Properties

Label 531.8.a.d.1.12
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 1.12
Root \(-10.1391\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.1391 q^{2} +19.3578 q^{4} -236.334 q^{5} +1426.66 q^{7} -1318.82 q^{8} +O(q^{10})\) \(q+12.1391 q^{2} +19.3578 q^{4} -236.334 q^{5} +1426.66 q^{7} -1318.82 q^{8} -2868.89 q^{10} +5472.58 q^{11} -8451.67 q^{13} +17318.4 q^{14} -18487.1 q^{16} +6087.08 q^{17} -12974.5 q^{19} -4574.91 q^{20} +66432.2 q^{22} +53979.7 q^{23} -22271.1 q^{25} -102596. q^{26} +27617.0 q^{28} +14368.8 q^{29} -48485.8 q^{31} -55607.6 q^{32} +73891.7 q^{34} -337169. q^{35} +82881.8 q^{37} -157499. q^{38} +311682. q^{40} +782250. q^{41} -369462. q^{43} +105937. q^{44} +655265. q^{46} -368463. q^{47} +1.21182e6 q^{49} -270351. q^{50} -163605. q^{52} -836648. q^{53} -1.29336e6 q^{55} -1.88151e6 q^{56} +174424. q^{58} +205379. q^{59} +37349.1 q^{61} -588573. q^{62} +1.69132e6 q^{64} +1.99742e6 q^{65} +2.64158e6 q^{67} +117832. q^{68} -4.09293e6 q^{70} +2.03139e6 q^{71} -3.64005e6 q^{73} +1.00611e6 q^{74} -251158. q^{76} +7.80751e6 q^{77} -7.63186e6 q^{79} +4.36913e6 q^{80} +9.49581e6 q^{82} +6.69291e6 q^{83} -1.43859e6 q^{85} -4.48494e6 q^{86} -7.21734e6 q^{88} -717966. q^{89} -1.20577e7 q^{91} +1.04493e6 q^{92} -4.47281e6 q^{94} +3.06632e6 q^{95} +1.06996e7 q^{97} +1.47104e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.1391 1.07296 0.536478 0.843915i \(-0.319755\pi\)
0.536478 + 0.843915i \(0.319755\pi\)
\(3\) 0 0
\(4\) 19.3578 0.151233
\(5\) −236.334 −0.845536 −0.422768 0.906238i \(-0.638941\pi\)
−0.422768 + 0.906238i \(0.638941\pi\)
\(6\) 0 0
\(7\) 1426.66 1.57209 0.786045 0.618169i \(-0.212125\pi\)
0.786045 + 0.618169i \(0.212125\pi\)
\(8\) −1318.82 −0.910689
\(9\) 0 0
\(10\) −2868.89 −0.907222
\(11\) 5472.58 1.23970 0.619851 0.784719i \(-0.287193\pi\)
0.619851 + 0.784719i \(0.287193\pi\)
\(12\) 0 0
\(13\) −8451.67 −1.06694 −0.533471 0.845819i \(-0.679113\pi\)
−0.533471 + 0.845819i \(0.679113\pi\)
\(14\) 17318.4 1.68678
\(15\) 0 0
\(16\) −18487.1 −1.12836
\(17\) 6087.08 0.300495 0.150248 0.988648i \(-0.451993\pi\)
0.150248 + 0.988648i \(0.451993\pi\)
\(18\) 0 0
\(19\) −12974.5 −0.433964 −0.216982 0.976176i \(-0.569621\pi\)
−0.216982 + 0.976176i \(0.569621\pi\)
\(20\) −4574.91 −0.127873
\(21\) 0 0
\(22\) 66432.2 1.33015
\(23\) 53979.7 0.925087 0.462544 0.886597i \(-0.346937\pi\)
0.462544 + 0.886597i \(0.346937\pi\)
\(24\) 0 0
\(25\) −22271.1 −0.285070
\(26\) −102596. −1.14478
\(27\) 0 0
\(28\) 27617.0 0.237751
\(29\) 14368.8 0.109402 0.0547011 0.998503i \(-0.482579\pi\)
0.0547011 + 0.998503i \(0.482579\pi\)
\(30\) 0 0
\(31\) −48485.8 −0.292313 −0.146157 0.989261i \(-0.546690\pi\)
−0.146157 + 0.989261i \(0.546690\pi\)
\(32\) −55607.6 −0.299992
\(33\) 0 0
\(34\) 73891.7 0.322418
\(35\) −337169. −1.32926
\(36\) 0 0
\(37\) 82881.8 0.269001 0.134500 0.990914i \(-0.457057\pi\)
0.134500 + 0.990914i \(0.457057\pi\)
\(38\) −157499. −0.465624
\(39\) 0 0
\(40\) 311682. 0.770020
\(41\) 782250. 1.77256 0.886282 0.463146i \(-0.153280\pi\)
0.886282 + 0.463146i \(0.153280\pi\)
\(42\) 0 0
\(43\) −369462. −0.708647 −0.354324 0.935123i \(-0.615289\pi\)
−0.354324 + 0.935123i \(0.615289\pi\)
\(44\) 105937. 0.187483
\(45\) 0 0
\(46\) 655265. 0.992577
\(47\) −368463. −0.517668 −0.258834 0.965922i \(-0.583338\pi\)
−0.258834 + 0.965922i \(0.583338\pi\)
\(48\) 0 0
\(49\) 1.21182e6 1.47147
\(50\) −270351. −0.305867
\(51\) 0 0
\(52\) −163605. −0.161356
\(53\) −836648. −0.771929 −0.385964 0.922514i \(-0.626131\pi\)
−0.385964 + 0.922514i \(0.626131\pi\)
\(54\) 0 0
\(55\) −1.29336e6 −1.04821
\(56\) −1.88151e6 −1.43169
\(57\) 0 0
\(58\) 174424. 0.117384
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 37349.1 0.0210681 0.0105341 0.999945i \(-0.496647\pi\)
0.0105341 + 0.999945i \(0.496647\pi\)
\(62\) −588573. −0.313639
\(63\) 0 0
\(64\) 1.69132e6 0.806484
\(65\) 1.99742e6 0.902137
\(66\) 0 0
\(67\) 2.64158e6 1.07300 0.536502 0.843899i \(-0.319745\pi\)
0.536502 + 0.843899i \(0.319745\pi\)
\(68\) 117832. 0.0454447
\(69\) 0 0
\(70\) −4.09293e6 −1.42623
\(71\) 2.03139e6 0.673579 0.336790 0.941580i \(-0.390659\pi\)
0.336790 + 0.941580i \(0.390659\pi\)
\(72\) 0 0
\(73\) −3.64005e6 −1.09516 −0.547580 0.836753i \(-0.684451\pi\)
−0.547580 + 0.836753i \(0.684451\pi\)
\(74\) 1.00611e6 0.288625
\(75\) 0 0
\(76\) −251158. −0.0656295
\(77\) 7.80751e6 1.94892
\(78\) 0 0
\(79\) −7.63186e6 −1.74155 −0.870774 0.491683i \(-0.836382\pi\)
−0.870774 + 0.491683i \(0.836382\pi\)
\(80\) 4.36913e6 0.954070
\(81\) 0 0
\(82\) 9.49581e6 1.90188
