Properties

Label 531.8.a.d.1.11
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-4.85375\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +O(q^{10})\) \(q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +1307.20 q^{10} -972.348 q^{11} -1278.77 q^{13} -5478.12 q^{14} +552.586 q^{16} -36467.5 q^{17} +1900.72 q^{19} -15453.9 q^{20} -6664.23 q^{22} -80181.7 q^{23} -41748.1 q^{25} -8764.35 q^{26} +64763.2 q^{28} +201337. q^{29} +106900. q^{31} +187162. q^{32} -249939. q^{34} -152446. q^{35} -246438. q^{37} +13027.1 q^{38} -273238. q^{40} +348348. q^{41} +303791. q^{43} +78785.6 q^{44} -549545. q^{46} -1.12223e6 q^{47} -184682. q^{49} -286131. q^{50} +103614. q^{52} -534787. q^{53} -185453. q^{55} +1.14507e6 q^{56} +1.37991e6 q^{58} +205379. q^{59} -2.14348e6 q^{61} +732666. q^{62} +1.21203e6 q^{64} -243896. q^{65} -158432. q^{67} +2.95482e6 q^{68} -1.04483e6 q^{70} +3.94945e6 q^{71} +3.25842e6 q^{73} -1.68903e6 q^{74} -154008. q^{76} +777186. q^{77} -5.28930e6 q^{79} +105393. q^{80} +2.38749e6 q^{82} +3.63337e6 q^{83} -6.95534e6 q^{85} +2.08211e6 q^{86} +1.39300e6 q^{88} +1.34439e6 q^{89} +1.02210e6 q^{91} +6.49682e6 q^{92} -7.69146e6 q^{94} +362519. q^{95} +2.79048e6 q^{97} -1.26576e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.85375 0.605791 0.302896 0.953024i \(-0.402047\pi\)
0.302896 + 0.953024i \(0.402047\pi\)
\(3\) 0 0
\(4\) −81.0262 −0.633017
\(5\) 190.727 0.682367 0.341183 0.939997i \(-0.389172\pi\)
0.341183 + 0.939997i \(0.389172\pi\)
\(6\) 0 0
\(7\) −799.288 −0.880765 −0.440383 0.897810i \(-0.645157\pi\)
−0.440383 + 0.897810i \(0.645157\pi\)
\(8\) −1432.61 −0.989267
\(9\) 0 0
\(10\) 1307.20 0.413372
\(11\) −972.348 −0.220266 −0.110133 0.993917i \(-0.535128\pi\)
−0.110133 + 0.993917i \(0.535128\pi\)
\(12\) 0 0
\(13\) −1278.77 −0.161432 −0.0807161 0.996737i \(-0.525721\pi\)
−0.0807161 + 0.996737i \(0.525721\pi\)
\(14\) −5478.12 −0.533560
\(15\) 0 0
\(16\) 552.586 0.0337272
\(17\) −36467.5 −1.80026 −0.900128 0.435625i \(-0.856527\pi\)
−0.900128 + 0.435625i \(0.856527\pi\)
\(18\) 0 0
\(19\) 1900.72 0.0635742 0.0317871 0.999495i \(-0.489880\pi\)
0.0317871 + 0.999495i \(0.489880\pi\)
\(20\) −15453.9 −0.431950
\(21\) 0 0
\(22\) −6664.23 −0.133435
\(23\) −80181.7 −1.37413 −0.687065 0.726596i \(-0.741101\pi\)
−0.687065 + 0.726596i \(0.741101\pi\)
\(24\) 0 0
\(25\) −41748.1 −0.534376
\(26\) −8764.35 −0.0977942
\(27\) 0 0
\(28\) 64763.2 0.557539
\(29\) 201337. 1.53296 0.766479 0.642270i \(-0.222007\pi\)
0.766479 + 0.642270i \(0.222007\pi\)
\(30\) 0 0
\(31\) 106900. 0.644484 0.322242 0.946657i \(-0.395564\pi\)
0.322242 + 0.946657i \(0.395564\pi\)
\(32\) 187162. 1.00970
\(33\) 0 0
\(34\) −249939. −1.09058
\(35\) −152446. −0.601005
\(36\) 0 0
\(37\) −246438. −0.799838 −0.399919 0.916550i \(-0.630962\pi\)
−0.399919 + 0.916550i \(0.630962\pi\)
\(38\) 13027.1 0.0385127
\(39\) 0 0
\(40\) −273238. −0.675043
\(41\) 348348. 0.789350 0.394675 0.918821i \(-0.370857\pi\)
0.394675 + 0.918821i \(0.370857\pi\)
\(42\) 0 0
\(43\) 303791. 0.582687 0.291343 0.956619i \(-0.405898\pi\)
0.291343 + 0.956619i \(0.405898\pi\)
\(44\) 78785.6 0.139432
\(45\) 0 0
\(46\) −549545. −0.832436
\(47\) −1.12223e6 −1.57666 −0.788330 0.615252i \(-0.789054\pi\)
−0.788330 + 0.615252i \(0.789054\pi\)
\(48\) 0 0
\(49\) −184682. −0.224253
\(50\) −286131. −0.323720
\(51\) 0 0
\(52\) 103614. 0.102189
\(53\) −534787. −0.493418 −0.246709 0.969090i \(-0.579349\pi\)
−0.246709 + 0.969090i \(0.579349\pi\)
\(54\) 0 0
\(55\) −185453. −0.150302
\(56\) 1.14507e6 0.871312
\(57\) 0 0
\(58\) 1.37991e6 0.928652
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −2.14348e6 −1.20911 −0.604555 0.796564i \(-0.706649\pi\)
−0.604555 + 0.796564i \(0.706649\pi\)
\(62\) 732666. 0.390423
\(63\) 0 0
\(64\) 1.21203e6 0.577940
\(65\) −243896. −0.110156
\(66\) 0 0
\(67\) −158432. −0.0643548 −0.0321774 0.999482i \(-0.510244\pi\)
−0.0321774 + 0.999482i \(0.510244\pi\)
\(68\) 2.95482e6 1.13959
\(69\) 0 0
\(70\) −1.04483e6 −0.364084
\(71\) 3.94945e6 1.30958 0.654790 0.755811i \(-0.272757\pi\)
0.654790 + 0.755811i \(0.272757\pi\)
\(72\) 0 0
\(73\) 3.25842e6 0.980342 0.490171 0.871626i \(-0.336934\pi\)
0.490171 + 0.871626i \(0.336934\pi\)
\(74\) −1.68903e6 −0.484535
\(75\) 0 0
\(76\) −154008. −0.0402435
\(77\) 777186. 0.194003
\(78\) 0 0
\(79\) −5.28930e6 −1.20699 −0.603494 0.797367i \(-0.706225\pi\)
−0.603494 + 0.797367i \(0.706225\pi\)
\(80\) 105393. 0.0230143
\(81\) 0 0
\(82\) 2.38749e6 0.478181
\(83\) 3.63337e6 0.697487 0.348744 0.937218i \(-0.386608\pi\)
0.348744 + 0.937218i \(0.386608\pi\)
\(84\) 0 0
