Properties

Label 150.12.a.a.1.1
Level $150$
Weight $12$
Character 150.1
Self dual yes
Analytic conductor $115.251$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,12,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(115.251477084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{2} -243.000 q^{3} +1024.00 q^{4} +7776.00 q^{6} -32936.0 q^{7} -32768.0 q^{8} +59049.0 q^{9} -758748. q^{11} -248832. q^{12} +2.48286e6 q^{13} +1.05395e6 q^{14} +1.04858e6 q^{16} -8.29039e6 q^{17} -1.88957e6 q^{18} -1.08673e7 q^{19} +8.00345e6 q^{21} +2.42799e7 q^{22} -2.05393e7 q^{23} +7.96262e6 q^{24} -7.94515e7 q^{26} -1.43489e7 q^{27} -3.37265e7 q^{28} +2.88146e7 q^{29} +1.50501e8 q^{31} -3.35544e7 q^{32} +1.84376e8 q^{33} +2.65292e8 q^{34} +6.04662e7 q^{36} +3.19892e8 q^{37} +3.47754e8 q^{38} -6.03334e8 q^{39} -3.68009e8 q^{41} -2.56110e8 q^{42} -6.20470e8 q^{43} -7.76958e8 q^{44} +6.57257e8 q^{46} -2.76311e9 q^{47} -2.54804e8 q^{48} -8.92547e8 q^{49} +2.01456e9 q^{51} +2.54245e9 q^{52} +2.68284e8 q^{53} +4.59165e8 q^{54} +1.07925e9 q^{56} +2.64075e9 q^{57} -9.22066e8 q^{58} +1.67289e9 q^{59} -7.78720e9 q^{61} -4.81604e9 q^{62} -1.94484e9 q^{63} +1.07374e9 q^{64} -5.90002e9 q^{66} -1.87067e10 q^{67} -8.48936e9 q^{68} +4.99104e9 q^{69} -8.34699e9 q^{71} -1.93492e9 q^{72} -1.96417e10 q^{73} -1.02365e10 q^{74} -1.11281e10 q^{76} +2.49901e10 q^{77} +1.93067e10 q^{78} -5.87381e9 q^{79} +3.48678e9 q^{81} +1.17763e10 q^{82} -8.49256e9 q^{83} +8.19553e9 q^{84} +1.98550e10 q^{86} -7.00194e9 q^{87} +2.48627e10 q^{88} +7.55279e10 q^{89} -8.17754e10 q^{91} -2.10322e10 q^{92} -3.65718e10 q^{93} +8.84195e10 q^{94} +8.15373e9 q^{96} +8.23568e10 q^{97} +2.85615e10 q^{98} -4.48033e10 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32.0000 −0.707107
\(3\) −243.000 −0.577350
\(4\) 1024.00 0.500000
\(5\) 0 0
\(6\) 7776.00 0.408248
\(7\) −32936.0 −0.740682 −0.370341 0.928896i \(-0.620759\pi\)
−0.370341 + 0.928896i \(0.620759\pi\)
\(8\) −32768.0 −0.353553
\(9\) 59049.0 0.333333
\(10\) 0 0
\(11\) −758748. −1.42049 −0.710244 0.703955i \(-0.751416\pi\)
−0.710244 + 0.703955i \(0.751416\pi\)
\(12\) −248832. −0.288675
\(13\) 2.48286e6 1.85466 0.927328 0.374249i \(-0.122100\pi\)
0.927328 + 0.374249i \(0.122100\pi\)
\(14\) 1.05395e6 0.523741
\(15\) 0 0
\(16\) 1.04858e6 0.250000
\(17\) −8.29039e6 −1.41614 −0.708069 0.706143i \(-0.750434\pi\)
−0.708069 + 0.706143i \(0.750434\pi\)
\(18\) −1.88957e6 −0.235702
\(19\) −1.08673e7 −1.00688 −0.503439 0.864031i \(-0.667932\pi\)
−0.503439 + 0.864031i \(0.667932\pi\)
\(20\) 0 0
\(21\) 8.00345e6 0.427633
\(22\) 2.42799e7 1.00444
\(23\) −2.05393e7 −0.665399 −0.332699 0.943033i \(-0.607959\pi\)
−0.332699 + 0.943033i \(0.607959\pi\)
\(24\) 7.96262e6 0.204124
\(25\) 0 0
\(26\) −7.94515e7 −1.31144
\(27\) −1.43489e7 −0.192450
\(28\) −3.37265e7 −0.370341
\(29\) 2.88146e7 0.260869 0.130435 0.991457i \(-0.458363\pi\)
0.130435 + 0.991457i \(0.458363\pi\)
\(30\) 0 0
\(31\) 1.50501e8 0.944172 0.472086 0.881552i \(-0.343501\pi\)
0.472086 + 0.881552i \(0.343501\pi\)
\(32\) −3.35544e7 −0.176777
\(33\) 1.84376e8 0.820120
\(34\) 2.65292e8 1.00136
\(35\) 0 0
\(36\) 6.04662e7 0.166667
\(37\) 3.19892e8 0.758392 0.379196 0.925316i \(-0.376201\pi\)
0.379196 + 0.925316i \(0.376201\pi\)
\(38\) 3.47754e8 0.711970
\(39\) −6.03334e8 −1.07079
\(40\) 0 0
\(41\) −3.68009e8 −0.496075 −0.248037 0.968750i \(-0.579786\pi\)
−0.248037 + 0.968750i \(0.579786\pi\)
\(42\) −2.56110e8 −0.302382
\(43\) −6.20470e8 −0.643641 −0.321821 0.946801i \(-0.604295\pi\)
−0.321821 + 0.946801i \(0.604295\pi\)
\(44\) −7.76958e8 −0.710244
\(45\) 0 0
\(46\) 6.57257e8 0.470508
\(47\) −2.76311e9 −1.75736 −0.878679 0.477414i \(-0.841574\pi\)
−0.878679 + 0.477414i \(0.841574\pi\)
\(48\) −2.54804e8 −0.144338
\(49\) −8.92547e8 −0.451391
\(50\) 0 0
\(51\) 2.01456e9 0.817608
\(52\) 2.54245e9 0.927328
\(53\) 2.68284e8 0.0881207 0.0440603 0.999029i \(-0.485971\pi\)
0.0440603 + 0.999029i \(0.485971\pi\)
\(54\) 4.59165e8 0.136083
\(55\) 0 0
\(56\) 1.07925e9 0.261871
\(57\) 2.64075e9 0.581321
\(58\) −9.22066e8 −0.184462
\(59\) 1.67289e9 0.304637 0.152318 0.988331i \(-0.451326\pi\)
0.152318 + 0.988331i \(0.451326\pi\)
\(60\) 0 0
\(61\) −7.78720e9 −1.18050 −0.590252 0.807219i \(-0.700971\pi\)
−0.590252 + 0.807219i \(0.700971\pi\)
\(62\) −4.81604e9 −0.667631
\(63\) −1.94484e9 −0.246894
\(64\) 1.07374e9 0.125000
\(65\) 0 0
\(66\) −5.90002e9 −0.579912
\(67\) −1.87067e10 −1.69272 −0.846361 0.532610i \(-0.821211\pi\)
−0.846361 + 0.532610i \(0.821211\pi\)
\(68\) −8.48936e9 −0.708069
\(69\) 4.99104e9 0.384168
\(70\) 0 0
\(71\) −8.34699e9 −0.549046 −0.274523 0.961580i \(-0.588520\pi\)
−0.274523 + 0.961580i \(0.588520\pi\)
\(72\) −1.93492e9 −0.117851
\(73\) −1.96417e10 −1.10893 −0.554465 0.832207i \(-0.687077\pi\)
−0.554465 + 0.832207i \(0.687077\pi\)
\(74\) −1.02365e10 −0.536264
\(75\) 0 0
\(76\) −1.11281e10 −0.503439
\(77\) 2.49901e10 1.05213
\(78\) 1.93067e10 0.757160
\(79\) −5.87381e9 −0.214769 −0.107384 0.994218i \(-0.534248\pi\)
−0.107384 + 0.994218i \(0.534248\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) 1.17763e10 0.350778
