Properties

Label 531.8.a.d.1.17
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 1.17
Root \(-20.3182\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+22.3182 q^{2} +370.104 q^{4} +84.4166 q^{5} +1532.13 q^{7} +5403.34 q^{8} +O(q^{10})\) \(q+22.3182 q^{2} +370.104 q^{4} +84.4166 q^{5} +1532.13 q^{7} +5403.34 q^{8} +1884.03 q^{10} -4639.41 q^{11} -1710.47 q^{13} +34194.5 q^{14} +73219.7 q^{16} +14226.6 q^{17} -21902.8 q^{19} +31242.9 q^{20} -103543. q^{22} +107697. q^{23} -70998.8 q^{25} -38174.6 q^{26} +567048. q^{28} +195131. q^{29} -68117.7 q^{31} +942507. q^{32} +317512. q^{34} +129337. q^{35} -309640. q^{37} -488831. q^{38} +456131. q^{40} -873409. q^{41} +159630. q^{43} -1.71706e6 q^{44} +2.40360e6 q^{46} +615386. q^{47} +1.52388e6 q^{49} -1.58457e6 q^{50} -633051. q^{52} +745946. q^{53} -391643. q^{55} +8.27862e6 q^{56} +4.35498e6 q^{58} +205379. q^{59} -2.69167e6 q^{61} -1.52027e6 q^{62} +1.16630e7 q^{64} -144392. q^{65} +2.11862e6 q^{67} +5.26531e6 q^{68} +2.88658e6 q^{70} +1.64750e6 q^{71} -2.91422e6 q^{73} -6.91062e6 q^{74} -8.10630e6 q^{76} -7.10818e6 q^{77} +171784. q^{79} +6.18095e6 q^{80} -1.94930e7 q^{82} -4.94728e6 q^{83} +1.20096e6 q^{85} +3.56266e6 q^{86} -2.50683e7 q^{88} +1.83391e6 q^{89} -2.62066e6 q^{91} +3.98589e7 q^{92} +1.37343e7 q^{94} -1.84896e6 q^{95} -1.09662e7 q^{97} +3.40104e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.3182 1.97267 0.986336 0.164744i \(-0.0526798\pi\)
0.986336 + 0.164744i \(0.0526798\pi\)
\(3\) 0 0
\(4\) 370.104 2.89144
\(5\) 84.4166 0.302018 0.151009 0.988532i \(-0.451748\pi\)
0.151009 + 0.988532i \(0.451748\pi\)
\(6\) 0 0
\(7\) 1532.13 1.68831 0.844156 0.536097i \(-0.180102\pi\)
0.844156 + 0.536097i \(0.180102\pi\)
\(8\) 5403.34 3.73119
\(9\) 0 0
\(10\) 1884.03 0.595782
\(11\) −4639.41 −1.05097 −0.525483 0.850804i \(-0.676115\pi\)
−0.525483 + 0.850804i \(0.676115\pi\)
\(12\) 0 0
\(13\) −1710.47 −0.215930 −0.107965 0.994155i \(-0.534433\pi\)
−0.107965 + 0.994155i \(0.534433\pi\)
\(14\) 34194.5 3.33049
\(15\) 0 0
\(16\) 73219.7 4.46897
\(17\) 14226.6 0.702310 0.351155 0.936317i \(-0.385789\pi\)
0.351155 + 0.936317i \(0.385789\pi\)
\(18\) 0 0
\(19\) −21902.8 −0.732590 −0.366295 0.930499i \(-0.619374\pi\)
−0.366295 + 0.930499i \(0.619374\pi\)
\(20\) 31242.9 0.873266
\(21\) 0 0
\(22\) −103543. −2.07321
\(23\) 107697. 1.84567 0.922836 0.385194i \(-0.125865\pi\)
0.922836 + 0.385194i \(0.125865\pi\)
\(24\) 0 0
\(25\) −70998.8 −0.908785
\(26\) −38174.6 −0.425959
\(27\) 0 0
\(28\) 567048. 4.88165
\(29\) 195131. 1.48571 0.742853 0.669454i \(-0.233472\pi\)
0.742853 + 0.669454i \(0.233472\pi\)
\(30\) 0 0
\(31\) −68117.7 −0.410671 −0.205336 0.978692i \(-0.565829\pi\)
−0.205336 + 0.978692i \(0.565829\pi\)
\(32\) 942507. 5.08463
\(33\) 0 0
\(34\) 317512. 1.38543
\(35\) 129337. 0.509901
\(36\) 0 0
\(37\) −309640. −1.00496 −0.502482 0.864588i \(-0.667580\pi\)
−0.502482 + 0.864588i \(0.667580\pi\)
\(38\) −488831. −1.44516
\(39\) 0 0
\(40\) 456131. 1.12689
\(41\) −873409. −1.97913 −0.989565 0.144086i \(-0.953976\pi\)
−0.989565 + 0.144086i \(0.953976\pi\)
\(42\) 0 0
\(43\) 159630. 0.306179 0.153089 0.988212i \(-0.451078\pi\)
0.153089 + 0.988212i \(0.451078\pi\)
\(44\) −1.71706e6 −3.03880
\(45\) 0 0
\(46\) 2.40360e6 3.64091
\(47\) 615386. 0.864580 0.432290 0.901735i \(-0.357706\pi\)
0.432290 + 0.901735i \(0.357706\pi\)
\(48\) 0 0
\(49\) 1.52388e6 1.85040
\(50\) −1.58457e6 −1.79274
\(51\) 0 0
\(52\) −633051. −0.624348
\(53\) 745946. 0.688243 0.344121 0.938925i \(-0.388177\pi\)
0.344121 + 0.938925i \(0.388177\pi\)
\(54\) 0 0
\(55\) −391643. −0.317410
\(56\) 8.27862e6 6.29941
\(57\) 0 0
\(58\) 4.35498e6 2.93081
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −2.69167e6 −1.51833 −0.759167 0.650896i \(-0.774393\pi\)
−0.759167 + 0.650896i \(0.774393\pi\)
\(62\) −1.52027e6 −0.810119
\(63\) 0 0
\(64\) 1.16630e7 5.56135
\(65\) −144392. −0.0652147
\(66\) 0 0
\(67\) 2.11862e6 0.860581 0.430290 0.902691i \(-0.358411\pi\)
0.430290 + 0.902691i \(0.358411\pi\)
\(68\) 5.26531e6 2.03068
\(69\) 0 0
\(70\) 2.88658e6 1.00587
\(71\) 1.64750e6 0.546289 0.273144 0.961973i \(-0.411936\pi\)
0.273144 + 0.961973i \(0.411936\pi\)
\(72\) 0 0
\(73\) −2.91422e6 −0.876783 −0.438391 0.898784i \(-0.644452\pi\)
−0.438391 + 0.898784i \(0.644452\pi\)
\(74\) −6.91062e6 −1.98247
\(75\) 0 0
\(76\) −8.10630e6 −2.11824
\(77\) −7.10818e6 −1.77436
\(78\) 0 0
\(79\) 171784. 0.0392002 0.0196001 0.999808i \(-0.493761\pi\)
0.0196001 + 0.999808i \(0.493761\pi\)
\(80\) 6.18095e6 1.34971
\(81\) 0 0
\(82\) −1.94930e7 −3.90418