\(85\) −6.95534e6 −1.22844
\(86\) 2.08211e6 0.352987
\(87\) 0 0
\(88\) 1.39300e6 0.217902
\(89\) 1.34439e6 0.202143 0.101072 0.994879i \(-0.467773\pi\)
0.101072 + 0.994879i \(0.467773\pi\)
\(90\) 0 0
\(91\) 1.02210e6 0.142184
\(92\) 6.49682e6 0.869848
\(93\) 0 0
\(94\) −7.69146e6 −0.955128
\(95\) 362519. 0.0433809
\(96\) 0 0
\(97\) 2.79048e6 0.310441 0.155220 0.987880i \(-0.450391\pi\)
0.155220 + 0.987880i \(0.450391\pi\)
\(98\) −1.26576e6 −0.135850
\(99\) 0 0
\(100\) 3.38269e6 0.338269
\(101\) 1.43415e7 1.38506 0.692532 0.721387i \(-0.256495\pi\)
0.692532 + 0.721387i \(0.256495\pi\)
\(102\) 0 0
\(103\) 1.62771e6 0.146773 0.0733867 0.997304i \(-0.476619\pi\)
0.0733867 + 0.997304i \(0.476619\pi\)
\(104\) 1.83198e6 0.159700
\(105\) 0 0
\(106\) −3.66529e6 −0.298908
\(107\) 7.86386e6 0.620572 0.310286 0.950643i \(-0.399575\pi\)
0.310286 + 0.950643i \(0.399575\pi\)
\(108\) 0 0
\(109\) 2.63958e6 0.195228 0.0976141 0.995224i \(-0.468879\pi\)
0.0976141 + 0.995224i \(0.468879\pi\)
\(110\) −1.27105e6 −0.0910518
\(111\) 0 0
\(112\) −441675. −0.0297057
\(113\) 2.91448e7 1.90015 0.950074 0.312026i \(-0.101008\pi\)
0.950074 + 0.312026i \(0.101008\pi\)
\(114\) 0 0
\(115\) −1.52928e7 −0.937661
\(116\) −1.63135e7 −0.970388
\(117\) 0 0
\(118\) 1.40762e6 0.0788673
\(119\) 2.91480e7 1.58560
\(120\) 0 0
\(121\) −1.85417e7 −0.951483
\(122\) −1.46909e7 −0.732468
\(123\) 0 0
\(124\) −8.66170e6 −0.407969
\(125\) −2.28631e7 −1.04701
\(126\) 0 0
\(127\) −3.71615e7 −1.60983 −0.804915 0.593390i \(-0.797789\pi\)
−0.804915 + 0.593390i \(0.797789\pi\)
\(128\) −1.56498e7 −0.659588
\(129\) 0 0
\(130\) −1.67160e6 −0.0667315
\(131\) −1.50104e7 −0.583369 −0.291684 0.956515i \(-0.594216\pi\)
−0.291684 + 0.956515i \(0.594216\pi\)
\(132\) 0 0
\(133\) −1.51922e6 −0.0559939
\(134\) −1.08585e6 −0.0389856
\(135\) 0 0
\(136\) 5.22437e7 1.78094
\(137\) −1.75306e7 −0.582470 −0.291235 0.956652i \(-0.594066\pi\)
−0.291235 + 0.956652i \(0.594066\pi\)
\(138\) 0 0
\(139\) −1.16871e7 −0.369110 −0.184555 0.982822i \(-0.559084\pi\)
−0.184555 + 0.982822i \(0.559084\pi\)
\(140\) 1.23521e7 0.380446
\(141\) 0 0
\(142\) 2.70685e7 0.793332
\(143\) 1.24341e6 0.0355580
\(144\) 0 0
\(145\) 3.84004e7 1.04604
\(146\) 2.23324e7 0.593883
\(147\) 0 0
\(148\) 1.99679e7 0.506311
\(149\) 6.76391e7 1.67512 0.837559 0.546346i \(-0.183982\pi\)
0.837559 + 0.546346i \(0.183982\pi\)
\(150\) 0 0
\(151\) 4.31830e7 1.02069 0.510345 0.859970i \(-0.329518\pi\)
0.510345 + 0.859970i \(0.329518\pi\)
\(152\) −2.72300e6 −0.0628919
\(153\) 0 0
\(154\) 5.32664e6 0.117525
\(155\) 2.03888e7 0.439775
\(156\) 0 0
\(157\) 4.62920e7 0.954679 0.477340 0.878719i \(-0.341601\pi\)
0.477340 + 0.878719i \(0.341601\pi\)
\(158\) −3.62515e7 −0.731183
\(159\) 0 0
\(160\) 3.56968e7 0.688985
\(161\) 6.40883e7 1.21029
\(162\) 0 0
\(163\) 7.22501e7 1.30672 0.653359 0.757048i \(-0.273359\pi\)
0.653359 + 0.757048i \(0.273359\pi\)
\(164\) −2.82253e7 −0.499672
\(165\) 0 0
\(166\) 2.49022e7 0.422532
\(167\) −3.24667e7 −0.539425 −0.269713 0.962941i \(-0.586929\pi\)
−0.269713 + 0.962941i \(0.586929\pi\)
\(168\) 0 0
\(169\) −6.11133e7 −0.973940
\(170\) −4.76702e7 −0.744175
\(171\) 0 0
\(172\) −2.46150e7 −0.368851
\(173\) −1.31381e7 −0.192918 −0.0964589 0.995337i \(-0.530752\pi\)
−0.0964589 + 0.995337i \(0.530752\pi\)
\(174\) 0 0
\(175\) 3.33687e7 0.470659
\(176\) −537306. −0.00742895
\(177\) 0 0
\(178\) 9.21409e6 0.122457
\(179\) 1.99708e7 0.260261 0.130131 0.991497i \(-0.458460\pi\)
0.130131 + 0.991497i \(0.458460\pi\)
\(180\) 0 0
\(181\) 9.38499e7 1.17641 0.588205 0.808712i \(-0.299835\pi\)
0.588205 + 0.808712i \(0.299835\pi\)
\(182\) 7.00524e6 0.0861337
\(183\) 0 0
\(184\) 1.14869e8 1.35938
\(185\) −4.70025e7 −0.545783
\(186\) 0 0
\(187\) 3.54591e7 0.396535
\(188\) 9.09298e7 0.998053
\(189\) 0 0
\(190\) 2.48462e6 0.0262798
\(191\) 7.50591e7 0.779447 0.389724 0.920932i \(-0.372571\pi\)
0.389724 + 0.920932i \(0.372571\pi\)
\(192\) 0 0
\(193\) −7.13691e7 −0.714594 −0.357297 0.933991i \(-0.616302\pi\)
−0.357297 + 0.933991i \(0.616302\pi\)
\(194\) 1.91253e7 0.188062
\(195\) 0 0
\(196\) 1.49641e7 0.141956
\(197\) 1.40656e8 1.31077 0.655384 0.755296i \(-0.272507\pi\)
0.655384 + 0.755296i \(0.272507\pi\)
\(198\) 0 0
\(199\) 2.79140e7 0.251094 0.125547 0.992088i \(-0.459931\pi\)
0.125547 + 0.992088i \(0.459931\pi\)
\(200\) 5.98088e7 0.528640
\(201\) 0 0
\(202\) 9.82931e7 0.839060
\(203\) −1.60926e8 −1.35018
\(204\) 0 0
\(205\) 6.64394e7 0.538626
\(206\) 1.11559e7 0.0889140
\(207\) 0 0
\(208\) −706629. −0.00544465
\(209\) −1.84816e6 −0.0140032
\(210\) 0 0
