Properties

Label 546.8.a.q.1.2
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 1.2
Root \(-309.150\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -307.150 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -307.150 q^{5} -216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -2457.20 q^{10} -7421.86 q^{11} -1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +8293.06 q^{15} +4096.00 q^{16} -5217.71 q^{17} +5832.00 q^{18} -610.200 q^{19} -19657.6 q^{20} -9261.00 q^{21} -59374.9 q^{22} -53724.9 q^{23} -13824.0 q^{24} +16216.4 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +55488.3 q^{29} +66344.5 q^{30} -13695.2 q^{31} +32768.0 q^{32} +200390. q^{33} -41741.7 q^{34} -105353. q^{35} +46656.0 q^{36} -143013. q^{37} -4881.60 q^{38} +59319.0 q^{39} -157261. q^{40} -303729. q^{41} -74088.0 q^{42} -68421.4 q^{43} -474999. q^{44} -223913. q^{45} -429799. q^{46} -927094. q^{47} -110592. q^{48} +117649. q^{49} +129731. q^{50} +140878. q^{51} -140608. q^{52} +1.71806e6 q^{53} -157464. q^{54} +2.27963e6 q^{55} +175616. q^{56} +16475.4 q^{57} +443906. q^{58} -316898. q^{59} +530756. q^{60} +1.21928e6 q^{61} -109562. q^{62} +250047. q^{63} +262144. q^{64} +674809. q^{65} +1.60312e6 q^{66} -2.28233e6 q^{67} -333934. q^{68} +1.45057e6 q^{69} -842821. q^{70} -2.74579e6 q^{71} +373248. q^{72} -3.47442e6 q^{73} -1.14410e6 q^{74} -437841. q^{75} -39052.8 q^{76} -2.54570e6 q^{77} +474552. q^{78} +4.82863e6 q^{79} -1.25809e6 q^{80} +531441. q^{81} -2.42984e6 q^{82} -2.84197e6 q^{83} -592704. q^{84} +1.60262e6 q^{85} -547372. q^{86} -1.49818e6 q^{87} -3.79999e6 q^{88} -419990. q^{89} -1.79130e6 q^{90} -753571. q^{91} -3.43839e6 q^{92} +369771. q^{93} -7.41675e6 q^{94} +187423. q^{95} -884736. q^{96} -5.79829e6 q^{97} +941192. q^{98} -5.41053e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 13 q^{5} - 1296 q^{6} + 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 104 q^{10} + 10054 q^{11} - 10368 q^{12} - 13182 q^{13} + 16464 q^{14} - 351 q^{15} + 24576 q^{16} + 21222 q^{17} + 34992 q^{18} + 9527 q^{19} + 832 q^{20} - 55566 q^{21} + 80432 q^{22} + 33229 q^{23} - 82944 q^{24} + 265321 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} + 174185 q^{29} - 2808 q^{30} + 119045 q^{31} + 196608 q^{32} - 271458 q^{33} + 169776 q^{34} + 4459 q^{35} + 279936 q^{36} + 56562 q^{37} + 76216 q^{38} + 355914 q^{39} + 6656 q^{40} + 101632 q^{41} - 444528 q^{42} + 441323 q^{43} + 643456 q^{44} + 9477 q^{45} + 265832 q^{46} - 892849 q^{47} - 663552 q^{48} + 705894 q^{49} + 2122568 q^{50} - 572994 q^{51} - 843648 q^{52} + 2093965 q^{53} - 944784 q^{54} - 331222 q^{55} + 1053696 q^{56} - 257229 q^{57} + 1393480 q^{58} - 136204 q^{59} - 22464 q^{60} - 3989946 q^{61} + 952360 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 28561 q^{65} - 2171664 q^{66} - 2218250 q^{67} + 1358208 q^{68} - 897183 q^{69} + 35672 q^{70} + 2045928 q^{71} + 2239488 q^{72} - 8557479 q^{73} + 452496 q^{74} - 7163667 q^{75} + 609728 q^{76} + 3448522 q^{77} + 2847312 q^{78} - 8559709 q^{79} + 53248 q^{80} + 3188646 q^{81} + 813056 q^{82} + 2496351 q^{83} - 3556224 q^{84} + 5335304 q^{85} + 3530584 q^{86} - 4702995 q^{87} + 5147648 q^{88} - 2446683 q^{89} + 75816 q^{90} - 4521426 q^{91} + 2126656 q^{92} - 3214215 q^{93} - 7142792 q^{94} + 16410211 q^{95} - 5308416 q^{96} + 5786889 q^{97} + 5647152 q^{98} + 7329366 q^{99} + O(q^{100}) \)

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
\(5\) −307.150 −1.09889 −0.549447 0.835528i \(-0.685162\pi\)
−0.549447 + 0.835528i \(0.685162\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −2457.20 −0.777036
\(11\) −7421.86 −1.68127 −0.840636 0.541600i \(-0.817819\pi\)
−0.840636 + 0.541600i \(0.817819\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) 8293.06 0.634447
\(16\) 4096.00 0.250000
\(17\) −5217.71 −0.257578 −0.128789 0.991672i \(-0.541109\pi\)
−0.128789 + 0.991672i \(0.541109\pi\)
\(18\) 5832.00 0.235702
\(19\) −610.200 −0.0204096 −0.0102048 0.999948i \(-0.503248\pi\)
−0.0102048 + 0.999948i \(0.503248\pi\)
\(20\) −19657.6 −0.549447
\(21\) −9261.00 −0.218218
\(22\) −59374.9 −1.18884
\(23\) −53724.9 −0.920721 −0.460360 0.887732i \(-0.652280\pi\)
−0.460360 + 0.887732i \(0.652280\pi\)
\(24\) −13824.0 −0.204124
\(25\) 16216.4 0.207569
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 55488.3 0.422482 0.211241 0.977434i \(-0.432250\pi\)
0.211241 + 0.977434i \(0.432250\pi\)
\(30\) 66344.5 0.448622
\(31\) −13695.2 −0.0825664 −0.0412832 0.999147i \(-0.513145\pi\)
−0.0412832 + 0.999147i \(0.513145\pi\)
\(32\) 32768.0 0.176777
\(33\) 200390. 0.970683
\(34\) −41741.7 −0.182135
\(35\) −105353. −0.415343
\(36\) 46656.0 0.166667
\(37\) −143013. −0.464161 −0.232080 0.972697i \(-0.574553\pi\)
−0.232080 + 0.972697i \(0.574553\pi\)
\(38\) −4881.60 −0.0144318
\(39\) 59319.0 0.160128
\(40\) −157261. −0.388518
\(41\) −303729. −0.688246 −0.344123 0.938925i \(-0.611824\pi\)
−0.344123 + 0.938925i \(0.611824\pi\)
\(42\) −74088.0 −0.154303
\(43\) −68421.4 −0.131236 −0.0656180 0.997845i \(-0.520902\pi\)
−0.0656180 + 0.997845i \(0.520902\pi\)
\(44\) −474999. −0.840636
\(45\) −223913. −0.366298
\(46\) −429799. −0.651048
\(47\) −927094. −1.30251 −0.651255 0.758859i \(-0.725757\pi\)
−0.651255 + 0.758859i \(0.725757\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 129731. 0.146774
\(51\) 140878. 0.148713
\(52\) −140608. −0.138675
\(53\) 1.71806e6 1.58515 0.792577 0.609772i \(-0.208739\pi\)
0.792577 + 0.609772i \(0.208739\pi\)
\(54\) −157464. −0.136083
