Properties

Label 570.8.a.b.1.2
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 1.2
Root \(32.6184\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} -677.988 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} -677.988 q^{7} +512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} -1909.33 q^{11} +1728.00 q^{12} +6012.82 q^{13} -5423.90 q^{14} -3375.00 q^{15} +4096.00 q^{16} -8038.71 q^{17} +5832.00 q^{18} +6859.00 q^{19} -8000.00 q^{20} -18305.7 q^{21} -15274.6 q^{22} +39335.6 q^{23} +13824.0 q^{24} +15625.0 q^{25} +48102.6 q^{26} +19683.0 q^{27} -43391.2 q^{28} -91955.5 q^{29} -27000.0 q^{30} +14215.0 q^{31} +32768.0 q^{32} -51551.8 q^{33} -64309.6 q^{34} +84748.5 q^{35} +46656.0 q^{36} -1520.80 q^{37} +54872.0 q^{38} +162346. q^{39} -64000.0 q^{40} -41340.7 q^{41} -146445. q^{42} -180882. q^{43} -122197. q^{44} -91125.0 q^{45} +314685. q^{46} -782103. q^{47} +110592. q^{48} -363875. q^{49} +125000. q^{50} -217045. q^{51} +384821. q^{52} +1.44685e6 q^{53} +157464. q^{54} +238666. q^{55} -347130. q^{56} +185193. q^{57} -735644. q^{58} +235153. q^{59} -216000. q^{60} -1.56702e6 q^{61} +113720. q^{62} -494253. q^{63} +262144. q^{64} -751603. q^{65} -412414. q^{66} -4.09999e6 q^{67} -514477. q^{68} +1.06206e6 q^{69} +677988. q^{70} +3.75097e6 q^{71} +373248. q^{72} -275065. q^{73} -12166.4 q^{74} +421875. q^{75} +438976. q^{76} +1.29450e6 q^{77} +1.29877e6 q^{78} -3.83583e6 q^{79} -512000. q^{80} +531441. q^{81} -330726. q^{82} -8.56078e6 q^{83} -1.17156e6 q^{84} +1.00484e6 q^{85} -1.44706e6 q^{86} -2.48280e6 q^{87} -977575. q^{88} -1.20821e7 q^{89} -729000. q^{90} -4.07662e6 q^{91} +2.51748e6 q^{92} +383805. q^{93} -6.25683e6 q^{94} -857375. q^{95} +884736. q^{96} -4.23201e6 q^{97} -2.91100e6 q^{98} -1.39190e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100}) \)

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
\(7\) −677.988 −0.747100 −0.373550 0.927610i \(-0.621860\pi\)
−0.373550 + 0.927610i \(0.621860\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1000.00 −0.316228
\(11\) −1909.33 −0.432519 −0.216260 0.976336i \(-0.569386\pi\)
−0.216260 + 0.976336i \(0.569386\pi\)
\(12\) 1728.00 0.288675
\(13\) 6012.82 0.759061 0.379530 0.925179i \(-0.376086\pi\)
0.379530 + 0.925179i \(0.376086\pi\)
\(14\) −5423.90 −0.528280
\(15\) −3375.00 −0.258199
\(16\) 4096.00 0.250000
\(17\) −8038.71 −0.396839 −0.198420 0.980117i \(-0.563581\pi\)
−0.198420 + 0.980117i \(0.563581\pi\)
\(18\) 5832.00 0.235702
\(19\) 6859.00 0.229416
\(20\) −8000.00 −0.223607
\(21\) −18305.7 −0.431339
\(22\) −15274.6 −0.305837
\(23\) 39335.6 0.674122 0.337061 0.941483i \(-0.390567\pi\)
0.337061 + 0.941483i \(0.390567\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) 48102.6 0.536737
\(27\) 19683.0 0.192450
\(28\) −43391.2 −0.373550
\(29\) −91955.5 −0.700140 −0.350070 0.936724i \(-0.613842\pi\)
−0.350070 + 0.936724i \(0.613842\pi\)
\(30\) −27000.0 −0.182574
\(31\) 14215.0 0.0857000 0.0428500 0.999082i \(-0.486356\pi\)
0.0428500 + 0.999082i \(0.486356\pi\)
\(32\) 32768.0 0.176777
\(33\) −51551.8 −0.249715
\(34\) −64309.6 −0.280608
\(35\) 84748.5 0.334113
\(36\) 46656.0 0.166667
\(37\) −1520.80 −0.00493590 −0.00246795 0.999997i \(-0.500786\pi\)
−0.00246795 + 0.999997i \(0.500786\pi\)
\(38\) 54872.0 0.162221
\(39\) 162346. 0.438244
\(40\) −64000.0 −0.158114
\(41\) −41340.7 −0.0936774 −0.0468387 0.998902i \(-0.514915\pi\)
−0.0468387 + 0.998902i \(0.514915\pi\)
\(42\) −146445. −0.305002
\(43\) −180882. −0.346941 −0.173470 0.984839i \(-0.555498\pi\)
−0.173470 + 0.984839i \(0.555498\pi\)
\(44\) −122197. −0.216260
\(45\) −91125.0 −0.149071
\(46\) 314685. 0.476676
\(47\) −782103. −1.09881 −0.549404 0.835557i \(-0.685145\pi\)
−0.549404 + 0.835557i \(0.685145\pi\)
\(48\) 110592. 0.144338
\(49\) −363875. −0.441841
\(50\) 125000. 0.141421
\(51\) −217045. −0.229115
\(52\) 384821. 0.379530
\(53\) 1.44685e6 1.33493 0.667465 0.744641i \(-0.267379\pi\)
0.667465 + 0.744641i \(0.267379\pi\)
\(54\) 157464. 0.136083
\(55\) 238666. 0.193429
\(56\) −347130. −0.264140
\(57\) 185193. 0.132453
\(58\) −735644. −0.495074
\(59\) 235153. 0.149063 0.0745313 0.997219i \(-0.476254\pi\)
0.0745313 + 0.997219i \(0.476254\pi\)
\(60\) −216000. −0.129099
\(61\) −1.56702e6 −0.883932 −0.441966 0.897032i \(-0.645719\pi\)
−0.441966 + 0.897032i \(0.645719\pi\)
\(62\) 113720. 0.0605991
\(63\) −494253. −0.249033
\(64\) 262144. 0.125000
\(65\) −751603. −0.339462
\(66\) −412414. −0.176575
\(67\) −4.09999e6 −1.66541 −0.832704 0.553718i \(-0.813208\pi\)
−0.832704 + 0.553718i \(0.813208\pi\)
\(68\) −514477. −0.198420
\(69\) 1.06206e6 0.389204
\(70\) 677988. 0.236254
\(71\) 3.75097e6 1.24377 0.621884 0.783109i \(-0.286367\pi\)
0.621884 + 0.783109i \(0.286367\pi\)
\(72\) 373248. 0.117851
\(73\) −275065. −0.0827571 −0.0413786 0.999144i \(-0.513175\pi\)
−0.0413786 + 0.999144i \(0.513175\pi\)
\(74\) −12166.4 −0.00349021
\(75\) 421875. 0.115470
\(76\) 438976. 0.114708
\(77\) 1.29450e6 0.323135
\(78\) 1.29877e6 0.309885
\(79\) −3.83583e6 −0.875315 −0.437657 0.899142i \(-0.644192\pi\)
−0.437657 + 0.899142i \(0.644192\pi\)
\(80\) −512000. −0.111803
\(81\) 531441. 0.111111
\(82\) −330726. −0.0662399
\(83\) −8.56078e6 −1.64339 −0.821694 0.569929i \(-0.806971\pi\)
−0.821694 + 0.569929i \(0.806971\pi\)
\(84\) −1.17156e6 −0.215669
\(85\) 1.00484e6 0.177472
