Properties

Label 50.26.b.e.49.3
Level $50$
Weight $26$
Character 50.49
Analytic conductor $197.998$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,26,Mod(49,50)]
 
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([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(197.998389976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 53353x^{2} + 711608976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(162.829i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.26.b.e.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4096.00i q^{2} -1.75788e6i q^{3} -1.67772e7 q^{4} +7.20027e9 q^{6} -3.04153e10i q^{7} -6.87195e10i q^{8} -2.24285e12 q^{9} +2.58704e12 q^{11} +2.94923e13i q^{12} +9.57327e13i q^{13} +1.24581e14 q^{14} +2.81475e14 q^{16} -1.64685e15i q^{17} -9.18671e15i q^{18} -4.95030e15 q^{19} -5.34664e16 q^{21} +1.05965e16i q^{22} +1.07650e16i q^{23} -1.20801e17 q^{24} -3.92121e17 q^{26} +2.45323e18i q^{27} +5.10284e17i q^{28} +1.36741e18 q^{29} -4.42000e18 q^{31} +1.15292e18i q^{32} -4.54771e18i q^{33} +6.74549e18 q^{34} +3.76288e19 q^{36} +1.01944e19i q^{37} -2.02764e19i q^{38} +1.68287e20 q^{39} +1.58687e20 q^{41} -2.18999e20i q^{42} -1.83575e20i q^{43} -4.34034e19 q^{44} -4.40934e19 q^{46} +1.40203e21i q^{47} -4.94799e20i q^{48} +4.15978e20 q^{49} -2.89496e21 q^{51} -1.60613e21i q^{52} +1.99903e21i q^{53} -1.00484e22 q^{54} -2.09012e21 q^{56} +8.70202e21i q^{57} +5.60091e21i q^{58} +4.16691e21 q^{59} +3.42128e22 q^{61} -1.81043e22i q^{62} +6.82170e22i q^{63} -4.72237e21 q^{64} +1.86274e22 q^{66} +8.67051e22i q^{67} +2.76295e22i q^{68} +1.89236e22 q^{69} -5.13159e22 q^{71} +1.54128e23i q^{72} -3.49147e22i q^{73} -4.17563e22 q^{74} +8.30522e22 q^{76} -7.86857e22i q^{77} +6.89302e23i q^{78} -2.91588e23 q^{79} +2.41214e24 q^{81} +6.49981e23i q^{82} +1.64916e24i q^{83} +8.97018e23 q^{84} +7.51925e23 q^{86} -2.40374e24i q^{87} -1.77780e23i q^{88} -8.74435e23 q^{89} +2.91174e24 q^{91} -1.80607e23i q^{92} +7.76982e24i q^{93} -5.74273e24 q^{94} +2.02670e24 q^{96} +1.00608e25i q^{97} +1.70384e24i q^{98} -5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 67108864 q^{4} + 3111714816 q^{6} - 6589062865332 q^{9} + 16646069220528 q^{11} + 3084590645248 q^{14} + 11\!\cdots\!24 q^{16} + 954158485898800 q^{19} - 18\!\cdots\!12 q^{21} - 52\!\cdots\!56 q^{24}+ \cdots - 23\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00i 0.707107i
\(3\) − 1.75788e6i − 1.90974i −0.297031 0.954868i \(-0.595996\pi\)
0.297031 0.954868i \(-0.404004\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) 7.20027e9 1.35039
\(7\) − 3.04153e10i − 0.830552i −0.909696 0.415276i \(-0.863685\pi\)
0.909696 0.415276i \(-0.136315\pi\)
\(8\) − 6.87195e10i − 0.353553i
\(9\) −2.24285e12 −2.64709
\(10\) 0 0
\(11\) 2.58704e12 0.248539 0.124270 0.992248i \(-0.460341\pi\)
0.124270 + 0.992248i \(0.460341\pi\)
\(12\) 2.94923e13i 0.954868i
\(13\) 9.57327e13i 1.13964i 0.821769 + 0.569821i \(0.192988\pi\)
−0.821769 + 0.569821i \(0.807012\pi\)
\(14\) 1.24581e14 0.587289
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) − 1.64685e15i − 0.685555i −0.939417 0.342777i \(-0.888632\pi\)
0.939417 0.342777i \(-0.111368\pi\)
\(18\) − 9.18671e15i − 1.87178i
\(19\) −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
\(22\) 1.05965e16i 0.175744i
\(23\) 1.07650e16i 0.102427i 0.998688 + 0.0512136i \(0.0163089\pi\)
−0.998688 + 0.0512136i \(0.983691\pi\)
\(24\) −1.20801e17 −0.675194
\(25\) 0 0
\(26\) −3.92121e17 −0.805849
\(27\) 2.45323e18i 3.14551i
\(28\) 5.10284e17i 0.415276i
\(29\) 1.36741e18 0.717668 0.358834 0.933401i \(-0.383174\pi\)
0.358834 + 0.933401i \(0.383174\pi\)
\(30\) 0 0
\(31\) −4.42000e18 −1.00786 −0.503931 0.863744i \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) 1.15292e18i 0.176777i
\(33\) − 4.54771e18i − 0.474645i
\(34\) 6.74549e18 0.484760
\(35\) 0 0
\(36\) 3.76288e19 1.32355
\(37\) 1.01944e19i 0.254590i 0.991865 + 0.127295i \(0.0406294\pi\)
−0.991865 + 0.127295i \(0.959371\pi\)
\(38\) − 2.02764e19i − 0.362824i
\(39\) 1.68287e20 2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) − 2.18999e20i − 1.12157i
\(43\) − 1.83575e20i − 0.700582i −0.936641 0.350291i \(-0.886083\pi\)
0.936641 0.350291i \(-0.113917\pi\)
\(44\) −4.34034e19 −0.124270
\(45\) 0 0
\(46\) −4.40934e19 −0.0724270
\(47\) 1.40203e21i 1.76009i 0.474890 + 0.880045i \(0.342488\pi\)
−0.474890 + 0.880045i \(0.657512\pi\)
\(48\) − 4.94799e20i − 0.477434i
\(49\) 4.15978e20 0.310184
\(50\) 0 0
\(51\) −2.89496e21 −1.30923
\(52\) − 1.60613e21i − 0.569821i
\(53\) 1.99903e21i 0.558946i 0.960154 + 0.279473i \(0.0901598\pi\)
−0.960154 + 0.279473i \(0.909840\pi\)
\(54\) −1.00484e22 −2.22421
\(55\) 0 0
\(56\) −2.09012e21 −0.293644
\(57\) 8.70202e21i 0.979906i
\(58\) 5.60091e21i 0.507468i
\(59\) 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 \(0.191087\pi\)
\(62\) − 1.81043e22i − 0.712666i
\(63\) 6.82170e22i 2.19855i
\(64\) −4.72237e21 −0.125000
\(65\) 0 0
\(66\) 1.86274e22 0.335624
