Properties

Label 91.8.a.e.1.12
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 1.12
Root \(-20.4545\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.4545 q^{2} -57.9549 q^{3} +332.295 q^{4} +368.020 q^{5} -1243.39 q^{6} +343.000 q^{7} +4383.05 q^{8} +1171.77 q^{9} +O(q^{10})\) \(q+21.4545 q^{2} -57.9549 q^{3} +332.295 q^{4} +368.020 q^{5} -1243.39 q^{6} +343.000 q^{7} +4383.05 q^{8} +1171.77 q^{9} +7895.68 q^{10} -3956.94 q^{11} -19258.1 q^{12} +2197.00 q^{13} +7358.89 q^{14} -21328.6 q^{15} +51502.3 q^{16} +18410.7 q^{17} +25139.8 q^{18} +49599.2 q^{19} +122291. q^{20} -19878.5 q^{21} -84894.2 q^{22} -65802.0 q^{23} -254019. q^{24} +57313.6 q^{25} +47135.5 q^{26} +58837.3 q^{27} +113977. q^{28} +139239. q^{29} -457593. q^{30} -80569.9 q^{31} +543925. q^{32} +229324. q^{33} +394991. q^{34} +126231. q^{35} +389375. q^{36} -331607. q^{37} +1.06413e6 q^{38} -127327. q^{39} +1.61305e6 q^{40} +600160. q^{41} -426484. q^{42} +407204. q^{43} -1.31487e6 q^{44} +431236. q^{45} -1.41175e6 q^{46} -403277. q^{47} -2.98481e6 q^{48} +117649. q^{49} +1.22963e6 q^{50} -1.06699e6 q^{51} +730052. q^{52} -766525. q^{53} +1.26233e6 q^{54} -1.45623e6 q^{55} +1.50339e6 q^{56} -2.87452e6 q^{57} +2.98731e6 q^{58} -2.94498e6 q^{59} -7.08738e6 q^{60} -772451. q^{61} -1.72859e6 q^{62} +401919. q^{63} +5.07735e6 q^{64} +808540. q^{65} +4.92004e6 q^{66} -4.03272e6 q^{67} +6.11777e6 q^{68} +3.81355e6 q^{69} +2.70822e6 q^{70} +2.55540e6 q^{71} +5.13594e6 q^{72} -2.53403e6 q^{73} -7.11445e6 q^{74} -3.32161e6 q^{75} +1.64816e7 q^{76} -1.35723e6 q^{77} -2.73174e6 q^{78} -2.00823e6 q^{79} +1.89539e7 q^{80} -5.97258e6 q^{81} +1.28761e7 q^{82} -8.63495e6 q^{83} -6.60554e6 q^{84} +6.77549e6 q^{85} +8.73635e6 q^{86} -8.06961e6 q^{87} -1.73435e7 q^{88} -4.56488e6 q^{89} +9.25195e6 q^{90} +753571. q^{91} -2.18657e7 q^{92} +4.66942e6 q^{93} -8.65210e6 q^{94} +1.82535e7 q^{95} -3.15232e7 q^{96} +2.73971e6 q^{97} +2.52410e6 q^{98} -4.63664e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.4545 1.89633 0.948163 0.317783i \(-0.102938\pi\)
0.948163 + 0.317783i \(0.102938\pi\)
\(3\) −57.9549 −1.23927 −0.619635 0.784890i \(-0.712719\pi\)
−0.619635 + 0.784890i \(0.712719\pi\)
\(4\) 332.295 2.59606
\(5\) 368.020 1.31667 0.658334 0.752726i \(-0.271262\pi\)
0.658334 + 0.752726i \(0.271262\pi\)
\(6\) −1243.39 −2.35006
\(7\) 343.000 0.377964
\(8\) 4383.05 3.02664
\(9\) 1171.77 0.535791
\(10\) 7895.68 2.49683
\(11\) −3956.94 −0.896366 −0.448183 0.893942i \(-0.647929\pi\)
−0.448183 + 0.893942i \(0.647929\pi\)
\(12\) −19258.1 −3.21721
\(13\) 2197.00 0.277350
\(14\) 7358.89 0.716744
\(15\) −21328.6 −1.63171
\(16\) 51502.3 3.14345
\(17\) 18410.7 0.908863 0.454431 0.890782i \(-0.349843\pi\)
0.454431 + 0.890782i \(0.349843\pi\)
\(18\) 25139.8 1.01603
\(19\) 49599.2 1.65896 0.829482 0.558534i \(-0.188636\pi\)
0.829482 + 0.558534i \(0.188636\pi\)
\(20\) 122291. 3.41814
\(21\) −19878.5 −0.468400
\(22\) −84894.2 −1.69980
\(23\) −65802.0 −1.12770 −0.563848 0.825879i \(-0.690679\pi\)
−0.563848 + 0.825879i \(0.690679\pi\)
\(24\) −254019. −3.75083
\(25\) 57313.6 0.733614
\(26\) 47135.5 0.525946
\(27\) 58837.3 0.575281
\(28\) 113977. 0.981217
\(29\) 139239. 1.06015 0.530077 0.847949i \(-0.322163\pi\)
0.530077 + 0.847949i \(0.322163\pi\)
\(30\) −457593. −3.09425
\(31\) −80569.9 −0.485743 −0.242872 0.970058i \(-0.578089\pi\)
−0.242872 + 0.970058i \(0.578089\pi\)
\(32\) 543925. 2.93437
\(33\) 229324. 1.11084
\(34\) 394991. 1.72350
\(35\) 126231. 0.497654
\(36\) 389375. 1.39094
\(37\) −331607. −1.07626 −0.538130 0.842862i \(-0.680869\pi\)
−0.538130 + 0.842862i \(0.680869\pi\)
\(38\) 1.06413e6 3.14594
\(39\) −127327. −0.343712
\(40\) 1.61305e6 3.98508
\(41\) 600160. 1.35995 0.679975 0.733235i \(-0.261990\pi\)
0.679975 + 0.733235i \(0.261990\pi\)
\(42\) −426484. −0.888240
\(43\) 407204. 0.781038 0.390519 0.920595i \(-0.372296\pi\)
0.390519 + 0.920595i \(0.372296\pi\)
\(44\) −1.31487e6 −2.32702
\(45\) 431236. 0.705458
\(46\) −1.41175e6 −2.13848
\(47\) −403277. −0.566580 −0.283290 0.959034i \(-0.591426\pi\)
−0.283290 + 0.959034i \(0.591426\pi\)
\(48\) −2.98481e6 −3.89558
\(49\) 117649. 0.142857
\(50\) 1.22963e6 1.39117
\(51\) −1.06699e6 −1.12633
\(52\) 730052. 0.720016
\(53\) −766525. −0.707231 −0.353615 0.935391i \(-0.615048\pi\)
−0.353615 + 0.935391i \(0.615048\pi\)
\(54\) 1.26233e6 1.09092
\(55\) −1.45623e6 −1.18022
\(56\) 1.50339e6 1.14396
\(57\) −2.87452e6 −2.05590
\(58\) 2.98731e6 2.01040
\(59\) −2.94498e6 −1.86681 −0.933405 0.358826i \(-0.883177\pi\)
−0.933405 + 0.358826i \(0.883177\pi\)
\(60\) −7.08738e6 −4.23600
\(61\) −772451. −0.435729 −0.217865 0.975979i \(-0.569909\pi\)
−0.217865 + 0.975979i \(0.569909\pi\)
\(62\) −1.72859e6 −0.921128
\(63\) 401919. 0.202510
\(64\) 5.07735e6 2.42107
\(65\) 808540. 0.365178
\(66\) 4.92004e6 2.10652
\(67\) −4.03272e6 −1.63809 −0.819043 0.573733i \(-0.805495\pi\)
−0.819043 + 0.573733i \(0.805495\pi\)
\(68\) 6.11777e6 2.35946
\(69\) 3.81355e6 1.39752
\(70\) 2.70822e6 0.943714
\(71\) 2.55540e6 0.847334 0.423667 0.905818i \(-0.360743\pi\)
0.423667 + 0.905818i \(0.360743\pi\)
\(72\) 5.13594e6 1.62165
\(73\) −2.53403e6 −0.762397 −0.381199 0.924493i \(-0.624489\pi\)
−0.381199 + 0.924493i \(0.624489\pi\)
\(74\) −7.11445e6 −2.04094
\(75\) −3.32161e6 −0.909146
