Properties

Label 531.8.a.d.1.2
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 1.2
Root \(18.2619\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.2619 q^{2} +136.450 q^{4} +30.6671 q^{5} -1356.62 q^{7} -137.421 q^{8} +O(q^{10})\) \(q-16.2619 q^{2} +136.450 q^{4} +30.6671 q^{5} -1356.62 q^{7} -137.421 q^{8} -498.706 q^{10} -1825.79 q^{11} -1173.55 q^{13} +22061.2 q^{14} -15230.9 q^{16} -13185.9 q^{17} -18595.5 q^{19} +4184.54 q^{20} +29690.8 q^{22} +14566.3 q^{23} -77184.5 q^{25} +19084.3 q^{26} -185111. q^{28} +183281. q^{29} -265230. q^{31} +265274. q^{32} +214428. q^{34} -41603.6 q^{35} -260898. q^{37} +302399. q^{38} -4214.29 q^{40} -719268. q^{41} -707870. q^{43} -249129. q^{44} -236876. q^{46} -1.30218e6 q^{47} +1.01687e6 q^{49} +1.25517e6 q^{50} -160132. q^{52} +1.80784e6 q^{53} -55991.6 q^{55} +186427. q^{56} -2.98051e6 q^{58} +205379. q^{59} +1.38182e6 q^{61} +4.31315e6 q^{62} -2.36431e6 q^{64} -35989.5 q^{65} +2.41963e6 q^{67} -1.79922e6 q^{68} +676555. q^{70} -2.72507e6 q^{71} +3.97598e6 q^{73} +4.24271e6 q^{74} -2.53736e6 q^{76} +2.47690e6 q^{77} -6.91787e6 q^{79} -467089. q^{80} +1.16967e7 q^{82} -8.32910e6 q^{83} -404374. q^{85} +1.15113e7 q^{86} +250901. q^{88} +6.71890e6 q^{89} +1.59207e6 q^{91} +1.98758e6 q^{92} +2.11760e7 q^{94} -570270. q^{95} -4.77557e6 q^{97} -1.65363e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.2619 −1.43737 −0.718683 0.695338i \(-0.755255\pi\)
−0.718683 + 0.695338i \(0.755255\pi\)
\(3\) 0 0
\(4\) 136.450 1.06602
\(5\) 30.6671 0.109718 0.0548590 0.998494i \(-0.482529\pi\)
0.0548590 + 0.998494i \(0.482529\pi\)
\(6\) 0 0
\(7\) −1356.62 −1.49491 −0.747455 0.664313i \(-0.768724\pi\)
−0.747455 + 0.664313i \(0.768724\pi\)
\(8\) −137.421 −0.0948936
\(9\) 0 0
\(10\) −498.706 −0.157705
\(11\) −1825.79 −0.413595 −0.206798 0.978384i \(-0.566304\pi\)
−0.206798 + 0.978384i \(0.566304\pi\)
\(12\) 0 0
\(13\) −1173.55 −0.148150 −0.0740750 0.997253i \(-0.523600\pi\)
−0.0740750 + 0.997253i \(0.523600\pi\)
\(14\) 22061.2 2.14873
\(15\) 0 0
\(16\) −15230.9 −0.929622
\(17\) −13185.9 −0.650937 −0.325468 0.945553i \(-0.605522\pi\)
−0.325468 + 0.945553i \(0.605522\pi\)
\(18\) 0 0
\(19\) −18595.5 −0.621971 −0.310985 0.950415i \(-0.600659\pi\)
−0.310985 + 0.950415i \(0.600659\pi\)
\(20\) 4184.54 0.116961
\(21\) 0 0
\(22\) 29690.8 0.594488
\(23\) 14566.3 0.249633 0.124817 0.992180i \(-0.460166\pi\)
0.124817 + 0.992180i \(0.460166\pi\)
\(24\) 0 0
\(25\) −77184.5 −0.987962
\(26\) 19084.3 0.212946
\(27\) 0 0
\(28\) −185111. −1.59360
\(29\) 183281. 1.39549 0.697743 0.716348i \(-0.254188\pi\)
0.697743 + 0.716348i \(0.254188\pi\)
\(30\) 0 0
\(31\) −265230. −1.59903 −0.799515 0.600646i \(-0.794910\pi\)
−0.799515 + 0.600646i \(0.794910\pi\)
\(32\) 265274. 1.43110
\(33\) 0 0
\(34\) 214428. 0.935634
\(35\) −41603.6 −0.164018
\(36\) 0 0
\(37\) −260898. −0.846769 −0.423385 0.905950i \(-0.639158\pi\)
−0.423385 + 0.905950i \(0.639158\pi\)
\(38\) 302399. 0.893999
\(39\) 0 0
\(40\) −4214.29 −0.0104115
\(41\) −719268. −1.62985 −0.814924 0.579568i \(-0.803221\pi\)
−0.814924 + 0.579568i \(0.803221\pi\)
\(42\) 0 0
\(43\) −707870. −1.35773 −0.678866 0.734262i \(-0.737528\pi\)
−0.678866 + 0.734262i \(0.737528\pi\)
\(44\) −249129. −0.440900
\(45\) 0 0
\(46\) −236876. −0.358814
\(47\) −1.30218e6 −1.82948 −0.914742 0.404038i \(-0.867606\pi\)
−0.914742 + 0.404038i \(0.867606\pi\)
\(48\) 0 0
\(49\) 1.01687e6 1.23475
\(50\) 1.25517e6 1.42006
\(51\) 0 0
\(52\) −160132. −0.157931
\(53\) 1.80784e6 1.66799 0.833996 0.551770i \(-0.186047\pi\)
0.833996 + 0.551770i \(0.186047\pi\)
\(54\) 0 0
\(55\) −55991.6 −0.0453788
\(56\) 186427. 0.141857
\(57\) 0 0
\(58\) −2.98051e6 −2.00582
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 1.38182e6 0.779468 0.389734 0.920928i \(-0.372567\pi\)
0.389734 + 0.920928i \(0.372567\pi\)
\(62\) 4.31315e6 2.29839
\(63\) 0 0
\(64\) −2.36431e6 −1.12739
\(65\) −35989.5 −0.0162547
\(66\) 0 0
\(67\) 2.41963e6 0.982849 0.491424 0.870920i \(-0.336476\pi\)
0.491424 + 0.870920i \(0.336476\pi\)
\(68\) −1.79922e6 −0.693911
\(69\) 0 0
\(70\) 676555. 0.235754
\(71\) −2.72507e6 −0.903595 −0.451797 0.892121i \(-0.649217\pi\)
−0.451797 + 0.892121i \(0.649217\pi\)
\(72\) 0 0
\(73\) 3.97598e6 1.19623 0.598115 0.801411i \(-0.295917\pi\)
0.598115 + 0.801411i \(0.295917\pi\)
\(74\) 4.24271e6 1.21712
\(75\) 0 0
\(76\) −2.53736e6 −0.663033
\(77\) 2.47690e6 0.618287
\(78\) 0 0
\(79\) −6.91787e6 −1.57862 −0.789310 0.613995i \(-0.789562\pi\)
−0.789310 + 0.613995i \(0.789562\pi\)
\(80\) −467089. −0.101996
\(81\) 0 0
\(82\) 1.16967e7 2.34269
