Properties

Label 177.8.a.b.1.2
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-19.6388\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-21.6388 q^{2} +27.0000 q^{3} +340.238 q^{4} -399.107 q^{5} -584.248 q^{6} -1780.57 q^{7} -4592.58 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-21.6388 q^{2} +27.0000 q^{3} +340.238 q^{4} -399.107 q^{5} -584.248 q^{6} -1780.57 q^{7} -4592.58 q^{8} +729.000 q^{9} +8636.21 q^{10} -204.067 q^{11} +9186.43 q^{12} +7466.22 q^{13} +38529.4 q^{14} -10775.9 q^{15} +55827.6 q^{16} +2070.83 q^{17} -15774.7 q^{18} +38611.6 q^{19} -135792. q^{20} -48075.4 q^{21} +4415.78 q^{22} -47604.4 q^{23} -124000. q^{24} +81161.6 q^{25} -161560. q^{26} +19683.0 q^{27} -605818. q^{28} +223646. q^{29} +233178. q^{30} +273826. q^{31} -620192. q^{32} -5509.82 q^{33} -44810.3 q^{34} +710639. q^{35} +248034. q^{36} -545039. q^{37} -835510. q^{38} +201588. q^{39} +1.83293e6 q^{40} -521804. q^{41} +1.04029e6 q^{42} +327902. q^{43} -69431.5 q^{44} -290949. q^{45} +1.03010e6 q^{46} -24173.5 q^{47} +1.50734e6 q^{48} +2.34689e6 q^{49} -1.75624e6 q^{50} +55912.4 q^{51} +2.54029e6 q^{52} -1.06061e6 q^{53} -425917. q^{54} +81444.8 q^{55} +8.17742e6 q^{56} +1.04251e6 q^{57} -4.83943e6 q^{58} -205379. q^{59} -3.66637e6 q^{60} +1.08933e6 q^{61} -5.92527e6 q^{62} -1.29804e6 q^{63} +6.27428e6 q^{64} -2.97982e6 q^{65} +119226. q^{66} -55751.2 q^{67} +704575. q^{68} -1.28532e6 q^{69} -1.53774e7 q^{70} -851859. q^{71} -3.34799e6 q^{72} +3.44512e6 q^{73} +1.17940e7 q^{74} +2.19136e6 q^{75} +1.31372e7 q^{76} +363356. q^{77} -4.36213e6 q^{78} -8.05252e6 q^{79} -2.22812e7 q^{80} +531441. q^{81} +1.12912e7 q^{82} +2.05020e6 q^{83} -1.63571e7 q^{84} -826483. q^{85} -7.09542e6 q^{86} +6.03843e6 q^{87} +937197. q^{88} -1.68320e6 q^{89} +6.29579e6 q^{90} -1.32941e7 q^{91} -1.61969e7 q^{92} +7.39330e6 q^{93} +523086. q^{94} -1.54102e7 q^{95} -1.67452e7 q^{96} +1.14133e7 q^{97} -5.07839e7 q^{98} -148765. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.6388 −1.91262 −0.956309 0.292356i \(-0.905561\pi\)
−0.956309 + 0.292356i \(0.905561\pi\)
\(3\) 27.0000 0.577350
\(4\) 340.238 2.65811
\(5\) −399.107 −1.42789 −0.713945 0.700202i \(-0.753093\pi\)
−0.713945 + 0.700202i \(0.753093\pi\)
\(6\) −584.248 −1.10425
\(7\) −1780.57 −1.96208 −0.981039 0.193812i \(-0.937915\pi\)
−0.981039 + 0.193812i \(0.937915\pi\)
\(8\) −4592.58 −3.17133
\(9\) 729.000 0.333333
\(10\) 8636.21 2.73101
\(11\) −204.067 −0.0462274 −0.0231137 0.999733i \(-0.507358\pi\)
−0.0231137 + 0.999733i \(0.507358\pi\)
\(12\) 9186.43 1.53466
\(13\) 7466.22 0.942539 0.471269 0.881989i \(-0.343796\pi\)
0.471269 + 0.881989i \(0.343796\pi\)
\(14\) 38529.4 3.75271
\(15\) −10775.9 −0.824392
\(16\) 55827.6 3.40744
\(17\) 2070.83 0.102229 0.0511144 0.998693i \(-0.483723\pi\)
0.0511144 + 0.998693i \(0.483723\pi\)
\(18\) −15774.7 −0.637540
\(19\) 38611.6 1.29146 0.645729 0.763566i \(-0.276553\pi\)
0.645729 + 0.763566i \(0.276553\pi\)
\(20\) −135792. −3.79549
\(21\) −48075.4 −1.13281
\(22\) 4415.78 0.0884154
\(23\) −47604.4 −0.815831 −0.407915 0.913020i \(-0.633744\pi\)
−0.407915 + 0.913020i \(0.633744\pi\)
\(24\) −124000. −1.83097
\(25\) 81161.6 1.03887
\(26\) −161560. −1.80272
\(27\) 19683.0 0.192450
\(28\) −605818. −5.21542
\(29\) 223646. 1.70282 0.851408 0.524505i \(-0.175749\pi\)
0.851408 + 0.524505i \(0.175749\pi\)
\(30\) 233178. 1.57675
\(31\) 273826. 1.65085 0.825427 0.564508i \(-0.190934\pi\)
0.825427 + 0.564508i \(0.190934\pi\)
\(32\) −620192. −3.34581
\(33\) −5509.82 −0.0266894
\(34\) −44810.3 −0.195525
\(35\) 710639. 2.80163
\(36\) 248034. 0.886037
\(37\) −545039. −1.76897 −0.884487 0.466565i \(-0.845491\pi\)
−0.884487 + 0.466565i \(0.845491\pi\)
\(38\) −835510. −2.47007
\(39\) 201588. 0.544175
\(40\) 1.83293e6 4.52832
\(41\) −521804. −1.18240 −0.591200 0.806525i \(-0.701346\pi\)
−0.591200 + 0.806525i \(0.701346\pi\)
\(42\) 1.04029e6 2.16663
\(43\) 327902. 0.628934 0.314467 0.949268i \(-0.398174\pi\)
0.314467 + 0.949268i \(0.398174\pi\)
\(44\) −69431.5 −0.122878
\(45\) −290949. −0.475963
\(46\) 1.03010e6 1.56037
\(47\) −24173.5 −0.0339623 −0.0169811 0.999856i \(-0.505406\pi\)
−0.0169811 + 0.999856i \(0.505406\pi\)
\(48\) 1.50734e6 1.96729
\(49\) 2.34689e6 2.84975
\(50\) −1.75624e6 −1.98696
\(51\) 55912.4 0.0590218
\(52\) 2.54029e6 2.50537
\(53\) −1.06061e6 −0.978570 −0.489285 0.872124i \(-0.662742\pi\)
−0.489285 + 0.872124i \(0.662742\pi\)
\(54\) −425917. −0.368084
\(55\) 81444.8 0.0660076
\(56\) 8.17742e6 6.22240
\(57\) 1.04251e6 0.745624
\(58\) −4.83943e6 −3.25684
\(59\) −205379. −0.130189
\(60\) −3.66637e6 −2.19133
\(61\) 1.08933e6 0.614475 0.307238 0.951633i \(-0.400595\pi\)
0.307238 + 0.951633i \(0.400595\pi\)
\(62\) −5.92527e6 −3.15746
\(63\) −1.29804e6 −0.654026
\(64\) 6.27428e6 2.99181
\(65\) −2.97982e6 −1.34584
\(66\) 119226. 0.0510466
\(67\) −55751.2 −0.0226461 −0.0113230 0.999936i \(-0.503604\pi\)
−0.0113230 + 0.999936i \(0.503604\pi\)
\(68\) 704575. 0.271735
\(69\) −1.28532e6 −0.471020
\(70\) −1.53774e7 −5.35845
\(71\) −851859. −0.282464 −0.141232 0.989976i \(-0.545106\pi\)
−0.141232 + 0.989976i \(0.545106\pi\)
\(72\) −3.34799e6 −1.05711
\(73\) 3.44512e6 1.03651 0.518256 0.855225i \(-0.326581\pi\)
0.518256 + 0.855225i \(0.326581\pi\)
\(74\) 1.17940e7 3.38337
\(75\) 2.19136e6 0.599791
\(76\) 1.31372e7 3.43284