\(86\) −1.44706e6 −0.245324
\(87\) −2.48280e6 −0.404226
\(88\) −977575. −0.152919
\(89\) −1.20821e7 −1.81667 −0.908337 0.418240i \(-0.862647\pi\)
−0.908337 + 0.418240i \(0.862647\pi\)
\(90\) −729000. −0.105409
\(91\) −4.07662e6 −0.567095
\(92\) 2.51748e6 0.337061
\(93\) 383805. 0.0494789
\(94\) −6.25683e6 −0.776974
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) −4.23201e6 −0.470810 −0.235405 0.971897i \(-0.575642\pi\)
−0.235405 + 0.971897i \(0.575642\pi\)
\(98\) −2.91100e6 −0.312429
\(99\) −1.39190e6 −0.144173
\(100\) 1.00000e6 0.100000
\(101\) 1.30277e7 1.25818 0.629091 0.777332i \(-0.283427\pi\)
0.629091 + 0.777332i \(0.283427\pi\)
\(102\) −1.73636e6 −0.162009
\(103\) 1.01658e7 0.916664 0.458332 0.888781i \(-0.348447\pi\)
0.458332 + 0.888781i \(0.348447\pi\)
\(104\) 3.07856e6 0.268369
\(105\) 2.28821e6 0.192900
\(106\) 1.15748e7 0.943938
\(107\) −1.87642e7 −1.48077 −0.740384 0.672184i \(-0.765356\pi\)
−0.740384 + 0.672184i \(0.765356\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 295607. 0.0218636 0.0109318 0.999940i \(-0.496520\pi\)
0.0109318 + 0.999940i \(0.496520\pi\)
\(110\) 1.90933e6 0.136775
\(111\) −41061.6 −0.00284974
\(112\) −2.77704e6 −0.186775
\(113\) −2.32869e7 −1.51823 −0.759114 0.650957i \(-0.774368\pi\)
−0.759114 + 0.650957i \(0.774368\pi\)
\(114\) 1.48154e6 0.0936586
\(115\) −4.91695e6 −0.301476
\(116\) −5.88515e6 −0.350070
\(117\) 4.38335e6 0.253020
\(118\) 1.88123e6 0.105403
\(119\) 5.45015e6 0.296479
\(120\) −1.72800e6 −0.0912871
\(121\) −1.58416e7 −0.812927
\(122\) −1.25361e7 −0.625035
\(123\) −1.11620e6 −0.0540847
\(124\) 909760. 0.0428500
\(125\) −1.95312e6 −0.0894427
\(126\) −3.95403e6 −0.176093
\(127\) −1.87557e7 −0.812492 −0.406246 0.913764i \(-0.633162\pi\)
−0.406246 + 0.913764i \(0.633162\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −4.88381e6 −0.200306
\(130\) −6.01282e6 −0.240036
\(131\) −5.42794e6 −0.210953 −0.105477 0.994422i \(-0.533637\pi\)
−0.105477 + 0.994422i \(0.533637\pi\)
\(132\) −3.29931e6 −0.124858
\(133\) −4.65032e6 −0.171397
\(134\) −3.27999e7 −1.17762
\(135\) −2.46038e6 −0.0860663
\(136\) −4.11582e6 −0.140304
\(137\) 2.11555e7 0.702913 0.351457 0.936204i \(-0.385687\pi\)
0.351457 + 0.936204i \(0.385687\pi\)
\(138\) 8.49649e6 0.275209
\(139\) −1.51956e7 −0.479916 −0.239958 0.970783i \(-0.577134\pi\)
−0.239958 + 0.970783i \(0.577134\pi\)
\(140\) 5.42390e6 0.167057
\(141\) −2.11168e7 −0.634397
\(142\) 3.00078e7 0.879477
\(143\) −1.14804e7 −0.328309
\(144\) 2.98598e6 0.0833333
\(145\) 1.14944e7 0.313112
\(146\) −2.20052e6 −0.0585181
\(147\) −9.82463e6 −0.255097
\(148\) −97331.3 −0.00246795
\(149\) −5.69981e7 −1.41159 −0.705794 0.708417i \(-0.749410\pi\)
−0.705794 + 0.708417i \(0.749410\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) 4.34796e7 1.02770 0.513850 0.857880i \(-0.328219\pi\)
0.513850 + 0.857880i \(0.328219\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) −5.86022e6 −0.132280
\(154\) 1.03560e7 0.228491
\(155\) −1.77688e6 −0.0383262
\(156\) 1.03902e7 0.219122
\(157\) −2.92610e7 −0.603448 −0.301724 0.953395i \(-0.597562\pi\)
−0.301724 + 0.953395i \(0.597562\pi\)
\(158\) −3.06866e7 −0.618941
\(159\) 3.90650e7 0.770722
\(160\) −4.09600e6 −0.0790569
\(161\) −2.66691e7 −0.503636
\(162\) 4.25153e6 0.0785674
\(163\) 3.54719e7 0.641546 0.320773 0.947156i \(-0.396057\pi\)
0.320773 + 0.947156i \(0.396057\pi\)
\(164\) −2.64581e6 −0.0468387
\(165\) 6.44397e6 0.111676
\(166\) −6.84862e7 −1.16205
\(167\) −1.91573e7 −0.318293 −0.159146 0.987255i \(-0.550874\pi\)
−0.159146 + 0.987255i \(0.550874\pi\)
\(168\) −9.37251e6 −0.152501
\(169\) −2.65945e7 −0.423827
\(170\) 8.03871e6 0.125492
\(171\) 5.00021e6 0.0764719
\(172\) −1.15764e7 −0.173470
\(173\) 9.37335e7 1.37636 0.688182 0.725538i \(-0.258409\pi\)
0.688182 + 0.725538i \(0.258409\pi\)
\(174\) −1.98624e7 −0.285831
\(175\) −1.05936e7 −0.149420
\(176\) −7.82060e6 −0.108130
\(177\) 6.34914e6 0.0860614
\(178\) −9.66567e7 −1.28458
\(179\) 9.43067e7 1.22901 0.614507 0.788911i \(-0.289355\pi\)
0.614507 + 0.788911i \(0.289355\pi\)
\(180\) −5.83200e6 −0.0745356
\(181\) 2.54494e7 0.319008 0.159504 0.987197i \(-0.449010\pi\)
0.159504 + 0.987197i \(0.449010\pi\)
\(182\) −3.26130e7 −0.400996
\(183\) −4.23094e7 −0.510339
\(184\) 2.01398e7 0.238338
\(185\) 190100. 0.00220740
\(186\) 3.07044e6 0.0349869
\(187\) 1.53485e7 0.171641
\(188\) −5.00546e7 −0.549404
\(189\) −1.33448e7 −0.143780
\(190\) −6.85900e6 −0.0725476
\(191\) 1.34906e8 1.40092 0.700460 0.713692i \(-0.252978\pi\)
0.700460 + 0.713692i \(0.252978\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −6.82785e7 −0.683649 −0.341825 0.939764i \(-0.611045\pi\)
−0.341825 + 0.939764i \(0.611045\pi\)
\(194\) −3.38561e7 −0.332913
\(195\) −2.02933e7 −0.195989
\(196\) −2.32880e7 −0.220921
\(197\) −1.91863e8 −1.78797 −0.893984 0.448100i \(-0.852101\pi\)
−0.893984 + 0.448100i \(0.852101\pi\)
\(198\) −1.11352e7 −0.101946
\(199\) −6.61263e7 −0.594824 −0.297412 0.954749i \(-0.596124\pi\)
−0.297412 + 0.954749i \(0.596124\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) −1.10700e8 −0.961524
\(202\) 1.04222e8 0.889668
\(203\) 6.23447e7 0.523075
\(204\) −1.38909e7 −0.114558
\(205\) 5.16759e6 0.0418938
\(206\) 8.13262e7 0.648180
\(207\) 2.86756e7 0.224707
\(208\) 2.46285e7 0.189765
\(209\) −1.30961e7 −0.0992268
\(210\) 1.83057e7 0.136401
