Properties

Label 531.8.a.d.1.4
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 1.4
Root \(15.6681\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.6681 q^{2} +58.8158 q^{4} +207.404 q^{5} +882.793 q^{7} +945.614 q^{8} +O(q^{10})\) \(q-13.6681 q^{2} +58.8158 q^{4} +207.404 q^{5} +882.793 q^{7} +945.614 q^{8} -2834.81 q^{10} +3637.21 q^{11} +208.023 q^{13} -12066.1 q^{14} -20453.1 q^{16} +13120.6 q^{17} -9914.89 q^{19} +12198.6 q^{20} -49713.6 q^{22} +37757.5 q^{23} -35108.6 q^{25} -2843.27 q^{26} +51922.2 q^{28} +98092.3 q^{29} +326883. q^{31} +158516. q^{32} -179333. q^{34} +183095. q^{35} +39434.0 q^{37} +135517. q^{38} +196124. q^{40} +719442. q^{41} -878099. q^{43} +213925. q^{44} -516072. q^{46} +158129. q^{47} -44219.7 q^{49} +479867. q^{50} +12235.0 q^{52} +1.83479e6 q^{53} +754372. q^{55} +834781. q^{56} -1.34073e6 q^{58} +205379. q^{59} +3.49764e6 q^{61} -4.46786e6 q^{62} +451395. q^{64} +43144.8 q^{65} -1.00103e6 q^{67} +771700. q^{68} -2.50255e6 q^{70} -331392. q^{71} -527191. q^{73} -538987. q^{74} -583152. q^{76} +3.21090e6 q^{77} -1.50960e6 q^{79} -4.24206e6 q^{80} -9.83337e6 q^{82} -2.48775e6 q^{83} +2.72127e6 q^{85} +1.20019e7 q^{86} +3.43940e6 q^{88} -6.37807e6 q^{89} +183641. q^{91} +2.22074e6 q^{92} -2.16132e6 q^{94} -2.05639e6 q^{95} -4.33428e6 q^{97} +604398. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.6681 −1.20810 −0.604049 0.796948i \(-0.706447\pi\)
−0.604049 + 0.796948i \(0.706447\pi\)
\(3\) 0 0
\(4\) 58.8158 0.459498
\(5\) 207.404 0.742031 0.371015 0.928627i \(-0.379010\pi\)
0.371015 + 0.928627i \(0.379010\pi\)
\(6\) 0 0
\(7\) 882.793 0.972782 0.486391 0.873741i \(-0.338313\pi\)
0.486391 + 0.873741i \(0.338313\pi\)
\(8\) 945.614 0.652978
\(9\) 0 0
\(10\) −2834.81 −0.896445
\(11\) 3637.21 0.823937 0.411969 0.911198i \(-0.364841\pi\)
0.411969 + 0.911198i \(0.364841\pi\)
\(12\) 0 0
\(13\) 208.023 0.0262609 0.0131305 0.999914i \(-0.495820\pi\)
0.0131305 + 0.999914i \(0.495820\pi\)
\(14\) −12066.1 −1.17522
\(15\) 0 0
\(16\) −20453.1 −1.24836
\(17\) 13120.6 0.647714 0.323857 0.946106i \(-0.395020\pi\)
0.323857 + 0.946106i \(0.395020\pi\)
\(18\) 0 0
\(19\) −9914.89 −0.331627 −0.165814 0.986157i \(-0.553025\pi\)
−0.165814 + 0.986157i \(0.553025\pi\)
\(20\) 12198.6 0.340962
\(21\) 0 0
\(22\) −49713.6 −0.995396
\(23\) 37757.5 0.647077 0.323538 0.946215i \(-0.395128\pi\)
0.323538 + 0.946215i \(0.395128\pi\)
\(24\) 0 0
\(25\) −35108.6 −0.449391
\(26\) −2843.27 −0.0317257
\(27\) 0 0
\(28\) 51922.2 0.446992
\(29\) 98092.3 0.746865 0.373432 0.927657i \(-0.378181\pi\)
0.373432 + 0.927657i \(0.378181\pi\)
\(30\) 0 0
\(31\) 326883. 1.97073 0.985364 0.170463i \(-0.0545265\pi\)
0.985364 + 0.170463i \(0.0545265\pi\)
\(32\) 158516. 0.855161
\(33\) 0 0
\(34\) −179333. −0.782501
\(35\) 183095. 0.721834
\(36\) 0 0
\(37\) 39434.0 0.127987 0.0639934 0.997950i \(-0.479616\pi\)
0.0639934 + 0.997950i \(0.479616\pi\)
\(38\) 135517. 0.400638
\(39\) 0 0
\(40\) 196124. 0.484530
\(41\) 719442. 1.63024 0.815121 0.579291i \(-0.196670\pi\)
0.815121 + 0.579291i \(0.196670\pi\)
\(42\) 0 0
\(43\) −878099. −1.68424 −0.842120 0.539290i \(-0.818693\pi\)
−0.842120 + 0.539290i \(0.818693\pi\)
\(44\) 213925. 0.378598
\(45\) 0 0
\(46\) −516072. −0.781731
\(47\) 158129. 0.222162 0.111081 0.993811i \(-0.464569\pi\)
0.111081 + 0.993811i \(0.464569\pi\)
\(48\) 0 0
\(49\) −44219.7 −0.0536945
\(50\) 479867. 0.542907
\(51\) 0 0
\(52\) 12235.0 0.0120669
\(53\) 1.83479e6 1.69286 0.846429 0.532502i \(-0.178748\pi\)
0.846429 + 0.532502i \(0.178748\pi\)
\(54\) 0 0
\(55\) 754372. 0.611387
\(56\) 834781. 0.635206
\(57\) 0 0
\(58\) −1.34073e6 −0.902285
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 3.49764e6 1.97297 0.986485 0.163851i \(-0.0523917\pi\)
0.986485 + 0.163851i \(0.0523917\pi\)
\(62\) −4.46786e6 −2.38083
\(63\) 0 0
\(64\) 451395. 0.215242
\(65\) 43144.8 0.0194864
\(66\) 0 0
\(67\) −1.00103e6 −0.406615 −0.203308 0.979115i \(-0.565169\pi\)
−0.203308 + 0.979115i \(0.565169\pi\)
\(68\) 771700. 0.297623
\(69\) 0 0
\(70\) −2.50255e6 −0.872046
\(71\) −331392. −0.109885 −0.0549423 0.998490i \(-0.517498\pi\)
−0.0549423 + 0.998490i \(0.517498\pi\)
\(72\) 0 0
\(73\) −527191. −0.158613 −0.0793063 0.996850i \(-0.525271\pi\)
−0.0793063 + 0.996850i \(0.525271\pi\)
\(74\) −538987. −0.154620
\(75\) 0 0
\(76\) −583152. −0.152382
\(77\) 3.21090e6 0.801512
\(78\) 0 0
\(79\) −1.50960e6 −0.344481 −0.172241 0.985055i \(-0.555101\pi\)
−0.172241 + 0.985055i \(0.555101\pi\)
\(80\) −4.24206e6 −0.926321
\(81\) 0 0
\(82\) −9.83337e6 −1.96949
\(83\) −2.48775e6 −0.477566 −0.238783 0.971073i \(-0.576748\pi\)
