Properties

Label 177.8.a.b.1.3
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-15.9892\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-17.9892 q^{2} +27.0000 q^{3} +195.612 q^{4} -98.9095 q^{5} -485.709 q^{6} +159.201 q^{7} -1216.28 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-17.9892 q^{2} +27.0000 q^{3} +195.612 q^{4} -98.9095 q^{5} -485.709 q^{6} +159.201 q^{7} -1216.28 q^{8} +729.000 q^{9} +1779.30 q^{10} -4884.97 q^{11} +5281.52 q^{12} +11664.5 q^{13} -2863.90 q^{14} -2670.56 q^{15} -3158.35 q^{16} -10463.7 q^{17} -13114.1 q^{18} +14966.8 q^{19} -19347.9 q^{20} +4298.42 q^{21} +87876.7 q^{22} -88.0105 q^{23} -32839.6 q^{24} -68341.9 q^{25} -209835. q^{26} +19683.0 q^{27} +31141.6 q^{28} -102600. q^{29} +48041.2 q^{30} -287624. q^{31} +212500. q^{32} -131894. q^{33} +188233. q^{34} -15746.5 q^{35} +142601. q^{36} +566144. q^{37} -269240. q^{38} +314941. q^{39} +120302. q^{40} +296117. q^{41} -77325.3 q^{42} +740471. q^{43} -955557. q^{44} -72105.0 q^{45} +1583.24 q^{46} +958168. q^{47} -85275.4 q^{48} -798198. q^{49} +1.22942e6 q^{50} -282519. q^{51} +2.28171e6 q^{52} -732872. q^{53} -354082. q^{54} +483170. q^{55} -193633. q^{56} +404103. q^{57} +1.84569e6 q^{58} -205379. q^{59} -522392. q^{60} +2.28293e6 q^{61} +5.17412e6 q^{62} +116057. q^{63} -3.41844e6 q^{64} -1.15373e6 q^{65} +2.37267e6 q^{66} -4.29840e6 q^{67} -2.04681e6 q^{68} -2376.28 q^{69} +283267. q^{70} -4.99901e6 q^{71} -886670. q^{72} +3.66974e6 q^{73} -1.01845e7 q^{74} -1.84523e6 q^{75} +2.92768e6 q^{76} -777691. q^{77} -5.66554e6 q^{78} +3.18719e6 q^{79} +312391. q^{80} +531441. q^{81} -5.32690e6 q^{82} -4.43945e6 q^{83} +840822. q^{84} +1.03496e6 q^{85} -1.33205e7 q^{86} -2.77020e6 q^{87} +5.94150e6 q^{88} -1.07216e7 q^{89} +1.29711e6 q^{90} +1.85700e6 q^{91} -17215.9 q^{92} -7.76584e6 q^{93} -1.72367e7 q^{94} -1.48036e6 q^{95} +5.73751e6 q^{96} -4.05589e6 q^{97} +1.43590e7 q^{98} -3.56114e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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 \(-0.792539\pi\)
−0.795018 + 0.606585i \(0.792539\pi\)
\(3\) 27.0000 0.577350
\(4\) 195.612 1.52822
\(5\) −98.9095 −0.353869 −0.176935 0.984223i \(-0.556618\pi\)
−0.176935 + 0.984223i \(0.556618\pi\)
\(6\) −485.709 −0.918008
\(7\) 159.201 0.175429 0.0877147 0.996146i \(-0.472044\pi\)
0.0877147 + 0.996146i \(0.472044\pi\)
\(8\) −1216.28 −0.839884
\(9\) 729.000 0.333333
\(10\) 1779.30 0.562665
\(11\) −4884.97 −1.10659 −0.553296 0.832985i \(-0.686630\pi\)
−0.553296 + 0.832985i \(0.686630\pi\)
\(12\) 5281.52 0.882316
\(13\) 11664.5 1.47253 0.736265 0.676694i \(-0.236588\pi\)
0.736265 + 0.676694i \(0.236588\pi\)
\(14\) −2863.90 −0.278939
\(15\) −2670.56 −0.204307
\(16\) −3158.35 −0.192770
\(17\) −10463.7 −0.516550 −0.258275 0.966071i \(-0.583154\pi\)
−0.258275 + 0.966071i \(0.583154\pi\)
\(18\) −13114.1 −0.530012
\(19\) 14966.8 0.500599 0.250300 0.968168i \(-0.419471\pi\)
0.250300 + 0.968168i \(0.419471\pi\)
\(20\) −19347.9 −0.540789
\(21\) 4298.42 0.101284
\(22\) 87876.7 1.75952
\(23\) −88.0105 −0.00150830 −0.000754148 1.00000i \(-0.500240\pi\)
−0.000754148 1.00000i \(0.500240\pi\)
\(24\) −32839.6 −0.484907
\(25\) −68341.9 −0.874776
\(26\) −209835. −2.34138
\(27\) 19683.0 0.192450
\(28\) 31141.6 0.268094
\(29\) −102600. −0.781185 −0.390592 0.920564i \(-0.627730\pi\)
−0.390592 + 0.920564i \(0.627730\pi\)
\(30\) 48041.2 0.324855
\(31\) −287624. −1.73404 −0.867019 0.498275i \(-0.833967\pi\)
−0.867019 + 0.498275i \(0.833967\pi\)
\(32\) 212500. 1.14640
\(33\) −131894. −0.638891
\(34\) 188233. 0.821333
\(35\) −15746.5 −0.0620791
\(36\) 142601. 0.509406
\(37\) 566144. 1.83747 0.918737 0.394870i \(-0.129210\pi\)
0.918737 + 0.394870i \(0.129210\pi\)
\(38\) −269240. −0.795971
\(39\) 314941. 0.850165
\(40\) 120302. 0.297209
\(41\) 296117. 0.670995 0.335497 0.942041i \(-0.391096\pi\)
0.335497 + 0.942041i \(0.391096\pi\)
\(42\) −77325.3 −0.161046
\(43\) 740471. 1.42026 0.710131 0.704070i \(-0.248636\pi\)
0.710131 + 0.704070i \(0.248636\pi\)
\(44\) −955557. −1.69111
\(45\) −72105.0 −0.117956
\(46\) 1583.24 0.00239825
\(47\) 958168. 1.34617 0.673084 0.739566i \(-0.264969\pi\)
0.673084 + 0.739566i \(0.264969\pi\)
\(48\) −85275.4 −0.111296
\(49\) −798198. −0.969225
\(50\) 1.22942e6 1.39093
\(51\) −282519. −0.298230
\(52\) 2.28171e6 2.25034
\(53\) −732872. −0.676181 −0.338090 0.941114i \(-0.609781\pi\)
−0.338090 + 0.941114i \(0.609781\pi\)
\(54\) −354082. −0.306003
\(55\) 483170. 0.391589
\(56\) −193633. −0.147340
\(57\) 404103. 0.289021
\(58\) 1.84569e6 1.24211
\(59\) −205379. −0.130189
\(60\) −522392. −0.312225
\(61\) 2.28293e6 1.28777 0.643884 0.765123i \(-0.277322\pi\)
0.643884 + 0.765123i \(0.277322\pi\)
\(62\) 5.17412e6 2.75718
\(63\) 116057. 0.0584765
\(64\) −3.41844e6 −1.63004
\(65\) −1.15373e6 −0.521083
\(66\) 2.37267e6 1.01586
\(67\) −4.29840e6 −1.74600 −0.873001 0.487719i \(-0.837829\pi\)
−0.873001 + 0.487719i \(0.837829\pi\)
\(68\) −2.04681e6 −0.789400
\(69\) −2376.28 −0.000870816 0
\(70\) 283267. 0.0987080
\(71\) −4.99901e6 −1.65760 −0.828800 0.559545i \(-0.810976\pi\)
−0.828800 + 0.559545i \(0.810976\pi\)
\(72\) −886670. −0.279961
\(73\) 3.66974e6 1.10409 0.552046 0.833814i \(-0.313847\pi\)
0.552046 + 0.833814i \(0.313847\pi\)
\(74\) −1.01845e7 −2.92165
\(75\) −1.84523e6 −0.505052
