Properties

Label 570.8.a.b.1.4
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 1.4
Root \(-60.6808\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +836.803 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +836.803 q^{7} +512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} -2850.39 q^{11} +1728.00 q^{12} -272.024 q^{13} +6694.43 q^{14} -3375.00 q^{15} +4096.00 q^{16} -18990.8 q^{17} +5832.00 q^{18} +6859.00 q^{19} -8000.00 q^{20} +22593.7 q^{21} -22803.1 q^{22} -83520.5 q^{23} +13824.0 q^{24} +15625.0 q^{25} -2176.19 q^{26} +19683.0 q^{27} +53555.4 q^{28} +39225.6 q^{29} -27000.0 q^{30} -70995.2 q^{31} +32768.0 q^{32} -76960.5 q^{33} -151927. q^{34} -104600. q^{35} +46656.0 q^{36} -342215. q^{37} +54872.0 q^{38} -7344.64 q^{39} -64000.0 q^{40} +12800.5 q^{41} +180749. q^{42} -555589. q^{43} -182425. q^{44} -91125.0 q^{45} -668164. q^{46} +675732. q^{47} +110592. q^{48} -123303. q^{49} +125000. q^{50} -512753. q^{51} -17409.5 q^{52} +597349. q^{53} +157464. q^{54} +356299. q^{55} +428443. q^{56} +185193. q^{57} +313805. q^{58} -2.38800e6 q^{59} -216000. q^{60} -2.70634e6 q^{61} -567961. q^{62} +610030. q^{63} +262144. q^{64} +34003.0 q^{65} -615684. q^{66} +1.27372e6 q^{67} -1.21541e6 q^{68} -2.25505e6 q^{69} -836803. q^{70} +2.54916e6 q^{71} +373248. q^{72} -311362. q^{73} -2.73772e6 q^{74} +421875. q^{75} +438976. q^{76} -2.38521e6 q^{77} -58757.1 q^{78} -1.44553e6 q^{79} -512000. q^{80} +531441. q^{81} +102404. q^{82} +4.27618e6 q^{83} +1.44600e6 q^{84} +2.37386e6 q^{85} -4.44471e6 q^{86} +1.05909e6 q^{87} -1.45940e6 q^{88} +3.07550e6 q^{89} -729000. q^{90} -227630. q^{91} -5.34531e6 q^{92} -1.91687e6 q^{93} +5.40585e6 q^{94} -857375. q^{95} +884736. q^{96} +5.31697e6 q^{97} -986427. q^{98} -2.07793e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100}) \)

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
\(7\) 836.803 0.922105 0.461052 0.887373i \(-0.347472\pi\)
0.461052 + 0.887373i \(0.347472\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1000.00 −0.316228
\(11\) −2850.39 −0.645698 −0.322849 0.946450i \(-0.604641\pi\)
−0.322849 + 0.946450i \(0.604641\pi\)
\(12\) 1728.00 0.288675
\(13\) −272.024 −0.0343404 −0.0171702 0.999853i \(-0.505466\pi\)
−0.0171702 + 0.999853i \(0.505466\pi\)
\(14\) 6694.43 0.652026
\(15\) −3375.00 −0.258199
\(16\) 4096.00 0.250000
\(17\) −18990.8 −0.937504 −0.468752 0.883330i \(-0.655296\pi\)
−0.468752 + 0.883330i \(0.655296\pi\)
\(18\) 5832.00 0.235702
\(19\) 6859.00 0.229416
\(20\) −8000.00 −0.223607
\(21\) 22593.7 0.532377
\(22\) −22803.1 −0.456578
\(23\) −83520.5 −1.43135 −0.715675 0.698433i \(-0.753881\pi\)
−0.715675 + 0.698433i \(0.753881\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) −2176.19 −0.0242823
\(27\) 19683.0 0.192450
\(28\) 53555.4 0.461052
\(29\) 39225.6 0.298659 0.149330 0.988787i \(-0.452288\pi\)
0.149330 + 0.988787i \(0.452288\pi\)
\(30\) −27000.0 −0.182574
\(31\) −70995.2 −0.428019 −0.214009 0.976832i \(-0.568652\pi\)
−0.214009 + 0.976832i \(0.568652\pi\)
\(32\) 32768.0 0.176777
\(33\) −76960.5 −0.372794
\(34\) −151927. −0.662915
\(35\) −104600. −0.412378
\(36\) 46656.0 0.166667
\(37\) −342215. −1.11069 −0.555345 0.831620i \(-0.687414\pi\)
−0.555345 + 0.831620i \(0.687414\pi\)
\(38\) 54872.0 0.162221
\(39\) −7344.64 −0.0198264
\(40\) −64000.0 −0.158114
\(41\) 12800.5 0.0290057 0.0145029 0.999895i \(-0.495383\pi\)
0.0145029 + 0.999895i \(0.495383\pi\)
\(42\) 180749. 0.376448
\(43\) −555589. −1.06565 −0.532824 0.846226i \(-0.678869\pi\)
−0.532824 + 0.846226i \(0.678869\pi\)
\(44\) −182425. −0.322849
\(45\) −91125.0 −0.149071
\(46\) −668164. −1.01212
\(47\) 675732. 0.949362 0.474681 0.880158i \(-0.342563\pi\)
0.474681 + 0.880158i \(0.342563\pi\)
\(48\) 110592. 0.144338
\(49\) −123303. −0.149723
\(50\) 125000. 0.141421
\(51\) −512753. −0.541268
\(52\) −17409.5 −0.0171702
\(53\) 597349. 0.551141 0.275570 0.961281i \(-0.411133\pi\)
0.275570 + 0.961281i \(0.411133\pi\)
\(54\) 157464. 0.136083
\(55\) 356299. 0.288765
\(56\) 428443. 0.326013
\(57\) 185193. 0.132453
\(58\) 313805. 0.211184
\(59\) −2.38800e6 −1.51374 −0.756871 0.653565i \(-0.773273\pi\)
−0.756871 + 0.653565i \(0.773273\pi\)
\(60\) −216000. −0.129099
\(61\) −2.70634e6 −1.52661 −0.763306 0.646038i \(-0.776425\pi\)
−0.763306 + 0.646038i \(0.776425\pi\)
\(62\) −567961. −0.302655
\(63\) 610030. 0.307368
\(64\) 262144. 0.125000
\(65\) 34003.0 0.0153575
\(66\) −615684. −0.263605
\(67\) 1.27372e6 0.517385 0.258692 0.965960i \(-0.416708\pi\)
0.258692 + 0.965960i \(0.416708\pi\)
\(68\) −1.21541e6 −0.468752
\(69\) −2.25505e6 −0.826390
\(70\) −836803. −0.291595
\(71\) 2.54916e6 0.845266 0.422633 0.906301i \(-0.361106\pi\)
0.422633 + 0.906301i \(0.361106\pi\)
\(72\) 373248. 0.117851
\(73\) −311362. −0.0936776 −0.0468388 0.998902i \(-0.514915\pi\)
−0.0468388 + 0.998902i \(0.514915\pi\)
\(74\) −2.73772e6 −0.785376
\(75\) 421875. 0.115470
\(76\) 438976. 0.114708
\(77\) −2.38521e6 −0.595402
\(78\) −58757.1 −0.0140194
\(79\) −1.44553e6 −0.329862 −0.164931 0.986305i \(-0.552740\pi\)
−0.164931 + 0.986305i \(0.552740\pi\)
\(80\) −512000. −0.111803
\(81\) 531441. 0.111111
\(82\) 102404. 0.0205101
\(83\) 4.27618e6 0.820885 0.410443 0.911886i \(-0.365374\pi\)
0.410443 + 0.911886i \(0.365374\pi\)
\(84\) 1.44600e6 0.266189
\(85\) 2.37386e6 0.419265
