Properties

Label 72.22.a.d.1.1
Level $72$
Weight $22$
Character 72.1
Self dual yes
Analytic conductor $201.224$
Analytic rank $1$
Dimension $3$
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] = [72,22,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(201.223687887\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2295485x - 828958533 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1671.27\) of defining polynomial
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.18179e7 q^{5} -6.87132e8 q^{7} +O(q^{10})\) \(q-4.18179e7 q^{5} -6.87132e8 q^{7} -1.49871e9 q^{11} -8.84777e9 q^{13} +9.41434e12 q^{17} -4.23870e13 q^{19} -1.09316e14 q^{23} +1.27190e15 q^{25} -1.18227e15 q^{29} -1.95307e15 q^{31} +2.87345e16 q^{35} +4.20325e16 q^{37} +6.92189e16 q^{41} +8.95366e16 q^{43} +5.26493e17 q^{47} -8.63955e16 q^{49} +2.24753e18 q^{53} +6.26728e16 q^{55} +2.41265e17 q^{59} +3.17652e18 q^{61} +3.69996e17 q^{65} -2.04213e19 q^{67} +3.92177e19 q^{71} -4.75993e19 q^{73} +1.02981e18 q^{77} +8.63125e18 q^{79} -2.65834e20 q^{83} -3.93688e20 q^{85} +4.57167e20 q^{89} +6.07959e18 q^{91} +1.77254e21 q^{95} -9.21686e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2080026 q^{5} - 1205282064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2080026 q^{5} - 1205282064 q^{7} - 13839247500 q^{11} + 718855551690 q^{13} - 2135189843046 q^{17} - 40122324686988 q^{19} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 15\!\cdots\!22 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −4.18179e7 −1.91504 −0.957520 0.288368i \(-0.906887\pi\)
−0.957520 + 0.288368i \(0.906887\pi\)
\(6\) 0 0
\(7\) −6.87132e8 −0.919413 −0.459707 0.888071i \(-0.652045\pi\)
−0.459707 + 0.888071i \(0.652045\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.49871e9 −0.0174218 −0.00871090 0.999962i \(-0.502773\pi\)
−0.00871090 + 0.999962i \(0.502773\pi\)
\(12\) 0 0
\(13\) −8.84777e9 −0.0178004 −0.00890018 0.999960i \(-0.502833\pi\)
−0.00890018 + 0.999960i \(0.502833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.41434e12 1.13260 0.566299 0.824200i \(-0.308375\pi\)
0.566299 + 0.824200i \(0.308375\pi\)
\(18\) 0 0
\(19\) −4.23870e13 −1.58606 −0.793030 0.609183i \(-0.791498\pi\)
−0.793030 + 0.609183i \(0.791498\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.09316e14 −0.550225 −0.275112 0.961412i \(-0.588715\pi\)
−0.275112 + 0.961412i \(0.588715\pi\)
\(24\) 0 0
\(25\) 1.27190e15 2.66738
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.18227e15 −0.521842 −0.260921 0.965360i \(-0.584026\pi\)
−0.260921 + 0.965360i \(0.584026\pi\)
\(30\) 0 0
\(31\) −1.95307e15 −0.427977 −0.213988 0.976836i \(-0.568645\pi\)
−0.213988 + 0.976836i \(0.568645\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.87345e16 1.76071
\(36\) 0 0
\(37\) 4.20325e16 1.43704 0.718518 0.695509i \(-0.244821\pi\)
0.718518 + 0.695509i \(0.244821\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92189e16 0.805367 0.402684 0.915339i \(-0.368077\pi\)
0.402684 + 0.915339i \(0.368077\pi\)
\(42\) 0 0
\(43\) 8.95366e16 0.631803 0.315902 0.948792i \(-0.397693\pi\)
0.315902 + 0.948792i \(0.397693\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.26493e17 1.46004 0.730021 0.683425i \(-0.239510\pi\)
0.730021 + 0.683425i \(0.239510\pi\)
\(48\) 0 0
\(49\) −8.63955e16 −0.154679
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.24753e18 1.76526 0.882629 0.470070i \(-0.155771\pi\)
0.882629 + 0.470070i \(0.155771\pi\)
\(54\) 0 0
\(55\) 6.26728e16 0.0333634
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.41265e17 0.0614538 0.0307269 0.999528i \(-0.490218\pi\)
0.0307269 + 0.999528i \(0.490218\pi\)
\(60\) 0 0
\(61\) 3.17652e18 0.570148 0.285074 0.958505i \(-0.407982\pi\)
0.285074 + 0.958505i \(0.407982\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.69996e17 0.0340884
\(66\) 0 0
\(67\) −2.04213e19 −1.36867 −0.684334 0.729169i \(-0.739907\pi\)
−0.684334 + 0.729169i \(0.739907\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.92177e19 1.42978 0.714891 0.699236i \(-0.246476\pi\)
0.714891 + 0.699236i \(0.246476\pi\)
\(72\) 0 0
