Properties

Label 50.26.a.c.1.1
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $1$
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] = [50,26,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} -1.75788e6 q^{3} +1.67772e7 q^{4} +7.20027e9 q^{6} +3.04153e10 q^{7} -6.87195e10 q^{8} +2.24285e12 q^{9} +2.58704e12 q^{11} -2.94923e13 q^{12} +9.57327e13 q^{13} -1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15 q^{17} -9.18671e15 q^{18} +4.95030e15 q^{19} -5.34664e16 q^{21} -1.05965e16 q^{22} +1.07650e16 q^{23} +1.20801e17 q^{24} -3.92121e17 q^{26} -2.45323e18 q^{27} +5.10284e17 q^{28} -1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18 q^{32} -4.54771e18 q^{33} -6.74549e18 q^{34} +3.76288e19 q^{36} -1.01944e19 q^{37} -2.02764e19 q^{38} -1.68287e20 q^{39} +1.58687e20 q^{41} +2.18999e20 q^{42} -1.83575e20 q^{43} +4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21 q^{47} -4.94799e20 q^{48} -4.15978e20 q^{49} -2.89496e21 q^{51} +1.60613e21 q^{52} +1.99903e21 q^{53} +1.00484e22 q^{54} -2.09012e21 q^{56} -8.70202e21 q^{57} +5.60091e21 q^{58} -4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22 q^{62} +6.82170e22 q^{63} +4.72237e21 q^{64} +1.86274e22 q^{66} -8.67051e22 q^{67} +2.76295e22 q^{68} -1.89236e22 q^{69} -5.13159e22 q^{71} -1.54128e23 q^{72} -3.49147e22 q^{73} +4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22 q^{77} +6.89302e23 q^{78} +2.91588e23 q^{79} +2.41214e24 q^{81} -6.49981e23 q^{82} +1.64916e24 q^{83} -8.97018e23 q^{84} +7.51925e23 q^{86} +2.40374e24 q^{87} -1.77780e23 q^{88} +8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23 q^{92} +7.76982e24 q^{93} +5.74273e24 q^{94} +2.02670e24 q^{96} -1.00608e25 q^{97} +1.70384e24 q^{98} +5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} - 379848 q^{3} + 33554432 q^{4} + 1555857408 q^{6} + 376536944 q^{7} - 137438953472 q^{8} + 3294531432666 q^{9} + 8323034610264 q^{11} - 6372791943168 q^{12} + 106467053152292 q^{13} - 1542295322624 q^{14}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) −1.75788e6 −1.90974 −0.954868 0.297031i \(-0.904004\pi\)
−0.954868 + 0.297031i \(0.904004\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) 7.20027e9 1.35039
\(7\) 3.04153e10 0.830552 0.415276 0.909696i \(-0.363685\pi\)
0.415276 + 0.909696i \(0.363685\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 2.24285e12 2.64709
\(10\) 0 0
\(11\) 2.58704e12 0.248539 0.124270 0.992248i \(-0.460341\pi\)
0.124270 + 0.992248i \(0.460341\pi\)
\(12\) −2.94923e13 −0.954868
\(13\) 9.57327e13 1.13964 0.569821 0.821769i \(-0.307012\pi\)
0.569821 + 0.821769i \(0.307012\pi\)
\(14\) −1.24581e14 −0.587289
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 1.64685e15 0.685555 0.342777 0.939417i \(-0.388632\pi\)
0.342777 + 0.939417i \(0.388632\pi\)
\(18\) −9.18671e15 −1.87178
\(19\) 4.95030e15 0.513111 0.256555 0.966530i \(-0.417412\pi\)
0.256555 + 0.966530i \(0.417412\pi\)
\(20\) 0 0
\(21\) −5.34664e16 −1.58613
\(22\) −1.05965e16 −0.175744
\(23\) 1.07650e16 0.102427 0.0512136 0.998688i \(-0.483691\pi\)
0.0512136 + 0.998688i \(0.483691\pi\)
\(24\) 1.20801e17 0.675194
\(25\) 0 0
\(26\) −3.92121e17 −0.805849
\(27\) −2.45323e18 −3.14551
\(28\) 5.10284e17 0.415276
\(29\) −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 \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) −4.54771e18 −0.474645
\(34\) −6.74549e18 −0.484760
\(35\) 0 0
\(36\) 3.76288e19 1.32355
\(37\) −1.01944e19 −0.254590 −0.127295 0.991865i \(-0.540629\pi\)
−0.127295 + 0.991865i \(0.540629\pi\)
\(38\) −2.02764e19 −0.362824
\(39\) −1.68287e20 −2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) 2.18999e20 1.12157
\(43\) −1.83575e20 −0.700582 −0.350291 0.936641i \(-0.613917\pi\)
−0.350291 + 0.936641i \(0.613917\pi\)
\(44\) 4.34034e19 0.124270
\(45\) 0 0
\(46\) −4.40934e19 −0.0724270
\(47\) −1.40203e21 −1.76009 −0.880045 0.474890i \(-0.842488\pi\)
−0.880045 + 0.474890i \(0.842488\pi\)
\(48\) −4.94799e20 −0.477434
\(49\) −4.15978e20 −0.310184
\(50\) 0 0
\(51\) −2.89496e21 −1.30923
\(52\) 1.60613e21 0.569821
\(53\) 1.99903e21 0.558946 0.279473 0.960154i \(-0.409840\pi\)
0.279473 + 0.960154i \(0.409840\pi\)
\(54\) 1.00484e22 2.22421
\(55\) 0 0
\(56\) −2.09012e21 −0.293644
\(57\) −8.70202e21 −0.979906
\(58\) 5.60091e21 0.507468
\(59\) −4.16691e21 −0.304905 −0.152452 0.988311i \(-0.548717\pi\)
−0.152452 + 0.988311i \(0.548717\pi\)
\(60\) 0 0
\(61\) 3.42128e22 1.65031 0.825156 0.564904i \(-0.191087\pi\)
0.825156 + 0.564904i \(0.191087\pi\)
\(62\) 1.81043e22 0.712666
\(63\) 6.82170e22 2.19855
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) 1.86274e22 0.335624
\(67\) −8.67051e22 −1.29452 −0.647261 0.762268i \(-0.724086\pi\)
−0.647261 + 0.762268i \(0.724086\pi\)
\(68\) 2.76295e22 0.342777
\(69\) −1.89236e22 −0.195609
\(70\) 0 0
\(71\) −5.13159e22 −0.371127 −0.185564 0.982632i \(-0.559411\pi\)
