Properties

Label 48.22.c.c.47.8
Level $48$
Weight $22$
Character 48.47
Analytic conductor $134.149$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,22,Mod(47,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.c (of order \(2\), degree \(1\), 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: \(134.149125258\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 47.8
Character \(\chi\) \(=\) 48.47
Dual form 48.22.c.c.47.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-66724.2 + 77512.8i) q^{3} +3.35931e6i q^{5} +5.38569e8i q^{7} +(-1.55613e9 - 1.03440e10i) q^{9} +O(q^{10})\) \(q+(-66724.2 + 77512.8i) q^{3} +3.35931e6i q^{5} +5.38569e8i q^{7} +(-1.55613e9 - 1.03440e10i) q^{9} -1.10838e10 q^{11} -1.02798e11 q^{13} +(-2.60390e11 - 2.24147e11i) q^{15} -7.33984e12i q^{17} +2.47425e13i q^{19} +(-4.17460e13 - 3.59356e13i) q^{21} +1.19781e13 q^{23} +4.65552e14 q^{25} +(9.05621e14 + 5.69572e14i) q^{27} +1.07027e15i q^{29} -2.35099e15i q^{31} +(7.39554e14 - 8.59133e14i) q^{33} -1.80922e15 q^{35} -9.92214e15 q^{37} +(6.85914e15 - 7.96820e15i) q^{39} -5.00256e16i q^{41} +2.92100e16i q^{43} +(3.47486e16 - 5.22752e15i) q^{45} +6.06543e17 q^{47} +2.68489e17 q^{49} +(5.68932e17 + 4.89745e17i) q^{51} -1.40099e18i q^{53} -3.72338e16i q^{55} +(-1.91786e18 - 1.65092e18i) q^{57} +2.94674e18 q^{59} +2.60888e18 q^{61} +(5.57094e18 - 8.38082e17i) q^{63} -3.45332e17i q^{65} -1.51456e18i q^{67} +(-7.99230e17 + 9.28459e17i) q^{69} +2.03995e19 q^{71} +1.87616e19 q^{73} +(-3.10636e19 + 3.60863e19i) q^{75} -5.96937e18i q^{77} -3.90268e19i q^{79} +(-1.04576e20 + 3.21930e19i) q^{81} -2.01616e20 q^{83} +2.46568e19 q^{85} +(-8.29597e19 - 7.14129e19i) q^{87} +3.22002e20i q^{89} -5.53641e19i q^{91} +(1.82232e20 + 1.56868e20i) q^{93} -8.31176e19 q^{95} -4.24560e20 q^{97} +(1.72477e19 + 1.14650e20i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 109254828 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 109254828 q^{9} + 285248048392 q^{13} + 247146979606248 q^{21} - 31\!\cdots\!84 q^{25}+ \cdots + 16\!\cdots\!20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −66724.2 + 77512.8i −0.652394 + 0.757880i
\(4\) 0 0
\(5\) 3.35931e6i 0.153839i 0.997037 + 0.0769193i \(0.0245084\pi\)
−0.997037 + 0.0769193i \(0.975492\pi\)
\(6\) 0 0
\(7\) 5.38569e8i 0.720630i 0.932831 + 0.360315i \(0.117331\pi\)
−0.932831 + 0.360315i \(0.882669\pi\)
\(8\) 0 0
\(9\) −1.55613e9 1.03440e10i −0.148764 0.988873i
\(10\) 0 0
\(11\) −1.10838e10 −0.128844 −0.0644219 0.997923i \(-0.520520\pi\)
−0.0644219 + 0.997923i \(0.520520\pi\)
\(12\) 0 0
\(13\) −1.02798e11 −0.206815 −0.103407 0.994639i \(-0.532975\pi\)
−0.103407 + 0.994639i \(0.532975\pi\)
\(14\) 0 0
\(15\) −2.60390e11 2.24147e11i −0.116591 0.100363i
\(16\) 0 0
\(17\) 7.33984e12i 0.883025i −0.897255 0.441513i \(-0.854442\pi\)
0.897255 0.441513i \(-0.145558\pi\)
\(18\) 0 0
\(19\) 2.47425e13i 0.925827i 0.886403 + 0.462914i \(0.153196\pi\)
−0.886403 + 0.462914i \(0.846804\pi\)
\(20\) 0 0
\(21\) −4.17460e13 3.59356e13i −0.546151 0.470134i
\(22\) 0 0
\(23\) 1.19781e13 0.0602902 0.0301451 0.999546i \(-0.490403\pi\)
0.0301451 + 0.999546i \(0.490403\pi\)
\(24\) 0 0
\(25\) 4.65552e14 0.976334
\(26\) 0 0
\(27\) 9.05621e14 + 5.69572e14i 0.846500 + 0.532389i
\(28\) 0 0
\(29\) 1.07027e15i 0.472405i 0.971704 + 0.236202i \(0.0759028\pi\)
−0.971704 + 0.236202i \(0.924097\pi\)
\(30\) 0 0
\(31\) 2.35099e15i 0.515173i −0.966255 0.257587i \(-0.917073\pi\)
0.966255 0.257587i \(-0.0829273\pi\)
\(32\) 0 0
\(33\) 7.39554e14 8.59133e14i 0.0840569 0.0976482i
\(34\) 0 0
\(35\) −1.80922e15 −0.110861
\(36\) 0 0
\(37\) −9.92214e15 −0.339224 −0.169612 0.985511i \(-0.554252\pi\)
−0.169612 + 0.985511i \(0.554252\pi\)
\(38\) 0 0
\(39\) 6.85914e15 7.96820e15i 0.134925 0.156741i
\(40\) 0 0
\(41\) 5.00256e16i 0.582052i −0.956715 0.291026i \(-0.906003\pi\)
0.956715 0.291026i \(-0.0939965\pi\)
\(42\) 0 0
\(43\) 2.92100e16i 0.206117i 0.994675 + 0.103058i \(0.0328628\pi\)
−0.994675 + 0.103058i \(0.967137\pi\)
\(44\) 0 0
\(45\) 3.47486e16 5.22752e15i 0.152127 0.0228857i
\(46\) 0 0
\(47\) 6.06543e17 1.68203 0.841016 0.541010i \(-0.181958\pi\)
0.841016 + 0.541010i \(0.181958\pi\)
\(48\) 0 0
\(49\) 2.68489e17 0.480693
\(50\) 0 0
\(51\) 5.68932e17 + 4.89745e17i 0.669227 + 0.576080i
\(52\) 0 0
\(53\) 1.40099e18i 1.10037i −0.835043 0.550185i \(-0.814557\pi\)
0.835043 0.550185i \(-0.185443\pi\)
\(54\) 0 0
\(55\) 3.72338e16i 0.0198212i
\(56\) 0 0
\(57\) −1.91786e18 1.65092e18i −0.701666 0.604004i
\(58\) 0 0
\(59\) 2.94674e18 0.750577 0.375288 0.926908i \(-0.377544\pi\)
0.375288 + 0.926908i \(0.377544\pi\)
\(60\) 0 0
\(61\) 2.60888e18 0.468264 0.234132 0.972205i \(-0.424775\pi\)
0.234132 + 0.972205i \(0.424775\pi\)
\(62\) 0 0
\(63\) 5.57094e18 8.38082e17i 0.712611 0.107204i
\(64\) 0 0
\(65\) 3.45332e17i 0.0318161i
\(66\) 0 0
\(67\) 1.51456e18i 0.101508i −0.998711 0.0507542i \(-0.983837\pi\)
0.998711 0.0507542i \(-0.0161625\pi\)
\(68\) 0 0
\(69\) −7.99230e17 + 9.28459e17i −0.0393329 + 0.0456927i
