Properties

Label 546.8.a.q.1.4
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 367021 x^{4} - 17702143 x^{3} + 34815194576 x^{2} + 1422988371620 x - 933871993059968\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(180.926\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.926 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.926 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +1463.41 q^{10} +3961.32 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} -4939.00 q^{15} +4096.00 q^{16} -35665.6 q^{17} +5832.00 q^{18} +46854.3 q^{19} +11707.3 q^{20} -9261.00 q^{21} +31690.5 q^{22} +49170.5 q^{23} -13824.0 q^{24} -44663.0 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +120077. q^{29} -39512.0 q^{30} +176117. q^{31} +32768.0 q^{32} -106956. q^{33} -285325. q^{34} +62743.7 q^{35} +46656.0 q^{36} +71904.5 q^{37} +374834. q^{38} +59319.0 q^{39} +93658.2 q^{40} -72808.6 q^{41} -74088.0 q^{42} +881838. q^{43} +253524. q^{44} +133353. q^{45} +393364. q^{46} -630654. q^{47} -110592. q^{48} +117649. q^{49} -357304. q^{50} +962971. q^{51} -140608. q^{52} -786307. q^{53} -157464. q^{54} +724628. q^{55} +175616. q^{56} -1.26507e6 q^{57} +960618. q^{58} -384980. q^{59} -316096. q^{60} -1.71564e6 q^{61} +1.40894e6 q^{62} +250047. q^{63} +262144. q^{64} -401889. q^{65} -855645. q^{66} -592009. q^{67} -2.28260e6 q^{68} -1.32760e6 q^{69} +501949. q^{70} -933329. q^{71} +373248. q^{72} -1.70766e6 q^{73} +575236. q^{74} +1.20590e6 q^{75} +2.99867e6 q^{76} +1.35873e6 q^{77} +474552. q^{78} +3.67639e6 q^{79} +749265. q^{80} +531441. q^{81} -582469. q^{82} -1.90923e6 q^{83} -592704. q^{84} -6.52417e6 q^{85} +7.05471e6 q^{86} -3.24209e6 q^{87} +2.02819e6 q^{88} +8.98136e6 q^{89} +1.06683e6 q^{90} -753571. q^{91} +3.14691e6 q^{92} -4.75516e6 q^{93} -5.04523e6 q^{94} +8.57087e6 q^{95} -884736. q^{96} -7.16753e6 q^{97} +941192. q^{98} +2.88780e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) 182.926 0.654456 0.327228 0.944945i \(-0.393885\pi\)
0.327228 + 0.944945i \(0.393885\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1463.41 0.462771
\(11\) 3961.32 0.897357 0.448679 0.893693i \(-0.351895\pi\)
0.448679 + 0.893693i \(0.351895\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) −4939.00 −0.377851
\(16\) 4096.00 0.250000
\(17\) −35665.6 −1.76067 −0.880336 0.474351i \(-0.842683\pi\)
−0.880336 + 0.474351i \(0.842683\pi\)
\(18\) 5832.00 0.235702
\(19\) 46854.3 1.56715 0.783577 0.621295i \(-0.213393\pi\)
0.783577 + 0.621295i \(0.213393\pi\)
\(20\) 11707.3 0.327228
\(21\) −9261.00 −0.218218
\(22\) 31690.5 0.634527
\(23\) 49170.5 0.842670 0.421335 0.906905i \(-0.361562\pi\)
0.421335 + 0.906905i \(0.361562\pi\)
\(24\) −13824.0 −0.204124
\(25\) −44663.0 −0.571687
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 120077. 0.914256 0.457128 0.889401i \(-0.348878\pi\)
0.457128 + 0.889401i \(0.348878\pi\)
\(30\) −39512.0 −0.267181
\(31\) 176117. 1.06178 0.530891 0.847440i \(-0.321857\pi\)
0.530891 + 0.847440i \(0.321857\pi\)
\(32\) 32768.0 0.176777
\(33\) −106956. −0.518089
\(34\) −285325. −1.24498
\(35\) 62743.7 0.247361
\(36\) 46656.0 0.166667
\(37\) 71904.5 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(38\) 374834. 1.10814
\(39\) 59319.0 0.160128
\(40\) 93658.2 0.231385
\(41\) −72808.6 −0.164983 −0.0824915 0.996592i \(-0.526288\pi\)
−0.0824915 + 0.996592i \(0.526288\pi\)
\(42\) −74088.0 −0.154303
\(43\) 881838. 1.69141 0.845706 0.533649i \(-0.179180\pi\)
0.845706 + 0.533649i \(0.179180\pi\)
\(44\) 253524. 0.448679
\(45\) 133353. 0.218152
\(46\) 393364. 0.595858
\(47\) −630654. −0.886031 −0.443015 0.896514i \(-0.646091\pi\)
−0.443015 + 0.896514i \(0.646091\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −357304. −0.404244
\(51\) 962971. 1.01652
\(52\) −140608. −0.138675
\(53\) −786307. −0.725482 −0.362741 0.931890i \(-0.618159\pi\)
−0.362741 + 0.931890i \(0.618159\pi\)
\(54\) −157464. −0.136083
\(55\) 724628. 0.587281
\(56\) 175616. 0.133631
\(57\) −1.26507e6 −0.904796
\(58\) 960618. 0.646476
\(59\) −384980. −0.244037 −0.122019 0.992528i \(-0.538937\pi\)
−0.122019 + 0.992528i \(0.538937\pi\)
\(60\) −316096. −0.188925
\(61\) −1.71564e6 −0.967771 −0.483885 0.875131i \(-0.660775\pi\)
−0.483885 + 0.875131i \(0.660775\pi\)
\(62\) 1.40894e6 0.750794
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −401889. −0.181514
\(66\) −855645. −0.366344
\(67\) −592009. −0.240473 −0.120237 0.992745i \(-0.538365\pi\)
−0.120237 + 0.992745i \(0.538365\pi\)
\(68\) −2.28260e6 −0.880336
\(69\) −1.32760e6 −0.486516
\(70\) 501949. 0.174911
\(71\) −933329. −0.309479 −0.154739 0.987955i \(-0.549454\pi\)
−0.154739 + 0.987955i \(0.549454\pi\)
\(72\) 373248. 0.117851
\(73\) −1.70766e6 −0.513775 −0.256887 0.966441i \(-0.582697\pi\)
−0.256887 + 0.966441i \(0.582697\pi\)
\(74\) 575236. 0.165019
\(75\) 1.20590e6 0.330064
\(76\) 2.99867e6 0.783577
\(77\) 1.35873e6 0.339169
\(78\) 474552. 0.113228
\(79\) 3.67639e6 0.838932 0.419466 0.907771i \(-0.362217\pi\)
0.419466 + 0.907771i \(0.362217\pi\)
\(80\) 749265. 0.163614
\(81\) 531441. 0.111111
\(82\) −582469. −0.116661
\(83\) −1.90923e6 −0.366510 −0.183255 0.983065i \(-0.558663\pi\)
−0.183255 + 0.983065i \(0.558663\pi\)
\(84\) −592704. −0.109109
\(85\) −6.52417e6 −1.15228
\(86\) 7.05471e6 1.19601
\(87\) −3.24209e6 −0.527846