\(83\) −8.32910e6 −1.59891 −0.799456 0.600725i \(-0.794879\pi\)
−0.799456 + 0.600725i \(0.794879\pi\)
\(84\) 0 0
\(85\) −404374. −0.0714195
\(86\) 1.15113e7 1.95156
\(87\) 0 0
\(88\) 250901. 0.0392476
\(89\) 6.71890e6 1.01026 0.505130 0.863043i \(-0.331444\pi\)
0.505130 + 0.863043i \(0.331444\pi\)
\(90\) 0 0
\(91\) 1.59207e6 0.221471
\(92\) 1.98758e6 0.266114
\(93\) 0 0
\(94\) 2.11760e7 2.62964
\(95\) −570270. −0.0682414
\(96\) 0 0
\(97\) −4.77557e6 −0.531281 −0.265640 0.964072i \(-0.585583\pi\)
−0.265640 + 0.964072i \(0.585583\pi\)
\(98\) −1.65363e7 −1.77479
\(99\) 0 0
\(100\) −1.05319e7 −1.05319
\(101\) −1.79352e7 −1.73213 −0.866066 0.499930i \(-0.833359\pi\)
−0.866066 + 0.499930i \(0.833359\pi\)
\(102\) 0 0
\(103\) −1.83344e7 −1.65324 −0.826619 0.562762i \(-0.809739\pi\)
−0.826619 + 0.562762i \(0.809739\pi\)
\(104\) 161271. 0.0140585
\(105\) 0 0
\(106\) −2.93990e7 −2.39752
\(107\) −1.58268e7 −1.24896 −0.624480 0.781041i \(-0.714689\pi\)
−0.624480 + 0.781041i \(0.714689\pi\)
\(108\) 0 0
\(109\) 4.43687e6 0.328159 0.164079 0.986447i \(-0.447535\pi\)
0.164079 + 0.986447i \(0.447535\pi\)
\(110\) 910531. 0.0652260
\(111\) 0 0
\(112\) 2.06626e7 1.38970
\(113\) −2.03544e7 −1.32704 −0.663520 0.748159i \(-0.730938\pi\)
−0.663520 + 0.748159i \(0.730938\pi\)
\(114\) 0 0
\(115\) 446707. 0.0273892
\(116\) 2.50088e7 1.48761
\(117\) 0 0
\(118\) −3.33986e6 −0.187129
\(119\) 1.78883e7 0.973091
\(120\) 0 0
\(121\) −1.61537e7 −0.828939
\(122\) −2.24711e7 −1.12038
\(123\) 0 0
\(124\) −3.61907e7 −1.70460
\(125\) −4.76289e6 −0.218115
\(126\) 0 0
\(127\) −9.80807e6 −0.424884 −0.212442 0.977174i \(-0.568142\pi\)
−0.212442 + 0.977174i \(0.568142\pi\)
\(128\) 4.49319e6 0.189374
\(129\) 0 0
\(130\) 585259. 0.0233640
\(131\) 2.55344e7 0.992377 0.496189 0.868215i \(-0.334732\pi\)
0.496189 + 0.868215i \(0.334732\pi\)
\(132\) 0 0
\(133\) 2.52270e7 0.929790
\(134\) −3.93478e7 −1.41271
\(135\) 0 0
\(136\) 1.81202e6 0.0617698
\(137\) 5.60036e6 0.186078 0.0930388 0.995662i \(-0.470342\pi\)
0.0930388 + 0.995662i \(0.470342\pi\)
\(138\) 0 0
\(139\) −994215. −0.0313999 −0.0157000 0.999877i \(-0.504998\pi\)
−0.0157000 + 0.999877i \(0.504998\pi\)
\(140\) −5.67683e6 −0.174847
\(141\) 0 0
\(142\) 4.43149e7 1.29880
\(143\) 2.14266e6 0.0612741
\(144\) 0 0
\(145\) 5.62071e6 0.153110
\(146\) −6.46572e7 −1.71942
\(147\) 0 0
\(148\) −3.55997e7 −0.902672
\(149\) −2.61359e7 −0.647269 −0.323635 0.946182i \(-0.604905\pi\)
−0.323635 + 0.946182i \(0.604905\pi\)
\(150\) 0 0
\(151\) −7.02885e7 −1.66136 −0.830682 0.556747i \(-0.812049\pi\)
−0.830682 + 0.556747i \(0.812049\pi\)
\(152\) 2.55540e6 0.0590211
\(153\) 0 0
\(154\) −4.02791e7 −0.888705
\(155\) −8.13383e6 −0.175442
\(156\) 0 0
\(157\) −2.28654e7 −0.471553 −0.235776 0.971807i \(-0.575763\pi\)
−0.235776 + 0.971807i \(0.575763\pi\)
\(158\) 1.12498e8 2.26905
\(159\) 0 0
\(160\) 8.13519e6 0.157017
\(161\) −1.97609e7 −0.373179
\(162\) 0 0
\(163\) −6.26397e7 −1.13290 −0.566452 0.824095i \(-0.691684\pi\)
−0.566452 + 0.824095i \(0.691684\pi\)
\(164\) −9.81444e7 −1.73745
\(165\) 0 0
\(166\) 1.35447e8 2.29822
\(167\) 2.02935e7 0.337171 0.168586 0.985687i \(-0.446080\pi\)
0.168586 + 0.985687i \(0.446080\pi\)
\(168\) 0 0
\(169\) −6.13713e7 −0.978052
\(170\) 6.57590e6 0.102656
\(171\) 0 0
\(172\) −9.65892e7 −1.44737
\(173\) 4.43065e7 0.650588 0.325294 0.945613i \(-0.394537\pi\)
0.325294 + 0.945613i \(0.394537\pi\)
\(174\) 0 0
\(175\) 1.04710e8 1.47691
\(176\) 2.78084e7 0.384487
\(177\) 0 0
\(178\) −1.09262e8 −1.45211
\(179\) −1.30339e8 −1.69858 −0.849292 0.527923i \(-0.822971\pi\)
−0.849292 + 0.527923i \(0.822971\pi\)
\(180\) 0 0
\(181\) −8.67394e7 −1.08728 −0.543640 0.839319i \(-0.682954\pi\)
−0.543640 + 0.839319i \(0.682954\pi\)
\(182\) −2.58901e7 −0.318334
\(183\) 0 0
\(184\) −2.00171e6 −0.0236886
\(185\) −8.00099e6 −0.0929058
\(186\) 0 0
\(187\) 2.40746e7 0.269224
\(188\) −1.77683e8 −1.95027
\(189\) 0 0
\(190\) 9.27369e6 0.0980878
\(191\) 1.00406e8 1.04266 0.521330 0.853355i \(-0.325436\pi\)
0.521330 + 0.853355i \(0.325436\pi\)
\(192\) 0 0
\(193\) 1.63066e8 1.63272 0.816361 0.577542i \(-0.195988\pi\)
0.816361 + 0.577542i \(0.195988\pi\)
\(194\) 7.76600e7 0.763645
\(195\) 0 0
\(196\) 1.38753e8 1.31627
\(197\) −1.22832e8 −1.14467 −0.572335 0.820020i \(-0.693962\pi\)
−0.572335 + 0.820020i \(0.693962\pi\)
\(198\) 0 0
\(199\) 1.85902e8 1.67224 0.836119 0.548549i \(-0.184820\pi\)
0.836119 + 0.548549i \(0.184820\pi\)
\(200\) 1.06067e7 0.0937513
\(201\) 0 0
\(202\) 2.91661e8 2.48971
\(203\) −2.48643e8 −2.08612
\(204\) 0 0
\(205\) −2.20579e7 −0.178824
