Properties

Label 138.8.a.h.1.1
Level $138$
Weight $8$
Character 138.1
Self dual yes
Analytic conductor $43.109$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1091335168\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(33.2734\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -427.795 q^{5} +216.000 q^{6} +345.429 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -427.795 q^{5} +216.000 q^{6} +345.429 q^{7} +512.000 q^{8} +729.000 q^{9} -3422.36 q^{10} +2181.54 q^{11} +1728.00 q^{12} -2610.39 q^{13} +2763.43 q^{14} -11550.5 q^{15} +4096.00 q^{16} +37979.0 q^{17} +5832.00 q^{18} -26348.1 q^{19} -27378.9 q^{20} +9326.57 q^{21} +17452.3 q^{22} -12167.0 q^{23} +13824.0 q^{24} +104884. q^{25} -20883.1 q^{26} +19683.0 q^{27} +22107.4 q^{28} +250851. q^{29} -92403.8 q^{30} +179145. q^{31} +32768.0 q^{32} +58901.7 q^{33} +303832. q^{34} -147773. q^{35} +46656.0 q^{36} +434281. q^{37} -210785. q^{38} -70480.6 q^{39} -219031. q^{40} -200429. q^{41} +74612.6 q^{42} +841346. q^{43} +139619. q^{44} -311863. q^{45} -97336.0 q^{46} -200025. q^{47} +110592. q^{48} -704222. q^{49} +839072. q^{50} +1.02543e6 q^{51} -167065. q^{52} -1.62417e6 q^{53} +157464. q^{54} -933254. q^{55} +176859. q^{56} -711399. q^{57} +2.00681e6 q^{58} -2.18662e6 q^{59} -739231. q^{60} +2.54771e6 q^{61} +1.43316e6 q^{62} +251817. q^{63} +262144. q^{64} +1.11671e6 q^{65} +471213. q^{66} +4.66306e6 q^{67} +2.43066e6 q^{68} -328509. q^{69} -1.18218e6 q^{70} +4.02892e6 q^{71} +373248. q^{72} -2.58030e6 q^{73} +3.47425e6 q^{74} +2.83187e6 q^{75} -1.68628e6 q^{76} +753568. q^{77} -563845. q^{78} +1.22032e6 q^{79} -1.75225e6 q^{80} +531441. q^{81} -1.60343e6 q^{82} -2.16585e6 q^{83} +596901. q^{84} -1.62473e7 q^{85} +6.73077e6 q^{86} +6.77298e6 q^{87} +1.11695e6 q^{88} +1.19998e6 q^{89} -2.49490e6 q^{90} -901704. q^{91} -778688. q^{92} +4.83690e6 q^{93} -1.60020e6 q^{94} +1.12716e7 q^{95} +884736. q^{96} +8.97814e6 q^{97} -5.63378e6 q^{98} +1.59035e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −427.795 −1.53053 −0.765264 0.643717i \(-0.777391\pi\)
−0.765264 + 0.643717i \(0.777391\pi\)
\(6\) 216.000 0.408248
\(7\) 345.429 0.380641 0.190320 0.981722i \(-0.439047\pi\)
0.190320 + 0.981722i \(0.439047\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −3422.36 −1.08225
\(11\) 2181.54 0.494185 0.247092 0.968992i \(-0.420525\pi\)
0.247092 + 0.968992i \(0.420525\pi\)
\(12\) 1728.00 0.288675
\(13\) −2610.39 −0.329537 −0.164768 0.986332i \(-0.552688\pi\)
−0.164768 + 0.986332i \(0.552688\pi\)
\(14\) 2763.43 0.269154
\(15\) −11550.5 −0.883651
\(16\) 4096.00 0.250000
\(17\) 37979.0 1.87488 0.937438 0.348152i \(-0.113191\pi\)
0.937438 + 0.348152i \(0.113191\pi\)
\(18\) 5832.00 0.235702
\(19\) −26348.1 −0.881276 −0.440638 0.897685i \(-0.645248\pi\)
−0.440638 + 0.897685i \(0.645248\pi\)
\(20\) −27378.9 −0.765264
\(21\) 9326.57 0.219763
\(22\) 17452.3 0.349442
\(23\) −12167.0 −0.208514
\(24\) 13824.0 0.204124
\(25\) 104884. 1.34251
\(26\) −20883.1 −0.233018
\(27\) 19683.0 0.192450
\(28\) 22107.4 0.190320
\(29\) 250851. 1.90995 0.954977 0.296679i \(-0.0958793\pi\)
0.954977 + 0.296679i \(0.0958793\pi\)
\(30\) −92403.8 −0.624835
\(31\) 179145. 1.08003 0.540017 0.841654i \(-0.318418\pi\)
0.540017 + 0.841654i \(0.318418\pi\)
\(32\) 32768.0 0.176777
\(33\) 58901.7 0.285318
\(34\) 303832. 1.32574
\(35\) −147773. −0.582581
\(36\) 46656.0 0.166667
\(37\) 434281. 1.40950 0.704749 0.709456i \(-0.251059\pi\)
0.704749 + 0.709456i \(0.251059\pi\)
\(38\) −210785. −0.623157
\(39\) −70480.6 −0.190258
\(40\) −219031. −0.541123
\(41\) −200429. −0.454168 −0.227084 0.973875i \(-0.572919\pi\)
−0.227084 + 0.973875i \(0.572919\pi\)
\(42\) 74612.6 0.155396
\(43\) 841346. 1.61375 0.806873 0.590725i \(-0.201158\pi\)
0.806873 + 0.590725i \(0.201158\pi\)
\(44\) 139619. 0.247092
\(45\) −311863. −0.510176
\(46\) −97336.0 −0.147442
\(47\) −200025. −0.281024 −0.140512 0.990079i \(-0.544875\pi\)
−0.140512 + 0.990079i \(0.544875\pi\)
\(48\) 110592. 0.144338
\(49\) −704222. −0.855113
\(50\) 839072. 0.949301
\(51\) 1.02543e6 1.08246
\(52\) −167065. −0.164768
\(53\) −1.62417e6 −1.49853 −0.749266 0.662269i \(-0.769593\pi\)
−0.749266 + 0.662269i \(0.769593\pi\)
\(54\) 157464. 0.136083
\(55\) −933254. −0.756364
\(56\) 176859. 0.134577
\(57\) −711399. −0.508805
\(58\) 2.00681e6 1.35054
\(59\) −2.18662e6 −1.38609 −0.693044 0.720895i \(-0.743731\pi\)
−0.693044 + 0.720895i \(0.743731\pi\)
\(60\) −739231. −0.441825
\(61\) 2.54771e6 1.43713 0.718564 0.695460i \(-0.244800\pi\)
0.718564 + 0.695460i \(0.244800\pi\)
\(62\) 1.43316e6 0.763700
\(63\) 251817. 0.126880
\(64\) 262144. 0.125000
\(65\) 1.11671e6 0.504365
\(66\) 471213. 0.201750
\(67\) 4.66306e6 1.89413 0.947063 0.321048i \(-0.104035\pi\)
0.947063 + 0.321048i \(0.104035\pi\)
\(68\) 2.43066e6 0.937438
\(69\) −328509. −0.120386
\(70\) −1.18218e6 −0.411947
\(71\) 4.02892e6 1.33593 0.667966 0.744192i \(-0.267165\pi\)
0.667966 + 0.744192i \(0.267165\pi\)
\(72\) 373248. 0.117851
\(73\) −2.58030e6 −0.776319 −0.388160 0.921592i \(-0.626889\pi\)
−0.388160 + 0.921592i \(0.626889\pi\)
\(74\) 3.47425e6 0.996666
\(75\) 2.83187e6 0.775101
\(76\) −1.68628e6 −0.440638
\(77\) 753568. 0.188107
\(78\) −563845. −0.134533
\(79\) 1.22032e6 0.278471 0.139236 0.990259i \(-0.455535\pi\)
0.139236 + 0.990259i \(0.455535\pi\)
\(80\) −1.75225e6 −0.382632
