Properties

Label 546.8.a.i.1.3
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 86504 x^{3} - 9117228 x^{2} + 89606664 x + 21810067776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(336.062\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -105.351 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -105.351 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +842.807 q^{10} +7421.28 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -2844.47 q^{15} +4096.00 q^{16} +16122.2 q^{17} -5832.00 q^{18} -36283.4 q^{19} -6742.45 q^{20} +9261.00 q^{21} -59370.3 q^{22} -68040.6 q^{23} -13824.0 q^{24} -67026.2 q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -194800. q^{29} +22755.8 q^{30} +90512.2 q^{31} -32768.0 q^{32} +200375. q^{33} -128977. q^{34} -36135.3 q^{35} +46656.0 q^{36} -438556. q^{37} +290267. q^{38} +59319.0 q^{39} +53939.6 q^{40} -262428. q^{41} -74088.0 q^{42} +164846. q^{43} +474962. q^{44} -76800.8 q^{45} +544325. q^{46} +188894. q^{47} +110592. q^{48} +117649. q^{49} +536210. q^{50} +435298. q^{51} +140608. q^{52} +1.49619e6 q^{53} -157464. q^{54} -781838. q^{55} -175616. q^{56} -979651. q^{57} +1.55840e6 q^{58} -292745. q^{59} -182046. q^{60} -2.08994e6 q^{61} -724098. q^{62} +250047. q^{63} +262144. q^{64} -231456. q^{65} -1.60300e6 q^{66} -929708. q^{67} +1.03182e6 q^{68} -1.83710e6 q^{69} +289083. q^{70} +317592. q^{71} -373248. q^{72} +315536. q^{73} +3.50845e6 q^{74} -1.80971e6 q^{75} -2.32213e6 q^{76} +2.54550e6 q^{77} -474552. q^{78} +5.30806e6 q^{79} -431517. q^{80} +531441. q^{81} +2.09942e6 q^{82} -4.17486e6 q^{83} +592704. q^{84} -1.69848e6 q^{85} -1.31876e6 q^{86} -5.25959e6 q^{87} -3.79970e6 q^{88} -7.81858e6 q^{89} +614406. q^{90} +753571. q^{91} -4.35460e6 q^{92} +2.44383e6 q^{93} -1.51115e6 q^{94} +3.82248e6 q^{95} -884736. q^{96} +8.73903e6 q^{97} -941192. q^{98} +5.41011e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99} + O(q^{100}) \)

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
\(5\) −105.351 −0.376915 −0.188457 0.982081i \(-0.560349\pi\)
−0.188457 + 0.982081i \(0.560349\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 842.807 0.266519
\(11\) 7421.28 1.68114 0.840571 0.541701i \(-0.182219\pi\)
0.840571 + 0.541701i \(0.182219\pi\)
\(12\) 1728.00 0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −2844.47 −0.217612
\(16\) 4096.00 0.250000
\(17\) 16122.2 0.795888 0.397944 0.917410i \(-0.369724\pi\)
0.397944 + 0.917410i \(0.369724\pi\)
\(18\) −5832.00 −0.235702
\(19\) −36283.4 −1.21358 −0.606792 0.794861i \(-0.707544\pi\)
−0.606792 + 0.794861i \(0.707544\pi\)
\(20\) −6742.45 −0.188457
\(21\) 9261.00 0.218218
\(22\) −59370.3 −1.18875
\(23\) −68040.6 −1.16606 −0.583030 0.812451i \(-0.698133\pi\)
−0.583030 + 0.812451i \(0.698133\pi\)
\(24\) −13824.0 −0.204124
\(25\) −67026.2 −0.857935
\(26\) −17576.0 −0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) −194800. −1.48318 −0.741592 0.670851i \(-0.765929\pi\)
−0.741592 + 0.670851i \(0.765929\pi\)
\(30\) 22755.8 0.153875
\(31\) 90512.2 0.545684 0.272842 0.962059i \(-0.412036\pi\)
0.272842 + 0.962059i \(0.412036\pi\)
\(32\) −32768.0 −0.176777
\(33\) 200375. 0.970608
\(34\) −128977. −0.562778
\(35\) −36135.3 −0.142460
\(36\) 46656.0 0.166667
\(37\) −438556. −1.42337 −0.711687 0.702497i \(-0.752068\pi\)
−0.711687 + 0.702497i \(0.752068\pi\)
\(38\) 290267. 0.858133
\(39\) 59319.0 0.160128
\(40\) 53939.6 0.133259
\(41\) −262428. −0.594656 −0.297328 0.954775i \(-0.596096\pi\)
−0.297328 + 0.954775i \(0.596096\pi\)
\(42\) −74088.0 −0.154303
\(43\) 164846. 0.316182 0.158091 0.987425i \(-0.449466\pi\)
0.158091 + 0.987425i \(0.449466\pi\)
\(44\) 474962. 0.840571
\(45\) −76800.8 −0.125638
\(46\) 544325. 0.824529
\(47\) 188894. 0.265384 0.132692 0.991157i \(-0.457638\pi\)
0.132692 + 0.991157i \(0.457638\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 536210. 0.606652
\(51\) 435298. 0.459506
\(52\) 140608. 0.138675
\(53\) 1.49619e6 1.38045 0.690224 0.723596i \(-0.257512\pi\)
0.690224 + 0.723596i \(0.257512\pi\)
\(54\) −157464. −0.136083
\(55\) −781838. −0.633647
\(56\) −175616. −0.133631
\(57\) −979651. −0.700663
\(58\) 1.55840e6 1.04877
\(59\) −292745. −0.185570 −0.0927851 0.995686i \(-0.529577\pi\)
−0.0927851 + 0.995686i \(0.529577\pi\)
\(60\) −182046. −0.108806
\(61\) −2.08994e6 −1.17891 −0.589453 0.807803i \(-0.700657\pi\)
−0.589453 + 0.807803i \(0.700657\pi\)
\(62\) −724098. −0.385857
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −231456. −0.104537
\(66\) −1.60300e6 −0.686324
\(67\) −929708. −0.377646 −0.188823 0.982011i \(-0.560467\pi\)
−0.188823 + 0.982011i \(0.560467\pi\)
\(68\) 1.03182e6 0.397944
\(69\) −1.83710e6 −0.673225
\(70\) 289083. 0.100735
\(71\) 317592. 0.105309 0.0526545 0.998613i \(-0.483232\pi\)
0.0526545 + 0.998613i \(0.483232\pi\)
\(72\) −373248. −0.117851
\(73\) 315536. 0.0949334 0.0474667 0.998873i \(-0.484885\pi\)
0.0474667 + 0.998873i \(0.484885\pi\)
\(74\) 3.50845e6 1.00648
\(75\) −1.80971e6 −0.495329
\(76\) −2.32213e6 −0.606792
\(77\) 2.54550e6 0.635412
\(78\) −474552. −0.113228
\(79\) 5.30806e6 1.21127 0.605635 0.795742i \(-0.292919\pi\)
0.605635 + 0.795742i \(0.292919\pi\)
\(80\) −431517. −0.0942287
\(81\) 531441. 0.111111
\(82\) 2.09942e6 0.420485
\(83\) −4.17486e6 −0.801436 −0.400718 0.916201i \(-0.631239\pi\)
