Properties

Label 531.8.a.b.1.12
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(14.7989\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+14.7989 q^{2} +91.0084 q^{4} +296.536 q^{5} -1410.76 q^{7} -547.437 q^{8} +O(q^{10})\) \(q+14.7989 q^{2} +91.0084 q^{4} +296.536 q^{5} -1410.76 q^{7} -547.437 q^{8} +4388.42 q^{10} +7538.45 q^{11} +7066.47 q^{13} -20877.7 q^{14} -19750.5 q^{16} -14832.8 q^{17} -24023.8 q^{19} +26987.3 q^{20} +111561. q^{22} -41590.5 q^{23} +9808.63 q^{25} +104576. q^{26} -128391. q^{28} -15791.6 q^{29} +108246. q^{31} -222215. q^{32} -219509. q^{34} -418341. q^{35} +331221. q^{37} -355527. q^{38} -162335. q^{40} -224396. q^{41} -134226. q^{43} +686063. q^{44} -615494. q^{46} -816259. q^{47} +1.16670e6 q^{49} +145157. q^{50} +643108. q^{52} -1.21786e6 q^{53} +2.23542e6 q^{55} +772301. q^{56} -233699. q^{58} -205379. q^{59} +1.88156e6 q^{61} +1.60193e6 q^{62} -760476. q^{64} +2.09546e6 q^{65} -2.09166e6 q^{67} -1.34991e6 q^{68} -6.19100e6 q^{70} +1.80830e6 q^{71} -4.93066e6 q^{73} +4.90172e6 q^{74} -2.18637e6 q^{76} -1.06349e7 q^{77} -8.26089e6 q^{79} -5.85675e6 q^{80} -3.32083e6 q^{82} +2.95933e6 q^{83} -4.39846e6 q^{85} -1.98640e6 q^{86} -4.12683e6 q^{88} -3.82628e6 q^{89} -9.96908e6 q^{91} -3.78508e6 q^{92} -1.20798e7 q^{94} -7.12393e6 q^{95} -6.24098e6 q^{97} +1.72659e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.7989 1.30805 0.654027 0.756472i \(-0.273078\pi\)
0.654027 + 0.756472i \(0.273078\pi\)
\(3\) 0 0
\(4\) 91.0084 0.711003
\(5\) 296.536 1.06092 0.530460 0.847710i \(-0.322019\pi\)
0.530460 + 0.847710i \(0.322019\pi\)
\(6\) 0 0
\(7\) −1410.76 −1.55457 −0.777284 0.629150i \(-0.783403\pi\)
−0.777284 + 0.629150i \(0.783403\pi\)
\(8\) −547.437 −0.378024
\(9\) 0 0
\(10\) 4388.42 1.38774
\(11\) 7538.45 1.70769 0.853843 0.520531i \(-0.174266\pi\)
0.853843 + 0.520531i \(0.174266\pi\)
\(12\) 0 0
\(13\) 7066.47 0.892073 0.446037 0.895015i \(-0.352835\pi\)
0.446037 + 0.895015i \(0.352835\pi\)
\(14\) −20877.7 −2.03346
\(15\) 0 0
\(16\) −19750.5 −1.20548
\(17\) −14832.8 −0.732237 −0.366118 0.930568i \(-0.619313\pi\)
−0.366118 + 0.930568i \(0.619313\pi\)
\(18\) 0 0
\(19\) −24023.8 −0.803535 −0.401767 0.915742i \(-0.631604\pi\)
−0.401767 + 0.915742i \(0.631604\pi\)
\(20\) 26987.3 0.754317
\(21\) 0 0
\(22\) 111561. 2.23374
\(23\) −41590.5 −0.712765 −0.356382 0.934340i \(-0.615990\pi\)
−0.356382 + 0.934340i \(0.615990\pi\)
\(24\) 0 0
\(25\) 9808.63 0.125550
\(26\) 104576. 1.16688
\(27\) 0 0
\(28\) −128391. −1.10530
\(29\) −15791.6 −0.120235 −0.0601177 0.998191i \(-0.519148\pi\)
−0.0601177 + 0.998191i \(0.519148\pi\)
\(30\) 0 0
\(31\) 108246. 0.652601 0.326300 0.945266i \(-0.394198\pi\)
0.326300 + 0.945266i \(0.394198\pi\)
\(32\) −222215. −1.19881
\(33\) 0 0
\(34\) −219509. −0.957805
\(35\) −418341. −1.64927
\(36\) 0 0
\(37\) 331221. 1.07501 0.537504 0.843261i \(-0.319367\pi\)
0.537504 + 0.843261i \(0.319367\pi\)
\(38\) −355527. −1.05107
\(39\) 0 0
\(40\) −162335. −0.401053
\(41\) −224396. −0.508478 −0.254239 0.967141i \(-0.581825\pi\)
−0.254239 + 0.967141i \(0.581825\pi\)
\(42\) 0 0
\(43\) −134226. −0.257453 −0.128726 0.991680i \(-0.541089\pi\)
−0.128726 + 0.991680i \(0.541089\pi\)
\(44\) 686063. 1.21417
\(45\) 0 0
\(46\) −615494. −0.932334
\(47\) −816259. −1.14679 −0.573397 0.819278i \(-0.694375\pi\)
−0.573397 + 0.819278i \(0.694375\pi\)
\(48\) 0 0
\(49\) 1.16670e6 1.41668
\(50\) 145157. 0.164227
\(51\) 0 0
\(52\) 643108. 0.634267
\(53\) −1.21786e6 −1.12365 −0.561825 0.827256i \(-0.689900\pi\)
−0.561825 + 0.827256i \(0.689900\pi\)
\(54\) 0 0
\(55\) 2.23542e6 1.81172
\(56\) 772301. 0.587663
\(57\) 0 0
\(58\) −233699. −0.157274
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 1.88156e6 1.06136 0.530681 0.847571i \(-0.321936\pi\)
0.530681 + 0.847571i \(0.321936\pi\)
\(62\) 1.60193e6 0.853637
\(63\) 0 0
\(64\) −760476. −0.362623
\(65\) 2.09546e6 0.946418
\(66\) 0 0
\(67\) −2.09166e6 −0.849627 −0.424814 0.905281i \(-0.639660\pi\)
−0.424814 + 0.905281i \(0.639660\pi\)
\(68\) −1.34991e6 −0.520622
\(69\) 0 0
\(70\) −6.19100e6 −2.15734
\(71\) 1.80830e6 0.599605 0.299802 0.954001i \(-0.403079\pi\)
0.299802 + 0.954001i \(0.403079\pi\)
\(72\) 0 0
\(73\) −4.93066e6 −1.48346 −0.741729 0.670699i \(-0.765994\pi\)
−0.741729 + 0.670699i \(0.765994\pi\)
\(74\) 4.90172e6 1.40617
\(75\) 0 0
\(76\) −2.18637e6 −0.571316
\(77\) −1.06349e7 −2.65471
\(78\) 0 0
\(79\) −8.26089e6 −1.88509 −0.942545 0.334078i \(-0.891575\pi\)
−0.942545 + 0.334078i \(0.891575\pi\)
\(80\) −5.85675e6 −1.27892
\(81\) 0 0
\(82\) −3.32083e6 −0.665116
