Properties

Label 531.8.a.d.1.1
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 1.1
Root \(24.0278\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-22.0278 q^{2} +357.226 q^{4} +492.460 q^{5} -1209.81 q^{7} -5049.35 q^{8} +O(q^{10})\) \(q-22.0278 q^{2} +357.226 q^{4} +492.460 q^{5} -1209.81 q^{7} -5049.35 q^{8} -10847.8 q^{10} +7759.91 q^{11} +1927.14 q^{13} +26649.5 q^{14} +65501.4 q^{16} +21871.8 q^{17} -31682.0 q^{19} +175919. q^{20} -170934. q^{22} -76017.8 q^{23} +164392. q^{25} -42450.8 q^{26} -432176. q^{28} +107474. q^{29} -56872.6 q^{31} -796537. q^{32} -481789. q^{34} -595783. q^{35} -72532.0 q^{37} +697885. q^{38} -2.48660e6 q^{40} +598173. q^{41} -127161. q^{43} +2.77204e6 q^{44} +1.67451e6 q^{46} -20753.9 q^{47} +640100. q^{49} -3.62120e6 q^{50} +688425. q^{52} -1.52906e6 q^{53} +3.82145e6 q^{55} +6.10876e6 q^{56} -2.36741e6 q^{58} +205379. q^{59} +2.40366e6 q^{61} +1.25278e6 q^{62} +9.16182e6 q^{64} +949040. q^{65} -606916. q^{67} +7.81319e6 q^{68} +1.31238e7 q^{70} +3.52312e6 q^{71} +1.27495e6 q^{73} +1.59772e6 q^{74} -1.13176e7 q^{76} -9.38803e6 q^{77} +3.89830e6 q^{79} +3.22568e7 q^{80} -1.31765e7 q^{82} +4.03800e6 q^{83} +1.07710e7 q^{85} +2.80108e6 q^{86} -3.91825e7 q^{88} -5.97046e6 q^{89} -2.33148e6 q^{91} -2.71555e7 q^{92} +457163. q^{94} -1.56021e7 q^{95} -1.98127e6 q^{97} -1.41000e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.0278 −1.94700 −0.973502 0.228677i \(-0.926560\pi\)
−0.973502 + 0.228677i \(0.926560\pi\)
\(3\) 0 0
\(4\) 357.226 2.79083
\(5\) 492.460 1.76188 0.880939 0.473230i \(-0.156912\pi\)
0.880939 + 0.473230i \(0.156912\pi\)
\(6\) 0 0
\(7\) −1209.81 −1.33314 −0.666568 0.745444i \(-0.732237\pi\)
−0.666568 + 0.745444i \(0.732237\pi\)
\(8\) −5049.35 −3.48675
\(9\) 0 0
\(10\) −10847.8 −3.43038
\(11\) 7759.91 1.75785 0.878926 0.476958i \(-0.158261\pi\)
0.878926 + 0.476958i \(0.158261\pi\)
\(12\) 0 0
\(13\) 1927.14 0.243283 0.121642 0.992574i \(-0.461184\pi\)
0.121642 + 0.992574i \(0.461184\pi\)
\(14\) 26649.5 2.59562
\(15\) 0 0
\(16\) 65501.4 3.99789
\(17\) 21871.8 1.07973 0.539864 0.841752i \(-0.318476\pi\)
0.539864 + 0.841752i \(0.318476\pi\)
\(18\) 0 0
\(19\) −31682.0 −1.05968 −0.529839 0.848098i \(-0.677748\pi\)
−0.529839 + 0.848098i \(0.677748\pi\)
\(20\) 175919. 4.91710
\(21\) 0 0
\(22\) −170934. −3.42255
\(23\) −76017.8 −1.30277 −0.651385 0.758747i \(-0.725812\pi\)
−0.651385 + 0.758747i \(0.725812\pi\)
\(24\) 0 0
\(25\) 164392. 2.10421
\(26\) −42450.8 −0.473673
\(27\) 0 0
\(28\) −432176. −3.72055
\(29\) 107474. 0.818293 0.409146 0.912469i \(-0.365827\pi\)
0.409146 + 0.912469i \(0.365827\pi\)
\(30\) 0 0
\(31\) −56872.6 −0.342876 −0.171438 0.985195i \(-0.554841\pi\)
−0.171438 + 0.985195i \(0.554841\pi\)
\(32\) −796537. −4.29715
\(33\) 0 0
\(34\) −481789. −2.10223
\(35\) −595783. −2.34882
\(36\) 0 0
\(37\) −72532.0 −0.235409 −0.117705 0.993049i \(-0.537554\pi\)
−0.117705 + 0.993049i \(0.537554\pi\)
\(38\) 697885. 2.06320
\(39\) 0 0
\(40\) −2.48660e6 −6.14322
\(41\) 598173. 1.35545 0.677725 0.735316i \(-0.262966\pi\)
0.677725 + 0.735316i \(0.262966\pi\)
\(42\) 0 0
\(43\) −127161. −0.243902 −0.121951 0.992536i \(-0.538915\pi\)
−0.121951 + 0.992536i \(0.538915\pi\)
\(44\) 2.77204e6 4.90586
\(45\) 0 0
\(46\) 1.67451e6 2.53650
\(47\) −20753.9 −0.0291579 −0.0145790 0.999894i \(-0.504641\pi\)
−0.0145790 + 0.999894i \(0.504641\pi\)
\(48\) 0 0
\(49\) 640100. 0.777252
\(50\) −3.62120e6 −4.09691
\(51\) 0 0
\(52\) 688425. 0.678961
\(53\) −1.52906e6 −1.41078 −0.705389 0.708821i \(-0.749228\pi\)
−0.705389 + 0.708821i \(0.749228\pi\)
\(54\) 0 0
\(55\) 3.82145e6 3.09712
\(56\) 6.10876e6 4.64831
\(57\) 0 0
\(58\) −2.36741e6 −1.59322
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 2.40366e6 1.35587 0.677935 0.735122i \(-0.262875\pi\)
0.677935 + 0.735122i \(0.262875\pi\)
\(62\) 1.25278e6 0.667582
\(63\) 0 0
\(64\) 9.16182e6 4.36869
\(65\) 949040. 0.428635
\(66\) 0 0
\(67\) −606916. −0.246528 −0.123264 0.992374i \(-0.539336\pi\)
−0.123264 + 0.992374i \(0.539336\pi\)
\(68\) 7.81319e6 3.01333
\(69\) 0 0
\(70\) 1.31238e7 4.57317
\(71\) 3.52312e6 1.16822 0.584108 0.811676i \(-0.301444\pi\)
0.584108 + 0.811676i \(0.301444\pi\)
\(72\) 0 0
\(73\) 1.27495e6 0.383585 0.191793 0.981435i \(-0.438570\pi\)
0.191793 + 0.981435i \(0.438570\pi\)
\(74\) 1.59772e6 0.458343
\(75\) 0 0
\(76\) −1.13176e7 −2.95738
\(77\) −9.38803e6 −2.34346
\(78\) 0 0
\(79\) 3.89830e6 0.889571 0.444786 0.895637i \(-0.353280\pi\)
0.444786 + 0.895637i \(0.353280\pi\)
\(80\) 3.22568e7 7.04379
\(81\) 0 0
\(82\) −1.31765e7 −2.63907
