Properties

Label 91.8.a.e.1.3
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(15.8833\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.8833 q^{2} -5.24717 q^{3} +93.5131 q^{4} -195.689 q^{5} +78.0952 q^{6} +343.000 q^{7} +513.280 q^{8} -2159.47 q^{9} +O(q^{10})\) \(q-14.8833 q^{2} -5.24717 q^{3} +93.5131 q^{4} -195.689 q^{5} +78.0952 q^{6} +343.000 q^{7} +513.280 q^{8} -2159.47 q^{9} +2912.49 q^{10} -3183.85 q^{11} -490.679 q^{12} +2197.00 q^{13} -5104.98 q^{14} +1026.81 q^{15} -19609.0 q^{16} +6858.42 q^{17} +32140.0 q^{18} -42564.0 q^{19} -18299.4 q^{20} -1799.78 q^{21} +47386.2 q^{22} -65607.9 q^{23} -2693.27 q^{24} -39831.0 q^{25} -32698.6 q^{26} +22806.6 q^{27} +32075.0 q^{28} +64709.1 q^{29} -15282.3 q^{30} -314040. q^{31} +226147. q^{32} +16706.2 q^{33} -102076. q^{34} -67121.2 q^{35} -201938. q^{36} +409330. q^{37} +633493. q^{38} -11528.0 q^{39} -100443. q^{40} +212866. q^{41} +26786.7 q^{42} +701159. q^{43} -297731. q^{44} +422583. q^{45} +976462. q^{46} +1.15821e6 q^{47} +102892. q^{48} +117649. q^{49} +592817. q^{50} -35987.3 q^{51} +205448. q^{52} +237438. q^{53} -339438. q^{54} +623042. q^{55} +176055. q^{56} +223340. q^{57} -963086. q^{58} +1.50451e6 q^{59} +96020.2 q^{60} -2.58681e6 q^{61} +4.67396e6 q^{62} -740697. q^{63} -855864. q^{64} -429928. q^{65} -248643. q^{66} -3.86220e6 q^{67} +641352. q^{68} +344255. q^{69} +998986. q^{70} +216598. q^{71} -1.10841e6 q^{72} +4.29213e6 q^{73} -6.09219e6 q^{74} +209000. q^{75} -3.98029e6 q^{76} -1.09206e6 q^{77} +171575. q^{78} +6.66778e6 q^{79} +3.83725e6 q^{80} +4.60308e6 q^{81} -3.16815e6 q^{82} +1.18185e6 q^{83} -168303. q^{84} -1.34211e6 q^{85} -1.04356e7 q^{86} -339539. q^{87} -1.63420e6 q^{88} -9.25815e6 q^{89} -6.28944e6 q^{90} +753571. q^{91} -6.13519e6 q^{92} +1.64782e6 q^{93} -1.72380e7 q^{94} +8.32929e6 q^{95} -1.18663e6 q^{96} -414756. q^{97} -1.75101e6 q^{98} +6.87541e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9} + 6668 q^{10} + 12168 q^{11} - 183 q^{12} + 26364 q^{13} + 2058 q^{14} - 28790 q^{15} + 85914 q^{16} + 82710 q^{17} - 44965 q^{18} - 10302 q^{19} + 141318 q^{20} + 28126 q^{21} - 97457 q^{22} + 98376 q^{23} - 519981 q^{24} + 272736 q^{25} + 13182 q^{26} + 306652 q^{27} + 338198 q^{28} + 350592 q^{29} + 231528 q^{30} + 55092 q^{31} + 114420 q^{32} + 609912 q^{33} + 812002 q^{34} + 351918 q^{35} + 1472143 q^{36} + 376310 q^{37} + 2825424 q^{38} + 180154 q^{39} + 2169290 q^{40} + 1387272 q^{41} + 105987 q^{42} + 568708 q^{43} + 3392031 q^{44} + 3556226 q^{45} - 1736829 q^{46} + 1359444 q^{47} + 4151249 q^{48} + 1411788 q^{49} + 3983712 q^{50} + 2709260 q^{51} + 2166242 q^{52} + 2061780 q^{53} + 2196651 q^{54} - 2112846 q^{55} + 78204 q^{56} + 2359902 q^{57} + 670268 q^{58} + 395964 q^{59} - 1052376 q^{60} + 444006 q^{61} + 2854353 q^{62} + 3739386 q^{63} + 12026858 q^{64} + 2254122 q^{65} - 4605681 q^{66} - 3094010 q^{67} + 4668954 q^{68} + 3839892 q^{69} + 2287124 q^{70} + 5694366 q^{71} - 9780585 q^{72} + 7052346 q^{73} - 4436259 q^{74} - 16288696 q^{75} - 3051830 q^{76} + 4173624 q^{77} + 678873 q^{78} + 4304160 q^{79} + 3807018 q^{80} - 6689556 q^{81} - 4733665 q^{82} + 2704554 q^{83} - 62769 q^{84} + 9301878 q^{85} + 1510998 q^{86} + 16231802 q^{87} - 70453923 q^{88} - 10986042 q^{89} - 12851300 q^{90} + 9042852 q^{91} - 16505451 q^{92} - 47230934 q^{93} - 24306151 q^{94} - 21839424 q^{95} - 86512741 q^{96} - 24462382 q^{97} + 705894 q^{98} + 11555078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.8833 −1.31551 −0.657756 0.753231i \(-0.728494\pi\)
−0.657756 + 0.753231i \(0.728494\pi\)
\(3\) −5.24717 −0.112202 −0.0561010 0.998425i \(-0.517867\pi\)
−0.0561010 + 0.998425i \(0.517867\pi\)
\(4\) 93.5131 0.730571
\(5\) −195.689 −0.700117 −0.350058 0.936728i \(-0.613838\pi\)
−0.350058 + 0.936728i \(0.613838\pi\)
\(6\) 78.0952 0.147603
\(7\) 343.000 0.377964
\(8\) 513.280 0.354437
\(9\) −2159.47 −0.987411
\(10\) 2912.49 0.921012
\(11\) −3183.85 −0.721236 −0.360618 0.932714i \(-0.617434\pi\)
−0.360618 + 0.932714i \(0.617434\pi\)
\(12\) −490.679 −0.0819715
\(13\) 2197.00 0.277350
\(14\) −5104.98 −0.497217
\(15\) 1026.81 0.0785545
\(16\) −19609.0 −1.19684
\(17\) 6858.42 0.338573 0.169287 0.985567i \(-0.445854\pi\)
0.169287 + 0.985567i \(0.445854\pi\)
\(18\) 32140.0 1.29895
\(19\) −42564.0 −1.42365 −0.711827 0.702354i \(-0.752132\pi\)
−0.711827 + 0.702354i \(0.752132\pi\)
\(20\) −18299.4 −0.511485
\(21\) −1799.78 −0.0424084
\(22\) 47386.2 0.948795
\(23\) −65607.9 −1.12437 −0.562184 0.827012i \(-0.690039\pi\)
−0.562184 + 0.827012i \(0.690039\pi\)
\(24\) −2693.27 −0.0397685
\(25\) −39831.0 −0.509837
\(26\) −32698.6 −0.364857
\(27\) 22806.6 0.222991
\(28\) 32075.0 0.276130
\(29\) 64709.1 0.492689 0.246344 0.969182i \(-0.420771\pi\)
0.246344 + 0.969182i \(0.420771\pi\)
\(30\) −15282.3 −0.103339
\(31\) −314040. −1.89330 −0.946649 0.322265i \(-0.895556\pi\)
−0.946649 + 0.322265i \(0.895556\pi\)
\(32\) 226147. 1.22002
\(33\) 16706.2 0.0809241
\(34\) −102076. −0.445397
\(35\) −67121.2 −0.264619
\(36\) −201938. −0.721373
\(37\) 409330. 1.32852 0.664259 0.747502i \(-0.268747\pi\)
0.664259 + 0.747502i \(0.268747\pi\)
\(38\) 633493. 1.87283
\(39\) −11528.0 −0.0311192
\(40\) −100443. −0.248147
\(41\) 212866. 0.482350 0.241175 0.970482i \(-0.422467\pi\)
0.241175 + 0.970482i \(0.422467\pi\)
\(42\) 26786.7 0.0557887
\(43\) 701159. 1.34486 0.672429 0.740161i \(-0.265251\pi\)
0.672429 + 0.740161i \(0.265251\pi\)
\(44\) −297731. −0.526914
\(45\) 422583. 0.691303
\(46\) 976462. 1.47912