\(81\) 531441. 0.111111
\(82\) −1.60343e6 −0.321145
\(83\) −2.16585e6 −0.415771 −0.207886 0.978153i \(-0.566658\pi\)
−0.207886 + 0.978153i \(0.566658\pi\)
\(84\) 596901. 0.109881
\(85\) −1.62473e7 −2.86955
\(86\) 6.73077e6 1.14109
\(87\) 6.77298e6 1.10271
\(88\) 1.11695e6 0.174721
\(89\) 1.19998e6 0.180430 0.0902148 0.995922i \(-0.471245\pi\)
0.0902148 + 0.995922i \(0.471245\pi\)
\(90\) −2.49490e6 −0.360749
\(91\) −901704. −0.125435
\(92\) −778688. −0.104257
\(93\) 4.83690e6 0.623558
\(94\) −1.60020e6 −0.198714
\(95\) 1.12716e7 1.34882
\(96\) 884736. 0.102062
\(97\) 8.97814e6 0.998815 0.499408 0.866367i \(-0.333551\pi\)
0.499408 + 0.866367i \(0.333551\pi\)
\(98\) −5.63378e6 −0.604656
\(99\) 1.59035e6 0.164728
\(100\) 6.71257e6 0.671257
\(101\) −9.74940e6 −0.941571 −0.470785 0.882248i \(-0.656029\pi\)
−0.470785 + 0.882248i \(0.656029\pi\)
\(102\) 8.20347e6 0.765415
\(103\) −1.50223e7 −1.35459 −0.677294 0.735713i \(-0.736847\pi\)
−0.677294 + 0.735713i \(0.736847\pi\)
\(104\) −1.33652e6 −0.116509
\(105\) −3.98987e6 −0.336353
\(106\) −1.29934e7 −1.05962
\(107\) −1.75798e7 −1.38730 −0.693650 0.720312i \(-0.743999\pi\)
−0.693650 + 0.720312i \(0.743999\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.64602e7 1.21742 0.608711 0.793392i \(-0.291687\pi\)
0.608711 + 0.793392i \(0.291687\pi\)
\(110\) −7.46604e6 −0.534830
\(111\) 1.17256e7 0.813775
\(112\) 1.41488e6 0.0951602
\(113\) 1.70487e6 0.111152 0.0555761 0.998454i \(-0.482300\pi\)
0.0555761 + 0.998454i \(0.482300\pi\)
\(114\) −5.69120e6 −0.359780
\(115\) 5.20499e6 0.319137
\(116\) 1.60545e7 0.954977
\(117\) −1.90298e6 −0.109846
\(118\) −1.74929e7 −0.980112
\(119\) 1.31190e7 0.713654
\(120\) −5.91384e6 −0.312418
\(121\) −1.47280e7 −0.755781
\(122\) 2.03817e7 1.01620
\(123\) −5.41157e6 −0.262214
\(124\) 1.14653e7 0.540017
\(125\) −1.14474e7 −0.524228
\(126\) 2.01454e6 0.0897179
\(127\) 1.90880e7 0.826889 0.413444 0.910529i \(-0.364326\pi\)
0.413444 + 0.910529i \(0.364326\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 2.27163e7 0.931696
\(130\) 8.93371e6 0.356640
\(131\) 3.31709e6 0.128916 0.0644582 0.997920i \(-0.479468\pi\)
0.0644582 + 0.997920i \(0.479468\pi\)
\(132\) 3.76971e6 0.142659
\(133\) −9.10140e6 −0.335450
\(134\) 3.73045e7 1.33935
\(135\) −8.42030e6 −0.294550
\(136\) 1.94453e7 0.662869
\(137\) −5.17700e7 −1.72011 −0.860055 0.510202i \(-0.829571\pi\)
−0.860055 + 0.510202i \(0.829571\pi\)
\(138\) −2.62807e6 −0.0851257
\(139\) −1.27587e7 −0.402954 −0.201477 0.979493i \(-0.564574\pi\)
−0.201477 + 0.979493i \(0.564574\pi\)
\(140\) −9.45746e6 −0.291291
\(141\) −5.40069e6 −0.162249
\(142\) 3.22314e7 0.944647
\(143\) −5.69468e6 −0.162852
\(144\) 2.98598e6 0.0833333
\(145\) −1.07313e8 −2.92324
\(146\) −2.06424e7 −0.548941
\(147\) −1.90140e7 −0.493700
\(148\) 2.77940e7 0.704749
\(149\) −4.76667e7 −1.18049 −0.590246 0.807223i \(-0.700969\pi\)
−0.590246 + 0.807223i \(0.700969\pi\)
\(150\) 2.26549e7 0.548079
\(151\) 4.87918e7 1.15326 0.576631 0.817005i \(-0.304367\pi\)
0.576631 + 0.817005i \(0.304367\pi\)
\(152\) −1.34902e7 −0.311578
\(153\) 2.76867e7 0.624959
\(154\) 6.02854e6 0.133012
\(155\) −7.66372e7 −1.65302
\(156\) −4.51076e6 −0.0951291
\(157\) 1.52646e7 0.314801 0.157401 0.987535i \(-0.449689\pi\)
0.157401 + 0.987535i \(0.449689\pi\)
\(158\) 9.76259e6 0.196909
\(159\) −4.38526e7 −0.865178
\(160\) −1.40180e7 −0.270562
\(161\) −4.20283e6 −0.0793691
\(162\) 4.25153e6 0.0785674
\(163\) −7.21694e7 −1.30526 −0.652629 0.757678i \(-0.726334\pi\)
−0.652629 + 0.757678i \(0.726334\pi\)
\(164\) −1.28274e7 −0.227084
\(165\) −2.51979e7 −0.436687
\(166\) −1.73268e7 −0.293995
\(167\) 3.34621e7 0.555962 0.277981 0.960587i \(-0.410335\pi\)
0.277981 + 0.960587i \(0.410335\pi\)
\(168\) 4.77521e6 0.0776979
\(169\) −5.59344e7 −0.891405
\(170\) −1.29978e8 −2.02908
\(171\) −1.92078e7 −0.293759
\(172\) 5.38461e7 0.806873
\(173\) −4.16573e7 −0.611688 −0.305844 0.952082i \(-0.598939\pi\)
−0.305844 + 0.952082i \(0.598939\pi\)
\(174\) 5.41838e7 0.779736
\(175\) 3.62299e7 0.511016
\(176\) 8.93560e6 0.123546
\(177\) −5.90387e7 −0.800258
\(178\) 9.59982e6 0.127583
\(179\) −6.94962e7 −0.905682 −0.452841 0.891591i \(-0.649589\pi\)
−0.452841 + 0.891591i \(0.649589\pi\)
\(180\) −1.99592e7 −0.255088
\(181\) 3.54533e7 0.444408 0.222204 0.975000i \(-0.428675\pi\)
0.222204 + 0.975000i \(0.428675\pi\)
\(182\) −7.21363e6 −0.0886960
\(183\) 6.87882e7 0.829727
\(184\) −6.22950e6 −0.0737210
\(185\) −1.85783e8 −2.15728
\(186\) 3.86952e7 0.440922
\(187\) 8.28529e7 0.926536
\(188\) −1.28016e7 −0.140512
\(189\) 6.79907e6 0.0732543
\(190\) 9.01729e7 0.953758
\(191\) 5.59759e7 0.581279 0.290639 0.956833i \(-0.406132\pi\)
0.290639 + 0.956833i \(0.406132\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 8.02723e7 0.803740 0.401870 0.915697i \(-0.368360\pi\)
0.401870 + 0.915697i \(0.368360\pi\)
\(194\) 7.18251e7 0.706269
\(195\) 3.01513e7 0.291195
\(196\) −4.50702e7 −0.427556
\(197\) 3.52484e7 0.328479 0.164240 0.986420i \(-0.447483\pi\)
0.164240 + 0.986420i \(0.447483\pi\)
\(198\) 1.27228e7 0.116481
\(199\) 4.36909e7 0.393012 0.196506 0.980503i \(-0.437041\pi\)
0.196506 + 0.980503i \(0.437041\pi\)
\(200\) 5.37006e7 0.474651
\(201\) 1.25903e8 1.09357
\(202\) −7.79952e7 −0.665791
\(203\) 8.66511e7 0.727006
\(204\) 6.56278e7 0.541230
\(205\) 8.57425e7 0.695116