\(73\) −4.75993e19 −1.29631 −0.648157 0.761507i \(-0.724460\pi\)
−0.648157 + 0.761507i \(0.724460\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.02981e18 0.0160178
\(78\) 0 0
\(79\) 8.63125e18 0.102563 0.0512813 0.998684i \(-0.483669\pi\)
0.0512813 + 0.998684i \(0.483669\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.65834e20 −1.88057 −0.940287 0.340382i \(-0.889444\pi\)
−0.940287 + 0.340382i \(0.889444\pi\)
\(84\) 0 0
\(85\) −3.93688e20 −2.16897
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.57167e20 1.55410 0.777051 0.629438i \(-0.216715\pi\)
0.777051 + 0.629438i \(0.216715\pi\)
\(90\) 0 0
\(91\) 6.07959e18 0.0163659
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.77254e21 3.03737
\(96\) 0 0
\(97\) −9.21686e20 −1.26905 −0.634527 0.772901i \(-0.718805\pi\)
−0.634527 + 0.772901i \(0.718805\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.57744e21 −1.42095 −0.710475 0.703722i \(-0.751520\pi\)
−0.710475 + 0.703722i \(0.751520\pi\)
\(102\) 0 0
\(103\) 1.20014e21 0.879919 0.439959 0.898018i \(-0.354993\pi\)
0.439959 + 0.898018i \(0.354993\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.40780e21 1.67473 0.837365 0.546644i \(-0.184095\pi\)
0.837365 + 0.546644i \(0.184095\pi\)
\(108\) 0 0
\(109\) 1.46243e21 0.591693 0.295846 0.955236i \(-0.404398\pi\)
0.295846 + 0.955236i \(0.404398\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.53398e21 0.425105 0.212552 0.977150i \(-0.431822\pi\)
0.212552 + 0.977150i \(0.431822\pi\)
\(114\) 0 0
\(115\) 4.57136e21 1.05370
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.46889e21 −1.04133
\(120\) 0 0
\(121\) −7.39800e21 −0.999696
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.32481e22 −3.19309
\(126\) 0 0
\(127\) 1.28428e21 0.104405 0.0522024 0.998637i \(-0.483376\pi\)
0.0522024 + 0.998637i \(0.483376\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.22923e22 −0.721577 −0.360789 0.932648i \(-0.617492\pi\)
−0.360789 + 0.932648i \(0.617492\pi\)
\(132\) 0 0
\(133\) 2.91254e22 1.45824
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.02076e22 0.374420 0.187210 0.982320i \(-0.440055\pi\)
0.187210 + 0.982320i \(0.440055\pi\)
\(138\) 0 0
\(139\) −5.48653e22 −1.72839 −0.864197 0.503154i \(-0.832173\pi\)
−0.864197 + 0.503154i \(0.832173\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.32602e19 0.000310114 0
\(144\) 0 0
\(145\) 4.94403e22 0.999348
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.46082e22 0.525682 0.262841 0.964839i \(-0.415341\pi\)
0.262841 + 0.964839i \(0.415341\pi\)
\(150\) 0 0
\(151\) −3.56145e21 −0.0470294 −0.0235147 0.999723i \(-0.507486\pi\)
−0.0235147 + 0.999723i \(0.507486\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.16734e22 0.819592
\(156\) 0 0
\(157\) 2.21166e23 1.93987 0.969933 0.243374i \(-0.0782541\pi\)
0.969933 + 0.243374i \(0.0782541\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.51143e22 0.505884
\(162\) 0 0
\(163\) −3.21431e23 −1.90159 −0.950797 0.309814i \(-0.899733\pi\)
−0.950797 + 0.309814i \(0.899733\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.92776e23 0.884158 0.442079 0.896976i \(-0.354241\pi\)
0.442079 + 0.896976i \(0.354241\pi\)
\(168\) 0 0
\(169\) −2.46986e23 −0.999683
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.48749e23 −0.470946 −0.235473 0.971881i \(-0.575664\pi\)
−0.235473 + 0.971881i \(0.575664\pi\)
\(174\) 0 0
\(175\) −8.73966e23 −2.45242
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.04507e23 0.673970 0.336985 0.941510i \(-0.390593\pi\)
0.336985 + 0.941510i \(0.390593\pi\)
\(180\) 0 0
\(181\) −2.74401e23 −0.540457 −0.270228 0.962796i \(-0.587099\pi\)
−0.270228 + 0.962796i \(0.587099\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.75771e24 −2.75198
\(186\) 0 0
\(187\) −1.41093e22 −0.0197319
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.88356e22 −0.0434890 −0.0217445 0.999764i \(-0.506922\pi\)
−0.0217445 + 0.999764i \(0.506922\pi\)
\(192\) 0 0
\(193\) 4.72989e23 0.474787 0.237394 0.971414i \(-0.423707\pi\)
0.237394 + 0.971414i \(0.423707\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.09009e22 −0.0735653 −0.0367826 0.999323i \(-0.511711\pi\)
