Properties

Label 531.8.a.d.1.5
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(12.7630\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +O(q^{10})\) \(q-10.7630 q^{2} -12.1577 q^{4} -451.863 q^{5} -504.672 q^{7} +1508.52 q^{8} +4863.40 q^{10} +8070.72 q^{11} -4674.69 q^{13} +5431.79 q^{14} -14680.0 q^{16} +13499.0 q^{17} -11364.6 q^{19} +5493.61 q^{20} -86865.2 q^{22} +66475.4 q^{23} +126055. q^{25} +50313.7 q^{26} +6135.65 q^{28} +174542. q^{29} +62482.4 q^{31} -35089.3 q^{32} -145289. q^{34} +228043. q^{35} +277403. q^{37} +122318. q^{38} -681643. q^{40} -529842. q^{41} +675111. q^{43} -98121.3 q^{44} -715475. q^{46} -175110. q^{47} -568849. q^{49} -1.35673e6 q^{50} +56833.4 q^{52} +313480. q^{53} -3.64686e6 q^{55} -761307. q^{56} -1.87860e6 q^{58} +205379. q^{59} -3.01895e6 q^{61} -672499. q^{62} +2.25671e6 q^{64} +2.11232e6 q^{65} -885587. q^{67} -164116. q^{68} -2.45442e6 q^{70} +1.15026e6 q^{71} +33498.4 q^{73} -2.98569e6 q^{74} +138168. q^{76} -4.07307e6 q^{77} -694337. q^{79} +6.63335e6 q^{80} +5.70270e6 q^{82} +672731. q^{83} -6.09968e6 q^{85} -7.26622e6 q^{86} +1.21748e7 q^{88} -7.67580e6 q^{89} +2.35918e6 q^{91} -808188. q^{92} +1.88471e6 q^{94} +5.13525e6 q^{95} +4.89911e6 q^{97} +6.12253e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.7630 −0.951324 −0.475662 0.879628i \(-0.657792\pi\)
−0.475662 + 0.879628i \(0.657792\pi\)
\(3\) 0 0
\(4\) −12.1577 −0.0949820
\(5\) −451.863 −1.61663 −0.808317 0.588748i \(-0.799621\pi\)
−0.808317 + 0.588748i \(0.799621\pi\)
\(6\) 0 0
\(7\) −504.672 −0.556117 −0.278058 0.960564i \(-0.589691\pi\)
−0.278058 + 0.960564i \(0.589691\pi\)
\(8\) 1508.52 1.04168
\(9\) 0 0
\(10\) 4863.40 1.53794
\(11\) 8070.72 1.82826 0.914129 0.405422i \(-0.132876\pi\)
0.914129 + 0.405422i \(0.132876\pi\)
\(12\) 0 0
\(13\) −4674.69 −0.590134 −0.295067 0.955477i \(-0.595342\pi\)
−0.295067 + 0.955477i \(0.595342\pi\)
\(14\) 5431.79 0.529048
\(15\) 0 0
\(16\) −14680.0 −0.895996
\(17\) 13499.0 0.666391 0.333196 0.942858i \(-0.391873\pi\)
0.333196 + 0.942858i \(0.391873\pi\)
\(18\) 0 0
\(19\) −11364.6 −0.380117 −0.190059 0.981773i \(-0.560868\pi\)
−0.190059 + 0.981773i \(0.560868\pi\)
\(20\) 5493.61 0.153551
\(21\) 0 0
\(22\) −86865.2 −1.73927
\(23\) 66475.4 1.13924 0.569618 0.821910i \(-0.307091\pi\)
0.569618 + 0.821910i \(0.307091\pi\)
\(24\) 0 0
\(25\) 126055. 1.61350
\(26\) 50313.7 0.561409
\(27\) 0 0
\(28\) 6135.65 0.0528211
\(29\) 174542. 1.32894 0.664472 0.747313i \(-0.268657\pi\)
0.664472 + 0.747313i \(0.268657\pi\)
\(30\) 0 0
\(31\) 62482.4 0.376697 0.188348 0.982102i \(-0.439687\pi\)
0.188348 + 0.982102i \(0.439687\pi\)
\(32\) −35089.3 −0.189300
\(33\) 0 0
\(34\) −145289. −0.633954
\(35\) 228043. 0.899038
\(36\) 0 0
\(37\) 277403. 0.900336 0.450168 0.892944i \(-0.351364\pi\)
0.450168 + 0.892944i \(0.351364\pi\)
\(38\) 122318. 0.361615
\(39\) 0 0
\(40\) −681643. −1.68402
\(41\) −529842. −1.20061 −0.600307 0.799770i \(-0.704955\pi\)
−0.600307 + 0.799770i \(0.704955\pi\)
\(42\) 0 0
\(43\) 675111. 1.29490 0.647449 0.762109i \(-0.275836\pi\)
0.647449 + 0.762109i \(0.275836\pi\)
\(44\) −98121.3 −0.173652
\(45\) 0 0
\(46\) −715475. −1.08378
\(47\) −175110. −0.246019 −0.123010 0.992405i \(-0.539255\pi\)
−0.123010 + 0.992405i \(0.539255\pi\)
\(48\) 0 0
\(49\) −568849. −0.690734
\(50\) −1.35673e6 −1.53497
\(51\) 0 0
\(52\) 56833.4 0.0560521
\(53\) 313480. 0.289231 0.144615 0.989488i \(-0.453805\pi\)
0.144615 + 0.989488i \(0.453805\pi\)
\(54\) 0 0
\(55\) −3.64686e6 −2.95563
\(56\) −761307. −0.579298
\(57\) 0 0
\(58\) −1.87860e6 −1.26426
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −3.01895e6 −1.70295 −0.851474 0.524396i \(-0.824291\pi\)
−0.851474 + 0.524396i \(0.824291\pi\)
\(62\) −672499. −0.358361
\(63\) 0 0
\(64\) 2.25671e6 1.07608
\(65\) 2.11232e6 0.954031
\(66\) 0 0
\(67\) −885587. −0.359724 −0.179862 0.983692i \(-0.557565\pi\)
−0.179862 + 0.983692i \(0.557565\pi\)
\(68\) −164116. −0.0632952
\(69\) 0 0
\(70\) −2.45442e6 −0.855276
\(71\) 1.15026e6 0.381411 0.190706 0.981647i \(-0.438922\pi\)
0.190706 + 0.981647i \(0.438922\pi\)
\(72\) 0 0
\(73\) 33498.4 0.0100785 0.00503923 0.999987i \(-0.498396\pi\)
0.00503923 + 0.999987i \(0.498396\pi\)
\(74\) −2.98569e6 −0.856511
\(75\) 0 0
\(76\) 138168. 0.0361043
\(77\) −4.07307e6 −1.01673
\(78\) 0 0
\(79\) −694337. −0.158444 −0.0792219 0.996857i \(-0.525244\pi\)
−0.0792219 + 0.996857i \(0.525244\pi\)
\(80\) 6.63335e6 1.44850
\(81\) 0 0
\(82\) 5.70270e6 1.14217
\(83\) 672731. 0.129142 0.0645711 0.997913i \(-0.479432\pi\)
0.0645711 + 0.997913i \(0.479432\pi\)