−0.400718 + 0.916201i \(0.631239\pi\)
\(84\) 592704. 0.109109
\(85\) −1.69848e6 −0.299982
\(86\) −1.31876e6 −0.223575
\(87\) −5.25959e6 −0.856317
\(88\) −3.79970e6 −0.594374
\(89\) −7.81858e6 −1.17561 −0.587805 0.809003i \(-0.700008\pi\)
−0.587805 + 0.809003i \(0.700008\pi\)
\(90\) 614406. 0.0888396
\(91\) 753571. 0.104828
\(92\) −4.35460e6 −0.583030
\(93\) 2.44383e6 0.315051
\(94\) −1.51115e6 −0.187655
\(95\) 3.82248e6 0.457418
\(96\) −884736. −0.102062
\(97\) 8.73903e6 0.972214 0.486107 0.873899i \(-0.338416\pi\)
0.486107 + 0.873899i \(0.338416\pi\)
\(98\) −941192. −0.101015
\(99\) 5.41011e6 0.560381
\(100\) −4.28968e6 −0.428968
\(101\) −5.72755e6 −0.553151 −0.276576 0.960992i \(-0.589200\pi\)
−0.276576 + 0.960992i \(0.589200\pi\)
\(102\) −3.48238e6 −0.324920
\(103\) −7.25175e6 −0.653902 −0.326951 0.945041i \(-0.606021\pi\)
−0.326951 + 0.945041i \(0.606021\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −975654. −0.0822495
\(106\) −1.19695e7 −0.976124
\(107\) −8.87675e6 −0.700504 −0.350252 0.936655i \(-0.613904\pi\)
−0.350252 + 0.936655i \(0.613904\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.77400e7 1.31208 0.656041 0.754725i \(-0.272230\pi\)
0.656041 + 0.754725i \(0.272230\pi\)
\(110\) 6.25471e6 0.448056
\(111\) −1.18410e7 −0.821785
\(112\) 1.40493e6 0.0944911
\(113\) −2.99498e7 −1.95263 −0.976315 0.216355i \(-0.930583\pi\)
−0.976315 + 0.216355i \(0.930583\pi\)
\(114\) 7.83720e6 0.495444
\(115\) 7.16814e6 0.439505
\(116\) −1.24672e7 −0.741592
\(117\) 1.60161e6 0.0924500
\(118\) 2.34196e6 0.131218
\(119\) 5.52990e6 0.300817
\(120\) 1.45637e6 0.0769374
\(121\) 3.55883e7 1.82624
\(122\) 1.67195e7 0.833612
\(123\) −7.08554e6 −0.343325
\(124\) 5.79278e6 0.272842
\(125\) 1.52918e7 0.700283
\(126\) −2.00038e6 −0.0890871
\(127\) −1.47859e7 −0.640523 −0.320262 0.947329i \(-0.603771\pi\)
−0.320262 + 0.947329i \(0.603771\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 4.45083e6 0.182548
\(130\) 1.85165e6 0.0739190
\(131\) −7.56713e6 −0.294091 −0.147045 0.989130i \(-0.546976\pi\)
−0.147045 + 0.989130i \(0.546976\pi\)
\(132\) 1.28240e7 0.485304
\(133\) −1.24452e7 −0.458692
\(134\) 7.43766e6 0.267036
\(135\) −2.07362e6 −0.0725373
\(136\) −8.25454e6 −0.281389
\(137\) −2.40391e6 −0.0798723 −0.0399361 0.999202i \(-0.512715\pi\)
−0.0399361 + 0.999202i \(0.512715\pi\)
\(138\) 1.46968e7 0.476042
\(139\) 9.00434e6 0.284381 0.142190 0.989839i \(-0.454585\pi\)
0.142190 + 0.989839i \(0.454585\pi\)
\(140\) −2.31266e6 −0.0712302
\(141\) 5.10013e6 0.153220
\(142\) −2.54074e6 −0.0744648
\(143\) 1.63046e7 0.466265
\(144\) 2.98598e6 0.0833333
\(145\) 2.05223e7 0.559034
\(146\) −2.52429e6 −0.0671280
\(147\) 3.17652e6 0.0824786
\(148\) −2.80676e7 −0.711687
\(149\) 3.02022e7 0.747974 0.373987 0.927434i \(-0.377990\pi\)
0.373987 + 0.927434i \(0.377990\pi\)
\(150\) 1.44777e7 0.350251
\(151\) 3.29383e6 0.0778543 0.0389271 0.999242i \(-0.487606\pi\)
0.0389271 + 0.999242i \(0.487606\pi\)
\(152\) 1.85771e7 0.429067
\(153\) 1.17530e7 0.265296
\(154\) −2.03640e7 −0.449304
\(155\) −9.53554e6 −0.205676
\(156\) 3.79642e6 0.0800641
\(157\) 7.22652e7 1.49032 0.745161 0.666884i \(-0.232373\pi\)
0.745161 + 0.666884i \(0.232373\pi\)
\(158\) −4.24645e7 −0.856498
\(159\) 4.03970e7 0.797002
\(160\) 3.45214e6 0.0666297
\(161\) −2.33379e7 −0.440729
\(162\) −4.25153e6 −0.0785674
\(163\) −4.21472e7 −0.762277 −0.381138 0.924518i \(-0.624468\pi\)
−0.381138 + 0.924518i \(0.624468\pi\)
\(164\) −1.67954e7 −0.297328
\(165\) −2.11096e7 −0.365836
\(166\) 3.33989e7 0.566701
\(167\) −4.54441e7 −0.755040 −0.377520 0.926001i \(-0.623223\pi\)
−0.377520 + 0.926001i \(0.623223\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 1.35879e7 0.212119
\(171\) −2.64506e7 −0.404528
\(172\) 1.05501e7 0.158091
\(173\) −7.57885e7 −1.11286 −0.556432 0.830893i \(-0.687830\pi\)
−0.556432 + 0.830893i \(0.687830\pi\)
\(174\) 4.20767e7 0.605507
\(175\) −2.29900e7 −0.324269
\(176\) 3.03976e7 0.420286
\(177\) −7.90413e6 −0.107139
\(178\) 6.25487e7 0.831281
\(179\) 7.18129e7 0.935874 0.467937 0.883762i \(-0.344997\pi\)
0.467937 + 0.883762i \(0.344997\pi\)
\(180\) −4.91525e6 −0.0628191
\(181\) −1.39510e8 −1.74877 −0.874383 0.485237i \(-0.838733\pi\)
−0.874383 + 0.485237i \(0.838733\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −5.64283e7 −0.680642
\(184\) 3.48368e7 0.412264
\(185\) 4.62022e7 0.536490
\(186\) −1.95506e7 −0.222775
\(187\) 1.19647e8 1.33800
\(188\) 1.20892e7 0.132692
\(189\) 6.75127e6 0.0727393
\(190\) −3.05799e7 −0.323443
\(191\) −1.21685e8 −1.26363 −0.631814 0.775120i \(-0.717689\pi\)
−0.631814 + 0.775120i \(0.717689\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 5.52196e6 0.0552895 0.0276447 0.999618i \(-0.491199\pi\)
0.0276447 + 0.999618i \(0.491199\pi\)
\(194\) −6.99122e7 −0.687459
\(195\) −6.24931e6 −0.0603546
\(196\) 7.52954e6 0.0714286
\(197\) −9.72068e6 −0.0905868 −0.0452934 0.998974i \(-0.514422\pi\)
−0.0452934 + 0.998974i \(0.514422\pi\)
\(198\) −4.32809e7 −0.396249
\(199\) 4.36161e7 0.392339 0.196169 0.980570i \(-0.437150\pi\)
0.196169 + 0.980570i \(0.437150\pi\)
\(200\) 3.43174e7 0.303326
\(201\) −2.51021e7 −0.218034
\(202\) 4.58204e7 0.391137
\(203\) −6.68163e7 −0.560591
\(204\) 2.78591e7 0.229753
\(205\) 2.76470e7 0.224135
\(206\) 5.80140e7 0.462378
\(207\) −4.96016e7 −0.388687
\(208\) 8.99891e6 0.0693375