\(83\) −4.94728e6 −0.949714 −0.474857 0.880063i \(-0.657500\pi\)
−0.474857 + 0.880063i \(0.657500\pi\)
\(84\) 0 0
\(85\) 1.20096e6 0.212110
\(86\) 3.56266e6 0.603990
\(87\) 0 0
\(88\) −2.50683e7 −3.92135
\(89\) 1.83391e6 0.275748 0.137874 0.990450i \(-0.455973\pi\)
0.137874 + 0.990450i \(0.455973\pi\)
\(90\) 0 0
\(91\) −2.62066e6 −0.364557
\(92\) 3.98589e7 5.33664
\(93\) 0 0
\(94\) 1.37343e7 1.70553
\(95\) −1.84896e6 −0.221255
\(96\) 0 0
\(97\) −1.09662e7 −1.21998 −0.609992 0.792408i \(-0.708827\pi\)
−0.609992 + 0.792408i \(0.708827\pi\)
\(98\) 3.40104e7 3.65023
\(99\) 0 0
\(100\) −2.62770e7 −2.62770
\(101\) 7.80317e6 0.753610 0.376805 0.926293i \(-0.377023\pi\)
0.376805 + 0.926293i \(0.377023\pi\)
\(102\) 0 0
\(103\) −150244. −0.0135477 −0.00677386 0.999977i \(-0.502156\pi\)
−0.00677386 + 0.999977i \(0.502156\pi\)
\(104\) −9.24223e6 −0.805675
\(105\) 0 0
\(106\) 1.66482e7 1.35768
\(107\) −8.15970e6 −0.643918 −0.321959 0.946754i \(-0.604341\pi\)
−0.321959 + 0.946754i \(0.604341\pi\)
\(108\) 0 0
\(109\) 6.65386e6 0.492131 0.246066 0.969253i \(-0.420862\pi\)
0.246066 + 0.969253i \(0.420862\pi\)
\(110\) −8.74078e6 −0.626147
\(111\) 0 0
\(112\) 1.12182e8 7.54502
\(113\) −1.76235e7 −1.14899 −0.574496 0.818507i \(-0.694802\pi\)
−0.574496 + 0.818507i \(0.694802\pi\)
\(114\) 0 0
\(115\) 9.09137e6 0.557426
\(116\) 7.22187e7 4.29583
\(117\) 0 0
\(118\) 4.58370e6 0.256820
\(119\) 2.17970e7 1.18572
\(120\) 0 0
\(121\) 2.03695e6 0.104528
\(122\) −6.00733e7 −2.99517
\(123\) 0 0
\(124\) −2.52106e7 −1.18743
\(125\) −1.25885e7 −0.576487
\(126\) 0 0
\(127\) −2.94003e7 −1.27362 −0.636808 0.771023i \(-0.719745\pi\)
−0.636808 + 0.771023i \(0.719745\pi\)
\(128\) 1.39656e8 5.88608
\(129\) 0 0
\(130\) −3.22257e6 −0.128647
\(131\) −4.56665e6 −0.177480 −0.0887398 0.996055i \(-0.528284\pi\)
−0.0887398 + 0.996055i \(0.528284\pi\)
\(132\) 0 0
\(133\) −3.35579e7 −1.23684
\(134\) 4.72839e7 1.69764
\(135\) 0 0
\(136\) 7.68709e7 2.62045
\(137\) −3.33611e6 −0.110846 −0.0554228 0.998463i \(-0.517651\pi\)
−0.0554228 + 0.998463i \(0.517651\pi\)
\(138\) 0 0
\(139\) 2.57204e7 0.812317 0.406158 0.913803i \(-0.366868\pi\)
0.406158 + 0.913803i \(0.366868\pi\)
\(140\) 4.78682e7 1.47435
\(141\) 0 0
\(142\) 3.67694e7 1.07765
\(143\) 7.93556e6 0.226935
\(144\) 0 0
\(145\) 1.64723e7 0.448710
\(146\) −6.50402e7 −1.72961
\(147\) 0 0
\(148\) −1.14599e8 −2.90579
\(149\) 7.15213e7 1.77127 0.885633 0.464387i \(-0.153725\pi\)
0.885633 + 0.464387i \(0.153725\pi\)
\(150\) 0 0
\(151\) 3.70730e7 0.876271 0.438135 0.898909i \(-0.355639\pi\)
0.438135 + 0.898909i \(0.355639\pi\)
\(152\) −1.18348e8 −2.73343
\(153\) 0 0
\(154\) −1.58642e8 −3.50023
\(155\) −5.75026e6 −0.124030
\(156\) 0 0
\(157\) −5.84600e7 −1.20562 −0.602810 0.797885i \(-0.705952\pi\)
−0.602810 + 0.797885i \(0.705952\pi\)
\(158\) 3.83392e6 0.0773291
\(159\) 0 0
\(160\) 7.95632e7 1.53565
\(161\) 1.65005e8 3.11607
\(162\) 0 0
\(163\) 1.08299e8 1.95869 0.979346 0.202190i \(-0.0648057\pi\)
0.979346 + 0.202190i \(0.0648057\pi\)
\(164\) −3.23252e8 −5.72253
\(165\) 0 0
\(166\) −1.10415e8 −1.87347
\(167\) 6.99062e6 0.116147 0.0580735 0.998312i \(-0.481504\pi\)
0.0580735 + 0.998312i \(0.481504\pi\)
\(168\) 0 0
\(169\) −5.98228e7 −0.953374
\(170\) 2.68033e7 0.418424
\(171\) 0 0
\(172\) 5.90797e7 0.885296
\(173\) 9.78233e7 1.43642 0.718209 0.695827i \(-0.244962\pi\)
0.718209 + 0.695827i \(0.244962\pi\)
\(174\) 0 0
\(175\) −1.08780e8 −1.53431
\(176\) −3.39696e8 −4.69674
\(177\) 0 0
\(178\) 4.09296e7 0.543961
\(179\) −7.71690e7 −1.00567 −0.502837 0.864381i \(-0.667710\pi\)
−0.502837 + 0.864381i \(0.667710\pi\)
\(180\) 0 0
\(181\) −6.26316e7 −0.785088 −0.392544 0.919733i \(-0.628405\pi\)
−0.392544 + 0.919733i \(0.628405\pi\)
\(182\) −5.84885e7 −0.719152
\(183\) 0 0
\(184\) 5.81921e8 6.88655
\(185\) −2.61387e7 −0.303517
\(186\) 0 0
\(187\) −6.60028e7 −0.738103
\(188\) 2.27757e8 2.49988
\(189\) 0 0
\(190\) −4.12654e7 −0.436464
\(191\) 2.82767e7 0.293638 0.146819 0.989163i \(-0.453097\pi\)
0.146819 + 0.989163i \(0.453097\pi\)
\(192\) 0 0
\(193\) −3.31268e7 −0.331688 −0.165844 0.986152i \(-0.553035\pi\)
−0.165844 + 0.986152i \(0.553035\pi\)
\(194\) −2.44746e8 −2.40663
\(195\) 0 0
\(196\) 5.63995e8 5.35031
\(197\) 1.09797e8 1.02320 0.511598 0.859225i \(-0.329054\pi\)
0.511598 + 0.859225i \(0.329054\pi\)
\(198\) 0 0
\(199\) 1.14654e8 1.03134 0.515671 0.856787i \(-0.327543\pi\)
0.515671 + 0.856787i \(0.327543\pi\)
\(200\) −3.83631e8 −3.39085
\(201\) 0 0
\(202\) 1.74153e8 1.48663
\(203\) 2.98966e8 2.50834
\(204\) 0 0
\(205\) −7.37302e7 −0.597733
\(206\) −3.35318e6 −0.0267252