\(47\) 1.15821e6 1.62721 0.813607 0.581415i \(-0.197501\pi\)
0.813607 + 0.581415i \(0.197501\pi\)
\(48\) 102892. 0.134287
\(49\) 117649. 0.142857
\(50\) 592817. 0.670696
\(51\) −35987.3 −0.0379886
\(52\) 205448. 0.202624
\(53\) 237438. 0.219071 0.109535 0.993983i \(-0.465064\pi\)
0.109535 + 0.993983i \(0.465064\pi\)
\(54\) −339438. −0.293348
\(55\) 623042. 0.504950
\(56\) 176055. 0.133965
\(57\) 223340. 0.159737
\(58\) −963086. −0.648138
\(59\) 1.50451e6 0.953700 0.476850 0.878985i \(-0.341778\pi\)
0.476850 + 0.878985i \(0.341778\pi\)
\(60\) 96020.2 0.0573896
\(61\) −2.58681e6 −1.45918 −0.729592 0.683883i \(-0.760290\pi\)
−0.729592 + 0.683883i \(0.760290\pi\)
\(62\) 4.67396e6 2.49066
\(63\) −740697. −0.373206
\(64\) −855864. −0.408108
\(65\) −429928. −0.194177
\(66\) −248643. −0.106457
\(67\) −3.86220e6 −1.56882 −0.784410 0.620242i \(-0.787034\pi\)
−0.784410 + 0.620242i \(0.787034\pi\)
\(68\) 641352. 0.247352
\(69\) 344255. 0.126156
\(70\) 998986. 0.348110
\(71\) 216598. 0.0718207 0.0359104 0.999355i \(-0.488567\pi\)
0.0359104 + 0.999355i \(0.488567\pi\)
\(72\) −1.10841e6 −0.349975
\(73\) 4.29213e6 1.29135 0.645673 0.763614i \(-0.276577\pi\)
0.645673 + 0.763614i \(0.276577\pi\)
\(74\) −6.09219e6 −1.74768
\(75\) 209000. 0.0572047
\(76\) −3.98029e6 −1.04008
\(77\) −1.09206e6 −0.272602
\(78\) 171575. 0.0409377
\(79\) 6.66778e6 1.52155 0.760775 0.649015i \(-0.224819\pi\)
0.760775 + 0.649015i \(0.224819\pi\)
\(80\) 3.83725e6 0.837926
\(81\) 4.60308e6 0.962391
\(82\) −3.16815e6 −0.634537
\(83\) 1.18185e6 0.226877 0.113438 0.993545i \(-0.463814\pi\)
0.113438 + 0.993545i \(0.463814\pi\)
\(84\) −168303. −0.0309823
\(85\) −1.34211e6 −0.237041
\(86\) −1.04356e7 −1.76918
\(87\) −339539. −0.0552806
\(88\) −1.63420e6 −0.255633
\(89\) −9.25815e6 −1.39206 −0.696032 0.718011i \(-0.745053\pi\)
−0.696032 + 0.718011i \(0.745053\pi\)
\(90\) −6.28944e6 −0.909417
\(91\) 753571. 0.104828
\(92\) −6.13519e6 −0.821430
\(93\) 1.64782e6 0.212432
\(94\) −1.72380e7 −2.14062
\(95\) 8.32929e6 0.996724
\(96\) −1.18663e6 −0.136888
\(97\) −414756. −0.0461415 −0.0230708 0.999734i \(-0.507344\pi\)
−0.0230708 + 0.999734i \(0.507344\pi\)
\(98\) −1.75101e6 −0.187930
\(99\) 6.87541e6 0.712156
\(100\) −3.72472e6 −0.372472
\(101\) −86743.8 −0.00837749 −0.00418874 0.999991i \(-0.501333\pi\)
−0.00418874 + 0.999991i \(0.501333\pi\)
\(102\) 535610. 0.0499744
\(103\) 1.24751e7 1.12490 0.562449 0.826832i \(-0.309859\pi\)
0.562449 + 0.826832i \(0.309859\pi\)
\(104\) 1.12768e6 0.0983032
\(105\) 352196. 0.0296908
\(106\) −3.53386e6 −0.288190
\(107\) −5.56743e6 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(108\) 2.13272e6 0.162911
\(109\) −2.35923e7 −1.74493 −0.872464 0.488678i \(-0.837479\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(110\) −9.27293e6 −0.664267
\(111\) −2.14782e6 −0.149062
\(112\) −6.72588e6 −0.452362
\(113\) 2.53551e7 1.65307 0.826535 0.562885i \(-0.190309\pi\)
0.826535 + 0.562885i \(0.190309\pi\)
\(114\) −3.32404e6 −0.210136
\(115\) 1.28387e7 0.787189
\(116\) 6.05115e6 0.359944
\(117\) −4.74435e6 −0.273858
\(118\) −2.23920e7 −1.25460
\(119\) 2.35244e6 0.127969
\(120\) 527041. 0.0278426
\(121\) −9.35030e6 −0.479818
\(122\) 3.85003e7 1.91957
\(123\) −1.11694e6 −0.0541206
\(124\) −2.93668e7 −1.38319
\(125\) 2.30826e7 1.05706
\(126\) 1.10240e7 0.490957
\(127\) 2.31858e7 1.00441 0.502203 0.864750i \(-0.332523\pi\)
0.502203 + 0.864750i \(0.332523\pi\)
\(128\) −1.62087e7 −0.683145
\(129\) −3.67910e6 −0.150896
\(130\) 6.39875e6 0.255443
\(131\) 2.60882e7 1.01390 0.506949 0.861976i \(-0.330773\pi\)
0.506949 + 0.861976i \(0.330773\pi\)
\(132\) 1.56224e6 0.0591208
\(133\) −1.45994e7 −0.538091
\(134\) 5.74824e7 2.06380
\(135\) −4.46300e6 −0.156120
\(136\) 3.52029e6 0.120003
\(137\) 3.05155e7 1.01391 0.506953 0.861974i \(-0.330772\pi\)
0.506953 + 0.861974i \(0.330772\pi\)
\(138\) −5.12366e6 −0.165960
\(139\) −2.88820e7 −0.912168 −0.456084 0.889937i \(-0.650748\pi\)
−0.456084 + 0.889937i \(0.650748\pi\)
\(140\) −6.27671e6 −0.193323
\(141\) −6.07732e6 −0.182577
\(142\) −3.22369e6 −0.0944810
\(143\) −6.99491e6 −0.200035
\(144\) 4.23449e7 1.18177
\(145\) −1.26628e7 −0.344939
\(146\) −6.38811e7 −1.69878
\(147\) −617324. −0.0160289
\(148\) 3.82777e7 0.970577
\(149\) 2.59242e7 0.642027 0.321013 0.947075i \(-0.395977\pi\)
0.321013 + 0.947075i \(0.395977\pi\)
\(150\) −3.11061e6 −0.0752534
\(151\) 1.39638e7 0.330054 0.165027 0.986289i \(-0.447229\pi\)
0.165027 + 0.986289i \(0.447229\pi\)
\(152\) −2.18472e7 −0.504596
\(153\) −1.48105e7 −0.334311
\(154\) 1.62535e7 0.358611
\(155\) 6.14541e7 1.32553
\(156\) −1.07802e6 −0.0227348
\(157\) 2.23968e7 0.461888 0.230944 0.972967i \(-0.425819\pi\)
0.230944 + 0.972967i \(0.425819\pi\)
\(158\) −9.92387e7 −2.00162
\(159\) −1.24588e6 −0.0245802
\(160\) −4.42543e7 −0.854153
\(161\) −2.25035e7 −0.424971
\(162\) −6.85092e7 −1.26604
\(163\) −7.88430e7 −1.42596 −0.712979 0.701185i \(-0.752655\pi\)
−0.712979 + 0.701185i \(0.752655\pi\)
\(164\) 1.99057e7 0.352391
\(165\) −3.26921e6 −0.0566563
\(166\) −1.75899e7 −0.298459
\(167\) −1.17393e8 −1.95044 −0.975221 0.221231i \(-0.928993\pi\)
−0.975221 + 0.221231i \(0.928993\pi\)
\(168\) −923790. −0.0150311
\(169\) 4.82681e6 0.0769231
\(170\) 1.99751e7 0.311830
\(171\) 9.19155e7 1.40573
\(172\) 6.55675e7 0.982514
\(173\) 8.56556e7 1.25775 0.628875 0.777507i \(-0.283516\pi\)
0.628875 + 0.777507i \(0.283516\pi\)
\(174\) 5.05347e6 0.0727223
\(175\) −1.36620e7 −0.192700
\(176\) 6.24320e7 0.863202
\(177\) −7.89439e6 −0.107007
\(178\) 1.37792e8 1.83128