\(67\) 8.67051e22i 1.29452i 0.762268 + 0.647261i \(0.224086\pi\)
−0.762268 + 0.647261i \(0.775914\pi\)
\(68\) 2.76295e22i 0.342777i
\(69\) 1.89236e22 0.195609
\(70\) 0 0
\(71\) −5.13159e22 −0.371127 −0.185564 0.982632i \(-0.559411\pi\)
−0.185564 + 0.982632i \(0.559411\pi\)
\(72\) 1.54128e23i 0.935888i
\(73\) − 3.49147e22i − 0.178432i −0.996012 0.0892159i \(-0.971564\pi\)
0.996012 0.0892159i \(-0.0284361\pi\)
\(74\) −4.17563e22 −0.180022
\(75\) 0 0
\(76\) 8.30522e22 0.256555
\(77\) − 7.86857e22i − 0.206425i
\(78\) 6.89302e23i 1.53896i
\(79\) −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
\(82\) 6.49981e23i 0.776654i
\(83\) 1.64916e24i 1.69351i 0.531986 + 0.846753i \(0.321446\pi\)
−0.531986 + 0.846753i \(0.678554\pi\)
\(84\) 8.97018e23 0.793067
\(85\) 0 0
\(86\) 7.51925e23 0.495386
\(87\) − 2.40374e24i − 1.37056i
\(88\) − 1.77780e23i − 0.0878720i
\(89\) −8.74435e23 −0.375277 −0.187639 0.982238i \(-0.560083\pi\)
−0.187639 + 0.982238i \(0.560083\pi\)
\(90\) 0 0
\(91\) 2.91174e24 0.946532
\(92\) − 1.80607e23i − 0.0512136i
\(93\) 7.76982e24i 1.92475i
\(94\) −5.74273e24 −1.24457
\(95\) 0 0
\(96\) 2.02670e24 0.337597
\(97\) 1.00608e25i 1.47227i 0.676835 + 0.736134i \(0.263351\pi\)
−0.676835 + 0.736134i \(0.736649\pi\)
\(98\) 1.70384e24i 0.219333i
\(99\) −5.80235e24 −0.657907
\(100\) 0 0
\(101\) −1.42151e25 −1.25526 −0.627628 0.778513i \(-0.715974\pi\)
−0.627628 + 0.778513i \(0.715974\pi\)
\(102\) − 1.18578e25i − 0.925764i
\(103\) − 1.03211e25i − 0.713283i −0.934241 0.356642i \(-0.883922\pi\)
0.934241 0.356642i \(-0.116078\pi\)
\(104\) 6.57870e24 0.402924
\(105\) 0 0
\(106\) −8.18801e24 −0.395235
\(107\) − 1.74396e25i − 0.748582i −0.927311 0.374291i \(-0.877886\pi\)
0.927311 0.374291i \(-0.122114\pi\)
\(108\) − 4.11583e25i − 1.57275i
\(109\) 1.65117e25 0.562292 0.281146 0.959665i \(-0.409286\pi\)
0.281146 + 0.959665i \(0.409286\pi\)
\(110\) 0 0
\(111\) 1.79205e25 0.486199
\(112\) − 8.56115e24i − 0.207638i
\(113\) − 6.96039e25i − 1.51061i −0.655372 0.755306i \(-0.727488\pi\)
0.655372 0.755306i \(-0.272512\pi\)
\(114\) −3.56435e25 −0.692898
\(115\) 0 0
\(116\) −2.29413e25 −0.358834
\(117\) − 2.14714e26i − 3.01674i
\(118\) 1.70677e25i 0.215600i
\(119\) −5.00894e25 −0.569389
\(120\) 0 0
\(121\) −1.01654e26 −0.938228
\(122\) 1.40136e26i 1.16695i
\(123\) − 2.78952e26i − 2.09757i
\(124\) 7.41552e25 0.503931
\(125\) 0 0
\(126\) −2.79417e26 −1.55461
\(127\) − 6.18922e25i − 0.311953i −0.987761 0.155977i \(-0.950148\pi\)
0.987761 0.155977i \(-0.0498524\pi\)
\(128\) − 1.93428e25i − 0.0883883i
\(129\) −3.22703e26 −1.33793
\(130\) 0 0
\(131\) −1.39682e26 −0.477805 −0.238902 0.971044i \(-0.576788\pi\)
−0.238902 + 0.971044i \(0.576788\pi\)
\(132\) 7.62979e25i 0.237322i
\(133\) 1.50565e26i 0.426165i
\(134\) −3.55144e26 −0.915366
\(135\) 0 0
\(136\) −1.13171e26 −0.242380
\(137\) − 6.92418e26i − 1.35320i −0.736352 0.676599i \(-0.763453\pi\)
0.736352 0.676599i \(-0.236547\pi\)
\(138\) 7.75109e25i 0.138317i
\(139\) 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
\(142\) − 2.10190e26i − 0.262427i
\(143\) 2.47665e26i 0.283246i
\(144\) −6.31306e26 −0.661773
\(145\) 0 0
\(146\) 1.43010e26 0.126170
\(147\) − 7.31239e26i − 0.592369i
\(148\) − 1.71034e26i − 0.127295i
\(149\) 4.24039e26 0.290120 0.145060 0.989423i \(-0.453663\pi\)
0.145060 + 0.989423i \(0.453663\pi\)
\(150\) 0 0
\(151\) 2.51382e27 1.45587 0.727934 0.685647i \(-0.240481\pi\)
0.727934 + 0.685647i \(0.240481\pi\)
\(152\) 3.40182e26i 0.181412i
\(153\) 3.69363e27i 1.81473i
\(154\) 3.22297e26 0.145964
\(155\) 0 0
\(156\) −2.82338e27 −1.08821
\(157\) 4.75882e27i 1.69338i 0.532088 + 0.846689i \(0.321407\pi\)
−0.532088 + 0.846689i \(0.678593\pi\)
\(158\) − 1.19434e27i − 0.392569i
\(159\) 3.51404e27 1.06744
\(160\) 0 0
\(161\) 3.27420e26 0.0850712
\(162\) 9.88012e27i 2.37588i
\(163\) − 7.59899e26i − 0.169204i −0.996415 0.0846020i \(-0.973038\pi\)
0.996415 0.0846020i \(-0.0269619\pi\)
\(164\) −2.66232e27 −0.549177
\(165\) 0 0
\(166\) −6.75496e27 −1.19749
\(167\) − 1.00207e28i − 1.64794i −0.566634 0.823970i \(-0.691755\pi\)
0.566634 0.823970i \(-0.308245\pi\)
\(168\) 3.67419e27i 0.560783i
\(169\) −2.10835e27 −0.298785
\(170\) 0 0
\(171\) 1.11028e28 1.35825
\(172\) 3.07989e27i 0.350291i
\(173\) 7.89989e27i 0.835689i 0.908518 + 0.417845i \(0.137214\pi\)
−0.908518 + 0.417845i \(0.862786\pi\)
\(174\) 9.84571e27 0.969129
\(175\) 0 0
\(176\) 7.28188e26 0.0621349
\(177\) − 7.32493e27i − 0.582288i
\(178\) − 3.58168e27i − 0.265361i
\(179\) 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 \(0.482704\pi\)
\(182\) 1.19265e28i 0.669299i
\(183\) − 6.01420e28i − 3.15166i
\(184\) 7.39765e26 0.0362135
\(185\) 0 0
\(186\) −3.18252e28 −1.36100
\(187\) − 4.26047e27i − 0.170387i
\(188\) − 2.35222e28i − 0.880045i
\(189\) 7.46157e28 2.61251
\(190\) 0 0
\(191\) −4.93091e28 −1.51360 −0.756798 0.653649i \(-0.773237\pi\)
−0.756798 + 0.653649i \(0.773237\pi\)
\(192\) 8.30135e27i 0.238717i
\(193\) − 1.42443e28i − 0.383861i −0.981408 0.191931i \(-0.938525\pi\)
