Properties

Label 91.8.a.e.1.2
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 1.2
Root \(19.0596\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-18.0596 q^{2} -64.0898 q^{3} +198.149 q^{4} +13.5880 q^{5} +1157.44 q^{6} +343.000 q^{7} -1266.87 q^{8} +1920.50 q^{9} +O(q^{10})\) \(q-18.0596 q^{2} -64.0898 q^{3} +198.149 q^{4} +13.5880 q^{5} +1157.44 q^{6} +343.000 q^{7} -1266.87 q^{8} +1920.50 q^{9} -245.394 q^{10} +6489.06 q^{11} -12699.3 q^{12} +2197.00 q^{13} -6194.44 q^{14} -870.853 q^{15} -2484.01 q^{16} -31064.5 q^{17} -34683.4 q^{18} -6937.84 q^{19} +2692.46 q^{20} -21982.8 q^{21} -117190. q^{22} +6153.33 q^{23} +81193.1 q^{24} -77940.4 q^{25} -39676.9 q^{26} +17080.0 q^{27} +67965.2 q^{28} -86998.5 q^{29} +15727.3 q^{30} +282367. q^{31} +207019. q^{32} -415883. q^{33} +561012. q^{34} +4660.69 q^{35} +380545. q^{36} -232493. q^{37} +125295. q^{38} -140805. q^{39} -17214.2 q^{40} +412653. q^{41} +397000. q^{42} -603630. q^{43} +1.28580e6 q^{44} +26095.8 q^{45} -111127. q^{46} -217376. q^{47} +159199. q^{48} +117649. q^{49} +1.40757e6 q^{50} +1.99092e6 q^{51} +435334. q^{52} +1.08757e6 q^{53} -308457. q^{54} +88173.5 q^{55} -434535. q^{56} +444645. q^{57} +1.57116e6 q^{58} -641874. q^{59} -172559. q^{60} -576801. q^{61} -5.09944e6 q^{62} +658731. q^{63} -3.42073e6 q^{64} +29852.9 q^{65} +7.51067e6 q^{66} -3.17002e6 q^{67} -6.15540e6 q^{68} -394366. q^{69} -84170.2 q^{70} +1.13223e6 q^{71} -2.43301e6 q^{72} +4.75764e6 q^{73} +4.19872e6 q^{74} +4.99518e6 q^{75} -1.37473e6 q^{76} +2.22575e6 q^{77} +2.54289e6 q^{78} -5.25839e6 q^{79} -33752.7 q^{80} -5.29478e6 q^{81} -7.45234e6 q^{82} +5.20649e6 q^{83} -4.35587e6 q^{84} -422105. q^{85} +1.09013e7 q^{86} +5.57572e6 q^{87} -8.22077e6 q^{88} +6.22177e6 q^{89} -471279. q^{90} +753571. q^{91} +1.21928e6 q^{92} -1.80969e7 q^{93} +3.92573e6 q^{94} -94271.6 q^{95} -1.32678e7 q^{96} -4.76838e6 q^{97} -2.12469e6 q^{98} +1.24622e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −18.0596 −1.59626 −0.798129 0.602486i \(-0.794177\pi\)
−0.798129 + 0.602486i \(0.794177\pi\)
\(3\) −64.0898 −1.37045 −0.685227 0.728330i \(-0.740297\pi\)
−0.685227 + 0.728330i \(0.740297\pi\)
\(4\) 198.149 1.54804
\(5\) 13.5880 0.0486140 0.0243070 0.999705i \(-0.492262\pi\)
0.0243070 + 0.999705i \(0.492262\pi\)
\(6\) 1157.44 2.18760
\(7\) 343.000 0.377964
\(8\) −1266.87 −0.874814
\(9\) 1920.50 0.878143
\(10\) −245.394 −0.0776005
\(11\) 6489.06 1.46997 0.734983 0.678085i \(-0.237190\pi\)
0.734983 + 0.678085i \(0.237190\pi\)
\(12\) −12699.3 −2.12152
\(13\) 2197.00 0.277350
\(14\) −6194.44 −0.603329
\(15\) −870.853 −0.0666232
\(16\) −2484.01 −0.151612
\(17\) −31064.5 −1.53353 −0.766766 0.641926i \(-0.778136\pi\)
−0.766766 + 0.641926i \(0.778136\pi\)
\(18\) −34683.4 −1.40174
\(19\) −6937.84 −0.232053 −0.116026 0.993246i \(-0.537016\pi\)
−0.116026 + 0.993246i \(0.537016\pi\)
\(20\) 2692.46 0.0752564
\(21\) −21982.8 −0.517983
\(22\) −117190. −2.34645
\(23\) 6153.33 0.105454 0.0527270 0.998609i \(-0.483209\pi\)
0.0527270 + 0.998609i \(0.483209\pi\)
\(24\) 81193.1 1.19889
\(25\) −77940.4 −0.997637
\(26\) −39676.9 −0.442722
\(27\) 17080.0 0.166999
\(28\) 67965.2 0.585104
\(29\) −86998.5 −0.662398 −0.331199 0.943561i \(-0.607453\pi\)
−0.331199 + 0.943561i \(0.607453\pi\)
\(30\) 15727.3 0.106348
\(31\) 282367. 1.70235 0.851174 0.524883i \(-0.175891\pi\)
0.851174 + 0.524883i \(0.175891\pi\)
\(32\) 207019. 1.11683
\(33\) −415883. −2.01452
\(34\) 561012. 2.44791
\(35\) 4660.69 0.0183744
\(36\) 380545. 1.35940
\(37\) −232493. −0.754576 −0.377288 0.926096i \(-0.623143\pi\)
−0.377288 + 0.926096i \(0.623143\pi\)
\(38\) 125295. 0.370416
\(39\) −140805. −0.380095
\(40\) −17214.2 −0.0425282
\(41\) 412653. 0.935064 0.467532 0.883976i \(-0.345143\pi\)
0.467532 + 0.883976i \(0.345143\pi\)
\(42\) 397000. 0.826834
\(43\) −603630. −1.15779 −0.578897 0.815401i \(-0.696517\pi\)
−0.578897 + 0.815401i \(0.696517\pi\)
\(44\) 1.28580e6 2.27557
\(45\) 26095.8 0.0426900
\(46\) −111127. −0.168332
\(47\) −217376. −0.305401 −0.152700 0.988273i \(-0.548797\pi\)
−0.152700 + 0.988273i \(0.548797\pi\)
\(48\) 159199. 0.207777
\(49\) 117649. 0.142857
\(50\) 1.40757e6 1.59249
\(51\) 1.99092e6 2.10164
\(52\) 435334. 0.429349
\(53\) 1.08757e6 1.00344 0.501721 0.865029i \(-0.332700\pi\)
0.501721 + 0.865029i \(0.332700\pi\)
\(54\) −308457. −0.266574
\(55\) 88173.5 0.0714610
\(56\) −434535. −0.330648
\(57\) 444645. 0.318018
\(58\) 1.57116e6 1.05736
\(59\) −641874. −0.406881 −0.203441 0.979087i \(-0.565212\pi\)
−0.203441 + 0.979087i \(0.565212\pi\)
\(60\) −172559. −0.103135
\(61\) −576801. −0.325366 −0.162683 0.986678i \(-0.552015\pi\)
−0.162683 + 0.986678i \(0.552015\pi\)
\(62\) −5.09944e6 −2.71739
\(63\) 658731. 0.331907
\(64\) −3.42073e6 −1.63113
\(65\) 29852.9 0.0134831
\(66\) 7.51067e6 3.21570
\(67\) −3.17002e6 −1.28766 −0.643828 0.765170i \(-0.722655\pi\)
−0.643828 + 0.765170i \(0.722655\pi\)
\(68\) −6.15540e6 −2.37397
\(69\) −394366. −0.144520
\(70\) −84170.2 −0.0293302
\(71\) 1.13223e6 0.375432 0.187716 0.982223i \(-0.439891\pi\)
0.187716 + 0.982223i \(0.439891\pi\)
\(72\) −2.43301e6 −0.768212
\(73\) 4.75764e6 1.43140 0.715701 0.698406i \(-0.246107\pi\)
0.715701 + 0.698406i \(0.246107\pi\)
\(74\) 4.19872e6 1.20450
\(75\) 4.99518e6 1.36721
\(76\) −1.37473e6 −0.359227
