Properties

Label 10.22.a.c.1.2
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
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] = [10,22,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(27.9477344287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{474529}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 118632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-343.930\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +194815. q^{3} +1.04858e6 q^{4} +9.76562e6 q^{5} -1.99490e8 q^{6} +9.63461e8 q^{7} -1.07374e9 q^{8} +2.74924e10 q^{9} -1.00000e10 q^{10} +6.10882e10 q^{11} +2.04278e11 q^{12} +5.25383e10 q^{13} -9.86584e11 q^{14} +1.90249e12 q^{15} +1.09951e12 q^{16} -1.13179e13 q^{17} -2.81522e13 q^{18} -4.53240e13 q^{19} +1.02400e13 q^{20} +1.87696e14 q^{21} -6.25543e13 q^{22} +2.47091e14 q^{23} -2.09181e14 q^{24} +9.53674e13 q^{25} -5.37992e13 q^{26} +3.31810e15 q^{27} +1.01026e15 q^{28} -2.58718e15 q^{29} -1.94815e15 q^{30} -1.35601e15 q^{31} -1.12590e15 q^{32} +1.19009e16 q^{33} +1.15895e16 q^{34} +9.40880e15 q^{35} +2.88279e16 q^{36} -1.35920e16 q^{37} +4.64117e16 q^{38} +1.02352e16 q^{39} -1.04858e16 q^{40} +6.72480e16 q^{41} -1.92201e17 q^{42} -3.70641e16 q^{43} +6.40556e16 q^{44} +2.68481e17 q^{45} -2.53021e17 q^{46} +3.17182e17 q^{47} +2.14201e17 q^{48} +3.69712e17 q^{49} -9.76562e16 q^{50} -2.20489e18 q^{51} +5.50904e16 q^{52} +1.44909e18 q^{53} -3.39773e18 q^{54} +5.96564e17 q^{55} -1.03451e18 q^{56} -8.82978e18 q^{57} +2.64928e18 q^{58} -1.08957e18 q^{59} +1.99490e18 q^{60} -1.59498e17 q^{61} +1.38855e18 q^{62} +2.64879e19 q^{63} +1.15292e18 q^{64} +5.13069e17 q^{65} -1.21865e19 q^{66} +8.71262e18 q^{67} -1.18677e19 q^{68} +4.81370e19 q^{69} -9.63461e18 q^{70} -3.06845e19 q^{71} -2.95198e19 q^{72} +2.36491e19 q^{73} +1.39182e19 q^{74} +1.85790e19 q^{75} -4.75256e19 q^{76} +5.88561e19 q^{77} -1.04809e19 q^{78} -9.35354e19 q^{79} +1.07374e19 q^{80} +3.58834e20 q^{81} -6.88619e19 q^{82} -2.00855e20 q^{83} +1.96814e20 q^{84} -1.10526e20 q^{85} +3.79536e19 q^{86} -5.04022e20 q^{87} -6.55929e19 q^{88} +3.85510e19 q^{89} -2.74924e20 q^{90} +5.06186e19 q^{91} +2.59094e20 q^{92} -2.64170e20 q^{93} -3.24795e20 q^{94} -4.42617e20 q^{95} -2.19342e20 q^{96} -5.46564e20 q^{97} -3.78585e20 q^{98} +1.67946e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{2} + 100308 q^{3} + 2097152 q^{4} + 19531250 q^{5} - 102715392 q^{6} + 1328895316 q^{7} - 2147483648 q^{8} + 25963598826 q^{9} - 20000000000 q^{10} + 21869194224 q^{11} + 105180561408 q^{12}+ \cdots + 17\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 194815. 1.90480 0.952398 0.304856i \(-0.0986084\pi\)
0.952398 + 0.304856i \(0.0986084\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 9.76562e6 0.447214
\(6\) −1.99490e8 −1.34689
\(7\) 9.63461e8 1.28915 0.644577 0.764539i \(-0.277033\pi\)
0.644577 + 0.764539i \(0.277033\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 2.74924e10 2.62825
\(10\) −1.00000e10 −0.316228
\(11\) 6.10882e10 0.710124 0.355062 0.934843i \(-0.384460\pi\)
0.355062 + 0.934843i \(0.384460\pi\)
\(12\) 2.04278e11 0.952398
\(13\) 5.25383e10 0.105699 0.0528495 0.998602i \(-0.483170\pi\)
0.0528495 + 0.998602i \(0.483170\pi\)
\(14\) −9.86584e11 −0.911570
\(15\) 1.90249e12 0.851851
\(16\) 1.09951e12 0.250000
\(17\) −1.13179e13 −1.36161 −0.680803 0.732466i \(-0.738369\pi\)
−0.680803 + 0.732466i \(0.738369\pi\)
\(18\) −2.81522e13 −1.85845
\(19\) −4.53240e13 −1.69596 −0.847979 0.530030i \(-0.822181\pi\)
−0.847979 + 0.530030i \(0.822181\pi\)
\(20\) 1.02400e13 0.223607
\(21\) 1.87696e14 2.45558
\(22\) −6.25543e13 −0.502133
\(23\) 2.47091e14 1.24370 0.621849 0.783137i \(-0.286382\pi\)
0.621849 + 0.783137i \(0.286382\pi\)
\(24\) −2.09181e14 −0.673447
\(25\) 9.53674e13 0.200000
\(26\) −5.37992e13 −0.0747404
\(27\) 3.31810e15 3.10149
\(28\) 1.01026e15 0.644577
\(29\) −2.58718e15 −1.14195 −0.570977 0.820966i \(-0.693435\pi\)
−0.570977 + 0.820966i \(0.693435\pi\)
\(30\) −1.94815e15 −0.602350
\(31\) −1.35601e15 −0.297142 −0.148571 0.988902i \(-0.547467\pi\)
−0.148571 + 0.988902i \(0.547467\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 1.19009e16 1.35264
\(34\) 1.15895e16 0.962801
\(35\) 9.40880e15 0.576527
\(36\) 2.88279e16 1.31413
\(37\) −1.35920e16 −0.464692 −0.232346 0.972633i \(-0.574640\pi\)
−0.232346 + 0.972633i \(0.574640\pi\)
\(38\) 4.64117e16 1.19922
\(39\) 1.02352e16 0.201335
\(40\) −1.04858e16 −0.158114
\(41\) 6.72480e16 0.782436 0.391218 0.920298i \(-0.372054\pi\)
0.391218 + 0.920298i \(0.372054\pi\)
\(42\) −1.92201e17 −1.73636
\(43\) −3.70641e16 −0.261538 −0.130769 0.991413i \(-0.541745\pi\)
−0.130769 + 0.991413i \(0.541745\pi\)
\(44\) 6.40556e16 0.355062
\(45\) 2.68481e17 1.17539
\(46\) −2.53021e17 −0.879427
\(47\) 3.17182e17 0.879592 0.439796 0.898098i \(-0.355051\pi\)
0.439796 + 0.898098i \(0.355051\pi\)
\(48\) 2.14201e17 0.476199
\(49\) 3.69712e17 0.661919
\(50\) −9.76562e16 −0.141421
\(51\) −2.20489e18 −2.59358
\(52\) 5.50904e16 0.0528495
\(53\) 1.44909e18 1.13815 0.569074 0.822286i \(-0.307302\pi\)
0.569074 + 0.822286i \(0.307302\pi\)
\(54\) −3.39773e18 −2.19308
\(55\) 5.96564e17 0.317577
\(56\) −1.03451e18 −0.455785
\(57\) −8.82978e18 −3.23046
\(58\) 2.64928e18 0.807483
\(59\) −1.08957e18 −0.277529 −0.138764 0.990325i \(-0.544313\pi\)
−0.138764 + 0.990325i \(0.544313\pi\)
\(60\) 1.99490e18 0.425925
\(61\) −1.59498e17 −0.0286281 −0.0143140 0.999898i \(-0.504556\pi\)
−0.0143140 + 0.999898i \(0.504556\pi\)
\(62\) 1.38855e18 0.210111
\(63\) 2.64879e19 3.38822