\(84\) 0 0
\(85\) −6.09968e6 −1.07731
\(86\) −7.26622e6 −1.23187
\(87\) 0 0
\(88\) 1.21748e7 1.90447
\(89\) −7.67580e6 −1.15414 −0.577070 0.816695i \(-0.695804\pi\)
−0.577070 + 0.816695i \(0.695804\pi\)
\(90\) 0 0
\(91\) 2.35918e6 0.328184
\(92\) −808188. −0.108207
\(93\) 0 0
\(94\) 1.88471e6 0.234044
\(95\) 5.13525e6 0.614510
\(96\) 0 0
\(97\) 4.89911e6 0.545025 0.272512 0.962152i \(-0.412145\pi\)
0.272512 + 0.962152i \(0.412145\pi\)
\(98\) 6.12253e6 0.657112
\(99\) 0 0
\(100\) −1.53254e6 −0.153254
\(101\) 1.84644e7 1.78324 0.891621 0.452782i \(-0.149568\pi\)
0.891621 + 0.452782i \(0.149568\pi\)
\(102\) 0 0
\(103\) −5.21469e6 −0.470217 −0.235109 0.971969i \(-0.575545\pi\)
−0.235109 + 0.971969i \(0.575545\pi\)
\(104\) −7.05185e6 −0.614733
\(105\) 0 0
\(106\) −3.37399e6 −0.275152
\(107\) 2.16312e6 0.170701 0.0853506 0.996351i \(-0.472799\pi\)
0.0853506 + 0.996351i \(0.472799\pi\)
\(108\) 0 0
\(109\) −1.59770e7 −1.18169 −0.590845 0.806785i \(-0.701206\pi\)
−0.590845 + 0.806785i \(0.701206\pi\)
\(110\) 3.92511e7 2.81176
\(111\) 0 0
\(112\) 7.40859e6 0.498279
\(113\) −9.29263e6 −0.605849 −0.302924 0.953015i \(-0.597963\pi\)
−0.302924 + 0.953015i \(0.597963\pi\)
\(114\) 0 0
\(115\) −3.00378e7 −1.84173
\(116\) −2.12203e6 −0.126226
\(117\) 0 0
\(118\) −2.21050e6 −0.123852
\(119\) −6.81255e6 −0.370591
\(120\) 0 0
\(121\) 4.56493e7 2.34253
\(122\) 3.24930e7 1.62006
\(123\) 0 0
\(124\) −759642. −0.0357794
\(125\) −2.16578e7 −0.991813
\(126\) 0 0
\(127\) 8.17874e6 0.354302 0.177151 0.984184i \(-0.443312\pi\)
0.177151 + 0.984184i \(0.443312\pi\)
\(128\) −1.97975e7 −0.834403
\(129\) 0 0
\(130\) −2.27349e7 −0.907593
\(131\) −4.40912e6 −0.171357 −0.0856787 0.996323i \(-0.527306\pi\)
−0.0856787 + 0.996323i \(0.527306\pi\)
\(132\) 0 0
\(133\) 5.73541e6 0.211390
\(134\) 9.53158e6 0.342214
\(135\) 0 0
\(136\) 2.03634e7 0.694168
\(137\) 8.21867e6 0.273073 0.136537 0.990635i \(-0.456403\pi\)
0.136537 + 0.990635i \(0.456403\pi\)
\(138\) 0 0
\(139\) −5.56805e7 −1.75854 −0.879268 0.476328i \(-0.841968\pi\)
−0.879268 + 0.476328i \(0.841968\pi\)
\(140\) −2.77247e6 −0.0853924
\(141\) 0 0
\(142\) −1.23803e7 −0.362846
\(143\) −3.77281e7 −1.07892
\(144\) 0 0
\(145\) −7.88690e7 −2.14842
\(146\) −360544. −0.00958789
\(147\) 0 0
\(148\) −3.37258e6 −0.0855157
\(149\) −2.85493e7 −0.707039 −0.353520 0.935427i \(-0.615015\pi\)
−0.353520 + 0.935427i \(0.615015\pi\)
\(150\) 0 0
\(151\) −1.41537e7 −0.334541 −0.167271 0.985911i \(-0.553495\pi\)
−0.167271 + 0.985911i \(0.553495\pi\)
\(152\) −1.71437e7 −0.395962
\(153\) 0 0
\(154\) 4.38384e7 0.967236
\(155\) −2.82335e7 −0.608981
\(156\) 0 0
\(157\) 6.04002e7 1.24563 0.622816 0.782368i \(-0.285989\pi\)
0.622816 + 0.782368i \(0.285989\pi\)
\(158\) 7.47315e6 0.150732
\(159\) 0 0
\(160\) 1.58556e7 0.306029
\(161\) −3.35483e7 −0.633548
\(162\) 0 0
\(163\) 7.29006e7 1.31848 0.659241 0.751931i \(-0.270878\pi\)
0.659241 + 0.751931i \(0.270878\pi\)
\(164\) 6.44166e6 0.114037
\(165\) 0 0
\(166\) −7.24061e6 −0.122856
\(167\) 9.50243e7 1.57880 0.789400 0.613879i \(-0.210392\pi\)
0.789400 + 0.613879i \(0.210392\pi\)
\(168\) 0 0
\(169\) −4.08958e7 −0.651742
\(170\) 6.56509e7 1.02487
\(171\) 0 0
\(172\) −8.20779e6 −0.122992
\(173\) −5.86358e7 −0.860996 −0.430498 0.902592i \(-0.641662\pi\)
−0.430498 + 0.902592i \(0.641662\pi\)
\(174\) 0 0
\(175\) −6.36165e7 −0.897298
\(176\) −1.18478e8 −1.63811
\(177\) 0 0
\(178\) 8.26147e7 1.09796
\(179\) 3.55324e7 0.463061 0.231531 0.972828i \(-0.425627\pi\)
0.231531 + 0.972828i \(0.425627\pi\)
\(180\) 0 0
\(181\) 1.28354e8 1.60892 0.804459 0.594008i \(-0.202455\pi\)
0.804459 + 0.594008i \(0.202455\pi\)
\(182\) −2.53919e7 −0.312209
\(183\) 0 0
\(184\) 1.00279e8 1.18672
\(185\) −1.25348e8 −1.45551
\(186\) 0 0
\(187\) 1.08946e8 1.21834
\(188\) 2.12894e6 0.0233674
\(189\) 0 0
\(190\) −5.52708e7 −0.584599
\(191\) 8.71206e7 0.904699 0.452350 0.891841i \(-0.350586\pi\)
0.452350 + 0.891841i \(0.350586\pi\)
\(192\) 0 0
\(193\) 3.34478e7 0.334902 0.167451 0.985880i \(-0.446446\pi\)
0.167451 + 0.985880i \(0.446446\pi\)
\(194\) −5.27292e7 −0.518495
\(195\) 0 0
\(196\) 6.91589e6 0.0656073
\(197\) −1.43873e8 −1.34074 −0.670372 0.742025i \(-0.733866\pi\)
−0.670372 + 0.742025i \(0.733866\pi\)
\(198\) 0 0
\(199\) −3.39107e7 −0.305036 −0.152518 0.988301i \(-0.548738\pi\)
−0.152518 + 0.988301i \(0.548738\pi\)
\(200\) 1.90156e8 1.68076
\(201\) 0 0
\(202\) −1.98732e8 −1.69644
\(203\) −8.80864e7 −0.739048
\(204\) 0 0
\(205\) 2.39416e8 1.94095
\(206\) 5.61258e7 0.447329
\(207\) 0 0
\(208\) 6.86244e7 0.528758
\(209\) −9.17207e7 −0.694953