\(207\) 0 0
\(208\) −1.25240e8 −0.964985
\(209\) 1.01616e8 0.769927
\(210\) 0 0
\(211\) −1.35295e8 −0.991502 −0.495751 0.868465i \(-0.665107\pi\)
−0.495751 + 0.868465i \(0.665107\pi\)
\(212\) 2.76077e8 1.99001
\(213\) 0 0
\(214\) −1.82110e8 −1.27024
\(215\) 1.34754e7 0.0924714
\(216\) 0 0
\(217\) −1.04365e8 −0.693341
\(218\) 1.48502e8 0.970814
\(219\) 0 0
\(220\) −1.44949e8 −0.917772
\(221\) −2.43341e7 −0.151650
\(222\) 0 0
\(223\) 9.14456e7 0.552199 0.276100 0.961129i \(-0.410958\pi\)
0.276100 + 0.961129i \(0.410958\pi\)
\(224\) 1.44404e9 8.58445
\(225\) 0 0
\(226\) −3.93325e8 −2.26659
\(227\) 8.92462e7 0.506406 0.253203 0.967413i \(-0.418516\pi\)
0.253203 + 0.967413i \(0.418516\pi\)
\(228\) 0 0
\(229\) 2.51758e8 1.38535 0.692673 0.721251i \(-0.256433\pi\)
0.692673 + 0.721251i \(0.256433\pi\)
\(230\) 2.02903e8 1.09962
\(231\) 0 0
\(232\) 1.05436e9 5.54345
\(233\) −3.50359e8 −1.81455 −0.907273 0.420542i \(-0.861840\pi\)
−0.907273 + 0.420542i \(0.861840\pi\)
\(234\) 0 0
\(235\) 5.19488e7 0.261119
\(236\) 7.60116e7 0.376433
\(237\) 0 0
\(238\) 4.86470e8 2.33903
\(239\) −1.94400e8 −0.921092 −0.460546 0.887636i \(-0.652346\pi\)
−0.460546 + 0.887636i \(0.652346\pi\)
\(240\) 0 0
\(241\) 6.19386e7 0.285038 0.142519 0.989792i \(-0.454480\pi\)
0.142519 + 0.989792i \(0.454480\pi\)
\(242\) 4.54612e7 0.206199
\(243\) 0 0
\(244\) −9.96197e8 −4.39017
\(245\) 1.28641e8 0.558853
\(246\) 0 0
\(247\) 3.74639e7 0.158188
\(248\) −3.68063e8 −1.53229
\(249\) 0 0
\(250\) −2.80954e8 −1.13722
\(251\) −2.42156e8 −0.966577 −0.483289 0.875461i \(-0.660558\pi\)
−0.483289 + 0.875461i \(0.660558\pi\)
\(252\) 0 0
\(253\) −4.99648e8 −1.93974
\(254\) −6.56162e8 −2.51243
\(255\) 0 0
\(256\) 1.62403e9 6.04996
\(257\) 3.80530e8 1.39837 0.699187 0.714939i \(-0.253546\pi\)
0.699187 + 0.714939i \(0.253546\pi\)
\(258\) 0 0
\(259\) −4.74409e8 −1.69669
\(260\) −5.34400e7 −0.188564
\(261\) 0 0
\(262\) −1.01920e8 −0.350109
\(263\) 1.88914e8 0.640353 0.320176 0.947358i \(-0.396258\pi\)
0.320176 + 0.947358i \(0.396258\pi\)
\(264\) 0 0
\(265\) 6.29702e7 0.207862
\(266\) −7.48953e8 −2.43988
\(267\) 0 0
\(268\) 7.84110e8 2.48832
\(269\) −2.07575e8 −0.650194 −0.325097 0.945681i \(-0.605397\pi\)
−0.325097 + 0.945681i \(0.605397\pi\)
\(270\) 0 0
\(271\) −7.81832e7 −0.238628 −0.119314 0.992857i \(-0.538069\pi\)
−0.119314 + 0.992857i \(0.538069\pi\)
\(272\) 1.04166e9 3.13860
\(273\) 0 0
\(274\) −7.44562e7 −0.218662
\(275\) 3.29393e8 0.955102
\(276\) 0 0
\(277\) −1.30983e8 −0.370284 −0.185142 0.982712i \(-0.559274\pi\)
−0.185142 + 0.982712i \(0.559274\pi\)
\(278\) 5.74033e8 1.60243
\(279\) 0 0
\(280\) 6.98853e8 1.90253
\(281\) −1.69682e8 −0.456209 −0.228105 0.973637i \(-0.573253\pi\)
−0.228105 + 0.973637i \(0.573253\pi\)
\(282\) 0 0
\(283\) 2.01304e8 0.527958 0.263979 0.964528i \(-0.414965\pi\)
0.263979 + 0.964528i \(0.414965\pi\)
\(284\) 6.09748e8 1.57956
\(285\) 0 0
\(286\) 1.77108e8 0.447668
\(287\) −1.33818e9 −3.34139
\(288\) 0 0
\(289\) −2.07944e8 −0.506761
\(290\) 3.67632e8 0.885158
\(291\) 0 0
\(292\) −1.07856e9 −2.53516
\(293\) 4.63019e7 0.107538 0.0537691 0.998553i \(-0.482877\pi\)
0.0537691 + 0.998553i \(0.482877\pi\)
\(294\) 0 0
\(295\) 1.73374e7 0.0393194
\(296\) −1.67309e9 −3.74971
\(297\) 0 0
\(298\) 1.59623e9 3.49413
\(299\) −1.84211e8 −0.398536
\(300\) 0 0
\(301\) 2.44574e8 0.516925
\(302\) 8.27404e8 1.72859
\(303\) 0 0
\(304\) −1.60371e9 −3.27393
\(305\) −2.27221e8 −0.458564
\(306\) 0 0
\(307\) −8.24546e8 −1.62641 −0.813206 0.581975i \(-0.802280\pi\)
−0.813206 + 0.581975i \(0.802280\pi\)
\(308\) −2.63077e9 −5.13044
\(309\) 0 0
\(310\) −1.28336e8 −0.244671
\(311\) −8.21297e8 −1.54824 −0.774121 0.633038i \(-0.781808\pi\)
−0.774121 + 0.633038i \(0.781808\pi\)
\(312\) 0 0
\(313\) 4.00974e8 0.739113 0.369557 0.929208i \(-0.379510\pi\)
0.369557 + 0.929208i \(0.379510\pi\)
\(314\) −1.30472e9 −2.37829
\(315\) 0 0
\(316\) 6.35779e7 0.113345
\(317\) −2.03257e8 −0.358375 −0.179187 0.983815i \(-0.557347\pi\)
−0.179187 + 0.983815i \(0.557347\pi\)
\(318\) 0 0
\(319\) −9.05292e8 −1.56143
\(320\) 9.84549e8 1.67963
\(321\) 0 0
\(322\) 3.68263e9 6.14699
\(323\) −3.11601e8 −0.514505
\(324\) 0 0
\(325\) 1.21441e8 0.196234
\(326\) 2.41704e9 3.86386
\(327\) 0 0
\(328\) −4.71932e9 −7.38451
\(329\) 9.42853e8 1.45968
\(330\) 0 0
\(331\) 4.59053e8 0.695769 0.347884 0.937537i \(-0.386900\pi\)
0.347884 + 0.937537i \(0.386900\pi\)
\(332\) −1.83101e9 −2.74604
\(333\) 0 0
\(334\) 1.56018e8 0.229120
\(335\) 1.78847e8 0.259911
\(336\) 0 0
\(337\) −1.03100e9 −1.46741 −0.733706 0.679467i \(-0.762211\pi\)
