Properties

Label 9.24.a.b.1.2
Level $9$
Weight $24$
Character 9.1
Self dual yes
Analytic conductor $30.168$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{144169}) \)
Defining polynomial: \(x^{2} - x - 36042\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-189.348\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+4016.35 q^{2} +7.74247e6 q^{4} -1.05062e8 q^{5} +3.81217e9 q^{7} -2.59512e9 q^{8} +O(q^{10})\) \(q+4016.35 q^{2} +7.74247e6 q^{4} -1.05062e8 q^{5} +3.81217e9 q^{7} -2.59512e9 q^{8} -4.21966e11 q^{10} -2.52200e11 q^{11} -3.59099e12 q^{13} +1.53110e13 q^{14} -7.53715e13 q^{16} -2.34190e14 q^{17} -6.23086e14 q^{19} -8.13439e14 q^{20} -1.01292e15 q^{22} +3.58786e15 q^{23} -8.82898e14 q^{25} -1.44227e16 q^{26} +2.95156e16 q^{28} +2.05923e16 q^{29} +1.36357e17 q^{31} -2.80949e17 q^{32} -9.40588e17 q^{34} -4.00514e17 q^{35} -1.23898e18 q^{37} -2.50253e18 q^{38} +2.72649e17 q^{40} -1.40074e18 q^{41} +2.18793e17 q^{43} -1.95265e18 q^{44} +1.44101e19 q^{46} +8.67836e18 q^{47} -1.28361e19 q^{49} -3.54603e18 q^{50} -2.78032e19 q^{52} +7.63436e19 q^{53} +2.64966e19 q^{55} -9.89304e18 q^{56} +8.27059e19 q^{58} +1.01862e18 q^{59} +2.87337e20 q^{61} +5.47658e20 q^{62} -4.96127e20 q^{64} +3.77277e20 q^{65} +1.47683e21 q^{67} -1.81321e21 q^{68} -1.60861e21 q^{70} -7.64346e20 q^{71} -3.49433e21 q^{73} -4.97617e21 q^{74} -4.82422e21 q^{76} -9.61427e20 q^{77} +1.02350e22 q^{79} +7.91868e21 q^{80} -5.62587e21 q^{82} +7.71597e21 q^{83} +2.46044e22 q^{85} +8.78750e20 q^{86} +6.54489e20 q^{88} -4.58518e21 q^{89} -1.36895e22 q^{91} +2.77789e22 q^{92} +3.48553e22 q^{94} +6.54627e22 q^{95} -1.13703e23 q^{97} -5.15544e22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1080q^{2} + 25326656q^{4} - 73069020q^{5} - 1359184400q^{7} - 49459023360q^{8} + O(q^{10}) \) \( 2q - 1080q^{2} + 25326656q^{4} - 73069020q^{5} - 1359184400q^{7} - 49459023360q^{8} - 585013636080q^{10} - 856801968264q^{11} + 4376109322060q^{13} + 41666034529728q^{14} + 15956586401792q^{16} - 254028147597540q^{17} + 4260600979960q^{19} - 250868387468160q^{20} + 2068343882177760q^{22} + 8144713079008560q^{23} - 11780274628800850q^{25} - 55025854658735184q^{26} - 61418438819709440q^{28} - 20818433601623340q^{29} + 137714017177000384q^{31} - 353265663781601280q^{32} - 839483655961325328q^{34} - 565961271250425120q^{35} - 897721264408967780q^{37} - 5699708971590961440q^{38} - 1226668524414336000q^{40} + 2294435477168314956q^{41} - 1750760768619855800q^{43} - 12584088840033038592q^{44} - 8813206018050221376q^{46} - 15759744217656780960q^{47} - 13461981704376200814q^{49} + 51990825483785316600q^{50} + 112291883783912022400q^{52} + 140287253401646796420q^{53} + 7153550955060182640q^{55} + 232456712054288117760q^{56} + 293749486923568689360q^{58} - 280872989971340771880q^{59} - 180452892516502223636q^{61} + 540743475843874103040q^{62} - 893690254469352914944q^{64} + 632168834809440380760q^{65} + 1754233163431557625240q^{67} - 2162050190142944330880q^{68} - 765428657799921252480q^{70} - 3055033510194143328624q^{71} - 8063408253877606149260q^{73} - 6715344283148807757072q^{74} + 6207154294513080590080q^{76} + 2165184764357449665600q^{77} + 6244916814559639980640q^{79} + 10840537585501794017280q^{80} - 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 23969743087870314902520q^{85} + 10916288812918999243296q^{86} + 28988514668199273707520q^{88} - 6395093086173070004820q^{89} - 54890178162704560146016q^{91} + 107907439017191756981760q^{92} + 159400518006534931827072q^{94} + 85533361066700858502000q^{95} - 31147288846254030500540q^{97} - 48364767616374671003640q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4016.35 1.38671 0.693357 0.720595i \(-0.256131\pi\)
0.693357 + 0.720595i \(0.256131\pi\)
\(3\) 0 0
\(4\) 7.74247e6 0.922974
\(5\) −1.05062e8 −0.962256 −0.481128 0.876650i \(-0.659773\pi\)
−0.481128 + 0.876650i \(0.659773\pi\)
\(6\) 0 0
\(7\) 3.81217e9 0.728693 0.364346 0.931263i \(-0.381292\pi\)
0.364346 + 0.931263i \(0.381292\pi\)
\(8\) −2.59512e9 −0.106813
\(9\) 0 0
\(10\) −4.21966e11 −1.33437
\(11\) −2.52200e11 −0.266519 −0.133260 0.991081i \(-0.542544\pi\)
−0.133260 + 0.991081i \(0.542544\pi\)
\(12\) 0 0
\(13\) −3.59099e12 −0.555733 −0.277867 0.960620i \(-0.589627\pi\)
−0.277867 + 0.960620i \(0.589627\pi\)
\(14\) 1.53110e13 1.01049
\(15\) 0 0
\(16\) −7.53715e13 −1.07109
\(17\) −2.34190e14 −1.65731 −0.828657 0.559756i \(-0.810895\pi\)
−0.828657 + 0.559756i \(0.810895\pi\)
\(18\) 0 0
\(19\) −6.23086e14 −1.22710 −0.613552 0.789654i \(-0.710260\pi\)
−0.613552 + 0.789654i \(0.710260\pi\)
\(20\) −8.13439e14 −0.888138
\(21\) 0 0
\(22\) −1.01292e15 −0.369586
\(23\) 3.58786e15 0.785173 0.392587 0.919715i \(-0.371580\pi\)
0.392587 + 0.919715i \(0.371580\pi\)
\(24\) 0 0
\(25\) −8.82898e14 −0.0740629
\(26\) −1.44227e16 −0.770642
\(27\) 0 0
\(28\) 2.95156e16 0.672565
\(29\) 2.05923e16 0.313421 0.156710 0.987645i \(-0.449911\pi\)
0.156710 + 0.987645i \(0.449911\pi\)
\(30\) 0 0
\(31\) 1.36357e17 0.963870 0.481935 0.876207i \(-0.339934\pi\)