\(206\) 2.98152e8 2.37631
\(207\) 0 0
\(208\) 1.78743e7 0.137724
\(209\) 3.39514e7 0.257244
\(210\) 0 0
\(211\) 4.25639e6 0.0311927 0.0155963 0.999878i \(-0.495035\pi\)
0.0155963 + 0.999878i \(0.495035\pi\)
\(212\) 2.46680e8 1.77811
\(213\) 0 0
\(214\) 2.57374e8 1.79521
\(215\) −2.17083e7 −0.148968
\(216\) 0 0
\(217\) 3.59816e8 2.39040
\(218\) −7.21521e7 −0.471684
\(219\) 0 0
\(220\) −7.64008e6 −0.0483747
\(221\) 1.54744e7 0.0964362
\(222\) 0 0
\(223\) −1.01090e8 −0.610437 −0.305219 0.952282i \(-0.598730\pi\)
−0.305219 + 0.952282i \(0.598730\pi\)
\(224\) −3.59876e8 −2.13937
\(225\) 0 0
\(226\) 3.31002e8 1.90744
\(227\) 1.56725e8 0.889299 0.444650 0.895705i \(-0.353328\pi\)
0.444650 + 0.895705i \(0.353328\pi\)
\(228\) 0 0
\(229\) 2.30104e8 1.26619 0.633097 0.774072i \(-0.281783\pi\)
0.633097 + 0.774072i \(0.281783\pi\)
\(230\) −7.26431e6 −0.0393683
\(231\) 0 0
\(232\) −2.51866e7 −0.132423
\(233\) −9.45434e7 −0.489649 −0.244825 0.969567i \(-0.578730\pi\)
−0.244825 + 0.969567i \(0.578730\pi\)
\(234\) 0 0
\(235\) −3.99341e7 −0.200727
\(236\) 2.80241e7 0.138784
\(237\) 0 0
\(238\) −2.90898e8 −1.39869
\(239\) 2.18199e8 1.03385 0.516927 0.856029i \(-0.327076\pi\)
0.516927 + 0.856029i \(0.327076\pi\)
\(240\) 0 0
\(241\) −7.52814e7 −0.346440 −0.173220 0.984883i \(-0.555417\pi\)
−0.173220 + 0.984883i \(0.555417\pi\)
\(242\) 2.62690e8 1.19149
\(243\) 0 0
\(244\) 1.88550e8 0.830927
\(245\) 3.11845e7 0.135475
\(246\) 0 0
\(247\) 2.18228e7 0.0921450
\(248\) 3.64481e7 0.151738
\(249\) 0 0
\(250\) 7.74539e7 0.313511
\(251\) 2.48536e8 0.992046 0.496023 0.868309i \(-0.334793\pi\)
0.496023 + 0.868309i \(0.334793\pi\)
\(252\) 0 0
\(253\) −2.65950e7 −0.103247
\(254\) 1.59498e8 0.610714
\(255\) 0 0
\(256\) 2.29564e8 0.855193
\(257\) −9.48986e7 −0.348734 −0.174367 0.984681i \(-0.555788\pi\)
−0.174367 + 0.984681i \(0.555788\pi\)
\(258\) 0 0
\(259\) 3.53939e8 1.26584
\(260\) −4.91079e6 −0.0173278
\(261\) 0 0
\(262\) −4.15239e8 −1.42641
\(263\) −7.74187e7 −0.262422 −0.131211 0.991354i \(-0.541887\pi\)
−0.131211 + 0.991354i \(0.541887\pi\)
\(264\) 0 0
\(265\) 5.54412e7 0.183009
\(266\) −4.10240e8 −1.33645
\(267\) 0 0
\(268\) 3.30159e8 1.04774
\(269\) 1.48368e8 0.464738 0.232369 0.972628i \(-0.425352\pi\)
0.232369 + 0.972628i \(0.425352\pi\)
\(270\) 0 0
\(271\) 2.14037e8 0.653275 0.326637 0.945150i \(-0.394084\pi\)
0.326637 + 0.945150i \(0.394084\pi\)
\(272\) 2.00834e8 0.605125
\(273\) 0 0
\(274\) −9.10727e7 −0.267462
\(275\) 1.40922e8 0.408616
\(276\) 0 0
\(277\) −4.49384e8 −1.27039 −0.635196 0.772351i \(-0.719081\pi\)
−0.635196 + 0.772351i \(0.719081\pi\)
\(278\) 1.61679e7 0.0451331
\(279\) 0 0
\(280\) 5.71719e6 0.0155643
\(281\) −5.47292e7 −0.147145 −0.0735727 0.997290i \(-0.523440\pi\)
−0.0735727 + 0.997290i \(0.523440\pi\)
\(282\) 0 0
\(283\) −6.96310e8 −1.82621 −0.913104 0.407727i \(-0.866322\pi\)
−0.913104 + 0.407727i \(0.866322\pi\)
\(284\) −3.71837e8 −0.963249
\(285\) 0 0
\(286\) −3.48438e7 −0.0880733
\(287\) 9.75773e8 2.43648
\(288\) 0 0
\(289\) −2.36471e8 −0.576281
\(290\) −9.14036e7 −0.220075
\(291\) 0 0
\(292\) 5.42525e8 1.27520
\(293\) 1.58823e8 0.368874 0.184437 0.982844i \(-0.440954\pi\)
0.184437 + 0.982844i \(0.440954\pi\)
\(294\) 0 0
\(295\) 6.29838e6 0.0142841
\(296\) 3.58528e7 0.0803530
\(297\) 0 0
\(298\) 4.25020e8 0.930363
\(299\) −1.70944e7 −0.0369831
\(300\) 0 0
\(301\) 9.60310e8 2.02969
\(302\) 1.14303e9 2.38799
\(303\) 0 0
\(304\) 2.83227e8 0.578198
\(305\) 4.23765e7 0.0855216
\(306\) 0 0
\(307\) 2.01551e8 0.397558 0.198779 0.980044i \(-0.436302\pi\)
0.198779 + 0.980044i \(0.436302\pi\)
\(308\) 3.37974e8 0.659106
\(309\) 0 0
\(310\) 1.32272e8 0.252175
\(311\) −3.80395e8 −0.717090 −0.358545 0.933513i \(-0.616727\pi\)
−0.358545 + 0.933513i \(0.616727\pi\)
\(312\) 0 0
\(313\) 1.43971e8 0.265382 0.132691 0.991157i \(-0.457638\pi\)
0.132691 + 0.991157i \(0.457638\pi\)
\(314\) 3.71836e8 0.677794
\(315\) 0 0
\(316\) −9.43947e8 −1.68284
\(317\) 3.85345e8 0.679426 0.339713 0.940529i \(-0.389670\pi\)
0.339713 + 0.940529i \(0.389670\pi\)
\(318\) 0 0
\(319\) −3.34633e8 −0.577166
\(320\) −7.25066e7 −0.123695
\(321\) 0 0
\(322\) 3.21351e8 0.536394
\(323\) 2.45198e8 0.404864
\(324\) 0 0
\(325\) 9.05803e7 0.146367
\(326\) 1.01864e9 1.62840
\(327\) 0 0
\(328\) 9.88423e7 0.154662
\(329\) 1.76656e9 2.73491
\(330\) 0 0
\(331\) −5.09634e8 −0.772432 −0.386216 0.922408i \(-0.626218\pi\)
−0.386216 + 0.922408i \(0.626218\pi\)
\(332\) −1.13651e9 −1.70447
\(333\) 0 0
\(334\) −3.30012e8 −0.484638
\(335\) 7.42030e7 0.107836
\(336\) 0 0
\(337\) 3.29994e8 0.469679 0.234840 0.972034i \(-0.424543\pi\)
