Properties

Label 49.24.a.b.1.1
Level $49$
Weight $24$
Character 49.1
Self dual yes
Analytic conductor $164.250$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(164.249978299\)
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 1.1
Root \(190.348\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q-4016.35 q^{2} -388445. q^{3} +7.74247e6 q^{4} -1.05062e8 q^{5} +1.56013e9 q^{6} +2.59512e9 q^{8} +5.67462e10 q^{9} +O(q^{10})\) \(q-4016.35 q^{2} -388445. q^{3} +7.74247e6 q^{4} -1.05062e8 q^{5} +1.56013e9 q^{6} +2.59512e9 q^{8} +5.67462e10 q^{9} +4.21966e11 q^{10} +2.52200e11 q^{11} -3.00752e12 q^{12} +3.59099e12 q^{13} +4.08108e13 q^{15} -7.53715e13 q^{16} -2.34190e14 q^{17} -2.27913e14 q^{18} +6.23086e14 q^{19} -8.13439e14 q^{20} -1.01292e15 q^{22} -3.58786e15 q^{23} -1.00806e15 q^{24} -8.82898e14 q^{25} -1.44227e16 q^{26} +1.45267e16 q^{27} -2.05923e16 q^{29} -1.63911e17 q^{30} -1.36357e17 q^{31} +2.80949e17 q^{32} -9.79656e16 q^{33} +9.40588e17 q^{34} +4.39356e17 q^{36} -1.23898e18 q^{37} -2.50253e18 q^{38} -1.39490e18 q^{39} -2.72649e17 q^{40} -1.40074e18 q^{41} +2.18793e17 q^{43} +1.95265e18 q^{44} -5.96187e18 q^{45} +1.44101e19 q^{46} +8.67836e18 q^{47} +2.92777e19 q^{48} +3.54603e18 q^{50} +9.09698e19 q^{51} +2.78032e19 q^{52} -7.63436e19 q^{53} -5.83441e19 q^{54} -2.64966e19 q^{55} -2.42034e20 q^{57} +8.27059e19 q^{58} +1.01862e18 q^{59} +3.15976e20 q^{60} -2.87337e20 q^{61} +5.47658e20 q^{62} -4.96127e20 q^{64} -3.77277e20 q^{65} +3.93464e20 q^{66} +1.47683e21 q^{67} -1.81321e21 q^{68} +1.39369e21 q^{69} +7.64346e20 q^{71} +1.47263e20 q^{72} +3.49433e21 q^{73} +4.97617e21 q^{74} +3.42957e20 q^{75} +4.82422e21 q^{76} +5.60242e21 q^{78} +1.02350e22 q^{79} +7.91868e21 q^{80} -1.09851e22 q^{81} +5.62587e21 q^{82} +7.71597e21 q^{83} +2.46044e22 q^{85} -8.78750e20 q^{86} +7.99897e21 q^{87} +6.54489e20 q^{88} -4.58518e21 q^{89} +2.39450e22 q^{90} -2.77789e22 q^{92} +5.29672e22 q^{93} -3.48553e22 q^{94} -6.54627e22 q^{95} -1.09133e23 q^{96} +1.13703e23 q^{97} +1.43114e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 1080q^{2} - 339480q^{3} + 25326656q^{4} - 73069020q^{5} + 1809673056q^{6} + 49459023360q^{8} - 34999394166q^{9} + O(q^{10}) \) \( 2q + 1080q^{2} - 339480q^{3} + 25326656q^{4} - 73069020q^{5} + 1809673056q^{6} + 49459023360q^{8} - 34999394166q^{9} + 585013636080q^{10} + 856801968264q^{11} - 2146514952960q^{12} - 4376109322060q^{13} + 42377338985040q^{15} + 15956586401792q^{16} - 254028147597540q^{17} - 695480683916520q^{18} - 4260600979960q^{19} - 250868387468160q^{20} + 2068343882177760q^{22} - 8144713079008560q^{23} + 1286622315141120q^{24} - 11780274628800850q^{25} - 55025854658735184q^{26} + 5424634982716560q^{27} + 20818433601623340q^{29} - 155926924188644160q^{30} - 137714017177000384q^{31} + 353265663781601280q^{32} - 68361366766001760q^{33} + 839483655961325328q^{34} - 1173916300077574848q^{36} - 897721264408967780q^{37} - 5699708971590961440q^{38} - 1785011473665029232q^{39} + 1226668524414336000q^{40} + 2294435477168314956q^{41} - 1750760768619855800q^{43} + 12584088840033038592q^{44} - 8897092690294206540q^{45} - 8813206018050221376q^{46} - 15759744217656780960q^{47} + 33749519399576616960q^{48} - 51990825483785316600q^{50} + 89998362845078292144q^{51} - 112291883783912022400q^{52} - 140287253401646796420q^{53} - 104731223417039799360q^{54} - 7153550955060182640q^{55} - 272752401448627175520q^{57} + 293749486923568689360q^{58} - 280872989971340771880q^{59} + 343522601114937592320q^{60} + 180452892516502223636q^{61} + 540743475843874103040q^{62} - 893690254469352914944q^{64} - 632168834809440380760q^{65} + 544338140913651883392q^{66} + 1754233163431557625240q^{67} - 2162050190142944330880q^{68} + 1170560672172404223552q^{69} + 3055033510194143328624q^{71} - 4152294352103038548480q^{72} + 8063408253877606149260q^{73} + 6715344283148807757072q^{74} - 190631089350520885800q^{75} - 6207154294513080590080q^{76} + 3614293948840808587200q^{78} + 6244916814559639980640q^{79} + 10840537585501794017280q^{80} - 2793528580929833975598q^{81} + 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 23969743087870314902520q^{85} - 10916288812918999243296q^{86} + 10026640653837674384880q^{87} + 28988514668199273707520q^{88} - 6395093086173070004820q^{89} + 8986073954327865866160q^{90} - 107907439017191756981760q^{92} + 52900811441357852079360q^{93} - 159400518006534931827072q^{94} - 85533361066700858502000q^{95} - 105592121669584394256384q^{96} + 31147288846254030500540q^{97} - 41158245132312135981912q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4016.35 −1.38671 −0.693357 0.720595i \(-0.743869\pi\)
−0.693357 + 0.720595i \(0.743869\pi\)
\(3\) −388445. −1.26600 −0.633002 0.774150i \(-0.718177\pi\)
−0.633002 + 0.774150i \(0.718177\pi\)
\(4\) 7.74247e6 0.922974
\(5\) −1.05062e8 −0.962256 −0.481128 0.876650i \(-0.659773\pi\)
−0.481128 + 0.876650i \(0.659773\pi\)
\(6\) 1.56013e9 1.75558
\(7\) 0 0
\(8\) 2.59512e9 0.106813
\(9\) 5.67462e10 0.602765
\(10\) 4.21966e11 1.33437
