Properties

Label 570.8.a.c.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: \(x^{4} - x^{3} - 2087 x^{2} + 44517 x - 205110\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(6.63333\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -1226.02 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -1226.02 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} -4664.78 q^{11} +1728.00 q^{12} +9141.59 q^{13} -9808.17 q^{14} +3375.00 q^{15} +4096.00 q^{16} -5655.69 q^{17} +5832.00 q^{18} -6859.00 q^{19} +8000.00 q^{20} -33102.6 q^{21} -37318.3 q^{22} +28546.1 q^{23} +13824.0 q^{24} +15625.0 q^{25} +73132.7 q^{26} +19683.0 q^{27} -78465.3 q^{28} -91499.2 q^{29} +27000.0 q^{30} +257718. q^{31} +32768.0 q^{32} -125949. q^{33} -45245.5 q^{34} -153253. q^{35} +46656.0 q^{36} -349283. q^{37} -54872.0 q^{38} +246823. q^{39} +64000.0 q^{40} +16623.6 q^{41} -264821. q^{42} -873064. q^{43} -298546. q^{44} +91125.0 q^{45} +228369. q^{46} +927779. q^{47} +110592. q^{48} +679585. q^{49} +125000. q^{50} -152704. q^{51} +585062. q^{52} -101382. q^{53} +157464. q^{54} -583098. q^{55} -627723. q^{56} -185193. q^{57} -731993. q^{58} -985529. q^{59} +216000. q^{60} -756404. q^{61} +2.06174e6 q^{62} -893769. q^{63} +262144. q^{64} +1.14270e6 q^{65} -1.00759e6 q^{66} -4.75212e6 q^{67} -361964. q^{68} +770744. q^{69} -1.22602e6 q^{70} -4.47915e6 q^{71} +373248. q^{72} -3.53609e6 q^{73} -2.79427e6 q^{74} +421875. q^{75} -438976. q^{76} +5.71912e6 q^{77} +1.97458e6 q^{78} -1.32434e6 q^{79} +512000. q^{80} +531441. q^{81} +132989. q^{82} -71365.3 q^{83} -2.11856e6 q^{84} -706962. q^{85} -6.98451e6 q^{86} -2.47048e6 q^{87} -2.38837e6 q^{88} -3.73761e6 q^{89} +729000. q^{90} -1.12078e7 q^{91} +1.82695e6 q^{92} +6.95839e6 q^{93} +7.42223e6 q^{94} -857375. q^{95} +884736. q^{96} -5.37895e6 q^{97} +5.43668e6 q^{98} -3.40063e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 6570q^{11} + 6912q^{12} - 17578q^{13} - 10528q^{14} + 13500q^{15} + 16384q^{16} - 28466q^{17} + 23328q^{18} - 27436q^{19} + 32000q^{20} - 35532q^{21} - 52560q^{22} - 32136q^{23} + 55296q^{24} + 62500q^{25} - 140624q^{26} + 78732q^{27} - 84224q^{28} - 159122q^{29} + 108000q^{30} - 67974q^{31} + 131072q^{32} - 177390q^{33} - 227728q^{34} - 164500q^{35} + 186624q^{36} - 823702q^{37} - 219488q^{38} - 474606q^{39} + 256000q^{40} - 781924q^{41} - 284256q^{42} - 1115638q^{43} - 420480q^{44} + 364500q^{45} - 257088q^{46} - 209160q^{47} + 442368q^{48} - 1159308q^{49} + 500000q^{50} - 768582q^{51} - 1124992q^{52} - 848424q^{53} + 629856q^{54} - 821250q^{55} - 673792q^{56} - 740772q^{57} - 1272976q^{58} - 3677830q^{59} + 864000q^{60} - 1161072q^{61} - 543792q^{62} - 959364q^{63} + 1048576q^{64} - 2197250q^{65} - 1419120q^{66} - 6154740q^{67} - 1821824q^{68} - 867672q^{69} - 1316000q^{70} - 4456224q^{71} + 1492992q^{72} + 1057792q^{73} - 6589616q^{74} + 1687500q^{75} - 1755904q^{76} + 2402388q^{77} - 3796848q^{78} + 2910090q^{79} + 2048000q^{80} + 2125764q^{81} - 6255392q^{82} - 1767198q^{83} - 2274048q^{84} - 3558250q^{85} - 8925104q^{86} - 4296294q^{87} - 3363840q^{88} - 3677360q^{89} + 2916000q^{90} - 6727732q^{91} - 2056704q^{92} - 1835298q^{93} - 1673280q^{94} - 3429500q^{95} + 3538944q^{96} - 10419094q^{97} - 9274464q^{98} - 4789530q^{99} + O(q^{100}) \)

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
\(7\) −1226.02 −1.35100 −0.675499 0.737361i \(-0.736072\pi\)
−0.675499 + 0.737361i \(0.736072\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1000.00 0.316228
\(11\) −4664.78 −1.05671 −0.528357 0.849022i \(-0.677192\pi\)
−0.528357 + 0.849022i \(0.677192\pi\)
\(12\) 1728.00 0.288675
\(13\) 9141.59 1.15404 0.577019 0.816731i \(-0.304216\pi\)
0.577019 + 0.816731i \(0.304216\pi\)
\(14\) −9808.17 −0.955300
\(15\) 3375.00 0.258199
\(16\) 4096.00 0.250000
\(17\) −5655.69 −0.279199 −0.139600 0.990208i \(-0.544582\pi\)
−0.139600 + 0.990208i \(0.544582\pi\)
\(18\) 5832.00 0.235702
\(19\) −6859.00 −0.229416
\(20\) 8000.00 0.223607
\(21\) −33102.6 −0.779999
\(22\) −37318.3 −0.747209
\(23\) 28546.1 0.489214 0.244607 0.969622i \(-0.421341\pi\)
0.244607 + 0.969622i \(0.421341\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) 73132.7 0.816028
\(27\) 19683.0 0.192450
\(28\) −78465.3 −0.675499
\(29\) −91499.2 −0.696665 −0.348333 0.937371i \(-0.613252\pi\)
−0.348333 + 0.937371i \(0.613252\pi\)
\(30\) 27000.0 0.182574
\(31\) 257718. 1.55374 0.776871 0.629660i \(-0.216806\pi\)
0.776871 + 0.629660i \(0.216806\pi\)
\(32\) 32768.0 0.176777
\(33\) −125949. −0.610094
\(34\) −45245.5 −0.197424
\(35\) −153253. −0.604185
\(36\) 46656.0 0.166667
\(37\) −349283. −1.13363 −0.566815 0.823845i \(-0.691825\pi\)
−0.566815 + 0.823845i \(0.691825\pi\)
\(38\) −54872.0 −0.162221
\(39\) 246823. 0.666284
\(40\) 64000.0 0.158114
\(41\) 16623.6 0.0376689 0.0188344 0.999823i \(-0.494004\pi\)
0.0188344 + 0.999823i \(0.494004\pi\)
\(42\) −264821. −0.551543
\(43\) −873064. −1.67458 −0.837291 0.546757i \(-0.815862\pi\)
−0.837291 + 0.546757i \(0.815862\pi\)
\(44\) −298546. −0.528357
\(45\) 91125.0 0.149071
\(46\) 228369. 0.345927
\(47\) 927779. 1.30347 0.651737 0.758445i \(-0.274041\pi\)
0.651737 + 0.758445i \(0.274041\pi\)
\(48\) 110592. 0.144338
\(49\) 679585. 0.825196
\(50\) 125000. 0.141421
\(51\) −152704. −0.161196
\(52\) 585062. 0.577019
\(53\) −101382. −0.0935399 −0.0467700 0.998906i \(-0.514893\pi\)
−0.0467700 + 0.998906i \(0.514893\pi\)
\(54\) 157464. 0.136083
\(55\) −583098. −0.472577
\(56\) −627723. −0.477650