−0.185564 + 0.982632i \(0.559411\pi\)
\(72\) −1.54128e23 −0.935888
\(73\) −3.49147e22 −0.178432 −0.0892159 0.996012i \(-0.528436\pi\)
−0.0892159 + 0.996012i \(0.528436\pi\)
\(74\) 4.17563e22 0.180022
\(75\) 0 0
\(76\) 8.30522e22 0.256555
\(77\) 7.86857e22 0.206425
\(78\) 6.89302e23 1.53896
\(79\) 2.91588e23 0.555176 0.277588 0.960700i \(-0.410465\pi\)
0.277588 + 0.960700i \(0.410465\pi\)
\(80\) 0 0
\(81\) 2.41214e24 3.36000
\(82\) −6.49981e23 −0.776654
\(83\) 1.64916e24 1.69351 0.846753 0.531986i \(-0.178554\pi\)
0.846753 + 0.531986i \(0.178554\pi\)
\(84\) −8.97018e23 −0.793067
\(85\) 0 0
\(86\) 7.51925e23 0.495386
\(87\) 2.40374e24 1.37056
\(88\) −1.77780e23 −0.0878720
\(89\) 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
\(92\) 1.80607e23 0.0512136
\(93\) 7.76982e24 1.92475
\(94\) 5.74273e24 1.24457
\(95\) 0 0
\(96\) 2.02670e24 0.337597
\(97\) −1.00608e25 −1.47227 −0.736134 0.676835i \(-0.763351\pi\)
−0.736134 + 0.676835i \(0.763351\pi\)
\(98\) 1.70384e24 0.219333
\(99\) 5.80235e24 0.657907
\(100\) 0 0
\(101\) −1.42151e25 −1.25526 −0.627628 0.778513i \(-0.715974\pi\)
−0.627628 + 0.778513i \(0.715974\pi\)
\(102\) 1.18578e25 0.925764
\(103\) −1.03211e25 −0.713283 −0.356642 0.934241i \(-0.616078\pi\)
−0.356642 + 0.934241i \(0.616078\pi\)
\(104\) −6.57870e24 −0.402924
\(105\) 0 0
\(106\) −8.18801e24 −0.395235
\(107\) 1.74396e25 0.748582 0.374291 0.927311i \(-0.377886\pi\)
0.374291 + 0.927311i \(0.377886\pi\)
\(108\) −4.11583e25 −1.57275
\(109\) −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
\(112\) 8.56115e24 0.207638
\(113\) −6.96039e25 −1.51061 −0.755306 0.655372i \(-0.772512\pi\)
−0.755306 + 0.655372i \(0.772512\pi\)
\(114\) 3.56435e25 0.692898
\(115\) 0 0
\(116\) −2.29413e25 −0.358834
\(117\) 2.14714e26 3.01674
\(118\) 1.70677e25 0.215600
\(119\) 5.00894e25 0.569389
\(120\) 0 0
\(121\) −1.01654e26 −0.938228
\(122\) −1.40136e26 −1.16695
\(123\) −2.78952e26 −2.09757
\(124\) −7.41552e25 −0.503931
\(125\) 0 0
\(126\) −2.79417e26 −1.55461
\(127\) 6.18922e25 0.311953 0.155977 0.987761i \(-0.450148\pi\)
0.155977 + 0.987761i \(0.450148\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 3.22703e26 1.33793
\(130\) 0 0
\(131\) −1.39682e26 −0.477805 −0.238902 0.971044i \(-0.576788\pi\)
−0.238902 + 0.971044i \(0.576788\pi\)
\(132\) −7.62979e25 −0.237322
\(133\) 1.50565e26 0.426165
\(134\) 3.55144e26 0.915366
\(135\) 0 0
\(136\) −1.13171e26 −0.242380
\(137\) 6.92418e26 1.35320 0.676599 0.736352i \(-0.263453\pi\)
0.676599 + 0.736352i \(0.263453\pi\)
\(138\) 7.75109e25 0.138317
\(139\) −8.06682e26 −1.31528 −0.657639 0.753333i \(-0.728445\pi\)
−0.657639 + 0.753333i \(0.728445\pi\)
\(140\) 0 0
\(141\) 2.46460e27 3.36131
\(142\) 2.10190e26 0.262427
\(143\) 2.47665e26 0.283246
\(144\) 6.31306e26 0.661773
\(145\) 0 0
\(146\) 1.43010e26 0.126170
\(147\) 7.31239e26 0.592369
\(148\) −1.71034e26 −0.127295
\(149\) −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 \(-0.240481\pi\)
0.727934 + 0.685647i \(0.240481\pi\)
\(152\) −3.40182e26 −0.181412
\(153\) 3.69363e27 1.81473
\(154\) −3.22297e26 −0.145964
\(155\) 0 0
\(156\) −2.82338e27 −1.08821
\(157\) −4.75882e27 −1.69338 −0.846689 0.532088i \(-0.821407\pi\)
−0.846689 + 0.532088i \(0.821407\pi\)
\(158\) −1.19434e27 −0.392569
\(159\) −3.51404e27 −1.06744
\(160\) 0 0
\(161\) 3.27420e26 0.0850712
\(162\) −9.88012e27 −2.37588
\(163\) −7.59899e26 −0.169204 −0.0846020 0.996415i \(-0.526962\pi\)
−0.0846020 + 0.996415i \(0.526962\pi\)
\(164\) 2.66232e27 0.549177
\(165\) 0 0
\(166\) −6.75496e27 −1.19749
\(167\) 1.00207e28 1.64794 0.823970 0.566634i \(-0.191755\pi\)
0.823970 + 0.566634i \(0.191755\pi\)
\(168\) 3.67419e27 0.560783
\(169\) 2.10835e27 0.298785
\(170\) 0 0
\(171\) 1.11028e28 1.35825
\(172\) −3.07989e27 −0.350291
\(173\) 7.89989e27 0.835689 0.417845 0.908518i \(-0.362786\pi\)
0.417845 + 0.908518i \(0.362786\pi\)
\(174\) −9.84571e27 −0.969129
\(175\) 0 0
\(176\) 7.28188e26 0.0621349
\(177\) 7.32493e27 0.582288
\(178\) −3.58168e27 −0.265361
\(179\) −8.08087e27 −0.558207 −0.279103 0.960261i \(-0.590037\pi\)
−0.279103 + 0.960261i \(0.590037\pi\)
\(180\) 0 0
\(181\) 1.80674e27 0.108621 0.0543104 0.998524i \(-0.482704\pi\)
0.0543104 + 0.998524i \(0.482704\pi\)
\(182\) −1.19265e28 −0.669299
\(183\) −6.01420e28 −3.15166
\(184\) −7.39765e26 −0.0362135
\(185\) 0 0
\(186\) −3.18252e28 −1.36100
\(187\) 4.26047e27 0.170387
\(188\) −2.35222e28 −0.880045
\(189\) −7.46157e28 −2.61251
\(190\) 0 0
\(191\) −4.93091e28 −1.51360 −0.756798 0.653649i \(-0.773237\pi\)
−0.756798 + 0.653649i \(0.773237\pi\)
\(192\) −8.30135e27 −0.238717
\(193\) −1.42443e28 −0.383861 −0.191931 0.981408i \(-0.561475\pi\)
−0.191931 + 0.981408i \(0.561475\pi\)
\(194\) 4.12092e28 1.04105
\(195\) 0 0
\(196\) −6.97895e27 −0.155092
\(197\) −3.59073e28 −0.748781 −0.374390 0.927271i \(-0.622148\pi\)