\(179\) 1.16626e8 1.51989 0.759943 0.649990i \(-0.225227\pi\)
0.759943 + 0.649990i \(0.225227\pi\)
\(180\) 3.95170e7 0.505046
\(181\) 6.33015e6 0.0793485 0.0396743 0.999213i \(-0.487368\pi\)
0.0396743 + 0.999213i \(0.487368\pi\)
\(182\) −1.12156e7 −0.137903
\(183\) 1.35734e7 0.163723
\(184\) −3.36752e7 −0.398518
\(185\) −8.01012e7 −0.930118
\(186\) −2.45250e7 −0.279457
\(187\) −2.18361e7 −0.244191
\(188\) 1.08308e8 1.18880
\(189\) 7.82268e6 0.0842828
\(190\) −1.23967e8 −1.31120
\(191\) 8.90254e7 0.924480 0.462240 0.886755i \(-0.347046\pi\)
0.462240 + 0.886755i \(0.347046\pi\)
\(192\) 4.49086e6 0.0457905
\(193\) 9.90402e7 0.991655 0.495828 0.868421i \(-0.334865\pi\)
0.495828 + 0.868421i \(0.334865\pi\)
\(194\) 6.17295e6 0.0606997
\(195\) 2.25590e6 0.0217871
\(196\) 1.10017e7 0.104367
\(197\) −4.54182e7 −0.423251 −0.211625 0.977351i \(-0.567876\pi\)
−0.211625 + 0.977351i \(0.567876\pi\)
\(198\) −1.02329e8 −0.936850
\(199\) −1.42078e7 −0.127803 −0.0639016 0.997956i \(-0.520354\pi\)
−0.0639016 + 0.997956i \(0.520354\pi\)
\(200\) −2.04444e7 −0.180705
\(201\) 2.02656e7 0.176025
\(202\) 1.29104e6 0.0110207
\(203\) 2.21952e7 0.186219
\(204\) −3.36528e6 −0.0277533
\(205\) −4.16554e7 −0.337701
\(206\) −1.85671e8 −1.47982
\(207\) 1.41678e8 1.11021
\(208\) −4.30809e7 −0.331943
\(209\) 1.35517e8 1.02679
\(210\) −5.24184e6 −0.0390586
\(211\) 1.16959e8 0.857127 0.428563 0.903512i \(-0.359020\pi\)
0.428563 + 0.903512i \(0.359020\pi\)
\(212\) 2.22035e7 0.160047
\(213\) −1.13652e6 −0.00805843
\(214\) 8.28618e7 0.577971
\(215\) −1.37209e8 −0.941558
\(216\) 1.17062e7 0.0790364
\(217\) −1.07716e8 −0.715600
\(218\) 3.51132e8 2.29547
\(219\) −2.25215e7 −0.144892
\(220\) 5.82626e7 0.368901
\(221\) 1.50679e7 0.0939033
\(222\) 3.19667e7 0.196093
\(223\) −3.09118e8 −1.86663 −0.933314 0.359062i \(-0.883097\pi\)
−0.933314 + 0.359062i \(0.883097\pi\)
\(224\) 7.75683e7 0.461123
\(225\) 8.60137e7 0.503418
\(226\) −3.77368e8 −2.17463
\(227\) 2.24510e8 1.27393 0.636964 0.770894i \(-0.280190\pi\)
0.636964 + 0.770894i \(0.280190\pi\)
\(228\) 2.08852e7 0.116699
\(229\) −1.31746e8 −0.724957 −0.362479 0.931992i \(-0.618069\pi\)
−0.362479 + 0.931992i \(0.618069\pi\)
\(230\) −1.91083e8 −1.03556
\(231\) 5.73021e6 0.0305864
\(232\) 3.32139e7 0.174627
\(233\) −1.88267e8 −0.975055 −0.487527 0.873108i \(-0.662101\pi\)
−0.487527 + 0.873108i \(0.662101\pi\)
\(234\) 7.06116e7 0.360264
\(235\) −2.26648e8 −1.13924
\(236\) 1.40691e8 0.696745
\(237\) −3.49869e7 −0.170721
\(238\) −3.50121e7 −0.168344
\(239\) −4.18684e8 −1.98378 −0.991890 0.127097i \(-0.959434\pi\)
−0.991890 + 0.127097i \(0.959434\pi\)
\(240\) −2.01347e7 −0.0940169
\(241\) 2.90747e8 1.33800 0.669000 0.743262i \(-0.266723\pi\)
0.669000 + 0.743262i \(0.266723\pi\)
\(242\) 1.39163e8 0.631206
\(243\) −7.40313e7 −0.330973
\(244\) −2.41900e8 −1.06604
\(245\) −2.30226e7 −0.100017
\(246\) 1.66238e7 0.0711963
\(247\) −9.35131e7 −0.394851
\(248\) −1.61190e8 −0.671056
\(249\) −6.20138e6 −0.0254560
\(250\) −3.43546e8 −1.39058
\(251\) −1.79203e7 −0.0715297 −0.0357649 0.999360i \(-0.511387\pi\)
−0.0357649 + 0.999360i \(0.511387\pi\)
\(252\) −6.92649e7 −0.272654
\(253\) 2.08885e8 0.810935
\(254\) −3.45082e8 −1.32131
\(255\) 7.04229e6 0.0265964
\(256\) 3.50790e8 1.30679
\(257\) 3.42514e8 1.25867 0.629336 0.777133i \(-0.283327\pi\)
0.629336 + 0.777133i \(0.283327\pi\)
\(258\) 5.47571e7 0.198505
\(259\) 1.40400e8 0.502133
\(260\) −4.02039e7 −0.141860
\(261\) −1.39737e8 −0.486486
\(262\) −3.88279e8 −1.33379
\(263\) −3.70759e8 −1.25674 −0.628371 0.777914i \(-0.716278\pi\)
−0.628371 + 0.777914i \(0.716278\pi\)
\(264\) 8.57494e6 0.0286825
\(265\) −4.64639e7 −0.153375
\(266\) 2.17288e8 0.707865
\(267\) 4.85791e7 0.156192
\(268\) −3.61166e8 −1.14613
\(269\) 4.50824e8 1.41213 0.706064 0.708148i \(-0.250469\pi\)
0.706064 + 0.708148i \(0.250469\pi\)
\(270\) 6.64242e7 0.205378
\(271\) −7.41871e7 −0.226431 −0.113216 0.993570i \(-0.536115\pi\)
−0.113216 + 0.993570i \(0.536115\pi\)
\(272\) −1.34487e8 −0.405217
\(273\) −3.95411e6 −0.0117620
\(274\) −4.54171e8 −1.33381
\(275\) 1.26816e8 0.367713
\(276\) 3.21924e7 0.0921661
\(277\) −5.25911e8 −1.48673 −0.743367 0.668883i \(-0.766773\pi\)
−0.743367 + 0.668883i \(0.766773\pi\)
\(278\) 4.29859e8 1.19997
\(279\) 6.78159e8 1.86946
\(280\) −3.44520e7 −0.0937909
\(281\) −5.04666e7 −0.135685 −0.0678426 0.997696i \(-0.521612\pi\)
−0.0678426 + 0.997696i \(0.521612\pi\)
\(282\) 9.04507e7 0.240182
\(283\) −3.57978e7 −0.0938867 −0.0469433 0.998898i \(-0.514948\pi\)
−0.0469433 + 0.998898i \(0.514948\pi\)
\(284\) 2.02547e7 0.0524701
\(285\) −4.37051e7 −0.111834
\(286\) 1.04107e8 0.263148
\(287\) 7.30129e7 0.182311
\(288\) −4.88357e8 −1.20466
\(289\) −3.63301e8 −0.885368
\(290\) 1.88465e8 0.453772
\(291\) 2.17629e6 0.00517717
\(292\) 4.01370e8 0.943420
\(293\) 5.77150e7 0.134045 0.0670227 0.997751i \(-0.478650\pi\)
0.0670227 + 0.997751i \(0.478650\pi\)
\(294\) 9.18783e6 0.0210861
\(295\) −2.94415e8 −0.667701
\(296\) 2.10101e8 0.470877
\(297\) −7.26128e7 −0.160829
\(298\) −3.85838e8 −0.844594
\(299\) −1.44140e8 −0.311844
\(300\) 1.95442e7 0.0417921
\(301\) 2.40497e8 0.508309
\(302\) −2.07828e8 −0.434190
\(303\) 455159. 0.000939970 0
\(304\) 8.34636e8 1.70388
\(305\) 5.06209e8 1.02160
\(306\) 2.20430e8 0.439790
\(307\) 8.63985e7 0.170421 0.0852103 0.996363i \(-0.472844\pi\)
0.0852103 + 0.996363i \(0.472844\pi\)
\(308\) −1.02122e8 −0.199155
\(309\) −6.54589e7 −0.126216
\(310\) −9.14640e8 −1.74375
\(311\) 7.60275e8 1.43321 0.716604 0.697480i \(-0.245695\pi\)