−0.0367826 + 0.999323i \(0.511711\pi\)
\(198\) 0 0
\(199\) 9.81119e23 0.714109 0.357055 0.934084i \(-0.383781\pi\)
0.357055 + 0.934084i \(0.383781\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 8.12378e23 0.479789
\(204\) 0 0
\(205\) −2.89459e24 −1.54231
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.35256e22 0.0276320
\(210\) 0 0
\(211\) 1.28388e24 0.505312 0.252656 0.967556i \(-0.418696\pi\)
0.252656 + 0.967556i \(0.418696\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.74424e24 −1.20993
\(216\) 0 0
\(217\) 1.34202e24 0.393488
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.32959e22 −0.0201607
\(222\) 0 0
\(223\) 1.11219e24 0.244893 0.122446 0.992475i \(-0.460926\pi\)
0.122446 + 0.992475i \(0.460926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.79086e24 −0.327182 −0.163591 0.986528i \(-0.552308\pi\)
−0.163591 + 0.986528i \(0.552308\pi\)
\(228\) 0 0
\(229\) −5.98363e24 −0.996993 −0.498496 0.866892i \(-0.666114\pi\)
−0.498496 + 0.866892i \(0.666114\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.20293e24 −1.13955 −0.569773 0.821802i \(-0.692969\pi\)
−0.569773 + 0.821802i \(0.692969\pi\)
\(234\) 0 0
\(235\) −2.20169e25 −2.79604
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.01227e24 −0.745900 −0.372950 0.927851i \(-0.621654\pi\)
−0.372950 + 0.927851i \(0.621654\pi\)
\(240\) 0 0
\(241\) −1.03693e25 −1.01058 −0.505288 0.862951i \(-0.668614\pi\)
−0.505288 + 0.862951i \(0.668614\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.61288e24 0.296217
\(246\) 0 0
\(247\) 3.75030e23 0.0282324
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.28168e25 −1.45104 −0.725522 0.688199i \(-0.758402\pi\)
−0.725522 + 0.688199i \(0.758402\pi\)
\(252\) 0 0
\(253\) 1.63832e23 0.00958591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.57165e25 0.779935 0.389968 0.920829i \(-0.372486\pi\)
0.389968 + 0.920829i \(0.372486\pi\)
\(258\) 0 0
\(259\) −2.88819e25 −1.32123
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.60373e24 −0.0624593 −0.0312296 0.999512i \(-0.509942\pi\)
−0.0312296 + 0.999512i \(0.509942\pi\)
\(264\) 0 0
\(265\) −9.39870e25 −3.38054
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.92003e25 1.51206 0.756029 0.654538i \(-0.227137\pi\)
0.756029 + 0.654538i \(0.227137\pi\)
\(270\) 0 0
\(271\) −7.98786e24 −0.227119 −0.113559 0.993531i \(-0.536225\pi\)
−0.113559 + 0.993531i \(0.536225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.90621e24 −0.0464705
\(276\) 0 0
\(277\) 2.26388e25 0.511465 0.255733 0.966748i \(-0.417683\pi\)
0.255733 + 0.966748i \(0.417683\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.09532e24 −0.137897 −0.0689486 0.997620i \(-0.521964\pi\)
−0.0689486 + 0.997620i \(0.521964\pi\)
\(282\) 0 0
\(283\) 1.98336e24 0.0357804 0.0178902 0.999840i \(-0.494305\pi\)
0.0178902 + 0.999840i \(0.494305\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.75625e25 −0.740465
\(288\) 0 0
\(289\) 1.95378e25 0.282780
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.17672e26 −1.47422 −0.737109 0.675774i \(-0.763810\pi\)
−0.737109 + 0.675774i \(0.763810\pi\)
\(294\) 0 0
\(295\) −1.00892e25 −0.117687
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.67200e23 0.00979419
\(300\) 0 0
\(301\) −6.15235e25 −0.580888
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.32835e26 −1.09186
\(306\) 0 0
\(307\) 7.71265e25 0.591903 0.295952 0.955203i \(-0.404363\pi\)
0.295952 + 0.955203i \(0.404363\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.96322e25 −0.533463 −0.266732 0.963771i \(-0.585944\pi\)
−0.266732 + 0.963771i \(0.585944\pi\)
\(312\) 0 0
\(313\) −5.68844e25 −0.356269 −0.178134 0.984006i \(-0.557006\pi\)
−0.178134 + 0.984006i \(0.557006\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.51016e26 0.827755 0.413878 0.910333i \(-0.364174\pi\)
0.413878 + 0.910333i \(0.364174\pi\)
\(318\) 0 0
\(319\) 1.77188e24 0.00909143
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.99045e26 −1.79637
\(324\) 0 0
\(325\) −1.12535e25 −0.0474802
