Properties

Label 64.24.a.g.1.2
Level $64$
Weight $24$
Character 64.1
Self dual yes
Analytic conductor $214.531$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(214.530583901\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{144169}) \)
Defining polynomial: \(x^{2} - x - 36042\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(\chi\) \(=\) 64.1

$q$-expansion

\(f(q)\) \(=\) \(q+388445. q^{3} -1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} +O(q^{10})\) \(q+388445. q^{3} -1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} +2.52200e11 q^{11} +3.59099e12 q^{13} -4.08108e13 q^{15} +2.34190e14 q^{17} -6.23086e14 q^{19} -1.48082e15 q^{21} +3.58786e15 q^{23} -8.82898e14 q^{25} -1.45267e16 q^{27} +2.05923e16 q^{29} -1.36357e17 q^{31} +9.79656e16 q^{33} +4.00514e17 q^{35} +1.23898e18 q^{37} +1.39490e18 q^{39} +1.40074e18 q^{41} +2.18793e17 q^{43} -5.96187e18 q^{45} +8.67836e18 q^{47} -1.28361e19 q^{49} +9.09698e19 q^{51} +7.63436e19 q^{53} -2.64966e19 q^{55} -2.42034e20 q^{57} -1.01862e18 q^{59} -2.87337e20 q^{61} -2.16326e20 q^{63} -3.77277e20 q^{65} +1.47683e21 q^{67} +1.39369e21 q^{69} -7.64346e20 q^{71} -3.49433e21 q^{73} -3.42957e20 q^{75} -9.61427e20 q^{77} -1.02350e22 q^{79} -1.09851e22 q^{81} -7.71597e21 q^{83} -2.46044e22 q^{85} +7.99897e21 q^{87} +4.58518e21 q^{89} -1.36895e22 q^{91} -5.29672e22 q^{93} +6.54627e22 q^{95} -1.13703e23 q^{97} +1.43114e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 339480q^{3} - 73069020q^{5} + 1359184400q^{7} - 34999394166q^{9} + O(q^{10}) \) \( 2q + 339480q^{3} - 73069020q^{5} + 1359184400q^{7} - 34999394166q^{9} + 856801968264q^{11} - 4376109322060q^{13} - 42377338985040q^{15} + 254028147597540q^{17} + 4260600979960q^{19} - 1734031637722944q^{21} + 8144713079008560q^{23} - 11780274628800850q^{25} - 5424634982716560q^{27} - 20818433601623340q^{29} - 137714017177000384q^{31} + 68361366766001760q^{33} + 565961271250425120q^{35} + 897721264408967780q^{37} + 1785011473665029232q^{39} - 2294435477168314956q^{41} - 1750760768619855800q^{43} - 8897092690294206540q^{45} - 15759744217656780960q^{47} - 13461981704376200814q^{49} + 89998362845078292144q^{51} + 140287253401646796420q^{53} - 7153550955060182640q^{55} - 272752401448627175520q^{57} + 280872989971340771880q^{59} + 180452892516502223636q^{61} - 690775113933935014320q^{63} - 632168834809440380760q^{65} + 1754233163431557625240q^{67} + 1170560672172404223552q^{69} - 3055033510194143328624q^{71} - 8063408253877606149260q^{73} + 190631089350520885800q^{75} + 2165184764357449665600q^{77} - 6244916814559639980640q^{79} - 2793528580929833975598q^{81} + 6875994082418498976120q^{83} - 23969743087870314902520q^{85} + 10026640653837674384880q^{87} + 6395093086173070004820q^{89} - 54890178162704560146016q^{91} - 52900811441357852079360q^{93} + 85533361066700858502000q^{95} - 31147288846254030500540q^{97} - 41158245132312135981912q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 388445. 1.26600 0.633002 0.774150i \(-0.281823\pi\)
0.633002 + 0.774150i \(0.281823\pi\)
\(4\) 0 0
\(5\) −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 \(-0.618708\pi\)
−0.364346 + 0.931263i \(0.618708\pi\)
\(8\) 0 0
\(9\) 5.67462e10 0.602765
\(10\) 0 0
\(11\) 2.52200e11 0.266519 0.133260 0.991081i \(-0.457456\pi\)
0.133260 + 0.991081i \(0.457456\pi\)
\(12\) 0 0
\(13\) 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
\(16\) 0 0
\(17\) 2.34190e14 1.65731 0.828657 0.559756i \(-0.189105\pi\)
0.828657 + 0.559756i \(0.189105\pi\)
\(18\) 0 0
\(19\) −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
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) −1.45267e16 −0.502901
\(28\) 0 0
\(29\) 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 \(-0.660066\pi\)
−0.481935 + 0.876207i \(0.660066\pi\)
\(32\) 0 0