0.234840 + 0.972034i \(0.424543\pi\)
\(338\) 9.98016e8 1.40582
\(339\) 0 0
\(340\) −5.51770e7 −0.0761345
\(341\) 4.84253e8 0.661351
\(342\) 0 0
\(343\) −2.62274e8 −0.350935
\(344\) 9.72760e7 0.128840
\(345\) 0 0
\(346\) −7.20510e8 −0.935133
\(347\) 1.54205e9 1.98128 0.990642 0.136488i \(-0.0435816\pi\)
0.990642 + 0.136488i \(0.0435816\pi\)
\(348\) 0 0
\(349\) −3.37260e8 −0.424694 −0.212347 0.977194i \(-0.568111\pi\)
−0.212347 + 0.977194i \(0.568111\pi\)
\(350\) −1.70279e9 −2.12286
\(351\) 0 0
\(352\) −4.84334e8 −0.591896
\(353\) 8.38966e8 1.01516 0.507578 0.861606i \(-0.330541\pi\)
0.507578 + 0.861606i \(0.330541\pi\)
\(354\) 0 0
\(355\) −8.35701e7 −0.0991406
\(356\) 9.16797e8 1.07696
\(357\) 0 0
\(358\) 2.11956e9 2.44149
\(359\) 5.75290e8 0.656231 0.328115 0.944638i \(-0.393587\pi\)
0.328115 + 0.944638i \(0.393587\pi\)
\(360\) 0 0
\(361\) −5.48079e8 −0.613152
\(362\) 1.41055e9 1.56282
\(363\) 0 0
\(364\) 2.17238e8 0.236092
\(365\) 1.21932e8 0.131248
\(366\) 0 0
\(367\) −1.63254e8 −0.172398 −0.0861990 0.996278i \(-0.527472\pi\)
−0.0861990 + 0.996278i \(0.527472\pi\)
\(368\) −2.21859e8 −0.232065
\(369\) 0 0
\(370\) 1.30112e8 0.133540
\(371\) −2.45255e9 −2.49350
\(372\) 0 0
\(373\) −5.02935e8 −0.501801 −0.250900 0.968013i \(-0.580727\pi\)
−0.250900 + 0.968013i \(0.580727\pi\)
\(374\) −3.91500e8 −0.386974
\(375\) 0 0
\(376\) 1.78947e8 0.173606
\(377\) −2.15091e8 −0.206741
\(378\) 0 0
\(379\) 1.84658e9 1.74234 0.871168 0.490986i \(-0.163363\pi\)
0.871168 + 0.490986i \(0.163363\pi\)
\(380\) −7.78136e7 −0.0727466
\(381\) 0 0
\(382\) −1.63279e9 −1.49868
\(383\) −78579.0 −7.14679e−5 0 −3.57339e−5 1.00000i \(-0.500011\pi\)
−3.57339e−5 1.00000i \(0.500011\pi\)
\(384\) 0 0
\(385\) 7.59593e7 0.0678372
\(386\) −2.65176e9 −2.34682
\(387\) 0 0
\(388\) −6.51629e8 −0.566355
\(389\) −1.01883e9 −0.877561 −0.438780 0.898594i \(-0.644589\pi\)
−0.438780 + 0.898594i \(0.644589\pi\)
\(390\) 0 0
\(391\) −1.92070e8 −0.162495
\(392\) −1.39739e8 −0.117170
\(393\) 0 0
\(394\) 1.99749e9 1.64531
\(395\) −2.12151e8 −0.173203
\(396\) 0 0
\(397\) 3.45551e8 0.277170 0.138585 0.990351i \(-0.455745\pi\)
0.138585 + 0.990351i \(0.455745\pi\)
\(398\) −3.02312e9 −2.40362
\(399\) 0 0
\(400\) 1.17559e9 0.918431
\(401\) 2.15091e9 1.66578 0.832890 0.553439i \(-0.186685\pi\)
0.832890 + 0.553439i \(0.186685\pi\)
\(402\) 0 0
\(403\) 3.11262e8 0.236896
\(404\) −2.44726e9 −1.84649
\(405\) 0 0
\(406\) 4.04342e9 2.99852
\(407\) 4.76344e8 0.350220
\(408\) 0 0
\(409\) −2.23244e8 −0.161342 −0.0806710 0.996741i \(-0.525706\pi\)
−0.0806710 + 0.996741i \(0.525706\pi\)
\(410\) 3.58704e8 0.257035
\(411\) 0 0
\(412\) −2.50173e9 −1.76238
\(413\) −2.78621e8 −0.194621
\(414\) 0 0
\(415\) −2.55429e8 −0.175429
\(416\) −3.11314e8 −0.212017
\(417\) 0 0
\(418\) −5.52115e8 −0.369754
\(419\) 1.16588e9 0.774291 0.387146 0.922019i \(-0.373461\pi\)
0.387146 + 0.922019i \(0.373461\pi\)
\(420\) 0 0
\(421\) −9.44097e8 −0.616637 −0.308318 0.951283i \(-0.599766\pi\)
−0.308318 + 0.951283i \(0.599766\pi\)
\(422\) −6.92171e7 −0.0448353
\(423\) 0 0
\(424\) −2.48434e8 −0.158282
\(425\) 1.01775e9 0.643101
\(426\) 0 0
\(427\) −1.87461e9 −1.16523
\(428\) −2.15957e9 −1.33142
\(429\) 0 0
\(430\) 3.53019e8 0.214121
\(431\) 1.01990e9 0.613600 0.306800 0.951774i \(-0.400742\pi\)
0.306800 + 0.951774i \(0.400742\pi\)
\(432\) 0 0
\(433\) −4.96072e8 −0.293655 −0.146827 0.989162i \(-0.546906\pi\)
−0.146827 + 0.989162i \(0.546906\pi\)
\(434\) −5.85130e9 −3.43588
\(435\) 0 0
\(436\) 6.05413e8 0.349824
\(437\) −2.70868e8 −0.155265
\(438\) 0 0
\(439\) 1.84581e9 1.04126 0.520632 0.853781i \(-0.325697\pi\)
0.520632 + 0.853781i \(0.325697\pi\)
\(440\) 7.69440e6 0.00430616
\(441\) 0 0
\(442\) −2.51643e8 −0.138614
\(443\) −2.12406e8 −0.116079 −0.0580394 0.998314i \(-0.518485\pi\)
−0.0580394 + 0.998314i \(0.518485\pi\)
\(444\) 0 0
\(445\) 2.06049e8 0.110844
\(446\) 1.64392e9 0.877421
\(447\) 0 0
\(448\) 3.20747e9 1.68535
\(449\) 3.53221e9 1.84155 0.920777 0.390090i \(-0.127556\pi\)
0.920777 + 0.390090i \(0.127556\pi\)
\(450\) 0 0
\(451\) 1.31323e9 0.674098
\(452\) −2.77737e9 −1.41465
\(453\) 0 0
\(454\) −2.54865e9 −1.27825
\(455\) 4.88241e7 0.0242993
\(456\) 0 0
\(457\) 2.05595e9 1.00764 0.503821 0.863808i \(-0.331927\pi\)
0.503821 + 0.863808i \(0.331927\pi\)
\(458\) −3.74194e9 −1.81998
\(459\) 0 0
\(460\) 6.09533e7 0.0291975
\(461\) 1.34533e9 0.639551 0.319776 0.947493i \(-0.396392\pi\)
0.319776 + 0.947493i \(0.396392\pi\)
\(462\) 0 0
\(463\) 1.16277e9 0.544452 0.272226 0.962233i \(-0.412240\pi\)
0.272226 + 0.962233i \(0.412240\pi\)