\(70\) 0 0
\(71\) 2.03995e19 0.743716 0.371858 0.928290i \(-0.378721\pi\)
0.371858 + 0.928290i \(0.378721\pi\)
\(72\) 0 0
\(73\) 1.87616e19 0.510951 0.255475 0.966816i \(-0.417768\pi\)
0.255475 + 0.966816i \(0.417768\pi\)
\(74\) 0 0
\(75\) −3.10636e19 + 3.60863e19i −0.636954 + 0.739944i
\(76\) 0 0
\(77\) 5.96937e18i 0.0928487i
\(78\) 0 0
\(79\) 3.90268e19i 0.463744i −0.972746 0.231872i \(-0.925515\pi\)
0.972746 0.231872i \(-0.0744851\pi\)
\(80\) 0 0
\(81\) −1.04576e20 + 3.21930e19i −0.955738 + 0.294218i
\(82\) 0 0
\(83\) −2.01616e20 −1.42628 −0.713141 0.701021i \(-0.752728\pi\)
−0.713141 + 0.701021i \(0.752728\pi\)
\(84\) 0 0
\(85\) 2.46568e19 0.135843
\(86\) 0 0
\(87\) −8.29597e19 7.14129e19i −0.358026 0.308194i
\(88\) 0 0
\(89\) 3.22002e20i 1.09462i 0.836930 + 0.547310i \(0.184348\pi\)
−0.836930 + 0.547310i \(0.815652\pi\)
\(90\) 0 0
\(91\) 5.53641e19i 0.149037i
\(92\) 0 0
\(93\) 1.82232e20 + 1.56868e20i 0.390440 + 0.336096i
\(94\) 0 0
\(95\) −8.31176e19 −0.142428
\(96\) 0 0
\(97\) −4.24560e20 −0.584569 −0.292285 0.956331i \(-0.594415\pi\)
−0.292285 + 0.956331i \(0.594415\pi\)
\(98\) 0 0
\(99\) 1.72477e19 + 1.14650e20i 0.0191674 + 0.127410i
\(100\) 0 0
\(101\) 7.28599e20i 0.656318i −0.944622 0.328159i \(-0.893572\pi\)
0.944622 0.328159i \(-0.106428\pi\)
\(102\) 0 0
\(103\) 1.10166e21i 0.807714i 0.914822 + 0.403857i \(0.132331\pi\)
−0.914822 + 0.403857i \(0.867669\pi\)
\(104\) 0 0
\(105\) 1.20719e20 1.40238e20i 0.0723248 0.0840191i
\(106\) 0 0
\(107\) 2.86116e21 1.40609 0.703043 0.711148i \(-0.251824\pi\)
0.703043 + 0.711148i \(0.251824\pi\)
\(108\) 0 0
\(109\) −6.14210e20 −0.248507 −0.124254 0.992250i \(-0.539654\pi\)
−0.124254 + 0.992250i \(0.539654\pi\)
\(110\) 0 0
\(111\) 6.62046e20 7.69093e20i 0.221308 0.257091i
\(112\) 0 0
\(113\) 6.66572e20i 0.184724i 0.995725 + 0.0923620i \(0.0294417\pi\)
−0.995725 + 0.0923620i \(0.970558\pi\)
\(114\) 0 0
\(115\) 4.02383e19i 0.00927496i
\(116\) 0 0
\(117\) 1.59967e20 + 1.06334e21i 0.0307666 + 0.204513i
\(118\) 0 0
\(119\) 3.95301e21 0.636334
\(120\) 0 0
\(121\) −7.27740e21 −0.983399
\(122\) 0 0
\(123\) 3.87762e21 + 3.33791e21i 0.441125 + 0.379727i
\(124\) 0 0
\(125\) 3.16578e21i 0.304036i
\(126\) 0 0
\(127\) 8.06323e21i 0.655496i 0.944765 + 0.327748i \(0.106290\pi\)
−0.944765 + 0.327748i \(0.893710\pi\)
\(128\) 0 0
\(129\) −2.26415e21 1.94901e21i −0.156212 0.134469i
\(130\) 0 0
\(131\) −5.11320e21 −0.300154 −0.150077 0.988674i \(-0.547952\pi\)
−0.150077 + 0.988674i \(0.547952\pi\)
\(132\) 0 0
\(133\) −1.33255e22 −0.667179
\(134\) 0 0
\(135\) −1.91337e21 + 3.04226e21i −0.0819020 + 0.130224i
\(136\) 0 0
\(137\) 3.00101e22i 1.10078i 0.834907 + 0.550392i \(0.185522\pi\)
−0.834907 + 0.550392i \(0.814478\pi\)
\(138\) 0 0
\(139\) 1.17753e22i 0.370951i 0.982649 + 0.185476i \(0.0593826\pi\)
−0.982649 + 0.185476i \(0.940617\pi\)
\(140\) 0 0
\(141\) −4.04711e22 + 4.70149e22i −1.09735 + 1.27478i
\(142\) 0 0
\(143\) 1.13939e21 0.0266468
\(144\) 0 0
\(145\) −3.59537e21 −0.0726741
\(146\) 0 0
\(147\) −1.79147e22 + 2.08114e22i −0.313601 + 0.364308i
\(148\) 0 0
\(149\) 1.04192e23i 1.58262i 0.611415 + 0.791310i \(0.290600\pi\)
−0.611415 + 0.791310i \(0.709400\pi\)
\(150\) 0 0
\(151\) 2.10277e22i 0.277673i −0.990315 0.138836i \(-0.955664\pi\)
0.990315 0.138836i \(-0.0443362\pi\)
\(152\) 0 0
\(153\) −7.59230e22 + 1.14217e22i −0.873199 + 0.131363i
\(154\) 0 0
\(155\) 7.89772e21 0.0792536
\(156\) 0 0
\(157\) −1.94849e23 −1.70904 −0.854521 0.519417i \(-0.826149\pi\)
−0.854521 + 0.519417i \(0.826149\pi\)
\(158\) 0 0
\(159\) 1.08595e23 + 9.34799e22i 0.833948 + 0.717874i
\(160\) 0 0
\(161\) 6.45105e21i 0.0434469i
\(162\) 0 0
\(163\) 2.58356e23i 1.52844i 0.644956 + 0.764220i \(0.276876\pi\)
−0.644956 + 0.764220i \(0.723124\pi\)
\(164\) 0 0
\(165\) 2.88610e21 + 2.48439e21i 0.0150221 + 0.0129312i
\(166\) 0 0
\(167\) 3.21326e23 1.47375 0.736874 0.676030i \(-0.236301\pi\)
0.736874 + 0.676030i \(0.236301\pi\)
\(168\) 0 0
\(169\) −2.36497e23 −0.957228
\(170\) 0 0
\(171\) 2.55935e23 3.85024e22i 0.915525 0.137730i
\(172\) 0 0
\(173\) 4.34693e23i 1.37625i 0.725591 + 0.688126i \(0.241566\pi\)
−0.725591 + 0.688126i \(0.758434\pi\)
\(174\) 0 0
\(175\) 2.50732e23i 0.703575i
\(176\) 0 0
\(177\) −1.96618e23 + 2.28410e23i −0.489672 + 0.568847i
\(178\) 0 0
\(179\) 5.46180e23 1.20887 0.604435 0.796655i \(-0.293399\pi\)
0.604435 + 0.796655i \(0.293399\pi\)
\(180\) 0 0
\(181\) −5.49682e23 −1.08265 −0.541323 0.840815i \(-0.682076\pi\)
−0.541323 + 0.840815i \(0.682076\pi\)
\(182\) 0 0
\(183\) −1.74075e23 + 2.02222e23i −0.305493 + 0.354888i
\(184\) 0 0
\(185\) 3.33316e22i 0.0521858i
\(186\) 0 0
\(187\) 8.13530e22i 0.113772i
\(188\) 0 0
\(189\) −3.06754e23 + 4.87739e23i −0.383655 + 0.610013i
\(190\) 0 0
\(191\) −3.75346e23 −0.420321 −0.210160 0.977667i \(-0.567399\pi\)
−0.210160 + 0.977667i \(0.567399\pi\)
\(192\) 0 0
\(193\) −1.23478e24 −1.23947 −0.619736 0.784810i \(-0.712760\pi\)
−0.619736 + 0.784810i \(0.712760\pi\)
\(194\) 0 0
