Properties

Label 45.12.a.d.1.1
Level $45$
Weight $12$
Character 45.1
Self dual yes
Analytic conductor $34.575$
Analytic rank $0$
Dimension $2$
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] = [45,12,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, 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("45.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(34.5754431252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 151 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-12.2882\) of defining polynomial
Character \(\chi\) \(=\) 45.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-63.7292 q^{2} +2013.42 q^{4} +3125.00 q^{5} +41926.3 q^{7} +2204.06 q^{8} +O(q^{10})\) \(q-63.7292 q^{2} +2013.42 q^{4} +3125.00 q^{5} +41926.3 q^{7} +2204.06 q^{8} -199154. q^{10} +957905. q^{11} +1.39098e6 q^{13} -2.67193e6 q^{14} -4.26394e6 q^{16} -3.76857e6 q^{17} +9.41036e6 q^{19} +6.29192e6 q^{20} -6.10466e7 q^{22} -3.02942e7 q^{23} +9.76562e6 q^{25} -8.86460e7 q^{26} +8.44151e7 q^{28} -1.03553e8 q^{29} -5.48554e7 q^{31} +2.67224e8 q^{32} +2.40168e8 q^{34} +1.31020e8 q^{35} +4.78282e8 q^{37} -5.99715e8 q^{38} +6.88767e6 q^{40} +9.29557e8 q^{41} -2.68608e7 q^{43} +1.92866e9 q^{44} +1.93063e9 q^{46} +1.20497e9 q^{47} -2.19508e8 q^{49} -6.22356e8 q^{50} +2.80062e9 q^{52} +4.02058e9 q^{53} +2.99345e9 q^{55} +9.24080e7 q^{56} +6.59933e9 q^{58} -7.97972e9 q^{59} -2.07472e9 q^{61} +3.49589e9 q^{62} -8.29741e9 q^{64} +4.34681e9 q^{65} +5.61370e9 q^{67} -7.58770e9 q^{68} -8.34979e9 q^{70} -1.51224e10 q^{71} -6.64484e9 q^{73} -3.04806e10 q^{74} +1.89470e10 q^{76} +4.01615e10 q^{77} -1.57985e10 q^{79} -1.33248e10 q^{80} -5.92399e10 q^{82} +2.04046e10 q^{83} -1.17768e10 q^{85} +1.71182e9 q^{86} +2.11128e9 q^{88} +4.21030e10 q^{89} +5.83187e10 q^{91} -6.09948e10 q^{92} -7.67919e10 q^{94} +2.94074e10 q^{95} +1.10181e11 q^{97} +1.39891e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{2} + 6976 q^{4} + 6250 q^{5} + 57900 q^{7} + 246240 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{2} + 6976 q^{4} + 6250 q^{5} + 57900 q^{7} + 246240 q^{8} + 62500 q^{10} + 618176 q^{11} + 3414260 q^{13} - 1334472 q^{14} + 6005632 q^{16} - 1317940 q^{17} + 5325320 q^{19} + 21800000 q^{20} - 89491840 q^{22} - 58943940 q^{23} + 19531250 q^{25} + 80761736 q^{26} + 163685760 q^{28} - 94140380 q^{29} + 244543464 q^{31} + 627301120 q^{32} + 445358072 q^{34} + 180937500 q^{35} + 21003220 q^{37} - 941752240 q^{38} + 769500000 q^{40} + 745743316 q^{41} + 629950100 q^{43} + 242725888 q^{44} - 468194856 q^{46} + 1402061540 q^{47} - 1941677414 q^{49} + 195312500 q^{50} + 12841321600 q^{52} - 1138320580 q^{53} + 1931800000 q^{55} + 3990553920 q^{56} + 7387417960 q^{58} - 7317515560 q^{59} - 1516425676 q^{61} + 28564327440 q^{62} + 819531776 q^{64} + 10669562500 q^{65} + 15734290140 q^{67} + 4573774720 q^{68} - 4170225000 q^{70} - 32938471544 q^{71} - 29982848860 q^{73} - 68768198072 q^{74} - 1325392640 q^{76} + 34734748800 q^{77} - 3302823120 q^{79} + 18767600000 q^{80} - 74630515640 q^{82} - 13299102420 q^{83} - 4118562500 q^{85} + 56706093896 q^{86} - 80794874880 q^{88} + 12674770860 q^{89} + 90637859064 q^{91} - 203171571840 q^{92} - 60289765528 q^{94} + 16641625000 q^{95} - 3080703740 q^{97} - 130206802940 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −63.7292 −1.40823 −0.704115 0.710086i \(-0.748656\pi\)
−0.704115 + 0.710086i \(0.748656\pi\)
\(3\) 0 0
\(4\) 2013.42 0.983113
\(5\) 3125.00 0.447214
\(6\) 0 0
\(7\) 41926.3 0.942861 0.471431 0.881903i \(-0.343738\pi\)
0.471431 + 0.881903i \(0.343738\pi\)
\(8\) 2204.06 0.0237809
\(9\) 0 0
\(10\) −199154. −0.629780
\(11\) 957905. 1.79334 0.896670 0.442699i \(-0.145979\pi\)
0.896670 + 0.442699i \(0.145979\pi\)
\(12\) 0 0
\(13\) 1.39098e6 1.03904 0.519520 0.854458i \(-0.326111\pi\)
0.519520 + 0.854458i \(0.326111\pi\)
\(14\) −2.67193e6 −1.32777
\(15\) 0 0
\(16\) −4.26394e6 −1.01660
\(17\) −3.76857e6 −0.643736 −0.321868 0.946785i \(-0.604311\pi\)
−0.321868 + 0.946785i \(0.604311\pi\)
\(18\) 0 0
\(19\) 9.41036e6 0.871889 0.435945 0.899973i \(-0.356414\pi\)
0.435945 + 0.899973i \(0.356414\pi\)
\(20\) 6.29192e6 0.439661
\(21\) 0 0
\(22\) −6.10466e7 −2.52544
\(23\) −3.02942e7 −0.981423 −0.490712 0.871322i \(-0.663263\pi\)
−0.490712 + 0.871322i \(0.663263\pi\)
\(24\) 0 0
\(25\) 9.76562e6 0.200000
\(26\) −8.86460e7 −1.46321
\(27\) 0 0
\(28\) 8.44151e7 0.926939
\(29\) −1.03553e8 −0.937502 −0.468751 0.883330i \(-0.655296\pi\)
−0.468751 + 0.883330i \(0.655296\pi\)
\(30\) 0 0
\(31\) −5.48554e7 −0.344136 −0.172068 0.985085i \(-0.555045\pi\)
−0.172068 + 0.985085i \(0.555045\pi\)
\(32\) 2.67224e8 1.40783
\(33\) 0 0
\(34\) 2.40168e8 0.906529
\(35\) 1.31020e8 0.421660
\(36\) 0 0
\(37\) 4.78282e8 1.13390 0.566950 0.823752i \(-0.308123\pi\)
0.566950 + 0.823752i \(0.308123\pi\)
\(38\) −5.99715e8 −1.22782
\(39\) 0 0
\(40\) 6.88767e6 0.0106351
\(41\) 9.29557e8 1.25304 0.626520 0.779406i \(-0.284479\pi\)