\(83\) 2.95933e6 0.568093 0.284047 0.958810i \(-0.408323\pi\)
0.284047 + 0.958810i \(0.408323\pi\)
\(84\) 0 0
\(85\) −4.39846e6 −0.776844
\(86\) −1.98640e6 −0.336762
\(87\) 0 0
\(88\) −4.12683e6 −0.645545
\(89\) −3.82628e6 −0.575323 −0.287661 0.957732i \(-0.592878\pi\)
−0.287661 + 0.957732i \(0.592878\pi\)
\(90\) 0 0
\(91\) −9.96908e6 −1.38679
\(92\) −3.78508e6 −0.506778
\(93\) 0 0
\(94\) −1.20798e7 −1.50007
\(95\) −7.12393e6 −0.852486
\(96\) 0 0
\(97\) −6.24098e6 −0.694308 −0.347154 0.937808i \(-0.612852\pi\)
−0.347154 + 0.937808i \(0.612852\pi\)
\(98\) 1.72659e7 1.85310
\(99\) 0 0
\(100\) 892667. 0.0892667
\(101\) −1.50527e7 −1.45375 −0.726873 0.686772i \(-0.759027\pi\)
−0.726873 + 0.686772i \(0.759027\pi\)
\(102\) 0 0
\(103\) 1.06416e7 0.959572 0.479786 0.877386i \(-0.340714\pi\)
0.479786 + 0.877386i \(0.340714\pi\)
\(104\) −3.86844e6 −0.337225
\(105\) 0 0
\(106\) −1.80230e7 −1.46979
\(107\) 1.71933e7 1.35680 0.678402 0.734691i \(-0.262673\pi\)
0.678402 + 0.734691i \(0.262673\pi\)
\(108\) 0 0
\(109\) −2.61258e7 −1.93231 −0.966154 0.257968i \(-0.916947\pi\)
−0.966154 + 0.257968i \(0.916947\pi\)
\(110\) 3.30819e7 2.36982
\(111\) 0 0
\(112\) 2.78633e7 1.87400
\(113\) −1.49158e7 −0.972463 −0.486231 0.873830i \(-0.661629\pi\)
−0.486231 + 0.873830i \(0.661629\pi\)
\(114\) 0 0
\(115\) −1.23331e7 −0.756186
\(116\) −1.43717e6 −0.0854878
\(117\) 0 0
\(118\) −3.03939e6 −0.170294
\(119\) 2.09255e7 1.13831
\(120\) 0 0
\(121\) 3.73411e7 1.91619
\(122\) 2.78451e7 1.38832
\(123\) 0 0
\(124\) 9.85133e6 0.464001
\(125\) −2.02583e7 −0.927721
\(126\) 0 0
\(127\) −1.99083e7 −0.862426 −0.431213 0.902250i \(-0.641914\pi\)
−0.431213 + 0.902250i \(0.641914\pi\)
\(128\) 1.71893e7 0.724475
\(129\) 0 0
\(130\) 3.10106e7 1.23797
\(131\) −2.18462e7 −0.849038 −0.424519 0.905419i \(-0.639557\pi\)
−0.424519 + 0.905419i \(0.639557\pi\)
\(132\) 0 0
\(133\) 3.38918e7 1.24915
\(134\) −3.09543e7 −1.11136
\(135\) 0 0
\(136\) 8.12001e6 0.276803
\(137\) 1.96904e7 0.654233 0.327117 0.944984i \(-0.393923\pi\)
0.327117 + 0.944984i \(0.393923\pi\)
\(138\) 0 0
\(139\) 2.42247e6 0.0765078 0.0382539 0.999268i \(-0.487820\pi\)
0.0382539 + 0.999268i \(0.487820\pi\)
\(140\) −3.80725e7 −1.17264
\(141\) 0 0
\(142\) 2.67608e7 0.784315
\(143\) 5.32702e7 1.52338
\(144\) 0 0
\(145\) −4.68277e6 −0.127560
\(146\) −7.29686e7 −1.94044
\(147\) 0 0
\(148\) 3.01439e7 0.764334
\(149\) −2.09513e7 −0.518871 −0.259436 0.965760i \(-0.583537\pi\)
−0.259436 + 0.965760i \(0.583537\pi\)
\(150\) 0 0
\(151\) 4.77834e6 0.112943 0.0564713 0.998404i \(-0.482015\pi\)
0.0564713 + 0.998404i \(0.482015\pi\)
\(152\) 1.31515e7 0.303755
\(153\) 0 0
\(154\) −1.57386e8 −3.47251
\(155\) 3.20990e7 0.692357
\(156\) 0 0
\(157\) −6.17818e7 −1.27412 −0.637062 0.770813i \(-0.719850\pi\)
−0.637062 + 0.770813i \(0.719850\pi\)
\(158\) −1.22252e8 −2.46580
\(159\) 0 0
\(160\) −6.58948e7 −1.27184
\(161\) 5.86741e7 1.10804
\(162\) 0 0
\(163\) 8.27535e7 1.49668 0.748342 0.663313i \(-0.230850\pi\)
0.748342 + 0.663313i \(0.230850\pi\)
\(164\) −2.04219e7 −0.361529
\(165\) 0 0
\(166\) 4.37949e7 0.743096
\(167\) 1.39381e7 0.231578 0.115789 0.993274i \(-0.463060\pi\)
0.115789 + 0.993274i \(0.463060\pi\)
\(168\) 0 0
\(169\) −1.28136e7 −0.204205
\(170\) −6.50925e7 −1.01615
\(171\) 0 0
\(172\) −1.22157e7 −0.183049
\(173\) −9.03037e7 −1.32600 −0.663001 0.748618i \(-0.730718\pi\)
−0.663001 + 0.748618i \(0.730718\pi\)
\(174\) 0 0
\(175\) −1.38376e7 −0.195177
\(176\) −1.48889e8 −2.05858
\(177\) 0 0
\(178\) −5.66248e7 −0.752553
\(179\) −9.61633e7 −1.25321 −0.626605 0.779337i \(-0.715556\pi\)
−0.626605 + 0.779337i \(0.715556\pi\)
\(180\) 0 0
\(181\) 1.27453e8 1.59763 0.798816 0.601575i \(-0.205460\pi\)
0.798816 + 0.601575i \(0.205460\pi\)
\(182\) −1.47532e8 −1.81399
\(183\) 0 0
\(184\) 2.27681e7 0.269442
\(185\) 9.82189e7 1.14050
\(186\) 0 0
\(187\) −1.11816e8 −1.25043
\(188\) −7.42864e7 −0.815374
\(189\) 0 0
\(190\) −1.05427e8 −1.11510
\(191\) −8.72604e6 −0.0906151 −0.0453075 0.998973i \(-0.514427\pi\)
−0.0453075 + 0.998973i \(0.514427\pi\)
\(192\) 0 0
\(193\) 6.92616e7 0.693492 0.346746 0.937959i \(-0.387287\pi\)
0.346746 + 0.937959i \(0.387287\pi\)
\(194\) −9.23599e7 −0.908191
\(195\) 0 0
\(196\) 1.06179e8 1.00726
\(197\) −1.99367e7 −0.185790 −0.0928949 0.995676i \(-0.529612\pi\)
−0.0928949 + 0.995676i \(0.529612\pi\)
\(198\) 0 0
\(199\) −1.15995e8 −1.04341 −0.521705 0.853126i \(-0.674704\pi\)
−0.521705 + 0.853126i \(0.674704\pi\)
\(200\) −5.36960e6 −0.0474610
\(201\) 0 0
\(202\) −2.22763e8 −1.90158
\(203\) 2.22781e7 0.186914
\(204\) 0 0
\(205\) −6.65416e7 −0.539454