\(86\) −4.44471e6 −0.753527
\(87\) 1.05909e6 0.172431
\(88\) −1.45940e6 −0.228289
\(89\) 3.07550e6 0.462435 0.231218 0.972902i \(-0.425729\pi\)
0.231218 + 0.972902i \(0.425729\pi\)
\(90\) −729000. −0.105409
\(91\) −227630. −0.0316654
\(92\) −5.34531e6 −0.715675
\(93\) −1.91687e6 −0.247117
\(94\) 5.40585e6 0.671300
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) 5.31697e6 0.591512 0.295756 0.955264i \(-0.404429\pi\)
0.295756 + 0.955264i \(0.404429\pi\)
\(98\) −986427. −0.105870
\(99\) −2.07793e6 −0.215233
\(100\) 1.00000e6 0.100000
\(101\) 1.36841e6 0.132158 0.0660789 0.997814i \(-0.478951\pi\)
0.0660789 + 0.997814i \(0.478951\pi\)
\(102\) −4.10202e6 −0.382734
\(103\) −1.59596e7 −1.43910 −0.719551 0.694440i \(-0.755652\pi\)
−0.719551 + 0.694440i \(0.755652\pi\)
\(104\) −139276. −0.0121412
\(105\) −2.82421e6 −0.238086
\(106\) 4.77879e6 0.389715
\(107\) 2.23193e7 1.76131 0.880656 0.473756i \(-0.157102\pi\)
0.880656 + 0.473756i \(0.157102\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −1.97882e7 −1.46357 −0.731787 0.681534i \(-0.761313\pi\)
−0.731787 + 0.681534i \(0.761313\pi\)
\(110\) 2.85039e6 0.204188
\(111\) −9.23980e6 −0.641257
\(112\) 3.42755e6 0.230526
\(113\) −5.44310e6 −0.354872 −0.177436 0.984132i \(-0.556780\pi\)
−0.177436 + 0.984132i \(0.556780\pi\)
\(114\) 1.48154e6 0.0936586
\(115\) 1.04401e7 0.640119
\(116\) 2.51044e6 0.149330
\(117\) −198305. −0.0114468
\(118\) −1.91040e7 −1.07038
\(119\) −1.58916e7 −0.864477
\(120\) −1.72800e6 −0.0912871
\(121\) −1.13625e7 −0.583073
\(122\) −2.16507e7 −1.07948
\(123\) 345613. 0.0167464
\(124\) −4.54369e6 −0.214009
\(125\) −1.95312e6 −0.0894427
\(126\) 4.88024e6 0.217342
\(127\) −2.66077e7 −1.15264 −0.576321 0.817224i \(-0.695512\pi\)
−0.576321 + 0.817224i \(0.695512\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.50009e7 −0.615253
\(130\) 272024. 0.0108594
\(131\) −6.56953e6 −0.255320 −0.127660 0.991818i \(-0.540747\pi\)
−0.127660 + 0.991818i \(0.540747\pi\)
\(132\) −4.92547e6 −0.186397
\(133\) 5.73963e6 0.211545
\(134\) 1.01898e7 0.365846
\(135\) −2.46038e6 −0.0860663
\(136\) −9.72331e6 −0.331458
\(137\) −2.40376e7 −0.798675 −0.399337 0.916804i \(-0.630760\pi\)
−0.399337 + 0.916804i \(0.630760\pi\)
\(138\) −1.80404e7 −0.584346
\(139\) −1.09363e7 −0.345396 −0.172698 0.984975i \(-0.555249\pi\)
−0.172698 + 0.984975i \(0.555249\pi\)
\(140\) −6.69443e6 −0.206189
\(141\) 1.82448e7 0.548114
\(142\) 2.03933e7 0.597694
\(143\) 775373. 0.0221735
\(144\) 2.98598e6 0.0833333
\(145\) −4.90320e6 −0.133565
\(146\) −2.49090e6 −0.0662400
\(147\) −3.32919e6 −0.0864427
\(148\) −2.19017e7 −0.555345
\(149\) 5.07363e7 1.25651 0.628256 0.778007i \(-0.283769\pi\)
0.628256 + 0.778007i \(0.283769\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) −2.92446e7 −0.691236 −0.345618 0.938375i \(-0.612331\pi\)
−0.345618 + 0.938375i \(0.612331\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) −1.38443e7 −0.312501
\(154\) −1.90817e7 −0.421012
\(155\) 8.87440e6 0.191416
\(156\) −470057. −0.00991321
\(157\) −5.62135e7 −1.15929 −0.579645 0.814869i \(-0.696809\pi\)
−0.579645 + 0.814869i \(0.696809\pi\)
\(158\) −1.15642e7 −0.233247
\(159\) 1.61284e7 0.318201
\(160\) −4.09600e6 −0.0790569
\(161\) −6.98903e7 −1.31985
\(162\) 4.25153e6 0.0785674
\(163\) −9.71809e7 −1.75762 −0.878809 0.477174i \(-0.841661\pi\)
−0.878809 + 0.477174i \(0.841661\pi\)
\(164\) 819232. 0.0145029
\(165\) 9.62006e6 0.166719
\(166\) 3.42094e7 0.580454
\(167\) 2.50137e7 0.415595 0.207797 0.978172i \(-0.433371\pi\)
0.207797 + 0.978172i \(0.433371\pi\)
\(168\) 1.15680e7 0.188224
\(169\) −6.26745e7 −0.998821
\(170\) 1.89908e7 0.296465
\(171\) 5.00021e6 0.0764719
\(172\) −3.55577e7 −0.532824
\(173\) −7.86742e7 −1.15524 −0.577619 0.816307i \(-0.696018\pi\)
−0.577619 + 0.816307i \(0.696018\pi\)
\(174\) 8.47272e6 0.121927
\(175\) 1.30751e7 0.184421
\(176\) −1.16752e7 −0.161425
\(177\) −6.44759e7 −0.873959
\(178\) 2.46040e7 0.326991
\(179\) 3.16655e7 0.412669 0.206334 0.978482i \(-0.433847\pi\)
0.206334 + 0.978482i \(0.433847\pi\)
\(180\) −5.83200e6 −0.0745356
\(181\) 1.43164e8 1.79456 0.897282 0.441458i \(-0.145539\pi\)
0.897282 + 0.441458i \(0.145539\pi\)
\(182\) −1.82104e6 −0.0223908
\(183\) −7.30713e7 −0.881389
\(184\) −4.27625e7 −0.506059
\(185\) 4.27768e7 0.496715
\(186\) −1.53350e7 −0.174738
\(187\) 5.41313e7 0.605345
\(188\) 4.32468e7 0.474681
\(189\) 1.64708e7 0.177459
\(190\) −6.85900e6 −0.0725476
\(191\) 2.56884e7 0.266760 0.133380 0.991065i \(-0.457417\pi\)
0.133380 + 0.991065i \(0.457417\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.15293e6 0.0115439 0.00577196 0.999983i \(-0.498163\pi\)
0.00577196 + 0.999983i \(0.498163\pi\)
\(194\) 4.25358e7 0.418262
\(195\) 918080. 0.00886665
\(196\) −7.89142e6 −0.0748615
\(197\) 1.25465e8 1.16920 0.584601 0.811321i \(-0.301251\pi\)
0.584601 + 0.811321i \(0.301251\pi\)
\(198\) −1.66235e7 −0.152193
\(199\) 7.86052e7 0.707075 0.353537 0.935420i \(-0.384979\pi\)
0.353537 + 0.935420i \(0.384979\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) 3.43906e7 0.298712
\(202\) 1.09473e7 0.0934497
\(203\) 3.28241e7 0.275395
\(204\) −3.28162e7 −0.270634
\(205\) −1.60006e6 −0.0129717
\(206\) −1.27677e8 −1.01760
\(207\) −6.08865e7 −0.477117
\(208\) −1.11421e6 −0.00858509
\(209\) −1.95508e7 −0.148133
\(210\) −2.25937e7 −0.168353