0.626520 + 0.779406i \(0.284479\pi\)
\(42\) 0 0
\(43\) −2.68608e7 −0.0278639 −0.0139320 0.999903i \(-0.504435\pi\)
−0.0139320 + 0.999903i \(0.504435\pi\)
\(44\) 1.92866e9 1.76306
\(45\) 0 0
\(46\) 1.93063e9 1.38207
\(47\) 1.20497e9 0.766370 0.383185 0.923672i \(-0.374827\pi\)
0.383185 + 0.923672i \(0.374827\pi\)
\(48\) 0 0
\(49\) −2.19508e8 −0.111013
\(50\) −6.22356e8 −0.281646
\(51\) 0 0
\(52\) 2.80062e9 1.02149
\(53\) 4.02058e9 1.32060 0.660301 0.751001i \(-0.270429\pi\)
0.660301 + 0.751001i \(0.270429\pi\)
\(54\) 0 0
\(55\) 2.99345e9 0.802006
\(56\) 9.24080e7 0.0224221
\(57\) 0 0
\(58\) 6.59933e9 1.32022
\(59\) −7.97972e9 −1.45312 −0.726560 0.687103i \(-0.758882\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(60\) 0 0
\(61\) −2.07472e9 −0.314518 −0.157259 0.987557i \(-0.550266\pi\)
−0.157259 + 0.987557i \(0.550266\pi\)
\(62\) 3.49589e9 0.484623
\(63\) 0 0
\(64\) −8.29741e9 −0.965946
\(65\) 4.34681e9 0.464673
\(66\) 0 0
\(67\) 5.61370e9 0.507969 0.253985 0.967208i \(-0.418259\pi\)
0.253985 + 0.967208i \(0.418259\pi\)
\(68\) −7.58770e9 −0.632865
\(69\) 0 0
\(70\) −8.34979e9 −0.593795
\(71\) −1.51224e10 −0.994714 −0.497357 0.867546i \(-0.665696\pi\)
−0.497357 + 0.867546i \(0.665696\pi\)
\(72\) 0 0
\(73\) −6.64484e9 −0.375153 −0.187577 0.982250i \(-0.560063\pi\)
−0.187577 + 0.982250i \(0.560063\pi\)
\(74\) −3.04806e10 −1.59679
\(75\) 0 0
\(76\) 1.89470e10 0.857166
\(77\) 4.01615e10 1.69087
\(78\) 0 0
\(79\) −1.57985e10 −0.577652 −0.288826 0.957382i \(-0.593265\pi\)
−0.288826 + 0.957382i \(0.593265\pi\)
\(80\) −1.33248e10 −0.454638
\(81\) 0 0
\(82\) −5.92399e10 −1.76457
\(83\) 2.04046e10 0.568589 0.284295 0.958737i \(-0.408241\pi\)
0.284295 + 0.958737i \(0.408241\pi\)
\(84\) 0 0
\(85\) −1.17768e10 −0.287888
\(86\) 1.71182e9 0.0392389
\(87\) 0 0
\(88\) 2.11128e9 0.0426472
\(89\) 4.21030e10 0.799223 0.399611 0.916685i \(-0.369145\pi\)
0.399611 + 0.916685i \(0.369145\pi\)
\(90\) 0 0
\(91\) 5.83187e10 0.979670
\(92\) −6.09948e10 −0.964850
\(93\) 0 0
\(94\) −7.67919e10 −1.07923
\(95\) 2.94074e10 0.389921
\(96\) 0 0
\(97\) 1.10181e11 1.30275 0.651376 0.758755i \(-0.274192\pi\)
0.651376 + 0.758755i \(0.274192\pi\)
\(98\) 1.39891e10 0.156331
\(99\) 0 0
\(100\) 1.96623e10 0.196623
\(101\) 3.33271e10 0.315522 0.157761 0.987477i \(-0.449572\pi\)
0.157761 + 0.987477i \(0.449572\pi\)
\(102\) 0 0
\(103\) 8.49419e10 0.721966 0.360983 0.932572i \(-0.382441\pi\)
0.360983 + 0.932572i \(0.382441\pi\)
\(104\) 3.06580e9 0.0247093
\(105\) 0 0
\(106\) −2.56229e11 −1.85971
\(107\) −1.43774e11 −0.990989 −0.495494 0.868611i \(-0.665013\pi\)
−0.495494 + 0.868611i \(0.665013\pi\)
\(108\) 0 0
\(109\) −3.01296e11 −1.87563 −0.937816 0.347134i \(-0.887155\pi\)
−0.937816 + 0.347134i \(0.887155\pi\)
\(110\) −1.90771e11 −1.12941
\(111\) 0 0
\(112\) −1.78771e11 −0.958515
\(113\) 2.22891e11 1.13805 0.569026 0.822320i \(-0.307320\pi\)
0.569026 + 0.822320i \(0.307320\pi\)
\(114\) 0 0
\(115\) −9.46694e10 −0.438906
\(116\) −2.08495e11 −0.921671
\(117\) 0 0
\(118\) 5.08541e11 2.04633
\(119\) −1.58003e11 −0.606954
\(120\) 0 0
\(121\) 6.32271e11 2.21607
\(122\) 1.32220e11 0.442914
\(123\) 0 0
\(124\) −1.10447e11 −0.338324
\(125\) 3.05176e10 0.0894427
\(126\) 0 0
\(127\) −1.79939e11 −0.483287 −0.241644 0.970365i \(-0.577687\pi\)
−0.241644 + 0.970365i \(0.577687\pi\)
\(128\) −1.84863e10 −0.0475549
\(129\) 0 0
\(130\) −2.77019e11 −0.654366
\(131\) 5.65153e11 1.27989 0.639946 0.768420i \(-0.278957\pi\)
0.639946 + 0.768420i \(0.278957\pi\)
\(132\) 0 0
\(133\) 3.94542e11 0.822071
\(134\) −3.57757e11 −0.715338
\(135\) 0 0
\(136\) −8.30615e9 −0.0153086
\(137\) −1.24499e9 −0.00220396 −0.00110198 0.999999i \(-0.500351\pi\)
−0.00110198 + 0.999999i \(0.500351\pi\)
\(138\) 0 0
\(139\) 4.68084e11 0.765142 0.382571 0.923926i \(-0.375039\pi\)
0.382571 + 0.923926i \(0.375039\pi\)
\(140\) 2.63797e11 0.414540
\(141\) 0 0
\(142\) 9.63736e11 1.40079
\(143\) 1.33243e12 1.86335
\(144\) 0 0
\(145\) −3.23602e11 −0.419264
\(146\) 4.23470e11 0.528302
\(147\) 0 0
\(148\) 9.62981e11 1.11475
\(149\) −1.05439e12 −1.17619 −0.588096 0.808791i \(-0.700122\pi\)
−0.588096 + 0.808791i \(0.700122\pi\)
\(150\) 0 0
\(151\) 4.91301e11 0.509301 0.254651 0.967033i \(-0.418040\pi\)
0.254651 + 0.967033i \(0.418040\pi\)
\(152\) 2.07410e10 0.0207343
\(153\) 0 0
\(154\) −2.55946e12 −2.38114
\(155\) −1.71423e11 −0.153902
\(156\) 0 0
\(157\) 6.17375e11 0.516536 0.258268 0.966073i \(-0.416848\pi\)
0.258268 + 0.966073i \(0.416848\pi\)
\(158\) 1.00683e12 0.813467
\(159\) 0 0
\(160\) 8.35074e11 0.629600
\(161\) −1.27013e12 −0.925346
\(162\) 0 0
\(163\) 2.97234e11 0.202333 0.101166 0.994870i \(-0.467743\pi\)
0.101166 + 0.994870i \(0.467743\pi\)
\(164\) 1.87158e12 1.23188
\(165\) 0 0
\(166\) −1.30037e12 −0.800705
\(167\) 2.53524e11 0.151035 0.0755177 0.997144i \(-0.475939\pi\)
0.0755177 + 0.997144i \(0.475939\pi\)
\(168\) 0 0
\(169\) 1.42662e11 0.0796035
\(170\) 7.50526e11 0.405412
\(171\) 0 0
\(172\) −5.40820e10 −0.0273934