\(209\) −2.69269e8 −2.04021
\(210\) 7.80523e6 0.0581592
\(211\) 1.40587e8 1.03028 0.515140 0.857106i \(-0.327740\pi\)
0.515140 + 0.857106i \(0.327740\pi\)
\(212\) 9.57559e7 0.690224
\(213\) 8.57499e6 0.0608002
\(214\) 7.10140e7 0.495331
\(215\) −1.73666e7 −0.119174
\(216\) −1.00777e7 −0.0680414
\(217\) 3.10457e7 0.206249
\(218\) −1.41920e8 −0.927782
\(219\) 8.51947e6 0.0548098
\(220\) −5.00377e7 −0.316824
\(221\) 3.54204e7 0.220740
\(222\) 9.47280e7 0.581090
\(223\) 1.15747e8 0.698946 0.349473 0.936946i \(-0.386361\pi\)
0.349473 + 0.936946i \(0.386361\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −4.88621e7 −0.285978
\(226\) 2.39599e8 1.38072
\(227\) 1.94933e8 1.10610 0.553052 0.833147i \(-0.313463\pi\)
0.553052 + 0.833147i \(0.313463\pi\)
\(228\) −6.26976e7 −0.350332
\(229\) −2.66973e8 −1.46907 −0.734537 0.678569i \(-0.762600\pi\)
−0.734537 + 0.678569i \(0.762600\pi\)
\(230\) −5.73451e7 −0.310777
\(231\) 6.87285e7 0.366855
\(232\) 9.97374e7 0.524385
\(233\) −1.69401e8 −0.877345 −0.438673 0.898647i \(-0.644551\pi\)
−0.438673 + 0.898647i \(0.644551\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) −1.99001e7 −0.100027
\(236\) −1.87357e7 −0.0927851
\(237\) 1.43318e8 0.699327
\(238\) −4.42392e7 −0.212710
\(239\) −4.63140e7 −0.219442 −0.109721 0.993962i \(-0.534996\pi\)
−0.109721 + 0.993962i \(0.534996\pi\)
\(240\) −1.16510e7 −0.0544029
\(241\) −4.43464e7 −0.204079 −0.102040 0.994780i \(-0.532537\pi\)
−0.102040 + 0.994780i \(0.532537\pi\)
\(242\) −2.84706e8 −1.29135
\(243\) 1.43489e7 0.0641500
\(244\) −1.33756e8 −0.589453
\(245\) −1.23944e7 −0.0538450
\(246\) 5.66843e7 0.242767
\(247\) −7.97145e7 −0.336588
\(248\) −4.63423e7 −0.192929
\(249\) −1.12721e8 −0.462710
\(250\) −1.22334e8 −0.495175
\(251\) 1.25315e8 0.500200 0.250100 0.968220i \(-0.419536\pi\)
0.250100 + 0.968220i \(0.419536\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −5.04949e8 −1.96031
\(254\) 1.18287e8 0.452918
\(255\) −4.58590e7 −0.173195
\(256\) 1.67772e7 0.0625000
\(257\) −2.67887e7 −0.0984432 −0.0492216 0.998788i \(-0.515674\pi\)
−0.0492216 + 0.998788i \(0.515674\pi\)
\(258\) −3.56066e7 −0.129081
\(259\) −1.50425e8 −0.537985
\(260\) −1.48132e7 −0.0522687
\(261\) −1.42009e8 −0.494395
\(262\) 6.05370e7 0.207954
\(263\) 3.76131e8 1.27495 0.637476 0.770470i \(-0.279978\pi\)
0.637476 + 0.770470i \(0.279978\pi\)
\(264\) −1.02592e8 −0.343162
\(265\) −1.57624e8 −0.520311
\(266\) 9.95615e7 0.324344
\(267\) −2.11102e8 −0.678738
\(268\) −5.95013e7 −0.188823
\(269\) −8.14871e7 −0.255244 −0.127622 0.991823i \(-0.540734\pi\)
−0.127622 + 0.991823i \(0.540734\pi\)
\(270\) 1.65890e7 0.0512916
\(271\) −5.77820e8 −1.76360 −0.881801 0.471622i \(-0.843669\pi\)
−0.881801 + 0.471622i \(0.843669\pi\)
\(272\) 6.60363e7 0.198972
\(273\) 2.03464e7 0.0605228
\(274\) 1.92313e7 0.0564782
\(275\) −4.97420e8 −1.44231
\(276\) −1.17574e8 −0.336612
\(277\) −6.41391e8 −1.81319 −0.906595 0.422001i \(-0.861328\pi\)
−0.906595 + 0.422001i \(0.861328\pi\)
\(278\) −7.20348e7 −0.201088
\(279\) 6.59834e7 0.181895
\(280\) 1.85013e7 0.0503673
\(281\) −6.07996e8 −1.63467 −0.817333 0.576166i \(-0.804548\pi\)
−0.817333 + 0.576166i \(0.804548\pi\)
\(282\) −4.08011e7 −0.108343
\(283\) 2.84310e8 0.745658 0.372829 0.927900i \(-0.378388\pi\)
0.372829 + 0.927900i \(0.378388\pi\)
\(284\) 2.03259e7 0.0526545
\(285\) 1.03207e8 0.264090
\(286\) −1.30436e8 −0.329699
\(287\) −9.00126e7 −0.224759
\(288\) −2.38879e7 −0.0589256
\(289\) −1.50415e8 −0.366563
\(290\) −1.64178e8 −0.395297
\(291\) 2.35954e8 0.561308
\(292\) 2.01943e7 0.0474667
\(293\) −7.62074e8 −1.76995 −0.884973 0.465642i \(-0.845824\pi\)
−0.884973 + 0.465642i \(0.845824\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 3.08410e7 0.0699441
\(296\) 2.24541e8 0.503238
\(297\) 1.46073e8 0.323536
\(298\) −2.41618e8 −0.528898
\(299\) −1.49485e8 −0.323407
\(300\) −1.15821e8 −0.247665
\(301\) 5.65420e7 0.119506
\(302\) −2.63507e7 −0.0550513
\(303\) −1.54644e8 −0.319362
\(304\) −1.48617e8 −0.303396
\(305\) 2.20177e8 0.444347
\(306\) −9.40244e7 −0.187593
\(307\) 1.69532e8 0.334401 0.167201 0.985923i \(-0.446527\pi\)
0.167201 + 0.985923i \(0.446527\pi\)
\(308\) 1.62912e8 0.317706
\(309\) −1.95797e8 −0.377530
\(310\) 7.62843e7 0.145435
\(311\) −2.89535e8 −0.545808 −0.272904 0.962041i \(-0.587984\pi\)
−0.272904 + 0.962041i \(0.587984\pi\)
\(312\) −3.03713e7 −0.0566139
\(313\) 1.39146e8 0.256488 0.128244 0.991743i \(-0.459066\pi\)
0.128244 + 0.991743i \(0.459066\pi\)
\(314\) −5.78121e8 −1.05382
\(315\) −2.63427e7 −0.0474868
\(316\) 3.39716e8 0.605635
\(317\) −5.53876e8 −0.976573 −0.488287 0.872683i \(-0.662378\pi\)
−0.488287 + 0.872683i \(0.662378\pi\)
\(318\) −3.23176e8 −0.563565
\(319\) −1.44566e9 −2.49344
\(320\) −2.76171e7 −0.0471143
\(321\) −2.39672e8 −0.404436
\(322\) 1.86703e8 0.311643
\(323\) −5.84966e8 −0.965877
\(324\) 3.40122e7 0.0555556
\(325\) −1.47257e8 −0.237948
\(326\) 3.37178e8 0.539011
\(327\) 4.78980e8 0.757531
\(328\) 1.34363e8 0.210243
\(329\) 6.47906e7 0.100306
\(330\) 1.68877e8 0.258685
\(331\) 7.68265e8 1.16443 0.582214 0.813035i \(-0.302186\pi\)
0.582214 + 0.813035i \(0.302186\pi\)
\(332\) −2.67191e8 −0.400718
\(333\) −3.19707e8 −0.474458
\(334\) 3.63553e8 0.533894
\(335\) 9.79455e7 0.142340
\(336\) 3.79331e7 0.0545545
\(337\) 1.08991e8 0.155127 0.0775633 0.996987i \(-0.475286\pi\)
