Properties

Label 177.8.a.d.1.4
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 1.4
Root \(-16.0158\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-15.0158 q^{2} +27.0000 q^{3} +97.4757 q^{4} +6.55078 q^{5} -405.428 q^{6} +1466.16 q^{7} +458.348 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-15.0158 q^{2} +27.0000 q^{3} +97.4757 q^{4} +6.55078 q^{5} -405.428 q^{6} +1466.16 q^{7} +458.348 q^{8} +729.000 q^{9} -98.3656 q^{10} +4599.84 q^{11} +2631.84 q^{12} +15226.3 q^{13} -22015.6 q^{14} +176.871 q^{15} -19359.4 q^{16} +38992.1 q^{17} -10946.6 q^{18} -499.547 q^{19} +638.542 q^{20} +39586.2 q^{21} -69070.4 q^{22} +58259.9 q^{23} +12375.4 q^{24} -78082.1 q^{25} -228636. q^{26} +19683.0 q^{27} +142915. q^{28} +211255. q^{29} -2655.87 q^{30} +61846.8 q^{31} +232029. q^{32} +124196. q^{33} -585500. q^{34} +9604.48 q^{35} +71059.8 q^{36} -62940.4 q^{37} +7501.12 q^{38} +411110. q^{39} +3002.54 q^{40} -619876. q^{41} -594421. q^{42} -356192. q^{43} +448372. q^{44} +4775.52 q^{45} -874822. q^{46} -953515. q^{47} -522703. q^{48} +1.32607e6 q^{49} +1.17247e6 q^{50} +1.05279e6 q^{51} +1.48419e6 q^{52} -1.97760e6 q^{53} -295557. q^{54} +30132.5 q^{55} +672010. q^{56} -13487.8 q^{57} -3.17218e6 q^{58} +205379. q^{59} +17240.6 q^{60} -2.98506e6 q^{61} -928683. q^{62} +1.06883e6 q^{63} -1.00611e6 q^{64} +99744.2 q^{65} -1.86490e6 q^{66} -1.04862e6 q^{67} +3.80078e6 q^{68} +1.57302e6 q^{69} -144219. q^{70} +4.10180e6 q^{71} +334136. q^{72} -260342. q^{73} +945104. q^{74} -2.10822e6 q^{75} -48693.7 q^{76} +6.74408e6 q^{77} -6.17317e6 q^{78} -1.33466e6 q^{79} -126819. q^{80} +531441. q^{81} +9.30797e6 q^{82} +8.30823e6 q^{83} +3.85870e6 q^{84} +255429. q^{85} +5.34852e6 q^{86} +5.70389e6 q^{87} +2.10832e6 q^{88} +3.49331e6 q^{89} -71708.5 q^{90} +2.23242e7 q^{91} +5.67893e6 q^{92} +1.66986e6 q^{93} +1.43178e7 q^{94} -3272.42 q^{95} +6.26478e6 q^{96} -1.00211e7 q^{97} -1.99121e7 q^{98} +3.35328e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.0158 −1.32723 −0.663613 0.748076i \(-0.730978\pi\)
−0.663613 + 0.748076i \(0.730978\pi\)
\(3\) 27.0000 0.577350
\(4\) 97.4757 0.761529
\(5\) 6.55078 0.0234368 0.0117184 0.999931i \(-0.496270\pi\)
0.0117184 + 0.999931i \(0.496270\pi\)
\(6\) −405.428 −0.766274
\(7\) 1466.16 1.61561 0.807807 0.589448i \(-0.200655\pi\)
0.807807 + 0.589448i \(0.200655\pi\)
\(8\) 458.348 0.316505
\(9\) 729.000 0.333333
\(10\) −98.3656 −0.0311059
\(11\) 4599.84 1.04200 0.521000 0.853556i \(-0.325559\pi\)
0.521000 + 0.853556i \(0.325559\pi\)
\(12\) 2631.84 0.439669
\(13\) 15226.3 1.92217 0.961087 0.276246i \(-0.0890903\pi\)
0.961087 + 0.276246i \(0.0890903\pi\)
\(14\) −22015.6 −2.14428
\(15\) 176.871 0.0135312
\(16\) −19359.4 −1.18160
\(17\) 38992.1 1.92489 0.962444 0.271481i \(-0.0875133\pi\)
0.962444 + 0.271481i \(0.0875133\pi\)
\(18\) −10946.6 −0.442409
\(19\) −499.547 −0.0167085 −0.00835427 0.999965i \(-0.502659\pi\)
−0.00835427 + 0.999965i \(0.502659\pi\)
\(20\) 638.542 0.0178478
\(21\) 39586.2 0.932775
\(22\) −69070.4 −1.38297
\(23\) 58259.9 0.998441 0.499220 0.866475i \(-0.333620\pi\)
0.499220 + 0.866475i \(0.333620\pi\)
\(24\) 12375.4 0.182734
\(25\) −78082.1 −0.999451
\(26\) −228636. −2.55116
\(27\) 19683.0 0.192450
\(28\) 142915. 1.23034
\(29\) 211255. 1.60848 0.804238 0.594307i \(-0.202574\pi\)
0.804238 + 0.594307i \(0.202574\pi\)
\(30\) −2655.87 −0.0179590
\(31\) 61846.8 0.372865 0.186432 0.982468i \(-0.440307\pi\)
0.186432 + 0.982468i \(0.440307\pi\)
\(32\) 232029. 1.25175
\(33\) 124196. 0.601599
\(34\) −585500. −2.55476
\(35\) 9604.48 0.0378648
\(36\) 71059.8 0.253843
\(37\) −62940.4 −0.204279 −0.102139 0.994770i \(-0.532569\pi\)
−0.102139 + 0.994770i \(0.532569\pi\)
\(38\) 7501.12 0.0221760
\(39\) 411110. 1.10977
\(40\) 3002.54 0.00741785
\(41\) −619876. −1.40463 −0.702314 0.711867i \(-0.747850\pi\)
−0.702314 + 0.711867i \(0.747850\pi\)
\(42\) −594421. −1.23800
\(43\) −356192. −0.683194 −0.341597 0.939846i \(-0.610968\pi\)
−0.341597 + 0.939846i \(0.610968\pi\)
\(44\) 448372. 0.793514
\(45\) 4775.52 0.00781226
\(46\) −874822. −1.32516
\(47\) −953515. −1.33963 −0.669815 0.742528i \(-0.733627\pi\)
−0.669815 + 0.742528i \(0.733627\pi\)
\(48\) −522703. −0.682199
\(49\) 1.32607e6 1.61021
\(50\) 1.17247e6 1.32650
\(51\) 1.05279e6 1.11133
\(52\) 1.48419e6 1.46379
\(53\) −1.97760e6 −1.82462 −0.912310 0.409501i \(-0.865703\pi\)
−0.912310 + 0.409501i \(0.865703\pi\)
\(54\) −295557. −0.255425
\(55\) 30132.5 0.0244211
\(56\) 672010. 0.511349
\(57\) −13487.8 −0.00964668
\(58\) −3.17218e6 −2.13481
\(59\) 205379. 0.130189
\(60\) 17240.6 0.0103044
\(61\) −2.98506e6 −1.68383 −0.841917 0.539607i \(-0.818573\pi\)
−0.841917 + 0.539607i \(0.818573\pi\)
\(62\) −928683. −0.494876
\(63\) 1.06883e6 0.538538
\(64\) −1.00611e6 −0.479751
\(65\) 99744.2 0.0450496
\(66\) −1.86490e6 −0.798458
\(67\) −1.04862e6 −0.425948 −0.212974 0.977058i \(-0.568315\pi\)
−0.212974 + 0.977058i \(0.568315\pi\)
\(68\) 3.80078e6 1.46586
\(69\) 1.57302e6 0.576450
\(70\) −144219. −0.0502551
\(71\) 4.10180e6 1.36010 0.680049 0.733166i \(-0.261958\pi\)
0.680049 + 0.733166i \(0.261958\pi\)
\(72\) 334136. 0.105502
\(73\) −260342. −0.0783273 −0.0391637 0.999233i \(-0.512469\pi\)
−0.0391637 + 0.999233i \(0.512469\pi\)
\(74\) 945104. 0.271124
\(75\) −2.10822e6 −0.577033
\(76\) −48693.7 −0.0127240
\(77\) 6.74408e6 1.68347