\(88\) 2.02819e6 0.317264
\(89\) 8.98136e6 1.35045 0.675223 0.737614i \(-0.264048\pi\)
0.675223 + 0.737614i \(0.264048\pi\)
\(90\) 1.06683e6 0.154257
\(91\) −753571. −0.104828
\(92\) 3.14691e6 0.421335
\(93\) −4.75516e6 −0.613021
\(94\) −5.04523e6 −0.626518
\(95\) 8.57087e6 1.02563
\(96\) −884736. −0.102062
\(97\) −7.16753e6 −0.797386 −0.398693 0.917084i \(-0.630536\pi\)
−0.398693 + 0.917084i \(0.630536\pi\)
\(98\) 941192. 0.101015
\(99\) 2.88780e6 0.299119
\(100\) −2.85843e6 −0.285843
\(101\) 5.83931e6 0.563945 0.281973 0.959422i \(-0.409011\pi\)
0.281973 + 0.959422i \(0.409011\pi\)
\(102\) 7.70377e6 0.718791
\(103\) 1.84954e7 1.66776 0.833878 0.551949i \(-0.186116\pi\)
0.833878 + 0.551949i \(0.186116\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −1.69408e6 −0.142814
\(106\) −6.29046e6 −0.512993
\(107\) 1.19931e7 0.946429 0.473215 0.880947i \(-0.343094\pi\)
0.473215 + 0.880947i \(0.343094\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) −830639. −0.0614355 −0.0307178 0.999528i \(-0.509779\pi\)
−0.0307178 + 0.999528i \(0.509779\pi\)
\(110\) 5.79703e6 0.415270
\(111\) −1.94142e6 −0.134738
\(112\) 1.40493e6 0.0944911
\(113\) 1.07261e6 0.0699303 0.0349652 0.999389i \(-0.488868\pi\)
0.0349652 + 0.999389i \(0.488868\pi\)
\(114\) −1.01205e7 −0.639788
\(115\) 8.99457e6 0.551491
\(116\) 7.68494e6 0.457128
\(117\) −1.60161e6 −0.0924500
\(118\) −3.07984e6 −0.172561
\(119\) −1.22333e7 −0.665471
\(120\) −2.52877e6 −0.133590
\(121\) −3.79513e6 −0.194750
\(122\) −1.37251e7 −0.684317
\(123\) 1.96583e6 0.0952529
\(124\) 1.12715e7 0.530891
\(125\) −2.24611e7 −1.02860
\(126\) 2.00038e6 0.0890871
\(127\) −1.12134e7 −0.485762 −0.242881 0.970056i \(-0.578092\pi\)
−0.242881 + 0.970056i \(0.578092\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.38096e7 −0.976537
\(130\) −3.21511e6 −0.128349
\(131\) 1.37935e7 0.536073 0.268037 0.963409i \(-0.413625\pi\)
0.268037 + 0.963409i \(0.413625\pi\)
\(132\) −6.84516e6 −0.259045
\(133\) 1.60710e7 0.592328
\(134\) −4.73607e6 −0.170040
\(135\) −3.60053e6 −0.125950
\(136\) −1.82608e7 −0.622491
\(137\) −8.84906e6 −0.294019 −0.147009 0.989135i \(-0.546965\pi\)
−0.147009 + 0.989135i \(0.546965\pi\)
\(138\) −1.06208e7 −0.344019
\(139\) 3.94389e7 1.24558 0.622792 0.782387i \(-0.285998\pi\)
0.622792 + 0.782387i \(0.285998\pi\)
\(140\) 4.01559e6 0.123681
\(141\) 1.70277e7 0.511550
\(142\) −7.46664e6 −0.218834
\(143\) −8.70301e6 −0.248882
\(144\) 2.98598e6 0.0833333
\(145\) 2.19653e7 0.598340
\(146\) −1.36613e7 −0.363294
\(147\) −3.17652e6 −0.0824786
\(148\) 4.60189e6 0.116686
\(149\) 3.04585e7 0.754322 0.377161 0.926148i \(-0.376900\pi\)
0.377161 + 0.926148i \(0.376900\pi\)
\(150\) 9.64722e6 0.233390
\(151\) 6.68784e7 1.58076 0.790381 0.612615i \(-0.209882\pi\)
0.790381 + 0.612615i \(0.209882\pi\)
\(152\) 2.39894e7 0.554072
\(153\) −2.60002e7 −0.586891
\(154\) 1.08699e7 0.239829
\(155\) 3.22164e7 0.694890
\(156\) 3.79642e6 0.0800641
\(157\) −9.11088e7 −1.87893 −0.939467 0.342640i \(-0.888679\pi\)
−0.939467 + 0.342640i \(0.888679\pi\)
\(158\) 2.94111e7 0.593215
\(159\) 2.12303e7 0.418857
\(160\) 5.99412e6 0.115693
\(161\) 1.68655e7 0.318499
\(162\) 4.25153e6 0.0785674
\(163\) 6.97149e7 1.26087 0.630433 0.776244i \(-0.282877\pi\)
0.630433 + 0.776244i \(0.282877\pi\)
\(164\) −4.65975e6 −0.0824915
\(165\) −1.95650e7 −0.339067
\(166\) −1.52739e7 −0.259162
\(167\) 8.71160e7 1.44741 0.723703 0.690112i \(-0.242439\pi\)
0.723703 + 0.690112i \(0.242439\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −5.21934e7 −0.814787
\(171\) 3.41568e7 0.522384
\(172\) 5.64376e7 0.845706
\(173\) −3.43576e7 −0.504500 −0.252250 0.967662i \(-0.581170\pi\)
−0.252250 + 0.967662i \(0.581170\pi\)
\(174\) −2.59367e7 −0.373243
\(175\) −1.53194e7 −0.216077
\(176\) 1.62256e7 0.224339
\(177\) 1.03945e7 0.140895
\(178\) 7.18509e7 0.954909
\(179\) −1.26368e8 −1.64684 −0.823420 0.567432i \(-0.807937\pi\)
−0.823420 + 0.567432i \(0.807937\pi\)
\(180\) 8.53460e6 0.109076
\(181\) 1.22054e8 1.52995 0.764974 0.644061i \(-0.222752\pi\)
0.764974 + 0.644061i \(0.222752\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 4.63224e7 0.558743
\(184\) 2.51753e7 0.297929
\(185\) 1.31532e7 0.152732
\(186\) −3.80413e7 −0.433471
\(187\) −1.41283e8 −1.57995
\(188\) −4.03619e7 −0.443015
\(189\) −6.75127e6 −0.0727393
\(190\) 6.85669e7 0.725232
\(191\) 2.62440e6 0.0272530 0.0136265 0.999907i \(-0.495662\pi\)
0.0136265 + 0.999907i \(0.495662\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 7.07430e7 0.708325 0.354163 0.935184i \(-0.384766\pi\)
0.354163 + 0.935184i \(0.384766\pi\)
\(194\) −5.73403e7 −0.563837
\(195\) 1.08510e7 0.104797
\(196\) 7.52954e6 0.0714286
\(197\) 1.07462e8 1.00144 0.500719 0.865610i \(-0.333069\pi\)
0.500719 + 0.865610i \(0.333069\pi\)
\(198\) 2.31024e7 0.211509
\(199\) −1.19494e8 −1.07488 −0.537439 0.843303i \(-0.680608\pi\)
−0.537439 + 0.843303i \(0.680608\pi\)
\(200\) −2.28675e7 −0.202122
\(201\) 1.59843e7 0.138837
\(202\) 4.67145e7 0.398769
\(203\) 4.11865e7 0.345556
\(204\) 6.16302e7 0.508262
\(205\) −1.33186e7 −0.107974
\(206\) 1.47963e8 1.17928
\(207\) 3.58453e7 0.280890
\(208\) −8.99891e6 −0.0693375
\(209\) 1.85605e8 1.40630
\(210\) −1.35526e7 −0.100985
\(211\) 9.51023e7 0.696951 0.348476 0.937318i \(-0.386699\pi\)
0.348476 + 0.937318i \(0.386699\pi\)