0.0775633 + 0.996987i \(0.475286\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −8.08645e8 −1.12735
\(340\) −1.08703e8 −0.149991
\(341\) 6.71717e8 0.917373
\(342\) 2.11605e8 0.286044
\(343\) 4.03536e7 0.0539949
\(344\) −8.44009e7 −0.111787
\(345\) 1.93540e8 0.253748
\(346\) 6.06308e8 0.786913
\(347\) 3.72515e8 0.478620 0.239310 0.970943i \(-0.423079\pi\)
0.239310 + 0.970943i \(0.423079\pi\)
\(348\) −3.36614e8 −0.428158
\(349\) 5.27370e8 0.664090 0.332045 0.943264i \(-0.392261\pi\)
0.332045 + 0.943264i \(0.392261\pi\)
\(350\) 1.83920e8 0.229293
\(351\) 4.32436e7 0.0533761
\(352\) −2.43181e8 −0.297187
\(353\) −1.16172e9 −1.40569 −0.702845 0.711343i \(-0.748087\pi\)
−0.702845 + 0.711343i \(0.748087\pi\)
\(354\) 6.32330e7 0.0757587
\(355\) −3.34586e7 −0.0396925
\(356\) −5.00389e8 −0.587805
\(357\) 1.49307e8 0.173677
\(358\) −5.74504e8 −0.661763
\(359\) −1.01521e9 −1.15805 −0.579025 0.815310i \(-0.696567\pi\)
−0.579025 + 0.815310i \(0.696567\pi\)
\(360\) 3.93220e7 0.0444198
\(361\) 4.22610e8 0.472786
\(362\) 1.11608e9 1.23656
\(363\) 9.60883e8 1.05438
\(364\) 4.82285e7 0.0524142
\(365\) −3.32420e7 −0.0357818
\(366\) 4.51427e8 0.481286
\(367\) 1.60357e7 0.0169339 0.00846693 0.999964i \(-0.497305\pi\)
0.00846693 + 0.999964i \(0.497305\pi\)
\(368\) −2.78694e8 −0.291515
\(369\) −1.91310e8 −0.198219
\(370\) −3.69618e8 −0.379356
\(371\) 5.13192e8 0.521760
\(372\) 1.56405e8 0.157525
\(373\) −6.94347e8 −0.692780 −0.346390 0.938091i \(-0.612593\pi\)
−0.346390 + 0.938091i \(0.612593\pi\)
\(374\) −9.57176e8 −0.946109
\(375\) 4.12879e8 0.404309
\(376\) −9.67136e7 −0.0938275
\(377\) −4.27975e8 −0.411361
\(378\) −5.40102e7 −0.0514344
\(379\) −1.36052e7 −0.0128371 −0.00641855 0.999979i \(-0.502043\pi\)
−0.00641855 + 0.999979i \(0.502043\pi\)
\(380\) 2.44639e8 0.228709
\(381\) −3.99219e8 −0.369806
\(382\) 9.73478e8 0.893520
\(383\) 3.86385e8 0.351419 0.175709 0.984442i \(-0.443778\pi\)
0.175709 + 0.984442i \(0.443778\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −2.68171e8 −0.239496
\(386\) −4.41757e7 −0.0390956
\(387\) 1.20172e8 0.105394
\(388\) 5.59298e8 0.486107
\(389\) 1.37966e9 1.18836 0.594180 0.804332i \(-0.297477\pi\)
0.594180 + 0.804332i \(0.297477\pi\)
\(390\) 4.99945e7 0.0426772
\(391\) −1.09696e9 −0.928052
\(392\) −6.02363e7 −0.0505076
\(393\) −2.04313e8 −0.169794
\(394\) 7.77655e7 0.0640546
\(395\) −5.59209e8 −0.456546
\(396\) 3.46247e8 0.280190
\(397\) −9.24476e8 −0.741530 −0.370765 0.928727i \(-0.620904\pi\)
−0.370765 + 0.928727i \(0.620904\pi\)
\(398\) −3.48929e8 −0.277425
\(399\) −3.36020e8 −0.264826
\(400\) −2.74539e8 −0.214484
\(401\) −9.49026e8 −0.734975 −0.367488 0.930028i \(-0.619782\pi\)
−0.367488 + 0.930028i \(0.619782\pi\)
\(402\) 2.00817e8 0.154173
\(403\) 1.98855e8 0.151346
\(404\) −3.66563e8 −0.276576
\(405\) −5.59878e7 −0.0418794
\(406\) 5.34530e8 0.396398
\(407\) −3.25465e9 −2.39289
\(408\) −2.22873e8 −0.162460
\(409\) 5.82559e8 0.421026 0.210513 0.977591i \(-0.432487\pi\)
0.210513 + 0.977591i \(0.432487\pi\)
\(410\) −2.21176e8 −0.158487
\(411\) −6.49056e7 −0.0461143
\(412\) −4.64112e8 −0.326951
\(413\) −1.00412e8 −0.0701389
\(414\) 3.96813e8 0.274843
\(415\) 4.39826e8 0.302073
\(416\) −7.19913e7 −0.0490290
\(417\) 2.43117e8 0.164187
\(418\) 2.15415e9 1.44264
\(419\) −5.08829e8 −0.337927 −0.168964 0.985622i \(-0.554042\pi\)
−0.168964 + 0.985622i \(0.554042\pi\)
\(420\) −6.24419e7 −0.0411248
\(421\) 6.62011e8 0.432392 0.216196 0.976350i \(-0.430635\pi\)
0.216196 + 0.976350i \(0.430635\pi\)
\(422\) −1.12469e9 −0.728518
\(423\) 1.37704e8 0.0884614
\(424\) −7.66047e8 −0.488062
\(425\) −1.08061e9 −0.682820
\(426\) −6.85999e7 −0.0429922
\(427\) −7.16849e8 −0.445585
\(428\) −5.68112e8 −0.350252
\(429\) 4.40223e8 0.269198
\(430\) 1.38933e8 0.0842686
\(431\) −7.24096e8 −0.435638 −0.217819 0.975989i \(-0.569894\pi\)
−0.217819 + 0.975989i \(0.569894\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.33205e8 0.0788523 0.0394262 0.999222i \(-0.487447\pi\)
0.0394262 + 0.999222i \(0.487447\pi\)
\(434\) −2.48366e8 −0.145840
\(435\) 5.54102e8 0.322758
\(436\) 1.13536e9 0.656041
\(437\) 2.46874e9 1.41511
\(438\) −6.81558e7 −0.0387564
\(439\) 3.07683e9 1.73571 0.867856 0.496816i \(-0.165498\pi\)
0.867856 + 0.496816i \(0.165498\pi\)
\(440\) 4.00301e8 0.224028
\(441\) 8.57661e7 0.0476190
\(442\) −2.83363e8 −0.156086
\(443\) −1.24948e9 −0.682837 −0.341418 0.939911i \(-0.610907\pi\)
−0.341418 + 0.939911i \(0.610907\pi\)
\(444\) −7.57824e8 −0.410892
\(445\) 8.23694e8 0.443104
\(446\) −9.25978e8 −0.494229
\(447\) 8.15459e8 0.431843
\(448\) 8.99154e7 0.0472456
\(449\) 1.01013e9 0.526639 0.263320 0.964709i \(-0.415183\pi\)
0.263320 + 0.964709i \(0.415183\pi\)
\(450\) 3.90897e8 0.202217
\(451\) −1.94755e9 −0.999702
\(452\) −1.91679e9 −0.976315
\(453\) 8.89335e7 0.0449492
\(454\) −1.55947e9 −0.782133
\(455\) −7.93893e7 −0.0395114
\(456\) 5.01581e8 0.247722
\(457\) 2.57025e9 1.25971 0.629853 0.776715i \(-0.283115\pi\)
0.629853 + 0.776715i \(0.283115\pi\)
\(458\) 2.13579e9 1.03879
\(459\) 3.17332e8 0.153169
\(460\) 4.58761e8 0.219752
\(461\) 3.08524e9 1.46668 0.733340 0.679863i \(-0.237961\pi\)
0.733340 + 0.679863i \(0.237961\pi\)
\(462\) −5.49828e8 −0.259406
\(463\) 1.35716e9 0.635475 0.317737 0.948179i \(-0.397077\pi\)
0.317737 + 0.948179i \(0.397077\pi\)
