Properties

Label 25.24.a.a.1.2
Level $25$
Weight $24$
Character 25.1
Self dual yes
Analytic conductor $83.801$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(83.8010093363\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q+4016.35 q^{2} -388445. q^{3} +7.74247e6 q^{4} -1.56013e9 q^{6} -3.81217e9 q^{7} -2.59512e9 q^{8} +5.67462e10 q^{9} +O(q^{10})\) \(q+4016.35 q^{2} -388445. q^{3} +7.74247e6 q^{4} -1.56013e9 q^{6} -3.81217e9 q^{7} -2.59512e9 q^{8} +5.67462e10 q^{9} +2.52200e11 q^{11} -3.00752e12 q^{12} +3.59099e12 q^{13} -1.53110e13 q^{14} -7.53715e13 q^{16} -2.34190e14 q^{17} +2.27913e14 q^{18} -6.23086e14 q^{19} +1.48082e15 q^{21} +1.01292e15 q^{22} +3.58786e15 q^{23} +1.00806e15 q^{24} +1.44227e16 q^{26} +1.45267e16 q^{27} -2.95156e16 q^{28} -2.05923e16 q^{29} +1.36357e17 q^{31} -2.80949e17 q^{32} -9.79656e16 q^{33} -9.40588e17 q^{34} +4.39356e17 q^{36} +1.23898e18 q^{37} -2.50253e18 q^{38} -1.39490e18 q^{39} +1.40074e18 q^{41} +5.94748e18 q^{42} -2.18793e17 q^{43} +1.95265e18 q^{44} +1.44101e19 q^{46} +8.67836e18 q^{47} +2.92777e19 q^{48} -1.28361e19 q^{49} +9.09698e19 q^{51} +2.78032e19 q^{52} +7.63436e19 q^{53} +5.83441e19 q^{54} +9.89304e18 q^{56} +2.42034e20 q^{57} -8.27059e19 q^{58} -1.01862e18 q^{59} +2.87337e20 q^{61} +5.47658e20 q^{62} -2.16326e20 q^{63} -4.96127e20 q^{64} -3.93464e20 q^{66} -1.47683e21 q^{67} -1.81321e21 q^{68} -1.39369e21 q^{69} +7.64346e20 q^{71} -1.47263e20 q^{72} +3.49433e21 q^{73} +4.97617e21 q^{74} -4.82422e21 q^{76} -9.61427e20 q^{77} -5.60242e21 q^{78} +1.02350e22 q^{79} -1.09851e22 q^{81} +5.62587e21 q^{82} +7.71597e21 q^{83} +1.14652e22 q^{84} -8.78750e20 q^{86} +7.99897e21 q^{87} -6.54489e20 q^{88} +4.58518e21 q^{89} -1.36895e22 q^{91} +2.77789e22 q^{92} -5.29672e22 q^{93} +3.48553e22 q^{94} +1.09133e23 q^{96} +1.13703e23 q^{97} -5.15544e22 q^{98} +1.43114e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1080q^{2} - 339480q^{3} + 25326656q^{4} - 1809673056q^{6} + 1359184400q^{7} - 49459023360q^{8} - 34999394166q^{9} + O(q^{10}) \) \( 2q - 1080q^{2} - 339480q^{3} + 25326656q^{4} - 1809673056q^{6} + 1359184400q^{7} - 49459023360q^{8} - 34999394166q^{9} + 856801968264q^{11} - 2146514952960q^{12} - 4376109322060q^{13} - 41666034529728q^{14} + 15956586401792q^{16} - 254028147597540q^{17} + 695480683916520q^{18} + 4260600979960q^{19} + 1734031637722944q^{21} - 2068343882177760q^{22} + 8144713079008560q^{23} - 1286622315141120q^{24} + 55025854658735184q^{26} + 5424634982716560q^{27} + 61418438819709440q^{28} + 20818433601623340q^{29} + 137714017177000384q^{31} - 353265663781601280q^{32} - 68361366766001760q^{33} - 839483655961325328q^{34} - 1173916300077574848q^{36} + 897721264408967780q^{37} - 5699708971590961440q^{38} - 1785011473665029232q^{39} - 2294435477168314956q^{41} + 4657011326437397760q^{42} + 1750760768619855800q^{43} + 12584088840033038592q^{44} - 8813206018050221376q^{46} - 15759744217656780960q^{47} + 33749519399576616960q^{48} - 13461981704376200814q^{49} + 89998362845078292144q^{51} - 112291883783912022400q^{52} + 140287253401646796420q^{53} + 104731223417039799360q^{54} - 232456712054288117760q^{56} + 272752401448627175520q^{57} - 293749486923568689360q^{58} + 280872989971340771880q^{59} - 180452892516502223636q^{61} + 540743475843874103040q^{62} - 690775113933935014320q^{63} - 893690254469352914944q^{64} - 544338140913651883392q^{66} - 1754233163431557625240q^{67} - 2162050190142944330880q^{68} - 1170560672172404223552q^{69} + 3055033510194143328624q^{71} + 4152294352103038548480q^{72} + 8063408253877606149260q^{73} + 6715344283148807757072q^{74} + 6207154294513080590080q^{76} + 2165184764357449665600q^{77} - 3614293948840808587200q^{78} + 6244916814559639980640q^{79} - 2793528580929833975598q^{81} + 24457792891615712450640q^{82} - 6875994082418498976120q^{83} + 15917751907190402476032q^{84} - 10916288812918999243296q^{86} + 10026640653837674384880q^{87} - 28988514668199273707520q^{88} + 6395093086173070004820q^{89} - 54890178162704560146016q^{91} + 107907439017191756981760q^{92} - 52900811441357852079360q^{93} + 159400518006534931827072q^{94} + 105592121669584394256384q^{96} + 31147288846254030500540q^{97} - 48364767616374671003640q^{98} - 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 \(-0.256131\pi\)
