Properties

Label 16.26.a.c.1.2
Level $16$
Weight $26$
Character 16.1
Self dual yes
Analytic conductor $63.359$
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] = [16,26,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(63.3594847924\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-162.829\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.37803e6 q^{3} +8.78994e8 q^{5} -3.00388e10 q^{7} +1.05168e12 q^{9} -5.73599e12 q^{11} -1.07343e13 q^{13} +1.21128e15 q^{15} +2.97473e15 q^{17} +5.42738e15 q^{19} -4.13944e16 q^{21} +1.04540e17 q^{23} +4.74607e17 q^{25} +2.81659e17 q^{27} +3.09182e18 q^{29} +4.26809e18 q^{31} -7.90437e18 q^{33} -2.64039e19 q^{35} -4.51028e19 q^{37} -1.47922e19 q^{39} -7.56724e19 q^{41} +1.39036e20 q^{43} +9.24421e20 q^{45} -4.67328e20 q^{47} -4.38741e20 q^{49} +4.09927e21 q^{51} -1.24315e21 q^{53} -5.04190e21 q^{55} +7.47909e21 q^{57} +1.32468e22 q^{59} -2.36984e20 q^{61} -3.15912e22 q^{63} -9.43539e21 q^{65} +5.36922e22 q^{67} +1.44059e23 q^{69} +2.27032e23 q^{71} +2.78528e23 q^{73} +6.54024e23 q^{75} +1.72302e23 q^{77} -6.36958e23 q^{79} -5.02942e23 q^{81} -1.19662e24 q^{83} +2.61477e24 q^{85} +4.26063e24 q^{87} -3.22134e24 q^{89} +3.22445e23 q^{91} +5.88156e24 q^{93} +4.77063e24 q^{95} +3.58465e24 q^{97} -6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.37803e6 1.49707 0.748537 0.663093i \(-0.230757\pi\)
0.748537 + 0.663093i \(0.230757\pi\)
\(4\) 0 0
\(5\) 8.78994e8 1.61013 0.805065 0.593187i \(-0.202130\pi\)
0.805065 + 0.593187i \(0.202130\pi\)
\(6\) 0 0
\(7\) −3.00388e10 −0.820270 −0.410135 0.912025i \(-0.634518\pi\)
−0.410135 + 0.912025i \(0.634518\pi\)
\(8\) 0 0
\(9\) 1.05168e12 1.24123
\(10\) 0 0
\(11\) −5.73599e12 −0.551061 −0.275531 0.961292i \(-0.588854\pi\)
−0.275531 + 0.961292i \(0.588854\pi\)
\(12\) 0 0
\(13\) −1.07343e13 −0.127786 −0.0638928 0.997957i \(-0.520352\pi\)
−0.0638928 + 0.997957i \(0.520352\pi\)
\(14\) 0 0
\(15\) 1.21128e15 2.41048
\(16\) 0 0
\(17\) 2.97473e15 1.23833 0.619164 0.785262i \(-0.287472\pi\)
0.619164 + 0.785262i \(0.287472\pi\)
\(18\) 0 0
\(19\) 5.42738e15 0.562561 0.281281 0.959626i \(-0.409241\pi\)
0.281281 + 0.959626i \(0.409241\pi\)
\(20\) 0 0
\(21\) −4.13944e16 −1.22800
\(22\) 0 0
\(23\) 1.04540e17 0.994683 0.497341 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(24\) 0 0
\(25\) 4.74607e17 1.59252
\(26\) 0 0
\(27\) 2.81659e17 0.361141
\(28\) 0 0
\(29\) 3.09182e18 1.62270 0.811352 0.584558i \(-0.198732\pi\)
0.811352 + 0.584558i \(0.198732\pi\)
\(30\) 0 0
\(31\) 4.26809e18 0.973222 0.486611 0.873619i \(-0.338233\pi\)
0.486611 + 0.873619i \(0.338233\pi\)
\(32\) 0 0
\(33\) −7.90437e18 −0.824980
\(34\) 0 0
\(35\) −2.64039e19 −1.32074
\(36\) 0 0
\(37\) −4.51028e19 −1.12637 −0.563186 0.826330i \(-0.690424\pi\)
−0.563186 + 0.826330i \(0.690424\pi\)
\(38\) 0 0
\(39\) −1.47922e19 −0.191305
\(40\) 0 0
\(41\) −7.56724e19 −0.523769 −0.261884 0.965099i \(-0.584344\pi\)
−0.261884 + 0.965099i \(0.584344\pi\)
\(42\) 0 0
\(43\) 1.39036e20 0.530605 0.265302 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(44\) 0 0
\(45\) 9.24421e20 1.99854
\(46\) 0 0
\(47\) −4.67328e20 −0.586676 −0.293338 0.956009i \(-0.594766\pi\)
−0.293338 + 0.956009i \(0.594766\pi\)
\(48\) 0 0
\(49\) −4.38741e20 −0.327158
\(50\) 0 0
\(51\) 4.09927e21 1.85387
\(52\) 0 0
\(53\) −1.24315e21 −0.347595 −0.173797 0.984781i \(-0.555604\pi\)
−0.173797 + 0.984781i \(0.555604\pi\)
\(54\) 0 0
\(55\) −5.04190e21 −0.887280
\(56\) 0 0
\(57\) 7.47909e21 0.842196
\(58\) 0 0
\(59\) 1.32468e22 0.969303 0.484652 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 0 0
\(63\) −3.15912e22 −1.01814
\(64\) 0 0
\(65\) −9.43539e21 −0.205752
\(66\) 0 0
\(67\) 5.36922e22 0.801634 0.400817 0.916158i \(-0.368726\pi\)
0.400817 + 0.916158i \(0.368726\pi\)
\(68\) 0 0
\(69\) 1.44059e23 1.48911
\(70\) 0 0
\(71\) 2.27032e23 1.64194 0.820971 0.570970i \(-0.193433\pi\)
0.820971 + 0.570970i \(0.193433\pi\)
\(72\) 0 0
\(73\) 2.78528e23 1.42342 0.711710 0.702473i \(-0.247921\pi\)
0.711710 + 0.702473i \(0.247921\pi\)
\(74\) 0 0
\(75\) 6.54024e23 2.38412
\(76\) 0 0
\(77\) 1.72302e23 0.452019
\(78\) 0 0
\(79\) −6.36958e23 −1.21275 −0.606376 0.795178i \(-0.707378\pi\)
−0.606376 + 0.795178i \(0.707378\pi\)
\(80\) 0 0
\(81\) −5.02942e23 −0.700576
\(82\) 0 0
