Properties

Label 91.8.a.e.1.8
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 1.8
Root \(-4.08138\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.08138 q^{2} -25.2372 q^{3} -102.180 q^{4} +250.262 q^{5} -128.240 q^{6} +343.000 q^{7} -1169.63 q^{8} -1550.08 q^{9} +O(q^{10})\) \(q+5.08138 q^{2} -25.2372 q^{3} -102.180 q^{4} +250.262 q^{5} -128.240 q^{6} +343.000 q^{7} -1169.63 q^{8} -1550.08 q^{9} +1271.68 q^{10} +79.1782 q^{11} +2578.73 q^{12} +2197.00 q^{13} +1742.91 q^{14} -6315.91 q^{15} +7135.66 q^{16} -19865.7 q^{17} -7876.57 q^{18} +6411.64 q^{19} -25571.7 q^{20} -8656.36 q^{21} +402.335 q^{22} +108189. q^{23} +29518.2 q^{24} -15494.0 q^{25} +11163.8 q^{26} +94313.5 q^{27} -35047.6 q^{28} +249746. q^{29} -32093.5 q^{30} +60879.1 q^{31} +185972. q^{32} -1998.24 q^{33} -100945. q^{34} +85839.8 q^{35} +158387. q^{36} +182166. q^{37} +32580.0 q^{38} -55446.1 q^{39} -292714. q^{40} -89925.4 q^{41} -43986.2 q^{42} +346302. q^{43} -8090.40 q^{44} -387927. q^{45} +549751. q^{46} +631857. q^{47} -180084. q^{48} +117649. q^{49} -78730.7 q^{50} +501354. q^{51} -224489. q^{52} -861465. q^{53} +479243. q^{54} +19815.3 q^{55} -401183. q^{56} -161812. q^{57} +1.26906e6 q^{58} -157822. q^{59} +645357. q^{60} +2.00450e6 q^{61} +309350. q^{62} -531679. q^{63} +31628.0 q^{64} +549825. q^{65} -10153.8 q^{66} -999718. q^{67} +2.02987e6 q^{68} -2.73039e6 q^{69} +436185. q^{70} -315139. q^{71} +1.81302e6 q^{72} +3.40160e6 q^{73} +925657. q^{74} +391024. q^{75} -655139. q^{76} +27158.1 q^{77} -281743. q^{78} -646177. q^{79} +1.78578e6 q^{80} +1.00983e6 q^{81} -456945. q^{82} -8.03418e6 q^{83} +884503. q^{84} -4.97163e6 q^{85} +1.75969e6 q^{86} -6.30290e6 q^{87} -92609.2 q^{88} +3.26867e6 q^{89} -1.97120e6 q^{90} +753571. q^{91} -1.10547e7 q^{92} -1.53642e6 q^{93} +3.21070e6 q^{94} +1.60459e6 q^{95} -4.69340e6 q^{96} +3.88681e6 q^{97} +597819. q^{98} -122733. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.08138 0.449135 0.224567 0.974459i \(-0.427903\pi\)
0.224567 + 0.974459i \(0.427903\pi\)
\(3\) −25.2372 −0.539655 −0.269828 0.962909i \(-0.586967\pi\)
−0.269828 + 0.962909i \(0.586967\pi\)
\(4\) −102.180 −0.798278
\(5\) 250.262 0.895364 0.447682 0.894193i \(-0.352250\pi\)
0.447682 + 0.894193i \(0.352250\pi\)
\(6\) −128.240 −0.242378
\(7\) 343.000 0.377964
\(8\) −1169.63 −0.807669
\(9\) −1550.08 −0.708772
\(10\) 1271.68 0.402139
\(11\) 79.1782 0.0179362 0.00896812 0.999960i \(-0.497145\pi\)
0.00896812 + 0.999960i \(0.497145\pi\)
\(12\) 2578.73 0.430795
\(13\) 2197.00 0.277350
\(14\) 1742.91 0.169757
\(15\) −6315.91 −0.483188
\(16\) 7135.66 0.435526
\(17\) −19865.7 −0.980692 −0.490346 0.871528i \(-0.663129\pi\)
−0.490346 + 0.871528i \(0.663129\pi\)
\(18\) −7876.57 −0.318334
\(19\) 6411.64 0.214453 0.107226 0.994235i \(-0.465803\pi\)
0.107226 + 0.994235i \(0.465803\pi\)
\(20\) −25571.7 −0.714750
\(21\) −8656.36 −0.203971
\(22\) 402.335 0.00805579
\(23\) 108189. 1.85411 0.927057 0.374919i \(-0.122330\pi\)
0.927057 + 0.374919i \(0.122330\pi\)
\(24\) 29518.2 0.435863
\(25\) −15494.0 −0.198323
\(26\) 11163.8 0.124568
\(27\) 94313.5 0.922148
\(28\) −35047.6 −0.301721
\(29\) 249746. 1.90154 0.950772 0.309892i \(-0.100293\pi\)
0.950772 + 0.309892i \(0.100293\pi\)
\(30\) −32093.5 −0.217017
\(31\) 60879.1 0.367030 0.183515 0.983017i \(-0.441252\pi\)
0.183515 + 0.983017i \(0.441252\pi\)
\(32\) 185972. 1.00328
\(33\) −1998.24 −0.00967939
\(34\) −100945. −0.440463
\(35\) 85839.8 0.338416
\(36\) 158387. 0.565797
\(37\) 182166. 0.591238 0.295619 0.955306i \(-0.404474\pi\)
0.295619 + 0.955306i \(0.404474\pi\)
\(38\) 32580.0 0.0963181
\(39\) −55446.1 −0.149673
\(40\) −292714. −0.723158
\(41\) −89925.4 −0.203769 −0.101885 0.994796i \(-0.532487\pi\)
−0.101885 + 0.994796i \(0.532487\pi\)
\(42\) −43986.2 −0.0916103
\(43\) 346302. 0.664225 0.332112 0.943240i \(-0.392239\pi\)
0.332112 + 0.943240i \(0.392239\pi\)
\(44\) −8090.40 −0.0143181
\(45\) −387927. −0.634609
\(46\) 549751. 0.832747
\(47\) 631857. 0.887720 0.443860 0.896096i \(-0.353609\pi\)
0.443860 + 0.896096i \(0.353609\pi\)
\(48\) −180084. −0.235034
\(49\) 117649. 0.142857
\(50\) −78730.7 −0.0890736
\(51\) 501354. 0.529236
\(52\) −224489. −0.221402
\(53\) −861465. −0.794826 −0.397413 0.917640i \(-0.630092\pi\)
−0.397413 + 0.917640i \(0.630092\pi\)
\(54\) 479243. 0.414169
\(55\) 19815.3 0.0160595
\(56\) −401183. −0.305270
\(57\) −161812. −0.115731
\(58\) 1.26906e6 0.854049
\(59\) −157822. −0.100042 −0.0500212 0.998748i \(-0.515929\pi\)
−0.0500212 + 0.998748i \(0.515929\pi\)
\(60\) 645357. 0.385719
\(61\) 2.00450e6 1.13071 0.565354 0.824848i \(-0.308739\pi\)
0.565354 + 0.824848i \(0.308739\pi\)
\(62\) 309350. 0.164846
\(63\) −531679. −0.267891
\(64\) 31628.0 0.0150814
\(65\) 549825. 0.248329
\(66\) −10153.8 −0.00434735
\(67\) −999718. −0.406084 −0.203042 0.979170i \(-0.565083\pi\)
−0.203042 + 0.979170i \(0.565083\pi\)
\(68\) 2.02987e6 0.782864
\(69\) −2.73039e6 −1.00058
\(70\) 436185. 0.151994
\(71\) −315139. −0.104496 −0.0522478 0.998634i \(-0.516639\pi\)
−0.0522478 + 0.998634i \(0.516639\pi\)
\(72\) 1.81302e6 0.572453
\(73\) 3.40160e6 1.02342 0.511710 0.859158i \(-0.329012\pi\)
0.511710 + 0.859158i \(0.329012\pi\)
\(74\) 925657. 0.265545
\(75\) 391024. 0.107026
\(76\) −655139. −0.171193
\(77\) 27158.1 0.00677926
\(78\) −281743. −0.0672236