\(173\) −1.80555e12 −0.885843 −0.442922 0.896560i \(-0.646058\pi\)
−0.442922 + 0.896560i \(0.646058\pi\)
\(174\) 0 0
\(175\) 4.09437e11 0.188572
\(176\) −4.08445e12 −1.82311
\(177\) 0 0
\(178\) −2.68319e12 −1.12549
\(179\) −1.63020e12 −0.663054 −0.331527 0.943446i \(-0.607564\pi\)
−0.331527 + 0.943446i \(0.607564\pi\)
\(180\) 0 0
\(181\) −4.16028e12 −1.59181 −0.795904 0.605423i \(-0.793004\pi\)
−0.795904 + 0.605423i \(0.793004\pi\)
\(182\) −3.71660e12 −1.37960
\(183\) 0 0
\(184\) −6.67701e10 −0.0233391
\(185\) 1.49463e12 0.507096
\(186\) 0 0
\(187\) −3.60994e12 −1.15444
\(188\) 2.42611e12 0.753429
\(189\) 0 0
\(190\) −1.87411e12 −0.549098
\(191\) 3.03600e12 0.864208 0.432104 0.901824i \(-0.357771\pi\)
0.432104 + 0.901824i \(0.357771\pi\)
\(192\) 0 0
\(193\) 3.77397e12 1.01445 0.507227 0.861812i \(-0.330670\pi\)
0.507227 + 0.861812i \(0.330670\pi\)
\(194\) −7.02174e12 −1.83457
\(195\) 0 0
\(196\) −4.41961e11 −0.109138
\(197\) 1.03079e12 0.247518 0.123759 0.992312i \(-0.460505\pi\)
0.123759 + 0.992312i \(0.460505\pi\)
\(198\) 0 0
\(199\) 1.14293e12 0.259614 0.129807 0.991539i \(-0.458564\pi\)
0.129807 + 0.991539i \(0.458564\pi\)
\(200\) 2.15240e10 0.00475617
\(201\) 0 0
\(202\) −2.12391e12 −0.444328
\(203\) −4.34159e12 −0.883935
\(204\) 0 0
\(205\) 2.90487e12 0.560376
\(206\) −5.41328e12 −1.01670
\(207\) 0 0
\(208\) −5.93105e12 −1.05629
\(209\) 9.01423e12 1.56359
\(210\) 0 0
\(211\) 6.96492e12 1.14647 0.573235 0.819391i \(-0.305688\pi\)
0.573235 + 0.819391i \(0.305688\pi\)
\(212\) 8.09510e12 1.29830
\(213\) 0 0
\(214\) 9.16259e12 1.39554
\(215\) −8.39400e10 −0.0124611
\(216\) 0 0
\(217\) −2.29989e12 −0.324472
\(218\) 1.92014e13 2.64132
\(219\) 0 0
\(220\) 6.02707e12 0.788463
\(221\) −5.24201e12 −0.668868
\(222\) 0 0
\(223\) 6.80403e12 0.826208 0.413104 0.910684i \(-0.364445\pi\)
0.413104 + 0.910684i \(0.364445\pi\)
\(224\) 1.12037e13 1.32739
\(225\) 0 0
\(226\) −1.42047e13 −1.60264
\(227\) −8.00368e12 −0.881348 −0.440674 0.897667i \(-0.645261\pi\)
−0.440674 + 0.897667i \(0.645261\pi\)
\(228\) 0 0
\(229\) 1.20624e13 1.26572 0.632862 0.774265i \(-0.281880\pi\)
0.632862 + 0.774265i \(0.281880\pi\)
\(230\) 6.03321e12 0.618081
\(231\) 0 0
\(232\) −2.28236e11 −0.0222946
\(233\) 1.05744e13 1.00878 0.504390 0.863476i \(-0.331717\pi\)
0.504390 + 0.863476i \(0.331717\pi\)
\(234\) 0 0
\(235\) 3.76554e12 0.342731
\(236\) −1.60665e13 −1.42858
\(237\) 0 0
\(238\) 1.00694e13 0.854731
\(239\) 4.81329e12 0.399258 0.199629 0.979872i \(-0.436026\pi\)
0.199629 + 0.979872i \(0.436026\pi\)
\(240\) 0 0
\(241\) −1.29674e13 −1.02745 −0.513724 0.857956i \(-0.671734\pi\)
−0.513724 + 0.857956i \(0.671734\pi\)
\(242\) −4.02941e13 −3.12074
\(243\) 0 0
\(244\) −4.17728e12 −0.309207
\(245\) −6.85963e11 −0.0496464
\(246\) 0 0
\(247\) 1.30896e13 0.905928
\(248\) −1.20904e11 −0.00818385
\(249\) 0 0
\(250\) −1.94486e12 −0.125956
\(251\) −5.10147e12 −0.323214 −0.161607 0.986855i \(-0.551668\pi\)
−0.161607 + 0.986855i \(0.551668\pi\)
\(252\) 0 0
\(253\) −2.90190e13 −1.76003
\(254\) 1.14674e13 0.680580
\(255\) 0 0
\(256\) 1.81712e13 1.03291
\(257\) 1.20976e13 0.673079 0.336539 0.941669i \(-0.390744\pi\)
0.336539 + 0.941669i \(0.390744\pi\)
\(258\) 0 0
\(259\) 2.00526e13 1.06911
\(260\) 8.75193e12 0.456826
\(261\) 0 0
\(262\) −3.60167e13 −1.80238
\(263\) 1.09431e13 0.536271 0.268135 0.963381i \(-0.413593\pi\)
0.268135 + 0.963381i \(0.413593\pi\)
\(264\) 0 0
\(265\) 1.25643e13 0.590591
\(266\) −2.51439e13 −1.15767
\(267\) 0 0
\(268\) 1.13027e13 0.499391
\(269\) −1.23621e13 −0.535125 −0.267562 0.963541i \(-0.586218\pi\)
−0.267562 + 0.963541i \(0.586218\pi\)
\(270\) 0 0
\(271\) 1.34683e13 0.559734 0.279867 0.960039i \(-0.409710\pi\)
0.279867 + 0.960039i \(0.409710\pi\)
\(272\) 1.60690e13 0.654424
\(273\) 0 0
\(274\) 7.93424e10 0.00310368
\(275\) 9.35454e12 0.358668
\(276\) 0 0
\(277\) 2.20976e13 0.814153 0.407077 0.913394i \(-0.366548\pi\)
0.407077 + 0.913394i \(0.366548\pi\)
\(278\) −2.98306e13 −1.07750
\(279\) 0 0
\(280\) 2.88775e11 0.0100274
\(281\) −7.15247e12 −0.243541 −0.121770 0.992558i \(-0.538857\pi\)
−0.121770 + 0.992558i \(0.538857\pi\)
\(282\) 0 0
\(283\) −9.90996e12 −0.324524 −0.162262 0.986748i \(-0.551879\pi\)
−0.162262 + 0.986748i \(0.551879\pi\)
\(284\) −3.04476e13 −0.977917
\(285\) 0 0
\(286\) −8.49145e13 −2.62403
\(287\) 3.89729e13 1.18144
\(288\) 0 0
\(289\) −2.00697e13 −0.585604
\(290\) 2.06229e13 0.590420
\(291\) 0 0
\(292\) −1.33788e13 −0.368818
\(293\) 6.76585e13 1.83042 0.915209 0.402980i \(-0.132026\pi\)
0.915209 + 0.402980i \(0.132026\pi\)
\(294\) 0 0
\(295\) −2.49366e13 −0.649855
\(296\) 1.05416e12 0.0269651
\(297\) 0 0
\(298\) 6.71957e13 1.65635
\(299\) −4.21386e13 −1.01974
\(300\) 0 0
\(301\) −1.12618e12 −0.0262718
\(302\) −3.13103e13 −0.717213
\(303\) 0 0
\(304\) −4.01252e13 −0.886364
\(305\) −6.48351e12 −0.140657
\(306\) 0 0
\(307\) −2.76335e13 −0.578328 −0.289164 0.957280i \(-0.593377\pi\)
−0.289164 + 0.957280i \(0.593377\pi\)