\(78\) −6.17317e6 −1.47291
\(79\) −1.33466e6 −0.304562 −0.152281 0.988337i \(-0.548662\pi\)
−0.152281 + 0.988337i \(0.548662\pi\)
\(80\) −126819. −0.0276930
\(81\) 531441. 0.111111
\(82\) 9.30797e6 1.86426
\(83\) 8.30823e6 1.59491 0.797454 0.603380i \(-0.206180\pi\)
0.797454 + 0.603380i \(0.206180\pi\)
\(84\) 3.85870e6 0.710335
\(85\) 255429. 0.0451132
\(86\) 5.34852e6 0.906753
\(87\) 5.70389e6 0.928654
\(88\) 2.10832e6 0.329798
\(89\) 3.49331e6 0.525257 0.262628 0.964897i \(-0.415411\pi\)
0.262628 + 0.964897i \(0.415411\pi\)
\(90\) −71708.5 −0.0103686
\(91\) 2.23242e7 3.10549
\(92\) 5.67893e6 0.760342
\(93\) 1.66986e6 0.215274
\(94\) 1.43178e7 1.77799
\(95\) −3272.42 −0.000391594 0
\(96\) 6.26478e6 0.722698
\(97\) −1.00211e7 −1.11484 −0.557420 0.830230i \(-0.688209\pi\)
−0.557420 + 0.830230i \(0.688209\pi\)
\(98\) −1.99121e7 −2.13711
\(99\) 3.35328e6 0.347334
\(100\) −7.61111e6 −0.761111
\(101\) 9.44535e6 0.912206 0.456103 0.889927i \(-0.349245\pi\)
0.456103 + 0.889927i \(0.349245\pi\)
\(102\) −1.58085e7 −1.47499
\(103\) −4.07761e6 −0.367684 −0.183842 0.982956i \(-0.558854\pi\)
−0.183842 + 0.982956i \(0.558854\pi\)
\(104\) 6.97894e6 0.608377
\(105\) 259321. 0.0218612
\(106\) 2.96953e7 2.42168
\(107\) −2.17048e7 −1.71282 −0.856410 0.516296i \(-0.827310\pi\)
−0.856410 + 0.516296i \(0.827310\pi\)
\(108\) 1.91861e6 0.146556
\(109\) −4.43849e6 −0.328278 −0.164139 0.986437i \(-0.552485\pi\)
−0.164139 + 0.986437i \(0.552485\pi\)
\(110\) −452465. −0.0324124
\(111\) −1.69939e6 −0.117941
\(112\) −2.83839e7 −1.90901
\(113\) 7.13404e6 0.465116 0.232558 0.972583i \(-0.425290\pi\)
0.232558 + 0.972583i \(0.425290\pi\)
\(114\) 202530. 0.0128033
\(115\) 381648. 0.0234002
\(116\) 2.05923e7 1.22490
\(117\) 1.11000e7 0.640725
\(118\) −3.08394e6 −0.172790
\(119\) 5.71686e7 3.10987
\(120\) 81068.5 0.00428270
\(121\) 1.67132e6 0.0857652
\(122\) 4.48233e7 2.23483
\(123\) −1.67367e7 −0.810963
\(124\) 6.02857e6 0.283948
\(125\) −1.02328e6 −0.0468607
\(126\) −1.60494e7 −0.714761
\(127\) −1.23488e7 −0.534948 −0.267474 0.963565i \(-0.586189\pi\)
−0.267474 + 0.963565i \(0.586189\pi\)
\(128\) −1.45921e7 −0.615010
\(129\) −9.61718e6 −0.394442
\(130\) −1.49774e6 −0.0597910
\(131\) −1.76969e7 −0.687779 −0.343889 0.939010i \(-0.611744\pi\)
−0.343889 + 0.939010i \(0.611744\pi\)
\(132\) 1.21061e7 0.458135
\(133\) −732414. −0.0269945
\(134\) 1.57459e7 0.565329
\(135\) 128939. 0.00451041
\(136\) 1.78719e7 0.609236
\(137\) −2.34535e6 −0.0779265 −0.0389632 0.999241i \(-0.512406\pi\)
−0.0389632 + 0.999241i \(0.512406\pi\)
\(138\) −2.36202e7 −0.765079
\(139\) −1.84954e7 −0.584134 −0.292067 0.956398i \(-0.594343\pi\)
−0.292067 + 0.956398i \(0.594343\pi\)
\(140\) 936203. 0.0288351
\(141\) −2.57449e7 −0.773436
\(142\) −6.15920e7 −1.80516
\(143\) 7.00385e7 2.00291
\(144\) −1.41130e7 −0.393868
\(145\) 1.38389e6 0.0376975
\(146\) 3.90925e6 0.103958
\(147\) 3.58040e7 0.929653
\(148\) −6.13516e6 −0.155564
\(149\) −1.87066e7 −0.463278 −0.231639 0.972802i \(-0.574409\pi\)
−0.231639 + 0.972802i \(0.574409\pi\)
\(150\) 3.16567e7 0.765853
\(151\) −3.23257e7 −0.764062 −0.382031 0.924150i \(-0.624775\pi\)
−0.382031 + 0.924150i \(0.624775\pi\)
\(152\) −228966. −0.00528833
\(153\) 2.84252e7 0.641629
\(154\) −1.01268e8 −2.23435
\(155\) 405145. 0.00873876
\(156\) 4.00733e7 0.845120
\(157\) −5.46991e7 −1.12806 −0.564029 0.825755i \(-0.690749\pi\)
−0.564029 + 0.825755i \(0.690749\pi\)
\(158\) 2.00411e7 0.404223
\(159\) −5.33951e7 −1.05344
\(160\) 1.51997e6 0.0293370
\(161\) 8.54182e7 1.61309
\(162\) −7.98004e6 −0.147470
\(163\) −4.22142e7 −0.763487 −0.381744 0.924268i \(-0.624676\pi\)
−0.381744 + 0.924268i \(0.624676\pi\)
\(164\) −6.04229e7 −1.06967
\(165\) 813578. 0.0140996
\(166\) −1.24755e8 −2.11680
\(167\) −8.74223e6 −0.145250 −0.0726248 0.997359i \(-0.523138\pi\)
−0.0726248 + 0.997359i \(0.523138\pi\)
\(168\) 1.81443e7 0.295228
\(169\) 1.69092e8 2.69475
\(170\) −3.83548e6 −0.0598754
\(171\) −364169. −0.00556951
\(172\) −3.47201e7 −0.520272
\(173\) 4.41610e7 0.648451 0.324225 0.945980i \(-0.394896\pi\)
0.324225 + 0.945980i \(0.394896\pi\)
\(174\) −8.56488e7 −1.23253
\(175\) −1.14481e8 −1.61473
\(176\) −8.90500e7 −1.23123
\(177\) 5.54523e6 0.0751646
\(178\) −5.24550e7 −0.697135
\(179\) −3.77776e7 −0.492322 −0.246161 0.969229i \(-0.579169\pi\)
−0.246161 + 0.969229i \(0.579169\pi\)
\(180\) 465497. 0.00594927
\(181\) 1.26649e8 1.58755 0.793774 0.608213i \(-0.208113\pi\)
0.793774 + 0.608213i \(0.208113\pi\)
\(182\) −3.35216e8 −4.12169
\(183\) −8.05967e7 −0.972162
\(184\) 2.67033e7 0.316011
\(185\) −412309. −0.00478764
\(186\) −2.50744e7 −0.285717
\(187\) 1.79357e8 2.00573
\(188\) −9.29445e7 −1.02017
\(189\) 2.88584e7 0.310925
\(190\) 49138.2 0.000519734 0
\(191\) −2.15055e7 −0.223323 −0.111661 0.993746i \(-0.535617\pi\)
−0.111661 + 0.993746i \(0.535617\pi\)
\(192\) −2.71650e7 −0.276985
\(193\) 2.18519e7 0.218796 0.109398 0.993998i \(-0.465108\pi\)
0.109398 + 0.993998i \(0.465108\pi\)
\(194\) 1.50475e8 1.47965
\(195\) 2.69309e6 0.0260094
\(196\) 1.29260e8 1.22622
\(197\) −8.73312e7 −0.813837 −0.406919 0.913464i \(-0.633397\pi\)
−0.406919 + 0.913464i \(0.633397\pi\)
\(198\) −5.03524e7 −0.460990
\(199\) −2.03499e8 −1.83053 −0.915266 0.402850i \(-0.868020\pi\)
−0.915266 + 0.402850i \(0.868020\pi\)
\(200\) −3.57888e7 −0.316331
\(201\) −2.83127e7 −0.245921