\(64\) 1.15292e18 0.125000
\(65\) 5.13069e17 0.0472700
\(66\) −1.21865e19 −0.956462
\(67\) 8.71262e18 0.583934 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(68\) −1.18677e19 −0.680803
\(69\) 4.81370e19 2.36899
\(70\) −9.63461e18 −0.407666
\(71\) −3.06845e19 −1.11868 −0.559340 0.828938i \(-0.688946\pi\)
−0.559340 + 0.828938i \(0.688946\pi\)
\(72\) −2.95198e19 −0.929227
\(73\) 2.36491e19 0.644059 0.322029 0.946730i \(-0.395635\pi\)
0.322029 + 0.946730i \(0.395635\pi\)
\(74\) 1.39182e19 0.328587
\(75\) 1.85790e19 0.380959
\(76\) −4.75256e19 −0.847979
\(77\) 5.88561e19 0.915459
\(78\) −1.04809e19 −0.142365
\(79\) −9.35354e19 −1.11145 −0.555727 0.831365i \(-0.687560\pi\)
−0.555727 + 0.831365i \(0.687560\pi\)
\(80\) 1.07374e19 0.111803
\(81\) 3.58834e20 3.27945
\(82\) −6.88619e19 −0.553266
\(83\) −2.00855e20 −1.42090 −0.710448 0.703750i \(-0.751508\pi\)
−0.710448 + 0.703750i \(0.751508\pi\)
\(84\) 1.96814e20 1.22779
\(85\) −1.10526e20 −0.608929
\(86\) 3.79536e19 0.184935
\(87\) −5.04022e20 −2.17519
\(88\) −6.55929e19 −0.251067
\(89\) 3.85510e19 0.131051 0.0655255 0.997851i \(-0.479128\pi\)
0.0655255 + 0.997851i \(0.479128\pi\)
\(90\) −2.74924e20 −0.831126
\(91\) 5.06186e19 0.136262
\(92\) 2.59094e20 0.621849
\(93\) −2.64170e20 −0.565996
\(94\) −3.24795e20 −0.621965
\(95\) −4.42617e20 −0.758456
\(96\) −2.19342e20 −0.336724
\(97\) −5.46564e20 −0.752554 −0.376277 0.926507i \(-0.622796\pi\)
−0.376277 + 0.926507i \(0.622796\pi\)
\(98\) −3.78585e20 −0.468047
\(99\) 1.67946e21 1.86638
\(100\) 1.00000e20 0.100000
\(101\) 1.87063e21 1.68505 0.842524 0.538658i \(-0.181069\pi\)
0.842524 + 0.538658i \(0.181069\pi\)
\(102\) 2.25781e21 1.83394
\(103\) −8.63482e20 −0.633086 −0.316543 0.948578i \(-0.602522\pi\)
−0.316543 + 0.948578i \(0.602522\pi\)
\(104\) −5.64125e19 −0.0373702
\(105\) 1.83297e21 1.09817
\(106\) −1.48387e21 −0.804792
\(107\) −3.65685e21 −1.79712 −0.898561 0.438848i \(-0.855387\pi\)
−0.898561 + 0.438848i \(0.855387\pi\)
\(108\) 3.47928e21 1.55074
\(109\) −6.39944e20 −0.258919 −0.129459 0.991585i \(-0.541324\pi\)
−0.129459 + 0.991585i \(0.541324\pi\)
\(110\) −6.10882e20 −0.224561
\(111\) −2.64792e21 −0.885143
\(112\) 1.05934e21 0.322289
\(113\) 6.17070e20 0.171006 0.0855029 0.996338i \(-0.472750\pi\)
0.0855029 + 0.996338i \(0.472750\pi\)
\(114\) 9.04169e21 2.28428
\(115\) 2.41300e21 0.556199
\(116\) −2.71286e21 −0.570977
\(117\) 1.44440e21 0.277803
\(118\) 1.11572e21 0.196242
\(119\) −1.09043e22 −1.75532
\(120\) −2.04278e21 −0.301175
\(121\) −3.66848e21 −0.495724
\(122\) 1.63326e20 0.0202431
\(123\) 1.31009e22 1.49038
\(124\) −1.42188e21 −0.148571
\(125\) 9.31323e20 0.0894427
\(126\) −2.71236e22 −2.39583
\(127\) −1.40405e22 −1.14142 −0.570709 0.821152i \(-0.693332\pi\)
−0.570709 + 0.821152i \(0.693332\pi\)
\(128\) −1.18059e21 −0.0883883
\(129\) −7.22063e21 −0.498176
\(130\) −5.25383e20 −0.0334249
\(131\) 2.03410e22 1.19405 0.597025 0.802222i \(-0.296349\pi\)
0.597025 + 0.802222i \(0.296349\pi\)
\(132\) 1.24790e22 0.676321
\(133\) −4.36679e22 −2.18635
\(134\) −8.92173e21 −0.412903
\(135\) 3.24033e22 1.38703
\(136\) 1.21525e22 0.481401
\(137\) −1.10091e21 −0.0403819 −0.0201909 0.999796i \(-0.506427\pi\)
−0.0201909 + 0.999796i \(0.506427\pi\)
\(138\) −4.92923e22 −1.67513
\(139\) 4.14583e22 1.30604 0.653019 0.757342i \(-0.273502\pi\)
0.653019 + 0.757342i \(0.273502\pi\)
\(140\) 9.86584e21 0.288264
\(141\) 6.17918e22 1.67544
\(142\) 3.14209e22 0.791027
\(143\) 3.20947e21 0.0750593
\(144\) 3.02282e22 0.657063
\(145\) −2.52655e22 −0.510697
\(146\) −2.42167e22 −0.455418
\(147\) 7.20253e22 1.26082
\(148\) −1.42522e22 −0.232346
\(149\) 9.61610e22 1.46064 0.730320 0.683106i \(-0.239371\pi\)
0.730320 + 0.683106i \(0.239371\pi\)
\(150\) −1.90249e22 −0.269379
\(151\) −1.13093e23 −1.49340 −0.746700 0.665161i \(-0.768363\pi\)
−0.746700 + 0.665161i \(0.768363\pi\)
\(152\) 4.86662e22 0.599612
\(153\) −3.11156e23 −3.57864
\(154\) −6.02687e22 −0.647327
\(155\) −1.32423e22 −0.132886
\(156\) 1.07324e22 0.100667
\(157\) 3.75753e22 0.329576 0.164788 0.986329i \(-0.447306\pi\)
0.164788 + 0.986329i \(0.447306\pi\)
\(158\) 9.57802e22 0.785917
\(159\) 2.82304e23 2.16794
\(160\) −1.09951e22 −0.0790569
\(161\) 2.38063e23 1.60332
\(162\) −3.67446e23 −2.31892
\(163\) 2.53934e23 1.50228 0.751140 0.660143i \(-0.229504\pi\)
0.751140 + 0.660143i \(0.229504\pi\)
\(164\) 7.05146e22 0.391218
\(165\) 1.16220e23 0.604920
\(166\) 2.05675e23 1.00472
\(167\) −7.88087e22 −0.361452 −0.180726 0.983533i \(-0.557845\pi\)
−0.180726 + 0.983533i \(0.557845\pi\)
\(168\) −2.01538e23 −0.868178
\(169\) −2.44304e23 −0.988828
\(170\) 1.13179e23 0.430578
\(171\) −1.24607e24 −4.45740
\(172\) −3.88645e22 −0.130769
\(173\) −3.30387e23 −1.04602 −0.523009 0.852327i \(-0.675191\pi\)
−0.523009 + 0.852327i \(0.675191\pi\)
\(174\) 5.16118e23 1.53809
\(175\) 9.18828e22 0.257831
\(176\) 6.71672e22 0.177531
\(177\) −2.12264e23 −0.528636
\(178\) −3.94762e22 −0.0926671
\(179\) 4.40276e23 0.974470 0.487235 0.873271i \(-0.338005\pi\)
0.487235 + 0.873271i \(0.338005\pi\)
\(180\) 2.81522e23 0.587695
\(181\) −8.24093e23 −1.62312 −0.811561 0.584268i \(-0.801382\pi\)
−0.811561 + 0.584268i \(0.801382\pi\)
\(182\) −5.18334e22 −0.0963519
\(183\) −3.10726e22 −0.0545306
\(184\) −2.65312e23 −0.439714
\(185\) −1.32734e23 −0.207816
\(186\) 2.70510e23 0.400219
\(187\) −6.91389e23 −0.966909
\(188\) 3.32590e23 0.439796
\(189\) 3.19686e24 3.99829
\(190\) 4.53240e23 0.536309
\(191\) 2.97957e23 0.333659 0.166830 0.985986i \(-0.446647\pi\)
0.166830 + 0.985986i \(0.446647\pi\)