\(77\) 363356. 0.0907017
\(78\) −4.36213e6 −1.04080
\(79\) −8.05252e6 −1.83754 −0.918771 0.394791i \(-0.870817\pi\)
−0.918771 + 0.394791i \(0.870817\pi\)
\(80\) −2.22812e7 −4.86545
\(81\) 531441. 0.111111
\(82\) 1.12912e7 2.26148
\(83\) 2.05020e6 0.393571 0.196785 0.980447i \(-0.436950\pi\)
0.196785 + 0.980447i \(0.436950\pi\)
\(84\) −1.63571e7 −3.01112
\(85\) −826483. −0.145971
\(86\) −7.09542e6 −1.20291
\(87\) 6.03843e6 0.983121
\(88\) 937197. 0.146602
\(89\) −1.68320e6 −0.253087 −0.126544 0.991961i \(-0.540388\pi\)
−0.126544 + 0.991961i \(0.540388\pi\)
\(90\) 6.29579e6 0.910336
\(91\) −1.32941e7 −1.84933
\(92\) −1.61969e7 −2.16857
\(93\) 7.39330e6 0.953121
\(94\) 523086. 0.0649569
\(95\) −1.54102e7 −1.84406
\(96\) −1.67452e7 −1.93170
\(97\) 1.14133e7 1.26972 0.634861 0.772627i \(-0.281058\pi\)
0.634861 + 0.772627i \(0.281058\pi\)
\(98\) −5.07839e7 −5.45048
\(99\) −148765. −0.0154091
\(100\) 2.76143e7 2.76143
\(101\) 7.19660e6 0.695029 0.347514 0.937675i \(-0.387026\pi\)
0.347514 + 0.937675i \(0.387026\pi\)
\(102\) −1.20988e6 −0.112886
\(103\) −2.77089e6 −0.249856 −0.124928 0.992166i \(-0.539870\pi\)
−0.124928 + 0.992166i \(0.539870\pi\)
\(104\) −3.42893e7 −2.98911
\(105\) 1.91872e7 1.61752
\(106\) 2.29504e7 1.87163
\(107\) −937525. −0.0739843 −0.0369922 0.999316i \(-0.511778\pi\)
−0.0369922 + 0.999316i \(0.511778\pi\)
\(108\) 6.69691e6 0.511554
\(109\) −1.52436e7 −1.12744 −0.563722 0.825965i \(-0.690631\pi\)
−0.563722 + 0.825965i \(0.690631\pi\)
\(110\) −1.76237e6 −0.126247
\(111\) −1.47161e7 −1.02132
\(112\) −9.94049e7 −6.68567
\(113\) −1.00374e7 −0.654402 −0.327201 0.944955i \(-0.606105\pi\)
−0.327201 + 0.944955i \(0.606105\pi\)
\(114\) −2.25588e7 −1.42609
\(115\) 1.89993e7 1.16492
\(116\) 7.60928e7 4.52627
\(117\) 5.44288e6 0.314180
\(118\) 4.44416e6 0.249002
\(119\) −3.68726e6 −0.200581
\(120\) 4.94892e7 2.61442
\(121\) −1.94455e7 −0.997863
\(122\) −2.35718e7 −1.17526
\(123\) −1.40887e7 −0.682659
\(124\) 9.31661e7 4.38816
\(125\) −1.21192e6 −0.0554993
\(126\) 2.80880e7 1.25090
\(127\) 9.06146e6 0.392541 0.196270 0.980550i \(-0.437117\pi\)
0.196270 + 0.980550i \(0.437117\pi\)
\(128\) −5.63835e7 −2.37639
\(129\) 8.85336e6 0.363115
\(130\) 6.44799e7 2.57408
\(131\) 2.09570e7 0.814478 0.407239 0.913322i \(-0.366492\pi\)
0.407239 + 0.913322i \(0.366492\pi\)
\(132\) −1.87465e6 −0.0709434
\(133\) −6.87507e7 −2.53394
\(134\) 1.20639e6 0.0433133
\(135\) −7.85563e6 −0.274797
\(136\) −9.51046e6 −0.324202
\(137\) 2.19828e7 0.730399 0.365199 0.930929i \(-0.381001\pi\)
0.365199 + 0.930929i \(0.381001\pi\)
\(138\) 2.78128e7 0.900882
\(139\) −6.44663e6 −0.203601 −0.101801 0.994805i \(-0.532460\pi\)
−0.101801 + 0.994805i \(0.532460\pi\)
\(140\) 2.41786e8 7.44704
\(141\) −652684. −0.0196081
\(142\) 1.84332e7 0.540247
\(143\) −1.52361e6 −0.0435711
\(144\) 4.06983e7 1.13581
\(145\) −8.92586e7 −2.43143
\(146\) −7.45484e7 −1.98245
\(147\) 6.33660e7 1.64530
\(148\) −1.85443e8 −4.70213
\(149\) −3.38724e7 −0.838869 −0.419434 0.907786i \(-0.637772\pi\)
−0.419434 + 0.907786i \(0.637772\pi\)
\(150\) −4.74185e7 −1.14717
\(151\) −6.27255e7 −1.48260 −0.741301 0.671172i \(-0.765791\pi\)
−0.741301 + 0.671172i \(0.765791\pi\)
\(152\) −1.77327e8 −4.09565
\(153\) 1.50963e6 0.0340762
\(154\) −7.86260e6 −0.173478
\(155\) −1.09286e8 −2.35724
\(156\) 6.85880e7 1.44648
\(157\) 4.72239e7 0.973898 0.486949 0.873430i \(-0.338110\pi\)
0.486949 + 0.873430i \(0.338110\pi\)
\(158\) 1.74247e8 3.51452
\(159\) −2.86366e7 −0.564977
\(160\) 2.47523e8 4.77744
\(161\) 8.47631e7 1.60072
\(162\) −1.14998e7 −0.212513
\(163\) −2.85412e7 −0.516198 −0.258099 0.966119i \(-0.583096\pi\)
−0.258099 + 0.966119i \(0.583096\pi\)
\(164\) −1.77538e8 −3.14295
\(165\) 2.19901e6 0.0381095
\(166\) −4.43639e7 −0.752751
\(167\) −6.05397e7 −1.00585 −0.502924 0.864331i \(-0.667742\pi\)
−0.502924 + 0.864331i \(0.667742\pi\)
\(168\) 2.20790e8 3.59251
\(169\) −7.00401e6 −0.111620
\(170\) 1.78841e7 0.279188
\(171\) 2.81479e7 0.430486
\(172\) 1.11565e8 1.67178
\(173\) −8.84147e7 −1.29826 −0.649132 0.760676i \(-0.724868\pi\)
−0.649132 + 0.760676i \(0.724868\pi\)
\(174\) −1.30665e8 −1.88034
\(175\) −1.44514e8 −2.03834
\(176\) −1.13926e7 −0.157517
\(177\) −5.54523e6 −0.0751646
\(178\) 3.64224e7 0.484059
\(179\) 9.25607e7 1.20626 0.603130 0.797643i \(-0.293920\pi\)
0.603130 + 0.797643i \(0.293920\pi\)
\(180\) −9.89920e7 −1.26516
\(181\) 7.46139e6 0.0935287 0.0467644 0.998906i \(-0.485109\pi\)
0.0467644 + 0.998906i \(0.485109\pi\)
\(182\) 2.87669e8 3.53707
\(183\) 2.94119e7 0.354767
\(184\) 2.18627e8 2.58727
\(185\) 2.17529e8 2.52590
\(186\) −1.59982e8 −1.82296
\(187\) −422589. −0.00472577
\(188\) −8.22475e6 −0.0902755
\(189\) −3.50470e7 −0.377602
\(190\) 3.33458e8 3.52698
\(191\) 6.59363e7 0.684712 0.342356 0.939570i \(-0.388775\pi\)
0.342356 + 0.939570i \(0.388775\pi\)
\(192\) 1.69406e8 1.72732
\(193\) 1.34481e7 0.134651 0.0673257 0.997731i \(-0.478553\pi\)
0.0673257 + 0.997731i \(0.478553\pi\)
\(194\) −2.46969e8 −2.42849
\(195\) −8.04552e7 −0.777022
\(196\) 7.98502e8 7.57495
\(197\) 1.59146e7 0.148308 0.0741539 0.997247i \(-0.476374\pi\)
0.0741539 + 0.997247i \(0.476374\pi\)
\(198\) 3.21910e6 0.0294718
\(199\) 3.59450e7 0.323335 0.161668 0.986845i \(-0.448313\pi\)
0.161668 + 0.986845i \(0.448313\pi\)
\(200\) −3.72741e8 −3.29460
\(201\) −1.50528e6 −0.0130747