\(76\) 2.92768e6 0.765024
\(77\) −777691. −0.194129
\(78\) −5.66554e6 −1.35179
\(79\) 3.18719e6 0.727300 0.363650 0.931536i \(-0.381530\pi\)
0.363650 + 0.931536i \(0.381530\pi\)
\(80\) 312391. 0.0682155
\(81\) 531441. 0.111111
\(82\) −5.32690e6 −1.06691
\(83\) −4.43945e6 −0.852228 −0.426114 0.904670i \(-0.640118\pi\)
−0.426114 + 0.904670i \(0.640118\pi\)
\(84\) 840822. 0.154784
\(85\) 1.03496e6 0.182791
\(86\) −1.33205e7 −2.25827
\(87\) −2.77020e6 −0.451017
\(88\) 5.94150e6 0.929409
\(89\) −1.07216e7 −1.61211 −0.806056 0.591839i \(-0.798402\pi\)
−0.806056 + 0.591839i \(0.798402\pi\)
\(90\) 1.29711e6 0.187555
\(91\) 1.85700e6 0.258325
\(92\) −17215.9 −0.00230500
\(93\) −7.76584e6 −1.00115
\(94\) −1.72367e7 −2.14046
\(95\) −1.48036e6 −0.177147
\(96\) 5.73751e6 0.661872
\(97\) −4.05589e6 −0.451216 −0.225608 0.974218i \(-0.572437\pi\)
−0.225608 + 0.974218i \(0.572437\pi\)
\(98\) 1.43590e7 1.54110
\(99\) −3.56114e6 −0.368864
\(100\) −1.33685e7 −1.33685
\(101\) −1.05360e7 −1.01754 −0.508768 0.860904i \(-0.669899\pi\)
−0.508768 + 0.860904i \(0.669899\pi\)
\(102\) 5.08229e6 0.474197
\(103\) −9.49129e6 −0.855845 −0.427922 0.903816i \(-0.640754\pi\)
−0.427922 + 0.903816i \(0.640754\pi\)
\(104\) −1.41873e7 −1.23675
\(105\) −425155. −0.0358414
\(106\) 1.31838e7 1.07515
\(107\) −7.68612e6 −0.606547 −0.303273 0.952904i \(-0.598079\pi\)
−0.303273 + 0.952904i \(0.598079\pi\)
\(108\) 3.85023e6 0.294105
\(109\) 1.46696e7 1.08499 0.542494 0.840059i \(-0.317480\pi\)
0.542494 + 0.840059i \(0.317480\pi\)
\(110\) −8.69185e6 −0.622641
\(111\) 1.52859e7 1.06087
\(112\) −502812. −0.0338176
\(113\) −1.52191e7 −0.992234 −0.496117 0.868256i \(-0.665241\pi\)
−0.496117 + 0.868256i \(0.665241\pi\)
\(114\) −7.26949e6 −0.459554
\(115\) 8705.07 0.000533740 0
\(116\) −2.00697e7 −1.19382
\(117\) 8.50341e6 0.490843
\(118\) 3.69461e6 0.207005
\(119\) −1.66582e6 −0.0906180
\(120\) 3.24815e6 0.171594
\(121\) 4.37575e6 0.224545
\(122\) −4.10680e7 −2.04760
\(123\) 7.99515e6 0.387399
\(124\) −5.62626e7 −2.64999
\(125\) 1.44870e7 0.663426
\(126\) −2.08778e6 −0.0929797
\(127\) 1.49590e7 0.648021 0.324011 0.946053i \(-0.394969\pi\)
0.324011 + 0.946053i \(0.394969\pi\)
\(128\) 3.42951e7 1.44543
\(129\) 1.99927e7 0.819988
\(130\) 2.07547e7 0.828541
\(131\) −2.82534e7 −1.09805 −0.549024 0.835807i \(-0.685000\pi\)
−0.549024 + 0.835807i \(0.685000\pi\)
\(132\) −2.58000e7 −0.976364
\(133\) 2.38272e6 0.0878198
\(134\) 7.73248e7 2.77621
\(135\) −1.94684e6 −0.0681022
\(136\) 1.27268e7 0.433842
\(137\) 4.47666e7 1.48741 0.743707 0.668506i \(-0.233066\pi\)
0.743707 + 0.668506i \(0.233066\pi\)
\(138\) 42747.4 0.00138463
\(139\) −4.23885e7 −1.33874 −0.669370 0.742929i \(-0.733436\pi\)
−0.669370 + 0.742929i \(0.733436\pi\)
\(140\) −3.08020e6 −0.0948703
\(141\) 2.58705e7 0.777210
\(142\) 8.99283e7 2.63565
\(143\) −5.69807e7 −1.62949
\(144\) −2.30244e6 −0.0642567
\(145\) 1.01481e7 0.276437
\(146\) −6.60157e7 −1.75555
\(147\) −2.15513e7 −0.559582
\(148\) 1.10744e8 2.80806
\(149\) −3.46535e7 −0.858214 −0.429107 0.903254i \(-0.641172\pi\)
−0.429107 + 0.903254i \(0.641172\pi\)
\(150\) 3.31943e7 0.803052
\(151\) −7.67555e7 −1.81422 −0.907110 0.420893i \(-0.861717\pi\)
−0.907110 + 0.420893i \(0.861717\pi\)
\(152\) −1.82038e7 −0.420445
\(153\) −7.62801e6 −0.172183
\(154\) 1.39901e7 0.308672
\(155\) 2.84487e7 0.613623
\(156\) 6.16062e7 1.29924
\(157\) −6.08764e7 −1.25545 −0.627726 0.778434i \(-0.716014\pi\)
−0.627726 + 0.778434i \(0.716014\pi\)
\(158\) −5.73351e7 −1.15643
\(159\) −1.97875e7 −0.390393
\(160\) −2.10183e7 −0.405675
\(161\) −14011.3 −0.000264600 0
\(162\) −9.56020e6 −0.176671
\(163\) −9.26090e7 −1.67493 −0.837465 0.546491i \(-0.815963\pi\)
−0.837465 + 0.546491i \(0.815963\pi\)
\(164\) 5.79239e7 1.02543
\(165\) 1.30456e7 0.226084
\(166\) 7.98622e7 1.35507
\(167\) 6.05390e7 1.00584 0.502918 0.864334i \(-0.332260\pi\)
0.502918 + 0.864334i \(0.332260\pi\)
\(168\) −5.22810e6 −0.0850670
\(169\) 7.33117e7 1.16834
\(170\) −1.86180e7 −0.290645
\(171\) 1.09108e7 0.166866
\(172\) 1.44845e8 2.17047
\(173\) 6.39593e6 0.0939166 0.0469583 0.998897i \(-0.485047\pi\)
0.0469583 + 0.998897i \(0.485047\pi\)
\(174\) 4.98336e7 0.717134
\(175\) −1.08801e7 −0.153461
\(176\) 1.54284e7 0.213318
\(177\) −5.54523e6 −0.0751646
\(178\) 1.92873e8 2.56332
\(179\) −1.31141e8 −1.70904 −0.854519 0.519421i \(-0.826148\pi\)
−0.854519 + 0.519421i \(0.826148\pi\)
\(180\) −1.41046e7 −0.180263
\(181\) −3.39901e7 −0.426066 −0.213033 0.977045i \(-0.568334\pi\)
−0.213033 + 0.977045i \(0.568334\pi\)
\(182\) −3.34059e7 −0.410746
\(183\) 6.16390e7 0.743493
\(184\) 107046. 0.00126679
\(185\) −5.59971e7 −0.650226
\(186\) 1.39701e8 1.59186
\(187\) 5.11147e7 0.571610
\(188\) 1.87429e8 2.05724
\(189\) 3.13355e6 0.0337614
\(190\) 2.66304e7 0.281670
\(191\) 1.52054e8 1.57899 0.789497 0.613755i \(-0.210342\pi\)
0.789497 + 0.613755i \(0.210342\pi\)
\(192\) −9.22980e7 −0.941105
\(193\) 5.77384e6 0.0578115 0.0289057 0.999582i \(-0.490798\pi\)
0.0289057 + 0.999582i \(0.490798\pi\)
\(194\) 7.29622e7 0.717450
\(195\) −3.11507e7 −0.300847
\(196\) −1.56137e8 −1.48119
\(197\) −1.02503e8 −0.955221 −0.477611 0.878572i \(-0.658497\pi\)
−0.477611 + 0.878572i \(0.658497\pi\)
\(198\) 6.40622e7 0.586507
\(199\) 8.62017e7 0.775408 0.387704 0.921784i \(-0.373268\pi\)
