Properties

Label 72.22.a.f.1.2
Level $72$
Weight $22$
Character 72.1
Self dual yes
Analytic conductor $201.224$
Analytic rank $0$
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: \(0\)
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: \( x^{3} - x^{2} - 4963x + 96223 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(21.2235\) 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)
 
\(f(q)\) \(=\) \(q+2.11555e7 q^{5} +7.32981e8 q^{7} +O(q^{10})\) \(q+2.11555e7 q^{5} +7.32981e8 q^{7} +5.59634e9 q^{11} -6.30207e10 q^{13} -1.35490e13 q^{17} +1.39028e13 q^{19} +2.80711e14 q^{23} -2.92803e13 q^{25} -1.19637e15 q^{29} +3.88487e15 q^{31} +1.55066e16 q^{35} +2.72004e16 q^{37} +6.89865e16 q^{41} +3.24557e16 q^{43} +2.07417e17 q^{47} -2.12844e16 q^{49} +6.38971e17 q^{53} +1.18394e17 q^{55} +3.04610e18 q^{59} -5.64497e18 q^{61} -1.33324e18 q^{65} -3.96718e18 q^{67} +2.61878e19 q^{71} +1.37659e19 q^{73} +4.10201e18 q^{77} -1.18258e20 q^{79} +1.60950e19 q^{83} -2.86637e20 q^{85} +2.30103e20 q^{89} -4.61930e19 q^{91} +2.94120e20 q^{95} +2.72050e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) \) 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
\(5\) 2.11555e7 0.968811 0.484406 0.874844i \(-0.339036\pi\)
0.484406 + 0.874844i \(0.339036\pi\)
\(6\) 0 0
\(7\) 7.32981e8 0.980761 0.490381 0.871508i \(-0.336858\pi\)
0.490381 + 0.871508i \(0.336858\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.59634e9 0.0650551 0.0325275 0.999471i \(-0.489644\pi\)
0.0325275 + 0.999471i \(0.489644\pi\)
\(12\) 0 0
\(13\) −6.30207e10 −0.126788 −0.0633940 0.997989i \(-0.520192\pi\)
−0.0633940 + 0.997989i \(0.520192\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.35490e13 −1.63003 −0.815013 0.579443i \(-0.803270\pi\)
−0.815013 + 0.579443i \(0.803270\pi\)
\(18\) 0 0
\(19\) 1.39028e13 0.520221 0.260111 0.965579i \(-0.416241\pi\)
0.260111 + 0.965579i \(0.416241\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.80711e14 1.41292 0.706458 0.707754i \(-0.250292\pi\)
0.706458 + 0.707754i \(0.250292\pi\)
\(24\) 0 0
\(25\) −2.92803e13 −0.0614052
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.19637e15 −0.528063 −0.264031 0.964514i \(-0.585052\pi\)
−0.264031 + 0.964514i \(0.585052\pi\)
\(30\) 0 0
\(31\) 3.88487e15 0.851292 0.425646 0.904890i \(-0.360047\pi\)
0.425646 + 0.904890i \(0.360047\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.55066e16 0.950173
\(36\) 0 0
\(37\) 2.72004e16 0.929944 0.464972 0.885325i \(-0.346064\pi\)
0.464972 + 0.885325i \(0.346064\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.89865e16 0.802663 0.401332 0.915933i \(-0.368547\pi\)
0.401332 + 0.915933i \(0.368547\pi\)
\(42\) 0 0
\(43\) 3.24557e16 0.229020 0.114510 0.993422i \(-0.463470\pi\)
0.114510 + 0.993422i \(0.463470\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.07417e17 0.575196 0.287598 0.957751i \(-0.407143\pi\)
0.287598 + 0.957751i \(0.407143\pi\)
\(48\) 0 0
\(49\) −2.12844e16 −0.0381069
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.38971e17 0.501862 0.250931 0.968005i \(-0.419263\pi\)
0.250931 + 0.968005i \(0.419263\pi\)
\(54\) 0 0
\(55\) 1.18394e17 0.0630261
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.04610e18 0.775887 0.387944 0.921683i \(-0.373186\pi\)
