Properties

Label 531.8.a.d.1.10
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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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 1.10
Root \(-4.01497\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.01497 q^{2} -91.8201 q^{4} +385.807 q^{5} +847.649 q^{7} -1322.21 q^{8} +O(q^{10})\) \(q+6.01497 q^{2} -91.8201 q^{4} +385.807 q^{5} +847.649 q^{7} -1322.21 q^{8} +2320.62 q^{10} +5526.47 q^{11} +2987.05 q^{13} +5098.58 q^{14} +3799.90 q^{16} +6651.02 q^{17} +47421.5 q^{19} -35424.9 q^{20} +33241.6 q^{22} -16902.5 q^{23} +70722.2 q^{25} +17967.0 q^{26} -77831.2 q^{28} -2638.69 q^{29} +138735. q^{31} +192099. q^{32} +40005.7 q^{34} +327029. q^{35} -91219.9 q^{37} +285239. q^{38} -510119. q^{40} -549406. q^{41} -240843. q^{43} -507441. q^{44} -101668. q^{46} +1.17880e6 q^{47} -105035. q^{49} +425392. q^{50} -274271. q^{52} +100058. q^{53} +2.13215e6 q^{55} -1.12077e6 q^{56} -15871.6 q^{58} +205379. q^{59} +71464.3 q^{61} +834486. q^{62} +669085. q^{64} +1.15243e6 q^{65} -912586. q^{67} -610697. q^{68} +1.96707e6 q^{70} +274897. q^{71} -5.34005e6 q^{73} -548685. q^{74} -4.35425e6 q^{76} +4.68451e6 q^{77} +1.12488e6 q^{79} +1.46603e6 q^{80} -3.30466e6 q^{82} +4.95003e6 q^{83} +2.56601e6 q^{85} -1.44866e6 q^{86} -7.30717e6 q^{88} +8.64528e6 q^{89} +2.53197e6 q^{91} +1.55199e6 q^{92} +7.09048e6 q^{94} +1.82956e7 q^{95} -2.69412e6 q^{97} -631782. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.01497 0.531654 0.265827 0.964021i \(-0.414355\pi\)
0.265827 + 0.964021i \(0.414355\pi\)
\(3\) 0 0
\(4\) −91.8201 −0.717345
\(5\) 385.807 1.38031 0.690153 0.723664i \(-0.257543\pi\)
0.690153 + 0.723664i \(0.257543\pi\)
\(6\) 0 0
\(7\) 847.649 0.934055 0.467028 0.884243i \(-0.345325\pi\)
0.467028 + 0.884243i \(0.345325\pi\)
\(8\) −1322.21 −0.913032
\(9\) 0 0
\(10\) 2320.62 0.733844
\(11\) 5526.47 1.25191 0.625956 0.779859i \(-0.284709\pi\)
0.625956 + 0.779859i \(0.284709\pi\)
\(12\) 0 0
\(13\) 2987.05 0.377087 0.188543 0.982065i \(-0.439623\pi\)
0.188543 + 0.982065i \(0.439623\pi\)
\(14\) 5098.58 0.496594
\(15\) 0 0
\(16\) 3799.90 0.231928
\(17\) 6651.02 0.328335 0.164167 0.986432i \(-0.447506\pi\)
0.164167 + 0.986432i \(0.447506\pi\)
\(18\) 0 0
\(19\) 47421.5 1.58613 0.793064 0.609139i \(-0.208485\pi\)
0.793064 + 0.609139i \(0.208485\pi\)
\(20\) −35424.9 −0.990155
\(21\) 0 0
\(22\) 33241.6 0.665583
\(23\) −16902.5 −0.289670 −0.144835 0.989456i \(-0.546265\pi\)
−0.144835 + 0.989456i \(0.546265\pi\)
\(24\) 0 0
\(25\) 70722.2 0.905244
\(26\) 17967.0 0.200479
\(27\) 0 0
\(28\) −77831.2 −0.670040
\(29\) −2638.69 −0.0200907 −0.0100454 0.999950i \(-0.503198\pi\)
−0.0100454 + 0.999950i \(0.503198\pi\)
\(30\) 0 0
\(31\) 138735. 0.836410 0.418205 0.908353i \(-0.362659\pi\)
0.418205 + 0.908353i \(0.362659\pi\)
\(32\) 192099. 1.03634
\(33\) 0 0
\(34\) 40005.7 0.174560
\(35\) 327029. 1.28928
\(36\) 0 0
\(37\) −91219.9 −0.296063 −0.148031 0.988983i \(-0.547294\pi\)
−0.148031 + 0.988983i \(0.547294\pi\)
\(38\) 285239. 0.843270
\(39\) 0 0
\(40\) −510119. −1.26026
\(41\) −549406. −1.24494 −0.622472 0.782642i \(-0.713871\pi\)
−0.622472 + 0.782642i \(0.713871\pi\)
\(42\) 0 0
\(43\) −240843. −0.461949 −0.230974 0.972960i \(-0.574191\pi\)
−0.230974 + 0.972960i \(0.574191\pi\)
\(44\) −507441. −0.898052
\(45\) 0 0
\(46\) −101668. −0.154004
\(47\) 1.17880e6 1.65615 0.828074 0.560618i \(-0.189437\pi\)
0.828074 + 0.560618i \(0.189437\pi\)
\(48\) 0 0
\(49\) −105035. −0.127540
\(50\) 425392. 0.481276
\(51\) 0 0
\(52\) −274271. −0.270501
\(53\) 100058. 0.0923184 0.0461592 0.998934i \(-0.485302\pi\)
0.0461592 + 0.998934i \(0.485302\pi\)
\(54\) 0 0
\(55\) 2.13215e6 1.72802
\(56\) −1.12077e6 −0.852823
\(57\) 0 0
\(58\) −15871.6 −0.0106813
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 71464.3 0.0403120 0.0201560 0.999797i \(-0.493584\pi\)
0.0201560 + 0.999797i \(0.493584\pi\)
\(62\) 834486. 0.444680
\(63\) 0 0
\(64\) 669085. 0.319045
\(65\) 1.15243e6 0.520495
\(66\) 0 0
\(67\) −912586. −0.370691 −0.185345 0.982673i \(-0.559340\pi\)
−0.185345 + 0.982673i \(0.559340\pi\)
\(68\) −610697. −0.235529
\(69\) 0 0
\(70\) 1.96707e6 0.685451
\(71\) 274897. 0.0911520 0.0455760 0.998961i \(-0.485488\pi\)
0.0455760 + 0.998961i \(0.485488\pi\)
\(72\) 0 0
\(73\) −5.34005e6 −1.60663 −0.803315 0.595555i \(-0.796932\pi\)
−0.803315 + 0.595555i \(0.796932\pi\)
\(74\) −548685. −0.157403
\(75\) 0 0
\(76\) −4.35425e6 −1.13780
\(77\) 4.68451e6 1.16935
\(78\) 0 0
\(79\) 1.12488e6 0.256692 0.128346 0.991729i \(-0.459033\pi\)
0.128346 + 0.991729i \(0.459033\pi\)
