Properties

Label 177.8.a.a.1.13
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-14.0604\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+14.0604 q^{2} -27.0000 q^{3} +69.6949 q^{4} +153.219 q^{5} -379.631 q^{6} -215.221 q^{7} -819.793 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+14.0604 q^{2} -27.0000 q^{3} +69.6949 q^{4} +153.219 q^{5} -379.631 q^{6} -215.221 q^{7} -819.793 q^{8} +729.000 q^{9} +2154.32 q^{10} +7869.22 q^{11} -1881.76 q^{12} -14239.1 q^{13} -3026.10 q^{14} -4136.91 q^{15} -20447.6 q^{16} -14439.7 q^{17} +10250.0 q^{18} +52274.2 q^{19} +10678.6 q^{20} +5810.97 q^{21} +110644. q^{22} -20917.2 q^{23} +22134.4 q^{24} -54648.9 q^{25} -200207. q^{26} -19683.0 q^{27} -14999.8 q^{28} -136874. q^{29} -58166.7 q^{30} +9325.59 q^{31} -182567. q^{32} -212469. q^{33} -203028. q^{34} -32976.0 q^{35} +50807.6 q^{36} -525467. q^{37} +734996. q^{38} +384455. q^{39} -125608. q^{40} +197767. q^{41} +81704.6 q^{42} +410661. q^{43} +548444. q^{44} +111697. q^{45} -294105. q^{46} -946517. q^{47} +552084. q^{48} -777223. q^{49} -768386. q^{50} +389871. q^{51} -992389. q^{52} -1.55829e6 q^{53} -276751. q^{54} +1.20571e6 q^{55} +176437. q^{56} -1.41140e6 q^{57} -1.92450e6 q^{58} +205379. q^{59} -288322. q^{60} -659307. q^{61} +131121. q^{62} -156896. q^{63} +50317.0 q^{64} -2.18169e6 q^{65} -2.98740e6 q^{66} -2.30296e6 q^{67} -1.00637e6 q^{68} +564765. q^{69} -463656. q^{70} -1.68822e6 q^{71} -597629. q^{72} +2.53442e6 q^{73} -7.38828e6 q^{74} +1.47552e6 q^{75} +3.64324e6 q^{76} -1.69362e6 q^{77} +5.40559e6 q^{78} -1.26627e6 q^{79} -3.13296e6 q^{80} +531441. q^{81} +2.78068e6 q^{82} +2.22258e6 q^{83} +404995. q^{84} -2.21243e6 q^{85} +5.77406e6 q^{86} +3.69558e6 q^{87} -6.45113e6 q^{88} -1.04798e7 q^{89} +1.57050e6 q^{90} +3.06455e6 q^{91} -1.45782e6 q^{92} -251791. q^{93} -1.33084e7 q^{94} +8.00940e6 q^{95} +4.92932e6 q^{96} +5.41316e6 q^{97} -1.09281e7 q^{98} +5.73666e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.0604 1.24278 0.621388 0.783503i \(-0.286569\pi\)
0.621388 + 0.783503i \(0.286569\pi\)
\(3\) −27.0000 −0.577350
\(4\) 69.6949 0.544491
\(5\) 153.219 0.548173 0.274087 0.961705i \(-0.411625\pi\)
0.274087 + 0.961705i \(0.411625\pi\)
\(6\) −379.631 −0.717517
\(7\) −215.221 −0.237160 −0.118580 0.992944i \(-0.537834\pi\)
−0.118580 + 0.992944i \(0.537834\pi\)
\(8\) −819.793 −0.566095
\(9\) 729.000 0.333333
\(10\) 2154.32 0.681256
\(11\) 7869.22 1.78261 0.891307 0.453401i \(-0.149789\pi\)
0.891307 + 0.453401i \(0.149789\pi\)
\(12\) −1881.76 −0.314362
\(13\) −14239.1 −1.79754 −0.898772 0.438416i \(-0.855540\pi\)
−0.898772 + 0.438416i \(0.855540\pi\)
\(14\) −3026.10 −0.294737
\(15\) −4136.91 −0.316488
\(16\) −20447.6 −1.24802
\(17\) −14439.7 −0.712830 −0.356415 0.934328i \(-0.616001\pi\)
−0.356415 + 0.934328i \(0.616001\pi\)
\(18\) 10250.0 0.414259
\(19\) 52274.2 1.74844 0.874218 0.485533i \(-0.161375\pi\)
0.874218 + 0.485533i \(0.161375\pi\)
\(20\) 10678.6 0.298475
\(21\) 5810.97 0.136925
\(22\) 110644. 2.21539
\(23\) −20917.2 −0.358473 −0.179237 0.983806i \(-0.557363\pi\)
−0.179237 + 0.983806i \(0.557363\pi\)
\(24\) 22134.4 0.326835
\(25\) −54648.9 −0.699506
\(26\) −200207. −2.23394
\(27\) −19683.0 −0.192450
\(28\) −14999.8 −0.129132
\(29\) −136874. −1.04214 −0.521070 0.853514i \(-0.674467\pi\)
−0.521070 + 0.853514i \(0.674467\pi\)
\(30\) −58166.7 −0.393323
\(31\) 9325.59 0.0562225 0.0281113 0.999605i \(-0.491051\pi\)
0.0281113 + 0.999605i \(0.491051\pi\)
\(32\) −182567. −0.984914
\(33\) −212469. −1.02919
\(34\) −203028. −0.885888
\(35\) −32976.0 −0.130005
\(36\) 50807.6 0.181497
\(37\) −525467. −1.70545 −0.852726 0.522358i \(-0.825052\pi\)
−0.852726 + 0.522358i \(0.825052\pi\)
\(38\) 734996. 2.17291
\(39\) 384455. 1.03781
\(40\) −125608. −0.310318
\(41\) 197767. 0.448136 0.224068 0.974574i \(-0.428066\pi\)
0.224068 + 0.974574i \(0.428066\pi\)
\(42\) 81704.6 0.170166
\(43\) 410661. 0.787669 0.393835 0.919181i \(-0.371148\pi\)
0.393835 + 0.919181i \(0.371148\pi\)
\(44\) 548444. 0.970617
\(45\) 111697. 0.182724
\(46\) −294105. −0.445502
\(47\) −946517. −1.32980 −0.664899 0.746933i \(-0.731525\pi\)
−0.664899 + 0.746933i \(0.731525\pi\)
\(48\) 552084. 0.720545
\(49\) −777223. −0.943755
\(50\) −768386. −0.869329
\(51\) 389871. 0.411553
\(52\) −992389. −0.978747
\(53\) −1.55829e6 −1.43775 −0.718873 0.695141i \(-0.755342\pi\)
−0.718873 + 0.695141i \(0.755342\pi\)
\(54\) −276751. −0.239172
\(55\) 1.20571e6 0.977181
\(56\) 176437. 0.134255
\(57\) −1.41140e6 −1.00946
\(58\) −1.92450e6 −1.29515
\(59\) 205379. 0.130189
\(60\) −288322. −0.172325
\(61\) −659307. −0.371906 −0.185953 0.982559i \(-0.559537\pi\)
−0.185953 + 0.982559i \(0.559537\pi\)
\(62\) 131121. 0.0698720
\(63\) −156896. −0.0790534
\(64\) 50317.0 0.0239930
\(65\) −2.18169e6 −0.985365
\(66\) −2.98740e6 −1.27906
\(67\) −2.30296e6 −0.935457 −0.467729 0.883872i \(-0.654928\pi\)
−0.467729 + 0.883872i \(0.654928\pi\)
\(68\) −1.00637e6 −0.388130
\(69\) 564765. 0.206965
\(70\) −463656. −0.161567
\(71\) −1.68822e6 −0.559788 −0.279894 0.960031i \(-0.590299\pi\)
−0.279894 + 0.960031i \(0.590299\pi\)
\(72\) −597629. −0.188698
\(73\) 2.53442e6 0.762514 0.381257 0.924469i \(-0.375491\pi\)
0.381257 + 0.924469i \(0.375491\pi\)
\(74\) −7.38828e6 −2.11949
\(75\) 1.47552e6 0.403860
\(76\) 3.64324e6 0.952008