0.693357 + 0.720595i \(0.256131\pi\)
\(3\) −388445. −1.26600 −0.633002 0.774150i \(-0.718177\pi\)
−0.633002 + 0.774150i \(0.718177\pi\)
\(4\) 7.74247e6 0.922974
\(5\) 0 0
\(6\) −1.56013e9 −1.75558
\(7\) −3.81217e9 −0.728693 −0.364346 0.931263i \(-0.618708\pi\)
−0.364346 + 0.931263i \(0.618708\pi\)
\(8\) −2.59512e9 −0.106813
\(9\) 5.67462e10 0.602765
\(10\) 0 0
\(11\) 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 \(0.410373\pi\)
\(14\) −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 \(0.810895\pi\)
\(18\) 2.27913e14 0.835863
\(19\) −6.23086e14 −1.22710 −0.613552 0.789654i \(-0.710260\pi\)
−0.613552 + 0.789654i \(0.710260\pi\)
\(20\) 0 0
\(21\) 1.48082e15 0.922528
\(22\) 1.01292e15 0.369586
\(23\) 3.58786e15 0.785173 0.392587 0.919715i \(-0.371580\pi\)
0.392587 + 0.919715i \(0.371580\pi\)
\(24\) 1.00806e15 0.135225
\(25\) 0 0
\(26\) 1.44227e16 0.770642
\(27\) 1.45267e16 0.502901
\(28\) −2.95156e16 −0.672565
\(29\) −2.05923e16 −0.313421 −0.156710 0.987645i \(-0.550089\pi\)
−0.156710 + 0.987645i \(0.550089\pi\)
\(30\) 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
\(33\) −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 \(-0.306005\pi\)
0.572419 + 0.819961i \(0.306005\pi\)
\(38\) −2.50253e18 −1.70164
\(39\) −1.39490e18 −0.703560
\(40\) 0 0
\(41\) 1.40074e18 0.397506 0.198753 0.980050i \(-0.436311\pi\)
0.198753 + 0.980050i \(0.436311\pi\)
\(42\) 5.94748e18 1.27928
\(43\) −2.18793e17 −0.0359043 −0.0179522 0.999839i \(-0.505715\pi\)
−0.0179522 + 0.999839i \(0.505715\pi\)
\(44\) 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 \(0.417587\pi\)
\(48\) 2.92777e19 1.35601
\(49\) −1.28361e19 −0.469007
\(50\) 0 0
\(51\) 9.09698e19 2.09817
\(52\) 2.78032e19 0.512927
\(53\) 7.63436e19 1.13136 0.565679 0.824626i \(-0.308614\pi\)
0.565679 + 0.824626i \(0.308614\pi\)
\(54\) 5.83441e19 0.697379
\(55\) 0 0
\(56\) 9.89304e18 0.0778337
\(57\) 2.42034e20 1.55352
\(58\) −8.27059e19 −0.434625
\(59\) −1.01862e18 −0.00439760 −0.00219880 0.999998i \(-0.500700\pi\)
−0.00219880 + 0.999998i \(0.500700\pi\)
\(60\) 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
\(63\) −2.16326e20 −0.439231
\(64\) −4.96127e20 −0.840472
\(65\) 0 0
\(66\) −3.93464e20 −0.467897
\(67\) −1.47683e21 −1.47730 −0.738652 0.674088i \(-0.764537\pi\)
−0.738652 + 0.674088i \(0.764537\pi\)
\(68\) −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
\(75\) 0 0
\(76\) −4.82422e21 −1.13259
\(77\) −9.61427e20 −0.194211
\(78\) −5.60242e21 −0.975636
\(79\) 1.02350e22 1.53948 0.769742 0.638356i \(-0.220385\pi\)
0.769742 + 0.638356i \(0.220385\pi\)
\(80\) 0 0
\(81\) −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 \(0.393348\pi\)
\(84\) 1.14652e22 0.851469
\(85\) 0 0
\(86\) −8.78750e20 −0.0497890
\(87\) 7.99897e21 0.396792
\(88\) −6.54489e20 −0.0284676
\(89\) 4.58518e21 0.175134 0.0875672 0.996159i \(-0.472091\pi\)
0.0875672 + 0.996159i \(0.472091\pi\)
\(90\) 0 0
\(91\) −1.36895e22 −0.404959
\(92\) 2.77789e22 0.724695
\(93\) −5.29672e22 −1.22026
\(94\) 3.48553e22 0.710067
\(95\) 0 0
\(96\) 1.09133e23 1.74517
\(97\) 1.13703e23 1.61398 0.806991 0.590564i \(-0.201095\pi\)
0.806991 + 0.590564i \(0.201095\pi\)
\(98\) −5.15544e22 −0.650378
\(99\) 1.43114e22 0.160648
\(100\) 0 0
\(101\) 1.36243e23 1.21512 0.607558 0.794275i \(-0.292149\pi\)
0.607558 + 0.794275i \(0.292149\pi\)
\(102\) 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 \(-0.544218\pi\)
−0.138469 + 0.990367i \(0.544218\pi\)
\(108\) 1.12472e23 0.464164
\(109\) −5.58169e22 −0.207186 −0.103593 0.994620i \(-0.533034\pi\)
−0.103593 + 0.994620i \(0.533034\pi\)
\(110\) 0 0
\(111\) −4.81275e23 −1.44937