0.716604 + 0.697480i \(0.245695\pi\)
\(312\) −5.91710e6 −0.0110298
\(313\) 4.80335e8 0.885400 0.442700 0.896670i \(-0.354021\pi\)
0.442700 + 0.896670i \(0.354021\pi\)
\(314\) −3.33339e8 −0.607620
\(315\) 1.44946e8 0.261288
\(316\) 6.23525e8 1.11160
\(317\) 7.68019e8 1.35414 0.677071 0.735917i \(-0.263249\pi\)
0.677071 + 0.735917i \(0.263249\pi\)
\(318\) 1.85428e7 0.0323355
\(319\) −2.06024e8 −0.355345
\(320\) 1.67483e8 0.285723
\(321\) 2.92132e7 0.0492960
\(322\) 3.34927e8 0.559054
\(323\) −2.91922e8 −0.482011
\(324\) 4.30449e8 0.703095
\(325\) −8.75087e7 −0.141403
\(326\) 1.17345e9 1.87586
\(327\) 1.23793e8 0.195784
\(328\) 1.09260e8 0.170963
\(329\) 3.97266e8 0.615029
\(330\) 4.86566e7 0.0745321
\(331\) 4.70357e8 0.712902 0.356451 0.934314i \(-0.383987\pi\)
0.356451 + 0.934314i \(0.383987\pi\)
\(332\) 1.10519e8 0.165750
\(333\) −8.83935e8 −1.31179
\(334\) 1.74719e9 2.56583
\(335\) 7.55789e8 1.09836
\(336\) 3.52918e7 0.0507559
\(337\) −6.68238e7 −0.0951100 −0.0475550 0.998869i \(-0.515143\pi\)
−0.0475550 + 0.998869i \(0.515143\pi\)
\(338\) −7.18389e7 −0.101193
\(339\) −1.33043e8 −0.185478
\(340\) −1.25505e8 −0.173175
\(341\) 9.99855e8 1.36552
\(342\) −1.36801e9 −1.84926
\(343\) 4.03536e7 0.0539949
\(344\) 3.59891e8 0.476668
\(345\) −6.73668e7 −0.0883241
\(346\) −1.27484e9 −1.65458
\(347\) 5.50538e8 0.707350 0.353675 0.935368i \(-0.384932\pi\)
0.353675 + 0.935368i \(0.384932\pi\)
\(348\) −3.17514e7 −0.0403864
\(349\) −2.08530e8 −0.262591 −0.131296 0.991343i \(-0.541914\pi\)
−0.131296 + 0.991343i \(0.541914\pi\)
\(350\) 2.03336e8 0.253499
\(351\) 5.01062e7 0.0618467
\(352\) −7.20016e8 −0.879920
\(353\) −2.60990e8 −0.315800 −0.157900 0.987455i \(-0.550472\pi\)
−0.157900 + 0.987455i \(0.550472\pi\)
\(354\) 1.17495e8 0.140769
\(355\) −4.23857e7 −0.0502829
\(356\) −8.65758e8 −1.01700
\(357\) −1.23436e7 −0.0143583
\(358\) −1.73579e9 −1.99943
\(359\) −7.98028e8 −0.910306 −0.455153 0.890413i \(-0.650416\pi\)
−0.455153 + 0.890413i \(0.650416\pi\)
\(360\) 2.16903e8 0.245023
\(361\) 9.17821e8 1.02679
\(362\) −9.42136e7 −0.104384
\(363\) 4.90626e7 0.0538365
\(364\) 7.04687e7 0.0765846
\(365\) −8.39920e8 −0.904093
\(366\) −2.02017e8 −0.215380
\(367\) −1.76979e8 −0.186892 −0.0934458 0.995624i \(-0.529788\pi\)
−0.0934458 + 0.995624i \(0.529788\pi\)
\(368\) 1.28650e9 1.34569
\(369\) −4.59676e8 −0.476277
\(370\) 1.19217e9 1.22358
\(371\) 8.14412e7 0.0828010
\(372\) 1.54093e8 0.155196
\(373\) −2.26523e8 −0.226012 −0.113006 0.993594i \(-0.536048\pi\)
−0.113006 + 0.993594i \(0.536048\pi\)
\(374\) 3.24994e8 0.321237
\(375\) −1.21118e8 −0.118604
\(376\) 5.94486e8 0.576745
\(377\) 1.42166e8 0.136647
\(378\) −1.16427e8 −0.110875
\(379\) −9.03686e8 −0.852668 −0.426334 0.904566i \(-0.640195\pi\)
−0.426334 + 0.904566i \(0.640195\pi\)
\(380\) 7.78897e8 0.728178
\(381\) −1.21660e8 −0.112696
\(382\) −1.32499e9 −1.21616
\(383\) 1.05887e9 0.963043 0.481521 0.876434i \(-0.340084\pi\)
0.481521 + 0.876434i \(0.340084\pi\)
\(384\) 8.50497e7 0.0766502
\(385\) 2.13703e8 0.190853
\(386\) −1.47405e9 −1.30453
\(387\) −1.51413e9 −1.32793
\(388\) −3.87851e7 −0.0337096
\(389\) −1.85890e9 −1.60115 −0.800574 0.599234i \(-0.795472\pi\)
−0.800574 + 0.599234i \(0.795472\pi\)
\(390\) −3.35753e7 −0.0286612
\(391\) −4.49966e8 −0.380681
\(392\) 6.03869e7 0.0506339
\(393\) −1.36889e8 −0.113761
\(394\) 6.75973e8 0.556792
\(395\) −1.30481e9 −1.06526
\(396\) 6.42941e8 0.520281
\(397\) −1.60381e8 −0.128643 −0.0643216 0.997929i \(-0.520488\pi\)
−0.0643216 + 0.997929i \(0.520488\pi\)
\(398\) 2.11459e8 0.168127
\(399\) 7.66057e7 0.0603749
\(400\) 7.81045e8 0.610191
\(401\) −1.03733e9 −0.803363 −0.401681 0.915779i \(-0.631574\pi\)
−0.401681 + 0.915779i \(0.631574\pi\)
\(402\) −3.01620e8 −0.231563
\(403\) −6.89946e8 −0.525107
\(404\) −8.11168e6 −0.00612035
\(405\) −9.00771e8 −0.673786
\(406\) −3.30339e8 −0.244973
\(407\) −1.30324e9 −0.958176
\(408\) −1.84715e7 −0.0134646
\(409\) 1.72511e9 1.24677 0.623383 0.781917i \(-0.285758\pi\)
0.623383 + 0.781917i \(0.285758\pi\)
\(410\) 6.19970e8 0.444250
\(411\) −1.60120e8 −0.113762
\(412\) 1.16658e9 0.821818
\(413\) 5.16045e8 0.360465
\(414\) −2.10864e9 −1.46050
\(415\) −2.31275e8 −0.158840
\(416\) 4.96844e8 0.338372
\(417\) 1.51548e8 0.102347
\(418\) −2.01694e9 −1.35076
\(419\) −1.79579e9 −1.19263 −0.596317 0.802749i \(-0.703370\pi\)
−0.596317 + 0.802749i \(0.703370\pi\)
\(420\) 3.29349e7 0.0216912
\(421\) 1.26746e9 0.827842 0.413921 0.910313i \(-0.364159\pi\)
0.413921 + 0.910313i \(0.364159\pi\)
\(422\) −1.74074e9 −1.12756
\(423\) −2.50112e9 −1.60673
\(424\) 1.21872e8 0.0776468
\(425\) −2.73177e8 −0.172617
\(426\) 1.69153e7 0.0106010
\(427\) −8.87275e8 −0.551520
\(428\) −5.20628e8 −0.320977
\(429\) 3.67034e7 0.0224443
\(430\) 2.04212e9 1.23863
\(431\) −5.65790e8 −0.340396 −0.170198 0.985410i \(-0.554441\pi\)
−0.170198 + 0.985410i \(0.554441\pi\)
\(432\) −4.47215e8 −0.266884
\(433\) 8.33904e8 0.493638 0.246819 0.969062i \(-0.420615\pi\)
0.246819 + 0.969062i \(0.420615\pi\)
\(434\) 1.60317e9 0.941380
\(435\) 6.64440e7 0.0387029
\(436\) −2.20619e9 −1.27479
\(437\) 2.79253e9 1.60071
\(438\) 3.35195e8 0.190607
\(439\) 2.20687e7 0.0124495 0.00622474 0.999981i \(-0.498019\pi\)
0.00622474 + 0.999981i \(0.498019\pi\)
\(440\) 3.19795e8 0.178973
\(441\) −2.54059e8 −0.141059
\(442\) −2.24261e8 −0.123531
\(443\) 3.44831e9 1.88449 0.942244 0.334928i \(-0.108712\pi\)
0.942244 + 0.334928i \(0.108712\pi\)
\(444\) −2.00850e8 −0.108901
\(445\) 1.81171e9 0.974608
\(446\) 4.60070e9 2.45557