\(83\) 6.69291e6 1.28482 0.642409 0.766362i \(-0.277935\pi\)
0.642409 + 0.766362i \(0.277935\pi\)
\(84\) 0 0
\(85\) −1.43859e6 −0.254079
\(86\) −4.48494e6 −0.760347
\(87\) 0 0
\(88\) −7.21734e6 −1.12898
\(89\) −717966. −0.107954 −0.0539770 0.998542i \(-0.517190\pi\)
−0.0539770 + 0.998542i \(0.517190\pi\)
\(90\) 0 0
\(91\) −1.20577e7 −1.67733
\(92\) 1.04493e6 0.139903
\(93\) 0 0
\(94\) −4.47281e6 −0.555435
\(95\) 3.06632e6 0.366932
\(96\) 0 0
\(97\) 1.06996e7 1.19033 0.595165 0.803604i \(-0.297087\pi\)
0.595165 + 0.803604i \(0.297087\pi\)
\(98\) 1.47104e7 1.57882
\(99\) 0 0
\(100\) −431118. −0.0431118
\(101\) 1.14739e7 1.10812 0.554058 0.832478i \(-0.313079\pi\)
0.554058 + 0.832478i \(0.313079\pi\)
\(102\) 0 0
\(103\) 7.95531e6 0.717343 0.358671 0.933464i \(-0.383230\pi\)
0.358671 + 0.933464i \(0.383230\pi\)
\(104\) 1.11462e7 0.971652
\(105\) 0 0
\(106\) −1.01562e7 −0.828245
\(107\) −1.10513e7 −0.872107 −0.436054 0.899921i \(-0.643624\pi\)
−0.436054 + 0.899921i \(0.643624\pi\)
\(108\) 0 0
\(109\) 8.08293e6 0.597828 0.298914 0.954280i \(-0.403376\pi\)
0.298914 + 0.954280i \(0.403376\pi\)
\(110\) −1.57002e7 −1.12469
\(111\) 0 0
\(112\) −2.63748e7 −1.77389
\(113\) 9.96488e6 0.649677 0.324839 0.945769i \(-0.394690\pi\)
0.324839 + 0.945769i \(0.394690\pi\)
\(114\) 0 0
\(115\) −1.27572e7 −0.782194
\(116\) 278147. 0.0165452
\(117\) 0 0
\(118\) 2.49312e6 0.139687
\(119\) 8.68420e6 0.472406
\(120\) 0 0
\(121\) 1.04619e7 0.536863
\(122\) 453384. 0.0226051
\(123\) 0 0
\(124\) −938576. −0.0442073
\(125\) 2.37270e7 1.08657
\(126\) 0 0
\(127\) 3.01471e7 1.30597 0.652984 0.757372i \(-0.273517\pi\)
0.652984 + 0.757372i \(0.273517\pi\)
\(128\) 2.76489e7 1.16531
\(129\) 0 0
\(130\) 2.42469e7 0.967952
\(131\) 4.85605e7 1.88727 0.943634 0.330990i \(-0.107383\pi\)
0.943634 + 0.330990i \(0.107383\pi\)
\(132\) 0 0
\(133\) −1.85102e7 −0.682231
\(134\) 3.20664e7 1.15129
\(135\) 0 0
\(136\) −8.02775e6 −0.273658
\(137\) 1.46792e7 0.487732 0.243866 0.969809i \(-0.421584\pi\)
0.243866 + 0.969809i \(0.421584\pi\)
\(138\) 0 0
\(139\) 3.93077e7 1.24144 0.620721 0.784032i \(-0.286840\pi\)
0.620721 + 0.784032i \(0.286840\pi\)
\(140\) −6.52684e6 −0.201027
\(141\) 0 0
\(142\) 2.46592e7 0.722720
\(143\) −4.62524e7 −1.32269
\(144\) 0 0
\(145\) −3.39583e6 −0.0925035
\(146\) −4.41870e7 −1.17506
\(147\) 0 0
\(148\) 1.60441e6 0.0406817
\(149\) 6.99960e7 1.73349 0.866745 0.498752i \(-0.166208\pi\)
0.866745 + 0.498752i \(0.166208\pi\)
\(150\) 0 0
\(151\) 5.84052e7 1.38049 0.690243 0.723577i \(-0.257503\pi\)
0.690243 + 0.723577i \(0.257503\pi\)
\(152\) 1.71110e7 0.395206
\(153\) 0 0
\(154\) 9.47762e7 2.09111
\(155\) 1.14588e7 0.247161
\(156\) 0 0
\(157\) 5.09364e7 1.05046 0.525230 0.850960i \(-0.323979\pi\)
0.525230 + 0.850960i \(0.323979\pi\)
\(158\) −9.26439e7 −1.86860
\(159\) 0 0
\(160\) 1.31420e7 0.253654
\(161\) 7.70107e7 1.45432
\(162\) 0 0
\(163\) −2.83386e7 −0.512532 −0.256266 0.966606i \(-0.582492\pi\)
−0.256266 + 0.966606i \(0.582492\pi\)
\(164\) 1.51426e7 0.268069
\(165\) 0 0
\(166\) 8.12459e7 1.37855
\(167\) −9.62396e6 −0.159899 −0.0799496 0.996799i \(-0.525476\pi\)
−0.0799496 + 0.996799i \(0.525476\pi\)
\(168\) 0 0
\(169\) 8.68213e6 0.138364
\(170\) −1.74631e7 −0.272616
\(171\) 0 0
\(172\) −7.15196e6 −0.107171
\(173\) −7.21139e7 −1.05891 −0.529454 0.848339i \(-0.677603\pi\)
−0.529454 + 0.848339i \(0.677603\pi\)
\(174\) 0 0
\(175\) −3.17733e7 −0.448155
\(176\) −1.01172e8 −1.39883
\(177\) 0 0
\(178\) −8.71546e6 −0.115830
\(179\) 2.09311e7 0.272777 0.136388 0.990655i \(-0.456451\pi\)
0.136388 + 0.990655i \(0.456451\pi\)
\(180\) 0 0
\(181\) 4.58129e7 0.574266 0.287133 0.957891i \(-0.407298\pi\)
0.287133 + 0.957891i \(0.407298\pi\)
\(182\) −1.46369e8 −1.79970
\(183\) 0 0
\(184\) −7.11894e7 −0.842467
\(185\) −1.95878e7 −0.227450
\(186\) 0 0
\(187\) 3.33120e7 0.372525
\(188\) −7.13262e6 −0.0782883
\(189\) 0 0
\(190\) 3.72224e7 0.393701
\(191\) 1.58678e8 1.64779 0.823893 0.566746i \(-0.191798\pi\)
0.823893 + 0.566746i \(0.191798\pi\)
\(192\) 0 0
\(193\) 6.81944e7 0.682807 0.341404 0.939917i \(-0.389098\pi\)
0.341404 + 0.939917i \(0.389098\pi\)
\(194\) 1.29884e8 1.27717
\(195\) 0 0
\(196\) 2.34581e7 0.222534
\(197\) −8.04764e7 −0.749957 −0.374979 0.927033i \(-0.622350\pi\)
−0.374979 + 0.927033i \(0.622350\pi\)
\(198\) 0 0
\(199\) −6.86011e7 −0.617085 −0.308543 0.951211i \(-0.599841\pi\)
−0.308543 + 0.951211i \(0.599841\pi\)
\(200\) 2.93715e7 0.259610
\(201\) 0 0
\(202\) 1.39283e8 1.18896
\(203\) 2.04993e7 0.171990
\(204\) 0 0
\(205\) −1.84872e8 −1.49877
\(206\) 9.65703e7 0.769677
\(207\) 0 0
\(208\) 1.56247e8 1.20390
