Properties

Label 546.8.a.r.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: \( x^{6} - x^{5} - 264981x^{4} + 17519669x^{3} + 15113237808x^{2} - 1787613752904x - 21984668630064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-11.2455\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +45.2455 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +45.2455 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +361.964 q^{10} +3722.93 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -1221.63 q^{15} +4096.00 q^{16} +24360.1 q^{17} +5832.00 q^{18} +16145.2 q^{19} +2895.71 q^{20} +9261.00 q^{21} +29783.4 q^{22} -19748.3 q^{23} -13824.0 q^{24} -76077.8 q^{25} +17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} +226116. q^{29} -9773.02 q^{30} -120387. q^{31} +32768.0 q^{32} -100519. q^{33} +194881. q^{34} -15519.2 q^{35} +46656.0 q^{36} -470925. q^{37} +129162. q^{38} -59319.0 q^{39} +23165.7 q^{40} -526838. q^{41} +74088.0 q^{42} +343410. q^{43} +238268. q^{44} +32984.0 q^{45} -157987. q^{46} +861916. q^{47} -110592. q^{48} +117649. q^{49} -608623. q^{50} -657722. q^{51} +140608. q^{52} +181808. q^{53} -157464. q^{54} +168446. q^{55} -175616. q^{56} -435920. q^{57} +1.80893e6 q^{58} -704361. q^{59} -78184.2 q^{60} +852775. q^{61} -963094. q^{62} -250047. q^{63} +262144. q^{64} +99404.3 q^{65} -804153. q^{66} -96440.5 q^{67} +1.55905e6 q^{68} +533205. q^{69} -124154. q^{70} +5.35527e6 q^{71} +373248. q^{72} -3.41008e6 q^{73} -3.76740e6 q^{74} +2.05410e6 q^{75} +1.03329e6 q^{76} -1.27697e6 q^{77} -474552. q^{78} -5.28067e6 q^{79} +185326. q^{80} +531441. q^{81} -4.21470e6 q^{82} -3.16918e6 q^{83} +592704. q^{84} +1.10218e6 q^{85} +2.74728e6 q^{86} -6.10513e6 q^{87} +1.90614e6 q^{88} +1.09078e7 q^{89} +263872. q^{90} -753571. q^{91} -1.26389e6 q^{92} +3.25044e6 q^{93} +6.89533e6 q^{94} +730497. q^{95} -884736. q^{96} +1.10372e7 q^{97} +941192. q^{98} +2.71402e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 1624 q^{10} - 2690 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} - 5481 q^{15} + 24576 q^{16} + 2910 q^{17} + 34992 q^{18} - 13055 q^{19} + 12992 q^{20} + 55566 q^{21} - 21520 q^{22} + 11581 q^{23} - 82944 q^{24} + 68081 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} - 92335 q^{29} - 43848 q^{30} - 83081 q^{31} + 196608 q^{32} + 72630 q^{33} + 23280 q^{34} - 69629 q^{35} + 279936 q^{36} - 265114 q^{37} - 104440 q^{38} - 355914 q^{39} + 103936 q^{40} - 367468 q^{41} + 444528 q^{42} + 454955 q^{43} - 172160 q^{44} + 147987 q^{45} + 92648 q^{46} + 733973 q^{47} - 663552 q^{48} + 705894 q^{49} + 544648 q^{50} - 78570 q^{51} + 843648 q^{52} - 1577379 q^{53} - 944784 q^{54} + 2231118 q^{55} - 1053696 q^{56} + 352485 q^{57} - 738680 q^{58} + 2062708 q^{59} - 350784 q^{60} - 271270 q^{61} - 664648 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 445991 q^{65} + 581040 q^{66} - 758674 q^{67} + 186240 q^{68} - 312687 q^{69} - 557032 q^{70} - 6138216 q^{71} + 2239488 q^{72} + 6361979 q^{73} - 2120912 q^{74} - 1838187 q^{75} - 835520 q^{76} + 922670 q^{77} - 2847312 q^{78} - 899781 q^{79} + 831488 q^{80} + 3188646 q^{81} - 2939744 q^{82} + 3313561 q^{83} + 3556224 q^{84} + 5307940 q^{85} + 3639640 q^{86} + 2493045 q^{87} - 1377280 q^{88} + 11210703 q^{89} + 1183896 q^{90} - 4521426 q^{91} + 741184 q^{92} + 2243187 q^{93} + 5871784 q^{94} + 12912395 q^{95} - 5308416 q^{96} + 28682643 q^{97} + 5647152 q^{98} - 1961010 q^{99}+O(q^{100}) \) 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
\(5\) 45.2455 0.161875 0.0809376 0.996719i \(-0.474209\pi\)
0.0809376 + 0.996719i \(0.474209\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 361.964 0.114463
\(11\) 3722.93 0.843355 0.421678 0.906746i \(-0.361441\pi\)
0.421678 + 0.906746i \(0.361441\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −1221.63 −0.0934587
\(16\) 4096.00 0.250000
\(17\) 24360.1 1.20256 0.601281 0.799038i \(-0.294657\pi\)
0.601281 + 0.799038i \(0.294657\pi\)
\(18\) 5832.00 0.235702
\(19\) 16145.2 0.540015 0.270007 0.962858i \(-0.412974\pi\)
0.270007 + 0.962858i \(0.412974\pi\)
\(20\) 2895.71 0.0809376
\(21\) 9261.00 0.218218
\(22\) 29783.4 0.596342
\(23\) −19748.3 −0.338441 −0.169221 0.985578i \(-0.554125\pi\)
−0.169221 + 0.985578i \(0.554125\pi\)
\(24\) −13824.0 −0.204124
\(25\) −76077.8 −0.973796
\(26\) 17576.0 0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) 226116. 1.72162 0.860812 0.508923i \(-0.169956\pi\)
0.860812 + 0.508923i \(0.169956\pi\)
\(30\) −9773.02 −0.0660853
\(31\) −120387. −0.725793 −0.362896 0.931829i \(-0.618212\pi\)
−0.362896 + 0.931829i \(0.618212\pi\)
\(32\) 32768.0 0.176777
\(33\) −100519. −0.486911
\(34\) 194881. 0.850340
\(35\) −15519.2 −0.0611831
\(36\) 46656.0 0.166667
\(37\) −470925. −1.52843 −0.764215 0.644962i \(-0.776873\pi\)
−0.764215 + 0.644962i \(0.776873\pi\)
\(38\) 129162. 0.381848
\(39\) −59319.0 −0.160128
\(40\) 23165.7 0.0572315
\(41\) −526838. −1.19380 −0.596902 0.802314i \(-0.703602\pi\)
−0.596902 + 0.802314i \(0.703602\pi\)
\(42\) 74088.0 0.154303
\(43\) 343410. 0.658678 0.329339 0.944212i \(-0.393174\pi\)
0.329339 + 0.944212i \(0.393174\pi\)
\(44\) 238268. 0.421678
\(45\) 32984.0 0.0539584
\(46\) −157987. −0.239314
\(47\) 861916. 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −608623. −0.688578
\(51\) −657722. −0.694300
\(52\) 140608. 0.138675
\(53\) 181808. 0.167744 0.0838720 0.996477i \(-0.473271\pi\)
0.0838720 + 0.996477i \(0.473271\pi\)
\(54\) −157464. −0.136083
\(55\) 168446. 0.136518
