Properties

Label 462.8.a.a.1.1
Level $462$
Weight $8$
Character 462.1
Self dual yes
Analytic conductor $144.322$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.321881774\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.000 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.000 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1456.00 q^{10} +1331.00 q^{11} -1728.00 q^{12} +13758.0 q^{13} -2744.00 q^{14} -4914.00 q^{15} +4096.00 q^{16} +14738.0 q^{17} -5832.00 q^{18} +34516.0 q^{19} +11648.0 q^{20} -9261.00 q^{21} -10648.0 q^{22} +49816.0 q^{23} +13824.0 q^{24} -45001.0 q^{25} -110064. q^{26} -19683.0 q^{27} +21952.0 q^{28} +141174. q^{29} +39312.0 q^{30} -86112.0 q^{31} -32768.0 q^{32} -35937.0 q^{33} -117904. q^{34} +62426.0 q^{35} +46656.0 q^{36} +535038. q^{37} -276128. q^{38} -371466. q^{39} -93184.0 q^{40} +708282. q^{41} +74088.0 q^{42} +162292. q^{43} +85184.0 q^{44} +132678. q^{45} -398528. q^{46} +397296. q^{47} -110592. q^{48} +117649. q^{49} +360008. q^{50} -397926. q^{51} +880512. q^{52} +224798. q^{53} +157464. q^{54} +242242. q^{55} -175616. q^{56} -931932. q^{57} -1.12939e6 q^{58} +694668. q^{59} -314496. q^{60} +50110.0 q^{61} +688896. q^{62} +250047. q^{63} +262144. q^{64} +2.50396e6 q^{65} +287496. q^{66} -3.22840e6 q^{67} +943232. q^{68} -1.34503e6 q^{69} -499408. q^{70} -1.87150e6 q^{71} -373248. q^{72} -937542. q^{73} -4.28030e6 q^{74} +1.21503e6 q^{75} +2.20902e6 q^{76} +456533. q^{77} +2.97173e6 q^{78} +1.26603e6 q^{79} +745472. q^{80} +531441. q^{81} -5.66626e6 q^{82} -2.54500e6 q^{83} -592704. q^{84} +2.68232e6 q^{85} -1.29834e6 q^{86} -3.81170e6 q^{87} -681472. q^{88} -9.90983e6 q^{89} -1.06142e6 q^{90} +4.71899e6 q^{91} +3.18822e6 q^{92} +2.32502e6 q^{93} -3.17837e6 q^{94} +6.28191e6 q^{95} +884736. q^{96} -1.40108e7 q^{97} -941192. q^{98} +970299. q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) 182.000 0.651143 0.325571 0.945517i \(-0.394443\pi\)
0.325571 + 0.945517i \(0.394443\pi\)
\(6\) 216.000 0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) −1456.00 −0.460428
\(11\) 1331.00 0.301511
\(12\) −1728.00 −0.288675
\(13\) 13758.0 1.73682 0.868408 0.495851i \(-0.165144\pi\)
0.868408 + 0.495851i \(0.165144\pi\)
\(14\) −2744.00 −0.267261
\(15\) −4914.00 −0.375938
\(16\) 4096.00 0.250000
\(17\) 14738.0 0.727558 0.363779 0.931485i \(-0.381486\pi\)
0.363779 + 0.931485i \(0.381486\pi\)
\(18\) −5832.00 −0.235702
\(19\) 34516.0 1.15447 0.577235 0.816578i \(-0.304132\pi\)
0.577235 + 0.816578i \(0.304132\pi\)
\(20\) 11648.0 0.325571
\(21\) −9261.00 −0.218218
\(22\) −10648.0 −0.213201
\(23\) 49816.0 0.853732 0.426866 0.904315i \(-0.359618\pi\)
0.426866 + 0.904315i \(0.359618\pi\)
\(24\) 13824.0 0.204124
\(25\) −45001.0 −0.576013
\(26\) −110064. −1.22811
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 141174. 1.07488 0.537442 0.843301i \(-0.319391\pi\)
0.537442 + 0.843301i \(0.319391\pi\)
\(30\) 39312.0 0.265828
\(31\) −86112.0 −0.519156 −0.259578 0.965722i \(-0.583583\pi\)
−0.259578 + 0.965722i \(0.583583\pi\)
\(32\) −32768.0 −0.176777
\(33\) −35937.0 −0.174078
\(34\) −117904. −0.514461
\(35\) 62426.0 0.246109
\(36\) 46656.0 0.166667
\(37\) 535038. 1.73652 0.868258 0.496114i \(-0.165240\pi\)
0.868258 + 0.496114i \(0.165240\pi\)
\(38\) −276128. −0.816334
\(39\) −371466. −1.00275
\(40\) −93184.0 −0.230214
\(41\) 708282. 1.60495 0.802477 0.596683i \(-0.203515\pi\)
0.802477 + 0.596683i \(0.203515\pi\)
\(42\) 74088.0 0.154303
\(43\) 162292. 0.311285 0.155642 0.987813i \(-0.450255\pi\)
0.155642 + 0.987813i \(0.450255\pi\)
\(44\) 85184.0 0.150756
\(45\) 132678. 0.217048
\(46\) −398528. −0.603680
\(47\) 397296. 0.558177 0.279088 0.960265i \(-0.409968\pi\)
0.279088 + 0.960265i \(0.409968\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 360008. 0.407303
\(51\) −397926. −0.420056
\(52\) 880512. 0.868408
\(53\) 224798. 0.207409 0.103704 0.994608i \(-0.466930\pi\)
0.103704 + 0.994608i \(0.466930\pi\)
\(54\) 157464. 0.136083
\(55\) 242242. 0.196327
\(56\) −175616. −0.133631
\(57\) −931932. −0.666534
\(58\) −1.12939e6 −0.760058
\(59\) 694668. 0.440347 0.220174 0.975461i \(-0.429338\pi\)
0.220174 + 0.975461i \(0.429338\pi\)
\(60\) −314496. −0.187969
\(61\) 50110.0 0.0282664 0.0141332 0.999900i \(-0.495501\pi\)
0.0141332 + 0.999900i \(0.495501\pi\)
\(62\) 688896. 0.367099
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) 2.50396e6 1.13091
\(66\) 287496. 0.123091
\(67\) −3.22840e6 −1.31137 −0.655686 0.755033i \(-0.727621\pi\)
−0.655686 + 0.755033i \(0.727621\pi\)
\(68\) 943232. 0.363779
\(69\) −1.34503e6 −0.492902
\(70\) −499408. −0.174025
\(71\) −1.87150e6 −0.620561 −0.310281 0.950645i \(-0.600423\pi\)
−0.310281 + 0.950645i \(0.600423\pi\)
\(72\) −373248. −0.117851
\(73\) −937542. −0.282072 −0.141036 0.990004i \(-0.545043\pi\)
−0.141036 + 0.990004i \(0.545043\pi\)
\(74\) −4.28030e6 −1.22790
\(75\) 1.21503e6 0.332561
\(76\) 2.20902e6 0.577235
\(77\) 456533. 0.113961
\(78\) 2.97173e6 0.709052
\(79\) 1.26603e6 0.288902 0.144451 0.989512i \(-0.453858\pi\)
0.144451 + 0.989512i \(0.453858\pi\)
\(80\) 745472. 0.162786
\(81\) 531441. 0.111111
\(82\) −5.66626e6 −1.13487
\(83\) −2.54500e6 −0.488557 −0.244278 0.969705i \(-0.578551\pi\)
−0.244278 + 0.969705i \(0.578551\pi\)
\(84\) −592704. −0.109109
\(85\) 2.68232e6 0.473744