\(211\) −5.18257e7 −0.379802 −0.189901 0.981803i \(-0.560817\pi\)
−0.189901 + 0.981803i \(0.560817\pi\)
\(212\) 9.25985e7 0.667465
\(213\) 1.01276e8 0.718090
\(214\) −1.50114e8 −1.04706
\(215\) 2.26102e7 0.155157
\(216\) 1.00777e7 0.0680414
\(217\) −9.63760e6 −0.0640265
\(218\) 2.36485e6 0.0154599
\(219\) −7.42676e6 −0.0477798
\(220\) 1.52746e7 0.0967143
\(221\) −4.83353e7 −0.301225
\(222\) −328493. −0.00201507
\(223\) −2.22035e7 −0.134077 −0.0670385 0.997750i \(-0.521355\pi\)
−0.0670385 + 0.997750i \(0.521355\pi\)
\(224\) −2.22163e7 −0.132070
\(225\) 1.13906e7 0.0666667
\(226\) −1.86295e8 −1.07355
\(227\) 9.17879e7 0.520829 0.260414 0.965497i \(-0.416141\pi\)
0.260414 + 0.965497i \(0.416141\pi\)
\(228\) 1.18524e7 0.0662266
\(229\) −2.34716e8 −1.29157 −0.645786 0.763519i \(-0.723470\pi\)
−0.645786 + 0.763519i \(0.723470\pi\)
\(230\) −3.93356e7 −0.213176
\(231\) 3.49515e7 0.186562
\(232\) −4.70812e7 −0.247537
\(233\) 3.16284e8 1.63806 0.819032 0.573748i \(-0.194511\pi\)
0.819032 + 0.573748i \(0.194511\pi\)
\(234\) 3.50668e7 0.178912
\(235\) 9.77629e7 0.491402
\(236\) 1.50498e7 0.0745313
\(237\) −1.03567e8 −0.505363
\(238\) 4.36012e7 0.209642
\(239\) −1.97926e8 −0.937798 −0.468899 0.883252i \(-0.655349\pi\)
−0.468899 + 0.883252i \(0.655349\pi\)
\(240\) −1.38240e7 −0.0645497
\(241\) −9.41243e6 −0.0433154 −0.0216577 0.999765i \(-0.506894\pi\)
−0.0216577 + 0.999765i \(0.506894\pi\)
\(242\) −1.26733e8 −0.574826
\(243\) 1.43489e7 0.0641500
\(244\) −1.00289e8 −0.441966
\(245\) 4.54844e7 0.197597
\(246\) −8.92960e6 −0.0382436
\(247\) 4.12419e7 0.174140
\(248\) 7.27808e6 0.0302995
\(249\) −2.31141e8 −0.948810
\(250\) −1.56250e7 −0.0632456
\(251\) 7.66045e7 0.305771 0.152886 0.988244i \(-0.451143\pi\)
0.152886 + 0.988244i \(0.451143\pi\)
\(252\) −3.16322e7 −0.124517
\(253\) −7.51044e7 −0.291571
\(254\) −1.50045e8 −0.574519
\(255\) 2.71306e7 0.102464
\(256\) 1.67772e7 0.0625000
\(257\) 4.45670e8 1.63775 0.818875 0.573972i \(-0.194598\pi\)
0.818875 + 0.573972i \(0.194598\pi\)
\(258\) −3.90705e7 −0.141638
\(259\) 1.03109e6 0.00368761
\(260\) −4.81026e7 −0.169731
\(261\) −6.70356e7 −0.233380
\(262\) −4.34235e7 −0.149166
\(263\) −5.32723e8 −1.80574 −0.902872 0.429910i \(-0.858545\pi\)
−0.902872 + 0.429910i \(0.858545\pi\)
\(264\) −2.63945e7 −0.0882876
\(265\) −1.80856e8 −0.596999
\(266\) −3.72026e7 −0.121196
\(267\) −3.26216e8 −1.04886
\(268\) −2.62399e8 −0.832704
\(269\) −3.33760e8 −1.04545 −0.522723 0.852502i \(-0.675084\pi\)
−0.522723 + 0.852502i \(0.675084\pi\)
\(270\) −1.96830e7 −0.0608581
\(271\) 6.42931e7 0.196233 0.0981164 0.995175i \(-0.468718\pi\)
0.0981164 + 0.995175i \(0.468718\pi\)
\(272\) −3.29265e7 −0.0992099
\(273\) −1.10069e8 −0.327412
\(274\) 1.69244e8 0.497035
\(275\) −2.98332e7 −0.0865039
\(276\) 6.79719e7 0.194602
\(277\) 2.93971e8 0.831047 0.415524 0.909582i \(-0.363598\pi\)
0.415524 + 0.909582i \(0.363598\pi\)
\(278\) −1.21565e8 −0.339352
\(279\) 1.03627e7 0.0285667
\(280\) 4.33912e7 0.118127
\(281\) −2.51358e8 −0.675803 −0.337901 0.941182i \(-0.609717\pi\)
−0.337901 + 0.941182i \(0.609717\pi\)
\(282\) −1.68934e8 −0.448586
\(283\) 5.80527e8 1.52254 0.761272 0.648433i \(-0.224575\pi\)
0.761272 + 0.648433i \(0.224575\pi\)
\(284\) 2.40062e8 0.621884
\(285\) −2.31491e7 −0.0592349
\(286\) −9.18435e7 −0.232149
\(287\) 2.80285e7 0.0699864
\(288\) 2.38879e7 0.0589256
\(289\) −3.45718e8 −0.842518
\(290\) 9.19555e7 0.221404
\(291\) −1.14264e8 −0.271822
\(292\) −1.76042e7 −0.0413786
\(293\) −6.98957e7 −0.162336 −0.0811678 0.996700i \(-0.525865\pi\)
−0.0811678 + 0.996700i \(0.525865\pi\)
\(294\) −7.85971e7 −0.180381
\(295\) −2.93942e7 −0.0666628
\(296\) −778651. −0.00174511
\(297\) −3.75813e7 −0.0832384
\(298\) −4.55985e8 −0.998144
\(299\) 2.36518e8 0.511699
\(300\) 2.70000e7 0.0577350
\(301\) 1.22636e8 0.259200
\(302\) 3.47837e8 0.726693
\(303\) 3.51748e8 0.726411
\(304\) 2.80945e7 0.0573539
\(305\) 1.95877e8 0.395307
\(306\) −4.68817e7 −0.0935360
\(307\) −2.24066e8 −0.441969 −0.220985 0.975277i \(-0.570927\pi\)
−0.220985 + 0.975277i \(0.570927\pi\)
\(308\) 8.28480e7 0.161568
\(309\) 2.74476e8 0.529236
\(310\) −1.42150e7 −0.0271007
\(311\) −6.03655e6 −0.0113796 −0.00568981 0.999984i \(-0.501811\pi\)
−0.00568981 + 0.999984i \(0.501811\pi\)
\(312\) 8.31212e7 0.154943
\(313\) 6.43097e7 0.118542 0.0592709 0.998242i \(-0.481122\pi\)
0.0592709 + 0.998242i \(0.481122\pi\)
\(314\) −2.34088e8 −0.426702
\(315\) 6.17817e7 0.111371
\(316\) −2.45493e8 −0.437657
\(317\) −3.70653e8 −0.653521 −0.326761 0.945107i \(-0.605957\pi\)
−0.326761 + 0.945107i \(0.605957\pi\)
\(318\) 3.12520e8 0.544983
\(319\) 1.75573e8 0.302824
\(320\) −3.27680e7 −0.0559017
\(321\) −5.06634e8 −0.854922
\(322\) −2.13352e8 −0.356125
\(323\) −5.51375e7 −0.0910412
\(324\) 3.40122e7 0.0555556
\(325\) 9.39503e7 0.151812
\(326\) 2.83775e8 0.453642
\(327\) 7.98138e6 0.0126229
\(328\) −2.11665e7 −0.0331200
\(329\) 5.30257e8 0.820919
\(330\) 5.15518e7 0.0789669
\(331\) 3.54795e7 0.0537749 0.0268875 0.999638i \(-0.491440\pi\)
0.0268875 + 0.999638i \(0.491440\pi\)
\(332\) −5.47890e8 −0.821694
\(333\) −1.10866e6 −0.00164530
\(334\) −1.53258e8 −0.225067
\(335\) 5.12498e8 0.744793
\(336\) −7.49800e7 −0.107835
\(337\) −5.58314e8 −0.794646 −0.397323 0.917679i \(-0.630061\pi\)
−0.397323 + 0.917679i \(0.630061\pi\)
\(338\) −2.12756e8 −0.299691
\(339\) −6.28746e8 −0.876550