\(112\) 2.87329e23 0.780498
\(113\) −3.51523e23 −0.862089 −0.431044 0.902331i \(-0.641855\pi\)
−0.431044 + 0.902331i \(0.641855\pi\)
\(114\) 9.72095e23 2.15428
\(115\) 0 0
\(116\) −1.59435e23 −0.289279
\(117\) 2.03775e23 0.334976
\(118\) −4.09115e21 −0.00609821
\(119\) 8.92770e23 1.20767
\(120\) 0 0
\(121\) −8.31826e23 −0.928968
\(122\) 1.15405e24 1.17243
\(123\) −5.44111e23 −0.503244
\(124\) 1.05574e24 0.889627
\(125\) 0 0
\(126\) −8.68842e23 −0.609087
\(127\) −2.32044e24 −1.48535 −0.742674 0.669653i \(-0.766443\pi\)
−0.742674 + 0.669653i \(0.766443\pi\)
\(128\) 3.64148e23 0.212992
\(129\) 8.49891e22 0.0454550
\(130\) 0 0
\(131\) 8.70825e23 0.390221 0.195111 0.980781i \(-0.437493\pi\)
0.195111 + 0.980781i \(0.437493\pi\)
\(132\) −7.58496e23 −0.311425
\(133\) 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 \(0.702534\pi\)
\(138\) −5.59753e24 −1.37844
\(139\) −4.97229e23 −0.112690 −0.0563452 0.998411i \(-0.517945\pi\)
−0.0563452 + 0.998411i \(0.517945\pi\)
\(140\) 0 0
\(141\) −3.37106e24 −0.648257
\(142\) 3.06988e24 0.544259
\(143\) 9.05647e23 0.148114
\(144\) −4.27705e24 −0.645617
\(145\) 0 0
\(146\) 1.40345e25 1.80775
\(147\) 4.98613e24 0.593764
\(148\) 9.59276e24 1.05666
\(149\) 1.10598e24 0.112748 0.0563738 0.998410i \(-0.482046\pi\)
0.0563738 + 0.998410i \(0.482046\pi\)
\(150\) 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
\(153\) −1.32894e25 −0.998972
\(154\) −3.86143e24 −0.269315
\(155\) 0 0
\(156\) −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
\(160\) 0 0
\(161\) −1.36775e25 −0.572150
\(162\) −4.41199e25 −1.71875
\(163\) 2.61170e25 0.947907 0.473953 0.880550i \(-0.342827\pi\)
0.473953 + 0.880550i \(0.342827\pi\)
\(164\) 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 \(0.578333\pi\)
\(168\) −3.84290e24 −0.0985377
\(169\) −2.88587e25 −0.691161
\(170\) 0 0
\(171\) −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 \(0.519720\pi\)
\(180\) 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
\(183\) −1.11615e26 −1.07037
\(184\) −9.31094e24 −0.0838665
\(185\) 0 0
\(186\) −2.12735e26 −1.69216
\(187\) −5.90625e25 −0.441706
\(188\) 6.71919e25 0.472609
\(189\) −5.53780e25 −0.366460
\(190\) 0 0
\(191\) 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 \(-0.391823\pi\)
0.333344 + 0.942805i \(0.391823\pi\)
\(194\) 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 \(0.204298\pi\)
\(198\) 5.74795e25 0.222773
\(199\) −1.25611e26 −0.459426 −0.229713 0.973258i \(-0.573779\pi\)
−0.229713 + 0.973258i \(0.573779\pi\)
\(200\) 0 0
\(201\) 5.73666e26 1.87027
\(202\) 5.47199e26 1.68502
\(203\) 7.85013e25 0.228387
\(204\) 7.04330e26 1.93655
\(205\) 0 0
\(206\) −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
\(215\) 0 0
\(216\) −3.76984e25 −0.0537162
\(217\) −5.19817e26 −0.702365
\(218\) −2.24180e26 −0.287308
\(219\) −1.35735e27 −1.65039
\(220\) 0 0
\(221\) −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 \(0.740356\pi\)
\(224\) 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 \(0.167304\pi\)
\(228\) 1.87394e27 1.43386
\(229\) −6.39851e26 −0.465554 −0.232777 0.972530i \(-0.574781\pi\)
−0.232777 + 0.972530i \(0.574781\pi\)
\(230\) 0 0
\(231\) 3.73461e26 0.245871
\(232\) 5.34395e25 0.0334773
\(233\) −2.19149e27 −1.30661 −0.653306 0.757094i \(-0.726619\pi\)
−0.653306 + 0.757094i \(0.726619\pi\)
\(234\) 8.18434e26 0.464516
\(235\) 0 0
\(236\) −7.88666e24 −0.00405887
\(237\) −3.97572e27 −1.94899
\(238\) 3.58568e27 1.67470
\(239\) 1.09944e27 0.489322 0.244661 0.969609i \(-0.421323\pi\)
0.244661 + 0.969609i \(0.421323\pi\)
\(240\) 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
\(243\) 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
\(250\) 0 0
\(251\) 2.24453e26 0.0568693 0.0284346 0.999596i \(-0.490948\pi\)
0.0284346 + 0.999596i \(0.490948\pi\)
\(252\) −1.67490e27 −0.405399
\(253\) 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 \(0.375454\pi\)
\(258\) 3.41346e26 0.0630330
\(259\) −4.72320e27 −0.834236
\(260\) 0 0