\(77\) 2.22575e6 0.555595
\(78\) 2.54289e6 0.606730
\(79\) −5.25839e6 −1.19994 −0.599968 0.800024i \(-0.704820\pi\)
−0.599968 + 0.800024i \(0.704820\pi\)
\(80\) −33752.7 −0.00737045
\(81\) −5.29478e6 −1.10701
\(82\) −7.45234e6 −1.49260
\(83\) 5.20649e6 0.999475 0.499737 0.866177i \(-0.333430\pi\)
0.499737 + 0.866177i \(0.333430\pi\)
\(84\) −4.35587e6 −0.801858
\(85\) −422105. −0.0745511
\(86\) 1.09013e7 1.84814
\(87\) 5.57572e6 0.907786
\(88\) −8.22077e6 −1.28595
\(89\) 6.22177e6 0.935511 0.467756 0.883858i \(-0.345063\pi\)
0.467756 + 0.883858i \(0.345063\pi\)
\(90\) −471279. −0.0681443
\(91\) 753571. 0.104828
\(92\) 1.21928e6 0.163247
\(93\) −1.80969e7 −2.33299
\(94\) 3.92573e6 0.487498
\(95\) −94271.6 −0.0112810
\(96\) −1.32678e7 −1.53056
\(97\) −4.76838e6 −0.530481 −0.265241 0.964182i \(-0.585451\pi\)
−0.265241 + 0.964182i \(0.585451\pi\)
\(98\) −2.12469e6 −0.228037
\(99\) 1.24622e7 1.29084
\(100\) −1.54438e7 −1.54438
\(101\) 1.50570e7 1.45417 0.727084 0.686548i \(-0.240875\pi\)
0.727084 + 0.686548i \(0.240875\pi\)
\(102\) −3.59552e7 −3.35475
\(103\) −1.44181e7 −1.30010 −0.650050 0.759891i \(-0.725252\pi\)
−0.650050 + 0.759891i \(0.725252\pi\)
\(104\) −2.78330e6 −0.242630
\(105\) −298703. −0.0251812
\(106\) −1.96411e7 −1.60175
\(107\) −1.56711e7 −1.23668 −0.618339 0.785911i \(-0.712194\pi\)
−0.618339 + 0.785911i \(0.712194\pi\)
\(108\) 3.38438e6 0.258521
\(109\) 1.05853e7 0.782909 0.391454 0.920198i \(-0.371972\pi\)
0.391454 + 0.920198i \(0.371972\pi\)
\(110\) −1.59238e6 −0.114070
\(111\) 1.49004e7 1.03411
\(112\) −852014. −0.0573038
\(113\) 1.71706e7 1.11946 0.559732 0.828673i \(-0.310904\pi\)
0.559732 + 0.828673i \(0.310904\pi\)
\(114\) −8.03011e6 −0.507638
\(115\) 83611.6 0.00512654
\(116\) −1.72387e7 −1.02542
\(117\) 4.21934e6 0.243553
\(118\) 1.15920e7 0.649487
\(119\) −1.06551e7 −0.579621
\(120\) 1.10325e6 0.0582829
\(121\) 2.26208e7 1.16080
\(122\) 1.04168e7 0.519368
\(123\) −2.64468e7 −1.28146
\(124\) 5.59508e7 2.63530
\(125\) −2.12062e6 −0.0971131
\(126\) −1.18964e7 −0.529809
\(127\) 1.15826e7 0.501758 0.250879 0.968018i \(-0.419280\pi\)
0.250879 + 0.968018i \(0.419280\pi\)
\(128\) 3.52785e7 1.48688
\(129\) 3.86865e7 1.58670
\(130\) −539131. −0.0215225
\(131\) 3.85740e7 1.49915 0.749576 0.661919i \(-0.230258\pi\)
0.749576 + 0.661919i \(0.230258\pi\)
\(132\) −8.24068e7 −3.11856
\(133\) −2.37968e6 −0.0877077
\(134\) 5.72492e7 2.05543
\(135\) 232083. 0.00811849
\(136\) 3.93545e7 1.34156
\(137\) 3.91337e7 1.30026 0.650128 0.759825i \(-0.274715\pi\)
0.650128 + 0.759825i \(0.274715\pi\)
\(138\) 7.12208e6 0.230691
\(139\) 2.36964e7 0.748396 0.374198 0.927349i \(-0.377918\pi\)
0.374198 + 0.927349i \(0.377918\pi\)
\(140\) 923512. 0.0284442
\(141\) 1.39316e7 0.418537
\(142\) −2.04477e7 −0.599287
\(143\) 1.42565e7 0.407695
\(144\) −4.77053e6 −0.133137
\(145\) −1.18214e6 −0.0322018
\(146\) −8.59211e7 −2.28489
\(147\) −7.54010e6 −0.195779
\(148\) −4.60682e7 −1.16811
\(149\) 4.32940e6 0.107220 0.0536100 0.998562i \(-0.482927\pi\)
0.0536100 + 0.998562i \(0.482927\pi\)
\(150\) −9.02110e7 −2.18243
\(151\) 4.42889e7 1.04683 0.523415 0.852078i \(-0.324658\pi\)
0.523415 + 0.852078i \(0.324658\pi\)
\(152\) 8.78931e6 0.203003
\(153\) −5.96593e7 −1.34666
\(154\) −4.01961e7 −0.886873
\(155\) 3.83681e6 0.0827580
\(156\) −2.79004e7 −0.588403
\(157\) 1.88511e7 0.388765 0.194383 0.980926i \(-0.437730\pi\)
0.194383 + 0.980926i \(0.437730\pi\)
\(158\) 9.49644e7 1.91541
\(159\) −6.97022e7 −1.37517
\(160\) 2.81298e6 0.0542933
\(161\) 2.11059e6 0.0398578
\(162\) 9.56217e7 1.76707
\(163\) −1.12952e7 −0.204285 −0.102143 0.994770i \(-0.532570\pi\)
−0.102143 + 0.994770i \(0.532570\pi\)
\(164\) 8.17668e7 1.44752
\(165\) −5.65102e6 −0.0979339
\(166\) −9.40272e7 −1.59542
\(167\) 4.71435e7 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(168\) 2.78492e7 0.453138
\(169\) 4.82681e6 0.0769231
\(170\) 7.62305e6 0.119003
\(171\) −1.33241e7 −0.203776
\(172\) −1.19609e8 −1.79231
\(173\) −8.73356e7 −1.28242 −0.641210 0.767366i \(-0.721567\pi\)
−0.641210 + 0.767366i \(0.721567\pi\)
\(174\) −1.00695e8 −1.44906
\(175\) −2.67335e7 −0.377071
\(176\) −1.61189e7 −0.222864
\(177\) 4.11376e7 0.557612
\(178\) −1.12363e8 −1.49332
\(179\) 1.44092e8 1.87782 0.938910 0.344164i \(-0.111838\pi\)
0.938910 + 0.344164i \(0.111838\pi\)
\(180\) 5.17086e6 0.0660859
\(181\) 8.95747e7 1.12282 0.561410 0.827538i \(-0.310259\pi\)
0.561410 + 0.827538i \(0.310259\pi\)
\(182\) −1.36092e7 −0.167333
\(183\) 3.69671e7 0.445898
\(184\) −7.79544e6 −0.0922525
\(185\) −3.15911e6 −0.0366830
\(186\) 3.26822e8 3.72405
\(187\) −2.01579e8 −2.25424
\(188\) −4.30729e7 −0.472772
\(189\) 5.85843e6 0.0631197
\(190\) 1.70251e6 0.0180074
\(191\) −1.40233e8 −1.45624 −0.728119 0.685450i \(-0.759605\pi\)
−0.728119 + 0.685450i \(0.759605\pi\)
\(192\) 2.19234e8 2.23539
\(193\) 4.24144e7 0.424681 0.212341 0.977196i \(-0.431891\pi\)
0.212341 + 0.977196i \(0.431891\pi\)
\(194\) 8.61151e7 0.846785
\(195\) −1.91326e6 −0.0184780
\(196\) 2.33120e7 0.221149
\(197\) 2.59251e7 0.241595 0.120798 0.992677i \(-0.461455\pi\)
0.120798 + 0.992677i \(0.461455\pi\)
\(198\) −2.25063e8 −2.06052
\(199\) 1.49127e8 1.34144 0.670721 0.741710i \(-0.265985\pi\)
0.670721 + 0.741710i \(0.265985\pi\)
\(200\) 9.87400e7 0.872746
\(201\) 2.03166e8 1.76467
\(202\) −2.71924e8 −2.32123