−0.374390 + 0.927271i \(0.622148\pi\)
\(198\) −2.37664e28 −0.465210
\(199\) −8.11996e28 −1.49242 −0.746209 0.665712i \(-0.768128\pi\)
−0.746209 + 0.665712i \(0.768128\pi\)
\(200\) 0 0
\(201\) 1.52417e29 2.47220
\(202\) 5.82250e28 0.887601
\(203\) −4.15902e28 −0.596060
\(204\) −4.85694e28 −0.654614
\(205\) 0 0
\(206\) 4.22753e28 0.504367
\(207\) 2.41443e28 0.271134
\(208\) 2.69464e28 0.284911
\(209\) 1.28066e28 0.127528
\(210\) 0 0
\(211\) −1.03285e29 −0.913075 −0.456537 0.889704i \(-0.650911\pi\)
−0.456537 + 0.889704i \(0.650911\pi\)
\(212\) 3.35381e28 0.279473
\(213\) 9.02072e28 0.708755
\(214\) −7.14326e28 −0.529327
\(215\) 0 0
\(216\) 1.68585e29 1.11211
\(217\) −1.34436e29 −0.837081
\(218\) 6.76321e28 0.397600
\(219\) 6.13758e28 0.340757
\(220\) 0 0
\(221\) 1.57657e29 0.781287
\(222\) −7.34025e28 −0.343795
\(223\) −1.15542e29 −0.511597 −0.255799 0.966730i \(-0.582338\pi\)
−0.255799 + 0.966730i \(0.582338\pi\)
\(224\) −3.50665e28 −0.146822
\(225\) 0 0
\(226\) 2.85098e29 1.06816
\(227\) 9.01151e28 0.319502 0.159751 0.987157i \(-0.448931\pi\)
0.159751 + 0.987157i \(0.448931\pi\)
\(228\) −1.45996e29 −0.489953
\(229\) 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
\(232\) 9.39676e28 0.253734
\(233\) −4.10753e28 −0.105107 −0.0525536 0.998618i \(-0.516736\pi\)
−0.0525536 + 0.998618i \(0.516736\pi\)
\(234\) −8.79469e29 −2.13316
\(235\) 0 0
\(236\) −6.99092e28 −0.152452
\(237\) −5.12576e29 −1.06024
\(238\) −2.05166e29 −0.402618
\(239\) −6.49097e29 −1.20875 −0.604374 0.796701i \(-0.706577\pi\)
−0.604374 + 0.796701i \(0.706577\pi\)
\(240\) 0 0
\(241\) 1.68781e29 0.283212 0.141606 0.989923i \(-0.454773\pi\)
0.141606 + 0.989923i \(0.454773\pi\)
\(242\) 4.16376e29 0.663427
\(243\) −2.16165e30 −3.27121
\(244\) 5.73996e29 0.825156
\(245\) 0 0
\(246\) 1.14259e30 1.48320
\(247\) 4.73905e29 0.584763
\(248\) 3.03740e29 0.356333
\(249\) −2.89903e30 −3.23415
\(250\) 0 0
\(251\) 2.65029e29 0.267530 0.133765 0.991013i \(-0.457293\pi\)
0.133765 + 0.991013i \(0.457293\pi\)
\(252\) 1.14449e30 1.09927
\(253\) 2.78495e28 0.0254572
\(254\) −2.53510e29 −0.220584
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −1.11592e30 −0.838433 −0.419217 0.907886i \(-0.637695\pi\)
−0.419217 + 0.907886i \(0.637695\pi\)
\(258\) −1.32179e30 −0.946057
\(259\) −3.10066e29 −0.211450
\(260\) 0 0
\(261\) −3.06689e30 −1.89973
\(262\) 5.72139e29 0.337859
\(263\) −1.53721e30 −0.865541 −0.432770 0.901504i \(-0.642464\pi\)
−0.432770 + 0.901504i \(0.642464\pi\)
\(264\) 3.12516e29 0.167812
\(265\) 0 0
\(266\) −6.16713e29 −0.301344
\(267\) −1.53715e30 −0.716681
\(268\) −1.45467e30 −0.647261
\(269\) 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 \(-0.596729\pi\)
−0.299227 + 0.954182i \(0.596729\pi\)
\(272\) 4.63547e29 0.171389
\(273\) −5.11849e30 −1.80763
\(274\) −2.83614e30 −0.956855
\(275\) 0 0
\(276\) −3.17485e29 −0.0978045
\(277\) 1.21614e30 0.358085 0.179042 0.983841i \(-0.442700\pi\)
0.179042 + 0.983841i \(0.442700\pi\)
\(278\) 3.30417e30 0.930042
\(279\) −9.91339e30 −2.66790
\(280\) 0 0
\(281\) 5.09885e30 1.25500 0.627499 0.778617i \(-0.284079\pi\)
0.627499 + 0.778617i \(0.284079\pi\)
\(282\) −1.00950e31 −2.37680
\(283\) 1.56288e30 0.352042 0.176021 0.984386i \(-0.443677\pi\)
0.176021 + 0.984386i \(0.443677\pi\)
\(284\) −8.60939e29 −0.185564
\(285\) 0 0
\(286\) −1.01444e30 −0.200285
\(287\) 4.82651e30 0.912240
\(288\) −2.58583e30 −0.467944
\(289\) −3.05852e30 −0.530015
\(290\) 0 0
\(291\) 1.76857e31 2.81164
\(292\) −5.85771e29 −0.0892159
\(293\) 4.17112e30 0.608706 0.304353 0.952559i \(-0.401560\pi\)
0.304353 + 0.952559i \(0.401560\pi\)
\(294\) −2.99515e30 −0.418868
\(295\) 0 0
\(296\) 7.00555e29 0.0900110
\(297\) −6.34661e30 −0.781783
\(298\) 1.73686e30 0.205146
\(299\) 1.03056e30 0.116730
\(300\) 0 0
\(301\) −5.58350e30 −0.581870
\(302\) −1.02966e31 −1.02945
\(303\) 2.49884e31 2.39721
\(304\) 1.39338e30 0.128278
\(305\) 0 0
\(306\) −1.51291e31 −1.28320
\(307\) −7.69994e30 −0.626986 −0.313493 0.949591i \(-0.601499\pi\)
−0.313493 + 0.949591i \(0.601499\pi\)
\(308\) 1.32013e30 0.103212
\(309\) 1.81433e31 1.36218
\(310\) 0 0
\(311\) 1.54260e31 1.06843 0.534217 0.845347i \(-0.320606\pi\)
0.534217 + 0.845347i \(0.320606\pi\)
\(312\) 1.15646e31 0.769479
\(313\) −1.62517e31 −1.03895 −0.519476 0.854485i \(-0.673873\pi\)
−0.519476 + 0.854485i \(0.673873\pi\)
\(314\) 1.94921e31 1.19740
\(315\) 0 0
\(316\) 4.89203e30 0.277588
\(317\) 2.29098e31 1.24963 0.624814 0.780773i \(-0.285175\pi\)
0.624814 + 0.780773i \(0.285175\pi\)
\(318\) 1.43935e31 0.754794
\(319\) −3.53755e30 −0.178369
\(320\) 0 0
\(321\) −3.06567e31 −1.42959
\(322\) −1.34111e30 −0.0601544
\(323\) 8.15239e30 0.351765
\(324\) 4.04690e31 1.68000
\(325\) 0 0
\(326\) 3.11255e30 0.119645
\(327\) 2.90256e31 1.07383
\(328\) −1.09049e31 −0.388327
\(329\) −4.26433e31 −1.46185
\(330\) 0 0
\(331\) 3.45199e31 1.09703 0.548517 0.836140i \(-0.315193\pi\)