\(206\) −1.20179e8 −0.957838
\(207\) −8.86974e6 −0.0695048
\(208\) −1.06922e7 −0.0823842
\(209\) −5.74796e7 −0.435514
\(210\) −3.19189e7 −0.237838
\(211\) −1.86224e8 −1.36473 −0.682365 0.731012i \(-0.739048\pi\)
−0.682365 + 0.731012i \(0.739048\pi\)
\(212\) −1.03947e8 −0.749266
\(213\) 1.08781e8 0.771301
\(214\) −1.40638e8 −0.980970
\(215\) −3.59924e8 −2.46988
\(216\) 1.00777e7 0.0680414
\(217\) 6.18817e7 0.411105
\(218\) 1.31681e8 0.860848
\(219\) −6.96681e7 −0.448208
\(220\) −5.97283e7 −0.378182
\(221\) −9.91401e7 −0.617841
\(222\) 9.38047e7 0.575425
\(223\) 4.97859e7 0.300635 0.150317 0.988638i \(-0.451970\pi\)
0.150317 + 0.988638i \(0.451970\pi\)
\(224\) 1.13190e7 0.0672884
\(225\) 7.64604e7 0.447505
\(226\) 1.36390e7 0.0785964
\(227\) 1.75842e8 0.997773 0.498886 0.866667i \(-0.333743\pi\)
0.498886 + 0.866667i \(0.333743\pi\)
\(228\) −4.55296e7 −0.254403
\(229\) 4.00032e7 0.220125 0.110063 0.993925i \(-0.464895\pi\)
0.110063 + 0.993925i \(0.464895\pi\)
\(230\) 4.16399e7 0.225664
\(231\) 2.03463e7 0.108604
\(232\) 1.28436e8 0.675271
\(233\) 1.16608e8 0.603923 0.301962 0.953320i \(-0.402359\pi\)
0.301962 + 0.953320i \(0.402359\pi\)
\(234\) −1.52238e7 −0.0776726
\(235\) 8.55700e7 0.430114
\(236\) −1.39944e8 −0.693044
\(237\) 3.29487e7 0.160775
\(238\) 1.04952e8 0.504630
\(239\) −7.54954e7 −0.357707 −0.178854 0.983876i \(-0.557239\pi\)
−0.178854 + 0.983876i \(0.557239\pi\)
\(240\) −4.73108e7 −0.220913
\(241\) 4.32182e8 1.98887 0.994436 0.105346i \(-0.0335949\pi\)
0.994436 + 0.105346i \(0.0335949\pi\)
\(242\) −1.17824e8 −0.534418
\(243\) 1.43489e7 0.0641500
\(244\) 1.63053e8 0.718564
\(245\) 3.01263e8 1.30877
\(246\) −4.32926e7 −0.185413
\(247\) 6.87789e7 0.290413
\(248\) 9.17220e7 0.381850
\(249\) −5.84779e7 −0.240046
\(250\) −9.15789e7 −0.370685
\(251\) −1.77661e8 −0.709141 −0.354571 0.935029i \(-0.615373\pi\)
−0.354571 + 0.935029i \(0.615373\pi\)
\(252\) 1.61163e7 0.0634401
\(253\) −2.65428e7 −0.103045
\(254\) 1.52704e8 0.584699
\(255\) −4.38676e8 −1.65674
\(256\) 1.67772e7 0.0625000
\(257\) −3.32079e8 −1.22033 −0.610163 0.792276i \(-0.708896\pi\)
−0.610163 + 0.792276i \(0.708896\pi\)
\(258\) 1.81731e8 0.658809
\(259\) 1.50013e8 0.536513
\(260\) 7.14697e7 0.252183
\(261\) 1.82870e8 0.636651
\(262\) 2.65368e7 0.0911577
\(263\) 3.55827e8 1.20613 0.603064 0.797693i \(-0.293946\pi\)
0.603064 + 0.797693i \(0.293946\pi\)
\(264\) 3.01577e7 0.100875
\(265\) 6.94812e8 2.29354
\(266\) −7.28112e7 −0.237199
\(267\) 3.23994e7 0.104171
\(268\) 2.98436e8 0.947063
\(269\) 2.51996e6 0.00789332 0.00394666 0.999992i \(-0.498744\pi\)
0.00394666 + 0.999992i \(0.498744\pi\)
\(270\) −6.73624e7 −0.208278
\(271\) −4.75682e7 −0.145186 −0.0725929 0.997362i \(-0.523127\pi\)
−0.0725929 + 0.997362i \(0.523127\pi\)
\(272\) 1.55562e8 0.468719
\(273\) −2.43460e7 −0.0724200
\(274\) −4.14160e8 −1.21630
\(275\) 2.28809e8 0.663451
\(276\) −2.10246e7 −0.0601929
\(277\) −6.57962e7 −0.186004 −0.0930018 0.995666i \(-0.529646\pi\)
−0.0930018 + 0.995666i \(0.529646\pi\)
\(278\) −1.02070e8 −0.284932
\(279\) 1.30596e8 0.360012
\(280\) −7.56597e7 −0.205973
\(281\) 1.33552e8 0.359069 0.179535 0.983752i \(-0.442541\pi\)
0.179535 + 0.983752i \(0.442541\pi\)
\(282\) −4.32055e7 −0.114727
\(283\) −1.61478e8 −0.423508 −0.211754 0.977323i \(-0.567917\pi\)
−0.211754 + 0.977323i \(0.567917\pi\)
\(284\) 2.57851e8 0.667966
\(285\) 3.04333e8 0.778740
\(286\) −4.55575e7 −0.115154
\(287\) −6.92338e7 −0.172875
\(288\) 2.38879e7 0.0589256
\(289\) 1.03207e9 2.51516
\(290\) −8.58504e8 −2.06704
\(291\) 2.42410e8 0.576666
\(292\) −1.65139e8 −0.388160
\(293\) −1.53715e8 −0.357009 −0.178505 0.983939i \(-0.557126\pi\)
−0.178505 + 0.983939i \(0.557126\pi\)
\(294\) −1.52112e8 −0.349098
\(295\) 9.35425e8 2.12145
\(296\) 2.22352e8 0.498333
\(297\) 4.29393e7 0.0951059
\(298\) −3.81333e8 −0.834734
\(299\) 3.17606e7 0.0687132
\(300\) 1.81239e8 0.387551
\(301\) 2.90625e8 0.614257
\(302\) 3.90335e8 0.815479
\(303\) −2.63234e8 −0.543616
\(304\) −1.07922e8 −0.220319
\(305\) −1.08990e9 −2.19957
\(306\) 2.21494e8 0.441913
\(307\) −4.52238e8 −0.892037 −0.446019 0.895024i \(-0.647159\pi\)
−0.446019 + 0.895024i \(0.647159\pi\)
\(308\) 4.82283e7 0.0940534
\(309\) −4.05603e8 −0.782071
\(310\) −6.13098e8 −1.16886
\(311\) 7.24988e8 1.36669 0.683344 0.730096i \(-0.260525\pi\)
0.683344 + 0.730096i \(0.260525\pi\)
\(312\) −3.60861e7 −0.0672664
\(313\) −5.66818e8 −1.04481 −0.522407 0.852696i \(-0.674966\pi\)
−0.522407 + 0.852696i \(0.674966\pi\)
\(314\) 1.22117e8 0.222598
\(315\) −1.07726e8 −0.194194
\(316\) 7.81007e7 0.139236
\(317\) −9.63349e8 −1.69854 −0.849271 0.527958i \(-0.822958\pi\)
−0.849271 + 0.527958i \(0.822958\pi\)
\(318\) −3.50821e8 −0.611773
\(319\) 5.47243e8 0.943871
\(320\) −1.12144e8 −0.191316
\(321\) −4.74654e8 −0.800958
\(322\) −3.36226e7 −0.0561224
\(323\) −1.00068e9 −1.65228
\(324\) 3.40122e7 0.0555556
\(325\) −2.73788e8 −0.442408
\(326\) −5.77355e8 −0.922957
\(327\) 4.44424e8 0.702879
\(328\) −1.02619e8 −0.160573
\(329\) −6.90945e7 −0.106969
\(330\) −2.01583e8 −0.308784
\(331\) −4.26221e8 −0.646006 −0.323003 0.946398i \(-0.604692\pi\)
−0.323003 + 0.946398i \(0.604692\pi\)
\(332\) −1.38614e8 −0.207886
\(333\) 3.16591e8 0.469833
\(334\) 2.67697e8 0.393125
\(335\) −1.99483e9 −2.89901
\(336\) 3.82016e7 0.0549407
\(337\) −2.41775e8 −0.344118 −0.172059 0.985087i \(-0.555042\pi\)