\(79\) −646177. −0.147454 −0.0737271 0.997278i \(-0.523489\pi\)
−0.0737271 + 0.997278i \(0.523489\pi\)
\(80\) 1.78578e6 0.389954
\(81\) 1.00983e6 0.211130
\(82\) −456945. −0.0915199
\(83\) −8.03418e6 −1.54230 −0.771149 0.636655i \(-0.780318\pi\)
−0.771149 + 0.636655i \(0.780318\pi\)
\(84\) 884503. 0.162825
\(85\) −4.97163e6 −0.878076
\(86\) 1.75969e6 0.298326
\(87\) −6.30290e6 −1.02618
\(88\) −92609.2 −0.0144865
\(89\) 3.26867e6 0.491480 0.245740 0.969336i \(-0.420969\pi\)
0.245740 + 0.969336i \(0.420969\pi\)
\(90\) −1.97120e6 −0.285025
\(91\) 753571. 0.104828
\(92\) −1.10547e7 −1.48010
\(93\) −1.53642e6 −0.198070
\(94\) 3.21070e6 0.398706
\(95\) 1.60459e6 0.192013
\(96\) −4.69340e6 −0.541425
\(97\) 3.88681e6 0.432406 0.216203 0.976348i \(-0.430633\pi\)
0.216203 + 0.976348i \(0.430633\pi\)
\(98\) 597819. 0.0641621
\(99\) −122733. −0.0127127
\(100\) 1.58317e6 0.158317
\(101\) −1.32661e7 −1.28120 −0.640601 0.767874i \(-0.721315\pi\)
−0.640601 + 0.767874i \(0.721315\pi\)
\(102\) 2.54757e6 0.237698
\(103\) 1.37517e7 1.24001 0.620006 0.784597i \(-0.287130\pi\)
0.620006 + 0.784597i \(0.287130\pi\)
\(104\) −2.56968e6 −0.224007
\(105\) −2.16636e6 −0.182628
\(106\) −4.37743e6 −0.356984
\(107\) −3.46186e6 −0.273191 −0.136595 0.990627i \(-0.543616\pi\)
−0.136595 + 0.990627i \(0.543616\pi\)
\(108\) −9.63691e6 −0.736131
\(109\) 3.12654e6 0.231245 0.115622 0.993293i \(-0.463114\pi\)
0.115622 + 0.993293i \(0.463114\pi\)
\(110\) 100689. 0.00721286
\(111\) −4.59737e6 −0.319065
\(112\) 2.44753e6 0.164613
\(113\) 4.84294e6 0.315744 0.157872 0.987460i \(-0.449537\pi\)
0.157872 + 0.987460i \(0.449537\pi\)
\(114\) −822227. −0.0519786
\(115\) 2.70756e7 1.66011
\(116\) −2.55190e7 −1.51796
\(117\) −3.40554e6 −0.196578
\(118\) −801951. −0.0449325
\(119\) −6.81393e6 −0.370667
\(120\) 7.38727e6 0.390256
\(121\) −1.94809e7 −0.999678
\(122\) 1.01856e7 0.507841
\(123\) 2.26946e6 0.109965
\(124\) −6.22060e6 −0.292992
\(125\) −2.34293e7 −1.07294
\(126\) −2.70166e6 −0.120319
\(127\) −3.32978e7 −1.44246 −0.721228 0.692697i \(-0.756422\pi\)
−0.721228 + 0.692697i \(0.756422\pi\)
\(128\) −2.36436e7 −0.996505
\(129\) −8.73968e6 −0.358452
\(130\) 2.79387e6 0.111533
\(131\) −2.65056e7 −1.03012 −0.515061 0.857154i \(-0.672231\pi\)
−0.515061 + 0.857154i \(0.672231\pi\)
\(132\) 204179. 0.00772684
\(133\) 2.19919e6 0.0810555
\(134\) −5.07995e6 −0.182386
\(135\) 2.36031e7 0.825659
\(136\) 2.32355e7 0.792074
\(137\) 4.90663e7 1.63028 0.815138 0.579267i \(-0.196661\pi\)
0.815138 + 0.579267i \(0.196661\pi\)
\(138\) −1.38742e7 −0.449397
\(139\) 4.79488e7 1.51435 0.757174 0.653213i \(-0.226579\pi\)
0.757174 + 0.653213i \(0.226579\pi\)
\(140\) −8.77108e6 −0.270150
\(141\) −1.59463e7 −0.479063
\(142\) −1.60134e6 −0.0469326
\(143\) 173955. 0.00497462
\(144\) −1.10609e7 −0.308689
\(145\) 6.25020e7 1.70257
\(146\) 1.72848e7 0.459653
\(147\) −2.96913e6 −0.0770936
\(148\) −1.86137e7 −0.471972
\(149\) 7.72401e7 1.91289 0.956447 0.291907i \(-0.0942897\pi\)
0.956447 + 0.291907i \(0.0942897\pi\)
\(150\) 1.98694e6 0.0480691
\(151\) −2.45116e7 −0.579364 −0.289682 0.957123i \(-0.593550\pi\)
−0.289682 + 0.957123i \(0.593550\pi\)
\(152\) −7.49924e6 −0.173207
\(153\) 3.07935e7 0.695087
\(154\) 138001. 0.00304480
\(155\) 1.52357e7 0.328626
\(156\) 5.66546e6 0.119481
\(157\) 2.83667e7 0.585005 0.292503 0.956265i \(-0.405512\pi\)
0.292503 + 0.956265i \(0.405512\pi\)
\(158\) −3.28347e6 −0.0662268
\(159\) 2.17409e7 0.428932
\(160\) 4.65416e7 0.898300
\(161\) 3.71089e7 0.700790
\(162\) 5.13131e6 0.0948257
\(163\) −2.08358e7 −0.376837 −0.188419 0.982089i \(-0.560336\pi\)
−0.188419 + 0.982089i \(0.560336\pi\)
\(164\) 9.18854e6 0.162665
\(165\) −500082. −0.00866658
\(166\) −4.08247e7 −0.692700
\(167\) 1.02968e8 1.71078 0.855391 0.517982i \(-0.173317\pi\)
0.855391 + 0.517982i \(0.173317\pi\)
\(168\) 1.01247e7 0.164741
\(169\) 4.82681e6 0.0769231
\(170\) −2.52627e7 −0.394374
\(171\) −9.93858e6 −0.151998
\(172\) −3.53850e7 −0.530236
\(173\) 7.84611e7 1.15211 0.576054 0.817412i \(-0.304592\pi\)
0.576054 + 0.817412i \(0.304592\pi\)
\(174\) −3.20274e7 −0.460892
\(175\) −5.31443e6 −0.0749590
\(176\) 564989. 0.00781170
\(177\) 3.98297e6 0.0539885
\(178\) 1.66093e7 0.220741
\(179\) −4.58627e7 −0.597688 −0.298844 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(180\) 3.96382e7 0.506595
\(181\) −4.79671e7 −0.601268 −0.300634 0.953740i \(-0.597198\pi\)
−0.300634 + 0.953740i \(0.597198\pi\)
\(182\) 3.82918e6 0.0470821
\(183\) −5.05878e7 −0.610193
\(184\) −1.26541e8 −1.49751
\(185\) 4.55893e7 0.529373
\(186\) −7.80711e6 −0.0889601
\(187\) −1.57293e6 −0.0175899
\(188\) −6.45629e7 −0.708648
\(189\) 3.23495e7 0.348539
\(190\) 8.15352e6 0.0862398
\(191\) −9.77001e7 −1.01456 −0.507281 0.861781i \(-0.669349\pi\)
−0.507281 + 0.861781i \(0.669349\pi\)
\(192\) −798203. −0.00813877
\(193\) 1.59457e8 1.59658 0.798292 0.602270i \(-0.205737\pi\)
0.798292 + 0.602270i \(0.205737\pi\)
\(194\) 1.97504e7 0.194209
\(195\) −1.38760e7 −0.134012
\(196\) −1.20213e7 −0.114040
\(197\) 1.18003e7 0.109967 0.0549833 0.998487i \(-0.482489\pi\)
0.0549833 + 0.998487i \(0.482489\pi\)
\(198\) −623653. −0.00570972
\(199\) −2.19765e7 −0.197684 −0.0988422 0.995103i \(-0.531514\pi\)
−0.0988422 + 0.995103i \(0.531514\pi\)
\(200\) 1.81222e7 0.160179
\(201\) 2.52301e7 0.219145
\(202\) −6.74100e7 −0.575432
\(203\) 8.56630e7 0.718716
