Properties

Label 570.8.a.b.1.1
Level $570$
Weight $8$
Character 570.1
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 3046 x^{2} + 50476 x + 497070\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.96971\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} -1119.70 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} -1119.70 q^{7} +512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} +1725.45 q^{11} +1728.00 q^{12} +47.9718 q^{13} -8957.58 q^{14} -3375.00 q^{15} +4096.00 q^{16} +28333.6 q^{17} +5832.00 q^{18} +6859.00 q^{19} -8000.00 q^{20} -30231.8 q^{21} +13803.6 q^{22} -85045.6 q^{23} +13824.0 q^{24} +15625.0 q^{25} +383.775 q^{26} +19683.0 q^{27} -71660.7 q^{28} -17328.4 q^{29} -27000.0 q^{30} -185488. q^{31} +32768.0 q^{32} +46587.2 q^{33} +226669. q^{34} +139962. q^{35} +46656.0 q^{36} +308884. q^{37} +54872.0 q^{38} +1295.24 q^{39} -64000.0 q^{40} +221162. q^{41} -241855. q^{42} -237096. q^{43} +110429. q^{44} -91125.0 q^{45} -680364. q^{46} +666086. q^{47} +110592. q^{48} +430180. q^{49} +125000. q^{50} +765008. q^{51} +3070.20 q^{52} -1.91062e6 q^{53} +157464. q^{54} -215681. q^{55} -573285. q^{56} +185193. q^{57} -138627. q^{58} +112866. q^{59} -216000. q^{60} -1.13216e6 q^{61} -1.48390e6 q^{62} -816260. q^{63} +262144. q^{64} -5996.48 q^{65} +372697. q^{66} +142005. q^{67} +1.81335e6 q^{68} -2.29623e6 q^{69} +1.11970e6 q^{70} -2.61842e6 q^{71} +373248. q^{72} -1.27471e6 q^{73} +2.47107e6 q^{74} +421875. q^{75} +438976. q^{76} -1.93198e6 q^{77} +10361.9 q^{78} +5.39395e6 q^{79} -512000. q^{80} +531441. q^{81} +1.76930e6 q^{82} -2.72417e6 q^{83} -1.93484e6 q^{84} -3.54170e6 q^{85} -1.89677e6 q^{86} -467866. q^{87} +883431. q^{88} -8.03487e6 q^{89} -729000. q^{90} -53714.0 q^{91} -5.44292e6 q^{92} -5.00818e6 q^{93} +5.32869e6 q^{94} -857375. q^{95} +884736. q^{96} -9.96801e6 q^{97} +3.44144e6 q^{98} +1.25785e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100}) \)

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
\(5\) −125.000 −0.447214
\(6\) 216.000 0.408248
\(7\) −1119.70 −1.23384 −0.616918 0.787027i \(-0.711619\pi\)
−0.616918 + 0.787027i \(0.711619\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1000.00 −0.316228
\(11\) 1725.45 0.390866 0.195433 0.980717i \(-0.437389\pi\)
0.195433 + 0.980717i \(0.437389\pi\)
\(12\) 1728.00 0.288675
\(13\) 47.9718 0.00605598 0.00302799 0.999995i \(-0.499036\pi\)
0.00302799 + 0.999995i \(0.499036\pi\)
\(14\) −8957.58 −0.872454
\(15\) −3375.00 −0.258199
\(16\) 4096.00 0.250000
\(17\) 28333.6 1.39872 0.699360 0.714770i \(-0.253468\pi\)
0.699360 + 0.714770i \(0.253468\pi\)
\(18\) 5832.00 0.235702
\(19\) 6859.00 0.229416
\(20\) −8000.00 −0.223607
\(21\) −30231.8 −0.712356
\(22\) 13803.6 0.276384
\(23\) −85045.6 −1.45749 −0.728743 0.684788i \(-0.759895\pi\)
−0.728743 + 0.684788i \(0.759895\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) 383.775 0.00428223
\(27\) 19683.0 0.192450
\(28\) −71660.7 −0.616918
\(29\) −17328.4 −0.131936 −0.0659682 0.997822i \(-0.521014\pi\)
−0.0659682 + 0.997822i \(0.521014\pi\)
\(30\) −27000.0 −0.182574
\(31\) −185488. −1.11828 −0.559139 0.829074i \(-0.688868\pi\)
−0.559139 + 0.829074i \(0.688868\pi\)
\(32\) 32768.0 0.176777
\(33\) 46587.2 0.225667
\(34\) 226669. 0.989044
\(35\) 139962. 0.551789
\(36\) 46656.0 0.166667
\(37\) 308884. 1.00251 0.501255 0.865299i \(-0.332872\pi\)
0.501255 + 0.865299i \(0.332872\pi\)
\(38\) 54872.0 0.162221
\(39\) 1295.24 0.00349642
\(40\) −64000.0 −0.158114
\(41\) 221162. 0.501149 0.250575 0.968097i \(-0.419380\pi\)
0.250575 + 0.968097i \(0.419380\pi\)
\(42\) −241855. −0.503712
\(43\) −237096. −0.454762 −0.227381 0.973806i \(-0.573016\pi\)
−0.227381 + 0.973806i \(0.573016\pi\)
\(44\) 110429. 0.195433
\(45\) −91125.0 −0.149071
\(46\) −680364. −1.03060
\(47\) 666086. 0.935811 0.467905 0.883779i \(-0.345009\pi\)
0.467905 + 0.883779i \(0.345009\pi\)
\(48\) 110592. 0.144338
\(49\) 430180. 0.522353
\(50\) 125000. 0.141421
\(51\) 765008. 0.807551
\(52\) 3070.20 0.00302799
\(53\) −1.91062e6 −1.76282 −0.881411 0.472351i \(-0.843405\pi\)
−0.881411 + 0.472351i \(0.843405\pi\)
\(54\) 157464. 0.136083
\(55\) −215681. −0.174801
\(56\) −573285. −0.436227
\(57\) 185193. 0.132453
\(58\) −138627. −0.0932931
\(59\) 112866. 0.0715452 0.0357726 0.999360i \(-0.488611\pi\)
0.0357726 + 0.999360i \(0.488611\pi\)
\(60\) −216000. −0.129099
\(61\) −1.13216e6 −0.638634 −0.319317 0.947648i \(-0.603453\pi\)
−0.319317 + 0.947648i \(0.603453\pi\)
\(62\) −1.48390e6 −0.790743
\(63\) −816260. −0.411279
\(64\) 262144. 0.125000
\(65\) −5996.48 −0.00270832
\(66\) 372697. 0.159571
\(67\) 142005. 0.0576820 0.0288410 0.999584i \(-0.490818\pi\)
0.0288410 + 0.999584i \(0.490818\pi\)
\(68\) 1.81335e6 0.699360
\(69\) −2.29623e6 −0.841480
\(70\) 1.11970e6 0.390173
\(71\) −2.61842e6 −0.868232 −0.434116 0.900857i \(-0.642939\pi\)
−0.434116 + 0.900857i \(0.642939\pi\)
\(72\) 373248. 0.117851
\(73\) −1.27471e6 −0.383513 −0.191756 0.981443i \(-0.561418\pi\)
−0.191756 + 0.981443i \(0.561418\pi\)
\(74\) 2.47107e6 0.708882
\(75\) 421875. 0.115470
\(76\) 438976. 0.114708
\(77\) −1.93198e6 −0.482265
\(78\) 10361.9 0.00247234
\(79\) 5.39395e6 1.23087 0.615434 0.788188i \(-0.288981\pi\)
0.615434 + 0.788188i \(0.288981\pi\)
\(80\) −512000. −0.111803
\(81\) 531441. 0.111111
\(82\) 1.76930e6 0.354366
