Properties

Label 177.8.a.d.1.1
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 1.1
Root \(-20.9501\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.9501 q^{2} +27.0000 q^{3} +270.005 q^{4} +505.865 q^{5} -538.652 q^{6} +222.970 q^{7} -2833.01 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-19.9501 q^{2} +27.0000 q^{3} +270.005 q^{4} +505.865 q^{5} -538.652 q^{6} +222.970 q^{7} -2833.01 q^{8} +729.000 q^{9} -10092.0 q^{10} -1831.62 q^{11} +7290.13 q^{12} +688.848 q^{13} -4448.27 q^{14} +13658.4 q^{15} +21958.1 q^{16} +2843.94 q^{17} -14543.6 q^{18} +34416.2 q^{19} +136586. q^{20} +6020.19 q^{21} +36540.9 q^{22} -3272.29 q^{23} -76491.2 q^{24} +177775. q^{25} -13742.6 q^{26} +19683.0 q^{27} +60203.0 q^{28} +128507. q^{29} -272485. q^{30} -16509.0 q^{31} -75439.5 q^{32} -49453.7 q^{33} -56736.8 q^{34} +112793. q^{35} +196834. q^{36} -424491. q^{37} -686606. q^{38} +18598.9 q^{39} -1.43312e6 q^{40} -12500.6 q^{41} -120103. q^{42} +338393. q^{43} -494546. q^{44} +368776. q^{45} +65282.4 q^{46} +560303. q^{47} +592867. q^{48} -773827. q^{49} -3.54662e6 q^{50} +76786.4 q^{51} +185992. q^{52} +2.04079e6 q^{53} -392677. q^{54} -926551. q^{55} -631676. q^{56} +929239. q^{57} -2.56371e6 q^{58} +205379. q^{59} +3.68783e6 q^{60} -2.42610e6 q^{61} +329355. q^{62} +162545. q^{63} -1.30561e6 q^{64} +348464. q^{65} +986603. q^{66} -2.63298e6 q^{67} +767879. q^{68} -88351.8 q^{69} -2.25022e6 q^{70} +2.62379e6 q^{71} -2.06526e6 q^{72} +3.21761e6 q^{73} +8.46862e6 q^{74} +4.79992e6 q^{75} +9.29256e6 q^{76} -408396. q^{77} -371049. q^{78} +3.42920e6 q^{79} +1.11078e7 q^{80} +531441. q^{81} +249387. q^{82} +1.20365e6 q^{83} +1.62548e6 q^{84} +1.43865e6 q^{85} -6.75097e6 q^{86} +3.46968e6 q^{87} +5.18899e6 q^{88} -2.50843e6 q^{89} -7.35710e6 q^{90} +153592. q^{91} -883534. q^{92} -445743. q^{93} -1.11781e7 q^{94} +1.74100e7 q^{95} -2.03687e6 q^{96} +8.73888e6 q^{97} +1.54379e7 q^{98} -1.33525e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.9501 −1.76335 −0.881677 0.471854i \(-0.843585\pi\)
−0.881677 + 0.471854i \(0.843585\pi\)
\(3\) 27.0000 0.577350
\(4\) 270.005 2.10941
\(5\) 505.865 1.80984 0.904919 0.425583i \(-0.139931\pi\)
0.904919 + 0.425583i \(0.139931\pi\)
\(6\) −538.652 −1.01807
\(7\) 222.970 0.245699 0.122849 0.992425i \(-0.460797\pi\)
0.122849 + 0.992425i \(0.460797\pi\)
\(8\) −2833.01 −1.95629
\(9\) 729.000 0.333333
\(10\) −10092.0 −3.19138
\(11\) −1831.62 −0.414916 −0.207458 0.978244i \(-0.566519\pi\)
−0.207458 + 0.978244i \(0.566519\pi\)
\(12\) 7290.13 1.21787
\(13\) 688.848 0.0869604 0.0434802 0.999054i \(-0.486155\pi\)
0.0434802 + 0.999054i \(0.486155\pi\)
\(14\) −4448.27 −0.433254
\(15\) 13658.4 1.04491
\(16\) 21958.1 1.34021
\(17\) 2843.94 0.140394 0.0701972 0.997533i \(-0.477637\pi\)
0.0701972 + 0.997533i \(0.477637\pi\)
\(18\) −14543.6 −0.587784
\(19\) 34416.2 1.15113 0.575567 0.817755i \(-0.304781\pi\)
0.575567 + 0.817755i \(0.304781\pi\)
\(20\) 136586. 3.81770
\(21\) 6020.19 0.141854
\(22\) 36540.9 0.731643
\(23\) −3272.29 −0.0560795 −0.0280398 0.999607i \(-0.508927\pi\)
−0.0280398 + 0.999607i \(0.508927\pi\)
\(24\) −76491.2 −1.12946
\(25\) 177775. 2.27552
\(26\) −13742.6 −0.153342
\(27\) 19683.0 0.192450
\(28\) 60203.0 0.518281
\(29\) 128507. 0.978436 0.489218 0.872162i \(-0.337282\pi\)
0.489218 + 0.872162i \(0.337282\pi\)
\(30\) −272485. −1.84255
\(31\) −16509.0 −0.0995301 −0.0497651 0.998761i \(-0.515847\pi\)
−0.0497651 + 0.998761i \(0.515847\pi\)
\(32\) −75439.5 −0.406981
\(33\) −49453.7 −0.239552
\(34\) −56736.8 −0.247565
\(35\) 112793. 0.444675
\(36\) 196834. 0.703138
\(37\) −424491. −1.37772 −0.688862 0.724892i \(-0.741889\pi\)
−0.688862 + 0.724892i \(0.741889\pi\)
\(38\) −686606. −2.02986
\(39\) 18598.9 0.0502066
\(40\) −1.43312e6 −3.54057
\(41\) −12500.6 −0.0283261 −0.0141630 0.999900i \(-0.504508\pi\)
−0.0141630 + 0.999900i \(0.504508\pi\)
\(42\) −120103. −0.250139
\(43\) 338393. 0.649056 0.324528 0.945876i \(-0.394795\pi\)
0.324528 + 0.945876i \(0.394795\pi\)
\(44\) −494546. −0.875230
\(45\) 368776. 0.603280
\(46\) 65282.4 0.0988880
\(47\) 560303. 0.787192 0.393596 0.919284i \(-0.371231\pi\)
0.393596 + 0.919284i \(0.371231\pi\)
\(48\) 592867. 0.773772
\(49\) −773827. −0.939632
\(50\) −3.54662e6 −4.01254
\(51\) 76786.4 0.0810567
\(52\) 185992. 0.183436
\(53\) 2.04079e6 1.88293 0.941463 0.337116i \(-0.109452\pi\)
0.941463 + 0.337116i \(0.109452\pi\)
\(54\) −392677. −0.339357
\(55\) −926551. −0.750931
\(56\) −631676. −0.480658
\(57\) 929239. 0.664608
\(58\) −2.56371e6 −1.72533
\(59\) 205379. 0.130189
\(60\) 3.68783e6 2.20415
\(61\) −2.42610e6 −1.36853 −0.684265 0.729233i \(-0.739877\pi\)
−0.684265 + 0.729233i \(0.739877\pi\)
\(62\) 329355. 0.175507
\(63\) 162545. 0.0818997
\(64\) −1.30561e6 −0.622563
\(65\) 348464. 0.157384
\(66\) 986603. 0.422415
\(67\) −2.63298e6 −1.06951 −0.534757 0.845006i \(-0.679597\pi\)
−0.534757 + 0.845006i \(0.679597\pi\)
\(68\) 767879. 0.296150
\(69\) −88351.8 −0.0323775
\(70\) −2.25022e6 −0.784120
\(71\) 2.62379e6 0.870013 0.435006 0.900427i \(-0.356746\pi\)
0.435006 + 0.900427i \(0.356746\pi\)
\(72\) −2.06526e6 −0.652096
\(73\) 3.21761e6 0.968062 0.484031 0.875051i \(-0.339172\pi\)
0.484031 + 0.875051i \(0.339172\pi\)
\(74\) 8.46862e6 2.42941
\(75\) 4.79992e6 1.31377
\(76\) 9.29256e6 2.42822
\(77\) −408396. −0.101944