\(206\) 1.57485e8 1.25517
\(207\) 0 0
\(208\) −1.39567e8 −1.07537
\(209\) −1.81103e8 −1.37218
\(210\) 0 0
\(211\) 6.67529e7 0.489195 0.244597 0.969625i \(-0.421344\pi\)
0.244597 + 0.969625i \(0.421344\pi\)
\(212\) −1.10835e8 −0.798918
\(213\) 0 0
\(214\) 2.54443e8 1.77477
\(215\) −3.98029e7 −0.273136
\(216\) 0 0
\(217\) −1.52710e8 −1.01451
\(218\) −3.86633e8 −2.52756
\(219\) 0 0
\(220\) 2.03442e8 1.28814
\(221\) −1.04815e8 −0.653209
\(222\) 0 0
\(223\) 1.76404e8 1.06522 0.532611 0.846360i \(-0.321211\pi\)
0.532611 + 0.846360i \(0.321211\pi\)
\(224\) 3.13492e8 1.86362
\(225\) 0 0
\(226\) −2.20738e8 −1.27203
\(227\) 1.70858e8 0.969491 0.484745 0.874655i \(-0.338912\pi\)
0.484745 + 0.874655i \(0.338912\pi\)
\(228\) 0 0
\(229\) −1.06631e8 −0.586757 −0.293379 0.955996i \(-0.594780\pi\)
−0.293379 + 0.955996i \(0.594780\pi\)
\(230\) −1.82516e8 −0.989132
\(231\) 0 0
\(232\) 8.64489e6 0.0454518
\(233\) 2.36647e8 1.22562 0.612809 0.790231i \(-0.290040\pi\)
0.612809 + 0.790231i \(0.290040\pi\)
\(234\) 0 0
\(235\) −2.42050e8 −1.21666
\(236\) −1.86912e7 −0.0925647
\(237\) 0 0
\(238\) 3.09675e8 1.48897
\(239\) −4.70420e6 −0.0222891 −0.0111446 0.999938i \(-0.503547\pi\)
−0.0111446 + 0.999938i \(0.503547\pi\)
\(240\) 0 0
\(241\) −1.57232e8 −0.723573 −0.361786 0.932261i \(-0.617833\pi\)
−0.361786 + 0.932261i \(0.617833\pi\)
\(242\) 5.52609e8 2.50648
\(243\) 0 0
\(244\) 1.71238e8 0.754632
\(245\) 3.45968e8 1.50299
\(246\) 0 0
\(247\) −1.69764e8 −0.716812
\(248\) −5.92581e7 −0.246699
\(249\) 0 0
\(250\) −2.99801e8 −1.21351
\(251\) −1.41210e7 −0.0563647 −0.0281823 0.999603i \(-0.508972\pi\)
−0.0281823 + 0.999603i \(0.508972\pi\)
\(252\) 0 0
\(253\) −3.13528e8 −1.21718
\(254\) −2.94622e8 −1.12810
\(255\) 0 0
\(256\) 3.51724e8 1.31027
\(257\) 3.85019e8 1.41487 0.707434 0.706780i \(-0.249853\pi\)
0.707434 + 0.706780i \(0.249853\pi\)
\(258\) 0 0
\(259\) −4.67273e8 −1.67117
\(260\) 1.90705e8 0.672906
\(261\) 0 0
\(262\) −3.23301e8 −1.11059
\(263\) −1.74319e8 −0.590882 −0.295441 0.955361i \(-0.595467\pi\)
−0.295441 + 0.955361i \(0.595467\pi\)
\(264\) 0 0
\(265\) −3.61138e8 −1.19210
\(266\) 5.01563e8 1.63395
\(267\) 0 0
\(268\) −1.90358e8 −0.604087
\(269\) −2.39784e7 −0.0751083 −0.0375541 0.999295i \(-0.511957\pi\)
−0.0375541 + 0.999295i \(0.511957\pi\)
\(270\) 0 0
\(271\) 5.61954e8 1.71518 0.857588 0.514338i \(-0.171962\pi\)
0.857588 + 0.514338i \(0.171962\pi\)
\(272\) 2.92956e8 0.882695
\(273\) 0 0
\(274\) 2.91397e8 0.855772
\(275\) 7.39419e7 0.214401
\(276\) 0 0
\(277\) 6.23564e7 0.176280 0.0881398 0.996108i \(-0.471908\pi\)
0.0881398 + 0.996108i \(0.471908\pi\)
\(278\) 3.58499e7 0.100076
\(279\) 0 0
\(280\) 2.29015e8 0.623463
\(281\) −5.33351e8 −1.43397 −0.716986 0.697087i \(-0.754479\pi\)
−0.716986 + 0.697087i \(0.754479\pi\)
\(282\) 0 0
\(283\) −7.38516e8 −1.93690 −0.968450 0.249207i \(-0.919830\pi\)
−0.968450 + 0.249207i \(0.919830\pi\)
\(284\) 1.64570e8 0.426321
\(285\) 0 0
\(286\) 7.88343e8 1.99266
\(287\) 3.16569e8 0.790464
\(288\) 0 0
\(289\) −1.90327e8 −0.463829
\(290\) −6.93001e7 −0.166856
\(291\) 0 0
\(292\) −4.48732e8 −1.05474
\(293\) −4.24434e8 −0.985764 −0.492882 0.870096i \(-0.664057\pi\)
−0.492882 + 0.870096i \(0.664057\pi\)
\(294\) 0 0
\(295\) −6.09023e7 −0.138120
\(296\) −1.81322e8 −0.406378
\(297\) 0 0
\(298\) −3.10057e8 −0.678711
\(299\) −2.93898e8 −0.635839
\(300\) 0 0
\(301\) 1.89361e8 0.400227
\(302\) 7.07143e7 0.147735
\(303\) 0 0
\(304\) 4.74484e8 0.968643
\(305\) 5.57951e8 1.12602
\(306\) 0 0
\(307\) 5.81873e8 1.14774 0.573871 0.818946i \(-0.305441\pi\)
0.573871 + 0.818946i \(0.305441\pi\)
\(308\) −9.67869e8 −1.88751
\(309\) 0 0
\(310\) 4.75031e8 0.905640
\(311\) −4.40429e8 −0.830260 −0.415130 0.909762i \(-0.636264\pi\)
−0.415130 + 0.909762i \(0.636264\pi\)
\(312\) 0 0
\(313\) 1.33282e8 0.245678 0.122839 0.992427i \(-0.460800\pi\)
0.122839 + 0.992427i \(0.460800\pi\)
\(314\) −9.14304e8 −1.66662
\(315\) 0 0
\(316\) −7.51811e8 −1.34031
\(317\) 5.31810e8 0.937668 0.468834 0.883286i \(-0.344674\pi\)
0.468834 + 0.883286i \(0.344674\pi\)
\(318\) 0 0
\(319\) −1.19044e8 −0.205324
\(320\) −2.25509e8 −0.384714
\(321\) 0 0
\(322\) 8.68314e8 1.44938
\(323\) 3.56340e8 0.588378
\(324\) 0 0
\(325\) 6.93123e7 0.112000
\(326\) 1.22466e9 1.95774
\(327\) 0 0
\(328\) 1.22843e8 0.192217
\(329\) 1.15154e9 1.78277
\(330\) 0 0
\(331\) −1.41095e8 −0.213853 −0.106926 0.994267i \(-0.534101\pi\)
−0.106926 + 0.994267i \(0.534101\pi\)
\(332\) 2.69323e8 0.403916
\(333\) 0 0
\(334\) 2.06269e8 0.302916
\(335\) −6.20251e8 −0.901386
\(336\) 0 0
\(337\) −1.15891e9 −1.64948 −0.824739 0.565513i \(-0.808678\pi\)