\(11\) 2.52200e11 0.266519 0.133260 0.991081i \(-0.457456\pi\)
0.133260 + 0.991081i \(0.457456\pi\)
\(12\) −3.00752e12 −1.16849
\(13\) 3.59099e12 0.555733 0.277867 0.960620i \(-0.410373\pi\)
0.277867 + 0.960620i \(0.410373\pi\)
\(14\) 0 0
\(15\) 4.08108e13 1.21822
\(16\) −7.53715e13 −1.07109
\(17\) −2.34190e14 −1.65731 −0.828657 0.559756i \(-0.810895\pi\)
−0.828657 + 0.559756i \(0.810895\pi\)
\(18\) −2.27913e14 −0.835863
\(19\) 6.23086e14 1.22710 0.613552 0.789654i \(-0.289740\pi\)
0.613552 + 0.789654i \(0.289740\pi\)
\(20\) −8.13439e14 −0.888138
\(21\) 0 0
\(22\) −1.01292e15 −0.369586
\(23\) −3.58786e15 −0.785173 −0.392587 0.919715i \(-0.628420\pi\)
−0.392587 + 0.919715i \(0.628420\pi\)
\(24\) −1.00806e15 −0.135225
\(25\) −8.82898e14 −0.0740629
\(26\) −1.44227e16 −0.770642
\(27\) 1.45267e16 0.502901
\(28\) 0 0
\(29\) −2.05923e16 −0.313421 −0.156710 0.987645i \(-0.550089\pi\)
−0.156710 + 0.987645i \(0.550089\pi\)
\(30\) −1.63911e17 −1.68932
\(31\) −1.36357e17 −0.963870 −0.481935 0.876207i \(-0.660066\pi\)
−0.481935 + 0.876207i \(0.660066\pi\)
\(32\) 2.80949e17 1.37849
\(33\) −9.79656e16 −0.337414
\(34\) 9.40588e17 2.29822
\(35\) 0 0
\(36\) 4.39356e17 0.556337
\(37\) −1.23898e18 −1.14484 −0.572419 0.819961i \(-0.693995\pi\)
−0.572419 + 0.819961i \(0.693995\pi\)
\(38\) −2.50253e18 −1.70164
\(39\) −1.39490e18 −0.703560
\(40\) −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
\(45\) −5.96187e18 −0.580015
\(46\) 1.44101e19 1.08881
\(47\) 8.67836e18 0.512050 0.256025 0.966670i \(-0.417587\pi\)
0.256025 + 0.966670i \(0.417587\pi\)
\(48\) 2.92777e19 1.35601
\(49\) 0 0
\(50\) 3.54603e18 0.102704
\(51\) 9.09698e19 2.09817
\(52\) 2.78032e19 0.512927
\(53\) −7.63436e19 −1.13136 −0.565679 0.824626i \(-0.691386\pi\)
−0.565679 + 0.824626i \(0.691386\pi\)
\(54\) −5.83441e19 −0.697379
\(55\) −2.64966e19 −0.256460
\(56\) 0 0
\(57\) −2.42034e20 −1.55352
\(58\) 8.27059e19 0.434625
\(59\) 1.01862e18 0.00439760 0.00219880 0.999998i \(-0.499300\pi\)
0.00219880 + 0.999998i \(0.499300\pi\)
\(60\) 3.15976e20 1.12439
\(61\) −2.87337e20 −0.845472 −0.422736 0.906253i \(-0.638930\pi\)
−0.422736 + 0.906253i \(0.638930\pi\)
\(62\) 5.47658e20 1.33661
\(63\) 0 0
\(64\) −4.96127e20 −0.840472
\(65\) −3.77277e20 −0.534758
\(66\) 3.93464e20 0.467897
\(67\) 1.47683e21 1.47730 0.738652 0.674088i \(-0.235463\pi\)
0.738652 + 0.674088i \(0.235463\pi\)
\(68\) −1.81321e21 −1.52966
\(69\) 1.39369e21 0.994032
\(70\) 0 0
\(71\) 7.64346e20 0.392481 0.196241 0.980556i \(-0.437127\pi\)
0.196241 + 0.980556i \(0.437127\pi\)
\(72\) 1.47263e20 0.0643830
\(73\) 3.49433e21 1.30362 0.651810 0.758382i \(-0.274010\pi\)
0.651810 + 0.758382i \(0.274010\pi\)
\(74\) 4.97617e21 1.58756
\(75\) 3.42957e20 0.0937639
\(76\) 4.82422e21 1.13259
\(77\) 0 0
\(78\) 5.60242e21 0.975636
\(79\) 1.02350e22 1.53948 0.769742 0.638356i \(-0.220385\pi\)
0.769742 + 0.638356i \(0.220385\pi\)
\(80\) 7.91868e21 1.03067
\(81\) −1.09851e22 −1.23944
\(82\) 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
\(87\) 7.99897e21 0.396792
\(88\) 6.54489e20 0.0284676
\(89\) −4.58518e21 −0.175134 −0.0875672 0.996159i \(-0.527909\pi\)
−0.0875672 + 0.996159i \(0.527909\pi\)
\(90\) 2.39450e22 0.804314
\(91\) 0 0
\(92\) −2.77789e22 −0.724695
\(93\) 5.29672e22 1.22026
\(94\) −3.48553e22 −0.710067
\(95\) −6.54627e22 −1.18079
\(96\) −1.09133e23 −1.74517
\(97\) 1.13703e23 1.61398 0.806991 0.590564i \(-0.201095\pi\)
0.806991 + 0.590564i \(0.201095\pi\)
\(98\) 0 0
\(99\) 1.43114e22 0.160648
\(100\) −6.83581e21 −0.0683581
\(101\) −1.36243e23 −1.21512 −0.607558 0.794275i \(-0.707851\pi\)
−0.607558 + 0.794275i \(0.707851\pi\)
\(102\) −3.65366e23 −2.90956
\(103\) −1.41401e22 −0.100653 −0.0503264 0.998733i \(-0.516026\pi\)
−0.0503264 + 0.998733i \(0.516026\pi\)
\(104\) 9.31907e21 0.0593594
\(105\) 0 0
\(106\) 3.06623e23 1.56887
\(107\) 6.02971e22 0.276938 0.138469 0.990367i \(-0.455782\pi\)
0.138469 + 0.990367i \(0.455782\pi\)
\(108\) 1.12472e23 0.464164
\(109\) −5.58169e22 −0.207186 −0.103593 0.994620i \(-0.533034\pi\)
−0.103593 + 0.994620i \(0.533034\pi\)
\(110\) 1.06420e23 0.355636
\(111\) 4.81275e23 1.44937
\(112\) 0 0
\(113\) 3.51523e23 0.862089 0.431044 0.902331i \(-0.358145\pi\)
0.431044 + 0.902331i \(0.358145\pi\)
\(114\) 9.72095e23 2.15428
\(115\) 3.76948e23 0.755538
\(116\) −1.59435e23 −0.289279
\(117\) 2.03775e23 0.334976
\(118\) −4.09115e21 −0.00609821
\(119\) 0 0
\(120\) 1.05909e23 0.130121
\(121\) −8.31826e23 −0.928968
\(122\) 1.15405e24 1.17243
\(123\) 5.44111e23 0.503244
\(124\) −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
\(129\) −8.49891e22 −0.0454550
\(130\) 1.51528e24 0.741556
\(131\) −8.70825e23 −0.390221 −0.195111 0.980781i \(-0.562507\pi\)
−0.195111 + 0.980781i \(0.562507\pi\)
\(132\) −7.58496e23 −0.311425
\(133\) 0 0
\(134\) −5.93146e24 −2.04860
\(135\) −1.52620e24 −0.483919
\(136\) −6.07751e23 −0.177022
\(137\) 4.43869e24 1.18841 0.594207 0.804312i \(-0.297466\pi\)
0.594207 + 0.804312i \(0.297466\pi\)
\(138\) −5.59753e24 −1.37844
\(139\) 4.97229e23 0.112690 0.0563452 0.998411i \(-0.482055\pi\)