\(308\) 8.08617e13 1.66232
\(309\) 0 0
\(310\) 1.09247e13 0.216730
\(311\) 1.21728e13 0.237251 0.118626 0.992939i \(-0.462151\pi\)
0.118626 + 0.992939i \(0.462151\pi\)
\(312\) 0 0
\(313\) 8.17271e13 1.53770 0.768851 0.639428i \(-0.220829\pi\)
0.768851 + 0.639428i \(0.220829\pi\)
\(314\) −3.93448e13 −0.727402
\(315\) 0 0
\(316\) −3.18089e13 −0.567897
\(317\) −7.61714e13 −1.33649 −0.668246 0.743941i \(-0.732955\pi\)
−0.668246 + 0.743941i \(0.732955\pi\)
\(318\) 0 0
\(319\) −9.91937e13 −1.68126
\(320\) −2.59294e13 −0.431984
\(321\) 0 0
\(322\) 8.09441e13 1.30310
\(323\) −3.54636e13 −0.561267
\(324\) 0 0
\(325\) 1.35838e13 0.207808
\(326\) −1.89425e13 −0.284931
\(327\) 0 0
\(328\) 2.04880e12 0.0297984
\(329\) 5.05201e13 0.722581
\(330\) 0 0
\(331\) 8.05093e13 1.11376 0.556881 0.830592i \(-0.311998\pi\)
0.556881 + 0.830592i \(0.311998\pi\)
\(332\) 4.10830e13 0.558988
\(333\) 0 0
\(334\) −1.61569e13 −0.212693
\(335\) 1.75428e13 0.227171
\(336\) 0 0
\(337\) −7.90669e13 −0.990901 −0.495451 0.868636i \(-0.664997\pi\)
−0.495451 + 0.868636i \(0.664997\pi\)
\(338\) −9.09176e12 −0.112100
\(339\) 0 0
\(340\) −2.37116e13 −0.283026
\(341\) −5.25463e13 −0.617153
\(342\) 0 0
\(343\) −9.21053e13 −1.04753
\(344\) −5.92027e10 −0.000662629 0
\(345\) 0 0
\(346\) 1.15067e14 1.24747
\(347\) −1.40156e14 −1.49554 −0.747772 0.663956i \(-0.768876\pi\)
−0.747772 + 0.663956i \(0.768876\pi\)
\(348\) 0 0
\(349\) −9.88347e13 −1.02181 −0.510905 0.859637i \(-0.670689\pi\)
−0.510905 + 0.859637i \(0.670689\pi\)
\(350\) −2.60931e13 −0.265553
\(351\) 0 0
\(352\) 2.55975e14 2.52472
\(353\) 2.74254e13 0.266313 0.133156 0.991095i \(-0.457489\pi\)
0.133156 + 0.991095i \(0.457489\pi\)
\(354\) 0 0
\(355\) −4.72573e13 −0.444850
\(356\) 8.47708e13 0.785726
\(357\) 0 0
\(358\) 1.03891e14 0.933733
\(359\) −1.57856e14 −1.39715 −0.698574 0.715538i \(-0.746182\pi\)
−0.698574 + 0.715538i \(0.746182\pi\)
\(360\) 0 0
\(361\) −2.79354e13 −0.239809
\(362\) 2.65132e14 2.24163
\(363\) 0 0
\(364\) 1.17420e14 0.963127
\(365\) −2.07651e13 −0.167774
\(366\) 0 0
\(367\) −1.95534e14 −1.53306 −0.766529 0.642210i \(-0.778018\pi\)
−0.766529 + 0.642210i \(0.778018\pi\)
\(368\) 1.29173e14 0.997717
\(369\) 0 0
\(370\) −9.52518e13 −0.714107
\(371\) 1.68568e14 1.24514
\(372\) 0 0
\(373\) 8.35940e13 0.599482 0.299741 0.954021i \(-0.403100\pi\)
0.299741 + 0.954021i \(0.403100\pi\)
\(374\) 2.30059e14 1.62572
\(375\) 0 0
\(376\) 2.65583e12 0.0182249
\(377\) −1.44040e14 −0.974102
\(378\) 0 0
\(379\) 2.81905e14 1.85177 0.925887 0.377802i \(-0.123320\pi\)
0.925887 + 0.377802i \(0.123320\pi\)
\(380\) 5.92093e13 0.383336
\(381\) 0 0
\(382\) −1.93482e14 −1.21700
\(383\) −2.51689e13 −0.156053 −0.0780264 0.996951i \(-0.524862\pi\)
−0.0780264 + 0.996951i \(0.524862\pi\)
\(384\) 0 0
\(385\) 1.25505e14 0.756181
\(386\) −2.40512e14 −1.42859
\(387\) 0 0
\(388\) 2.21840e14 1.28075
\(389\) 1.44036e14 0.819879 0.409939 0.912113i \(-0.365550\pi\)
0.409939 + 0.912113i \(0.365550\pi\)
\(390\) 0 0
\(391\) 1.14166e14 0.631778
\(392\) −4.83809e11 −0.00263998
\(393\) 0 0
\(394\) −6.56916e13 −0.348563
\(395\) −4.93703e13 −0.258334
\(396\) 0 0
\(397\) 3.78848e13 0.192804 0.0964021 0.995342i \(-0.469267\pi\)
0.0964021 + 0.995342i \(0.469267\pi\)
\(398\) −7.28380e13 −0.365596
\(399\) 0 0
\(400\) −4.16400e13 −0.203320
\(401\) −1.24111e13 −0.0597747 −0.0298874 0.999553i \(-0.509515\pi\)
−0.0298874 + 0.999553i \(0.509515\pi\)
\(402\) 0 0
\(403\) −7.63027e13 −0.357571
\(404\) 6.71012e13 0.310194
\(405\) 0 0
\(406\) 2.76686e14 1.24478
\(407\) 4.58149e14 2.03347
\(408\) 0 0
\(409\) −4.37138e14 −1.88860 −0.944301 0.329082i \(-0.893261\pi\)
−0.944301 + 0.329082i \(0.893261\pi\)
\(410\) −1.85125e14 −0.789139
\(411\) 0 0
\(412\) 1.71023e14 0.709775
\(413\) −3.34560e14 −1.37009
\(414\) 0 0
\(415\) 6.37644e13 0.254281
\(416\) 3.71702e14 1.46279
\(417\) 0 0
\(418\) −5.74470e14 −2.20190
\(419\) −5.14822e14 −1.94751 −0.973755 0.227598i \(-0.926913\pi\)
−0.973755 + 0.227598i \(0.926913\pi\)
\(420\) 0 0
\(421\) −3.90682e14 −1.43970 −0.719850 0.694130i \(-0.755789\pi\)
−0.719850 + 0.694130i \(0.755789\pi\)
\(422\) −4.43869e14 −1.61449
\(423\) 0 0
\(424\) 8.86159e12 0.0314050
\(425\) −3.68025e13 −0.128747
\(426\) 0 0
\(427\) −8.69855e13 −0.296547
\(428\) −2.89476e14 −0.974254
\(429\) 0 0
\(430\) 5.34943e12 0.0175482
\(431\) 3.29050e14 1.06571 0.532853 0.846208i \(-0.321120\pi\)
0.532853 + 0.846208i \(0.321120\pi\)
\(432\) 0 0
\(433\) −5.59793e14 −1.76744 −0.883718 0.468019i \(-0.844968\pi\)
−0.883718 + 0.468019i \(0.844968\pi\)
\(434\) 1.46570e14 0.456932
\(435\) 0 0
\(436\) −6.06634e14 −1.84396
\(437\) −2.85079e14 −0.855693
\(438\) 0 0
\(439\) 1.25752e14 0.368095 0.184047 0.982917i \(-0.441080\pi\)
0.184047 + 0.982917i \(0.441080\pi\)
\(440\) 6.59774e12 0.0190724
\(441\) 0 0
\(442\) 3.34069e14 0.941920
\(443\) −3.33211e14 −0.927894 −0.463947 0.885863i \(-0.653567\pi\)
−0.463947 + 0.885863i \(0.653567\pi\)
\(444\) 0 0