\(202\) −1.41830e8 −1.21070
\(203\) 3.09734e8 2.59868
\(204\) 1.02621e8 0.846314
\(205\) −4.06067e6 −0.0329200
\(206\) 6.12288e7 0.488000
\(207\) 4.24715e7 0.332814
\(208\) −2.94772e8 −2.27125
\(209\) −2.29783e6 −0.0174103
\(210\) −3.89392e6 −0.0290148
\(211\) −7.64544e7 −0.560291 −0.280146 0.959958i \(-0.590383\pi\)
−0.280146 + 0.959958i \(0.590383\pi\)
\(212\) −1.92768e8 −1.38950
\(213\) 1.10749e8 0.785253
\(214\) 3.25915e8 2.27330
\(215\) −2.33333e6 −0.0160119
\(216\) 9.02166e6 0.0609114
\(217\) 9.06772e7 0.602406
\(218\) 6.66477e7 0.435700
\(219\) −7.02922e6 −0.0452223
\(220\) 2.93719e6 0.0185974
\(221\) 5.93706e8 3.69997
\(222\) 2.55178e7 0.156534
\(223\) −5.24257e7 −0.316575 −0.158288 0.987393i \(-0.550597\pi\)
−0.158288 + 0.987393i \(0.550597\pi\)
\(224\) 3.40191e8 2.02234
\(225\) −5.69218e7 −0.333150
\(226\) −1.07124e8 −0.617314
\(227\) 3.76491e6 0.0213631 0.0106816 0.999943i \(-0.496600\pi\)
0.0106816 + 0.999943i \(0.496600\pi\)
\(228\) −1.31473e6 −0.00734623
\(229\) −1.48786e8 −0.818723 −0.409362 0.912372i \(-0.634249\pi\)
−0.409362 + 0.912372i \(0.634249\pi\)
\(230\) −5.73077e6 −0.0310574
\(231\) 1.82090e8 0.971952
\(232\) 9.68284e7 0.509090
\(233\) 1.35403e8 0.701265 0.350632 0.936513i \(-0.385967\pi\)
0.350632 + 0.936513i \(0.385967\pi\)
\(234\) −1.66676e8 −0.850387
\(235\) −6.24627e6 −0.0313966
\(236\) 2.00195e7 0.0991427
\(237\) −3.60358e7 −0.175839
\(238\) −8.58434e8 −4.12751
\(239\) −3.35966e8 −1.59185 −0.795926 0.605394i \(-0.793016\pi\)
−0.795926 + 0.605394i \(0.793016\pi\)
\(240\) −3.42411e6 −0.0159885
\(241\) −2.48239e8 −1.14238 −0.571189 0.820818i \(-0.693518\pi\)
−0.571189 + 0.820818i \(0.693518\pi\)
\(242\) −2.50963e7 −0.113830
\(243\) 1.43489e7 0.0641500
\(244\) −2.90971e8 −1.28229
\(245\) 8.68682e6 0.0377381
\(246\) 2.51315e8 1.07633
\(247\) −7.60625e6 −0.0321167
\(248\) 2.83474e7 0.118014
\(249\) 2.24322e8 0.920820
\(250\) 1.53654e7 0.0621948
\(251\) 9.71369e7 0.387727 0.193864 0.981028i \(-0.437898\pi\)
0.193864 + 0.981028i \(0.437898\pi\)
\(252\) 1.04185e8 0.410112
\(253\) 2.67986e8 1.04038
\(254\) 1.85428e8 0.709997
\(255\) 6.89658e6 0.0260461
\(256\) 3.47895e8 1.29601
\(257\) 5.01702e6 0.0184366 0.00921829 0.999958i \(-0.497066\pi\)
0.00921829 + 0.999958i \(0.497066\pi\)
\(258\) 1.44410e8 0.523514
\(259\) −9.22806e7 −0.330036
\(260\) 9.72264e6 0.0343066
\(261\) 1.54005e8 0.536159
\(262\) 2.65735e8 0.912838
\(263\) −3.81564e8 −1.29337 −0.646685 0.762757i \(-0.723845\pi\)
−0.646685 + 0.762757i \(0.723845\pi\)
\(264\) 5.69248e7 0.190409
\(265\) −1.29548e7 −0.0427632
\(266\) 1.09978e7 0.0358278
\(267\) 9.43193e7 0.303257
\(268\) −1.02215e8 −0.324372
\(269\) −2.82835e8 −0.885931 −0.442965 0.896539i \(-0.646074\pi\)
−0.442965 + 0.896539i \(0.646074\pi\)
\(270\) −1.93613e6 −0.00598634
\(271\) 2.76884e7 0.0845095 0.0422547 0.999107i \(-0.486546\pi\)
0.0422547 + 0.999107i \(0.486546\pi\)
\(272\) −7.54863e8 −2.27445
\(273\) 6.02752e8 1.79296
\(274\) 3.52174e7 0.103426
\(275\) −3.59165e8 −1.04143
\(276\) 1.53331e8 0.438983
\(277\) 1.63786e7 0.0463018 0.0231509 0.999732i \(-0.492630\pi\)
0.0231509 + 0.999732i \(0.492630\pi\)
\(278\) 2.77725e8 0.775278
\(279\) 4.50863e7 0.124288
\(280\) 4.40219e6 0.0119844
\(281\) −1.44402e8 −0.388240 −0.194120 0.980978i \(-0.562185\pi\)
−0.194120 + 0.980978i \(0.562185\pi\)
\(282\) 3.86582e8 1.02652
\(283\) 1.29642e8 0.340011 0.170005 0.985443i \(-0.445622\pi\)
0.170005 + 0.985443i \(0.445622\pi\)
\(284\) 3.99826e8 1.03575
\(285\) −88355.3 −0.000226087 0
\(286\) −1.05169e9 −2.65831
\(287\) −9.08836e8 −2.26934
\(288\) 1.69149e8 0.417250
\(289\) 1.11005e9 2.70519
\(290\) −2.07802e7 −0.0500331
\(291\) −2.70569e8 −0.643654
\(292\) −2.53770e7 −0.0596486
\(293\) −5.48210e8 −1.27324 −0.636620 0.771178i \(-0.719668\pi\)
−0.636620 + 0.771178i \(0.719668\pi\)
\(294\) −5.37628e8 −1.23386
\(295\) 1.34539e6 0.00305121
\(296\) −2.88486e7 −0.0646553
\(297\) 9.05386e7 0.200533
\(298\) 2.80895e8 0.614875
\(299\) 8.87083e8 1.91918
\(300\) −2.05500e8 −0.439428
\(301\) −5.22233e8 −1.10378
\(302\) 4.85398e8 1.01408
\(303\) 2.55024e8 0.526663
\(304\) 9.67091e6 0.0197428
\(305\) −1.95545e7 −0.0394637
\(306\) −4.26829e8 −0.851587
\(307\) 7.57326e8 1.49382 0.746910 0.664925i \(-0.231536\pi\)
0.746910 + 0.664925i \(0.231536\pi\)
\(308\) 6.57384e8 1.28201
\(309\) −1.10095e8 −0.212283
\(310\) −6.08360e6 −0.0115983
\(311\) 3.48341e8 0.656665 0.328332 0.944562i \(-0.393513\pi\)
0.328332 + 0.944562i \(0.393513\pi\)
\(312\) 1.88431e8 0.351247
\(313\) −7.49639e8 −1.38180 −0.690902 0.722948i \(-0.742787\pi\)
−0.690902 + 0.722948i \(0.742787\pi\)
\(314\) 8.21354e8 1.49719
\(315\) 7.00166e6 0.0126216
\(316\) −1.30097e8 −0.231933
\(317\) 7.39103e7 0.130316 0.0651580 0.997875i \(-0.479245\pi\)
0.0651580 + 0.997875i \(0.479245\pi\)
\(318\) 8.01773e8 1.39816
\(319\) 9.71740e8 1.67603
\(320\) −6.59082e6 −0.0112438
\(321\) −5.86029e8 −0.988897
\(322\) −1.28263e9 −2.14094
\(323\) −1.94784e7 −0.0321621
\(324\) 5.18026e7 0.0846144
\(325\) −1.18890e9 −1.92112
\(326\) 6.33882e8 1.01332
\(327\) −1.19839e8 −0.189532
\(328\) −2.84119e8 −0.444572
\(329\) −1.39800e9 −2.16432
\(330\) −1.22166e7 −0.0187133
\(331\) −6.07136e8 −0.920212 −0.460106 0.887864i \(-0.652189\pi\)
−0.460106 + 0.887864i \(0.652189\pi\)
\(332\) 8.09851e8 1.21457
\(333\) −4.58836e7 −0.0680930
\(334\) 1.31272e8 0.192779
\(335\) −6.86928e6 −0.00998285