\(204\) −5.12282e7 −0.422477
\(205\) −2.25049e7 −0.182448
\(206\) 6.98776e7 0.556933
\(207\) −1.67702e8 −1.31414
\(208\) 1.56770e7 0.120793
\(209\) 507662. 0.00384647
\(210\) −1.10081e7 −0.0820246
\(211\) 3.53463e7 0.259033 0.129517 0.991577i \(-0.458657\pi\)
0.129517 + 0.991577i \(0.458657\pi\)
\(212\) 8.80241e7 0.634492
\(213\) 7.95323e6 0.0563917
\(214\) −1.75910e7 −0.122699
\(215\) 8.66661e7 0.594723
\(216\) −1.10312e8 −0.744790
\(217\) 2.08815e7 0.138724
\(218\) 1.58872e7 0.103860
\(219\) −8.58469e7 −0.552294
\(220\) −2.02472e6 −0.0128199
\(221\) −4.36449e7 −0.271995
\(222\) −2.33610e7 −0.143303
\(223\) −3.81868e7 −0.230593 −0.115297 0.993331i \(-0.536782\pi\)
−0.115297 + 0.993331i \(0.536782\pi\)
\(224\) 6.37882e7 0.379204
\(225\) 2.40170e7 0.140566
\(226\) 2.46088e7 0.141811
\(227\) −1.30865e8 −0.742561 −0.371280 0.928521i \(-0.621081\pi\)
−0.371280 + 0.928521i \(0.621081\pi\)
\(228\) 1.65339e7 0.0923851
\(229\) 1.80392e8 0.992645 0.496322 0.868138i \(-0.334683\pi\)
0.496322 + 0.868138i \(0.334683\pi\)
\(230\) 1.37582e8 0.745612
\(231\) −685395. −0.00365847
\(232\) −2.92111e8 −1.53582
\(233\) −3.43754e8 −1.78034 −0.890169 0.455630i \(-0.849414\pi\)
−0.890169 + 0.455630i \(0.849414\pi\)
\(234\) −1.73048e7 −0.0882900
\(235\) 1.58130e8 0.794833
\(236\) 1.61261e7 0.0798617
\(237\) 1.63077e7 0.0795744
\(238\) −3.46242e7 −0.166479
\(239\) 1.55197e8 0.735346 0.367673 0.929955i \(-0.380155\pi\)
0.367673 + 0.929955i \(0.380155\pi\)
\(240\) −4.50681e7 −0.210441
\(241\) −1.44456e8 −0.664776 −0.332388 0.943143i \(-0.607854\pi\)
−0.332388 + 0.943143i \(0.607854\pi\)
\(242\) −9.89898e7 −0.448990
\(243\) −2.31749e8 −1.03609
\(244\) −2.04819e8 −0.902620
\(245\) 2.94431e7 0.127909
\(246\) 1.15320e7 0.0493892
\(247\) 1.40864e7 0.0594785
\(248\) −7.12060e7 −0.296439
\(249\) 2.02760e8 0.832310
\(250\) −1.19053e8 −0.481893
\(251\) −1.27311e8 −0.508168 −0.254084 0.967182i \(-0.581774\pi\)
−0.254084 + 0.967182i \(0.581774\pi\)
\(252\) 5.43267e7 0.213851
\(253\) 8.56623e6 0.0332559
\(254\) −1.69199e8 −0.647857
\(255\) 1.25470e8 0.473859
\(256\) −1.24191e8 −0.462646
\(257\) 2.52010e8 0.926086 0.463043 0.886336i \(-0.346757\pi\)
0.463043 + 0.886336i \(0.346757\pi\)
\(258\) −4.44096e7 −0.160993
\(259\) 6.24831e7 0.223467
\(260\) −5.61809e7 −0.198236
\(261\) −3.87128e8 −1.34776
\(262\) −1.34685e8 −0.462663
\(263\) 1.19269e8 0.404280 0.202140 0.979357i \(-0.435210\pi\)
0.202140 + 0.979357i \(0.435210\pi\)
\(264\) 2.33720e6 0.00781774
\(265\) −2.15592e8 −0.711659
\(266\) 1.11749e7 0.0364048
\(267\) −8.24919e7 −0.265230
\(268\) 1.02151e8 0.324168
\(269\) 8.10971e7 0.254023 0.127011 0.991901i \(-0.459462\pi\)
0.127011 + 0.991901i \(0.459462\pi\)
\(270\) 1.19936e8 0.370832
\(271\) −6.68806e7 −0.204130 −0.102065 0.994778i \(-0.532545\pi\)
−0.102065 + 0.994778i \(0.532545\pi\)
\(272\) −1.41755e8 −0.427117
\(273\) −1.90180e7 −0.0565713
\(274\) 2.49324e8 0.732214
\(275\) −1.22678e6 −0.00355716
\(276\) 2.78990e8 0.798744
\(277\) 4.93720e8 1.39573 0.697865 0.716230i \(-0.254134\pi\)
0.697865 + 0.716230i \(0.254134\pi\)
\(278\) 2.43646e8 0.680147
\(279\) −9.43677e7 −0.260141
\(280\) −1.00401e8 −0.273328
\(281\) 4.72884e8 1.27140 0.635701 0.771935i \(-0.280711\pi\)
0.635701 + 0.771935i \(0.280711\pi\)
\(282\) −8.10291e7 −0.215164
\(283\) −4.54591e7 −0.119225 −0.0596126 0.998222i \(-0.518987\pi\)
−0.0596126 + 0.998222i \(0.518987\pi\)
\(284\) 3.22008e7 0.0834166
\(285\) −4.04953e7 −0.103621
\(286\) 883929. 0.00223427
\(287\) −3.08444e7 −0.0770176
\(288\) −2.88272e8 −0.711096
\(289\) −1.56931e7 −0.0382442
\(290\) 3.17596e8 0.764685
\(291\) −9.80921e7 −0.233350
\(292\) −3.47574e8 −0.816973
\(293\) −6.32640e8 −1.46933 −0.734666 0.678429i \(-0.762661\pi\)
−0.734666 + 0.678429i \(0.762661\pi\)
\(294\) −1.50873e7 −0.0346254
\(295\) −3.94967e7 −0.0895744
\(296\) −2.13067e8 −0.477525
\(297\) 7.46758e6 0.0165399
\(298\) 3.92486e8 0.859147
\(299\) 2.37692e8 0.514239
\(300\) −3.99547e7 −0.0854365
\(301\) 1.18781e8 0.251053
\(302\) −1.24553e8 −0.260213
\(303\) 3.34798e8 0.691408
\(304\) 4.57512e7 0.0933997
\(305\) 5.01649e8 1.01240
\(306\) 1.56473e8 0.312188
\(307\) 7.75564e8 1.52980 0.764898 0.644151i \(-0.222789\pi\)
0.764898 + 0.644151i \(0.222789\pi\)
\(308\) −2.77501e6 −0.00541174
\(309\) −3.47054e8 −0.669180
\(310\) 7.74184e7 0.147597
\(311\) 3.83572e8 0.723078 0.361539 0.932357i \(-0.382251\pi\)
0.361539 + 0.932357i \(0.382251\pi\)
\(312\) 6.48514e7 0.120887
\(313\) 9.26958e8 1.70866 0.854329 0.519733i \(-0.173969\pi\)
0.854329 + 0.519733i \(0.173969\pi\)
\(314\) 1.44142e8 0.262746
\(315\) −1.33059e8 −0.239860
\(316\) 6.60261e7 0.117709
\(317\) −3.22691e8 −0.568957 −0.284479 0.958682i \(-0.591821\pi\)
−0.284479 + 0.958682i \(0.591821\pi\)
\(318\) 1.10474e8 0.192648
\(319\) 1.97745e7 0.0341065
\(320\) 7.91530e6 0.0135034
\(321\) 8.73675e7 0.147429
\(322\) 1.88564e8 0.314749
\(323\) −1.27372e8 −0.210312
\(324\) −1.03184e8 −0.168540
\(325\) −3.40402e7 −0.0550048
\(326\) −1.05875e8 −0.169251
\(327\) −7.89052e7 −0.124792
\(328\) 1.05179e8 0.164578
\(329\) 2.16727e8 0.335527
\(330\) −2.54111e6 −0.00389246
\(331\) −1.10934e9 −1.68139 −0.840693 0.541512i \(-0.817852\pi\)
−0.840693 + 0.541512i \(0.817852\pi\)
\(332\) 8.20930e8 1.23118
\(333\) −2.82373e8 −0.419053
\(334\) 5.23220e8 0.768372
\(335\) −2.50191e8 −0.363593
\(336\) −6.17688e7 −0.0888345