\(76\) 1.64816e7 4.30676
\(77\) −1.35723e6 −0.338795
\(78\) −2.73174e6 −0.651790
\(79\) −2.00823e6 −0.458266 −0.229133 0.973395i \(-0.573589\pi\)
−0.229133 + 0.973395i \(0.573589\pi\)
\(80\) 1.89539e7 4.13888
\(81\) −5.97258e6 −1.24872
\(82\) 1.28761e7 2.57891
\(83\) −8.63495e6 −1.65763 −0.828813 0.559526i \(-0.810983\pi\)
−0.828813 + 0.559526i \(0.810983\pi\)
\(84\) −6.60554e6 −1.21599
\(85\) 6.77549e6 1.19667
\(86\) 8.73635e6 1.48110
\(87\) −8.06961e6 −1.31382
\(88\) −1.73435e7 −2.71298
\(89\) −4.56488e6 −0.686380 −0.343190 0.939266i \(-0.611507\pi\)
−0.343190 + 0.939266i \(0.611507\pi\)
\(90\) 9.25195e6 1.33778
\(91\) 753571. 0.104828
\(92\) −2.18657e7 −2.92756
\(93\) 4.66942e6 0.601967
\(94\) −8.65210e6 −1.07442
\(95\) 1.82535e7 2.18430
\(96\) −3.15232e7 −3.63647
\(97\) 2.73971e6 0.304792 0.152396 0.988319i \(-0.451301\pi\)
0.152396 + 0.988319i \(0.451301\pi\)
\(98\) 2.52410e6 0.270904
\(99\) −4.63664e6 −0.480265
\(100\) 1.90450e7 1.90450
\(101\) 1.86139e7 1.79768 0.898839 0.438280i \(-0.144412\pi\)
0.898839 + 0.438280i \(0.144412\pi\)
\(102\) −2.28917e7 −2.13588
\(103\) −4.55736e6 −0.410945 −0.205472 0.978663i \(-0.565873\pi\)
−0.205472 + 0.978663i \(0.565873\pi\)
\(104\) 9.62956e6 0.839440
\(105\) −7.31570e6 −0.616727
\(106\) −1.64454e7 −1.34114
\(107\) −1.70465e6 −0.134521 −0.0672606 0.997735i \(-0.521426\pi\)
−0.0672606 + 0.997735i \(0.521426\pi\)
\(108\) 1.95514e7 1.49346
\(109\) −1.42288e7 −1.05239 −0.526195 0.850364i \(-0.676382\pi\)
−0.526195 + 0.850364i \(0.676382\pi\)
\(110\) −3.12428e7 −2.23808
\(111\) 1.92182e7 1.33378
\(112\) 1.76653e7 1.18811
\(113\) 2.43770e7 1.58930 0.794651 0.607066i \(-0.207654\pi\)
0.794651 + 0.607066i \(0.207654\pi\)
\(114\) −6.16713e7 −3.89867
\(115\) −2.42164e7 −1.48480
\(116\) 4.62686e7 2.75222
\(117\) 2.57439e6 0.148602
\(118\) −6.31830e7 −3.54008
\(119\) 6.31486e6 0.343518
\(120\) −9.34841e7 −4.93860
\(121\) −3.82977e6 −0.196528
\(122\) −1.65726e7 −0.826285
\(123\) −3.47822e7 −1.68535
\(124\) −2.67730e7 −1.26102
\(125\) −7.65901e6 −0.350742
\(126\) 8.62296e6 0.384025
\(127\) 3.65627e7 1.58389 0.791946 0.610592i \(-0.209068\pi\)
0.791946 + 0.610592i \(0.209068\pi\)
\(128\) 3.93094e7 1.65677
\(129\) −2.35995e7 −0.967917
\(130\) 1.73468e7 0.692497
\(131\) 4.30483e7 1.67304 0.836520 0.547936i \(-0.184586\pi\)
0.836520 + 0.547936i \(0.184586\pi\)
\(132\) 7.62034e7 2.88380
\(133\) 1.70125e7 0.627029
\(134\) −8.65200e7 −3.10635
\(135\) 2.16533e7 0.757454
\(136\) 8.06948e7 2.75080
\(137\) −6.08723e6 −0.202254 −0.101127 0.994874i \(-0.532245\pi\)
−0.101127 + 0.994874i \(0.532245\pi\)
\(138\) 8.18178e7 2.65015
\(139\) −1.53650e7 −0.485266 −0.242633 0.970118i \(-0.578011\pi\)
−0.242633 + 0.970118i \(0.578011\pi\)
\(140\) 4.19459e7 1.29194
\(141\) 2.33719e7 0.702145
\(142\) 5.48248e7 1.60682
\(143\) −8.69341e6 −0.248607
\(144\) 6.03491e7 1.68423
\(145\) 5.12429e7 1.39587
\(146\) −5.43663e7 −1.44575
\(147\) −6.81834e6 −0.177039
\(148\) −1.10191e8 −2.79403
\(149\) −6.92801e7 −1.71576 −0.857880 0.513851i \(-0.828219\pi\)
−0.857880 + 0.513851i \(0.828219\pi\)
\(150\) −7.12634e7 −1.72404
\(151\) −4.72439e7 −1.11667 −0.558337 0.829614i \(-0.688560\pi\)
−0.558337 + 0.829614i \(0.688560\pi\)
\(152\) 2.17396e8 5.02109
\(153\) 2.15731e7 0.486960
\(154\) −2.91187e7 −0.642465
\(155\) −2.96513e7 −0.639562
\(156\) −4.23101e7 −0.892295
\(157\) 2.88947e7 0.595895 0.297948 0.954582i \(-0.403698\pi\)
0.297948 + 0.954582i \(0.403698\pi\)
\(158\) −4.30854e7 −0.869022
\(159\) 4.44239e7 0.876450
\(160\) 2.00175e8 3.86359
\(161\) −2.25701e7 −0.426229
\(162\) −1.28139e8 −2.36798
\(163\) 3.43157e7 0.620635 0.310318 0.950633i \(-0.399565\pi\)
0.310318 + 0.950633i \(0.399565\pi\)
\(164\) 1.99430e8 3.53051
\(165\) 8.43959e7 1.46261
\(166\) −1.85258e8 −3.14340
\(167\) −4.49299e7 −0.746497 −0.373249 0.927731i \(-0.621756\pi\)
−0.373249 + 0.927731i \(0.621756\pi\)
\(168\) −8.71286e7 −1.41768
\(169\) 4.82681e6 0.0769231
\(170\) 1.45365e8 2.26928
\(171\) 5.81190e7 0.888857
\(172\) 1.35312e8 2.02762
\(173\) 5.77523e7 0.848023 0.424012 0.905657i \(-0.360622\pi\)
0.424012 + 0.905657i \(0.360622\pi\)
\(174\) −1.73129e8 −2.49143
\(175\) 1.96586e7 0.277280
\(176\) −2.03792e8 −2.81768
\(177\) 1.70676e8 2.31348
\(178\) −9.79372e7 −1.30160
\(179\) −1.11756e8 −1.45641 −0.728205 0.685360i \(-0.759645\pi\)
−0.728205 + 0.685360i \(0.759645\pi\)
\(180\) 1.43298e8 1.83141
\(181\) 8.62628e7 1.08131 0.540653 0.841246i \(-0.318177\pi\)
0.540653 + 0.841246i \(0.318177\pi\)
\(182\) 1.61675e7 0.198789
\(183\) 4.47674e7 0.539986
\(184\) −2.88413e8 −3.41313
\(185\) −1.22038e8 −1.41708
\(186\) 1.00180e8 1.14153
\(187\) −7.28500e7 −0.814674
\(188\) −1.34007e8 −1.47087
\(189\) 2.01812e7 0.217436
\(190\) 3.91619e8 4.14215
\(191\) −1.53573e8 −1.59477 −0.797387 0.603468i \(-0.793785\pi\)
−0.797387 + 0.603468i \(0.793785\pi\)
\(192\) −2.94257e8 −3.00036
\(193\) 1.62516e8 1.62722 0.813609 0.581412i \(-0.197499\pi\)
0.813609 + 0.581412i \(0.197499\pi\)
\(194\) 5.87792e7 0.577986
\(195\) −4.68589e7 −0.452554
\(196\) 3.90942e7 0.370865
\(197\) −1.21457e6 −0.0113185 −0.00565925 0.999984i \(-0.501801\pi\)
−0.00565925 + 0.999984i \(0.501801\pi\)
\(198\) −9.94768e7 −0.910739
\(199\) −3.17081e7 −0.285223 −0.142612 0.989779i \(-0.545550\pi\)
−0.142612 + 0.989779i \(0.545550\pi\)