\(211\) −4.92739e7 −0.361100 −0.180550 0.983566i \(-0.557788\pi\)
−0.180550 + 0.983566i \(0.557788\pi\)
\(212\) 3.82303e7 0.275570
\(213\) 6.88274e7 0.488015
\(214\) 1.78554e8 1.24544
\(215\) 6.94486e7 0.476573
\(216\) 1.00777e7 0.0680414
\(217\) −5.94090e7 −0.394678
\(218\) −1.58306e8 −1.03490
\(219\) −8.40677e6 −0.0540848
\(220\) 2.28031e7 0.144383
\(221\) 5.16596e6 0.0321942
\(222\) −7.39184e7 −0.453437
\(223\) −2.10879e8 −1.27340 −0.636702 0.771110i \(-0.719702\pi\)
−0.636702 + 0.771110i \(0.719702\pi\)
\(224\) 2.74204e7 0.163007
\(225\) 1.13906e7 0.0666667
\(226\) −4.35448e7 −0.250932
\(227\) 1.61754e7 0.0917837 0.0458918 0.998946i \(-0.485387\pi\)
0.0458918 + 0.998946i \(0.485387\pi\)
\(228\) 1.18524e7 0.0662266
\(229\) 2.84435e8 1.56516 0.782580 0.622550i \(-0.213903\pi\)
0.782580 + 0.622550i \(0.213903\pi\)
\(230\) 8.35205e7 0.452633
\(231\) −6.44008e7 −0.343755
\(232\) 2.00835e7 0.105592
\(233\) 1.92691e8 0.997966 0.498983 0.866612i \(-0.333707\pi\)
0.498983 + 0.866612i \(0.333707\pi\)
\(234\) −1.58644e6 −0.00809410
\(235\) −8.44664e7 −0.424567
\(236\) −1.52832e8 −0.756871
\(237\) −3.90293e7 −0.190446
\(238\) −1.27133e8 −0.611277
\(239\) −2.22040e8 −1.05206 −0.526028 0.850468i \(-0.676319\pi\)
−0.526028 + 0.850468i \(0.676319\pi\)
\(240\) −1.38240e7 −0.0645497
\(241\) −4.24494e8 −1.95349 −0.976747 0.214395i \(-0.931222\pi\)
−0.976747 + 0.214395i \(0.931222\pi\)
\(242\) −9.08996e7 −0.412295
\(243\) 1.43489e7 0.0641500
\(244\) −1.73206e8 −0.763306
\(245\) 1.54129e7 0.0669582
\(246\) 2.76491e6 0.0118415
\(247\) −1.86581e6 −0.00787822
\(248\) −3.63495e7 −0.151328
\(249\) 1.15457e8 0.473938
\(250\) −1.56250e7 −0.0632456
\(251\) −3.69876e8 −1.47638 −0.738190 0.674593i \(-0.764319\pi\)
−0.738190 + 0.674593i \(0.764319\pi\)
\(252\) 3.90419e7 0.153684
\(253\) 2.38066e8 0.924220
\(254\) −2.12862e8 −0.815041
\(255\) 6.40941e7 0.242062
\(256\) 1.67772e7 0.0625000
\(257\) −1.34218e8 −0.493223 −0.246612 0.969114i \(-0.579317\pi\)
−0.246612 + 0.969114i \(0.579317\pi\)
\(258\) −1.20007e8 −0.435049
\(259\) −2.86366e8 −1.02417
\(260\) 2.17619e6 0.00767874
\(261\) 2.85954e7 0.0995532
\(262\) −5.25563e7 −0.180539
\(263\) 2.90630e8 0.985133 0.492566 0.870275i \(-0.336059\pi\)
0.492566 + 0.870275i \(0.336059\pi\)
\(264\) −3.94038e7 −0.131803
\(265\) −7.46686e7 −0.246478
\(266\) 4.59171e7 0.149585
\(267\) 8.30386e7 0.266987
\(268\) 8.15184e7 0.258692
\(269\) −2.34158e8 −0.733458 −0.366729 0.930328i \(-0.619522\pi\)
−0.366729 + 0.930328i \(0.619522\pi\)
\(270\) −1.96830e7 −0.0608581
\(271\) 4.75402e8 1.45100 0.725502 0.688220i \(-0.241608\pi\)
0.725502 + 0.688220i \(0.241608\pi\)
\(272\) −7.77865e7 −0.234376
\(273\) −6.14602e6 −0.0182820
\(274\) −1.92301e8 −0.564748
\(275\) −4.45373e7 −0.129140
\(276\) −1.44323e8 −0.413195
\(277\) −5.18595e8 −1.46605 −0.733026 0.680201i \(-0.761893\pi\)
−0.733026 + 0.680201i \(0.761893\pi\)
\(278\) −8.74902e7 −0.244232
\(279\) −5.17555e7 −0.142673
\(280\) −5.35554e7 −0.145798
\(281\) 3.79068e8 1.01917 0.509583 0.860421i \(-0.329800\pi\)
0.509583 + 0.860421i \(0.329800\pi\)
\(282\) 1.45958e8 0.387575
\(283\) −1.54943e8 −0.406367 −0.203184 0.979141i \(-0.565129\pi\)
−0.203184 + 0.979141i \(0.565129\pi\)
\(284\) 1.63146e8 0.422633
\(285\) −2.31491e7 −0.0592349
\(286\) 6.20299e6 0.0156791
\(287\) 1.07115e7 0.0267463
\(288\) 2.38879e7 0.0589256
\(289\) −4.96864e7 −0.121086
\(290\) −3.92256e7 −0.0944444
\(291\) 1.43558e8 0.341509
\(292\) −1.99272e7 −0.0468388
\(293\) −4.92640e8 −1.14418 −0.572089 0.820192i \(-0.693867\pi\)
−0.572089 + 0.820192i \(0.693867\pi\)
\(294\) −2.66335e7 −0.0611242
\(295\) 2.98500e8 0.676966
\(296\) −1.75214e8 −0.392688
\(297\) −5.61042e7 −0.124265
\(298\) 4.05890e8 0.888488
\(299\) 2.27196e7 0.0491531
\(300\) 2.70000e7 0.0577350
\(301\) −4.64919e8 −0.982640
\(302\) −2.33957e8 −0.488778
\(303\) 3.69472e7 0.0763013
\(304\) 2.80945e7 0.0573539
\(305\) 3.38293e8 0.682721
\(306\) −1.10755e8 −0.220972
\(307\) 1.42736e8 0.281547 0.140773 0.990042i \(-0.455041\pi\)
0.140773 + 0.990042i \(0.455041\pi\)
\(308\) −1.52654e8 −0.297701
\(309\) −4.30909e8 −0.830866
\(310\) 7.09952e7 0.135351
\(311\) 8.89524e7 0.167686 0.0838429 0.996479i \(-0.473281\pi\)
0.0838429 + 0.996479i \(0.473281\pi\)
\(312\) −3.76046e6 −0.00700970
\(313\) 8.07590e8 1.48863 0.744314 0.667830i \(-0.232777\pi\)
0.744314 + 0.667830i \(0.232777\pi\)
\(314\) −4.49708e8 −0.819742
\(315\) −7.62537e7 −0.137459
\(316\) −9.25138e7 −0.164931
\(317\) −6.88675e8 −1.21425 −0.607123 0.794608i \(-0.707677\pi\)
−0.607123 + 0.794608i \(0.707677\pi\)
\(318\) 1.29027e8 0.225002
\(319\) −1.11808e8 −0.192844
\(320\) −3.27680e7 −0.0559017
\(321\) 6.02620e8 1.01689
\(322\) −5.59122e8 −0.933278
\(323\) −1.30258e8 −0.215078
\(324\) 3.40122e7 0.0555556
\(325\) −4.25037e6 −0.00686807
\(326\) −7.77447e8 −1.24282
\(327\) −5.34283e8 −0.844995
\(328\) 6.55386e6 0.0102551
\(329\) 5.65454e8 0.875411
\(330\) 7.69605e7 0.117888
\(331\) 1.48744e8 0.225445 0.112723 0.993626i \(-0.464043\pi\)
0.112723 + 0.993626i \(0.464043\pi\)
\(332\) 2.73675e8 0.410443
\(333\) −2.49474e8 −0.370230
\(334\) 2.00110e8 0.293870
\(335\) −1.59216e8 −0.231382
\(336\) 9.25437e7 0.133094
\(337\) −3.35266e6 −0.00477183 −0.00238592 0.999997i \(-0.500759\pi\)
−0.00238592 + 0.999997i \(0.500759\pi\)
\(338\) −5.01396e8 −0.706273
