Properties

Label 16.26.a.c.1.1
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 1.1
Root \(163.829\) 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)
 
\(f(q)\) \(=\) \(q-1.75788e6 q^{3} -1.37041e8 q^{5} +3.04153e10 q^{7} +2.24285e12 q^{9} -2.58704e12 q^{11} -9.57327e13 q^{13} +2.40901e14 q^{15} -1.64685e15 q^{17} -4.95030e15 q^{19} -5.34664e16 q^{21} +1.07650e16 q^{23} -2.79243e17 q^{25} -2.45323e18 q^{27} -1.36741e18 q^{29} +4.42000e18 q^{31} +4.54771e18 q^{33} -4.16814e18 q^{35} +1.01944e19 q^{37} +1.68287e20 q^{39} +1.58687e20 q^{41} -1.83575e20 q^{43} -3.07362e20 q^{45} -1.40203e21 q^{47} -4.15978e20 q^{49} +2.89496e21 q^{51} -1.99903e21 q^{53} +3.54531e20 q^{55} +8.70202e21 q^{57} +4.16691e21 q^{59} +3.42128e22 q^{61} +6.82170e22 q^{63} +1.31193e22 q^{65} -8.67051e22 q^{67} -1.89236e22 q^{69} +5.13159e22 q^{71} +3.49147e22 q^{73} +4.90875e23 q^{75} -7.86857e22 q^{77} -2.91588e23 q^{79} +2.41214e24 q^{81} +1.64916e24 q^{83} +2.25686e23 q^{85} +2.40374e24 q^{87} +8.74435e23 q^{89} -2.91174e24 q^{91} -7.76982e24 q^{93} +6.78393e23 q^{95} +1.00608e25 q^{97} -5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 379848 q^{3} + 741953100 q^{5} + 376536944 q^{7} + 3294531432666 q^{9} - 8323034610264 q^{11} - 106467053152292 q^{13} + 14\!\cdots\!00 q^{15} + 13\!\cdots\!56 q^{17} + 477079242949400 q^{19} - 94\!\cdots\!56 q^{21}+ \cdots - 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) −1.75788e6 −1.90974 −0.954868 0.297031i \(-0.904004\pi\)
−0.954868 + 0.297031i \(0.904004\pi\)
\(4\) 0 0
\(5\) −1.37041e8 −0.251030 −0.125515 0.992092i \(-0.540058\pi\)
−0.125515 + 0.992092i \(0.540058\pi\)
\(6\) 0 0
\(7\) 3.04153e10 0.830552 0.415276 0.909696i \(-0.363685\pi\)
0.415276 + 0.909696i \(0.363685\pi\)
\(8\) 0 0
\(9\) 2.24285e12 2.64709
\(10\) 0 0
\(11\) −2.58704e12 −0.248539 −0.124270 0.992248i \(-0.539659\pi\)
−0.124270 + 0.992248i \(0.539659\pi\)
\(12\) 0 0
\(13\) −9.57327e13 −1.13964 −0.569821 0.821769i \(-0.692988\pi\)
−0.569821 + 0.821769i \(0.692988\pi\)
\(14\) 0 0
\(15\) 2.40901e14 0.479400
\(16\) 0 0
\(17\) −1.64685e15 −0.685555 −0.342777 0.939417i \(-0.611368\pi\)
−0.342777 + 0.939417i \(0.611368\pi\)
\(18\) 0 0
\(19\) −4.95030e15 −0.513111 −0.256555 0.966530i \(-0.582588\pi\)
−0.256555 + 0.966530i \(0.582588\pi\)
\(20\) 0 0
\(21\) −5.34664e16 −1.58613
\(22\) 0 0
\(23\) 1.07650e16 0.102427 0.0512136 0.998688i \(-0.483691\pi\)
0.0512136 + 0.998688i \(0.483691\pi\)
\(24\) 0 0
\(25\) −2.79243e17 −0.936984
\(26\) 0 0
\(27\) −2.45323e18 −3.14551
\(28\) 0 0
\(29\) −1.36741e18 −0.717668 −0.358834 0.933401i \(-0.616826\pi\)
−0.358834 + 0.933401i \(0.616826\pi\)
\(30\) 0 0
\(31\) 4.42000e18 1.00786 0.503931 0.863744i \(-0.331887\pi\)
0.503931 + 0.863744i \(0.331887\pi\)
\(32\) 0 0
\(33\) 4.54771e18 0.474645
\(34\) 0 0
\(35\) −4.16814e18 −0.208493
\(36\) 0 0
\(37\) 1.01944e19 0.254590 0.127295 0.991865i \(-0.459371\pi\)
0.127295 + 0.991865i \(0.459371\pi\)
\(38\) 0 0
\(39\) 1.68287e20 2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) 0 0
\(43\) −1.83575e20 −0.700582 −0.350291 0.936641i \(-0.613917\pi\)
−0.350291 + 0.936641i \(0.613917\pi\)
\(44\) 0 0
\(45\) −3.07362e20 −0.664499
\(46\) 0 0
\(47\) −1.40203e21 −1.76009 −0.880045 0.474890i \(-0.842488\pi\)
−0.880045 + 0.474890i \(0.842488\pi\)
\(48\) 0 0
\(49\) −4.15978e20 −0.310184
\(50\) 0 0
\(51\) 2.89496e21 1.30923
\(52\) 0 0
\(53\) −1.99903e21 −0.558946 −0.279473 0.960154i \(-0.590160\pi\)
−0.279473 + 0.960154i \(0.590160\pi\)
\(54\) 0 0
\(55\) 3.54531e20 0.0623908
\(56\) 0 0
\(57\) 8.70202e21 0.979906
\(58\) 0 0
\(59\) 4.16691e21 0.304905 0.152452 0.988311i \(-0.451283\pi\)
0.152452 + 0.988311i \(0.451283\pi\)
\(60\) 0 0
\(61\) 3.42128e22 1.65031 0.825156 0.564904i \(-0.191087\pi\)
0.825156 + 0.564904i \(0.191087\pi\)
\(62\) 0 0
\(63\) 6.82170e22 2.19855
\(64\) 0 0
\(65\) 1.31193e22 0.286084
\(66\) 0 0
\(67\) −8.67051e22 −1.29452 −0.647261 0.762268i \(-0.724086\pi\)
−0.647261 + 0.762268i \(0.724086\pi\)
\(68\) 0 0
\(69\) −1.89236e22 −0.195609
\(70\) 0 0
\(71\) 5.13159e22 0.371127 0.185564 0.982632i \(-0.440589\pi\)
0.185564 + 0.982632i \(0.440589\pi\)
\(72\) 0 0
\(73\) 3.49147e22 0.178432 0.0892159 0.996012i \(-0.471564\pi\)
0.0892159 + 0.996012i \(0.471564\pi\)
\(74\) 0 0
\(75\) 4.90875e23 1.78939
\(76\) 0 0
\(77\) −7.86857e22 −0.206425
\(78\) 0 0