\(192\) 2.24606e23 0.238100
\(193\) 9.58444e23 0.962088 0.481044 0.876696i \(-0.340258\pi\)
0.481044 + 0.876696i \(0.340258\pi\)
\(194\) 5.59682e23 0.532136
\(195\) 9.99534e22 0.0900397
\(196\) 3.87671e23 0.330959
\(197\) −7.67214e23 −0.620899 −0.310450 0.950590i \(-0.600480\pi\)
−0.310450 + 0.950590i \(0.600480\pi\)
\(198\) −1.71977e24 −1.31973
\(199\) −2.93227e23 −0.213426 −0.106713 0.994290i \(-0.534033\pi\)
−0.106713 + 0.994290i \(0.534033\pi\)
\(200\) −1.02400e23 −0.0707107
\(201\) 1.69735e24 1.11227
\(202\) −1.91552e24 −1.19151
\(203\) −2.49265e24 −1.47215
\(204\) −2.31200e24 −1.29679
\(205\) 6.56719e23 0.349916
\(206\) 8.84206e23 0.447659
\(207\) 6.79314e24 3.26875
\(208\) 5.77664e22 0.0264247
\(209\) −2.76876e24 −1.20434
\(210\) −1.87696e24 −0.776522
\(211\) −2.05920e24 −0.810464 −0.405232 0.914214i \(-0.632809\pi\)
−0.405232 + 0.914214i \(0.632809\pi\)
\(212\) 1.51948e24 0.569074
\(213\) −5.97779e24 −2.13086
\(214\) 3.74462e24 1.27076
\(215\) −3.61954e23 −0.116963
\(216\) −3.56278e24 −1.09654
\(217\) −1.30646e24 −0.383062
\(218\) 6.55303e23 0.183083
\(219\) 4.60720e24 1.22680
\(220\) 6.25543e23 0.158788
\(221\) −5.94622e23 −0.143920
\(222\) 2.71147e24 0.625891
\(223\) −3.95841e24 −0.871605 −0.435803 0.900042i \(-0.643535\pi\)
−0.435803 + 0.900042i \(0.643535\pi\)
\(224\) −1.08476e24 −0.227892
\(225\) 2.62188e24 0.525650
\(226\) −6.31880e23 −0.120919
\(227\) −6.66876e23 −0.121835 −0.0609177 0.998143i \(-0.519403\pi\)
−0.0609177 + 0.998143i \(0.519403\pi\)
\(228\) −9.25869e24 −1.61523
\(229\) −7.22731e24 −1.20421 −0.602107 0.798415i \(-0.705672\pi\)
−0.602107 + 0.798415i \(0.705672\pi\)
\(230\) −2.47091e24 −0.393292
\(231\) 1.14660e25 1.74376
\(232\) 2.77797e24 0.403742
\(233\) 1.07597e25 1.49474 0.747368 0.664410i \(-0.231317\pi\)
0.747368 + 0.664410i \(0.231317\pi\)
\(234\) −1.47907e24 −0.196437
\(235\) 3.09748e24 0.393365
\(236\) −1.14249e24 −0.138764
\(237\) −1.82221e25 −2.11709
\(238\) 1.11661e25 1.24120
\(239\) 3.16911e24 0.337101 0.168550 0.985693i \(-0.446091\pi\)
0.168550 + 0.985693i \(0.446091\pi\)
\(240\) 2.09181e24 0.212963
\(241\) −1.75468e24 −0.171009 −0.0855043 0.996338i \(-0.527250\pi\)
−0.0855043 + 0.996338i \(0.527250\pi\)
\(242\) 3.75653e24 0.350530
\(243\) 3.51977e25 3.14520
\(244\) −1.67246e23 −0.0143140
\(245\) 3.61047e24 0.296019
\(246\) −1.34153e25 −1.05386
\(247\) −2.38124e24 −0.179261
\(248\) 1.45600e24 0.105056
\(249\) −3.91295e25 −2.70652
\(250\) −9.53674e23 −0.0632456
\(251\) 1.75868e25 1.11844 0.559221 0.829019i \(-0.311100\pi\)
0.559221 + 0.829019i \(0.311100\pi\)
\(252\) 2.77746e25 1.69411
\(253\) 1.50944e25 0.883179
\(254\) 1.43775e25 0.807104
\(255\) −2.15321e25 −1.15989
\(256\) 1.20893e24 0.0625000
\(257\) −9.88264e24 −0.490428 −0.245214 0.969469i \(-0.578858\pi\)
−0.245214 + 0.969469i \(0.578858\pi\)
\(258\) 7.39392e24 0.352264
\(259\) −1.30954e25 −0.599059
\(260\) 5.37992e23 0.0236350
\(261\) −7.11280e25 −3.00134
\(262\) −2.08291e25 −0.844321
\(263\) −1.60390e25 −0.624656 −0.312328 0.949974i \(-0.601109\pi\)
−0.312328 + 0.949974i \(0.601109\pi\)
\(264\) −1.27785e25 −0.478231
\(265\) 1.41513e25 0.508995
\(266\) 4.47159e25 1.54598
\(267\) 7.51030e24 0.249626
\(268\) 9.13585e24 0.291967
\(269\) 1.76073e25 0.541122 0.270561 0.962703i \(-0.412791\pi\)
0.270561 + 0.962703i \(0.412791\pi\)
\(270\) −3.31810e25 −0.980776
\(271\) 3.45015e25 0.980981 0.490490 0.871447i \(-0.336818\pi\)
0.490490 + 0.871447i \(0.336818\pi\)
\(272\) −1.24441e25 −0.340402
\(273\) 9.86125e24 0.259552
\(274\) 1.12733e24 0.0285543
\(275\) 5.82582e24 0.142025
\(276\) 5.04753e25 1.18450
\(277\) −2.59482e25 −0.586231 −0.293116 0.956077i \(-0.594692\pi\)
−0.293116 + 0.956077i \(0.594692\pi\)
\(278\) −4.24533e25 −0.923508
\(279\) −3.72799e25 −0.780964
\(280\) −1.01026e25 −0.203833
\(281\) 7.82254e25 1.52031 0.760154 0.649743i \(-0.225124\pi\)
0.760154 + 0.649743i \(0.225124\pi\)
\(282\) −6.32748e25 −1.18472
\(283\) 8.38167e24 0.151207 0.0756037 0.997138i \(-0.475912\pi\)
0.0756037 + 0.997138i \(0.475912\pi\)
\(284\) −3.21750e25 −0.559340
\(285\) −8.62283e25 −1.44470
\(286\) −3.28649e24 −0.0530749
\(287\) 6.47908e25 1.00868
\(288\) −3.09537e25 −0.464613
\(289\) 5.90026e25 0.853973
\(290\) 2.58718e25 0.361117
\(291\) −1.06479e26 −1.43346
\(292\) 2.47979e25 0.322029
\(293\) 4.97478e25 0.623252 0.311626 0.950205i \(-0.399126\pi\)
0.311626 + 0.950205i \(0.399126\pi\)
\(294\) −7.37540e25 −0.891535
\(295\) −1.06403e25 −0.124115
\(296\) 1.45943e25 0.164293
\(297\) 2.02697e26 2.20244
\(298\) −9.84688e25 −1.03283
\(299\) 1.29817e25 0.131458
\(300\) 1.94815e25 0.190480
\(301\) −3.57098e25 −0.337163
\(302\) 1.15807e26 1.05599
\(303\) 3.64425e26 3.20967
\(304\) −4.98342e25 −0.423990
\(305\) −1.55760e24 −0.0128029
\(306\) 3.18624e26 2.53048
\(307\) −1.41280e26 −1.08425 −0.542123 0.840299i \(-0.682379\pi\)
−0.542123 + 0.840299i \(0.682379\pi\)
\(308\) 6.17151e25 0.457730
\(309\) −1.68219e26 −1.20590
\(310\) 1.35601e25 0.0939646
\(311\) 1.39904e26 0.937232 0.468616 0.883402i \(-0.344753\pi\)
0.468616 + 0.883402i \(0.344753\pi\)
\(312\) −1.09900e25 −0.0711827
\(313\) −2.79550e26 −1.75083 −0.875414 0.483373i \(-0.839411\pi\)
−0.875414 + 0.483373i \(0.839411\pi\)
\(314\) −3.84771e25 −0.233045
\(315\) 2.58671e26 1.51526
\(316\) −9.80789e25 −0.555727
\(317\) −1.16992e26 −0.641259 −0.320629 0.947205i \(-0.603894\pi\)
−0.320629 + 0.947205i \(0.603894\pi\)
\(318\) −2.89079e26 −1.53297
\(319\) −1.58046e26 −0.810928
\(320\) 1.12590e25 0.0559017
\(321\) −7.12409e26 −3.42315
\(322\) −2.43776e26 −1.13372
