Properties

Label 546.8.a.j.1.4
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: \(x^{5} - 5672 x^{3} - 117684 x^{2} + 1695035 x + 39011360\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-21.9822\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1997.82 q^{10} +3233.99 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +6742.63 q^{15} +4096.00 q^{16} +9015.75 q^{17} -5832.00 q^{18} -35945.0 q^{19} +15982.5 q^{20} -9261.00 q^{21} -25871.9 q^{22} -38564.1 q^{23} -13824.0 q^{24} -15761.4 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +68241.3 q^{29} -53941.0 q^{30} +118835. q^{31} -32768.0 q^{32} +87317.8 q^{33} -72126.0 q^{34} -85656.4 q^{35} +46656.0 q^{36} -590446. q^{37} +287560. q^{38} -59319.0 q^{39} -127860. q^{40} +414223. q^{41} +74088.0 q^{42} +47634.0 q^{43} +206975. q^{44} +182051. q^{45} +308513. q^{46} -347512. q^{47} +110592. q^{48} +117649. q^{49} +126091. q^{50} +243425. q^{51} -140608. q^{52} -547089. q^{53} -157464. q^{54} +807615. q^{55} +175616. q^{56} -970514. q^{57} -545931. q^{58} +995629. q^{59} +431528. q^{60} -2.70586e6 q^{61} -950680. q^{62} -250047. q^{63} +262144. q^{64} -548650. q^{65} -698542. q^{66} -3.28358e6 q^{67} +577008. q^{68} -1.04123e6 q^{69} +685251. q^{70} -5.43613e6 q^{71} -373248. q^{72} -618715. q^{73} +4.72357e6 q^{74} -425558. q^{75} -2.30048e6 q^{76} -1.10926e6 q^{77} +474552. q^{78} -6.29147e6 q^{79} +1.02288e6 q^{80} +531441. q^{81} -3.31378e6 q^{82} +5.10103e6 q^{83} -592704. q^{84} +2.25148e6 q^{85} -381072. q^{86} +1.84252e6 q^{87} -1.65580e6 q^{88} +6.51911e6 q^{89} -1.45641e6 q^{90} +753571. q^{91} -2.46810e6 q^{92} +3.20854e6 q^{93} +2.78010e6 q^{94} -8.97643e6 q^{95} -884736. q^{96} +9.09144e6 q^{97} -941192. q^{98} +2.35758e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99} + O(q^{100}) \)

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
\(5\) 249.727 0.893451 0.446725 0.894671i \(-0.352590\pi\)
0.446725 + 0.894671i \(0.352590\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) −1997.82 −0.631765
\(11\) 3233.99 0.732596 0.366298 0.930498i \(-0.380625\pi\)
0.366298 + 0.930498i \(0.380625\pi\)
\(12\) 1728.00 0.288675
\(13\) −2197.00 −0.277350
\(14\) 2744.00 0.267261
\(15\) 6742.63 0.515834
\(16\) 4096.00 0.250000
\(17\) 9015.75 0.445072 0.222536 0.974924i \(-0.428566\pi\)
0.222536 + 0.974924i \(0.428566\pi\)
\(18\) −5832.00 −0.235702
\(19\) −35945.0 −1.20227 −0.601133 0.799149i \(-0.705284\pi\)
−0.601133 + 0.799149i \(0.705284\pi\)
\(20\) 15982.5 0.446725
\(21\) −9261.00 −0.218218
\(22\) −25871.9 −0.518023
\(23\) −38564.1 −0.660899 −0.330450 0.943824i \(-0.607200\pi\)
−0.330450 + 0.943824i \(0.607200\pi\)
\(24\) −13824.0 −0.204124
\(25\) −15761.4 −0.201746
\(26\) 17576.0 0.196116
\(27\) 19683.0 0.192450
\(28\) −21952.0 −0.188982
\(29\) 68241.3 0.519582 0.259791 0.965665i \(-0.416346\pi\)
0.259791 + 0.965665i \(0.416346\pi\)
\(30\) −53941.0 −0.364750
\(31\) 118835. 0.716438 0.358219 0.933638i \(-0.383384\pi\)
0.358219 + 0.933638i \(0.383384\pi\)
\(32\) −32768.0 −0.176777
\(33\) 87317.8 0.422964
\(34\) −72126.0 −0.314714
\(35\) −85656.4 −0.337693
\(36\) 46656.0 0.166667
\(37\) −590446. −1.91635 −0.958174 0.286186i \(-0.907612\pi\)
−0.958174 + 0.286186i \(0.907612\pi\)
\(38\) 287560. 0.850130
\(39\) −59319.0 −0.160128
\(40\) −127860. −0.315882
\(41\) 414223. 0.938621 0.469311 0.883033i \(-0.344503\pi\)
0.469311 + 0.883033i \(0.344503\pi\)
\(42\) 74088.0 0.154303
\(43\) 47634.0 0.0913645 0.0456822 0.998956i \(-0.485454\pi\)
0.0456822 + 0.998956i \(0.485454\pi\)
\(44\) 206975. 0.366298
\(45\) 182051. 0.297817
\(46\) 308513. 0.467326
\(47\) −347512. −0.488233 −0.244117 0.969746i \(-0.578498\pi\)
−0.244117 + 0.969746i \(0.578498\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 126091. 0.142656
\(51\) 243425. 0.256963
\(52\) −140608. −0.138675
\(53\) −547089. −0.504769 −0.252385 0.967627i \(-0.581215\pi\)
−0.252385 + 0.967627i \(0.581215\pi\)
\(54\) −157464. −0.136083
\(55\) 807615. 0.654538
\(56\) 175616. 0.133631
\(57\) −970514. −0.694128
\(58\) −545931. −0.367400
\(59\) 995629. 0.631125 0.315563 0.948905i \(-0.397807\pi\)
0.315563 + 0.948905i \(0.397807\pi\)
\(60\) 431528. 0.257917
\(61\) −2.70586e6 −1.52634 −0.763168 0.646200i \(-0.776357\pi\)
−0.763168 + 0.646200i \(0.776357\pi\)
\(62\) −950680. −0.506598
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) −548650. −0.247799
\(66\) −698542. −0.299081
\(67\) −3.28358e6 −1.33379 −0.666893 0.745154i \(-0.732376\pi\)
−0.666893 + 0.745154i \(0.732376\pi\)
\(68\) 577008. 0.222536
\(69\) −1.04123e6 −0.381570
\(70\) 685251. 0.238785
\(71\) −5.43613e6 −1.80254 −0.901272 0.433253i \(-0.857366\pi\)
−0.901272 + 0.433253i \(0.857366\pi\)
\(72\) −373248. −0.117851
\(73\) −618715. −0.186149 −0.0930744 0.995659i \(-0.529669\pi\)
−0.0930744 + 0.995659i \(0.529669\pi\)
\(74\) 4.72357e6 1.35506
\(75\) −425558. −0.116478
\(76\) −2.30048e6 −0.601133
\(77\) −1.10926e6 −0.276895
\(78\) 474552. 0.113228
\(79\) −6.29147e6 −1.43568 −0.717839 0.696209i \(-0.754869\pi\)
−0.717839 + 0.696209i \(0.754869\pi\)
\(80\) 1.02288e6 0.223363
\(81\) 531441. 0.111111
\(82\) −3.31378e6 −0.663705
\(83\) 5.10103e6 0.979230 0.489615 0.871939i \(-0.337137\pi\)
0.489615 + 0.871939i \(0.337137\pi\)
\(84\) −592704. −0.109109
\(85\) 2.25148e6 0.397650