\(79\) −2.91588e23 −0.555176 −0.277588 0.960700i \(-0.589535\pi\)
−0.277588 + 0.960700i \(0.589535\pi\)
\(80\) 0 0
\(81\) 2.41214e24 3.36000
\(82\) 0 0
\(83\) 1.64916e24 1.69351 0.846753 0.531986i \(-0.178554\pi\)
0.846753 + 0.531986i \(0.178554\pi\)
\(84\) 0 0
\(85\) 2.25686e23 0.172095
\(86\) 0 0
\(87\) 2.40374e24 1.37056
\(88\) 0 0
\(89\) 8.74435e23 0.375277 0.187639 0.982238i \(-0.439917\pi\)
0.187639 + 0.982238i \(0.439917\pi\)
\(90\) 0 0
\(91\) −2.91174e24 −0.946532
\(92\) 0 0
\(93\) −7.76982e24 −1.92475
\(94\) 0 0
\(95\) 6.78393e23 0.128806
\(96\) 0 0
\(97\) 1.00608e25 1.47227 0.736134 0.676835i \(-0.236649\pi\)
0.736134 + 0.676835i \(0.236649\pi\)
\(98\) 0 0
\(99\) −5.80235e24 −0.657907
\(100\) 0 0
\(101\) −1.42151e25 −1.25526 −0.627628 0.778513i \(-0.715974\pi\)
−0.627628 + 0.778513i \(0.715974\pi\)
\(102\) 0 0
\(103\) −1.03211e25 −0.713283 −0.356642 0.934241i \(-0.616078\pi\)
−0.356642 + 0.934241i \(0.616078\pi\)
\(104\) 0 0
\(105\) 7.32709e24 0.398167
\(106\) 0 0
\(107\) 1.74396e25 0.748582 0.374291 0.927311i \(-0.377886\pi\)
0.374291 + 0.927311i \(0.377886\pi\)
\(108\) 0 0
\(109\) −1.65117e25 −0.562292 −0.281146 0.959665i \(-0.590714\pi\)
−0.281146 + 0.959665i \(0.590714\pi\)
\(110\) 0 0
\(111\) −1.79205e25 −0.486199
\(112\) 0 0
\(113\) 6.96039e25 1.51061 0.755306 0.655372i \(-0.227488\pi\)
0.755306 + 0.655372i \(0.227488\pi\)
\(114\) 0 0
\(115\) −1.47524e24 −0.0257123
\(116\) 0 0
\(117\) −2.14714e26 −3.01674
\(118\) 0 0
\(119\) −5.00894e25 −0.569389
\(120\) 0 0
\(121\) −1.01654e26 −0.938228
\(122\) 0 0
\(123\) −2.78952e26 −2.09757
\(124\) 0 0
\(125\) 7.91091e25 0.486241
\(126\) 0 0
\(127\) 6.18922e25 0.311953 0.155977 0.987761i \(-0.450148\pi\)
0.155977 + 0.987761i \(0.450148\pi\)
\(128\) 0 0
\(129\) 3.22703e26 1.33793
\(130\) 0 0
\(131\) 1.39682e26 0.477805 0.238902 0.971044i \(-0.423212\pi\)
0.238902 + 0.971044i \(0.423212\pi\)
\(132\) 0 0
\(133\) −1.50565e26 −0.426165
\(134\) 0 0
\(135\) 3.36193e26 0.789616
\(136\) 0 0
\(137\) −6.92418e26 −1.35320 −0.676599 0.736352i \(-0.736547\pi\)
−0.676599 + 0.736352i \(0.736547\pi\)
\(138\) 0 0
\(139\) 8.06682e26 1.31528 0.657639 0.753333i \(-0.271555\pi\)
0.657639 + 0.753333i \(0.271555\pi\)
\(140\) 0 0
\(141\) 2.46460e27 3.36131
\(142\) 0 0
\(143\) 2.47665e26 0.283246
\(144\) 0 0
\(145\) 1.87391e26 0.180156
\(146\) 0 0
\(147\) 7.31239e26 0.592369
\(148\) 0 0
\(149\) −4.24039e26 −0.290120 −0.145060 0.989423i \(-0.546337\pi\)
−0.145060 + 0.989423i \(0.546337\pi\)
\(150\) 0 0
\(151\) −2.51382e27 −1.45587 −0.727934 0.685647i \(-0.759519\pi\)
−0.727934 + 0.685647i \(0.759519\pi\)
\(152\) 0 0
\(153\) −3.69363e27 −1.81473
\(154\) 0 0
\(155\) −6.05720e26 −0.253003
\(156\) 0 0
\(157\) 4.75882e27 1.69338 0.846689 0.532088i \(-0.178593\pi\)
0.846689 + 0.532088i \(0.178593\pi\)
\(158\) 0 0
\(159\) 3.51404e27 1.06744
\(160\) 0 0
\(161\) 3.27420e26 0.0850712
\(162\) 0 0
\(163\) −7.59899e26 −0.169204 −0.0846020 0.996415i \(-0.526962\pi\)
−0.0846020 + 0.996415i \(0.526962\pi\)
\(164\) 0 0
\(165\) −6.23222e26 −0.119150
\(166\) 0 0
\(167\) 1.00207e28 1.64794 0.823970 0.566634i \(-0.191755\pi\)
0.823970 + 0.566634i \(0.191755\pi\)
\(168\) 0 0
\(169\) 2.10835e27 0.298785
\(170\) 0 0
\(171\) −1.11028e28 −1.35825
\(172\) 0 0
\(173\) −7.89989e27 −0.835689 −0.417845 0.908518i \(-0.637214\pi\)
−0.417845 + 0.908518i \(0.637214\pi\)
\(174\) 0 0
\(175\) −8.49326e27 −0.778214
\(176\) 0 0
\(177\) −7.32493e27 −0.582288
\(178\) 0 0
\(179\) 8.08087e27 0.558207 0.279103 0.960261i \(-0.409963\pi\)
0.279103 + 0.960261i \(0.409963\pi\)
\(180\) 0 0
\(181\) 1.80674e27 0.108621 0.0543104 0.998524i \(-0.482704\pi\)
0.0543104 + 0.998524i \(0.482704\pi\)
\(182\) 0 0
\(183\) −6.01420e28 −3.15166
\(184\) 0 0
\(185\) −1.39705e27 −0.0639096
\(186\) 0 0
\(187\) 4.26047e27 0.170387
\(188\) 0 0
\(189\) −7.46157e28 −2.61251
\(190\) 0 0
\(191\) 4.93091e28 1.51360 0.756798 0.653649i \(-0.226763\pi\)
0.756798 + 0.653649i \(0.226763\pi\)
\(192\) 0 0
\(193\) 1.42443e28 0.383861 0.191931 0.981408i \(-0.438525\pi\)
0.191931 + 0.981408i \(0.438525\pi\)
\(194\) 0 0
\(195\) −2.30621e28 −0.546345
\(196\) 0 0
\(197\) 3.59073e28 0.748781 0.374390 0.927271i \(-0.377852\pi\)
0.374390 + 0.927271i \(0.377852\pi\)
\(198\) 0 0
\(199\) 8.11996e28 1.49242 0.746209 0.665712i \(-0.231872\pi\)
0.746209 + 0.665712i \(0.231872\pi\)