\(83\) 4.03800e6 0.775164 0.387582 0.921835i \(-0.373310\pi\)
0.387582 + 0.921835i \(0.373310\pi\)
\(84\) 0 0
\(85\) 1.07710e7 1.90235
\(86\) 2.80108e6 0.474877
\(87\) 0 0
\(88\) −3.91825e7 −6.12919
\(89\) −5.97046e6 −0.897724 −0.448862 0.893601i \(-0.648170\pi\)
−0.448862 + 0.893601i \(0.648170\pi\)
\(90\) 0 0
\(91\) −2.33148e6 −0.324330
\(92\) −2.71555e7 −3.63581
\(93\) 0 0
\(94\) 457163. 0.0567706
\(95\) −1.56021e7 −1.86703
\(96\) 0 0
\(97\) −1.98127e6 −0.220415 −0.110208 0.993909i \(-0.535152\pi\)
−0.110208 + 0.993909i \(0.535152\pi\)
\(98\) −1.41000e7 −1.51331
\(99\) 0 0
\(100\) 5.87250e7 5.87250
\(101\) 1.54871e7 1.49570 0.747851 0.663867i \(-0.231086\pi\)
0.747851 + 0.663867i \(0.231086\pi\)
\(102\) 0 0
\(103\) −1.20275e7 −1.08454 −0.542269 0.840205i \(-0.682435\pi\)
−0.542269 + 0.840205i \(0.682435\pi\)
\(104\) −9.73081e6 −0.848267
\(105\) 0 0
\(106\) 3.36819e7 2.74679
\(107\) 7.69022e6 0.606870 0.303435 0.952852i \(-0.401866\pi\)
0.303435 + 0.952852i \(0.401866\pi\)
\(108\) 0 0
\(109\) −2.11383e6 −0.156342 −0.0781711 0.996940i \(-0.524908\pi\)
−0.0781711 + 0.996940i \(0.524908\pi\)
\(110\) −8.41782e7 −6.03011
\(111\) 0 0
\(112\) −7.92443e7 −5.32973
\(113\) −2.03050e6 −0.132382 −0.0661909 0.997807i \(-0.521085\pi\)
−0.0661909 + 0.997807i \(0.521085\pi\)
\(114\) 0 0
\(115\) −3.74357e7 −2.29532
\(116\) 3.83923e7 2.28371
\(117\) 0 0
\(118\) −4.52406e6 −0.253478
\(119\) −2.64608e7 −1.43942
\(120\) 0 0
\(121\) 4.07291e7 2.09005
\(122\) −5.29474e7 −2.63989
\(123\) 0 0
\(124\) −2.03164e7 −0.956908
\(125\) 4.24829e7 1.94549
\(126\) 0 0
\(127\) 1.37874e7 0.597269 0.298635 0.954367i \(-0.403469\pi\)
0.298635 + 0.954367i \(0.403469\pi\)
\(128\) −9.98583e7 −4.20871
\(129\) 0 0
\(130\) −2.09053e7 −0.834555
\(131\) 845013. 0.0328408 0.0164204 0.999865i \(-0.494773\pi\)
0.0164204 + 0.999865i \(0.494773\pi\)
\(132\) 0 0
\(133\) 3.83292e7 1.41270
\(134\) 1.33690e7 0.479992
\(135\) 0 0
\(136\) −1.10439e8 −3.76474
\(137\) 4.84595e7 1.61012 0.805058 0.593196i \(-0.202134\pi\)
0.805058 + 0.593196i \(0.202134\pi\)
\(138\) 0 0
\(139\) −2.05210e6 −0.0648106 −0.0324053 0.999475i \(-0.510317\pi\)
−0.0324053 + 0.999475i \(0.510317\pi\)
\(140\) −2.12829e8 −6.55516
\(141\) 0 0
\(142\) −7.76067e7 −2.27452
\(143\) 1.49545e7 0.427656
\(144\) 0 0
\(145\) 5.29264e7 1.44173
\(146\) −2.80843e7 −0.746842
\(147\) 0 0
\(148\) −2.59103e7 −0.656987
\(149\) 2.53433e6 0.0627641 0.0313821 0.999507i \(-0.490009\pi\)
0.0313821 + 0.999507i \(0.490009\pi\)
\(150\) 0 0
\(151\) −5.49477e7 −1.29876 −0.649382 0.760462i \(-0.724972\pi\)
−0.649382 + 0.760462i \(0.724972\pi\)
\(152\) 1.59973e8 3.69483
\(153\) 0 0
\(154\) 2.06798e8 4.56272
\(155\) −2.80075e7 −0.604106
\(156\) 0 0
\(157\) 7.27529e7 1.50038 0.750191 0.661221i \(-0.229962\pi\)
0.750191 + 0.661221i \(0.229962\pi\)
\(158\) −8.58712e7 −1.73200
\(159\) 0 0
\(160\) −3.92263e8 −7.57106
\(161\) 9.19672e7 1.73677
\(162\) 0 0
\(163\) 1.99055e6 0.0360011 0.0180006 0.999838i \(-0.494270\pi\)
0.0180006 + 0.999838i \(0.494270\pi\)
\(164\) 2.13683e8 3.78283
\(165\) 0 0
\(166\) −8.89485e7 −1.50925
\(167\) 3.49476e7 0.580643 0.290322 0.956929i \(-0.406238\pi\)
0.290322 + 0.956929i \(0.406238\pi\)
\(168\) 0 0
\(169\) −5.90346e7 −0.940813
\(170\) −2.37262e8 −3.70388
\(171\) 0 0
\(172\) −4.54252e7 −0.680687
\(173\) 2.91994e7 0.428759 0.214380 0.976750i \(-0.431227\pi\)
0.214380 + 0.976750i \(0.431227\pi\)
\(174\) 0 0
\(175\) −1.98883e8 −2.80520
\(176\) 5.08285e8 7.02769
\(177\) 0 0
\(178\) 1.31516e8 1.74787
\(179\) −1.31626e8 −1.71536 −0.857680 0.514184i \(-0.828095\pi\)
−0.857680 + 0.514184i \(0.828095\pi\)
\(180\) 0 0
\(181\) −1.51957e8 −1.90479 −0.952393 0.304872i \(-0.901386\pi\)
−0.952393 + 0.304872i \(0.901386\pi\)
\(182\) 5.13574e7 0.631471
\(183\) 0 0
\(184\) 3.83840e8 4.54243
\(185\) −3.57191e7 −0.414763
\(186\) 0 0
\(187\) 1.69724e8 1.89800
\(188\) −7.41382e6 −0.0813747
\(189\) 0 0
\(190\) 3.43680e8 3.63511
\(191\) 1.21713e8 1.26392 0.631962 0.775000i \(-0.282250\pi\)
0.631962 + 0.775000i \(0.282250\pi\)
\(192\) 0 0
\(193\) 1.32612e8 1.32780 0.663900 0.747821i \(-0.268900\pi\)
0.663900 + 0.747821i \(0.268900\pi\)
\(194\) 4.36430e7 0.429149
\(195\) 0 0
\(196\) 2.28660e8 2.16917
\(197\) −1.11991e8 −1.04364 −0.521819 0.853056i \(-0.674746\pi\)
−0.521819 + 0.853056i \(0.674746\pi\)
\(198\) 0 0
\(199\) −1.81688e8 −1.63433 −0.817165 0.576403i \(-0.804456\pi\)
−0.817165 + 0.576403i \(0.804456\pi\)
\(200\) −8.30071e8 −7.33686
\(201\) 0 0
\(202\) −3.41147e8 −2.91214
\(203\) −1.30023e8 −1.09090
\(204\) 0 0
\(205\) 2.94576e8 2.38814