−0.238783 + 0.971073i \(0.576748\pi\)
\(84\) 0 0
\(85\) 2.72127e6 0.480623
\(86\) 1.20019e7 2.03472
\(87\) 0 0
\(88\) 3.43940e6 0.538013
\(89\) −6.37807e6 −0.959013 −0.479506 0.877538i \(-0.659184\pi\)
−0.479506 + 0.877538i \(0.659184\pi\)
\(90\) 0 0
\(91\) 183641. 0.0255462
\(92\) 2.22074e6 0.297331
\(93\) 0 0
\(94\) −2.16132e6 −0.268393
\(95\) −2.05639e6 −0.246078
\(96\) 0 0
\(97\) −4.33428e6 −0.482187 −0.241094 0.970502i \(-0.577506\pi\)
−0.241094 + 0.970502i \(0.577506\pi\)
\(98\) 604398. 0.0648682
\(99\) 0 0
\(100\) −2.06494e6 −0.206494
\(101\) −6.87221e6 −0.663700 −0.331850 0.943332i \(-0.607673\pi\)
−0.331850 + 0.943332i \(0.607673\pi\)
\(102\) 0 0
\(103\) 1.71578e6 0.154715 0.0773574 0.997003i \(-0.475352\pi\)
0.0773574 + 0.997003i \(0.475352\pi\)
\(104\) 196710. 0.0171478
\(105\) 0 0
\(106\) −2.50780e7 −2.04514
\(107\) 7.08886e6 0.559414 0.279707 0.960085i \(-0.409763\pi\)
0.279707 + 0.960085i \(0.409763\pi\)
\(108\) 0 0
\(109\) 2.29535e7 1.69768 0.848840 0.528650i \(-0.177301\pi\)
0.848840 + 0.528650i \(0.177301\pi\)
\(110\) −1.03108e7 −0.738614
\(111\) 0 0
\(112\) −1.80559e7 −1.21438
\(113\) 1.69616e7 1.10584 0.552919 0.833235i \(-0.313514\pi\)
0.552919 + 0.833235i \(0.313514\pi\)
\(114\) 0 0
\(115\) 7.83105e6 0.480151
\(116\) 5.76938e6 0.343183
\(117\) 0 0
\(118\) −2.80713e6 −0.157281
\(119\) 1.15828e7 0.630084
\(120\) 0 0
\(121\) −6.25787e6 −0.321127
\(122\) −4.78059e7 −2.38354
\(123\) 0 0
\(124\) 1.92259e7 0.905546
\(125\) −2.34851e7 −1.07549
\(126\) 0 0
\(127\) −1.24664e7 −0.540044 −0.270022 0.962854i \(-0.587031\pi\)
−0.270022 + 0.962854i \(0.587031\pi\)
\(128\) −2.64597e7 −1.11519
\(129\) 0 0
\(130\) −589706. −0.0235415
\(131\) −4.68201e7 −1.81963 −0.909815 0.415015i \(-0.863776\pi\)
−0.909815 + 0.415015i \(0.863776\pi\)
\(132\) 0 0
\(133\) −8.75279e6 −0.322601
\(134\) 1.36821e7 0.491231
\(135\) 0 0
\(136\) 1.24070e7 0.422943
\(137\) 1.03397e7 0.343545 0.171773 0.985137i \(-0.445051\pi\)
0.171773 + 0.985137i \(0.445051\pi\)
\(138\) 0 0
\(139\) −3.07437e7 −0.970965 −0.485483 0.874246i \(-0.661356\pi\)
−0.485483 + 0.874246i \(0.661356\pi\)
\(140\) 1.07689e7 0.331682
\(141\) 0 0
\(142\) 4.52948e6 0.132751
\(143\) 756624. 0.0216374
\(144\) 0 0
\(145\) 2.03447e7 0.554196
\(146\) 7.20567e6 0.191619
\(147\) 0 0
\(148\) 2.31934e6 0.0588097
\(149\) −3.66220e6 −0.0906963 −0.0453482 0.998971i \(-0.514440\pi\)
−0.0453482 + 0.998971i \(0.514440\pi\)
\(150\) 0 0
\(151\) 7.28781e7 1.72257 0.861286 0.508120i \(-0.169659\pi\)
0.861286 + 0.508120i \(0.169659\pi\)
\(152\) −9.37565e6 −0.216545
\(153\) 0 0
\(154\) −4.38868e7 −0.968304
\(155\) 6.77968e7 1.46234
\(156\) 0 0
\(157\) −9.09496e7 −1.87565 −0.937826 0.347106i \(-0.887164\pi\)
−0.937826 + 0.347106i \(0.887164\pi\)
\(158\) 2.06332e7 0.416167
\(159\) 0 0
\(160\) 3.28768e7 0.634556
\(161\) 3.33320e7 0.629465
\(162\) 0 0
\(163\) 5.62746e7 1.01778 0.508892 0.860830i \(-0.330055\pi\)
0.508892 + 0.860830i \(0.330055\pi\)
\(164\) 4.23145e7 0.749094
\(165\) 0 0
\(166\) 3.40027e7 0.576945
\(167\) −3.21050e7 −0.533415 −0.266707 0.963778i \(-0.585936\pi\)
−0.266707 + 0.963778i \(0.585936\pi\)
\(168\) 0 0
\(169\) −6.27052e7 −0.999310
\(170\) −3.71944e7 −0.580640
\(171\) 0 0
\(172\) −5.16461e7 −0.773905
\(173\) −7.85878e7 −1.15397 −0.576984 0.816756i \(-0.695770\pi\)
−0.576984 + 0.816756i \(0.695770\pi\)
\(174\) 0 0
\(175\) −3.09937e7 −0.437159
\(176\) −7.43923e7 −1.02857
\(177\) 0 0
\(178\) 8.71759e7 1.15858
\(179\) 5.87666e7 0.765853 0.382926 0.923779i \(-0.374916\pi\)
0.382926 + 0.923779i \(0.374916\pi\)
\(180\) 0 0
\(181\) −4.81650e7 −0.603749 −0.301875 0.953348i \(-0.597612\pi\)
−0.301875 + 0.953348i \(0.597612\pi\)
\(182\) −2.51002e6 −0.0308622
\(183\) 0 0
\(184\) 3.57040e7 0.422527
\(185\) 8.17877e6 0.0949701
\(186\) 0 0
\(187\) 4.77225e7 0.533675
\(188\) 9.30049e6 0.102083
\(189\) 0 0
\(190\) 2.81068e7 0.297286
\(191\) −1.65487e8 −1.71849 −0.859247 0.511560i \(-0.829068\pi\)
−0.859247 + 0.511560i \(0.829068\pi\)
\(192\) 0 0
\(193\) −1.11360e8 −1.11501 −0.557504 0.830174i \(-0.688241\pi\)
−0.557504 + 0.830174i \(0.688241\pi\)
\(194\) 5.92412e7 0.582529
\(195\) 0 0
\(196\) −2.60082e6 −0.0246725
\(197\) −3.64560e7 −0.339733 −0.169866 0.985467i \(-0.554334\pi\)
−0.169866 + 0.985467i \(0.554334\pi\)
\(198\) 0 0
\(199\) 7.61265e7 0.684779 0.342389 0.939558i \(-0.388764\pi\)
0.342389 + 0.939558i \(0.388764\pi\)
\(200\) −3.31992e7 −0.293442
\(201\) 0 0
\(202\) 9.39298e7 0.801814
\(203\) 8.65952e7 0.726537
\(204\) 0 0
\(205\) 1.49215e8 1.20969
\(206\) −2.34514e7 −0.186911
\(207\) 0 0