0.481935 + 0.876207i \(0.339934\pi\)
\(32\) −2.80949e17 −1.37849
\(33\) 0 0
\(34\) −9.40588e17 −2.29822
\(35\) −4.00514e17 −0.701189
\(36\) 0 0
\(37\) −1.23898e18 −1.14484 −0.572419 0.819961i \(-0.693995\pi\)
−0.572419 + 0.819961i \(0.693995\pi\)
\(38\) −2.50253e18 −1.70164
\(39\) 0 0
\(40\) 2.72649e17 0.102781
\(41\) −1.40074e18 −0.397506 −0.198753 0.980050i \(-0.563689\pi\)
−0.198753 + 0.980050i \(0.563689\pi\)
\(42\) 0 0
\(43\) 2.18793e17 0.0359043 0.0179522 0.999839i \(-0.494285\pi\)
0.0179522 + 0.999839i \(0.494285\pi\)
\(44\) −1.95265e18 −0.245990
\(45\) 0 0
\(46\) 1.44101e19 1.08881
\(47\) 8.67836e18 0.512050 0.256025 0.966670i \(-0.417587\pi\)
0.256025 + 0.966670i \(0.417587\pi\)
\(48\) 0 0
\(49\) −1.28361e19 −0.469007
\(50\) −3.54603e18 −0.102704
\(51\) 0 0
\(52\) −2.78032e19 −0.512927
\(53\) 7.63436e19 1.13136 0.565679 0.824626i \(-0.308614\pi\)
0.565679 + 0.824626i \(0.308614\pi\)
\(54\) 0 0
\(55\) 2.64966e19 0.256460
\(56\) −9.89304e18 −0.0778337
\(57\) 0 0
\(58\) 8.27059e19 0.434625
\(59\) 1.01862e18 0.00439760 0.00219880 0.999998i \(-0.499300\pi\)
0.00219880 + 0.999998i \(0.499300\pi\)
\(60\) 0 0
\(61\) 2.87337e20 0.845472 0.422736 0.906253i \(-0.361070\pi\)
0.422736 + 0.906253i \(0.361070\pi\)
\(62\) 5.47658e20 1.33661
\(63\) 0 0
\(64\) −4.96127e20 −0.840472
\(65\) 3.77277e20 0.534758
\(66\) 0 0
\(67\) 1.47683e21 1.47730 0.738652 0.674088i \(-0.235463\pi\)
0.738652 + 0.674088i \(0.235463\pi\)
\(68\) −1.81321e21 −1.52966
\(69\) 0 0
\(70\) −1.60861e21 −0.972349
\(71\) −7.64346e20 −0.392481 −0.196241 0.980556i \(-0.562873\pi\)
−0.196241 + 0.980556i \(0.562873\pi\)
\(72\) 0 0
\(73\) −3.49433e21 −1.30362 −0.651810 0.758382i \(-0.725990\pi\)
−0.651810 + 0.758382i \(0.725990\pi\)
\(74\) −4.97617e21 −1.58756
\(75\) 0 0
\(76\) −4.82422e21 −1.13259
\(77\) −9.61427e20 −0.194211
\(78\) 0 0
\(79\) 1.02350e22 1.53948 0.769742 0.638356i \(-0.220385\pi\)
0.769742 + 0.638356i \(0.220385\pi\)
\(80\) 7.91868e21 1.03067
\(81\) 0 0
\(82\) −5.62587e21 −0.551227
\(83\) 7.71597e21 0.657646 0.328823 0.944392i \(-0.393348\pi\)
0.328823 + 0.944392i \(0.393348\pi\)
\(84\) 0 0
\(85\) 2.46044e22 1.59476
\(86\) 8.78750e20 0.0497890
\(87\) 0 0
\(88\) 6.54489e20 0.0284676
\(89\) −4.58518e21 −0.175134 −0.0875672 0.996159i \(-0.527909\pi\)
−0.0875672 + 0.996159i \(0.527909\pi\)
\(90\) 0 0
\(91\) −1.36895e22 −0.404959
\(92\) 2.77789e22 0.724695
\(93\) 0 0
\(94\) 3.48553e22 0.710067
\(95\) 6.54627e22 1.18079
\(96\) 0 0
\(97\) −1.13703e23 −1.61398 −0.806991 0.590564i \(-0.798905\pi\)
−0.806991 + 0.590564i \(0.798905\pi\)
\(98\) −5.15544e22 −0.650378
\(99\) 0 0
\(100\) −6.83581e21 −0.0683581
\(101\) −1.36243e23 −1.21512 −0.607558 0.794275i \(-0.707851\pi\)
−0.607558 + 0.794275i \(0.707851\pi\)
\(102\) 0 0
\(103\) 1.41401e22 0.100653 0.0503264 0.998733i \(-0.483974\pi\)
0.0503264 + 0.998733i \(0.483974\pi\)
\(104\) 9.31907e21 0.0593594
\(105\) 0 0
\(106\) 3.06623e23 1.56887
\(107\) −6.02971e22 −0.276938 −0.138469 0.990367i \(-0.544218\pi\)
−0.138469 + 0.990367i \(0.544218\pi\)
\(108\) 0 0
\(109\) −5.58169e22 −0.207186 −0.103593 0.994620i \(-0.533034\pi\)
−0.103593 + 0.994620i \(0.533034\pi\)
\(110\) 1.06420e23 0.355636
\(111\) 0 0
\(112\) −2.87329e23 −0.780498
\(113\) −3.51523e23 −0.862089 −0.431044 0.902331i \(-0.641855\pi\)
−0.431044 + 0.902331i \(0.641855\pi\)
\(114\) 0 0
\(115\) −3.76948e23 −0.755538
\(116\) 1.59435e23 0.289279
\(117\) 0 0
\(118\) 4.09115e21 0.00609821
\(119\) −8.92770e23 −1.20767
\(120\) 0 0
\(121\) −8.31826e23 −0.928968
\(122\) 1.15405e24 1.17243
\(123\) 0 0
\(124\) 1.05574e24 0.889627
\(125\) 1.34520e24 1.03352
\(126\) 0 0
\(127\) 2.32044e24 1.48535 0.742674 0.669653i \(-0.233557\pi\)
0.742674 + 0.669653i \(0.233557\pi\)
\(128\) 3.64148e23 0.212992
\(129\) 0 0
\(130\) 1.51528e24 0.741556
\(131\) −8.70825e23 −0.390221 −0.195111 0.980781i \(-0.562507\pi\)
−0.195111 + 0.980781i \(0.562507\pi\)
\(132\) 0 0
\(133\) −2.37531e24 −0.894182
\(134\) 5.93146e24 2.04860
\(135\) 0 0
\(136\) 6.07751e23 0.177022
\(137\) −4.43869e24 −1.18841 −0.594207 0.804312i \(-0.702534\pi\)
−0.594207 + 0.804312i \(0.702534\pi\)
\(138\) 0 0
\(139\) −4.97229e23 −0.112690 −0.0563452 0.998411i \(-0.517945\pi\)
−0.0563452 + 0.998411i \(0.517945\pi\)
\(140\) −3.10097e24 −0.647180
\(141\) 0 0
\(142\) −3.06988e24 −0.544259
\(143\) 9.05647e23 0.148114
\(144\) 0 0
\(145\) −2.16347e24 −0.301591
\(146\) −1.40345e25 −1.80775
\(147\) 0 0
\(148\) −9.59276e24 −1.05666
\(149\) −1.10598e24 −0.112748 −0.0563738 0.998410i \(-0.517954\pi\)
−0.0563738 + 0.998410i \(0.517954\pi\)
\(150\) 0 0
\(151\) 3.76304e24 0.329082 0.164541 0.986370i \(-0.447386\pi\)
0.164541 + 0.986370i \(0.447386\pi\)
\(152\) 1.61698e24 0.131070
\(153\) 0 0
\(154\) −3.86143e24 −0.269315
\(155\) −1.43260e25 −0.927490
\(156\) 0 0
\(157\) 1.50090e25 0.838504 0.419252 0.907870i \(-0.362292\pi\)
0.419252 + 0.907870i \(0.362292\pi\)
\(158\) 4.11072e25 2.13482
\(159\) 0 0
\(160\) 2.95170e25 1.32646
\(161\) 1.36775e25 0.572150
\(162\) 0 0
