Properties

Label 9.42.a.c.1.1
Level $9$
Weight $42$
Character 9.1
Self dual yes
Analytic conductor $95.825$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(95.8245034108\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 196497525461 x^{2} + 10360343667016365 x + 6095744045744274504000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{10}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-433547.\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.58383e6 q^{2} +4.47715e12 q^{4} -2.55548e14 q^{5} +1.98945e17 q^{7} -5.88630e18 q^{8} +O(q^{10})\) \(q-2.58383e6 q^{2} +4.47715e12 q^{4} -2.55548e14 q^{5} +1.98945e17 q^{7} -5.88630e18 q^{8} +6.60294e20 q^{10} +2.76444e21 q^{11} -8.56075e22 q^{13} -5.14041e23 q^{14} +5.36384e24 q^{16} +3.38977e24 q^{17} +2.99407e26 q^{19} -1.14413e27 q^{20} -7.14285e27 q^{22} +1.40889e28 q^{23} +1.98303e28 q^{25} +2.21195e29 q^{26} +8.90709e29 q^{28} +4.94074e28 q^{29} +5.76050e30 q^{31} -9.15126e29 q^{32} -8.75860e30 q^{34} -5.08402e31 q^{35} -1.16818e32 q^{37} -7.73616e32 q^{38} +1.50424e33 q^{40} -1.48580e32 q^{41} +2.64197e32 q^{43} +1.23768e34 q^{44} -3.64032e34 q^{46} -2.16318e34 q^{47} -4.98839e33 q^{49} -5.12381e34 q^{50} -3.83278e35 q^{52} -5.57170e34 q^{53} -7.06449e35 q^{55} -1.17105e36 q^{56} -1.27660e35 q^{58} +2.05748e36 q^{59} +1.01936e36 q^{61} -1.48842e37 q^{62} -9.43067e36 q^{64} +2.18769e37 q^{65} -2.42878e37 q^{67} +1.51765e37 q^{68} +1.31362e38 q^{70} -5.77043e36 q^{71} -1.22382e38 q^{73} +3.01839e38 q^{74} +1.34049e39 q^{76} +5.49973e38 q^{77} -7.70707e38 q^{79} -1.37072e39 q^{80} +3.83906e38 q^{82} -1.70727e39 q^{83} -8.66251e38 q^{85} -6.82641e38 q^{86} -1.62723e40 q^{88} +4.08353e39 q^{89} -1.70312e40 q^{91} +6.30780e40 q^{92} +5.58928e40 q^{94} -7.65129e40 q^{95} +1.60920e40 q^{97} +1.28891e40 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 69822q^{2} + 5352947588932q^{4} - 118536963776280q^{5} + 150256264888927136q^{7} - 6279626248119395784q^{8} + O(q^{10}) \) \( 4q + 69822q^{2} + 5352947588932q^{4} - 118536963776280q^{5} + 150256264888927136q^{7} - 6279626248119395784q^{8} + \)\(72\!\cdots\!40\)\(q^{10} - \)\(72\!\cdots\!56\)\(q^{11} - \)\(88\!\cdots\!08\)\(q^{13} - \)\(49\!\cdots\!68\)\(q^{14} - \)\(59\!\cdots\!00\)\(q^{16} + \)\(38\!\cdots\!88\)\(q^{17} + \)\(26\!\cdots\!48\)\(q^{19} - \)\(78\!\cdots\!60\)\(q^{20} - \)\(63\!\cdots\!76\)\(q^{22} + \)\(15\!\cdots\!32\)\(q^{23} + \)\(11\!\cdots\!00\)\(q^{25} + \)\(62\!\cdots\!52\)\(q^{26} + \)\(68\!\cdots\!12\)\(q^{28} + \)\(10\!\cdots\!64\)\(q^{29} + \)\(92\!\cdots\!04\)\(q^{31} - \)\(19\!\cdots\!56\)\(q^{32} + \)\(92\!\cdots\!04\)\(q^{34} - \)\(20\!\cdots\!40\)\(q^{35} + \)\(20\!\cdots\!56\)\(q^{37} - \)\(11\!\cdots\!28\)\(q^{38} + \)\(25\!\cdots\!00\)\(q^{40} - \)\(23\!\cdots\!04\)\(q^{41} + \)\(39\!\cdots\!60\)\(q^{43} + \)\(18\!\cdots\!04\)\(q^{44} - \)\(44\!\cdots\!16\)\(q^{46} + \)\(88\!\cdots\!20\)\(q^{47} - \)\(33\!\cdots\!80\)\(q^{49} + \)\(21\!\cdots\!50\)\(q^{50} - \)\(59\!\cdots\!08\)\(q^{52} - \)\(95\!\cdots\!28\)\(q^{53} + \)\(12\!\cdots\!60\)\(q^{55} - \)\(19\!\cdots\!20\)\(q^{56} + \)\(38\!\cdots\!16\)\(q^{58} + \)\(18\!\cdots\!08\)\(q^{59} + \)\(53\!\cdots\!40\)\(q^{61} - \)\(14\!\cdots\!52\)\(q^{62} - \)\(38\!\cdots\!92\)\(q^{64} + \)\(97\!\cdots\!80\)\(q^{65} - \)\(73\!\cdots\!28\)\(q^{67} + \)\(89\!\cdots\!24\)\(q^{68} - \)\(12\!\cdots\!80\)\(q^{70} + \)\(84\!\cdots\!52\)\(q^{71} - \)\(44\!\cdots\!32\)\(q^{73} + \)\(29\!\cdots\!12\)\(q^{74} + \)\(11\!\cdots\!72\)\(q^{76} - \)\(83\!\cdots\!52\)\(q^{77} + \)\(14\!\cdots\!80\)\(q^{79} - \)\(15\!\cdots\!80\)\(q^{80} + \)\(61\!\cdots\!12\)\(q^{82} - \)\(15\!\cdots\!04\)\(q^{83} - \)\(28\!\cdots\!60\)\(q^{85} + \)\(12\!\cdots\!24\)\(q^{86} - \)\(13\!\cdots\!96\)\(q^{88} + \)\(39\!\cdots\!72\)\(q^{89} - \)\(88\!\cdots\!16\)\(q^{91} + \)\(65\!\cdots\!44\)\(q^{92} + \)\(98\!\cdots\!32\)\(q^{94} - \)\(78\!\cdots\!00\)\(q^{95} + \)\(36\!\cdots\!52\)\(q^{97} + \)\(68\!\cdots\!22\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58383e6 −1.74240 −0.871202 0.490924i \(-0.836659\pi\)
−0.871202 + 0.490924i \(0.836659\pi\)
\(3\) 0 0
\(4\) 4.47715e12 2.03597
\(5\) −2.55548e14 −1.19836 −0.599181 0.800613i \(-0.704507\pi\)
−0.599181 + 0.800613i \(0.704507\pi\)
\(6\) 0 0
\(7\) 1.98945e17 0.942375 0.471188 0.882033i \(-0.343825\pi\)
0.471188 + 0.882033i \(0.343825\pi\)
\(8\) −5.88630e18 −1.80509
\(9\) 0 0
\(10\) 6.60294e20 2.08803
\(11\) 2.76444e21 1.23896 0.619480 0.785012i \(-0.287343\pi\)
0.619480 + 0.785012i \(0.287343\pi\)
\(12\) 0 0
\(13\) −8.56075e22 −1.24932 −0.624659 0.780898i \(-0.714762\pi\)
−0.624659 + 0.780898i \(0.714762\pi\)
\(14\) −5.14041e23 −1.64200
\(15\) 0 0
\(16\) 5.36384e24 1.10922
\(17\) 3.38977e24 0.202287 0.101143 0.994872i \(-0.467750\pi\)
0.101143 + 0.994872i \(0.467750\pi\)
\(18\) 0 0
\(19\) 2.99407e26 1.82731 0.913656 0.406488i \(-0.133247\pi\)
0.913656 + 0.406488i \(0.133247\pi\)
\(20\) −1.14413e27 −2.43983
\(21\) 0 0
