Properties

Label 531.8.a.d.1.15
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 1.15
Root \(-15.9892\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.9892 q^{2} +195.612 q^{4} +98.9095 q^{5} +159.201 q^{7} +1216.28 q^{8} +O(q^{10})\) \(q+17.9892 q^{2} +195.612 q^{4} +98.9095 q^{5} +159.201 q^{7} +1216.28 q^{8} +1779.30 q^{10} +4884.97 q^{11} +11664.5 q^{13} +2863.90 q^{14} -3158.35 q^{16} +10463.7 q^{17} +14966.8 q^{19} +19347.9 q^{20} +87876.7 q^{22} +88.0105 q^{23} -68341.9 q^{25} +209835. q^{26} +31141.6 q^{28} +102600. q^{29} -287624. q^{31} -212500. q^{32} +188233. q^{34} +15746.5 q^{35} +566144. q^{37} +269240. q^{38} +120302. q^{40} -296117. q^{41} +740471. q^{43} +955557. q^{44} +1583.24 q^{46} -958168. q^{47} -798198. q^{49} -1.22942e6 q^{50} +2.28171e6 q^{52} +732872. q^{53} +483170. q^{55} +193633. q^{56} +1.84569e6 q^{58} +205379. q^{59} +2.28293e6 q^{61} -5.17412e6 q^{62} -3.41844e6 q^{64} +1.15373e6 q^{65} -4.29840e6 q^{67} +2.04681e6 q^{68} +283267. q^{70} +4.99901e6 q^{71} +3.66974e6 q^{73} +1.01845e7 q^{74} +2.92768e6 q^{76} +777691. q^{77} +3.18719e6 q^{79} -312391. q^{80} -5.32690e6 q^{82} +4.43945e6 q^{83} +1.03496e6 q^{85} +1.33205e7 q^{86} +5.94150e6 q^{88} +1.07216e7 q^{89} +1.85700e6 q^{91} +17215.9 q^{92} -1.72367e7 q^{94} +1.48036e6 q^{95} -4.05589e6 q^{97} -1.43590e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.9892 1.59004 0.795018 0.606585i \(-0.207461\pi\)
0.795018 + 0.606585i \(0.207461\pi\)
\(3\) 0 0
\(4\) 195.612 1.52822
\(5\) 98.9095 0.353869 0.176935 0.984223i \(-0.443382\pi\)
0.176935 + 0.984223i \(0.443382\pi\)
\(6\) 0 0
\(7\) 159.201 0.175429 0.0877147 0.996146i \(-0.472044\pi\)
0.0877147 + 0.996146i \(0.472044\pi\)
\(8\) 1216.28 0.839884
\(9\) 0 0
\(10\) 1779.30 0.562665
\(11\) 4884.97 1.10659 0.553296 0.832985i \(-0.313370\pi\)
0.553296 + 0.832985i \(0.313370\pi\)
\(12\) 0 0
\(13\) 11664.5 1.47253 0.736265 0.676694i \(-0.236588\pi\)
0.736265 + 0.676694i \(0.236588\pi\)
\(14\) 2863.90 0.278939
\(15\) 0 0
\(16\) −3158.35 −0.192770
\(17\) 10463.7 0.516550 0.258275 0.966071i \(-0.416846\pi\)
0.258275 + 0.966071i \(0.416846\pi\)
\(18\) 0 0
\(19\) 14966.8 0.500599 0.250300 0.968168i \(-0.419471\pi\)
0.250300 + 0.968168i \(0.419471\pi\)
\(20\) 19347.9 0.540789
\(21\) 0 0
\(22\) 87876.7 1.75952
\(23\) 88.0105 0.00150830 0.000754148 1.00000i \(-0.499760\pi\)
0.000754148 1.00000i \(0.499760\pi\)
\(24\) 0 0
\(25\) −68341.9 −0.874776
\(26\) 209835. 2.34138
\(27\) 0 0
\(28\) 31141.6 0.268094
\(29\) 102600. 0.781185 0.390592 0.920564i \(-0.372270\pi\)
0.390592 + 0.920564i \(0.372270\pi\)
\(30\) 0 0
\(31\) −287624. −1.73404 −0.867019 0.498275i \(-0.833967\pi\)
−0.867019 + 0.498275i \(0.833967\pi\)
\(32\) −212500. −1.14640
\(33\) 0 0
\(34\) 188233. 0.821333
\(35\) 15746.5 0.0620791
\(36\) 0 0
\(37\) 566144. 1.83747 0.918737 0.394870i \(-0.129210\pi\)
0.918737 + 0.394870i \(0.129210\pi\)
\(38\) 269240. 0.795971
\(39\) 0 0
\(40\) 120302. 0.297209
\(41\) −296117. −0.670995 −0.335497 0.942041i \(-0.608904\pi\)
−0.335497 + 0.942041i \(0.608904\pi\)
\(42\) 0 0
\(43\) 740471. 1.42026 0.710131 0.704070i \(-0.248636\pi\)
0.710131 + 0.704070i \(0.248636\pi\)
\(44\) 955557. 1.69111
\(45\) 0 0
\(46\) 1583.24 0.00239825
\(47\) −958168. −1.34617 −0.673084 0.739566i \(-0.735031\pi\)
−0.673084 + 0.739566i \(0.735031\pi\)
\(48\) 0 0
\(49\) −798198. −0.969225
\(50\) −1.22942e6 −1.39093
\(51\) 0 0
\(52\) 2.28171e6 2.25034
\(53\) 732872. 0.676181 0.338090 0.941114i \(-0.390219\pi\)
0.338090 + 0.941114i \(0.390219\pi\)
\(54\) 0 0
\(55\) 483170. 0.391589
\(56\) 193633. 0.147340
\(57\) 0 0
\(58\) 1.84569e6 1.24211
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 2.28293e6 1.28777 0.643884 0.765123i \(-0.277322\pi\)
0.643884 + 0.765123i \(0.277322\pi\)
\(62\) −5.17412e6 −2.75718
\(63\) 0 0
\(64\) −3.41844e6 −1.63004
\(65\) 1.15373e6 0.521083
\(66\) 0 0
\(67\) −4.29840e6 −1.74600 −0.873001 0.487719i \(-0.837829\pi\)
−0.873001 + 0.487719i \(0.837829\pi\)
\(68\) 2.04681e6 0.789400
\(69\) 0 0
\(70\) 283267. 0.0987080
\(71\) 4.99901e6 1.65760 0.828800 0.559545i \(-0.189024\pi\)
0.828800 + 0.559545i \(0.189024\pi\)
\(72\) 0 0
\(73\) 3.66974e6 1.10409 0.552046 0.833814i \(-0.313847\pi\)
0.552046 + 0.833814i \(0.313847\pi\)
\(74\) 1.01845e7 2.92165
\(75\) 0 0
\(76\) 2.92768e6 0.765024
\(77\) 777691. 0.194129
\(78\) 0 0
\(79\) 3.18719e6 0.727300 0.363650 0.931536i \(-0.381530\pi\)
0.363650 + 0.931536i \(0.381530\pi\)
\(80\) −312391. −0.0682155
\(81\) 0 0
\(82\) −5.32690e6 −1.06691
\(83\) 4.43945e6 0.852228 0.426114 0.904670i \(-0.359882\pi\)