\(86\) −1.29834e6 −0.220111
\(87\) −3.81170e6 −0.620585
\(88\) −681472. −0.106600
\(89\) −9.90983e6 −1.49005 −0.745026 0.667036i \(-0.767563\pi\)
−0.745026 + 0.667036i \(0.767563\pi\)
\(90\) −1.06142e6 −0.153476
\(91\) 4.71899e6 0.656454
\(92\) 3.18822e6 0.426866
\(93\) 2.32502e6 0.299735
\(94\) −3.17837e6 −0.394691
\(95\) 6.28191e6 0.751725
\(96\) 884736. 0.102062
\(97\) −1.40108e7 −1.55870 −0.779350 0.626588i \(-0.784451\pi\)
−0.779350 + 0.626588i \(0.784451\pi\)
\(98\) −941192. −0.101015
\(99\) 970299. 0.100504
\(100\) −2.88006e6 −0.288006
\(101\) −7.34281e6 −0.709149 −0.354574 0.935028i \(-0.615374\pi\)
−0.354574 + 0.935028i \(0.615374\pi\)
\(102\) 3.18341e6 0.297024
\(103\) −8.86030e6 −0.798947 −0.399473 0.916745i \(-0.630807\pi\)
−0.399473 + 0.916745i \(0.630807\pi\)
\(104\) −7.04410e6 −0.614057
\(105\) −1.68550e6 −0.142091
\(106\) −1.79838e6 −0.146660
\(107\) −1.50185e7 −1.18518 −0.592588 0.805506i \(-0.701894\pi\)
−0.592588 + 0.805506i \(0.701894\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 1.06251e7 0.785851 0.392925 0.919570i \(-0.371463\pi\)
0.392925 + 0.919570i \(0.371463\pi\)
\(110\) −1.93794e6 −0.138824
\(111\) −1.44460e7 −1.00258
\(112\) 1.40493e6 0.0944911
\(113\) −7.09310e6 −0.462447 −0.231223 0.972901i \(-0.574273\pi\)
−0.231223 + 0.972901i \(0.574273\pi\)
\(114\) 7.45546e6 0.471311
\(115\) 9.06651e6 0.555901
\(116\) 9.03514e6 0.537442
\(117\) 1.00296e7 0.578938
\(118\) −5.55734e6 −0.311373
\(119\) 5.05513e6 0.274991
\(120\) 2.51597e6 0.132914
\(121\) 1.77156e6 0.0909091
\(122\) −400880. −0.0199873
\(123\) −1.91236e7 −0.926621
\(124\) −5.51117e6 −0.259578
\(125\) −2.24089e7 −1.02621
\(126\) −2.00038e6 −0.0890871
\(127\) 3.60394e7 1.56122 0.780611 0.625017i \(-0.214908\pi\)
0.780611 + 0.625017i \(0.214908\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −4.38188e6 −0.179720
\(130\) −2.00316e7 −0.799678
\(131\) 2.25637e7 0.876923 0.438462 0.898750i \(-0.355523\pi\)
0.438462 + 0.898750i \(0.355523\pi\)
\(132\) −2.29997e6 −0.0870388
\(133\) 1.18390e7 0.436349
\(134\) 2.58272e7 0.927280
\(135\) −3.58231e6 −0.125313
\(136\) −7.54586e6 −0.257230
\(137\) 1.24461e7 0.413535 0.206768 0.978390i \(-0.433706\pi\)
0.206768 + 0.978390i \(0.433706\pi\)
\(138\) 1.07603e7 0.348535
\(139\) 1.63300e6 0.0515746 0.0257873 0.999667i \(-0.491791\pi\)
0.0257873 + 0.999667i \(0.491791\pi\)
\(140\) 3.99526e6 0.123054
\(141\) −1.07270e7 −0.322263
\(142\) 1.49720e7 0.438803
\(143\) 1.83119e7 0.523669
\(144\) 2.98598e6 0.0833333
\(145\) 2.56937e7 0.699903
\(146\) 7.50034e6 0.199455
\(147\) −3.17652e6 −0.0824786
\(148\) 3.42424e7 0.868258
\(149\) 6.75131e7 1.67200 0.835999 0.548730i \(-0.184889\pi\)
0.835999 + 0.548730i \(0.184889\pi\)
\(150\) −9.72022e6 −0.235156
\(151\) −6.23218e7 −1.47306 −0.736530 0.676405i \(-0.763537\pi\)
−0.736530 + 0.676405i \(0.763537\pi\)
\(152\) −1.76722e7 −0.408167
\(153\) 1.07440e7 0.242519
\(154\) −3.65226e6 −0.0805823
\(155\) −1.56724e7 −0.338045
\(156\) −2.37738e7 −0.501375
\(157\) −8.02772e7 −1.65555 −0.827777 0.561058i \(-0.810395\pi\)
−0.827777 + 0.561058i \(0.810395\pi\)
\(158\) −1.01283e7 −0.204284
\(159\) −6.06955e6 −0.119747
\(160\) −5.96378e6 −0.115107
\(161\) 1.70869e7 0.322680
\(162\) −4.25153e6 −0.0785674
\(163\) −1.17849e7 −0.213143 −0.106571 0.994305i \(-0.533987\pi\)
−0.106571 + 0.994305i \(0.533987\pi\)
\(164\) 4.53300e7 0.802477
\(165\) −6.54053e6 −0.113349
\(166\) 2.03600e7 0.345462
\(167\) −5.51803e7 −0.916805 −0.458402 0.888745i \(-0.651578\pi\)
−0.458402 + 0.888745i \(0.651578\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 1.26534e8 2.01653
\(170\) −2.14585e7 −0.334988
\(171\) 2.51622e7 0.384824
\(172\) 1.03867e7 0.155642
\(173\) 5.63952e7 0.828096 0.414048 0.910255i \(-0.364114\pi\)
0.414048 + 0.910255i \(0.364114\pi\)
\(174\) 3.04936e7 0.438820
\(175\) −1.54353e7 −0.217712
\(176\) 5.45178e6 0.0753778
\(177\) −1.87560e7 −0.254235
\(178\) 7.92786e7 1.05363
\(179\) 1.03312e7 0.134637 0.0673185 0.997732i \(-0.478556\pi\)
0.0673185 + 0.997732i \(0.478556\pi\)
\(180\) 8.49139e6 0.108524
\(181\) 6.67241e7 0.836388 0.418194 0.908358i \(-0.362663\pi\)
0.418194 + 0.908358i \(0.362663\pi\)
\(182\) −3.77520e7 −0.464183
\(183\) −1.35297e6 −0.0163196
\(184\) −2.55058e7 −0.301840
\(185\) 9.73769e7 1.13072
\(186\) −1.86002e7 −0.211944
\(187\) 1.96163e7 0.219367
\(188\) 2.54269e7 0.279088
\(189\) −6.75127e6 −0.0727393
\(190\) −5.02553e7 −0.531550
\(191\) −4.53862e7 −0.471310 −0.235655 0.971837i \(-0.575724\pi\)
−0.235655 + 0.971837i \(0.575724\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −2.92030e7 −0.292399 −0.146200 0.989255i \(-0.546704\pi\)
−0.146200 + 0.989255i \(0.546704\pi\)
\(194\) 1.12087e8 1.10217
\(195\) −6.76068e7 −0.652934
\(196\) 7.52954e6 0.0714286
\(197\) 1.17638e8 1.09626 0.548131 0.836393i \(-0.315340\pi\)
0.548131 + 0.836393i \(0.315340\pi\)
\(198\) −7.76239e6 −0.0710669
\(199\) −6.21347e7 −0.558918 −0.279459 0.960158i \(-0.590155\pi\)
−0.279459 + 0.960158i \(0.590155\pi\)
\(200\) 2.30405e7 0.203651
\(201\) 8.71669e7 0.757121
\(202\) 5.87425e7 0.501444
\(203\) 4.84227e7 0.406268
\(204\) −2.54673e7 −0.210028
\(205\) 1.28907e8 1.04505
\(206\) 7.08824e7 0.564941
\(207\) 3.63159e7 0.284577
\(208\) 5.63528e7 0.434204
\(209\) 4.59408e7 0.348086
\(210\) 1.34840e7 0.100474