\(33\) 9.79656e16 0.337414
\(34\) 0 0
\(35\) 4.00514e17 0.701189
\(36\) 0 0
\(37\) 1.23898e18 1.14484 0.572419 0.819961i \(-0.306005\pi\)
0.572419 + 0.819961i \(0.306005\pi\)
\(38\) 0 0
\(39\) 1.39490e18 0.703560
\(40\) 0 0
\(41\) 1.40074e18 0.397506 0.198753 0.980050i \(-0.436311\pi\)
0.198753 + 0.980050i \(0.436311\pi\)
\(42\) 0 0
\(43\) 2.18793e17 0.0359043 0.0179522 0.999839i \(-0.494285\pi\)
0.0179522 + 0.999839i \(0.494285\pi\)
\(44\) 0 0
\(45\) −5.96187e18 −0.580015
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 9.09698e19 2.09817
\(52\) 0 0
\(53\) 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
\(56\) 0 0
\(57\) −2.42034e20 −1.55352
\(58\) 0 0
\(59\) −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 \(-0.638930\pi\)
−0.422736 + 0.906253i \(0.638930\pi\)
\(62\) 0 0
\(63\) −2.16326e20 −0.439231
\(64\) 0 0
\(65\) −3.77277e20 −0.534758
\(66\) 0 0
\(67\) 1.47683e21 1.47730 0.738652 0.674088i \(-0.235463\pi\)
0.738652 + 0.674088i \(0.235463\pi\)
\(68\) 0 0
\(69\) 1.39369e21 0.994032
\(70\) 0 0
\(71\) −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 \(0.725990\pi\)
\(74\) 0 0
\(75\) −3.42957e20 −0.0937639
\(76\) 0 0
\(77\) −9.61427e20 −0.194211
\(78\) 0 0
\(79\) −1.02350e22 −1.53948 −0.769742 0.638356i \(-0.779615\pi\)
−0.769742 + 0.638356i \(0.779615\pi\)
\(80\) 0 0
\(81\) −1.09851e22 −1.23944
\(82\) 0 0
\(83\) −7.71597e21 −0.657646 −0.328823 0.944392i \(-0.606652\pi\)
−0.328823 + 0.944392i \(0.606652\pi\)
\(84\) 0 0
\(85\) −2.46044e22 −1.59476
\(86\) 0 0
\(87\) 7.99897e21 0.396792
\(88\) 0 0
\(89\) 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
\(92\) 0 0
\(93\) −5.29672e22 −1.22026
\(94\) 0 0
\(95\) 6.54627e22 1.18079
\(96\) 0 0
\(97\) −1.13703e23 −1.61398 −0.806991 0.590564i \(-0.798905\pi\)
−0.806991 + 0.590564i \(0.798905\pi\)
\(98\) 0 0
\(99\) 1.43114e22 0.160648
\(100\) 0 0
\(101\) −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 \(-0.516026\pi\)
−0.0503264 + 0.998733i \(0.516026\pi\)
\(104\) 0 0
\(105\) 1.55578e23 0.887708
\(106\) 0 0
\(107\) 6.02971e22 0.276938 0.138469 0.990367i \(-0.455782\pi\)
0.138469 + 0.990367i \(0.455782\pi\)
\(108\) 0 0
\(109\) 5.58169e22 0.207186 0.103593 0.994620i \(-0.466966\pi\)
0.103593 + 0.994620i \(0.466966\pi\)
\(110\) 0 0
\(111\) 4.81275e23 1.44937
\(112\) 0 0
\(113\) 3.51523e23 0.862089 0.431044 0.902331i \(-0.358145\pi\)
0.431044 + 0.902331i \(0.358145\pi\)
\(114\) 0 0
\(115\) −3.76948e23 −0.755538
\(116\) 0 0
\(117\) 2.03775e23 0.334976
\(118\) 0 0
\(119\) −8.92770e23 −1.20767
\(120\) 0 0
\(121\) −8.31826e23 −0.928968
\(122\) 0 0
\(123\) 5.44111e23 0.503244
\(124\) 0 0
\(125\) 1.34520e24 1.03352
\(126\) 0 0
\(127\) −2.32044e24 −1.48535 −0.742674 0.669653i \(-0.766443\pi\)
−0.742674 + 0.669653i \(0.766443\pi\)
\(128\) 0 0
\(129\) 8.49891e22 0.0454550
\(130\) 0 0
\(131\) 8.70825e23 0.390221 0.195111 0.980781i \(-0.437493\pi\)
0.195111 + 0.980781i \(0.437493\pi\)
\(132\) 0 0
\(133\) 2.37531e24 0.894182
\(134\) 0 0
\(135\) 1.52620e24 0.483919
\(136\) 0 0
\(137\) 4.43869e24 1.18841 0.594207 0.804312i \(-0.297466\pi\)
0.594207 + 0.804312i \(0.297466\pi\)
\(138\) 0 0
\(139\) −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
\(142\) 0 0
\(143\) 9.05647e23 0.148114
\(144\) 0 0
\(145\) −2.16347e24 −0.301591
\(146\) 0 0
\(147\) −4.98613e24 −0.593764
\(148\) 0 0
\(149\) −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 \(-0.552614\pi\)
−0.164541 + 0.986370i \(0.552614\pi\)
\(152\) 0 0
\(153\) 1.32894e25 0.998972
\(154\) 0 0
\(155\) 1.43260e25 0.927490
\(156\) 0 0
\(157\) −1.50090e25 −0.838504 −0.419252 0.907870i \(-0.637708\pi\)
−0.419252 + 0.907870i \(0.637708\pi\)
\(158\) 0 0
\(159\) 2.96553e25 1.43230
\(160\) 0 0
\(161\) −1.36775e25 −0.572150
\(162\) 0 0
\(163\) −2.61170e25 −0.947907 −0.473953 0.880550i \(-0.657173\pi\)