\(202\) −1.55726e8 −1.32932
\(203\) −3.98217e8 −3.34105
\(204\) 1.90235e7 0.156886
\(205\) 2.08256e8 1.68834
\(206\) 5.99588e7 0.477878
\(207\) −3.47036e7 −0.271944
\(208\) 4.16821e8 3.21165
\(209\) −7.87937e6 −0.0597007
\(210\) −4.15189e8 −3.09370
\(211\) −2.03281e8 −1.48973 −0.744866 0.667214i \(-0.767486\pi\)
−0.744866 + 0.667214i \(0.767486\pi\)
\(212\) −3.60861e8 −2.60115
\(213\) −2.30002e7 −0.163081
\(214\) 2.02869e7 0.141504
\(215\) −1.30868e8 −0.898048
\(216\) −9.03958e7 −0.610324
\(217\) −4.87567e8 −3.23910
\(218\) 3.29853e8 2.15637
\(219\) 9.30183e7 0.598431
\(220\) 2.77106e7 0.175455
\(221\) 1.54613e7 0.0963546
\(222\) 3.18438e8 1.95339
\(223\) −1.24671e8 −0.752830 −0.376415 0.926451i \(-0.622843\pi\)
−0.376415 + 0.926451i \(0.622843\pi\)
\(224\) 1.10429e9 6.56473
\(225\) 5.91668e7 0.346289
\(226\) 2.17196e8 1.25162
\(227\) −1.63994e7 −0.0930547 −0.0465273 0.998917i \(-0.514815\pi\)
−0.0465273 + 0.998917i \(0.514815\pi\)
\(228\) 3.54703e8 1.98195
\(229\) −1.34148e8 −0.738179 −0.369089 0.929394i \(-0.620330\pi\)
−0.369089 + 0.929394i \(0.620330\pi\)
\(230\) −4.11122e8 −2.22804
\(231\) 9.81062e6 0.0523666
\(232\) −1.02711e9 −5.40020
\(233\) −6.54300e7 −0.338869 −0.169434 0.985541i \(-0.554194\pi\)
−0.169434 + 0.985541i \(0.554194\pi\)
\(234\) −1.17777e8 −0.600906
\(235\) 9.64781e6 0.0484944
\(236\) −6.98778e7 −0.346057
\(237\) −2.17418e8 −1.06091
\(238\) 7.97879e7 0.383634
\(239\) −1.59121e8 −0.753936 −0.376968 0.926226i \(-0.623033\pi\)
−0.376968 + 0.926226i \(0.623033\pi\)
\(240\) −6.01592e8 −2.80907
\(241\) 1.38094e8 0.635501 0.317750 0.948174i \(-0.397073\pi\)
0.317750 + 0.948174i \(0.397073\pi\)
\(242\) 4.20778e8 1.90853
\(243\) 1.43489e7 0.0641500
\(244\) 3.70631e8 1.63334
\(245\) −9.36661e8 −4.06912
\(246\) 3.04863e8 1.30567
\(247\) 2.88283e8 1.21725
\(248\) −1.25757e9 −5.23541
\(249\) 5.53554e7 0.227228
\(250\) 2.62244e7 0.106149
\(251\) 1.39109e8 0.555262 0.277631 0.960688i \(-0.410451\pi\)
0.277631 + 0.960688i \(0.410451\pi\)
\(252\) −4.41642e8 −1.73847
\(253\) 9.71452e6 0.0377137
\(254\) −1.96079e8 −0.750781
\(255\) −2.23150e7 −0.0842766
\(256\) 4.16963e8 1.55331
\(257\) 4.09751e8 1.50575 0.752877 0.658162i \(-0.228666\pi\)
0.752877 + 0.658162i \(0.228666\pi\)
\(258\) −1.91576e8 −0.694501
\(259\) 9.70480e8 3.47086
\(260\) −1.01385e9 −3.57740
\(261\) 1.63038e8 0.567605
\(262\) −4.53484e8 −1.55779
\(263\) 1.19894e8 0.406400 0.203200 0.979137i \(-0.434866\pi\)
0.203200 + 0.979137i \(0.434866\pi\)
\(264\) 2.53043e7 0.0846410
\(265\) 4.23299e8 1.39729
\(266\) 1.48768e9 4.84647
\(267\) −4.54463e7 −0.146120
\(268\) −1.89687e7 −0.0601958
\(269\) −2.45594e8 −0.769282 −0.384641 0.923066i \(-0.625675\pi\)
−0.384641 + 0.923066i \(0.625675\pi\)
\(270\) 1.69986e8 0.525583
\(271\) 2.52445e8 0.770503 0.385252 0.922812i \(-0.374115\pi\)
0.385252 + 0.922812i \(0.374115\pi\)
\(272\) 1.15609e8 0.348339
\(273\) −3.58942e8 −1.06771
\(274\) −4.75681e8 −1.39697
\(275\) −1.65624e7 −0.0480241
\(276\) −4.37315e8 −1.25202
\(277\) 3.24074e8 0.916147 0.458074 0.888914i \(-0.348540\pi\)
0.458074 + 0.888914i \(0.348540\pi\)
\(278\) 1.39497e8 0.389412
\(279\) 1.99619e8 0.550285
\(280\) −3.26367e9 −8.88491
\(281\) −1.78256e8 −0.479260 −0.239630 0.970864i \(-0.577026\pi\)
−0.239630 + 0.970864i \(0.577026\pi\)
\(282\) 1.41233e7 0.0375029
\(283\) −4.64602e8 −1.21851 −0.609254 0.792975i \(-0.708531\pi\)
−0.609254 + 0.792975i \(0.708531\pi\)
\(284\) −2.89835e8 −0.750822
\(285\) −4.16075e8 −1.06467
\(286\) 3.29692e7 0.0833349
\(287\) 9.29110e8 2.31996
\(288\) −4.52120e8 −1.11527
\(289\) −4.06050e8 −0.989549
\(290\) 1.93145e9 4.65040
\(291\) 3.08158e8 0.733074
\(292\) 1.17216e9 2.75517
\(293\) −7.63943e8 −1.77429 −0.887145 0.461492i \(-0.847314\pi\)
−0.887145 + 0.461492i \(0.847314\pi\)
\(294\) −1.37117e9 −3.14684
\(295\) 8.19682e7 0.185895
\(296\) 2.50314e9 5.61001
\(297\) −4.01666e6 −0.00889646
\(298\) 7.32959e8 1.60444
\(299\) −3.55425e8 −0.768952
\(300\) 7.45585e8 1.59431
\(301\) −5.83853e8 −1.23402
\(302\) 1.35731e9 2.83565
\(303\) 1.94308e8 0.401275
\(304\) 2.15559e9 4.40057
\(305\) −4.34759e8 −0.877402
\(306\) −3.26667e7 −0.0651749
\(307\) −5.34126e8 −1.05356 −0.526780 0.850001i \(-0.676601\pi\)
−0.526780 + 0.850001i \(0.676601\pi\)
\(308\) 1.23628e8 0.241095
\(309\) −7.48140e7 −0.144254
\(310\) 2.36482e9 4.50850
\(311\) 7.32126e8 1.38014 0.690072 0.723741i \(-0.257579\pi\)
0.690072 + 0.723741i \(0.257579\pi\)
\(312\) −9.25810e8 −1.72576
\(313\) 5.31534e8 0.979774 0.489887 0.871786i \(-0.337038\pi\)
0.489887 + 0.871786i \(0.337038\pi\)
\(314\) −1.02187e9 −1.86270
\(315\) 5.18056e8 0.933876
\(316\) −2.73978e9 −4.88439
\(317\) 1.07205e9 1.89020 0.945102 0.326775i \(-0.105962\pi\)
0.945102 + 0.326775i \(0.105962\pi\)
\(318\) 6.19662e8 1.08059
\(319\) −4.56388e7 −0.0787167
\(320\) −2.50411e9 −4.27197
\(321\) −2.53132e7 −0.0427149
\(322\) −1.83417e9 −3.06157
\(323\) 7.99581e7 0.132024
\(324\) 1.80817e8 0.295346
\(325\) 6.05970e8 0.979174
\(326\) 6.17598e8 0.987289
\(327\) −4.11577e8 −0.650930
\(328\) 2.39643e9 3.74979
\(329\) 4.30426e7 0.0666366
\(330\) −4.75839e7 −0.0728889
\(331\) −4.36057e7 −0.0660914 −0.0330457 0.999454i \(-0.510521\pi\)
−0.0330457 + 0.999454i \(0.510521\pi\)
\(332\) 6.97556e8 1.04615
\(333\) −3.97333e8 −0.589658
\(334\) 1.31001e9 1.92380
\(335\) 2.22507e7 0.0323361