\(200\) 0 0
\(201\) 1.52417e29 2.47220
\(202\) 0 0
\(203\) −4.15902e28 −0.596060
\(204\) 0 0
\(205\) −2.17466e28 −0.275720
\(206\) 0 0
\(207\) 2.41443e28 0.271134
\(208\) 0 0
\(209\) 1.28066e28 0.127528
\(210\) 0 0
\(211\) 1.03285e29 0.913075 0.456537 0.889704i \(-0.349089\pi\)
0.456537 + 0.889704i \(0.349089\pi\)
\(212\) 0 0
\(213\) −9.02072e28 −0.708755
\(214\) 0 0
\(215\) 2.51573e28 0.175867
\(216\) 0 0
\(217\) 1.34436e29 0.837081
\(218\) 0 0
\(219\) −6.13758e28 −0.340757
\(220\) 0 0
\(221\) 1.57657e29 0.781287
\(222\) 0 0
\(223\) −1.15542e29 −0.511597 −0.255799 0.966730i \(-0.582338\pi\)
−0.255799 + 0.966730i \(0.582338\pi\)
\(224\) 0 0
\(225\) −6.26300e29 −2.48028
\(226\) 0 0
\(227\) 9.01151e28 0.319502 0.159751 0.987157i \(-0.448931\pi\)
0.159751 + 0.987157i \(0.448931\pi\)
\(228\) 0 0
\(229\) 3.69415e29 1.17374 0.586869 0.809682i \(-0.300360\pi\)
0.586869 + 0.809682i \(0.300360\pi\)
\(230\) 0 0
\(231\) 1.38320e29 0.394217
\(232\) 0 0
\(233\) 4.10753e28 0.105107 0.0525536 0.998618i \(-0.483264\pi\)
0.0525536 + 0.998618i \(0.483264\pi\)
\(234\) 0 0
\(235\) 1.92136e29 0.441835
\(236\) 0 0
\(237\) 5.12576e29 1.06024
\(238\) 0 0
\(239\) 6.49097e29 1.20875 0.604374 0.796701i \(-0.293423\pi\)
0.604374 + 0.796701i \(0.293423\pi\)
\(240\) 0 0
\(241\) 1.68781e29 0.283212 0.141606 0.989923i \(-0.454773\pi\)
0.141606 + 0.989923i \(0.454773\pi\)
\(242\) 0 0
\(243\) −2.16165e30 −3.27121
\(244\) 0 0
\(245\) 5.70060e28 0.0778653
\(246\) 0 0
\(247\) 4.73905e29 0.584763
\(248\) 0 0
\(249\) −2.89903e30 −3.23415
\(250\) 0 0
\(251\) −2.65029e29 −0.267530 −0.133765 0.991013i \(-0.542707\pi\)
−0.133765 + 0.991013i \(0.542707\pi\)
\(252\) 0 0
\(253\) −2.78495e28 −0.0254572
\(254\) 0 0
\(255\) −3.96728e29 −0.328655
\(256\) 0 0
\(257\) 1.11592e30 0.838433 0.419217 0.907886i \(-0.362305\pi\)
0.419217 + 0.907886i \(0.362305\pi\)
\(258\) 0 0
\(259\) 3.10066e29 0.211450
\(260\) 0 0
\(261\) −3.06689e30 −1.89973
\(262\) 0 0
\(263\) −1.53721e30 −0.865541 −0.432770 0.901504i \(-0.642464\pi\)
−0.432770 + 0.901504i \(0.642464\pi\)
\(264\) 0 0
\(265\) 2.73948e29 0.140312
\(266\) 0 0
\(267\) −1.53715e30 −0.716681
\(268\) 0 0
\(269\) 2.34612e30 0.996431 0.498216 0.867053i \(-0.333989\pi\)
0.498216 + 0.867053i \(0.333989\pi\)
\(270\) 0 0
\(271\) 1.54578e30 0.598454 0.299227 0.954182i \(-0.403271\pi\)
0.299227 + 0.954182i \(0.403271\pi\)
\(272\) 0 0
\(273\) 5.11849e30 1.80763
\(274\) 0 0
\(275\) 7.22414e29 0.232877
\(276\) 0 0
\(277\) −1.21614e30 −0.358085 −0.179042 0.983841i \(-0.557300\pi\)
−0.179042 + 0.983841i \(0.557300\pi\)
\(278\) 0 0
\(279\) 9.91339e30 2.66790
\(280\) 0 0
\(281\) 5.09885e30 1.25500 0.627499 0.778617i \(-0.284079\pi\)
0.627499 + 0.778617i \(0.284079\pi\)
\(282\) 0 0
\(283\) 1.56288e30 0.352042 0.176021 0.984386i \(-0.443677\pi\)
0.176021 + 0.984386i \(0.443677\pi\)
\(284\) 0 0
\(285\) −1.19253e30 −0.245986
\(286\) 0 0
\(287\) 4.82651e30 0.912240
\(288\) 0 0
\(289\) −3.05852e30 −0.530015
\(290\) 0 0
\(291\) −1.76857e31 −2.81164
\(292\) 0 0
\(293\) −4.17112e30 −0.608706 −0.304353 0.952559i \(-0.598440\pi\)
−0.304353 + 0.952559i \(0.598440\pi\)
\(294\) 0 0
\(295\) −5.71038e29 −0.0765402
\(296\) 0 0
\(297\) 6.34661e30 0.781783
\(298\) 0 0
\(299\) −1.03056e30 −0.116730
\(300\) 0 0
\(301\) −5.58350e30 −0.581870
\(302\) 0 0
\(303\) 2.49884e31 2.39721
\(304\) 0 0
\(305\) −4.68856e30 −0.414278
\(306\) 0 0
\(307\) −7.69994e30 −0.626986 −0.313493 0.949591i \(-0.601499\pi\)
−0.313493 + 0.949591i \(0.601499\pi\)
\(308\) 0 0
\(309\) 1.81433e31 1.36218
\(310\) 0 0
\(311\) −1.54260e31 −1.06843 −0.534217 0.845347i \(-0.679394\pi\)
−0.534217 + 0.845347i \(0.679394\pi\)
\(312\) 0 0
\(313\) 1.62517e31 1.03895 0.519476 0.854485i \(-0.326127\pi\)
0.519476 + 0.854485i \(0.326127\pi\)
\(314\) 0 0
\(315\) −9.34852e30 −0.551900
\(316\) 0 0
\(317\) −2.29098e31 −1.24963 −0.624814 0.780773i \(-0.714825\pi\)
−0.624814 + 0.780773i \(0.714825\pi\)
\(318\) 0 0
\(319\) 3.53755e30 0.178369
\(320\) 0 0
\(321\) −3.06567e31 −1.42959
\(322\) 0 0
\(323\) 8.15239e30 0.351765
\(324\) 0 0
\(325\) 2.67327e31 1.06783
\(326\) 0 0
\(327\) 2.90256e31 1.07383
\(328\) 0 0
\(329\) −4.26433e31 −1.46185
\(330\) 0 0
\(331\) −3.45199e31 −1.09703 −0.548517 0.836140i \(-0.684807\pi\)
−0.548517 + 0.836140i \(0.684807\pi\)
\(332\) 0 0
\(333\) 2.28645e31 0.673922
\(334\) 0 0