\(195\) 2.67677e22 + 2.30420e22i 0.0241128 + 0.0207566i
\(196\) 0 0
\(197\) 1.87914e24i 1.52077i 0.649472 + 0.760386i \(0.274990\pi\)
−0.649472 + 0.760386i \(0.725010\pi\)
\(198\) 0 0
\(199\) 1.12227e24i 0.816843i 0.912793 + 0.408421i \(0.133921\pi\)
−0.912793 + 0.408421i \(0.866079\pi\)
\(200\) 0 0
\(201\) 1.17398e23 + 1.01058e23i 0.0769312 + 0.0662235i
\(202\) 0 0
\(203\) −5.76414e23 −0.340429
\(204\) 0 0
\(205\) 1.68051e23 0.0895420
\(206\) 0 0
\(207\) −1.86395e22 1.23901e23i −0.00896903 0.0596193i
\(208\) 0 0
\(209\) 2.74239e23i 0.119287i
\(210\) 0 0
\(211\) 3.45410e23i 0.135947i −0.997687 0.0679734i \(-0.978347\pi\)
0.997687 0.0679734i \(-0.0216533\pi\)
\(212\) 0 0
\(213\) −1.36114e24 + 1.58122e24i −0.485196 + 0.563647i
\(214\) 0 0
\(215\) −9.81256e22 −0.0317087
\(216\) 0 0
\(217\) 1.26617e24 0.371249
\(218\) 0 0
\(219\) −1.25185e24 + 1.45426e24i −0.333341 + 0.387239i
\(220\) 0 0
\(221\) 7.54524e23i 0.182623i
\(222\) 0 0
\(223\) 4.46788e24i 0.983785i −0.870656 0.491892i \(-0.836305\pi\)
0.870656 0.491892i \(-0.163695\pi\)
\(224\) 0 0
\(225\) −7.24458e23 4.81565e24i −0.145244 0.965470i
\(226\) 0 0
\(227\) 5.79434e23 0.105860 0.0529301 0.998598i \(-0.483144\pi\)
0.0529301 + 0.998598i \(0.483144\pi\)
\(228\) 0 0
\(229\) 2.40712e24 0.401075 0.200538 0.979686i \(-0.435731\pi\)
0.200538 + 0.979686i \(0.435731\pi\)
\(230\) 0 0
\(231\) 4.62703e23 + 3.98301e23i 0.0703682 + 0.0605739i
\(232\) 0 0
\(233\) 1.26681e25i 1.75984i −0.475122 0.879920i \(-0.657596\pi\)
0.475122 0.879920i \(-0.342404\pi\)
\(234\) 0 0
\(235\) 2.03757e24i 0.258761i
\(236\) 0 0
\(237\) 3.02508e24 + 2.60403e24i 0.351463 + 0.302544i
\(238\) 0 0
\(239\) 7.20140e24 0.766018 0.383009 0.923745i \(-0.374888\pi\)
0.383009 + 0.923745i \(0.374888\pi\)
\(240\) 0 0
\(241\) 1.59925e25 1.55861 0.779305 0.626645i \(-0.215573\pi\)
0.779305 + 0.626645i \(0.215573\pi\)
\(242\) 0 0
\(243\) 4.48237e24 1.02540e25i 0.400536 0.916281i
\(244\) 0 0
\(245\) 9.01939e23i 0.0739491i
\(246\) 0 0
\(247\) 2.54349e24i 0.191475i
\(248\) 0 0
\(249\) 1.34527e25 1.56278e25i 0.930497 1.08095i
\(250\) 0 0
\(251\) −1.04461e25 −0.664323 −0.332162 0.943223i \(-0.607778\pi\)
−0.332162 + 0.943223i \(0.607778\pi\)
\(252\) 0 0
\(253\) −1.32763e23 −0.00776802
\(254\) 0 0
\(255\) −1.64521e24 + 1.91122e24i −0.0886234 + 0.102953i
\(256\) 0 0
\(257\) 2.38723e25i 1.18467i −0.805692 0.592335i \(-0.798206\pi\)
0.805692 0.592335i \(-0.201794\pi\)
\(258\) 0 0
\(259\) 5.34376e24i 0.244455i
\(260\) 0 0
\(261\) 1.10708e25 1.66548e24i 0.467148 0.0702770i
\(262\) 0 0
\(263\) −4.88198e25 −1.90134 −0.950672 0.310197i \(-0.899605\pi\)
−0.950672 + 0.310197i \(0.899605\pi\)
\(264\) 0 0
\(265\) 4.70636e24 0.169279
\(266\) 0 0
\(267\) −2.49593e25 2.14853e25i −0.829590 0.714123i
\(268\) 0 0
\(269\) 8.09476e23i 0.0248774i 0.999923 + 0.0124387i \(0.00395946\pi\)
−0.999923 + 0.0124387i \(0.996041\pi\)
\(270\) 0 0
\(271\) 4.04241e25i 1.14938i −0.818372 0.574689i \(-0.805123\pi\)
0.818372 0.574689i \(-0.194877\pi\)
\(272\) 0 0
\(273\) 4.29143e24 + 3.69412e24i 0.112952 + 0.0972307i
\(274\) 0 0
\(275\) −5.16007e24 −0.125795
\(276\) 0 0
\(277\) 5.50913e25 1.24465 0.622323 0.782761i \(-0.286189\pi\)
0.622323 + 0.782761i \(0.286189\pi\)
\(278\) 0 0
\(279\) −2.43186e25 + 3.65844e24i −0.509441 + 0.0766394i
\(280\) 0 0
\(281\) 6.47626e24i 0.125866i 0.998018 + 0.0629330i \(0.0200454\pi\)
−0.998018 + 0.0629330i \(0.979955\pi\)
\(282\) 0 0
\(283\) 2.84729e25i 0.513658i 0.966457 + 0.256829i \(0.0826778\pi\)
−0.966457 + 0.256829i \(0.917322\pi\)
\(284\) 0 0
\(285\) 5.54595e24 6.44268e24i 0.0929192 0.107943i
\(286\) 0 0
\(287\) 2.69422e25 0.419444
\(288\) 0 0
\(289\) 1.52187e25 0.220267
\(290\) 0 0
\(291\) 2.83284e25 3.29089e25i 0.381369 0.443033i
\(292\) 0 0
\(293\) 4.97655e25i 0.623474i −0.950168 0.311737i \(-0.899089\pi\)
0.950168 0.311737i \(-0.100911\pi\)
\(294\) 0 0
\(295\) 9.89901e24i 0.115468i
\(296\) 0 0
\(297\) −1.00377e25 6.31300e24i −0.109066 0.0685950i
\(298\) 0 0
\(299\) −1.23133e24 −0.0124689
\(300\) 0 0
\(301\) −1.57316e25 −0.148534
\(302\) 0 0
\(303\) 5.64758e25 + 4.86152e25i 0.497410 + 0.428178i
\(304\) 0 0
\(305\) 8.76404e24i 0.0720371i
\(306\) 0 0
\(307\) 4.07188e25i 0.312494i 0.987718 + 0.156247i \(0.0499396\pi\)
−0.987718 + 0.156247i \(0.950060\pi\)
\(308\) 0 0
\(309\) −8.53929e25 7.35075e25i −0.612150 0.526948i
\(310\) 0 0
\(311\) 1.17735e26 0.788720 0.394360 0.918956i \(-0.370966\pi\)
0.394360 + 0.918956i \(0.370966\pi\)
\(312\) 0 0
\(313\) −3.96270e25 −0.248185 −0.124092 0.992271i \(-0.539602\pi\)
−0.124092 + 0.992271i \(0.539602\pi\)
\(314\) 0 0
\(315\) 2.81538e24 + 1.87145e25i 0.0164921 + 0.109627i
\(316\) 0 0
\(317\) 1.71567e25i 0.0940398i 0.998894 + 0.0470199i \(0.0149724\pi\)
−0.998894 + 0.0470199i \(0.985028\pi\)
\(318\) 0 0
\(319\) 1.18626e25i 0.0608665i
\(320\) 0 0
\(321\) −1.90908e26 + 2.21776e26i −0.917322 + 1.06564i
\(322\) 0 0
\(323\) 1.81606e26 0.817529
\(324\) 0 0
\(325\) −4.78581e25 −0.201920
\(326\) 0 0