−0.172059 + 0.985087i \(0.555042\pi\)
\(338\) −4.47475e8 −0.630319
\(339\) 4.60316e7 0.0641737
\(340\) −1.03982e9 −1.43477
\(341\) 3.90812e8 0.533737
\(342\) −1.53662e8 −0.207719
\(343\) −5.27734e8 −0.706131
\(344\) 4.30769e8 0.570545
\(345\) 1.40535e8 0.184254
\(346\) −3.33258e8 −0.432528
\(347\) 1.34412e9 1.72697 0.863484 0.504375i \(-0.168277\pi\)
0.863484 + 0.504375i \(0.168277\pi\)
\(348\) 4.33471e8 0.551356
\(349\) 3.27029e8 0.411811 0.205905 0.978572i \(-0.433986\pi\)
0.205905 + 0.978572i \(0.433986\pi\)
\(350\) 2.89839e8 0.361343
\(351\) −5.13803e7 −0.0634194
\(352\) 7.14848e7 0.0873604
\(353\) 1.52857e9 1.84958 0.924789 0.380480i \(-0.124241\pi\)
0.924789 + 0.380480i \(0.124241\pi\)
\(354\) −4.72309e8 −0.565868
\(355\) −1.72355e9 −2.04468
\(356\) 7.67985e7 0.0902148
\(357\) 3.54214e8 0.412028
\(358\) −5.55970e8 −0.640414
\(359\) −4.65294e7 −0.0530759 −0.0265379 0.999648i \(-0.508448\pi\)
−0.0265379 + 0.999648i \(0.508448\pi\)
\(360\) −1.59674e8 −0.180374
\(361\) −1.99648e8 −0.223352
\(362\) 2.83627e8 0.314244
\(363\) −3.97657e8 −0.436351
\(364\) −5.77091e7 −0.0627176
\(365\) 1.10384e9 1.18818
\(366\) 5.50306e8 0.586705
\(367\) 4.58322e8 0.483993 0.241997 0.970277i \(-0.422198\pi\)
0.241997 + 0.970277i \(0.422198\pi\)
\(368\) −4.98360e7 −0.0521286
\(369\) −1.46112e8 −0.151389
\(370\) −1.48627e9 −1.52543
\(371\) −5.61035e8 −0.570402
\(372\) 3.09562e8 0.311779
\(373\) 7.39771e8 0.738102 0.369051 0.929409i \(-0.379683\pi\)
0.369051 + 0.929409i \(0.379683\pi\)
\(374\) 6.62823e8 0.655160
\(375\) −3.09079e8 −0.302663
\(376\) −1.02413e8 −0.0993568
\(377\) −6.54820e8 −0.629400
\(378\) 5.43926e7 0.0517986
\(379\) 3.93984e8 0.371742 0.185871 0.982574i \(-0.440489\pi\)
0.185871 + 0.982574i \(0.440489\pi\)
\(380\) 7.21383e8 0.674409
\(381\) 5.15376e8 0.477404
\(382\) 4.47807e8 0.411026
\(383\) 1.33816e9 1.21706 0.608530 0.793531i \(-0.291760\pi\)
0.608530 + 0.793531i \(0.291760\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −3.22373e8 −0.287903
\(386\) 6.42179e8 0.568330
\(387\) 6.13341e8 0.537915
\(388\) 5.74601e8 0.499408
\(389\) −1.71698e9 −1.47891 −0.739455 0.673206i \(-0.764917\pi\)
−0.739455 + 0.673206i \(0.764917\pi\)
\(390\) 2.41210e8 0.205906
\(391\) −4.62091e8 −0.390939
\(392\) −3.60562e8 −0.302328
\(393\) 8.95615e7 0.0744299
\(394\) 2.81987e8 0.232270
\(395\) −5.22049e8 −0.426208
\(396\) 1.01782e8 0.0823642
\(397\) −9.91563e7 −0.0795341 −0.0397671 0.999209i \(-0.512662\pi\)
−0.0397671 + 0.999209i \(0.512662\pi\)
\(398\) 3.49528e8 0.277901
\(399\) −2.45738e8 −0.193672
\(400\) 4.29605e8 0.335629
\(401\) 1.41606e9 1.09667 0.548334 0.836259i \(-0.315262\pi\)
0.548334 + 0.836259i \(0.315262\pi\)
\(402\) 1.00722e9 0.773274
\(403\) −4.67638e8 −0.355911
\(404\) −6.23961e8 −0.470785
\(405\) −2.27348e8 −0.170059
\(406\) 6.93209e8 0.514071
\(407\) 9.47403e8 0.696553
\(408\) 5.25022e8 0.382707
\(409\) −1.39619e9 −1.00905 −0.504524 0.863397i \(-0.668332\pi\)
−0.504524 + 0.863397i \(0.668332\pi\)
\(410\) 6.85940e8 0.491521
\(411\) −1.39779e9 −0.993106
\(412\) −9.61429e8 −0.677294
\(413\) −7.55320e8 −0.527601
\(414\) −7.09579e7 −0.0491473
\(415\) 9.26540e8 0.636350
\(416\) −8.55373e7 −0.0582544
\(417\) −3.44486e8 −0.232646
\(418\) −4.59837e8 −0.307955
\(419\) −9.81338e8 −0.651733 −0.325866 0.945416i \(-0.605656\pi\)
−0.325866 + 0.945416i \(0.605656\pi\)
\(420\) −2.55351e8 −0.168177
\(421\) −1.16824e9 −0.763037 −0.381518 0.924361i \(-0.624599\pi\)
−0.381518 + 0.924361i \(0.624599\pi\)
\(422\) −1.48979e9 −0.965010
\(423\) −1.45819e8 −0.0936745
\(424\) −8.31575e8 −0.529811
\(425\) 3.98339e9 2.51705
\(426\) 8.70247e8 0.545392
\(427\) 8.80052e8 0.547030
\(428\) −1.12511e9 −0.693650
\(429\) −1.53756e8 −0.0940227
\(430\) −2.87939e9 −1.74647
\(431\) 1.91210e9 1.15037 0.575187 0.818022i \(-0.304929\pi\)
0.575187 + 0.818022i \(0.304929\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.67730e9 0.992894 0.496447 0.868067i \(-0.334638\pi\)
0.496447 + 0.868067i \(0.334638\pi\)
\(434\) 4.95053e8 0.290695
\(435\) −2.89745e9 −1.68773
\(436\) 1.05345e9 0.608711
\(437\) 3.20578e8 0.183759
\(438\) −5.57345e8 −0.316931
\(439\) −1.78469e8 −0.100679 −0.0503393 0.998732i \(-0.516030\pi\)
−0.0503393 + 0.998732i \(0.516030\pi\)
\(440\) −4.77826e8 −0.267415
\(441\) −5.13378e8 −0.285038
\(442\) −7.93121e8 −0.436879
\(443\) 1.00078e7 0.00546921 0.00273461 0.999996i \(-0.499130\pi\)
0.00273461 + 0.999996i \(0.499130\pi\)
\(444\) 7.50437e8 0.406887
\(445\) −5.13345e8 −0.276153
\(446\) 3.98287e8 0.212581
\(447\) −1.28700e9 −0.681557
\(448\) 9.05520e7 0.0475801
\(449\) 2.38685e9 1.24441 0.622203 0.782856i \(-0.286238\pi\)
0.622203 + 0.782856i \(0.286238\pi\)
\(450\) 6.11683e8 0.316434
\(451\) −4.37244e8 −0.224443
\(452\) 1.09112e8 0.0555761
\(453\) 1.31738e9 0.665836
\(454\) 1.40673e9 0.705532
\(455\) 3.85745e8 0.191982
\(456\) −3.64237e8 −0.179890
\(457\) 1.40659e9 0.689385 0.344692 0.938716i \(-0.387983\pi\)
0.344692 + 0.938716i \(0.387983\pi\)
\(458\) 3.20025e8 0.155652
\(459\) 7.47541e8 0.360820
\(460\) 3.33119e8 0.159569
\(461\) −8.75198e8 −0.416057 −0.208029 0.978123i \(-0.566705\pi\)
−0.208029 + 0.978123i \(0.566705\pi\)
\(462\) 1.62771e8 0.0767943
\(463\) −8.95572e8 −0.419341 −0.209670 0.977772i \(-0.567239\pi\)
−0.209670 + 0.977772i \(0.567239\pi\)
\(464\) 1.02749e9 0.477489