\(203\) −2.98405e7 −0.250363
\(204\) 3.94498e8 3.25342
\(205\) 5.60714e6 0.0454572
\(206\) 2.60385e8 2.07530
\(207\) 1.18175e7 0.0926037
\(208\) −5.45736e6 −0.0420495
\(209\) −4.50201e7 −0.341110
\(210\) 5.39445e6 0.0401957
\(211\) −2.18089e8 −1.59825 −0.799127 0.601163i \(-0.794704\pi\)
−0.799127 + 0.601163i \(0.794704\pi\)
\(212\) 2.15501e8 1.55337
\(213\) −7.25646e7 −0.514513
\(214\) 2.83014e8 1.97406
\(215\) −8.20213e6 −0.0562849
\(216\) −2.16380e7 −0.146093
\(217\) 9.68520e7 0.643427
\(218\) −1.91167e8 −1.24972
\(219\) −3.04916e8 −1.96167
\(220\) 1.74715e7 0.110624
\(221\) −6.82487e7 −0.425325
\(222\) −2.69095e8 −1.65071
\(223\) −8.12974e7 −0.490919 −0.245459 0.969407i \(-0.578939\pi\)
−0.245459 + 0.969407i \(0.578939\pi\)
\(224\) 7.10075e7 0.422120
\(225\) −1.49684e8 −0.876068
\(226\) −3.10094e8 −1.78695
\(227\) −1.99361e8 −1.13123 −0.565614 0.824670i \(-0.691361\pi\)
−0.565614 + 0.824670i \(0.691361\pi\)
\(228\) 8.81060e7 0.492304
\(229\) −5.12300e7 −0.281903 −0.140952 0.990016i \(-0.545016\pi\)
−0.140952 + 0.990016i \(0.545016\pi\)
\(230\) −1.50999e6 −0.00818328
\(231\) −1.42648e8 −0.761418
\(232\) 1.10215e8 0.579475
\(233\) 1.32927e8 0.688443 0.344221 0.938889i \(-0.388143\pi\)
0.344221 + 0.938889i \(0.388143\pi\)
\(234\) −7.61995e7 −0.388774
\(235\) −2.95371e6 −0.0148467
\(236\) −1.27187e8 −0.629869
\(237\) 3.37009e8 1.64446
\(238\) 1.92427e8 0.925225
\(239\) 3.79964e8 1.80032 0.900161 0.435558i \(-0.143449\pi\)
0.900161 + 0.435558i \(0.143449\pi\)
\(240\) 2.16320e6 0.0101009
\(241\) −2.17702e8 −1.00185 −0.500925 0.865491i \(-0.667007\pi\)
−0.500925 + 0.865491i \(0.667007\pi\)
\(242\) −4.08522e8 −1.85294
\(243\) 3.01988e8 1.35010
\(244\) −1.14293e8 −0.503679
\(245\) 1.59862e6 0.00694486
\(246\) 4.77619e8 2.04554
\(247\) −1.52424e7 −0.0643599
\(248\) −3.57721e8 −1.48924
\(249\) −3.33683e8 −1.36973
\(250\) 3.82975e7 0.155018
\(251\) 2.96487e6 0.0118344 0.00591721 0.999982i \(-0.498116\pi\)
0.00591721 + 0.999982i \(0.498116\pi\)
\(252\) 1.30527e8 0.513805
\(253\) 3.99293e7 0.155014
\(254\) −2.09178e8 −0.800935
\(255\) 2.70526e7 0.102169
\(256\) −1.99263e8 −0.742313
\(257\) 6.08740e7 0.223700 0.111850 0.993725i \(-0.464322\pi\)
0.111850 + 0.993725i \(0.464322\pi\)
\(258\) −6.98663e8 −2.53279
\(259\) −7.97450e7 −0.285203
\(260\) 5.91532e6 0.0208724
\(261\) −1.67081e8 −0.581680
\(262\) −6.96632e8 −2.39303
\(263\) −2.65978e8 −0.901573 −0.450787 0.892632i \(-0.648856\pi\)
−0.450787 + 0.892632i \(0.648856\pi\)
\(264\) 5.26867e8 1.76233
\(265\) 1.47779e7 0.0487813
\(266\) 4.29761e7 0.140004
\(267\) −3.98752e8 −1.28208
\(268\) −6.28136e8 −1.99334
\(269\) 3.71244e8 1.16286 0.581429 0.813597i \(-0.302494\pi\)
0.581429 + 0.813597i \(0.302494\pi\)
\(270\) −4.19133e6 −0.0129592
\(271\) −3.00232e7 −0.0916357 −0.0458179 0.998950i \(-0.514589\pi\)
−0.0458179 + 0.998950i \(0.514589\pi\)
\(272\) 7.71644e7 0.232502
\(273\) −4.82962e7 −0.143663
\(274\) −7.06739e8 −2.07554
\(275\) −5.05760e8 −1.46649
\(276\) −7.81432e7 −0.223722
\(277\) 2.27552e8 0.643283 0.321642 0.946861i \(-0.395765\pi\)
0.321642 + 0.946861i \(0.395765\pi\)
\(278\) −4.27948e8 −1.19463
\(279\) 5.42286e8 1.49491
\(280\) −5.90447e6 −0.0160741
\(281\) 2.09229e8 0.562536 0.281268 0.959629i \(-0.409245\pi\)
0.281268 + 0.959629i \(0.409245\pi\)
\(282\) −2.51599e8 −0.668094
\(283\) 4.77356e8 1.25196 0.625979 0.779840i \(-0.284699\pi\)
0.625979 + 0.779840i \(0.284699\pi\)
\(284\) 2.24351e8 0.581184
\(285\) 6.04184e6 0.0154601
\(286\) −2.57466e8 −0.650787
\(287\) 1.41540e8 0.353421
\(288\) 3.97580e8 0.980732
\(289\) 5.54664e8 1.35172
\(290\) 2.13489e7 0.0514024
\(291\) 3.05605e8 0.727000
\(292\) 9.42723e8 2.21587
\(293\) 7.52718e8 1.74822 0.874109 0.485730i \(-0.161446\pi\)
0.874109 + 0.485730i \(0.161446\pi\)
\(294\) 1.36171e8 0.312514
\(295\) −8.72180e6 −0.0197801
\(296\) 2.94537e8 0.660114
\(297\) 1.10833e8 0.245483
\(298\) −7.81873e7 −0.171151
\(299\) 1.35189e7 0.0292477
\(300\) 9.89791e8 2.11650
\(301\) −2.07045e8 −0.437605
\(302\) −7.99840e8 −1.67101
\(303\) −9.65002e8 −1.99287
\(304\) 1.72336e7 0.0351819
\(305\) −7.83759e6 −0.0158173
\(306\) 1.07742e9 2.14962
\(307\) 1.70871e8 0.337042 0.168521 0.985698i \(-0.446101\pi\)
0.168521 + 0.985698i \(0.446101\pi\)
\(308\) 4.41030e8 0.860084
\(309\) 9.24051e8 1.78173
\(310\) −6.92913e7 −0.132103
\(311\) 3.68376e8 0.694432 0.347216 0.937785i \(-0.387127\pi\)
0.347216 + 0.937785i \(0.387127\pi\)
\(312\) 1.78381e8 0.332513
\(313\) 2.94356e8 0.542586 0.271293 0.962497i \(-0.412549\pi\)
0.271293 + 0.962497i \(0.412549\pi\)
\(314\) −3.40443e8 −0.620570
\(315\) 8.95086e6 0.0161353
\(316\) −1.04195e9 −1.85755
\(317\) −2.29062e8 −0.403873 −0.201937 0.979399i \(-0.564724\pi\)
−0.201937 + 0.979399i \(0.564724\pi\)
\(318\) 1.25879e9 2.19513
\(319\) −5.64539e8 −0.973703
\(320\) −4.64809e7 −0.0792957
\(321\) 1.00436e9 1.69481
\(322\) −3.81165e7 −0.0636234
\(323\) 2.15521e8 0.355861
\(324\) −1.04916e9 −1.71369
\(325\) −1.71235e8 −0.276695
\(326\) 2.03987e8 0.326092
\(327\) −6.78411e8 −1.07294
\(328\) −5.22776e8 −0.818007
\(329\) −7.45601e7 −0.115431
\(330\) 1.02055e8 0.156328
\(331\) 1.31268e9 1.98958 0.994789 0.101959i \(-0.0325112\pi\)
0.994789 + 0.101959i \(0.0325112\pi\)
\(332\) 1.03166e9 1.54723
\(333\) −4.46502e8 −0.662626
\(334\) −8.51392e8 −1.25031
\(335\) −4.30743e7 −0.0625981