\(339\) −1.46964e8 −0.204885
\(340\) 1.51927e8 0.209632
\(341\) 2.02364e8 0.276371
\(342\) 4.00017e7 0.0540738
\(343\) −7.92324e8 −1.06016
\(344\) −2.84462e8 −0.376764
\(345\) 2.81882e8 0.369573
\(346\) −6.29394e8 −0.816876
\(347\) 8.54721e8 1.09817 0.549087 0.835765i \(-0.314976\pi\)
0.549087 + 0.835765i \(0.314976\pi\)
\(348\) 6.77818e7 0.0862156
\(349\) −9.51858e7 −0.119862 −0.0599312 0.998203i \(-0.519088\pi\)
−0.0599312 + 0.998203i \(0.519088\pi\)
\(350\) 1.04600e8 0.130405
\(351\) −5.35424e6 −0.00660881
\(352\) −9.34016e7 −0.114144
\(353\) −9.20946e8 −1.11435 −0.557176 0.830394i \(-0.688115\pi\)
−0.557176 + 0.830394i \(0.688115\pi\)
\(354\) −5.15807e8 −0.617982
\(355\) −3.18646e8 −0.378015
\(356\) 1.96832e8 0.231218
\(357\) −4.29073e8 −0.499106
\(358\) 2.53324e8 0.291801
\(359\) −2.97767e8 −0.339661 −0.169830 0.985473i \(-0.554322\pi\)
−0.169830 + 0.985473i \(0.554322\pi\)
\(360\) −4.66560e7 −0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) 1.14531e9 1.26895
\(363\) −3.06786e8 −0.336638
\(364\) −1.45683e7 −0.0158327
\(365\) 3.89203e7 0.0418939
\(366\) −5.84570e8 −0.623236
\(367\) 3.30475e7 0.0348986 0.0174493 0.999848i \(-0.494445\pi\)
0.0174493 + 0.999848i \(0.494445\pi\)
\(368\) −3.42100e8 −0.357837
\(369\) 9.33156e6 0.00966857
\(370\) 3.42215e8 0.351231
\(371\) 4.99864e8 0.508210
\(372\) −1.22680e8 −0.123558
\(373\) 4.57835e8 0.456802 0.228401 0.973567i \(-0.426650\pi\)
0.228401 + 0.973567i \(0.426650\pi\)
\(374\) 4.33050e8 0.428043
\(375\) −5.27344e7 −0.0516398
\(376\) 3.45975e8 0.335650
\(377\) −1.06703e7 −0.0102561
\(378\) 1.31766e8 0.125483
\(379\) −1.10898e9 −1.04637 −0.523185 0.852219i \(-0.675256\pi\)
−0.523185 + 0.852219i \(0.675256\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) −7.18408e8 −0.665478
\(382\) 2.05507e8 0.188628
\(383\) −1.36304e9 −1.23969 −0.619843 0.784726i \(-0.712804\pi\)
−0.619843 + 0.784726i \(0.712804\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 2.98152e8 0.266272
\(386\) 9.22345e6 0.00816278
\(387\) −4.05024e8 −0.355216
\(388\) 3.40286e8 0.295756
\(389\) 1.22100e9 1.05170 0.525850 0.850577i \(-0.323747\pi\)
0.525850 + 0.850577i \(0.323747\pi\)
\(390\) 7.34464e6 0.00626967
\(391\) 1.58613e9 1.34190
\(392\) −6.31313e7 −0.0529351
\(393\) −1.77377e8 −0.147409
\(394\) 1.00372e9 0.826751
\(395\) 1.80691e8 0.147519
\(396\) −1.32988e8 −0.107616
\(397\) 1.95045e9 1.56447 0.782236 0.622982i \(-0.214079\pi\)
0.782236 + 0.622982i \(0.214079\pi\)
\(398\) 6.28842e8 0.499977
\(399\) 1.54970e8 0.122136
\(400\) 6.40000e7 0.0500000
\(401\) −3.28954e8 −0.254759 −0.127380 0.991854i \(-0.540657\pi\)
−0.127380 + 0.991854i \(0.540657\pi\)
\(402\) 2.75124e8 0.211221
\(403\) 1.93124e7 0.0146983
\(404\) 8.75785e7 0.0660789
\(405\) −6.64301e7 −0.0496904
\(406\) 2.62593e8 0.194734
\(407\) 9.75445e8 0.717170
\(408\) −2.62529e8 −0.191367
\(409\) 8.61975e8 0.622964 0.311482 0.950252i \(-0.399175\pi\)
0.311482 + 0.950252i \(0.399175\pi\)
\(410\) −1.28005e7 −0.00917241
\(411\) −6.49016e8 −0.461115
\(412\) −1.02141e9 −0.719551
\(413\) −1.99828e9 −1.39583
\(414\) −4.87092e8 −0.337372
\(415\) −5.34522e8 −0.367111
\(416\) −8.91367e6 −0.00607058
\(417\) −2.95279e8 −0.199415
\(418\) −1.56407e8 −0.104746
\(419\) −1.76742e9 −1.17379 −0.586894 0.809664i \(-0.699649\pi\)
−0.586894 + 0.809664i \(0.699649\pi\)
\(420\) −1.80749e8 −0.119043
\(421\) −2.80577e8 −0.183259 −0.0916295 0.995793i \(-0.529208\pi\)
−0.0916295 + 0.995793i \(0.529208\pi\)
\(422\) −3.94191e8 −0.255337
\(423\) 4.92608e8 0.316454
\(424\) 3.05843e8 0.194858
\(425\) −2.96732e8 −0.187501
\(426\) 5.50619e8 0.345079
\(427\) −2.26468e9 −1.40770
\(428\) 1.42843e9 0.880656
\(429\) 2.09351e7 0.0128019
\(430\) 5.55589e8 0.336988
\(431\) 8.32322e8 0.500750 0.250375 0.968149i \(-0.419446\pi\)
0.250375 + 0.968149i \(0.419446\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 2.14254e9 1.26830 0.634148 0.773211i \(-0.281351\pi\)
0.634148 + 0.773211i \(0.281351\pi\)
\(434\) −4.75272e8 −0.279080
\(435\) −1.32386e8 −0.0771135
\(436\) −1.26645e9 −0.731787
\(437\) −5.72867e8 −0.328374
\(438\) −6.72542e7 −0.0382437
\(439\) −2.21507e9 −1.24957 −0.624785 0.780797i \(-0.714813\pi\)
−0.624785 + 0.780797i \(0.714813\pi\)
\(440\) 1.82425e8 0.102094
\(441\) −8.98882e7 −0.0499077
\(442\) 4.13277e7 0.0227648
\(443\) −1.40738e9 −0.769128 −0.384564 0.923098i \(-0.625648\pi\)
−0.384564 + 0.923098i \(0.625648\pi\)
\(444\) −5.91347e8 −0.320628
\(445\) −3.84438e8 −0.206807
\(446\) −1.68703e9 −0.900433
\(447\) 1.36988e9 0.725447
\(448\) 2.19363e8 0.115263
\(449\) 1.42630e9 0.743618 0.371809 0.928309i \(-0.378738\pi\)
0.371809 + 0.928309i \(0.378738\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) −3.64864e7 −0.0187289
\(452\) −3.48358e8 −0.177436
\(453\) −7.89605e8 −0.399086
\(454\) 1.29404e8 0.0649009
\(455\) 2.84538e7 0.0141612
\(456\) 9.48188e7 0.0468293
\(457\) 2.56687e9 1.25805 0.629024 0.777386i \(-0.283455\pi\)
0.629024 + 0.777386i \(0.283455\pi\)
\(458\) 2.27548e9 1.10673
\(459\) −3.73797e8 −0.180423
\(460\) 6.68164e8 0.320060
\(461\) 1.12083e9 0.532826 0.266413 0.963859i \(-0.414161\pi\)
0.266413 + 0.963859i \(0.414161\pi\)
\(462\) −5.15206e8 −0.243072
\(463\) 1.77884e9 0.832922 0.416461 0.909153i \(-0.363270\pi\)
0.416461 + 0.909153i \(0.363270\pi\)
\(464\) 1.60668e8 0.0746649
\(465\) 2.39609e8 0.110514