\(78\) −371049. −0.0885320
\(79\) 3.42920e6 0.782525 0.391263 0.920279i \(-0.372038\pi\)
0.391263 + 0.920279i \(0.372038\pi\)
\(80\) 1.11078e7 2.42557
\(81\) 531441. 0.111111
\(82\) 249387. 0.0499489
\(83\) 1.20365e6 0.231060 0.115530 0.993304i \(-0.463143\pi\)
0.115530 + 0.993304i \(0.463143\pi\)
\(84\) 1.62548e6 0.299230
\(85\) 1.43865e6 0.254091
\(86\) −6.75097e6 −1.14451
\(87\) 3.46968e6 0.564900
\(88\) 5.18899e6 0.811695
\(89\) −2.50843e6 −0.377170 −0.188585 0.982057i \(-0.560390\pi\)
−0.188585 + 0.982057i \(0.560390\pi\)
\(90\) −7.35710e6 −1.06379
\(91\) 153592. 0.0213661
\(92\) −883534. −0.118295
\(93\) −445743. −0.0574637
\(94\) −1.11781e7 −1.38810
\(95\) 1.74100e7 2.08337
\(96\) −2.03687e6 −0.234970
\(97\) 8.73888e6 0.972198 0.486099 0.873904i \(-0.338420\pi\)
0.486099 + 0.873904i \(0.338420\pi\)
\(98\) 1.54379e7 1.65690
\(99\) −1.33525e6 −0.138305
\(100\) 4.80000e7 4.80000
\(101\) −814288. −0.0786418 −0.0393209 0.999227i \(-0.512519\pi\)
−0.0393209 + 0.999227i \(0.512519\pi\)
\(102\) −1.53189e6 −0.142932
\(103\) −1.34032e7 −1.20859 −0.604296 0.796760i \(-0.706546\pi\)
−0.604296 + 0.796760i \(0.706546\pi\)
\(104\) −1.95151e6 −0.170120
\(105\) 3.04541e6 0.256734
\(106\) −4.07139e7 −3.32026
\(107\) −5.36199e6 −0.423139 −0.211569 0.977363i \(-0.567857\pi\)
−0.211569 + 0.977363i \(0.567857\pi\)
\(108\) 5.31451e6 0.405957
\(109\) 2.36393e6 0.174841 0.0874203 0.996172i \(-0.472138\pi\)
0.0874203 + 0.996172i \(0.472138\pi\)
\(110\) 1.84848e7 1.32416
\(111\) −1.14613e7 −0.795429
\(112\) 4.89599e6 0.329289
\(113\) 1.26518e7 0.824855 0.412428 0.910990i \(-0.364681\pi\)
0.412428 + 0.910990i \(0.364681\pi\)
\(114\) −1.85384e7 −1.17194
\(115\) −1.65534e6 −0.101495
\(116\) 3.46974e7 2.06393
\(117\) 502170. 0.0289868
\(118\) −4.09732e6 −0.229569
\(119\) 634114. 0.0344947
\(120\) −3.86943e7 −2.04415
\(121\) −1.61324e7 −0.827845
\(122\) 4.84009e7 2.41320
\(123\) −337516. −0.0163541
\(124\) −4.45751e6 −0.209950
\(125\) 5.04093e7 2.30848
\(126\) −3.24279e6 −0.144418
\(127\) −2.94888e7 −1.27745 −0.638726 0.769434i \(-0.720538\pi\)
−0.638726 + 0.769434i \(0.720538\pi\)
\(128\) 3.57032e7 1.50478
\(129\) 9.13662e6 0.374733
\(130\) −6.95188e6 −0.277524
\(131\) 2.80740e7 1.09107 0.545537 0.838087i \(-0.316326\pi\)
0.545537 + 0.838087i \(0.316326\pi\)
\(132\) −1.33527e7 −0.505314
\(133\) 7.67379e6 0.282832
\(134\) 5.25282e7 1.88593
\(135\) 9.95695e6 0.348304
\(136\) −8.05691e6 −0.274652
\(137\) −5.12815e7 −1.70388 −0.851939 0.523641i \(-0.824573\pi\)
−0.851939 + 0.523641i \(0.824573\pi\)
\(138\) 1.76262e6 0.0570930
\(139\) 3.88050e6 0.122556 0.0612782 0.998121i \(-0.480482\pi\)
0.0612782 + 0.998121i \(0.480482\pi\)
\(140\) 3.04546e7 0.938005
\(141\) 1.51282e7 0.454485
\(142\) −5.23449e7 −1.53414
\(143\) −1.26171e6 −0.0360813
\(144\) 1.60074e7 0.446738
\(145\) 6.50070e7 1.77081
\(146\) −6.41915e7 −1.70704
\(147\) −2.08933e7 −0.542497
\(148\) −1.14615e8 −2.90619
\(149\) −3.38987e6 −0.0839520 −0.0419760 0.999119i \(-0.513365\pi\)
−0.0419760 + 0.999119i \(0.513365\pi\)
\(150\) −9.57586e7 −2.31664
\(151\) 4.99988e7 1.18179 0.590895 0.806748i \(-0.298775\pi\)
0.590895 + 0.806748i \(0.298775\pi\)
\(152\) −9.75015e7 −2.25195
\(153\) 2.07323e6 0.0467981
\(154\) 8.14752e6 0.179764
\(155\) −8.35132e6 −0.180133
\(156\) 5.02179e6 0.105907
\(157\) −4.76455e7 −0.982591 −0.491295 0.870993i \(-0.663476\pi\)
−0.491295 + 0.870993i \(0.663476\pi\)
\(158\) −6.84128e7 −1.37987
\(159\) 5.51014e7 1.08711
\(160\) −3.81622e7 −0.736569
\(161\) −729622. −0.0137787
\(162\) −1.06023e7 −0.195928
\(163\) 7.64073e7 1.38191 0.690953 0.722900i \(-0.257191\pi\)
0.690953 + 0.722900i \(0.257191\pi\)
\(164\) −3.37522e6 −0.0597514
\(165\) −2.50169e7 −0.433550
\(166\) −2.40128e7 −0.407441
\(167\) −6.91463e6 −0.114885 −0.0574423 0.998349i \(-0.518295\pi\)
−0.0574423 + 0.998349i \(0.518295\pi\)
\(168\) −1.70552e7 −0.277508
\(169\) −6.22740e7 −0.992438
\(170\) −2.87012e7 −0.448052
\(171\) 2.50894e7 0.383711
\(172\) 9.13679e7 1.36913
\(173\) 9.54381e7 1.40139 0.700697 0.713459i \(-0.252872\pi\)
0.700697 + 0.713459i \(0.252872\pi\)
\(174\) −6.92203e7 −0.996118
\(175\) 3.96384e7 0.559092
\(176\) −4.02187e7 −0.556076
\(177\) 5.54523e6 0.0751646
\(178\) 5.00434e7 0.665085
\(179\) 3.26295e7 0.425231 0.212616 0.977136i \(-0.431802\pi\)
0.212616 + 0.977136i \(0.431802\pi\)
\(180\) 9.95713e7 1.27257
\(181\) −8.74090e7 −1.09567 −0.547837 0.836585i \(-0.684548\pi\)
−0.547837 + 0.836585i \(0.684548\pi\)
\(182\) −3.06418e6 −0.0376760
\(183\) −6.55048e7 −0.790122
\(184\) 9.27042e6 0.109708
\(185\) −2.14735e8 −2.49346
\(186\) 8.89259e6 0.101329
\(187\) −5.20901e6 −0.0582519
\(188\) 1.51285e8 1.66051
\(189\) 4.38872e6 0.0472848
\(190\) −3.47330e8 −3.67371
\(191\) 2.62452e7 0.272542 0.136271 0.990672i \(-0.456488\pi\)
0.136271 + 0.990672i \(0.456488\pi\)
\(192\) −3.52514e7 −0.359437
\(193\) 3.00614e7 0.300995 0.150497 0.988610i \(-0.451912\pi\)
0.150497 + 0.988610i \(0.451912\pi\)
\(194\) −1.74341e8 −1.71433
\(195\) 9.40853e6 0.0908659
\(196\) −2.08937e8 −1.98207
\(197\) −6.36038e7 −0.592722 −0.296361 0.955076i \(-0.595773\pi\)
−0.296361 + 0.955076i \(0.595773\pi\)
\(198\) 2.66383e7 0.243881
\(199\) 1.34497e8 1.20984 0.604919 0.796287i \(-0.293205\pi\)
0.604919 + 0.796287i \(0.293205\pi\)
\(200\) −5.03637e8 −4.45157
\(201\) −7.10906e7 −0.617484
\(202\) 1.62451e7 0.138673