0.981408 0.191931i \(-0.0614749\pi\)
\(194\) −4.12092e28 −1.04105
\(195\) 0 0
\(196\) −6.97895e27 −0.155092
\(197\) 3.59073e28i 0.748781i 0.927271 + 0.374390i \(0.122148\pi\)
−0.927271 + 0.374390i \(0.877852\pi\)
\(198\) − 2.37664e28i − 0.465210i
\(199\) 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
\(202\) − 5.82250e28i − 0.887601i
\(203\) − 4.15902e28i − 0.596060i
\(204\) 4.85694e28 0.654614
\(205\) 0 0
\(206\) 4.22753e28 0.504367
\(207\) − 2.41443e28i − 0.271134i
\(208\) 2.69464e28i 0.284911i
\(209\) −1.28066e28 −0.127528
\(210\) 0 0
\(211\) −1.03285e29 −0.913075 −0.456537 0.889704i \(-0.650911\pi\)
−0.456537 + 0.889704i \(0.650911\pi\)
\(212\) − 3.35381e28i − 0.279473i
\(213\) 9.02072e28i 0.708755i
\(214\) 7.14326e28 0.529327
\(215\) 0 0
\(216\) 1.68585e29 1.11211
\(217\) 1.34436e29i 0.837081i
\(218\) 6.76321e28i 0.397600i
\(219\) −6.13758e28 −0.340757
\(220\) 0 0
\(221\) 1.57657e29 0.781287
\(222\) 7.34025e28i 0.343795i
\(223\) − 1.15542e29i − 0.511597i −0.966730 0.255799i \(-0.917662\pi\)
0.966730 0.255799i \(-0.0823384\pi\)
\(224\) 3.50665e28 0.146822
\(225\) 0 0
\(226\) 2.85098e29 1.06816
\(227\) − 9.01151e28i − 0.319502i −0.987157 0.159751i \(-0.948931\pi\)
0.987157 0.159751i \(-0.0510692\pi\)
\(228\) − 1.45996e29i − 0.489953i
\(229\) −3.69415e29 −1.17374 −0.586869 0.809682i \(-0.699640\pi\)
−0.586869 + 0.809682i \(0.699640\pi\)
\(230\) 0 0
\(231\) −1.38320e29 −0.394217
\(232\) − 9.39676e28i − 0.253734i
\(233\) − 4.10753e28i − 0.105107i −0.998618 0.0525536i \(-0.983264\pi\)
0.998618 0.0525536i \(-0.0167360\pi\)
\(234\) 8.79469e29 2.13316
\(235\) 0 0
\(236\) −6.99092e28 −0.152452
\(237\) 5.12576e29i 1.06024i
\(238\) − 2.05166e29i − 0.402618i
\(239\) 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 \(0.454773\pi\)
\(242\) − 4.16376e29i − 0.663427i
\(243\) − 2.16165e30i − 3.27121i
\(244\) −5.73996e29 −0.825156
\(245\) 0 0
\(246\) 1.14259e30 1.48320
\(247\) − 4.73905e29i − 0.584763i
\(248\) 3.03740e29i 0.356333i
\(249\) 2.89903e30 3.23415
\(250\) 0 0
\(251\) 2.65029e29 0.267530 0.133765 0.991013i \(-0.457293\pi\)
0.133765 + 0.991013i \(0.457293\pi\)
\(252\) − 1.14449e30i − 1.09927i
\(253\) 2.78495e28i 0.0254572i
\(254\) 2.53510e29 0.220584
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) 1.11592e30i 0.838433i 0.907886 + 0.419217i \(0.137695\pi\)
−0.907886 + 0.419217i \(0.862305\pi\)
\(258\) − 1.32179e30i − 0.946057i
\(259\) 3.10066e29 0.211450
\(260\) 0 0
\(261\) −3.06689e30 −1.89973
\(262\) − 5.72139e29i − 0.337859i
\(263\) − 1.53721e30i − 0.865541i −0.901504 0.432770i \(-0.857536\pi\)
0.901504 0.432770i \(-0.142464\pi\)
\(264\) −3.12516e29 −0.167812
\(265\) 0 0
\(266\) −6.16713e29 −0.301344
\(267\) 1.53715e30i 0.716681i
\(268\) − 1.45467e30i − 0.647261i
\(269\) −2.34612e30 −0.996431 −0.498216 0.867053i \(-0.666011\pi\)
−0.498216 + 0.867053i \(0.666011\pi\)
\(270\) 0 0
\(271\) −1.54578e30 −0.598454 −0.299227 0.954182i \(-0.596729\pi\)
−0.299227 + 0.954182i \(0.596729\pi\)
\(272\) − 4.63547e29i − 0.171389i
\(273\) − 5.11849e30i − 1.80763i
\(274\) 2.83614e30 0.956855
\(275\) 0 0
\(276\) −3.17485e29 −0.0978045
\(277\) − 1.21614e30i − 0.358085i −0.983841 0.179042i \(-0.942700\pi\)
0.983841 0.179042i \(-0.0572999\pi\)
\(278\) 3.30417e30i 0.930042i
\(279\) 9.91339e30 2.66790
\(280\) 0 0
\(281\) 5.09885e30 1.25500 0.627499 0.778617i \(-0.284079\pi\)
0.627499 + 0.778617i \(0.284079\pi\)
\(282\) 1.00950e31i 2.37680i
\(283\) 1.56288e30i 0.352042i 0.984386 + 0.176021i \(0.0563227\pi\)
−0.984386 + 0.176021i \(0.943677\pi\)
\(284\) 8.60939e29 0.185564
\(285\) 0 0
\(286\) −1.01444e30 −0.200285
\(287\) − 4.82651e30i − 0.912240i
\(288\) − 2.58583e30i − 0.467944i
\(289\) 3.05852e30 0.530015
\(290\) 0 0
\(291\) 1.76857e31 2.81164
\(292\) 5.85771e29i 0.0892159i
\(293\) 4.17112e30i 0.608706i 0.952559 + 0.304353i \(0.0984403\pi\)
−0.952559 + 0.304353i \(0.901560\pi\)
\(294\) 2.99515e30 0.418868
\(295\) 0 0
\(296\) 7.00555e29 0.0900110
\(297\) 6.34661e30i 0.781783i
\(298\) 1.73686e30i 0.205146i
\(299\) −1.03056e30 −0.116730
\(300\) 0 0
\(301\) −5.58350e30 −0.581870
\(302\) 1.02966e31i 1.02945i
\(303\) 2.49884e31i 2.39721i
\(304\) −1.39338e30 −0.128278
\(305\) 0 0
\(306\) −1.51291e31 −1.28320
\(307\) 7.69994e30i 0.626986i 0.949591 + 0.313493i \(0.101499\pi\)
−0.949591 + 0.313493i \(0.898501\pi\)
\(308\) 1.32013e30i 0.103212i
\(309\) −1.81433e31 −1.36218
\(310\) 0 0
\(311\) 1.54260e31 1.06843 0.534217 0.845347i \(-0.320606\pi\)
0.534217 + 0.845347i \(0.320606\pi\)
\(312\) − 1.15646e31i − 0.769479i
\(313\) − 1.62517e31i − 1.03895i −0.854485 0.519476i \(-0.826127\pi\)
0.854485 0.519476i \(-0.173873\pi\)
\(314\) −1.94921e31 −1.19740
\(315\) 0 0
\(316\) 4.89203e30 0.277588
\(317\) − 2.29098e31i − 1.24963i −0.780773 0.624814i \(-0.785175\pi\)
0.780773 0.624814i \(-0.214825\pi\)
\(318\) 1.43935e31i 0.754794i
\(319\) 3.53755e30 0.178369
\(320\) 0 0
\(321\) −3.06567e31 −1.42959
\(322\) 1.34111e30i 0.0601544i
\(323\) 8.15239e30i 0.351765i