\(80\) 1.46603e6 0.320131
\(81\) 0 0
\(82\) −3.30466e6 −0.661878
\(83\) 4.95003e6 0.950243 0.475122 0.879920i \(-0.342404\pi\)
0.475122 + 0.879920i \(0.342404\pi\)
\(84\) 0 0
\(85\) 2.56601e6 0.453202
\(86\) −1.44866e6 −0.245597
\(87\) 0 0
\(88\) −7.30717e6 −1.14304
\(89\) 8.64528e6 1.29991 0.649956 0.759972i \(-0.274787\pi\)
0.649956 + 0.759972i \(0.274787\pi\)
\(90\) 0 0
\(91\) 2.53197e6 0.352220
\(92\) 1.55199e6 0.207793
\(93\) 0 0
\(94\) 7.09048e6 0.880497
\(95\) 1.82956e7 2.18934
\(96\) 0 0
\(97\) −2.69412e6 −0.299720 −0.149860 0.988707i \(-0.547882\pi\)
−0.149860 + 0.988707i \(0.547882\pi\)
\(98\) −631782. −0.0678073
\(99\) 0 0
\(100\) −6.49372e6 −0.649372
\(101\) −1.51552e7 −1.46365 −0.731826 0.681492i \(-0.761332\pi\)
−0.731826 + 0.681492i \(0.761332\pi\)
\(102\) 0 0
\(103\) −1.40028e7 −1.26266 −0.631328 0.775516i \(-0.717490\pi\)
−0.631328 + 0.775516i \(0.717490\pi\)
\(104\) −3.94952e6 −0.344292
\(105\) 0 0
\(106\) 601849. 0.0490814
\(107\) 8.87562e6 0.700415 0.350208 0.936672i \(-0.386111\pi\)
0.350208 + 0.936672i \(0.386111\pi\)
\(108\) 0 0
\(109\) −2.49654e7 −1.84648 −0.923242 0.384219i \(-0.874471\pi\)
−0.923242 + 0.384219i \(0.874471\pi\)
\(110\) 1.28248e7 0.918708
\(111\) 0 0
\(112\) 3.22098e6 0.216633
\(113\) −1.44492e7 −0.942038 −0.471019 0.882123i \(-0.656114\pi\)
−0.471019 + 0.882123i \(0.656114\pi\)
\(114\) 0 0
\(115\) −6.52110e6 −0.399833
\(116\) 242285. 0.0144120
\(117\) 0 0
\(118\) 1.23535e6 0.0692154
\(119\) 5.63773e6 0.306683
\(120\) 0 0
\(121\) 1.10547e7 0.567283
\(122\) 429856. 0.0214320
\(123\) 0 0
\(124\) −1.27386e7 −0.599994
\(125\) −2.85605e6 −0.130792
\(126\) 0 0
\(127\) 1.71357e7 0.742315 0.371157 0.928570i \(-0.378961\pi\)
0.371157 + 0.928570i \(0.378961\pi\)
\(128\) −2.05642e7 −0.866716
\(129\) 0 0
\(130\) 6.93181e6 0.276723
\(131\) 1.82887e7 0.710777 0.355388 0.934719i \(-0.384349\pi\)
0.355388 + 0.934719i \(0.384349\pi\)
\(132\) 0 0
\(133\) 4.01968e7 1.48153
\(134\) −5.48918e6 −0.197079
\(135\) 0 0
\(136\) −8.79406e6 −0.299780
\(137\) 4.79157e7 1.59205 0.796023 0.605266i \(-0.206933\pi\)
0.796023 + 0.605266i \(0.206933\pi\)
\(138\) 0 0
\(139\) 2.76336e7 0.872740 0.436370 0.899767i \(-0.356264\pi\)
0.436370 + 0.899767i \(0.356264\pi\)
\(140\) −3.00278e7 −0.924860
\(141\) 0 0
\(142\) 1.65350e6 0.0484613
\(143\) 1.65079e7 0.472079
\(144\) 0 0
\(145\) −1.01803e6 −0.0277313
\(146\) −3.21203e7 −0.854170
\(147\) 0 0
\(148\) 8.37582e6 0.212379
\(149\) −6.17248e6 −0.152865 −0.0764325 0.997075i \(-0.524353\pi\)
−0.0764325 + 0.997075i \(0.524353\pi\)
\(150\) 0 0
\(151\) −2.68093e7 −0.633673 −0.316837 0.948480i \(-0.602621\pi\)
−0.316837 + 0.948480i \(0.602621\pi\)
\(152\) −6.27013e7 −1.44819
\(153\) 0 0
\(154\) 2.81772e7 0.621692
\(155\) 5.35249e7 1.15450
\(156\) 0 0
\(157\) 1.87147e7 0.385952 0.192976 0.981203i \(-0.438186\pi\)
0.192976 + 0.981203i \(0.438186\pi\)
\(158\) 6.76614e6 0.136471
\(159\) 0 0
\(160\) 7.41134e7 1.43046
\(161\) −1.43274e7 −0.270568
\(162\) 0 0
\(163\) 9.00787e7 1.62917 0.814583 0.580047i \(-0.196966\pi\)
0.814583 + 0.580047i \(0.196966\pi\)
\(164\) 5.04465e7 0.893053
\(165\) 0 0
\(166\) 2.97743e7 0.505200
\(167\) 6.92715e7 1.15093 0.575463 0.817828i \(-0.304822\pi\)
0.575463 + 0.817828i \(0.304822\pi\)
\(168\) 0 0
\(169\) −5.38260e7 −0.857806
\(170\) 1.54345e7 0.240947
\(171\) 0 0
\(172\) 2.21142e7 0.331376
\(173\) 1.01646e8 1.49255 0.746277 0.665635i \(-0.231839\pi\)
0.746277 + 0.665635i \(0.231839\pi\)
\(174\) 0 0
\(175\) 5.99476e7 0.845548
\(176\) 2.10001e7 0.290353
\(177\) 0 0
\(178\) 5.20011e7 0.691103
\(179\) −1.59338e7 −0.207650 −0.103825 0.994596i \(-0.533108\pi\)
−0.103825 + 0.994596i \(0.533108\pi\)
\(180\) 0 0
\(181\) 8.29983e7 1.04038 0.520192 0.854049i \(-0.325860\pi\)
0.520192 + 0.854049i \(0.325860\pi\)
\(182\) 1.52297e7 0.187259
\(183\) 0 0
\(184\) 2.23487e7 0.264478
\(185\) −3.51933e7 −0.408657
\(186\) 0 0
\(187\) 3.67567e7 0.411046
\(188\) −1.08238e8 −1.18803
\(189\) 0 0
\(190\) 1.10047e8 1.16397
\(191\) 2.94448e7 0.305768 0.152884 0.988244i \(-0.451144\pi\)
0.152884 + 0.988244i \(0.451144\pi\)
\(192\) 0 0
\(193\) 8.61854e7 0.862945 0.431473 0.902126i \(-0.357994\pi\)
0.431473 + 0.902126i \(0.357994\pi\)
\(194\) −1.62051e7 −0.159347
\(195\) 0 0
\(196\) 9.64432e6 0.0914904
\(197\) 1.74170e8 1.62308 0.811541 0.584296i \(-0.198629\pi\)
0.811541 + 0.584296i \(0.198629\pi\)
\(198\) 0 0
\(199\) 9.07182e7 0.816035 0.408017 0.912974i \(-0.366220\pi\)
0.408017 + 0.912974i \(0.366220\pi\)
\(200\) −9.35098e7 −0.826517
\(201\) 0 0
\(202\) −9.11583e7 −0.778155
\(203\) −2.23668e6 −0.0187658
\(204\) 0 0