\(211\) −1.84022e8 −1.34860 −0.674299 0.738459i \(-0.735554\pi\)
−0.674299 + 0.738459i \(0.735554\pi\)
\(212\) 1.43871e7 0.103704
\(213\) 5.05304e7 0.358281
\(214\) 1.20148e8 0.838046
\(215\) 2.95371e7 0.202691
\(216\) 1.00777e7 0.0680414
\(217\) −2.95364e7 −0.196222
\(218\) −8.50008e7 −0.555680
\(219\) 2.53136e7 0.162855
\(220\) 1.55035e7 0.0981635
\(221\) 2.02765e8 1.26363
\(222\) 1.15568e8 0.708929
\(223\) 8.81072e7 0.532040 0.266020 0.963967i \(-0.414291\pi\)
0.266020 + 0.963967i \(0.414291\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −3.28057e7 −0.192004
\(226\) 5.67448e7 0.326999
\(227\) 2.75878e8 1.56541 0.782703 0.622395i \(-0.213840\pi\)
0.782703 + 0.622395i \(0.213840\pi\)
\(228\) −5.96436e7 −0.333267
\(229\) −8.28916e7 −0.456128 −0.228064 0.973646i \(-0.573239\pi\)
−0.228064 + 0.973646i \(0.573239\pi\)
\(230\) −7.25321e7 −0.393082
\(231\) −1.23264e7 −0.0657952
\(232\) −7.22811e7 −0.380029
\(233\) −2.36402e8 −1.22435 −0.612175 0.790722i \(-0.709705\pi\)
−0.612175 + 0.790722i \(0.709705\pi\)
\(234\) −8.02367e7 −0.409371
\(235\) 7.23079e7 0.363453
\(236\) 4.44588e7 0.220174
\(237\) −3.41829e7 −0.166797
\(238\) −4.04411e7 −0.194448
\(239\) 2.78052e8 1.31745 0.658725 0.752384i \(-0.271096\pi\)
0.658725 + 0.752384i \(0.271096\pi\)
\(240\) −2.01277e7 −0.0939844
\(241\) −1.48269e8 −0.682324 −0.341162 0.940004i \(-0.610821\pi\)
−0.341162 + 0.940004i \(0.610821\pi\)
\(242\) −1.41725e7 −0.0642824
\(243\) −1.43489e7 −0.0641500
\(244\) 3.20704e6 0.0141332
\(245\) 2.14121e7 0.0930204
\(246\) 1.52989e8 0.655220
\(247\) 4.74871e8 2.00510
\(248\) 4.40893e7 0.183549
\(249\) 6.87151e7 0.282068
\(250\) 1.79271e8 0.725640
\(251\) 2.89064e8 1.15381 0.576906 0.816810i \(-0.304260\pi\)
0.576906 + 0.816810i \(0.304260\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 6.63051e7 0.257410
\(254\) −2.88315e8 −1.10395
\(255\) −7.24225e7 −0.273516
\(256\) 1.67772e7 0.0625000
\(257\) −1.99415e8 −0.732813 −0.366406 0.930455i \(-0.619412\pi\)
−0.366406 + 0.930455i \(0.619412\pi\)
\(258\) 3.50551e7 0.127081
\(259\) 1.83518e8 0.656341
\(260\) 1.60253e8 0.565457
\(261\) 1.02916e8 0.358295
\(262\) −1.80510e8 −0.620079
\(263\) −4.04763e8 −1.37200 −0.686002 0.727600i \(-0.740636\pi\)
−0.686002 + 0.727600i \(0.740636\pi\)
\(264\) 1.83997e7 0.0615457
\(265\) 4.09132e7 0.135053
\(266\) −9.47119e7 −0.308545
\(267\) 2.67565e8 0.860281
\(268\) −2.06618e8 −0.655686
\(269\) 4.35356e7 0.136368 0.0681838 0.997673i \(-0.478280\pi\)
0.0681838 + 0.997673i \(0.478280\pi\)
\(270\) 2.86584e7 0.0886093
\(271\) −2.64981e8 −0.808765 −0.404382 0.914590i \(-0.632514\pi\)
−0.404382 + 0.914590i \(0.632514\pi\)
\(272\) 6.03668e7 0.181889
\(273\) −1.27413e8 −0.379004
\(274\) −9.95691e7 −0.292414
\(275\) −5.98963e7 −0.173674
\(276\) −8.60820e7 −0.246451
\(277\) 6.49499e8 1.83611 0.918057 0.396449i \(-0.129758\pi\)
0.918057 + 0.396449i \(0.129758\pi\)
\(278\) −1.30640e7 −0.0364687
\(279\) −6.27756e7 −0.173052
\(280\) −3.19621e7 −0.0870126
\(281\) 1.75806e8 0.472674 0.236337 0.971671i \(-0.424053\pi\)
0.236337 + 0.971671i \(0.424053\pi\)
\(282\) 8.58159e7 0.227875
\(283\) −1.21587e8 −0.318885 −0.159443 0.987207i \(-0.550970\pi\)
−0.159443 + 0.987207i \(0.550970\pi\)
\(284\) −1.19776e8 −0.310281
\(285\) −1.69612e8 −0.434009
\(286\) −1.46495e8 −0.370290
\(287\) 2.42941e8 0.606616
\(288\) −2.38879e7 −0.0589256
\(289\) −1.93130e8 −0.470660
\(290\) −2.05549e8 −0.494906
\(291\) 3.78292e8 0.899916
\(292\) −6.00027e7 −0.141036
\(293\) 5.27431e8 1.22498 0.612490 0.790478i \(-0.290168\pi\)
0.612490 + 0.790478i \(0.290168\pi\)
\(294\) 2.54122e7 0.0583212
\(295\) 1.26430e8 0.286729
\(296\) −2.73939e8 −0.613951
\(297\) −2.61981e7 −0.0580259
\(298\) −5.40105e8 −1.18228
\(299\) 6.85369e8 1.48277
\(300\) 7.77617e7 0.166281
\(301\) 5.56662e7 0.117654
\(302\) 4.98574e8 1.04161
\(303\) 1.98256e8 0.409427
\(304\) 1.41378e8 0.288618
\(305\) 9.12002e6 0.0184054
\(306\) −8.59520e7 −0.171487
\(307\) 1.23804e8 0.244203 0.122101 0.992518i \(-0.461037\pi\)
0.122101 + 0.992518i \(0.461037\pi\)
\(308\) 2.92181e7 0.0569803
\(309\) 2.39228e8 0.461272
\(310\) 1.25379e8 0.239034
\(311\) 5.17448e8 0.975451 0.487726 0.872997i \(-0.337827\pi\)
0.487726 + 0.872997i \(0.337827\pi\)
\(312\) 1.90191e8 0.354526
\(313\) −9.84678e7 −0.181505 −0.0907526 0.995873i \(-0.528927\pi\)
−0.0907526 + 0.995873i \(0.528927\pi\)
\(314\) 6.42217e8 1.17065
\(315\) 4.55086e7 0.0820363
\(316\) 8.10260e7 0.144451
\(317\) 3.52106e8 0.620820 0.310410 0.950603i \(-0.399534\pi\)
0.310410 + 0.950603i \(0.399534\pi\)
\(318\) 4.85564e7 0.0846742
\(319\) 1.87903e8 0.324090
\(320\) 4.77102e7 0.0813929
\(321\) 4.05499e8 0.684261
\(322\) −1.36695e8 −0.228169
\(323\) 5.08697e8 0.839944
\(324\) 3.40122e7 0.0555556
\(325\) −6.19124e8 −1.00043
\(326\) 9.42795e7 0.150715
\(327\) −2.86878e8 −0.453711
\(328\) −3.62640e8 −0.567437
\(329\) 1.36273e8 0.210971
\(330\) 5.23243e7 0.0801502
\(331\) −5.36200e8 −0.812697 −0.406349 0.913718i \(-0.633198\pi\)
−0.406349 + 0.913718i \(0.633198\pi\)
\(332\) −1.62880e8 −0.244278
\(333\) 3.90043e8 0.578838
\(334\) 4.41443e8 0.648279
\(335\) −5.87570e8 −0.853891
\(336\) −3.79331e7 −0.0545545
\(337\) −7.28923e8 −1.03747 −0.518737 0.854934i \(-0.673598\pi\)
−0.518737 + 0.854934i \(0.673598\pi\)
\(338\) −1.01227e9 −1.42590
\(339\) 1.91514e8 0.266994
\(340\) 1.71668e8 0.236872