\(211\) 1.66531e8 1.22041 0.610207 0.792242i \(-0.291086\pi\)
0.610207 + 0.792242i \(0.291086\pi\)
\(212\) 4.33317e7 0.312342
\(213\) 0 0
\(214\) 5.38969e7 0.375937
\(215\) 5.79412e7 0.397606
\(216\) 0 0
\(217\) −8.54439e7 −0.567639
\(218\) 1.80910e7 0.118268
\(219\) 0 0
\(220\) 1.50266e7 0.0951438
\(221\) 4.66334e7 0.290619
\(222\) 0 0
\(223\) 8.94297e6 0.0540026 0.0270013 0.999635i \(-0.491404\pi\)
0.0270013 + 0.999635i \(0.491404\pi\)
\(224\) −1.49596e8 −0.889308
\(225\) 0 0
\(226\) 1.99751e8 1.15109
\(227\) 6.88172e7 0.390487 0.195243 0.980755i \(-0.437450\pi\)
0.195243 + 0.980755i \(0.437450\pi\)
\(228\) 0 0
\(229\) 1.97507e8 1.08682 0.543411 0.839467i \(-0.317132\pi\)
0.543411 + 0.839467i \(0.317132\pi\)
\(230\) −1.04813e8 −0.568027
\(231\) 0 0
\(232\) −2.88438e8 −1.51651
\(233\) −2.46365e8 −1.27595 −0.637973 0.770058i \(-0.720227\pi\)
−0.637973 + 0.770058i \(0.720227\pi\)
\(234\) 0 0
\(235\) −2.14039e8 −1.07586
\(236\) −1.66411e7 −0.0824118
\(237\) 0 0
\(238\) 1.99773e8 0.960545
\(239\) 3.78259e8 1.79224 0.896120 0.443812i \(-0.146374\pi\)
0.896120 + 0.443812i \(0.146374\pi\)
\(240\) 0 0
\(241\) −2.71983e8 −1.25165 −0.625824 0.779964i \(-0.715237\pi\)
−0.625824 + 0.779964i \(0.715237\pi\)
\(242\) −1.27080e8 −0.576400
\(243\) 0 0
\(244\) 1.73678e8 0.765387
\(245\) −3.52239e7 −0.153023
\(246\) 0 0
\(247\) −2.43058e6 −0.0102629
\(248\) −1.53146e8 −0.637567
\(249\) 0 0
\(250\) −1.56698e8 −0.634268
\(251\) −1.67816e8 −0.669847 −0.334923 0.942245i \(-0.608710\pi\)
−0.334923 + 0.942245i \(0.608710\pi\)
\(252\) 0 0
\(253\) 7.79645e7 0.302674
\(254\) −2.54695e8 −0.975221
\(255\) 0 0
\(256\) −2.62399e8 −0.977513
\(257\) 3.77370e8 1.38676 0.693381 0.720571i \(-0.256120\pi\)
0.693381 + 0.720571i \(0.256120\pi\)
\(258\) 0 0
\(259\) 1.96975e8 0.704470
\(260\) 1.97619e7 0.0697305
\(261\) 0 0
\(262\) −1.02878e8 −0.353400
\(263\) 2.37395e8 0.804686 0.402343 0.915489i \(-0.368196\pi\)
0.402343 + 0.915489i \(0.368196\pi\)
\(264\) 0 0
\(265\) −1.01998e8 −0.336692
\(266\) −1.04124e7 −0.0339206
\(267\) 0 0
\(268\) 1.28371e7 0.0407377
\(269\) −2.56851e8 −0.804542 −0.402271 0.915521i \(-0.631779\pi\)
−0.402271 + 0.915521i \(0.631779\pi\)
\(270\) 0 0
\(271\) −2.99436e8 −0.913927 −0.456964 0.889485i \(-0.651063\pi\)
−0.456964 + 0.889485i \(0.651063\pi\)
\(272\) −2.01514e7 −0.0607176
\(273\) 0 0
\(274\) −1.20150e8 −0.352855
\(275\) 4.05937e7 0.117705
\(276\) 0 0
\(277\) −1.07342e8 −0.303453 −0.151727 0.988422i \(-0.548483\pi\)
−0.151727 + 0.988422i \(0.548483\pi\)
\(278\) −8.01006e7 −0.223604
\(279\) 0 0
\(280\) 2.18396e8 0.594555
\(281\) −1.26999e8 −0.341451 −0.170726 0.985319i \(-0.554611\pi\)
−0.170726 + 0.985319i \(0.554611\pi\)
\(282\) 0 0
\(283\) 6.79420e8 1.78191 0.890955 0.454091i \(-0.150036\pi\)
0.890955 + 0.454091i \(0.150036\pi\)
\(284\) −3.20009e8 −0.828986
\(285\) 0 0
\(286\) 8.52200e6 0.0215407
\(287\) −2.78430e8 −0.695232
\(288\) 0 0
\(289\) 9.19537e8 2.24092
\(290\) 2.63187e8 0.633682
\(291\) 0 0
\(292\) −2.64018e8 −0.620573
\(293\) −3.54246e8 −0.822752 −0.411376 0.911466i \(-0.634952\pi\)
−0.411376 + 0.911466i \(0.634952\pi\)
\(294\) 0 0
\(295\) 3.91714e7 0.0888366
\(296\) 3.53051e8 0.791254
\(297\) 0 0
\(298\) 4.63581e8 1.01477
\(299\) 1.02534e8 0.221829
\(300\) 0 0
\(301\) −2.42816e8 −0.513210
\(302\) 2.95966e8 0.618325
\(303\) 0 0
\(304\) 1.05031e6 0.00214418
\(305\) −4.08821e8 −0.825056
\(306\) 0 0
\(307\) −4.47178e8 −0.882056 −0.441028 0.897493i \(-0.645386\pi\)
−0.441028 + 0.897493i \(0.645386\pi\)
\(308\) −6.29724e7 −0.122807
\(309\) 0 0
\(310\) 1.39739e8 0.266412
\(311\) 1.03177e9 1.94501 0.972506 0.232879i \(-0.0748145\pi\)
0.972506 + 0.232879i \(0.0748145\pi\)
\(312\) 0 0
\(313\) −1.47667e8 −0.272194 −0.136097 0.990695i \(-0.543456\pi\)
−0.136097 + 0.990695i \(0.543456\pi\)
\(314\) 3.17274e8 0.578336
\(315\) 0 0
\(316\) 4.28571e8 0.764044
\(317\) 8.08129e8 1.42486 0.712431 0.701742i \(-0.247594\pi\)
0.712431 + 0.701742i \(0.247594\pi\)
\(318\) 0 0
\(319\) −1.95769e8 −0.337658
\(320\) 2.31167e8 0.394367
\(321\) 0 0
\(322\) 4.39245e8 0.733181
\(323\) −6.93145e7 −0.114450
\(324\) 0 0
\(325\) 5.33861e7 0.0862654
\(326\) 4.95184e8 0.791599
\(327\) 0 0
\(328\) −4.99047e8 −0.780878
\(329\) 8.96983e8 1.38867
\(330\) 0 0
\(331\) −1.71934e8 −0.260594 −0.130297 0.991475i \(-0.541593\pi\)
−0.130297 + 0.991475i \(0.541593\pi\)
\(332\) −2.94398e8 −0.441521
\(333\) 0 0
\(334\) −2.22519e8 −0.326779
\(335\) −3.02173e7 −0.0439136
\(336\) 0 0
\(337\) 2.64986e8 0.377154 0.188577 0.982058i \(-0.439613\pi\)
0.188577 + 0.982058i \(0.439613\pi\)
\(338\) −4.18855e8 −0.590004
