Properties

Label 50.22.b.d.49.1
Level $50$
Weight $22$
Character 50.49
Analytic conductor $139.739$
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] = [50,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
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: \(139.738672144\)
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: \( x^{4} + 237265x^{2} + 14073551424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(344.930i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.d.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} -194815. i q^{3} -1.04858e6 q^{4} -1.99490e8 q^{6} +9.63461e8i q^{7} +1.07374e9i q^{8} -2.74924e10 q^{9} +6.10882e10 q^{11} +2.04278e11i q^{12} -5.25383e10i q^{13} +9.86584e11 q^{14} +1.09951e12 q^{16} -1.13179e13i q^{17} +2.81522e13i q^{18} +4.53240e13 q^{19} +1.87696e14 q^{21} -6.25543e13i q^{22} -2.47091e14i q^{23} +2.09181e14 q^{24} -5.37992e13 q^{26} +3.31810e15i q^{27} -1.01026e15i q^{28} +2.58718e15 q^{29} -1.35601e15 q^{31} -1.12590e15i q^{32} -1.19009e16i q^{33} -1.15895e16 q^{34} +2.88279e16 q^{36} -1.35920e16i q^{37} -4.64117e16i q^{38} -1.02352e16 q^{39} +6.72480e16 q^{41} -1.92201e17i q^{42} +3.70641e16i q^{43} -6.40556e16 q^{44} -2.53021e17 q^{46} +3.17182e17i q^{47} -2.14201e17i q^{48} -3.69712e17 q^{49} -2.20489e18 q^{51} +5.50904e16i q^{52} -1.44909e18i q^{53} +3.39773e18 q^{54} -1.03451e18 q^{56} -8.82978e18i q^{57} -2.64928e18i q^{58} +1.08957e18 q^{59} -1.59498e17 q^{61} +1.38855e18i q^{62} -2.64879e19i q^{63} -1.15292e18 q^{64} -1.21865e19 q^{66} +8.71262e18i q^{67} +1.18677e19i q^{68} -4.81370e19 q^{69} -3.06845e19 q^{71} -2.95198e19i q^{72} -2.36491e19i q^{73} -1.39182e19 q^{74} -4.75256e19 q^{76} +5.88561e19i q^{77} +1.04809e19i q^{78} +9.35354e19 q^{79} +3.58834e20 q^{81} -6.88619e19i q^{82} +2.00855e20i q^{83} -1.96814e20 q^{84} +3.79536e19 q^{86} -5.04022e20i q^{87} +6.55929e19i q^{88} -3.85510e19 q^{89} +5.06186e19 q^{91} +2.59094e20i q^{92} +2.64170e20i q^{93} +3.24795e20 q^{94} -2.19342e20 q^{96} -5.46564e20i q^{97} +3.78585e20i q^{98} -1.67946e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} - 205430784 q^{6} - 51927197652 q^{9} + 43738388448 q^{11} + 2721577607168 q^{14} + 4398046511104 q^{16} + 104145722023120 q^{19} + 306321025902528 q^{21} + 215409789763584 q^{24} + 843899164205056 q^{26}+ \cdots - 34\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1024.00i − 0.707107i
\(3\) − 194815.i − 1.90480i −0.304856 0.952398i \(-0.598608\pi\)
0.304856 0.952398i \(-0.401392\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −1.99490e8 −1.34689
\(7\) 9.63461e8i 1.28915i 0.764539 + 0.644577i \(0.222967\pi\)
−0.764539 + 0.644577i \(0.777033\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −2.74924e10 −2.62825
\(10\) 0 0
\(11\) 6.10882e10 0.710124 0.355062 0.934843i \(-0.384460\pi\)
0.355062 + 0.934843i \(0.384460\pi\)
\(12\) 2.04278e11i 0.952398i
\(13\) − 5.25383e10i − 0.105699i −0.998602 0.0528495i \(-0.983170\pi\)
0.998602 0.0528495i \(-0.0168303\pi\)
\(14\) 9.86584e11 0.911570
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 1.13179e13i − 1.36161i −0.732466 0.680803i \(-0.761631\pi\)
0.732466 0.680803i \(-0.238369\pi\)
\(18\) 2.81522e13i 1.85845i
\(19\) 4.53240e13 1.69596 0.847979 0.530030i \(-0.177819\pi\)
0.847979 + 0.530030i \(0.177819\pi\)
\(20\) 0 0
\(21\) 1.87696e14 2.45558
\(22\) − 6.25543e13i − 0.502133i
\(23\) − 2.47091e14i − 1.24370i −0.783137 0.621849i \(-0.786382\pi\)
0.783137 0.621849i \(-0.213618\pi\)
\(24\) 2.09181e14 0.673447
\(25\) 0 0
\(26\) −5.37992e13 −0.0747404
\(27\) 3.31810e15i 3.10149i
\(28\) − 1.01026e15i − 0.644577i
\(29\) 2.58718e15 1.14195 0.570977 0.820966i \(-0.306565\pi\)
0.570977 + 0.820966i \(0.306565\pi\)
\(30\) 0 0
\(31\) −1.35601e15 −0.297142 −0.148571 0.988902i \(-0.547467\pi\)
−0.148571 + 0.988902i \(0.547467\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 1.19009e16i − 1.35264i
\(34\) −1.15895e16 −0.962801
\(35\) 0 0
\(36\) 2.88279e16 1.31413
\(37\) − 1.35920e16i − 0.464692i −0.972633 0.232346i \(-0.925360\pi\)
0.972633 0.232346i \(-0.0746401\pi\)
\(38\) − 4.64117e16i − 1.19922i
\(39\) −1.02352e16 −0.201335
\(40\) 0 0
\(41\) 6.72480e16 0.782436 0.391218 0.920298i \(-0.372054\pi\)
0.391218 + 0.920298i \(0.372054\pi\)
\(42\) − 1.92201e17i − 1.73636i
\(43\) 3.70641e16i 0.261538i 0.991413 + 0.130769i \(0.0417446\pi\)
−0.991413 + 0.130769i \(0.958255\pi\)
\(44\) −6.40556e16 −0.355062
\(45\) 0 0
\(46\) −2.53021e17 −0.879427
\(47\) 3.17182e17i 0.879592i 0.898098 + 0.439796i \(0.144949\pi\)
−0.898098 + 0.439796i \(0.855051\pi\)
\(48\) − 2.14201e17i − 0.476199i
\(49\) −3.69712e17 −0.661919
\(50\) 0 0
\(51\) −2.20489e18 −2.59358
\(52\) 5.50904e16i 0.0528495i
\(53\) − 1.44909e18i − 1.13815i −0.822286 0.569074i \(-0.807302\pi\)
0.822286 0.569074i \(-0.192698\pi\)
\(54\) 3.39773e18 2.19308
\(55\) 0 0
\(56\) −1.03451e18 −0.455785
\(57\) − 8.82978e18i − 3.23046i
\(58\) − 2.64928e18i − 0.807483i
\(59\) 1.08957e18 0.277529 0.138764 0.990325i \(-0.455687\pi\)
0.138764 + 0.990325i \(0.455687\pi\)
\(60\) 0 0
\(61\) −1.59498e17 −0.0286281 −0.0143140 0.999898i \(-0.504556\pi\)
−0.0143140 + 0.999898i \(0.504556\pi\)
\(62\) 1.38855e18i 0.210111i