\(445\) 1.31572e14 0.357423
\(446\) −4.33616e14 −1.16349
\(447\) 0 0
\(448\) −3.47880e14 −0.910753
\(449\) 1.08196e14 0.279804 0.139902 0.990165i \(-0.455321\pi\)
0.139902 + 0.990165i \(0.455321\pi\)
\(450\) 0 0
\(451\) 8.90427e14 2.24713
\(452\) 4.48773e14 1.11883
\(453\) 0 0
\(454\) 5.10068e14 1.24114
\(455\) 1.82246e14 0.438122
\(456\) 0 0
\(457\) −1.81057e14 −0.424890 −0.212445 0.977173i \(-0.568143\pi\)
−0.212445 + 0.977173i \(0.568143\pi\)
\(458\) −7.68728e14 −1.78243
\(459\) 0 0
\(460\) −1.90609e14 −0.431494
\(461\) 3.74610e14 0.837962 0.418981 0.907995i \(-0.362387\pi\)
0.418981 + 0.907995i \(0.362387\pi\)
\(462\) 0 0
\(463\) 2.33341e13 0.0509678 0.0254839 0.999675i \(-0.491887\pi\)
0.0254839 + 0.999675i \(0.491887\pi\)
\(464\) 4.41542e14 0.953067
\(465\) 0 0
\(466\) −6.73896e14 −1.42060
\(467\) 7.37382e14 1.53621 0.768103 0.640326i \(-0.221201\pi\)
0.768103 + 0.640326i \(0.221201\pi\)
\(468\) 0 0
\(469\) 2.35362e14 0.478945
\(470\) −2.39975e14 −0.482645
\(471\) 0 0
\(472\) −1.75877e13 −0.0345564
\(473\) −2.57301e13 −0.0499695
\(474\) 0 0
\(475\) 9.18980e13 0.174378
\(476\) −3.18125e14 −0.596704
\(477\) 0 0
\(478\) −3.06747e14 −0.562247
\(479\) 3.39424e14 0.615030 0.307515 0.951543i \(-0.400503\pi\)
0.307515 + 0.951543i \(0.400503\pi\)
\(480\) 0 0
\(481\) 6.65281e14 1.17817
\(482\) 8.26404e14 1.44688
\(483\) 0 0
\(484\) 1.27302e15 2.17865
\(485\) 3.44315e14 0.582608
\(486\) 0 0
\(487\) −3.27468e14 −0.541700 −0.270850 0.962622i \(-0.587305\pi\)
−0.270850 + 0.962622i \(0.587305\pi\)
\(488\) −4.57280e12 −0.00747952
\(489\) 0 0
\(490\) 4.37159e13 0.0699135
\(491\) 3.28187e14 0.519006 0.259503 0.965742i \(-0.416441\pi\)
0.259503 + 0.965742i \(0.416441\pi\)
\(492\) 0 0
\(493\) 3.90246e14 0.603504
\(494\) −8.34191e14 −1.27576
\(495\) 0 0
\(496\) 2.33900e14 0.349849
\(497\) −6.34025e14 −0.937878
\(498\) 0 0
\(499\) 2.84812e14 0.412103 0.206052 0.978541i \(-0.433939\pi\)
0.206052 + 0.978541i \(0.433939\pi\)
\(500\) 6.14446e13 0.0879323
\(501\) 0 0
\(502\) 3.25112e14 0.455159
\(503\) 2.01563e13 0.0279117 0.0139559 0.999903i \(-0.495558\pi\)
0.0139559 + 0.999903i \(0.495558\pi\)
\(504\) 0 0
\(505\) 1.04147e14 0.141106
\(506\) 1.84936e15 2.47852
\(507\) 0 0
\(508\) −3.62292e14 −0.475126
\(509\) 6.91697e14 0.897362 0.448681 0.893692i \(-0.351894\pi\)
0.448681 + 0.893692i \(0.351894\pi\)
\(510\) 0 0
\(511\) −2.78594e14 −0.353717
\(512\) −1.12018e15 −1.40703
\(513\) 0 0
\(514\) −7.70969e14 −0.947850
\(515\) 2.65443e14 0.322873
\(516\) 0 0
\(517\) 1.15425e15 1.37436
\(518\) −1.27794e15 −1.50555
\(519\) 0 0
\(520\) 9.58061e12 0.0110503
\(521\) −5.00013e14 −0.570656 −0.285328 0.958430i \(-0.592102\pi\)
−0.285328 + 0.958430i \(0.592102\pi\)
\(522\) 0 0
\(523\) −2.89272e14 −0.323256 −0.161628 0.986852i \(-0.551674\pi\)
−0.161628 + 0.986852i \(0.551674\pi\)
\(524\) 1.13789e15 1.25828
\(525\) 0 0
\(526\) −6.97396e14 −0.755193
\(527\) 2.06727e14 0.221533
\(528\) 0 0
\(529\) −3.50713e13 −0.0368083
\(530\) −8.00715e14 −0.831688
\(531\) 0 0
\(532\) 7.94377e14 0.808188
\(533\) 1.29299e15 1.30196
\(534\) 0 0
\(535\) −4.49293e14 −0.443184
\(536\) 1.23729e13 0.0120800
\(537\) 0 0
\(538\) 7.87828e14 0.753579
\(539\) −2.10268e14 −0.199084
\(540\) 0 0
\(541\) 1.20798e15 1.12066 0.560331 0.828269i \(-0.310674\pi\)
0.560331 + 0.828269i \(0.310674\pi\)
\(542\) −8.58325e14 −0.788235
\(543\) 0 0
\(544\) −1.00705e15 −0.906271
\(545\) −9.41550e14 −0.838808
\(546\) 0 0
\(547\) −1.74686e15 −1.52521 −0.762603 0.646867i \(-0.776079\pi\)
−0.762603 + 0.646867i \(0.776079\pi\)
\(548\) −2.50668e12 −0.00216674
\(549\) 0 0
\(550\) −5.96158e14 −0.505087
\(551\) −9.74468e14 −0.817398
\(552\) 0 0
\(553\) −6.62373e14 −0.544646
\(554\) −1.40826e15 −1.14652
\(555\) 0 0
\(556\) 9.42447e14 0.752221
\(557\) −8.14516e14 −0.643719 −0.321859 0.946787i \(-0.604308\pi\)
−0.321859 + 0.946787i \(0.604308\pi\)
\(558\) 0 0
\(559\) −3.73628e13 −0.0289517
\(560\) −5.58660e14 −0.428661
\(561\) 0 0
\(562\) 4.55821e14 0.342961
\(563\) 8.94179e14 0.666237 0.333118 0.942885i \(-0.391899\pi\)
0.333118 + 0.942885i \(0.391899\pi\)
\(564\) 0 0
\(565\) 6.96536e14 0.508952
\(566\) 6.31554e14 0.457004
\(567\) 0 0
\(568\) −3.33305e13 −0.0236552
\(569\) −5.58478e14 −0.392544 −0.196272 0.980549i \(-0.562884\pi\)
−0.196272 + 0.980549i \(0.562884\pi\)
\(570\) 0 0
\(571\) 1.19826e15 0.826136 0.413068 0.910700i \(-0.364457\pi\)
0.413068 + 0.910700i \(0.364457\pi\)
\(572\) 2.68273e15 1.83189
\(573\) 0 0
\(574\) −2.48371e15 −1.66374
\(575\) −2.95842e14 −0.196285
\(576\) 0 0
\(577\) −1.70453e15 −1.10953 −0.554764 0.832008i \(-0.687192\pi\)
−0.554764 + 0.832008i \(0.687192\pi\)
\(578\) 1.27903e15 0.824665
\(579\) 0 0
\(580\) −6.51545e14 −0.412184
\(581\) 8.55491e14 0.536101
\(582\) 0 0
\(583\) 3.85134e15 2.36829
\(584\) −1.46456e13 −0.00892147
\(585\) 0 0
\(586\) −4.31182e15 −2.57765
\(587\) −8.77553e14 −0.519713 −0.259857 0.965647i \(-0.583675\pi\)
−0.259857 + 0.965647i \(0.583675\pi\)
\(588\) 0 0