\(83\) −2.72417e6 −0.522951 −0.261475 0.965210i \(-0.584209\pi\)
−0.261475 + 0.965210i \(0.584209\pi\)
\(84\) −1.93484e6 −0.356178
\(85\) −3.54170e6 −0.625527
\(86\) −1.89677e6 −0.321566
\(87\) −467866. −0.0761735
\(88\) 883431. 0.138192
\(89\) −8.03487e6 −1.20813 −0.604065 0.796935i \(-0.706453\pi\)
−0.604065 + 0.796935i \(0.706453\pi\)
\(90\) −729000. −0.105409
\(91\) −53714.0 −0.00747209
\(92\) −5.44292e6 −0.728743
\(93\) −5.00818e6 −0.645639
\(94\) 5.32869e6 0.661718
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) −9.96801e6 −1.10894 −0.554469 0.832204i \(-0.687079\pi\)
−0.554469 + 0.832204i \(0.687079\pi\)
\(98\) 3.44144e6 0.369359
\(99\) 1.25785e6 0.130289
\(100\) 1.00000e6 0.100000
\(101\) −1.36343e7 −1.31677 −0.658384 0.752682i \(-0.728760\pi\)
−0.658384 + 0.752682i \(0.728760\pi\)
\(102\) 6.12006e6 0.571025
\(103\) −1.13321e7 −1.02183 −0.510915 0.859631i \(-0.670693\pi\)
−0.510915 + 0.859631i \(0.670693\pi\)
\(104\) 24561.6 0.00214111
\(105\) 3.77898e6 0.318575
\(106\) −1.52849e7 −1.24650
\(107\) −1.30823e7 −1.03239 −0.516193 0.856472i \(-0.672651\pi\)
−0.516193 + 0.856472i \(0.672651\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 364392. 0.0269510 0.0134755 0.999909i \(-0.495710\pi\)
0.0134755 + 0.999909i \(0.495710\pi\)
\(110\) −1.72545e6 −0.123603
\(111\) 8.33986e6 0.578800
\(112\) −4.58628e6 −0.308459
\(113\) 7.56438e6 0.493173 0.246586 0.969121i \(-0.420691\pi\)
0.246586 + 0.969121i \(0.420691\pi\)
\(114\) 1.48154e6 0.0936586
\(115\) 1.06307e7 0.651807
\(116\) −1.10902e6 −0.0659682
\(117\) 34971.5 0.00201866
\(118\) 902927. 0.0505901
\(119\) −3.17251e7 −1.72579
\(120\) −1.72800e6 −0.0912871
\(121\) −1.65100e7 −0.847224
\(122\) −9.05725e6 −0.451582
\(123\) 5.97138e6 0.289339
\(124\) −1.18712e7 −0.559139
\(125\) −1.95312e6 −0.0894427
\(126\) −6.53008e6 −0.290818
\(127\) −1.18095e7 −0.511586 −0.255793 0.966732i \(-0.582337\pi\)
−0.255793 + 0.966732i \(0.582337\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −6.40159e6 −0.262557
\(130\) −47971.8 −0.00191507
\(131\) 1.45149e7 0.564109 0.282055 0.959398i \(-0.408984\pi\)
0.282055 + 0.959398i \(0.408984\pi\)
\(132\) 2.98158e6 0.112833
\(133\) −7.68001e6 −0.283062
\(134\) 1.13604e6 0.0407874
\(135\) −2.46038e6 −0.0860663
\(136\) 1.45068e7 0.494522
\(137\) −5.09194e7 −1.69185 −0.845924 0.533304i \(-0.820950\pi\)
−0.845924 + 0.533304i \(0.820950\pi\)
\(138\) −1.83698e7 −0.595016
\(139\) 1.66451e7 0.525695 0.262848 0.964837i \(-0.415338\pi\)
0.262848 + 0.964837i \(0.415338\pi\)
\(140\) 8.95758e6 0.275894
\(141\) 1.79843e7 0.540290
\(142\) −2.09474e7 −0.613933
\(143\) 82773.1 0.00236708
\(144\) 2.98598e6 0.0833333
\(145\) 2.16605e6 0.0590038
\(146\) −1.01976e7 −0.271185
\(147\) 1.16149e7 0.301581
\(148\) 1.97686e7 0.501255
\(149\) −1.37603e7 −0.340782 −0.170391 0.985377i \(-0.554503\pi\)
−0.170391 + 0.985377i \(0.554503\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) 2.19402e7 0.518585 0.259293 0.965799i \(-0.416511\pi\)
0.259293 + 0.965799i \(0.416511\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) 2.06552e7 0.466240
\(154\) −1.54559e7 −0.341013
\(155\) 2.31860e7 0.500110
\(156\) 82895.3 0.00174821
\(157\) 5.42084e7 1.11794 0.558969 0.829189i \(-0.311197\pi\)
0.558969 + 0.829189i \(0.311197\pi\)
\(158\) 4.31516e7 0.870356
\(159\) −5.15867e7 −1.01777
\(160\) −4.09600e6 −0.0790569
\(161\) 9.52253e7 1.79830
\(162\) 4.25153e6 0.0785674
\(163\) 4.59104e7 0.830336 0.415168 0.909745i \(-0.363723\pi\)
0.415168 + 0.909745i \(0.363723\pi\)
\(164\) 1.41544e7 0.250575
\(165\) −5.82340e6 −0.100921
\(166\) −2.17934e7 −0.369782
\(167\) −2.68258e7 −0.445703 −0.222851 0.974852i \(-0.571536\pi\)
−0.222851 + 0.974852i \(0.571536\pi\)
\(168\) −1.54787e7 −0.251856
\(169\) −6.27462e7 −0.999963
\(170\) −2.83336e7 −0.442314
\(171\) 5.00021e6 0.0764719
\(172\) −1.51741e7 −0.227381
\(173\) −1.06963e6 −0.0157063 −0.00785314 0.999969i \(-0.502500\pi\)
−0.00785314 + 0.999969i \(0.502500\pi\)
\(174\) −3.74293e6 −0.0538628
\(175\) −1.74953e7 −0.246767
\(176\) 7.06745e6 0.0977166
\(177\) 3.04738e6 0.0413066
\(178\) −6.42789e7 −0.854277
\(179\) −1.03267e8 −1.34578 −0.672892 0.739740i \(-0.734948\pi\)
−0.672892 + 0.739740i \(0.734948\pi\)
\(180\) −5.83200e6 −0.0745356
\(181\) −1.56751e8 −1.96488 −0.982440 0.186577i \(-0.940261\pi\)
−0.982440 + 0.186577i \(0.940261\pi\)
\(182\) −429712. −0.00528357
\(183\) −3.05682e7 −0.368715
\(184\) −4.35433e7 −0.515299
\(185\) −3.86105e7 −0.448336
\(186\) −4.00654e7 −0.456535
\(187\) 4.88883e7 0.546713
\(188\) 4.26295e7 0.467905
\(189\) −2.20390e7 −0.237452
\(190\) −6.85900e6 −0.0725476
\(191\) −7.99566e7 −0.830305 −0.415153 0.909752i \(-0.636272\pi\)
−0.415153 + 0.909752i \(0.636272\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 4.45266e7 0.445830 0.222915 0.974838i \(-0.428443\pi\)
0.222915 + 0.974838i \(0.428443\pi\)
\(194\) −7.97441e7 −0.784138
\(195\) −161905. −0.00156365
\(196\) 2.75315e7 0.261176
\(197\) 5.51428e7 0.513875 0.256937 0.966428i \(-0.417287\pi\)
0.256937 + 0.966428i \(0.417287\pi\)
\(198\) 1.00628e7 0.0921281
\(199\) −1.74844e7 −0.157277 −0.0786385 0.996903i \(-0.525057\pi\)
−0.0786385 + 0.996903i \(0.525057\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) 3.83412e6 0.0333027
\(202\) −1.09075e8 −0.931096
\(203\) 1.94025e7 0.162788
\(204\) 4.89605e7 0.403776
\(205\) −2.76453e7 −0.224121
\(206\) −9.06566e7 −0.722543