\(83\) −8.49256e9 −0.236651 −0.118326 0.992975i \(-0.537753\pi\)
−0.118326 + 0.992975i \(0.537753\pi\)
\(84\) 8.19553e9 0.213816
\(85\) 0 0
\(86\) 1.98550e10 0.455123
\(87\) −7.00194e9 −0.150613
\(88\) 2.48627e10 0.502219
\(89\) 7.55279e10 1.43371 0.716856 0.697221i \(-0.245580\pi\)
0.716856 + 0.697221i \(0.245580\pi\)
\(90\) 0 0
\(91\) −8.17754e10 −1.37371
\(92\) −2.10322e10 −0.332699
\(93\) −3.65718e10 −0.545118
\(94\) 8.84195e10 1.24264
\(95\) 0 0
\(96\) 8.15373e9 0.102062
\(97\) 8.23568e10 0.973767 0.486883 0.873467i \(-0.338134\pi\)
0.486883 + 0.873467i \(0.338134\pi\)
\(98\) 2.85615e10 0.319181
\(99\) −4.48033e10 −0.473496
\(100\) 0 0
\(101\) −4.13141e10 −0.391138 −0.195569 0.980690i \(-0.562655\pi\)
−0.195569 + 0.980690i \(0.562655\pi\)
\(102\) −6.44660e10 −0.578136
\(103\) 6.45183e10 0.548376 0.274188 0.961676i \(-0.411591\pi\)
0.274188 + 0.961676i \(0.411591\pi\)
\(104\) −8.13583e10 −0.655720
\(105\) 0 0
\(106\) −8.58510e9 −0.0623107
\(107\) −1.10219e11 −0.759706 −0.379853 0.925047i \(-0.624025\pi\)
−0.379853 + 0.925047i \(0.624025\pi\)
\(108\) −1.46933e10 −0.0962250
\(109\) 1.77778e11 1.10671 0.553353 0.832947i \(-0.313348\pi\)
0.553353 + 0.832947i \(0.313348\pi\)
\(110\) 0 0
\(111\) −7.77337e10 −0.437858
\(112\) −3.45359e10 −0.185170
\(113\) −7.89206e10 −0.402957 −0.201479 0.979493i \(-0.564575\pi\)
−0.201479 + 0.979493i \(0.564575\pi\)
\(114\) −8.45041e10 −0.411056
\(115\) 0 0
\(116\) 2.95061e10 0.130435
\(117\) 1.46610e11 0.618219
\(118\) −5.35326e10 −0.215411
\(119\) 2.73052e11 1.04891
\(120\) 0 0
\(121\) 2.90387e11 1.01779
\(122\) 2.49190e11 0.834742
\(123\) 8.94262e10 0.286409
\(124\) 1.54113e11 0.472086
\(125\) 0 0
\(126\) 6.22348e10 0.174580
\(127\) 2.76649e11 0.743035 0.371518 0.928426i \(-0.378838\pi\)
0.371518 + 0.928426i \(0.378838\pi\)
\(128\) −3.43597e10 −0.0883883
\(129\) 1.50774e11 0.371607
\(130\) 0 0
\(131\) −3.40872e11 −0.771968 −0.385984 0.922505i \(-0.626138\pi\)
−0.385984 + 0.922505i \(0.626138\pi\)
\(132\) 1.88801e11 0.410060
\(133\) 3.57925e11 0.745776
\(134\) 5.98614e11 1.19694
\(135\) 0 0
\(136\) 2.71659e11 0.500681
\(137\) 9.44784e11 1.67251 0.836256 0.548339i \(-0.184740\pi\)
0.836256 + 0.548339i \(0.184740\pi\)
\(138\) −1.59713e11 −0.271648
\(139\) 1.03770e11 0.169626 0.0848128 0.996397i \(-0.472971\pi\)
0.0848128 + 0.996397i \(0.472971\pi\)
\(140\) 0 0
\(141\) 6.71436e11 1.01461
\(142\) 2.67104e11 0.388234
\(143\) −1.88386e12 −2.63452
\(144\) 6.19174e10 0.0833333
\(145\) 0 0
\(146\) 6.28536e11 0.784132
\(147\) 2.16889e11 0.260610
\(148\) 3.27569e11 0.379196
\(149\) −1.22852e12 −1.37043 −0.685216 0.728340i \(-0.740292\pi\)
−0.685216 + 0.728340i \(0.740292\pi\)
\(150\) 0 0
\(151\) 2.12466e11 0.220250 0.110125 0.993918i \(-0.464875\pi\)
0.110125 + 0.993918i \(0.464875\pi\)
\(152\) 3.56100e11 0.355985
\(153\) −4.89539e11 −0.472046
\(154\) −7.99684e11 −0.743968
\(155\) 0 0
\(156\) −6.17815e11 −0.535393
\(157\) 1.58632e12 1.32722 0.663612 0.748077i \(-0.269023\pi\)
0.663612 + 0.748077i \(0.269023\pi\)
\(158\) 1.87962e11 0.151864
\(159\) −6.51931e10 −0.0508765
\(160\) 0 0
\(161\) 6.76481e11 0.492849
\(162\) −1.11577e11 −0.0785674
\(163\) −8.65020e11 −0.588836 −0.294418 0.955677i \(-0.595126\pi\)
−0.294418 + 0.955677i \(0.595126\pi\)
\(164\) −3.76841e11 −0.248037
\(165\) 0 0
\(166\) 2.71762e11 0.167338
\(167\) 2.25802e12 1.34520 0.672600 0.740006i \(-0.265177\pi\)
0.672600 + 0.740006i \(0.265177\pi\)
\(168\) −2.62257e11 −0.151191
\(169\) 4.37242e12 2.43975
\(170\) 0 0
\(171\) −6.41703e11 −0.335626
\(172\) −6.35361e11 −0.321821
\(173\) 2.15960e12 1.05955 0.529773 0.848139i \(-0.322277\pi\)
0.529773 + 0.848139i \(0.322277\pi\)
\(174\) 2.24062e11 0.106499
\(175\) 0 0
\(176\) −7.95605e11 −0.355122
\(177\) −4.06513e11 −0.175882
\(178\) −2.41689e12 −1.01379
\(179\) 4.50786e12 1.83349 0.916745 0.399472i \(-0.130807\pi\)
0.916745 + 0.399472i \(0.130807\pi\)
\(180\) 0 0
\(181\) 4.34053e11 0.166077 0.0830387 0.996546i \(-0.473537\pi\)
0.0830387 + 0.996546i \(0.473537\pi\)
\(182\) 2.61681e12 0.971360
\(183\) 1.89229e12 0.681564
\(184\) 6.73031e11 0.235254
\(185\) 0 0
\(186\) 1.17030e12 0.385457
\(187\) 6.29031e12 2.01161
\(188\) −2.82942e12 −0.878679
\(189\) 4.72596e11 0.142544
\(190\) 0 0
\(191\) 8.67702e11 0.246994 0.123497 0.992345i \(-0.460589\pi\)
0.123497 + 0.992345i \(0.460589\pi\)
\(192\) −2.60919e11 −0.0721688
\(193\) −1.86946e11 −0.0502516 −0.0251258 0.999684i \(-0.507999\pi\)
−0.0251258 + 0.999684i \(0.507999\pi\)
\(194\) −2.63542e12 −0.688557
\(195\) 0 0
\(196\) −9.13968e11 −0.225695
\(197\) −3.32798e12 −0.799129 −0.399565 0.916705i \(-0.630839\pi\)
−0.399565 + 0.916705i \(0.630839\pi\)
\(198\) 1.43371e12 0.334812
\(199\) −1.58581e12 −0.360212 −0.180106 0.983647i \(-0.557644\pi\)
−0.180106 + 0.983647i \(0.557644\pi\)
\(200\) 0 0
\(201\) 4.54573e12 0.977293
\(202\) 1.32205e12 0.276577
\(203\) −9.49036e11 −0.193221
\(204\) 2.06291e12 0.408804
\(205\) 0 0
\(206\) −2.06459e12 −0.387760
\(207\) −1.21282e12 −0.221800
\(208\) 2.60347e12 0.463664