\(464\) −7.97899e8 −0.370796
\(465\) −2.57460e8 −0.118747
\(466\) 1.35521e9 0.620377
\(467\) 1.80963e9 0.822205 0.411103 0.911589i \(-0.365144\pi\)
0.411103 + 0.911589i \(0.365144\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −3.18890e8 −0.142737
\(470\) 1.59201e8 0.0707299
\(471\) 1.95116e9 0.860438
\(472\) 1.49886e8 0.0656089
\(473\) 1.22337e9 0.531548
\(474\) −1.14654e9 −0.494499
\(475\) 2.43194e9 1.04118
\(476\) 3.53913e8 0.150409
\(477\) 1.09072e9 0.460149
\(478\) 3.70512e8 0.155169
\(479\) −3.09084e9 −1.28500 −0.642499 0.766287i \(-0.722102\pi\)
−0.642499 + 0.766287i \(0.722102\pi\)
\(480\) 9.32077e7 0.0384687
\(481\) −9.63507e8 −0.394773
\(482\) 3.54771e8 0.144306
\(483\) −6.30124e8 −0.254455
\(484\) 2.27765e9 0.913120
\(485\) −9.20664e8 −0.366442
\(486\) −1.14791e8 −0.0453609
\(487\) 7.18778e8 0.281996 0.140998 0.990010i \(-0.454969\pi\)
0.140998 + 0.990010i \(0.454969\pi\)
\(488\) 1.07005e9 0.416806
\(489\) −1.13798e9 −0.440101
\(490\) 9.91554e7 0.0380741
\(491\) −2.07699e9 −0.791861 −0.395930 0.918281i \(-0.629578\pi\)
−0.395930 + 0.918281i \(0.629578\pi\)
\(492\) −4.53475e8 −0.171662
\(493\) −3.14059e9 −1.18045
\(494\) 6.37716e8 0.238003
\(495\) −5.69960e8 −0.211216
\(496\) 3.70738e8 0.136421
\(497\) 1.08934e8 0.0398031
\(498\) 9.01771e8 0.327185
\(499\) −2.03246e9 −0.732268 −0.366134 0.930562i \(-0.619319\pi\)
−0.366134 + 0.930562i \(0.619319\pi\)
\(500\) 9.78675e8 0.350142
\(501\) −1.22699e9 −0.435923
\(502\) −1.00252e9 −0.353695
\(503\) −2.05667e9 −0.720570 −0.360285 0.932842i \(-0.617320\pi\)
−0.360285 + 0.932842i \(0.617320\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 6.03402e8 0.208491
\(506\) 4.03959e9 1.38615
\(507\) 1.30324e8 0.0444116
\(508\) −9.46298e8 −0.320262
\(509\) −3.44086e9 −1.15652 −0.578262 0.815851i \(-0.696269\pi\)
−0.578262 + 0.815851i \(0.696269\pi\)
\(510\) 3.66872e8 0.122467
\(511\) 1.08229e8 0.0358815
\(512\) −1.34218e8 −0.0441942
\(513\) −7.14165e8 −0.233554
\(514\) 2.14310e8 0.0696099
\(515\) 7.63978e8 0.246465
\(516\) 2.84853e8 0.0912740
\(517\) 1.40183e9 0.446149
\(518\) 1.20340e9 0.380412
\(519\) −2.04629e9 −0.642512
\(520\) 1.18505e8 0.0369595
\(521\) 5.21852e9 1.61665 0.808323 0.588739i \(-0.200376\pi\)
0.808323 + 0.588739i \(0.200376\pi\)
\(522\) 1.13607e9 0.349590
\(523\) 1.33645e9 0.408504 0.204252 0.978918i \(-0.434524\pi\)
0.204252 + 0.978918i \(0.434524\pi\)
\(524\) −4.84296e8 −0.147045
\(525\) −6.20730e8 −0.187217
\(526\) −3.00905e9 −0.901528
\(527\) 1.45925e9 0.434303
\(528\) 8.20734e8 0.242652
\(529\) 1.22470e9 0.359695
\(530\) 1.26100e9 0.367915
\(531\) −2.13411e8 −0.0618567
\(532\) −7.96492e8 −0.229346
\(533\) −5.76553e8 −0.164928
\(534\) 1.68881e9 0.479941
\(535\) 9.35173e8 0.264030
\(536\) 4.76010e8 0.133518
\(537\) 1.93895e9 0.540327
\(538\) 6.51897e8 0.180485
\(539\) 8.73106e8 0.240163
\(540\) −1.32712e8 −0.0362686
\(541\) −2.67688e9 −0.726839 −0.363419 0.931626i \(-0.618391\pi\)
−0.363419 + 0.931626i \(0.618391\pi\)
\(542\) 4.62256e9 1.24705
\(543\) −3.76678e9 −1.00965
\(544\) −5.28291e8 −0.140694
\(545\) −1.86892e9 −0.494543
\(546\) −1.62771e8 −0.0427960
\(547\) 2.98950e9 0.780984 0.390492 0.920606i \(-0.372305\pi\)
0.390492 + 0.920606i \(0.372305\pi\)
\(548\) −1.53850e8 −0.0399361
\(549\) −1.52357e9 −0.392969
\(550\) 3.97936e9 1.01987
\(551\) 7.06798e9 1.79997
\(552\) 9.40593e8 0.238021
\(553\) 1.82067e9 0.457817
\(554\) 5.13112e9 1.28212
\(555\) 1.24746e9 0.309743
\(556\) 5.76278e8 0.142190
\(557\) −4.47645e9 −1.09759 −0.548795 0.835957i \(-0.684913\pi\)
−0.548795 + 0.835957i \(0.684913\pi\)
\(558\) −5.27867e8 −0.128619
\(559\) 3.62166e8 0.0876932
\(560\) −1.48010e8 −0.0356151
\(561\) 3.23047e9 0.772495
\(562\) 4.86397e9 1.15588
\(563\) 1.87324e9 0.442400 0.221200 0.975229i \(-0.429003\pi\)
0.221200 + 0.975229i \(0.429003\pi\)
\(564\) 3.26408e8 0.0766098
\(565\) 3.15524e9 0.735975
\(566\) −2.27448e9 −0.527260
\(567\) 1.82284e8 0.0419961
\(568\) −1.62607e8 −0.0372324
\(569\) 1.86104e9 0.423509 0.211755 0.977323i \(-0.432082\pi\)
0.211755 + 0.977323i \(0.432082\pi\)
\(570\) −8.25656e8 −0.186740
\(571\) −2.58928e9 −0.582039 −0.291020 0.956717i \(-0.593994\pi\)
−0.291020 + 0.956717i \(0.593994\pi\)
\(572\) 1.04349e9 0.233133
\(573\) −3.28549e9 −0.729556
\(574\) 7.20101e8 0.158929
\(575\) 4.56050e9 1.00040
\(576\) 1.91103e8 0.0416667
\(577\) −1.50996e9 −0.327228 −0.163614 0.986524i \(-0.552315\pi\)
−0.163614 + 0.986524i \(0.552315\pi\)
\(578\) 1.20332e9 0.259199
\(579\) 1.49093e8 0.0319214
\(580\) 1.31343e9 0.279517
\(581\) −1.43198e9 −0.302914
\(582\) −1.88763e9 −0.396905
\(583\) 1.11036e10 2.32073
\(584\) −1.61554e8 −0.0335640
\(585\) −1.68731e8 −0.0348458
\(586\) 6.09659e9 1.25154
\(587\) −3.57414e9 −0.729353 −0.364677 0.931134i \(-0.618820\pi\)
−0.364677 + 0.931134i \(0.618820\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) −3.28409e9 −0.662234
\(590\) −2.46728e8 −0.0494579
\(591\) −2.62458e8 −0.0523003
\(592\) −1.79632e9 −0.355843
\(593\) −5.87623e9 −1.15720 −0.578599 0.815612i \(-0.696400\pi\)
−0.578599 + 0.815612i \(0.696400\pi\)
\(594\) −1.16858e9 −0.228775
\(595\) −5.82579e8 −0.113382
\(596\) 1.93294e9 0.373987
\(597\) 1.17763e9 0.226517
\(598\) 1.19588e9 0.228683
\(599\) −3.85934e9 −0.733700 −0.366850 0.930280i \(-0.619564\pi\)
−0.366850 + 0.930280i \(0.619564\pi\)
\(600\) 9.26570e8 0.175125