\(323\) 5.12972e26 2.30923
\(324\) 3.76265e26 1.63973
\(325\) 5.01044e24 0.0211398
\(326\) −2.60028e26 −1.06227
\(327\) −1.24671e26 −0.493188
\(328\) −7.22070e25 −0.276633
\(329\) 3.05593e26 1.13393
\(330\) −1.19009e26 −0.427743
\(331\) −3.83627e26 −1.33572 −0.667859 0.744288i \(-0.732789\pi\)
−0.667859 + 0.744288i \(0.732789\pi\)
\(332\) −2.10612e26 −0.710448
\(333\) −3.73677e26 −1.22133
\(334\) 8.07001e25 0.255585
\(335\) 8.50842e25 0.261143
\(336\) 2.06374e26 0.613894
\(337\) 1.36118e26 0.392465 0.196233 0.980557i \(-0.437129\pi\)
0.196233 + 0.980557i \(0.437129\pi\)
\(338\) 2.50168e26 0.699207
\(339\) 1.20214e26 0.325731
\(340\) −1.15895e26 −0.304465
\(341\) −8.28360e25 −0.211008
\(342\) 1.27597e27 3.15186
\(343\) −1.81934e26 −0.435839
\(344\) 3.97973e25 0.0924676
\(345\) 4.70088e26 1.05945
\(346\) 3.38316e26 0.739646
\(347\) 2.49928e26 0.530097 0.265048 0.964235i \(-0.414612\pi\)
0.265048 + 0.964235i \(0.414612\pi\)
\(348\) −5.28505e26 −1.08759
\(349\) 1.05322e26 0.210307 0.105153 0.994456i \(-0.466467\pi\)
0.105153 + 0.994456i \(0.466467\pi\)
\(350\) −9.40880e25 −0.182314
\(351\) 1.74327e26 0.327824
\(352\) −6.87792e25 −0.125533
\(353\) −1.44843e26 −0.256604 −0.128302 0.991735i \(-0.540953\pi\)
−0.128302 + 0.991735i \(0.540953\pi\)
\(354\) 2.17358e26 0.373802
\(355\) −2.99653e26 −0.500289
\(356\) 4.04236e25 0.0655255
\(357\) −2.12433e27 −3.34353
\(358\) −4.50843e26 −0.689055
\(359\) 5.68219e25 0.0843382 0.0421691 0.999110i \(-0.486573\pi\)
0.0421691 + 0.999110i \(0.486573\pi\)
\(360\) −2.88279e26 −0.415563
\(361\) 1.34005e27 1.87627
\(362\) 8.43871e26 1.14772
\(363\) −7.14675e26 −0.944254
\(364\) 5.30774e25 0.0681311
\(365\) 2.30949e26 0.288032
\(366\) 3.18183e25 0.0385590
\(367\) 3.29729e25 0.0388296 0.0194148 0.999812i \(-0.493820\pi\)
0.0194148 + 0.999812i \(0.493820\pi\)
\(368\) 2.71680e26 0.310924
\(369\) 1.84881e27 2.05644
\(370\) 1.35920e26 0.146948
\(371\) 1.39614e27 1.46725
\(372\) −2.77003e26 −0.282998
\(373\) −6.24564e25 −0.0620346 −0.0310173 0.999519i \(-0.509875\pi\)
−0.0310173 + 0.999519i \(0.509875\pi\)
\(374\) 7.07983e26 0.683708
\(375\) 1.81435e26 0.170370
\(376\) −3.40572e26 −0.310983
\(377\) −1.35926e26 −0.120703
\(378\) −3.27359e27 −2.82722
\(379\) 1.65025e27 1.38624 0.693119 0.720823i \(-0.256236\pi\)
0.693119 + 0.720823i \(0.256236\pi\)
\(380\) −4.64117e26 −0.379228
\(381\) −2.73530e27 −2.17417
\(382\) −3.05108e26 −0.235933
\(383\) 1.20420e27 0.905967 0.452983 0.891519i \(-0.350360\pi\)
0.452983 + 0.891519i \(0.350360\pi\)
\(384\) −2.29997e26 −0.168362
\(385\) 5.74767e26 0.409406
\(386\) −9.81446e26 −0.680299
\(387\) −1.01898e27 −0.687387
\(388\) −5.73114e26 −0.376277
\(389\) 1.68483e26 0.107667 0.0538337 0.998550i \(-0.482856\pi\)
0.0538337 + 0.998550i \(0.482856\pi\)
\(390\) −1.02352e26 −0.0636677
\(391\) −2.79655e27 −1.69343
\(392\) −3.96975e26 −0.234024
\(393\) 3.96272e27 2.27442
\(394\) 7.85627e26 0.439042
\(395\) −9.13431e26 −0.497057
\(396\) 1.76104e27 0.933192
\(397\) 1.48612e27 0.766925 0.383462 0.923557i \(-0.374732\pi\)
0.383462 + 0.923557i \(0.374732\pi\)
\(398\) 3.00264e26 0.150915
\(399\) −8.50715e27 −4.16456
\(400\) 1.04858e26 0.0500000
\(401\) −6.25371e26 −0.290484 −0.145242 0.989396i \(-0.546396\pi\)
−0.145242 + 0.989396i \(0.546396\pi\)
\(402\) −1.73808e27 −0.786497
\(403\) −7.12423e25 −0.0314076
\(404\) 1.96149e27 0.842524
\(405\) 3.50424e27 1.46661
\(406\) 2.55248e27 1.04097
\(407\) −8.30310e26 −0.329989
\(408\) 2.36748e27 0.916970
\(409\) 3.35895e27 1.26797 0.633985 0.773345i \(-0.281418\pi\)
0.633985 + 0.773345i \(0.281418\pi\)
\(410\) −6.72480e26 −0.247428
\(411\) −2.14474e26 −0.0769192
\(412\) −9.05427e26 −0.316543
\(413\) −1.04976e27 −0.357777
\(414\) −6.95617e27 −2.31136
\(415\) −1.96147e27 −0.635444
\(416\) −5.91528e25 −0.0186851
\(417\) 8.07668e27 2.48774
\(418\) 2.83521e27 0.851597
\(419\) −1.39454e27 −0.408492 −0.204246 0.978920i \(-0.565474\pi\)
−0.204246 + 0.978920i \(0.565474\pi\)
\(420\) 1.92201e27 0.549084
\(421\) −5.26111e27 −1.46594 −0.732968 0.680263i \(-0.761866\pi\)
−0.732968 + 0.680263i \(0.761866\pi\)
\(422\) 2.10862e27 0.573084
\(423\) 8.72011e27 2.31179
\(424\) −1.55595e27 −0.402396
\(425\) −1.07936e27 −0.272321
\(426\) 6.12126e27 1.50675
\(427\) −1.53670e26 −0.0369060
\(428\) −3.83449e27 −0.898561
\(429\) 6.25252e26 0.142973
\(430\) 3.70641e26 0.0827055
\(431\) −3.99327e27 −0.869595 −0.434797 0.900528i \(-0.643180\pi\)
−0.434797 + 0.900528i \(0.643180\pi\)
\(432\) 3.64829e27 0.775372
\(433\) 1.71947e27 0.356674 0.178337 0.983969i \(-0.442928\pi\)
0.178337 + 0.983969i \(0.442928\pi\)
\(434\) 1.33782e27 0.270866
\(435\) −4.92209e27 −0.972774
\(436\) −6.71030e26 −0.129459
\(437\) −1.11992e28 −2.10926
\(438\) −4.71777e27 −0.867479
\(439\) 2.57641e27 0.462528 0.231264 0.972891i \(-0.425714\pi\)
0.231264 + 0.972891i \(0.425714\pi\)
\(440\) −6.40556e26 −0.112280
\(441\) 1.01643e28 1.73969
\(442\) 6.08893e26 0.101767
\(443\) −1.26523e27 −0.206504 −0.103252 0.994655i \(-0.532925\pi\)
−0.103252 + 0.994655i \(0.532925\pi\)
\(444\) −2.77655e27 −0.442572
\(445\) 3.76475e26 0.0586078
\(446\) 4.05341e27 0.616318
\(447\) 1.87336e28 2.78222
\(448\) 1.11080e27 0.161144
\(449\) 3.20858e26 0.0454702 0.0227351 0.999742i \(-0.492763\pi\)
0.0227351 + 0.999742i \(0.492763\pi\)
\(450\) −2.68481e27 −0.371691
\(451\) 4.10806e27 0.555626
\(452\) 6.47045e26 0.0855029
\(453\) −2.20321e28 −2.84462
\(454\) 6.82881e26 0.0861506
\(455\) 4.94322e26 0.0609383
\(456\) 9.48090e27 1.14214
\(457\) −1.41432e28 −1.66506 −0.832528 0.553983i \(-0.813107\pi\)