\(465\) −2.06921e9 −0.954373
\(466\) 9.32862e8 0.427038
\(467\) −2.18071e9 −0.990807 −0.495403 0.868663i \(-0.664980\pi\)
−0.495403 + 0.868663i \(0.664980\pi\)
\(468\) −1.21790e8 −0.0549228
\(469\) 1.61075e9 0.720981
\(470\) 6.84560e8 0.304137
\(471\) 4.12144e8 0.181751
\(472\) −1.11955e9 −0.490056
\(473\) 1.83543e9 0.797489
\(474\) 2.63590e8 0.113685
\(475\) −2.76350e9 −1.18313
\(476\) 8.39619e8 0.356827
\(477\) −1.18402e9 −0.499510
\(478\) −6.03963e8 −0.252937
\(479\) 1.86608e9 0.775812 0.387906 0.921699i \(-0.373199\pi\)
0.387906 + 0.921699i \(0.373199\pi\)
\(480\) −3.78486e8 −0.156209
\(481\) −1.13364e9 −0.464482
\(482\) 3.45745e9 1.40634
\(483\) −1.13476e8 −0.0458237
\(484\) −9.42594e8 −0.377891
\(485\) −3.84081e9 −1.52871
\(486\) 1.14791e8 0.0453609
\(487\) −9.54001e8 −0.374281 −0.187140 0.982333i \(-0.559922\pi\)
−0.187140 + 0.982333i \(0.559922\pi\)
\(488\) 1.30443e9 0.508102
\(489\) −1.94857e9 −0.753591
\(490\) 2.41010e9 0.925443
\(491\) 1.21183e9 0.462017 0.231008 0.972952i \(-0.425798\pi\)
0.231008 + 0.972952i \(0.425798\pi\)
\(492\) −3.46341e8 −0.131107
\(493\) 9.52708e9 3.58093
\(494\) 5.50232e8 0.205353
\(495\) −6.80343e8 −0.252121
\(496\) 7.33776e8 0.270009
\(497\) 1.39170e9 0.508510
\(498\) −4.67823e8 −0.169738
\(499\) −1.17097e9 −0.421884 −0.210942 0.977499i \(-0.567653\pi\)
−0.210942 + 0.977499i \(0.567653\pi\)
\(500\) −7.32631e8 −0.262114
\(501\) 9.03476e8 0.320985
\(502\) −1.42128e9 −0.501439
\(503\) −3.36243e9 −1.17806 −0.589028 0.808113i \(-0.700489\pi\)
−0.589028 + 0.808113i \(0.700489\pi\)
\(504\) 1.28931e8 0.0448589
\(505\) 4.17075e9 1.44110
\(506\) −2.12343e8 −0.0728636
\(507\) −1.51023e9 −0.514653
\(508\) 1.22163e9 0.413444
\(509\) −3.89948e9 −1.31067 −0.655336 0.755337i \(-0.727473\pi\)
−0.655336 + 0.755337i \(0.727473\pi\)
\(510\) −3.50941e9 −1.17149
\(511\) −8.91310e8 −0.295499
\(512\) 1.34218e8 0.0441942
\(513\) −5.18610e8 −0.169602
\(514\) −2.65663e9 −0.862900
\(515\) 6.42648e9 2.07323
\(516\) 1.45385e9 0.465848
\(517\) −4.36364e8 −0.138878
\(518\) 1.20010e9 0.379372
\(519\) −1.12475e9 −0.353158
\(520\) 5.71758e8 0.178320
\(521\) −4.08573e9 −1.26572 −0.632860 0.774266i \(-0.718119\pi\)
−0.632860 + 0.774266i \(0.718119\pi\)
\(522\) 1.46296e9 0.450181
\(523\) −6.17299e9 −1.88686 −0.943430 0.331571i \(-0.892421\pi\)
−0.943430 + 0.331571i \(0.892421\pi\)
\(524\) 2.12294e8 0.0644582
\(525\) 9.78208e8 0.295035
\(526\) 2.84661e9 0.852861
\(527\) 6.80374e9 2.02493
\(528\) 2.41261e8 0.0713295
\(529\) 1.48036e8 0.0434783
\(530\) 5.55850e9 1.62178
\(531\) −1.59404e9 −0.462029
\(532\) −5.82489e8 −0.167725
\(533\) 5.23197e8 0.149665
\(534\) 2.59195e8 0.0736601
\(535\) 7.52056e9 2.12330
\(536\) 2.38748e9 0.669675
\(537\) −1.87640e9 −0.522896
\(538\) 2.01596e7 0.00558142
\(539\) −1.53629e9 −0.422584
\(540\) −5.38899e8 −0.147275
\(541\) 6.18398e9 1.67910 0.839552 0.543280i \(-0.182818\pi\)
0.839552 + 0.543280i \(0.182818\pi\)
\(542\) −3.80546e8 −0.102662
\(543\) 9.57240e8 0.256579
\(544\) 1.24450e9 0.331434
\(545\) −7.04158e9 −1.86330
\(546\) −1.94768e8 −0.0512087
\(547\) −3.11011e9 −0.812494 −0.406247 0.913763i \(-0.633163\pi\)
−0.406247 + 0.913763i \(0.633163\pi\)
\(548\) −3.31328e9 −0.860055
\(549\) 1.85728e9 0.479043
\(550\) 1.83047e9 0.469130
\(551\) −6.60946e9 −1.68320
\(552\) −1.68197e8 −0.0425628
\(553\) 4.21535e8 0.105997
\(554\) −5.26369e8 −0.131524
\(555\) −5.01615e9 −1.24550
\(556\) −8.16559e8 −0.201477
\(557\) 5.56754e9 1.36512 0.682559 0.730831i \(-0.260867\pi\)
0.682559 + 0.730831i \(0.260867\pi\)
\(558\) 1.04477e9 0.254567
\(559\) −2.19624e9 −0.531789
\(560\) −6.05277e8 −0.145645
\(561\) 2.23703e9 0.534936
\(562\) 1.06842e9 0.253900
\(563\) −6.88023e9 −1.62489 −0.812445 0.583038i \(-0.801864\pi\)
−0.812445 + 0.583038i \(0.801864\pi\)
\(564\) −3.45644e8 −0.0811245
\(565\) −7.29337e8 −0.170121
\(566\) −1.29182e9 −0.299465
\(567\) 1.83575e8 0.0422934
\(568\) 2.06281e9 0.472323
\(569\) 8.40627e9 1.91298 0.956490 0.291765i \(-0.0942425\pi\)
0.956490 + 0.291765i \(0.0942425\pi\)
\(570\) 2.43467e9 0.550653
\(571\) −5.25925e8 −0.118222 −0.0591109 0.998251i \(-0.518827\pi\)
−0.0591109 + 0.998251i \(0.518827\pi\)
\(572\) −3.64460e8 −0.0814261
\(573\) 1.51135e9 0.335602
\(574\) −5.53870e8 −0.122241
\(575\) −1.27612e9 −0.279934
\(576\) 1.91103e8 0.0416667
\(577\) −1.66741e9 −0.361349 −0.180674 0.983543i \(-0.557828\pi\)
−0.180674 + 0.983543i \(0.557828\pi\)
\(578\) 8.25654e9 1.77849
\(579\) 2.16735e9 0.464039
\(580\) −6.86803e9 −1.46162
\(581\) −7.48146e8 −0.158259
\(582\) 1.93928e9 0.407765
\(583\) −3.54320e9 −0.740552
\(584\) −1.32111e9 −0.274470
\(585\) 8.14084e8 0.168122
\(586\) −1.22972e9 −0.252444
\(587\) −1.24841e9 −0.254756 −0.127378 0.991854i \(-0.540656\pi\)
−0.127378 + 0.991854i \(0.540656\pi\)
\(588\) −1.21690e9 −0.246850
\(589\) −4.72012e9 −0.951809
\(590\) 7.48340e9 1.50009
\(591\) 9.51707e8 0.189648
\(592\) 1.77881e9 0.352375
\(593\) −2.61091e9 −0.514163 −0.257081 0.966390i \(-0.582761\pi\)
−0.257081 + 0.966390i \(0.582761\pi\)
\(594\) 3.43515e8 0.0672501
\(595\) −5.61227e9 −1.09227
\(596\) −3.05067e9 −0.590246
\(597\) 1.17966e9 0.226905
\(598\) 2.54085e8 0.0485876
\(599\) −6.36475e9 −1.21001 −0.605003 0.796223i \(-0.706828\pi\)
−0.605003 + 0.796223i \(0.706828\pi\)
\(600\) 1.44992e9 0.274040
\(601\) −1.44570e9 −0.271656 −0.135828 0.990732i \(-0.543369\pi\)