\(466\) 1.54153e9 0.705669
\(467\) 1.47455e9 0.669962 0.334981 0.942225i \(-0.391270\pi\)
0.334981 + 0.942225i \(0.391270\pi\)
\(468\) −1.26915e7 −0.00572340
\(469\) 1.06586e9 0.477083
\(470\) −6.75732e8 −0.300215
\(471\) −1.51777e9 −0.669317
\(472\) −1.22265e9 −0.535188
\(473\) 1.58364e9 0.688088
\(474\) −3.12234e8 −0.134665
\(475\) 1.07172e8 0.0458831
\(476\) −1.01706e9 −0.432238
\(477\) 4.35467e8 0.183714
\(478\) −1.77632e9 −0.743915
\(479\) −2.31447e9 −0.962226 −0.481113 0.876658i \(-0.659767\pi\)
−0.481113 + 0.876658i \(0.659767\pi\)
\(480\) −1.10592e8 −0.0456435
\(481\) 9.30905e7 0.0381415
\(482\) −3.39595e9 −1.38133
\(483\) −1.88704e9 −0.762018
\(484\) −7.27197e8 −0.291537
\(485\) −6.64622e8 −0.264532
\(486\) 1.14791e8 0.0453609
\(487\) 2.23123e9 0.875372 0.437686 0.899128i \(-0.355798\pi\)
0.437686 + 0.899128i \(0.355798\pi\)
\(488\) −1.38565e9 −0.539739
\(489\) −2.62388e9 −1.01476
\(490\) 1.23303e8 0.0473466
\(491\) 5.31217e7 0.0202529 0.0101264 0.999949i \(-0.496777\pi\)
0.0101264 + 0.999949i \(0.496777\pi\)
\(492\) 2.21193e7 0.00837322
\(493\) −7.44927e8 −0.279994
\(494\) −1.49265e7 −0.00557074
\(495\) 2.59742e8 0.0962550
\(496\) −2.90796e8 −0.107005
\(497\) 2.13315e9 0.779424
\(498\) 9.23654e8 0.335125
\(499\) 3.92892e9 1.41554 0.707769 0.706444i \(-0.249702\pi\)
0.707769 + 0.706444i \(0.249702\pi\)
\(500\) −1.25000e8 −0.0447214
\(501\) 6.75370e8 0.239944
\(502\) −2.95901e9 −1.04396
\(503\) −3.54976e9 −1.24369 −0.621843 0.783142i \(-0.713616\pi\)
−0.621843 + 0.783142i \(0.713616\pi\)
\(504\) 3.12335e8 0.108671
\(505\) −1.71052e8 −0.0591028
\(506\) 1.90453e9 0.653523
\(507\) −1.69221e9 −0.576669
\(508\) −1.70289e9 −0.576321
\(509\) 5.26540e9 1.76978 0.884890 0.465800i \(-0.154233\pi\)
0.884890 + 0.465800i \(0.154233\pi\)
\(510\) 5.12753e8 0.171164
\(511\) −2.60549e8 −0.0863805
\(512\) 1.34218e8 0.0441942
\(513\) 1.35006e8 0.0441511
\(514\) −1.07374e9 −0.348762
\(515\) 1.99495e9 0.643586
\(516\) −9.60058e8 −0.307626
\(517\) −1.92610e9 −0.613001
\(518\) −2.29093e9 −0.724199
\(519\) −2.12420e9 −0.666977
\(520\) 1.74095e7 0.00542969
\(521\) 2.83042e9 0.876836 0.438418 0.898771i \(-0.355539\pi\)
0.438418 + 0.898771i \(0.355539\pi\)
\(522\) 2.28764e8 0.0703947
\(523\) −2.22245e9 −0.679322 −0.339661 0.940548i \(-0.610312\pi\)
−0.339661 + 0.940548i \(0.610312\pi\)
\(524\) −4.20450e8 −0.127660
\(525\) 3.53026e8 0.106475
\(526\) 2.32504e9 0.696594
\(527\) 1.34826e9 0.401269
\(528\) −3.15230e8 −0.0931985
\(529\) 3.57085e9 1.04876
\(530\) −5.97349e8 −0.174286
\(531\) −1.74085e9 −0.504580
\(532\) 3.67337e8 0.105773
\(533\) −3.48204e6 −0.000996067 0
\(534\) 6.64309e8 0.188788
\(535\) −2.78991e9 −0.787683
\(536\) 6.52147e8 0.182923
\(537\) 8.54970e8 0.238254
\(538\) −1.87326e9 −0.518633
\(539\) 3.51463e8 0.0966760
\(540\) −1.57464e8 −0.0430331
\(541\) 4.33154e9 1.17612 0.588060 0.808817i \(-0.299892\pi\)
0.588060 + 0.808817i \(0.299892\pi\)
\(542\) 3.80322e9 1.02601
\(543\) 3.86543e9 1.03609
\(544\) −6.22292e8 −0.165729
\(545\) 2.47353e9 0.654530
\(546\) −4.91681e7 −0.0129274
\(547\) −4.41256e9 −1.15275 −0.576375 0.817186i \(-0.695533\pi\)
−0.576375 + 0.817186i \(0.695533\pi\)
\(548\) −1.53841e9 −0.399337
\(549\) −1.97292e9 −0.508870
\(550\) −3.56299e8 −0.0913156
\(551\) 2.69048e8 0.0685172
\(552\) −1.15459e9 −0.292173
\(553\) −1.20962e9 −0.304167
\(554\) −4.14876e9 −1.03666
\(555\) 1.15497e9 0.286779
\(556\) −6.99922e8 −0.172698
\(557\) −3.89753e9 −0.955644 −0.477822 0.878457i \(-0.658574\pi\)
−0.477822 + 0.878457i \(0.658574\pi\)
\(558\) −4.14044e8 −0.100885
\(559\) 1.51133e8 0.0365948
\(560\) −4.28443e8 −0.103094
\(561\) 1.46155e9 0.349496
\(562\) 3.03255e9 0.720660
\(563\) 7.98467e8 0.188572 0.0942860 0.995545i \(-0.469943\pi\)
0.0942860 + 0.995545i \(0.469943\pi\)
\(564\) 1.16766e9 0.274057
\(565\) 6.80387e8 0.158704
\(566\) −1.23954e9 −0.287345
\(567\) 4.44712e8 0.102456
\(568\) 1.30517e9 0.298847
\(569\) −8.96244e8 −0.203955 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) 2.09621e9 0.471204 0.235602 0.971850i \(-0.424294\pi\)
0.235602 + 0.971850i \(0.424294\pi\)
\(572\) 4.96239e7 0.0110868
\(573\) 6.93587e8 0.154014
\(574\) 8.56920e7 0.0189125
\(575\) −1.30501e9 −0.286270
\(576\) 1.91103e8 0.0416667
\(577\) −6.21510e9 −1.34689 −0.673446 0.739237i \(-0.735187\pi\)
−0.673446 + 0.739237i \(0.735187\pi\)
\(578\) −3.97491e8 −0.0856209
\(579\) 3.11292e7 0.00666488
\(580\) −3.13805e8 −0.0667823
\(581\) 3.57832e9 0.756942
\(582\) 1.14847e9 0.241484
\(583\) −1.70268e9 −0.355871
\(584\) −1.59417e8 −0.0331200
\(585\) 2.47882e7 0.00511916
\(586\) −3.94112e9 −0.809056
\(587\) −3.38000e9 −0.689736 −0.344868 0.938651i \(-0.612076\pi\)
−0.344868 + 0.938651i \(0.612076\pi\)
\(588\) −2.13068e8 −0.0432213
\(589\) −4.86956e8 −0.0981943
\(590\) 2.38800e9 0.478687
\(591\) 3.38755e9 0.675039
\(592\) −1.40171e9 −0.277672
\(593\) 7.87465e9 1.55074 0.775371 0.631506i \(-0.217563\pi\)
0.775371 + 0.631506i \(0.217563\pi\)
\(594\) −4.48834e8 −0.0878684
\(595\) 1.98645e9 0.386606
\(596\) 3.24712e9 0.628256
\(597\) 2.12234e9 0.408230
\(598\) 1.81757e8 0.0347565
\(599\) 7.17992e9 1.36498 0.682489 0.730895i \(-0.260897\pi\)
0.682489 + 0.730895i \(0.260897\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) −2.45321e9 −0.460971 −0.230486 0.973076i \(-0.574031\pi\)
−0.230486 + 0.973076i \(0.574031\pi\)