\(340\) 6.43096e7 0.0887360
\(341\) −2.71411e7 −0.0370669
\(342\) 4.00017e7 0.0540738
\(343\) 8.05055e8 1.07720
\(344\) −9.26115e7 −0.122662
\(345\) −1.32758e8 −0.174057
\(346\) 7.49868e8 0.973237
\(347\) −1.11678e9 −1.43488 −0.717439 0.696621i \(-0.754686\pi\)
−0.717439 + 0.696621i \(0.754686\pi\)
\(348\) −1.58899e8 −0.202113
\(349\) −1.10534e8 −0.139190 −0.0695951 0.997575i \(-0.522171\pi\)
−0.0695951 + 0.997575i \(0.522171\pi\)
\(350\) −8.47485e7 −0.105656
\(351\) 1.18350e8 0.146081
\(352\) −6.25648e7 −0.0764593
\(353\) 1.48318e8 0.179466 0.0897329 0.995966i \(-0.471399\pi\)
0.0897329 + 0.995966i \(0.471399\pi\)
\(354\) 5.07931e7 0.0608546
\(355\) −4.68871e8 −0.556230
\(356\) −7.73253e8 −0.908337
\(357\) 1.47154e8 0.171172
\(358\) 7.54454e8 0.869044
\(359\) −1.08840e9 −1.24154 −0.620768 0.783995i \(-0.713179\pi\)
−0.620768 + 0.783995i \(0.713179\pi\)
\(360\) −4.66560e7 −0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) 2.03595e8 0.225573
\(363\) −4.27724e8 −0.469344
\(364\) −2.60904e8 −0.283547
\(365\) 3.43831e7 0.0370101
\(366\) −3.38476e8 −0.360864
\(367\) −1.75870e7 −0.0185721 −0.00928603 0.999957i \(-0.502956\pi\)
−0.00928603 + 0.999957i \(0.502956\pi\)
\(368\) 1.61119e8 0.168530
\(369\) −3.01374e7 −0.0312258
\(370\) 1.52080e6 0.00156087
\(371\) −9.80948e8 −0.997326
\(372\) 2.45635e7 0.0247395
\(373\) −1.26160e9 −1.25876 −0.629379 0.777098i \(-0.716691\pi\)
−0.629379 + 0.777098i \(0.716691\pi\)
\(374\) 1.22788e8 0.121368
\(375\) −5.27344e7 −0.0516398
\(376\) −4.00437e8 −0.388487
\(377\) −5.52912e8 −0.531449
\(378\) −1.06759e8 −0.101667
\(379\) 1.30499e9 1.23132 0.615660 0.788012i \(-0.288890\pi\)
0.615660 + 0.788012i \(0.288890\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) −5.06403e8 −0.469093
\(382\) 1.07924e9 0.990600
\(383\) −9.84769e8 −0.895651 −0.447825 0.894121i \(-0.647801\pi\)
−0.447825 + 0.894121i \(0.647801\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −1.61812e8 −0.144511
\(386\) −5.46228e8 −0.483413
\(387\) −1.31863e8 −0.115647
\(388\) −2.70849e8 −0.235405
\(389\) 8.49975e8 0.732121 0.366060 0.930591i \(-0.380706\pi\)
0.366060 + 0.930591i \(0.380706\pi\)
\(390\) −1.62346e8 −0.138585
\(391\) −3.16207e8 −0.267518
\(392\) −1.86304e8 −0.156214
\(393\) −1.46554e8 −0.121794
\(394\) −1.53490e9 −1.26428
\(395\) 4.79478e8 0.391453
\(396\) −8.90815e7 −0.0720866
\(397\) 2.60959e7 0.0209318 0.0104659 0.999945i \(-0.496669\pi\)
0.0104659 + 0.999945i \(0.496669\pi\)
\(398\) −5.29011e8 −0.420604
\(399\) −1.25559e8 −0.0989558
\(400\) 6.40000e7 0.0500000
\(401\) 6.41330e8 0.496680 0.248340 0.968673i \(-0.420115\pi\)
0.248340 + 0.968673i \(0.420115\pi\)
\(402\) −8.85597e8 −0.679900
\(403\) 8.54723e7 0.0650515
\(404\) 8.33773e8 0.629091
\(405\) −6.64301e7 −0.0496904
\(406\) 4.98758e8 0.369870
\(407\) 2.90371e6 0.00213487
\(408\) −1.11127e8 −0.0810045
\(409\) 2.15806e9 1.55967 0.779835 0.625985i \(-0.215303\pi\)
0.779835 + 0.625985i \(0.215303\pi\)
\(410\) 4.13407e7 0.0296234
\(411\) 5.71199e8 0.405827
\(412\) 6.50610e8 0.458332
\(413\) −1.59431e8 −0.111365
\(414\) 2.29405e8 0.158892
\(415\) 1.07010e9 0.734945
\(416\) 1.97028e8 0.134184
\(417\) −4.10280e8 −0.277079
\(418\) −1.04769e8 −0.0701639
\(419\) −9.22754e8 −0.612825 −0.306413 0.951899i \(-0.599129\pi\)
−0.306413 + 0.951899i \(0.599129\pi\)
\(420\) 1.46445e8 0.0964502
\(421\) 2.23810e9 1.46181 0.730907 0.682478i \(-0.239098\pi\)
0.730907 + 0.682478i \(0.239098\pi\)
\(422\) −4.14606e8 −0.268560
\(423\) −5.70153e8 −0.366269
\(424\) 7.40788e8 0.471969
\(425\) −1.25605e8 −0.0793679
\(426\) 8.10210e8 0.507766
\(427\) 1.06242e9 0.660386
\(428\) −1.20091e9 −0.740384
\(429\) −3.09972e8 −0.189549
\(430\) 1.80882e8 0.109712
\(431\) −8.65471e8 −0.520694 −0.260347 0.965515i \(-0.583837\pi\)
−0.260347 + 0.965515i \(0.583837\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −1.89196e9 −1.11996 −0.559982 0.828505i \(-0.689192\pi\)
−0.559982 + 0.828505i \(0.689192\pi\)
\(434\) −7.71008e7 −0.0452736
\(435\) 3.10350e8 0.180775
\(436\) 1.89188e7 0.0109318
\(437\) 2.69803e8 0.154654
\(438\) −5.94140e7 −0.0337855
\(439\) −3.18378e9 −1.79605 −0.898023 0.439949i \(-0.854996\pi\)
−0.898023 + 0.439949i \(0.854996\pi\)
\(440\) 1.22197e8 0.0683873
\(441\) −2.65265e8 −0.147280
\(442\) −3.86682e8 −0.212998
\(443\) 1.58476e9 0.866066 0.433033 0.901378i \(-0.357443\pi\)
0.433033 + 0.901378i \(0.357443\pi\)
\(444\) −2.62795e6 −0.00142487
\(445\) 1.51026e9 0.812441
\(446\) −1.77628e8 −0.0948067
\(447\) −1.53895e9 −0.814981
\(448\) −1.77730e8 −0.0933875
\(449\) 1.90931e9 0.995437 0.497718 0.867339i \(-0.334171\pi\)
0.497718 + 0.867339i \(0.334171\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 7.89329e7 0.0405173
\(452\) −1.49036e9 −0.759114
\(453\) 1.17395e9 0.593343
\(454\) 7.34303e8 0.368281
\(455\) 5.09578e8 0.253612
\(456\) 9.48188e7 0.0468293
\(457\) 2.13286e9 1.04534 0.522668 0.852536i \(-0.324937\pi\)
0.522668 + 0.852536i \(0.324937\pi\)
\(458\) −1.87773e9 −0.913279
\(459\) −1.58226e8 −0.0763718
\(460\) −3.14685e8 −0.150738
\(461\) −6.19615e7 −0.0294556 −0.0147278 0.999892i \(-0.504688\pi\)
−0.0147278 + 0.999892i \(0.504688\pi\)
\(462\) 2.79612e8 0.131919
\(463\) −1.84782e9 −0.865218 −0.432609 0.901582i \(-0.642407\pi\)
−0.432609 + 0.901582i \(0.642407\pi\)
\(464\) −3.76650e8 −0.175035
\(465\) −4.79756e7 −0.0221277