0.387704 + 0.921784i \(0.373268\pi\)
\(200\) 8.31230e7 0.734711
\(201\) −1.16057e8 −1.00805
\(202\) 1.89534e8 1.61792
\(203\) −1.63340e7 −0.137043
\(204\) −5.52640e7 −0.455761
\(205\) −2.92887e7 −0.237445
\(206\) 1.70741e8 1.36082
\(207\) −64159.6 −0.000502766 0
\(208\) −3.68405e7 −0.283860
\(209\) −7.31122e7 −0.553959
\(210\) 7.64820e6 0.0569891
\(211\) 2.46914e7 0.180949 0.0904746 0.995899i \(-0.471162\pi\)
0.0904746 + 0.995899i \(0.471162\pi\)
\(212\) −1.43358e8 −1.03335
\(213\) −1.34973e8 −0.957016
\(214\) 1.38267e8 0.964431
\(215\) −7.32396e7 −0.502587
\(216\) −2.39401e7 −0.161636
\(217\) −4.57899e7 −0.304201
\(218\) −2.63894e8 −1.72517
\(219\) 9.90830e7 0.637448
\(220\) 9.45137e7 0.598433
\(221\) −1.22053e8 −0.760635
\(222\) −2.74981e8 −1.68682
\(223\) 4.55302e7 0.274936 0.137468 0.990506i \(-0.456104\pi\)
0.137468 + 0.990506i \(0.456104\pi\)
\(224\) 3.38302e7 0.201111
\(225\) −4.98212e7 −0.291592
\(226\) 2.73780e8 1.57769
\(227\) −2.56384e8 −1.45479 −0.727395 0.686219i \(-0.759269\pi\)
−0.727395 + 0.686219i \(0.759269\pi\)
\(228\) 7.90472e7 0.441687
\(229\) −3.44709e8 −1.89683 −0.948415 0.317033i \(-0.897314\pi\)
−0.948415 + 0.317033i \(0.897314\pi\)
\(230\) −156597. −0.000848667 0
\(231\) −2.09977e7 −0.112080
\(232\) 1.24790e8 0.656104
\(233\) −1.44767e8 −0.749762 −0.374881 0.927073i \(-0.622316\pi\)
−0.374881 + 0.927073i \(0.622316\pi\)
\(234\) −1.52970e8 −0.780458
\(235\) −9.47720e7 −0.476368
\(236\) −4.01745e7 −0.198957
\(237\) 8.60542e7 0.419907
\(238\) 2.99669e7 0.144086
\(239\) −2.45941e8 −1.16530 −0.582650 0.812723i \(-0.697984\pi\)
−0.582650 + 0.812723i \(0.697984\pi\)
\(240\) 8.43455e6 0.0393842
\(241\) 2.54396e8 1.17071 0.585357 0.810775i \(-0.300954\pi\)
0.585357 + 0.810775i \(0.300954\pi\)
\(242\) −7.87164e7 −0.357035
\(243\) 1.43489e7 0.0641500
\(244\) 4.46567e8 1.96799
\(245\) 7.89494e7 0.342979
\(246\) −1.43826e8 −0.615979
\(247\) 1.74580e8 0.737147
\(248\) 3.49831e8 1.45639
\(249\) −1.19865e8 −0.492034
\(250\) −2.60609e8 −1.05487
\(251\) 3.25172e8 1.29794 0.648971 0.760813i \(-0.275200\pi\)
0.648971 + 0.760813i \(0.275200\pi\)
\(252\) 2.27022e7 0.0893647
\(253\) 429928. 0.00166907
\(254\) −2.69100e8 −1.03038
\(255\) 2.79438e7 0.105535
\(256\) −1.79381e8 −0.668245
\(257\) 2.54981e8 0.937003 0.468502 0.883463i \(-0.344794\pi\)
0.468502 + 0.883463i \(0.344794\pi\)
\(258\) −3.59653e8 −1.30381
\(259\) 9.01307e7 0.322347
\(260\) −2.25683e8 −0.796328
\(261\) −7.47953e7 −0.260395
\(262\) 5.08256e8 1.74594
\(263\) 4.19400e8 1.42162 0.710809 0.703385i \(-0.248329\pi\)
0.710809 + 0.703385i \(0.248329\pi\)
\(264\) 1.60421e8 0.536594
\(265\) 7.24880e7 0.239280
\(266\) −4.28633e7 −0.139637
\(267\) −2.89484e8 −0.930754
\(268\) −8.40817e8 −2.66827
\(269\) −4.15320e8 −1.30092 −0.650459 0.759541i \(-0.725424\pi\)
−0.650459 + 0.759541i \(0.725424\pi\)
\(270\) 3.50220e7 0.108285
\(271\) −5.50963e7 −0.168163 −0.0840814 0.996459i \(-0.526796\pi\)
−0.0840814 + 0.996459i \(0.526796\pi\)
\(272\) 3.30479e7 0.0995755
\(273\) 5.01389e7 0.149144
\(274\) −8.05315e8 −2.36504
\(275\) 3.33848e8 0.968020
\(276\) −464829. −0.00133080
\(277\) −2.16584e8 −0.612275 −0.306138 0.951987i \(-0.599037\pi\)
−0.306138 + 0.951987i \(0.599037\pi\)
\(278\) 7.62536e8 2.12865
\(279\) −2.09678e8 −0.578013
\(280\) 1.91522e7 0.0521392
\(281\) −1.30596e8 −0.351123 −0.175561 0.984468i \(-0.556174\pi\)
−0.175561 + 0.984468i \(0.556174\pi\)
\(282\) −4.65391e8 −1.23579
\(283\) 4.03750e8 1.05891 0.529457 0.848337i \(-0.322396\pi\)
0.529457 + 0.848337i \(0.322396\pi\)
\(284\) −9.77865e8 −2.53317
\(285\) −3.99696e7 −0.102276
\(286\) 1.02504e9 2.59095
\(287\) 4.71420e7 0.117712
\(288\) 1.54913e8 0.382132
\(289\) −3.00851e8 −0.733176
\(290\) −1.82556e8 −0.439546
\(291\) −1.09509e8 −0.260510
\(292\) 7.17844e8 1.68729
\(293\) 1.61200e8 0.374394 0.187197 0.982322i \(-0.440060\pi\)
0.187197 + 0.982322i \(0.440060\pi\)
\(294\) 3.87692e8 0.889756
\(295\) 2.03139e7 0.0460699
\(296\) −6.88591e8 −1.54327
\(297\) −9.61509e7 −0.212964
\(298\) 6.23390e8 1.36459
\(299\) −1.02660e6 −0.00222101
\(300\) −3.60949e8 −0.771830
\(301\) 1.17884e8 0.249156
\(302\) 1.38077e9 2.88468
\(303\) −2.84471e8 −0.587474
\(304\) −4.72702e7 −0.0965007
\(305\) −2.25803e8 −0.455701
\(306\) 1.37222e8 0.273778
\(307\) −7.32355e8 −1.44457 −0.722283 0.691597i \(-0.756907\pi\)
−0.722283 + 0.691597i \(0.756907\pi\)
\(308\) −1.52126e8 −0.296671
\(309\) −2.56265e8 −0.494122
\(310\) −5.11770e8 −0.975683
\(311\) 1.98727e8 0.374624 0.187312 0.982300i \(-0.440022\pi\)
0.187312 + 0.982300i \(0.440022\pi\)
\(312\) −3.83057e8 −0.714040
\(313\) 2.25644e8 0.415929 0.207964 0.978136i \(-0.433316\pi\)
0.207964 + 0.978136i \(0.433316\pi\)
\(314\) 1.09512e9 1.99622
\(315\) −1.14792e7 −0.0206930
\(316\) 6.23453e8 1.11147
\(317\) 5.60321e8 0.987937 0.493968 0.869480i \(-0.335546\pi\)
0.493968 + 0.869480i \(0.335546\pi\)
\(318\) 3.55962e8 0.620739
\(319\) 5.01197e8 0.864452
\(320\) 3.38117e8 0.576822
\(321\) −2.07525e8 −0.350190
\(322\) 252053. 0.000420723 0
\(323\) −1.56607e8 −0.258585
\(324\) 1.03956e8 0.169802
\(325\) −7.97173e8 −1.28813
\(326\) 1.66596e9 2.66320
\(327\) 3.96079e8 0.626419
\(328\) −3.60161e8 −0.563558
\(329\) 1.52541e8 0.236157
\(330\) −2.34680e8 −0.359482
\(331\) 4.88144e7 0.0739861 0.0369930 0.999316i \(-0.488222\pi\)
0.0369930 + 0.999316i \(0.488222\pi\)