\(200\) 2.51208e8 2.22039
\(201\) 2.33716e8 2.03003
\(202\) 3.99351e8 3.40898
\(203\) 4.77591e7 0.400701
\(204\) −3.54555e8 −2.92401
\(205\) 2.20871e8 1.79060
\(206\) −9.77759e7 −0.779285
\(207\) −7.71051e7 −0.604209
\(208\) 1.13151e8 0.871836
\(209\) −1.96261e8 −1.48704
\(210\) −1.56955e8 −1.16952
\(211\) −1.29291e8 −0.947499 −0.473749 0.880660i \(-0.657100\pi\)
−0.473749 + 0.880660i \(0.657100\pi\)
\(212\) −2.54713e8 −1.83601
\(213\) −1.48098e8 −1.05008
\(214\) −3.65723e7 −0.255096
\(215\) 1.49859e8 1.02837
\(216\) 2.57887e8 1.74117
\(217\) −2.76355e7 −0.183594
\(218\) −3.05273e8 −1.99568
\(219\) 1.46859e8 0.944816
\(220\) −4.83899e8 −3.06391
\(221\) 4.04482e7 0.252073
\(222\) 4.12318e8 2.52928
\(223\) 1.08405e8 0.654611 0.327306 0.944919i \(-0.393859\pi\)
0.327306 + 0.944919i \(0.393859\pi\)
\(224\) 1.86566e8 1.10909
\(225\) 6.71586e7 0.393064
\(226\) 5.22997e8 3.01384
\(227\) −2.18762e8 −1.24131 −0.620656 0.784083i \(-0.713134\pi\)
−0.620656 + 0.784083i \(0.713134\pi\)
\(228\) −9.55188e8 −5.33724
\(229\) 2.06490e7 0.113625 0.0568126 0.998385i \(-0.481906\pi\)
0.0568126 + 0.998385i \(0.481906\pi\)
\(230\) −5.19551e8 −2.81567
\(231\) 7.86583e7 0.419858
\(232\) 6.10293e8 3.20871
\(233\) 4.41473e7 0.228643 0.114322 0.993444i \(-0.463531\pi\)
0.114322 + 0.993444i \(0.463531\pi\)
\(234\) 5.52322e7 0.281797
\(235\) −1.48414e8 −0.745997
\(236\) −9.78602e8 −4.84634
\(237\) 1.16387e8 0.567915
\(238\) 1.35482e8 0.651422
\(239\) 2.96903e8 1.40677 0.703383 0.710811i \(-0.251672\pi\)
0.703383 + 0.710811i \(0.251672\pi\)
\(240\) −1.09847e9 −5.12919
\(241\) 6.20616e7 0.285604 0.142802 0.989751i \(-0.454389\pi\)
0.142802 + 0.989751i \(0.454389\pi\)
\(242\) −8.21657e7 −0.372681
\(243\) 2.17463e8 0.972219
\(244\) −2.56682e8 −1.13118
\(245\) 4.32972e7 0.188095
\(246\) −7.46235e8 −3.19597
\(247\) 1.08969e8 0.460114
\(248\) −3.53142e8 −1.47017
\(249\) 5.00438e8 2.05425
\(250\) −1.64320e8 −0.665121
\(251\) 1.46602e8 0.585171 0.292585 0.956239i \(-0.405484\pi\)
0.292585 + 0.956239i \(0.405484\pi\)
\(252\) 1.33556e8 0.525727
\(253\) 2.60375e8 1.01083
\(254\) 7.84434e8 3.00358
\(255\) −3.92673e8 −1.48300
\(256\) 1.93464e8 0.720709
\(257\) −1.74246e8 −0.640319 −0.320160 0.947364i \(-0.603737\pi\)
−0.320160 + 0.947364i \(0.603737\pi\)
\(258\) −5.06315e8 −1.83549
\(259\) −1.13741e8 −0.406788
\(260\) 2.68674e8 0.948022
\(261\) 1.63157e8 0.568021
\(262\) 9.23579e8 3.17263
\(263\) 6.67763e7 0.226348 0.113174 0.993575i \(-0.463898\pi\)
0.113174 + 0.993575i \(0.463898\pi\)
\(264\) 1.00514e9 3.36212
\(265\) −2.82097e8 −0.931188
\(266\) 3.64995e8 1.18905
\(267\) 2.64557e8 0.850610
\(268\) −1.34005e9 −4.25256
\(269\) 1.25914e8 0.394404 0.197202 0.980363i \(-0.436814\pi\)
0.197202 + 0.980363i \(0.436814\pi\)
\(270\) 4.64561e8 1.43638
\(271\) −2.29438e8 −0.700281 −0.350140 0.936697i \(-0.613866\pi\)
−0.350140 + 0.936697i \(0.613866\pi\)
\(272\) 9.48192e8 2.85696
\(273\) −4.36732e7 −0.129911
\(274\) −1.30598e8 −0.383540
\(275\) −2.26787e8 −0.657587
\(276\) 1.26722e9 3.62804
\(277\) 2.44611e8 0.691508 0.345754 0.938325i \(-0.387623\pi\)
0.345754 + 0.938325i \(0.387623\pi\)
\(278\) −3.29648e8 −0.920224
\(279\) −9.44097e7 −0.260257
\(280\) 5.53276e8 1.50622
\(281\) 3.03651e8 0.816398 0.408199 0.912893i \(-0.366157\pi\)
0.408199 + 0.912893i \(0.366157\pi\)
\(282\) 5.01432e8 1.33150
\(283\) 6.42389e8 1.68479 0.842394 0.538862i \(-0.181145\pi\)
0.842394 + 0.538862i \(0.181145\pi\)
\(284\) 8.49147e8 2.19973
\(285\) −1.05788e9 −2.70694
\(286\) −1.86513e8 −0.471441
\(287\) 2.05855e8 0.514013
\(288\) 6.37358e8 1.57221
\(289\) −7.13861e7 −0.173969
\(290\) 1.09939e9 2.64703
\(291\) −1.58780e8 −0.377720
\(292\) −8.42045e8 −1.97923
\(293\) −1.82926e8 −0.424852 −0.212426 0.977177i \(-0.568136\pi\)
−0.212426 + 0.977177i \(0.568136\pi\)
\(294\) −1.46284e8 −0.335723
\(295\) −1.08381e9 −2.45797
\(296\) −1.45345e9 −3.25746
\(297\) −2.32816e8 −0.515662
\(298\) −1.48637e9 −3.25364
\(299\) −1.44567e8 −0.312766
\(300\) −1.10375e9 −2.36019
\(301\) 1.39671e8 0.295205
\(302\) −1.01359e9 −2.11758
\(303\) −1.07877e9 −2.22781
\(304\) 2.55447e9 5.21487
\(305\) −2.84277e8 −0.573711
\(306\) 4.62841e8 0.923436
\(307\) −2.22234e8 −0.438356 −0.219178 0.975685i \(-0.570338\pi\)
−0.219178 + 0.975685i \(0.570338\pi\)
\(308\) −4.51002e8 −0.879530
\(309\) 2.64122e8 0.509271
\(310\) −6.36154e8 −1.21282
\(311\) 3.37759e8 0.636715 0.318358 0.947971i \(-0.396869\pi\)
0.318358 + 0.947971i \(0.396869\pi\)
\(312\) −5.58080e8 −1.04029
\(313\) −7.44664e8 −1.37264 −0.686318 0.727301i \(-0.740774\pi\)
−0.686318 + 0.727301i \(0.740774\pi\)
\(314\) 6.19922e8 1.13001
\(315\) 1.47914e8 0.266638
\(316\) −6.67323e8 −1.18968
\(317\) 2.03277e7 0.0358411 0.0179205 0.999839i \(-0.494295\pi\)
0.0179205 + 0.999839i \(0.494295\pi\)
\(318\) 9.53093e8 1.66204
\(319\) −5.50963e8 −0.950287
\(320\) 1.86856e9 3.18774
\(321\) 9.87926e7 0.166708
\(322\) −4.84230e8 −0.808269
\(323\) 9.13154e8 1.50777
\(324\) −1.98466e9 −3.24174
\(325\) 1.25918e8 0.203468
\(326\) 7.36226e8 1.17693
\(327\) 8.24632e8 1.30420
\(328\) 2.63053e9 4.11609
\(329\) −1.38324e8 −0.214147
\(330\) 1.81067e9 2.77358
\(331\) 8.04493e8 1.21934 0.609669 0.792656i \(-0.291302\pi\)
0.609669 + 0.792656i \(0.291302\pi\)
\(332\) −2.86935e9 −4.30329
\(333\) −3.88568e8 −0.576650
\(334\) −9.63949e8 −1.41560