\(207\) −6.19982e7 −0.485828
\(208\) 196493. 0.00151400
\(209\) 1.18349e7 0.0896709
\(210\) 3.02318e7 0.225267
\(211\) 1.44125e8 1.05621 0.528107 0.849178i \(-0.322902\pi\)
0.528107 + 0.849178i \(0.322902\pi\)
\(212\) −1.22280e8 −0.881411
\(213\) −7.06975e7 −0.501274
\(214\) −1.04659e8 −0.730007
\(215\) 2.96370e7 0.203376
\(216\) 1.00777e7 0.0680414
\(217\) 2.07691e8 1.37977
\(218\) 2.91513e6 0.0190573
\(219\) −3.44171e7 −0.221421
\(220\) −1.38036e7 −0.0874004
\(221\) 1.35922e6 0.00847062
\(222\) 6.67189e7 0.409273
\(223\) 2.00665e8 1.21173 0.605864 0.795568i \(-0.292828\pi\)
0.605864 + 0.795568i \(0.292828\pi\)
\(224\) −3.66903e7 −0.218114
\(225\) 1.13906e7 0.0666667
\(226\) 6.05151e7 0.348726
\(227\) 2.62691e8 1.49058 0.745288 0.666742i \(-0.232312\pi\)
0.745288 + 0.666742i \(0.232312\pi\)
\(228\) 1.18524e7 0.0662266
\(229\) −3.03629e8 −1.67078 −0.835388 0.549660i \(-0.814757\pi\)
−0.835388 + 0.549660i \(0.814757\pi\)
\(230\) 8.50456e7 0.460897
\(231\) −5.21636e7 −0.278436
\(232\) −8.87212e6 −0.0466466
\(233\) 7.52452e7 0.389703 0.194851 0.980833i \(-0.437578\pi\)
0.194851 + 0.980833i \(0.437578\pi\)
\(234\) 279772. 0.00142741
\(235\) −8.32608e7 −0.418507
\(236\) 7.22341e6 0.0357726
\(237\) 1.45637e8 0.710642
\(238\) −2.53801e8 −1.22032
\(239\) 1.17291e8 0.555741 0.277870 0.960619i \(-0.410371\pi\)
0.277870 + 0.960619i \(0.410371\pi\)
\(240\) −1.38240e7 −0.0645497
\(241\) −1.81339e8 −0.834509 −0.417254 0.908790i \(-0.637008\pi\)
−0.417254 + 0.908790i \(0.637008\pi\)
\(242\) −1.32080e8 −0.599077
\(243\) 1.43489e7 0.0641500
\(244\) −7.24580e7 −0.319317
\(245\) −5.37725e7 −0.233603
\(246\) 4.77710e7 0.204593
\(247\) 329039. 0.00138934
\(248\) −9.49699e7 −0.395371
\(249\) −7.35526e7 −0.301926
\(250\) −1.56250e7 −0.0632456
\(251\) 2.98853e8 1.19289 0.596445 0.802654i \(-0.296579\pi\)
0.596445 + 0.802654i \(0.296579\pi\)
\(252\) −5.22406e7 −0.205639
\(253\) −1.46742e8 −0.569682
\(254\) −9.44761e7 −0.361746
\(255\) −9.56260e7 −0.361148
\(256\) 1.67772e7 0.0625000
\(257\) −3.78646e8 −1.39145 −0.695725 0.718308i \(-0.744917\pi\)
−0.695725 + 0.718308i \(0.744917\pi\)
\(258\) −5.12127e7 −0.185656
\(259\) −3.45856e8 −1.23693
\(260\) −383775. −0.00135416
\(261\) −1.26324e7 −0.0439788
\(262\) 1.16119e8 0.398885
\(263\) 5.17351e8 1.75364 0.876819 0.480821i \(-0.159661\pi\)
0.876819 + 0.480821i \(0.159661\pi\)
\(264\) 2.38526e7 0.0797853
\(265\) 2.38827e8 0.788358
\(266\) −6.14401e7 −0.200155
\(267\) −2.16941e8 −0.697514
\(268\) 9.08829e6 0.0288410
\(269\) 1.92975e8 0.604461 0.302230 0.953235i \(-0.402269\pi\)
0.302230 + 0.953235i \(0.402269\pi\)
\(270\) −1.96830e7 −0.0608581
\(271\) −2.99352e8 −0.913669 −0.456835 0.889552i \(-0.651017\pi\)
−0.456835 + 0.889552i \(0.651017\pi\)
\(272\) 1.16055e8 0.349680
\(273\) −1.45028e6 −0.00431402
\(274\) −4.07355e8 −1.19632
\(275\) 2.69602e7 0.0781733
\(276\) −1.46959e8 −0.420740
\(277\) −5.02737e8 −1.42122 −0.710610 0.703586i \(-0.751581\pi\)
−0.710610 + 0.703586i \(0.751581\pi\)
\(278\) 1.33161e8 0.371723
\(279\) −1.35221e8 −0.372760
\(280\) 7.16607e7 0.195087
\(281\) 3.05398e8 0.821095 0.410548 0.911839i \(-0.365338\pi\)
0.410548 + 0.911839i \(0.365338\pi\)
\(282\) 1.43875e8 0.382043
\(283\) −5.40221e8 −1.41683 −0.708417 0.705795i \(-0.750590\pi\)
−0.708417 + 0.705795i \(0.750590\pi\)
\(284\) −1.67579e8 −0.434116
\(285\) −2.31491e7 −0.0592349
\(286\) 662184. 0.00167378
\(287\) −2.47635e8 −0.618337
\(288\) 2.38879e7 0.0589256
\(289\) 3.92455e8 0.956418
\(290\) 1.73284e7 0.0417220
\(291\) −2.69136e8 −0.640246
\(292\) −8.15812e7 −0.191756
\(293\) −2.78333e8 −0.646440 −0.323220 0.946324i \(-0.604765\pi\)
−0.323220 + 0.946324i \(0.604765\pi\)
\(294\) 9.29189e7 0.213250
\(295\) −1.41082e7 −0.0319960
\(296\) 1.58148e8 0.354441
\(297\) 3.39621e7 0.0752223
\(298\) −1.10082e8 −0.240969
\(299\) −4.07979e6 −0.00882651
\(300\) 2.70000e7 0.0577350
\(301\) 2.65476e8 0.561102
\(302\) 1.75521e8 0.366695
\(303\) −3.68127e8 −0.760237
\(304\) 2.80945e7 0.0573539
\(305\) 1.41520e8 0.285606
\(306\) 1.65242e8 0.329681
\(307\) 4.49571e8 0.886777 0.443388 0.896330i \(-0.353776\pi\)
0.443388 + 0.896330i \(0.353776\pi\)
\(308\) −1.23647e8 −0.241133
\(309\) −3.05966e8 −0.589954
\(310\) 1.85488e8 0.353631
\(311\) −7.75155e8 −1.46126 −0.730629 0.682775i \(-0.760773\pi\)
−0.730629 + 0.682775i \(0.760773\pi\)
\(312\) 663163. 0.00123617
\(313\) 1.99952e8 0.368570 0.184285 0.982873i \(-0.441003\pi\)
0.184285 + 0.982873i \(0.441003\pi\)
\(314\) 4.33667e8 0.790502
\(315\) 1.02032e8 0.183930
\(316\) 3.45213e8 0.615434
\(317\) −7.21290e8 −1.27175 −0.635876 0.771791i \(-0.719361\pi\)
−0.635876 + 0.771791i \(0.719361\pi\)
\(318\) −4.12693e8 −0.719669
\(319\) −2.98993e7 −0.0515695
\(320\) −3.27680e7 −0.0559017
\(321\) −3.53223e8 −0.596048
\(322\) 7.61803e8 1.27159
\(323\) 1.94340e8 0.320888
\(324\) 3.40122e7 0.0555556
\(325\) 749560. 0.00121120
\(326\) 3.67283e8 0.587136
\(327\) 9.83857e6 0.0155602
\(328\) 1.13235e8 0.177183
\(329\) −7.45815e8 −1.15464
\(330\) −4.65872e7 −0.0713621
\(331\) 2.46711e8 0.373930 0.186965 0.982367i \(-0.440135\pi\)
0.186965 + 0.982367i \(0.440135\pi\)
\(332\) −1.74347e8 −0.261475
\(333\) 2.25176e8 0.334170
\(334\) −2.14607e8 −0.315160
\(335\) −1.77506e7 −0.0257962
\(336\) −1.23830e8 −0.178089
\(337\) 8.10307e8 1.15331 0.576654 0.816989i \(-0.304358\pi\)
0.576654 + 0.816989i \(0.304358\pi\)