\(22\) −7.14285e27 −2.15877
\(23\) 1.40889e28 1.71181 0.855904 0.517134i \(-0.173001\pi\)
0.855904 + 0.517134i \(0.173001\pi\)
\(24\) 0 0
\(25\) 1.98303e28 0.436072
\(26\) 2.21195e29 2.17682
\(27\) 0 0
\(28\) 8.90709e29 1.91865
\(29\) 4.94074e28 0.0518361 0.0259181 0.999664i \(-0.491749\pi\)
0.0259181 + 0.999664i \(0.491749\pi\)
\(30\) 0 0
\(31\) 5.76050e30 1.54009 0.770044 0.637991i \(-0.220234\pi\)
0.770044 + 0.637991i \(0.220234\pi\)
\(32\) −9.15126e29 −0.127616
\(33\) 0 0
\(34\) −8.75860e30 −0.352466
\(35\) −5.08402e31 −1.12931
\(36\) 0 0
\(37\) −1.16818e32 −0.830569 −0.415284 0.909692i \(-0.636318\pi\)
−0.415284 + 0.909692i \(0.636318\pi\)
\(38\) −7.73616e32 −3.18392
\(39\) 0 0
\(40\) 1.50424e33 2.16315
\(41\) −1.48580e32 −0.128793 −0.0643964 0.997924i \(-0.520512\pi\)
−0.0643964 + 0.997924i \(0.520512\pi\)
\(42\) 0 0
\(43\) 2.64197e32 0.0862631 0.0431315 0.999069i \(-0.486267\pi\)
0.0431315 + 0.999069i \(0.486267\pi\)
\(44\) 1.23768e34 2.52249
\(45\) 0 0
\(46\) −3.64032e34 −2.98266
\(47\) −2.16318e34 −1.14048 −0.570238 0.821480i \(-0.693149\pi\)
−0.570238 + 0.821480i \(0.693149\pi\)
\(48\) 0 0
\(49\) −4.98839e33 −0.111928
\(50\) −5.12381e34 −0.759814
\(51\) 0 0
\(52\) −3.83278e35 −2.54358
\(53\) −5.57170e34 −0.250226 −0.125113 0.992142i \(-0.539929\pi\)
−0.125113 + 0.992142i \(0.539929\pi\)
\(54\) 0 0
\(55\) −7.06449e35 −1.48472
\(56\) −1.17105e36 −1.70107
\(57\) 0 0
\(58\) −1.27660e35 −0.0903195
\(59\) 2.05748e36 1.02534 0.512672 0.858584i \(-0.328656\pi\)
0.512672 + 0.858584i \(0.328656\pi\)
\(60\) 0 0
\(61\) 1.01936e36 0.256489 0.128244 0.991743i \(-0.459066\pi\)
0.128244 + 0.991743i \(0.459066\pi\)
\(62\) −1.48842e37 −2.68346
\(63\) 0 0
\(64\) −9.43067e36 −0.886856
\(65\) 2.18769e37 1.49714
\(66\) 0 0
\(67\) −2.42878e37 −0.893006 −0.446503 0.894782i \(-0.647331\pi\)
−0.446503 + 0.894782i \(0.647331\pi\)
\(68\) 1.51765e37 0.411851
\(69\) 0 0
\(70\) 1.31362e38 1.96771
\(71\) −5.77043e36 −0.0646268 −0.0323134 0.999478i \(-0.510287\pi\)
−0.0323134 + 0.999478i \(0.510287\pi\)
\(72\) 0 0
\(73\) −1.22382e38 −0.775532 −0.387766 0.921758i \(-0.626753\pi\)
−0.387766 + 0.921758i \(0.626753\pi\)
\(74\) 3.01839e38 1.44719
\(75\) 0 0
\(76\) 1.34049e39 3.72036
\(77\) 5.49973e38 1.16757
\(78\) 0 0
\(79\) −7.70707e38 −0.967240 −0.483620 0.875278i \(-0.660678\pi\)
−0.483620 + 0.875278i \(0.660678\pi\)
\(80\) −1.37072e39 −1.32924
\(81\) 0 0
\(82\) 3.83906e38 0.224409
\(83\) −1.70727e39 −0.778397 −0.389198 0.921154i \(-0.627248\pi\)
−0.389198 + 0.921154i \(0.627248\pi\)
\(84\) 0 0
\(85\) −8.66251e38 −0.242413
\(86\) −6.82641e38 −0.150305
\(87\) 0 0
\(88\) −1.62723e40 −2.23643
\(89\) 4.08353e39 0.445185 0.222592 0.974912i \(-0.428548\pi\)
0.222592 + 0.974912i \(0.428548\pi\)
\(90\) 0 0
\(91\) −1.70312e40 −1.17733
\(92\) 6.30780e40 3.48520
\(93\) 0 0
\(94\) 5.58928e40 1.98717
\(95\) −7.65129e40 −2.18978
\(96\) 0 0
\(97\) 1.60920e40 0.300462 0.150231 0.988651i \(-0.451998\pi\)
0.150231 + 0.988651i \(0.451998\pi\)
\(98\) 1.28891e40 0.195025
\(99\) 0 0
\(100\) 8.87832e40 0.887832
\(101\) −8.47485e39 −0.0691105 −0.0345552 0.999403i \(-0.511001\pi\)
−0.0345552 + 0.999403i \(0.511001\pi\)
\(102\) 0 0
\(103\) 2.25909e41 1.23246 0.616228 0.787568i \(-0.288660\pi\)
0.616228 + 0.787568i \(0.288660\pi\)
\(104\) 5.03911e41 2.25513
\(105\) 0 0
\(106\) 1.43963e41 0.435995
\(107\) 6.78077e41 1.69399 0.846996 0.531599i \(-0.178409\pi\)
0.846996 + 0.531599i \(0.178409\pi\)
\(108\) 0 0
\(109\) 7.17358e41 1.22601 0.613003 0.790080i \(-0.289961\pi\)
0.613003 + 0.790080i \(0.289961\pi\)
\(110\) 1.82534e42 2.58699
\(111\) 0 0
\(112\) 1.06711e42 1.04530
\(113\) −1.60519e42 −1.31045 −0.655223 0.755435i \(-0.727425\pi\)
−0.655223 + 0.755435i \(0.727425\pi\)
\(114\) 0 0
\(115\) −3.60039e42 −2.05137
\(116\) 2.21204e41 0.105537
\(117\) 0 0
\(118\) −5.31619e42 −1.78657
\(119\) 6.74380e41 0.190630
\(120\) 0 0
\(121\) 2.66363e42 0.535024
\(122\) −2.63386e42 −0.446907
\(123\) 0 0
\(124\) 2.57906e43 3.13558
\(125\) 6.55340e42 0.675790
\(126\) 0 0
\(127\) 2.84948e42 0.212222 0.106111 0.994354i \(-0.466160\pi\)
0.106111 + 0.994354i \(0.466160\pi\)
\(128\) 2.63796e43 1.67288
\(129\) 0 0
\(130\) −5.65261e43 −2.60862
\(131\) −2.19486e43 −0.865656 −0.432828 0.901477i \(-0.642484\pi\)
−0.432828 + 0.901477i \(0.642484\pi\)
\(132\) 0 0
\(133\) 5.95656e43 1.72201
\(134\) 6.27555e43 1.55598
\(135\) 0 0
\(136\) −1.99532e43 −0.365145
\(137\) −3.35892e43 −0.528966 −0.264483 0.964390i \(-0.585201\pi\)
−0.264483 + 0.964390i \(0.585201\pi\)
\(138\) 0 0
\(139\) 4.50230e43 0.526782 0.263391 0.964689i \(-0.415159\pi\)
0.263391 + 0.964689i \(0.415159\pi\)
\(140\) −2.27619e44 −2.29924
\(141\) 0 0
\(142\) 1.49098e43 0.112606
\(143\) −2.36657e44 −1.54786
\(144\) 0 0
\(145\) −1.26260e43 −0.0621185
\(146\) 3.16215e44 1.35129
\(147\) 0 0
\(148\) −5.23013e44 −1.69102
\(149\) 2.36545e44 0.666186 0.333093 0.942894i \(-0.391908\pi\)
0.333093 + 0.942894i \(0.391908\pi\)
\(150\) 0 0