\(336\) −7.66365e8 −1.10217
\(337\) 1.21856e9 1.73437 0.867186 0.497985i \(-0.165926\pi\)
0.867186 + 0.497985i \(0.165926\pi\)
\(338\) −2.53906e9 −3.57655
\(339\) 1.92619e8 0.268535
\(340\) 2.48981e7 0.0343550
\(341\) 2.84485e8 0.388526
\(342\) 5.46831e6 0.00739200
\(343\) 7.36790e8 0.985858
\(344\) −1.63260e8 −0.216234
\(345\) 1.03045e7 0.0135101
\(346\) −6.63114e8 −0.860641
\(347\) 8.27617e8 1.06335 0.531675 0.846948i \(-0.321563\pi\)
0.531675 + 0.846948i \(0.321563\pi\)
\(348\) 5.55991e8 0.707197
\(349\) −9.88709e8 −1.24503 −0.622515 0.782608i \(-0.713889\pi\)
−0.622515 + 0.782608i \(0.713889\pi\)
\(350\) 1.71902e9 2.14311
\(351\) 2.99699e8 0.369923
\(352\) 1.06730e9 1.30432
\(353\) 2.48535e7 0.0300730 0.0150365 0.999887i \(-0.495214\pi\)
0.0150365 + 0.999887i \(0.495214\pi\)
\(354\) −8.32664e7 −0.0997604
\(355\) 2.68700e7 0.0318763
\(356\) 3.40513e8 0.399998
\(357\) 1.54355e9 1.79549
\(358\) 5.67263e8 0.653423
\(359\) 6.66922e8 0.760755 0.380377 0.924831i \(-0.375794\pi\)
0.380377 + 0.924831i \(0.375794\pi\)
\(360\) 2.18885e6 0.00247262
\(361\) −8.93622e8 −0.999721
\(362\) −1.90174e9 −2.10703
\(363\) 4.51257e7 0.0495166
\(364\) 2.17606e9 2.36492
\(365\) −1.70544e6 −0.00183574
\(366\) 1.21023e9 1.29028
\(367\) 4.68476e8 0.494716 0.247358 0.968924i \(-0.420438\pi\)
0.247358 + 0.968924i \(0.420438\pi\)
\(368\) −1.12788e9 −1.17976
\(369\) −4.51890e8 −0.468209
\(370\) 6.19117e6 0.00635429
\(371\) −2.89947e9 −2.94788
\(372\) 1.62771e8 0.163937
\(373\) 5.89698e8 0.588368 0.294184 0.955749i \(-0.404952\pi\)
0.294184 + 0.955749i \(0.404952\pi\)
\(374\) −2.69320e9 −2.66206
\(375\) −2.76285e7 −0.0270550
\(376\) −4.37041e8 −0.423999
\(377\) 3.21664e9 3.09177
\(378\) −4.33333e8 −0.412668
\(379\) 7.55066e8 0.712439 0.356220 0.934402i \(-0.384066\pi\)
0.356220 + 0.934402i \(0.384066\pi\)
\(380\) −318982. −0.000298211 0
\(381\) −3.33418e8 −0.308853
\(382\) 3.22924e8 0.296400
\(383\) −5.16218e8 −0.469502 −0.234751 0.972056i \(-0.575427\pi\)
−0.234751 + 0.972056i \(0.575427\pi\)
\(384\) −3.93986e8 −0.355076
\(385\) 4.41790e7 0.0394551
\(386\) −3.28125e8 −0.290392
\(387\) −2.59664e8 −0.227731
\(388\) −9.76811e8 −0.848984
\(389\) 2.02594e9 1.74503 0.872514 0.488589i \(-0.162488\pi\)
0.872514 + 0.488589i \(0.162488\pi\)
\(390\) −4.04391e7 −0.0345203
\(391\) 2.27168e9 1.92189
\(392\) 6.07803e8 0.509638
\(393\) −4.77817e8 −0.397089
\(394\) 1.31135e9 1.08015
\(395\) −8.74307e6 −0.00713796
\(396\) 3.26863e8 0.264505
\(397\) 1.11905e9 0.897601 0.448801 0.893632i \(-0.351851\pi\)
0.448801 + 0.893632i \(0.351851\pi\)
\(398\) 3.05572e9 2.42953
\(399\) −1.97752e7 −0.0155853
\(400\) 1.51162e9 1.18095
\(401\) −2.75487e8 −0.213352 −0.106676 0.994294i \(-0.534021\pi\)
−0.106676 + 0.994294i \(0.534021\pi\)
\(402\) 4.25140e8 0.326393
\(403\) 9.41699e8 0.716711
\(404\) 9.20692e8 0.694672
\(405\) 3.48135e6 0.00260409
\(406\) −4.65091e9 −3.44903
\(407\) −2.89516e8 −0.212859
\(408\) 4.82543e8 0.351743
\(409\) −7.49535e8 −0.541702 −0.270851 0.962621i \(-0.587305\pi\)
−0.270851 + 0.962621i \(0.587305\pi\)
\(410\) 6.09745e7 0.0436923
\(411\) −6.33244e7 −0.0449909
\(412\) −3.97468e8 −0.280002
\(413\) 3.01118e8 0.210335
\(414\) −6.37745e8 −0.441719
\(415\) 5.44254e7 0.0373795
\(416\) 3.53294e9 2.40608
\(417\) −4.99377e8 −0.337250
\(418\) 3.45039e7 0.0231074
\(419\) 2.09732e9 1.39289 0.696445 0.717611i \(-0.254764\pi\)
0.696445 + 0.717611i \(0.254764\pi\)
\(420\) 2.52775e7 0.0166480
\(421\) 8.22045e8 0.536919 0.268459 0.963291i \(-0.413486\pi\)
0.268459 + 0.963291i \(0.413486\pi\)
\(422\) 1.14803e9 0.743633
\(423\) −6.95112e8 −0.446543
\(424\) −9.06427e8 −0.577501
\(425\) −3.04458e9 −1.92383
\(426\) −1.66298e9 −1.04221
\(427\) −4.37657e9 −2.72042
\(428\) −2.11569e9 −1.30436
\(429\) 1.89104e9 1.15638
\(430\) 3.50370e7 0.0212514
\(431\) −9.73123e8 −0.585460 −0.292730 0.956195i \(-0.594564\pi\)
−0.292730 + 0.956195i \(0.594564\pi\)
\(432\) −3.81051e8 −0.227400
\(433\) 2.68877e9 1.59165 0.795823 0.605529i \(-0.207039\pi\)
0.795823 + 0.605529i \(0.207039\pi\)
\(434\) −1.36160e9 −0.799528
\(435\) 3.73650e7 0.0217647
\(436\) −4.32645e8 −0.249994
\(437\) −2.91035e7 −0.0166825
\(438\) 1.05550e8 0.0600202
\(439\) 2.20359e9 1.24309 0.621547 0.783377i \(-0.286504\pi\)
0.621547 + 0.783377i \(0.286504\pi\)
\(440\) 1.38112e7 0.00772941
\(441\) 9.66708e8 0.536736
\(442\) −8.91499e9 −4.91070
\(443\) −1.04965e9 −0.573630 −0.286815 0.957986i \(-0.592597\pi\)
−0.286815 + 0.957986i \(0.592597\pi\)
\(444\) −1.65649e8 −0.0898151
\(445\) 2.28839e7 0.0123103
\(446\) 7.87216e8 0.420167
\(447\) −5.05077e8 −0.267474
\(448\) −1.47512e9 −0.775093
\(449\) −2.02286e9 −1.05464 −0.527320 0.849667i \(-0.676803\pi\)
−0.527320 + 0.849667i \(0.676803\pi\)
\(450\) 8.54730e8 0.442166
\(451\) −2.85133e9 −1.46362
\(452\) 6.95396e8 0.354199
\(453\) −8.72794e8 −0.441131
\(454\) −5.65334e7 −0.0283537
\(455\) 1.46241e8 0.0727827
\(456\) −6.18208e6 −0.00305322
\(457\) −3.01761e9 −1.47896 −0.739480 0.673178i \(-0.764929\pi\)
−0.739480 + 0.673178i \(0.764929\pi\)
\(458\) 2.23414e9 1.08663
\(459\) 7.67482e8 0.370445
\(460\) 3.72014e7 0.0178200
\(461\) 3.13368e9 1.48971 0.744854 0.667228i \(-0.232519\pi\)
0.744854 + 0.667228i \(0.232519\pi\)
\(462\) −2.73424e9 −1.29000
\(463\) −2.30769e9 −1.08055 −0.540275 0.841489i \(-0.681680\pi\)
−0.540275 + 0.841489i \(0.681680\pi\)