\(83\) −1.19662e24 −1.22880 −0.614398 0.788996i \(-0.710601\pi\)
−0.614398 + 0.788996i \(0.710601\pi\)
\(84\) 0 0
\(85\) 2.61477e24 1.99387
\(86\) 0 0
\(87\) 4.26063e24 2.42931
\(88\) 0 0
\(89\) −3.22134e24 −1.38249 −0.691244 0.722621i \(-0.742937\pi\)
−0.691244 + 0.722621i \(0.742937\pi\)
\(90\) 0 0
\(91\) 3.22445e23 0.104819
\(92\) 0 0
\(93\) 5.88156e24 1.45699
\(94\) 0 0
\(95\) 4.77063e24 0.905797
\(96\) 0 0
\(97\) 3.58465e24 0.524566 0.262283 0.964991i \(-0.415525\pi\)
0.262283 + 0.964991i \(0.415525\pi\)
\(98\) 0 0
\(99\) −6.03243e24 −0.683994
\(100\) 0 0
\(101\) −3.40630e24 −0.300792 −0.150396 0.988626i \(-0.548055\pi\)
−0.150396 + 0.988626i \(0.548055\pi\)
\(102\) 0 0
\(103\) −8.27075e24 −0.571583 −0.285792 0.958292i \(-0.592257\pi\)
−0.285792 + 0.958292i \(0.592257\pi\)
\(104\) 0 0
\(105\) −3.63854e25 −1.97725
\(106\) 0 0
\(107\) −1.57986e25 −0.678142 −0.339071 0.940761i \(-0.610113\pi\)
−0.339071 + 0.940761i \(0.610113\pi\)
\(108\) 0 0
\(109\) 3.00869e25 1.02458 0.512290 0.858813i \(-0.328797\pi\)
0.512290 + 0.858813i \(0.328797\pi\)
\(110\) 0 0
\(111\) −6.21530e25 −1.68626
\(112\) 0 0
\(113\) −8.06745e25 −1.75088 −0.875439 0.483329i \(-0.839428\pi\)
−0.875439 + 0.483329i \(0.839428\pi\)
\(114\) 0 0
\(115\) 9.18901e25 1.60157
\(116\) 0 0
\(117\) −1.12891e25 −0.158612
\(118\) 0 0
\(119\) −8.93571e25 −1.01576
\(120\) 0 0
\(121\) −7.54455e25 −0.696332
\(122\) 0 0
\(123\) −1.04279e26 −0.784121
\(124\) 0 0
\(125\) 1.55216e26 0.954030
\(126\) 0 0
\(127\) 1.32973e26 0.670221 0.335111 0.942179i \(-0.391226\pi\)
0.335111 + 0.942179i \(0.391226\pi\)
\(128\) 0 0
\(129\) 1.91596e26 0.794355
\(130\) 0 0
\(131\) −6.73908e25 −0.230520 −0.115260 0.993335i \(-0.536770\pi\)
−0.115260 + 0.993335i \(0.536770\pi\)
\(132\) 0 0
\(133\) −1.63032e26 −0.461452
\(134\) 0 0
\(135\) 2.47577e26 0.581484
\(136\) 0 0
\(137\) −3.22963e26 −0.631170 −0.315585 0.948897i \(-0.602201\pi\)
−0.315585 + 0.948897i \(0.602201\pi\)
\(138\) 0 0
\(139\) 9.21581e25 0.150262 0.0751309 0.997174i \(-0.476063\pi\)
0.0751309 + 0.997174i \(0.476063\pi\)
\(140\) 0 0
\(141\) −6.43992e26 −0.878298
\(142\) 0 0
\(143\) 6.15719e25 0.0704177
\(144\) 0 0
\(145\) 2.71769e27 2.61276
\(146\) 0 0
\(147\) −6.04599e26 −0.489779
\(148\) 0 0
\(149\) −7.12667e26 −0.487594 −0.243797 0.969826i \(-0.578393\pi\)
−0.243797 + 0.969826i \(0.578393\pi\)
\(150\) 0 0
\(151\) 3.06161e27 1.77312 0.886559 0.462616i \(-0.153089\pi\)
0.886559 + 0.462616i \(0.153089\pi\)
\(152\) 0 0
\(153\) 3.12846e27 1.53705
\(154\) 0 0
\(155\) 3.75162e27 1.56701
\(156\) 0 0
\(157\) 2.38960e27 0.850315 0.425157 0.905119i \(-0.360219\pi\)
0.425157 + 0.905119i \(0.360219\pi\)
\(158\) 0 0
\(159\) −1.71309e27 −0.520375
\(160\) 0 0
\(161\) −3.14025e27 −0.815908
\(162\) 0 0
\(163\) −6.63092e27 −1.47648 −0.738242 0.674536i \(-0.764344\pi\)
−0.738242 + 0.674536i \(0.764344\pi\)
\(164\) 0 0
\(165\) −6.94790e27 −1.32832
\(166\) 0 0
\(167\) −4.06997e27 −0.669323 −0.334662 0.942338i \(-0.608622\pi\)
−0.334662 + 0.942338i \(0.608622\pi\)
\(168\) 0 0
\(169\) −6.94118e27 −0.983671
\(170\) 0 0
\(171\) 5.70787e27 0.698269
\(172\) 0 0
\(173\) −8.68916e27 −0.919182 −0.459591 0.888131i \(-0.652004\pi\)
−0.459591 + 0.888131i \(0.652004\pi\)
\(174\) 0 0
\(175\) −1.42566e28 −1.30629
\(176\) 0 0
\(177\) 1.82545e28 1.45112
\(178\) 0 0
\(179\) 4.31406e27 0.298005 0.149002 0.988837i \(-0.452394\pi\)
0.149002 + 0.988837i \(0.452394\pi\)
\(180\) 0 0
\(181\) 2.22969e28 1.34049 0.670243 0.742142i \(-0.266190\pi\)
0.670243 + 0.742142i \(0.266190\pi\)
\(182\) 0 0
\(183\) −3.26571e26 −0.0171135
\(184\) 0 0
\(185\) −3.96451e28 −1.81360
\(186\) 0 0
\(187\) −1.70630e28 −0.682394
\(188\) 0 0
\(189\) −8.46070e27 −0.296233
\(190\) 0 0
\(191\) 1.91337e28 0.587329 0.293665 0.955908i \(-0.405125\pi\)
0.293665 + 0.955908i \(0.405125\pi\)
\(192\) 0 0
\(193\) −5.96748e28 −1.60814 −0.804071 0.594533i \(-0.797337\pi\)
−0.804071 + 0.594533i \(0.797337\pi\)
\(194\) 0 0
\(195\) −1.30023e28 −0.308025
\(196\) 0 0
\(197\) 6.24204e28 1.30166 0.650832 0.759222i \(-0.274420\pi\)
0.650832 + 0.759222i \(0.274420\pi\)
\(198\) 0 0
\(199\) 3.24229e28 0.595921 0.297960 0.954578i \(-0.403694\pi\)
0.297960 + 0.954578i \(0.403694\pi\)
\(200\) 0 0
\(201\) 7.39895e28 1.20011
\(202\) 0 0
\(203\) −9.28745e28 −1.33106
\(204\) 0 0
\(205\) −6.65156e28 −0.843335
\(206\) 0 0