\(151\) −4.16293e44 −0.892015 −0.446007 0.895029i \(-0.647154\pi\)
−0.446007 + 0.895029i \(0.647154\pi\)
\(152\) −1.76240e45 −3.29845
\(153\) 0 0
\(154\) −1.42104e45 −2.03437
\(155\) −1.47209e45 −1.84558
\(156\) 0 0
\(157\) 1.20032e45 1.15706 0.578528 0.815663i \(-0.303627\pi\)
0.578528 + 0.815663i \(0.303627\pi\)
\(158\) 1.99138e45 1.68532
\(159\) 0 0
\(160\) 2.33859e44 0.152931
\(161\) 2.80291e45 1.61317
\(162\) 0 0
\(163\) 2.08406e45 0.931246 0.465623 0.884983i \(-0.345830\pi\)
0.465623 + 0.884983i \(0.345830\pi\)
\(164\) −6.65216e44 −0.262219
\(165\) 0 0
\(166\) 4.41129e45 1.35628
\(167\) 3.55110e45 0.965328 0.482664 0.875806i \(-0.339669\pi\)
0.482664 + 0.875806i \(0.339669\pi\)
\(168\) 0 0
\(169\) 2.63319e45 0.560795
\(170\) 2.23825e45 0.422382
\(171\) 0 0
\(172\) 1.18285e45 0.175629
\(173\) −1.13689e46 −1.49890 −0.749449 0.662062i \(-0.769681\pi\)
−0.749449 + 0.662062i \(0.769681\pi\)
\(174\) 0 0
\(175\) 3.94514e45 0.410944
\(176\) 1.48280e46 1.37427
\(177\) 0 0
\(178\) −1.05511e46 −0.775692
\(179\) −7.48686e45 −0.490696 −0.245348 0.969435i \(-0.578902\pi\)
−0.245348 + 0.969435i \(0.578902\pi\)
\(180\) 0 0
\(181\) −5.11394e45 −0.266898 −0.133449 0.991056i \(-0.542605\pi\)
−0.133449 + 0.991056i \(0.542605\pi\)
\(182\) 4.40057e46 2.05138
\(183\) 0 0
\(184\) −8.29313e46 −3.08996
\(185\) 2.98527e46 0.995322
\(186\) 0 0
\(187\) 9.37084e45 0.250626
\(188\) −9.68487e46 −2.32198
\(189\) 0 0
\(190\) 1.97696e47 3.81549
\(191\) 6.00314e46 1.04039 0.520193 0.854049i \(-0.325860\pi\)
0.520193 + 0.854049i \(0.325860\pi\)
\(192\) 0 0
\(193\) −9.10745e45 −0.127489 −0.0637444 0.997966i \(-0.520304\pi\)
−0.0637444 + 0.997966i \(0.520304\pi\)
\(194\) −4.15790e46 −0.523526
\(195\) 0 0
\(196\) −2.23338e46 −0.227883
\(197\) 3.20025e46 0.294189 0.147094 0.989122i \(-0.453008\pi\)
0.147094 + 0.989122i \(0.453008\pi\)
\(198\) 0 0
\(199\) −4.23622e46 −0.316585 −0.158293 0.987392i \(-0.550599\pi\)
−0.158293 + 0.987392i \(0.550599\pi\)
\(200\) −1.16727e47 −0.787148
\(201\) 0 0
\(202\) 2.18976e46 0.120418
\(203\) 9.82936e45 0.0488491
\(204\) 0 0
\(205\) 3.79694e46 0.154340
\(206\) −5.83712e47 −2.14744
\(207\) 0 0
\(208\) −4.59184e47 −1.38576
\(209\) 8.27693e47 2.26397
\(210\) 0 0
\(211\) 5.34998e47 1.20382 0.601911 0.798563i \(-0.294406\pi\)
0.601911 + 0.798563i \(0.294406\pi\)
\(212\) −2.49454e47 −0.509454
\(213\) 0 0
\(214\) −1.75204e48 −2.95162
\(215\) −6.75152e46 −0.103374
\(216\) 0 0
\(217\) 1.14602e48 1.45134
\(218\) −1.85353e48 −2.13620
\(219\) 0 0
\(220\) −3.16288e48 −3.02286
\(221\) −2.90190e47 −0.252721
\(222\) 0 0
\(223\) 2.57714e48 1.86590 0.932950 0.360006i \(-0.117225\pi\)
0.932950 + 0.360006i \(0.117225\pi\)
\(224\) −1.82060e47 −0.120263
\(225\) 0 0
\(226\) 4.14755e48 2.28333
\(227\) −2.96815e48 −1.49264 −0.746320 0.665587i \(-0.768181\pi\)
−0.746320 + 0.665587i \(0.768181\pi\)
\(228\) 0 0
\(229\) −2.39386e48 −1.00571 −0.502854 0.864371i \(-0.667717\pi\)
−0.502854 + 0.864371i \(0.667717\pi\)
\(230\) 9.30279e48 3.57431
\(231\) 0 0
\(232\) −2.90827e47 −0.0935686
\(233\) 2.35236e48 0.692960 0.346480 0.938057i \(-0.387377\pi\)
0.346480 + 0.938057i \(0.387377\pi\)
\(234\) 0 0
\(235\) 5.52796e48 1.36670
\(236\) 9.21167e48 2.08758
\(237\) 0 0
\(238\) −1.74248e48 −0.332155
\(239\) 5.65009e48 0.988322 0.494161 0.869370i \(-0.335475\pi\)
0.494161 + 0.869370i \(0.335475\pi\)
\(240\) 0 0
\(241\) 1.15148e49 1.69788 0.848940 0.528490i \(-0.177241\pi\)
0.848940 + 0.528490i \(0.177241\pi\)
\(242\) −6.88236e48 −0.932229
\(243\) 0 0
\(244\) 4.56385e48 0.522204
\(245\) 1.27477e48 0.134131
\(246\) 0 0
\(247\) −2.56315e49 −2.28289
\(248\) −3.39080e49 −2.77999
\(249\) 0 0
\(250\) −1.69329e49 −1.17750
\(251\) −1.49455e48 −0.0957634 −0.0478817 0.998853i \(-0.515247\pi\)
−0.0478817 + 0.998853i \(0.515247\pi\)
\(252\) 0 0
\(253\) 3.89479e49 2.12086
\(254\) −7.36258e48 −0.369776
\(255\) 0 0
\(256\) −4.74222e49 −2.02797
\(257\) 5.98105e48 0.236129 0.118064 0.993006i \(-0.462331\pi\)
0.118064 + 0.993006i \(0.462331\pi\)
\(258\) 0 0
\(259\) −2.32405e49 −0.782707
\(260\) 9.79460e49 3.04813
\(261\) 0 0
\(262\) 5.67115e49 1.50832
\(263\) −3.57534e49 −0.879476 −0.439738 0.898126i \(-0.644929\pi\)
−0.439738 + 0.898126i \(0.644929\pi\)
\(264\) 0 0
\(265\) 1.42384e49 0.299861
\(266\) −1.53907e50 −3.00045
\(267\) 0 0
\(268\) −1.08740e50 −1.81814
\(269\) −2.52785e48 −0.0391589 −0.0195794 0.999808i \(-0.506233\pi\)
−0.0195794 + 0.999808i \(0.506233\pi\)
\(270\) 0 0
\(271\) 5.81899e49 0.774421 0.387211 0.921991i \(-0.373439\pi\)
0.387211 + 0.921991i \(0.373439\pi\)
\(272\) 1.81822e49 0.224380
\(273\) 0 0
\(274\) 8.67887e49 0.921673
\(275\) 5.48197e49 0.540277
\(276\) 0 0
\(277\) −1.51629e50 −1.28810 −0.644048 0.764985i \(-0.722746\pi\)
−0.644048 + 0.764985i \(0.722746\pi\)
\(278\) −1.16332e50 −0.917867
\(279\) 0 0
\(280\) 2.99261e50 2.03850
\(281\) 1.12012e50 0.709226 0.354613 0.935013i \(-0.384613\pi\)
0.354613 + 0.935013i \(0.384613\pi\)
\(282\) 0 0
\(283\) −2.86308e50 −1.56752 −0.783758 0.621066i \(-0.786700\pi\)