\(337\) −1.09216e9 −1.55446 −0.777231 0.629216i \(-0.783376\pi\)
−0.777231 + 0.629216i \(0.783376\pi\)
\(338\) 2.45268e7 0.0345488
\(339\) −1.22222e8 −0.170393
\(340\) 5.07999e8 0.700949
\(341\) 4.82030e6 0.00658315
\(342\) −5.05017e7 −0.0682676
\(343\) 4.03536e7 0.0539949
\(344\) −4.05045e8 −0.536474
\(345\) −6.83313e8 −0.895886
\(346\) 3.98691e8 0.517451
\(347\) 7.78256e8 0.999929 0.499965 0.866046i \(-0.333346\pi\)
0.499965 + 0.866046i \(0.333346\pi\)
\(348\) 6.44027e8 0.819176
\(349\) −1.03537e9 −1.30379 −0.651896 0.758309i \(-0.726026\pi\)
−0.651896 + 0.758309i \(0.726026\pi\)
\(350\) −2.70046e7 −0.0336667
\(351\) 2.07207e8 0.255758
\(352\) 1.47249e7 0.0179950
\(353\) 9.16514e8 1.10899 0.554495 0.832187i \(-0.312911\pi\)
0.554495 + 0.832187i \(0.312911\pi\)
\(354\) 2.02390e7 0.0242481
\(355\) −7.88674e7 −0.0935617
\(356\) −3.33991e8 −0.392337
\(357\) 1.71964e8 0.200032
\(358\) −2.33046e8 −0.268442
\(359\) −1.26975e9 −1.44840 −0.724200 0.689589i \(-0.757791\pi\)
−0.724200 + 0.689589i \(0.757791\pi\)
\(360\) 4.53731e8 0.512554
\(361\) −8.52763e8 −0.954010
\(362\) −2.43739e8 −0.270050
\(363\) 4.91643e8 0.539482
\(364\) −7.69996e7 −0.0836823
\(365\) 8.51292e8 0.916333
\(366\) −2.57056e8 −0.274059
\(367\) 1.57507e8 0.166330 0.0831648 0.996536i \(-0.473497\pi\)
0.0831648 + 0.996536i \(0.473497\pi\)
\(368\) 7.72001e8 0.807515
\(369\) 1.39392e8 0.144426
\(370\) 2.31657e8 0.237760
\(371\) −2.95482e8 −0.300416
\(372\) 1.56990e8 0.158115
\(373\) −1.01070e9 −1.00841 −0.504207 0.863583i \(-0.668215\pi\)
−0.504207 + 0.863583i \(0.668215\pi\)
\(374\) −7.99265e6 −0.00790024
\(375\) 5.91289e8 0.579015
\(376\) −7.39038e8 −0.716984
\(377\) 5.48693e8 0.527393
\(378\) 1.64380e8 0.156541
\(379\) 8.65372e8 0.816518 0.408259 0.912866i \(-0.366136\pi\)
0.408259 + 0.912866i \(0.366136\pi\)
\(380\) −1.63956e8 −0.153280
\(381\) 8.40343e8 0.778430
\(382\) −4.96451e8 −0.455675
\(383\) 4.93362e8 0.448714 0.224357 0.974507i \(-0.427972\pi\)
0.224357 + 0.974507i \(0.427972\pi\)
\(384\) 5.96699e8 0.537769
\(385\) 6.79665e6 0.00606991
\(386\) 8.10259e8 0.717081
\(387\) −5.36797e8 −0.470784
\(388\) −3.97153e8 −0.345181
\(389\) −9.07436e8 −0.781614 −0.390807 0.920473i \(-0.627804\pi\)
−0.390807 + 0.920473i \(0.627804\pi\)
\(390\) −7.05095e7 −0.0601896
\(391\) −2.14925e9 −1.81831
\(392\) −1.37606e8 −0.115381
\(393\) 6.68927e8 0.555910
\(394\) 5.99618e7 0.0493899
\(395\) −1.61714e8 −0.132025
\(396\) 1.25408e7 0.0101483
\(397\) −1.06013e9 −0.850336 −0.425168 0.905115i \(-0.639785\pi\)
−0.425168 + 0.905115i \(0.639785\pi\)
\(398\) −1.11671e8 −0.0887869
\(399\) −5.55014e7 −0.0437420
\(400\) −1.10560e8 −0.0863747
\(401\) 2.81346e8 0.217889 0.108945 0.994048i \(-0.465253\pi\)
0.108945 + 0.994048i \(0.465253\pi\)
\(402\) 1.28204e8 0.0984258
\(403\) 1.33751e8 0.101796
\(404\) 1.35552e9 1.02276
\(405\) 2.52721e8 0.189038
\(406\) 4.35286e8 0.322800
\(407\) 1.44236e7 0.0106046
\(408\) −5.86399e8 −0.427447
\(409\) 2.45583e9 1.77487 0.887436 0.460932i \(-0.152485\pi\)
0.887436 + 0.460932i \(0.152485\pi\)
\(410\) −1.14356e8 −0.0819437
\(411\) −1.23829e9 −0.879787
\(412\) −1.40514e9 −0.989875
\(413\) −5.41328e7 −0.0378125
\(414\) −8.52160e8 −0.590228
\(415\) −2.01065e9 −1.38092
\(416\) 4.08580e8 0.278259
\(417\) −1.21009e9 −0.817227
\(418\) 2.57962e6 0.00172759
\(419\) 1.12187e8 0.0745066 0.0372533 0.999306i \(-0.488139\pi\)
0.0372533 + 0.999306i \(0.488139\pi\)
\(420\) 2.21357e8 0.145788
\(421\) −1.09698e9 −0.716491 −0.358245 0.933627i \(-0.616625\pi\)
−0.358245 + 0.933627i \(0.616625\pi\)
\(422\) 1.79608e8 0.116341
\(423\) −9.79431e8 −0.629191
\(424\) 1.00759e9 0.641956
\(425\) 3.07798e8 0.194493
\(426\) 4.04134e7 0.0253274
\(427\) 6.87542e8 0.427368
\(428\) 3.53731e8 0.218082
\(429\) −4.39012e6 −0.00268458
\(430\) 4.40383e8 0.267111
\(431\) −1.34997e9 −0.812184 −0.406092 0.913832i \(-0.633109\pi\)
−0.406092 + 0.913832i \(0.633109\pi\)
\(432\) 6.72989e8 0.401619
\(433\) −2.79237e9 −1.65297 −0.826487 0.562956i \(-0.809664\pi\)
−0.826487 + 0.562956i \(0.809664\pi\)
\(434\) 1.06107e8 0.0623060
\(435\) −1.57738e9 −0.918803
\(436\) −3.19469e8 −0.184598
\(437\) 6.93670e8 0.397620
\(438\) −4.36221e8 −0.248054
\(439\) 1.73244e9 0.977309 0.488654 0.872477i \(-0.337488\pi\)
0.488654 + 0.872477i \(0.337488\pi\)
\(440\) −2.31766e7 −0.0129707
\(441\) −1.82366e8 −0.101253
\(442\) −2.21776e8 −0.122162
\(443\) 9.19646e8 0.502583 0.251291 0.967911i \(-0.419145\pi\)
0.251291 + 0.967911i \(0.419145\pi\)
\(444\) 4.69757e8 0.254702
\(445\) 8.18023e8 0.440053
\(446\) −1.94042e8 −0.103567
\(447\) −1.94932e9 −1.03230
\(448\) 1.08484e7 0.00570024
\(449\) 9.29747e8 0.484733 0.242367 0.970185i \(-0.422076\pi\)
0.242367 + 0.970185i \(0.422076\pi\)
\(450\) 1.22039e8 0.0631329
\(451\) −7.12014e6 −0.00365486
\(452\) −4.94850e8 −0.252051
\(453\) 6.18603e8 0.312657
\(454\) −6.64973e8 −0.333510
\(455\) 1.88590e8 0.0938597
\(456\) 1.89260e8 0.0934720
\(457\) −3.81360e8 −0.186908 −0.0934542 0.995624i \(-0.529791\pi\)
−0.0934542 + 0.995624i \(0.529791\pi\)
\(458\) 9.16642e8 0.445831
\(459\) −1.87360e9 −0.904343
\(460\) −2.76658e9 −1.32523
\(461\) 2.33633e9 1.11066 0.555330 0.831630i \(-0.312592\pi\)
0.555330 + 0.831630i \(0.312592\pi\)
\(462\) −3.48275e6 −0.00164314
\(463\) −3.70132e9 −1.73310 −0.866550 0.499090i \(-0.833668\pi\)
−0.866550 + 0.499090i \(0.833668\pi\)