\(209\) 8.24554e12 1.43026
\(210\) 0 0
\(211\) 3.54958e12 0.584283 0.292142 0.956375i \(-0.405632\pi\)
0.292142 + 0.956375i \(0.405632\pi\)
\(212\) 2.74723e11 0.0440603
\(213\) 2.02832e12 0.316992
\(214\) 3.52701e12 0.537193
\(215\) 0 0
\(216\) 4.70185e11 0.0680414
\(217\) −4.95691e12 −0.699331
\(218\) −5.68890e12 −0.782560
\(219\) 4.77294e12 0.640241
\(220\) 0 0
\(221\) −2.05839e13 −2.62645
\(222\) 2.48748e12 0.309612
\(223\) 5.18022e12 0.629030 0.314515 0.949252i \(-0.398158\pi\)
0.314515 + 0.949252i \(0.398158\pi\)
\(224\) 1.10515e12 0.130935
\(225\) 0 0
\(226\) 2.52546e12 0.284934
\(227\) 1.33665e13 1.47189 0.735943 0.677044i \(-0.236739\pi\)
0.735943 + 0.677044i \(0.236739\pi\)
\(228\) 2.70413e12 0.290661
\(229\) −2.03080e12 −0.213095 −0.106547 0.994308i \(-0.533980\pi\)
−0.106547 + 0.994308i \(0.533980\pi\)
\(230\) 0 0
\(231\) −6.07260e12 −0.607448
\(232\) −9.44195e11 −0.0922312
\(233\) −7.13322e12 −0.680500 −0.340250 0.940335i \(-0.610512\pi\)
−0.340250 + 0.940335i \(0.610512\pi\)
\(234\) −4.69153e12 −0.437147
\(235\) 0 0
\(236\) 1.71304e12 0.152318
\(237\) 1.42734e12 0.123997
\(238\) −8.73767e12 −0.741690
\(239\) 3.30363e12 0.274033 0.137016 0.990569i \(-0.456249\pi\)
0.137016 + 0.990569i \(0.456249\pi\)
\(240\) 0 0
\(241\) 4.16938e11 0.0330353 0.0165176 0.999864i \(-0.494742\pi\)
0.0165176 + 0.999864i \(0.494742\pi\)
\(242\) −9.29238e12 −0.719685
\(243\) −8.47289e11 −0.0641500
\(244\) −7.97409e12 −0.590252
\(245\) 0 0
\(246\) −2.86164e12 −0.202522
\(247\) −2.69820e13 −1.86741
\(248\) −4.93163e12 −0.333815
\(249\) 2.06369e12 0.136631
\(250\) 0 0
\(251\) −1.91970e12 −0.121626 −0.0608131 0.998149i \(-0.519369\pi\)
−0.0608131 + 0.998149i \(0.519369\pi\)
\(252\) −1.99151e12 −0.123447
\(253\) 1.55841e13 0.945191
\(254\) −8.85278e12 −0.525405
\(255\) 0 0
\(256\) 1.09951e12 0.0625000
\(257\) −9.76031e12 −0.543040 −0.271520 0.962433i \(-0.587526\pi\)
−0.271520 + 0.962433i \(0.587526\pi\)
\(258\) −4.82477e12 −0.262766
\(259\) −1.05360e13 −0.561727
\(260\) 0 0
\(261\) 1.70147e12 0.0869564
\(262\) 1.09079e13 0.545864
\(263\) 4.04845e12 0.198396 0.0991980 0.995068i \(-0.468372\pi\)
0.0991980 + 0.995068i \(0.468372\pi\)
\(264\) −6.04163e12 −0.289956
\(265\) 0 0
\(266\) −1.14536e13 −0.527343
\(267\) −1.83533e13 −0.827754
\(268\) −1.91557e13 −0.846361
\(269\) −1.13455e13 −0.491118 −0.245559 0.969382i \(-0.578972\pi\)
−0.245559 + 0.969382i \(0.578972\pi\)
\(270\) 0 0
\(271\) −2.35252e12 −0.0977691 −0.0488846 0.998804i \(-0.515567\pi\)
−0.0488846 + 0.998804i \(0.515567\pi\)
\(272\) −8.69310e12 −0.354035
\(273\) 1.98714e13 0.793112
\(274\) −3.02331e13 −1.18264
\(275\) 0 0
\(276\) 5.11083e12 0.192084
\(277\) 4.36327e13 1.60758 0.803791 0.594912i \(-0.202813\pi\)
0.803791 + 0.594912i \(0.202813\pi\)
\(278\) −3.32065e12 −0.119943
\(279\) 8.88696e12 0.314724
\(280\) 0 0
\(281\) 2.75099e13 0.936707 0.468353 0.883541i \(-0.344847\pi\)
0.468353 + 0.883541i \(0.344847\pi\)
\(282\) −2.14859e13 −0.717438
\(283\) 2.57815e13 0.844273 0.422136 0.906532i \(-0.361280\pi\)
0.422136 + 0.906532i \(0.361280\pi\)
\(284\) −8.54732e12 −0.274523
\(285\) 0 0
\(286\) 6.02836e13 1.86289
\(287\) 1.21207e13 0.367434
\(288\) −1.98136e12 −0.0589256
\(289\) 3.44586e13 1.00545
\(290\) 0 0
\(291\) −2.00127e13 −0.562204
\(292\) −2.01131e13 −0.554465
\(293\) −5.59324e13 −1.51318 −0.756591 0.653888i \(-0.773137\pi\)
−0.756591 + 0.653888i \(0.773137\pi\)
\(294\) −6.94044e12 −0.184279
\(295\) 0 0
\(296\) −1.04822e13 −0.268132
\(297\) 1.08872e13 0.273373
\(298\) 3.93126e13 0.969041
\(299\) −5.09961e13 −1.23409
\(300\) 0 0
\(301\) 2.04358e13 0.476733
\(302\) −6.79890e12 −0.155740
\(303\) 1.00393e13 0.225824
\(304\) −1.13952e13 −0.251719
\(305\) 0 0
\(306\) 1.56652e13 0.333787
\(307\) −5.68056e13 −1.18886 −0.594429 0.804148i \(-0.702622\pi\)
−0.594429 + 0.804148i \(0.702622\pi\)
\(308\) 2.55899e13 0.526065
\(309\) −1.56779e13 −0.316605
\(310\) 0 0
\(311\) −6.71193e13 −1.30817 −0.654087 0.756419i \(-0.726947\pi\)
−0.654087 + 0.756419i \(0.726947\pi\)
\(312\) 1.97701e13 0.378580
\(313\) 3.66685e13 0.689920 0.344960 0.938617i \(-0.387892\pi\)
0.344960 + 0.938617i \(0.387892\pi\)
\(314\) −5.07624e13 −0.938488
\(315\) 0 0
\(316\) −6.01478e12 −0.107384
\(317\) −1.01210e14 −1.77581 −0.887905 0.460027i \(-0.847840\pi\)
−0.887905 + 0.460027i \(0.847840\pi\)
\(318\) 2.08618e12 0.0359751
\(319\) −2.18630e13 −0.370562
\(320\) 0 0
\(321\) 2.67832e13 0.438616
\(322\) −2.16474e13 −0.348497
\(323\) 9.00941e13 1.42588
\(324\) 3.57047e12 0.0555556
\(325\) 0 0
\(326\) 2.76806e13 0.416370
\(327\) −4.32001e13 −0.638957
\(328\) 1.20589e13 0.175389
\(329\) 9.10058e13 1.30164
\(330\) 0 0
\(331\) 4.96614e13 0.687013 0.343506 0.939150i \(-0.388385\pi\)
0.343506 + 0.939150i \(0.388385\pi\)
\(332\) −8.69638e12 −0.118326
\(333\) 1.88893e13 0.252797
\(334\) −7.22566e13 −0.951200
\(335\) 0 0
\(336\) 8.39222e12 0.106908
\(337\) 8.40508e13 1.05336 0.526681 0.850063i \(-0.323436\pi\)
0.526681 + 0.850063i \(0.323436\pi\)
\(338\) −1.39918e14 −1.72516
\(339\) 1.91777e13 0.232647
\(340\) 0 0
\(341\) −1.14193e14 −1.34119