−0.783758 + 0.621066i \(0.786700\pi\)
\(284\) −2.58351e49 −0.131578
\(285\) 0 0
\(286\) 6.11481e50 2.69699
\(287\) −2.95593e49 −0.121371
\(288\) 0 0
\(289\) −2.69315e50 −0.959080
\(290\) 3.26234e49 0.108236
\(291\) 0 0
\(292\) −5.47924e50 −1.57896
\(293\) 1.89439e50 0.508960 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(294\) 0 0
\(295\) −5.25787e50 −1.22873
\(296\) 6.87628e50 1.49925
\(297\) 0 0
\(298\) −6.11193e50 −1.16077
\(299\) −1.20611e51 −2.13859
\(300\) 0 0
\(301\) 5.25608e49 0.0812922
\(302\) 1.07563e51 1.55425
\(303\) 0 0
\(304\) 1.60597e51 2.02688
\(305\) −2.60497e50 −0.307366
\(306\) 0 0
\(307\) 4.49477e50 0.463843 0.231921 0.972735i \(-0.425499\pi\)
0.231921 + 0.972735i \(0.425499\pi\)
\(308\) 2.46231e51 2.37713
\(309\) 0 0
\(310\) 3.80362e51 3.21575
\(311\) 4.89729e50 0.387586 0.193793 0.981042i \(-0.437921\pi\)
0.193793 + 0.981042i \(0.437921\pi\)
\(312\) 0 0
\(313\) 1.99055e51 1.38139 0.690693 0.723148i \(-0.257306\pi\)
0.690693 + 0.723148i \(0.257306\pi\)
\(314\) −3.10142e51 −2.01606
\(315\) 0 0
\(316\) −3.45057e51 −1.96927
\(317\) −8.77183e50 −0.469219 −0.234609 0.972090i \(-0.575381\pi\)
−0.234609 + 0.972090i \(0.575381\pi\)
\(318\) 0 0
\(319\) 1.36584e50 0.0642230
\(320\) 2.40999e51 1.06277
\(321\) 0 0
\(322\) −7.24225e51 −2.81079
\(323\) 1.01492e51 0.369641
\(324\) 0 0
\(325\) −1.69762e51 −0.544793
\(326\) −5.38486e51 −1.62261
\(327\) 0 0
\(328\) 8.74587e50 0.232482
\(329\) −4.30354e51 −1.07476
\(330\) 0 0
\(331\) 2.35505e51 0.519429 0.259715 0.965685i \(-0.416372\pi\)
0.259715 + 0.965685i \(0.416372\pi\)
\(332\) −7.64370e51 −1.58479
\(333\) 0 0
\(334\) −9.17543e51 −1.68199
\(335\) 6.20670e51 1.07014
\(336\) 0 0
\(337\) 2.75468e51 0.420396 0.210198 0.977659i \(-0.432589\pi\)
0.210198 + 0.977659i \(0.432589\pi\)
\(338\) −6.80371e51 −0.977132
\(339\) 0 0
\(340\) −3.87834e51 −0.493547
\(341\) 1.59246e52 1.90811
\(342\) 0 0
\(343\) −9.85894e51 −1.04785
\(344\) −1.55514e51 −0.155712
\(345\) 0 0
\(346\) 2.93752e52 2.61169
\(347\) −8.25043e51 −0.691389 −0.345695 0.938347i \(-0.612357\pi\)
−0.345695 + 0.938347i \(0.612357\pi\)
\(348\) 0 0
\(349\) 8.65779e51 0.644891 0.322445 0.946588i \(-0.395495\pi\)
0.322445 + 0.946588i \(0.395495\pi\)
\(350\) −1.01936e52 −0.716031
\(351\) 0 0
\(352\) −2.52981e51 −0.158112
\(353\) 6.01269e51 0.354558 0.177279 0.984161i \(-0.443270\pi\)
0.177279 + 0.984161i \(0.443270\pi\)
\(354\) 0 0
\(355\) 1.47463e51 0.0774463
\(356\) 1.82826e52 0.906385
\(357\) 0 0
\(358\) 1.93448e52 0.854991
\(359\) 1.54293e52 0.644035 0.322017 0.946734i \(-0.395639\pi\)
0.322017 + 0.946734i \(0.395639\pi\)
\(360\) 0 0
\(361\) 6.27973e52 2.33907
\(362\) 1.32136e52 0.465044
\(363\) 0 0
\(364\) −7.62513e52 −2.39701
\(365\) 3.12746e52 0.929369
\(366\) 0 0
\(367\) −3.80556e52 −1.01103 −0.505515 0.862818i \(-0.668698\pi\)
−0.505515 + 0.862818i \(0.668698\pi\)
\(368\) 7.55703e52 1.89876
\(369\) 0 0
\(370\) −7.71344e52 −1.73425
\(371\) −1.10846e52 −0.235807
\(372\) 0 0
\(373\) 4.94905e52 0.942955 0.471478 0.881878i \(-0.343721\pi\)
0.471478 + 0.881878i \(0.343721\pi\)
\(374\) −2.42126e52 −0.436691
\(375\) 0 0
\(376\) 1.27331e53 2.05865
\(377\) −4.22964e51 −0.0647598
\(378\) 0 0
\(379\) 4.34671e52 0.597113 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(380\) −3.42560e53 −4.45834
\(381\) 0 0
\(382\) −1.55111e53 −1.81277
\(383\) 2.09548e52 0.232118 0.116059 0.993242i \(-0.462974\pi\)
0.116059 + 0.993242i \(0.462974\pi\)
\(384\) 0 0
\(385\) −1.40545e53 −1.39917
\(386\) 2.35321e52 0.222137
\(387\) 0 0
\(388\) 7.20463e52 0.611732
\(389\) 1.49021e53 1.20028 0.600139 0.799896i \(-0.295112\pi\)
0.600139 + 0.799896i \(0.295112\pi\)
\(390\) 0 0
\(391\) 4.77581e52 0.346277
\(392\) 2.93631e52 0.202040
\(393\) 0 0
\(394\) −8.26890e52 −0.512596
\(395\) 1.96953e53 1.15910
\(396\) 0 0
\(397\) 3.46598e53 1.83916 0.919582 0.392898i \(-0.128528\pi\)
0.919582 + 0.392898i \(0.128528\pi\)
\(398\) 1.09457e53 0.551620
\(399\) 0 0
\(400\) 1.06366e53 0.483698
\(401\) −3.81967e53 −1.65031 −0.825157 0.564904i \(-0.808913\pi\)
−0.825157 + 0.564904i \(0.808913\pi\)
\(402\) 0 0
\(403\) −4.93142e53 −1.92406
\(404\) −3.79432e52 −0.140707
\(405\) 0 0
\(406\) −2.53974e52 −0.0851149
\(407\) −3.22938e53 −1.02904
\(408\) 0 0
\(409\) −1.49738e53 −0.431523 −0.215761 0.976446i \(-0.569223\pi\)
−0.215761 + 0.976446i \(0.569223\pi\)
\(410\) −9.81065e52 −0.268923
\(411\) 0 0
\(412\) 1.01143e54 2.50925
\(413\) 4.09327e53 0.966260
\(414\) 0 0
\(415\) 4.36290e53 0.932801
\(416\) 7.83416e52 0.159433
\(417\) 0 0
\(418\) −2.13862e54 −3.94475
\(419\) −5.57082e53 −0.978434 −0.489217 0.872162i \(-0.662717\pi\)
−0.489217 + 0.872162i \(0.662717\pi\)
\(420\) 0 0
\(421\) −4.02757e53 −0.641593 −0.320797 0.947148i \(-0.603951\pi\)
−0.320797 + 0.947148i \(0.603951\pi\)
\(422\) −1.38234e54 −2.09754
\(423\) 0 0
\(424\) 3.27967e53 0.451679
\(425\) 6.72201e52 0.0882117
\(426\) 0 0
\(427\) 2.02798e53 0.241708
\(428\) 3.03585e54 3.44892
\(429\) 0 0