\(335\) 1.18821e31 0.324964
\(336\) 0 0
\(337\) −5.27735e31 −1.33981 −0.669904 0.742448i \(-0.733665\pi\)
−0.669904 + 0.742448i \(0.733665\pi\)
\(338\) 0 0
\(339\) −1.22355e32 −2.88487
\(340\) 0 0
\(341\) −1.14347e31 −0.250493
\(342\) 0 0
\(343\) −5.34411e31 −1.08818
\(344\) 0 0
\(345\) 2.59330e30 0.0491037
\(346\) 0 0
\(347\) −4.40093e30 −0.0775222 −0.0387611 0.999249i \(-0.512341\pi\)
−0.0387611 + 0.999249i \(0.512341\pi\)
\(348\) 0 0
\(349\) 2.98069e31 0.488651 0.244326 0.969693i \(-0.421433\pi\)
0.244326 + 0.969693i \(0.421433\pi\)
\(350\) 0 0
\(351\) 2.34854e32 3.58476
\(352\) 0 0
\(353\) 4.46084e31 0.634209 0.317105 0.948391i \(-0.397289\pi\)
0.317105 + 0.948391i \(0.397289\pi\)
\(354\) 0 0
\(355\) −7.03238e30 −0.0931640
\(356\) 0 0
\(357\) 8.80511e31 1.08738
\(358\) 0 0
\(359\) −4.38054e31 −0.504483 −0.252242 0.967664i \(-0.581168\pi\)
−0.252242 + 0.967664i \(0.581168\pi\)
\(360\) 0 0
\(361\) −6.85711e31 −0.736717
\(362\) 0 0
\(363\) 1.78696e32 1.79177
\(364\) 0 0
\(365\) −4.78474e30 −0.0447917
\(366\) 0 0
\(367\) 7.76735e31 0.679121 0.339560 0.940584i \(-0.389722\pi\)
0.339560 + 0.940584i \(0.389722\pi\)
\(368\) 0 0
\(369\) 3.55911e32 2.90744
\(370\) 0 0
\(371\) −6.08010e31 −0.464234
\(372\) 0 0
\(373\) −2.37205e32 −1.69342 −0.846708 0.532058i \(-0.821419\pi\)
−0.846708 + 0.532058i \(0.821419\pi\)
\(374\) 0 0
\(375\) −1.39064e32 −0.928591
\(376\) 0 0
\(377\) 1.30906e32 0.817884
\(378\) 0 0
\(379\) 1.38944e32 0.812551 0.406276 0.913751i \(-0.366827\pi\)
0.406276 + 0.913751i \(0.366827\pi\)
\(380\) 0 0
\(381\) −1.08799e32 −0.595748
\(382\) 0 0
\(383\) 3.79085e31 0.194425 0.0972125 0.995264i \(-0.469007\pi\)
0.0972125 + 0.995264i \(0.469007\pi\)
\(384\) 0 0
\(385\) 1.07832e31 0.0518188
\(386\) 0 0
\(387\) −4.11732e32 −1.85450
\(388\) 0 0
\(389\) −2.20209e32 −0.929963 −0.464982 0.885320i \(-0.653939\pi\)
−0.464982 + 0.885320i \(0.653939\pi\)
\(390\) 0 0
\(391\) −1.77283e31 −0.0702195
\(392\) 0 0
\(393\) −2.45545e32 −0.912481
\(394\) 0 0
\(395\) 3.99595e31 0.139366
\(396\) 0 0
\(397\) 2.18941e32 0.716877 0.358439 0.933553i \(-0.383309\pi\)
0.358439 + 0.933553i \(0.383309\pi\)
\(398\) 0 0
\(399\) 2.64675e32 0.813863
\(400\) 0 0
\(401\) 2.99036e32 0.863810 0.431905 0.901919i \(-0.357842\pi\)
0.431905 + 0.901919i \(0.357842\pi\)
\(402\) 0 0
\(403\) −4.23138e32 −1.14860
\(404\) 0 0
\(405\) −3.30562e32 −0.843460
\(406\) 0 0
\(407\) −2.63734e31 −0.0632755
\(408\) 0 0
\(409\) 7.12330e31 0.160746 0.0803730 0.996765i \(-0.474389\pi\)
0.0803730 + 0.996765i \(0.474389\pi\)
\(410\) 0 0
\(411\) 1.21719e33 2.58425
\(412\) 0 0
\(413\) 1.26738e32 0.253239
\(414\) 0 0
\(415\) −2.26003e32 −0.425121
\(416\) 0 0
\(417\) −1.41805e33 −2.51183
\(418\) 0 0
\(419\) −2.94461e32 −0.491307 −0.245654 0.969358i \(-0.579003\pi\)
−0.245654 + 0.969358i \(0.579003\pi\)
\(420\) 0 0
\(421\) 1.01892e33 1.60182 0.800908 0.598787i \(-0.204350\pi\)
0.800908 + 0.598787i \(0.204350\pi\)
\(422\) 0 0
\(423\) −3.14455e33 −4.65912
\(424\) 0 0
\(425\) 4.59871e32 0.642354
\(426\) 0 0
\(427\) 1.04059e33 1.37067
\(428\) 0 0
\(429\) −4.35365e32 −0.540925
\(430\) 0 0
\(431\) 6.79497e32 0.796565 0.398283 0.917263i \(-0.369606\pi\)
0.398283 + 0.917263i \(0.369606\pi\)
\(432\) 0 0
\(433\) −7.24164e32 −0.801195 −0.400597 0.916254i \(-0.631197\pi\)
−0.400597 + 0.916254i \(0.631197\pi\)
\(434\) 0 0
\(435\) −3.29411e32 −0.344050
\(436\) 0 0
\(437\) −5.32899e31 −0.0525566
\(438\) 0 0
\(439\) 9.13459e32 0.850908 0.425454 0.904980i \(-0.360114\pi\)
0.425454 + 0.904980i \(0.360114\pi\)
\(440\) 0 0
\(441\) −9.32976e32 −0.821085
\(442\) 0 0
\(443\) 3.52555e32 0.293211 0.146605 0.989195i \(-0.453165\pi\)
0.146605 + 0.989195i \(0.453165\pi\)
\(444\) 0 0
\(445\) −1.19833e32 −0.0942058
\(446\) 0 0
\(447\) 7.45409e32 0.554052
\(448\) 0 0
\(449\) 3.70918e32 0.260735 0.130367 0.991466i \(-0.458384\pi\)
0.130367 + 0.991466i \(0.458384\pi\)
\(450\) 0 0
\(451\) −4.10530e32 −0.272984
\(452\) 0 0
\(453\) 4.41899e33 2.78032
\(454\) 0 0
\(455\) 3.99028e32 0.237608
\(456\) 0 0
\(457\) 1.76958e33 0.997512 0.498756 0.866742i \(-0.333790\pi\)
0.498756 + 0.866742i \(0.333790\pi\)
\(458\) 0 0
\(459\) 4.04009e33 2.15642
\(460\) 0 0
\(461\) 7.01018e32 0.354379 0.177189 0.984177i \(-0.443299\pi\)
0.177189 + 0.984177i \(0.443299\pi\)
\(462\) 0 0
\(463\) 9.20521e32 0.440830 0.220415 0.975406i \(-0.429259\pi\)