\(342\) 2.05345e13 0.237323
\(343\) 9.45221e13 1.07502
\(344\) 2.03315e13 0.227562
\(345\) 0 0
\(346\) −6.91072e13 −0.749212
\(347\) −1.74097e13 −0.185772 −0.0928858 0.995677i \(-0.529609\pi\)
−0.0928858 + 0.995677i \(0.529609\pi\)
\(348\) −7.16998e12 −0.0753065
\(349\) −6.60920e13 −0.683296 −0.341648 0.939828i \(-0.610985\pi\)
−0.341648 + 0.939828i \(0.610985\pi\)
\(350\) 0 0
\(351\) −3.56263e13 −0.356929
\(352\) 2.54594e13 0.251109
\(353\) 1.74556e14 1.69502 0.847508 0.530783i \(-0.178102\pi\)
0.847508 + 0.530783i \(0.178102\pi\)
\(354\) 1.30084e13 0.124368
\(355\) 0 0
\(356\) 7.73405e13 0.716856
\(357\) −6.63517e13 −0.605587
\(358\) −1.44251e14 −1.29647
\(359\) 3.60488e13 0.319059 0.159530 0.987193i \(-0.449002\pi\)
0.159530 + 0.987193i \(0.449002\pi\)
\(360\) 0 0
\(361\) 1.60795e12 0.0138033
\(362\) −1.38897e13 −0.117434
\(363\) −7.05640e13 −0.587620
\(364\) −8.37380e13 −0.686855
\(365\) 0 0
\(366\) −6.05532e13 −0.481938
\(367\) 5.04151e13 0.395273 0.197636 0.980275i \(-0.436673\pi\)
0.197636 + 0.980275i \(0.436673\pi\)
\(368\) −2.15370e13 −0.166350
\(369\) −2.17306e13 −0.165358
\(370\) 0 0
\(371\) −8.83621e12 −0.0652694
\(372\) −3.74496e13 −0.272559
\(373\) −2.09367e14 −1.50144 −0.750722 0.660619i \(-0.770294\pi\)
−0.750722 + 0.660619i \(0.770294\pi\)
\(374\) −2.01290e14 −1.42242
\(375\) 0 0
\(376\) 9.05416e13 0.621320
\(377\) 7.15424e13 0.483823
\(378\) −1.51231e13 −0.100794
\(379\) 2.95197e14 1.93908 0.969540 0.244931i \(-0.0787655\pi\)
0.969540 + 0.244931i \(0.0787655\pi\)
\(380\) 0 0
\(381\) −6.72258e13 −0.428992
\(382\) −2.77664e13 −0.174651
\(383\) −1.28924e14 −0.799356 −0.399678 0.916656i \(-0.630878\pi\)
−0.399678 + 0.916656i \(0.630878\pi\)
\(384\) 8.34942e12 0.0510310
\(385\) 0 0
\(386\) 5.98226e12 0.0355333
\(387\) −3.66381e13 −0.214547
\(388\) 8.43333e13 0.486883
\(389\) 1.04145e14 0.592810 0.296405 0.955062i \(-0.404212\pi\)
0.296405 + 0.955062i \(0.404212\pi\)
\(390\) 0 0
\(391\) 1.70278e14 0.942297
\(392\) 2.92470e13 0.159591
\(393\) 8.28319e13 0.445696
\(394\) 1.06496e14 0.565070
\(395\) 0 0
\(396\) −4.58786e13 −0.236748
\(397\) 1.15408e14 0.587338 0.293669 0.955907i \(-0.405124\pi\)
0.293669 + 0.955907i \(0.405124\pi\)
\(398\) 5.07458e13 0.254709
\(399\) −8.69759e13 −0.430574
\(400\) 0 0
\(401\) −2.20847e14 −1.06365 −0.531824 0.846855i \(-0.678493\pi\)
−0.531824 + 0.846855i \(0.678493\pi\)
\(402\) −1.45463e14 −0.691051
\(403\) 3.73674e14 1.75111
\(404\) −4.23056e13 −0.195569
\(405\) 0 0
\(406\) 3.03692e13 0.136628
\(407\) −2.42717e14 −1.07729
\(408\) −6.60132e13 −0.289068
\(409\) −8.56852e13 −0.370193 −0.185096 0.982720i \(-0.559260\pi\)
−0.185096 + 0.982720i \(0.559260\pi\)
\(410\) 0 0
\(411\) −2.29582e14 −0.965626
\(412\) 6.60667e13 0.274188
\(413\) −5.50985e13 −0.225639
\(414\) 3.88104e13 0.156836
\(415\) 0 0
\(416\) −8.33109e13 −0.327860
\(417\) −2.52162e13 −0.0979334
\(418\) −2.63857e14 −1.01135
\(419\) 2.58945e14 0.979560 0.489780 0.871846i \(-0.337077\pi\)
0.489780 + 0.871846i \(0.337077\pi\)
\(420\) 0 0
\(421\) 1.09351e14 0.402968 0.201484 0.979492i \(-0.435424\pi\)
0.201484 + 0.979492i \(0.435424\pi\)
\(422\) −1.13587e14 −0.413151
\(423\) −1.63159e14 −0.585786
\(424\) −8.79114e12 −0.0311554
\(425\) 0 0
\(426\) −6.49062e13 −0.224147
\(427\) 2.56479e14 0.874377
\(428\) −1.12864e14 −0.379853
\(429\) 4.57779e14 1.52104
\(430\) 0 0
\(431\) 2.92180e14 0.946292 0.473146 0.880984i \(-0.343118\pi\)
0.473146 + 0.880984i \(0.343118\pi\)
\(432\) −1.50459e13 −0.0481125
\(433\) −9.53988e13 −0.301203 −0.150602 0.988595i \(-0.548121\pi\)
−0.150602 + 0.988595i \(0.548121\pi\)
\(434\) 1.58621e14 0.494502
\(435\) 0 0
\(436\) 1.82045e14 0.553353
\(437\) 2.23206e14 0.669975
\(438\) −1.52734e14 −0.452719
\(439\) −4.80574e14 −1.40671 −0.703357 0.710837i \(-0.748316\pi\)
−0.703357 + 0.710837i \(0.748316\pi\)
\(440\) 0 0
\(441\) −5.27040e13 −0.150464
\(442\) 6.58683e14 1.85718
\(443\) 3.15148e14 0.877596 0.438798 0.898586i \(-0.355404\pi\)
0.438798 + 0.898586i \(0.355404\pi\)
\(444\) −7.95993e13 −0.218929
\(445\) 0 0
\(446\) −1.65767e14 −0.444792
\(447\) 2.98530e14 0.791219
\(448\) −3.53648e13 −0.0925852
\(449\) 5.03692e14 1.30260 0.651299 0.758821i \(-0.274224\pi\)
0.651299 + 0.758821i \(0.274224\pi\)
\(450\) 0 0
\(451\) 2.79226e14 0.704669
\(452\) −8.08147e13 −0.201479
\(453\) −5.16292e13 −0.127161
\(454\) −4.27726e14 −1.04078
\(455\) 0 0
\(456\) −8.65322e13 −0.205528
\(457\) 2.96302e14 0.695338 0.347669 0.937617i \(-0.386973\pi\)
0.347669 + 0.937617i \(0.386973\pi\)
\(458\) 6.49857e13 0.150681
\(459\) 1.18958e14 0.272536
\(460\) 0 0
\(461\) 1.00284e14 0.224325 0.112163 0.993690i \(-0.464222\pi\)
0.112163 + 0.993690i \(0.464222\pi\)
\(462\) 1.94323e14 0.429530
\(463\) −2.49998e14 −0.546061 −0.273030 0.962005i \(-0.588026\pi\)
−0.273030 + 0.962005i \(0.588026\pi\)
\(464\) 3.02142e13 0.0652173
\(465\) 0 0
\(466\) 2.28263e14 0.481186
\(467\) 7.16272e13 0.149223 0.0746114 0.997213i \(-0.476228\pi\)
0.0746114 + 0.997213i \(0.476228\pi\)
\(468\) 1.50129e14 0.309109
\(469\) 6.16124e14 1.25377
\(470\) 0 0
\(471\) −3.85477e14 −0.766273