\(207\) 1.09943e29 1.23463
\(208\) 0 0
\(209\) −3.11314e28 −0.310006
\(210\) 0 0
\(211\) −3.82197e28 −0.337875 −0.168938 0.985627i \(-0.554034\pi\)
−0.168938 + 0.985627i \(0.554034\pi\)
\(212\) 0 0
\(213\) 3.12857e29 2.45811
\(214\) 0 0
\(215\) 1.22212e29 0.854342
\(216\) 0 0
\(217\) −1.28208e29 −0.798305
\(218\) 0 0
\(219\) 3.83820e29 2.13097
\(220\) 0 0
\(221\) −3.19316e28 −0.158241
\(222\) 0 0
\(223\) −3.11100e29 −1.37749 −0.688746 0.725003i \(-0.741839\pi\)
−0.688746 + 0.725003i \(0.741839\pi\)
\(224\) 0 0
\(225\) 4.99135e29 1.97668
\(226\) 0 0
\(227\) 1.05572e29 0.374304 0.187152 0.982331i \(-0.440074\pi\)
0.187152 + 0.982331i \(0.440074\pi\)
\(228\) 0 0
\(229\) −3.70322e29 −1.17662 −0.588310 0.808635i \(-0.700207\pi\)
−0.588310 + 0.808635i \(0.700207\pi\)
\(230\) 0 0
\(231\) 2.37438e29 0.676706
\(232\) 0 0
\(233\) −6.84781e28 −0.175228 −0.0876138 0.996155i \(-0.527924\pi\)
−0.0876138 + 0.996155i \(0.527924\pi\)
\(234\) 0 0
\(235\) −4.10778e29 −0.944625
\(236\) 0 0
\(237\) −8.77748e29 −1.81558
\(238\) 0 0
\(239\) −3.87773e29 −0.722111 −0.361055 0.932544i \(-0.617583\pi\)
−0.361055 + 0.932544i \(0.617583\pi\)
\(240\) 0 0
\(241\) 1.03723e30 1.74046 0.870230 0.492646i \(-0.163970\pi\)
0.870230 + 0.492646i \(0.163970\pi\)
\(242\) 0 0
\(243\) −9.31717e29 −1.40996
\(244\) 0 0
\(245\) −3.85651e29 −0.526766
\(246\) 0 0
\(247\) −5.82591e28 −0.0718873
\(248\) 0 0
\(249\) −1.64898e30 −1.83960
\(250\) 0 0
\(251\) −1.47990e30 −1.49386 −0.746930 0.664903i \(-0.768473\pi\)
−0.746930 + 0.664903i \(0.768473\pi\)
\(252\) 0 0
\(253\) −5.99641e29 −0.548131
\(254\) 0 0
\(255\) 3.60323e30 2.98497
\(256\) 0 0
\(257\) 6.75384e29 0.507442 0.253721 0.967277i \(-0.418345\pi\)
0.253721 + 0.967277i \(0.418345\pi\)
\(258\) 0 0
\(259\) 1.35483e30 0.923929
\(260\) 0 0
\(261\) 3.25161e30 2.01415
\(262\) 0 0
\(263\) −1.42102e30 −0.800119 −0.400059 0.916489i \(-0.631011\pi\)
−0.400059 + 0.916489i \(0.631011\pi\)
\(264\) 0 0
\(265\) −1.09272e30 −0.559673
\(266\) 0 0
\(267\) −4.43911e30 −2.06969
\(268\) 0 0
\(269\) −1.90129e30 −0.807504 −0.403752 0.914868i \(-0.632294\pi\)
−0.403752 + 0.914868i \(0.632294\pi\)
\(270\) 0 0
\(271\) 1.45800e30 0.564471 0.282236 0.959345i \(-0.408924\pi\)
0.282236 + 0.959345i \(0.408924\pi\)
\(272\) 0 0
\(273\) 4.44340e29 0.156921
\(274\) 0 0
\(275\) −2.72234e30 −0.877575
\(276\) 0 0
\(277\) −5.31088e29 −0.156376 −0.0781879 0.996939i \(-0.524913\pi\)
−0.0781879 + 0.996939i \(0.524913\pi\)
\(278\) 0 0
\(279\) 4.48867e30 1.20799
\(280\) 0 0
\(281\) −3.64974e30 −0.898323 −0.449161 0.893451i \(-0.648277\pi\)
−0.449161 + 0.893451i \(0.648277\pi\)
\(282\) 0 0
\(283\) 1.43965e30 0.324285 0.162143 0.986767i \(-0.448160\pi\)
0.162143 + 0.986767i \(0.448160\pi\)
\(284\) 0 0
\(285\) 6.57408e30 1.35604
\(286\) 0 0
\(287\) 2.27311e30 0.429632
\(288\) 0 0
\(289\) 3.07837e30 0.533456
\(290\) 0 0
\(291\) 4.93976e30 0.785315
\(292\) 0 0
\(293\) 1.24504e31 1.81692 0.908462 0.417967i \(-0.137257\pi\)
0.908462 + 0.417967i \(0.137257\pi\)
\(294\) 0 0
\(295\) 1.16438e31 1.56070
\(296\) 0 0
\(297\) −1.61559e30 −0.199011
\(298\) 0 0
\(299\) −1.12217e30 −0.127106
\(300\) 0 0
\(301\) −4.17647e30 −0.435239
\(302\) 0 0
\(303\) −4.69399e30 −0.450307
\(304\) 0 0
\(305\) −2.08307e29 −0.0184059
\(306\) 0 0
\(307\) −2.27193e31 −1.84997 −0.924986 0.380001i \(-0.875924\pi\)
−0.924986 + 0.380001i \(0.875924\pi\)
\(308\) 0 0
\(309\) −1.13974e31 −0.855703
\(310\) 0 0
\(311\) 1.84336e31 1.27675 0.638374 0.769726i \(-0.279607\pi\)
0.638374 + 0.769726i \(0.279607\pi\)
\(312\) 0 0
\(313\) 4.98693e30 0.318808 0.159404 0.987213i \(-0.449043\pi\)
0.159404 + 0.987213i \(0.449043\pi\)
\(314\) 0 0
\(315\) −2.77685e31 −1.63934
\(316\) 0 0
\(317\) −8.79478e30 −0.479716 −0.239858 0.970808i \(-0.577101\pi\)
−0.239858 + 0.970808i \(0.577101\pi\)
\(318\) 0 0
\(319\) −1.77347e31 −0.894209
\(320\) 0 0
\(321\) −2.17709e31 −1.01523
\(322\) 0 0
\(323\) 1.61450e31 0.696635
\(324\) 0 0
\(325\) −5.09458e30 −0.203501
\(326\) 0 0
\(327\) 4.14606e31 1.53387
\(328\) 0 0
\(329\) 1.40380e31 0.481233
\(330\) 0 0
\(331\) −5.97122e31 −1.89764 −0.948818 0.315822i \(-0.897720\pi\)
−0.948818 + 0.315822i \(0.897720\pi\)
\(332\) 0 0
\(333\) −4.74337e31 −1.39809
\(334\) 0 0
\(335\) 4.71951e31 1.29073
\(336\) 0 0
\(337\) 1.08428e31 0.275275 0.137638 0.990483i \(-0.456049\pi\)