−0.733706 + 0.679467i \(0.762211\pi\)
\(338\) −1.33514e9 −1.88070
\(339\) 0 0
\(340\) 4.44479e8 0.613303
\(341\) 3.16026e8 0.431601
\(342\) 0 0
\(343\) 1.07301e9 1.43574
\(344\) 8.62534e8 1.14241
\(345\) 0 0
\(346\) 2.18324e9 2.83358
\(347\) −8.50881e8 −1.09324 −0.546620 0.837381i \(-0.684086\pi\)
−0.546620 + 0.837381i \(0.684086\pi\)
\(348\) 0 0
\(349\) −7.28476e8 −0.917331 −0.458666 0.888609i \(-0.651672\pi\)
−0.458666 + 0.888609i \(0.651672\pi\)
\(350\) −2.42777e9 −3.02670
\(351\) 0 0
\(352\) −4.37268e9 −5.34377
\(353\) −1.14502e9 −1.38548 −0.692741 0.721186i \(-0.743597\pi\)
−0.692741 + 0.721186i \(0.743597\pi\)
\(354\) 0 0
\(355\) 1.39077e8 0.164989
\(356\) 6.78737e8 0.797309
\(357\) 0 0
\(358\) −1.72228e9 −1.98387
\(359\) −4.66157e8 −0.531743 −0.265871 0.964008i \(-0.585660\pi\)
−0.265871 + 0.964008i \(0.585660\pi\)
\(360\) 0 0
\(361\) −4.14141e8 −0.463312
\(362\) −1.39783e9 −1.54872
\(363\) 0 0
\(364\) −9.69917e8 −1.05409
\(365\) −2.46008e8 −0.264804
\(366\) 0 0
\(367\) 1.98730e8 0.209861 0.104930 0.994480i \(-0.466538\pi\)
0.104930 + 0.994480i \(0.466538\pi\)
\(368\) 7.88551e9 8.24826
\(369\) 0 0
\(370\) −5.83370e8 −0.598740
\(371\) 1.14289e9 1.16197
\(372\) 0 0
\(373\) 9.45866e8 0.943732 0.471866 0.881670i \(-0.343581\pi\)
0.471866 + 0.881670i \(0.343581\pi\)
\(374\) −1.47307e9 −1.45604
\(375\) 0 0
\(376\) 3.32514e9 3.22591
\(377\) −3.33765e8 −0.320809
\(378\) 0 0
\(379\) −1.07925e9 −1.01832 −0.509162 0.860671i \(-0.670044\pi\)
−0.509162 + 0.860671i \(0.670044\pi\)
\(380\) −6.84306e8 −0.639746
\(381\) 0 0
\(382\) 6.31086e8 0.579251
\(383\) 5.46503e8 0.497046 0.248523 0.968626i \(-0.420055\pi\)
0.248523 + 0.968626i \(0.420055\pi\)
\(384\) 0 0
\(385\) −6.00048e8 −0.535888
\(386\) −7.39333e8 −0.654311
\(387\) 0 0
\(388\) −4.05863e9 −3.52751
\(389\) 6.83446e8 0.588682 0.294341 0.955700i \(-0.404900\pi\)
0.294341 + 0.955700i \(0.404900\pi\)
\(390\) 0 0
\(391\) 1.53215e9 1.29623
\(392\) 8.23405e9 6.90418
\(393\) 0 0
\(394\) 2.45048e9 2.01843
\(395\) 1.45014e7 0.0118391
\(396\) 0 0
\(397\) −1.21932e9 −0.978030 −0.489015 0.872275i \(-0.662644\pi\)
−0.489015 + 0.872275i \(0.662644\pi\)
\(398\) 2.55887e9 2.03450
\(399\) 0 0
\(400\) −5.19851e9 −4.06134
\(401\) 1.27738e9 0.989268 0.494634 0.869101i \(-0.335302\pi\)
0.494634 + 0.869101i \(0.335302\pi\)
\(402\) 0 0
\(403\) 1.16513e8 0.0886762
\(404\) 2.88799e9 2.17902
\(405\) 0 0
\(406\) 6.67240e9 4.94813
\(407\) 1.43655e9 1.05618
\(408\) 0 0
\(409\) −1.52758e9 −1.10401 −0.552004 0.833842i \(-0.686137\pi\)
−0.552004 + 0.833842i \(0.686137\pi\)
\(410\) −1.64553e9 −1.17913
\(411\) 0 0
\(412\) −5.56058e7 −0.0391724
\(413\) 3.14668e8 0.219800
\(414\) 0 0
\(415\) −4.17632e8 −0.286831
\(416\) −1.61213e9 −1.09792
\(417\) 0 0
\(418\) 2.26789e9 1.51881
\(419\) −2.15816e9 −1.43329 −0.716645 0.697438i \(-0.754323\pi\)
−0.716645 + 0.697438i \(0.754323\pi\)
\(420\) 0 0
\(421\) −4.86121e8 −0.317509 −0.158755 0.987318i \(-0.550748\pi\)
−0.158755 + 0.987318i \(0.550748\pi\)
\(422\) −3.01955e9 −1.95591
\(423\) 0 0
\(424\) 4.03059e9 2.56796
\(425\) −1.01007e9 −0.638249
\(426\) 0 0
\(427\) −4.12399e9 −2.56342
\(428\) −3.01994e9 −1.86185
\(429\) 0 0
\(430\) 3.00747e8 0.182416
\(431\) −8.97584e8 −0.540013 −0.270007 0.962858i \(-0.587026\pi\)
−0.270007 + 0.962858i \(0.587026\pi\)
\(432\) 0 0
\(433\) −1.07413e8 −0.0635843 −0.0317921 0.999495i \(-0.510121\pi\)
−0.0317921 + 0.999495i \(0.510121\pi\)
\(434\) −2.32925e9 −1.36773
\(435\) 0 0
\(436\) 2.46262e9 1.42297
\(437\) −2.35885e9 −1.35212
\(438\) 0 0
\(439\) −4.23839e8 −0.239097 −0.119549 0.992828i \(-0.538145\pi\)
−0.119549 + 0.992828i \(0.538145\pi\)
\(440\) −2.11618e9 −1.18432
\(441\) 0 0
\(442\) −5.43094e8 −0.299155
\(443\) 1.56918e9 0.857552 0.428776 0.903411i \(-0.358945\pi\)
0.428776 + 0.903411i \(0.358945\pi\)
\(444\) 0 0
\(445\) 1.54812e8 0.0832809
\(446\) 2.04091e9 1.08931
\(447\) 0 0
\(448\) 1.78692e10 9.38929
\(449\) 6.52660e8 0.340271 0.170135 0.985421i \(-0.445580\pi\)
0.170135 + 0.985421i \(0.445580\pi\)
\(450\) 0 0
\(451\) 4.05210e9 2.08000
\(452\) −6.52252e9 −3.32224
\(453\) 0 0
\(454\) 1.99182e9 0.998974
\(455\) −2.21227e8 −0.110103
\(456\) 0 0
\(457\) −3.14784e9 −1.54278 −0.771392 0.636360i \(-0.780439\pi\)
−0.771392 + 0.636360i \(0.780439\pi\)
\(458\) 5.61879e9 2.73284
\(459\) 0 0
\(460\) 3.36475e9 1.61176
\(461\) −3.81603e9 −1.81409 −0.907044 0.421037i \(-0.861666\pi\)
−0.907044 + 0.421037i \(0.861666\pi\)
\(462\) 0 0
\(463\) 7.32034e8 0.342766 0.171383 0.985204i \(-0.445176\pi\)
0.171383 + 0.985204i \(0.445176\pi\)
\(464\) 1.42874e10 6.63958
\(465\) 0 0