\(206\) 2.64940e8 2.11160
\(207\) 0 0
\(208\) 1.26230e8 0.972618
\(209\) −2.45849e8 −1.86276
\(210\) 0 0
\(211\) −3.34639e6 −0.0245238 −0.0122619 0.999925i \(-0.503903\pi\)
−0.0122619 + 0.999925i \(0.503903\pi\)
\(212\) −5.46219e8 −3.93724
\(213\) 0 0
\(214\) −1.69399e8 −1.18158
\(215\) −6.26217e7 −0.429725
\(216\) 0 0
\(217\) 6.88051e7 0.457101
\(218\) 4.65630e7 0.304399
\(219\) 0 0
\(220\) 1.36512e9 8.64353
\(221\) 4.21501e7 0.262679
\(222\) 0 0
\(223\) 3.23603e7 0.195410 0.0977048 0.995215i \(-0.468850\pi\)
0.0977048 + 0.995215i \(0.468850\pi\)
\(224\) 9.63659e8 5.72869
\(225\) 0 0
\(226\) 4.47275e7 0.257748
\(227\) −1.60575e8 −0.911145 −0.455573 0.890199i \(-0.650565\pi\)
−0.455573 + 0.890199i \(0.650565\pi\)
\(228\) 0 0
\(229\) 4.34106e7 0.238876 0.119438 0.992842i \(-0.461891\pi\)
0.119438 + 0.992842i \(0.461891\pi\)
\(230\) 8.24628e8 4.46900
\(231\) 0 0
\(232\) −5.42672e8 −2.85318
\(233\) 2.86270e7 0.148262 0.0741310 0.997249i \(-0.476382\pi\)
0.0741310 + 0.997249i \(0.476382\pi\)
\(234\) 0 0
\(235\) −1.02204e7 −0.0513727
\(236\) 7.33667e7 0.363335
\(237\) 0 0
\(238\) 5.82874e8 2.80256
\(239\) 6.06584e7 0.287408 0.143704 0.989621i \(-0.454099\pi\)
0.143704 + 0.989621i \(0.454099\pi\)
\(240\) 0 0
\(241\) 5.03046e7 0.231498 0.115749 0.993278i \(-0.463073\pi\)
0.115749 + 0.993278i \(0.463073\pi\)
\(242\) −8.97173e8 −4.06933
\(243\) 0 0
\(244\) 8.58648e8 3.78400
\(245\) 3.15224e8 1.36942
\(246\) 0 0
\(247\) −6.10556e7 −0.257802
\(248\) 2.87170e8 1.19552
\(249\) 0 0
\(250\) −9.35807e8 −3.78788
\(251\) −1.79753e8 −0.717494 −0.358747 0.933435i \(-0.616796\pi\)
−0.358747 + 0.933435i \(0.616796\pi\)
\(252\) 0 0
\(253\) −5.89891e8 −2.29008
\(254\) −3.03707e8 −1.16289
\(255\) 0 0
\(256\) 1.02695e9 3.82569
\(257\) −8.52960e7 −0.313446 −0.156723 0.987643i \(-0.550093\pi\)
−0.156723 + 0.987643i \(0.550093\pi\)
\(258\) 0 0
\(259\) 8.77500e7 0.313833
\(260\) 3.39022e8 1.19625
\(261\) 0 0
\(262\) −1.86138e7 −0.0639412
\(263\) −4.14430e8 −1.40477 −0.702386 0.711796i \(-0.747882\pi\)
−0.702386 + 0.711796i \(0.747882\pi\)
\(264\) 0 0
\(265\) −7.53000e8 −2.48562
\(266\) −8.44309e8 −2.75053
\(267\) 0 0
\(268\) −2.16806e8 −0.688018
\(269\) 3.07821e8 0.964196 0.482098 0.876117i \(-0.339875\pi\)
0.482098 + 0.876117i \(0.339875\pi\)
\(270\) 0 0
\(271\) −2.35783e8 −0.719646 −0.359823 0.933021i \(-0.617163\pi\)
−0.359823 + 0.933021i \(0.617163\pi\)
\(272\) 1.43264e9 4.31663
\(273\) 0 0
\(274\) −1.06746e9 −3.13490
\(275\) 1.27567e9 3.69890
\(276\) 0 0
\(277\) −2.15483e8 −0.609165 −0.304582 0.952486i \(-0.598517\pi\)
−0.304582 + 0.952486i \(0.598517\pi\)
\(278\) 4.52032e7 0.126186
\(279\) 0 0
\(280\) 3.00832e9 8.18975
\(281\) 2.06356e8 0.554810 0.277405 0.960753i \(-0.410526\pi\)
0.277405 + 0.960753i \(0.410526\pi\)
\(282\) 0 0
\(283\) 6.73678e8 1.76685 0.883426 0.468571i \(-0.155231\pi\)
0.883426 + 0.468571i \(0.155231\pi\)
\(284\) 1.25855e9 3.26029
\(285\) 0 0
\(286\) −3.29414e8 −0.832648
\(287\) −7.23677e8 −1.80700
\(288\) 0 0
\(289\) 6.80388e7 0.165811
\(290\) −1.16586e9 −2.80706
\(291\) 0 0
\(292\) 4.55444e8 1.07052
\(293\) 2.68265e8 0.623056 0.311528 0.950237i \(-0.399159\pi\)
0.311528 + 0.950237i \(0.399159\pi\)
\(294\) 0 0
\(295\) 1.01141e8 0.229377
\(296\) 3.66239e8 0.820813
\(297\) 0 0
\(298\) −5.58259e7 −0.122202
\(299\) −1.46497e8 −0.316942
\(300\) 0 0
\(301\) 1.53841e8 0.325154
\(302\) 1.21038e9 2.52870
\(303\) 0 0
\(304\) −2.07521e9 −4.23648
\(305\) 1.18370e9 2.38888
\(306\) 0 0
\(307\) 6.59253e8 1.30037 0.650186 0.759775i \(-0.274691\pi\)
0.650186 + 0.759775i \(0.274691\pi\)
\(308\) −3.35365e9 −6.54018
\(309\) 0 0
\(310\) 6.16945e8 1.17620
\(311\) 8.73613e8 1.64686 0.823432 0.567415i \(-0.192056\pi\)
0.823432 + 0.567415i \(0.192056\pi\)
\(312\) 0 0
\(313\) 2.84022e8 0.523537 0.261768 0.965131i \(-0.415694\pi\)
0.261768 + 0.965131i \(0.415694\pi\)
\(314\) −1.60259e9 −2.92125
\(315\) 0 0
\(316\) 1.39257e9 2.48264
\(317\) 2.16935e8 0.382491 0.191246 0.981542i \(-0.438747\pi\)
0.191246 + 0.981542i \(0.438747\pi\)
\(318\) 0 0
\(319\) 8.33986e8 1.43844
\(320\) 4.51183e9 7.69711
\(321\) 0 0
\(322\) −2.02584e9 −3.38150
\(323\) −6.92943e8 −1.14416
\(324\) 0 0
\(325\) 3.16806e8 0.511920
\(326\) −4.38475e7 −0.0700943
\(327\) 0 0
\(328\) −3.02039e9 −4.72611
\(329\) 2.51083e7 0.0388715
\(330\) 0 0
\(331\) −9.98498e8 −1.51338 −0.756692 0.653772i \(-0.773186\pi\)
−0.756692 + 0.653772i \(0.773186\pi\)
\(332\) 1.44248e9 2.16335
\(333\) 0 0
\(334\) −7.69820e8 −1.13052
\(335\) −2.98882e8 −0.434353
\(336\) 0 0
\(337\) −1.17882e9 −1.67780 −0.838902 0.544282i \(-0.816802\pi\)