\(209\) −7.10040e7 −0.537986
\(210\) 0 0
\(211\) 2.88847e7 0.211679 0.105840 0.994383i \(-0.466247\pi\)
0.105840 + 0.994383i \(0.466247\pi\)
\(212\) −1.61956e7 −0.116741
\(213\) 0 0
\(214\) −1.34153e8 −0.935732
\(215\) 8.73166e7 0.599187
\(216\) 0 0
\(217\) −6.91727e7 −0.459543
\(218\) 9.81195e7 0.641442
\(219\) 0 0
\(220\) −2.50365e7 −0.158524
\(221\) −5.14459e7 −0.320611
\(222\) 0 0
\(223\) 2.59394e8 1.56637 0.783184 0.621790i \(-0.213594\pi\)
0.783184 + 0.621790i \(0.213594\pi\)
\(224\) −7.93332e7 −0.471614
\(225\) 0 0
\(226\) 1.20965e8 0.697074
\(227\) 2.03673e8 1.15570 0.577848 0.816144i \(-0.303893\pi\)
0.577848 + 0.816144i \(0.303893\pi\)
\(228\) 0 0
\(229\) −3.48601e8 −1.91825 −0.959124 0.282986i \(-0.908675\pi\)
−0.959124 + 0.282986i \(0.908675\pi\)
\(230\) −1.54862e8 −0.839259
\(231\) 0 0
\(232\) −1.89498e7 −0.0996315
\(233\) −1.56687e8 −0.811495 −0.405748 0.913985i \(-0.632989\pi\)
−0.405748 + 0.913985i \(0.632989\pi\)
\(234\) 0 0
\(235\) 8.70805e7 0.437707
\(236\) 3.97568e6 0.0196888
\(237\) 0 0
\(238\) 1.05418e8 0.506870
\(239\) −2.08708e8 −0.988885 −0.494443 0.869210i \(-0.664628\pi\)
−0.494443 + 0.869210i \(0.664628\pi\)
\(240\) 0 0
\(241\) −3.80494e8 −1.75101 −0.875505 0.483209i \(-0.839471\pi\)
−0.875505 + 0.483209i \(0.839471\pi\)
\(242\) 1.26998e8 0.576029
\(243\) 0 0
\(244\) 722995. 0.00318619
\(245\) −2.86394e8 −1.24418
\(246\) 0 0
\(247\) 1.09656e8 0.463014
\(248\) 6.39439e7 0.266206
\(249\) 0 0
\(250\) 2.88025e8 1.16584
\(251\) 1.88517e8 0.752474 0.376237 0.926523i \(-0.377218\pi\)
0.376237 + 0.926523i \(0.377218\pi\)
\(252\) 0 0
\(253\) 2.95408e8 1.14683
\(254\) 3.65959e8 1.40125
\(255\) 0 0
\(256\) 1.19144e8 0.443844
\(257\) −3.79943e8 −1.39621 −0.698107 0.715993i \(-0.745974\pi\)
−0.698107 + 0.715993i \(0.745974\pi\)
\(258\) 0 0
\(259\) 1.18244e8 0.422893
\(260\) 3.86656e7 0.136433
\(261\) 0 0
\(262\) 5.89481e8 2.02495
\(263\) −1.63239e8 −0.553322 −0.276661 0.960968i \(-0.589228\pi\)
−0.276661 + 0.960968i \(0.589228\pi\)
\(264\) 0 0
\(265\) 1.97729e8 0.652693
\(266\) −2.24698e8 −0.732003
\(267\) 0 0
\(268\) 5.11350e7 0.162273
\(269\) 5.26713e8 1.64984 0.824918 0.565252i \(-0.191221\pi\)
0.824918 + 0.565252i \(0.191221\pi\)
\(270\) 0 0
\(271\) 1.21168e8 0.369824 0.184912 0.982755i \(-0.440800\pi\)
0.184912 + 0.982755i \(0.440800\pi\)
\(272\) −1.12532e8 −0.339067
\(273\) 0 0
\(274\) 1.78193e8 0.523315
\(275\) −1.21880e8 −0.353402
\(276\) 0 0
\(277\) 3.39227e8 0.958985 0.479492 0.877546i \(-0.340821\pi\)
0.479492 + 0.877546i \(0.340821\pi\)
\(278\) 4.77161e8 1.33201
\(279\) 0 0
\(280\) 4.44665e8 1.21054
\(281\) 2.48935e8 0.669289 0.334644 0.942344i \(-0.391384\pi\)
0.334644 + 0.942344i \(0.391384\pi\)
\(282\) 0 0
\(283\) −6.01535e8 −1.57764 −0.788821 0.614623i \(-0.789308\pi\)
−0.788821 + 0.614623i \(0.789308\pi\)
\(284\) 3.93232e7 0.101867
\(285\) 0 0
\(286\) −5.61462e8 −1.41919
\(287\) 1.11600e9 2.78663
\(288\) 0 0
\(289\) −3.73286e8 −0.909703
\(290\) −4.12224e7 −0.0992521
\(291\) 0 0
\(292\) −7.04633e7 −0.165624
\(293\) −2.39333e8 −0.555860 −0.277930 0.960601i \(-0.589648\pi\)
−0.277930 + 0.960601i \(0.589648\pi\)
\(294\) 0 0
\(295\) −4.85381e7 −0.110079
\(296\) −1.09306e8 −0.244976
\(297\) 0 0
\(298\) 8.49688e8 1.85996
\(299\) −4.56218e8 −0.987014
\(300\) 0 0
\(301\) −5.27097e8 −1.11406
\(302\) 7.08987e8 1.48120
\(303\) 0 0
\(304\) 2.39861e8 0.489668
\(305\) −8.82687e6 −0.0178138
\(306\) 0 0
\(307\) 1.00262e8 0.197767 0.0988833 0.995099i \(-0.468473\pi\)
0.0988833 + 0.995099i \(0.468473\pi\)
\(308\) 1.51136e8 0.294741
\(309\) 0 0
\(310\) 1.39100e8 0.265193
\(311\) −7.10291e8 −1.33898 −0.669491 0.742820i \(-0.733488\pi\)
−0.669491 + 0.742820i \(0.733488\pi\)
\(312\) 0 0
\(313\) −6.55081e8 −1.20751 −0.603753 0.797171i \(-0.706329\pi\)
−0.603753 + 0.797171i \(0.706329\pi\)
\(314\) 6.18322e8 1.12710
\(315\) 0 0
\(316\) −1.47736e8 −0.263379
\(317\) 1.70031e8 0.299793 0.149896 0.988702i \(-0.452106\pi\)
0.149896 + 0.988702i \(0.452106\pi\)
\(318\) 0 0
\(319\) 7.86342e7 0.135626
\(320\) −3.99717e8 −0.681911
\(321\) 0 0
\(322\) 9.34840e8 1.56042
\(323\) −7.89769e7 −0.130404
\(324\) 0 0
\(325\) 1.88228e8 0.304153
\(326\) −3.44005e8 −0.549924
\(327\) 0 0
\(328\) −1.03165e9 −1.61425
\(329\) −5.25672e8 −0.813821
\(330\) 0 0
\(331\) 1.29730e8 0.196626 0.0983131 0.995156i \(-0.468655\pi\)
0.0983131 + 0.995156i \(0.468655\pi\)
\(332\) 1.29560e8 0.194306
\(333\) 0 0
\(334\) −1.16826e8 −0.171565
\(335\) −6.24295e8 −0.907263
\(336\) 0 0
\(337\) 9.55271e7 0.135963 0.0679817 0.997687i \(-0.478344\pi\)
0.0679817 + 0.997687i \(0.478344\pi\)