\(464\) 1.78210e9 0.828171
\(465\) −3.84507e8 −0.177345
\(466\) −1.74675e9 −0.799612
\(467\) −5.19950e8 −0.236239 −0.118120 0.992999i \(-0.537687\pi\)
−0.118120 + 0.992999i \(0.537687\pi\)
\(468\) 3.47976e8 0.156924
\(469\) −3.42903e8 −0.153485
\(470\) 8.03517e8 0.356987
\(471\) −7.15895e8 −0.315701
\(472\) 1.84593e8 0.0808012
\(473\) 2.74196e7 0.0119137
\(474\) 8.28656e7 0.0357396
\(475\) −9.93417e7 −0.0425308
\(476\) 6.96245e8 0.295895
\(477\) 1.33534e9 0.563350
\(478\) 7.88617e8 0.330269
\(479\) −1.76687e9 −0.734565 −0.367282 0.930110i \(-0.619712\pi\)
−0.367282 + 0.930110i \(0.619712\pi\)
\(480\) −1.17458e9 −0.484772
\(481\) 4.00220e8 0.163980
\(482\) −7.34035e8 −0.298574
\(483\) −9.36524e8 −0.378185
\(484\) 1.99055e9 0.798021
\(485\) 9.72720e8 0.387161
\(486\) −1.17760e9 −0.465342
\(487\) 2.04299e9 0.801521 0.400760 0.916183i \(-0.368746\pi\)
0.400760 + 0.916183i \(0.368746\pi\)
\(488\) −2.34452e9 −0.913239
\(489\) 5.25837e8 0.203362
\(490\) 1.49611e8 0.0574485
\(491\) 3.47893e9 1.32636 0.663179 0.748461i \(-0.269207\pi\)
0.663179 + 0.748461i \(0.269207\pi\)
\(492\) −2.31893e8 −0.0877829
\(493\) −4.96138e9 −1.86483
\(494\) 7.15782e7 0.0267138
\(495\) −3.07154e7 −0.0113825
\(496\) 4.34412e8 0.159851
\(497\) −1.08093e8 −0.0394957
\(498\) 1.03030e9 0.373819
\(499\) 1.97195e9 0.710466 0.355233 0.934778i \(-0.384401\pi\)
0.355233 + 0.934778i \(0.384401\pi\)
\(500\) 2.39399e9 0.856501
\(501\) −2.59862e9 −0.923233
\(502\) −6.46914e8 −0.228236
\(503\) 2.02086e9 0.708025 0.354012 0.935241i \(-0.384817\pi\)
0.354012 + 0.935241i \(0.384817\pi\)
\(504\) 6.21867e8 0.216367
\(505\) −3.31999e9 −1.14714
\(506\) 4.35283e7 0.0149364
\(507\) −1.21815e8 −0.0415120
\(508\) 3.40236e9 1.15148
\(509\) 5.59050e9 1.87905 0.939525 0.342481i \(-0.111267\pi\)
0.939525 + 0.342481i \(0.111267\pi\)
\(510\) 6.37560e8 0.212826
\(511\) 1.16675e9 0.386816
\(512\) 2.39533e9 0.788715
\(513\) 6.04704e8 0.197757
\(514\) 1.28056e9 0.415937
\(515\) 3.44153e9 1.11026
\(516\) 8.93017e8 0.286145
\(517\) 5.00293e7 0.0159224
\(518\) 3.17500e8 0.100367
\(519\) −1.98014e9 −0.621741
\(520\) −6.43092e8 −0.200568
\(521\) 4.67025e7 0.0144680 0.00723400 0.999974i \(-0.497697\pi\)
0.00723400 + 0.999974i \(0.497697\pi\)
\(522\) −1.96714e9 −0.605326
\(523\) 6.32147e9 1.93224 0.966122 0.258086i \(-0.0830917\pi\)
0.966122 + 0.258086i \(0.0830917\pi\)
\(524\) 2.70833e9 0.822323
\(525\) 1.34121e8 0.0404520
\(526\) 6.06050e8 0.181576
\(527\) −1.20940e9 −0.359944
\(528\) −1.42587e7 −0.00421562
\(529\) 8.30009e9 2.43774
\(530\) −1.09550e9 −0.319631
\(531\) 2.44637e8 0.0709073
\(532\) −2.24713e8 −0.0647048
\(533\) −1.97566e8 −0.0565155
\(534\) −4.19173e8 −0.119124
\(535\) −8.66371e8 −0.244605
\(536\) 1.16930e9 0.327981
\(537\) 1.15745e9 0.322545
\(538\) 4.12085e8 0.114090
\(539\) 9.31524e6 0.00256232
\(540\) −2.41175e9 −0.659105
\(541\) 2.16904e9 0.588948 0.294474 0.955660i \(-0.404856\pi\)
0.294474 + 0.955660i \(0.404856\pi\)
\(542\) −3.39845e8 −0.0916820
\(543\) 1.21055e9 0.324478
\(544\) −3.69445e9 −0.983907
\(545\) 7.82455e8 0.207048
\(546\) −9.66377e7 −0.0254081
\(547\) −4.63439e9 −1.21070 −0.605351 0.795959i \(-0.706967\pi\)
−0.605351 + 0.795959i \(0.706967\pi\)
\(548\) −5.01357e9 −1.30141
\(549\) −3.10714e9 −0.801415
\(550\) −6.23376e6 −0.00159765
\(551\) 1.60128e9 0.407791
\(552\) 3.19355e9 0.808140
\(553\) −2.21639e8 −0.0557324
\(554\) 2.50878e9 0.626870
\(555\) −1.15055e9 −0.285679
\(556\) −4.89939e9 −1.20887
\(557\) 3.36947e9 0.826167 0.413084 0.910693i \(-0.364452\pi\)
0.413084 + 0.910693i \(0.364452\pi\)
\(558\) −4.79518e8 −0.116838
\(559\) 7.60825e8 0.184223
\(560\) 6.12524e8 0.147389
\(561\) 3.96963e7 0.00949250
\(562\) 2.40291e9 0.571031
\(563\) 4.34631e9 1.02646 0.513229 0.858252i \(-0.328449\pi\)
0.513229 + 0.858252i \(0.328449\pi\)
\(564\) 1.62939e9 0.382426
\(565\) 1.21200e9 0.282706
\(566\) −2.30995e8 −0.0535482
\(567\) 3.46371e8 0.0797996
\(568\) 3.68596e8 0.0843979
\(569\) 4.26627e9 0.970857 0.485429 0.874276i \(-0.338664\pi\)
0.485429 + 0.874276i \(0.338664\pi\)
\(570\) −2.05772e8 −0.0465398
\(571\) −5.43149e9 −1.22094 −0.610468 0.792041i \(-0.709019\pi\)
−0.610468 + 0.792041i \(0.709019\pi\)
\(572\) −1.77746e7 −0.00397113
\(573\) 2.46568e9 0.547514
\(574\) −1.56732e8 −0.0345913
\(575\) −1.67628e9 −0.367713
\(576\) −4.90261e7 −0.0106893
\(577\) 6.01348e9 1.30320 0.651599 0.758563i \(-0.274098\pi\)
0.651599 + 0.758563i \(0.274098\pi\)
\(578\) −7.97424e7 −0.0171768
\(579\) −4.02424e9 −0.861605
\(580\) −6.38643e9 −1.35913
\(581\) −2.75572e9 −0.582934
\(582\) −4.98443e8 −0.104806
\(583\) −6.82093e7 −0.0142562
\(584\) −3.97862e9 −0.826584
\(585\) −8.52276e8 −0.176009
\(586\) −3.21468e9 −0.659928
\(587\) −4.21371e9 −0.859866 −0.429933 0.902861i \(-0.641463\pi\)
−0.429933 + 0.902861i \(0.641463\pi\)
\(588\) 3.03384e8 0.0615422
\(589\) 3.90335e8 0.0787107
\(590\) −2.00698e8 −0.0402310
\(591\) −2.97806e8 −0.0593441
\(592\) 1.29988e9 0.257499
\(593\) −3.07960e9 −0.606461 −0.303231 0.952917i \(-0.598065\pi\)
−0.303231 + 0.952917i \(0.598065\pi\)
\(594\) 3.79456e7 0.00742863
\(595\) −1.70527e9 −0.331882
\(596\) −7.89236e9 −1.52702
\(597\) 5.54625e8 0.106681
\(598\) 1.20780e9 0.230963
\(599\) 5.05994e8 0.0961948 0.0480974 0.998843i \(-0.484684\pi\)
0.0480974 + 0.998843i \(0.484684\pi\)
\(600\) −4.57353e8 −0.0864415