\(338\) −5.01970e8 −0.707081
\(339\) 2.04238e8 0.284733
\(340\) −2.26669e8 −0.312763
\(341\) −3.20051e8 −0.437098
\(342\) 4.00017e7 0.0540738
\(343\) 4.40448e8 0.589338
\(344\) −1.21393e8 −0.160783
\(345\) 2.87029e8 0.376321
\(346\) −8.55706e6 −0.0111060
\(347\) −1.29901e9 −1.66901 −0.834503 0.551004i \(-0.814245\pi\)
−0.834503 + 0.551004i \(0.814245\pi\)
\(348\) −2.99434e7 −0.0380868
\(349\) 1.12893e7 0.0142160 0.00710802 0.999975i \(-0.497737\pi\)
0.00710802 + 0.999975i \(0.497737\pi\)
\(350\) −1.39962e8 −0.174491
\(351\) 944230. 0.00116547
\(352\) 5.65396e7 0.0690961
\(353\) −2.91341e8 −0.352525 −0.176262 0.984343i \(-0.556401\pi\)
−0.176262 + 0.984343i \(0.556401\pi\)
\(354\) 2.43790e7 0.0292082
\(355\) 3.27303e8 0.388285
\(356\) −5.14231e8 −0.604065
\(357\) −8.56577e8 −0.996387
\(358\) −8.26135e8 −0.951614
\(359\) 1.27308e9 1.45219 0.726097 0.687593i \(-0.241332\pi\)
0.726097 + 0.687593i \(0.241332\pi\)
\(360\) −4.66560e7 −0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) −1.25401e9 −1.38938
\(363\) −4.45770e8 −0.489145
\(364\) −3.43769e6 −0.00373605
\(365\) 1.59338e8 0.171512
\(366\) −2.44546e8 −0.260721
\(367\) −5.18592e8 −0.547640 −0.273820 0.961781i \(-0.588287\pi\)
−0.273820 + 0.961781i \(0.588287\pi\)
\(368\) −3.48347e8 −0.364371
\(369\) 1.61227e8 0.167050
\(370\) −3.08884e8 −0.317022
\(371\) 2.13931e9 2.17503
\(372\) −3.20523e8 −0.322819
\(373\) 1.45203e9 1.44876 0.724379 0.689402i \(-0.242127\pi\)
0.724379 + 0.689402i \(0.242127\pi\)
\(374\) 3.91106e8 0.386584
\(375\) −5.27344e7 −0.0516398
\(376\) 3.41036e8 0.330859
\(377\) −831274. −0.000799005 0
\(378\) −1.76312e8 −0.167904
\(379\) 9.42327e8 0.889128 0.444564 0.895747i \(-0.353359\pi\)
0.444564 + 0.895747i \(0.353359\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) −3.18857e8 −0.295365
\(382\) −6.39653e8 −0.587114
\(383\) −1.39307e8 −0.126700 −0.0633502 0.997991i \(-0.520178\pi\)
−0.0633502 + 0.997991i \(0.520178\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 2.41498e8 0.215676
\(386\) 3.56213e8 0.315249
\(387\) −1.72843e8 −0.151587
\(388\) −6.37953e8 −0.554469
\(389\) −1.31288e9 −1.13084 −0.565421 0.824803i \(-0.691286\pi\)
−0.565421 + 0.824803i \(0.691286\pi\)
\(390\) −1.29524e6 −0.00110567
\(391\) −2.40965e9 −2.03861
\(392\) 2.20252e8 0.184680
\(393\) 3.91901e8 0.325689
\(394\) 4.41143e8 0.363364
\(395\) −6.74243e8 −0.550461
\(396\) 8.05027e7 0.0651444
\(397\) 1.47480e8 0.118295 0.0591475 0.998249i \(-0.481162\pi\)
0.0591475 + 0.998249i \(0.481162\pi\)
\(398\) −1.39875e8 −0.111212
\(399\) −2.07360e8 −0.163426
\(400\) 6.40000e7 0.0500000
\(401\) −1.79473e9 −1.38994 −0.694968 0.719041i \(-0.744581\pi\)
−0.694968 + 0.719041i \(0.744581\pi\)
\(402\) 3.06730e7 0.0235486
\(403\) −8.89821e6 −0.00677228
\(404\) −8.72598e8 −0.658384
\(405\) −6.64301e7 −0.0496904
\(406\) 1.55220e8 0.115108
\(407\) 5.32964e8 0.391848
\(408\) 3.91684e8 0.285513
\(409\) −1.25791e8 −0.0909111 −0.0454556 0.998966i \(-0.514474\pi\)
−0.0454556 + 0.998966i \(0.514474\pi\)
\(410\) −2.21162e8 −0.158477
\(411\) −1.37482e9 −0.976788
\(412\) −7.25252e8 −0.510915
\(413\) −1.26376e8 −0.0882751
\(414\) −4.95986e8 −0.343533
\(415\) 3.40521e8 0.233871
\(416\) 1.57194e6 0.00107056
\(417\) 4.49417e8 0.303510
\(418\) 9.46790e7 0.0634069
\(419\) −8.86373e8 −0.588664 −0.294332 0.955703i \(-0.595097\pi\)
−0.294332 + 0.955703i \(0.595097\pi\)
\(420\) 2.41855e8 0.159288
\(421\) −1.84141e9 −1.20272 −0.601360 0.798979i \(-0.705374\pi\)
−0.601360 + 0.798979i \(0.705374\pi\)
\(422\) 1.15300e9 0.746856
\(423\) 4.85577e8 0.311937
\(424\) −9.78236e8 −0.623251
\(425\) 4.42713e8 0.279744
\(426\) −5.65580e8 −0.354454
\(427\) 1.26767e9 0.787970
\(428\) −8.37269e8 −0.516193
\(429\) 2.23487e6 0.00136663
\(430\) 2.37096e8 0.143808
\(431\) 1.87364e9 1.12724 0.563620 0.826034i \(-0.309408\pi\)
0.563620 + 0.826034i \(0.309408\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 7.91307e7 0.0468422 0.0234211 0.999726i \(-0.492544\pi\)
0.0234211 + 0.999726i \(0.492544\pi\)
\(434\) 1.66153e9 0.975647
\(435\) 5.84832e7 0.0340658
\(436\) 2.33211e7 0.0134755
\(437\) −5.83327e8 −0.334370
\(438\) −2.75336e8 −0.156568
\(439\) 1.65113e9 0.931439 0.465720 0.884932i \(-0.345796\pi\)
0.465720 + 0.884932i \(0.345796\pi\)
\(440\) −1.10429e8 −0.0618014
\(441\) 3.13601e8 0.174118
\(442\) 1.08737e7 0.00598964
\(443\) −6.31651e8 −0.345195 −0.172597 0.984992i \(-0.555216\pi\)
−0.172597 + 0.984992i \(0.555216\pi\)
\(444\) 5.33751e8 0.289400
\(445\) 1.00436e9 0.540292
\(446\) 1.60532e9 0.856821
\(447\) −3.71528e8 −0.196750
\(448\) −2.93522e8 −0.154230
\(449\) 1.33505e9 0.696042 0.348021 0.937487i \(-0.386854\pi\)
0.348021 + 0.937487i \(0.386854\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 3.81604e8 0.195882
\(452\) 4.84121e8 0.246586
\(453\) 5.92384e8 0.299405
\(454\) 2.10153e9 1.05400
\(455\) 6.71424e6 0.00334162
\(456\) 9.48188e7 0.0468293
\(457\) 1.49804e9 0.734205 0.367103 0.930180i \(-0.380350\pi\)
0.367103 + 0.930180i \(0.380350\pi\)
\(458\) −2.42903e9 −1.18142
\(459\) 5.57691e8 0.269184
\(460\) 6.80364e8 0.325904
\(461\) −1.75477e9 −0.834195 −0.417098 0.908862i \(-0.636953\pi\)
−0.417098 + 0.908862i \(0.636953\pi\)
\(462\) −4.17309e8 −0.196884
\(463\) −4.17524e8 −0.195500 −0.0977502 0.995211i \(-0.531165\pi\)
−0.0977502 + 0.995211i \(0.531165\pi\)
\(464\) −7.09770e7 −0.0329841