0.426114 + 0.904670i \(0.359882\pi\)
\(84\) 0 0
\(85\) 1.03496e6 0.182791
\(86\) 1.33205e7 2.25827
\(87\) 0 0
\(88\) 5.94150e6 0.929409
\(89\) 1.07216e7 1.61211 0.806056 0.591839i \(-0.201598\pi\)
0.806056 + 0.591839i \(0.201598\pi\)
\(90\) 0 0
\(91\) 1.85700e6 0.258325
\(92\) 17215.9 0.00230500
\(93\) 0 0
\(94\) −1.72367e7 −2.14046
\(95\) 1.48036e6 0.177147
\(96\) 0 0
\(97\) −4.05589e6 −0.451216 −0.225608 0.974218i \(-0.572437\pi\)
−0.225608 + 0.974218i \(0.572437\pi\)
\(98\) −1.43590e7 −1.54110
\(99\) 0 0
\(100\) −1.33685e7 −1.33685
\(101\) 1.05360e7 1.01754 0.508768 0.860904i \(-0.330101\pi\)
0.508768 + 0.860904i \(0.330101\pi\)
\(102\) 0 0
\(103\) −9.49129e6 −0.855845 −0.427922 0.903816i \(-0.640754\pi\)
−0.427922 + 0.903816i \(0.640754\pi\)
\(104\) 1.41873e7 1.23675
\(105\) 0 0
\(106\) 1.31838e7 1.07515
\(107\) 7.68612e6 0.606547 0.303273 0.952904i \(-0.401921\pi\)
0.303273 + 0.952904i \(0.401921\pi\)
\(108\) 0 0
\(109\) 1.46696e7 1.08499 0.542494 0.840059i \(-0.317480\pi\)
0.542494 + 0.840059i \(0.317480\pi\)
\(110\) 8.69185e6 0.622641
\(111\) 0 0
\(112\) −502812. −0.0338176
\(113\) 1.52191e7 0.992234 0.496117 0.868256i \(-0.334759\pi\)
0.496117 + 0.868256i \(0.334759\pi\)
\(114\) 0 0
\(115\) 8705.07 0.000533740 0
\(116\) 2.00697e7 1.19382
\(117\) 0 0
\(118\) 3.69461e6 0.207005
\(119\) 1.66582e6 0.0906180
\(120\) 0 0
\(121\) 4.37575e6 0.224545
\(122\) 4.10680e7 2.04760
\(123\) 0 0
\(124\) −5.62626e7 −2.64999
\(125\) −1.44870e7 −0.663426
\(126\) 0 0
\(127\) 1.49590e7 0.648021 0.324011 0.946053i \(-0.394969\pi\)
0.324011 + 0.946053i \(0.394969\pi\)
\(128\) −3.42951e7 −1.44543
\(129\) 0 0
\(130\) 2.07547e7 0.828541
\(131\) 2.82534e7 1.09805 0.549024 0.835807i \(-0.315000\pi\)
0.549024 + 0.835807i \(0.315000\pi\)
\(132\) 0 0
\(133\) 2.38272e6 0.0878198
\(134\) −7.73248e7 −2.77621
\(135\) 0 0
\(136\) 1.27268e7 0.433842
\(137\) −4.47666e7 −1.48741 −0.743707 0.668506i \(-0.766934\pi\)
−0.743707 + 0.668506i \(0.766934\pi\)
\(138\) 0 0
\(139\) −4.23885e7 −1.33874 −0.669370 0.742929i \(-0.733436\pi\)
−0.669370 + 0.742929i \(0.733436\pi\)
\(140\) 3.08020e6 0.0948703
\(141\) 0 0
\(142\) 8.99283e7 2.63565
\(143\) 5.69807e7 1.62949
\(144\) 0 0
\(145\) 1.01481e7 0.276437
\(146\) 6.60157e7 1.75555
\(147\) 0 0
\(148\) 1.10744e8 2.80806
\(149\) 3.46535e7 0.858214 0.429107 0.903254i \(-0.358828\pi\)
0.429107 + 0.903254i \(0.358828\pi\)
\(150\) 0 0
\(151\) −7.67555e7 −1.81422 −0.907110 0.420893i \(-0.861717\pi\)
−0.907110 + 0.420893i \(0.861717\pi\)
\(152\) 1.82038e7 0.420445
\(153\) 0 0
\(154\) 1.39901e7 0.308672
\(155\) −2.84487e7 −0.613623
\(156\) 0 0
\(157\) −6.08764e7 −1.25545 −0.627726 0.778434i \(-0.716014\pi\)
−0.627726 + 0.778434i \(0.716014\pi\)
\(158\) 5.73351e7 1.15643
\(159\) 0 0
\(160\) −2.10183e7 −0.405675
\(161\) 14011.3 0.000264600 0
\(162\) 0 0
\(163\) −9.26090e7 −1.67493 −0.837465 0.546491i \(-0.815963\pi\)
−0.837465 + 0.546491i \(0.815963\pi\)
\(164\) −5.79239e7 −1.02543
\(165\) 0 0
\(166\) 7.98622e7 1.35507
\(167\) −6.05390e7 −1.00584 −0.502918 0.864334i \(-0.667740\pi\)
−0.502918 + 0.864334i \(0.667740\pi\)
\(168\) 0 0
\(169\) 7.33117e7 1.16834
\(170\) 1.86180e7 0.290645
\(171\) 0 0
\(172\) 1.44845e8 2.17047
\(173\) −6.39593e6 −0.0939166 −0.0469583 0.998897i \(-0.514953\pi\)
−0.0469583 + 0.998897i \(0.514953\pi\)
\(174\) 0 0
\(175\) −1.08801e7 −0.153461
\(176\) −1.54284e7 −0.213318
\(177\) 0 0
\(178\) 1.92873e8 2.56332
\(179\) 1.31141e8 1.70904 0.854519 0.519421i \(-0.173852\pi\)
0.854519 + 0.519421i \(0.173852\pi\)
\(180\) 0 0
\(181\) −3.39901e7 −0.426066 −0.213033 0.977045i \(-0.568334\pi\)
−0.213033 + 0.977045i \(0.568334\pi\)
\(182\) 3.34059e7 0.410746
\(183\) 0 0
\(184\) 107046. 0.00126679
\(185\) 5.59971e7 0.650226
\(186\) 0 0
\(187\) 5.11147e7 0.571610
\(188\) −1.87429e8 −2.05724
\(189\) 0 0
\(190\) 2.66304e7 0.281670
\(191\) −1.52054e8 −1.57899 −0.789497 0.613755i \(-0.789658\pi\)
−0.789497 + 0.613755i \(0.789658\pi\)
\(192\) 0 0
\(193\) 5.77384e6 0.0578115 0.0289057 0.999582i \(-0.490798\pi\)
0.0289057 + 0.999582i \(0.490798\pi\)
\(194\) −7.29622e7 −0.717450
\(195\) 0 0
\(196\) −1.56137e8 −1.48119
\(197\) 1.02503e8 0.955221 0.477611 0.878572i \(-0.341503\pi\)
0.477611 + 0.878572i \(0.341503\pi\)
\(198\) 0 0
\(199\) 8.62017e7 0.775408 0.387704 0.921784i \(-0.373268\pi\)
0.387704 + 0.921784i \(0.373268\pi\)
\(200\) −8.31230e7 −0.734711
\(201\) 0 0
\(202\) 1.89534e8 1.61792
\(203\) 1.63340e7 0.137043
\(204\) 0 0
\(205\) −2.92887e7 −0.237445
\(206\) −1.70741e8 −1.36082
\(207\) 0 0