−0.824739 + 0.565513i \(0.808678\pi\)
\(338\) −1.89627e8 −0.267111
\(339\) 0 0
\(340\) −4.00296e8 −0.552339
\(341\) 8.16011e8 1.11444
\(342\) 0 0
\(343\) −4.84110e8 −0.647760
\(344\) 7.34802e7 0.0973231
\(345\) 0 0
\(346\) −1.33640e9 −1.73448
\(347\) −4.17462e8 −0.536369 −0.268184 0.963368i \(-0.586424\pi\)
−0.268184 + 0.963368i \(0.586424\pi\)
\(348\) 0 0
\(349\) −9.78142e7 −0.123172 −0.0615861 0.998102i \(-0.519616\pi\)
−0.0615861 + 0.998102i \(0.519616\pi\)
\(350\) −2.04782e8 −0.255301
\(351\) 0 0
\(352\) −1.67516e9 −2.04718
\(353\) 1.27204e9 1.53918 0.769590 0.638539i \(-0.220461\pi\)
0.769590 + 0.638539i \(0.220461\pi\)
\(354\) 0 0
\(355\) 5.36225e8 0.636133
\(356\) −3.48223e8 −0.409056
\(357\) 0 0
\(358\) −1.42311e9 −1.63927
\(359\) 2.70429e8 0.308477 0.154238 0.988034i \(-0.450708\pi\)
0.154238 + 0.988034i \(0.450708\pi\)
\(360\) 0 0
\(361\) −3.16727e8 −0.354332
\(362\) 1.88618e9 2.08979
\(363\) 0 0
\(364\) −9.07270e8 −0.986011
\(365\) −1.46212e9 −1.57383
\(366\) 0 0
\(367\) 1.26226e9 1.33296 0.666481 0.745522i \(-0.267800\pi\)
0.666481 + 0.745522i \(0.267800\pi\)
\(368\) 8.21434e8 0.859222
\(369\) 0 0
\(370\) 1.45354e9 1.49183
\(371\) 1.71810e9 1.74679
\(372\) 0 0
\(373\) −4.77047e7 −0.0475971 −0.0237985 0.999717i \(-0.507576\pi\)
−0.0237985 + 0.999717i \(0.507576\pi\)
\(374\) −1.65476e9 −1.63563
\(375\) 0 0
\(376\) 4.46850e8 0.433515
\(377\) −1.11591e8 −0.107259
\(378\) 0 0
\(379\) 1.65053e9 1.55735 0.778674 0.627429i \(-0.215893\pi\)
0.778674 + 0.627429i \(0.215893\pi\)
\(380\) −6.48338e8 −0.606120
\(381\) 0 0
\(382\) −1.29136e8 −0.118529
\(383\) −4.84522e7 −0.0440674 −0.0220337 0.999757i \(-0.507014\pi\)
−0.0220337 + 0.999757i \(0.507014\pi\)
\(384\) 0 0
\(385\) −3.15364e9 −2.81644
\(386\) 1.02500e9 0.907125
\(387\) 0 0
\(388\) −5.67982e8 −0.493655
\(389\) −2.13932e9 −1.84269 −0.921345 0.388745i \(-0.872909\pi\)
−0.921345 + 0.388745i \(0.872909\pi\)
\(390\) 0 0
\(391\) 6.16902e8 0.521913
\(392\) −6.38693e8 −0.535539
\(393\) 0 0
\(394\) −2.95042e8 −0.243023
\(395\) −2.44965e9 −1.99993
\(396\) 0 0
\(397\) 1.67380e9 1.34257 0.671284 0.741200i \(-0.265743\pi\)
0.671284 + 0.741200i \(0.265743\pi\)
\(398\) −1.71661e9 −1.36484
\(399\) 0 0
\(400\) −1.93726e8 −0.151348
\(401\) 5.90271e8 0.457137 0.228568 0.973528i \(-0.426596\pi\)
0.228568 + 0.973528i \(0.426596\pi\)
\(402\) 0 0
\(403\) 7.64920e8 0.582168
\(404\) −1.36992e9 −1.03362
\(405\) 0 0
\(406\) 3.29692e8 0.244494
\(407\) 2.49689e9 1.83578
\(408\) 0 0
\(409\) −2.01375e8 −0.145537 −0.0727685 0.997349i \(-0.523183\pi\)
−0.0727685 + 0.997349i \(0.523183\pi\)
\(410\) −9.84745e8 −0.705635
\(411\) 0 0
\(412\) 9.68476e8 0.682258
\(413\) 2.89740e8 0.202388
\(414\) 0 0
\(415\) 8.77547e8 0.602701
\(416\) −1.57028e9 −1.06942
\(417\) 0 0
\(418\) −2.68012e9 −1.79489
\(419\) −1.53310e9 −1.01817 −0.509086 0.860716i \(-0.670017\pi\)
−0.509086 + 0.860716i \(0.670017\pi\)
\(420\) 0 0
\(421\) 1.10242e9 0.720044 0.360022 0.932944i \(-0.382769\pi\)
0.360022 + 0.932944i \(0.382769\pi\)
\(422\) 9.87872e8 0.639892
\(423\) 0 0
\(424\) 6.66699e8 0.424766
\(425\) −1.45489e8 −0.0919326
\(426\) 0 0
\(427\) −2.65443e9 −1.64996
\(428\) 1.56474e9 0.964691
\(429\) 0 0
\(430\) −5.89040e8 −0.357277
\(431\) 2.15146e9 1.29438 0.647192 0.762327i \(-0.275943\pi\)
0.647192 + 0.762327i \(0.275943\pi\)
\(432\) 0 0
\(433\) 4.65835e8 0.275756 0.137878 0.990449i \(-0.455972\pi\)
0.137878 + 0.990449i \(0.455972\pi\)
\(434\) −2.25994e9 −1.32704
\(435\) 0 0
\(436\) −2.37766e9 −1.37388
\(437\) 9.99162e8 0.572732
\(438\) 0 0
\(439\) 2.25997e9 1.27490 0.637451 0.770491i \(-0.279989\pi\)
0.637451 + 0.770491i \(0.279989\pi\)
\(440\) −1.22375e9 −0.684872
\(441\) 0 0
\(442\) −1.55116e9 −0.854432
\(443\) 1.51398e9 0.827383 0.413691 0.910417i \(-0.364239\pi\)
0.413691 + 0.910417i \(0.364239\pi\)
\(444\) 0 0
\(445\) −1.13463e9 −0.610371
\(446\) 2.61058e9 1.39337
\(447\) 0 0
\(448\) 1.07285e9 0.563723
\(449\) 2.99785e9 1.56296 0.781480 0.623930i \(-0.214465\pi\)
0.781480 + 0.623930i \(0.214465\pi\)
\(450\) 0 0
\(451\) −1.69160e9 −0.868321
\(452\) −1.35747e9 −0.691424
\(453\) 0 0
\(454\) 2.52851e9 1.26815
\(455\) −2.95619e9 −1.47127
\(456\) 0 0
\(457\) 2.58428e9 1.26658 0.633289 0.773915i \(-0.281704\pi\)
0.633289 + 0.773915i \(0.281704\pi\)
\(458\) −1.57802e9 −0.767510
\(459\) 0 0
\(460\) −1.12241e9 −0.537651
\(461\) 1.75068e9 0.832247 0.416124 0.909308i \(-0.363388\pi\)
0.416124 + 0.909308i \(0.363388\pi\)
\(462\) 0 0
\(463\) −3.28077e9 −1.53618 −0.768091 0.640341i \(-0.778793\pi\)
−0.768091 + 0.640341i \(0.778793\pi\)