\(339\) 0 0
\(340\) 5.63565e8 0.777620
\(341\) −1.03944e8 −0.141958
\(342\) 0 0
\(343\) 8.05862e8 1.07828
\(344\) −4.35215e8 −0.576433
\(345\) 0 0
\(346\) −9.00455e7 −0.116868
\(347\) −5.14421e8 −0.660945 −0.330473 0.943816i \(-0.607208\pi\)
−0.330473 + 0.943816i \(0.607208\pi\)
\(348\) 0 0
\(349\) −6.09218e7 −0.0767157 −0.0383578 0.999264i \(-0.512213\pi\)
−0.0383578 + 0.999264i \(0.512213\pi\)
\(350\) 2.28701e8 0.285121
\(351\) 0 0
\(352\) −1.81986e8 −0.222402
\(353\) 1.90611e8 0.230641 0.115320 0.993328i \(-0.463211\pi\)
0.115320 + 0.993328i \(0.463211\pi\)
\(354\) 0 0
\(355\) 7.53268e8 0.893614
\(356\) −1.08931e8 −0.127960
\(357\) 0 0
\(358\) 1.36875e8 0.157664
\(359\) 1.39955e9 1.59646 0.798229 0.602354i \(-0.205770\pi\)
0.798229 + 0.602354i \(0.205770\pi\)
\(360\) 0 0
\(361\) −8.90259e8 −0.995958
\(362\) 6.43224e8 0.712659
\(363\) 0 0
\(364\) −8.28171e7 −0.0900047
\(365\) 6.21471e8 0.668953
\(366\) 0 0
\(367\) −1.50266e9 −1.58683 −0.793413 0.608684i \(-0.791698\pi\)
−0.793413 + 0.608684i \(0.791698\pi\)
\(368\) −4.43073e7 −0.0463455
\(369\) 0 0
\(370\) −3.22143e8 −0.330631
\(371\) 4.27449e8 0.434585
\(372\) 0 0
\(373\) 8.89561e7 0.0887554 0.0443777 0.999015i \(-0.485869\pi\)
0.0443777 + 0.999015i \(0.485869\pi\)
\(374\) 2.43027e8 0.240218
\(375\) 0 0
\(376\) 1.60772e9 1.55974
\(377\) −2.57463e8 −0.247469
\(378\) 0 0
\(379\) −7.27075e8 −0.686028 −0.343014 0.939330i \(-0.611448\pi\)
−0.343014 + 0.939330i \(0.611448\pi\)
\(380\) −2.93736e7 −0.0274608
\(381\) 0 0
\(382\) 5.14436e8 0.472182
\(383\) −2.06745e9 −1.88036 −0.940178 0.340683i \(-0.889342\pi\)
−0.940178 + 0.340683i \(0.889342\pi\)
\(384\) 0 0
\(385\) 1.48231e8 0.132381
\(386\) −4.89145e8 −0.432895
\(387\) 0 0
\(388\) −2.26102e8 −0.196514
\(389\) 8.24531e7 0.0710205 0.0355102 0.999369i \(-0.488694\pi\)
0.0355102 + 0.999369i \(0.488694\pi\)
\(390\) 0 0
\(391\) 2.92402e9 2.47379
\(392\) 2.64578e8 0.221846
\(393\) 0 0
\(394\) 9.64019e8 0.794051
\(395\) −1.00881e9 −0.823609
\(396\) 0 0
\(397\) 2.60792e8 0.209183 0.104592 0.994515i \(-0.466646\pi\)
0.104592 + 0.994515i \(0.466646\pi\)
\(398\) 1.91316e8 0.152111
\(399\) 0 0
\(400\) −2.30694e7 −0.0180230
\(401\) 5.57165e8 0.431498 0.215749 0.976449i \(-0.430781\pi\)
0.215749 + 0.976449i \(0.430781\pi\)
\(402\) 0 0
\(403\) −1.36700e8 −0.104040
\(404\) −1.16204e9 −0.876769
\(405\) 0 0
\(406\) −1.10295e9 −0.817925
\(407\) 2.39624e8 0.176177
\(408\) 0 0
\(409\) −1.59001e9 −1.14913 −0.574564 0.818460i \(-0.694828\pi\)
−0.574564 + 0.818460i \(0.694828\pi\)
\(410\) 4.55359e8 0.326295
\(411\) 0 0
\(412\) −1.31887e8 −0.0929100
\(413\) −1.64157e8 −0.114666
\(414\) 0 0
\(415\) 6.92983e8 0.475942
\(416\) −2.39336e8 −0.162998
\(417\) 0 0
\(418\) −1.26668e7 −0.00848303
\(419\) −5.88021e8 −0.390520 −0.195260 0.980751i \(-0.562555\pi\)
−0.195260 + 0.980751i \(0.562555\pi\)
\(420\) 0 0
\(421\) −7.50924e8 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(422\) 1.14136e9 0.739316
\(423\) 0 0
\(424\) 7.66142e8 0.488123
\(425\) 1.52245e9 0.962013
\(426\) 0 0
\(427\) 1.71326e9 1.06494
\(428\) −6.37178e8 −0.392833
\(429\) 0 0
\(430\) 3.97114e8 0.240866
\(431\) 2.36435e9 1.42247 0.711233 0.702957i \(-0.248137\pi\)
0.711233 + 0.702957i \(0.248137\pi\)
\(432\) 0 0
\(433\) 6.97398e7 0.0412832 0.0206416 0.999787i \(-0.493429\pi\)
0.0206416 + 0.999787i \(0.493429\pi\)
\(434\) −5.85611e8 −0.343871
\(435\) 0 0
\(436\) −2.13875e8 −0.123583
\(437\) −1.52403e8 −0.0873592
\(438\) 0 0
\(439\) −2.54774e9 −1.43724 −0.718620 0.695403i \(-0.755226\pi\)
−0.718620 + 0.695403i \(0.755226\pi\)
\(440\) 2.65683e8 0.148689
\(441\) 0 0
\(442\) 3.19614e8 0.176055
\(443\) −1.15431e9 −0.630825 −0.315412 0.948955i \(-0.602143\pi\)
−0.315412 + 0.948955i \(0.602143\pi\)
\(444\) 0 0
\(445\) 2.56411e8 0.137936
\(446\) 6.12929e7 0.0327143
\(447\) 0 0
\(448\) −9.68759e8 −0.509029
\(449\) −5.55994e8 −0.289873 −0.144937 0.989441i \(-0.546298\pi\)
−0.144937 + 0.989441i \(0.546298\pi\)
\(450\) 0 0
\(451\) −3.38715e8 −0.173867
\(452\) −2.36150e9 −1.20283
\(453\) 0 0
\(454\) 4.71655e8 0.236554
\(455\) 1.94943e8 0.0970215
\(456\) 0 0
\(457\) 3.93559e8 0.192887 0.0964435 0.995338i \(-0.469253\pi\)
0.0964435 + 0.995338i \(0.469253\pi\)
\(458\) 1.35366e9 0.658388
\(459\) 0 0
\(460\) 1.23912e9 0.593555
\(461\) 2.99382e9 1.42322 0.711611 0.702574i \(-0.247966\pi\)
0.711611 + 0.702574i \(0.247966\pi\)
\(462\) 0 0
\(463\) −1.18369e9 −0.554248 −0.277124 0.960834i \(-0.589381\pi\)
−0.277124 + 0.960834i \(0.589381\pi\)
\(464\) 1.11256e8 0.0517023
\(465\) 0 0
\(466\) −1.68852e9 −0.772957