\(208\) −3.68405e7 −0.283860
\(209\) 7.31122e7 0.553959
\(210\) 0 0
\(211\) 2.46914e7 0.180949 0.0904746 0.995899i \(-0.471162\pi\)
0.0904746 + 0.995899i \(0.471162\pi\)
\(212\) 1.43358e8 1.03335
\(213\) 0 0
\(214\) 1.38267e8 0.964431
\(215\) 7.32396e7 0.502587
\(216\) 0 0
\(217\) −4.57899e7 −0.304201
\(218\) 2.63894e8 1.72517
\(219\) 0 0
\(220\) 9.45137e7 0.598433
\(221\) 1.22053e8 0.760635
\(222\) 0 0
\(223\) 4.55302e7 0.274936 0.137468 0.990506i \(-0.456104\pi\)
0.137468 + 0.990506i \(0.456104\pi\)
\(224\) −3.38302e7 −0.201111
\(225\) 0 0
\(226\) 2.73780e8 1.57769
\(227\) 2.56384e8 1.45479 0.727395 0.686219i \(-0.240731\pi\)
0.727395 + 0.686219i \(0.240731\pi\)
\(228\) 0 0
\(229\) −3.44709e8 −1.89683 −0.948415 0.317033i \(-0.897314\pi\)
−0.948415 + 0.317033i \(0.897314\pi\)
\(230\) 156597. 0.000848667 0
\(231\) 0 0
\(232\) 1.24790e8 0.656104
\(233\) 1.44767e8 0.749762 0.374881 0.927073i \(-0.377684\pi\)
0.374881 + 0.927073i \(0.377684\pi\)
\(234\) 0 0
\(235\) −9.47720e7 −0.476368
\(236\) 4.01745e7 0.198957
\(237\) 0 0
\(238\) 2.99669e7 0.144086
\(239\) 2.45941e8 1.16530 0.582650 0.812723i \(-0.302016\pi\)
0.582650 + 0.812723i \(0.302016\pi\)
\(240\) 0 0
\(241\) 2.54396e8 1.17071 0.585357 0.810775i \(-0.300954\pi\)
0.585357 + 0.810775i \(0.300954\pi\)
\(242\) 7.87164e7 0.357035
\(243\) 0 0
\(244\) 4.46567e8 1.96799
\(245\) −7.89494e7 −0.342979
\(246\) 0 0
\(247\) 1.74580e8 0.737147
\(248\) −3.49831e8 −1.45639
\(249\) 0 0
\(250\) −2.60609e8 −1.05487
\(251\) −3.25172e8 −1.29794 −0.648971 0.760813i \(-0.724800\pi\)
−0.648971 + 0.760813i \(0.724800\pi\)
\(252\) 0 0
\(253\) 429928. 0.00166907
\(254\) 2.69100e8 1.03038
\(255\) 0 0
\(256\) −1.79381e8 −0.668245
\(257\) −2.54981e8 −0.937003 −0.468502 0.883463i \(-0.655206\pi\)
−0.468502 + 0.883463i \(0.655206\pi\)
\(258\) 0 0
\(259\) 9.01307e7 0.322347
\(260\) 2.25683e8 0.796328
\(261\) 0 0
\(262\) 5.08256e8 1.74594
\(263\) −4.19400e8 −1.42162 −0.710809 0.703385i \(-0.751671\pi\)
−0.710809 + 0.703385i \(0.751671\pi\)
\(264\) 0 0
\(265\) 7.24880e7 0.239280
\(266\) 4.28633e7 0.139637
\(267\) 0 0
\(268\) −8.40817e8 −2.66827
\(269\) 4.15320e8 1.30092 0.650459 0.759541i \(-0.274576\pi\)
0.650459 + 0.759541i \(0.274576\pi\)
\(270\) 0 0
\(271\) −5.50963e7 −0.168163 −0.0840814 0.996459i \(-0.526796\pi\)
−0.0840814 + 0.996459i \(0.526796\pi\)
\(272\) −3.30479e7 −0.0995755
\(273\) 0 0
\(274\) −8.05315e8 −2.36504
\(275\) −3.33848e8 −0.968020
\(276\) 0 0
\(277\) −2.16584e8 −0.612275 −0.306138 0.951987i \(-0.599037\pi\)
−0.306138 + 0.951987i \(0.599037\pi\)
\(278\) −7.62536e8 −2.12865
\(279\) 0 0
\(280\) 1.91522e7 0.0521392
\(281\) 1.30596e8 0.351123 0.175561 0.984468i \(-0.443826\pi\)
0.175561 + 0.984468i \(0.443826\pi\)
\(282\) 0 0
\(283\) 4.03750e8 1.05891 0.529457 0.848337i \(-0.322396\pi\)
0.529457 + 0.848337i \(0.322396\pi\)
\(284\) 9.77865e8 2.53317
\(285\) 0 0
\(286\) 1.02504e9 2.59095
\(287\) −4.71420e7 −0.117712
\(288\) 0 0
\(289\) −3.00851e8 −0.733176
\(290\) 1.82556e8 0.439546
\(291\) 0 0
\(292\) 7.17844e8 1.68729
\(293\) −1.61200e8 −0.374394 −0.187197 0.982322i \(-0.559940\pi\)
−0.187197 + 0.982322i \(0.559940\pi\)
\(294\) 0 0
\(295\) 2.03139e7 0.0460699
\(296\) 6.88591e8 1.54327
\(297\) 0 0
\(298\) 6.23390e8 1.36459
\(299\) 1.02660e6 0.00222101
\(300\) 0 0
\(301\) 1.17884e8 0.249156
\(302\) −1.38077e9 −2.88468
\(303\) 0 0
\(304\) −4.72702e7 −0.0965007
\(305\) 2.25803e8 0.455701
\(306\) 0 0
\(307\) −7.32355e8 −1.44457 −0.722283 0.691597i \(-0.756907\pi\)
−0.722283 + 0.691597i \(0.756907\pi\)
\(308\) 1.52126e8 0.296671
\(309\) 0 0
\(310\) −5.11770e8 −0.975683
\(311\) −1.98727e8 −0.374624 −0.187312 0.982300i \(-0.559978\pi\)
−0.187312 + 0.982300i \(0.559978\pi\)
\(312\) 0 0
\(313\) 2.25644e8 0.415929 0.207964 0.978136i \(-0.433316\pi\)
0.207964 + 0.978136i \(0.433316\pi\)
\(314\) −1.09512e9 −1.99622
\(315\) 0 0
\(316\) 6.23453e8 1.11147
\(317\) −5.60321e8 −0.987937 −0.493968 0.869480i \(-0.664454\pi\)
−0.493968 + 0.869480i \(0.664454\pi\)
\(318\) 0 0
\(319\) 5.01197e8 0.864452
\(320\) −3.38117e8 −0.576822
\(321\) 0 0
\(322\) 252053. 0.000420723 0
\(323\) 1.56607e8 0.258585
\(324\) 0 0
\(325\) −7.97173e8 −1.28813
\(326\) −1.66596e9 −2.66320
\(327\) 0 0
\(328\) −3.60161e8 −0.563558
\(329\) −1.52541e8 −0.236157
\(330\) 0 0
\(331\) 4.88144e7 0.0739861 0.0369930 0.999316i \(-0.488222\pi\)
0.0369930 + 0.999316i \(0.488222\pi\)
\(332\) 8.68408e8 1.30239
\(333\) 0 0
\(334\) −1.08905e9 −1.59932
\(335\) −4.25152e8 −0.617857
\(336\) 0 0
\(337\) −4.98448e8 −0.709439 −0.354720 0.934973i \(-0.615424\pi\)