\(56\) −175616. −0.133631
\(57\) −435920. −0.311778
\(58\) 1.80893e6 1.21737
\(59\) −704361. −0.446492 −0.223246 0.974762i \(-0.571665\pi\)
−0.223246 + 0.974762i \(0.571665\pi\)
\(60\) −78184.2 −0.0467293
\(61\) 852775. 0.481039 0.240519 0.970644i \(-0.422682\pi\)
0.240519 + 0.970644i \(0.422682\pi\)
\(62\) −963094. −0.513213
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 99404.3 0.0448961
\(66\) −804153. −0.344298
\(67\) −96440.5 −0.0391740 −0.0195870 0.999808i \(-0.506235\pi\)
−0.0195870 + 0.999808i \(0.506235\pi\)
\(68\) 1.55905e6 0.601281
\(69\) 533205. 0.195399
\(70\) −124154. −0.0432630
\(71\) 5.35527e6 1.77573 0.887865 0.460103i \(-0.152188\pi\)
0.887865 + 0.460103i \(0.152188\pi\)
\(72\) 373248. 0.117851
\(73\) −3.41008e6 −1.02597 −0.512986 0.858397i \(-0.671461\pi\)
−0.512986 + 0.858397i \(0.671461\pi\)
\(74\) −3.76740e6 −1.08076
\(75\) 2.05410e6 0.562222
\(76\) 1.03329e6 0.270007
\(77\) −1.27697e6 −0.318758
\(78\) −474552. −0.113228
\(79\) −5.28067e6 −1.20502 −0.602510 0.798111i \(-0.705833\pi\)
−0.602510 + 0.798111i \(0.705833\pi\)
\(80\) 185326. 0.0404688
\(81\) 531441. 0.111111
\(82\) −4.21470e6 −0.844147
\(83\) −3.16918e6 −0.608377 −0.304189 0.952612i \(-0.598385\pi\)
−0.304189 + 0.952612i \(0.598385\pi\)
\(84\) 592704. 0.109109
\(85\) 1.10218e6 0.194665
\(86\) 2.74728e6 0.465756
\(87\) −6.10513e6 −0.993980
\(88\) 1.90614e6 0.298171
\(89\) 1.09078e7 1.64010 0.820052 0.572289i \(-0.193945\pi\)
0.820052 + 0.572289i \(0.193945\pi\)
\(90\) 263872. 0.0381543
\(91\) −753571. −0.104828
\(92\) −1.26389e6 −0.169221
\(93\) 3.25044e6 0.419037
\(94\) 6.89533e6 0.856263
\(95\) 730497. 0.0874150
\(96\) −884736. −0.102062
\(97\) 1.10372e7 1.22788 0.613941 0.789352i \(-0.289583\pi\)
0.613941 + 0.789352i \(0.289583\pi\)
\(98\) 941192. 0.101015
\(99\) 2.71402e6 0.281118
\(100\) −4.86898e6 −0.486898
\(101\) 1.06988e7 1.03327 0.516633 0.856207i \(-0.327185\pi\)
0.516633 + 0.856207i \(0.327185\pi\)
\(102\) −5.26178e6 −0.490944
\(103\) 2.78324e6 0.250969 0.125484 0.992096i \(-0.459951\pi\)
0.125484 + 0.992096i \(0.459951\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 419018. 0.0353241
\(106\) 1.45446e6 0.118613
\(107\) 1.03594e7 0.817509 0.408754 0.912644i \(-0.365963\pi\)
0.408754 + 0.912644i \(0.365963\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 5.32933e6 0.394166 0.197083 0.980387i \(-0.436853\pi\)
0.197083 + 0.980387i \(0.436853\pi\)
\(110\) 1.34757e6 0.0965330
\(111\) 1.27150e7 0.882440
\(112\) −1.40493e6 −0.0944911
\(113\) −397565. −0.0259199 −0.0129599 0.999916i \(-0.504125\pi\)
−0.0129599 + 0.999916i \(0.504125\pi\)
\(114\) −3.48736e6 −0.220460
\(115\) −893524. −0.0547852
\(116\) 1.44714e7 0.860812
\(117\) 1.60161e6 0.0924500
\(118\) −5.63489e6 −0.315717
\(119\) −8.35551e6 −0.454526
\(120\) −625474. −0.0330426
\(121\) −5.62696e6 −0.288752
\(122\) 6.82220e6 0.340146
\(123\) 1.42246e7 0.689244
\(124\) −7.70475e6 −0.362896
\(125\) −6.97698e6 −0.319509
\(126\) −2.00038e6 −0.0890871
\(127\) −1.69684e7 −0.735069 −0.367534 0.930010i \(-0.619798\pi\)
−0.367534 + 0.930010i \(0.619798\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −9.27207e6 −0.380288
\(130\) 795235. 0.0317463
\(131\) −3.03552e7 −1.17973 −0.589867 0.807501i \(-0.700820\pi\)
−0.589867 + 0.807501i \(0.700820\pi\)
\(132\) −6.43322e6 −0.243456
\(133\) −5.53780e6 −0.204106
\(134\) −771524. −0.0277002
\(135\) −890567. −0.0311529
\(136\) 1.24724e7 0.425170
\(137\) 1.87177e7 0.621915 0.310958 0.950424i \(-0.399350\pi\)
0.310958 + 0.950424i \(0.399350\pi\)
\(138\) 4.26564e6 0.138168
\(139\) −2.13020e7 −0.672774 −0.336387 0.941724i \(-0.609205\pi\)
−0.336387 + 0.941724i \(0.609205\pi\)
\(140\) −993229. −0.0305915
\(141\) −2.32717e7 −0.699136
\(142\) 4.28422e7 1.25563
\(143\) 8.17928e6 0.233905
\(144\) 2.98598e6 0.0833333
\(145\) 1.02307e7 0.278688
\(146\) −2.72807e7 −0.725471
\(147\) −3.17652e6 −0.0824786
\(148\) −3.01392e7 −0.764215
\(149\) 1.39114e7 0.344523 0.172261 0.985051i \(-0.444893\pi\)
0.172261 + 0.985051i \(0.444893\pi\)
\(150\) 1.64328e7 0.397551
\(151\) 7.53171e7 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(152\) 8.26634e6 0.190924
\(153\) 1.77585e7 0.400854
\(154\) −1.02157e7 −0.225396
\(155\) −5.44696e6 −0.117488
\(156\) −3.79642e6 −0.0800641
\(157\) 5.67481e7 1.17031 0.585157 0.810920i \(-0.301033\pi\)
0.585157 + 0.810920i \(0.301033\pi\)
\(158\) −4.22454e7 −0.852078
\(159\) −4.90881e6 −0.0968471
\(160\) 1.48260e6 0.0286158
\(161\) 6.77368e6 0.127919
\(162\) 4.25153e6 0.0785674
\(163\) −2.58947e7 −0.468332 −0.234166 0.972197i \(-0.575236\pi\)
−0.234166 + 0.972197i \(0.575236\pi\)
\(164\) −3.37176e7 −0.596902
\(165\) −4.54804e6 −0.0788189
\(166\) −2.53534e7 −0.430188
\(167\) 1.75791e7 0.292072 0.146036 0.989279i \(-0.453349\pi\)
0.146036 + 0.989279i \(0.453349\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 8.81747e6 0.137649
\(171\) 1.17698e7 0.180005
\(172\) 2.19782e7 0.329339
\(173\) 9.70629e7 1.42525 0.712626 0.701544i \(-0.247506\pi\)
0.712626 + 0.701544i \(0.247506\pi\)
\(174\) −4.88411e7 −0.702850
\(175\) 2.60947e7 0.368060
\(176\) 1.52491e7 0.210839
\(177\) 1.90178e7 0.257782
\(178\) 8.72622e7 1.15973
\(179\) 4.75406e6 0.0619554 0.0309777 0.999520i \(-0.490138\pi\)
0.0309777 + 0.999520i \(0.490138\pi\)
\(180\) 2.11097e6 0.0269792
\(181\) 1.49793e7 0.187766 0.0938830 0.995583i \(-0.470072\pi\)
0.0938830 + 0.995583i \(0.470072\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −2.30249e7 −0.277728
\(184\) −1.01112e7 −0.119657