\(466\) −7.81941e9 −3.57951
\(467\) 9.56937e8 0.434785 0.217392 0.976084i \(-0.430245\pi\)
0.217392 + 0.976084i \(0.430245\pi\)
\(468\) 0 0
\(469\) 3.24601e9 1.45293
\(470\) 1.15941e9 0.515102
\(471\) 0 0
\(472\) 1.10973e9 0.485759
\(473\) −7.40589e8 −0.321783
\(474\) 0 0
\(475\) 1.55507e9 0.665767
\(476\) 8.06714e9 3.42843
\(477\) 0 0
\(478\) −4.33866e9 −1.81701
\(479\) −3.19983e9 −1.33031 −0.665154 0.746706i \(-0.731634\pi\)
−0.665154 + 0.746706i \(0.731634\pi\)
\(480\) 0 0
\(481\) 5.29629e8 0.217002
\(482\) 1.38236e9 0.562286
\(483\) 0 0
\(484\) 7.53884e8 0.302236
\(485\) −9.25727e8 −0.368457
\(486\) 0 0
\(487\) −3.55741e8 −0.139567 −0.0697834 0.997562i \(-0.522231\pi\)
−0.0697834 + 0.997562i \(0.522231\pi\)
\(488\) −1.45440e10 −5.66518
\(489\) 0 0
\(490\) 2.87104e9 1.10243
\(491\) 1.77516e7 0.00676788 0.00338394 0.999994i \(-0.498923\pi\)
0.00338394 + 0.999994i \(0.498923\pi\)
\(492\) 0 0
\(493\) 2.77604e9 1.04343
\(494\) 8.36129e8 0.312054
\(495\) 0 0
\(496\) −4.98755e9 −1.83528
\(497\) 2.52419e9 0.922306
\(498\) 0 0
\(499\) 1.97045e9 0.709927 0.354963 0.934880i \(-0.384493\pi\)
0.354963 + 0.934880i \(0.384493\pi\)
\(500\) −4.65906e9 −1.66688
\(501\) 0 0
\(502\) −5.40449e9 −1.90674
\(503\) 4.77882e9 1.67430 0.837148 0.546976i \(-0.184221\pi\)
0.837148 + 0.546976i \(0.184221\pi\)
\(504\) 0 0
\(505\) 6.58717e8 0.227604
\(506\) −1.11513e10 −3.82647
\(507\) 0 0
\(508\) −1.08812e10 −3.68258
\(509\) 2.21746e9 0.745321 0.372660 0.927968i \(-0.378446\pi\)
0.372660 + 0.927968i \(0.378446\pi\)
\(510\) 0 0
\(511\) −4.46496e9 −1.48028
\(512\) 1.83694e10 6.04852
\(513\) 0 0
\(514\) 8.49277e9 2.75853
\(515\) −1.26831e7 −0.00409165
\(516\) 0 0
\(517\) −2.85503e9 −0.908644
\(518\) −1.05880e10 −3.34702
\(519\) 0 0
\(520\) −7.80197e8 −0.243328
\(521\) 2.05916e9 0.637908 0.318954 0.947770i \(-0.396668\pi\)
0.318954 + 0.947770i \(0.396668\pi\)
\(522\) 0 0
\(523\) 4.60384e9 1.40723 0.703614 0.710582i \(-0.251568\pi\)
0.703614 + 0.710582i \(0.251568\pi\)
\(524\) −1.69014e9 −0.513171
\(525\) 0 0
\(526\) 4.21623e9 1.26321
\(527\) −9.69081e8 −0.288418
\(528\) 0 0
\(529\) 8.19372e9 2.40650
\(530\) 1.40538e9 0.410043
\(531\) 0 0
\(532\) −1.24199e10 −3.57625
\(533\) 1.49394e9 0.427354
\(534\) 0 0
\(535\) −6.88814e8 −0.194475
\(536\) 1.14476e10 3.21099
\(537\) 0 0
\(538\) −4.63272e9 −1.28262
\(539\) −7.06992e9 −1.94470
\(540\) 0 0
\(541\) −5.44524e9 −1.47852 −0.739258 0.673422i \(-0.764824\pi\)
−0.739258 + 0.673422i \(0.764824\pi\)
\(542\) −1.74491e9 −0.470735
\(543\) 0 0
\(544\) 1.34086e10 3.57099
\(545\) 5.61696e8 0.148632
\(546\) 0 0
\(547\) −2.09397e9 −0.547035 −0.273518 0.961867i \(-0.588187\pi\)
−0.273518 + 0.961867i \(0.588187\pi\)
\(548\) −1.23471e9 −0.320503
\(549\) 0 0
\(550\) 7.35147e9 1.88410
\(551\) −4.27390e9 −1.08841
\(552\) 0 0
\(553\) 2.63196e8 0.0661821
\(554\) −2.92330e9 −0.730449
\(555\) 0 0
\(556\) 9.51921e9 2.34876
\(557\) −4.55263e9 −1.11627 −0.558135 0.829750i \(-0.688483\pi\)
−0.558135 + 0.829750i \(0.688483\pi\)
\(558\) 0 0
\(559\) −2.73042e8 −0.0661131
\(560\) 9.47003e9 2.27873
\(561\) 0 0
\(562\) −3.78701e9 −0.899952
\(563\) 2.34649e9 0.554166 0.277083 0.960846i \(-0.410632\pi\)
0.277083 + 0.960846i \(0.410632\pi\)
\(564\) 0 0
\(565\) −1.48771e9 −0.347016
\(566\) 4.49274e9 1.04149
\(567\) 0 0
\(568\) 8.90202e9 2.03831
\(569\) 2.75800e9 0.627626 0.313813 0.949485i \(-0.398394\pi\)
0.313813 + 0.949485i \(0.398394\pi\)
\(570\) 0 0
\(571\) −1.49430e9 −0.335901 −0.167950 0.985795i \(-0.553715\pi\)
−0.167950 + 0.985795i \(0.553715\pi\)
\(572\) 2.93698e9 0.656168
\(573\) 0 0
\(574\) −2.98658e10 −6.59147
\(575\) −7.64633e9 −1.67732
\(576\) 0 0
\(577\) −5.27606e8 −0.114339 −0.0571695 0.998364i \(-0.518208\pi\)
−0.0571695 + 0.998364i \(0.518208\pi\)
\(578\) −4.64094e9 −0.999673
\(579\) 0 0
\(580\) 6.09646e9 1.29742
\(581\) −7.57987e9 −1.60341
\(582\) 0 0
\(583\) −3.46075e9 −0.723319
\(584\) −1.57465e10 −3.27144
\(585\) 0 0
\(586\) 1.03338e9 0.212138
\(587\) 3.02681e9 0.617664 0.308832 0.951117i \(-0.400062\pi\)
0.308832 + 0.951117i \(0.400062\pi\)
\(588\) 0 0
\(589\) 1.49197e9 0.300854
\(590\) 3.86940e8 0.0775643
\(591\) 0 0
\(592\) −2.26717e10 −4.49116
\(593\) 7.55687e9 1.48816 0.744081 0.668089i \(-0.232887\pi\)
0.744081 + 0.668089i \(0.232887\pi\)
\(594\) 0 0
\(595\) 1.84002e9 0.358108
\(596\) 2.64703e10 5.12150
\(597\) 0 0
\(598\) −4.11127e9 −0.786181
\(599\) 7.21826e9 1.37227 0.686133 0.727476i \(-0.259307\pi\)
0.686133 + 0.727476i \(0.259307\pi\)
\(600\) 0 0
\(601\) −4.11700e9 −0.773607 −0.386803 0.922162i \(-0.626421\pi\)