−0.354720 + 0.934973i \(0.615424\pi\)
\(338\) 1.31882e9 1.85771
\(339\) 0 0
\(340\) 2.02449e8 0.279345
\(341\) −1.40503e9 −1.91887
\(342\) 0 0
\(343\) −2.58183e8 −0.345460
\(344\) 9.00621e8 1.19286
\(345\) 0 0
\(346\) −1.15058e8 −0.149331
\(347\) −4.64305e8 −0.596554 −0.298277 0.954479i \(-0.596412\pi\)
−0.298277 + 0.954479i \(0.596412\pi\)
\(348\) 0 0
\(349\) −3.75860e8 −0.473301 −0.236651 0.971595i \(-0.576050\pi\)
−0.236651 + 0.971595i \(0.576050\pi\)
\(350\) −1.95724e8 −0.244009
\(351\) 0 0
\(352\) −1.03806e9 −1.26859
\(353\) 6.79230e8 0.821874 0.410937 0.911664i \(-0.365202\pi\)
0.410937 + 0.911664i \(0.365202\pi\)
\(354\) 0 0
\(355\) 4.94450e8 0.586574
\(356\) 2.09727e9 2.46366
\(357\) 0 0
\(358\) 2.35912e9 2.71743
\(359\) −4.73258e8 −0.539843 −0.269922 0.962882i \(-0.586998\pi\)
−0.269922 + 0.962882i \(0.586998\pi\)
\(360\) 0 0
\(361\) −6.69868e8 −0.749400
\(362\) −6.11455e8 −0.677461
\(363\) 0 0
\(364\) 3.63250e8 0.394776
\(365\) 3.62972e8 0.390704
\(366\) 0 0
\(367\) 1.29401e9 1.36649 0.683245 0.730189i \(-0.260568\pi\)
0.683245 + 0.730189i \(0.260568\pi\)
\(368\) −277968. −0.000290755 0
\(369\) 0 0
\(370\) 1.00734e9 1.03388
\(371\) 1.16674e8 0.118622
\(372\) 0 0
\(373\) −6.91706e8 −0.690146 −0.345073 0.938576i \(-0.612146\pi\)
−0.345073 + 0.938576i \(0.612146\pi\)
\(374\) 9.19512e8 0.908881
\(375\) 0 0
\(376\) −1.16540e9 −1.13062
\(377\) 1.19677e9 1.15032
\(378\) 0 0
\(379\) −9.69203e8 −0.914487 −0.457243 0.889342i \(-0.651163\pi\)
−0.457243 + 0.889342i \(0.651163\pi\)
\(380\) 2.89575e8 0.270719
\(381\) 0 0
\(382\) −2.73533e9 −2.51066
\(383\) 1.28084e9 1.16493 0.582465 0.812856i \(-0.302088\pi\)
0.582465 + 0.812856i \(0.302088\pi\)
\(384\) 0 0
\(385\) 7.69211e7 0.0686962
\(386\) 1.03867e8 0.0919224
\(387\) 0 0
\(388\) −7.93379e8 −0.689556
\(389\) −1.19302e9 −1.02760 −0.513802 0.857909i \(-0.671763\pi\)
−0.513802 + 0.857909i \(0.671763\pi\)
\(390\) 0 0
\(391\) 920911. 0.000779111 0
\(392\) −9.70834e8 −0.814036
\(393\) 0 0
\(394\) 1.84394e9 1.51884
\(395\) 3.15244e8 0.257369
\(396\) 0 0
\(397\) −4.46557e7 −0.0358187 −0.0179094 0.999840i \(-0.505701\pi\)
−0.0179094 + 0.999840i \(0.505701\pi\)
\(398\) 1.55070e9 1.23293
\(399\) 0 0
\(400\) 2.15847e8 0.168631
\(401\) −5.52185e8 −0.427641 −0.213821 0.976873i \(-0.568591\pi\)
−0.213821 + 0.976873i \(0.568591\pi\)
\(402\) 0 0
\(403\) −3.35498e9 −2.55342
\(404\) 2.06096e9 1.55501
\(405\) 0 0
\(406\) 2.93835e8 0.217903
\(407\) 2.76560e9 2.03333
\(408\) 0 0
\(409\) 8.01492e8 0.579252 0.289626 0.957140i \(-0.406469\pi\)
0.289626 + 0.957140i \(0.406469\pi\)
\(410\) −5.26881e8 −0.377546
\(411\) 0 0
\(412\) −1.85661e9 −1.30792
\(413\) 3.26965e7 0.0228390
\(414\) 0 0
\(415\) 4.39104e8 0.301577
\(416\) −2.47871e9 −1.68810
\(417\) 0 0
\(418\) 1.31523e9 0.880815
\(419\) 3.06611e8 0.203629 0.101814 0.994803i \(-0.467535\pi\)
0.101814 + 0.994803i \(0.467535\pi\)
\(420\) 0 0
\(421\) 1.08219e9 0.706832 0.353416 0.935466i \(-0.385020\pi\)
0.353416 + 0.935466i \(0.385020\pi\)
\(422\) 4.44178e8 0.287716
\(423\) 0 0
\(424\) 8.91379e8 0.567913
\(425\) −7.15106e8 −0.451866
\(426\) 0 0
\(427\) 3.63444e8 0.225912
\(428\) 1.50350e9 0.926935
\(429\) 0 0
\(430\) 1.31752e9 0.799132
\(431\) −1.67605e9 −1.00836 −0.504182 0.863597i \(-0.668206\pi\)
−0.504182 + 0.863597i \(0.668206\pi\)
\(432\) 0 0
\(433\) 2.51807e9 1.49060 0.745300 0.666730i \(-0.232306\pi\)
0.745300 + 0.666730i \(0.232306\pi\)
\(434\) −8.23725e8 −0.483691
\(435\) 0 0
\(436\) 2.86954e9 1.65810
\(437\) 1.31723e6 0.000755053 0
\(438\) 0 0
\(439\) −1.36932e9 −0.772465 −0.386232 0.922402i \(-0.626224\pi\)
−0.386232 + 0.922402i \(0.626224\pi\)
\(440\) 5.87671e8 0.328889
\(441\) 0 0
\(442\) 2.19564e9 1.20944
\(443\) 3.62914e7 0.0198331 0.00991656 0.999951i \(-0.496843\pi\)
0.00991656 + 0.999951i \(0.496843\pi\)
\(444\) 0 0
\(445\) 1.06047e9 0.570477
\(446\) 8.19052e8 0.437159
\(447\) 0 0
\(448\) −5.44219e8 −0.285957
\(449\) 2.01372e9 1.04987 0.524937 0.851141i \(-0.324089\pi\)
0.524937 + 0.851141i \(0.324089\pi\)
\(450\) 0 0
\(451\) −1.44652e9 −0.742517
\(452\) 2.97703e9 1.51635
\(453\) 0 0
\(454\) 4.61214e9 2.31317
\(455\) 1.83675e8 0.0914133
\(456\) 0 0
\(457\) −1.10905e8 −0.0543554 −0.0271777 0.999631i \(-0.508652\pi\)
−0.0271777 + 0.999631i \(0.508652\pi\)
\(458\) −6.20104e9 −3.01603
\(459\) 0 0
\(460\) 1.70281e6 0.000815671 0
\(461\) −3.59226e9 −1.70771 −0.853856 0.520510i \(-0.825742\pi\)
−0.853856 + 0.520510i \(0.825742\pi\)
\(462\) 0 0
\(463\) −2.38603e9 −1.11723 −0.558614 0.829427i \(-0.688667\pi\)
−0.558614 + 0.829427i \(0.688667\pi\)