−0.135828 + 0.990732i \(0.543369\pi\)
\(602\) 2.32500e9 0.434345
\(603\) 3.39937e9 0.631375
\(604\) 3.12268e9 0.576631
\(605\) 6.30059e9 1.15674
\(606\) −2.10587e9 −0.384395
\(607\) −4.03112e9 −0.731586 −0.365793 0.930696i \(-0.619202\pi\)
−0.365793 + 0.930696i \(0.619202\pi\)
\(608\) −8.63375e8 −0.155789
\(609\) 2.33958e9 0.419737
\(610\) −8.71919e9 −1.55533
\(611\) 5.22145e8 0.0926076
\(612\) 1.77195e9 0.312479
\(613\) 2.16682e9 0.379936 0.189968 0.981790i \(-0.439162\pi\)
0.189968 + 0.981790i \(0.439162\pi\)
\(614\) −3.61791e9 −0.630766
\(615\) 2.31505e9 0.401325
\(616\) 3.85827e8 0.0665058
\(617\) −3.74148e9 −0.641276 −0.320638 0.947202i \(-0.603897\pi\)
−0.320638 + 0.947202i \(0.603897\pi\)
\(618\) −3.24482e9 −0.553008
\(619\) 1.01384e10 1.71811 0.859054 0.511885i \(-0.171053\pi\)
0.859054 + 0.511885i \(0.171053\pi\)
\(620\) −4.90478e9 −0.826511
\(621\) −2.39483e8 −0.0401286
\(622\) 5.79990e9 0.966394
\(623\) 4.14506e8 0.0686789
\(624\) −2.88688e8 −0.0475645
\(625\) −3.29693e9 −0.540169
\(626\) −4.53455e9 −0.738795
\(627\) −1.55195e9 −0.251444
\(628\) 9.76934e8 0.157401
\(629\) 1.64936e10 2.64264
\(630\) −8.61811e8 −0.137316
\(631\) 8.21596e9 1.30183 0.650917 0.759149i \(-0.274385\pi\)
0.650917 + 0.759149i \(0.274385\pi\)
\(632\) 6.24806e8 0.0984544
\(633\) −5.02804e9 −0.787927
\(634\) −7.70679e9 −1.20105
\(635\) −8.16575e9 −1.26558
\(636\) −2.80657e9 −0.432589
\(637\) 1.83830e9 0.281791
\(638\) 4.37794e9 0.667417
\(639\) 2.93708e9 0.445311
\(640\) −8.97152e8 −0.135281
\(641\) 8.20911e8 0.123110 0.0615549 0.998104i \(-0.480394\pi\)
0.0615549 + 0.998104i \(0.480394\pi\)
\(642\) −3.79724e9 −0.566363
\(643\) 5.92103e9 0.878333 0.439166 0.898406i \(-0.355274\pi\)
0.439166 + 0.898406i \(0.355274\pi\)
\(644\) −2.68981e8 −0.0396845
\(645\) −9.71795e9 −1.42599
\(646\) −8.00541e9 −1.16834
\(647\) −1.09657e10 −1.59174 −0.795870 0.605468i \(-0.792986\pi\)
−0.795870 + 0.605468i \(0.792986\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −4.77020e9 −0.684984
\(650\) −2.19031e9 −0.312830
\(651\) 1.67080e9 0.237352
\(652\) −4.61884e9 −0.652629
\(653\) −6.73255e9 −0.946201 −0.473101 0.881008i \(-0.656865\pi\)
−0.473101 + 0.881008i \(0.656865\pi\)
\(654\) 3.55539e9 0.497011
\(655\) −1.41904e9 −0.197310
\(656\) −8.20956e8 −0.113542
\(657\) −1.88104e9 −0.258773
\(658\) −5.52756e8 −0.0756385
\(659\) −1.25637e10 −1.71009 −0.855043 0.518557i \(-0.826469\pi\)
−0.855043 + 0.518557i \(0.826469\pi\)
\(660\) −1.61266e9 −0.218343
\(661\) 7.27312e9 0.979526 0.489763 0.871856i \(-0.337083\pi\)
0.489763 + 0.871856i \(0.337083\pi\)
\(662\) −3.40977e9 −0.456796
\(663\) −2.67678e9 −0.356711
\(664\) −1.10891e9 −0.146997
\(665\) 3.89354e9 0.513415
\(666\) 2.53273e9 0.332222
\(667\) −3.05210e9 −0.398253
\(668\) 2.14157e9 0.277981
\(669\) 1.34422e9 0.173572
\(670\) −1.59587e10 −2.04991
\(671\) 5.55794e9 0.710207
\(672\) 3.05613e8 0.0388490
\(673\) −2.38576e9 −0.301699 −0.150849 0.988557i \(-0.548201\pi\)
−0.150849 + 0.988557i \(0.548201\pi\)
\(674\) −1.93420e9 −0.243328
\(675\) 2.06443e9 0.258367
\(676\) −3.57980e9 −0.445703
\(677\) −1.25080e10 −1.54928 −0.774639 0.632404i \(-0.782068\pi\)
−0.774639 + 0.632404i \(0.782068\pi\)
\(678\) 3.68253e8 0.0453777
\(679\) 3.10131e9 0.380190
\(680\) −8.31859e9 −1.01454
\(681\) 4.74773e9 0.576064
\(682\) 3.12649e9 0.377409
\(683\) −6.33284e9 −0.760548 −0.380274 0.924874i \(-0.624170\pi\)
−0.380274 + 0.924874i \(0.624170\pi\)
\(684\) −1.22930e9 −0.146879
\(685\) 2.21470e10 2.63267
\(686\) −4.22187e9 −0.499310
\(687\) 1.08009e9 0.127089
\(688\) 3.44615e9 0.403436
\(689\) 4.23972e9 0.493821
\(690\) 1.12428e9 0.130287
\(691\) −2.30362e9 −0.265606 −0.132803 0.991142i \(-0.542398\pi\)
−0.132803 + 0.991142i \(0.542398\pi\)
\(692\) −2.66607e9 −0.305844
\(693\) 5.49351e8 0.0627023
\(694\) 1.07530e10 1.22115
\(695\) 5.45813e9 0.616733
\(696\) 3.46777e9 0.389868
\(697\) −7.61208e9 −0.851508
\(698\) 2.61624e9 0.291194
\(699\) 3.14841e9 0.348675
\(700\) 2.31871e9 0.255508
\(701\) −5.07693e9 −0.556657 −0.278329 0.960486i \(-0.589780\pi\)
−0.278329 + 0.960486i \(0.589780\pi\)
\(702\) −4.11043e8 −0.0448443
\(703\) −1.14425e10 −1.24216
\(704\) 5.71879e8 0.0617731
\(705\) 2.31039e9 0.248327
\(706\) 1.22285e10 1.30785
\(707\) −3.36772e9 −0.358400
\(708\) −3.77848e9 −0.400129
\(709\) 8.96481e8 0.0944669 0.0472334 0.998884i \(-0.484960\pi\)
0.0472334 + 0.998884i \(0.484960\pi\)
\(710\) −1.37884e10 −1.44581
\(711\) 8.89616e8 0.0928237
\(712\) 6.14388e8 0.0637915
\(713\) −2.17965e9 −0.225203
\(714\) 2.83371e9 0.291348
\(715\) 2.43616e9 0.249250
\(716\) −4.44776e9 −0.452841
\(717\) −2.03838e9 −0.206522
\(718\) −3.72235e8 −0.0375303
\(719\) −4.13828e9 −0.415210 −0.207605 0.978213i \(-0.566567\pi\)
−0.207605 + 0.978213i \(0.566567\pi\)
\(720\) −1.27739e9 −0.127544
\(721\) −5.18914e9 −0.515611
\(722\) −1.59718e9 −0.157934
\(723\) 1.16689e10 1.14828
\(724\) 2.26901e9 0.222204
\(725\) 2.63103e10 2.56414
\(726\) −3.18126e9 −0.308546
\(727\) 1.34281e10 1.29611 0.648057 0.761591i \(-0.275582\pi\)
0.648057 + 0.761591i \(0.275582\pi\)
\(728\) −4.61672e8 −0.0443480
\(729\) 3.87420e8 0.0370370
\(730\) 8.83073e9 0.840169
\(731\) 3.19535e10 3.02557
\(732\) 4.40244e9 0.414863
\(733\) 2.59875e8 0.0243725 0.0121863 0.999926i \(-0.496121\pi\)
0.0121863 + 0.999926i \(0.496121\pi\)
\(734\) 3.66658e9 0.342235
\(735\) 8.13410e9 0.755621