\(205\) −2.11965e8 −1.71840
\(206\) −8.42266e7 −0.671296
\(207\) 0 0
\(208\) 1.13505e7 0.0874568
\(209\) 2.62074e8 1.98569
\(210\) 0 0
\(211\) −1.22589e8 −0.898387 −0.449194 0.893434i \(-0.648289\pi\)
−0.449194 + 0.893434i \(0.648289\pi\)
\(212\) −9.18737e6 −0.0662241
\(213\) 0 0
\(214\) 5.33866e7 0.372378
\(215\) −9.29188e7 −0.637631
\(216\) 0 0
\(217\) 1.17598e8 0.781254
\(218\) −1.50166e8 −0.981690
\(219\) 0 0
\(220\) −1.95775e8 −1.23959
\(221\) 1.98669e7 0.123811
\(222\) 0 0
\(223\) −5.48773e7 −0.331380 −0.165690 0.986178i \(-0.552985\pi\)
−0.165690 + 0.986178i \(0.552985\pi\)
\(224\) 1.62833e8 0.967997
\(225\) 0 0
\(226\) −8.69114e7 −0.500838
\(227\) −1.64956e8 −0.936003 −0.468001 0.883728i \(-0.655026\pi\)
−0.468001 + 0.883728i \(0.655026\pi\)
\(228\) 0 0
\(229\) 1.18120e8 0.649980 0.324990 0.945717i \(-0.394639\pi\)
0.324990 + 0.945717i \(0.394639\pi\)
\(230\) −3.92243e7 −0.212573
\(231\) 0 0
\(232\) 3.48891e6 0.0183435
\(233\) 1.12335e8 0.581795 0.290898 0.956754i \(-0.406046\pi\)
0.290898 + 0.956754i \(0.406046\pi\)
\(234\) 0 0
\(235\) 4.54791e8 2.28599
\(236\) −1.88579e7 −0.0933903
\(237\) 0 0
\(238\) 3.39108e7 0.163049
\(239\) −2.96755e8 −1.40607 −0.703033 0.711158i \(-0.748171\pi\)
−0.703033 + 0.711158i \(0.748171\pi\)
\(240\) 0 0
\(241\) 1.37243e7 0.0631583 0.0315791 0.999501i \(-0.489946\pi\)
0.0315791 + 0.999501i \(0.489946\pi\)
\(242\) 6.64939e7 0.301598
\(243\) 0 0
\(244\) −6.56186e6 −0.0289176
\(245\) −4.05232e7 −0.176045
\(246\) 0 0
\(247\) 1.41651e8 0.598107
\(248\) −1.83437e8 −0.763670
\(249\) 0 0
\(250\) −1.71791e7 −0.0695359
\(251\) 2.78781e7 0.111277 0.0556385 0.998451i \(-0.482281\pi\)
0.0556385 + 0.998451i \(0.482281\pi\)
\(252\) 0 0
\(253\) −9.34112e7 −0.362641
\(254\) 1.03071e8 0.394654
\(255\) 0 0
\(256\) −2.09336e8 −0.779837
\(257\) −1.86791e8 −0.686419 −0.343209 0.939259i \(-0.611514\pi\)
−0.343209 + 0.939259i \(0.611514\pi\)
\(258\) 0 0
\(259\) −7.73224e7 −0.276539
\(260\) −1.05816e8 −0.373374
\(261\) 0 0
\(262\) 1.10006e8 0.377887
\(263\) −2.51415e8 −0.852210 −0.426105 0.904674i \(-0.640115\pi\)
−0.426105 + 0.904674i \(0.640115\pi\)
\(264\) 0 0
\(265\) 3.86033e7 0.127428
\(266\) 2.41783e8 0.787661
\(267\) 0 0
\(268\) 8.37937e7 0.265913
\(269\) 4.66642e8 1.46168 0.730838 0.682551i \(-0.239130\pi\)
0.730838 + 0.682551i \(0.239130\pi\)
\(270\) 0 0
\(271\) 4.71085e8 1.43783 0.718914 0.695099i \(-0.244640\pi\)
0.718914 + 0.695099i \(0.244640\pi\)
\(272\) 2.52732e7 0.0761500
\(273\) 0 0
\(274\) 2.88212e8 0.846417
\(275\) 3.90844e8 1.13329
\(276\) 0 0
\(277\) −4.94424e8 −1.39772 −0.698860 0.715259i \(-0.746309\pi\)
−0.698860 + 0.715259i \(0.746309\pi\)
\(278\) 1.66215e8 0.463995
\(279\) 0 0
\(280\) −4.32402e8 −1.17716
\(281\) 1.88387e8 0.506499 0.253249 0.967401i \(-0.418501\pi\)
0.253249 + 0.967401i \(0.418501\pi\)
\(282\) 0 0
\(283\) 3.45526e8 0.906208 0.453104 0.891458i \(-0.350317\pi\)
0.453104 + 0.891458i \(0.350317\pi\)
\(284\) −2.52411e7 −0.0653874
\(285\) 0 0
\(286\) 9.92943e7 0.250982
\(287\) −4.65703e8 −1.16285
\(288\) 0 0
\(289\) −3.66103e8 −0.892196
\(290\) −6.12340e6 −0.0147435
\(291\) 0 0
\(292\) 4.90324e8 1.15251
\(293\) −4.35989e8 −1.01260 −0.506301 0.862357i \(-0.668988\pi\)
−0.506301 + 0.862357i \(0.668988\pi\)
\(294\) 0 0
\(295\) 7.92367e7 0.179701
\(296\) 1.20612e8 0.270315
\(297\) 0 0
\(298\) −3.71273e7 −0.0812712
\(299\) −5.04886e7 −0.109231
\(300\) 0 0
\(301\) −2.04150e8 −0.431486
\(302\) −1.61257e8 −0.336895
\(303\) 0 0
\(304\) 1.80197e8 0.367867
\(305\) 2.75714e7 0.0556429
\(306\) 0 0
\(307\) 9.90146e8 1.95306 0.976529 0.215387i \(-0.0691013\pi\)
0.976529 + 0.215387i \(0.0691013\pi\)
\(308\) −4.30132e8 −0.838830
\(309\) 0 0
\(310\) 3.21951e8 0.613795
\(311\) 2.78364e8 0.524750 0.262375 0.964966i \(-0.415494\pi\)
0.262375 + 0.964966i \(0.415494\pi\)
\(312\) 0 0
\(313\) −7.58142e8 −1.39748 −0.698740 0.715376i \(-0.746256\pi\)
−0.698740 + 0.715376i \(0.746256\pi\)
\(314\) 1.12568e8 0.205193
\(315\) 0 0
\(316\) −1.03287e8 −0.184137
\(317\) 3.93211e8 0.693296 0.346648 0.937995i \(-0.387320\pi\)
0.346648 + 0.937995i \(0.387320\pi\)
\(318\) 0 0
\(319\) −1.45826e7 −0.0251518
\(320\) 2.58138e8 0.440379
\(321\) 0 0
\(322\) −8.61788e7 −0.143848
\(323\) 3.15402e8 0.520781
\(324\) 0 0
\(325\) 2.11251e8 0.341355
\(326\) 5.41821e8 0.866152
\(327\) 0 0
\(328\) 7.26431e8 1.13667
\(329\) 9.99212e8 1.54693
\(330\) 0 0
\(331\) −9.50914e7 −0.144126 −0.0720632 0.997400i \(-0.522958\pi\)
−0.0720632 + 0.997400i \(0.522958\pi\)
\(332\) −4.54512e8 −0.681652
\(333\) 0 0
\(334\) 4.16666e8 0.611894
\(335\) −3.52082e8 −0.511667
\(336\) 0 0