\(163\) −2.61170e25 −0.947907 −0.473953 0.880550i \(-0.657173\pi\)
−0.473953 + 0.880550i \(0.657173\pi\)
\(164\) −1.08452e25 −0.366888
\(165\) 0 0
\(166\) 3.09900e25 0.911967
\(167\) −1.77408e25 −0.487230 −0.243615 0.969872i \(-0.578333\pi\)
−0.243615 + 0.969872i \(0.578333\pi\)
\(168\) 0 0
\(169\) −2.88587e25 −0.691161
\(170\) 9.88201e25 2.21148
\(171\) 0 0
\(172\) 1.69400e24 0.0331387
\(173\) −1.04109e26 −1.90528 −0.952640 0.304100i \(-0.901644\pi\)
−0.952640 + 0.304100i \(0.901644\pi\)
\(174\) 0 0
\(175\) −3.36576e24 −0.0539691
\(176\) 1.90087e25 0.285467
\(177\) 0 0
\(178\) −1.84157e25 −0.242861
\(179\) 1.00142e25 0.123824 0.0619122 0.998082i \(-0.480280\pi\)
0.0619122 + 0.998082i \(0.480280\pi\)
\(180\) 0 0
\(181\) −5.17169e25 −0.562768 −0.281384 0.959595i \(-0.590793\pi\)
−0.281384 + 0.959595i \(0.590793\pi\)
\(182\) −5.49817e25 −0.561562
\(183\) 0 0
\(184\) −9.31094e24 −0.0838665
\(185\) 1.30170e26 1.10163
\(186\) 0 0
\(187\) 5.90625e25 0.441706
\(188\) 6.71919e25 0.472609
\(189\) 0 0
\(190\) 2.62921e26 1.63742
\(191\) −3.10126e26 −1.81825 −0.909127 0.416520i \(-0.863250\pi\)
−0.909127 + 0.416520i \(0.863250\pi\)
\(192\) 0 0
\(193\) −1.28183e26 −0.666687 −0.333344 0.942805i \(-0.608177\pi\)
−0.333344 + 0.942805i \(0.608177\pi\)
\(194\) −4.56673e26 −2.23813
\(195\) 0 0
\(196\) −9.93833e25 −0.432881
\(197\) 3.89967e26 1.60201 0.801007 0.598655i \(-0.204298\pi\)
0.801007 + 0.598655i \(0.204298\pi\)
\(198\) 0 0
\(199\) −1.25611e26 −0.459426 −0.229713 0.973258i \(-0.573779\pi\)
−0.229713 + 0.973258i \(0.573779\pi\)
\(200\) 2.29123e24 0.00791086
\(201\) 0 0
\(202\) −5.47199e26 −1.68502
\(203\) 7.85013e25 0.228387
\(204\) 0 0
\(205\) 1.47165e26 0.382503
\(206\) 5.67918e25 0.139577
\(207\) 0 0
\(208\) 2.70658e26 0.595242
\(209\) 1.57142e26 0.327047
\(210\) 0 0
\(211\) 5.88286e26 1.09734 0.548669 0.836040i \(-0.315135\pi\)
0.548669 + 0.836040i \(0.315135\pi\)
\(212\) 5.91088e26 1.04421
\(213\) 0 0
\(214\) −2.42174e26 −0.384034
\(215\) −2.29869e25 −0.0345491
\(216\) 0 0
\(217\) 5.19817e26 0.702365
\(218\) −2.24180e26 −0.287308
\(219\) 0 0
\(220\) 2.05149e26 0.236706
\(221\) 8.40974e26 0.921025
\(222\) 0 0
\(223\) 1.38821e27 1.37072 0.685361 0.728203i \(-0.259644\pi\)
0.685361 + 0.728203i \(0.259644\pi\)
\(224\) −1.07102e27 −1.00449
\(225\) 0 0
\(226\) −1.41184e27 −1.19547
\(227\) 2.14958e27 1.73004 0.865022 0.501734i \(-0.167304\pi\)
0.865022 + 0.501734i \(0.167304\pi\)
\(228\) 0 0
\(229\) −6.39851e26 −0.465554 −0.232777 0.972530i \(-0.574781\pi\)
−0.232777 + 0.972530i \(0.574781\pi\)
\(230\) −1.51396e27 −1.04771
\(231\) 0 0
\(232\) −5.34395e25 −0.0334773
\(233\) −2.19149e27 −1.30661 −0.653306 0.757094i \(-0.726619\pi\)
−0.653306 + 0.757094i \(0.726619\pi\)
\(234\) 0 0
\(235\) −9.11766e26 −0.492723
\(236\) 7.88666e24 0.00405887
\(237\) 0 0
\(238\) −3.58568e27 −1.67470
\(239\) −1.09944e27 −0.489322 −0.244661 0.969609i \(-0.578677\pi\)
−0.244661 + 0.969609i \(0.578677\pi\)
\(240\) 0 0
\(241\) −1.44651e27 −0.584961 −0.292480 0.956272i \(-0.594481\pi\)
−0.292480 + 0.956272i \(0.594481\pi\)
\(242\) −3.34090e27 −1.28821
\(243\) 0 0
\(244\) 2.22470e27 0.780349
\(245\) 1.34859e27 0.451304
\(246\) 0 0
\(247\) 2.23750e27 0.681942
\(248\) −3.53864e26 −0.102954
\(249\) 0 0
\(250\) 5.40278e27 1.43320
\(251\) −2.24453e26 −0.0568693 −0.0284346 0.999596i \(-0.509052\pi\)
−0.0284346 + 0.999596i \(0.509052\pi\)
\(252\) 0 0
\(253\) −9.04857e26 −0.209264
\(254\) 9.31972e27 2.05975
\(255\) 0 0
\(256\) 5.62436e27 1.13583
\(257\) 3.95005e27 0.762732 0.381366 0.924424i \(-0.375454\pi\)
0.381366 + 0.924424i \(0.375454\pi\)
\(258\) 0 0
\(259\) −4.72320e27 −0.834236
\(260\) 2.92106e27 0.493567
\(261\) 0 0
\(262\) −3.49754e27 −0.541125
\(263\) −2.01521e27 −0.298420 −0.149210 0.988805i \(-0.547673\pi\)
−0.149210 + 0.988805i \(0.547673\pi\)
\(264\) 0 0
\(265\) −8.02081e27 −1.08866
\(266\) −9.54007e27 −1.23997
\(267\) 0 0
\(268\) 1.14343e28 1.36351
\(269\) −5.72063e27 −0.653571 −0.326785 0.945099i \(-0.605965\pi\)
−0.326785 + 0.945099i \(0.605965\pi\)
\(270\) 0 0
\(271\) 5.18050e27 0.543531 0.271766 0.962363i \(-0.412392\pi\)
0.271766 + 0.962363i \(0.412392\pi\)
\(272\) 1.76512e28 1.77514
\(273\) 0 0
\(274\) −1.78273e28 −1.64799
\(275\) 2.22667e26 0.0197392
\(276\) 0 0
\(277\) 1.29611e28 1.05712 0.528560 0.848896i \(-0.322732\pi\)
0.528560 + 0.848896i \(0.322732\pi\)
\(278\) −1.99705e27 −0.156269
\(279\) 0 0
\(280\) 1.03938e27 0.0748959
\(281\) −1.24154e28 −0.858691 −0.429346 0.903140i \(-0.641256\pi\)
−0.429346 + 0.903140i \(0.641256\pi\)
\(282\) 0 0
\(283\) −2.15031e28 −1.37074 −0.685372 0.728193i \(-0.740360\pi\)
−0.685372 + 0.728193i \(0.740360\pi\)
\(284\) −5.91792e27 −0.362250
\(285\) 0 0
\(286\) 3.63740e27 0.205391
\(287\) −5.33986e27 −0.289660
\(288\) 0 0
\(289\) 3.48772e28 1.74669
\(290\) −8.68925e27 −0.418220
\(291\) 0 0
\(292\) −2.70547e28 −1.20321
\(293\) 1.20307e28 0.514416 0.257208 0.966356i \(-0.417197\pi\)
0.257208 + 0.966356i \(0.417197\pi\)
\(294\) 0 0
\(295\) −1.07019e26 −0.00423161