\(185\) −2.13072e7 −0.247415
\(186\) 2.60035e7 0.296304
\(187\) 9.06909e7 1.01419
\(188\) 5.51626e7 0.605470
\(189\) 6.75127e6 0.0727393
\(190\) 5.84398e6 0.0618117
\(191\) 1.53951e8 1.59870 0.799349 0.600867i \(-0.205178\pi\)
0.799349 + 0.600867i \(0.205178\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −1.74528e8 −1.74749 −0.873747 0.486381i \(-0.838317\pi\)
−0.873747 + 0.486381i \(0.838317\pi\)
\(194\) 8.82974e7 0.868244
\(195\) −2.68392e6 −0.0259208
\(196\) 7.52954e6 0.0714286
\(197\) 1.21338e8 1.13075 0.565373 0.824835i \(-0.308732\pi\)
0.565373 + 0.824835i \(0.308732\pi\)
\(198\) 2.17121e7 0.198781
\(199\) 1.00380e8 0.902941 0.451470 0.892286i \(-0.350900\pi\)
0.451470 + 0.892286i \(0.350900\pi\)
\(200\) −3.89519e7 −0.344289
\(201\) 2.60389e6 0.0226171
\(202\) 8.55907e7 0.730629
\(203\) −7.75578e7 −0.650713
\(204\) −4.20942e7 −0.347150
\(205\) −2.38370e7 −0.193247
\(206\) 2.22659e7 0.177462
\(207\) −1.43965e7 −0.112814
\(208\) 8.99891e6 0.0693375
\(209\) 6.01074e7 0.455424
\(210\) 3.35215e6 0.0249779
\(211\) −6.30327e7 −0.461931 −0.230965 0.972962i \(-0.574188\pi\)
−0.230965 + 0.972962i \(0.574188\pi\)
\(212\) 1.16357e7 0.0838720
\(213\) −1.44592e8 −1.02522
\(214\) 8.28754e7 0.578066
\(215\) 1.55378e7 0.106624
\(216\) −1.00777e7 −0.0680414
\(217\) 4.12927e7 0.274324
\(218\) 4.26346e7 0.278718
\(219\) 9.20723e7 0.592345
\(220\) 1.07805e7 0.0682591
\(221\) 5.35191e7 0.333531
\(222\) 1.01720e8 0.623979
\(223\) 2.82615e8 1.70658 0.853292 0.521434i \(-0.174603\pi\)
0.853292 + 0.521434i \(0.174603\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −5.54607e7 −0.324599
\(226\) −3.18052e6 −0.0183281
\(227\) 2.19590e8 1.24601 0.623005 0.782218i \(-0.285912\pi\)
0.623005 + 0.782218i \(0.285912\pi\)
\(228\) −2.78989e7 −0.155889
\(229\) −4.19625e7 −0.230907 −0.115453 0.993313i \(-0.536832\pi\)
−0.115453 + 0.993313i \(0.536832\pi\)
\(230\) −7.14819e6 −0.0387390
\(231\) 3.44781e7 0.184035
\(232\) 1.15771e8 0.608686
\(233\) −9.08700e7 −0.470625 −0.235312 0.971920i \(-0.575611\pi\)
−0.235312 + 0.971920i \(0.575611\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) 3.89978e7 0.196021
\(236\) −4.50791e7 −0.223246
\(237\) 1.42578e8 0.695718
\(238\) −6.68441e7 −0.321398
\(239\) −8.05843e7 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(240\) −5.00379e6 −0.0233647
\(241\) 3.77861e8 1.73889 0.869446 0.494027i \(-0.164476\pi\)
0.869446 + 0.494027i \(0.164476\pi\)
\(242\) −4.50157e7 −0.204178
\(243\) −1.43489e7 −0.0641500
\(244\) 5.45776e7 0.240519
\(245\) 5.32309e6 0.0231250
\(246\) 1.13797e8 0.487369
\(247\) 3.54710e7 0.149773
\(248\) −6.16380e7 −0.256607
\(249\) 8.55677e7 0.351247
\(250\) −5.58159e7 −0.225927
\(251\) 5.13358e6 0.0204910 0.0102455 0.999948i \(-0.496739\pi\)
0.0102455 + 0.999948i \(0.496739\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) −7.35217e7 −0.285426
\(254\) −1.35747e8 −0.519772
\(255\) −2.97590e7 −0.112390
\(256\) 1.67772e7 0.0625000
\(257\) 2.41968e8 0.889183 0.444592 0.895733i \(-0.353349\pi\)
0.444592 + 0.895733i \(0.353349\pi\)
\(258\) −7.41766e7 −0.268904
\(259\) 1.61527e8 0.577692
\(260\) 6.36188e6 0.0224480
\(261\) 1.64839e8 0.573875
\(262\) −2.42842e8 −0.834197
\(263\) −4.62882e8 −1.56901 −0.784504 0.620124i \(-0.787082\pi\)
−0.784504 + 0.620124i \(0.787082\pi\)
\(264\) −5.14658e7 −0.172149
\(265\) 8.22599e6 0.0271536
\(266\) −4.43024e7 −0.144325
\(267\) −2.94510e8 −0.946914
\(268\) −6.17219e6 −0.0195870
\(269\) 3.60270e8 1.12848 0.564241 0.825610i \(-0.309169\pi\)
0.564241 + 0.825610i \(0.309169\pi\)
\(270\) −7.12454e6 −0.0220284
\(271\) 3.52906e8 1.07713 0.538564 0.842585i \(-0.318967\pi\)
0.538564 + 0.842585i \(0.318967\pi\)
\(272\) 9.97789e7 0.300641
\(273\) 2.03464e7 0.0605228
\(274\) 1.49742e8 0.439761
\(275\) −2.83233e8 −0.821256
\(276\) 3.41251e7 0.0976996
\(277\) −6.88485e7 −0.194633 −0.0973163 0.995254i \(-0.531026\pi\)
−0.0973163 + 0.995254i \(0.531026\pi\)
\(278\) −1.70416e8 −0.475723
\(279\) −8.77619e7 −0.241931
\(280\) −7.94583e6 −0.0216315
\(281\) 7.22937e8 1.94370 0.971848 0.235610i \(-0.0757088\pi\)
0.971848 + 0.235610i \(0.0757088\pi\)
\(282\) −1.86174e8 −0.494364
\(283\) 1.66278e8 0.436097 0.218048 0.975938i \(-0.430031\pi\)
0.218048 + 0.975938i \(0.430031\pi\)
\(284\) 3.42737e8 0.887865
\(285\) −1.97234e7 −0.0504691
\(286\) 6.54342e7 0.165396
\(287\) 1.80705e8 0.451216
\(288\) 2.38879e7 0.0589256
\(289\) 1.83075e8 0.446155
\(290\) 8.18458e7 0.197062
\(291\) −2.98004e8 −0.708918
\(292\) −2.18245e8 −0.512986
\(293\) 9.85330e7 0.228847 0.114423 0.993432i \(-0.463498\pi\)
0.114423 + 0.993432i \(0.463498\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −3.18692e7 −0.0722759
\(296\) −2.41114e8 −0.540382
\(297\) −7.32784e7 −0.162304
\(298\) 1.11291e8 0.243614
\(299\) −4.33871e7 −0.0938667
\(300\) 1.31463e8 0.281111
\(301\) −1.17790e8 −0.248957
\(302\) 6.02537e8 1.25881
\(303\) −2.88869e8 −0.596556
\(304\) 6.61307e7 0.135004
\(305\) 3.85842e7 0.0778682
\(306\) 1.42068e8 0.283447
\(307\) −6.20104e8 −1.22315 −0.611575 0.791186i \(-0.709464\pi\)
−0.611575 + 0.791186i \(0.709464\pi\)
\(308\) −8.17258e7 −0.159379
\(309\) −7.51474e7 −0.144897
\(310\) −4.35757e7 −0.0830765
\(311\) −4.03113e8 −0.759915 −0.379958 0.925004i \(-0.624061\pi\)
−0.379958 + 0.925004i \(0.624061\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) 4.74610e8 0.874846 0.437423 0.899256i \(-0.355891\pi\)
0.437423 + 0.899256i \(0.355891\pi\)
\(314\) 4.53985e8 0.827537
\(315\) −1.13135e7 −0.0203944