\(63\) − 2.64879e19i − 3.38822i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −1.21865e19 −0.956462
\(67\) 8.71262e18i 0.583934i 0.956428 + 0.291967i \(0.0943097\pi\)
−0.956428 + 0.291967i \(0.905690\pi\)
\(68\) 1.18677e19i 0.680803i
\(69\) −4.81370e19 −2.36899
\(70\) 0 0
\(71\) −3.06845e19 −1.11868 −0.559340 0.828938i \(-0.688946\pi\)
−0.559340 + 0.828938i \(0.688946\pi\)
\(72\) − 2.95198e19i − 0.929227i
\(73\) − 2.36491e19i − 0.644059i −0.946730 0.322029i \(-0.895635\pi\)
0.946730 0.322029i \(-0.104365\pi\)
\(74\) −1.39182e19 −0.328587
\(75\) 0 0
\(76\) −4.75256e19 −0.847979
\(77\) 5.88561e19i 0.915459i
\(78\) 1.04809e19i 0.142365i
\(79\) 9.35354e19 1.11145 0.555727 0.831365i \(-0.312440\pi\)
0.555727 + 0.831365i \(0.312440\pi\)
\(80\) 0 0
\(81\) 3.58834e20 3.27945
\(82\) − 6.88619e19i − 0.553266i
\(83\) 2.00855e20i 1.42090i 0.703750 + 0.710448i \(0.251508\pi\)
−0.703750 + 0.710448i \(0.748492\pi\)
\(84\) −1.96814e20 −1.22779
\(85\) 0 0
\(86\) 3.79536e19 0.184935
\(87\) − 5.04022e20i − 2.17519i
\(88\) 6.55929e19i 0.251067i
\(89\) −3.85510e19 −0.131051 −0.0655255 0.997851i \(-0.520872\pi\)
−0.0655255 + 0.997851i \(0.520872\pi\)
\(90\) 0 0
\(91\) 5.06186e19 0.136262
\(92\) 2.59094e20i 0.621849i
\(93\) 2.64170e20i 0.565996i
\(94\) 3.24795e20 0.621965
\(95\) 0 0
\(96\) −2.19342e20 −0.336724
\(97\) − 5.46564e20i − 0.752554i −0.926507 0.376277i \(-0.877204\pi\)
0.926507 0.376277i \(-0.122796\pi\)
\(98\) 3.78585e20i 0.468047i
\(99\) −1.67946e21 −1.86638
\(100\) 0 0
\(101\) 1.87063e21 1.68505 0.842524 0.538658i \(-0.181069\pi\)
0.842524 + 0.538658i \(0.181069\pi\)
\(102\) 2.25781e21i 1.83394i
\(103\) 8.63482e20i 0.633086i 0.948578 + 0.316543i \(0.102522\pi\)
−0.948578 + 0.316543i \(0.897478\pi\)
\(104\) 5.64125e19 0.0373702
\(105\) 0 0
\(106\) −1.48387e21 −0.804792
\(107\) − 3.65685e21i − 1.79712i −0.438848 0.898561i \(-0.644613\pi\)
0.438848 0.898561i \(-0.355387\pi\)
\(108\) − 3.47928e21i − 1.55074i
\(109\) 6.39944e20 0.258919 0.129459 0.991585i \(-0.458676\pi\)
0.129459 + 0.991585i \(0.458676\pi\)
\(110\) 0 0
\(111\) −2.64792e21 −0.885143
\(112\) 1.05934e21i 0.322289i
\(113\) − 6.17070e20i − 0.171006i −0.996338 0.0855029i \(-0.972750\pi\)
0.996338 0.0855029i \(-0.0272497\pi\)
\(114\) −9.04169e21 −2.28428
\(115\) 0 0
\(116\) −2.71286e21 −0.570977
\(117\) 1.44440e21i 0.277803i
\(118\) − 1.11572e21i − 0.196242i
\(119\) 1.09043e22 1.75532
\(120\) 0 0
\(121\) −3.66848e21 −0.495724
\(122\) 1.63326e20i 0.0202431i
\(123\) − 1.31009e22i − 1.49038i
\(124\) 1.42188e21 0.148571
\(125\) 0 0
\(126\) −2.71236e22 −2.39583
\(127\) − 1.40405e22i − 1.14142i −0.821152 0.570709i \(-0.806668\pi\)
0.821152 0.570709i \(-0.193332\pi\)
\(128\) 1.18059e21i 0.0883883i
\(129\) 7.22063e21 0.498176
\(130\) 0 0
\(131\) 2.03410e22 1.19405 0.597025 0.802222i \(-0.296349\pi\)
0.597025 + 0.802222i \(0.296349\pi\)
\(132\) 1.24790e22i 0.676321i
\(133\) 4.36679e22i 2.18635i
\(134\) 8.92173e21 0.412903
\(135\) 0 0
\(136\) 1.21525e22 0.481401
\(137\) − 1.10091e21i − 0.0403819i −0.999796 0.0201909i \(-0.993573\pi\)
0.999796 0.0201909i \(-0.00642741\pi\)
\(138\) 4.92923e22i 1.67513i
\(139\) −4.14583e22 −1.30604 −0.653019 0.757342i \(-0.726498\pi\)
−0.653019 + 0.757342i \(0.726498\pi\)
\(140\) 0 0
\(141\) 6.17918e22 1.67544
\(142\) 3.14209e22i 0.791027i
\(143\) − 3.20947e21i − 0.0750593i
\(144\) −3.02282e22 −0.657063
\(145\) 0 0
\(146\) −2.42167e22 −0.455418
\(147\) 7.20253e22i 1.26082i
\(148\) 1.42522e22i 0.232346i
\(149\) −9.61610e22 −1.46064 −0.730320 0.683106i \(-0.760629\pi\)
−0.730320 + 0.683106i \(0.760629\pi\)
\(150\) 0 0
\(151\) −1.13093e23 −1.49340 −0.746700 0.665161i \(-0.768363\pi\)
−0.746700 + 0.665161i \(0.768363\pi\)
\(152\) 4.86662e22i 0.599612i
\(153\) 3.11156e23i 3.57864i
\(154\) 6.02687e22 0.647327
\(155\) 0 0
\(156\) 1.07324e22 0.100667
\(157\) 3.75753e22i 0.329576i 0.986329 + 0.164788i \(0.0526940\pi\)
−0.986329 + 0.164788i \(0.947306\pi\)
\(158\) − 9.57802e22i − 0.785917i
\(159\) −2.82304e23 −2.16794
\(160\) 0 0
\(161\) 2.38063e23 1.60332
\(162\) − 3.67446e23i − 2.31892i
\(163\) − 2.53934e23i − 1.50228i −0.660143 0.751140i \(-0.729504\pi\)
0.660143 0.751140i \(-0.270496\pi\)
\(164\) −7.05146e22 −0.391218
\(165\) 0 0
\(166\) 2.05675e23 1.00472
\(167\) − 7.88087e22i − 0.361452i −0.983533 0.180726i \(-0.942155\pi\)
0.983533 0.180726i \(-0.0578448\pi\)
\(168\) 2.01538e23i 0.868178i
\(169\) 2.44304e23 0.988828
\(170\) 0 0
\(171\) −1.24607e24 −4.45740
\(172\) − 3.88645e22i − 0.130769i
\(173\) 3.30387e23i 1.04602i 0.852327 + 0.523009i \(0.175191\pi\)
−0.852327 + 0.523009i \(0.824809\pi\)
\(174\) −5.16118e23 −1.53809
\(175\) 0 0
\(176\) 6.71672e22 0.177531
\(177\) − 2.12264e23i − 0.528636i
\(178\) 3.94762e22i 0.0926671i
\(179\) −4.40276e23 −0.974470 −0.487235 0.873271i \(-0.661995\pi\)
−0.487235 + 0.873271i \(0.661995\pi\)
\(180\) 0 0
\(181\) −8.24093e23 −1.62312 −0.811561 0.584268i \(-0.801382\pi\)
−0.811561 + 0.584268i \(0.801382\pi\)
\(182\) − 5.18334e22i − 0.0963519i
\(183\) 3.10726e22i 0.0545306i
\(184\) 2.65312e23 0.439714
\(185\) 0 0
\(186\) 2.70510e23 0.400219
\(187\) − 6.91389e23i − 0.966909i
\(188\) − 3.32590e23i − 0.439796i
\(189\) −3.19686e24 −3.99829
\(190\) 0 0
\(191\) 2.97957e23 0.333659 0.166830 0.985986i \(-0.446647\pi\)
0.166830 + 0.985986i \(0.446647\pi\)