\(296\) 3.21530e27 0.122283
\(297\) 0 0
\(298\) −4.44202e27 −0.156349
\(299\) −1.28840e28 −0.436347
\(300\) 0 0
\(301\) 8.34076e26 0.0261632
\(302\) 1.51137e28 0.456342
\(303\) 0 0
\(304\) 4.69629e28 1.31434
\(305\) −3.01883e28 −0.813560
\(306\) 0 0
\(307\) −4.70428e28 −1.17598 −0.587992 0.808867i \(-0.700081\pi\)
−0.587992 + 0.808867i \(0.700081\pi\)
\(308\) −7.44382e27 −0.179251
\(309\) 0 0
\(310\) −5.75381e28 −1.28616
\(311\) 8.99672e26 0.0193794 0.00968968 0.999953i \(-0.496916\pi\)
0.00968968 + 0.999953i \(0.496916\pi\)
\(312\) 0 0
\(313\) −1.13525e28 −0.227159 −0.113579 0.993529i \(-0.536232\pi\)
−0.113579 + 0.993529i \(0.536232\pi\)
\(314\) 6.02814e28 1.16277
\(315\) 0 0
\(316\) 7.92439e28 1.42090
\(317\) −1.37896e28 −0.238436 −0.119218 0.992868i \(-0.538039\pi\)
−0.119218 + 0.992868i \(0.538039\pi\)
\(318\) 0 0
\(319\) −5.19337e27 −0.0835326
\(320\) 5.21241e28 0.808750
\(321\) 0 0
\(322\) 5.49338e28 0.793408
\(323\) 1.45920e29 2.03370
\(324\) 0 0
\(325\) 3.17048e27 0.0411592
\(326\) −1.04895e29 −1.31448
\(327\) 0 0
\(328\) 3.63510e27 0.0424587
\(329\) 3.30834e28 0.373127
\(330\) 0 0
\(331\) 1.62668e29 1.71112 0.855559 0.517706i \(-0.173214\pi\)
0.855559 + 0.517706i \(0.173214\pi\)
\(332\) 5.97406e28 0.606990
\(333\) 0 0
\(334\) −7.12534e28 −0.675649
\(335\) −1.55159e29 −1.42154
\(336\) 0 0
\(337\) −1.64138e29 −1.40432 −0.702160 0.712019i \(-0.747781\pi\)
−0.702160 + 0.712019i \(0.747781\pi\)
\(338\) −1.15907e29 −0.958442
\(339\) 0 0
\(340\) 1.90499e29 1.47192
\(341\) −3.43892e28 −0.256890
\(342\) 0 0
\(343\) −1.53268e29 −1.07045
\(344\) −5.67795e26 −0.00383504
\(345\) 0 0
\(346\) −4.18139e29 −2.64208
\(347\) −1.45586e29 −0.889876 −0.444938 0.895561i \(-0.646774\pi\)
−0.444938 + 0.895561i \(0.646774\pi\)
\(348\) 0 0
\(349\) 2.22110e28 0.127080 0.0635398 0.997979i \(-0.479761\pi\)
0.0635398 + 0.997979i \(0.479761\pi\)
\(350\) −1.35181e28 −0.0748397
\(351\) 0 0
\(352\) 7.08552e28 0.367393
\(353\) −2.57166e28 −0.129064 −0.0645320 0.997916i \(-0.520555\pi\)
−0.0645320 + 0.997916i \(0.520555\pi\)
\(354\) 0 0
\(355\) 8.03037e28 0.377668
\(356\) −3.55006e28 −0.161644
\(357\) 0 0
\(358\) 4.02205e28 0.171709
\(359\) 3.34157e29 1.38154 0.690771 0.723074i \(-0.257271\pi\)
0.690771 + 0.723074i \(0.257271\pi\)
\(360\) 0 0
\(361\) 1.30406e29 0.505785
\(362\) −2.07713e29 −0.780398
\(363\) 0 0
\(364\) −1.05990e29 −0.373766
\(365\) 3.67122e29 1.25442
\(366\) 0 0
\(367\) 1.62664e29 0.521954 0.260977 0.965345i \(-0.415955\pi\)
0.260977 + 0.965345i \(0.415955\pi\)
\(368\) −2.70422e29 −0.840994
\(369\) 0 0
\(370\) 5.22807e29 1.52764
\(371\) 2.91034e29 0.824412
\(372\) 0 0
\(373\) 1.03540e29 0.275712 0.137856 0.990452i \(-0.455979\pi\)
0.137856 + 0.990452i \(0.455979\pi\)
\(374\) 2.37216e29 0.612520
\(375\) 0 0
\(376\) −2.25214e28 −0.0546935
\(377\) −7.39468e28 −0.174178
\(378\) 0 0
\(379\) 6.62210e29 1.46772 0.733862 0.679298i \(-0.237716\pi\)
0.733862 + 0.679298i \(0.237716\pi\)
\(380\) 5.06843e29 1.08984
\(381\) 0 0
\(382\) −1.24557e30 −2.52140
\(383\) −2.96585e29 −0.582591 −0.291295 0.956633i \(-0.594086\pi\)
−0.291295 + 0.956633i \(0.594086\pi\)
\(384\) 0 0
\(385\) 1.01010e29 0.186880
\(386\) −5.14829e29 −0.924504
\(387\) 0 0
\(388\) −8.80345e29 −1.48966
\(389\) 1.06609e30 1.75134 0.875672 0.482906i \(-0.160419\pi\)
0.875672 + 0.482906i \(0.160419\pi\)
\(390\) 0 0
\(391\) −8.40240e29 −1.30128
\(392\) 3.33113e28 0.0500959
\(393\) 0 0
\(394\) 1.56624e30 2.22153
\(395\) −1.07531e30 −1.48138
\(396\) 0 0
\(397\) 1.67462e28 0.0217683 0.0108842 0.999941i \(-0.496535\pi\)
0.0108842 + 0.999941i \(0.496535\pi\)
\(398\) −5.04496e29 −0.637093
\(399\) 0 0
\(400\) 6.65453e28 0.0793282
\(401\) −3.87121e29 −0.448421 −0.224211 0.974541i \(-0.571980\pi\)
−0.224211 + 0.974541i \(0.571980\pi\)
\(402\) 0 0
\(403\) −4.89658e29 −0.535654
\(404\) −1.05486e30 −1.12152
\(405\) 0 0
\(406\) 3.15289e29 0.316708
\(407\) 3.12470e29 0.305121
\(408\) 0 0
\(409\) 5.00511e29 0.461950 0.230975 0.972960i \(-0.425808\pi\)
0.230975 + 0.972960i \(0.425808\pi\)
\(410\) 5.91065e29 0.530422
\(411\) 0 0
\(412\) 1.09480e29 0.0928999
\(413\) 3.88316e27 0.00320450
\(414\) 0 0
\(415\) −8.10655e29 −0.632824
\(416\) 1.00889e30 0.766070
\(417\) 0 0
\(418\) 6.31137e29 0.453520
\(419\) −1.70760e30 −1.19378 −0.596891 0.802322i \(-0.703598\pi\)
−0.596891 + 0.802322i \(0.703598\pi\)
\(420\) 0 0
\(421\) −5.97218e29 −0.395265 −0.197633 0.980276i \(-0.563325\pi\)
−0.197633 + 0.980276i \(0.563325\pi\)
\(422\) 2.36276e30 1.52169
\(423\) 0 0
\(424\) −1.98121e29 −0.120843
\(425\) 2.06766e29 0.122746
\(426\) 0 0
\(427\) 1.09538e30 0.616089
\(428\) −4.66848e29 −0.255607
\(429\) 0 0
\(430\) −9.23233e28 −0.0479098
\(431\) −1.91854e30 −0.969353 −0.484676 0.874694i \(-0.661063\pi\)
−0.484676 + 0.874694i \(0.661063\pi\)
\(432\) 0 0
\(433\) −2.68456e30 −1.28606 −0.643032 0.765839i \(-0.722324\pi\)
−0.643032 + 0.765839i \(0.722324\pi\)
\(434\) 2.08777e30 0.973980
\(435\) 0 0
\(436\) −4.32160e29 −0.191228
\(437\) −2.23555e30 −0.963489
\(438\) 0 0