\(601\) 9.12497e9 1.71463 0.857315 0.514792i \(-0.172131\pi\)
0.857315 + 0.514792i \(0.172131\pi\)
\(602\) 6.03574e8 0.112757
\(603\) 1.54965e9 0.287821
\(604\) 2.50458e9 0.462494
\(605\) −4.87533e9 −0.895076
\(606\) 1.70124e9 0.310535
\(607\) −6.00486e9 −1.08979 −0.544894 0.838505i \(-0.683430\pi\)
−0.544894 + 0.838505i \(0.683430\pi\)
\(608\) 1.19238e9 0.215156
\(609\) −2.16189e9 −0.387859
\(610\) 2.54907e9 0.454702
\(611\) 1.38819e9 0.246209
\(612\) −3.14647e9 −0.554872
\(613\) −6.37941e9 −1.11858 −0.559292 0.828971i \(-0.688927\pi\)
−0.559292 + 0.828971i \(0.688927\pi\)
\(614\) 3.94094e9 0.687085
\(615\) 5.67961e8 0.0984590
\(616\) −3.17650e7 −0.00547540
\(617\) 8.45037e9 1.44836 0.724182 0.689609i \(-0.242217\pi\)
0.724182 + 0.689609i \(0.242217\pi\)
\(618\) −1.76351e9 −0.300552
\(619\) −9.26189e9 −1.56958 −0.784788 0.619764i \(-0.787228\pi\)
−0.784788 + 0.619764i \(0.787228\pi\)
\(620\) −1.55678e9 −0.262335
\(621\) 1.02037e10 1.70977
\(622\) 1.94907e9 0.324759
\(623\) 1.12115e9 0.185762
\(624\) −3.95644e8 −0.0651867
\(625\) −4.65299e9 −0.762345
\(626\) 4.71023e9 0.767417
\(627\) −1.28120e7 −0.00207577
\(628\) −2.89849e9 −0.466997
\(629\) −3.61886e9 −0.579822
\(630\) −6.76123e8 −0.107729
\(631\) −4.80371e9 −0.761156 −0.380578 0.924749i \(-0.624275\pi\)
−0.380578 + 0.924749i \(0.624275\pi\)
\(632\) 7.55788e8 0.119094
\(633\) −8.92041e8 −0.139789
\(634\) −1.63972e9 −0.255539
\(635\) −8.33318e9 −1.29152
\(636\) −2.22148e9 −0.342407
\(637\) 2.58475e8 0.0396214
\(638\) 1.00482e8 0.0153184
\(639\) 4.88492e8 0.0740636
\(640\) −5.91710e9 −0.892235
\(641\) 3.90672e9 0.585881 0.292940 0.956131i \(-0.405366\pi\)
0.292940 + 0.956131i \(0.405366\pi\)
\(642\) 4.43947e8 0.0662154
\(643\) 2.35459e9 0.349282 0.174641 0.984632i \(-0.444124\pi\)
0.174641 + 0.984632i \(0.444124\pi\)
\(644\) −3.79177e9 −0.559425
\(645\) −2.18721e9 −0.320945
\(646\) −6.47223e8 −0.0944584
\(647\) 1.09365e9 0.158750 0.0793750 0.996845i \(-0.474708\pi\)
0.0793750 + 0.996845i \(0.474708\pi\)
\(648\) −1.18112e9 −0.170523
\(649\) −1.24960e7 −0.00179439
\(650\) −1.72971e8 −0.0247046
\(651\) −5.26991e8 −0.0748634
\(652\) 2.12900e9 0.300821
\(653\) −6.19178e8 −0.0870201 −0.0435100 0.999053i \(-0.513854\pi\)
−0.0435100 + 0.999053i \(0.513854\pi\)
\(654\) −4.00947e8 −0.0560486
\(655\) −6.63335e9 −0.922334
\(656\) −6.41677e8 −0.0887468
\(657\) −5.27277e9 −0.725371
\(658\) 1.10127e9 0.150697
\(659\) 8.13367e9 1.10710 0.553551 0.832815i \(-0.313272\pi\)
0.553551 + 0.832815i \(0.313272\pi\)
\(660\) 5.10982e7 0.00691834
\(661\) −3.23653e9 −0.435887 −0.217944 0.975961i \(-0.569935\pi\)
−0.217944 + 0.975961i \(0.569935\pi\)
\(662\) −5.63699e9 −0.755169
\(663\) 1.10147e9 0.146784
\(664\) 9.39702e9 1.24567
\(665\) 5.50374e8 0.0725742
\(666\) −1.43485e9 −0.188211
\(667\) 2.70199e10 3.52568
\(668\) −1.05212e10 −1.36568
\(669\) 9.63728e8 0.124441
\(670\) −1.27132e9 −0.163302
\(671\) 1.58712e8 0.0202807
\(672\) −1.60984e9 −0.204639
\(673\) −8.09061e9 −1.02313 −0.511563 0.859246i \(-0.670933\pi\)
−0.511563 + 0.859246i \(0.670933\pi\)
\(674\) −5.54966e9 −0.698163
\(675\) −1.46129e9 −0.182883
\(676\) −4.93201e8 −0.0614060
\(677\) 5.86699e9 0.726700 0.363350 0.931653i \(-0.381633\pi\)
0.363350 + 0.931653i \(0.381633\pi\)
\(678\) −6.21057e8 −0.0765293
\(679\) 1.33318e9 0.163434
\(680\) 5.81496e9 0.709195
\(681\) 3.30266e9 0.400727
\(682\) 2.44938e7 0.00295672
\(683\) −1.37044e10 −1.64584 −0.822922 0.568155i \(-0.807658\pi\)
−0.822922 + 0.568155i \(0.807658\pi\)
\(684\) 1.01552e9 0.121337
\(685\) 1.22794e10 1.45969
\(686\) 2.05052e8 0.0242510
\(687\) −4.55259e9 −0.535686
\(688\) 2.47109e9 0.289287
\(689\) −1.89264e9 −0.220445
\(690\) −3.47217e9 −0.402374
\(691\) −1.60119e10 −1.84617 −0.923083 0.384600i \(-0.874339\pi\)
−0.923083 + 0.384600i \(0.874339\pi\)
\(692\) −8.01712e9 −0.919702
\(693\) −4.20974e7 −0.00480495
\(694\) 3.95461e9 0.449103
\(695\) 1.19998e10 1.35589
\(696\) 7.37205e9 0.828812
\(697\) 1.78643e9 0.199835
\(698\) −5.26113e9 −0.585578
\(699\) 8.67539e9 0.960769
\(700\) 5.43026e8 0.0598381
\(701\) −1.24919e10 −1.36967 −0.684834 0.728699i \(-0.740125\pi\)
−0.684834 + 0.728699i \(0.740125\pi\)
\(702\) 1.05290e9 0.114870
\(703\) 1.16799e9 0.126793
\(704\) 2.50425e6 0.000270504 0
\(705\) −3.99075e9 −0.428936
\(706\) 4.65716e9 0.498086
\(707\) −4.55026e9 −0.484249
\(708\) −4.06979e8 −0.0430978
\(709\) 6.42382e9 0.676911 0.338455 0.940982i \(-0.390096\pi\)
0.338455 + 0.940982i \(0.390096\pi\)
\(710\) −4.00755e8 −0.0420218
\(711\) 1.00163e9 0.104511
\(712\) −3.82313e9 −0.396953
\(713\) 6.58646e9 0.680517
\(714\) 8.73817e8 0.0898414
\(715\) 4.35342e7 0.00445410
\(716\) 4.68623e9 0.477121
\(717\) −3.91675e9 −0.396834
\(718\) −6.45210e9 −0.650527
\(719\) −5.20098e9 −0.521836 −0.260918 0.965361i \(-0.584025\pi\)
−0.260918 + 0.965361i \(0.584025\pi\)
\(720\) −2.76811e9 −0.276389
\(721\) 4.71683e9 0.468681
\(722\) −4.33321e9 −0.428479
\(723\) 3.64566e9 0.358750
\(724\) 4.90126e9 0.479979
\(725\) −3.86956e9 −0.377119
\(726\) 2.49823e9 0.242300
\(727\) −1.29835e10 −1.25321 −0.626603 0.779339i \(-0.715555\pi\)
−0.626603 + 0.779339i \(0.715555\pi\)
\(728\) −8.81399e8 −0.0846667
\(729\) 3.64020e9 0.347999
\(730\) 4.32574e9 0.411557
\(731\) −6.87952e9 −0.651399
\(732\) 5.16905e9 0.487104
\(733\) −2.17698e9 −0.204169 −0.102085 0.994776i \(-0.532551\pi\)
−0.102085 + 0.994776i \(0.532551\pi\)