\(261\) −1.16853e27 −0.188919
\(262\) 3.49754e27 0.541125
\(263\) −2.01521e27 −0.298420 −0.149210 0.988805i \(-0.547673\pi\)
−0.149210 + 0.988805i \(0.547673\pi\)
\(264\) 2.54233e26 0.0360401
\(265\) 0 0
\(266\) 9.54007e27 1.23997
\(267\) −1.78109e27 −0.221721
\(268\) −1.14343e28 −1.36351
\(269\) 5.72063e27 0.653571 0.326785 0.945099i \(-0.394035\pi\)
0.326785 + 0.945099i \(0.394035\pi\)
\(270\) 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
\(273\) 5.31761e27 0.512679
\(274\) −1.78273e28 −1.64799
\(275\) 0 0
\(276\) −1.07906e28 −0.917466
\(277\) −1.29611e28 −1.05712 −0.528560 0.848896i \(-0.677268\pi\)
−0.528560 + 0.848896i \(0.677268\pi\)
\(278\) −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
\(285\) 0 0
\(286\) 3.63740e27 0.205391
\(287\) −5.33986e27 −0.289660
\(288\) −1.59428e28 −0.830903
\(289\) 3.48772e28 1.74669
\(290\) 0 0
\(291\) −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 \(0.417197\pi\)
\(294\) 2.00260e28 0.823380
\(295\) 0 0
\(296\) −3.21530e27 −0.122283
\(297\) 3.66362e27 0.134033
\(298\) 4.44202e27 0.156349
\(299\) 1.28840e28 0.436347
\(300\) 0 0
\(301\) 8.34076e26 0.0261632
\(302\) 1.51137e28 0.456342
\(303\) −5.29228e28 −1.53834
\(304\) 4.69629e28 1.31434
\(305\) 0 0
\(306\) −5.33748e28 −1.38529
\(307\) 4.70428e28 1.17598 0.587992 0.808867i \(-0.299919\pi\)
0.587992 + 0.808867i \(0.299919\pi\)
\(308\) −7.44382e27 −0.179251
\(309\) 5.49267e27 0.127427
\(310\) 0 0
\(311\) −8.99672e26 −0.0193794 −0.00968968 0.999953i \(-0.503084\pi\)
−0.00968968 + 0.999953i \(0.503084\pi\)
\(312\) 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 \(-0.538039\pi\)
−0.119218 + 0.992868i \(0.538039\pi\)
\(318\) −1.19106e29 −1.98619
\(319\) −5.19337e27 −0.0835326
\(320\) 0 0
\(321\) 2.34221e28 0.350605
\(322\) −5.49338e28 −0.793408
\(323\) 1.45920e29 2.03370
\(324\) −8.50516e28 −1.14397
\(325\) 0 0
\(326\) 1.04895e29 1.31448
\(327\) 2.16818e28 0.262298
\(328\) −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
\(333\) 7.03074e28 0.690069
\(334\) −7.12534e28 −0.675649
\(335\) 0 0
\(336\) −1.11611e29 −0.988113
\(337\) 1.64138e29 1.40432 0.702160 0.712019i \(-0.252219\pi\)
0.702160 + 0.712019i \(0.252219\pi\)
\(338\) −1.15907e29 −0.958442
\(339\) 1.36547e29 1.09141
\(340\) 0 0
\(341\) 3.43892e28 0.256890
\(342\) −1.42009e29 −1.02569
\(343\) 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 \(0.646774\pi\)
\(348\) 6.19318e28 0.366229
\(349\) 2.22110e28 0.127080 0.0635398 0.997979i \(-0.479761\pi\)
0.0635398 + 0.997979i \(0.479761\pi\)
\(350\) 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
\(355\) 0 0
\(356\) 3.55006e28 0.161644
\(357\) −3.46792e29 −1.52892
\(358\) −4.02205e28 −0.171709
\(359\) −3.34157e29 −1.38154 −0.690771 0.723074i \(-0.742729\pi\)
−0.690771 + 0.723074i \(0.742729\pi\)
\(360\) 0 0
\(361\) 1.30406e29 0.505785
\(362\) −2.07713e29 −0.780398
\(363\) 3.23118e29 1.17608
\(364\) −1.05990e29 −0.373766
\(365\) 0 0
\(366\) −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
\(370\) 0 0
\(371\) −2.91034e29 −0.824412
\(372\) −4.10097e29 −1.12627
\(373\) −1.03540e29 −0.275712 −0.137856 0.990452i \(-0.544021\pi\)
−0.137856 + 0.990452i \(0.544021\pi\)
\(374\) −2.37216e29 −0.612520
\(375\) 0 0
\(376\) −2.25214e28 −0.0546935
\(377\) −7.39468e28 −0.174178
\(378\) −2.22418e29 −0.508175
\(379\) 6.62210e29 1.46772 0.733862 0.679298i \(-0.237716\pi\)
0.733862 + 0.679298i \(0.237716\pi\)
\(380\) 0 0
\(381\) 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 \(0.839581\pi\)
\(390\) 0 0
\(391\) −8.40240e29 −1.30128
\(392\) 3.33113e28 0.0500959
\(393\) −3.38267e29 −0.494021
\(394\) 1.56624e30 2.22153
\(395\) 0 0
\(396\) 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
\(399\) −9.22676e29 −1.13204
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 3.15289e29 0.316708
\(407\) 3.12470e29 0.305121
\(408\) −2.36078e29 −0.224111
\(409\) 5.00511e29 0.461950 0.230975 0.972960i \(-0.425808\pi\)
0.230975 + 0.972960i \(0.425808\pi\)