\(316\) −3.37963e8 −0.602510
\(317\) −6.83280e8 −1.20473 −0.602367 0.798219i \(-0.705776\pi\)
−0.602367 + 0.798219i \(0.705776\pi\)
\(318\) −3.92705e7 −0.0684812
\(319\) 8.41814e8 1.45194
\(320\) 1.18608e7 0.0202344
\(321\) −2.79705e8 −0.471989
\(322\) 5.41895e7 0.0904522
\(323\) 3.93298e8 0.649401
\(324\) 3.40122e7 0.0555556
\(325\) −1.67143e8 −0.270083
\(326\) −2.07157e8 −0.331161
\(327\) −1.43892e8 −0.227572
\(328\) −2.69741e8 −0.422074
\(329\) −2.95637e8 −0.457692
\(330\) −3.63843e7 −0.0557333
\(331\) −7.91703e8 −1.19995 −0.599977 0.800018i \(-0.704823\pi\)
−0.599977 + 0.800018i \(0.704823\pi\)
\(332\) −2.02827e8 −0.304189
\(333\) −3.43304e8 −0.509477
\(334\) 1.40633e8 0.206526
\(335\) −4.36350e6 −0.00634129
\(336\) 3.79331e7 0.0545545
\(337\) −5.23438e8 −0.745008 −0.372504 0.928031i \(-0.621501\pi\)
−0.372504 + 0.928031i \(0.621501\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 1.07342e7 0.0149649
\(340\) 7.05398e7 0.0973325
\(341\) −4.48191e8 −0.612101
\(342\) 9.41588e7 0.127283
\(343\) −4.03536e7 −0.0539949
\(344\) 1.75826e8 0.232878
\(345\) 2.41251e7 0.0316303
\(346\) 7.76503e8 1.00781
\(347\) 2.91343e8 0.374327 0.187164 0.982329i \(-0.440071\pi\)
0.187164 + 0.982329i \(0.440071\pi\)
\(348\) −3.90728e8 −0.496990
\(349\) −1.53659e9 −1.93495 −0.967474 0.252969i \(-0.918593\pi\)
−0.967474 + 0.252969i \(0.918593\pi\)
\(350\) 2.08758e8 0.260258
\(351\) −4.32436e7 −0.0533761
\(352\) 1.21993e8 0.149086
\(353\) 4.16447e8 0.503904 0.251952 0.967740i \(-0.418927\pi\)
0.251952 + 0.967740i \(0.418927\pi\)
\(354\) 1.52142e8 0.182279
\(355\) 2.42302e8 0.287447
\(356\) 6.98098e8 0.820052
\(357\) 2.25599e8 0.262421
\(358\) 3.80325e7 0.0438091
\(359\) 8.22125e8 0.937794 0.468897 0.883253i \(-0.344652\pi\)
0.468897 + 0.883253i \(0.344652\pi\)
\(360\) 1.68878e7 0.0190772
\(361\) −6.33204e8 −0.708384
\(362\) 1.19835e8 0.132771
\(363\) 1.51928e8 0.166711
\(364\) −4.82285e7 −0.0524142
\(365\) −1.54291e8 −0.166079
\(366\) −1.84199e8 −0.196383
\(367\) −2.14976e8 −0.227018 −0.113509 0.993537i \(-0.536209\pi\)
−0.113509 + 0.993537i \(0.536209\pi\)
\(368\) −8.08892e7 −0.0846103
\(369\) −3.84065e8 −0.397935
\(370\) −1.70458e8 −0.174949
\(371\) −6.23601e7 −0.0634013
\(372\) 2.08028e8 0.209518
\(373\) −9.51709e7 −0.0949561 −0.0474781 0.998872i \(-0.515118\pi\)
−0.0474781 + 0.998872i \(0.515118\pi\)
\(374\) 7.25527e8 0.717138
\(375\) 1.88379e8 0.184468
\(376\) 4.41301e8 0.428132
\(377\) 4.96777e8 0.477493
\(378\) 5.40102e7 0.0514344
\(379\) 3.52162e8 0.332281 0.166140 0.986102i \(-0.446870\pi\)
0.166140 + 0.986102i \(0.446870\pi\)
\(380\) 4.67518e7 0.0437075
\(381\) 4.58147e8 0.424392
\(382\) 1.23161e9 1.13045
\(383\) 9.50839e8 0.864791 0.432395 0.901684i \(-0.357668\pi\)
0.432395 + 0.901684i \(0.357668\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −5.77769e7 −0.0515991
\(386\) −1.39623e9 −1.23566
\(387\) 2.50346e8 0.219559
\(388\) 7.06379e8 0.613941
\(389\) 3.05702e7 0.0263314 0.0131657 0.999913i \(-0.495809\pi\)
0.0131657 + 0.999913i \(0.495809\pi\)
\(390\) −2.14713e7 −0.0183288
\(391\) −4.81071e8 −0.406997
\(392\) 6.02363e7 0.0505076
\(393\) 8.19591e8 0.681119
\(394\) 9.70704e8 0.799558
\(395\) −2.38926e8 −0.195063
\(396\) 1.73697e8 0.140559
\(397\) −1.02611e9 −0.823052 −0.411526 0.911398i \(-0.635004\pi\)
−0.411526 + 0.911398i \(0.635004\pi\)
\(398\) 8.03036e8 0.638476
\(399\) 1.49521e8 0.117841
\(400\) −3.11615e8 −0.243449
\(401\) 1.95089e9 1.51087 0.755434 0.655225i \(-0.227426\pi\)
0.755434 + 0.655225i \(0.227426\pi\)
\(402\) 2.08312e7 0.0159927
\(403\) −2.64490e8 −0.201299
\(404\) 6.84726e8 0.516633
\(405\) 2.40453e7 0.0179861
\(406\) −6.20462e8 −0.460123
\(407\) −1.75322e9 −1.28901
\(408\) −3.36754e8 −0.245472
\(409\) 1.53654e8 0.111049 0.0555243 0.998457i \(-0.482317\pi\)
0.0555243 + 0.998457i \(0.482317\pi\)
\(410\) −1.90696e8 −0.136647
\(411\) −5.05379e8 −0.359063
\(412\) 1.78127e8 0.125484
\(413\) 2.41596e8 0.168758
\(414\) −1.15172e8 −0.0797714
\(415\) −1.43391e8 −0.0984812
\(416\) 7.19913e7 0.0490290
\(417\) 5.75154e8 0.388426
\(418\) 4.80860e8 0.322034
\(419\) −8.57510e8 −0.569495 −0.284747 0.958603i \(-0.591910\pi\)
−0.284747 + 0.958603i \(0.591910\pi\)
\(420\) 2.68172e7 0.0176620
\(421\) 1.72844e9 1.12893 0.564464 0.825458i \(-0.309083\pi\)
0.564464 + 0.825458i \(0.309083\pi\)
\(422\) −5.04261e8 −0.326634
\(423\) 6.28337e8 0.403646
\(424\) 9.30856e7 0.0593065
\(425\) −1.85326e9 −1.17105
\(426\) −1.15674e9 −0.724939
\(427\) −2.92502e8 −0.181816
\(428\) 6.63003e8 0.408754
\(429\) −2.20841e8 −0.135045
\(430\) 1.24302e8 0.0753943
\(431\) −2.49520e9 −1.50119 −0.750594 0.660763i \(-0.770233\pi\)
−0.750594 + 0.660763i \(0.770233\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 2.13733e9 1.26521 0.632606 0.774474i \(-0.281985\pi\)
0.632606 + 0.774474i \(0.281985\pi\)
\(434\) 3.30341e8 0.193976
\(435\) −2.76230e8 −0.160901
\(436\) 3.41077e8 0.197083
\(437\) −3.18841e8 −0.182763
\(438\) 7.36578e8 0.418851
\(439\) 1.64700e8 0.0929112 0.0464556 0.998920i \(-0.485207\pi\)
0.0464556 + 0.998920i \(0.485207\pi\)
\(440\) 8.62442e7 0.0482665
\(441\) 8.57661e7 0.0476190
\(442\) 4.28153e8 0.235842
\(443\) 8.32466e8 0.454939 0.227470 0.973785i \(-0.426955\pi\)
0.227470 + 0.973785i \(0.426955\pi\)
\(444\) 8.13758e8 0.441220
\(445\) 4.93528e8 0.265492
\(446\) 2.26092e9 1.20674
\(447\) −3.75607e8 −0.198910
\(448\) −8.99154e7 −0.0472456
\(449\) −2.55539e8 −0.133228 −0.0666140 0.997779i \(-0.521220\pi\)
−0.0666140 + 0.997779i \(0.521220\pi\)
\(450\) −4.43686e8 −0.229526