\(447\) −1.36028e8 −0.0720367
\(448\) −2.93561e8 −0.154250
\(449\) 4.12114e8 0.214860 0.107430 0.994213i \(-0.465738\pi\)
0.107430 + 0.994213i \(0.465738\pi\)
\(450\) −1.28017e9 −0.662252
\(451\) −6.77731e8 −0.347888
\(452\) 2.37104e9 1.20768
\(453\) −7.32705e7 −0.0370327
\(454\) −3.34145e9 −1.67587
\(455\) −1.47465e8 −0.0733922
\(456\) 1.14636e8 0.0566167
\(457\) −3.31019e9 −1.62236 −0.811178 0.584799i \(-0.801173\pi\)
−0.811178 + 0.584799i \(0.801173\pi\)
\(458\) 1.96081e9 0.953690
\(459\) 1.56417e8 0.0754989
\(460\) 1.20059e9 0.575097
\(461\) −1.89375e8 −0.0900263 −0.0450132 0.998986i \(-0.514333\pi\)
−0.0450132 + 0.998986i \(0.514333\pi\)
\(462\) −8.52846e7 −0.0402368
\(463\) −8.71775e8 −0.408198 −0.204099 0.978950i \(-0.565426\pi\)
−0.204099 + 0.978950i \(0.565426\pi\)
\(464\) −1.26888e9 −0.589668
\(465\) −3.22460e8 −0.148727
\(466\) 2.80204e9 1.28270
\(467\) −1.01015e8 −0.0458964 −0.0229482 0.999737i \(-0.507305\pi\)
−0.0229482 + 0.999737i \(0.507305\pi\)
\(468\) −4.43659e8 −0.200073
\(469\) −1.32474e9 −0.592959
\(470\) 3.37328e9 1.49868
\(471\) −1.17520e8 −0.0518248
\(472\) 7.72233e8 0.338027
\(473\) −2.23238e9 −0.969961
\(474\) 5.20722e8 0.224585
\(475\) 1.69537e9 0.725831
\(476\) 2.19984e8 0.0934902
\(477\) −5.12739e8 −0.216313
\(478\) 6.23140e9 2.60969
\(479\) −2.58810e9 −1.07599 −0.537993 0.842949i \(-0.680817\pi\)
−0.537993 + 0.842949i \(0.680817\pi\)
\(480\) 2.32210e8 0.0958377
\(481\) 8.99298e8 0.368465
\(482\) −4.32729e9 −1.76016
\(483\) 1.18080e8 0.0476826
\(484\) −8.74375e8 −0.350541
\(485\) 8.11630e7 0.0323044
\(486\) 1.10183e9 0.435399
\(487\) −2.71918e9 −1.06681 −0.533404 0.845860i \(-0.679088\pi\)
−0.533404 + 0.845860i \(0.679088\pi\)
\(488\) −1.32776e9 −0.517189
\(489\) 4.13702e8 0.159995
\(490\) 3.42652e8 0.131573
\(491\) −3.74748e8 −0.142874 −0.0714372 0.997445i \(-0.522759\pi\)
−0.0714372 + 0.997445i \(0.522759\pi\)
\(492\) −1.04449e8 −0.0395389
\(493\) 4.43802e8 0.166811
\(494\) 1.39178e9 0.519431
\(495\) −1.34544e9 −0.498593
\(496\) 6.15801e9 2.26597
\(497\) 7.42931e7 0.0271457
\(498\) 9.22971e7 0.0334877
\(499\) −9.73363e7 −0.0350690 −0.0175345 0.999846i \(-0.505582\pi\)
−0.0175345 + 0.999846i \(0.505582\pi\)
\(500\) 2.15853e9 0.772259
\(501\) 6.15979e8 0.218844
\(502\) 2.66713e8 0.0940982
\(503\) 5.25273e9 1.84034 0.920168 0.391525i \(-0.128052\pi\)
0.920168 + 0.391525i \(0.128052\pi\)
\(504\) −3.80185e8 −0.132278
\(505\) 1.69748e7 0.00586522
\(506\) −3.10891e9 −1.06679
\(507\) −2.53271e7 −0.00863092
\(508\) 2.16818e9 0.733789
\(509\) 1.08707e8 0.0365381 0.0182690 0.999833i \(-0.494184\pi\)
0.0182690 + 0.999833i \(0.494184\pi\)
\(510\) −1.04813e8 −0.0349879
\(511\) 1.47220e9 0.488083
\(512\) −3.14620e9 −1.03596
\(513\) −9.70741e8 −0.317463
\(514\) −5.09775e9 −1.65580
\(515\) −2.44123e9 −0.787560
\(516\) −3.44043e8 −0.110240
\(517\) −3.68756e9 −1.17361
\(518\) −2.08962e9 −0.660562
\(519\) −4.49449e8 −0.141122
\(520\) −2.20673e8 −0.0688237
\(521\) 2.85963e8 0.0885886 0.0442943 0.999019i \(-0.485896\pi\)
0.0442943 + 0.999019i \(0.485896\pi\)
\(522\) 2.07975e9 0.639978
\(523\) 4.73315e8 0.144675 0.0723376 0.997380i \(-0.476954\pi\)
0.0723376 + 0.997380i \(0.476954\pi\)
\(524\) 2.43959e9 0.740724
\(525\) 7.16869e7 0.0216213
\(526\) 5.51812e9 1.65326
\(527\) −2.15382e9 −0.641020
\(528\) −3.27591e8 −0.0968530
\(529\) 8.99566e8 0.264203
\(530\) 6.91537e8 0.201767
\(531\) −3.24893e9 −0.941694
\(532\) −1.36524e9 −0.393114
\(533\) 4.67666e8 0.133780
\(534\) −7.23018e8 −0.205473
\(535\) 1.08948e9 0.307597
\(536\) −1.98239e9 −0.556049
\(537\) −6.11958e8 −0.170534
\(538\) −6.70975e9 −1.85767
\(539\) −3.74576e8 −0.103034
\(540\) −4.17349e8 −0.114057
\(541\) 3.49250e9 0.948300 0.474150 0.880444i \(-0.342755\pi\)
0.474150 + 0.880444i \(0.342755\pi\)
\(542\) 1.10415e9 0.297873
\(543\) −3.32153e7 −0.00890306
\(544\) 1.55101e9 0.413065
\(545\) 4.61675e9 1.22165
\(546\) 5.88503e7 0.0154730
\(547\) 6.26068e9 1.63556 0.817778 0.575533i \(-0.195205\pi\)
0.817778 + 0.575533i \(0.195205\pi\)
\(548\) 2.85359e9 0.740730
\(549\) 5.58613e9 1.44081
\(550\) −1.88744e9 −0.483730
\(551\) −2.75428e9 −0.701418
\(552\) 1.76699e8 0.0447145
\(553\) 2.28705e9 0.575092
\(554\) 7.82731e9 1.95582
\(555\) 4.20305e8 0.104361
\(556\) −2.70084e9 −0.666403
\(557\) −2.34785e9 −0.575675 −0.287837 0.957679i \(-0.592936\pi\)
−0.287837 + 0.957679i \(0.592936\pi\)
\(558\) −1.00933e10 −2.45930
\(559\) 1.54045e9 0.372997
\(560\) 1.31618e9 0.316706
\(561\) 1.14578e8 0.0273987
\(562\) 7.51111e8 0.178495
\(563\) −2.39497e8 −0.0565616 −0.0282808 0.999600i \(-0.509003\pi\)
−0.0282808 + 0.999600i \(0.509003\pi\)
\(564\) −5.68309e8 −0.133385
\(565\) −4.96171e9 −1.15734
\(566\) 5.32790e8 0.123509
\(567\) 1.57886e9 0.363749
\(568\) 1.11175e8 0.0254559
\(569\) −5.41348e9 −1.23192 −0.615961 0.787776i \(-0.711232\pi\)
−0.615961 + 0.787776i \(0.711232\pi\)
\(570\) 6.50477e8 0.147119
\(571\) −3.87667e9 −0.871431 −0.435715 0.900085i \(-0.643505\pi\)
−0.435715 + 0.900085i \(0.643505\pi\)
\(572\) −6.54115e8 −0.146140
\(573\) −4.67131e8 −0.103728
\(574\) −1.08667e9 −0.239832
\(575\) 2.61323e9 0.573244
\(576\) 1.84821e9 0.402970
\(577\) −2.76934e9 −0.600152 −0.300076 0.953915i \(-0.597012\pi\)
−0.300076 + 0.953915i \(0.597012\pi\)
\(578\) 5.40712e9 1.16471
\(579\) −5.19680e8 −0.111266
\(580\) −1.18414e9 −0.252003
\(581\) 4.05376e8 0.0857514
\(582\) −3.23905e7 −0.00681062
\(583\) −7.55966e8 −0.158002
\(584\) 2.20306e9 0.457701
\(585\) 9.28415e8 0.191733
\(586\) −8.58990e8 −0.176338
\(587\) −4.12702e8 −0.0842176 −0.0421088 0.999113i \(-0.513408\pi\)