\(336\) −2.68393e9 −3.85997
\(337\) −6.67140e8 −0.949538 −0.474769 0.880110i \(-0.657468\pi\)
−0.474769 + 0.880110i \(0.657468\pi\)
\(338\) 1.51559e8 0.213487
\(339\) −2.71009e8 −0.377819
\(340\) −2.81201e8 −0.388008
\(341\) −5.58790e7 −0.0763147
\(342\) −6.09087e8 −0.823356
\(343\) −2.71243e9 −3.62935
\(344\) −1.50592e9 −1.99456
\(345\) 5.12980e8 0.672565
\(346\) 1.91319e9 2.48308
\(347\) 3.15540e7 0.0405417 0.0202708 0.999795i \(-0.493547\pi\)
0.0202708 + 0.999795i \(0.493547\pi\)
\(348\) 2.05451e9 2.61324
\(349\) 7.24543e8 0.912379 0.456189 0.889883i \(-0.349214\pi\)
0.456189 + 0.889883i \(0.349214\pi\)
\(350\) 3.12711e9 3.89857
\(351\) 1.46958e8 0.181392
\(352\) 1.26561e8 0.154668
\(353\) 9.26501e8 1.12107 0.560537 0.828129i \(-0.310595\pi\)
0.560537 + 0.828129i \(0.310595\pi\)
\(354\) 1.19992e8 0.143761
\(355\) 3.39983e8 0.403328
\(356\) −5.72688e8 −0.672734
\(357\) −9.95560e7 −0.115805
\(358\) −2.00290e9 −2.30712
\(359\) −1.08624e9 −1.23906 −0.619532 0.784971i \(-0.712678\pi\)
−0.619532 + 0.784971i \(0.712678\pi\)
\(360\) 1.33621e9 1.50944
\(361\) 5.96986e8 0.667866
\(362\) −1.61456e8 −0.178885
\(363\) −5.25029e8 −0.576116
\(364\) −4.52318e9 −4.91574
\(365\) −1.37497e9 −1.48003
\(366\) −6.36438e8 −0.678535
\(367\) 8.58587e8 0.906678 0.453339 0.891338i \(-0.350233\pi\)
0.453339 + 0.891338i \(0.350233\pi\)
\(368\) −2.65764e9 −2.77990
\(369\) −3.80395e8 −0.394133
\(370\) −4.70707e9 −4.83108
\(371\) 1.88850e9 1.92003
\(372\) 2.51548e9 2.53350
\(373\) −1.80912e8 −0.180504 −0.0902518 0.995919i \(-0.528767\pi\)
−0.0902518 + 0.995919i \(0.528767\pi\)
\(374\) 9.14432e6 0.00903859
\(375\) −3.27217e7 −0.0320425
\(376\) 1.11019e8 0.107706
\(377\) 1.66979e9 1.60497
\(378\) 7.58375e8 0.722209
\(379\) −1.73658e9 −1.63854 −0.819270 0.573408i \(-0.805621\pi\)
−0.819270 + 0.573408i \(0.805621\pi\)
\(380\) −5.24313e9 −4.90172
\(381\) 2.44659e8 0.226634
\(382\) −1.42678e9 −1.30959
\(383\) 1.24471e9 1.13207 0.566035 0.824381i \(-0.308477\pi\)
0.566035 + 0.824381i \(0.308477\pi\)
\(384\) −1.52235e9 −1.37201
\(385\) −1.45018e8 −0.129512
\(386\) −2.91001e8 −0.257537
\(387\) 2.39041e8 0.209645
\(388\) 3.88322e9 3.37506
\(389\) −1.41664e9 −1.22021 −0.610106 0.792320i \(-0.708873\pi\)
−0.610106 + 0.792320i \(0.708873\pi\)
\(390\) 1.74096e9 1.48615
\(391\) −9.85807e7 −0.0834014
\(392\) −1.07783e10 −9.03750
\(393\) 5.65839e8 0.470239
\(394\) −3.44373e8 −0.283656
\(395\) 3.21382e9 2.62381
\(396\) −5.06156e7 −0.0409592
\(397\) −1.99168e9 −1.59755 −0.798773 0.601633i \(-0.794517\pi\)
−0.798773 + 0.601633i \(0.794517\pi\)
\(398\) −7.77808e8 −0.618417
\(399\) −1.85627e9 −1.46297
\(400\) 4.53105e9 3.53988
\(401\) −9.22799e8 −0.714664 −0.357332 0.933977i \(-0.616313\pi\)
−0.357332 + 0.933977i \(0.616313\pi\)
\(402\) 3.25725e7 0.0250069
\(403\) 2.04445e9 1.55599
\(404\) 2.44856e9 1.84746
\(405\) −2.12102e8 −0.158654
\(406\) 8.61694e9 6.39017
\(407\) 1.11225e8 0.0817750
\(408\) −2.56782e8 −0.187178
\(409\) −3.58871e7 −0.0259362 −0.0129681 0.999916i \(-0.504128\pi\)
−0.0129681 + 0.999916i \(0.504128\pi\)
\(410\) −4.50641e9 −3.22914
\(411\) 5.93534e8 0.421696
\(412\) −9.42763e8 −0.664144
\(413\) 3.65692e8 0.255441
\(414\) 7.50946e8 0.520124
\(415\) −8.18249e8 −0.561975
\(416\) −4.63049e9 −3.15355
\(417\) −1.74059e8 −0.117549
\(418\) 1.70500e8 0.114185
\(419\) −2.26805e8 −0.150627 −0.0753135 0.997160i \(-0.523996\pi\)
−0.0753135 + 0.997160i \(0.523996\pi\)
\(420\) 6.52823e9 4.29955
\(421\) 7.22132e8 0.471660 0.235830 0.971794i \(-0.424219\pi\)
0.235830 + 0.971794i \(0.424219\pi\)
\(422\) 4.39876e9 2.84929
\(423\) −1.76225e7 −0.0113208
\(424\) 4.87096e9 3.10337
\(425\) 1.68072e8 0.106202
\(426\) 4.97697e8 0.311912
\(427\) −1.93963e9 −1.20565
\(428\) −3.18982e8 −0.196659
\(429\) −4.11375e7 −0.0251558
\(430\) 2.83183e9 1.71762
\(431\) 2.37550e8 0.142917 0.0714586 0.997444i \(-0.477235\pi\)
0.0714586 + 0.997444i \(0.477235\pi\)
\(432\) 1.09885e9 0.655763
\(433\) −2.62934e8 −0.155646 −0.0778231 0.996967i \(-0.524797\pi\)
−0.0778231 + 0.996967i \(0.524797\pi\)
\(434\) 1.05504e10 6.19517
\(435\) −2.40998e9 −1.40379
\(436\) −5.18645e9 −2.99687
\(437\) −1.83809e9 −1.05361
\(438\) −2.01281e9 −1.14457
\(439\) 9.64067e8 0.543853 0.271926 0.962318i \(-0.412339\pi\)
0.271926 + 0.962318i \(0.412339\pi\)
\(440\) −3.74042e8 −0.209332
\(441\) 1.71088e9 0.949916
\(442\) −3.34564e8 −0.184290
\(443\) 2.26896e9 1.23998 0.619989 0.784611i \(-0.287137\pi\)
0.619989 + 0.784611i \(0.287137\pi\)
\(444\) −5.00696e9 −2.71478
\(445\) 6.71776e8 0.361380
\(446\) 2.69772e9 1.43988
\(447\) −9.14555e8 −0.484321
\(448\) −1.11718e10 −5.87016
\(449\) −6.29447e8 −0.328168 −0.164084 0.986446i \(-0.552467\pi\)
−0.164084 + 0.986446i \(0.552467\pi\)
\(450\) −1.28030e9 −0.662320
\(451\) 1.06483e8 0.0546592
\(452\) −3.41509e9 −1.73947
\(453\) −1.69359e9 −0.855981
\(454\) 3.54864e8 0.177978
\(455\) 5.30579e9 2.64064
\(456\) −4.78783e9 −2.36462
\(457\) −2.73973e9 −1.34277 −0.671383 0.741111i \(-0.734299\pi\)
−0.671383 + 0.741111i \(0.734299\pi\)
\(458\) 2.90281e9 1.41185
\(459\) 4.07601e7 0.0196739
\(460\) 6.46428e9 3.09648
\(461\) 1.50755e9 0.716668 0.358334 0.933593i \(-0.383345\pi\)
0.358334 + 0.933593i \(0.383345\pi\)
\(462\) −2.12290e8 −0.100157
\(463\) −7.07432e8 −0.331246 −0.165623 0.986189i \(-0.552964\pi\)
−0.165623 + 0.986189i \(0.552964\pi\)