\(465\) 6.26022e8 0.288738
\(466\) 6.01962e8 0.275561
\(467\) 2.65812e9 1.20772 0.603858 0.797092i \(-0.293629\pi\)
0.603858 + 0.797092i \(0.293629\pi\)
\(468\) 2.23817e6 0.00100933
\(469\) −1.59002e8 −0.0711702
\(470\) −6.66086e8 −0.295929
\(471\) 1.46363e9 0.645442
\(472\) 5.77873e7 0.0252950
\(473\) −4.09097e8 −0.177751
\(474\) 1.16509e9 0.502500
\(475\) 1.07172e8 0.0458831
\(476\) −2.03041e9 −0.862896
\(477\) −1.39284e9 −0.587607
\(478\) 9.38329e8 0.392968
\(479\) −1.77848e9 −0.739391 −0.369696 0.929153i \(-0.620538\pi\)
−0.369696 + 0.929153i \(0.620538\pi\)
\(480\) −1.10592e8 −0.0456435
\(481\) 1.48177e7 0.00607119
\(482\) −1.45071e9 −0.590087
\(483\) 2.57108e9 1.03825
\(484\) −1.05664e9 −0.423612
\(485\) 1.24600e9 0.495932
\(486\) 1.14791e8 0.0453609
\(487\) −1.46602e9 −0.575161 −0.287580 0.957756i \(-0.592851\pi\)
−0.287580 + 0.957756i \(0.592851\pi\)
\(488\) −5.79664e8 −0.225791
\(489\) 1.23958e9 0.479395
\(490\) −4.30180e8 −0.165183
\(491\) 2.85504e9 1.08850 0.544248 0.838924i \(-0.316815\pi\)
0.544248 + 0.838924i \(0.316815\pi\)
\(492\) 3.82168e8 0.144669
\(493\) −4.90975e8 −0.184542
\(494\) 2.63231e6 0.000982410 0
\(495\) −1.57232e8 −0.0582669
\(496\) −7.59759e8 −0.279570
\(497\) 2.93184e9 1.07126
\(498\) −5.88421e8 −0.213494
\(499\) −3.21188e8 −0.115720 −0.0578599 0.998325i \(-0.518428\pi\)
−0.0578599 + 0.998325i \(0.518428\pi\)
\(500\) −1.25000e8 −0.0447214
\(501\) −7.24297e8 −0.257327
\(502\) 2.39083e9 0.843500
\(503\) −3.27732e9 −1.14823 −0.574117 0.818773i \(-0.694655\pi\)
−0.574117 + 0.818773i \(0.694655\pi\)
\(504\) −4.17925e8 −0.145409
\(505\) 1.70429e9 0.588877
\(506\) −1.17394e9 −0.402826
\(507\) −1.69415e9 −0.577329
\(508\) −7.55809e8 −0.255793
\(509\) 8.75728e8 0.294345 0.147173 0.989111i \(-0.452983\pi\)
0.147173 + 0.989111i \(0.452983\pi\)
\(510\) −7.65008e8 −0.255370
\(511\) 1.42729e9 0.473192
\(512\) 1.34218e8 0.0441942
\(513\) 1.35006e8 0.0441511
\(514\) −3.02917e9 −0.983904
\(515\) 1.41651e9 0.456977
\(516\) −4.09702e8 −0.131279
\(517\) 1.14930e9 0.365777
\(518\) −2.76685e9 −0.874645
\(519\) −2.88801e7 −0.00906803
\(520\) −3.07020e6 −0.000957535 0
\(521\) 5.04023e9 1.56141 0.780707 0.624897i \(-0.214859\pi\)
0.780707 + 0.624897i \(0.214859\pi\)
\(522\) −1.01059e8 −0.0310977
\(523\) −3.57359e9 −1.09232 −0.546158 0.837682i \(-0.683910\pi\)
−0.546158 + 0.837682i \(0.683910\pi\)
\(524\) 9.28951e8 0.282055
\(525\) −4.72372e8 −0.142471
\(526\) 4.13880e9 1.24001
\(527\) −5.25555e9 −1.56416
\(528\) 1.90821e8 0.0564167
\(529\) 3.82792e9 1.12426
\(530\) 1.91062e9 0.557453
\(531\) 8.22792e7 0.0238484
\(532\) −4.91520e8 −0.141531
\(533\) 1.06096e7 0.00303495
\(534\) −1.73553e9 −0.493217
\(535\) 1.63529e9 0.461697
\(536\) 7.27064e7 0.0203937
\(537\) −2.78821e9 −0.776989
\(538\) 1.54380e9 0.427418
\(539\) 7.42255e8 0.204170
\(540\) −1.57464e8 −0.0430331
\(541\) −7.01044e9 −1.90351 −0.951755 0.306860i \(-0.900722\pi\)
−0.951755 + 0.306860i \(0.900722\pi\)
\(542\) −2.39481e9 −0.646062
\(543\) −4.23229e9 −1.13442
\(544\) 9.28436e8 0.247261
\(545\) −4.55490e7 −0.0120529
\(546\) −1.16022e7 −0.00305047
\(547\) 1.11829e9 0.292146 0.146073 0.989274i \(-0.453336\pi\)
0.146073 + 0.989274i \(0.453336\pi\)
\(548\) −3.25884e9 −0.845924
\(549\) −8.25342e8 −0.212878
\(550\) 2.15681e8 0.0552768
\(551\) −1.18855e8 −0.0302683
\(552\) −1.17567e9 −0.297508
\(553\) −6.03959e9 −1.51869
\(554\) −4.02190e9 −1.00495
\(555\) −1.04248e9 −0.258847
\(556\) 1.06529e9 0.262848
\(557\) 4.38631e9 1.07549 0.537744 0.843108i \(-0.319277\pi\)
0.537744 + 0.843108i \(0.319277\pi\)
\(558\) −1.08177e9 −0.263581
\(559\) −1.13739e7 −0.00275403
\(560\) 5.73285e8 0.137947
\(561\) 1.31998e9 0.315645
\(562\) 2.44318e9 0.580602
\(563\) 3.15070e9 0.744093 0.372046 0.928214i \(-0.378656\pi\)
0.372046 + 0.928214i \(0.378656\pi\)
\(564\) 1.15100e9 0.270145
\(565\) −9.45548e8 −0.220554
\(566\) −4.32177e9 −1.00185
\(567\) −5.95053e8 −0.137093
\(568\) −1.34063e9 −0.306966
\(569\) −1.19494e9 −0.271929 −0.135964 0.990714i \(-0.543413\pi\)
−0.135964 + 0.990714i \(0.543413\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) 7.01962e9 1.57793 0.788964 0.614440i \(-0.210618\pi\)
0.788964 + 0.614440i \(0.210618\pi\)
\(572\) 5.29748e6 0.00118354
\(573\) −2.15883e9 −0.479377
\(574\) −1.98108e9 −0.437230
\(575\) −1.32884e9 −0.291497
\(576\) 1.91103e8 0.0416667
\(577\) 4.83242e9 1.04725 0.523624 0.851950i \(-0.324580\pi\)
0.523624 + 0.851950i \(0.324580\pi\)
\(578\) 3.13964e9 0.676290
\(579\) 1.20222e9 0.257400
\(580\) 1.38627e8 0.0295019
\(581\) 3.05025e9 0.645236
\(582\) −2.15309e9 −0.452722
\(583\) −3.29668e9 −0.689027
\(584\) −6.52649e8 −0.135592
\(585\) −4.37143e6 −0.000902773 0
\(586\) −2.22667e9 −0.457102
\(587\) −3.82091e9 −0.779711 −0.389855 0.920876i \(-0.627475\pi\)
−0.389855 + 0.920876i \(0.627475\pi\)
\(588\) 7.43351e8 0.150790
\(589\) −1.27226e9 −0.256551
\(590\) −1.12866e8 −0.0226246
\(591\) 1.48886e9 0.296686
\(592\) 1.26519e9 0.250628
\(593\) 8.54084e9 1.68193 0.840967 0.541086i \(-0.181987\pi\)
0.840967 + 0.541086i \(0.181987\pi\)
\(594\) 2.71696e8 0.0531902
\(595\) 3.96564e9 0.771798
\(596\) −8.80659e8 −0.170391
\(597\) −4.72079e8 −0.0908040
\(598\) −3.26383e7 −0.00624128
\(599\) 5.83250e8 0.110882 0.0554409 0.998462i \(-0.482344\pi\)
0.0554409 + 0.998462i \(0.482344\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) 2.26108e9 0.424869 0.212435 0.977175i \(-0.431861\pi\)