−0.473953 + 0.880550i \(0.657173\pi\)
\(164\) 0 0
\(165\) −1.02925e25 −0.324679
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) −3.53578e25 −0.739656
\(172\) 0 0
\(173\) −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
\(176\) 0 0
\(177\) −3.95679e23 −0.00556737
\(178\) 0 0
\(179\) −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 \(-0.409207\pi\)
0.281384 + 0.959595i \(0.409207\pi\)
\(182\) 0 0
\(183\) −1.11615e26 −1.07037
\(184\) 0 0
\(185\) −1.30170e26 −1.10163
\(186\) 0 0
\(187\) 5.90625e25 0.441706
\(188\) 0 0
\(189\) 5.53780e25 0.366460
\(190\) 0 0
\(191\) −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 \(0.608177\pi\)
\(194\) 0 0
\(195\) −1.46551e26 −0.677005
\(196\) 0 0
\(197\) 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 \(-0.426221\pi\)
0.229713 + 0.973258i \(0.426221\pi\)
\(200\) 0 0
\(201\) 5.73666e26 1.87027
\(202\) 0 0
\(203\) −7.85013e25 −0.228387
\(204\) 0 0
\(205\) −1.47165e26 −0.382503
\(206\) 0 0
\(207\) 2.03598e26 0.473275
\(208\) 0 0
\(209\) −1.57142e26 −0.327047
\(210\) 0 0
\(211\) 5.88286e26 1.09734 0.548669 0.836040i \(-0.315135\pi\)
0.548669 + 0.836040i \(0.315135\pi\)
\(212\) 0 0
\(213\) −2.96906e26 −0.496883
\(214\) 0 0
\(215\) −2.29869e25 −0.0345491
\(216\) 0 0
\(217\) 5.19817e26 0.702365
\(218\) 0 0
\(219\) −1.35735e27 −1.65039
\(220\) 0 0
\(221\) 8.40974e26 0.921025
\(222\) 0 0
\(223\) −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
\(226\) 0 0
\(227\) −2.14958e27 −1.73004 −0.865022 0.501734i \(-0.832696\pi\)
−0.865022 + 0.501734i \(0.832696\pi\)
\(228\) 0 0
\(229\) 6.39851e26 0.465554 0.232777 0.972530i \(-0.425219\pi\)
0.232777 + 0.972530i \(0.425219\pi\)
\(230\) 0 0
\(231\) −3.73461e26 −0.245871
\(232\) 0 0
\(233\) 2.19149e27 1.30661 0.653306 0.757094i \(-0.273381\pi\)
0.653306 + 0.757094i \(0.273381\pi\)
\(234\) 0 0
\(235\) −9.11766e26 −0.492723
\(236\) 0 0
\(237\) −3.97572e27 −1.94899
\(238\) 0 0
\(239\) −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 \(0.594481\pi\)
\(242\) 0 0
\(243\) −2.89951e27 −1.06623
\(244\) 0 0
\(245\) 1.34859e27 0.451304
\(246\) 0 0
\(247\) −2.23750e27 −0.681942
\(248\) 0 0
\(249\) −2.99723e27 −0.832582
\(250\) 0 0
\(251\) 2.24453e26 0.0568693 0.0284346 0.999596i \(-0.490948\pi\)
0.0284346 + 0.999596i \(0.490948\pi\)
\(252\) 0 0
\(253\) 9.04857e26 0.209264
\(254\) 0 0
\(255\) −9.55747e27 −2.01897
\(256\) 0 0
\(257\) −3.95005e27 −0.762732 −0.381366 0.924424i \(-0.624546\pi\)
−0.381366 + 0.924424i \(0.624546\pi\)
\(258\) 0 0
\(259\) −4.72320e27 −0.834236
\(260\) 0 0
\(261\) 1.16853e27 0.188919
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) 1.78109e27 0.221721
\(268\) 0 0
\(269\) −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 \(-0.587608\pi\)
−0.271766 + 0.962363i \(0.587608\pi\)
\(272\) 0 0
\(273\) −5.31761e27 −0.512679
\(274\) 0 0
\(275\) −2.22667e26 −0.0197392
\(276\) 0 0
\(277\) −1.29611e28 −1.05712 −0.528560 0.848896i \(-0.677268\pi\)
−0.528560 + 0.848896i \(0.677268\pi\)
\(278\) 0 0
\(279\) −7.73776e27 −0.580987
\(280\) 0 0
\(281\) 1.24154e28 0.858691 0.429346 0.903140i \(-0.358744\pi\)
0.429346 + 0.903140i \(0.358744\pi\)
\(282\) 0 0
\(283\) −2.15031e28 −1.37074 −0.685372 0.728193i \(-0.740360\pi\)
−0.685372 + 0.728193i \(0.740360\pi\)
\(284\) 0 0
\(285\) 2.54286e28 1.49488
\(286\) 0 0
\(287\) −5.33986e27 −0.289660
\(288\) 0 0
\(289\) 3.48772e28 1.74669
\(290\) 0 0
\(291\) −4.41675e28 −2.04331
\(292\) 0 0
\(293\) 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
\(296\) 0 0
\(297\) −3.66362e27 −0.134033
\(298\) 0 0
\(299\) 1.28840e28 0.436347
\(300\) 0 0
\(301\) −8.34076e26 −0.0261632
\(302\) 0 0
\(303\) −5.29228e28 −1.53834
\(304\) 0 0
\(305\) 3.01883e28 0.813560
\(306\) 0 0
\(307\) −4.70428e28 −1.17598 −0.587992 0.808867i \(-0.700081\pi\)
−0.587992 + 0.808867i \(0.700081\pi\)