\(466\) 2.53027e9 1.15829
\(467\) 5.47593e8 0.248799 0.124399 0.992232i \(-0.460300\pi\)
0.124399 + 0.992232i \(0.460300\pi\)
\(468\) 2.80534e8 0.126510
\(469\) 2.77974e9 1.24423
\(470\) 7.82103e8 0.347474
\(471\) −7.90046e8 −0.348401
\(472\) 1.20398e8 0.0527016
\(473\) 3.45362e8 0.150059
\(474\) −8.28539e8 −0.357346
\(475\) 1.07172e8 0.0458831
\(476\) 3.48809e8 0.148239
\(477\) 1.05475e9 0.444977
\(478\) −1.58340e9 −0.663123
\(479\) 3.01854e9 1.25494 0.627470 0.778641i \(-0.284090\pi\)
0.627470 + 0.778641i \(0.284090\pi\)
\(480\) −1.10592e8 −0.0456435
\(481\) −9.14431e6 −0.00374665
\(482\) −7.52995e7 −0.0306286
\(483\) −7.20064e8 −0.290775
\(484\) −1.01387e9 −0.406463
\(485\) 5.29001e8 0.210553
\(486\) 1.14791e8 0.0453609
\(487\) 1.19500e8 0.0468831 0.0234416 0.999725i \(-0.492538\pi\)
0.0234416 + 0.999725i \(0.492538\pi\)
\(488\) −8.02312e8 −0.312517
\(489\) 9.57742e8 0.370397
\(490\) 3.63875e8 0.139722
\(491\) 5.15480e9 1.96529 0.982644 0.185501i \(-0.0593907\pi\)
0.982644 + 0.185501i \(0.0593907\pi\)
\(492\) −7.14368e7 −0.0270423
\(493\) 7.39203e8 0.277843
\(494\) 3.29936e8 0.123136
\(495\) 1.73987e8 0.0644762
\(496\) 5.82247e7 0.0214250
\(497\) −2.54311e9 −0.929220
\(498\) −1.84913e9 −0.670910
\(499\) −2.28035e9 −0.821578 −0.410789 0.911730i \(-0.634747\pi\)
−0.410789 + 0.911730i \(0.634747\pi\)
\(500\) −1.25000e8 −0.0447214
\(501\) −5.17247e8 −0.183766
\(502\) 6.12836e8 0.216213
\(503\) −3.05998e8 −0.107209 −0.0536044 0.998562i \(-0.517071\pi\)
−0.0536044 + 0.998562i \(0.517071\pi\)
\(504\) −2.53058e8 −0.0880466
\(505\) −1.62846e9 −0.562676
\(506\) −6.00836e8 −0.206172
\(507\) −7.18051e8 −0.244696
\(508\) −1.20036e9 −0.406246
\(509\) −4.39986e9 −1.47886 −0.739429 0.673235i \(-0.764904\pi\)
−0.739429 + 0.673235i \(0.764904\pi\)
\(510\) 2.17045e8 0.0724526
\(511\) 1.86491e8 0.0618279
\(512\) 1.34218e8 0.0441942
\(513\) 1.35006e8 0.0441511
\(514\) 3.56536e9 1.15806
\(515\) −1.27072e9 −0.409945
\(516\) −3.12564e8 −0.100153
\(517\) 1.49329e9 0.475256
\(518\) 8.24868e6 0.00260754
\(519\) 2.53080e9 0.794644
\(520\) −3.84821e8 −0.120018
\(521\) 9.87814e8 0.306015 0.153008 0.988225i \(-0.451104\pi\)
0.153008 + 0.988225i \(0.451104\pi\)
\(522\) −5.36285e8 −0.165025
\(523\) 9.73811e8 0.297659 0.148829 0.988863i \(-0.452449\pi\)
0.148829 + 0.988863i \(0.452449\pi\)
\(524\) −3.47388e8 −0.105477
\(525\) −2.86026e8 −0.0862677
\(526\) −4.26178e9 −1.27685
\(527\) −1.14270e8 −0.0340092
\(528\) −2.11156e8 −0.0624288
\(529\) −1.85754e9 −0.545560
\(530\) −1.44685e9 −0.422142
\(531\) 1.71427e8 0.0496876
\(532\) −2.97620e8 −0.0856983
\(533\) −2.48574e8 −0.0711068
\(534\) −2.60973e9 −0.741654
\(535\) 2.34553e9 0.662219
\(536\) −2.09919e9 −0.588811
\(537\) 2.54628e9 0.709572
\(538\) −2.67008e9 −0.739242
\(539\) 6.94756e8 0.191105
\(540\) −1.57464e8 −0.0430331
\(541\) −4.23358e9 −1.14952 −0.574761 0.818321i \(-0.694905\pi\)
−0.574761 + 0.818321i \(0.694905\pi\)
\(542\) 5.14345e8 0.138758
\(543\) 6.87133e8 0.184180
\(544\) −2.63412e8 −0.0701520
\(545\) −3.69508e7 −0.00977769
\(546\) −8.80550e8 −0.231515
\(547\) 1.07096e9 0.279782 0.139891 0.990167i \(-0.455325\pi\)
0.139891 + 0.990167i \(0.455325\pi\)
\(548\) 1.35395e9 0.351457
\(549\) −1.14235e9 −0.294644
\(550\) −2.38666e8 −0.0611675
\(551\) −6.30723e8 −0.160623
\(552\) 5.43775e8 0.137604
\(553\) 2.60064e9 0.653948
\(554\) 2.35177e9 0.587639
\(555\) 5.13271e6 0.00127444
\(556\) −9.72516e8 −0.239958
\(557\) 1.69046e9 0.414487 0.207243 0.978289i \(-0.433551\pi\)
0.207243 + 0.978289i \(0.433551\pi\)
\(558\) 8.29019e7 0.0201997
\(559\) −1.08761e9 −0.263349
\(560\) 3.47130e8 0.0835283
\(561\) 4.14410e8 0.0990968
\(562\) −2.01086e9 −0.477865
\(563\) −7.93600e9 −1.87423 −0.937113 0.349026i \(-0.886513\pi\)
−0.937113 + 0.349026i \(0.886513\pi\)
\(564\) −1.35147e9 −0.317198
\(565\) 2.91086e9 0.678973
\(566\) 4.64422e9 1.07660
\(567\) −3.60311e8 −0.0830111
\(568\) 1.92050e9 0.439739
\(569\) 4.83397e9 1.10005 0.550024 0.835149i \(-0.314619\pi\)
0.550024 + 0.835149i \(0.314619\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) −2.11248e9 −0.474861 −0.237430 0.971405i \(-0.576305\pi\)
−0.237430 + 0.971405i \(0.576305\pi\)
\(572\) −7.34748e8 −0.164154
\(573\) 3.64245e9 0.808821
\(574\) 2.24228e8 0.0494879
\(575\) 6.14619e8 0.134824
\(576\) 1.91103e8 0.0416667
\(577\) −4.06100e9 −0.880070 −0.440035 0.897981i \(-0.645034\pi\)
−0.440035 + 0.897981i \(0.645034\pi\)
\(578\) −2.76574e9 −0.595750
\(579\) −1.84352e9 −0.394705
\(580\) 7.35644e8 0.156556
\(581\) 5.80411e9 1.22778
\(582\) −9.14114e8 −0.192207
\(583\) −2.76251e9 −0.577383
\(584\) −1.40833e8 −0.0292591
\(585\) −5.47918e8 −0.113154
\(586\) −5.59165e8 −0.114789
\(587\) 1.49589e9 0.305257 0.152628 0.988284i \(-0.451226\pi\)
0.152628 + 0.988284i \(0.451226\pi\)
\(588\) −6.28777e8 −0.127549
\(589\) 9.75007e7 0.0196609
\(590\) −2.35153e8 −0.0471378
\(591\) −5.18030e9 −1.03228
\(592\) −6.22920e6 −0.00123398
\(593\) 4.28641e8 0.0844116 0.0422058 0.999109i \(-0.486561\pi\)
0.0422058 + 0.999109i \(0.486561\pi\)
\(594\) −3.00650e8 −0.0588584
\(595\) −6.81268e8 −0.132589
\(596\) −3.64788e9 −0.705794
\(597\) −1.78541e9 −0.343422
\(598\) 1.89214e9 0.361826
\(599\) 9.27348e9 1.76299 0.881493 0.472197i \(-0.156539\pi\)
0.881493 + 0.472197i \(0.156539\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) 4.15328e9 0.780424 0.390212 0.920725i \(-0.372402\pi\)