\(327\) 4.09827e25 4.76092e25i 0.162125 0.188339i
\(328\) 0 0
\(329\) 3.26665e26i 1.21212i
\(330\) 0 0
\(331\) 3.33714e26i 1.16193i 0.813928 + 0.580966i \(0.197325\pi\)
−0.813928 + 0.580966i \(0.802675\pi\)
\(332\) 0 0
\(333\) 1.54401e25 + 1.02634e26i 0.0504645 + 0.335450i
\(334\) 0 0
\(335\) 5.08789e24 0.0156159
\(336\) 0 0
\(337\) −2.88797e26 −0.832682 −0.416341 0.909209i \(-0.636688\pi\)
−0.416341 + 0.909209i \(0.636688\pi\)
\(338\) 0 0
\(339\) −5.16679e25 4.44764e25i −0.139999 0.120513i
\(340\) 0 0
\(341\) 2.60578e25i 0.0663769i
\(342\) 0 0
\(343\) 4.45416e26i 1.06703i
\(344\) 0 0
\(345\) −3.11898e24 2.68486e24i −0.00702931 0.00605093i
\(346\) 0 0
\(347\) −2.15406e26 −0.456876 −0.228438 0.973558i \(-0.573362\pi\)
−0.228438 + 0.973558i \(0.573362\pi\)
\(348\) 0 0
\(349\) −7.99890e26 −1.59721 −0.798607 0.601853i \(-0.794429\pi\)
−0.798607 + 0.601853i \(0.794429\pi\)
\(350\) 0 0
\(351\) −9.30964e25 5.85511e25i −0.175069 0.110106i
\(352\) 0 0
\(353\) 9.63826e26i 1.70751i 0.520671 + 0.853757i \(0.325682\pi\)
−0.520671 + 0.853757i \(0.674318\pi\)
\(354\) 0 0
\(355\) 6.85283e25i 0.114412i
\(356\) 0 0
\(357\) −2.63761e26 + 3.06409e26i −0.415140 + 0.482265i
\(358\) 0 0
\(359\) −4.26259e26 −0.632677 −0.316339 0.948646i \(-0.602454\pi\)
−0.316339 + 0.948646i \(0.602454\pi\)
\(360\) 0 0
\(361\) 1.02020e26 0.142844
\(362\) 0 0
\(363\) 4.85578e26 5.64092e26i 0.641564 0.745299i
\(364\) 0 0
\(365\) 6.30259e25i 0.0786039i
\(366\) 0 0
\(367\) 3.28445e26i 0.386784i 0.981122 + 0.193392i \(0.0619489\pi\)
−0.981122 + 0.193392i \(0.938051\pi\)
\(368\) 0 0
\(369\) −5.17462e26 + 7.78461e25i −0.575575 + 0.0865885i
\(370\) 0 0
\(371\) 7.54530e26 0.792959
\(372\) 0 0
\(373\) 9.10674e26 0.904524 0.452262 0.891885i \(-0.350617\pi\)
0.452262 + 0.891885i \(0.350617\pi\)
\(374\) 0 0
\(375\) −2.45389e26 2.11234e26i −0.230423 0.198352i
\(376\) 0 0
\(377\) 1.10022e26i 0.0977003i
\(378\) 0 0
\(379\) 1.08412e27i 0.910677i 0.890318 + 0.455339i \(0.150482\pi\)
−0.890318 + 0.455339i \(0.849518\pi\)
\(380\) 0 0
\(381\) −6.25004e26 5.38012e26i −0.496787 0.427642i
\(382\) 0 0
\(383\) 8.00460e26 0.602216 0.301108 0.953590i \(-0.402643\pi\)
0.301108 + 0.953590i \(0.402643\pi\)
\(384\) 0 0
\(385\) 2.00530e25 0.0142837
\(386\) 0 0
\(387\) 3.02147e26 4.54545e25i 0.203823 0.0306628i
\(388\) 0 0
\(389\) 2.52183e27i 1.61156i −0.592218 0.805778i \(-0.701747\pi\)
0.592218 0.805778i \(-0.298253\pi\)
\(390\) 0 0
\(391\) 8.79175e25i 0.0532377i
\(392\) 0 0
\(393\) 3.41174e26 3.96339e26i 0.195819 0.227481i
\(394\) 0 0
\(395\) 1.31103e26 0.0713418
\(396\) 0 0
\(397\) 2.36092e27 1.21838 0.609188 0.793026i \(-0.291495\pi\)
0.609188 + 0.793026i \(0.291495\pi\)
\(398\) 0 0
\(399\) 8.89134e26 1.03290e27i 0.435263 0.505641i
\(400\) 0 0
\(401\) 7.73414e26i 0.359249i −0.983735 0.179625i \(-0.942512\pi\)
0.983735 0.179625i \(-0.0574883\pi\)
\(402\) 0 0
\(403\) 2.41678e26i 0.106545i
\(404\) 0 0
\(405\) −1.08146e26 3.51303e26i −0.0452621 0.147029i
\(406\) 0 0
\(407\) 1.09975e26 0.0437070
\(408\) 0 0
\(409\) 2.53526e27 0.957035 0.478517 0.878078i \(-0.341174\pi\)
0.478517 + 0.878078i \(0.341174\pi\)
\(410\) 0 0
\(411\) −2.32617e27 2.00240e27i −0.834262 0.718144i
\(412\) 0 0
\(413\) 1.58702e27i 0.540888i
\(414\) 0 0
\(415\) 6.77291e26i 0.219417i
\(416\) 0 0
\(417\) −9.12738e26 7.85698e26i −0.281137 0.242007i
\(418\) 0 0
\(419\) −4.48106e27 −1.31260 −0.656302 0.754498i \(-0.727880\pi\)
−0.656302 + 0.754498i \(0.727880\pi\)
\(420\) 0 0
\(421\) 3.95661e27 1.10246 0.551228 0.834354i \(-0.314159\pi\)
0.551228 + 0.834354i \(0.314159\pi\)
\(422\) 0 0
\(423\) −9.43858e26 6.27406e27i −0.250226 1.66332i
\(424\) 0 0
\(425\) 3.41708e27i 0.862127i
\(426\) 0 0
\(427\) 1.40506e27i 0.337445i
\(428\) 0 0
\(429\) −7.60250e25 + 8.83176e25i −0.0173842 + 0.0201951i
\(430\) 0 0
\(431\) −5.76889e27 −1.25626 −0.628132 0.778106i \(-0.716180\pi\)
−0.628132 + 0.778106i \(0.716180\pi\)
\(432\) 0 0
\(433\) 4.74142e27 0.983524 0.491762 0.870730i \(-0.336353\pi\)
0.491762 + 0.870730i \(0.336353\pi\)
\(434\) 0 0
\(435\) 2.39898e26 2.78687e26i 0.0474121 0.0550783i
\(436\) 0 0
\(437\) 2.96368e26i 0.0558183i
\(438\) 0 0
\(439\) 5.57774e27i 1.00134i 0.865639 + 0.500669i \(0.166913\pi\)
−0.865639 + 0.500669i \(0.833087\pi\)
\(440\) 0 0
\(441\) −4.17803e26 2.77724e27i −0.0715100 0.475344i
\(442\) 0 0
\(443\) 2.92820e27 0.477927 0.238964 0.971028i \(-0.423192\pi\)
0.238964 + 0.971028i \(0.423192\pi\)
\(444\) 0 0
\(445\) −1.08170e27 −0.168395
\(446\) 0 0
\(447\) −8.07618e27 6.95209e27i −1.19944 1.03249i
\(448\) 0 0
\(449\) 9.39845e27i 1.33189i −0.745999 0.665947i \(-0.768028\pi\)
0.745999 0.665947i \(-0.231972\pi\)
\(450\) 0 0
\(451\) 5.54471e26i 0.0749938i
\(452\) 0 0
\(453\) 1.62991e27 + 1.40305e27i 0.210443 + 0.181152i
\(454\) 0 0
\(455\) 1.85985e26 0.0229276
\(456\) 0 0
\(457\) 7.76565e27 0.914234 0.457117 0.889407i \(-0.348882\pi\)
0.457117 + 0.889407i \(0.348882\pi\)
\(458\) 0 0
\(459\) 4.18057e27 6.64711e27i 0.470113 0.747480i
\(460\) 0 0