0.220415 + 0.975406i \(0.429259\pi\)
\(464\) 0 0
\(465\) 1.06478e33 0.483169
\(466\) 0 0
\(467\) 1.84600e33 0.793908 0.396954 0.917838i \(-0.370067\pi\)
0.396954 + 0.917838i \(0.370067\pi\)
\(468\) 0 0
\(469\) −2.63716e33 −1.07517
\(470\) 0 0
\(471\) −8.36543e33 −3.23390
\(472\) 0 0
\(473\) 4.74918e32 0.174122
\(474\) 0 0
\(475\) 1.38234e33 0.480777
\(476\) 0 0
\(477\) −4.48351e33 −1.47958
\(478\) 0 0
\(479\) 3.88481e32 0.121668 0.0608339 0.998148i \(-0.480624\pi\)
0.0608339 + 0.998148i \(0.480624\pi\)
\(480\) 0 0
\(481\) −9.75939e32 −0.290141
\(482\) 0 0
\(483\) −5.75566e32 −0.162463
\(484\) 0 0
\(485\) −1.37875e33 −0.369583
\(486\) 0 0
\(487\) −4.26102e33 −1.08493 −0.542465 0.840078i \(-0.682509\pi\)
−0.542465 + 0.840078i \(0.682509\pi\)
\(488\) 0 0
\(489\) 1.33581e33 0.323135
\(490\) 0 0
\(491\) 4.52548e33 1.04027 0.520134 0.854084i \(-0.325882\pi\)
0.520134 + 0.854084i \(0.325882\pi\)
\(492\) 0 0
\(493\) 2.25191e33 0.492000
\(494\) 0 0
\(495\) 7.95160e32 0.165154
\(496\) 0 0
\(497\) 1.56079e33 0.308241
\(498\) 0 0
\(499\) −4.45020e33 −0.835840 −0.417920 0.908484i \(-0.637241\pi\)
−0.417920 + 0.908484i \(0.637241\pi\)
\(500\) 0 0
\(501\) −1.76151e34 −3.14713
\(502\) 0 0
\(503\) −2.29032e33 −0.389311 −0.194656 0.980872i \(-0.562359\pi\)
−0.194656 + 0.980872i \(0.562359\pi\)
\(504\) 0 0
\(505\) 1.94805e33 0.315107
\(506\) 0 0
\(507\) −3.70622e33 −0.570600
\(508\) 0 0
\(509\) 5.35889e33 0.785421 0.392711 0.919662i \(-0.371537\pi\)
0.392711 + 0.919662i \(0.371537\pi\)
\(510\) 0 0
\(511\) 1.06194e33 0.148197
\(512\) 0 0
\(513\) 1.21442e34 1.61400
\(514\) 0 0
\(515\) 1.41442e33 0.179055
\(516\) 0 0
\(517\) 3.62712e33 0.437452
\(518\) 0 0
\(519\) 1.38871e34 1.59595
\(520\) 0 0
\(521\) −1.14090e34 −1.24961 −0.624807 0.780779i \(-0.714822\pi\)
−0.624807 + 0.780779i \(0.714822\pi\)
\(522\) 0 0
\(523\) −1.37717e34 −1.43786 −0.718928 0.695085i \(-0.755367\pi\)
−0.718928 + 0.695085i \(0.755367\pi\)
\(524\) 0 0
\(525\) 1.49301e34 1.48618
\(526\) 0 0
\(527\) −7.27906e33 −0.690944
\(528\) 0 0
\(529\) −1.09299e34 −0.989509
\(530\) 0 0
\(531\) 9.34576e33 0.807111
\(532\) 0 0
\(533\) −1.51915e34 −1.25173
\(534\) 0 0
\(535\) −2.38994e33 −0.187916
\(536\) 0 0
\(537\) −1.42052e34 −1.06603
\(538\) 0 0
\(539\) 1.07615e33 0.0770929
\(540\) 0 0
\(541\) −1.58682e34 −1.08533 −0.542667 0.839948i \(-0.682585\pi\)
−0.542667 + 0.839948i \(0.682585\pi\)
\(542\) 0 0
\(543\) −3.17603e33 −0.207437
\(544\) 0 0
\(545\) 2.26278e33 0.141152
\(546\) 0 0
\(547\) 8.29322e33 0.494176 0.247088 0.968993i \(-0.420526\pi\)
0.247088 + 0.968993i \(0.420526\pi\)
\(548\) 0 0
\(549\) 7.67343e34 4.36853
\(550\) 0 0
\(551\) 6.76908e33 0.368243
\(552\) 0 0
\(553\) −8.86874e33 −0.461103
\(554\) 0 0
\(555\) 2.45585e33 0.122050
\(556\) 0 0
\(557\) 3.56496e34 1.69381 0.846905 0.531744i \(-0.178463\pi\)
0.846905 + 0.531744i \(0.178463\pi\)
\(558\) 0 0
\(559\) 1.75742e34 0.798413
\(560\) 0 0
\(561\) −7.48939e33 −0.325395
\(562\) 0 0
\(563\) −1.62059e34 −0.673471 −0.336735 0.941599i \(-0.609323\pi\)
−0.336735 + 0.941599i \(0.609323\pi\)
\(564\) 0 0
\(565\) −9.53858e33 −0.379209
\(566\) 0 0
\(567\) 7.33659e34 2.79065
\(568\) 0 0
\(569\) 4.19926e34 1.52851 0.764257 0.644912i \(-0.223106\pi\)
0.764257 + 0.644912i \(0.223106\pi\)
\(570\) 0 0
\(571\) −2.27922e34 −0.794026 −0.397013 0.917813i \(-0.629953\pi\)
−0.397013 + 0.917813i \(0.629953\pi\)
\(572\) 0 0
\(573\) −8.66794e34 −2.89057
\(574\) 0 0
\(575\) −3.00605e33 −0.0959727
\(576\) 0 0
\(577\) −1.30064e34 −0.397613 −0.198807 0.980039i \(-0.563707\pi\)
−0.198807 + 0.980039i \(0.563707\pi\)
\(578\) 0 0
\(579\) −2.50397e34 −0.733074
\(580\) 0 0
\(581\) 5.01597e34 1.40655
\(582\) 0 0
\(583\) 5.17157e33 0.138920
\(584\) 0 0
\(585\) 2.94246e34 0.757291
\(586\) 0 0
\(587\) −3.13619e34 −0.773440 −0.386720 0.922197i \(-0.626392\pi\)
−0.386720 + 0.922197i \(0.626392\pi\)
\(588\) 0 0
\(589\) −2.18803e34 −0.517145
\(590\) 0 0
\(591\) −6.31206e34 −1.42997
\(592\) 0 0
\(593\) −6.43753e34 −1.39809 −0.699046 0.715077i \(-0.746392\pi\)
−0.699046 + 0.715077i \(0.746392\pi\)
\(594\) 0 0
\(595\) 6.86429e33 0.142933
\(596\) 0 0
\(597\) −1.42739e35 −2.85012
\(598\) 0 0
\(599\) 9.82124e34 1.88075 0.940375 0.340140i \(-0.110474\pi\)
0.940375 + 0.340140i \(0.110474\pi\)
\(600\) 0 0
\(601\) 1.10144e34 0.202316 0.101158 0.994870i \(-0.467745\pi\)