\(192\) 2.24606e23i 0.238100i
\(193\) − 9.58444e23i − 0.962088i −0.876696 0.481044i \(-0.840258\pi\)
0.876696 0.481044i \(-0.159742\pi\)
\(194\) −5.59682e23 −0.532136
\(195\) 0 0
\(196\) 3.87671e23 0.330959
\(197\) − 7.67214e23i − 0.620899i −0.950590 0.310450i \(-0.899520\pi\)
0.950590 0.310450i \(-0.100480\pi\)
\(198\) 1.71977e24i 1.31973i
\(199\) 2.93227e23 0.213426 0.106713 0.994290i \(-0.465967\pi\)
0.106713 + 0.994290i \(0.465967\pi\)
\(200\) 0 0
\(201\) 1.69735e24 1.11227
\(202\) − 1.91552e24i − 1.19151i
\(203\) 2.49265e24i 1.47215i
\(204\) 2.31200e24 1.29679
\(205\) 0 0
\(206\) 8.84206e23 0.447659
\(207\) 6.79314e24i 3.26875i
\(208\) − 5.77664e22i − 0.0264247i
\(209\) 2.76876e24 1.20434
\(210\) 0 0
\(211\) −2.05920e24 −0.810464 −0.405232 0.914214i \(-0.632809\pi\)
−0.405232 + 0.914214i \(0.632809\pi\)
\(212\) 1.51948e24i 0.569074i
\(213\) 5.97779e24i 2.13086i
\(214\) −3.74462e24 −1.27076
\(215\) 0 0
\(216\) −3.56278e24 −1.09654
\(217\) − 1.30646e24i − 0.383062i
\(218\) − 6.55303e23i − 0.183083i
\(219\) −4.60720e24 −1.22680
\(220\) 0 0
\(221\) −5.94622e23 −0.143920
\(222\) 2.71147e24i 0.625891i
\(223\) 3.95841e24i 0.871605i 0.900042 + 0.435803i \(0.143535\pi\)
−0.900042 + 0.435803i \(0.856465\pi\)
\(224\) 1.08476e24 0.227892
\(225\) 0 0
\(226\) −6.31880e23 −0.120919
\(227\) − 6.66876e23i − 0.121835i −0.998143 0.0609177i \(-0.980597\pi\)
0.998143 0.0609177i \(-0.0194027\pi\)
\(228\) 9.25869e24i 1.61523i
\(229\) 7.22731e24 1.20421 0.602107 0.798415i \(-0.294328\pi\)
0.602107 + 0.798415i \(0.294328\pi\)
\(230\) 0 0
\(231\) 1.14660e25 1.74376
\(232\) 2.77797e24i 0.403742i
\(233\) − 1.07597e25i − 1.49474i −0.664410 0.747368i \(-0.731317\pi\)
0.664410 0.747368i \(-0.268683\pi\)
\(234\) 1.47907e24 0.196437
\(235\) 0 0
\(236\) −1.14249e24 −0.138764
\(237\) − 1.82221e25i − 2.11709i
\(238\) − 1.11661e25i − 1.24120i
\(239\) −3.16911e24 −0.337101 −0.168550 0.985693i \(-0.553909\pi\)
−0.168550 + 0.985693i \(0.553909\pi\)
\(240\) 0 0
\(241\) −1.75468e24 −0.171009 −0.0855043 0.996338i \(-0.527250\pi\)
−0.0855043 + 0.996338i \(0.527250\pi\)
\(242\) 3.75653e24i 0.350530i
\(243\) − 3.51977e25i − 3.14520i
\(244\) 1.67246e23 0.0143140
\(245\) 0 0
\(246\) −1.34153e25 −1.05386
\(247\) − 2.38124e24i − 0.179261i
\(248\) − 1.45600e24i − 0.105056i
\(249\) 3.91295e25 2.70652
\(250\) 0 0
\(251\) 1.75868e25 1.11844 0.559221 0.829019i \(-0.311100\pi\)
0.559221 + 0.829019i \(0.311100\pi\)
\(252\) 2.77746e25i 1.69411i
\(253\) − 1.50944e25i − 0.883179i
\(254\) −1.43775e25 −0.807104
\(255\) 0 0
\(256\) 1.20893e24 0.0625000
\(257\) − 9.88264e24i − 0.490428i −0.969469 0.245214i \(-0.921142\pi\)
0.969469 0.245214i \(-0.0788582\pi\)
\(258\) − 7.39392e24i − 0.352264i
\(259\) 1.30954e25 0.599059
\(260\) 0 0
\(261\) −7.11280e25 −3.00134
\(262\) − 2.08291e25i − 0.844321i
\(263\) 1.60390e25i 0.624656i 0.949974 + 0.312328i \(0.101109\pi\)
−0.949974 + 0.312328i \(0.898891\pi\)
\(264\) 1.27785e25 0.478231
\(265\) 0 0
\(266\) 4.47159e25 1.54598
\(267\) 7.51030e24i 0.249626i
\(268\) − 9.13585e24i − 0.291967i
\(269\) −1.76073e25 −0.541122 −0.270561 0.962703i \(-0.587209\pi\)
−0.270561 + 0.962703i \(0.587209\pi\)
\(270\) 0 0
\(271\) 3.45015e25 0.980981 0.490490 0.871447i \(-0.336818\pi\)
0.490490 + 0.871447i \(0.336818\pi\)
\(272\) − 1.24441e25i − 0.340402i
\(273\) − 9.86125e24i − 0.259552i
\(274\) −1.12733e24 −0.0285543
\(275\) 0 0
\(276\) 5.04753e25 1.18450
\(277\) − 2.59482e25i − 0.586231i −0.956077 0.293116i \(-0.905308\pi\)
0.956077 0.293116i \(-0.0946921\pi\)
\(278\) 4.24533e25i 0.923508i
\(279\) 3.72799e25 0.780964
\(280\) 0 0
\(281\) 7.82254e25 1.52031 0.760154 0.649743i \(-0.225124\pi\)
0.760154 + 0.649743i \(0.225124\pi\)
\(282\) − 6.32748e25i − 1.18472i
\(283\) − 8.38167e24i − 0.151207i −0.997138 0.0756037i \(-0.975912\pi\)
0.997138 0.0756037i \(-0.0240884\pi\)
\(284\) 3.21750e25 0.559340
\(285\) 0 0
\(286\) −3.28649e24 −0.0530749
\(287\) 6.47908e25i 1.00868i
\(288\) 3.09537e25i 0.464613i
\(289\) −5.90026e25 −0.853973
\(290\) 0 0
\(291\) −1.06479e26 −1.43346
\(292\) 2.47979e25i 0.322029i
\(293\) − 4.97478e25i − 0.623252i −0.950205 0.311626i \(-0.899126\pi\)
0.950205 0.311626i \(-0.100874\pi\)
\(294\) 7.37540e25 0.891535
\(295\) 0 0
\(296\) 1.45943e25 0.164293
\(297\) 2.02697e26i 2.20244i
\(298\) 9.84688e25i 1.03283i
\(299\) −1.29817e25 −0.131458
\(300\) 0 0
\(301\) −3.57098e25 −0.337163
\(302\) 1.15807e26i 1.05599i
\(303\) − 3.64425e26i − 3.20967i
\(304\) 4.98342e25 0.423990
\(305\) 0 0
\(306\) 3.18624e26 2.53048
\(307\) − 1.41280e26i − 1.08425i −0.840299 0.542123i \(-0.817621\pi\)
0.840299 0.542123i \(-0.182379\pi\)
\(308\) − 6.17151e25i − 0.457730i
\(309\) 1.68219e26 1.20590
\(310\) 0 0
\(311\) 1.39904e26 0.937232 0.468616 0.883402i \(-0.344753\pi\)
0.468616 + 0.883402i \(0.344753\pi\)
\(312\) − 1.09900e25i − 0.0711827i
\(313\) 2.79550e26i 1.75083i 0.483373 + 0.875414i \(0.339411\pi\)
−0.483373 + 0.875414i \(0.660589\pi\)
\(314\) 3.84771e25 0.233045
\(315\) 0 0
\(316\) −9.80789e25 −0.555727
\(317\) − 1.16992e26i − 0.641259i −0.947205 0.320629i \(-0.896106\pi\)
0.947205 0.320629i \(-0.103894\pi\)
\(318\) 2.89079e26i 1.53297i
\(319\) 1.58046e26 0.810928
\(320\) 0 0
\(321\) −7.12409e26 −3.42315
\(322\) − 2.43776e26i − 1.13372i