\(341\) −1.14615e8 −0.156531
\(342\) −2.01297e8 −0.272111
\(343\) 4.03536e7 0.0539949
\(344\) −8.30935e7 −0.110056
\(345\) −2.44796e8 −0.320950
\(346\) −4.51162e8 −0.585553
\(347\) 8.08560e8 1.03887 0.519433 0.854511i \(-0.326143\pi\)
0.519433 + 0.854511i \(0.326143\pi\)
\(348\) −2.43949e8 −0.310292
\(349\) −6.65358e8 −0.837850 −0.418925 0.908021i \(-0.637593\pi\)
−0.418925 + 0.908021i \(0.637593\pi\)
\(350\) 1.23483e8 0.153946
\(351\) −2.70799e8 −0.334250
\(352\) −4.36142e7 −0.0533002
\(353\) −6.66617e7 −0.0806613 −0.0403306 0.999186i \(-0.512841\pi\)
−0.0403306 + 0.999186i \(0.512841\pi\)
\(354\) 1.50048e8 0.179771
\(355\) −3.40612e8 −0.404074
\(356\) −6.34229e8 −0.745026
\(357\) −1.36489e8 −0.158766
\(358\) −8.26494e7 −0.0952027
\(359\) −4.00001e8 −0.456279 −0.228140 0.973628i \(-0.573264\pi\)
−0.228140 + 0.973628i \(0.573264\pi\)
\(360\) −6.79311e7 −0.0767379
\(361\) 2.97483e8 0.332802
\(362\) −5.33793e8 −0.591416
\(363\) −4.78321e7 −0.0524864
\(364\) 3.02016e8 0.328227
\(365\) −1.70633e8 −0.183670
\(366\) 1.08238e7 0.0115397
\(367\) −3.49648e8 −0.369232 −0.184616 0.982811i \(-0.559104\pi\)
−0.184616 + 0.982811i \(0.559104\pi\)
\(368\) 2.04046e8 0.213433
\(369\) 5.16338e8 0.534985
\(370\) −7.79015e8 −0.799540
\(371\) 7.71057e7 0.0783931
\(372\) 1.48802e8 0.149867
\(373\) 3.23621e8 0.322891 0.161445 0.986882i \(-0.448384\pi\)
0.161445 + 0.986882i \(0.448384\pi\)
\(374\) −1.56930e8 −0.155116
\(375\) 6.05041e8 0.592482
\(376\) −2.03416e8 −0.197345
\(377\) 1.94227e9 1.86688
\(378\) 5.40102e7 0.0514344
\(379\) −6.49660e8 −0.612984 −0.306492 0.951873i \(-0.599155\pi\)
−0.306492 + 0.951873i \(0.599155\pi\)
\(380\) 4.02042e8 0.375863
\(381\) −9.73064e8 −0.901372
\(382\) 3.63090e8 0.333267
\(383\) 1.11697e9 1.01589 0.507946 0.861389i \(-0.330405\pi\)
0.507946 + 0.861389i \(0.330405\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 8.30890e7 0.0742046
\(386\) 2.33624e8 0.206758
\(387\) 1.18311e8 0.103762
\(388\) −8.96693e8 −0.779350
\(389\) 1.18772e9 1.02303 0.511517 0.859273i \(-0.329084\pi\)
0.511517 + 0.859273i \(0.329084\pi\)
\(390\) 5.40854e8 0.461694
\(391\) 7.34188e8 0.621139
\(392\) −6.02363e7 −0.0505076
\(393\) −6.09221e8 −0.506292
\(394\) −9.41100e8 −0.775174
\(395\) 2.30418e8 0.188116
\(396\) 6.20991e7 0.0502519
\(397\) 2.20243e9 1.76659 0.883293 0.468822i \(-0.155321\pi\)
0.883293 + 0.468822i \(0.155321\pi\)
\(398\) 4.97078e8 0.395215
\(399\) −3.19653e8 −0.251926
\(400\) −1.84324e8 −0.144003
\(401\) 8.22757e8 0.637186 0.318593 0.947892i \(-0.396790\pi\)
0.318593 + 0.947892i \(0.396790\pi\)
\(402\) −6.97335e8 −0.535366
\(403\) −1.18473e9 −0.901678
\(404\) −4.69940e8 −0.354574
\(405\) 9.67223e7 0.0723492
\(406\) −3.87381e8 −0.287275
\(407\) 7.12136e8 0.523579
\(408\) 2.03738e8 0.148512
\(409\) 9.01095e8 0.651237 0.325618 0.945501i \(-0.394428\pi\)
0.325618 + 0.945501i \(0.394428\pi\)
\(410\) −1.03126e9 −0.738965
\(411\) −3.36046e8 −0.238755
\(412\) −5.67059e8 −0.399473
\(413\) 2.38271e8 0.166436
\(414\) −2.90527e8 −0.201227
\(415\) −4.63191e8 −0.318120
\(416\) −4.50822e8 −0.307028
\(417\) −4.40911e7 −0.0297766
\(418\) −3.67526e8 −0.246134
\(419\) −1.76829e9 −1.17437 −0.587184 0.809453i \(-0.699764\pi\)
−0.587184 + 0.809453i \(0.699764\pi\)
\(420\) −1.07872e8 −0.0710455
\(421\) 4.56926e8 0.298441 0.149220 0.988804i \(-0.452324\pi\)
0.149220 + 0.988804i \(0.452324\pi\)
\(422\) 1.47218e9 0.953602
\(423\) 2.89629e8 0.186059
\(424\) −1.15097e8 −0.0733300
\(425\) −6.63225e8 −0.419082
\(426\) −4.04243e8 −0.253343
\(427\) 1.71877e7 0.0106837
\(428\) −9.61182e8 −0.592588
\(429\) −4.94421e8 −0.302341
\(430\) −2.36297e8 −0.143324
\(431\) −1.56635e9 −0.942366 −0.471183 0.882036i \(-0.656173\pi\)
−0.471183 + 0.882036i \(0.656173\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) −1.70307e9 −1.00815 −0.504075 0.863660i \(-0.668166\pi\)
−0.504075 + 0.863660i \(0.668166\pi\)
\(434\) 2.36291e8 0.138750
\(435\) −6.93729e8 −0.404089
\(436\) 6.80006e8 0.392925
\(437\) 1.71945e9 0.985608
\(438\) −2.02509e8 −0.115156
\(439\) 1.64535e9 0.928179 0.464090 0.885788i \(-0.346382\pi\)
0.464090 + 0.885788i \(0.346382\pi\)
\(440\) −1.24028e8 −0.0694121
\(441\) 8.57661e7 0.0476190
\(442\) −1.62212e9 −0.893523
\(443\) 1.37306e9 0.750372 0.375186 0.926950i \(-0.377579\pi\)
0.375186 + 0.926950i \(0.377579\pi\)
\(444\) −9.24546e8 −0.501289
\(445\) −1.80359e9 −0.970236
\(446\) −7.04858e8 −0.376209
\(447\) −1.82285e9 −0.965329
\(448\) 8.99154e7 0.0472456
\(449\) 2.61157e9 1.36157 0.680785 0.732483i \(-0.261639\pi\)
0.680785 + 0.732483i \(0.261639\pi\)
\(450\) 2.62446e8 0.135768
\(451\) 9.42723e8 0.483912
\(452\) −4.53959e8 −0.231223
\(453\) 1.68269e9 0.850472
\(454\) −2.20703e9 −1.10691
\(455\) 8.58857e8 0.427446
\(456\) 4.77149e8 0.235655
\(457\) −3.13057e9 −1.53432 −0.767162 0.641453i \(-0.778332\pi\)
−0.767162 + 0.641453i \(0.778332\pi\)
\(458\) 6.63133e8 0.322531
\(459\) −2.90088e8 −0.140019
\(460\) 5.80257e8 0.277951
\(461\) −1.69540e9 −0.805968 −0.402984 0.915207i \(-0.632027\pi\)
−0.402984 + 0.915207i \(0.632027\pi\)
\(462\) 9.86111e7 0.0465242
\(463\) −8.34761e8 −0.390867 −0.195433 0.980717i \(-0.562611\pi\)
−0.195433 + 0.980717i \(0.562611\pi\)
\(464\) 5.78249e8 0.268721
\(465\) 4.23154e8 0.195170
\(466\) 1.89122e9 0.865746
\(467\) −1.12459e9 −0.510960 −0.255480 0.966814i \(-0.582233\pi\)