0.101158 + 0.994870i \(0.467745\pi\)
\(602\) 0 0
\(603\) −1.94467e35 −3.42672
\(604\) 0 0
\(605\) 1.39308e34 0.235523
\(606\) 0 0
\(607\) −9.06735e34 −1.47103 −0.735516 0.677508i \(-0.763060\pi\)
−0.735516 + 0.677508i \(0.763060\pi\)
\(608\) 0 0
\(609\) 7.31105e34 1.13832
\(610\) 0 0
\(611\) 1.34220e35 2.00587
\(612\) 0 0
\(613\) 2.62922e34 0.377199 0.188599 0.982054i \(-0.439605\pi\)
0.188599 + 0.982054i \(0.439605\pi\)
\(614\) 0 0
\(615\) 3.82279e34 0.526552
\(616\) 0 0
\(617\) 6.11422e34 0.808679 0.404340 0.914609i \(-0.367501\pi\)
0.404340 + 0.914609i \(0.367501\pi\)
\(618\) 0 0
\(619\) 4.66687e34 0.592779 0.296389 0.955067i \(-0.404217\pi\)
0.296389 + 0.955067i \(0.404217\pi\)
\(620\) 0 0
\(621\) −2.64090e34 −0.322186
\(622\) 0 0
\(623\) 2.65962e34 0.311687
\(624\) 0 0
\(625\) 7.23797e34 0.814923
\(626\) 0 0
\(627\) −2.25125e34 −0.243545
\(628\) 0 0
\(629\) −1.67886e34 −0.174535
\(630\) 0 0
\(631\) 7.88083e33 0.0787417 0.0393709 0.999225i \(-0.487465\pi\)
0.0393709 + 0.999225i \(0.487465\pi\)
\(632\) 0 0
\(633\) −1.81562e35 −1.74373
\(634\) 0 0
\(635\) −8.48176e33 −0.0783095
\(636\) 0 0
\(637\) 3.98227e34 0.353499
\(638\) 0 0
\(639\) 1.15094e35 0.982408
\(640\) 0 0
\(641\) 3.14908e34 0.258499 0.129250 0.991612i \(-0.458743\pi\)
0.129250 + 0.991612i \(0.458743\pi\)
\(642\) 0 0
\(643\) −8.32480e34 −0.657259 −0.328630 0.944459i \(-0.606587\pi\)
−0.328630 + 0.944459i \(0.606587\pi\)
\(644\) 0 0
\(645\) −4.42236e34 −0.335859
\(646\) 0 0
\(647\) 1.33124e35 0.972641 0.486321 0.873780i \(-0.338339\pi\)
0.486321 + 0.873780i \(0.338339\pi\)
\(648\) 0 0
\(649\) −1.07800e34 −0.0757809
\(650\) 0 0
\(651\) −2.36321e35 −1.59860
\(652\) 0 0
\(653\) 2.06756e35 1.34599 0.672997 0.739645i \(-0.265007\pi\)
0.672997 + 0.739645i \(0.265007\pi\)
\(654\) 0 0
\(655\) −1.91422e34 −0.119943
\(656\) 0 0
\(657\) 7.83084e34 0.472325
\(658\) 0 0
\(659\) 2.57849e35 1.49726 0.748632 0.662986i \(-0.230711\pi\)
0.748632 + 0.662986i \(0.230711\pi\)
\(660\) 0 0
\(661\) −3.31671e34 −0.185434 −0.0927170 0.995692i \(-0.529555\pi\)
−0.0927170 + 0.995692i \(0.529555\pi\)
\(662\) 0 0
\(663\) −2.77142e35 −1.49205
\(664\) 0 0
\(665\) 2.06335e34 0.106980
\(666\) 0 0
\(667\) −1.47201e34 −0.0735087
\(668\) 0 0
\(669\) 2.03109e35 0.977016
\(670\) 0 0
\(671\) −8.85101e34 −0.410168
\(672\) 0 0
\(673\) −2.87165e35 −1.28216 −0.641082 0.767472i \(-0.721514\pi\)
−0.641082 + 0.767472i \(0.721514\pi\)
\(674\) 0 0
\(675\) 6.85047e35 2.94729
\(676\) 0 0
\(677\) 3.20244e35 1.32777 0.663885 0.747835i \(-0.268907\pi\)
0.663885 + 0.747835i \(0.268907\pi\)
\(678\) 0 0
\(679\) 3.06003e35 1.22280
\(680\) 0 0
\(681\) −1.58411e35 −0.610165
\(682\) 0 0
\(683\) 4.34050e35 1.61169 0.805843 0.592129i \(-0.201712\pi\)
0.805843 + 0.592129i \(0.201712\pi\)
\(684\) 0 0
\(685\) 9.48896e34 0.339693
\(686\) 0 0
\(687\) −6.49388e35 −2.24153
\(688\) 0 0
\(689\) 1.91372e35 0.636999
\(690\) 0 0
\(691\) −3.59659e35 −1.15456 −0.577279 0.816547i \(-0.695886\pi\)
−0.577279 + 0.816547i \(0.695886\pi\)
\(692\) 0 0
\(693\) −1.76480e35 −0.546425
\(694\) 0 0
\(695\) −1.10548e35 −0.330174
\(696\) 0 0
\(697\) −2.61333e35 −0.752982
\(698\) 0 0
\(699\) −7.22055e34 −0.200727
\(700\) 0 0
\(701\) −4.01671e34 −0.107745 −0.0538723 0.998548i \(-0.517156\pi\)
−0.0538723 + 0.998548i \(0.517156\pi\)
\(702\) 0 0
\(703\) −5.04654e34 −0.130633
\(704\) 0 0
\(705\) −3.37752e35 −0.843788
\(706\) 0 0
\(707\) −4.32357e35 −1.04256
\(708\) 0 0
\(709\) −2.37128e34 −0.0551957 −0.0275979 0.999619i \(-0.508786\pi\)
−0.0275979 + 0.999619i \(0.508786\pi\)
\(710\) 0 0
\(711\) −6.53988e35 −1.46960
\(712\) 0 0
\(713\) 4.75812e34 0.103232
\(714\) 0 0
\(715\) −3.39402e34 −0.0711032
\(716\) 0 0
\(717\) −1.14103e36 −2.30839
\(718\) 0 0
\(719\) 8.41947e35 1.64503 0.822514 0.568744i \(-0.192571\pi\)
0.822514 + 0.568744i \(0.192571\pi\)
\(720\) 0 0
\(721\) −3.13920e35 −0.592419
\(722\) 0 0
\(723\) −2.96697e35 −0.540859
\(724\) 0 0
\(725\) 3.81839e35 0.672443
\(726\) 0 0
\(727\) −8.07486e35 −1.37390 −0.686950 0.726705i \(-0.741051\pi\)
−0.686950 + 0.726705i \(0.741051\pi\)
\(728\) 0 0
\(729\) 1.75615e36 2.88714
\(730\) 0 0
\(731\) 3.02321e35 0.480287
\(732\) 0 0
\(733\) −9.32041e35 −1.43099 −0.715493 0.698620i \(-0.753798\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(734\) 0 0
\(735\) −1.00210e35 −0.148702
\(736\) 0 0