\(323\) − 5.12972e26i − 2.30923i
\(324\) −3.76265e26 −1.63973
\(325\) 0 0
\(326\) −2.60028e26 −1.06227
\(327\) − 1.24671e26i − 0.493188i
\(328\) 7.22070e25i 0.276633i
\(329\) −3.05593e26 −1.13393
\(330\) 0 0
\(331\) −3.83627e26 −1.33572 −0.667859 0.744288i \(-0.732789\pi\)
−0.667859 + 0.744288i \(0.732789\pi\)
\(332\) − 2.10612e26i − 0.710448i
\(333\) 3.73677e26i 1.22133i
\(334\) −8.07001e25 −0.255585
\(335\) 0 0
\(336\) 2.06374e26 0.613894
\(337\) 1.36118e26i 0.392465i 0.980557 + 0.196233i \(0.0628708\pi\)
−0.980557 + 0.196233i \(0.937129\pi\)
\(338\) − 2.50168e26i − 0.699207i
\(339\) −1.20214e26 −0.325731
\(340\) 0 0
\(341\) −8.28360e25 −0.211008
\(342\) 1.27597e27i 3.15186i
\(343\) 1.81934e26i 0.435839i
\(344\) −3.97973e25 −0.0924676
\(345\) 0 0
\(346\) 3.38316e26 0.739646
\(347\) 2.49928e26i 0.530097i 0.964235 + 0.265048i \(0.0853879\pi\)
−0.964235 + 0.265048i \(0.914612\pi\)
\(348\) 5.28505e26i 1.08759i
\(349\) −1.05322e26 −0.210307 −0.105153 0.994456i \(-0.533533\pi\)
−0.105153 + 0.994456i \(0.533533\pi\)
\(350\) 0 0
\(351\) 1.74327e26 0.327824
\(352\) − 6.87792e25i − 0.125533i
\(353\) 1.44843e26i 0.256604i 0.991735 + 0.128302i \(0.0409527\pi\)
−0.991735 + 0.128302i \(0.959047\pi\)
\(354\) −2.17358e26 −0.373802
\(355\) 0 0
\(356\) 4.04236e25 0.0655255
\(357\) − 2.12433e27i − 3.34353i
\(358\) 4.50843e26i 0.689055i
\(359\) −5.68219e25 −0.0843382 −0.0421691 0.999110i \(-0.513427\pi\)
−0.0421691 + 0.999110i \(0.513427\pi\)
\(360\) 0 0
\(361\) 1.34005e27 1.87627
\(362\) 8.43871e26i 1.14772i
\(363\) 7.14675e26i 0.944254i
\(364\) −5.30774e25 −0.0681311
\(365\) 0 0
\(366\) 3.18183e25 0.0385590
\(367\) 3.29729e25i 0.0388296i 0.999812 + 0.0194148i \(0.00618032\pi\)
−0.999812 + 0.0194148i \(0.993820\pi\)
\(368\) − 2.71680e26i − 0.310924i
\(369\) −1.84881e27 −2.05644
\(370\) 0 0
\(371\) 1.39614e27 1.46725
\(372\) − 2.77003e26i − 0.282998i
\(373\) 6.24564e25i 0.0620346i 0.999519 + 0.0310173i \(0.00987470\pi\)
−0.999519 + 0.0310173i \(0.990125\pi\)
\(374\) −7.07983e26 −0.683708
\(375\) 0 0
\(376\) −3.40572e26 −0.310983
\(377\) − 1.35926e26i − 0.120703i
\(378\) 3.27359e27i 2.82722i
\(379\) −1.65025e27 −1.38624 −0.693119 0.720823i \(-0.743764\pi\)
−0.693119 + 0.720823i \(0.743764\pi\)
\(380\) 0 0
\(381\) −2.73530e27 −2.17417
\(382\) − 3.05108e26i − 0.235933i
\(383\) − 1.20420e27i − 0.905967i −0.891519 0.452983i \(-0.850360\pi\)
0.891519 0.452983i \(-0.149640\pi\)
\(384\) 2.29997e26 0.168362
\(385\) 0 0
\(386\) −9.81446e26 −0.680299
\(387\) − 1.01898e27i − 0.687387i
\(388\) 5.73114e26i 0.376277i
\(389\) −1.68483e26 −0.107667 −0.0538337 0.998550i \(-0.517144\pi\)
−0.0538337 + 0.998550i \(0.517144\pi\)
\(390\) 0 0
\(391\) −2.79655e27 −1.69343
\(392\) − 3.96975e26i − 0.234024i
\(393\) − 3.96272e27i − 2.27442i
\(394\) −7.85627e26 −0.439042
\(395\) 0 0
\(396\) 1.76104e27 0.933192
\(397\) 1.48612e27i 0.766925i 0.923557 + 0.383462i \(0.125268\pi\)
−0.923557 + 0.383462i \(0.874732\pi\)
\(398\) − 3.00264e26i − 0.150915i
\(399\) 8.50715e27 4.16456
\(400\) 0 0
\(401\) −6.25371e26 −0.290484 −0.145242 0.989396i \(-0.546396\pi\)
−0.145242 + 0.989396i \(0.546396\pi\)
\(402\) − 1.73808e27i − 0.786497i
\(403\) 7.12423e25i 0.0314076i
\(404\) −1.96149e27 −0.842524
\(405\) 0 0
\(406\) 2.55248e27 1.04097
\(407\) − 8.30310e26i − 0.329989i
\(408\) − 2.36748e27i − 0.916970i
\(409\) −3.35895e27 −1.26797 −0.633985 0.773345i \(-0.718582\pi\)
−0.633985 + 0.773345i \(0.718582\pi\)
\(410\) 0 0
\(411\) −2.14474e26 −0.0769192
\(412\) − 9.05427e26i − 0.316543i
\(413\) 1.04976e27i 0.357777i
\(414\) 6.95617e27 2.31136
\(415\) 0 0
\(416\) −5.91528e25 −0.0186851
\(417\) 8.07668e27i 2.48774i
\(418\) − 2.83521e27i − 0.851597i
\(419\) 1.39454e27 0.408492 0.204246 0.978920i \(-0.434526\pi\)
0.204246 + 0.978920i \(0.434526\pi\)
\(420\) 0 0
\(421\) −5.26111e27 −1.46594 −0.732968 0.680263i \(-0.761866\pi\)
−0.732968 + 0.680263i \(0.761866\pi\)
\(422\) 2.10862e27i 0.573084i
\(423\) − 8.72011e27i − 2.31179i
\(424\) 1.55595e27 0.402396
\(425\) 0 0
\(426\) 6.12126e27 1.50675
\(427\) − 1.53670e26i − 0.0369060i
\(428\) 3.83449e27i 0.898561i
\(429\) −6.25252e26 −0.142973
\(430\) 0 0
\(431\) −3.99327e27 −0.869595 −0.434797 0.900528i \(-0.643180\pi\)
−0.434797 + 0.900528i \(0.643180\pi\)
\(432\) 3.64829e27i 0.775372i
\(433\) − 1.71947e27i − 0.356674i −0.983969 0.178337i \(-0.942928\pi\)
0.983969 0.178337i \(-0.0570717\pi\)
\(434\) −1.33782e27 −0.270866
\(435\) 0 0
\(436\) −6.71030e26 −0.129459
\(437\) − 1.11992e28i − 2.10926i
\(438\) 4.71777e27i 0.867479i
\(439\) −2.57641e27 −0.462528 −0.231264 0.972891i \(-0.574286\pi\)
−0.231264 + 0.972891i \(0.574286\pi\)
\(440\) 0 0
\(441\) 1.01643e28 1.73969
\(442\) 6.08893e26i 0.101767i
\(443\) 1.26523e27i 0.206504i 0.994655 + 0.103252i \(0.0329249\pi\)
−0.994655 + 0.103252i \(0.967075\pi\)
\(444\) 2.77655e27 0.442572
\(445\) 0 0
\(446\) 4.05341e27 0.616318
\(447\) 1.87336e28i 2.78222i
\(448\) − 1.11080e27i − 0.161144i
\(449\) −3.20858e26 −0.0454702 −0.0227351 0.999742i \(-0.507237\pi\)
−0.0227351 + 0.999742i \(0.507237\pi\)
\(450\) 0 0
\(451\) 4.10806e27 0.555626
\(452\) 6.47045e26i 0.0855029i
\(453\) 2.20321e28i 2.84462i
\(454\) −6.82881e26 −0.0861506
\(455\) 0 0
\(456\) 9.48090e27 1.14214