−0.0421088 + 0.999113i \(0.513408\pi\)
\(588\) −5.77278e7 −0.0117102
\(589\) 1.33668e10 2.69540
\(590\) 4.38186e9 0.878369
\(591\) 2.38317e8 0.0474896
\(592\) −8.02655e9 −1.59002
\(593\) −3.60387e9 −0.709705 −0.354853 0.934922i \(-0.615469\pi\)
−0.354853 + 0.934922i \(0.615469\pi\)
\(594\) 1.08072e9 0.211573
\(595\) −4.60345e8 −0.0895930
\(596\) 2.42425e9 0.469046
\(597\) 7.45508e7 0.0143398
\(598\) 2.14529e9 0.410234
\(599\) −3.92164e9 −0.745544 −0.372772 0.927923i \(-0.621593\pi\)
−0.372772 + 0.927923i \(0.621593\pi\)
\(600\) 1.07275e8 0.0202755
\(601\) 2.34111e9 0.439907 0.219954 0.975510i \(-0.429409\pi\)
0.219954 + 0.975510i \(0.429409\pi\)
\(602\) −3.57940e9 −0.668686
\(603\) 8.34030e9 1.54907
\(604\) 1.30580e9 0.241128
\(605\) 1.82975e9 0.335929
\(606\) −6.77428e6 −0.00123654
\(607\) 3.44961e9 0.626051 0.313025 0.949745i \(-0.398658\pi\)
0.313025 + 0.949745i \(0.398658\pi\)
\(608\) −9.62571e9 −1.73688
\(609\) −1.16462e8 −0.0208941
\(610\) −7.53407e9 −1.34393
\(611\) 2.54459e9 0.451308
\(612\) −1.38498e9 −0.244238
\(613\) 6.38422e9 1.11943 0.559714 0.828686i \(-0.310911\pi\)
0.559714 + 0.828686i \(0.310911\pi\)
\(614\) −1.28590e9 −0.224190
\(615\) 2.18573e8 0.0378907
\(616\) −5.60532e8 −0.0966202
\(617\) −3.65335e9 −0.626172 −0.313086 0.949725i \(-0.601363\pi\)
−0.313086 + 0.949725i \(0.601363\pi\)
\(618\) 9.74245e8 0.166038
\(619\) 3.34534e9 0.566922 0.283461 0.958984i \(-0.408517\pi\)
0.283461 + 0.958984i \(0.408517\pi\)
\(620\) 5.74676e9 0.968394
\(621\) −1.49629e9 −0.250724
\(622\) −1.13154e10 −1.88540
\(623\) −3.17555e9 −0.526151
\(624\) 2.26053e8 0.0372446
\(625\) −1.40521e9 −0.230230
\(626\) −7.14898e9 −1.16475
\(627\) −7.11081e8 −0.115208
\(628\) 2.09439e9 0.337442
\(629\) 2.80736e9 0.449801
\(630\) −2.15728e9 −0.343727
\(631\) 1.52494e9 0.241630 0.120815 0.992675i \(-0.461449\pi\)
0.120815 + 0.992675i \(0.461449\pi\)
\(632\) 3.42244e9 0.539294
\(633\) −6.13703e8 −0.0961713
\(634\) −1.14307e10 −1.78139
\(635\) −4.53720e9 −0.703201
\(636\) −1.16506e8 −0.0179576
\(637\) 2.58475e8 0.0396214
\(638\) 3.06632e9 0.467460
\(639\) −4.67736e8 −0.0709166
\(640\) 3.17185e9 0.478281
\(641\) 8.56554e9 1.28455 0.642276 0.766473i \(-0.277990\pi\)
0.642276 + 0.766473i \(0.277990\pi\)
\(642\) −4.34790e8 −0.0648495
\(643\) 6.96729e9 1.03354 0.516768 0.856125i \(-0.327135\pi\)
0.516768 + 0.856125i \(0.327135\pi\)
\(644\) −2.10437e9 −0.310472
\(645\) 7.19957e8 0.105645
\(646\) 4.34476e9 0.634092
\(647\) 1.11034e9 0.161173 0.0805865 0.996748i \(-0.474321\pi\)
0.0805865 + 0.996748i \(0.474321\pi\)
\(648\) 2.36267e9 0.341107
\(649\) −4.79011e9 −0.687843
\(650\) 1.30242e9 0.186018
\(651\) 5.65202e8 0.0802917
\(652\) −7.37285e9 −1.04176
\(653\) 8.09862e9 1.13819 0.569095 0.822272i \(-0.307294\pi\)
0.569095 + 0.822272i \(0.307294\pi\)
\(654\) −1.84245e9 −0.257557
\(655\) −5.10516e9 −0.709847
\(656\) −4.17408e9 −0.577294
\(657\) −9.26871e9 −1.27509
\(658\) −5.91263e9 −0.809078
\(659\) −8.67590e9 −1.18091 −0.590453 0.807072i \(-0.701051\pi\)
−0.590453 + 0.807072i \(0.701051\pi\)
\(660\) −3.05713e8 −0.0413915
\(661\) 1.14859e10 1.54689 0.773446 0.633862i \(-0.218531\pi\)
0.773446 + 0.633862i \(0.218531\pi\)
\(662\) −7.00048e9 −0.937831
\(663\) −7.90640e7 −0.0105361
\(664\) 6.06621e8 0.0804136
\(665\) 2.85694e9 0.376726
\(666\) 1.31559e10 1.72568
\(667\) −4.24543e9 −0.553963
\(668\) −1.09777e10 −1.42494
\(669\) 1.62199e9 0.209439
\(670\) −1.12486e10 −1.44490
\(671\) 8.23600e9 1.05242
\(672\) −4.07014e8 −0.0517389
\(673\) 1.36770e9 0.172957 0.0864785 0.996254i \(-0.472439\pi\)
0.0864785 + 0.996254i \(0.472439\pi\)
\(674\) 9.94559e8 0.125118
\(675\) −9.08411e8 −0.113689
\(676\) 4.51370e8 0.0561978
\(677\) 3.23186e9 0.400306 0.200153 0.979765i \(-0.435856\pi\)
0.200153 + 0.979765i \(0.435856\pi\)
\(678\) 1.98011e9 0.243998
\(679\) −1.42261e8 −0.0174399
\(680\) −6.88880e8 −0.0840161
\(681\) −1.17804e9 −0.142937
\(682\) −1.48812e10 −1.79635
\(683\) 1.12350e9 0.134927 0.0674637 0.997722i \(-0.478509\pi\)
0.0674637 + 0.997722i \(0.478509\pi\)
\(684\) 8.59530e9 1.02699
\(685\) −5.97152e9 −0.709853
\(686\) −6.00595e8 −0.0710310
\(687\) 6.91292e8 0.0813416
\(688\) −1.37490e10 −1.60958
\(689\) 5.21651e8 0.0607593
\(690\) 1.00264e9 0.116191
\(691\) −6.98149e8 −0.0804961 −0.0402480 0.999190i \(-0.512815\pi\)
−0.0402480 + 0.999190i \(0.512815\pi\)
\(692\) 8.00991e9 0.918875
\(693\) 2.35827e9 0.269170
\(694\) −8.19384e9 −0.930527
\(695\) 5.65187e9 0.638624
\(696\) −1.74279e8 −0.0195935
\(697\) 1.45992e9 0.163311
\(698\) 3.10362e9 0.345442
\(699\) 9.87869e8 0.109403
\(700\) −1.27758e9 −0.140781
\(701\) −2.98839e9 −0.327661 −0.163830 0.986489i \(-0.552385\pi\)
−0.163830 + 0.986489i \(0.552385\pi\)
\(702\) −7.45746e8 −0.0813600
\(703\) −1.74227e10 −1.89135
\(704\) 2.72494e9 0.294342
\(705\) 1.18926e9 0.127825
\(706\) 3.88440e9 0.415439
\(707\) −2.97531e7 −0.00316639
\(708\) −7.38229e8 −0.0781762
\(709\) 3.32007e9 0.349853 0.174927 0.984581i \(-0.444031\pi\)
0.174927 + 0.984581i \(0.444031\pi\)
\(710\) 6.30840e8 0.0661477
\(711\) −1.43989e10 −1.50240
\(712\) −4.75202e9 −0.493399
\(713\) 2.06035e10 2.12876
\(714\) 1.83714e8 0.0188886
\(715\) 1.36882e9 0.140048
\(716\) 1.09061e10 1.11038
\(717\) 2.19690e9 0.222584
\(718\) 1.18773e10 1.19752
\(719\) −4.91147e9 −0.492788 −0.246394 0.969170i \(-0.579246\pi\)
−0.246394 + 0.969170i \(0.579246\pi\)
\(720\) −8.28642e9 −0.827377
\(721\) 4.27896e9 0.425172
\(722\) −1.36602e10 −1.35076
\(723\) −1.52560e9 −0.150126
\(724\) 5.91951e8 0.0579697