\(737\) 2.24310e35 0.321740
\(738\) 0 0
\(739\) 9.24490e34 0.128188 0.0640939 0.997944i \(-0.479584\pi\)
0.0640939 + 0.997944i \(0.479584\pi\)
\(740\) 0 0
\(741\) −8.33068e35 −1.11674
\(742\) 0 0
\(743\) −3.86129e35 −0.500463 −0.250232 0.968186i \(-0.580507\pi\)
−0.250232 + 0.968186i \(0.580507\pi\)
\(744\) 0 0
\(745\) 5.81106e34 0.0728287
\(746\) 0 0
\(747\) 3.69882e36 4.48287
\(748\) 0 0
\(749\) 5.30431e35 0.621736
\(750\) 0 0
\(751\) −5.31819e35 −0.602927 −0.301463 0.953478i \(-0.597475\pi\)
−0.301463 + 0.953478i \(0.597475\pi\)
\(752\) 0 0
\(753\) 4.65889e35 0.510911
\(754\) 0 0
\(755\) 3.44496e35 0.365466
\(756\) 0 0
\(757\) 1.46803e36 1.50674 0.753368 0.657599i \(-0.228428\pi\)
0.753368 + 0.657599i \(0.228428\pi\)
\(758\) 0 0
\(759\) 4.89561e34 0.0486166
\(760\) 0 0
\(761\) 1.26153e36 1.21225 0.606123 0.795371i \(-0.292724\pi\)
0.606123 + 0.795371i \(0.292724\pi\)
\(762\) 0 0
\(763\) −5.02210e35 −0.467012
\(764\) 0 0
\(765\) 5.06179e35 0.455550
\(766\) 0 0
\(767\) −3.98910e35 −0.347482
\(768\) 0 0
\(769\) −9.78483e35 −0.825039 −0.412519 0.910949i \(-0.635351\pi\)
−0.412519 + 0.910949i \(0.635351\pi\)
\(770\) 0 0
\(771\) −1.96165e36 −1.60119
\(772\) 0 0
\(773\) −1.97217e36 −1.55848 −0.779238 0.626728i \(-0.784394\pi\)
−0.779238 + 0.626728i \(0.784394\pi\)
\(774\) 0 0
\(775\) −1.23425e36 −0.944350
\(776\) 0 0
\(777\) −5.45059e35 −0.403813
\(778\) 0 0
\(779\) −7.85547e35 −0.563578
\(780\) 0 0
\(781\) −1.32757e35 −0.0922398
\(782\) 0 0
\(783\) 3.35457e36 2.25743
\(784\) 0 0
\(785\) −6.52153e35 −0.425088
\(786\) 0 0
\(787\) 2.25189e35 0.142188 0.0710940 0.997470i \(-0.477351\pi\)
0.0710940 + 0.997470i \(0.477351\pi\)
\(788\) 0 0
\(789\) 2.70224e36 1.65295
\(790\) 0 0
\(791\) 2.11702e36 1.25464
\(792\) 0 0
\(793\) −3.27529e36 −1.88077
\(794\) 0 0
\(795\) −4.81568e35 −0.267959
\(796\) 0 0
\(797\) 3.69113e36 1.99035 0.995177 0.0980978i \(-0.0312758\pi\)
0.995177 + 0.0980978i \(0.0312758\pi\)
\(798\) 0 0
\(799\) 2.30893e36 1.20664
\(800\) 0 0
\(801\) 1.96123e36 0.993393
\(802\) 0 0
\(803\) −9.03258e34 −0.0443473
\(804\) 0 0
\(805\) −4.48700e34 −0.0213554
\(806\) 0 0
\(807\) −4.12420e36 −1.90292
\(808\) 0 0
\(809\) −2.93815e36 −1.31437 −0.657185 0.753730i \(-0.728253\pi\)
−0.657185 + 0.753730i \(0.728253\pi\)
\(810\) 0 0
\(811\) 2.44583e36 1.06088 0.530439 0.847723i \(-0.322027\pi\)
0.530439 + 0.847723i \(0.322027\pi\)
\(812\) 0 0
\(813\) −2.71729e36 −1.14289
\(814\) 0 0
\(815\) 1.04137e35 0.0424752
\(816\) 0 0
\(817\) 9.08753e35 0.359476
\(818\) 0 0
\(819\) −6.53060e36 −2.50556
\(820\) 0 0
\(821\) 3.33815e36 1.24227 0.621136 0.783703i \(-0.286672\pi\)
0.621136 + 0.783703i \(0.286672\pi\)
\(822\) 0 0
\(823\) −2.54036e36 −0.917058 −0.458529 0.888679i \(-0.651623\pi\)
−0.458529 + 0.888679i \(0.651623\pi\)
\(824\) 0 0
\(825\) −1.26992e36 −0.444734
\(826\) 0 0
\(827\) 3.63276e36 1.23429 0.617146 0.786849i \(-0.288289\pi\)
0.617146 + 0.786849i \(0.288289\pi\)
\(828\) 0 0
\(829\) 1.07727e36 0.355135 0.177567 0.984109i \(-0.443177\pi\)
0.177567 + 0.984109i \(0.443177\pi\)
\(830\) 0 0
\(831\) 2.13782e36 0.683847
\(832\) 0 0
\(833\) 6.85052e35 0.212648
\(834\) 0 0
\(835\) −1.37324e36 −0.413682
\(836\) 0 0
\(837\) −1.08433e37 −3.17024
\(838\) 0 0
\(839\) 2.96808e36 0.842269 0.421134 0.906998i \(-0.361632\pi\)
0.421134 + 0.906998i \(0.361632\pi\)
\(840\) 0 0
\(841\) −1.76056e36 −0.484953
\(842\) 0 0
\(843\) −8.96316e36 −2.39671
\(844\) 0 0
\(845\) −2.88930e35 −0.0750039
\(846\) 0 0
\(847\) −3.09185e36 −0.779247
\(848\) 0 0
\(849\) −2.74735e36 −0.672308
\(850\) 0 0
\(851\) 1.09743e35 0.0260769
\(852\) 0 0
\(853\) 6.13786e36 1.41630 0.708148 0.706064i \(-0.249531\pi\)
0.708148 + 0.706064i \(0.249531\pi\)
\(854\) 0 0
\(855\) 1.52153e36 0.340961
\(856\) 0 0
\(857\) 3.16481e36 0.688791 0.344395 0.938825i \(-0.388084\pi\)
0.344395 + 0.938825i \(0.388084\pi\)
\(858\) 0 0
\(859\) −5.65370e36 −1.19514 −0.597569 0.801818i \(-0.703866\pi\)
−0.597569 + 0.801818i \(0.703866\pi\)
\(860\) 0 0
\(861\) −8.48442e36 −1.74214
\(862\) 0 0
\(863\) 8.00828e36 1.59736 0.798682 0.601753i \(-0.205531\pi\)
0.798682 + 0.601753i \(0.205531\pi\)
\(864\) 0 0
\(865\) 1.08261e36 0.209783
\(866\) 0 0
\(867\) 5.37651e36 1.01219
\(868\) 0 0
\(869\) 7.54351e35 0.137983
\(870\) 0 0
\(871\) 8.30052e36 1.47529
\(872\) 0 0
\(873\) 2.25649e37 3.89723
\(874\) 0 0