\(324\) −4.04690e31 −1.68000
\(325\) 0 0
\(326\) 3.11255e30 0.119645
\(327\) − 2.90256e31i − 1.07383i
\(328\) − 1.09049e31i − 0.388327i
\(329\) 4.26433e31 1.46185
\(330\) 0 0
\(331\) 3.45199e31 1.09703 0.548517 0.836140i \(-0.315193\pi\)
0.548517 + 0.836140i \(0.315193\pi\)
\(332\) − 2.76683e31i − 0.846753i
\(333\) − 2.28645e31i − 0.673922i
\(334\) 4.10447e31 1.16527
\(335\) 0 0
\(336\) −1.50495e31 −0.396534
\(337\) − 5.27735e31i − 1.33981i −0.742448 0.669904i \(-0.766335\pi\)
0.742448 0.669904i \(-0.233665\pi\)
\(338\) − 8.63579e30i − 0.211273i
\(339\) −1.22355e32 −2.88487
\(340\) 0 0
\(341\) −1.14347e31 −0.250493
\(342\) 4.54770e31i 0.960429i
\(343\) − 5.34411e31i − 1.08818i
\(344\) −1.26152e31 −0.247693
\(345\) 0 0
\(346\) −3.23580e31 −0.590922
\(347\) 4.40093e30i 0.0775222i 0.999249 + 0.0387611i \(0.0123411\pi\)
−0.999249 + 0.0387611i \(0.987659\pi\)
\(348\) 4.03280e31i 0.685278i
\(349\) −2.98069e31 −0.488651 −0.244326 0.969693i \(-0.578567\pi\)
−0.244326 + 0.969693i \(0.578567\pi\)
\(350\) 0 0
\(351\) −2.34854e32 −3.58476
\(352\) 2.98266e30i 0.0439360i
\(353\) − 4.46084e31i − 0.634209i −0.948391 0.317105i \(-0.897289\pi\)
0.948391 0.317105i \(-0.102711\pi\)
\(354\) 3.00029e31 0.411740
\(355\) 0 0
\(356\) 1.46706e31 0.187639
\(357\) 8.80511e31i 1.08738i
\(358\) 3.30992e31i 0.394712i
\(359\) −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
\(362\) 7.40039e30i 0.0768065i
\(363\) 1.78696e32i 1.79177i
\(364\) −4.88509e31 −0.473266
\(365\) 0 0
\(366\) 2.46342e32 2.22856
\(367\) − 7.76735e31i − 0.679121i −0.940584 0.339560i \(-0.889722\pi\)
0.940584 0.339560i \(-0.110278\pi\)
\(368\) 3.03008e30i 0.0256068i
\(369\) −3.55911e32 −2.90744
\(370\) 0 0
\(371\) 6.08010e31 0.464234
\(372\) − 1.30356e32i − 0.962374i
\(373\) 2.37205e32i 1.69342i 0.532058 + 0.846708i \(0.321419\pi\)
−0.532058 + 0.846708i \(0.678581\pi\)
\(374\) 1.74509e31 0.120482
\(375\) 0 0
\(376\) 9.63470e31 0.622286
\(377\) 1.30906e32i 0.817884i
\(378\) 3.05626e32i 1.84732i
\(379\) 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
\(382\) − 2.01970e32i − 1.07027i
\(383\) 3.79085e31i 0.194425i 0.995264 + 0.0972125i \(0.0309926\pi\)
−0.995264 + 0.0972125i \(0.969007\pi\)
\(384\) −3.40023e31 −0.168798
\(385\) 0 0
\(386\) 5.83446e31 0.271431
\(387\) 4.11732e32i 1.85450i
\(388\) − 1.68793e32i − 0.736134i
\(389\) 2.20209e32 0.929963 0.464982 0.885320i \(-0.346061\pi\)
0.464982 + 0.885320i \(0.346061\pi\)
\(390\) 0 0
\(391\) 1.77283e31 0.0702195
\(392\) − 2.85858e31i − 0.109667i
\(393\) 2.45545e32i 0.912481i
\(394\) −1.47076e32 −0.529468
\(395\) 0 0
\(396\) 9.73473e31 0.328953
\(397\) 2.18941e32i 0.716877i 0.933553 + 0.358439i \(0.116691\pi\)
−0.933553 + 0.358439i \(0.883309\pi\)
\(398\) 3.32594e32i 1.05530i
\(399\) 2.64675e32 0.813863
\(400\) 0 0
\(401\) 2.99036e32 0.863810 0.431905 0.901919i \(-0.357842\pi\)
0.431905 + 0.901919i \(0.357842\pi\)
\(402\) 6.24300e32i 1.74811i
\(403\) − 4.23138e32i − 1.14860i
\(404\) 2.38490e32 0.627628
\(405\) 0 0
\(406\) 1.70353e32 0.421478
\(407\) 2.63734e31i 0.0632755i
\(408\) 1.98940e32i 0.462882i
\(409\) −7.12330e31 −0.160746 −0.0803730 0.996765i \(-0.525611\pi\)
−0.0803730 + 0.996765i \(0.525611\pi\)
\(410\) 0 0
\(411\) −1.21719e33 −2.58425
\(412\) 1.73160e32i 0.356642i
\(413\) − 1.26738e32i − 0.253239i
\(414\) 9.88949e31 0.191721
\(415\) 0 0
\(416\) −1.10372e32 −0.201462
\(417\) − 1.41805e33i − 2.51183i
\(418\) − 5.24560e31i − 0.0901761i
\(419\) −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 \(0.204350\pi\)
\(422\) − 4.23055e32i − 0.645641i
\(423\) − 3.14455e33i − 4.65912i
\(424\) 1.37372e32 0.197617
\(425\) 0 0
\(426\) −3.69489e32 −0.501166
\(427\) − 1.04059e33i − 1.37067i
\(428\) 2.92588e32i 0.374291i
\(429\) 4.35365e32 0.540925
\(430\) 0 0
\(431\) −6.79497e32 −0.796565 −0.398283 0.917263i \(-0.630394\pi\)
−0.398283 + 0.917263i \(0.630394\pi\)
\(432\) 6.90523e32i 0.786377i
\(433\) 7.24164e32i 0.801195i 0.916254 + 0.400597i \(0.131197\pi\)
−0.916254 + 0.400597i \(0.868803\pi\)
\(434\) −5.50648e32 −0.591906
\(435\) 0 0
\(436\) −2.77021e32 −0.281146
\(437\) − 5.32899e31i − 0.0525566i
\(438\) − 2.51395e32i − 0.240952i
\(439\) 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
\(442\) 6.45764e32i 0.552453i
\(443\) 3.52555e32i 0.293211i 0.989195 + 0.146605i \(0.0468347\pi\)
−0.989195 + 0.146605i \(0.953165\pi\)
\(444\) −3.00657e32 −0.243099
\(445\) 0 0
\(446\) 4.73259e32 0.361754
\(447\) − 7.45409e32i − 0.554052i
\(448\) 1.43632e32i 0.103819i
\(449\) −3.70918e32 −0.260735 −0.130367 0.991466i \(-0.541616\pi\)
−0.130367 + 0.991466i \(0.541616\pi\)
\(450\) 0 0
\(451\) 4.10530e32 0.272984
\(452\) 1.16776e33i 0.755306i
\(453\) − 4.41899e33i − 2.78032i
\(454\) 3.69111e32 0.225922
\(455\) 0 0
\(456\) 5.97998e32 0.346449
\(457\) 1.76958e33i 0.997512i 0.866742 + 0.498756i \(0.166210\pi\)
−0.866742 + 0.498756i \(0.833790\pi\)
\(458\) − 1.51313e33i − 0.829959i
\(459\) 4.04009e33 2.15642
\(460\) 0 0
\(461\) 7.01018e32 0.354379 0.177189 0.984177i \(-0.443299\pi\)