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.61770e26 −1.34238
\(330\) 0 0
\(331\) −3.29882e26 −1.14859 −0.574295 0.818649i \(-0.694724\pi\)
−0.574295 + 0.818649i \(0.694724\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.53977e26 2.62105
\(336\) 0 0
\(337\) 4.68433e26 1.35062 0.675310 0.737534i \(-0.264010\pi\)
0.675310 + 0.737534i \(0.264010\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.92708e24 0.00745613
\(342\) 0 0
\(343\) 4.43160e26 1.06163
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.05861e26 1.28503 0.642515 0.766273i \(-0.277891\pi\)
0.642515 + 0.766273i \(0.277891\pi\)
\(348\) 0 0
\(349\) 8.48298e25 0.169388 0.0846938 0.996407i \(-0.473009\pi\)
0.0846938 + 0.996407i \(0.473009\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.10413e26 1.08141 0.540704 0.841213i \(-0.318158\pi\)
0.540704 + 0.841213i \(0.318158\pi\)
\(354\) 0 0
\(355\) −1.64000e27 −2.73809
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.20073e26 0.920346 0.460173 0.887829i \(-0.347787\pi\)
0.460173 + 0.887829i \(0.347787\pi\)
\(360\) 0 0
\(361\) 1.08245e27 1.51558
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.99050e27 2.48249
\(366\) 0 0
\(367\) 1.05311e27 1.24017 0.620084 0.784536i \(-0.287099\pi\)
0.620084 + 0.784536i \(0.287099\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.54435e27 −1.62300
\(372\) 0 0
\(373\) 9.54206e26 0.947762 0.473881 0.880589i \(-0.342853\pi\)
0.473881 + 0.880589i \(0.342853\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.04605e25 0.00928897
\(378\) 0 0
\(379\) 7.26138e26 0.609968 0.304984 0.952357i \(-0.401349\pi\)
0.304984 + 0.952357i \(0.401349\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.16288e27 −0.874879 −0.437439 0.899248i \(-0.644115\pi\)
−0.437439 + 0.899248i \(0.644115\pi\)
\(384\) 0 0
\(385\) −4.30645e25 −0.0306748
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.26901e27 −0.810951 −0.405476 0.914106i \(-0.632894\pi\)
−0.405476 + 0.914106i \(0.632894\pi\)
\(390\) 0 0
\(391\) −1.02913e27 −0.623184
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.60941e26 −0.196412
\(396\) 0 0
\(397\) 3.15060e27 1.62590 0.812948 0.582337i \(-0.197862\pi\)
0.812948 + 0.582337i \(0.197862\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.77564e26 0.128928 0.0644640 0.997920i \(-0.479466\pi\)
0.0644640 + 0.997920i \(0.479466\pi\)
\(402\) 0 0
\(403\) 1.72803e25 0.00761814
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.29944e25 −0.0250357
\(408\) 0 0
\(409\) −1.19865e27 −0.452478 −0.226239 0.974072i \(-0.572643\pi\)
−0.226239 + 0.974072i \(0.572643\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.65781e26 −0.0565015
\(414\) 0 0
\(415\) 1.11166e28 3.60137
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.80111e27 −1.69928 −0.849638 0.527366i \(-0.823180\pi\)
−0.849638 + 0.527366i \(0.823180\pi\)
\(420\) 0 0
\(421\) 5.69555e26 0.158699 0.0793494 0.996847i \(-0.474716\pi\)
0.0793494 + 0.996847i \(0.474716\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.19741e28 3.02107
\(426\) 0 0
\(427\) −2.18269e27 −0.524202
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.82453e27 0.397320 0.198660 0.980069i \(-0.436341\pi\)
0.198660 + 0.980069i \(0.436341\pi\)
\(432\) 0 0
\(433\) 4.12366e27 0.855381 0.427690 0.903925i \(-0.359327\pi\)
0.427690 + 0.903925i \(0.359327\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.63356e27 0.872689
\(438\) 0 0
\(439\) 6.93325e27 1.24468 0.622342 0.782745i \(-0.286181\pi\)
0.622342 + 0.782745i \(0.286181\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.53594e27 0.250689 0.125345 0.992113i \(-0.459996\pi\)
0.125345 + 0.992113i \(0.459996\pi\)
\(444\) 0 0
\(445\) −1.91178e28 −2.97617
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.32087e27 0.187186 0.0935929 0.995611i \(-0.470165\pi\)
0.0935929 + 0.995611i \(0.470165\pi\)
\(450\) 0 0
\(451\) −1.03739e26 −0.0140310
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.54236e26 −0.0313413
\(456\) 0 0
\(457\) 2.28794e27 0.269354 0.134677 0.990890i \(-0.457000\pi\)
