −0.795362
\(306\) 4.37792e8 0.873461
\(307\) −2.51384e8 −0.495853 −0.247927 0.968779i \(-0.579749\pi\)
−0.247927 + 0.968779i \(0.579749\pi\)
\(308\) −6.47586e8 −1.26290
\(309\) 4.83769e8 0.932788
\(310\) 1.94534e9 3.70877
\(311\) 3.46715e8 0.653599 0.326799 0.945094i \(-0.394030\pi\)
0.326799 + 0.945094i \(0.394030\pi\)
\(312\) −7.51110e8 −1.40011
\(313\) 8.59091e8 1.58356 0.791779 0.610807i \(-0.209155\pi\)
0.791779 + 0.610807i \(0.209155\pi\)
\(314\) 1.17122e9 2.13493
\(315\) −1.60259e8 −0.288892
\(316\) 1.85827e8 0.331287
\(317\) 6.36316e8 1.12193 0.560964 0.827840i \(-0.310431\pi\)
0.560964 + 0.827840i \(0.310431\pi\)
\(318\) −6.46258e8 −1.12697
\(319\) 3.64791e8 0.629182
\(320\) 2.69047e9 4.58990
\(321\) 2.29977e6 0.00388076
\(322\) 1.61160e9 2.69005
\(323\) −1.44563e9 −2.38698
\(324\) 1.86381e8 0.304435
\(325\) −1.99681e8 −0.322660
\(326\) −1.84146e9 −2.94375
\(327\) 4.28486e8 0.677672
\(328\) −2.78132e9 −4.35204
\(329\) 2.94852e8 0.456477
\(330\) 5.61205e8 0.859653
\(331\) 4.08883e8 0.619728 0.309864 0.950781i \(-0.399716\pi\)
0.309864 + 0.950781i \(0.399716\pi\)
\(332\) 2.64409e9 3.96546
\(333\) −1.18622e8 −0.176039
\(334\) 1.37559e8 0.202012
\(335\) −4.19842e8 −0.610139
\(336\) −1.08935e9 −1.56668
\(337\) 6.00455e8 0.854626 0.427313 0.904104i \(-0.359460\pi\)
0.427313 + 0.904104i \(0.359460\pi\)
\(338\) −6.59774e8 −0.929366
\(339\) −5.41897e8 −0.755471
\(340\) 3.23732e9 4.46693
\(341\) 7.46818e8 1.01994
\(342\) −8.40072e8 −1.13560
\(343\) 7.97349e8 1.06689
\(344\) 2.28362e9 3.02461
\(345\) −1.02319e9 −1.34150
\(346\) −1.01709e9 −1.32005
\(347\) 1.81629e8 0.233363 0.116681 0.993169i \(-0.462774\pi\)
0.116681 + 0.993169i \(0.462774\pi\)
\(348\) 1.22283e9 1.55539
\(349\) −7.06742e8 −0.889963 −0.444981 0.895540i \(-0.646790\pi\)
−0.444981 + 0.895540i \(0.646790\pi\)
\(350\) −5.00229e8 −0.623635
\(351\) −1.12372e8 −0.138702
\(352\) 2.05288e9 2.50880
\(353\) 2.31567e8 0.280199 0.140099 0.990137i \(-0.455258\pi\)
0.140099 + 0.990137i \(0.455258\pi\)
\(354\) 1.21327e8 0.145360
\(355\) 9.78384e8 1.16067
\(356\) 2.13194e9 2.50438
\(357\) −4.84429e8 −0.563497
\(358\) 2.47022e9 2.84541
\(359\) 1.13982e9 1.30019 0.650093 0.759854i \(-0.274730\pi\)
0.650093 + 0.759854i \(0.274730\pi\)
\(360\) 1.19463e9 1.34951
\(361\) 1.88013e9 2.10335
\(362\) −2.28477e9 −2.53141
\(363\) −3.10706e8 −0.340939
\(364\) 1.30880e9 1.42239
\(365\) 6.55805e8 0.705911
\(366\) −6.92278e8 −0.738069
\(367\) 6.44750e8 0.680864 0.340432 0.940269i \(-0.389427\pi\)
0.340432 + 0.940269i \(0.389427\pi\)
\(368\) −6.95506e9 −7.27500
\(369\) −4.16107e8 −0.431134
\(370\) −1.19731e9 −1.22885
\(371\) 7.15101e8 0.727041
\(372\) 2.50344e9 2.52137
\(373\) 1.69263e8 0.168881 0.0844404 0.996429i \(-0.473090\pi\)
0.0844404 + 0.996429i \(0.473090\pi\)
\(374\) 1.69640e9 1.67679
\(375\) −3.91802e8 −0.383670
\(376\) −2.19795e9 −2.13236
\(377\) −7.37260e8 −0.708640
\(378\) −2.81507e8 −0.268082
\(379\) −1.01373e9 −0.956503 −0.478252 0.878223i \(-0.658729\pi\)
−0.478252 + 0.878223i \(0.658729\pi\)
\(380\) −6.21203e9 −5.80753
\(381\) 3.23508e8 0.299673
\(382\) −3.03751e9 −2.78802
\(383\) 5.54096e8 0.503953 0.251976 0.967733i \(-0.418920\pi\)
0.251976 + 0.967733i \(0.418920\pi\)
\(384\) 2.21442e9 1.99572
\(385\) −6.20988e8 −0.554589
\(386\) 6.71435e8 0.594221
\(387\) 3.41647e8 0.299633
\(388\) −9.64033e8 −0.837879
\(389\) 1.17131e9 1.00890 0.504450 0.863441i \(-0.331695\pi\)
0.504450 + 0.863441i \(0.331695\pi\)