\(86\) −381072. −0.0646045
\(87\) 1.84252e6 0.299981
\(88\) −1.65580e6 −0.259012
\(89\) 6.51911e6 0.980220 0.490110 0.871661i \(-0.336957\pi\)
0.490110 + 0.871661i \(0.336957\pi\)
\(90\) −1.45641e6 −0.210588
\(91\) 753571. 0.104828
\(92\) −2.46810e6 −0.330450
\(93\) 3.20854e6 0.413635
\(94\) 2.78010e6 0.345233
\(95\) −8.97643e6 −1.07416
\(96\) −884736. −0.102062
\(97\) 9.09144e6 1.01142 0.505710 0.862703i \(-0.331231\pi\)
0.505710 + 0.862703i \(0.331231\pi\)
\(98\) −941192. −0.101015
\(99\) 2.35758e6 0.244199
\(100\) −1.00873e6 −0.100873
\(101\) −1.64041e7 −1.58426 −0.792130 0.610353i \(-0.791028\pi\)
−0.792130 + 0.610353i \(0.791028\pi\)
\(102\) −1.94740e6 −0.181700
\(103\) −3.47734e6 −0.313557 −0.156778 0.987634i \(-0.550111\pi\)
−0.156778 + 0.987634i \(0.550111\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) −2.31272e6 −0.194967
\(106\) 4.37672e6 0.356926
\(107\) 9.49661e6 0.749420 0.374710 0.927142i \(-0.377742\pi\)
0.374710 + 0.927142i \(0.377742\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 9.41138e6 0.696082 0.348041 0.937479i \(-0.386847\pi\)
0.348041 + 0.937479i \(0.386847\pi\)
\(110\) −6.46092e6 −0.462828
\(111\) −1.59421e7 −1.10640
\(112\) −1.40493e6 −0.0944911
\(113\) 1.22161e7 0.796448 0.398224 0.917288i \(-0.369627\pi\)
0.398224 + 0.917288i \(0.369627\pi\)
\(114\) 7.76411e6 0.490823
\(115\) −9.63049e6 −0.590481
\(116\) 4.36744e6 0.259791
\(117\) −1.60161e6 −0.0924500
\(118\) −7.96503e6 −0.446273
\(119\) −3.09240e6 −0.168222
\(120\) −3.45223e6 −0.182375
\(121\) −9.02847e6 −0.463303
\(122\) 2.16469e7 1.07928
\(123\) 1.11840e7 0.541913
\(124\) 7.60544e6 0.358219
\(125\) −2.34460e7 −1.07370
\(126\) 2.00038e6 0.0890871
\(127\) 3.04755e7 1.32019 0.660097 0.751180i \(-0.270515\pi\)
0.660097 + 0.751180i \(0.270515\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 1.28612e6 0.0527493
\(130\) 4.38920e6 0.175220
\(131\) 1.52623e7 0.593159 0.296580 0.955008i \(-0.404154\pi\)
0.296580 + 0.955008i \(0.404154\pi\)
\(132\) 5.58834e6 0.211482
\(133\) 1.23291e7 0.454414
\(134\) 2.62687e7 0.943129
\(135\) 4.91538e6 0.171945
\(136\) −4.61606e6 −0.157357
\(137\) 5.04710e6 0.167695 0.0838475 0.996479i \(-0.473279\pi\)
0.0838475 + 0.996479i \(0.473279\pi\)
\(138\) 8.32984e6 0.269811
\(139\) −2.25126e7 −0.711007 −0.355504 0.934675i \(-0.615691\pi\)
−0.355504 + 0.934675i \(0.615691\pi\)
\(140\) −5.48201e6 −0.168846
\(141\) −9.38283e6 −0.281882
\(142\) 4.34891e7 1.27459
\(143\) −7.10508e6 −0.203186
\(144\) 2.98598e6 0.0833333
\(145\) 1.70417e7 0.464221
\(146\) 4.94972e6 0.131627
\(147\) 3.17652e6 0.0824786
\(148\) −3.77886e7 −0.958174
\(149\) −3.32626e7 −0.823766 −0.411883 0.911237i \(-0.635129\pi\)
−0.411883 + 0.911237i \(0.635129\pi\)
\(150\) 3.40446e6 0.0823625
\(151\) −3.66037e7 −0.865179 −0.432590 0.901591i \(-0.642400\pi\)
−0.432590 + 0.901591i \(0.642400\pi\)
\(152\) 1.84038e7 0.425065
\(153\) 6.57248e6 0.148357
\(154\) 8.87407e6 0.195794
\(155\) 2.96763e7 0.640102
\(156\) −3.79642e6 −0.0800641
\(157\) −4.81360e6 −0.0992708 −0.0496354 0.998767i \(-0.515806\pi\)
−0.0496354 + 0.998767i \(0.515806\pi\)
\(158\) 5.03317e7 1.01518
\(159\) −1.47714e7 −0.291429
\(160\) −8.18306e6 −0.157941
\(161\) 1.32275e7 0.249796
\(162\) −4.25153e6 −0.0785674
\(163\) 1.76309e7 0.318873 0.159437 0.987208i \(-0.449032\pi\)
0.159437 + 0.987208i \(0.449032\pi\)
\(164\) 2.65102e7 0.469311
\(165\) 2.18056e7 0.377898
\(166\) −4.08083e7 −0.692420
\(167\) 5.12049e7 0.850754 0.425377 0.905016i \(-0.360141\pi\)
0.425377 + 0.905016i \(0.360141\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −1.80118e7 −0.281181
\(171\) −2.62039e7 −0.400755
\(172\) 3.04858e6 0.0456822
\(173\) −6.62195e6 −0.0972355 −0.0486177 0.998817i \(-0.515482\pi\)
−0.0486177 + 0.998817i \(0.515482\pi\)
\(174\) −1.47401e7 −0.212119
\(175\) 5.40616e6 0.0762528
\(176\) 1.32464e7 0.183149
\(177\) 2.68820e7 0.364380
\(178\) −5.21529e7 −0.693120
\(179\) 281431. 0.00366764 0.00183382 0.999998i \(-0.499416\pi\)
0.00183382 + 0.999998i \(0.499416\pi\)
\(180\) 1.16513e7 0.148908
\(181\) 1.25701e8 1.57567 0.787835 0.615886i \(-0.211202\pi\)
0.787835 + 0.615886i \(0.211202\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −7.30581e7 −0.881231
\(184\) 1.97448e7 0.233663
\(185\) −1.47450e8 −1.71216
\(186\) −2.56684e7 −0.292484
\(187\) 2.91569e7 0.326058
\(188\) −2.22408e7 −0.244117
\(189\) −6.75127e6 −0.0727393
\(190\) 7.18114e7 0.759549
\(191\) −1.56684e7 −0.162708 −0.0813538 0.996685i \(-0.525924\pi\)
−0.0813538 + 0.996685i \(0.525924\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −9.89310e6 −0.0990563 −0.0495281 0.998773i \(-0.515772\pi\)
−0.0495281 + 0.998773i \(0.515772\pi\)
\(194\) −7.27315e7 −0.715182
\(195\) −1.48136e7 −0.143067
\(196\) 7.52954e6 0.0714286
\(197\) −7.26858e7 −0.677357 −0.338679 0.940902i \(-0.609980\pi\)
−0.338679 + 0.940902i \(0.609980\pi\)
\(198\) −1.88606e7 −0.172674
\(199\) −1.22132e8 −1.09861 −0.549303 0.835623i \(-0.685107\pi\)
−0.549303 + 0.835623i \(0.685107\pi\)
\(200\) 8.06984e6 0.0713280
\(201\) −8.86567e7 −0.770061
\(202\) 1.31232e8 1.12024
\(203\) −2.34068e7 −0.196384
\(204\) 1.55792e7 0.128481
\(205\) 1.03443e8 0.838612
\(206\) 2.78187e7 0.221718
\(207\) −2.81132e7 −0.220300
\(208\) −8.99891e6 −0.0693375
\(209\) −1.16246e8 −0.880775
\(210\) 1.85018e7 0.137862