\(464\) 3.11892e8 0.144941
\(465\) 0 0
\(466\) 3.50212e9 1.60317
\(467\) −2.15325e9 −0.978329 −0.489164 0.872192i \(-0.662698\pi\)
−0.489164 + 0.872192i \(0.662698\pi\)
\(468\) 0 0
\(469\) 2.95082e9 1.32080
\(470\) −3.58208e9 −1.59145
\(471\) 0 0
\(472\) 1.12432e8 0.0492145
\(473\) −1.01186e9 −0.439648
\(474\) 0 0
\(475\) −2.35641e8 −0.100884
\(476\) 1.90439e9 0.809343
\(477\) 0 0
\(478\) −6.96171e7 −0.0291554
\(479\) 4.22659e9 1.75718 0.878589 0.477578i \(-0.158485\pi\)
0.878589 + 0.477578i \(0.158485\pi\)
\(480\) 0 0
\(481\) 2.34056e9 0.958986
\(482\) −2.32687e9 −0.946471
\(483\) 0 0
\(484\) 3.39836e9 1.36242
\(485\) −1.85068e9 −0.736605
\(486\) 0 0
\(487\) −3.71716e9 −1.45835 −0.729173 0.684330i \(-0.760095\pi\)
−0.729173 + 0.684330i \(0.760095\pi\)
\(488\) −1.03004e9 −0.401220
\(489\) 0 0
\(490\) 5.11996e9 1.96599
\(491\) −3.64965e9 −1.39144 −0.695722 0.718311i \(-0.744915\pi\)
−0.695722 + 0.718311i \(0.744915\pi\)
\(492\) 0 0
\(493\) 2.34233e8 0.0880409
\(494\) −2.51232e9 −0.937628
\(495\) 0 0
\(496\) −2.13793e9 −0.786696
\(497\) −2.55107e9 −0.932127
\(498\) 0 0
\(499\) −8.31205e8 −0.299472 −0.149736 0.988726i \(-0.547842\pi\)
−0.149736 + 0.988726i \(0.547842\pi\)
\(500\) −1.84367e9 −0.659612
\(501\) 0 0
\(502\) −2.08976e8 −0.0737280
\(503\) 3.38642e9 1.18646 0.593230 0.805033i \(-0.297853\pi\)
0.593230 + 0.805033i \(0.297853\pi\)
\(504\) 0 0
\(505\) −4.46366e9 −1.54231
\(506\) −4.63988e9 −1.59213
\(507\) 0 0
\(508\) −1.81183e9 −0.613188
\(509\) 2.52186e9 0.847633 0.423817 0.905748i \(-0.360690\pi\)
0.423817 + 0.905748i \(0.360690\pi\)
\(510\) 0 0
\(511\) 6.95598e9 2.30614
\(512\) 3.00491e9 0.989434
\(513\) 0 0
\(514\) 5.69786e9 1.85072
\(515\) 3.15562e9 1.01803
\(516\) 0 0
\(517\) −6.15333e9 −1.95836
\(518\) −6.91514e9 −2.18598
\(519\) 0 0
\(520\) −1.14713e9 −0.357768
\(521\) −4.55029e9 −1.40964 −0.704818 0.709388i \(-0.748971\pi\)
−0.704818 + 0.709388i \(0.748971\pi\)
\(522\) 0 0
\(523\) −1.39010e9 −0.424903 −0.212451 0.977172i \(-0.568145\pi\)
−0.212451 + 0.977172i \(0.568145\pi\)
\(524\) −1.98819e9 −0.603668
\(525\) 0 0
\(526\) −2.57974e9 −0.772905
\(527\) −1.60560e9 −0.477858
\(528\) 0 0
\(529\) −1.67506e9 −0.491966
\(530\) −5.34446e9 −1.55933
\(531\) 0 0
\(532\) 3.08444e9 0.888149
\(533\) −1.58569e9 −0.453600
\(534\) 0 0
\(535\) 5.09844e9 1.43946
\(536\) 1.14505e9 0.321179
\(537\) 0 0
\(538\) −3.54855e8 −0.0982456
\(539\) 8.79510e9 2.41925
\(540\) 0 0
\(541\) 3.45589e9 0.938360 0.469180 0.883103i \(-0.344550\pi\)
0.469180 + 0.883103i \(0.344550\pi\)
\(542\) 8.31632e9 2.24354
\(543\) 0 0
\(544\) 3.29607e9 0.877809
\(545\) −7.74723e9 −2.05002
\(546\) 0 0
\(547\) 5.69696e9 1.48829 0.744145 0.668018i \(-0.232857\pi\)
0.744145 + 0.668018i \(0.232857\pi\)
\(548\) 1.79199e9 0.465162
\(549\) 0 0
\(550\) 1.09426e9 0.280447
\(551\) 3.79374e8 0.0966134
\(552\) 0 0
\(553\) 1.16541e10 2.93050
\(554\) 9.22809e8 0.230583
\(555\) 0 0
\(556\) 2.20465e8 0.0543973
\(557\) 7.56400e9 1.85464 0.927318 0.374275i \(-0.122108\pi\)
0.927318 + 0.374275i \(0.122108\pi\)
\(558\) 0 0
\(559\) −9.48504e8 −0.229667
\(560\) 8.26246e9 1.98816
\(561\) 0 0
\(562\) −7.89302e9 −1.87571
\(563\) 5.67154e9 1.33943 0.669717 0.742616i \(-0.266415\pi\)
0.669717 + 0.742616i \(0.266415\pi\)
\(564\) 0 0
\(565\) −4.42308e9 −1.03170
\(566\) −1.09292e10 −2.53357
\(567\) 0 0
\(568\) −9.89927e8 −0.226665
\(569\) −1.36290e9 −0.310149 −0.155074 0.987903i \(-0.549562\pi\)
−0.155074 + 0.987903i \(0.549562\pi\)
\(570\) 0 0
\(571\) −2.64928e9 −0.595528 −0.297764 0.954640i \(-0.596241\pi\)
−0.297764 + 0.954640i \(0.596241\pi\)
\(572\) 4.84804e9 1.08313
\(573\) 0 0
\(574\) 4.68489e9 1.03397
\(575\) −4.07945e8 −0.0894880
\(576\) 0 0
\(577\) 7.60073e9 1.64718 0.823588 0.567188i \(-0.191969\pi\)
0.823588 + 0.567188i \(0.191969\pi\)
\(578\) −2.81664e9 −0.606713
\(579\) 0 0
\(580\) −4.26172e8 −0.0906957
\(581\) −4.17490e9 −0.883139
\(582\) 0 0
\(583\) −9.18076e9 −1.91884
\(584\) 2.69923e9 0.560782
\(585\) 0 0
\(586\) −6.28116e9 −1.28943
\(587\) 3.30015e9 0.673443 0.336721 0.941604i \(-0.390682\pi\)
0.336721 + 0.941604i \(0.390682\pi\)
\(588\) 0 0
\(589\) −2.60049e9 −0.524388
\(590\) −9.01289e8 −0.180668
\(591\) 0 0
\(592\) −6.54179e9 −1.29590
\(593\) −5.80009e9 −1.14220 −0.571102 0.820879i \(-0.693484\pi\)
−0.571102 + 0.820879i \(0.693484\pi\)
\(594\) 0 0
\(595\) 6.20516e9 1.20766
\(596\) −1.90675e9 −0.368919
\(597\) 0 0
\(598\) −4.34937e9 −0.831711
\(599\) −3.53769e9 −0.672551 −0.336276 0.941764i \(-0.609167\pi\)
−0.336276 + 0.941764i \(0.609167\pi\)
\(600\) 0 0
\(601\) −7.08661e9 −1.33161 −0.665806 0.746125i \(-0.731912\pi\)