\(390\) −1.13422e9 −0.968217
\(391\) −3.09289e9 −2.61665
\(392\) −1.93084e9 −1.61900
\(393\) 2.41771e8 0.200923
\(394\) −6.10048e8 −0.502490
\(395\) 1.78195e8 0.145481
\(396\) 7.22210e8 0.584427
\(397\) 1.86752e9 1.49796 0.748978 0.662594i \(-0.230545\pi\)
0.748978 + 0.662594i \(0.230545\pi\)
\(398\) 2.86756e9 2.27993
\(399\) 9.29562e8 0.732610
\(400\) 2.15880e9 1.68656
\(401\) 2.23006e9 1.72708 0.863539 0.504282i \(-0.168243\pi\)
0.863539 + 0.504282i \(0.168243\pi\)
\(402\) −7.37483e8 −0.566188
\(403\) −1.50936e9 −1.14875
\(404\) −6.66635e9 −5.02983
\(405\) 1.78726e8 0.133689
\(406\) −1.84694e9 −1.36965
\(407\) −4.59649e8 −0.337945
\(408\) 3.61113e9 2.63228
\(409\) −1.03440e9 −0.747575 −0.373788 0.927514i \(-0.621941\pi\)
−0.373788 + 0.927514i \(0.621941\pi\)
\(410\) −4.19997e9 −3.00956
\(411\) −4.59473e8 −0.326448
\(412\) 6.28378e9 4.42671
\(413\) −1.34251e8 −0.0937761
\(414\) −1.79731e9 −1.24486
\(415\) 2.53550e9 1.74138
\(416\) −4.14898e9 −2.82563
\(417\) −4.03837e8 −0.272728
\(418\) −3.25520e9 −2.18002
\(419\) −1.07822e9 −0.716075 −0.358038 0.933707i \(-0.616554\pi\)
−0.358038 + 0.933707i \(0.616554\pi\)
\(420\) −2.08164e9 −1.37099
\(421\) −1.88974e9 −1.23428 −0.617140 0.786853i \(-0.711709\pi\)
−0.617140 + 0.786853i \(0.711709\pi\)
\(422\) 1.53285e9 0.992904
\(423\) −3.28829e8 −0.211241
\(424\) −5.33065e9 −3.39625
\(425\) 9.60010e8 0.606617
\(426\) 1.71860e9 1.07707
\(427\) 7.66024e8 0.476151
\(428\) 2.98723e7 0.0184168
\(429\) −4.35430e8 −0.266267
\(430\) 3.44841e9 2.09160
\(431\) −2.24349e9 −1.34975 −0.674876 0.737931i \(-0.735803\pi\)
−0.674876 + 0.737931i \(0.735803\pi\)
\(432\) 1.21488e9 0.725003
\(433\) 1.57772e9 0.933948 0.466974 0.884271i \(-0.345344\pi\)
0.466974 + 0.884271i \(0.345344\pi\)
\(434\) −3.78115e9 −2.22029
\(435\) 1.17261e9 0.683030
\(436\) 5.56570e9 3.21601
\(437\) 5.93488e9 3.40194
\(438\) 1.15197e9 0.655061
\(439\) 2.36005e9 1.33136 0.665681 0.746236i \(-0.268141\pi\)
0.665681 + 0.746236i \(0.268141\pi\)
\(440\) 4.62910e9 2.59067
\(441\) −2.88868e8 −0.160385
\(442\) −3.42852e9 −1.88855
\(443\) −4.93538e8 −0.269717 −0.134858 0.990865i \(-0.543058\pi\)
−0.134858 + 0.990865i \(0.543058\pi\)
\(444\) −1.54081e9 −0.835426
\(445\) 2.04438e9 1.09977
\(446\) 3.12708e9 1.66904
\(447\) −1.40234e9 −0.742640
\(448\) −5.22945e9 −2.74779
\(449\) −2.07688e9 −1.08280 −0.541401 0.840764i \(-0.682106\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(450\) 5.57872e8 0.288596
\(451\) −1.61238e9 −0.827653
\(452\) −7.03882e9 −3.58522
\(453\) −1.29418e8 −0.0654112
\(454\) −2.54591e9 −1.27687
\(455\) 1.25505e9 0.624627
\(456\) −6.92933e9 −3.42227
\(457\) −1.48289e9 −0.726777 −0.363389 0.931638i \(-0.618380\pi\)
−0.363389 + 0.931638i \(0.618380\pi\)
\(458\) 5.43814e9 2.64497
\(459\) 5.40252e8 0.260766
\(460\) −1.32905e10 −6.36630
\(461\) 2.57673e9 1.22494 0.612471 0.790493i \(-0.290175\pi\)
0.612471 + 0.790493i \(0.290175\pi\)
\(462\) −1.09081e9 −0.514639
\(463\) −4.54013e8 −0.212586 −0.106293 0.994335i \(-0.533898\pi\)
−0.106293 + 0.994335i \(0.533898\pi\)
\(464\) 7.97070e9 3.70411
\(465\) 2.40062e9 1.10723
\(466\) −2.80784e9 −1.28535
\(467\) 3.63008e9 1.64933 0.824663 0.565624i \(-0.191365\pi\)
0.824663 + 0.565624i \(0.191365\pi\)
\(468\) −1.45962e9 −0.658233
\(469\) 8.16044e8 0.365265
\(470\) −3.31904e9 −1.47458
\(471\) 1.44532e9 0.637370
\(472\) 1.00076e9 0.438060
\(473\) 1.32385e9 0.575208
\(474\) 3.13013e8 0.135001
\(475\) −1.84215e9 −0.788672
\(476\) −6.29236e9 −2.67417