0.0563452 + 0.998411i \(0.482055\pi\)
\(140\) 0 0
\(141\) −3.37106e24 −0.648257
\(142\) −3.06988e24 −0.544259
\(143\) 9.05647e23 0.148114
\(144\) −4.27705e24 −0.645617
\(145\) 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 \(-0.482046\pi\)
0.0563738 + 0.998410i \(0.482046\pi\)
\(150\) −1.37744e24 −0.130024
\(151\) 3.76304e24 0.329082 0.164541 0.986370i \(-0.447386\pi\)
0.164541 + 0.986370i \(0.447386\pi\)
\(152\) 1.61698e24 0.131070
\(153\) −1.32894e25 −0.998972
\(154\) 0 0
\(155\) 1.43260e25 0.927490
\(156\) −1.08000e25 −0.649368
\(157\) −1.50090e25 −0.838504 −0.419252 0.907870i \(-0.637708\pi\)
−0.419252 + 0.907870i \(0.637708\pi\)
\(158\) −4.11072e25 −2.13482
\(159\) 2.96553e25 1.43230
\(160\) −2.95170e25 −1.32646
\(161\) 0 0
\(162\) 4.41199e25 1.71875
\(163\) −2.61170e25 −0.947907 −0.473953 0.880550i \(-0.657173\pi\)
−0.473953 + 0.880550i \(0.657173\pi\)
\(164\) −1.08452e25 −0.366888
\(165\) 1.02925e25 0.324679
\(166\) −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
\(171\) 3.53578e25 0.739656
\(172\) 1.69400e24 0.0331387
\(173\) −1.04109e26 −1.90528 −0.952640 0.304100i \(-0.901644\pi\)
−0.952640 + 0.304100i \(0.901644\pi\)
\(174\) −3.21267e25 −0.550237
\(175\) 0 0
\(176\) −1.90087e25 −0.285467
\(177\) −3.95679e23 −0.00556737
\(178\) 1.84157e25 0.242861
\(179\) −1.00142e25 −0.123824 −0.0619122 0.998082i \(-0.519720\pi\)
−0.0619122 + 0.998082i \(0.519720\pi\)
\(180\) −4.61596e25 −0.535338
\(181\) 5.17169e25 0.562768 0.281384 0.959595i \(-0.409207\pi\)
0.281384 + 0.959595i \(0.409207\pi\)
\(182\) 0 0
\(183\) 1.11615e26 1.07037
\(184\) −9.31094e24 −0.0838665
\(185\) 1.30170e26 1.10163
\(186\) −2.12735e26 −1.69216
\(187\) −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 \(-0.136750\pi\)
0.909127 + 0.416520i \(0.136750\pi\)
\(192\) 1.92718e26 1.06404
\(193\) −1.28183e26 −0.666687 −0.333344 0.942805i \(-0.608177\pi\)
−0.333344 + 0.942805i \(0.608177\pi\)
\(194\) −4.56673e26 −2.23813
\(195\) 1.46551e26 0.677005
\(196\) 0 0
\(197\) −3.89967e26 −1.60201 −0.801007 0.598655i \(-0.795702\pi\)
−0.801007 + 0.598655i \(0.795702\pi\)
\(198\) −5.74795e25 −0.222773
\(199\) 1.25611e26 0.459426 0.229713 0.973258i \(-0.426221\pi\)
0.229713 + 0.973258i \(0.426221\pi\)
\(200\) −2.29123e24 −0.00791086
\(201\) −5.73666e26 −1.87027
\(202\) 5.47199e26 1.68502
\(203\) 0 0
\(204\) 7.04330e26 1.93655
\(205\) 1.47165e26 0.382503
\(206\) 5.67918e25 0.139577
\(207\) −2.03598e26 −0.473275
\(208\) −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
\(213\) −2.96906e26 −0.496883
\(214\) −2.42174e26 −0.384034
\(215\) −2.29869e25 −0.0345491
\(216\) 3.76984e25 0.0537162
\(217\) 0 0
\(218\) 2.24180e26 0.287308
\(219\) −1.35735e27 −1.65039
\(220\) −2.05149e26 −0.236706
\(221\) −8.40974e26 −0.921025
\(222\) −1.93297e27 −2.00986
\(223\) −1.38821e27 −1.37072 −0.685361 0.728203i \(-0.740356\pi\)
−0.685361 + 0.728203i \(0.740356\pi\)
\(224\) 0 0
\(225\) −5.01012e25 −0.0446425
\(226\) −1.41184e27 −1.19547
\(227\) 2.14958e27 1.73004 0.865022 0.501734i \(-0.167304\pi\)
0.865022 + 0.501734i \(0.167304\pi\)
\(228\) −1.87394e27 −1.43386
\(229\) 6.39851e26 0.465554 0.232777 0.972530i \(-0.425219\pi\)
0.232777 + 0.972530i \(0.425219\pi\)
\(230\) −1.51396e27 −1.04771
\(231\) 0 0
\(232\) −5.34395e25 −0.0334773
\(233\) 2.19149e27 1.30661 0.653306 0.757094i \(-0.273381\pi\)
0.653306 + 0.757094i \(0.273381\pi\)
\(234\) −8.18434e26 −0.464516
\(235\) −9.11766e26 −0.492723
\(236\) 7.88666e24 0.00405887
\(237\) −3.97572e27 −1.94899
\(238\) 0 0
\(239\) 1.09944e27 0.489322 0.244661 0.969609i \(-0.421323\pi\)
0.244661 + 0.969609i \(0.421323\pi\)
\(240\) −3.07597e27 −1.30483
\(241\) 1.44651e27 0.584961 0.292480 0.956272i \(-0.405519\pi\)
0.292480 + 0.956272i \(0.405519\pi\)
\(242\) 3.34090e27 1.28821
\(243\) 2.89951e27 1.06623
\(244\) −2.22470e27 −0.780349
\(245\) 0 0
\(246\) −2.18534e27 −0.697855
\(247\) 2.23750e27 0.681942
\(248\) −3.53864e26 −0.102954
\(249\) −2.99723e27 −0.832582
\(250\) −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
\(255\) −9.55747e27 −2.01897
\(256\) 5.62436e27 1.13583
\(257\) 3.95005e27 0.762732 0.381366 0.924424i \(-0.375454\pi\)
0.381366 + 0.924424i \(0.375454\pi\)
\(258\) 3.41346e26 0.0630330
\(259\) 0 0
\(260\) −2.92106e27 −0.493567
\(261\) −1.16853e27 −0.188919
\(262\) 3.49754e27 0.541125
\(263\) 2.01521e27 0.298420 0.149210 0.988805i \(-0.452327\pi\)
0.149210 + 0.988805i \(0.452327\pi\)
\(264\) −2.54233e26 −0.0360401
\(265\) 8.02081e27 1.08866
\(266\) 0 0
\(267\) 1.78109e27 0.221721
\(268\) 1.14343e28 1.36351
\(269\) −5.72063e27 −0.653571 −0.326785 0.945099i \(-0.605965\pi\)
−0.326785 + 0.945099i \(0.605965\pi\)
\(270\) 6.12975e27 0.671058
\(271\) −5.18050e27 −0.543531 −0.271766 0.962363i \(-0.587608\pi\)
−0.271766 + 0.962363i \(0.587608\pi\)
\(272\) 1.76512e28 1.77514
\(273\) 0 0
\(274\) −1.78273e28 −1.64799
\(275\) −2.22667e26 −0.0197392
\(276\) 1.07906e28 0.917466
\(277\) 1.29611e28 1.05712 0.528560 0.848896i \(-0.322732\pi\)
0.528560 + 0.848896i \(0.322732\pi\)
\(278\) −1.99705e27 −0.156269