\(308\) 0 0
\(309\) −5.49267e27 −0.127427
\(310\) 0 0
\(311\) 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 \(0.536232\pi\)
\(314\) 0 0
\(315\) 2.27277e28 0.422653
\(316\) 0 0
\(317\) −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
\(320\) 0 0
\(321\) 2.34221e28 0.350605
\(322\) 0 0
\(323\) −1.45920e29 −2.03370
\(324\) 0 0
\(325\) −3.17048e27 −0.0411592
\(326\) 0 0
\(327\) 2.16818e28 0.262298
\(328\) 0 0
\(329\) −3.30834e28 −0.373127
\(330\) 0 0
\(331\) 1.62668e29 1.71112 0.855559 0.517706i \(-0.173214\pi\)
0.855559 + 0.517706i \(0.173214\pi\)
\(332\) 0 0
\(333\) 7.03074e28 0.690069
\(334\) 0 0
\(335\) −1.55159e29 −1.42154
\(336\) 0 0
\(337\) −1.64138e29 −1.40432 −0.702160 0.712019i \(-0.747781\pi\)
−0.702160 + 0.712019i \(0.747781\pi\)
\(338\) 0 0
\(339\) 1.36547e29 1.09141
\(340\) 0 0
\(341\) −3.43892e28 −0.256890
\(342\) 0 0
\(343\) 1.53268e29 1.07045
\(344\) 0 0
\(345\) −1.46424e29 −0.956514
\(346\) 0 0
\(347\) 1.45586e29 0.889876 0.444938 0.895561i \(-0.353226\pi\)
0.444938 + 0.895561i \(0.353226\pi\)
\(348\) 0 0
\(349\) −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
\(352\) 0 0
\(353\) 2.57166e28 0.129064 0.0645320 0.997916i \(-0.479445\pi\)
0.0645320 + 0.997916i \(0.479445\pi\)
\(354\) 0 0
\(355\) 8.03037e28 0.377668
\(356\) 0 0
\(357\) −3.46792e29 −1.52892
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) −3.23118e29 −1.17608
\(364\) 0 0
\(365\) 3.67122e29 1.25442
\(366\) 0 0
\(367\) −1.62664e29 −0.521954 −0.260977 0.965345i \(-0.584045\pi\)
−0.260977 + 0.965345i \(0.584045\pi\)
\(368\) 0 0
\(369\) 7.94868e28 0.239603
\(370\) 0 0
\(371\) −2.91034e29 −0.824412
\(372\) 0 0
\(373\) −1.03540e29 −0.275712 −0.137856 0.990452i \(-0.544021\pi\)
−0.137856 + 0.990452i \(0.544021\pi\)
\(374\) 0 0
\(375\) 5.22535e29 1.30844
\(376\) 0 0
\(377\) 7.39468e28 0.174178
\(378\) 0 0
\(379\) 6.62210e29 1.46772 0.733862 0.679298i \(-0.237716\pi\)
0.733862 + 0.679298i \(0.237716\pi\)
\(380\) 0 0
\(381\) −9.01364e29 −1.88046
\(382\) 0 0
\(383\) −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
\(386\) 0 0
\(387\) 1.24157e28 0.0216419
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) 3.38267e29 0.494021
\(394\) 0 0
\(395\) 1.07531e30 1.48138
\(396\) 0 0
\(397\) −1.67462e28 −0.0217683 −0.0108842 0.999941i \(-0.503465\pi\)
−0.0108842 + 0.999941i \(0.503465\pi\)
\(398\) 0 0
\(399\) 9.22676e29 1.13204
\(400\) 0 0
\(401\) 3.87121e29 0.448421 0.224211 0.974541i \(-0.428020\pi\)
0.224211 + 0.974541i \(0.428020\pi\)
\(402\) 0 0
\(403\) −4.89658e29 −0.535654
\(404\) 0 0
\(405\) 1.15411e30 1.19266
\(406\) 0 0
\(407\) 3.12470e29 0.305121
\(408\) 0 0
\(409\) 5.00511e29 0.461950 0.230975 0.972960i \(-0.425808\pi\)
0.230975 + 0.972960i \(0.425808\pi\)
\(410\) 0 0
\(411\) 1.72418e30 1.50454
\(412\) 0 0
\(413\) 3.88316e27 0.00320450
\(414\) 0 0
\(415\) 8.10655e29 0.632824
\(416\) 0 0
\(417\) −1.93146e29 −0.142666
\(418\) 0 0
\(419\) 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 \(-0.436675\pi\)
0.197633 + 0.980276i \(0.436675\pi\)
\(422\) 0 0
\(423\) 4.92464e29 0.308646
\(424\) 0 0
\(425\) −2.06766e29 −0.122746
\(426\) 0 0
\(427\) 1.09538e30 0.616089
\(428\) 0 0
\(429\) 3.51794e29 0.187512
\(430\) 0 0
\(431\) −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 \(0.722324\pi\)
\(434\) 0 0
\(435\) −8.40388e29 −0.381815
\(436\) 0 0
\(437\) −2.23555e30 −0.963489
\(438\) 0 0
\(439\) −2.11168e29 −0.0863549 −0.0431775 0.999067i \(-0.513748\pi\)
−0.0431775 + 0.999067i \(0.513748\pi\)
\(440\) 0 0
\(441\) −7.28401e29 −0.282701
\(442\) 0 0
\(443\) −8.12623e29 −0.299396 −0.149698 0.988732i \(-0.547830\pi\)
−0.149698 + 0.988732i \(0.547830\pi\)
\(444\) 0 0
\(445\) −4.81728e29 −0.168524
\(446\) 0 0
\(447\) −4.29614e29 −0.142739
\(448\) 0 0
\(449\) −3.76596e30 −1.18862 −0.594310 0.804236i \(-0.702575\pi\)