\(430\) 1.74448e53 0.180120
\(431\) 1.10307e54 1.08597 0.542986 0.839742i \(-0.317294\pi\)
0.542986 + 0.839742i \(0.317294\pi\)
\(432\) 0 0
\(433\) 2.59468e53 0.232317 0.116159 0.993231i \(-0.462942\pi\)
0.116159 + 0.993231i \(0.462942\pi\)
\(434\) −2.96113e54 −2.52882
\(435\) 0 0
\(436\) 3.21172e54 2.49612
\(437\) 4.21830e54 3.12801
\(438\) 0 0
\(439\) −6.51584e53 −0.439995 −0.219997 0.975500i \(-0.570605\pi\)
−0.219997 + 0.975500i \(0.570605\pi\)
\(440\) 4.15837e54 2.68005
\(441\) 0 0
\(442\) 7.49801e53 0.440342
\(443\) 8.31973e53 0.466480 0.233240 0.972419i \(-0.425067\pi\)
0.233240 + 0.972419i \(0.425067\pi\)
\(444\) 0 0
\(445\) −1.04354e54 −0.533493
\(446\) −6.65889e54 −3.25115
\(447\) 0 0
\(448\) −1.87619e54 −0.835751
\(449\) −1.75134e54 −0.745280 −0.372640 0.927976i \(-0.621547\pi\)
−0.372640 + 0.927976i \(0.621547\pi\)
\(450\) 0 0
\(451\) −4.10741e53 −0.159569
\(452\) −7.18670e54 −2.66803
\(453\) 0 0
\(454\) 7.66919e54 2.60078
\(455\) 4.35230e54 1.41086
\(456\) 0 0
\(457\) 5.74991e54 1.70364 0.851822 0.523831i \(-0.175498\pi\)
0.851822 + 0.523831i \(0.175498\pi\)
\(458\) 6.18534e54 1.75235
\(459\) 0 0
\(460\) −1.61195e55 −4.17653
\(461\) −7.61685e54 −1.88758 −0.943792 0.330540i \(-0.892769\pi\)
−0.943792 + 0.330540i \(0.892769\pi\)
\(462\) 0 0
\(463\) 4.82618e54 1.09444 0.547221 0.836988i \(-0.315686\pi\)
0.547221 + 0.836988i \(0.315686\pi\)
\(464\) 2.65013e53 0.0574974
\(465\) 0 0
\(466\) −6.07811e54 −1.20742
\(467\) 8.48997e54 1.61402 0.807012 0.590535i \(-0.201083\pi\)
0.807012 + 0.590535i \(0.201083\pi\)
\(468\) 0 0
\(469\) −4.83194e54 −0.841547
\(470\) −1.42833e55 −2.38135
\(471\) 0 0
\(472\) −1.21110e55 −1.85084
\(473\) 7.30358e53 0.106877
\(474\) 0 0
\(475\) 5.93732e54 0.796840
\(476\) 3.01930e54 0.388118
\(477\) 0 0
\(478\) −1.45989e55 −1.72206
\(479\) 7.54248e54 0.852385 0.426193 0.904632i \(-0.359855\pi\)
0.426193 + 0.904632i \(0.359855\pi\)
\(480\) 0 0
\(481\) 1.00005e55 1.03764
\(482\) −2.97523e55 −2.95839
\(483\) 0 0
\(484\) 1.19255e55 1.08930
\(485\) −4.11228e54 −0.360062
\(486\) 0 0
\(487\) 2.26646e55 1.82392 0.911958 0.410283i \(-0.134570\pi\)
0.911958 + 0.410283i \(0.134570\pi\)
\(488\) −6.00029e54 −0.462984
\(489\) 0 0
\(490\) −3.29380e54 −0.233710
\(491\) 9.02019e54 0.613826 0.306913 0.951738i \(-0.400704\pi\)
0.306913 + 0.951738i \(0.400704\pi\)
\(492\) 0 0
\(493\) 1.67480e53 0.0104858
\(494\) 6.62273e55 3.97772
\(495\) 0 0
\(496\) 3.08984e55 1.70829
\(497\) −1.14800e54 −0.0609027
\(498\) 0 0
\(499\) −4.11207e54 −0.200909 −0.100454 0.994942i \(-0.532030\pi\)
−0.100454 + 0.994942i \(0.532030\pi\)
\(500\) 2.93406e55 1.37589
\(501\) 0 0
\(502\) 3.86167e54 0.166859
\(503\) 3.08721e55 1.28063 0.640313 0.768114i \(-0.278805\pi\)
0.640313 + 0.768114i \(0.278805\pi\)
\(504\) 0 0
\(505\) 2.16573e54 0.0828194
\(506\) −1.00635e56 −3.69540
\(507\) 0 0
\(508\) 1.27576e55 0.432078
\(509\) −2.73650e54 −0.0890185 −0.0445092 0.999009i \(-0.514172\pi\)
−0.0445092 + 0.999009i \(0.514172\pi\)
\(510\) 0 0
\(511\) −2.43474e55 −0.730843
\(512\) 6.45215e55 1.86067
\(513\) 0 0
\(514\) −1.54540e55 −0.411432
\(515\) −5.77308e55 −1.47693
\(516\) 0 0
\(517\) −5.97998e55 −1.41300
\(518\) 6.00494e55 1.36379
\(519\) 0 0
\(520\) −1.28774e56 −2.70246
\(521\) 7.68480e54 0.155046 0.0775228 0.996991i \(-0.475299\pi\)
0.0775228 + 0.996991i \(0.475299\pi\)
\(522\) 0 0
\(523\) −5.42535e55 −1.01191 −0.505957 0.862559i \(-0.668860\pi\)
−0.505957 + 0.862559i \(0.668860\pi\)
\(524\) −9.82673e55 −1.76245
\(525\) 0 0
\(526\) 9.23806e55 1.53240
\(527\) 1.95268e55 0.311540
\(528\) 0 0
\(529\) 1.30757e56 1.93029
\(530\) −3.67896e55 −0.522480
\(531\) 0 0
\(532\) 2.66684e56 3.50598
\(533\) 1.27196e55 0.160903
\(534\) 0 0
\(535\) −1.73282e56 −2.03002
\(536\) 1.42965e56 1.61195
\(537\) 0 0
\(538\) 6.53154e54 0.0682306
\(539\) −1.37901e55 −0.138675
\(540\) 0 0
\(541\) 1.06824e56 0.995687 0.497844 0.867267i \(-0.334125\pi\)
0.497844 + 0.867267i \(0.334125\pi\)
\(542\) −1.50353e56 −1.34936
\(543\) 0 0
\(544\) −3.10207e54 −0.0258151
\(545\) −1.83320e56 −1.46920
\(546\) 0 0
\(547\) −2.25958e56 −1.67991 −0.839956 0.542654i \(-0.817419\pi\)
−0.839956 + 0.542654i \(0.817419\pi\)
\(548\) −1.50384e56 −1.07696
\(549\) 0 0
\(550\) −1.41645e56 −0.941381
\(551\) 1.47929e55 0.0947208
\(552\) 0 0
\(553\) −1.53329e56 −0.911503
\(554\) 3.91785e56 2.24438
\(555\) 0 0
\(556\) 2.01575e56 1.07251
\(557\) 9.26630e55 0.475198 0.237599 0.971363i \(-0.423640\pi\)
0.237599 + 0.971363i \(0.423640\pi\)
\(558\) 0 0
\(559\) −2.26173e55 −0.107770
\(560\) −2.72698e56 −1.25264
\(561\) 0 0
\(562\) −2.89420e56 −1.23576
\(563\) −1.20733e56 −0.497053 −0.248526 0.968625i \(-0.579946\pi\)
−0.248526 + 0.968625i \(0.579946\pi\)
\(564\) 0 0
\(565\) 4.10205e56 1.57039
\(566\) 7.39772e56 2.73125
\(567\) 0 0
\(568\) 3.39665e55 0.116657
\(569\) −2.31976e56 −0.768495 −0.384248 0.923230i \(-0.625539\pi\)
−0.384248 + 0.923230i \(0.625539\pi\)
\(570\) 0 0
\(571\) 2.57033e55 0.0792409 0.0396204 0.999215i \(-0.487385\pi\)