\(77\) −1.69362e6 −0.422765
\(78\) 5.40559e6 1.28977
\(79\) −1.26627e6 −0.288957 −0.144478 0.989508i \(-0.546150\pi\)
−0.144478 + 0.989508i \(0.546150\pi\)
\(80\) −3.13296e6 −0.684131
\(81\) 531441. 0.111111
\(82\) 2.78068e6 0.556932
\(83\) 2.22258e6 0.426662 0.213331 0.976980i \(-0.431569\pi\)
0.213331 + 0.976980i \(0.431569\pi\)
\(84\) 404995. 0.0745542
\(85\) −2.21243e6 −0.390754
\(86\) 5.77406e6 0.978896
\(87\) 3.69558e6 0.601680
\(88\) −6.45113e6 −1.00913
\(89\) −1.04798e7 −1.57575 −0.787876 0.615834i \(-0.788819\pi\)
−0.787876 + 0.615834i \(0.788819\pi\)
\(90\) 1.57050e6 0.227085
\(91\) 3.06455e6 0.426306
\(92\) −1.45782e6 −0.195186
\(93\) −251791. −0.0324601
\(94\) −1.33084e7 −1.65264
\(95\) 8.00940e6 0.958446
\(96\) 4.92932e6 0.568641
\(97\) 5.41316e6 0.602212 0.301106 0.953591i \(-0.402644\pi\)
0.301106 + 0.953591i \(0.402644\pi\)
\(98\) −1.09281e7 −1.17288
\(99\) 5.73666e6 0.594204
\(100\) −3.80875e6 −0.380875
\(101\) −8.00726e6 −0.773320 −0.386660 0.922222i \(-0.626371\pi\)
−0.386660 + 0.922222i \(0.626371\pi\)
\(102\) 5.48174e6 0.511468
\(103\) 2.41569e6 0.217826 0.108913 0.994051i \(-0.465263\pi\)
0.108913 + 0.994051i \(0.465263\pi\)
\(104\) 1.16731e7 1.01758
\(105\) 890352. 0.0750584
\(106\) −2.19102e7 −1.78680
\(107\) 1.50841e7 1.19035 0.595175 0.803596i \(-0.297083\pi\)
0.595175 + 0.803596i \(0.297083\pi\)
\(108\) −1.37180e6 −0.104787
\(109\) 1.30931e7 0.968386 0.484193 0.874961i \(-0.339113\pi\)
0.484193 + 0.874961i \(0.339113\pi\)
\(110\) 1.69528e7 1.21442
\(111\) 1.41876e7 0.984643
\(112\) 4.40075e6 0.295981
\(113\) −8.64628e6 −0.563709 −0.281854 0.959457i \(-0.590950\pi\)
−0.281854 + 0.959457i \(0.590950\pi\)
\(114\) −1.98449e7 −1.25453
\(115\) −3.20492e6 −0.196505
\(116\) −9.53938e6 −0.567437
\(117\) −1.03803e7 −0.599181
\(118\) 2.88771e6 0.161796
\(119\) 3.10772e6 0.169055
\(120\) 3.39141e6 0.179162
\(121\) 4.24374e7 2.17771
\(122\) −9.27012e6 −0.462196
\(123\) −5.33970e6 −0.258731
\(124\) 649946. 0.0306127
\(125\) −2.03435e7 −0.931624
\(126\) −2.20602e6 −0.0982457
\(127\) 2.77631e7 1.20269 0.601347 0.798988i \(-0.294631\pi\)
0.601347 + 0.798988i \(0.294631\pi\)
\(128\) 2.40761e7 1.01473
\(129\) −1.10878e7 −0.454761
\(130\) −3.06755e7 −1.22459
\(131\) 2.08878e7 0.811788 0.405894 0.913920i \(-0.366960\pi\)
0.405894 + 0.913920i \(0.366960\pi\)
\(132\) −1.48080e7 −0.560386
\(133\) −1.12505e7 −0.414660
\(134\) −3.23805e7 −1.16256
\(135\) −3.01581e6 −0.105496
\(136\) 1.18375e7 0.403530
\(137\) −1.47003e7 −0.488432 −0.244216 0.969721i \(-0.578531\pi\)
−0.244216 + 0.969721i \(0.578531\pi\)
\(138\) 7.94083e6 0.257211
\(139\) −3.38613e7 −1.06943 −0.534715 0.845033i \(-0.679581\pi\)
−0.534715 + 0.845033i \(0.679581\pi\)
\(140\) −2.29826e6 −0.0707865
\(141\) 2.55559e7 0.767759
\(142\) −2.37370e7 −0.695691
\(143\) −1.12050e8 −3.20433
\(144\) −1.49063e7 −0.416007
\(145\) −2.09716e7 −0.571274
\(146\) 3.56349e7 0.947634
\(147\) 2.09850e7 0.544877
\(148\) −3.66224e7 −0.928604
\(149\) −3.52382e7 −0.872694 −0.436347 0.899778i \(-0.643728\pi\)
−0.436347 + 0.899778i \(0.643728\pi\)
\(150\) 2.07464e7 0.501907
\(151\) 4.31481e7 1.01986 0.509932 0.860215i \(-0.329671\pi\)
0.509932 + 0.860215i \(0.329671\pi\)
\(152\) −4.28541e7 −0.989781
\(153\) −1.05265e7 −0.237610
\(154\) −2.38130e7 −0.525402
\(155\) 1.42886e6 0.0308197
\(156\) 2.67945e7 0.565080
\(157\) 7.00495e6 0.144463 0.0722314 0.997388i \(-0.476988\pi\)
0.0722314 + 0.997388i \(0.476988\pi\)
\(158\) −1.78043e7 −0.359108
\(159\) 4.20738e7 0.830083
\(160\) −2.79728e7 −0.539904
\(161\) 4.50183e6 0.0850156
\(162\) 7.47227e6 0.138086
\(163\) 5.15447e7 0.932240 0.466120 0.884722i \(-0.345652\pi\)
0.466120 + 0.884722i \(0.345652\pi\)
\(164\) 1.37833e7 0.244006
\(165\) −3.25543e7 −0.564176
\(166\) 3.12503e7 0.530245
\(167\) −8.79382e6 −0.146107 −0.0730533 0.997328i \(-0.523274\pi\)
−0.0730533 + 0.997328i \(0.523274\pi\)
\(168\) −4.76380e6 −0.0775123
\(169\) 1.40002e8 2.23116
\(170\) −3.11077e7 −0.485620
\(171\) 3.81079e7 0.582812
\(172\) 2.86210e7 0.428879
\(173\) 5.52334e7 0.811037 0.405518 0.914087i \(-0.367091\pi\)
0.405518 + 0.914087i \(0.367091\pi\)
\(174\) 5.19614e7 0.747754
\(175\) 1.17616e7 0.165895
\(176\) −1.60906e8 −2.22474
\(177\) −5.54523e6 −0.0751646
\(178\) −1.47350e8 −1.95831
\(179\) 1.14162e8 1.48776 0.743882 0.668311i \(-0.232982\pi\)
0.743882 + 0.668311i \(0.232982\pi\)
\(180\) 7.78469e6 0.0994918
\(181\) 7.45147e6 0.0934044 0.0467022 0.998909i \(-0.485129\pi\)
0.0467022 + 0.998909i \(0.485129\pi\)
\(182\) 4.30888e7 0.529803
\(183\) 1.78013e7 0.214720
\(184\) 1.71478e7 0.202930
\(185\) −8.05116e7 −0.934883
\(186\) −3.54028e6 −0.0403406
\(187\) −1.13629e8 −1.27070
\(188\) −6.59674e7 −0.724063
\(189\) 4.23620e6 0.0456415
\(190\) 1.12615e8 1.19113
\(191\) −2.65020e7 −0.275208 −0.137604 0.990487i \(-0.543940\pi\)
−0.137604 + 0.990487i \(0.543940\pi\)
\(192\) −1.35856e6 −0.0138524
\(193\) −7.95433e7 −0.796440 −0.398220 0.917290i \(-0.630372\pi\)
−0.398220 + 0.917290i \(0.630372\pi\)
\(194\) 7.61112e7 0.748415
\(195\) 5.89058e7 0.568901
\(196\) −5.41685e7 −0.513866
\(197\) 6.69673e7 0.624067 0.312033 0.950071i \(-0.398990\pi\)
0.312033 + 0.950071i \(0.398990\pi\)
\(198\) 8.06597e7 0.738463
\(199\) −5.47720e7 −0.492689 −0.246345 0.969182i \(-0.579229\pi\)
−0.246345 + 0.969182i \(0.579229\pi\)
\(200\) 4.48008e7 0.395987