\(57\) −185193. −0.132453
\(58\) −731993. −0.492617
\(59\) −985529. −0.624723 −0.312361 0.949963i \(-0.601120\pi\)
−0.312361 + 0.949963i \(0.601120\pi\)
\(60\) 216000. 0.129099
\(61\) −756404. −0.426677 −0.213339 0.976978i \(-0.568434\pi\)
−0.213339 + 0.976978i \(0.568434\pi\)
\(62\) 2.06174e6 1.09866
\(63\) −893769. −0.450333
\(64\) 262144. 0.125000
\(65\) 1.14270e6 0.516101
\(66\) −1.00759e6 −0.431401
\(67\) −4.75212e6 −1.93030 −0.965151 0.261693i \(-0.915719\pi\)
−0.965151 + 0.261693i \(0.915719\pi\)
\(68\) −361964. −0.139600
\(69\) 770744. 0.282448
\(70\) −1.22602e6 −0.427223
\(71\) −4.47915e6 −1.48522 −0.742612 0.669722i \(-0.766413\pi\)
−0.742612 + 0.669722i \(0.766413\pi\)
\(72\) 373248. 0.117851
\(73\) −3.53609e6 −1.06388 −0.531941 0.846781i \(-0.678537\pi\)
−0.531941 + 0.846781i \(0.678537\pi\)
\(74\) −2.79427e6 −0.801598
\(75\) 421875. 0.115470
\(76\) −438976. −0.114708
\(77\) 5.71912e6 1.42762
\(78\) 1.97458e6 0.471134
\(79\) −1.32434e6 −0.302206 −0.151103 0.988518i \(-0.548283\pi\)
−0.151103 + 0.988518i \(0.548283\pi\)
\(80\) 512000. 0.111803
\(81\) 531441. 0.111111
\(82\) 132989. 0.0266359
\(83\) −71365.3 −0.0136998 −0.00684989 0.999977i \(-0.502180\pi\)
−0.00684989 + 0.999977i \(0.502180\pi\)
\(84\) −2.11856e6 −0.390000
\(85\) −706962. −0.124862
\(86\) −6.98451e6 −1.18411
\(87\) −2.47048e6 −0.402220
\(88\) −2.38837e6 −0.373605
\(89\) −3.73761e6 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(90\) 729000. 0.105409
\(91\) −1.12078e7 −1.55910
\(92\) 1.82695e6 0.244607
\(93\) 6.95839e6 0.897054
\(94\) 7.42223e6 0.921695
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) −5.37895e6 −0.598406 −0.299203 0.954189i \(-0.596721\pi\)
−0.299203 + 0.954189i \(0.596721\pi\)
\(98\) 5.43668e6 0.583502
\(99\) −3.40063e6 −0.352238
\(100\) 1.00000e6 0.100000
\(101\) −1.36707e7 −1.32028 −0.660139 0.751143i \(-0.729503\pi\)
−0.660139 + 0.751143i \(0.729503\pi\)
\(102\) −1.22163e6 −0.113983
\(103\) 252601. 0.0227775 0.0113887 0.999935i \(-0.496375\pi\)
0.0113887 + 0.999935i \(0.496375\pi\)
\(104\) 4.68049e6 0.408014
\(105\) −4.13782e6 −0.348826
\(106\) −811059. −0.0661427
\(107\) 656153. 0.0517800 0.0258900 0.999665i \(-0.491758\pi\)
0.0258900 + 0.999665i \(0.491758\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −8.93726e6 −0.661016 −0.330508 0.943803i \(-0.607220\pi\)
−0.330508 + 0.943803i \(0.607220\pi\)
\(110\) −4.66478e6 −0.334162
\(111\) −9.43065e6 −0.654502
\(112\) −5.02178e6 −0.337750
\(113\) −4.69004e6 −0.305775 −0.152887 0.988244i \(-0.548857\pi\)
−0.152887 + 0.988244i \(0.548857\pi\)
\(114\) −1.48154e6 −0.0936586
\(115\) 3.56826e6 0.218783
\(116\) −5.85595e6 −0.348333
\(117\) 6.66422e6 0.384679
\(118\) −7.88423e6 −0.441746
\(119\) 6.93400e6 0.377198
\(120\) 1.72800e6 0.0912871
\(121\) 2.27305e6 0.116643
\(122\) −6.05123e6 −0.301706
\(123\) 448838. 0.0217481
\(124\) 1.64940e7 0.776871
\(125\) 1.95312e6 0.0894427
\(126\) −7.15015e6 −0.318433
\(127\) 3.91285e7 1.69504 0.847521 0.530762i \(-0.178094\pi\)
0.847521 + 0.530762i \(0.178094\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.35727e7 −0.966821
\(130\) 9.14159e6 0.364939
\(131\) −3.14409e7 −1.22193 −0.610963 0.791659i \(-0.709218\pi\)
−0.610963 + 0.791659i \(0.709218\pi\)
\(132\) −8.06075e6 −0.305047
\(133\) 8.40928e6 0.309940
\(134\) −3.80169e7 −1.36493
\(135\) 2.46038e6 0.0860663
\(136\) −2.89571e6 −0.0987119
\(137\) −4.41045e7 −1.46542 −0.732709 0.680543i \(-0.761744\pi\)
−0.732709 + 0.680543i \(0.761744\pi\)
\(138\) 6.16595e6 0.199721
\(139\) 4.26450e7 1.34684 0.673421 0.739260i \(-0.264824\pi\)
0.673421 + 0.739260i \(0.264824\pi\)
\(140\) −9.80817e6 −0.302092
\(141\) 2.50500e7 0.752561
\(142\) −3.58332e7 −1.05021
\(143\) −4.26436e7 −1.21949
\(144\) 2.98598e6 0.0833333
\(145\) −1.14374e7 −0.311558
\(146\) −2.82887e7 −0.752278
\(147\) 1.83488e7 0.476427
\(148\) −2.23541e7 −0.566815
\(149\) 8.55796e6 0.211943 0.105971 0.994369i \(-0.466205\pi\)
0.105971 + 0.994369i \(0.466205\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) 4.01350e7 0.948645 0.474322 0.880351i \(-0.342693\pi\)
0.474322 + 0.880351i \(0.342693\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) −4.12300e6 −0.0930665
\(154\) 4.57530e7 1.00948
\(155\) 3.22148e7 0.694855
\(156\) 1.57967e7 0.333142
\(157\) 6.26893e7 1.29284 0.646419 0.762982i \(-0.276266\pi\)
0.646419 + 0.762982i \(0.276266\pi\)
\(158\) −1.05947e7 −0.213692
\(159\) −2.73733e6 −0.0540053
\(160\) 4.09600e6 0.0790569
\(161\) −3.49981e7 −0.660928
\(162\) 4.25153e6 0.0785674
\(163\) 4.59226e7 0.830557 0.415279 0.909694i \(-0.363684\pi\)
0.415279 + 0.909694i \(0.363684\pi\)
\(164\) 1.06391e6 0.0188344
\(165\) −1.57436e7 −0.272842
\(166\) −570922. −0.00968721
\(167\) 2.20343e7 0.366093 0.183046 0.983104i \(-0.441404\pi\)
0.183046 + 0.983104i \(0.441404\pi\)
\(168\) −1.69485e7 −0.275771
\(169\) 2.08202e7 0.331803
\(170\) −5.65569e6 −0.0882906
\(171\) −5.00021e6 −0.0764719
\(172\) −5.58761e7 −0.837291
\(173\) −6.47515e7 −0.950799 −0.475400 0.879770i \(-0.657696\pi\)
−0.475400 + 0.879770i \(0.657696\pi\)
\(174\) −1.97638e7 −0.284412
\(175\) −1.91566e7 −0.270200
\(176\) −1.91070e7 −0.264178
\(177\) −2.66093e7 −0.360684
\(178\) −2.99008e7 −0.397387
\(179\) 5.05932e7 0.659335 0.329668 0.944097i \(-0.393063\pi\)
0.329668 + 0.944097i \(0.393063\pi\)
\(180\) 5.83200e6 0.0745356
\(181\) −1.39134e8 −1.74405 −0.872026 0.489459i \(-0.837194\pi\)
−0.872026 + 0.489459i \(0.837194\pi\)
\(182\) −8.96623e7 −1.10245
\(183\) −2.04229e7 −0.246342
\(184\) 1.46156e7 0.172963