\(602\) −3.71935e9 −0.694831
\(603\) 9.28545e8 0.172462
\(604\) −1.87166e9 −0.345618
\(605\) 1.42031e9 0.260758
\(606\) 2.95577e8 0.0539532
\(607\) 7.61815e9 1.38258 0.691288 0.722579i \(-0.257044\pi\)
0.691288 + 0.722579i \(0.257044\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) 8.86250e8 0.159000
\(610\) 2.70634e9 0.482757
\(611\) −1.83815e8 −0.0326014
\(612\) −8.86037e8 −0.156251
\(613\) 6.97473e9 1.22297 0.611485 0.791256i \(-0.290572\pi\)
0.611485 + 0.791256i \(0.290572\pi\)
\(614\) 1.14189e9 0.199083
\(615\) −4.32017e7 −0.00748924
\(616\) −1.22123e9 −0.210506
\(617\) 5.04278e9 0.864315 0.432157 0.901798i \(-0.357753\pi\)
0.432157 + 0.901798i \(0.357753\pi\)
\(618\) −3.44727e9 −0.587511
\(619\) 8.53979e8 0.144720 0.0723602 0.997379i \(-0.476947\pi\)
0.0723602 + 0.997379i \(0.476947\pi\)
\(620\) 5.67961e8 0.0957079
\(621\) −1.64393e9 −0.275463
\(622\) 7.11619e8 0.118572
\(623\) 2.57359e9 0.426414
\(624\) −3.00836e7 −0.00495661
\(625\) 2.44141e8 0.0400000
\(626\) 6.46072e9 1.05262
\(627\) −5.27872e8 −0.0855249
\(628\) −3.59767e9 −0.579645
\(629\) 6.49895e9 1.04128
\(630\) −6.10030e8 −0.0971984
\(631\) 5.87967e8 0.0931645 0.0465823 0.998914i \(-0.485167\pi\)
0.0465823 + 0.998914i \(0.485167\pi\)
\(632\) −7.40111e8 −0.116624
\(633\) −1.33039e9 −0.208481
\(634\) −5.50940e9 −0.858602
\(635\) 3.32596e9 0.515477
\(636\) 1.03222e9 0.159101
\(637\) 3.35414e7 0.00514155
\(638\) −8.94465e8 −0.136361
\(639\) 1.85834e9 0.281755
\(640\) −2.62144e8 −0.0395285
\(641\) −1.94543e9 −0.291751 −0.145876 0.989303i \(-0.546600\pi\)
−0.145876 + 0.989303i \(0.546600\pi\)
\(642\) 4.82096e9 0.719053
\(643\) 9.67317e9 1.43493 0.717465 0.696595i \(-0.245303\pi\)
0.717465 + 0.696595i \(0.245303\pi\)
\(644\) −4.47298e9 −0.659927
\(645\) 1.87511e9 0.275149
\(646\) −1.04207e9 −0.152083
\(647\) −6.66124e9 −0.966919 −0.483459 0.875367i \(-0.660620\pi\)
−0.483459 + 0.875367i \(0.660620\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 6.80672e9 0.977420
\(650\) −3.40030e7 −0.00485646
\(651\) −1.60404e9 −0.227868
\(652\) −6.21958e9 −0.878809
\(653\) −1.14296e9 −0.160633 −0.0803167 0.996769i \(-0.525593\pi\)
−0.0803167 + 0.996769i \(0.525593\pi\)
\(654\) −4.27426e9 −0.597501
\(655\) 8.21192e8 0.114183
\(656\) 5.24308e7 0.00725143
\(657\) −2.26983e8 −0.0312259
\(658\) 4.52363e9 0.619009
\(659\) 6.76019e9 0.920152 0.460076 0.887879i \(-0.347822\pi\)
0.460076 + 0.887879i \(0.347822\pi\)
\(660\) 6.15684e8 0.0833593
\(661\) 1.05365e10 1.41903 0.709515 0.704691i \(-0.248914\pi\)
0.709515 + 0.704691i \(0.248914\pi\)
\(662\) 1.18995e9 0.159414
\(663\) 1.39481e8 0.0185874
\(664\) 2.18940e9 0.290227
\(665\) −7.17454e8 −0.0946059
\(666\) −1.99580e9 −0.261792
\(667\) −3.27614e9 −0.427486
\(668\) 1.60088e9 0.207797
\(669\) −5.69373e9 −0.735200
\(670\) −1.27372e9 −0.163611
\(671\) 7.71413e9 0.985731
\(672\) 7.40350e8 0.0941119
\(673\) −8.07302e9 −1.02090 −0.510450 0.859907i \(-0.670521\pi\)
−0.510450 + 0.859907i \(0.670521\pi\)
\(674\) −2.68213e7 −0.00337419
\(675\) 3.07547e8 0.0384900
\(676\) −4.01117e9 −0.499410
\(677\) 7.42536e9 0.919724 0.459862 0.887991i \(-0.347899\pi\)
0.459862 + 0.887991i \(0.347899\pi\)
\(678\) −1.17571e9 −0.144876
\(679\) 4.44926e9 0.545436
\(680\) 1.21541e9 0.148232
\(681\) 4.36737e8 0.0529913
\(682\) 1.61891e9 0.195424
\(683\) 8.93289e9 1.07280 0.536401 0.843963i \(-0.319783\pi\)
0.536401 + 0.843963i \(0.319783\pi\)
\(684\) 3.20014e8 0.0382360
\(685\) 3.00471e9 0.357178
\(686\) −6.33859e9 −0.749650
\(687\) 7.67974e9 0.903645
\(688\) −2.27569e9 −0.266412
\(689\) −1.62493e8 −0.0189264
\(690\) 2.25505e9 0.261328
\(691\) 4.10109e9 0.472853 0.236426 0.971649i \(-0.424024\pi\)
0.236426 + 0.971649i \(0.424024\pi\)
\(692\) −5.03515e9 −0.577619
\(693\) −1.73882e9 −0.198467
\(694\) 6.83776e9 0.776526
\(695\) 1.36703e9 0.154466
\(696\) 5.42254e8 0.0609636
\(697\) −2.43092e8 −0.0271930
\(698\) −7.61487e8 −0.0847556
\(699\) 5.20266e9 0.576176
\(700\) 8.36803e8 0.0922105
\(701\) −1.01608e10 −1.11408 −0.557040 0.830486i \(-0.688063\pi\)
−0.557040 + 0.830486i \(0.688063\pi\)
\(702\) −4.28339e7 −0.00467313
\(703\) −2.34725e9 −0.254810
\(704\) −7.47212e8 −0.0807123
\(705\) −2.28059e9 −0.245124
\(706\) −7.36757e9 −0.787967
\(707\) 1.14509e9 0.121863
\(708\) −4.12646e9 −0.436979
\(709\) 1.15232e10 1.21426 0.607129 0.794603i \(-0.292321\pi\)
0.607129 + 0.794603i \(0.292321\pi\)
\(710\) −2.54916e9 −0.267297
\(711\) −1.05379e9 −0.109954
\(712\) 1.57466e9 0.163496
\(713\) 5.92956e9 0.612645
\(714\) −3.43259e9 −0.352921
\(715\) −9.69217e7 −0.00991630
\(716\) 2.02660e9 0.206334
\(717\) −5.99508e9 −0.607404
\(718\) −2.38213e9 −0.240177
\(719\) 4.00451e9 0.401789 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(720\) −3.73248e8 −0.0372678
\(721\) −1.33550e10 −1.32700
\(722\) 3.76367e8 0.0372161
\(723\) −1.14613e10 −1.12785
\(724\) 9.16250e9 0.897282
\(725\) 6.12900e8 0.0597319
\(726\) −2.45429e9 −0.238039
\(727\) 9.31051e9 0.898676 0.449338 0.893362i \(-0.351660\pi\)
0.449338 + 0.893362i \(0.351660\pi\)
\(728\) −1.16547e8 −0.0111954
\(729\) 3.87420e8 0.0370370
\(730\) 3.11362e8 0.0296234
\(731\) 1.05511e10 0.999050
\(732\) −4.67656e9 −0.440695
\(733\) 1.75464e10 1.64560 0.822798 0.568334i \(-0.192412\pi\)
0.822798 + 0.568334i \(0.192412\pi\)
\(734\) 2.64380e8 0.0246770
\(735\) 4.16149e8 0.0386583