\(332\) −8.68408e8 −1.30239
\(333\) 4.12719e8 0.612491
\(334\) −1.08905e9 −1.59932
\(335\) 4.25152e8 0.617857
\(336\) −1.35759e7 −0.0195246
\(337\) −4.98448e8 −0.709439 −0.354720 0.934973i \(-0.615424\pi\)
−0.354720 + 0.934973i \(0.615424\pi\)
\(338\) −1.31882e9 −1.85771
\(339\) −4.10916e8 −0.572867
\(340\) 2.02449e8 0.279345
\(341\) 1.40503e9 1.91887
\(342\) −1.96276e8 −0.265324
\(343\) −2.58183e8 −0.345460
\(344\) −9.00621e8 −1.19286
\(345\) 235037. 0.000308155 0
\(346\) −1.15058e8 −0.149331
\(347\) 4.64305e8 0.596554 0.298277 0.954479i \(-0.403588\pi\)
0.298277 + 0.954479i \(0.403588\pi\)
\(348\) −5.41883e8 −0.689252
\(349\) −3.75860e8 −0.473301 −0.236651 0.971595i \(-0.576050\pi\)
−0.236651 + 0.971595i \(0.576050\pi\)
\(350\) 1.95724e8 0.244009
\(351\) 2.29592e8 0.283388
\(352\) −1.03806e9 −1.26859
\(353\) −6.79230e8 −0.821874 −0.410937 0.911664i \(-0.634798\pi\)
−0.410937 + 0.911664i \(0.634798\pi\)
\(354\) 9.97544e7 0.119514
\(355\) 4.94450e8 0.586574
\(356\) −2.09727e9 −2.46366
\(357\) −4.49772e7 −0.0523184
\(358\) 2.35912e9 2.71743
\(359\) 4.73258e8 0.539843 0.269922 0.962882i \(-0.413002\pi\)
0.269922 + 0.962882i \(0.413002\pi\)
\(360\) 8.77001e7 0.0990698
\(361\) −6.69868e8 −0.749400
\(362\) 6.11455e8 0.677461
\(363\) 1.18145e8 0.129641
\(364\) 3.63250e8 0.394776
\(365\) −3.62972e8 −0.390704
\(366\) −1.10884e9 −1.18218
\(367\) 1.29401e9 1.36649 0.683245 0.730189i \(-0.260568\pi\)
0.683245 + 0.730189i \(0.260568\pi\)
\(368\) 277968. 0.000290755 0
\(369\) 2.15869e8 0.223665
\(370\) 1.00734e9 1.03388
\(371\) −1.16674e8 −0.118622
\(372\) −1.51909e9 −1.52997
\(373\) −6.91706e8 −0.690146 −0.345073 0.938576i \(-0.612146\pi\)
−0.345073 + 0.938576i \(0.612146\pi\)
\(374\) −9.19512e8 −0.908881
\(375\) 3.91148e8 0.383029
\(376\) −1.16540e9 −1.13062
\(377\) −1.19677e9 −1.15032
\(378\) −5.63701e7 −0.0536819
\(379\) −9.69203e8 −0.914487 −0.457243 0.889342i \(-0.651163\pi\)
−0.457243 + 0.889342i \(0.651163\pi\)
\(380\) −2.89575e8 −0.270719
\(381\) 4.03893e8 0.374135
\(382\) −2.73533e9 −2.51066
\(383\) −1.28084e9 −1.16493 −0.582465 0.812856i \(-0.697912\pi\)
−0.582465 + 0.812856i \(0.697912\pi\)
\(384\) 9.25967e8 0.834519
\(385\) 7.69211e7 0.0686962
\(386\) −1.03867e8 −0.0919224
\(387\) 5.39803e8 0.473421
\(388\) −7.93379e8 −0.689556
\(389\) 1.19302e9 1.02760 0.513802 0.857909i \(-0.328237\pi\)
0.513802 + 0.857909i \(0.328237\pi\)
\(390\) 5.60376e8 0.478359
\(391\) 920911. 0.000779111 0
\(392\) 9.70834e8 0.814036
\(393\) −7.62842e8 −0.633958
\(394\) 1.84394e9 1.51884
\(395\) −3.15244e8 −0.257369
\(396\) −6.96601e8 −0.563704
\(397\) −4.46557e7 −0.0358187 −0.0179094 0.999840i \(-0.505701\pi\)
−0.0179094 + 0.999840i \(0.505701\pi\)
\(398\) −1.55070e9 −1.23293
\(399\) 6.43335e7 0.0507028
\(400\) 2.15847e8 0.168631
\(401\) 5.52185e8 0.427641 0.213821 0.976873i \(-0.431409\pi\)
0.213821 + 0.976873i \(0.431409\pi\)
\(402\) 2.08777e9 1.60284
\(403\) −3.35498e9 −2.55342
\(404\) −2.06096e9 −1.55501
\(405\) −5.25646e7 −0.0393188
\(406\) 2.93835e8 0.217903
\(407\) −2.76560e9 −2.03333
\(408\) 3.43623e8 0.250479
\(409\) 8.01492e8 0.579252 0.289626 0.957140i \(-0.406469\pi\)
0.289626 + 0.957140i \(0.406469\pi\)
\(410\) 5.26881e8 0.377546
\(411\) 1.20870e9 0.858759
\(412\) −1.85661e9 −1.30792
\(413\) −3.26965e7 −0.0228390
\(414\) 1.15418e6 0.000799416 0
\(415\) 4.39104e8 0.301577
\(416\) 2.47871e9 1.68810
\(417\) −1.14449e9 −0.772922
\(418\) 1.31523e9 0.880815
\(419\) −3.06611e8 −0.203629 −0.101814 0.994803i \(-0.532465\pi\)
−0.101814 + 0.994803i \(0.532465\pi\)
\(420\) −8.31653e7 −0.0547734
\(421\) 1.08219e9 0.706832 0.353416 0.935466i \(-0.385020\pi\)
0.353416 + 0.935466i \(0.385020\pi\)
\(422\) −4.44178e8 −0.287716
\(423\) 6.98505e8 0.448723
\(424\) 8.91379e8 0.567913
\(425\) 7.15106e8 0.451866
\(426\) 2.42806e9 1.52169
\(427\) 3.63444e8 0.225912
\(428\) −1.50350e9 −0.926935
\(429\) −1.53848e9 −0.940786
\(430\) 1.31752e9 0.799132
\(431\) 1.67605e9 1.00836 0.504182 0.863597i \(-0.331794\pi\)
0.504182 + 0.863597i \(0.331794\pi\)
\(432\) −6.21658e7 −0.0370986
\(433\) 2.51807e9 1.49060 0.745300 0.666730i \(-0.232306\pi\)
0.745300 + 0.666730i \(0.232306\pi\)
\(434\) 8.23725e8 0.483691
\(435\) 2.73999e8 0.159601
\(436\) 2.86954e9 1.65810
\(437\) −1.31723e6 −0.000755053 0
\(438\) −1.78242e9 −1.01357
\(439\) −1.36932e9 −0.772465 −0.386232 0.922402i \(-0.626224\pi\)
−0.386232 + 0.922402i \(0.626224\pi\)
\(440\) −5.87671e8 −0.328889
\(441\) −5.81886e8 −0.323075
\(442\) 2.19564e9 1.20944
\(443\) −3.62914e7 −0.0198331 −0.00991656 0.999951i \(-0.503157\pi\)
−0.00991656 + 0.999951i \(0.503157\pi\)
\(444\) 2.99010e9 1.62123
\(445\) 1.06047e9 0.570477
\(446\) −8.19052e8 −0.437159
\(447\) −9.35646e8 −0.495490
\(448\) −5.44219e8 −0.285957
\(449\) −2.01372e9 −1.04987 −0.524937 0.851141i \(-0.675911\pi\)
−0.524937 + 0.851141i \(0.675911\pi\)
\(450\) 8.96245e8 0.463642
\(451\) −1.44652e9 −0.742517
\(452\) −2.97703e9 −1.51635
\(453\) −2.07240e9 −1.04744
\(454\) 4.61214e9 2.31317
\(455\) −1.83675e8 −0.0914133
\(456\) −4.91503e8 −0.242744
\(457\) −1.10905e8 −0.0543554 −0.0271777 0.999631i \(-0.508652\pi\)
−0.0271777 + 0.999631i \(0.508652\pi\)
\(458\) 6.20104e9 3.01603
\(459\) −2.05956e8 −0.0994101
\(460\) 1.70281e6 0.000815671 0
\(461\) 3.59226e9 1.70771 0.853856 0.520510i \(-0.174258\pi\)