0.137638 + 0.990483i \(0.456049\pi\)
\(338\) 0 0
\(339\) −1.11172e32 −2.62119
\(340\) 0 0
\(341\) −2.44817e31 −0.536305
\(342\) 0 0
\(343\) 5.34633e31 1.08863
\(344\) 0 0
\(345\) 1.26627e32 2.39767
\(346\) 0 0
\(347\) −7.20193e30 −0.126862 −0.0634308 0.997986i \(-0.520204\pi\)
−0.0634308 + 0.997986i \(0.520204\pi\)
\(348\) 0 0
\(349\) 5.27779e31 0.865235 0.432618 0.901577i \(-0.357590\pi\)
0.432618 + 0.901577i \(0.357590\pi\)
\(350\) 0 0
\(351\) −3.02342e30 −0.0461487
\(352\) 0 0
\(353\) −5.67715e31 −0.807135 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(354\) 0 0
\(355\) 1.99560e32 2.64374
\(356\) 0 0
\(357\) −1.23137e32 −1.52067
\(358\) 0 0
\(359\) −2.18925e31 −0.252125 −0.126062 0.992022i \(-0.540234\pi\)
−0.126062 + 0.992022i \(0.540234\pi\)
\(360\) 0 0
\(361\) −6.36201e31 −0.683525
\(362\) 0 0
\(363\) −1.03966e32 −1.04246
\(364\) 0 0
\(365\) 2.44824e32 2.29189
\(366\) 0 0
\(367\) −1.00500e32 −0.878699 −0.439349 0.898316i \(-0.644791\pi\)
−0.439349 + 0.898316i \(0.644791\pi\)
\(368\) 0 0
\(369\) −7.95833e31 −0.650118
\(370\) 0 0
\(371\) 3.73425e31 0.285122
\(372\) 0 0
\(373\) 6.31052e31 0.450510 0.225255 0.974300i \(-0.427678\pi\)
0.225255 + 0.974300i \(0.427678\pi\)
\(374\) 0 0
\(375\) 2.13893e32 1.42825
\(376\) 0 0
\(377\) −3.31886e31 −0.207358
\(378\) 0 0
\(379\) 8.85755e31 0.517993 0.258996 0.965878i \(-0.416608\pi\)
0.258996 + 0.965878i \(0.416608\pi\)
\(380\) 0 0
\(381\) 1.83241e32 1.00337
\(382\) 0 0
\(383\) −9.08315e31 −0.465856 −0.232928 0.972494i \(-0.574831\pi\)
−0.232928 + 0.972494i \(0.574831\pi\)
\(384\) 0 0
\(385\) 1.51453e32 0.727809
\(386\) 0 0
\(387\) 1.46221e32 0.658603
\(388\) 0 0
\(389\) 7.13043e31 0.301124 0.150562 0.988601i \(-0.451892\pi\)
0.150562 + 0.988601i \(0.451892\pi\)
\(390\) 0 0
\(391\) 3.10978e32 1.23174
\(392\) 0 0
\(393\) −9.28666e31 −0.345106
\(394\) 0 0
\(395\) −5.59883e32 −1.95269
\(396\) 0 0
\(397\) 2.99427e31 0.0980413 0.0490206 0.998798i \(-0.484390\pi\)
0.0490206 + 0.998798i \(0.484390\pi\)
\(398\) 0 0
\(399\) −2.24663e32 −0.690828
\(400\) 0 0
\(401\) −5.00588e32 −1.44603 −0.723013 0.690835i \(-0.757243\pi\)
−0.723013 + 0.690835i \(0.757243\pi\)
\(402\) 0 0
\(403\) −4.58150e31 −0.124364
\(404\) 0 0
\(405\) −4.42083e32 −1.12802
\(406\) 0 0
\(407\) 2.58709e32 0.620700
\(408\) 0 0
\(409\) −2.78379e32 −0.628195 −0.314097 0.949391i \(-0.601702\pi\)
−0.314097 + 0.949391i \(0.601702\pi\)
\(410\) 0 0
\(411\) −4.45054e32 −0.944908
\(412\) 0 0
\(413\) −3.97916e32 −0.795090
\(414\) 0 0
\(415\) −1.05182e33 −1.97852
\(416\) 0 0
\(417\) 1.26997e32 0.224953
\(418\) 0 0
\(419\) 6.36828e32 1.06254 0.531272 0.847201i \(-0.321714\pi\)
0.531272 + 0.847201i \(0.321714\pi\)
\(420\) 0 0
\(421\) −3.92275e32 −0.616686 −0.308343 0.951275i \(-0.599775\pi\)
−0.308343 + 0.951275i \(0.599775\pi\)
\(422\) 0 0
\(423\) −4.91480e32 −0.728201
\(424\) 0 0
\(425\) 1.41183e33 1.97206
\(426\) 0 0
\(427\) 7.11870e30 0.00937675
\(428\) 0 0
\(429\) 8.48480e31 0.105421
\(430\) 0 0
\(431\) −2.44178e32 −0.286247 −0.143124 0.989705i \(-0.545715\pi\)
−0.143124 + 0.989705i \(0.545715\pi\)
\(432\) 0 0
\(433\) 9.12872e32 1.00998 0.504988 0.863126i \(-0.331497\pi\)
0.504988 + 0.863126i \(0.331497\pi\)
\(434\) 0 0
\(435\) 3.74506e33 3.91150
\(436\) 0 0
\(437\) 5.67378e32 0.559570
\(438\) 0 0
\(439\) −6.07601e32 −0.565995 −0.282997 0.959121i \(-0.591329\pi\)
−0.282997 + 0.959121i \(0.591329\pi\)
\(440\) 0 0
\(441\) −4.61416e32 −0.406078
\(442\) 0 0
\(443\) −9.06289e32 −0.753737 −0.376869 0.926267i \(-0.622999\pi\)
−0.376869 + 0.926267i \(0.622999\pi\)
\(444\) 0 0
\(445\) −2.83154e33 −2.22598
\(446\) 0 0
\(447\) −9.82078e32 −0.729965
\(448\) 0 0
\(449\) −1.21579e33 −0.854630 −0.427315 0.904103i \(-0.640540\pi\)
−0.427315 + 0.904103i \(0.640540\pi\)
\(450\) 0 0
\(451\) 4.34056e32 0.288629
\(452\) 0 0
\(453\) 4.21899e33 2.65449
\(454\) 0 0
\(455\) 2.83428e32 0.168772
\(456\) 0 0
\(457\) −5.59530e32 −0.315407 −0.157703 0.987487i \(-0.550409\pi\)
−0.157703 + 0.987487i \(0.550409\pi\)
\(458\) 0 0
\(459\) 8.37859e32 0.447211
\(460\) 0 0
\(461\) −3.78592e32 −0.191386 −0.0956930 0.995411i \(-0.530507\pi\)
−0.0956930 + 0.995411i \(0.530507\pi\)
\(462\) 0 0
\(463\) −2.77039e33 −1.32672 −0.663359 0.748301i \(-0.730870\pi\)
−0.663359 + 0.748301i \(0.730870\pi\)
\(464\) 0 0