0.134677 + 0.990890i \(0.457000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.47625e27 −0.910635 −0.455317 0.890329i \(-0.650474\pi\)
−0.455317 + 0.890329i \(0.650474\pi\)
\(462\) 0 0
\(463\) −9.11529e27 −0.935772 −0.467886 0.883789i \(-0.654984\pi\)
−0.467886 + 0.883789i \(0.654984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.41181e28 −1.32419 −0.662093 0.749421i \(-0.730332\pi\)
−0.662093 + 0.749421i \(0.730332\pi\)
\(468\) 0 0
\(469\) 1.40321e28 1.25837
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.34189e26 −0.0110072
\(474\) 0 0
\(475\) −5.39121e28 −4.23062
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.20208e28 −1.58238 −0.791189 0.611572i \(-0.790537\pi\)
−0.791189 + 0.611572i \(0.790537\pi\)
\(480\) 0 0
\(481\) −3.71894e26 −0.0255797
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.85430e28 2.43029
\(486\) 0 0
\(487\) −1.76771e28 −1.06748 −0.533738 0.845650i \(-0.679213\pi\)
−0.533738 + 0.845650i \(0.679213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.25882e27 0.291429 0.145715 0.989327i \(-0.453452\pi\)
0.145715 + 0.989327i \(0.453452\pi\)
\(492\) 0 0
\(493\) −1.11303e28 −0.591038
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.69478e28 −1.31456
\(498\) 0 0
\(499\) −1.70312e28 −0.796508 −0.398254 0.917275i \(-0.630384\pi\)
−0.398254 + 0.917275i \(0.630384\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.74942e27 −0.333277 −0.166639 0.986018i \(-0.553291\pi\)
−0.166639 + 0.986018i \(0.553291\pi\)
\(504\) 0 0
\(505\) 6.59654e28 2.72118
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.30034e28 0.873483 0.436742 0.899587i \(-0.356132\pi\)
0.436742 + 0.899587i \(0.356132\pi\)
\(510\) 0 0
\(511\) 3.27070e28 1.19185
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.01876e28 −1.68508
\(516\) 0 0
\(517\) −7.89058e26 −0.0254366
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.09418e27 0.181184 0.0905918 0.995888i \(-0.471124\pi\)
0.0905918 + 0.995888i \(0.471124\pi\)
\(522\) 0 0
\(523\) 2.18205e28 0.623155 0.311578 0.950221i \(-0.399143\pi\)
0.311578 + 0.950221i \(0.399143\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.83869e28 −0.484726
\(528\) 0 0
\(529\) −2.75217e28 −0.697253
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.12433e26 −0.0143358
\(534\) 0 0
\(535\) −1.42507e29 −3.20718
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.29481e26 0.00269479
\(540\) 0 0
\(541\) −6.59608e27 −0.132043 −0.0660214 0.997818i \(-0.521031\pi\)
−0.0660214 + 0.997818i \(0.521031\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.11557e28 −1.13311
\(546\) 0 0
\(547\) 4.42458e28 0.788870 0.394435 0.918924i \(-0.370940\pi\)
0.394435 + 0.918924i \(0.370940\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.01130e28 0.827673
\(552\) 0 0
\(553\) −5.93081e27 −0.0942975
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.40727e28 −0.649666 −0.324833 0.945771i \(-0.605308\pi\)
−0.324833 + 0.945771i \(0.605308\pi\)
\(558\) 0 0
\(559\) −7.92199e26 −0.0112463
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.93890e28 −0.914013 −0.457006 0.889463i \(-0.651078\pi\)
−0.457006 + 0.889463i \(0.651078\pi\)
\(564\) 0 0
\(565\) −6.41479e28 −0.814093
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.87409e28 0.220858 0.110429 0.993884i \(-0.464778\pi\)
0.110429 + 0.993884i \(0.464778\pi\)
\(570\) 0 0
\(571\) −3.73641e28 −0.424400 −0.212200 0.977226i \(-0.568063\pi\)
−0.212200 + 0.977226i \(0.568063\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.39039e29 −1.46766
\(576\) 0 0
\(577\) −1.05657e29 −1.07536 −0.537680 0.843149i \(-0.680699\pi\)
−0.537680 + 0.843149i \(0.680699\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.82663e29 1.72903
\(582\) 0 0
\(583\) −3.36838e27 −0.0307540
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.31403e29 1.11662 0.558310 0.829632i \(-0.311450\pi\)
0.558310 + 0.829632i \(0.311450\pi\)
\(588\) 0 0
\(589\) 8.27847e28 0.678797
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.29428e29 1.75215 0.876075 0.482174i \(-0.160153\pi\)
0.876075 + 0.482174i \(0.160153\pi\)
\(594\) 0 0
\(595\) 2.70516e29 1.99418