\(185\) −4.36604e7 −0.506975
\(186\) 5.56671e7 0.634313
\(187\) 2.63826e7 0.295034
\(188\) 5.93779e7 0.651737
\(189\) −2.41318e7 −0.260000
\(190\) −6.85900e6 −0.0725476
\(191\) 6.61827e7 0.687270 0.343635 0.939103i \(-0.388342\pi\)
0.343635 + 0.939103i \(0.388342\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −5.81508e7 −0.582244 −0.291122 0.956686i \(-0.594029\pi\)
−0.291122 + 0.956686i \(0.594029\pi\)
\(194\) −4.30316e7 −0.423137
\(195\) 3.08529e7 0.297971
\(196\) 4.34934e7 0.412598
\(197\) 1.32939e8 1.23885 0.619426 0.785055i \(-0.287366\pi\)
0.619426 + 0.785055i \(0.287366\pi\)
\(198\) −2.72050e7 −0.249070
\(199\) −7.83654e7 −0.704918 −0.352459 0.935827i \(-0.614654\pi\)
−0.352459 + 0.935827i \(0.614654\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) −1.28307e8 −1.11446
\(202\) −1.09365e8 −0.933578
\(203\) 1.12180e8 0.941194
\(204\) −9.77304e6 −0.0805979
\(205\) 2.07795e6 0.0168460
\(206\) 2.02081e6 0.0161061
\(207\) 2.08101e7 0.163071
\(208\) 3.74440e7 0.288509
\(209\) 3.19958e7 0.242427
\(210\) −3.31026e7 −0.246657
\(211\) 5.07664e7 0.372039 0.186019 0.982546i \(-0.440441\pi\)
0.186019 + 0.982546i \(0.440441\pi\)
\(212\) −6.48847e6 −0.0467700
\(213\) −1.20937e8 −0.857494
\(214\) 5.24922e6 0.0366140
\(215\) −1.09133e8 −0.748896
\(216\) 1.00777e7 0.0680414
\(217\) −3.15968e8 −2.09910
\(218\) −7.14981e7 −0.467409
\(219\) −9.54745e7 −0.614233
\(220\) −3.73183e7 −0.236288
\(221\) −5.17020e7 −0.322207
\(222\) −7.54452e7 −0.462803
\(223\) −1.46701e8 −0.885861 −0.442930 0.896556i \(-0.646061\pi\)
−0.442930 + 0.896556i \(0.646061\pi\)
\(224\) −4.01743e7 −0.238825
\(225\) 1.13906e7 0.0666667
\(226\) −3.75203e7 −0.216215
\(227\) −5.87583e7 −0.333410 −0.166705 0.986007i \(-0.553313\pi\)
−0.166705 + 0.986007i \(0.553313\pi\)
\(228\) −1.18524e7 −0.0662266
\(229\) 3.04392e8 1.67498 0.837488 0.546456i \(-0.184024\pi\)
0.837488 + 0.546456i \(0.184024\pi\)
\(230\) 2.85461e7 0.154703
\(231\) 1.54416e8 0.824236
\(232\) −4.68476e7 −0.246308
\(233\) −2.17333e8 −1.12559 −0.562793 0.826598i \(-0.690273\pi\)
−0.562793 + 0.826598i \(0.690273\pi\)
\(234\) 5.33138e7 0.272009
\(235\) 1.15972e8 0.582931
\(236\) −6.30739e7 −0.312361
\(237\) −3.57571e7 −0.174479
\(238\) 5.54720e7 0.266719
\(239\) 2.48827e8 1.17897 0.589487 0.807778i \(-0.299330\pi\)
0.589487 + 0.807778i \(0.299330\pi\)
\(240\) 1.38240e7 0.0645497
\(241\) −3.68119e8 −1.69406 −0.847030 0.531544i \(-0.821612\pi\)
−0.847030 + 0.531544i \(0.821612\pi\)
\(242\) 1.81844e7 0.0824793
\(243\) 1.43489e7 0.0641500
\(244\) −4.84099e7 −0.213339
\(245\) 8.49481e7 0.369039
\(246\) 3.59071e6 0.0153783
\(247\) −6.27022e7 −0.264754
\(248\) 1.31952e8 0.549331
\(249\) −1.92686e6 −0.00790957
\(250\) 1.56250e7 0.0632456
\(251\) −1.97259e8 −0.787370 −0.393685 0.919245i \(-0.628800\pi\)
−0.393685 + 0.919245i \(0.628800\pi\)
\(252\) −5.72012e7 −0.225166
\(253\) −1.33161e8 −0.516959
\(254\) 3.13028e8 1.19858
\(255\) −1.90880e7 −0.0720890
\(256\) 1.67772e7 0.0625000
\(257\) −2.06192e8 −0.757713 −0.378857 0.925455i \(-0.623683\pi\)
−0.378857 + 0.925455i \(0.623683\pi\)
\(258\) −1.88582e8 −0.683646
\(259\) 4.28229e8 1.53153
\(260\) 7.31327e7 0.258051
\(261\) −6.67029e7 −0.232222
\(262\) −2.51527e8 −0.864032
\(263\) −1.32978e8 −0.450749 −0.225375 0.974272i \(-0.572361\pi\)
−0.225375 + 0.974272i \(0.572361\pi\)
\(264\) −6.44860e7 −0.215701
\(265\) −1.26728e7 −0.0418323
\(266\) 6.72742e7 0.219161
\(267\) −1.00915e8 −0.324465
\(268\) −3.04136e8 −0.965151
\(269\) 1.76304e8 0.552241 0.276120 0.961123i \(-0.410951\pi\)
0.276120 + 0.961123i \(0.410951\pi\)
\(270\) 1.96830e7 0.0608581
\(271\) −2.81735e8 −0.859902 −0.429951 0.902852i \(-0.641469\pi\)
−0.429951 + 0.902852i \(0.641469\pi\)
\(272\) −2.31657e7 −0.0697999
\(273\) −3.02610e8 −0.900149
\(274\) −3.52836e8 −1.03621
\(275\) −7.28873e7 −0.211343
\(276\) 4.93276e7 0.141224
\(277\) 3.32412e8 0.939718 0.469859 0.882742i \(-0.344305\pi\)
0.469859 + 0.882742i \(0.344305\pi\)
\(278\) 3.41160e8 0.952360
\(279\) 1.87877e8 0.517914
\(280\) −7.84653e7 −0.213612
\(281\) 3.64016e8 0.978698 0.489349 0.872088i \(-0.337234\pi\)
0.489349 + 0.872088i \(0.337234\pi\)
\(282\) 2.00400e8 0.532141
\(283\) 4.69965e8 1.23257 0.616287 0.787522i \(-0.288636\pi\)
0.616287 + 0.787522i \(0.288636\pi\)
\(284\) −2.86666e8 −0.742612
\(285\) −2.31491e7 −0.0592349
\(286\) −3.41148e8 −0.862308
\(287\) −2.03809e7 −0.0508906
\(288\) 2.38879e7 0.0589256
\(289\) −3.78352e8 −0.922048
\(290\) −9.14992e7 −0.220305
\(291\) −1.45232e8 −0.345490
\(292\) −2.26310e8 −0.531941
\(293\) −7.25312e8 −1.68457 −0.842284 0.539035i \(-0.818789\pi\)
−0.842284 + 0.539035i \(0.818789\pi\)
\(294\) 1.46790e8 0.336885
\(295\) −1.23191e8 −0.279385
\(296\) −1.78833e8 −0.400799
\(297\) −9.18170e7 −0.203365
\(298\) 6.84636e7 0.149866
\(299\) 2.60957e8 0.564572
\(300\) 2.70000e7 0.0577350
\(301\) 1.07040e9 2.26236
\(302\) 3.21080e8 0.670793
\(303\) −3.69109e8 −0.762263
\(304\) −2.80945e7 −0.0573539
\(305\) −9.45505e7 −0.190816
\(306\) −3.29840e7 −0.0658079
\(307\) 1.97615e8 0.389795 0.194898 0.980824i \(-0.437563\pi\)
0.194898 + 0.980824i \(0.437563\pi\)
\(308\) 3.66024e8 0.713809
\(309\) 6.82023e6 0.0131506
\(310\) 2.57718e8 0.491337
\(311\) 8.09420e8 1.52585 0.762927 0.646485i \(-0.223762\pi\)
0.762927 + 0.646485i \(0.223762\pi\)
\(312\) 1.26373e8 0.235567
\(313\) −3.54187e8 −0.652870 −0.326435 0.945220i \(-0.605847\pi\)
−0.326435 + 0.945220i \(0.605847\pi\)
\(314\) 5.01514e8 0.914175
\(315\) −1.11721e8 −0.201395
\(316\) −8.47576e7 −0.151103