\(465\) 5.16985e33 2.34594
\(466\) 0 0
\(467\) 2.31922e33 0.997428 0.498714 0.866767i \(-0.333806\pi\)
0.498714 + 0.866767i \(0.333806\pi\)
\(468\) 0 0
\(469\) −1.61285e33 −0.657556
\(470\) 0 0
\(471\) 3.29294e33 1.27298
\(472\) 0 0
\(473\) −7.97508e32 −0.292396
\(474\) 0 0
\(475\) 2.57587e33 0.895889
\(476\) 0 0
\(477\) −1.30739e33 −0.431446
\(478\) 0 0
\(479\) −2.24376e33 −0.702720 −0.351360 0.936240i \(-0.614281\pi\)
−0.351360 + 0.936240i \(0.614281\pi\)
\(480\) 0 0
\(481\) 4.84147e32 0.143934
\(482\) 0 0
\(483\) −4.32737e33 −1.22148
\(484\) 0 0
\(485\) 3.15089e33 0.844620
\(486\) 0 0
\(487\) −3.71393e32 −0.0945632 −0.0472816 0.998882i \(-0.515056\pi\)
−0.0472816 + 0.998882i \(0.515056\pi\)
\(488\) 0 0
\(489\) −9.13761e33 −2.21041
\(490\) 0 0
\(491\) 2.65508e33 0.610323 0.305161 0.952301i \(-0.401290\pi\)
0.305161 + 0.952301i \(0.401290\pi\)
\(492\) 0 0
\(493\) 9.19732e33 2.00944
\(494\) 0 0
\(495\) −5.30247e33 −1.10132
\(496\) 0 0
\(497\) −6.81976e33 −1.34684
\(498\) 0 0
\(499\) 2.39117e33 0.449111 0.224556 0.974461i \(-0.427907\pi\)
0.224556 + 0.974461i \(0.427907\pi\)
\(500\) 0 0
\(501\) −5.60855e33 −1.00203
\(502\) 0 0
\(503\) 6.34146e33 1.07793 0.538964 0.842329i \(-0.318816\pi\)
0.538964 + 0.842329i \(0.318816\pi\)
\(504\) 0 0
\(505\) −2.99412e33 −0.484314
\(506\) 0 0
\(507\) −9.56517e33 −1.47263
\(508\) 0 0
\(509\) −3.62528e32 −0.0531337 −0.0265668 0.999647i \(-0.508457\pi\)
−0.0265668 + 0.999647i \(0.508457\pi\)
\(510\) 0 0
\(511\) −8.36664e33 −1.16759
\(512\) 0 0
\(513\) 1.52867e33 0.203164
\(514\) 0 0
\(515\) −7.26994e33 −0.920323
\(516\) 0 0
\(517\) 2.68059e33 0.323295
\(518\) 0 0
\(519\) −1.19739e34 −1.37608
\(520\) 0 0
\(521\) 8.23573e33 0.902050 0.451025 0.892511i \(-0.351058\pi\)
0.451025 + 0.892511i \(0.351058\pi\)
\(522\) 0 0
\(523\) 1.20443e34 1.25750 0.628752 0.777606i \(-0.283566\pi\)
0.628752 + 0.777606i \(0.283566\pi\)
\(524\) 0 0
\(525\) −1.96461e34 −1.95562
\(526\) 0 0
\(527\) 1.26964e34 1.20517
\(528\) 0 0
\(529\) −1.17149e32 −0.0106057
\(530\) 0 0
\(531\) 1.39314e34 1.20313
\(532\) 0 0
\(533\) 8.12291e32 0.0669301
\(534\) 0 0
\(535\) −1.38868e34 −1.09190
\(536\) 0 0
\(537\) 5.94490e33 0.446135
\(538\) 0 0
\(539\) 2.51661e33 0.180284
\(540\) 0 0
\(541\) 1.30366e34 0.891657 0.445829 0.895118i \(-0.352909\pi\)
0.445829 + 0.895118i \(0.352909\pi\)
\(542\) 0 0
\(543\) 3.07258e34 2.00681
\(544\) 0 0
\(545\) 2.64462e34 1.64971
\(546\) 0 0
\(547\) 1.78334e34 1.06266 0.531328 0.847166i \(-0.321693\pi\)
0.531328 + 0.847166i \(0.321693\pi\)
\(548\) 0 0
\(549\) −2.49231e32 −0.0141889
\(550\) 0 0
\(551\) 1.67805e34 0.912871
\(552\) 0 0
\(553\) 1.91334e34 0.994784
\(554\) 0 0
\(555\) −5.46321e34 −2.71510
\(556\) 0 0
\(557\) 8.73339e33 0.414947 0.207474 0.978241i \(-0.433476\pi\)
0.207474 + 0.978241i \(0.433476\pi\)
\(558\) 0 0
\(559\) −1.49245e33 −0.0678037
\(560\) 0 0
\(561\) −2.35134e34 −1.02160
\(562\) 0 0
\(563\) −3.82416e34 −1.58921 −0.794606 0.607126i \(-0.792322\pi\)
−0.794606 + 0.607126i \(0.792322\pi\)
\(564\) 0 0
\(565\) −7.09124e34 −2.81914
\(566\) 0 0
\(567\) 1.51078e34 0.574661
\(568\) 0 0
\(569\) 3.49591e34 1.27250 0.636248 0.771484i \(-0.280485\pi\)
0.636248 + 0.771484i \(0.280485\pi\)
\(570\) 0 0
\(571\) 3.60524e34 1.25598 0.627989 0.778222i \(-0.283878\pi\)
0.627989 + 0.778222i \(0.283878\pi\)
\(572\) 0 0
\(573\) 2.63668e34 0.879276
\(574\) 0 0
\(575\) 4.96155e34 1.58405
\(576\) 0 0
\(577\) −4.45926e34 −1.36322 −0.681609 0.731717i \(-0.738719\pi\)
−0.681609 + 0.731717i \(0.738719\pi\)
\(578\) 0 0
\(579\) −8.22337e34 −2.40751
\(580\) 0 0
\(581\) 3.59450e34 1.00794
\(582\) 0 0
\(583\) 7.13067e33 0.191546
\(584\) 0 0
\(585\) −9.92303e33 −0.255385
\(586\) 0 0
\(587\) 2.50958e34 0.618905 0.309452 0.950915i \(-0.399854\pi\)
0.309452 + 0.950915i \(0.399854\pi\)
\(588\) 0 0
\(589\) 2.31645e34 0.547497
\(590\) 0 0
\(591\) 8.60173e34 1.94869
\(592\) 0 0
\(593\) −4.88078e33 −0.106000 −0.0530000 0.998595i \(-0.516878\pi\)
−0.0530000 + 0.998595i \(0.516878\pi\)
\(594\) 0 0
\(595\) −7.85444e34 −1.63551
\(596\) 0 0
\(597\) 4.46798e34 0.892138
\(598\) 0 0
\(599\) −4.61838e34 −0.884412 −0.442206 0.896914i \(-0.645804\pi\)
−0.442206 + 0.896914i \(0.645804\pi\)
\(600\) 0 0
\(601\) −3.07405e34 −0.564651 −0.282325 0.959319i \(-0.591106\pi\)
−0.282325 + 0.959319i \(0.591106\pi\)