\(601\) −9.90147e9 −1.86054 −0.930270 0.366876i \(-0.880427\pi\)
−0.930270 + 0.366876i \(0.880427\pi\)
\(602\) −4.52336e8 −0.0845033
\(603\) −6.77757e8 −0.125882
\(604\) 2.10805e8 0.0389271
\(605\) −3.74925e9 −0.688337
\(606\) 1.23715e9 0.225823
\(607\) 7.17550e9 1.30224 0.651121 0.758974i \(-0.274299\pi\)
0.651121 + 0.758974i \(0.274299\pi\)
\(608\) 1.18893e9 0.214533
\(609\) −1.80404e9 −0.323657
\(610\) −1.76141e9 −0.314201
\(611\) 4.15000e8 0.0736044
\(612\) 7.52195e8 0.132648
\(613\) 4.19039e9 0.734755 0.367378 0.930072i \(-0.380256\pi\)
0.367378 + 0.930072i \(0.380256\pi\)
\(614\) −1.35626e9 −0.236457
\(615\) 7.46468e8 0.129404
\(616\) −1.30330e9 −0.224652
\(617\) −3.31392e9 −0.567995 −0.283997 0.958825i \(-0.591661\pi\)
−0.283997 + 0.958825i \(0.591661\pi\)
\(618\) 1.56638e9 0.266954
\(619\) 1.16365e10 1.97199 0.985993 0.166788i \(-0.0533395\pi\)
0.985993 + 0.166788i \(0.0533395\pi\)
\(620\) −6.10275e8 −0.102838
\(621\) −1.33924e9 −0.224408
\(622\) 2.31628e9 0.385944
\(623\) −2.68177e9 −0.444339
\(624\) 2.42971e8 0.0400320
\(625\) 3.62542e9 0.593988
\(626\) −1.11317e9 −0.181364
\(627\) −7.27026e9 −1.17791
\(628\) 4.62497e9 0.745161
\(629\) −7.07046e9 −1.13285
\(630\) 2.10741e8 0.0335782
\(631\) −2.75326e9 −0.436259 −0.218129 0.975920i \(-0.569995\pi\)
−0.218129 + 0.975920i \(0.569995\pi\)
\(632\) −2.71773e9 −0.428249
\(633\) 3.79584e9 0.594832
\(634\) 4.43100e9 0.690542
\(635\) 1.55771e9 0.241423
\(636\) 2.58541e9 0.398501
\(637\) 2.58475e8 0.0396214
\(638\) 1.15653e10 1.76313
\(639\) 2.31525e8 0.0351030
\(640\) 2.20937e8 0.0333149
\(641\) 5.67146e8 0.0850535 0.0425267 0.999095i \(-0.486459\pi\)
0.0425267 + 0.999095i \(0.486459\pi\)
\(642\) 1.91738e9 0.285980
\(643\) −7.33487e9 −1.08806 −0.544032 0.839065i \(-0.683103\pi\)
−0.544032 + 0.839065i \(0.683103\pi\)
\(644\) −1.49363e9 −0.220365
\(645\) −4.68899e8 −0.0688050
\(646\) 4.67973e9 0.682978
\(647\) 1.03775e10 1.50635 0.753176 0.657819i \(-0.228521\pi\)
0.753176 + 0.657819i \(0.228521\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −2.17255e9 −0.311970
\(650\) 1.17805e9 0.168255
\(651\) 8.38234e8 0.119078
\(652\) −2.69742e9 −0.381138
\(653\) −1.24513e10 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(654\) −3.83184e9 −0.535655
\(655\) 7.97204e8 0.110847
\(656\) −1.07490e9 −0.148664
\(657\) 2.30026e8 0.0316445
\(658\) −5.18325e8 −0.0709269
\(659\) −8.80623e9 −1.19865 −0.599323 0.800507i \(-0.704564\pi\)
−0.599323 + 0.800507i \(0.704564\pi\)
\(660\) −1.35102e9 −0.182918
\(661\) −9.54926e9 −1.28607 −0.643035 0.765837i \(-0.722325\pi\)
−0.643035 + 0.765837i \(0.722325\pi\)
\(662\) −6.14612e9 −0.823376
\(663\) 9.56350e8 0.127444
\(664\) 2.13753e9 0.283351
\(665\) 1.31111e9 0.172888
\(666\) 2.55766e9 0.335492
\(667\) 1.32543e10 1.72948
\(668\) −2.90842e9 −0.377520
\(669\) 3.12517e9 0.403537
\(670\) −7.83564e8 −0.100650
\(671\) −1.55100e10 −1.98191
\(672\) −3.03464e8 −0.0385758
\(673\) 1.29949e10 1.64332 0.821658 0.569982i \(-0.193050\pi\)
0.821658 + 0.569982i \(0.193050\pi\)
\(674\) −8.71929e8 −0.109691
\(675\) −1.31928e9 −0.165110
\(676\) 3.08916e8 0.0384615
\(677\) 1.11442e10 1.38035 0.690174 0.723644i \(-0.257534\pi\)
0.690174 + 0.723644i \(0.257534\pi\)
\(678\) 6.46916e9 0.797158
\(679\) 2.99749e9 0.367462
\(680\) 8.69623e8 0.106060
\(681\) 5.26320e9 0.638609
\(682\) −5.37374e9 −0.648681
\(683\) −1.17435e10 −1.41034 −0.705172 0.709036i \(-0.749130\pi\)
−0.705172 + 0.709036i \(0.749130\pi\)
\(684\) −1.69284e9 −0.202264
\(685\) 2.53254e8 0.0301050
\(686\) −3.22829e8 −0.0381802
\(687\) −7.20828e9 −0.848170
\(688\) 6.75207e8 0.0790456
\(689\) 3.28712e9 0.382867
\(690\) −1.54832e9 −0.179427
\(691\) 4.75656e8 0.0548428 0.0274214 0.999624i \(-0.491270\pi\)
0.0274214 + 0.999624i \(0.491270\pi\)
\(692\) −4.85046e9 −0.556432
\(693\) 1.85567e9 0.211804
\(694\) −2.98012e9 −0.338435
\(695\) −9.48615e8 −0.107187
\(696\) 2.69291e9 0.302754
\(697\) −4.23090e9 −0.473280
\(698\) −4.21896e9 −0.469582
\(699\) −4.57383e9 −0.506536
\(700\) −1.47136e9 −0.162135
\(701\) 2.54623e8 0.0279181 0.0139590 0.999903i \(-0.495557\pi\)
0.0139590 + 0.999903i \(0.495557\pi\)
\(702\) −3.45948e8 −0.0377426
\(703\) 1.59123e10 1.72738
\(704\) 1.94544e9 0.210143
\(705\) −5.37303e8 −0.0577507
\(706\) 9.29375e9 0.993973
\(707\) −1.96455e9 −0.209072
\(708\) −5.05864e8 −0.0535695
\(709\) 4.28666e9 0.451708 0.225854 0.974161i \(-0.427483\pi\)
0.225854 + 0.974161i \(0.427483\pi\)
\(710\) 2.67669e8 0.0280669
\(711\) 3.86958e9 0.403757
\(712\) 4.00312e9 0.415641
\(713\) −6.15851e9 −0.636300
\(714\) −1.19446e9 −0.122808
\(715\) −1.71770e9 −0.175742
\(716\) 4.59603e9 0.467937
\(717\) −1.25048e9 −0.126695
\(718\) 8.12171e9 0.818865
\(719\) 1.07444e10 1.07803 0.539014 0.842297i \(-0.318797\pi\)
0.539014 + 0.842297i \(0.318797\pi\)
\(720\) −3.14576e8 −0.0314096
\(721\) −2.48735e9 −0.247152
\(722\) −3.38088e9 −0.334310
\(723\) −1.19735e9 −0.117825
\(724\) −8.92867e9 −0.874383
\(725\) 1.30567e10 1.27248
\(726\) −7.68706e9 −0.745560
\(727\) 1.13786e10 1.09829 0.549146 0.835726i \(-0.314953\pi\)
0.549146 + 0.835726i \(0.314953\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 2.65936e8 0.0253015
\(731\) 2.65766e9 0.251646
\(732\) −3.61141e9 −0.340321
\(733\) −1.53993e10 −1.44423 −0.722115 0.691773i \(-0.756830\pi\)
−0.722115 + 0.691773i \(0.756830\pi\)