\(589\) −5.16209e14 −0.300048
\(590\) 1.58919e15 0.915145
\(591\) 0 0
\(592\) −2.03937e15 −1.15273
\(593\) −3.16216e15 −1.77086 −0.885428 0.464776i \(-0.846135\pi\)
−0.885428 + 0.464776i \(0.846135\pi\)
\(594\) 0 0
\(595\) −4.93758e14 −0.271438
\(596\) −2.12293e15 −1.15633
\(597\) 0 0
\(598\) 2.68546e15 1.43603
\(599\) −1.86633e15 −0.988873 −0.494437 0.869214i \(-0.664626\pi\)
−0.494437 + 0.869214i \(0.664626\pi\)
\(600\) 0 0
\(601\) 8.57728e14 0.446211 0.223105 0.974794i \(-0.428381\pi\)
0.223105 + 0.974794i \(0.428381\pi\)
\(602\) 7.17703e13 0.0369968
\(603\) 0 0
\(604\) 9.89193e14 0.500701
\(605\) 1.97585e15 0.991057
\(606\) 0 0
\(607\) 2.90938e15 1.43306 0.716528 0.697558i \(-0.245730\pi\)
0.716528 + 0.697558i \(0.245730\pi\)
\(608\) 2.51467e15 1.22747
\(609\) 0 0
\(610\) 4.13189e14 0.198077
\(611\) 1.67609e15 0.796289
\(612\) 0 0
\(613\) −3.95415e15 −1.84510 −0.922551 0.385875i \(-0.873900\pi\)
−0.922551 + 0.385875i \(0.873900\pi\)
\(614\) 1.76106e15 0.814419
\(615\) 0 0
\(616\) 8.85181e13 0.0402104
\(617\) 1.44545e15 0.650778 0.325389 0.945580i \(-0.394505\pi\)
0.325389 + 0.945580i \(0.394505\pi\)
\(618\) 0 0
\(619\) −4.27132e15 −1.88914 −0.944569 0.328314i \(-0.893519\pi\)
−0.944569 + 0.328314i \(0.893519\pi\)
\(620\) −3.45146e14 −0.151303
\(621\) 0 0
\(622\) −7.75764e14 −0.334105
\(623\) 1.76522e15 0.753556
\(624\) 0 0
\(625\) 9.53674e13 0.0400000
\(626\) −5.20840e15 −2.16544
\(627\) 0 0
\(628\) 1.24303e15 0.507813
\(629\) −1.80244e15 −0.729933
\(630\) 0 0
\(631\) 2.19799e15 0.874711 0.437355 0.899289i \(-0.355915\pi\)
0.437355 + 0.899289i \(0.355915\pi\)
\(632\) −3.48207e13 −0.0137371
\(633\) 0 0
\(634\) 4.85435e15 1.88209
\(635\) −5.62310e14 −0.216133
\(636\) 0 0
\(637\) −3.05331e14 −0.115347
\(638\) 6.32154e15 2.36760
\(639\) 0 0
\(640\) −5.77698e13 −0.0212672
\(641\) 6.69744e14 0.244450 0.122225 0.992502i \(-0.460997\pi\)
0.122225 + 0.992502i \(0.460997\pi\)
\(642\) 0 0
\(643\) 3.43086e15 1.23096 0.615478 0.788154i \(-0.288963\pi\)
0.615478 + 0.788154i \(0.288963\pi\)
\(644\) −2.55729e15 −0.909720
\(645\) 0 0
\(646\) 2.26007e15 0.790393
\(647\) 2.12502e15 0.736869 0.368435 0.929654i \(-0.379894\pi\)
0.368435 + 0.929654i \(0.379894\pi\)
\(648\) 0 0
\(649\) −7.64381e15 −2.60594
\(650\) −8.65684e14 −0.292641
\(651\) 0 0
\(652\) 5.98455e14 0.198916
\(653\) −4.76025e15 −1.56894 −0.784471 0.620166i \(-0.787065\pi\)
−0.784471 + 0.620166i \(0.787065\pi\)
\(654\) 0 0
\(655\) 1.76610e15 0.572385
\(656\) −3.96357e15 −1.27384
\(657\) 0 0
\(658\) −3.21961e15 −1.01756
\(659\) −1.31734e15 −0.412886 −0.206443 0.978459i \(-0.566189\pi\)
−0.206443 + 0.978459i \(0.566189\pi\)
\(660\) 0 0
\(661\) −2.45528e15 −0.756822 −0.378411 0.925638i \(-0.623529\pi\)
−0.378411 + 0.925638i \(0.623529\pi\)
\(662\) −5.13080e15 −1.56843
\(663\) 0 0
\(664\) 4.49729e13 0.0135215
\(665\) 1.23294e15 0.367641
\(666\) 0 0
\(667\) 3.13705e15 0.920087
\(668\) 5.10449e14 0.148485
\(669\) 0 0
\(670\) −1.11799e15 −0.319909
\(671\) −1.98739e15 −0.564038
\(672\) 0 0
\(673\) −8.60705e14 −0.240310 −0.120155 0.992755i \(-0.538339\pi\)
−0.120155 + 0.992755i \(0.538339\pi\)
\(674\) 5.03888e15 1.39542
\(675\) 0 0
\(676\) 2.87238e14 0.0782593
\(677\) −1.90003e15 −0.513479 −0.256739 0.966481i \(-0.582648\pi\)
−0.256739 + 0.966481i \(0.582648\pi\)
\(678\) 0 0
\(679\) 4.61948e15 1.22831
\(680\) −2.59567e13 −0.00684622
\(681\) 0 0
\(682\) 3.34873e15 0.869093
\(683\) 2.35084e15 0.605214 0.302607 0.953115i \(-0.402143\pi\)
0.302607 + 0.953115i \(0.402143\pi\)
\(684\) 0 0
\(685\) −3.89060e12 −0.000985640 0
\(686\) 5.86980e15 1.47516
\(687\) 0 0
\(688\) 1.14533e14 0.0283265
\(689\) 5.59255e15 1.37216
\(690\) 0 0
\(691\) −6.60668e15 −1.59534 −0.797672 0.603092i \(-0.793935\pi\)
−0.797672 + 0.603092i \(0.793935\pi\)
\(692\) −3.63533e15 −0.870884
\(693\) 0 0
\(694\) 8.93203e15 2.10607
\(695\) 1.46276e15 0.342182
\(696\) 0 0
\(697\) −3.50310e15 −0.806627
\(698\) 6.29866e15 1.43894
\(699\) 0 0
\(700\) 8.24367e14 0.185388
\(701\) 8.01300e14 0.178791 0.0893956 0.995996i \(-0.471506\pi\)
0.0893956 + 0.995996i \(0.471506\pi\)
\(702\) 0 0
\(703\) 4.50081e15 0.988636
\(704\) −7.94813e15 −1.73227
\(705\) 0 0
\(706\) −1.74780e15 −0.375030
\(707\) 1.39728e15 0.297493
\(708\) 0 0
\(709\) 4.10789e15 0.861121 0.430560 0.902562i \(-0.358316\pi\)
0.430560 + 0.902562i \(0.358316\pi\)
\(710\) 3.01167e15 0.626451
\(711\) 0 0
\(712\) 9.27973e13 0.0190062
\(713\) 1.66180e15 0.337743
\(714\) 0 0
\(715\) 4.16383e15 0.833316
\(716\) −3.28227e15 −0.651857
\(717\) 0 0
\(718\) 1.00601e16 1.96751
\(719\) −2.12875e15 −0.413157 −0.206578 0.978430i \(-0.566233\pi\)
−0.206578 + 0.978430i \(0.566233\pi\)
\(720\) 0 0
\(721\) 3.56130e15 0.680714
\(722\) 1.78030e15 0.337706
\(723\) 0 0
\(724\) −8.37637e15 −1.56493
\(725\) −1.01126e15 −0.187500
\(726\) 0 0
\(727\) −4.60450e15 −0.840898 −0.420449 0.907316i \(-0.638127\pi\)
−0.420449 + 0.907316i \(0.638127\pi\)
\(728\) 1.28538e14 0.0232974
\(729\) 0 0