\(602\) 0 0
\(603\) 5.64671e34 0.995013
\(604\) 0 0
\(605\) −6.63161e34 −1.12118
\(606\) 0 0
\(607\) −1.03737e35 −1.68297 −0.841484 0.540282i \(-0.818318\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(608\) 0 0
\(609\) −1.27984e35 −1.99269
\(610\) 0 0
\(611\) 5.01644e33 0.0749688
\(612\) 0 0
\(613\) 2.85744e34 0.409941 0.204970 0.978768i \(-0.434290\pi\)
0.204970 + 0.978768i \(0.434290\pi\)
\(614\) 0 0
\(615\) −9.16606e34 −1.26254
\(616\) 0 0
\(617\) 5.36336e34 0.709370 0.354685 0.934986i \(-0.384588\pi\)
0.354685 + 0.934986i \(0.384588\pi\)
\(618\) 0 0
\(619\) −6.60909e34 −0.839477 −0.419738 0.907645i \(-0.637878\pi\)
−0.419738 + 0.907645i \(0.637878\pi\)
\(620\) 0 0
\(621\) 2.94447e34 0.359221
\(622\) 0 0
\(623\) 9.67651e34 1.13401
\(624\) 0 0
\(625\) −5.00979e33 −0.0564053
\(626\) 0 0
\(627\) −4.29000e34 −0.464102
\(628\) 0 0
\(629\) −1.34168e35 −1.39482
\(630\) 0 0
\(631\) −8.62816e34 −0.862087 −0.431043 0.902331i \(-0.641854\pi\)
−0.431043 + 0.902331i \(0.641854\pi\)
\(632\) 0 0
\(633\) −5.26679e34 −0.505824
\(634\) 0 0
\(635\) 1.16883e35 1.07914
\(636\) 0 0
\(637\) 4.70958e33 0.0418061
\(638\) 0 0
\(639\) 2.38765e35 2.03803
\(640\) 0 0
\(641\) 1.77093e35 1.45370 0.726850 0.686796i \(-0.240983\pi\)
0.726850 + 0.686796i \(0.240983\pi\)
\(642\) 0 0
\(643\) 7.98617e34 0.630524 0.315262 0.949005i \(-0.397908\pi\)
0.315262 + 0.949005i \(0.397908\pi\)
\(644\) 0 0
\(645\) 1.68412e35 1.27901
\(646\) 0 0
\(647\) 2.16671e35 1.58306 0.791530 0.611130i \(-0.209285\pi\)
0.791530 + 0.611130i \(0.209285\pi\)
\(648\) 0 0
\(649\) −7.59833e34 −0.534145
\(650\) 0 0
\(651\) −1.76675e35 −1.19512
\(652\) 0 0
\(653\) 2.41184e35 1.57013 0.785063 0.619416i \(-0.212631\pi\)
0.785063 + 0.619416i \(0.212631\pi\)
\(654\) 0 0
\(655\) −5.92361e34 −0.371168
\(656\) 0 0
\(657\) 2.92923e35 1.76679
\(658\) 0 0
\(659\) −2.18788e35 −1.27045 −0.635223 0.772329i \(-0.719092\pi\)
−0.635223 + 0.772329i \(0.719092\pi\)
\(660\) 0 0
\(661\) −8.90411e34 −0.497820 −0.248910 0.968527i \(-0.580072\pi\)
−0.248910 + 0.968527i \(0.580072\pi\)
\(662\) 0 0
\(663\) −4.40028e34 −0.236898
\(664\) 0 0
\(665\) −1.43304e35 −0.742998
\(666\) 0 0
\(667\) 3.23219e35 1.61408
\(668\) 0 0
\(669\) −4.28705e35 −2.06221
\(670\) 0 0
\(671\) 1.35934e33 0.00629934
\(672\) 0 0
\(673\) 2.39665e35 1.07008 0.535040 0.844827i \(-0.320297\pi\)
0.535040 + 0.844827i \(0.320297\pi\)
\(674\) 0 0
\(675\) 1.33678e35 0.575124
\(676\) 0 0
\(677\) −3.79216e34 −0.157227 −0.0786137 0.996905i \(-0.525049\pi\)
−0.0786137 + 0.996905i \(0.525049\pi\)
\(678\) 0 0
\(679\) −1.07679e35 −0.430286
\(680\) 0 0
\(681\) 1.45481e35 0.560361
\(682\) 0 0
\(683\) 3.21208e35 1.19269 0.596345 0.802729i \(-0.296619\pi\)
0.596345 + 0.802729i \(0.296619\pi\)
\(684\) 0 0
\(685\) −2.83883e35 −1.01627
\(686\) 0 0
\(687\) −5.10316e35 −1.76149
\(688\) 0 0
\(689\) 1.33443e34 0.0444177
\(690\) 0 0
\(691\) −8.34540e34 −0.267899 −0.133950 0.990988i \(-0.542766\pi\)
−0.133950 + 0.990988i \(0.542766\pi\)
\(692\) 0 0
\(693\) 1.81207e35 0.561060
\(694\) 0 0
\(695\) 8.10064e34 0.241941
\(696\) 0 0
\(697\) −2.25105e35 −0.648597
\(698\) 0 0
\(699\) −9.43649e34 −0.262329
\(700\) 0 0
\(701\) 1.50008e35 0.402383 0.201191 0.979552i \(-0.435519\pi\)
0.201191 + 0.979552i \(0.435519\pi\)
\(702\) 0 0
\(703\) −2.44790e35 −0.633653
\(704\) 0 0
\(705\) −5.66066e35 −1.41417
\(706\) 0 0
\(707\) 1.02321e35 0.246730
\(708\) 0 0
\(709\) −2.12042e35 −0.493564 −0.246782 0.969071i \(-0.579373\pi\)
−0.246782 + 0.969071i \(0.579373\pi\)
\(710\) 0 0
\(711\) −6.69877e35 −1.50531
\(712\) 0 0
\(713\) 4.46186e35 0.968048
\(714\) 0 0
\(715\) 5.41213e34 0.113382
\(716\) 0 0
\(717\) −5.34363e35 −1.08105
\(718\) 0 0
\(719\) −3.27911e35 −0.640686 −0.320343 0.947302i \(-0.603798\pi\)
−0.320343 + 0.947302i \(0.603798\pi\)
\(720\) 0 0
\(721\) 2.48443e35 0.468853
\(722\) 0 0
\(723\) 1.42934e36 2.60560
\(724\) 0 0
\(725\) 1.46740e36 2.58419
\(726\) 0 0
\(727\) −2.40795e35 −0.409701 −0.204850 0.978793i \(-0.565671\pi\)
−0.204850 + 0.978793i \(0.565671\pi\)
\(728\) 0 0
\(729\) −8.57797e35 −1.41023
\(730\) 0 0
\(731\) 4.13594e35 0.657063
\(732\) 0 0
\(733\) −3.39196e35 −0.520776 −0.260388 0.965504i \(-0.583850\pi\)
−0.260388 + 0.965504i \(0.583850\pi\)
\(734\) 0 0
\(735\) −5.31439e35 −0.788608
\(736\) 0 0
\(737\) −3.07978e35 −0.441749