−0.255480 + 0.966814i \(0.582233\pi\)
\(468\) 6.41893e8 0.289469
\(469\) −1.10734e9 −0.495652
\(470\) −5.78463e8 −0.257000
\(471\) 2.16748e9 0.955834
\(472\) −3.55670e8 −0.155686
\(473\) 2.16011e8 0.0938558
\(474\) 2.73463e8 0.117944
\(475\) −1.55325e9 −0.664990
\(476\) 3.23529e8 0.137495
\(477\) 1.63878e8 0.0691362
\(478\) −2.22442e9 −0.931578
\(479\) −3.77226e9 −1.56830 −0.784148 0.620574i \(-0.786899\pi\)
−0.784148 + 0.620574i \(0.786899\pi\)
\(480\) 1.61022e8 0.0664570
\(481\) 7.36105e9 3.01601
\(482\) 1.18615e9 0.482476
\(483\) −4.61346e8 −0.186300
\(484\) 1.13380e8 0.0454545
\(485\) −2.54997e9 −1.01494
\(486\) 1.14791e8 0.0453609
\(487\) −4.40349e9 −1.72761 −0.863805 0.503827i \(-0.831925\pi\)
−0.863805 + 0.503827i \(0.831925\pi\)
\(488\) −2.56563e7 −0.00999367
\(489\) 3.18193e8 0.123058
\(490\) −1.71297e8 −0.0657754
\(491\) −2.61532e9 −0.997102 −0.498551 0.866860i \(-0.666134\pi\)
−0.498551 + 0.866860i \(0.666134\pi\)
\(492\) −1.22391e9 −0.463310
\(493\) 2.08062e9 0.782040
\(494\) −3.79897e9 −1.41782
\(495\) 1.76594e8 0.0654423
\(496\) −3.52715e8 −0.129789
\(497\) −6.41923e8 −0.234550
\(498\) −5.49721e8 −0.199453
\(499\) 2.15541e9 0.776566 0.388283 0.921540i \(-0.373068\pi\)
0.388283 + 0.921540i \(0.373068\pi\)
\(500\) −1.43417e9 −0.513105
\(501\) 1.48987e9 0.529317
\(502\) −2.31251e9 −0.815869
\(503\) −4.75527e9 −1.66605 −0.833023 0.553238i \(-0.813392\pi\)
−0.833023 + 0.553238i \(0.813392\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) −1.33639e9 −0.461757
\(506\) −5.30441e8 −0.182016
\(507\) −3.41642e9 −1.16424
\(508\) 2.30652e9 0.780611
\(509\) −3.04257e9 −1.02265 −0.511326 0.859387i \(-0.670845\pi\)
−0.511326 + 0.859387i \(0.670845\pi\)
\(510\) 5.79380e8 0.193405
\(511\) −3.21577e8 −0.106613
\(512\) −1.34218e8 −0.0441942
\(513\) −6.79378e8 −0.222178
\(514\) 1.59532e9 0.518177
\(515\) −1.61257e9 −0.520229
\(516\) −2.80441e8 −0.0898601
\(517\) 5.28801e8 0.168297
\(518\) −1.46814e9 −0.464103
\(519\) −1.52267e9 −0.478102
\(520\) −1.28203e9 −0.399839
\(521\) −1.81380e9 −0.561897 −0.280949 0.959723i \(-0.590649\pi\)
−0.280949 + 0.959723i \(0.590649\pi\)
\(522\) −8.23327e8 −0.253353
\(523\) 4.85164e8 0.148297 0.0741485 0.997247i \(-0.476376\pi\)
0.0741485 + 0.997247i \(0.476376\pi\)
\(524\) 1.44408e9 0.438462
\(525\) 4.16754e8 0.125696
\(526\) 3.23810e9 0.970153
\(527\) −1.26912e9 −0.377716
\(528\) −1.47198e8 −0.0435194
\(529\) −9.23192e8 −0.271142
\(530\) −3.27306e8 −0.0954967
\(531\) 5.06413e8 0.146782
\(532\) 7.57695e8 0.218174
\(533\) 9.74454e9 2.78751
\(534\) −2.14052e9 −0.608311
\(535\) −2.73336e9 −0.771719
\(536\) 1.65294e9 0.463640
\(537\) −2.78942e8 −0.0777327
\(538\) −3.48285e8 −0.0964265
\(539\) 1.56591e8 0.0430730
\(540\) −2.29268e8 −0.0626563
\(541\) −4.45922e9 −1.21079 −0.605394 0.795926i \(-0.706985\pi\)
−0.605394 + 0.795926i \(0.706985\pi\)
\(542\) 2.11985e9 0.571883
\(543\) −1.80155e9 −0.482889
\(544\) −4.82935e8 −0.128615
\(545\) 1.93377e9 0.511701
\(546\) 1.01930e9 0.267996
\(547\) −3.95780e9 −1.03395 −0.516973 0.856002i \(-0.672941\pi\)
−0.516973 + 0.856002i \(0.672941\pi\)
\(548\) 7.96553e8 0.206768
\(549\) 3.65302e7 0.00942212
\(550\) 4.79171e8 0.122806
\(551\) 4.87276e9 1.24092
\(552\) 6.88656e8 0.174267
\(553\) 4.34249e8 0.109195
\(554\) −5.19599e9 −1.29833
\(555\) −2.62918e9 −0.652821
\(556\) 1.04512e8 0.0257873
\(557\) 2.30018e9 0.563987 0.281994 0.959416i \(-0.409004\pi\)
0.281994 + 0.959416i \(0.409004\pi\)
\(558\) 5.02205e8 0.122366
\(559\) 2.23281e9 0.540644
\(560\) 2.55697e8 0.0615272
\(561\) −5.29640e8 −0.126652
\(562\) −1.40645e9 −0.334231
\(563\) 3.77133e9 0.890667 0.445333 0.895365i \(-0.353085\pi\)
0.445333 + 0.895365i \(0.353085\pi\)
\(564\) −6.86527e8 −0.161132
\(565\) −1.29094e9 −0.301119
\(566\) 9.72695e8 0.225486
\(567\) 1.82284e8 0.0419961
\(568\) 9.58206e8 0.219402
\(569\) −3.86882e9 −0.880412 −0.440206 0.897897i \(-0.645095\pi\)
−0.440206 + 0.897897i \(0.645095\pi\)
\(570\) 1.35689e9 0.306891
\(571\) 2.65511e9 0.596837 0.298418 0.954435i \(-0.403541\pi\)
0.298418 + 0.954435i \(0.403541\pi\)
\(572\) 1.17196e9 0.261835
\(573\) 1.22543e9 0.272111
\(574\) −1.94353e9 −0.428942
\(575\) −2.24177e9 −0.491760
\(576\) 1.91103e8 0.0416667
\(577\) −6.50529e9 −1.40978 −0.704890 0.709317i \(-0.749004\pi\)
−0.704890 + 0.709317i \(0.749004\pi\)
\(578\) 1.54504e9 0.332807
\(579\) 7.88480e8 0.168817
\(580\) 1.64439e9 0.349952
\(581\) −8.72936e8 −0.184657
\(582\) −3.02634e9 −0.636337
\(583\) 2.99206e8 0.0625361
\(584\) 4.80022e8 0.0997277
\(585\) 1.82538e9 0.376972
\(586\) −4.21945e9 −0.866192
\(587\) 5.82831e9 1.18935 0.594675 0.803966i \(-0.297281\pi\)
0.594675 + 0.803966i \(0.297281\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −2.97224e9 −0.599350
\(590\) −1.01144e9 −0.202748
\(591\) −3.17621e9 −0.632927
\(592\) 2.19152e9 0.434129
\(593\) 9.21565e9 1.81482 0.907412 0.420243i \(-0.138055\pi\)
0.907412 + 0.420243i \(0.138055\pi\)
\(594\) 2.09585e8 0.0410305
\(595\) 9.20034e8 0.179058
\(596\) 4.32084e9 0.835999
\(597\) 1.67764e9 0.322692
\(598\) −5.48295e9 −1.04848
\(599\) 8.32457e9 1.58259 0.791294 0.611436i \(-0.209408\pi\)
0.791294 + 0.611436i \(0.209408\pi\)
\(600\) −6.22094e8 −0.117578
\(601\) −1.80533e9 −0.339230 −0.169615 0.985510i \(-0.554252\pi\)
−0.169615 + 0.985510i \(0.554252\pi\)