\(335\) −1.48412e9 −2.15681
\(336\) −1.02379e9 −1.47239
\(337\) 1.45891e8 0.207646 0.103823 0.994596i \(-0.466892\pi\)
0.103823 + 0.994596i \(0.466892\pi\)
\(338\) 1.03557e8 0.145871
\(339\) −1.41277e9 −1.96958
\(340\) 2.25146e9 3.10662
\(341\) 3.18810e8 0.435404
\(342\) 1.24691e9 1.68556
\(343\) 4.03536e7 0.0539949
\(344\) 1.78479e9 2.36392
\(345\) 1.40346e9 1.84007
\(346\) 1.23905e9 1.60813
\(347\) −6.38212e8 −0.819996 −0.409998 0.912086i \(-0.634471\pi\)
−0.409998 + 0.912086i \(0.634471\pi\)
\(348\) −2.68149e9 −3.41075
\(349\) 5.47535e8 0.689483 0.344741 0.938698i \(-0.387967\pi\)
0.344741 + 0.938698i \(0.387967\pi\)
\(350\) 4.21764e8 0.525814
\(351\) 1.29266e8 0.159554
\(352\) −2.15228e9 −2.63027
\(353\) −1.18800e8 −0.143749 −0.0718746 0.997414i \(-0.522898\pi\)
−0.0718746 + 0.997414i \(0.522898\pi\)
\(354\) 3.66177e9 4.38712
\(355\) 9.40438e8 1.11566
\(356\) −1.51689e9 −1.78188
\(357\) −3.65977e8 −0.425711
\(358\) −2.39766e9 −2.76183
\(359\) −3.95266e8 −0.450878 −0.225439 0.974257i \(-0.572382\pi\)
−0.225439 + 0.974257i \(0.572382\pi\)
\(360\) 1.89013e9 2.13517
\(361\) 1.56621e9 1.75216
\(362\) 1.85072e9 2.05051
\(363\) 2.21954e8 0.243551
\(364\) 2.50408e8 0.272141
\(365\) −9.32573e8 −1.00382
\(366\) 9.60461e8 1.02399
\(367\) −7.08721e8 −0.748417 −0.374209 0.927345i \(-0.622086\pi\)
−0.374209 + 0.927345i \(0.622086\pi\)
\(368\) −3.38895e9 −3.54485
\(369\) 7.03251e8 0.728649
\(370\) −2.61826e9 −2.68724
\(371\) −2.62918e8 −0.267308
\(372\) 1.55163e9 1.56274
\(373\) −1.04724e9 −1.04488 −0.522438 0.852677i \(-0.674978\pi\)
−0.522438 + 0.852677i \(0.674978\pi\)
\(374\) −1.56296e9 −1.54489
\(375\) 4.43877e8 0.434664
\(376\) −1.76758e9 −1.71483
\(377\) 3.05909e8 0.294034
\(378\) 4.32978e8 0.412329
\(379\) 1.38424e9 1.30609 0.653046 0.757318i \(-0.273491\pi\)
0.653046 + 0.757318i \(0.273491\pi\)
\(380\) 6.06554e9 5.67058
\(381\) −2.11899e9 −1.96287
\(382\) −3.29484e9 −3.02421
\(383\) −6.78514e7 −0.0617111 −0.0308555 0.999524i \(-0.509823\pi\)
−0.0308555 + 0.999524i \(0.509823\pi\)
\(384\) −2.27818e9 −2.05318
\(385\) −4.99488e8 −0.446080
\(386\) 3.48670e9 3.08574
\(387\) 4.77151e8 0.418473
\(388\) 9.10394e8 0.791258
\(389\) 4.66805e8 0.402080 0.201040 0.979583i \(-0.435568\pi\)
0.201040 + 0.979583i \(0.435568\pi\)
\(390\) −1.00533e9 −0.858191
\(391\) −1.21146e9 −1.02492
\(392\) 5.15661e8 0.432378
\(393\) −2.49486e9 −2.07335
\(394\) −2.60579e7 −0.0214636
\(395\) −7.39067e8 −0.603384
\(396\) −1.54073e9 −1.24679
\(397\) 1.13367e9 0.909324 0.454662 0.890664i \(-0.349760\pi\)
0.454662 + 0.890664i \(0.349760\pi\)
\(398\) −6.80282e8 −0.540877
\(399\) −9.85959e8 −0.777059
\(400\) 2.95178e9 2.30608
\(401\) −2.42658e8 −0.187927 −0.0939634 0.995576i \(-0.529954\pi\)
−0.0939634 + 0.995576i \(0.529954\pi\)
\(402\) 5.01426e9 3.84960
\(403\) −1.77012e8 −0.134721
\(404\) 6.18530e9 4.66687
\(405\) −2.19803e9 −1.64415
\(406\) 1.02465e9 0.759860
\(407\) 1.31215e9 0.964723
\(408\) −4.67666e9 −3.40899
\(409\) 8.57583e8 0.619790 0.309895 0.950771i \(-0.399706\pi\)
0.309895 + 0.950771i \(0.399706\pi\)
\(410\) 4.73867e9 3.39557
\(411\) 3.52785e8 0.250648
\(412\) −1.51439e9 −1.06683
\(413\) −1.01013e9 −0.705588
\(414\) −1.65425e9 −1.14578
\(415\) −3.17783e9 −2.18254
\(416\) 1.19500e9 0.813847
\(417\) 8.90476e8 0.601376
\(418\) −4.21068e9 −2.81991
\(419\) 2.01001e9 1.33490 0.667452 0.744653i \(-0.267385\pi\)
0.667452 + 0.744653i \(0.267385\pi\)
\(420\) −2.43097e9 −1.60106
\(421\) 1.21619e9 0.794351 0.397176 0.917743i \(-0.369990\pi\)
0.397176 + 0.917743i \(0.369990\pi\)
\(422\) −2.77387e9 −1.79677
\(423\) −4.72550e8 −0.303568
\(424\) −3.35972e9 −2.14054
\(425\) 1.05518e9 0.666754
\(426\) −3.17737e9 −1.99129
\(427\) −2.64951e8 −0.164690
\(428\) −5.66445e8 −0.349225
\(429\) 5.03826e8 0.308092
\(430\) 3.21515e9 1.95012
\(431\) 1.80990e9 1.08889 0.544444 0.838797i \(-0.316741\pi\)
0.544444 + 0.838797i \(0.316741\pi\)
\(432\) 3.03026e9 1.80837
\(433\) 1.31114e9 0.776144 0.388072 0.921629i \(-0.373141\pi\)
0.388072 + 0.921629i \(0.373141\pi\)
\(434\) −5.92905e8 −0.348154
\(435\) −2.96978e9 −1.72986
\(436\) −4.72818e9 −2.73206
\(437\) −3.26373e9 −1.87081
\(438\) 3.15079e9 1.79168
\(439\) −9.08435e8 −0.512470 −0.256235 0.966615i \(-0.582482\pi\)
−0.256235 + 0.966615i \(0.582482\pi\)
\(440\) −6.38274e9 −3.57210
\(441\) 1.37858e8 0.0765415
\(442\) 8.67796e8 0.478013
\(443\) 1.75993e9 0.961796 0.480898 0.876777i \(-0.340311\pi\)
0.480898 + 0.876777i \(0.340311\pi\)
\(444\) 6.38613e9 3.46256
\(445\) −1.67997e9 −0.903734
\(446\) 2.32578e9 1.24136
\(447\) 4.01512e9 2.12629
\(448\) 1.74153e9 0.915078
\(449\) −1.48143e9 −0.772358 −0.386179 0.922424i \(-0.626205\pi\)
−0.386179 + 0.922424i \(0.626205\pi\)
\(450\) 1.44085e9 0.745377
\(451\) −2.37480e9 −1.21901
\(452\) 8.10037e9 4.12592
\(453\) 2.73802e9 1.38386
\(454\) −4.69342e9 −2.35393
\(455\) 2.77329e8 0.138024
\(456\) −1.25991e10 −6.22249
\(457\) −3.76896e9 −1.84721 −0.923603 0.383351i \(-0.874770\pi\)
−0.923603 + 0.383351i \(0.874770\pi\)
\(458\) 4.43014e8 0.215471
\(459\) 1.08323e9 0.522851
\(460\) −8.04701e9 −3.85462
\(461\) 2.52786e9 1.20171 0.600855 0.799358i \(-0.294827\pi\)
0.600855 + 0.799358i \(0.294827\pi\)
\(462\) 1.68757e9 0.796188
\(463\) −1.87727e8 −0.0879007 −0.0439504 0.999034i \(-0.513994\pi\)