\(410\) 0 0
\(411\) 1.72418e30 1.50454
\(412\) −1.09480e29 −0.0928999
\(413\) 3.88316e27 0.00320450
\(414\) 8.17719e29 0.656297
\(415\) 0 0
\(416\) −1.00889e30 −0.766070
\(417\) 1.93146e29 0.142666
\(418\) −6.31137e29 −0.453520
\(419\) 1.70760e30 1.19378 0.596891 0.802322i \(-0.296402\pi\)
0.596891 + 0.802322i \(0.296402\pi\)
\(420\) 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
\(425\) 0 0
\(426\) −1.19248e30 −0.689034
\(427\) −1.09538e30 −0.616089
\(428\) −4.66848e29 −0.255607
\(429\) −3.51794e29 −0.187512
\(430\) 0 0
\(431\) 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 \(0.277676\pi\)
\(434\) −2.08777e30 −0.973980
\(435\) 0 0
\(436\) −4.32160e29 −0.191228
\(437\) −2.23555e30 −0.963489
\(438\) −5.45161e30 −2.28862
\(439\) 2.11168e29 0.0863549 0.0431775 0.999067i \(-0.486252\pi\)
0.0431775 + 0.999067i \(0.486252\pi\)
\(440\) 0 0
\(441\) −7.28401e29 −0.282701
\(442\) −3.37765e30 −1.27720
\(443\) 8.12623e29 0.299396 0.149698 0.988732i \(-0.452170\pi\)
0.149698 + 0.988732i \(0.452170\pi\)
\(444\) −3.72626e30 −1.33773
\(445\) 0 0
\(446\) −5.57554e30 −1.90080
\(447\) −4.29614e29 −0.142739
\(448\) 1.89132e30 0.612446
\(449\) −3.76596e30 −1.18862 −0.594310 0.804236i \(-0.702575\pi\)
−0.594310 + 0.804236i \(0.702575\pi\)
\(450\) 0 0
\(451\) 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 \(-0.588929\pi\)
−0.275759 + 0.961227i \(0.588929\pi\)
\(458\) −2.56986e30 −0.645591
\(459\) −3.40199e30 −0.833465
\(460\) 0 0
\(461\) 4.76379e30 1.11018 0.555088 0.831792i \(-0.312685\pi\)
0.555088 + 0.831792i \(0.312685\pi\)
\(462\) 1.49995e30 0.340953
\(463\) 7.67574e30 1.70192 0.850959 0.525232i \(-0.176021\pi\)
0.850959 + 0.525232i \(0.176021\pi\)
\(464\) 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 \(0.406645\pi\)
\(468\) 1.57772e30 0.309175
\(469\) 5.62992e30 1.07650
\(470\) 0 0
\(471\) 5.83016e30 1.06155
\(472\) 2.64345e27 0.000469719 0
\(473\) −5.51796e28 −0.00956919
\(474\) −1.59679e31 −2.70269
\(475\) 0 0
\(476\) 6.91225e30 1.11465
\(477\) 4.33221e30 0.681943
\(478\) 4.41573e30 0.678549
\(479\) 6.82143e30 1.02333 0.511666 0.859184i \(-0.329028\pi\)
0.511666 + 0.859184i \(0.329028\pi\)
\(480\) 0 0
\(481\) 4.44917e30 0.636224
\(482\) −5.80971e30 −0.811173
\(483\) 5.31297e30 0.724344
\(484\) −6.44038e30 −0.857413
\(485\) 0 0
\(486\) 1.16455e31 1.47856
\(487\) −8.59361e29 −0.106560 −0.0532798 0.998580i \(-0.516968\pi\)
−0.0532798 + 0.998580i \(0.516968\pi\)
\(488\) −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
\(495\) 0 0
\(496\) −1.02774e31 −1.03239
\(497\) −2.91382e30 −0.285998
\(498\) −1.20379e31 −1.15455
\(499\) 1.77913e31 1.66744 0.833722 0.552185i \(-0.186206\pi\)
0.833722 + 0.552185i \(0.186206\pi\)
\(500\) 0 0
\(501\) 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 \(0.590503\pi\)
\(504\) 5.61393e29 0.0469154
\(505\) 0 0
\(506\) 3.63422e30 0.290189
\(507\) 1.12100e31 0.875012
\(508\) −1.79660e31 −1.37094
\(509\) −1.19826e31 −0.893917 −0.446959 0.894555i \(-0.647493\pi\)
−0.446959 + 0.894555i \(0.647493\pi\)
\(510\) 0 0
\(511\) −1.33210e31 −0.949939
\(512\) 1.95347e31 1.36208
\(513\) −9.05135e30 −0.617112
\(514\) 1.58648e31 1.05769
\(515\) 0 0
\(516\) 6.58025e29 0.0419538
\(517\) 2.18868e30 0.136471
\(518\) −1.89700e31 −1.15685
\(519\) 4.04407e31 2.41209
\(520\) 0 0
\(521\) −2.26240e31 −1.29103 −0.645514 0.763749i \(-0.723357\pi\)
−0.645514 + 0.763749i \(0.723357\pi\)
\(522\) −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
\(530\) 0 0
\(531\) −5.78030e28 −0.00265072
\(532\) 1.83907e31 0.825307
\(533\) 5.03006e30 0.220907
\(534\) −7.15348e30 −0.307463
\(535\) 0 0
\(536\) 3.83255e30 0.157795
\(537\) 3.88996e30 0.156762
\(538\) 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
\(543\) 2.00892e31 0.712466
\(544\) 6.57953e31 2.28459
\(545\) 0 0
\(546\) 2.13574e31 0.710939
\(547\) 1.03197e31 0.336367 0.168184 0.985756i \(-0.446210\pi\)
0.168184 + 0.985756i \(0.446210\pi\)
\(548\) −3.43664e31 −1.09688