\(461\) 7.60358e27i 0.816881i 0.912785 + 0.408440i \(0.133927\pi\)
−0.912785 + 0.408440i \(0.866073\pi\)
\(462\) 0 0
\(463\) 1.13806e28i 1.16833i 0.811634 + 0.584166i \(0.198578\pi\)
−0.811634 + 0.584166i \(0.801422\pi\)
\(464\) 0 0
\(465\) −5.26969e26 + 6.12174e26i −0.0517045 + 0.0600647i
\(466\) 0 0
\(467\) 7.98186e27 0.748647 0.374323 0.927298i \(-0.377875\pi\)
0.374323 + 0.927298i \(0.377875\pi\)
\(468\) 0 0
\(469\) 8.15698e26 0.0731500
\(470\) 0 0
\(471\) 1.30012e28 1.51033e28i 1.11497 1.29525i
\(472\) 0 0
\(473\) 3.23757e26i 0.0265569i
\(474\) 0 0
\(475\) 1.15189e28i 0.903916i
\(476\) 0 0
\(477\) −1.44918e28 + 2.18012e27i −1.08813 + 0.163696i
\(478\) 0 0
\(479\) −4.21612e27 −0.302963 −0.151482 0.988460i \(-0.548404\pi\)
−0.151482 + 0.988460i \(0.548404\pi\)
\(480\) 0 0
\(481\) 1.01998e27 0.0701566
\(482\) 0 0
\(483\) −5.00039e26 4.30441e26i −0.0329275 0.0283445i
\(484\) 0 0
\(485\) 1.42623e27i 0.0899293i
\(486\) 0 0
\(487\) 6.41939e27i 0.387650i 0.981036 + 0.193825i \(0.0620894\pi\)
−0.981036 + 0.193825i \(0.937911\pi\)
\(488\) 0 0
\(489\) −2.00259e28 1.72386e28i −1.15837 0.997145i
\(490\) 0 0
\(491\) 1.36908e28 0.758704 0.379352 0.925252i \(-0.376147\pi\)
0.379352 + 0.925252i \(0.376147\pi\)
\(492\) 0 0
\(493\) 7.85561e27 0.417145
\(494\) 0 0
\(495\) −3.85145e26 + 5.79405e25i −0.0196006 + 0.00294868i
\(496\) 0 0
\(497\) 1.09865e28i 0.535943i
\(498\) 0 0
\(499\) 3.87160e28i 1.81065i 0.424718 + 0.905326i \(0.360373\pi\)
−0.424718 + 0.905326i \(0.639627\pi\)
\(500\) 0 0
\(501\) −2.14402e28 + 2.49069e28i −0.961464 + 1.11692i
\(502\) 0 0
\(503\) 8.82865e27 0.379691 0.189846 0.981814i \(-0.439201\pi\)
0.189846 + 0.981814i \(0.439201\pi\)
\(504\) 0 0
\(505\) 2.44759e27 0.100967
\(506\) 0 0
\(507\) 1.57801e28 1.83316e28i 0.624490 0.725464i
\(508\) 0 0
\(509\) 1.32243e28i 0.502153i 0.967967 + 0.251077i \(0.0807846\pi\)
−0.967967 + 0.251077i \(0.919215\pi\)
\(510\) 0 0
\(511\) 1.01044e28i 0.368206i
\(512\) 0 0
\(513\) −1.40926e28 + 2.24073e28i −0.492900 + 0.783713i
\(514\) 0 0
\(515\) −3.70083e27 −0.124258
\(516\) 0 0
\(517\) −6.72278e27 −0.216719
\(518\) 0 0
\(519\) −3.36943e28 2.90045e28i −1.04303 0.897859i
\(520\) 0 0
\(521\) 5.45445e28i 1.62164i 0.585295 + 0.810820i \(0.300979\pi\)
−0.585295 + 0.810820i \(0.699021\pi\)
\(522\) 0 0
\(523\) 5.79200e28i 1.65410i 0.562130 + 0.827049i \(0.309982\pi\)
−0.562130 + 0.827049i \(0.690018\pi\)
\(524\) 0 0
\(525\) −1.94350e28 1.67299e28i −0.533225 0.459008i
\(526\) 0 0
\(527\) −1.72559e28 −0.454911
\(528\) 0 0
\(529\) −3.93281e28 −0.996365
\(530\) 0 0
\(531\) −4.58550e27 3.04809e28i −0.111659 0.742225i
\(532\) 0 0
\(533\) 5.14255e27i 0.120377i
\(534\) 0 0
\(535\) 9.61151e27i 0.216310i
\(536\) 0 0
\(537\) −3.64434e28 + 4.23360e28i −0.788659 + 0.916178i
\(538\) 0 0
\(539\) −2.97587e27 −0.0619343
\(540\) 0 0
\(541\) 3.05578e28 0.611718 0.305859 0.952077i \(-0.401056\pi\)
0.305859 + 0.952077i \(0.401056\pi\)
\(542\) 0 0
\(543\) 3.66771e28 4.26074e28i 0.706312 0.820516i
\(544\) 0 0
\(545\) 2.06332e27i 0.0382300i
\(546\) 0 0
\(547\) 7.14206e28i 1.27338i 0.771121 + 0.636688i \(0.219696\pi\)
−0.771121 + 0.636688i \(0.780304\pi\)
\(548\) 0 0
\(549\) −4.05975e27 2.69861e28i −0.0696610 0.463054i
\(550\) 0 0
\(551\) −2.64811e28 −0.437365
\(552\) 0 0
\(553\) 2.10186e28 0.334188
\(554\) 0 0
\(555\) 2.58362e27 + 2.22402e27i 0.0395506 + 0.0340457i
\(556\) 0 0
\(557\) 3.93564e28i 0.580145i −0.957005 0.290072i \(-0.906321\pi\)
0.957005 0.290072i \(-0.0936794\pi\)
\(558\) 0 0
\(559\) 3.00275e27i 0.0426280i
\(560\) 0 0
\(561\) −6.30590e27 5.42821e27i −0.0862258 0.0742244i
\(562\) 0 0
\(563\) 1.33412e29 1.75734 0.878669 0.477431i \(-0.158432\pi\)
0.878669 + 0.477431i \(0.158432\pi\)
\(564\) 0 0
\(565\) −2.23922e27 −0.0284177
\(566\) 0 0
\(567\) −1.73382e28 5.63214e28i −0.212022 0.688733i
\(568\) 0 0
\(569\) 1.93784e27i 0.0228370i 0.999935 + 0.0114185i \(0.00363470\pi\)
−0.999935 + 0.0114185i \(0.996365\pi\)
\(570\) 0 0
\(571\) 1.02725e29i 1.16680i 0.812183 + 0.583402i \(0.198279\pi\)
−0.812183 + 0.583402i \(0.801721\pi\)
\(572\) 0 0
\(573\) 2.50446e28 2.90941e28i 0.274215 0.318553i
\(574\) 0 0
\(575\) 5.57644e27 0.0588633
\(576\) 0 0
\(577\) 1.55640e29 1.58408 0.792038 0.610472i \(-0.209020\pi\)
0.792038 + 0.610472i \(0.209020\pi\)
\(578\) 0 0
\(579\) 8.23894e28 9.57110e28i 0.808624 0.939371i
\(580\) 0 0
\(581\) 1.08584e29i 1.02782i
\(582\) 0 0
\(583\) 1.55282e28i 0.141776i
\(584\) 0 0
\(585\) −3.57210e27 + 5.37381e26i −0.0314621 + 0.00473310i
\(586\) 0 0
\(587\) 6.21099e27 0.0527790 0.0263895 0.999652i \(-0.491599\pi\)
0.0263895 + 0.999652i \(0.491599\pi\)
\(588\) 0 0
\(589\) 5.81693e28 0.476962
\(590\) 0 0
\(591\) −1.45658e29 1.25384e29i −1.15256 0.992142i
\(592\) 0 0
\(593\) 1.08708e29i 0.830207i −0.909774 0.415104i \(-0.863745\pi\)
0.909774 0.415104i \(-0.136255\pi\)
\(594\) 0 0
\(595\) 1.32794e28i 0.0978927i
\(596\) 0 0
\(597\) −8.69900e28 7.48823e28i −0.619069 0.532903i
\(598\) 0 0
\(599\) −2.09707e29 −1.44089 −0.720445 0.693512i \(-0.756063\pi\)