\(451\) −1.96138e9 −1.00680
\(452\) −2.54441e7 −0.0129599
\(453\) −2.03356e9 −1.02781
\(454\) 1.75672e9 0.881062
\(455\) −3.40957e7 −0.0169691
\(456\) −2.23191e8 −0.110230
\(457\) 7.17790e7 0.0351796 0.0175898 0.999845i \(-0.494401\pi\)
0.0175898 + 0.999845i \(0.494401\pi\)
\(458\) −3.35700e8 −0.163276
\(459\) −4.79479e8 −0.231433
\(460\) −5.71855e7 −0.0273926
\(461\) 6.01481e8 0.285936 0.142968 0.989727i \(-0.454335\pi\)
0.142968 + 0.989727i \(0.454335\pi\)
\(462\) 2.75824e8 0.130133
\(463\) 1.28593e9 0.602120 0.301060 0.953605i \(-0.402660\pi\)
0.301060 + 0.953605i \(0.402660\pi\)
\(464\) 9.26171e8 0.430406
\(465\) 1.47068e8 0.0678316
\(466\) −7.26960e8 −0.332782
\(467\) 2.45950e9 1.11747 0.558736 0.829345i \(-0.311286\pi\)
0.558736 + 0.829345i \(0.311286\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 3.30791e7 0.0148064
\(470\) 3.11982e8 0.138608
\(471\) −1.53220e9 −0.675681
\(472\) −3.60633e8 −0.157859
\(473\) 1.27849e9 0.555500
\(474\) 1.14062e9 0.491947
\(475\) −1.22829e9 −0.525865
\(476\) −5.34752e8 −0.227263
\(477\) 1.32538e8 0.0559147
\(478\) −6.44675e8 −0.269987
\(479\) −2.53463e7 −0.0105376 −0.00526878 0.999986i \(-0.501677\pi\)
−0.00526878 + 0.999986i \(0.501677\pi\)
\(480\) −4.00303e7 −0.0165213
\(481\) −1.03462e9 −0.423910
\(482\) 3.02289e9 1.22958
\(483\) −1.82889e8 −0.0738539
\(484\) −3.60125e8 −0.144376
\(485\) 4.99382e8 0.198764
\(486\) −1.14791e8 −0.0453609
\(487\) 3.70470e8 0.145345 0.0726727 0.997356i \(-0.476847\pi\)
0.0726727 + 0.997356i \(0.476847\pi\)
\(488\) 4.36621e8 0.170073
\(489\) 6.99156e8 0.270392
\(490\) 4.25847e7 0.0163519
\(491\) 2.53115e9 0.965011 0.482506 0.875893i \(-0.339727\pi\)
0.482506 + 0.875893i \(0.339727\pi\)
\(492\) 9.10375e8 0.344622
\(493\) 5.50820e9 2.07036
\(494\) 2.83768e8 0.105906
\(495\) 1.22797e8 0.0455061
\(496\) −4.93104e8 −0.181448
\(497\) −1.83686e9 −0.671163
\(498\) 6.84542e8 0.248369
\(499\) −4.12645e9 −1.48670 −0.743352 0.668901i \(-0.766765\pi\)
−0.743352 + 0.668901i \(0.766765\pi\)
\(500\) −4.46527e8 −0.159754
\(501\) −4.74636e8 −0.168628
\(502\) 4.10686e7 0.0144893
\(503\) −1.78396e9 −0.625025 −0.312512 0.949914i \(-0.601171\pi\)
−0.312512 + 0.949914i \(0.601171\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 4.84074e8 0.167260
\(506\) −5.88174e8 −0.201827
\(507\) −1.30324e8 −0.0444116
\(508\) −1.08598e9 −0.367534
\(509\) −1.87446e9 −0.630034 −0.315017 0.949086i \(-0.602010\pi\)
−0.315017 + 0.949086i \(0.602010\pi\)
\(510\) −2.38072e8 −0.0794716
\(511\) 1.16966e9 0.387781
\(512\) 1.34218e8 0.0441942
\(513\) −3.17786e8 −0.103926
\(514\) 1.93574e9 0.628747
\(515\) 1.25929e8 0.0406256
\(516\) −5.93413e8 −0.190144
\(517\) 3.20885e9 1.02125
\(518\) 1.29222e9 0.408490
\(519\) −2.62070e9 −0.822870
\(520\) 5.08950e7 0.0158732
\(521\) 5.40774e9 1.67527 0.837633 0.546234i \(-0.183939\pi\)
0.837633 + 0.546234i \(0.183939\pi\)
\(522\) 1.31871e9 0.405791
\(523\) −4.58115e9 −1.40029 −0.700147 0.713999i \(-0.746882\pi\)
−0.700147 + 0.713999i \(0.746882\pi\)
\(524\) −1.94273e9 −0.589867
\(525\) −7.04557e8 −0.212500
\(526\) −3.70306e9 −1.10946
\(527\) −2.93263e9 −0.872811
\(528\) −4.11726e8 −0.121728
\(529\) −3.01483e9 −0.885458
\(530\) 6.58079e7 0.0192005
\(531\) −5.13479e8 −0.148831
\(532\) −3.54419e8 −0.102053
\(533\) −1.15746e9 −0.331102
\(534\) −2.35608e9 −0.669570
\(535\) 4.68717e8 0.132334
\(536\) −4.93775e7 −0.0138501
\(537\) −1.28360e8 −0.0357700
\(538\) 2.88216e9 0.797957
\(539\) 4.37999e8 0.120479
\(540\) −5.69963e7 −0.0155764
\(541\) −4.46285e9 −1.21177 −0.605887 0.795550i \(-0.707182\pi\)
−0.605887 + 0.795550i \(0.707182\pi\)
\(542\) 2.82325e9 0.761644
\(543\) −4.04442e8 −0.108407
\(544\) 7.98231e8 0.212585
\(545\) 2.41128e8 0.0638057
\(546\) 1.62771e8 0.0427960
\(547\) −1.06071e9 −0.277102 −0.138551 0.990355i \(-0.544245\pi\)
−0.138551 + 0.990355i \(0.544245\pi\)
\(548\) 1.19793e9 0.310958
\(549\) 6.21673e8 0.160346
\(550\) −2.26586e9 −0.580716
\(551\) 3.65069e9 0.929703
\(552\) 2.73001e8 0.0690840
\(553\) 1.81127e9 0.455455
\(554\) −5.50788e8 −0.137626
\(555\) 5.75295e8 0.142845
\(556\) −1.36333e9 −0.336387
\(557\) 7.57958e8 0.185846 0.0929228 0.995673i \(-0.470379\pi\)
0.0929228 + 0.995673i \(0.470379\pi\)
\(558\) −7.02095e8 −0.171071
\(559\) 7.54472e8 0.182685
\(560\) −6.35666e7 −0.0152958
\(561\) −2.44865e9 −0.585541
\(562\) 5.78349e9 1.37440
\(563\) −7.07097e9 −1.66993 −0.834967 0.550300i \(-0.814513\pi\)
−0.834967 + 0.550300i \(0.814513\pi\)
\(564\) −1.48939e9 −0.349568
\(565\) −1.79880e7 −0.00419579
\(566\) 1.33023e9 0.308367
\(567\) −1.82284e8 −0.0419961
\(568\) 2.74190e9 0.627816
\(569\) 2.97246e9 0.676431 0.338216 0.941069i \(-0.390177\pi\)
0.338216 + 0.941069i \(0.390177\pi\)
\(570\) −1.57787e8 −0.0356870
\(571\) −1.74612e9 −0.392508 −0.196254 0.980553i \(-0.562878\pi\)
−0.196254 + 0.980553i \(0.562878\pi\)
\(572\) 5.23474e8 0.116952
\(573\) −4.15668e9 −0.923009
\(574\) 1.44564e9 0.319058
\(575\) 1.50241e9 0.329573
\(576\) 1.91103e8 0.0416667
\(577\) −2.37230e9 −0.514107 −0.257053 0.966397i \(-0.582752\pi\)
−0.257053 + 0.966397i \(0.582752\pi\)
\(578\) 1.46460e9 0.315480
\(579\) 4.71227e9 1.00892
\(580\) 6.54767e8 0.139344
\(581\) 1.08703e9 0.229945
\(582\) −2.38403e9 −0.501281
\(583\) 6.76858e8 0.141468
\(584\) −1.74596e9 −0.362736
\(585\) 7.24658e7 0.0149654
\(586\) 7.88264e8 0.161819
\(587\) −2.43530e9 −0.496958 −0.248479 0.968637i \(-0.579931\pi\)
−0.248479 + 0.968637i \(0.579931\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −1.94367e9 −0.391939