\(464\) −2.79155e9 −1.29727
\(465\) 0 0
\(466\) 1.53746e9 0.703805
\(467\) −3.37053e9 −1.53140 −0.765700 0.643198i \(-0.777607\pi\)
−0.765700 + 0.643198i \(0.777607\pi\)
\(468\) 0 0
\(469\) −3.28251e9 −1.46927
\(470\) 6.49406e8 0.288519
\(471\) 0 0
\(472\) −2.82233e7 −0.0123541
\(473\) 1.29242e9 0.561552
\(474\) 0 0
\(475\) 1.43528e9 0.614484
\(476\) 2.44086e9 1.03733
\(477\) 0 0
\(478\) −3.54833e9 −1.48603
\(479\) 1.66586e9 0.692571 0.346286 0.938129i \(-0.387443\pi\)
0.346286 + 0.938129i \(0.387443\pi\)
\(480\) 0 0
\(481\) 3.06178e8 0.125449
\(482\) 1.22422e9 0.497961
\(483\) 0 0
\(484\) −2.20418e9 −0.883665
\(485\) −1.46453e8 −0.0582910
\(486\) 0 0
\(487\) −1.02449e9 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(488\) −1.89891e8 −0.0739665
\(489\) 0 0
\(490\) −5.07121e8 −0.194727
\(491\) −2.27469e9 −0.867235 −0.433618 0.901097i \(-0.642763\pi\)
−0.433618 + 0.901097i \(0.642763\pi\)
\(492\) 0 0
\(493\) −2.41673e9 −0.908373
\(494\) −3.54881e8 −0.132446
\(495\) 0 0
\(496\) 4.03970e9 1.48649
\(497\) 3.69688e9 1.35079
\(498\) 0 0
\(499\) 2.02609e9 0.729971 0.364986 0.931013i \(-0.381074\pi\)
0.364986 + 0.931013i \(0.381074\pi\)
\(500\) −6.49899e8 −0.232515
\(501\) 0 0
\(502\) −4.04168e9 −1.42593
\(503\) −1.54149e9 −0.540073 −0.270037 0.962850i \(-0.587036\pi\)
−0.270037 + 0.962850i \(0.587036\pi\)
\(504\) 0 0
\(505\) −5.50020e8 −0.190046
\(506\) 4.32486e8 0.148404
\(507\) 0 0
\(508\) −1.33832e9 −0.452935
\(509\) −3.87817e9 −1.30351 −0.651755 0.758430i \(-0.725967\pi\)
−0.651755 + 0.758430i \(0.725967\pi\)
\(510\) 0 0
\(511\) −5.39389e9 −1.78825
\(512\) −4.30828e9 −1.41860
\(513\) 0 0
\(514\) 1.54323e9 0.501258
\(515\) −5.62262e8 −0.181390
\(516\) 0 0
\(517\) 2.37750e9 0.756666
\(518\) −5.75574e9 −1.81948
\(519\) 0 0
\(520\) 4.94570e6 0.00154247
\(521\) 4.39564e8 0.136173 0.0680864 0.997679i \(-0.478311\pi\)
0.0680864 + 0.997679i \(0.478311\pi\)
\(522\) 0 0
\(523\) −6.36830e9 −1.94656 −0.973280 0.229623i \(-0.926251\pi\)
−0.973280 + 0.229623i \(0.926251\pi\)
\(524\) 3.48419e9 1.05789
\(525\) 0 0
\(526\) 1.25898e9 0.377197
\(527\) 3.49730e9 1.04087
\(528\) 0 0
\(529\) −3.19265e9 −0.937683
\(530\) −9.01581e8 −0.263051
\(531\) 0 0
\(532\) 3.44224e9 0.991174
\(533\) 8.44100e8 0.241462
\(534\) 0 0
\(535\) −4.85361e8 −0.137033
\(536\) −3.32507e8 −0.0932661
\(537\) 0 0
\(538\) −2.41276e9 −0.667999
\(539\) −1.85659e9 −0.510688
\(540\) 0 0
\(541\) −1.34859e9 −0.366175 −0.183088 0.983097i \(-0.558609\pi\)
−0.183088 + 0.983097i \(0.558609\pi\)
\(542\) −3.48065e9 −0.938995
\(543\) 0 0
\(544\) −3.49788e9 −0.931556
\(545\) 1.36066e8 0.0360049
\(546\) 0 0
\(547\) −2.38331e9 −0.622622 −0.311311 0.950308i \(-0.600768\pi\)
−0.311311 + 0.950308i \(0.600768\pi\)
\(548\) 7.64172e8 0.198362
\(549\) 0 0
\(550\) −2.29167e9 −0.587331
\(551\) −3.40821e9 −0.867951
\(552\) 0 0
\(553\) 9.38492e9 2.35989
\(554\) 7.30785e9 1.82602
\(555\) 0 0
\(556\) −1.35661e8 −0.0334729
\(557\) 2.39992e8 0.0588442 0.0294221 0.999567i \(-0.490633\pi\)
0.0294221 + 0.999567i \(0.490633\pi\)
\(558\) 0 0
\(559\) 8.30724e8 0.201148
\(560\) 6.33661e8 0.152475
\(561\) 0 0
\(562\) 8.90002e8 0.211502
\(563\) 1.44852e9 0.342093 0.171047 0.985263i \(-0.445285\pi\)
0.171047 + 0.985263i \(0.445285\pi\)
\(564\) 0 0
\(565\) −6.24211e8 −0.145600
\(566\) 1.13234e10 2.62493
\(567\) 0 0
\(568\) 3.74481e8 0.0857454
\(569\) −7.46267e9 −1.69825 −0.849124 0.528193i \(-0.822870\pi\)
−0.849124 + 0.528193i \(0.822870\pi\)
\(570\) 0 0
\(571\) 1.56608e9 0.352036 0.176018 0.984387i \(-0.443678\pi\)
0.176018 + 0.984387i \(0.443678\pi\)
\(572\) 2.92367e8 0.0653194
\(573\) 0 0
\(574\) −1.58679e10 −3.50211
\(575\) −1.12429e9 −0.246628
\(576\) 0 0
\(577\) −6.11728e7 −0.0132569 −0.00662847 0.999978i \(-0.502110\pi\)
−0.00662847 + 0.999978i \(0.502110\pi\)
\(578\) 3.84547e9 0.828327
\(579\) 0 0
\(580\) 7.66948e8 0.163218
\(581\) 1.12994e10 2.39023
\(582\) 0 0
\(583\) −3.30073e9 −0.689874
\(584\) −5.46382e8 −0.113515
\(585\) 0 0
\(586\) −2.58277e9 −0.530206
\(587\) −2.95616e9 −0.603245 −0.301623 0.953427i \(-0.597528\pi\)
−0.301623 + 0.953427i \(0.597528\pi\)
\(588\) 0 0
\(589\) 4.93208e9 0.994550
\(590\) −1.02424e8 −0.0205314
\(591\) 0 0
\(592\) 3.97372e9 0.787175
\(593\) −1.97406e8 −0.0388748 −0.0194374 0.999811i \(-0.506188\pi\)
−0.0194374 + 0.999811i \(0.506188\pi\)
\(594\) 0 0
\(595\) 5.48581e8 0.106766
\(596\) −3.56625e9 −0.690002
\(597\) 0 0
\(598\) 2.77987e8 0.0531583
\(599\) −8.20815e8 −0.156046 −0.0780228 0.996952i \(-0.524861\pi\)
−0.0780228 + 0.996952i \(0.524861\pi\)
\(600\) 0 0
\(601\) 9.75118e9 1.83230 0.916150 0.400837i \(-0.131280\pi\)