−0.720445 + 0.693512i \(0.756063\pi\)
\(600\) 0 0
\(601\) −9.73455e26 −0.00645853 −0.00322927 0.999995i \(-0.501028\pi\)
−0.00322927 + 0.999995i \(0.501028\pi\)
\(602\) 0 0
\(603\) −1.56666e28 + 2.35685e27i −0.100379 + 0.0151008i
\(604\) 0 0
\(605\) 2.44471e28i 0.151285i
\(606\) 0 0
\(607\) 2.06415e29i 1.23384i −0.787024 0.616922i \(-0.788379\pi\)
0.787024 0.616922i \(-0.211621\pi\)
\(608\) 0 0
\(609\) 3.84608e28 4.46795e28i 0.222094 0.258004i
\(610\) 0 0
\(611\) −6.23517e28 −0.347869
\(612\) 0 0
\(613\) 3.35444e29 1.80836 0.904181 0.427149i \(-0.140482\pi\)
0.904181 + 0.427149i \(0.140482\pi\)
\(614\) 0 0
\(615\) −1.12131e28 + 1.30261e28i −0.0584167 + 0.0678621i
\(616\) 0 0
\(617\) 7.58509e28i 0.381915i 0.981598 + 0.190957i \(0.0611592\pi\)
−0.981598 + 0.190957i \(0.938841\pi\)
\(618\) 0 0
\(619\) 1.69637e29i 0.825599i 0.910822 + 0.412800i \(0.135449\pi\)
−0.910822 + 0.412800i \(0.864551\pi\)
\(620\) 0 0
\(621\) 1.08476e28 + 6.82241e27i 0.0510356 + 0.0320978i
\(622\) 0 0
\(623\) −1.73420e29 −0.788815
\(624\) 0 0
\(625\) 2.11358e29 0.929561
\(626\) 0 0
\(627\) 2.12571e28 + 1.82984e28i 0.0904054 + 0.0778222i
\(628\) 0 0
\(629\) 7.28269e28i 0.299544i
\(630\) 0 0
\(631\) 4.18428e29i 1.66461i −0.554318 0.832305i \(-0.687021\pi\)
0.554318 0.832305i \(-0.312979\pi\)
\(632\) 0 0
\(633\) 2.67737e28 + 2.30472e28i 0.103031 + 0.0886908i
\(634\) 0 0
\(635\) −2.70869e28 −0.100841
\(636\) 0 0
\(637\) −2.76003e28 −0.0994144
\(638\) 0 0
\(639\) −3.17442e28 2.11012e29i −0.110638 0.735440i
\(640\) 0 0
\(641\) 2.30463e29i 0.777306i −0.921384 0.388653i \(-0.872940\pi\)
0.921384 0.388653i \(-0.127060\pi\)
\(642\) 0 0
\(643\) 3.90657e29i 1.27521i 0.770365 + 0.637603i \(0.220074\pi\)
−0.770365 + 0.637603i \(0.779926\pi\)
\(644\) 0 0
\(645\) 6.54735e27 7.60599e27i 0.0206866 0.0240314i
\(646\) 0 0
\(647\) −2.10310e29 −0.643229 −0.321615 0.946871i \(-0.604226\pi\)
−0.321615 + 0.946871i \(0.604226\pi\)
\(648\) 0 0
\(649\) −3.26609e28 −0.0967072
\(650\) 0 0
\(651\) −8.44843e28 + 9.81446e28i −0.242201 + 0.281362i
\(652\) 0 0
\(653\) 3.73996e28i 0.103819i −0.998652 0.0519096i \(-0.983469\pi\)
0.998652 0.0519096i \(-0.0165308\pi\)
\(654\) 0 0
\(655\) 1.71768e28i 0.0461753i
\(656\) 0 0
\(657\) −2.91954e28 1.94069e29i −0.0760112 0.505265i
\(658\) 0 0
\(659\) 5.10909e29 1.28839 0.644195 0.764862i \(-0.277193\pi\)
0.644195 + 0.764862i \(0.277193\pi\)
\(660\) 0 0
\(661\) −4.57481e29 −1.11753 −0.558763 0.829327i \(-0.688724\pi\)
−0.558763 + 0.829327i \(0.688724\pi\)
\(662\) 0 0
\(663\) −5.84853e28 5.03450e28i −0.138406 0.119142i
\(664\) 0 0
\(665\) 4.47646e28i 0.102638i
\(666\) 0 0
\(667\) 1.28198e28i 0.0284814i
\(668\) 0 0
\(669\) 3.46318e29 + 2.98115e29i 0.745591 + 0.641815i
\(670\) 0 0
\(671\) −2.89162e28 −0.0603329
\(672\) 0 0
\(673\) −4.21002e29 −0.851386 −0.425693 0.904868i \(-0.639970\pi\)
−0.425693 + 0.904868i \(0.639970\pi\)
\(674\) 0 0
\(675\) 4.21614e29 + 2.65165e29i 0.826466 + 0.519789i
\(676\) 0 0
\(677\) 9.75613e28i 0.185394i 0.995694 + 0.0926972i \(0.0295489\pi\)
−0.995694 + 0.0926972i \(0.970451\pi\)
\(678\) 0 0
\(679\) 2.28655e29i 0.421258i
\(680\) 0 0
\(681\) −3.86622e28 + 4.49136e28i −0.0690625 + 0.0802293i
\(682\) 0 0
\(683\) 8.90606e29 1.54265 0.771326 0.636440i \(-0.219594\pi\)
0.771326 + 0.636440i \(0.219594\pi\)
\(684\) 0 0
\(685\) −1.00813e29 −0.169343
\(686\) 0 0
\(687\) −1.60613e29 + 1.86583e29i −0.261659 + 0.303967i
\(688\) 0 0
\(689\) 1.44020e29i 0.227573i
\(690\) 0 0
\(691\) 4.37221e29i 0.670165i 0.942189 + 0.335082i \(0.108764\pi\)
−0.942189 + 0.335082i \(0.891236\pi\)
\(692\) 0 0
\(693\) −6.17469e28 + 9.28910e27i −0.0918155 + 0.0138126i
\(694\) 0 0
\(695\) −3.95570e28 −0.0570667
\(696\) 0 0
\(697\) −3.67180e29 −0.513966
\(698\) 0 0
\(699\) 9.81938e29 + 8.45266e29i 1.33375 + 1.14811i
\(700\) 0 0
\(701\) 1.09604e30i 1.44473i 0.691514 + 0.722363i \(0.256944\pi\)
−0.691514 + 0.722363i \(0.743056\pi\)
\(702\) 0 0
\(703\) 2.45498e29i 0.314063i
\(704\) 0 0
\(705\) −1.57938e29 1.35955e29i −0.196110 0.168814i
\(706\) 0 0
\(707\) 3.92401e29 0.472962
\(708\) 0 0
\(709\) −6.48518e29 −0.758816 −0.379408 0.925229i \(-0.623872\pi\)
−0.379408 + 0.925229i \(0.623872\pi\)
\(710\) 0 0
\(711\) −4.03692e29 + 6.07307e28i −0.458584 + 0.0689886i
\(712\) 0 0
\(713\) 2.81605e28i 0.0310599i
\(714\) 0 0
\(715\) 3.82758e27i 0.00409931i
\(716\) 0 0
\(717\) −4.80507e29 + 5.58201e29i −0.499745 + 0.580549i
\(718\) 0 0
\(719\) −1.11310e30 −1.12430 −0.562149 0.827036i \(-0.690025\pi\)
−0.562149 + 0.827036i \(0.690025\pi\)
\(720\) 0 0
\(721\) −5.93321e29 −0.582062
\(722\) 0 0
\(723\) −1.06709e30 + 1.23962e30i −1.01683 + 1.18124i
\(724\) 0 0
\(725\) 4.98267e29i 0.461225i
\(726\) 0 0
\(727\) 1.24390e30i 1.11860i −0.828966 0.559299i \(-0.811070\pi\)
0.828966 0.559299i \(-0.188930\pi\)
\(728\) 0 0
\(729\) 4.95737e29 + 1.03163e30i 0.433124 + 0.901334i
\(730\) 0 0
\(731\) 2.14397e29 0.182006
\(732\) 0 0
\(733\) −1.82256e30 −1.50346 −0.751729 0.659473i \(-0.770780\pi\)