\(208\) −4.25472e6 −0.0327831
\(209\) −3.60625e7 −0.273240
\(210\) 0 0
\(211\) 2.35187e8 1.72355 0.861775 0.507291i \(-0.169353\pi\)
0.861775 + 0.507291i \(0.169353\pi\)
\(212\) 1.07915e8 0.777866
\(213\) 0 0
\(214\) −9.68910e7 −0.675826
\(215\) −1.82121e8 −1.24976
\(216\) 0 0
\(217\) 2.88570e8 1.91709
\(218\) −3.13730e8 −2.05096
\(219\) 0 0
\(220\) 4.43690e7 0.280931
\(221\) 2.72939e6 0.0170096
\(222\) 0 0
\(223\) 1.57494e8 0.951036 0.475518 0.879706i \(-0.342261\pi\)
0.475518 + 0.879706i \(0.342261\pi\)
\(224\) 1.39937e8 0.831886
\(225\) 0 0
\(226\) −2.31832e8 −1.33596
\(227\) 4.25579e7 0.241485 0.120742 0.992684i \(-0.461473\pi\)
0.120742 + 0.992684i \(0.461473\pi\)
\(228\) 0 0
\(229\) −5.07210e7 −0.279103 −0.139551 0.990215i \(-0.544566\pi\)
−0.139551 + 0.990215i \(0.544566\pi\)
\(230\) −1.07035e8 −0.580069
\(231\) 0 0
\(232\) 9.27574e7 0.487686
\(233\) 2.44954e8 1.26864 0.634321 0.773069i \(-0.281280\pi\)
0.634321 + 0.773069i \(0.281280\pi\)
\(234\) 0 0
\(235\) 3.27966e7 0.164851
\(236\) 1.20795e7 0.0598216
\(237\) 0 0
\(238\) −1.58314e8 −0.761203
\(239\) 3.38900e6 0.0160575 0.00802877 0.999968i \(-0.497444\pi\)
0.00802877 + 0.999968i \(0.497444\pi\)
\(240\) 0 0
\(241\) 2.42976e8 1.11816 0.559080 0.829114i \(-0.311155\pi\)
0.559080 + 0.829114i \(0.311155\pi\)
\(242\) 8.55329e7 0.387953
\(243\) 0 0
\(244\) 2.05716e8 0.906577
\(245\) −9.17134e6 −0.0398430
\(246\) 0 0
\(247\) −2.06253e6 −0.00870884
\(248\) 3.09105e8 1.28684
\(249\) 0 0
\(250\) 3.20996e8 1.29930
\(251\) 1.00242e7 0.0400120 0.0200060 0.999800i \(-0.493631\pi\)
0.0200060 + 0.999800i \(0.493631\pi\)
\(252\) 0 0
\(253\) 1.37332e8 0.533150
\(254\) 1.70392e8 0.652425
\(255\) 0 0
\(256\) 3.03875e8 1.13202
\(257\) 1.65390e8 0.607774 0.303887 0.952708i \(-0.401715\pi\)
0.303887 + 0.952708i \(0.401715\pi\)
\(258\) 0 0
\(259\) 3.48121e7 0.124503
\(260\) 2.53760e6 0.00895397
\(261\) 0 0
\(262\) 6.39940e8 2.19829
\(263\) 3.86525e8 1.31019 0.655093 0.755549i \(-0.272630\pi\)
0.655093 + 0.755549i \(0.272630\pi\)
\(264\) 0 0
\(265\) 3.80542e8 1.25615
\(266\) 1.19634e8 0.389734
\(267\) 0 0
\(268\) −5.88762e7 −0.186839
\(269\) −2.13282e8 −0.668070 −0.334035 0.942561i \(-0.608410\pi\)
−0.334035 + 0.942561i \(0.608410\pi\)
\(270\) 0 0
\(271\) −5.08751e7 −0.155279 −0.0776396 0.996981i \(-0.524738\pi\)
−0.0776396 + 0.996981i \(0.524738\pi\)
\(272\) −2.68358e8 −0.808580
\(273\) 0 0
\(274\) −1.41323e8 −0.415036
\(275\) −1.27698e8 −0.370270
\(276\) 0 0
\(277\) −4.14639e8 −1.17217 −0.586086 0.810249i \(-0.699332\pi\)
−0.586086 + 0.810249i \(0.699332\pi\)
\(278\) 4.20206e8 1.17302
\(279\) 0 0
\(280\) 1.73137e8 0.471342
\(281\) 3.84206e8 1.03298 0.516491 0.856293i \(-0.327238\pi\)
0.516491 + 0.856293i \(0.327238\pi\)
\(282\) 0 0
\(283\) 4.66433e8 1.22331 0.611656 0.791124i \(-0.290504\pi\)
0.611656 + 0.791124i \(0.290504\pi\)
\(284\) −1.94911e7 −0.0504918
\(285\) 0 0
\(286\) −1.03416e7 −0.0261400
\(287\) 6.35118e8 1.58587
\(288\) 0 0
\(289\) −2.38188e8 −0.580467
\(290\) −2.78073e8 −0.669523
\(291\) 0 0
\(292\) −3.10071e7 −0.0728823
\(293\) 7.57454e8 1.75922 0.879609 0.475697i \(-0.157804\pi\)
0.879609 + 0.475697i \(0.157804\pi\)
\(294\) 0 0
\(295\) 4.25964e7 0.0966042
\(296\) 3.72894e7 0.0835726
\(297\) 0 0
\(298\) 5.00551e7 0.109570
\(299\) 7.85443e6 0.0169928
\(300\) 0 0
\(301\) −7.75180e8 −1.63840
\(302\) −9.96101e8 −2.08103
\(303\) 0 0
\(304\) 2.02790e8 0.413990
\(305\) 7.25424e8 1.46400
\(306\) 0 0
\(307\) −5.11428e8 −1.00879 −0.504394 0.863473i \(-0.668284\pi\)
−0.504394 + 0.863473i \(0.668284\pi\)
\(308\) 1.88852e8 0.368293
\(309\) 0 0
\(310\) −9.26651e8 −1.76665
\(311\) −4.47095e8 −0.842826 −0.421413 0.906869i \(-0.638466\pi\)
−0.421413 + 0.906869i \(0.638466\pi\)
\(312\) 0 0
\(313\) −3.05704e7 −0.0563503 −0.0281751 0.999603i \(-0.508970\pi\)
−0.0281751 + 0.999603i \(0.508970\pi\)
\(314\) 1.24310e9 2.26597
\(315\) 0 0
\(316\) −8.87881e7 −0.158289
\(317\) 2.57462e8 0.453948 0.226974 0.973901i \(-0.427117\pi\)
0.226974 + 0.973901i \(0.427117\pi\)
\(318\) 0 0
\(319\) 3.56782e8 0.615370
\(320\) 9.36211e7 0.159716
\(321\) 0 0
\(322\) −4.55584e8 −0.760454
\(323\) −1.30089e8 −0.214800
\(324\) 0 0
\(325\) −7.30341e6 −0.0118014
\(326\) −7.69164e8 −1.22958
\(327\) 0 0
\(328\) 6.80314e8 1.06451
\(329\) 1.39595e8 0.216115
\(330\) 0 0
\(331\) 6.68825e8 1.01371 0.506856 0.862031i \(-0.330808\pi\)
0.506856 + 0.862031i \(0.330808\pi\)
\(332\) −1.46319e8 −0.219441
\(333\) 0 0
\(334\) 4.38813e8 0.644417
\(335\) −2.07617e8 −0.301721
\(336\) 0 0
\(337\) 1.47419e7 0.0209821 0.0104911 0.999945i \(-0.496661\pi\)
0.0104911 + 0.999945i \(0.496661\pi\)