\(211\) −7.04312e7 −0.516151 −0.258075 0.966125i \(-0.583088\pi\)
−0.258075 + 0.966125i \(0.583088\pi\)
\(212\) −3.50137e7 −0.252385
\(213\) −1.46776e8 −1.04070
\(214\) −7.59729e7 −0.529920
\(215\) 1.18955e7 0.0816297
\(216\) −1.00777e7 −0.0680414
\(217\) −4.07604e7 −0.270788
\(218\) −7.52910e7 −0.492204
\(219\) −1.67053e7 −0.107473
\(220\) 5.16874e7 0.327269
\(221\) −1.98076e7 −0.123441
\(222\) 1.27536e8 0.782346
\(223\) −4.01526e7 −0.242463 −0.121232 0.992624i \(-0.538684\pi\)
−0.121232 + 0.992624i \(0.538684\pi\)
\(224\) 1.12394e7 0.0668153
\(225\) −1.14901e7 −0.0672487
\(226\) −9.77287e7 −0.563174
\(227\) 4.39233e7 0.249232 0.124616 0.992205i \(-0.460230\pi\)
0.124616 + 0.992205i \(0.460230\pi\)
\(228\) −6.21129e7 −0.347064
\(229\) 7.28347e7 0.400788 0.200394 0.979715i \(-0.435778\pi\)
0.200394 + 0.979715i \(0.435778\pi\)
\(230\) 7.70439e7 0.417533
\(231\) −2.99500e7 −0.159866
\(232\) −3.49396e7 −0.183700
\(233\) 2.54186e8 1.31646 0.658228 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −8.67832e7 −0.436212
\(236\) 6.37203e7 0.315563
\(237\) −1.69870e8 −0.828889
\(238\) 2.47392e7 0.118951
\(239\) −2.59103e8 −1.22767 −0.613833 0.789436i \(-0.710373\pi\)
−0.613833 + 0.789436i \(0.710373\pi\)
\(240\) 2.76178e7 0.128958
\(241\) −3.73599e8 −1.71928 −0.859639 0.510903i \(-0.829311\pi\)
−0.859639 + 0.510903i \(0.829311\pi\)
\(242\) 7.22278e7 0.327605
\(243\) 1.43489e7 0.0641500
\(244\) −1.73175e8 −0.763168
\(245\) 2.93801e7 0.127636
\(246\) −8.94721e7 −0.383190
\(247\) 7.89711e7 0.333448
\(248\) −6.08435e7 −0.253299
\(249\) 1.37728e8 0.565359
\(250\) 1.87568e8 0.759221
\(251\) 3.26441e8 1.30301 0.651503 0.758646i \(-0.274139\pi\)
0.651503 + 0.758646i \(0.274139\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) −1.24716e8 −0.484172
\(254\) −2.43804e8 −0.933518
\(255\) 6.07899e7 0.229583
\(256\) 1.67772e7 0.0625000
\(257\) −4.20813e8 −1.54640 −0.773202 0.634160i \(-0.781346\pi\)
−0.773202 + 0.634160i \(0.781346\pi\)
\(258\) −1.02889e7 −0.0372994
\(259\) 2.02523e8 0.724312
\(260\) −3.51136e7 −0.123899
\(261\) 4.97479e7 0.173194
\(262\) −1.22099e8 −0.419427
\(263\) −4.68224e8 −1.58711 −0.793557 0.608496i \(-0.791773\pi\)
−0.793557 + 0.608496i \(0.791773\pi\)
\(264\) −4.47067e7 −0.149541
\(265\) −1.36623e8 −0.450986
\(266\) −9.86330e7 −0.321319
\(267\) 1.76016e8 0.565930
\(268\) −2.10149e8 −0.666893
\(269\) −1.75612e8 −0.550076 −0.275038 0.961433i \(-0.588690\pi\)
−0.275038 + 0.961433i \(0.588690\pi\)
\(270\) −3.93230e7 −0.121583
\(271\) 1.22913e8 0.375150 0.187575 0.982250i \(-0.439937\pi\)
0.187575 + 0.982250i \(0.439937\pi\)
\(272\) 3.69285e7 0.111268
\(273\) 2.03464e7 0.0605228
\(274\) −4.03768e7 −0.118578
\(275\) −5.09723e7 −0.147798
\(276\) −6.66387e7 −0.190785
\(277\) 3.46024e8 0.978197 0.489099 0.872228i \(-0.337326\pi\)
0.489099 + 0.872228i \(0.337326\pi\)
\(278\) 1.80101e8 0.502758
\(279\) 8.66307e7 0.238813
\(280\) 4.38561e7 0.119392
\(281\) −1.60442e8 −0.431367 −0.215683 0.976463i \(-0.569198\pi\)
−0.215683 + 0.976463i \(0.569198\pi\)
\(282\) 7.50626e7 0.199320
\(283\) −4.63629e8 −1.21596 −0.607979 0.793953i \(-0.708019\pi\)
−0.607979 + 0.793953i \(0.708019\pi\)
\(284\) −3.47913e8 −0.901272
\(285\) −2.42364e8 −0.620169
\(286\) 5.68406e7 0.143674
\(287\) −1.42078e8 −0.354765
\(288\) −2.38879e7 −0.0589256
\(289\) −3.29055e8 −0.801911
\(290\) −1.36334e8 −0.328254
\(291\) 2.45469e8 0.583944
\(292\) −3.95977e7 −0.0930744
\(293\) −7.31152e7 −0.169813 −0.0849066 0.996389i \(-0.527059\pi\)
−0.0849066 + 0.996389i \(0.527059\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 2.48635e8 0.563879
\(296\) 3.02309e8 0.677531
\(297\) 6.36547e7 0.140988
\(298\) 2.66101e8 0.582491
\(299\) 8.47252e7 0.183301
\(300\) −2.72357e7 −0.0582391
\(301\) −1.63385e7 −0.0345325
\(302\) 2.92830e8 0.611774
\(303\) −4.42909e8 −0.914673
\(304\) −1.47231e8 −0.300566
\(305\) −6.75726e8 −1.36371
\(306\) −5.25798e7 −0.104905
\(307\) −7.31387e8 −1.44266 −0.721328 0.692594i \(-0.756468\pi\)
−0.721328 + 0.692594i \(0.756468\pi\)
\(308\) −7.09926e7 −0.138448
\(309\) −9.38881e7 −0.181032
\(310\) −2.37410e8 −0.452620
\(311\) −9.69470e8 −1.82757 −0.913783 0.406203i \(-0.866853\pi\)
−0.913783 + 0.406203i \(0.866853\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 7.36168e8 1.35697 0.678487 0.734612i \(-0.262636\pi\)
0.678487 + 0.734612i \(0.262636\pi\)
\(314\) 3.85088e7 0.0701950
\(315\) −6.24435e7 −0.112564
\(316\) −4.02654e8 −0.717839
\(317\) −1.27245e8 −0.224355 −0.112177 0.993688i \(-0.535782\pi\)
−0.112177 + 0.993688i \(0.535782\pi\)
\(318\) 1.18171e8 0.206071
\(319\) 2.20692e8 0.380644
\(320\) 6.54644e7 0.111681
\(321\) 2.56409e8 0.432678
\(322\) −1.05820e8 −0.176633
\(323\) −3.24071e8 −0.535095
\(324\) 3.40122e7 0.0555556
\(325\) 3.46278e7 0.0559543
\(326\) −1.41047e8 −0.225477
\(327\) 2.54107e8 0.401883
\(328\) −2.12082e8 −0.331853
\(329\) 1.19197e8 0.184535
\(330\) −1.74445e8 −0.267214
\(331\) 1.22358e8 0.185452 0.0927262 0.995692i \(-0.470442\pi\)
0.0927262 + 0.995692i \(0.470442\pi\)
\(332\) 3.26466e8 0.489615
\(333\) −4.30435e8 −0.638783
\(334\) −4.09640e8 −0.601574
\(335\) −8.19999e8 −1.19167
\(336\) −3.79331e7 −0.0545545
\(337\) −1.06041e8 −0.150928 −0.0754639 0.997149i \(-0.524044\pi\)
−0.0754639 + 0.997149i \(0.524044\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 3.29834e8 0.459830