\(602\) −4.45329e8 −0.0831943
\(603\) −2.35351e9 −0.437124
\(604\) −3.98859e9 −0.736530
\(605\) 3.22424e8 0.0591948
\(606\) −1.58605e9 −0.289509
\(607\) 9.91327e9 1.79911 0.899553 0.436812i \(-0.143892\pi\)
0.899553 + 0.436812i \(0.143892\pi\)
\(608\) −1.13102e9 −0.204083
\(609\) −1.30741e9 −0.234559
\(610\) −7.29602e7 −0.0130146
\(611\) 5.46600e9 0.969450
\(612\) 6.87616e8 0.121260
\(613\) −1.21157e9 −0.212440 −0.106220 0.994343i \(-0.533875\pi\)
−0.106220 + 0.994343i \(0.533875\pi\)
\(614\) −9.90432e8 −0.172677
\(615\) −3.48050e9 −0.603363
\(616\) −2.33745e8 −0.0402911
\(617\) −9.57641e8 −0.164136 −0.0820681 0.996627i \(-0.526153\pi\)
−0.0820681 + 0.996627i \(0.526153\pi\)
\(618\) −1.91382e9 −0.326169
\(619\) −4.81235e9 −0.815531 −0.407765 0.913087i \(-0.633692\pi\)
−0.407765 + 0.913087i \(0.633692\pi\)
\(620\) −1.00303e9 −0.169022
\(621\) −9.80528e8 −0.164301
\(622\) −4.13959e9 −0.689748
\(623\) −3.39907e9 −0.563186
\(624\) −1.52152e9 −0.250688
\(625\) −5.62722e8 −0.0921965
\(626\) 7.87743e8 0.128344
\(627\) −1.24040e9 −0.200968
\(628\) −5.13774e9 −0.827777
\(629\) 7.88539e9 1.26341
\(630\) −3.64068e8 −0.0580084
\(631\) −5.18280e9 −0.821224 −0.410612 0.911810i \(-0.634685\pi\)
−0.410612 + 0.911810i \(0.634685\pi\)
\(632\) −6.48208e8 −0.102142
\(633\) 4.96861e9 0.778613
\(634\) −2.81685e9 −0.438986
\(635\) 6.55917e9 1.01658
\(636\) −3.88451e8 −0.0598737
\(637\) 1.61861e9 0.248116
\(638\) −1.50322e9 −0.229166
\(639\) −1.36432e9 −0.206854
\(640\) −3.81682e8 −0.0575535
\(641\) −2.63106e9 −0.394574 −0.197287 0.980346i \(-0.563213\pi\)
−0.197287 + 0.980346i \(0.563213\pi\)
\(642\) −3.24399e9 −0.483846
\(643\) −4.32469e9 −0.641530 −0.320765 0.947159i \(-0.603940\pi\)
−0.320765 + 0.947159i \(0.603940\pi\)
\(644\) 1.09356e9 0.161340
\(645\) −7.97503e8 −0.117024
\(646\) −4.06957e9 −0.593930
\(647\) −4.34323e9 −0.630446 −0.315223 0.949018i \(-0.602079\pi\)
−0.315223 + 0.949018i \(0.602079\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) 9.24603e8 0.132770
\(650\) 4.95299e9 0.707409
\(651\) 7.97483e8 0.113289
\(652\) −7.54236e8 −0.106571
\(653\) 4.59143e9 0.645285 0.322643 0.946521i \(-0.395429\pi\)
0.322643 + 0.946521i \(0.395429\pi\)
\(654\) 2.29502e9 0.320822
\(655\) 4.10660e9 0.571003
\(656\) 2.90112e9 0.401239
\(657\) −6.83468e8 −0.0940242
\(658\) −1.09018e9 −0.149179
\(659\) 3.99244e9 0.543424 0.271712 0.962379i \(-0.412410\pi\)
0.271712 + 0.962379i \(0.412410\pi\)
\(660\) −4.18594e8 −0.0566747
\(661\) −4.95805e9 −0.667737 −0.333869 0.942620i \(-0.608354\pi\)
−0.333869 + 0.942620i \(0.608354\pi\)
\(662\) 4.28960e9 0.574664
\(663\) −5.47467e9 −0.729559
\(664\) 1.30304e9 0.172731
\(665\) 2.15470e9 0.284125
\(666\) −3.12034e9 −0.409301
\(667\) 7.03272e9 0.917663
\(668\) −3.53154e9 −0.458402
\(669\) −2.37889e9 −0.307174
\(670\) 4.70056e9 0.603792
\(671\) 6.66964e7 0.00852263
\(672\) 3.03464e8 0.0385758
\(673\) 2.86855e8 0.0362751 0.0181376 0.999836i \(-0.494226\pi\)
0.0181376 + 0.999836i \(0.494226\pi\)
\(674\) 5.83138e9 0.733605
\(675\) 8.85755e8 0.110854
\(676\) 8.09818e9 1.00826
\(677\) 7.36156e8 0.0911821 0.0455911 0.998960i \(-0.485483\pi\)
0.0455911 + 0.998960i \(0.485483\pi\)
\(678\) −1.53211e9 −0.188793
\(679\) −4.80571e9 −0.589134
\(680\) −1.37335e9 −0.167494
\(681\) −7.44872e9 −0.903788
\(682\) 9.16921e8 0.110684
\(683\) 2.38050e8 0.0285888 0.0142944 0.999898i \(-0.495450\pi\)
0.0142944 + 0.999898i \(0.495450\pi\)
\(684\) 1.61038e9 0.192412
\(685\) 2.26520e9 0.269271
\(686\) −3.22829e8 −0.0381802
\(687\) 2.23807e9 0.263345
\(688\) 6.64748e8 0.0778211
\(689\) 3.09277e9 0.360230
\(690\) 1.95837e9 0.226946
\(691\) 2.18434e9 0.251852 0.125926 0.992040i \(-0.459810\pi\)
0.125926 + 0.992040i \(0.459810\pi\)
\(692\) 3.60929e9 0.414048
\(693\) 3.32813e8 0.0379869
\(694\) −6.46848e9 −0.734589
\(695\) 2.97207e8 0.0335824
\(696\) 1.95159e9 0.219410
\(697\) 1.04387e10 1.16770
\(698\) 5.32286e9 0.592449
\(699\) 6.38286e9 0.706879
\(700\) −9.87862e8 −0.108856
\(701\) 1.33891e10 1.46804 0.734019 0.679129i \(-0.237643\pi\)
0.734019 + 0.679129i \(0.237643\pi\)
\(702\) 2.16639e9 0.236351
\(703\) 1.84674e10 2.00476
\(704\) 3.48914e8 0.0376889
\(705\) −1.95231e9 −0.209840
\(706\) 5.33294e8 0.0570361
\(707\) −2.51858e9 −0.268033
\(708\) −1.20039e9 −0.127117
\(709\) 1.20468e10 1.26944 0.634718 0.772744i \(-0.281116\pi\)
0.634718 + 0.772744i \(0.281116\pi\)
\(710\) 2.72490e9 0.285724
\(711\) 9.22937e8 0.0963005
\(712\) 5.07383e9 0.526813
\(713\) −4.28976e9 −0.443220
\(714\) 1.09191e9 0.112265
\(715\) 3.33277e9 0.340984
\(716\) 6.61196e8 0.0673185
\(717\) −7.50742e9 −0.760630
\(718\) 3.20001e9 0.322638
\(719\) −1.40604e9 −0.141074 −0.0705368 0.997509i \(-0.522471\pi\)
−0.0705368 + 0.997509i \(0.522471\pi\)
\(720\) 5.43449e8 0.0542619
\(721\) −3.03908e9 −0.301974
\(722\) −2.37986e9 −0.235327
\(723\) 4.00327e9 0.393940
\(724\) 4.27034e9 0.418194
\(725\) −6.35297e9 −0.619147
\(726\) 3.82657e8 0.0371135
\(727\) 2.21138e9 0.213448 0.106724 0.994289i \(-0.465964\pi\)
0.106724 + 0.994289i \(0.465964\pi\)
\(728\) −2.41612e9 −0.232092
\(729\) 3.87420e8 0.0370370
\(730\) 1.36506e9 0.129874
\(731\) 2.39186e9 0.226477
\(732\) −8.65901e7 −0.00815980
\(733\) −6.28279e9 −0.589235 −0.294618 0.955615i \(-0.595192\pi\)
−0.294618 + 0.955615i \(0.595192\pi\)
\(734\) 2.79718e9 0.261087
\(735\) −5.78127e8 −0.0537054