\(338\) 1.05393e8 0.148458
\(339\) 0 0
\(340\) −2.78478e7 −0.0384251
\(341\) −2.65342e8 −0.362381
\(342\) 0 0
\(343\) 5.53936e8 0.741191
\(344\) 4.87254e8 0.645358
\(345\) 0 0
\(346\) −8.75398e8 −1.13616
\(347\) 1.13393e8 0.145690 0.0728452 0.997343i \(-0.476792\pi\)
0.0728452 + 0.997343i \(0.476792\pi\)
\(348\) 0 0
\(349\) −3.72221e8 −0.468719 −0.234359 0.972150i \(-0.575299\pi\)
−0.234359 + 0.972150i \(0.575299\pi\)
\(350\) −3.85699e8 −0.480850
\(351\) 0 0
\(352\) −3.04317e8 −0.371901
\(353\) −5.74672e8 −0.695358 −0.347679 0.937614i \(-0.613030\pi\)
−0.347679 + 0.937614i \(0.613030\pi\)
\(354\) 0 0
\(355\) −4.80087e8 −0.569535
\(356\) −1.38982e7 −0.0163262
\(357\) 0 0
\(358\) 2.54085e8 0.292677
\(359\) 5.65130e7 0.0644640 0.0322320 0.999480i \(-0.489738\pi\)
0.0322320 + 0.999480i \(0.489738\pi\)
\(360\) 0 0
\(361\) −7.25534e8 −0.811675
\(362\) 5.56127e8 0.616161
\(363\) 0 0
\(364\) −2.33409e8 −0.253667
\(365\) 8.60270e8 0.925997
\(366\) 0 0
\(367\) −2.25564e8 −0.238198 −0.119099 0.992882i \(-0.538001\pi\)
−0.119099 + 0.992882i \(0.538001\pi\)
\(368\) −9.97926e8 −1.04383
\(369\) 0 0
\(370\) −2.37779e8 −0.244043
\(371\) −1.19361e9 −1.21354
\(372\) 0 0
\(373\) 5.86159e8 0.584837 0.292418 0.956290i \(-0.405540\pi\)
0.292418 + 0.956290i \(0.405540\pi\)
\(374\) 4.04378e8 0.399702
\(375\) 0 0
\(376\) 4.85936e8 0.471435
\(377\) −1.21440e8 −0.116726
\(378\) 0 0
\(379\) 4.01728e8 0.379048 0.189524 0.981876i \(-0.439305\pi\)
0.189524 + 0.981876i \(0.439305\pi\)
\(380\) 5.93572e7 0.0554921
\(381\) 0 0
\(382\) 1.92621e9 1.76800
\(383\) 9.90140e8 0.900536 0.450268 0.892894i \(-0.351329\pi\)
0.450268 + 0.892894i \(0.351329\pi\)
\(384\) 0 0
\(385\) −1.84518e9 −1.64789
\(386\) 8.27819e8 0.732622
\(387\) 0 0
\(388\) 2.07121e8 0.180017
\(389\) −9.31063e8 −0.801965 −0.400982 0.916086i \(-0.631331\pi\)
−0.400982 + 0.916086i \(0.631331\pi\)
\(390\) 0 0
\(391\) 3.28578e8 0.277984
\(392\) −1.59817e9 −1.34005
\(393\) 0 0
\(394\) −9.76911e8 −0.804671
\(395\) 1.80367e9 1.47254
\(396\) 0 0
\(397\) 4.18272e8 0.335500 0.167750 0.985830i \(-0.446350\pi\)
0.167750 + 0.985830i \(0.446350\pi\)
\(398\) −8.32755e8 −0.662105
\(399\) 0 0
\(400\) 4.11727e8 0.321662
\(401\) 1.89613e9 1.46846 0.734230 0.678900i \(-0.237543\pi\)
0.734230 + 0.678900i \(0.237543\pi\)
\(402\) 0 0
\(403\) 4.09785e8 0.311881
\(404\) 2.22109e8 0.167583
\(405\) 0 0
\(406\) 2.48844e8 0.184538
\(407\) 4.53577e8 0.333481
\(408\) 0 0
\(409\) −2.41716e9 −1.74693 −0.873463 0.486890i \(-0.838131\pi\)
−0.873463 + 0.486890i \(0.838131\pi\)
\(410\) −2.24419e9 −1.60811
\(411\) 0 0
\(412\) 1.53997e8 0.108486
\(413\) 2.93006e8 0.204669
\(414\) 0 0
\(415\) −1.58176e9 −1.08636
\(416\) 4.69977e8 0.320074
\(417\) 0 0
\(418\) −8.61925e8 −0.577235
\(419\) 1.73070e9 1.14941 0.574703 0.818362i \(-0.305118\pi\)
0.574703 + 0.818362i \(0.305118\pi\)
\(420\) 0 0
\(421\) 2.38954e9 1.56073 0.780363 0.625327i \(-0.215034\pi\)
0.780363 + 0.625327i \(0.215034\pi\)
\(422\) 3.50634e8 0.227123
\(423\) 0 0
\(424\) 1.10339e9 0.702987
\(425\) −1.35566e8 −0.0856621
\(426\) 0 0
\(427\) 5.32845e7 0.0331210
\(428\) −2.13928e8 −0.131891
\(429\) 0 0
\(430\) 1.05994e9 0.642900
\(431\) 1.06409e9 0.640188 0.320094 0.947386i \(-0.396286\pi\)
0.320094 + 0.947386i \(0.396286\pi\)
\(432\) 0 0
\(433\) −8.34289e8 −0.493866 −0.246933 0.969033i \(-0.579423\pi\)
−0.246933 + 0.969033i \(0.579423\pi\)
\(434\) −8.39695e8 −0.493069
\(435\) 0 0
\(436\) 1.56467e8 0.0904110
\(437\) −7.00360e8 −0.401454
\(438\) 0 0
\(439\) 1.91632e9 1.08104 0.540521 0.841331i \(-0.318227\pi\)
0.540521 + 0.841331i \(0.318227\pi\)
\(440\) 1.70571e9 0.954596
\(441\) 0 0
\(442\) −6.24508e8 −0.344001
\(443\) −1.43118e9 −0.782136 −0.391068 0.920362i \(-0.627894\pi\)
−0.391068 + 0.920362i \(0.627894\pi\)
\(444\) 0 0
\(445\) 1.69680e8 0.0912789
\(446\) 3.14882e9 1.68064
\(447\) 0 0
\(448\) 2.41294e9 1.26787
\(449\) 1.00483e9 0.523877 0.261938 0.965085i \(-0.415638\pi\)
0.261938 + 0.965085i \(0.415638\pi\)
\(450\) 0 0
\(451\) 4.28092e9 2.19745
\(452\) 1.92898e8 0.0982524
\(453\) 0 0
\(454\) 2.47241e9 1.24001
\(455\) 2.84964e9 1.41824
\(456\) 0 0
\(457\) 3.31474e9 1.62459 0.812293 0.583250i \(-0.198219\pi\)
0.812293 + 0.583250i \(0.198219\pi\)
\(458\) −4.23171e9 −2.05819
\(459\) 0 0
\(460\) −2.46952e8 −0.118293
\(461\) −2.98497e7 −0.0141901 −0.00709507 0.999975i \(-0.502258\pi\)
−0.00709507 + 0.999975i \(0.502258\pi\)
\(462\) 0 0
\(463\) 5.11925e8 0.239702 0.119851 0.992792i \(-0.461758\pi\)
0.119851 + 0.992792i \(0.461758\pi\)
\(464\) −2.65636e8 −0.123445
\(465\) 0 0
\(466\) −1.90203e9 −0.870698