\(55\) 2.27963e6 1.84754
\(56\) 175616. 0.133631
\(57\) 16475.4 0.0117835
\(58\) 443906. 0.298740
\(59\) −316898. −0.200880 −0.100440 0.994943i \(-0.532025\pi\)
−0.100440 + 0.994943i \(0.532025\pi\)
\(60\) 530756. 0.317224
\(61\) 1.21928e6 0.687778 0.343889 0.939010i \(-0.388256\pi\)
0.343889 + 0.939010i \(0.388256\pi\)
\(62\) −109562. −0.0583833
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) 674809. 0.304779
\(66\) 1.60312e6 0.686377
\(67\) −2.28233e6 −0.927081 −0.463540 0.886076i \(-0.653421\pi\)
−0.463540 + 0.886076i \(0.653421\pi\)
\(68\) −333934. −0.128789
\(69\) 1.45057e6 0.531578
\(70\) −842821. −0.293692
\(71\) −2.74579e6 −0.910466 −0.455233 0.890372i \(-0.650444\pi\)
−0.455233 + 0.890372i \(0.650444\pi\)
\(72\) 373248. 0.117851
\(73\) −3.47442e6 −1.04533 −0.522664 0.852539i \(-0.675062\pi\)
−0.522664 + 0.852539i \(0.675062\pi\)
\(74\) −1.14410e6 −0.328211
\(75\) −437841. −0.119840
\(76\) −39052.8 −0.0102048
\(77\) −2.54570e6 −0.635461
\(78\) 474552. 0.113228
\(79\) 4.82863e6 1.10187 0.550933 0.834549i \(-0.314272\pi\)
0.550933 + 0.834549i \(0.314272\pi\)
\(80\) −1.25809e6 −0.274724
\(81\) 531441. 0.111111
\(82\) −2.42984e6 −0.486663
\(83\) −2.84197e6 −0.545564 −0.272782 0.962076i \(-0.587944\pi\)
−0.272782 + 0.962076i \(0.587944\pi\)
\(84\) −592704. −0.109109
\(85\) 1.60262e6 0.283051
\(86\) −547372. −0.0927978
\(87\) −1.49818e6 −0.243920
\(88\) −3.79999e6 −0.594420
\(89\) −419990. −0.0631501 −0.0315751 0.999501i \(-0.510052\pi\)
−0.0315751 + 0.999501i \(0.510052\pi\)
\(90\) −1.79130e6 −0.259012
\(91\) −753571. −0.104828
\(92\) −3.43839e6 −0.460360
\(93\) 369771. 0.0476697
\(94\) −7.41675e6 −0.921014
\(95\) 187423. 0.0224280
\(96\) −884736. −0.102062
\(97\) −5.79829e6 −0.645058 −0.322529 0.946560i \(-0.604533\pi\)
−0.322529 + 0.946560i \(0.604533\pi\)
\(98\) 941192. 0.101015
\(99\) −5.41053e6 −0.560424
\(100\) 1.03785e6 0.103785
\(101\) 1.36733e7 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(102\) 1.12703e6 0.105156
\(103\) 8.83659e6 0.796809 0.398405 0.917210i \(-0.369564\pi\)
0.398405 + 0.917210i \(0.369564\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 2.84452e6 0.239798
\(106\) 1.37444e7 1.12087
\(107\) −8.19534e6 −0.646731 −0.323366 0.946274i \(-0.604814\pi\)
−0.323366 + 0.946274i \(0.604814\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 7.31130e6 0.540757 0.270378 0.962754i \(-0.412851\pi\)
0.270378 + 0.962754i \(0.412851\pi\)
\(110\) 1.82370e7 1.30641
\(111\) 3.86134e6 0.267983
\(112\) 1.40493e6 0.0944911
\(113\) −9.07202e6 −0.591466 −0.295733 0.955271i \(-0.595564\pi\)
−0.295733 + 0.955271i \(0.595564\pi\)
\(114\) 131803. 0.00833219
\(115\) 1.65016e7 1.01177
\(116\) 3.55125e6 0.211241
\(117\) −1.60161e6 −0.0924500
\(118\) −2.53519e6 −0.142044
\(119\) −1.78968e6 −0.0973553
\(120\) 4.24605e6 0.224311
\(121\) 3.55968e7 1.82668
\(122\) 9.75423e6 0.486333
\(123\) 8.20069e6 0.397359
\(124\) −876495. −0.0412832
\(125\) 1.90153e7 0.870798
\(126\) 2.00038e6 0.0890871
\(127\) 3.47437e7 1.50509 0.752545 0.658541i \(-0.228826\pi\)
0.752545 + 0.658541i \(0.228826\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 1.84738e6 0.0757691
\(130\) 5.39847e6 0.215511
\(131\) 928307. 0.0360780 0.0180390 0.999837i \(-0.494258\pi\)
0.0180390 + 0.999837i \(0.494258\pi\)
\(132\) 1.28250e7 0.485342
\(133\) −209299. −0.00771411
\(134\) −1.82587e7 −0.655545
\(135\) 6.04564e6 0.211482
\(136\) −2.67147e6 −0.0910676
\(137\) 8.16328e6 0.271233 0.135617 0.990761i \(-0.456698\pi\)
0.135617 + 0.990761i \(0.456698\pi\)
\(138\) 1.16046e7 0.375883
\(139\) −1.18165e7 −0.373196 −0.186598 0.982436i \(-0.559746\pi\)
−0.186598 + 0.982436i \(0.559746\pi\)
\(140\) −6.74256e6 −0.207672
\(141\) 2.50315e7 0.752005
\(142\) −2.19664e7 −0.643797
\(143\) 1.63058e7 0.466301
\(144\) 2.98598e6 0.0833333
\(145\) −1.70432e7 −0.464263
\(146\) −2.77954e7 −0.739158
\(147\) −3.17652e6 −0.0824786
\(148\) −9.15281e6 −0.232080
\(149\) 2.13672e7 0.529170 0.264585 0.964362i \(-0.414765\pi\)
0.264585 + 0.964362i \(0.414765\pi\)
\(150\) −3.50273e6 −0.0847398
\(151\) 4.40020e7 1.04005 0.520023 0.854152i \(-0.325923\pi\)
0.520023 + 0.854152i \(0.325923\pi\)
\(152\) −312422. −0.00721588
\(153\) −3.80371e6 −0.0858593
\(154\) −2.03656e7 −0.449339
\(155\) 4.20650e6 0.0907318
\(156\) 3.79642e6 0.0800641
\(157\) 8.55801e7 1.76492 0.882458 0.470391i \(-0.155887\pi\)
0.882458 + 0.470391i \(0.155887\pi\)
\(158\) 3.86290e7 0.779137
\(159\) −4.63875e7 −0.915189
\(160\) −1.00647e7 −0.194259
\(161\) −1.84276e7 −0.348000
\(162\) 4.25153e6 0.0785674
\(163\) 1.02567e8 1.85503 0.927513 0.373790i \(-0.121942\pi\)
0.927513 + 0.373790i \(0.121942\pi\)
\(164\) −1.94387e7 −0.344123
\(165\) −6.15499e7 −1.06668
\(166\) −2.27357e7 −0.385772
\(167\) −4.69828e7 −0.780605 −0.390303 0.920687i \(-0.627630\pi\)
−0.390303 + 0.920687i \(0.627630\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 1.28210e7 0.200147
\(171\) −444836. −0.00680320
\(172\) −4.37897e6 −0.0656180
\(173\) 2.45043e7 0.359816 0.179908 0.983683i \(-0.442420\pi\)
0.179908 + 0.983683i \(0.442420\pi\)
\(174\) −1.19855e7 −0.172478
\(175\) 5.56221e6 0.0784538
\(176\) −3.03999e7 −0.420318
\(177\) 8.55625e6 0.115978
\(178\) −3.35992e6 −0.0446539
\(179\) −2.71333e7 −0.353604 −0.176802 0.984246i \(-0.556575\pi\)
−0.176802 + 0.984246i \(0.556575\pi\)
\(180\) −1.43304e7 −0.183149
\(181\) −9.04524e7 −1.13382 −0.566911 0.823779i \(-0.691862\pi\)
−0.566911 + 0.823779i \(0.691862\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −3.29205e7 −0.397089