−0.832528 + 0.553983i \(0.813107\pi\)
\(458\) 7.40076e27 0.851508
\(459\) −3.75539e28 −4.22300
\(460\) 2.53021e27 0.278099
\(461\) 6.08545e27 0.653782 0.326891 0.945062i \(-0.393999\pi\)
0.326891 + 0.945062i \(0.393999\pi\)
\(462\) −1.17412e28 −1.23303
\(463\) 6.53768e27 0.671155 0.335578 0.942013i \(-0.391069\pi\)
0.335578 + 0.942013i \(0.391069\pi\)
\(464\) −2.84464e27 −0.285488
\(465\) −2.57979e27 −0.253121
\(466\) −1.10180e28 −1.05694
\(467\) −9.77307e27 −0.916651 −0.458325 0.888784i \(-0.651551\pi\)
−0.458325 + 0.888784i \(0.651551\pi\)
\(468\) 1.51457e27 0.138902
\(469\) 8.39428e27 0.752781
\(470\) −3.17182e27 −0.278151
\(471\) 7.32022e27 0.627775
\(472\) 1.16991e27 0.0981212
\(473\) −2.26418e27 −0.185724
\(474\) 1.86594e28 1.49701
\(475\) −4.32243e27 −0.339192
\(476\) −1.14340e28 −0.877661
\(477\) 3.98390e28 2.99134
\(478\) −3.24517e27 −0.238366
\(479\) −1.53019e28 −1.09957 −0.549784 0.835307i \(-0.685290\pi\)
−0.549784 + 0.835307i \(0.685290\pi\)
\(480\) −2.14201e27 −0.150587
\(481\) −7.14100e26 −0.0491174
\(482\) 1.79679e27 0.120921
\(483\) 4.63781e28 3.05400
\(484\) −3.84668e27 −0.247862
\(485\) −5.33754e27 −0.336553
\(486\) −3.60424e28 −2.22399
\(487\) −2.24122e28 −1.35341 −0.676705 0.736254i \(-0.736593\pi\)
−0.676705 + 0.736254i \(0.736593\pi\)
\(488\) 1.71260e26 0.0101215
\(489\) 4.94701e28 2.86154
\(490\) −3.69712e27 −0.209317
\(491\) 1.97103e28 1.09229 0.546145 0.837691i \(-0.316095\pi\)
0.546145 + 0.837691i \(0.316095\pi\)
\(492\) 1.37373e28 0.745191
\(493\) 2.92815e28 1.55489
\(494\) 2.43839e27 0.126757
\(495\) 1.64010e28 0.834672
\(496\) −1.49095e27 −0.0742856
\(497\) −2.95633e28 −1.44215
\(498\) 4.00686e28 1.91380
\(499\) 3.08705e28 1.44374 0.721869 0.692030i \(-0.243283\pi\)
0.721869 + 0.692030i \(0.243283\pi\)
\(500\) 9.76563e26 0.0447214
\(501\) −1.53531e28 −0.688493
\(502\) −1.80089e28 −0.790858
\(503\) −2.52448e28 −1.08570 −0.542848 0.839831i \(-0.682654\pi\)
−0.542848 + 0.839831i \(0.682654\pi\)
\(504\) −2.84412e28 −1.19792
\(505\) 1.82678e28 0.753577
\(506\) −1.54566e28 −0.624502
\(507\) −4.75941e28 −1.88352
\(508\) −1.47226e28 −0.570709
\(509\) 2.42632e28 0.921323 0.460661 0.887576i \(-0.347612\pi\)
0.460661 + 0.887576i \(0.347612\pi\)
\(510\) 2.20489e28 0.820163
\(511\) 2.27850e28 0.830291
\(512\) −1.23794e27 −0.0441942
\(513\) −1.50389e29 −5.25999
\(514\) 1.01198e28 0.346785
\(515\) −8.43244e27 −0.283125
\(516\) −7.57138e27 −0.249088
\(517\) 1.93761e28 0.624619
\(518\) 1.34096e28 0.423599
\(519\) −6.43643e28 −1.99245
\(520\) −5.50904e26 −0.0167125
\(521\) −1.28803e28 −0.382939 −0.191470 0.981499i \(-0.561325\pi\)
−0.191470 + 0.981499i \(0.561325\pi\)
\(522\) 7.28351e28 2.12227
\(523\) 4.72268e28 1.34872 0.674359 0.738404i \(-0.264420\pi\)
0.674359 + 0.738404i \(0.264420\pi\)
\(524\) 2.13290e28 0.597025
\(525\) 1.79001e28 0.491115
\(526\) 1.64239e28 0.441699
\(527\) 1.53471e28 0.404591
\(528\) 1.30852e28 0.338160
\(529\) 2.15825e28 0.546785
\(530\) −1.44909e28 −0.359914
\(531\) −2.99548e28 −0.729415
\(532\) −4.57891e28 −1.09318
\(533\) 3.53309e27 0.0827026
\(534\) −7.69055e27 −0.176512
\(535\) −3.57114e28 −0.803698
\(536\) −9.35511e27 −0.206452
\(537\) 8.57723e28 1.85617
\(538\) −1.80299e28 −0.382631
\(539\) 2.25850e28 0.470044
\(540\) 3.39773e28 0.693513
\(541\) 1.70978e28 0.342270 0.171135 0.985248i \(-0.445257\pi\)
0.171135 + 0.985248i \(0.445257\pi\)
\(542\) −3.53295e28 −0.693658
\(543\) −1.60545e29 −3.09172
\(544\) 1.27428e28 0.240700
\(545\) −6.24945e27 −0.115792
\(546\) −1.00979e28 −0.183531
\(547\) 6.17365e28 1.10072 0.550358 0.834929i \(-0.314491\pi\)
0.550358 + 0.834929i \(0.314491\pi\)
\(548\) −1.15439e27 −0.0201909
\(549\) −4.38499e27 −0.0752417
\(550\) −5.96564e27 −0.100427
\(551\) 1.17262e29 1.93671
\(552\) −5.16867e28 −0.837565
\(553\) −9.01177e28 −1.43284
\(554\) 2.65709e28 0.414528
\(555\) −2.58586e28 −0.395848
\(556\) 4.34721e28 0.653019
\(557\) −1.03162e29 −1.52069 −0.760346 0.649518i \(-0.774971\pi\)
−0.760346 + 0.649518i \(0.774971\pi\)
\(558\) 3.81747e28 0.552225
\(559\) −1.94728e27 −0.0276443
\(560\) 1.03451e28 0.144132
\(561\) −1.34693e29 −1.84177
\(562\) −8.01028e28 −1.07502
\(563\) −3.65428e28 −0.481353 −0.240677 0.970605i \(-0.577369\pi\)
−0.240677 + 0.970605i \(0.577369\pi\)
\(564\) 6.47934e28 0.837722
\(565\) 6.02608e27 0.0764761
\(566\) −8.58283e27 −0.106920
\(567\) 3.45723e29 4.22772
\(568\) 3.29472e28 0.395513
\(569\) 4.54287e28 0.535367 0.267683 0.963507i \(-0.413742\pi\)
0.267683 + 0.963507i \(0.413742\pi\)
\(570\) 8.82978e28 1.02156
\(571\) 1.49843e29 1.70200 0.850998 0.525168i \(-0.175998\pi\)
0.850998 + 0.525168i \(0.175998\pi\)
\(572\) 3.36537e27 0.0375297
\(573\) 5.80464e28 0.635553
\(574\) −6.63458e28 −0.713245
\(575\) 2.35644e28 0.248740
\(576\) 3.16966e28 0.328531
\(577\) −1.44438e29 −1.47006 −0.735029 0.678035i \(-0.762832\pi\)
−0.735029 + 0.678035i \(0.762832\pi\)
\(578\) −6.04187e28 −0.603850
\(579\) 1.86719e29 1.83258
\(580\) −2.64928e28 −0.255349
\(581\) −1.93516e29 −1.83175
\(582\) 1.09034e29 1.01361
\(583\) 8.85223e28 0.808226
\(584\) −2.53931e28 −0.227709
\(585\) 1.41055e28 0.124237
\(586\) −5.09417e28 −0.440706
\(587\) −1.82681e29 −1.55237 −0.776183 0.630508i \(-0.782846\pi\)
−0.776183 + 0.630508i \(0.782846\pi\)
\(588\) 7.55241e28 0.630410
\(589\) 6.14597e28 0.503941
\(590\) 1.08957e28 0.0877623
\(591\) −1.49465e29 −1.18269
\(592\) −1.49446e28 −0.116173
\(593\) 1.28812e29 0.983742 0.491871 0.870668i \(-0.336313\pi\)
0.491871 + 0.870668i \(0.336313\pi\)
\(594\) −2.07561e29 −1.55736
\(595\) −1.06488e29 −0.785004