\(203\) 2.86531e7 0.240401
\(204\) 2.07327e7 0.170982
\(205\) −6.32361e6 −0.0512656
\(206\) 2.67396e8 2.13117
\(207\) −2.38550e6 −0.0186932
\(208\) 1.51258e7 0.116545
\(209\) −6.30374e7 −0.477624
\(210\) −6.07560e7 −0.452712
\(211\) 1.72469e8 1.26393 0.631965 0.774997i \(-0.282249\pi\)
0.631965 + 0.774997i \(0.282249\pi\)
\(212\) 5.51024e8 3.97187
\(213\) 7.08425e7 0.502302
\(214\) 1.06972e8 0.746143
\(215\) 1.71181e8 1.17469
\(216\) −5.57621e7 −0.376488
\(217\) −3.68101e6 −0.0244544
\(218\) −4.71606e7 −0.308306
\(219\) 8.68754e7 0.558911
\(220\) −2.50173e8 −1.58402
\(221\) 1.95904e6 0.0122088
\(222\) 2.28653e8 1.40262
\(223\) 1.98662e8 1.19963 0.599815 0.800138i \(-0.295241\pi\)
0.599815 + 0.800138i \(0.295241\pi\)
\(224\) −1.68207e7 −0.0999947
\(225\) 1.29598e8 0.758505
\(226\) −2.52404e8 −1.45451
\(227\) −1.97341e8 −1.11977 −0.559884 0.828571i \(-0.689154\pi\)
−0.559884 + 0.828571i \(0.689154\pi\)
\(228\) 2.50899e8 1.40193
\(229\) 2.77784e8 1.52856 0.764281 0.644883i \(-0.223094\pi\)
0.764281 + 0.644883i \(0.223094\pi\)
\(230\) 3.30241e7 0.178971
\(231\) −1.10267e7 −0.0588576
\(232\) −3.64060e8 −1.91410
\(233\) 3.29662e8 1.70735 0.853677 0.520803i \(-0.174367\pi\)
0.853677 + 0.520803i \(0.174367\pi\)
\(234\) −1.00183e7 −0.0511140
\(235\) 2.83438e8 1.42469
\(236\) 5.54534e7 0.274622
\(237\) 9.25885e7 0.451791
\(238\) −1.26506e7 −0.0608264
\(239\) 3.52106e8 1.66833 0.834163 0.551518i \(-0.185951\pi\)
0.834163 + 0.551518i \(0.185951\pi\)
\(240\) 2.99911e8 1.40040
\(241\) −1.57129e8 −0.723097 −0.361549 0.932353i \(-0.617752\pi\)
−0.361549 + 0.932353i \(0.617752\pi\)
\(242\) 3.21841e8 1.45978
\(243\) 1.43489e7 0.0641500
\(244\) −6.55060e8 −2.88680
\(245\) −3.91452e8 −1.70058
\(246\) 6.73346e6 0.0288380
\(247\) 2.37076e7 0.100103
\(248\) 4.67701e7 0.194710
\(249\) 3.24985e7 0.133403
\(250\) −1.00567e9 −4.07066
\(251\) −2.31577e8 −0.924353 −0.462177 0.886788i \(-0.652931\pi\)
−0.462177 + 0.886788i \(0.652931\pi\)
\(252\) 4.38880e7 0.172760
\(253\) 5.99358e6 0.0232683
\(254\) 5.88304e8 2.25260
\(255\) 3.88436e7 0.146700
\(256\) −5.45164e8 −2.03089
\(257\) 3.21230e8 1.18046 0.590228 0.807237i \(-0.299038\pi\)
0.590228 + 0.807237i \(0.299038\pi\)
\(258\) −1.82276e8 −0.660786
\(259\) −9.46487e7 −0.338505
\(260\) 9.40871e7 0.331989
\(261\) 9.36813e7 0.326145
\(262\) −5.60078e8 −1.92395
\(263\) 3.68323e8 1.24849 0.624243 0.781230i \(-0.285408\pi\)
0.624243 + 0.781230i \(0.285408\pi\)
\(264\) 1.40103e8 0.468633
\(265\) 1.03237e9 3.40779
\(266\) −1.53093e8 −0.498733
\(267\) −6.77277e7 −0.217759
\(268\) −7.10919e8 −2.25605
\(269\) −2.17400e8 −0.680967 −0.340484 0.940250i \(-0.610591\pi\)
−0.340484 + 0.940250i \(0.610591\pi\)
\(270\) −1.98642e8 −0.614182
\(271\) 1.69294e8 0.516713 0.258356 0.966050i \(-0.416819\pi\)
0.258356 + 0.966050i \(0.416819\pi\)
\(272\) 6.24474e7 0.188158
\(273\) 4.14700e6 0.0123357
\(274\) 1.02307e9 3.00454
\(275\) −3.25615e8 −0.944148
\(276\) −2.38554e7 −0.0682976
\(277\) 5.18960e6 0.0146708 0.00733541 0.999973i \(-0.497665\pi\)
0.00733541 + 0.999973i \(0.497665\pi\)
\(278\) −7.74162e7 −0.216110
\(279\) −1.20351e7 −0.0331767
\(280\) −3.19543e8 −0.869913
\(281\) −4.92517e8 −1.32419 −0.662093 0.749421i \(-0.730332\pi\)
−0.662093 + 0.749421i \(0.730332\pi\)
\(282\) −3.01808e8 −0.801418
\(283\) −5.84814e8 −1.53379 −0.766893 0.641775i \(-0.778198\pi\)
−0.766893 + 0.641775i \(0.778198\pi\)
\(284\) 7.08438e8 1.83522
\(285\) 4.70070e8 1.20283
\(286\) 2.51711e7 0.0636240
\(287\) −2.78725e6 −0.00695969
\(288\) −5.49954e7 −0.135660
\(289\) −4.02251e8 −0.980289
\(290\) −1.29689e9 −3.12256
\(291\) 2.35950e8 0.561299
\(292\) 8.68770e8 2.04204
\(293\) 6.94020e8 1.61189 0.805945 0.591990i \(-0.201658\pi\)
0.805945 + 0.591990i \(0.201658\pi\)
\(294\) 4.16823e8 0.956613
\(295\) 1.03894e8 0.235621
\(296\) 1.20259e9 2.69523
\(297\) −3.60517e7 −0.0798506
\(298\) 6.76281e7 0.148037
\(299\) −2.25411e6 −0.00487670
\(300\) 1.29600e9 2.77128
\(301\) 7.54515e7 0.159472
\(302\) −9.97480e8 −2.08391
\(303\) −2.19858e7 −0.0454038
\(304\) 7.55714e8 1.54276
\(305\) −1.22728e9 −2.47682
\(306\) −4.13611e7 −0.0825216
\(307\) 6.33917e8 1.25040 0.625199 0.780465i \(-0.285018\pi\)
0.625199 + 0.780465i \(0.285018\pi\)
\(308\) −1.10269e8 −0.215043
\(309\) −3.61888e8 −0.697781
\(310\) 1.66609e8 0.317639
\(311\) −2.89601e8 −0.545932 −0.272966 0.962024i \(-0.588005\pi\)
−0.272966 + 0.962024i \(0.588005\pi\)
\(312\) −5.26908e7 −0.0982186
\(313\) 5.90653e8 1.08875 0.544374 0.838843i \(-0.316767\pi\)
0.544374 + 0.838843i \(0.316767\pi\)
\(314\) 9.50530e8 1.73265
\(315\) 8.22259e7 0.148225
\(316\) 9.25902e8 1.65067
\(317\) −9.69791e8 −1.70990 −0.854950 0.518710i \(-0.826413\pi\)
−0.854950 + 0.518710i \(0.826413\pi\)
\(318\) −1.09928e9 −1.91695
\(319\) −2.35375e8 −0.405969
\(320\) −6.60462e8 −1.12674
\(321\) −1.44774e8 −0.244299
\(322\) 1.45560e7 0.0242967
\(323\) 9.78778e7 0.161613
\(324\) 1.43492e8 0.234379
\(325\) 1.22460e8 0.197880
\(326\) −1.52433e9 −2.43679
\(327\) 6.38262e7 0.100944
\(328\) 3.54142e7 0.0554140
\(329\) 1.24931e8 0.193412
\(330\) 4.99088e8 0.764502
\(331\) −1.23241e9 −1.86792 −0.933959 0.357381i \(-0.883670\pi\)
−0.933959 + 0.357381i \(0.883670\pi\)
\(332\) 3.24991e8 0.487402
\(333\) −3.09454e8 −0.459241
\(334\) 1.37947e8 0.202582
\(335\) −1.33194e9 −1.93565
\(336\) 1.32192e8 0.190115