\(734\) −1.28285e8 −0.0119740
\(735\) −3.34649e8 −0.0310874
\(736\) 2.22955e9 0.206132
\(737\) −6.89962e9 −0.634876
\(738\) 1.53048e9 0.140162
\(739\) 7.52468e9 0.685855 0.342928 0.939362i \(-0.388581\pi\)
0.342928 + 0.939362i \(0.388581\pi\)
\(740\) 2.95694e9 0.268245
\(741\) −2.15229e9 −0.194329
\(742\) −4.10553e9 −0.368940
\(743\) −1.28963e10 −1.15347 −0.576733 0.816933i \(-0.695673\pi\)
−0.576733 + 0.816933i \(0.695673\pi\)
\(744\) −1.25124e9 −0.111387
\(745\) −3.18183e9 −0.281922
\(746\) 5.55477e9 0.489870
\(747\) −3.04348e9 −0.267145
\(748\) 7.65741e9 0.669000
\(749\) −3.04473e9 −0.264766
\(750\) −3.30303e9 −0.285889
\(751\) −1.43684e10 −1.23785 −0.618925 0.785450i \(-0.712431\pi\)
−0.618925 + 0.785450i \(0.712431\pi\)
\(752\) 7.73709e8 0.0663461
\(753\) 3.38350e9 0.288791
\(754\) 3.42380e9 0.290876
\(755\) −3.47008e8 −0.0293444
\(756\) 4.32081e8 0.0363696
\(757\) 8.75812e8 0.0733796 0.0366898 0.999327i \(-0.488319\pi\)
0.0366898 + 0.999327i \(0.488319\pi\)
\(758\) 1.08841e8 0.00907720
\(759\) −1.36336e10 −1.13179
\(760\) −1.95711e9 −0.161722
\(761\) 3.48167e9 0.286379 0.143190 0.989695i \(-0.454264\pi\)
0.143190 + 0.989695i \(0.454264\pi\)
\(762\) 3.19376e9 0.261493
\(763\) 6.08482e9 0.495920
\(764\) −7.78783e9 −0.631814
\(765\) −1.23819e9 −0.0999939
\(766\) −3.09108e9 −0.248490
\(767\) −6.43162e8 −0.0514679
\(768\) 4.52985e8 0.0360844
\(769\) 5.48045e9 0.434584 0.217292 0.976107i \(-0.430278\pi\)
0.217292 + 0.976107i \(0.430278\pi\)
\(770\) 2.14536e9 0.169349
\(771\) −7.23295e8 −0.0568362
\(772\) 3.53405e8 0.0276447
\(773\) −8.36540e9 −0.651416 −0.325708 0.945470i \(-0.605603\pi\)
−0.325708 + 0.945470i \(0.605603\pi\)
\(774\) −9.61379e8 −0.0745249
\(775\) −6.06669e9 −0.468162
\(776\) −4.47438e9 −0.343730
\(777\) −4.06146e9 −0.310606
\(778\) −1.10373e10 −0.840298
\(779\) 9.52175e9 0.721665
\(780\) −3.99956e8 −0.0301773
\(781\) 2.35694e9 0.177040
\(782\) 8.77569e9 0.656232
\(783\) −3.83424e9 −0.285439
\(784\) 4.81890e8 0.0357143
\(785\) −7.61320e9 −0.561724
\(786\) 1.63450e9 0.120062
\(787\) −1.86597e10 −1.36456 −0.682282 0.731089i \(-0.739012\pi\)
−0.682282 + 0.731089i \(0.739012\pi\)
\(788\) −6.22124e8 −0.0452934
\(789\) 1.01555e10 0.736094
\(790\) 4.47367e9 0.322827
\(791\) −1.02728e10 −0.738025
\(792\) −2.76998e9 −0.198125
\(793\) −4.59160e9 −0.326970
\(794\) 7.39581e9 0.524341
\(795\) −4.25586e9 −0.300402
\(796\) 2.79143e9 0.196169
\(797\) 6.32532e9 0.442567 0.221283 0.975210i \(-0.428975\pi\)
0.221283 + 0.975210i \(0.428975\pi\)
\(798\) 2.68816e9 0.187260
\(799\) 3.04537e9 0.211216
\(800\) 2.19631e9 0.151663
\(801\) −5.69975e9 −0.391870
\(802\) 7.59221e9 0.519706
\(803\) 2.34168e9 0.159597
\(804\) −1.60654e9 −0.109017
\(805\) 2.45867e9 0.166117
\(806\) −1.59084e9 −0.107017
\(807\) −2.20015e9 −0.147365
\(808\) 2.93250e9 0.195569
\(809\) 8.57316e9 0.569274 0.284637 0.958635i \(-0.408127\pi\)
0.284637 + 0.958635i \(0.408127\pi\)
\(810\) 4.47902e8 0.0296132
\(811\) 9.98151e9 0.657087 0.328544 0.944489i \(-0.393442\pi\)
0.328544 + 0.944489i \(0.393442\pi\)
\(812\) −4.27624e9 −0.280295
\(813\) −1.56012e10 −1.01822
\(814\) 2.60372e10 1.69203
\(815\) 4.44025e9 0.287313
\(816\) 1.78298e9 0.114876
\(817\) −5.98115e9 −0.383714
\(818\) −4.66048e9 −0.297710
\(819\) 5.49353e8 0.0349428
\(820\) 1.76941e9 0.112067
\(821\) −1.87762e10 −1.18415 −0.592076 0.805882i \(-0.701691\pi\)
−0.592076 + 0.805882i \(0.701691\pi\)
\(822\) 5.19244e8 0.0326077
\(823\) −1.24917e10 −0.781127 −0.390563 0.920576i \(-0.627720\pi\)
−0.390563 + 0.920576i \(0.627720\pi\)
\(824\) 3.71290e9 0.231189
\(825\) −1.34303e10 −0.832719
\(826\) 8.03293e8 0.0495957
\(827\) 2.74099e10 1.68515 0.842574 0.538580i \(-0.181039\pi\)
0.842574 + 0.538580i \(0.181039\pi\)
\(828\) −3.17450e9 −0.194343
\(829\) 3.76459e9 0.229497 0.114748 0.993395i \(-0.463394\pi\)
0.114748 + 0.993395i \(0.463394\pi\)
\(830\) −3.51860e9 −0.213598
\(831\) −1.73175e10 −1.04685
\(832\) 5.75930e8 0.0346688
\(833\) 1.89676e9 0.113698
\(834\) −1.94494e9 −0.116098
\(835\) 4.78758e9 0.284586
\(836\) −1.72332e10 −1.02010
\(837\) 1.78155e9 0.105017
\(838\) 4.07064e9 0.238951
\(839\) −3.12089e10 −1.82436 −0.912182 0.409786i \(-0.865603\pi\)
−0.912182 + 0.409786i \(0.865603\pi\)
\(840\) 4.99535e8 0.0290796
\(841\) 2.06970e10 1.19983
\(842\) −5.29609e9 −0.305747
\(843\) −1.64159e10 −0.943775
\(844\) 8.99754e9 0.515140
\(845\) −5.08508e8 −0.0289934
\(846\) −1.10163e9 −0.0625517
\(847\) 1.22068e10 0.690254
\(848\) 6.12838e9 0.345112
\(849\) 7.67638e9 0.430506
\(850\) 8.64485e9 0.482827
\(851\) 2.98396e10 1.65974
\(852\) 5.48800e8 0.0304001
\(853\) −1.73617e10 −0.957792 −0.478896 0.877872i \(-0.658963\pi\)
−0.478896 + 0.877872i \(0.658963\pi\)
\(854\) 5.73479e9 0.315076
\(855\) 2.78659e9 0.152473
\(856\) 4.54490e9 0.247666
\(857\) 6.93173e8 0.0376191 0.0188096 0.999823i \(-0.494012\pi\)
0.0188096 + 0.999823i \(0.494012\pi\)
\(858\) −3.52178e9 −0.190352
\(859\) 8.13662e9 0.437994 0.218997 0.975726i \(-0.429722\pi\)
0.218997 + 0.975726i \(0.429722\pi\)
\(860\) −1.11146e9 −0.0595869
\(861\) −2.43034e9 −0.129765
\(862\) 5.79276e9 0.308042
\(863\) 2.00601e10 1.06242 0.531209 0.847241i \(-0.321738\pi\)
0.531209 + 0.847241i \(0.321738\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 7.98438e9 0.419455
\(866\) −1.06564e9 −0.0557570
\(867\) −4.06120e9 −0.211635