\(457\) − 1.41432e28i − 1.66506i −0.553983 0.832528i \(-0.686893\pi\)
0.553983 0.832528i \(-0.313107\pi\)
\(458\) − 7.40076e27i − 0.851508i
\(459\) 3.75539e28 4.22300
\(460\) 0 0
\(461\) 6.08545e27 0.653782 0.326891 0.945062i \(-0.393999\pi\)
0.326891 + 0.945062i \(0.393999\pi\)
\(462\) − 1.17412e28i − 1.23303i
\(463\) − 6.53768e27i − 0.671155i −0.942013 0.335578i \(-0.891069\pi\)
0.942013 0.335578i \(-0.108931\pi\)
\(464\) 2.84464e27 0.285488
\(465\) 0 0
\(466\) −1.10180e28 −1.05694
\(467\) − 9.77307e27i − 0.916651i −0.888784 0.458325i \(-0.848449\pi\)
0.888784 0.458325i \(-0.151551\pi\)
\(468\) − 1.51457e27i − 0.138902i
\(469\) −8.39428e27 −0.752781
\(470\) 0 0
\(471\) 7.32022e27 0.627775
\(472\) 1.16991e27i 0.0981212i
\(473\) 2.26418e27i 0.185724i
\(474\) −1.86594e28 −1.49701
\(475\) 0 0
\(476\) −1.14340e28 −0.877661
\(477\) 3.98390e28i 2.99134i
\(478\) 3.24517e27i 0.238366i
\(479\) 1.53019e28 1.09957 0.549784 0.835307i \(-0.314710\pi\)
0.549784 + 0.835307i \(0.314710\pi\)
\(480\) 0 0
\(481\) −7.14100e26 −0.0491174
\(482\) 1.79679e27i 0.120921i
\(483\) − 4.63781e28i − 3.05400i
\(484\) 3.84668e27 0.247862
\(485\) 0 0
\(486\) −3.60424e28 −2.22399
\(487\) − 2.24122e28i − 1.35341i −0.736254 0.676705i \(-0.763407\pi\)
0.736254 0.676705i \(-0.236593\pi\)
\(488\) − 1.71260e26i − 0.0101215i
\(489\) −4.94701e28 −2.86154
\(490\) 0 0
\(491\) 1.97103e28 1.09229 0.546145 0.837691i \(-0.316095\pi\)
0.546145 + 0.837691i \(0.316095\pi\)
\(492\) 1.37373e28i 0.745191i
\(493\) − 2.92815e28i − 1.55489i
\(494\) −2.43839e27 −0.126757
\(495\) 0 0
\(496\) −1.49095e27 −0.0742856
\(497\) − 2.95633e28i − 1.44215i
\(498\) − 4.00686e28i − 1.91380i
\(499\) −3.08705e28 −1.44374 −0.721869 0.692030i \(-0.756717\pi\)
−0.721869 + 0.692030i \(0.756717\pi\)
\(500\) 0 0
\(501\) −1.53531e28 −0.688493
\(502\) − 1.80089e28i − 0.790858i
\(503\) 2.52448e28i 1.08570i 0.839831 + 0.542848i \(0.182654\pi\)
−0.839831 + 0.542848i \(0.817346\pi\)
\(504\) 2.84412e28 1.19792
\(505\) 0 0
\(506\) −1.54566e28 −0.624502
\(507\) − 4.75941e28i − 1.88352i
\(508\) 1.47226e28i 0.570709i
\(509\) −2.42632e28 −0.921323 −0.460661 0.887576i \(-0.652388\pi\)
−0.460661 + 0.887576i \(0.652388\pi\)
\(510\) 0 0
\(511\) 2.27850e28 0.830291
\(512\) − 1.23794e27i − 0.0441942i
\(513\) 1.50389e29i 5.25999i
\(514\) −1.01198e28 −0.346785
\(515\) 0 0
\(516\) −7.57138e27 −0.249088
\(517\) 1.93761e28i 0.624619i
\(518\) − 1.34096e28i − 0.423599i
\(519\) 6.43643e28 1.99245
\(520\) 0 0
\(521\) −1.28803e28 −0.382939 −0.191470 0.981499i \(-0.561325\pi\)
−0.191470 + 0.981499i \(0.561325\pi\)
\(522\) 7.28351e28i 2.12227i
\(523\) − 4.72268e28i − 1.34872i −0.738404 0.674359i \(-0.764420\pi\)
0.738404 0.674359i \(-0.235580\pi\)
\(524\) −2.13290e28 −0.597025
\(525\) 0 0
\(526\) 1.64239e28 0.441699
\(527\) 1.53471e28i 0.404591i
\(528\) − 1.30852e28i − 0.338160i
\(529\) −2.15825e28 −0.546785
\(530\) 0 0
\(531\) −2.99548e28 −0.729415
\(532\) − 4.57891e28i − 1.09318i
\(533\) − 3.53309e27i − 0.0827026i
\(534\) 7.69055e27 0.176512
\(535\) 0 0
\(536\) −9.35511e27 −0.206452
\(537\) 8.57723e28i 1.85617i
\(538\) 1.80299e28i 0.382631i
\(539\) −2.25850e28 −0.470044
\(540\) 0 0
\(541\) 1.70978e28 0.342270 0.171135 0.985248i \(-0.445257\pi\)
0.171135 + 0.985248i \(0.445257\pi\)
\(542\) − 3.53295e28i − 0.693658i
\(543\) 1.60545e29i 3.09172i
\(544\) −1.27428e28 −0.240700
\(545\) 0 0
\(546\) −1.00979e28 −0.183531
\(547\) 6.17365e28i 1.10072i 0.834929 + 0.550358i \(0.185509\pi\)
−0.834929 + 0.550358i \(0.814491\pi\)
\(548\) 1.15439e27i 0.0201909i
\(549\) 4.38499e27 0.0752417
\(550\) 0 0
\(551\) 1.17262e29 1.93671
\(552\) − 5.16867e28i − 0.837565i
\(553\) 9.01177e28i 1.43284i
\(554\) −2.65709e28 −0.414528
\(555\) 0 0
\(556\) 4.34721e28 0.653019
\(557\) − 1.03162e29i − 1.52069i −0.649518 0.760346i \(-0.725029\pi\)
0.649518 0.760346i \(-0.274971\pi\)
\(558\) − 3.81747e28i − 0.552225i
\(559\) 1.94728e27 0.0276443
\(560\) 0 0
\(561\) −1.34693e29 −1.84177
\(562\) − 8.01028e28i − 1.07502i
\(563\) 3.65428e28i 0.481353i 0.970605 + 0.240677i \(0.0773693\pi\)
−0.970605 + 0.240677i \(0.922631\pi\)
\(564\) −6.47934e28 −0.837722
\(565\) 0 0
\(566\) −8.58283e27 −0.106920
\(567\) 3.45723e29i 4.22772i
\(568\) − 3.29472e28i − 0.395513i
\(569\) −4.54287e28 −0.535367 −0.267683 0.963507i \(-0.586258\pi\)
−0.267683 + 0.963507i \(0.586258\pi\)
\(570\) 0 0
\(571\) 1.49843e29 1.70200 0.850998 0.525168i \(-0.175998\pi\)
0.850998 + 0.525168i \(0.175998\pi\)
\(572\) 3.36537e27i 0.0375297i
\(573\) − 5.80464e28i − 0.635553i
\(574\) 6.63458e28 0.713245
\(575\) 0 0
\(576\) 3.16966e28 0.328531
\(577\) − 1.44438e29i − 1.47006i −0.678035 0.735029i \(-0.737168\pi\)
0.678035 0.735029i \(-0.262832\pi\)
\(578\) 6.04187e28i 0.603850i
\(579\) −1.86719e29 −1.83258
\(580\) 0 0
\(581\) −1.93516e29 −1.83175
\(582\) 1.09034e29i 1.01361i
\(583\) − 8.85223e28i − 0.808226i
\(584\) 2.53931e28 0.227709
\(585\) 0 0
\(586\) −5.09417e28 −0.440706
\(587\) − 1.82681e29i − 1.55237i −0.630508 0.776183i \(-0.717154\pi\)
0.630508 0.776183i \(-0.282846\pi\)
\(588\) − 7.55241e28i − 0.630410i
\(589\) −6.14597e28 −0.503941
\(590\) 0 0
\(591\) −1.49465e29 −1.18269
\(592\) − 1.49446e28i − 0.116173i
\(593\) − 1.28812e29i − 0.983742i −0.870668 0.491871i \(-0.836313\pi\)
0.870668 0.491871i \(-0.163687\pi\)