\(212\) −5.03236e7 −0.362741
\(213\) 2.51999e7 0.178678
\(214\) 9.59448e7 0.669226
\(215\) 1.61311e8 1.10696
\(216\) −1.00777e7 −0.0680414
\(217\) 6.04082e7 0.401316
\(218\) −6.64511e6 −0.0434415
\(219\) 4.61070e7 0.296628
\(220\) 4.63762e7 0.293640
\(221\) 7.83573e7 0.488322
\(222\) −1.55314e7 −0.0952740
\(223\) 2.98610e8 1.80317 0.901587 0.432597i \(-0.142403\pi\)
0.901587 + 0.432597i \(0.142403\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) −3.25594e7 −0.190562
\(226\) 8.58085e6 0.0494482
\(227\) 4.11865e7 0.233703 0.116851 0.993149i \(-0.462720\pi\)
0.116851 + 0.993149i \(0.462720\pi\)
\(228\) −8.09642e7 −0.452398
\(229\) −2.28974e8 −1.25997 −0.629986 0.776606i \(-0.716940\pi\)
−0.629986 + 0.776606i \(0.716940\pi\)
\(230\) 7.19566e7 0.389963
\(231\) −3.66858e7 −0.195819
\(232\) 6.14795e7 0.323238
\(233\) 3.16684e8 1.64014 0.820069 0.572264i \(-0.193935\pi\)
0.820069 + 0.572264i \(0.193935\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) −1.15363e8 −0.579869
\(236\) −2.46387e7 −0.122019
\(237\) −9.92626e7 −0.484358
\(238\) −9.78664e7 −0.470559
\(239\) 2.77147e8 1.31316 0.656580 0.754257i \(-0.272003\pi\)
0.656580 + 0.754257i \(0.272003\pi\)
\(240\) −2.02302e7 −0.0944626
\(241\) 1.08077e8 0.497361 0.248681 0.968586i \(-0.420003\pi\)
0.248681 + 0.968586i \(0.420003\pi\)
\(242\) −3.03611e7 −0.137709
\(243\) −1.43489e7 −0.0641500
\(244\) −1.09801e8 −0.483885
\(245\) 2.15211e7 0.0934938
\(246\) 1.57267e7 0.0673540
\(247\) −1.02939e8 −0.434650
\(248\) 9.01720e7 0.375397
\(249\) 5.15493e7 0.211605
\(250\) −1.79689e8 −0.727330
\(251\) 5.96208e7 0.237980 0.118990 0.992895i \(-0.462034\pi\)
0.118990 + 0.992895i \(0.462034\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 1.94780e8 0.756176
\(254\) −8.97070e7 −0.343485
\(255\) 1.76153e8 0.665271
\(256\) 1.67772e7 0.0625000
\(257\) 4.27487e8 1.57093 0.785466 0.618905i \(-0.212424\pi\)
0.785466 + 0.618905i \(0.212424\pi\)
\(258\) −1.90477e8 −0.690516
\(259\) 2.46632e7 0.0882066
\(260\) −2.57209e7 −0.0907568
\(261\) 8.75363e7 0.304752
\(262\) 1.10348e8 0.379061
\(263\) −2.17144e8 −0.736041 −0.368021 0.929818i \(-0.619964\pi\)
−0.368021 + 0.929818i \(0.619964\pi\)
\(264\) −5.47613e7 −0.183172
\(265\) −1.43836e8 −0.474796
\(266\) 1.28568e8 0.418839
\(267\) −2.42497e8 −0.779680
\(268\) −3.78886e7 −0.120237
\(269\) −6.07894e7 −0.190412 −0.0952061 0.995458i \(-0.530351\pi\)
−0.0952061 + 0.995458i \(0.530351\pi\)
\(270\) −2.88043e7 −0.0890602
\(271\) 7.05492e7 0.215327 0.107664 0.994187i \(-0.465663\pi\)
0.107664 + 0.994187i \(0.465663\pi\)
\(272\) −1.46086e8 −0.440168
\(273\) 2.03464e7 0.0605228
\(274\) −7.07925e7 −0.207903
\(275\) −1.76924e8 −0.513007
\(276\) −8.49667e7 −0.243258
\(277\) −2.77984e6 −0.00785852 −0.00392926 0.999992i \(-0.501251\pi\)
−0.00392926 + 0.999992i \(0.501251\pi\)
\(278\) 3.15511e8 0.880761
\(279\) 1.28389e8 0.353928
\(280\) 3.21248e7 0.0874554
\(281\) −1.06623e8 −0.286667 −0.143334 0.989674i \(-0.545782\pi\)
−0.143334 + 0.989674i \(0.545782\pi\)
\(282\) 1.36221e8 0.361721
\(283\) −5.32045e8 −1.39539 −0.697695 0.716394i \(-0.745791\pi\)
−0.697695 + 0.716394i \(0.745791\pi\)
\(284\) −5.97331e7 −0.154739
\(285\) −2.31413e8 −0.592150
\(286\) −6.96241e7 −0.175986
\(287\) −2.49733e7 −0.0623577
\(288\) 2.38879e7 0.0589256
\(289\) 8.61697e8 2.09996
\(290\) 1.75722e8 0.423091
\(291\) 1.93523e8 0.460371
\(292\) −1.09291e8 −0.256887
\(293\) 4.50077e8 1.04532 0.522661 0.852540i \(-0.324939\pi\)
0.522661 + 0.852540i \(0.324939\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −7.04230e7 −0.159712
\(296\) 3.68151e7 0.0825097
\(297\) −7.79706e7 −0.172696
\(298\) 2.43668e8 0.533386
\(299\) −1.08028e8 −0.233715
\(300\) 7.71777e7 0.165032
\(301\) 3.02471e8 0.639294
\(302\) 5.35027e8 1.11777
\(303\) −1.57661e8 −0.325594
\(304\) 1.91915e8 0.391788
\(305\) −3.13836e8 −0.633364
\(306\) −2.08002e8 −0.414994
\(307\) 5.17742e7 0.102124 0.0510621 0.998695i \(-0.483739\pi\)
0.0510621 + 0.998695i \(0.483739\pi\)
\(308\) 8.69588e7 0.169585
\(309\) −4.99375e8 −0.962879
\(310\) 2.57731e8 0.491362
\(311\) −1.24416e8 −0.234539 −0.117270 0.993100i \(-0.537414\pi\)
−0.117270 + 0.993100i \(0.537414\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) −6.96323e8 −1.28353 −0.641764 0.766902i \(-0.721797\pi\)
−0.641764 + 0.766902i \(0.721797\pi\)
\(314\) −7.28870e8 −1.32861
\(315\) 4.57401e7 0.0824537
\(316\) 2.35289e8 0.419466
\(317\) −2.96628e8 −0.523004 −0.261502 0.965203i \(-0.584218\pi\)
−0.261502 + 0.965203i \(0.584218\pi\)
\(318\) 1.69842e8 0.296177
\(319\) 4.75664e8 0.820414
\(320\) 4.79530e7 0.0818070
\(321\) −3.23814e8 −0.546421
\(322\) 1.34924e8 0.225213
\(323\) −1.67109e9 −2.75924
\(324\) 3.40122e7 0.0555556
\(325\) 9.81247e7 0.158557
\(326\) 5.57719e8 0.891567
\(327\) 2.24273e7 0.0354698
\(328\) −3.72780e7 −0.0583303
\(329\) −2.16314e8 −0.334888
\(330\) −1.56520e8 −0.239756
\(331\) 4.70178e8 0.712630 0.356315 0.934366i \(-0.384033\pi\)
0.356315 + 0.934366i \(0.384033\pi\)
\(332\) −1.22191e8 −0.183255
\(333\) 5.24184e7 0.0777909
\(334\) 6.96928e8 1.02347
\(335\) −1.08294e8 −0.157379
\(336\) −3.79331e7 −0.0545545
\(337\) −2.50685e8 −0.356799 −0.178400 0.983958i \(-0.557092\pi\)
−0.178400 + 0.983958i \(0.557092\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −2.89604e7 −0.0403743
\(340\) −4.17547e8 −0.576141