−0.665806 + 0.746125i \(0.731912\pi\)
\(602\) 2.80233e9 0.523519
\(603\) 0 0
\(604\) 4.34869e8 0.0803025
\(605\) 1.10730e10 2.03292
\(606\) 0 0
\(607\) −9.13138e8 −0.165720 −0.0828602 0.996561i \(-0.526406\pi\)
−0.0828602 + 0.996561i \(0.526406\pi\)
\(608\) 5.33846e9 0.963282
\(609\) 0 0
\(610\) 8.25707e9 1.47289
\(611\) −5.76807e9 −1.02302
\(612\) 0 0
\(613\) 6.24314e8 0.109469 0.0547346 0.998501i \(-0.482569\pi\)
0.0547346 + 0.998501i \(0.482569\pi\)
\(614\) 8.61110e9 1.50131
\(615\) 0 0
\(616\) 5.82196e9 1.00354
\(617\) 1.00319e10 1.71943 0.859714 0.510775i \(-0.170642\pi\)
0.859714 + 0.510775i \(0.170642\pi\)
\(618\) 0 0
\(619\) −1.02470e10 −1.73652 −0.868258 0.496114i \(-0.834760\pi\)
−0.868258 + 0.496114i \(0.834760\pi\)
\(620\) 2.92128e9 0.492268
\(621\) 0 0
\(622\) −6.51787e9 −1.08602
\(623\) 5.39796e9 0.894378
\(624\) 0 0
\(625\) −6.77361e9 −1.10979
\(626\) 1.97243e9 0.321360
\(627\) 0 0
\(628\) −5.62266e9 −0.905906
\(629\) −4.91293e9 −0.787160
\(630\) 0 0
\(631\) 3.58757e9 0.568457 0.284228 0.958757i \(-0.408263\pi\)
0.284228 + 0.958757i \(0.408263\pi\)
\(632\) 4.52232e9 0.712609
\(633\) 0 0
\(634\) 7.87022e9 1.22652
\(635\) −5.90354e9 −0.914965
\(636\) 0 0
\(637\) 8.24444e9 1.26378
\(638\) −1.76173e9 −0.268575
\(639\) 0 0
\(640\) 5.09725e9 0.768609
\(641\) 2.92622e9 0.438837 0.219419 0.975631i \(-0.429584\pi\)
0.219419 + 0.975631i \(0.429584\pi\)
\(642\) 0 0
\(643\) 5.85273e9 0.868200 0.434100 0.900865i \(-0.357066\pi\)
0.434100 + 0.900865i \(0.357066\pi\)
\(644\) 5.33984e9 0.787821
\(645\) 0 0
\(646\) 5.27346e9 0.769629
\(647\) 7.40954e9 1.07554 0.537770 0.843092i \(-0.319267\pi\)
0.537770 + 0.843092i \(0.319267\pi\)
\(648\) 0 0
\(649\) −1.54824e9 −0.222322
\(650\) 1.02575e9 0.146502
\(651\) 0 0
\(652\) 7.53127e9 1.06415
\(653\) 4.89548e9 0.688018 0.344009 0.938966i \(-0.388215\pi\)
0.344009 + 0.938966i \(0.388215\pi\)
\(654\) 0 0
\(655\) −6.47820e9 −0.900761
\(656\) 4.43195e9 0.612959
\(657\) 0 0
\(658\) 1.70416e10 2.33196
\(659\) 1.26032e10 1.71547 0.857736 0.514091i \(-0.171871\pi\)
0.857736 + 0.514091i \(0.171871\pi\)
\(660\) 0 0
\(661\) 3.60198e8 0.0485106 0.0242553 0.999706i \(-0.492279\pi\)
0.0242553 + 0.999706i \(0.492279\pi\)
\(662\) −2.08806e9 −0.279731
\(663\) 0 0
\(664\) −1.62004e9 −0.214753
\(665\) 1.00502e10 1.32525
\(666\) 0 0
\(667\) 6.56779e8 0.0856996
\(668\) 1.26849e9 0.164652
\(669\) 0 0
\(670\) −9.17906e9 −1.17906
\(671\) 1.41841e10 1.81247
\(672\) 0 0
\(673\) 1.40565e9 0.177756 0.0888782 0.996043i \(-0.471672\pi\)
0.0888782 + 0.996043i \(0.471672\pi\)
\(674\) −1.71507e10 −2.15761
\(675\) 0 0
\(676\) −1.16614e9 −0.145190
\(677\) −5.12306e9 −0.634555 −0.317278 0.948333i \(-0.602769\pi\)
−0.317278 + 0.948333i \(0.602769\pi\)
\(678\) 0 0
\(679\) 8.80452e9 1.07935
\(680\) 2.40788e9 0.293665
\(681\) 0 0
\(682\) 1.20761e10 1.45774
\(683\) 1.70981e9 0.205341 0.102671 0.994715i \(-0.467261\pi\)
0.102671 + 0.994715i \(0.467261\pi\)
\(684\) 0 0
\(685\) 5.83891e9 0.694089
\(686\) −7.16431e9 −0.847305
\(687\) 0 0
\(688\) 2.65104e9 0.310353
\(689\) −8.60594e9 −1.00238
\(690\) 0 0
\(691\) −1.38466e10 −1.59651 −0.798253 0.602323i \(-0.794242\pi\)
−0.798253 + 0.602323i \(0.794242\pi\)
\(692\) −8.21839e9 −0.942791
\(693\) 0 0
\(694\) −6.17799e9 −0.701599
\(695\) 7.18348e8 0.0811686
\(696\) 0 0
\(697\) 3.32842e9 0.372326
\(698\) −1.44755e9 −0.161116
\(699\) 0 0
\(700\) −1.25934e9 −0.138771
\(701\) 1.16579e10 1.27823 0.639115 0.769111i \(-0.279301\pi\)
0.639115 + 0.769111i \(0.279301\pi\)
\(702\) 0 0
\(703\) −7.95720e9 −0.863806
\(704\) −5.73282e9 −0.619247
\(705\) 0 0
\(706\) 1.88248e10 2.01333
\(707\) 2.12357e10 2.25995
\(708\) 0 0
\(709\) −1.12659e10 −1.18714 −0.593572 0.804781i \(-0.702283\pi\)
−0.593572 + 0.804781i \(0.702283\pi\)
\(710\) 7.93555e9 0.832095
\(711\) 0 0
\(712\) 2.09464e9 0.217486
\(713\) −4.50202e9 −0.465151
\(714\) 0 0
\(715\) 1.57965e10 1.61618
\(716\) −8.75167e9 −0.891036
\(717\) 0 0
\(718\) 4.00206e9 0.403504
\(719\) 9.57915e9 0.961116 0.480558 0.876963i \(-0.340434\pi\)
0.480558 + 0.876963i \(0.340434\pi\)
\(720\) 0 0
\(721\) −1.50128e10 −1.49172
\(722\) −4.68722e9 −0.463485
\(723\) 0 0
\(724\) 1.15993e10 1.13592
\(725\) −1.54894e8 −0.0150956
\(726\) 0 0
\(727\) 1.06396e10 1.02697 0.513484 0.858099i \(-0.328355\pi\)
0.513484 + 0.858099i \(0.328355\pi\)
\(728\) 5.45744e9 0.524239
\(729\) 0 0
\(730\) −2.16378e10 −2.05865
\(731\) 1.99095e9 0.188516
\(732\) 0 0
\(733\) −1.85210e10 −1.73700 −0.868500 0.495689i \(-0.834916\pi\)
−0.868500 + 0.495689i \(0.834916\pi\)
\(734\) 1.86801e10 1.74359
\(735\) 0 0