\(596\) 1.00832e29 0.730320
\(597\) −5.71249e28 −0.406532
\(598\) −1.32933e28 −0.0929545
\(599\) −1.90622e28 −0.130976 −0.0654879 0.997853i \(-0.520860\pi\)
−0.0654879 + 0.997853i \(0.520860\pi\)
\(600\) −1.99490e28 −0.134689
\(601\) −4.02359e28 −0.266951 −0.133476 0.991052i \(-0.542614\pi\)
−0.133476 + 0.991052i \(0.542614\pi\)
\(602\) 3.65668e28 0.238410
\(603\) 2.39531e29 1.53472
\(604\) −1.18586e29 −0.746700
\(605\) −3.58250e28 −0.221695
\(606\) −3.73172e29 −2.26958
\(607\) 1.34619e29 0.804681 0.402340 0.915490i \(-0.368197\pi\)
0.402340 + 0.915490i \(0.368197\pi\)
\(608\) 5.10303e28 0.299806
\(609\) −4.85606e29 −2.80416
\(610\) 1.59498e27 0.00905299
\(611\) 1.66642e28 0.0929719
\(612\) −3.26271e29 −1.78932
\(613\) 6.61541e28 0.356633 0.178316 0.983973i \(-0.442935\pi\)
0.178316 + 0.983973i \(0.442935\pi\)
\(614\) 1.44671e29 0.766677
\(615\) 1.27938e29 0.666519
\(616\) −6.31963e28 −0.323664
\(617\) −5.23241e28 −0.263456 −0.131728 0.991286i \(-0.542052\pi\)
−0.131728 + 0.991286i \(0.542052\pi\)
\(618\) 1.72256e29 0.852700
\(619\) 1.15227e29 0.560793 0.280397 0.959884i \(-0.409534\pi\)
0.280397 + 0.959884i \(0.409534\pi\)
\(620\) −1.38855e28 −0.0664430
\(621\) 8.19873e29 3.85731
\(622\) −1.43262e29 −0.662723
\(623\) 3.71424e28 0.168945
\(624\) 1.12538e28 0.0503337
\(625\) 9.09495e27 0.0400000
\(626\) 2.86259e29 1.23802
\(627\) −5.39395e29 −2.29402
\(628\) 3.94005e28 0.164788
\(629\) 1.53833e29 0.632727
\(630\) −2.64879e29 −1.07145
\(631\) −1.82557e29 −0.726257 −0.363128 0.931739i \(-0.618291\pi\)
−0.363128 + 0.931739i \(0.618291\pi\)
\(632\) 1.00433e29 0.392958
\(633\) −4.01163e29 −1.54377
\(634\) 1.19800e29 0.453438
\(635\) −1.37115e29 −0.510458
\(636\) 2.96017e29 1.08397
\(637\) 1.94240e28 0.0699641
\(638\) 1.61840e29 0.573413
\(639\) −8.43591e29 −2.94017
\(640\) −1.15292e28 −0.0395285
\(641\) 4.06750e29 1.37189 0.685943 0.727655i \(-0.259390\pi\)
0.685943 + 0.727655i \(0.259390\pi\)
\(642\) 7.29506e29 2.42053
\(643\) 1.10344e29 0.360192 0.180096 0.983649i \(-0.442359\pi\)
0.180096 + 0.983649i \(0.442359\pi\)
\(644\) 2.49627e29 0.801659
\(645\) −7.05140e28 −0.222791
\(646\) −5.25283e29 −1.63287
\(647\) 5.65654e29 1.73004 0.865020 0.501737i \(-0.167306\pi\)
0.865020 + 0.501737i \(0.167306\pi\)
\(648\) −3.85295e29 −1.15946
\(649\) −6.65597e28 −0.197080
\(650\) −5.13069e27 −0.0149481
\(651\) −2.54518e29 −0.729656
\(652\) 2.66269e29 0.751140
\(653\) 9.98139e28 0.277078 0.138539 0.990357i \(-0.455759\pi\)
0.138539 + 0.990357i \(0.455759\pi\)
\(654\) 1.27663e29 0.348736
\(655\) 1.98642e29 0.533996
\(656\) 7.39399e28 0.195609
\(657\) 6.50172e29 1.69275
\(658\) −3.12927e29 −0.801809
\(659\) 4.27384e29 1.07776 0.538880 0.842383i \(-0.318848\pi\)
0.538880 + 0.842383i \(0.318848\pi\)
\(660\) 1.21865e29 0.302460
\(661\) 1.06580e29 0.260352 0.130176 0.991491i \(-0.458446\pi\)
0.130176 + 0.991491i \(0.458446\pi\)
\(662\) 3.92834e29 0.944495
\(663\) −1.15841e29 −0.274139
\(664\) 2.15666e29 0.502362
\(665\) −4.26444e29 −0.977766
\(666\) 3.82645e29 0.863608
\(667\) −6.39271e29 −1.42025
\(668\) −8.26369e28 −0.180726
\(669\) −7.71157e29 −1.66023
\(670\) −8.71262e28 −0.184656
\(671\) −9.74344e27 −0.0203295
\(672\) −2.11327e29 −0.434089
\(673\) 9.20636e29 1.86179 0.930893 0.365292i \(-0.119031\pi\)
0.930893 + 0.365292i \(0.119031\pi\)
\(674\) −1.39385e29 −0.277515
\(675\) 3.16439e29 0.620297
\(676\) −2.56172e29 −0.494414
\(677\) 1.03701e29 0.197062 0.0985309 0.995134i \(-0.468586\pi\)
0.0985309 + 0.995134i \(0.468586\pi\)
\(678\) −1.23100e29 −0.230327
\(679\) −5.26593e29 −0.970159
\(680\) 1.18677e29 0.215289
\(681\) −1.29917e29 −0.232072
\(682\) 8.48241e28 0.149205
\(683\) −1.05066e30 −1.81988 −0.909942 0.414735i \(-0.863874\pi\)
−0.909942 + 0.414735i \(0.863874\pi\)
\(684\) −1.30660e30 −2.22870
\(685\) −1.07511e28 −0.0180593
\(686\) 1.86301e29 0.308185
\(687\) −1.40799e30 −2.29378
\(688\) −4.07524e28 −0.0653844
\(689\) 7.61327e28 0.120301
\(690\) −4.81370e29 −0.749141
\(691\) −2.09599e29 −0.321270 −0.160635 0.987014i \(-0.551354\pi\)
−0.160635 + 0.987014i \(0.551354\pi\)
\(692\) −3.46436e29 −0.523009
\(693\) 1.61810e30 2.40606
\(694\) −2.55926e29 −0.374835
\(695\) 4.04866e29 0.584078
\(696\) 5.41189e29 0.769046
\(697\) −7.61105e29 −1.06537
\(698\) −1.07850e29 −0.148709
\(699\) 2.09616e30 2.84717
\(700\) 9.63461e28 0.128915
\(701\) −9.09765e29 −1.19920 −0.599598 0.800301i \(-0.704673\pi\)
−0.599598 + 0.800301i \(0.704673\pi\)
\(702\) −1.78511e29 −0.231806
\(703\) 6.16043e29 0.788098
\(704\) 7.04299e28 0.0887655
\(705\) 6.03435e29 0.749281
\(706\) 1.48319e29 0.181446
\(707\) 1.80228e30 2.17229
\(708\) −2.22575e29 −0.264318
\(709\) −1.17415e30 −1.37385 −0.686925 0.726728i \(-0.741040\pi\)
−0.686925 + 0.726728i \(0.741040\pi\)
\(710\) 3.06845e29 0.353758
\(711\) −2.57151e30 −2.92118
\(712\) −4.13938e28 −0.0463335
\(713\) −3.35057e29 −0.369555
\(714\) 2.17531e30 2.36423
\(715\) 3.13425e28 0.0335675
\(716\) 4.61663e29 0.487235
\(717\) 6.17390e29 0.642108
\(718\) −5.81857e28 −0.0596361
\(719\) 2.38367e29 0.240765 0.120382 0.992728i \(-0.461588\pi\)
0.120382 + 0.992728i \(0.461588\pi\)
\(720\) 2.95198e29 0.293847
\(721\) −8.31932e29 −0.816145
\(722\) −1.37221e30 −1.32673
\(723\) −3.41837e29 −0.325737
\(724\) −8.64124e29 −0.811561
\(725\) −2.46733e29 −0.228391
\(726\) 7.31827e29 0.667688
\(727\) 1.60120e30 1.43991 0.719954 0.694022i \(-0.244163\pi\)
0.719954 + 0.694022i \(0.244163\pi\)
\(728\) −5.43513e28 −0.0481760
\(729\) 3.10350e30 2.71152
\(730\) −2.36491e29 −0.203669
\(731\) 4.19487e29 0.356112