\(337\) −6.91531e8 −0.984254 −0.492127 0.870524i \(-0.663780\pi\)
−0.492127 + 0.870524i \(0.663780\pi\)
\(338\) −3.23762e8 −0.456055
\(339\) 0 0
\(340\) −2.35611e8 −0.325102
\(341\) 7.66714e8 1.04711
\(342\) 0 0
\(343\) −7.87108e8 −1.05319
\(344\) 3.18445e8 0.421774
\(345\) 0 0
\(346\) 6.11400e8 0.793522
\(347\) 5.15844e8 0.662774 0.331387 0.943495i \(-0.392484\pi\)
0.331387 + 0.943495i \(0.392484\pi\)
\(348\) 0 0
\(349\) −8.89896e8 −1.12060 −0.560300 0.828290i \(-0.689314\pi\)
−0.560300 + 0.828290i \(0.689314\pi\)
\(350\) 3.60583e8 0.449539
\(351\) 0 0
\(352\) 1.06163e9 1.29740
\(353\) 3.85325e8 0.466246 0.233123 0.972447i \(-0.425105\pi\)
0.233123 + 0.972447i \(0.425105\pi\)
\(354\) 0 0
\(355\) 1.06057e8 0.125818
\(356\) −7.93810e8 −0.932485
\(357\) 0 0
\(358\) −9.58411e7 −0.110398
\(359\) 1.51447e9 1.72755 0.863777 0.503875i \(-0.168093\pi\)
0.863777 + 0.503875i \(0.168093\pi\)
\(360\) 0 0
\(361\) 1.35493e9 1.51580
\(362\) 4.99232e8 0.553124
\(363\) 0 0
\(364\) −2.32486e8 −0.252663
\(365\) −2.06023e9 −2.21764
\(366\) 0 0
\(367\) −8.13769e8 −0.859350 −0.429675 0.902984i \(-0.641372\pi\)
−0.429675 + 0.902984i \(0.641372\pi\)
\(368\) −6.42279e7 −0.0671825
\(369\) 0 0
\(370\) −2.11687e8 −0.217264
\(371\) 8.48144e7 0.0862305
\(372\) 0 0
\(373\) −1.20212e9 −1.19940 −0.599702 0.800223i \(-0.704714\pi\)
−0.599702 + 0.800223i \(0.704714\pi\)
\(374\) 2.21090e8 0.218534
\(375\) 0 0
\(376\) −1.55863e9 −1.51212
\(377\) −7.88190e6 −0.00757594
\(378\) 0 0
\(379\) −4.34931e8 −0.410377 −0.205189 0.978722i \(-0.565781\pi\)
−0.205189 + 0.978722i \(0.565781\pi\)
\(380\) −1.67990e9 −1.57051
\(381\) 0 0
\(382\) 1.77110e8 0.162563
\(383\) −3.58726e7 −0.0326263 −0.0163131 0.999867i \(-0.505193\pi\)
−0.0163131 + 0.999867i \(0.505193\pi\)
\(384\) 0 0
\(385\) 1.80732e9 1.61407
\(386\) 5.18403e8 0.458788
\(387\) 0 0
\(388\) 2.47374e8 0.215003
\(389\) 1.13638e9 0.978815 0.489407 0.872055i \(-0.337213\pi\)
0.489407 + 0.872055i \(0.337213\pi\)
\(390\) 0 0
\(391\) −1.12419e8 −0.0951087
\(392\) 1.38878e8 0.116448
\(393\) 0 0
\(394\) 1.04762e9 0.862917
\(395\) 4.33988e8 0.354314
\(396\) 0 0
\(397\) −5.01657e8 −0.402383 −0.201192 0.979552i \(-0.564481\pi\)
−0.201192 + 0.979552i \(0.564481\pi\)
\(398\) 5.45667e8 0.433848
\(399\) 0 0
\(400\) 2.68738e8 0.209951
\(401\) −6.17044e8 −0.477871 −0.238936 0.971035i \(-0.576798\pi\)
−0.238936 + 0.971035i \(0.576798\pi\)
\(402\) 0 0
\(403\) 4.14408e8 0.315399
\(404\) 1.39155e9 1.04994
\(405\) 0 0
\(406\) −1.34536e7 −0.00997692
\(407\) −5.04124e8 −0.370644
\(408\) 0 0
\(409\) 1.67124e9 1.20783 0.603915 0.797048i \(-0.293606\pi\)
0.603915 + 0.797048i \(0.293606\pi\)
\(410\) −1.27496e9 −0.913595
\(411\) 0 0
\(412\) 1.28574e9 0.905760
\(413\) 1.74089e8 0.121604
\(414\) 0 0
\(415\) 1.90976e9 1.31163
\(416\) 5.73811e8 0.390789
\(417\) 0 0
\(418\) 1.57637e9 1.05570
\(419\) 6.96670e7 0.0462677 0.0231339 0.999732i \(-0.492636\pi\)
0.0231339 + 0.999732i \(0.492636\pi\)
\(420\) 0 0
\(421\) −2.02237e9 −1.32091 −0.660457 0.750864i \(-0.729637\pi\)
−0.660457 + 0.750864i \(0.729637\pi\)
\(422\) −7.37371e8 −0.477631
\(423\) 0 0
\(424\) −1.32298e8 −0.0842896
\(425\) 4.70375e8 0.297223
\(426\) 0 0
\(427\) 6.05766e7 0.0376537
\(428\) −8.14960e8 −0.502439
\(429\) 0 0
\(430\) −5.58904e8 −0.338999
\(431\) 2.20133e9 1.32439 0.662193 0.749333i \(-0.269626\pi\)
0.662193 + 0.749333i \(0.269626\pi\)
\(432\) 0 0
\(433\) −1.11885e9 −0.662316 −0.331158 0.943575i \(-0.607439\pi\)
−0.331158 + 0.943575i \(0.607439\pi\)
\(434\) 7.07351e8 0.415356
\(435\) 0 0
\(436\) 2.29232e9 1.32457
\(437\) −8.01542e8 −0.459453
\(438\) 0 0
\(439\) 3.23226e8 0.182340 0.0911698 0.995835i \(-0.470939\pi\)
0.0911698 + 0.995835i \(0.470939\pi\)
\(440\) −2.81916e9 −1.57774
\(441\) 0 0
\(442\) 1.19499e8 0.0658244
\(443\) 2.28214e9 1.24718 0.623591 0.781751i \(-0.285673\pi\)
0.623591 + 0.781751i \(0.285673\pi\)
\(444\) 0 0
\(445\) 3.33541e9 1.79428
\(446\) −3.30086e8 −0.176179
\(447\) 0 0
\(448\) 5.67149e8 0.298005
\(449\) −2.89839e9 −1.51111 −0.755553 0.655087i \(-0.772632\pi\)
−0.755553 + 0.655087i \(0.772632\pi\)
\(450\) 0 0
\(451\) −3.03627e9 −1.55856
\(452\) 1.32672e9 0.675766
\(453\) 0 0
\(454\) −9.92205e8 −0.497629
\(455\) 9.76852e8 0.486171
\(456\) 0 0
\(457\) −1.61157e9 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(458\) 7.10490e8 0.345564
\(459\) 0 0
\(460\) 5.98768e8 0.286818
\(461\) −4.98049e8 −0.236766 −0.118383 0.992968i \(-0.537771\pi\)
−0.118383 + 0.992968i \(0.537771\pi\)
\(462\) 0 0
\(463\) −1.13218e9 −0.530131 −0.265065 0.964230i \(-0.585394\pi\)
−0.265065 + 0.964230i \(0.585394\pi\)