−0.838902 + 0.544282i \(0.816802\pi\)
\(338\) 1.30041e9 1.83177
\(339\) 0 0
\(340\) 3.84768e9 5.30912
\(341\) −4.41327e8 −0.602726
\(342\) 0 0
\(343\) 2.21931e8 0.296954
\(344\) 6.42081e8 0.850423
\(345\) 0 0
\(346\) −6.43201e8 −0.834796
\(347\) 8.46545e8 1.08767 0.543835 0.839192i \(-0.316972\pi\)
0.543835 + 0.839192i \(0.316972\pi\)
\(348\) 0 0
\(349\) 6.62378e8 0.834098 0.417049 0.908884i \(-0.363064\pi\)
0.417049 + 0.908884i \(0.363064\pi\)
\(350\) 4.38096e9 5.46174
\(351\) 0 0
\(352\) −6.18106e9 −7.55376
\(353\) −8.83004e8 −1.06844 −0.534221 0.845345i \(-0.679395\pi\)
−0.534221 + 0.845345i \(0.679395\pi\)
\(354\) 0 0
\(355\) 1.73500e9 2.05825
\(356\) −2.13280e9 −2.50539
\(357\) 0 0
\(358\) 2.89943e9 3.33981
\(359\) −7.02849e7 −0.0801736 −0.0400868 0.999196i \(-0.512763\pi\)
−0.0400868 + 0.999196i \(0.512763\pi\)
\(360\) 0 0
\(361\) 1.09874e8 0.122919
\(362\) 3.34729e9 3.70863
\(363\) 0 0
\(364\) −8.32864e8 −0.905147
\(365\) 6.27860e8 0.675830
\(366\) 0 0
\(367\) −1.46843e9 −1.55068 −0.775340 0.631544i \(-0.782421\pi\)
−0.775340 + 0.631544i \(0.782421\pi\)
\(368\) −4.97927e9 −5.20833
\(369\) 0 0
\(370\) 7.86815e8 0.807544
\(371\) 1.84987e9 1.88076
\(372\) 0 0
\(373\) 1.19545e9 1.19275 0.596376 0.802705i \(-0.296607\pi\)
0.596376 + 0.802705i \(0.296607\pi\)
\(374\) −3.73864e9 −3.69542
\(375\) 0 0
\(376\) 1.04794e8 0.101666
\(377\) 2.07117e8 0.199077
\(378\) 0 0
\(379\) 1.21004e9 1.14173 0.570863 0.821045i \(-0.306609\pi\)
0.570863 + 0.821045i \(0.306609\pi\)
\(380\) −5.57347e9 −5.21054
\(381\) 0 0
\(382\) −2.68108e9 −2.46086
\(383\) 1.02900e9 0.935876 0.467938 0.883761i \(-0.344997\pi\)
0.467938 + 0.883761i \(0.344997\pi\)
\(384\) 0 0
\(385\) −4.62323e9 −4.12888
\(386\) −2.92116e9 −2.58523
\(387\) 0 0
\(388\) −7.07759e8 −0.615141
\(389\) 2.22572e9 1.91711 0.958556 0.284904i \(-0.0919616\pi\)
0.958556 + 0.284904i \(0.0919616\pi\)
\(390\) 0 0
\(391\) −1.66265e9 −1.40664
\(392\) −3.23209e9 −2.71008
\(393\) 0 0
\(394\) 2.46691e9 2.03197
\(395\) 1.91976e9 1.56732
\(396\) 0 0
\(397\) 9.68593e8 0.776917 0.388458 0.921466i \(-0.373008\pi\)
0.388458 + 0.921466i \(0.373008\pi\)
\(398\) 4.00219e9 3.18205
\(399\) 0 0
\(400\) 1.07679e10 8.41241
\(401\) −9.25332e6 −0.00716626 −0.00358313 0.999994i \(-0.501141\pi\)
−0.00358313 + 0.999994i \(0.501141\pi\)
\(402\) 0 0
\(403\) −1.09602e8 −0.0834160
\(404\) 5.53239e9 4.17425
\(405\) 0 0
\(406\) 2.86412e9 2.12398
\(407\) −5.62842e8 −0.413815
\(408\) 0 0
\(409\) 4.22397e8 0.305274 0.152637 0.988282i \(-0.451224\pi\)
0.152637 + 0.988282i \(0.451224\pi\)
\(410\) −6.48888e9 −4.64971
\(411\) 0 0
\(412\) −4.29653e9 −3.02676
\(413\) −2.48470e8 −0.173560
\(414\) 0 0
\(415\) 1.98856e9 1.36574
\(416\) −1.53504e9 −1.04543
\(417\) 0 0
\(418\) 5.41553e9 3.62680
\(419\) 1.30988e9 0.869925 0.434962 0.900449i \(-0.356762\pi\)
0.434962 + 0.900449i \(0.356762\pi\)
\(420\) 0 0
\(421\) 2.15584e9 1.40809 0.704043 0.710158i \(-0.251376\pi\)
0.704043 + 0.710158i \(0.251376\pi\)
\(422\) 7.37137e7 0.0477479
\(423\) 0 0
\(424\) 7.72075e9 4.91902
\(425\) 3.59555e9 2.27198
\(426\) 0 0
\(427\) −2.90797e9 −1.80756
\(428\) 2.74715e9 1.69367
\(429\) 0 0
\(430\) 1.37942e9 0.836676
\(431\) −8.50792e8 −0.511862 −0.255931 0.966695i \(-0.582382\pi\)
−0.255931 + 0.966695i \(0.582382\pi\)
\(432\) 0 0
\(433\) −1.01714e9 −0.602109 −0.301054 0.953607i \(-0.597339\pi\)
−0.301054 + 0.953607i \(0.597339\pi\)
\(434\) −1.51563e9 −0.889977
\(435\) 0 0
\(436\) −7.55113e8 −0.436324
\(437\) 2.40839e9 1.38052
\(438\) 0 0
\(439\) −1.42920e9 −0.806245 −0.403123 0.915146i \(-0.632075\pi\)
−0.403123 + 0.915146i \(0.632075\pi\)
\(440\) −1.92958e10 −10.7989
\(441\) 0 0
\(442\) −9.28477e8 −0.511438
\(443\) −2.71763e8 −0.148517 −0.0742586 0.997239i \(-0.523659\pi\)
−0.0742586 + 0.997239i \(0.523659\pi\)
\(444\) 0 0
\(445\) −2.94021e9 −1.58168
\(446\) −7.12828e8 −0.380463
\(447\) 0 0
\(448\) −1.10841e10 −5.82406
\(449\) 1.24125e9 0.647138 0.323569 0.946205i \(-0.395117\pi\)
0.323569 + 0.946205i \(0.395117\pi\)
\(450\) 0 0
\(451\) 4.64177e9 2.38268
\(452\) −7.25346e8 −0.369454
\(453\) 0 0
\(454\) 3.53712e9 1.77400
\(455\) −1.14816e9 −0.571429
\(456\) 0 0
\(457\) −3.37198e9 −1.65264 −0.826320 0.563201i \(-0.809570\pi\)
−0.826320 + 0.563201i \(0.809570\pi\)
\(458\) −9.56242e8 −0.465092
\(459\) 0 0
\(460\) −1.33730e10 −6.40585
\(461\) 2.86356e9 1.36130 0.680650 0.732609i \(-0.261698\pi\)
0.680650 + 0.732609i \(0.261698\pi\)
\(462\) 0 0
\(463\) 4.00910e8 0.187721 0.0938607 0.995585i \(-0.470079\pi\)
0.0938607 + 0.995585i \(0.470079\pi\)
\(464\) 7.03967e9 3.27144