\(201\) 6.21798e7 0.540087
\(202\) −1.12585e8 −0.961063
\(203\) 2.94581e7 0.247154
\(204\) 2.71720e7 0.224087
\(205\) 3.03016e7 0.245656
\(206\) 3.39655e7 0.270709
\(207\) −1.52487e7 −0.119491
\(208\) 2.91154e8 2.24337
\(209\) 4.11357e8 3.11679
\(210\) 1.25187e7 0.0932807
\(211\) −6.45829e7 −0.473292 −0.236646 0.971596i \(-0.576048\pi\)
−0.236646 + 0.971596i \(0.576048\pi\)
\(212\) −1.08605e8 −0.782840
\(213\) 4.55818e7 0.323194
\(214\) 2.12088e8 1.47934
\(215\) 6.29211e7 0.431779
\(216\) 1.61360e7 0.108945
\(217\) −2.00706e6 −0.0133337
\(218\) 1.84094e8 1.20349
\(219\) −6.84292e7 −0.440238
\(220\) 8.40321e7 0.532066
\(221\) 2.05607e8 1.28134
\(222\) 1.99484e8 1.22369
\(223\) −2.02022e8 −1.21992 −0.609961 0.792431i \(-0.708815\pi\)
−0.609961 + 0.792431i \(0.708815\pi\)
\(224\) 3.92924e7 0.233583
\(225\) −3.98391e7 −0.233169
\(226\) −1.21570e8 −0.700564
\(227\) −2.76130e8 −1.56683 −0.783416 0.621497i \(-0.786525\pi\)
−0.783416 + 0.621497i \(0.786525\pi\)
\(228\) −9.83676e7 −0.549642
\(229\) −1.40164e8 −0.771280 −0.385640 0.922649i \(-0.626019\pi\)
−0.385640 + 0.922649i \(0.626019\pi\)
\(230\) −4.50624e7 −0.244212
\(231\) 4.57278e7 0.244084
\(232\) 1.12208e8 0.589951
\(233\) 8.67302e7 0.449185 0.224592 0.974453i \(-0.427895\pi\)
0.224592 + 0.974453i \(0.427895\pi\)
\(234\) −1.45951e8 −0.744648
\(235\) −1.45024e8 −0.728960
\(236\) 1.43139e7 0.0708867
\(237\) 3.41894e7 0.166829
\(238\) 4.36958e7 0.210097
\(239\) 2.22648e8 1.05494 0.527469 0.849574i \(-0.323141\pi\)
0.527469 + 0.849574i \(0.323141\pi\)
\(240\) 8.45898e7 0.394983
\(241\) 4.27798e8 1.96870 0.984350 0.176226i \(-0.0563888\pi\)
0.984350 + 0.176226i \(0.0563888\pi\)
\(242\) 5.96687e8 2.70641
\(243\) −1.43489e7 −0.0641500
\(244\) −4.59503e7 −0.202500
\(245\) −1.19085e8 −0.517341
\(246\) −7.50783e7 −0.321545
\(247\) −7.44335e8 −3.14289
\(248\) −7.64505e6 −0.0318273
\(249\) −6.00096e7 −0.246333
\(250\) −2.86038e8 −1.15780
\(251\) 4.79689e7 0.191471 0.0957353 0.995407i \(-0.469480\pi\)
0.0957353 + 0.995407i \(0.469480\pi\)
\(252\) −1.09349e7 −0.0430439
\(253\) −1.64602e8 −0.639019
\(254\) 3.90360e8 1.49468
\(255\) 5.97357e7 0.225602
\(256\) 3.32079e8 1.23709
\(257\) 1.81282e8 0.666175 0.333087 0.942896i \(-0.391910\pi\)
0.333087 + 0.942896i \(0.391910\pi\)
\(258\) −1.55900e8 −0.565166
\(259\) 1.13092e8 0.404466
\(260\) −1.52053e8 −0.536523
\(261\) −9.97808e7 −0.347380
\(262\) 2.93691e8 1.00887
\(263\) −2.00422e8 −0.679359 −0.339680 0.940541i \(-0.610319\pi\)
−0.339680 + 0.940541i \(0.610319\pi\)
\(264\) 1.74181e8 0.582621
\(265\) −2.38759e8 −0.788134
\(266\) −1.58187e8 −0.515329
\(267\) 2.82955e8 0.909761
\(268\) −1.60504e8 −0.509348
\(269\) 2.62947e8 0.823637 0.411819 0.911266i \(-0.364894\pi\)
0.411819 + 0.911266i \(0.364894\pi\)
\(270\) −4.24035e7 −0.131108
\(271\) 3.69328e8 1.12725 0.563625 0.826031i \(-0.309406\pi\)
0.563625 + 0.826031i \(0.309406\pi\)
\(272\) 2.95256e8 0.889627
\(273\) −8.27428e7 −0.246128
\(274\) −2.06692e8 −0.607011
\(275\) −4.30044e8 −1.24695
\(276\) 3.93613e7 0.112690
\(277\) −2.94313e8 −0.832014 −0.416007 0.909361i \(-0.636571\pi\)
−0.416007 + 0.909361i \(0.636571\pi\)
\(278\) −4.76104e8 −1.32906
\(279\) 6.79835e6 0.0187408
\(280\) 2.70335e7 0.0735951
\(281\) 4.33874e8 1.16652 0.583258 0.812287i \(-0.301777\pi\)
0.583258 + 0.812287i \(0.301777\pi\)
\(282\) 3.59327e8 0.954153
\(283\) −8.12492e7 −0.213092 −0.106546 0.994308i \(-0.533979\pi\)
−0.106546 + 0.994308i \(0.533979\pi\)
\(284\) −1.17660e8 −0.304800
\(285\) −2.16254e8 −0.553359
\(286\) −1.57547e9 −3.98226
\(287\) −4.25636e7 −0.106280
\(288\) −1.33092e8 −0.328305
\(289\) −2.01835e8 −0.491873
\(290\) −2.94869e8 −0.709965
\(291\) −1.46155e8 −0.347687
\(292\) 1.76636e8 0.415182
\(293\) −6.42476e8 −1.49218 −0.746089 0.665847i \(-0.768071\pi\)
−0.746089 + 0.665847i \(0.768071\pi\)
\(294\) 2.95058e8 0.677160
\(295\) 3.14680e7 0.0713661
\(296\) 4.30775e8 0.965448
\(297\) −1.54890e8 −0.343064
\(298\) −4.95463e8 −1.08456
\(299\) 2.97842e8 0.644372
\(300\) 1.02836e8 0.219898
\(301\) −8.83829e7 −0.186804
\(302\) 6.06679e8 1.26746
\(303\) 2.16196e8 0.446477
\(304\) −1.06888e9 −2.18208
\(305\) −1.01018e8 −0.203869
\(306\) −1.48007e8 −0.295296
\(307\) −6.93624e8 −1.36817 −0.684084 0.729403i \(-0.739798\pi\)
−0.684084 + 0.729403i \(0.739798\pi\)
\(308\) −1.18037e8 −0.230192
\(309\) −6.52235e7 −0.125762
\(310\) 2.00903e7 0.0383019
\(311\) −1.86724e8 −0.351997 −0.175999 0.984390i \(-0.556315\pi\)
−0.175999 + 0.984390i \(0.556315\pi\)
\(312\) −3.15173e8 −0.587501
\(313\) 4.50446e8 0.830304 0.415152 0.909752i \(-0.363728\pi\)
0.415152 + 0.909752i \(0.363728\pi\)
\(314\) 9.84924e7 0.179535
\(315\) −2.40395e7 −0.0433350
\(316\) −8.82527e7 −0.157334
\(317\) −3.97194e8 −0.700318 −0.350159 0.936690i \(-0.613872\pi\)
−0.350159 + 0.936690i \(0.613872\pi\)
\(318\) 5.91574e8 1.03161
\(319\) −1.07709e9 −1.85773
\(320\) 7.70953e6 0.0131523
\(321\) −4.07270e8 −0.687249
\(322\) 6.32976e7 0.105655
\(323\) −7.54822e8 −1.24634
\(324\) 3.70387e7 0.0604990
\(325\) 7.78149e8 1.25739
\(326\) 7.24740e8 1.15856
\(327\) −3.53513e8 −0.559098
\(328\) −1.62128e8 −0.253687
\(329\) 2.03710e8 0.315375
\(330\) −4.57726e8 −0.701144
\(331\) −7.65695e8 −1.16053 −0.580267 0.814426i \(-0.697052\pi\)
−0.580267 + 0.814426i \(0.697052\pi\)
\(332\) 1.54902e8 0.232314
\(333\) −3.83066e8 −0.568484