0.212435 + 0.977175i \(0.431861\pi\)
\(602\) 2.12381e9 0.396759
\(603\) 1.03521e8 0.0192273
\(604\) 1.40417e9 0.259293
\(605\) 2.06375e9 0.378890
\(606\) −2.94502e9 −0.537568
\(607\) −6.08814e9 −1.10490 −0.552452 0.833545i \(-0.686308\pi\)
−0.552452 + 0.833545i \(0.686308\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 5.23868e8 0.0939857
\(610\) 1.13216e9 0.201954
\(611\) 3.19534e7 0.00566725
\(612\) 1.32193e9 0.233120
\(613\) 4.36248e9 0.764929 0.382465 0.923970i \(-0.375075\pi\)
0.382465 + 0.923970i \(0.375075\pi\)
\(614\) 3.59657e9 0.627046
\(615\) −7.46422e8 −0.129396
\(616\) −9.89176e8 −0.170507
\(617\) −4.74419e9 −0.813138 −0.406569 0.913620i \(-0.633275\pi\)
−0.406569 + 0.913620i \(0.633275\pi\)
\(618\) −2.44773e9 −0.417161
\(619\) 5.39704e9 0.914615 0.457307 0.889309i \(-0.348814\pi\)
0.457307 + 0.889309i \(0.348814\pi\)
\(620\) 1.48390e9 0.250055
\(621\) −1.67395e9 −0.280493
\(622\) −6.20124e9 −1.03327
\(623\) 8.99662e9 1.49064
\(624\) 5.30530e6 0.000874106 0
\(625\) 2.44141e8 0.0400000
\(626\) 1.59961e9 0.260618
\(627\) 3.19541e8 0.0517715
\(628\) 3.46934e9 0.558969
\(629\) 8.75179e9 1.40223
\(630\) 8.16260e8 0.130058
\(631\) −5.46391e9 −0.865767 −0.432884 0.901450i \(-0.642504\pi\)
−0.432884 + 0.901450i \(0.642504\pi\)
\(632\) 2.76170e9 0.435178
\(633\) 3.89139e9 0.609806
\(634\) −5.77032e9 −0.899265
\(635\) 1.47619e9 0.228788
\(636\) −3.30155e9 −0.508883
\(637\) 2.06365e7 0.00316336
\(638\) −2.39194e8 −0.0364651
\(639\) −1.90883e9 −0.289411
\(640\) −2.62144e8 −0.0395285
\(641\) −4.72360e8 −0.0708386 −0.0354193 0.999373i \(-0.511277\pi\)
−0.0354193 + 0.999373i \(0.511277\pi\)
\(642\) −2.82578e9 −0.421470
\(643\) 2.26551e8 0.0336069 0.0168034 0.999859i \(-0.494651\pi\)
0.0168034 + 0.999859i \(0.494651\pi\)
\(644\) 6.09442e9 0.899149
\(645\) 8.00199e8 0.117419
\(646\) 1.55472e9 0.226902
\(647\) 3.86399e9 0.560882 0.280441 0.959871i \(-0.409519\pi\)
0.280441 + 0.959871i \(0.409519\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 1.94744e8 0.0279646
\(650\) 5.99648e6 0.000856445 0
\(651\) 5.60765e9 0.796613
\(652\) 2.93826e9 0.415168
\(653\) 7.63350e9 1.07282 0.536411 0.843957i \(-0.319780\pi\)
0.536411 + 0.843957i \(0.319780\pi\)
\(654\) 7.87086e7 0.0110027
\(655\) −1.81436e9 −0.252277
\(656\) 9.05880e8 0.125287
\(657\) −9.29260e8 −0.127838
\(658\) −5.96652e9 −0.816452
\(659\) 9.98595e9 1.35922 0.679611 0.733573i \(-0.262149\pi\)
0.679611 + 0.733573i \(0.262149\pi\)
\(660\) −3.72697e8 −0.0504606
\(661\) 3.98575e9 0.536791 0.268395 0.963309i \(-0.413507\pi\)
0.268395 + 0.963309i \(0.413507\pi\)
\(662\) 1.97369e9 0.264408
\(663\) 3.66988e7 0.00489052
\(664\) −1.39478e9 −0.184891
\(665\) 9.60001e8 0.126589
\(666\) 1.80141e9 0.236294
\(667\) 1.47370e9 0.192295
\(668\) −1.71685e9 −0.222851
\(669\) 5.41796e9 0.699591
\(670\) −1.42005e8 −0.0182407
\(671\) −1.95348e9 −0.249620
\(672\) −9.90637e8 −0.125928
\(673\) 1.74903e9 0.221180 0.110590 0.993866i \(-0.464726\pi\)
0.110590 + 0.993866i \(0.464726\pi\)
\(674\) 6.48246e9 0.815512
\(675\) 3.07547e8 0.0384900
\(676\) −4.01576e9 −0.499982
\(677\) 3.50541e9 0.434189 0.217094 0.976151i \(-0.430342\pi\)
0.217094 + 0.976151i \(0.430342\pi\)
\(678\) 1.63391e9 0.201337
\(679\) 1.11612e10 1.36825
\(680\) −1.81335e9 −0.221157
\(681\) 7.09265e9 0.860585
\(682\) −2.56041e9 −0.309075
\(683\) 3.54964e8 0.0426297 0.0213148 0.999773i \(-0.493215\pi\)
0.0213148 + 0.999773i \(0.493215\pi\)
\(684\) 3.20014e8 0.0382360
\(685\) 6.36492e9 0.756617
\(686\) 3.52358e9 0.416725
\(687\) −8.19797e9 −0.964624
\(688\) −9.71145e8 −0.113691
\(689\) −9.16558e7 −0.0106756
\(690\) 2.29623e9 0.266099
\(691\) −5.81392e9 −0.670342 −0.335171 0.942157i \(-0.608794\pi\)
−0.335171 + 0.942157i \(0.608794\pi\)
\(692\) −6.84565e7 −0.00785314
\(693\) −1.40842e9 −0.160755
\(694\) −1.03920e10 −1.18016
\(695\) −2.08064e9 −0.235098
\(696\) −2.39547e8 −0.0269314
\(697\) 6.26632e9 0.700968
\(698\) 9.03145e7 0.0100523
\(699\) 2.03162e9 0.224995
\(700\) −1.11970e9 −0.123384
\(701\) 1.45880e10 1.59949 0.799747 0.600338i \(-0.204967\pi\)
0.799747 + 0.600338i \(0.204967\pi\)
\(702\) 7.55384e6 0.000824115 0
\(703\) 2.11863e9 0.229992
\(704\) 4.52317e8 0.0488583
\(705\) −2.24804e9 −0.241625
\(706\) −2.33073e9 −0.249273
\(707\) 1.52663e10 1.62468
\(708\) 1.95032e8 0.0206533
\(709\) −1.06100e10 −1.11803 −0.559015 0.829158i \(-0.688821\pi\)
−0.559015 + 0.829158i \(0.688821\pi\)
\(710\) 2.61842e9 0.274559
\(711\) 3.93219e9 0.410290
\(712\) −4.11385e9 −0.427138
\(713\) 1.57749e10 1.62988
\(714\) −6.85262e9 −0.704552
\(715\) −1.03466e7 −0.00105859
\(716\) −6.60908e9 −0.672892
\(717\) 3.16686e9 0.320857
\(718\) 1.01846e10 1.02686
\(719\) 1.40920e10 1.41391 0.706955 0.707259i \(-0.250068\pi\)
0.706955 + 0.707259i \(0.250068\pi\)
\(720\) −3.73248e8 −0.0372678
\(721\) 1.26885e10 1.26077
\(722\) 3.76367e8 0.0372161
\(723\) −4.89614e9 −0.481804
\(724\) −1.00321e10 −0.982440
\(725\) −2.70756e8 −0.0263873
\(726\) −3.56616e9 −0.345878
\(727\) −8.87157e9 −0.856308 −0.428154 0.903706i \(-0.640836\pi\)
−0.428154 + 0.903706i \(0.640836\pi\)
\(728\) −2.75015e7 −0.00264178
\(729\) 3.87420e8 0.0370370
\(730\) 1.27471e9 0.121277
\(731\) −6.71778e9 −0.636085
\(732\) −1.95637e9 −0.184358
\(733\) 9.54110e7 0.00894818 0.00447409 0.999990i \(-0.498576\pi\)
0.00447409 + 0.999990i \(0.498576\pi\)