0.548517 + 0.836140i \(0.315193\pi\)
\(332\) 2.76683e31 0.846753
\(333\) −2.28645e31 −0.673922
\(334\) −4.10447e31 −1.16527
\(335\) 0 0
\(336\) −1.50495e31 −0.396534
\(337\) 5.27735e31 1.33981 0.669904 0.742448i \(-0.266335\pi\)
0.669904 + 0.742448i \(0.266335\pi\)
\(338\) −8.63579e30 −0.211273
\(339\) 1.22355e32 2.88487
\(340\) 0 0
\(341\) −1.14347e31 −0.250493
\(342\) −4.54770e31 −0.960429
\(343\) −5.34411e31 −1.08818
\(344\) 1.26152e31 0.247693
\(345\) 0 0
\(346\) −3.23580e31 −0.590922
\(347\) −4.40093e30 −0.0775222 −0.0387611 0.999249i \(-0.512341\pi\)
−0.0387611 + 0.999249i \(0.512341\pi\)
\(348\) 4.03280e31 0.685278
\(349\) 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
\(352\) −2.98266e30 −0.0439360
\(353\) −4.46084e31 −0.634209 −0.317105 0.948391i \(-0.602711\pi\)
−0.317105 + 0.948391i \(0.602711\pi\)
\(354\) −3.00029e31 −0.411740
\(355\) 0 0
\(356\) 1.46706e31 0.187639
\(357\) −8.80511e31 −1.08738
\(358\) 3.30992e31 0.394712
\(359\) 4.38054e31 0.504483 0.252242 0.967664i \(-0.418832\pi\)
0.252242 + 0.967664i \(0.418832\pi\)
\(360\) 0 0
\(361\) −6.85711e31 −0.736717
\(362\) −7.40039e30 −0.0768065
\(363\) 1.78696e32 1.79177
\(364\) 4.88509e31 0.473266
\(365\) 0 0
\(366\) 2.46342e32 2.22856
\(367\) 7.76735e31 0.679121 0.339560 0.940584i \(-0.389722\pi\)
0.339560 + 0.940584i \(0.389722\pi\)
\(368\) 3.03008e30 0.0256068
\(369\) 3.55911e32 2.90744
\(370\) 0 0
\(371\) 6.08010e31 0.464234
\(372\) 1.30356e32 0.962374
\(373\) 2.37205e32 1.69342 0.846708 0.532058i \(-0.178581\pi\)
0.846708 + 0.532058i \(0.178581\pi\)
\(374\) −1.74509e31 −0.120482
\(375\) 0 0
\(376\) 9.63470e31 0.622286
\(377\) −1.30906e32 −0.817884
\(378\) 3.05626e32 1.84732
\(379\) −1.38944e32 −0.812551 −0.406276 0.913751i \(-0.633173\pi\)
−0.406276 + 0.913751i \(0.633173\pi\)
\(380\) 0 0
\(381\) −1.08799e32 −0.595748
\(382\) 2.01970e32 1.07027
\(383\) 3.79085e31 0.194425 0.0972125 0.995264i \(-0.469007\pi\)
0.0972125 + 0.995264i \(0.469007\pi\)
\(384\) 3.40023e31 0.168798
\(385\) 0 0
\(386\) 5.83446e31 0.271431
\(387\) −4.11732e32 −1.85450
\(388\) −1.68793e32 −0.736134
\(389\) −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
\(392\) 2.85858e31 0.109667
\(393\) 2.45545e32 0.912481
\(394\) 1.47076e32 0.529468
\(395\) 0 0
\(396\) 9.73473e31 0.328953
\(397\) −2.18941e32 −0.716877 −0.358439 0.933553i \(-0.616691\pi\)
−0.358439 + 0.933553i \(0.616691\pi\)
\(398\) 3.32594e32 1.05530
\(399\) −2.64675e32 −0.813863
\(400\) 0 0
\(401\) 2.99036e32 0.863810 0.431905 0.901919i \(-0.357842\pi\)
0.431905 + 0.901919i \(0.357842\pi\)
\(402\) −6.24300e32 −1.74811
\(403\) −4.23138e32 −1.14860
\(404\) −2.38490e32 −0.627628
\(405\) 0 0
\(406\) 1.70353e32 0.421478
\(407\) −2.63734e31 −0.0632755
\(408\) 1.98940e32 0.462882
\(409\) 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
\(412\) −1.73160e32 −0.356642
\(413\) −1.26738e32 −0.253239
\(414\) −9.88949e31 −0.191721
\(415\) 0 0
\(416\) −1.10372e32 −0.201462
\(417\) 1.41805e33 2.51183
\(418\) −5.24560e31 −0.0901761
\(419\) 2.94461e32 0.491307 0.245654 0.969358i \(-0.420997\pi\)
0.245654 + 0.969358i \(0.420997\pi\)
\(420\) 0 0
\(421\) 1.01892e33 1.60182 0.800908 0.598787i \(-0.204350\pi\)
0.800908 + 0.598787i \(0.204350\pi\)
\(422\) 4.23055e32 0.645641
\(423\) −3.14455e33 −4.65912
\(424\) −1.37372e32 −0.197617
\(425\) 0 0
\(426\) −3.69489e32 −0.501166
\(427\) 1.04059e33 1.37067
\(428\) 2.92588e32 0.374291
\(429\) −4.35365e32 −0.540925
\(430\) 0 0
\(431\) −6.79497e32 −0.796565 −0.398283 0.917263i \(-0.630394\pi\)
−0.398283 + 0.917263i \(0.630394\pi\)
\(432\) −6.90523e32 −0.786377
\(433\) 7.24164e32 0.801195 0.400597 0.916254i \(-0.368803\pi\)
0.400597 + 0.916254i \(0.368803\pi\)
\(434\) 5.50648e32 0.591906
\(435\) 0 0
\(436\) −2.77021e32 −0.281146
\(437\) 5.32899e31 0.0525566
\(438\) −2.51395e32 −0.240952
\(439\) −9.13459e32 −0.850908 −0.425454 0.904980i \(-0.639886\pi\)
−0.425454 + 0.904980i \(0.639886\pi\)
\(440\) 0 0
\(441\) −9.32976e32 −0.821085
\(442\) −6.45764e32 −0.552453
\(443\) 3.52555e32 0.293211 0.146605 0.989195i \(-0.453165\pi\)
0.146605 + 0.989195i \(0.453165\pi\)
\(444\) 3.00657e32 0.243099
\(445\) 0 0
\(446\) 4.73259e32 0.361754
\(447\) 7.45409e32 0.554052
\(448\) 1.43632e32 0.103819
\(449\) 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
\(452\) −1.16776e33 −0.755306
\(453\) −4.41899e33 −2.78032
\(454\) −3.69111e32 −0.225922
\(455\) 0 0
\(456\) 5.97998e32 0.346449
\(457\) −1.76958e33 −0.997512 −0.498756 0.866742i \(-0.666210\pi\)
−0.498756 + 0.866742i \(0.666210\pi\)
\(458\) −1.51313e33 −0.829959
\(459\) −4.04009e33 −2.15642
\(460\) 0 0
\(461\) 7.01018e32 0.354379 0.177189 0.984177i \(-0.443299\pi\)
0.177189 + 0.984177i \(0.443299\pi\)
\(462\) 5.66559e32 0.278753
\(463\) 9.20521e32 0.440830 0.220415 0.975406i \(-0.429259\pi\)
0.220415 + 0.975406i \(0.429259\pi\)