\(549\) 1.63053e31 0.509621
\(550\) 0 0
\(551\) 1.28308e31 0.384600
\(552\) 3.61679e30 0.106175
\(553\) −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 \(0.557709\pi\)
\(558\) 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
\(565\) 0 0
\(566\) 8.63638e31 1.90083
\(567\) 4.18769e31 0.903171
\(568\) −1.98357e30 −0.0419220
\(569\) −6.93051e31 −1.43540 −0.717702 0.696350i \(-0.754806\pi\)
−0.717702 + 0.696350i \(0.754806\pi\)
\(570\) 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
\(573\) −1.20467e32 −2.30192
\(574\) −2.14468e31 −0.401675
\(575\) 0 0
\(576\) −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
\(580\) 0 0
\(581\) −2.94146e31 −0.479222
\(582\) −1.77392e32 −2.83348
\(583\) 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 \(0.701306\pi\)
\(588\) 3.86049e31 0.548029
\(589\) −8.49622e31 −1.18277
\(590\) 0 0
\(591\) −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 \(0.129010\pi\)
\(600\) 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
\(603\) −8.38044e31 −0.890467
\(604\) 2.91352e31 0.303734
\(605\) 0 0
\(606\) −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
\(609\) −3.04934e31 −0.289139
\(610\) 0 0
\(611\) 3.11639e31 0.284563
\(612\) −1.02893e32 −0.922025
\(613\) −1.93558e31 −0.170222 −0.0851110 0.996371i \(-0.527124\pi\)
−0.0851110 + 0.996371i \(0.527124\pi\)
\(614\) 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 \(0.568801\pi\)
\(618\) 2.20605e31 0.176704
\(619\) −9.42009e31 −0.740650 −0.370325 0.928902i \(-0.620754\pi\)
−0.370325 + 0.928902i \(0.620754\pi\)
\(620\) 0 0
\(621\) 5.21196e31 0.394864
\(622\) −3.61340e30 −0.0268736
\(623\) −1.74795e31 −0.127619
\(624\) 1.05136e32 0.753578
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) −2.29605e32 −1.32198
\(637\) −4.60944e31 −0.260642
\(638\) −2.08584e31 −0.115836
\(639\) 4.33738e31 0.236574
\(640\) 0 0
\(641\) 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 \(0.530053\pi\)
\(644\) −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 \(0.149035\pi\)
\(648\) 2.85076e31 0.132388
\(649\) −2.56896e29 −0.00117204
\(650\) 0 0
\(651\) 2.01920e32 0.889197
\(652\) 2.02210e32 0.874893
\(653\) 8.58042e31 0.364760 0.182380 0.983228i \(-0.441620\pi\)
0.182380 + 0.983228i \(0.441620\pi\)
\(654\) 8.70816e31 0.363733
\(655\) 0 0
\(656\) −1.05576e32 −0.425766
\(657\) 1.98290e32 0.785777
\(658\) −1.32874e32 −0.517421
\(659\) 1.46687e32 0.561319 0.280660 0.959807i \(-0.409447\pi\)
0.280660 + 0.959807i \(0.409447\pi\)
\(660\) 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
\(663\) 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
\(670\) 0 0
\(671\) 7.24664e31 0.225334
\(672\) −4.16034e32 −1.27169
\(673\) −5.34692e32 −1.60668 −0.803341 0.595519i \(-0.796947\pi\)
−0.803341 + 0.595519i \(0.796947\pi\)
\(674\) 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 \(0.549360\pi\)
\(678\) 5.48422e32 1.51347
\(679\) −4.33457e32 −1.17610
\(680\) 0 0
\(681\) −8.34995e32 −2.19024
\(682\) 1.38119e32 0.356233
\(683\) 2.97297e32 0.753967 0.376983 0.926220i \(-0.376961\pi\)
0.376983 + 0.926220i \(0.376961\pi\)
\(684\) −2.73756e32 −0.682683
\(685\) 0 0
\(686\) 6.15577e32 1.48441
\(687\) 2.48547e32 0.589394
\(688\) 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
\(693\) −5.45574e31 −0.117063
\(694\) −5.84723e32 −1.23400
\(695\) 0 0
\(696\) −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
\(705\) 0 0
\(706\) −1.03287e32 −0.178975
\(707\) −5.19380e32 −0.885446
\(708\) 3.06353e30 0.00513854
\(709\) −2.50942e32 −0.414134 −0.207067 0.978327i \(-0.566392\pi\)
−0.207067 + 0.978327i \(0.566392\pi\)
\(710\) 0 0
\(711\) 5.80796e32 0.927947
\(712\) −1.18991e31 −0.0187066
\(713\) 4.89231e32 0.756805
\(714\) −1.39284e33 −2.12017
\(715\) 0 0
\(716\) −7.75346e31 −0.114287
\(717\) −4.27071e32 −0.619483
\(718\) −1.34209e33 −1.91580
\(719\) 7.09247e32 0.996358 0.498179 0.867074i \(-0.334002\pi\)
0.498179 + 0.867074i \(0.334002\pi\)
\(720\) 0 0
\(721\) 5.39046e31 0.0733450