−0.0439504 + 0.999034i \(0.513994\pi\)
\(464\) 7.17115e9 3.33254
\(465\) 1.71844e9 0.792591
\(466\) 9.47159e8 0.433583
\(467\) 3.11191e9 1.41390 0.706948 0.707266i \(-0.250072\pi\)
0.706948 + 0.707266i \(0.250072\pi\)
\(468\) 8.55457e8 0.385778
\(469\) −1.38322e9 −0.619138
\(470\) −3.18415e9 −1.41465
\(471\) −1.67459e9 −0.738475
\(472\) −1.29080e10 −5.65017
\(473\) −1.61128e9 −0.700096
\(474\) 2.49701e9 1.07695
\(475\) 2.84271e9 1.21704
\(476\) 2.09840e9 0.891791
\(477\) −8.98195e8 −0.378928
\(478\) 6.36990e9 2.66769
\(479\) 2.73570e8 0.113735 0.0568675 0.998382i \(-0.481889\pi\)
0.0568675 + 0.998382i \(0.481889\pi\)
\(480\) −1.16011e10 −4.78803
\(481\) −7.28540e8 −0.298501
\(482\) 1.33150e9 0.541598
\(483\) 1.30805e9 0.528213
\(484\) −1.27261e9 −0.510197
\(485\) 1.00827e9 0.401310
\(486\) 4.66557e9 1.84365
\(487\) 4.11444e8 0.161421 0.0807104 0.996738i \(-0.474281\pi\)
0.0807104 + 0.996738i \(0.474281\pi\)
\(488\) −3.38569e9 −1.31880
\(489\) −1.98876e9 −0.769134
\(490\) 9.28919e8 0.356690
\(491\) −4.56732e9 −1.74131 −0.870655 0.491895i \(-0.836304\pi\)
−0.870655 + 0.491895i \(0.836304\pi\)
\(492\) −1.15580e10 −4.37525
\(493\) 2.56349e9 0.963535
\(494\) 2.33788e9 0.872526
\(495\) −1.70638e9 −0.632349
\(496\) −4.14953e9 −1.52691
\(497\) 8.76502e8 0.320262
\(498\) 1.07366e10 3.89552
\(499\) 2.29418e9 0.826562 0.413281 0.910604i \(-0.364383\pi\)
0.413281 + 0.910604i \(0.364383\pi\)
\(500\) −2.54505e9 −0.910546
\(501\) 2.60391e9 0.925112
\(502\) 3.14528e9 1.10968
\(503\) 1.58482e8 0.0555254 0.0277627 0.999615i \(-0.491162\pi\)
0.0277627 + 0.999615i \(0.491162\pi\)
\(504\) 1.76163e9 0.612925
\(505\) 6.85027e9 2.36694
\(506\) 5.58621e9 1.91686
\(507\) −2.79737e8 −0.0953285
\(508\) 1.21496e10 4.11187
\(509\) 1.66384e9 0.559242 0.279621 0.960110i \(-0.409791\pi\)
0.279621 + 0.960110i \(0.409791\pi\)
\(510\) −8.42460e9 −2.81225
\(511\) −8.69172e8 −0.288159
\(512\) −8.80942e8 −0.290070
\(513\) 2.91828e9 0.954370
\(514\) −3.73836e9 −1.21425
\(515\) −1.67720e9 −0.541077
\(516\) −7.84199e9 −2.51277
\(517\) 1.59574e9 0.507863
\(518\) −2.44026e9 −0.771403
\(519\) −3.34703e9 −1.05093
\(520\) 3.54387e9 1.10526
\(521\) 4.75766e9 1.47388 0.736939 0.675959i \(-0.236271\pi\)
0.736939 + 0.675959i \(0.236271\pi\)
\(522\) 3.50045e9 1.07715
\(523\) −6.17825e9 −1.88847 −0.944235 0.329274i \(-0.893196\pi\)
−0.944235 + 0.329274i \(0.893196\pi\)
\(524\) 1.43047e10 4.34331
\(525\) −1.13931e9 −0.343625
\(526\) 1.43265e9 0.429230
\(527\) −1.48334e9 −0.441474
\(528\) 1.18107e10 3.49187
\(529\) 9.25079e8 0.271697
\(530\) −6.05224e9 −1.76584
\(531\) −3.45085e9 −1.00022
\(532\) 5.65318e9 1.62780
\(533\) 1.31855e9 0.377183
\(534\) 5.67594e9 1.61303
\(535\) −6.27343e8 −0.177120
\(536\) −1.76756e10 −4.95790
\(537\) 6.47679e9 1.80489
\(538\) 2.70142e9 0.747920
\(539\) −4.65530e8 −0.128052
\(540\) 7.19529e9 1.96639
\(541\) −6.89344e9 −1.87174 −0.935871 0.352344i \(-0.885385\pi\)
−0.935871 + 0.352344i \(0.885385\pi\)
\(542\) −4.92247e9 −1.32796
\(543\) −4.99935e9 −1.34003
\(544\) 1.00140e10 2.66694
\(545\) −5.23650e9 −1.38565
\(546\) −9.36985e8 −0.246353
\(547\) 2.50406e9 0.654169 0.327084 0.944995i \(-0.393934\pi\)
0.327084 + 0.944995i \(0.393934\pi\)
\(548\) −2.02276e9 −0.525063
\(549\) −9.05139e8 −0.233460
\(550\) −4.86559e9 −1.24700
\(551\) 6.90616e9 1.75876
\(552\) 1.67150e10 4.22979
\(553\) −6.88821e8 −0.173208
\(554\) 5.24801e9 1.31132
\(555\) 7.07269e9 1.75614
\(556\) −5.10571e9 −1.25978
\(557\) 1.68359e9 0.412804 0.206402 0.978467i \(-0.433825\pi\)
0.206402 + 0.978467i \(0.433825\pi\)
\(558\) −2.02551e9 −0.493532
\(559\) 8.94627e8 0.216621
\(560\) 6.50118e9 1.56435
\(561\) 4.22201e9 1.00960
\(562\) 6.51467e9 1.54816
\(563\) 6.19180e8 0.146230 0.0731152 0.997324i \(-0.476706\pi\)
0.0731152 + 0.997324i \(0.476706\pi\)
\(564\) 7.76637e9 1.82281
\(565\) 8.97124e9 2.09258
\(566\) 1.37821e10 3.19491
\(567\) −2.04860e9 −0.471971
\(568\) 1.12004e10 2.56458
\(569\) 3.42963e9 0.780468 0.390234 0.920716i \(-0.372394\pi\)
0.390234 + 0.920716i \(0.372394\pi\)
\(570\) −2.26963e10 −5.13325
\(571\) −4.28991e9 −0.964322 −0.482161 0.876083i \(-0.660148\pi\)
−0.482161 + 0.876083i \(0.660148\pi\)
\(572\) −2.88878e9 −0.645398
\(573\) 8.90034e9 1.97636
\(574\) 4.41651e9 0.974737
\(575\) −3.77135e9 −0.827293
\(576\) 5.94950e9 1.29719
\(577\) −5.28606e9 −1.14556 −0.572778 0.819710i \(-0.694134\pi\)
−0.572778 + 0.819710i \(0.694134\pi\)
\(578\) −1.53155e9 −0.329902
\(579\) −9.41861e9 −2.01656
\(580\) 1.70278e10 3.62376
\(581\) −2.96179e9 −0.626524
\(582\) −3.40654e9 −0.716281
\(583\) 3.03310e9 0.633938
\(584\) −1.11068e10 −2.30751
\(585\) 9.47426e8 0.195659
\(586\) −3.92458e9 −0.805659
\(587\) −1.76980e9 −0.361153 −0.180576 0.983561i \(-0.557796\pi\)
−0.180576 + 0.983561i \(0.557796\pi\)
\(588\) −2.26570e9 −0.459602
\(589\) −3.99620e9 −0.805830
\(590\) −2.32526e10 −4.66111
\(591\) 7.03901e7 0.0140267
\(592\) −1.70785e10 −3.38317
\(593\) −2.25369e9 −0.443816 −0.221908 0.975068i \(-0.571228\pi\)
−0.221908 + 0.975068i \(0.571228\pi\)
\(594\) −4.99495e9 −0.977864
\(595\) 2.32399e9 0.452299
\(596\) −2.30214e10 −4.45421
\(597\) 1.83764e9 0.353469
\(598\) −3.10161e9 −0.593107
\(599\) 8.55681e9 1.62674 0.813369 0.581747i \(-0.197631\pi\)
0.813369 + 0.581747i \(0.197631\pi\)
\(600\) −1.45588e10 −2.75166