\(334\) −1.23645e8 −0.181578
\(335\) −3.52857e8 −0.512793
\(336\) −1.18820e8 −0.170885
\(337\) 1.34277e8 0.191116 0.0955581 0.995424i \(-0.469536\pi\)
0.0955581 + 0.995424i \(0.469536\pi\)
\(338\) 1.96849e9 2.77284
\(339\) 2.33450e8 0.325457
\(340\) −1.54195e8 −0.212762
\(341\) 7.33851e7 0.100223
\(342\) 5.35812e8 0.724305
\(343\) 3.44519e8 0.460981
\(344\) −3.36657e8 −0.445896
\(345\) 8.65328e7 0.113452
\(346\) 7.76604e8 1.00794
\(347\) 1.31108e9 1.68451 0.842257 0.539076i \(-0.181227\pi\)
0.842257 + 0.539076i \(0.181227\pi\)
\(348\) 2.57563e8 0.327610
\(349\) −1.04850e9 −1.32032 −0.660160 0.751125i \(-0.729511\pi\)
−0.660160 + 0.751125i \(0.729511\pi\)
\(350\) 1.65373e8 0.206170
\(351\) 2.80267e8 0.345937
\(352\) −1.43666e9 −1.75572
\(353\) −1.06498e9 −1.28863 −0.644317 0.764758i \(-0.722858\pi\)
−0.644317 + 0.764758i \(0.722858\pi\)
\(354\) −7.79682e7 −0.0934127
\(355\) −2.58667e8 −0.306861
\(356\) −7.30388e8 −0.857983
\(357\) −8.39085e7 −0.0976039
\(358\) 1.60516e9 1.84896
\(359\) −8.86207e8 −1.01089 −0.505446 0.862858i \(-0.668672\pi\)
−0.505446 + 0.862858i \(0.668672\pi\)
\(360\) −9.15682e7 −0.103439
\(361\) 1.83872e9 2.05703
\(362\) 1.04771e8 0.116081
\(363\) −1.14581e9 −1.25730
\(364\) 2.13583e8 0.232120
\(365\) 3.88321e8 0.417990
\(366\) 2.50293e8 0.266849
\(367\) −9.92035e8 −1.04760 −0.523801 0.851841i \(-0.675486\pi\)
−0.523801 + 0.851841i \(0.675486\pi\)
\(368\) 4.27707e8 0.447382
\(369\) 1.44172e8 0.149379
\(370\) −1.13203e9 −1.16185
\(371\) 3.35377e8 0.340976
\(372\) −1.75485e7 −0.0176742
\(373\) 9.84405e8 0.982184 0.491092 0.871108i \(-0.336598\pi\)
0.491092 + 0.871108i \(0.336598\pi\)
\(374\) −1.59767e9 −1.57920
\(375\) 5.49274e8 0.537873
\(376\) 7.75948e8 0.752792
\(377\) 1.94895e9 1.87329
\(378\) 5.95627e7 0.0567222
\(379\) −1.03216e9 −0.973888 −0.486944 0.873433i \(-0.661888\pi\)
−0.486944 + 0.873433i \(0.661888\pi\)
\(380\) 5.58214e8 0.521865
\(381\) −7.49604e8 −0.694375
\(382\) −3.72629e8 −0.342022
\(383\) 1.11996e9 1.01861 0.509303 0.860587i \(-0.329903\pi\)
0.509303 + 0.860587i \(0.329903\pi\)
\(384\) −6.50055e8 −0.585856
\(385\) −2.59495e8 −0.231748
\(386\) −1.11841e9 −0.989796
\(387\) 2.99372e8 0.262556
\(388\) 3.77269e8 0.327899
\(389\) −2.19361e9 −1.88945 −0.944725 0.327865i \(-0.893671\pi\)
−0.944725 + 0.327865i \(0.893671\pi\)
\(390\) 8.28239e8 0.707016
\(391\) 3.02038e8 0.255531
\(392\) 6.37162e8 0.534255
\(393\) −5.63970e8 −0.468686
\(394\) 9.41587e8 0.775575
\(395\) −1.94017e8 −0.158398
\(396\) 3.99816e8 0.323539
\(397\) −1.87970e9 −1.50772 −0.753862 0.657033i \(-0.771811\pi\)
−0.753862 + 0.657033i \(0.771811\pi\)
\(398\) −7.70117e8 −0.612302
\(399\) 3.03764e8 0.239404
\(400\) 1.11744e9 0.872998
\(401\) 1.82808e9 1.41576 0.707881 0.706332i \(-0.249651\pi\)
0.707881 + 0.706332i \(0.249651\pi\)
\(402\) 8.74273e8 0.671206
\(403\) −1.32788e8 −0.101062
\(404\) −5.58065e8 −0.421066
\(405\) 8.14269e7 0.0609081
\(406\) 4.14192e8 0.307157
\(407\) −4.13502e9 −3.04016
\(408\) −3.19614e8 −0.232978
\(409\) −2.05110e9 −1.48237 −0.741183 0.671303i \(-0.765735\pi\)
−0.741183 + 0.671303i \(0.765735\pi\)
\(410\) 4.26053e8 0.305295
\(411\) 3.96908e8 0.281996
\(412\) 1.68361e8 0.118604
\(413\) −4.42019e7 −0.0308756
\(414\) −2.14402e8 −0.148501
\(415\) 3.40541e8 0.233884
\(416\) 2.59959e9 1.77043
\(417\) 9.14255e8 0.617435
\(418\) 5.78385e9 3.87347
\(419\) −1.07209e8 −0.0712005 −0.0356002 0.999366i \(-0.511334\pi\)
−0.0356002 + 0.999366i \(0.511334\pi\)
\(420\) 6.20530e7 0.0408686
\(421\) −2.81280e9 −1.83718 −0.918589 0.395213i \(-0.870671\pi\)
−0.918589 + 0.395213i \(0.870671\pi\)
\(422\) −9.08061e8 −0.588195
\(423\) −6.90011e8 −0.443266
\(424\) 1.27747e9 0.813901
\(425\) 7.89113e8 0.498629
\(426\) 6.40899e8 0.401657
\(427\) 1.41897e8 0.0882013
\(428\) 1.05128e9 0.648135
\(429\) 3.02536e9 1.85002
\(430\) 8.84696e8 0.536604
\(431\) 1.62988e9 0.980583 0.490292 0.871558i \(-0.336890\pi\)
0.490292 + 0.871558i \(0.336890\pi\)
\(432\) 4.02469e8 0.240182
\(433\) −8.30649e8 −0.491711 −0.245856 0.969306i \(-0.579069\pi\)
−0.245856 + 0.969306i \(0.579069\pi\)
\(434\) −2.82201e7 −0.0165709
\(435\) 5.66234e8 0.329825
\(436\) 9.12520e8 0.527278
\(437\) −1.09343e9 −0.626768
\(438\) −9.62142e8 −0.547117
\(439\) 2.22387e9 1.25454 0.627270 0.778802i \(-0.284172\pi\)
0.627270 + 0.778802i \(0.284172\pi\)
\(440\) −9.88436e8 −0.553177
\(441\) −5.66595e8 −0.314585
\(442\) 2.89092e9 1.59242
\(443\) 1.30013e9 0.710516 0.355258 0.934768i \(-0.384393\pi\)
0.355258 + 0.934768i \(0.384393\pi\)
\(444\) 9.88804e8 0.536130
\(445\) −1.60570e9 −0.863785
\(446\) −2.84052e9 −1.51609
\(447\) 9.51432e8 0.503850
\(448\) −1.08293e7 −0.00569019
\(449\) 1.41883e9 0.739720 0.369860 0.929088i \(-0.379406\pi\)
0.369860 + 0.929088i \(0.379406\pi\)
\(450\) −5.60153e8 −0.289776
\(451\) 1.55627e9 0.798852
\(452\) −6.02602e8 −0.306935
\(453\) −1.16500e9 −0.588818
\(454\) −3.88249e9 −1.94722
\(455\) 4.69547e8 0.233689
\(456\) 1.15706e9 0.571451
\(457\) −2.30832e9 −1.13133 −0.565664 0.824636i \(-0.691380\pi\)
−0.565664 + 0.824636i \(0.691380\pi\)
\(458\) −1.97076e9 −0.958528
\(459\) 2.84216e8 0.137184
\(460\) −2.23366e8 −0.106995
\(461\) −2.65874e8 −0.126393 −0.0631965 0.998001i \(-0.520129\pi\)
−0.0631965 + 0.998001i \(0.520129\pi\)
\(462\) 6.42951e8 0.303341
\(463\) −1.76799e9 −0.827840 −0.413920 0.910313i \(-0.635841\pi\)