\(736\) −3.98688e8 −0.0368605
\(737\) 1.01727e10 0.936049
\(738\) −1.16890e9 −0.107048
\(739\) −1.08950e10 −0.993056 −0.496528 0.868021i \(-0.665392\pi\)
−0.496528 + 0.868021i \(0.665392\pi\)
\(740\) −1.18901e10 −1.07864
\(741\) 1.85703e9 0.167670
\(742\) −4.48828e9 −0.403335
\(743\) −1.77102e10 −1.58403 −0.792014 0.610503i \(-0.790967\pi\)
−0.792014 + 0.610503i \(0.790967\pi\)
\(744\) 2.47649e9 0.220461
\(745\) 2.03916e10 1.80678
\(746\) 5.91817e9 0.521917
\(747\) −1.57890e9 −0.138590
\(748\) 5.30259e9 0.463268
\(749\) −6.07256e9 −0.528063
\(750\) −2.47263e9 −0.214015
\(751\) −2.21088e10 −1.90470 −0.952348 0.305013i \(-0.901339\pi\)
−0.952348 + 0.305013i \(0.901339\pi\)
\(752\) −8.19304e8 −0.0702559
\(753\) −4.79683e9 −0.409423
\(754\) −5.23856e9 −0.445053
\(755\) −2.08729e10 −1.76510
\(756\) 4.35141e8 0.0366272
\(757\) −1.59200e10 −1.33385 −0.666925 0.745125i \(-0.732390\pi\)
−0.666925 + 0.745125i \(0.732390\pi\)
\(758\) 3.15188e9 0.262861
\(759\) −7.16657e8 −0.0594929
\(760\) 5.77106e9 0.476879
\(761\) −6.68228e7 −0.00549640 −0.00274820 0.999996i \(-0.500875\pi\)
−0.00274820 + 0.999996i \(0.500875\pi\)
\(762\) 4.12301e9 0.337576
\(763\) 5.68581e9 0.463400
\(764\) 3.58246e9 0.290639
\(765\) −1.18442e10 −0.956517
\(766\) 1.07053e10 0.860591
\(767\) 5.70793e9 0.456767
\(768\) 4.52985e8 0.0360844
\(769\) −6.62635e9 −0.525451 −0.262726 0.964871i \(-0.584621\pi\)
−0.262726 + 0.964871i \(0.584621\pi\)
\(770\) −2.57898e9 −0.203578
\(771\) −8.96613e9 −0.704555
\(772\) 5.13743e9 0.401870
\(773\) −3.61939e9 −0.281843 −0.140922 0.990021i \(-0.545007\pi\)
−0.140922 + 0.990021i \(0.545007\pi\)
\(774\) 4.90673e9 0.380363
\(775\) 1.87894e10 1.44996
\(776\) 4.59681e9 0.353135
\(777\) 4.05035e9 0.309756
\(778\) −1.37359e10 −1.04575
\(779\) 5.28092e9 0.400247
\(780\) 1.92968e9 0.145598
\(781\) 8.78926e9 0.660197
\(782\) −3.69673e9 −0.276435
\(783\) 4.93750e9 0.367571
\(784\) −2.88449e9 −0.213778
\(785\) −6.53012e9 −0.481812
\(786\) 7.16492e8 0.0526299
\(787\) 1.01537e10 0.742525 0.371263 0.928528i \(-0.378925\pi\)
0.371263 + 0.928528i \(0.378925\pi\)
\(788\) 2.25590e9 0.164240
\(789\) 9.60732e9 0.696358
\(790\) −4.17639e9 −0.301374
\(791\) 5.88912e8 0.0423090
\(792\) 8.14257e8 0.0582403
\(793\) −6.65052e9 −0.473587
\(794\) −7.93250e8 −0.0562391
\(795\) 1.87599e10 1.32418
\(796\) 2.79622e9 0.196506
\(797\) 1.25904e10 0.880916 0.440458 0.897773i \(-0.354816\pi\)
0.440458 + 0.897773i \(0.354816\pi\)
\(798\) −1.96590e9 −0.136947
\(799\) −7.59677e9 −0.526884
\(800\) 3.43684e9 0.237325
\(801\) 8.74783e8 0.0601432
\(802\) 1.13285e10 0.775461
\(803\) −5.62904e9 −0.383645
\(804\) 8.05776e9 0.546787
\(805\) 1.79795e9 0.121477
\(806\) −3.74110e9 −0.251667
\(807\) 6.80388e7 0.00455721
\(808\) −4.99169e9 −0.332896
\(809\) 2.20668e10 1.46528 0.732638 0.680618i \(-0.238289\pi\)
0.732638 + 0.680618i \(0.238289\pi\)
\(810\) −1.81878e9 −0.120250
\(811\) 7.09047e9 0.466769 0.233384 0.972385i \(-0.425020\pi\)
0.233384 + 0.972385i \(0.425020\pi\)
\(812\) 5.54567e9 0.363503
\(813\) −1.28434e9 −0.0838231
\(814\) 7.57922e9 0.492537
\(815\) 3.08737e10 1.99773
\(816\) 4.20018e9 0.270615
\(817\) −2.21679e10 −1.42216
\(818\) −1.11695e10 −0.713505
\(819\) −6.57342e8 −0.0418117
\(820\) 5.48752e9 0.347558
\(821\) 3.26295e9 0.205783 0.102891 0.994693i \(-0.467191\pi\)
0.102891 + 0.994693i \(0.467191\pi\)
\(822\) −1.11823e10 −0.702232
\(823\) 4.72161e9 0.295250 0.147625 0.989043i \(-0.452837\pi\)
0.147625 + 0.989043i \(0.452837\pi\)
\(824\) −7.69143e9 −0.478919
\(825\) 6.17784e9 0.383043
\(826\) −6.04256e9 −0.373071
\(827\) 1.78452e10 1.09712 0.548558 0.836112i \(-0.315177\pi\)
0.548558 + 0.836112i \(0.315177\pi\)
\(828\) −5.67664e8 −0.0347524
\(829\) −2.93379e10 −1.78850 −0.894248 0.447572i \(-0.852289\pi\)
−0.894248 + 0.447572i \(0.852289\pi\)
\(830\) 7.41232e9 0.449967
\(831\) −1.77650e9 −0.107389
\(832\) −6.84299e8 −0.0411921
\(833\) −2.67457e10 −1.60323
\(834\) −2.75589e9 −0.164505
\(835\) −1.43149e10 −0.850916
\(836\) −3.67869e9 −0.217757
\(837\) 3.52610e9 0.207853
\(838\) −7.85070e9 −0.460845
\(839\) −1.90554e9 −0.111391 −0.0556957 0.998448i \(-0.517738\pi\)
−0.0556957 + 0.998448i \(0.517738\pi\)
\(840\) −2.04281e9 −0.118919
\(841\) 4.56764e10 2.64793
\(842\) −9.34594e9 −0.539549
\(843\) 3.60590e9 0.207309
\(844\) −1.19183e10 −0.682365
\(845\) 2.39285e10 1.36432
\(846\) −1.16655e9 −0.0662379
\(847\) −5.08749e9 −0.287681
\(848\) −6.65260e9 −0.374633
\(849\) −4.35991e9 −0.244512
\(850\) 3.18671e10 1.77982
\(851\) −5.28390e9 −0.293901
\(852\) 6.96197e9 0.385650
\(853\) 1.63652e8 0.00902818 0.00451409 0.999990i \(-0.498563\pi\)
0.00451409 + 0.999990i \(0.498563\pi\)
\(854\) 7.04042e9 0.386808
\(855\) 8.21700e9 0.449606
\(856\) −9.00085e9 −0.490485
\(857\) −6.70099e9 −0.363669 −0.181834 0.983329i \(-0.558203\pi\)
−0.181834 + 0.983329i \(0.558203\pi\)
\(858\) −1.23005e9 −0.0664841
\(859\) 1.52632e10 0.821620 0.410810 0.911721i \(-0.365246\pi\)
0.410810 + 0.911721i \(0.365246\pi\)
\(860\) −2.30351e10 −1.23494
\(861\) −1.86931e9 −0.0998092
\(862\) 1.52968e10 0.813437
\(863\) −1.65285e9 −0.0875376 −0.0437688 0.999042i \(-0.513936\pi\)
−0.0437688 + 0.999042i \(0.513936\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 1.78208e10 0.936205
\(866\) 1.34184e10 0.702082
\(867\) 2.78658e10 1.45213
\(868\) 3.96043e9 0.205553