\(341\) 6.97656e8 0.952798
\(342\) 2.73254e8 0.369382
\(343\) 4.03536e7 0.0539949
\(344\) 4.51501e8 0.598004
\(345\) −2.42853e8 −0.318403
\(346\) −2.74860e8 −0.356735
\(347\) −7.17287e8 −0.921594 −0.460797 0.887505i \(-0.652436\pi\)
−0.460797 + 0.887505i \(0.652436\pi\)
\(348\) −2.07493e8 −0.263923
\(349\) 9.32096e8 1.17374 0.586869 0.809682i \(-0.300360\pi\)
0.586869 + 0.809682i \(0.300360\pi\)
\(350\) −1.22555e8 −0.152790
\(351\) 4.32436e7 0.0533761
\(352\) 1.29804e8 0.158632
\(353\) −6.86106e8 −0.830195 −0.415097 0.909777i \(-0.636252\pi\)
−0.415097 + 0.909777i \(0.636252\pi\)
\(354\) 8.31557e7 0.0996279
\(355\) −1.70730e8 −0.202540
\(356\) 5.74807e8 0.675223
\(357\) 3.30299e8 0.384210
\(358\) −1.01094e9 −1.16449
\(359\) 3.38800e8 0.386467 0.193233 0.981153i \(-0.438103\pi\)
0.193233 + 0.981153i \(0.438103\pi\)
\(360\) 6.82768e7 0.0771284
\(361\) 1.30145e9 1.45597
\(362\) 9.76431e8 1.08184
\(363\) 1.02469e8 0.112439
\(364\) −4.82285e7 −0.0524142
\(365\) −3.12376e8 −0.336243
\(366\) 3.70579e8 0.395091
\(367\) −1.09475e9 −1.15607 −0.578033 0.816013i \(-0.696180\pi\)
−0.578033 + 0.816013i \(0.696180\pi\)
\(368\) 2.01402e8 0.210667
\(369\) −5.30774e7 −0.0549943
\(370\) 1.05226e8 0.107998
\(371\) −2.69703e8 −0.274206
\(372\) −3.04330e8 −0.306510
\(373\) −5.25220e8 −0.524035 −0.262018 0.965063i \(-0.584388\pi\)
−0.262018 + 0.965063i \(0.584388\pi\)
\(374\) −1.13026e9 −1.11719
\(375\) 6.06451e8 0.593863
\(376\) −3.22895e8 −0.313259
\(377\) −2.63810e8 −0.253569
\(378\) −5.40102e7 −0.0514344
\(379\) −4.96096e8 −0.468089 −0.234045 0.972226i \(-0.575196\pi\)
−0.234045 + 0.972226i \(0.575196\pi\)
\(380\) 5.48536e8 0.512817
\(381\) 3.02761e8 0.280455
\(382\) 2.09952e7 0.0192708
\(383\) 1.52741e9 1.38919 0.694594 0.719402i \(-0.255584\pi\)
0.694594 + 0.719402i \(0.255584\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 2.48548e8 0.221971
\(386\) 5.65944e8 0.500861
\(387\) 6.42860e8 0.563804
\(388\) −4.58722e8 −0.398693
\(389\) −1.30037e9 −1.12007 −0.560033 0.828470i \(-0.689211\pi\)
−0.560033 + 0.828470i \(0.689211\pi\)
\(390\) 8.68080e7 0.0741026
\(391\) −1.75370e9 −1.48366
\(392\) 6.02363e7 0.0505076
\(393\) −3.72424e8 −0.309502
\(394\) 8.59698e8 0.708124
\(395\) 6.72508e8 0.549045
\(396\) 1.84819e8 0.149560
\(397\) 4.21261e8 0.337897 0.168949 0.985625i \(-0.445963\pi\)
0.168949 + 0.985625i \(0.445963\pi\)
\(398\) −9.55949e8 −0.760053
\(399\) −4.33917e8 −0.341981
\(400\) −1.82940e8 −0.142922
\(401\) −1.05933e9 −0.820397 −0.410199 0.911996i \(-0.634541\pi\)
−0.410199 + 0.911996i \(0.634541\pi\)
\(402\) 1.27874e8 0.0981728
\(403\) −3.86929e8 −0.294486
\(404\) 3.73716e8 0.281973
\(405\) 9.72144e7 0.0727174
\(406\) 3.29492e8 0.244345
\(407\) 2.84837e8 0.209419
\(408\) 4.93041e8 0.359396
\(409\) −1.90948e9 −1.38001 −0.690006 0.723804i \(-0.742392\pi\)
−0.690006 + 0.723804i \(0.742392\pi\)
\(410\) −1.06549e8 −0.0763492
\(411\) 2.38925e8 0.169752
\(412\) 1.18370e9 0.833878
\(413\) −1.32048e8 −0.0922375
\(414\) 2.86763e8 0.198619
\(415\) −3.49249e8 −0.239865
\(416\) −7.19913e7 −0.0490290
\(417\) −1.06485e9 −0.719138
\(418\) 1.48484e9 0.994402
\(419\) 1.68175e9 1.11689 0.558446 0.829541i \(-0.311398\pi\)
0.558446 + 0.829541i \(0.311398\pi\)
\(420\) −1.08421e8 −0.0714070
\(421\) −2.89394e8 −0.189018 −0.0945089 0.995524i \(-0.530128\pi\)
−0.0945089 + 0.995524i \(0.530128\pi\)
\(422\) 7.60818e8 0.492819
\(423\) −4.59747e8 −0.295344
\(424\) −4.02589e8 −0.256497
\(425\) 1.59293e9 1.00655
\(426\) 2.01599e8 0.126344
\(427\) −5.88466e8 −0.365783
\(428\) 7.67558e8 0.473215
\(429\) 2.34981e8 0.143692
\(430\) 1.29049e9 0.782735
\(431\) 2.53501e9 1.52514 0.762569 0.646907i \(-0.223938\pi\)
0.762569 + 0.646907i \(0.223938\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 2.03208e9 1.20291 0.601455 0.798907i \(-0.294588\pi\)
0.601455 + 0.798907i \(0.294588\pi\)
\(434\) 4.83265e8 0.283773
\(435\) −5.93062e8 −0.345452
\(436\) −5.31609e7 −0.0307178
\(437\) 2.30385e9 1.32059
\(438\) 3.68856e8 0.209748
\(439\) 1.48051e9 0.835189 0.417595 0.908633i \(-0.362873\pi\)
0.417595 + 0.908633i \(0.362873\pi\)
\(440\) 3.71010e8 0.207635
\(441\) 8.57661e7 0.0476190
\(442\) 6.26859e8 0.345296
\(443\) 7.70309e8 0.420971 0.210485 0.977597i \(-0.432496\pi\)
0.210485 + 0.977597i \(0.432496\pi\)
\(444\) −1.24251e8 −0.0673689
\(445\) 1.64293e9 0.883808
\(446\) 2.38888e9 1.27504
\(447\) −8.22380e8 −0.435508
\(448\) 8.99154e7 0.0472456
\(449\) 7.82205e8 0.407810 0.203905 0.978991i \(-0.434637\pi\)
0.203905 + 0.978991i \(0.434637\pi\)
\(450\) −2.60475e8 −0.134748
\(451\) −2.88418e8 −0.148049
\(452\) 6.86468e7 0.0349652
\(453\) −1.80572e9 −0.912654
\(454\) 3.29492e8 0.165253
\(455\) −1.37848e8 −0.0686057
\(456\) −6.47713e8 −0.319894
\(457\) −1.73248e8 −0.0849106 −0.0424553 0.999098i \(-0.513518\pi\)
−0.0424553 + 0.999098i \(0.513518\pi\)
\(458\) −1.83179e9 −0.890935
\(459\) 7.02006e8 0.338841
\(460\) 5.75653e8 0.275745
\(461\) 9.33126e8 0.443595 0.221798 0.975093i \(-0.428808\pi\)
0.221798 + 0.975093i \(0.428808\pi\)
\(462\) −2.93486e8 −0.138465
\(463\) 9.33958e8 0.437315 0.218657 0.975802i \(-0.429832\pi\)
0.218657 + 0.975802i \(0.429832\pi\)
\(464\) 4.91836e8 0.228564
\(465\) −8.69843e8 −0.401195
\(466\) 2.53347e9 1.15975
\(467\) −4.19058e9 −1.90399 −0.951996 0.306112i \(-0.900972\pi\)