\(464\) −4.08977e9 −1.90058
\(465\) 1.09389e7 0.00504532
\(466\) −2.03319e9 −0.930737
\(467\) −3.23147e9 −1.46822 −0.734110 0.679030i \(-0.762400\pi\)
−0.734110 + 0.679030i \(0.762400\pi\)
\(468\) 1.08198e9 0.487931
\(469\) −1.53744e9 −0.688167
\(470\) 9.37930e7 0.0416704
\(471\) −1.47688e9 −0.651285
\(472\) 9.41350e7 0.0412054
\(473\) −1.63842e9 −0.711889
\(474\) 5.41109e8 0.233378
\(475\) 3.90056e7 0.0166994
\(476\) 5.57255e9 2.36826
\(477\) −1.44167e9 −0.608206
\(478\) 5.04482e9 2.11275
\(479\) 3.12158e9 1.29778 0.648888 0.760884i \(-0.275234\pi\)
0.648888 + 0.760884i \(0.275234\pi\)
\(480\) 4.10392e7 0.0169377
\(481\) −9.58350e8 −0.392660
\(482\) 3.72752e9 1.51619
\(483\) 2.30629e9 0.931320
\(484\) 1.62913e8 0.0653127
\(485\) −6.56458e7 −0.0261283
\(486\) −2.15461e8 −0.0851416
\(487\) −1.98211e9 −0.777636 −0.388818 0.921315i \(-0.627116\pi\)
−0.388818 + 0.921315i \(0.627116\pi\)
\(488\) −1.36820e9 −0.532941
\(489\) −1.13978e9 −0.440800
\(490\) −1.30440e8 −0.0500870
\(491\) 2.49858e9 0.952594 0.476297 0.879285i \(-0.341979\pi\)
0.476297 + 0.879285i \(0.341979\pi\)
\(492\) −1.63142e9 −0.617572
\(493\) 8.23729e9 3.09614
\(494\) 1.14214e8 0.0426261
\(495\) 2.19666e7 0.00814038
\(496\) −1.19732e9 −0.440578
\(497\) 6.01389e9 2.19739
\(498\) −3.36839e9 −1.22214
\(499\) 4.80008e9 1.72940 0.864702 0.502285i \(-0.167507\pi\)
0.864702 + 0.502285i \(0.167507\pi\)
\(500\) −9.97448e7 −0.0356858
\(501\) −2.36040e8 −0.0838599
\(502\) −1.45859e9 −0.514602
\(503\) −3.37167e9 −1.18129 −0.590645 0.806931i \(-0.701127\pi\)
−0.590645 + 0.806931i \(0.701127\pi\)
\(504\) 4.89895e8 0.170450
\(505\) 6.18744e7 0.0213792
\(506\) −4.02404e9 −1.38081
\(507\) 4.56548e9 1.55582
\(508\) −1.20371e9 −0.407379
\(509\) −2.96439e9 −0.996376 −0.498188 0.867069i \(-0.666001\pi\)
−0.498188 + 0.867069i \(0.666001\pi\)
\(510\) −1.03558e8 −0.0345691
\(511\) −3.81702e8 −0.126547
\(512\) −3.35615e9 −1.10509
\(513\) −9.83257e6 −0.00321556
\(514\) −7.53349e7 −0.0244695
\(515\) −2.67115e7 −0.00861734
\(516\) −9.37441e8 −0.300379
\(517\) −4.38601e9 −1.39590
\(518\) 1.38567e9 0.438032
\(519\) 1.19235e9 0.374383
\(520\) 4.57175e7 0.0142584
\(521\) −2.91409e9 −0.902757 −0.451378 0.892333i \(-0.649067\pi\)
−0.451378 + 0.892333i \(0.649067\pi\)
\(522\) −2.31252e9 −0.711604
\(523\) −4.99881e9 −1.52796 −0.763978 0.645243i \(-0.776756\pi\)
−0.763978 + 0.645243i \(0.776756\pi\)
\(524\) −1.72502e9 −0.523763
\(525\) −3.09098e9 −0.932262
\(526\) 5.72951e9 1.71659
\(527\) 2.41154e9 0.717723
\(528\) −2.40435e9 −0.710851
\(529\) −1.06102e7 −0.00311622
\(530\) 1.94527e8 0.0567565
\(531\) 1.49721e8 0.0433963
\(532\) −7.13926e7 −0.0205571
\(533\) −9.43842e9 −2.69994
\(534\) −1.41628e9 −0.402491
\(535\) −1.42183e8 −0.0401430
\(536\) −4.80633e8 −0.134814
\(537\) −1.02000e9 −0.284242
\(538\) 4.24700e9 1.17583
\(539\) 6.09972e9 1.67784
\(540\) 1.25684e7 0.00343481
\(541\) −1.32516e9 −0.359813 −0.179906 0.983684i \(-0.557580\pi\)
−0.179906 + 0.983684i \(0.557580\pi\)
\(542\) −4.15765e8 −0.112163
\(543\) 3.41952e9 0.916571
\(544\) 9.04730e9 2.40948
\(545\) −2.90756e7 −0.00769379
\(546\) −9.05084e9 −2.37966
\(547\) 1.49203e9 0.389782 0.194891 0.980825i \(-0.437565\pi\)
0.194891 + 0.980825i \(0.437565\pi\)
\(548\) −2.28614e8 −0.0593433
\(549\) −2.17611e9 −0.561278
\(550\) 5.39316e9 1.38221
\(551\) −1.05532e8 −0.0268753
\(552\) 7.20989e8 0.182449
\(553\) −1.95682e9 −0.492055
\(554\) −2.45939e8 −0.0614530
\(555\) −1.11323e7 −0.00276415
\(556\) −1.80286e9 −0.444835
\(557\) −1.87638e9 −0.460074 −0.230037 0.973182i \(-0.573885\pi\)
−0.230037 + 0.973182i \(0.573885\pi\)
\(558\) −6.77010e8 −0.164959
\(559\) −5.42348e9 −1.31322
\(560\) −1.85937e8 −0.0447411
\(561\) 4.84265e9 1.15801
\(562\) 2.16831e9 0.515282
\(563\) −2.13285e9 −0.503711 −0.251855 0.967765i \(-0.581041\pi\)
−0.251855 + 0.967765i \(0.581041\pi\)
\(564\) −2.50950e9 −0.588994
\(565\) 4.67336e7 0.0109008
\(566\) −1.94668e9 −0.451271
\(567\) 7.79176e8 0.179513
\(568\) 1.88005e9 0.430478
\(569\) −1.42082e9 −0.323331 −0.161666 0.986846i \(-0.551687\pi\)
−0.161666 + 0.986846i \(0.551687\pi\)
\(570\) 1.32673e6 0.000300069 0
\(571\) 1.63133e9 0.366703 0.183352 0.983047i \(-0.441305\pi\)
0.183352 + 0.983047i \(0.441305\pi\)
\(572\) 6.82705e9 1.52527
\(573\) −5.80649e8 −0.128936
\(574\) 1.36469e10 3.01192
\(575\) −4.54905e9 −0.997892
\(576\) −7.33455e8 −0.159917
\(577\) 7.83244e9 1.69739 0.848695 0.528882i \(-0.177389\pi\)
0.848695 + 0.528882i \(0.177389\pi\)
\(578\) −1.66683e10 −3.59040
\(579\) 5.90002e8 0.126322
\(580\) 1.34895e8 0.0287078
\(581\) 1.21812e10 2.57675
\(582\) 4.06282e9 0.854274
\(583\) −9.09662e9 −1.90125
\(584\) −1.19327e8 −0.0247910
\(585\) 7.27135e7 0.0150165
\(586\) 8.23184e9 1.68988
\(587\) 5.25336e9 1.07202 0.536011 0.844211i \(-0.319931\pi\)
0.536011 + 0.844211i \(0.319931\pi\)
\(588\) 3.49002e9 0.707958
\(589\) −3.08954e7 −0.00623003
\(590\) −2.02022e7 −0.00404965
\(591\) −2.35794e9 −0.469869
\(592\) 1.21849e9 0.241377
\(593\) 3.96594e9 0.781007 0.390503 0.920601i \(-0.372301\pi\)
0.390503 + 0.920601i \(0.372301\pi\)
\(594\) −1.35951e9 −0.266153
\(595\) 3.74499e8 0.0728855
\(596\) −1.82344e9 −0.352800
\(597\) −5.49448e9 −1.05686
\(598\) −1.33203e10 −2.54718
\(599\) 4.82067e9 0.916461 0.458230 0.888833i \(-0.348483\pi\)
0.458230 + 0.888833i \(0.348483\pi\)
\(600\) −9.66296e8 −0.182634