−0.751729 + 0.659473i \(0.770780\pi\)
\(734\) 0 0
\(735\) −6.99118e28 6.01811e28i −0.0560446 0.0482440i
\(736\) 0 0
\(737\) 1.67871e28i 0.0130787i
\(738\) 0 0
\(739\) 3.87087e29i 0.293118i −0.989202 0.146559i \(-0.953180\pi\)
0.989202 0.146559i \(-0.0468198\pi\)
\(740\) 0 0
\(741\) 1.97153e29 + 1.69712e29i 0.145115 + 0.124917i
\(742\) 0 0
\(743\) −1.65148e29 −0.118166 −0.0590828 0.998253i \(-0.518818\pi\)
−0.0590828 + 0.998253i \(0.518818\pi\)
\(744\) 0 0
\(745\) −3.50012e29 −0.243468
\(746\) 0 0
\(747\) 3.13740e29 + 2.08551e30i 0.212180 + 1.41041i
\(748\) 0 0
\(749\) 1.54093e30i 1.01327i
\(750\) 0 0
\(751\) 1.67639e30i 1.07191i −0.844248 0.535953i \(-0.819952\pi\)
0.844248 0.535953i \(-0.180048\pi\)
\(752\) 0 0
\(753\) 6.97006e29 8.09706e29i 0.433400 0.503477i
\(754\) 0 0
\(755\) 7.06385e28 0.0427168
\(756\) 0 0
\(757\) 1.60704e30 0.945195 0.472597 0.881278i \(-0.343317\pi\)
0.472597 + 0.881278i \(0.343317\pi\)
\(758\) 0 0
\(759\) 8.85847e27 1.02908e28i 0.00506781 0.00588723i
\(760\) 0 0
\(761\) 2.22744e28i 0.0123956i 0.999981 + 0.00619779i \(0.00197283\pi\)
−0.999981 + 0.00619779i \(0.998027\pi\)
\(762\) 0 0
\(763\) 3.30795e29i 0.179082i
\(764\) 0 0
\(765\) −3.83691e28 2.55049e29i −0.0202086 0.134332i
\(766\) 0 0
\(767\) −3.02920e29 −0.155230
\(768\) 0 0
\(769\) −3.43319e29 −0.171187 −0.0855937 0.996330i \(-0.527279\pi\)
−0.0855937 + 0.996330i \(0.527279\pi\)
\(770\) 0 0
\(771\) 1.85041e30 + 1.59286e30i 0.897837 + 0.772871i
\(772\) 0 0
\(773\) 3.61088e30i 1.70501i −0.522715 0.852507i \(-0.675081\pi\)
0.522715 0.852507i \(-0.324919\pi\)
\(774\) 0 0
\(775\) 1.09451e30i 0.502981i
\(776\) 0 0
\(777\) 4.14210e29 + 3.56558e29i 0.185268 + 0.159481i
\(778\) 0 0
\(779\) 1.23776e30 0.538879
\(780\) 0 0
\(781\) −2.26103e29 −0.0958232
\(782\) 0 0
\(783\) −6.09596e29 + 9.69259e29i −0.251503 + 0.399891i
\(784\) 0 0
\(785\) 6.54560e29i 0.262917i
\(786\) 0 0
\(787\) 3.41302e29i 0.133476i −0.997771 0.0667381i \(-0.978741\pi\)
0.997771 0.0667381i \(-0.0212592\pi\)
\(788\) 0 0
\(789\) 3.25746e30 3.78416e30i 1.24043 1.44099i
\(790\) 0 0
\(791\) −3.58995e29 −0.133118
\(792\) 0 0
\(793\) −2.68189e29 −0.0968439
\(794\) 0 0
\(795\) −3.14028e29 + 3.64804e29i −0.110437 + 0.128293i
\(796\) 0 0
\(797\) 2.68234e30i 0.918759i −0.888240 0.459379i \(-0.848072\pi\)
0.888240 0.459379i \(-0.151928\pi\)
\(798\) 0 0
\(799\) 4.45193e30i 1.48528i
\(800\) 0 0
\(801\) 3.33077e30 5.01076e29i 1.08244 0.162840i
\(802\) 0 0
\(803\) −2.07949e29 −0.0658328
\(804\) 0 0
\(805\) −2.16711e28 −0.00668381
\(806\) 0 0
\(807\) −6.27448e28 5.40116e28i −0.0188541 0.0162299i
\(808\) 0 0
\(809\) 3.53882e30i 1.03609i 0.855352 + 0.518046i \(0.173341\pi\)
−0.855352 + 0.518046i \(0.826659\pi\)
\(810\) 0 0
\(811\) 2.99263e30i 0.853755i −0.904309 0.426878i \(-0.859613\pi\)
0.904309 0.426878i \(-0.140387\pi\)
\(812\) 0 0
\(813\) 3.13339e30 + 2.69727e30i 0.871091 + 0.749847i
\(814\) 0 0
\(815\) −8.67897e29 −0.235133
\(816\) 0 0
\(817\) −7.22728e29 −0.190828
\(818\) 0 0
\(819\) −5.72684e29 + 8.61536e28i −0.147378 + 0.0221714i
\(820\) 0 0
\(821\) 5.35892e30i 1.34423i −0.740446 0.672116i \(-0.765386\pi\)
0.740446 0.672116i \(-0.234614\pi\)
\(822\) 0 0
\(823\) 3.16207e30i 0.773168i −0.922254 0.386584i \(-0.873655\pi\)
0.922254 0.386584i \(-0.126345\pi\)
\(824\) 0 0
\(825\) 3.44301e29 3.99971e29i 0.0820676 0.0953372i
\(826\) 0 0
\(827\) 3.29159e29 0.0764889 0.0382444 0.999268i \(-0.487823\pi\)
0.0382444 + 0.999268i \(0.487823\pi\)
\(828\) 0 0
\(829\) 1.48817e30 0.337155 0.168578 0.985688i \(-0.446083\pi\)
0.168578 + 0.985688i \(0.446083\pi\)
\(830\) 0 0
\(831\) −3.67592e30 + 4.27029e30i −0.811999 + 0.943292i
\(832\) 0 0
\(833\) 1.97067e30i 0.424464i
\(834\) 0 0
\(835\) 1.07943e30i 0.226719i
\(836\) 0 0
\(837\) 1.33906e30 2.12911e30i 0.274273 0.436094i
\(838\) 0 0
\(839\) −7.74493e30 −1.54709 −0.773547 0.633739i \(-0.781519\pi\)
−0.773547 + 0.633739i \(0.781519\pi\)
\(840\) 0 0
\(841\) 3.98736e30 0.776834
\(842\) 0 0
\(843\) −5.01994e29 4.32123e29i −0.0953913 0.0821142i
\(844\) 0 0
\(845\) 7.94467e29i 0.147259i
\(846\) 0 0
\(847\) 3.91938e30i 0.708667i
\(848\) 0 0
\(849\) −2.20702e30 1.89983e30i −0.389291 0.335108i
\(850\) 0 0
\(851\) −1.18849e29 −0.0204519
\(852\) 0 0
\(853\) 2.25888e29 0.0379253 0.0189626 0.999820i \(-0.493964\pi\)
0.0189626 + 0.999820i \(0.493964\pi\)
\(854\) 0 0
\(855\) 1.29342e29 + 8.59765e29i 0.0211882 + 0.140843i
\(856\) 0 0
\(857\) 5.81104e30i 0.928871i 0.885607 + 0.464435i \(0.153743\pi\)
−0.885607 + 0.464435i \(0.846257\pi\)
\(858\) 0 0
\(859\) 9.98116e29i 0.155687i −0.996966 0.0778436i \(-0.975197\pi\)
0.996966 0.0778436i \(-0.0248035\pi\)
\(860\) 0 0
\(861\) −1.79770e30 + 2.08837e30i −0.273642 + 0.317888i
\(862\) 0 0
\(863\) −1.39637e30 −0.207438 −0.103719 0.994607i \(-0.533074\pi\)
−0.103719 + 0.994607i \(0.533074\pi\)
\(864\) 0 0
\(865\) −1.46027e30 −0.211721
\(866\) 0 0
\(867\) −1.01545e30 + 1.17964e30i −0.143701 + 0.166936i
\(868\) 0 0
\(869\) 4.32564e29i 0.0597506i
\(870\) 0 0