\(734\) 8.00355e8 0.0747044
\(735\) −7.43060e8 −0.0690269
\(736\) 2.01201e10 1.86019
\(737\) −7.91559e7 −0.00728362
\(738\) 7.08304e8 0.0648667
\(739\) −3.15646e9 −0.287703 −0.143852 0.989599i \(-0.545949\pi\)
−0.143852 + 0.989599i \(0.545949\pi\)
\(740\) −4.65830e9 −0.422587
\(741\) −3.55500e8 −0.0320979
\(742\) −1.50146e9 −0.134927
\(743\) 1.71071e10 1.53008 0.765042 0.643980i \(-0.222718\pi\)
0.765042 + 0.643980i \(0.222718\pi\)
\(744\) 1.79704e9 0.159975
\(745\) 1.93303e10 1.71274
\(746\) −5.13572e9 −0.452914
\(747\) 1.24537e10 1.09314
\(748\) 1.60721e8 0.0140416
\(749\) −1.18742e9 −0.103256
\(750\) 3.00456e9 0.260056
\(751\) 4.81930e9 0.415188 0.207594 0.978215i \(-0.433437\pi\)
0.207594 + 0.978215i \(0.433437\pi\)
\(752\) 4.50871e9 0.386625
\(753\) 3.21296e9 0.274235
\(754\) 2.78812e9 0.236871
\(755\) −6.13431e9 −0.518742
\(756\) −3.30546e9 −0.278231
\(757\) −1.31139e9 −0.109875 −0.0549373 0.998490i \(-0.517496\pi\)
−0.0549373 + 0.998490i \(0.517496\pi\)
\(758\) 4.39728e9 0.366726
\(759\) −2.16188e8 −0.0179467
\(760\) −1.87678e9 −0.155083
\(761\) 7.59828e9 0.624984 0.312492 0.949920i \(-0.398836\pi\)
0.312492 + 0.949920i \(0.398836\pi\)
\(762\) 4.27010e9 0.349620
\(763\) 1.07240e9 0.0874023
\(764\) 9.98296e9 0.809902
\(765\) 7.70644e9 0.622356
\(766\) 2.50696e9 0.201533
\(767\) −3.46734e8 −0.0277468
\(768\) 3.13422e9 0.249670
\(769\) −5.74291e9 −0.455397 −0.227698 0.973732i \(-0.573120\pi\)
−0.227698 + 0.973732i \(0.573120\pi\)
\(770\) 3.45363e7 0.00272621
\(771\) −6.36002e9 −0.499768
\(772\) −1.62932e10 −1.27452
\(773\) 9.84863e9 0.766916 0.383458 0.923558i \(-0.374733\pi\)
0.383458 + 0.923558i \(0.374733\pi\)
\(774\) −2.72767e9 −0.211445
\(775\) −9.43258e8 −0.0727905
\(776\) −4.54613e9 −0.349241
\(777\) −1.57690e9 −0.120595
\(778\) −4.61102e9 −0.351050
\(779\) −5.76569e8 −0.0436989
\(780\) 1.41785e9 0.106979
\(781\) −2.49522e7 −0.00187426
\(782\) −1.09212e10 −0.816668
\(783\) 2.35545e10 1.75350
\(784\) 8.39503e8 0.0622180
\(785\) 7.09910e9 0.523793
\(786\) 3.39907e9 0.249679
\(787\) −1.37136e10 −1.00286 −0.501429 0.865199i \(-0.667192\pi\)
−0.501429 + 0.865199i \(0.667192\pi\)
\(788\) −1.20575e9 −0.0877840
\(789\) −3.01001e9 −0.218172
\(790\) −8.21728e8 −0.0592971
\(791\) 1.66113e9 0.119340
\(792\) 1.43552e8 0.0102677
\(793\) 4.40388e9 0.313602
\(794\) −5.38690e9 −0.381915
\(795\) 5.44093e9 0.384050
\(796\) 2.24555e9 0.157807
\(797\) 1.65594e9 0.115862 0.0579308 0.998321i \(-0.481550\pi\)
0.0579308 + 0.998321i \(0.481550\pi\)
\(798\) −2.82024e8 −0.0196461
\(799\) −1.25523e10 −0.870580
\(800\) −2.88144e9 −0.198973
\(801\) −5.06671e9 −0.348347
\(802\) 1.42963e9 0.0978616
\(803\) 2.69333e8 0.0183563
\(804\) −2.57800e9 −0.174939
\(805\) 9.28695e9 0.627462
\(806\) 6.79641e8 0.0457201
\(807\) −2.04666e9 −0.137085
\(808\) 1.55164e10 1.03479
\(809\) −4.41839e9 −0.293389 −0.146695 0.989182i \(-0.546863\pi\)
−0.146695 + 0.989182i \(0.546863\pi\)
\(810\) 1.28417e9 0.0849036
\(811\) −1.42306e10 −0.936806 −0.468403 0.883515i \(-0.655170\pi\)
−0.468403 + 0.883515i \(0.655170\pi\)
\(812\) −8.75301e9 −0.573735
\(813\) 1.68788e9 0.110160
\(814\) 7.32919e7 0.00476289
\(815\) −5.21441e9 −0.337407
\(816\) 3.57749e9 0.230496
\(817\) 2.22036e9 0.142445
\(818\) 1.24790e10 0.797156
\(819\) −1.16810e9 −0.0742995
\(820\) 2.29954e9 0.145644
\(821\) −6.36868e9 −0.401650 −0.200825 0.979627i \(-0.564362\pi\)
−0.200825 + 0.979627i \(0.564362\pi\)
\(822\) −6.29225e9 −0.395143
\(823\) −1.52881e10 −0.955993 −0.477997 0.878362i \(-0.658637\pi\)
−0.477997 + 0.878362i \(0.658637\pi\)
\(824\) −1.60844e10 −1.00152
\(825\) 3.09606e7 0.00191964
\(826\) −2.75069e8 −0.0169829
\(827\) −8.42914e7 −0.00518220 −0.00259110 0.999997i \(-0.500825\pi\)
−0.00259110 + 0.999997i \(0.500825\pi\)
\(828\) 1.71358e10 1.04905
\(829\) 2.68377e9 0.163608 0.0818040 0.996648i \(-0.473932\pi\)
0.0818040 + 0.996648i \(0.473932\pi\)
\(830\) −1.02169e10 −0.620219
\(831\) −1.24601e10 −0.753213
\(832\) 6.94868e7 0.00418284
\(833\) −2.33718e9 −0.140099
\(834\) −6.14894e9 −0.367045
\(835\) 2.57690e10 1.53177
\(836\) −5.18727e7 −0.00307056
\(837\) 5.74172e9 0.338456
\(838\) 5.70066e8 0.0334635
\(839\) −1.82317e10 −1.06576 −0.532881 0.846190i \(-0.678891\pi\)
−0.532881 + 0.846190i \(0.678891\pi\)
\(840\) 2.53383e9 0.147503
\(841\) 4.51234e10 2.61587
\(842\) −5.57416e9 −0.321801
\(843\) −1.19343e10 −0.686119
\(844\) −3.61167e9 −0.206780
\(845\) 1.20797e9 0.0688742
\(846\) −4.97686e9 −0.282592
\(847\) −6.68195e9 −0.377843
\(848\) −6.14712e9 −0.346167
\(849\) 1.14726e9 0.0643406
\(850\) 1.56404e9 0.0873537
\(851\) 1.97084e10 1.09622
\(852\) −8.12658e8 −0.0450162
\(853\) 2.15367e10 1.18811 0.594055 0.804424i \(-0.297526\pi\)
0.594055 + 0.804424i \(0.297526\pi\)
\(854\) 3.49366e9 0.191946
\(855\) −2.48725e9 −0.136094
\(856\) 4.04909e9 0.220648
\(857\) −1.59483e10 −0.865526 −0.432763 0.901508i \(-0.642461\pi\)
−0.432763 + 0.901508i \(0.642461\pi\)
\(858\) −2.23079e7 −0.00120574
\(859\) 1.52450e10 0.820639 0.410319 0.911942i \(-0.365417\pi\)
0.410319 + 0.911942i \(0.365417\pi\)
\(860\) −8.85551e9 −0.474754
\(861\) 7.78426e8 0.0415630
\(862\) −6.85972e9 −0.364780
\(863\) −2.76238e9 −0.146300 −0.0731502 0.997321i \(-0.523305\pi\)
−0.0731502 + 0.997321i \(0.523305\pi\)
\(864\) 1.75396e10 0.925172
\(865\) 1.96358e10 1.03156
\(866\) −1.41891e10 −0.742408