\(601\) 1.29063e9 0.242516 0.121258 0.992621i \(-0.461307\pi\)
0.121258 + 0.992621i \(0.461307\pi\)
\(602\) 7.84177e9 1.46496
\(603\) −7.64444e8 −0.141983
\(604\) −3.15097e9 −0.581855
\(605\) 1.09485e7 0.00201006
\(606\) −3.82941e9 −0.699000
\(607\) 7.37737e9 1.33888 0.669440 0.742866i \(-0.266534\pi\)
0.669440 + 0.742866i \(0.266534\pi\)
\(608\) −1.15909e8 −0.0209149
\(609\) 8.36281e9 1.50035
\(610\) 2.93627e8 0.0523772
\(611\) −1.45185e10 −2.57500
\(612\) 2.77077e9 0.488619
\(613\) −8.76333e9 −1.53659 −0.768294 0.640097i \(-0.778894\pi\)
−0.768294 + 0.640097i \(0.778894\pi\)
\(614\) −1.13719e10 −1.98264
\(615\) −1.09638e8 −0.0190064
\(616\) 3.09114e9 0.532826
\(617\) 9.05070e9 1.55126 0.775629 0.631189i \(-0.217433\pi\)
0.775629 + 0.631189i \(0.217433\pi\)
\(618\) 1.65318e9 0.281747
\(619\) 7.25987e9 1.23030 0.615151 0.788409i \(-0.289095\pi\)
0.615151 + 0.788409i \(0.289095\pi\)
\(620\) 3.94918e7 0.00665482
\(621\) 1.14673e9 0.192150
\(622\) −5.23064e9 −0.871543
\(623\) 5.12174e9 0.848612
\(624\) −7.95884e9 −1.31130
\(625\) 6.09346e9 0.998352
\(626\) 1.12565e10 1.83397
\(627\) −6.20415e7 −0.0100518
\(628\) −5.33184e9 −0.859049
\(629\) −2.45418e9 −0.393214
\(630\) −1.05136e8 −0.0167517
\(631\) 6.82668e9 1.08170 0.540850 0.841119i \(-0.318103\pi\)
0.540850 + 0.841119i \(0.318103\pi\)
\(632\) −6.11739e8 −0.0963954
\(633\) −2.06427e9 −0.323484
\(634\) −1.10983e9 −0.172959
\(635\) −8.08943e7 −0.0125375
\(636\) −5.20473e9 −0.802229
\(637\) 2.01912e10 3.09510
\(638\) −1.45915e10 −2.22448
\(639\) 2.99021e9 0.453366
\(640\) −9.55896e7 −0.0144139
\(641\) 2.79576e9 0.419273 0.209637 0.977779i \(-0.432772\pi\)
0.209637 + 0.977779i \(0.432772\pi\)
\(642\) 8.79972e9 1.31249
\(643\) −2.72738e9 −0.404583 −0.202292 0.979325i \(-0.564839\pi\)
−0.202292 + 0.979325i \(0.564839\pi\)
\(644\) 8.32620e9 1.22842
\(645\) −6.30000e7 −0.00924446
\(646\) 2.92484e8 0.0426863
\(647\) −4.23656e9 −0.614962 −0.307481 0.951554i \(-0.599486\pi\)
−0.307481 + 0.951554i \(0.599486\pi\)
\(648\) 2.43585e8 0.0351672
\(649\) 9.44710e8 0.135657
\(650\) 1.78524e10 2.54976
\(651\) 2.44828e9 0.347799
\(652\) −4.11486e9 −0.581418
\(653\) −5.22693e9 −0.734599 −0.367300 0.930103i \(-0.619718\pi\)
−0.367300 + 0.930103i \(0.619718\pi\)
\(654\) 1.79949e9 0.251551
\(655\) −1.15929e8 −0.0161193
\(656\) 1.20004e10 1.65971
\(657\) −1.89789e8 −0.0261091
\(658\) 2.09922e10 2.87255
\(659\) 3.27514e9 0.445791 0.222896 0.974842i \(-0.428449\pi\)
0.222896 + 0.974842i \(0.428449\pi\)
\(660\) 7.93041e7 0.0107372
\(661\) 1.03921e10 1.39958 0.699788 0.714351i \(-0.253278\pi\)
0.699788 + 0.714351i \(0.253278\pi\)
\(662\) 9.11666e9 1.22133
\(663\) 1.60300e10 2.13618
\(664\) 3.80806e9 0.504796
\(665\) −4.79788e6 −0.000632665 0
\(666\) 6.88981e8 0.0903748
\(667\) 1.23077e10 1.60597
\(668\) −8.52156e8 −0.110612
\(669\) −1.41549e9 −0.182775
\(670\) 1.03148e8 0.0132495
\(671\) −1.37308e10 −1.75456
\(672\) 9.18516e9 1.16760
\(673\) 1.08020e10 1.36601 0.683003 0.730416i \(-0.260674\pi\)
0.683003 + 0.730416i \(0.260674\pi\)
\(674\) −1.82977e10 −2.30190
\(675\) −1.53689e9 −0.192344
\(676\) 1.64823e10 2.05213
\(677\) 1.40245e8 0.0173711 0.00868555 0.999962i \(-0.497235\pi\)
0.00868555 + 0.999962i \(0.497235\pi\)
\(678\) −2.89234e9 −0.356406
\(679\) −1.46925e10 −1.80115
\(680\) 1.17075e8 0.0142785
\(681\) 1.01653e8 0.0123340
\(682\) −4.27179e9 −0.515661
\(683\) −5.38006e9 −0.646122 −0.323061 0.946378i \(-0.604712\pi\)
−0.323061 + 0.946378i \(0.604712\pi\)
\(684\) −3.54977e7 −0.00424135
\(685\) −1.53639e7 −0.00182635
\(686\) −1.10635e10 −1.30846
\(687\) −4.01721e9 −0.472690
\(688\) 6.89565e9 0.807264
\(689\) −3.01115e10 −3.50724
\(690\) −1.54731e8 −0.0179310
\(691\) −8.51475e9 −0.981745 −0.490873 0.871231i \(-0.663322\pi\)
−0.490873 + 0.871231i \(0.663322\pi\)
\(692\) 4.30462e9 0.493814
\(693\) 4.91644e9 0.561157
\(694\) −1.24274e10 −1.41131
\(695\) −1.21160e8 −0.0136902
\(696\) 2.61437e9 0.293923
\(697\) −2.41703e10 −2.70375
\(698\) 1.48463e10 1.65244
\(699\) 3.65588e9 0.404875
\(700\) −1.11591e10 −1.22966
\(701\) 1.29311e10 1.41782 0.708911 0.705298i \(-0.249187\pi\)
0.708911 + 0.705298i \(0.249187\pi\)
\(702\) −4.50024e9 −0.490971
\(703\) 3.14417e7 0.00341320
\(704\) −4.62795e9 −0.499901
\(705\) −1.68649e8 −0.0181269
\(706\) −3.73197e8 −0.0399137
\(707\) 1.38484e10 1.47377
\(708\) 5.40526e8 0.0572400
\(709\) 5.39507e9 0.568506 0.284253 0.958749i \(-0.408254\pi\)
0.284253 + 0.958749i \(0.408254\pi\)
\(710\) −4.03476e8 −0.0423071
\(711\) −9.72968e8 −0.101521
\(712\) 1.60115e9 0.166246
\(713\) 3.60319e9 0.372284
\(714\) −2.31777e10 −2.38302
\(715\) 4.58807e8 0.0469417
\(716\) −3.68240e9 −0.374918
\(717\) −9.07108e9 −0.919056
\(718\) −1.00144e10 −1.00969
\(719\) −8.62160e9 −0.865041 −0.432521 0.901624i \(-0.642376\pi\)
−0.432521 + 0.901624i \(0.642376\pi\)
\(720\) −9.24511e7 −0.00923099
\(721\) −5.97842e9 −0.594036
\(722\) 1.34185e10 1.32686
\(723\) −6.70245e9 −0.659553
\(724\) 1.23452e10 1.20896
\(725\) −1.64953e10 −1.60759
\(726\) −6.77601e8 −0.0657197
\(727\) −7.76855e9 −0.749841 −0.374921 0.927057i \(-0.622330\pi\)
−0.374921 + 0.927057i \(0.622330\pi\)
\(728\) 1.02322e10 0.982902
\(729\) 3.87420e8 0.0370370
\(730\) 2.56086e7 0.00243644
\(731\) −1.38887e10 −1.31507
\(732\) −7.85622e9 −0.740330
\(733\) −2.43007e9 −0.227906 −0.113953 0.993486i \(-0.536351\pi\)
−0.113953 + 0.993486i \(0.536351\pi\)