\(732\) −3.25819e28 −0.0272653
\(733\) −8.02614e28 −0.0662087 −0.0331043 0.999452i \(-0.510539\pi\)
−0.0331043 + 0.999452i \(0.510539\pi\)
\(734\) −3.37642e28 −0.0274567
\(735\) 7.03373e29 0.563856
\(736\) −2.78200e29 −0.219857
\(737\) 5.32238e29 0.414665
\(738\) −1.89318e30 −1.45412
\(739\) 2.07121e30 1.56840 0.784200 0.620508i \(-0.213073\pi\)
0.784200 + 0.620508i \(0.213073\pi\)
\(740\) −1.39182e29 −0.103908
\(741\) −4.63901e29 −0.341456
\(742\) −1.42965e30 −1.03750
\(743\) −2.00787e30 −1.43666 −0.718329 0.695704i \(-0.755093\pi\)
−0.718329 + 0.695704i \(0.755093\pi\)
\(744\) 2.83651e29 0.200110
\(745\) 9.39072e29 0.653218
\(746\) 6.39554e28 0.0438651
\(747\) −5.52199e30 −3.73447
\(748\) −7.24974e29 −0.483455
\(749\) −3.52324e30 −2.31677
\(750\) −1.85790e29 −0.120470
\(751\) 2.68196e30 1.71488 0.857440 0.514584i \(-0.172054\pi\)
0.857440 + 0.514584i \(0.172054\pi\)
\(752\) 3.48745e29 0.219898
\(753\) 3.42617e30 2.13040
\(754\) 1.39188e29 0.0853501
\(755\) −1.10442e30 −0.667869
\(756\) 3.35215e30 1.99915
\(757\) 2.29494e30 1.34979 0.674894 0.737915i \(-0.264189\pi\)
0.674894 + 0.737915i \(0.264189\pi\)
\(758\) −1.68986e30 −0.980219
\(759\) 2.94060e30 1.68228
\(760\) 4.75256e29 0.268155
\(761\) 3.17369e29 0.176614 0.0883072 0.996093i \(-0.471854\pi\)
0.0883072 + 0.996093i \(0.471854\pi\)
\(762\) 2.80095e30 1.53737
\(763\) −6.16561e29 −0.333786
\(764\) 3.12430e29 0.166830
\(765\) −3.03863e30 −1.60042
\(766\) −1.23310e30 −0.640615
\(767\) −5.72440e28 −0.0293345
\(768\) 2.35517e29 0.119050
\(769\) 3.31592e30 1.65340 0.826699 0.562644i \(-0.190216\pi\)
0.826699 + 0.562644i \(0.190216\pi\)
\(770\) −5.88561e29 −0.289494
\(771\) −1.92528e30 −0.934166
\(772\) 1.00500e30 0.481044
\(773\) −2.86833e29 −0.135439 −0.0677196 0.997704i \(-0.521572\pi\)
−0.0677196 + 0.997704i \(0.521572\pi\)
\(774\) 1.04344e30 0.486056
\(775\) −1.29319e29 −0.0594284
\(776\) 5.86869e29 0.266068
\(777\) −2.55117e30 −1.14109
\(778\) −1.72526e29 −0.0761323
\(779\) −3.04795e30 −1.32698
\(780\) 1.04809e29 0.0450199
\(781\) −1.87446e30 −0.794402
\(782\) 2.86367e30 1.19743
\(783\) −8.58454e30 −3.54175
\(784\) 4.06503e29 0.165480
\(785\) 3.66946e29 0.147391
\(786\) −4.05782e30 −1.60826
\(787\) 1.37451e30 0.537544 0.268772 0.963204i \(-0.413382\pi\)
0.268772 + 0.963204i \(0.413382\pi\)
\(788\) −8.04482e29 −0.310450
\(789\) −3.12463e30 −1.18984
\(790\) 9.35354e29 0.351473
\(791\) 5.94523e29 0.220453
\(792\) −1.80331e30 −0.659866
\(793\) −8.37975e27 −0.00302596
\(794\) −1.52178e30 −0.542298
\(795\) 2.75688e30 0.969533
\(796\) −3.07471e29 −0.106713
\(797\) −4.30808e30 −1.47561 −0.737805 0.675014i \(-0.764138\pi\)
−0.737805 + 0.675014i \(0.764138\pi\)
\(798\) 8.71132e30 2.94479
\(799\) −3.58983e30 −1.19766
\(800\) −1.07374e29 −0.0353553
\(801\) 1.05986e30 0.344435
\(802\) 6.40380e29 0.205403
\(803\) 1.44468e30 0.457361
\(804\) 1.77980e30 0.556137
\(805\) 2.32483e30 0.717026
\(806\) 7.29521e28 0.0222085
\(807\) 3.43017e30 1.03073
\(808\) −2.00857e30 −0.595755
\(809\) −8.36783e29 −0.244993 −0.122496 0.992469i \(-0.539090\pi\)
−0.122496 + 0.992469i \(0.539090\pi\)
\(810\) −3.58834e30 −1.03705
\(811\) 3.30696e30 0.943432 0.471716 0.881751i \(-0.343635\pi\)
0.471716 + 0.881751i \(0.343635\pi\)
\(812\) −2.61374e30 −0.736077
\(813\) 6.72140e30 1.86857
\(814\) 8.50237e29 0.233337
\(815\) 2.47982e30 0.671840
\(816\) −2.42430e30 −0.648396
\(817\) 1.67989e30 0.443557
\(818\) −3.43957e30 −0.896591
\(819\) 1.39163e30 0.358131
\(820\) 6.88619e29 0.174958
\(821\) 7.58969e30 1.90380 0.951899 0.306413i \(-0.0991290\pi\)
0.951899 + 0.306413i \(0.0991290\pi\)
\(822\) 2.19621e29 0.0543901
\(823\) 5.61508e30 1.37296 0.686479 0.727149i \(-0.259155\pi\)
0.686479 + 0.727149i \(0.259155\pi\)
\(824\) 9.27157e29 0.223830
\(825\) 1.13496e30 0.270528
\(826\) 1.07495e30 0.252987
\(827\) 5.39629e30 1.25397 0.626985 0.779031i \(-0.284289\pi\)
0.626985 + 0.779031i \(0.284289\pi\)
\(828\) 7.12312e30 1.63437
\(829\) −4.47440e30 −1.01371 −0.506853 0.862032i \(-0.669191\pi\)
−0.506853 + 0.862032i \(0.669191\pi\)
\(830\) 2.00855e30 0.449327
\(831\) −5.05508e30 −1.11665
\(832\) 6.05725e28 0.0132124
\(833\) −4.18436e30 −0.901273
\(834\) −8.27052e30 −1.75910
\(835\) −7.69616e29 −0.161646
\(836\) −2.90325e30 −0.602170
\(837\) −4.49937e30 −0.921582
\(838\) 1.42801e30 0.288847
\(839\) −1.16605e30 −0.232925 −0.116462 0.993195i \(-0.537155\pi\)
−0.116462 + 0.993195i \(0.537155\pi\)
\(840\) −1.96814e30 −0.388261
\(841\) 1.56068e30 0.304058
\(842\) 5.38738e30 1.03657
\(843\) 1.52395e31 2.89588
\(844\) −2.15923e30 −0.405232
\(845\) −2.38578e30 −0.442217
\(846\) −8.92939e30 −1.63468
\(847\) −3.53444e30 −0.639065
\(848\) 1.59329e30 0.284537
\(849\) 1.63287e30 0.288019
\(850\) 1.10526e30 0.192560
\(851\) −3.35846e30 −0.577936
\(852\) −6.26817e30 −1.06543
\(853\) −1.08909e31 −1.82852 −0.914262 0.405123i \(-0.867229\pi\)
−0.914262 + 0.405123i \(0.867229\pi\)
\(854\) 1.57358e29 0.0260965
\(855\) −1.21686e31 −1.99341
\(856\) 3.92651e30 0.635379
\(857\) 1.32603e30 0.211960 0.105980 0.994368i \(-0.466202\pi\)
0.105980 + 0.994368i \(0.466202\pi\)
\(858\) −6.40258e29 −0.101097
\(859\) −6.84458e30 −1.06762 −0.533812 0.845603i \(-0.679241\pi\)
−0.533812 + 0.845603i \(0.679241\pi\)
\(860\) −3.79536e29 −0.0584816
\(861\) 1.26222e31 1.92133
\(862\) 4.08910e30 0.614896
\(863\) 3.45729e30 0.513597 0.256799 0.966465i \(-0.417332\pi\)
0.256799 + 0.966465i \(0.417332\pi\)
\(864\) −3.73585e30 −0.548270
\(865\) −3.22644e30 −0.467793
\(866\) −1.76074e30 −0.252206
\(867\) 1.14946e31 1.62664