0.387944 + 0.921683i \(0.373186\pi\)
\(60\) 0 0
\(61\) −5.64497e18 −1.01321 −0.506604 0.862179i \(-0.669099\pi\)
−0.506604 + 0.862179i \(0.669099\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.33324e18 −0.122834
\(66\) 0 0
\(67\) −3.96718e18 −0.265886 −0.132943 0.991124i \(-0.542443\pi\)
−0.132943 + 0.991124i \(0.542443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.61878e19 0.954743 0.477371 0.878702i \(-0.341590\pi\)
0.477371 + 0.878702i \(0.341590\pi\)
\(72\) 0 0
\(73\) 1.37659e19 0.374899 0.187449 0.982274i \(-0.439978\pi\)
0.187449 + 0.982274i \(0.439978\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.10201e18 0.0638035
\(78\) 0 0
\(79\) −1.18258e20 −1.40523 −0.702615 0.711570i \(-0.747984\pi\)
−0.702615 + 0.711570i \(0.747984\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.60950e19 0.113860 0.0569301 0.998378i \(-0.481869\pi\)
0.0569301 + 0.998378i \(0.481869\pi\)
\(84\) 0 0
\(85\) −2.86637e20 −1.57919
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.30103e20 0.782217 0.391109 0.920345i \(-0.372092\pi\)
0.391109 + 0.920345i \(0.372092\pi\)
\(90\) 0 0
\(91\) −4.61930e19 −0.124349
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.94120e20 0.503996
\(96\) 0 0
\(97\) 2.72050e20 0.374581 0.187290 0.982305i \(-0.440029\pi\)
0.187290 + 0.982305i \(0.440029\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.18197e21 1.06471 0.532357 0.846520i \(-0.321307\pi\)
0.532357 + 0.846520i \(0.321307\pi\)
\(102\) 0 0
\(103\) −2.39849e21 −1.75852 −0.879258 0.476345i \(-0.841961\pi\)
−0.879258 + 0.476345i \(0.841961\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.01203e20 −0.197167 −0.0985835 0.995129i \(-0.531431\pi\)
−0.0985835 + 0.995129i \(0.531431\pi\)
\(108\) 0 0
\(109\) 4.25100e21 1.71994 0.859970 0.510345i \(-0.170482\pi\)
0.859970 + 0.510345i \(0.170482\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.36941e21 0.933748 0.466874 0.884324i \(-0.345380\pi\)
0.466874 + 0.884324i \(0.345380\pi\)
\(114\) 0 0
\(115\) 5.93859e21 1.36885
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.93118e21 −1.59867
\(120\) 0 0
\(121\) −7.36893e21 −0.995768
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.07072e22 −1.02830
\(126\) 0 0
\(127\) −1.15394e20 −0.00938093 −0.00469046 0.999989i \(-0.501493\pi\)
−0.00469046 + 0.999989i \(0.501493\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.13322e22 −0.665223 −0.332611 0.943064i \(-0.607930\pi\)
−0.332611 + 0.943064i \(0.607930\pi\)
\(132\) 0 0
\(133\) 1.01905e22 0.510213
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.18696e21 −0.263621 −0.131810 0.991275i \(-0.542079\pi\)
−0.131810 + 0.991275i \(0.542079\pi\)
\(138\) 0 0
\(139\) 2.34620e22 0.739109 0.369555 0.929209i \(-0.379510\pi\)
0.369555 + 0.929209i \(0.379510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.52686e20 −0.00824820
\(144\) 0 0
\(145\) −2.53098e22 −0.511593
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.57122e22 0.390557 0.195278 0.980748i \(-0.437439\pi\)
0.195278 + 0.980748i \(0.437439\pi\)
\(150\) 0 0
\(151\) 1.38339e23 1.82679 0.913393 0.407078i \(-0.133452\pi\)
0.913393 + 0.407078i \(0.133452\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.21865e22 0.824741