−0.951996 + 0.306112i \(0.900972\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −2.03059e8 −0.0908903
\(470\) −9.22905e8 −0.410029
\(471\) 2.45994e9 1.08480
\(472\) −1.97110e8 −0.0862803
\(473\) 3.49324e9 1.51780
\(474\) −7.94101e8 −0.342493
\(475\) −2.09265e9 −0.895921
\(476\) −7.82931e8 −0.332736
\(477\) −5.73218e8 −0.241827
\(478\) 2.21718e9 0.928544
\(479\) −1.47647e9 −0.613832 −0.306916 0.951737i \(-0.599297\pi\)
−0.306916 + 0.951737i \(0.599297\pi\)
\(480\) −1.61841e8 −0.0667952
\(481\) −1.57974e8 −0.0647259
\(482\) 8.64612e8 0.351687
\(483\) −4.55368e8 −0.183886
\(484\) −2.42889e8 −0.0973752
\(485\) −1.31113e9 −0.521854
\(486\) −1.14791e8 −0.0453609
\(487\) 4.68279e7 0.0183719 0.00918594 0.999958i \(-0.497076\pi\)
0.00918594 + 0.999958i \(0.497076\pi\)
\(488\) −8.78409e8 −0.342159
\(489\) −1.88230e9 −0.727961
\(490\) 1.72169e8 0.0661101
\(491\) 1.13424e9 0.432433 0.216217 0.976345i \(-0.430628\pi\)
0.216217 + 0.976345i \(0.430628\pi\)
\(492\) 1.25813e8 0.0476265
\(493\) −4.28263e9 −1.60970
\(494\) −8.23511e8 −0.307344
\(495\) 5.28254e8 0.195760
\(496\) 7.21376e8 0.265446
\(497\) −3.20132e8 −0.116972
\(498\) 4.12395e8 0.149627
\(499\) −3.52134e9 −1.26869 −0.634345 0.773050i \(-0.718730\pi\)
−0.634345 + 0.773050i \(0.718730\pi\)
\(500\) −1.43751e9 −0.514300
\(501\) −2.35213e9 −0.835660
\(502\) 4.76967e8 0.168277
\(503\) −2.27544e9 −0.797218 −0.398609 0.917121i \(-0.630507\pi\)
−0.398609 + 0.917121i \(0.630507\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) 1.06816e9 0.369077
\(506\) 1.55824e9 0.534697
\(507\) −1.30324e8 −0.0444116
\(508\) −7.17656e8 −0.242881
\(509\) 9.06030e7 0.0304530 0.0152265 0.999884i \(-0.495153\pi\)
0.0152265 + 0.999884i \(0.495153\pi\)
\(510\) 1.40922e9 0.470417
\(511\) −5.85729e8 −0.194189
\(512\) 1.34218e8 0.0441942
\(513\) −9.22233e8 −0.301599
\(514\) 3.41990e9 1.11082
\(515\) 3.38328e9 1.09147
\(516\) −1.52382e9 −0.488268
\(517\) −2.49822e9 −0.795086
\(518\) 1.97306e8 0.0623715
\(519\) 9.27654e8 0.291273
\(520\) −2.05767e8 −0.0641747
\(521\) 2.46442e9 0.763454 0.381727 0.924275i \(-0.375330\pi\)
0.381727 + 0.924275i \(0.375330\pi\)
\(522\) 7.00290e8 0.215492
\(523\) −4.61350e9 −1.41018 −0.705090 0.709118i \(-0.749093\pi\)
−0.705090 + 0.709118i \(0.749093\pi\)
\(524\) 8.82783e8 0.268037
\(525\) 4.13624e8 0.124752
\(526\) −1.73715e9 −0.520460
\(527\) −6.28132e9 −1.86945
\(528\) −4.38090e8 −0.129522
\(529\) −9.87085e8 −0.289908
\(530\) −1.15069e9 −0.335732
\(531\) −2.80651e8 −0.0813458
\(532\) 1.02854e9 0.296164
\(533\) 1.59960e8 0.0457580
\(534\) −1.93997e9 −0.551317
\(535\) 2.19385e9 0.619396
\(536\) −3.03109e8 −0.0850201
\(537\) 3.41193e9 0.950803
\(538\) −4.86315e8 −0.134642
\(539\) 4.66045e8 0.128194
\(540\) −2.30434e8 −0.0629751
\(541\) 5.72039e9 1.55323 0.776614 0.629976i \(-0.216935\pi\)
0.776614 + 0.629976i \(0.216935\pi\)
\(542\) 5.64393e8 0.152260
\(543\) −3.29546e9 −0.883316
\(544\) −1.16869e9 −0.311246
\(545\) −1.51946e8 −0.0402069
\(546\) 1.62771e8 0.0427960
\(547\) −1.98970e9 −0.519796 −0.259898 0.965636i \(-0.583689\pi\)
−0.259898 + 0.965636i \(0.583689\pi\)
\(548\) −5.66340e8 −0.147009
\(549\) −1.25070e9 −0.322590
\(550\) −1.41540e9 −0.362751
\(551\) 5.62613e9 1.43278
\(552\) −6.79733e8 −0.172009
\(553\) 1.26100e9 0.317087
\(554\) −2.22387e7 −0.00555681
\(555\) −3.55137e8 −0.0881800
\(556\) 2.52409e9 0.622792
\(557\) −5.84384e9 −1.43286 −0.716432 0.697657i \(-0.754226\pi\)
−0.716432 + 0.697657i \(0.754226\pi\)
\(558\) 1.02712e9 0.250265
\(559\) −1.93740e9 −0.469113
\(560\) 2.56998e8 0.0618403
\(561\) 3.81464e9 0.912185
\(562\) −8.52983e8 −0.202704
\(563\) −3.09890e9 −0.731860 −0.365930 0.930642i \(-0.619249\pi\)
−0.365930 + 0.930642i \(0.619249\pi\)
\(564\) 1.08977e9 0.255775
\(565\) 1.96208e8 0.0457663
\(566\) −4.25636e9 −0.986690
\(567\) 1.82284e8 0.0419961
\(568\) −4.77865e8 −0.109417
\(569\) 1.40217e9 0.319087 0.159543 0.987191i \(-0.448998\pi\)
0.159543 + 0.987191i \(0.448998\pi\)
\(570\) −1.85131e9 −0.418713
\(571\) 5.22715e8 0.117500 0.0587501 0.998273i \(-0.481288\pi\)
0.0587501 + 0.998273i \(0.481288\pi\)
\(572\) −5.56993e8 −0.124441
\(573\) −7.08589e7 −0.0157345
\(574\) −1.99787e8 −0.0440935
\(575\) −2.19611e9 −0.481743
\(576\) 1.91103e8 0.0416667
\(577\) −3.10717e9 −0.673364 −0.336682 0.941618i \(-0.609305\pi\)
−0.336682 + 0.941618i \(0.609305\pi\)
\(578\) 6.89357e9 1.48490
\(579\) −1.91006e9 −0.408952
\(580\) 1.40578e9 0.299170
\(581\) −6.54868e8 −0.138528
\(582\) 1.54819e9 0.325532
\(583\) −3.11481e9 −0.651016
\(584\) −8.74324e8 −0.181647
\(585\) −2.92977e8 −0.0605045
\(586\) 3.60062e9 0.739155
\(587\) −7.02518e9 −1.43359 −0.716794 0.697285i \(-0.754391\pi\)
−0.716794 + 0.697285i \(0.754391\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 8.25184e9 1.66398
\(590\) −5.63384e8 −0.112933
\(591\) −2.90148e9 −0.578181
\(592\) 2.94521e8 0.0583432
\(593\) 2.56327e9 0.504782 0.252391 0.967625i \(-0.418783\pi\)
0.252391 + 0.967625i \(0.418783\pi\)
\(594\) −6.23765e8 −0.122115
\(595\) −2.23779e9 −0.435522
\(596\) 1.94935e9 0.377161
\(597\) 3.22633e9 0.620581
\(598\) −8.64221e8 −0.165261
\(599\) −3.77344e9 −0.717370 −0.358685 0.933459i \(-0.616775\pi\)
−0.358685 + 0.933459i \(0.616775\pi\)
\(600\) 6.17422e8 0.116695
\(601\) 6.94724e9 1.30542 0.652712 0.757606i \(-0.273631\pi\)