\(464\) 1.24856e10 5.80225
\(465\) −2.95072e9 −1.36095
\(466\) 1.41583e9 0.648126
\(467\) −3.62529e9 −1.64715 −0.823575 0.567207i \(-0.808024\pi\)
−0.823575 + 0.567207i \(0.808024\pi\)
\(468\) 1.85188e9 0.835124
\(469\) 9.92690e7 0.0444333
\(470\) −2.08767e8 −0.0927513
\(471\) 1.27505e9 0.562280
\(472\) 9.43220e8 0.412873
\(473\) −6.69142e7 −0.0290740
\(474\) 4.70467e9 2.02911
\(475\) 3.13378e9 1.34166
\(476\) −1.25455e9 −0.533166
\(477\) −7.73188e8 −0.326190
\(478\) 3.44319e9 1.44199
\(479\) 1.79818e9 0.747580 0.373790 0.927513i \(-0.378058\pi\)
0.373790 + 0.927513i \(0.378058\pi\)
\(480\) 6.68312e9 2.75826
\(481\) −4.06938e9 −1.66733
\(482\) −2.98820e9 −1.21547
\(483\) 2.28860e9 0.924178
\(484\) −6.61611e9 −2.65243
\(485\) −4.55511e9 −1.81302
\(486\) −3.10493e8 −0.122695
\(487\) −1.58236e9 −0.620802 −0.310401 0.950606i \(-0.600463\pi\)
−0.310401 + 0.950606i \(0.600463\pi\)
\(488\) −5.00283e9 −1.94871
\(489\) −7.70613e8 −0.298027
\(490\) 2.02682e10 7.78268
\(491\) −2.99676e8 −0.114253 −0.0571264 0.998367i \(-0.518194\pi\)
−0.0571264 + 0.998367i \(0.518194\pi\)
\(492\) −4.79352e9 −1.81458
\(493\) 4.63132e8 0.174077
\(494\) −6.23810e9 −2.32814
\(495\) 5.93732e7 0.0220025
\(496\) 1.52870e10 5.62520
\(497\) 1.51680e9 0.554217
\(498\) −1.19782e9 −0.434601
\(499\) 1.35009e9 0.486419 0.243210 0.969974i \(-0.421800\pi\)
0.243210 + 0.969974i \(0.421800\pi\)
\(500\) −4.12340e8 −0.147523
\(501\) −1.63457e9 −0.580727
\(502\) −3.01016e9 −1.06200
\(503\) 1.94191e9 0.680365 0.340183 0.940359i \(-0.389511\pi\)
0.340183 + 0.940359i \(0.389511\pi\)
\(504\) 5.96134e9 2.07413
\(505\) −2.87222e9 −0.992424
\(506\) −2.10211e8 −0.0721320
\(507\) −1.89108e8 −0.0644441
\(508\) 3.08305e9 1.04342
\(509\) −1.85092e9 −0.622120 −0.311060 0.950390i \(-0.600684\pi\)
−0.311060 + 0.950390i \(0.600684\pi\)
\(510\) 4.82871e8 0.161189
\(511\) −6.13429e9 −2.03372
\(512\) −1.80551e9 −0.594504
\(513\) 7.59993e8 0.248541
\(514\) −8.86652e9 −2.87993
\(515\) 1.10588e9 0.356766
\(516\) 3.01225e9 0.965200
\(517\) 4.93302e6 0.00156999
\(518\) −2.10000e10 −6.63844
\(519\) −2.38720e9 −0.749553
\(520\) 1.36851e10 4.26811
\(521\) 2.35474e9 0.729475 0.364738 0.931110i \(-0.381159\pi\)
0.364738 + 0.931110i \(0.381159\pi\)
\(522\) −3.52794e9 −1.08561
\(523\) 1.86850e9 0.571133 0.285566 0.958359i \(-0.407818\pi\)
0.285566 + 0.958359i \(0.407818\pi\)
\(524\) 7.13037e9 2.16497
\(525\) −3.90188e9 −1.17684
\(526\) −2.59437e9 −0.777288
\(527\) 5.67047e8 0.168765
\(528\) −3.07600e8 −0.0909426
\(529\) −1.13864e9 −0.334420
\(530\) −9.15968e9 −2.67248
\(531\) −1.49721e8 −0.0433963
\(532\) −2.33916e10 −6.73550
\(533\) −3.89591e9 −1.11446
\(534\) 9.83405e8 0.279472
\(535\) 3.74173e8 0.105641
\(536\) 2.56042e8 0.0718183
\(537\) 2.49914e9 0.696434
\(538\) 5.31437e9 1.47134
\(539\) −4.78924e8 −0.131736
\(540\) −2.67278e9 −0.730442
\(541\) −1.58014e8 −0.0429046 −0.0214523 0.999770i \(-0.506829\pi\)
−0.0214523 + 0.999770i \(0.506829\pi\)
\(542\) −5.46261e9 −1.47368
\(543\) 2.01458e8 0.0539988
\(544\) −1.28431e9 −0.342038
\(545\) 6.08383e9 1.60986
\(546\) 7.76707e9 2.04213
\(547\) −4.19427e8 −0.109572 −0.0547862 0.998498i \(-0.517448\pi\)
−0.0547862 + 0.998498i \(0.517448\pi\)
\(548\) 7.47937e9 1.94148
\(549\) 7.94120e8 0.204825
\(550\) 3.58391e8 0.0918519
\(551\) 8.63532e9 2.19912
\(552\) 5.90294e9 1.49376
\(553\) 1.43381e10 3.60540
\(554\) −7.01258e9 −1.75224
\(555\) 5.87328e9 1.45833
\(556\) −2.19339e9 −0.541195
\(557\) −7.37086e9 −1.80728 −0.903639 0.428295i \(-0.859114\pi\)
−0.903639 + 0.428295i \(0.859114\pi\)
\(558\) −4.31952e9 −1.05249
\(559\) 2.44819e9 0.592795
\(560\) 3.96732e10 9.54640
\(561\) −1.14099e7 −0.00272842
\(562\) 3.85724e9 0.916642
\(563\) −5.40369e9 −1.27618 −0.638089 0.769963i \(-0.720275\pi\)
−0.638089 + 0.769963i \(0.720275\pi\)
\(564\) −2.22068e8 −0.0521206
\(565\) 4.00598e9 0.934414
\(566\) 1.00534e10 2.33054
\(567\) −9.46268e8 −0.218009
\(568\) 3.91224e9 0.895789
\(569\) −2.39098e9 −0.544104 −0.272052 0.962283i \(-0.587702\pi\)
−0.272052 + 0.962283i \(0.587702\pi\)
\(570\) 9.00337e9 2.03631
\(571\) −4.97857e9 −1.11912 −0.559562 0.828789i \(-0.689031\pi\)
−0.559562 + 0.828789i \(0.689031\pi\)
\(572\) −5.18391e8 −0.115817
\(573\) 1.78028e9 0.395318
\(574\) −2.01048e10 −4.43720
\(575\) −3.86365e9 −0.847541
\(576\) 4.57395e9 0.997270
\(577\) 3.57830e9 0.775464 0.387732 0.921772i \(-0.373259\pi\)
0.387732 + 0.921772i \(0.373259\pi\)
\(578\) 8.78645e9 1.89263
\(579\) 3.63099e8 0.0777410
\(580\) −3.03692e10 −6.46302
\(581\) −3.65053e9 −0.772216
\(582\) −6.66817e9 −1.40209
\(583\) 2.16437e8 0.0452367
\(584\) −1.58220e10 −3.28713
\(585\) −2.17229e9 −0.448614
\(586\) 1.65308e10 3.39354
\(587\) −2.26324e9 −0.461845 −0.230923 0.972972i \(-0.574174\pi\)
−0.230923 + 0.972972i \(0.574174\pi\)
\(588\) 2.15595e10 4.37340
\(589\) 1.05729e10 2.13201
\(590\) −1.77370e9 −0.355547
\(591\) 4.29694e8 0.0856255
\(592\) −3.04282e10 −6.02768
\(593\) 2.71803e9 0.535257 0.267628 0.963522i \(-0.413760\pi\)
0.267628 + 0.963522i \(0.413760\pi\)
\(594\) 8.69157e7 0.0170155
\(595\) 1.47161e9 0.286407
\(596\) −1.15247e10 −2.22981
\(597\) 9.70516e8 0.186678
\(598\) 7.69099e9 1.47071
\(599\) 5.48589e9 1.04293 0.521463 0.853274i \(-0.325386\pi\)
0.521463 + 0.853274i \(0.325386\pi\)
\(600\) −1.00640e10 −1.90214
\(601\) 5.20845e9 0.978695 0.489348 0.872089i \(-0.337235\pi\)