\(464\) −1.00268e7 −0.00465959
\(465\) 0 0
\(466\) 6.75693e8 0.309313
\(467\) −3.04377e9 −1.38294 −0.691470 0.722405i \(-0.743037\pi\)
−0.691470 + 0.722405i \(0.743037\pi\)
\(468\) 0 0
\(469\) −7.73552e8 −0.346246
\(470\) 2.73556e9 1.21536
\(471\) 0 0
\(472\) −2.71555e8 −0.118867
\(473\) −1.33101e9 −0.578319
\(474\) 0 0
\(475\) 3.35376e9 1.43583
\(476\) −5.17657e8 −0.219997
\(477\) 0 0
\(478\) −1.78497e9 −0.747539
\(479\) −1.85843e9 −0.772630 −0.386315 0.922367i \(-0.626252\pi\)
−0.386315 + 0.922367i \(0.626252\pi\)
\(480\) 0 0
\(481\) −2.72479e8 −0.111641
\(482\) 8.25512e7 0.0335783
\(483\) 0 0
\(484\) −1.01505e9 −0.406937
\(485\) −1.03941e9 −0.413705
\(486\) 0 0
\(487\) −3.95871e8 −0.155311 −0.0776555 0.996980i \(-0.524743\pi\)
−0.0776555 + 0.996980i \(0.524743\pi\)
\(488\) −9.44909e7 −0.0368062
\(489\) 0 0
\(490\) −2.43746e8 −0.0935948
\(491\) −8.60610e8 −0.328111 −0.164056 0.986451i \(-0.552458\pi\)
−0.164056 + 0.986451i \(0.552458\pi\)
\(492\) 0 0
\(493\) −1.75500e7 −0.00659648
\(494\) 8.52025e8 0.317986
\(495\) 0 0
\(496\) 5.27179e8 0.193987
\(497\) 2.33016e8 0.0851410
\(498\) 0 0
\(499\) 2.92320e9 1.05319 0.526595 0.850116i \(-0.323469\pi\)
0.526595 + 0.850116i \(0.323469\pi\)
\(500\) 2.62243e8 0.0938228
\(501\) 0 0
\(502\) 1.67686e8 0.0591609
\(503\) −2.74156e9 −0.960529 −0.480264 0.877124i \(-0.659459\pi\)
−0.480264 + 0.877124i \(0.659459\pi\)
\(504\) 0 0
\(505\) −5.84700e9 −2.02029
\(506\) −5.61866e8 −0.192799
\(507\) 0 0
\(508\) −1.57340e9 −0.532495
\(509\) −3.66404e9 −1.23154 −0.615769 0.787927i \(-0.711154\pi\)
−0.615769 + 0.787927i \(0.711154\pi\)
\(510\) 0 0
\(511\) −4.52649e9 −1.50068
\(512\) 1.37307e9 0.452113
\(513\) 0 0
\(514\) −1.12354e9 −0.364937
\(515\) −5.40239e9 −1.74285
\(516\) 0 0
\(517\) 6.51463e9 2.07335
\(518\) −4.65092e8 −0.147023
\(519\) 0 0
\(520\) −1.52375e9 −0.475229
\(521\) 4.43075e9 1.37260 0.686302 0.727317i \(-0.259233\pi\)
0.686302 + 0.727317i \(0.259233\pi\)
\(522\) 0 0
\(523\) −5.16773e9 −1.57959 −0.789794 0.613372i \(-0.789813\pi\)
−0.789794 + 0.613372i \(0.789813\pi\)
\(524\) −1.67927e9 −0.509872
\(525\) 0 0
\(526\) −1.51226e9 −0.453080
\(527\) 9.22728e8 0.274623
\(528\) 0 0
\(529\) −3.11913e9 −0.916091
\(530\) 2.32198e8 0.0677473
\(531\) 0 0
\(532\) −3.69087e9 −1.06277
\(533\) −1.64110e9 −0.469451
\(534\) 0 0
\(535\) 3.42428e9 0.966787
\(536\) 1.20663e9 0.338453
\(537\) 0 0
\(538\) 2.80684e9 0.777105
\(539\) −5.80473e8 −0.159669
\(540\) 0 0
\(541\) 5.16094e9 1.40132 0.700662 0.713493i \(-0.252888\pi\)
0.700662 + 0.713493i \(0.252888\pi\)
\(542\) 2.83356e9 0.764426
\(543\) 0 0
\(544\) 1.27766e9 0.340266
\(545\) −9.63183e9 −2.54871
\(546\) 0 0
\(547\) −1.77725e9 −0.464295 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(548\) −4.39962e9 −1.14205
\(549\) 0 0
\(550\) 2.35092e9 0.602515
\(551\) −1.25131e8 −0.0318664
\(552\) 0 0
\(553\) 9.53505e8 0.239765
\(554\) −2.97394e9 −0.743102
\(555\) 0 0
\(556\) −2.53732e9 −0.626055
\(557\) 2.65766e9 0.651637 0.325819 0.945432i \(-0.394360\pi\)
0.325819 + 0.945432i \(0.394360\pi\)
\(558\) 0 0
\(559\) −7.19409e8 −0.174195
\(560\) 1.24268e9 0.299020
\(561\) 0 0
\(562\) 1.13314e9 0.269282
\(563\) 2.85664e9 0.674646 0.337323 0.941389i \(-0.390478\pi\)
0.337323 + 0.941389i \(0.390478\pi\)
\(564\) 0 0
\(565\) −5.57459e9 −1.30030
\(566\) 2.07833e9 0.481789
\(567\) 0 0
\(568\) −3.63472e8 −0.0832247
\(569\) 1.18310e9 0.269233 0.134617 0.990898i \(-0.457020\pi\)
0.134617 + 0.990898i \(0.457020\pi\)
\(570\) 0 0
\(571\) 7.04152e8 0.158285 0.0791426 0.996863i \(-0.474782\pi\)
0.0791426 + 0.996863i \(0.474782\pi\)
\(572\) −1.51575e9 −0.338643
\(573\) 0 0
\(574\) −2.80119e9 −0.618231
\(575\) −1.19538e9 −0.262222
\(576\) 0 0
\(577\) −5.26118e9 −1.14017 −0.570083 0.821587i \(-0.693089\pi\)
−0.570083 + 0.821587i \(0.693089\pi\)
\(578\) −2.20210e9 −0.474339
\(579\) 0 0
\(580\) 9.34752e7 0.0198929
\(581\) 4.19589e9 0.887580
\(582\) 0 0
\(583\) 5.52970e8 0.115574
\(584\) 7.06068e9 1.46690
\(585\) 0 0
\(586\) −2.62246e9 −0.538354
\(587\) −7.97467e9 −1.62734 −0.813672 0.581324i \(-0.802535\pi\)
−0.813672 + 0.581324i \(0.802535\pi\)
\(588\) 0 0
\(589\) 6.57902e9 1.32665
\(590\) 4.76607e8 0.0955384
\(591\) 0 0
\(592\) −3.46627e8 −0.0686651
\(593\) 5.84550e9 1.15115 0.575573 0.817750i \(-0.304779\pi\)
0.575573 + 0.817750i \(0.304779\pi\)
\(594\) 0 0
\(595\) 2.17508e9 0.423316
\(596\) 5.66758e8 0.109657
\(597\) 0 0
\(598\) −3.03688e8 −0.0580729
\(599\) 2.93990e9 0.558906 0.279453 0.960159i \(-0.409847\pi\)
0.279453 + 0.960159i \(0.409847\pi\)
\(600\) 0 0
\(601\) −6.68201e9 −1.25559 −0.627793 0.778380i \(-0.716042\pi\)