\(336\) 5.46054e7 0.0785322
\(337\) −4.44497e8 −0.632651 −0.316326 0.948651i \(-0.602449\pi\)
−0.316326 + 0.948651i \(0.602449\pi\)
\(338\) −8.71702e7 −0.122789
\(339\) −1.10046e9 −1.53417
\(340\) −8.36398e7 −0.115408
\(341\) 1.83230e9 2.50240
\(342\) 2.40628e8 0.325278
\(343\) 4.03536e7 0.0539949
\(344\) 7.64718e8 1.01285
\(345\) −5.35865e6 −0.00702568
\(346\) 1.57725e9 2.04707
\(347\) −6.01032e7 −0.0772225 −0.0386113 0.999254i \(-0.512293\pi\)
−0.0386113 + 0.999254i \(0.512293\pi\)
\(348\) 1.10482e9 1.40529
\(349\) −1.29834e9 −1.63493 −0.817463 0.575980i \(-0.804620\pi\)
−0.817463 + 0.575980i \(0.804620\pi\)
\(350\) 4.82797e8 0.601903
\(351\) 3.75247e7 0.0463172
\(352\) 1.34336e9 1.64170
\(353\) −8.17130e7 −0.0988734 −0.0494367 0.998777i \(-0.515743\pi\)
−0.0494367 + 0.998777i \(0.515743\pi\)
\(354\) −7.42928e8 −0.890093
\(355\) 1.53848e7 0.0182513
\(356\) 1.23284e9 1.44821
\(357\) 6.82884e8 0.794344
\(358\) −2.60224e9 −2.99748
\(359\) 7.04389e8 0.803493 0.401747 0.915751i \(-0.368403\pi\)
0.401747 + 0.915751i \(0.368403\pi\)
\(360\) −3.30599e7 −0.0373458
\(361\) −8.45738e8 −0.946152
\(362\) −1.61768e9 −1.79231
\(363\) −1.44976e9 −1.59083
\(364\) 1.49319e8 0.162279
\(365\) 6.46470e7 0.0695862
\(366\) −6.67610e8 −0.711769
\(367\) 7.55308e6 0.00797614 0.00398807 0.999992i \(-0.498731\pi\)
0.00398807 + 0.999992i \(0.498731\pi\)
\(368\) −1.52849e7 −0.0159880
\(369\) 7.92499e8 0.821120
\(370\) 5.70523e7 0.0585555
\(371\) 3.73037e8 0.379265
\(372\) −3.58588e9 −3.61156
\(373\) −4.06459e8 −0.405542 −0.202771 0.979226i \(-0.564995\pi\)
−0.202771 + 0.979226i \(0.564995\pi\)
\(374\) 3.64044e9 3.59835
\(375\) 1.35910e8 0.133089
\(376\) 2.75387e8 0.267169
\(377\) −1.91136e8 −0.183716
\(378\) −1.05801e8 −0.100755
\(379\) −4.82455e8 −0.455218 −0.227609 0.973753i \(-0.573091\pi\)
−0.227609 + 0.973753i \(0.573091\pi\)
\(380\) −1.86798e7 −0.0174635
\(381\) −7.42328e8 −0.687636
\(382\) 2.53255e9 2.32453
\(383\) −3.62016e8 −0.329255 −0.164628 0.986356i \(-0.552642\pi\)
−0.164628 + 0.986356i \(0.552642\pi\)
\(384\) −2.26099e9 −2.03770
\(385\) 3.02435e7 0.0270097
\(386\) −7.65987e8 −0.677901
\(387\) −1.15927e9 −1.01671
\(388\) −9.44851e8 −0.821206
\(389\) 9.51268e8 0.819368 0.409684 0.912227i \(-0.365639\pi\)
0.409684 + 0.912227i \(0.365639\pi\)
\(390\) 3.45528e7 0.0294956
\(391\) −1.91150e8 −0.161717
\(392\) −1.49045e8 −0.124973
\(393\) −2.47220e9 −2.05452
\(394\) −4.68197e8 −0.385649
\(395\) −7.14511e7 −0.0583337
\(396\) 2.46938e9 1.99827
\(397\) 1.01269e9 0.812291 0.406145 0.913808i \(-0.366873\pi\)
0.406145 + 0.913808i \(0.366873\pi\)
\(398\) −2.69318e9 −2.14129
\(399\) 1.52513e8 0.120199
\(400\) 1.93604e8 0.151253
\(401\) 6.31867e8 0.489351 0.244676 0.969605i \(-0.421319\pi\)
0.244676 + 0.969605i \(0.421319\pi\)
\(402\) −3.66909e9 −2.81687
\(403\) 6.20361e8 0.472147
\(404\) 2.98354e9 2.25111
\(405\) −7.19456e7 −0.0538161
\(406\) 5.38908e8 0.399644
\(407\) −1.50866e9 −1.10920
\(408\) −2.52222e9 −1.83854
\(409\) −1.02626e9 −0.741695 −0.370847 0.928694i \(-0.620933\pi\)
−0.370847 + 0.928694i \(0.620933\pi\)
\(410\) −1.01263e8 −0.0725614
\(411\) −2.50807e9 −1.78194
\(412\) −2.85693e9 −2.01261
\(413\) −2.20163e8 −0.153787
\(414\) −2.13419e8 −0.147819
\(415\) 7.07459e7 0.0485885
\(416\) 4.54821e8 0.309752
\(417\) −1.51870e9 −1.02564
\(418\) 8.13045e8 0.544500
\(419\) 1.68359e9 1.11811 0.559057 0.829129i \(-0.311163\pi\)
0.559057 + 0.829129i \(0.311163\pi\)
\(420\) −5.91877e7 −0.0389815
\(421\) 7.56670e8 0.494219 0.247109 0.968988i \(-0.420519\pi\)
0.247109 + 0.968988i \(0.420519\pi\)
\(422\) 3.93860e9 2.55122
\(423\) −4.17471e8 −0.268185
\(424\) −1.37781e9 −0.877825
\(425\) 2.42118e9 1.52991
\(426\) 1.31049e9 0.821295
\(427\) −1.97843e8 −0.122977
\(428\) −3.10522e9 −1.91443
\(429\) −9.13694e8 −0.558728
\(430\) 1.48127e8 0.0898453
\(431\) −2.68350e9 −1.61447 −0.807237 0.590228i \(-0.799038\pi\)
−0.807237 + 0.590228i \(0.799038\pi\)
\(432\) −4.24267e7 −0.0253190
\(433\) 1.52805e8 0.0904546 0.0452273 0.998977i \(-0.485599\pi\)
0.0452273 + 0.998977i \(0.485599\pi\)
\(434\) −1.74911e9 −1.02708
\(435\) 7.57630e7 0.0441311
\(436\) 2.09747e9 1.21197
\(437\) −4.26908e7 −0.0244709
\(438\) 5.50667e9 3.13133
\(439\) 3.28997e9 1.85595 0.927976 0.372641i \(-0.121548\pi\)
0.927976 + 0.372641i \(0.121548\pi\)
\(440\) −1.11704e8 −0.0625150
\(441\) 2.25945e8 0.125449
\(442\) 1.23254e9 0.678929
\(443\) −1.89428e9 −1.03522 −0.517609 0.855617i \(-0.673178\pi\)
−0.517609 + 0.855617i \(0.673178\pi\)
\(444\) 2.95250e9 1.60085
\(445\) 8.45416e7 0.0454789
\(446\) 1.46820e9 0.783633
\(447\) −2.77470e8 −0.146940
\(448\) −1.17331e9 −0.616509
\(449\) −3.53879e9 −1.84498 −0.922491 0.386019i \(-0.873850\pi\)
−0.922491 + 0.386019i \(0.873850\pi\)
\(450\) 2.70324e9 1.39843
\(451\) 2.67773e9 1.37451
\(452\) 3.40234e9 1.73298
\(453\) −2.83847e9 −1.43463
\(454\) 3.60038e9 1.80573
\(455\) 1.02395e7 0.00509613
\(456\) −5.63305e8 −0.278206
\(457\) 1.43128e9 0.701485 0.350743 0.936472i \(-0.385929\pi\)
0.350743 + 0.936472i \(0.385929\pi\)
\(458\) 9.25194e8 0.449991
\(459\) −5.30581e8 −0.256099
\(460\) 1.65676e7 0.00793608
\(461\) −1.19485e9 −0.568018 −0.284009 0.958822i \(-0.591664\pi\)
−0.284009 + 0.958822i \(0.591664\pi\)
\(462\) 2.57616e9 1.21542
\(463\) 2.80198e9 1.31199 0.655996 0.754765i \(-0.272249\pi\)