0.0396204 + 0.999215i \(0.487385\pi\)
\(572\) −1.05955e57 −3.15139
\(573\) 0 0
\(574\) 7.63763e55 0.211478
\(575\) 2.79386e56 0.746472
\(576\) 0 0
\(577\) −2.83608e56 −0.705690 −0.352845 0.935682i \(-0.614786\pi\)
−0.352845 + 0.935682i \(0.614786\pi\)
\(578\) 6.95864e56 1.67111
\(579\) 0 0
\(580\) −5.65284e55 −0.126472
\(581\) −3.39653e56 −0.733542
\(582\) 0 0
\(583\) −1.54027e56 −0.310020
\(584\) 7.20379e56 1.39990
\(585\) 0 0
\(586\) −4.89479e56 −0.886814
\(587\) −3.96279e56 −0.693298 −0.346649 0.937995i \(-0.612681\pi\)
−0.346649 + 0.937995i \(0.612681\pi\)
\(588\) 0 0
\(589\) 1.72473e57 2.81422
\(590\) 1.35854e57 2.14095
\(591\) 0 0
\(592\) −6.26594e56 −0.921279
\(593\) −1.25467e57 −1.78200 −0.891001 0.454001i \(-0.849996\pi\)
−0.891001 + 0.454001i \(0.849996\pi\)
\(594\) 0 0
\(595\) −1.72337e56 −0.228444
\(596\) 1.05905e57 1.35634
\(597\) 0 0
\(598\) 3.11639e57 3.72629
\(599\) 1.06361e57 1.22895 0.614475 0.788936i \(-0.289368\pi\)
0.614475 + 0.788936i \(0.289368\pi\)
\(600\) 0 0
\(601\) −4.74011e56 −0.511519 −0.255759 0.966740i \(-0.582325\pi\)
−0.255759 + 0.966740i \(0.582325\pi\)
\(602\) −1.35808e56 −0.141644
\(603\) 0 0
\(604\) −1.86381e57 −1.81612
\(605\) −6.80686e56 −0.641153
\(606\) 0 0
\(607\) −3.75437e56 −0.330498 −0.165249 0.986252i \(-0.552843\pi\)
−0.165249 + 0.986252i \(0.552843\pi\)
\(608\) −2.73995e56 −0.233195
\(609\) 0 0
\(610\) 6.73080e56 0.535556
\(611\) 1.85184e57 1.42482
\(612\) 0 0
\(613\) 1.97052e57 1.41788 0.708941 0.705267i \(-0.249173\pi\)
0.708941 + 0.705267i \(0.249173\pi\)
\(614\) −1.16137e57 −0.808202
\(615\) 0 0
\(616\) −3.23731e57 −2.10756
\(617\) −1.04592e57 −0.658649 −0.329325 0.944217i \(-0.606821\pi\)
−0.329325 + 0.944217i \(0.606821\pi\)
\(618\) 0 0
\(619\) −3.26806e57 −1.92589 −0.962947 0.269692i \(-0.913078\pi\)
−0.962947 + 0.269692i \(0.913078\pi\)
\(620\) −6.59076e57 −3.75756
\(621\) 0 0
\(622\) −1.26538e57 −0.675332
\(623\) 8.12399e56 0.419531
\(624\) 0 0
\(625\) −2.57649e57 −1.24591
\(626\) −5.14325e57 −2.40693
\(627\) 0 0
\(628\) 5.37402e57 2.35574
\(629\) −3.95988e56 −0.168013
\(630\) 0 0
\(631\) 7.88492e54 0.00313470 0.00156735 0.999999i \(-0.499501\pi\)
0.00156735 + 0.999999i \(0.499501\pi\)
\(632\) 4.53662e57 1.74595
\(633\) 0 0
\(634\) 2.26649e57 0.817569
\(635\) −7.28181e56 −0.254318
\(636\) 0 0
\(637\) 4.27043e56 0.139834
\(638\) −3.52909e56 −0.111902
\(639\) 0 0
\(640\) −6.74127e57 −2.00471
\(641\) 5.60781e57 1.61511 0.807557 0.589789i \(-0.200789\pi\)
0.807557 + 0.589789i \(0.200789\pi\)
\(642\) 0 0
\(643\) −2.39387e57 −0.646810 −0.323405 0.946261i \(-0.604828\pi\)
−0.323405 + 0.946261i \(0.604828\pi\)
\(644\) 1.25491e58 3.28436
\(645\) 0 0
\(646\) −2.62238e57 −0.644065
\(647\) −5.56725e57 −1.32466 −0.662328 0.749214i \(-0.730431\pi\)
−0.662328 + 0.749214i \(0.730431\pi\)
\(648\) 0 0
\(649\) 5.68779e57 1.27036
\(650\) 4.38636e57 0.949250
\(651\) 0 0
\(652\) 9.33066e57 1.89599
\(653\) −6.52017e57 −1.28392 −0.641961 0.766737i \(-0.721879\pi\)
−0.641961 + 0.766737i \(0.721879\pi\)
\(654\) 0 0
\(655\) 5.60893e57 1.03737
\(656\) −7.96959e56 −0.142859
\(657\) 0 0
\(658\) 1.11196e58 1.87266
\(659\) 7.55343e57 1.23309 0.616543 0.787321i \(-0.288533\pi\)
0.616543 + 0.787321i \(0.288533\pi\)
\(660\) 0 0
\(661\) −7.41557e57 −1.13767 −0.568833 0.822453i \(-0.692605\pi\)
−0.568833 + 0.822453i \(0.692605\pi\)
\(662\) −6.08505e57 −0.905056
\(663\) 0 0
\(664\) 1.00495e58 1.40507
\(665\) −1.52219e58 −2.06360
\(666\) 0 0
\(667\) 6.96093e56 0.0887336
\(668\) 1.58988e58 1.96538
\(669\) 0 0
\(670\) −1.60371e58 −1.86463
\(671\) 2.81798e57 0.317779
\(672\) 0 0
\(673\) −1.69942e58 −1.80298 −0.901490 0.432800i \(-0.857526\pi\)
−0.901490 + 0.432800i \(0.857526\pi\)
\(674\) −7.11763e57 −0.732500
\(675\) 0 0
\(676\) 1.17892e58 1.14176
\(677\) 1.85953e57 0.174717 0.0873585 0.996177i \(-0.472157\pi\)
0.0873585 + 0.996177i \(0.472157\pi\)
\(678\) 0 0
\(679\) 3.20143e57 0.283148
\(680\) 5.09902e57 0.437576
\(681\) 0 0
\(682\) −4.11464e58 −3.32470
\(683\) 1.28888e58 1.01062 0.505312 0.862937i \(-0.331377\pi\)
0.505312 + 0.862937i \(0.331377\pi\)
\(684\) 0 0
\(685\) 8.58366e57 0.633893
\(686\) 2.54738e58 1.82579
\(687\) 0 0
\(688\) 1.41711e57 0.0956843
\(689\) 4.76979e57 0.312612
\(690\) 0 0
\(691\) 5.69871e57 0.351947 0.175973 0.984395i \(-0.443693\pi\)
0.175973 + 0.984395i \(0.443693\pi\)
\(692\) −5.09001e58 −3.05172
\(693\) 0 0
\(694\) 2.13177e58 1.20468
\(695\) −1.15056e58 −0.631276
\(696\) 0 0
\(697\) −5.03653e56 −0.0260531
\(698\) −2.23703e58 −1.12366
\(699\) 0 0
\(700\) 1.76630e58 0.836671
\(701\) 8.78499e57 0.404131 0.202065 0.979372i \(-0.435235\pi\)
0.202065 + 0.979372i \(0.435235\pi\)
\(702\) 0 0
\(703\) −3.49762e58 −1.51771
\(704\) −2.60706e58 −1.09878
\(705\) 0 0
\(706\) −1.55358e58 −0.617784
\(707\) −1.68603e57 −0.0651280
\(708\) 0 0
\(709\) 4.95487e58 1.80628 0.903139 0.429347i \(-0.141256\pi\)
0.903139 + 0.429347i \(0.141256\pi\)
\(710\) −3.81018e57 −0.134943
\(711\) 0 0
\(712\) −2.40369e58 −0.803597