0.853856 + 0.520510i \(0.174258\pi\)
\(462\) 3.77731e8 0.178212
\(463\) −2.38603e9 −1.11723 −0.558614 0.829427i \(-0.688667\pi\)
−0.558614 + 0.829427i \(0.688667\pi\)
\(464\) 3.24046e8 0.150589
\(465\) 7.68115e8 0.354275
\(466\) 2.60424e9 1.19215
\(467\) 1.97044e9 0.895271 0.447636 0.894216i \(-0.352266\pi\)
0.447636 + 0.894216i \(0.352266\pi\)
\(468\) 1.66337e9 0.750115
\(469\) −6.84308e8 −0.306300
\(470\) 1.70487e9 0.757442
\(471\) −1.64366e9 −0.724836
\(472\) 2.49799e8 0.109344
\(473\) −3.61718e9 −1.57165
\(474\) −1.54805e9 −0.667667
\(475\) −1.02286e9 −0.437913
\(476\) −3.25855e8 −0.138484
\(477\) −5.34264e8 −0.225394
\(478\) 4.42428e9 1.85287
\(479\) 4.56276e9 1.89694 0.948469 0.316869i \(-0.102632\pi\)
0.948469 + 0.316869i \(0.102632\pi\)
\(480\) −5.67494e8 −0.234216
\(481\) 6.60378e9 2.70573
\(482\) −4.57639e9 −1.86148
\(483\) −378306. −0.000152767 0
\(484\) 8.55949e8 0.343154
\(485\) 4.01166e8 0.159672
\(486\) −2.58126e8 −0.102001
\(487\) 2.91958e9 1.14543 0.572715 0.819754i \(-0.305890\pi\)
0.572715 + 0.819754i \(0.305890\pi\)
\(488\) −2.77668e9 −1.08158
\(489\) −2.50044e9 −0.967021
\(490\) −1.42024e9 −0.545349
\(491\) −8.15321e8 −0.310845 −0.155422 0.987848i \(-0.549674\pi\)
−0.155422 + 0.987848i \(0.549674\pi\)
\(492\) 1.56394e9 0.592030
\(493\) 1.07357e9 0.403521
\(494\) −3.14055e9 −1.17209
\(495\) 3.52231e8 0.130530
\(496\) 9.08415e8 0.334271
\(497\) −7.95847e8 −0.290792
\(498\) 2.15628e9 0.782352
\(499\) −3.24554e9 −1.16932 −0.584662 0.811277i \(-0.698773\pi\)
−0.584662 + 0.811277i \(0.698773\pi\)
\(500\) 2.83382e9 1.01386
\(501\) 1.63455e9 0.580720
\(502\) −5.84959e9 −2.06378
\(503\) −4.45372e9 −1.56040 −0.780198 0.625532i \(-0.784882\pi\)
−0.780198 + 0.625532i \(0.784882\pi\)
\(504\) −1.41159e8 −0.0491134
\(505\) 1.04211e9 0.360075
\(506\) −7.73407e6 −0.00265388
\(507\) 1.97942e9 0.674542
\(508\) 2.92615e9 0.990317
\(509\) 4.82014e9 1.62012 0.810060 0.586347i \(-0.199435\pi\)
0.810060 + 0.586347i \(0.199435\pi\)
\(510\) −5.02687e8 −0.167804
\(511\) 5.84226e8 0.193690
\(512\) −1.16285e9 −0.382896
\(513\) 2.94591e8 0.0963404
\(514\) −4.58690e9 −1.48987
\(515\) 9.38779e8 0.302857
\(516\) 3.91081e9 1.25312
\(517\) −4.68062e9 −1.48966
\(518\) −1.62138e9 −0.512543
\(519\) 1.72690e8 0.0542228
\(520\) 1.40326e9 0.437649
\(521\) −3.16646e9 −0.980939 −0.490469 0.871458i \(-0.663175\pi\)
−0.490469 + 0.871458i \(0.663175\pi\)
\(522\) 1.34551e9 0.414037
\(523\) −1.33084e9 −0.406790 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(524\) −5.52670e9 −1.67806
\(525\) −2.93762e8 −0.0886010
\(526\) −7.54467e9 −2.26042
\(527\) 3.00960e9 0.895717
\(528\) 4.16568e8 0.123159
\(529\) −3.40482e9 −0.999998
\(530\) −1.30400e9 −0.380463
\(531\) −1.49721e8 −0.0433963
\(532\) 4.66088e8 0.134208
\(533\) 3.45405e9 0.988059
\(534\) 5.20758e9 1.47993
\(535\) 7.60231e8 0.214638
\(536\) 5.22806e9 1.46644
\(537\) −3.54080e9 −0.986713
\(538\) 7.47128e9 2.06851
\(539\) 3.89917e9 1.07254
\(540\) −3.80824e8 −0.104075
\(541\) 2.45311e8 0.0666081 0.0333040 0.999445i \(-0.489397\pi\)
0.0333040 + 0.999445i \(0.489397\pi\)
\(542\) 9.91139e8 0.267385
\(543\) −9.17732e8 −0.245990
\(544\) −2.22353e9 −0.592171
\(545\) −1.45096e9 −0.383944
\(546\) −9.01959e8 −0.237144
\(547\) 1.35454e9 0.353865 0.176933 0.984223i \(-0.443383\pi\)
0.176933 + 0.984223i \(0.443383\pi\)
\(548\) 8.75687e9 2.27309
\(549\) 1.66425e9 0.429256
\(550\) −6.00566e9 −1.53919
\(551\) −1.53559e9 −0.391061
\(552\) 2.89023e6 0.000731384 0
\(553\) 5.07404e8 0.127590
\(554\) 3.89617e9 0.973540
\(555\) −1.51192e9 −0.375408
\(556\) −8.29169e9 −2.04589
\(557\) −2.79389e9 −0.685041 −0.342520 0.939510i \(-0.611281\pi\)
−0.342520 + 0.939510i \(0.611281\pi\)
\(558\) 3.77194e9 0.919061
\(559\) 8.63721e9 2.09138
\(560\) 4.97329e7 0.0119670
\(561\) 1.38010e9 0.330019
\(562\) 2.34933e9 0.558298
\(563\) 1.29186e9 0.305095 0.152548 0.988296i \(-0.451252\pi\)
0.152548 + 0.988296i \(0.451252\pi\)
\(564\) 5.06058e9 1.18775
\(565\) 1.50531e9 0.351121
\(566\) −7.26315e9 −1.68371
\(567\) 8.46059e7 0.0194922
\(568\) 6.08021e9 1.39219
\(569\) 2.83891e9 0.646040 0.323020 0.946392i \(-0.395302\pi\)
0.323020 + 0.946392i \(0.395302\pi\)
\(570\) 7.19022e8 0.162622
\(571\) −2.52009e9 −0.566487 −0.283243 0.959048i \(-0.591410\pi\)
−0.283243 + 0.959048i \(0.591410\pi\)
\(572\) −1.11461e10 −2.49021
\(573\) 4.10545e9 0.911632
\(574\) −8.48048e8 −0.187167
\(575\) 6.01480e6 0.00131942
\(576\) −2.49205e9 −0.543347
\(577\) −7.41099e9 −1.60606 −0.803028 0.595941i \(-0.796779\pi\)
−0.803028 + 0.595941i \(0.796779\pi\)
\(578\) 5.41206e9 1.16578
\(579\) 1.55894e8 0.0333775
\(580\) 1.98509e9 0.422456
\(581\) −7.06764e8 −0.149506
\(582\) 1.96998e9 0.414220
\(583\) 3.58006e9 0.748256
\(584\) −4.46344e9 −0.927309
\(585\) −8.41068e8 −0.173694
\(586\) −2.89987e9 −0.595301
\(587\) −2.54336e9 −0.519008 −0.259504 0.965742i \(-0.583559\pi\)
−0.259504 + 0.965742i \(0.583559\pi\)
\(588\) −4.21570e9 −0.855163
\(589\) −4.30480e9 −0.868058
\(590\) −3.65432e8 −0.0732528
\(591\) −2.76758e9 −0.551497
\(592\) −1.78808e9 −0.354210
\(593\) 9.16399e9 1.80465 0.902326 0.431055i \(-0.141858\pi\)
0.902326 + 0.431055i \(0.141858\pi\)
\(594\) 1.72968e9 0.338620
\(595\) 1.64766e8 0.0320670
\(596\) −6.77864e9 −1.31154
\(597\) 2.32745e9 0.447682
\(598\) 1.84677e7 0.00353149