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.80553e27 −0.0124058 −0.00620288 0.999981i \(-0.501974\pi\)
−0.00620288 + 0.999981i \(0.501974\pi\)
\(600\) 0 0
\(601\) 9.79897e28 0.650128 0.325064 0.945692i \(-0.394614\pi\)
0.325064 + 0.945692i \(0.394614\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.09369e29 1.91446
\(606\) 0 0
\(607\) −9.51352e28 −0.568670 −0.284335 0.958725i \(-0.591773\pi\)
−0.284335 + 0.958725i \(0.591773\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.65829e27 −0.0259893
\(612\) 0 0
\(613\) −1.06132e29 −0.572151 −0.286075 0.958207i \(-0.592351\pi\)
−0.286075 + 0.958207i \(0.592351\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.24152e29 −0.625113 −0.312557 0.949899i \(-0.601185\pi\)
−0.312557 + 0.949899i \(0.601185\pi\)
\(618\) 0 0
\(619\) 3.70280e28 0.180210 0.0901049 0.995932i \(-0.471280\pi\)
0.0901049 + 0.995932i \(0.471280\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.14134e29 −1.42886
\(624\) 0 0
\(625\) 7.83875e29 3.44752
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.95708e29 1.62758
\(630\) 0 0
\(631\) −3.98518e29 −1.58540 −0.792702 0.609610i \(-0.791326\pi\)
−0.792702 + 0.609610i \(0.791326\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.37060e28 −0.199939
\(636\) 0 0
\(637\) 7.64408e26 0.00275335
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.11810e29 −1.05167 −0.525836 0.850586i \(-0.676247\pi\)
−0.525836 + 0.850586i \(0.676247\pi\)
\(642\) 0 0
\(643\) 5.91629e29 1.93123 0.965616 0.259973i \(-0.0837138\pi\)
0.965616 + 0.259973i \(0.0837138\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.57623e29 −0.787934 −0.393967 0.919125i \(-0.628898\pi\)
−0.393967 + 0.919125i \(0.628898\pi\)
\(648\) 0 0
\(649\) −3.61586e26 −0.00107064
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.06486e29 1.12838 0.564192 0.825644i \(-0.309188\pi\)
0.564192 + 0.825644i \(0.309188\pi\)
\(654\) 0 0
\(655\) 5.14037e29 1.38185
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.76184e29 0.444294 0.222147 0.975013i \(-0.428694\pi\)
0.222147 + 0.975013i \(0.428694\pi\)
\(660\) 0 0
\(661\) −2.99970e29 −0.732760 −0.366380 0.930465i \(-0.619403\pi\)
−0.366380 + 0.930465i \(0.619403\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.21797e30 −2.79259
\(666\) 0 0
\(667\) 1.29241e29 0.287130
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.76066e27 −0.00993301
\(672\) 0 0
\(673\) 5.25225e29 1.06215 0.531077 0.847324i \(-0.321788\pi\)
0.531077 + 0.847324i \(0.321788\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.62662e29 −0.689162 −0.344581 0.938757i \(-0.611979\pi\)
−0.344581 + 0.938757i \(0.611979\pi\)
\(678\) 0 0
\(679\) 6.33320e29 1.16678
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.66967e29 0.462424 0.231212 0.972903i \(-0.425731\pi\)
0.231212 + 0.972903i \(0.425731\pi\)
\(684\) 0 0
\(685\) −4.26863e29 −0.717030
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.98856e28 −0.0314222
\(690\) 0 0
\(691\) −5.15096e29 −0.789531 −0.394765 0.918782i \(-0.629174\pi\)
−0.394765 + 0.918782i \(0.629174\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.29436e30 3.30994
\(696\) 0 0
\(697\) 6.51650e29 0.912158
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.60995e29 0.607655 0.303827 0.952727i \(-0.401735\pi\)
0.303827 + 0.952727i \(0.401735\pi\)
\(702\) 0 0
\(703\) −1.78163e30 −2.27922
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.08391e30 1.30644
\(708\) 0 0
\(709\) −1.05247e30 −1.23147 −0.615733 0.787955i \(-0.711140\pi\)
−0.615733 + 0.787955i \(0.711140\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.13501e29 0.235483
\(714\) 0 0
\(715\) −5.54514e26 −0.000593881 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.88703e29 −0.291607 −0.145804 0.989314i \(-0.546577\pi\)
−0.145804 + 0.989314i \(0.546577\pi\)
\(720\) 0 0
\(721\) −8.24658e29 −0.809009
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.50374e30 −1.39195
\(726\) 0 0
\(727\) 1.96219e30 1.76454 0.882268 0.470748i \(-0.156016\pi\)
0.882268 + 0.470748i \(0.156016\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.42928e29 0.715580
\(732\) 0 0