\(734\) −7.03456e9 −0.656600
\(735\) 2.34544e8 0.0217881
\(736\) 1.35180e10 1.24980
\(737\) −4.82348e9 −0.443838
\(738\) 6.78551e9 0.621420
\(739\) −7.09023e9 −0.646256 −0.323128 0.946355i \(-0.604734\pi\)
−0.323128 + 0.946355i \(0.604734\pi\)
\(740\) −4.01901e7 −0.00364593
\(741\) −2.05369e8 −0.0185426
\(742\) 4.35380e10 3.91250
\(743\) 6.11412e9 0.546857 0.273428 0.961892i \(-0.411842\pi\)
0.273428 + 0.961892i \(0.411842\pi\)
\(744\) 7.65379e8 0.0681351
\(745\) −1.22543e8 −0.0108578
\(746\) −8.85482e9 −0.780897
\(747\) 6.05670e9 0.531636
\(748\) 1.74830e10 1.52743
\(749\) −3.18226e10 −2.76726
\(750\) 4.14866e8 0.0359082
\(751\) 1.01871e10 0.877629 0.438815 0.898578i \(-0.355398\pi\)
0.438815 + 0.898578i \(0.355398\pi\)
\(752\) 1.84594e10 1.58291
\(753\) 2.62270e9 0.223854
\(754\) −4.83005e10 −4.10348
\(755\) −2.11759e8 −0.0179072
\(756\) 2.81299e9 0.236778
\(757\) 2.04917e10 1.71689 0.858446 0.512903i \(-0.171430\pi\)
0.858446 + 0.512903i \(0.171430\pi\)
\(758\) −1.13380e10 −0.945568
\(759\) 7.23562e9 0.600661
\(760\) −1.49991e6 −0.000123941 0
\(761\) 4.69491e9 0.386172 0.193086 0.981182i \(-0.438150\pi\)
0.193086 + 0.981182i \(0.438150\pi\)
\(762\) 5.00655e9 0.409917
\(763\) −6.50752e9 −0.530371
\(764\) −2.09627e9 −0.170067
\(765\) 1.86208e8 0.0150377
\(766\) 7.75145e9 0.623135
\(767\) 3.12716e9 0.250246
\(768\) 9.39316e9 0.748251
\(769\) 1.90779e10 1.51282 0.756412 0.654095i \(-0.226951\pi\)
0.756412 + 0.654095i \(0.226951\pi\)
\(770\) −6.63386e8 −0.0523659
\(771\) 1.35460e8 0.0106444
\(772\) 2.13003e9 0.166619
\(773\) −1.43599e10 −1.11821 −0.559105 0.829097i \(-0.688855\pi\)
−0.559105 + 0.829097i \(0.688855\pi\)
\(774\) 3.89907e9 0.302251
\(775\) −4.82913e9 −0.372660
\(776\) −4.59313e9 −0.352852
\(777\) −2.49158e9 −0.190546
\(778\) −3.04212e10 −2.31605
\(779\) 3.09657e8 0.0234693
\(780\) 2.62511e8 0.0198069
\(781\) 1.88676e10 1.41722
\(782\) −3.41111e10 −2.55078
\(783\) 4.15814e9 0.309551
\(784\) −2.56720e10 −1.90262
\(785\) −3.58322e8 −0.0264381
\(786\) 7.17483e9 0.527027
\(787\) −4.56113e9 −0.333550 −0.166775 0.985995i \(-0.553335\pi\)
−0.166775 + 0.985995i \(0.553335\pi\)
\(788\) −8.51267e9 −0.619761
\(789\) −1.03022e10 −0.746727
\(790\) 1.31285e8 0.00947369
\(791\) 1.04596e10 0.751447
\(792\) 1.53697e9 0.109933
\(793\) −4.54515e10 −3.23662
\(794\) −1.68035e10 −1.19132
\(795\) −3.49780e8 −0.0246894
\(796\) −1.98363e10 −1.39400
\(797\) 1.19252e10 0.834374 0.417187 0.908821i \(-0.363016\pi\)
0.417187 + 0.908821i \(0.363016\pi\)
\(798\) 2.96941e8 0.0206852
\(799\) −3.71795e10 −2.57864
\(800\) −1.81173e10 −1.25106
\(801\) 2.54662e9 0.175086
\(802\) 4.13667e9 0.283166
\(803\) −1.19753e9 −0.0816171
\(804\) −2.75980e9 −0.187276
\(805\) 5.59556e8 0.0378057
\(806\) −1.41404e10 −0.951238
\(807\) −7.63654e9 −0.511492
\(808\) 4.32925e9 0.288718
\(809\) −1.14473e10 −0.760121 −0.380061 0.924962i \(-0.624097\pi\)
−0.380061 + 0.924962i \(0.624097\pi\)
\(810\) −5.22755e7 −0.00345621
\(811\) −1.28875e10 −0.848389 −0.424195 0.905571i \(-0.639443\pi\)
−0.424195 + 0.905571i \(0.639443\pi\)
\(812\) 3.01915e10 1.97897
\(813\) 7.47587e8 0.0487916
\(814\) 4.34732e9 0.282512
\(815\) −2.76536e8 −0.0178937
\(816\) −2.03813e10 −1.31316
\(817\) 1.77934e8 0.0114152
\(818\) 1.12549e10 0.718961
\(819\) 1.62743e10 1.03516
\(820\) −3.95817e8 −0.0250695
\(821\) −1.25024e10 −0.788481 −0.394240 0.919007i \(-0.628992\pi\)
−0.394240 + 0.919007i \(0.628992\pi\)
\(822\) 9.50869e8 0.0597131
\(823\) 1.32497e10 0.828528 0.414264 0.910157i \(-0.364039\pi\)
0.414264 + 0.910157i \(0.364039\pi\)
\(824\) −1.86896e9 −0.116374
\(825\) −9.69745e9 −0.601269
\(826\) −4.52154e9 −0.279162
\(827\) 3.06111e9 0.188196 0.0940980 0.995563i \(-0.470003\pi\)
0.0940980 + 0.995563i \(0.470003\pi\)
\(828\) 4.13994e9 0.253447
\(829\) −1.74744e10 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(830\) −8.17244e8 −0.0496111
\(831\) 4.42223e8 0.0267324
\(832\) −1.53194e10 −0.922166
\(833\) 5.17064e10 3.09947
\(834\) 7.49856e9 0.447607
\(835\) −5.72685e7 −0.00340418
\(836\) −2.23983e8 −0.0132585
\(837\) 1.21733e9 0.0717579
\(838\) −3.14931e10 −1.84868
\(839\) −3.10592e9 −0.181562 −0.0907808 0.995871i \(-0.528936\pi\)
−0.0907808 + 0.995871i \(0.528936\pi\)
\(840\) 1.18859e8 0.00691919
\(841\) 2.73789e10 1.58720
\(842\) −1.23437e10 −0.712612
\(843\) −3.89885e9 −0.224150
\(844\) −7.45245e9 −0.426678
\(845\) 1.10768e9 0.0631564
\(846\) 1.04377e10 0.592664
\(847\) 2.45042e9 0.138563
\(848\) 3.82850e10 2.15597
\(849\) 3.50033e9 0.196305
\(850\) 4.57170e10 2.55336
\(851\) −3.66690e9 −0.203960
\(852\) 1.07953e10 0.597993
\(853\) 1.91007e10 1.05372 0.526862 0.849951i \(-0.323368\pi\)
0.526862 + 0.849951i \(0.323368\pi\)
\(854\) 6.57180e10 3.61062
\(855\) −2.38559e6 −0.000130531 0
\(856\) −9.94833e9 −0.542116
\(857\) 3.07421e9 0.166840 0.0834202 0.996514i \(-0.473416\pi\)
0.0834202 + 0.996514i \(0.473416\pi\)
\(858\) −2.83956e10 −1.53478
\(859\) −7.57219e9 −0.407610 −0.203805 0.979011i \(-0.565331\pi\)
−0.203805 + 0.979011i \(0.565331\pi\)
\(860\) −2.27443e8 −0.0121935
\(861\) −2.45386e10 −1.31020
\(862\) 1.46123e10 0.777038
\(863\) 6.48477e9 0.343445 0.171722 0.985145i \(-0.445067\pi\)
0.171722 + 0.985145i \(0.445067\pi\)
\(864\) 4.56703e9 0.240899
\(865\) 2.89289e8 0.0151976
\(866\) −4.03742e10 −2.11247
\(867\) 2.99712e10 1.56184
\(868\) 8.83883e9 0.458749