−0.594310 + 0.804236i \(0.702575\pi\)
\(450\) 0 0
\(451\) 3.53267e29 0.105943
\(452\) 0 0
\(453\) −1.46173e30 −0.416619
\(454\) 0 0
\(455\) 1.43824e30 0.389674
\(456\) 0 0
\(457\) 2.14090e30 0.551518 0.275759 0.961227i \(-0.411071\pi\)
0.275759 + 0.961227i \(0.411071\pi\)
\(458\) 0 0
\(459\) −3.40199e30 −0.833465
\(460\) 0 0
\(461\) −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 \(-0.176021\pi\)
0.850959 + 0.525232i \(0.176021\pi\)
\(464\) 0 0
\(465\) 5.56485e30 1.17421
\(466\) 0 0
\(467\) −2.87884e30 −0.578194 −0.289097 0.957300i \(-0.593355\pi\)
−0.289097 + 0.957300i \(0.593355\pi\)
\(468\) 0 0
\(469\) −5.62992e30 −1.07650
\(470\) 0 0
\(471\) −5.83016e30 −1.06155
\(472\) 0 0
\(473\) 5.51796e28 0.00956919
\(474\) 0 0
\(475\) 5.50121e29 0.0908829
\(476\) 0 0
\(477\) 4.33221e30 0.681943
\(478\) 0 0
\(479\) −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
\(482\) 0 0
\(483\) −5.31297e30 −0.724344
\(484\) 0 0
\(485\) 1.19459e31 1.55306
\(486\) 0 0
\(487\) −8.59361e29 −0.106560 −0.0532798 0.998580i \(-0.516968\pi\)
−0.0532798 + 0.998580i \(0.516968\pi\)
\(488\) 0 0
\(489\) −1.01450e31 −1.20005
\(490\) 0 0
\(491\) −7.15140e30 −0.807149 −0.403575 0.914947i \(-0.632232\pi\)
−0.403575 + 0.914947i \(0.632232\pi\)
\(492\) 0 0
\(493\) 4.82250e30 0.519437
\(494\) 0 0
\(495\) −1.50358e30 −0.154585
\(496\) 0 0
\(497\) 2.91382e30 0.285998
\(498\) 0 0
\(499\) 1.77913e31 1.66744 0.833722 0.552185i \(-0.186206\pi\)
0.833722 + 0.552185i \(0.186206\pi\)
\(500\) 0 0
\(501\) −6.89133e30 −0.616835
\(502\) 0 0
\(503\) −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
\(506\) 0 0
\(507\) −1.12100e31 −0.875012
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) 9.05135e30 0.617112
\(514\) 0 0
\(515\) 1.48559e30 0.0968537
\(516\) 0 0
\(517\) 2.18868e30 0.136471
\(518\) 0 0
\(519\) −4.04407e31 −2.41209
\(520\) 0 0
\(521\) −2.26240e31 −1.29103 −0.645514 0.763749i \(-0.723357\pi\)
−0.645514 + 0.763749i \(0.723357\pi\)
\(522\) 0 0
\(523\) −3.41914e31 −1.86702 −0.933508 0.358557i \(-0.883269\pi\)
−0.933508 + 0.358557i \(0.883269\pi\)
\(524\) 0 0
\(525\) 1.30741e30 0.0683251
\(526\) 0 0
\(527\) −3.19334e31 −1.59744
\(528\) 0 0
\(529\) −8.00772e30 −0.383503
\(530\) 0 0
\(531\) −5.78030e28 −0.00265072
\(532\) 0 0
\(533\) 5.03006e30 0.220907
\(534\) 0 0
\(535\) −6.33494e30 −0.266486
\(536\) 0 0
\(537\) −3.88996e30 −0.156762
\(538\) 0 0
\(539\) −3.23726e30 −0.124999
\(540\) 0 0
\(541\) 2.75024e30 0.101766 0.0508829 0.998705i \(-0.483796\pi\)
0.0508829 + 0.998705i \(0.483796\pi\)
\(542\) 0 0
\(543\) 2.00892e31 0.712466
\(544\) 0 0
\(545\) −5.86424e30 −0.199366
\(546\) 0 0
\(547\) −1.03197e31 −0.336367 −0.168184 0.985756i \(-0.553790\pi\)
−0.168184 + 0.985756i \(0.553790\pi\)
\(548\) 0 0
\(549\) −1.63053e31 −0.509621
\(550\) 0 0
\(551\) −1.28308e31 −0.384600
\(552\) 0 0
\(553\) 3.90174e31 1.12181
\(554\) 0 0
\(555\) −5.05637e31 −1.39466
\(556\) 0 0
\(557\) −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
\(560\) 0 0
\(561\) 2.29425e31 0.559202
\(562\) 0 0
\(563\) 3.09091e31 0.723169 0.361585 0.932339i \(-0.382236\pi\)
0.361585 + 0.932339i \(0.382236\pi\)
\(564\) 0 0
\(565\) −3.69317e31 −0.829550
\(566\) 0 0
\(567\) 4.18769e31 0.903171
\(568\) 0 0
\(569\) −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 \(0.519799\pi\)
\(572\) 0 0
\(573\) −1.20467e32 −2.30192
\(574\) 0 0
\(575\) −3.16772e30 −0.0581522
\(576\) 0 0
\(577\) −1.59740e31 −0.281768 −0.140884 0.990026i \(-0.544994\pi\)
−0.140884 + 0.990026i \(0.544994\pi\)
\(578\) 0 0
\(579\) −4.97921e31 −0.844029
\(580\) 0 0
\(581\) 2.94146e31 0.479222
\(582\) 0 0
\(583\) 1.92538e31 0.301529
\(584\) 0 0
\(585\) −2.14091e31 −0.322333
\(586\) 0 0
\(587\) 8.16635e31 1.18220 0.591099 0.806599i \(-0.298694\pi\)
0.591099 + 0.806599i \(0.298694\pi\)
\(588\) 0 0