\(477\) −7.97506e8 −0.336449
\(478\) 3.33426e9 1.39637
\(479\) −9.05516e7 −0.0376463 −0.0188231 0.999823i \(-0.505992\pi\)
−0.0188231 + 0.999823i \(0.505992\pi\)
\(480\) 6.59892e9 2.72351
\(481\) 9.28973e8 0.380623
\(482\) −5.39427e9 −2.19416
\(483\) 1.98877e9 0.803099
\(484\) −4.03584e9 −1.61799
\(485\) −9.24439e8 −0.367944
\(486\) 3.13946e8 0.124059
\(487\) 2.49820e8 0.0980112 0.0490056 0.998799i \(-0.484395\pi\)
0.0490056 + 0.998799i \(0.484395\pi\)
\(488\) −5.71025e9 −2.22426
\(489\) −2.27243e9 −0.878839
\(490\) −2.91569e9 −1.11958
\(491\) 4.25867e8 0.162364 0.0811818 0.996699i \(-0.474131\pi\)
0.0811818 + 0.996699i \(0.474131\pi\)
\(492\) −5.40491e9 −2.04602
\(493\) 3.54454e9 1.33228
\(494\) 6.57892e9 2.45533
\(495\) 6.92548e8 0.256644
\(496\) 1.63180e10 6.00457
\(497\) −1.90168e9 −0.694849
\(498\) 4.45379e9 1.61595
\(499\) −4.17328e9 −1.50358 −0.751789 0.659404i \(-0.770809\pi\)
−0.751789 + 0.659404i \(0.770809\pi\)
\(500\) −5.08921e9 −1.82077
\(501\) 1.69753e8 0.0603094
\(502\) −3.36826e8 −0.118835
\(503\) 4.99769e9 1.75098 0.875491 0.483234i \(-0.160538\pi\)
0.875491 + 0.483234i \(0.160538\pi\)
\(504\) −2.32201e9 −0.807898
\(505\) −6.39255e9 −2.20879
\(506\) −6.96440e9 −2.38977
\(507\) −8.14185e8 −0.277457
\(508\) 4.20212e9 1.42215
\(509\) −3.79733e9 −1.27634 −0.638170 0.769895i \(-0.720308\pi\)
−0.638170 + 0.769895i \(0.720308\pi\)
\(510\) 5.45303e9 1.82030
\(511\) −1.27468e9 −0.422600
\(512\) 6.35877e9 2.09377
\(513\) −1.03668e9 −0.339026
\(514\) 3.17841e8 0.103238
\(515\) 6.02570e9 1.94394
\(516\) 4.43773e9 1.42196
\(517\) −1.27418e9 −0.405522
\(518\) 2.32721e9 0.735666
\(519\) −1.25512e9 −0.394094
\(520\) −9.35563e9 −2.91784
\(521\) 4.83903e9 1.49909 0.749543 0.661956i \(-0.230273\pi\)
0.749543 + 0.661956i \(0.230273\pi\)
\(522\) 2.05977e9 0.633828
\(523\) −1.38035e9 −0.421922 −0.210961 0.977494i \(-0.567659\pi\)
−0.210961 + 0.977494i \(0.567659\pi\)
\(524\) 3.14042e9 0.953515
\(525\) −6.17300e8 −0.186182
\(526\) 8.94325e9 2.67945
\(527\) 7.25657e9 2.15970
\(528\) 4.70754e9 1.39180
\(529\) 9.29267e9 2.72926
\(530\) −8.04962e9 −2.34860
\(531\) 1.49721e8 0.0433963
\(532\) 1.20743e10 3.47673
\(533\) 3.25869e9 0.932175
\(534\) 3.59110e9 1.02055
\(535\) 2.86454e7 0.00808754
\(536\) −6.08312e9 −1.70628
\(537\) 3.04834e9 0.849480
\(538\) 4.44490e7 0.0123062
\(539\) −1.11934e9 −0.307894
\(540\) 2.32152e9 0.634445
\(541\) 3.09741e9 0.841023 0.420512 0.907287i \(-0.361851\pi\)
0.420512 + 0.907287i \(0.361851\pi\)
\(542\) 2.23203e9 0.602146
\(543\) −2.81949e9 −0.755736
\(544\) 1.99471e10 5.31232
\(545\) 5.33711e9 1.41227
\(546\) 2.20458e9 0.579632
\(547\) 4.36175e9 1.13948 0.569738 0.821826i \(-0.307045\pi\)
0.569738 + 0.821826i \(0.307045\pi\)
\(548\) −5.96820e9 −1.54921
\(549\) −8.54296e8 −0.220346
\(550\) 2.16170e9 0.554021
\(551\) −6.80155e9 −1.73212
\(552\) −1.48251e10 −3.75155
\(553\) −3.46357e8 −0.0870935
\(554\) −3.94645e9 −0.986103
\(555\) −1.47752e9 −0.366867
\(556\) −5.24553e9 −1.29428
\(557\) 7.54573e9 1.85016 0.925078 0.379778i \(-0.124000\pi\)
0.925078 + 0.379778i \(0.124000\pi\)
\(558\) 4.21687e9 1.02747
\(559\) −2.67556e9 −0.647849
\(560\) −1.35686e10 −3.26496
\(561\) 2.09342e9 0.500596
\(562\) 7.41999e9 1.76330
\(563\) 5.42932e9 1.28223 0.641115 0.767445i \(-0.278472\pi\)
0.641115 + 0.767445i \(0.278472\pi\)
\(564\) −4.27124e9 −1.00248
\(565\) −6.74973e9 −1.57441
\(566\) −2.68834e9 −0.623199
\(567\) −3.47389e8 −0.0800342
\(568\) 1.41759e10 3.24587