−0.413920 + 0.910313i \(0.635841\pi\)
\(464\) 2.79873e9 1.30061
\(465\) −3.85792e7 −0.0177937
\(466\) 1.21946e9 0.558236
\(467\) 3.74007e9 1.69930 0.849652 0.527344i \(-0.176812\pi\)
0.849652 + 0.527344i \(0.176812\pi\)
\(468\) −7.23452e8 −0.326249
\(469\) 4.95645e8 0.221853
\(470\) −2.03910e9 −0.905933
\(471\) −1.89134e8 −0.0834057
\(472\) −1.68368e8 −0.0736993
\(473\) 3.23158e9 1.40411
\(474\) 4.80716e8 0.207331
\(475\) −2.85673e9 −1.22304
\(476\) 2.16592e8 0.0920490
\(477\) −1.13599e9 −0.479249
\(478\) 3.13052e9 1.31105
\(479\) −4.49040e8 −0.186686 −0.0933428 0.995634i \(-0.529755\pi\)
−0.0933428 + 0.995634i \(0.529755\pi\)
\(480\) 7.55266e8 0.311713
\(481\) 7.48216e9 3.06563
\(482\) 6.01502e9 2.44665
\(483\) −1.21549e8 −0.0490838
\(484\) 2.95767e9 1.18574
\(485\) 8.29399e8 0.330117
\(486\) −2.01751e8 −0.0797241
\(487\) −3.09706e9 −1.21506 −0.607531 0.794296i \(-0.707840\pi\)
−0.607531 + 0.794296i \(0.707840\pi\)
\(488\) 5.40495e8 0.210534
\(489\) −1.39171e9 −0.538229
\(490\) −1.67439e9 −0.642939
\(491\) −2.15480e9 −0.821527 −0.410763 0.911742i \(-0.634738\pi\)
−0.410763 + 0.911742i \(0.634738\pi\)
\(492\) −3.72150e8 −0.140877
\(493\) 1.97641e9 0.742869
\(494\) −1.04657e10 −3.90591
\(495\) 8.78965e8 0.325727
\(496\) −1.90686e8 −0.0701669
\(497\) 3.63340e8 0.132759
\(498\) −8.43759e8 −0.306137
\(499\) 5.11219e9 1.84185 0.920926 0.389738i \(-0.127434\pi\)
0.920926 + 0.389738i \(0.127434\pi\)
\(500\) −1.41784e9 −0.507261
\(501\) 2.37433e8 0.0843547
\(502\) 6.74462e8 0.237955
\(503\) −3.11720e9 −1.09214 −0.546069 0.837740i \(-0.683876\pi\)
−0.546069 + 0.837740i \(0.683876\pi\)
\(504\) 1.28623e8 0.0447518
\(505\) −1.22687e9 −0.423913
\(506\) −2.31437e9 −0.794158
\(507\) −3.78006e9 −1.28816
\(508\) 1.93495e9 0.654856
\(509\) −3.33182e9 −1.11988 −0.559938 0.828535i \(-0.689175\pi\)
−0.559938 + 0.828535i \(0.689175\pi\)
\(510\) 8.39908e8 0.280373
\(511\) −5.45460e8 −0.180838
\(512\) 1.58742e9 0.522695
\(513\) −1.02891e9 −0.336487
\(514\) 2.54889e9 0.827905
\(515\) 3.70129e8 0.119406
\(516\) −7.72766e8 −0.247613
\(517\) −7.44834e9 −2.37052
\(518\) 1.59011e9 0.502660
\(519\) −1.49130e9 −0.468252
\(520\) 1.78854e9 0.557811
\(521\) −1.20322e8 −0.0372745 −0.0186373 0.999826i \(-0.505933\pi\)
−0.0186373 + 0.999826i \(0.505933\pi\)
\(522\) −1.40296e9 −0.431716
\(523\) 1.56158e9 0.477318 0.238659 0.971103i \(-0.423292\pi\)
0.238659 + 0.971103i \(0.423292\pi\)
\(524\) 1.45577e9 0.442012
\(525\) −3.17563e8 −0.0957796
\(526\) −2.81801e9 −0.844291
\(527\) −1.34658e8 −0.0400771
\(528\) 4.34447e9 1.28445
\(529\) −2.96729e9 −0.871497
\(530\) −3.35705e9 −0.979473
\(531\) 1.49721e8 0.0433963
\(532\) −7.84103e8 −0.225779
\(533\) −2.81601e9 −0.805543
\(534\) 3.97845e9 1.13063
\(535\) 2.31116e9 0.652518
\(536\) 1.88795e9 0.529558
\(537\) −3.08236e9 −0.858961
\(538\) 3.69715e9 1.02360
\(539\) −6.11614e9 −1.68235
\(540\) −2.10187e8 −0.0574416
\(541\) 2.94601e9 0.799914 0.399957 0.916534i \(-0.369025\pi\)
0.399957 + 0.916534i \(0.369025\pi\)
\(542\) 5.19290e9 1.40092
\(543\) −2.01190e8 −0.0539270
\(544\) 2.63621e9 0.702077
\(545\) 2.00611e9 0.530843
\(546\) −1.16340e9 −0.305882
\(547\) 5.07000e9 1.32450 0.662251 0.749282i \(-0.269601\pi\)
0.662251 + 0.749282i \(0.269601\pi\)
\(548\) −1.02453e9 −0.265947
\(549\) −4.80635e8 −0.123969
\(550\) −6.04659e9 −1.54968
\(551\) −7.15495e9 −1.82212
\(552\) −4.62991e8 −0.117162
\(553\) 2.72529e8 0.0685290
\(554\) −4.13816e9 −1.03401
\(555\) 2.17381e9 0.539755
\(556\) −2.35996e9 −0.582295
\(557\) 7.03628e9 1.72524 0.862621 0.505851i \(-0.168821\pi\)
0.862621 + 0.505851i \(0.168821\pi\)
\(558\) 9.55876e7 0.0232907
\(559\) −5.84742e9 −1.41587
\(560\) 6.74279e8 0.162249
\(561\) 3.06798e9 0.733639
\(562\) 6.10044e9 1.44972
\(563\) −4.94078e9 −1.16685 −0.583426 0.812166i \(-0.698288\pi\)
−0.583426 + 0.812166i \(0.698288\pi\)
\(564\) 1.78112e9 0.418038
\(565\) −1.32478e9 −0.309010
\(566\) −1.14240e9 −0.264825
\(567\) −1.14377e8 −0.0263511
\(568\) 1.38399e9 0.316893
\(569\) 2.66776e9 0.607090 0.303545 0.952817i \(-0.401830\pi\)
0.303545 + 0.952817i \(0.401830\pi\)
\(570\) −3.04062e9 −0.687701
\(571\) 6.77696e9 1.52338 0.761690 0.647941i \(-0.224370\pi\)
0.761690 + 0.647941i \(0.224370\pi\)
\(572\) −7.80933e9 −1.74473
\(573\) 7.15554e8 0.158892
\(574\) −5.98461e8 −0.132082
\(575\) 1.14310e9 0.250754
\(576\) 3.66811e7 0.00799768
\(577\) −5.53458e9 −1.19941 −0.599707 0.800219i \(-0.704716\pi\)
−0.599707 + 0.800219i \(0.704716\pi\)
\(578\) −2.83787e9 −0.611288
\(579\) 2.14767e9 0.459825
\(580\) −1.46162e9 −0.311053
\(581\) −4.78346e8 −0.101187
\(582\) −2.05500e9 −0.432098
\(583\) −1.22625e10 −2.56294
\(584\) −2.07770e9 −0.431655
\(585\) −1.59046e9 −0.328455
\(586\) −9.03347e9 −1.85444
\(587\) 5.22919e9 1.06709 0.533545 0.845771i \(-0.320859\pi\)
0.533545 + 0.845771i \(0.320859\pi\)
\(588\) 1.46255e9 0.296681
\(589\) 4.87488e8 0.0983015
\(590\) 4.42452e8 0.0886920
\(591\) −1.80812e9 −0.360305
\(592\) 1.07445e10 2.12844
\(593\) 8.33930e8 0.164224 0.0821122 0.996623i \(-0.473833\pi\)
0.0821122 + 0.996623i \(0.473833\pi\)
\(594\) −2.17781e9 −0.426352
\(595\) 4.76162e8 0.0926714
\(596\) −2.45592e9 −0.475174
\(597\) 1.47884e9 0.284454
\(598\) 4.18777e9 0.800809
\(599\) 3.80759e9 0.723863 0.361931 0.932205i \(-0.382117\pi\)
0.361931 + 0.932205i \(0.382117\pi\)
\(600\) −1.20962e9 −0.228623