\(590\) −2.54953e8 −0.0511068
\(591\) −3.27613e9 −0.652837
\(592\) −1.92891e9 −0.382108
\(593\) −9.78771e9 −1.92748 −0.963740 0.266844i \(-0.914019\pi\)
−0.963740 + 0.266844i \(0.914019\pi\)
\(594\) −5.86228e8 −0.114766
\(595\) −3.78049e8 −0.0735764
\(596\) 8.90328e8 0.172261
\(597\) −2.71025e9 −0.521313
\(598\) −3.47097e8 −0.0663738
\(599\) −6.71217e9 −1.27605 −0.638027 0.770014i \(-0.720249\pi\)
−0.638027 + 0.770014i \(0.720249\pi\)
\(600\) 1.05170e9 0.198775
\(601\) 6.12890e9 1.15165 0.575827 0.817571i \(-0.304680\pi\)
0.575827 + 0.817571i \(0.304680\pi\)
\(602\) −9.42317e8 −0.176039
\(603\) −7.03051e7 −0.0130580
\(604\) 4.82029e9 0.890111
\(605\) −2.54595e8 −0.0467418
\(606\) −2.31095e9 −0.421829
\(607\) −7.16389e9 −1.30013 −0.650067 0.759877i \(-0.725259\pi\)
−0.650067 + 0.759877i \(0.725259\pi\)
\(608\) 5.29046e8 0.0954620
\(609\) 2.09406e9 0.375689
\(610\) 3.08674e8 0.0550612
\(611\) 1.89363e9 0.335854
\(612\) 1.13654e9 0.200427
\(613\) 2.29204e9 0.401893 0.200946 0.979602i \(-0.435598\pi\)
0.200946 + 0.979602i \(0.435598\pi\)
\(614\) −4.96083e9 −0.864898
\(615\) 6.43600e8 0.111571
\(616\) −6.53806e8 −0.112698
\(617\) −3.39556e9 −0.581988 −0.290994 0.956725i \(-0.593986\pi\)
−0.290994 + 0.956725i \(0.593986\pi\)
\(618\) −6.01179e8 −0.102458
\(619\) 1.75199e9 0.296904 0.148452 0.988920i \(-0.452571\pi\)
0.148452 + 0.988920i \(0.452571\pi\)
\(620\) −3.48605e8 −0.0587439
\(621\) 3.88707e8 0.0651331
\(622\) −3.22490e9 −0.537341
\(623\) −3.74137e9 −0.619901
\(624\) −2.42971e8 −0.0400320
\(625\) 5.62790e9 0.922076
\(626\) 3.79688e9 0.618609
\(627\) −1.62290e9 −0.262939
\(628\) 3.63188e9 0.585157
\(629\) −1.14718e10 −1.83803
\(630\) −9.05080e7 −0.0144210
\(631\) −8.12154e9 −1.28687 −0.643436 0.765500i \(-0.722492\pi\)
−0.643436 + 0.765500i \(0.722492\pi\)
\(632\) −2.70370e9 −0.426039
\(633\) 1.70188e9 0.266696
\(634\) −5.46624e9 −0.851876
\(635\) −7.67744e8 −0.118989
\(636\) −3.14164e8 −0.0484235
\(637\) 2.58475e8 0.0396214
\(638\) 6.73451e9 1.02668
\(639\) 3.90399e9 0.591910
\(640\) 9.48867e7 0.0143079
\(641\) −1.03353e10 −1.54996 −0.774981 0.631985i \(-0.782241\pi\)
−0.774981 + 0.631985i \(0.782241\pi\)
\(642\) −2.23764e9 −0.333747
\(643\) 8.14924e9 1.20887 0.604434 0.796655i \(-0.293399\pi\)
0.604434 + 0.796655i \(0.293399\pi\)
\(644\) 4.33516e8 0.0639594
\(645\) −4.19519e8 −0.0615592
\(646\) 3.14639e9 0.459196
\(647\) −3.31452e9 −0.481122 −0.240561 0.970634i \(-0.577331\pi\)
−0.240561 + 0.970634i \(0.577331\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −2.62229e9 −0.376551
\(650\) −1.33714e9 −0.190977
\(651\) −1.11490e9 −0.158381
\(652\) −1.65726e9 −0.234166
\(653\) −2.59491e9 −0.364692 −0.182346 0.983234i \(-0.558369\pi\)
−0.182346 + 0.983234i \(0.558369\pi\)
\(654\) −1.15113e9 −0.160918
\(655\) −1.37344e9 −0.190970
\(656\) −2.15793e9 −0.298451
\(657\) −2.48595e9 −0.341990
\(658\) −2.36510e9 −0.323637
\(659\) 4.26176e9 0.580083 0.290042 0.957014i \(-0.406331\pi\)
0.290042 + 0.957014i \(0.406331\pi\)
\(660\) −2.91074e8 −0.0394094
\(661\) 1.39258e10 1.87549 0.937743 0.347329i \(-0.112911\pi\)
0.937743 + 0.347329i \(0.112911\pi\)
\(662\) −6.33362e9 −0.848495
\(663\) −1.44502e9 −0.192564
\(664\) −1.62262e9 −0.215094
\(665\) −2.50561e8 −0.0330398
\(666\) −2.74643e9 −0.360254
\(667\) −4.46542e9 −0.582669
\(668\) 1.12506e9 0.146036
\(669\) −7.63059e9 −0.985296
\(670\) −3.49080e7 −0.00448397
\(671\) 3.17482e9 0.405687
\(672\) 3.03464e8 0.0385758
\(673\) −6.26914e9 −0.792785 −0.396392 0.918081i \(-0.629738\pi\)
−0.396392 + 0.918081i \(0.629738\pi\)
\(674\) −4.18750e9 −0.526800
\(675\) 1.49744e9 0.187407
\(676\) 3.08916e8 0.0384615
\(677\) −9.16513e9 −1.13521 −0.567607 0.823299i \(-0.692131\pi\)
−0.567607 + 0.823299i \(0.692131\pi\)
\(678\) 8.58739e7 0.0105817
\(679\) −3.78575e9 −0.464096
\(680\) 5.64318e8 0.0688244
\(681\) −5.92892e9 −0.719384
\(682\) −3.58553e9 −0.432821
\(683\) 9.24488e9 1.11027 0.555135 0.831760i \(-0.312667\pi\)
0.555135 + 0.831760i \(0.312667\pi\)
\(684\) 7.53270e8 0.0900025
\(685\) 8.46893e8 0.100673
\(686\) −3.22829e8 −0.0381802
\(687\) 1.13299e9 0.133314
\(688\) 1.40661e9 0.164670
\(689\) 3.99432e8 0.0465238
\(690\) 1.93001e8 0.0223660
\(691\) 8.26194e9 0.952597 0.476298 0.879284i \(-0.341978\pi\)
0.476298 + 0.879284i \(0.341978\pi\)
\(692\) 6.21202e9 0.712626
\(693\) −9.30908e8 −0.106253
\(694\) 2.33074e9 0.264689
\(695\) −9.63820e8 −0.108905
\(696\) −3.12583e9 −0.351425
\(697\) −1.28338e10 −1.43562
\(698\) −1.22927e10 −1.36822
\(699\) 2.45349e9 0.271715
\(700\) 1.67006e9 0.184030
\(701\) −1.01497e10 −1.11286 −0.556430 0.830894i \(-0.687829\pi\)
−0.556430 + 0.830894i \(0.687829\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) −7.60317e9 −0.825375
\(704\) 9.75944e8 0.105419
\(705\) −1.05294e9 −0.113173
\(706\) 3.33157e9 0.356314
\(707\) −3.66970e9 −0.390538
\(708\) 1.21714e9 0.128891
\(709\) −1.11510e10 −1.17504 −0.587518 0.809211i \(-0.699895\pi\)
−0.587518 + 0.809211i \(0.699895\pi\)
\(710\) 1.93841e9 0.203256
\(711\) −3.84961e9 −0.401673
\(712\) 5.58478e9 0.579864
\(713\) 2.37744e9 0.245638
\(714\) 1.80479e9 0.185559
\(715\) 3.70075e8 0.0378634
\(716\) 3.04260e8 0.0309777
\(717\) 2.17578e9 0.220443
\(718\) 6.57700e9 0.663120
\(719\) −1.37090e10 −1.37549 −0.687743 0.725954i \(-0.741398\pi\)
−0.687743 + 0.725954i \(0.741398\pi\)
\(720\) 1.35102e8 0.0134896
\(721\) −9.54650e8 −0.0948573
\(722\) −5.06563e9 −0.500903
\(723\) −1.02023e10 −1.00395
\(724\) 9.58677e8 0.0938830
\(725\) −1.72024e10 −1.67651