\(589\) 8.49622e31 1.18277
\(590\) 0 0
\(591\) 1.51481e32 2.02816
\(592\) 0 0
\(593\) −5.61370e30 −0.0722970 −0.0361485 0.999346i \(-0.511509\pi\)
−0.0361485 + 0.999346i \(0.511509\pi\)
\(594\) 0 0
\(595\) 9.37963e31 1.16209
\(596\) 0 0
\(597\) 4.87928e31 0.581635
\(598\) 0 0
\(599\) −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 \(0.345021\pi\)
\(602\) 0 0
\(603\) 8.38044e31 0.890467
\(604\) 0 0
\(605\) 8.73933e31 0.893905
\(606\) 0 0
\(607\) 1.63793e31 0.161296 0.0806481 0.996743i \(-0.474301\pi\)
0.0806481 + 0.996743i \(0.474301\pi\)
\(608\) 0 0
\(609\) −3.04934e31 −0.289139
\(610\) 0 0
\(611\) 3.11639e31 0.284563
\(612\) 0 0
\(613\) −1.93558e31 −0.170222 −0.0851110 0.996371i \(-0.527124\pi\)
−0.0851110 + 0.996371i \(0.527124\pi\)
\(614\) 0 0
\(615\) −5.71654e31 −0.484250
\(616\) 0 0
\(617\) 5.25618e31 0.428934 0.214467 0.976731i \(-0.431199\pi\)
0.214467 + 0.976731i \(0.431199\pi\)
\(618\) 0 0
\(619\) −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
\(622\) 0 0
\(623\) −1.74795e31 −0.127619
\(624\) 0 0
\(625\) −1.30804e32 −0.920452
\(626\) 0 0
\(627\) −6.10410e31 −0.414042
\(628\) 0 0
\(629\) 2.90156e32 1.89736
\(630\) 0 0
\(631\) 2.22784e32 1.40458 0.702288 0.711892i \(-0.252162\pi\)
0.702288 + 0.711892i \(0.252162\pi\)
\(632\) 0 0
\(633\) 2.28517e32 1.38923
\(634\) 0 0
\(635\) 2.43790e32 1.42929
\(636\) 0 0
\(637\) −4.60944e31 −0.260642
\(638\) 0 0
\(639\) −4.33738e31 −0.236574
\(640\) 0 0
\(641\) 1.38602e32 0.729295 0.364648 0.931146i \(-0.381189\pi\)
0.364648 + 0.931146i \(0.381189\pi\)
\(642\) 0 0
\(643\) 3.71401e31 0.188546 0.0942729 0.995546i \(-0.469947\pi\)
0.0942729 + 0.995546i \(0.469947\pi\)
\(644\) 0 0
\(645\) −8.92913e30 −0.0437393
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) 2.01920e32 0.889197
\(652\) 0 0
\(653\) 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
\(656\) 0 0
\(657\) −1.98290e32 −0.785777
\(658\) 0 0
\(659\) 1.46687e32 0.561319 0.280660 0.959807i \(-0.409447\pi\)
0.280660 + 0.959807i \(0.409447\pi\)
\(660\) 0 0
\(661\) −2.02066e32 −0.746752 −0.373376 0.927680i \(-0.621800\pi\)
−0.373376 + 0.927680i \(0.621800\pi\)
\(662\) 0 0
\(663\) 3.26672e32 1.16602
\(664\) 0 0
\(665\) −2.49555e32 −0.860432
\(666\) 0 0
\(667\) 7.38823e31 0.246090
\(668\) 0 0
\(669\) −5.39243e32 −1.73534
\(670\) 0 0
\(671\) −7.24664e31 −0.225334
\(672\) 0 0
\(673\) 5.34692e32 1.60668 0.803341 0.595519i \(-0.203053\pi\)
0.803341 + 0.595519i \(0.203053\pi\)
\(674\) 0 0
\(675\) 1.28256e31 0.0372463
\(676\) 0 0
\(677\) −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
\(680\) 0 0
\(681\) −8.34995e32 −2.19024
\(682\) 0 0
\(683\) −2.97297e32 −0.753967 −0.376983 0.926220i \(-0.623039\pi\)
−0.376983 + 0.926220i \(0.623039\pi\)
\(684\) 0 0
\(685\) −4.66337e32 −1.14356
\(686\) 0 0
\(687\) 2.48547e32 0.589394
\(688\) 0 0
\(689\) 2.74149e32 0.628733
\(690\) 0 0
\(691\) −7.18838e32 −1.59453 −0.797266 0.603628i \(-0.793721\pi\)
−0.797266 + 0.603628i \(0.793721\pi\)
\(692\) 0 0
\(693\) −5.45574e31 −0.117063
\(694\) 0 0
\(695\) 5.22399e31 0.108437
\(696\) 0 0
\(697\) 3.28039e32 0.658793
\(698\) 0 0
\(699\) 8.51274e32 1.65418
\(700\) 0 0
\(701\) 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
\(704\) 0 0
\(705\) −3.54171e32 −0.623790
\(706\) 0 0
\(707\) 5.19380e32 0.885446
\(708\) 0 0
\(709\) 2.50942e32 0.414134 0.207067 0.978327i \(-0.433608\pi\)
0.207067 + 0.978327i \(0.433608\pi\)
\(710\) 0 0
\(711\) −5.80796e32 −0.927947
\(712\) 0 0
\(713\) −4.89231e32 −0.756805
\(714\) 0 0
\(715\) −9.51492e31 −0.142523
\(716\) 0 0
\(717\) −4.27071e32 −0.619483
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) −5.61891e32 −0.740562
\(724\) 0 0
\(725\) −1.81809e31 −0.0232128
\(726\) 0 0
\(727\) −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
\(730\) 0 0
\(731\) 5.12391e31 0.0595047