\(738\) 0 0
\(739\) −1.23366e36 −1.71057 −0.855284 0.518159i \(-0.826617\pi\)
−0.855284 + 0.518159i \(0.826617\pi\)
\(740\) 0 0
\(741\) −8.02829e34 −0.107621
\(742\) 0 0
\(743\) 6.24478e35 0.809389 0.404694 0.914452i \(-0.367378\pi\)
0.404694 + 0.914452i \(0.367378\pi\)
\(744\) 0 0
\(745\) −6.26430e35 −0.785090
\(746\) 0 0
\(747\) −1.25846e36 −1.52522
\(748\) 0 0
\(749\) 4.74570e35 0.556259
\(750\) 0 0
\(751\) 4.41291e35 0.500294 0.250147 0.968208i \(-0.419521\pi\)
0.250147 + 0.968208i \(0.419521\pi\)
\(752\) 0 0
\(753\) −2.03934e36 −2.23642
\(754\) 0 0
\(755\) 2.69113e36 2.85495
\(756\) 0 0
\(757\) 1.83266e35 0.188098 0.0940489 0.995568i \(-0.470019\pi\)
0.0940489 + 0.995568i \(0.470019\pi\)
\(758\) 0 0
\(759\) −8.26323e35 −0.820593
\(760\) 0 0
\(761\) 1.88027e36 1.80681 0.903406 0.428787i \(-0.141059\pi\)
0.903406 + 0.428787i \(0.141059\pi\)
\(762\) 0 0
\(763\) −9.03772e35 −0.840431
\(764\) 0 0
\(765\) 2.74990e36 2.47485
\(766\) 0 0
\(767\) −1.42195e35 −0.123863
\(768\) 0 0
\(769\) 1.28453e36 1.08309 0.541546 0.840671i \(-0.317839\pi\)
0.541546 + 0.840671i \(0.317839\pi\)
\(770\) 0 0
\(771\) 9.30700e35 0.759679
\(772\) 0 0
\(773\) −1.04808e36 −0.828226 −0.414113 0.910225i \(-0.635908\pi\)
−0.414113 + 0.910225i \(0.635908\pi\)
\(774\) 0 0
\(775\) 2.02566e36 1.54987
\(776\) 0 0
\(777\) 1.86700e36 1.38319
\(778\) 0 0
\(779\) −4.10703e35 −0.294652
\(780\) 0 0
\(781\) −1.30225e36 −0.904811
\(782\) 0 0
\(783\) 8.70840e35 0.586025
\(784\) 0 0
\(785\) 2.10044e36 1.36912
\(786\) 0 0
\(787\) 8.93146e35 0.563947 0.281974 0.959422i \(-0.409011\pi\)
0.281974 + 0.959422i \(0.409011\pi\)
\(788\) 0 0
\(789\) −1.95822e36 −1.19784
\(790\) 0 0
\(791\) 2.42336e36 1.43619
\(792\) 0 0
\(793\) 2.54386e33 0.00146076
\(794\) 0 0
\(795\) −1.50580e36 −0.837872
\(796\) 0 0
\(797\) −1.02840e36 −0.554540 −0.277270 0.960792i \(-0.589430\pi\)
−0.277270 + 0.960792i \(0.589430\pi\)
\(798\) 0 0
\(799\) −1.39017e36 −0.726498
\(800\) 0 0
\(801\) −3.38782e36 −1.71599
\(802\) 0 0
\(803\) −1.59763e36 −0.784392
\(804\) 0 0
\(805\) −2.76026e36 −1.31372
\(806\) 0 0
\(807\) −2.62004e36 −1.20889
\(808\) 0 0
\(809\) 1.33897e36 0.598981 0.299490 0.954099i \(-0.403183\pi\)
0.299490 + 0.954099i \(0.403183\pi\)
\(810\) 0 0
\(811\) −1.29485e36 −0.561643 −0.280821 0.959760i \(-0.590607\pi\)
−0.280821 + 0.959760i \(0.590607\pi\)
\(812\) 0 0
\(813\) 2.00917e36 0.845055
\(814\) 0 0
\(815\) −5.82854e36 −2.37733
\(816\) 0 0
\(817\) 7.54600e35 0.298498
\(818\) 0 0
\(819\) 3.39110e35 0.130104
\(820\) 0 0
\(821\) 3.91664e34 0.0145755 0.00728777 0.999973i \(-0.497680\pi\)
0.00728777 + 0.999973i \(0.497680\pi\)
\(822\) 0 0
\(823\) 3.40400e36 1.22883 0.614414 0.788984i \(-0.289392\pi\)
0.614414 + 0.788984i \(0.289392\pi\)
\(824\) 0 0
\(825\) −3.75147e36 −1.31379
\(826\) 0 0
\(827\) 5.02186e36 1.70626 0.853132 0.521696i \(-0.174700\pi\)
0.853132 + 0.521696i \(0.174700\pi\)
\(828\) 0 0
\(829\) 8.20485e35 0.270482 0.135241 0.990813i \(-0.456819\pi\)
0.135241 + 0.990813i \(0.456819\pi\)
\(830\) 0 0
\(831\) −7.31856e35 −0.234106
\(832\) 0 0
\(833\) −1.30513e36 −0.405128
\(834\) 0 0
\(835\) −3.57748e36 −1.07770
\(836\) 0 0
\(837\) 1.20215e36 0.351471
\(838\) 0 0
\(839\) 5.38664e36 1.52860 0.764300 0.644861i \(-0.223085\pi\)
0.764300 + 0.644861i \(0.223085\pi\)
\(840\) 0 0
\(841\) 5.92900e36 1.63317
\(842\) 0 0
\(843\) −5.02945e36 −1.34486
\(844\) 0 0
\(845\) −6.10126e36 −1.58384
\(846\) 0 0
\(847\) 2.26629e36 0.571180
\(848\) 0 0
\(849\) 1.98389e36 0.485479
\(850\) 0 0
\(851\) −4.71505e36 −1.12038
\(852\) 0 0
\(853\) 9.55369e35 0.220449 0.110225 0.993907i \(-0.464843\pi\)
0.110225 + 0.993907i \(0.464843\pi\)
\(854\) 0 0
\(855\) 5.01718e36 1.12430
\(856\) 0 0
\(857\) −5.39118e36 −1.17334 −0.586669 0.809827i \(-0.699561\pi\)
−0.586669 + 0.809827i \(0.699561\pi\)
\(858\) 0 0
\(859\) −5.48992e36 −1.16052 −0.580258 0.814432i \(-0.697048\pi\)
−0.580258 + 0.814432i \(0.697048\pi\)
\(860\) 0 0
\(861\) 3.13241e36 0.643190
\(862\) 0 0
\(863\) 2.89510e36 0.577469 0.288735 0.957409i \(-0.406766\pi\)
0.288735 + 0.957409i \(0.406766\pi\)
\(864\) 0 0
\(865\) −7.63772e36 −1.48000
\(866\) 0 0
\(867\) 4.24209e36 0.798623
\(868\) 0 0
\(869\) 3.65359e36 0.668301
\(870\) 0 0
\(871\) −5.76349e35 −0.102437
\(872\) 0 0
\(873\) 3.76991e36 0.651108
\(874\) 0 0