\(594\) 2.07561e29 1.55736
\(595\) 0 0
\(596\) 1.00832e29 0.730320
\(597\) − 5.71249e28i − 0.406532i
\(598\) 1.32933e28i 0.0929545i
\(599\) 1.90622e28 0.130976 0.0654879 0.997853i \(-0.479140\pi\)
0.0654879 + 0.997853i \(0.479140\pi\)
\(600\) 0 0
\(601\) −4.02359e28 −0.266951 −0.133476 0.991052i \(-0.542614\pi\)
−0.133476 + 0.991052i \(0.542614\pi\)
\(602\) 3.65668e28i 0.238410i
\(603\) − 2.39531e29i − 1.53472i
\(604\) 1.18586e29 0.746700
\(605\) 0 0
\(606\) −3.73172e29 −2.26958
\(607\) 1.34619e29i 0.804681i 0.915490 + 0.402340i \(0.131803\pi\)
−0.915490 + 0.402340i \(0.868197\pi\)
\(608\) − 5.10303e28i − 0.299806i
\(609\) 4.85606e29 2.80416
\(610\) 0 0
\(611\) 1.66642e28 0.0929719
\(612\) − 3.26271e29i − 1.78932i
\(613\) − 6.61541e28i − 0.356633i −0.983973 0.178316i \(-0.942935\pi\)
0.983973 0.178316i \(-0.0570651\pi\)
\(614\) −1.44671e29 −0.766677
\(615\) 0 0
\(616\) −6.31963e28 −0.323664
\(617\) − 5.23241e28i − 0.263456i −0.991286 0.131728i \(-0.957948\pi\)
0.991286 0.131728i \(-0.0420525\pi\)
\(618\) − 1.72256e29i − 0.852700i
\(619\) −1.15227e29 −0.560793 −0.280397 0.959884i \(-0.590466\pi\)
−0.280397 + 0.959884i \(0.590466\pi\)
\(620\) 0 0
\(621\) 8.19873e29 3.85731
\(622\) − 1.43262e29i − 0.662723i
\(623\) − 3.71424e28i − 0.168945i
\(624\) −1.12538e28 −0.0503337
\(625\) 0 0
\(626\) 2.86259e29 1.23802
\(627\) − 5.39395e29i − 2.29402i
\(628\) − 3.94005e28i − 0.164788i
\(629\) −1.53833e29 −0.632727
\(630\) 0 0
\(631\) −1.82557e29 −0.726257 −0.363128 0.931739i \(-0.618291\pi\)
−0.363128 + 0.931739i \(0.618291\pi\)
\(632\) 1.00433e29i 0.392958i
\(633\) 4.01163e29i 1.54377i
\(634\) −1.19800e29 −0.453438
\(635\) 0 0
\(636\) 2.96017e29 1.08397
\(637\) 1.94240e28i 0.0699641i
\(638\) − 1.61840e29i − 0.573413i
\(639\) 8.43591e29 2.94017
\(640\) 0 0
\(641\) 4.06750e29 1.37189 0.685943 0.727655i \(-0.259390\pi\)
0.685943 + 0.727655i \(0.259390\pi\)
\(642\) 7.29506e29i 2.42053i
\(643\) − 1.10344e29i − 0.360192i −0.983649 0.180096i \(-0.942359\pi\)
0.983649 0.180096i \(-0.0576409\pi\)
\(644\) −2.49627e29 −0.801659
\(645\) 0 0
\(646\) −5.25283e29 −1.63287
\(647\) 5.65654e29i 1.73004i 0.501737 + 0.865020i \(0.332694\pi\)
−0.501737 + 0.865020i \(0.667306\pi\)
\(648\) 3.85295e29i 1.15946i
\(649\) 6.65597e28 0.197080
\(650\) 0 0
\(651\) −2.54518e29 −0.729656
\(652\) 2.66269e29i 0.751140i
\(653\) − 9.98139e28i − 0.277078i −0.990357 0.138539i \(-0.955759\pi\)
0.990357 0.138539i \(-0.0442407\pi\)
\(654\) −1.27663e29 −0.348736
\(655\) 0 0
\(656\) 7.39399e28 0.195609
\(657\) 6.50172e29i 1.69275i
\(658\) 3.12927e29i 0.801809i
\(659\) −4.27384e29 −1.07776 −0.538880 0.842383i \(-0.681152\pi\)
−0.538880 + 0.842383i \(0.681152\pi\)
\(660\) 0 0
\(661\) 1.06580e29 0.260352 0.130176 0.991491i \(-0.458446\pi\)
0.130176 + 0.991491i \(0.458446\pi\)
\(662\) 3.92834e29i 0.944495i
\(663\) 1.15841e29i 0.274139i
\(664\) −2.15666e29 −0.502362
\(665\) 0 0
\(666\) 3.82645e29 0.863608
\(667\) − 6.39271e29i − 1.42025i
\(668\) 8.26369e28i 0.180726i
\(669\) 7.71157e29 1.66023
\(670\) 0 0
\(671\) −9.74344e27 −0.0203295
\(672\) − 2.11327e29i − 0.434089i
\(673\) − 9.20636e29i − 1.86179i −0.365292 0.930893i \(-0.619031\pi\)
0.365292 0.930893i \(-0.380969\pi\)
\(674\) 1.39385e29 0.277515
\(675\) 0 0
\(676\) −2.56172e29 −0.494414
\(677\) 1.03701e29i 0.197062i 0.995134 + 0.0985309i \(0.0314143\pi\)
−0.995134 + 0.0985309i \(0.968586\pi\)
\(678\) 1.23100e29i 0.230327i
\(679\) 5.26593e29 0.970159
\(680\) 0 0
\(681\) −1.29917e29 −0.232072
\(682\) 8.48241e28i 0.149205i
\(683\) 1.05066e30i 1.81988i 0.414735 + 0.909942i \(0.363874\pi\)
−0.414735 + 0.909942i \(0.636126\pi\)
\(684\) 1.30660e30 2.22870
\(685\) 0 0
\(686\) 1.86301e29 0.308185
\(687\) − 1.40799e30i − 2.29378i
\(688\) 4.07524e28i 0.0653844i
\(689\) −7.61327e28 −0.120301
\(690\) 0 0
\(691\) −2.09599e29 −0.321270 −0.160635 0.987014i \(-0.551354\pi\)
−0.160635 + 0.987014i \(0.551354\pi\)
\(692\) − 3.46436e29i − 0.523009i
\(693\) − 1.61810e30i − 2.40606i
\(694\) 2.55926e29 0.374835
\(695\) 0 0
\(696\) 5.41189e29 0.769046
\(697\) − 7.61105e29i − 1.06537i
\(698\) 1.07850e29i 0.148709i
\(699\) −2.09616e30 −2.84717
\(700\) 0 0
\(701\) −9.09765e29 −1.19920 −0.599598 0.800301i \(-0.704673\pi\)
−0.599598 + 0.800301i \(0.704673\pi\)
\(702\) − 1.78511e29i − 0.231806i
\(703\) − 6.16043e29i − 0.788098i
\(704\) −7.04299e28 −0.0887655
\(705\) 0 0
\(706\) 1.48319e29 0.181446
\(707\) 1.80228e30i 2.17229i
\(708\) 2.22575e29i 0.264318i
\(709\) 1.17415e30 1.37385 0.686925 0.726728i \(-0.258960\pi\)
0.686925 + 0.726728i \(0.258960\pi\)
\(710\) 0 0
\(711\) −2.57151e30 −2.92118
\(712\) − 4.13938e28i − 0.0463335i
\(713\) 3.35057e29i 0.369555i
\(714\) −2.17531e30 −2.36423
\(715\) 0 0
\(716\) 4.61663e29 0.487235
\(717\) 6.17390e29i 0.642108i
\(718\) 5.81857e28i 0.0596361i
\(719\) −2.38367e29 −0.240765 −0.120382 0.992728i \(-0.538412\pi\)
−0.120382 + 0.992728i \(0.538412\pi\)
\(720\) 0 0
\(721\) −8.31932e29 −0.816145
\(722\) − 1.37221e30i − 1.32673i
\(723\) 3.41837e29i 0.325737i
\(724\) 8.64124e29 0.811561
\(725\) 0 0
\(726\) 7.31827e29 0.667688
\(727\) 1.60120e30i 1.43991i 0.694022 + 0.719954i \(0.255837\pi\)
−0.694022 + 0.719954i \(0.744163\pi\)
\(728\) 5.43513e28i 0.0481760i
\(729\) −3.10350e30 −2.71152
\(730\) 0 0