0.177189 + 0.984177i \(0.443299\pi\)
\(462\) − 5.66559e32i − 0.278753i
\(463\) 9.20521e32i 0.440830i 0.975406 + 0.220415i \(0.0707412\pi\)
−0.975406 + 0.220415i \(0.929259\pi\)
\(464\) 3.84891e32 0.179417
\(465\) 0 0
\(466\) 1.68245e32 0.0743219
\(467\) − 1.84600e33i − 0.793908i −0.917838 0.396954i \(-0.870067\pi\)
0.917838 0.396954i \(-0.129933\pi\)
\(468\) 3.60231e33i 1.50837i
\(469\) 2.63716e33 1.07517
\(470\) 0 0
\(471\) 8.36543e33 3.23390
\(472\) − 2.86348e32i − 0.107800i
\(473\) − 4.74918e32i − 0.174122i
\(474\) −2.09951e33 −0.749703
\(475\) 0 0
\(476\) 8.40360e32 0.284694
\(477\) − 4.48351e33i − 1.47958i
\(478\) 2.65870e33i 0.854714i
\(479\) 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
\(482\) 6.91328e32i 0.200261i
\(483\) − 5.75566e32i − 0.162463i
\(484\) 1.70548e33 0.469114
\(485\) 0 0
\(486\) 8.85414e33 2.31309
\(487\) 4.26102e33i 1.08493i 0.840078 + 0.542465i \(0.182509\pi\)
−0.840078 + 0.542465i \(0.817491\pi\)
\(488\) − 2.35109e33i − 0.583474i
\(489\) −1.33581e33 −0.323135
\(490\) 0 0
\(491\) −4.52548e33 −1.04027 −0.520134 0.854084i \(-0.674118\pi\)
−0.520134 + 0.854084i \(0.674118\pi\)
\(492\) 4.68004e33i 1.04878i
\(493\) − 2.25191e33i − 0.492000i
\(494\) 1.94112e33 0.413490
\(495\) 0 0
\(496\) −1.24412e33 −0.251965
\(497\) 1.56079e33i 0.308241i
\(498\) 1.18744e34i 2.28689i
\(499\) −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
\(502\) 1.08556e33i 0.189172i
\(503\) − 2.29032e33i − 0.389311i −0.980872 0.194656i \(-0.937641\pi\)
0.980872 0.194656i \(-0.0623590\pi\)
\(504\) 4.68784e33 0.777304
\(505\) 0 0
\(506\) −1.14072e32 −0.0180010
\(507\) 3.70622e33i 0.570600i
\(508\) 1.03838e33i 0.155977i
\(509\) −5.35889e33 −0.785421 −0.392711 0.919662i \(-0.628463\pi\)
−0.392711 + 0.919662i \(0.628463\pi\)
\(510\) 0 0
\(511\) −1.06194e33 −0.148197
\(512\) 3.24519e32i 0.0441942i
\(513\) − 1.21442e34i − 1.61400i
\(514\) −4.57080e33 −0.592862
\(515\) 0 0
\(516\) 5.41407e33 0.668963
\(517\) 3.62712e33i 0.437452i
\(518\) 1.27003e33i 0.149518i
\(519\) 1.38871e34 1.59595
\(520\) 0 0
\(521\) −1.14090e34 −1.24961 −0.624807 0.780779i \(-0.714822\pi\)
−0.624807 + 0.780779i \(0.714822\pi\)
\(522\) − 1.25620e34i − 1.34331i
\(523\) − 1.37717e34i − 1.43786i −0.695085 0.718928i \(-0.744633\pi\)
0.695085 0.718928i \(-0.255367\pi\)
\(524\) 2.34348e33 0.238902
\(525\) 0 0
\(526\) 6.29643e33 0.612030
\(527\) 7.27906e33i 0.690944i
\(528\) − 1.28007e33i − 0.118661i
\(529\) 1.09299e34 0.989509
\(530\) 0 0
\(531\) −9.34576e33 −0.807111
\(532\) − 2.52606e33i − 0.213083i
\(533\) 1.51915e34i 1.25173i
\(534\) −6.29617e33 −0.506770
\(535\) 0 0
\(536\) 5.95833e33 0.457683
\(537\) − 1.42052e34i − 1.06603i
\(538\) − 9.60972e33i − 0.704583i
\(539\) 1.07615e33 0.0770929
\(540\) 0 0
\(541\) −1.58682e34 −1.08533 −0.542667 0.839948i \(-0.682585\pi\)
−0.542667 + 0.839948i \(0.682585\pi\)
\(542\) − 6.33151e33i − 0.423171i
\(543\) − 3.17603e33i − 0.207437i
\(544\) 1.89869e33 0.121190
\(545\) 0 0
\(546\) 2.09653e34 1.27818
\(547\) − 8.29322e33i − 0.494176i −0.968993 0.247088i \(-0.920526\pi\)
0.968993 0.247088i \(-0.0794737\pi\)
\(548\) 1.16168e34i 0.676599i
\(549\) −7.67343e34 −4.36853
\(550\) 0 0
\(551\) −6.76908e33 −0.368243
\(552\) − 1.30042e33i − 0.0691583i
\(553\) 8.86874e33i 0.461103i
\(554\) 4.98130e33 0.253204
\(555\) 0 0
\(556\) −1.35339e34 −0.657639
\(557\) 3.56496e34i 1.69381i 0.531744 + 0.846905i \(0.321537\pi\)
−0.531744 + 0.846905i \(0.678463\pi\)
\(558\) 4.06052e34i 1.88649i
\(559\) 1.75742e34 0.798413
\(560\) 0 0
\(561\) −7.48939e33 −0.325395
\(562\) 2.08849e34i 0.887417i
\(563\) − 1.62059e34i − 0.673471i −0.941599 0.336735i \(-0.890677\pi\)
0.941599 0.336735i \(-0.109323\pi\)
\(564\) −4.13492e34 −1.68065
\(565\) 0 0
\(566\) −6.40155e33 −0.248931
\(567\) − 7.33659e34i − 2.79065i
\(568\) 3.52641e33i 0.131213i
\(569\) −4.19926e34 −1.52851 −0.764257 0.644912i \(-0.776894\pi\)
−0.764257 + 0.644912i \(0.776894\pi\)
\(570\) 0 0
\(571\) 2.27922e34 0.794026 0.397013 0.917813i \(-0.370047\pi\)
0.397013 + 0.917813i \(0.370047\pi\)
\(572\) − 4.15513e33i − 0.141623i
\(573\) 8.66794e34i 2.89057i
\(574\) 1.97694e34 0.645051
\(575\) 0 0
\(576\) 1.05916e34 0.330886
\(577\) − 1.30064e34i − 0.397613i −0.980039 0.198807i \(-0.936293\pi\)
0.980039 0.198807i \(-0.0637066\pi\)
\(578\) 1.25277e34i 0.374777i
\(579\) −2.50397e34 −0.733074
\(580\) 0 0
\(581\) 5.01597e34 1.40655
\(582\) 7.24407e34i 1.98813i
\(583\) 5.17157e33i 0.138920i
\(584\) −2.39932e33 −0.0630851
\(585\) 0 0
\(586\) −1.70849e34 −0.430420
\(587\) 3.13619e34i 0.773440i 0.922197 + 0.386720i \(0.126392\pi\)
−0.922197 + 0.386720i \(0.873608\pi\)
\(588\) 1.22681e34i 0.296185i
\(589\) 2.18803e34 0.517145
\(590\) 0 0
\(591\) 6.31206e34 1.42997
\(592\) 2.86947e33i 0.0636474i
\(593\) 6.43753e34i 1.39809i 0.715077 + 0.699046i \(0.246392\pi\)
−0.715077 + 0.699046i \(0.753608\pi\)
\(594\) −2.59957e34 −0.552804
\(595\) 0 0
\(596\) −7.11419e33 −0.145060
\(597\) − 1.42739e35i − 2.85012i