\(464\) −3.84891e32 −0.179417
\(465\) 0 0
\(466\) 1.68245e32 0.0743219
\(467\) 1.84600e33 0.793908 0.396954 0.917838i \(-0.370067\pi\)
0.396954 + 0.917838i \(0.370067\pi\)
\(468\) 3.60231e33 1.50837
\(469\) −2.63716e33 −1.07517
\(470\) 0 0
\(471\) 8.36543e33 3.23390
\(472\) 2.86348e32 0.107800
\(473\) −4.74918e32 −0.174122
\(474\) 2.09951e33 0.749703
\(475\) 0 0
\(476\) 8.40360e32 0.284694
\(477\) 4.48351e33 1.47958
\(478\) 2.65870e33 0.854714
\(479\) −3.88481e32 −0.121668 −0.0608339 0.998148i \(-0.519376\pi\)
−0.0608339 + 0.998148i \(0.519376\pi\)
\(480\) 0 0
\(481\) −9.75939e32 −0.290141
\(482\) −6.91328e32 −0.200261
\(483\) −5.75566e32 −0.162463
\(484\) −1.70548e33 −0.469114
\(485\) 0 0
\(486\) 8.85414e33 2.31309
\(487\) −4.26102e33 −1.08493 −0.542465 0.840078i \(-0.682509\pi\)
−0.542465 + 0.840078i \(0.682509\pi\)
\(488\) −2.35109e33 −0.583474
\(489\) 1.33581e33 0.323135
\(490\) 0 0
\(491\) −4.52548e33 −1.04027 −0.520134 0.854084i \(-0.674118\pi\)
−0.520134 + 0.854084i \(0.674118\pi\)
\(492\) −4.68004e33 −1.04878
\(493\) −2.25191e33 −0.492000
\(494\) −1.94112e33 −0.413490
\(495\) 0 0
\(496\) −1.24412e33 −0.251965
\(497\) −1.56079e33 −0.308241
\(498\) 1.18744e34 2.28689
\(499\) 4.45020e33 0.835840 0.417920 0.908484i \(-0.362759\pi\)
0.417920 + 0.908484i \(0.362759\pi\)
\(500\) 0 0
\(501\) −1.76151e34 −3.14713
\(502\) −1.08556e33 −0.189172
\(503\) −2.29032e33 −0.389311 −0.194656 0.980872i \(-0.562359\pi\)
−0.194656 + 0.980872i \(0.562359\pi\)
\(504\) −4.68784e33 −0.777304
\(505\) 0 0
\(506\) −1.14072e32 −0.0180010
\(507\) −3.70622e33 −0.570600
\(508\) 1.03838e33 0.155977
\(509\) 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
\(512\) −3.24519e32 −0.0441942
\(513\) −1.21442e34 −1.61400
\(514\) 4.57080e33 0.592862
\(515\) 0 0
\(516\) 5.41407e33 0.668963
\(517\) −3.62712e33 −0.437452
\(518\) 1.27003e33 0.149518
\(519\) −1.38871e34 −1.59595
\(520\) 0 0
\(521\) −1.14090e34 −1.24961 −0.624807 0.780779i \(-0.714822\pi\)
−0.624807 + 0.780779i \(0.714822\pi\)
\(522\) 1.25620e34 1.34331
\(523\) −1.37717e34 −1.43786 −0.718928 0.695085i \(-0.755367\pi\)
−0.718928 + 0.695085i \(0.755367\pi\)
\(524\) −2.34348e33 −0.238902
\(525\) 0 0
\(526\) 6.29643e33 0.612030
\(527\) −7.27906e33 −0.690944
\(528\) −1.28007e33 −0.118661
\(529\) −1.09299e34 −0.989509
\(530\) 0 0
\(531\) −9.34576e33 −0.807111
\(532\) 2.52606e33 0.213083
\(533\) 1.51915e34 1.25173
\(534\) 6.29617e33 0.506770
\(535\) 0 0
\(536\) 5.95833e33 0.457683
\(537\) 1.42052e34 1.06603
\(538\) −9.60972e33 −0.704583
\(539\) −1.07615e33 −0.0770929
\(540\) 0 0
\(541\) −1.58682e34 −1.08533 −0.542667 0.839948i \(-0.682585\pi\)
−0.542667 + 0.839948i \(0.682585\pi\)
\(542\) 6.33151e33 0.423171
\(543\) −3.17603e33 −0.207437
\(544\) −1.89869e33 −0.121190
\(545\) 0 0
\(546\) 2.09653e34 1.27818
\(547\) 8.29322e33 0.494176 0.247088 0.968993i \(-0.420526\pi\)
0.247088 + 0.968993i \(0.420526\pi\)
\(548\) 1.16168e34 0.676599
\(549\) 7.67343e34 4.36853
\(550\) 0 0
\(551\) −6.76908e33 −0.368243
\(552\) 1.30042e33 0.0691583
\(553\) 8.86874e33 0.461103
\(554\) −4.98130e33 −0.253204
\(555\) 0 0
\(556\) −1.35339e34 −0.657639
\(557\) −3.56496e34 −1.69381 −0.846905 0.531744i \(-0.821537\pi\)
−0.846905 + 0.531744i \(0.821537\pi\)
\(558\) 4.06052e34 1.88649
\(559\) −1.75742e34 −0.798413
\(560\) 0 0
\(561\) −7.48939e33 −0.325395
\(562\) −2.08849e34 −0.887417
\(563\) −1.62059e34 −0.673471 −0.336735 0.941599i \(-0.609323\pi\)
−0.336735 + 0.941599i \(0.609323\pi\)
\(564\) 4.13492e34 1.68065
\(565\) 0 0
\(566\) −6.40155e33 −0.248931
\(567\) 7.33659e34 2.79065
\(568\) 3.52641e33 0.131213
\(569\) 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 \(-0.370047\pi\)
0.397013 + 0.917813i \(0.370047\pi\)
\(572\) 4.15513e33 0.141623
\(573\) 8.66794e34 2.89057
\(574\) −1.97694e34 −0.645051
\(575\) 0 0
\(576\) 1.05916e34 0.330886
\(577\) 1.30064e34 0.397613 0.198807 0.980039i \(-0.436293\pi\)
0.198807 + 0.980039i \(0.436293\pi\)
\(578\) 1.25277e34 0.374777
\(579\) 2.50397e34 0.733074
\(580\) 0 0
\(581\) 5.01597e34 1.40655
\(582\) −7.24407e34 −1.98813
\(583\) 5.17157e33 0.138920
\(584\) 2.39932e33 0.0630851
\(585\) 0 0
\(586\) −1.70849e34 −0.430420
\(587\) −3.13619e34 −0.773440 −0.386720 0.922197i \(-0.626392\pi\)
−0.386720 + 0.922197i \(0.626392\pi\)
\(588\) 1.22681e34 0.296185
\(589\) −2.18803e34 −0.517145
\(590\) 0 0
\(591\) 6.31206e34 1.42997
\(592\) −2.86947e33 −0.0636474
\(593\) 6.43753e34 1.39809 0.699046 0.715077i \(-0.253608\pi\)
0.699046 + 0.715077i \(0.253608\pi\)
\(594\) 2.59957e34 0.552804
\(595\) 0 0
\(596\) −7.11419e33 −0.145060
\(597\) 1.42739e35 2.85012
\(598\) −4.22118e33 −0.0825409
\(599\) −9.82124e34 −1.88075 −0.940375 0.340140i \(-0.889526\pi\)
−0.940375 + 0.340140i \(0.889526\pi\)
\(600\) 0 0
\(601\) 1.10144e34 0.202316 0.101158 0.994870i \(-0.467745\pi\)
0.101158 + 0.994870i \(0.467745\pi\)