\(317\) 1.06264e9 1.87361 0.936803 0.349857i \(-0.113770\pi\)
0.936803 + 0.349857i \(0.113770\pi\)
\(318\) −2.18986e7 −0.0381875
\(319\) 4.26824e8 0.736176
\(320\) 3.27680e7 0.0559017
\(321\) 1.77161e7 0.0298952
\(322\) −2.79985e8 −0.467346
\(323\) 3.87924e7 0.0640527
\(324\) 3.40122e7 0.0555556
\(325\) 1.42837e8 0.230808
\(326\) 3.67381e8 0.587293
\(327\) −2.41306e8 −0.381638
\(328\) 8.51130e6 0.0133180
\(329\) −1.13748e9 −1.76099
\(330\) −1.25949e8 −0.192929
\(331\) 2.26602e8 0.343452 0.171726 0.985145i \(-0.445066\pi\)
0.171726 + 0.985145i \(0.445066\pi\)
\(332\) −4.56738e6 −0.00684989
\(333\) −2.54627e8 −0.377877
\(334\) 1.76274e8 0.258867
\(335\) −5.94015e8 −0.863258
\(336\) −1.35588e8 −0.195000
\(337\) 3.83591e8 0.545964 0.272982 0.962019i \(-0.411990\pi\)
0.272982 + 0.962019i \(0.411990\pi\)
\(338\) 1.66561e8 0.234620
\(339\) −1.26631e8 −0.176539
\(340\) −4.52455e7 −0.0624309
\(341\) −1.20220e9 −1.64186
\(342\) −4.00017e7 −0.0540738
\(343\) 1.76496e8 0.236160
\(344\) −4.47009e8 −0.592054
\(345\) 9.63430e7 0.126315
\(346\) −5.18012e8 −0.672317
\(347\) 5.75945e8 0.739993 0.369997 0.929033i \(-0.379359\pi\)
0.369997 + 0.929033i \(0.379359\pi\)
\(348\) −1.58111e8 −0.201110
\(349\) −1.03015e9 −1.29721 −0.648606 0.761124i \(-0.724648\pi\)
−0.648606 + 0.761124i \(0.724648\pi\)
\(350\) −1.53253e8 −0.191060
\(351\) 1.79934e8 0.222095
\(352\) −1.52856e8 −0.186802
\(353\) −8.91753e8 −1.07903 −0.539514 0.841976i \(-0.681392\pi\)
−0.539514 + 0.841976i \(0.681392\pi\)
\(354\) −2.12874e8 −0.255042
\(355\) −5.59894e8 −0.664212
\(356\) −2.39207e8 −0.280995
\(357\) 1.87218e8 0.217775
\(358\) 4.04745e8 0.466220
\(359\) −1.42951e9 −1.63063 −0.815316 0.579017i \(-0.803437\pi\)
−0.815316 + 0.579017i \(0.803437\pi\)
\(360\) 4.66560e7 0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) −1.11308e9 −1.23323
\(363\) 6.13723e7 0.0673440
\(364\) −7.17298e8 −0.779552
\(365\) −4.42011e8 −0.475782
\(366\) −1.63383e8 −0.174190
\(367\) −1.08198e9 −1.14259 −0.571293 0.820746i \(-0.693558\pi\)
−0.571293 + 0.820746i \(0.693558\pi\)
\(368\) 1.16925e8 0.122304
\(369\) 1.21186e7 0.0125563
\(370\) −3.49283e8 −0.358486
\(371\) 1.24297e8 0.126372
\(372\) 4.45337e8 0.448527
\(373\) −1.41493e9 −1.41174 −0.705870 0.708341i \(-0.749444\pi\)
−0.705870 + 0.708341i \(0.749444\pi\)
\(374\) 2.11061e8 0.208620
\(375\) 5.27344e7 0.0516398
\(376\) 4.75023e8 0.460847
\(377\) −8.36448e8 −0.803978
\(378\) −1.93054e8 −0.183848
\(379\) −1.51393e8 −0.142846 −0.0714232 0.997446i \(-0.522754\pi\)
−0.0714232 + 0.997446i \(0.522754\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) 1.05647e9 0.978633
\(382\) 5.29461e8 0.485973
\(383\) 5.80373e8 0.527851 0.263926 0.964543i \(-0.414983\pi\)
0.263926 + 0.964543i \(0.414983\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 7.14891e8 0.638450
\(386\) −4.65206e8 −0.411709
\(387\) −6.36464e8 −0.558194
\(388\) −3.44252e8 −0.299203
\(389\) 1.59614e8 0.137482 0.0687411 0.997635i \(-0.478102\pi\)
0.0687411 + 0.997635i \(0.478102\pi\)
\(390\) 2.46823e8 0.210698
\(391\) −1.61448e8 −0.136588
\(392\) 3.47947e8 0.291751
\(393\) −8.48904e8 −0.705480
\(394\) 1.06351e9 0.876000
\(395\) −1.65542e8 −0.135151
\(396\) −2.17640e8 −0.176119
\(397\) 2.30750e8 0.185087 0.0925435 0.995709i \(-0.470500\pi\)
0.0925435 + 0.995709i \(0.470500\pi\)
\(398\) −6.26923e8 −0.498453
\(399\) 2.27051e8 0.178944
\(400\) 6.40000e7 0.0500000
\(401\) −7.76649e8 −0.601477 −0.300739 0.953707i \(-0.597233\pi\)
−0.300739 + 0.953707i \(0.597233\pi\)
\(402\) −1.02646e9 −0.788043
\(403\) 2.35595e9 1.79308
\(404\) −8.74924e8 −0.660139
\(405\) 6.64301e7 0.0496904
\(406\) 8.97439e8 0.665524
\(407\) 1.62933e9 1.19792
\(408\) −7.81843e7 −0.0569913
\(409\) 5.26460e8 0.380482 0.190241 0.981737i \(-0.439073\pi\)
0.190241 + 0.981737i \(0.439073\pi\)
\(410\) 1.66236e7 0.0119119
\(411\) −1.19082e9 −0.846059
\(412\) 1.61665e7 0.0113887
\(413\) 1.20828e9 0.844000
\(414\) 1.66481e8 0.115309
\(415\) −8.92066e6 −0.00612673
\(416\) 2.99552e8 0.204007
\(417\) 1.15142e9 0.777599
\(418\) 2.55966e8 0.171422
\(419\) −2.31838e9 −1.53970 −0.769850 0.638225i \(-0.779669\pi\)
−0.769850 + 0.638225i \(0.779669\pi\)
\(420\) −2.64821e8 −0.174413
\(421\) −2.16319e9 −1.41288 −0.706442 0.707771i \(-0.749701\pi\)
−0.706442 + 0.707771i \(0.749701\pi\)
\(422\) 4.06131e8 0.263071
\(423\) 6.76351e8 0.434491
\(424\) −5.19078e7 −0.0330714
\(425\) −8.83702e7 −0.0558399
\(426\) −9.67497e8 −0.606340
\(427\) 9.27368e8 0.576440
\(428\) 4.19938e7 0.0258900
\(429\) −1.15138e9 −0.704071
\(430\) −8.73064e8 −0.529550
\(431\) −2.19064e8 −0.131796 −0.0658979 0.997826i \(-0.520991\pi\)
−0.0658979 + 0.997826i \(0.520991\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 3.08103e9 1.82384 0.911922 0.410363i \(-0.134598\pi\)
0.911922 + 0.410363i \(0.134598\pi\)
\(434\) −2.52774e9 −1.48429
\(435\) −3.08810e8 −0.179878
\(436\) −5.71985e8 −0.330508
\(437\) −1.95798e8 −0.112233
\(438\) −7.63796e8 −0.434328
\(439\) 2.69086e9 1.51798 0.758988 0.651105i \(-0.225694\pi\)
0.758988 + 0.651105i \(0.225694\pi\)
\(440\) −2.98546e8 −0.167081
\(441\) 4.95417e8 0.275065
\(442\) −4.13616e8 −0.227835
\(443\) 3.73882e7 0.0204325 0.0102162 0.999948i \(-0.496748\pi\)
0.0102162 + 0.999948i \(0.496748\pi\)
\(444\) −6.03561e8 −0.327251
\(445\) −4.67201e8 −0.251330
\(446\) −1.17361e9 −0.626398
\(447\) 2.31065e8 0.122365
\(448\) −3.21394e8 −0.168875
\(449\) 1.53922e9 0.802489 0.401244 0.915971i \(-0.368578\pi\)