\(210\) 0 0
\(211\) −1.13428e8 −0.831249 −0.415624 0.909536i \(-0.636437\pi\)
−0.415624 + 0.909536i \(0.636437\pi\)
\(212\) −3.81120e6 −0.0274717
\(213\) 0 0
\(214\) −2.32816e7 −0.162392
\(215\) −3.05058e8 −2.09338
\(216\) 0 0
\(217\) −3.15331e7 −0.209487
\(218\) 1.71961e8 1.12417
\(219\) 0 0
\(220\) 4.43374e7 0.280731
\(221\) −6.31034e7 −0.393260
\(222\) 0 0
\(223\) −1.58906e8 −0.959565 −0.479782 0.877387i \(-0.659284\pi\)
−0.479782 + 0.877387i \(0.659284\pi\)
\(224\) 1.77086e7 0.105273
\(225\) 0 0
\(226\) 1.00017e8 0.576358
\(227\) 2.46698e7 0.139983 0.0699915 0.997548i \(-0.477703\pi\)
0.0699915 + 0.997548i \(0.477703\pi\)
\(228\) 0 0
\(229\) 2.79166e6 0.0153617 0.00768084 0.999971i \(-0.497555\pi\)
0.00768084 + 0.999971i \(0.497555\pi\)
\(230\) 3.23297e8 1.75208
\(231\) 0 0
\(232\) 2.63300e8 1.38434
\(233\) −1.72078e8 −0.891207 −0.445604 0.895230i \(-0.647011\pi\)
−0.445604 + 0.895230i \(0.647011\pi\)
\(234\) 0 0
\(235\) 7.91259e7 0.397723
\(236\) −2.49694e6 −0.0123656
\(237\) 0 0
\(238\) 7.33235e7 0.352553
\(239\) 3.26143e8 1.54531 0.772656 0.634825i \(-0.218928\pi\)
0.772656 + 0.634825i \(0.218928\pi\)
\(240\) 0 0
\(241\) 4.12780e8 1.89958 0.949792 0.312882i \(-0.101294\pi\)
0.949792 + 0.312882i \(0.101294\pi\)
\(242\) −4.91324e8 −2.22851
\(243\) 0 0
\(244\) 3.67035e7 0.161750
\(245\) 2.57042e8 1.11666
\(246\) 0 0
\(247\) 5.31261e7 0.224320
\(248\) 9.42559e7 0.392399
\(249\) 0 0
\(250\) 2.33103e8 0.943536
\(251\) −4.66684e8 −1.86279 −0.931397 0.364005i \(-0.881409\pi\)
−0.931397 + 0.364005i \(0.881409\pi\)
\(252\) 0 0
\(253\) 5.36504e8 2.08282
\(254\) −8.80278e7 −0.337056
\(255\) 0 0
\(256\) −7.57777e7 −0.282294
\(257\) 9.36406e7 0.344111 0.172055 0.985087i \(-0.444959\pi\)
0.172055 + 0.985087i \(0.444959\pi\)
\(258\) 0 0
\(259\) −1.39997e8 −0.500692
\(260\) −2.56809e7 −0.0906158
\(261\) 0 0
\(262\) 4.74554e7 0.163016
\(263\) 5.00867e8 1.69776 0.848882 0.528582i \(-0.177276\pi\)
0.848882 + 0.528582i \(0.177276\pi\)
\(264\) 0 0
\(265\) −1.41650e8 −0.467581
\(266\) −6.17303e7 −0.201100
\(267\) 0 0
\(268\) 1.07667e7 0.0341673
\(269\) −7.84305e7 −0.245670 −0.122835 0.992427i \(-0.539199\pi\)
−0.122835 + 0.992427i \(0.539199\pi\)
\(270\) 0 0
\(271\) 3.97558e8 1.21341 0.606705 0.794927i \(-0.292491\pi\)
0.606705 + 0.794927i \(0.292491\pi\)
\(272\) −1.98165e8 −0.597084
\(273\) 0 0
\(274\) −8.84576e7 −0.259781
\(275\) 1.01735e9 2.94990
\(276\) 0 0
\(277\) −9.64793e7 −0.272744 −0.136372 0.990658i \(-0.543544\pi\)
−0.136372 + 0.990658i \(0.543544\pi\)
\(278\) 5.99289e8 1.67294
\(279\) 0 0
\(280\) 3.44006e8 0.936512
\(281\) 5.37086e8 1.44402 0.722008 0.691885i \(-0.243219\pi\)
0.722008 + 0.691885i \(0.243219\pi\)
\(282\) 0 0
\(283\) −6.18129e8 −1.62116 −0.810582 0.585626i \(-0.800849\pi\)
−0.810582 + 0.585626i \(0.800849\pi\)
\(284\) −1.39846e7 −0.0362272
\(285\) 0 0
\(286\) 4.06067e8 1.02640
\(287\) 2.67397e8 0.667682
\(288\) 0 0
\(289\) −2.28117e8 −0.555923
\(290\) 8.48867e8 2.04384
\(291\) 0 0
\(292\) −407264. −0.000957273 0
\(293\) −3.39223e8 −0.787858 −0.393929 0.919141i \(-0.628884\pi\)
−0.393929 + 0.919141i \(0.628884\pi\)
\(294\) 0 0
\(295\) −9.28031e7 −0.210468
\(296\) 4.18467e8 0.937864
\(297\) 0 0
\(298\) 3.07276e8 0.672623
\(299\) −3.10752e8 −0.672302
\(300\) 0 0
\(301\) −3.40710e8 −0.720115
\(302\) 1.52336e8 0.318257
\(303\) 0 0
\(304\) 1.66833e8 0.340584
\(305\) 1.36415e9 2.75304
\(306\) 0 0
\(307\) −9.62816e7 −0.189915 −0.0949575 0.995481i \(-0.530272\pi\)
−0.0949575 + 0.995481i \(0.530272\pi\)
\(308\) 4.95191e7 0.0965707
\(309\) 0 0
\(310\) 3.03877e8 0.579338
\(311\) 7.91196e8 1.49150 0.745749 0.666227i \(-0.232092\pi\)
0.745749 + 0.666227i \(0.232092\pi\)
\(312\) 0 0
\(313\) −8.83580e7 −0.162870 −0.0814349 0.996679i \(-0.525950\pi\)
−0.0814349 + 0.996679i \(0.525950\pi\)
\(314\) −6.50088e8 −1.18500
\(315\) 0 0
\(316\) 8.44154e6 0.0150493
\(317\) 3.65056e8 0.643654 0.321827 0.946799i \(-0.395703\pi\)
0.321827 + 0.946799i \(0.395703\pi\)
\(318\) 0 0
\(319\) 1.40868e9 2.42965
\(320\) −1.01972e9 −1.73963
\(321\) 0 0
\(322\) 3.61080e8 0.602710
\(323\) −1.53411e8 −0.253307
\(324\) 0 0
\(325\) −5.89268e8 −0.952184
\(326\) −7.84630e8 −1.25430
\(327\) 0 0
\(328\) −7.99277e8 −1.25066
\(329\) 8.83733e7 0.136816
\(330\) 0 0
\(331\) 8.84610e8 1.34077 0.670384 0.742014i \(-0.266129\pi\)
0.670384 + 0.742014i \(0.266129\pi\)
\(332\) −8.17886e6 −0.0122662
\(333\) 0 0
\(334\) −1.02275e9 −1.50195
\(335\) 4.00164e8 0.581542
\(336\) 0 0
\(337\) −1.01312e9 −1.44197 −0.720983 0.692953i \(-0.756309\pi\)
−0.720983 + 0.692953i \(0.756309\pi\)