\(337\) 3.81254e8 0.542638 0.271319 0.962490i \(-0.412540\pi\)
0.271319 + 0.962490i \(0.412540\pi\)
\(338\) 1.24237e9 1.75002
\(339\) 3.41599e8 0.476230
\(340\) 3.88443e8 0.535983
\(341\) 3.02381e7 0.0412966
\(342\) −5.00536e8 −0.676619
\(343\) −3.56166e8 −0.476566
\(344\) −9.58671e8 −1.26974
\(345\) −4.46941e7 −0.0585981
\(346\) −1.90400e9 −2.47115
\(347\) −2.23058e7 −0.0286592 −0.0143296 0.999897i \(-0.504561\pi\)
−0.0143296 + 0.999897i \(0.504561\pi\)
\(348\) 9.36830e8 1.19161
\(349\) −1.22256e9 −1.53951 −0.769755 0.638340i \(-0.779622\pi\)
−0.769755 + 0.638340i \(0.779622\pi\)
\(350\) −7.90789e8 −0.985876
\(351\) 1.35586e7 0.0167355
\(352\) 1.38176e8 0.168863
\(353\) −1.53975e9 −1.86311 −0.931557 0.363595i \(-0.881549\pi\)
−0.931557 + 0.363595i \(0.881549\pi\)
\(354\) −1.10628e8 −0.132542
\(355\) 1.32729e9 1.57458
\(356\) −6.77290e8 −0.795609
\(357\) 1.71211e7 0.0199155
\(358\) −6.50961e8 −0.749832
\(359\) −6.36027e7 −0.0725513 −0.0362757 0.999342i \(-0.511549\pi\)
−0.0362757 + 0.999342i \(0.511549\pi\)
\(360\) −1.04474e9 −1.18019
\(361\) 2.90606e8 0.325110
\(362\) 1.74381e9 1.93206
\(363\) −4.35573e8 −0.477956
\(364\) 4.14707e7 0.0450699
\(365\) 1.62768e9 1.75204
\(366\) 1.30682e9 1.39326
\(367\) 2.43871e8 0.257531 0.128765 0.991675i \(-0.458899\pi\)
0.128765 + 0.991675i \(0.458899\pi\)
\(368\) −7.18531e7 −0.0751585
\(369\) −9.11292e6 −0.00944203
\(370\) 4.28398e9 4.39685
\(371\) 4.55035e8 0.462633
\(372\) −1.20353e8 −0.121215
\(373\) −5.36181e8 −0.534972 −0.267486 0.963562i \(-0.586193\pi\)
−0.267486 + 0.963562i \(0.586193\pi\)
\(374\) 1.03920e8 0.102719
\(375\) 1.36105e9 1.33280
\(376\) −1.58734e9 −1.53997
\(377\) 8.85215e7 0.0850852
\(378\) −8.75552e7 −0.0833798
\(379\) −9.43440e8 −0.890178 −0.445089 0.895486i \(-0.646828\pi\)
−0.445089 + 0.895486i \(0.646828\pi\)
\(380\) 4.70078e9 4.39468
\(381\) −7.96198e8 −0.737537
\(382\) −5.23593e8 −0.480587
\(383\) 7.47419e8 0.679780 0.339890 0.940465i \(-0.389610\pi\)
0.339890 + 0.940465i \(0.389610\pi\)
\(384\) 9.63987e8 0.868784
\(385\) −2.06593e8 −0.184503
\(386\) −5.99727e8 −0.530760
\(387\) 2.46689e8 0.216352
\(388\) 2.35954e9 2.05077
\(389\) 5.07334e8 0.436989 0.218494 0.975838i \(-0.429885\pi\)
0.218494 + 0.975838i \(0.429885\pi\)
\(390\) −1.87701e8 −0.160229
\(391\) −9.30620e6 −0.00787324
\(392\) 2.19226e9 1.83819
\(393\) 7.57997e8 0.629932
\(394\) 1.26890e9 1.04518
\(395\) 1.73471e9 1.41624
\(396\) −3.60524e8 −0.291743
\(397\) 3.30439e8 0.265048 0.132524 0.991180i \(-0.457692\pi\)
0.132524 + 0.991180i \(0.457692\pi\)
\(398\) −2.68323e9 −2.13337
\(399\) 2.07192e8 0.163293
\(400\) 3.90359e9 3.04968
\(401\) −8.76898e8 −0.679116 −0.339558 0.940585i \(-0.610277\pi\)
−0.339558 + 0.940585i \(0.610277\pi\)
\(402\) 1.41826e9 1.08884
\(403\) −1.13722e7 −0.00865518
\(404\) −2.19862e8 −0.165888
\(405\) 2.68838e8 0.201093
\(406\) −5.71631e8 −0.423911
\(407\) 7.77505e8 0.571640
\(408\) −2.17537e8 −0.158570
\(409\) 1.20367e9 0.869915 0.434958 0.900451i \(-0.356763\pi\)
0.434958 + 0.900451i \(0.356763\pi\)
\(410\) 1.26156e8 0.0903994
\(411\) −1.38460e9 −0.983734
\(412\) −3.61894e9 −2.54942
\(413\) 4.57934e7 0.0319873
\(414\) 4.75908e7 0.0329627
\(415\) 6.08883e8 0.418182
\(416\) −5.19663e7 −0.0353912
\(417\) 1.04774e8 0.0707580
\(418\) 1.25760e9 0.842220
\(419\) 1.30342e9 0.865638 0.432819 0.901481i \(-0.357519\pi\)
0.432819 + 0.901481i \(0.357519\pi\)
\(420\) 8.22275e8 0.541557
\(421\) −2.37970e9 −1.55430 −0.777150 0.629315i \(-0.783336\pi\)
−0.777150 + 0.629315i \(0.783336\pi\)
\(422\) −3.44077e9 −2.22875
\(423\) 4.08461e8 0.262397
\(424\) −5.78158e9 −3.68355
\(425\) 5.05581e8 0.319470
\(426\) −1.41331e9 −0.885736
\(427\) −5.40948e8 −0.336247
\(428\) −1.44776e9 −0.892574
\(429\) −3.40660e7 −0.0208315
\(430\) −3.41508e9 −2.07139
\(431\) 1.38543e9 0.833515 0.416758 0.909018i \(-0.363166\pi\)
0.416758 + 0.909018i \(0.363166\pi\)
\(432\) 4.32200e8 0.257924
\(433\) 2.85632e9 1.69082 0.845412 0.534114i \(-0.179355\pi\)
0.845412 + 0.534114i \(0.179355\pi\)
\(434\) 7.34364e7 0.0431218
\(435\) 1.75519e9 1.02238
\(436\) 6.38274e8 0.368811
\(437\) −1.12620e8 −0.0645550
\(438\) −1.73317e9 −0.985557
\(439\) −3.08827e9 −1.74217 −0.871083 0.491137i \(-0.836582\pi\)
−0.871083 + 0.491137i \(0.836582\pi\)
\(440\) 2.62493e9 1.46904
\(441\) −5.64120e8 −0.313211
\(442\) −3.90830e7 −0.0215283
\(443\) 2.10701e9 1.15147 0.575737 0.817635i \(-0.304715\pi\)
0.575737 + 0.817635i \(0.304715\pi\)
\(444\) −3.09460e9 −1.67789
\(445\) −1.26893e9 −0.682618
\(446\) −3.96332e9 −2.11537
\(447\) −9.15264e7 −0.0484697
\(448\) −2.91112e8 −0.152963
\(449\) −5.27169e8 −0.274845 −0.137422 0.990513i \(-0.543882\pi\)
−0.137422 + 0.990513i \(0.543882\pi\)
\(450\) −2.58548e9 −1.33751
\(451\) 2.28963e7 0.0117529
\(452\) 3.41605e9 1.73996
\(453\) 1.34997e9 0.682307
\(454\) 3.93697e9 1.97454
\(455\) 7.76971e7 0.0386692
\(456\) −2.63254e9 −1.30016
\(457\) 1.47995e9 0.725337 0.362668 0.931918i \(-0.381866\pi\)
0.362668 + 0.931918i \(0.381866\pi\)
\(458\) −5.54181e9 −2.69540
\(459\) 5.59773e7 0.0270189
\(460\) −4.46949e8 −0.214095
\(461\) −3.37266e9 −1.60332 −0.801658 0.597783i \(-0.796048\pi\)
−0.801658 + 0.597783i \(0.796048\pi\)
\(462\) 2.19983e8 0.103787
\(463\) −2.58669e9 −1.21119 −0.605593 0.795774i \(-0.707064\pi\)
−0.605593 + 0.795774i \(0.707064\pi\)
\(464\) 2.82175e9 1.31131