\(184\) −2.75071e7 −0.325524
\(185\) 4.39264e7 0.510064
\(186\) 2.95817e6 0.0337076
\(187\) 3.87251e7 0.433059
\(188\) −5.93340e7 −0.651255
\(189\) −6.75127e6 −0.0727393
\(190\) 1.49939e6 0.0158590
\(191\) 1.61288e8 1.67489 0.837443 0.546525i \(-0.184049\pi\)
0.837443 + 0.546525i \(0.184049\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 4.29648e7 0.430192 0.215096 0.976593i \(-0.430994\pi\)
0.215096 + 0.976593i \(0.430994\pi\)
\(194\) −4.63863e7 −0.456125
\(195\) −1.82199e7 −0.175964
\(196\) 7.52954e6 0.0714286
\(197\) −1.01679e8 −0.947548 −0.473774 0.880646i \(-0.657109\pi\)
−0.473774 + 0.880646i \(0.657109\pi\)
\(198\) −4.32843e7 −0.396280
\(199\) 7.99607e7 0.719268 0.359634 0.933093i \(-0.382901\pi\)
0.359634 + 0.933093i \(0.382901\pi\)
\(200\) 8.30277e6 0.0733868
\(201\) 6.16230e7 0.535250
\(202\) 1.09387e8 0.933758
\(203\) 1.90325e7 0.159683
\(204\) 9.01621e6 0.0743564
\(205\) 9.32906e7 0.756309
\(206\) 7.06927e7 0.563429
\(207\) −3.91654e7 −0.306907
\(208\) −8.99891e6 −0.0693375
\(209\) 4.52882e6 0.0343141
\(210\) 2.27562e7 0.169563
\(211\) 2.05436e7 0.150552 0.0752761 0.997163i \(-0.476016\pi\)
0.0752761 + 0.997163i \(0.476016\pi\)
\(212\) 1.09956e8 0.792577
\(213\) 7.41365e7 0.525658
\(214\) −6.55627e7 −0.457308
\(215\) 2.10157e7 0.144214
\(216\) −1.00777e7 −0.0680414
\(217\) −4.69746e6 −0.0312072
\(218\) 5.84904e7 0.382373
\(219\) 9.38094e7 0.603520
\(220\) 1.45896e8 0.923771
\(221\) 1.14633e7 0.0714393
\(222\) 3.08907e7 0.189493
\(223\) 660460. 0.00398822 0.00199411 0.999998i \(-0.499365\pi\)
0.00199411 + 0.999998i \(0.499365\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) 1.18217e7 0.0691898
\(226\) −7.25762e7 −0.418229
\(227\) −2.60405e8 −1.47761 −0.738804 0.673920i \(-0.764609\pi\)
−0.738804 + 0.673920i \(0.764609\pi\)
\(228\) 1.05443e6 0.00589175
\(229\) 1.41759e8 0.780057 0.390028 0.920803i \(-0.372465\pi\)
0.390028 + 0.920803i \(0.372465\pi\)
\(230\) 1.32013e8 0.715433
\(231\) 6.87338e7 0.366884
\(232\) 2.84100e7 0.149370
\(233\) 1.36610e7 0.0707518 0.0353759 0.999374i \(-0.488737\pi\)
0.0353759 + 0.999374i \(0.488737\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 2.84757e8 1.43132
\(236\) −2.02815e7 −0.100440
\(237\) −1.30373e8 −0.636163
\(238\) −1.43174e7 −0.0688406
\(239\) −7.19950e7 −0.341122 −0.170561 0.985347i \(-0.554558\pi\)
−0.170561 + 0.985347i \(0.554558\pi\)
\(240\) 3.39684e7 0.158612
\(241\) 3.34392e8 1.53885 0.769425 0.638737i \(-0.220543\pi\)
0.769425 + 0.638737i \(0.220543\pi\)
\(242\) 2.84774e8 1.29166
\(243\) −1.43489e7 −0.0641500
\(244\) 7.80338e7 0.343889
\(245\) −3.61359e7 −0.156985
\(246\) 6.56056e7 0.280975
\(247\) 1.34061e6 0.00566061
\(248\) −7.01196e6 −0.0291916
\(249\) 7.67331e7 0.314981
\(250\) 1.52122e8 0.615747
\(251\) −8.40775e6 −0.0335600 −0.0167800 0.999859i \(-0.505341\pi\)
−0.0167800 + 0.999859i \(0.505341\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 3.98738e8 1.54798
\(254\) 2.77949e8 1.06426
\(255\) −4.32708e7 −0.163420
\(256\) 1.67772e7 0.0625000
\(257\) 1.07446e8 0.394842 0.197421 0.980319i \(-0.436743\pi\)
0.197421 + 0.980319i \(0.436743\pi\)
\(258\) 1.47790e7 0.0535768
\(259\) −4.90534e7 −0.175436
\(260\) 4.31878e7 0.152389
\(261\) 4.04509e7 0.140827
\(262\) 7.42646e6 0.0255110
\(263\) 6.92726e7 0.234810 0.117405 0.993084i \(-0.462542\pi\)
0.117405 + 0.993084i \(0.462542\pi\)
\(264\) 1.02600e8 0.343188
\(265\) −5.27701e8 −1.74192
\(266\) −1.67439e6 −0.00545470
\(267\) 1.13397e7 0.0364597
\(268\) −1.46069e8 −0.463540
\(269\) 1.25605e8 0.393435 0.196718 0.980460i \(-0.436972\pi\)
0.196718 + 0.980460i \(0.436972\pi\)
\(270\) 4.83651e7 0.149541
\(271\) −3.20934e8 −0.979543 −0.489771 0.871851i \(-0.662920\pi\)
−0.489771 + 0.871851i \(0.662920\pi\)
\(272\) −2.13717e7 −0.0643945
\(273\) 2.03464e7 0.0605228
\(274\) 6.53062e7 0.191791
\(275\) −1.20355e8 −0.348981
\(276\) 9.28366e7 0.265789
\(277\) 6.25623e7 0.176861 0.0884307 0.996082i \(-0.471815\pi\)
0.0884307 + 0.996082i \(0.471815\pi\)
\(278\) −9.45321e7 −0.263890
\(279\) −9.98382e6 −0.0275221
\(280\) −5.39405e7 −0.146846
\(281\) 5.22648e8 1.40520 0.702599 0.711586i \(-0.252023\pi\)
0.702599 + 0.711586i \(0.252023\pi\)
\(282\) 2.00252e8 0.531748
\(283\) −1.06892e8 −0.280346 −0.140173 0.990127i \(-0.544766\pi\)
−0.140173 + 0.990127i \(0.544766\pi\)
\(284\) −1.75731e8 −0.455233
\(285\) −5.06042e6 −0.0129488
\(286\) 1.30447e8 0.329725
\(287\) −1.04179e8 −0.260132
\(288\) 2.38879e7 0.0589256
\(289\) −3.83114e8 −0.933654
\(290\) −1.36346e8 −0.328284
\(291\) 1.56554e8 0.372425
\(292\) −2.22363e8 −0.522664
\(293\) −2.66459e8 −0.618862 −0.309431 0.950922i \(-0.600139\pi\)
−0.309431 + 0.950922i \(0.600139\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 9.73354e7 0.220746
\(296\) −7.32225e7 −0.164106
\(297\) 1.46084e8 0.323561
\(298\) 1.70937e8 0.374180
\(299\) 1.18034e8 0.255362
\(300\) −2.80219e7 −0.0599201
\(301\) −2.34686e7 −0.0496025
\(302\) 3.52016e8 0.735424
\(303\) −3.69180e8 −0.762410
\(304\) −2.49938e6 −0.00510240
\(305\) −3.74502e8 −0.755796
\(306\) −3.04297e7 −0.0607117
\(307\) 5.30742e8 1.04688 0.523442 0.852061i \(-0.324648\pi\)
0.523442 + 0.852061i \(0.324648\pi\)
\(308\) −1.62925e8 −0.317731
\(309\) −2.38588e8 −0.460038
\(310\) 3.36520e7 0.0641571
\(311\) −3.44691e8 −0.649784 −0.324892 0.945751i \(-0.605328\pi\)
−0.324892 + 0.945751i \(0.605328\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 1.35845e8 0.250402 0.125201 0.992131i \(-0.460042\pi\)
0.125201 + 0.992131i \(0.460042\pi\)
\(314\) 6.84641e8 1.24798