\(730\) 1.32334e15 0.236264
\(731\) 1.01227e14 0.0179370
\(732\) 0 0
\(733\) −3.94830e15 −0.689190 −0.344595 0.938752i \(-0.611984\pi\)
−0.344595 + 0.938752i \(0.611984\pi\)
\(734\) 1.24612e16 2.15890
\(735\) 0 0
\(736\) −8.09532e15 −1.38168
\(737\) 5.37739e15 0.910962
\(738\) 0 0
\(739\) 2.01640e15 0.336537 0.168269 0.985741i \(-0.446182\pi\)
0.168269 + 0.985741i \(0.446182\pi\)
\(740\) 3.00932e15 0.498532
\(741\) 0 0
\(742\) −1.07427e16 −1.75345
\(743\) 7.15627e15 1.15944 0.579720 0.814816i \(-0.303162\pi\)
0.579720 + 0.814816i \(0.303162\pi\)
\(744\) 0 0
\(745\) −3.29498e15 −0.526009
\(746\) −5.32738e15 −0.844209
\(747\) 0 0
\(748\) −7.26830e15 −1.13494
\(749\) −6.02791e15 −0.934365
\(750\) 0 0
\(751\) −2.55257e14 −0.0389904 −0.0194952 0.999810i \(-0.506206\pi\)
−0.0194952 + 0.999810i \(0.506206\pi\)
\(752\) −5.13792e15 −0.779094
\(753\) 0 0
\(754\) 9.17953e15 1.37176
\(755\) 1.53532e15 0.227766
\(756\) 0 0
\(757\) −5.67357e15 −0.829524 −0.414762 0.909930i \(-0.636135\pi\)
−0.414762 + 0.909930i \(0.636135\pi\)
\(758\) −1.79656e16 −2.60772
\(759\) 0 0
\(760\) 6.48155e13 0.00927265
\(761\) −8.59220e15 −1.22036 −0.610181 0.792262i \(-0.708903\pi\)
−0.610181 + 0.792262i \(0.708903\pi\)
\(762\) 0 0
\(763\) −1.26322e16 −1.76846
\(764\) 6.11273e15 0.849614
\(765\) 0 0
\(766\) 1.60400e15 0.219758
\(767\) −1.10996e16 −1.50985
\(768\) 0 0
\(769\) 1.23682e16 1.65849 0.829243 0.558888i \(-0.188772\pi\)
0.829243 + 0.558888i \(0.188772\pi\)
\(770\) −7.99831e15 −1.06488
\(771\) 0 0
\(772\) 7.59856e15 0.997324
\(773\) −8.62674e15 −1.12424 −0.562120 0.827055i \(-0.690014\pi\)
−0.562120 + 0.827055i \(0.690014\pi\)
\(774\) 0 0
\(775\) −5.35697e14 −0.0688272
\(776\) 2.42845e14 0.0309806
\(777\) 0 0
\(778\) −9.17933e15 −1.15458
\(779\) 8.74746e15 1.09251
\(780\) 0 0
\(781\) −1.44858e16 −1.78386
\(782\) −7.27571e15 −0.889689
\(783\) 0 0
\(784\) 9.35970e14 0.112856
\(785\) 1.92930e15 0.231002
\(786\) 0 0
\(787\) 1.18931e16 1.40421 0.702106 0.712073i \(-0.252243\pi\)
0.702106 + 0.712073i \(0.252243\pi\)
\(788\) 2.07541e15 0.243338
\(789\) 0 0
\(790\) 3.14633e15 0.363794
\(791\) 9.34502e15 1.07302
\(792\) 0 0
\(793\) −2.88589e15 −0.326797
\(794\) −2.41437e15 −0.271513
\(795\) 0 0
\(796\) 2.30119e15 0.255230
\(797\) 6.33100e15 0.697351 0.348676 0.937243i \(-0.386631\pi\)
0.348676 + 0.937243i \(0.386631\pi\)
\(798\) 0 0
\(799\) −4.54103e15 −0.493340
\(800\) 2.60961e15 0.281566
\(801\) 0 0
\(802\) 7.90952e14 0.0841766
\(803\) −6.36512e15 −0.672777
\(804\) 0 0
\(805\) −3.96914e15 −0.413827
\(806\) 4.86271e15 0.503542
\(807\) 0 0
\(808\) 7.34547e13 0.00750339
\(809\) −1.06496e16 −1.08048 −0.540238 0.841512i \(-0.681666\pi\)
−0.540238 + 0.841512i \(0.681666\pi\)
\(810\) 0 0
\(811\) 6.79444e15 0.680047 0.340024 0.940417i \(-0.389565\pi\)
0.340024 + 0.940417i \(0.389565\pi\)
\(812\) −8.74141e15 −0.869008
\(813\) 0 0
\(814\) −2.91975e16 −2.86359
\(815\) 9.28855e14 0.0904860
\(816\) 0 0
\(817\) −2.52770e14 −0.0242943
\(818\) 2.78585e16 2.65959
\(819\) 0 0
\(820\) 5.84870e15 0.550913
\(821\) −2.03099e16 −1.90029 −0.950145 0.311807i \(-0.899066\pi\)
−0.950145 + 0.311807i \(0.899066\pi\)
\(822\) 0 0
\(823\) −1.88821e15 −0.174322 −0.0871609 0.996194i \(-0.527779\pi\)
−0.0871609 + 0.996194i \(0.527779\pi\)
\(824\) 1.87217e14 0.0171690
\(825\) 0 0
\(826\) 2.13213e16 1.92940
\(827\) 1.10204e16 0.990644 0.495322 0.868710i \(-0.335050\pi\)
0.495322 + 0.868710i \(0.335050\pi\)
\(828\) 0 0
\(829\) −8.45180e15 −0.749720 −0.374860 0.927081i \(-0.622309\pi\)
−0.374860 + 0.927081i \(0.622309\pi\)
\(830\) −4.06366e15 −0.358086
\(831\) 0 0
\(832\) −1.15415e16 −1.00366
\(833\) 8.27233e14 0.0714629
\(834\) 0 0
\(835\) 7.92263e14 0.0675451
\(836\) 1.81494e16 1.53719
\(837\) 0 0
\(838\) 3.28092e16 2.74254
\(839\) −5.30367e15 −0.440439 −0.220219 0.975450i \(-0.570677\pi\)
−0.220219 + 0.975450i \(0.570677\pi\)
\(840\) 0 0
\(841\) −1.47735e15 −0.121089
\(842\) 2.48979e16 2.02743
\(843\) 0 0
\(844\) 1.40233e16 1.12711
\(845\) 4.45820e14 0.0355998
\(846\) 0 0
\(847\) 2.65088e16 2.08945
\(848\) −1.71435e16 −1.34253
\(849\) 0 0
\(850\) 2.34539e15 0.181306
\(851\) −1.44892e16 −1.11284
\(852\) 0 0
\(853\) −4.43767e15 −0.336462 −0.168231 0.985748i \(-0.553805\pi\)
−0.168231 + 0.985748i \(0.553805\pi\)
\(854\) 5.54352e15 0.417607
\(855\) 0 0
\(856\) −3.16885e14 −0.0235666
\(857\) 2.12803e16 1.57247 0.786236 0.617926i \(-0.212027\pi\)
0.786236 + 0.617926i \(0.212027\pi\)
\(858\) 0 0
\(859\) −4.61602e15 −0.336748 −0.168374 0.985723i \(-0.553852\pi\)
−0.168374 + 0.985723i \(0.553852\pi\)
\(860\) −1.69006e14 −0.0122507
\(861\) 0 0
\(862\) −2.09701e16 −1.50076
\(863\) −6.10081e15 −0.433839 −0.216919 0.976190i \(-0.569601\pi\)
−0.216919 + 0.976190i \(0.569601\pi\)
\(864\) 0 0
\(865\) −5.64235e15 −0.396161
\(866\) 3.56752e16 2.48896
\(867\) 0 0
\(868\) −4.63062e15 −0.318993
\(869\) −1.51334e16 −1.03593
\(870\) 0 0
\(871\) 7.80854e15 0.527800
\(872\) −6.64073e14 −0.0446041
\(873\) 0 0