\(338\) 4.40162e8 0.620018
\(339\) 0 0
\(340\) 7.41581e7 0.102325
\(341\) 5.04278e8 0.688699
\(342\) 0 0
\(343\) 7.02701e8 0.940246
\(344\) 1.01842e9 1.34887
\(345\) 0 0
\(346\) 6.31097e8 0.819086
\(347\) 4.59045e8 0.589796 0.294898 0.955529i \(-0.404714\pi\)
0.294898 + 0.955529i \(0.404714\pi\)
\(348\) 0 0
\(349\) 6.65833e8 0.838448 0.419224 0.907883i \(-0.362302\pi\)
0.419224 + 0.907883i \(0.362302\pi\)
\(350\) 6.84705e8 0.853621
\(351\) 0 0
\(352\) −2.83196e8 −0.346089
\(353\) 7.46486e7 0.0903255 0.0451627 0.998980i \(-0.485619\pi\)
0.0451627 + 0.998980i \(0.485619\pi\)
\(354\) 0 0
\(355\) −5.19762e8 −0.616603
\(356\) 9.33201e7 0.109623
\(357\) 0 0
\(358\) −3.82435e8 −0.440521
\(359\) 1.55696e9 1.77601 0.888005 0.459833i \(-0.152091\pi\)
0.888005 + 0.459833i \(0.152091\pi\)
\(360\) 0 0
\(361\) −7.64717e8 −0.855511
\(362\) −1.38147e9 −1.53060
\(363\) 0 0
\(364\) −2.86822e7 −0.0311715
\(365\) −1.51367e7 −0.0162932
\(366\) 0 0
\(367\) −4.78302e7 −0.0505092 −0.0252546 0.999681i \(-0.508040\pi\)
−0.0252546 + 0.999681i \(0.508040\pi\)
\(368\) −9.75859e8 −1.02075
\(369\) 0 0
\(370\) 1.34912e9 1.38467
\(371\) −1.58205e8 −0.160846
\(372\) 0 0
\(373\) −1.46088e9 −1.45759 −0.728793 0.684735i \(-0.759918\pi\)
−0.728793 + 0.684735i \(0.759918\pi\)
\(374\) −1.17259e9 −1.15903
\(375\) 0 0
\(376\) −2.64157e8 −0.256274
\(377\) −8.15928e8 −0.784255
\(378\) 0 0
\(379\) 8.99699e8 0.848907 0.424453 0.905450i \(-0.360466\pi\)
0.424453 + 0.905450i \(0.360466\pi\)
\(380\) −6.24329e7 −0.0583674
\(381\) 0 0
\(382\) −9.37680e8 −0.860662
\(383\) −7.51806e8 −0.683770 −0.341885 0.939742i \(-0.611065\pi\)
−0.341885 + 0.939742i \(0.611065\pi\)
\(384\) 0 0
\(385\) 1.84047e9 1.64367
\(386\) −3.59999e8 −0.318600
\(387\) 0 0
\(388\) −5.95619e7 −0.0517675
\(389\) −1.98944e9 −1.71359 −0.856794 0.515660i \(-0.827547\pi\)
−0.856794 + 0.515660i \(0.827547\pi\)
\(390\) 0 0
\(391\) 8.97349e8 0.759176
\(392\) −8.58119e8 −0.719526
\(393\) 0 0
\(394\) 1.54850e9 1.27548
\(395\) 3.13745e8 0.256146
\(396\) 0 0
\(397\) −2.39171e9 −1.91841 −0.959207 0.282704i \(-0.908769\pi\)
−0.959207 + 0.282704i \(0.908769\pi\)
\(398\) 3.64981e8 0.290188
\(399\) 0 0
\(400\) −1.85049e9 −1.44569
\(401\) −1.23670e9 −0.957767 −0.478883 0.877879i \(-0.658958\pi\)
−0.478883 + 0.877879i \(0.658958\pi\)
\(402\) 0 0
\(403\) −2.92086e8 −0.222302
\(404\) −2.24485e8 −0.169376
\(405\) 0 0
\(406\) 9.48075e8 0.703075
\(407\) 2.23884e9 1.64605
\(408\) 0 0
\(409\) 1.08001e9 0.780545 0.390272 0.920699i \(-0.372381\pi\)
0.390272 + 0.920699i \(0.372381\pi\)
\(410\) −2.57684e9 −1.84648
\(411\) 0 0
\(412\) 6.33986e7 0.0446622
\(413\) −1.03649e8 −0.0724003
\(414\) 0 0
\(415\) −3.03982e8 −0.208776
\(416\) 1.64032e8 0.111712
\(417\) 0 0
\(418\) 9.87190e8 0.661125
\(419\) 1.95040e9 1.29531 0.647656 0.761933i \(-0.275749\pi\)
0.647656 + 0.761933i \(0.275749\pi\)
\(420\) 0 0
\(421\) 4.11047e8 0.268475 0.134238 0.990949i \(-0.457141\pi\)
0.134238 + 0.990949i \(0.457141\pi\)
\(422\) 1.22082e9 0.790787
\(423\) 0 0
\(424\) 4.72891e8 0.301287
\(425\) 1.70161e9 1.07523
\(426\) 0 0
\(427\) 1.52358e9 0.947039
\(428\) −2.62985e7 −0.0162135
\(429\) 0 0
\(430\) 3.28334e9 1.99148
\(431\) 2.36621e9 1.42358 0.711792 0.702390i \(-0.247884\pi\)
0.711792 + 0.702390i \(0.247884\pi\)
\(432\) 0 0
\(433\) −2.22967e8 −0.131987 −0.0659937 0.997820i \(-0.521022\pi\)
−0.0659937 + 0.997820i \(0.521022\pi\)
\(434\) 3.39391e8 0.199291
\(435\) 0 0
\(436\) 1.94244e8 0.112239
\(437\) −7.55468e8 −0.433043
\(438\) 0 0
\(439\) −6.34602e8 −0.357994 −0.178997 0.983850i \(-0.557285\pi\)
−0.178997 + 0.983850i \(0.557285\pi\)
\(440\) −5.50135e9 −3.07882
\(441\) 0 0
\(442\) 6.79182e8 0.374118
\(443\) 1.44884e9 0.791788 0.395894 0.918296i \(-0.370435\pi\)
0.395894 + 0.918296i \(0.370435\pi\)
\(444\) 0 0
\(445\) 3.46841e9 1.86582
\(446\) 1.71031e9 0.912857
\(447\) 0 0
\(448\) −1.13890e9 −0.598427
\(449\) −3.21927e9 −1.67840 −0.839199 0.543825i \(-0.816976\pi\)
−0.839199 + 0.543825i \(0.816976\pi\)
\(450\) 0 0
\(451\) −4.27621e9 −2.19503
\(452\) 1.12977e8 0.0575447
\(453\) 0 0
\(454\) −2.65521e8 −0.133169
\(455\) −1.06603e9 −0.530553
\(456\) 0 0
\(457\) −1.03131e9 −0.505454 −0.252727 0.967538i \(-0.581328\pi\)
−0.252727 + 0.967538i \(0.581328\pi\)
\(458\) −3.00467e7 −0.0146139
\(459\) 0 0
\(460\) 3.65190e8 0.174931
\(461\) −3.21305e9 −1.52744 −0.763720 0.645547i \(-0.776629\pi\)
−0.763720 + 0.645547i \(0.776629\pi\)
\(462\) 0 0
\(463\) −1.66945e9 −0.781698 −0.390849 0.920455i \(-0.627819\pi\)
−0.390849 + 0.920455i \(0.627819\pi\)
\(464\) −2.56228e9 −1.19073
\(465\) 0 0