\(722\) 5.23757e32 0.701378
\(723\) 5.61891e32 0.740562
\(724\) −4.00417e32 −0.519420
\(725\) 0 0
\(726\) 1.29776e33 1.63088
\(727\) −4.63712e32 −0.573591 −0.286795 0.957992i \(-0.592590\pi\)
−0.286795 + 0.957992i \(0.592590\pi\)
\(728\) 3.55259e31 0.0432547
\(729\) −9.21300e31 −0.110417
\(730\) 0 0
\(731\) 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 \(0.777094\pi\)
\(740\) 0 0
\(741\) 8.69144e32 0.863341
\(742\) −1.16890e33 −1.14322
\(743\) −1.19934e32 −0.115497 −0.0577487 0.998331i \(-0.518392\pi\)
−0.0577487 + 0.998331i \(0.518392\pi\)
\(744\) 1.37456e32 0.130340
\(745\) 0 0
\(746\) −4.15852e32 −0.382334
\(747\) 4.37852e32 0.396406
\(748\) −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
\(753\) −8.71877e31 −0.0719967
\(754\) −2.96996e32 −0.241535
\(755\) 0 0
\(756\) −4.28763e32 −0.338233
\(757\) 1.56208e33 1.21367 0.606835 0.794828i \(-0.292439\pi\)
0.606835 + 0.794828i \(0.292439\pi\)
\(758\) 2.65967e33 2.03531
\(759\) −3.51487e32 −0.264929
\(760\) 0 0
\(761\) 1.92115e33 1.40488 0.702439 0.711744i \(-0.252094\pi\)
0.702439 + 0.711744i \(0.252094\pi\)
\(762\) 3.62020e33 2.60765
\(763\) 2.12783e32 0.150975
\(764\) 2.40114e33 1.67820
\(765\) 0 0
\(766\) −1.19119e33 −0.807886
\(767\) −3.65787e30 −0.00244389
\(768\) −2.18475e33 −1.43797
\(769\) 1.22615e33 0.795044 0.397522 0.917593i \(-0.369870\pi\)
0.397522 + 0.917593i \(0.369870\pi\)
\(770\) 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
\(775\) 0 0
\(776\) −2.95074e32 −0.172394
\(777\) 1.83470e33 1.05615
\(778\) −4.28177e33 −2.42861
\(779\) −8.72782e32 −0.487781
\(780\) 0 0
\(781\) 1.92768e32 0.104604
\(782\) −3.37470e33 −1.80450
\(783\) −2.99137e32 −0.157620
\(784\) 9.67477e32 0.502350
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) 1.34007e33 0.628198
\(792\) −3.71398e31 −0.0171593
\(793\) 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 \(0.609871\pi\)
\(798\) −3.70579e33 −1.56981
\(799\) −2.03238e33 −0.848628
\(800\) 0 0
\(801\) 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 \(0.653924\pi\)
\(810\) 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
\(813\) −2.01234e33 −0.688112
\(814\) 1.25499e33 0.423116
\(815\) 0 0
\(816\) −6.85652e33 −2.24733
\(817\) 1.36327e32 0.0440583
\(818\) 2.01023e33 0.640593
\(819\) −7.76826e32 −0.244095
\(820\) 0 0
\(821\) −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 \(-0.832830\pi\)
−0.865233 + 0.501370i \(0.832830\pi\)
\(824\) 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 \(0.330621\pi\)
\(828\) 1.57635e33 0.436821
\(829\) 6.27935e33 1.71608 0.858040 0.513584i \(-0.171682\pi\)
0.858040 + 0.513584i \(0.171682\pi\)
\(830\) 0 0
\(831\) 5.03467e33 1.33832
\(832\) −1.78159e33 −0.467078
\(833\) 3.00609e33 0.777292
\(834\) 7.75743e32 0.197837
\(835\) 0 0
\(836\) −1.21667e33 −0.301856
\(837\) 1.98081e33 0.484731
\(838\) 6.85832e33 1.65543
\(839\) 7.51696e32 0.178970 0.0894849 0.995988i \(-0.471478\pi\)
0.0894849 + 0.995988i \(0.471478\pi\)
\(840\) 0 0
\(841\) −3.89268e33 −0.901767
\(842\) −2.39864e33 −0.548120
\(843\) −4.82269e33 −1.08711
\(844\) 4.55478e33 1.01281
\(845\) 0 0
\(846\) 1.97791e33 0.428003
\(847\) 3.17106e33 0.676932
\(848\) −5.75412e33 −1.21179
\(849\) −8.35275e33 −1.73537
\(850\) 0 0
\(851\) 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 \(0.145860\pi\)
\(854\) −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 \(0.695310\pi\)
\(858\) −1.41293e33 −0.260026
\(859\) −1.86175e33 −0.338065 −0.169033 0.985610i \(-0.554064\pi\)
−0.169033 + 0.985610i \(0.554064\pi\)
\(860\) 0 0
\(861\) 2.07424e33 0.366710
\(862\) 7.70552e33 1.34421
\(863\) 5.22144e33 0.898806 0.449403 0.893329i \(-0.351637\pi\)
0.449403 + 0.893329i \(0.351637\pi\)
\(864\) −4.08125e33 −0.693242
\(865\) 0 0
\(866\) 1.07821e34 1.78340
\(867\) −1.35479e34 −2.21132
\(868\) −4.02466e33 −0.648265
\(869\) 2.58126e33 0.410302
\(870\) 0 0
\(871\) −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 \(-0.484653\pi\)