\(733\) −7.69033e28 −0.0634386 −0.0317193 0.999497i \(-0.510098\pi\)
−0.0317193 + 0.999497i \(0.510098\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.06055e28 0.0238447
\(738\) 0 0
\(739\) −4.37247e29 −0.331101 −0.165550 0.986201i \(-0.552940\pi\)
−0.165550 + 0.986201i \(0.552940\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.93922e30 −1.38754 −0.693770 0.720197i \(-0.744052\pi\)
−0.693770 + 0.720197i \(0.744052\pi\)
\(744\) 0 0
\(745\) −1.44724e30 −1.00670
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.34161e30 −1.53977
\(750\) 0 0
\(751\) −9.91312e29 −0.633857 −0.316928 0.948449i \(-0.602651\pi\)
−0.316928 + 0.948449i \(0.602651\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.48933e29 0.0900631
\(756\) 0 0
\(757\) −2.12609e30 −1.25048 −0.625238 0.780434i \(-0.714998\pi\)
−0.625238 + 0.780434i \(0.714998\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.13487e29 0.452701 0.226351 0.974046i \(-0.427320\pi\)
0.226351 + 0.974046i \(0.427320\pi\)
\(762\) 0 0
\(763\) −1.00488e30 −0.544010
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.13466e27 −0.00109390
\(768\) 0 0
\(769\) 3.51201e30 1.75117 0.875587 0.483061i \(-0.160475\pi\)
0.875587 + 0.483061i \(0.160475\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.85796e30 0.877308 0.438654 0.898656i \(-0.355455\pi\)
0.438654 + 0.898656i \(0.355455\pi\)
\(774\) 0 0
\(775\) −2.48412e30 −1.14157
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.93398e30 −1.27736
\(780\) 0 0
\(781\) −5.87758e28 −0.0249094
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.24870e30 −3.71492
\(786\) 0 0
\(787\) −3.95934e30 −1.54842 −0.774209 0.632930i \(-0.781852\pi\)
−0.774209 + 0.632930i \(0.781852\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.05405e30 −0.390847
\(792\) 0 0
\(793\) −2.81051e28 −0.0101488
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.59540e30 −0.546460 −0.273230 0.961949i \(-0.588092\pi\)
−0.273230 + 0.961949i \(0.588092\pi\)
\(798\) 0 0
\(799\) 4.95658e30 1.65364
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.13373e28 0.0225841
\(804\) 0 0
\(805\) −3.14112e30 −0.968787
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.96445e30 0.575152 0.287576 0.957758i \(-0.407151\pi\)
0.287576 + 0.957758i \(0.407151\pi\)
\(810\) 0 0
\(811\) 5.85777e30 1.67114 0.835571 0.549382i \(-0.185137\pi\)
0.835571 + 0.549382i \(0.185137\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.34416e31 3.64163
\(816\) 0 0
\(817\) −3.79518e30 −1.00208
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.07728e30 1.02274 0.511372 0.859359i \(-0.329137\pi\)
0.511372 + 0.859359i \(0.329137\pi\)
\(822\) 0 0
\(823\) 7.57530e30 1.85226 0.926129 0.377206i \(-0.123115\pi\)
0.926129 + 0.377206i \(0.123115\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.09644e30 −1.88142 −0.940711 0.339209i \(-0.889841\pi\)
−0.940711 + 0.339209i \(0.889841\pi\)
\(828\) 0 0
\(829\) −4.67371e30 −1.05886 −0.529431 0.848353i \(-0.677594\pi\)
−0.529431 + 0.848353i \(0.677594\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.13356e29 −0.175190
\(834\) 0 0
\(835\) −8.06149e30 −1.69320
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.78537e30 0.356637 0.178319 0.983973i \(-0.442934\pi\)
0.178319 + 0.983973i \(0.442934\pi\)
\(840\) 0 0
\(841\) −3.73507e30 −0.727681
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.03285e31 1.91443
\(846\) 0 0
\(847\) 5.08341e30 0.919134
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.59481e30 −0.790692
\(852\) 0 0
\(853\) −3.78820e29 −0.0636017 −0.0318008 0.999494i \(-0.510124\pi\)
−0.0318008 + 0.999494i \(0.510124\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.68323e30 −1.06829 −0.534144 0.845394i \(-0.679366\pi\)
−0.534144 + 0.845394i \(0.679366\pi\)
\(858\) 0 0
\(859\) 1.08420e30 0.169114 0.0845569 0.996419i \(-0.473053\pi\)
0.0845569 + 0.996419i \(0.473053\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.95381e29 −0.0438803 −0.0219401 0.999759i \(-0.506984\pi\)
−0.0219401 + 0.999759i \(0.506984\pi\)
\(864\) 0 0
\(865\) 6.22039e30 0.901880