\(725\) −2.57743e9 −0.251191
\(726\) −7.30214e8 −0.0708226
\(727\) −4.76748e9 −0.460171 −0.230085 0.973170i \(-0.573901\pi\)
−0.230085 + 0.973170i \(0.573901\pi\)
\(728\) 3.86793e8 0.0371551
\(729\) −9.67849e9 −0.925255
\(730\) 1.25008e10 1.18934
\(731\) 4.80884e9 0.455333
\(732\) 1.26929e9 0.119611
\(733\) 1.66873e10 1.56503 0.782513 0.622634i \(-0.213938\pi\)
0.782513 + 0.622634i \(0.213938\pi\)
\(734\) 2.63403e9 0.245858
\(735\) 1.20803e8 0.0112221
\(736\) −1.48370e10 −1.37175
\(737\) 1.22967e10 1.13149
\(738\) 6.84151e9 0.626548
\(739\) −1.59056e10 −1.44976 −0.724878 0.688877i \(-0.758104\pi\)
−0.724878 + 0.688877i \(0.758104\pi\)
\(740\) −7.49051e9 −0.679517
\(741\) 4.90679e8 0.0443030
\(742\) −1.21212e9 −0.108926
\(743\) −7.24535e9 −0.648035 −0.324018 0.946051i \(-0.605034\pi\)
−0.324018 + 0.946051i \(0.605034\pi\)
\(744\) 8.45793e8 0.0752937
\(745\) −5.07307e9 −0.449494
\(746\) 3.37141e9 0.297321
\(747\) −2.55217e9 −0.224021
\(748\) −2.04196e9 −0.178399
\(749\) −1.90963e9 −0.166059
\(750\) 1.80264e9 0.156025
\(751\) 1.00599e10 0.866670 0.433335 0.901233i \(-0.357337\pi\)
0.433335 + 0.901233i \(0.357337\pi\)
\(752\) −2.27113e10 −1.94751
\(753\) 9.40306e7 0.00802577
\(754\) −2.11590e9 −0.179761
\(755\) −2.73256e9 −0.231076
\(756\) 7.31522e8 0.0615746
\(757\) 9.99365e9 0.837314 0.418657 0.908144i \(-0.362501\pi\)
0.418657 + 0.908144i \(0.362501\pi\)
\(758\) 1.34498e10 1.12170
\(759\) −1.09606e9 −0.0909885
\(760\) 4.27526e9 0.353276
\(761\) 8.16710e9 0.671771 0.335886 0.941903i \(-0.390964\pi\)
0.335886 + 0.941903i \(0.390964\pi\)
\(762\) 1.81070e9 0.148253
\(763\) −8.09216e9 −0.659521
\(764\) 8.32504e9 0.675398
\(765\) 2.89825e9 0.234057
\(766\) −1.57594e10 −1.26689
\(767\) 3.30540e9 0.264509
\(768\) −1.84065e9 −0.146625
\(769\) 2.08195e10 1.65093 0.825466 0.564452i \(-0.190912\pi\)
0.825466 + 0.564452i \(0.190912\pi\)
\(770\) −3.18062e9 −0.251069
\(771\) −1.79723e9 −0.141225
\(772\) 9.26155e9 0.724474
\(773\) −2.67717e9 −0.208472 −0.104236 0.994553i \(-0.533240\pi\)
−0.104236 + 0.994553i \(0.533240\pi\)
\(774\) 2.25353e10 1.74690
\(775\) 1.25085e10 0.965273
\(776\) −2.12886e8 −0.0163543
\(777\) −7.36703e8 −0.0563403
\(778\) 2.76665e10 2.10633
\(779\) −9.06041e9 −0.686700
\(780\) 2.10956e8 0.0159170
\(781\) −6.89614e8 −0.0517997
\(782\) 6.69699e9 0.500790
\(783\) 1.47580e9 0.109865
\(784\) −2.30698e9 −0.170977
\(785\) −4.38280e9 −0.323376
\(786\) 2.03736e9 0.149654
\(787\) −1.58180e10 −1.15675 −0.578374 0.815771i \(-0.696313\pi\)
−0.578374 + 0.815771i \(0.696313\pi\)
\(788\) −4.24719e9 −0.309215
\(789\) 1.94543e9 0.141009
\(790\) 1.94199e10 1.40137
\(791\) 8.69681e9 0.624802
\(792\) 3.52901e9 0.252415
\(793\) −5.68322e9 −0.404705
\(794\) 2.38700e9 0.169232
\(795\) 2.43804e8 0.0172090
\(796\) −1.32862e9 −0.0933693
\(797\) 1.63509e10 1.14403 0.572015 0.820243i \(-0.306162\pi\)
0.572015 + 0.820243i \(0.306162\pi\)
\(798\) −1.14015e9 −0.0794238
\(799\) 7.94349e9 0.550931
\(800\) −9.00765e9 −0.622009
\(801\) 1.99927e10 1.37454
\(802\) 1.54389e10 1.05683
\(803\) −1.36655e10 −0.931366
\(804\) 1.89510e9 0.128599
\(805\) 4.40368e9 0.297529
\(806\) 1.02687e10 0.690784
\(807\) −2.36555e9 −0.158443
\(808\) −4.45239e7 −0.00296929
\(809\) −5.30679e9 −0.352380 −0.176190 0.984356i \(-0.556377\pi\)
−0.176190 + 0.984356i \(0.556377\pi\)
\(810\) 1.34065e10 0.886373
\(811\) 1.92993e10 1.27048 0.635241 0.772314i \(-0.280901\pi\)
0.635241 + 0.772314i \(0.280901\pi\)
\(812\) 2.07554e9 0.136046
\(813\) 3.89272e8 0.0254060
\(814\) 1.93966e10 1.26049
\(815\) 1.54287e10 0.998337
\(816\) 7.05673e8 0.0454661
\(817\) −2.98441e10 −1.91461
\(818\) −2.56753e10 −1.64014
\(819\) −1.62731e9 −0.103509
\(820\) −3.89532e9 −0.246715
\(821\) 1.61623e10 1.01930 0.509649 0.860382i \(-0.329775\pi\)
0.509649 + 0.860382i \(0.329775\pi\)
\(822\) 2.38311e9 0.149656
\(823\) −1.93984e10 −1.21301 −0.606507 0.795078i \(-0.707430\pi\)
−0.606507 + 0.795078i \(0.707430\pi\)
\(824\) 6.40321e9 0.398706
\(825\) −6.65423e8 −0.0412581
\(826\) −7.68047e9 −0.474195
\(827\) −2.83991e10 −1.74596 −0.872982 0.487752i \(-0.837817\pi\)
−0.872982 + 0.487752i \(0.837817\pi\)
\(828\) 1.32487e10 0.811089
\(829\) 3.08750e9 0.188220 0.0941100 0.995562i \(-0.469999\pi\)
0.0941100 + 0.995562i \(0.469999\pi\)
\(830\) 3.44214e9 0.208956
\(831\) 2.75954e9 0.166815
\(832\) −1.88033e9 −0.113189
\(833\) 8.06886e8 0.0483676
\(834\) −2.25554e9 −0.134639
\(835\) 2.29724e10 1.36554
\(836\) 1.26726e10 0.750144
\(837\) −7.16220e9 −0.422189
\(838\) 2.67273e10 1.56892
\(839\) 2.32722e9 0.136041 0.0680207 0.997684i \(-0.478332\pi\)
0.0680207 + 0.997684i \(0.478332\pi\)
\(840\) 1.80775e8 0.0105235
\(841\) −1.30626e10 −0.757258
\(842\) −1.88640e10 −1.08904
\(843\) 2.64807e8 0.0152241
\(844\) 1.09372e10 0.626192
\(845\) −9.44551e8 −0.0538551
\(846\) 3.72249e10 2.11367
\(847\) −3.20715e9 −0.181354
\(848\) −4.65591e9 −0.262192
\(849\) 1.87837e8 0.0105343
\(850\) 4.06579e9 0.227080
\(851\) −2.68553e10 −1.49374
\(852\) −1.06280e8 −0.00588725
\(853\) 4.59126e9 0.253285 0.126643 0.991948i \(-0.459580\pi\)
0.126643 + 0.991948i \(0.459580\pi\)
\(854\) 1.32056e10 0.725530
\(855\) −1.79868e10 −0.984176
\(856\) −2.85765e9 −0.155722
\(857\) −2.38629e10 −1.29506 −0.647530 0.762040i \(-0.724198\pi\)
−0.647530 + 0.762040i \(0.724198\pi\)
\(858\) −5.46269e8 −0.0295258
\(859\) −7.81358e9 −0.420605 −0.210302 0.977636i \(-0.567445\pi\)
−0.210302 + 0.977636i \(0.567445\pi\)
\(860\) −1.28308e10 −0.687875
\(861\) −3.83111e8 −0.0204557
\(862\) 8.42083e9 0.447795