0.401244 + 0.915971i \(0.368578\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) −7.75457e7 −0.0398052
\(452\) −3.00162e8 −0.152887
\(453\) 1.08364e9 0.547700
\(454\) −4.70066e8 −0.235757
\(455\) −1.40097e9 −0.697252
\(456\) −9.48188e7 −0.0468293
\(457\) −3.29252e9 −1.61370 −0.806848 0.590759i \(-0.798828\pi\)
−0.806848 + 0.590759i \(0.798828\pi\)
\(458\) 2.43513e9 1.18439
\(459\) −1.11321e8 −0.0537320
\(460\) 2.28369e8 0.109392
\(461\) 2.59005e9 1.23128 0.615638 0.788029i \(-0.288898\pi\)
0.615638 + 0.788029i \(0.288898\pi\)
\(462\) 1.23533e9 0.582823
\(463\) 2.10724e9 0.986689 0.493344 0.869834i \(-0.335774\pi\)
0.493344 + 0.869834i \(0.335774\pi\)
\(464\) −3.74781e8 −0.174166
\(465\) 8.69799e8 0.401175
\(466\) −1.73866e9 −0.795910
\(467\) 9.33516e8 0.424143 0.212072 0.977254i \(-0.431979\pi\)
0.212072 + 0.977254i \(0.431979\pi\)
\(468\) 4.26510e8 0.192340
\(469\) 5.82620e9 2.60784
\(470\) 9.27779e8 0.412194
\(471\) 1.69261e9 0.746421
\(472\) −5.04591e8 −0.220873
\(473\) 4.07266e9 1.76955
\(474\) −2.86057e8 −0.123375
\(475\) −1.07172e8 −0.0458831
\(476\) 4.43776e8 0.188599
\(477\) −7.39078e7 −0.0311800
\(478\) 1.99061e9 0.833660
\(479\) 9.27378e8 0.385552 0.192776 0.981243i \(-0.438251\pi\)
0.192776 + 0.981243i \(0.438251\pi\)
\(480\) 1.10592e8 0.0456435
\(481\) −3.19300e9 −1.30825
\(482\) −2.94495e9 −1.19788
\(483\) −9.44949e8 −0.381587
\(484\) 1.45475e8 0.0583216
\(485\) −6.72368e8 −0.267615
\(486\) 1.14791e8 0.0453609
\(487\) 3.59857e7 0.0141182 0.00705909 0.999975i \(-0.497753\pi\)
0.00705909 + 0.999975i \(0.497753\pi\)
\(488\) −3.87279e8 −0.150853
\(489\) 1.23991e9 0.479523
\(490\) 6.79585e8 0.260950
\(491\) −4.85651e9 −1.85157 −0.925783 0.378056i \(-0.876593\pi\)
−0.925783 + 0.378056i \(0.876593\pi\)
\(492\) 2.87256e7 0.0108741
\(493\) 5.17491e8 0.194509
\(494\) −5.01617e8 −0.187210
\(495\) −4.25079e8 −0.157526
\(496\) 1.05561e9 0.388436
\(497\) 5.49154e9 2.00653
\(498\) −1.54149e7 −0.00559291
\(499\) −5.20186e8 −0.187416 −0.0937080 0.995600i \(-0.529872\pi\)
−0.0937080 + 0.995600i \(0.529872\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) 5.94926e8 0.211364
\(502\) −1.57807e9 −0.556754
\(503\) −4.48316e8 −0.157071 −0.0785355 0.996911i \(-0.525024\pi\)
−0.0785355 + 0.996911i \(0.525024\pi\)
\(504\) −4.57610e8 −0.159217
\(505\) −1.70884e9 −0.590446
\(506\) −1.06529e9 −0.365545
\(507\) 5.62144e8 0.191567
\(508\) 2.50423e9 0.847521
\(509\) −1.93635e9 −0.650837 −0.325419 0.945570i \(-0.605505\pi\)
−0.325419 + 0.945570i \(0.605505\pi\)
\(510\) −1.52704e8 −0.0509746
\(511\) 4.33532e9 1.43730
\(512\) 1.34218e8 0.0441942
\(513\) −1.35006e8 −0.0441511
\(514\) −1.64953e9 −0.535784
\(515\) 3.15752e7 0.0101864
\(516\) −1.50866e9 −0.483410
\(517\) −4.32789e9 −1.37740
\(518\) 3.42583e9 1.08296
\(519\) −1.74829e9 −0.548944
\(520\) 5.85062e8 0.182469
\(521\) 3.32620e9 1.03042 0.515212 0.857063i \(-0.327713\pi\)
0.515212 + 0.857063i \(0.327713\pi\)
\(522\) −5.33623e8 −0.164206
\(523\) −2.66642e9 −0.815029 −0.407515 0.913199i \(-0.633604\pi\)
−0.407515 + 0.913199i \(0.633604\pi\)
\(524\) −2.01222e9 −0.610963
\(525\) −5.17228e8 −0.156000
\(526\) −1.06382e9 −0.318728
\(527\) −1.45757e9 −0.433804
\(528\) −5.15888e8 −0.152523
\(529\) −2.58995e9 −0.760669
\(530\) −1.01382e8 −0.0295799
\(531\) −7.18451e8 −0.208241
\(532\) 5.38194e8 0.154970
\(533\) 1.51966e8 0.0434713
\(534\) −8.07323e8 −0.229431
\(535\) 8.20191e7 0.0231567
\(536\) −2.43308e9 −0.682465
\(537\) 1.36602e9 0.380667
\(538\) 1.41043e9 0.390493
\(539\) −3.17012e9 −0.871996
\(540\) 1.57464e8 0.0430331
\(541\) 3.53887e9 0.960892 0.480446 0.877024i \(-0.340475\pi\)
0.480446 + 0.877024i \(0.340475\pi\)
\(542\) −2.25388e9 −0.608042
\(543\) −3.75663e9 −1.00693
\(544\) −1.85326e8 −0.0493560
\(545\) −1.11716e9 −0.295615
\(546\) −2.42088e9 −0.636501
\(547\) 6.17856e9 1.61410 0.807052 0.590481i \(-0.201062\pi\)
0.807052 + 0.590481i \(0.201062\pi\)
\(548\) −2.82269e9 −0.732709
\(549\) −5.51419e8 −0.142226
\(550\) −5.83098e8 −0.149442
\(551\) 6.27593e8 0.159826
\(552\) 3.94621e8 0.0998604
\(553\) 1.62367e9 0.408280
\(554\) 2.65930e9 0.664481
\(555\) −1.17883e9 −0.292702
\(556\) 2.72928e9 0.673421
\(557\) 4.64096e8 0.113793 0.0568964 0.998380i \(-0.481880\pi\)
0.0568964 + 0.998380i \(0.481880\pi\)
\(558\) 1.50301e9 0.366221
\(559\) −7.98120e9 −1.93253
\(560\) −6.27723e8 −0.151046
\(561\) 7.12330e8 0.170338
\(562\) 2.91213e9 0.692044
\(563\) 6.20879e9 1.46632 0.733158 0.680059i \(-0.238046\pi\)
0.733158 + 0.680059i \(0.238046\pi\)
\(564\) 1.60320e9 0.376280
\(565\) −5.86255e8 −0.136747
\(566\) 3.75972e9 0.871561
\(567\) −6.51558e8 −0.150111
\(568\) −2.29333e9 −0.525106
\(569\) −3.12134e8 −0.0710311 −0.0355155 0.999369i \(-0.511307\pi\)
−0.0355155 + 0.999369i \(0.511307\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) 5.24046e9 1.17799 0.588997 0.808135i \(-0.299523\pi\)
0.588997 + 0.808135i \(0.299523\pi\)
\(572\) −2.72919e9 −0.609744
\(573\) 1.78693e9 0.396796
\(574\) −1.63047e8 −0.0359851
\(575\) 4.46033e8 0.0978429
\(576\) 1.91103e8 0.0416667
\(577\) −6.89035e9 −1.49323 −0.746614 0.665257i \(-0.768322\pi\)
−0.746614 + 0.665257i \(0.768322\pi\)
\(578\) −3.02681e9 −0.651986
\(579\) −1.57007e9 −0.336159
\(580\) −7.31993e8 −0.155779
\(581\) 8.74954e7 0.0185084
\(582\) −1.16185e9 −0.244298
\(583\) 4.72927e8 0.0988449
\(584\) −1.81048e9 −0.376139
\(585\) 8.33027e8 0.172034
\(586\) −5.80250e9 −1.19117
\(587\) 1.37892e9 0.281388 0.140694 0.990053i \(-0.455067\pi\)
0.140694 + 0.990053i \(0.455067\pi\)
\(588\) 1.17432e9 0.238214