0.390212 + 0.920725i \(0.372402\pi\)
\(602\) 9.81086e8 0.183282
\(603\) −2.98889e9 −0.555136
\(604\) 2.78269e9 0.513850
\(605\) 1.98021e9 0.363552
\(606\) 2.81398e9 0.513650
\(607\) 6.43465e9 1.16779 0.583895 0.811829i \(-0.301528\pi\)
0.583895 + 0.811829i \(0.301528\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 1.68331e9 0.301997
\(610\) 1.56702e9 0.279524
\(611\) −4.70265e9 −0.834062
\(612\) −3.75054e8 −0.0661399
\(613\) −3.92088e9 −0.687498 −0.343749 0.939062i \(-0.611697\pi\)
−0.343749 + 0.939062i \(0.611697\pi\)
\(614\) −1.79253e9 −0.312520
\(615\) 1.39525e8 0.0241874
\(616\) 6.62784e8 0.114246
\(617\) 1.83220e9 0.314032 0.157016 0.987596i \(-0.449813\pi\)
0.157016 + 0.987596i \(0.449813\pi\)
\(618\) 2.19581e9 0.374227
\(619\) 3.24063e9 0.549177 0.274588 0.961562i \(-0.411458\pi\)
0.274588 + 0.961562i \(0.411458\pi\)
\(620\) −1.13720e8 −0.0191631
\(621\) 7.74242e8 0.129735
\(622\) −4.82924e7 −0.00804660
\(623\) 8.19151e9 1.35724
\(624\) 6.64970e8 0.109561
\(625\) 2.44141e8 0.0400000
\(626\) 5.14478e8 0.0838217
\(627\) −3.53594e8 −0.0572886
\(628\) −1.87270e9 −0.301724
\(629\) 1.22253e7 0.00195876
\(630\) 4.94253e8 0.0787513
\(631\) 2.92548e9 0.463547 0.231773 0.972770i \(-0.425547\pi\)
0.231773 + 0.972770i \(0.425547\pi\)
\(632\) −1.96394e9 −0.309470
\(633\) −1.39929e9 −0.219279
\(634\) −2.96522e9 −0.462109
\(635\) 2.34446e9 0.363358
\(636\) 2.50016e9 0.385361
\(637\) −2.18792e9 −0.335384
\(638\) 1.40458e9 0.214129
\(639\) 2.73446e9 0.414589
\(640\) −2.62144e8 −0.0395285
\(641\) 5.41303e9 0.811778 0.405889 0.913922i \(-0.366962\pi\)
0.405889 + 0.913922i \(0.366962\pi\)
\(642\) −4.05307e9 −0.604521
\(643\) 1.03186e10 1.53067 0.765335 0.643632i \(-0.222573\pi\)
0.765335 + 0.643632i \(0.222573\pi\)
\(644\) −1.70682e9 −0.251818
\(645\) 6.10476e8 0.0895798
\(646\) −4.41100e8 −0.0643759
\(647\) 3.63563e9 0.527733 0.263867 0.964559i \(-0.415002\pi\)
0.263867 + 0.964559i \(0.415002\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −4.48984e8 −0.0644725
\(650\) 7.51603e8 0.107347
\(651\) −2.60215e8 −0.0369657
\(652\) 2.27020e9 0.320773
\(653\) 1.19529e10 1.67988 0.839939 0.542680i \(-0.182590\pi\)
0.839939 + 0.542680i \(0.182590\pi\)
\(654\) 6.38510e7 0.00892577
\(655\) 6.78493e8 0.0943411
\(656\) −1.69332e8 −0.0234193
\(657\) −2.00522e8 −0.0275857
\(658\) 4.24205e9 0.580478
\(659\) 1.35429e9 0.184337 0.0921686 0.995743i \(-0.470620\pi\)
0.0921686 + 0.995743i \(0.470620\pi\)
\(660\) 4.12414e8 0.0558380
\(661\) −2.19950e9 −0.296223 −0.148111 0.988971i \(-0.547319\pi\)
−0.148111 + 0.988971i \(0.547319\pi\)
\(662\) 2.83836e8 0.0380246
\(663\) −1.30505e9 −0.173913
\(664\) −4.38312e9 −0.581025
\(665\) 5.81290e8 0.0766509
\(666\) −8.86932e6 −0.00116340
\(667\) −3.61712e9 −0.471979
\(668\) −1.22607e9 −0.159146
\(669\) −5.99494e8 −0.0774093
\(670\) 4.09999e9 0.526648
\(671\) 2.99194e9 0.382318
\(672\) −5.99840e8 −0.0762506
\(673\) 1.18276e10 1.49570 0.747850 0.663868i \(-0.231086\pi\)
0.747850 + 0.663868i \(0.231086\pi\)
\(674\) −4.46651e9 −0.561900
\(675\) 3.07547e8 0.0384900
\(676\) −1.70205e9 −0.211913
\(677\) 6.48567e9 0.803331 0.401665 0.915787i \(-0.368432\pi\)
0.401665 + 0.915787i \(0.368432\pi\)
\(678\) −5.02997e9 −0.619814
\(679\) 2.86925e9 0.351742
\(680\) 5.14477e8 0.0627458
\(681\) 2.47827e9 0.300701
\(682\) −2.17129e8 −0.0262103
\(683\) 1.06736e10 1.28186 0.640929 0.767600i \(-0.278549\pi\)
0.640929 + 0.767600i \(0.278549\pi\)
\(684\) 3.20014e8 0.0382360
\(685\) −2.64444e9 −0.314352
\(686\) 6.44044e9 0.761695
\(687\) −6.33733e9 −0.745689
\(688\) −7.40892e8 −0.0867352
\(689\) 8.69966e9 1.01329
\(690\) −1.06206e9 −0.123077
\(691\) 2.06366e9 0.237939 0.118969 0.992898i \(-0.462041\pi\)
0.118969 + 0.992898i \(0.462041\pi\)
\(692\) 5.99894e9 0.688182
\(693\) 9.43690e8 0.107712
\(694\) −8.93425e9 −1.01461
\(695\) 1.89945e9 0.214625
\(696\) −1.27119e9 −0.142915
\(697\) 3.32326e8 0.0371749
\(698\) −8.84276e8 −0.0984223
\(699\) 8.53966e9 0.945737
\(700\) −6.77988e8 −0.0747100
\(701\) 1.07951e10 1.18362 0.591810 0.806078i \(-0.298414\pi\)
0.591810 + 0.806078i \(0.298414\pi\)
\(702\) 9.46803e8 0.103295
\(703\) −1.04312e7 −0.00113237
\(704\) −5.00518e8 −0.0540649
\(705\) 2.63960e9 0.283711
\(706\) 1.18654e9 0.126901
\(707\) −8.83263e9 −0.939987
\(708\) 4.06345e8 0.0430307
\(709\) 6.74599e9 0.710860 0.355430 0.934703i \(-0.384335\pi\)
0.355430 + 0.934703i \(0.384335\pi\)
\(710\) −3.75097e9 −0.393314
\(711\) −2.79632e9 −0.291772
\(712\) −6.18603e9 −0.642291
\(713\) 5.59155e8 0.0577722
\(714\) 1.17723e9 0.121037
\(715\) 1.43505e9 0.146824
\(716\) 6.03563e9 0.614507
\(717\) −5.34399e9 −0.541438
\(718\) −8.70722e9 −0.877898
\(719\) 1.38406e10 1.38869 0.694344 0.719643i \(-0.255695\pi\)
0.694344 + 0.719643i \(0.255695\pi\)
\(720\) −3.73248e8 −0.0372678
\(721\) −6.89228e9 −0.684840
\(722\) 3.76367e8 0.0372161
\(723\) −2.54136e8 −0.0250082
\(724\) 1.62876e9 0.159504
\(725\) −1.43681e9 −0.140028
\(726\) −3.42180e9 −0.331876
\(727\) 5.52113e9 0.532915 0.266457 0.963847i \(-0.414147\pi\)
0.266457 + 0.963847i \(0.414147\pi\)
\(728\) −2.08723e9 −0.200498
\(729\) 3.87420e8 0.0370370
\(730\) 2.75065e8 0.0261701
\(731\) 1.45406e9 0.137680
\(732\) −2.70780e9 −0.255169
\(733\) −6.01727e9 −0.564333 −0.282166 0.959365i \(-0.591053\pi\)
−0.282166 + 0.959365i \(0.591053\pi\)