\(467\) 3.33639e9 1.51589 0.757946 0.652317i \(-0.226203\pi\)
0.757946 + 0.652317i \(0.226203\pi\)
\(468\) 0 0
\(469\) 3.76863e9 1.68686
\(470\) 1.05708e9 0.469640
\(471\) 0 0
\(472\) −2.70858e8 −0.118562
\(473\) −2.02191e9 −0.878512
\(474\) 0 0
\(475\) 2.88956e8 0.123710
\(476\) 1.68107e8 0.0714432
\(477\) 0 0
\(478\) −2.53352e9 −1.06103
\(479\) 2.42772e8 0.100931 0.0504655 0.998726i \(-0.483930\pi\)
0.0504655 + 0.998726i \(0.483930\pi\)
\(480\) 0 0
\(481\) −7.00489e8 −0.287008
\(482\) −4.61886e9 −1.87875
\(483\) 0 0
\(484\) 2.02520e8 0.0811911
\(485\) −2.52869e9 −1.00647
\(486\) 0 0
\(487\) −1.18746e9 −0.465872 −0.232936 0.972492i \(-0.574833\pi\)
−0.232936 + 0.972492i \(0.574833\pi\)
\(488\) −4.92567e7 −0.0191865
\(489\) 0 0
\(490\) −3.47657e9 −1.33495
\(491\) −1.93067e9 −0.736077 −0.368039 0.929811i \(-0.619971\pi\)
−0.368039 + 0.929811i \(0.619971\pi\)
\(492\) 0 0
\(493\) 8.74638e7 0.0328749
\(494\) 1.33113e9 0.496793
\(495\) 0 0
\(496\) 8.96360e8 0.329835
\(497\) 2.89810e9 1.05893
\(498\) 0 0
\(499\) 3.11720e9 1.12308 0.561542 0.827448i \(-0.310208\pi\)
0.561542 + 0.827448i \(0.310208\pi\)
\(500\) 4.59303e8 0.164325
\(501\) 0 0
\(502\) 2.28842e9 0.807371
\(503\) 3.19404e8 0.111906 0.0559529 0.998433i \(-0.482180\pi\)
0.0559529 + 0.998433i \(0.482180\pi\)
\(504\) 0 0
\(505\) −2.71167e9 −0.936952
\(506\) 3.58599e9 1.23050
\(507\) 0 0
\(508\) 5.83581e8 0.197505
\(509\) −2.35968e9 −0.793124 −0.396562 0.918008i \(-0.629797\pi\)
−0.396562 + 0.918008i \(0.629797\pi\)
\(510\) 0 0
\(511\) −5.19312e9 −1.72169
\(512\) −2.09276e9 −0.689088
\(513\) 0 0
\(514\) −4.61216e9 −1.49808
\(515\) −1.88011e9 −0.606539
\(516\) 0 0
\(517\) −2.01644e9 −0.641754
\(518\) 1.43538e9 0.453745
\(519\) 0 0
\(520\) −2.63423e9 −0.821566
\(521\) −6.10979e9 −1.89276 −0.946378 0.323062i \(-0.895288\pi\)
−0.946378 + 0.323062i \(0.895288\pi\)
\(522\) 0 0
\(523\) 3.79857e8 0.116109 0.0580543 0.998313i \(-0.481510\pi\)
0.0580543 + 0.998313i \(0.481510\pi\)
\(524\) 9.40023e8 0.285417
\(525\) 0 0
\(526\) −1.98157e9 −0.593690
\(527\) −2.95137e8 −0.0878387
\(528\) 0 0
\(529\) −4.91022e8 −0.144214
\(530\) 2.40025e9 0.700310
\(531\) 0 0
\(532\) −3.58317e8 −0.103176
\(533\) −6.61131e9 −1.89122
\(534\) 0 0
\(535\) 2.61180e9 0.737398
\(536\) −3.48376e9 −0.977173
\(537\) 0 0
\(538\) 6.39382e9 1.77020
\(539\) 6.63177e9 1.82418
\(540\) 0 0
\(541\) −1.86189e8 −0.0505549 −0.0252774 0.999680i \(-0.508047\pi\)
−0.0252774 + 0.999680i \(0.508047\pi\)
\(542\) 1.47087e9 0.396805
\(543\) 0 0
\(544\) −3.38488e8 −0.0901461
\(545\) −1.91027e9 −0.505484
\(546\) 0 0
\(547\) 2.56203e8 0.0669311 0.0334656 0.999440i \(-0.489346\pi\)
0.0334656 + 0.999440i \(0.489346\pi\)
\(548\) 2.84157e8 0.0737611
\(549\) 0 0
\(550\) −1.47952e9 −0.379184
\(551\) −1.86428e8 −0.0474766
\(552\) 0 0
\(553\) −1.08881e10 −2.73787
\(554\) 4.11792e9 1.02895
\(555\) 0 0
\(556\) 7.60911e8 0.187746
\(557\) −3.54346e9 −0.868829 −0.434415 0.900713i \(-0.643045\pi\)
−0.434415 + 0.900713i \(0.643045\pi\)
\(558\) 0 0
\(559\) 3.12257e9 0.756085
\(560\) 6.23327e9 1.49988
\(561\) 0 0
\(562\) 3.02184e9 0.718117
\(563\) 5.23102e9 1.23540 0.617699 0.786414i \(-0.288065\pi\)
0.617699 + 0.786414i \(0.288065\pi\)
\(564\) 0 0
\(565\) −2.35504e9 −0.549325
\(566\) −7.30209e9 −1.69274
\(567\) 0 0
\(568\) −2.67903e9 −0.613421
\(569\) 2.02667e8 0.0461200 0.0230600 0.999734i \(-0.492659\pi\)
0.0230600 + 0.999734i \(0.492659\pi\)
\(570\) 0 0
\(571\) 1.61917e9 0.363969 0.181985 0.983301i \(-0.441748\pi\)
0.181985 + 0.983301i \(0.441748\pi\)
\(572\) −8.95343e8 −0.200034
\(573\) 0 0
\(574\) 1.35473e10 2.98993
\(575\) −1.20218e9 −0.263714
\(576\) 0 0
\(577\) −8.09171e8 −0.175358 −0.0876789 0.996149i \(-0.527945\pi\)
−0.0876789 + 0.996149i \(0.527945\pi\)
\(578\) −4.53136e9 −0.976070
\(579\) 0 0
\(580\) −6.57358e7 −0.0139895
\(581\) 9.54851e9 2.01985
\(582\) 0 0
\(583\) −4.57862e9 −0.956962
\(584\) 4.80057e9 0.997351
\(585\) 0 0
\(586\) −2.90528e9 −0.596413
\(587\) 3.34084e9 0.681746 0.340873 0.940109i \(-0.389277\pi\)
0.340873 + 0.940109i \(0.389277\pi\)
\(588\) 0 0
\(589\) 6.29079e8 0.126853
\(590\) −5.89209e8 −0.118110
\(591\) 0 0
\(592\) −1.53224e9 −0.303530
\(593\) −4.75989e9 −0.937358 −0.468679 0.883369i \(-0.655270\pi\)
−0.468679 + 0.883369i \(0.655270\pi\)
\(594\) 0 0
\(595\) −2.05237e9 −0.399436
\(596\) 1.35497e9 0.262160
\(597\) 0 0
\(598\) −5.53808e9 −1.05902
\(599\) −5.11303e9 −0.972041 −0.486020 0.873947i \(-0.661552\pi\)
−0.486020 + 0.873947i \(0.661552\pi\)
\(600\) 0 0
\(601\) −6.88520e9 −1.29377 −0.646883 0.762589i \(-0.723928\pi\)