\(340\) 1.44094e8 0.198825
\(341\) 3.84311e8 0.524859
\(342\) 2.09631e8 0.283377
\(343\) −4.03536e7 −0.0539949
\(344\) −2.43886e7 −0.0323022
\(345\) −2.60023e8 −0.340914
\(346\) 5.29756e7 0.0687559
\(347\) −1.27113e9 −1.63319 −0.816596 0.577210i \(-0.804141\pi\)
−0.816596 + 0.577210i \(0.804141\pi\)
\(348\) 1.17921e8 0.149991
\(349\) 1.17229e9 1.47621 0.738103 0.674688i \(-0.235722\pi\)
0.738103 + 0.674688i \(0.235722\pi\)
\(350\) −4.32493e7 −0.0539189
\(351\) −4.32436e7 −0.0533761
\(352\) −1.05971e8 −0.129506
\(353\) 9.12868e8 1.10458 0.552289 0.833653i \(-0.313754\pi\)
0.552289 + 0.833653i \(0.313754\pi\)
\(354\) −2.15056e8 −0.257656
\(355\) −1.35755e9 −1.61048
\(356\) 4.17223e8 0.490110
\(357\) −8.34948e7 −0.0971227
\(358\) −2.25145e6 −0.00259341
\(359\) −9.38327e8 −1.07035 −0.535173 0.844743i \(-0.679753\pi\)
−0.535173 + 0.844743i \(0.679753\pi\)
\(360\) −9.32101e7 −0.105294
\(361\) 3.98168e8 0.445442
\(362\) −1.00561e9 −1.11417
\(363\) −2.43769e8 −0.267488
\(364\) 4.82285e7 0.0524142
\(365\) −1.54510e8 −0.166315
\(366\) 5.84465e8 0.623124
\(367\) 1.09083e8 0.115193 0.0575967 0.998340i \(-0.481656\pi\)
0.0575967 + 0.998340i \(0.481656\pi\)
\(368\) −1.57958e8 −0.165225
\(369\) 3.01968e8 0.312874
\(370\) 1.17960e9 1.21068
\(371\) 1.87652e8 0.190785
\(372\) 2.05347e8 0.206818
\(373\) −6.53066e8 −0.651593 −0.325796 0.945440i \(-0.605632\pi\)
−0.325796 + 0.945440i \(0.605632\pi\)
\(374\) −2.33255e8 −0.230558
\(375\) −6.33041e8 −0.619901
\(376\) 1.77926e8 0.172617
\(377\) −1.49926e8 −0.144106
\(378\) 5.40102e7 0.0514344
\(379\) −2.65598e8 −0.250604 −0.125302 0.992119i \(-0.539990\pi\)
−0.125302 + 0.992119i \(0.539990\pi\)
\(380\) −5.74491e8 −0.537082
\(381\) 8.22839e8 0.762215
\(382\) 1.25347e8 0.115052
\(383\) −1.11095e9 −1.01041 −0.505205 0.863000i \(-0.668583\pi\)
−0.505205 + 0.863000i \(0.668583\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −2.77012e8 −0.247392
\(386\) 7.91448e7 0.0700434
\(387\) 3.47252e7 0.0304548
\(388\) 5.81852e8 0.505710
\(389\) −1.13929e9 −0.981323 −0.490661 0.871350i \(-0.663245\pi\)
−0.490661 + 0.871350i \(0.663245\pi\)
\(390\) 1.18508e8 0.101163
\(391\) −3.47684e8 −0.294148
\(392\) −6.02363e7 −0.0505076
\(393\) 4.12083e8 0.342461
\(394\) 5.81487e8 0.478964
\(395\) −1.57115e9 −1.28271
\(396\) 1.50885e8 0.122099
\(397\) −1.23511e9 −0.990690 −0.495345 0.868696i \(-0.664958\pi\)
−0.495345 + 0.868696i \(0.664958\pi\)
\(398\) 9.77053e8 0.776832
\(399\) 3.32886e8 0.262356
\(400\) −6.45587e7 −0.0504365
\(401\) −1.44292e9 −1.11747 −0.558736 0.829346i \(-0.688713\pi\)
−0.558736 + 0.829346i \(0.688713\pi\)
\(402\) 7.09254e8 0.544516
\(403\) −2.61080e8 −0.198704
\(404\) −1.04986e9 −0.792130
\(405\) 1.32715e8 0.0992723
\(406\) 1.87254e8 0.138864
\(407\) −1.90950e9 −1.40391
\(408\) −1.24634e8 −0.0908500
\(409\) −1.83879e9 −1.32892 −0.664462 0.747322i \(-0.731339\pi\)
−0.664462 + 0.747322i \(0.731339\pi\)
\(410\) −8.27541e8 −0.592988
\(411\) 1.36272e8 0.0968188
\(412\) −2.22550e8 −0.156778
\(413\) −3.41501e8 −0.238543
\(414\) 2.24906e8 0.155775
\(415\) 1.27387e9 0.874894
\(416\) 7.19913e7 0.0490290
\(417\) −6.07840e8 −0.410500
\(418\) 9.29965e8 0.622802
\(419\) −1.01820e9 −0.676215 −0.338107 0.941108i \(-0.609787\pi\)
−0.338107 + 0.941108i \(0.609787\pi\)
\(420\) −1.48014e8 −0.0974835
\(421\) −5.82161e8 −0.380238 −0.190119 0.981761i \(-0.560887\pi\)
−0.190119 + 0.981761i \(0.560887\pi\)
\(422\) 5.63450e8 0.364974
\(423\) −2.53336e8 −0.162744
\(424\) 2.80110e8 0.178463
\(425\) −1.42101e8 −0.0897916
\(426\) 1.17421e9 0.735886
\(427\) 9.28109e8 0.576901
\(428\) 6.07783e8 0.374710
\(429\) −1.91837e8 −0.117309
\(430\) −9.51640e7 −0.0577209
\(431\) 1.89524e8 0.114023 0.0570117 0.998374i \(-0.481843\pi\)
0.0570117 + 0.998374i \(0.481843\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 2.17028e9 1.28472 0.642359 0.766404i \(-0.277956\pi\)
0.642359 + 0.766404i \(0.277956\pi\)
\(434\) 3.26083e8 0.191476
\(435\) 4.60126e8 0.268018
\(436\) 6.02328e8 0.348041
\(437\) 1.38618e9 0.794577
\(438\) 1.33642e8 0.0759950
\(439\) 5.12061e8 0.288866 0.144433 0.989515i \(-0.453864\pi\)
0.144433 + 0.989515i \(0.453864\pi\)
\(440\) −4.13499e8 −0.231414
\(441\) 8.57661e7 0.0476190
\(442\) 1.58461e8 0.0872859
\(443\) 9.82509e8 0.536937 0.268469 0.963288i \(-0.413482\pi\)
0.268469 + 0.963288i \(0.413482\pi\)
\(444\) −1.02029e9 −0.553202
\(445\) 1.62800e9 0.875778
\(446\) 3.21221e8 0.171448
\(447\) −8.98090e8 −0.475602
\(448\) −8.99154e7 −0.0472456
\(449\) −2.04698e9 −1.06721 −0.533607 0.845732i \(-0.679164\pi\)
−0.533607 + 0.845732i \(0.679164\pi\)
\(450\) 9.19205e7 0.0475520
\(451\) 1.33959e9 0.687630
\(452\) 7.81830e8 0.398224
\(453\) −9.88301e8 −0.499511
\(454\) −3.51386e8 −0.176234
\(455\) 1.88187e8 0.0936591
\(456\) 4.96903e8 0.245411
\(457\) 2.14052e9 1.04909 0.524544 0.851383i \(-0.324236\pi\)
0.524544 + 0.851383i \(0.324236\pi\)
\(458\) −5.82678e8 −0.283400
\(459\) 1.77457e8 0.0856542
\(460\) −6.16351e8 −0.295240
\(461\) 1.84663e9 0.877862 0.438931 0.898521i \(-0.355357\pi\)
0.438931 + 0.898521i \(0.355357\pi\)
\(462\) 2.39600e8 0.113042
\(463\) 2.49284e9 1.16724 0.583622 0.812026i \(-0.301635\pi\)
0.583622 + 0.812026i \(0.301635\pi\)
\(464\) 2.79516e8 0.129896
\(465\) 8.01260e8 0.369563
\(466\) −2.03349e9 −0.930874