\(338\) 8.57059e8 1.20726
\(339\) 0 0
\(340\) 1.60053e8 0.220846
\(341\) 1.18894e9 1.62376
\(342\) 0 0
\(343\) −7.66055e8 −1.02502
\(344\) −8.30342e8 −1.09977
\(345\) 0 0
\(346\) 1.07414e9 1.39410
\(347\) 6.05234e8 0.777625 0.388812 0.921317i \(-0.372885\pi\)
0.388812 + 0.921317i \(0.372885\pi\)
\(348\) 0 0
\(349\) −1.02679e9 −1.29298 −0.646491 0.762922i \(-0.723764\pi\)
−0.646491 + 0.762922i \(0.723764\pi\)
\(350\) 4.23623e8 0.528131
\(351\) 0 0
\(352\) 5.76556e8 0.704599
\(353\) 1.41214e9 1.70870 0.854352 0.519695i \(-0.173954\pi\)
0.854352 + 0.519695i \(0.173954\pi\)
\(354\) 0 0
\(355\) −6.87319e7 −0.0815378
\(356\) −3.75131e8 −0.440665
\(357\) 0 0
\(358\) −8.03226e8 −0.925224
\(359\) 2.89775e8 0.330545 0.165272 0.986248i \(-0.447150\pi\)
0.165272 + 0.986248i \(0.447150\pi\)
\(360\) 0 0
\(361\) −7.95567e8 −0.890023
\(362\) 6.58322e8 0.729388
\(363\) 0 0
\(364\) 1.08010e7 0.0117384
\(365\) −1.09341e8 −0.117695
\(366\) 0 0
\(367\) −6.11428e8 −0.645676 −0.322838 0.946454i \(-0.604637\pi\)
−0.322838 + 0.946454i \(0.604637\pi\)
\(368\) −7.72259e8 −0.807784
\(369\) 0 0
\(370\) −1.11788e8 −0.114733
\(371\) 1.61974e9 1.64678
\(372\) 0 0
\(373\) 6.09388e7 0.0608013 0.0304006 0.999538i \(-0.490322\pi\)
0.0304006 + 0.999538i \(0.490322\pi\)
\(374\) −6.52273e8 −0.644732
\(375\) 0 0
\(376\) 1.49529e8 0.145067
\(377\) 2.04055e7 0.0196134
\(378\) 0 0
\(379\) −9.94919e8 −0.938751 −0.469376 0.882999i \(-0.655521\pi\)
−0.469376 + 0.882999i \(0.655521\pi\)
\(380\) −1.20948e8 −0.113072
\(381\) 0 0
\(382\) 2.26189e9 2.07611
\(383\) 1.69498e8 0.154159 0.0770794 0.997025i \(-0.475440\pi\)
0.0770794 + 0.997025i \(0.475440\pi\)
\(384\) 0 0
\(385\) 6.65954e8 0.594746
\(386\) 1.52207e9 1.34704
\(387\) 0 0
\(388\) −2.54924e8 −0.221564
\(389\) 1.92883e8 0.166139 0.0830694 0.996544i \(-0.473528\pi\)
0.0830694 + 0.996544i \(0.473528\pi\)
\(390\) 0 0
\(391\) 4.95402e8 0.419120
\(392\) −4.18148e7 −0.0350613
\(393\) 0 0
\(394\) 4.98283e8 0.410430
\(395\) −3.13096e8 −0.255616
\(396\) 0 0
\(397\) 1.91282e9 1.53429 0.767146 0.641472i \(-0.221676\pi\)
0.767146 + 0.641472i \(0.221676\pi\)
\(398\) −1.04050e9 −0.827279
\(399\) 0 0
\(400\) 7.18081e8 0.561001
\(401\) −2.01241e8 −0.155852 −0.0779258 0.996959i \(-0.524830\pi\)
−0.0779258 + 0.996959i \(0.524830\pi\)
\(402\) 0 0
\(403\) 6.79993e7 0.0517531
\(404\) −4.04195e8 −0.304969
\(405\) 0 0
\(406\) −1.18359e9 −0.877727
\(407\) 1.43430e8 0.105453
\(408\) 0 0
\(409\) 3.51133e8 0.253770 0.126885 0.991917i \(-0.459502\pi\)
0.126885 + 0.991917i \(0.459502\pi\)
\(410\) −2.03948e9 −1.46142
\(411\) 0 0
\(412\) 1.00915e8 0.0710912
\(413\) 1.81307e8 0.126645
\(414\) 0 0
\(415\) −5.15968e8 −0.354368
\(416\) 3.29750e7 0.0224573
\(417\) 0 0
\(418\) 4.92905e8 0.330101
\(419\) −3.67310e8 −0.243940 −0.121970 0.992534i \(-0.538921\pi\)
−0.121970 + 0.992534i \(0.538921\pi\)
\(420\) 0 0
\(421\) 1.20983e9 0.790202 0.395101 0.918638i \(-0.370710\pi\)
0.395101 + 0.918638i \(0.370710\pi\)
\(422\) −3.21454e9 −2.08222
\(423\) 0 0
\(424\) 1.73500e9 1.10540
\(425\) −4.60647e8 −0.291076
\(426\) 0 0
\(427\) 3.08769e9 1.91927
\(428\) 4.16937e8 0.257050
\(429\) 0 0
\(430\) 2.48924e9 1.50983
\(431\) −6.04724e8 −0.363820 −0.181910 0.983315i \(-0.558228\pi\)
−0.181910 + 0.983315i \(0.558228\pi\)
\(432\) 0 0
\(433\) −8.36784e8 −0.495343 −0.247671 0.968844i \(-0.579665\pi\)
−0.247671 + 0.968844i \(0.579665\pi\)
\(434\) −3.94419e9 −2.31603
\(435\) 0 0
\(436\) 1.35003e9 0.780081
\(437\) −3.74361e8 −0.214588
\(438\) 0 0
\(439\) −2.70885e9 −1.52813 −0.764064 0.645140i \(-0.776799\pi\)
−0.764064 + 0.645140i \(0.776799\pi\)
\(440\) 7.13344e8 0.399222
\(441\) 0 0
\(442\) −3.73055e7 −0.0205492
\(443\) 9.58898e8 0.524034 0.262017 0.965063i \(-0.415612\pi\)
0.262017 + 0.965063i \(0.415612\pi\)
\(444\) 0 0
\(445\) −1.32284e9 −0.711617
\(446\) −2.15264e9 −1.14894
\(447\) 0 0
\(448\) 3.98488e8 0.209384
\(449\) −1.53949e9 −0.802630 −0.401315 0.915940i \(-0.631447\pi\)
−0.401315 + 0.915940i \(0.631447\pi\)
\(450\) 0 0
\(451\) 2.61676e9 1.34322
\(452\) 9.97607e8 0.508130
\(453\) 0 0
\(454\) −5.81683e8 −0.291737
\(455\) 3.80879e7 0.0189560
\(456\) 0 0
\(457\) −3.33563e9 −1.63483 −0.817413 0.576052i \(-0.804592\pi\)
−0.817413 + 0.576052i \(0.804592\pi\)
\(458\) 6.93258e8 0.337183
\(459\) 0 0
\(460\) 4.60589e8 0.220628
\(461\) −2.62350e9 −1.24718 −0.623589 0.781753i \(-0.714326\pi\)
−0.623589 + 0.781753i \(0.714326\pi\)
\(462\) 0 0
\(463\) −6.85295e8 −0.320881 −0.160441 0.987045i \(-0.551292\pi\)
−0.160441 + 0.987045i \(0.551292\pi\)
\(464\) −2.00629e9 −0.932356
\(465\) 0 0