\(464\) −3.24046e8 −0.150589
\(465\) 0 0
\(466\) 2.60424e9 1.19215
\(467\) −1.97044e9 −0.895271 −0.447636 0.894216i \(-0.647734\pi\)
−0.447636 + 0.894216i \(0.647734\pi\)
\(468\) 0 0
\(469\) −6.84308e8 −0.306300
\(470\) −1.70487e9 −0.757442
\(471\) 0 0
\(472\) 2.49799e8 0.109344
\(473\) 3.61718e9 1.57165
\(474\) 0 0
\(475\) −1.02286e9 −0.437913
\(476\) 3.25855e8 0.138484
\(477\) 0 0
\(478\) 4.42428e9 1.85287
\(479\) −4.56276e9 −1.89694 −0.948469 0.316869i \(-0.897368\pi\)
−0.948469 + 0.316869i \(0.897368\pi\)
\(480\) 0 0
\(481\) 6.60378e9 2.70573
\(482\) 4.57639e9 1.86148
\(483\) 0 0
\(484\) 8.55949e8 0.343154
\(485\) −4.01166e8 −0.159672
\(486\) 0 0
\(487\) 2.91958e9 1.14543 0.572715 0.819754i \(-0.305890\pi\)
0.572715 + 0.819754i \(0.305890\pi\)
\(488\) 2.77668e9 1.08158
\(489\) 0 0
\(490\) −1.42024e9 −0.545349
\(491\) 8.15321e8 0.310845 0.155422 0.987848i \(-0.450326\pi\)
0.155422 + 0.987848i \(0.450326\pi\)
\(492\) 0 0
\(493\) 1.07357e9 0.403521
\(494\) 3.14055e9 1.17209
\(495\) 0 0
\(496\) 9.08415e8 0.334271
\(497\) 7.95847e8 0.290792
\(498\) 0 0
\(499\) −3.24554e9 −1.16932 −0.584662 0.811277i \(-0.698773\pi\)
−0.584662 + 0.811277i \(0.698773\pi\)
\(500\) −2.83382e9 −1.01386
\(501\) 0 0
\(502\) −5.84959e9 −2.06378
\(503\) 4.45372e9 1.56040 0.780198 0.625532i \(-0.215118\pi\)
0.780198 + 0.625532i \(0.215118\pi\)
\(504\) 0 0
\(505\) 1.04211e9 0.360075
\(506\) 7.73407e6 0.00265388
\(507\) 0 0
\(508\) 2.92615e9 0.990317
\(509\) −4.82014e9 −1.62012 −0.810060 0.586347i \(-0.800565\pi\)
−0.810060 + 0.586347i \(0.800565\pi\)
\(510\) 0 0
\(511\) 5.84226e8 0.193690
\(512\) 1.16285e9 0.382896
\(513\) 0 0
\(514\) −4.58690e9 −1.48987
\(515\) −9.38779e8 −0.302857
\(516\) 0 0
\(517\) −4.68062e9 −1.48966
\(518\) 1.62138e9 0.512543
\(519\) 0 0
\(520\) 1.40326e9 0.437649
\(521\) 3.16646e9 0.980939 0.490469 0.871458i \(-0.336825\pi\)
0.490469 + 0.871458i \(0.336825\pi\)
\(522\) 0 0
\(523\) −1.33084e9 −0.406790 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(524\) 5.52670e9 1.67806
\(525\) 0 0
\(526\) −7.54467e9 −2.26042
\(527\) −3.00960e9 −0.895717
\(528\) 0 0
\(529\) −3.40482e9 −0.999998
\(530\) 1.30400e9 0.380463
\(531\) 0 0
\(532\) 4.66088e8 0.134208
\(533\) −3.45405e9 −0.988059
\(534\) 0 0
\(535\) 7.60231e8 0.214638
\(536\) −5.22806e9 −1.46644
\(537\) 0 0
\(538\) 7.47128e9 2.06851
\(539\) −3.89917e9 −1.07254
\(540\) 0 0
\(541\) 2.45311e8 0.0666081 0.0333040 0.999445i \(-0.489397\pi\)
0.0333040 + 0.999445i \(0.489397\pi\)
\(542\) −9.91139e8 −0.267385
\(543\) 0 0
\(544\) −2.22353e9 −0.592171
\(545\) 1.45096e9 0.383944
\(546\) 0 0
\(547\) 1.35454e9 0.353865 0.176933 0.984223i \(-0.443383\pi\)
0.176933 + 0.984223i \(0.443383\pi\)
\(548\) −8.75687e9 −2.27309
\(549\) 0 0
\(550\) −6.00566e9 −1.53919
\(551\) 1.53559e9 0.391061
\(552\) 0 0
\(553\) 5.07404e8 0.127590
\(554\) −3.89617e9 −0.973540
\(555\) 0 0
\(556\) −8.29169e9 −2.04589
\(557\) 2.79389e9 0.685041 0.342520 0.939510i \(-0.388719\pi\)
0.342520 + 0.939510i \(0.388719\pi\)
\(558\) 0 0
\(559\) 8.63721e9 2.09138
\(560\) −4.97329e7 −0.0119670
\(561\) 0 0
\(562\) 2.34933e9 0.558298
\(563\) −1.29186e9 −0.305095 −0.152548 0.988296i \(-0.548748\pi\)
−0.152548 + 0.988296i \(0.548748\pi\)
\(564\) 0 0
\(565\) 1.50531e9 0.351121
\(566\) 7.26315e9 1.68371
\(567\) 0 0
\(568\) 6.08021e9 1.39219
\(569\) −2.83891e9 −0.646040 −0.323020 0.946392i \(-0.604698\pi\)
−0.323020 + 0.946392i \(0.604698\pi\)
\(570\) 0 0
\(571\) −2.52009e9 −0.566487 −0.283243 0.959048i \(-0.591410\pi\)
−0.283243 + 0.959048i \(0.591410\pi\)
\(572\) 1.11461e10 2.49021
\(573\) 0 0
\(574\) −8.48048e8 −0.187167
\(575\) −6.01480e6 −0.00131942
\(576\) 0 0
\(577\) −7.41099e9 −1.60606 −0.803028 0.595941i \(-0.796779\pi\)
−0.803028 + 0.595941i \(0.796779\pi\)
\(578\) −5.41206e9 −1.16578
\(579\) 0 0
\(580\) 1.98509e9 0.422456
\(581\) 7.06764e8 0.149506
\(582\) 0 0
\(583\) 3.58006e9 0.748256
\(584\) 4.46344e9 0.927309
\(585\) 0 0
\(586\) −2.89987e9 −0.595301
\(587\) 2.54336e9 0.519008 0.259504 0.965742i \(-0.416441\pi\)
0.259504 + 0.965742i \(0.416441\pi\)
\(588\) 0 0
\(589\) −4.30480e9 −0.868058
\(590\) 3.65432e8 0.0732528
\(591\) 0 0
\(592\) −1.78808e9 −0.354210
\(593\) −9.16399e9 −1.80465 −0.902326 0.431055i \(-0.858142\pi\)
−0.902326 + 0.431055i \(0.858142\pi\)
\(594\) 0 0
\(595\) 1.64766e8 0.0320670
\(596\) 6.77864e9 1.31154
\(597\) 0 0
\(598\) 1.84677e7 0.00353149
\(599\) 5.47291e9 1.04046 0.520229 0.854027i \(-0.325847\pi\)
0.520229 + 0.854027i \(0.325847\pi\)
\(600\) 0 0
\(601\) −4.19336e9 −0.787955 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(602\) 2.12063e9 0.396167