\(713\) 8.11589e58 2.63634
\(714\) 0 0
\(715\) 6.04773e58 1.85489
\(716\) −3.35198e58 −0.999044
\(717\) 0 0
\(718\) −3.98666e58 −1.12217
\(719\) 4.51451e58 1.23500 0.617501 0.786570i \(-0.288145\pi\)
0.617501 + 0.786570i \(0.288145\pi\)
\(720\) 0 0
\(721\) 4.49436e58 1.16144
\(722\) −1.62258e59 −4.07561
\(723\) 0 0
\(724\) −2.28959e58 −0.543397
\(725\) 9.79761e56 0.0226043
\(726\) 0 0
\(727\) 1.85132e58 0.403669 0.201834 0.979420i \(-0.435310\pi\)
0.201834 + 0.979420i \(0.435310\pi\)
\(728\) 1.00251e59 2.12517
\(729\) 0 0
\(730\) −8.08083e58 −1.61934
\(731\) 8.95569e56 0.0174499
\(732\) 0 0
\(733\) −7.75891e57 −0.142945 −0.0714725 0.997443i \(-0.522770\pi\)
−0.0714725 + 0.997443i \(0.522770\pi\)
\(734\) 9.83292e58 1.76162
\(735\) 0 0
\(736\) −1.28931e58 −0.218455
\(737\) −6.71422e58 −1.10640
\(738\) 0 0
\(739\) −1.13939e59 −1.77608 −0.888041 0.459764i \(-0.847934\pi\)
−0.888041 + 0.459764i \(0.847934\pi\)
\(740\) 1.33655e59 2.02645
\(741\) 0 0
\(742\) 2.86408e58 0.410871
\(743\) −9.08942e58 −1.26843 −0.634213 0.773158i \(-0.718676\pi\)
−0.634213 + 0.773158i \(0.718676\pi\)
\(744\) 0 0
\(745\) −6.04488e58 −0.798333
\(746\) −1.27875e59 −1.64301
\(747\) 0 0
\(748\) 4.19547e58 0.510267
\(749\) 1.34900e59 1.59638
\(750\) 0 0
\(751\) 1.09322e59 1.22487 0.612433 0.790522i \(-0.290191\pi\)
0.612433 + 0.790522i \(0.290191\pi\)
\(752\) −1.16029e59 −1.26503
\(753\) 0 0
\(754\) 1.09287e58 0.112838
\(755\) 1.06383e59 1.06896
\(756\) 0 0
\(757\) −1.50195e59 −1.42951 −0.714757 0.699372i \(-0.753463\pi\)
−0.714757 + 0.699372i \(0.753463\pi\)
\(758\) −1.12312e59 −1.04041
\(759\) 0 0
\(760\) 4.50378e59 3.95274
\(761\) 1.91836e58 0.163887 0.0819436 0.996637i \(-0.473887\pi\)
0.0819436 + 0.996637i \(0.473887\pi\)
\(762\) 0 0
\(763\) 1.42715e59 1.15536
\(764\) 2.68770e59 2.11820
\(765\) 0 0
\(766\) −5.41438e58 −0.404444
\(767\) −1.76136e59 −1.28098
\(768\) 0 0
\(769\) 8.09865e58 0.558371 0.279185 0.960237i \(-0.409936\pi\)
0.279185 + 0.960237i \(0.409936\pi\)
\(770\) 3.63144e59 2.43792
\(771\) 0 0
\(772\) −4.07755e58 −0.259564
\(773\) 1.75935e59 1.09062 0.545310 0.838234i \(-0.316412\pi\)
0.545310 + 0.838234i \(0.316412\pi\)
\(774\) 0 0
\(775\) 1.14232e59 0.671590
\(776\) −9.47223e58 −0.542359
\(777\) 0 0
\(778\) −3.85046e59 −2.09137
\(779\) −4.44859e58 −0.235345
\(780\) 0 0
\(781\) −1.59520e58 −0.0800700
\(782\) −1.23399e59 −0.603354
\(783\) 0 0
\(784\) −2.67569e58 −0.124153
\(785\) −3.06740e59 −1.38657
\(786\) 0 0
\(787\) −1.42029e59 −0.609390 −0.304695 0.952450i \(-0.598555\pi\)
−0.304695 + 0.952450i \(0.598555\pi\)
\(788\) 1.43280e59 0.598961
\(789\) 0 0
\(790\) −5.08893e59 −2.01963
\(791\) −3.19346e59 −1.23493
\(792\) 0 0
\(793\) −8.72652e58 −0.320436
\(794\) −8.95550e59 −3.20457
\(795\) 0 0
\(796\) −1.89662e59 −0.644559
\(797\) −1.77809e59 −0.588922 −0.294461 0.955664i \(-0.595140\pi\)
−0.294461 + 0.955664i \(0.595140\pi\)
\(798\) 0 0
\(799\) −7.33268e58 −0.230703
\(800\) −1.81472e58 −0.0556500
\(801\) 0 0
\(802\) 9.86938e59 2.87551
\(803\) −3.38319e59 −0.960854
\(804\) 0 0
\(805\) −7.16280e59 −1.93316
\(806\) 1.27419e60 3.35249
\(807\) 0 0
\(808\) 4.98855e58 0.124750
\(809\) 4.67029e59 1.13867 0.569336 0.822105i \(-0.307200\pi\)
0.569336 + 0.822105i \(0.307200\pi\)
\(810\) 0 0
\(811\) −1.45390e59 −0.336983 −0.168492 0.985703i \(-0.553890\pi\)
−0.168492 + 0.985703i \(0.553890\pi\)
\(812\) 4.40076e58 0.0994555
\(813\) 0 0
\(814\) 8.34416e59 1.79301
\(815\) −5.32578e59 −1.11597
\(816\) 0 0
\(817\) 7.91024e58 0.157630
\(818\) 3.86897e59 0.751887
\(819\) 0 0
\(820\) 1.69995e59 0.314233
\(821\) 1.07845e59 0.194432 0.0972158 0.995263i \(-0.469006\pi\)
0.0972158 + 0.995263i \(0.469006\pi\)
\(822\) 0 0
\(823\) 2.28591e59 0.392069 0.196035 0.980597i \(-0.437193\pi\)
0.196035 + 0.980597i \(0.437193\pi\)
\(824\) −1.32977e60 −2.22469
\(825\) 0 0
\(826\) −1.05763e60 −1.68362
\(827\) 5.76088e59 0.894594 0.447297 0.894385i \(-0.352387\pi\)
0.447297 + 0.894385i \(0.352387\pi\)
\(828\) 0 0
\(829\) 2.25613e59 0.333424 0.166712 0.986006i \(-0.446685\pi\)
0.166712 + 0.986006i \(0.446685\pi\)
\(830\) −1.12730e60 −1.62532
\(831\) 0 0
\(832\) 8.07336e59 1.10796
\(833\) −1.69095e58 −0.0226417
\(834\) 0 0
\(835\) −9.07477e59 −1.15681
\(836\) 3.70571e60 4.60938
\(837\) 0 0
\(838\) 1.43940e60 1.70483
\(839\) 1.01240e60 1.17012 0.585060 0.810990i \(-0.301071\pi\)
0.585060 + 0.810990i \(0.301071\pi\)
\(840\) 0 0
\(841\) −9.06044e59 −0.997313
\(842\) 1.04065e60 1.11791
\(843\) 0 0
\(844\) 2.39527e60 2.45095
\(845\) −6.72907e59 −0.672036
\(846\) 0 0
\(847\) 5.29916e59 0.504194
\(848\) −2.98857e59 −0.277555
\(849\) 0 0
\(850\) −1.73685e59 −0.153701
\(851\) −1.64584e60 −1.42177
\(852\) 0 0
\(853\) −1.19199e60 −0.981337 −0.490669 0.871346i \(-0.663247\pi\)
−0.490669 + 0.871346i \(0.663247\pi\)
\(854\) −5.23995e59 −0.421154
\(855\) 0 0
\(856\) −3.99136e60 −3.05780
\(857\) 1.98782e60 1.48686 0.743431 0.668813i \(-0.233197\pi\)
0.743431 + 0.668813i \(0.233197\pi\)