\(439\) 2.11168e29 0.0863549 0.0431775 0.999067i \(-0.486252\pi\)
0.0431775 + 0.999067i \(0.486252\pi\)
\(440\) −6.87619e28 −0.0273932
\(441\) 0 0
\(442\) 3.37765e30 1.27720
\(443\) 8.12623e29 0.299396 0.149698 0.988732i \(-0.452170\pi\)
0.149698 + 0.988732i \(0.452170\pi\)
\(444\) 0 0
\(445\) 4.81728e29 0.168524
\(446\) 5.57554e30 1.90080
\(447\) 0 0
\(448\) −1.89132e30 −0.612446
\(449\) 3.76596e30 1.18862 0.594310 0.804236i \(-0.297425\pi\)
0.594310 + 0.804236i \(0.297425\pi\)
\(450\) 0 0
\(451\) 3.53267e29 0.105943
\(452\) −2.72166e30 −0.795686
\(453\) 0 0
\(454\) 8.63348e30 2.39907
\(455\) 1.43824e30 0.389674
\(456\) 0 0
\(457\) 2.14090e30 0.551518 0.275759 0.961227i \(-0.411071\pi\)
0.275759 + 0.961227i \(0.411071\pi\)
\(458\) −2.56986e30 −0.645591
\(459\) 0 0
\(460\) −2.91851e30 −0.697342
\(461\) −4.76379e30 −1.11018 −0.555088 0.831792i \(-0.687315\pi\)
−0.555088 + 0.831792i \(0.687315\pi\)
\(462\) 0 0
\(463\) −7.67574e30 −1.70192 −0.850959 0.525232i \(-0.823979\pi\)
−0.850959 + 0.525232i \(0.823979\pi\)
\(464\) −1.55207e30 −0.335703
\(465\) 0 0
\(466\) −8.80180e30 −1.81190
\(467\) 2.87884e30 0.578194 0.289097 0.957300i \(-0.406645\pi\)
0.289097 + 0.957300i \(0.406645\pi\)
\(468\) 0 0
\(469\) 5.62992e30 1.07650
\(470\) −3.66197e30 −0.683266
\(471\) 0 0
\(472\) −2.64345e27 −0.000469719 0
\(473\) −5.51796e28 −0.00956919
\(474\) 0 0
\(475\) 5.50121e29 0.0908829
\(476\) −6.91225e30 −1.11465
\(477\) 0 0
\(478\) −4.41573e30 −0.678549
\(479\) −6.82143e30 −1.02333 −0.511666 0.859184i \(-0.670972\pi\)
−0.511666 + 0.859184i \(0.670972\pi\)
\(480\) 0 0
\(481\) 4.44917e30 0.636224
\(482\) −5.80971e30 −0.811173
\(483\) 0 0
\(484\) −6.44038e30 −0.857413
\(485\) 1.19459e31 1.55306
\(486\) 0 0
\(487\) 8.59361e29 0.106560 0.0532798 0.998580i \(-0.483032\pi\)
0.0532798 + 0.998580i \(0.483032\pi\)
\(488\) −7.45676e29 −0.0903071
\(489\) 0 0
\(490\) 5.41641e30 0.625830
\(491\) 7.15140e30 0.807149 0.403575 0.914947i \(-0.367768\pi\)
0.403575 + 0.914947i \(0.367768\pi\)
\(492\) 0 0
\(493\) −4.82250e30 −0.519437
\(494\) 8.98658e30 0.945659
\(495\) 0 0
\(496\) −1.02774e31 −1.03239
\(497\) −2.91382e30 −0.285998
\(498\) 0 0
\(499\) 1.77913e31 1.66744 0.833722 0.552185i \(-0.186206\pi\)
0.833722 + 0.552185i \(0.186206\pi\)
\(500\) 1.04151e31 0.953916
\(501\) 0 0
\(502\) −9.01483e29 −0.0788614
\(503\) −6.56155e30 −0.561015 −0.280507 0.959852i \(-0.590503\pi\)
−0.280507 + 0.959852i \(0.590503\pi\)
\(504\) 0 0
\(505\) 1.43139e31 1.16925
\(506\) −3.63422e30 −0.290189
\(507\) 0 0
\(508\) 1.79660e31 1.37094
\(509\) 1.19826e31 0.893917 0.446959 0.894555i \(-0.352507\pi\)
0.446959 + 0.894555i \(0.352507\pi\)
\(510\) 0 0
\(511\) −1.33210e31 −0.949939
\(512\) 1.95347e31 1.36208
\(513\) 0 0
\(514\) 1.58648e31 1.05769
\(515\) −1.48559e30 −0.0968537
\(516\) 0 0
\(517\) −2.18868e30 −0.136471
\(518\) −1.89700e31 −1.15685
\(519\) 0 0
\(520\) −9.79080e29 −0.0571189
\(521\) 2.26240e31 1.29103 0.645514 0.763749i \(-0.276643\pi\)
0.645514 + 0.763749i \(0.276643\pi\)
\(522\) 0 0
\(523\) −3.41914e31 −1.86702 −0.933508 0.358557i \(-0.883269\pi\)
−0.933508 + 0.358557i \(0.883269\pi\)
\(524\) −6.74233e30 −0.360164
\(525\) 0 0
\(526\) −8.09378e30 −0.413824
\(527\) −3.19334e31 −1.59744
\(528\) 0 0
\(529\) −8.00772e30 −0.383503
\(530\) −3.22144e31 −1.50965
\(531\) 0 0
\(532\) −1.83907e31 −0.825307
\(533\) 5.03006e30 0.220907
\(534\) 0 0
\(535\) 6.33494e30 0.266486
\(536\) −3.83255e30 −0.157795
\(537\) 0 0
\(538\) −2.29761e31 −0.906315
\(539\) 3.23726e30 0.124999
\(540\) 0 0
\(541\) −2.75024e30 −0.101766 −0.0508829 0.998705i \(-0.516204\pi\)
−0.0508829 + 0.998705i \(0.516204\pi\)
\(542\) 2.08067e31 0.753722
\(543\) 0 0
\(544\) 6.57953e31 2.28459
\(545\) 5.86424e30 0.199366
\(546\) 0 0
\(547\) −1.03197e31 −0.336367 −0.168184 0.985756i \(-0.553790\pi\)
−0.168184 + 0.985756i \(0.553790\pi\)
\(548\) −3.43664e31 −1.09688
\(549\) 0 0
\(550\) 8.94307e29 0.0273726
\(551\) −1.28308e31 −0.384600
\(552\) 0 0
\(553\) 3.90174e31 1.12181
\(554\) 5.20563e31 1.46592
\(555\) 0 0
\(556\) −3.84978e30 −0.104010
\(557\) −1.36262e31 −0.360613 −0.180307 0.983610i \(-0.557709\pi\)
−0.180307 + 0.983610i \(0.557709\pi\)
\(558\) 0 0
\(559\) −7.85685e29 −0.0199532
\(560\) 3.01873e31 0.751039
\(561\) 0 0
\(562\) −4.98645e31 −1.19076
\(563\) −3.09091e31 −0.723169 −0.361585 0.932339i \(-0.617764\pi\)
−0.361585 + 0.932339i \(0.617764\pi\)
\(564\) 0 0
\(565\) 3.69317e31 0.829550
\(566\) −8.63638e31 −1.90083
\(567\) 0 0
\(568\) 1.98357e30 0.0419220
\(569\) 6.93051e31 1.43540 0.717702 0.696350i \(-0.245194\pi\)
0.717702 + 0.696350i \(0.245194\pi\)
\(570\) 0 0
\(571\) −6.24961e30 −0.124319 −0.0621595 0.998066i \(-0.519799\pi\)
−0.0621595 + 0.998066i \(0.519799\pi\)
\(572\) 7.01195e30 0.136705
\(573\) 0 0
\(574\) −2.14468e31 −0.401675
\(575\) −3.16772e30 −0.0581522
\(576\) 0 0
\(577\) −1.59740e31 −0.281768 −0.140884 0.990026i \(-0.544994\pi\)
−0.140884 + 0.990026i \(0.544994\pi\)
\(578\) 1.40079e32 2.42216
\(579\) 0 0
\(580\) −1.67506e31 −0.278361
\(581\) 2.94146e31 0.479222
\(582\) 0 0