\(603\) 0 0
\(604\) −1.50143e10 −2.77252
\(605\) 4.32804e8 0.0794598
\(606\) 0 0
\(607\) 6.57768e9 1.19375 0.596874 0.802335i \(-0.296409\pi\)
0.596874 + 0.802335i \(0.296409\pi\)
\(608\) −3.18044e9 −0.573885
\(609\) 0 0
\(610\) 4.06202e9 0.724582
\(611\) −1.11765e10 −1.98227
\(612\) 0 0
\(613\) −7.04485e9 −1.23527 −0.617633 0.786467i \(-0.711908\pi\)
−0.617633 + 0.786467i \(0.711908\pi\)
\(614\) −1.31745e10 −2.29691
\(615\) 0 0
\(616\) 9.45892e8 0.163046
\(617\) 4.68870e9 0.803627 0.401813 0.915722i \(-0.368380\pi\)
0.401813 + 0.915722i \(0.368380\pi\)
\(618\) 0 0
\(619\) −6.66470e9 −1.12944 −0.564720 0.825282i \(-0.691016\pi\)
−0.564720 + 0.825282i \(0.691016\pi\)
\(620\) −5.56490e9 −0.937749
\(621\) 0 0
\(622\) −3.57495e9 −0.595667
\(623\) 1.70689e9 0.282812
\(624\) 0 0
\(625\) 3.90631e9 0.640010
\(626\) 4.05916e9 0.661342
\(627\) 0 0
\(628\) −1.19081e10 −1.91860
\(629\) 5.92394e9 0.949147
\(630\) 0 0
\(631\) 6.09161e8 0.0965227 0.0482614 0.998835i \(-0.484632\pi\)
0.0482614 + 0.998835i \(0.484632\pi\)
\(632\) 3.87653e9 0.610848
\(633\) 0 0
\(634\) −1.00797e10 −1.57086
\(635\) 1.47959e9 0.229315
\(636\) 0 0
\(637\) −9.31057e9 −1.42721
\(638\) 9.01614e9 1.37451
\(639\) 0 0
\(640\) −3.39211e9 −0.511493
\(641\) 3.31465e7 0.00497089 0.00248545 0.999997i \(-0.499209\pi\)
0.00248545 + 0.999997i \(0.499209\pi\)
\(642\) 0 0
\(643\) −4.47226e9 −0.663421 −0.331710 0.943381i \(-0.607626\pi\)
−0.331710 + 0.943381i \(0.607626\pi\)
\(644\) 2.74078e6 0.000404366 0
\(645\) 0 0
\(646\) 2.81724e9 0.411159
\(647\) 4.33561e9 0.629340 0.314670 0.949201i \(-0.398106\pi\)
0.314670 + 0.949201i \(0.398106\pi\)
\(648\) 0 0
\(649\) 1.00327e9 0.144066
\(650\) −1.43405e10 −2.04818
\(651\) 0 0
\(652\) −1.81154e10 −2.55966
\(653\) 5.43227e9 0.763458 0.381729 0.924274i \(-0.375329\pi\)
0.381729 + 0.924274i \(0.375329\pi\)
\(654\) 0 0
\(655\) 2.79453e9 0.388566
\(656\) 9.35239e8 0.129348
\(657\) 0 0
\(658\) −2.74410e9 −0.375499
\(659\) −5.26358e9 −0.716444 −0.358222 0.933636i \(-0.616617\pi\)
−0.358222 + 0.933636i \(0.616617\pi\)
\(660\) 0 0
\(661\) 1.51100e9 0.203497 0.101749 0.994810i \(-0.467556\pi\)
0.101749 + 0.994810i \(0.467556\pi\)
\(662\) 8.78133e8 0.117641
\(663\) 0 0
\(664\) 5.39962e9 0.715772
\(665\) 2.35674e8 0.0310768
\(666\) 0 0
\(667\) 9.02986e6 0.00117826
\(668\) −1.18421e10 −1.53714
\(669\) 0 0
\(670\) −7.64816e9 −0.982415
\(671\) 1.11520e10 1.42503
\(672\) 0 0
\(673\) 4.92429e8 0.0622717 0.0311359 0.999515i \(-0.490088\pi\)
0.0311359 + 0.999515i \(0.490088\pi\)
\(674\) −8.96668e9 −1.12803
\(675\) 0 0
\(676\) 1.43406e10 1.78548
\(677\) −1.38348e10 −1.71361 −0.856805 0.515640i \(-0.827554\pi\)
−0.856805 + 0.515640i \(0.827554\pi\)
\(678\) 0 0
\(679\) −6.45701e8 −0.0791566
\(680\) 1.25880e9 0.153523
\(681\) 0 0
\(682\) −2.52754e10 −3.05108
\(683\) −8.01482e8 −0.0962545 −0.0481273 0.998841i \(-0.515325\pi\)
−0.0481273 + 0.998841i \(0.515325\pi\)
\(684\) 0 0
\(685\) −4.42784e9 −0.526350
\(686\) −4.64450e9 −0.549294
\(687\) 0 0
\(688\) −2.33866e9 −0.273784
\(689\) 8.54858e9 0.995696
\(690\) 0 0
\(691\) −2.59241e9 −0.298903 −0.149452 0.988769i \(-0.547751\pi\)
−0.149452 + 0.988769i \(0.547751\pi\)
\(692\) −1.25112e9 −0.143525
\(693\) 0 0
\(694\) −8.35247e9 −0.948543
\(695\) −4.19263e9 −0.473739
\(696\) 0 0
\(697\) −3.09846e9 −0.346602
\(698\) −6.76143e9 −0.752566
\(699\) 0 0
\(700\) −2.12827e9 −0.234522
\(701\) −1.40491e10 −1.54040 −0.770201 0.637801i \(-0.779844\pi\)
−0.770201 + 0.637801i \(0.779844\pi\)
\(702\) 0 0
\(703\) 8.47335e9 0.919838
\(704\) −1.66990e10 −1.80379
\(705\) 0 0
\(706\) 1.22188e10 1.30681
\(707\) 1.67733e9 0.178506
\(708\) 0 0
\(709\) −9.39937e9 −0.990460 −0.495230 0.868762i \(-0.664916\pi\)
−0.495230 + 0.868762i \(0.664916\pi\)
\(710\) 8.89476e9 0.932675
\(711\) 0 0
\(712\) 1.30405e10 1.35399
\(713\) −2.53139e7 −0.00261544
\(714\) 0 0
\(715\) 5.63593e9 0.576626
\(716\) 2.56526e10 2.61178
\(717\) 0 0
\(718\) −8.51354e9 −0.858371
\(719\) −1.11397e10 −1.11769 −0.558846 0.829272i \(-0.688756\pi\)
−0.558846 + 0.829272i \(0.688756\pi\)
\(720\) 0 0
\(721\) −1.51102e9 −0.150140
\(722\) −1.20504e10 −1.19157
\(723\) 0 0
\(724\) −6.64886e9 −0.651122
\(725\) −7.01187e9 −0.683362
\(726\) 0 0
\(727\) −9.70625e9 −0.936874 −0.468437 0.883497i \(-0.655183\pi\)
−0.468437 + 0.883497i \(0.655183\pi\)
\(728\) 2.25863e9 0.216963
\(729\) 0 0
\(730\) 6.52958e9 0.621234
\(731\) 7.74803e9 0.733636
\(732\) 0 0
\(733\) 4.02746e9 0.377718 0.188859 0.982004i \(-0.439521\pi\)
0.188859 + 0.982004i \(0.439521\pi\)