\(156\) 0 0
\(157\) −1.14060e23 −1.00043 −0.500215 0.865901i \(-0.666746\pi\)
−0.500215 + 0.865901i \(0.666746\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.05756e23 1.38573
\(162\) 0 0
\(163\) 6.59877e22 0.390385 0.195192 0.980765i \(-0.437467\pi\)
0.195192 + 0.980765i \(0.437467\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.09416e22 −0.0960477 −0.0480239 0.998846i \(-0.515292\pi\)
−0.0480239 + 0.998846i \(0.515292\pi\)
\(168\) 0 0
\(169\) −2.43093e23 −0.983925
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.69739e23 0.537400 0.268700 0.963224i \(-0.413406\pi\)
0.268700 + 0.963224i \(0.413406\pi\)
\(174\) 0 0
\(175\) −2.14619e22 −0.0602238
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.58531e23 −1.23620 −0.618102 0.786098i \(-0.712098\pi\)
−0.618102 + 0.786098i \(0.712098\pi\)
\(180\) 0 0
\(181\) −2.50056e23 −0.492506 −0.246253 0.969206i \(-0.579199\pi\)
−0.246253 + 0.969206i \(0.579199\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.75439e23 0.900940
\(186\) 0 0
\(187\) −7.58250e22 −0.106041
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.66807e23 0.186794 0.0933971 0.995629i \(-0.470227\pi\)
0.0933971 + 0.995629i \(0.470227\pi\)
\(192\) 0 0
\(193\) 1.33244e24 1.33750 0.668751 0.743487i \(-0.266829\pi\)
0.668751 + 0.743487i \(0.266829\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.38870e23 0.517032 0.258516 0.966007i \(-0.416767\pi\)
0.258516 + 0.966007i \(0.416767\pi\)
\(198\) 0 0
\(199\) 1.84623e24 1.34378 0.671889 0.740652i \(-0.265483\pi\)
0.671889 + 0.740652i \(0.265483\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.76915e23 −0.517904
\(204\) 0 0
\(205\) 1.45945e24 0.777629
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.78046e22 0.0338430
\(210\) 0 0
\(211\) 3.35146e24 1.31907 0.659536 0.751673i \(-0.270753\pi\)
0.659536 + 0.751673i \(0.270753\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.86619e23 0.221877
\(216\) 0 0
\(217\) 2.84754e24 0.834915
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.53869e23 0.206668
\(222\) 0 0
\(223\) 4.56926e22 0.0100611 0.00503054 0.999987i \(-0.498399\pi\)
0.00503054 + 0.999987i \(0.498399\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.05295e24 1.10585 0.552924 0.833232i \(-0.313512\pi\)
0.552924 + 0.833232i \(0.313512\pi\)
\(228\) 0 0
\(229\) 5.16931e24 0.861311 0.430655 0.902516i \(-0.358282\pi\)
0.430655 + 0.902516i \(0.358282\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.21547e24 −0.724530 −0.362265 0.932075i \(-0.617997\pi\)
−0.362265 + 0.932075i \(0.617997\pi\)
\(234\) 0 0
\(235\) 4.38801e24 0.557257
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.57703e24 −0.167749 −0.0838747 0.996476i \(-0.526730\pi\)
−0.0838747 + 0.996476i \(0.526730\pi\)
\(240\) 0 0
\(241\) 9.93088e24 0.967851 0.483926 0.875109i \(-0.339211\pi\)
0.483926 + 0.875109i \(0.339211\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.50284e23 −0.0369184
\(246\) 0 0
\(247\) −8.76162e23 −0.0659578
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.38505e25 1.51679 0.758393 0.651798i \(-0.225985\pi\)
0.758393 + 0.651798i \(0.225985\pi\)
\(252\) 0 0
\(253\) 1.57095e24 0.0919174