\(315\) −7.68020e7 −0.138448
\(316\) 3.09032e8 0.550933
\(317\) 1.85065e8 0.326301 0.163150 0.986601i \(-0.447834\pi\)
0.163150 + 0.986601i \(0.447834\pi\)
\(318\) −3.71100e8 −0.647137
\(319\) −4.11826e8 −0.710307
\(320\) −8.05176e7 −0.137362
\(321\) 2.21274e8 0.373390
\(322\) −1.47421e8 −0.246073
\(323\) 3.18385e6 0.00525707
\(324\) 3.40122e7 0.0555556
\(325\) −3.56273e7 −0.0575694
\(326\) 8.20534e8 1.31170
\(327\) −1.97405e8 −0.312206
\(328\) −1.55509e8 −0.243332
\(329\) −3.17993e8 −0.492303
\(330\) −4.92399e8 −0.754256
\(331\) −9.81458e7 −0.148756 −0.0743779 0.997230i \(-0.523697\pi\)
−0.0743779 + 0.997230i \(0.523697\pi\)
\(332\) −1.81886e8 −0.272782
\(333\) −1.04256e8 −0.154720
\(334\) −3.75863e8 −0.551971
\(335\) 7.01020e8 1.01876
\(336\) −3.79331e7 −0.0545545
\(337\) 9.10127e8 1.29538 0.647690 0.761904i \(-0.275735\pi\)
0.647690 + 0.761904i \(0.275735\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 2.44945e8 0.341483
\(340\) 1.02568e8 0.141526
\(341\) 1.01644e8 0.138817
\(342\) −3.55869e6 −0.00481059
\(343\) 4.03536e7 0.0539949
\(344\) −3.50318e7 −0.0463989
\(345\) −4.45544e8 −0.584149
\(346\) 1.96034e8 0.254428
\(347\) 2.62561e8 0.337348 0.168674 0.985672i \(-0.446052\pi\)
0.168674 + 0.985672i \(0.446052\pi\)
\(348\) −9.58837e7 −0.121960
\(349\) −5.47050e8 −0.688871 −0.344435 0.938810i \(-0.611930\pi\)
−0.344435 + 0.938810i \(0.611930\pi\)
\(350\) 4.44977e7 0.0554752
\(351\) 4.32436e7 0.0533761
\(352\) −2.43199e8 −0.297210
\(353\) 2.61210e8 0.316067 0.158033 0.987434i \(-0.449485\pi\)
0.158033 + 0.987434i \(0.449485\pi\)
\(354\) 6.84500e7 0.0820091
\(355\) 8.43372e8 1.00051
\(356\) −2.68794e7 −0.0315751
\(357\) 4.83212e7 0.0562081
\(358\) −2.17066e8 −0.250036
\(359\) 1.25452e9 1.43103 0.715513 0.698600i \(-0.246193\pi\)
0.715513 + 0.698600i \(0.246193\pi\)
\(360\) −1.14643e8 −0.129506
\(361\) −8.93499e8 −0.999583
\(362\) −7.23619e8 −0.801733
\(363\) −9.61113e8 −1.05463
\(364\) −4.82285e7 −0.0524142
\(365\) 1.06717e9 1.14870
\(366\) −2.63364e8 −0.280784
\(367\) −1.59118e9 −1.68030 −0.840150 0.542354i \(-0.817533\pi\)
−0.840150 + 0.542354i \(0.817533\pi\)
\(368\) −2.20057e8 −0.230180
\(369\) −2.21419e8 −0.229415
\(370\) 3.51411e8 0.360670
\(371\) 5.89293e8 0.599132
\(372\) 2.36654e7 0.0238349
\(373\) −8.48507e8 −0.846592 −0.423296 0.905991i \(-0.639127\pi\)
−0.423296 + 0.905991i \(0.639127\pi\)
\(374\) 3.09801e8 0.306219
\(375\) −5.13412e8 −0.502755
\(376\) −4.74672e8 −0.460507
\(377\) −1.21908e8 −0.117175
\(378\) −5.40102e7 −0.0514344
\(379\) 9.05904e8 0.854762 0.427381 0.904072i \(-0.359436\pi\)
0.427381 + 0.904072i \(0.359436\pi\)
\(380\) 1.19951e7 0.0112140
\(381\) −9.38079e8 −0.868964
\(382\) 1.29030e9 1.18432
\(383\) 2.52031e8 0.229223 0.114612 0.993410i \(-0.463438\pi\)
0.114612 + 0.993410i \(0.463438\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 7.81912e8 0.698305
\(386\) 3.43718e8 0.304192
\(387\) −4.98792e7 −0.0437453
\(388\) −3.71091e8 −0.322529
\(389\) 1.22239e9 1.05290 0.526449 0.850207i \(-0.323523\pi\)
0.526449 + 0.850207i \(0.323523\pi\)
\(390\) −1.45759e8 −0.124425
\(391\) 2.80321e8 0.237157
\(392\) 6.02363e7 0.0505076
\(393\) −2.50643e7 −0.0208296
\(394\) −8.13435e8 −0.670018
\(395\) −1.48312e9 −1.21084
\(396\) −3.46274e8 −0.280212
\(397\) −2.28816e9 −1.83536 −0.917678 0.397325i \(-0.869939\pi\)
−0.917678 + 0.397325i \(0.869939\pi\)
\(398\) 6.39686e8 0.508600
\(399\) 5.65106e6 0.00445374
\(400\) 6.64222e7 0.0518923
\(401\) −7.96685e8 −0.616994 −0.308497 0.951225i \(-0.599826\pi\)
−0.308497 + 0.951225i \(0.599826\pi\)
\(402\) 4.92984e8 0.378479
\(403\) 3.00884e7 0.0228998
\(404\) 8.75093e8 0.660267
\(405\) −1.63232e8 −0.122099
\(406\) 1.52260e8 0.112913
\(407\) 1.06142e9 0.780381
\(408\) 7.21296e7 0.0525779
\(409\) −1.31730e9 −0.952039 −0.476019 0.879435i \(-0.657921\pi\)
−0.476019 + 0.879435i \(0.657921\pi\)
\(410\) 7.46325e8 0.534791
\(411\) −2.20408e8 −0.156596
\(412\) 5.65542e8 0.398405
\(413\) −1.08696e8 −0.0759257
\(414\) −3.13323e8 −0.217016
\(415\) 8.72911e8 0.599517
\(416\) −7.19913e7 −0.0490290
\(417\) 3.19046e8 0.215465
\(418\) 3.62305e7 0.0242637
\(419\) −4.85663e8 −0.322541 −0.161271 0.986910i \(-0.551559\pi\)
−0.161271 + 0.986910i \(0.551559\pi\)
\(420\) 1.82049e8 0.119899
\(421\) 1.18576e9 0.774482 0.387241 0.921979i \(-0.373428\pi\)
0.387241 + 0.921979i \(0.373428\pi\)
\(422\) 1.64349e8 0.106457
\(423\) −6.75852e8 −0.434170
\(424\) 8.79644e8 0.560437
\(425\) −8.46122e7 −0.0534653
\(426\) 5.93092e8 0.371696
\(427\) 4.18213e8 0.259956
\(428\) −5.24502e8 −0.323366
\(429\) −4.40257e8 −0.269219
\(430\) 1.68125e8 0.101975
\(431\) 1.24407e9 0.748472 0.374236 0.927334i \(-0.377905\pi\)
0.374236 + 0.927334i \(0.377905\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −2.23177e9 −1.32112 −0.660560 0.750773i \(-0.729681\pi\)
−0.660560 + 0.750773i \(0.729681\pi\)
\(434\) −3.75797e7 −0.0220668
\(435\) 4.60168e8 0.268042
\(436\) 4.67923e8 0.270378
\(437\) 3.27829e7 0.0187915
\(438\) 7.50475e8 0.426753
\(439\) 8.05249e8 0.454260 0.227130 0.973864i \(-0.427066\pi\)
0.227130 + 0.973864i \(0.427066\pi\)
\(440\) 1.16717e9 0.653205
\(441\) 8.57661e7 0.0476190
\(442\) 9.17065e7 0.0505152
\(443\) 2.79456e9 1.52722 0.763608 0.645680i \(-0.223426\pi\)
0.763608 + 0.645680i \(0.223426\pi\)
\(444\) 2.47126e8 0.133992
\(445\) 1.29000e8 0.0693953
\(446\) 5.28368e6 0.00282010
\(447\) −5.76914e8 −0.305516
\(448\) 8.99154e7 0.0472456
\(449\) −7.24424e8 −0.377686 −0.188843 0.982007i \(-0.560474\pi\)
−0.188843 + 0.982007i \(0.560474\pi\)