\(467\) −5.56531e8 −0.252860 −0.126430 0.991976i \(-0.540352\pi\)
−0.126430 + 0.991976i \(0.540352\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) 1.12627e9 0.504124
\(470\) 6.94265e8 0.308449
\(471\) −1.29967e8 −0.0573140
\(472\) −5.09762e8 −0.223136
\(473\) 1.54048e8 0.0669333
\(474\) 1.35896e9 0.586113
\(475\) 5.66543e8 0.242552
\(476\) −1.97914e8 −0.0841108
\(477\) −3.98828e8 −0.168256
\(478\) 2.07283e9 0.868091
\(479\) 4.38428e8 0.182274 0.0911368 0.995838i \(-0.470950\pi\)
0.0911368 + 0.995838i \(0.470950\pi\)
\(480\) −2.20943e8 −0.0911874
\(481\) 1.29721e9 0.531499
\(482\) 2.98879e9 1.21571
\(483\) 3.57142e8 0.144220
\(484\) −5.77822e8 −0.231652
\(485\) 2.27038e9 0.903654
\(486\) −1.14791e8 −0.0453609
\(487\) −1.05882e9 −0.415402 −0.207701 0.978192i \(-0.566598\pi\)
−0.207701 + 0.978192i \(0.566598\pi\)
\(488\) 1.38540e9 0.539642
\(489\) 4.76034e8 0.184101
\(490\) −2.35041e8 −0.0902521
\(491\) 5.19176e8 0.197938 0.0989690 0.995091i \(-0.468446\pi\)
0.0989690 + 0.995091i \(0.468446\pi\)
\(492\) 7.15777e8 0.270957
\(493\) 6.15247e8 0.231252
\(494\) −6.31769e8 −0.235784
\(495\) 5.88751e8 0.218179
\(496\) 4.86748e8 0.179109
\(497\) 1.86459e9 0.681298
\(498\) −1.10182e9 −0.399769
\(499\) 1.25064e8 0.0450590 0.0225295 0.999746i \(-0.492828\pi\)
0.0225295 + 0.999746i \(0.492828\pi\)
\(500\) −1.50054e9 −0.536850
\(501\) 1.38253e9 0.491183
\(502\) −2.61153e9 −0.921364
\(503\) 2.84960e9 0.998380 0.499190 0.866492i \(-0.333631\pi\)
0.499190 + 0.866492i \(0.333631\pi\)
\(504\) 1.28024e8 0.0445435
\(505\) −4.09654e9 −1.41546
\(506\) 9.97727e8 0.342361
\(507\) 1.30324e8 0.0444116
\(508\) 1.95043e9 0.660097
\(509\) 4.26498e9 1.43352 0.716761 0.697319i \(-0.245624\pi\)
0.716761 + 0.697319i \(0.245624\pi\)
\(510\) −4.86319e8 −0.162340
\(511\) 2.12219e8 0.0703577
\(512\) −1.34218e8 −0.0441942
\(513\) −7.07505e8 −0.231376
\(514\) 3.36650e9 1.09347
\(515\) −8.68385e8 −0.280148
\(516\) 8.23116e7 0.0263747
\(517\) −1.12385e9 −0.357678
\(518\) −1.62018e9 −0.512166
\(519\) −1.78793e8 −0.0561389
\(520\) 2.80909e8 0.0876100
\(521\) −6.08449e8 −0.188492 −0.0942458 0.995549i \(-0.530044\pi\)
−0.0942458 + 0.995549i \(0.530044\pi\)
\(522\) −3.97983e8 −0.122467
\(523\) −5.94178e9 −1.81619 −0.908093 0.418768i \(-0.862462\pi\)
−0.908093 + 0.418768i \(0.862462\pi\)
\(524\) 9.76789e8 0.296580
\(525\) 1.45966e8 0.0440246
\(526\) 3.74579e9 1.12226
\(527\) 1.07139e9 0.318867
\(528\) 3.57654e8 0.105741
\(529\) −1.91764e9 −0.563212
\(530\) 1.09298e9 0.318895
\(531\) 7.25813e8 0.210375
\(532\) 7.89064e8 0.227207
\(533\) −9.10047e8 −0.260327
\(534\) −1.40813e9 −0.400173
\(535\) 2.37156e9 0.669570
\(536\) 1.68119e9 0.471564
\(537\) 7.59864e6 0.00211751
\(538\) 1.40490e9 0.388962
\(539\) 3.80476e8 0.104657
\(540\) 3.14584e8 0.0859723
\(541\) −1.89699e9 −0.515081 −0.257541 0.966267i \(-0.582912\pi\)
−0.257541 + 0.966267i \(0.582912\pi\)
\(542\) −9.83303e8 −0.265271
\(543\) 3.39394e9 0.909714
\(544\) −2.95428e8 −0.0786784
\(545\) 2.35027e9 0.621915
\(546\) −1.62771e8 −0.0427960
\(547\) 3.82199e9 0.998468 0.499234 0.866467i \(-0.333615\pi\)
0.499234 + 0.866467i \(0.333615\pi\)
\(548\) 3.23015e8 0.0838475
\(549\) −1.97257e9 −0.508779
\(550\) 4.07778e8 0.104509
\(551\) −2.45293e9 −0.624676
\(552\) 5.33110e8 0.134906
\(553\) 2.15797e9 0.542635
\(554\) −2.76819e9 −0.691690
\(555\) −3.98116e9 −0.988517
\(556\) −1.44081e9 −0.355504
\(557\) 5.09004e9 1.24804 0.624019 0.781409i \(-0.285499\pi\)
0.624019 + 0.781409i \(0.285499\pi\)
\(558\) −6.93046e8 −0.168866
\(559\) −1.04652e8 −0.0253400
\(560\) −3.50849e8 −0.0844231
\(561\) 7.87235e8 0.188250
\(562\) 1.28354e9 0.305022
\(563\) −6.69908e9 −1.58211 −0.791053 0.611748i \(-0.790467\pi\)
−0.791053 + 0.611748i \(0.790467\pi\)
\(564\) −6.00501e8 −0.140941
\(565\) 3.05069e9 0.711587
\(566\) 3.70904e9 0.859812
\(567\) −1.82284e8 −0.0419961
\(568\) 2.78330e9 0.637296
\(569\) −1.92335e9 −0.437689 −0.218845 0.975760i \(-0.570229\pi\)
−0.218845 + 0.975760i \(0.570229\pi\)
\(570\) 1.93891e9 0.438526
\(571\) 4.72517e9 1.06216 0.531081 0.847321i \(-0.321786\pi\)
0.531081 + 0.847321i \(0.321786\pi\)
\(572\) −4.54725e8 −0.101593
\(573\) −4.23047e8 −0.0939393
\(574\) 1.13663e9 0.250857
\(575\) 6.07824e8 0.133334
\(576\) 1.91103e8 0.0416667
\(577\) 7.32990e9 1.58848 0.794241 0.607602i \(-0.207869\pi\)
0.794241 + 0.607602i \(0.207869\pi\)
\(578\) 2.63244e9 0.567036
\(579\) −2.67114e8 −0.0571902
\(580\) 1.09067e9 0.232111
\(581\) −1.74965e9 −0.370114
\(582\) −1.96375e9 −0.412911
\(583\) −1.76928e9 −0.369792
\(584\) 3.16782e8 0.0658136
\(585\) −3.99966e8 −0.0825995
\(586\) 5.84922e8 0.120076
\(587\) 6.87279e9 1.40249 0.701245 0.712921i \(-0.252628\pi\)
0.701245 + 0.712921i \(0.252628\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) −4.27152e9 −0.861348
\(590\) −1.98908e9 −0.398723
\(591\) −1.96252e9 −0.391073
\(592\) −2.41847e9 −0.479087
\(593\) −4.80630e9 −0.946497 −0.473249 0.880929i \(-0.656919\pi\)
−0.473249 + 0.880929i \(0.656919\pi\)
\(594\) −5.09237e8 −0.0996937
\(595\) −7.72256e8 −0.150298
\(596\) −2.12881e9 −0.411883
\(597\) −3.29755e9 −0.634281
\(598\) −6.77802e8 −0.129613
\(599\) −1.59452e9 −0.303134 −0.151567 0.988447i \(-0.548432\pi\)
−0.151567 + 0.988447i \(0.548432\pi\)
\(600\) 2.17886e8 0.0411812
\(601\) 6.26679e9 1.17756 0.588782 0.808292i \(-0.299608\pi\)