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.76190e25 1.86685 0.933425 0.358772i \(-0.116805\pi\)
0.933425 + 0.358772i \(0.116805\pi\)
\(258\) 0 0
\(259\) 1.99374e25 0.912054
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.52809e25 0.984592 0.492296 0.870428i \(-0.336158\pi\)
0.492296 + 0.870428i \(0.336158\pi\)
\(264\) 0 0
\(265\) 1.35178e25 0.486209
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.38239e25 −1.34683 −0.673414 0.739266i \(-0.735173\pi\)
−0.673414 + 0.739266i \(0.735173\pi\)
\(270\) 0 0
\(271\) 2.43576e24 0.0692560 0.0346280 0.999400i \(-0.488975\pi\)
0.0346280 + 0.999400i \(0.488975\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.63862e23 −0.00399472
\(276\) 0 0
\(277\) 4.43475e25 1.00192 0.500958 0.865472i \(-0.332981\pi\)
0.500958 + 0.865472i \(0.332981\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.89794e25 1.72931 0.864655 0.502365i \(-0.167537\pi\)
0.864655 + 0.502365i \(0.167537\pi\)
\(282\) 0 0
\(283\) −2.34320e25 −0.422719 −0.211359 0.977408i \(-0.567789\pi\)
−0.211359 + 0.977408i \(0.567789\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.05658e25 0.787221
\(288\) 0 0
\(289\) 1.14484e26 1.65698
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.82209e25 −0.729405 −0.364703 0.931124i \(-0.618829\pi\)
−0.364703 + 0.931124i \(0.618829\pi\)
\(294\) 0 0
\(295\) 6.44420e25 0.751688
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.76906e25 −0.179141
\(300\) 0 0
\(301\) 2.37894e25 0.224614
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.19422e26 −0.981607
\(306\) 0 0
\(307\) 1.92535e26 1.47760 0.738799 0.673926i \(-0.235393\pi\)
0.738799 + 0.673926i \(0.235393\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.71812e26 −1.15098 −0.575491 0.817808i \(-0.695189\pi\)
−0.575491 + 0.817808i \(0.695189\pi\)
\(312\) 0 0
\(313\) −3.10727e25 −0.194609 −0.0973045 0.995255i \(-0.531022\pi\)
−0.0973045 + 0.995255i \(0.531022\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.10055e26 1.15136 0.575680 0.817675i \(-0.304738\pi\)
0.575680 + 0.817675i \(0.304738\pi\)
\(318\) 0 0
\(319\) −6.69528e24 −0.0343532
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.88369e26 −0.847974
\(324\) 0 0
\(325\) 1.84526e24 0.00778544
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.52033e26 0.564130
\(330\) 0 0
\(331\) −2.36914e26 −0.824891 −0.412446 0.910982i \(-0.635325\pi\)
−0.412446 + 0.910982i \(0.635325\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.39278e25 −0.257594
\(336\) 0 0
\(337\) −6.34353e25 −0.182902 −0.0914508 0.995810i \(-0.529150\pi\)
−0.0914508 + 0.995810i \(0.529150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.17411e25 0.0553809
\(342\) 0 0
\(343\) −4.25005e26 −1.01814
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.30760e26 0.489442 0.244721 0.969593i \(-0.421304\pi\)
0.244721 + 0.969593i \(0.421304\pi\)
\(348\) 0 0
\(349\) −6.64125e26 −1.32612 −0.663060 0.748566i \(-0.730742\pi\)
−0.663060 + 0.748566i \(0.730742\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.72081e26 1.36782 0.683909 0.729567i \(-0.260278\pi\)
0.683909 + 0.729567i \(0.260278\pi\)
\(354\) 0 0
\(355\) 5.54017e26 0.924965
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.35001e26 −1.23935 −0.619677 0.784857i \(-0.712736\pi\)