\(279\) −7.73776e27 −0.580987
\(280\) 0 0
\(281\) 1.24154e28 0.858691 0.429346 0.903140i \(-0.358744\pi\)
0.429346 + 0.903140i \(0.358744\pi\)
\(282\) 1.35394e28 0.898947
\(283\) 2.15031e28 1.37074 0.685372 0.728193i \(-0.259640\pi\)
0.685372 + 0.728193i \(0.259640\pi\)
\(284\) 5.91792e27 0.362250
\(285\) 2.54286e28 1.49488
\(286\) −3.63740e27 −0.205391
\(287\) 0 0
\(288\) 1.59428e28 0.830903
\(289\) 3.48772e28 1.74669
\(290\) −8.68925e27 −0.418220
\(291\) −4.41675e28 −2.04331
\(292\) 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
\(297\) 3.66362e27 0.134033
\(298\) −4.44202e27 −0.156349
\(299\) −1.28840e28 −0.436347
\(300\) 2.65534e27 0.0865416
\(301\) 0 0
\(302\) −1.51137e28 −0.456342
\(303\) 5.29228e28 1.53834
\(304\) −4.69629e28 −1.31434
\(305\) 3.01883e28 0.813560
\(306\) 5.33748e28 1.38529
\(307\) 4.70428e28 1.17598 0.587992 0.808867i \(-0.299919\pi\)
0.587992 + 0.808867i \(0.299919\pi\)
\(308\) 0 0
\(309\) 5.49267e27 0.127427
\(310\) −5.75381e28 −1.28616
\(311\) 8.99672e26 0.0193794 0.00968968 0.999953i \(-0.496916\pi\)
0.00968968 + 0.999953i \(0.496916\pi\)
\(312\) −3.61994e27 −0.0751492
\(313\) 1.13525e28 0.227159 0.113579 0.993529i \(-0.463768\pi\)
0.113579 + 0.993529i \(0.463768\pi\)
\(314\) 6.02814e28 1.16277
\(315\) 0 0
\(316\) 7.92439e28 1.42090
\(317\) 1.37896e28 0.238436 0.119218 0.992868i \(-0.461961\pi\)
0.119218 + 0.992868i \(0.461961\pi\)
\(318\) −1.19106e29 −1.98619
\(319\) −5.19337e27 −0.0835326
\(320\) 5.21241e28 0.808750
\(321\) −2.34221e28 −0.350605
\(322\) 0 0
\(323\) −1.45920e29 −2.03370
\(324\) −8.50516e28 −1.14397
\(325\) −3.17048e27 −0.0411592
\(326\) 1.04895e29 1.31448
\(327\) 2.16818e28 0.262298
\(328\) −3.63510e27 −0.0424587
\(329\) 0 0
\(330\) −4.13382e28 −0.450237
\(331\) 1.62668e29 1.71112 0.855559 0.517706i \(-0.173214\pi\)
0.855559 + 0.517706i \(0.173214\pi\)
\(332\) 5.97406e28 0.606990
\(333\) −7.03074e28 −0.690069
\(334\) 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
\(339\) −1.36547e29 −1.09141
\(340\) 1.90499e29 1.47192
\(341\) −3.43892e28 −0.256890
\(342\) −1.42009e29 −1.02569
\(343\) 0 0
\(344\) 5.67795e26 0.00383504
\(345\) −1.46424e29 −0.956514
\(346\) 4.18139e29 2.64208
\(347\) 1.45586e29 0.889876 0.444938 0.895561i \(-0.353226\pi\)
0.444938 + 0.895561i \(0.353226\pi\)
\(348\) 6.19318e28 0.366229
\(349\) −2.22110e28 −0.127080 −0.0635398 0.997979i \(-0.520239\pi\)
−0.0635398 + 0.997979i \(0.520239\pi\)
\(350\) 0 0
\(351\) 5.21651e28 0.279479
\(352\) 7.08552e28 0.367393
\(353\) −2.57166e28 −0.129064 −0.0645320 0.997916i \(-0.520555\pi\)
−0.0645320 + 0.997916i \(0.520555\pi\)
\(354\) 1.58919e27 0.00772035
\(355\) −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 \(-0.742729\pi\)
−0.690771 + 0.723074i \(0.742729\pi\)
\(360\) −1.54718e28 −0.0619529
\(361\) 1.30406e29 0.505785
\(362\) −2.07713e29 −0.780398
\(363\) 3.23118e29 1.17608
\(364\) 0 0
\(365\) −3.67122e29 −1.25442
\(366\) −4.48284e29 −1.48430
\(367\) −1.62664e29 −0.521954 −0.260977 0.965345i \(-0.584045\pi\)
−0.260977 + 0.965345i \(0.584045\pi\)
\(368\) 2.70422e29 0.840994
\(369\) −7.94868e28 −0.239603
\(370\) −5.22807e29 −1.52764
\(371\) 0 0
\(372\) 4.10097e29 1.12627
\(373\) 1.03540e29 0.275712 0.137856 0.990452i \(-0.455979\pi\)
0.137856 + 0.990452i \(0.455979\pi\)
\(374\) 2.37216e29 0.612520
\(375\) −5.22535e29 −1.30844
\(376\) 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
\(381\) −9.01364e29 −1.88046
\(382\) −1.24557e30 −2.52140
\(383\) −2.96585e29 −0.582591 −0.291295 0.956633i \(-0.594086\pi\)
−0.291295 + 0.956633i \(0.594086\pi\)
\(384\) 1.41451e29 0.269648
\(385\) 0 0
\(386\) 5.14829e29 0.924504
\(387\) 1.24157e28 0.0216419
\(388\) 8.80345e29 1.48966
\(389\) −1.06609e30 −1.75134 −0.875672 0.482906i \(-0.839581\pi\)
−0.875672 + 0.482906i \(0.839581\pi\)
\(390\) −5.88602e29 −0.938812
\(391\) 8.40240e29 1.30128
\(392\) 0 0
\(393\) 3.38267e29 0.494021
\(394\) 1.56624e30 2.22153
\(395\) −1.07531e30 −1.48138
\(396\) 1.10805e29 0.148274
\(397\) −1.67462e28 −0.0217683 −0.0108842 0.999941i \(-0.503465\pi\)
−0.0108842 + 0.999941i \(0.503465\pi\)
\(398\) −5.04496e29 −0.637093
\(399\) 0 0
\(400\) 6.65453e28 0.0793282
\(401\) 3.87121e29 0.448421 0.224211 0.974541i \(-0.428020\pi\)
0.224211 + 0.974541i \(0.428020\pi\)
\(402\) 2.30404e30 2.59353
\(403\) −4.89658e29 −0.535654
\(404\) −1.05486e30 −1.12152
\(405\) 1.15411e30 1.19266
\(406\) 0 0
\(407\) −3.12470e29 −0.305121
\(408\) 2.36078e29 0.224111
\(409\) −5.00511e29 −0.461950 −0.230975 0.972960i \(-0.574192\pi\)
−0.230975 + 0.972960i \(0.574192\pi\)
\(410\) −5.91065e29 −0.530422
\(411\) −1.72418e30 −1.50454
\(412\) −1.09480e29 −0.0928999
\(413\) 0 0
\(414\) 8.17719e29 0.656297
\(415\) −8.10655e29 −0.632824
\(416\) 1.00889e30 0.766070
\(417\) −1.93146e29 −0.142666
\(418\) −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
\(423\) 4.92464e29 0.308646
\(424\) −1.98121e29 −0.120843
\(425\) 2.06766e29 0.122746
\(426\) 1.19248e30 0.689034
\(427\) 0 0
\(428\) 4.66848e29 0.255607
\(429\) −3.51794e29 −0.187512
\(430\) 9.23233e28 0.0479098