0.655996 + 0.754765i \(0.272249\pi\)
\(464\) 2.16105e8 0.100427
\(465\) −2.45901e8 −0.113416
\(466\) −2.40061e9 −1.09893
\(467\) 1.71210e9 0.777894 0.388947 0.921260i \(-0.372839\pi\)
0.388947 + 0.921260i \(0.372839\pi\)
\(468\) 8.36058e8 0.377030
\(469\) −1.08732e9 −0.486688
\(470\) 5.33429e7 0.0236992
\(471\) −1.20816e9 −0.532785
\(472\) 8.13168e8 0.355945
\(473\) −3.91699e9 −1.70192
\(474\) −6.08625e9 −2.62498
\(475\) 5.40738e8 0.231504
\(476\) −2.11130e9 −0.897277
\(477\) 2.08868e9 0.881166
\(478\) −6.86200e9 −2.87378
\(479\) −1.28285e9 −0.533335 −0.266668 0.963789i \(-0.585923\pi\)
−0.266668 + 0.963789i \(0.585923\pi\)
\(480\) −1.80283e8 −0.0744065
\(481\) −5.10786e8 −0.209282
\(482\) 3.93161e9 1.59921
\(483\) −1.35267e8 −0.0546233
\(484\) 4.48228e9 1.79697
\(485\) −6.47929e7 −0.0257888
\(486\) −5.45377e9 −2.15511
\(487\) −2.20830e9 −0.866377 −0.433188 0.901303i \(-0.642611\pi\)
−0.433188 + 0.901303i \(0.642611\pi\)
\(488\) 7.30729e8 0.284634
\(489\) 7.23906e8 0.279963
\(490\) −2.88704e7 −0.0110858
\(491\) 2.82900e9 1.07857 0.539283 0.842124i \(-0.318695\pi\)
0.539283 + 0.842124i \(0.318695\pi\)
\(492\) −5.24042e9 −1.98375
\(493\) 2.70257e9 1.01581
\(494\) 2.75272e8 0.102735
\(495\) 1.69337e8 0.0627530
\(496\) −7.01402e8 −0.258096
\(497\) 3.88356e8 0.141900
\(498\) 6.02618e9 2.18645
\(499\) −3.18287e9 −1.14675 −0.573373 0.819295i \(-0.694365\pi\)
−0.573373 + 0.819295i \(0.694365\pi\)
\(500\) −4.20199e8 −0.150335
\(501\) −3.02141e9 −1.07344
\(502\) −5.35443e7 −0.0188908
\(503\) −5.34031e9 −1.87102 −0.935510 0.353301i \(-0.885059\pi\)
−0.935510 + 0.353301i \(0.885059\pi\)
\(504\) −8.34524e8 −0.290357
\(505\) 2.04595e8 0.0706929
\(506\) −7.21108e8 −0.247442
\(507\) −3.09349e8 −0.105420
\(508\) 2.29509e9 0.776741
\(509\) 1.21801e9 0.409391 0.204696 0.978826i \(-0.434380\pi\)
0.204696 + 0.978826i \(0.434380\pi\)
\(510\) −4.88559e8 −0.163088
\(511\) 1.63187e9 0.541019
\(512\) −9.17040e8 −0.301956
\(513\) −1.18498e8 −0.0387526
\(514\) −1.09936e9 −0.357083
\(515\) −1.95913e8 −0.0632030
\(516\) 7.66570e9 2.45628
\(517\) −1.41057e9 −0.448929
\(518\) 1.44016e9 0.455258
\(519\) 5.59732e9 1.75750
\(520\) −3.78196e7 −0.0117952
\(521\) −4.13286e9 −1.28032 −0.640161 0.768241i \(-0.721132\pi\)
−0.640161 + 0.768241i \(0.721132\pi\)
\(522\) 3.01741e9 0.928512
\(523\) 5.27698e9 1.61298 0.806491 0.591247i \(-0.201364\pi\)
0.806491 + 0.591247i \(0.201364\pi\)
\(524\) 7.64341e9 2.32075
\(525\) 1.71335e9 0.516759
\(526\) 4.80346e9 1.43914
\(527\) −8.77160e9 −2.61061
\(528\) 1.03305e9 0.305425
\(529\) −3.36696e9 −0.988879
\(530\) −2.66884e8 −0.0778676
\(531\) −1.23272e9 −0.357300
\(532\) −4.71532e8 −0.135775
\(533\) 9.06598e8 0.259340
\(534\) 7.20130e9 2.04652
\(535\) −2.12939e8 −0.0601199
\(536\) 4.01598e9 1.12646
\(537\) −9.23481e9 −2.57346
\(538\) −6.70453e9 −1.85622
\(539\) 7.63432e8 0.209995
\(540\) 4.59871e7 0.0125677
\(541\) 3.81762e9 1.03658 0.518290 0.855205i \(-0.326569\pi\)
0.518290 + 0.855205i \(0.326569\pi\)
\(542\) 5.42207e8 0.146274
\(543\) −5.74082e9 −1.53877
\(544\) −6.43094e9 −1.71269
\(545\) 1.43834e8 0.0380603
\(546\) 8.72210e8 0.229323
\(547\) 5.52169e9 1.44250 0.721251 0.692674i \(-0.243567\pi\)
0.721251 + 0.692674i \(0.243567\pi\)
\(548\) 7.75431e9 2.01285
\(549\) −1.10775e9 −0.285718
\(550\) 9.13382e9 2.34090
\(551\) 6.03582e8 0.153711
\(552\) 4.99608e8 0.126428
\(553\) −1.80363e9 −0.453533
\(554\) −4.10951e9 −1.02685
\(555\) 2.02467e8 0.0502723
\(556\) 4.69543e9 1.15855
\(557\) 1.32102e9 0.323904 0.161952 0.986799i \(-0.448221\pi\)
0.161952 + 0.986799i \(0.448221\pi\)
\(558\) −9.79347e9 −2.38626
\(559\) −1.32617e9 −0.321114
\(560\) −1.15772e7 −0.00278577
\(561\) 1.29192e10 3.08933
\(562\) −3.77860e9 −0.897953
\(563\) 5.95845e9 1.40719 0.703597 0.710599i \(-0.251576\pi\)
0.703597 + 0.710599i \(0.251576\pi\)
\(564\) 2.76054e9 0.647913
\(565\) 2.33314e8 0.0544216
\(566\) −8.62086e9 −1.99845
\(567\) −1.81611e9 −0.418410
\(568\) −1.43439e9 −0.328433
\(569\) 2.49678e9 0.568182 0.284091 0.958797i \(-0.408308\pi\)
0.284091 + 0.958797i \(0.408308\pi\)
\(570\) −1.09113e8 −0.0246783
\(571\) −5.66005e8 −0.127231 −0.0636156 0.997974i \(-0.520263\pi\)
−0.0636156 + 0.997974i \(0.520263\pi\)
\(572\) 2.82491e9 0.631129
\(573\) 8.98748e9 1.99571
\(574\) −2.55615e9 −0.564151
\(575\) −4.79593e8 −0.105205
\(576\) −6.56950e9 −1.43237
\(577\) 5.17728e9 1.12198 0.560992 0.827821i \(-0.310420\pi\)
0.560992 + 0.827821i \(0.310420\pi\)
\(578\) −1.00170e10 −2.15770
\(579\) −2.71833e9 −0.582006
\(580\) −2.34240e8 −0.0498497
\(581\) 1.78583e9 0.377766
\(582\) −5.51910e9 −1.16048
\(583\) 7.05732e9 1.47503
\(584\) −6.02730e9 −1.25221
\(585\) 5.73324e7 0.0118401
\(586\) −1.35938e10 −2.79061
\(587\) 6.60553e9 1.34795 0.673975 0.738754i \(-0.264585\pi\)
0.673975 + 0.738754i \(0.264585\pi\)
\(588\) −1.49406e9 −0.303074
\(589\) −1.95902e9 −0.395035
\(590\) 1.57512e8 0.0315742
\(591\) −1.66153e9 −0.331095
\(592\) 5.77513e8 0.114403
\(593\) 9.46379e9 1.86369 0.931845 0.362857i \(-0.118199\pi\)
0.931845 + 0.362857i \(0.118199\pi\)
\(594\) −2.00160e9 −0.391854
\(595\) −1.44782e8 −0.0281777
\(596\) 8.57867e8 0.165981
\(597\) −9.55755e9 −1.83838
\(598\) −2.44145e8 −0.0466868
\(599\) −5.88369e9 −1.11855 −0.559275 0.828982i \(-0.688920\pi\)
−0.559275 + 0.828982i \(0.688920\pi\)
\(600\) −6.32822e9 −1.19606