−0.646883 + 0.762589i \(0.723928\pi\)
\(602\) −6.39848e9 −1.19533
\(603\) 0 0
\(604\) 1.13059e9 0.208775
\(605\) −2.47251e9 −0.453936
\(606\) 0 0
\(607\) −9.35321e9 −1.69746 −0.848732 0.528824i \(-0.822633\pi\)
−0.848732 + 0.528824i \(0.822633\pi\)
\(608\) 7.21482e8 0.130186
\(609\) 0 0
\(610\) −1.07150e8 −0.0191134
\(611\) 3.11413e9 0.552321
\(612\) 0 0
\(613\) 5.18941e9 0.909927 0.454963 0.890510i \(-0.349652\pi\)
0.454963 + 0.890510i \(0.349652\pi\)
\(614\) 1.21709e9 0.212195
\(615\) 0 0
\(616\) −1.02967e10 −1.77486
\(617\) −6.12264e9 −1.04940 −0.524700 0.851287i \(-0.675822\pi\)
−0.524700 + 0.851287i \(0.675822\pi\)
\(618\) 0 0
\(619\) 2.63490e9 0.446527 0.223263 0.974758i \(-0.428329\pi\)
0.223263 + 0.974758i \(0.428329\pi\)
\(620\) 2.21818e8 0.0373788
\(621\) 0 0
\(622\) −8.62230e9 −1.43667
\(623\) −1.02429e9 −0.169713
\(624\) 0 0
\(625\) −3.86759e9 −0.633666
\(626\) −7.95209e9 −1.29560
\(627\) 0 0
\(628\) 9.86016e8 0.158864
\(629\) 5.04508e8 0.0808334
\(630\) 0 0
\(631\) 2.67647e9 0.424092 0.212046 0.977260i \(-0.431987\pi\)
0.212046 + 0.977260i \(0.431987\pi\)
\(632\) 1.00650e10 1.58601
\(633\) 0 0
\(634\) 2.06403e9 0.321664
\(635\) −7.12480e9 −1.10424
\(636\) 0 0
\(637\) −1.02419e10 −1.56997
\(638\) 9.54548e8 0.145521
\(639\) 0 0
\(640\) −6.53438e9 −0.985313
\(641\) 1.12657e10 1.68949 0.844747 0.535166i \(-0.179751\pi\)
0.844747 + 0.535166i \(0.179751\pi\)
\(642\) 0 0
\(643\) 5.64942e9 0.838041 0.419021 0.907977i \(-0.362373\pi\)
0.419021 + 0.907977i \(0.362373\pi\)
\(644\) 1.49075e9 0.219941
\(645\) 0 0
\(646\) −9.58709e8 −0.139918
\(647\) −8.48556e9 −1.23173 −0.615865 0.787852i \(-0.711193\pi\)
−0.615865 + 0.787852i \(0.711193\pi\)
\(648\) 0 0
\(649\) 1.12395e9 0.161396
\(650\) 2.28491e9 0.326342
\(651\) 0 0
\(652\) −5.48572e8 −0.0775116
\(653\) 5.48317e9 0.770611 0.385306 0.922789i \(-0.374096\pi\)
0.385306 + 0.922789i \(0.374096\pi\)
\(654\) 0 0
\(655\) −1.14765e10 −1.59575
\(656\) −1.44615e10 −2.00009
\(657\) 0 0
\(658\) −6.38118e9 −0.873193
\(659\) 5.95072e9 0.809974 0.404987 0.914323i \(-0.367276\pi\)
0.404987 + 0.914323i \(0.367276\pi\)
\(660\) 0 0
\(661\) −8.08136e9 −1.08838 −0.544188 0.838963i \(-0.683162\pi\)
−0.544188 + 0.838963i \(0.683162\pi\)
\(662\) 1.57480e9 0.210971
\(663\) 0 0
\(664\) −8.82673e9 −1.17007
\(665\) 4.37460e9 0.576850
\(666\) 0 0
\(667\) 7.75621e8 0.101207
\(668\) −1.86299e8 −0.0241820
\(669\) 0 0
\(670\) −7.57838e9 −0.973453
\(671\) 2.04396e8 0.0261182
\(672\) 0 0
\(673\) −9.07433e9 −1.14752 −0.573762 0.819022i \(-0.694517\pi\)
−0.573762 + 0.819022i \(0.694517\pi\)
\(674\) 1.15961e9 0.145883
\(675\) 0 0
\(676\) 1.68067e8 0.0209251
\(677\) −1.13075e10 −1.40057 −0.700286 0.713862i \(-0.746944\pi\)
−0.700286 + 0.713862i \(0.746944\pi\)
\(678\) 0 0
\(679\) 1.52647e10 1.87131
\(680\) 1.89723e9 0.231387
\(681\) 0 0
\(682\) −3.22101e9 −0.388819
\(683\) −1.42517e10 −1.71157 −0.855783 0.517336i \(-0.826924\pi\)
−0.855783 + 0.517336i \(0.826924\pi\)
\(684\) 0 0
\(685\) −3.46921e9 −0.412395
\(686\) 6.72429e9 0.795265
\(687\) 0 0
\(688\) 6.83027e9 0.799610
\(689\) 7.07107e9 0.823603
\(690\) 0 0
\(691\) 1.15882e10 1.33612 0.668058 0.744109i \(-0.267126\pi\)
0.668058 + 0.744109i \(0.267126\pi\)
\(692\) −1.39597e9 −0.160141
\(693\) 0 0
\(694\) 1.37648e9 0.156319
\(695\) −9.28977e9 −1.04968
\(696\) 0 0
\(697\) 4.76161e9 0.532647
\(698\) −4.51843e9 −0.502914
\(699\) 0 0
\(700\) −6.15059e8 −0.0677757
\(701\) −3.16850e9 −0.347408 −0.173704 0.984798i \(-0.555574\pi\)
−0.173704 + 0.984798i \(0.555574\pi\)
\(702\) 0 0
\(703\) −1.07535e9 −0.116737
\(704\) 9.25587e9 0.999800
\(705\) 0 0
\(706\) −6.97600e9 −0.746088
\(707\) 1.63693e10 1.74206
\(708\) 0 0
\(709\) 7.17426e9 0.755988 0.377994 0.925808i \(-0.376614\pi\)
0.377994 + 0.925808i \(0.376614\pi\)
\(710\) −5.82782e9 −0.611086
\(711\) 0 0
\(712\) 9.46867e8 0.0983125
\(713\) −2.61724e9 −0.270415
\(714\) 0 0
\(715\) 1.09310e10 1.11838
\(716\) 4.05180e8 0.0412527
\(717\) 0 0
\(718\) 6.86017e8 0.0691670
\(719\) −3.09019e9 −0.310052 −0.155026 0.987910i \(-0.549546\pi\)
−0.155026 + 0.987910i \(0.549546\pi\)
\(720\) 0 0
\(721\) 1.13495e10 1.12773
\(722\) −8.80733e9 −0.870891
\(723\) 0 0
\(724\) 8.86836e8 0.0868477
\(725\) −3.20008e8 −0.0311873
\(726\) 0 0
\(727\) 1.26127e9 0.121741 0.0608707 0.998146i \(-0.480612\pi\)
0.0608707 + 0.998146i \(0.480612\pi\)
\(728\) 1.59019e10 1.52753
\(729\) 0 0
\(730\) 1.04429e10 0.993553
\(731\) −2.24894e9 −0.212945
\(732\) 0 0
\(733\) −1.05469e10 −0.989146 −0.494573 0.869136i \(-0.664676\pi\)
−0.494573 + 0.869136i \(0.664676\pi\)
\(734\) −2.73815e9 −0.255576