−0.386803 + 0.922162i \(0.626421\pi\)
\(602\) 5.45846e9 1.01972
\(603\) 0 0
\(604\) 1.37209e10 2.53368
\(605\) 1.71952e8 0.0315693
\(606\) 0 0
\(607\) 3.42535e9 0.621649 0.310824 0.950467i \(-0.399395\pi\)
0.310824 + 0.950467i \(0.399395\pi\)
\(608\) −2.06435e10 −3.72495
\(609\) 0 0
\(610\) −5.07118e9 −0.904596
\(611\) −1.05260e9 −0.186689
\(612\) 0 0
\(613\) −8.07359e9 −1.41565 −0.707823 0.706390i \(-0.750323\pi\)
−0.707823 + 0.706390i \(0.750323\pi\)
\(614\) −1.84024e10 −3.20838
\(615\) 0 0
\(616\) −3.84079e10 −6.62046
\(617\) 7.36421e9 1.26220 0.631100 0.775702i \(-0.282604\pi\)
0.631100 + 0.775702i \(0.282604\pi\)
\(618\) 0 0
\(619\) 9.33576e9 1.58210 0.791048 0.611755i \(-0.209536\pi\)
0.791048 + 0.611755i \(0.209536\pi\)
\(620\) −2.12820e9 −0.358625
\(621\) 0 0
\(622\) −1.83299e10 −3.05417
\(623\) 2.80979e9 0.465549
\(624\) 0 0
\(625\) 4.48410e9 0.734676
\(626\) 8.94903e9 1.45803
\(627\) 0 0
\(628\) −2.16363e10 −3.48597
\(629\) −4.40511e9 −0.705796
\(630\) 0 0
\(631\) 1.58549e9 0.251225 0.125612 0.992079i \(-0.459910\pi\)
0.125612 + 0.992079i \(0.459910\pi\)
\(632\) 9.28206e8 0.146263
\(633\) 0 0
\(634\) −4.53633e9 −0.706956
\(635\) −2.48187e9 −0.384655
\(636\) 0 0
\(637\) −2.60655e9 −0.399556
\(638\) −2.02045e10 −3.08018
\(639\) 0 0
\(640\) 1.17893e10 1.77770
\(641\) −7.99377e9 −1.19880 −0.599402 0.800448i \(-0.704595\pi\)
−0.599402 + 0.800448i \(0.704595\pi\)
\(642\) 0 0
\(643\) −2.26242e9 −0.335610 −0.167805 0.985820i \(-0.553668\pi\)
−0.167805 + 0.985820i \(0.553668\pi\)
\(644\) 6.10691e10 9.00992
\(645\) 0 0
\(646\) −6.95438e9 −1.01495
\(647\) −4.58044e8 −0.0664879 −0.0332439 0.999447i \(-0.510584\pi\)
−0.0332439 + 0.999447i \(0.510584\pi\)
\(648\) 0 0
\(649\) −9.52837e8 −0.136824
\(650\) 2.71035e9 0.387105
\(651\) 0 0
\(652\) 4.00818e10 5.66344
\(653\) −8.67155e9 −1.21871 −0.609355 0.792897i \(-0.708572\pi\)
−0.609355 + 0.792897i \(0.708572\pi\)
\(654\) 0 0
\(655\) −3.85501e8 −0.0536020
\(656\) −6.39507e10 −8.84468
\(657\) 0 0
\(658\) 2.10428e10 2.87947
\(659\) −8.76494e9 −1.19303 −0.596513 0.802603i \(-0.703448\pi\)
−0.596513 + 0.802603i \(0.703448\pi\)
\(660\) 0 0
\(661\) 1.58045e9 0.212850 0.106425 0.994321i \(-0.466060\pi\)
0.106425 + 0.994321i \(0.466060\pi\)
\(662\) 1.02453e10 1.37252
\(663\) 0 0
\(664\) −2.67318e10 −3.54356
\(665\) −2.83284e9 −0.373548
\(666\) 0 0
\(667\) 2.10149e10 2.74213
\(668\) 2.58726e9 0.335832
\(669\) 0 0
\(670\) 3.99155e9 0.512719
\(671\) 1.24877e10 1.59572
\(672\) 0 0
\(673\) −2.26086e9 −0.285905 −0.142952 0.989730i \(-0.545660\pi\)
−0.142952 + 0.989730i \(0.545660\pi\)
\(674\) −2.30100e10 −2.89472
\(675\) 0 0
\(676\) −2.21407e10 −2.75662
\(677\) −6.78561e9 −0.840483 −0.420241 0.907412i \(-0.638055\pi\)
−0.420241 + 0.907412i \(0.638055\pi\)
\(678\) 0 0
\(679\) −1.68016e10 −2.05971
\(680\) 6.48918e9 0.791423
\(681\) 0 0
\(682\) 7.05314e9 0.851407
\(683\) 7.74627e9 0.930294 0.465147 0.885234i \(-0.346002\pi\)
0.465147 + 0.885234i \(0.346002\pi\)
\(684\) 0 0
\(685\) −2.81623e8 −0.0334774
\(686\) 2.39477e10 2.83224
\(687\) 0 0
\(688\) 1.16880e10 1.36830
\(689\) −1.27592e9 −0.148612
\(690\) 0 0
\(691\) −5.90227e9 −0.680528 −0.340264 0.940330i \(-0.610516\pi\)
−0.340264 + 0.940330i \(0.610516\pi\)
\(692\) 3.62048e10 4.15331
\(693\) 0 0
\(694\) −1.89902e10 −2.15660
\(695\) 2.17122e9 0.245334
\(696\) 0 0
\(697\) −1.24256e10 −1.38996
\(698\) −1.62583e10 −1.80959
\(699\) 0 0
\(700\) −4.02597e10 −4.43637
\(701\) 1.01648e10 1.11452 0.557259 0.830339i \(-0.311853\pi\)
0.557259 + 0.830339i \(0.311853\pi\)
\(702\) 0 0
\(703\) 6.78196e9 0.736227
\(704\) −5.41094e10 −5.84478
\(705\) 0 0
\(706\) −2.55548e10 −2.73310
\(707\) 1.19555e10 1.27233
\(708\) 0 0
\(709\) −7.24332e9 −0.763266 −0.381633 0.924314i \(-0.624638\pi\)
−0.381633 + 0.924314i \(0.624638\pi\)
\(710\) 3.10395e9 0.325469
\(711\) 0 0
\(712\) 9.90922e9 1.02887
\(713\) −7.33604e9 −0.757964
\(714\) 0 0
\(715\) 6.69892e8 0.0685384
\(716\) −2.85606e10 −2.90784
\(717\) 0 0
\(718\) −1.04038e10 −1.04895
\(719\) −1.37810e10 −1.38270 −0.691350 0.722520i \(-0.742984\pi\)
−0.691350 + 0.722520i \(0.742984\pi\)
\(720\) 0 0
\(721\) −2.30193e8 −0.0228728
\(722\) −9.24290e9 −0.913962
\(723\) 0 0
\(724\) −2.31802e10 −2.27003
\(725\) −1.38541e10 −1.35019
\(726\) 0 0
\(727\) −1.04638e10 −1.00999 −0.504996 0.863122i \(-0.668506\pi\)
−0.504996 + 0.863122i \(0.668506\pi\)
\(728\) −1.41603e10 −1.36023
\(729\) 0 0
\(730\) −5.49047e9 −0.522372
\(731\) 2.27099e9 0.215032
\(732\) 0 0
\(733\) −6.09625e9 −0.571740 −0.285870 0.958268i \(-0.592283\pi\)
−0.285870 + 0.958268i \(0.592283\pi\)
\(734\) 4.43530e9 0.413987