\(465\) −2.25486e8 −0.104000
\(466\) −6.57678e9 −3.01067
\(467\) −7.12728e8 −0.323828 −0.161914 0.986805i \(-0.551767\pi\)
−0.161914 + 0.986805i \(0.551767\pi\)
\(468\) 1.35588e8 0.0611452
\(469\) −5.87077e8 −0.262778
\(470\) −5.65460e9 −2.51223
\(471\) −1.28643e9 −0.567299
\(472\) −5.81840e8 −0.254687
\(473\) −6.19807e8 −0.269304
\(474\) −1.84715e9 −0.796667
\(475\) 6.11834e9 2.61942
\(476\) 1.71214e8 0.0727637
\(477\) 1.48774e9 0.627642
\(478\) −7.02454e9 −2.94185
\(479\) −1.28719e9 −0.535140 −0.267570 0.963538i \(-0.586221\pi\)
−0.267570 + 0.963538i \(0.586221\pi\)
\(480\) −1.03038e9 −0.425258
\(481\) −2.92410e8 −0.119807
\(482\) 3.13473e9 1.27508
\(483\) −1.96998e7 −0.00795512
\(484\) −4.35582e9 −1.74627
\(485\) 4.42069e9 1.75952
\(486\) −2.86262e8 −0.113119
\(487\) −4.87502e8 −0.191260 −0.0956301 0.995417i \(-0.530487\pi\)
−0.0956301 + 0.995417i \(0.530487\pi\)
\(488\) 6.87317e9 2.67724
\(489\) 2.06300e9 0.797843
\(490\) 7.80950e9 2.99873
\(491\) −5.36881e8 −0.204688 −0.102344 0.994749i \(-0.532634\pi\)
−0.102344 + 0.994749i \(0.532634\pi\)
\(492\) −9.11309e7 −0.0344975
\(493\) 3.65465e8 0.137367
\(494\) −4.72967e8 −0.176517
\(495\) −6.75456e8 −0.250310
\(496\) −3.62505e8 −0.133392
\(497\) 5.85028e8 0.213761
\(498\) −6.48346e8 −0.235236
\(499\) −4.34284e8 −0.156467 −0.0782334 0.996935i \(-0.524928\pi\)
−0.0782334 + 0.996935i \(0.524928\pi\)
\(500\) 1.36108e10 4.86954
\(501\) −1.86695e8 −0.0663286
\(502\) 4.61998e9 1.62996
\(503\) −3.59458e9 −1.25939 −0.629696 0.776842i \(-0.716820\pi\)
−0.629696 + 0.776842i \(0.716820\pi\)
\(504\) −4.60492e8 −0.160219
\(505\) −4.11920e8 −0.142329
\(506\) −1.19572e8 −0.0410302
\(507\) −1.68140e9 −0.572984
\(508\) −7.96213e9 −2.69467
\(509\) −3.79038e9 −1.27400 −0.637002 0.770862i \(-0.719826\pi\)
−0.637002 + 0.770862i \(0.719826\pi\)
\(510\) −7.74932e8 −0.258683
\(511\) 7.17430e8 0.237852
\(512\) 6.30604e9 2.07640
\(513\) 6.77415e8 0.221536
\(514\) −6.40855e9 −2.08156
\(515\) −6.78024e9 −2.18736
\(516\) 2.46693e9 0.790466
\(517\) −1.02626e9 −0.326619
\(518\) 1.88825e9 0.596905
\(519\) 2.57683e9 0.809096
\(520\) −9.87202e8 −0.307889
\(521\) 2.93658e9 0.909724 0.454862 0.890562i \(-0.349688\pi\)
0.454862 + 0.890562i \(0.349688\pi\)
\(522\) −1.86895e9 −0.575109
\(523\) 2.13933e8 0.0653917 0.0326959 0.999465i \(-0.489591\pi\)
0.0326959 + 0.999465i \(0.489591\pi\)
\(524\) 7.58011e9 2.30153
\(525\) 1.07024e9 0.322792
\(526\) −7.34806e9 −2.20152
\(527\) −4.69506e7 −0.0139735
\(528\) −1.08591e9 −0.321051
\(529\) −3.39412e9 −0.996855
\(530\) −2.05958e10 −6.00914
\(531\) 1.49721e8 0.0433963
\(532\) 2.07196e9 0.596611
\(533\) −8.61100e6 −0.00246325
\(534\) 1.35117e9 0.383987
\(535\) −2.71244e9 −0.765813
\(536\) 7.45927e9 2.09228
\(537\) 8.80997e8 0.245507
\(538\) 4.33714e9 1.20079
\(539\) 1.41736e9 0.389868
\(540\) 2.68843e9 0.734716
\(541\) 2.06559e9 0.560860 0.280430 0.959874i \(-0.409523\pi\)
0.280430 + 0.959874i \(0.409523\pi\)
\(542\) −3.37743e9 −0.911147
\(543\) −2.36004e9 −0.632587
\(544\) −2.14545e8 −0.0571378
\(545\) 1.19583e9 0.316433
\(546\) −8.27328e7 −0.0217522
\(547\) 2.12658e9 0.555555 0.277777 0.960645i \(-0.410402\pi\)
0.277777 + 0.960645i \(0.410402\pi\)
\(548\) −1.38463e10 −3.59418
\(549\) −1.76863e9 −0.456177
\(550\) 6.49604e9 1.66487
\(551\) 4.42271e9 1.12631
\(552\) 2.50301e8 0.0633398
\(553\) 7.64609e8 0.192266
\(554\) −1.03533e8 −0.0258698
\(555\) −5.79785e9 −1.43960
\(556\) 1.04775e9 0.258522
\(557\) 3.52559e9 0.864448 0.432224 0.901766i \(-0.357729\pi\)
0.432224 + 0.901766i \(0.357729\pi\)
\(558\) 2.40100e8 0.0585022
\(559\) 2.33101e8 0.0564422
\(560\) 2.47671e9 0.595960
\(561\) −1.40643e8 −0.0336317
\(562\) 9.82575e9 2.33501
\(563\) 3.62920e9 0.857100 0.428550 0.903518i \(-0.359025\pi\)
0.428550 + 0.903518i \(0.359025\pi\)
\(564\) 4.08469e9 0.958698
\(565\) 6.40011e9 1.49285
\(566\) 1.16671e10 2.70461
\(567\) 1.18495e8 0.0272999
\(568\) −7.43323e9 −1.70200
\(569\) −1.16789e9 −0.265772 −0.132886 0.991131i \(-0.542424\pi\)
−0.132886 + 0.991131i \(0.542424\pi\)
\(570\) −9.37792e9 −2.12102
\(571\) 4.02884e8 0.0905635 0.0452818 0.998974i \(-0.485581\pi\)
0.0452818 + 0.998974i \(0.485581\pi\)
\(572\) −3.40667e8 −0.0761103
\(573\) 7.08620e8 0.157352
\(574\) 5.56059e7 0.0122724
\(575\) −5.81730e8 −0.127610
\(576\) −9.51789e8 −0.207521
\(577\) 3.29893e9 0.714921 0.357461 0.933928i \(-0.383643\pi\)
0.357461 + 0.933928i \(0.383643\pi\)
\(578\) 8.02493e9 1.72860
\(579\) 8.11658e8 0.173779
\(580\) 1.75522e10 3.73537
\(581\) 2.68377e8 0.0567713
\(582\) −4.70721e9 −0.989768
\(583\) −3.73795e9 −0.781256
\(584\) −9.11551e9 −1.89381
\(585\) 2.54030e8 0.0524614
\(586\) −1.38457e10 −2.84233
\(587\) 4.64184e9 0.947233 0.473617 0.880731i \(-0.342948\pi\)
0.473617 + 0.880731i \(0.342948\pi\)
\(588\) −5.64131e9 −1.14435
\(589\) −5.68177e8 −0.114572
\(590\) −2.07269e9 −0.415483
\(591\) −1.71730e9 −0.342208
\(592\) −9.32099e9 −1.84644
\(593\) −4.01516e7 −0.00790699 −0.00395349 0.999992i \(-0.501258\pi\)
−0.00395349 + 0.999992i \(0.501258\pi\)
\(594\) 7.19234e8 0.140805
\(595\) 3.20776e8 0.0624299
\(596\) −9.15281e8 −0.177089
\(597\) 3.63142e9 0.698500
\(598\) 4.49696e7 0.00859934
\(599\) 6.08823e9 1.15744 0.578719 0.815527i \(-0.303553\pi\)
0.578719 + 0.815527i \(0.303553\pi\)
\(600\) −1.35982e10 −2.57011
\(601\) 8.96979e9 1.68547 0.842736 0.538327i \(-0.180944\pi\)