−0.627793 + 0.778380i \(0.716042\pi\)
\(602\) −1.22796e9 −0.229401
\(603\) 0 0
\(604\) 2.46163e9 0.454562
\(605\) 4.26500e9 0.783024
\(606\) 0 0
\(607\) 4.48841e8 0.0814577 0.0407289 0.999170i \(-0.487032\pi\)
0.0407289 + 0.999170i \(0.487032\pi\)
\(608\) 9.10965e9 1.64376
\(609\) 0 0
\(610\) 1.65841e8 0.0295827
\(611\) 3.52115e9 0.624511
\(612\) 0 0
\(613\) 7.78075e9 1.36430 0.682150 0.731212i \(-0.261045\pi\)
0.682150 + 0.731212i \(0.261045\pi\)
\(614\) 5.95570e9 1.03835
\(615\) 0 0
\(616\) −6.19391e9 −1.06766
\(617\) 6.97925e9 1.19622 0.598109 0.801414i \(-0.295919\pi\)
0.598109 + 0.801414i \(0.295919\pi\)
\(618\) 0 0
\(619\) −3.49030e9 −0.591488 −0.295744 0.955267i \(-0.595568\pi\)
−0.295744 + 0.955267i \(0.595568\pi\)
\(620\) −4.91466e9 −0.828176
\(621\) 0 0
\(622\) 1.67435e9 0.278985
\(623\) 7.32816e9 1.21419
\(624\) 0 0
\(625\) −6.62706e9 −1.08578
\(626\) −4.56021e9 −0.742975
\(627\) 0 0
\(628\) −1.71838e9 −0.276861
\(629\) −6.06705e8 −0.0972077
\(630\) 0 0
\(631\) −7.42461e9 −1.17644 −0.588222 0.808700i \(-0.700172\pi\)
−0.588222 + 0.808700i \(0.700172\pi\)
\(632\) −1.48733e9 −0.234368
\(633\) 0 0
\(634\) 2.36516e9 0.368593
\(635\) 6.61107e9 1.02462
\(636\) 0 0
\(637\) −3.13745e8 −0.0480938
\(638\) −8.77142e7 −0.0133720
\(639\) 0 0
\(640\) −7.93382e9 −1.19633
\(641\) 2.52065e9 0.378015 0.189007 0.981976i \(-0.439473\pi\)
0.189007 + 0.981976i \(0.439473\pi\)
\(642\) 0 0
\(643\) −7.35494e9 −1.09104 −0.545520 0.838098i \(-0.683668\pi\)
−0.545520 + 0.838098i \(0.683668\pi\)
\(644\) 1.31554e9 0.194090
\(645\) 0 0
\(646\) 1.89713e9 0.276875
\(647\) 9.35379e8 0.135776 0.0678879 0.997693i \(-0.478374\pi\)
0.0678879 + 0.997693i \(0.478374\pi\)
\(648\) 0 0
\(649\) 1.13502e9 0.162985
\(650\) 1.27067e9 0.181483
\(651\) 0 0
\(652\) −8.27103e9 −1.16867
\(653\) −1.22407e10 −1.72032 −0.860161 0.510022i \(-0.829637\pi\)
−0.860161 + 0.510022i \(0.829637\pi\)
\(654\) 0 0
\(655\) 7.05591e9 0.981089
\(656\) −2.08769e9 −0.288737
\(657\) 0 0
\(658\) 6.01023e9 0.822433
\(659\) −5.21215e9 −0.709444 −0.354722 0.934972i \(-0.615425\pi\)
−0.354722 + 0.934972i \(0.615425\pi\)
\(660\) 0 0
\(661\) −1.11350e10 −1.49964 −0.749820 0.661642i \(-0.769860\pi\)
−0.749820 + 0.661642i \(0.769860\pi\)
\(662\) −5.71972e8 −0.0766253
\(663\) 0 0
\(664\) −6.54499e9 −0.867603
\(665\) 1.55082e10 2.04497
\(666\) 0 0
\(667\) 4.46004e7 0.00581967
\(668\) −6.36052e9 −0.825610
\(669\) 0 0
\(670\) −2.11776e9 −0.272029
\(671\) 3.94945e8 0.0504671
\(672\) 0 0
\(673\) 9.15611e9 1.15787 0.578933 0.815375i \(-0.303469\pi\)
0.578933 + 0.815375i \(0.303469\pi\)
\(674\) −4.15954e9 −0.523282
\(675\) 0 0
\(676\) 4.94231e9 0.615342
\(677\) −1.14733e10 −1.42111 −0.710557 0.703639i \(-0.751557\pi\)
−0.710557 + 0.703639i \(0.751557\pi\)
\(678\) 0 0
\(679\) −2.28367e9 −0.279955
\(680\) −3.39281e9 −0.413788
\(681\) 0 0
\(682\) 4.61176e9 0.556701
\(683\) 5.63555e9 0.676806 0.338403 0.941001i \(-0.390113\pi\)
0.338403 + 0.941001i \(0.390113\pi\)
\(684\) 0 0
\(685\) 1.84862e10 2.19751
\(686\) −4.73443e9 −0.559930
\(687\) 0 0
\(688\) −9.15179e8 −0.107139
\(689\) 2.98880e8 0.0348120
\(690\) 0 0
\(691\) −6.22407e9 −0.717632 −0.358816 0.933408i \(-0.616819\pi\)
−0.358816 + 0.933408i \(0.616819\pi\)
\(692\) −9.33317e9 −1.07068
\(693\) 0 0
\(694\) 3.10279e9 0.352366
\(695\) 1.06612e10 1.20465
\(696\) 0 0
\(697\) −3.65411e9 −0.408758
\(698\) −5.35270e9 −0.595771
\(699\) 0 0
\(700\) −5.50439e9 −0.606550
\(701\) −8.57820e9 −0.940553 −0.470276 0.882519i \(-0.655846\pi\)
−0.470276 + 0.882519i \(0.655846\pi\)
\(702\) 0 0
\(703\) −4.32579e9 −0.469593
\(704\) 3.69768e9 0.399416
\(705\) 0 0
\(706\) 2.31772e9 0.247881
\(707\) −1.28463e10 −1.36713
\(708\) 0 0
\(709\) 2.72444e9 0.287088 0.143544 0.989644i \(-0.454150\pi\)
0.143544 + 0.989644i \(0.454150\pi\)
\(710\) 6.37932e8 0.0668914
\(711\) 0 0
\(712\) −1.14309e10 −1.18686
\(713\) −2.34496e9 −0.242283
\(714\) 0 0
\(715\) 6.36885e9 0.651613
\(716\) 1.46304e9 0.148957
\(717\) 0 0
\(718\) 9.10952e9 0.918460
\(719\) 2.39574e9 0.240375 0.120188 0.992751i \(-0.461650\pi\)
0.120188 + 0.992751i \(0.461650\pi\)
\(720\) 0 0
\(721\) −1.18695e10 −1.17939
\(722\) 8.14988e9 0.805881
\(723\) 0 0
\(724\) −7.62091e9 −0.746314
\(725\) −1.86614e8 −0.0181870
\(726\) 0 0
\(727\) 1.12427e9 0.108517 0.0542586 0.998527i \(-0.482720\pi\)
0.0542586 + 0.998527i \(0.482720\pi\)
\(728\) −3.34780e9 −0.321588
\(729\) 0 0
\(730\) −1.23922e10 −1.17902
\(731\) −1.60185e9 −0.151674
\(732\) 0 0
\(733\) 1.04255e10 0.977759 0.488880 0.872351i \(-0.337406\pi\)
0.488880 + 0.872351i \(0.337406\pi\)