\(466\) 1.85207e9 0.847827
\(467\) −3.79498e8 −0.172425 −0.0862125 0.996277i \(-0.527476\pi\)
−0.0862125 + 0.996277i \(0.527476\pi\)
\(468\) 0 0
\(469\) 4.46931e8 0.200049
\(470\) −8.51632e8 −0.378364
\(471\) 0 0
\(472\) 3.09818e8 0.135616
\(473\) 5.44863e9 2.36741
\(474\) 0 0
\(475\) −1.43257e9 −0.613321
\(476\) 8.28249e7 0.0351995
\(477\) 0 0
\(478\) −3.51028e9 −1.47009
\(479\) −1.60692e9 −0.668068 −0.334034 0.942561i \(-0.608410\pi\)
−0.334034 + 0.942561i \(0.608410\pi\)
\(480\) 0 0
\(481\) −1.29677e9 −0.531319
\(482\) −4.44275e9 −1.80712
\(483\) 0 0
\(484\) −5.54990e8 −0.222498
\(485\) −2.21373e9 −0.881105
\(486\) 0 0
\(487\) 4.45492e9 1.74779 0.873893 0.486118i \(-0.161587\pi\)
0.873893 + 0.486118i \(0.161587\pi\)
\(488\) −4.55414e9 −1.77393
\(489\) 0 0
\(490\) −2.76654e9 −1.06231
\(491\) 3.60524e9 1.37451 0.687256 0.726415i \(-0.258815\pi\)
0.687256 + 0.726415i \(0.258815\pi\)
\(492\) 0 0
\(493\) 2.35613e9 0.885596
\(494\) −5.71796e8 −0.213401
\(495\) 0 0
\(496\) −9.17242e8 −0.337519
\(497\) −5.80507e8 −0.212109
\(498\) 0 0
\(499\) −2.51714e9 −0.906893 −0.453446 0.891284i \(-0.649806\pi\)
−0.453446 + 0.891284i \(0.649806\pi\)
\(500\) 2.63309e8 0.0942044
\(501\) 0 0
\(502\) 5.02292e9 1.77212
\(503\) −3.07735e9 −1.07817 −0.539087 0.842250i \(-0.681230\pi\)
−0.539087 + 0.842250i \(0.681230\pi\)
\(504\) 0 0
\(505\) −8.34338e9 −2.88285
\(506\) −5.77440e9 −1.98143
\(507\) 0 0
\(508\) −9.94346e7 −0.0336523
\(509\) 5.06424e9 1.70217 0.851083 0.525031i \(-0.175946\pi\)
0.851083 + 0.525031i \(0.175946\pi\)
\(510\) 0 0
\(511\) −1.69057e7 −0.00560481
\(512\) 3.34968e9 1.10296
\(513\) 0 0
\(514\) −1.00785e9 −0.327361
\(515\) 2.35633e9 0.760169
\(516\) 0 0
\(517\) −1.41327e9 −0.449787
\(518\) 1.50679e9 0.476320
\(519\) 0 0
\(520\) 3.18647e9 0.993798
\(521\) 3.99436e9 1.23741 0.618707 0.785622i \(-0.287657\pi\)
0.618707 + 0.785622i \(0.287657\pi\)
\(522\) 0 0
\(523\) 6.31987e9 1.93175 0.965877 0.259001i \(-0.0833931\pi\)
0.965877 + 0.259001i \(0.0833931\pi\)
\(524\) 5.36048e7 0.0162759
\(525\) 0 0
\(526\) −5.39084e9 −1.61513
\(527\) 8.43448e8 0.251027
\(528\) 0 0
\(529\) 1.01415e9 0.297857
\(530\) 1.52458e9 0.444821
\(531\) 0 0
\(532\) −6.97294e7 −0.0200782
\(533\) 2.47685e9 0.708523
\(534\) 0 0
\(535\) −9.77432e8 −0.275961
\(536\) −1.33592e9 −0.374718
\(537\) 0 0
\(538\) 8.44148e8 0.233712
\(539\) −4.59102e9 −1.26284
\(540\) 0 0
\(541\) −4.36436e9 −1.18503 −0.592516 0.805559i \(-0.701865\pi\)
−0.592516 + 0.805559i \(0.701865\pi\)
\(542\) −4.27892e9 −1.15435
\(543\) 0 0
\(544\) −4.73670e8 −0.126148
\(545\) 7.21943e9 1.91036
\(546\) 0 0
\(547\) −2.77236e9 −0.724258 −0.362129 0.932128i \(-0.617950\pi\)
−0.362129 + 0.932128i \(0.617950\pi\)
\(548\) −9.99201e7 −0.0259371
\(549\) 0 0
\(550\) −1.09498e10 −2.80632
\(551\) −1.98360e9 −0.505154
\(552\) 0 0
\(553\) 3.50412e8 0.0881133
\(554\) 1.03841e9 0.259468
\(555\) 0 0
\(556\) 6.76946e8 0.167029
\(557\) −6.99835e9 −1.71594 −0.857970 0.513699i \(-0.828275\pi\)
−0.857970 + 0.513699i \(0.828275\pi\)
\(558\) 0 0
\(559\) −3.15593e9 −0.764163
\(560\) −3.34767e9 −0.805534
\(561\) 0 0
\(562\) −5.78066e9 −1.37373
\(563\) −1.13060e9 −0.267012 −0.133506 0.991048i \(-0.542624\pi\)
−0.133506 + 0.991048i \(0.542624\pi\)
\(564\) 0 0
\(565\) 4.19899e9 0.979435
\(566\) 6.65293e9 1.54225
\(567\) 0 0
\(568\) 1.73520e9 0.397310
\(569\) 4.34076e9 0.987810 0.493905 0.869516i \(-0.335569\pi\)
0.493905 + 0.869516i \(0.335569\pi\)
\(570\) 0 0
\(571\) 1.60830e9 0.361526 0.180763 0.983527i \(-0.442143\pi\)
0.180763 + 0.983527i \(0.442143\pi\)
\(572\) 4.58686e8 0.102478
\(573\) 0 0
\(574\) −2.87799e9 −0.635182
\(575\) 8.37956e9 1.83816
\(576\) 0 0
\(577\) 4.44207e9 0.962654 0.481327 0.876541i \(-0.340155\pi\)
0.481327 + 0.876541i \(0.340155\pi\)
\(578\) 2.45522e9 0.528863
\(579\) 0 0
\(580\) 9.58865e8 0.204061
\(581\) −3.39509e8 −0.0718181
\(582\) 0 0
\(583\) 2.53001e9 0.528789
\(584\) 5.05330e7 0.0104986
\(585\) 0 0
\(586\) 3.65106e9 0.749509
\(587\) −5.12698e8 −0.104623 −0.0523116 0.998631i \(-0.516659\pi\)
−0.0523116 + 0.998631i \(0.516659\pi\)
\(588\) 0 0
\(589\) −7.10089e8 −0.143189
\(590\) 9.98841e8 0.200223
\(591\) 0 0
\(592\) −4.07227e9 −0.806698
\(593\) −4.79247e9 −0.943774 −0.471887 0.881659i \(-0.656427\pi\)
−0.471887 + 0.881659i \(0.656427\pi\)
\(594\) 0 0
\(595\) 3.07834e9 0.599111
\(596\) 3.47094e8 0.0671560
\(597\) 0 0
\(598\) 3.34462e9 0.639577
\(599\) −6.08958e9 −1.15769 −0.578846 0.815437i \(-0.696497\pi\)
−0.578846 + 0.815437i \(0.696497\pi\)
\(600\) 0 0
\(601\) −5.35978e9 −1.00713 −0.503566 0.863957i \(-0.667979\pi\)