\(735\) 0 0
\(736\) −3.00168e9 −0.277519
\(737\) 1.44562e10 1.33021
\(738\) 0 0
\(739\) 7.89796e9 0.719879 0.359940 0.932976i \(-0.382797\pi\)
0.359940 + 0.932976i \(0.382797\pi\)
\(740\) −3.79177e8 −0.0343978
\(741\) 0 0
\(742\) −1.44894e10 −1.30208
\(743\) 7.69335e8 0.0688105 0.0344053 0.999408i \(-0.489046\pi\)
0.0344053 + 0.999408i \(0.489046\pi\)
\(744\) 0 0
\(745\) −1.65425e10 −1.46573
\(746\) 7.11545e9 0.627504
\(747\) 0 0
\(748\) 6.44846e8 0.0563379
\(749\) −1.57664e10 −1.37103
\(750\) 0 0
\(751\) 1.79359e10 1.54519 0.772596 0.634898i \(-0.218958\pi\)
0.772596 + 0.634898i \(0.218958\pi\)
\(752\) 6.81180e9 0.584117
\(753\) 0 0
\(754\) −1.47417e9 −0.125242
\(755\) −1.38032e10 −1.16725
\(756\) 0 0
\(757\) 4.00304e9 0.335394 0.167697 0.985839i \(-0.446367\pi\)
0.167697 + 0.985839i \(0.446367\pi\)
\(758\) 4.87661e9 0.406702
\(759\) 0 0
\(760\) −4.04393e9 −0.334161
\(761\) −3.91126e9 −0.321714 −0.160857 0.986978i \(-0.551426\pi\)
−0.160857 + 0.986978i \(0.551426\pi\)
\(762\) 0 0
\(763\) 1.15316e10 0.939839
\(764\) 3.07166e9 0.249199
\(765\) 0 0
\(766\) 1.20194e10 0.966234
\(767\) −1.73579e9 −0.138904
\(768\) 0 0
\(769\) 1.77498e10 1.40751 0.703756 0.710441i \(-0.251505\pi\)
0.703756 + 0.710441i \(0.251505\pi\)
\(770\) −2.23989e10 −1.76811
\(771\) 0 0
\(772\) 1.32009e9 0.103263
\(773\) −2.00177e10 −1.55879 −0.779393 0.626536i \(-0.784472\pi\)
−0.779393 + 0.626536i \(0.784472\pi\)
\(774\) 0 0
\(775\) 1.07983e9 0.0833296
\(776\) −1.41109e10 −1.08402
\(777\) 0 0
\(778\) −1.13023e10 −0.860472
\(779\) −1.01493e10 −0.769229
\(780\) 0 0
\(781\) 1.11169e10 0.835038
\(782\) 3.98865e9 0.298265
\(783\) 0 0
\(784\) −2.24030e10 −1.66035
\(785\) −1.20380e10 −0.888201
\(786\) 0 0
\(787\) −1.57588e10 −1.15242 −0.576212 0.817300i \(-0.695470\pi\)
−0.576212 + 0.817300i \(0.695470\pi\)
\(788\) −1.55784e9 −0.113418
\(789\) 0 0
\(790\) 2.18949e10 1.57997
\(791\) 1.42165e10 1.02135
\(792\) 0 0
\(793\) −3.15662e8 −0.0224784
\(794\) 5.07745e9 0.359976
\(795\) 0 0
\(796\) −1.32796e9 −0.0933234
\(797\) 9.19325e9 0.643228 0.321614 0.946871i \(-0.395775\pi\)
0.321614 + 0.946871i \(0.395775\pi\)
\(798\) 0 0
\(799\) −2.24286e9 −0.155557
\(800\) 1.23844e9 0.0855185
\(801\) 0 0
\(802\) 2.30173e10 1.57559
\(803\) −1.99205e10 −1.35767
\(804\) 0 0
\(805\) −1.82003e10 −1.22968
\(806\) 4.97443e9 0.334634
\(807\) 0 0
\(808\) −1.51320e10 −1.00915
\(809\) 1.69661e10 1.12658 0.563290 0.826259i \(-0.309535\pi\)
0.563290 + 0.826259i \(0.309535\pi\)
\(810\) 0 0
\(811\) −2.19205e10 −1.44303 −0.721517 0.692397i \(-0.756555\pi\)
−0.721517 + 0.692397i \(0.756555\pi\)
\(812\) 3.96822e8 0.0260105
\(813\) 0 0
\(814\) 5.50602e9 0.357810
\(815\) 6.69738e9 0.433364
\(816\) 0 0
\(817\) 4.79359e9 0.307527
\(818\) −2.93422e10 −1.87437
\(819\) 0 0
\(820\) −3.57872e9 −0.226662
\(821\) −1.97496e10 −1.24554 −0.622770 0.782405i \(-0.713993\pi\)
−0.622770 + 0.782405i \(0.713993\pi\)
\(822\) 0 0
\(823\) 3.26341e9 0.204067 0.102033 0.994781i \(-0.467465\pi\)
0.102033 + 0.994781i \(0.467465\pi\)
\(824\) −1.04916e10 −0.653276
\(825\) 0 0
\(826\) 3.55683e9 0.219600
\(827\) −7.08713e9 −0.435713 −0.217857 0.975981i \(-0.569907\pi\)
−0.217857 + 0.975981i \(0.569907\pi\)
\(828\) 0 0
\(829\) 3.03587e10 1.85073 0.925365 0.379078i \(-0.123759\pi\)
0.925365 + 0.379078i \(0.123759\pi\)
\(830\) −1.92012e10 −1.16561
\(831\) 0 0
\(832\) −1.42945e10 −0.860471
\(833\) 7.37643e9 0.442169
\(834\) 0 0
\(835\) 2.27447e9 0.135200
\(836\) −1.37448e9 −0.0813611
\(837\) 0 0
\(838\) 2.10092e10 1.23326
\(839\) −1.35904e10 −0.794448 −0.397224 0.917722i \(-0.630026\pi\)
−0.397224 + 0.917722i \(0.630026\pi\)
\(840\) 0 0
\(841\) −1.70434e10 −0.988031
\(842\) 2.90069e10 1.67459
\(843\) 0 0
\(844\) 5.59143e8 0.0320128
\(845\) −2.05189e9 −0.116992
\(846\) 0 0
\(847\) 1.49256e10 0.843997
\(848\) 1.54672e10 0.871014
\(849\) 0 0
\(850\) −1.64565e9 −0.0919116
\(851\) 4.47393e9 0.248849
\(852\) 0 0
\(853\) 1.23490e10 0.681255 0.340628 0.940198i \(-0.389361\pi\)
0.340628 + 0.940198i \(0.389361\pi\)
\(854\) 6.46826e8 0.0355373
\(855\) 0 0
\(856\) 1.45747e10 0.794219
\(857\) 1.91652e10 1.04011 0.520057 0.854132i \(-0.325911\pi\)
0.520057 + 0.854132i \(0.325911\pi\)
\(858\) 0 0
\(859\) −2.69718e10 −1.45189 −0.725946 0.687751i \(-0.758598\pi\)
−0.725946 + 0.687751i \(0.758598\pi\)
\(860\) 1.69025e9 0.0906166
\(861\) 0 0
\(862\) 1.29171e10 0.686893
\(863\) 1.15124e9 0.0609715 0.0304858 0.999535i \(-0.490295\pi\)
0.0304858 + 0.999535i \(0.490295\pi\)
\(864\) 0 0
\(865\) 1.70430e10 0.895344
\(866\) −1.01275e10 −0.529896
\(867\) 0 0
\(868\) −1.33903e9 −0.0694978