0.916150 + 0.400837i \(0.131280\pi\)
\(602\) −1.56165e10 −2.91740
\(603\) 0 0
\(604\) −9.59089e9 −1.77105
\(605\) −4.95386e8 −0.0909495
\(606\) 0 0
\(607\) −5.21137e9 −0.945784 −0.472892 0.881120i \(-0.656790\pi\)
−0.472892 + 0.881120i \(0.656790\pi\)
\(608\) −4.93290e9 −0.890103
\(609\) 0 0
\(610\) −6.89124e8 −0.122926
\(611\) 1.52818e9 0.271038
\(612\) 0 0
\(613\) −1.02466e9 −0.179666 −0.0898330 0.995957i \(-0.528633\pi\)
−0.0898330 + 0.995957i \(0.528633\pi\)
\(614\) −3.27761e9 −0.571437
\(615\) 0 0
\(616\) −3.40377e8 −0.0586715
\(617\) 1.71355e9 0.293696 0.146848 0.989159i \(-0.453087\pi\)
0.146848 + 0.989159i \(0.453087\pi\)
\(618\) 0 0
\(619\) 3.38211e9 0.573153 0.286577 0.958057i \(-0.407483\pi\)
0.286577 + 0.958057i \(0.407483\pi\)
\(620\) −1.10987e9 −0.187025
\(621\) 0 0
\(622\) 6.18596e9 1.03072
\(623\) −9.11499e9 −1.51025
\(624\) 0 0
\(625\) 5.88398e9 0.964031
\(626\) −2.34125e9 −0.381451
\(627\) 0 0
\(628\) −3.12000e9 −0.502684
\(629\) 3.44018e9 0.551193
\(630\) 0 0
\(631\) 2.96207e8 0.0469345 0.0234673 0.999725i \(-0.492529\pi\)
0.0234673 + 0.999725i \(0.492529\pi\)
\(632\) 9.50658e8 0.149801
\(633\) 0 0
\(634\) −6.26645e9 −0.976583
\(635\) −3.00785e8 −0.0466174
\(636\) 0 0
\(637\) −1.19336e9 −0.182929
\(638\) 5.44177e9 0.829599
\(639\) 0 0
\(640\) 1.37793e8 0.0207777
\(641\) −5.82615e9 −0.873732 −0.436866 0.899526i \(-0.643912\pi\)
−0.436866 + 0.899526i \(0.643912\pi\)
\(642\) 0 0
\(643\) −5.33573e9 −0.791508 −0.395754 0.918357i \(-0.629517\pi\)
−0.395754 + 0.918357i \(0.629517\pi\)
\(644\) −2.69639e9 −0.397816
\(645\) 0 0
\(646\) −3.98740e9 −0.581937
\(647\) −1.10418e10 −1.60279 −0.801393 0.598138i \(-0.795907\pi\)
−0.801393 + 0.598138i \(0.795907\pi\)
\(648\) 0 0
\(649\) −3.74978e8 −0.0538455
\(650\) −1.47301e9 −0.210382
\(651\) 0 0
\(652\) −8.54722e9 −1.20770
\(653\) 7.56490e9 1.06318 0.531590 0.847002i \(-0.321595\pi\)
0.531590 + 0.847002i \(0.321595\pi\)
\(654\) 0 0
\(655\) 7.83068e8 0.108882
\(656\) 1.09551e10 1.51514
\(657\) 0 0
\(658\) −2.87277e10 −3.93107
\(659\) 5.59162e9 0.761095 0.380547 0.924761i \(-0.375736\pi\)
0.380547 + 0.924761i \(0.375736\pi\)
\(660\) 0 0
\(661\) −1.88074e9 −0.253293 −0.126647 0.991948i \(-0.540421\pi\)
−0.126647 + 0.991948i \(0.540421\pi\)
\(662\) 8.28763e9 1.11027
\(663\) 0 0
\(664\) 1.14459e9 0.151727
\(665\) 7.73639e8 0.102015
\(666\) 0 0
\(667\) 2.66973e9 0.348359
\(668\) 2.76906e9 0.359431
\(669\) 0 0
\(670\) −1.20668e9 −0.155000
\(671\) −2.52291e9 −0.322384
\(672\) 0 0
\(673\) 8.51715e9 1.07706 0.538532 0.842605i \(-0.318979\pi\)
0.538532 + 0.842605i \(0.318979\pi\)
\(674\) −5.36634e9 −0.675101
\(675\) 0 0
\(676\) −8.37414e9 −1.04262
\(677\) 1.19888e10 1.48497 0.742484 0.669864i \(-0.233648\pi\)
0.742484 + 0.669864i \(0.233648\pi\)
\(678\) 0 0
\(679\) 6.47863e9 0.794216
\(680\) 5.55693e7 0.00677725
\(681\) 0 0
\(682\) −7.87489e9 −0.950603
\(683\) 7.29599e9 0.876217 0.438109 0.898922i \(-0.355649\pi\)
0.438109 + 0.898922i \(0.355649\pi\)
\(684\) 0 0
\(685\) 1.71747e8 0.0204161
\(686\) 4.26509e9 0.504422
\(687\) 0 0
\(688\) 1.07815e10 1.26218
\(689\) −2.12160e9 −0.247113
\(690\) 0 0
\(691\) 2.48922e9 0.287006 0.143503 0.989650i \(-0.454163\pi\)
0.143503 + 0.989650i \(0.454163\pi\)
\(692\) 6.04564e9 0.693540
\(693\) 0 0
\(694\) −2.50768e10 −2.84783
\(695\) −3.04897e7 −0.00344513
\(696\) 0 0
\(697\) 9.48420e9 1.06093
\(698\) 5.48450e9 0.610440
\(699\) 0 0
\(700\) 1.42877e10 1.57442
\(701\) −1.04516e10 −1.14596 −0.572980 0.819570i \(-0.694213\pi\)
−0.572980 + 0.819570i \(0.694213\pi\)
\(702\) 0 0
\(703\) 4.85153e9 0.526666
\(704\) 4.31673e9 0.466284
\(705\) 0 0
\(706\) −1.36432e10 −1.45915
\(707\) 2.43312e10 2.58938
\(708\) 0 0
\(709\) 4.68714e9 0.493908 0.246954 0.969027i \(-0.420570\pi\)
0.246954 + 0.969027i \(0.420570\pi\)
\(710\) 1.35901e9 0.142501
\(711\) 0 0
\(712\) −9.23316e8 −0.0958673
\(713\) −3.86342e9 −0.399171
\(714\) 0 0
\(715\) 6.57092e7 0.00672287
\(716\) −1.77847e10 −1.81072
\(717\) 0 0
\(718\) −9.35533e9 −0.943243
\(719\) 1.88048e10 1.88676 0.943381 0.331710i \(-0.107626\pi\)
0.943381 + 0.331710i \(0.107626\pi\)
\(720\) 0 0
\(721\) 2.48727e10 2.47144
\(722\) 8.91283e9 0.881324
\(723\) 0 0
\(724\) −1.18356e10 −1.15906
\(725\) −1.41465e10 −1.37869
\(726\) 0 0
\(727\) −1.41494e10 −1.36574 −0.682869 0.730541i \(-0.739268\pi\)
−0.682869 + 0.730541i \(0.739268\pi\)
\(728\) −2.18783e8 −0.0210162
\(729\) 0 0
\(730\) −1.98285e9 −0.188651
\(731\) 9.33391e9 0.883798
\(732\) 0 0
\(733\) −1.16913e10 −1.09647 −0.548237 0.836323i \(-0.684701\pi\)
−0.548237 + 0.836323i \(0.684701\pi\)