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.29357e28 −0.00178683
\(870\) 0 0
\(871\) 1.80683e29 0.0243628
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.28458e31 2.93577
\(876\) 0 0
\(877\) −1.13789e31 −1.42759 −0.713797 0.700353i \(-0.753026\pi\)
−0.713797 + 0.700353i \(0.753026\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.24451e30 −0.148851 −0.0744257 0.997227i \(-0.523712\pi\)
−0.0744257 + 0.997227i \(0.523712\pi\)
\(882\) 0 0
\(883\) 3.51456e30 0.410472 0.205236 0.978712i \(-0.434204\pi\)
0.205236 + 0.978712i \(0.434204\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.29130e30 −1.03485 −0.517426 0.855728i \(-0.673110\pi\)
−0.517426 + 0.855728i \(0.673110\pi\)
\(888\) 0 0
\(889\) −8.82470e29 −0.0959912
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.23164e31 −2.31571
\(894\) 0 0
\(895\) −1.27339e31 −1.29068
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.30906e30 0.223336
\(900\) 0 0
\(901\) 2.11590e31 1.99933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.14749e31 1.03500
\(906\) 0 0
\(907\) −1.98499e29 −0.0174937 −0.00874685 0.999962i \(-0.502784\pi\)
−0.00874685 + 0.999962i \(0.502784\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.61422e29 0.0640741 0.0320371 0.999487i \(-0.489801\pi\)
0.0320371 + 0.999487i \(0.489801\pi\)
\(912\) 0 0
\(913\) 3.98407e29 0.0327630
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.44640e30 0.663428
\(918\) 0 0
\(919\) 2.14472e31 1.64648 0.823242 0.567691i \(-0.192163\pi\)
0.823242 + 0.567691i \(0.192163\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.46989e29 −0.0254506
\(924\) 0 0
\(925\) 5.34613e31 3.83311
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.20678e31 0.826916 0.413458 0.910523i \(-0.364321\pi\)
0.413458 + 0.910523i \(0.364321\pi\)
\(930\) 0 0
\(931\) 3.66204e30 0.245331
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.90023e29 0.0377874
\(936\) 0 0
\(937\) 1.48955e31 0.932803 0.466401 0.884573i \(-0.345550\pi\)
0.466401 + 0.884573i \(0.345550\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.63339e29 0.0517000 0.0258500 0.999666i \(-0.491771\pi\)
0.0258500 + 0.999666i \(0.491771\pi\)
\(942\) 0 0
\(943\) −7.56671e30 −0.443133
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.29303e31 0.724328 0.362164 0.932114i \(-0.382038\pi\)
0.362164 + 0.932114i \(0.382038\pi\)
\(948\) 0 0
\(949\) 4.21147e29 0.0230749
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.85893e31 0.974513 0.487257 0.873259i \(-0.337998\pi\)
0.487257 + 0.873259i \(0.337998\pi\)
\(954\) 0 0
\(955\) 1.62402e30 0.0832831
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7.01400e30 −0.344247
\(960\) 0 0
\(961\) −1.70110e31 −0.816836
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.97794e31 −0.909236
\(966\) 0 0
\(967\) −2.45390e31 −1.10377 −0.551886 0.833919i \(-0.686092\pi\)
−0.551886 + 0.833919i \(0.686092\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.38258e31 −1.45696 −0.728478 0.685069i \(-0.759772\pi\)
−0.728478 + 0.685069i \(0.759772\pi\)
\(972\) 0 0
\(973\) 3.76997e31 1.58911
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.12303e30 0.247214 0.123607 0.992331i \(-0.460554\pi\)
0.123607 + 0.992331i \(0.460554\pi\)
\(978\) 0 0
\(979\) −6.85158e29 −0.0270753
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.33411e31 0.883710 0.441855 0.897087i \(-0.354321\pi\)
0.441855 + 0.897087i \(0.354321\pi\)
\(984\) 0 0
\(985\) 3.80129e30 0.140880
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.78775e30 −0.347634
\(990\) 0 0
\(991\) −1.93811e31 −0.673913 −0.336957 0.941520i \(-0.609398\pi\)
−0.336957 + 0.941520i \(0.609398\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.10284e31 −1.36755
\(996\) 0 0
\(997\) −4.93033e31 −1.60908 −0.804538 0.593901i \(-0.797587\pi\)
−0.804538 + 0.593901i \(0.797587\pi\)
\(998\) 0 0
\(999\) 0 0
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 72.22.a.d.1.1 3
3.2 odd 2 24.22.a.c.1.3 3
12.11 even 2 48.22.a.l.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.3 3 3.2 odd 2
48.22.a.l.1.3 3 12.11 even 2
72.22.a.d.1.1 3 1.1 even 1 trivial