\(875\) −4.66251e36 −0.782562
\(876\) 0 0
\(877\) 2.04429e36 0.333463 0.166731 0.986002i \(-0.446679\pi\)
0.166731 + 0.986002i \(0.446679\pi\)
\(878\) 0 0
\(879\) 1.71570e37 2.72007
\(880\) 0 0
\(881\) −1.76163e36 −0.271466 −0.135733 0.990745i \(-0.543339\pi\)
−0.135733 + 0.990745i \(0.543339\pi\)
\(882\) 0 0
\(883\) −1.25672e37 −1.88248 −0.941240 0.337738i \(-0.890338\pi\)
−0.941240 + 0.337738i \(0.890338\pi\)
\(884\) 0 0
\(885\) 1.60456e37 2.33649
\(886\) 0 0
\(887\) −8.81003e36 −1.24719 −0.623595 0.781748i \(-0.714328\pi\)
−0.623595 + 0.781748i \(0.714328\pi\)
\(888\) 0 0
\(889\) −3.99436e36 −0.549762
\(890\) 0 0
\(891\) 2.88487e36 0.386060
\(892\) 0 0
\(893\) −2.53636e36 −0.330041
\(894\) 0 0
\(895\) 3.79203e36 0.479826
\(896\) 0 0
\(897\) −1.54638e36 −0.190288
\(898\) 0 0
\(899\) 1.31962e37 1.57925
\(900\) 0 0
\(901\) −3.69802e36 −0.430436
\(902\) 0 0
\(903\) −5.75530e36 −0.651585
\(904\) 0 0
\(905\) 1.95988e37 2.15835
\(906\) 0 0
\(907\) 1.59523e37 1.70896 0.854482 0.519481i \(-0.173875\pi\)
0.854482 + 0.519481i \(0.173875\pi\)
\(908\) 0 0
\(909\) −3.58234e36 −0.373352
\(910\) 0 0
\(911\) 1.71775e36 0.174172 0.0870862 0.996201i \(-0.472244\pi\)
0.0870862 + 0.996201i \(0.472244\pi\)
\(912\) 0 0
\(913\) 6.86379e36 0.677142
\(914\) 0 0
\(915\) −2.87054e35 −0.0275550
\(916\) 0 0
\(917\) 2.02434e36 0.189089
\(918\) 0 0
\(919\) 7.70373e35 0.0700256 0.0350128 0.999387i \(-0.488853\pi\)
0.0350128 + 0.999387i \(0.488853\pi\)
\(920\) 0 0
\(921\) −3.13079e37 −2.76955
\(922\) 0 0
\(923\) −2.43703e36 −0.209817
\(924\) 0 0
\(925\) −2.14061e37 −1.79377
\(926\) 0 0
\(927\) −8.69819e36 −0.709467
\(928\) 0 0
\(929\) 1.66420e37 1.32132 0.660662 0.750684i \(-0.270276\pi\)
0.660662 + 0.750684i \(0.270276\pi\)
\(930\) 0 0
\(931\) −2.38121e36 −0.184046
\(932\) 0 0
\(933\) 2.54021e37 1.91139
\(934\) 0 0
\(935\) −1.49983e37 −1.09874
\(936\) 0 0
\(937\) 9.66427e36 0.689325 0.344663 0.938727i \(-0.387993\pi\)
0.344663 + 0.938727i \(0.387993\pi\)
\(938\) 0 0
\(939\) 6.87214e36 0.477279
\(940\) 0 0
\(941\) −1.33808e37 −0.904923 −0.452461 0.891784i \(-0.649454\pi\)
−0.452461 + 0.891784i \(0.649454\pi\)
\(942\) 0 0
\(943\) −7.91080e36 −0.520984
\(944\) 0 0
\(945\) −7.43690e36 −0.476974
\(946\) 0 0
\(947\) −2.63078e37 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(948\) 0 0
\(949\) −2.98981e36 −0.181893
\(950\) 0 0
\(951\) −1.21195e37 −0.718170
\(952\) 0 0
\(953\) −9.10378e36 −0.525485 −0.262742 0.964866i \(-0.584627\pi\)
−0.262742 + 0.964866i \(0.584627\pi\)
\(954\) 0 0
\(955\) 1.68184e37 0.945677
\(956\) 0 0
\(957\) −2.44389e37 −1.33870
\(958\) 0 0
\(959\) 9.70142e36 0.517729
\(960\) 0 0
\(961\) −1.01623e36 −0.0528386
\(962\) 0 0
\(963\) −1.66151e37 −0.841731
\(964\) 0 0
\(965\) −5.24538e37 −2.58932
\(966\) 0 0
\(967\) 1.34794e36 0.0648394 0.0324197 0.999474i \(-0.489679\pi\)
0.0324197 + 0.999474i \(0.489679\pi\)
\(968\) 0 0
\(969\) 2.22483e37 1.04291
\(970\) 0 0
\(971\) −3.82504e36 −0.174741 −0.0873707 0.996176i \(-0.527846\pi\)
−0.0873707 + 0.996176i \(0.527846\pi\)
\(972\) 0 0
\(973\) −2.76831e36 −0.123255
\(974\) 0 0
\(975\) −7.02049e36 −0.304656
\(976\) 0 0
\(977\) 1.16914e37 0.494522 0.247261 0.968949i \(-0.420470\pi\)
0.247261 + 0.968949i \(0.420470\pi\)
\(978\) 0 0
\(979\) 1.84776e37 0.761835
\(980\) 0 0
\(981\) 3.16418e37 1.27174
\(982\) 0 0
\(983\) −1.38138e37 −0.541244 −0.270622 0.962686i \(-0.587229\pi\)
−0.270622 + 0.962686i \(0.587229\pi\)
\(984\) 0 0
\(985\) 5.48672e37 2.09585
\(986\) 0 0
\(987\) 1.93447e37 0.720441
\(988\) 0 0
\(989\) 1.45348e37 0.527784
\(990\) 0 0
\(991\) 3.79186e37 1.34255 0.671276 0.741207i \(-0.265746\pi\)
0.671276 + 0.741207i \(0.265746\pi\)
\(992\) 0 0
\(993\) −8.22853e37 −2.84090
\(994\) 0 0
\(995\) 2.84995e37 0.959510
\(996\) 0 0
\(997\) 5.34619e37 1.75531 0.877657 0.479289i \(-0.159106\pi\)
0.877657 + 0.479289i \(0.159106\pi\)
\(998\) 0 0
\(999\) −1.27036e37 −0.406779
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 16.26.a.c.1.2 2
4.3 odd 2 2.26.a.b.1.1 2
12.11 even 2 18.26.a.e.1.1 2
20.3 even 4 50.26.b.e.49.1 4
20.7 even 4 50.26.b.e.49.4 4
20.19 odd 2 50.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 2 4.3 odd 2
16.26.a.c.1.2 2 1.1 even 1 trivial
18.26.a.e.1.1 2 12.11 even 2
50.26.a.c.1.2 2 20.19 odd 2
50.26.b.e.49.1 4 20.3 even 4
50.26.b.e.49.4 4 20.7 even 4