\(736\) −2.73680e9 −0.253029
\(737\) −3.63061e9 −0.334075
\(738\) 7.46525e7 0.00683671
\(739\) 9.97408e9 0.909112 0.454556 0.890718i \(-0.349798\pi\)
0.454556 + 0.890718i \(0.349798\pi\)
\(740\) 2.73772e9 0.248358
\(741\) −5.03769e7 −0.00454849
\(742\) 3.99891e9 0.359358
\(743\) 1.46948e10 1.31433 0.657163 0.753749i \(-0.271756\pi\)
0.657163 + 0.753749i \(0.271756\pi\)
\(744\) −9.81437e8 −0.0873690
\(745\) −6.34203e9 −0.561929
\(746\) 3.66268e9 0.323008
\(747\) 3.11733e9 0.273628
\(748\) 3.46440e9 0.302672
\(749\) 1.86768e10 1.62411
\(750\) −4.21875e8 −0.0365148
\(751\) −5.42860e9 −0.467679 −0.233839 0.972275i \(-0.575129\pi\)
−0.233839 + 0.972275i \(0.575129\pi\)
\(752\) 2.76780e9 0.237340
\(753\) −9.98665e9 −0.852388
\(754\) −8.53623e7 −0.00725214
\(755\) 3.65558e9 0.309130
\(756\) 1.05413e9 0.0887296
\(757\) −1.31707e10 −1.10350 −0.551751 0.834009i \(-0.686040\pi\)
−0.551751 + 0.834009i \(0.686040\pi\)
\(758\) −8.87182e9 −0.739896
\(759\) 6.42778e9 0.533599
\(760\) −4.38976e8 −0.0362738
\(761\) −1.00210e10 −0.824260 −0.412130 0.911125i \(-0.635215\pi\)
−0.412130 + 0.911125i \(0.635215\pi\)
\(762\) −5.74726e9 −0.470564
\(763\) −1.65589e10 −1.34957
\(764\) 1.64406e9 0.133380
\(765\) 1.73054e9 0.139755
\(766\) −1.09043e10 −0.876590
\(767\) 6.49592e8 0.0519824
\(768\) 4.52985e8 0.0360844
\(769\) 9.00079e9 0.713737 0.356869 0.934155i \(-0.383844\pi\)
0.356869 + 0.934155i \(0.383844\pi\)
\(770\) 2.38521e9 0.188282
\(771\) −3.62388e9 −0.284763
\(772\) 7.37876e7 0.00577196
\(773\) −5.57639e9 −0.434235 −0.217118 0.976145i \(-0.569666\pi\)
−0.217118 + 0.976145i \(0.569666\pi\)
\(774\) −3.24020e9 −0.251176
\(775\) −1.10930e9 −0.0856038
\(776\) 2.72229e9 0.209131
\(777\) −7.73189e9 −0.591306
\(778\) 9.76800e9 0.743664
\(779\) 8.77986e7 0.00665436
\(780\) 5.87571e7 0.00443332
\(781\) −7.26611e9 −0.545787
\(782\) 1.26890e10 0.948864
\(783\) 7.72077e8 0.0574770
\(784\) −5.05051e8 −0.0374308
\(785\) 7.02669e9 0.518450
\(786\) −1.41902e9 −0.104234
\(787\) 3.98455e7 0.00291385 0.00145693 0.999999i \(-0.499536\pi\)
0.00145693 + 0.999999i \(0.499536\pi\)
\(788\) 8.02974e9 0.584601
\(789\) 7.84700e9 0.568767
\(790\) 1.44553e9 0.104311
\(791\) −4.55480e9 −0.327229
\(792\) −1.06390e9 −0.0760963
\(793\) 7.36190e8 0.0524244
\(794\) 1.56036e10 1.10625
\(795\) −2.01605e9 −0.142304
\(796\) 5.03073e9 0.353537
\(797\) −5.19745e9 −0.363652 −0.181826 0.983331i \(-0.558201\pi\)
−0.181826 + 0.983331i \(0.558201\pi\)
\(798\) 1.23976e9 0.0863630
\(799\) −1.28327e10 −0.890030
\(800\) 5.12000e8 0.0353553
\(801\) 2.24204e9 0.154145
\(802\) −2.63163e9 −0.180142
\(803\) 8.87503e8 0.0604875
\(804\) 2.20100e9 0.149356
\(805\) 8.73628e9 0.590257
\(806\) 1.54499e8 0.0103933
\(807\) −6.32225e9 −0.423462
\(808\) 7.00628e8 0.0467248
\(809\) 2.93360e9 0.194796 0.0973981 0.995246i \(-0.468948\pi\)
0.0973981 + 0.995246i \(0.468948\pi\)
\(810\) −5.31441e8 −0.0351364
\(811\) −2.05022e10 −1.34967 −0.674833 0.737970i \(-0.735784\pi\)
−0.674833 + 0.737970i \(0.735784\pi\)
\(812\) 2.10074e9 0.137698
\(813\) 1.28359e10 0.837738
\(814\) 7.80356e9 0.507116
\(815\) 1.21476e10 0.786030
\(816\) −2.10024e9 −0.135317
\(817\) −3.81079e9 −0.244477
\(818\) 6.89580e9 0.440502
\(819\) −1.65942e8 −0.0105551
\(820\) −1.02404e8 −0.00648587
\(821\) 4.09630e9 0.258339 0.129170 0.991623i \(-0.458769\pi\)
0.129170 + 0.991623i \(0.458769\pi\)
\(822\) −5.19213e9 −0.326058
\(823\) −2.13738e10 −1.33654 −0.668270 0.743919i \(-0.732965\pi\)
−0.668270 + 0.743919i \(0.732965\pi\)
\(824\) −8.17131e9 −0.508799
\(825\) −1.20251e9 −0.0745588
\(826\) −1.59863e10 −0.986999
\(827\) 1.24488e10 0.765346 0.382673 0.923884i \(-0.375004\pi\)
0.382673 + 0.923884i \(0.375004\pi\)
\(828\) −3.89673e9 −0.238558
\(829\) 2.19927e10 1.34072 0.670358 0.742038i \(-0.266141\pi\)
0.670358 + 0.742038i \(0.266141\pi\)
\(830\) −4.27618e9 −0.259587
\(831\) −1.40021e10 −0.846426
\(832\) −7.13094e7 −0.00429255
\(833\) 2.34164e9 0.140366
\(834\) −2.36224e9 −0.141007
\(835\) −3.12671e9 −0.185860
\(836\) −1.25125e9 −0.0740667
\(837\) −1.39740e9 −0.0823723
\(838\) −1.41393e10 −0.829993
\(839\) 2.85756e10 1.67043 0.835215 0.549923i \(-0.185343\pi\)
0.835215 + 0.549923i \(0.185343\pi\)
\(840\) −1.44600e9 −0.0841763
\(841\) −1.57112e10 −0.910803
\(842\) −2.24462e9 −0.129584
\(843\) 1.02348e10 0.588416
\(844\) −3.15353e9 −0.180550
\(845\) 7.83432e9 0.446686
\(846\) 3.94087e9 0.223767
\(847\) −9.50814e9 −0.537655
\(848\) 2.44674e9 0.137785
\(849\) −4.18345e9 −0.234616
\(850\) −2.37386e9 −0.132583
\(851\) 2.85819e10 1.58978
\(852\) 4.40496e9 0.244007
\(853\) −4.93353e9 −0.272168 −0.136084 0.990697i \(-0.543452\pi\)
−0.136084 + 0.990697i \(0.543452\pi\)
\(854\) −1.81174e10 −0.995391
\(855\) −6.25026e8 −0.0341993
\(856\) 1.14275e10 0.622718
\(857\) −2.34984e9 −0.127528 −0.0637641 0.997965i \(-0.520311\pi\)
−0.0637641 + 0.997965i \(0.520311\pi\)
\(858\) 1.67481e8 0.00905230
\(859\) −9.50673e9 −0.511747 −0.255873 0.966710i \(-0.582363\pi\)
−0.255873 + 0.966710i \(0.582363\pi\)
\(860\) 4.44471e9 0.238286
\(861\) 2.89210e8 0.0154420
\(862\) 6.65858e9 0.354084
\(863\) 1.22499e10 0.648778 0.324389 0.945924i \(-0.394841\pi\)
0.324389 + 0.945924i \(0.394841\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 9.83428e9 0.516638
\(866\) 1.71403e10 0.896821
\(867\) −1.34153e9 −0.0699092
\(868\) −3.80218e9 −0.197339