\(602\) 2.28700e34 0.411444
\(603\) −1.94467e35 −3.42672
\(604\) 4.21749e34 0.727934
\(605\) 0 0
\(606\) −1.02353e35 −1.69508
\(607\) −9.06735e34 −1.47103 −0.735516 0.677508i \(-0.763060\pi\)
−0.735516 + 0.677508i \(0.763060\pi\)
\(608\) −5.70730e33 −0.0907060
\(609\) 7.31105e34 1.13832
\(610\) 0 0
\(611\) −1.34220e35 −2.00587
\(612\) 6.19689e34 0.907363
\(613\) −2.62922e34 −0.377199 −0.188599 0.982054i \(-0.560395\pi\)
−0.188599 + 0.982054i \(0.560395\pi\)
\(614\) 3.15390e34 0.443346
\(615\) 0 0
\(616\) −5.40724e33 −0.0729822
\(617\) −6.11422e34 −0.808679 −0.404340 0.914609i \(-0.632499\pi\)
−0.404340 + 0.914609i \(0.632499\pi\)
\(618\) −7.43150e34 −0.963208
\(619\) −4.66687e34 −0.592779 −0.296389 0.955067i \(-0.595783\pi\)
−0.296389 + 0.955067i \(0.595783\pi\)
\(620\) 0 0
\(621\) −2.64090e34 −0.322186
\(622\) −6.31848e34 −0.755497
\(623\) 2.65962e34 0.311687
\(624\) −4.73685e34 −0.544104
\(625\) 0 0
\(626\) 6.65672e34 0.734651
\(627\) −2.25125e34 −0.243545
\(628\) −7.98398e34 −0.846689
\(629\) −1.67886e34 −0.174535
\(630\) 0 0
\(631\) −7.88083e33 −0.0787417 −0.0393709 0.999225i \(-0.512535\pi\)
−0.0393709 + 0.999225i \(0.512535\pi\)
\(632\) −2.00378e34 −0.196284
\(633\) 1.81562e35 1.74373
\(634\) −9.38386e34 −0.883621
\(635\) 0 0
\(636\) −5.89559e34 −0.533720
\(637\) −3.98227e34 −0.353499
\(638\) 1.44898e34 0.126126
\(639\) −1.15094e35 −0.982408
\(640\) 0 0
\(641\) 3.14908e34 0.258499 0.129250 0.991612i \(-0.458743\pi\)
0.129250 + 0.991612i \(0.458743\pi\)
\(642\) 1.25570e35 1.01088
\(643\) −8.32480e34 −0.657259 −0.328630 0.944459i \(-0.606587\pi\)
−0.328630 + 0.944459i \(0.606587\pi\)
\(644\) 5.49320e33 0.0425356
\(645\) 0 0
\(646\) −3.33922e34 −0.248736
\(647\) 1.33124e35 0.972641 0.486321 0.873780i \(-0.338339\pi\)
0.486321 + 0.873780i \(0.338339\pi\)
\(648\) −1.65761e35 −1.18794
\(649\) −1.07800e34 −0.0757809
\(650\) 0 0
\(651\) 2.36321e35 1.59860
\(652\) −1.27490e34 −0.0846020
\(653\) −2.06756e35 −1.34599 −0.672997 0.739645i \(-0.734993\pi\)
−0.672997 + 0.739645i \(0.734993\pi\)
\(654\) −1.18889e35 −0.759311
\(655\) 0 0
\(656\) 4.46664e34 0.274589
\(657\) −7.83084e34 −0.472325
\(658\) 1.74667e35 1.03368
\(659\) −2.57849e35 −1.49726 −0.748632 0.662986i \(-0.769289\pi\)
−0.748632 + 0.662986i \(0.769289\pi\)
\(660\) 0 0
\(661\) −3.31671e34 −0.185434 −0.0927170 0.995692i \(-0.529555\pi\)
−0.0927170 + 0.995692i \(0.529555\pi\)
\(662\) −1.41394e35 −0.775720
\(663\) −2.77142e35 −1.49205
\(664\) −1.13329e35 −0.598745
\(665\) 0 0
\(666\) 9.36532e34 0.476535
\(667\) −1.47201e34 −0.0735087
\(668\) 1.68119e35 0.823970
\(669\) 2.03109e35 0.977016
\(670\) 0 0
\(671\) 8.85101e34 0.410168
\(672\) 6.16426e34 0.280392
\(673\) 2.87165e35 1.28216 0.641082 0.767472i \(-0.278486\pi\)
0.641082 + 0.767472i \(0.278486\pi\)
\(674\) −2.16160e35 −0.947387
\(675\) 0 0
\(676\) 3.53722e34 0.149392
\(677\) −3.20244e35 −1.32777 −0.663885 0.747835i \(-0.731093\pi\)
−0.663885 + 0.747835i \(0.731093\pi\)
\(678\) −5.01167e35 −2.03991
\(679\) −3.06003e35 −1.22280
\(680\) 0 0
\(681\) −1.58411e35 −0.610165
\(682\) 4.68366e34 0.177125
\(683\) 4.34050e35 1.61169 0.805843 0.592129i \(-0.201712\pi\)
0.805843 + 0.592129i \(0.201712\pi\)
\(684\) 1.86274e35 0.679126
\(685\) 0 0
\(686\) 2.18895e35 0.769456
\(687\) −6.49388e35 −2.24153
\(688\) −5.16719e34 −0.175145
\(689\) 1.91372e35 0.636999
\(690\) 0 0
\(691\) 3.59659e35 1.15456 0.577279 0.816547i \(-0.304114\pi\)
0.577279 + 0.816547i \(0.304114\pi\)
\(692\) 1.32538e35 0.417845
\(693\) 1.76480e35 0.546425
\(694\) 1.80262e34 0.0548165
\(695\) 0 0
\(696\) −1.65184e35 −0.484565
\(697\) 2.61333e35 0.752982
\(698\) −1.22089e35 −0.345529
\(699\) 7.22055e34 0.200727
\(700\) 0 0
\(701\) −4.01671e34 −0.107745 −0.0538723 0.998548i \(-0.517156\pi\)
−0.0538723 + 0.998548i \(0.517156\pi\)
\(702\) 9.61963e35 2.53481
\(703\) −5.04654e34 −0.130633
\(704\) 1.22170e34 0.0310674
\(705\) 0 0
\(706\) 1.82716e35 0.448454
\(707\) −4.32357e35 −1.04256
\(708\) 1.22892e35 0.291144
\(709\) −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
\(712\) −6.00907e34 −0.132681
\(713\) −4.75812e34 −0.103232
\(714\) 3.60657e35 0.768895
\(715\) 0 0
\(716\) −1.35574e35 −0.279103
\(717\) 1.14103e36 2.30839
\(718\) −1.79427e35 −0.356724
\(719\) −8.41947e35 −1.64503 −0.822514 0.568744i \(-0.807429\pi\)
−0.822514 + 0.568744i \(0.807429\pi\)
\(720\) 0 0
\(721\) −3.13920e35 −0.592419
\(722\) 2.80867e35 0.520938
\(723\) −2.96697e35 −0.540859
\(724\) 3.03120e34 0.0543104
\(725\) 0 0
\(726\) −7.31938e35 −1.26697
\(727\) −8.07486e35 −1.37390 −0.686950 0.726705i \(-0.741051\pi\)
−0.686950 + 0.726705i \(0.741051\pi\)
\(728\) −2.00093e35 −0.334650
\(729\) 1.75615e36 2.88714
\(730\) 0 0
\(731\) −3.02321e35 −0.480287
\(732\) −1.00902e36 −1.57583
\(733\) 9.32041e35 1.43099 0.715493 0.698620i \(-0.246202\pi\)
0.715493 + 0.698620i \(0.246202\pi\)