\(734\) −1.40696e8 −0.0131324
\(735\) 1.22808e9 0.114083
\(736\) 1.28895e9 0.119169
\(737\) 7.82821e9 0.720321
\(738\) −2.41099e8 −0.0220800
\(739\) −4.20651e9 −0.383413 −0.191706 0.981452i \(-0.561402\pi\)
−0.191706 + 0.981452i \(0.561402\pi\)
\(740\) 1.21664e7 0.00110370
\(741\) 1.11353e9 0.100540
\(742\) −7.84758e9 −0.705216
\(743\) 2.06456e9 0.184658 0.0923288 0.995729i \(-0.470569\pi\)
0.0923288 + 0.995729i \(0.470569\pi\)
\(744\) 1.96508e8 0.0174934
\(745\) 7.12476e9 0.631282
\(746\) −1.00928e10 −0.890077
\(747\) −6.24081e9 −0.547796
\(748\) 9.82304e8 0.0858204
\(749\) 1.27219e10 1.10628
\(750\) −4.21875e8 −0.0365148
\(751\) −8.64553e9 −0.744821 −0.372410 0.928068i \(-0.621469\pi\)
−0.372410 + 0.928068i \(0.621469\pi\)
\(752\) −3.20350e9 −0.274702
\(753\) 2.06832e9 0.176537
\(754\) −4.42330e9 −0.375791
\(755\) −5.43495e9 −0.459601
\(756\) −8.54070e8 −0.0718898
\(757\) 8.70124e9 0.729030 0.364515 0.931197i \(-0.381235\pi\)
0.364515 + 0.931197i \(0.381235\pi\)
\(758\) 1.04399e10 0.870675
\(759\) −2.02782e9 −0.168338
\(760\) −4.38976e8 −0.0362738
\(761\) 6.24533e9 0.513700 0.256850 0.966451i \(-0.417315\pi\)
0.256850 + 0.966451i \(0.417315\pi\)
\(762\) −4.05122e9 −0.331699
\(763\) −2.00418e8 −0.0163343
\(764\) 8.63396e9 0.700460
\(765\) 7.32527e8 0.0591573
\(766\) −7.87815e9 −0.633321
\(767\) 1.41393e9 0.113148
\(768\) 4.52985e8 0.0360844
\(769\) 1.20439e10 0.955046 0.477523 0.878619i \(-0.341535\pi\)
0.477523 + 0.878619i \(0.341535\pi\)
\(770\) −1.29450e9 −0.102184
\(771\) 1.20331e10 0.945555
\(772\) −4.36982e9 −0.341825
\(773\) 1.43546e10 1.11780 0.558899 0.829235i \(-0.311224\pi\)
0.558899 + 0.829235i \(0.311224\pi\)
\(774\) −1.05490e9 −0.0817748
\(775\) 2.22109e8 0.0171400
\(776\) −2.16679e9 −0.166456
\(777\) 2.78393e7 0.00212904
\(778\) 6.79980e9 0.517687
\(779\) −2.83556e8 −0.0214911
\(780\) −1.29877e9 −0.0979943
\(781\) −7.16183e9 −0.537954
\(782\) −2.52966e9 −0.189164
\(783\) −1.80996e9 −0.134742
\(784\) −1.49043e9 −0.110460
\(785\) 3.65762e9 0.269870
\(786\) −1.17244e9 −0.0861212
\(787\) 4.17514e7 0.00305323 0.00152661 0.999999i \(-0.499514\pi\)
0.00152661 + 0.999999i \(0.499514\pi\)
\(788\) −1.22792e10 −0.893984
\(789\) −1.43835e10 −1.04255
\(790\) 3.83583e9 0.276799
\(791\) 1.57882e10 1.13427
\(792\) −7.12652e8 −0.0509729
\(793\) −9.42219e9 −0.670959
\(794\) 2.08767e8 0.0148010
\(795\) −4.88312e9 −0.344677
\(796\) −4.23208e9 −0.297412
\(797\) −2.62024e10 −1.83331 −0.916657 0.399676i \(-0.869123\pi\)
−0.916657 + 0.399676i \(0.869123\pi\)
\(798\) −1.00447e9 −0.0699723
\(799\) 6.28710e9 0.436050
\(800\) 5.12000e8 0.0353553
\(801\) −8.80784e9 −0.605558
\(802\) 5.13064e9 0.351206
\(803\) 5.25189e8 0.0357941
\(804\) −7.08478e9 −0.480762
\(805\) 3.33363e9 0.225233
\(806\) 6.83778e8 0.0459984
\(807\) −9.01153e9 −0.603589
\(808\) 6.67019e9 0.444834
\(809\) 2.07368e10 1.37696 0.688481 0.725255i \(-0.258278\pi\)
0.688481 + 0.725255i \(0.258278\pi\)
\(810\) −5.31441e8 −0.0351364
\(811\) −2.92477e8 −0.0192539 −0.00962697 0.999954i \(-0.503064\pi\)
−0.00962697 + 0.999954i \(0.503064\pi\)
\(812\) 3.99006e9 0.261537
\(813\) 1.73591e9 0.113295
\(814\) 2.32296e7 0.00150958
\(815\) −4.43399e9 −0.286908
\(816\) −8.89016e8 −0.0572788
\(817\) −1.24067e9 −0.0795937
\(818\) 1.72645e10 1.10285
\(819\) −2.97186e9 −0.189032
\(820\) 3.30726e8 0.0209469
\(821\) 6.82856e9 0.430654 0.215327 0.976542i \(-0.430918\pi\)
0.215327 + 0.976542i \(0.430918\pi\)
\(822\) 4.56959e9 0.286963
\(823\) −8.19085e8 −0.0512188 −0.0256094 0.999672i \(-0.508153\pi\)
−0.0256094 + 0.999672i \(0.508153\pi\)
\(824\) 5.20488e9 0.324090
\(825\) −8.05497e8 −0.0499430
\(826\) −1.27545e9 −0.0787468
\(827\) −4.29716e9 −0.264187 −0.132094 0.991237i \(-0.542170\pi\)
−0.132094 + 0.991237i \(0.542170\pi\)
\(828\) 1.83524e9 0.112354
\(829\) −5.14740e9 −0.313796 −0.156898 0.987615i \(-0.550149\pi\)
−0.156898 + 0.987615i \(0.550149\pi\)
\(830\) 8.56078e9 0.519685
\(831\) 7.93722e9 0.479805
\(832\) 1.57623e9 0.0948826
\(833\) 2.92509e9 0.175340
\(834\) −3.28224e9 −0.195925
\(835\) 2.39466e9 0.142345
\(836\) −8.38148e8 −0.0496134
\(837\) 2.79794e8 0.0164930
\(838\) −7.38203e9 −0.433333
\(839\) −1.10362e10 −0.645138 −0.322569 0.946546i \(-0.604547\pi\)
−0.322569 + 0.946546i \(0.604547\pi\)
\(840\) 1.17156e9 0.0682006
\(841\) −8.79406e9 −0.509804
\(842\) 1.79048e10 1.03366
\(843\) −6.78666e9 −0.390175
\(844\) −3.31685e9 −0.189901
\(845\) 3.32431e9 0.189541
\(846\) −4.56123e9 −0.258991
\(847\) 1.07404e10 0.607338
\(848\) 5.92630e9 0.333732
\(849\) 1.56742e10 0.879041
\(850\) −1.00484e9 −0.0561216
\(851\) −5.98216e7 −0.00332740
\(852\) 6.48168e9 0.359045
\(853\) −5.99095e9 −0.330502 −0.165251 0.986252i \(-0.552843\pi\)
−0.165251 + 0.986252i \(0.552843\pi\)
\(854\) 8.49935e9 0.466964
\(855\) −6.25026e8 −0.0341993
\(856\) −9.60727e9 −0.523530
\(857\) −1.22804e10 −0.666470 −0.333235 0.942844i \(-0.608140\pi\)
−0.333235 + 0.942844i \(0.608140\pi\)
\(858\) −2.47977e9 −0.134031
\(859\) 3.91925e9 0.210973 0.105486 0.994421i \(-0.466360\pi\)
0.105486 + 0.994421i \(0.466360\pi\)
\(860\) 1.44706e9 0.0775784
\(861\) 7.56770e8 0.0404067
\(862\) −6.92377e9 −0.368186
\(863\) −7.89892e9 −0.418341 −0.209170 0.977879i \(-0.567076\pi\)
−0.209170 + 0.977879i \(0.567076\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −1.17167e10 −0.615529
\(866\) −1.51357e10 −0.791934