\(598\) − 4.22118e33i − 0.0825409i
\(599\) 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 \(0.467745\pi\)
\(602\) − 2.28700e34i − 0.411444i
\(603\) − 1.94467e35i − 3.42672i
\(604\) −4.21749e34 −0.727934
\(605\) 0 0
\(606\) −1.02353e35 −1.69508
\(607\) 9.06735e34i 1.47103i 0.677508 + 0.735516i \(0.263060\pi\)
−0.677508 + 0.735516i \(0.736940\pi\)
\(608\) − 5.70730e33i − 0.0907060i
\(609\) −7.31105e34 −1.13832
\(610\) 0 0
\(611\) −1.34220e35 −2.00587
\(612\) − 6.19689e34i − 0.907363i
\(613\) − 2.62922e34i − 0.377199i −0.982054 0.188599i \(-0.939605\pi\)
0.982054 0.188599i \(-0.0603948\pi\)
\(614\) −3.15390e34 −0.443346
\(615\) 0 0
\(616\) −5.40724e33 −0.0729822
\(617\) 6.11422e34i 0.808679i 0.914609 + 0.404340i \(0.132499\pi\)
−0.914609 + 0.404340i \(0.867501\pi\)
\(618\) − 7.43150e34i − 0.963208i
\(619\) 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
\(622\) 6.31848e34i 0.755497i
\(623\) 2.65962e34i 0.311687i
\(624\) 4.73685e34 0.544104
\(625\) 0 0
\(626\) 6.65672e34 0.734651
\(627\) 2.25125e34i 0.243545i
\(628\) − 7.98398e34i − 0.846689i
\(629\) 1.67886e34 0.174535
\(630\) 0 0
\(631\) −7.88083e33 −0.0787417 −0.0393709 0.999225i \(-0.512535\pi\)
−0.0393709 + 0.999225i \(0.512535\pi\)
\(632\) 2.00378e34i 0.196284i
\(633\) 1.81562e35i 1.74373i
\(634\) 9.38386e34 0.883621
\(635\) 0 0
\(636\) −5.89559e34 −0.533720
\(637\) 3.98227e34i 0.353499i
\(638\) 1.44898e34i 0.126126i
\(639\) 1.15094e35 0.982408
\(640\) 0 0
\(641\) 3.14908e34 0.258499 0.129250 0.991612i \(-0.458743\pi\)
0.129250 + 0.991612i \(0.458743\pi\)
\(642\) − 1.25570e35i − 1.01088i
\(643\) − 8.32480e34i − 0.657259i −0.944459 0.328630i \(-0.893413\pi\)
0.944459 0.328630i \(-0.106587\pi\)
\(644\) −5.49320e33 −0.0425356
\(645\) 0 0
\(646\) −3.33922e34 −0.248736
\(647\) − 1.33124e35i − 0.972641i −0.873780 0.486321i \(-0.838339\pi\)
0.873780 0.486321i \(-0.161661\pi\)
\(648\) − 1.65761e35i − 1.18794i
\(649\) 1.07800e34 0.0757809
\(650\) 0 0
\(651\) 2.36321e35 1.59860
\(652\) 1.27490e34i 0.0846020i
\(653\) − 2.06756e35i − 1.34599i −0.739645 0.672997i \(-0.765007\pi\)
0.739645 0.672997i \(-0.234993\pi\)
\(654\) 1.18889e35 0.759311
\(655\) 0 0
\(656\) 4.46664e34 0.274589
\(657\) 7.83084e34i 0.472325i
\(658\) 1.74667e35i 1.03368i
\(659\) 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 \(0.529555\pi\)
\(662\) 1.41394e35i 0.775720i
\(663\) − 2.77142e35i − 1.49205i
\(664\) 1.13329e35 0.598745
\(665\) 0 0
\(666\) 9.36532e34 0.476535
\(667\) 1.47201e34i 0.0735087i
\(668\) 1.68119e35i 0.823970i
\(669\) −2.03109e35 −0.977016
\(670\) 0 0
\(671\) 8.85101e34 0.410168
\(672\) − 6.16426e34i − 0.280392i
\(673\) 2.87165e35i 1.28216i 0.767472 + 0.641082i \(0.221514\pi\)
−0.767472 + 0.641082i \(0.778486\pi\)
\(674\) 2.16160e35 0.947387
\(675\) 0 0
\(676\) 3.53722e34 0.149392
\(677\) 3.20244e35i 1.32777i 0.747835 + 0.663885i \(0.231093\pi\)
−0.747835 + 0.663885i \(0.768907\pi\)
\(678\) − 5.01167e35i − 2.03991i
\(679\) 3.06003e35 1.22280
\(680\) 0 0
\(681\) −1.58411e35 −0.610165
\(682\) − 4.68366e34i − 0.177125i
\(683\) 4.34050e35i 1.61169i 0.592129 + 0.805843i \(0.298288\pi\)
−0.592129 + 0.805843i \(0.701712\pi\)
\(684\) −1.86274e35 −0.679126
\(685\) 0 0
\(686\) 2.18895e35 0.769456
\(687\) 6.49388e35i 2.24153i
\(688\) − 5.16719e34i − 0.175145i
\(689\) −1.91372e35 −0.636999
\(690\) 0 0
\(691\) 3.59659e35 1.15456 0.577279 0.816547i \(-0.304114\pi\)
0.577279 + 0.816547i \(0.304114\pi\)
\(692\) − 1.32538e35i − 0.417845i
\(693\) 1.76480e35i 0.546425i
\(694\) −1.80262e34 −0.0548165
\(695\) 0 0
\(696\) −1.65184e35 −0.484565
\(697\) − 2.61333e35i − 0.752982i
\(698\) − 1.22089e35i − 0.345529i
\(699\) −7.22055e34 −0.200727
\(700\) 0 0
\(701\) −4.01671e34 −0.107745 −0.0538723 0.998548i \(-0.517156\pi\)
−0.0538723 + 0.998548i \(0.517156\pi\)
\(702\) − 9.61963e35i − 2.53481i
\(703\) − 5.04654e34i − 0.130633i
\(704\) −1.22170e34 −0.0310674
\(705\) 0 0
\(706\) 1.82716e35 0.448454
\(707\) 4.32357e35i 1.04256i
\(708\) 1.22892e35i 0.291144i
\(709\) 2.37128e34 0.0551957 0.0275979 0.999619i \(-0.491214\pi\)
0.0275979 + 0.999619i \(0.491214\pi\)
\(710\) 0 0
\(711\) 6.53988e35 1.46960
\(712\) 6.00907e34i 0.132681i
\(713\) − 4.75812e34i − 0.103232i
\(714\) −3.60657e35 −0.768895
\(715\) 0 0
\(716\) −1.35574e35 −0.279103
\(717\) − 1.14103e36i − 2.30839i
\(718\) − 1.79427e35i − 0.356724i
\(719\) 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
\(722\) − 2.80867e35i − 0.520938i
\(723\) − 2.96697e35i − 0.540859i
\(724\) −3.03120e34 −0.0543104
\(725\) 0 0
\(726\) −7.31938e35 −1.26697
\(727\) 8.07486e35i 1.37390i 0.726705 + 0.686950i \(0.241051\pi\)
−0.726705 + 0.686950i \(0.758949\pi\)
\(728\) − 2.00093e35i − 0.334650i
\(729\) −1.75615e36 −2.88714
\(730\) 0 0
\(731\) −3.02321e35 −0.480287
\(732\) 1.00902e36i 1.57583i
\(733\) 9.32041e35i 1.43099i 0.698620 + 0.715493i \(0.253798\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(734\) 3.18151e35 0.480211
\(735\) 0 0
\(736\) −1.24112e34 −0.0181068