\(569\) −7.43477e9 −1.69190 −0.845950 0.533262i \(-0.820966\pi\)
−0.845950 + 0.533262i \(0.820966\pi\)
\(570\) −1.04637e10 −2.36660
\(571\) −6.57051e9 −1.47697 −0.738487 0.674268i \(-0.764459\pi\)
−0.738487 + 0.674268i \(0.764459\pi\)
\(572\) −5.65590e9 −1.26362
\(573\) −3.74840e9 −0.832347
\(574\) 8.16347e9 1.80170
\(575\) −3.94122e9 −0.864555
\(576\) 5.83207e9 1.27158
\(577\) 3.17169e9 0.687347 0.343673 0.939089i \(-0.388329\pi\)
0.343673 + 0.939089i \(0.388329\pi\)
\(578\) 7.50538e9 1.61668
\(579\) 8.28575e8 0.177401
\(580\) 1.52313e10 3.24144
\(581\) −4.92823e9 −1.04250
\(582\) −1.62384e9 −0.341440
\(583\) −3.09026e9 −0.645885
\(584\) 9.50201e9 1.97411
\(585\) −1.39967e9 −0.289055
\(586\) −5.99321e9 −1.23032
\(587\) −4.96828e9 −1.01385 −0.506923 0.861991i \(-0.669217\pi\)
−0.506923 + 0.861991i \(0.669217\pi\)
\(588\) −3.75218e9 −0.761137
\(589\) −1.39245e10 −2.80786
\(590\) 1.51121e9 0.302931
\(591\) −7.52821e8 −0.150015
\(592\) −1.00434e10 −1.98954
\(593\) −1.11991e9 −0.220542 −0.110271 0.993902i \(-0.535172\pi\)
−0.110271 + 0.993902i \(0.535172\pi\)
\(594\) 1.21651e9 0.238157
\(595\) −6.03392e9 −1.17433
\(596\) −1.82154e10 −3.52433
\(597\) 3.53867e9 0.680660
\(598\) 1.40754e10 2.69157
\(599\) 1.35680e9 0.257942 0.128971 0.991648i \(-0.458833\pi\)
0.128971 + 0.991648i \(0.458833\pi\)
\(600\) 4.60160e9 0.869721
\(601\) −8.43823e9 −1.58559 −0.792795 0.609489i \(-0.791375\pi\)
−0.792795 + 0.609489i \(0.791375\pi\)
\(602\) −6.70266e9 −1.25216
\(603\) −9.10080e8 −0.169032
\(604\) −1.68105e9 −0.310420
\(605\) −3.87008e9 −0.710519
\(606\) −1.12290e10 −2.04968
\(607\) 2.84233e9 0.515839 0.257920 0.966166i \(-0.416963\pi\)
0.257920 + 0.966166i \(0.416963\pi\)
\(608\) −3.82762e10 −6.90663
\(609\) −2.27919e9 −0.408902
\(610\) −8.62284e9 −1.53814
\(611\) 2.57519e9 0.456735
\(612\) 7.01746e9 1.23751
\(613\) −5.43477e8 −0.0952949 −0.0476475 0.998864i \(-0.515172\pi\)
−0.0476475 + 0.998864i \(0.515172\pi\)
\(614\) −5.50014e9 −0.958924
\(615\) −5.18292e9 −0.898487
\(616\) −8.99755e9 −1.55093
\(617\) 5.89919e8 0.101110 0.0505550 0.998721i \(-0.483901\pi\)
0.0505550 + 0.998721i \(0.483901\pi\)
\(618\) 1.05846e10 1.80391
\(619\) −8.29034e9 −1.40493 −0.702466 0.711717i \(-0.747918\pi\)
−0.702466 + 0.711717i \(0.747918\pi\)
\(620\) 3.11822e10 5.25456
\(621\) −2.21794e9 −0.371646
\(622\) 7.58592e9 1.26399
\(623\) −3.97364e9 −0.658386
\(624\) −9.51417e9 −1.56756
\(625\) −7.61270e9 −1.24726
\(626\) 1.87964e10 3.06242
\(627\) −4.01703e9 −0.650832
\(628\) 1.87736e10 3.02475
\(629\) −4.46624e9 −0.715591
\(630\) −3.50637e9 −0.558684
\(631\) 3.36985e9 0.533959 0.266980 0.963702i \(-0.413974\pi\)
0.266980 + 0.963702i \(0.413974\pi\)
\(632\) 2.58188e9 0.406843
\(633\) 1.89160e9 0.296426
\(634\) 1.39222e10 2.16968
\(635\) 4.02953e9 0.624520
\(636\) −1.03590e10 −1.59668
\(637\) 2.26224e9 0.346777
\(638\) 7.98141e9 1.21677
\(639\) 2.12082e9 0.321552
\(640\) 2.75822e10 4.15909
\(641\) −6.77698e9 −1.01633 −0.508163 0.861261i \(-0.669675\pi\)
−0.508163 + 0.861261i \(0.669675\pi\)
\(642\) 5.03177e7 0.00750496
\(643\) 5.27093e9 0.781895 0.390948 0.920413i \(-0.372147\pi\)
0.390948 + 0.920413i \(0.372147\pi\)
\(644\) 2.58326e10 3.81125
\(645\) 4.25546e9 0.624436
\(646\) −3.16296e10 −4.61615
\(647\) −1.36234e10 −1.97752 −0.988762 0.149498i \(-0.952234\pi\)
−0.988762 + 0.149498i \(0.952234\pi\)
\(648\) 2.58958e9 0.373867
\(649\) 5.80156e8 0.0833083
\(650\) −4.36890e9 −0.623987
\(651\) −4.66607e9 −0.662854