\(583\) −1.92538e31 −0.301529
\(584\) 9.06822e30 0.139243
\(585\) 0 0
\(586\) 4.83197e31 0.713348
\(587\) −8.16635e31 −1.18220 −0.591099 0.806599i \(-0.701306\pi\)
−0.591099 + 0.806599i \(0.701306\pi\)
\(588\) 0 0
\(589\) −8.49622e31 −1.18277
\(590\) −4.29824e29 −0.00586804
\(591\) 0 0
\(592\) 9.33837e31 1.22623
\(593\) 5.61370e30 0.0722970 0.0361485 0.999346i \(-0.488491\pi\)
0.0361485 + 0.999346i \(0.488491\pi\)
\(594\) 0 0
\(595\) 9.37963e31 1.16209
\(596\) −8.56305e30 −0.104063
\(597\) 0 0
\(598\) −5.17466e31 −0.605088
\(599\) −1.60231e32 −1.83797 −0.918985 0.394293i \(-0.870990\pi\)
−0.918985 + 0.394293i \(0.870990\pi\)
\(600\) 0 0
\(601\) 8.47641e31 0.935741 0.467870 0.883797i \(-0.345021\pi\)
0.467870 + 0.883797i \(0.345021\pi\)
\(602\) 3.34994e30 0.0362809
\(603\) 0 0
\(604\) 2.91352e31 0.303734
\(605\) 8.73933e31 0.893905
\(606\) 0 0
\(607\) −1.63793e31 −0.161296 −0.0806481 0.996743i \(-0.525699\pi\)
−0.0806481 + 0.996743i \(0.525699\pi\)
\(608\) 1.75055e32 1.69155
\(609\) 0 0
\(610\) −1.21247e32 −1.12818
\(611\) −3.11639e31 −0.284563
\(612\) 0 0
\(613\) 1.93558e31 0.170222 0.0851110 0.996371i \(-0.472876\pi\)
0.0851110 + 0.996371i \(0.472876\pi\)
\(614\) −1.88940e32 −1.63075
\(615\) 0 0
\(616\) 2.49502e30 0.0207442
\(617\) −5.25618e31 −0.428934 −0.214467 0.976731i \(-0.568801\pi\)
−0.214467 + 0.976731i \(0.568801\pi\)
\(618\) 0 0
\(619\) −9.42009e31 −0.740650 −0.370325 0.928902i \(-0.620754\pi\)
−0.370325 + 0.928902i \(0.620754\pi\)
\(620\) −1.10918e32 −0.856049
\(621\) 0 0
\(622\) 3.61340e30 0.0268736
\(623\) −1.74795e31 −0.127619
\(624\) 0 0
\(625\) −1.30804e32 −0.920452
\(626\) −4.55954e31 −0.315004
\(627\) 0 0
\(628\) 1.16207e32 0.773918
\(629\) 2.90156e32 1.89736
\(630\) 0 0
\(631\) −2.22784e32 −1.40458 −0.702288 0.711892i \(-0.747838\pi\)
−0.702288 + 0.711892i \(0.747838\pi\)
\(632\) −2.65610e31 −0.164436
\(633\) 0 0
\(634\) −5.53840e31 −0.330642
\(635\) −2.43790e32 −1.42929
\(636\) 0 0
\(637\) 4.60944e31 0.260642
\(638\) −2.08584e31 −0.115836
\(639\) 0 0
\(640\) −3.82581e31 −0.204953
\(641\) −1.38602e32 −0.729295 −0.364648 0.931146i \(-0.618811\pi\)
−0.364648 + 0.931146i \(0.618811\pi\)
\(642\) 0 0
\(643\) 3.71401e31 0.188546 0.0942729 0.995546i \(-0.469947\pi\)
0.0942729 + 0.995546i \(0.469947\pi\)
\(644\) 1.05898e32 0.528080
\(645\) 0 0
\(646\) 5.86067e32 2.82016
\(647\) 3.77554e32 1.78476 0.892379 0.451286i \(-0.149035\pi\)
0.892379 + 0.451286i \(0.149035\pi\)
\(648\) 0 0
\(649\) −2.56896e29 −0.00117204
\(650\) 1.27338e31 0.0570760
\(651\) 0 0
\(652\) −2.02210e32 −0.874893
\(653\) 8.58042e31 0.364760 0.182380 0.983228i \(-0.441620\pi\)
0.182380 + 0.983228i \(0.441620\pi\)
\(654\) 0 0
\(655\) 9.14906e31 0.375493
\(656\) 1.05576e32 0.425766
\(657\) 0 0
\(658\) 1.32874e32 0.517421
\(659\) −1.46687e32 −0.561319 −0.280660 0.959807i \(-0.590553\pi\)
−0.280660 + 0.959807i \(0.590553\pi\)
\(660\) 0 0
\(661\) 2.02066e32 0.746752 0.373376 0.927680i \(-0.378200\pi\)
0.373376 + 0.927680i \(0.378200\pi\)
\(662\) 6.53331e32 2.37283
\(663\) 0 0
\(664\) −2.00239e31 −0.0702450
\(665\) 2.49555e32 0.860432
\(666\) 0 0
\(667\) 7.38823e31 0.246090
\(668\) −1.37358e32 −0.449701
\(669\) 0 0
\(670\) −6.23171e32 −1.97127
\(671\) −7.24664e31 −0.225334
\(672\) 0 0
\(673\) 5.34692e32 1.60668 0.803341 0.595519i \(-0.203053\pi\)
0.803341 + 0.595519i \(0.203053\pi\)
\(674\) −6.59237e32 −1.94739
\(675\) 0 0
\(676\) −2.23437e32 −0.637924
\(677\) −1.10049e32 −0.308899 −0.154449 0.988001i \(-0.549360\pi\)
−0.154449 + 0.988001i \(0.549360\pi\)
\(678\) 0 0
\(679\) −4.33457e32 −1.17610
\(680\) −6.38515e31 −0.170341
\(681\) 0 0
\(682\) −1.38119e32 −0.356233
\(683\) 2.97297e32 0.753967 0.376983 0.926220i \(-0.376961\pi\)
0.376983 + 0.926220i \(0.376961\pi\)
\(684\) 0 0
\(685\) 4.66337e32 1.14356
\(686\) −6.15577e32 −1.48441
\(687\) 0 0
\(688\) −1.64908e31 −0.0384568
\(689\) −2.74149e32 −0.628733
\(690\) 0 0
\(691\) −7.18838e32 −1.59453 −0.797266 0.603628i \(-0.793721\pi\)
−0.797266 + 0.603628i \(0.793721\pi\)
\(692\) −8.06062e32 −1.75852
\(693\) 0 0
\(694\) −5.84723e32 −1.23400
\(695\) 5.22399e31 0.108437
\(696\) 0 0
\(697\) 3.28039e32 0.658793
\(698\) 8.92072e31 0.176223
\(699\) 0 0
\(700\) −2.60593e31 −0.0498121
\(701\) 1.02832e33 1.93362 0.966810 0.255497i \(-0.0822390\pi\)
0.966810 + 0.255497i \(0.0822390\pi\)
\(702\) 0 0
\(703\) 7.71990e32 1.40484
\(704\) 1.25123e32 0.224002
\(705\) 0 0
\(706\) −1.03287e32 −0.178975
\(707\) −5.19380e32 −0.885446
\(708\) 0 0
\(709\) −2.50942e32 −0.414134 −0.207067 0.978327i \(-0.566392\pi\)
−0.207067 + 0.978327i \(0.566392\pi\)
\(710\) 3.22528e32 0.523717
\(711\) 0 0
\(712\) 1.18991e31 0.0187066
\(713\) 4.89231e32 0.756805
\(714\) 0 0
\(715\) −9.51492e31 −0.142523
\(716\) 7.75346e31 0.114287
\(717\) 0 0
\(718\) 1.34209e33 1.91580
\(719\) −7.09247e32 −0.996358 −0.498179 0.867074i \(-0.665998\pi\)
−0.498179 + 0.867074i \(0.665998\pi\)
\(720\) 0 0
\(721\) 5.39046e31 0.0733450
\(722\) 5.23757e32 0.701378
\(723\) 0 0
\(724\) −4.00417e32 −0.519420