\(734\) −4.14874e9 −0.387240
\(735\) −1.45186e9 −0.134871
\(736\) −2.78677e9 −0.257649
\(737\) 2.45022e8 0.0225460
\(738\) 1.28982e9 0.118122
\(739\) 1.27194e10 1.15934 0.579671 0.814851i \(-0.303181\pi\)
0.579671 + 0.814851i \(0.303181\pi\)
\(740\) −2.47107e9 −0.224168
\(741\) 8.88405e6 0.000802134 0
\(742\) 1.71145e10 1.53798
\(743\) −1.15304e10 −1.03130 −0.515650 0.856799i \(-0.672450\pi\)
−0.515650 + 0.856799i \(0.672450\pi\)
\(744\) −2.56419e9 −0.228268
\(745\) 1.72004e9 0.152402
\(746\) 1.16163e10 1.02443
\(747\) −1.98592e9 −0.174317
\(748\) 3.12885e9 0.273356
\(749\) 1.46483e10 1.27380
\(750\) −4.21875e8 −0.0365148
\(751\) −1.90878e9 −0.164443 −0.0822216 0.996614i \(-0.526202\pi\)
−0.0822216 + 0.996614i \(0.526202\pi\)
\(752\) 2.72829e9 0.233953
\(753\) 8.06904e9 0.688715
\(754\) −6.65019e6 −0.000564982 0
\(755\) −2.74252e9 −0.231918
\(756\) −1.41050e9 −0.118726
\(757\) 7.08476e9 0.593594 0.296797 0.954941i \(-0.404082\pi\)
0.296797 + 0.954941i \(0.404082\pi\)
\(758\) 7.53862e9 0.628709
\(759\) −3.96203e9 −0.328906
\(760\) −4.38976e8 −0.0362738
\(761\) 2.32756e10 1.91450 0.957249 0.289264i \(-0.0934106\pi\)
0.957249 + 0.289264i \(0.0934106\pi\)
\(762\) −2.55086e9 −0.208854
\(763\) −4.08008e8 −0.0332532
\(764\) −5.11723e9 −0.415153
\(765\) −2.58190e9 −0.208509
\(766\) −1.11446e9 −0.0895907
\(767\) 5.41438e6 0.000433276 0
\(768\) 4.52985e8 0.0360844
\(769\) −2.51570e10 −1.99488 −0.997441 0.0714999i \(-0.977221\pi\)
−0.997441 + 0.0714999i \(0.977221\pi\)
\(770\) 1.93198e9 0.152506
\(771\) −1.02234e10 −0.803354
\(772\) 2.84970e9 0.222915
\(773\) 1.03014e10 0.802170 0.401085 0.916041i \(-0.368633\pi\)
0.401085 + 0.916041i \(0.368633\pi\)
\(774\) −1.38274e9 −0.107189
\(775\) −2.89825e9 −0.223656
\(776\) −5.10362e9 −0.392069
\(777\) −9.33812e9 −0.714144
\(778\) −1.05030e10 −0.799625
\(779\) 1.51695e9 0.114972
\(780\) −1.03619e7 −0.000781824 0
\(781\) −4.51796e9 −0.339363
\(782\) −1.92772e10 −1.44152
\(783\) −3.41074e8 −0.0253912
\(784\) 1.76202e9 0.130588
\(785\) −6.77605e9 −0.499957
\(786\) 3.13521e9 0.230297
\(787\) −2.63516e10 −1.92706 −0.963529 0.267604i \(-0.913768\pi\)
−0.963529 + 0.267604i \(0.913768\pi\)
\(788\) 3.52914e9 0.256937
\(789\) 1.39685e10 1.01246
\(790\) −5.39395e9 −0.389235
\(791\) −8.46982e9 −0.608495
\(792\) 6.44021e8 0.0460640
\(793\) −5.43116e7 −0.00386755
\(794\) 1.17984e9 0.0836472
\(795\) 6.44833e9 0.455158
\(796\) −1.11900e9 −0.0786385
\(797\) 2.63384e10 1.84283 0.921416 0.388579i \(-0.127034\pi\)
0.921416 + 0.388579i \(0.127034\pi\)
\(798\) −1.65888e9 −0.115559
\(799\) 1.88726e10 1.30894
\(800\) 5.12000e8 0.0353553
\(801\) −5.85742e9 −0.402710
\(802\) −1.43579e10 −0.982833
\(803\) −2.19944e9 −0.149902
\(804\) 2.45384e8 0.0166514
\(805\) −1.19032e10 −0.804224
\(806\) −7.11856e7 −0.00478872
\(807\) 5.21032e9 0.348986
\(808\) −6.98078e9 −0.465548
\(809\) 4.24126e9 0.281628 0.140814 0.990036i \(-0.455028\pi\)
0.140814 + 0.990036i \(0.455028\pi\)
\(810\) −5.31441e8 −0.0351364
\(811\) 5.56399e9 0.366280 0.183140 0.983087i \(-0.441374\pi\)
0.183140 + 0.983087i \(0.441374\pi\)
\(812\) 1.24176e9 0.0813940
\(813\) −8.08249e9 −0.527507
\(814\) 4.26371e9 0.277078
\(815\) −5.73879e9 −0.371338
\(816\) 3.13347e9 0.201888
\(817\) −1.62624e9 −0.104330
\(818\) −1.00633e9 −0.0642839
\(819\) −3.91575e7 −0.00249070
\(820\) −1.76930e9 −0.112060
\(821\) −2.99915e9 −0.189146 −0.0945729 0.995518i \(-0.530149\pi\)
−0.0945729 + 0.995518i \(0.530149\pi\)
\(822\) −1.09986e10 −0.690694
\(823\) 1.82277e10 1.13981 0.569904 0.821711i \(-0.306980\pi\)
0.569904 + 0.821711i \(0.306980\pi\)
\(824\) −5.80202e9 −0.361272
\(825\) 7.27925e8 0.0451334
\(826\) −1.01100e9 −0.0624199
\(827\) −3.03052e10 −1.86315 −0.931576 0.363546i \(-0.881566\pi\)
−0.931576 + 0.363546i \(0.881566\pi\)
\(828\) −3.96789e9 −0.242914
\(829\) −9.22068e9 −0.562111 −0.281055 0.959692i \(-0.590684\pi\)
−0.281055 + 0.959692i \(0.590684\pi\)
\(830\) 2.72417e9 0.165372
\(831\) −1.35739e10 −0.820542
\(832\) 1.25755e7 0.000756998 0
\(833\) 1.21886e10 0.730626
\(834\) 3.59534e9 0.214614
\(835\) 3.35323e9 0.199324
\(836\) 7.57432e8 0.0448354
\(837\) −3.65096e9 −0.215213
\(838\) −7.09098e9 −0.416248
\(839\) 2.26505e9 0.132407 0.0662034 0.997806i \(-0.478911\pi\)
0.0662034 + 0.997806i \(0.478911\pi\)
\(840\) 1.93484e9 0.112633
\(841\) −1.69496e10 −0.982593
\(842\) −1.47313e10 −0.850451
\(843\) 8.24574e9 0.474060
\(844\) 9.22403e9 0.528107
\(845\) 7.84328e9 0.447197
\(846\) 3.88461e9 0.220573
\(847\) 1.84862e10 1.04534
\(848\) −7.82589e9 −0.440705
\(849\) −1.45860e10 −0.818009
\(850\) 3.54170e9 0.197809
\(851\) −2.62692e10 −1.46114
\(852\) −4.52464e9 −0.250637
\(853\) 7.21996e9 0.398303 0.199151 0.979969i \(-0.436181\pi\)
0.199151 + 0.979969i \(0.436181\pi\)
\(854\) 1.01414e10 0.557179
\(855\) −6.25026e8 −0.0341993
\(856\) −6.69815e9 −0.365003
\(857\) −1.16026e10 −0.629685 −0.314842 0.949144i \(-0.601952\pi\)
−0.314842 + 0.949144i \(0.601952\pi\)
\(858\) 1.78790e7 0.000966356 0
\(859\) −2.98667e9 −0.160772 −0.0803861 0.996764i \(-0.525615\pi\)
−0.0803861 + 0.996764i \(0.525615\pi\)
\(860\) 1.89677e9 0.101688
\(861\) −6.68614e9 −0.356997
\(862\) 1.49891e10 0.797079
\(863\) 1.27942e10 0.677604 0.338802 0.940858i \(-0.389978\pi\)
0.338802 + 0.940858i \(0.389978\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 1.33704e8 0.00702406
\(866\) 6.33046e8 0.0331224