0.0481954 + 0.998838i \(0.484653\pi\)
\(878\) 8.48127e32 0.119750
\(879\) −4.67328e33 −0.651253
\(880\) 0 0
\(881\) 1.36978e34 1.85963 0.929815 0.368028i \(-0.119967\pi\)
0.929815 + 0.368028i \(0.119967\pi\)
\(882\) −2.92552e33 −0.392025
\(883\) −1.49590e33 −0.197859 −0.0989294 0.995094i \(-0.531542\pi\)
−0.0989294 + 0.995094i \(0.531542\pi\)
\(884\) −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 \(0.367308\pi\)
\(888\) 1.24897e33 0.154811
\(889\) 8.84592e33 1.08236
\(890\) 0 0
\(891\) −2.77043e33 −0.330334
\(892\) −1.07482e34 −1.26514
\(893\) −5.40736e33 −0.628339
\(894\) −1.72548e33 −0.197938
\(895\) 0 0
\(896\) −1.38819e33 −0.155206
\(897\) −5.00472e33 −0.552417
\(898\) −1.51254e34 −1.64827
\(899\) −2.80791e33 −0.302097
\(900\) 0 0
\(901\) −1.78789e34 −1.87502
\(902\) 1.41884e33 0.146913
\(903\) −3.23993e32 −0.0331227
\(904\) 9.12246e32 0.0920820
\(905\) 0 0
\(906\) −5.87083e33 −0.577731
\(907\) −6.42282e33 −0.624083 −0.312041 0.950069i \(-0.601013\pi\)
−0.312041 + 0.950069i \(0.601013\pi\)
\(908\) 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
\(915\) 0 0
\(916\) −4.95402e33 −0.429695
\(917\) −3.31973e33 −0.284351
\(918\) −1.36636e34 −1.15578
\(919\) 3.10701e33 0.259546 0.129773 0.991544i \(-0.458575\pi\)
0.129773 + 0.991544i \(0.458575\pi\)
\(920\) 0 0
\(921\) −1.82735e34 −1.48880
\(922\) 1.91330e34 1.53950
\(923\) 2.74476e33 0.218115
\(924\) 2.89151e33 0.226933
\(925\) 0 0
\(926\) 3.08285e34 2.36007
\(927\) −8.02400e32 −0.0606700
\(928\) 5.78538e33 0.432046
\(929\) 1.72846e34 1.27490 0.637452 0.770490i \(-0.279988\pi\)
0.637452 + 0.770490i \(0.279988\pi\)
\(930\) 0 0
\(931\) 7.99801e33 0.575520
\(932\) −1.69676e34 −1.20597
\(933\) 3.49473e32 0.0245343
\(934\) 1.15624e34 0.801790
\(935\) 0 0
\(936\) −5.28822e32 −0.0357797
\(937\) 1.61548e34 1.07969 0.539843 0.841766i \(-0.318484\pi\)
0.539843 + 0.841766i \(0.318484\pi\)
\(938\) 2.26117e34 1.49280
\(939\) −4.40980e33 −0.287584
\(940\) 0 0
\(941\) 7.69275e33 0.489554 0.244777 0.969579i \(-0.421285\pi\)
0.244777 + 0.969579i \(0.421285\pi\)
\(942\) 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 \(-0.163778\pi\)
0.870528 + 0.492119i \(0.163778\pi\)
\(948\) −3.07819e34 −1.79887
\(949\) 1.25481e34 0.724465
\(950\) 0 0
\(951\) 5.35651e33 0.301860
\(952\) −2.31685e33 −0.128995
\(953\) −1.97292e34 −1.08528 −0.542641 0.839965i \(-0.682575\pi\)
−0.542641 + 0.839965i \(0.682575\pi\)
\(954\) 1.73997e34 0.945659
\(955\) 0 0
\(956\) 8.51236e33 0.451632
\(957\) 2.01734e33 0.105753
\(958\) 2.73973e34 1.41907
\(959\) 1.69210e34 0.865989
\(960\) 0 0
\(961\) −1.42003e33 −0.0709542
\(962\) 1.78694e34 0.882261
\(963\) −3.42163e33 −0.166929
\(964\) −1.11996e34 −0.539904
\(965\) 0 0
\(966\) 2.13387e34 1.00446
\(967\) 2.03266e34 0.945496 0.472748 0.881198i \(-0.343262\pi\)
0.472748 + 0.881198i \(0.343262\pi\)
\(968\) 2.15869e33 0.0992255
\(969\) −5.66820e34 −2.57467
\(970\) 0 0
\(971\) 3.40470e34 1.51028 0.755141 0.655563i \(-0.227569\pi\)
0.755141 + 0.655563i \(0.227569\pi\)
\(972\) 2.24494e34 0.984106
\(973\) 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 \(0.497579\pi\)
\(978\) −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
\(985\) 0 0
\(986\) 1.93689e34 0.720310
\(987\) 1.28511e34 0.472380
\(988\) −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
\(993\) −6.31875e34 −2.16628
\(994\) −1.17029e34 −0.396598
\(995\) 0 0
\(996\) −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 25.24.a.a.1.2 2
5.2 odd 4 25.24.b.a.24.3 4
5.3 odd 4 25.24.b.a.24.2 4
5.4 even 2 1.24.a.a.1.1 2
15.14 odd 2 9.24.a.b.1.2 2
20.19 odd 2 16.24.a.b.1.1 2
35.34 odd 2 49.24.a.b.1.1 2
40.19 odd 2 64.24.a.g.1.2 2
40.29 even 2 64.24.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.24.a.a.1.1 2 5.4 even 2
9.24.a.b.1.2 2 15.14 odd 2
16.24.a.b.1.1 2 20.19 odd 2
25.24.a.a.1.2 2 1.1 even 1 trivial
25.24.b.a.24.2 4 5.3 odd 4
25.24.b.a.24.3 4 5.2 odd 4
49.24.a.b.1.1 2 35.34 odd 2
64.24.a.d.1.1 2 40.29 even 2
64.24.a.g.1.2 2 40.19 odd 2