\(734\) 2.65482e9 0.247799
\(735\) 0 0
\(736\) 3.86407e9 0.357250
\(737\) −4.41772e9 −0.406502
\(738\) 0 0
\(739\) 8.53323e9 0.777782 0.388891 0.921284i \(-0.372858\pi\)
0.388891 + 0.921284i \(0.372858\pi\)
\(740\) −1.09174e9 −0.0990393
\(741\) 0 0
\(742\) 3.98832e10 3.58407
\(743\) 2.88722e9 0.258237 0.129119 0.991629i \(-0.458785\pi\)
0.129119 + 0.991629i \(0.458785\pi\)
\(744\) 0 0
\(745\) −8.01512e8 −0.0710171
\(746\) 8.17870e9 0.721271
\(747\) 0 0
\(748\) 3.28500e9 0.286998
\(749\) 2.14709e10 1.86708
\(750\) 0 0
\(751\) 1.71155e10 1.47452 0.737258 0.675611i \(-0.236120\pi\)
0.737258 + 0.675611i \(0.236120\pi\)
\(752\) 1.98334e10 1.70073
\(753\) 0 0
\(754\) 3.49779e9 0.297162
\(755\) −2.15554e9 −0.182281
\(756\) 0 0
\(757\) 1.12521e10 0.942751 0.471375 0.881933i \(-0.343758\pi\)
0.471375 + 0.881933i \(0.343758\pi\)
\(758\) −3.00290e10 −2.50437
\(759\) 0 0
\(760\) 7.83669e7 0.00647567
\(761\) −9.66355e9 −0.794859 −0.397430 0.917633i \(-0.630098\pi\)
−0.397430 + 0.917633i \(0.630098\pi\)
\(762\) 0 0
\(763\) −6.01915e9 −0.490568
\(764\) 1.37004e10 1.11150
\(765\) 0 0
\(766\) 1.27785e6 0.000102725 0
\(767\) −2.41023e8 −0.0192875
\(768\) 0 0
\(769\) 6.93012e8 0.0549539 0.0274769 0.999622i \(-0.491253\pi\)
0.0274769 + 0.999622i \(0.491253\pi\)
\(770\) −1.23524e9 −0.0975069
\(771\) 0 0
\(772\) 2.22504e10 1.74051
\(773\) 6.71030e8 0.0522533 0.0261266 0.999659i \(-0.491683\pi\)
0.0261266 + 0.999659i \(0.491683\pi\)
\(774\) 0 0
\(775\) 2.04716e10 1.57978
\(776\) 6.56262e8 0.0504152
\(777\) 0 0
\(778\) 1.65681e10 1.26138
\(779\) 1.33751e10 1.01372
\(780\) 0 0
\(781\) 4.97540e9 0.373723
\(782\) 3.12343e9 0.233565
\(783\) 0 0
\(784\) −1.54879e10 −1.14785
\(785\) −7.01216e8 −0.0517378
\(786\) 0 0
\(787\) −1.62943e10 −1.19158 −0.595790 0.803141i \(-0.703161\pi\)
−0.595790 + 0.803141i \(0.703161\pi\)
\(788\) −1.67605e10 −1.22024
\(789\) 0 0
\(790\) 3.44999e9 0.248956
\(791\) 2.76132e10 1.98380
\(792\) 0 0
\(793\) −1.62165e9 −0.115478
\(794\) −5.61933e9 −0.398394
\(795\) 0 0
\(796\) 2.53664e10 1.78264
\(797\) −6.31608e9 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(798\) 0 0
\(799\) 1.71704e10 1.19088
\(800\) −2.04751e10 −1.41387
\(801\) 0 0
\(802\) −3.49780e10 −2.39433
\(803\) −7.25929e9 −0.494755
\(804\) 0 0
\(805\) −6.06011e8 −0.0409444
\(806\) −5.06172e9 −0.340506
\(807\) 0 0
\(808\) 2.46466e9 0.164368
\(809\) 5.93171e9 0.393876 0.196938 0.980416i \(-0.436900\pi\)
0.196938 + 0.980416i \(0.436900\pi\)
\(810\) 0 0
\(811\) 3.74376e9 0.246454 0.123227 0.992379i \(-0.460676\pi\)
0.123227 + 0.992379i \(0.460676\pi\)
\(812\) −3.39274e10 −2.22385
\(813\) 0 0
\(814\) −7.74628e9 −0.503394
\(815\) −1.92098e9 −0.124300
\(816\) 0 0
\(817\) 1.31632e10 0.844470
\(818\) 3.63037e9 0.231907
\(819\) 0 0
\(820\) −3.00981e9 −0.190629
\(821\) −2.22338e10 −1.40221 −0.701103 0.713060i \(-0.747309\pi\)
−0.701103 + 0.713060i \(0.747309\pi\)
\(822\) 0 0
\(823\) −2.57937e10 −1.61292 −0.806462 0.591286i \(-0.798620\pi\)
−0.806462 + 0.591286i \(0.798620\pi\)
\(824\) 2.51952e9 0.156882
\(825\) 0 0
\(826\) 4.53092e9 0.279741
\(827\) −2.36853e10 −1.45616 −0.728082 0.685490i \(-0.759588\pi\)
−0.728082 + 0.685490i \(0.759588\pi\)
\(828\) 0 0
\(829\) −3.07672e10 −1.87563 −0.937815 0.347137i \(-0.887154\pi\)
−0.937815 + 0.347137i \(0.887154\pi\)
\(830\) 4.15377e9 0.252156
\(831\) 0 0
\(832\) 2.77465e9 0.167023
\(833\) −1.34084e10 −0.803746
\(834\) 0 0
\(835\) 6.22344e8 0.0369937
\(836\) 4.63268e9 0.274227
\(837\) 0 0
\(838\) −1.89594e10 −1.11294
\(839\) 9.92051e9 0.579919 0.289959 0.957039i \(-0.406358\pi\)
0.289959 + 0.957039i \(0.406358\pi\)
\(840\) 0 0
\(841\) 1.63422e10 0.947379
\(842\) 1.53528e10 0.886332
\(843\) 0 0
\(844\) 5.80786e8 0.0332520
\(845\) −1.88208e9 −0.107310
\(846\) 0 0
\(847\) 2.19144e10 1.23919
\(848\) −2.75351e10 −1.55060
\(849\) 0 0
\(850\) −1.65505e10 −0.924371
\(851\) −3.80032e9 −0.211382
\(852\) 0 0
\(853\) 1.89035e10 1.04285 0.521424 0.853298i \(-0.325401\pi\)
0.521424 + 0.853298i \(0.325401\pi\)
\(854\) 3.04847e10 1.67487
\(855\) 0 0
\(856\) 2.17492e9 0.118518
\(857\) −1.56380e10 −0.848688 −0.424344 0.905501i \(-0.639495\pi\)
−0.424344 + 0.905501i \(0.639495\pi\)
\(858\) 0 0
\(859\) 3.59002e10 1.93251 0.966254 0.257592i \(-0.0829291\pi\)
0.966254 + 0.257592i \(0.0829291\pi\)
\(860\) −2.96211e9 −0.158802
\(861\) 0 0
\(862\) −1.65855e10 −0.881968
\(863\) 3.31160e10 1.75388 0.876941 0.480599i \(-0.159581\pi\)
0.876941 + 0.480599i \(0.159581\pi\)
\(864\) 0 0
\(865\) 1.35875e9 0.0713812
\(866\) 8.06708e9 0.422089
\(867\) 0 0
\(868\) 4.90970e10 2.54822