\(467\) −3.15650e9 −1.43416 −0.717079 0.696992i \(-0.754521\pi\)
−0.717079 + 0.696992i \(0.754521\pi\)
\(468\) 0 0
\(469\) 1.26633e8 0.0566815
\(470\) −1.46697e9 −0.651747
\(471\) 0 0
\(472\) −2.94228e8 −0.128792
\(473\) −2.95390e8 −0.128346
\(474\) 0 0
\(475\) −7.93515e7 −0.0339725
\(476\) −2.36175e9 −1.00371
\(477\) 0 0
\(478\) 2.59249e9 1.08572
\(479\) −2.27050e9 −0.943945 −0.471972 0.881613i \(-0.656458\pi\)
−0.471972 + 0.881613i \(0.656458\pi\)
\(480\) 0 0
\(481\) 3.15137e8 0.129120
\(482\) −1.86410e9 −0.758238
\(483\) 0 0
\(484\) 1.50236e9 0.602305
\(485\) 5.32222e8 0.211834
\(486\) 0 0
\(487\) 2.04995e8 0.0804252 0.0402126 0.999191i \(-0.487196\pi\)
0.0402126 + 0.999191i \(0.487196\pi\)
\(488\) 3.07078e9 1.19613
\(489\) 0 0
\(490\) −2.41416e8 −0.0926999
\(491\) −1.08333e9 −0.413023 −0.206512 0.978444i \(-0.566211\pi\)
−0.206512 + 0.978444i \(0.566211\pi\)
\(492\) 0 0
\(493\) −7.34224e9 −2.75972
\(494\) −1.66586e7 −0.00621718
\(495\) 0 0
\(496\) 5.90715e7 0.0217366
\(497\) −3.15675e9 −1.15343
\(498\) 0 0
\(499\) 1.23774e9 0.445942 0.222971 0.974825i \(-0.428425\pi\)
0.222971 + 0.974825i \(0.428425\pi\)
\(500\) 1.85251e9 0.662773
\(501\) 0 0
\(502\) −1.15017e9 −0.405787
\(503\) −3.42069e9 −1.19847 −0.599234 0.800574i \(-0.704528\pi\)
−0.599234 + 0.800574i \(0.704528\pi\)
\(504\) 0 0
\(505\) 2.73532e9 0.945122
\(506\) 5.34349e8 0.183357
\(507\) 0 0
\(508\) 3.01105e9 1.01905
\(509\) 3.62806e9 1.21944 0.609722 0.792616i \(-0.291281\pi\)
0.609722 + 0.792616i \(0.291281\pi\)
\(510\) 0 0
\(511\) −2.60442e9 −0.863451
\(512\) 2.04753e8 0.0674195
\(513\) 0 0
\(514\) 2.58640e9 0.840088
\(515\) 3.10449e8 0.100153
\(516\) 0 0
\(517\) 1.09120e9 0.347285
\(518\) 1.35002e9 0.426762
\(519\) 0 0
\(520\) 3.49408e8 0.108974
\(521\) 1.36162e9 0.421817 0.210909 0.977506i \(-0.432358\pi\)
0.210909 + 0.977506i \(0.432358\pi\)
\(522\) 0 0
\(523\) 1.07415e9 0.328329 0.164165 0.986433i \(-0.447507\pi\)
0.164165 + 0.986433i \(0.447507\pi\)
\(524\) 1.21624e9 0.369282
\(525\) 0 0
\(526\) 1.62705e9 0.487472
\(527\) −3.89838e9 −1.16024
\(528\) 0 0
\(529\) 3.02428e9 0.888234
\(530\) −6.99072e8 −0.203965
\(531\) 0 0
\(532\) 1.23097e8 0.0354451
\(533\) −4.45456e8 −0.127426
\(534\) 0 0
\(535\) 1.49985e9 0.423458
\(536\) 2.26972e8 0.0636642
\(537\) 0 0
\(538\) −1.76039e9 −0.487385
\(539\) 1.79575e8 0.0493953
\(540\) 0 0
\(541\) −6.13348e9 −1.66539 −0.832696 0.553730i \(-0.813204\pi\)
−0.832696 + 0.553730i \(0.813204\pi\)
\(542\) −2.05226e9 −0.553649
\(543\) 0 0
\(544\) −6.82531e9 −1.81772
\(545\) 5.03440e8 0.133217
\(546\) 0 0
\(547\) −6.39180e9 −1.66981 −0.834906 0.550392i \(-0.814478\pi\)
−0.834906 + 0.550392i \(0.814478\pi\)
\(548\) 1.42043e9 0.368713
\(549\) 0 0
\(550\) 2.78219e8 0.0713045
\(551\) 3.82685e8 0.0974565
\(552\) 0 0
\(553\) 4.22767e9 1.06307
\(554\) −7.35697e8 −0.183829
\(555\) 0 0
\(556\) 9.46963e8 0.233653
\(557\) −5.20568e9 −1.27639 −0.638197 0.769873i \(-0.720319\pi\)
−0.638197 + 0.769873i \(0.720319\pi\)
\(558\) 0 0
\(559\) −3.88478e8 −0.0940643
\(560\) −8.42396e7 −0.0202702
\(561\) 0 0
\(562\) −8.70420e8 −0.206848
\(563\) −3.32429e9 −0.785089 −0.392545 0.919733i \(-0.628405\pi\)
−0.392545 + 0.919733i \(0.628405\pi\)
\(564\) 0 0
\(565\) 5.55872e9 1.29660
\(566\) 4.65657e9 1.07947
\(567\) 0 0
\(568\) −5.65803e9 −1.29553
\(569\) −1.61858e9 −0.368333 −0.184166 0.982895i \(-0.558959\pi\)
−0.184166 + 0.982895i \(0.558959\pi\)
\(570\) 0 0
\(571\) −6.26853e8 −0.140909 −0.0704546 0.997515i \(-0.522445\pi\)
−0.0704546 + 0.997515i \(0.522445\pi\)
\(572\) −1.00749e8 −0.0225088
\(573\) 0 0
\(574\) −1.90829e9 −0.421165
\(575\) 3.34743e9 0.734302
\(576\) 0 0
\(577\) 2.33845e9 0.506772 0.253386 0.967365i \(-0.418456\pi\)
0.253386 + 0.967365i \(0.418456\pi\)
\(578\) 6.30228e9 1.35753
\(579\) 0 0
\(580\) −3.11144e9 −0.662161
\(581\) −2.90411e9 −0.614323
\(582\) 0 0
\(583\) 5.19999e8 0.108683
\(584\) −4.66806e9 −0.969820
\(585\) 0 0
\(586\) −2.42791e9 −0.498416
\(587\) 5.68734e9 1.16058 0.580291 0.814409i \(-0.302939\pi\)
0.580291 + 0.814409i \(0.302939\pi\)
\(588\) 0 0
\(589\) 2.03187e8 0.0409725
\(590\) 2.68471e8 0.0538164
\(591\) 0 0
\(592\) −1.36178e8 −0.0269763
\(593\) 6.92748e9 1.36422 0.682109 0.731250i \(-0.261063\pi\)
0.682109 + 0.731250i \(0.261063\pi\)
\(594\) 0 0
\(595\) 5.55932e9 1.08196
\(596\) −5.48053e9 −1.06038
\(597\) 0 0
\(598\) 7.02741e8 0.134382
\(599\) −1.04577e7 −0.00198812 −0.000994062 1.00000i \(-0.500316\pi\)
−0.000994062 1.00000i \(0.500316\pi\)
\(600\) 0 0
\(601\) 5.50968e9 1.03530 0.517649 0.855593i \(-0.326807\pi\)
0.517649 + 0.855593i \(0.326807\pi\)