\(589\) −1.76769e9 −0.356453
\(590\) −9.85529e8 −0.197555
\(591\) 3.58934e9 0.715251
\(592\) −1.43066e9 −0.283408
\(593\) −3.35153e9 −0.660012 −0.330006 0.943979i \(-0.607051\pi\)
−0.330006 + 0.943979i \(0.607051\pi\)
\(594\) −7.34536e8 −0.143800
\(595\) 8.66750e8 0.168688
\(596\) 5.47709e8 0.105971
\(597\) −2.11587e9 −0.406985
\(598\) 2.08765e9 0.399213
\(599\) 6.74203e9 1.28173 0.640865 0.767654i \(-0.278576\pi\)
0.640865 + 0.767654i \(0.278576\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) 4.63680e9 0.871281 0.435640 0.900121i \(-0.356522\pi\)
0.435640 + 0.900121i \(0.356522\pi\)
\(602\) 8.56316e9 1.59973
\(603\) −3.46429e9 −0.643434
\(604\) 2.56864e9 0.474322
\(605\) 2.84131e8 0.0521645
\(606\) −2.95287e9 −0.539001
\(607\) −4.15246e9 −0.753607 −0.376804 0.926293i \(-0.622977\pi\)
−0.376804 + 0.926293i \(0.622977\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) 3.02886e9 0.543398
\(610\) −7.56404e8 −0.134927
\(611\) 8.48138e9 1.50426
\(612\) −2.63872e8 −0.0465332
\(613\) −5.87720e9 −1.03053 −0.515263 0.857032i \(-0.672306\pi\)
−0.515263 + 0.857032i \(0.672306\pi\)
\(614\) 1.58092e9 0.275627
\(615\) 5.61048e7 0.00972606
\(616\) 2.92819e9 0.504739
\(617\) 4.26910e9 0.731709 0.365854 0.930672i \(-0.380777\pi\)
0.365854 + 0.930672i \(0.380777\pi\)
\(618\) 5.45619e7 0.00929886
\(619\) 6.08008e9 1.03037 0.515183 0.857080i \(-0.327724\pi\)
0.515183 + 0.857080i \(0.327724\pi\)
\(620\) 2.06174e9 0.347427
\(621\) 5.61873e8 0.0941493
\(622\) 6.47536e9 1.07894
\(623\) 4.58238e9 0.759247
\(624\) 1.01099e9 0.166571
\(625\) 2.44141e8 0.0400000
\(626\) −2.83349e9 −0.461649
\(627\) 8.63886e8 0.139965
\(628\) 4.01211e9 0.646419
\(629\) 1.97544e9 0.316509
\(630\) −8.93769e8 −0.142408
\(631\) −2.76268e9 −0.437751 −0.218876 0.975753i \(-0.570239\pi\)
−0.218876 + 0.975753i \(0.570239\pi\)
\(632\) −6.78061e8 −0.106846
\(633\) 1.37069e9 0.214797
\(634\) 8.50111e9 1.32484
\(635\) 4.89107e9 0.758046
\(636\) −1.75189e8 −0.0270027
\(637\) 6.21248e9 0.952308
\(638\) 3.41459e9 0.520555
\(639\) −3.26530e9 −0.495074
\(640\) 2.62144e8 0.0395285
\(641\) −4.56044e9 −0.683918 −0.341959 0.939715i \(-0.611090\pi\)
−0.341959 + 0.939715i \(0.611090\pi\)
\(642\) 1.41729e8 0.0211391
\(643\) −5.32233e9 −0.789521 −0.394760 0.918784i \(-0.629172\pi\)
−0.394760 + 0.918784i \(0.629172\pi\)
\(644\) −2.23988e9 −0.330464
\(645\) −2.94659e9 −0.432375
\(646\) 3.10339e8 0.0452921
\(647\) 3.18208e9 0.461897 0.230949 0.972966i \(-0.425817\pi\)
0.230949 + 0.972966i \(0.425817\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 4.59728e9 0.660153
\(650\) 1.14270e9 0.163206
\(651\) −8.53113e9 −1.21192
\(652\) 2.93905e9 0.415279
\(653\) 9.07611e9 1.27557 0.637784 0.770215i \(-0.279851\pi\)
0.637784 + 0.770215i \(0.279851\pi\)
\(654\) −1.93045e9 −0.269859
\(655\) −3.93011e9 −0.546462
\(656\) 6.80904e7 0.00941722
\(657\) −2.57781e9 −0.354627
\(658\) −9.09981e9 −1.24521
\(659\) 5.36153e9 0.729777 0.364888 0.931051i \(-0.381107\pi\)
0.364888 + 0.931051i \(0.381107\pi\)
\(660\) −1.00759e9 −0.136421
\(661\) 6.20884e9 0.836191 0.418096 0.908403i \(-0.362698\pi\)
0.418096 + 0.908403i \(0.362698\pi\)
\(662\) 1.81282e9 0.242857
\(663\) −1.39595e9 −0.186026
\(664\) −3.65390e7 −0.00484361
\(665\) 1.05116e9 0.138609
\(666\) −2.03702e9 −0.267199
\(667\) −2.61194e9 −0.340819
\(668\) 1.41019e9 0.183046
\(669\) −3.96092e9 −0.511452
\(670\) −4.75212e9 −0.610415
\(671\) 3.52846e9 0.450876
\(672\) −1.08470e9 −0.137886
\(673\) 3.64417e9 0.460836 0.230418 0.973092i \(-0.425991\pi\)
0.230418 + 0.973092i \(0.425991\pi\)
\(674\) 3.06873e9 0.386055
\(675\) 3.07547e8 0.0384900
\(676\) 1.33249e9 0.165902
\(677\) 9.03739e9 1.11939 0.559697 0.828697i \(-0.310918\pi\)
0.559697 + 0.828697i \(0.310918\pi\)
\(678\) −1.01305e9 −0.124832
\(679\) 6.59470e9 0.808446
\(680\) −3.61964e8 −0.0441453
\(681\) −1.58647e9 −0.192494
\(682\) −9.61760e9 −1.16097
\(683\) −4.80985e9 −0.577643 −0.288821 0.957383i \(-0.593263\pi\)
−0.288821 + 0.957383i \(0.593263\pi\)
\(684\) −3.20014e8 −0.0382360
\(685\) −5.51307e9 −0.655354
\(686\) 1.41197e9 0.166990
\(687\) 8.21857e9 0.967047
\(688\) −3.57607e9 −0.418646
\(689\) −9.26797e8 −0.107949
\(690\) 7.70744e8 0.0893179
\(691\) 8.74228e9 1.00798 0.503990 0.863710i \(-0.331865\pi\)
0.503990 + 0.863710i \(0.331865\pi\)
\(692\) −4.14410e9 −0.475400
\(693\) 4.16924e9 0.475873
\(694\) 4.60756e9 0.523254
\(695\) 5.33063e9 0.602326
\(696\) −1.26488e9 −0.142206
\(697\) −9.40182e7 −0.0105171
\(698\) −8.24120e9 −0.917268
\(699\) −5.86798e9 −0.649858
\(700\) −1.22602e9 −0.135100
\(701\) 1.06528e10 1.16803 0.584013 0.811744i \(-0.301482\pi\)
0.584013 + 0.811744i \(0.301482\pi\)
\(702\) 1.43947e9 0.157045
\(703\) 2.39573e9 0.260073
\(704\) −1.22285e9 −0.132089
\(705\) 3.13125e9 0.336555
\(706\) −7.13402e9 −0.762988
\(707\) 1.67605e10 1.78369
\(708\) −1.70299e9 −0.180342
\(709\) 1.03896e10 1.09481 0.547403 0.836869i \(-0.315617\pi\)
0.547403 + 0.836869i \(0.315617\pi\)
\(710\) −4.47915e9 −0.469669
\(711\) −9.65442e8 −0.100735
\(712\) −1.91365e9 −0.198693
\(713\) 7.35684e9 0.760113
\(714\) 1.49774e9 0.153990
\(715\) −5.33044e9 −0.545371
\(716\) 3.23796e9 0.329668
\(717\) 6.71832e9 0.680681
\(718\) −1.14361e10 −1.15303
\(719\) 7.96840e9 0.799503 0.399752 0.916624i \(-0.369096\pi\)
0.399752 + 0.916624i \(0.369096\pi\)
\(720\) 3.73248e8 0.0372678
\(721\) −3.09694e8 −0.0307723
\(722\) 3.76367e8 0.0372161
\(723\) −9.93922e9 −0.978066
\(724\) −8.90460e9 −0.872026
\(725\) −1.42967e9 −0.139333
\(726\) 4.90978e8 0.0476194