\(450\) 9.45738e7 0.0489245
\(451\) 2.25424e9 1.15713
\(452\) −5.80609e8 −0.295733
\(453\) −1.18805e9 −0.600471
\(454\) −2.08324e9 −1.04483
\(455\) 2.31460e8 0.115195
\(456\) 8.43540e6 0.00416609
\(457\) −1.03355e9 −0.506551 −0.253275 0.967394i \(-0.581508\pi\)
−0.253275 + 0.967394i \(0.581508\pi\)
\(458\) 1.13407e9 0.551583
\(459\) 1.02700e8 0.0495709
\(460\) 1.05610e9 0.505887
\(461\) −7.46265e8 −0.354764 −0.177382 0.984142i \(-0.556763\pi\)
−0.177382 + 0.984142i \(0.556763\pi\)
\(462\) 5.49871e8 0.259426
\(463\) 3.01290e9 1.41075 0.705377 0.708832i \(-0.250778\pi\)
0.705377 + 0.708832i \(0.250778\pi\)
\(464\) 2.27280e8 0.105620
\(465\) −1.13575e8 −0.0523840
\(466\) 1.09288e8 0.0500290
\(467\) −2.46352e9 −1.11930 −0.559650 0.828729i \(-0.689064\pi\)
−0.559650 + 0.828729i \(0.689064\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −7.82841e8 −0.350404
\(470\) 2.27806e9 1.01210
\(471\) −2.31066e9 −1.01897
\(472\) −1.62252e8 −0.0710220
\(473\) 5.07814e8 0.220643
\(474\) −1.04298e9 −0.449835
\(475\) −9.89522e6 −0.00423641
\(476\) −1.14539e8 −0.0486777
\(477\) 1.25246e9 0.528385
\(478\) −5.75960e8 −0.241210
\(479\) 1.83805e7 0.00764158 0.00382079 0.999993i \(-0.498784\pi\)
0.00382079 + 0.999993i \(0.498784\pi\)
\(480\) 2.71747e8 0.112155
\(481\) 3.14199e8 0.128735
\(482\) 2.67514e9 1.08813
\(483\) 4.97546e8 0.200918
\(484\) 2.27819e9 0.913339
\(485\) 1.78095e9 0.708851
\(486\) −1.14791e8 −0.0453609
\(487\) 4.42038e8 0.173424 0.0867118 0.996233i \(-0.472364\pi\)
0.0867118 + 0.996233i \(0.472364\pi\)
\(488\) 6.24271e8 0.243166
\(489\) −2.76930e9 −1.07100
\(490\) −2.89087e8 −0.111005
\(491\) 2.35072e9 0.896223 0.448112 0.893978i \(-0.352097\pi\)
0.448112 + 0.893978i \(0.352097\pi\)
\(492\) 5.24844e8 0.198679
\(493\) −2.89522e8 −0.108822
\(494\) 1.07249e7 0.00400265
\(495\) 1.66185e9 0.615847
\(496\) −5.60957e7 −0.0206416
\(497\) −9.41808e8 −0.344124
\(498\) 6.13865e8 0.222725
\(499\) 1.15163e9 0.414917 0.207459 0.978244i \(-0.433481\pi\)
0.207459 + 0.978244i \(0.433481\pi\)
\(500\) 1.21698e9 0.435399
\(501\) 1.26854e9 0.450683
\(502\) −6.72620e7 −0.0237305
\(503\) −3.31726e9 −1.16223 −0.581114 0.813822i \(-0.697383\pi\)
−0.581114 + 0.813822i \(0.697383\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) −4.19977e9 −1.45113
\(506\) 3.18991e9 1.09459
\(507\) −1.30324e8 −0.0444116
\(508\) 2.22359e9 0.752545
\(509\) −4.32515e9 −1.45375 −0.726873 0.686772i \(-0.759027\pi\)
−0.726873 + 0.686772i \(0.759027\pi\)
\(510\) −3.46166e8 −0.115555
\(511\) −1.19173e9 −0.395097
\(512\) 1.34218e8 0.0441942
\(513\) 1.20106e7 0.00392783
\(514\) 8.59566e8 0.279195
\(515\) −2.71416e9 −0.875609
\(516\) 1.18232e8 0.0378845
\(517\) 6.88076e9 2.18988
\(518\) −3.92427e8 −0.124052
\(519\) −6.61615e8 −0.207740
\(520\) 3.45502e8 0.107755
\(521\) 1.40342e9 0.434766 0.217383 0.976086i \(-0.430248\pi\)
0.217383 + 0.976086i \(0.430248\pi\)
\(522\) 3.23608e8 0.0995800
\(523\) 3.52019e9 1.07599 0.537997 0.842947i \(-0.319181\pi\)
0.537997 + 0.842947i \(0.319181\pi\)
\(524\) 5.94117e7 0.0180390
\(525\) −1.50180e8 −0.0452953
\(526\) 5.54181e8 0.166036
\(527\) 7.14578e7 0.0212673
\(528\) 8.20798e8 0.242671
\(529\) −5.18464e8 −0.152273
\(530\) −4.22161e9 −1.23172
\(531\) −2.31019e8 −0.0669602
\(532\) −1.33951e7 −0.00385705
\(533\) 6.67294e8 0.190885
\(534\) 9.07179e7 0.0257809
\(535\) 2.51720e9 0.710689
\(536\) −1.16856e9 −0.327773
\(537\) 7.32599e8 0.204153
\(538\) 1.00484e9 0.278201
\(539\) −8.73174e8 −0.240182
\(540\) 3.86921e8 0.105741
\(541\) −2.84184e9 −0.771629 −0.385814 0.922576i \(-0.626080\pi\)
−0.385814 + 0.922576i \(0.626080\pi\)
\(542\) −2.56747e9 −0.692641
\(543\) 2.44221e9 0.654612
\(544\) −1.70974e8 −0.0455338
\(545\) −2.24567e9 −0.594235
\(546\) 1.62771e8 0.0427960
\(547\) −1.55988e9 −0.407507 −0.203754 0.979022i \(-0.565314\pi\)
−0.203754 + 0.979022i \(0.565314\pi\)
\(548\) 5.22450e8 0.135617
\(549\) 8.88854e8 0.229259
\(550\) −9.62843e8 −0.246767
\(551\) −3.38589e7 −0.00862269
\(552\) 7.42693e8 0.187941
\(553\) 1.65622e9 0.416466
\(554\) 5.00498e8 0.125060
\(555\) −1.18601e9 −0.294486
\(556\) −7.56256e8 −0.186598
\(557\) 1.60240e9 0.392897 0.196448 0.980514i \(-0.437059\pi\)
0.196448 + 0.980514i \(0.437059\pi\)
\(558\) −7.98706e7 −0.0194611
\(559\) 1.50322e8 0.0363983
\(560\) −4.31524e8 −0.103836
\(561\) −1.04558e9 −0.250027
\(562\) 4.18119e9 0.993625
\(563\) −6.56271e9 −1.54990 −0.774950 0.632023i \(-0.782225\pi\)
−0.774950 + 0.632023i \(0.782225\pi\)
\(564\) 1.60202e9 0.376002
\(565\) 2.78647e9 0.649958
\(566\) −8.55139e8 −0.198235
\(567\) 1.82284e8 0.0419961
\(568\) −1.40585e9 −0.321898
\(569\) −4.75022e9 −1.08099 −0.540493 0.841348i \(-0.681762\pi\)
−0.540493 + 0.841348i \(0.681762\pi\)
\(570\) −4.04834e7 −0.00915619
\(571\) 4.04874e9 0.910108 0.455054 0.890464i \(-0.349620\pi\)
0.455054 + 0.890464i \(0.349620\pi\)
\(572\) 1.04357e9 0.233151
\(573\) −4.35478e9 −0.966996
\(574\) −8.33434e8 −0.183941
\(575\) −8.71221e8 −0.191113
\(576\) 1.91103e8 0.0416667
\(577\) 2.39361e9 0.518726 0.259363 0.965780i \(-0.416487\pi\)
0.259363 + 0.965780i \(0.416487\pi\)
\(578\) −3.06491e9 −0.660193
\(579\) −1.16005e9 −0.248371
\(580\) −1.09077e9 −0.232132
\(581\) −9.74794e8 −0.206204
\(582\) 1.25243e9 0.263344
\(583\) −1.27512e10 −2.66508
\(584\) −1.77890e9 −0.369579
\(585\) 4.91936e8 0.101593
\(586\) −2.13167e9 −0.437601
\(587\) −2.96727e9 −0.605513 −0.302757 0.953068i \(-0.597907\pi\)
−0.302757 + 0.953068i \(0.597907\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 8.35683e6 0.00168515