0.652712 + 0.757606i \(0.273631\pi\)
\(602\) 2.41976e9 0.452049
\(603\) −4.31575e8 −0.0801577
\(604\) 4.28022e9 0.790381
\(605\) −6.94229e8 −0.127456
\(606\) −1.26129e9 −0.230230
\(607\) −4.38390e9 −0.795611 −0.397805 0.917470i \(-0.630228\pi\)
−0.397805 + 0.917470i \(0.630228\pi\)
\(608\) 1.53532e9 0.277036
\(609\) −1.11204e9 −0.199507
\(610\) −2.51069e9 −0.447856
\(611\) 1.38555e9 0.245741
\(612\) −1.66401e9 −0.293445
\(613\) −2.26333e9 −0.396858 −0.198429 0.980115i \(-0.563584\pi\)
−0.198429 + 0.980115i \(0.563584\pi\)
\(614\) 4.14193e8 0.0722127
\(615\) 3.59602e8 0.0623389
\(616\) 6.95671e8 0.119914
\(617\) −4.55416e9 −0.780566 −0.390283 0.920695i \(-0.627623\pi\)
−0.390283 + 0.920695i \(0.627623\pi\)
\(618\) −3.99500e9 −0.680858
\(619\) 6.48895e9 1.09966 0.549828 0.835278i \(-0.314693\pi\)
0.549828 + 0.835278i \(0.314693\pi\)
\(620\) 2.06185e9 0.347445
\(621\) −9.67823e8 −0.162172
\(622\) −9.95330e8 −0.165844
\(623\) 3.08061e9 0.510420
\(624\) 2.42971e8 0.0400320
\(625\) −6.19429e8 −0.101487
\(626\) −5.57058e9 −0.907591
\(627\) −5.01132e9 −0.811925
\(628\) −5.83096e9 −0.939467
\(629\) −2.56452e9 −0.410893
\(630\) 3.65921e8 0.0583036
\(631\) −4.12166e9 −0.653084 −0.326542 0.945183i \(-0.605883\pi\)
−0.326542 + 0.945183i \(0.605883\pi\)
\(632\) 1.88231e9 0.296607
\(633\) −2.56776e9 −0.402385
\(634\) −2.37302e9 −0.369820
\(635\) −2.05122e9 −0.317910
\(636\) 1.35874e9 0.209429
\(637\) −2.58475e8 −0.0396214
\(638\) 3.80531e9 0.580120
\(639\) −6.80397e8 −0.103160
\(640\) 3.83624e8 0.0578463
\(641\) −3.92731e9 −0.588968 −0.294484 0.955656i \(-0.595148\pi\)
−0.294484 + 0.955656i \(0.595148\pi\)
\(642\) −2.59051e9 −0.386378
\(643\) −3.10296e9 −0.460296 −0.230148 0.973156i \(-0.573921\pi\)
−0.230148 + 0.973156i \(0.573921\pi\)
\(644\) 1.07939e9 0.159250
\(645\) −4.35540e9 −0.639101
\(646\) −1.33687e10 −1.95108
\(647\) −1.28622e10 −1.86702 −0.933510 0.358552i \(-0.883271\pi\)
−0.933510 + 0.358552i \(0.883271\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −1.52503e9 −0.218989
\(650\) 7.84998e8 0.112117
\(651\) −1.63102e9 −0.231700
\(652\) 4.46175e9 0.630433
\(653\) −9.22985e9 −1.29717 −0.648587 0.761140i \(-0.724640\pi\)
−0.648587 + 0.761140i \(0.724640\pi\)
\(654\) 1.79418e8 0.0250809
\(655\) 2.52319e9 0.350837
\(656\) −2.98224e8 −0.0412457
\(657\) −1.24489e9 −0.171258
\(658\) −1.73052e9 −0.236802
\(659\) −2.95484e9 −0.402193 −0.201097 0.979571i \(-0.564450\pi\)
−0.201097 + 0.979571i \(0.564450\pi\)
\(660\) −1.25216e9 −0.169533
\(661\) −1.03249e10 −1.39053 −0.695265 0.718754i \(-0.744713\pi\)
−0.695265 + 0.718754i \(0.744713\pi\)
\(662\) 3.76142e9 0.503906
\(663\) −2.11565e9 −0.281933
\(664\) −9.77528e8 −0.129581
\(665\) 2.93981e9 0.387653
\(666\) 4.19347e8 0.0550065
\(667\) 5.90426e9 0.770416
\(668\) 5.57543e9 0.723703
\(669\) −8.06248e9 −1.04106
\(670\) −8.66352e8 −0.111284
\(671\) −6.79621e9 −0.868436
\(672\) −3.03464e8 −0.0385758
\(673\) 6.90398e9 0.873065 0.436533 0.899688i \(-0.356206\pi\)
0.436533 + 0.899688i \(0.356206\pi\)
\(674\) −2.00548e9 −0.252295
\(675\) 8.79103e8 0.110021
\(676\) 3.08916e8 0.0384615
\(677\) −7.94792e9 −0.984449 −0.492224 0.870468i \(-0.663816\pi\)
−0.492224 + 0.870468i \(0.663816\pi\)
\(678\) −2.31683e8 −0.0285489
\(679\) −2.45846e9 −0.301384
\(680\) −3.34038e9 −0.407393
\(681\) −1.11203e9 −0.134928
\(682\) 5.58125e9 0.673730
\(683\) 1.08255e10 1.30009 0.650047 0.759894i \(-0.274749\pi\)
0.650047 + 0.759894i \(0.274749\pi\)
\(684\) 2.18603e9 0.261192
\(685\) −1.61872e9 −0.192422
\(686\) 3.22829e8 0.0381802
\(687\) 6.18229e9 0.727445
\(688\) 3.61201e9 0.422853
\(689\) 1.72752e9 0.201212
\(690\) −1.94283e9 −0.225145
\(691\) 3.22486e9 0.371824 0.185912 0.982566i \(-0.440476\pi\)
0.185912 + 0.982566i \(0.440476\pi\)
\(692\) −2.19888e9 −0.252250
\(693\) 9.90516e8 0.113056
\(694\) −5.73830e9 −0.651666
\(695\) 7.21441e9 0.815181
\(696\) −1.65995e9 −0.186622
\(697\) 2.59676e9 0.290481
\(698\) 7.45676e9 0.829959
\(699\) −8.55047e9 −0.946935
\(700\) −9.80443e8 −0.108039
\(701\) −4.66557e9 −0.511554 −0.255777 0.966736i \(-0.582331\pi\)
−0.255777 + 0.966736i \(0.582331\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 3.36903e9 0.365731
\(704\) 1.03844e9 0.112170
\(705\) 3.11480e9 0.334787
\(706\) −5.48885e9 −0.587036
\(707\) 2.00288e9 0.213151
\(708\) 6.65246e8 0.0704475
\(709\) 8.75909e9 0.922991 0.461495 0.887143i \(-0.347313\pi\)
0.461495 + 0.887143i \(0.347313\pi\)
\(710\) −1.36584e9 −0.143218
\(711\) 2.68009e9 0.279644
\(712\) 4.59846e9 0.477455
\(713\) 8.65977e9 0.894732
\(714\) 2.64239e9 0.271678
\(715\) −1.59201e9 −0.162882
\(716\) −8.08755e9 −0.823420
\(717\) −7.48297e9 −0.758153
\(718\) 2.71040e9 0.273273
\(719\) −1.13424e10 −1.13803 −0.569013 0.822329i \(-0.692674\pi\)
−0.569013 + 0.822329i \(0.692674\pi\)
\(720\) 5.46214e8 0.0545380
\(721\) 6.34391e9 0.630352
\(722\) 1.04116e10 1.02953
\(723\) −2.91807e9 −0.287152
\(724\) 7.81145e9 0.764974
\(725\) −5.36301e9 −0.522668
\(726\) 8.19749e8 0.0795065
\(727\) −8.56279e9 −0.826503 −0.413252 0.910617i \(-0.635607\pi\)
−0.413252 + 0.910617i \(0.635607\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −2.49901e9 −0.237760
\(731\) −3.14513e10 −2.97802
\(732\) 2.96463e9 0.279371
\(733\) −5.77607e9 −0.541712 −0.270856 0.962620i \(-0.587307\pi\)
−0.270856 + 0.962620i \(0.587307\pi\)
\(734\) −8.75798e9 −0.817462