0.588782 + 0.808292i \(0.299608\pi\)
\(602\) 1.30708e8 0.0244182
\(603\) −2.39373e9 −0.444595
\(604\) −2.34264e9 −0.432590
\(605\) −2.25465e9 −0.413939
\(606\) 3.54327e9 0.646771
\(607\) 2.80434e9 0.508944 0.254472 0.967080i \(-0.418098\pi\)
0.254472 + 0.967080i \(0.418098\pi\)
\(608\) 1.17784e9 0.212533
\(609\) −6.31983e8 −0.113382
\(610\) 5.40580e9 0.964286
\(611\) 7.63484e8 0.135412
\(612\) 4.20639e8 0.0741787
\(613\) 3.19476e9 0.560178 0.280089 0.959974i \(-0.409636\pi\)
0.280089 + 0.959974i \(0.409636\pi\)
\(614\) 5.85109e9 1.02011
\(615\) 2.79295e9 0.484173
\(616\) 5.67941e8 0.0978972
\(617\) −4.24149e9 −0.726977 −0.363488 0.931599i \(-0.618414\pi\)
−0.363488 + 0.931599i \(0.618414\pi\)
\(618\) 7.51105e8 0.128009
\(619\) 5.76481e9 0.976939 0.488470 0.872581i \(-0.337555\pi\)
0.488470 + 0.872581i \(0.337555\pi\)
\(620\) 1.89928e9 0.320051
\(621\) −7.59056e8 −0.127190
\(622\) 7.75576e9 1.29228
\(623\) −2.23606e9 −0.370488
\(624\) −2.42971e8 −0.0400320
\(625\) −4.62373e9 −0.757553
\(626\) −5.88934e9 −0.959526
\(627\) −3.13863e9 −0.508516
\(628\) −3.08071e8 −0.0496354
\(629\) −5.32332e9 −0.852913
\(630\) 4.99548e8 0.0795949
\(631\) −2.32202e9 −0.367929 −0.183964 0.982933i \(-0.558893\pi\)
−0.183964 + 0.982933i \(0.558893\pi\)
\(632\) 3.22123e9 0.507589
\(633\) −1.90164e9 −0.298000
\(634\) 1.01796e9 0.158643
\(635\) 7.61056e9 1.17953
\(636\) −9.45371e8 −0.145714
\(637\) −2.58475e8 −0.0396214
\(638\) −1.76553e9 −0.269156
\(639\) −3.96294e9 −0.600848
\(640\) −5.23716e8 −0.0789706
\(641\) −1.21454e10 −1.82141 −0.910706 0.413055i \(-0.864462\pi\)
−0.910706 + 0.413055i \(0.864462\pi\)
\(642\) −2.05127e9 −0.305950
\(643\) 4.30565e9 0.638705 0.319353 0.947636i \(-0.396535\pi\)
0.319353 + 0.947636i \(0.396535\pi\)
\(644\) 8.46558e8 0.124898
\(645\) 3.21178e8 0.0471289
\(646\) 2.59257e9 0.378369
\(647\) 5.95403e9 0.864263 0.432131 0.901811i \(-0.357762\pi\)
0.432131 + 0.901811i \(0.357762\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 3.21986e9 0.462360
\(650\) −2.77022e8 −0.0395656
\(651\) −1.10053e9 −0.156339
\(652\) 1.12838e9 0.159437
\(653\) 2.52831e9 0.355332 0.177666 0.984091i \(-0.443145\pi\)
0.177666 + 0.984091i \(0.443145\pi\)
\(654\) −2.03286e9 −0.284174
\(655\) 3.81142e9 0.529958
\(656\) 1.69666e9 0.234655
\(657\) −4.51043e8 −0.0620496
\(658\) −9.53573e8 −0.130486
\(659\) 1.19368e10 1.62475 0.812376 0.583134i \(-0.198174\pi\)
0.812376 + 0.583134i \(0.198174\pi\)
\(660\) 1.39556e9 0.188949
\(661\) −1.39313e10 −1.87624 −0.938119 0.346313i \(-0.887433\pi\)
−0.938119 + 0.346313i \(0.887433\pi\)
\(662\) −9.78860e8 −0.131135
\(663\) −5.34805e8 −0.0712686
\(664\) −2.61173e9 −0.346210
\(665\) 3.07891e9 0.405996
\(666\) 3.44348e9 0.451688
\(667\) −2.63166e9 −0.343392
\(668\) 3.27712e9 0.425377
\(669\) −1.08412e9 −0.139986
\(670\) 6.55999e9 0.842639
\(671\) −8.75072e9 −1.11819
\(672\) 3.03464e8 0.0385758
\(673\) −6.16910e9 −0.780134 −0.390067 0.920787i \(-0.627548\pi\)
−0.390067 + 0.920787i \(0.627548\pi\)
\(674\) 8.48328e8 0.106722
\(675\) −3.10232e8 −0.0388260
\(676\) 3.08916e8 0.0384615
\(677\) −1.17438e10 −1.45461 −0.727306 0.686314i \(-0.759228\pi\)
−0.727306 + 0.686314i \(0.759228\pi\)
\(678\) −2.63867e9 −0.325149
\(679\) −3.11836e9 −0.382281
\(680\) −1.15276e9 −0.140591
\(681\) 1.18593e9 0.143894
\(682\) −3.07449e9 −0.371131
\(683\) 4.19103e9 0.503325 0.251662 0.967815i \(-0.419023\pi\)
0.251662 + 0.967815i \(0.419023\pi\)
\(684\) −1.67705e9 −0.200378
\(685\) 1.26040e9 0.149827
\(686\) 3.22829e8 0.0381802
\(687\) 1.96654e9 0.231395
\(688\) 1.95109e8 0.0228411
\(689\) 1.20196e9 0.139998
\(690\) 2.08019e9 0.241063
\(691\) −4.69866e9 −0.541752 −0.270876 0.962614i \(-0.587313\pi\)
−0.270876 + 0.962614i \(0.587313\pi\)
\(692\) −4.23805e8 −0.0486177
\(693\) −8.08650e8 −0.0922984
\(694\) 1.01690e10 1.15484
\(695\) −5.62201e9 −0.635250
\(696\) −9.43368e8 −0.106059
\(697\) 3.73453e9 0.417754
\(698\) −9.37834e9 −1.04384
\(699\) 6.86303e9 0.760056
\(700\) 3.45994e8 0.0381264
\(701\) 3.88240e9 0.425683 0.212842 0.977087i \(-0.431728\pi\)
0.212842 + 0.977087i \(0.431728\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 2.12236e10 2.30396
\(704\) 8.47771e8 0.0915745
\(705\) −2.34315e9 −0.251847
\(706\) −7.30294e9 −0.781055
\(707\) 5.62659e9 0.598794
\(708\) 1.72045e9 0.182190
\(709\) 9.11818e9 0.960830 0.480415 0.877041i \(-0.340486\pi\)
0.480415 + 0.877041i \(0.340486\pi\)
\(710\) 1.08604e10 1.13878
\(711\) −4.58648e9 −0.478560
\(712\) −3.33779e9 −0.346560
\(713\) −4.58276e9 −0.473493
\(714\) 6.67959e8 0.0686761
\(715\) −1.77433e9 −0.181536
\(716\) 1.80116e7 0.00183382
\(717\) −6.99578e9 −0.708793
\(718\) 7.50662e9 0.756848
\(719\) −1.22833e10 −1.23243 −0.616216 0.787577i \(-0.711335\pi\)
−0.616216 + 0.787577i \(0.711335\pi\)
\(720\) 7.45681e8 0.0744542
\(721\) 1.19273e9 0.118513
\(722\) −3.18535e9 −0.314975
\(723\) −1.00872e10 −0.992625
\(724\) 8.04489e9 0.787835
\(725\) −1.07558e9 −0.104824
\(726\) 1.95015e9 0.189143
\(727\) 4.84128e8 0.0467293 0.0233647 0.999727i \(-0.492562\pi\)
0.0233647 + 0.999727i \(0.492562\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 1.23608e9 0.117602
\(731\) 4.29456e8 0.0406638
\(732\) −4.67572e9 −0.440615
\(733\) −7.90496e9 −0.741371 −0.370686 0.928758i \(-0.620877\pi\)
−0.370686 + 0.928758i \(0.620877\pi\)
\(734\) −8.72667e8 −0.0814540