\(868\) 1.98692e9 0.103125
\(869\) 3.93926e10 2.03632
\(870\) −4.43282e9 −0.228225
\(871\) −2.04257e9 −0.104740
\(872\) −9.08288e9 −0.463891
\(873\) 6.37075e9 0.324071
\(874\) −1.97499e10 −1.00063
\(875\) 5.24509e9 0.264682
\(876\) 5.45246e8 0.0274049
\(877\) −1.82633e10 −0.914281 −0.457141 0.889394i \(-0.651126\pi\)
−0.457141 + 0.889394i \(0.651126\pi\)
\(878\) −2.46146e10 −1.22733
\(879\) −2.05760e10 −1.02188
\(880\) −3.20241e9 −0.158412
\(881\) 1.30945e10 0.645168 0.322584 0.946541i \(-0.395448\pi\)
0.322584 + 0.946541i \(0.395448\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) 2.51147e10 1.22763 0.613813 0.789451i \(-0.289635\pi\)
0.613813 + 0.789451i \(0.289635\pi\)
\(884\) 2.26690e9 0.110370
\(885\) 8.32706e8 0.0403822
\(886\) 9.99586e9 0.482839
\(887\) 2.97056e10 1.42924 0.714621 0.699512i \(-0.246599\pi\)
0.714621 + 0.699512i \(0.246599\pi\)
\(888\) 6.06259e9 0.290545
\(889\) −5.07157e9 −0.242095
\(890\) −6.58956e9 −0.313322
\(891\) 3.94397e9 0.186794
\(892\) 7.40782e9 0.349473
\(893\) −6.85370e9 −0.322066
\(894\) −6.52368e9 −0.305359
\(895\) −7.56556e9 −0.352744
\(896\) −7.19323e8 −0.0334077
\(897\) −4.03610e9 −0.186719
\(898\) −8.08100e9 −0.372390
\(899\) −1.76317e10 −0.809350
\(900\) −3.12717e9 −0.142989
\(901\) 2.41217e10 1.09868
\(902\) 1.55804e10 0.706896
\(903\) 1.52663e9 0.0689966
\(904\) 1.53343e10 0.690359
\(905\) 1.46975e10 0.659135
\(906\) −7.11468e8 −0.0317839
\(907\) −6.96329e9 −0.309877 −0.154938 0.987924i \(-0.549518\pi\)
−0.154938 + 0.987924i \(0.549518\pi\)
\(908\) 1.24757e10 0.553052
\(909\) −4.17538e9 −0.184384
\(910\) 6.35115e8 0.0279388
\(911\) 1.54935e10 0.678948 0.339474 0.940615i \(-0.389751\pi\)
0.339474 + 0.940615i \(0.389751\pi\)
\(912\) −4.01265e9 −0.175166
\(913\) −3.09828e10 −1.34733
\(914\) −2.05620e10 −0.890746
\(915\) 5.94477e9 0.256544
\(916\) −1.70863e10 −0.734537
\(917\) −2.59553e9 −0.111156
\(918\) −2.53866e9 −0.108307
\(919\) −2.78052e10 −1.18174 −0.590870 0.806767i \(-0.701215\pi\)
−0.590870 + 0.806767i \(0.701215\pi\)
\(920\) −3.67009e9 −0.155388
\(921\) 4.57737e9 0.193067
\(922\) −2.46819e10 −1.03710
\(923\) 6.97750e8 0.0292075
\(924\) 4.39862e9 0.183428
\(925\) 2.93947e10 1.22116
\(926\) −1.08573e10 −0.449348
\(927\) −5.28653e9 −0.217967
\(928\) 6.38319e9 0.262192
\(929\) −2.02174e10 −0.827313 −0.413656 0.910433i \(-0.635749\pi\)
−0.413656 + 0.910433i \(0.635749\pi\)
\(930\) 2.05968e9 0.0839670
\(931\) −4.26870e9 −0.173369
\(932\) −1.08417e10 −0.438673
\(933\) −7.81744e9 −0.315122
\(934\) −1.44770e10 −0.581387
\(935\) −1.26049e10 −0.504312
\(936\) −8.20026e8 −0.0326860
\(937\) −6.44768e9 −0.256044 −0.128022 0.991771i \(-0.540863\pi\)
−0.128022 + 0.991771i \(0.540863\pi\)
\(938\) 2.55112e9 0.100930
\(939\) 3.75696e9 0.148083
\(940\) −1.27361e9 −0.0500136
\(941\) −2.37514e10 −0.929235 −0.464618 0.885511i \(-0.653808\pi\)
−0.464618 + 0.885511i \(0.653808\pi\)
\(942\) −1.56093e10 −0.608422
\(943\) 1.78557e10 0.693404
\(944\) −1.19909e9 −0.0463925
\(945\) −7.11252e8 −0.0274165
\(946\) −9.78692e9 −0.375861
\(947\) 3.20929e10 1.22796 0.613980 0.789321i \(-0.289567\pi\)
0.613980 + 0.789321i \(0.289567\pi\)
\(948\) 9.17233e9 0.349664
\(949\) 6.93233e8 0.0263298
\(950\) −1.94555e10 −0.736223
\(951\) −1.49546e10 −0.563825
\(952\) −2.83131e9 −0.106355
\(953\) 2.67052e10 0.999471 0.499736 0.866178i \(-0.333431\pi\)
0.499736 + 0.866178i \(0.333431\pi\)
\(954\) −8.72575e9 −0.325375
\(955\) 1.28196e10 0.476280
\(956\) −2.96410e9 −0.109721
\(957\) −3.90329e10 −1.43959
\(958\) 2.47267e10 0.908630
\(959\) −8.24541e8 −0.0301889
\(960\) −7.45661e8 −0.0272015
\(961\) −1.93201e10 −0.702229
\(962\) 7.70806e9 0.279146
\(963\) −6.47115e9 −0.233501
\(964\) −2.83817e9 −0.102040
\(965\) −5.81743e8 −0.0208394
\(966\) 5.04099e9 0.179927
\(967\) 4.84920e10 1.72456 0.862278 0.506436i \(-0.169037\pi\)
0.862278 + 0.506436i \(0.169037\pi\)
\(968\) −1.82212e10 −0.645674
\(969\) −1.57941e10 −0.557649
\(970\) 7.36531e9 0.259113
\(971\) −2.36498e10 −0.829010 −0.414505 0.910047i \(-0.636045\pi\)
−0.414505 + 0.910047i \(0.636045\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 3.08849e9 0.107486
\(974\) −5.75022e9 −0.199401
\(975\) −3.97593e9 −0.137380
\(976\) −8.56039e9 −0.294726
\(977\) 1.45041e10 0.497578 0.248789 0.968558i \(-0.419967\pi\)
0.248789 + 0.968558i \(0.419967\pi\)
\(978\) 9.10380e9 0.311198
\(979\) −5.80239e10 −1.97637
\(980\) −7.93243e8 −0.0269225
\(981\) 1.29325e10 0.437361
\(982\) 1.66159e10 0.559930
\(983\) −4.38465e10 −1.47230 −0.736152 0.676816i \(-0.763359\pi\)
−0.736152 + 0.676816i \(0.763359\pi\)
\(984\) 3.62780e9 0.121384
\(985\) 1.02408e9 0.0341435
\(986\) 2.51247e10 0.834703
\(987\) 1.74935e9 0.0579116
\(988\) −5.10173e9 −0.168294
\(989\) −1.12162e10 −0.368687
\(990\) 4.55968e9 0.149352
\(991\) −3.00532e10 −0.980920 −0.490460 0.871464i \(-0.663171\pi\)
−0.490460 + 0.871464i \(0.663171\pi\)
\(992\) −2.96591e9 −0.0964643
\(993\) 2.07431e10 0.672283
\(994\) −8.71473e8 −0.0281450
\(995\) −4.59499e9 −0.147878
\(996\) −7.21417e9 −0.231355
\(997\) 1.48434e10 0.474352 0.237176 0.971467i \(-0.423778\pi\)
0.237176 + 0.971467i \(0.423778\pi\)
\(998\) 1.62597e10 0.517792
\(999\) −8.63209e9 −0.273928
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.i.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.i.1.3 5 1.1 even 1 trivial