\(472\) −5.48174e13 −0.107705
\(473\) 4.70780e14 0.914285
\(474\) −4.56747e13 −0.0876789
\(475\) 0 0
\(476\) 2.79605e14 0.524454
\(477\) 1.58419e13 0.0293736
\(478\) −1.05716e14 −0.193771
\(479\) 2.34866e14 0.425574 0.212787 0.977099i \(-0.431746\pi\)
0.212787 + 0.977099i \(0.431746\pi\)
\(480\) 0 0
\(481\) 7.94246e14 1.40656
\(482\) −1.33420e13 −0.0233595
\(483\) −1.64385e14 −0.284546
\(484\) 2.97356e14 0.508894
\(485\) 0 0
\(486\) 2.71132e13 0.0453609
\(487\) 9.89828e14 1.63738 0.818692 0.574233i \(-0.194700\pi\)
0.818692 + 0.574233i \(0.194700\pi\)
\(488\) 2.55171e14 0.417371
\(489\) 2.10200e14 0.339965
\(490\) 0 0
\(491\) −8.70345e14 −1.37639 −0.688197 0.725523i \(-0.741598\pi\)
−0.688197 + 0.725523i \(0.741598\pi\)
\(492\) 9.15724e13 0.143204
\(493\) −2.38884e14 −0.369427
\(494\) 8.63423e14 1.32046
\(495\) 0 0
\(496\) 1.57812e14 0.236043
\(497\) 2.74916e14 0.406669
\(498\) −6.60381e13 −0.0966125
\(499\) −5.38314e14 −0.778902 −0.389451 0.921047i \(-0.627335\pi\)
−0.389451 + 0.921047i \(0.627335\pi\)
\(500\) 0 0
\(501\) −5.48698e14 −0.776652
\(502\) 6.14303e13 0.0860027
\(503\) −8.38818e14 −1.16157 −0.580783 0.814059i \(-0.697253\pi\)
−0.580783 + 0.814059i \(0.697253\pi\)
\(504\) 6.37284e13 0.0872902
\(505\) 0 0
\(506\) −4.98692e14 −0.668351
\(507\) −1.06250e15 −1.40859
\(508\) 2.83289e14 0.371518
\(509\) −5.97231e14 −0.774809 −0.387404 0.921910i \(-0.626628\pi\)
−0.387404 + 0.921910i \(0.626628\pi\)
\(510\) 0 0
\(511\) 6.46921e14 0.821365
\(512\) −3.51844e13 −0.0441942
\(513\) 1.55934e14 0.193774
\(514\) 3.12330e14 0.383987
\(515\) 0 0
\(516\) 1.54393e14 0.185803
\(517\) 2.09650e15 2.49631
\(518\) 3.37151e14 0.397201
\(519\) −5.24783e14 −0.611729
\(520\) 0 0
\(521\) 1.36712e15 1.56027 0.780133 0.625613i \(-0.215151\pi\)
0.780133 + 0.625613i \(0.215151\pi\)
\(522\) −5.44471e13 −0.0614875
\(523\) −6.90497e14 −0.771619 −0.385809 0.922579i \(-0.626078\pi\)
−0.385809 + 0.922579i \(0.626078\pi\)
\(524\) −3.49053e14 −0.385984
\(525\) 0 0
\(526\) −1.29551e14 −0.140287
\(527\) −1.24771e15 −1.33708
\(528\) 1.93332e14 0.205030
\(529\) −5.30948e14 −0.557245
\(530\) 0 0
\(531\) 9.87828e13 0.101546
\(532\) 3.66516e14 0.372888
\(533\) −9.13714e14 −0.920048
\(534\) 5.87305e14 0.585311
\(535\) 0 0
\(536\) 6.12981e14 0.598468
\(537\) −1.09541e15 −1.05857
\(538\) 3.63056e14 0.347273
\(539\) 6.77218e14 0.641195
\(540\) 0 0
\(541\) −4.48077e14 −0.415688 −0.207844 0.978162i \(-0.566645\pi\)
−0.207844 + 0.978162i \(0.566645\pi\)
\(542\) 7.52806e13 0.0691332
\(543\) −1.05475e14 −0.0958848
\(544\) 2.78179e14 0.250340
\(545\) 0 0
\(546\) −6.35886e14 −0.560815
\(547\) −8.45293e14 −0.738035 −0.369017 0.929423i \(-0.620306\pi\)
−0.369017 + 0.929423i \(0.620306\pi\)
\(548\) 9.67459e14 0.836256
\(549\) −4.59826e14 −0.393501
\(550\) 0 0
\(551\) −3.13136e14 −0.262663
\(552\) −1.63547e14 −0.135824
\(553\) 1.93460e14 0.159075
\(554\) −1.39625e15 −1.13673
\(555\) 0 0
\(556\) 1.06261e14 0.0848128
\(557\) 6.21109e14 0.490868 0.245434 0.969413i \(-0.421070\pi\)
0.245434 + 0.969413i \(0.421070\pi\)
\(558\) −2.84383e14 −0.222544
\(559\) −1.54054e15 −1.19373
\(560\) 0 0
\(561\) −1.52855e15 −1.16140
\(562\) −8.80316e14 −0.662352
\(563\) 2.31809e15 1.72717 0.863585 0.504203i \(-0.168214\pi\)
0.863585 + 0.504203i \(0.168214\pi\)
\(564\) 6.87550e14 0.507305
\(565\) 0 0
\(566\) −8.25008e14 −0.596991
\(567\) −1.14841e14 −0.0822980
\(568\) 2.73514e14 0.194117
\(569\) −1.54963e15 −1.08921 −0.544605 0.838693i \(-0.683320\pi\)
−0.544605 + 0.838693i \(0.683320\pi\)
\(570\) 0 0
\(571\) 1.59408e15 1.09903 0.549516 0.835483i \(-0.314812\pi\)
0.549516 + 0.835483i \(0.314812\pi\)
\(572\) −1.92908e15 −1.31726
\(573\) −2.10851e14 −0.142602
\(574\) −3.87864e14 −0.259815
\(575\) 0 0
\(576\) 6.34034e13 0.0416667
\(577\) −1.77831e15 −1.15755 −0.578777 0.815486i \(-0.696470\pi\)
−0.578777 + 0.815486i \(0.696470\pi\)
\(578\) −1.10268e15 −0.710959
\(579\) 4.54278e13 0.0290128
\(580\) 0 0
\(581\) 2.79711e14 0.175283
\(582\) 6.40406e14 0.397539
\(583\) −2.03560e14 −0.125174
\(584\) 6.43621e14 0.392066
\(585\) 0 0
\(586\) 1.78984e15 1.06998
\(587\) −2.98810e14 −0.176964 −0.0884821 0.996078i \(-0.528202\pi\)
−0.0884821 + 0.996078i \(0.528202\pi\)
\(588\) 2.22094e14 0.130305
\(589\) −1.63554e15 −0.950666
\(590\) 0 0
\(591\) 8.08700e14 0.461378
\(592\) 3.35431e14 0.189598
\(593\) −2.13060e15 −1.19317 −0.596585 0.802550i \(-0.703476\pi\)
−0.596585 + 0.802550i \(0.703476\pi\)
\(594\) −3.48391e14 −0.193304
\(595\) 0 0
\(596\) −1.25800e15 −0.685216
\(597\) 3.85351e14 0.207969
\(598\) 1.63188e15 0.872631
\(599\) 6.91981e14 0.366646 0.183323 0.983053i \(-0.441315\pi\)
0.183323 + 0.983053i \(0.441315\pi\)
\(600\) 0 0
\(601\) 1.90167e15 0.989296 0.494648 0.869094i \(-0.335297\pi\)
0.494648 + 0.869094i \(0.335297\pi\)
\(602\) −6.53945e14 −0.337101
\(603\) −1.10461e15 −0.564241
\(604\) 2.17565e14 0.110125
\(605\) 0 0
\(606\) −3.21258e14 −0.159682
\(607\) −2.60368e14 −0.128248 −0.0641238 0.997942i \(-0.520425\pi\)
−0.0641238 + 0.997942i \(0.520425\pi\)
\(608\) 3.64646e14 0.177993