\(737\) 2.24310e35i 0.321740i
\(738\) − 1.45781e36i − 2.05587i
\(739\) 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
\(742\) 2.49041e35i 0.328263i
\(743\) − 3.86129e35i − 0.500463i −0.968186 0.250232i \(-0.919493\pi\)
0.968186 0.250232i \(-0.0805068\pi\)
\(744\) 5.33938e35 0.680501
\(745\) 0 0
\(746\) −9.71592e35 −1.19743
\(747\) − 3.69882e36i − 4.48287i
\(748\) 7.14788e34i 0.0851937i
\(749\) −5.30431e35 −0.621736
\(750\) 0 0
\(751\) 5.31819e35 0.602927 0.301463 0.953478i \(-0.402525\pi\)
0.301463 + 0.953478i \(0.402525\pi\)
\(752\) 3.94637e35i 0.440023i
\(753\) − 4.65889e35i − 0.510911i
\(754\) −5.36190e35 −0.578332
\(755\) 0 0
\(756\) −1.25184e36 −1.30625
\(757\) 1.46803e36i 1.50674i 0.657599 + 0.753368i \(0.271572\pi\)
−0.657599 + 0.753368i \(0.728428\pi\)
\(758\) 5.69116e35i 0.574560i
\(759\) 4.89561e34 0.0486166
\(760\) 0 0
\(761\) 1.26153e36 1.21225 0.606123 0.795371i \(-0.292724\pi\)
0.606123 + 0.795371i \(0.292724\pi\)
\(762\) − 4.45641e35i − 0.421257i
\(763\) − 5.02210e35i − 0.467012i
\(764\) 8.27269e35 0.756798
\(765\) 0 0
\(766\) −1.55273e35 −0.137479
\(767\) 3.98910e35i 0.347482i
\(768\) − 1.39274e35i − 0.119358i
\(769\) 9.78483e35 0.825039 0.412519 0.910949i \(-0.364649\pi\)
0.412519 + 0.910949i \(0.364649\pi\)
\(770\) 0 0
\(771\) 1.96165e36 1.60119
\(772\) 2.38980e35i 0.191931i
\(773\) 1.97217e36i 1.55848i 0.626728 + 0.779238i \(0.284394\pi\)
−0.626728 + 0.779238i \(0.715606\pi\)
\(774\) −1.68646e36 −1.31133
\(775\) 0 0
\(776\) 6.91375e35 0.520526
\(777\) − 5.45059e35i − 0.403813i
\(778\) 9.01978e35i 0.657583i
\(779\) −7.85547e35 −0.563578
\(780\) 0 0
\(781\) −1.32757e35 −0.0922398
\(782\) 7.26151e34i 0.0496527i
\(783\) 3.35457e36i 2.25743i
\(784\) 1.17087e35 0.0775459
\(785\) 0 0
\(786\) −1.00575e36 −0.645222
\(787\) − 2.25189e35i − 0.142188i −0.997470 0.0710940i \(-0.977351\pi\)
0.997470 0.0710940i \(-0.0226490\pi\)
\(788\) − 6.02424e35i − 0.374390i
\(789\) −2.70224e36 −1.65295
\(790\) 0 0
\(791\) −2.11702e36 −1.25464
\(792\) 3.98735e35i 0.232605i
\(793\) 3.27529e36i 1.88077i
\(794\) −8.96781e35 −0.506909
\(795\) 0 0
\(796\) −1.36230e36 −0.746209
\(797\) 3.69113e36i 1.99035i 0.0980978 + 0.995177i \(0.468724\pi\)
−0.0980978 + 0.995177i \(0.531276\pi\)
\(798\) 1.08411e36i 0.575488i
\(799\) 2.30893e36 1.20664
\(800\) 0 0
\(801\) 1.96123e36 0.993393
\(802\) 1.22485e36i 0.610806i
\(803\) − 9.03258e34i − 0.0443473i
\(804\) −2.55713e36 −1.23610
\(805\) 0 0
\(806\) 1.73317e36 0.812184
\(807\) 4.12420e36i 1.90292i
\(808\) 9.76854e35i 0.443800i
\(809\) 2.93815e36 1.31437 0.657185 0.753730i \(-0.271747\pi\)
0.657185 + 0.753730i \(0.271747\pi\)
\(810\) 0 0
\(811\) −2.44583e36 −1.06088 −0.530439 0.847723i \(-0.677973\pi\)
−0.530439 + 0.847723i \(0.677973\pi\)
\(812\) 6.97767e35i 0.298030i
\(813\) 2.71729e36i 1.14289i
\(814\) −1.08025e35 −0.0447426
\(815\) 0 0
\(816\) −8.14859e35 −0.327307
\(817\) 9.08753e35i 0.359476i
\(818\) − 2.91771e35i − 0.113665i
\(819\) −6.53060e36 −2.50556
\(820\) 0 0
\(821\) 3.33815e36 1.24227 0.621136 0.783703i \(-0.286672\pi\)
0.621136 + 0.783703i \(0.286672\pi\)
\(822\) − 4.98560e36i − 1.82734i
\(823\) − 2.54036e36i − 0.917058i −0.888679 0.458529i \(-0.848377\pi\)
0.888679 0.458529i \(-0.151623\pi\)
\(824\) −7.09263e35 −0.252184
\(825\) 0 0
\(826\) 5.19119e35 0.179067
\(827\) − 3.63276e36i − 1.23429i −0.786849 0.617146i \(-0.788289\pi\)
0.786849 0.617146i \(-0.211711\pi\)
\(828\) 4.05074e35i 0.135567i
\(829\) −1.07727e36 −0.355135 −0.177567 0.984109i \(-0.556823\pi\)
−0.177567 + 0.984109i \(0.556823\pi\)
\(830\) 0 0
\(831\) −2.13782e36 −0.683847
\(832\) − 4.52085e35i − 0.142455i
\(833\) − 6.85052e35i − 0.212648i
\(834\) 5.80833e36 1.77613
\(835\) 0 0
\(836\) 2.14860e35 0.0637641
\(837\) − 1.08433e37i − 3.17024i
\(838\) − 1.20611e36i − 0.347407i
\(839\) 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
\(842\) 4.17349e36i 1.13266i
\(843\) − 8.96316e36i − 2.39671i
\(844\) 1.73283e36 0.456537
\(845\) 0 0
\(846\) 1.28801e37 3.29450
\(847\) 3.09185e36i 0.779247i
\(848\) 5.62676e35i 0.139737i
\(849\) 2.74735e36 0.672308
\(850\) 0 0
\(851\) −1.09743e35 −0.0260769
\(852\) − 1.51343e36i − 0.354378i
\(853\) − 6.13786e36i − 1.41630i −0.706064 0.708148i \(-0.749531\pi\)
0.706064 0.708148i \(-0.250469\pi\)
\(854\) 4.26227e36 0.969210
\(855\) 0 0
\(856\) −1.19844e36 −0.264664
\(857\) 3.16481e36i 0.688791i 0.938825 + 0.344395i \(0.111916\pi\)
−0.938825 + 0.344395i \(0.888084\pi\)
\(858\) 1.78325e36i 0.382492i
\(859\) −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
\(862\) − 2.78322e36i − 0.563257i
\(863\) 8.00828e36i 1.59736i 0.601753 + 0.798682i \(0.294469\pi\)
−0.601753 + 0.798682i \(0.705531\pi\)
\(864\) −2.82838e36 −0.556053
\(865\) 0 0
\(866\) −2.96617e36 −0.566530
\(867\) − 5.37651e36i − 1.01219i
\(868\) − 2.25545e36i − 0.418540i
\(869\) −7.54351e35 −0.137983
\(870\) 0 0
\(871\) −8.30052e36 −1.47529
\(872\) − 1.13468e36i − 0.198800i
\(873\) − 2.25649e37i − 3.89723i
\(874\) 2.18275e35 0.0371631
\(875\) 0 0