−0.503566 + 0.863957i \(0.667979\pi\)
\(602\) 3.66706e9 0.685063
\(603\) 0 0
\(604\) 1.72076e8 0.0317754
\(605\) −2.06272e10 −3.78701
\(606\) 0 0
\(607\) 9.02137e9 1.63724 0.818619 0.574337i \(-0.194740\pi\)
0.818619 + 0.574337i \(0.194740\pi\)
\(608\) 3.98777e8 0.0719561
\(609\) 0 0
\(610\) −1.46824e10 −2.61904
\(611\) 8.18586e8 0.145184
\(612\) 0 0
\(613\) 7.72450e9 1.35444 0.677219 0.735782i \(-0.263185\pi\)
0.677219 + 0.735782i \(0.263185\pi\)
\(614\) 1.03628e9 0.180671
\(615\) 0 0
\(616\) −6.14429e9 −1.05911
\(617\) 4.39185e9 0.752748 0.376374 0.926468i \(-0.377171\pi\)
0.376374 + 0.926468i \(0.377171\pi\)
\(618\) 0 0
\(619\) 3.33323e9 0.564870 0.282435 0.959286i \(-0.408858\pi\)
0.282435 + 0.959286i \(0.408858\pi\)
\(620\) 3.43254e8 0.0578422
\(621\) 0 0
\(622\) −8.51565e9 −1.41890
\(623\) 3.87376e9 0.641837
\(624\) 0 0
\(625\) −6.16859e7 −0.0101066
\(626\) 9.50998e8 0.154942
\(627\) 0 0
\(628\) −7.34327e8 −0.118313
\(629\) 3.74465e9 0.599976
\(630\) 0 0
\(631\) −7.87439e9 −1.24771 −0.623856 0.781540i \(-0.714435\pi\)
−0.623856 + 0.781540i \(0.714435\pi\)
\(632\) −1.04742e9 −0.165048
\(633\) 0 0
\(634\) −3.92910e9 −0.612323
\(635\) −3.69567e9 −0.572776
\(636\) 0 0
\(637\) 2.65919e9 0.407626
\(638\) −1.51616e10 −2.31139
\(639\) 0 0
\(640\) 8.94577e9 1.34892
\(641\) −2.66936e8 −0.0400316 −0.0200158 0.999800i \(-0.506372\pi\)
−0.0200158 + 0.999800i \(0.506372\pi\)
\(642\) 0 0
\(643\) −9.18242e9 −1.36213 −0.681065 0.732223i \(-0.738483\pi\)
−0.681065 + 0.732223i \(0.738483\pi\)
\(644\) 4.07870e8 0.0601757
\(645\) 0 0
\(646\) 1.65116e9 0.240977
\(647\) −7.79548e9 −1.13156 −0.565781 0.824556i \(-0.691425\pi\)
−0.565781 + 0.824556i \(0.691425\pi\)
\(648\) 0 0
\(649\) 1.65756e9 0.238019
\(650\) 6.34229e9 0.905836
\(651\) 0 0
\(652\) −8.86303e8 −0.125232
\(653\) 7.64228e8 0.107406 0.0537028 0.998557i \(-0.482898\pi\)
0.0537028 + 0.998557i \(0.482898\pi\)
\(654\) 0 0
\(655\) 1.99232e9 0.277022
\(656\) 7.77809e9 1.07575
\(657\) 0 0
\(658\) −9.51162e8 −0.130156
\(659\) 4.42497e9 0.602299 0.301149 0.953577i \(-0.402630\pi\)
0.301149 + 0.953577i \(0.402630\pi\)
\(660\) 0 0
\(661\) −1.21787e9 −0.164019 −0.0820097 0.996632i \(-0.526134\pi\)
−0.0820097 + 0.996632i \(0.526134\pi\)
\(662\) −9.52106e9 −1.27551
\(663\) 0 0
\(664\) 1.01483e9 0.134525
\(665\) −2.59162e9 −0.341740
\(666\) 0 0
\(667\) 1.16027e10 1.51398
\(668\) −1.15528e9 −0.149958
\(669\) 0 0
\(670\) −4.30697e9 −0.553235
\(671\) −2.43651e10 −3.11343
\(672\) 0 0
\(673\) −6.63758e9 −0.839377 −0.419689 0.907668i \(-0.637861\pi\)
−0.419689 + 0.907668i \(0.637861\pi\)
\(674\) 1.09042e10 1.37178
\(675\) 0 0
\(676\) 4.97199e8 0.0619037
\(677\) −1.51749e9 −0.187961 −0.0939803 0.995574i \(-0.529959\pi\)
−0.0939803 + 0.995574i \(0.529959\pi\)
\(678\) 0 0
\(679\) −2.47244e9 −0.303098
\(680\) −9.20148e9 −1.12222
\(681\) 0 0
\(682\) −5.42755e9 −0.655176
\(683\) 1.25832e10 1.51119 0.755595 0.655039i \(-0.227348\pi\)
0.755595 + 0.655039i \(0.227348\pi\)
\(684\) 0 0
\(685\) −3.71371e9 −0.441460
\(686\) −7.56318e9 −0.894479
\(687\) 0 0
\(688\) −9.91063e9 −1.16022
\(689\) −1.46542e9 −0.170685
\(690\) 0 0
\(691\) −5.99235e9 −0.690914 −0.345457 0.938435i \(-0.612276\pi\)
−0.345457 + 0.938435i \(0.612276\pi\)
\(692\) 7.12876e8 0.0817791
\(693\) 0 0
\(694\) −4.94070e9 −0.561088
\(695\) 2.51599e10 2.84291
\(696\) 0 0
\(697\) −7.15232e9 −0.800078
\(698\) −7.16636e9 −0.797636
\(699\) 0 0
\(700\) 7.73430e8 0.0852271
\(701\) 1.16633e10 1.27881 0.639406 0.768869i \(-0.279180\pi\)
0.639406 + 0.768869i \(0.279180\pi\)
\(702\) 0 0
\(703\) −3.15258e9 −0.342233
\(704\) 1.82132e10 1.96736
\(705\) 0 0
\(706\) −8.03443e8 −0.0859288
\(707\) −9.31847e9 −0.991691
\(708\) 0 0
\(709\) −5.32106e9 −0.560707 −0.280353 0.959897i \(-0.590452\pi\)
−0.280353 + 0.959897i \(0.590452\pi\)
\(710\) 5.59420e9 0.586589
\(711\) 0 0
\(712\) −1.15791e10 −1.20225
\(713\) 4.15354e9 0.429146
\(714\) 0 0
\(715\) 1.70479e10 1.74422
\(716\) −4.31992e8 −0.0439825
\(717\) 0 0
\(718\) −1.67575e10 −1.68956
\(719\) 1.68037e9 0.168599 0.0842993 0.996440i \(-0.473135\pi\)
0.0842993 + 0.996440i \(0.473135\pi\)
\(720\) 0 0
\(721\) 2.63171e9 0.261496
\(722\) 8.23065e9 0.813868
\(723\) 0 0
\(724\) −1.56049e9 −0.152818
\(725\) 2.20019e10 2.14426
\(726\) 0 0
\(727\) 6.43210e9 0.620844 0.310422 0.950599i \(-0.399530\pi\)
0.310422 + 0.950599i \(0.399530\pi\)
\(728\) 3.55887e9 0.341863
\(729\) 0 0
\(730\) 1.62916e8 0.0155001
\(731\) 9.11330e9 0.862908
\(732\) 0 0
\(733\) 9.99078e9 0.936991 0.468495 0.883466i \(-0.344796\pi\)
0.468495 + 0.883466i \(0.344796\pi\)
\(734\) 5.14796e8 0.0480507