\(465\) 0 0
\(466\) −6.30591e8 −0.288667
\(467\) 8.05487e8 0.365973 0.182987 0.983115i \(-0.441423\pi\)
0.182987 + 0.983115i \(0.441423\pi\)
\(468\) 0 0
\(469\) 7.34254e8 0.328656
\(470\) 2.25134e8 0.100023
\(471\) 0 0
\(472\) −1.03703e9 −0.453936
\(473\) −9.86759e8 −0.428743
\(474\) 0 0
\(475\) −5.20825e9 −2.22979
\(476\) −9.45248e9 −4.01718
\(477\) 0 0
\(478\) −1.33617e9 −0.559584
\(479\) 1.45641e9 0.605495 0.302747 0.953071i \(-0.402096\pi\)
0.302747 + 0.953071i \(0.402096\pi\)
\(480\) 0 0
\(481\) −1.39779e8 −0.0572711
\(482\) −1.10810e9 −0.450728
\(483\) 0 0
\(484\) 1.45495e10 5.83295
\(485\) −9.75694e8 −0.388345
\(486\) 0 0
\(487\) 1.91160e9 0.749973 0.374986 0.927030i \(-0.377647\pi\)
0.374986 + 0.927030i \(0.377647\pi\)
\(488\) −1.21369e10 −4.72758
\(489\) 0 0
\(490\) −6.94370e9 −2.66627
\(491\) −3.11951e8 −0.118933 −0.0594663 0.998230i \(-0.518940\pi\)
−0.0594663 + 0.998230i \(0.518940\pi\)
\(492\) 0 0
\(493\) 2.35065e9 0.883533
\(494\) 1.34492e9 0.501942
\(495\) 0 0
\(496\) −3.72524e9 −1.37078
\(497\) −4.26231e9 −1.55739
\(498\) 0 0
\(499\) −1.10424e9 −0.397844 −0.198922 0.980015i \(-0.563744\pi\)
−0.198922 + 0.980015i \(0.563744\pi\)
\(500\) 1.51760e10 5.42953
\(501\) 0 0
\(502\) 3.95957e9 1.39696
\(503\) 3.17067e9 1.11087 0.555434 0.831560i \(-0.312552\pi\)
0.555434 + 0.831560i \(0.312552\pi\)
\(504\) 0 0
\(505\) 7.62677e9 2.63524
\(506\) 1.29940e10 4.45879
\(507\) 0 0
\(508\) 4.92523e9 1.66688
\(509\) −1.80020e9 −0.605074 −0.302537 0.953138i \(-0.597834\pi\)
−0.302537 + 0.953138i \(0.597834\pi\)
\(510\) 0 0
\(511\) −1.54244e9 −0.511371
\(512\) −9.83963e9 −3.23992
\(513\) 0 0
\(514\) 1.87889e9 0.610281
\(515\) −5.92306e9 −1.91082
\(516\) 0 0
\(517\) −1.61048e8 −0.0512553
\(518\) −1.93294e9 −0.611034
\(519\) 0 0
\(520\) −4.79204e9 −1.49454
\(521\) 1.90764e9 0.590968 0.295484 0.955348i \(-0.404519\pi\)
0.295484 + 0.955348i \(0.404519\pi\)
\(522\) 0 0
\(523\) 1.01205e9 0.309347 0.154673 0.987966i \(-0.450567\pi\)
0.154673 + 0.987966i \(0.450567\pi\)
\(524\) 3.01860e8 0.0916530
\(525\) 0 0
\(526\) 9.12900e9 2.73510
\(527\) −1.24391e9 −0.370213
\(528\) 0 0
\(529\) 2.37388e9 0.697211
\(530\) 1.65870e10 4.83951
\(531\) 0 0
\(532\) 1.36922e10 3.94259
\(533\) 1.15276e9 0.329758
\(534\) 0 0
\(535\) 3.78713e9 1.06923
\(536\) 3.06453e9 0.859582
\(537\) 0 0
\(538\) −6.78063e9 −1.87729
\(539\) 4.96712e9 1.36629
\(540\) 0 0
\(541\) −5.50100e9 −1.49366 −0.746829 0.665016i \(-0.768425\pi\)
−0.746829 + 0.665016i \(0.768425\pi\)
\(542\) 5.19378e9 1.40115
\(543\) 0 0
\(544\) −1.74217e10 −4.63976
\(545\) −1.04097e9 −0.275456
\(546\) 0 0
\(547\) 4.40039e9 1.14957 0.574786 0.818304i \(-0.305085\pi\)
0.574786 + 0.818304i \(0.305085\pi\)
\(548\) 1.73110e10 4.49355
\(549\) 0 0
\(550\) −2.81002e10 −7.20177
\(551\) −3.40497e9 −0.867128
\(552\) 0 0
\(553\) −4.71621e9 −1.18592
\(554\) 4.74664e9 1.18605
\(555\) 0 0
\(556\) −7.33062e8 −0.180875
\(557\) −5.20574e9 −1.27641 −0.638203 0.769868i \(-0.720322\pi\)
−0.638203 + 0.769868i \(0.720322\pi\)
\(558\) 0 0
\(559\) −2.45057e8 −0.0593371
\(560\) −3.90246e10 −9.39033
\(561\) 0 0
\(562\) −4.54557e9 −1.08022
\(563\) −6.20253e9 −1.46484 −0.732419 0.680854i \(-0.761609\pi\)
−0.732419 + 0.680854i \(0.761609\pi\)
\(564\) 0 0
\(565\) −9.99939e8 −0.233240
\(566\) −1.48397e10 −3.44007
\(567\) 0 0
\(568\) −1.77895e10 −4.07328
\(569\) −6.72094e9 −1.52946 −0.764728 0.644353i \(-0.777127\pi\)
−0.764728 + 0.644353i \(0.777127\pi\)
\(570\) 0 0
\(571\) 3.74441e8 0.0841700 0.0420850 0.999114i \(-0.486600\pi\)
0.0420850 + 0.999114i \(0.486600\pi\)
\(572\) 5.34212e9 1.19351
\(573\) 0 0
\(574\) 1.59410e10 3.51824
\(575\) −1.24967e10 −2.74131
\(576\) 0 0
\(577\) 1.91890e9 0.415851 0.207925 0.978145i \(-0.433329\pi\)
0.207925 + 0.978145i \(0.433329\pi\)
\(578\) −1.49875e9 −0.322835
\(579\) 0 0
\(580\) 1.89067e10 4.02362
\(581\) −4.88522e9 −1.03340
\(582\) 0 0
\(583\) −1.18654e10 −2.47994
\(584\) −6.43765e9 −1.33746
\(585\) 0 0
\(586\) −5.90929e9 −1.21309
\(587\) 9.08556e9 1.85404 0.927018 0.375016i \(-0.122363\pi\)
0.927018 + 0.375016i \(0.122363\pi\)
\(588\) 0 0
\(589\) 1.80184e9 0.363339
\(590\) −2.22792e9 −0.446598
\(591\) 0 0
\(592\) −4.75095e9 −0.941140
\(593\) 1.90930e9 0.375997 0.187998 0.982169i \(-0.439800\pi\)
0.187998 + 0.982169i \(0.439800\pi\)
\(594\) 0 0
\(595\) −1.30309e10 −2.53609
\(596\) 9.05329e8 0.175164
\(597\) 0 0
\(598\) 3.22701e9 0.617088
\(599\) 5.47139e9 1.04017 0.520084 0.854115i \(-0.325901\pi\)
0.520084 + 0.854115i \(0.325901\pi\)
\(600\) 0 0
\(601\) 3.55169e9 0.667381 0.333691 0.942683i \(-0.391706\pi\)