\(590\) 7.78683e8 0.156091
\(591\) 2.74534e9 0.547067
\(592\) −5.85780e8 −0.116040
\(593\) −6.32481e8 −0.124554 −0.0622768 0.998059i \(-0.519836\pi\)
−0.0622768 + 0.998059i \(0.519836\pi\)
\(594\) 1.16868e9 0.228792
\(595\) 5.49699e8 0.106983
\(596\) 1.36750e9 0.264585
\(597\) −2.15894e9 −0.415270
\(598\) 9.44268e8 0.180568
\(599\) −4.58618e9 −0.871882 −0.435941 0.899975i \(-0.643584\pi\)
−0.435941 + 0.899975i \(0.643584\pi\)
\(600\) −2.24175e8 −0.0423699
\(601\) −6.49684e9 −1.22079 −0.610395 0.792097i \(-0.708989\pi\)
−0.610395 + 0.792097i \(0.708989\pi\)
\(602\) −1.87748e8 −0.0350743
\(603\) −1.66382e9 −0.309027
\(604\) 2.81612e9 0.520023
\(605\) −1.09336e10 −2.00733
\(606\) −2.95344e9 −0.539105
\(607\) 3.47548e9 0.630747 0.315373 0.948968i \(-0.397870\pi\)
0.315373 + 0.948968i \(0.397870\pi\)
\(608\) −1.99950e7 −0.00360794
\(609\) −5.13877e8 −0.0921931
\(610\) −2.99602e9 −0.534428
\(611\) 2.03683e9 0.361252
\(612\) −2.43438e8 −0.0429297
\(613\) −7.66711e8 −0.134437 −0.0672187 0.997738i \(-0.521413\pi\)
−0.0672187 + 0.997738i \(0.521413\pi\)
\(614\) 4.24593e9 0.740259
\(615\) −2.51885e9 −0.436655
\(616\) −1.30340e9 −0.224670
\(617\) −1.19940e9 −0.205573 −0.102787 0.994703i \(-0.532776\pi\)
−0.102787 + 0.994703i \(0.532776\pi\)
\(618\) −1.90870e9 −0.325296
\(619\) −9.43634e9 −1.59914 −0.799569 0.600574i \(-0.794939\pi\)
−0.799569 + 0.600574i \(0.794939\pi\)
\(620\) 2.69216e8 0.0453659
\(621\) 1.05747e9 0.177193
\(622\) −2.75753e9 −0.459467
\(623\) −1.44057e8 −0.0238685
\(624\) 2.42971e8 0.0400320
\(625\) −7.10745e9 −1.16448
\(626\) 1.08676e9 0.177061
\(627\) −1.22278e8 −0.0198113
\(628\) 5.47713e9 0.882458
\(629\) 7.46199e8 0.119558
\(630\) −6.14416e8 −0.0978973
\(631\) −3.35561e9 −0.531703 −0.265851 0.964014i \(-0.585653\pi\)
−0.265851 + 0.964014i \(0.585653\pi\)
\(632\) 2.47226e9 0.389569
\(633\) −5.54676e8 −0.0869214
\(634\) 1.48052e9 0.230729
\(635\) −1.06715e10 −1.65394
\(636\) −2.96880e9 −0.457595
\(637\) −2.58475e8 −0.0396214
\(638\) −3.29461e9 −0.502263
\(639\) −2.00168e9 −0.303489
\(640\) −6.44141e8 −0.0971295
\(641\) −7.13903e9 −1.07062 −0.535311 0.844655i \(-0.679805\pi\)
−0.535311 + 0.844655i \(0.679805\pi\)
\(642\) 1.77019e9 0.264027
\(643\) −1.31859e9 −0.195601 −0.0978006 0.995206i \(-0.531181\pi\)
−0.0978006 + 0.995206i \(0.531181\pi\)
\(644\) −1.17937e9 −0.174000
\(645\) −5.67423e8 −0.0832622
\(646\) 2.54708e7 0.00371731
\(647\) 1.11425e10 1.61741 0.808703 0.588217i \(-0.200170\pi\)
0.808703 + 0.588217i \(0.200170\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 2.35197e9 0.337735
\(650\) −2.85019e8 −0.0407077
\(651\) 1.26832e8 0.0180175
\(652\) 6.56427e9 0.927513
\(653\) −5.38939e8 −0.0757432 −0.0378716 0.999283i \(-0.512058\pi\)
−0.0378716 + 0.999283i \(0.512058\pi\)
\(654\) −1.57924e9 −0.220763
\(655\) −2.85130e8 −0.0396459
\(656\) −1.24408e9 −0.172061
\(657\) −2.53285e9 −0.348443
\(658\) −2.54395e9 −0.348111
\(659\) −9.61970e9 −1.30937 −0.654685 0.755902i \(-0.727199\pi\)
−0.654685 + 0.755902i \(0.727199\pi\)
\(660\) −3.93919e9 −0.533339
\(661\) −3.90947e9 −0.526517 −0.263258 0.964725i \(-0.584797\pi\)
−0.263258 + 0.964725i \(0.584797\pi\)
\(662\) −7.85166e8 −0.105186
\(663\) −3.09509e8 −0.0412455
\(664\) −1.45509e9 −0.192886
\(665\) 6.42861e7 0.00847699
\(666\) −8.34050e8 −0.109404
\(667\) −2.98110e9 −0.388988
\(668\) −3.00690e9 −0.390303
\(669\) −1.78324e7 −0.00230260
\(670\) 5.60816e9 0.720375
\(671\) −9.04931e9 −1.15634
\(672\) −3.03464e8 −0.0385758
\(673\) −1.11640e10 −1.41178 −0.705889 0.708322i \(-0.749452\pi\)
−0.705889 + 0.708322i \(0.749452\pi\)
\(674\) 7.28101e9 0.915972
\(675\) −3.19186e8 −0.0399467
\(676\) 3.08916e8 0.0384615
\(677\) 1.02373e10 1.26801 0.634007 0.773327i \(-0.281409\pi\)
0.634007 + 0.773327i \(0.281409\pi\)
\(678\) 1.95956e9 0.241465
\(679\) −1.98881e9 −0.243809
\(680\) 8.20542e8 0.100074
\(681\) 7.03094e9 0.853097
\(682\) 8.13152e8 0.0981582
\(683\) −5.61057e9 −0.673806 −0.336903 0.941539i \(-0.609379\pi\)
−0.336903 + 0.941539i \(0.609379\pi\)
\(684\) −2.84695e7 −0.00340160
\(685\) −2.50735e9 −0.298057
\(686\) 3.22829e8 0.0381802
\(687\) −3.82749e9 −0.450366
\(688\) −2.80254e8 −0.0328090
\(689\) −3.77457e9 −0.439643
\(690\) −3.56435e9 −0.413055
\(691\) 1.11068e10 1.28061 0.640305 0.768121i \(-0.278808\pi\)
0.640305 + 0.768121i \(0.278808\pi\)
\(692\) 1.56827e9 0.179908
\(693\) −1.85581e9 −0.211820
\(694\) 2.10049e9 0.238541
\(695\) 3.62944e9 0.410103
\(696\) −7.67070e8 −0.0862388
\(697\) 1.58477e9 0.177277
\(698\) −4.37640e9 −0.487105
\(699\) −3.68847e8 −0.0408485
\(700\) 3.55981e8 0.0392269
\(701\) 3.53649e8 0.0387757 0.0193879 0.999812i \(-0.493828\pi\)
0.0193879 + 0.999812i \(0.493828\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 8.72663e7 0.00947334
\(704\) −1.94560e9 −0.210159
\(705\) −7.68845e9 −0.826374
\(706\) 2.08968e9 0.223493
\(707\) 4.68995e9 0.499115
\(708\) 5.47600e8 0.0579892
\(709\) 1.32419e10 1.39537 0.697683 0.716406i \(-0.254214\pi\)
0.697683 + 0.716406i \(0.254214\pi\)
\(710\) 6.74697e9 0.707465
\(711\) 3.52007e9 0.367289
\(712\) −2.15035e8 −0.0223269
\(713\) 7.35775e8 0.0760206
\(714\) 3.86570e8 0.0397452
\(715\) −5.00834e9 −0.512416
\(716\) −1.73653e9 −0.176802
\(717\) 1.94387e9 0.196947
\(718\) 1.00362e10 1.01189
\(719\) −1.19576e10 −1.19975 −0.599877 0.800092i \(-0.704784\pi\)
−0.599877 + 0.800092i \(0.704784\pi\)
\(720\) −9.17146e8 −0.0915745
\(721\) 3.03095e9 0.301166
\(722\) −7.14800e9 −0.706812
\(723\) −9.02859e9 −0.888456
\(724\) −5.78895e9 −0.566911