\(736\) −1.63237e9 −0.150920
\(737\) −4.29701e9 −0.395394
\(738\) −4.13070e9 −0.378291
\(739\) −2.04236e10 −1.86156 −0.930779 0.365582i \(-0.880870\pi\)
−0.930779 + 0.365582i \(0.880870\pi\)
\(740\) 6.23212e9 0.565360
\(741\) −1.28215e10 −1.15765
\(742\) −6.16846e8 −0.0554323
\(743\) −1.04573e10 −0.935314 −0.467657 0.883910i \(-0.654902\pi\)
−0.467657 + 0.883910i \(0.654902\pi\)
\(744\) −1.19041e9 −0.105972
\(745\) 1.22874e10 1.08871
\(746\) −2.58897e9 −0.228318
\(747\) −1.85531e9 −0.162852
\(748\) 1.25544e9 0.109683
\(749\) −5.15134e9 −0.447954
\(750\) −4.84033e9 −0.418948
\(751\) −3.96979e9 −0.342001 −0.171001 0.985271i \(-0.554700\pi\)
−0.171001 + 0.985271i \(0.554700\pi\)
\(752\) 1.62732e9 0.139544
\(753\) −7.80472e9 −0.666154
\(754\) −1.55382e10 −1.32008
\(755\) −1.13426e10 −0.959173
\(756\) −4.32081e8 −0.0363696
\(757\) −2.09521e9 −0.175546 −0.0877731 0.996140i \(-0.527975\pi\)
−0.0877731 + 0.996140i \(0.527975\pi\)
\(758\) 5.19728e9 0.433445
\(759\) −1.79024e9 −0.148616
\(760\) −3.21634e9 −0.265775
\(761\) 2.98963e9 0.245907 0.122954 0.992412i \(-0.460763\pi\)
0.122954 + 0.992412i \(0.460763\pi\)
\(762\) 7.78451e9 0.637366
\(763\) 3.64441e9 0.297024
\(764\) −2.90472e9 −0.235655
\(765\) 1.95541e9 0.157915
\(766\) −8.93580e9 −0.718344
\(767\) 9.55724e9 0.764802
\(768\) −4.52985e8 −0.0360844
\(769\) 2.28863e10 1.81482 0.907411 0.420245i \(-0.138056\pi\)
0.907411 + 0.420245i \(0.138056\pi\)
\(770\) −6.64712e8 −0.0524706
\(771\) 5.38422e9 0.423090
\(772\) −1.86899e9 −0.146200
\(773\) 1.50121e10 1.16900 0.584499 0.811394i \(-0.301291\pi\)
0.584499 + 0.811394i \(0.301291\pi\)
\(774\) −9.46487e8 −0.0733705
\(775\) 3.87513e9 0.299040
\(776\) 7.17354e9 0.551084
\(777\) −4.95499e9 −0.378939
\(778\) −9.50175e9 −0.723394
\(779\) 2.44471e10 1.85287
\(780\) −4.32684e9 −0.326467
\(781\) −2.49096e9 −0.187106
\(782\) −5.87351e9 −0.439212
\(783\) −2.77873e9 −0.206862
\(784\) 4.81890e8 0.0357143
\(785\) −1.46104e10 −1.07800
\(786\) 4.87377e9 0.358003
\(787\) 1.23722e10 0.904762 0.452381 0.891825i \(-0.350575\pi\)
0.452381 + 0.891825i \(0.350575\pi\)
\(788\) 7.52880e9 0.548131
\(789\) 1.09286e10 0.792127
\(790\) −1.84334e9 −0.133018
\(791\) −2.43293e9 −0.174788
\(792\) −4.96793e8 −0.0355335
\(793\) 6.89413e8 0.0490934
\(794\) −1.76194e10 −1.24916
\(795\) −1.10466e9 −0.0779727
\(796\) −3.97662e9 −0.279459
\(797\) 1.59988e10 1.11939 0.559696 0.828698i \(-0.310918\pi\)
0.559696 + 0.828698i \(0.310918\pi\)
\(798\) 2.55722e9 0.178139
\(799\) 5.85535e9 0.406106
\(800\) 1.47459e9 0.101826
\(801\) −7.22427e9 −0.496684
\(802\) −6.58206e9 −0.450559
\(803\) −1.24787e9 −0.0850481
\(804\) 5.57868e9 0.378561
\(805\) 3.10981e9 0.210111
\(806\) 9.47783e9 0.637582
\(807\) −1.17546e9 −0.0787319
\(808\) 3.75952e9 0.250722
\(809\) 1.19166e10 0.791281 0.395641 0.918405i \(-0.370523\pi\)
0.395641 + 0.918405i \(0.370523\pi\)
\(810\) −7.73778e8 −0.0511586
\(811\) −2.03252e10 −1.33802 −0.669010 0.743254i \(-0.733282\pi\)
−0.669010 + 0.743254i \(0.733282\pi\)
\(812\) 3.09905e9 0.203134
\(813\) 7.15449e9 0.466941
\(814\) −5.69708e9 −0.370226
\(815\) −2.14486e9 −0.138786
\(816\) −1.62990e9 −0.105014
\(817\) 5.60167e9 0.359369
\(818\) −7.20876e9 −0.460494
\(819\) 3.44015e9 0.218818
\(820\) 8.25007e9 0.522527
\(821\) −8.13232e9 −0.512877 −0.256439 0.966561i \(-0.582549\pi\)
−0.256439 + 0.966561i \(0.582549\pi\)
\(822\) 2.68837e9 0.168825
\(823\) 1.27514e10 0.797366 0.398683 0.917089i \(-0.369467\pi\)
0.398683 + 0.917089i \(0.369467\pi\)
\(824\) 4.53647e9 0.282470
\(825\) 1.61720e9 0.100271
\(826\) −1.90617e9 −0.117688
\(827\) −3.43433e9 −0.211141 −0.105570 0.994412i \(-0.533667\pi\)
−0.105570 + 0.994412i \(0.533667\pi\)
\(828\) 2.32422e9 0.142289
\(829\) 3.10696e10 1.89407 0.947033 0.321136i \(-0.104065\pi\)
0.947033 + 0.321136i \(0.104065\pi\)
\(830\) 3.70553e9 0.224945
\(831\) −1.75365e10 −1.06008
\(832\) 3.60658e9 0.217102
\(833\) 1.73391e9 0.103937
\(834\) 3.52729e8 0.0210552
\(835\) −1.00428e10 −0.596971
\(836\) 2.94021e9 0.174043
\(837\) 1.69494e9 0.0999116
\(838\) 1.41463e10 0.830404
\(839\) −2.69261e10 −1.57401 −0.787003 0.616949i \(-0.788368\pi\)
−0.787003 + 0.616949i \(0.788368\pi\)
\(840\) 8.62977e8 0.0502368
\(841\) 2.68022e9 0.155376
\(842\) −3.65541e9 −0.211030
\(843\) −4.74676e9 −0.272898
\(844\) −1.17774e10 −0.674299
\(845\) 2.30292e10 1.31305
\(846\) −2.31703e9 −0.131564
\(847\) 6.07645e8 0.0343604
\(848\) 9.20773e8 0.0518522
\(849\) 3.28285e9 0.184108
\(850\) 5.30580e9 0.296336
\(851\) 2.66535e10 1.48252
\(852\) 3.23395e9 0.179141
\(853\) −2.28862e10 −1.26256 −0.631281 0.775554i \(-0.717470\pi\)
−0.631281 + 0.775554i \(0.717470\pi\)
\(854\) −1.37502e8 −0.00755450
\(855\) 4.57951e9 0.250575
\(856\) 7.68946e9 0.419023
\(857\) −2.94221e10 −1.59676 −0.798382 0.602151i \(-0.794311\pi\)
−0.798382 + 0.602151i \(0.794311\pi\)
\(858\) 3.95537e9 0.213787
\(859\) −1.03932e10 −0.559465 −0.279733 0.960078i \(-0.590246\pi\)
−0.279733 + 0.960078i \(0.590246\pi\)
\(860\) 1.89038e9 0.101345
\(861\) −6.55940e9 −0.350230
\(862\) 1.25308e10 0.666353
\(863\) 2.21900e10 1.17522 0.587610 0.809144i \(-0.300069\pi\)
0.587610 + 0.809144i \(0.300069\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 1.02639e10 0.539209
\(866\) 1.36246e10 0.712870
\(867\) 5.21451e9 0.271736
\(868\) −1.89033e9 −0.0981112