\(734\) 2.32782e10 2.17277
\(735\) 0 0
\(736\) −1.87022e7 −0.00172911
\(737\) −2.09975e10 −1.93211
\(738\) 0 0
\(739\) 1.14088e10 1.03989 0.519943 0.854201i \(-0.325953\pi\)
0.519943 + 0.854201i \(0.325953\pi\)
\(740\) 1.09537e10 0.993686
\(741\) 0 0
\(742\) 2.09887e9 0.188613
\(743\) 3.48068e9 0.311317 0.155659 0.987811i \(-0.450250\pi\)
0.155659 + 0.987811i \(0.450250\pi\)
\(744\) 0 0
\(745\) 3.42757e9 0.303696
\(746\) −1.24432e10 −1.09736
\(747\) 0 0
\(748\) 9.99863e9 0.873544
\(749\) 1.22364e9 0.106406
\(750\) 0 0
\(751\) −1.06375e10 −0.916433 −0.458217 0.888841i \(-0.651512\pi\)
−0.458217 + 0.888841i \(0.651512\pi\)
\(752\) 3.02623e9 0.259501
\(753\) 0 0
\(754\) 2.15290e10 1.82905
\(755\) −7.59185e9 −0.641997
\(756\) 0 0
\(757\) 2.08009e10 1.74279 0.871397 0.490578i \(-0.163214\pi\)
0.871397 + 0.490578i \(0.163214\pi\)
\(758\) −1.74352e10 −1.45407
\(759\) 0 0
\(760\) 1.80053e9 0.148783
\(761\) −1.33150e9 −0.109521 −0.0547604 0.998500i \(-0.517439\pi\)
−0.0547604 + 0.998500i \(0.517439\pi\)
\(762\) 0 0
\(763\) 2.33541e9 0.190339
\(764\) −2.97435e10 −2.41304
\(765\) 0 0
\(766\) 2.30413e10 1.85228
\(767\) 2.39564e9 0.191707
\(768\) 0 0
\(769\) −2.19213e10 −1.73830 −0.869148 0.494553i \(-0.835332\pi\)
−0.869148 + 0.494553i \(0.835332\pi\)
\(770\) 1.38375e9 0.109230
\(771\) 0 0
\(772\) 1.12943e9 0.0883485
\(773\) −3.83517e9 −0.298646 −0.149323 0.988788i \(-0.547709\pi\)
−0.149323 + 0.988788i \(0.547709\pi\)
\(774\) 0 0
\(775\) 1.96567e10 1.51690
\(776\) −4.93310e9 −0.378969
\(777\) 0 0
\(778\) −2.14616e10 −1.63393
\(779\) −4.43191e9 −0.335900
\(780\) 0 0
\(781\) 2.44200e10 1.83429
\(782\) 1.65665e7 0.00123881
\(783\) 0 0
\(784\) 2.52099e9 0.186838
\(785\) −6.02126e9 −0.444266
\(786\) 0 0
\(787\) 1.79274e10 1.31101 0.655504 0.755192i \(-0.272456\pi\)
0.655504 + 0.755192i \(0.272456\pi\)
\(788\) 2.00508e10 1.45979
\(789\) 0 0
\(790\) 5.67099e9 0.409227
\(791\) 2.42289e9 0.174067
\(792\) 0 0
\(793\) 2.66292e10 1.89627
\(794\) −8.03321e8 −0.0569531
\(795\) 0 0
\(796\) 1.68621e10 1.18499
\(797\) 5.47185e9 0.382851 0.191426 0.981507i \(-0.438689\pi\)
0.191426 + 0.981507i \(0.438689\pi\)
\(798\) 0 0
\(799\) −1.00259e10 −0.695363
\(800\) 1.45227e10 1.00284
\(801\) 0 0
\(802\) −9.93338e9 −0.679965
\(803\) 1.79266e10 1.22178
\(804\) 0 0
\(805\) 1.38586e6 9.36337e−5 0
\(806\) −6.03535e10 −4.06003
\(807\) 0 0
\(808\) 1.28147e10 0.854612
\(809\) −2.20226e9 −0.146234 −0.0731171 0.997323i \(-0.523295\pi\)
−0.0731171 + 0.997323i \(0.523295\pi\)
\(810\) 0 0
\(811\) −5.11433e9 −0.336679 −0.168339 0.985729i \(-0.553840\pi\)
−0.168339 + 0.985729i \(0.553840\pi\)
\(812\) 3.19512e9 0.209431
\(813\) 0 0
\(814\) 4.97509e10 3.23307
\(815\) −9.15991e9 −0.592707
\(816\) 0 0
\(817\) 1.10825e10 0.710982
\(818\) 1.44182e10 0.921032
\(819\) 0 0
\(820\) −5.72922e9 −0.362867
\(821\) 4.81471e9 0.303647 0.151824 0.988408i \(-0.451485\pi\)
0.151824 + 0.988408i \(0.451485\pi\)
\(822\) 0 0
\(823\) −2.90096e10 −1.81402 −0.907011 0.421107i \(-0.861642\pi\)
−0.907011 + 0.421107i \(0.861642\pi\)
\(824\) −1.15441e10 −0.718810
\(825\) 0 0
\(826\) 5.88185e8 0.0363148
\(827\) −5.95215e9 −0.365936 −0.182968 0.983119i \(-0.558570\pi\)
−0.182968 + 0.983119i \(0.558570\pi\)
\(828\) 0 0
\(829\) 1.00332e10 0.611643 0.305821 0.952089i \(-0.401069\pi\)
0.305821 + 0.952089i \(0.401069\pi\)
\(830\) 7.89913e9 0.479519
\(831\) 0 0
\(832\) −3.98744e10 −2.40028
\(833\) −8.35207e9 −0.500653
\(834\) 0 0
\(835\) −5.98788e9 −0.355935
\(836\) 1.43016e10 0.846570
\(837\) 0 0
\(838\) 5.51569e9 0.323777
\(839\) −7.75973e9 −0.453607 −0.226803 0.973941i \(-0.572827\pi\)
−0.226803 + 0.973941i \(0.572827\pi\)
\(840\) 0 0
\(841\) −6.72315e9 −0.389751
\(842\) 1.94677e10 1.12389
\(843\) 0 0
\(844\) 4.82992e9 0.276529
\(845\) 7.25123e9 0.413441
\(846\) 0 0
\(847\) 6.96624e8 0.0393919
\(848\) −2.31466e9 −0.130347
\(849\) 0 0
\(850\) −1.28642e10 −0.718483
\(851\) 4.98266e7 0.00277146
\(852\) 0 0
\(853\) 3.15852e10 1.74246 0.871229 0.490877i \(-0.163324\pi\)
0.871229 + 0.490877i \(0.163324\pi\)
\(854\) 6.53807e9 0.359209
\(855\) 0 0
\(856\) 9.34849e9 0.509429
\(857\) 1.81369e10 0.984304 0.492152 0.870509i \(-0.336210\pi\)
0.492152 + 0.870509i \(0.336210\pi\)
\(858\) 0 0
\(859\) 2.80979e10 1.51251 0.756253 0.654279i \(-0.227028\pi\)
0.756253 + 0.654279i \(0.227028\pi\)
\(860\) 1.43265e10 0.768062
\(861\) 0 0
\(862\) −3.01509e10 −1.60334
\(863\) 1.12434e10 0.595467 0.297734 0.954649i \(-0.403769\pi\)
0.297734 + 0.954649i \(0.403769\pi\)
\(864\) 0 0
\(865\) −6.32618e8 −0.0332342
\(866\) 4.52982e10 2.37011