\(609\) 2.30616e14 0.111556
\(610\) 0 0
\(611\) −6.86041e15 −3.25929
\(612\) −5.01288e14 −0.236023
\(613\) 3.20683e15 1.49638 0.748192 0.663482i \(-0.230922\pi\)
0.748192 + 0.663482i \(0.230922\pi\)
\(614\) 1.81778e15 0.840649
\(615\) 0 0
\(616\) −8.18876e14 −0.371984
\(617\) −3.99520e15 −1.79875 −0.899374 0.437180i \(-0.855977\pi\)
−0.899374 + 0.437180i \(0.855977\pi\)
\(618\) 5.01694e14 0.223873
\(619\) −1.00277e15 −0.443508 −0.221754 0.975103i \(-0.571178\pi\)
−0.221754 + 0.975103i \(0.571178\pi\)
\(620\) 0 0
\(621\) 2.94716e14 0.128056
\(622\) 2.14782e15 0.925019
\(623\) −2.48759e15 −1.06192
\(624\) −6.32642e14 −0.267697
\(625\) 0 0
\(626\) −1.17339e15 −0.487847
\(627\) −2.00367e15 −0.825760
\(628\) 1.62440e15 0.663612
\(629\) −2.65203e15 −1.07399
\(630\) 0 0
\(631\) −2.06951e15 −0.823579 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(632\) 1.92473e14 0.0759321
\(633\) −8.62548e14 −0.337336
\(634\) 3.23871e15 1.25569
\(635\) 0 0
\(636\) −6.67577e13 −0.0254383
\(637\) −2.21607e15 −0.837174
\(638\) 6.99615e14 0.262027
\(639\) −4.92881e14 −0.183015
\(640\) 0 0
\(641\) −1.00036e14 −0.0365122 −0.0182561 0.999833i \(-0.505811\pi\)
−0.0182561 + 0.999833i \(0.505811\pi\)
\(642\) −8.57063e14 −0.310149
\(643\) 3.03786e14 0.108995 0.0544976 0.998514i \(-0.482644\pi\)
0.0544976 + 0.998514i \(0.482644\pi\)
\(644\) 6.92717e14 0.246424
\(645\) 0 0
\(646\) −2.88301e15 −1.00825
\(647\) −1.90746e14 −0.0661426 −0.0330713 0.999453i \(-0.510529\pi\)
−0.0330713 + 0.999453i \(0.510529\pi\)
\(648\) −1.14255e14 −0.0392837
\(649\) −1.26931e15 −0.432733
\(650\) 0 0
\(651\) 1.20453e15 0.403759
\(652\) −8.85780e14 −0.294418
\(653\) −1.64793e15 −0.543145 −0.271573 0.962418i \(-0.587544\pi\)
−0.271573 + 0.962418i \(0.587544\pi\)
\(654\) 1.38240e15 0.451811
\(655\) 0 0
\(656\) −3.85885e14 −0.124019
\(657\) −1.15983e15 −0.369644
\(658\) −2.91219e15 −0.920400
\(659\) −2.62861e15 −0.823866 −0.411933 0.911214i \(-0.635146\pi\)
−0.411933 + 0.911214i \(0.635146\pi\)
\(660\) 0 0
\(661\) 4.80241e15 1.48031 0.740153 0.672439i \(-0.234753\pi\)
0.740153 + 0.672439i \(0.234753\pi\)
\(662\) −1.58916e15 −0.485791
\(663\) 5.00188e15 1.51638
\(664\) 2.78284e14 0.0836689
\(665\) 0 0
\(666\) −6.04457e14 −0.178755
\(667\) −5.91830e14 −0.173582
\(668\) 2.31221e15 0.672600
\(669\) −1.25879e15 −0.363171
\(670\) 0 0
\(671\) 5.90852e15 1.67689
\(672\) −2.68551e14 −0.0755955
\(673\) 1.25325e15 0.349909 0.174954 0.984577i \(-0.444022\pi\)
0.174954 + 0.984577i \(0.444022\pi\)
\(674\) −2.68963e15 −0.744839
\(675\) 0 0
\(676\) 4.47736e15 1.21988
\(677\) 1.92035e15 0.518969 0.259485 0.965747i \(-0.416447\pi\)
0.259485 + 0.965747i \(0.416447\pi\)
\(678\) −6.13686e14 −0.164507
\(679\) −2.71250e15 −0.721251
\(680\) 0 0
\(681\) −3.24805e15 −0.849793
\(682\) 3.65416e15 0.948362
\(683\) −6.12548e14 −0.157698 −0.0788490 0.996887i \(-0.525125\pi\)
−0.0788490 + 0.996887i \(0.525125\pi\)
\(684\) −6.57104e14 −0.167813
\(685\) 0 0
\(686\) −3.02471e15 −0.760153
\(687\) 4.93485e14 0.123030
\(688\) −6.50610e14 −0.160910
\(689\) 6.66112e14 0.163434
\(690\) 0 0
\(691\) 5.08990e15 1.22908 0.614539 0.788886i \(-0.289342\pi\)
0.614539 + 0.788886i \(0.289342\pi\)
\(692\) 2.21143e15 0.529773
\(693\) 1.47564e15 0.350710
\(694\) 5.57111e14 0.131360
\(695\) 0 0
\(696\) 2.29439e14 0.0532497
\(697\) 3.05094e15 0.702511
\(698\) 2.11494e15 0.483163
\(699\) 1.73337e15 0.392887
\(700\) 0 0
\(701\) −1.78664e15 −0.398647 −0.199323 0.979934i \(-0.563874\pi\)
−0.199323 + 0.979934i \(0.563874\pi\)
\(702\) 1.14004e15 0.252387
\(703\) −3.47636e15 −0.763608
\(704\) −8.14699e14 −0.177561
\(705\) 0 0
\(706\) −5.58579e15 −1.19856
\(707\) 1.36072e15 0.289709
\(708\) −4.16270e14 −0.0879411
\(709\) 4.18187e15 0.876631 0.438315 0.898821i \(-0.355575\pi\)
0.438315 + 0.898821i \(0.355575\pi\)
\(710\) 0 0
\(711\) −3.46842e14 −0.0715895
\(712\) −2.47490e15 −0.506894
\(713\) −3.09119e15 −0.628251
\(714\) 2.12325e15 0.428215
\(715\) 0 0
\(716\) 4.61605e15 0.916745
\(717\) −8.02782e14 −0.158213
\(718\) −1.15356e15 −0.225609
\(719\) 8.02554e15 1.55763 0.778817 0.627251i \(-0.215820\pi\)
0.778817 + 0.627251i \(0.215820\pi\)
\(720\) 0 0
\(721\) −2.12497e15 −0.406172
\(722\) −5.14544e13 −0.00976041
\(723\) −1.01316e14 −0.0190729
\(724\) 4.44470e14 0.0830387
\(725\) 0 0
\(726\) 2.25805e15 0.415510
\(727\) 2.93804e15 0.536560 0.268280 0.963341i \(-0.413545\pi\)
0.268280 + 0.963341i \(0.413545\pi\)
\(728\) 2.67962e15 0.485680
\(729\) 2.05891e14 0.0370370
\(730\) 0 0
\(731\) 5.14393e15 0.911485
\(732\) 1.93770e15 0.340782
\(733\) 4.42319e15 0.772083 0.386042 0.922481i \(-0.373842\pi\)
0.386042 + 0.922481i \(0.373842\pi\)
\(734\) −1.61328e15 −0.279500
\(735\) 0 0
\(736\) 6.89184e14 0.117627
\(737\) 1.41937e16 2.40449
\(738\) 6.95378e14 0.116926
\(739\) −1.11732e16 −1.86481 −0.932405 0.361416i \(-0.882293\pi\)
−0.932405 + 0.361416i \(0.882293\pi\)
\(740\) 0 0
\(741\) 6.55662e15 1.07815
\(742\) 2.82759e14 0.0461524
\(743\) −1.98624e15 −0.321805 −0.160903 0.986970i \(-0.551441\pi\)
−0.160903 + 0.986970i \(0.551441\pi\)
\(744\) 1.19839e15 0.192728