\(867\) 1.05963e10 0.552188
\(868\) 1.32922e10 0.689887
\(869\) 9.30699e9 0.481105
\(870\) 4.67866e8 0.0240882
\(871\) 6.81222e6 0.000349321 0
\(872\) 1.86569e8 0.00952863
\(873\) −7.26668e9 −0.369646
\(874\) −4.66662e9 −0.236435
\(875\) 2.18691e9 0.110358
\(876\) −2.20269e9 −0.110711
\(877\) 2.24309e9 0.112292 0.0561459 0.998423i \(-0.482119\pi\)
0.0561459 + 0.998423i \(0.482119\pi\)
\(878\) 1.32090e10 0.658627
\(879\) −7.51500e9 −0.373222
\(880\) −8.83431e8 −0.0437002
\(881\) 1.15239e10 0.567787 0.283893 0.958856i \(-0.408374\pi\)
0.283893 + 0.958856i \(0.408374\pi\)
\(882\) 2.50881e9 0.123120
\(883\) −9.27109e9 −0.453177 −0.226589 0.973991i \(-0.572757\pi\)
−0.226589 + 0.973991i \(0.572757\pi\)
\(884\) 8.69898e7 0.00423531
\(885\) −3.80922e8 −0.0184729
\(886\) −5.05321e9 −0.244089
\(887\) −2.13154e10 −1.02556 −0.512779 0.858521i \(-0.671384\pi\)
−0.512779 + 0.858521i \(0.671384\pi\)
\(888\) 4.27001e9 0.204637
\(889\) 1.32231e10 0.631214
\(890\) 8.03487e9 0.382044
\(891\) 9.16976e8 0.0434296
\(892\) 1.28426e10 0.605864
\(893\) 4.56869e9 0.214690
\(894\) −2.97223e9 −0.139123
\(895\) 1.29084e10 0.601853
\(896\) −2.34818e9 −0.109057
\(897\) −1.10154e8 −0.00509599
\(898\) 1.06804e10 0.492176
\(899\) 3.21421e9 0.147542
\(900\) 7.29000e8 0.0333333
\(901\) −5.41347e10 −2.46569
\(902\) 3.05284e9 0.138510
\(903\) 7.16785e9 0.323953
\(904\) 3.87296e9 0.174363
\(905\) 1.95939e10 0.878721
\(906\) 4.73907e9 0.211712
\(907\) −4.13216e8 −0.0183887 −0.00919435 0.999958i \(-0.502927\pi\)
−0.00919435 + 0.999958i \(0.502927\pi\)
\(908\) 1.68122e10 0.745288
\(909\) −9.93944e9 −0.438923
\(910\) 5.37140e7 0.00236288
\(911\) 3.46681e9 0.151920 0.0759602 0.997111i \(-0.475798\pi\)
0.0759602 + 0.997111i \(0.475798\pi\)
\(912\) 7.58551e8 0.0331133
\(913\) −4.70042e9 −0.204404
\(914\) 1.19843e10 0.519161
\(915\) 3.82103e9 0.164895
\(916\) −1.94322e10 −0.835388
\(917\) −1.62522e10 −0.696018
\(918\) 4.46152e9 0.190342
\(919\) −2.19386e10 −0.932406 −0.466203 0.884678i \(-0.654378\pi\)
−0.466203 + 0.884678i \(0.654378\pi\)
\(920\) 5.44292e9 0.230449
\(921\) 1.21384e10 0.511981
\(922\) −1.40382e10 −0.589865
\(923\) −1.25611e8 −0.00525800
\(924\) −3.33847e9 −0.139218
\(925\) 4.82631e9 0.200502
\(926\) −3.34019e9 −0.138240
\(927\) −8.26108e9 −0.340610
\(928\) −5.67816e8 −0.0233233
\(929\) 1.61947e10 0.662703 0.331352 0.943507i \(-0.392495\pi\)
0.331352 + 0.943507i \(0.392495\pi\)
\(930\) 5.00818e9 0.204169
\(931\) 2.95061e9 0.119836
\(932\) 4.81569e9 0.194851
\(933\) −2.09292e10 −0.843658
\(934\) 2.12649e10 0.853985
\(935\) −6.11103e9 −0.244497
\(936\) 1.79054e7 0.000713704 0
\(937\) 9.44456e9 0.375053 0.187527 0.982260i \(-0.439953\pi\)
0.187527 + 0.982260i \(0.439953\pi\)
\(938\) −1.27202e9 −0.0503249
\(939\) 5.39870e9 0.212794
\(940\) −5.32869e9 −0.209254
\(941\) −2.59333e10 −1.01460 −0.507299 0.861770i \(-0.669356\pi\)
−0.507299 + 0.861770i \(0.669356\pi\)
\(942\) 1.17090e10 0.456396
\(943\) −1.88089e10 −0.730418
\(944\) 4.62298e8 0.0178863
\(945\) 2.75488e9 0.106192
\(946\) −3.27278e9 −0.125689
\(947\) 2.40537e9 0.0920359 0.0460180 0.998941i \(-0.485347\pi\)
0.0460180 + 0.998941i \(0.485347\pi\)
\(948\) 9.32074e9 0.355321
\(949\) −6.11500e7 −0.00232255
\(950\) 8.57375e8 0.0324443
\(951\) −1.94748e10 −0.734246
\(952\) −1.62432e10 −0.610160
\(953\) −3.83728e10 −1.43614 −0.718072 0.695969i \(-0.754975\pi\)
−0.718072 + 0.695969i \(0.754975\pi\)
\(954\) −1.11427e10 −0.415501
\(955\) 9.99458e9 0.371324
\(956\) 7.50663e9 0.277870
\(957\) −8.07280e8 −0.0297737
\(958\) −1.42278e10 −0.522829
\(959\) 5.70143e10 2.08746
\(960\) −8.84736e8 −0.0322749
\(961\) 6.89323e9 0.250548
\(962\) 1.18542e8 0.00429298
\(963\) −9.53702e9 −0.344129
\(964\) −1.16057e10 −0.417254
\(965\) −5.56583e9 −0.199381
\(966\) 2.05687e10 0.734152
\(967\) 1.51477e10 0.538707 0.269353 0.963041i \(-0.413190\pi\)
0.269353 + 0.963041i \(0.413190\pi\)
\(968\) −8.45311e9 −0.299539
\(969\) 5.24719e9 0.185265
\(970\) 9.96801e9 0.350677
\(971\) −5.97035e9 −0.209282 −0.104641 0.994510i \(-0.533369\pi\)
−0.104641 + 0.994510i \(0.533369\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.86375e10 −0.648622
\(974\) −1.17282e10 −0.406700
\(975\) 2.02381e7 0.000699285 0
\(976\) −4.63731e9 −0.159658
\(977\) 2.54311e10 0.872436 0.436218 0.899841i \(-0.356318\pi\)
0.436218 + 0.899841i \(0.356318\pi\)
\(978\) 9.91664e9 0.338983
\(979\) −1.38638e10 −0.472217
\(980\) −3.44144e9 −0.116802
\(981\) 2.65641e8 0.00898368
\(982\) 2.28403e10 0.769683
\(983\) 2.08515e10 0.700163 0.350082 0.936719i \(-0.386154\pi\)
0.350082 + 0.936719i \(0.386154\pi\)
\(984\) 3.05735e9 0.102297
\(985\) −6.89285e9 −0.229812
\(986\) −3.92780e9 −0.130491
\(987\) −2.01370e10 −0.666630
\(988\) 2.10585e7 0.000694669 0
\(989\) 2.01640e10 0.662809
\(990\) −1.25785e9 −0.0412009
\(991\) −4.40793e10 −1.43872 −0.719360 0.694637i \(-0.755565\pi\)
−0.719360 + 0.694637i \(0.755565\pi\)
\(992\) −6.07807e9 −0.197686
\(993\) 6.66119e9 0.215889
\(994\) 2.34548e10 0.757493
\(995\) 2.18555e9 0.0703364
\(996\) −4.70737e9 −0.150963
\(997\) −3.76004e10 −1.20160 −0.600800 0.799400i \(-0.705151\pi\)
−0.600800 + 0.799400i \(0.705151\pi\)
\(998\) −2.56950e9 −0.0818262
\(999\) 6.07976e9 0.192933
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 570.8.a.b.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.b.1.1 4 1.1 even 1 trivial