\(735\) 0 0
\(736\) 1.01505e11 9.38456
\(737\) −9.82915e9 −0.904440
\(738\) 0 0
\(739\) 1.62223e10 1.47862 0.739308 0.673367i \(-0.235153\pi\)
0.739308 + 0.673367i \(0.235153\pi\)
\(740\) −9.67405e9 −0.877601
\(741\) 0 0
\(742\) 2.55072e10 2.29218
\(743\) 2.03461e10 1.81978 0.909891 0.414847i \(-0.136165\pi\)
0.909891 + 0.414847i \(0.136165\pi\)
\(744\) 0 0
\(745\) 6.03758e9 0.534954
\(746\) 2.11101e10 1.86167
\(747\) 0 0
\(748\) −2.44279e10 −2.13418
\(749\) −1.25017e10 −1.08714
\(750\) 0 0
\(751\) 6.71015e9 0.578086 0.289043 0.957316i \(-0.406663\pi\)
0.289043 + 0.957316i \(0.406663\pi\)
\(752\) 4.50584e10 3.86379
\(753\) 0 0
\(754\) −7.44905e9 −0.632850
\(755\) 3.12957e9 0.264649
\(756\) 0 0
\(757\) 7.51213e9 0.629401 0.314700 0.949191i \(-0.398096\pi\)
0.314700 + 0.949191i \(0.398096\pi\)
\(758\) −2.40870e10 −2.00882
\(759\) 0 0
\(760\) −9.99053e9 −0.825545
\(761\) −1.17043e10 −0.962721 −0.481361 0.876523i \(-0.659857\pi\)
−0.481361 + 0.876523i \(0.659857\pi\)
\(762\) 0 0
\(763\) 1.01946e10 0.830871
\(764\) 1.04653e10 0.849035
\(765\) 0 0
\(766\) 1.21970e10 0.980509
\(767\) −3.51294e8 −0.0281117
\(768\) 0 0
\(769\) −2.01241e9 −0.159579 −0.0797894 0.996812i \(-0.525425\pi\)
−0.0797894 + 0.996812i \(0.525425\pi\)
\(770\) −1.33920e10 −1.05713
\(771\) 0 0
\(772\) −1.22604e10 −0.959055
\(773\) 5.06105e9 0.394105 0.197053 0.980393i \(-0.436863\pi\)
0.197053 + 0.980393i \(0.436863\pi\)
\(774\) 0 0
\(775\) 4.83628e9 0.373212
\(776\) −5.92539e10 −4.55199
\(777\) 0 0
\(778\) 1.52533e10 1.16128
\(779\) 1.91301e10 1.44989
\(780\) 0 0
\(781\) −7.64344e9 −0.574130
\(782\) 3.41949e10 2.55704
\(783\) 0 0
\(784\) 1.11578e11 8.26938
\(785\) −4.93499e9 −0.364119
\(786\) 0 0
\(787\) −1.29130e10 −0.944311 −0.472155 0.881515i \(-0.656524\pi\)
−0.472155 + 0.881515i \(0.656524\pi\)
\(788\) 4.06364e10 2.95851
\(789\) 0 0
\(790\) 3.23646e8 0.0233548
\(791\) −2.70015e10 −1.93986
\(792\) 0 0
\(793\) 4.60401e9 0.327854
\(794\) −2.72132e10 −1.92933
\(795\) 0 0
\(796\) 4.24338e10 2.98206
\(797\) 1.75345e10 1.22684 0.613422 0.789755i \(-0.289792\pi\)
0.613422 + 0.789755i \(0.289792\pi\)
\(798\) 0 0
\(799\) 8.75483e9 0.607203
\(800\) −6.69169e10 −4.62084
\(801\) 0 0
\(802\) 2.85088e10 1.95150
\(803\) 1.35203e10 0.921468
\(804\) 0 0
\(805\) 1.39292e10 0.941109
\(806\) 2.60037e9 0.174929
\(807\) 0 0
\(808\) 4.21632e10 2.81186
\(809\) −3.03994e9 −0.201858 −0.100929 0.994894i \(-0.532181\pi\)
−0.100929 + 0.994894i \(0.532181\pi\)
\(810\) 0 0
\(811\) 1.35418e10 0.891465 0.445733 0.895166i \(-0.352943\pi\)
0.445733 + 0.895166i \(0.352943\pi\)
\(812\) 1.10649e11 7.25270
\(813\) 0 0
\(814\) 3.20612e10 2.08350
\(815\) 9.14220e9 0.591560
\(816\) 0 0
\(817\) −3.49634e9 −0.224303
\(818\) −3.40929e10 −2.17785
\(819\) 0 0
\(820\) −2.72879e10 −1.72831
\(821\) 1.44424e10 0.910834 0.455417 0.890278i \(-0.349490\pi\)
0.455417 + 0.890278i \(0.349490\pi\)
\(822\) 0 0
\(823\) 1.58161e10 0.989009 0.494504 0.869175i \(-0.335350\pi\)
0.494504 + 0.869175i \(0.335350\pi\)
\(824\) −8.11818e8 −0.0505491
\(825\) 0 0
\(826\) 7.02283e9 0.433593
\(827\) 2.91022e10 1.78919 0.894594 0.446879i \(-0.147465\pi\)
0.894594 + 0.446879i \(0.147465\pi\)
\(828\) 0 0
\(829\) −1.38090e10 −0.841825 −0.420913 0.907101i \(-0.638290\pi\)
−0.420913 + 0.907101i \(0.638290\pi\)
\(830\) −9.32081e9 −0.565823
\(831\) 0 0
\(832\) −1.99492e10 −1.20086
\(833\) 2.16796e10 1.29955
\(834\) 0 0
\(835\) 5.90124e8 0.0350785
\(836\) 3.76084e10 2.22620
\(837\) 0 0
\(838\) −4.81663e10 −2.82741
\(839\) −2.26960e10 −1.32673 −0.663366 0.748295i \(-0.730873\pi\)
−0.663366 + 0.748295i \(0.730873\pi\)
\(840\) 0 0
\(841\) 2.08262e10 1.20732
\(842\) −1.08494e10 −0.626342
\(843\) 0 0
\(844\) −5.00733e10 −2.86687
\(845\) −5.05004e9 −0.287936
\(846\) 0 0
\(847\) 3.12088e9 0.176476
\(848\) 5.46179e10 3.07574
\(849\) 0 0
\(850\) −2.25430e10 −1.25906
\(851\) −3.33471e10 −1.85483
\(852\) 0 0
\(853\) −2.76986e9 −0.152804 −0.0764022 0.997077i \(-0.524343\pi\)
−0.0764022 + 0.997077i \(0.524343\pi\)
\(854\) −9.20402e10 −5.05679
\(855\) 0 0
\(856\) −4.40896e10 −2.40258
\(857\) −8.53606e9 −0.463260 −0.231630 0.972804i \(-0.574406\pi\)
−0.231630 + 0.972804i \(0.574406\pi\)
\(858\) 0 0
\(859\) −7.27081e9 −0.391387 −0.195694 0.980665i \(-0.562696\pi\)
−0.195694 + 0.980665i \(0.562696\pi\)
\(860\) 4.98730e9 0.267375
\(861\) 0 0
\(862\) −2.00325e10 −1.06527
\(863\) −2.06289e10 −1.09254 −0.546270 0.837609i \(-0.683953\pi\)
−0.546270 + 0.837609i \(0.683953\pi\)
\(864\) 0 0
\(865\) 8.25791e9 0.433824
\(866\) −2.39727e9 −0.125431
\(867\) 0 0