\(867\) 0 0
\(868\) −8.95705e9 −0.464885
\(869\) 1.55693e10 0.804824
\(870\) 0 0
\(871\) −5.01386e10 −2.57104
\(872\) 1.78424e10 0.911265
\(873\) 0 0
\(874\) 2.36960e7 0.00120056
\(875\) −2.30634e9 −0.116384
\(876\) 0 0
\(877\) −1.39725e10 −0.699481 −0.349740 0.936847i \(-0.613730\pi\)
−0.349740 + 0.936847i \(0.613730\pi\)
\(878\) −2.46330e10 −1.22825
\(879\) 0 0
\(880\) −1.52602e9 −0.0754867
\(881\) 1.05370e10 0.519158 0.259579 0.965722i \(-0.416416\pi\)
0.259579 + 0.965722i \(0.416416\pi\)
\(882\) 0 0
\(883\) 2.48119e10 1.21282 0.606412 0.795151i \(-0.292608\pi\)
0.606412 + 0.795151i \(0.292608\pi\)
\(884\) 2.38750e10 1.16242
\(885\) 0 0
\(886\) 6.52854e8 0.0315354
\(887\) −3.02069e10 −1.45336 −0.726680 0.686976i \(-0.758938\pi\)
−0.726680 + 0.686976i \(0.758938\pi\)
\(888\) 0 0
\(889\) 2.38148e9 0.113682
\(890\) 1.90770e10 0.907080
\(891\) 0 0
\(892\) 8.90624e9 0.420162
\(893\) −1.43407e10 −0.673891
\(894\) 0 0
\(895\) 1.29711e10 0.604776
\(896\) −5.45981e9 −0.253571
\(897\) 0 0
\(898\) 3.62253e10 1.66934
\(899\) −2.95101e10 −1.35460
\(900\) 0 0
\(901\) 7.66853e9 0.349281
\(902\) −2.60218e10 −1.18063
\(903\) 0 0
\(904\) 1.85107e10 0.833362
\(905\) −3.36194e9 −0.150772
\(906\) 0 0
\(907\) 1.15712e10 0.514937 0.257468 0.966287i \(-0.417112\pi\)
0.257468 + 0.966287i \(0.417112\pi\)
\(908\) 5.01517e10 2.22323
\(909\) 0 0
\(910\) 3.30416e9 0.145350
\(911\) 1.60273e10 0.702338 0.351169 0.936312i \(-0.385784\pi\)
0.351169 + 0.936312i \(0.385784\pi\)
\(912\) 0 0
\(913\) 2.16866e10 0.943068
\(914\) −1.99509e9 −0.0864271
\(915\) 0 0
\(916\) −6.74291e10 −2.89877
\(917\) 4.49797e9 0.192630
\(918\) 0 0
\(919\) 9.52685e9 0.404897 0.202449 0.979293i \(-0.435110\pi\)
0.202449 + 0.979293i \(0.435110\pi\)
\(920\) 1.05878e7 0.000448280 0
\(921\) 0 0
\(922\) −6.46219e10 −2.71532
\(923\) 5.83109e10 2.44086
\(924\) 0 0
\(925\) −3.86914e10 −1.60738
\(926\) −4.29228e10 −1.77643
\(927\) 0 0
\(928\) −2.18025e10 −0.895547
\(929\) −7.62148e9 −0.311878 −0.155939 0.987767i \(-0.549840\pi\)
−0.155939 + 0.987767i \(0.549840\pi\)
\(930\) 0 0
\(931\) −1.19464e10 −0.485193
\(932\) 2.83181e10 1.14580
\(933\) 0 0
\(934\) −3.54467e10 −1.42351
\(935\) 5.05573e9 0.202275
\(936\) 0 0
\(937\) 2.55429e9 0.101434 0.0507168 0.998713i \(-0.483849\pi\)
0.0507168 + 0.998713i \(0.483849\pi\)
\(938\) −1.23102e10 −0.487028
\(939\) 0 0
\(940\) −1.85385e10 −0.727993
\(941\) 1.11902e10 0.437797 0.218898 0.975748i \(-0.429754\pi\)
0.218898 + 0.975748i \(0.429754\pi\)
\(942\) 0 0
\(943\) −2.60613e7 −0.00101206
\(944\) −6.48658e8 −0.0250965
\(945\) 0 0
\(946\) 6.50702e10 2.49898
\(947\) −2.91042e10 −1.11361 −0.556803 0.830645i \(-0.687972\pi\)
−0.556803 + 0.830645i \(0.687972\pi\)
\(948\) 0 0
\(949\) 4.28056e10 1.62581
\(950\) −1.84004e10 −0.696297
\(951\) 0 0
\(952\) 2.02611e9 0.0761087
\(953\) −1.18538e10 −0.443643 −0.221821 0.975087i \(-0.571200\pi\)
−0.221821 + 0.975087i \(0.571200\pi\)
\(954\) 0 0
\(955\) −1.50396e10 −0.558758
\(956\) 4.81089e10 1.78083
\(957\) 0 0
\(958\) −8.20804e10 −3.01620
\(959\) −7.12688e9 −0.260936
\(960\) 0 0
\(961\) 5.52147e10 2.00689
\(962\) 1.18797e11 4.30222
\(963\) 0 0
\(964\) 4.97629e10 1.78911
\(965\) 5.71088e8 0.0204577
\(966\) 0 0
\(967\) −3.91605e10 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(968\) 5.32215e9 0.188592
\(969\) 0 0
\(970\) −7.21666e9 −0.253884
\(971\) −4.49711e10 −1.57640 −0.788200 0.615419i \(-0.788987\pi\)
−0.788200 + 0.615419i \(0.788987\pi\)
\(972\) 0 0
\(973\) −6.74829e9 −0.234854
\(974\) 5.25209e10 1.82128
\(975\) 0 0
\(976\) −7.21027e9 −0.248243
\(977\) 3.26239e10 1.11919 0.559597 0.828765i \(-0.310956\pi\)
0.559597 + 0.828765i \(0.310956\pi\)
\(978\) 0 0
\(979\) 5.23748e10 1.78395
\(980\) −1.54434e10 −0.524146
\(981\) 0 0
\(982\) 1.46670e10 0.494254
\(983\) 4.19068e10 1.40717 0.703585 0.710611i \(-0.251581\pi\)
0.703585 + 0.710611i \(0.251581\pi\)
\(984\) 0 0
\(985\) 1.01385e10 0.338024
\(986\) 1.93127e10 0.641613
\(987\) 0 0
\(988\) 3.41498e10 1.12652
\(989\) 6.51692e7 0.00214218
\(990\) 0 0
\(991\) 1.39919e10 0.456687 0.228343 0.973581i \(-0.426669\pi\)
0.228343 + 0.973581i \(0.426669\pi\)
\(992\) 6.11201e10 1.98789
\(993\) 0 0
\(994\) 1.43167e10 0.462370
\(995\) 8.52617e9 0.274393
\(996\) 0 0
\(997\) 5.75080e10 1.83779 0.918893 0.394506i \(-0.129084\pi\)
0.918893 + 0.394506i \(0.129084\pi\)
\(998\) −5.83846e10 −1.85927
\(999\) 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.1.15 17
3.2 odd 2 177.8.a.b.1.3 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.3 17 3.2 odd 2
531.8.a.d.1.15 17 1.1 even 1 trivial