\(726\) 1.21542e9 0.117883
\(727\) 1.87696e9 0.181170 0.0905848 0.995889i \(-0.471126\pi\)
0.0905848 + 0.995889i \(0.471126\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −1.23433e9 −0.117436
\(731\) 8.36550e9 0.792102
\(732\) −1.47360e9 −0.138864
\(733\) 1.20773e10 1.13267 0.566336 0.824174i \(-0.308360\pi\)
0.566336 + 0.824174i \(0.308360\pi\)
\(734\) −1.71981e9 −0.160526
\(735\) −1.43723e8 −0.0133512
\(736\) −6.47114e8 −0.0598285
\(737\) −3.59041e8 −0.0330376
\(738\) −3.07252e9 −0.281382
\(739\) 1.37380e10 1.25218 0.626091 0.779750i \(-0.284654\pi\)
0.626091 + 0.779750i \(0.284654\pi\)
\(740\) −1.36366e9 −0.123707
\(741\) −9.57717e8 −0.0864716
\(742\) −4.98881e8 −0.0448315
\(743\) −2.15574e10 −1.92813 −0.964065 0.265667i \(-0.914408\pi\)
−0.964065 + 0.265667i \(0.914408\pi\)
\(744\) 1.66423e9 0.148152
\(745\) 6.29427e8 0.0557697
\(746\) −7.61367e8 −0.0671441
\(747\) −2.31033e9 −0.202792
\(748\) 5.80422e9 0.507093
\(749\) −3.55328e9 −0.308989
\(750\) 1.50703e9 0.130439
\(751\) −1.91870e9 −0.165298 −0.0826488 0.996579i \(-0.526338\pi\)
−0.0826488 + 0.996579i \(0.526338\pi\)
\(752\) 3.53041e9 0.302735
\(753\) −1.38607e8 −0.0118305
\(754\) 3.97421e9 0.337638
\(755\) 3.40776e9 0.288174
\(756\) 4.32081e8 0.0363696
\(757\) 1.18531e9 0.0993110 0.0496555 0.998766i \(-0.484188\pi\)
0.0496555 + 0.998766i \(0.484188\pi\)
\(758\) 2.81730e9 0.234958
\(759\) 1.98509e9 0.164791
\(760\) 3.74015e8 0.0309059
\(761\) 9.79789e9 0.805910 0.402955 0.915220i \(-0.367983\pi\)
0.402955 + 0.915220i \(0.367983\pi\)
\(762\) 3.66518e9 0.300091
\(763\) −1.82796e9 −0.148981
\(764\) 9.85288e9 0.799349
\(765\) 8.03492e8 0.0648883
\(766\) 7.60671e9 0.611500
\(767\) −1.54748e9 −0.123835
\(768\) −4.52985e8 −0.0360844
\(769\) 4.65848e9 0.369404 0.184702 0.982795i \(-0.440868\pi\)
0.184702 + 0.982795i \(0.440868\pi\)
\(770\) −4.62215e8 −0.0364860
\(771\) −6.53312e9 −0.513370
\(772\) −1.11698e10 −0.873747
\(773\) −2.61317e9 −0.203488 −0.101744 0.994811i \(-0.532442\pi\)
−0.101744 + 0.994811i \(0.532442\pi\)
\(774\) 2.00277e9 0.155252
\(775\) 9.15876e9 0.706775
\(776\) 5.65103e9 0.434122
\(777\) −4.36124e9 −0.333531
\(778\) 2.44561e8 0.0186191
\(779\) −8.50590e9 −0.644672
\(780\) −1.71771e8 −0.0129604
\(781\) 1.99373e10 1.49757
\(782\) −3.84857e9 −0.287790
\(783\) −4.45064e9 −0.331327
\(784\) 4.81890e8 0.0357143
\(785\) 2.56760e9 0.189445
\(786\) 6.55673e9 0.481624
\(787\) −2.85339e9 −0.208665 −0.104332 0.994542i \(-0.533271\pi\)
−0.104332 + 0.994542i \(0.533271\pi\)
\(788\) 7.76563e9 0.565373
\(789\) 1.24978e10 0.905867
\(790\) −1.91141e9 −0.137930
\(791\) 1.36365e8 0.00979680
\(792\) 1.38958e9 0.0993904
\(793\) 1.87355e9 0.133416
\(794\) −8.20888e9 −0.581986
\(795\) −2.22102e8 −0.0156771
\(796\) 6.42429e9 0.451470
\(797\) 2.55168e9 0.178534 0.0892671 0.996008i \(-0.471548\pi\)
0.0892671 + 0.996008i \(0.471548\pi\)
\(798\) 1.19617e9 0.0833261
\(799\) 2.09963e10 1.45623
\(800\) −2.49292e9 −0.172145
\(801\) 7.95177e9 0.546701
\(802\) 1.56071e10 1.06834
\(803\) −1.26955e10 −0.865258
\(804\) 1.66649e8 0.0113086
\(805\) 3.06479e8 0.0207069
\(806\) −2.11592e9 −0.142340
\(807\) −9.72728e9 −0.651529
\(808\) 5.47780e9 0.365314
\(809\) 9.89115e9 0.656791 0.328395 0.944540i \(-0.393492\pi\)
0.328395 + 0.944540i \(0.393492\pi\)
\(810\) 1.92362e8 0.0127181
\(811\) 2.28009e10 1.50099 0.750497 0.660874i \(-0.229814\pi\)
0.750497 + 0.660874i \(0.229814\pi\)
\(812\) −4.96370e9 −0.325356
\(813\) −9.52847e9 −0.621880
\(814\) −1.40258e10 −0.911467
\(815\) −1.17162e9 −0.0758114
\(816\) −2.69403e9 −0.173575
\(817\) 5.54442e9 0.355696
\(818\) 1.22924e9 0.0785233
\(819\) −5.49353e8 −0.0349428
\(820\) −1.52557e9 −0.0966237
\(821\) −1.42769e10 −0.900396 −0.450198 0.892929i \(-0.648647\pi\)
−0.450198 + 0.892929i \(0.648647\pi\)
\(822\) −4.04303e9 −0.253896
\(823\) −2.44680e10 −1.53003 −0.765014 0.644013i \(-0.777268\pi\)
−0.765014 + 0.644013i \(0.777268\pi\)
\(824\) 1.42502e9 0.0887309
\(825\) 7.64728e9 0.474153
\(826\) 1.93277e9 0.119330
\(827\) 5.12659e9 0.315181 0.157590 0.987505i \(-0.449627\pi\)
0.157590 + 0.987505i \(0.449627\pi\)
\(828\) −9.21379e8 −0.0564069
\(829\) −1.62718e10 −0.991962 −0.495981 0.868333i \(-0.665191\pi\)
−0.495981 + 0.868333i \(0.665191\pi\)
\(830\) −1.14713e9 −0.0696367
\(831\) 1.85891e9 0.112371
\(832\) 5.75930e8 0.0346688
\(833\) 2.86594e9 0.171795
\(834\) 4.60124e9 0.274659
\(835\) 7.95376e8 0.0472792
\(836\) 3.84688e9 0.227712
\(837\) 2.36957e9 0.139679
\(838\) −6.86008e9 −0.402694
\(839\) 1.65243e10 0.965953 0.482976 0.875633i \(-0.339556\pi\)
0.482976 + 0.875633i \(0.339556\pi\)
\(840\) 2.14537e8 0.0124889
\(841\) 3.38786e10 1.96399
\(842\) 1.38275e10 0.798273
\(843\) −1.95193e10 −1.12219
\(844\) −4.03409e9 −0.230965
\(845\) 2.18391e8 0.0124519
\(846\) 5.02669e9 0.285421
\(847\) 1.93005e9 0.109138
\(848\) 7.44685e8 0.0419360
\(849\) −4.48951e9 −0.251781
\(850\) −1.48261e10 −0.828058
\(851\) 9.29999e9 0.517284
\(852\) −9.25391e9 −0.512609
\(853\) 5.24311e9 0.289246 0.144623 0.989487i \(-0.453803\pi\)
0.144623 + 0.989487i \(0.453803\pi\)
\(854\) −2.34002e9 −0.128563
\(855\) 5.32533e8 0.0291383
\(856\) 5.30403e9 0.289033
\(857\) 3.67505e10 1.99448 0.997241 0.0742354i \(-0.0236516\pi\)
0.997241 + 0.0742354i \(0.0236516\pi\)
\(858\) −1.76672e9 −0.0954912
\(859\) −1.21564e10 −0.654380 −0.327190 0.944959i \(-0.606102\pi\)
−0.327190 + 0.944959i \(0.606102\pi\)
\(860\) 9.94416e8 0.0533118
\(861\) −4.87904e9 −0.260510
\(862\) −1.99616e10 −1.06150