\(601\) 6.91141e9 1.29869 0.649346 0.760493i \(-0.275043\pi\)
0.649346 + 0.760493i \(0.275043\pi\)
\(602\) 3.73915e9 0.698530
\(603\) −6.08801e9 −1.13075
\(604\) 8.77581e9 1.62053
\(605\) 3.07371e8 0.0564312
\(606\) 1.74276e10 3.18114
\(607\) −1.03483e10 −1.87805 −0.939024 0.343852i \(-0.888268\pi\)
−0.939024 + 0.343852i \(0.888268\pi\)
\(608\) −1.43626e9 −0.259162
\(609\) 1.91247e9 0.343111
\(610\) 1.41544e8 0.0252485
\(611\) −4.77576e8 −0.0847029
\(612\) −1.18214e10 −2.08469
\(613\) −6.87166e9 −1.20490 −0.602448 0.798158i \(-0.705808\pi\)
−0.602448 + 0.798158i \(0.705808\pi\)
\(614\) −3.08586e9 −0.538006
\(615\) −3.59360e8 −0.0622970
\(616\) −2.81972e9 −0.486042
\(617\) 2.90376e9 0.497694 0.248847 0.968543i \(-0.419948\pi\)
0.248847 + 0.968543i \(0.419948\pi\)
\(618\) −1.66880e10 −2.84410
\(619\) 4.97057e9 0.842343 0.421171 0.906981i \(-0.361619\pi\)
0.421171 + 0.906981i \(0.361619\pi\)
\(620\) 7.60261e8 0.128113
\(621\) 1.05099e8 0.0176107
\(622\) −6.65272e9 −1.10849
\(623\) 2.13407e9 0.353590
\(624\) 3.49761e8 0.0576269
\(625\) 6.06028e9 0.992916
\(626\) −5.31596e9 −0.866107
\(627\) 2.88533e9 0.467475
\(628\) 3.73533e9 0.601824
\(629\) 7.22226e9 1.15717
\(630\) −1.61649e8 −0.0257561
\(631\) 1.15484e10 1.82987 0.914936 0.403600i \(-0.132241\pi\)
0.914936 + 0.403600i \(0.132241\pi\)
\(632\) 6.66167e9 1.04972
\(633\) 1.39773e10 2.19033
\(634\) 4.13676e9 0.644686
\(635\) 1.57385e8 0.0243924
\(636\) −1.38114e10 −2.12882
\(637\) 2.58475e8 0.0396214
\(638\) 1.01953e10 1.55428
\(639\) 2.17445e9 0.329683
\(640\) 4.79365e8 0.0722831
\(641\) 4.99629e9 0.749281 0.374641 0.927170i \(-0.377766\pi\)
0.374641 + 0.927170i \(0.377766\pi\)
\(642\) −1.81383e10 −2.70535
\(643\) 8.74939e9 1.29789 0.648947 0.760833i \(-0.275210\pi\)
0.648947 + 0.760833i \(0.275210\pi\)
\(644\) 4.18212e8 0.0617015
\(645\) 5.25673e8 0.0771359
\(646\) −3.89222e9 −0.568045
\(647\) −9.81643e9 −1.42491 −0.712457 0.701716i \(-0.752418\pi\)
−0.712457 + 0.701716i \(0.752418\pi\)
\(648\) 6.70778e9 0.968425
\(649\) −4.16516e9 −0.598102
\(650\) 3.09244e9 0.441676
\(651\) −6.20722e9 −0.881787
\(652\) −2.23813e9 −0.316242
\(653\) 1.05908e10 1.48844 0.744221 0.667934i \(-0.232821\pi\)
0.744221 + 0.667934i \(0.232821\pi\)
\(654\) 1.22518e10 1.71269
\(655\) 5.24145e8 0.0728797
\(656\) −1.02503e9 −0.141767
\(657\) 9.13705e9 1.25698
\(658\) 1.34653e9 0.184257
\(659\) −1.02716e10 −1.39810 −0.699050 0.715073i \(-0.746394\pi\)
−0.699050 + 0.715073i \(0.746394\pi\)
\(660\) −1.11975e9 −0.151606
\(661\) −2.47221e9 −0.332951 −0.166475 0.986046i \(-0.553239\pi\)
−0.166475 + 0.986046i \(0.553239\pi\)
\(662\) −2.37065e10 −3.17588
\(663\) 4.37404e9 0.582889
\(664\) −6.59593e9 −0.874354
\(665\) −3.23351e7 −0.00426382
\(666\) 8.06364e9 1.05772
\(667\) −5.35331e8 −0.0698525
\(668\) 9.34144e9 1.21254
\(669\) 5.21033e9 0.672781
\(670\) 7.77904e8 0.0999227
\(671\) −3.74290e9 −0.478277
\(672\) −4.55085e9 −0.578496
\(673\) −8.81539e9 −1.11478 −0.557389 0.830251i \(-0.688197\pi\)
−0.557389 + 0.830251i \(0.688197\pi\)
\(674\) 8.02744e9 1.00987
\(675\) −1.33122e9 −0.166604
\(676\) 9.56428e8 0.119080
\(677\) −1.46015e10 −1.80858 −0.904289 0.426921i \(-0.859598\pi\)
−0.904289 + 0.426921i \(0.859598\pi\)
\(678\) 1.98738e10 2.44894
\(679\) −1.63556e9 −0.200503
\(680\) 5.34750e8 0.0652184
\(681\) 1.27770e10 1.55030
\(682\) −3.30906e10 −3.99447
\(683\) 1.09423e10 1.31413 0.657064 0.753835i \(-0.271798\pi\)
0.657064 + 0.753835i \(0.271798\pi\)
\(684\) −2.64016e9 −0.315453
\(685\) 5.31749e8 0.0632106
\(686\) −7.28770e8 −0.0861898
\(687\) 3.28332e9 0.386336
\(688\) 1.49942e9 0.175535
\(689\) 2.38939e9 0.278305
\(690\) 9.67750e7 0.0112148
\(691\) −6.77060e9 −0.780646 −0.390323 0.920678i \(-0.627637\pi\)
−0.390323 + 0.920678i \(0.627637\pi\)
\(692\) −1.73055e10 −1.98524
\(693\) 4.27455e9 0.487892
\(694\) 1.08544e9 0.123267
\(695\) 3.21988e8 0.0363825
\(696\) −7.06368e9 −0.794143
\(697\) −1.28189e10 −1.43395
\(698\) 2.34474e10 2.60977
\(699\) −8.51927e9 −0.943479
\(700\) −5.29723e9 −0.583721
\(701\) −1.55109e10 −1.70068 −0.850342 0.526231i \(-0.823605\pi\)
−0.850342 + 0.526231i \(0.823605\pi\)
\(702\) −6.77681e8 −0.0739342
\(703\) 1.61300e9 0.175101
\(704\) −2.21973e10 −2.39771
\(705\) 1.89303e8 0.0203468
\(706\) 1.47570e9 0.157827
\(707\) 5.16457e9 0.549624
\(708\) 8.15137e9 0.863206
\(709\) −3.96173e9 −0.417468 −0.208734 0.977972i \(-0.566934\pi\)
−0.208734 + 0.977972i \(0.566934\pi\)
\(710\) −2.77844e8 −0.0291337
\(711\) −1.00987e10 −1.05372
\(712\) −7.88215e9 −0.818398
\(713\) 1.73750e9 0.179519
\(714\) −1.23326e10 −1.26798
\(715\) 1.93717e8 0.0198197
\(716\) 2.85517e10 2.90694
\(717\) −2.43518e10 −2.46726
\(718\) −1.27210e10 −1.28258
\(719\) 1.68875e9 0.169439 0.0847195 0.996405i \(-0.473001\pi\)
0.0847195 + 0.996405i \(0.473001\pi\)
\(720\) −6.48221e7 −0.00647231
\(721\) −4.94540e9 −0.491392
\(722\) 1.52737e10 1.51030
\(723\) 1.39525e10 1.37299
\(724\) 1.77491e10 1.73817
\(725\) 6.78070e9 0.660833
\(726\) 2.61821e10 2.53937
\(727\) −1.12769e10 −1.08848 −0.544240 0.838930i \(-0.683182\pi\)
−0.544240 + 0.838930i \(0.683182\pi\)
\(728\) −9.54673e8 −0.0917054
\(729\) −7.77462e9 −0.743247
\(730\) −1.16750e9 −0.111078
\(731\) 1.87515e10 1.77551
\(732\) 7.32499e9 0.690269
\(733\) 1.62161e10 1.52084 0.760419 0.649433i \(-0.224994\pi\)