\(735\) −5.81069e8 −0.0539786
\(736\) 1.61122e9 0.148964
\(737\) −2.34514e9 −0.215790
\(738\) −4.24620e8 −0.0388868
\(739\) 1.28018e10 1.16685 0.583425 0.812167i \(-0.301712\pi\)
0.583425 + 0.812167i \(0.301712\pi\)
\(740\) 8.41806e8 0.0763661
\(741\) 2.77935e9 0.250945
\(742\) −2.15763e9 −0.193893
\(743\) −1.96648e10 −1.75885 −0.879425 0.476038i \(-0.842072\pi\)
−0.879425 + 0.476038i \(0.842072\pi\)
\(744\) −2.43464e9 −0.216736
\(745\) 5.57166e9 0.493671
\(746\) −4.20176e9 −0.370549
\(747\) −1.39183e9 −0.122170
\(748\) −9.04210e9 −0.789976
\(749\) 4.11363e9 0.357717
\(750\) 4.85161e9 0.419924
\(751\) 9.14519e9 0.787867 0.393934 0.919139i \(-0.371114\pi\)
0.393934 + 0.919139i \(0.371114\pi\)
\(752\) −2.58316e9 −0.221508
\(753\) −1.60976e9 −0.137398
\(754\) −2.11048e9 −0.179300
\(755\) 1.22338e10 1.03454
\(756\) −4.32081e8 −0.0363696
\(757\) −1.46791e9 −0.122989 −0.0614943 0.998107i \(-0.519587\pi\)
−0.0614943 + 0.998107i \(0.519587\pi\)
\(758\) −3.96877e9 −0.330989
\(759\) −5.25906e9 −0.436578
\(760\) 4.38828e9 0.362616
\(761\) 4.96452e9 0.408348 0.204174 0.978935i \(-0.434549\pi\)
0.204174 + 0.978935i \(0.434549\pi\)
\(762\) 2.42209e9 0.198311
\(763\) −2.84909e8 −0.0232204
\(764\) 1.67962e8 0.0136265
\(765\) −4.75612e9 −0.384094
\(766\) 1.22193e10 0.982305
\(767\) 8.45802e8 0.0676838
\(768\) −4.52985e8 −0.0360844
\(769\) −8.77630e9 −0.695936 −0.347968 0.937506i \(-0.613128\pi\)
−0.347968 + 0.937506i \(0.613128\pi\)
\(770\) 1.98838e9 0.156957
\(771\) −1.15422e10 −0.906978
\(772\) 4.52755e9 0.354163
\(773\) −9.51150e9 −0.740663 −0.370332 0.928900i \(-0.620756\pi\)
−0.370332 + 0.928900i \(0.620756\pi\)
\(774\) 5.14288e9 0.398670
\(775\) −7.86593e9 −0.607007
\(776\) −3.66978e9 −0.281919
\(777\) −6.65908e8 −0.0509261
\(778\) −1.04030e10 −0.792006
\(779\) −3.41139e9 −0.258554
\(780\) 6.94464e8 0.0523984
\(781\) −3.69721e9 −0.277713
\(782\) −1.40296e10 −1.04911
\(783\) −2.36348e9 −0.175949
\(784\) 4.81890e8 0.0357143
\(785\) −1.66662e10 −1.22968
\(786\) −2.97939e9 −0.218851
\(787\) 1.22117e10 0.893024 0.446512 0.894778i \(-0.352666\pi\)
0.446512 + 0.894778i \(0.352666\pi\)
\(788\) 6.87758e9 0.500719
\(789\) 5.86288e9 0.424953
\(790\) 5.38006e9 0.388233
\(791\) 3.67904e8 0.0264312
\(792\) 1.47855e9 0.105755
\(793\) 3.76927e9 0.268411
\(794\) 3.37009e9 0.238929
\(795\) 3.88357e9 0.274124
\(796\) −7.64759e9 −0.537439
\(797\) −5.98021e9 −0.418420 −0.209210 0.977871i \(-0.567089\pi\)
−0.209210 + 0.977871i \(0.567089\pi\)
\(798\) −3.47134e9 −0.241817
\(799\) 2.24927e10 1.56001
\(800\) −1.46352e9 −0.101061
\(801\) 6.54741e9 0.450148
\(802\) −8.47461e9 −0.580108
\(803\) −6.76460e9 −0.461039
\(804\) 1.02299e9 0.0694186
\(805\) 3.08514e9 0.208444
\(806\) −3.09543e9 −0.208233
\(807\) 1.64131e9 0.109935
\(808\) 2.98973e9 0.199385
\(809\) 8.49553e9 0.564119 0.282060 0.959397i \(-0.408982\pi\)
0.282060 + 0.959397i \(0.408982\pi\)
\(810\) 7.77715e8 0.0514189
\(811\) −2.73867e10 −1.80288 −0.901441 0.432903i \(-0.857489\pi\)
−0.901441 + 0.432903i \(0.857489\pi\)
\(812\) 2.63594e9 0.172778
\(813\) −1.90483e9 −0.124319
\(814\) 2.27869e9 0.148081
\(815\) 1.27527e10 0.825181
\(816\) 3.94433e9 0.254131
\(817\) 4.13179e10 2.65070
\(818\) −1.52758e10 −0.975816
\(819\) −5.49353e8 −0.0349428
\(820\) −8.52390e8 −0.0539871
\(821\) −8.80748e9 −0.555457 −0.277729 0.960660i \(-0.589582\pi\)
−0.277729 + 0.960660i \(0.589582\pi\)
\(822\) 1.91140e9 0.120033
\(823\) 2.89145e10 1.80807 0.904037 0.427455i \(-0.140590\pi\)
0.904037 + 0.427455i \(0.140590\pi\)
\(824\) 9.46962e9 0.589641
\(825\) 4.77696e9 0.296185
\(826\) −1.05639e9 −0.0652218
\(827\) 2.50014e9 0.153708 0.0768539 0.997042i \(-0.475513\pi\)
0.0768539 + 0.997042i \(0.475513\pi\)
\(828\) 2.29410e9 0.140445
\(829\) −1.64084e10 −1.00029 −0.500144 0.865942i \(-0.666720\pi\)
−0.500144 + 0.865942i \(0.666720\pi\)
\(830\) −2.79399e9 −0.169610
\(831\) 7.50557e7 0.00453712
\(832\) −5.75930e8 −0.0346688
\(833\) −4.19602e9 −0.251525
\(834\) −8.51881e9 −0.508508
\(835\) 1.59358e10 0.947264
\(836\) 1.18787e10 0.703148
\(837\) −3.46651e9 −0.204340
\(838\) 1.34540e10 0.789763
\(839\) −2.11082e10 −1.23391 −0.616956 0.786997i \(-0.711634\pi\)
−0.616956 + 0.786997i \(0.711634\pi\)
\(840\) −8.67368e8 −0.0504924
\(841\) −2.83133e9 −0.164136
\(842\) −2.31515e9 −0.133656
\(843\) 2.87882e9 0.165507
\(844\) 6.08655e9 0.348476
\(845\) 8.82949e8 0.0503428
\(846\) −3.67798e9 −0.208839
\(847\) −1.30173e9 −0.0736087
\(848\) −3.22071e9 −0.181370
\(849\) 1.43652e10 0.805629
\(850\) 1.27435e10 0.711740
\(851\) 3.53558e9 0.196656
\(852\) 1.61279e9 0.0893388
\(853\) 1.55639e10 0.858614 0.429307 0.903159i \(-0.358758\pi\)
0.429307 + 0.903159i \(0.358758\pi\)
\(854\) −4.70773e9 −0.258648
\(855\) 6.24816e9 0.341878
\(856\) 6.14046e9 0.334613
\(857\) −2.12614e10 −1.15388 −0.576939 0.816787i \(-0.695753\pi\)
−0.576939 + 0.816787i \(0.695753\pi\)
\(858\) 1.87985e9 0.101606
\(859\) 3.32648e10 1.79064 0.895322 0.445420i \(-0.146945\pi\)
0.895322 + 0.445420i \(0.146945\pi\)
\(860\) 1.03239e10 0.553478
\(861\) 6.74280e8 0.0360022
\(862\) 2.02801e10 1.07844
\(863\) 3.43661e10 1.82009 0.910044 0.414512i \(-0.136048\pi\)
0.910044 + 0.414512i \(0.136048\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −6.28489e9 −0.330173
\(866\) 1.62566e10 0.850586
\(867\) −2.32658e10 −1.21242