\(869\) 2.66219e9 0.137616
\(870\) −2.31796e10 −1.19341
\(871\) −1.21724e10 −0.624184
\(872\) 8.42760e9 0.430424
\(873\) 6.54506e9 0.332938
\(874\) 2.56462e9 0.129937
\(875\) −3.95425e9 −0.199543
\(876\) −4.45876e9 −0.224104
\(877\) −1.13923e10 −0.570313 −0.285156 0.958481i \(-0.592045\pi\)
−0.285156 + 0.958481i \(0.592045\pi\)
\(878\) −1.42775e9 −0.0711905
\(879\) −4.15030e9 −0.206119
\(880\) −3.82261e9 −0.189091
\(881\) −1.15694e10 −0.570024 −0.285012 0.958524i \(-0.591998\pi\)
−0.285012 + 0.958524i \(0.591998\pi\)
\(882\) −4.10702e9 −0.201552
\(883\) −2.51966e10 −1.23163 −0.615814 0.787892i \(-0.711173\pi\)
−0.615814 + 0.787892i \(0.711173\pi\)
\(884\) −6.34497e9 −0.308920
\(885\) 2.52565e10 1.22482
\(886\) 8.00622e7 0.00386732
\(887\) −3.78563e10 −1.82140 −0.910700 0.413069i \(-0.864457\pi\)
−0.910700 + 0.413069i \(0.864457\pi\)
\(888\) 6.00350e9 0.287713
\(889\) 6.59354e9 0.314747
\(890\) −4.10676e9 −0.195269
\(891\) 1.15936e9 0.0549094
\(892\) 3.18630e9 0.150317
\(893\) 5.27030e9 0.247659
\(894\) −1.02960e10 −0.481934
\(895\) 2.97302e10 1.38617
\(896\) 7.24416e8 0.0336442
\(897\) 8.57537e8 0.0396716
\(898\) 1.90948e10 0.879928
\(899\) 4.49386e10 2.06282
\(900\) 4.89347e9 0.223752
\(901\) −6.16844e10 −2.80956
\(902\) −3.49795e9 −0.158705
\(903\) 7.84687e9 0.354641
\(904\) 8.72895e8 0.0392982
\(905\) −1.51668e10 −0.680179
\(906\) 1.05390e10 0.470817
\(907\) −9.07644e9 −0.403915 −0.201957 0.979394i \(-0.564730\pi\)
−0.201957 + 0.979394i \(0.564730\pi\)
\(908\) 1.12539e10 0.498886
\(909\) −7.10731e9 −0.313857
\(910\) 3.08596e9 0.135752
\(911\) −3.63398e10 −1.59246 −0.796229 0.604995i \(-0.793175\pi\)
−0.796229 + 0.604995i \(0.793175\pi\)
\(912\) −2.91389e9 −0.127201
\(913\) −4.72489e9 −0.205468
\(914\) 1.12527e10 0.487469
\(915\) −2.94273e10 −1.26992
\(916\) 2.56020e9 0.110063
\(917\) 1.14582e9 0.0490708
\(918\) 5.98033e9 0.255138
\(919\) 2.16445e10 0.919904 0.459952 0.887944i \(-0.347867\pi\)
0.459952 + 0.887944i \(0.347867\pi\)
\(920\) 2.66495e9 0.112832
\(921\) −1.22104e10 −0.515018
\(922\) −7.00158e9 −0.294197
\(923\) −1.05171e10 −0.440239
\(924\) 1.30216e9 0.0543018
\(925\) 4.55491e10 1.89227
\(926\) −7.16458e9 −0.296519
\(927\) −1.09513e10 −0.451529
\(928\) 8.21989e9 0.337635
\(929\) 7.00431e9 0.286623 0.143311 0.989678i \(-0.454225\pi\)
0.143311 + 0.989678i \(0.454225\pi\)
\(930\) −1.65536e10 −0.674844
\(931\) 1.85549e10 0.753591
\(932\) 7.46289e9 0.301962
\(933\) 1.95747e10 0.789058
\(934\) −1.74457e10 −0.700606
\(935\) −3.54441e10 −1.41809
\(936\) −9.74324e8 −0.0388363
\(937\) −3.16263e10 −1.25591 −0.627957 0.778248i \(-0.716109\pi\)
−0.627957 + 0.778248i \(0.716109\pi\)
\(938\) 1.28860e10 0.509811
\(939\) −1.53041e10 −0.603223
\(940\) 5.47648e9 0.215057
\(941\) −2.31615e10 −0.906155 −0.453077 0.891471i \(-0.649674\pi\)
−0.453077 + 0.891471i \(0.649674\pi\)
\(942\) 3.29715e9 0.128517
\(943\) 2.43861e9 0.0947005
\(944\) −8.95639e9 −0.346522
\(945\) −2.90861e9 −0.112118
\(946\) 1.46835e10 0.563910
\(947\) 1.11874e10 0.428060 0.214030 0.976827i \(-0.431341\pi\)
0.214030 + 0.976827i \(0.431341\pi\)
\(948\) 2.10872e9 0.0803877
\(949\) 6.73560e9 0.255826
\(950\) −2.21080e10 −0.836597
\(951\) −2.60104e10 −0.980653
\(952\) 6.71695e9 0.252315
\(953\) −1.85383e10 −0.693817 −0.346908 0.937899i \(-0.612768\pi\)
−0.346908 + 0.937899i \(0.612768\pi\)
\(954\) −9.47216e9 −0.353207
\(955\) −2.39463e10 −0.889663
\(956\) −4.83170e9 −0.178854
\(957\) 1.47755e10 0.544944
\(958\) 1.49287e10 0.548582
\(959\) −1.78828e10 −0.654743
\(960\) −3.02789e9 −0.110456
\(961\) 4.58016e9 0.166475
\(962\) −9.06915e9 −0.328438
\(963\) −1.28157e10 −0.462433
\(964\) 2.76596e10 0.994436
\(965\) −3.43401e10 −1.23015
\(966\) −9.07811e8 −0.0324023
\(967\) −7.91753e9 −0.281577 −0.140788 0.990040i \(-0.544964\pi\)
−0.140788 + 0.990040i \(0.544964\pi\)
\(968\) −7.54076e9 −0.267209
\(969\) −2.70183e10 −0.953947
\(970\) −3.07265e10 −1.08096
\(971\) −3.25866e10 −1.14228 −0.571139 0.820853i \(-0.693498\pi\)
−0.571139 + 0.820853i \(0.693498\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −4.40723e9 −0.153381
\(974\) −7.63201e9 −0.264656
\(975\) −7.39228e9 −0.255424
\(976\) 1.04354e10 0.359282
\(977\) 6.25658e9 0.214638 0.107319 0.994225i \(-0.465773\pi\)
0.107319 + 0.994225i \(0.465773\pi\)
\(978\) −1.55886e10 −0.532869
\(979\) 2.61780e9 0.0891656
\(980\) 1.92808e10 0.654387
\(981\) 1.19995e10 0.405807
\(982\) 9.69467e9 0.326695
\(983\) 8.04489e9 0.270136 0.135068 0.990836i \(-0.456875\pi\)
0.135068 + 0.990836i \(0.456875\pi\)
\(984\) −2.77073e9 −0.0927066
\(985\) −1.50791e10 −0.502746
\(986\) 7.62166e10 2.53210
\(987\) −1.86555e9 −0.0617586
\(988\) 4.40185e9 0.145207
\(989\) −1.02367e10 −0.336489
\(990\) −5.44274e9 −0.178277
\(991\) −3.99983e10 −1.30552 −0.652761 0.757564i \(-0.726389\pi\)
−0.652761 + 0.757564i \(0.726389\pi\)
\(992\) 5.87021e9 0.190925
\(993\) −1.15080e10 −0.372972
\(994\) 1.11336e10 0.359571
\(995\) −1.86908e10 −0.601515
\(996\) −3.74258e9 −0.120023
\(997\) 3.00799e10 0.961267 0.480633 0.876922i \(-0.340407\pi\)
0.480633 + 0.876922i \(0.340407\pi\)
\(998\) −9.36774e9 −0.298317
\(999\) 8.54795e9 0.271258
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 138.8.a.h.1.1 4
3.2 odd 2 414.8.a.i.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.8.a.h.1.1 4 1.1 even 1 trivial
414.8.a.i.1.4 4 3.2 odd 2