\(602\) −1.66420e9 −0.310898
\(603\) 0 0
\(604\) −3.49896e9 −0.646114
\(605\) −3.53641e9 −0.649260
\(606\) 0 0
\(607\) −1.08515e10 −1.96939 −0.984693 0.174298i \(-0.944234\pi\)
−0.984693 + 0.174298i \(0.944234\pi\)
\(608\) 3.55742e8 0.0641908
\(609\) 0 0
\(610\) −2.80195e9 −0.499812
\(611\) 1.43507e9 0.254524
\(612\) 0 0
\(613\) −5.00263e9 −0.877177 −0.438588 0.898688i \(-0.644521\pi\)
−0.438588 + 0.898688i \(0.644521\pi\)
\(614\) −3.06485e9 −0.534342
\(615\) 0 0
\(616\) −1.11341e9 −0.191920
\(617\) −1.47593e9 −0.252970 −0.126485 0.991969i \(-0.540370\pi\)
−0.126485 + 0.991969i \(0.540370\pi\)
\(618\) 0 0
\(619\) −9.36559e9 −1.58715 −0.793575 0.608472i \(-0.791783\pi\)
−0.793575 + 0.608472i \(0.791783\pi\)
\(620\) −1.65202e9 −0.278385
\(621\) 0 0
\(622\) 7.07150e9 1.17827
\(623\) −1.07455e9 −0.178041
\(624\) 0 0
\(625\) −1.09904e9 −0.180067
\(626\) −1.01207e9 −0.164893
\(627\) 0 0
\(628\) −3.75087e9 −0.604328
\(629\) 8.98698e9 1.43991
\(630\) 0 0
\(631\) −1.19966e10 −1.90088 −0.950438 0.310914i \(-0.899365\pi\)
−0.950438 + 0.310914i \(0.899365\pi\)
\(632\) 7.57751e9 1.19403
\(633\) 0 0
\(634\) 5.53871e9 0.863170
\(635\) −7.08771e9 −1.09849
\(636\) 0 0
\(637\) 2.36165e8 0.0362016
\(638\) −1.34175e9 −0.204551
\(639\) 0 0
\(640\) −2.98484e9 −0.450081
\(641\) 1.25626e10 1.88398 0.941989 0.335644i \(-0.108954\pi\)
0.941989 + 0.335644i \(0.108954\pi\)
\(642\) 0 0
\(643\) −4.11211e9 −0.609995 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(644\) −5.19283e9 −0.766131
\(645\) 0 0
\(646\) −4.75064e8 −0.0693327
\(647\) 8.36038e9 1.21356 0.606780 0.794870i \(-0.292461\pi\)
0.606780 + 0.794870i \(0.292461\pi\)
\(648\) 0 0
\(649\) −1.99700e8 −0.0286762
\(650\) 3.65895e8 0.0522588
\(651\) 0 0
\(652\) −5.85415e9 −0.827175
\(653\) −5.92578e8 −0.0832817 −0.0416409 0.999133i \(-0.513259\pi\)
−0.0416409 + 0.999133i \(0.513259\pi\)
\(654\) 0 0
\(655\) −2.86290e9 −0.398072
\(656\) 1.92492e8 0.0266225
\(657\) 0 0
\(658\) 6.14769e9 0.841243
\(659\) −3.61916e9 −0.492617 −0.246308 0.969192i \(-0.579218\pi\)
−0.246308 + 0.969192i \(0.579218\pi\)
\(660\) 0 0
\(661\) 1.49686e9 0.201593 0.100797 0.994907i \(-0.467861\pi\)
0.100797 + 0.994907i \(0.467861\pi\)
\(662\) −1.17839e9 −0.157866
\(663\) 0 0
\(664\) −5.20521e9 −0.690002
\(665\) −2.89757e8 −0.0382084
\(666\) 0 0
\(667\) −1.61435e10 −2.10648
\(668\) 2.63066e9 0.341465
\(669\) 0 0
\(670\) −2.07102e8 −0.0266025
\(671\) 2.08421e9 0.266326
\(672\) 0 0
\(673\) 1.55009e10 1.96022 0.980109 0.198461i \(-0.0635943\pi\)
0.980109 + 0.198461i \(0.0635943\pi\)
\(674\) 1.81615e9 0.228476
\(675\) 0 0
\(676\) 4.95177e9 0.616520
\(677\) 7.97429e9 0.987715 0.493857 0.869543i \(-0.335586\pi\)
0.493857 + 0.869543i \(0.335586\pi\)
\(678\) 0 0
\(679\) −2.23040e9 −0.273425
\(680\) 9.96431e9 1.21525
\(681\) 0 0
\(682\) −7.12407e8 −0.0859969
\(683\) −7.37364e9 −0.885542 −0.442771 0.896635i \(-0.646005\pi\)
−0.442771 + 0.896635i \(0.646005\pi\)
\(684\) 0 0
\(685\) −3.34356e9 −0.397458
\(686\) 5.52317e9 0.653212
\(687\) 0 0
\(688\) 1.67871e8 0.0196524
\(689\) 6.83868e8 0.0796535
\(690\) 0 0
\(691\) 5.70889e9 0.658232 0.329116 0.944290i \(-0.393249\pi\)
0.329116 + 0.944290i \(0.393249\pi\)
\(692\) 1.06453e9 0.122120
\(693\) 0 0
\(694\) −3.52571e9 −0.400395
\(695\) −2.22905e9 −0.251868
\(696\) 0 0
\(697\) −1.27034e10 −1.42103
\(698\) −4.17543e8 −0.0464737
\(699\) 0 0
\(700\) −2.70374e9 −0.297935
\(701\) −1.23928e10 −1.35880 −0.679402 0.733767i \(-0.737761\pi\)
−0.679402 + 0.733767i \(0.737761\pi\)
\(702\) 0 0
\(703\) −4.68410e8 −0.0508491
\(704\) −1.17851e9 −0.127300
\(705\) 0 0
\(706\) 1.30640e9 0.139720
\(707\) −1.14630e10 −1.21992
\(708\) 0 0
\(709\) 1.39979e10 1.47504 0.737518 0.675328i \(-0.235998\pi\)
0.737518 + 0.675328i \(0.235998\pi\)
\(710\) 5.16271e9 0.541344
\(711\) 0 0
\(712\) −1.92599e9 −0.199974
\(713\) −8.57143e9 −0.885605
\(714\) 0 0
\(715\) 2.37152e8 0.0242636
\(716\) −1.61815e9 −0.164750
\(717\) 0 0
\(718\) 9.59215e9 0.967120
\(719\) −7.66012e9 −0.768572 −0.384286 0.923214i \(-0.625552\pi\)
−0.384286 + 0.923214i \(0.625552\pi\)
\(720\) 0 0
\(721\) −1.30101e9 −0.129273
\(722\) −6.10161e9 −0.603343
\(723\) 0 0
\(724\) −7.60430e9 −0.744688
\(725\) −8.40543e9 −0.819175
\(726\) 0 0
\(727\) 1.02618e10 0.990493 0.495247 0.868752i \(-0.335078\pi\)
0.495247 + 0.868752i \(0.335078\pi\)
\(728\) −1.46428e9 −0.140658
\(729\) 0 0
\(730\) 4.25940e9 0.405246
\(731\) −1.10785e10 −1.04899
\(732\) 0 0
\(733\) −1.67714e10 −1.57292 −0.786459 0.617643i \(-0.788088\pi\)
−0.786459 + 0.617643i \(0.788088\pi\)
\(734\) −1.02988e10 −0.961285
\(735\) 0 0