\(599\) −5.47291e9 −1.04046 −0.520229 0.854027i \(-0.674153\pi\)
−0.520229 + 0.854027i \(0.674153\pi\)
\(600\) 2.24432e9 0.424185
\(601\) −4.19336e9 −0.787955 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(602\) −2.12063e9 −0.396167
\(603\) −3.13353e9 −0.582001
\(604\) −1.50143e10 −2.77252
\(605\) −4.32804e8 −0.0794598
\(606\) 5.11741e9 0.934106
\(607\) 6.57768e9 1.19375 0.596874 0.802335i \(-0.296409\pi\)
0.596874 + 0.802335i \(0.296409\pi\)
\(608\) 3.18044e9 0.573885
\(609\) −4.41018e8 −0.0791216
\(610\) 4.06202e9 0.724582
\(611\) 1.11765e10 1.98227
\(612\) −1.49213e9 −0.263133
\(613\) −7.04485e9 −1.23527 −0.617633 0.786467i \(-0.711908\pi\)
−0.617633 + 0.786467i \(0.711908\pi\)
\(614\) 1.31745e10 2.29691
\(615\) −7.90796e8 −0.137089
\(616\) 9.45892e8 0.163046
\(617\) −4.68870e9 −0.803627 −0.401813 0.915722i \(-0.631620\pi\)
−0.401813 + 0.915722i \(0.631620\pi\)
\(618\) 4.61000e9 0.785672
\(619\) −6.66470e9 −1.12944 −0.564720 0.825282i \(-0.691016\pi\)
−0.564720 + 0.825282i \(0.691016\pi\)
\(620\) 5.56490e9 0.937749
\(621\) −1.73231e6 −0.000290272 0
\(622\) −3.57495e9 −0.595667
\(623\) −1.70689e9 −0.282812
\(624\) −9.94693e8 −0.163887
\(625\) 3.90631e9 0.640010
\(626\) −4.05916e9 −0.661342
\(627\) −1.97403e9 −0.319828
\(628\) −1.19081e10 −1.91860
\(629\) −5.92394e9 −0.949147
\(630\) 2.06502e8 0.0329027
\(631\) 6.09161e8 0.0965227 0.0482614 0.998835i \(-0.484632\pi\)
0.0482614 + 0.998835i \(0.484632\pi\)
\(632\) −3.87653e9 −0.610848
\(633\) 6.66667e8 0.104471
\(634\) −1.00797e10 −1.57086
\(635\) −1.47959e9 −0.229315
\(636\) −3.87068e9 −0.596605
\(637\) −9.31057e9 −1.42721
\(638\) −9.01614e9 −1.37451
\(639\) −3.64428e9 −0.552534
\(640\) −3.39211e9 −0.511493
\(641\) −3.31465e7 −0.00497089 −0.00248545 0.999997i \(-0.500791\pi\)
−0.00248545 + 0.999997i \(0.500791\pi\)
\(642\) 3.73322e9 0.556815
\(643\) −4.47226e9 −0.663421 −0.331710 0.943381i \(-0.607626\pi\)
−0.331710 + 0.943381i \(0.607626\pi\)
\(644\) −2.74078e6 −0.000404366 0
\(645\) −1.97747e9 −0.290169
\(646\) 2.81724e9 0.411159
\(647\) −4.33561e9 −0.629340 −0.314670 0.949201i \(-0.601894\pi\)
−0.314670 + 0.949201i \(0.601894\pi\)
\(648\) −6.46382e8 −0.0933205
\(649\) 1.00327e9 0.144066
\(650\) 1.43405e10 2.04818
\(651\) −1.23633e9 −0.175631
\(652\) −1.81154e10 −2.55966
\(653\) −5.43227e9 −0.763458 −0.381729 0.924274i \(-0.624671\pi\)
−0.381729 + 0.924274i \(0.624671\pi\)
\(654\) −7.12515e9 −0.996029
\(655\) 2.79453e9 0.388566
\(656\) −9.35239e8 −0.129348
\(657\) 2.67524e9 0.368031
\(658\) −2.74410e9 −0.375499
\(659\) 5.26358e9 0.716444 0.358222 0.933636i \(-0.383383\pi\)
0.358222 + 0.933636i \(0.383383\pi\)
\(660\) 2.55187e9 0.345505
\(661\) 1.51100e9 0.203497 0.101749 0.994810i \(-0.467556\pi\)
0.101749 + 0.994810i \(0.467556\pi\)
\(662\) −8.78133e8 −0.117641
\(663\) −3.29544e9 −0.439153
\(664\) 5.39962e9 0.715772
\(665\) −2.35674e8 −0.0310768
\(666\) −7.42449e9 −0.973884
\(667\) 9.02986e6 0.00117826
\(668\) 1.18421e10 1.53714
\(669\) 1.22931e9 0.158735
\(670\) −7.64816e9 −0.982415
\(671\) −1.11520e10 −1.42503
\(672\) 9.13416e8 0.116112
\(673\) 4.92429e8 0.0622717 0.0311359 0.999515i \(-0.490088\pi\)
0.0311359 + 0.999515i \(0.490088\pi\)
\(674\) 8.96668e9 1.12803
\(675\) −1.34517e9 −0.168351
\(676\) 1.43406e10 1.78548
\(677\) 1.38348e10 1.71361 0.856805 0.515640i \(-0.172446\pi\)
0.856805 + 0.515640i \(0.172446\pi\)
\(678\) 7.39205e9 0.910879
\(679\) −6.45701e8 −0.0791566
\(680\) −1.25880e9 −0.153523
\(681\) −6.92236e9 −0.839923
\(682\) −2.52754e10 −3.05108
\(683\) 8.01482e8 0.0962545 0.0481273 0.998841i \(-0.484675\pi\)
0.0481273 + 0.998841i \(0.484675\pi\)
\(684\) 2.13428e9 0.255008
\(685\) −4.42784e9 −0.526350
\(686\) 4.64450e9 0.549294
\(687\) −9.30714e9 −1.09513
\(688\) −2.33866e9 −0.273784
\(689\) −8.54858e9 −0.995696
\(690\) −4.22813e6 −0.000489978 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
\(693\) −5.66937e8 −0.0647096
\(694\) −8.35247e9 −0.948543
\(695\) 4.19263e9 0.473739
\(696\) 3.36934e9 0.378802
\(697\) −3.09846e9 −0.346602
\(698\) 6.76143e9 0.752566
\(699\) −3.90871e9 −0.432876
\(700\) −2.12827e9 −0.234522
\(701\) 1.40491e10 1.54040 0.770201 0.637801i \(-0.220156\pi\)
0.770201 + 0.637801i \(0.220156\pi\)
\(702\) −4.13018e9 −0.450598
\(703\) 8.47335e9 0.919838
\(704\) 1.66990e10 1.80379
\(705\) −2.55884e9 −0.275031
\(706\) 1.22188e10 1.30681
\(707\) −1.67733e9 −0.178506
\(708\) −1.08471e9 −0.114868
\(709\) −9.39937e9 −0.990460 −0.495230 0.868762i \(-0.664916\pi\)
−0.495230 + 0.868762i \(0.664916\pi\)
\(710\) −8.89476e9 −0.932675
\(711\) 2.32346e9 0.242433
\(712\) 1.30405e10 1.35399
\(713\) 2.53139e7 0.00261544
\(714\) 8.09105e8 0.0831881
\(715\) 5.63593e9 0.576626
\(716\) −2.56526e10 −2.61178
\(717\) −6.64040e9 −0.672786
\(718\) −8.51354e9 −0.858371
\(719\) 1.11397e10 1.11769 0.558846 0.829272i \(-0.311244\pi\)
0.558846 + 0.829272i \(0.311244\pi\)
\(720\) 2.27733e8 0.0227385
\(721\) −1.51102e9 −0.150140
\(722\) 1.20504e10 1.19157
\(723\) 6.86870e9 0.675912
\(724\) −6.64886e9 −0.651122
\(725\) 7.01187e9 0.683362
\(726\) −2.12534e9 −0.206134
\(727\) −9.70625e9 −0.936874 −0.468437 0.883497i \(-0.655183\pi\)
−0.468437 + 0.883497i \(0.655183\pi\)
\(728\) −2.25863e9 −0.216963
\(729\) 3.87420e8 0.0370370
\(730\) 6.52958e9 0.621234
\(731\) −7.74803e9 −0.733636
\(732\) 1.20573e10 1.13622