\(869\) 1.26306e10 0.652910
\(870\) 0 0
\(871\) −2.83956e9 −0.145609
\(872\) −6.09718e8 −0.0311402
\(873\) 0 0
\(874\) 4.40483e9 0.223172
\(875\) 6.46143e9 0.326062
\(876\) 0 0
\(877\) 3.12644e10 1.56513 0.782567 0.622566i \(-0.213910\pi\)
0.782567 + 0.622566i \(0.213910\pi\)
\(878\) −3.00164e10 −1.49668
\(879\) 0 0
\(880\) 8.52804e8 0.0421852
\(881\) 1.15281e9 0.0567991 0.0283996 0.999597i \(-0.490959\pi\)
0.0283996 + 0.999597i \(0.490959\pi\)
\(882\) 0 0
\(883\) −7.36328e9 −0.359922 −0.179961 0.983674i \(-0.557597\pi\)
−0.179961 + 0.983674i \(0.557597\pi\)
\(884\) 2.11149e9 0.102803
\(885\) 0 0
\(886\) 3.45413e9 0.166848
\(887\) −2.01350e10 −0.968765 −0.484383 0.874856i \(-0.660956\pi\)
−0.484383 + 0.874856i \(0.660956\pi\)
\(888\) 0 0
\(889\) 1.33058e10 0.635163
\(890\) −3.35076e9 −0.159323
\(891\) 0 0
\(892\) −1.37938e10 −0.650738
\(893\) 2.42147e10 1.13789
\(894\) 0 0
\(895\) −3.99710e9 −0.186365
\(896\) −6.09554e9 −0.283096
\(897\) 0 0
\(898\) −5.74406e10 −2.64698
\(899\) −4.86117e10 −2.23142
\(900\) 0 0
\(901\) −2.38380e10 −1.08576
\(902\) −2.13557e10 −0.968925
\(903\) 0 0
\(904\) 2.79712e9 0.125928
\(905\) −2.66005e9 −0.119294
\(906\) 0 0
\(907\) −4.37196e10 −1.94559 −0.972793 0.231677i \(-0.925579\pi\)
−0.972793 + 0.231677i \(0.925579\pi\)
\(908\) 2.13852e10 0.948010
\(909\) 0 0
\(910\) −7.93974e8 −0.0349270
\(911\) −2.71783e10 −1.19099 −0.595494 0.803360i \(-0.703044\pi\)
−0.595494 + 0.803360i \(0.703044\pi\)
\(912\) 0 0
\(913\) 1.52071e10 0.661302
\(914\) −3.34337e10 −1.44835
\(915\) 0 0
\(916\) 3.13978e10 1.34979
\(917\) −3.46405e10 −1.48351
\(918\) 0 0
\(919\) −1.97736e10 −0.840393 −0.420196 0.907433i \(-0.638039\pi\)
−0.420196 + 0.907433i \(0.638039\pi\)
\(920\) −6.13867e7 −0.00259906
\(921\) 0 0
\(922\) −2.18777e10 −0.919269
\(923\) 3.19802e9 0.133868
\(924\) 0 0
\(925\) 2.01373e10 0.836576
\(926\) −1.89088e10 −0.782576
\(927\) 0 0
\(928\) 4.86198e10 1.99708
\(929\) 3.68821e10 1.50925 0.754625 0.656156i \(-0.227819\pi\)
0.754625 + 0.656156i \(0.227819\pi\)
\(930\) 0 0
\(931\) −1.89092e10 −0.767981
\(932\) −1.29005e10 −0.521976
\(933\) 0 0
\(934\) 5.48113e10 2.20118
\(935\) 7.38300e8 0.0295388
\(936\) 0 0
\(937\) −2.54302e10 −1.00986 −0.504930 0.863160i \(-0.668482\pi\)
−0.504930 + 0.863160i \(0.668482\pi\)
\(938\) 5.33800e10 2.11188
\(939\) 0 0
\(940\) −5.44903e9 −0.213979
\(941\) 1.08449e10 0.424288 0.212144 0.977238i \(-0.431955\pi\)
0.212144 + 0.977238i \(0.431955\pi\)
\(942\) 0 0
\(943\) −1.04771e10 −0.406864
\(944\) −3.12811e9 −0.121027
\(945\) 0 0
\(946\) −2.10172e10 −0.807155
\(947\) 2.44245e9 0.0934547 0.0467273 0.998908i \(-0.485121\pi\)
0.0467273 + 0.998908i \(0.485121\pi\)
\(948\) 0 0
\(949\) −4.66603e9 −0.177221
\(950\) −2.33405e10 −0.883237
\(951\) 0 0
\(952\) −2.45822e9 −0.0923402
\(953\) 3.90381e10 1.46104 0.730522 0.682890i \(-0.239277\pi\)
0.730522 + 0.682890i \(0.239277\pi\)
\(954\) 0 0
\(955\) 3.07916e9 0.114398
\(956\) 2.97733e10 1.10211
\(957\) 0 0
\(958\) −2.70901e10 −0.995478
\(959\) −7.59756e9 −0.278169
\(960\) 0 0
\(961\) 4.28343e10 1.55690
\(962\) −4.97905e9 −0.180316
\(963\) 0 0
\(964\) −1.02722e10 −0.369312
\(965\) 5.00076e9 0.179139
\(966\) 0 0
\(967\) −2.52051e10 −0.896388 −0.448194 0.893936i \(-0.647933\pi\)
−0.448194 + 0.893936i \(0.647933\pi\)
\(968\) 2.21985e9 0.0786610
\(969\) 0 0
\(970\) 2.38161e9 0.0837855
\(971\) 6.22835e9 0.218326 0.109163 0.994024i \(-0.465183\pi\)
0.109163 + 0.994024i \(0.465183\pi\)
\(972\) 0 0
\(973\) 1.34877e9 0.0469400
\(974\) 1.66601e10 0.577726
\(975\) 0 0
\(976\) −2.10465e10 −0.724611
\(977\) −1.86878e10 −0.641103 −0.320551 0.947231i \(-0.603868\pi\)
−0.320551 + 0.947231i \(0.603868\pi\)
\(978\) 0 0
\(979\) −1.22673e10 −0.417839
\(980\) 4.25514e9 0.144419
\(981\) 0 0
\(982\) 3.69908e10 1.24653
\(983\) −4.26187e10 −1.43108 −0.715538 0.698574i \(-0.753818\pi\)
−0.715538 + 0.698574i \(0.753818\pi\)
\(984\) 0 0
\(985\) −3.76691e9 −0.125591
\(986\) 3.93007e10 1.30566
\(987\) 0 0
\(988\) 2.97773e9 0.0982283
\(989\) −1.03111e10 −0.338935
\(990\) 0 0
\(991\) −3.72023e9 −0.121426 −0.0607130 0.998155i \(-0.519337\pi\)
−0.0607130 + 0.998155i \(0.519337\pi\)
\(992\) −7.03586e10 −2.28837
\(993\) 0 0
\(994\) −6.01185e10 −1.94158
\(995\) 5.70107e9 0.183474
\(996\) 0 0
\(997\) 3.00026e10 0.958794 0.479397 0.877598i \(-0.340855\pi\)
0.479397 + 0.877598i \(0.340855\pi\)
\(998\) −3.29481e10 −1.04924
\(999\) 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.1.2 17
3.2 odd 2 177.8.a.b.1.16 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.16 17 3.2 odd 2
531.8.a.d.1.2 17 1.1 even 1 trivial