\(652\) −2.95171e10 −4.17068
\(653\) 7.55574e9 1.06189 0.530947 0.847405i \(-0.321836\pi\)
0.530947 + 0.847405i \(0.321836\pi\)
\(654\) 9.37502e9 1.31054
\(655\) 3.01143e9 0.418725
\(656\) −3.52305e10 −4.87254
\(657\) 1.42157e9 0.195565
\(658\) 6.45120e9 0.882774
\(659\) 6.65668e9 0.906063 0.453032 0.891494i \(-0.350342\pi\)
0.453032 + 0.891494i \(0.350342\pi\)
\(660\) 8.99566e9 1.21795
\(661\) −9.57345e9 −1.28933 −0.644664 0.764466i \(-0.723003\pi\)
−0.644664 + 0.764466i \(0.723003\pi\)
\(662\) 8.94613e9 1.19848
\(663\) −4.23091e9 −0.563815
\(664\) 3.67370e10 4.86985
\(665\) 1.15784e10 1.52676
\(666\) −2.59538e9 −0.340440
\(667\) −1.45517e10 −1.89878
\(668\) 2.20496e9 0.286209
\(669\) 3.85892e9 0.498282
\(670\) −9.18589e9 −1.17994
\(671\) −3.31032e9 −0.423001
\(672\) −1.28263e10 −1.63045
\(673\) 8.00895e9 1.01280 0.506399 0.862299i \(-0.330976\pi\)
0.506399 + 0.862299i \(0.330976\pi\)
\(674\) 1.31376e10 1.65275
\(675\) 6.88434e8 0.0861587
\(676\) −1.05756e10 −1.31672
\(677\) −3.66833e8 −0.0454368 −0.0227184 0.999742i \(-0.507232\pi\)
−0.0227184 + 0.999742i \(0.507232\pi\)
\(678\) −1.18564e10 −1.46099
\(679\) 1.79683e9 0.220273
\(680\) 4.49793e10 5.48569
\(681\) −3.14174e9 −0.381202
\(682\) 1.63400e10 1.97245
\(683\) −9.58167e9 −1.15072 −0.575359 0.817901i \(-0.695137\pi\)
−0.575359 + 0.817901i \(0.695137\pi\)
\(684\) −1.34657e10 −1.60891
\(685\) −5.72307e9 −0.680319
\(686\) 1.74455e10 2.06324
\(687\) 6.71086e9 0.789641
\(688\) 2.89262e10 3.38635
\(689\) 6.24557e9 0.727452
\(690\) −2.23868e10 −2.59430
\(691\) 1.25570e10 1.44781 0.723907 0.689898i \(-0.242345\pi\)
0.723907 + 0.689898i \(0.242345\pi\)
\(692\) −1.63031e10 −1.87024
\(693\) −1.34610e9 −0.153643
\(694\) 3.97393e9 0.451297
\(695\) −5.03008e9 −0.568367
\(696\) 1.69900e10 1.91012
\(697\) −1.56669e10 −1.75254
\(698\) −1.54631e10 −1.72109
\(699\) −3.46497e9 −0.383733
\(700\) −8.01825e9 −0.883561
\(701\) −2.90472e9 −0.318487 −0.159243 0.987239i \(-0.550905\pi\)
−0.159243 + 0.987239i \(0.550905\pi\)
\(702\) −2.45863e9 −0.268233
\(703\) 8.57019e9 0.930351
\(704\) 2.25987e10 2.44107
\(705\) −4.09581e9 −0.440229
\(706\) 5.06656e9 0.541872
\(707\) 1.24252e10 1.32231
\(708\) 1.94476e9 0.205945
\(709\) 4.07802e9 0.429721 0.214861 0.976645i \(-0.431070\pi\)
0.214861 + 0.976645i \(0.431070\pi\)
\(710\) 2.14065e10 2.24461
\(711\) 3.86269e8 0.0403038
\(712\) 2.96211e10 3.07554
\(713\) −2.97910e10 −3.07802
\(714\) −1.05990e10 −1.08974
\(715\) −5.42360e9 −0.554902
\(716\) 3.95956e10 4.03136
\(717\) 4.11460e9 0.416879
\(718\) 2.49386e10 2.51441
\(719\) −6.67220e8 −0.0669450 −0.0334725 0.999440i \(-0.510657\pi\)
−0.0334725 + 0.999440i \(0.510657\pi\)
\(720\) 1.51322e10 1.51091
\(721\) −1.17121e10 −1.16376
\(722\) 4.11361e10 4.06764
\(723\) −6.65672e9 −0.655052
\(724\) −3.66229e10 −3.58648
\(725\) 4.51675e9 0.440193
\(726\) −6.79807e9 −0.659337
\(727\) −1.54917e10 −1.49530 −0.747651 0.664092i \(-0.768819\pi\)
−0.747651 + 0.664092i \(0.768819\pi\)
\(728\) 1.81845e10 1.74679
\(729\) 3.87420e8 0.0370370
\(730\) 1.43486e10 1.36515
\(731\) 1.28634e10 1.21799
\(732\) −1.10967e10 −1.04569
\(733\) 1.32907e10 1.24647 0.623237 0.782033i \(-0.285817\pi\)
0.623237 + 0.782033i \(0.285817\pi\)
\(734\) 1.41068e10 1.31671
\(735\) −3.59807e9 −0.334244
\(736\) −8.18908e10 −7.57116
\(737\) −3.52648e9 −0.324493
\(738\) −9.10418e9 −0.833765
\(739\) 1.38709e10 1.26430 0.632148 0.774847i \(-0.282173\pi\)
0.632148 + 0.774847i \(0.282173\pi\)