\(725\) −1.81809e31 −0.0232128
\(726\) 0 0
\(727\) 4.63712e32 0.573591 0.286795 0.957992i \(-0.407410\pi\)
0.286795 + 0.957992i \(0.407410\pi\)
\(728\) 3.55259e31 0.0432547
\(729\) 0 0
\(730\) 1.47449e33 1.73952
\(731\) −5.12391e31 −0.0595047
\(732\) 0 0
\(733\) −1.54780e33 −1.74188 −0.870941 0.491387i \(-0.836490\pi\)
−0.870941 + 0.491387i \(0.836490\pi\)
\(734\) 6.53317e32 0.723801
\(735\) 0 0
\(736\) −1.00801e33 −1.08235
\(737\) −3.72455e32 −0.393730
\(738\) 0 0
\(739\) −1.49249e33 −1.52932 −0.764662 0.644431i \(-0.777094\pi\)
−0.764662 + 0.644431i \(0.777094\pi\)
\(740\) 1.00783e33 1.01677
\(741\) 0 0
\(742\) 1.16890e33 1.14322
\(743\) −1.19934e32 −0.115497 −0.0577487 0.998331i \(-0.518392\pi\)
−0.0577487 + 0.998331i \(0.518392\pi\)
\(744\) 0 0
\(745\) 1.16197e32 0.108492
\(746\) 4.15852e32 0.382334
\(747\) 0 0
\(748\) 4.57290e32 0.407683
\(749\) −2.29863e32 −0.201803
\(750\) 0 0
\(751\) 1.42812e33 1.21592 0.607959 0.793969i \(-0.291989\pi\)
0.607959 + 0.793969i \(0.291989\pi\)
\(752\) −6.54101e32 −0.548453
\(753\) 0 0
\(754\) −2.96996e32 −0.241535
\(755\) −3.95352e32 −0.316661
\(756\) 0 0
\(757\) −1.56208e33 −1.21367 −0.606835 0.794828i \(-0.707561\pi\)
−0.606835 + 0.794828i \(0.707561\pi\)
\(758\) 2.65967e33 2.03531
\(759\) 0 0
\(760\) −1.69884e32 −0.126123
\(761\) −1.92115e33 −1.40488 −0.702439 0.711744i \(-0.747906\pi\)
−0.702439 + 0.711744i \(0.747906\pi\)
\(762\) 0 0
\(763\) −2.12783e32 −0.150975
\(764\) −2.40114e33 −1.67820
\(765\) 0 0
\(766\) −1.19119e33 −0.807886
\(767\) −3.65787e30 −0.00244389
\(768\) 0 0
\(769\) 1.22615e33 0.795044 0.397522 0.917593i \(-0.369870\pi\)
0.397522 + 0.917593i \(0.369870\pi\)
\(770\) 4.05690e32 0.259150
\(771\) 0 0
\(772\) −9.92455e32 −0.615335
\(773\) 8.43914e32 0.515507 0.257753 0.966211i \(-0.417018\pi\)
0.257753 + 0.966211i \(0.417018\pi\)
\(774\) 0 0
\(775\) −1.20390e32 −0.0713870
\(776\) 2.95074e32 0.172394
\(777\) 0 0
\(778\) 4.28177e33 2.42861
\(779\) 8.72782e32 0.487781
\(780\) 0 0
\(781\) 1.92768e32 0.104604
\(782\) −3.37470e33 −1.80450
\(783\) 0 0
\(784\) 9.67477e32 0.502350
\(785\) −1.57687e33 −0.806856
\(786\) 0 0
\(787\) 1.37027e33 0.680919 0.340460 0.940259i \(-0.389417\pi\)
0.340460 + 0.940259i \(0.389417\pi\)
\(788\) 3.01931e33 1.47862
\(789\) 0 0
\(790\) −4.31881e33 −2.05425
\(791\) −1.34007e33 −0.628198
\(792\) 0 0
\(793\) −1.03183e33 −0.469857
\(794\) 6.72586e31 0.0301865
\(795\) 0 0
\(796\) −9.72536e32 −0.424039
\(797\) −1.57462e33 −0.676714 −0.338357 0.941018i \(-0.609871\pi\)
−0.338357 + 0.941018i \(0.609871\pi\)
\(798\) 0 0
\(799\) −2.03238e33 −0.848628
\(800\) 2.48049e32 0.102095
\(801\) 0 0
\(802\) −1.55481e33 −0.621832
\(803\) 8.81269e32 0.347440
\(804\) 0 0
\(805\) −1.43699e33 −0.550555
\(806\) −1.96664e33 −0.742799
\(807\) 0 0
\(808\) 3.53567e32 0.129790
\(809\) 2.56942e33 0.929882 0.464941 0.885342i \(-0.346076\pi\)
0.464941 + 0.885342i \(0.346076\pi\)
\(810\) 0 0
\(811\) 1.25770e33 0.442423 0.221212 0.975226i \(-0.428999\pi\)
0.221212 + 0.975226i \(0.428999\pi\)
\(812\) 6.07794e32 0.210796
\(813\) 0 0
\(814\) 1.25499e33 0.423116
\(815\) 2.74390e33 0.912129
\(816\) 0 0
\(817\) −1.36327e32 −0.0440583
\(818\) 2.01023e33 0.640593
\(819\) 0 0
\(820\) 1.13942e33 0.353040
\(821\) 2.38117e32 0.0727519 0.0363759 0.999338i \(-0.488419\pi\)
0.0363759 + 0.999338i \(0.488419\pi\)
\(822\) 0 0
\(823\) 5.82453e33 1.73047 0.865233 0.501370i \(-0.167170\pi\)
0.865233 + 0.501370i \(0.167170\pi\)
\(824\) −3.66954e31 −0.0107510
\(825\) 0 0
\(826\) 1.55961e31 0.00444372
\(827\) 3.61128e33 1.01472 0.507362 0.861733i \(-0.330621\pi\)
0.507362 + 0.861733i \(0.330621\pi\)
\(828\) 0 0
\(829\) 6.27935e33 1.71608 0.858040 0.513584i \(-0.171682\pi\)
0.858040 + 0.513584i \(0.171682\pi\)
\(830\) −3.25588e33 −0.877546
\(831\) 0 0
\(832\) 1.78159e33 0.467078
\(833\) 3.00609e33 0.777292
\(834\) 0 0
\(835\) 1.86389e33 0.468841
\(836\) 1.21667e33 0.301856
\(837\) 0 0
\(838\) −6.85832e33 −1.65543
\(839\) −7.51696e32 −0.178970 −0.0894849 0.995988i \(-0.528522\pi\)
−0.0894849 + 0.995988i \(0.528522\pi\)
\(840\) 0 0
\(841\) −3.89268e33 −0.901767
\(842\) −2.39864e33 −0.548120
\(843\) 0 0
\(844\) 4.55478e33 1.01281
\(845\) 3.03195e33 0.665074
\(846\) 0 0
\(847\) −3.17106e33 −0.676932
\(848\) −5.75412e33 −1.21179
\(849\) 0 0
\(850\) 8.30443e32 0.170213
\(851\) −4.44528e33 −0.898896
\(852\) 0 0
\(853\) −9.11290e33 −1.79367 −0.896836 0.442364i \(-0.854140\pi\)
−0.896836 + 0.442364i \(0.854140\pi\)
\(854\) 4.39943e33 0.854339
\(855\) 0 0
\(856\) 1.56478e32 0.0295805
\(857\) −6.17423e33 −1.15160 −0.575802 0.817589i \(-0.695310\pi\)
−0.575802 + 0.817589i \(0.695310\pi\)
\(858\) 0 0
\(859\) −1.86175e33 −0.338065 −0.169033 0.985610i \(-0.554064\pi\)
−0.169033 + 0.985610i \(0.554064\pi\)
\(860\) −1.77975e32 −0.0318880
\(861\) 0 0
\(862\) −7.70552e33 −1.34421
\(863\) 5.22144e33 0.898806 0.449403 0.893329i \(-0.351637\pi\)
0.449403 + 0.893329i \(0.351637\pi\)
\(864\) 0 0
\(865\) 1.09379e34 1.83337
\(866\) −1.07821e34 −1.78340
\(867\) 0 0
\(868\) 4.02466e33 0.648265