\(431\) 1.91854e30 0.969353 0.484676 0.874694i \(-0.338937\pi\)
0.484676 + 0.874694i \(0.338937\pi\)
\(432\) −1.09490e30 −0.538653
\(433\) 2.68456e30 1.28606 0.643032 0.765839i \(-0.277676\pi\)
0.643032 + 0.765839i \(0.277676\pi\)
\(434\) 0 0
\(435\) −8.40388e29 −0.381815
\(436\) −4.32160e29 −0.191228
\(437\) −2.23555e30 −0.963489
\(438\) 5.45161e30 2.28862
\(439\) −2.11168e29 −0.0863549 −0.0431775 0.999067i \(-0.513748\pi\)
−0.0431775 + 0.999067i \(0.513748\pi\)
\(440\) −6.87619e28 −0.0273932
\(441\) 0 0
\(442\) 3.37765e30 1.27720
\(443\) −8.12623e29 −0.299396 −0.149698 0.988732i \(-0.547830\pi\)
−0.149698 + 0.988732i \(0.547830\pi\)
\(444\) 3.72626e30 1.33773
\(445\) 4.81728e29 0.168524
\(446\) 5.57554e30 1.90080
\(447\) −4.29614e29 −0.142739
\(448\) 0 0
\(449\) −3.76596e30 −1.18862 −0.594310 0.804236i \(-0.702575\pi\)
−0.594310 + 0.804236i \(0.702575\pi\)
\(450\) 2.01224e29 0.0619064
\(451\) −3.53267e29 −0.105943
\(452\) 2.72166e30 0.795686
\(453\) −1.46173e30 −0.416619
\(454\) −8.63348e30 −2.39907
\(455\) 0 0
\(456\) −6.28109e29 −0.165936
\(457\) 2.14090e30 0.551518 0.275759 0.961227i \(-0.411071\pi\)
0.275759 + 0.961227i \(0.411071\pi\)
\(458\) −2.56986e30 −0.645591
\(459\) −3.40199e30 −0.833465
\(460\) 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
\(465\) −5.56485e30 −1.17421
\(466\) −8.80180e30 −1.81190
\(467\) 2.87884e30 0.578194 0.289097 0.957300i \(-0.406645\pi\)
0.289097 + 0.957300i \(0.406645\pi\)
\(468\) 1.57772e30 0.309175
\(469\) 0 0
\(470\) 3.66197e30 0.683266
\(471\) 5.83016e30 1.06155
\(472\) 2.64345e27 0.000469719 0
\(473\) 5.51796e28 0.00956919
\(474\) 1.59679e31 2.70269
\(475\) −5.50121e29 −0.0908829
\(476\) 0 0
\(477\) −4.33221e30 −0.681943
\(478\) −4.41573e30 −0.678549
\(479\) −6.82143e30 −1.02333 −0.511666 0.859184i \(-0.670972\pi\)
−0.511666 + 0.859184i \(0.670972\pi\)
\(480\) 1.14657e31 1.67930
\(481\) −4.44917e30 −0.636224
\(482\) −5.80971e30 −0.811173
\(483\) 0 0
\(484\) −6.44038e30 −0.857413
\(485\) −1.19459e31 −1.55306
\(486\) −1.16455e31 −1.47856
\(487\) 8.59361e29 0.106560 0.0532798 0.998580i \(-0.483032\pi\)
0.0532798 + 0.998580i \(0.483032\pi\)
\(488\) −7.45676e29 −0.0903071
\(489\) 1.01450e31 1.20005
\(490\) 0 0
\(491\) −7.15140e30 −0.807149 −0.403575 0.914947i \(-0.632232\pi\)
−0.403575 + 0.914947i \(0.632232\pi\)
\(492\) 4.21276e30 0.464481
\(493\) 4.82250e30 0.519437
\(494\) −8.98658e30 −0.945659
\(495\) −1.50358e30 −0.154585
\(496\) 1.02774e31 1.03239
\(497\) 0 0
\(498\) 1.20379e31 1.15455
\(499\) 1.77913e31 1.66744 0.833722 0.552185i \(-0.186206\pi\)
0.833722 + 0.552185i \(0.186206\pi\)
\(500\) 1.04151e31 0.953916
\(501\) 6.89133e30 0.616835
\(502\) 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
\(507\) 1.12100e31 0.875012
\(508\) 1.79660e31 1.37094
\(509\) 1.19826e31 0.893917 0.446959 0.894555i \(-0.352507\pi\)
0.446959 + 0.894555i \(0.352507\pi\)
\(510\) 3.83861e31 2.79974
\(511\) 0 0
\(512\) −1.95347e31 −1.36208
\(513\) 9.05135e30 0.617112
\(514\) −1.58648e31 −1.05769
\(515\) 1.48559e30 0.0968537
\(516\) −6.58025e29 −0.0419538
\(517\) 2.18868e30 0.136471
\(518\) 0 0
\(519\) 4.04407e31 2.41209
\(520\) −9.79080e29 −0.0571189
\(521\) 2.26240e31 1.29103 0.645514 0.763749i \(-0.276643\pi\)
0.645514 + 0.763749i \(0.276643\pi\)
\(522\) 4.69325e30 0.261977
\(523\) 3.41914e31 1.86702 0.933508 0.358557i \(-0.116731\pi\)
0.933508 + 0.358557i \(0.116731\pi\)
\(524\) −6.74233e30 −0.360164
\(525\) 0 0
\(526\) −8.09378e30 −0.413824
\(527\) 3.19334e31 1.59744
\(528\) 7.38381e30 0.361402
\(529\) −8.00772e30 −0.383503
\(530\) −3.22144e31 −1.50965
\(531\) 5.78030e28 0.00265072
\(532\) 0 0
\(533\) −5.03006e30 −0.220907
\(534\) −7.15348e30 −0.307463
\(535\) −6.33494e30 −0.266486
\(536\) 3.83255e30 0.157795
\(537\) 3.88996e30 0.156762
\(538\) 2.29761e31 0.906315
\(539\) 0 0
\(540\) −1.18166e31 −0.446645
\(541\) −2.75024e30 −0.101766 −0.0508829 0.998705i \(-0.516204\pi\)
−0.0508829 + 0.998705i \(0.516204\pi\)
\(542\) 2.08067e31 0.753722
\(543\) −2.00892e31 −0.712466
\(544\) −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
\(549\) −1.63053e31 −0.509621
\(550\) 8.94307e29 0.0273726
\(551\) −1.28308e31 −0.384600
\(552\) 3.61679e30 0.106175
\(553\) 0 0
\(554\) −5.20563e31 −1.46592
\(555\) −5.05637e31 −1.39466
\(556\) 3.84978e30 0.104010
\(557\) 1.36262e31 0.360613 0.180307 0.983610i \(-0.442291\pi\)
0.180307 + 0.983610i \(0.442291\pi\)
\(558\) 3.10775e31 0.805663
\(559\) 7.85685e29 0.0199532
\(560\) 0 0
\(561\) 2.29425e31 0.559202
\(562\) −4.98645e31 −1.19076
\(563\) −3.09091e31 −0.723169 −0.361585 0.932339i \(-0.617764\pi\)
−0.361585 + 0.932339i \(0.617764\pi\)
\(564\) −2.61004e31 −0.598325
\(565\) −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 \(-0.754806\pi\)
−0.717702 + 0.696350i \(0.754806\pi\)
\(570\) −1.02130e32 −2.07297
\(571\) −6.24961e30 −0.124319 −0.0621595 0.998066i \(-0.519799\pi\)
−0.0621595 + 0.998066i \(0.519799\pi\)
\(572\) 7.01195e30 0.136705
\(573\) −1.20467e32 −2.30192
\(574\) 0 0
\(575\) 3.16772e30 0.0581522
\(576\) −2.81534e31 −0.506607