\(735\) 7.93264e8 0.0736906
\(736\) 1.26367e9 0.116832
\(737\) −1.06191e10 −0.977126
\(738\) −2.41575e9 −0.221235
\(739\) 4.35082e9 0.396566 0.198283 0.980145i \(-0.436463\pi\)
0.198283 + 0.980145i \(0.436463\pi\)
\(740\) −9.43683e9 −0.856081
\(741\) 2.13222e9 0.192517
\(742\) −1.50121e9 −0.134905
\(743\) 2.89756e9 0.259162 0.129581 0.991569i \(-0.458637\pi\)
0.129581 + 0.991569i \(0.458637\pi\)
\(744\) −1.64277e9 −0.146242
\(745\) −8.30657e9 −0.735995
\(746\) 5.22453e9 0.460746
\(747\) 3.71865e9 0.326410
\(748\) 1.86604e9 0.163029
\(749\) −3.25734e9 −0.283254
\(750\) 5.06433e9 0.438336
\(751\) −2.15345e10 −1.85522 −0.927609 0.373554i \(-0.878139\pi\)
−0.927609 + 0.373554i \(0.878139\pi\)
\(752\) −1.42341e9 −0.122058
\(753\) 8.81390e9 0.752290
\(754\) 1.19941e9 0.101898
\(755\) −9.14094e9 −0.772995
\(756\) −4.32081e8 −0.0363696
\(757\) 1.64305e10 1.37663 0.688313 0.725414i \(-0.258351\pi\)
0.688313 + 0.725414i \(0.258351\pi\)
\(758\) 2.12479e9 0.177204
\(759\) −3.36733e9 −0.279537
\(760\) 4.59593e9 0.379775
\(761\) 1.79037e10 1.47264 0.736319 0.676635i \(-0.236562\pi\)
0.736319 + 0.676635i \(0.236562\pi\)
\(762\) −6.58271e9 −0.538967
\(763\) −3.22810e9 −0.263094
\(764\) −1.00278e9 −0.0813538
\(765\) 1.64133e9 0.132550
\(766\) 8.88757e9 0.714467
\(767\) −2.18740e9 −0.175043
\(768\) 4.52985e8 0.0360844
\(769\) 5.11934e9 0.405949 0.202974 0.979184i \(-0.434939\pi\)
0.202974 + 0.979184i \(0.434939\pi\)
\(770\) 2.21610e9 0.174933
\(771\) −1.13619e10 −0.892817
\(772\) −6.33159e8 −0.0495281
\(773\) 1.08379e10 0.843954 0.421977 0.906606i \(-0.361336\pi\)
0.421977 + 0.906606i \(0.361336\pi\)
\(774\) −2.77802e8 −0.0215348
\(775\) −1.87301e9 −0.144538
\(776\) −4.65482e9 −0.357591
\(777\) 5.46812e9 0.418181
\(778\) 9.11434e9 0.693900
\(779\) −1.48892e10 −1.12847
\(780\) −9.48068e8 −0.0715333
\(781\) −1.75804e10 −1.32054
\(782\) 2.78147e9 0.207994
\(783\) 1.34319e9 0.0999937
\(784\) 4.81890e8 0.0357143
\(785\) −1.20209e9 −0.0886935
\(786\) −3.29666e9 −0.242156
\(787\) −3.56535e8 −0.0260730 −0.0130365 0.999915i \(-0.504150\pi\)
−0.0130365 + 0.999915i \(0.504150\pi\)
\(788\) −4.65189e9 −0.338679
\(789\) −1.26420e10 −0.916321
\(790\) 1.25692e10 0.907011
\(791\) −4.19012e9 −0.301029
\(792\) −1.20708e9 −0.0863372
\(793\) 5.94477e9 0.423330
\(794\) 9.88085e9 0.700524
\(795\) −3.68882e9 −0.260377
\(796\) −7.81642e9 −0.549303
\(797\) −1.99101e10 −1.39306 −0.696528 0.717530i \(-0.745273\pi\)
−0.696528 + 0.717530i \(0.745273\pi\)
\(798\) −2.66309e9 −0.185514
\(799\) −3.13308e9 −0.217299
\(800\) 5.16470e8 0.0356640
\(801\) 4.75243e9 0.326740
\(802\) 1.15433e10 0.790172
\(803\) −2.00092e9 −0.136372
\(804\) −5.67403e9 −0.385031
\(805\) 3.30326e9 0.223181
\(806\) 2.08864e9 0.140505
\(807\) −4.74154e9 −0.317586
\(808\) 8.39887e9 0.560120
\(809\) −2.43083e10 −1.61411 −0.807057 0.590474i \(-0.798941\pi\)
−0.807057 + 0.590474i \(0.798941\pi\)
\(810\) −1.06172e9 −0.0701961
\(811\) 6.52755e9 0.429712 0.214856 0.976646i \(-0.431072\pi\)
0.214856 + 0.976646i \(0.431072\pi\)
\(812\) −1.49803e9 −0.0981918
\(813\) 3.31865e9 0.216593
\(814\) 1.52760e10 0.992713
\(815\) 4.40291e9 0.284897
\(816\) 9.97070e8 0.0642407
\(817\) −1.71220e9 −0.109844
\(818\) 1.47103e10 0.939691
\(819\) 5.49353e8 0.0349428
\(820\) 6.62033e9 0.419306
\(821\) −5.20602e9 −0.328326 −0.164163 0.986433i \(-0.552492\pi\)
−0.164163 + 0.986433i \(0.552492\pi\)
\(822\) −1.09017e9 −0.0684612
\(823\) 1.61413e10 1.00934 0.504671 0.863311i \(-0.331614\pi\)
0.504671 + 0.863311i \(0.331614\pi\)
\(824\) 1.78040e9 0.110859
\(825\) −1.37625e9 −0.0853314
\(826\) 2.73201e9 0.168675
\(827\) 1.20070e10 0.738182 0.369091 0.929393i \(-0.379669\pi\)
0.369091 + 0.929393i \(0.379669\pi\)
\(828\) −1.79924e9 −0.110150
\(829\) 1.61936e10 0.987192 0.493596 0.869691i \(-0.335682\pi\)
0.493596 + 0.869691i \(0.335682\pi\)
\(830\) −1.01909e10 −0.618643
\(831\) 9.34264e9 0.564763
\(832\) −5.75930e8 −0.0346688
\(833\) 1.06069e9 0.0635818
\(834\) 4.86272e9 0.290267
\(835\) 1.27873e10 0.760107
\(836\) −7.43972e9 −0.440387
\(837\) 2.33903e9 0.137878
\(838\) 8.14561e9 0.478156
\(839\) 1.91601e10 1.12003 0.560017 0.828481i \(-0.310795\pi\)
0.560017 + 0.828481i \(0.310795\pi\)
\(840\) 1.18411e9 0.0689312
\(841\) −1.25930e10 −0.730034
\(842\) 4.65729e9 0.268869
\(843\) −4.33194e9 −0.249050
\(844\) −4.50760e9 −0.258075
\(845\) 1.20538e9 0.0687270
\(846\) 2.02669e9 0.115078
\(847\) 3.09677e9 0.175112
\(848\) −2.24088e9 −0.126192
\(849\) −1.25180e10 −0.702033
\(850\) 1.13681e9 0.0634922
\(851\) 2.27700e10 1.26651
\(852\) −9.39364e9 −0.520350
\(853\) −8.97603e9 −0.495179 −0.247590 0.968865i \(-0.579639\pi\)
−0.247590 + 0.968865i \(0.579639\pi\)
\(854\) −7.42487e9 −0.407931
\(855\) −6.54382e9 −0.358055
\(856\) −4.86227e9 −0.264960
\(857\) 1.85639e10 1.00748 0.503740 0.863855i \(-0.331957\pi\)
0.503740 + 0.863855i \(0.331957\pi\)
\(858\) 1.53470e9 0.0829501
\(859\) 1.30357e10 0.701709 0.350855 0.936430i \(-0.385891\pi\)
0.350855 + 0.936430i \(0.385891\pi\)
\(860\) 7.61312e8 0.0408148
\(861\) −3.83612e9 −0.204824
\(862\) −1.51619e9 −0.0806268
\(863\) 4.32991e8 0.0229320 0.0114660 0.999934i \(-0.496350\pi\)
0.0114660 + 0.999934i \(0.496350\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −1.65368e9 −0.0868751
\(866\) −1.73622e10 −0.908433
\(867\) −8.88448e9 −0.462983