\(863\) −1.50179e10 −0.795372 −0.397686 0.917522i \(-0.630187\pi\)
−0.397686 + 0.917522i \(0.630187\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 4.39166e9 0.230713
\(866\) 1.70986e10 0.894639
\(867\) −4.94302e9 −0.257588
\(868\) 2.64273e9 0.137162
\(869\) −1.96596e10 −1.01626
\(870\) −2.20984e9 −0.113774
\(871\) −2.11880e8 −0.0108649
\(872\) 2.72862e9 0.139359
\(873\) 8.04610e9 0.409294
\(874\) −2.55073e9 −0.129233
\(875\) 2.39311e9 0.120763
\(876\) 5.89263e9 0.296172
\(877\) 2.52800e10 1.26555 0.632773 0.774338i \(-0.281917\pi\)
0.632773 + 0.774338i \(0.281917\pi\)
\(878\) 1.31760e9 0.0656981
\(879\) −2.66039e9 −0.132125
\(880\) 6.89954e8 0.0341296
\(881\) −1.19057e10 −0.586595 −0.293298 0.956021i \(-0.594753\pi\)
−0.293298 + 0.956021i \(0.594753\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 1.07352e10 0.524744 0.262372 0.964967i \(-0.415495\pi\)
0.262372 + 0.964967i \(0.415495\pi\)
\(884\) 3.42522e9 0.166765
\(885\) 8.60468e8 0.0417285
\(886\) 6.65973e9 0.321691
\(887\) 8.48147e9 0.408073 0.204037 0.978963i \(-0.434594\pi\)
0.204037 + 0.978963i \(0.434594\pi\)
\(888\) 6.51007e9 0.311989
\(889\) 5.82016e9 0.277830
\(890\) 3.94822e9 0.187731
\(891\) 1.97852e9 0.0937061
\(892\) 1.80873e10 0.853292
\(893\) 1.39158e10 0.653925
\(894\) −3.00486e9 −0.140651
\(895\) 2.15100e8 0.0100290
\(896\) −7.19323e8 −0.0334077
\(897\) 1.17145e9 0.0541940
\(898\) −2.04431e9 −0.0942064
\(899\) −2.72214e10 −1.24954
\(900\) −3.54949e9 −0.162299
\(901\) 4.42885e9 0.201723
\(902\) −1.56910e10 −0.711916
\(903\) 3.18032e9 0.143735
\(904\) −2.03553e8 −0.00916406
\(905\) 6.77747e8 0.0303947
\(906\) −1.62685e10 −0.726772
\(907\) 2.28935e10 1.01879 0.509397 0.860532i \(-0.329868\pi\)
0.509397 + 0.860532i \(0.329868\pi\)
\(908\) 1.40537e10 0.623005
\(909\) 7.79945e9 0.344422
\(910\) −2.72765e8 −0.0119990
\(911\) 6.83396e9 0.299473 0.149737 0.988726i \(-0.452157\pi\)
0.149737 + 0.988726i \(0.452157\pi\)
\(912\) −1.78553e9 −0.0779444
\(913\) −1.17986e10 −0.513078
\(914\) 5.74232e8 0.0248757
\(915\) −1.04177e9 −0.0449573
\(916\) −2.68560e9 −0.115453
\(917\) 1.04118e10 0.445897
\(918\) −3.83584e9 −0.163648
\(919\) 2.41526e10 1.02650 0.513251 0.858239i \(-0.328441\pi\)
0.513251 + 0.858239i \(0.328441\pi\)
\(920\) −4.57484e8 −0.0193695
\(921\) 1.67428e10 0.706186
\(922\) 4.81185e9 0.202187
\(923\) 1.17655e10 0.492499
\(924\) 2.20660e9 0.0920176
\(925\) 3.58269e10 1.48838
\(926\) 1.02874e10 0.425763
\(927\) 2.02898e9 0.0836563
\(928\) 7.40937e9 0.304343
\(929\) −3.37778e10 −1.38222 −0.691110 0.722750i \(-0.742878\pi\)
−0.691110 + 0.722750i \(0.742878\pi\)
\(930\) 1.17654e9 0.0479642
\(931\) 1.89947e9 0.0771450
\(932\) −5.81568e9 −0.235312
\(933\) 1.08840e10 0.438737
\(934\) 1.96760e10 0.790173
\(935\) 4.10335e9 0.164172
\(936\) 8.20026e8 0.0326860
\(937\) 3.03888e10 1.20677 0.603385 0.797450i \(-0.293818\pi\)
0.603385 + 0.797450i \(0.293818\pi\)
\(938\) 2.64633e8 0.0104697
\(939\) −1.28145e10 −0.505092
\(940\) 2.49586e9 0.0980105
\(941\) −2.49435e10 −0.975873 −0.487937 0.872879i \(-0.662250\pi\)
−0.487937 + 0.872879i \(0.662250\pi\)
\(942\) −1.22576e10 −0.477779
\(943\) 1.04042e10 0.404033
\(944\) −2.88506e9 −0.111623
\(945\) 3.05464e8 0.0117747
\(946\) 1.02279e10 0.392798
\(947\) −4.07511e10 −1.55925 −0.779623 0.626249i \(-0.784589\pi\)
−0.779623 + 0.626249i \(0.784589\pi\)
\(948\) 9.12500e9 0.347859
\(949\) −7.49196e9 −0.284553
\(950\) −9.82633e9 −0.371842
\(951\) 1.84486e10 0.695553
\(952\) −4.27802e9 −0.160699
\(953\) −4.32313e8 −0.0161798 −0.00808989 0.999967i \(-0.502575\pi\)
−0.00808989 + 0.999967i \(0.502575\pi\)
\(954\) 1.06030e9 0.0395377
\(955\) 6.96560e9 0.258789
\(956\) −5.15740e9 −0.190910
\(957\) −2.27290e10 −0.838278
\(958\) −2.02770e8 −0.00745118
\(959\) −6.42018e9 −0.235062
\(960\) −3.20242e8 −0.0116823
\(961\) −1.30196e10 −0.473225
\(962\) −8.27698e9 −0.299750
\(963\) 7.55202e9 0.272503
\(964\) 2.41831e10 0.869446
\(965\) −7.89663e9 −0.282876
\(966\) −1.46312e9 −0.0522226
\(967\) 2.43671e9 0.0866583 0.0433292 0.999061i \(-0.486204\pi\)
0.0433292 + 0.999061i \(0.486204\pi\)
\(968\) −2.88100e9 −0.102089
\(969\) −1.06191e10 −0.374932
\(970\) 3.99506e9 0.140547
\(971\) −1.69523e10 −0.594240 −0.297120 0.954840i \(-0.596026\pi\)
−0.297120 + 0.954840i \(0.596026\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 7.30659e9 0.254284
\(974\) 2.96376e9 0.102775
\(975\) 4.51286e9 0.155932
\(976\) 3.49297e9 0.120260
\(977\) −1.45782e10 −0.500119 −0.250059 0.968231i \(-0.580450\pi\)
−0.250059 + 0.968231i \(0.580450\pi\)
\(978\) 5.59325e9 0.191196
\(979\) 4.06089e10 1.38319
\(980\) 3.40678e8 0.0115625
\(981\) 3.88508e9 0.131389
\(982\) 2.02492e10 0.682366
\(983\) −3.82318e10 −1.28377 −0.641886 0.766800i \(-0.721848\pi\)
−0.641886 + 0.766800i \(0.721848\pi\)
\(984\) 7.28300e9 0.243684
\(985\) 5.49000e9 0.183040
\(986\) 4.40656e10 1.46397
\(987\) 7.98220e9 0.264249
\(988\) 2.27014e9 0.0748866
\(989\) −6.78178e9 −0.222924
\(990\) 9.82376e8 0.0321777
\(991\) 1.67284e9 0.0546005 0.0273002 0.999627i \(-0.491309\pi\)
0.0273002 + 0.999627i \(0.491309\pi\)
\(992\) −3.94483e9 −0.128303
\(993\) 2.13760e10 0.692793
\(994\) −1.46949e10 −0.474584
\(995\) 4.54172e9 0.146164
\(996\) 5.47634e9 0.175623
\(997\) 4.99020e10 1.59472 0.797360 0.603504i \(-0.206229\pi\)
0.797360 + 0.603504i \(0.206229\pi\)
\(998\) −3.30116e10 −1.05126
\(999\) 9.26921e9 0.294147
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.r.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.r.1.4 6 1.1 even 1 trivial