\(732\) 0 0
\(733\) 1.54780e33 1.74188 0.870941 0.491387i \(-0.163510\pi\)
0.870941 + 0.491387i \(0.163510\pi\)
\(734\) 0 0
\(735\) 5.23852e32 0.571353
\(736\) 0 0
\(737\) 3.72455e32 0.393730
\(738\) 0 0
\(739\) −1.49249e33 −1.52932 −0.764662 0.644431i \(-0.777094\pi\)
−0.764662 + 0.644431i \(0.777094\pi\)
\(740\) 0 0
\(741\) −8.69144e32 −0.863341
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) −4.37852e32 −0.396406
\(748\) 0 0
\(749\) −2.29863e32 −0.201803
\(750\) 0 0
\(751\) −1.42812e33 −1.21592 −0.607959 0.793969i \(-0.708011\pi\)
−0.607959 + 0.793969i \(0.708011\pi\)
\(752\) 0 0
\(753\) 8.71877e31 0.0719967
\(754\) 0 0
\(755\) 3.95352e32 0.316661
\(756\) 0 0
\(757\) 1.56208e33 1.21367 0.606835 0.794828i \(-0.292439\pi\)
0.606835 + 0.794828i \(0.292439\pi\)
\(758\) 0 0
\(759\) 3.51487e32 0.264929
\(760\) 0 0
\(761\) 1.92115e33 1.40488 0.702439 0.711744i \(-0.252094\pi\)
0.702439 + 0.711744i \(0.252094\pi\)
\(762\) 0 0
\(763\) −2.12783e32 −0.150975
\(764\) 0 0
\(765\) −1.39621e33 −0.961267
\(766\) 0 0
\(767\) −3.65787e30 −0.00244389
\(768\) 0 0
\(769\) 1.22615e33 0.795044 0.397522 0.917593i \(-0.369870\pi\)
0.397522 + 0.917593i \(0.369870\pi\)
\(770\) 0 0
\(771\) −1.53438e33 −0.965621
\(772\) 0 0
\(773\) 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
\(776\) 0 0
\(777\) −1.83470e33 −1.05615
\(778\) 0 0
\(779\) −8.72782e32 −0.487781
\(780\) 0 0
\(781\) −1.92768e32 −0.104604
\(782\) 0 0
\(783\) −2.99137e32 −0.157620
\(784\) 0 0
\(785\) 1.57687e33 0.806856
\(786\) 0 0
\(787\) 1.37027e33 0.680919 0.340460 0.940259i \(-0.389417\pi\)
0.340460 + 0.940259i \(0.389417\pi\)
\(788\) 0 0
\(789\) −7.82797e32 −0.377801
\(790\) 0 0
\(791\) −1.34007e33 −0.628198
\(792\) 0 0
\(793\) −1.03183e33 −0.469857
\(794\) 0 0
\(795\) −3.11564e33 −1.37824
\(796\) 0 0
\(797\) −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
\(800\) 0 0
\(801\) 2.60192e32 0.105565
\(802\) 0 0
\(803\) −8.81269e32 −0.347440
\(804\) 0 0
\(805\) 1.43699e33 0.550555
\(806\) 0 0
\(807\) −2.22215e33 −0.827423
\(808\) 0 0
\(809\) −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 \(0.428999\pi\)
\(812\) 0 0
\(813\) −2.01234e33 −0.688112
\(814\) 0 0
\(815\) 2.74390e33 0.912129
\(816\) 0 0
\(817\) −1.36327e32 −0.0440583
\(818\) 0 0
\(819\) −7.76826e32 −0.244095
\(820\) 0 0
\(821\) 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 \(-0.832830\pi\)
−0.865233 + 0.501370i \(0.832830\pi\)
\(824\) 0 0
\(825\) −8.64937e31 −0.0249899
\(826\) 0 0
\(827\) −3.61128e33 −1.01472 −0.507362 0.861733i \(-0.669379\pi\)
−0.507362 + 0.861733i \(0.669379\pi\)
\(828\) 0 0
\(829\) −6.27935e33 −1.71608 −0.858040 0.513584i \(-0.828318\pi\)
−0.858040 + 0.513584i \(0.828318\pi\)
\(830\) 0 0
\(831\) −5.03467e33 −1.33832
\(832\) 0 0
\(833\) −3.00609e33 −0.777292
\(834\) 0 0
\(835\) 1.86389e33 0.468841
\(836\) 0 0
\(837\) 1.98081e33 0.484731
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) 4.82269e33 1.08711
\(844\) 0 0
\(845\) 3.03195e33 0.665074
\(846\) 0 0
\(847\) 3.17106e33 0.676932
\(848\) 0 0
\(849\) −8.35275e33 −1.73537
\(850\) 0 0
\(851\) 4.44528e33 0.898896
\(852\) 0 0
\(853\) 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
\(856\) 0 0
\(857\) 6.17423e33 1.15160 0.575802 0.817589i \(-0.304690\pi\)
0.575802 + 0.817589i \(0.304690\pi\)
\(858\) 0 0
\(859\) −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
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 1.35479e34 2.21132
\(868\) 0 0
\(869\) −2.58126e33 −0.410302
\(870\) 0 0
\(871\) 5.30328e33 0.820986
\(872\) 0 0
\(873\) −6.45224e33 −0.972852
\(874\) 0 0
\(875\) −5.12811e33 −0.753121
\(876\) 0 0
\(877\) 6.73800e32 0.0963908 0.0481954 0.998838i \(-0.484653\pi\)
0.0481954 + 0.998838i \(0.484653\pi\)
\(878\) 0 0
\(879\) 4.67328e33 0.651253
\(880\) 0 0