0.489348 + 0.872089i \(0.337235\pi\)
\(602\) 1.26339e10 2.36020
\(603\) −4.06427e7 −0.00754869
\(604\) −2.13416e10 −3.94092
\(605\) 7.76085e9 1.42484
\(606\) −4.20460e9 −0.767486
\(607\) −8.43683e9 −1.53115 −0.765577 0.643345i \(-0.777546\pi\)
−0.765577 + 0.643345i \(0.777546\pi\)
\(608\) −2.39466e10 −4.32097
\(609\) −1.07519e10 −1.92896
\(610\) 9.40766e9 1.67814
\(611\) −1.80485e8 −0.0320108
\(612\) 5.13635e8 0.0905785
\(613\) −6.65621e9 −1.16712 −0.583560 0.812070i \(-0.698341\pi\)
−0.583560 + 0.812070i \(0.698341\pi\)
\(614\) 1.15579e10 2.01506
\(615\) 5.62291e9 0.974761
\(616\) −1.66874e9 −0.287645
\(617\) 1.08431e10 1.85846 0.929231 0.369498i \(-0.120470\pi\)
0.929231 + 0.369498i \(0.120470\pi\)
\(618\) 1.61889e9 0.275903
\(619\) −1.05882e10 −1.79434 −0.897172 0.441681i \(-0.854382\pi\)
−0.897172 + 0.441681i \(0.854382\pi\)
\(620\) −3.71833e10 −6.26580
\(621\) −9.36998e8 −0.157007
\(622\) −1.58423e10 −2.63969
\(623\) 2.99705e9 0.496576
\(624\) 1.12542e10 1.85425
\(625\) −5.85706e9 −0.959621
\(626\) −1.15018e10 −1.87394
\(627\) −2.12743e8 −0.0344682
\(628\) 1.60674e10 2.58873
\(629\) −1.12868e9 −0.180840
\(630\) −1.12101e10 −1.78615
\(631\) 9.06740e9 1.43675 0.718373 0.695658i \(-0.244887\pi\)
0.718373 + 0.695658i \(0.244887\pi\)
\(632\) 3.69819e10 5.82746
\(633\) −5.48858e9 −0.860097
\(634\) −2.31980e10 −3.61524
\(635\) −3.61649e9 −0.560505
\(636\) −9.74326e9 −1.50177
\(637\) 1.75224e10 2.68600
\(638\) 9.87569e8 0.150555
\(639\) −6.21006e8 −0.0941548
\(640\) 2.25030e10 3.39322
\(641\) −6.94011e9 −1.04079 −0.520395 0.853926i \(-0.674215\pi\)
−0.520395 + 0.853926i \(0.674215\pi\)
\(642\) 5.47747e8 0.0816973
\(643\) −4.50631e9 −0.668471 −0.334235 0.942490i \(-0.608478\pi\)
−0.334235 + 0.942490i \(0.608478\pi\)
\(644\) 2.88396e10 4.25490
\(645\) −3.53344e9 −0.518488
\(646\) −1.73020e9 −0.252512
\(647\) 5.05823e9 0.734233 0.367116 0.930175i \(-0.380345\pi\)
0.367116 + 0.930175i \(0.380345\pi\)
\(648\) −2.44069e9 −0.352371
\(649\) 4.19112e7 0.00601829
\(650\) −1.31125e10 −1.87279
\(651\) −1.31643e10 −1.87010
\(652\) −9.71082e9 −1.37211
\(653\) −3.31489e9 −0.465879 −0.232940 0.972491i \(-0.574834\pi\)
−0.232940 + 0.972491i \(0.574834\pi\)
\(654\) 8.90604e9 1.24498
\(655\) −8.36409e9 −1.16298
\(656\) −2.91311e10 −4.02896
\(657\) 2.51149e9 0.345504
\(658\) −9.31391e8 −0.127450
\(659\) −9.88658e9 −1.34570 −0.672849 0.739780i \(-0.734929\pi\)
−0.672849 + 0.739780i \(0.734929\pi\)
\(660\) 7.48187e8 0.101299
\(661\) −1.76683e9 −0.237952 −0.118976 0.992897i \(-0.537961\pi\)
−0.118976 + 0.992897i \(0.537961\pi\)
\(662\) 9.43576e8 0.126408
\(663\) 4.17454e8 0.0556303
\(664\) −9.41571e9 −1.24814
\(665\) 2.74389e10 3.61819
\(666\) 8.59782e9 1.12779
\(667\) −1.06465e10 −1.38921
\(668\) −2.05979e10 −2.67366
\(669\) −3.36611e9 −0.434647
\(670\) −4.81479e8 −0.0618466
\(671\) −2.22296e8 −0.0284056
\(672\) 2.98160e10 3.79015
\(673\) −8.78145e9 −1.11049 −0.555244 0.831688i \(-0.687375\pi\)
−0.555244 + 0.831688i \(0.687375\pi\)
\(674\) 1.44361e10 1.81610
\(675\) 1.59750e9 0.199930
\(676\) −2.38303e9 −0.296699
\(677\) 1.26024e9 0.156096 0.0780482 0.996950i \(-0.475131\pi\)
0.0780482 + 0.996950i \(0.475131\pi\)
\(678\) 5.86430e9 0.722624
\(679\) −2.03221e10 −2.49129
\(680\) 3.79569e9 0.462924
\(681\) −4.42785e8 −0.0537251
\(682\) 1.20915e9 0.145961
\(683\) 1.20509e10 1.44726 0.723630 0.690188i \(-0.242472\pi\)
0.723630 + 0.690188i \(0.242472\pi\)
\(684\) 9.57698e9 1.14428
\(685\) −8.77348e9 −1.04293
\(686\) 5.86937e10 6.94156
\(687\) −3.62201e9 −0.426188
\(688\) 1.83060e10 2.14306
\(689\) −7.91878e9 −0.922340
\(690\) −1.11003e10 −1.28636
\(691\) −1.63179e9 −0.188144 −0.0940720 0.995565i \(-0.529988\pi\)
−0.0940720 + 0.995565i \(0.529988\pi\)
\(692\) −3.00820e10 −3.45093
\(693\) 2.64887e8 0.0302339
\(694\) −6.82792e8 −0.0775408
\(695\) 2.57290e9 0.290720
\(696\) −2.77320e10 −3.11781
\(697\) −1.08057e9 −0.120875
\(698\) −1.56782e10 −1.74503
\(699\) −1.76661e9 −0.195646
\(700\) −4.91692e10 −5.41813
\(701\) −4.92646e9 −0.540159 −0.270080 0.962838i \(-0.587050\pi\)
−0.270080 + 0.962838i \(0.587050\pi\)
\(702\) −3.17999e9 −0.346933
\(703\) −2.10448e10 −2.28456
\(704\) −1.28038e9 −0.138304
\(705\) 2.60491e8 0.0279982
\(706\) −2.00484e10 −2.14419
\(707\) −1.28141e10 −1.36370
\(708\) −1.88670e9 −0.199796
\(709\) 1.23998e10 1.30663 0.653315 0.757087i \(-0.273378\pi\)
0.653315 + 0.757087i \(0.273378\pi\)
\(710\) −7.35683e9 −0.771413
\(711\) −5.87029e9 −0.612514
\(712\) 7.73022e9 0.802624
\(713\) −1.30353e10 −1.34682
\(714\) 2.15427e9 0.221491
\(715\) 6.08085e8 0.0622147
\(716\) 3.14927e10 3.20637
\(717\) −4.29627e9 −0.435285
\(718\) 2.35049e10 2.36986
\(719\) −7.30887e9 −0.733329 −0.366665 0.930353i \(-0.619500\pi\)
−0.366665 + 0.930353i \(0.619500\pi\)
\(720\) −1.62430e10 −1.62182
\(721\) 4.93377e9 0.490236
\(722\) −1.29181e10 −1.27737
\(723\) 3.72854e9 0.366906
\(724\) 2.53865e9 0.248610
\(725\) 1.81514e10 1.76900
\(726\) 1.13610e10 1.10189
\(727\) −1.04087e9 −0.100467 −0.0502337 0.998737i \(-0.515997\pi\)
−0.0502337 + 0.998737i \(0.515997\pi\)
\(728\) 6.10545e10 5.86486
\(729\) 3.87420e8 0.0370370
\(730\) 2.97528e10 2.83073
\(731\) 6.79030e8 0.0642951
\(732\) 1.00070e10 0.943011
\(733\) 1.03455e10 0.970262 0.485131 0.874441i \(-0.338772\pi\)
0.485131 + 0.874441i \(0.338772\pi\)