\(745\) 0 0
\(746\) 6.69973e15 1.06168
\(747\) −5.01477e14 −0.0788838
\(748\) 6.44128e15 1.00580
\(749\) 3.63017e15 0.562700
\(750\) 0 0
\(751\) −7.63503e15 −1.16625 −0.583125 0.812383i \(-0.698170\pi\)
−0.583125 + 0.812383i \(0.698170\pi\)
\(752\) −2.89733e15 −0.439339
\(753\) 4.66486e14 0.0702209
\(754\) −2.28936e15 −0.342114
\(755\) 0 0
\(756\) 4.83938e14 0.0712721
\(757\) −7.89241e15 −1.15394 −0.576969 0.816766i \(-0.695765\pi\)
−0.576969 + 0.816766i \(0.695765\pi\)
\(758\) −9.44630e15 −1.37114
\(759\) −3.78694e15 −0.545706
\(760\) 0 0
\(761\) −1.10602e16 −1.57089 −0.785446 0.618930i \(-0.787566\pi\)
−0.785446 + 0.618930i \(0.787566\pi\)
\(762\) 2.15123e15 0.303343
\(763\) −5.85530e15 −0.819717
\(764\) 8.88526e14 0.123497
\(765\) 0 0
\(766\) 4.12557e15 0.565230
\(767\) 4.15356e15 0.564997
\(768\) −2.67181e14 −0.0360844
\(769\) 1.66698e15 0.223529 0.111765 0.993735i \(-0.464350\pi\)
0.111765 + 0.993735i \(0.464350\pi\)
\(770\) 0 0
\(771\) 2.37176e15 0.313524
\(772\) −1.91432e14 −0.0251258
\(773\) −6.92568e15 −0.902558 −0.451279 0.892383i \(-0.649032\pi\)
−0.451279 + 0.892383i \(0.649032\pi\)
\(774\) 1.17242e15 0.151708
\(775\) 0 0
\(776\) −2.69867e15 −0.344278
\(777\) 2.56024e15 0.324313
\(778\) −3.33264e15 −0.419180
\(779\) 3.99926e15 0.499487
\(780\) 0 0
\(781\) 6.33326e15 0.779914
\(782\) −5.44891e15 −0.666304
\(783\) −4.13457e14 −0.0502043
\(784\) −9.35903e14 −0.112848
\(785\) 0 0
\(786\) −2.65062e15 −0.315155
\(787\) 2.99770e15 0.353939 0.176969 0.984216i \(-0.443371\pi\)
0.176969 + 0.984216i \(0.443371\pi\)
\(788\) −3.40786e15 −0.399565
\(789\) −9.83775e14 −0.114544
\(790\) 0 0
\(791\) 2.59933e15 0.298463
\(792\) 1.46811e15 0.167406
\(793\) −1.93345e16 −2.18943
\(794\) −3.69306e15 −0.415311
\(795\) 0 0
\(796\) −1.62387e15 −0.180106
\(797\) 5.67794e15 0.625417 0.312709 0.949849i \(-0.398764\pi\)
0.312709 + 0.949849i \(0.398764\pi\)
\(798\) 2.78323e15 0.304462
\(799\) 2.29073e16 2.48866
\(800\) 0 0
\(801\) 4.45984e15 0.477904
\(802\) 7.06711e15 0.752113
\(803\) 1.49031e16 1.57522
\(804\) 4.65482e15 0.488647
\(805\) 0 0
\(806\) −1.19576e16 −1.23823
\(807\) 2.75696e15 0.283547
\(808\) 1.35378e15 0.138288
\(809\) 4.43159e15 0.449617 0.224809 0.974403i \(-0.427824\pi\)
0.224809 + 0.974403i \(0.427824\pi\)
\(810\) 0 0
\(811\) −8.99323e15 −0.900121 −0.450060 0.892998i \(-0.648598\pi\)
−0.450060 + 0.892998i \(0.648598\pi\)
\(812\) −9.71813e14 −0.0966105
\(813\) 5.71662e14 0.0564470
\(814\) 7.76695e15 0.761757
\(815\) 0 0
\(816\) 2.11242e15 0.204402
\(817\) 6.74283e15 0.648068
\(818\) 2.74193e15 0.261766
\(819\) −4.82876e15 −0.457903
\(820\) 0 0
\(821\) 9.08013e15 0.849581 0.424790 0.905292i \(-0.360348\pi\)
0.424790 + 0.905292i \(0.360348\pi\)
\(822\) 7.34664e15 0.682800
\(823\) 1.29654e16 1.19698 0.598488 0.801132i \(-0.295768\pi\)
0.598488 + 0.801132i \(0.295768\pi\)
\(824\) −2.11414e15 −0.193880
\(825\) 0 0
\(826\) 1.76315e15 0.159551
\(827\) 5.40262e15 0.485651 0.242826 0.970070i \(-0.421926\pi\)
0.242826 + 0.970070i \(0.421926\pi\)
\(828\) −1.24193e15 −0.110900
\(829\) −7.97388e14 −0.0707326 −0.0353663 0.999374i \(-0.511260\pi\)
−0.0353663 + 0.999374i \(0.511260\pi\)
\(830\) 0 0
\(831\) −1.06027e16 −0.928138
\(832\) 2.66595e15 0.231832
\(833\) 7.39956e15 0.639231
\(834\) 8.06918e14 0.0692494
\(835\) 0 0
\(836\) 8.44344e15 0.715129
\(837\) −2.15953e15 −0.181706
\(838\) −8.28625e15 −0.692653
\(839\) 1.02313e16 0.849646 0.424823 0.905276i \(-0.360336\pi\)
0.424823 + 0.905276i \(0.360336\pi\)
\(840\) 0 0
\(841\) −1.13702e16 −0.931947
\(842\) −3.49922e15 −0.284941
\(843\) −6.68490e15 −0.540808
\(844\) 3.63477e15 0.292142
\(845\) 0 0
\(846\) 5.22108e15 0.414213
\(847\) −9.56418e15 −0.753857
\(848\) 2.81316e14 0.0220302
\(849\) −6.26490e15 −0.487441
\(850\) 0 0
\(851\) −6.57034e15 −0.504633
\(852\) 2.07700e15 0.158496
\(853\) 1.33010e16 1.00847 0.504237 0.863565i \(-0.331774\pi\)
0.504237 + 0.863565i \(0.331774\pi\)
\(854\) −8.20733e15 −0.618278
\(855\) 0 0
\(856\) 3.61166e15 0.268597
\(857\) −3.38483e15 −0.250116 −0.125058 0.992149i \(-0.539912\pi\)
−0.125058 + 0.992149i \(0.539912\pi\)
\(858\) −1.46489e16 −1.07554
\(859\) 2.32376e16 1.69523 0.847615 0.530612i \(-0.178038\pi\)
0.847615 + 0.530612i \(0.178038\pi\)
\(860\) 0 0
\(861\) −2.94534e15 −0.212138
\(862\) −9.34976e15 −0.669130
\(863\) −2.44743e16 −1.74041 −0.870203 0.492693i \(-0.836013\pi\)
−0.870203 + 0.492693i \(0.836013\pi\)
\(864\) 4.81469e14 0.0340207
\(865\) 0 0
\(866\) 3.05276e15 0.212983
\(867\) −8.37344e15 −0.580496
\(868\) −5.07588e15 −0.349666
\(869\) 4.45674e15 0.305076
\(870\) 0 0
\(871\) −4.64461e16 −3.13942
\(872\) −5.82543e15 −0.391280
\(873\) 4.86309e15 0.324589
\(874\) −7.14261e15 −0.473744
\(875\) 0 0
\(876\) 4.88750e15 0.320121
\(877\) −2.15305e16 −1.40138 −0.700689 0.713467i \(-0.747124\pi\)
−0.700689 + 0.713467i \(0.747124\pi\)
\(878\) 1.53784e16 0.994697
\(879\) 1.35916e16 0.873636
\(880\) 0 0
\(881\) 2.77094e16 1.75898 0.879488 0.475922i \(-0.157885\pi\)
0.879488 + 0.475922i \(0.157885\pi\)
\(882\) 1.68653e15 0.106394