\(734\) −4.89480e9 −0.456876
\(735\) 0 0
\(736\) −3.24696e9 −0.300196
\(737\) −5.04338e9 −0.464072
\(738\) 0 0
\(739\) −1.61439e8 −0.0147147 −0.00735737 0.999973i \(-0.502342\pi\)
−0.00735737 + 0.999973i \(0.502342\pi\)
\(740\) 3.23145e9 0.293148
\(741\) 0 0
\(742\) 5.10156e8 0.0458447
\(743\) −1.59100e10 −1.42302 −0.711508 0.702678i \(-0.751987\pi\)
−0.711508 + 0.702678i \(0.751987\pi\)
\(744\) 0 0
\(745\) −2.38139e9 −0.211000
\(746\) −7.23070e9 −0.637668
\(747\) 0 0
\(748\) −3.37500e9 −0.294862
\(749\) 7.52341e9 0.654226
\(750\) 0 0
\(751\) 9.63052e9 0.829679 0.414839 0.909895i \(-0.363838\pi\)
0.414839 + 0.909895i \(0.363838\pi\)
\(752\) 4.47934e9 0.384107
\(753\) 0 0
\(754\) −4.74094e7 −0.00402777
\(755\) −1.03432e10 −0.874663
\(756\) 0 0
\(757\) −1.23984e10 −1.03880 −0.519398 0.854532i \(-0.673844\pi\)
−0.519398 + 0.854532i \(0.673844\pi\)
\(758\) −2.61610e9 −0.218178
\(759\) 0 0
\(760\) −2.41906e10 −1.99894
\(761\) 3.84482e9 0.316249 0.158125 0.987419i \(-0.449455\pi\)
0.158125 + 0.987419i \(0.449455\pi\)
\(762\) 0 0
\(763\) −2.11619e10 −1.72472
\(764\) −2.70363e9 −0.219341
\(765\) 0 0
\(766\) −2.15773e8 −0.0173459
\(767\) 6.13478e8 0.0490925
\(768\) 0 0
\(769\) 1.85468e10 1.47071 0.735354 0.677683i \(-0.237016\pi\)
0.735354 + 0.677683i \(0.237016\pi\)
\(770\) 1.08710e10 0.858125
\(771\) 0 0
\(772\) −7.91355e9 −0.619029
\(773\) −9.18489e9 −0.715230 −0.357615 0.933869i \(-0.616410\pi\)
−0.357615 + 0.933869i \(0.616410\pi\)
\(774\) 0 0
\(775\) 9.81163e9 0.757156
\(776\) 3.56220e9 0.273654
\(777\) 0 0
\(778\) 6.83530e9 0.520390
\(779\) −2.60537e10 −1.97464
\(780\) 0 0
\(781\) 1.51921e9 0.114114
\(782\) −6.76196e8 −0.0505649
\(783\) 0 0
\(784\) −3.99123e8 −0.0295801
\(785\) 7.22025e9 0.532732
\(786\) 0 0
\(787\) −1.00821e10 −0.737291 −0.368646 0.929570i \(-0.620178\pi\)
−0.368646 + 0.929570i \(0.620178\pi\)
\(788\) −1.59923e10 −1.16431
\(789\) 0 0
\(790\) 2.61043e9 0.188372
\(791\) −1.22478e10 −0.879916
\(792\) 0 0
\(793\) 2.13467e8 0.0152011
\(794\) −3.01745e9 −0.213928
\(795\) 0 0
\(796\) −8.32975e9 −0.585378
\(797\) 1.03385e10 0.723361 0.361681 0.932302i \(-0.382203\pi\)
0.361681 + 0.932302i \(0.382203\pi\)
\(798\) 0 0
\(799\) 7.84025e9 0.543771
\(800\) 1.35857e10 0.938139
\(801\) 0 0
\(802\) −3.71150e9 −0.254062
\(803\) −2.95117e10 −2.01136
\(804\) 0 0
\(805\) −5.52760e9 −0.373466
\(806\) 2.49265e9 0.167683
\(807\) 0 0
\(808\) 2.00384e10 1.33636
\(809\) −2.64371e10 −1.75547 −0.877735 0.479146i \(-0.840947\pi\)
−0.877735 + 0.479146i \(0.840947\pi\)
\(810\) 0 0
\(811\) 2.22378e10 1.46392 0.731961 0.681346i \(-0.238605\pi\)
0.731961 + 0.681346i \(0.238605\pi\)
\(812\) 2.05372e8 0.0134616
\(813\) 0 0
\(814\) −3.03229e9 −0.197054
\(815\) 3.47530e10 2.24875
\(816\) 0 0
\(817\) −1.14211e10 −0.732710
\(818\) 1.00524e10 0.642147
\(819\) 0 0
\(820\) 1.94626e10 1.23269
\(821\) −1.19117e10 −0.751229 −0.375615 0.926776i \(-0.622568\pi\)
−0.375615 + 0.926776i \(0.622568\pi\)
\(822\) 0 0
\(823\) 1.73151e9 0.108274 0.0541372 0.998534i \(-0.482759\pi\)
0.0541372 + 0.998534i \(0.482759\pi\)
\(824\) 1.85147e10 1.15285
\(825\) 0 0
\(826\) 1.04714e9 0.0646510
\(827\) −2.70946e10 −1.66576 −0.832882 0.553451i \(-0.813311\pi\)
−0.832882 + 0.553451i \(0.813311\pi\)
\(828\) 0 0
\(829\) 1.95311e10 1.19065 0.595326 0.803484i \(-0.297023\pi\)
0.595326 + 0.803484i \(0.297023\pi\)
\(830\) 1.14871e10 0.697331
\(831\) 0 0
\(832\) 1.99859e9 0.120307
\(833\) −6.98589e8 −0.0418759
\(834\) 0 0
\(835\) 2.67255e10 1.58863
\(836\) −2.40637e10 −1.42442
\(837\) 0 0
\(838\) 4.19045e8 0.0245984
\(839\) 2.20343e10 1.28805 0.644024 0.765005i \(-0.277263\pi\)
0.644024 + 0.765005i \(0.277263\pi\)
\(840\) 0 0
\(841\) −1.72429e10 −0.999596
\(842\) −1.21645e10 −0.702268
\(843\) 0 0
\(844\) 1.12562e10 0.644453
\(845\) −2.07665e10 −1.18403
\(846\) 0 0
\(847\) 9.37053e9 0.529873
\(848\) 3.80212e8 0.0214112
\(849\) 0 0
\(850\) 2.82929e9 0.158020
\(851\) 1.54184e9 0.0857604
\(852\) 0 0
\(853\) 2.56385e10 1.41440 0.707199 0.707014i \(-0.249958\pi\)
0.707199 + 0.707014i \(0.249958\pi\)
\(854\) 3.64366e8 0.0200187
\(855\) 0 0
\(856\) −1.17354e10 −0.639502
\(857\) 1.90552e10 1.03414 0.517072 0.855942i \(-0.327022\pi\)
0.517072 + 0.855942i \(0.327022\pi\)
\(858\) 0 0
\(859\) 1.94648e10 1.04779 0.523896 0.851782i \(-0.324478\pi\)
0.523896 + 0.851782i \(0.324478\pi\)
\(860\) 8.53182e9 0.457401
\(861\) 0 0
\(862\) 1.32409e10 0.704115
\(863\) 6.07650e9 0.321822 0.160911 0.986969i \(-0.448557\pi\)
0.160911 + 0.986969i \(0.448557\pi\)
\(864\) 0 0
\(865\) 3.92159e10 2.06018
\(866\) −6.72987e9 −0.352123
\(867\) 0 0