\(727\) 1.46339e10 1.41251 0.706253 0.707960i \(-0.250384\pi\)
0.706253 + 0.707960i \(0.250384\pi\)
\(728\) −5.73838e9 −0.551226
\(729\) 3.87420e8 0.0370370
\(730\) −3.53609e9 −0.336429
\(731\) 4.93778e9 0.467543
\(732\) −1.30707e9 −0.123171
\(733\) 1.27362e10 1.19447 0.597234 0.802067i \(-0.296266\pi\)
0.597234 + 0.802067i \(0.296266\pi\)
\(734\) −8.65586e9 −0.807930
\(735\) 2.29360e9 0.213065
\(736\) 9.35398e8 0.0864817
\(737\) 2.21676e10 2.03978
\(738\) 9.69491e7 0.00887864
\(739\) −1.95071e10 −1.77802 −0.889009 0.457890i \(-0.848605\pi\)
−0.889009 + 0.457890i \(0.848605\pi\)
\(740\) −2.79427e9 −0.253488
\(741\) −1.69296e9 −0.152856
\(742\) 9.94376e8 0.0893587
\(743\) 1.07928e10 0.965326 0.482663 0.875806i \(-0.339670\pi\)
0.482663 + 0.875806i \(0.339670\pi\)
\(744\) 3.56270e9 0.317156
\(745\) 1.06974e9 0.0947836
\(746\) −1.13195e10 −0.998251
\(747\) −5.20253e7 −0.00456660
\(748\) 1.68849e9 0.147517
\(749\) −8.04457e8 −0.0699547
\(750\) 4.21875e8 0.0365148
\(751\) 1.57746e10 1.35899 0.679497 0.733679i \(-0.262198\pi\)
0.679497 + 0.733679i \(0.262198\pi\)
\(752\) 3.80018e9 0.325868
\(753\) −5.32599e9 −0.454588
\(754\) −6.69158e9 −0.568498
\(755\) 5.01687e9 0.424247
\(756\) −1.54443e9 −0.130000
\(757\) −1.16114e9 −0.0972855 −0.0486427 0.998816i \(-0.515490\pi\)
−0.0486427 + 0.998816i \(0.515490\pi\)
\(758\) −1.21115e9 −0.101008
\(759\) −3.59536e9 −0.298467
\(760\) −4.38976e8 −0.0362738
\(761\) −7.77889e9 −0.639840 −0.319920 0.947445i \(-0.603656\pi\)
−0.319920 + 0.947445i \(0.603656\pi\)
\(762\) 8.45176e9 0.691998
\(763\) 1.09573e10 0.893031
\(764\) 4.23569e9 0.343635
\(765\) −5.15375e8 −0.0416206
\(766\) 4.64299e9 0.373247
\(767\) −9.00930e9 −0.720954
\(768\) 4.52985e8 0.0360844
\(769\) 6.68999e9 0.530497 0.265249 0.964180i \(-0.414546\pi\)
0.265249 + 0.964180i \(0.414546\pi\)
\(770\) 5.71912e9 0.451452
\(771\) −5.56717e9 −0.437466
\(772\) −3.72165e9 −0.291122
\(773\) −2.23508e10 −1.74046 −0.870230 0.492645i \(-0.836030\pi\)
−0.870230 + 0.492645i \(0.836030\pi\)
\(774\) −5.09171e9 −0.394703
\(775\) 4.02685e9 0.310748
\(776\) −2.75402e9 −0.211569
\(777\) 1.15622e10 0.884231
\(778\) 1.27691e9 0.0972147
\(779\) −1.14022e8 −0.00864183
\(780\) 1.97458e9 0.148986
\(781\) 2.08943e10 1.56946
\(782\) −1.29158e9 −0.0965826
\(783\) −1.80098e9 −0.134073
\(784\) 2.78358e9 0.206299
\(785\) 7.83616e9 0.578175
\(786\) −6.79123e9 −0.498849
\(787\) −9.69306e9 −0.708841 −0.354421 0.935086i \(-0.615322\pi\)
−0.354421 + 0.935086i \(0.615322\pi\)
\(788\) 8.50807e9 0.619426
\(789\) −3.59041e9 −0.260240
\(790\) −1.32434e9 −0.0955661
\(791\) 5.75008e9 0.413101
\(792\) −1.74112e9 −0.124535
\(793\) −6.91474e9 −0.492402
\(794\) 1.84600e9 0.130876
\(795\) −3.42166e8 −0.0241519
\(796\) −5.01539e9 −0.352459
\(797\) 1.58293e10 1.10754 0.553769 0.832670i \(-0.313189\pi\)
0.553769 + 0.832670i \(0.313189\pi\)
\(798\) 1.81640e9 0.126533
\(799\) −5.24723e9 −0.363929
\(800\) 5.12000e8 0.0353553
\(801\) −2.72471e9 −0.187330
\(802\) −6.21319e9 −0.425309
\(803\) 1.64951e10 1.12422
\(804\) −8.21166e9 −0.557230
\(805\) −4.37476e9 −0.295576
\(806\) 1.88476e10 1.26790
\(807\) 4.76020e9 0.318836
\(808\) −6.99939e9 −0.466789
\(809\) −7.03323e9 −0.467020 −0.233510 0.972354i \(-0.575021\pi\)
−0.233510 + 0.972354i \(0.575021\pi\)
\(810\) 5.31441e8 0.0351364
\(811\) −2.22464e10 −1.46449 −0.732244 0.681042i \(-0.761527\pi\)
−0.732244 + 0.681042i \(0.761527\pi\)
\(812\) 7.17951e9 0.470597
\(813\) −7.60685e9 −0.496464
\(814\) 1.30346e10 0.847059
\(815\) 5.74032e9 0.371437
\(816\) −6.25474e8 −0.0402990
\(817\) 5.98835e9 0.384176
\(818\) 4.21168e9 0.269041
\(819\) −8.17047e9 −0.519701
\(820\) 1.32989e8 0.00842301
\(821\) 1.45023e10 0.914611 0.457305 0.889310i \(-0.348815\pi\)
0.457305 + 0.889310i \(0.348815\pi\)
\(822\) −9.52658e9 −0.598254
\(823\) −1.39647e9 −0.0873237 −0.0436619 0.999046i \(-0.513902\pi\)
−0.0436619 + 0.999046i \(0.513902\pi\)
\(824\) 1.29332e8 0.00805305
\(825\) −1.96796e9 −0.122019
\(826\) 9.66624e9 0.596798
\(827\) 6.31315e9 0.388130 0.194065 0.980989i \(-0.437833\pi\)
0.194065 + 0.980989i \(0.437833\pi\)
\(828\) 1.33185e9 0.0815357
\(829\) −2.65187e9 −0.161663 −0.0808317 0.996728i \(-0.525758\pi\)
−0.0808317 + 0.996728i \(0.525758\pi\)
\(830\) −7.13653e7 −0.00433225
\(831\) 8.97512e9 0.542546
\(832\) 2.39641e9 0.144255
\(833\) −3.84352e9 −0.230394
\(834\) 9.21132e9 0.549846
\(835\) 2.75429e9 0.163722
\(836\) 2.04773e9 0.121213
\(837\) 5.07267e9 0.299018
\(838\) −1.85471e10 −1.08873
\(839\) 2.35103e10 1.37433 0.687165 0.726502i \(-0.258855\pi\)
0.687165 + 0.726502i \(0.258855\pi\)
\(840\) −2.11856e9 −0.123329
\(841\) −8.87778e9 −0.514657
\(842\) −1.73055e10 −0.999060
\(843\) 9.82844e9 0.565051
\(844\) 3.24905e9 0.186019
\(845\) 2.60252e9 0.148387
\(846\) 5.41081e9 0.307232
\(847\) −2.78680e9 −0.157585
\(848\) −4.15262e8 −0.0233850
\(849\) 1.26890e10 0.711627
\(850\) −7.06962e8 −0.0394848
\(851\) −9.97067e9 −0.554588
\(852\) −7.73998e9 −0.428747
\(853\) 1.25648e10 0.693160 0.346580 0.938020i \(-0.387343\pi\)
0.346580 + 0.938020i \(0.387343\pi\)
\(854\) 7.41894e9 0.407605
\(855\) −6.25026e8 −0.0341993
\(856\) 3.35950e8 0.0183070
\(857\) 2.24742e10 1.21970 0.609848 0.792519i \(-0.291231\pi\)
0.609848 + 0.792519i \(0.291231\pi\)
\(858\) −9.21101e9 −0.497854
\(859\) −1.20621e10 −0.649302 −0.324651 0.945834i \(-0.605247\pi\)
−0.324651 + 0.945834i \(0.605247\pi\)
\(860\) −6.98451e9 −0.374448
\(861\) −5.50285e8 −0.0293817
\(862\) −1.75252e9 −0.0931937