\(881\) 1.36978e34 1.85963 0.929815 0.368028i \(-0.119967\pi\)
0.929815 + 0.368028i \(0.119967\pi\)
\(882\) 0 0
\(883\) 1.49590e33 0.197859 0.0989294 0.995094i \(-0.468458\pi\)
0.0989294 + 0.995094i \(0.468458\pi\)
\(884\) 0 0
\(885\) 4.15708e31 0.00535724
\(886\) 0 0
\(887\) 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
\(890\) 0 0
\(891\) −2.77043e33 −0.330334
\(892\) 0 0
\(893\) −5.40736e33 −0.628339
\(894\) 0 0
\(895\) 1.05211e33 0.119151
\(896\) 0 0
\(897\) 5.00472e33 0.552417
\(898\) 0 0
\(899\) −2.80791e33 −0.302097
\(900\) 0 0
\(901\) 1.78789e34 1.87502
\(902\) 0 0
\(903\) −3.23993e32 −0.0331227
\(904\) 0 0
\(905\) −5.43349e33 −0.541527
\(906\) 0 0
\(907\) 6.42282e33 0.624083 0.312041 0.950069i \(-0.398987\pi\)
0.312041 + 0.950069i \(0.398987\pi\)
\(908\) 0 0
\(909\) −7.73126e33 −0.732430
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) 1.17265e34 1.02997
\(916\) 0 0
\(917\) −3.31973e33 −0.284351
\(918\) 0 0
\(919\) −3.10701e33 −0.259546 −0.129773 0.991544i \(-0.541425\pi\)
−0.129773 + 0.991544i \(0.541425\pi\)
\(920\) 0 0
\(921\) −1.82735e34 −1.48880
\(922\) 0 0
\(923\) −2.74476e33 −0.218115
\(924\) 0 0
\(925\) −1.09389e33 −0.0847900
\(926\) 0 0
\(927\) −8.02400e32 −0.0606700
\(928\) 0 0
\(929\) 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
\(932\) 0 0
\(933\) 3.49473e32 0.0245343
\(934\) 0 0
\(935\) −6.20523e33 −0.425035
\(936\) 0 0
\(937\) −1.61548e34 −1.07969 −0.539843 0.841766i \(-0.681516\pi\)
−0.539843 + 0.841766i \(0.681516\pi\)
\(938\) 0 0
\(939\) −4.40980e33 −0.287584
\(940\) 0 0
\(941\) −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
\(944\) 0 0
\(945\) −5.81813e33 −0.352629
\(946\) 0 0
\(947\) −2.94332e34 −1.74106 −0.870528 0.492119i \(-0.836222\pi\)
−0.870528 + 0.492119i \(0.836222\pi\)
\(948\) 0 0
\(949\) −1.25481e34 −0.724465
\(950\) 0 0
\(951\) −5.35651e33 −0.301860
\(952\) 0 0
\(953\) 1.97292e34 1.08528 0.542641 0.839965i \(-0.317425\pi\)
0.542641 + 0.839965i \(0.317425\pi\)
\(954\) 0 0
\(955\) 3.25825e34 1.74963
\(956\) 0 0
\(957\) 2.01734e33 0.105753
\(958\) 0 0
\(959\) −1.69210e34 −0.865989
\(960\) 0 0
\(961\) −1.42003e33 −0.0709542
\(962\) 0 0
\(963\) 3.42163e33 0.166929
\(964\) 0 0
\(965\) 1.34672e34 0.641524
\(966\) 0 0
\(967\) 2.03266e34 0.945496 0.472748 0.881198i \(-0.343262\pi\)
0.472748 + 0.881198i \(0.343262\pi\)
\(968\) 0 0
\(969\) −5.66820e34 −2.57467
\(970\) 0 0
\(971\) 3.40470e34 1.51028 0.755141 0.655563i \(-0.227569\pi\)
0.755141 + 0.655563i \(0.227569\pi\)
\(972\) 0 0
\(973\) 1.89552e33 0.0821166
\(974\) 0 0
\(975\) −1.23156e33 −0.0521077
\(976\) 0 0
\(977\) −3.68167e32 −0.0152145 −0.00760725 0.999971i \(-0.502421\pi\)
−0.00760725 + 0.999971i \(0.502421\pi\)
\(978\) 0 0
\(979\) 1.15638e33 0.0466767
\(980\) 0 0
\(981\) 3.16740e33 0.124885
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) −1.28511e34 −0.472380
\(988\) 0 0
\(989\) 7.85000e32 0.0281911
\(990\) 0 0
\(991\) −2.70349e33 −0.0948590 −0.0474295 0.998875i \(-0.515103\pi\)
−0.0474295 + 0.998875i \(0.515103\pi\)
\(992\) 0 0
\(993\) 6.31875e34 2.16628
\(994\) 0 0
\(995\) −1.31969e34 −0.442086
\(996\) 0 0
\(997\) 4.64692e33 0.152115 0.0760573 0.997103i \(-0.475767\pi\)
0.0760573 + 0.997103i \(0.475767\pi\)
\(998\) 0 0
\(999\) −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 64.24.a.g.1.2 2
4.3 odd 2 64.24.a.d.1.1 2
8.3 odd 2 1.24.a.a.1.1 2
8.5 even 2 16.24.a.b.1.1 2
24.11 even 2 9.24.a.b.1.2 2
40.3 even 4 25.24.b.a.24.3 4
40.19 odd 2 25.24.a.a.1.2 2
40.27 even 4 25.24.b.a.24.2 4
56.27 even 2 49.24.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.24.a.a.1.1 2 8.3 odd 2
9.24.a.b.1.2 2 24.11 even 2
16.24.a.b.1.1 2 8.5 even 2
25.24.a.a.1.2 2 40.19 odd 2
25.24.b.a.24.2 4 40.27 even 4
25.24.b.a.24.3 4 40.3 even 4
49.24.a.b.1.1 2 56.27 even 2
64.24.a.d.1.1 2 4.3 odd 2
64.24.a.g.1.2 2 1.1 even 1 trivial