\(868\) 3.86612e9 0.200658
\(869\) 1.45634e10 0.752822
\(870\) −4.74450e9 −0.244271
\(871\) 1.30064e9 0.0666953
\(872\) −4.25287e8 −0.0217207
\(873\) −5.22513e9 −0.265795
\(874\) 1.84308e10 0.933800
\(875\) −7.70417e9 −0.388774
\(876\) 2.95084e9 0.148314
\(877\) 2.01140e10 1.00693 0.503466 0.864015i \(-0.332058\pi\)
0.503466 + 0.864015i \(0.332058\pi\)
\(878\) 1.18441e10 0.590568
\(879\) −1.21521e10 −0.603517
\(880\) 2.96808e9 0.146820
\(881\) −2.02562e10 −0.998029 −0.499014 0.866594i \(-0.666305\pi\)
−0.499014 + 0.866594i \(0.666305\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 5.52142e9 0.269891 0.134945 0.990853i \(-0.456914\pi\)
0.134945 + 0.990853i \(0.456914\pi\)
\(884\) 5.01487e9 0.244161
\(885\) 1.90142e9 0.0922097
\(886\) 6.16247e9 0.297671
\(887\) −3.10218e10 −1.49257 −0.746284 0.665628i \(-0.768164\pi\)
−0.746284 + 0.665628i \(0.768164\pi\)
\(888\) −9.94008e8 −0.0476370
\(889\) −3.84619e9 −0.183601
\(890\) 1.31434e10 0.624946
\(891\) 2.10521e9 0.0997063
\(892\) 1.91111e10 0.901587
\(893\) −2.95488e10 −1.38855
\(894\) −6.57904e9 −0.307951
\(895\) −2.31160e10 −1.07778
\(896\) 7.19323e8 0.0334077
\(897\) 2.91675e9 0.134935
\(898\) 6.25764e9 0.288366
\(899\) 2.11477e10 0.970741
\(900\) −2.08380e9 −0.0952811
\(901\) 2.80441e10 1.27734
\(902\) −2.30734e9 −0.104686
\(903\) −8.16670e9 −0.369096
\(904\) 5.49174e8 0.0247241
\(905\) 2.23268e10 1.00128
\(906\) −1.44457e10 −0.645344
\(907\) 5.10220e9 0.227056 0.113528 0.993535i \(-0.463785\pi\)
0.113528 + 0.993535i \(0.463785\pi\)
\(908\) 2.63593e9 0.116851
\(909\) 4.25686e9 0.187982
\(910\) −1.10278e9 −0.0485115
\(911\) −1.43685e10 −0.629646 −0.314823 0.949150i \(-0.601945\pi\)
−0.314823 + 0.949150i \(0.601945\pi\)
\(912\) −5.18171e9 −0.226199
\(913\) −7.56309e9 −0.328890
\(914\) −1.38598e9 −0.0600408
\(915\) 8.47357e9 0.365673
\(916\) −1.46543e10 −0.629986
\(917\) 4.73116e9 0.202617
\(918\) 5.61605e9 0.239597
\(919\) 2.34317e10 0.995862 0.497931 0.867217i \(-0.334093\pi\)
0.497931 + 0.867217i \(0.334093\pi\)
\(920\) 4.60522e9 0.194981
\(921\) −1.39790e9 −0.0589614
\(922\) 7.46501e9 0.313669
\(923\) 2.05052e9 0.0858339
\(924\) −2.34789e9 −0.0979097
\(925\) −3.21147e9 −0.133416
\(926\) 7.47167e9 0.309228
\(927\) 1.34831e10 0.555919
\(928\) 3.93469e9 0.161619
\(929\) −2.86727e10 −1.17331 −0.586656 0.809836i \(-0.699556\pi\)
−0.586656 + 0.809836i \(0.699556\pi\)
\(930\) −6.95875e9 −0.283688
\(931\) 5.51236e9 0.223879
\(932\) 2.02678e10 0.820069
\(933\) 3.35924e9 0.135411
\(934\) −3.35246e10 −1.34633
\(935\) −2.58443e10 −1.03401
\(936\) −8.20026e8 −0.0326860
\(937\) −2.30512e10 −0.915388 −0.457694 0.889110i \(-0.651324\pi\)
−0.457694 + 0.889110i \(0.651324\pi\)
\(938\) −1.62447e9 −0.0642692
\(939\) 1.88007e10 0.741045
\(940\) −7.38324e9 −0.289934
\(941\) 3.90728e10 1.52866 0.764330 0.644825i \(-0.223070\pi\)
0.764330 + 0.644825i \(0.223070\pi\)
\(942\) 1.96795e10 0.767071
\(943\) −3.58004e9 −0.139026
\(944\) −1.57688e9 −0.0610094
\(945\) −1.23498e9 −0.0476047
\(946\) 2.79459e10 1.07325
\(947\) −2.53616e10 −0.970400 −0.485200 0.874403i \(-0.661253\pi\)
−0.485200 + 0.874403i \(0.661253\pi\)
\(948\) −6.35280e9 −0.242179
\(949\) 3.75174e9 0.142495
\(950\) −1.67412e10 −0.633512
\(951\) 8.00896e9 0.301956
\(952\) −6.26345e9 −0.235280
\(953\) 3.52515e10 1.31933 0.659664 0.751561i \(-0.270699\pi\)
0.659664 + 0.751561i \(0.270699\pi\)
\(954\) −4.58574e9 −0.170998
\(955\) 4.80072e8 0.0178359
\(956\) 1.77374e10 0.656580
\(957\) −1.28429e10 −0.473666
\(958\) −1.18117e10 −0.434045
\(959\) −3.03523e9 −0.111129
\(960\) −1.29473e9 −0.0472313
\(961\) 3.50463e9 0.127383
\(962\) −1.26379e9 −0.0457682
\(963\) 8.74297e9 0.315476
\(964\) 6.91690e9 0.248681
\(965\) 1.29407e10 0.463568
\(966\) −3.64295e9 −0.130027
\(967\) 5.11070e10 1.81756 0.908778 0.417280i \(-0.137017\pi\)
0.908778 + 0.417280i \(0.137017\pi\)
\(968\) −1.94311e9 −0.0688547
\(969\) 4.51193e10 1.59305
\(970\) −1.04890e10 −0.369007
\(971\) −3.58094e10 −1.25525 −0.627623 0.778517i \(-0.715972\pi\)
−0.627623 + 0.778517i \(0.715972\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 1.35275e10 0.470787
\(974\) 3.74623e8 0.0129909
\(975\) −2.64937e9 −0.0915432
\(976\) −7.02727e9 −0.241943
\(977\) 1.64617e10 0.564735 0.282368 0.959306i \(-0.408880\pi\)
0.282368 + 0.959306i \(0.408880\pi\)
\(978\) −1.50584e10 −0.514746
\(979\) 3.55780e10 1.21183
\(980\) 1.37735e9 0.0467469
\(981\) −6.05536e8 −0.0204785
\(982\) 9.07391e9 0.305777
\(983\) 2.31524e10 0.777425 0.388712 0.921359i \(-0.372920\pi\)
0.388712 + 0.921359i \(0.372920\pi\)
\(984\) 1.00651e9 0.0336770
\(985\) 1.96576e10 0.655397
\(986\) −3.42610e10 −1.13823
\(987\) 5.84049e9 0.193348
\(988\) −6.58808e9 −0.217325
\(989\) 4.33604e10 1.42530
\(990\) 4.22603e9 0.138423
\(991\) 5.71792e10 1.86629 0.933147 0.359495i \(-0.117051\pi\)
0.933147 + 0.359495i \(0.117051\pi\)
\(992\) 5.77101e9 0.187698
\(993\) −1.26948e10 −0.411437
\(994\) −2.56106e9 −0.0827117
\(995\) −2.18585e10 −0.703460
\(996\) 3.29916e9 0.105802
\(997\) 5.60393e10 1.79085 0.895425 0.445212i \(-0.146872\pi\)
0.895425 + 0.445212i \(0.146872\pi\)
\(998\) −2.81707e10 −0.897100
\(999\) −1.41530e9 −0.0449126
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 546.8.a.q.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.q.1.4 6 1.1 even 1 trivial