\(733\) 4.02746e9 0.377718 0.188859 0.982004i \(-0.439521\pi\)
0.188859 + 0.982004i \(0.439521\pi\)
\(734\) −2.32782e10 −2.17277
\(735\) 2.13163e9 0.198019
\(736\) −1.87022e7 −0.00172911
\(737\) 2.09975e10 1.93211
\(738\) −3.88331e9 −0.355635
\(739\) 1.14088e10 1.03989 0.519943 0.854201i \(-0.325953\pi\)
0.519943 + 0.854201i \(0.325953\pi\)
\(740\) −1.09537e10 −0.993686
\(741\) 4.71365e9 0.425592
\(742\) 2.09887e9 0.188613
\(743\) −3.48068e9 −0.311317 −0.155659 0.987811i \(-0.549750\pi\)
−0.155659 + 0.987811i \(0.549750\pi\)
\(744\) 9.44545e9 0.840848
\(745\) 3.42757e9 0.303696
\(746\) 1.24432e10 1.09736
\(747\) −3.23636e9 −0.284076
\(748\) 9.99863e9 0.873544
\(749\) −1.22364e9 −0.106406
\(750\) −7.03645e9 −0.609031
\(751\) −1.06375e10 −0.916433 −0.458217 0.888841i \(-0.651512\pi\)
−0.458217 + 0.888841i \(0.651512\pi\)
\(752\) −3.02623e9 −0.259501
\(753\) 8.77965e9 0.749367
\(754\) 2.15290e10 1.82905
\(755\) 7.59185e9 0.641997
\(756\) 6.12959e8 0.0515947
\(757\) 2.08009e10 1.74279 0.871397 0.490578i \(-0.163214\pi\)
0.871397 + 0.490578i \(0.163214\pi\)
\(758\) 1.74352e10 1.45407
\(759\) 1.16081e7 0.000963637 0
\(760\) 1.80053e9 0.148783
\(761\) 1.33150e9 0.109521 0.0547604 0.998500i \(-0.482561\pi\)
0.0547604 + 0.998500i \(0.482561\pi\)
\(762\) −7.26571e9 −0.594889
\(763\) 2.33541e9 0.190339
\(764\) 2.97435e10 2.41304
\(765\) 7.54483e8 0.0609304
\(766\) 2.30413e10 1.85228
\(767\) −2.39564e9 −0.191707
\(768\) −4.84328e9 −0.385811
\(769\) −2.19213e10 −1.73830 −0.869148 0.494553i \(-0.835332\pi\)
−0.869148 + 0.494553i \(0.835332\pi\)
\(770\) −1.38375e9 −0.109230
\(771\) 6.88448e9 0.540979
\(772\) 1.12943e9 0.0883485
\(773\) 3.83517e9 0.298646 0.149323 0.988788i \(-0.452291\pi\)
0.149323 + 0.988788i \(0.452291\pi\)
\(774\) −9.71064e9 −0.752756
\(775\) 1.96567e10 1.51690
\(776\) 4.93310e9 0.378969
\(777\) 2.43353e9 0.186107
\(778\) −2.14616e10 −1.63393
\(779\) 4.43191e9 0.335900
\(780\) −6.09344e9 −0.459760
\(781\) 2.44200e10 1.83429
\(782\) −1.65665e7 −0.00123881
\(783\) −2.01947e9 −0.150339
\(784\) 2.52099e9 0.186838
\(785\) 6.02126e9 0.444266
\(786\) 1.37229e10 1.00802
\(787\) 1.79274e10 1.31101 0.655504 0.755192i \(-0.272456\pi\)
0.655504 + 0.755192i \(0.272456\pi\)
\(788\) −2.00508e10 −1.45979
\(789\) 1.13238e10 0.820771
\(790\) 5.67099e9 0.409227
\(791\) −2.42289e9 −0.174067
\(792\) 4.33135e9 0.309803
\(793\) 2.66292e10 1.89627
\(794\) 8.03321e8 0.0569531
\(795\) 1.95718e9 0.138148
\(796\) 1.68621e10 1.18499
\(797\) −5.47185e9 −0.382851 −0.191426 0.981507i \(-0.561311\pi\)
−0.191426 + 0.981507i \(0.561311\pi\)
\(798\) −1.15731e9 −0.0806193
\(799\) −1.00259e10 −0.695363
\(800\) −1.45227e10 −1.00284
\(801\) −7.81606e9 −0.537371
\(802\) −9.93338e9 −0.679965
\(803\) −1.79266e10 −1.22178
\(804\) −2.27021e10 −1.54053
\(805\) 1.38586e6 9.36337e−5 0
\(806\) 6.03535e10 4.06003
\(807\) −1.12136e10 −0.751086
\(808\) 1.28147e10 0.854612
\(809\) 2.20226e9 0.146234 0.0731171 0.997323i \(-0.476705\pi\)
0.0731171 + 0.997323i \(0.476705\pi\)
\(810\) 9.45595e8 0.0625184
\(811\) −5.11433e9 −0.336679 −0.168339 0.985729i \(-0.553840\pi\)
−0.168339 + 0.985729i \(0.553840\pi\)
\(812\) −3.19512e9 −0.209431
\(813\) −1.48760e9 −0.0970889
\(814\) 4.97509e10 3.23307
\(815\) 9.15991e9 0.592707
\(816\) 8.92293e8 0.0574899
\(817\) 1.10825e10 0.710982
\(818\) −1.44182e10 −0.921032
\(819\) 1.35375e9 0.0861083
\(820\) −5.72922e9 −0.362867
\(821\) −4.81471e9 −0.303647 −0.151824 0.988408i \(-0.548515\pi\)
−0.151824 + 0.988408i \(0.548515\pi\)
\(822\) −2.17435e10 −1.36546
\(823\) −2.90096e10 −1.81402 −0.907011 0.421107i \(-0.861642\pi\)
−0.907011 + 0.421107i \(0.861642\pi\)
\(824\) 1.15441e10 0.718810
\(825\) 9.01390e9 0.558887
\(826\) 5.88185e8 0.0363148
\(827\) 5.95215e9 0.365936 0.182968 0.983119i \(-0.441430\pi\)
0.182968 + 0.983119i \(0.441430\pi\)
\(828\) −1.25504e7 −0.000768335 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
\(831\) −5.84776e9 −0.353497
\(832\) −3.98744e10 −2.40028
\(833\) 8.35207e9 0.500653
\(834\) 2.05885e10 1.22897
\(835\) −5.98788e9 −0.355935
\(836\) −1.43016e10 −0.846570
\(837\) −5.66130e9 −0.333716
\(838\) 5.51569e9 0.323777
\(839\) 7.75973e9 0.453607 0.226803 0.973941i \(-0.427173\pi\)
0.226803 + 0.973941i \(0.427173\pi\)
\(840\) 5.17108e8 0.0301026
\(841\) −6.72315e9 −0.389751
\(842\) −1.94677e10 −1.12389
\(843\) −3.52610e9 −0.202721
\(844\) 4.82992e9 0.276529
\(845\) −7.25123e9 −0.413441
\(846\) −1.25655e10 −0.713485
\(847\) 6.96624e8 0.0393919
\(848\) 2.31466e9 0.130347
\(849\) 1.09013e10 0.611364
\(850\) −1.28642e10 −0.718483
\(851\) −4.98266e7 −0.00277146
\(852\) −2.64024e10 −1.46253
\(853\) 3.15852e10 1.74246 0.871229 0.490877i \(-0.163324\pi\)
0.871229 + 0.490877i \(0.163324\pi\)
\(854\) −6.53807e9 −0.359209
\(855\) −1.07918e9 −0.0590490
\(856\) 9.34849e9 0.509429
\(857\) −1.81369e10 −0.984304 −0.492152 0.870509i \(-0.663790\pi\)
−0.492152 + 0.870509i \(0.663790\pi\)
\(858\) 2.76760e10 1.49588
\(859\) 2.80979e10 1.51251 0.756253 0.654279i \(-0.227028\pi\)
0.756253 + 0.654279i \(0.227028\pi\)
\(860\) −1.43265e10 −0.768062
\(861\) 1.27283e9 0.0679612
\(862\) −3.01509e10 −1.60334
\(863\) −1.12434e10 −0.595467 −0.297734 0.954649i \(-0.596231\pi\)
−0.297734 + 0.954649i \(0.596231\pi\)
\(864\) 4.18264e9 0.220624
\(865\) −6.32618e8 −0.0332342