\(601\) −7.40943e8 −0.139227 −0.0696136 0.997574i \(-0.522177\pi\)
−0.0696136 + 0.997574i \(0.522177\pi\)
\(602\) 2.99657e9 0.559805
\(603\) −4.72544e9 −0.877671
\(604\) −1.56989e10 −2.89895
\(605\) −1.40943e9 −0.258762
\(606\) −2.31444e10 −4.22465
\(607\) 2.78806e9 0.505989 0.252994 0.967468i \(-0.418585\pi\)
0.252994 + 0.967468i \(0.418585\pi\)
\(608\) 2.69782e10 4.86801
\(609\) −2.76788e9 −0.496577
\(610\) −6.09903e9 −1.08794
\(611\) −8.86000e8 −0.157141
\(612\) 7.16865e9 1.26418
\(613\) 3.41393e9 0.598609 0.299305 0.954158i \(-0.403245\pi\)
0.299305 + 0.954158i \(0.403245\pi\)
\(614\) −4.76793e9 −0.831266
\(615\) −1.28005e10 −2.21904
\(616\) −5.94881e9 −1.02541
\(617\) −4.91692e9 −0.842742 −0.421371 0.906888i \(-0.638451\pi\)
−0.421371 + 0.906888i \(0.638451\pi\)
\(618\) 5.66659e9 0.965745
\(619\) 4.67165e9 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(620\) −9.85299e9 −1.66034
\(621\) −3.87162e9 −0.648741
\(622\) 7.24644e9 1.20742
\(623\) −1.56575e9 −0.259427
\(624\) −6.55763e9 −1.08044
\(625\) −7.29629e9 −1.19542
\(626\) −1.59764e10 −2.60297
\(627\) 1.13743e10 1.84284
\(628\) 9.60158e9 1.54698
\(629\) −6.10510e9 −0.978173
\(630\) 3.17342e9 0.505633
\(631\) −4.89335e9 −0.775360 −0.387680 0.921794i \(-0.626723\pi\)
−0.387680 + 0.921794i \(0.626723\pi\)
\(632\) −8.80215e9 −1.38701
\(633\) 7.49303e9 1.17421
\(634\) 4.36121e8 0.0679664
\(635\) 1.34558e10 2.08546
\(636\) 1.47619e10 2.27531
\(637\) 2.58475e8 0.0396214
\(638\) −1.18206e10 −1.80205
\(639\) 2.99435e9 0.453994
\(640\) 1.44667e10 2.18141
\(641\) −3.13608e9 −0.470310 −0.235155 0.971958i \(-0.575560\pi\)
−0.235155 + 0.971958i \(0.575560\pi\)
\(642\) 2.11954e9 0.316133
\(643\) −6.75599e9 −1.00219 −0.501096 0.865392i \(-0.667070\pi\)
−0.501096 + 0.865392i \(0.667070\pi\)
\(644\) −7.49993e9 −1.10651
\(645\) −8.68507e9 −1.27443
\(646\) 1.95912e10 2.85922
\(647\) 8.28130e9 1.20208 0.601040 0.799219i \(-0.294753\pi\)
0.601040 + 0.799219i \(0.294753\pi\)
\(648\) −2.61781e10 −3.77943
\(649\) 1.16531e10 1.67334
\(650\) 2.70151e9 0.385842
\(651\) 1.60161e9 0.227522
\(652\) 1.14029e10 1.61120
\(653\) −4.00554e9 −0.562944 −0.281472 0.959569i \(-0.590823\pi\)
−0.281472 + 0.959569i \(0.590823\pi\)
\(654\) 1.76921e10 2.47318
\(655\) 1.58426e10 2.20284
\(656\) 3.09096e10 4.27494
\(657\) −2.96931e9 −0.408485
\(658\) −2.96767e9 −0.406093
\(659\) −7.39356e9 −1.00636 −0.503181 0.864181i \(-0.667837\pi\)
−0.503181 + 0.864181i \(0.667837\pi\)
\(660\) 2.80444e10 3.79701
\(661\) −2.42892e9 −0.327120 −0.163560 0.986533i \(-0.552298\pi\)
−0.163560 + 0.986533i \(0.552298\pi\)
\(662\) 1.72600e10 2.31227
\(663\) −2.34417e9 −0.312387
\(664\) −3.78474e10 −5.01704
\(665\) 6.26094e9 0.825589
\(666\) −8.33653e9 −1.09352
\(667\) −9.16224e9 −1.19553
\(668\) −1.49300e10 −1.93795
\(669\) −6.28262e9 −0.811240
\(670\) −3.18411e10 −4.09002
\(671\) 3.05655e9 0.390573
\(672\) −1.08124e10 −1.37446
\(673\) 1.63332e9 0.206547 0.103273 0.994653i \(-0.467068\pi\)
0.103273 + 0.994653i \(0.467068\pi\)
\(674\) 3.13002e9 0.393765
\(675\) 3.37218e9 0.422034
\(676\) 1.60393e9 0.199697
\(677\) −8.03237e9 −0.994909 −0.497454 0.867490i \(-0.665732\pi\)
−0.497454 + 0.867490i \(0.665732\pi\)
\(678\) −3.03103e10 −3.73496
\(679\) 9.39722e8 0.115201
\(680\) 2.96973e10 3.62189
\(681\) 1.26783e10 1.53832
\(682\) 6.83992e9 0.825668
\(683\) −8.15621e8 −0.0979526 −0.0489763 0.998800i \(-0.515596\pi\)
−0.0489763 + 0.998800i \(0.515596\pi\)
\(684\) 1.93127e10 2.30752
\(685\) −2.24022e9 −0.266302
\(686\) 8.65766e8 0.102392
\(687\) −1.19671e9 −0.140812
\(688\) 2.09719e10 2.45515
\(689\) −1.68406e9 −0.196150
\(690\) 3.01106e10 3.48937
\(691\) −2.74837e9 −0.316886 −0.158443 0.987368i \(-0.550647\pi\)
−0.158443 + 0.987368i \(0.550647\pi\)
\(692\) 1.91908e10 2.20152
\(693\) −1.59037e9 −0.181523
\(694\) −1.36925e10 −1.55498
\(695\) −5.65462e9 −0.638935
\(696\) −3.53695e10 −3.97646
\(697\) 1.10493e10 1.23601
\(698\) 1.17471e10 1.30748
\(699\) −2.55856e9 −0.283351
\(700\) 6.53245e9 0.719834
\(701\) 9.43733e9 1.03475 0.517375 0.855758i \(-0.326909\pi\)
0.517375 + 0.855758i \(0.326909\pi\)
\(702\) 2.77333e9 0.302567
\(703\) −1.64474e10 −1.78548
\(704\) −2.00908e10 −2.17016
\(705\) 8.60132e9 0.924492
\(706\) −2.54880e9 −0.272595
\(707\) 6.38455e9 0.679458
\(708\) 5.67148e10 6.00593
\(709\) 4.12614e9 0.434792 0.217396 0.976083i \(-0.430244\pi\)
0.217396 + 0.976083i \(0.430244\pi\)
\(710\) 2.01766e10 2.11565
\(711\) −2.35319e9 −0.245535
\(712\) −2.00081e10 −2.07743
\(713\) 5.30166e9 0.547770
\(714\) −7.85185e9 −0.807288
\(715\) −3.19935e9 −0.327333
\(716\) −3.71358e10 −3.78092
\(717\) −1.72070e10 −1.74336
\(718\) −8.48024e9 −0.855013
\(719\) 8.21156e9 0.823900 0.411950 0.911206i \(-0.364848\pi\)
0.411950 + 0.911206i \(0.364848\pi\)
\(720\) 2.22097e10 2.21757
\(721\) −1.56317e9 −0.155322
\(722\) 3.36022e10 3.32267
\(723\) −3.59678e9 −0.353940
\(724\) 2.86647e10 2.80713
\(725\) 7.98031e9 0.777744
\(726\) 4.76191e9 0.461852
\(727\) −1.95125e10 −1.88340 −0.941698 0.336460i \(-0.890770\pi\)
−0.941698 + 0.336460i \(0.890770\pi\)
\(728\) 3.30294e9 0.317279
\(729\) 4.58963e8 0.0438764
\(730\) −2.00079e10 −1.90358
\(731\) 7.49689e9 0.709856
\(732\) 1.48760e10 1.40183
\(733\) −5.50452e9 −0.516245 −0.258122 0.966112i \(-0.583104\pi\)
−0.258122 + 0.966112i \(0.583104\pi\)