\(863\) −3.12352e10 −1.65427 −0.827136 0.562001i \(-0.810032\pi\)
−0.827136 + 0.562001i \(0.810032\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −8.09394e9 −0.425210
\(866\) 2.46482e10 1.28965
\(867\) −1.02155e10 −0.532344
\(868\) −2.02219e10 −1.04955
\(869\) 6.17775e9 0.319346
\(870\) −2.47048e9 −0.127193
\(871\) −4.34419e10 −2.22764
\(872\) −4.57588e9 −0.233704
\(873\) −3.92125e9 −0.199469
\(874\) −1.56638e9 −0.0793610
\(875\) −2.39457e9 −0.120837
\(876\) −6.11037e9 −0.307116
\(877\) −1.16277e10 −0.582096 −0.291048 0.956709i \(-0.594004\pi\)
−0.291048 + 0.956709i \(0.594004\pi\)
\(878\) 2.15268e10 1.07337
\(879\) −1.95834e10 −0.972585
\(880\) −2.38837e9 −0.118144
\(881\) −6.80837e9 −0.335450 −0.167725 0.985834i \(-0.553642\pi\)
−0.167725 + 0.985834i \(0.553642\pi\)
\(882\) 3.96334e9 0.194501
\(883\) −4.66300e8 −0.0227931 −0.0113965 0.999935i \(-0.503628\pi\)
−0.0113965 + 0.999935i \(0.503628\pi\)
\(884\) −3.30893e9 −0.161103
\(885\) −3.32616e9 −0.161303
\(886\) 2.99105e8 0.0144480
\(887\) −3.49184e10 −1.68005 −0.840025 0.542548i \(-0.817460\pi\)
−0.840025 + 0.542548i \(0.817460\pi\)
\(888\) −4.82849e9 −0.231401
\(889\) −4.79724e10 −2.29000
\(890\) −3.73761e9 −0.177717
\(891\) −2.47906e9 −0.117413
\(892\) −9.38885e9 −0.442930
\(893\) −6.36364e9 −0.299037
\(894\) 1.84852e9 0.0865252
\(895\) 6.32414e9 0.294864
\(896\) −2.57115e9 −0.119413
\(897\) 7.04583e9 0.325956
\(898\) 1.23138e10 0.567445
\(899\) −2.35810e10 −1.08244
\(900\) 7.29000e8 0.0333333
\(901\) 5.73388e8 0.0261163
\(902\) −6.20366e8 −0.0281465
\(903\) 2.89007e10 1.30617
\(904\) −2.40130e9 −0.108108
\(905\) −1.73918e10 −0.779964
\(906\) 8.66915e9 0.387283
\(907\) −9.65664e9 −0.429735 −0.214867 0.976643i \(-0.568932\pi\)
−0.214867 + 0.976643i \(0.568932\pi\)
\(908\) −3.76053e9 −0.166705
\(909\) −9.96593e9 −0.440093
\(910\) −1.12078e10 −0.493032
\(911\) 5.54939e8 0.0243182 0.0121591 0.999926i \(-0.496130\pi\)
0.0121591 + 0.999926i \(0.496130\pi\)
\(912\) −7.58551e8 −0.0331133
\(913\) 3.32904e8 0.0144767
\(914\) −2.63402e10 −1.14106
\(915\) −2.55286e9 −0.110168
\(916\) 1.94811e10 0.837488
\(917\) 3.85472e10 1.65082
\(918\) −8.90568e8 −0.0379942
\(919\) −9.56305e9 −0.406436 −0.203218 0.979134i \(-0.565140\pi\)
−0.203218 + 0.979134i \(0.565140\pi\)
\(920\) 1.82695e9 0.0773516
\(921\) 5.33562e9 0.225048
\(922\) 2.07204e10 0.870644
\(923\) −4.09466e10 −1.71400
\(924\) 9.88265e9 0.412118
\(925\) −5.45755e9 −0.226726
\(926\) 1.68579e10 0.697694
\(927\) 1.84146e8 0.00759249
\(928\) −2.99824e9 −0.123154
\(929\) −2.73541e10 −1.11935 −0.559677 0.828711i \(-0.689075\pi\)
−0.559677 + 0.828711i \(0.689075\pi\)
\(930\) 6.95839e9 0.283673
\(931\) −4.66127e9 −0.189313
\(932\) −1.39093e10 −0.562793
\(933\) 2.18543e10 0.880952
\(934\) 7.46812e9 0.299915
\(935\) 3.29782e9 0.131943
\(936\) 3.41208e9 0.136005
\(937\) −1.94603e10 −0.772787 −0.386394 0.922334i \(-0.626279\pi\)
−0.386394 + 0.922334i \(0.626279\pi\)
\(938\) 4.66096e10 1.84402
\(939\) −9.56304e9 −0.376935
\(940\) 7.42223e9 0.291465
\(941\) 1.34447e10 0.526002 0.263001 0.964796i \(-0.415288\pi\)
0.263001 + 0.964796i \(0.415288\pi\)
\(942\) 1.35409e10 0.527799
\(943\) 4.74540e8 0.0184281
\(944\) −4.03673e9 −0.156181
\(945\) −3.01647e9 −0.116275
\(946\) 3.25813e10 1.25126
\(947\) 8.81228e9 0.337181 0.168591 0.985686i \(-0.446078\pi\)
0.168591 + 0.985686i \(0.446078\pi\)
\(948\) −2.28845e9 −0.0872395
\(949\) −3.23255e10 −1.22776
\(950\) −8.57375e8 −0.0324443
\(951\) 2.86912e10 1.08173
\(952\) 3.55021e9 0.133360
\(953\) 2.10715e10 0.788623 0.394312 0.918977i \(-0.370983\pi\)
0.394312 + 0.918977i \(0.370983\pi\)
\(954\) −5.91262e8 −0.0220476
\(955\) 8.27283e9 0.307356
\(956\) 1.59249e10 0.589487
\(957\) 1.15242e10 0.425031
\(958\) 7.41903e9 0.272626
\(959\) 5.40731e10 1.97978
\(960\) 8.84736e8 0.0322749
\(961\) 3.89060e10 1.41412
\(962\) −2.55440e10 −0.925074
\(963\) 4.78336e8 0.0172600
\(964\) −2.35596e10 −0.847030
\(965\) −7.26885e9 −0.260387
\(966\) −7.55959e9 −0.269823
\(967\) −8.38176e9 −0.298087 −0.149043 0.988831i \(-0.547619\pi\)
−0.149043 + 0.988831i \(0.547619\pi\)
\(968\) 1.16380e9 0.0412396
\(969\) 1.04739e9 0.0369809
\(970\) −5.37895e9 −0.189233
\(971\) 1.05194e10 0.368742 0.184371 0.982857i \(-0.440975\pi\)
0.184371 + 0.982857i \(0.440975\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −5.22837e10 −1.81958
\(974\) 2.87886e8 0.00998306
\(975\) 3.85661e9 0.133257
\(976\) −3.09823e9 −0.106669
\(977\) 1.62963e10 0.559059 0.279529 0.960137i \(-0.409822\pi\)
0.279529 + 0.960137i \(0.409822\pi\)
\(978\) 9.91928e9 0.339074
\(979\) 1.74351e10 0.593862
\(980\) 5.43668e9 0.184519
\(981\) −6.51527e9 −0.220339
\(982\) −3.88521e10 −1.30925
\(983\) 3.02334e9 0.101519 0.0507597 0.998711i \(-0.483836\pi\)
0.0507597 + 0.998711i \(0.483836\pi\)
\(984\) 2.29805e8 0.00768913
\(985\) 1.66173e10 0.554031
\(986\) 4.13993e9 0.137538
\(987\) −3.07119e10 −1.01671
\(988\) −4.01294e9 −0.132377
\(989\) −2.49226e10 −0.819230
\(990\) −3.40063e9 −0.111387
\(991\) −3.28205e10 −1.07124 −0.535620 0.844459i \(-0.679922\pi\)
−0.535620 + 0.844459i \(0.679922\pi\)
\(992\) 8.44491e9 0.274665
\(993\) 6.11825e9 0.198292
\(994\) 4.39323e10 1.41883
\(995\) −9.79568e9 −0.315249
\(996\) −1.23319e8 −0.00395479
\(997\) −4.27161e10 −1.36508 −0.682541 0.730848i \(-0.739125\pi\)
−0.682541 + 0.730848i \(0.739125\pi\)
\(998\) −4.16149e9 −0.132523
\(999\) −6.87494e9 −0.218167
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.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.c.1.1 4 1.1 even 1 trivial