\(740\) −1.91919e10 −1.74103
\(741\) 8.11862e9 0.733025
\(742\) 1.56460e10 1.40601
\(743\) −9.01349e9 −0.806181 −0.403090 0.915160i \(-0.632064\pi\)
−0.403090 + 0.915160i \(0.632064\pi\)
\(744\) 3.47828e10 3.09641
\(745\) −1.74672e10 −1.54767
\(746\) 3.70337e9 0.326596
\(747\) 5.49613e9 0.482431
\(748\) 2.71920e10 2.37567
\(749\) −5.56779e7 −0.00484168
\(750\) −8.57241e9 −0.741974
\(751\) −6.25646e9 −0.539000 −0.269500 0.963000i \(-0.586858\pi\)
−0.269500 + 0.963000i \(0.586858\pi\)
\(752\) −2.78410e10 −2.38738
\(753\) −4.15656e8 −0.0354774
\(754\) −1.61308e10 −1.37043
\(755\) −1.61200e9 −0.136317
\(756\) −4.51232e9 −0.379816
\(757\) −2.50426e9 −0.209819 −0.104909 0.994482i \(-0.533455\pi\)
−0.104909 + 0.994482i \(0.533455\pi\)
\(758\) −2.21799e10 −1.84977
\(759\) −8.59432e9 −0.713453
\(760\) −8.63099e10 −7.13203
\(761\) 6.57794e9 0.541057 0.270529 0.962712i \(-0.412802\pi\)
0.270529 + 0.962712i \(0.412802\pi\)
\(762\) 7.07816e9 0.579533
\(763\) −1.03737e10 −0.845470
\(764\) −4.86888e10 −3.95005
\(765\) 6.72924e9 0.543439
\(766\) 1.21233e10 0.974587
\(767\) −1.17252e9 −0.0938292
\(768\) 2.08018e10 1.65706
\(769\) 5.82071e9 0.461566 0.230783 0.973005i \(-0.425871\pi\)
0.230783 + 0.973005i \(0.425871\pi\)
\(770\) −1.35869e10 −1.07251
\(771\) 3.92227e8 0.0308210
\(772\) 1.07626e10 0.841889
\(773\) 5.98583e9 0.466118 0.233059 0.972463i \(-0.425126\pi\)
0.233059 + 0.972463i \(0.425126\pi\)
\(774\) 7.47504e9 0.579455
\(775\) 9.24693e9 0.713578
\(776\) −1.33943e10 −1.02897
\(777\) 2.87186e9 0.219629
\(778\) 2.56276e10 1.95110
\(779\) 3.00629e10 2.27850
\(780\) −1.81807e10 −1.37176
\(781\) 8.21798e9 0.617286
\(782\) −6.76706e10 −5.06030
\(783\) 2.54183e9 0.189226
\(784\) −2.44576e10 −1.81262
\(785\) 1.80026e10 1.32828
\(786\) 5.28980e9 0.388562
\(787\) −4.85827e9 −0.355279 −0.177640 0.984096i \(-0.556846\pi\)
−0.177640 + 0.984096i \(0.556846\pi\)
\(788\) −9.77857e9 −0.711924
\(789\) 1.10363e10 0.799933
\(790\) 3.89880e9 0.281343
\(791\) 1.31194e10 0.942533
\(792\) 1.00344e10 0.717716
\(793\) 6.69032e9 0.476421
\(794\) 4.08603e10 2.89688
\(795\) −9.93352e9 −0.701162
\(796\) 4.59647e10 3.23019
\(797\) −7.28406e9 −0.509647 −0.254823 0.966988i \(-0.582017\pi\)
−0.254823 + 0.966988i \(0.582017\pi\)
\(798\) 2.03383e10 1.41679
\(799\) −1.23808e10 −0.858685
\(800\) 2.54183e10 1.75522
\(801\) 4.43154e9 0.304678
\(802\) 4.87925e10 3.33997
\(803\) 5.50846e9 0.375427
\(804\) −1.18212e10 −0.802171
\(805\) 2.47716e10 1.67366
\(806\) −3.30239e10 −2.22155
\(807\) 5.48516e7 0.00367394
\(808\) −9.26222e10 −6.17697
\(809\) 3.79381e9 0.251916 0.125958 0.992036i \(-0.459799\pi\)
0.125958 + 0.992036i \(0.459799\pi\)
\(810\) 3.91043e9 0.258539
\(811\) −4.79601e9 −0.315724 −0.157862 0.987461i \(-0.550460\pi\)
−0.157862 + 0.987461i \(0.550460\pi\)
\(812\) −2.96049e10 −1.94052
\(813\) 2.75440e9 0.179767
\(814\) −1.00569e10 −0.653547
\(815\) −2.83048e10 −1.83150
\(816\) 4.57415e10 2.94710
\(817\) −2.46833e10 −1.58353
\(818\) −2.26320e10 −1.44573
\(819\) 2.72054e9 0.173046
\(820\) −6.73222e10 −4.26392
\(821\) 1.01365e10 0.639272 0.319636 0.947540i \(-0.396439\pi\)
0.319636 + 0.947540i \(0.396439\pi\)
\(822\) −1.00530e10 −0.631313
\(823\) −9.27062e9 −0.579708 −0.289854 0.957071i \(-0.593607\pi\)
−0.289854 + 0.957071i \(0.593607\pi\)
\(824\) 8.73068e10 5.43629
\(825\) 2.66762e9 0.165400
\(826\) −2.93733e9 −0.181352
\(827\) −1.15153e10 −0.707953 −0.353977 0.935254i \(-0.615171\pi\)