\(875\) 2.40613e36 0.403848
\(876\) 0 0
\(877\) 3.29560e36 0.537575 0.268788 0.963199i \(-0.413377\pi\)
0.268788 + 0.963199i \(0.413377\pi\)
\(878\) 0 0
\(879\) 7.33233e36 1.16247
\(880\) 0 0
\(881\) −8.05386e36 −1.24109 −0.620547 0.784169i \(-0.713090\pi\)
−0.620547 + 0.784169i \(0.713090\pi\)
\(882\) 0 0
\(883\) 4.91849e35 0.0736754 0.0368377 0.999321i \(-0.488272\pi\)
0.0368377 + 0.999321i \(0.488272\pi\)
\(884\) 0 0
\(885\) 1.00381e36 0.146172
\(886\) 0 0
\(887\) −7.79484e36 −1.10347 −0.551737 0.834018i \(-0.686035\pi\)
−0.551737 + 0.834018i \(0.686035\pi\)
\(888\) 0 0
\(889\) 1.88247e36 0.259093
\(890\) 0 0
\(891\) −6.24031e36 −0.835093
\(892\) 0 0
\(893\) 6.94048e36 0.903121
\(894\) 0 0
\(895\) −1.10741e36 −0.140127
\(896\) 0 0
\(897\) 1.81160e36 0.222924
\(898\) 0 0
\(899\) −6.04394e36 −0.723309
\(900\) 0 0
\(901\) 3.29209e36 0.383188
\(902\) 0 0
\(903\) 9.81513e36 1.11122
\(904\) 0 0
\(905\) −2.47597e35 −0.0272670
\(906\) 0 0
\(907\) 1.31417e37 1.40786 0.703932 0.710268i \(-0.251426\pi\)
0.703932 + 0.710268i \(0.251426\pi\)
\(908\) 0 0
\(909\) −3.18823e37 −3.32278
\(910\) 0 0
\(911\) −1.44767e37 −1.46788 −0.733940 0.679214i \(-0.762321\pi\)
−0.733940 + 0.679214i \(0.762321\pi\)
\(912\) 0 0
\(913\) −4.26645e36 −0.420903
\(914\) 0 0
\(915\) 8.24192e36 0.791161
\(916\) 0 0
\(917\) 4.24848e36 0.396842
\(918\) 0 0
\(919\) 1.03191e37 0.937990 0.468995 0.883201i \(-0.344616\pi\)
0.468995 + 0.883201i \(0.344616\pi\)
\(920\) 0 0
\(921\) 1.35356e37 1.19738
\(922\) 0 0
\(923\) −4.91262e36 −0.422953
\(924\) 0 0
\(925\) −2.84672e36 −0.238546
\(926\) 0 0
\(927\) −2.31487e37 −1.88813
\(928\) 0 0
\(929\) −2.17912e37 −1.73015 −0.865075 0.501642i \(-0.832729\pi\)
−0.865075 + 0.501642i \(0.832729\pi\)
\(930\) 0 0
\(931\) 2.05921e36 0.159159
\(932\) 0 0
\(933\) 2.71170e37 2.04043
\(934\) 0 0
\(935\) −5.83858e35 −0.0427723
\(936\) 0 0
\(937\) −2.08020e37 −1.48375 −0.741873 0.670540i \(-0.766062\pi\)
−0.741873 + 0.670540i \(0.766062\pi\)
\(938\) 0 0
\(939\) −2.85686e37 −1.98413
\(940\) 0 0
\(941\) 2.39454e37 1.61939 0.809695 0.586851i \(-0.199632\pi\)
0.809695 + 0.586851i \(0.199632\pi\)
\(942\) 0 0
\(943\) 1.70826e36 0.112501
\(944\) 0 0
\(945\) 1.02254e37 0.655817
\(946\) 0 0
\(947\) 8.06134e36 0.503539 0.251769 0.967787i \(-0.418988\pi\)
0.251769 + 0.967787i \(0.418988\pi\)
\(948\) 0 0
\(949\) −3.34248e36 −0.203348
\(950\) 0 0
\(951\) 4.02727e37 2.38646
\(952\) 0 0
\(953\) −1.92589e37 −1.11166 −0.555829 0.831297i \(-0.687599\pi\)
−0.555829 + 0.831297i \(0.687599\pi\)
\(954\) 0 0
\(955\) −6.75736e36 −0.379957
\(956\) 0 0
\(957\) −6.21858e36 −0.340637
\(958\) 0 0
\(959\) −2.10601e37 −1.12390
\(960\) 0 0
\(961\) 3.03575e35 0.0157842
\(962\) 0 0
\(963\) 3.91144e37 1.98157
\(964\) 0 0
\(965\) −1.95205e36 −0.0963606
\(966\) 0 0
\(967\) −2.29324e37 −1.10311 −0.551554 0.834139i \(-0.685965\pi\)
−0.551554 + 0.834139i \(0.685965\pi\)
\(968\) 0 0
\(969\) −1.43309e37 −0.671779
\(970\) 0 0
\(971\) −2.10199e37 −0.960265 −0.480132 0.877196i \(-0.659411\pi\)
−0.480132 + 0.877196i \(0.659411\pi\)
\(972\) 0 0
\(973\) 2.45355e37 1.09241
\(974\) 0 0
\(975\) −4.69929e37 −2.03927
\(976\) 0 0
\(977\) −3.52592e37 −1.49139 −0.745694 0.666289i \(-0.767882\pi\)
−0.745694 + 0.666289i \(0.767882\pi\)
\(978\) 0 0
\(979\) −2.26220e36 −0.0932712
\(980\) 0 0
\(981\) −3.70334e37 −1.48844
\(982\) 0 0
\(983\) −7.31962e36 −0.286794 −0.143397 0.989665i \(-0.545803\pi\)
−0.143397 + 0.989665i \(0.545803\pi\)
\(984\) 0 0
\(985\) −4.92076e36 −0.187966
\(986\) 0 0
\(987\) 7.49617e37 2.79174
\(988\) 0 0
\(989\) −1.97619e36 −0.0717587
\(990\) 0 0
\(991\) −1.16985e37 −0.414200 −0.207100 0.978320i \(-0.566403\pi\)
−0.207100 + 0.978320i \(0.566403\pi\)
\(992\) 0 0
\(993\) 6.06819e37 2.09504
\(994\) 0 0
\(995\) −1.11277e37 −0.374641
\(996\) 0 0
\(997\) 2.33411e37 0.766358 0.383179 0.923674i \(-0.374829\pi\)
0.383179 + 0.923674i \(0.374829\pi\)
\(998\) 0 0
\(999\) −2.50092e37 −0.800814
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.1.1 2
4.3 odd 2 2.26.a.b.1.2 2
12.11 even 2 18.26.a.e.1.2 2
20.3 even 4 50.26.b.e.49.2 4
20.7 even 4 50.26.b.e.49.3 4
20.19 odd 2 50.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 2 4.3 odd 2
16.26.a.c.1.1 2 1.1 even 1 trivial
18.26.a.e.1.2 2 12.11 even 2
50.26.a.c.1.1 2 20.19 odd 2
50.26.b.e.49.2 4 20.3 even 4
50.26.b.e.49.3 4 20.7 even 4