\(734\) −1.52052e10 −1.41924
\(735\) −2.50928e9 −0.233101
\(736\) −3.57914e10 −3.30907
\(737\) 1.59573e10 1.46832
\(738\) 1.50879e10 1.38176
\(739\) 6.54719e9 0.596759 0.298380 0.954447i \(-0.403554\pi\)
0.298380 + 0.954447i \(0.403554\pi\)
\(740\) −4.05526e10 −3.67881
\(741\) −6.31531e9 −0.570205
\(742\) −5.64078e9 −0.506903
\(743\) −2.12398e10 −1.89972 −0.949860 0.312676i \(-0.898775\pi\)
−0.949860 + 0.312676i \(0.898775\pi\)
\(744\) 2.04663e10 1.82194
\(745\) −2.54964e10 −2.25909
\(746\) −2.24680e10 −1.98143
\(747\) −1.01182e10 −0.888140
\(748\) −2.42077e10 −2.11494
\(749\) −5.84693e8 −0.0508442
\(750\) 9.52316e9 0.824265
\(751\) 1.73021e10 1.49059 0.745296 0.666734i \(-0.232308\pi\)
0.745296 + 0.666734i \(0.232308\pi\)
\(752\) −2.07697e10 −1.78102
\(753\) −8.49632e9 −0.725185
\(754\) 6.56312e9 0.557585
\(755\) −1.73867e10 −1.47029
\(756\) 6.70612e9 0.564475
\(757\) 6.08797e9 0.510078 0.255039 0.966931i \(-0.417912\pi\)
0.255039 + 0.966931i \(0.417912\pi\)
\(758\) 2.96981e10 2.47678
\(759\) −1.50900e10 −1.25269
\(760\) 8.00059e10 6.61111
\(761\) 1.92333e10 1.58201 0.791004 0.611811i \(-0.209559\pi\)
0.791004 + 0.611811i \(0.209559\pi\)
\(762\) −4.54618e10 −3.72224
\(763\) −4.88049e9 −0.397766
\(764\) −5.10317e10 −4.14012
\(765\) 7.93934e9 0.641165
\(766\) −1.45572e9 −0.117024
\(767\) −6.47012e9 −0.517760
\(768\) −1.12122e10 −0.893153
\(769\) 4.91707e9 0.389910 0.194955 0.980812i \(-0.437544\pi\)
0.194955 + 0.980812i \(0.437544\pi\)
\(770\) −1.07163e10 −0.845913
\(771\) 1.00984e10 0.793529
\(772\) 5.40033e10 4.22435
\(773\) 1.69143e10 1.31712 0.658559 0.752529i \(-0.271166\pi\)
0.658559 + 0.752529i \(0.271166\pi\)
\(774\) 1.02370e10 0.793561
\(775\) −4.61775e9 −0.356348
\(776\) 1.20083e10 0.922498
\(777\) 6.59186e9 0.504120
\(778\) 1.00151e10 0.762475
\(779\) 2.97674e10 2.25611
\(780\) −1.55710e10 −1.17486
\(781\) −1.01116e10 −0.759521
\(782\) −2.59912e10 −1.94358
\(783\) 8.19248e9 0.609887
\(784\) 6.05919e9 0.449064
\(785\) 1.06338e10 0.784596
\(786\) −5.35260e10 −3.93175
\(787\) −1.05413e10 −0.770871 −0.385436 0.922735i \(-0.625949\pi\)
−0.385436 + 0.922735i \(0.625949\pi\)
\(788\) −4.03594e8 −0.0293835
\(789\) −3.87002e9 −0.280507
\(790\) −1.58563e10 −1.14421
\(791\) 8.36133e9 0.600700
\(792\) −2.03226e10 −1.45359
\(793\) −1.69708e9 −0.120850
\(794\) 2.43222e10 1.72438
\(795\) 1.63489e10 1.15399
\(796\) −1.05365e10 −0.740456
\(797\) 1.03772e10 0.726065 0.363032 0.931777i \(-0.381741\pi\)
0.363032 + 0.931777i \(0.381741\pi\)
\(798\) −2.11533e10 −1.47356
\(799\) −7.42460e9 −0.514943
\(800\) 3.11743e10 2.15269
\(801\) −5.34901e9 −0.367756
\(802\) −5.20610e9 −0.356371
\(803\) 1.00270e10 0.683387
\(804\) 7.76627e10 5.27007
\(805\) −8.30624e9 −0.561202
\(806\) −3.79770e9 −0.255475
\(807\) −7.29735e9 −0.488774
\(808\) 8.15855e10 5.44093
\(809\) 1.94726e10 1.29302 0.646510 0.762906i \(-0.276228\pi\)
0.646510 + 0.762906i \(0.276228\pi\)
\(810\) −4.71576e10 −3.11784
\(811\) 1.31550e9 0.0865998 0.0432999 0.999062i \(-0.486213\pi\)
0.0432999 + 0.999062i \(0.486213\pi\)
\(812\) 1.58701e10 1.04024
\(813\) 1.32970e10 0.867837
\(814\) 2.81515e10 1.82943
\(815\) 1.26289e10 0.817170
\(816\) −5.49524e10 −3.54055
\(817\) 2.01970e10 1.29571
\(818\) 1.83990e10 1.17532
\(819\) 8.83015e8 0.0561661
\(820\) 7.33942e10 4.64851
\(821\) 2.19897e10 1.38682 0.693409 0.720545i \(-0.256108\pi\)
0.693409 + 0.720545i \(0.256108\pi\)
\(822\) 7.56882e9 0.475310
\(823\) 1.47417e10 0.921826 0.460913 0.887445i \(-0.347522\pi\)
0.460913 + 0.887445i \(0.347522\pi\)
\(824\) −1.99751e10 −1.24378
\(825\) 1.31434e10 0.814928
\(826\) −2.16718e10 −1.33802
\(827\) −1.94790e10 −1.19756 −0.598781 0.800913i \(-0.704348\pi\)
−0.598781 + 0.800913i \(0.704348\pi\)
\(828\) −2.56217e10 −1.56856
\(829\) −1.26005e10 −0.768154 −0.384077 0.923301i \(-0.625480\pi\)
−0.384077 + 0.923301i \(0.625480\pi\)
\(830\) −6.81788e10 −4.13881
\(831\) −1.41764e10 −0.856965
\(832\) 1.11549e10 0.671483
\(833\) 2.16600e9 0.129838
\(834\) 1.91047e10 1.14041
\(835\) −1.65351e10 −0.982889
\(836\) −6.52166e10 −3.86044
\(837\) −4.74052e9 −0.279439
\(838\) 4.31238e10 2.53141
\(839\) −1.39227e10 −0.813872 −0.406936 0.913457i \(-0.633403\pi\)
−0.406936 + 0.913457i \(0.633403\pi\)
\(840\) −3.20651e10 −1.86661
\(841\) 2.13775e9 0.123928
\(842\) 2.60927e10 1.50635
\(843\) −1.75980e10 −1.01174
\(844\) −4.29627e10 −2.45976
\(845\) 1.77636e9 0.101282
\(846\) −1.01383e10 −0.575664
\(847\) −1.31361e9 −0.0742805
\(848\) −3.94778e10 −2.22314
\(849\) −3.72296e10 −2.08791
\(850\) 2.26384e10 1.26438
\(851\) 2.18204e10 1.21369
\(852\) −4.92122e10 −2.72605
\(853\) −3.26613e10 −1.80182 −0.900911 0.434004i \(-0.857100\pi\)
−0.900911 + 0.434004i \(0.857100\pi\)
\(854\) −5.68439e9 −0.312306
\(855\) 2.13890e10 1.17033
\(856\) −7.47154e9 −0.407148
\(857\) −2.41294e10 −1.30952 −0.654761 0.755836i \(-0.727231\pi\)
−0.654761 + 0.755836i \(0.727231\pi\)
\(858\) 1.08093e10 0.584242
\(859\) 6.13516e9 0.330255 0.165128 0.986272i \(-0.447196\pi\)
0.165128 + 0.986272i \(0.447196\pi\)
\(860\) 4.97974e10 2.66970
\(861\) −1.19303e10 −0.637001
\(862\) 3.88304e10 2.06489
\(863\) −2.79398e10 −1.47974 −0.739870 0.672750i \(-0.765113\pi\)
−0.739870 + 0.672750i \(0.765113\pi\)
\(864\) 3.20031e10 1.68808
\(865\) 2.12540e10 1.11656
\(866\) 2.81299e10 1.47182