−0.353977 + 0.935254i \(0.615171\pi\)
\(828\) −2.88094e10 −1.76371
\(829\) −7.01951e9 −0.427923 −0.213962 0.976842i \(-0.568637\pi\)
−0.213962 + 0.976842i \(0.568637\pi\)
\(830\) 5.54752e10 3.36764
\(831\) −4.87006e9 −0.294395
\(832\) −4.56731e10 −2.74934
\(833\) −1.08762e10 −0.651958
\(834\) −8.83571e9 −0.527425
\(835\) 2.11440e9 0.125685
\(836\) −5.21782e10 −3.08864
\(837\) 5.20376e9 0.306746
\(838\) −2.35908e10 −1.38481
\(839\) 2.55234e10 1.49201 0.746004 0.665942i \(-0.231970\pi\)
0.746004 + 0.665942i \(0.231970\pi\)
\(840\) −2.89223e10 −1.68366
\(841\) −5.73184e8 −0.0332283
\(842\) −4.13463e10 −2.38696
\(843\) 9.15654e9 0.526423
\(844\) 2.45704e10 1.40674
\(845\) −1.01413e10 −0.578222
\(846\) −7.19460e9 −0.408517
\(847\) 7.52225e9 0.425359
\(848\) −6.75224e10 −3.80244
\(849\) −3.31751e9 −0.186052
\(850\) 2.10045e10 1.17313
\(851\) 1.83357e10 1.01987
\(852\) 2.75478e10 1.52598
\(853\) 1.30151e10 0.718004 0.359002 0.933337i \(-0.383117\pi\)
0.359002 + 0.933337i \(0.383117\pi\)
\(854\) 1.67602e10 0.920822
\(855\) −1.29126e10 −0.706533
\(856\) 4.15045e8 0.0226171
\(857\) −7.46263e9 −0.405004 −0.202502 0.979282i \(-0.564907\pi\)
−0.202502 + 0.979282i \(0.564907\pi\)
\(858\) −9.52695e9 −0.514930
\(859\) 2.06753e9 0.111295 0.0556476 0.998450i \(-0.482278\pi\)
0.0556476 + 0.998450i \(0.482278\pi\)
\(860\) 5.52752e10 2.96337
\(861\) 1.00740e10 0.537888
\(862\) −4.90863e10 −2.61027
\(863\) −2.78352e10 −1.47420 −0.737099 0.675785i \(-0.763805\pi\)
−0.737099 + 0.675785i \(0.763805\pi\)
\(864\) 1.43043e10 0.754517
\(865\) −1.56335e10 −0.821295
\(866\) 3.45196e10 1.80615
\(867\) 9.26191e9 0.482651
\(868\) −6.06087e10 −3.14569
\(869\) 1.49676e9 0.0773716
\(870\) 2.56559e10 1.32090
\(871\) 7.12718e9 0.365472
\(872\) 7.73298e10 3.94947
\(873\) −2.00388e9 −0.101935
\(874\) 1.29852e11 6.57897
\(875\) 9.48560e9 0.478670
\(876\) 1.84651e10 0.928086
\(877\) −1.33724e10 −0.669436 −0.334718 0.942318i \(-0.608641\pi\)
−0.334718 + 0.942318i \(0.608641\pi\)
\(878\) 5.16366e10 2.57470
\(879\) −7.39584e9 −0.367305
\(880\) 5.86359e10 2.90051
\(881\) −3.66543e10 −1.80596 −0.902982 0.429678i \(-0.858627\pi\)
−0.902982 + 0.429678i \(0.858627\pi\)
\(882\) −6.32027e9 −0.310167
\(883\) 3.23252e10 1.58008 0.790040 0.613055i \(-0.210060\pi\)
0.790040 + 0.613055i \(0.210060\pi\)
\(884\) −5.49563e10 −2.67568
\(885\) 1.86489e9 0.0904382
\(886\) −1.07983e10 −0.521602
\(887\) 2.43296e10 1.17058 0.585292 0.810823i \(-0.300980\pi\)
0.585292 + 0.810823i \(0.300980\pi\)
\(888\) −2.14080e10 −1.02596
\(889\) −7.83217e9 −0.373875
\(890\) 4.47298e10 2.12682
\(891\) 1.50122e9 0.0711004
\(892\) 5.01245e10 2.36468
\(893\) 2.37572e10 1.11639
\(894\) −3.06825e10 −1.43618
\(895\) 3.79693e10 1.77032
\(896\) −5.36113e10 −2.48988
\(897\) 1.73695e10 0.803554
\(898\) −4.54410e10 −2.09402
\(899\) 3.41414e10 1.56719
\(900\) 8.94223e9 0.408881
\(901\) −3.00269e10 −1.36765
\(902\) −3.52778e10 −1.60059
\(903\) −8.27132e9 −0.373825
\(904\) −9.77973e10 −4.40289
\(905\) −3.51188e10 −1.57496
\(906\) −2.83160e9 −0.126498
\(907\) −1.00143e10 −0.445653 −0.222827 0.974858i \(-0.571528\pi\)
−0.222827 + 0.974858i \(0.571528\pi\)
\(908\) −4.08088e10 −1.80906
\(909\) −1.38570e10 −0.611920
\(910\) 2.74597e10 1.20796
\(911\) −1.43309e10 −0.627998 −0.313999 0.949423i \(-0.601669\pi\)
−0.313999 + 0.949423i \(0.601669\pi\)
\(912\) −8.77725e10 −3.83157
\(913\) 2.12970e10 0.926127
\(914\) −3.24447e10 −1.40550
\(915\) −1.06409e10 −0.459203
\(916\) 8.71689e10 3.74738