\(601\) 8.81584e9 1.65654 0.828272 0.560326i \(-0.189324\pi\)
0.828272 + 0.560326i \(0.189324\pi\)
\(602\) −1.24270e9 −0.232155
\(603\) −1.67886e9 −0.311819
\(604\) 3.00720e9 0.555307
\(605\) 6.50222e9 1.19376
\(606\) 3.03980e9 0.554870
\(607\) 3.12869e9 0.567809 0.283904 0.958853i \(-0.408370\pi\)
0.283904 + 0.958853i \(0.408370\pi\)
\(608\) −9.54357e9 −1.72206
\(609\) −7.95368e8 −0.142695
\(610\) −1.42036e9 −0.253363
\(611\) 1.34775e10 2.39037
\(612\) −7.33645e8 −0.129377
\(613\) 6.07990e9 1.06607 0.533034 0.846094i \(-0.321052\pi\)
0.533034 + 0.846094i \(0.321052\pi\)
\(614\) −9.75263e9 −1.70033
\(615\) −8.18143e8 −0.141829
\(616\) 1.38842e9 0.239325
\(617\) −1.45081e9 −0.248664 −0.124332 0.992241i \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(618\) −9.17069e8 −0.156294
\(619\) −3.93245e9 −0.666416 −0.333208 0.942853i \(-0.608131\pi\)
−0.333208 + 0.942853i \(0.608131\pi\)
\(620\) 9.95841e7 0.0167810
\(621\) 4.11714e8 0.0689882
\(622\) −2.62542e9 −0.437454
\(623\) 2.25547e9 0.373706
\(624\) −7.86116e9 −1.29521
\(625\) 1.15244e9 0.188815
\(626\) 6.33345e9 1.03188
\(627\) −1.11066e10 −1.79948
\(628\) 4.88209e8 0.0786588
\(629\) 7.58757e9 1.21570
\(630\) −3.38005e8 −0.0538556
\(631\) 9.82173e9 1.55627 0.778135 0.628097i \(-0.216166\pi\)
0.778135 + 0.628097i \(0.216166\pi\)
\(632\) 1.03808e9 0.163577
\(633\) 1.74374e9 0.273255
\(634\) −5.58470e9 −0.870338
\(635\) 4.25383e9 0.659284
\(636\) 2.93233e9 0.451973
\(637\) 1.10669e10 1.69644
\(638\) −1.51443e10 −2.30875
\(639\) −1.23071e9 −0.186596
\(640\) 3.68892e9 0.556249
\(641\) −7.96630e9 −1.19469 −0.597343 0.801986i \(-0.703777\pi\)
−0.597343 + 0.801986i \(0.703777\pi\)
\(642\) −5.72637e9 −0.854097
\(643\) 7.54836e9 1.11973 0.559866 0.828583i \(-0.310852\pi\)
0.559866 + 0.828583i \(0.310852\pi\)
\(644\) 3.13755e8 0.0462903
\(645\) −1.69887e9 −0.249288
\(646\) −1.06131e10 −1.54892
\(647\) 1.12209e10 1.62878 0.814388 0.580321i \(-0.197073\pi\)
0.814388 + 0.580321i \(0.197073\pi\)
\(648\) −4.35672e8 −0.0628995
\(649\) 1.61617e9 0.232076
\(650\) 1.09411e10 1.56266
\(651\) 5.41907e7 0.00769824
\(652\) 3.59240e9 0.507596
\(653\) −6.06010e9 −0.851694 −0.425847 0.904795i \(-0.640024\pi\)
−0.425847 + 0.904795i \(0.640024\pi\)
\(654\) −4.97053e9 −0.694834
\(655\) 3.20041e9 0.445001
\(656\) −4.04385e9 −0.559282
\(657\) 1.84759e9 0.254171
\(658\) 2.86425e9 0.391941
\(659\) −1.39437e10 −1.89792 −0.948960 0.315398i \(-0.897862\pi\)
−0.948960 + 0.315398i \(0.897862\pi\)
\(660\) −2.26887e9 −0.307189
\(661\) −8.46218e9 −1.13966 −0.569832 0.821761i \(-0.692992\pi\)
−0.569832 + 0.821761i \(0.692992\pi\)
\(662\) −1.07660e10 −1.44228
\(663\) −5.55140e9 −0.739784
\(664\) −1.82205e9 −0.241531
\(665\) −1.72379e9 −0.227305
\(666\) −5.38606e9 −0.706498
\(667\) 2.86302e9 0.373580
\(668\) −6.12884e8 −0.0795538
\(669\) 5.45460e9 0.704323
\(670\) −4.96131e9 −0.637286
\(671\) −5.18823e9 −0.662965
\(672\) −1.06089e9 −0.134859
\(673\) 2.59986e9 0.328773 0.164387 0.986396i \(-0.447436\pi\)
0.164387 + 0.986396i \(0.447436\pi\)
\(674\) 1.88799e9 0.237514
\(675\) 1.07565e9 0.134620
\(676\) 9.75744e9 1.21485
\(677\) 2.24286e8 0.0277807 0.0138903 0.999904i \(-0.495578\pi\)
0.0138903 + 0.999904i \(0.495578\pi\)
\(678\) 3.28240e9 0.404471
\(679\) −1.16503e9 −0.142821
\(680\) 1.81374e9 0.221204
\(681\) 7.45550e9 0.904611
\(682\) 1.03182e9 0.124555
\(683\) −1.29921e9 −0.156029 −0.0780146 0.996952i \(-0.524858\pi\)
−0.0780146 + 0.996952i \(0.524858\pi\)
\(684\) 2.65593e9 0.317336
\(685\) −2.25236e9 −0.267745
\(686\) 4.84407e9 0.572896
\(687\) 3.78443e9 0.445299
\(688\) −8.39702e9 −0.983027
\(689\) 2.21886e10 2.58441
\(690\) 1.21669e9 0.140996
\(691\) −1.09890e10 −1.26703 −0.633514 0.773732i \(-0.718388\pi\)
−0.633514 + 0.773732i \(0.718388\pi\)
\(692\) 3.84949e9 0.441602
\(693\) −1.23465e9 −0.140922
\(694\) 1.84342e10 2.09347
\(695\) −5.18820e9 −0.586232
\(696\) −3.02962e9 −0.340608
\(697\) −2.85568e9 −0.319445
\(698\) −1.47423e10 −1.64086
\(699\) −2.34172e9 −0.259337
\(700\) 8.19724e8 0.0903284
\(701\) 1.12676e10 1.23543 0.617715 0.786402i \(-0.288059\pi\)
0.617715 + 0.786402i \(0.288059\pi\)
\(702\) 3.94067e9 0.429923
\(703\) −2.74684e10 −2.98188
\(704\) 3.95956e8 0.0427703
\(705\) 3.91566e9 0.420865
\(706\) −1.49740e10 −1.60148
\(707\) 1.72333e9 0.183401
\(708\) −3.86474e8 −0.0409265
\(709\) −8.10979e9 −0.854571 −0.427285 0.904117i \(-0.640530\pi\)
−0.427285 + 0.904117i \(0.640530\pi\)
\(710\) −3.63696e9 −0.381359
\(711\) −9.23113e8 −0.0963188
\(712\) 8.59127e9 0.892026
\(713\) −1.95065e8 −0.0201543
\(714\) −1.17979e9 −0.121300
\(715\) −1.71682e10 −1.75653
\(716\) 7.95647e9 0.810075
\(717\) −6.01150e9 −0.609069
\(718\) −1.24604e10 −1.25631
\(719\) 1.37914e10 1.38375 0.691873 0.722019i \(-0.256786\pi\)
0.691873 + 0.722019i \(0.256786\pi\)
\(720\) −2.28393e9 −0.228044
\(721\) −5.19907e8 −0.0516597
\(722\) 2.58532e10 2.55643
\(723\) −1.15506e10 −1.13663
\(724\) 5.19330e8 0.0508579
\(725\) 7.47999e9 0.728984
\(726\) −1.61105e10 −1.56254
\(727\) −8.25616e9 −0.796907 −0.398454 0.917189i \(-0.630453\pi\)
−0.398454 + 0.917189i \(0.630453\pi\)
\(728\) −2.51230e9 −0.241330
\(729\) 3.87420e8 0.0370370
\(730\) 5.45994e9 0.519467
\(731\) −5.92981e9 −0.561474
\(732\) 1.24066e9 0.116913
\(733\) 1.14387e10 1.07279 0.536395 0.843967i \(-0.319786\pi\)
0.536395 + 0.843967i \(0.319786\pi\)