\(871\) 1.55695e29i 0.0209934i
\(872\) 0 0
\(873\) 6.60669e29 + 4.39163e30i 0.0869630 + 0.578065i
\(874\) 0 0
\(875\) −1.70499e30 −0.219098
\(876\) 0 0
\(877\) 3.19505e29 0.0400850 0.0200425 0.999799i \(-0.493620\pi\)
0.0200425 + 0.999799i \(0.493620\pi\)
\(878\) 0 0
\(879\) 3.85747e30 + 3.32056e30i 0.472519 + 0.406751i
\(880\) 0 0
\(881\) 5.17345e30i 0.618776i −0.950936 0.309388i \(-0.899876\pi\)
0.950936 0.309388i \(-0.100124\pi\)
\(882\) 0 0
\(883\) 4.69512e30i 0.548352i −0.961679 0.274176i \(-0.911595\pi\)
0.961679 0.274176i \(-0.0884051\pi\)
\(884\) 0 0
\(885\) −7.67300e29 6.60503e29i −0.0875106 0.0753304i
\(886\) 0 0
\(887\) 1.10473e31 1.23043 0.615215 0.788359i \(-0.289069\pi\)
0.615215 + 0.788359i \(0.289069\pi\)
\(888\) 0 0
\(889\) −4.34261e30 −0.472370
\(890\) 0 0
\(891\) 1.15909e30 3.56820e29i 0.123141 0.0379082i
\(892\) 0 0
\(893\) 1.50074e31i 1.55727i
\(894\) 0 0
\(895\) 1.83479e30i 0.185971i
\(896\) 0 0
\(897\) 8.21597e28 9.54441e28i 0.00813463 0.00944993i
\(898\) 0 0
\(899\) 2.51620e30 0.243370
\(900\) 0 0
\(901\) −1.02830e31 −0.971654
\(902\) 0 0
\(903\) 1.04968e30 1.21940e30i 0.0969025 0.112571i
\(904\) 0 0
\(905\) 1.84655e30i 0.166553i
\(906\) 0 0
\(907\) 1.08940e31i 0.960092i 0.877243 + 0.480046i \(0.159380\pi\)
−0.877243 + 0.480046i \(0.840620\pi\)
\(908\) 0 0
\(909\) −7.53660e30 + 1.13379e30i −0.649015 + 0.0976367i
\(910\) 0 0
\(911\) 1.98401e31 1.66956 0.834778 0.550586i \(-0.185596\pi\)
0.834778 + 0.550586i \(0.185596\pi\)
\(912\) 0 0
\(913\) 2.23466e30 0.183768
\(914\) 0 0
\(915\) −6.79326e29 5.84773e29i −0.0545955 0.0469966i
\(916\) 0 0
\(917\) 2.75381e30i 0.216300i
\(918\) 0 0
\(919\) 2.28095e31i 1.75107i 0.483155 + 0.875535i \(0.339491\pi\)
−0.483155 + 0.875535i \(0.660509\pi\)
\(920\) 0 0
\(921\) −3.15623e30 2.71692e30i −0.236833 0.203869i
\(922\) 0 0
\(923\) −2.09704e30 −0.153811
\(924\) 0 0
\(925\) −4.61927e30 −0.331196
\(926\) 0 0
\(927\) 1.13955e31 1.71433e30i 0.798726 0.120159i
\(928\) 0 0
\(929\) 1.79281e31i 1.22848i −0.789118 0.614241i \(-0.789462\pi\)
0.789118 0.614241i \(-0.210538\pi\)
\(930\) 0 0
\(931\) 6.64308e30i 0.445039i
\(932\) 0 0
\(933\) −7.85580e30 + 9.12601e30i −0.514556 + 0.597755i
\(934\) 0 0
\(935\) −2.73290e29 −0.0175026
\(936\) 0 0
\(937\) −1.61944e31 −1.01414 −0.507070 0.861905i \(-0.669272\pi\)
−0.507070 + 0.861905i \(0.669272\pi\)
\(938\) 0 0
\(939\) 2.64408e30 3.07160e30i 0.161914 0.188094i
\(940\) 0 0
\(941\) 3.23862e31i 1.93940i 0.244290 + 0.969702i \(0.421445\pi\)
−0.244290 + 0.969702i \(0.578555\pi\)
\(942\) 0 0
\(943\) 5.99212e29i 0.0350920i
\(944\) 0 0
\(945\) −1.63847e30 1.03048e30i −0.0938435 0.0590210i
\(946\) 0 0
\(947\) 2.43921e29 0.0136639 0.00683194 0.999977i \(-0.497825\pi\)
0.00683194 + 0.999977i \(0.497825\pi\)
\(948\) 0 0
\(949\) −1.92866e30 −0.105672
\(950\) 0 0
\(951\) −1.32987e30 1.14477e30i −0.0712709 0.0613510i
\(952\) 0 0
\(953\) 3.54176e31i 1.85671i 0.371699 + 0.928353i \(0.378775\pi\)
−0.371699 + 0.928353i \(0.621225\pi\)
\(954\) 0 0
\(955\) 1.26090e30i 0.0646616i
\(956\) 0 0
\(957\) 9.19504e29 + 7.91523e29i 0.0461295 + 0.0397089i
\(958\) 0 0
\(959\) −1.61625e31 −0.793257
\(960\) 0 0
\(961\) 1.52983e31 0.734596
\(962\) 0 0
\(963\) −4.45232e30 2.95957e31i −0.209175 1.39044i
\(964\) 0 0
\(965\) 4.14800e30i 0.190679i
\(966\) 0 0
\(967\) 2.46691e31i 1.10962i 0.831977 + 0.554811i \(0.187209\pi\)
−0.831977 + 0.554811i \(0.812791\pi\)
\(968\) 0 0
\(969\) −1.21175e31 + 1.40768e31i −0.533351 + 0.619589i
\(970\) 0 0
\(971\) −2.25722e31 −0.972235 −0.486118 0.873893i \(-0.661587\pi\)
−0.486118 + 0.873893i \(0.661587\pi\)
\(972\) 0 0
\(973\) −6.34182e30 −0.267319
\(974\) 0 0
\(975\) 3.19329e30 3.70961e30i 0.131732 0.153031i
\(976\) 0 0
\(977\) 2.32937e31i 0.940471i −0.882541 0.470235i \(-0.844169\pi\)
0.882541 0.470235i \(-0.155831\pi\)
\(978\) 0 0
\(979\) 3.56899e30i 0.141035i
\(980\) 0 0
\(981\) 9.55790e29 + 6.35337e30i 0.0369690 + 0.245742i
\(982\) 0 0
\(983\) 4.85194e30 0.183697 0.0918487 0.995773i \(-0.470722\pi\)
0.0918487 + 0.995773i \(0.470722\pi\)
\(984\) 0 0
\(985\) −6.31262e30 −0.233953
\(986\) 0 0
\(987\) −2.53208e31 2.17965e31i −0.918643 0.790781i
\(988\) 0 0
\(989\) 3.49881e29i 0.0124268i
\(990\) 0 0
\(991\) 4.92717e31i 1.71326i 0.515927 + 0.856632i \(0.327448\pi\)
−0.515927 + 0.856632i \(0.672552\pi\)
\(992\) 0 0
\(993\) −2.58671e31 2.22668e31i −0.880605 0.758037i
\(994\) 0 0
\(995\) −3.77004e30 −0.125662
\(996\) 0 0
\(997\) 4.46075e31 1.45582 0.727911 0.685671i \(-0.240491\pi\)
0.727911 + 0.685671i \(0.240491\pi\)
\(998\) 0 0
\(999\) −8.98569e30 5.65137e30i −0.287153 0.180599i
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 48.22.c.c.47.8 yes 28
3.2 odd 2 inner 48.22.c.c.47.22 yes 28
4.3 odd 2 inner 48.22.c.c.47.21 yes 28
12.11 even 2 inner 48.22.c.c.47.7 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.22.c.c.47.7 28 12.11 even 2 inner
48.22.c.c.47.8 yes 28 1.1 even 1 trivial
48.22.c.c.47.21 yes 28 4.3 odd 2 inner
48.22.c.c.47.22 yes 28 3.2 odd 2 inner