\(876\) 1.02971e36 0.170379
\(877\) 3.29560e36i 0.537575i 0.963199 + 0.268788i \(0.0866230\pi\)
−0.963199 + 0.268788i \(0.913377\pi\)
\(878\) 3.74153e36i 0.601683i
\(879\) 7.33233e36 1.16247
\(880\) 0 0
\(881\) −8.05386e36 −1.24109 −0.620547 0.784169i \(-0.713090\pi\)
−0.620547 + 0.784169i \(0.713090\pi\)
\(882\) − 3.82147e36i − 0.580595i
\(883\) 4.91849e35i 0.0736754i 0.999321 + 0.0368377i \(0.0117285\pi\)
−0.999321 + 0.0368377i \(0.988272\pi\)
\(884\) −2.64505e36 −0.390644
\(885\) 0 0
\(886\) −1.44406e36 −0.207331
\(887\) 7.79484e36i 1.10347i 0.834018 + 0.551737i \(0.186035\pi\)
−0.834018 + 0.551737i \(0.813965\pi\)
\(888\) − 1.23149e36i − 0.171897i
\(889\) −1.88247e36 −0.259093
\(890\) 0 0
\(891\) 6.24031e36 0.835093
\(892\) 1.93847e36i 0.255799i
\(893\) − 6.94048e36i − 0.903121i
\(894\) 3.05319e36 0.391774
\(895\) 0 0
\(896\) −5.88318e35 −0.0734111
\(897\) 1.81160e36i 0.222924i
\(898\) − 1.51928e36i − 0.184367i
\(899\) −6.04394e36 −0.723309
\(900\) 0 0
\(901\) 3.29209e36 0.383188
\(902\) 1.68153e36i 0.193029i
\(903\) 9.81513e36i 1.11122i
\(904\) −4.78314e36 −0.534082
\(905\) 0 0
\(906\) 1.81002e37 1.96599
\(907\) − 1.31417e37i − 1.40786i −0.710268 0.703932i \(-0.751426\pi\)
0.710268 0.703932i \(-0.248574\pi\)
\(908\) 1.51188e36i 0.159751i
\(909\) 3.18823e37 3.32278
\(910\) 0 0
\(911\) 1.44767e37 1.46788 0.733940 0.679214i \(-0.237679\pi\)
0.733940 + 0.679214i \(0.237679\pi\)
\(912\) 2.44940e36i 0.244977i
\(913\) 4.26645e36i 0.420903i
\(914\) −7.24821e36 −0.705347
\(915\) 0 0
\(916\) 6.19776e36 0.586869
\(917\) 4.24848e36i 0.396842i
\(918\) 1.65482e37i 1.52482i
\(919\) 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
\(922\) 2.87137e36i 0.250584i
\(923\) − 4.91262e36i − 0.422953i
\(924\) 2.32062e36 0.197108
\(925\) 0 0
\(926\) −3.77045e36 −0.311714
\(927\) 2.31487e37i 1.88813i
\(928\) 1.57651e36i 0.126867i
\(929\) 2.17912e37 1.73015 0.865075 0.501642i \(-0.167271\pi\)
0.865075 + 0.501642i \(0.167271\pi\)
\(930\) 0 0
\(931\) −2.05921e36 −0.159159
\(932\) 6.89130e35i 0.0525536i
\(933\) − 2.71170e37i − 2.04043i
\(934\) 7.56121e36 0.561378
\(935\) 0 0
\(936\) −1.47550e37 −1.06658
\(937\) − 2.08020e37i − 1.48375i −0.670540 0.741873i \(-0.733938\pi\)
0.670540 0.741873i \(-0.266062\pi\)
\(938\) 1.08018e37i 0.760258i
\(939\) −2.85686e37 −1.98413
\(940\) 0 0
\(941\) 2.39454e37 1.61939 0.809695 0.586851i \(-0.199632\pi\)
0.809695 + 0.586851i \(0.199632\pi\)
\(942\) 3.42648e37i 2.28672i
\(943\) 1.70826e36i 0.112501i
\(944\) 1.17288e36 0.0762262
\(945\) 0 0
\(946\) 1.94526e36 0.123123
\(947\) − 8.06134e36i − 0.503539i −0.967787 0.251769i \(-0.918988\pi\)
0.967787 0.251769i \(-0.0810124\pi\)
\(948\) − 8.59960e36i − 0.530120i
\(949\) 3.34248e36 0.203348
\(950\) 0 0
\(951\) −4.02727e37 −2.38646
\(952\) 3.44212e36i 0.201309i
\(953\) 1.92589e37i 1.11166i 0.831297 + 0.555829i \(0.187599\pi\)
−0.831297 + 0.555829i \(0.812401\pi\)
\(954\) 1.83645e37 1.04622
\(955\) 0 0
\(956\) −1.08900e37 −0.604374
\(957\) − 6.21858e36i − 0.340637i
\(958\) 1.59122e36i 0.0860321i
\(959\) −2.10601e37 −1.12390
\(960\) 0 0
\(961\) 3.03575e35 0.0157842
\(962\) − 3.99745e36i − 0.205161i
\(963\) 3.91144e37i 1.98157i
\(964\) −2.83168e36 −0.141606
\(965\) 0 0
\(966\) 2.35752e36 0.114879
\(967\) 2.29324e37i 1.10311i 0.834139 + 0.551554i \(0.185965\pi\)
−0.834139 + 0.551554i \(0.814035\pi\)
\(968\) 6.98563e36i 0.331714i
\(969\) 1.43309e37 0.671779
\(970\) 0 0
\(971\) 2.10199e37 0.960265 0.480132 0.877196i \(-0.340589\pi\)
0.480132 + 0.877196i \(0.340589\pi\)
\(972\) 3.62665e37i 1.63560i
\(973\) − 2.45355e37i − 1.09241i
\(974\) −1.74531e37 −0.767161
\(975\) 0 0
\(976\) 9.63006e36 0.412578
\(977\) − 3.52592e37i − 1.49139i −0.666289 0.745694i \(-0.732118\pi\)
0.666289 0.745694i \(-0.267882\pi\)
\(978\) − 5.47148e36i − 0.228491i
\(979\) −2.26220e36 −0.0932712
\(980\) 0 0
\(981\) −3.70334e37 −1.48844
\(982\) − 1.85363e37i − 0.735581i
\(983\) − 7.31962e36i − 0.286794i −0.989665 0.143397i \(-0.954197\pi\)
0.989665 0.143397i \(-0.0458025\pi\)
\(984\) −1.91694e37 −0.741602
\(985\) 0 0
\(986\) 9.22384e36 0.347897
\(987\) − 7.49617e37i − 2.79174i
\(988\) 7.95081e36i 0.292381i
\(989\) 1.97619e36 0.0717587
\(990\) 0 0
\(991\) 1.16985e37 0.414200 0.207100 0.978320i \(-0.433597\pi\)
0.207100 + 0.978320i \(0.433597\pi\)
\(992\) − 5.09591e36i − 0.178166i
\(993\) − 6.06819e37i − 2.09504i
\(994\) −6.39300e36 −0.217959
\(995\) 0 0
\(996\) −4.86376e37 −1.61708
\(997\) 2.33411e37i 0.766358i 0.923674 + 0.383179i \(0.125171\pi\)
−0.923674 + 0.383179i \(0.874829\pi\)
\(998\) − 1.82280e37i − 0.591028i
\(999\) −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.b.e.49.3 4
5.2 odd 4 50.26.a.c.1.1 2
5.3 odd 4 2.26.a.b.1.2 2
5.4 even 2 inner 50.26.b.e.49.2 4
15.8 even 4 18.26.a.e.1.2 2
20.3 even 4 16.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 2 5.3 odd 4
16.26.a.c.1.1 2 20.3 even 4
18.26.a.e.1.2 2 15.8 even 4
50.26.a.c.1.1 2 5.2 odd 4
50.26.b.e.49.2 4 5.4 even 2 inner
50.26.b.e.49.3 4 1.1 even 1 trivial