\(869\) −6.13922e9 −0.317354
\(870\) −5.61067e8 −0.0288866
\(871\) −1.59666e10 −0.818746
\(872\) −2.03437e9 −0.103902
\(873\) −7.30536e9 −0.371614
\(874\) 4.37014e8 0.0221414
\(875\) −1.50029e9 −0.0757088
\(876\) −6.85178e8 −0.0344381
\(877\) −7.00447e9 −0.350652 −0.175326 0.984510i \(-0.556098\pi\)
−0.175326 + 0.984510i \(0.556098\pi\)
\(878\) −3.30887e10 −1.64987
\(879\) −1.48017e10 −0.735106
\(880\) −5.83347e8 −0.0288561
\(881\) −1.76566e10 −0.869944 −0.434972 0.900444i \(-0.643242\pi\)
−0.434972 + 0.900444i \(0.643242\pi\)
\(882\) −1.45159e10 −0.712369
\(883\) −1.27783e10 −0.624610 −0.312305 0.949982i \(-0.601101\pi\)
−0.312305 + 0.949982i \(0.601101\pi\)
\(884\) 5.78719e10 2.81763
\(885\) 3.63256e7 0.00176162
\(886\) 1.57614e10 0.761337
\(887\) 3.30434e10 1.58983 0.794917 0.606719i \(-0.207515\pi\)
0.794917 + 0.606719i \(0.207515\pi\)
\(888\) −7.78912e8 −0.0373287
\(889\) −1.81053e10 −0.864270
\(890\) −3.43621e8 −0.0163386
\(891\) 2.44454e9 0.115778
\(892\) −5.11023e9 −0.241081
\(893\) 4.76325e8 0.0223833
\(894\) 7.58416e9 0.354998
\(895\) −2.47473e8 −0.0115384
\(896\) −2.13943e10 −0.993619
\(897\) 2.39512e10 1.10804
\(898\) 3.03750e10 1.39975
\(899\) 1.30655e10 0.599744
\(900\) −5.54850e9 −0.253704
\(901\) −7.71107e10 −3.51219
\(902\) 4.28151e10 1.94256
\(903\) −1.41003e10 −0.637266
\(904\) 3.26987e9 0.147211
\(905\) 8.29650e8 0.0372070
\(906\) 1.31057e10 0.585481
\(907\) −1.42333e10 −0.633401 −0.316701 0.948526i \(-0.602575\pi\)
−0.316701 + 0.948526i \(0.602575\pi\)
\(908\) 3.66988e8 0.0162686
\(909\) 6.88566e9 0.304069
\(910\) −2.19593e9 −0.0965991
\(911\) 8.60227e9 0.376963 0.188482 0.982077i \(-0.439643\pi\)
0.188482 + 0.982077i \(0.439643\pi\)
\(912\) 2.61115e8 0.0113985
\(913\) 3.82165e10 1.66189
\(914\) 4.53120e10 1.96292
\(915\) −5.27971e8 −0.0227844
\(916\) −1.45030e10 −0.623482
\(917\) −2.59465e10 −1.11118
\(918\) −1.15244e10 −0.491664
\(919\) 2.07267e10 0.880898 0.440449 0.897778i \(-0.354819\pi\)
0.440449 + 0.897778i \(0.354819\pi\)
\(920\) 1.74927e8 0.00740629
\(921\) 2.04478e10 0.862458
\(922\) −4.70548e10 −1.97718
\(923\) 6.24553e10 2.61435
\(924\) 1.77494e10 0.740170
\(925\) 4.91452e9 0.204167
\(926\) 3.46520e10 1.43413
\(927\) −2.97258e9 −0.122561
\(928\) 4.90174e10 2.01341
\(929\) −4.14549e10 −1.69637 −0.848186 0.529699i \(-0.822305\pi\)
−0.848186 + 0.529699i \(0.822305\pi\)
\(930\) −1.64257e8 −0.00669629
\(931\) −6.62436e8 −0.0269042
\(932\) 1.31985e10 0.534034
\(933\) 9.40522e9 0.379126
\(934\) 4.85233e10 1.94866
\(935\) 1.17493e9 0.0470080
\(936\) 5.08765e9 0.202792
\(937\) −2.37792e10 −0.944296 −0.472148 0.881519i \(-0.656521\pi\)
−0.472148 + 0.881519i \(0.656521\pi\)
\(938\) 2.30860e10 0.913353
\(939\) −2.02402e10 −0.797785
\(940\) −6.08859e8 −0.0239094
\(941\) −1.90527e10 −0.745407 −0.372703 0.927951i \(-0.621569\pi\)
−0.372703 + 0.927951i \(0.621569\pi\)
\(942\) 2.21766e10 0.864402
\(943\) −3.61139e10 −1.40244
\(944\) −3.97601e9 −0.153832
\(945\) 1.89045e8 0.00728708
\(946\) 2.46023e10 0.944838
\(947\) 2.17206e10 0.831089 0.415544 0.909573i \(-0.363591\pi\)
0.415544 + 0.909573i \(0.363591\pi\)
\(948\) −3.51262e9 −0.133907
\(949\) −3.96404e9 −0.150559
\(950\) −5.85703e8 −0.0221638
\(951\) 1.99558e9 0.0752379
\(952\) 2.62031e10 0.984290
\(953\) −2.72331e9 −0.101923 −0.0509614 0.998701i \(-0.516229\pi\)
−0.0509614 + 0.998701i \(0.516229\pi\)
\(954\) 2.16479e10 0.807227
\(955\) −1.40878e8 −0.00523397
\(956\) −3.27485e10 −1.21224
\(957\) 2.62370e10 0.967658
\(958\) −4.68732e10 −1.72244
\(959\) −3.43865e9 −0.125899
\(960\) −1.77952e8 −0.00649163
\(961\) −2.36876e10 −0.860972
\(962\) 1.43904e10 0.521148
\(963\) −1.58228e10 −0.570940
\(964\) −2.41973e10 −0.869955
\(965\) 1.43147e8 0.00512787
\(966\) −3.46309e10 −1.23607
\(967\) −1.19767e10 −0.425935 −0.212967 0.977059i \(-0.568313\pi\)
−0.212967 + 0.977059i \(0.568313\pi\)
\(968\) 7.66047e8 0.0271451
\(969\) −5.25916e8 −0.0185688
\(970\) 9.85728e8 0.0346781
\(971\) 4.20855e9 0.147525 0.0737625 0.997276i \(-0.476499\pi\)
0.0737625 + 0.997276i \(0.476499\pi\)
\(972\) 1.39867e9 0.0488521
\(973\) −2.71172e10 −0.943735
\(974\) 2.97631e10 1.03210
\(975\) −3.21003e10 −1.10916
\(976\) 5.77890e10 1.98962
\(977\) 3.92707e10 1.34722 0.673608 0.739089i \(-0.264744\pi\)
0.673608 + 0.739089i \(0.264744\pi\)
\(978\) 1.71148e10 0.585041
\(979\) 1.60686e10 0.547318
\(980\) 8.46754e8 0.0287386
\(981\) −3.23566e9 −0.109426
\(982\) −3.75183e10 −1.26431
\(983\) 2.85690e10 0.959308 0.479654 0.877458i \(-0.340762\pi\)
0.479654 + 0.877458i \(0.340762\pi\)
\(984\) −7.67121e9 −0.256673
\(985\) −5.72087e8 −0.0190737
\(986\) −1.23690e11 −4.10927
\(987\) −3.77461e10 −1.24957
\(988\) −7.41424e8 −0.0244578
\(989\) −2.07517e10 −0.682129
\(990\) −3.29847e8 −0.0108041
\(991\) 2.53806e10 0.828408 0.414204 0.910184i \(-0.364060\pi\)
0.414204 + 0.910184i \(0.364060\pi\)
\(992\) 1.43503e10 0.466733
\(993\) −1.63927e10 −0.531285
\(994\) −9.03036e10 −2.91644
\(995\) −1.33308e9 −0.0429018
\(996\) 2.18660e10 0.701231
\(997\) 9.36748e9 0.299357 0.149679 0.988735i \(-0.452176\pi\)
0.149679 + 0.988735i \(0.452176\pi\)
\(998\) −7.20773e10 −2.29531
\(999\) −1.23886e9 −0.0393135
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.1.4 18
3.2 odd 2 531.8.a.e.1.15 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.4 18 1.1 even 1 trivial
531.8.a.e.1.15 18 3.2 odd 2