\(874\) 1.81679e16 1.20501
\(875\) 1.27949e15 0.0843321
\(876\) 0 0
\(877\) 9.32501e14 0.0606948 0.0303474 0.999539i \(-0.490339\pi\)
0.0303474 + 0.999539i \(0.490339\pi\)
\(878\) −8.01407e15 −0.518362
\(879\) 0 0
\(880\) −1.27639e16 −0.815321
\(881\) 1.26049e16 0.800150 0.400075 0.916482i \(-0.368984\pi\)
0.400075 + 0.916482i \(0.368984\pi\)
\(882\) 0 0
\(883\) 2.18236e16 1.36818 0.684090 0.729397i \(-0.260199\pi\)
0.684090 + 0.729397i \(0.260199\pi\)
\(884\) −1.05543e16 −0.657572
\(885\) 0 0
\(886\) 2.12353e16 1.30669
\(887\) 2.71175e16 1.65832 0.829162 0.559008i \(-0.188818\pi\)
0.829162 + 0.559008i \(0.188818\pi\)
\(888\) 0 0
\(889\) −7.54420e15 −0.455673
\(890\) −8.38497e15 −0.503334
\(891\) 0 0
\(892\) 1.36993e16 0.812256
\(893\) 1.13392e16 0.668190
\(894\) 0 0
\(895\) −5.09437e15 −0.296527
\(896\) −7.75064e14 −0.0448377
\(897\) 0 0
\(898\) −6.89522e15 −0.394029
\(899\) 5.68042e15 0.322628
\(900\) 0 0
\(901\) −1.51519e16 −0.850119
\(902\) −5.67463e16 −3.16447
\(903\) 0 0
\(904\) 4.91265e14 0.0270638
\(905\) −1.30009e16 −0.711878
\(906\) 0 0
\(907\) 2.41689e16 1.30742 0.653712 0.756744i \(-0.273211\pi\)
0.653712 + 0.756744i \(0.273211\pi\)
\(908\) −1.61147e16 −0.866465
\(909\) 0 0
\(910\) −1.16144e16 −0.616977
\(911\) 2.08024e16 1.09841 0.549203 0.835689i \(-0.314931\pi\)
0.549203 + 0.835689i \(0.314931\pi\)
\(912\) 0 0
\(913\) 1.95457e16 1.01967
\(914\) 1.15386e16 0.598342
\(915\) 0 0
\(916\) 2.42866e16 1.24435
\(917\) 2.36948e16 1.20676
\(918\) 0 0
\(919\) −1.27606e16 −0.642151 −0.321076 0.947054i \(-0.604044\pi\)
−0.321076 + 0.947054i \(0.604044\pi\)
\(920\) −2.08657e14 −0.0104376
\(921\) 0 0
\(922\) −2.38736e16 −1.18004
\(923\) −2.10349e16 −1.03355
\(924\) 0 0
\(925\) 4.67073e15 0.226780
\(926\) −1.48707e15 −0.0717744
\(927\) 0 0
\(928\) −2.76717e16 −1.31984
\(929\) −2.66057e16 −1.26150 −0.630751 0.775985i \(-0.717253\pi\)
−0.630751 + 0.775985i \(0.717253\pi\)
\(930\) 0 0
\(931\) −2.06565e15 −0.0967908
\(932\) 2.12906e16 0.991745
\(933\) 0 0
\(934\) −4.69928e16 −2.16333
\(935\) −1.12811e16 −0.516281
\(936\) 0 0
\(937\) −1.35312e16 −0.612023 −0.306012 0.952028i \(-0.598995\pi\)
−0.306012 + 0.952028i \(0.598995\pi\)
\(938\) −1.49994e16 −0.674465
\(939\) 0 0
\(940\) 7.58159e15 0.336944
\(941\) −2.75735e16 −1.21829 −0.609143 0.793061i \(-0.708486\pi\)
−0.609143 + 0.793061i \(0.708486\pi\)
\(942\) 0 0
\(943\) −2.81602e16 −1.22976
\(944\) 3.40250e16 1.47724
\(945\) 0 0
\(946\) 1.63976e15 0.0703686
\(947\) −4.64018e16 −1.97975 −0.989873 0.141955i \(-0.954661\pi\)
−0.989873 + 0.141955i \(0.954661\pi\)
\(948\) 0 0
\(949\) −9.24283e15 −0.389799
\(950\) −5.85659e15 −0.245564
\(951\) 0 0
\(952\) −3.48246e14 −0.0144339
\(953\) −5.27189e15 −0.217248 −0.108624 0.994083i \(-0.534644\pi\)
−0.108624 + 0.994083i \(0.534644\pi\)
\(954\) 0 0
\(955\) 9.48750e15 0.386485
\(956\) 9.69115e15 0.392516
\(957\) 0 0
\(958\) −2.16312e16 −0.866105
\(959\) −5.21979e13 −0.00207803
\(960\) 0 0
\(961\) −2.23994e16 −0.881571
\(962\) −4.23978e16 −1.65913
\(963\) 0 0
\(964\) −2.61088e16 −1.01010
\(965\) 1.17936e16 0.453678
\(966\) 0 0
\(967\) 3.13722e16 1.19316 0.596581 0.802553i \(-0.296526\pi\)
0.596581 + 0.802553i \(0.296526\pi\)
\(968\) 1.39356e15 0.0527001
\(969\) 0 0
\(970\) −2.19429e16 −0.820446
\(971\) 4.02261e16 1.49555 0.747776 0.663951i \(-0.231122\pi\)
0.747776 + 0.663951i \(0.231122\pi\)
\(972\) 0 0
\(973\) 1.96250e16 0.721423
\(974\) 2.08693e16 0.762839
\(975\) 0 0
\(976\) 8.84648e15 0.319740
\(977\) 1.61108e16 0.579025 0.289513 0.957174i \(-0.406507\pi\)
0.289513 + 0.957174i \(0.406507\pi\)
\(978\) 0 0
\(979\) 4.03307e16 1.43328
\(980\) −1.38113e15 −0.0488080
\(981\) 0 0
\(982\) −2.09151e16 −0.730880
\(983\) 3.39027e16 1.17812 0.589060 0.808089i \(-0.299498\pi\)
0.589060 + 0.808089i \(0.299498\pi\)
\(984\) 0 0
\(985\) 3.22123e15 0.110694
\(986\) −2.48701e16 −0.849873
\(987\) 0 0
\(988\) 2.63548e16 0.890629
\(989\) 8.13727e14 0.0273463
\(990\) 0 0
\(991\) −4.18340e16 −1.39035 −0.695174 0.718841i \(-0.744673\pi\)
−0.695174 + 0.718841i \(0.744673\pi\)
\(992\) −1.46586e16 −0.484484
\(993\) 0 0
\(994\) 4.04059e16 1.32075
\(995\) 3.57166e15 0.116103
\(996\) 0 0
\(997\) 3.94947e16 1.26974 0.634871 0.772618i \(-0.281053\pi\)
0.634871 + 0.772618i \(0.281053\pi\)
\(998\) −1.81509e16 −0.580336
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 45.12.a.d.1.1 2
3.2 odd 2 5.12.a.b.1.2 2
5.2 odd 4 225.12.b.f.199.2 4
5.3 odd 4 225.12.b.f.199.3 4
5.4 even 2 225.12.a.h.1.2 2
12.11 even 2 80.12.a.j.1.1 2
15.2 even 4 25.12.b.c.24.3 4
15.8 even 4 25.12.b.c.24.2 4
15.14 odd 2 25.12.a.c.1.1 2
21.20 even 2 245.12.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.12.a.b.1.2 2 3.2 odd 2
25.12.a.c.1.1 2 15.14 odd 2
25.12.b.c.24.2 4 15.8 even 4
25.12.b.c.24.3 4 15.2 even 4
45.12.a.d.1.1 2 1.1 even 1 trivial
80.12.a.j.1.1 2 12.11 even 2
225.12.a.h.1.2 2 5.4 even 2
225.12.b.f.199.2 4 5.2 odd 4
225.12.b.f.199.3 4 5.3 odd 4
245.12.a.b.1.2 2 21.20 even 2