\(867\) −9.33438e9 −0.486428
\(868\) −6.16806e8 −0.0320133
\(869\) 7.32384e9 0.378591
\(870\) 2.48280e9 0.127827
\(871\) −2.46525e10 −1.26415
\(872\) 1.51351e8 0.00772994
\(873\) −3.08513e9 −0.156937
\(874\) 2.15842e9 0.109357
\(875\) 1.32420e9 0.0668227
\(876\) −4.75312e8 −0.0238899
\(877\) −1.77306e10 −0.887617 −0.443808 0.896122i \(-0.646373\pi\)
−0.443808 + 0.896122i \(0.646373\pi\)
\(878\) −2.54702e10 −1.27000
\(879\) −1.88718e9 −0.0937244
\(880\) 9.77575e8 0.0483571
\(881\) −3.72452e10 −1.83508 −0.917540 0.397644i \(-0.869828\pi\)
−0.917540 + 0.397644i \(0.869828\pi\)
\(882\) −2.12212e9 −0.104143
\(883\) −2.05226e10 −1.00316 −0.501580 0.865111i \(-0.667248\pi\)
−0.501580 + 0.865111i \(0.667248\pi\)
\(884\) −3.09346e9 −0.150613
\(885\) −7.93642e8 −0.0384878
\(886\) 1.26781e10 0.612401
\(887\) 4.23295e9 0.203662 0.101831 0.994802i \(-0.467530\pi\)
0.101831 + 0.994802i \(0.467530\pi\)
\(888\) −2.10236e7 −0.00100754
\(889\) 1.27161e10 0.607013
\(890\) 1.20821e10 0.574483
\(891\) −1.01469e9 −0.0480577
\(892\) −1.42102e9 −0.0670385
\(893\) −5.36445e9 −0.252084
\(894\) −1.23116e10 −0.576279
\(895\) −1.17883e10 −0.549632
\(896\) −1.42184e9 −0.0660350
\(897\) 6.38598e9 0.295430
\(898\) 1.52745e10 0.703880
\(899\) −1.30715e9 −0.0600020
\(900\) 7.29000e8 0.0333333
\(901\) −1.16308e10 −0.529753
\(902\) 6.31463e8 0.0286500
\(903\) 3.31117e9 0.149649
\(904\) −1.19229e10 −0.536775
\(905\) −3.18117e9 −0.142665
\(906\) 9.39159e9 0.419557
\(907\) −2.38257e10 −1.06028 −0.530140 0.847910i \(-0.677861\pi\)
−0.530140 + 0.847910i \(0.677861\pi\)
\(908\) 5.87442e9 0.260414
\(909\) 9.49720e9 0.419394
\(910\) 4.07662e9 0.179331
\(911\) 1.39370e10 0.610738 0.305369 0.952234i \(-0.401220\pi\)
0.305369 + 0.952234i \(0.401220\pi\)
\(912\) 7.58551e8 0.0331133
\(913\) 1.63453e10 0.710797
\(914\) 1.70629e10 0.739164
\(915\) 5.28868e9 0.228230
\(916\) −1.50218e10 −0.645786
\(917\) 3.68008e9 0.157603
\(918\) −1.26581e9 −0.0540030
\(919\) −2.43549e9 −0.103510 −0.0517550 0.998660i \(-0.516481\pi\)
−0.0517550 + 0.998660i \(0.516481\pi\)
\(920\) −2.51748e9 −0.106588
\(921\) −6.04979e9 −0.255171
\(922\) −4.95692e8 −0.0208283
\(923\) 2.25539e10 0.944096
\(924\) 2.23690e9 0.0932811
\(925\) −2.37625e7 −0.000987181 0
\(926\) −1.47825e10 −0.611802
\(927\) 7.41085e9 0.305555
\(928\) −3.01320e9 −0.123768
\(929\) 2.23368e10 0.914040 0.457020 0.889457i \(-0.348917\pi\)
0.457020 + 0.889457i \(0.348917\pi\)
\(930\) −3.83805e8 −0.0156466
\(931\) −2.49582e9 −0.101365
\(932\) 2.02421e10 0.819032
\(933\) −1.62987e8 −0.00657003
\(934\) 4.38074e9 0.175927
\(935\) −1.91856e9 −0.0767601
\(936\) 2.24427e9 0.0894562
\(937\) −1.93999e10 −0.770392 −0.385196 0.922835i \(-0.625866\pi\)
−0.385196 + 0.922835i \(0.625866\pi\)
\(938\) 2.22379e10 0.879801
\(939\) 1.73636e9 0.0684401
\(940\) 6.25683e9 0.245701
\(941\) −6.09000e9 −0.238261 −0.119131 0.992879i \(-0.538011\pi\)
−0.119131 + 0.992879i \(0.538011\pi\)
\(942\) −6.32037e9 −0.246357
\(943\) −1.62616e9 −0.0631499
\(944\) 9.63188e8 0.0372657
\(945\) 1.66810e9 0.0643001
\(946\) 2.76290e9 0.106108
\(947\) −7.40768e9 −0.283437 −0.141719 0.989907i \(-0.545263\pi\)
−0.141719 + 0.989907i \(0.545263\pi\)
\(948\) −6.62831e9 −0.252682
\(949\) −1.65392e9 −0.0628177
\(950\) 8.57375e8 0.0324443
\(951\) −1.00076e10 −0.377311
\(952\) 2.79047e9 0.104821
\(953\) 2.45741e10 0.919713 0.459856 0.887993i \(-0.347901\pi\)
0.459856 + 0.887993i \(0.347901\pi\)
\(954\) 8.43804e9 0.314646
\(955\) −1.68632e10 −0.626510
\(956\) −1.26672e10 −0.468899
\(957\) 4.74047e9 0.174836
\(958\) 2.41483e10 0.887377
\(959\) −1.43432e10 −0.525146
\(960\) −8.84736e8 −0.0322749
\(961\) −2.73105e10 −0.992656
\(962\) −7.31545e7 −0.00264928
\(963\) −1.36791e10 −0.493589
\(964\) −6.02396e8 −0.0216577
\(965\) 8.53481e9 0.305737
\(966\) −5.76052e9 −0.205609
\(967\) 1.71505e9 0.0609937 0.0304968 0.999535i \(-0.490291\pi\)
0.0304968 + 0.999535i \(0.490291\pi\)
\(968\) −8.11092e9 −0.287413
\(969\) −1.48871e9 −0.0525627
\(970\) 4.23201e9 0.148883
\(971\) −6.80742e9 −0.238625 −0.119312 0.992857i \(-0.538069\pi\)
−0.119312 + 0.992857i \(0.538069\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 1.03024e10 0.358545
\(974\) 9.56000e8 0.0331514
\(975\) 2.53666e9 0.0876488
\(976\) −6.41850e9 −0.220983
\(977\) 5.77757e8 0.0198205 0.00991025 0.999951i \(-0.496845\pi\)
0.00991025 + 0.999951i \(0.496845\pi\)
\(978\) 7.66193e9 0.261910
\(979\) 2.30686e10 0.785746
\(980\) 2.91100e9 0.0987987
\(981\) 2.15497e8 0.00728786
\(982\) 4.12384e10 1.38967
\(983\) −2.41039e9 −0.0809375 −0.0404688 0.999181i \(-0.512885\pi\)
−0.0404688 + 0.999181i \(0.512885\pi\)
\(984\) −5.71494e8 −0.0191218
\(985\) 2.39829e10 0.799603
\(986\) 5.91363e9 0.196465
\(987\) 1.43169e10 0.473958
\(988\) 2.63948e9 0.0870702
\(989\) −7.11510e9 −0.233880
\(990\) 1.39190e9 0.0455915
\(991\) −2.52506e10 −0.824165 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(992\) 4.65797e8 0.0151498
\(993\) 9.57947e8 0.0310470
\(994\) −2.03449e10 −0.657058
\(995\) 8.26579e9 0.266013
\(996\) −1.47930e10 −0.474405
\(997\) 1.10682e10 0.353707 0.176853 0.984237i \(-0.443408\pi\)
0.176853 + 0.984237i \(0.443408\pi\)
\(998\) −1.82428e10 −0.580944
\(999\) −2.99339e7 −0.000949915 0
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.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.b.1.2 4 1.1 even 1 trivial