\(867\) 3.96049e8 0.0206387
\(868\) −2.13367e9 −0.110741
\(869\) −5.11632e7 −0.00264477
\(870\) −8.01524e9 −0.412666
\(871\) −2.19638e9 −0.112627
\(872\) −3.65690e9 −0.186769
\(873\) −6.02488e9 −0.306478
\(874\) 3.52480e9 0.178585
\(875\) −8.03624e9 −0.405531
\(876\) 8.77180e9 0.440884
\(877\) −1.72192e10 −0.862016 −0.431008 0.902348i \(-0.641842\pi\)
−0.431008 + 0.902348i \(0.641842\pi\)
\(878\) 8.80317e9 0.438943
\(879\) 1.59661e10 0.792933
\(880\) 1.41395e8 0.00699431
\(881\) 9.29977e8 0.0458201 0.0229101 0.999738i \(-0.492707\pi\)
0.0229101 + 0.999738i \(0.492707\pi\)
\(882\) −9.26670e8 −0.0454763
\(883\) −1.36965e10 −0.669494 −0.334747 0.942308i \(-0.608651\pi\)
−0.334747 + 0.942308i \(0.608651\pi\)
\(884\) 4.45962e9 0.217128
\(885\) 9.96786e8 0.0483393
\(886\) 4.67307e9 0.225727
\(887\) −2.01674e10 −0.970327 −0.485163 0.874424i \(-0.661240\pi\)
−0.485163 + 0.874424i \(0.661240\pi\)
\(888\) 5.37722e9 0.257699
\(889\) −1.14212e10 −0.545197
\(890\) 4.15668e9 0.197643
\(891\) 7.99563e7 0.00378687
\(892\) 3.90192e9 0.184078
\(893\) 4.05124e9 0.190374
\(894\) −9.90525e9 −0.463643
\(895\) −1.14777e10 −0.535148
\(896\) −8.10977e9 −0.376644
\(897\) −5.99867e9 −0.277512
\(898\) 4.72440e9 0.217711
\(899\) 1.52043e10 0.697924
\(900\) −2.45404e9 −0.112210
\(901\) 1.71136e10 0.779479
\(902\) −3.61801e7 −0.00164152
\(903\) −2.99771e9 −0.135482
\(904\) −5.66445e9 −0.255016
\(905\) −1.20043e10 −0.538354
\(906\) 3.14336e9 0.140425
\(907\) −1.69394e10 −0.753829 −0.376914 0.926248i \(-0.623015\pi\)
−0.376914 + 0.926248i \(0.623015\pi\)
\(908\) 1.33717e10 0.592770
\(909\) 2.05635e10 0.908080
\(910\) 9.58298e8 0.0421556
\(911\) −4.37632e9 −0.191776 −0.0958881 0.995392i \(-0.530569\pi\)
−0.0958881 + 0.995392i \(0.530569\pi\)
\(912\) −1.15463e9 −0.0504036
\(913\) −6.36132e8 −0.0276630
\(914\) −1.93784e9 −0.0839471
\(915\) −1.26602e10 −0.546345
\(916\) −1.84324e10 −0.792406
\(917\) −9.09142e9 −0.389349
\(918\) −9.52048e9 −0.406172
\(919\) −2.89737e10 −1.23140 −0.615702 0.787979i \(-0.711127\pi\)
−0.615702 + 0.787979i \(0.711127\pi\)
\(920\) −3.16685e10 −1.34082
\(921\) −1.95731e10 −0.825563
\(922\) 1.18718e10 0.498836
\(923\) −6.92361e8 −0.0289819
\(924\) 7.00334e7 0.00292047
\(925\) −2.82248e9 −0.117256
\(926\) −1.88078e10 −0.778395
\(927\) −2.13163e10 −0.878886
\(928\) 4.64457e10 1.90778
\(929\) 8.50621e9 0.348082 0.174041 0.984738i \(-0.444318\pi\)
0.174041 + 0.984738i \(0.444318\pi\)
\(930\) −1.95382e9 −0.0796517
\(931\) 7.54323e8 0.0306361
\(932\) 3.51247e10 1.42121
\(933\) −9.68027e9 −0.390213
\(934\) −2.64206e9 −0.106103
\(935\) −3.93645e8 −0.0157494
\(936\) 3.98322e9 0.158770
\(937\) −2.24557e10 −0.891741 −0.445870 0.895098i \(-0.647106\pi\)
−0.445870 + 0.895098i \(0.647106\pi\)
\(938\) −1.74242e9 −0.0689355
\(939\) −2.33938e10 −0.922086
\(940\) −1.61576e10 −0.634498
\(941\) −3.71711e9 −0.145426 −0.0727129 0.997353i \(-0.523166\pi\)
−0.0727129 + 0.997353i \(0.523166\pi\)
\(942\) −3.63773e9 −0.141792
\(943\) −9.72896e9 −0.377812
\(944\) −1.12616e9 −0.0435711
\(945\) 8.09586e9 0.312070
\(946\) 1.39329e8 0.00535085
\(947\) 1.48219e10 0.567126 0.283563 0.958954i \(-0.408483\pi\)
0.283563 + 0.958954i \(0.408483\pi\)
\(948\) −1.66631e9 −0.0635225
\(949\) 7.47332e9 0.283845
\(950\) −5.04793e8 −0.0191021
\(951\) 8.14382e9 0.307041
\(952\) 7.96978e9 0.299376
\(953\) −2.23056e10 −0.834811 −0.417405 0.908720i \(-0.637060\pi\)
−0.417405 + 0.908720i \(0.637060\pi\)
\(954\) 6.78538e9 0.253020
\(955\) −2.44506e10 −0.908402
\(956\) −1.58580e10 −0.587011
\(957\) −4.99052e8 −0.0184058
\(958\) −8.97813e9 −0.329918
\(959\) 1.68297e10 0.616186
\(960\) −1.99760e8 −0.00728717
\(961\) −2.38064e10 −0.865289
\(962\) 2.03367e9 0.0736491
\(963\) 5.36617e9 0.193630
\(964\) 1.47604e10 0.530676
\(965\) 3.99059e10 1.42952
\(966\) −4.75884e9 −0.169856
\(967\) −4.96874e10 −1.76707 −0.883534 0.468366i \(-0.844843\pi\)
−0.883534 + 0.468366i \(0.844843\pi\)
\(968\) 2.27854e10 0.807409
\(969\) 3.21450e9 0.113496
\(970\) 4.94276e9 0.173888
\(971\) −3.96345e10 −1.38933 −0.694665 0.719333i \(-0.744448\pi\)
−0.694665 + 0.719333i \(0.744448\pi\)
\(972\) 2.36800e10 0.827084
\(973\) 1.64464e10 0.572370
\(974\) 1.03812e10 0.359991
\(975\) 8.59080e8 0.0296837
\(976\) 1.43034e10 0.492453
\(977\) −2.94172e10 −1.00918 −0.504592 0.863358i \(-0.668357\pi\)
−0.504592 + 0.863358i \(0.668357\pi\)
\(978\) 2.67198e9 0.0913371
\(979\) 2.58807e8 0.00881530
\(980\) −3.00848e9 −0.102107
\(981\) −4.84641e9 −0.163900
\(982\) 1.76778e10 0.595713
\(983\) −4.51735e10 −1.51686 −0.758432 0.651752i \(-0.774034\pi\)
−0.758432 + 0.651752i \(0.774034\pi\)
\(984\) −2.65443e9 −0.0888155
\(985\) 2.95317e9 0.0984603
\(986\) −2.52107e10 −0.837559
\(987\) −5.46958e9 −0.181069
\(988\) −1.43934e9 −0.0474804
\(989\) 3.74661e10 1.23155
\(990\) −1.56077e8 −0.00511228
\(991\) 2.66926e10 0.871232 0.435616 0.900133i \(-0.356531\pi\)
0.435616 + 0.900133i \(0.356531\pi\)
\(992\) 1.13218e10 0.368234
\(993\) 2.79967e10 0.907369
\(994\) −5.49260e8 −0.0177389
\(995\) −5.49988e9 −0.177000
\(996\) −2.07180e10 −0.664415
\(997\) −4.73466e10 −1.51306 −0.756529 0.653960i \(-0.773107\pi\)
−0.756529 + 0.653960i \(0.773107\pi\)
\(998\) 1.00202e10 0.319095
\(999\) 1.71808e10 0.545209
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.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.8 12 1.1 even 1 trivial