\(734\) −1.39484e10 −1.30193
\(735\) 3.21530e9 0.298687
\(736\) 3.81881e9 0.353065
\(737\) −1.81225e10 −1.66756
\(738\) 2.02711e9 0.185644
\(739\) 1.48024e10 1.34920 0.674598 0.738185i \(-0.264317\pi\)
0.674598 + 0.738185i \(0.264317\pi\)
\(740\) −5.61125e9 −0.509036
\(741\) 2.00971e10 1.81455
\(742\) 4.71553e9 0.423757
\(743\) 1.21140e10 1.08349 0.541746 0.840542i \(-0.317764\pi\)
0.541746 + 0.840542i \(0.317764\pi\)
\(744\) 2.06416e8 0.0183755
\(745\) −5.39917e9 −0.478387
\(746\) 1.38411e10 1.22063
\(747\) 1.62026e9 0.142221
\(748\) −7.91935e9 −0.691885
\(749\) −3.24641e9 −0.282304
\(750\) 7.72302e9 0.668456
\(751\) −3.71223e9 −0.319812 −0.159906 0.987132i \(-0.551119\pi\)
−0.159906 + 0.987132i \(0.551119\pi\)
\(752\) 1.93540e10 1.65962
\(753\) −1.29516e9 −0.110546
\(754\) 2.74030e10 2.32808
\(755\) 6.61111e9 0.559062
\(756\) 2.95241e8 0.0248514
\(757\) 2.68089e9 0.224617 0.112309 0.993673i \(-0.464175\pi\)
0.112309 + 0.993673i \(0.464175\pi\)
\(758\) −1.45126e10 −1.21032
\(759\) 4.44426e9 0.368938
\(760\) −6.56606e9 −0.542572
\(761\) −8.48234e9 −0.697701 −0.348850 0.937178i \(-0.613428\pi\)
−0.348850 + 0.937178i \(0.613428\pi\)
\(762\) −1.05397e10 −0.862953
\(763\) −2.81791e9 −0.229663
\(764\) −1.84705e9 −0.149849
\(765\) −1.61286e9 −0.130251
\(766\) 1.57471e10 1.26590
\(767\) −2.92440e9 −0.234020
\(768\) −8.96614e9 −0.714235
\(769\) 4.34767e9 0.344758 0.172379 0.985031i \(-0.444855\pi\)
0.172379 + 0.985031i \(0.444855\pi\)
\(770\) −3.64861e9 −0.288011
\(771\) −4.89461e9 −0.384616
\(772\) −5.54376e9 −0.433655
\(773\) −1.06536e10 −0.829601 −0.414800 0.909912i \(-0.636149\pi\)
−0.414800 + 0.909912i \(0.636149\pi\)
\(774\) 4.20929e9 0.326299
\(775\) −5.09633e8 −0.0393280
\(776\) −4.43767e9 −0.340910
\(777\) −3.05348e9 −0.233518
\(778\) −3.08430e10 −2.34816
\(779\) 1.03381e10 0.783537
\(780\) 4.10543e9 0.309762
\(781\) −1.32849e10 −0.997886
\(782\) 4.24678e9 0.317567
\(783\) 2.69408e9 0.200560
\(784\) 1.58923e10 1.17783
\(785\) 1.07329e9 0.0791907
\(786\) −7.92965e9 −0.582472
\(787\) 2.49085e9 0.182153 0.0910763 0.995844i \(-0.470969\pi\)
0.0910763 + 0.995844i \(0.470969\pi\)
\(788\) 4.66728e9 0.339799
\(789\) 5.41138e9 0.392228
\(790\) −2.72796e9 −0.196853
\(791\) 1.86086e9 0.133689
\(792\) −4.70288e9 −0.336376
\(793\) 9.38791e9 0.668517
\(794\) −2.64293e10 −1.87376
\(795\) 6.44650e9 0.455029
\(796\) −3.81733e9 −0.268265
\(797\) 6.71594e9 0.469897 0.234949 0.972008i \(-0.424508\pi\)
0.234949 + 0.972008i \(0.424508\pi\)
\(798\) 4.27104e9 0.297525
\(799\) 1.36674e10 0.947920
\(800\) 9.97712e9 0.688954
\(801\) −7.63977e9 −0.525251
\(802\) 2.57036e10 1.75947
\(803\) 1.99439e10 1.35927
\(804\) 4.33362e9 0.294072
\(805\) 6.89766e8 0.0466033
\(806\) −1.86705e9 −0.125598
\(807\) −7.09958e9 −0.475527
\(808\) 6.56430e9 0.437773
\(809\) −5.26318e9 −0.349485 −0.174743 0.984614i \(-0.555909\pi\)
−0.174743 + 0.984614i \(0.555909\pi\)
\(810\) 1.14489e9 0.0756951
\(811\) 1.05418e10 0.693971 0.346985 0.937870i \(-0.387205\pi\)
0.346985 + 0.937870i \(0.387205\pi\)
\(812\) 2.05308e9 0.134573
\(813\) −9.97186e9 −0.650818
\(814\) −5.81400e10 −3.77824
\(815\) 7.89763e9 0.511029
\(816\) −7.97192e9 −0.513626
\(817\) 2.14670e10 1.37719
\(818\) −2.88393e10 −1.84225
\(819\) 2.23405e9 0.142102
\(820\) 2.11187e9 0.133757
\(821\) 2.59089e10 1.63399 0.816994 0.576646i \(-0.195639\pi\)
0.816994 + 0.576646i \(0.195639\pi\)
\(822\) 5.58068e9 0.350458
\(823\) −8.31491e9 −0.519946 −0.259973 0.965616i \(-0.583714\pi\)
−0.259973 + 0.965616i \(0.583714\pi\)
\(824\) −1.98036e9 −0.123310
\(825\) 1.16112e10 0.719926
\(826\) −6.21497e8 −0.0383715
\(827\) −6.36168e9 −0.391113 −0.195557 0.980692i \(-0.562651\pi\)
−0.195557 + 0.980692i \(0.562651\pi\)
\(828\) −1.06275e9 −0.0650619
\(829\) −2.31191e10 −1.40938 −0.704692 0.709513i \(-0.748915\pi\)
−0.704692 + 0.709513i \(0.748915\pi\)
\(830\) 4.78815e9 0.290666
\(831\) 7.94646e9 0.480364
\(832\) −7.16467e8 −0.0431285
\(833\) 1.12228e10 0.672737
\(834\) 1.28548e10 0.767333
\(835\) −1.34738e9 −0.0800918
\(836\) 2.86695e10 1.69706
\(837\) −1.83556e8 −0.0108200
\(838\) −1.50740e9 −0.0884862
\(839\) 1.96736e10 1.15005 0.575026 0.818135i \(-0.304992\pi\)
0.575026 + 0.818135i \(0.304992\pi\)
\(840\) −7.29904e8 −0.0424902
\(841\) 1.48448e9 0.0860574
\(842\) −3.95491e10 −2.28320
\(843\) −1.17146e10 −0.673489
\(844\) −4.50110e9 −0.257703
\(845\) 2.14510e10 1.22306
\(846\) −9.70183e9 −0.550880
\(847\) −9.13343e9 −0.516466
\(848\) 3.18632e10 1.79434
\(849\) 2.19373e9 0.123029
\(850\) 1.10952e10 0.619684
\(851\) 1.09913e10 0.611359
\(852\) 3.17682e9 0.175976
\(853\) −2.05583e10 −1.13413 −0.567067 0.823671i \(-0.691922\pi\)
−0.567067 + 0.823671i \(0.691922\pi\)
\(854\) 1.99513e9 0.109614
\(855\) 5.83886e9 0.319482
\(856\) −1.23658e10 −0.673852
\(857\) −1.22990e10 −0.667476 −0.333738 0.942666i \(-0.608310\pi\)
−0.333738 + 0.942666i \(0.608310\pi\)
\(858\) 4.25377e10 2.29916
\(859\) 6.11395e9 0.329114 0.164557 0.986368i \(-0.447381\pi\)
0.164557 + 0.986368i \(0.447381\pi\)
\(860\) 4.38528e9 0.235100
\(861\) 1.14922e9 0.0613608
\(862\) 2.29167e10 1.21864
\(863\) −5.85317e9 −0.309994 −0.154997 0.987915i \(-0.549537\pi\)
−0.154997 + 0.987915i \(0.549537\pi\)
\(864\) 3.59348e9 0.189547
\(865\) 8.46281e9 0.444589
\(866\) −1.16793e10 −0.611087
\(867\) 5.44953e9 0.283983
\(868\) −1.39882e8 −0.00726011