\(735\) 0 0
\(736\) −2.33258e9 −0.215657
\(737\) −7.14732e9 −0.657669
\(738\) 0 0
\(739\) 5.40356e9 0.492521 0.246260 0.969204i \(-0.420798\pi\)
0.246260 + 0.969204i \(0.420798\pi\)
\(740\) 1.52394e9 0.138248
\(741\) 0 0
\(742\) 1.70276e9 0.153017
\(743\) 1.37163e10 1.22680 0.613402 0.789771i \(-0.289801\pi\)
0.613402 + 0.789771i \(0.289801\pi\)
\(744\) 0 0
\(745\) 1.29004e10 1.14302
\(746\) 1.57235e10 1.38664
\(747\) 0 0
\(748\) −1.32454e9 −0.115720
\(749\) −1.09166e9 −0.0949299
\(750\) 0 0
\(751\) 1.22637e9 0.105653 0.0528264 0.998604i \(-0.483177\pi\)
0.0528264 + 0.998604i \(0.483177\pi\)
\(752\) 2.57062e9 0.220432
\(753\) 0 0
\(754\) 8.78184e9 0.746081
\(755\) 6.39552e9 0.540831
\(756\) 0 0
\(757\) −1.17911e10 −0.987912 −0.493956 0.869487i \(-0.664450\pi\)
−0.493956 + 0.869487i \(0.664450\pi\)
\(758\) −9.68347e9 −0.807586
\(759\) 0 0
\(760\) 7.74662e9 0.640125
\(761\) 1.03626e10 0.852359 0.426179 0.904639i \(-0.359859\pi\)
0.426179 + 0.904639i \(0.359859\pi\)
\(762\) 0 0
\(763\) 8.06317e9 0.657158
\(764\) −1.05919e9 −0.0859301
\(765\) 0 0
\(766\) 8.09169e9 0.650487
\(767\) −9.60082e8 −0.0768289
\(768\) 0 0
\(769\) 7.77986e9 0.616921 0.308460 0.951237i \(-0.400186\pi\)
0.308460 + 0.951237i \(0.400186\pi\)
\(770\) −1.98090e10 −1.56367
\(771\) 0 0
\(772\) −4.06648e8 −0.0318096
\(773\) −8.05208e9 −0.627018 −0.313509 0.949585i \(-0.601505\pi\)
−0.313509 + 0.949585i \(0.601505\pi\)
\(774\) 0 0
\(775\) 7.87623e9 0.607802
\(776\) 7.39040e9 0.567743
\(777\) 0 0
\(778\) 2.14123e10 1.63018
\(779\) 6.02146e9 0.456374
\(780\) 0 0
\(781\) 9.28346e9 0.697319
\(782\) −9.65817e9 −0.722223
\(783\) 0 0
\(784\) 8.35071e9 0.618895
\(785\) −2.72926e10 −2.01373
\(786\) 0 0
\(787\) 2.48655e10 1.81838 0.909192 0.416377i \(-0.136700\pi\)
0.909192 + 0.416377i \(0.136700\pi\)
\(788\) 1.74916e9 0.127347
\(789\) 0 0
\(790\) −3.37684e9 −0.243678
\(791\) 4.68973e9 0.336923
\(792\) 0 0
\(793\) 1.41126e10 1.00497
\(794\) 2.57420e10 1.82503
\(795\) 0 0
\(796\) 4.12276e8 0.0289729
\(797\) 1.33668e9 0.0935241 0.0467621 0.998906i \(-0.485110\pi\)
0.0467621 + 0.998906i \(0.485110\pi\)
\(798\) 0 0
\(799\) −2.36381e9 −0.163945
\(800\) −4.42319e9 −0.305436
\(801\) 0 0
\(802\) 1.33106e10 0.911147
\(803\) 2.70356e8 0.0184261
\(804\) 0 0
\(805\) 1.51592e10 1.02422
\(806\) 3.14372e9 0.211481
\(807\) 0 0
\(808\) 2.78539e10 1.85757
\(809\) 5.48115e9 0.363959 0.181979 0.983302i \(-0.441750\pi\)
0.181979 + 0.983302i \(0.441750\pi\)
\(810\) 0 0
\(811\) 1.52888e10 1.00647 0.503233 0.864151i \(-0.332144\pi\)
0.503233 + 0.864151i \(0.332144\pi\)
\(812\) 1.07093e9 0.0701963
\(813\) 0 0
\(814\) −2.40966e10 −1.56592
\(815\) −3.29411e10 −2.13150
\(816\) 0 0
\(817\) −7.67238e9 −0.492213
\(818\) −1.16242e10 −0.742551
\(819\) 0 0
\(820\) −2.91075e9 −0.184356
\(821\) −5.00980e9 −0.315951 −0.157975 0.987443i \(-0.550497\pi\)
−0.157975 + 0.987443i \(0.550497\pi\)
\(822\) 0 0
\(823\) −1.43300e10 −0.896077 −0.448038 0.894014i \(-0.647877\pi\)
−0.448038 + 0.894014i \(0.647877\pi\)
\(824\) −7.86646e9 −0.489817
\(825\) 0 0
\(826\) 1.11558e9 0.0688761
\(827\) −2.07445e10 −1.27536 −0.637682 0.770300i \(-0.720107\pi\)
−0.637682 + 0.770300i \(0.720107\pi\)
\(828\) 0 0
\(829\) −3.94434e9 −0.240455 −0.120227 0.992746i \(-0.538362\pi\)
−0.120227 + 0.992746i \(0.538362\pi\)
\(830\) 3.27176e9 0.198613
\(831\) 0 0
\(832\) −1.05494e10 −0.635033
\(833\) −7.67887e9 −0.460299
\(834\) 0 0
\(835\) −4.29380e10 −2.55234
\(836\) 1.11511e9 0.0660080
\(837\) 0 0
\(838\) −2.09922e10 −1.23226
\(839\) 1.52467e10 0.891268 0.445634 0.895215i \(-0.352978\pi\)
0.445634 + 0.895215i \(0.352978\pi\)
\(840\) 0 0
\(841\) 1.32150e10 0.766091
\(842\) −4.42410e9 −0.255407
\(843\) 0 0
\(844\) 1.37902e9 0.0789537
\(845\) 1.84793e10 1.05363
\(846\) 0 0
\(847\) −2.30379e10 −1.30272
\(848\) −4.60189e9 −0.259150
\(849\) 0 0
\(850\) −1.83145e10 −1.02289
\(851\) 1.84404e10 1.02569
\(852\) 0 0
\(853\) −2.14258e10 −1.18200 −0.590998 0.806673i \(-0.701266\pi\)
−0.590998 + 0.806673i \(0.701266\pi\)
\(854\) −1.63983e10 −0.900941
\(855\) 0 0
\(856\) 3.26310e9 0.177817
\(857\) −2.49225e10 −1.35257 −0.676283 0.736642i \(-0.736410\pi\)
−0.676283 + 0.736642i \(0.736410\pi\)
\(858\) 0 0
\(859\) −8.80692e9 −0.474076 −0.237038 0.971500i \(-0.576177\pi\)
−0.237038 + 0.971500i \(0.576177\pi\)
\(860\) 3.70880e9 0.198833
\(861\) 0 0
\(862\) −2.54675e10 −1.35429
\(863\) −5.86301e9 −0.310515 −0.155257 0.987874i \(-0.549621\pi\)
−0.155257 + 0.987874i \(0.549621\pi\)
\(864\) 0 0
\(865\) 2.64953e10 1.39192
\(866\) 2.39979e9 0.125563
\(867\) 0 0