\(868\) −1.36992e30 −0.191531
\(869\) −5.71391e30 −0.789270
\(870\) 5.04022e30 0.687855
\(871\) 4.57746e29 0.0617212
\(872\) 6.87135e29 0.0915416
\(873\) −1.50264e31 −1.97790
\(874\) 1.14679e31 1.49147
\(875\) 8.97293e29 0.115305
\(876\) 4.83100e30 0.613400
\(877\) −1.37025e30 −0.171912 −0.0859558 0.996299i \(-0.527394\pi\)
−0.0859558 + 0.996299i \(0.527394\pi\)
\(878\) −2.63825e30 −0.327057
\(879\) 9.69160e30 1.18717
\(880\) 6.55929e29 0.0793942
\(881\) −3.75611e30 −0.449253 −0.224627 0.974445i \(-0.572116\pi\)
−0.224627 + 0.974445i \(0.572116\pi\)
\(882\) −1.04082e31 −1.23015
\(883\) 6.64376e30 0.775937 0.387968 0.921673i \(-0.373177\pi\)
0.387968 + 0.921673i \(0.373177\pi\)
\(884\) −6.23507e29 −0.0719602
\(885\) −2.07289e30 −0.236413
\(886\) 1.29559e30 0.146021
\(887\) −6.01786e30 −0.670261 −0.335131 0.942172i \(-0.608780\pi\)
−0.335131 + 0.942172i \(0.608780\pi\)
\(888\) 2.84318e30 0.312945
\(889\) −1.35275e31 −1.47146
\(890\) −3.85510e29 −0.0414420
\(891\) 2.19205e31 2.32882
\(892\) −4.15070e30 −0.435803
\(893\) −1.43760e31 −1.49175
\(894\) −1.91832e31 −1.96733
\(895\) 4.29957e30 0.435796
\(896\) −1.13745e30 −0.113946
\(897\) 2.52903e30 0.250400
\(898\) −3.28559e29 −0.0321523
\(899\) 3.50824e30 0.339323
\(900\) 2.74924e30 0.262825
\(901\) −1.64006e31 −1.54971
\(902\) −4.20665e30 −0.392887
\(903\) −6.95680e30 −0.642226
\(904\) −6.62574e29 −0.0604597
\(905\) −8.04778e30 −0.725882
\(906\) 2.25609e31 2.01145
\(907\) 1.42699e31 1.25761 0.628804 0.777564i \(-0.283545\pi\)
0.628804 + 0.777564i \(0.283545\pi\)
\(908\) −6.99270e29 −0.0609177
\(909\) 5.14280e31 4.42873
\(910\) −5.06186e29 −0.0430899
\(911\) 1.18912e31 1.00065 0.500326 0.865837i \(-0.333214\pi\)
0.500326 + 0.865837i \(0.333214\pi\)
\(912\) −9.70844e30 −0.807614
\(913\) −1.22699e31 −1.00901
\(914\) 1.44827e31 1.17737
\(915\) −3.03443e29 −0.0243868
\(916\) −7.57838e30 −0.602107
\(917\) 1.95977e31 1.53932
\(918\) 3.84552e31 2.98612
\(919\) −3.13990e30 −0.241047 −0.120524 0.992710i \(-0.538457\pi\)
−0.120524 + 0.992710i \(0.538457\pi\)
\(920\) −2.59094e30 −0.196646
\(921\) −2.75234e31 −2.06527
\(922\) −6.23150e30 −0.462294
\(923\) −1.61211e30 −0.118243
\(924\) 1.20230e31 0.871882
\(925\) −1.29623e30 −0.0929383
\(926\) −6.69458e30 −0.474578
\(927\) −2.37392e31 −1.66391
\(928\) 2.91291e30 0.201871
\(929\) 1.39588e30 0.0956498 0.0478249 0.998856i \(-0.484771\pi\)
0.0478249 + 0.998856i \(0.484771\pi\)
\(930\) 2.64170e30 0.178983
\(931\) −1.67568e31 −1.12259
\(932\) 1.12824e31 0.747368
\(933\) 2.72554e31 1.78524
\(934\) 1.00076e31 0.648170
\(935\) −6.75185e30 −0.432415
\(936\) −1.55092e30 −0.0982183
\(937\) −2.22831e31 −1.39543 −0.697717 0.716374i \(-0.745801\pi\)
−0.697717 + 0.716374i \(0.745801\pi\)
\(938\) −8.59574e30 −0.532296
\(939\) −5.44604e31 −3.33497
\(940\) 3.24795e30 0.196683
\(941\) 2.09348e31 1.25366 0.626828 0.779157i \(-0.284353\pi\)
0.626828 + 0.779157i \(0.284353\pi\)
\(942\) −7.49590e30 −0.443904
\(943\) 1.66164e31 0.973114
\(944\) −1.19799e30 −0.0693822
\(945\) 3.12193e31 1.78809
\(946\) 2.31852e30 0.131327
\(947\) 2.96424e31 1.66050 0.830249 0.557393i \(-0.188198\pi\)
0.830249 + 0.557393i \(0.188198\pi\)
\(948\) −1.91072e31 −1.05855
\(949\) 1.24248e30 0.0680763
\(950\) 4.42617e30 0.239845
\(951\) −2.27917e31 −1.22147
\(952\) 1.17085e31 0.620600
\(953\) 3.78184e30 0.198257 0.0991283 0.995075i \(-0.468395\pi\)
0.0991283 + 0.995075i \(0.468395\pi\)
\(954\) −4.07952e31 −2.11520
\(955\) 2.90973e30 0.149217
\(956\) 3.32305e30 0.168550
\(957\) −3.07898e31 −1.54465
\(958\) 1.56691e31 0.777512
\(959\) −1.06069e30 −0.0520584
\(960\) 2.19342e30 0.106481
\(961\) −1.89867e31 −0.911707
\(962\) 7.31238e29 0.0347313
\(963\) −1.00536e32 −4.72329
\(964\) −1.83991e30 −0.0855043
\(965\) 9.35980e30 0.430259
\(966\) −4.74912e31 −2.15950
\(967\) −2.27343e31 −1.02259 −0.511297 0.859404i \(-0.670835\pi\)
−0.511297 + 0.859404i \(0.670835\pi\)
\(968\) 3.93900e30 0.175265
\(969\) 9.99344e31 4.39861
\(970\) 5.46564e30 0.237979
\(971\) 3.59388e31 1.54797 0.773983 0.633206i \(-0.218261\pi\)
0.773983 + 0.633206i \(0.218261\pi\)
\(972\) 3.69074e31 1.57260
\(973\) 3.99434e31 1.68368
\(974\) 2.29500e31 0.957006
\(975\) 9.76107e29 0.0402670
\(976\) −1.75370e29 −0.00715702
\(977\) 8.76227e30 0.353772 0.176886 0.984231i \(-0.443398\pi\)
0.176886 + 0.984231i \(0.443398\pi\)
\(978\) −5.06574e31 −2.02341
\(979\) 2.35501e30 0.0930625
\(980\) 3.78585e30 0.148010
\(981\) −1.75936e31 −0.680504
\(982\) −2.01834e31 −0.772365
\(983\) 3.99246e31 1.51157 0.755786 0.654819i \(-0.227255\pi\)
0.755786 + 0.654819i \(0.227255\pi\)
\(984\) −1.40670e31 −0.526929
\(985\) −7.49232e30 −0.277675
\(986\) −2.99842e31 −1.09947
\(987\) 5.95340e31 2.15991
\(988\) −2.49691e30 −0.0896305
\(989\) −9.15821e30 −0.325274
\(990\) −1.67946e31 −0.590202
\(991\) 4.03797e31 1.40407 0.702037 0.712141i \(-0.252274\pi\)
0.702037 + 0.712141i \(0.252274\pi\)
\(992\) 1.52673e30 0.0525278
\(993\) −7.47361e31 −2.54427
\(994\) 3.02728e31 1.01976
\(995\) −2.86354e30 −0.0954468
\(996\) −4.10302e31 −1.35326
\(997\) 1.29884e31 0.423894 0.211947 0.977281i \(-0.432020\pi\)
0.211947 + 0.977281i \(0.432020\pi\)
\(998\) −3.16114e31 −1.02088
\(999\) −4.50996e31 −1.44124
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 10.22.a.c.1.2 2
4.3 odd 2 80.22.a.b.1.1 2
5.2 odd 4 50.22.b.d.49.1 4
5.3 odd 4 50.22.b.d.49.4 4
5.4 even 2 50.22.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.2 2 1.1 even 1 trivial
50.22.a.e.1.1 2 5.4 even 2
50.22.b.d.49.1 4 5.2 odd 4
50.22.b.d.49.4 4 5.3 odd 4
80.22.a.b.1.1 2 4.3 odd 2