\(868\) −3.86260e10 −2.00475
\(869\) −7.96976e8 −0.0411980
\(870\) 0 0
\(871\) −3.62383e9 −0.185825
\(872\) 3.59530e10 1.83623
\(873\) 0 0
\(874\) −5.26454e10 −2.66729
\(875\) −1.92873e10 −0.973291
\(876\) 0 0
\(877\) 1.04224e10 0.521760 0.260880 0.965371i \(-0.415987\pi\)
0.260880 + 0.965371i \(0.415987\pi\)
\(878\) −9.45934e9 −0.471661
\(879\) 0 0
\(880\) −2.86760e10 −1.41850
\(881\) −3.28267e10 −1.61738 −0.808690 0.588235i \(-0.799823\pi\)
−0.808690 + 0.588235i \(0.799823\pi\)
\(882\) 0 0
\(883\) 2.09738e10 1.02522 0.512608 0.858623i \(-0.328680\pi\)
0.512608 + 0.858623i \(0.328680\pi\)
\(884\) −9.00613e9 −0.438486
\(885\) 0 0
\(886\) 3.50214e10 1.69167
\(887\) −4.36272e9 −0.209906 −0.104953 0.994477i \(-0.533469\pi\)
−0.104953 + 0.994477i \(0.533469\pi\)
\(888\) 0 0
\(889\) −4.50451e10 −2.15026
\(890\) 3.45514e9 0.164286
\(891\) 0 0
\(892\) 3.38444e10 1.59665
\(893\) −1.34787e10 −0.633383
\(894\) 0 0
\(895\) −6.51434e9 −0.303732
\(896\) 2.13972e11 9.93754
\(897\) 0 0
\(898\) 1.45662e10 0.671243
\(899\) −1.32919e10 −0.610137
\(900\) 0 0
\(901\) 1.06122e10 0.483360
\(902\) 9.04359e10 4.10315
\(903\) 0 0
\(904\) −9.52256e10 −4.28711
\(905\) −5.28714e9 −0.237111
\(906\) 0 0
\(907\) 9.67343e9 0.430482 0.215241 0.976561i \(-0.430946\pi\)
0.215241 + 0.976561i \(0.430946\pi\)
\(908\) 3.30304e10 1.46424
\(909\) 0 0
\(910\) −4.93740e9 −0.217197
\(911\) −1.13862e10 −0.498959 −0.249480 0.968380i \(-0.580260\pi\)
−0.249480 + 0.968380i \(0.580260\pi\)
\(912\) 0 0
\(913\) 2.29524e10 0.998116
\(914\) −7.02542e10 −3.04341
\(915\) 0 0
\(916\) 9.31765e10 4.00564
\(917\) −6.99671e9 −0.299641
\(918\) 0 0
\(919\) −1.74980e9 −0.0743676 −0.0371838 0.999308i \(-0.511839\pi\)
−0.0371838 + 0.999308i \(0.511839\pi\)
\(920\) 4.91237e10 2.07986
\(921\) 0 0
\(922\) −8.51670e10 −3.57860
\(923\) −2.81800e9 −0.117960
\(924\) 0 0
\(925\) 2.19841e10 0.913297
\(926\) 1.63377e10 0.676165
\(927\) 0 0
\(928\) 1.83912e11 7.55428
\(929\) 5.01633e9 0.205273 0.102636 0.994719i \(-0.467272\pi\)
0.102636 + 0.994719i \(0.467272\pi\)
\(930\) 0 0
\(931\) −3.33772e10 −1.35558
\(932\) −1.29669e11 −5.24665
\(933\) 0 0
\(934\) 2.13572e10 0.857688
\(935\) −5.57173e9 −0.222920
\(936\) 0 0
\(937\) 2.41581e10 0.959343 0.479672 0.877448i \(-0.340756\pi\)
0.479672 + 0.877448i \(0.340756\pi\)
\(938\) 7.24452e10 2.86615
\(939\) 0 0
\(940\) 1.92265e10 0.755009
\(941\) 2.00603e10 0.784826 0.392413 0.919789i \(-0.371640\pi\)
0.392413 + 0.919789i \(0.371640\pi\)
\(942\) 0 0
\(943\) −9.40632e10 −3.65282
\(944\) 1.50378e10 0.581811
\(945\) 0 0
\(946\) −1.65286e10 −0.634772
\(947\) 1.33454e10 0.510632 0.255316 0.966858i \(-0.417820\pi\)
0.255316 + 0.966858i \(0.417820\pi\)
\(948\) 0 0
\(949\) 4.98467e9 0.189324
\(950\) 3.47064e10 1.31334
\(951\) 0 0
\(952\) 1.17776e11 4.42414
\(953\) 4.58477e10 1.71590 0.857950 0.513733i \(-0.171738\pi\)
0.857950 + 0.513733i \(0.171738\pi\)
\(954\) 0 0
\(955\) 2.38702e9 0.0886838
\(956\) −7.19481e10 −2.66328
\(957\) 0 0
\(958\) −7.14146e10 −2.62426
\(959\) −5.11136e9 −0.187142
\(960\) 0 0
\(961\) −2.28726e10 −0.831349
\(962\) 1.18204e10 0.428074
\(963\) 0 0
\(964\) 2.29237e10 0.824168
\(965\) −2.79645e9 −0.100176
\(966\) 0 0
\(967\) 2.81641e10 1.00162 0.500810 0.865557i \(-0.333036\pi\)
0.500810 + 0.865557i \(0.333036\pi\)
\(968\) 1.10063e10 0.390013
\(969\) 0 0
\(970\) −2.06606e10 −0.726845
\(971\) −2.91527e10 −1.02191 −0.510954 0.859608i \(-0.670708\pi\)
−0.510954 + 0.859608i \(0.670708\pi\)
\(972\) 0 0
\(973\) 3.94070e10 1.37144
\(974\) −7.93951e9 −0.275320
\(975\) 0 0
\(976\) −1.97083e11 −6.78539
\(977\) −2.18798e10 −0.750607 −0.375303 0.926902i \(-0.622461\pi\)
−0.375303 + 0.926902i \(0.622461\pi\)
\(978\) 0 0
\(979\) −8.50825e9 −0.289802
\(980\) 4.76105e10 1.61589
\(981\) 0 0
\(982\) 3.96185e8 0.0133508
\(983\) −7.16781e9 −0.240685 −0.120342 0.992732i \(-0.538399\pi\)
−0.120342 + 0.992732i \(0.538399\pi\)
\(984\) 0 0
\(985\) 9.26870e9 0.309024
\(986\) 6.19564e10 2.05834
\(987\) 0 0
\(988\) 1.38656e10 0.457391
\(989\) 1.71916e10 0.565105
\(990\) 0 0
\(991\) −3.83800e9 −0.125270 −0.0626350 0.998037i \(-0.519950\pi\)
−0.0626350 + 0.998037i \(0.519950\pi\)
\(992\) −6.42014e10 −2.08811
\(993\) 0 0
\(994\) 5.63355e10 1.81941
\(995\) 9.67868e9 0.311484
\(996\) 0 0
\(997\) 5.63360e10 1.80033 0.900166 0.435547i \(-0.143445\pi\)
0.900166 + 0.435547i \(0.143445\pi\)
\(998\) 4.39770e10 1.40045
\(999\) 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.1.17 17
3.2 odd 2 177.8.a.b.1.1 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.1 17 3.2 odd 2
531.8.a.d.1.17 17 1.1 even 1 trivial