0.842736 + 0.538327i \(0.180944\pi\)
\(602\) −1.50526e9 −0.281206
\(603\) −1.91945e9 −0.356505
\(604\) 1.34999e10 2.49289
\(605\) −8.16080e9 −1.49827
\(606\) 4.38618e8 0.0800630
\(607\) 2.86102e9 0.519230 0.259615 0.965712i \(-0.416404\pi\)
0.259615 + 0.965712i \(0.416404\pi\)
\(608\) −2.59634e9 −0.468489
\(609\) 7.73634e8 0.138795
\(610\) 2.44843e10 4.36751
\(611\) 3.85964e8 0.0684545
\(612\) 5.59784e8 0.0987166
\(613\) −2.84576e9 −0.498984 −0.249492 0.968377i \(-0.580264\pi\)
−0.249492 + 0.968377i \(0.580264\pi\)
\(614\) −1.26467e10 −2.20489
\(615\) −1.70737e8 −0.0295982
\(616\) 1.15699e9 0.199433
\(617\) −9.53360e9 −1.63403 −0.817013 0.576619i \(-0.804372\pi\)
−0.817013 + 0.576619i \(0.804372\pi\)
\(618\) 7.21968e9 1.23043
\(619\) 4.09983e9 0.694783 0.347391 0.937720i \(-0.387068\pi\)
0.347391 + 0.937720i \(0.387068\pi\)
\(620\) −2.25490e9 −0.379976
\(621\) −6.44085e7 −0.0107925
\(622\) 5.77756e9 0.962672
\(623\) −5.59306e8 −0.0926704
\(624\) 4.08395e8 0.0672876
\(625\) 1.16117e10 1.90246
\(626\) −1.17836e10 −1.91985
\(627\) −1.70201e9 −0.275756
\(628\) −1.28645e10 −2.07269
\(629\) −1.20723e9 −0.193425
\(630\) −1.64041e9 −0.261373
\(631\) −1.39182e9 −0.220536 −0.110268 0.993902i \(-0.535171\pi\)
−0.110268 + 0.993902i \(0.535171\pi\)
\(632\) −9.71496e9 −1.53085
\(633\) 4.65667e9 0.729730
\(634\) 1.93474e10 3.01516
\(635\) −1.49174e10 −2.31198
\(636\) 1.48777e10 2.29316
\(637\) −5.33049e8 −0.0817108
\(638\) 4.69574e9 0.715866
\(639\) 1.91275e9 0.290004
\(640\) 1.80610e10 2.72341
\(641\) −5.13220e9 −0.769663 −0.384832 0.922987i \(-0.625740\pi\)
−0.384832 + 0.922987i \(0.625740\pi\)
\(642\) 2.88824e9 0.430786
\(643\) −3.56886e9 −0.529408 −0.264704 0.964330i \(-0.585274\pi\)
−0.264704 + 0.964330i \(0.585274\pi\)
\(644\) −1.97002e8 −0.0290649
\(645\) 4.62190e9 0.678205
\(646\) −1.95267e9 −0.284980
\(647\) −7.81292e9 −1.13409 −0.567046 0.823686i \(-0.691914\pi\)
−0.567046 + 0.823686i \(0.691914\pi\)
\(648\) −1.50558e9 −0.217365
\(649\) −3.76176e8 −0.0540175
\(650\) −2.44308e9 −0.348932
\(651\) −9.93872e7 −0.0141188
\(652\) 2.06304e10 2.91501
\(653\) −8.76103e9 −1.23129 −0.615643 0.788025i \(-0.711104\pi\)
−0.615643 + 0.788025i \(0.711104\pi\)
\(654\) −1.27334e9 −0.178000
\(655\) 1.42016e10 1.97467
\(656\) −2.74488e8 −0.0379630
\(657\) 2.34564e9 0.322687
\(658\) −2.49238e9 −0.341054
\(659\) −1.26819e10 −1.72618 −0.863089 0.505052i \(-0.831473\pi\)
−0.863089 + 0.505052i \(0.831473\pi\)
\(660\) −6.75468e9 −0.914537
\(661\) −1.96990e9 −0.265301 −0.132651 0.991163i \(-0.542349\pi\)
−0.132651 + 0.991163i \(0.542349\pi\)
\(662\) 2.45867e10 3.29380
\(663\) 5.28942e7 0.00704873
\(664\) −3.40994e9 −0.452021
\(665\) 3.88190e9 0.511881
\(666\) 6.17362e9 0.809805
\(667\) −4.20511e8 −0.0548702
\(668\) −1.86699e9 −0.242339
\(669\) 5.36387e9 0.692607
\(670\) 2.65722e10 3.41323
\(671\) 4.44369e9 0.567825
\(672\) −4.54160e8 −0.0577320
\(673\) 8.38158e9 1.05992 0.529960 0.848023i \(-0.322207\pi\)
0.529960 + 0.848023i \(0.322207\pi\)
\(674\) −7.60605e9 −0.956862
\(675\) 3.49914e9 0.437923
\(676\) −1.68143e10 −2.09346
\(677\) 8.10465e9 1.00386 0.501931 0.864908i \(-0.332623\pi\)
0.501931 + 0.864908i \(0.332623\pi\)
\(678\) −6.81491e9 −0.839762
\(679\) 1.94851e9 0.238868
\(680\) −4.07571e9 −0.497075
\(681\) −5.32822e9 −0.646498
\(682\) −6.03253e8 −0.0728206
\(683\) 8.52346e9 1.02363 0.511816 0.859095i \(-0.328973\pi\)
0.511816 + 0.859095i \(0.328973\pi\)
\(684\) 6.77427e9 0.809406
\(685\) −2.59415e10 −3.08374
\(686\) 7.10553e9 0.840353
\(687\) 7.50017e9 0.882516
\(688\) 7.43046e9 0.869873
\(689\) 1.40580e9 0.163740
\(690\) 8.91650e8 0.103329
\(691\) −6.80331e9 −0.784417 −0.392208 0.919876i \(-0.628289\pi\)
−0.392208 + 0.919876i \(0.628289\pi\)
\(692\) 2.57688e10 2.95612
\(693\) −2.97720e8 −0.0339815
\(694\) 4.45001e8 0.0505362
\(695\) 1.96301e9 0.221807
\(696\) −9.82962e9 −1.10511
\(697\) −3.55509e7 −0.00397682
\(698\) 2.43902e10 2.71470
\(699\) 8.90088e9 0.985741
\(700\) 1.07026e10 1.17936
\(701\) −1.06363e9 −0.116621 −0.0583106 0.998298i \(-0.518571\pi\)
−0.0583106 + 0.998298i \(0.518571\pi\)
\(702\) −2.70495e8 −0.0295107
\(703\) −1.46094e10 −1.58595
\(704\) 2.39138e9 0.258311
\(705\) 7.65282e9 0.822545
\(706\) 3.07182e10 3.28533
\(707\) −1.81562e8 −0.0193222
\(708\) 1.49724e9 0.158553
\(709\) 1.77993e10 1.87561 0.937803 0.347168i \(-0.112857\pi\)
0.937803 + 0.347168i \(0.112857\pi\)
\(710\) −2.64794e10 −2.77655
\(711\) 2.49989e9 0.260842
\(712\) 7.10641e9 0.737854
\(713\) 5.40222e7 0.00558160
\(714\) −3.41567e8 −0.0351181
\(715\) −6.38253e8 −0.0653013
\(716\) 8.81013e9 0.896988
\(717\) 9.50686e9 0.963208
\(718\) 1.26888e9 0.127934
\(719\) −1.95303e10 −1.95956 −0.979780 0.200078i \(-0.935880\pi\)
−0.979780 + 0.200078i \(0.935880\pi\)
\(720\) 8.09760e9 0.808523
\(721\) −2.98852e9 −0.296950
\(722\) −5.79761e9 −0.573283
\(723\) −4.24248e9 −0.417480
\(724\) −2.36009e10 −2.31123
\(725\) 2.28452e10 2.22645
\(726\) 8.68972e9 0.842806
\(727\) −1.51589e10 −1.46317 −0.731587 0.681748i \(-0.761220\pi\)
−0.731587 + 0.681748i \(0.761220\pi\)
\(728\) −4.35129e8 −0.0417982
\(729\) 3.87420e8 0.0370370
\(730\) −3.24723e10 −3.08946
\(731\) 9.62371e8 0.0911238
\(732\) −1.76866e10 −1.66669
\(733\) 1.19810e7 0.00112365 0.000561823 1.00000i \(-0.499821\pi\)
0.000561823 1.00000i \(0.499821\pi\)
\(734\) −4.86524e9 −0.454118