\(867\) 4.13718e9 0.215594
\(868\) −9.18313e9 −0.476619
\(869\) 7.94643e9 0.410774
\(870\) −6.37151e10 −3.28038
\(871\) −8.85989e9 −0.454323
\(872\) −6.23657e10 −3.18521
\(873\) 3.21033e9 0.163305
\(874\) −7.00216e10 −3.54766
\(875\) −2.62704e9 −0.132568
\(876\) 4.88007e10 2.45280
\(877\) 2.32137e10 1.16211 0.581054 0.813865i \(-0.302641\pi\)
0.581054 + 0.813865i \(0.302641\pi\)
\(878\) −1.94900e10 −0.971810
\(879\) 1.06014e10 0.526507
\(880\) −7.49994e10 −3.70995
\(881\) 1.29500e10 0.638050 0.319025 0.947746i \(-0.396645\pi\)
0.319025 + 0.947746i \(0.396645\pi\)
\(882\) 2.95767e9 0.145148
\(883\) −1.70222e10 −0.832057 −0.416029 0.909352i \(-0.636578\pi\)
−0.416029 + 0.909352i \(0.636578\pi\)
\(884\) 1.34408e10 0.654396
\(885\) 6.28121e10 3.04609
\(886\) 3.77585e10 1.82388
\(887\) 1.41891e10 0.682690 0.341345 0.939938i \(-0.389118\pi\)
0.341345 + 0.939938i \(0.389118\pi\)
\(888\) 8.42345e10 4.03687
\(889\) 1.25410e10 0.598655
\(890\) −3.60428e10 −1.71378
\(891\) 2.36332e10 1.11931
\(892\) 3.60225e10 1.69941
\(893\) −2.00022e10 −0.939935
\(894\) 8.61424e10 4.03214
\(895\) −4.11283e10 −1.91761
\(896\) 1.34831e10 0.626200
\(897\) 8.37837e9 0.387602
\(898\) −3.17833e10 −1.46464
\(899\) −1.12185e10 −0.514963
\(900\) 2.23165e10 1.02041
\(901\) −1.41122e10 −0.642775
\(902\) −5.09501e10 −2.31165
\(903\) −8.09462e9 −0.365838
\(904\) 1.06846e11 4.81025
\(905\) 3.17464e10 1.42372
\(906\) 5.87427e10 2.62425
\(907\) −2.21194e9 −0.0984345 −0.0492173 0.998788i \(-0.515673\pi\)
−0.0492173 + 0.998788i \(0.515673\pi\)
\(908\) −7.26935e10 −3.22251
\(909\) 2.18112e10 0.963179
\(910\) 5.94995e9 0.261739
\(911\) −1.35193e10 −0.592435 −0.296217 0.955121i \(-0.595725\pi\)
−0.296217 + 0.955121i \(0.595725\pi\)
\(912\) −1.48044e11 −6.46263
\(913\) 3.41680e10 1.48584
\(914\) −8.08612e10 −3.50290
\(915\) 1.64753e10 0.710983
\(916\) 6.86156e9 0.294977
\(917\) 1.47656e10 0.632350
\(918\) 2.32402e10 0.991497
\(919\) −2.13982e8 −0.00909437 −0.00454719 0.999990i \(-0.501447\pi\)
−0.00454719 + 0.999990i \(0.501447\pi\)
\(920\) −1.06142e11 −4.49396
\(921\) 1.28796e10 0.543242
\(922\) 5.42340e10 2.27884
\(923\) 5.61421e9 0.235008
\(924\) 2.61378e10 1.08997
\(925\) −1.90056e10 −0.789559
\(926\) −4.02758e9 −0.166689
\(927\) −5.34020e9 −0.220180
\(928\) 7.57359e10 3.11088
\(929\) −1.98996e10 −0.814308 −0.407154 0.913359i \(-0.633479\pi\)
−0.407154 + 0.913359i \(0.633479\pi\)
\(930\) 3.68682e10 1.50301
\(931\) 5.83529e9 0.236995
\(932\) 1.46699e10 0.593571
\(933\) −1.95748e10 −0.789062
\(934\) 6.67643e10 2.68121
\(935\) −2.68102e10 −1.07265
\(936\) 1.12837e10 0.449764
\(937\) 1.63524e10 0.649373 0.324686 0.945822i \(-0.394741\pi\)
0.324686 + 0.945822i \(0.394741\pi\)
\(938\) −2.96764e10 −1.17409
\(939\) 4.31570e10 1.70107
\(940\) −4.93172e10 −1.93665
\(941\) 2.48061e10 0.970500 0.485250 0.874376i \(-0.338729\pi\)
0.485250 + 0.874376i \(0.338729\pi\)
\(942\) −3.59275e10 −1.40039
\(943\) −3.94917e10 −1.53361
\(944\) −1.51673e11 −5.86822
\(945\) 7.42709e9 0.286291
\(946\) −3.45692e10 −1.32761
\(947\) 2.71990e10 1.04070 0.520352 0.853952i \(-0.325801\pi\)
0.520352 + 0.853952i \(0.325801\pi\)
\(948\) 3.86747e10 1.47434
\(949\) −5.56726e9 −0.211451
\(950\) 6.09888e10 2.30790
\(951\) −1.17809e9 −0.0444168
\(952\) 2.76783e10 1.03971
\(953\) −4.18779e9 −0.156733 −0.0783663 0.996925i \(-0.524970\pi\)
−0.0783663 + 0.996925i \(0.524970\pi\)
\(954\) −1.92703e10 −0.718571
\(955\) −5.65181e10 −2.09979
\(956\) 9.86594e10 3.65204
\(957\) 3.19310e10 1.17766
\(958\) 5.86931e9 0.215679
\(959\) −2.08792e9 −0.0764449
\(960\) −1.08293e11 −3.95047
\(961\) −2.10211e10 −0.764054
\(962\) −1.56305e10 −0.566055
\(963\) −1.99746e9 −0.0720752
\(964\) 2.06228e10 0.741443
\(965\) 5.98092e10 2.14251
\(966\) 2.80635e10 1.00166
\(967\) −4.05099e10 −1.44068 −0.720342 0.693619i \(-0.756015\pi\)
−0.720342 + 0.693619i \(0.756015\pi\)
\(968\) −1.67861e10 −0.594819
\(969\) −5.29218e10 −1.86853
\(970\) 2.16319e10 0.761016
\(971\) 2.12395e10 0.744523 0.372261 0.928128i \(-0.378583\pi\)
0.372261 + 0.928128i \(0.378583\pi\)
\(972\) 7.22621e10 2.52394
\(973\) −5.27019e9 −0.183413
\(974\) 8.82732e9 0.306107
\(975\) −7.29757e9 −0.252152
\(976\) −3.97830e10 −1.36969
\(977\) −4.95442e10 −1.69966 −0.849830 0.527056i \(-0.823296\pi\)
−0.849830 + 0.527056i \(0.823296\pi\)
\(978\) −4.26679e10 −1.45853
\(979\) 1.80630e10 0.615248
\(980\) 1.43874e10 0.488306
\(981\) −1.66730e10 −0.563861
\(982\) −9.79894e10 −3.30209
\(983\) 2.16480e10 0.726910 0.363455 0.931612i \(-0.381597\pi\)
0.363455 + 0.931612i \(0.381597\pi\)
\(984\) −1.52452e11 −5.10094
\(985\) −4.46984e8 −0.0149027
\(986\) 5.49984e10 1.82718
\(987\) 8.01656e9 0.265386
\(988\) 3.62100e10 1.19448
\(989\) −2.67948e10 −0.880773
\(990\) −3.66095e10 −1.19914
\(991\) −5.12814e10 −1.67379 −0.836897 0.547361i \(-0.815632\pi\)
−0.836897 + 0.547361i \(0.815632\pi\)
\(992\) −4.38240e10 −1.42535
\(993\) −4.66244e10 −1.51109
\(994\) 1.88049e10 0.607322
\(995\) −1.16692e10 −0.375544
\(996\) 1.66293e11 5.33294
\(997\) −1.54006e10 −0.492157 −0.246078 0.969250i \(-0.579142\pi\)
−0.246078 + 0.969250i \(0.579142\pi\)
\(998\) 4.92204e10 1.56743
\(999\) −1.95109e10 −0.619152
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.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.12 12 1.1 even 1 trivial