0.760419 + 0.649433i \(0.224994\pi\)
\(734\) −1.36406e8 −0.0127320
\(735\) −1.02455e8 −0.00951760
\(736\) 1.27386e9 0.117774
\(737\) −2.05704e10 −1.89281
\(738\) −1.43122e10 −1.31072
\(739\) −1.25803e10 −1.14666 −0.573329 0.819325i \(-0.694348\pi\)
−0.573329 + 0.819325i \(0.694348\pi\)
\(740\) −6.25976e8 −0.0567867
\(741\) 9.76884e8 0.0882022
\(742\) −6.73690e9 −0.605406
\(743\) −6.47067e9 −0.578747 −0.289373 0.957216i \(-0.593447\pi\)
−0.289373 + 0.957216i \(0.593447\pi\)
\(744\) 2.29263e10 2.04093
\(745\) 5.88280e7 0.00521239
\(746\) 7.34048e9 0.647349
\(747\) 9.99906e9 0.877682
\(748\) −3.99428e10 −3.48966
\(749\) −5.37519e9 −0.467420
\(750\) −2.45448e9 −0.212444
\(751\) −1.46699e10 −1.26383 −0.631913 0.775040i \(-0.717730\pi\)
−0.631913 + 0.775040i \(0.717730\pi\)
\(752\) 5.39964e8 0.0463023
\(753\) −1.90018e8 −0.0162185
\(754\) 3.45184e9 0.293258
\(755\) 6.01799e8 0.0508905
\(756\) 1.16084e9 0.0977118
\(757\) 1.26419e10 1.05920 0.529598 0.848249i \(-0.322343\pi\)
0.529598 + 0.848249i \(0.322343\pi\)
\(758\) 8.71294e9 0.726645
\(759\) −2.55906e9 −0.212439
\(760\) 1.19429e8 0.00986878
\(761\) 1.64848e8 0.0135593 0.00677964 0.999977i \(-0.497842\pi\)
0.00677964 + 0.999977i \(0.497842\pi\)
\(762\) 1.34061e10 1.09764
\(763\) 3.63076e9 0.295912
\(764\) −2.77870e10 −2.25432
\(765\) −8.10652e8 −0.0654666
\(766\) 6.53787e9 0.525576
\(767\) −1.41020e9 −0.112849
\(768\) 1.27707e10 1.01731
\(769\) 1.70983e10 1.35585 0.677925 0.735131i \(-0.262879\pi\)
0.677925 + 0.735131i \(0.262879\pi\)
\(770\) −5.46186e8 −0.0431145
\(771\) −3.90140e9 −0.306570
\(772\) 8.40438e9 0.657423
\(773\) −5.74494e9 −0.447360 −0.223680 0.974663i \(-0.571807\pi\)
−0.223680 + 0.974663i \(0.571807\pi\)
\(774\) 2.09360e10 1.62293
\(775\) −2.20078e10 −1.69833
\(776\) 6.04090e9 0.464072
\(777\) 5.11084e9 0.390857
\(778\) −1.71795e10 −1.30792
\(779\) −2.86292e9 −0.216984
\(780\) −3.79112e8 −0.0286046
\(781\) 7.34713e9 0.551873
\(782\) 3.45209e9 0.258142
\(783\) −1.48593e9 −0.110620
\(784\) −2.92241e8 −0.0216588
\(785\) 2.56149e8 0.0188994
\(786\) 4.46470e10 3.27954
\(787\) 7.62864e9 0.557873 0.278937 0.960309i \(-0.410018\pi\)
0.278937 + 0.960309i \(0.410018\pi\)
\(788\) 5.13704e9 0.373999
\(789\) 1.70465e10 1.23556
\(790\) 1.29038e9 0.0931156
\(791\) 5.88951e9 0.423118
\(792\) −1.57880e10 −1.12925
\(793\) −1.26723e9 −0.0902402
\(794\) −1.82889e10 −1.29663
\(795\) −9.47115e8 −0.0668525
\(796\) 2.95495e10 2.07661
\(797\) 9.18514e9 0.642661 0.321330 0.946967i \(-0.395870\pi\)
0.321330 + 0.946967i \(0.395870\pi\)
\(798\) −2.75433e9 −0.191869
\(799\) 6.75269e9 0.468342
\(800\) −1.61351e10 −1.11419
\(801\) 1.19489e10 0.821513
\(802\) −1.14113e10 −0.781131
\(803\) 3.08726e10 2.10411
\(804\) 4.02571e10 2.73178
\(805\) 2.86788e7 0.00193765
\(806\) −1.12035e10 −0.753668
\(807\) −2.37930e10 −1.59364
\(808\) −1.90752e10 −1.27213
\(809\) 6.69362e9 0.444469 0.222234 0.974993i \(-0.428665\pi\)
0.222234 + 0.974993i \(0.428665\pi\)
\(810\) 1.29931e9 0.0859043
\(811\) 9.00116e9 0.592551 0.296275 0.955103i \(-0.404255\pi\)
0.296275 + 0.955103i \(0.404255\pi\)
\(812\) −5.91287e9 −0.387572
\(813\) 1.92418e9 0.125582
\(814\) 2.72458e10 1.77057
\(815\) −1.53479e8 −0.00993112
\(816\) −4.94545e9 −0.318633
\(817\) 4.18789e9 0.268669
\(818\) 1.85338e10 1.18394
\(819\) 1.44723e9 0.0920544
\(820\) 1.11105e9 0.0703695
\(821\) 1.60680e10 1.01335 0.506677 0.862136i \(-0.330874\pi\)
0.506677 + 0.862136i \(0.330874\pi\)
\(822\) 4.52947e10 2.84444
\(823\) 2.07361e10 1.29667 0.648333 0.761357i \(-0.275467\pi\)
0.648333 + 0.761357i \(0.275467\pi\)
\(824\) 1.82658e10 1.13735
\(825\) 3.24140e10 2.00976
\(826\) 3.97605e9 0.245483
\(827\) −2.72719e10 −1.67667 −0.838334 0.545158i \(-0.816470\pi\)
−0.838334 + 0.545158i \(0.816470\pi\)
\(828\) 2.34162e9 0.143354
\(829\) 8.40379e9 0.512312 0.256156 0.966635i \(-0.417544\pi\)
0.256156 + 0.966635i \(0.417544\pi\)
\(830\) −1.27764e9 −0.0775597
\(831\) −1.45838e10 −0.881590
\(832\) −7.51534e9 −0.452394
\(833\) −3.65471e9 −0.219076
\(834\) 2.74271e10 1.63719
\(835\) 6.40586e8 0.0380781
\(836\) −8.92069e9 −0.528052
\(837\) 4.82283e9 0.284291
\(838\) −3.04049e10 −1.78480
\(839\) 3.20862e9 0.187565 0.0937825 0.995593i \(-0.470104\pi\)
0.0937825 + 0.995593i \(0.470104\pi\)
\(840\) 3.78416e8 0.0220289
\(841\) −9.68113e9 −0.561229
\(842\) −1.36652e10 −0.788900
\(843\) −1.34095e10 −0.770930
\(844\) −4.32142e10 −2.47416
\(845\) 6.55868e7 0.00373954
\(846\) 7.53936e9 0.428093
\(847\) 7.75892e9 0.438742
\(848\) −2.70153e9 −0.152134
\(849\) −3.05936e10 −1.71575
\(850\) −4.37255e10 −2.44213
\(851\) −1.43060e9 −0.0795730
\(852\) −1.43786e10 −0.796486
\(853\) 1.92263e10 1.06066 0.530329 0.847792i \(-0.322069\pi\)
0.530329 + 0.847792i \(0.322069\pi\)
\(854\) 3.57296e9 0.196302
\(855\) −1.81048e8 −0.00990634
\(856\) 1.98532e10 1.08186
\(857\) 2.53280e10 1.37457 0.687287 0.726386i \(-0.258801\pi\)
0.687287 + 0.726386i \(0.258801\pi\)
\(858\) 1.65009e10 0.891874
\(859\) −1.64704e10 −0.886598 −0.443299 0.896374i \(-0.646192\pi\)
−0.443299 + 0.896374i \(0.646192\pi\)
\(860\) −1.62525e9 −0.0871314
\(861\) −9.07126e9 −0.484347
\(862\) 4.84629e10 2.57712
\(863\) 2.95746e10 1.56632 0.783160 0.621820i \(-0.213607\pi\)
0.783160 + 0.621820i \(0.213607\pi\)
\(864\) 3.53588e9 0.186509
\(865\) −1.18672e9 −0.0623435
\(866\) −2.75960e9 −0.144389