\(466\) −3.34805e9 −1.53264
\(467\) 3.13987e9 1.42660 0.713301 0.700858i \(-0.247199\pi\)
0.713301 + 0.700858i \(0.247199\pi\)
\(468\) 0 0
\(469\) −8.83699e8 −0.395548
\(470\) −4.48266e8 −0.199156
\(471\) 0 0
\(472\) 1.94209e8 0.0850105
\(473\) −3.19383e9 −1.38771
\(474\) 0 0
\(475\) 3.48098e8 0.149030
\(476\) 6.81251e8 0.289523
\(477\) 0 0
\(478\) −4.63211e7 −0.0193991
\(479\) −1.33909e9 −0.556718 −0.278359 0.960477i \(-0.589790\pi\)
−0.278359 + 0.960477i \(0.589790\pi\)
\(480\) 0 0
\(481\) 8.20319e6 0.00336105
\(482\) −3.32101e9 −1.35084
\(483\) 0 0
\(484\) −3.68061e8 −0.147558
\(485\) −8.98946e8 −0.357798
\(486\) 0 0
\(487\) −3.02031e9 −1.18495 −0.592476 0.805588i \(-0.701849\pi\)
−0.592476 + 0.805588i \(0.701849\pi\)
\(488\) 3.30741e9 1.28831
\(489\) 0 0
\(490\) 1.25354e8 0.0481342
\(491\) 2.38195e9 0.908129 0.454065 0.890969i \(-0.349974\pi\)
0.454065 + 0.890969i \(0.349974\pi\)
\(492\) 0 0
\(493\) 1.28703e9 0.483754
\(494\) 2.81907e7 0.0105211
\(495\) 0 0
\(496\) −6.68578e9 −2.46018
\(497\) −2.92550e8 −0.106894
\(498\) 0 0
\(499\) 2.51861e9 0.907421 0.453711 0.891149i \(-0.350100\pi\)
0.453711 + 0.891149i \(0.350100\pi\)
\(500\) −1.38129e9 −0.494187
\(501\) 0 0
\(502\) −1.37011e8 −0.0483384
\(503\) −8.04366e8 −0.281816 −0.140908 0.990023i \(-0.545002\pi\)
−0.140908 + 0.990023i \(0.545002\pi\)
\(504\) 0 0
\(505\) −1.42532e9 −0.492486
\(506\) −1.87706e9 −0.644098
\(507\) 0 0
\(508\) −7.33223e8 −0.248149
\(509\) 5.19329e9 1.74554 0.872772 0.488128i \(-0.162320\pi\)
0.872772 + 0.488128i \(0.162320\pi\)
\(510\) 0 0
\(511\) −4.65400e8 −0.154296
\(512\) −7.66530e8 −0.252397
\(513\) 0 0
\(514\) −2.26055e9 −0.734250
\(515\) 3.55860e8 0.114803
\(516\) 0 0
\(517\) 5.75149e8 0.183047
\(518\) −4.75814e8 −0.150412
\(519\) 0 0
\(520\) 4.07983e7 0.0127242
\(521\) −4.19746e9 −1.30033 −0.650166 0.759792i \(-0.725301\pi\)
−0.650166 + 0.759792i \(0.725301\pi\)
\(522\) 0 0
\(523\) −3.75242e9 −1.14698 −0.573490 0.819213i \(-0.694411\pi\)
−0.573490 + 0.819213i \(0.694411\pi\)
\(524\) −2.75376e9 −0.836117
\(525\) 0 0
\(526\) −5.28305e9 −1.58283
\(527\) 4.28891e9 1.27647
\(528\) 0 0
\(529\) −1.97920e9 −0.581292
\(530\) −5.20127e9 −1.51755
\(531\) 0 0
\(532\) −5.14803e8 −0.148235
\(533\) 1.49661e8 0.0428117
\(534\) 0 0
\(535\) 1.47026e9 0.415102
\(536\) −9.46585e8 −0.265511
\(537\) 0 0
\(538\) 2.91516e9 0.807093
\(539\) −1.60837e8 −0.0442409
\(540\) 0 0
\(541\) −1.64171e9 −0.445765 −0.222882 0.974845i \(-0.571547\pi\)
−0.222882 + 0.974845i \(0.571547\pi\)
\(542\) 6.95364e8 0.187592
\(543\) 0 0
\(544\) 2.07983e9 0.553900
\(545\) 4.76064e9 1.25973
\(546\) 0 0
\(547\) 6.20746e9 1.62166 0.810828 0.585285i \(-0.199017\pi\)
0.810828 + 0.585285i \(0.199017\pi\)
\(548\) 6.08135e8 0.157858
\(549\) 0 0
\(550\) 1.74538e9 0.447322
\(551\) −9.72574e8 −0.247681
\(552\) 0 0
\(553\) −1.33266e9 −0.335106
\(554\) 5.66732e9 1.41610
\(555\) 0 0
\(556\) −1.80821e9 −0.446157
\(557\) −6.02362e9 −1.47694 −0.738472 0.674284i \(-0.764452\pi\)
−0.738472 + 0.674284i \(0.764452\pi\)
\(558\) 0 0
\(559\) −1.82665e8 −0.0442297
\(560\) −3.74486e9 −0.901109
\(561\) 0 0
\(562\) −5.25136e9 −1.24794
\(563\) 3.76918e9 0.890159 0.445079 0.895491i \(-0.353175\pi\)
0.445079 + 0.895491i \(0.353175\pi\)
\(564\) 0 0
\(565\) 3.51789e9 0.820565
\(566\) −6.37524e9 −1.47788
\(567\) 0 0
\(568\) −3.13368e8 −0.0717523
\(569\) 5.40701e9 1.23045 0.615226 0.788351i \(-0.289065\pi\)
0.615226 + 0.788351i \(0.289065\pi\)
\(570\) 0 0
\(571\) 3.44502e7 0.00774399 0.00387200 0.999993i \(-0.498768\pi\)
0.00387200 + 0.999993i \(0.498768\pi\)
\(572\) 4.45015e7 0.00994233
\(573\) 0 0
\(574\) −8.68083e9 −1.91589
\(575\) −1.32561e9 −0.290790
\(576\) 0 0
\(577\) 1.54181e8 0.0334131 0.0167065 0.999860i \(-0.494682\pi\)
0.0167065 + 0.999860i \(0.494682\pi\)
\(578\) 3.25557e9 0.701261
\(579\) 0 0
\(580\) 1.19659e9 0.254652
\(581\) −2.19617e9 −0.464567
\(582\) 0 0
\(583\) 6.67351e9 1.39481
\(584\) −4.98519e8 −0.103571
\(585\) 0 0
\(586\) −1.03529e10 −2.12531
\(587\) −7.89426e9 −1.61093 −0.805467 0.592640i \(-0.798086\pi\)
−0.805467 + 0.592640i \(0.798086\pi\)
\(588\) 0 0
\(589\) −3.24101e9 −0.653547
\(590\) −5.82210e8 −0.116707
\(591\) 0 0
\(592\) −8.06549e8 −0.159774
\(593\) −1.93894e9 −0.381833 −0.190917 0.981606i \(-0.561146\pi\)
−0.190917 + 0.981606i \(0.561146\pi\)
\(594\) 0 0
\(595\) 2.40231e9 0.467542
\(596\) −2.15395e8 −0.0416748
\(597\) 0 0
\(598\) −1.07355e8 −0.0205290
\(599\) 4.91194e9 0.933812 0.466906 0.884307i \(-0.345369\pi\)
0.466906 + 0.884307i \(0.345369\pi\)
\(600\) 0 0
\(601\) −2.50771e9 −0.471211 −0.235606 0.971849i \(-0.575707\pi\)