\(869\) −9.96458e9 −0.515098
\(870\) 7.96148e9 0.409898
\(871\) 3.27919e10 1.68153
\(872\) −1.07336e10 −0.548199
\(873\) 3.94619e9 0.200737
\(874\) −1.53741e10 −0.778932
\(875\) 4.37835e9 0.220944
\(876\) −4.76917e9 −0.239705
\(877\) −5.35250e9 −0.267953 −0.133976 0.990985i \(-0.542775\pi\)
−0.133976 + 0.990985i \(0.542775\pi\)
\(878\) 3.12686e10 1.55911
\(879\) 1.73469e10 0.861509
\(880\) −2.46539e10 −1.21954
\(881\) −9.28115e9 −0.457284 −0.228642 0.973511i \(-0.573429\pi\)
−0.228642 + 0.973511i \(0.573429\pi\)
\(882\) −7.96656e9 −0.390959
\(883\) −1.19702e10 −0.585114 −0.292557 0.956248i \(-0.594506\pi\)
−0.292557 + 0.956248i \(0.594506\pi\)
\(884\) 1.43298e10 0.697680
\(885\) −8.49635e8 −0.0412032
\(886\) 1.82804e10 0.883012
\(887\) 2.48052e10 1.19347 0.596734 0.802439i \(-0.296465\pi\)
0.596734 + 0.802439i \(0.296465\pi\)
\(888\) −1.16309e10 −0.557402
\(889\) −5.97521e9 −0.285231
\(890\) −2.25769e10 −1.07349
\(891\) 4.18202e9 0.198068
\(892\) −1.40799e10 −0.664237
\(893\) −4.94784e10 −2.32507
\(894\) 1.33775e10 0.626173
\(895\) 1.74917e10 0.815552
\(896\) −5.18169e9 −0.240654
\(897\) −8.04173e9 −0.372028
\(898\) 1.99493e10 0.919306
\(899\) −1.27643e9 −0.0585918
\(900\) −2.77658e9 −0.126958
\(901\) 2.25012e10 1.02487
\(902\) 2.18818e10 0.992794
\(903\) 2.38634e9 0.107851
\(904\) 7.08816e9 0.319113
\(905\) 1.14171e9 0.0512018
\(906\) −1.63803e10 −0.731769
\(907\) 9.90452e9 0.440766 0.220383 0.975413i \(-0.429269\pi\)
0.220383 + 0.975413i \(0.429269\pi\)
\(908\) −1.92448e10 −0.853127
\(909\) −5.83730e9 −0.257773
\(910\) 6.60202e9 0.290424
\(911\) −8.15910e8 −0.0357543 −0.0178771 0.999840i \(-0.505691\pi\)
−0.0178771 + 0.999840i \(0.505691\pi\)
\(912\) 2.88598e10 1.25983
\(913\) 1.74899e10 0.760573
\(914\) −3.24559e10 −1.40599
\(915\) 2.72750e9 0.117704
\(916\) −9.76871e9 −0.419955
\(917\) −4.49549e9 −0.192524
\(918\) 3.99619e9 0.170489
\(919\) −3.65375e9 −0.155287 −0.0776434 0.996981i \(-0.524740\pi\)
−0.0776434 + 0.996981i \(0.524740\pi\)
\(920\) 2.62737e9 0.111241
\(921\) 1.87278e10 0.789912
\(922\) −3.73830e9 −0.157078
\(923\) 2.40386e10 1.00624
\(924\) 3.18699e9 0.132901
\(925\) 2.87162e10 1.19297
\(926\) −2.48587e10 −1.02882
\(927\) 1.76103e9 0.0726087
\(928\) 2.49886e10 1.02642
\(929\) 3.46180e10 1.41660 0.708299 0.705913i \(-0.249463\pi\)
0.708299 + 0.705913i \(0.249463\pi\)
\(930\) −5.42438e8 −0.0221136
\(931\) −4.06287e10 −1.65010
\(932\) 6.04465e9 0.244577
\(933\) 5.04155e9 0.203226
\(934\) 5.25869e10 2.11185
\(935\) −1.74101e10 −0.696564
\(936\) 8.50968e9 0.339194
\(937\) −1.64978e10 −0.655143 −0.327572 0.944826i \(-0.606230\pi\)
−0.327572 + 0.944826i \(0.606230\pi\)
\(938\) 6.96897e9 0.275714
\(939\) −1.21620e10 −0.479376
\(940\) −1.01075e10 −0.396912
\(941\) −2.07806e10 −0.813009 −0.406504 0.913649i \(-0.633252\pi\)
−0.406504 + 0.913649i \(0.633252\pi\)
\(942\) −2.65930e9 −0.103655
\(943\) −4.13673e9 −0.160645
\(944\) −4.19950e9 −0.162478
\(945\) 6.49066e8 0.0250195
\(946\) 4.54373e10 1.74499
\(947\) −3.39662e10 −1.29964 −0.649818 0.760090i \(-0.725155\pi\)
−0.649818 + 0.760090i \(0.725155\pi\)
\(948\) 2.38282e9 0.0908370
\(949\) −3.60877e10 −1.37065
\(950\) −4.01668e10 −1.51997
\(951\) 1.07242e10 0.404329
\(952\) −2.54769e9 −0.0957012
\(953\) 1.25066e9 0.0468074 0.0234037 0.999726i \(-0.492550\pi\)
0.0234037 + 0.999726i \(0.492550\pi\)
\(954\) −1.59725e10 −0.595599
\(955\) −4.06061e9 −0.150862
\(956\) 1.55174e10 0.574404
\(957\) 2.90814e10 1.07256
\(958\) −6.31369e9 −0.232008
\(959\) 3.16381e9 0.115837
\(960\) −2.08157e8 −0.00759351
\(961\) −2.74256e10 −0.996839
\(962\) 1.05202e11 3.80989
\(963\) 1.09963e10 0.396784
\(964\) 2.98154e10 1.07194
\(965\) −1.21876e10 −0.436587
\(966\) −1.70903e9 −0.0610001
\(967\) 1.99659e10 0.710063 0.355032 0.934854i \(-0.384470\pi\)
0.355032 + 0.934854i \(0.384470\pi\)
\(968\) −3.47899e10 −1.23279
\(969\) 2.03802e10 0.719574
\(970\) 1.16617e10 0.410261
\(971\) −5.02413e10 −1.76114 −0.880569 0.473919i \(-0.842839\pi\)
−0.880569 + 0.473919i \(0.842839\pi\)
\(972\) −1.00005e9 −0.0349291
\(973\) 7.28767e9 0.253626
\(974\) −4.35459e10 −1.51005
\(975\) −2.10100e10 −0.725956
\(976\) 1.34812e10 0.464146
\(977\) −5.76202e10 −1.97671 −0.988357 0.152155i \(-0.951379\pi\)
−0.988357 + 0.152155i \(0.951379\pi\)
\(978\) −1.95680e10 −0.668898
\(979\) −8.24678e10 −2.80896
\(980\) −8.29964e9 −0.281688
\(981\) 9.54485e9 0.322795
\(982\) −3.02974e10 −1.02097
\(983\) 1.60520e10 0.539003 0.269502 0.963000i \(-0.413141\pi\)
0.269502 + 0.963000i \(0.413141\pi\)
\(984\) 4.37745e9 0.146466
\(985\) 1.02607e10 0.342096
\(986\) 2.77891e10 0.923220
\(987\) −5.50018e9 −0.182082
\(988\) −5.18764e10 −1.71128
\(989\) −8.58989e9 −0.282358
\(990\) 1.23586e10 0.404805
\(991\) −8.99460e9 −0.293578 −0.146789 0.989168i \(-0.546894\pi\)
−0.146789 + 0.989168i \(0.546894\pi\)
\(992\) −1.70255e9 −0.0553744
\(993\) 2.06738e10 0.670034
\(994\) 5.10870e9 0.164990
\(995\) −8.39212e9 −0.270079
\(996\) −4.18236e9 −0.134126
\(997\) 1.62638e9 0.0519744 0.0259872 0.999662i \(-0.491727\pi\)
0.0259872 + 0.999662i \(0.491727\pi\)
\(998\) 7.18794e10 2.28901
\(999\) 1.03428e10 0.328214
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.a.1.13 16
3.2 odd 2 531.8.a.b.1.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.13 16 1.1 even 1 trivial
531.8.a.b.1.4 16 3.2 odd 2