\(869\) −4.17659e10 −2.15900
\(870\) 0 0
\(871\) −2.23257e10 −1.14483
\(872\) −1.06599e10 −0.544435
\(873\) 0 0
\(874\) −8.50174e9 −0.430743
\(875\) 3.38504e10 1.70819
\(876\) 0 0
\(877\) −2.50007e9 −0.125157 −0.0625784 0.998040i \(-0.519932\pi\)
−0.0625784 + 0.998040i \(0.519932\pi\)
\(878\) 2.32624e10 1.15991
\(879\) 0 0
\(880\) 2.39104e10 1.18276
\(881\) −1.84978e10 −0.911392 −0.455696 0.890136i \(-0.650610\pi\)
−0.455696 + 0.890136i \(0.650610\pi\)
\(882\) 0 0
\(883\) 3.27036e10 1.59857 0.799287 0.600950i \(-0.205211\pi\)
0.799287 + 0.600950i \(0.205211\pi\)
\(884\) −9.95879e8 −0.0484868
\(885\) 0 0
\(886\) −1.73733e10 −0.839196
\(887\) 2.66571e10 1.28257 0.641285 0.767303i \(-0.278402\pi\)
0.641285 + 0.767303i \(0.278402\pi\)
\(888\) 0 0
\(889\) 4.30097e10 2.05310
\(890\) 2.05976e9 0.0979382
\(891\) 0 0
\(892\) 5.02130e9 0.236886
\(893\) 4.78063e9 0.224649
\(894\) 0 0
\(895\) −4.94674e9 −0.230642
\(896\) 3.94455e10 1.83198
\(897\) 0 0
\(898\) 1.21977e10 0.562096
\(899\) −6.96680e8 −0.0319797
\(900\) 0 0
\(901\) −5.09274e9 −0.231961
\(902\) 5.19665e10 2.35777
\(903\) 0 0
\(904\) −1.31419e10 −0.591654
\(905\) −1.08272e10 −0.485562
\(906\) 0 0
\(907\) −2.98097e10 −1.32658 −0.663289 0.748364i \(-0.730840\pi\)
−0.663289 + 0.748364i \(0.730840\pi\)
\(908\) 3.94266e9 0.174779
\(909\) 0 0
\(910\) 3.45921e10 1.52171
\(911\) 1.60230e10 0.702149 0.351074 0.936348i \(-0.385816\pi\)
0.351074 + 0.936348i \(0.385816\pi\)
\(912\) 0 0
\(913\) 3.66274e10 1.59279
\(914\) 4.02379e10 1.74311
\(915\) 0 0
\(916\) −6.74815e9 −0.290102
\(917\) 6.92794e10 2.96696
\(918\) 0 0
\(919\) 2.67157e10 1.13543 0.567717 0.823224i \(-0.307827\pi\)
0.567717 + 0.823224i \(0.307827\pi\)
\(920\) 1.68245e10 0.712336
\(921\) 0 0
\(922\) −3.62349e8 −0.0152254
\(923\) −1.71686e10 −0.718670
\(924\) 0 0
\(925\) −1.84587e9 −0.0766839
\(926\) 6.21430e9 0.257190
\(927\) 0 0
\(928\) −7.99012e8 −0.0328198
\(929\) 1.01936e10 0.417130 0.208565 0.978009i \(-0.433121\pi\)
0.208565 + 0.978009i \(0.433121\pi\)
\(930\) 0 0
\(931\) −1.57228e10 −0.638564
\(932\) −3.03310e9 −0.122725
\(933\) 0 0
\(934\) 4.05008e10 1.62648
\(935\) −7.87277e9 −0.314983
\(936\) 0 0
\(937\) −2.50793e10 −0.995923 −0.497962 0.867199i \(-0.665918\pi\)
−0.497962 + 0.867199i \(0.665918\pi\)
\(938\) 4.57478e10 1.80992
\(939\) 0 0
\(940\) 1.68568e9 0.0661955
\(941\) 5.38850e9 0.210816 0.105408 0.994429i \(-0.466385\pi\)
0.105408 + 0.994429i \(0.466385\pi\)
\(942\) 0 0
\(943\) 4.22256e10 1.63978
\(944\) −3.79686e9 −0.146900
\(945\) 0 0
\(946\) −2.45442e10 −0.942604
\(947\) 1.51551e10 0.579876 0.289938 0.957045i \(-0.406365\pi\)
0.289938 + 0.957045i \(0.406365\pi\)
\(948\) 0 0
\(949\) 3.07645e10 1.16847
\(950\) 3.50767e9 0.132735
\(951\) 0 0
\(952\) −1.14529e10 −0.430215
\(953\) −4.14430e9 −0.155105 −0.0775526 0.996988i \(-0.524711\pi\)
−0.0775526 + 0.996988i \(0.524711\pi\)
\(954\) 0 0
\(955\) −3.75011e10 −1.39326
\(956\) −4.04012e9 −0.149552
\(957\) 0 0
\(958\) 2.94703e9 0.108294
\(959\) 2.09423e10 0.766760
\(960\) 0 0
\(961\) −2.51617e10 −0.914553
\(962\) −8.50331e9 −0.307946
\(963\) 0 0
\(964\) −7.36552e9 −0.264810
\(965\) −1.61167e10 −0.577338
\(966\) 0 0
\(967\) −7.19424e9 −0.255854 −0.127927 0.991784i \(-0.540832\pi\)
−0.127927 + 0.991784i \(0.540832\pi\)
\(968\) −1.37974e10 −0.488915
\(969\) 0 0
\(970\) −3.06960e10 −1.07989
\(971\) 3.04216e10 1.06639 0.533193 0.845993i \(-0.320992\pi\)
0.533193 + 0.845993i \(0.320992\pi\)
\(972\) 0 0
\(973\) 5.60788e10 1.95166
\(974\) −1.44147e10 −0.499860
\(975\) 0 0
\(976\) −6.90475e8 −0.0237724
\(977\) 1.19680e9 0.0410572 0.0205286 0.999789i \(-0.493465\pi\)
0.0205286 + 0.999789i \(0.493465\pi\)
\(978\) 0 0
\(979\) −3.92912e9 −0.133831
\(980\) −5.54395e9 −0.188160
\(981\) 0 0
\(982\) −2.34366e10 −0.789778
\(983\) −5.01551e10 −1.68414 −0.842069 0.539370i \(-0.818663\pi\)
−0.842069 + 0.539370i \(0.818663\pi\)
\(984\) 0 0
\(985\) 1.90193e10 0.634116
\(986\) 1.06173e9 0.0352733
\(987\) 0 0
\(988\) 2.12270e9 0.0700228
\(989\) −1.99434e10 −0.655561
\(990\) 0 0
\(991\) 2.70432e10 0.882674 0.441337 0.897341i \(-0.354504\pi\)
0.441337 + 0.897341i \(0.354504\pi\)
\(992\) 2.69618e9 0.0876915
\(993\) 0 0
\(994\) 3.51803e10 1.13618
\(995\) 1.62128e10 0.521767
\(996\) 0 0
\(997\) 1.18645e10 0.379154 0.189577 0.981866i \(-0.439288\pi\)
0.189577 + 0.981866i \(0.439288\pi\)
\(998\) 3.78400e10 1.20502
\(999\) 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.1.12 17
3.2 odd 2 177.8.a.b.1.6 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.6 17 3.2 odd 2
531.8.a.d.1.12 17 1.1 even 1 trivial