\(917\) −5.85331e9 −0.250674
\(918\) 1.18204e10 0.504293
\(919\) 5.16709e9 0.219605 0.109802 0.993953i \(-0.464978\pi\)
0.109802 + 0.993953i \(0.464978\pi\)
\(920\) −1.84657e11 −7.81825
\(921\) −6.78737e9 −0.286281
\(922\) 5.63774e10 2.36890
\(923\) −1.66089e10 −0.695242
\(924\) −1.74848e10 −0.729137
\(925\) −5.69126e9 −0.236435
\(926\) −9.93354e9 −0.411117
\(927\) 1.30617e10 0.538545
\(928\) 9.38492e10 3.85490
\(929\) −3.79974e10 −1.55489 −0.777444 0.628953i \(-0.783484\pi\)
−0.777444 + 0.628953i \(0.783484\pi\)
\(930\) 5.25242e10 2.14126
\(931\) 2.08701e10 0.847621
\(932\) −4.50073e10 −1.82107
\(933\) 9.36130e9 0.377355
\(934\) 7.94240e10 3.18961
\(935\) 2.60752e10 1.04325
\(936\) −2.02800e10 −0.808355
\(937\) −3.56133e10 −1.41424 −0.707121 0.707093i \(-0.750006\pi\)
−0.707121 + 0.707093i \(0.750006\pi\)
\(938\) 1.78546e10 0.706382
\(939\) 2.31955e10 0.914268
\(940\) −5.32015e10 −2.08918
\(941\) 1.98729e10 0.777495 0.388748 0.921344i \(-0.372908\pi\)
0.388748 + 0.921344i \(0.372908\pi\)
\(942\) 3.16228e10 1.23260
\(943\) 6.43186e10 2.49773
\(944\) 1.26764e10 0.490451
\(945\) −4.32699e9 −0.166792
\(946\) 2.89651e10 1.11239
\(947\) 1.94063e10 0.742537 0.371269 0.928525i \(-0.378923\pi\)
0.371269 + 0.928525i \(0.378923\pi\)
\(948\) 5.01734e9 0.191269
\(949\) −1.11329e10 −0.422839
\(950\) −4.03051e10 −1.52520
\(951\) 1.71805e10 0.647746
\(952\) −8.74260e10 −3.28406
\(953\) −2.57497e10 −0.963711 −0.481855 0.876251i \(-0.660037\pi\)
−0.481855 + 0.876251i \(0.660037\pi\)
\(954\) −1.74490e10 −0.650654
\(955\) −4.66891e10 −1.73462
\(956\) 5.34454e10 1.97837
\(957\) 9.84935e9 0.363258
\(958\) −1.98122e9 −0.0728036
\(959\) 1.11239e10 0.407279
\(960\) 7.26427e10 2.64998
\(961\) 4.23834e10 1.54051
\(962\) 2.03254e10 0.736082
\(963\) 6.20939e7 0.00224056
\(964\) −8.64657e10 −3.10867
\(965\) 1.03205e10 0.369705
\(966\) 4.35131e10 1.55310
\(967\) −4.00078e10 −1.42283 −0.711413 0.702774i \(-0.751945\pi\)
−0.711413 + 0.702774i \(0.751945\pi\)
\(968\) −5.60739e10 −1.98699
\(969\) −3.90321e10 −1.37812
\(970\) −2.02262e10 −0.711562
\(971\) 1.34901e10 0.472878 0.236439 0.971646i \(-0.424020\pi\)
0.236439 + 0.971646i \(0.424020\pi\)
\(972\) 5.03230e9 0.175766
\(973\) 9.77695e9 0.340258
\(974\) 5.46592e9 0.189542
\(975\) −5.39139e9 −0.186288
\(976\) −7.23307e10 −2.49028
\(977\) 4.28309e9 0.146935 0.0734676 0.997298i \(-0.476593\pi\)
0.0734676 + 0.997298i \(0.476593\pi\)
\(978\) −4.97194e10 −1.69957
\(979\) 1.71718e10 0.584893
\(980\) −4.67362e10 −1.58621
\(981\) 1.15691e10 0.391254
\(982\) 9.31773e9 0.313993
\(983\) −1.23005e10 −0.413034 −0.206517 0.978443i \(-0.566213\pi\)
−0.206517 + 0.978443i \(0.566213\pi\)
\(984\) −7.50958e10 −2.51265
\(985\) −9.37695e9 −0.312633
\(986\) 7.75525e10 2.57648
\(987\) 7.96101e9 0.263547
\(988\) 1.05455e11 3.47870
\(989\) −5.28091e10 −1.73589
\(990\) 1.51525e10 0.496321
\(991\) −2.97106e10 −0.969737 −0.484868 0.874587i \(-0.661133\pi\)
−0.484868 + 0.874587i \(0.661133\pi\)
\(992\) 1.92133e11 6.24901
\(993\) 1.10398e10 0.357800
\(994\) −4.16077e10 −1.34376
\(995\) 4.40768e10 1.41850
\(996\) 7.13905e10 2.28946
\(997\) 1.27080e10 0.406112 0.203056 0.979167i \(-0.434913\pi\)
0.203056 + 0.979167i \(0.434913\pi\)
\(998\) −9.13090e10 −2.90775
\(999\) −3.20279e9 −0.101636
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.1.18 18
3.2 odd 2 531.8.a.e.1.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.18 18 1.1 even 1 trivial
531.8.a.e.1.1 18 3.2 odd 2