\(734\) −1.85788e10 −1.73413
\(735\) −2.52898e10 −2.34931
\(736\) 2.95239e10 2.72961
\(737\) 1.13770e7 0.00104687
\(738\) 8.23131e9 0.753827
\(739\) −8.36230e9 −0.762203 −0.381101 0.924533i \(-0.624455\pi\)
−0.381101 + 0.924533i \(0.624455\pi\)
\(740\) 7.40117e10 6.71412
\(741\) 7.78364e9 0.702780
\(742\) −4.08649e10 −3.67228
\(743\) −1.24177e10 −1.11066 −0.555330 0.831630i \(-0.687408\pi\)
−0.555330 + 0.831630i \(0.687408\pi\)
\(744\) −3.39544e10 −3.02267
\(745\) 1.35187e10 1.19781
\(746\) 3.91472e9 0.345235
\(747\) 1.49460e9 0.131190
\(748\) −1.43781e8 −0.0125616
\(749\) 1.66933e9 0.145163
\(750\) 7.08059e8 0.0612851
\(751\) −4.11718e9 −0.354699 −0.177349 0.984148i \(-0.556752\pi\)
−0.177349 + 0.984148i \(0.556752\pi\)
\(752\) −1.34955e9 −0.115725
\(753\) 3.75595e9 0.320580
\(754\) −3.61322e10 −3.06970
\(755\) 2.50342e10 2.11699
\(756\) −1.19243e10 −1.00371
\(757\) 7.02287e9 0.588409 0.294204 0.955743i \(-0.404945\pi\)
0.294204 + 0.955743i \(0.404945\pi\)
\(758\) 3.75775e10 3.13390
\(759\) 2.62292e8 0.0217740
\(760\) 7.07725e10 5.84813
\(761\) −3.65462e9 −0.300605 −0.150302 0.988640i \(-0.548025\pi\)
−0.150302 + 0.988640i \(0.548025\pi\)
\(762\) −5.29414e9 −0.433464
\(763\) 2.71423e10 2.21213
\(764\) 2.24340e10 1.82004
\(765\) −6.02506e8 −0.0486571
\(766\) −2.69341e10 −2.16522
\(767\) −1.53341e9 −0.122708
\(768\) 1.12580e10 0.896804
\(769\) 1.51627e10 1.20236 0.601179 0.799114i \(-0.294698\pi\)
0.601179 + 0.799114i \(0.294698\pi\)
\(770\) 3.13802e9 0.247707
\(771\) 1.10633e10 0.869347
\(772\) 4.57556e9 0.357918
\(773\) 1.27681e10 0.994253 0.497126 0.867678i \(-0.334388\pi\)
0.497126 + 0.867678i \(0.334388\pi\)
\(774\) −5.17256e9 −0.400970
\(775\) 2.22242e10 1.71502
\(776\) −5.24163e10 −4.02671
\(777\) 2.62030e10 2.00390
\(778\) 3.06543e10 2.33380
\(779\) −2.01477e10 −1.52702
\(780\) −2.73740e10 −2.06541
\(781\) 1.73837e8 0.0130576
\(782\) 2.13317e9 0.159515
\(783\) 4.40202e9 0.327707
\(784\) 1.31021e11 9.71036
\(785\) −1.88474e10 −1.39062
\(786\) −1.22441e10 −0.899388
\(787\) −3.81227e9 −0.278787 −0.139394 0.990237i \(-0.544515\pi\)
−0.139394 + 0.990237i \(0.544515\pi\)
\(788\) 5.41476e9 0.394219
\(789\) 3.23715e9 0.234635
\(790\) −6.95433e10 −5.01834
\(791\) 1.78722e10 1.28399
\(792\) 6.83216e8 0.0488675
\(793\) 8.13317e9 0.579167
\(794\) 4.30976e10 3.05550
\(795\) 1.14291e10 0.806725
\(796\) 1.22299e10 0.859461
\(797\) −9.63179e9 −0.673911 −0.336956 0.941521i \(-0.609397\pi\)
−0.336956 + 0.941521i \(0.609397\pi\)
\(798\) 4.01675e10 2.79811
\(799\) −5.00592e7 −0.00347192
\(800\) −5.03357e10 −3.47585
\(801\) −1.22705e9 −0.0843624
\(802\) 1.99683e10 1.36688
\(803\) −7.03037e8 −0.0479153
\(804\) −5.12155e8 −0.0347540
\(805\) −3.38296e10 −2.28566
\(806\) −4.42394e10 −2.97603
\(807\) −6.63105e9 −0.444145
\(808\) −3.30510e10 −2.20417
\(809\) 1.38532e9 0.0919878 0.0459939 0.998942i \(-0.485355\pi\)
0.0459939 + 0.998942i \(0.485355\pi\)
\(810\) 4.58963e9 0.303445
\(811\) 1.41931e10 0.934339 0.467170 0.884168i \(-0.345274\pi\)
0.467170 + 0.884168i \(0.345274\pi\)
\(812\) −1.35489e11 −8.88090
\(813\) 6.81602e9 0.444850
\(814\) −2.40677e9 −0.156404
\(815\) 1.13910e10 0.737073
\(816\) 3.12145e9 0.201113
\(817\) 1.26608e10 0.812242
\(818\) 7.76554e8 0.0496061
\(819\) −9.69143e9 −0.616445
\(820\) 7.08566e10 4.48778
\(821\) −1.94897e10 −1.22915 −0.614573 0.788860i \(-0.710672\pi\)
−0.614573 + 0.788860i \(0.710672\pi\)
\(822\) −1.28434e10 −0.806544
\(823\) 7.94309e9 0.496696 0.248348 0.968671i \(-0.420112\pi\)
0.248348 + 0.968671i \(0.420112\pi\)
\(824\) 1.27255e10 0.792376
\(825\) −4.47186e8 −0.0277268
\(826\) −7.91314e9 −0.488561
\(827\) 1.74454e10 1.07254 0.536268 0.844048i \(-0.319834\pi\)
0.536268 + 0.844048i \(0.319834\pi\)
\(828\) −1.18075e10 −0.722856
\(829\) 2.55780e10 1.55929 0.779644 0.626223i \(-0.215400\pi\)
0.779644 + 0.626223i \(0.215400\pi\)
\(830\) 1.77059e10 1.07484
\(831\) 8.75000e9 0.528938
\(832\) 4.68452e10 2.81990
\(833\) 4.86001e9 0.291326
\(834\) 3.76643e9 0.224827
\(835\) 2.41618e10 1.43624
\(836\) −2.68086e9 −0.158691
\(837\) 5.38972e9 0.317707
\(838\) 4.90778e9 0.288092
\(839\) −7.34489e9 −0.429357 −0.214678 0.976685i \(-0.568870\pi\)
−0.214678 + 0.976685i \(0.568870\pi\)
\(840\) −8.81190e10 −5.12970
\(841\) 3.27675e10 1.89958
\(842\) −1.56261e10 −0.902106
\(843\) −4.81290e9 −0.276701
\(844\) −6.91639e10 −3.95987
\(845\) 2.79535e9 0.159382
\(846\) 3.81329e8 0.0216523
\(847\) 3.46241e10 1.95788
\(848\) −5.92115e10 −3.33442
\(849\) −1.25443e10 −0.703506
\(850\) −3.63687e9 −0.203124
\(851\) 2.59463e10 1.44318
\(852\) −7.82555e9 −0.433487
\(853\) −3.22700e10 −1.78023 −0.890116 0.455734i \(-0.849377\pi\)
−0.890116 + 0.455734i \(0.849377\pi\)
\(854\) 4.19712e10 2.30594
\(855\) −1.12340e10 −0.614687
\(856\) 4.30566e9 0.234629
\(857\) −7.06953e9 −0.383670 −0.191835 0.981427i \(-0.561444\pi\)
−0.191835 + 0.981427i \(0.561444\pi\)
\(858\) 8.90168e8 0.0481134
\(859\) −2.05380e10 −1.10556 −0.552779 0.833328i \(-0.686432\pi\)
−0.552779 + 0.833328i \(0.686432\pi\)
\(860\) −4.45264e10 −2.38711
\(861\) 2.50860e10 1.33943
\(862\) −5.14030e9 −0.273346
\(863\) −3.50261e10 −1.85504 −0.927520 0.373773i \(-0.878064\pi\)
−0.927520 + 0.373773i \(0.878064\pi\)
\(864\) −1.22072e10 −0.643901
\(865\) 3.52869e10 1.85378
\(866\) 5.68957e9 0.297692
\(867\) −1.09634e10 −0.571317