\(734\) −3.18151e35 −0.480211
\(735\) 0 0
\(736\) −1.24112e34 −0.0181068
\(737\) −2.24310e35 −0.321740
\(738\) −1.45781e36 −2.05587
\(739\) −9.24490e34 −0.128188 −0.0640939 0.997944i \(-0.520416\pi\)
−0.0640939 + 0.997944i \(0.520416\pi\)
\(740\) 0 0
\(741\) −8.33068e35 −1.11674
\(742\) −2.49041e35 −0.328263
\(743\) −3.86129e35 −0.500463 −0.250232 0.968186i \(-0.580507\pi\)
−0.250232 + 0.968186i \(0.580507\pi\)
\(744\) −5.33938e35 −0.680501
\(745\) 0 0
\(746\) −9.71592e35 −1.19743
\(747\) 3.69882e36 4.48287
\(748\) 7.14788e34 0.0851937
\(749\) 5.30431e35 0.621736
\(750\) 0 0
\(751\) 5.31819e35 0.602927 0.301463 0.953478i \(-0.402525\pi\)
0.301463 + 0.953478i \(0.402525\pi\)
\(752\) −3.94637e35 −0.440023
\(753\) −4.65889e35 −0.510911
\(754\) 5.36190e35 0.578332
\(755\) 0 0
\(756\) −1.25184e36 −1.30625
\(757\) −1.46803e36 −1.50674 −0.753368 0.657599i \(-0.771572\pi\)
−0.753368 + 0.657599i \(0.771572\pi\)
\(758\) 5.69116e35 0.574560
\(759\) −4.89561e34 −0.0486166
\(760\) 0 0
\(761\) 1.26153e36 1.21225 0.606123 0.795371i \(-0.292724\pi\)
0.606123 + 0.795371i \(0.292724\pi\)
\(762\) 4.45641e35 0.421257
\(763\) −5.02210e35 −0.467012
\(764\) −8.27269e35 −0.756798
\(765\) 0 0
\(766\) −1.55273e35 −0.137479
\(767\) −3.98910e35 −0.347482
\(768\) −1.39274e35 −0.119358
\(769\) −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
\(772\) −2.38980e35 −0.191931
\(773\) 1.97217e36 1.55848 0.779238 0.626728i \(-0.215606\pi\)
0.779238 + 0.626728i \(0.215606\pi\)
\(774\) 1.68646e36 1.31133
\(775\) 0 0
\(776\) 6.91375e35 0.520526
\(777\) 5.45059e35 0.403813
\(778\) 9.01978e35 0.657583
\(779\) 7.85547e35 0.563578
\(780\) 0 0
\(781\) −1.32757e35 −0.0922398
\(782\) −7.26151e34 −0.0496527
\(783\) 3.35457e36 2.25743
\(784\) −1.17087e35 −0.0775459
\(785\) 0 0
\(786\) −1.00575e36 −0.645222
\(787\) 2.25189e35 0.142188 0.0710940 0.997470i \(-0.477351\pi\)
0.0710940 + 0.997470i \(0.477351\pi\)
\(788\) −6.02424e35 −0.374390
\(789\) 2.70224e36 1.65295
\(790\) 0 0
\(791\) −2.11702e36 −1.25464
\(792\) −3.98735e35 −0.232605
\(793\) 3.27529e36 1.88077
\(794\) 8.96781e35 0.506909
\(795\) 0 0
\(796\) −1.36230e36 −0.746209
\(797\) −3.69113e36 −1.99035 −0.995177 0.0980978i \(-0.968724\pi\)
−0.995177 + 0.0980978i \(0.968724\pi\)
\(798\) 1.08411e36 0.575488
\(799\) −2.30893e36 −1.20664
\(800\) 0 0
\(801\) 1.96123e36 0.993393
\(802\) −1.22485e36 −0.610806
\(803\) −9.03258e34 −0.0443473
\(804\) 2.55713e36 1.23610
\(805\) 0 0
\(806\) 1.73317e36 0.812184
\(807\) −4.12420e36 −1.90292
\(808\) 9.76854e35 0.443800
\(809\) −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 \(-0.677973\pi\)
−0.530439 + 0.847723i \(0.677973\pi\)
\(812\) −6.97767e35 −0.298030
\(813\) 2.71729e36 1.14289
\(814\) 1.08025e35 0.0447426
\(815\) 0 0
\(816\) −8.14859e35 −0.327307
\(817\) −9.08753e35 −0.359476
\(818\) −2.91771e35 −0.113665
\(819\) 6.53060e36 2.50556
\(820\) 0 0
\(821\) 3.33815e36 1.24227 0.621136 0.783703i \(-0.286672\pi\)
0.621136 + 0.783703i \(0.286672\pi\)
\(822\) 4.98560e36 1.82734
\(823\) −2.54036e36 −0.917058 −0.458529 0.888679i \(-0.651623\pi\)
−0.458529 + 0.888679i \(0.651623\pi\)
\(824\) 7.09263e35 0.252184
\(825\) 0 0
\(826\) 5.19119e35 0.179067
\(827\) 3.63276e36 1.23429 0.617146 0.786849i \(-0.288289\pi\)
0.617146 + 0.786849i \(0.288289\pi\)
\(828\) 4.05074e35 0.135567
\(829\) 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
\(832\) 4.52085e35 0.142455
\(833\) −6.85052e35 −0.212648
\(834\) −5.80833e36 −1.77613
\(835\) 0 0
\(836\) 2.14860e35 0.0637641
\(837\) 1.08433e37 3.17024
\(838\) −1.20611e36 −0.347407
\(839\) −2.96808e36 −0.842269 −0.421134 0.906998i \(-0.638368\pi\)
−0.421134 + 0.906998i \(0.638368\pi\)
\(840\) 0 0
\(841\) −1.76056e36 −0.484953
\(842\) −4.17349e36 −1.13266
\(843\) −8.96316e36 −2.39671
\(844\) −1.73283e36 −0.456537
\(845\) 0 0
\(846\) 1.28801e37 3.29450
\(847\) −3.09185e36 −0.779247
\(848\) 5.62676e35 0.139737
\(849\) −2.74735e36 −0.672308
\(850\) 0 0
\(851\) −1.09743e35 −0.0260769
\(852\) 1.51343e36 0.354378
\(853\) −6.13786e36 −1.41630 −0.708148 0.706064i \(-0.750469\pi\)
−0.708148 + 0.706064i \(0.750469\pi\)
\(854\) −4.26227e36 −0.969210
\(855\) 0 0
\(856\) −1.19844e36 −0.264664
\(857\) −3.16481e36 −0.688791 −0.344395 0.938825i \(-0.611916\pi\)
−0.344395 + 0.938825i \(0.611916\pi\)
\(858\) 1.78325e36 0.382492
\(859\) 5.65370e36 1.19514 0.597569 0.801818i \(-0.296134\pi\)
0.597569 + 0.801818i \(0.296134\pi\)
\(860\) 0 0
\(861\) −8.48442e36 −1.74214
\(862\) 2.78322e36 0.563257
\(863\) 8.00828e36 1.59736 0.798682 0.601753i \(-0.205531\pi\)
0.798682 + 0.601753i \(0.205531\pi\)
\(864\) 2.82838e36 0.556053
\(865\) 0 0
\(866\) −2.96617e36 −0.566530
\(867\) 5.37651e36 1.01219
\(868\) −2.25545e36 −0.418540
\(869\) 7.54351e35 0.137983
\(870\) 0 0
\(871\) −8.30052e36 −1.47529
\(872\) 1.13468e36 0.198800
\(873\) −2.25649e37 −3.89723
\(874\) −2.18275e35 −0.0371631