0.333691 + 0.942683i \(0.391706\pi\)
\(602\) −3.38878e9 −0.633076
\(603\) 0 0
\(604\) −1.96287e10 −3.62463
\(605\) 2.00574e10 3.68240
\(606\) 0 0
\(607\) 5.38370e9 0.977057 0.488529 0.872548i \(-0.337534\pi\)
0.488529 + 0.872548i \(0.337534\pi\)
\(608\) 2.52358e10 4.55360
\(609\) 0 0
\(610\) −2.60745e10 −4.65116
\(611\) −3.99956e7 −0.00709363
\(612\) 0 0
\(613\) 5.24608e9 0.919863 0.459932 0.887954i \(-0.347874\pi\)
0.459932 + 0.887954i \(0.347874\pi\)
\(614\) −1.45219e10 −2.53183
\(615\) 0 0
\(616\) 4.74034e10 8.17104
\(617\) −6.88172e9 −1.17950 −0.589751 0.807585i \(-0.700774\pi\)
−0.589751 + 0.807585i \(0.700774\pi\)
\(618\) 0 0
\(619\) 5.67281e9 0.961350 0.480675 0.876899i \(-0.340392\pi\)
0.480675 + 0.876899i \(0.340392\pi\)
\(620\) −1.00050e10 −1.68596
\(621\) 0 0
\(622\) −1.92438e10 −3.20645
\(623\) 7.22313e9 1.19679
\(624\) 0 0
\(625\) 8.07802e9 1.32350
\(626\) −6.25640e9 −1.01933
\(627\) 0 0
\(628\) 2.59892e10 4.18730
\(629\) −1.58641e9 −0.254178
\(630\) 0 0
\(631\) 1.88338e9 0.298425 0.149212 0.988805i \(-0.452326\pi\)
0.149212 + 0.988805i \(0.452326\pi\)
\(632\) −1.96839e10 −3.10171
\(633\) 0 0
\(634\) −4.77860e9 −0.744713
\(635\) 6.78976e9 1.05232
\(636\) 0 0
\(637\) 1.23356e9 0.189092
\(638\) −1.83709e10 −2.80065
\(639\) 0 0
\(640\) −4.91762e10 −7.41524
\(641\) −5.23503e9 −0.785084 −0.392542 0.919734i \(-0.628404\pi\)
−0.392542 + 0.919734i \(0.628404\pi\)
\(642\) 0 0
\(643\) −7.39762e8 −0.109737 −0.0548686 0.998494i \(-0.517474\pi\)
−0.0548686 + 0.998494i \(0.517474\pi\)
\(644\) 3.28530e10 4.84702
\(645\) 0 0
\(646\) 1.52640e10 2.22769
\(647\) 7.49507e9 1.08796 0.543978 0.839100i \(-0.316918\pi\)
0.543978 + 0.839100i \(0.316918\pi\)
\(648\) 0 0
\(649\) 1.59372e9 0.228853
\(650\) −6.97856e9 −0.996710
\(651\) 0 0
\(652\) 7.11075e8 0.100473
\(653\) −9.12022e9 −1.28177 −0.640884 0.767638i \(-0.721432\pi\)
−0.640884 + 0.767638i \(0.721432\pi\)
\(654\) 0 0
\(655\) 4.16135e8 0.0578615
\(656\) 3.91812e10 5.41893
\(657\) 0 0
\(658\) −5.53081e8 −0.0756829
\(659\) −8.11531e9 −1.10460 −0.552301 0.833645i \(-0.686250\pi\)
−0.552301 + 0.833645i \(0.686250\pi\)
\(660\) 0 0
\(661\) 6.72245e9 0.905362 0.452681 0.891672i \(-0.350468\pi\)
0.452681 + 0.891672i \(0.350468\pi\)
\(662\) 2.19947e10 2.94656
\(663\) 0 0
\(664\) −2.03893e10 −2.70280
\(665\) 1.88756e10 2.48900
\(666\) 0 0
\(667\) −8.16991e9 −1.06605
\(668\) 1.24842e10 1.62047
\(669\) 0 0
\(670\) 6.58372e9 0.845687
\(671\) 1.86522e10 2.38342
\(672\) 0 0
\(673\) 1.14461e10 1.44745 0.723725 0.690089i \(-0.242429\pi\)
0.723725 + 0.690089i \(0.242429\pi\)
\(674\) 2.59668e10 3.26669
\(675\) 0 0
\(676\) −2.10887e10 −2.62565
\(677\) 4.26750e9 0.528583 0.264292 0.964443i \(-0.414862\pi\)
0.264292 + 0.964443i \(0.414862\pi\)
\(678\) 0 0
\(679\) 2.39696e9 0.293843
\(680\) −5.43866e10 −6.63301
\(681\) 0 0
\(682\) 9.72147e9 1.17351
\(683\) −6.74829e9 −0.810441 −0.405220 0.914219i \(-0.632805\pi\)
−0.405220 + 0.914219i \(0.632805\pi\)
\(684\) 0 0
\(685\) 2.38644e10 2.83683
\(686\) −4.88867e9 −0.578171
\(687\) 0 0
\(688\) −8.32922e9 −0.975091
\(689\) −2.94671e9 −0.343218
\(690\) 0 0
\(691\) −6.41648e9 −0.739816 −0.369908 0.929068i \(-0.620611\pi\)
−0.369908 + 0.929068i \(0.620611\pi\)
\(692\) 1.04308e10 1.19659
\(693\) 0 0
\(694\) −1.86476e10 −2.11770
\(695\) −1.01057e9 −0.114188
\(696\) 0 0
\(697\) 1.30832e10 1.46352
\(698\) −1.45908e10 −1.62399
\(699\) 0 0
\(700\) −7.10461e10 −7.82884
\(701\) 1.49165e10 1.63552 0.817758 0.575562i \(-0.195216\pi\)
0.817758 + 0.575562i \(0.195216\pi\)
\(702\) 0 0
\(703\) 2.29796e9 0.249458
\(704\) 7.10949e10 7.67952
\(705\) 0 0
\(706\) 1.94507e10 2.08026
\(707\) −1.87365e10 −1.99397
\(708\) 0 0
\(709\) 5.36963e9 0.565826 0.282913 0.959146i \(-0.408699\pi\)
0.282913 + 0.959146i \(0.408699\pi\)
\(710\) −3.82182e10 −4.00743
\(711\) 0 0
\(712\) 3.01469e10 3.13014
\(713\) 4.32333e9 0.446689
\(714\) 0 0
\(715\) 7.36447e9 0.753477
\(716\) −4.70201e10 −4.78727
\(717\) 0 0
\(718\) 1.54822e9 0.156098
\(719\) 2.42754e9 0.243565 0.121783 0.992557i \(-0.461139\pi\)
0.121783 + 0.992557i \(0.461139\pi\)
\(720\) 0 0
\(721\) 1.45510e10 1.44584
\(722\) −2.42029e9 −0.239325
\(723\) 0 0
\(724\) −5.42830e10 −5.31593
\(725\) 1.76678e10 1.72186
\(726\) 0 0
\(727\) 1.20971e10 1.16764 0.583822 0.811882i \(-0.301557\pi\)
0.583822 + 0.811882i \(0.301557\pi\)
\(728\) 1.17724e10 1.13086
\(729\) 0 0
\(730\) −1.38304e10 −1.31584
\(731\) −2.78125e9 −0.263347
\(732\) 0 0
\(733\) 3.52480e7 0.00330575 0.00165288 0.999999i \(-0.499474\pi\)
0.00165288 + 0.999999i \(0.499474\pi\)
\(734\) 3.23463e10 3.01918