\(736\) 9.24203e9 0.854467
\(737\) −1.57679e10 −1.45090
\(738\) 0 0
\(739\) 2.17412e9 0.198166 0.0990828 0.995079i \(-0.468409\pi\)
0.0990828 + 0.995079i \(0.468409\pi\)
\(740\) 8.93875e9 0.810897
\(741\) 0 0
\(742\) 2.54261e10 2.28489
\(743\) −1.86262e10 −1.66595 −0.832977 0.553307i \(-0.813366\pi\)
−0.832977 + 0.553307i \(0.813366\pi\)
\(744\) 0 0
\(745\) −6.21282e9 −0.550481
\(746\) −7.05979e8 −0.0622595
\(747\) 0 0
\(748\) −1.01762e10 −0.889060
\(749\) −2.42557e10 −2.10924
\(750\) 0 0
\(751\) −2.24532e9 −0.193437 −0.0967185 0.995312i \(-0.530835\pi\)
−0.0967185 + 0.995312i \(0.530835\pi\)
\(752\) 1.61216e10 1.38243
\(753\) 0 0
\(754\) −1.65142e9 −0.140300
\(755\) 1.41695e9 0.119823
\(756\) 0 0
\(757\) 1.91646e10 1.60570 0.802852 0.596179i \(-0.203315\pi\)
0.802852 + 0.596179i \(0.203315\pi\)
\(758\) 2.44260e10 2.03709
\(759\) 0 0
\(760\) 3.89990e9 0.322260
\(761\) −1.33554e10 −1.09853 −0.549265 0.835648i \(-0.685092\pi\)
−0.549265 + 0.835648i \(0.685092\pi\)
\(762\) 0 0
\(763\) 3.68572e10 3.00390
\(764\) −7.94143e8 −0.0644276
\(765\) 0 0
\(766\) −7.17041e8 −0.0576426
\(767\) −1.45130e9 −0.116138
\(768\) 0 0
\(769\) −5.20020e9 −0.412361 −0.206181 0.978514i \(-0.566103\pi\)
−0.206181 + 0.978514i \(0.566103\pi\)
\(770\) −4.66706e10 −3.68405
\(771\) 0 0
\(772\) 6.30338e9 0.493075
\(773\) −2.37920e10 −1.85269 −0.926347 0.376672i \(-0.877068\pi\)
−0.926347 + 0.376672i \(0.877068\pi\)
\(774\) 0 0
\(775\) 1.06175e9 0.0819343
\(776\) 3.41654e9 0.262465
\(777\) 0 0
\(778\) −3.16597e10 −2.41034
\(779\) 5.39086e9 0.408580
\(780\) 0 0
\(781\) 1.36318e10 1.02394
\(782\) 9.12950e9 0.682690
\(783\) 0 0
\(784\) −2.30429e10 −1.70778
\(785\) −1.83205e10 −1.35174
\(786\) 0 0
\(787\) 5.45988e9 0.399275 0.199637 0.979870i \(-0.436024\pi\)
0.199637 + 0.979870i \(0.436024\pi\)
\(788\) −1.81441e9 −0.132097
\(789\) 0 0
\(790\) −3.62522e10 −2.61601
\(791\) 2.10427e10 1.51176
\(792\) 0 0
\(793\) 1.32960e10 0.946814
\(794\) 2.47704e10 1.75615
\(795\) 0 0
\(796\) −1.05566e10 −0.741868
\(797\) 4.12638e9 0.288712 0.144356 0.989526i \(-0.453889\pi\)
0.144356 + 0.989526i \(0.453889\pi\)
\(798\) 0 0
\(799\) 1.21074e10 0.839725
\(800\) −2.17963e9 −0.150511
\(801\) 0 0
\(802\) 8.73538e9 0.597959
\(803\) −3.71696e10 −2.53328
\(804\) 0 0
\(805\) 1.73990e10 1.17554
\(806\) 1.13200e10 0.761507
\(807\) 0 0
\(808\) 8.24037e9 0.549550
\(809\) −4.36775e9 −0.290027 −0.145013 0.989430i \(-0.546322\pi\)
−0.145013 + 0.989430i \(0.546322\pi\)
\(810\) 0 0
\(811\) −1.81321e10 −1.19364 −0.596822 0.802374i \(-0.703570\pi\)
−0.596822 + 0.802374i \(0.703570\pi\)
\(812\) 2.02750e9 0.132897
\(813\) 0 0
\(814\) 3.69514e10 2.40129
\(815\) 2.45394e10 1.58786
\(816\) 0 0
\(817\) 3.22462e9 0.206872
\(818\) −2.98013e9 −0.190370
\(819\) 0 0
\(820\) −6.05584e9 −0.383554
\(821\) −1.18439e10 −0.746951 −0.373475 0.927640i \(-0.621834\pi\)
−0.373475 + 0.927640i \(0.621834\pi\)
\(822\) 0 0
\(823\) 7.82921e8 0.0489574 0.0244787 0.999700i \(-0.492207\pi\)
0.0244787 + 0.999700i \(0.492207\pi\)
\(824\) −5.82561e9 −0.362741
\(825\) 0 0
\(826\) 4.28785e9 0.264734
\(827\) 1.09138e10 0.670974 0.335487 0.942045i \(-0.391099\pi\)
0.335487 + 0.942045i \(0.391099\pi\)
\(828\) 0 0
\(829\) −2.91147e10 −1.77489 −0.887446 0.460911i \(-0.847523\pi\)
−0.887446 + 0.460911i \(0.847523\pi\)
\(830\) 1.29868e10 0.788365
\(831\) 0 0
\(832\) −5.37388e9 −0.323487
\(833\) −1.73054e10 −1.03735
\(834\) 0 0
\(835\) 4.13316e9 0.245685
\(836\) −1.64819e10 −0.975628
\(837\) 0 0
\(838\) −2.26882e10 −1.33182
\(839\) 3.47596e9 0.203193 0.101596 0.994826i \(-0.467605\pi\)
0.101596 + 0.994826i \(0.467605\pi\)
\(840\) 0 0
\(841\) −1.70005e10 −0.985543
\(842\) 1.63146e10 0.941855
\(843\) 0 0
\(844\) 6.07507e9 0.347819
\(845\) −3.79969e9 −0.216645
\(846\) 0 0
\(847\) −5.26793e10 −2.97885
\(848\) 2.40533e10 1.35453
\(849\) 0 0
\(850\) −2.15309e9 −0.120253
\(851\) −1.37756e10 −0.766228
\(852\) 0 0
\(853\) −2.90613e10 −1.60322 −0.801610 0.597847i \(-0.796023\pi\)
−0.801610 + 0.597847i \(0.796023\pi\)
\(854\) −3.92827e10 −2.15824
\(855\) 0 0
\(856\) −9.41226e9 −0.512904
\(857\) −3.15674e10 −1.71319 −0.856597 0.515987i \(-0.827425\pi\)
−0.856597 + 0.515987i \(0.827425\pi\)
\(858\) 0 0
\(859\) 4.24010e9 0.228244 0.114122 0.993467i \(-0.463594\pi\)
0.114122 + 0.993467i \(0.463594\pi\)
\(860\) −3.62239e9 −0.194201
\(861\) 0 0
\(862\) 3.18393e10 1.69312
\(863\) 2.94013e9 0.155715 0.0778573 0.996965i \(-0.475192\pi\)
0.0778573 + 0.996965i \(0.475192\pi\)
\(864\) 0 0
\(865\) −2.67783e10 −1.40678
\(866\) 6.89387e9 0.360703
\(867\) 0 0
\(868\) −1.38979e10 −0.721321