\(725\) 8.99817e8 0.0876943
\(726\) −7.68891e9 −0.745738
\(727\) −1.41066e10 −1.36161 −0.680805 0.732465i \(-0.738370\pi\)
−0.680805 + 0.732465i \(0.738370\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 8.53736e9 0.812257
\(731\) 3.57003e8 0.0338035
\(732\) −2.10691e9 −0.198545
\(733\) 1.16573e10 1.09328 0.546642 0.837366i \(-0.315906\pi\)
0.546642 + 0.837366i \(0.315906\pi\)
\(734\) −1.27294e10 −1.18815
\(735\) 9.75670e8 0.0906353
\(736\) −1.76046e9 −0.162762
\(737\) 1.69392e10 1.55868
\(738\) −1.77135e9 −0.162221
\(739\) −2.02090e9 −0.184200 −0.0920998 0.995750i \(-0.529358\pi\)
−0.0920998 + 0.995750i \(0.529358\pi\)
\(740\) 2.81129e9 0.255032
\(741\) −3.61965e7 −0.00326815
\(742\) 4.71434e9 0.423650
\(743\) −1.27579e10 −1.14108 −0.570542 0.821268i \(-0.693267\pi\)
−0.570542 + 0.821268i \(0.693267\pi\)
\(744\) 1.89323e8 0.0168538
\(745\) −6.56293e9 −0.581502
\(746\) −6.78805e9 −0.598631
\(747\) −2.07179e9 −0.181855
\(748\) 2.47841e9 0.216529
\(749\) −2.81100e9 −0.244441
\(750\) −4.10730e9 −0.355502
\(751\) −1.64992e9 −0.142143 −0.0710713 0.997471i \(-0.522642\pi\)
−0.0710713 + 0.997471i \(0.522642\pi\)
\(752\) −3.79738e9 −0.325628
\(753\) 2.27009e8 0.0193759
\(754\) −9.75262e8 −0.0828555
\(755\) −1.35152e10 −1.14290
\(756\) −4.32081e8 −0.0363696
\(757\) 2.32674e10 1.94945 0.974725 0.223406i \(-0.0717177\pi\)
0.974725 + 0.223406i \(0.0717177\pi\)
\(758\) 7.24723e9 0.604408
\(759\) −1.07659e10 −0.893728
\(760\) 9.59607e7 0.00792950
\(761\) 7.25075e9 0.596399 0.298199 0.954504i \(-0.403614\pi\)
0.298199 + 0.954504i \(0.403614\pi\)
\(762\) −7.50463e9 −0.614451
\(763\) 2.50778e9 0.204387
\(764\) 1.03224e10 0.837443
\(765\) 1.16831e9 0.0943504
\(766\) 2.01625e9 0.162085
\(767\) 6.96225e8 0.0557142
\(768\) −4.52985e8 −0.0360844
\(769\) 6.28292e9 0.498218 0.249109 0.968475i \(-0.419862\pi\)
0.249109 + 0.968475i \(0.419862\pi\)
\(770\) 6.25529e9 0.493776
\(771\) −2.90104e9 −0.227962
\(772\) 2.74975e9 0.215096
\(773\) 1.19758e10 0.932560 0.466280 0.884637i \(-0.345594\pi\)
0.466280 + 0.884637i \(0.345594\pi\)
\(774\) −3.99034e8 −0.0309326
\(775\) −2.22087e8 −0.0171383
\(776\) −2.96872e9 −0.228063
\(777\) 1.32444e9 0.101288
\(778\) 9.77913e9 0.744512
\(779\) 1.85336e8 0.0140468
\(780\) −1.16607e9 −0.0879820
\(781\) 2.03789e10 1.53074
\(782\) 2.24257e9 0.167696
\(783\) −1.09218e9 −0.0813067
\(784\) 4.81890e8 0.0357143
\(785\) −2.62860e10 −1.93946
\(786\) −2.00514e8 −0.0147288
\(787\) 2.56707e9 0.187727 0.0938634 0.995585i \(-0.470078\pi\)
0.0938634 + 0.995585i \(0.470078\pi\)
\(788\) −6.50748e9 −0.473774
\(789\) −1.87036e9 −0.135568
\(790\) −1.18649e10 −0.856190
\(791\) −3.11170e9 −0.223553
\(792\) −2.77019e9 −0.198140
\(793\) −2.67876e9 −0.190755
\(794\) −1.83053e10 −1.29779
\(795\) 1.42479e10 1.00570
\(796\) 5.11749e9 0.359634
\(797\) 7.83653e9 0.548302 0.274151 0.961687i \(-0.411603\pi\)
0.274151 + 0.961687i \(0.411603\pi\)
\(798\) 4.52085e7 0.00314927
\(799\) 4.83731e9 0.335498
\(800\) 5.31377e8 0.0366934
\(801\) −3.06173e8 −0.0210500
\(802\) −6.37348e9 −0.436281
\(803\) 2.57867e10 1.75748
\(804\) 3.94387e9 0.267625
\(805\) 5.66005e9 0.382415
\(806\) 2.40707e8 0.0161926
\(807\) −3.39133e9 −0.227150
\(808\) 7.00074e9 0.466879
\(809\) 2.34797e10 1.55910 0.779548 0.626343i \(-0.215449\pi\)
0.779548 + 0.626343i \(0.215449\pi\)
\(810\) −1.30586e9 −0.0863373
\(811\) −8.10949e8 −0.0533851 −0.0266926 0.999644i \(-0.508498\pi\)
−0.0266926 + 0.999644i \(0.508498\pi\)
\(812\) 1.21808e9 0.0798416
\(813\) 8.66522e9 0.565539
\(814\) 8.49136e9 0.551813
\(815\) −3.15034e10 −2.03848
\(816\) 5.77037e8 0.0371782
\(817\) 4.17508e7 0.00267847
\(818\) −1.05384e10 −0.673193
\(819\) −5.49353e8 −0.0349428
\(820\) 5.97060e9 0.378155
\(821\) −3.19238e9 −0.201332 −0.100666 0.994920i \(-0.532097\pi\)
−0.100666 + 0.994920i \(0.532097\pi\)
\(822\) −1.76327e9 −0.110730
\(823\) 2.55539e9 0.159793 0.0798966 0.996803i \(-0.474541\pi\)
0.0798966 + 0.996803i \(0.474541\pi\)
\(824\) 4.52433e9 0.281715
\(825\) 3.24960e9 0.201484
\(826\) −8.69569e8 −0.0536876
\(827\) −1.25561e10 −0.771945 −0.385973 0.922510i \(-0.626134\pi\)
−0.385973 + 0.922510i \(0.626134\pi\)
\(828\) −2.50659e9 −0.153453
\(829\) 1.96008e10 1.19490 0.597451 0.801906i \(-0.296180\pi\)
0.597451 + 0.801906i \(0.296180\pi\)
\(830\) 6.98329e9 0.423923
\(831\) −1.68918e9 −0.102111
\(832\) −5.75930e8 −0.0346688
\(833\) −6.13859e8 −0.0367969
\(834\) 2.55237e9 0.152357
\(835\) 1.44308e10 0.857803
\(836\) 2.89844e8 0.0171571
\(837\) 2.69563e8 0.0158899
\(838\) −3.88530e9 −0.228071
\(839\) −3.31937e10 −1.94039 −0.970194 0.242330i \(-0.922088\pi\)
−0.970194 + 0.242330i \(0.922088\pi\)
\(840\) 1.45639e9 0.0847816
\(841\) −1.41709e10 −0.821509
\(842\) 9.48612e9 0.547641
\(843\) −1.41115e10 −0.811291
\(844\) 1.31479e9 0.0752761
\(845\) −1.48256e9 −0.0845304
\(846\) −5.40681e9 −0.307005
\(847\) 1.22097e10 0.690419
\(848\) 7.03715e9 0.396289
\(849\) 2.88609e9 0.161858
\(850\) −6.76898e8 −0.0378057
\(851\) 7.68334e9 0.427363
\(852\) 4.74473e9 0.262829
\(853\) 1.76675e10 0.974663 0.487332 0.873217i \(-0.337970\pi\)
0.487332 + 0.873217i \(0.337970\pi\)
\(854\) 3.34570e9 0.183817
\(855\) 1.36631e8 0.00747600
\(856\) −4.19601e9 −0.228654
\(857\) 1.64284e10 0.891585 0.445792 0.895136i \(-0.352922\pi\)
0.445792 + 0.895136i \(0.352922\pi\)
\(858\) −3.52206e9 −0.190367
\(859\) −2.49779e9 −0.134456 −0.0672280 0.997738i \(-0.521416\pi\)
−0.0672280 + 0.997738i \(0.521416\pi\)
\(860\) 1.34500e9 0.0721072
\(861\) 2.81284e9 0.150187
\(862\) 9.95259e9 0.529250