\(735\) 0 0
\(736\) 6.05510e10 5.59821
\(737\) −4.70962e9 −0.433360
\(738\) 0 0
\(739\) 9.27198e9 0.845118 0.422559 0.906336i \(-0.361132\pi\)
0.422559 + 0.906336i \(0.361132\pi\)
\(740\) −1.27598e10 −1.15753
\(741\) 0 0
\(742\) −4.07487e10 −3.66184
\(743\) 1.63793e10 1.46499 0.732494 0.680773i \(-0.238356\pi\)
0.732494 + 0.680773i \(0.238356\pi\)
\(744\) 0 0
\(745\) 1.24806e9 0.110583
\(746\) −2.63331e10 −2.32229
\(747\) 0 0
\(748\) 6.06296e10 5.29699
\(749\) −9.30372e9 −0.809040
\(750\) 0 0
\(751\) 2.46248e9 0.212145 0.106073 0.994358i \(-0.466172\pi\)
0.106073 + 0.994358i \(0.466172\pi\)
\(752\) −1.35941e9 −0.116570
\(753\) 0 0
\(754\) −4.56234e9 −0.387604
\(755\) −2.70596e10 −2.28826
\(756\) 0 0
\(757\) 7.88850e9 0.660935 0.330468 0.943817i \(-0.392794\pi\)
0.330468 + 0.943817i \(0.392794\pi\)
\(758\) −2.66545e10 −2.22295
\(759\) 0 0
\(760\) 7.87804e10 6.50984
\(761\) 2.28543e10 1.87984 0.939922 0.341390i \(-0.110898\pi\)
0.939922 + 0.341390i \(0.110898\pi\)
\(762\) 0 0
\(763\) 2.55733e9 0.208426
\(764\) 4.34791e10 3.52739
\(765\) 0 0
\(766\) −2.26666e10 −1.82216
\(767\) 3.95795e8 0.0316728
\(768\) 0 0
\(769\) 3.75836e9 0.298028 0.149014 0.988835i \(-0.452390\pi\)
0.149014 + 0.988835i \(0.452390\pi\)
\(770\) 1.01840e11 8.03896
\(771\) 0 0
\(772\) 4.73725e10 3.70566
\(773\) 2.31574e10 1.80327 0.901637 0.432494i \(-0.142366\pi\)
0.901637 + 0.432494i \(0.142366\pi\)
\(774\) 0 0
\(775\) −9.34939e9 −0.721485
\(776\) 1.00041e10 0.768532
\(777\) 0 0
\(778\) −4.90279e10 −3.73263
\(779\) −1.89513e10 −1.43634
\(780\) 0 0
\(781\) 2.73391e10 2.05355
\(782\) 3.66246e10 2.73873
\(783\) 0 0
\(784\) 4.19274e10 3.10736
\(785\) 3.58279e10 2.64349
\(786\) 0 0
\(787\) 7.24985e8 0.0530173 0.0265087 0.999649i \(-0.491561\pi\)
0.0265087 + 0.999649i \(0.491561\pi\)
\(788\) −4.00059e10 −2.91261
\(789\) 0 0
\(790\) −4.22881e10 −3.05157
\(791\) 2.45652e9 0.176483
\(792\) 0 0
\(793\) 4.63219e9 0.329860
\(794\) −2.13360e10 −1.51266
\(795\) 0 0
\(796\) −6.49036e10 −4.56113
\(797\) −1.66646e10 −1.16598 −0.582989 0.812480i \(-0.698117\pi\)
−0.582989 + 0.812480i \(0.698117\pi\)
\(798\) 0 0
\(799\) −4.53925e8 −0.0314826
\(800\) −1.30944e11 −9.04213
\(801\) 0 0
\(802\) 2.03831e8 0.0139527
\(803\) 9.89347e9 0.674286
\(804\) 0 0
\(805\) 4.52901e10 3.05998
\(806\) 2.41429e9 0.162411
\(807\) 0 0
\(808\) −7.81997e10 −5.21514
\(809\) 1.50299e10 0.998010 0.499005 0.866599i \(-0.333699\pi\)
0.499005 + 0.866599i \(0.333699\pi\)
\(810\) 0 0
\(811\) −1.62482e10 −1.06963 −0.534813 0.844971i \(-0.679618\pi\)
−0.534813 + 0.844971i \(0.679618\pi\)
\(812\) −4.64475e10 −3.04450
\(813\) 0 0
\(814\) 1.23982e10 0.805699
\(815\) 9.80265e8 0.0634296
\(816\) 0 0
\(817\) 4.02871e9 0.258457
\(818\) −9.30449e9 −0.594369
\(819\) 0 0
\(820\) 1.05230e11 6.66488
\(821\) 2.31309e10 1.45879 0.729393 0.684095i \(-0.239803\pi\)
0.729393 + 0.684095i \(0.239803\pi\)
\(822\) 0 0
\(823\) −2.00237e10 −1.25211 −0.626057 0.779777i \(-0.715332\pi\)
−0.626057 + 0.779777i \(0.715332\pi\)
\(824\) 6.07310e10 3.78151
\(825\) 0 0
\(826\) 5.47325e9 0.337921
\(827\) 1.75153e10 1.07683 0.538417 0.842679i \(-0.319023\pi\)
0.538417 + 0.842679i \(0.319023\pi\)
\(828\) 0 0
\(829\) 1.27473e10 0.777099 0.388550 0.921428i \(-0.372976\pi\)
0.388550 + 0.921428i \(0.372976\pi\)
\(830\) −4.38036e10 −2.65911
\(831\) 0 0
\(832\) 1.76561e10 1.06283
\(833\) 1.40002e10 0.839220
\(834\) 0 0
\(835\) 1.72103e10 1.02302
\(836\) −8.78237e10 −5.19864
\(837\) 0 0
\(838\) −2.88538e10 −1.69375
\(839\) −1.43905e10 −0.841217 −0.420609 0.907242i \(-0.638183\pi\)
−0.420609 + 0.907242i \(0.638183\pi\)
\(840\) 0 0
\(841\) −5.69930e9 −0.330397
\(842\) −4.74885e10 −2.74155
\(843\) 0 0
\(844\) −1.19542e9 −0.0684416
\(845\) −2.90722e10 −1.65760
\(846\) 0 0
\(847\) −4.92745e10 −2.78631
\(848\) −1.00155e11 −5.64013
\(849\) 0 0
\(850\) −7.92022e10 −4.42355
\(851\) 5.51372e9 0.306684
\(852\) 0 0
\(853\) −2.47811e10 −1.36710 −0.683549 0.729905i \(-0.739564\pi\)
−0.683549 + 0.729905i \(0.739564\pi\)
\(854\) 6.40563e10 3.51933
\(855\) 0 0
\(856\) −3.88306e10 −2.11600
\(857\) −1.34333e10 −0.729036 −0.364518 0.931196i \(-0.618766\pi\)
−0.364518 + 0.931196i \(0.618766\pi\)
\(858\) 0 0
\(859\) 1.54319e10 0.830701 0.415350 0.909662i \(-0.363659\pi\)
0.415350 + 0.909662i \(0.363659\pi\)
\(860\) −2.23701e10 −1.19929
\(861\) 0 0
\(862\) 1.87411e10 0.996598
\(863\) 1.32034e10 0.699273 0.349636 0.936885i \(-0.386305\pi\)
0.349636 + 0.936885i \(0.386305\pi\)
\(864\) 0 0
\(865\) 1.43796e10 0.755421
\(866\) 2.24055e10 1.17231
\(867\) 0 0
\(868\) 2.45790e10 1.27569