\(858\) 0 0
\(859\) 1.93547e60 1.38015 0.690076 0.723737i \(-0.257577\pi\)
0.690076 + 0.723737i \(0.257577\pi\)
\(860\) −3.02276e59 −0.210468
\(861\) 0 0
\(862\) −2.85014e60 −1.89220
\(863\) −1.93131e60 −1.25208 −0.626038 0.779793i \(-0.715325\pi\)
−0.626038 + 0.779793i \(0.715325\pi\)
\(864\) 0 0
\(865\) 2.90530e60 1.79622
\(866\) −6.70421e59 −0.404791
\(867\) 0 0
\(868\) 5.13093e60 2.95489
\(869\) −2.13058e60 −1.19837
\(870\) 0 0
\(871\) 2.07922e60 1.11565
\(872\) −4.22258e60 −2.21305
\(873\) 0 0
\(874\) −1.08994e61 −5.45026
\(875\) 1.30377e60 0.636848
\(876\) 0 0
\(877\) 4.31437e59 0.201106 0.100553 0.994932i \(-0.467939\pi\)
0.100553 + 0.994932i \(0.467939\pi\)
\(878\) 1.68358e60 0.766649
\(879\) 0 0
\(880\) −3.78928e60 −1.64688
\(881\) 1.33549e60 0.567066 0.283533 0.958962i \(-0.408493\pi\)
0.283533 + 0.958962i \(0.408493\pi\)
\(882\) 0 0
\(883\) −9.62321e59 −0.390055 −0.195027 0.980798i \(-0.562480\pi\)
−0.195027 + 0.980798i \(0.562480\pi\)
\(884\) −1.29922e60 −0.514533
\(885\) 0 0
\(886\) −2.14968e60 −0.812796
\(887\) 3.90527e60 1.44283 0.721417 0.692500i \(-0.243491\pi\)
0.721417 + 0.692500i \(0.243491\pi\)
\(888\) 0 0
\(889\) 5.66891e59 0.199993
\(890\) 2.69633e60 0.929560
\(891\) 0 0
\(892\) 1.15383e61 3.79892
\(893\) −6.47670e60 −2.08400
\(894\) 0 0
\(895\) 1.91325e60 0.588031
\(896\) 5.24810e60 1.57648
\(897\) 0 0
\(898\) 4.52515e60 1.29858
\(899\) 2.84611e59 0.0798322
\(900\) 0 0
\(901\) −1.88868e59 −0.0506175
\(902\) 1.06129e60 0.278034
\(903\) 0 0
\(904\) 9.44865e60 2.36547
\(905\) 1.30686e60 0.319840
\(906\) 0 0
\(907\) −2.96464e60 −0.693462 −0.346731 0.937965i \(-0.612708\pi\)
−0.346731 + 0.937965i \(0.612708\pi\)
\(908\) −1.32888e61 −3.03898
\(909\) 0 0
\(910\) −1.12456e61 −2.45830
\(911\) 7.01681e60 1.49973 0.749865 0.661591i \(-0.230118\pi\)
0.749865 + 0.661591i \(0.230118\pi\)
\(912\) 0 0
\(913\) −4.71965e60 −0.964403
\(914\) −1.48568e61 −2.96844
\(915\) 0 0
\(916\) −1.07177e61 −2.04760
\(917\) −4.36657e60 −0.815773
\(918\) 0 0
\(919\) −6.52121e60 −1.16509 −0.582546 0.812798i \(-0.697943\pi\)
−0.582546 + 0.812798i \(0.697943\pi\)
\(920\) 2.11930e61 3.70289
\(921\) 0 0
\(922\) 1.96806e61 3.28893
\(923\) 4.93992e59 0.0807394
\(924\) 0 0
\(925\) −2.31654e60 −0.362188
\(926\) −1.24700e61 −1.90696
\(927\) 0 0
\(928\) −4.52140e58 −0.00661514
\(929\) 4.97882e60 0.712532 0.356266 0.934385i \(-0.384050\pi\)
0.356266 + 0.934385i \(0.384050\pi\)
\(930\) 0 0
\(931\) −1.49356e60 −0.204528
\(932\) 1.05319e61 1.41085
\(933\) 0 0
\(934\) −2.19366e61 −2.81228
\(935\) −2.39470e60 −0.300340
\(936\) 0 0
\(937\) 9.62681e60 1.15564 0.577818 0.816166i \(-0.303904\pi\)
0.577818 + 0.816166i \(0.303904\pi\)
\(938\) 1.24849e61 1.46632
\(939\) 0 0
\(940\) 2.47495e61 2.78257
\(941\) 3.82719e60 0.421010 0.210505 0.977593i \(-0.432489\pi\)
0.210505 + 0.977593i \(0.432489\pi\)
\(942\) 0 0
\(943\) −2.09332e60 −0.220469
\(944\) 1.10360e61 1.13733
\(945\) 0 0
\(946\) −1.88712e60 −0.186222
\(947\) −8.12706e59 −0.0784800 −0.0392400 0.999230i \(-0.512494\pi\)
−0.0392400 + 0.999230i \(0.512494\pi\)
\(948\) 0 0
\(949\) 1.04768e61 0.968886
\(950\) −1.53410e61 −1.38842
\(951\) 0 0
\(952\) −3.96960e60 −0.344104
\(953\) 3.41043e60 0.289337 0.144669 0.989480i \(-0.453788\pi\)
0.144669 + 0.989480i \(0.453788\pi\)
\(954\) 0 0
\(955\) −1.53409e61 −1.24676
\(956\) 2.52963e61 2.01220
\(957\) 0 0
\(958\) −1.94885e61 −1.48520
\(959\) −6.68241e60 −0.498485
\(960\) 0 0
\(961\) 1.91930e61 1.37187
\(962\) −2.58396e61 −1.80800
\(963\) 0 0
\(964\) 5.15535e61 3.45684
\(965\) 2.32740e60 0.152778
\(966\) 0 0
\(967\) −2.57180e61 −1.61806 −0.809030 0.587767i \(-0.800007\pi\)
−0.809030 + 0.587767i \(0.800007\pi\)
\(968\) −1.56789e61 −0.965765
\(969\) 0 0
\(970\) 1.06254e61 0.627374
\(971\) 1.55166e61 0.897019 0.448509 0.893778i \(-0.351955\pi\)
0.448509 + 0.893778i \(0.351955\pi\)
\(972\) 0 0
\(973\) 8.95711e60 0.496426
\(974\) −5.85616e61 −3.17800
\(975\) 0 0
\(976\) 5.46770e60 0.284501
\(977\) −4.45394e60 −0.226937 −0.113469 0.993542i \(-0.536196\pi\)
−0.113469 + 0.993542i \(0.536196\pi\)
\(978\) 0 0
\(979\) 1.12887e61 0.551567
\(980\) 5.70736e60 0.273087
\(981\) 0 0
\(982\) −2.33066e61 −1.06953
\(983\) 3.36188e61 1.51090 0.755450 0.655207i \(-0.227418\pi\)
0.755450 + 0.655207i \(0.227418\pi\)
\(984\) 0 0
\(985\) −8.17819e60 −0.352545
\(986\) −4.32739e59 −0.0182705
\(987\) 0 0
\(988\) −1.14756e62 −4.64791
\(989\) 3.72224e60 0.147666
\(990\) 0 0
\(991\) 5.79150e60 0.220435 0.110218 0.993907i \(-0.464845\pi\)
0.110218 + 0.993907i \(0.464845\pi\)
\(992\) −5.27158e60 −0.196540
\(993\) 0 0
\(994\) 2.96624e60 0.106117
\(995\) 1.08256e61 0.379384
\(996\) 0 0
\(997\) 1.75910e61 0.591615 0.295808 0.955248i \(-0.404411\pi\)
0.295808 + 0.955248i \(0.404411\pi\)
\(998\) 1.06249e61 0.350065
\(999\) 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 9.42.a.c.1.1 4
3.2 odd 2 3.42.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.42.a.b.1.4 4 3.2 odd 2
9.42.a.c.1.1 4 1.1 even 1 trivial