\(735\) −1.05692e10 −0.981832
\(736\) 2.46860e8 0.0228233
\(737\) 4.82262e9 0.443758
\(738\) 1.81803e8 0.0166496
\(739\) −1.13380e10 −1.03343 −0.516715 0.856158i \(-0.672845\pi\)
−0.516715 + 0.856158i \(0.672845\pi\)
\(740\) −5.79796e10 −5.25974
\(741\) 6.40104e8 0.0577946
\(742\) −9.07799e9 −0.815785
\(743\) −2.12227e10 −1.89819 −0.949096 0.314987i \(-0.898000\pi\)
−0.949096 + 0.314987i \(0.898000\pi\)
\(744\) 1.26279e9 0.112416
\(745\) −1.71482e9 −0.151939
\(746\) 1.06969e10 0.943344
\(747\) 8.77458e8 0.0770202
\(748\) −1.40646e9 −0.122877
\(749\) −1.19556e9 −0.103965
\(750\) −2.71531e10 −2.35020
\(751\) −2.06229e10 −1.77669 −0.888343 0.459181i \(-0.848143\pi\)
−0.888343 + 0.459181i \(0.848143\pi\)
\(752\) 1.23032e10 1.05500
\(753\) −6.25259e9 −0.533676
\(754\) −1.76601e9 −0.150035
\(755\) 2.52927e10 2.13885
\(756\) 1.18498e9 0.0997432
\(757\) −2.12910e10 −1.78386 −0.891929 0.452175i \(-0.850648\pi\)
−0.891929 + 0.452175i \(0.850648\pi\)
\(758\) 1.88217e10 1.56970
\(759\) 1.61827e8 0.0134340
\(760\) −4.93226e10 −4.07567
\(761\) −4.78297e9 −0.393415 −0.196708 0.980462i \(-0.563025\pi\)
−0.196708 + 0.980462i \(0.563025\pi\)
\(762\) 1.58842e10 1.30054
\(763\) 5.27086e8 0.0429582
\(764\) 7.08633e9 0.574903
\(765\) 1.04878e9 0.0846970
\(766\) −1.49111e10 −1.19869
\(767\) 1.41475e8 0.0113213
\(768\) −1.47194e10 −1.17254
\(769\) −7.38239e9 −0.585403 −0.292702 0.956204i \(-0.594554\pi\)
−0.292702 + 0.956204i \(0.594554\pi\)
\(770\) 4.12155e9 0.325344
\(771\) 8.67320e9 0.681536
\(772\) 8.11673e9 0.634922
\(773\) −1.10993e10 −0.864305 −0.432153 0.901801i \(-0.642246\pi\)
−0.432153 + 0.901801i \(0.642246\pi\)
\(774\) −4.92145e9 −0.381505
\(775\) −2.93488e9 −0.226482
\(776\) −2.47573e10 −1.90190
\(777\) −2.55552e9 −0.195436
\(778\) −1.01213e10 −0.770566
\(779\) −4.30223e8 −0.0326071
\(780\) 2.54035e9 0.191674
\(781\) −4.80579e9 −0.360982
\(782\) 1.85659e8 0.0138833
\(783\) 2.52939e9 0.188300
\(784\) −1.69917e10 −1.25931
\(785\) −2.41022e10 −1.77833
\(786\) −1.51221e10 −1.11079
\(787\) 2.48619e10 1.81812 0.909060 0.416666i \(-0.136801\pi\)
0.909060 + 0.416666i \(0.136801\pi\)
\(788\) −1.71733e10 −1.25030
\(789\) 9.94472e9 0.720813
\(790\) −3.46077e10 −2.49734
\(791\) 2.82097e9 0.202666
\(792\) 3.78277e9 0.270565
\(793\) −1.67122e9 −0.119008
\(794\) −6.59229e9 −0.467374
\(795\) 2.78739e10 1.96749
\(796\) 3.63149e10 2.55205
\(797\) −9.78058e9 −0.684322 −0.342161 0.939641i \(-0.611159\pi\)
−0.342161 + 0.939641i \(0.611159\pi\)
\(798\) −4.13350e9 −0.287944
\(799\) 1.59347e9 0.110517
\(800\) −1.34112e10 −0.926091
\(801\) −1.82865e9 −0.125723
\(802\) 1.74942e10 1.19752
\(803\) −5.89343e9 −0.401665
\(804\) −1.91948e10 −1.30253
\(805\) −3.69091e8 −0.0249372
\(806\) 2.26876e8 0.0152621
\(807\) −5.86980e9 −0.393157
\(808\) 2.30688e9 0.153846
\(809\) −4.39665e9 −0.291945 −0.145973 0.989289i \(-0.546631\pi\)
−0.145973 + 0.989289i \(0.546631\pi\)
\(810\) −5.36333e9 −0.354598
\(811\) −9.64279e9 −0.634790 −0.317395 0.948293i \(-0.602808\pi\)
−0.317395 + 0.948293i \(0.602808\pi\)
\(812\) 7.73648e9 0.507104
\(813\) 4.57094e9 0.298324
\(814\) −1.55113e10 −1.00800
\(815\) 3.86518e10 2.50103
\(816\) 1.68608e9 0.108633
\(817\) 1.16462e10 0.747150
\(818\) −2.40133e10 −1.53397
\(819\) 1.11969e8 0.00712203
\(820\) −1.70741e9 −0.108140
\(821\) −1.47265e10 −0.928750 −0.464375 0.885639i \(-0.653721\pi\)
−0.464375 + 0.885639i \(0.653721\pi\)
\(822\) 2.76229e10 1.73467
\(823\) −1.30056e10 −0.813264 −0.406632 0.913592i \(-0.633297\pi\)
−0.406632 + 0.913592i \(0.633297\pi\)
\(824\) 3.79715e10 2.36435
\(825\) −8.79161e9 −0.545104
\(826\) −9.13580e8 −0.0564049
\(827\) 2.38555e10 1.46662 0.733312 0.679892i \(-0.237974\pi\)
0.733312 + 0.679892i \(0.237974\pi\)
\(828\) −6.44096e8 −0.0394316
\(829\) 2.02291e10 1.23321 0.616605 0.787273i \(-0.288508\pi\)
0.616605 + 0.787273i \(0.288508\pi\)
\(830\) −1.21473e10 −0.737403
\(831\) 1.40119e8 0.00847020
\(832\) −8.99366e8 −0.0541383
\(833\) −2.20072e9 −0.131919
\(834\) −2.09024e9 −0.124771
\(835\) −3.49787e9 −0.207922
\(836\) −1.70204e10 −1.00751
\(837\) −3.24946e8 −0.0191546
\(838\) −2.60034e10 −1.52643
\(839\) −1.07547e10 −0.628684 −0.314342 0.949310i \(-0.601784\pi\)
−0.314342 + 0.949310i \(0.601784\pi\)
\(840\) −8.62766e9 −0.502245
\(841\) −7.35941e8 −0.0426635
\(842\) 4.74752e10 2.74078
\(843\) −1.32980e10 −0.764519
\(844\) 4.65675e10 2.66615
\(845\) −3.15023e10 −1.79615
\(846\) −8.14882e9 −0.462699
\(847\) −3.59703e9 −0.203401
\(848\) 4.48118e10 2.52352
\(849\) −1.57900e10 −0.885532
\(850\) −1.00864e10 −0.563338
\(851\) 1.38906e9 0.0772621
\(852\) 1.91278e10 1.05956
\(853\) 3.60150e10 1.98683 0.993417 0.114556i \(-0.0365445\pi\)
0.993417 + 0.114556i \(0.0365445\pi\)
\(854\) 1.07919e10 0.592922
\(855\) 1.26919e10 0.694456
\(856\) 1.51906e10 0.827781
\(857\) −7.43401e9 −0.403450 −0.201725 0.979442i \(-0.564655\pi\)
−0.201725 + 0.979442i \(0.564655\pi\)
\(858\) 6.79620e8 0.0367333
\(859\) −2.19350e10 −1.18076 −0.590380 0.807125i \(-0.701022\pi\)
−0.590380 + 0.807125i \(0.701022\pi\)
\(860\) 4.62198e10 2.47790
\(861\) −7.52558e7 −0.00401818
\(862\) −2.76394e10 −1.46978
\(863\) 2.36338e10 1.25169 0.625843 0.779949i \(-0.284755\pi\)
0.625843 + 0.779949i \(0.284755\pi\)
\(864\) −1.48487e9 −0.0783234
\(865\) 4.82788e10 2.53630
\(866\) −5.69837e10 −2.98152
\(867\) −1.08608e10 −0.565970