\(577\) 1.59740e31 0.281768 0.140884 0.990026i \(-0.455006\pi\)
0.140884 + 0.990026i \(0.455006\pi\)
\(578\) −1.40079e32 −2.42216
\(579\) 4.97921e31 0.844029
\(580\) 1.67506e31 0.278361
\(581\) 0 0
\(582\) 1.77392e32 2.83348
\(583\) −1.92538e31 −0.301529
\(584\) 9.06822e30 0.139243
\(585\) −2.14091e31 −0.322333
\(586\) −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
\(591\) 1.51481e32 2.02816
\(592\) 9.33837e31 1.22623
\(593\) 5.61370e30 0.0722970 0.0361485 0.999346i \(-0.488491\pi\)
0.0361485 + 0.999346i \(0.488491\pi\)
\(594\) −1.47144e31 −0.185865
\(595\) 0 0
\(596\) 8.56305e30 0.104063
\(597\) −4.87928e31 −0.581635
\(598\) 5.17466e31 0.605088
\(599\) 1.60231e32 1.83797 0.918985 0.394293i \(-0.129010\pi\)
0.918985 + 0.394293i \(0.129010\pi\)
\(600\) 8.90016e29 0.0100152
\(601\) −8.47641e31 −0.935741 −0.467870 0.883797i \(-0.654979\pi\)
−0.467870 + 0.883797i \(0.654979\pi\)
\(602\) 0 0
\(603\) 8.38044e31 0.890467
\(604\) 2.91352e31 0.303734
\(605\) 8.73933e31 0.893905
\(606\) −2.12557e32 −2.13324
\(607\) 1.63793e31 0.161296 0.0806481 0.996743i \(-0.474301\pi\)
0.0806481 + 0.996743i \(0.474301\pi\)
\(608\) 1.75055e32 1.69155
\(609\) 0 0
\(610\) −1.21247e32 −1.12818
\(611\) 3.11639e31 0.284563
\(612\) −1.02893e32 −0.922025
\(613\) 1.93558e31 0.170222 0.0851110 0.996371i \(-0.472876\pi\)
0.0851110 + 0.996371i \(0.472876\pi\)
\(614\) −1.88940e32 −1.63075
\(615\) −5.71654e31 −0.484250
\(616\) 0 0
\(617\) 5.25618e31 0.428934 0.214467 0.976731i \(-0.431199\pi\)
0.214467 + 0.976731i \(0.431199\pi\)
\(618\) −2.20605e31 −0.176704
\(619\) 9.42009e31 0.740650 0.370325 0.928902i \(-0.379246\pi\)
0.370325 + 0.928902i \(0.379246\pi\)
\(620\) 1.10918e32 0.856049
\(621\) −5.21196e31 −0.394864
\(622\) −3.61340e30 −0.0268736
\(623\) 0 0
\(624\) 1.05136e32 0.753578
\(625\) −1.30804e32 −0.920452
\(626\) −4.55954e31 −0.315004
\(627\) −6.10410e31 −0.414042
\(628\) −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
\(633\) −2.28517e32 −1.38923
\(634\) −5.53840e31 −0.330642
\(635\) −2.43790e32 −1.42929
\(636\) 2.29605e32 1.32198
\(637\) 0 0
\(638\) 2.08584e31 0.115836
\(639\) 4.33738e31 0.236574
\(640\) 3.82581e31 0.204953
\(641\) 1.38602e32 0.729295 0.364648 0.931146i \(-0.381189\pi\)
0.364648 + 0.931146i \(0.381189\pi\)
\(642\) 9.40714e31 0.486188
\(643\) −3.71401e31 −0.188546 −0.0942729 0.995546i \(-0.530053\pi\)
−0.0942729 + 0.995546i \(0.530053\pi\)
\(644\) 0 0
\(645\) 8.92913e30 0.0437393
\(646\) 5.86067e32 2.82016
\(647\) 3.77554e32 1.78476 0.892379 0.451286i \(-0.149035\pi\)
0.892379 + 0.451286i \(0.149035\pi\)
\(648\) −2.85076e31 −0.132388
\(649\) 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 \(-0.558380\pi\)
−0.182380 + 0.983228i \(0.558380\pi\)
\(654\) −8.70816e31 −0.363733
\(655\) 9.14906e31 0.375493
\(656\) 1.05576e32 0.425766
\(657\) 1.98290e32 0.785777
\(658\) 0 0
\(659\) 1.46687e32 0.561319 0.280660 0.959807i \(-0.409447\pi\)
0.280660 + 0.959807i \(0.409447\pi\)
\(660\) 7.96891e31 0.299670
\(661\) −2.02066e32 −0.746752 −0.373376 0.927680i \(-0.621800\pi\)
−0.373376 + 0.927680i \(0.621800\pi\)
\(662\) −6.53331e32 −2.37283
\(663\) 3.26672e32 1.16602
\(664\) 2.00239e31 0.0702450
\(665\) 0 0
\(666\) 2.82379e32 0.956927
\(667\) 7.38823e31 0.246090
\(668\) −1.37358e32 −0.449701
\(669\) 5.39243e32 1.73534
\(670\) 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
\(675\) −1.28256e31 −0.0372463
\(676\) −2.23437e32 −0.637924
\(677\) −1.10049e32 −0.308899 −0.154449 0.988001i \(-0.549360\pi\)
−0.154449 + 0.988001i \(0.549360\pi\)
\(678\) 5.48422e32 1.51347
\(679\) 0 0
\(680\) 6.38515e31 0.170341
\(681\) −8.34995e32 −2.19024
\(682\) 1.38119e32 0.356233
\(683\) −2.97297e32 −0.753967 −0.376983 0.926220i \(-0.623039\pi\)
−0.376983 + 0.926220i \(0.623039\pi\)
\(684\) 2.73756e32 0.682683
\(685\) −4.66337e32 −1.14356
\(686\) 0 0
\(687\) −2.48547e32 −0.589394
\(688\) −1.64908e31 −0.0384568
\(689\) −2.74149e32 −0.628733
\(690\) 5.88088e32 1.32641
\(691\) 7.18838e32 1.59453 0.797266 0.603628i \(-0.206279\pi\)
0.797266 + 0.603628i \(0.206279\pi\)
\(692\) −8.06062e32 −1.75852
\(693\) 0 0
\(694\) −5.84723e32 −1.23400
\(695\) −5.22399e31 −0.108437
\(696\) 2.07583e31 0.0423824
\(697\) 3.28039e32 0.658793
\(698\) 8.92072e31 0.176223
\(699\) −8.51274e32 −1.65418
\(700\) 0 0
\(701\) −1.02832e33 −1.93362 −0.966810 0.255497i \(-0.917761\pi\)
−0.966810 + 0.255497i \(0.917761\pi\)
\(702\) −2.09513e32 −0.387557
\(703\) −7.71990e32 −1.40484
\(704\) −1.25123e32 −0.224002
\(705\) 3.54171e32 0.623790
\(706\) 1.03287e32 0.178975
\(707\) 0 0
\(708\) −3.06353e30 −0.00513854
\(709\) −2.50942e32 −0.414134 −0.207067 0.978327i \(-0.566392\pi\)
−0.207067 + 0.978327i \(0.566392\pi\)
\(710\) 3.22528e32 0.523717
\(711\) 5.80796e32 0.927947
\(712\) −1.18991e31 −0.0187066
\(713\) 4.89231e32 0.756805
\(714\) 0 0
\(715\) −9.51492e31 −0.142523
\(716\) −7.75346e31 −0.114287
\(717\) −4.27071e32 −0.619483
\(718\) 1.34209e33 1.91580
\(719\) −7.09247e32 −0.996358 −0.498179 0.867074i \(-0.665998\pi\)
−0.498179 + 0.867074i \(0.665998\pi\)