\(867\) −3.55483e10 −1.85247
\(868\) 1.91911e10 0.996051
\(869\) −3.41220e10 −1.76387
\(870\) −1.36825e9 −0.0704446
\(871\) −6.96453e9 −0.357131
\(872\) −1.34102e10 −0.684899
\(873\) −9.15768e9 −0.465839
\(874\) 7.70979e8 0.0390618
\(875\) −7.27373e8 −0.0367053
\(876\) −6.04189e10 −3.03675
\(877\) 1.56818e10 0.785049 0.392524 0.919742i \(-0.371602\pi\)
0.392524 + 0.919742i \(0.371602\pi\)
\(878\) −5.94156e10 −2.96258
\(879\) −4.82415e10 −2.39585
\(880\) −2.19024e8 −0.0108343
\(881\) 3.70407e9 0.182500 0.0912501 0.995828i \(-0.470914\pi\)
0.0912501 + 0.995828i \(0.470914\pi\)
\(882\) −4.08047e9 −0.200249
\(883\) −3.17603e10 −1.55247 −0.776234 0.630445i \(-0.782872\pi\)
−0.776234 + 0.630445i \(0.782872\pi\)
\(884\) −1.35234e10 −0.658421
\(885\) 5.58978e8 0.0271077
\(886\) 3.42100e10 1.65247
\(887\) −2.88472e10 −1.38794 −0.693970 0.720004i \(-0.744140\pi\)
−0.693970 + 0.720004i \(0.744140\pi\)
\(888\) −1.88768e10 −0.904655
\(889\) 3.97284e9 0.189647
\(890\) −1.52679e9 −0.0725961
\(891\) −3.43582e10 −1.62726
\(892\) −1.61090e10 −0.759962
\(893\) 1.50812e9 0.0708690
\(894\) 5.01100e9 0.234554
\(895\) 1.95792e9 0.0912883
\(896\) 1.21005e10 0.561987
\(897\) −8.66421e8 −0.0400826
\(898\) 6.39090e10 2.94507
\(899\) −2.45655e10 −1.12763
\(900\) −2.96598e10 −1.35619
\(901\) −3.37849e10 −1.53881
\(902\) −4.83587e10 −2.19408
\(903\) 1.32695e10 0.599717
\(904\) −2.17528e10 −0.979323
\(905\) 1.21714e9 0.0545848
\(906\) 5.12616e10 2.29004
\(907\) −3.44892e10 −1.53482 −0.767409 0.641158i \(-0.778454\pi\)
−0.767409 + 0.641158i \(0.778454\pi\)
\(908\) −3.95033e10 −1.75119
\(909\) 2.89170e10 1.27697
\(910\) −1.84922e8 −0.00813474
\(911\) −3.96476e10 −1.73741 −0.868705 0.495330i \(-0.835047\pi\)
−0.868705 + 0.495330i \(0.835047\pi\)
\(912\) −1.10450e9 −0.0482152
\(913\) 3.37853e10 1.46920
\(914\) −2.58484e10 −1.11975
\(915\) 5.02309e8 0.0216769
\(916\) −1.01512e10 −0.436398
\(917\) 1.32309e10 0.566626
\(918\) 9.58207e9 0.408799
\(919\) 6.76983e9 0.287722 0.143861 0.989598i \(-0.454048\pi\)
0.143861 + 0.989598i \(0.454048\pi\)
\(920\) −1.05925e8 −0.00448476
\(921\) −1.09511e10 −0.461901
\(922\) 2.15786e10 0.906703
\(923\) 2.48752e9 0.104126
\(924\) −2.82655e10 −1.17871
\(925\) 1.81206e10 0.752793
\(926\) −5.06026e10 −2.09428
\(927\) −2.76899e10 −1.14167
\(928\) −1.80103e10 −0.739783
\(929\) −2.26338e10 −0.926195 −0.463097 0.886307i \(-0.653262\pi\)
−0.463097 + 0.886307i \(0.653262\pi\)
\(930\) 4.44087e9 0.181041
\(931\) −8.16230e8 −0.0331504
\(932\) 2.63394e10 1.06574
\(933\) −2.36091e10 −0.951687
\(934\) −3.09199e10 −1.24172
\(935\) −2.73907e9 −0.109588
\(936\) −5.34533e9 −0.213064
\(937\) −7.17774e9 −0.285036 −0.142518 0.989792i \(-0.545520\pi\)
−0.142518 + 0.989792i \(0.545520\pi\)
\(938\) 1.96365e10 0.776880
\(939\) −1.88652e10 −0.743589
\(940\) −5.85276e8 −0.0229833
\(941\) 7.71375e9 0.301788 0.150894 0.988550i \(-0.451785\pi\)
0.150894 + 0.988550i \(0.451785\pi\)
\(942\) 2.18189e10 0.850462
\(943\) 2.53919e9 0.0986062
\(944\) 1.59442e9 0.0616879
\(945\) 7.96045e7 0.00306850
\(946\) 7.07393e10 2.71670
\(947\) 3.60811e10 1.38056 0.690280 0.723542i \(-0.257487\pi\)
0.690280 + 0.723542i \(0.257487\pi\)
\(948\) 6.67780e10 2.54568
\(949\) 1.04525e10 0.397000
\(950\) −9.76551e9 −0.369541
\(951\) 1.46805e10 0.553490
\(952\) 1.34986e10 0.507060
\(953\) 2.80356e10 1.04926 0.524632 0.851329i \(-0.324203\pi\)
0.524632 + 0.851329i \(0.324203\pi\)
\(954\) −3.77207e10 −1.40657
\(955\) −1.90549e9 −0.0707936
\(956\) 7.52896e10 2.78697
\(957\) 3.61812e10 1.33442
\(958\) 2.31677e10 0.851341
\(959\) 1.34229e10 0.491450
\(960\) 2.97895e9 0.108671
\(961\) 5.22187e10 1.89799
\(962\) 9.22459e9 0.334068
\(963\) −3.00964e10 −1.08598
\(964\) −4.31374e10 −1.55090
\(965\) 5.76328e8 0.0206454
\(966\) 2.44287e9 0.0871929
\(967\) 4.53973e10 1.61450 0.807248 0.590213i \(-0.200956\pi\)
0.807248 + 0.590213i \(0.200956\pi\)
\(968\) −2.86575e10 −1.01549
\(969\) −1.38127e10 −0.487690
\(970\) 1.17013e9 0.0411656
\(971\) −3.65836e10 −1.28239 −0.641194 0.767379i \(-0.721560\pi\)
−0.641194 + 0.767379i \(0.721560\pi\)
\(972\) 5.98386e10 2.09001
\(973\) 8.12788e9 0.282867
\(974\) 3.98810e10 1.38296
\(975\) 1.09744e10 0.379197
\(976\) 1.43278e9 0.0493292
\(977\) 3.46498e9 0.118869 0.0594347 0.998232i \(-0.481070\pi\)
0.0594347 + 0.998232i \(0.481070\pi\)
\(978\) −1.30735e10 −0.446894
\(979\) 4.03735e10 1.37517
\(980\) 3.16765e8 0.0107509
\(981\) 2.03291e10 0.687506
\(982\) −5.10905e10 −1.72167
\(983\) −2.52120e10 −0.846583 −0.423291 0.905994i \(-0.639125\pi\)
−0.423291 + 0.905994i \(0.639125\pi\)
\(984\) 3.35046e10 1.12104
\(985\) 3.52271e8 0.0117449
\(986\) −4.88073e10 −1.62149
\(987\) 4.77854e9 0.158192
\(988\) −3.02028e9 −0.0996317
\(989\) −3.71433e9 −0.122094
\(990\) −3.05816e9 −0.100170
\(991\) −5.79005e10 −1.88984 −0.944919 0.327303i \(-0.893860\pi\)
−0.944919 + 0.327303i \(0.893860\pi\)
\(992\) 5.84554e10 1.90123
\(993\) −8.41293e10 −2.72662
\(994\) −7.01355e9 −0.226509
\(995\) 2.02635e9 0.0652129
\(996\) −6.61190e10 −2.12040
\(997\) 1.71906e10 0.549360 0.274680 0.961536i \(-0.411428\pi\)
0.274680 + 0.961536i \(0.411428\pi\)
\(998\) 5.74814e10 1.83050
\(999\) −3.97097e9 −0.126013
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.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.2 12 1.1 even 1 trivial