\(883\) 8.61677e15 0.540208 0.270104 0.962831i \(-0.412942\pi\)
0.270104 + 0.962831i \(0.412942\pi\)
\(884\) −2.10779e16 −1.31322
\(885\) 0 0
\(886\) −1.00848e16 −0.620554
\(887\) 3.22651e16 1.97312 0.986558 0.163409i \(-0.0522490\pi\)
0.986558 + 0.163409i \(0.0522490\pi\)
\(888\) 2.54718e15 0.154806
\(889\) −9.11173e15 −0.550353
\(890\) 0 0
\(891\) −2.64559e15 −0.157832
\(892\) 5.30455e15 0.314515
\(893\) 3.00275e16 1.76944
\(894\) −9.55296e15 −0.559476
\(895\) 0 0
\(896\) 1.13167e15 0.0654676
\(897\) 1.23921e16 0.712500
\(898\) −1.61181e16 −0.921075
\(899\) 4.33663e15 0.246305
\(900\) 0 0
\(901\) −2.22418e15 −0.124791
\(902\) −8.93523e15 −0.498276
\(903\) −4.96590e15 −0.275242
\(904\) 2.58607e15 0.142467
\(905\) 0 0
\(906\) 1.65213e15 0.0899166
\(907\) −3.10436e16 −1.67932 −0.839658 0.543115i \(-0.817245\pi\)
−0.839658 + 0.543115i \(0.817245\pi\)
\(908\) 1.36872e16 0.735943
\(909\) −2.43955e15 −0.130379
\(910\) 0 0
\(911\) 1.78173e16 0.940786 0.470393 0.882457i \(-0.344112\pi\)
0.470393 + 0.882457i \(0.344112\pi\)
\(912\) 2.76903e15 0.145330
\(913\) 6.44371e15 0.336161
\(914\) −9.48167e15 −0.491678
\(915\) 0 0
\(916\) −2.07954e15 −0.106547
\(917\) 1.12270e16 0.571783
\(918\) −3.80666e15 −0.192712
\(919\) 2.90046e16 1.45959 0.729797 0.683664i \(-0.239615\pi\)
0.729797 + 0.683664i \(0.239615\pi\)
\(920\) 0 0
\(921\) 1.38038e16 0.686387
\(922\) −3.20910e15 −0.158622
\(923\) −2.07244e16 −1.01829
\(924\) −6.21834e15 −0.303724
\(925\) 0 0
\(926\) 7.99993e15 0.386123
\(927\) 3.80974e15 0.182792
\(928\) −9.66856e14 −0.0461156
\(929\) 1.09610e16 0.519713 0.259856 0.965647i \(-0.416325\pi\)
0.259856 + 0.965647i \(0.416325\pi\)
\(930\) 0 0
\(931\) 9.69957e15 0.454495
\(932\) −7.30442e15 −0.340250
\(933\) 1.63100e16 0.755274
\(934\) −2.29207e15 −0.105516
\(935\) 0 0
\(936\) −4.80413e15 −0.218573
\(937\) −9.13874e15 −0.413351 −0.206675 0.978410i \(-0.566264\pi\)
−0.206675 + 0.978410i \(0.566264\pi\)
\(938\) −1.97160e16 −0.886548
\(939\) −8.91043e15 −0.398325
\(940\) 0 0
\(941\) 6.97287e14 0.0308083 0.0154042 0.999881i \(-0.495097\pi\)
0.0154042 + 0.999881i \(0.495097\pi\)
\(942\) 1.23353e16 0.541837
\(943\) 7.55864e15 0.330088
\(944\) 1.75416e15 0.0761592
\(945\) 0 0
\(946\) −1.50650e16 −0.646497
\(947\) 2.17555e15 0.0928205 0.0464103 0.998922i \(-0.485222\pi\)
0.0464103 + 0.998922i \(0.485222\pi\)
\(948\) 1.46159e15 0.0619983
\(949\) −4.87677e16 −2.05669
\(950\) 0 0
\(951\) 2.45940e16 1.02526
\(952\) −8.94737e15 −0.370845
\(953\) 3.59675e16 1.48217 0.741087 0.671409i \(-0.234310\pi\)
0.741087 + 0.671409i \(0.234310\pi\)
\(954\) −5.06941e14 −0.0207702
\(955\) 0 0
\(956\) 3.38291e15 0.137016
\(957\) 5.31270e15 0.213944
\(958\) −7.51572e15 −0.300926
\(959\) −3.11174e16 −1.23880
\(960\) 0 0
\(961\) −2.75781e15 −0.108539
\(962\) −2.54159e16 −0.994585
\(963\) −6.50832e15 −0.253235
\(964\) 4.26945e14 0.0165176
\(965\) 0 0
\(966\) 5.26032e15 0.201205
\(967\) −3.64625e16 −1.38676 −0.693379 0.720573i \(-0.743879\pi\)
−0.693379 + 0.720573i \(0.743879\pi\)
\(968\) −9.51540e15 −0.359842
\(969\) −2.18929e16 −0.823231
\(970\) 0 0
\(971\) −1.01490e16 −0.377328 −0.188664 0.982042i \(-0.560416\pi\)
−0.188664 + 0.982042i \(0.560416\pi\)
\(972\) −8.67624e14 −0.0320750
\(973\) −3.41778e15 −0.125639
\(974\) −3.16745e16 −1.15781
\(975\) 0 0
\(976\) −8.16547e15 −0.295126
\(977\) −2.99278e16 −1.07561 −0.537804 0.843070i \(-0.680746\pi\)
−0.537804 + 0.843070i \(0.680746\pi\)
\(978\) −6.72639e15 −0.240391
\(979\) −5.73066e16 −2.03657
\(980\) 0 0
\(981\) 1.04976e16 0.368902
\(982\) 2.78510e16 0.973258
\(983\) 3.23592e16 1.12448 0.562242 0.826973i \(-0.309939\pi\)
0.562242 + 0.826973i \(0.309939\pi\)
\(984\) −2.93032e15 −0.101261
\(985\) 0 0
\(986\) 7.64428e15 0.261224
\(987\) −2.21144e16 −0.751503
\(988\) −2.76295e16 −0.933706
\(989\) 1.27440e16 0.428278
\(990\) 0 0
\(991\) −2.28589e16 −0.759716 −0.379858 0.925045i \(-0.624027\pi\)
−0.379858 + 0.925045i \(0.624027\pi\)
\(992\) −5.04999e15 −0.166908
\(993\) −1.20677e16 −0.396647
\(994\) −8.79733e15 −0.287558
\(995\) 0 0
\(996\) 2.11322e15 0.0683154
\(997\) 1.99784e16 0.642298 0.321149 0.947029i \(-0.395931\pi\)
0.321149 + 0.947029i \(0.395931\pi\)
\(998\) 1.72260e16 0.550767
\(999\) −4.59010e15 −0.145953
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 150.12.a.a.1.1 1
5.2 odd 4 150.12.c.e.49.1 2
5.3 odd 4 150.12.c.e.49.2 2
5.4 even 2 6.12.a.c.1.1 1
15.14 odd 2 18.12.a.a.1.1 1
20.19 odd 2 48.12.a.d.1.1 1
40.19 odd 2 192.12.a.m.1.1 1
40.29 even 2 192.12.a.c.1.1 1
45.4 even 6 162.12.c.b.55.1 2
45.14 odd 6 162.12.c.i.55.1 2
45.29 odd 6 162.12.c.i.109.1 2
45.34 even 6 162.12.c.b.109.1 2
60.59 even 2 144.12.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.12.a.c.1.1 1 5.4 even 2
18.12.a.a.1.1 1 15.14 odd 2
48.12.a.d.1.1 1 20.19 odd 2
144.12.a.e.1.1 1 60.59 even 2
150.12.a.a.1.1 1 1.1 even 1 trivial
150.12.c.e.49.1 2 5.2 odd 4
150.12.c.e.49.2 2 5.3 odd 4
162.12.c.b.55.1 2 45.4 even 6
162.12.c.b.109.1 2 45.34 even 6
162.12.c.i.55.1 2 45.14 odd 6
162.12.c.i.109.1 2 45.29 odd 6
192.12.a.c.1.1 1 40.29 even 2
192.12.a.m.1.1 1 40.19 odd 2