\(731\) 4.19487e29 0.356112
\(732\) − 3.25819e28i − 0.0272653i
\(733\) 8.02614e28i 0.0662087i 0.999452 + 0.0331043i \(0.0105394\pi\)
−0.999452 + 0.0331043i \(0.989461\pi\)
\(734\) 3.37642e28 0.0274567
\(735\) 0 0
\(736\) −2.78200e29 −0.219857
\(737\) 5.32238e29i 0.414665i
\(738\) 1.89318e30i 1.45412i
\(739\) −2.07121e30 −1.56840 −0.784200 0.620508i \(-0.786927\pi\)
−0.784200 + 0.620508i \(0.786927\pi\)
\(740\) 0 0
\(741\) −4.63901e29 −0.341456
\(742\) − 1.42965e30i − 1.03750i
\(743\) 2.00787e30i 1.43666i 0.695704 + 0.718329i \(0.255093\pi\)
−0.695704 + 0.718329i \(0.744907\pi\)
\(744\) −2.83651e29 −0.200110
\(745\) 0 0
\(746\) 6.39554e28 0.0438651
\(747\) − 5.52199e30i − 3.73447i
\(748\) 7.24974e29i 0.483455i
\(749\) 3.52324e30 2.31677
\(750\) 0 0
\(751\) 2.68196e30 1.71488 0.857440 0.514584i \(-0.172054\pi\)
0.857440 + 0.514584i \(0.172054\pi\)
\(752\) 3.48745e29i 0.219898i
\(753\) − 3.42617e30i − 2.13040i
\(754\) −1.39188e29 −0.0853501
\(755\) 0 0
\(756\) 3.35215e30 1.99915
\(757\) 2.29494e30i 1.34979i 0.737915 + 0.674894i \(0.235811\pi\)
−0.737915 + 0.674894i \(0.764189\pi\)
\(758\) 1.68986e30i 0.980219i
\(759\) −2.94060e30 −1.68228
\(760\) 0 0
\(761\) 3.17369e29 0.176614 0.0883072 0.996093i \(-0.471854\pi\)
0.0883072 + 0.996093i \(0.471854\pi\)
\(762\) 2.80095e30i 1.53737i
\(763\) 6.16561e29i 0.333786i
\(764\) −3.12430e29 −0.166830
\(765\) 0 0
\(766\) −1.23310e30 −0.640615
\(767\) − 5.72440e28i − 0.0293345i
\(768\) − 2.35517e29i − 0.119050i
\(769\) −3.31592e30 −1.65340 −0.826699 0.562644i \(-0.809784\pi\)
−0.826699 + 0.562644i \(0.809784\pi\)
\(770\) 0 0
\(771\) −1.92528e30 −0.934166
\(772\) 1.00500e30i 0.481044i
\(773\) 2.86833e29i 0.135439i 0.997704 + 0.0677196i \(0.0215723\pi\)
−0.997704 + 0.0677196i \(0.978428\pi\)
\(774\) −1.04344e30 −0.486056
\(775\) 0 0
\(776\) 5.86869e29 0.266068
\(777\) − 2.55117e30i − 1.14109i
\(778\) 1.72526e29i 0.0761323i
\(779\) 3.04795e30 1.32698
\(780\) 0 0
\(781\) −1.87446e30 −0.794402
\(782\) 2.86367e30i 1.19743i
\(783\) 8.58454e30i 3.54175i
\(784\) −4.06503e29 −0.165480
\(785\) 0 0
\(786\) −4.05782e30 −1.60826
\(787\) 1.37451e30i 0.537544i 0.963204 + 0.268772i \(0.0866179\pi\)
−0.963204 + 0.268772i \(0.913382\pi\)
\(788\) 8.04482e29i 0.310450i
\(789\) 3.12463e30 1.18984
\(790\) 0 0
\(791\) 5.94523e29 0.220453
\(792\) − 1.80331e30i − 0.659866i
\(793\) 8.37975e27i 0.00302596i
\(794\) 1.52178e30 0.542298
\(795\) 0 0
\(796\) −3.07471e29 −0.106713
\(797\) − 4.30808e30i − 1.47561i −0.675014 0.737805i \(-0.735862\pi\)
0.675014 0.737805i \(-0.264138\pi\)
\(798\) − 8.71132e30i − 2.94479i
\(799\) 3.58983e30 1.19766
\(800\) 0 0
\(801\) 1.05986e30 0.344435
\(802\) 6.40380e29i 0.205403i
\(803\) − 1.44468e30i − 0.457361i
\(804\) −1.77980e30 −0.556137
\(805\) 0 0
\(806\) 7.29521e28 0.0222085
\(807\) 3.43017e30i 1.03073i
\(808\) 2.00857e30i 0.595755i
\(809\) 8.36783e29 0.244993 0.122496 0.992469i \(-0.460910\pi\)
0.122496 + 0.992469i \(0.460910\pi\)
\(810\) 0 0
\(811\) 3.30696e30 0.943432 0.471716 0.881751i \(-0.343635\pi\)
0.471716 + 0.881751i \(0.343635\pi\)
\(812\) − 2.61374e30i − 0.736077i
\(813\) − 6.72140e30i − 1.86857i
\(814\) −8.50237e29 −0.233337
\(815\) 0 0
\(816\) −2.42430e30 −0.648396
\(817\) 1.67989e30i 0.443557i
\(818\) 3.43957e30i 0.896591i
\(819\) −1.39163e30 −0.358131
\(820\) 0 0
\(821\) 7.58969e30 1.90380 0.951899 0.306413i \(-0.0991290\pi\)
0.951899 + 0.306413i \(0.0991290\pi\)
\(822\) 2.19621e29i 0.0543901i
\(823\) − 5.61508e30i − 1.37296i −0.727149 0.686479i \(-0.759155\pi\)
0.727149 0.686479i \(-0.240845\pi\)
\(824\) −9.27157e29 −0.223830
\(825\) 0 0
\(826\) 1.07495e30 0.252987
\(827\) 5.39629e30i 1.25397i 0.779031 + 0.626985i \(0.215711\pi\)
−0.779031 + 0.626985i \(0.784289\pi\)
\(828\) − 7.12312e30i − 1.63437i
\(829\) 4.47440e30 1.01371 0.506853 0.862032i \(-0.330809\pi\)
0.506853 + 0.862032i \(0.330809\pi\)
\(830\) 0 0
\(831\) −5.05508e30 −1.11665
\(832\) 6.05725e28i 0.0132124i
\(833\) 4.18436e30i 0.901273i
\(834\) 8.27052e30 1.75910
\(835\) 0 0
\(836\) −2.90325e30 −0.602170
\(837\) − 4.49937e30i − 0.921582i
\(838\) − 1.42801e30i − 0.288847i
\(839\) 1.16605e30 0.232925 0.116462 0.993195i \(-0.462845\pi\)
0.116462 + 0.993195i \(0.462845\pi\)
\(840\) 0 0
\(841\) 1.56068e30 0.304058
\(842\) 5.38738e30i 1.03657i
\(843\) − 1.52395e31i − 2.89588i
\(844\) 2.15923e30 0.405232
\(845\) 0 0
\(846\) −8.92939e30 −1.63468
\(847\) − 3.53444e30i − 0.639065i
\(848\) − 1.59329e30i − 0.284537i
\(849\) −1.63287e30 −0.288019
\(850\) 0 0
\(851\) −3.35846e30 −0.577936
\(852\) − 6.26817e30i − 1.06543i
\(853\) 1.08909e31i 1.82852i 0.405123 + 0.914262i \(0.367229\pi\)
−0.405123 + 0.914262i \(0.632771\pi\)
\(854\) −1.57358e29 −0.0260965
\(855\) 0 0
\(856\) 3.92651e30 0.635379
\(857\) 1.32603e30i 0.211960i 0.994368 + 0.105980i \(0.0337979\pi\)
−0.994368 + 0.105980i \(0.966202\pi\)
\(858\) 6.40258e29i 0.101097i
\(859\) 6.84458e30 1.06762 0.533812 0.845603i \(-0.320759\pi\)
0.533812 + 0.845603i \(0.320759\pi\)
\(860\) 0 0
\(861\) 1.26222e31 1.92133
\(862\) 4.08910e30i 0.614896i
\(863\) − 3.45729e30i − 0.513597i −0.966465 0.256799i \(-0.917332\pi\)
0.966465 0.256799i \(-0.0826677\pi\)
\(864\) 3.73585e30 0.548270
\(865\) 0 0
\(866\) −1.76074e30 −0.252206
\(867\) 1.14946e31i 1.62664i