\(736\) −1.50069e10 −1.38746
\(737\) 1.54051e8 0.0141752
\(738\) 0 0
\(739\) 5.49527e9 0.500880 0.250440 0.968132i \(-0.419425\pi\)
0.250440 + 0.968132i \(0.419425\pi\)
\(740\) 3.80843e9 0.345490
\(741\) 0 0
\(742\) 2.92962e9 0.263268
\(743\) 1.77986e9 0.159193 0.0795966 0.996827i \(-0.474637\pi\)
0.0795966 + 0.996827i \(0.474637\pi\)
\(744\) 0 0
\(745\) 1.29006e10 1.14305
\(746\) 6.09683e8 0.0537673
\(747\) 0 0
\(748\) −2.87311e9 −0.251013
\(749\) −6.28548e9 −0.546578
\(750\) 0 0
\(751\) −7.12062e9 −0.613448 −0.306724 0.951798i \(-0.599233\pi\)
−0.306724 + 0.951798i \(0.599233\pi\)
\(752\) −6.20127e8 −0.0531763
\(753\) 0 0
\(754\) −1.76459e9 −0.149914
\(755\) 8.23619e9 0.696485
\(756\) 0 0
\(757\) 6.43143e9 0.538855 0.269428 0.963021i \(-0.413166\pi\)
0.269428 + 0.963021i \(0.413166\pi\)
\(758\) −4.98319e9 −0.415590
\(759\) 0 0
\(760\) −5.19350e8 −0.0429153
\(761\) −1.46912e10 −1.20840 −0.604202 0.796831i \(-0.706508\pi\)
−0.604202 + 0.796831i \(0.706508\pi\)
\(762\) 0 0
\(763\) −2.10979e9 −0.171950
\(764\) −6.08175e9 −0.493403
\(765\) 0 0
\(766\) −1.41698e10 −1.13910
\(767\) −2.62632e8 −0.0210167
\(768\) 0 0
\(769\) 1.24049e10 0.983677 0.491838 0.870687i \(-0.336325\pi\)
0.491838 + 0.870687i \(0.336325\pi\)
\(770\) 1.01594e9 0.0801952
\(771\) 0 0
\(772\) 5.78276e9 0.452350
\(773\) 2.65186e9 0.206501 0.103250 0.994655i \(-0.467076\pi\)
0.103250 + 0.994655i \(0.467076\pi\)
\(774\) 0 0
\(775\) −4.46287e9 −0.344396
\(776\) −3.99768e9 −0.307109
\(777\) 0 0
\(778\) 5.65113e8 0.0430236
\(779\) 6.62112e8 0.0501823
\(780\) 0 0
\(781\) −3.84024e9 −0.288456
\(782\) 2.00405e10 1.49860
\(783\) 0 0
\(784\) −1.02053e8 −0.00756342
\(785\) 8.82916e9 0.651441
\(786\) 0 0
\(787\) −4.05957e9 −0.296871 −0.148436 0.988922i \(-0.547424\pi\)
−0.148436 + 0.988922i \(0.547424\pi\)
\(788\) −1.13968e10 −0.829738
\(789\) 0 0
\(790\) −6.91415e9 −0.498935
\(791\) −2.32951e10 −1.67358
\(792\) 0 0
\(793\) 2.74102e9 0.195189
\(794\) 1.78740e9 0.126721
\(795\) 0 0
\(796\) −2.26177e9 −0.158947
\(797\) −2.36833e10 −1.65706 −0.828530 0.559944i \(-0.810823\pi\)
−0.828530 + 0.559944i \(0.810823\pi\)
\(798\) 0 0
\(799\) 4.09248e10 2.83839
\(800\) −7.81364e9 −0.539558
\(801\) 0 0
\(802\) 3.81867e9 0.261398
\(803\) −3.16832e9 −0.215936
\(804\) 0 0
\(805\) 1.22234e10 0.825859
\(806\) −9.36910e8 −0.0630268
\(807\) 0 0
\(808\) −2.05458e10 −1.37020
\(809\) 9.66394e9 0.641704 0.320852 0.947129i \(-0.396031\pi\)
0.320852 + 0.947129i \(0.396031\pi\)
\(810\) 0 0
\(811\) 1.27359e10 0.838413 0.419207 0.907891i \(-0.362308\pi\)
0.419207 + 0.907891i \(0.362308\pi\)
\(812\) 1.30392e10 0.854684
\(813\) 0 0
\(814\) 1.64232e9 0.106727
\(815\) 1.37801e10 0.891661
\(816\) 0 0
\(817\) 5.77422e8 0.0370438
\(818\) −1.08975e10 −0.696132
\(819\) 0 0
\(820\) −5.38333e9 −0.340959
\(821\) −1.18429e10 −0.746893 −0.373446 0.927652i \(-0.621824\pi\)
−0.373446 + 0.927652i \(0.621824\pi\)
\(822\) 0 0
\(823\) 1.12076e10 0.700829 0.350415 0.936595i \(-0.386041\pi\)
0.350415 + 0.936595i \(0.386041\pi\)
\(824\) −2.33188e9 −0.145198
\(825\) 0 0
\(826\) −1.12509e9 −0.0694636
\(827\) 1.69838e10 1.04416 0.522078 0.852898i \(-0.325157\pi\)
0.522078 + 0.852898i \(0.325157\pi\)
\(828\) 0 0
\(829\) −4.59293e9 −0.279994 −0.139997 0.990152i \(-0.544709\pi\)
−0.139997 + 0.990152i \(0.544709\pi\)
\(830\) 4.74953e9 0.288322
\(831\) 0 0
\(832\) −1.54990e9 −0.0932980
\(833\) 6.73488e9 0.403713
\(834\) 0 0
\(835\) −6.19230e9 −0.368086
\(836\) 1.49749e8 0.00886428
\(837\) 0 0
\(838\) −4.03015e9 −0.236574
\(839\) 2.30945e10 1.35002 0.675012 0.737806i \(-0.264138\pi\)
0.675012 + 0.737806i \(0.264138\pi\)
\(840\) 0 0
\(841\) 2.32866e10 1.34996
\(842\) −5.14664e9 −0.297120
\(843\) 0 0
\(844\) −1.34934e10 −0.772542
\(845\) −1.16560e10 −0.664584
\(846\) 0 0
\(847\) 1.48202e10 0.838033
\(848\) −2.95516e8 −0.0166416
\(849\) 0 0
\(850\) 1.04345e10 0.582779
\(851\) 1.97598e10 1.09908
\(852\) 0 0
\(853\) 2.22496e10 1.22744 0.613721 0.789523i \(-0.289672\pi\)
0.613721 + 0.789523i \(0.289672\pi\)
\(854\) 1.17422e10 0.645132
\(855\) 0 0
\(856\) −1.12659e10 −0.613912
\(857\) −1.07105e10 −0.581270 −0.290635 0.956834i \(-0.593867\pi\)
−0.290635 + 0.956834i \(0.593867\pi\)
\(858\) 0 0
\(859\) 3.19391e10 1.71928 0.859639 0.510902i \(-0.170688\pi\)
0.859639 + 0.510902i \(0.170688\pi\)
\(860\) −4.69475e9 −0.251691
\(861\) 0 0
\(862\) 1.62047e10 0.861717
\(863\) −2.90663e10 −1.53940 −0.769700 0.638406i \(-0.779594\pi\)
−0.769700 + 0.638406i \(0.779594\pi\)
\(864\) 0 0
\(865\) −2.50580e9 −0.131641
\(866\) 4.77979e8 0.0250090
\(867\) 0 0
\(868\) 6.92319e9 0.359325