\(868\) −9.93891e8 −0.0515845
\(869\) −6.28099e9 −0.324682
\(870\) −3.50161e10 −1.80281
\(871\) −1.81373e9 −0.0930054
\(872\) −6.69704e9 −0.342039
\(873\) 6.37064e9 0.324066
\(874\) 2.24677e9 0.113833
\(875\) 1.12398e10 0.567191
\(876\) 2.34568e10 1.17897
\(877\) −6.10352e9 −0.305550 −0.152775 0.988261i \(-0.548821\pi\)
−0.152775 + 0.988261i \(0.548821\pi\)
\(878\) 6.16112e10 3.07205
\(879\) 1.87386e10 0.930625
\(880\) −2.03453e10 −1.00641
\(881\) 7.45828e9 0.367471 0.183735 0.982976i \(-0.441181\pi\)
0.183735 + 0.982976i \(0.441181\pi\)
\(882\) 1.12542e10 0.552301
\(883\) 2.44182e10 1.19358 0.596789 0.802398i \(-0.296443\pi\)
0.596789 + 0.802398i \(0.296443\pi\)
\(884\) 5.28952e8 0.0257533
\(885\) 2.80514e9 0.136036
\(886\) −4.20350e10 −2.03046
\(887\) −1.25593e10 −0.604272 −0.302136 0.953265i \(-0.597700\pi\)
−0.302136 + 0.953265i \(0.597700\pi\)
\(888\) 3.24698e10 1.55609
\(889\) −6.57512e9 −0.313869
\(890\) 2.53152e10 1.20370
\(891\) −9.73396e8 −0.0461018
\(892\) 5.36397e10 2.53052
\(893\) 1.92835e10 0.906163
\(894\) 1.82596e9 0.0854692
\(895\) 1.65061e10 0.769600
\(896\) 7.96075e9 0.369723
\(897\) −6.08609e7 −0.00281556
\(898\) 1.05170e10 0.484648
\(899\) −2.12151e9 −0.0973838
\(900\) 3.49920e10 1.60000
\(901\) 5.80390e9 0.264352
\(902\) −4.56782e8 −0.0207246
\(903\) 2.03719e9 0.0920714
\(904\) −3.58426e10 −1.61365
\(905\) −4.42172e10 −1.98299
\(906\) −2.69320e10 −1.20315
\(907\) −1.35865e10 −0.604618 −0.302309 0.953210i \(-0.597758\pi\)
−0.302309 + 0.953210i \(0.597758\pi\)
\(908\) −5.32832e10 −2.36205
\(909\) −5.93616e8 −0.0262139
\(910\) −1.55006e9 −0.0681874
\(911\) 2.70365e9 0.118478 0.0592389 0.998244i \(-0.481133\pi\)
0.0592389 + 0.998244i \(0.481133\pi\)
\(912\) 2.04043e10 0.890716
\(913\) −2.20462e9 −0.0958707
\(914\) −2.95251e10 −1.27903
\(915\) −3.31366e10 −1.42999
\(916\) 7.50031e10 3.22437
\(917\) 6.25965e9 0.268076
\(918\) −1.11675e9 −0.0476439
\(919\) −1.61178e10 −0.685019 −0.342509 0.939514i \(-0.611277\pi\)
−0.342509 + 0.939514i \(0.611277\pi\)
\(920\) 4.68958e9 0.198553
\(921\) 1.71158e10 0.721918
\(922\) 6.72848e10 2.82721
\(923\) 1.80740e9 0.0756567
\(924\) −2.97726e9 −0.124155
\(925\) −7.54637e10 −3.13503
\(926\) 5.16046e10 2.13575
\(927\) −9.77097e9 −0.402864
\(928\) −9.69446e9 −0.398204
\(929\) −4.59174e10 −1.87898 −0.939491 0.342573i \(-0.888702\pi\)
−0.939491 + 0.342573i \(0.888702\pi\)
\(930\) 4.49845e9 0.183389
\(931\) −2.66322e10 −1.08164
\(932\) 8.90104e10 3.60152
\(933\) −7.81923e9 −0.315194
\(934\) 1.42190e10 0.571023
\(935\) −2.63506e9 −0.105426
\(936\) −1.42265e9 −0.0567066
\(937\) −2.98124e10 −1.18388 −0.591942 0.805981i \(-0.701638\pi\)
−0.591942 + 0.805981i \(0.701638\pi\)
\(938\) 1.17122e10 0.463371
\(939\) 1.59476e10 0.628589
\(940\) 7.65297e10 3.00526
\(941\) −3.77260e10 −1.47597 −0.737983 0.674819i \(-0.764222\pi\)
−0.737983 + 0.674819i \(0.764222\pi\)
\(942\) 2.56643e10 1.00035
\(943\) 4.09055e7 0.00158851
\(944\) 4.50972e9 0.174481
\(945\) 2.22010e9 0.0855778
\(946\) 1.23652e10 0.474877
\(947\) 2.08865e10 0.799173 0.399587 0.916695i \(-0.369154\pi\)
0.399587 + 0.916695i \(0.369154\pi\)
\(948\) 2.49993e10 0.953015
\(949\) 2.21644e9 0.0841831
\(950\) −1.22061e11 −4.61897
\(951\) −2.61844e10 −0.987212
\(952\) −1.79645e9 −0.0674817
\(953\) 3.06639e9 0.114763 0.0573815 0.998352i \(-0.481725\pi\)
0.0573815 + 0.998352i \(0.481725\pi\)
\(954\) −2.96805e10 −1.10675
\(955\) 1.32765e10 0.493256
\(956\) 9.50704e10 3.51919
\(957\) −6.35512e9 −0.234386
\(958\) 2.56795e10 0.943641
\(959\) −1.14342e10 −0.418641
\(960\) −1.78325e10 −0.650523
\(961\) −2.72401e10 −0.990094
\(962\) 5.83359e9 0.211263
\(963\) −3.90889e9 −0.141046
\(964\) −4.24256e10 −1.52531
\(965\) 1.52070e10 0.544752
\(966\) 3.93012e8 0.0140277
\(967\) 3.45519e10 1.22880 0.614398 0.788997i \(-0.289399\pi\)
0.614398 + 0.788997i \(0.289399\pi\)
\(968\) 4.57031e10 1.61950
\(969\) 2.64270e9 0.0933071
\(970\) −8.81931e10 −3.10266
\(971\) 3.51892e10 1.23351 0.616754 0.787156i \(-0.288447\pi\)
0.616754 + 0.787156i \(0.288447\pi\)
\(972\) 3.87428e9 0.135319
\(973\) 8.65235e8 0.0301120
\(974\) 9.72569e9 0.337259
\(975\) 3.30641e9 0.114246
\(976\) −5.32725e10 −1.83412
\(977\) −6.56102e9 −0.225082 −0.112541 0.993647i \(-0.535899\pi\)
−0.112541 + 0.993647i \(0.535899\pi\)
\(978\) −4.11569e10 −1.40688
\(979\) 4.59449e9 0.156494
\(980\) −1.05694e11 −3.58723
\(981\) 1.72331e9 0.0582802
\(982\) 1.07108e10 0.360938
\(983\) −1.57970e9 −0.0530442 −0.0265221 0.999648i \(-0.508443\pi\)
−0.0265221 + 0.999648i \(0.508443\pi\)
\(984\) 9.56184e8 0.0319933
\(985\) −3.21749e10 −1.07273
\(986\) −7.29105e9 −0.242226
\(987\) 3.37313e9 0.111667
\(988\) 6.40116e9 0.211159
\(989\) −1.10732e9 −0.0363987
\(990\) 1.34754e10 0.441386
\(991\) −5.75725e10 −1.87913 −0.939566 0.342369i \(-0.888771\pi\)
−0.939566 + 0.342369i \(0.888771\pi\)
\(992\) 1.24543e9 0.0405068
\(993\) −3.32751e10 −1.07844
\(994\) −1.16713e10 −0.376937
\(995\) 6.80374e10 2.18961
\(996\) 8.77475e9 0.281402
\(997\) −3.86631e10 −1.23556 −0.617780 0.786351i \(-0.711968\pi\)
−0.617780 + 0.786351i \(0.711968\pi\)
\(998\) 8.66400e9 0.275906
\(999\) −8.35525e9 −0.265143
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.1.1 18
3.2 odd 2 531.8.a.e.1.18 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.1 18 1.1 even 1 trivial
531.8.a.e.1.18 18 3.2 odd 2