\(868\) 1.36992e30i 0.191531i
\(869\) 5.71391e30 0.789270
\(870\) 0 0
\(871\) 4.57746e29 0.0617212
\(872\) 6.87135e29i 0.0915416i
\(873\) 1.50264e31i 1.97790i
\(874\) −1.14679e31 −1.49147
\(875\) 0 0
\(876\) 4.83100e30 0.613400
\(877\) − 1.37025e30i − 0.171912i −0.996299 0.0859558i \(-0.972606\pi\)
0.996299 0.0859558i \(-0.0273944\pi\)
\(878\) 2.63825e30i 0.327057i
\(879\) −9.69160e30 −1.18717
\(880\) 0 0
\(881\) −3.75611e30 −0.449253 −0.224627 0.974445i \(-0.572116\pi\)
−0.224627 + 0.974445i \(0.572116\pi\)
\(882\) − 1.04082e31i − 1.23015i
\(883\) − 6.64376e30i − 0.775937i −0.921673 0.387968i \(-0.873177\pi\)
0.921673 0.387968i \(-0.126823\pi\)
\(884\) 6.23507e29 0.0719602
\(885\) 0 0
\(886\) 1.29559e30 0.146021
\(887\) − 6.01786e30i − 0.670261i −0.942172 0.335131i \(-0.891220\pi\)
0.942172 0.335131i \(-0.108780\pi\)
\(888\) − 2.84318e30i − 0.312945i
\(889\) 1.35275e31 1.47146
\(890\) 0 0
\(891\) 2.19205e31 2.32882
\(892\) − 4.15070e30i − 0.435803i
\(893\) 1.43760e31i 1.49175i
\(894\) 1.91832e31 1.96733
\(895\) 0 0
\(896\) −1.13745e30 −0.113946
\(897\) 2.52903e30i 0.250400i
\(898\) 3.28559e29i 0.0321523i
\(899\) −3.50824e30 −0.339323
\(900\) 0 0
\(901\) −1.64006e31 −1.54971
\(902\) − 4.20665e30i − 0.392887i
\(903\) 6.95680e30i 0.642226i
\(904\) 6.62574e29 0.0604597
\(905\) 0 0
\(906\) 2.25609e31 2.01145
\(907\) 1.42699e31i 1.25761i 0.777564 + 0.628804i \(0.216455\pi\)
−0.777564 + 0.628804i \(0.783545\pi\)
\(908\) 6.99270e29i 0.0609177i
\(909\) −5.14280e31 −4.42873
\(910\) 0 0
\(911\) 1.18912e31 1.00065 0.500326 0.865837i \(-0.333214\pi\)
0.500326 + 0.865837i \(0.333214\pi\)
\(912\) − 9.70844e30i − 0.807614i
\(913\) 1.22699e31i 1.00901i
\(914\) −1.44827e31 −1.17737
\(915\) 0 0
\(916\) −7.57838e30 −0.602107
\(917\) 1.95977e31i 1.53932i
\(918\) − 3.84552e31i − 2.98612i
\(919\) 3.13990e30 0.241047 0.120524 0.992710i \(-0.461543\pi\)
0.120524 + 0.992710i \(0.461543\pi\)
\(920\) 0 0
\(921\) −2.75234e31 −2.06527
\(922\) − 6.23150e30i − 0.462294i
\(923\) 1.61211e30i 0.118243i
\(924\) −1.20230e31 −0.871882
\(925\) 0 0
\(926\) −6.69458e30 −0.474578
\(927\) − 2.37392e31i − 1.66391i
\(928\) − 2.91291e30i − 0.201871i
\(929\) −1.39588e30 −0.0956498 −0.0478249 0.998856i \(-0.515229\pi\)
−0.0478249 + 0.998856i \(0.515229\pi\)
\(930\) 0 0
\(931\) −1.67568e31 −1.12259
\(932\) 1.12824e31i 0.747368i
\(933\) − 2.72554e31i − 1.78524i
\(934\) −1.00076e31 −0.648170
\(935\) 0 0
\(936\) −1.55092e30 −0.0982183
\(937\) − 2.22831e31i − 1.39543i −0.716374 0.697717i \(-0.754199\pi\)
0.716374 0.697717i \(-0.245801\pi\)
\(938\) 8.59574e30i 0.532296i
\(939\) 5.44604e31 3.33497
\(940\) 0 0
\(941\) 2.09348e31 1.25366 0.626828 0.779157i \(-0.284353\pi\)
0.626828 + 0.779157i \(0.284353\pi\)
\(942\) − 7.49590e30i − 0.443904i
\(943\) − 1.66164e31i − 0.973114i
\(944\) 1.19799e30 0.0693822
\(945\) 0 0
\(946\) 2.31852e30 0.131327
\(947\) 2.96424e31i 1.66050i 0.557393 + 0.830249i \(0.311802\pi\)
−0.557393 + 0.830249i \(0.688198\pi\)
\(948\) 1.91072e31i 1.05855i
\(949\) −1.24248e30 −0.0680763
\(950\) 0 0
\(951\) −2.27917e31 −1.22147
\(952\) 1.17085e31i 0.620600i
\(953\) − 3.78184e30i − 0.198257i −0.995075 0.0991283i \(-0.968395\pi\)
0.995075 0.0991283i \(-0.0316054\pi\)
\(954\) 4.07952e31 2.11520
\(955\) 0 0
\(956\) 3.32305e30 0.168550
\(957\) − 3.07898e31i − 1.54465i
\(958\) − 1.56691e31i − 0.777512i
\(959\) 1.06069e30 0.0520584
\(960\) 0 0
\(961\) −1.89867e31 −0.911707
\(962\) 7.31238e29i 0.0347313i
\(963\) 1.00536e32i 4.72329i
\(964\) 1.83991e30 0.0855043
\(965\) 0 0
\(966\) −4.74912e31 −2.15950
\(967\) − 2.27343e31i − 1.02259i −0.859404 0.511297i \(-0.829165\pi\)
0.859404 0.511297i \(-0.170835\pi\)
\(968\) − 3.93900e30i − 0.175265i
\(969\) −9.99344e31 −4.39861
\(970\) 0 0
\(971\) 3.59388e31 1.54797 0.773983 0.633206i \(-0.218261\pi\)
0.773983 + 0.633206i \(0.218261\pi\)
\(972\) 3.69074e31i 1.57260i
\(973\) − 3.99434e31i − 1.68368i
\(974\) −2.29500e31 −0.957006
\(975\) 0 0
\(976\) −1.75370e29 −0.00715702
\(977\) 8.76227e30i 0.353772i 0.984231 + 0.176886i \(0.0566024\pi\)
−0.984231 + 0.176886i \(0.943398\pi\)
\(978\) 5.06574e31i 2.02341i
\(979\) −2.35501e30 −0.0930625
\(980\) 0 0
\(981\) −1.75936e31 −0.680504
\(982\) − 2.01834e31i − 0.772365i
\(983\) − 3.99246e31i − 1.51157i −0.654819 0.755786i \(-0.727255\pi\)
0.654819 0.755786i \(-0.272745\pi\)
\(984\) 1.40670e31 0.526929
\(985\) 0 0
\(986\) −2.99842e31 −1.09947
\(987\) 5.95340e31i 2.15991i
\(988\) 2.49691e30i 0.0896305i
\(989\) 9.15821e30 0.325274
\(990\) 0 0
\(991\) 4.03797e31 1.40407 0.702037 0.712141i \(-0.252274\pi\)
0.702037 + 0.712141i \(0.252274\pi\)
\(992\) 1.52673e30i 0.0525278i
\(993\) 7.47361e31i 2.54427i
\(994\) −3.02728e31 −1.01976
\(995\) 0 0
\(996\) −4.10302e31 −1.35326
\(997\) 1.29884e31i 0.423894i 0.977281 + 0.211947i \(0.0679803\pi\)
−0.977281 + 0.211947i \(0.932020\pi\)
\(998\) 3.16114e31i 1.02088i
\(999\) 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 50.22.b.d.49.1 4
5.2 odd 4 50.22.a.e.1.1 2
5.3 odd 4 10.22.a.c.1.2 2
5.4 even 2 inner 50.22.b.d.49.4 4
20.3 even 4 80.22.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.c.1.2 2 5.3 odd 4
50.22.a.e.1.1 2 5.2 odd 4
50.22.b.d.49.1 4 1.1 even 1 trivial
50.22.b.d.49.4 4 5.4 even 2 inner
80.22.a.b.1.1 2 20.3 even 4