\(869\) 5.14304e9 0.265858
\(870\) 0 0
\(871\) 2.02598e8 0.0103889
\(872\) −3.78150e9 −0.193133
\(873\) 0 0
\(874\) −1.04453e9 −0.0529215
\(875\) 1.82742e10 0.922167
\(876\) 0 0
\(877\) −1.87443e9 −0.0938364 −0.0469182 0.998899i \(-0.514940\pi\)
−0.0469182 + 0.998899i \(0.514940\pi\)
\(878\) −1.74616e10 −0.870668
\(879\) 0 0
\(880\) −1.02479e8 −0.00506927
\(881\) 3.84330e10 1.89360 0.946802 0.321818i \(-0.104294\pi\)
0.946802 + 0.321818i \(0.104294\pi\)
\(882\) 0 0
\(883\) 1.17045e10 0.572125 0.286063 0.958211i \(-0.407653\pi\)
0.286063 + 0.958211i \(0.407653\pi\)
\(884\) −3.77853e9 −0.183967
\(885\) 0 0
\(886\) −7.91134e9 −0.382148
\(887\) −3.09354e9 −0.148841 −0.0744205 0.997227i \(-0.523711\pi\)
−0.0744205 + 0.997227i \(0.523711\pi\)
\(888\) 0 0
\(889\) 2.97027e10 1.41788
\(890\) 1.75738e9 0.0835604
\(891\) 0 0
\(892\) −7.24615e8 −0.0341846
\(893\) −2.13304e9 −0.100235
\(894\) 0 0
\(895\) 3.80897e9 0.177594
\(896\) 1.25087e10 0.580942
\(897\) 0 0
\(898\) −3.81064e9 −0.175603
\(899\) 2.15229e10 0.987967
\(900\) 0 0
\(901\) 1.95023e10 0.888279
\(902\) −2.32147e9 −0.105327
\(903\) 0 0
\(904\) −4.17533e10 −1.87975
\(905\) 1.78997e10 0.802744
\(906\) 0 0
\(907\) −5.95597e9 −0.265050 −0.132525 0.991180i \(-0.542308\pi\)
−0.132525 + 0.991180i \(0.542308\pi\)
\(908\) −5.57599e9 −0.247185
\(909\) 0 0
\(910\) 1.33609e9 0.0587748
\(911\) 4.08258e10 1.78904 0.894521 0.447026i \(-0.147517\pi\)
0.894521 + 0.447026i \(0.147517\pi\)
\(912\) 0 0
\(913\) −3.53290e9 −0.153633
\(914\) 2.69735e9 0.116849
\(915\) 0 0
\(916\) −1.60032e10 −0.687977
\(917\) 1.19976e10 0.513811
\(918\) 0 0
\(919\) −1.12519e10 −0.478214 −0.239107 0.970993i \(-0.576855\pi\)
−0.239107 + 0.970993i \(0.576855\pi\)
\(920\) 2.19087e10 0.927598
\(921\) 0 0
\(922\) 2.05189e10 0.862176
\(923\) −5.05043e9 −0.211408
\(924\) 0 0
\(925\) 1.02883e10 0.427414
\(926\) −8.11271e9 −0.335759
\(927\) 0 0
\(928\) 3.76825e10 1.54783
\(929\) −1.44429e10 −0.591018 −0.295509 0.955340i \(-0.595489\pi\)
−0.295509 + 0.955340i \(0.595489\pi\)
\(930\) 0 0
\(931\) −3.51029e8 −0.0142567
\(932\) 1.99620e10 0.807696
\(933\) 0 0
\(934\) −2.16339e10 −0.868801
\(935\) 6.76301e9 0.270582
\(936\) 0 0
\(937\) 2.43718e10 0.967829 0.483915 0.875115i \(-0.339215\pi\)
0.483915 + 0.875115i \(0.339215\pi\)
\(938\) 8.67909e8 0.0343372
\(939\) 0 0
\(940\) 1.73428e10 0.681038
\(941\) 1.04613e10 0.409280 0.204640 0.978837i \(-0.434398\pi\)
0.204640 + 0.978837i \(0.434398\pi\)
\(942\) 0 0
\(943\) −2.79311e10 −1.08467
\(944\) 1.13490e8 0.00439090
\(945\) 0 0
\(946\) −2.02453e9 −0.0777509
\(947\) 6.80097e9 0.260223 0.130112 0.991499i \(-0.458466\pi\)
0.130112 + 0.991499i \(0.458466\pi\)
\(948\) 0 0
\(949\) −4.16677e9 −0.158259
\(950\) −5.43855e8 −0.0205802
\(951\) 0 0
\(952\) −4.17578e10 −1.56859
\(953\) 9.17306e8 0.0343312 0.0171656 0.999853i \(-0.494536\pi\)
0.0171656 + 0.999853i \(0.494536\pi\)
\(954\) 0 0
\(955\) 1.43158e10 0.531869
\(956\) −3.06488e10 −1.13452
\(957\) 0 0
\(958\) −1.55614e10 −0.571833
\(959\) 1.40120e10 0.513019
\(960\) 0 0
\(961\) −1.60850e10 −0.584640
\(962\) 2.15987e9 0.0782195
\(963\) 0 0
\(964\) 2.20377e10 0.792314
\(965\) −1.36120e10 −0.487615
\(966\) 0 0
\(967\) −3.69970e10 −1.31575 −0.657875 0.753127i \(-0.728544\pi\)
−0.657875 + 0.753127i \(0.728544\pi\)
\(968\) 2.65631e10 0.941271
\(969\) 0 0
\(970\) 3.64771e9 0.128327
\(971\) 1.31210e10 0.459937 0.229968 0.973198i \(-0.426138\pi\)
0.229968 + 0.973198i \(0.426138\pi\)
\(972\) 0 0
\(973\) 9.34138e9 0.325099
\(974\) 1.40498e9 0.0487209
\(975\) 0 0
\(976\) −1.18446e9 −0.0407798
\(977\) 3.90962e10 1.34123 0.670615 0.741806i \(-0.266030\pi\)
0.670615 + 0.741806i \(0.266030\pi\)
\(978\) 0 0
\(979\) −1.30721e9 −0.0445253
\(980\) 2.85406e9 0.0968660
\(981\) 0 0
\(982\) −7.42485e9 −0.250206
\(983\) 3.75059e10 1.25940 0.629699 0.776840i \(-0.283178\pi\)
0.629699 + 0.776840i \(0.283178\pi\)
\(984\) 0 0
\(985\) 2.68269e10 0.894424
\(986\) −5.03219e10 −1.67181
\(987\) 0 0
\(988\) 1.96941e8 0.00649660
\(989\) −2.43585e10 −0.800688
\(990\) 0 0
\(991\) 5.39247e10 1.76007 0.880036 0.474907i \(-0.157518\pi\)
0.880036 + 0.474907i \(0.157518\pi\)
\(992\) 2.00076e10 0.650735
\(993\) 0 0
\(994\) −2.16355e10 −0.698739
\(995\) 5.32397e9 0.171338
\(996\) 0 0
\(997\) 2.01581e10 0.644192 0.322096 0.946707i \(-0.395613\pi\)
0.322096 + 0.946707i \(0.395613\pi\)
\(998\) 8.48317e9 0.270148
\(999\) 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.d.1.11 17
3.2 odd 2 177.8.a.b.1.7 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.7 17 3.2 odd 2
531.8.a.d.1.11 17 1.1 even 1 trivial