\(869\) 4.12032e9 0.212991
\(870\) −1.05909e9 −0.0545275
\(871\) −3.46483e8 −0.0177672
\(872\) −1.01316e10 −0.517451
\(873\) 3.87607e9 0.197171
\(874\) −4.58294e9 −0.232196
\(875\) −1.63438e9 −0.0824755
\(876\) −5.38034e8 −0.0270424
\(877\) −2.77100e10 −1.38719 −0.693597 0.720363i \(-0.743975\pi\)
−0.693597 + 0.720363i \(0.743975\pi\)
\(878\) −1.77205e10 −0.883580
\(879\) −1.33013e10 −0.660591
\(880\) 1.45940e9 0.0721913
\(881\) 1.72189e10 0.848379 0.424190 0.905573i \(-0.360559\pi\)
0.424190 + 0.905573i \(0.360559\pi\)
\(882\) −7.19105e8 −0.0352901
\(883\) 2.21121e10 1.08085 0.540427 0.841391i \(-0.318263\pi\)
0.540427 + 0.841391i \(0.318263\pi\)
\(884\) 3.30621e8 0.0160971
\(885\) 8.05949e9 0.390846
\(886\) −1.12591e10 −0.543856
\(887\) 3.13319e10 1.50749 0.753745 0.657167i \(-0.228245\pi\)
0.753745 + 0.657167i \(0.228245\pi\)
\(888\) −4.73078e9 −0.226718
\(889\) −2.22654e10 −1.06286
\(890\) −3.07550e9 −0.146235
\(891\) −1.51481e9 −0.0717443
\(892\) −1.34963e10 −0.636702
\(893\) 4.63484e9 0.217799
\(894\) 1.09590e10 0.512969
\(895\) −3.95819e9 −0.184551
\(896\) 1.75490e9 0.0815033
\(897\) 6.13428e8 0.0283786
\(898\) 1.14104e10 0.525817
\(899\) −2.78483e9 −0.127832
\(900\) 7.29000e8 0.0333333
\(901\) −1.13442e10 −0.516697
\(902\) −2.91891e8 −0.0132434
\(903\) −1.25528e10 −0.567327
\(904\) −2.78687e9 −0.125466
\(905\) −1.78955e10 −0.802553
\(906\) −6.31684e9 −0.282196
\(907\) −2.96071e10 −1.31756 −0.658781 0.752335i \(-0.728927\pi\)
−0.658781 + 0.752335i \(0.728927\pi\)
\(908\) 1.03523e9 0.0458918
\(909\) 9.97574e8 0.0440526
\(910\) 2.27630e8 0.0100135
\(911\) −8.84001e9 −0.387381 −0.193690 0.981063i \(-0.562046\pi\)
−0.193690 + 0.981063i \(0.562046\pi\)
\(912\) 7.58551e8 0.0331133
\(913\) −1.21888e10 −0.530044
\(914\) 2.05350e10 0.889575
\(915\) 9.13391e9 0.394169
\(916\) 1.82038e10 0.782580
\(917\) −5.49741e9 −0.235432
\(918\) −2.99037e9 −0.127578
\(919\) −1.09414e10 −0.465018 −0.232509 0.972594i \(-0.574693\pi\)
−0.232509 + 0.972594i \(0.574693\pi\)
\(920\) 5.34531e9 0.226316
\(921\) 3.85388e9 0.162551
\(922\) 8.96662e9 0.376765
\(923\) −6.93433e8 −0.0290268
\(924\) −4.12165e9 −0.171878
\(925\) −5.34710e9 −0.222138
\(926\) 1.42308e10 0.588965
\(927\) −1.16345e10 −0.479701
\(928\) 1.28534e9 0.0527960
\(929\) 1.20285e10 0.492216 0.246108 0.969242i \(-0.420848\pi\)
0.246108 + 0.969242i \(0.420848\pi\)
\(930\) 1.91687e9 0.0781452
\(931\) −8.45738e8 −0.0343488
\(932\) 1.23322e10 0.498983
\(933\) 2.40171e9 0.0968134
\(934\) 1.17964e10 0.473735
\(935\) −6.76641e9 −0.270718
\(936\) −1.01532e8 −0.00404705
\(937\) −1.05850e10 −0.420341 −0.210171 0.977665i \(-0.567402\pi\)
−0.210171 + 0.977665i \(0.567402\pi\)
\(938\) 8.52685e9 0.337349
\(939\) 2.18049e10 0.859459
\(940\) −5.40585e9 −0.212284
\(941\) −2.76549e10 −1.08195 −0.540977 0.841037i \(-0.681945\pi\)
−0.540977 + 0.841037i \(0.681945\pi\)
\(942\) −1.21421e10 −0.473278
\(943\) −1.06910e9 −0.0415173
\(944\) −9.78123e9 −0.378435
\(945\) −2.05885e9 −0.0793621
\(946\) 1.26692e10 0.486552
\(947\) 4.89171e7 0.00187170 0.000935849 1.00000i \(-0.499702\pi\)
0.000935849 1.00000i \(0.499702\pi\)
\(948\) −2.49787e9 −0.0952229
\(949\) 8.46979e7 0.00321692
\(950\) 8.57375e8 0.0324443
\(951\) −1.85942e10 −0.701046
\(952\) −8.13650e9 −0.305639
\(953\) 2.42208e10 0.906489 0.453245 0.891386i \(-0.350266\pi\)
0.453245 + 0.891386i \(0.350266\pi\)
\(954\) 3.48374e9 0.129905
\(955\) −3.21105e9 −0.119299
\(956\) −1.42106e10 −0.526028
\(957\) −3.01882e9 −0.111339
\(958\) −1.85158e10 −0.680397
\(959\) −2.01148e10 −0.736462
\(960\) −8.84736e8 −0.0322749
\(961\) −2.24723e10 −0.816800
\(962\) 7.44724e8 0.0269701
\(963\) 1.62707e10 0.587104
\(964\) −2.71676e10 −0.976747
\(965\) −1.44116e8 −0.00516259
\(966\) −1.50963e10 −0.538828
\(967\) −4.40304e10 −1.56589 −0.782944 0.622093i \(-0.786283\pi\)
−0.782944 + 0.622093i \(0.786283\pi\)
\(968\) −5.81758e9 −0.206148
\(969\) −3.51697e9 −0.124175
\(970\) −5.31697e9 −0.187052
\(971\) −2.09837e10 −0.735554 −0.367777 0.929914i \(-0.619881\pi\)
−0.367777 + 0.929914i \(0.619881\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −9.15151e9 −0.318492
\(974\) 1.78498e10 0.618981
\(975\) −1.14760e8 −0.00396528
\(976\) −1.10852e10 −0.381653
\(977\) −3.82831e10 −1.31334 −0.656669 0.754179i \(-0.728035\pi\)
−0.656669 + 0.754179i \(0.728035\pi\)
\(978\) −2.09911e10 −0.717544
\(979\) −8.76638e9 −0.298594
\(980\) 9.86427e8 0.0334791
\(981\) −1.44256e10 −0.487858
\(982\) 4.24974e8 0.0143210
\(983\) 2.18431e9 0.0733460 0.0366730 0.999327i \(-0.488324\pi\)
0.0366730 + 0.999327i \(0.488324\pi\)
\(984\) 1.76954e8 0.00592076
\(985\) −1.56831e10 −0.522883
\(986\) −5.95941e9 −0.197986
\(987\) 1.52673e10 0.505419
\(988\) −1.19412e8 −0.00393911
\(989\) 4.64031e10 1.52532
\(990\) 2.07793e9 0.0680626
\(991\) −3.30368e10 −1.07830 −0.539151 0.842209i \(-0.681255\pi\)
−0.539151 + 0.842209i \(0.681255\pi\)
\(992\) −2.32637e9 −0.0756638
\(993\) 4.01609e9 0.130161
\(994\) 1.70652e10 0.551136
\(995\) −9.82565e9 −0.316214
\(996\) 7.38924e9 0.236969
\(997\) 4.25291e10 1.35911 0.679553 0.733627i \(-0.262174\pi\)
0.679553 + 0.733627i \(0.262174\pi\)
\(998\) 3.14314e10 1.00094
\(999\) −6.73581e9 −0.213752
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.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.b.1.4 4 1.1 even 1 trivial