−0.235606 + 0.971849i \(0.575707\pi\)
\(602\) 1.05952e10 1.97934
\(603\) 0 0
\(604\) 4.28638e9 0.791519
\(605\) −1.29791e9 −0.238286
\(606\) 0 0
\(607\) −5.17646e8 −0.0939446 −0.0469723 0.998896i \(-0.514957\pi\)
−0.0469723 + 0.998896i \(0.514957\pi\)
\(608\) −1.57167e9 −0.283595
\(609\) 0 0
\(610\) −9.91513e9 −1.76866
\(611\) 3.28945e7 0.00583417
\(612\) 0 0
\(613\) −2.24783e9 −0.394141 −0.197070 0.980389i \(-0.563143\pi\)
−0.197070 + 0.980389i \(0.563143\pi\)
\(614\) 6.99023e9 1.21871
\(615\) 0 0
\(616\) 3.03627e9 0.523370
\(617\) 3.82861e9 0.656211 0.328106 0.944641i \(-0.393590\pi\)
0.328106 + 0.944641i \(0.393590\pi\)
\(618\) 0 0
\(619\) 1.53060e9 0.259385 0.129693 0.991554i \(-0.458601\pi\)
0.129693 + 0.991554i \(0.458601\pi\)
\(620\) 3.98753e9 0.671943
\(621\) 0 0
\(622\) 6.11091e9 1.01822
\(623\) −5.63052e9 −0.932911
\(624\) 0 0
\(625\) −2.12804e9 −0.348658
\(626\) 4.17838e8 0.0680766
\(627\) 0 0
\(628\) −5.34928e9 −0.861859
\(629\) 5.17399e8 0.0828988
\(630\) 0 0
\(631\) 8.76195e9 1.38835 0.694174 0.719807i \(-0.255770\pi\)
0.694174 + 0.719807i \(0.255770\pi\)
\(632\) −1.42749e9 −0.224939
\(633\) 0 0
\(634\) −3.51900e9 −0.548413
\(635\) −2.58558e9 −0.400729
\(636\) 0 0
\(637\) −9.19873e6 −0.00141007
\(638\) −4.87652e9 −0.743426
\(639\) 0 0
\(640\) −5.48785e9 −0.827508
\(641\) 8.85004e9 1.32722 0.663609 0.748080i \(-0.269024\pi\)
0.663609 + 0.748080i \(0.269024\pi\)
\(642\) 0 0
\(643\) −1.79228e9 −0.265869 −0.132935 0.991125i \(-0.542440\pi\)
−0.132935 + 0.991125i \(0.542440\pi\)
\(644\) 1.96045e9 0.289238
\(645\) 0 0
\(646\) 1.77807e9 0.259499
\(647\) 7.01663e9 1.01851 0.509253 0.860617i \(-0.329922\pi\)
0.509253 + 0.860617i \(0.329922\pi\)
\(648\) 0 0
\(649\) 7.47007e8 0.107267
\(650\) 9.98234e7 0.0142572
\(651\) 0 0
\(652\) 3.30983e9 0.467670
\(653\) −5.63028e9 −0.791287 −0.395644 0.918404i \(-0.629478\pi\)
−0.395644 + 0.918404i \(0.629478\pi\)
\(654\) 0 0
\(655\) −9.71067e9 −1.35022
\(656\) −1.47148e10 −2.03513
\(657\) 0 0
\(658\) −1.90800e9 −0.261088
\(659\) −2.18357e9 −0.297214 −0.148607 0.988896i \(-0.547479\pi\)
−0.148607 + 0.988896i \(0.547479\pi\)
\(660\) 0 0
\(661\) 1.27138e10 1.71227 0.856134 0.516755i \(-0.172860\pi\)
0.856134 + 0.516755i \(0.172860\pi\)
\(662\) −9.14153e9 −1.22466
\(663\) 0 0
\(664\) −2.35245e9 −0.311840
\(665\) −1.81536e9 −0.239380
\(666\) 0 0
\(667\) 3.70372e9 0.483279
\(668\) −1.88828e9 −0.245103
\(669\) 0 0
\(670\) 2.83772e9 0.364508
\(671\) 1.27216e10 1.62560
\(672\) 0 0
\(673\) −5.04564e9 −0.638063 −0.319031 0.947744i \(-0.603358\pi\)
−0.319031 + 0.947744i \(0.603358\pi\)
\(674\) −2.01494e8 −0.0253485
\(675\) 0 0
\(676\) −3.68806e9 −0.459182
\(677\) 4.93561e9 0.611337 0.305668 0.952138i \(-0.401120\pi\)
0.305668 + 0.952138i \(0.401120\pi\)
\(678\) 0 0
\(679\) −3.82627e9 −0.469063
\(680\) 2.57327e9 0.313837
\(681\) 0 0
\(682\) −1.62505e10 −1.96165
\(683\) 8.54468e9 1.02618 0.513090 0.858335i \(-0.328501\pi\)
0.513090 + 0.858335i \(0.328501\pi\)
\(684\) 0 0
\(685\) 2.14448e9 0.254921
\(686\) 1.04705e10 1.23832
\(687\) 0 0
\(688\) 1.79599e10 2.10254
\(689\) 3.81679e8 0.0444560
\(690\) 0 0
\(691\) −2.27573e9 −0.262390 −0.131195 0.991357i \(-0.541881\pi\)
−0.131195 + 0.991357i \(0.541881\pi\)
\(692\) −4.62220e9 −0.530246
\(693\) 0 0
\(694\) −8.27237e9 −0.939446
\(695\) −6.37635e9 −0.720486
\(696\) 0 0
\(697\) 9.43952e9 1.05593
\(698\) 1.40342e10 1.56205
\(699\) 0 0
\(700\) −1.82292e9 −0.200874
\(701\) −2.18428e8 −0.0239494 −0.0119747 0.999928i \(-0.503812\pi\)
−0.0119747 + 0.999928i \(0.503812\pi\)
\(702\) 0 0
\(703\) −3.90984e8 −0.0424439
\(704\) 1.64182e9 0.177346
\(705\) 0 0
\(706\) −1.93012e10 −2.06428
\(707\) −6.06674e9 −0.645636
\(708\) 0 0
\(709\) 1.81988e10 1.91770 0.958849 0.283917i \(-0.0916341\pi\)
0.958849 + 0.283917i \(0.0916341\pi\)
\(710\) 9.39431e8 0.0985056
\(711\) 0 0
\(712\) −6.03119e9 −0.626215
\(713\) 1.23423e10 1.27521
\(714\) 0 0
\(715\) 1.56927e8 0.0160556
\(716\) 3.45641e9 0.351908
\(717\) 0 0
\(718\) −3.96066e9 −0.399330
\(719\) 6.16960e9 0.619022 0.309511 0.950896i \(-0.399835\pi\)
0.309511 + 0.950896i \(0.399835\pi\)
\(720\) 0 0
\(721\) 1.51468e9 0.150504
\(722\) 1.08739e10 1.07523
\(723\) 0 0
\(724\) −2.83286e9 −0.277422
\(725\) −3.44389e9 −0.335634
\(726\) 0 0
\(727\) 5.20930e9 0.502816 0.251408 0.967881i \(-0.419106\pi\)
0.251408 + 0.967881i \(0.419106\pi\)
\(728\) 1.73654e8 0.0166811
\(729\) 0 0
\(730\) 1.49448e9 0.142188
\(731\) −1.15212e10 −1.09090
\(732\) 0 0
\(733\) 9.86561e9 0.925252 0.462626 0.886554i \(-0.346907\pi\)
0.462626 + 0.886554i \(0.346907\pi\)
\(734\) 8.35704e9 0.780039