\(869\) 3.02505e10 1.56373
\(870\) 0 0
\(871\) −1.16961e9 −0.0599762
\(872\) 1.06734e10 0.545126
\(873\) 0 0
\(874\) −5.30517e10 −2.68788
\(875\) −5.13963e10 −2.59360
\(876\) 0 0
\(877\) −1.39419e10 −0.697949 −0.348975 0.937132i \(-0.613470\pi\)
−0.348975 + 0.937132i \(0.613470\pi\)
\(878\) 3.14822e10 1.56976
\(879\) 0 0
\(880\) 2.50310e11 12.3819
\(881\) −1.37479e10 −0.677360 −0.338680 0.940902i \(-0.609980\pi\)
−0.338680 + 0.940902i \(0.609980\pi\)
\(882\) 0 0
\(883\) 8.50641e9 0.415799 0.207900 0.978150i \(-0.433337\pi\)
0.207900 + 0.978150i \(0.433337\pi\)
\(884\) 1.50571e10 0.733093
\(885\) 0 0
\(886\) 5.98635e9 0.289164
\(887\) 1.27701e10 0.614417 0.307208 0.951642i \(-0.400605\pi\)
0.307208 + 0.951642i \(0.400605\pi\)
\(888\) 0 0
\(889\) −1.66802e10 −0.796241
\(890\) 6.47665e10 3.07954
\(891\) 0 0
\(892\) 1.15599e10 0.545354
\(893\) 6.57523e8 0.0308980
\(894\) 0 0
\(895\) −6.48204e10 −3.02225
\(896\) 1.20810e11 5.61079
\(897\) 0 0
\(898\) −2.73420e10 −1.25998
\(899\) −6.11231e9 −0.280573
\(900\) 0 0
\(901\) −3.34433e10 −1.52326
\(902\) −1.02248e11 −4.63909
\(903\) 0 0
\(904\) 1.02527e10 0.461582
\(905\) −7.48328e10 −3.35600
\(906\) 0 0
\(907\) 3.70884e10 1.65049 0.825244 0.564776i \(-0.191037\pi\)
0.825244 + 0.564776i \(0.191037\pi\)
\(908\) −5.73616e10 −2.54285
\(909\) 0 0
\(910\) 2.52915e10 1.11257
\(911\) −1.58470e10 −0.694436 −0.347218 0.937784i \(-0.612874\pi\)
−0.347218 + 0.937784i \(0.612874\pi\)
\(912\) 0 0
\(913\) 3.13346e10 1.36262
\(914\) 7.42774e10 3.21770
\(915\) 0 0
\(916\) 1.55074e10 0.666660
\(917\) −1.02231e9 −0.0437813
\(918\) 0 0
\(919\) 1.09730e10 0.466359 0.233180 0.972434i \(-0.425087\pi\)
0.233180 + 0.972434i \(0.425087\pi\)
\(920\) 1.89026e11 8.00321
\(921\) 0 0
\(922\) −6.30781e10 −2.65046
\(923\) 6.78955e9 0.284207
\(924\) 0 0
\(925\) −1.19237e10 −0.495352
\(926\) −8.83119e9 −0.365494
\(927\) 0 0
\(928\) −8.56067e10 −3.51633
\(929\) −3.30298e10 −1.35161 −0.675804 0.737081i \(-0.736203\pi\)
−0.675804 + 0.737081i \(0.736203\pi\)
\(930\) 0 0
\(931\) −2.02796e10 −0.823637
\(932\) 1.02263e10 0.413774
\(933\) 0 0
\(934\) −1.77431e10 −0.712552
\(935\) 8.35821e10 3.34405
\(936\) 0 0
\(937\) −1.81069e10 −0.719045 −0.359522 0.933136i \(-0.617060\pi\)
−0.359522 + 0.933136i \(0.617060\pi\)
\(938\) −1.61740e10 −0.639894
\(939\) 0 0
\(940\) −3.65101e9 −0.143372
\(941\) −2.38460e10 −0.932938 −0.466469 0.884538i \(-0.654474\pi\)
−0.466469 + 0.884538i \(0.654474\pi\)
\(942\) 0 0
\(943\) −4.54718e10 −1.76584
\(944\) 1.34526e10 0.520480
\(945\) 0 0
\(946\) 2.17362e10 0.834764
\(947\) −2.50480e10 −0.958404 −0.479202 0.877705i \(-0.659074\pi\)
−0.479202 + 0.877705i \(0.659074\pi\)
\(948\) 0 0
\(949\) 2.45700e9 0.0933198
\(950\) 1.14727e11 4.34141
\(951\) 0 0
\(952\) 1.33610e11 5.01891
\(953\) 2.27700e10 0.852194 0.426097 0.904677i \(-0.359888\pi\)
0.426097 + 0.904677i \(0.359888\pi\)
\(954\) 0 0
\(955\) 5.99389e10 2.22688
\(956\) 2.16688e10 0.802105
\(957\) 0 0
\(958\) −3.20816e10 −1.17890
\(959\) −5.86269e10 −2.14650
\(960\) 0 0
\(961\) −2.42781e10 −0.882436
\(962\) 3.07904e9 0.111507
\(963\) 0 0
\(964\) 1.79701e10 0.646072
\(965\) 6.53062e10 2.33942
\(966\) 0 0
\(967\) 1.53182e10 0.544772 0.272386 0.962188i \(-0.412187\pi\)
0.272386 + 0.962188i \(0.412187\pi\)
\(968\) −2.05655e11 −7.28746
\(969\) 0 0
\(970\) 2.14924e10 0.756109
\(971\) −1.35596e10 −0.475314 −0.237657 0.971349i \(-0.576379\pi\)
−0.237657 + 0.971349i \(0.576379\pi\)
\(972\) 0 0
\(973\) 2.48265e9 0.0864013
\(974\) −4.21084e10 −1.46020
\(975\) 0 0
\(976\) 1.57443e11 5.42061
\(977\) −5.16415e10 −1.77161 −0.885804 0.464060i \(-0.846392\pi\)
−0.885804 + 0.464060i \(0.846392\pi\)
\(978\) 0 0
\(979\) −4.63302e10 −1.57807
\(980\) 1.12606e11 3.82182
\(981\) 0 0
\(982\) 6.87160e9 0.231562
\(983\) −2.68451e10 −0.901421 −0.450711 0.892670i \(-0.648829\pi\)
−0.450711 + 0.892670i \(0.648829\pi\)
\(984\) 0 0
\(985\) −5.51509e10 −1.83876
\(986\) −5.17796e10 −1.72024
\(987\) 0 0
\(988\) −2.18106e10 −0.719481
\(989\) 9.66650e9 0.317748
\(990\) 0 0
\(991\) 3.98369e10 1.30025 0.650126 0.759826i \(-0.274716\pi\)
0.650126 + 0.759826i \(0.274716\pi\)
\(992\) 4.53012e10 1.47339
\(993\) 0 0
\(994\) 9.38895e10 3.03225
\(995\) −8.94740e10 −2.87949
\(996\) 0 0
\(997\) −5.65997e9 −0.180876 −0.0904380 0.995902i \(-0.528827\pi\)
−0.0904380 + 0.995902i \(0.528827\pi\)
\(998\) 2.43241e10 0.774605
\(999\) 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.1.1 17
3.2 odd 2 177.8.a.b.1.17 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.17 17 3.2 odd 2
531.8.a.d.1.1 17 1.1 even 1 trivial