\(869\) −6.22744e10 −3.21914
\(870\) 0 0
\(871\) −1.47806e10 −0.757930
\(872\) 1.43022e10 0.730458
\(873\) 0 0
\(874\) 1.47865e10 0.749163
\(875\) 2.85795e10 1.44220
\(876\) 0 0
\(877\) 2.78040e10 1.39190 0.695952 0.718088i \(-0.254983\pi\)
0.695952 + 0.718088i \(0.254983\pi\)
\(878\) 3.34451e10 1.66764
\(879\) 0 0
\(880\) −4.41508e10 −2.18398
\(881\) −2.13975e10 −1.05426 −0.527131 0.849784i \(-0.676732\pi\)
−0.527131 + 0.849784i \(0.676732\pi\)
\(882\) 0 0
\(883\) 2.73260e10 1.33572 0.667858 0.744289i \(-0.267211\pi\)
0.667858 + 0.744289i \(0.267211\pi\)
\(884\) −9.53908e9 −0.464433
\(885\) 0 0
\(886\) 2.24053e10 1.08226
\(887\) −2.35865e10 −1.13483 −0.567416 0.823431i \(-0.692057\pi\)
−0.567416 + 0.823431i \(0.692057\pi\)
\(888\) 0 0
\(889\) 2.80859e10 1.34070
\(890\) −1.67913e10 −0.798398
\(891\) 0 0
\(892\) 1.60542e10 0.757376
\(893\) 1.96097e10 0.921489
\(894\) 0 0
\(895\) −2.85159e10 −1.32955
\(896\) −2.42500e10 −1.12625
\(897\) 0 0
\(898\) 4.43650e10 2.04444
\(899\) −1.70938e9 −0.0784658
\(900\) 0 0
\(901\) 1.80642e10 0.822777
\(902\) −2.50339e10 −1.13581
\(903\) 0 0
\(904\) 8.16547e9 0.367614
\(905\) 3.77946e10 1.69496
\(906\) 0 0
\(907\) −1.48021e9 −0.0658716 −0.0329358 0.999457i \(-0.510486\pi\)
−0.0329358 + 0.999457i \(0.510486\pi\)
\(908\) 1.55495e10 0.689311
\(909\) 0 0
\(910\) −4.37485e10 −1.92450
\(911\) −1.06953e10 −0.468681 −0.234341 0.972155i \(-0.575293\pi\)
−0.234341 + 0.972155i \(0.575293\pi\)
\(912\) 0 0
\(913\) 2.23087e10 0.970124
\(914\) 3.82445e10 1.65675
\(915\) 0 0
\(916\) −9.70430e9 −0.417186
\(917\) 3.08198e10 1.31989
\(918\) 0 0
\(919\) −3.15248e9 −0.133983 −0.0669913 0.997754i \(-0.521340\pi\)
−0.0669913 + 0.997754i \(0.521340\pi\)
\(920\) 6.75157e9 0.285856
\(921\) 0 0
\(922\) 2.59081e10 1.08862
\(923\) 1.27783e10 0.534892
\(924\) 0 0
\(925\) 3.24882e9 0.134968
\(926\) −4.85519e10 −2.00941
\(927\) 0 0
\(928\) 3.50913e9 0.144139
\(929\) −4.53386e10 −1.85529 −0.927647 0.373458i \(-0.878172\pi\)
−0.927647 + 0.373458i \(0.878172\pi\)
\(930\) 0 0
\(931\) −2.80286e10 −1.13835
\(932\) 2.15368e10 0.871418
\(933\) 0 0
\(934\) −3.18658e10 −1.27971
\(935\) −3.31576e10 −1.32661
\(936\) 0 0
\(937\) 4.84194e9 0.192279 0.0961393 0.995368i \(-0.469351\pi\)
0.0961393 + 0.995368i \(0.469351\pi\)
\(938\) 4.36690e10 1.72768
\(939\) 0 0
\(940\) −2.20286e10 −0.865046
\(941\) 8.67653e9 0.339455 0.169728 0.985491i \(-0.445711\pi\)
0.169728 + 0.985491i \(0.445711\pi\)
\(942\) 0 0
\(943\) 9.33275e9 0.362425
\(944\) 4.05635e9 0.156940
\(945\) 0 0
\(946\) −1.49744e10 −0.575083
\(947\) −1.85330e10 −0.709121 −0.354561 0.935033i \(-0.615370\pi\)
−0.354561 + 0.935033i \(0.615370\pi\)
\(948\) 0 0
\(949\) −3.48424e10 −1.32335
\(950\) −3.48723e9 −0.131962
\(951\) 0 0
\(952\) −1.14554e10 −0.430309
\(953\) −1.33156e10 −0.498352 −0.249176 0.968458i \(-0.580160\pi\)
−0.249176 + 0.968458i \(0.580160\pi\)
\(954\) 0 0
\(955\) −2.58759e9 −0.0961353
\(956\) −4.28122e8 −0.0158476
\(957\) 0 0
\(958\) 6.25490e10 2.29848
\(959\) −2.77784e10 −1.01705
\(960\) 0 0
\(961\) −1.57953e10 −0.574112
\(962\) 3.46378e10 1.25440
\(963\) 0 0
\(964\) −1.43095e10 −0.514462
\(965\) 2.05385e10 0.735740
\(966\) 0 0
\(967\) 1.13617e10 0.404064 0.202032 0.979379i \(-0.435246\pi\)
0.202032 + 0.979379i \(0.435246\pi\)
\(968\) −2.04419e10 −0.724365
\(969\) 0 0
\(970\) −2.73880e10 −0.963518
\(971\) −1.97070e10 −0.690802 −0.345401 0.938455i \(-0.612257\pi\)
−0.345401 + 0.938455i \(0.612257\pi\)
\(972\) 0 0
\(973\) −3.41751e9 −0.118937
\(974\) −5.50100e10 −1.90759
\(975\) 0 0
\(976\) −3.71619e10 −1.27945
\(977\) 4.83923e10 1.66014 0.830072 0.557657i \(-0.188299\pi\)
0.830072 + 0.557657i \(0.188299\pi\)
\(978\) 0 0
\(979\) −2.88442e10 −0.982470
\(980\) 3.14860e10 1.06863
\(981\) 0 0
\(982\) −5.40109e10 −1.82008
\(983\) −2.12104e10 −0.712214 −0.356107 0.934445i \(-0.615896\pi\)
−0.356107 + 0.934445i \(0.615896\pi\)
\(984\) 0 0
\(985\) −5.91196e9 −0.197108
\(986\) 3.46640e9 0.115162
\(987\) 0 0
\(988\) −1.54499e10 −0.509655
\(989\) 5.58252e9 0.183503
\(990\) 0 0
\(991\) 3.64631e10 1.19013 0.595067 0.803676i \(-0.297125\pi\)
0.595067 + 0.803676i \(0.297125\pi\)
\(992\) −2.40540e10 −0.782342
\(993\) 0 0
\(994\) −3.77531e10 −1.21927
\(995\) −3.43968e10 −1.10697
\(996\) 0 0
\(997\) −3.41980e10 −1.09287 −0.546434 0.837502i \(-0.684015\pi\)
−0.546434 + 0.837502i \(0.684015\pi\)
\(998\) −1.23009e10 −0.391725
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.8.a.b.1.12 16
3.2 odd 2 177.8.a.a.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.5 16 3.2 odd 2
531.8.a.b.1.12 16 1.1 even 1 trivial