\(868\) −2.60867e9 −0.135394
\(869\) −2.03466e10 −1.05177
\(870\) −3.68101e9 −0.189518
\(871\) 7.21403e9 0.369926
\(872\) −4.81862e9 −0.246102
\(873\) 6.62766e9 0.337140
\(874\) −1.10895e10 −0.561850
\(875\) 8.04197e9 0.405821
\(876\) −1.06914e9 −0.0537365
\(877\) 1.50599e10 0.753915 0.376957 0.926231i \(-0.376970\pi\)
0.376957 + 0.926231i \(0.376970\pi\)
\(878\) −4.09649e9 −0.204259
\(879\) −1.97411e9 −0.0980417
\(880\) 3.30799e9 0.163635
\(881\) 3.03610e10 1.49589 0.747947 0.663758i \(-0.231040\pi\)
0.747947 + 0.663758i \(0.231040\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −2.74514e10 −1.34184 −0.670921 0.741529i \(-0.734101\pi\)
−0.670921 + 0.741529i \(0.734101\pi\)
\(884\) −1.26769e9 −0.0617204
\(885\) 6.71316e9 0.325556
\(886\) −7.86007e9 −0.379672
\(887\) −2.30605e10 −1.10952 −0.554762 0.832009i \(-0.687191\pi\)
−0.554762 + 0.832009i \(0.687191\pi\)
\(888\) 8.16233e9 0.391173
\(889\) −1.04531e10 −0.498987
\(890\) −1.30240e10 −0.619268
\(891\) 1.71868e9 0.0813995
\(892\) −2.56976e9 −0.121232
\(893\) 1.24913e10 0.586986
\(894\) 7.18472e9 0.336301
\(895\) 7.02809e7 0.00327685
\(896\) 7.19323e8 0.0334077
\(897\) 2.28758e9 0.105829
\(898\) 1.63759e10 0.754635
\(899\) 8.10946e9 0.372248
\(900\) −7.35364e8 −0.0336243
\(901\) −4.93242e9 −0.224659
\(902\) −1.07167e10 −0.486228
\(903\) −4.41138e8 −0.0199374
\(904\) −6.25464e9 −0.281587
\(905\) 3.13911e10 1.40778
\(906\) 7.90641e9 0.353208
\(907\) 1.74751e10 0.777667 0.388834 0.921308i \(-0.372878\pi\)
0.388834 + 0.921308i \(0.372878\pi\)
\(908\) 2.81109e9 0.124616
\(909\) −1.19586e10 −0.528086
\(910\) −1.50550e9 −0.0662270
\(911\) −3.45096e10 −1.51226 −0.756128 0.654424i \(-0.772911\pi\)
−0.756128 + 0.654424i \(0.772911\pi\)
\(912\) −3.97522e9 −0.173532
\(913\) 1.64967e10 0.717380
\(914\) −1.71241e10 −0.741817
\(915\) −1.82446e10 −0.787336
\(916\) 4.66142e9 0.200394
\(917\) −5.23498e9 −0.224193
\(918\) −1.41966e9 −0.0605667
\(919\) −2.03987e10 −0.866957 −0.433478 0.901164i \(-0.642714\pi\)
−0.433478 + 0.901164i \(0.642714\pi\)
\(920\) 4.93081e9 0.208767
\(921\) −1.97474e10 −0.832918
\(922\) −1.47730e10 −0.620742
\(923\) 1.19432e10 0.499936
\(924\) −1.91680e9 −0.0799328
\(925\) 9.30627e9 0.386616
\(926\) −1.99427e10 −0.825366
\(927\) −2.53498e9 −0.104519
\(928\) −2.23613e9 −0.0918501
\(929\) −4.32372e10 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(930\) −6.41008e9 −0.261320
\(931\) −4.22889e9 −0.171752
\(932\) 1.62679e10 0.658228
\(933\) −2.61757e10 −1.05515
\(934\) 4.45225e9 0.178799
\(935\) 7.28125e9 0.291317
\(936\) 8.20026e8 0.0326860
\(937\) 1.71234e10 0.679989 0.339995 0.940427i \(-0.389575\pi\)
0.339995 + 0.940427i \(0.389575\pi\)
\(938\) −9.01015e9 −0.356469
\(939\) 1.98765e10 0.783450
\(940\) −5.55412e9 −0.218106
\(941\) 1.40113e10 0.548169 0.274085 0.961706i \(-0.411625\pi\)
0.274085 + 0.961706i \(0.411625\pi\)
\(942\) 1.03974e9 0.0405271
\(943\) −1.59741e10 −0.620334
\(944\) 4.07810e9 0.157781
\(945\) −1.68597e9 −0.0649890
\(946\) −1.23238e9 −0.0473290
\(947\) −1.27534e10 −0.487978 −0.243989 0.969778i \(-0.578456\pi\)
−0.243989 + 0.969778i \(0.578456\pi\)
\(948\) −1.08717e10 −0.414445
\(949\) 1.35932e9 0.0516284
\(950\) −4.53234e9 −0.171510
\(951\) −3.43563e9 −0.129531
\(952\) 1.58331e9 0.0594753
\(953\) −2.20228e10 −0.824227 −0.412114 0.911132i \(-0.635209\pi\)
−0.412114 + 0.911132i \(0.635209\pi\)
\(954\) 3.19063e9 0.118975
\(955\) −3.91282e9 −0.145371
\(956\) −1.65826e10 −0.613833
\(957\) 5.95868e9 0.219765
\(958\) −3.50742e9 −0.128887
\(959\) −1.73116e9 −0.0633828
\(960\) 1.76754e9 0.0644792
\(961\) −1.33909e10 −0.486717
\(962\) −1.03777e10 −0.375827
\(963\) 6.92303e9 0.249807
\(964\) −2.39103e10 −0.859639
\(965\) −2.47058e9 −0.0885019
\(966\) −2.85713e9 −0.101979
\(967\) −1.58385e10 −0.563275 −0.281637 0.959521i \(-0.590878\pi\)
−0.281637 + 0.959521i \(0.590878\pi\)
\(968\) 4.62258e9 0.163802
\(969\) −8.74991e9 −0.308937
\(970\) −1.81630e10 −0.638980
\(971\) 5.83241e9 0.204447 0.102223 0.994761i \(-0.467404\pi\)
0.102223 + 0.994761i \(0.467404\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 7.72182e9 0.268735
\(974\) 8.47052e9 0.293734
\(975\) 9.34951e8 0.0323052
\(976\) −1.10832e10 −0.381584
\(977\) 2.30405e10 0.790426 0.395213 0.918589i \(-0.370671\pi\)
0.395213 + 0.918589i \(0.370671\pi\)
\(978\) −3.80827e9 −0.130179
\(979\) 2.10828e10 0.718105
\(980\) 1.88033e9 0.0638179
\(981\) 6.86089e9 0.232027
\(982\) −4.15340e9 −0.139963
\(983\) 1.46813e10 0.492979 0.246489 0.969145i \(-0.420723\pi\)
0.246489 + 0.969145i \(0.420723\pi\)
\(984\) −5.72621e9 −0.191595
\(985\) −1.81516e10 −0.605185
\(986\) −4.92197e9 −0.163520
\(987\) 3.21831e9 0.106541
\(988\) 5.05415e9 0.166724
\(989\) −1.83696e9 −0.0603827
\(990\) −4.71001e9 −0.154276
\(991\) 3.27312e10 1.06833 0.534164 0.845381i \(-0.320626\pi\)
0.534164 + 0.845381i \(0.320626\pi\)
\(992\) −3.89398e9 −0.126649
\(993\) 3.30365e9 0.107071
\(994\) −1.49168e10 −0.481750
\(995\) −3.04996e10 −0.981551
\(996\) 8.81458e9 0.282679
\(997\) −5.28138e10 −1.68778 −0.843888 0.536520i \(-0.819739\pi\)
−0.843888 + 0.536520i \(0.819739\pi\)
\(998\) −1.00051e9 −0.0318615
\(999\) −1.16218e10 −0.368801
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.j.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.j.1.4 5 1.1 even 1 trivial