\(868\) −1.07979e10 −0.560428
\(869\) 6.21663e9 0.321356
\(870\) 0 0
\(871\) −2.72594e9 −0.139783
\(872\) 3.30095e10 1.68590
\(873\) 0 0
\(874\) −4.82126e9 −0.244270
\(875\) −2.42092e9 −0.122167
\(876\) 0 0
\(877\) −1.82857e10 −0.915404 −0.457702 0.889106i \(-0.651327\pi\)
−0.457702 + 0.889106i \(0.651327\pi\)
\(878\) 1.94420e9 0.0969415
\(879\) 0 0
\(880\) 8.10198e9 0.400776
\(881\) −1.41923e10 −0.699259 −0.349629 0.936888i \(-0.613692\pi\)
−0.349629 + 0.936888i \(0.613692\pi\)
\(882\) 0 0
\(883\) 1.61031e10 0.787131 0.393566 0.919297i \(-0.371241\pi\)
0.393566 + 0.919297i \(0.371241\pi\)
\(884\) −1.82418e9 −0.0888149
\(885\) 0 0
\(886\) 1.37270e10 0.663068
\(887\) 1.49039e9 0.0717077 0.0358539 0.999357i \(-0.488585\pi\)
0.0358539 + 0.999357i \(0.488585\pi\)
\(888\) 0 0
\(889\) 1.45250e10 0.693363
\(890\) 2.00624e10 0.953933
\(891\) 0 0
\(892\) 5.03884e9 0.237713
\(893\) 5.59007e10 2.62686
\(894\) 0 0
\(895\) −6.14736e9 −0.286621
\(896\) −1.74312e10 −0.809561
\(897\) 0 0
\(898\) −1.74338e10 −0.803385
\(899\) −3.66078e8 −0.0168041
\(900\) 0 0
\(901\) 6.65490e8 0.0303113
\(902\) −1.82631e10 −0.828613
\(903\) 0 0
\(904\) 1.91049e10 0.860111
\(905\) 3.20213e10 1.43605
\(906\) 0 0
\(907\) −1.44651e10 −0.643717 −0.321858 0.946788i \(-0.604307\pi\)
−0.321858 + 0.946788i \(0.604307\pi\)
\(908\) 1.51463e10 0.671436
\(909\) 0 0
\(910\) 5.87574e9 0.258475
\(911\) 2.79187e10 1.22343 0.611717 0.791077i \(-0.290479\pi\)
0.611717 + 0.791077i \(0.290479\pi\)
\(912\) 0 0
\(913\) 2.73562e10 1.18962
\(914\) −9.69353e9 −0.419923
\(915\) 0 0
\(916\) −1.08458e10 −0.466259
\(917\) 1.55024e10 0.663905
\(918\) 0 0
\(919\) 1.89312e10 0.804590 0.402295 0.915510i \(-0.368213\pi\)
0.402295 + 0.915510i \(0.368213\pi\)
\(920\) 8.62228e9 0.365061
\(921\) 0 0
\(922\) −2.99575e9 −0.125877
\(923\) 8.21133e8 0.0343722
\(924\) 0 0
\(925\) −6.45127e9 −0.268009
\(926\) −6.81005e9 −0.281846
\(927\) 0 0
\(928\) −5.06891e8 −0.0208208
\(929\) 3.58709e10 1.46787 0.733934 0.679221i \(-0.237682\pi\)
0.733934 + 0.679221i \(0.237682\pi\)
\(930\) 0 0
\(931\) −4.98092e9 −0.202295
\(932\) −1.03146e10 −0.417348
\(933\) 0 0
\(934\) −1.83082e10 −0.735245
\(935\) 1.41810e10 0.567369
\(936\) 0 0
\(937\) 1.18752e10 0.471577 0.235789 0.971804i \(-0.424233\pi\)
0.235789 + 0.971804i \(0.424233\pi\)
\(938\) −4.65289e9 −0.184083
\(939\) 0 0
\(940\) −4.17590e10 −1.63984
\(941\) 2.99086e10 1.17013 0.585063 0.810988i \(-0.301070\pi\)
0.585063 + 0.810988i \(0.301070\pi\)
\(942\) 0 0
\(943\) 9.28632e9 0.360623
\(944\) 7.80420e8 0.0301944
\(945\) 0 0
\(946\) −8.00599e9 −0.307465
\(947\) 1.97326e10 0.755021 0.377510 0.926005i \(-0.376780\pi\)
0.377510 + 0.926005i \(0.376780\pi\)
\(948\) 0 0
\(949\) −1.59510e10 −0.605838
\(950\) 2.01728e10 0.763366
\(951\) 0 0
\(952\) −7.45427e9 −0.280011
\(953\) −3.05905e10 −1.14488 −0.572442 0.819945i \(-0.694004\pi\)
−0.572442 + 0.819945i \(0.694004\pi\)
\(954\) 0 0
\(955\) 1.13600e10 0.422053
\(956\) 2.72481e10 1.00863
\(957\) 0 0
\(958\) −1.11784e10 −0.410772
\(959\) 4.06157e10 1.48706
\(960\) 0 0
\(961\) −8.26528e9 −0.300418
\(962\) −1.63895e9 −0.0593545
\(963\) 0 0
\(964\) −1.26017e9 −0.0453062
\(965\) 3.32510e10 1.19113
\(966\) 0 0
\(967\) 1.80021e10 0.640221 0.320111 0.947380i \(-0.396280\pi\)
0.320111 + 0.947380i \(0.396280\pi\)
\(968\) −1.46167e10 −0.517947
\(969\) 0 0
\(970\) −6.25203e9 −0.219948
\(971\) 6.56410e9 0.230095 0.115048 0.993360i \(-0.463298\pi\)
0.115048 + 0.993360i \(0.463298\pi\)
\(972\) 0 0
\(973\) 2.34235e10 0.815188
\(974\) −2.38115e9 −0.0825716
\(975\) 0 0
\(976\) 2.71557e8 0.00934947
\(977\) −1.67236e10 −0.573718 −0.286859 0.957973i \(-0.592611\pi\)
−0.286859 + 0.957973i \(0.592611\pi\)
\(978\) 0 0
\(979\) 4.77779e10 1.62737
\(980\) 3.72085e9 0.126285
\(981\) 0 0
\(982\) −5.17655e9 −0.174442
\(983\) 6.60882e9 0.221915 0.110957 0.993825i \(-0.464608\pi\)
0.110957 + 0.993825i \(0.464608\pi\)
\(984\) 0 0
\(985\) 6.71959e10 2.24035
\(986\) −1.05563e8 −0.00350704
\(987\) 0 0
\(988\) −1.30064e10 −0.429049
\(989\) 4.07084e9 0.133813
\(990\) 0 0
\(991\) −2.45768e10 −0.802173 −0.401086 0.916040i \(-0.631367\pi\)
−0.401086 + 0.916040i \(0.631367\pi\)
\(992\) 2.66509e10 0.866803
\(993\) 0 0
\(994\) 1.40159e9 0.0452655
\(995\) 3.49997e10 1.12638
\(996\) 0 0
\(997\) 6.18690e10 1.97715 0.988575 0.150727i \(-0.0481614\pi\)
0.988575 + 0.150727i \(0.0481614\pi\)
\(998\) 1.75830e10 0.559932
\(999\) 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.1.10 17
3.2 odd 2 177.8.a.b.1.8 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.8 17 3.2 odd 2
531.8.a.d.1.10 17 1.1 even 1 trivial