\(875\) 0 0
\(876\) 1.02971e36 0.170379
\(877\) −3.29560e36 −0.537575 −0.268788 0.963199i \(-0.586623\pi\)
−0.268788 + 0.963199i \(0.586623\pi\)
\(878\) 3.74153e36 0.601683
\(879\) −7.33233e36 −1.16247
\(880\) 0 0
\(881\) −8.05386e36 −1.24109 −0.620547 0.784169i \(-0.713090\pi\)
−0.620547 + 0.784169i \(0.713090\pi\)
\(882\) 3.82147e36 0.580595
\(883\) 4.91849e35 0.0736754 0.0368377 0.999321i \(-0.488272\pi\)
0.0368377 + 0.999321i \(0.488272\pi\)
\(884\) 2.64505e36 0.390644
\(885\) 0 0
\(886\) −1.44406e36 −0.207331
\(887\) −7.79484e36 −1.10347 −0.551737 0.834018i \(-0.686035\pi\)
−0.551737 + 0.834018i \(0.686035\pi\)
\(888\) −1.23149e36 −0.171897
\(889\) 1.88247e36 0.259093
\(890\) 0 0
\(891\) 6.24031e36 0.835093
\(892\) −1.93847e36 −0.255799
\(893\) −6.94048e36 −0.903121
\(894\) −3.05319e36 −0.391774
\(895\) 0 0
\(896\) −5.88318e35 −0.0734111
\(897\) −1.81160e36 −0.222924
\(898\) −1.51928e36 −0.184367
\(899\) 6.04394e36 0.723309
\(900\) 0 0
\(901\) 3.29209e36 0.383188
\(902\) −1.68153e36 −0.193029
\(903\) 9.81513e36 1.11122
\(904\) 4.78314e36 0.534082
\(905\) 0 0
\(906\) 1.81002e37 1.96599
\(907\) 1.31417e37 1.40786 0.703932 0.710268i \(-0.251426\pi\)
0.703932 + 0.710268i \(0.251426\pi\)
\(908\) 1.51188e36 0.159751
\(909\) −3.18823e37 −3.32278
\(910\) 0 0
\(911\) 1.44767e37 1.46788 0.733940 0.679214i \(-0.237679\pi\)
0.733940 + 0.679214i \(0.237679\pi\)
\(912\) −2.44940e36 −0.244977
\(913\) 4.26645e36 0.420903
\(914\) 7.24821e36 0.705347
\(915\) 0 0
\(916\) 6.19776e36 0.586869
\(917\) −4.24848e36 −0.396842
\(918\) 1.65482e37 1.52482
\(919\) −1.03191e37 −0.937990 −0.468995 0.883201i \(-0.655384\pi\)
−0.468995 + 0.883201i \(0.655384\pi\)
\(920\) 0 0
\(921\) 1.35356e37 1.19738
\(922\) −2.87137e36 −0.250584
\(923\) −4.91262e36 −0.422953
\(924\) −2.32062e36 −0.197108
\(925\) 0 0
\(926\) −3.77045e36 −0.311714
\(927\) −2.31487e37 −1.88813
\(928\) 1.57651e36 0.126867
\(929\) −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
\(932\) −6.89130e35 −0.0525536
\(933\) −2.71170e37 −2.04043
\(934\) −7.56121e36 −0.561378
\(935\) 0 0
\(936\) −1.47550e37 −1.06658
\(937\) 2.08020e37 1.48375 0.741873 0.670540i \(-0.233938\pi\)
0.741873 + 0.670540i \(0.233938\pi\)
\(938\) 1.08018e37 0.760258
\(939\) 2.85686e37 1.98413
\(940\) 0 0
\(941\) 2.39454e37 1.61939 0.809695 0.586851i \(-0.199632\pi\)
0.809695 + 0.586851i \(0.199632\pi\)
\(942\) −3.42648e37 −2.28672
\(943\) 1.70826e36 0.112501
\(944\) −1.17288e36 −0.0762262
\(945\) 0 0
\(946\) 1.94526e36 0.123123
\(947\) 8.06134e36 0.503539 0.251769 0.967787i \(-0.418988\pi\)
0.251769 + 0.967787i \(0.418988\pi\)
\(948\) −8.59960e36 −0.530120
\(949\) −3.34248e36 −0.203348
\(950\) 0 0
\(951\) −4.02727e37 −2.38646
\(952\) −3.44212e36 −0.201309
\(953\) 1.92589e37 1.11166 0.555829 0.831297i \(-0.312401\pi\)
0.555829 + 0.831297i \(0.312401\pi\)
\(954\) −1.83645e37 −1.04622
\(955\) 0 0
\(956\) −1.08900e37 −0.604374
\(957\) 6.21858e36 0.340637
\(958\) 1.59122e36 0.0860321
\(959\) 2.10601e37 1.12390
\(960\) 0 0
\(961\) 3.03575e35 0.0157842
\(962\) 3.99745e36 0.205161
\(963\) 3.91144e37 1.98157
\(964\) 2.83168e36 0.141606
\(965\) 0 0
\(966\) 2.35752e36 0.114879
\(967\) −2.29324e37 −1.10311 −0.551554 0.834139i \(-0.685965\pi\)
−0.551554 + 0.834139i \(0.685965\pi\)
\(968\) 6.98563e36 0.331714
\(969\) −1.43309e37 −0.671779
\(970\) 0 0
\(971\) 2.10199e37 0.960265 0.480132 0.877196i \(-0.340589\pi\)
0.480132 + 0.877196i \(0.340589\pi\)
\(972\) −3.62665e37 −1.63560
\(973\) −2.45355e37 −1.09241
\(974\) 1.74531e37 0.767161
\(975\) 0 0
\(976\) 9.63006e36 0.412578
\(977\) 3.52592e37 1.49139 0.745694 0.666289i \(-0.232118\pi\)
0.745694 + 0.666289i \(0.232118\pi\)
\(978\) −5.47148e36 −0.228491
\(979\) 2.26220e36 0.0932712
\(980\) 0 0
\(981\) −3.70334e37 −1.48844
\(982\) 1.85363e37 0.735581
\(983\) −7.31962e36 −0.286794 −0.143397 0.989665i \(-0.545803\pi\)
−0.143397 + 0.989665i \(0.545803\pi\)
\(984\) 1.91694e37 0.741602
\(985\) 0 0
\(986\) 9.22384e36 0.347897
\(987\) 7.49617e37 2.79174
\(988\) 7.95081e36 0.292381
\(989\) −1.97619e36 −0.0717587
\(990\) 0 0
\(991\) 1.16985e37 0.414200 0.207100 0.978320i \(-0.433597\pi\)
0.207100 + 0.978320i \(0.433597\pi\)
\(992\) 5.09591e36 0.178166
\(993\) −6.06819e37 −2.09504
\(994\) 6.39300e36 0.217959
\(995\) 0 0
\(996\) −4.86376e37 −1.61708
\(997\) −2.33411e37 −0.766358 −0.383179 0.923674i \(-0.625171\pi\)
−0.383179 + 0.923674i \(0.625171\pi\)
\(998\) −1.82280e37 −0.591028
\(999\) 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 50.26.a.c.1.1 2
5.2 odd 4 50.26.b.e.49.2 4
5.3 odd 4 50.26.b.e.49.3 4
5.4 even 2 2.26.a.b.1.2 2
15.14 odd 2 18.26.a.e.1.2 2
20.19 odd 2 16.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 2 5.4 even 2
16.26.a.c.1.1 2 20.19 odd 2
18.26.a.e.1.2 2 15.14 odd 2
50.26.a.c.1.1 2 1.1 even 1 trivial
50.26.b.e.49.2 4 5.2 odd 4
50.26.b.e.49.3 4 5.3 odd 4