\(863\) −2.24625e10 −1.18965 −0.594826 0.803854i \(-0.702779\pi\)
−0.594826 + 0.803854i \(0.702779\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −7.52649e9 −0.395400
\(866\) −1.78542e10 −0.934173
\(867\) 1.03441e10 0.539045
\(868\) −3.00638e8 −0.0156036
\(869\) −3.58374e10 −1.85254
\(870\) 3.68134e9 0.189535
\(871\) 5.01429e9 0.257126
\(872\) 3.74339e9 0.191186
\(873\) −4.22695e9 −0.215019
\(874\) 2.62263e8 0.0132876
\(875\) 6.52224e9 0.329131
\(876\) 6.00380e9 0.301760
\(877\) 1.03251e10 0.516885 0.258443 0.966027i \(-0.416791\pi\)
0.258443 + 0.966027i \(0.416791\pi\)
\(878\) 6.44199e9 0.321210
\(879\) 7.19439e9 0.357300
\(880\) 9.33735e9 0.461885
\(881\) 1.26952e10 0.625494 0.312747 0.949836i \(-0.398751\pi\)
0.312747 + 0.949836i \(0.398751\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 3.26436e10 1.59564 0.797821 0.602895i \(-0.205986\pi\)
0.797821 + 0.602895i \(0.205986\pi\)
\(884\) 7.33652e8 0.0357196
\(885\) −2.62806e9 −0.127448
\(886\) 2.23565e10 1.07991
\(887\) 1.48765e10 0.715761 0.357880 0.933767i \(-0.383500\pi\)
0.357880 + 0.933767i \(0.383500\pi\)
\(888\) 1.97701e9 0.0947464
\(889\) 1.19171e10 0.568871
\(890\) 1.03200e9 0.0490699
\(891\) −3.94428e9 −0.186808
\(892\) 4.22694e7 0.00199411
\(893\) 5.65713e8 0.0265837
\(894\) −4.61531e9 −0.216033
\(895\) 8.33400e9 0.388573
\(896\) 7.19323e8 0.0334077
\(897\) −3.18691e9 −0.147433
\(898\) −5.79539e9 −0.267064
\(899\) −7.59925e8 −0.0348828
\(900\) 7.56590e8 0.0345949
\(901\) −8.96432e9 −0.408301
\(902\) 1.80339e10 0.818213
\(903\) 6.33651e8 0.0286380
\(904\) −4.64487e9 −0.209115
\(905\) 2.77825e10 1.24595
\(906\) −9.50442e9 −0.424597
\(907\) 2.62488e10 1.16811 0.584055 0.811714i \(-0.301465\pi\)
0.584055 + 0.811714i \(0.301465\pi\)
\(908\) −1.66659e10 −0.738804
\(909\) 9.96785e9 0.440178
\(910\) 1.85168e9 0.0814555
\(911\) 1.57989e10 0.692329 0.346164 0.938174i \(-0.387484\pi\)
0.346164 + 0.938174i \(0.387484\pi\)
\(912\) 6.74832e7 0.00294587
\(913\) 2.10927e10 0.917242
\(914\) −8.26836e9 −0.358185
\(915\) 1.01116e10 0.436359
\(916\) 9.07257e9 0.390028
\(917\) 3.18409e8 0.0136362
\(918\) 8.21602e8 0.0350519
\(919\) 8.27399e9 0.351650 0.175825 0.984421i \(-0.443741\pi\)
0.175825 + 0.984421i \(0.443741\pi\)
\(920\) 8.44883e9 0.357716
\(921\) −1.43300e10 −0.604419
\(922\) −5.97012e9 −0.250856
\(923\) 6.03251e9 0.252518
\(924\) 4.39896e9 0.183442
\(925\) −2.31914e9 −0.0963455
\(926\) 2.41032e10 0.997554
\(927\) 6.44187e9 0.265603
\(928\) 1.81824e9 0.0746850
\(929\) −1.08563e10 −0.444249 −0.222125 0.975018i \(-0.571299\pi\)
−0.222125 + 0.975018i \(0.571299\pi\)
\(930\) −9.08603e8 −0.0370411
\(931\) −7.17894e7 −0.00291566
\(932\) 8.74305e8 0.0353759
\(933\) 9.30666e9 0.375153
\(934\) −1.97081e10 −0.791464
\(935\) −1.18944e10 −0.475886
\(936\) −8.20026e8 −0.0326860
\(937\) −3.05749e10 −1.21416 −0.607080 0.794641i \(-0.707659\pi\)
−0.607080 + 0.794641i \(0.707659\pi\)
\(938\) −6.26273e9 −0.247773
\(939\) −3.66782e9 −0.144570
\(940\) 1.82245e10 0.715661
\(941\) 2.70986e10 1.06019 0.530094 0.847939i \(-0.322157\pi\)
0.530094 + 0.847939i \(0.322157\pi\)
\(942\) −1.84853e10 −0.720524
\(943\) 1.63178e10 0.633682
\(944\) −1.29802e9 −0.0502201
\(945\) 2.07365e9 0.0799328
\(946\) 4.06251e9 0.156018
\(947\) 1.35307e10 0.517721 0.258860 0.965915i \(-0.416653\pi\)
0.258860 + 0.965915i \(0.416653\pi\)
\(948\) −8.34387e9 −0.318081
\(949\) 7.63330e9 0.289922
\(950\) −7.91617e7 −0.00299559
\(951\) −4.99677e9 −0.188390
\(952\) −9.16314e8 −0.0344203
\(953\) −4.46129e10 −1.66969 −0.834844 0.550486i \(-0.814442\pi\)
−0.834844 + 0.550486i \(0.814442\pi\)
\(954\) 1.00197e10 0.373624
\(955\) −4.95397e10 −1.84052
\(956\) −4.60768e9 −0.170561
\(957\) 1.11193e10 0.410096
\(958\) 1.47044e8 0.00540341
\(959\) 2.80000e9 0.102516
\(960\) 2.17398e9 0.0793059
\(961\) −2.73251e10 −0.993183
\(962\) 2.51359e9 0.0910294
\(963\) −5.97440e9 −0.215577
\(964\) 2.14011e10 0.769425
\(965\) −1.31967e10 −0.472736
\(966\) 3.98037e9 0.142070
\(967\) 2.85038e10 1.01370 0.506851 0.862034i \(-0.330809\pi\)
0.506851 + 0.862034i \(0.330809\pi\)
\(968\) 1.82256e10 0.645828
\(969\) −8.59639e7 −0.00303517
\(970\) 1.42476e10 0.501233
\(971\) −2.06546e10 −0.724018 −0.362009 0.932175i \(-0.617909\pi\)
−0.362009 + 0.932175i \(0.617909\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −4.05306e9 −0.141055
\(974\) 3.53630e9 0.122629
\(975\) 9.61938e8 0.0332377
\(976\) 4.99417e9 0.171945
\(977\) −2.31006e10 −0.792488 −0.396244 0.918145i \(-0.629687\pi\)
−0.396244 + 0.918145i \(0.629687\pi\)
\(978\) −2.21544e10 −0.757311
\(979\) 3.11711e9 0.106173
\(980\) −2.31270e9 −0.0784925
\(981\) 5.32994e9 0.180252
\(982\) 1.88058e10 0.633726
\(983\) −1.78640e10 −0.599848 −0.299924 0.953963i \(-0.596961\pi\)
−0.299924 + 0.953963i \(0.596961\pi\)
\(984\) 4.19876e9 0.140488
\(985\) 3.12309e10 1.04126
\(986\) −2.31617e9 −0.0769488
\(987\) 8.58582e9 0.284231
\(988\) 8.57990e7 0.00283030
\(989\) 3.67593e9 0.120832
\(990\) 1.32948e10 0.435470
\(991\) −3.85854e10 −1.25940 −0.629702 0.776837i \(-0.716823\pi\)
−0.629702 + 0.776837i \(0.716823\pi\)
\(992\) −4.48765e8 −0.0145958
\(993\) 2.64994e9 0.0858842
\(994\) −7.53446e9 −0.243332
\(995\) −2.45600e10 −0.790400
\(996\) 4.91092e9 0.157491
\(997\) 4.66291e10 1.49013 0.745065 0.666992i \(-0.232418\pi\)
0.745065 + 0.666992i \(0.232418\pi\)
\(998\) 9.21305e9 0.293391
\(999\) 2.81492e9 0.0893278
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.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.q.1.2 6 1.1 even 1 trivial