\(868\) −1.65889e11 −8.60990
\(869\) 1.64326e9 0.0849447
\(870\) 5.21492e10 2.68491
\(871\) −4.16251e8 −0.0213448
\(872\) 7.00075e10 3.57550
\(873\) 8.32026e9 0.423240
\(874\) 3.97740e10 2.01516
\(875\) 2.15790e9 0.108894
\(876\) 3.16484e10 1.59070
\(877\) 2.05838e10 1.03045 0.515225 0.857055i \(-0.327708\pi\)
0.515225 + 0.857055i \(0.327708\pi\)
\(878\) −2.08613e10 −1.04018
\(879\) −2.06265e10 −1.02439
\(880\) 4.54686e9 0.224917
\(881\) 3.80144e9 0.187298 0.0936488 0.995605i \(-0.470147\pi\)
0.0936488 + 0.995605i \(0.470147\pi\)
\(882\) −3.70215e10 −1.81683
\(883\) −2.46327e10 −1.20406 −0.602032 0.798472i \(-0.705642\pi\)
−0.602032 + 0.798472i \(0.705642\pi\)
\(884\) 5.26052e9 0.256121
\(885\) 2.21314e9 0.107327
\(886\) −4.90976e10 −2.37161
\(887\) −2.28817e10 −1.10092 −0.550461 0.834861i \(-0.685548\pi\)
−0.550461 + 0.834861i \(0.685548\pi\)
\(888\) 6.75847e10 3.23894
\(889\) −1.61346e10 −0.770196
\(890\) −1.45364e10 −0.691183
\(891\) −1.08450e8 −0.00513637
\(892\) −4.24177e10 −2.00111
\(893\) −9.33378e8 −0.0438609
\(894\) 1.97899e10 0.926322
\(895\) −3.69416e10 −1.72241
\(896\) 1.00395e11 4.66265
\(897\) −9.59649e9 −0.443955
\(898\) 1.36205e10 0.627661
\(899\) 6.12400e10 2.81110
\(900\) 2.01308e10 0.920476
\(901\) −2.19635e9 −0.100038
\(902\) −2.30417e9 −0.104542
\(903\) −1.57640e10 −0.712460
\(904\) 4.60974e10 2.07533
\(905\) −2.97790e9 −0.133549
\(906\) 3.66472e10 1.63717
\(907\) 2.43988e10 1.08578 0.542892 0.839802i \(-0.317329\pi\)
0.542892 + 0.839802i \(0.317329\pi\)
\(908\) −5.57971e9 −0.247350
\(909\) 5.24632e9 0.231676
\(910\) −1.14811e11 −5.05055
\(911\) 7.15550e9 0.313563 0.156782 0.987633i \(-0.449888\pi\)
0.156782 + 0.987633i \(0.449888\pi\)
\(912\) 5.82010e10 2.54067
\(913\) −4.18379e8 −0.0181937
\(914\) 5.92844e10 2.56820
\(915\) −1.17385e10 −0.506569
\(916\) −4.56424e10 −1.96216
\(917\) −3.73154e10 −1.59807
\(918\) −8.82001e8 −0.0376287
\(919\) −3.31022e10 −1.40687 −0.703433 0.710761i \(-0.748351\pi\)
−0.703433 + 0.710761i \(0.748351\pi\)
\(920\) −8.72558e10 −3.69434
\(921\) −1.44214e10 −0.608274
\(922\) −3.26216e10 −1.37071
\(923\) −6.36017e9 −0.266234
\(924\) 3.33795e9 0.139196
\(925\) −4.42362e10 −1.83773
\(926\) 1.53080e10 0.633548
\(927\) −2.01998e9 −0.0832852
\(928\) −1.38703e11 −5.69729
\(929\) 4.50956e10 1.84535 0.922676 0.385577i \(-0.125997\pi\)
0.922676 + 0.385577i \(0.125997\pi\)
\(930\) 6.38501e10 2.60298
\(931\) 9.06172e10 3.68033
\(932\) −2.22618e10 −0.900750
\(933\) 1.97674e10 0.796827
\(934\) 7.84469e10 3.15037
\(935\) 1.68658e8 0.00674787
\(936\) −2.49969e10 −0.996369
\(937\) 2.71853e10 1.07956 0.539779 0.841807i \(-0.318508\pi\)
0.539779 + 0.841807i \(0.318508\pi\)
\(938\) −2.14806e9 −0.0849840
\(939\) 1.43514e10 0.565673
\(940\) 3.28256e9 0.128903
\(941\) −4.59907e10 −1.79931 −0.899655 0.436601i \(-0.856182\pi\)
−0.899655 + 0.436601i \(0.856182\pi\)
\(942\) −2.75905e10 −1.07543
\(943\) 2.48402e10 0.964638
\(944\) −1.14658e10 −0.443611
\(945\) 1.39875e10 0.539174
\(946\) 1.44794e9 0.0556074
\(947\) 4.47468e10 1.71213 0.856065 0.516868i \(-0.172902\pi\)
0.856065 + 0.516868i \(0.172902\pi\)
\(948\) −7.39740e10 −2.82000
\(949\) 2.57221e10 0.976954
\(950\) −6.78113e10 −2.56608
\(951\) 2.89454e10 1.09131
\(952\) 1.69340e10 0.636109
\(953\) −6.72893e9 −0.251838 −0.125919 0.992041i \(-0.540188\pi\)
−0.125919 + 0.992041i \(0.540188\pi\)
\(954\) 1.67309e10 0.623877
\(955\) −2.63156e10 −0.977692
\(956\) −5.41390e10 −2.00405
\(957\) −1.23225e9 −0.0454471
\(958\) −3.89104e10 −1.42984
\(959\) −3.91419e10 −1.43310
\(960\) −6.76110e10 −2.46643
\(961\) 4.74681e10 1.72532
\(962\) 8.80566e10 3.18896
\(963\) −6.83456e8 −0.0246614
\(964\) 4.69849e10 1.68923
\(965\) −5.36724e9 −0.192267
\(966\) −4.95227e10 −1.76760
\(967\) −1.86690e9 −0.0663941 −0.0331970 0.999449i \(-0.510569\pi\)
−0.0331970 + 0.999449i \(0.510569\pi\)
\(968\) 8.93052e10 3.16456
\(969\) 2.15887e9 0.0762242
\(970\) 9.85672e10 3.46762
\(971\) 2.50323e10 0.877471 0.438735 0.898616i \(-0.355427\pi\)
0.438735 + 0.898616i \(0.355427\pi\)
\(972\) 4.88205e9 0.170518
\(973\) 1.14787e10 0.399482
\(974\) 3.42403e10 1.18736
\(975\) 1.63612e10 0.565326
\(976\) 6.08145e10 2.09379
\(977\) 3.21353e10 1.10243 0.551216 0.834363i \(-0.314164\pi\)
0.551216 + 0.834363i \(0.314164\pi\)
\(978\) 1.66752e10 0.570012
\(979\) 3.43486e8 0.0116996
\(980\) −3.18688e11 −10.8162
\(981\) −1.11126e10 −0.375814
\(982\) 6.48464e9 0.218522
\(983\) −2.52657e10 −0.848387 −0.424194 0.905572i \(-0.639442\pi\)
−0.424194 + 0.905572i \(0.639442\pi\)
\(984\) 6.47036e10 2.16494
\(985\) −6.35163e9 −0.211767
\(986\) −1.00216e10 −0.332942
\(987\) 1.16215e9 0.0384727
\(988\) 9.80849e10 3.23559
\(989\) −1.56096e10 −0.513104
\(990\) −1.28477e9 −0.0420824
\(991\) −5.79422e10 −1.89120 −0.945600 0.325331i \(-0.894524\pi\)
−0.945600 + 0.325331i \(0.894524\pi\)
\(992\) −1.69825e11 −5.52344
\(993\) −1.17735e9 −0.0381579
\(994\) −3.28217e10 −1.06001
\(995\) −1.43459e10 −0.461687
\(996\) 1.88340e10 0.603998
\(997\) −4.76882e10 −1.52398 −0.761988 0.647591i \(-0.775777\pi\)
−0.761988 + 0.647591i \(0.775777\pi\)
\(998\) −2.92143e10 −0.930334
\(999\) −1.07280e10 −0.340439
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.b.1.2 17
3.2 odd 2 531.8.a.d.1.16 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.2 17 1.1 even 1 trivial
531.8.a.d.1.16 17 3.2 odd 2