\(735\) 0 0
\(736\) 5.98516e9 0.553355
\(737\) −3.64095e9 −0.335026
\(738\) 0 0
\(739\) 6.68716e9 0.609517 0.304759 0.952430i \(-0.401424\pi\)
0.304759 + 0.952430i \(0.401424\pi\)
\(740\) 4.81041e8 0.0436386
\(741\) 0 0
\(742\) −2.21387e10 −1.98947
\(743\) −1.17332e10 −1.04944 −0.524719 0.851275i \(-0.675830\pi\)
−0.524719 + 0.851275i \(0.675830\pi\)
\(744\) 0 0
\(745\) −7.59553e8 −0.0672994
\(746\) −8.32915e8 −0.0734539
\(747\) 0 0
\(748\) 2.80683e9 0.245223
\(749\) 6.25800e9 0.544188
\(750\) 0 0
\(751\) −1.63538e10 −1.40890 −0.704448 0.709756i \(-0.748805\pi\)
−0.704448 + 0.709756i \(0.748805\pi\)
\(752\) −3.23423e9 −0.277338
\(753\) 0 0
\(754\) −2.78903e8 −0.0236948
\(755\) 1.51152e10 1.27820
\(756\) 0 0
\(757\) −7.22904e9 −0.605683 −0.302841 0.953041i \(-0.597935\pi\)
−0.302841 + 0.953041i \(0.597935\pi\)
\(758\) 1.35986e10 1.13410
\(759\) 0 0
\(760\) −1.94455e9 −0.160683
\(761\) 1.16317e9 0.0956747 0.0478373 0.998855i \(-0.484767\pi\)
0.0478373 + 0.998855i \(0.484767\pi\)
\(762\) 0 0
\(763\) 2.02632e10 1.65147
\(764\) −9.73328e9 −0.789646
\(765\) 0 0
\(766\) −2.31671e9 −0.186239
\(767\) 4.27236e7 0.00341888
\(768\) 0 0
\(769\) 2.40619e10 1.90804 0.954022 0.299735i \(-0.0968984\pi\)
0.954022 + 0.299735i \(0.0968984\pi\)
\(770\) −9.10230e9 −0.718511
\(771\) 0 0
\(772\) −6.54972e9 −0.512344
\(773\) 1.37533e9 0.107097 0.0535486 0.998565i \(-0.482947\pi\)
0.0535486 + 0.998565i \(0.482947\pi\)
\(774\) 0 0
\(775\) −1.14764e10 −0.885627
\(776\) −4.09855e9 −0.314858
\(777\) 0 0
\(778\) −2.63634e9 −0.200712
\(779\) −7.13318e9 −0.540633
\(780\) 0 0
\(781\) −1.20534e9 −0.0905381
\(782\) −6.77118e9 −0.506338
\(783\) 0 0
\(784\) 9.04432e8 0.0670301
\(785\) −1.88633e10 −1.39179
\(786\) 0 0
\(787\) 1.62448e10 1.18796 0.593980 0.804480i \(-0.297556\pi\)
0.593980 + 0.804480i \(0.297556\pi\)
\(788\) −2.14419e9 −0.156107
\(789\) 0 0
\(790\) 4.27941e9 0.308809
\(791\) 1.49735e10 1.07574
\(792\) 0 0
\(793\) 7.27590e8 0.0518120
\(794\) −2.61446e10 −1.85357
\(795\) 0 0
\(796\) 4.47744e9 0.314655
\(797\) 1.10596e10 0.773814 0.386907 0.922119i \(-0.373543\pi\)
0.386907 + 0.922119i \(0.373543\pi\)
\(798\) 0 0
\(799\) 2.07475e9 0.143897
\(800\) −5.56528e9 −0.384301
\(801\) 0 0
\(802\) 2.75057e9 0.188284
\(803\) −1.91750e9 −0.130687
\(804\) 0 0
\(805\) 6.91319e9 0.467082
\(806\) −9.29418e8 −0.0625228
\(807\) 0 0
\(808\) −6.49846e9 −0.433382
\(809\) 2.52839e10 1.67890 0.839450 0.543436i \(-0.182877\pi\)
0.839450 + 0.543436i \(0.182877\pi\)
\(810\) 0 0
\(811\) 7.05869e9 0.464677 0.232339 0.972635i \(-0.425362\pi\)
0.232339 + 0.972635i \(0.425362\pi\)
\(812\) 5.09317e9 0.333843
\(813\) 0 0
\(814\) −1.96041e9 −0.127398
\(815\) 1.16716e10 0.755227
\(816\) 0 0
\(817\) 8.70625e9 0.558540
\(818\) −4.79930e9 −0.306578
\(819\) 0 0
\(820\) 8.77620e9 0.555850
\(821\) 1.72937e10 1.09066 0.545328 0.838223i \(-0.316405\pi\)
0.545328 + 0.838223i \(0.316405\pi\)
\(822\) 0 0
\(823\) 1.28299e10 0.802278 0.401139 0.916017i \(-0.368614\pi\)
0.401139 + 0.916017i \(0.368614\pi\)
\(824\) 1.62247e9 0.101025
\(825\) 0 0
\(826\) −2.47812e9 −0.153000
\(827\) −1.83922e10 −1.13074 −0.565372 0.824836i \(-0.691268\pi\)
−0.565372 + 0.824836i \(0.691268\pi\)
\(828\) 0 0
\(829\) −4.09547e9 −0.249668 −0.124834 0.992178i \(-0.539840\pi\)
−0.124834 + 0.992178i \(0.539840\pi\)
\(830\) 7.05229e9 0.428111
\(831\) 0 0
\(832\) 9.39006e7 0.00565245
\(833\) −5.80190e8 −0.0347787
\(834\) 0 0
\(835\) −6.65870e9 −0.395810
\(836\) −2.12105e9 −0.125553
\(837\) 0 0
\(838\) 5.02041e9 0.294704
\(839\) 1.88735e9 0.110328 0.0551641 0.998477i \(-0.482432\pi\)
0.0551641 + 0.998477i \(0.482432\pi\)
\(840\) 0 0
\(841\) −7.62778e9 −0.442193
\(842\) −1.65361e10 −0.954641
\(843\) 0 0
\(844\) 1.38327e10 0.791969
\(845\) −1.30053e10 −0.741519
\(846\) 0 0
\(847\) −5.52440e9 −0.312387
\(848\) −3.75272e10 −2.11330
\(849\) 0 0
\(850\) 6.29615e9 0.351649
\(851\) 1.48893e9 0.0828173
\(852\) 0 0
\(853\) 6.81026e8 0.0375701 0.0187850 0.999824i \(-0.494020\pi\)
0.0187850 + 0.999824i \(0.494020\pi\)
\(854\) −4.22027e10 −2.31866
\(855\) 0 0
\(856\) 6.70332e9 0.365285
\(857\) −6.43948e9 −0.349476 −0.174738 0.984615i \(-0.555908\pi\)
−0.174738 + 0.984615i \(0.555908\pi\)
\(858\) 0 0
\(859\) 1.88733e10 1.01595 0.507974 0.861373i \(-0.330395\pi\)
0.507974 + 0.861373i \(0.330395\pi\)
\(860\) −1.07116e10 −0.574262
\(861\) 0 0
\(862\) 8.26541e9 0.439530
\(863\) −1.84568e10 −0.977505 −0.488753 0.872422i \(-0.662548\pi\)
−0.488753 + 0.872422i \(0.662548\pi\)
\(864\) 0 0
\(865\) −1.62994e10 −0.856279
\(866\) 1.14372e10 0.598422
\(867\) 0 0
\(868\) 1.69725e10 0.880900