\(720\) 4.49355e32 0.621249
\(721\) 0 0
\(722\) −5.23757e32 −0.701378
\(723\) −5.61891e32 −0.740562
\(724\) 4.00417e32 0.519420
\(725\) 1.81809e31 0.0232128
\(726\) −1.29776e33 −1.63088
\(727\) −4.63712e32 −0.573591 −0.286795 0.957992i \(-0.592590\pi\)
−0.286795 + 0.957992i \(0.592590\pi\)
\(728\) 0 0
\(729\) −9.21300e31 −0.110417
\(730\) 1.47449e33 1.73952
\(731\) −5.12391e31 −0.0595047
\(732\) 8.64174e32 0.987924
\(733\) 1.54780e33 1.74188 0.870941 0.491387i \(-0.163510\pi\)
0.870941 + 0.491387i \(0.163510\pi\)
\(734\) 6.53317e32 0.723801
\(735\) 0 0
\(736\) −1.00801e33 −1.08235
\(737\) 3.72455e32 0.393730
\(738\) 3.19247e32 0.332260
\(739\) −1.49249e33 −1.52932 −0.764662 0.644431i \(-0.777094\pi\)
−0.764662 + 0.644431i \(0.777094\pi\)
\(740\) 1.00783e33 1.01677
\(741\) −8.69144e32 −0.863341
\(742\) 0 0
\(743\) 1.19934e32 0.115497 0.0577487 0.998331i \(-0.481608\pi\)
0.0577487 + 0.998331i \(0.481608\pi\)
\(744\) 1.37456e32 0.130340
\(745\) −1.16197e32 −0.108492
\(746\) −4.15852e32 −0.382334
\(747\) 4.37852e32 0.396406
\(748\) −4.57290e32 −0.407683
\(749\) 0 0
\(750\) 2.09868e33 1.81444
\(751\) 1.42812e33 1.21592 0.607959 0.793969i \(-0.291989\pi\)
0.607959 + 0.793969i \(0.291989\pi\)
\(752\) −6.54101e32 −0.548453
\(753\) 8.71877e31 0.0719967
\(754\) 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
\(759\) 3.51487e32 0.264929
\(760\) −1.69884e32 −0.126123
\(761\) −1.92115e33 −1.40488 −0.702439 0.711744i \(-0.747906\pi\)
−0.702439 + 0.711744i \(0.747906\pi\)
\(762\) 3.62020e33 2.60765
\(763\) 0 0
\(764\) 2.40114e33 1.67820
\(765\) 1.39621e33 0.961267
\(766\) 1.19119e33 0.807886
\(767\) 3.65787e30 0.00244389
\(768\) −2.18475e33 −1.43797
\(769\) −1.22615e33 −0.795044 −0.397522 0.917593i \(-0.630130\pi\)
−0.397522 + 0.917593i \(0.630130\pi\)
\(770\) 0 0
\(771\) −1.53438e33 −0.965621
\(772\) −9.92455e32 −0.615335
\(773\) 8.43914e32 0.515507 0.257753 0.966211i \(-0.417018\pi\)
0.257753 + 0.966211i \(0.417018\pi\)
\(774\) −4.98658e31 −0.0300111
\(775\) 1.20390e32 0.0713870
\(776\) 2.95074e32 0.172394
\(777\) 0 0
\(778\) 4.28177e33 2.42861
\(779\) −8.72782e32 −0.487781
\(780\) 1.13467e33 0.624858
\(781\) 1.92768e32 0.104604
\(782\) −3.37470e33 −1.80450
\(783\) −2.99137e32 −0.157620
\(784\) 0 0
\(785\) 1.57687e33 0.806856
\(786\) −1.35860e33 −0.685066
\(787\) −1.37027e33 −0.680919 −0.340460 0.940259i \(-0.610583\pi\)
−0.340460 + 0.940259i \(0.610583\pi\)
\(788\) −3.01931e33 −1.47862
\(789\) −7.82797e32 −0.377801
\(790\) 4.31881e33 2.05425
\(791\) 0 0
\(792\) 3.71398e31 0.0171593
\(793\) −1.03183e33 −0.469857
\(794\) 6.72586e31 0.0301865
\(795\) −3.11564e33 −1.37824
\(796\) 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
\(801\) −2.60192e32 −0.105565
\(802\) −1.55481e33 −0.621832
\(803\) 8.81269e32 0.347440
\(804\) −4.44159e33 −1.72621
\(805\) 0 0
\(806\) 1.96664e33 0.742799
\(807\) 2.22215e33 0.827423
\(808\) −3.53567e32 −0.129790
\(809\) −2.56942e33 −0.929882 −0.464941 0.885342i \(-0.653924\pi\)
−0.464941 + 0.885342i \(0.653924\pi\)
\(810\) −4.63533e33 −1.65388
\(811\) −1.25770e33 −0.442423 −0.221212 0.975226i \(-0.571001\pi\)
−0.221212 + 0.975226i \(0.571001\pi\)
\(812\) 0 0
\(813\) 2.01234e33 0.688112
\(814\) 1.25499e33 0.423116
\(815\) 2.74390e33 0.912129
\(816\) −6.85652e33 −2.24733
\(817\) 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 \(-0.511581\pi\)
−0.0363759 + 0.999338i \(0.511581\pi\)
\(822\) 6.92493e33 2.08636
\(823\) 5.82453e33 1.73047 0.865233 0.501370i \(-0.167170\pi\)
0.865233 + 0.501370i \(0.167170\pi\)
\(824\) −3.66954e31 −0.0107510
\(825\) 8.64937e31 0.0249899
\(826\) 0 0
\(827\) −3.61128e33 −1.01472 −0.507362 0.861733i \(-0.669379\pi\)
−0.507362 + 0.861733i \(0.669379\pi\)
\(828\) −1.57635e33 −0.436821
\(829\) −6.27935e33 −1.71608 −0.858040 0.513584i \(-0.828318\pi\)
−0.858040 + 0.513584i \(0.828318\pi\)
\(830\) 3.25588e33 0.877546
\(831\) −5.03467e33 −1.33832
\(832\) −1.78159e33 −0.467078
\(833\) 0 0
\(834\) 7.75743e32 0.197837
\(835\) 1.86389e33 0.468841
\(836\) 1.21667e33 0.301856
\(837\) −1.98081e33 −0.484731
\(838\) 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
\(843\) −4.82269e33 −1.08711
\(844\) 4.55478e33 1.01281
\(845\) 3.03195e33 0.665074
\(846\) −1.97791e33 −0.428003
\(847\) 0 0
\(848\) 5.75412e33 1.21179
\(849\) −8.35275e33 −1.73537
\(850\) −8.30443e32 −0.170213
\(851\) 4.44528e33 0.898896
\(852\) −2.29879e33 −0.458610
\(853\) 9.11290e33 1.79367 0.896836 0.442364i \(-0.145860\pi\)
0.896836 + 0.442364i \(0.145860\pi\)
\(854\) 0 0
\(855\) −3.71476e33 −0.711738
\(856\) 1.56478e32 0.0295805
\(857\) −6.17423e33 −1.15160 −0.575802 0.817589i \(-0.695310\pi\)
−0.575802 + 0.817589i \(0.695310\pi\)
\(858\) 1.41293e33 0.260026
\(859\) 1.86175e33 0.338065 0.169033 0.985610i \(-0.445936\pi\)
0.169033 + 0.985610i \(0.445936\pi\)
\(860\) −1.77975e32 −0.0318880
\(861\) 0 0
\(862\) −7.70552e33 −1.34421
\(863\) −5.22144e33 −0.898806 −0.449403 0.893329i \(-0.648363\pi\)
−0.449403 + 0.893329i \(0.648363\pi\)
\(864\) 4.08125e33 0.693242
\(865\) 1.09379e34 1.83337