\(869\) 1.68509e9 0.0871071
\(870\) 5.54983e9 0.285734
\(871\) −4.44164e10 −2.27761
\(872\) −5.44005e9 −0.277840
\(873\) −1.02139e10 −0.519567
\(874\) −1.37556e10 −0.696930
\(875\) −7.68626e9 −0.387871
\(876\) 1.62007e9 0.0814273
\(877\) 3.59473e10 1.79957 0.899783 0.436337i \(-0.143724\pi\)
0.899783 + 0.436337i \(0.143724\pi\)
\(878\) −1.31628e10 −0.656322
\(879\) −1.42406e10 −0.707243
\(880\) 9.92223e8 0.0490817
\(881\) −2.76295e10 −1.36131 −0.680655 0.732605i \(-0.738305\pi\)
−0.680655 + 0.732605i \(0.738305\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −2.32225e10 −1.13513 −0.567567 0.823328i \(-0.692115\pi\)
−0.567567 + 0.823328i \(0.692115\pi\)
\(884\) 1.29770e10 0.631816
\(885\) −3.41360e9 −0.165543
\(886\) −1.09845e10 −0.530593
\(887\) 2.93323e10 1.41128 0.705641 0.708569i \(-0.250659\pi\)
0.705641 + 0.708569i \(0.250659\pi\)
\(888\) 7.39637e9 0.354465
\(889\) 1.23615e10 0.590086
\(890\) 1.44287e10 0.686061
\(891\) 7.07348e8 0.0335013
\(892\) 5.63886e9 0.266020
\(893\) 1.37131e10 0.644399
\(894\) 1.45828e10 0.682591
\(895\) 1.88027e9 0.0876679
\(896\) −7.19323e8 −0.0334077
\(897\) −1.85050e10 −0.856080
\(898\) −2.08926e10 −0.962775
\(899\) −1.21568e10 −0.558032
\(900\) −2.09957e9 −0.0960021
\(901\) 3.31307e9 0.150902
\(902\) −7.54179e9 −0.342177
\(903\) −1.50299e9 −0.0679279
\(904\) 3.63167e9 0.163500
\(905\) 1.21438e10 0.544608
\(906\) −1.34615e10 −0.601374
\(907\) −1.38883e10 −0.618050 −0.309025 0.951054i \(-0.600003\pi\)
−0.309025 + 0.951054i \(0.600003\pi\)
\(908\) 1.76562e10 0.782703
\(909\) −5.35291e9 −0.236383
\(910\) −6.87086e9 −0.302250
\(911\) 4.83193e9 0.211742 0.105871 0.994380i \(-0.466237\pi\)
0.105871 + 0.994380i \(0.466237\pi\)
\(912\) −3.81719e9 −0.166633
\(913\) −3.38740e9 −0.147305
\(914\) 2.50446e10 1.08493
\(915\) −2.46241e8 −0.0106264
\(916\) −5.30506e9 −0.228064
\(917\) 7.73937e9 0.331446
\(918\) 2.32070e9 0.0990080
\(919\) 1.95192e10 0.829578 0.414789 0.909918i \(-0.363855\pi\)
0.414789 + 0.909918i \(0.363855\pi\)
\(920\) −4.64205e9 −0.196541
\(921\) −3.34271e9 −0.140990
\(922\) 1.35632e10 0.569906
\(923\) −2.57480e10 −1.07780
\(924\) −7.88889e8 −0.0328976
\(925\) −2.40772e10 −1.00025
\(926\) 6.67809e9 0.276385
\(927\) −6.45916e9 −0.266316
\(928\) −4.62599e9 −0.190014
\(929\) −3.96997e10 −1.62455 −0.812274 0.583277i \(-0.801770\pi\)
−0.812274 + 0.583277i \(0.801770\pi\)
\(930\) −3.38523e9 −0.138006
\(931\) 4.06077e9 0.164924
\(932\) −1.51297e10 −0.612175
\(933\) −1.39711e10 −0.563177
\(934\) 8.99675e9 0.361303
\(935\) 3.57016e9 0.142839
\(936\) −5.13515e9 −0.204686
\(937\) 3.04155e9 0.120783 0.0603916 0.998175i \(-0.480765\pi\)
0.0603916 + 0.998175i \(0.480765\pi\)
\(938\) 8.85874e9 0.350479
\(939\) 2.65863e9 0.104792
\(940\) 4.62770e9 0.181726
\(941\) −7.60593e9 −0.297570 −0.148785 0.988870i \(-0.547536\pi\)
−0.148785 + 0.988870i \(0.547536\pi\)
\(942\) −1.73399e10 −0.675877
\(943\) 3.52838e10 1.37020
\(944\) 2.84536e9 0.110087
\(945\) −1.22873e9 −0.0473637
\(946\) −1.72809e9 −0.0663661
\(947\) 2.22782e10 0.852424 0.426212 0.904623i \(-0.359848\pi\)
0.426212 + 0.904623i \(0.359848\pi\)
\(948\) −2.18770e9 −0.0833987
\(949\) −1.28987e10 −0.489908
\(950\) 1.24260e10 0.470219
\(951\) −9.50685e9 −0.358431
\(952\) −2.58823e9 −0.0972240
\(953\) −4.93836e10 −1.84824 −0.924119 0.382106i \(-0.875199\pi\)
−0.924119 + 0.382106i \(0.875199\pi\)
\(954\) −1.31102e9 −0.0488867
\(955\) −8.26029e9 −0.306891
\(956\) 1.77954e10 0.658725
\(957\) −5.07337e9 −0.187113
\(958\) 3.01781e10 1.10895
\(959\) 4.26903e9 0.156302
\(960\) −1.28818e9 −0.0469922
\(961\) −2.00973e10 −0.730477
\(962\) −5.88884e10 −2.13264
\(963\) −1.09485e10 −0.395058
\(964\) −9.48922e9 −0.341162
\(965\) −5.31494e9 −0.190394
\(966\) 3.69077e9 0.131734
\(967\) −4.59604e8 −0.0163452 −0.00817261 0.999967i \(-0.502601\pi\)
−0.00817261 + 0.999967i \(0.502601\pi\)
\(968\) −9.07039e8 −0.0321412
\(969\) −1.37348e10 −0.484942
\(970\) 2.03998e10 0.717669
\(971\) 2.51853e9 0.0882836 0.0441418 0.999025i \(-0.485945\pi\)
0.0441418 + 0.999025i \(0.485945\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 5.60120e8 0.0194933
\(974\) 3.52279e10 1.22160
\(975\) 1.67163e10 0.577597
\(976\) 2.05251e8 0.00706659
\(977\) 2.47811e10 0.850137 0.425069 0.905161i \(-0.360250\pi\)
0.425069 + 0.905161i \(0.360250\pi\)
\(978\) −2.54555e9 −0.0870151
\(979\) −1.31900e10 −0.449267
\(980\) 1.37038e9 0.0465102
\(981\) 7.74569e9 0.261950
\(982\) 2.09226e10 0.705058
\(983\) 4.99754e10 1.67810 0.839052 0.544052i \(-0.183110\pi\)
0.839052 + 0.544052i \(0.183110\pi\)
\(984\) 9.79129e9 0.327610
\(985\) 2.14100e10 0.713823
\(986\) −1.66450e10 −0.552986
\(987\) −3.67936e9 −0.121804
\(988\) 3.03918e10 1.00255
\(989\) 8.08474e9 0.265753
\(990\) −1.41276e9 −0.0462747
\(991\) 2.07635e9 0.0677708 0.0338854 0.999426i \(-0.489212\pi\)
0.0338854 + 0.999426i \(0.489212\pi\)
\(992\) 2.82172e9 0.0917747
\(993\) 1.44774e10 0.469211
\(994\) 5.13539e9 0.165852
\(995\) −1.13085e10 −0.363936
\(996\) 4.39777e9 0.141034
\(997\) −1.92580e10 −0.615428 −0.307714 0.951479i \(-0.599564\pi\)
−0.307714 + 0.951479i \(0.599564\pi\)
\(998\) −1.72433e10 −0.549115
\(999\) −1.05312e10 −0.334193
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 462.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.8.a.a.1.1 1 1.1 even 1 trivial