\(868\) 3.83370e8 0.0198975
\(869\) −5.60380e9 −0.289676
\(870\) 0 0
\(871\) 4.13984e9 0.212285
\(872\) −2.41017e10 −1.23095
\(873\) 0 0
\(874\) 8.13111e9 0.411964
\(875\) 1.09301e10 0.551564
\(876\) 0 0
\(877\) 3.56534e10 1.78485 0.892427 0.451193i \(-0.149001\pi\)
0.892427 + 0.451193i \(0.149001\pi\)
\(878\) 6.83022e9 0.340568
\(879\) 0 0
\(880\) 5.35359e10 2.64823
\(881\) −3.92292e10 −1.93283 −0.966414 0.256989i \(-0.917269\pi\)
−0.966414 + 0.256989i \(0.917269\pi\)
\(882\) 0 0
\(883\) 2.18077e10 1.06598 0.532988 0.846123i \(-0.321069\pi\)
0.532988 + 0.846123i \(0.321069\pi\)
\(884\) 7.67192e8 0.0373526
\(885\) 0 0
\(886\) −1.55939e10 −0.753247
\(887\) −4.76501e9 −0.229262 −0.114631 0.993408i \(-0.536569\pi\)
−0.114631 + 0.993408i \(0.536569\pi\)
\(888\) 0 0
\(889\) −4.12758e9 −0.197033
\(890\) −3.73305e10 −1.77500
\(891\) 0 0
\(892\) 1.93194e9 0.0911414
\(893\) 1.99006e9 0.0935162
\(894\) 0 0
\(895\) −1.60558e10 −0.748601
\(896\) 9.99126e9 0.464026
\(897\) 0 0
\(898\) 3.46490e10 1.59670
\(899\) 1.09058e10 0.500609
\(900\) 0 0
\(901\) 4.23166e9 0.192741
\(902\) 4.60249e10 2.08819
\(903\) 0 0
\(904\) −1.40181e10 −0.631102
\(905\) −5.79984e10 −2.60103
\(906\) 0 0
\(907\) −8.85912e9 −0.394244 −0.197122 0.980379i \(-0.563159\pi\)
−0.197122 + 0.980379i \(0.563159\pi\)
\(908\) −2.99928e8 −0.0132959
\(909\) 0 0
\(910\) 1.14737e10 0.504728
\(911\) 5.66060e9 0.248055 0.124028 0.992279i \(-0.460419\pi\)
0.124028 + 0.992279i \(0.460419\pi\)
\(912\) 0 0
\(913\) 5.42942e9 0.236105
\(914\) 1.11000e10 0.480851
\(915\) 0 0
\(916\) −3.39402e7 −0.00145908
\(917\) 2.22516e9 0.0952947
\(918\) 0 0
\(919\) 3.15687e10 1.34169 0.670846 0.741597i \(-0.265931\pi\)
0.670846 + 0.741597i \(0.265931\pi\)
\(920\) −4.53125e10 −1.91849
\(921\) 0 0
\(922\) 3.45821e10 1.45309
\(923\) −5.37713e9 −0.225084
\(924\) 0 0
\(925\) 3.49680e10 1.45270
\(926\) 1.79683e10 0.743649
\(927\) 0 0
\(928\) −6.12456e9 −0.251569
\(929\) −2.88328e10 −1.17986 −0.589932 0.807453i \(-0.700845\pi\)
−0.589932 + 0.807453i \(0.700845\pi\)
\(930\) 0 0
\(931\) 6.46476e9 0.262560
\(932\) 2.09207e9 0.0846487
\(933\) 0 0
\(934\) 4.08454e9 0.164032
\(935\) −4.92288e10 −1.96960
\(936\) 0 0
\(937\) −4.19782e10 −1.66700 −0.833500 0.552520i \(-0.813666\pi\)
−0.833500 + 0.552520i \(0.813666\pi\)
\(938\) −4.81032e9 −0.190311
\(939\) 0 0
\(940\) −9.61988e8 −0.0377766
\(941\) 4.83827e10 1.89289 0.946447 0.322860i \(-0.104644\pi\)
0.946447 + 0.322860i \(0.104644\pi\)
\(942\) 0 0
\(943\) −3.52215e10 −1.36778
\(944\) −3.01496e9 −0.116649
\(945\) 0 0
\(946\) −5.86436e10 −2.25217
\(947\) 7.76682e9 0.297179 0.148590 0.988899i \(-0.452527\pi\)
0.148590 + 0.988899i \(0.452527\pi\)
\(948\) 0 0
\(949\) −1.56595e8 −0.00594765
\(950\) 1.54187e10 0.583467
\(951\) 0 0
\(952\) −1.02769e10 −0.386039
\(953\) 4.85606e9 0.181743 0.0908717 0.995863i \(-0.471035\pi\)
0.0908717 + 0.995863i \(0.471035\pi\)
\(954\) 0 0
\(955\) −3.93666e10 −1.46257
\(956\) −3.96515e9 −0.146777
\(957\) 0 0
\(958\) 1.72953e10 0.635549
\(959\) −4.14773e9 −0.151861
\(960\) 0 0
\(961\) −2.36086e10 −0.858100
\(962\) 1.39571e10 0.505457
\(963\) 0 0
\(964\) −5.01845e9 −0.180426
\(965\) −1.51138e10 −0.541413
\(966\) 0 0
\(967\) −2.84618e10 −1.01221 −0.506103 0.862473i \(-0.668915\pi\)
−0.506103 + 0.862473i \(0.668915\pi\)
\(968\) 6.88628e10 2.44017
\(969\) 0 0
\(970\) 2.38264e10 0.838217
\(971\) 3.54110e10 1.24128 0.620642 0.784094i \(-0.286872\pi\)
0.620642 + 0.784094i \(0.286872\pi\)
\(972\) 0 0
\(973\) 2.81004e10 0.977952
\(974\) −4.79483e10 −1.66271
\(975\) 0 0
\(976\) 4.43182e10 1.52584
\(977\) −4.07835e10 −1.39911 −0.699557 0.714577i \(-0.746619\pi\)
−0.699557 + 0.714577i \(0.746619\pi\)
\(978\) 0 0
\(979\) −6.19492e10 −2.11007
\(980\) −3.12504e9 −0.106063
\(981\) 0 0
\(982\) −3.88032e10 −1.30761
\(983\) 4.57225e10 1.53530 0.767649 0.640871i \(-0.221427\pi\)
0.767649 + 0.640871i \(0.221427\pi\)
\(984\) 0 0
\(985\) 6.50107e10 2.16749
\(986\) −2.53591e10 −0.842489
\(987\) 0 0
\(988\) −6.45891e8 −0.0213064
\(989\) 4.48782e10 1.47519
\(990\) 0 0
\(991\) 7.18635e9 0.234558 0.117279 0.993099i \(-0.462583\pi\)
0.117279 + 0.993099i \(0.462583\pi\)
\(992\) −2.19247e9 −0.0713086
\(993\) 0 0
\(994\) 6.24800e9 0.201785
\(995\) 1.53230e10 0.493131
\(996\) 0 0
\(997\) 5.76916e10 1.84365 0.921827 0.387602i \(-0.126696\pi\)
0.921827 + 0.387602i \(0.126696\pi\)
\(998\) 2.70920e10 0.862749
\(999\) 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.1.5 17
3.2 odd 2 177.8.a.b.1.13 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.13 17 3.2 odd 2
531.8.a.d.1.5 17 1.1 even 1 trivial