\(866\) −1.07821e34 −1.78340
\(867\) −1.35479e34 −2.21132
\(868\) 0 0
\(869\) 2.58126e33 0.410302
\(870\) 3.37529e33 0.529469
\(871\) 5.30328e33 0.820986
\(872\) −1.44852e32 −0.0221301
\(873\) 6.45224e33 0.972852
\(874\) 8.97874e33 1.33608
\(875\) 0 0
\(876\) −1.05093e34 −1.52327
\(877\) −6.73800e32 −0.0963908 −0.0481954 0.998838i \(-0.515347\pi\)
−0.0481954 + 0.998838i \(0.515347\pi\)
\(878\) 8.48127e32 0.119750
\(879\) −4.67328e33 −0.651253
\(880\) 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
\(885\) 4.15708e31 0.00535724
\(886\) 3.26378e33 0.415177
\(887\) 6.44904e33 0.809792 0.404896 0.914363i \(-0.367308\pi\)
0.404896 + 0.914363i \(0.367308\pi\)
\(888\) 1.24897e33 0.154811
\(889\) 0 0
\(890\) −1.93479e33 −0.233695
\(891\) −2.77043e33 −0.330334
\(892\) −1.07482e34 −1.26514
\(893\) 5.40736e33 0.628339
\(894\) 1.72548e33 0.197938
\(895\) 1.05211e33 0.119151
\(896\) 0 0
\(897\) 5.00472e33 0.552417
\(898\) 1.51254e34 1.64827
\(899\) 2.80791e33 0.302097
\(900\) −3.87907e32 −0.0412039
\(901\) 1.78789e34 1.87502
\(902\) 1.41884e33 0.146913
\(903\) 0 0
\(904\) 9.12246e32 0.0920820
\(905\) −5.43349e33 −0.541527
\(906\) 5.87083e33 0.577731
\(907\) 6.42282e33 0.624083 0.312041 0.950069i \(-0.398987\pi\)
0.312041 + 0.950069i \(0.398987\pi\)
\(908\) 1.66431e34 1.59679
\(909\) −7.73126e33 −0.732430
\(910\) 0 0
\(911\) −1.48932e34 −1.37571 −0.687855 0.725848i \(-0.741447\pi\)
−0.687855 + 0.725848i \(0.741447\pi\)
\(912\) 1.82425e34 1.66396
\(913\) 1.94596e33 0.175275
\(914\) −8.59859e33 −0.764797
\(915\) −1.17265e34 −1.02997
\(916\) 4.95402e33 0.429695
\(917\) 0 0
\(918\) 1.36636e34 1.15578
\(919\) 3.10701e33 0.259546 0.129773 0.991544i \(-0.458575\pi\)
0.129773 + 0.991544i \(0.458575\pi\)
\(920\) 9.78226e32 0.0807011
\(921\) −1.82735e34 −1.48880
\(922\) 1.91330e34 1.53950
\(923\) 2.74476e33 0.218115
\(924\) 0 0
\(925\) 1.09389e33 0.0847900
\(926\) 3.08285e34 2.36007
\(927\) −8.02400e32 −0.0606700
\(928\) −5.78538e33 −0.432046
\(929\) −1.72846e34 −1.27490 −0.637452 0.770490i \(-0.720012\pi\)
−0.637452 + 0.770490i \(0.720012\pi\)
\(930\) 2.23504e34 1.62829
\(931\) 0 0
\(932\) 1.69676e34 1.20597
\(933\) −3.49473e32 −0.0245343
\(934\) −1.15624e34 −0.801790
\(935\) 6.20523e33 0.425035
\(936\) 5.28822e32 0.0357797
\(937\) 1.61548e34 1.07969 0.539843 0.841766i \(-0.318484\pi\)
0.539843 + 0.841766i \(0.318484\pi\)
\(938\) 0 0
\(939\) −4.40980e33 −0.287584
\(940\) −7.05932e33 −0.454771
\(941\) −7.69275e33 −0.489554 −0.244777 0.969579i \(-0.578715\pi\)
−0.244777 + 0.969579i \(0.578715\pi\)
\(942\) −2.34160e34 −1.47206
\(943\) 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 \(-0.836222\pi\)
−0.870528 + 0.492119i \(0.836222\pi\)
\(948\) −3.07819e34 −1.79887
\(949\) 1.25481e34 0.724465
\(950\) 2.20948e33 0.126029
\(951\) −5.35651e33 −0.301860
\(952\) 0 0
\(953\) 1.97292e34 1.08528 0.542641 0.839965i \(-0.317425\pi\)
0.542641 + 0.839965i \(0.317425\pi\)
\(954\) 1.73997e34 0.945659
\(955\) −3.25825e34 −1.74963
\(956\) 8.51236e33 0.451632
\(957\) 2.01734e33 0.105753
\(958\) 2.73973e34 1.41907
\(959\) 0 0
\(960\) −2.02474e34 −1.02388
\(961\) −1.42003e33 −0.0709542
\(962\) 1.78694e34 0.882261
\(963\) 3.42163e33 0.166929
\(964\) 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
\(969\) 5.66820e34 2.57467
\(970\) 4.79790e34 2.15365
\(971\) −3.40470e34 −1.51028 −0.755141 0.655563i \(-0.772431\pi\)
−0.755141 + 0.655563i \(0.772431\pi\)
\(972\) 2.24494e34 0.984106
\(973\) 0 0
\(974\) −3.45150e33 −0.147768
\(975\) 1.23156e33 0.0521077
\(976\) 2.16570e34 0.905579
\(977\) −3.68167e32 −0.0152145 −0.00760725 0.999971i \(-0.502421\pi\)
−0.00760725 + 0.999971i \(0.502421\pi\)
\(978\) −4.07459e34 −1.66413
\(979\) −1.15638e33 −0.0466767
\(980\) 0 0
\(981\) −3.16740e33 −0.124885
\(982\) 2.87225e34 1.11928
\(983\) 8.23724e33 0.317261 0.158630 0.987338i \(-0.449292\pi\)
0.158630 + 0.987338i \(0.449292\pi\)
\(984\) 1.41203e33 0.0537529
\(985\) 4.09707e34 1.54155
\(986\) −1.93689e34 −0.720310
\(987\) 0 0
\(988\) 1.73238e34 0.629415
\(989\) −7.85000e32 −0.0281911
\(990\) 6.03891e33 0.214365
\(991\) 2.70349e33 0.0948590 0.0474295 0.998875i \(-0.484897\pi\)
0.0474295 + 0.998875i \(0.484897\pi\)
\(992\) −3.83094e34 −1.32868
\(993\) −6.31875e34 −2.16628
\(994\) 0 0
\(995\) −1.31969e34 −0.442086
\(996\) −2.32059e34 −0.768452
\(997\) 4.64692e33 0.152115 0.0760573 0.997103i \(-0.475767\pi\)
0.0760573 + 0.997103i \(0.475767\pi\)
\(998\) −7.14561e34 −2.31227
\(999\) −1.79982e34 −0.575740
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 49.24.a.b.1.1 2
7.6 odd 2 1.24.a.a.1.1 2
21.20 even 2 9.24.a.b.1.2 2
28.27 even 2 16.24.a.b.1.1 2
35.13 even 4 25.24.b.a.24.3 4
35.27 even 4 25.24.b.a.24.2 4
35.34 odd 2 25.24.a.a.1.2 2
56.13 odd 2 64.24.a.d.1.1 2
56.27 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 7.6 odd 2
9.24.a.b.1.2 2 21.20 even 2
16.24.a.b.1.1 2 28.27 even 2
25.24.a.a.1.2 2 35.34 odd 2
25.24.b.a.24.2 4 35.27 even 4
25.24.b.a.24.3 4 35.13 even 4
49.24.a.b.1.1 2 1.1 even 1 trivial
64.24.a.d.1.1 2 56.13 odd 2
64.24.a.g.1.2 2 56.27 even 2