\(869\) −5.49072e9 −0.283831
\(870\) 0 0
\(871\) −2.08237e8 −0.0106781
\(872\) 2.17051e10 1.10855
\(873\) 0 0
\(874\) 5.11679e9 0.259243
\(875\) −2.07325e10 −1.04622
\(876\) 0 0
\(877\) 1.79041e10 0.896301 0.448151 0.893958i \(-0.352083\pi\)
0.448151 + 0.893958i \(0.352083\pi\)
\(878\) 3.70248e10 1.84613
\(879\) 0 0
\(880\) −1.54293e10 −0.763230
\(881\) 1.86101e10 0.916922 0.458461 0.888715i \(-0.348401\pi\)
0.458461 + 0.888715i \(0.348401\pi\)
\(882\) 0 0
\(883\) −1.45294e10 −0.710207 −0.355103 0.934827i \(-0.615554\pi\)
−0.355103 + 0.934827i \(0.615554\pi\)
\(884\) 1.60531e8 0.00781586
\(885\) 0 0
\(886\) −1.31063e10 −0.633084
\(887\) 3.46786e9 0.166851 0.0834256 0.996514i \(-0.473414\pi\)
0.0834256 + 0.996514i \(0.473414\pi\)
\(888\) 0 0
\(889\) −1.10053e10 −0.525345
\(890\) 1.80806e10 0.859702
\(891\) 0 0
\(892\) 9.26313e9 0.436999
\(893\) −1.56783e9 −0.0736749
\(894\) 0 0
\(895\) 1.21884e10 0.568286
\(896\) −2.33585e10 −1.08484
\(897\) 0 0
\(898\) 2.10419e10 0.969655
\(899\) 3.20647e10 1.47187
\(900\) 0 0
\(901\) 2.40736e10 1.09649
\(902\) −3.57660e10 −1.62274
\(903\) 0 0
\(904\) 1.60391e10 0.722088
\(905\) −9.98961e9 −0.448001
\(906\) 0 0
\(907\) −4.86854e9 −0.216657 −0.108329 0.994115i \(-0.534550\pi\)
−0.108329 + 0.994115i \(0.534550\pi\)
\(908\) 2.50308e9 0.110962
\(909\) 0 0
\(910\) −5.20588e8 −0.0229007
\(911\) −5.41157e9 −0.237142 −0.118571 0.992946i \(-0.537831\pi\)
−0.118571 + 0.992946i \(0.537831\pi\)
\(912\) 0 0
\(913\) −9.04846e9 −0.393484
\(914\) 4.55916e10 1.97503
\(915\) 0 0
\(916\) −2.98320e9 −0.128247
\(917\) −4.13325e10 −1.77010
\(918\) 0 0
\(919\) −4.36861e10 −1.85669 −0.928343 0.371724i \(-0.878767\pi\)
−0.928343 + 0.371724i \(0.878767\pi\)
\(920\) 7.40515e9 0.313528
\(921\) 0 0
\(922\) 3.58582e10 1.50671
\(923\) −6.89371e7 −0.00288567
\(924\) 0 0
\(925\) −1.38448e9 −0.0575161
\(926\) 9.36666e9 0.387656
\(927\) 0 0
\(928\) 1.55492e10 0.638690
\(929\) −1.09031e10 −0.446164 −0.223082 0.974800i \(-0.571612\pi\)
−0.223082 + 0.974800i \(0.571612\pi\)
\(930\) 0 0
\(931\) 4.38434e8 0.0178066
\(932\) 1.44072e10 0.582939
\(933\) 0 0
\(934\) −4.29159e10 −1.72347
\(935\) 9.89782e9 0.396003
\(936\) 0 0
\(937\) 3.25215e10 1.29146 0.645732 0.763564i \(-0.276552\pi\)
0.645732 + 0.763564i \(0.276552\pi\)
\(938\) 1.20785e10 0.477861
\(939\) 0 0
\(940\) 1.92896e9 0.0757487
\(941\) 2.70964e10 1.06010 0.530052 0.847965i \(-0.322172\pi\)
0.530052 + 0.847965i \(0.322172\pi\)
\(942\) 0 0
\(943\) 2.71643e10 1.05489
\(944\) −4.20064e9 −0.162523
\(945\) 0 0
\(946\) 4.36535e10 1.67649
\(947\) −2.06829e10 −0.791382 −0.395691 0.918384i \(-0.629495\pi\)
−0.395691 + 0.918384i \(0.629495\pi\)
\(948\) 0 0
\(949\) −1.09668e8 −0.00416531
\(950\) −4.75783e9 −0.180043
\(951\) 0 0
\(952\) 1.09528e10 0.411431
\(953\) 3.31031e10 1.23892 0.619460 0.785028i \(-0.287351\pi\)
0.619460 + 0.785028i \(0.287351\pi\)
\(954\) 0 0
\(955\) −3.43227e10 −1.27518
\(956\) 1.99327e8 0.00737841
\(957\) 0 0
\(958\) 1.83027e10 0.672569
\(959\) 9.12777e9 0.334195
\(960\) 0 0
\(961\) 7.93400e10 2.88377
\(962\) −1.12122e8 −0.00406048
\(963\) 0 0
\(964\) 1.42908e10 0.513792
\(965\) −2.30965e10 −0.827370
\(966\) 0 0
\(967\) −2.24560e10 −0.798618 −0.399309 0.916816i \(-0.630750\pi\)
−0.399309 + 0.916816i \(0.630750\pi\)
\(968\) −5.91752e9 −0.209689
\(969\) 0 0
\(970\) 1.22868e10 0.432254
\(971\) 1.83485e10 0.643182 0.321591 0.946879i \(-0.395782\pi\)
0.321591 + 0.946879i \(0.395782\pi\)
\(972\) 0 0
\(973\) −2.71403e10 −0.944538
\(974\) 4.12818e10 1.43154
\(975\) 0 0
\(976\) −7.15376e10 −2.46298
\(977\) −2.21334e10 −0.759306 −0.379653 0.925129i \(-0.623957\pi\)
−0.379653 + 0.925129i \(0.623957\pi\)
\(978\) 0 0
\(979\) −2.31984e10 −0.790166
\(980\) −5.39420e8 −0.0183078
\(981\) 0 0
\(982\) −3.25566e10 −1.09711
\(983\) −5.39105e10 −1.81024 −0.905121 0.425155i \(-0.860220\pi\)
−0.905121 + 0.425155i \(0.860220\pi\)
\(984\) 0 0
\(985\) −7.56112e9 −0.252092
\(986\) −1.75912e10 −0.584422
\(987\) 0 0
\(988\) −1.21309e8 −0.00400170
\(989\) −3.31548e10 −1.08983
\(990\) 0 0
\(991\) −1.06845e10 −0.348734 −0.174367 0.984681i \(-0.555788\pi\)
−0.174367 + 0.984681i \(0.555788\pi\)
\(992\) 5.18162e10 1.68529
\(993\) 0 0
\(994\) 3.99859e9 0.129138
\(995\) 1.57889e10 0.508127
\(996\) 0 0
\(997\) −3.63110e10 −1.16039 −0.580196 0.814477i \(-0.697024\pi\)
−0.580196 + 0.814477i \(0.697024\pi\)
\(998\) −3.44245e10 −1.09625
\(999\) 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.1.4 17
3.2 odd 2 177.8.a.b.1.14 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.14 17 3.2 odd 2
531.8.a.d.1.4 17 1.1 even 1 trivial