\(863\) −5.67070e9 −0.300330 −0.150165 0.988661i \(-0.547980\pi\)
−0.150165 + 0.988661i \(0.547980\pi\)
\(864\) 5.15765e9 0.272053
\(865\) −1.67618e10 −0.880571
\(866\) −1.24113e10 −0.649386
\(867\) 1.90630e9 0.0993400
\(868\) −1.00728e10 −0.522796
\(869\) −2.12292e10 −1.09740
\(870\) −9.88907e8 −0.0509141
\(871\) −8.48526e9 −0.435113
\(872\) −1.21095e10 −0.618468
\(873\) 8.95652e8 0.0455606
\(874\) −4.15621e10 −2.10575
\(875\) 7.91734e9 0.399532
\(876\) −2.10606e9 −0.105854
\(877\) 2.56397e10 1.28355 0.641777 0.766891i \(-0.278197\pi\)
0.641777 + 0.766891i \(0.278197\pi\)
\(878\) −3.28456e8 −0.0163774
\(879\) −3.02840e8 −0.0150402
\(880\) −1.22172e10 −0.604342
\(881\) −5.91729e9 −0.291546 −0.145773 0.989318i \(-0.546567\pi\)
−0.145773 + 0.989318i \(0.546567\pi\)
\(882\) 3.78124e9 0.185564
\(883\) 1.75203e9 0.0856404 0.0428202 0.999083i \(-0.486366\pi\)
0.0428202 + 0.999083i \(0.486366\pi\)
\(884\) 1.40905e9 0.0686030
\(885\) 1.54484e9 0.0749174
\(886\) −5.13223e10 −2.47906
\(887\) 9.80417e9 0.471713 0.235857 0.971788i \(-0.424210\pi\)
0.235857 + 0.971788i \(0.424210\pi\)
\(888\) −1.10243e9 −0.0528333
\(889\) 7.95273e9 0.379630
\(890\) −2.69643e10 −1.28211
\(891\) −1.46555e10 −0.694111
\(892\) −2.89066e10 −1.36370
\(893\) −4.92980e10 −2.31659
\(894\) 2.02455e9 0.0947651
\(895\) −2.28224e10 −1.06410
\(896\) −5.55958e9 −0.258205
\(897\) 7.56329e8 0.0349895
\(898\) −6.13362e9 −0.282650
\(899\) −2.03213e10 −0.932807
\(900\) 8.04341e9 0.367783
\(901\) 1.62845e9 0.0741715
\(902\) 1.00869e10 0.457651
\(903\) −1.26193e9 −0.0570332
\(904\) 1.30143e10 0.585910
\(905\) −1.23874e9 −0.0555532
\(906\) 1.09051e9 0.0487169
\(907\) 2.52800e10 1.12500 0.562499 0.826798i \(-0.309840\pi\)
0.562499 + 0.826798i \(0.309840\pi\)
\(908\) 2.09946e10 0.930695
\(909\) 1.87320e8 0.00827202
\(910\) 2.19477e9 0.0965483
\(911\) −1.01736e10 −0.445820 −0.222910 0.974839i \(-0.571556\pi\)
−0.222910 + 0.974839i \(0.571556\pi\)
\(912\) −4.37948e9 −0.191179
\(913\) −3.76284e9 −0.163632
\(914\) 4.92666e10 2.13423
\(915\) −2.65616e9 −0.114625
\(916\) −1.23199e10 −0.529633
\(917\) 8.94824e9 0.383217
\(918\) −2.32801e9 −0.0993197
\(919\) −1.23533e10 −0.525025 −0.262513 0.964929i \(-0.584551\pi\)
−0.262513 + 0.964929i \(0.584551\pi\)
\(920\) 6.58985e9 0.279009
\(921\) −4.53347e8 −0.0191215
\(922\) 2.81853e9 0.118431
\(923\) 4.75865e8 0.0199195
\(924\) 5.35850e8 0.0223456
\(925\) −1.63040e10 −0.677328
\(926\) 1.29749e10 0.536989
\(927\) −2.69395e10 −1.11074
\(928\) 1.46338e10 0.601088
\(929\) 1.97042e10 0.806313 0.403157 0.915131i \(-0.367913\pi\)
0.403157 + 0.915131i \(0.367913\pi\)
\(930\) 4.79927e9 0.195652
\(931\) −5.00761e9 −0.203379
\(932\) −1.76054e10 −0.712346
\(933\) −3.98929e9 −0.160809
\(934\) 1.50344e9 0.0603772
\(935\) 4.27308e9 0.170962
\(936\) −2.43518e9 −0.0970656
\(937\) 2.40983e10 0.956969 0.478485 0.878096i \(-0.341186\pi\)
0.478485 + 0.878096i \(0.341186\pi\)
\(938\) 1.97165e10 0.780044
\(939\) −2.52040e9 −0.0993436
\(940\) −2.11946e10 −0.832295
\(941\) −1.50546e10 −0.588986 −0.294493 0.955654i \(-0.595151\pi\)
−0.294493 + 0.955654i \(0.595151\pi\)
\(942\) 1.74908e9 0.0681761
\(943\) −1.39657e10 −0.542339
\(944\) −2.95018e10 −1.14142
\(945\) −1.53081e9 −0.0590078
\(946\) 3.32252e10 1.27599
\(947\) −4.37877e10 −1.67543 −0.837716 0.546106i \(-0.816110\pi\)
−0.837716 + 0.546106i \(0.816110\pi\)
\(948\) −3.27174e9 −0.124724
\(949\) 9.42980e9 0.358155
\(950\) −2.52327e10 −0.954840
\(951\) −4.02992e9 −0.151937
\(952\) 1.20746e9 0.0453569
\(953\) 1.58423e10 0.592914 0.296457 0.955046i \(-0.404195\pi\)
0.296457 + 0.955046i \(0.404195\pi\)
\(954\) 7.63126e9 0.284562
\(955\) −1.74213e10 −0.647244
\(956\) −3.91524e10 −1.44929
\(957\) 1.08104e9 0.0398704
\(958\) 3.85195e10 1.41547
\(959\) 1.04668e10 0.383221
\(960\) −8.78811e8 −0.0320587
\(961\) 7.11086e10 2.58458
\(962\) −1.33845e10 −0.484720
\(963\) 1.20227e10 0.433820
\(964\) 2.71887e10 0.977504
\(965\) −1.93810e10 −0.694274
\(966\) −1.75742e9 −0.0627270
\(967\) −3.32273e10 −1.18169 −0.590844 0.806786i \(-0.701205\pi\)
−0.590844 + 0.806786i \(0.701205\pi\)
\(968\) −4.79932e9 −0.170065
\(969\) 1.53176e9 0.0540826
\(970\) −1.20798e9 −0.0424969
\(971\) 3.66665e10 1.28529 0.642647 0.766162i \(-0.277836\pi\)
0.642647 + 0.766162i \(0.277836\pi\)
\(972\) −6.92289e9 −0.241800
\(973\) −9.90651e9 −0.344767
\(974\) 4.04704e10 1.40340
\(975\) 4.59173e8 0.0158657
\(976\) 5.07247e10 1.74641
\(977\) −1.24706e10 −0.427816 −0.213908 0.976854i \(-0.568619\pi\)
−0.213908 + 0.976854i \(0.568619\pi\)
\(978\) −6.15726e9 −0.210476
\(979\) 2.94765e10 1.00401
\(980\) −2.15291e9 −0.0730693
\(981\) 5.09468e10 1.72296
\(982\) 5.57750e9 0.187953
\(983\) 5.84965e10 1.96423 0.982116 0.188279i \(-0.0602910\pi\)
0.982116 + 0.188279i \(0.0602910\pi\)
\(984\) −5.73304e8 −0.0191824
\(985\) 8.88782e9 0.296325
\(986\) −6.60525e9 −0.219442
\(987\) −2.08452e9 −0.0690075
\(988\) −8.74469e9 −0.288466
\(989\) −4.60015e10 −1.51212
\(990\) 2.00246e10 0.655904
\(991\) −2.40516e10 −0.785030 −0.392515 0.919746i \(-0.628395\pi\)
−0.392515 + 0.919746i \(0.628395\pi\)
\(992\) −7.10192e10 −2.30985
\(993\) −2.46804e9 −0.0799890
\(994\) −1.10573e9 −0.0357105
\(995\) 2.78031e9 0.0894771
\(996\) −5.79910e8 −0.0185974
\(997\) 2.91141e10 0.930402 0.465201 0.885205i \(-0.345982\pi\)
0.465201 + 0.885205i \(0.345982\pi\)
\(998\) 1.44869e9 0.0461336
\(999\) 9.33544e9 0.296248
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 91.8.a.e.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.3 12 1.1 even 1 trivial