\(869\) −2.58126e33 −0.410302
\(870\) 0 0
\(871\) −5.30328e33 −0.820986
\(872\) 1.44852e32 0.0221301
\(873\) 0 0
\(874\) −8.97874e33 −1.33608
\(875\) 5.12811e33 0.753121
\(876\) 0 0
\(877\) −6.73800e32 −0.0963908 −0.0481954 0.998838i \(-0.515347\pi\)
−0.0481954 + 0.998838i \(0.515347\pi\)
\(878\) 8.48127e32 0.119750
\(879\) 0 0
\(880\) −1.99709e33 −0.274692
\(881\) −1.36978e34 −1.85963 −0.929815 0.368028i \(-0.880033\pi\)
−0.929815 + 0.368028i \(0.880033\pi\)
\(882\) 0 0
\(883\) 1.49590e33 0.197859 0.0989294 0.995094i \(-0.468458\pi\)
0.0989294 + 0.995094i \(0.468458\pi\)
\(884\) 6.51121e33 0.850082
\(885\) 0 0
\(886\) 3.26378e33 0.415177
\(887\) 6.44904e33 0.809792 0.404896 0.914363i \(-0.367308\pi\)
0.404896 + 0.914363i \(0.367308\pi\)
\(888\) 0 0
\(889\) 8.84592e33 1.08236
\(890\) 1.93479e33 0.233695
\(891\) 0 0
\(892\) 1.07482e34 1.26514
\(893\) −5.40736e33 −0.628339
\(894\) 0 0
\(895\) −1.05211e33 −0.119151
\(896\) 1.38819e33 0.155206
\(897\) 0 0
\(898\) 1.51254e34 1.64827
\(899\) 2.80791e33 0.302097
\(900\) 0 0
\(901\) −1.78789e34 −1.87502
\(902\) 1.41884e33 0.146913
\(903\) 0 0
\(904\) 9.12246e32 0.0920820
\(905\) 5.43349e33 0.541527
\(906\) 0 0
\(907\) 6.42282e33 0.624083 0.312041 0.950069i \(-0.398987\pi\)
0.312041 + 0.950069i \(0.398987\pi\)
\(908\) 1.66431e34 1.59679
\(909\) 0 0
\(910\) 5.77649e33 0.540366
\(911\) 1.48932e34 1.37571 0.687855 0.725848i \(-0.258553\pi\)
0.687855 + 0.725848i \(0.258553\pi\)
\(912\) 0 0
\(913\) −1.94596e33 −0.175275
\(914\) 8.59859e33 0.764797
\(915\) 0 0
\(916\) −4.95402e33 −0.429695
\(917\) −3.31973e33 −0.284351
\(918\) 0 0
\(919\) 3.10701e33 0.259546 0.129773 0.991544i \(-0.458575\pi\)
0.129773 + 0.991544i \(0.458575\pi\)
\(920\) 9.78226e32 0.0807011
\(921\) 0 0
\(922\) −1.91330e34 −1.53950
\(923\) 2.74476e33 0.218115
\(924\) 0 0
\(925\) 1.09389e33 0.0847900
\(926\) −3.08285e34 −2.36007
\(927\) 0 0
\(928\) −5.78538e33 −0.432046
\(929\) −1.72846e34 −1.27490 −0.637452 0.770490i \(-0.720012\pi\)
−0.637452 + 0.770490i \(0.720012\pi\)
\(930\) 0 0
\(931\) 7.99801e33 0.575520
\(932\) −1.69676e34 −1.20597
\(933\) 0 0
\(934\) 1.15624e34 0.801790
\(935\) −6.20523e33 −0.425035
\(936\) 0 0
\(937\) −1.61548e34 −1.07969 −0.539843 0.841766i \(-0.681516\pi\)
−0.539843 + 0.841766i \(0.681516\pi\)
\(938\) 2.26117e34 1.49280
\(939\) 0 0
\(940\) −7.05932e33 −0.454771
\(941\) −7.69275e33 −0.489554 −0.244777 0.969579i \(-0.578715\pi\)
−0.244777 + 0.969579i \(0.578715\pi\)
\(942\) 0 0
\(943\) −5.02567e33 −0.312111
\(944\) −7.67751e31 −0.00471023
\(945\) 0 0
\(946\) −2.21620e32 −0.0132697
\(947\) 2.94332e34 1.74106 0.870528 0.492119i \(-0.163778\pi\)
0.870528 + 0.492119i \(0.163778\pi\)
\(948\) 0 0
\(949\) 1.25481e34 0.724465
\(950\) 2.20948e33 0.126029
\(951\) 0 0
\(952\) 2.31685e33 0.128995
\(953\) −1.97292e34 −1.08528 −0.542641 0.839965i \(-0.682575\pi\)
−0.542641 + 0.839965i \(0.682575\pi\)
\(954\) 0 0
\(955\) 3.25825e34 1.74963
\(956\) −8.51236e33 −0.451632
\(957\) 0 0
\(958\) −2.73973e34 −1.41907
\(959\) −1.69210e34 −0.865989
\(960\) 0 0
\(961\) −1.42003e33 −0.0709542
\(962\) 1.78694e34 0.882261
\(963\) 0 0
\(964\) −1.11996e34 −0.539904
\(965\) 1.34672e34 0.641524
\(966\) 0 0
\(967\) −2.03266e34 −0.945496 −0.472748 0.881198i \(-0.656738\pi\)
−0.472748 + 0.881198i \(0.656738\pi\)
\(968\) 2.15869e33 0.0992255
\(969\) 0 0
\(970\) 4.79790e34 2.15365
\(971\) −3.40470e34 −1.51028 −0.755141 0.655563i \(-0.772431\pi\)
−0.755141 + 0.655563i \(0.772431\pi\)
\(972\) 0 0
\(973\) −1.89552e33 −0.0821166
\(974\) 3.45150e33 0.147768
\(975\) 0 0
\(976\) −2.16570e34 −0.905579
\(977\) 3.68167e32 0.0152145 0.00760725 0.999971i \(-0.497579\pi\)
0.00760725 + 0.999971i \(0.497579\pi\)
\(978\) 0 0
\(979\) 1.15638e33 0.0466767
\(980\) 1.04414e34 0.416542
\(981\) 0 0
\(982\) 2.87225e34 1.11928
\(983\) 8.23724e33 0.317261 0.158630 0.987338i \(-0.449292\pi\)
0.158630 + 0.987338i \(0.449292\pi\)
\(984\) 0 0
\(985\) −4.09707e34 −1.54155
\(986\) −1.93689e34 −0.720310
\(987\) 0 0
\(988\) 1.73238e34 0.629415
\(989\) 7.85000e32 0.0281911
\(990\) 0 0
\(991\) 2.70349e33 0.0948590 0.0474295 0.998875i \(-0.484897\pi\)
0.0474295 + 0.998875i \(0.484897\pi\)
\(992\) −3.83094e34 −1.32868
\(993\) 0 0
\(994\) −1.17029e34 −0.396598
\(995\) 1.31969e34 0.442086
\(996\) 0 0
\(997\) −4.64692e33 −0.152115 −0.0760573 0.997103i \(-0.524233\pi\)
−0.0760573 + 0.997103i \(0.524233\pi\)
\(998\) 7.14561e34 2.31227
\(999\) 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 9.24.a.b.1.2 2
3.2 odd 2 1.24.a.a.1.1 2
12.11 even 2 16.24.a.b.1.1 2
15.2 even 4 25.24.b.a.24.2 4
15.8 even 4 25.24.b.a.24.3 4
15.14 odd 2 25.24.a.a.1.2 2
21.20 even 2 49.24.a.b.1.1 2
24.5 odd 2 64.24.a.d.1.1 2
24.11 even 2 64.24.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.24.a.a.1.1 2 3.2 odd 2
9.24.a.b.1.2 2 1.1 even 1 trivial
16.24.a.b.1.1 2 12.11 even 2
25.24.a.a.1.2 2 15.14 odd 2
25.24.b.a.24.2 4 15.2 even 4
25.24.b.a.24.3 4 15.8 even 4
49.24.a.b.1.1 2 21.20 even 2
64.24.a.d.1.1 2 24.5 odd 2
64.24.a.g.1.2 2 24.11 even 2