\(866\) −4.52982e10 −2.37011
\(867\) −8.12296e9 −0.423299
\(868\) −8.95705e9 −0.464885
\(869\) −1.55693e10 −0.804824
\(870\) −4.92902e9 −0.253772
\(871\) −5.01386e10 −2.57104
\(872\) −1.78424e10 −0.911265
\(873\) −2.95674e9 −0.150405
\(874\) 2.36960e7 0.00120056
\(875\) 2.30634e9 0.116384
\(876\) 1.93818e10 0.974158
\(877\) −1.39725e10 −0.699481 −0.349740 0.936847i \(-0.613730\pi\)
−0.349740 + 0.936847i \(0.613730\pi\)
\(878\) 2.46330e10 1.22825
\(879\) 4.35241e9 0.216157
\(880\) −1.52602e9 −0.0754867
\(881\) −1.05370e10 −0.519158 −0.259579 0.965722i \(-0.583584\pi\)
−0.259579 + 0.965722i \(0.583584\pi\)
\(882\) 1.04677e10 0.513701
\(883\) 2.48119e10 1.21282 0.606412 0.795151i \(-0.292608\pi\)
0.606412 + 0.795151i \(0.292608\pi\)
\(884\) −2.38750e10 −1.16242
\(885\) 5.48476e8 0.0265985
\(886\) 6.52854e8 0.0315354
\(887\) 3.02069e10 1.45336 0.726680 0.686976i \(-0.241062\pi\)
0.726680 + 0.686976i \(0.241062\pi\)
\(888\) −1.85920e10 −0.891005
\(889\) 2.38148e9 0.113682
\(890\) −1.90770e10 −0.907080
\(891\) −2.59607e9 −0.122955
\(892\) 8.90624e9 0.420162
\(893\) 1.43407e10 0.673891
\(894\) 1.68315e10 0.787848
\(895\) 1.29711e10 0.604776
\(896\) 5.45981e9 0.253571
\(897\) −2.77181e7 −0.00128230
\(898\) 3.62253e10 1.66934
\(899\) 2.95101e10 1.35460
\(900\) −9.74562e9 −0.445616
\(901\) 7.66853e9 0.349281
\(902\) 2.60218e10 1.18063
\(903\) 3.18286e9 0.143850
\(904\) 1.85107e10 0.833362
\(905\) 3.36194e9 0.150772
\(906\) 3.72808e10 1.66547
\(907\) 1.15712e10 0.514937 0.257468 0.966287i \(-0.417112\pi\)
0.257468 + 0.966287i \(0.417112\pi\)
\(908\) −5.01517e10 −2.22323
\(909\) −7.68072e9 −0.339178
\(910\) 3.30416e9 0.145350
\(911\) −1.60273e10 −0.702338 −0.351169 0.936312i \(-0.614216\pi\)
−0.351169 + 0.936312i \(0.614216\pi\)
\(912\) −1.27630e9 −0.0557147
\(913\) 2.16866e10 0.943068
\(914\) 1.99509e9 0.0864271
\(915\) −6.09668e9 −0.263099
\(916\) −6.74291e10 −2.89877
\(917\) −4.49797e9 −0.192630
\(918\) 3.70499e9 0.158066
\(919\) 9.52685e9 0.404897 0.202449 0.979293i \(-0.435110\pi\)
0.202449 + 0.979293i \(0.435110\pi\)
\(920\) −1.05878e7 −0.000448280 0
\(921\) −1.97736e10 −0.834021
\(922\) −6.46219e10 −2.71532
\(923\) −5.83109e10 −2.44086
\(924\) −4.10739e9 −0.171283
\(925\) −3.86914e10 −1.60738
\(926\) 4.29228e10 1.77643
\(927\) −6.91915e9 −0.285282
\(928\) −2.18025e10 −0.895547
\(929\) 7.62148e9 0.311878 0.155939 0.987767i \(-0.450160\pi\)
0.155939 + 0.987767i \(0.450160\pi\)
\(930\) −1.38178e10 −0.563311
\(931\) −1.19464e10 −0.485193
\(932\) −2.83181e10 −1.14580
\(933\) 5.36564e9 0.216290
\(934\) −3.54467e10 −1.42351
\(935\) −5.05573e9 −0.202275
\(936\) −1.03425e10 −0.412251
\(937\) 2.55429e9 0.101434 0.0507168 0.998713i \(-0.483849\pi\)
0.0507168 + 0.998713i \(0.483849\pi\)
\(938\) 1.23102e10 0.487028
\(939\) 6.09239e9 0.240137
\(940\) −1.85385e10 −0.727993
\(941\) −1.11902e10 −0.437797 −0.218898 0.975748i \(-0.570246\pi\)
−0.218898 + 0.975748i \(0.570246\pi\)
\(942\) 2.95682e10 1.15252
\(943\) −2.60613e7 −0.00101206
\(944\) 6.48658e8 0.0250965
\(945\) −3.09938e8 −0.0119471
\(946\) 6.50702e10 2.49898
\(947\) 2.91042e10 1.11361 0.556803 0.830645i \(-0.312028\pi\)
0.556803 + 0.830645i \(0.312028\pi\)
\(948\) 1.68332e10 0.641709
\(949\) 4.28056e10 1.62581
\(950\) 1.84004e10 0.696297
\(951\) 1.51287e10 0.570386
\(952\) 2.02611e9 0.0761087
\(953\) 1.18538e10 0.443643 0.221821 0.975087i \(-0.428800\pi\)
0.221821 + 0.975087i \(0.428800\pi\)
\(954\) 9.61099e9 0.358384
\(955\) −1.50396e10 −0.558758
\(956\) −4.81089e10 −1.78083
\(957\) 1.35323e10 0.499092
\(958\) −8.20804e10 −3.01620
\(959\) 7.12688e9 0.260936
\(960\) 9.12915e9 0.333028
\(961\) 5.52147e10 2.00689
\(962\) −1.18797e11 −4.30222
\(963\) −5.60318e9 −0.202182
\(964\) 4.97629e10 1.78911
\(965\) −5.71088e8 −0.0204577
\(966\) 6.80543e6 0.000242905 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
\(969\) −4.22839e9 −0.149294
\(970\) −7.21666e9 −0.253884
\(971\) 4.49711e10 1.57640 0.788200 0.615419i \(-0.211013\pi\)
0.788200 + 0.615419i \(0.211013\pi\)
\(972\) 2.80681e9 0.0980352
\(973\) −6.74829e9 −0.234854
\(974\) −5.25209e10 −1.82128
\(975\) −2.15237e10 −0.743704
\(976\) −7.21027e9 −0.248243
\(977\) −3.26239e10 −1.11919 −0.559597 0.828765i \(-0.689044\pi\)
−0.559597 + 0.828765i \(0.689044\pi\)
\(978\) 4.49810e10 1.53760
\(979\) 5.23748e10 1.78395
\(980\) 1.54434e10 0.524146
\(981\) 1.06941e10 0.361663
\(982\) 1.46670e10 0.494254
\(983\) −4.19068e10 −1.40717 −0.703585 0.710611i \(-0.748419\pi\)
−0.703585 + 0.710611i \(0.748419\pi\)
\(984\) −9.72435e9 −0.325370
\(985\) 1.01385e10 0.338024
\(986\) −1.93127e10 −0.641613
\(987\) 4.11861e9 0.136346
\(988\) 3.41498e10 1.12652
\(989\) −6.51692e7 −0.00214218
\(990\) −6.33636e9 −0.207547
\(991\) 1.39919e10 0.456687 0.228343 0.973581i \(-0.426669\pi\)
0.228343 + 0.973581i \(0.426669\pi\)
\(992\) −6.11201e10 −1.98789
\(993\) 1.31799e9 0.0427159
\(994\) 1.43167e10 0.462370
\(995\) −8.52617e9 −0.274393
\(996\) −2.34470e10 −0.751934
\(997\) 5.75080e10 1.83779 0.918893 0.394506i \(-0.129084\pi\)
0.918893 + 0.394506i \(0.129084\pi\)
\(998\) 5.83846e10 1.85927
\(999\) 1.11434e10 0.353622
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 177.8.a.b.1.3 17
3.2 odd 2 531.8.a.d.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.3 17 1.1 even 1 trivial
531.8.a.d.1.15 17 3.2 odd 2