−0.619677 + 0.784857i \(0.712736\pi\)
\(360\) 0 0
\(361\) −5.20923e26 −0.729370
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.91225e26 0.363206
\(366\) 0 0
\(367\) −4.88336e26 −0.575075 −0.287538 0.957769i \(-0.592837\pi\)
−0.287538 + 0.957769i \(0.592837\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.68353e26 0.492207
\(372\) 0 0
\(373\) −1.33813e27 −1.32909 −0.664547 0.747247i \(-0.731375\pi\)
−0.664547 + 0.747247i \(0.731375\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.53960e25 0.0669520
\(378\) 0 0
\(379\) 7.26568e25 0.0610329 0.0305165 0.999534i \(-0.490285\pi\)
0.0305165 + 0.999534i \(0.490285\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.24472e27 −0.936449 −0.468225 0.883610i \(-0.655106\pi\)
−0.468225 + 0.883610i \(0.655106\pi\)
\(384\) 0 0
\(385\) 8.67803e25 0.0618135
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.14241e27 −1.36909 −0.684544 0.728972i \(-0.739998\pi\)
−0.684544 + 0.728972i \(0.739998\pi\)
\(390\) 0 0
\(391\) −3.80336e27 −2.30309
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.50182e27 −1.36140
\(396\) 0 0
\(397\) 2.44762e27 1.26312 0.631560 0.775327i \(-0.282415\pi\)
0.631560 + 0.775327i \(0.282415\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.80970e27 −1.30510 −0.652550 0.757746i \(-0.726301\pi\)
−0.652550 + 0.757746i \(0.726301\pi\)
\(402\) 0 0
\(403\) −2.44827e26 −0.107934
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.52223e26 0.0604976
\(408\) 0 0
\(409\) −3.81600e27 −1.44050 −0.720250 0.693714i \(-0.755973\pi\)
−0.720250 + 0.693714i \(0.755973\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.23274e27 0.760960
\(414\) 0 0
\(415\) 3.40499e26 0.110309
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.13782e26 0.121206 0.0606031 0.998162i \(-0.480698\pi\)
0.0606031 + 0.998162i \(0.480698\pi\)
\(420\) 0 0
\(421\) 4.16773e27 1.16128 0.580641 0.814160i \(-0.302802\pi\)
0.580641 + 0.814160i \(0.302802\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.96719e26 0.100092
\(426\) 0 0
\(427\) −4.13766e27 −0.993715
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.31148e27 −0.721125 −0.360563 0.932735i \(-0.617415\pi\)
−0.360563 + 0.932735i \(0.617415\pi\)
\(432\) 0 0
\(433\) 7.71199e27 1.59972 0.799859 0.600188i \(-0.204908\pi\)
0.799859 + 0.600188i \(0.204908\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.90265e27 0.735029
\(438\) 0 0
\(439\) −7.40474e27 −1.32933 −0.664665 0.747142i \(-0.731426\pi\)
−0.664665 + 0.747142i \(0.731426\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.59541e27 −0.423611 −0.211805 0.977312i \(-0.567934\pi\)
−0.211805 + 0.977312i \(0.567934\pi\)
\(444\) 0 0
\(445\) 4.86795e27 0.757821
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.63721e27 1.22402 0.612008 0.790852i \(-0.290362\pi\)
0.612008 + 0.790852i \(0.290362\pi\)
\(450\) 0 0
\(451\) 3.86072e26 0.0522173
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.77238e26 −0.120470
\(456\) 0 0
\(457\) −7.22330e27 −0.850385 −0.425193 0.905103i \(-0.639794\pi\)
−0.425193 + 0.905103i \(0.639794\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.67275e27 0.716878 0.358439 0.933553i \(-0.383309\pi\)
0.358439 + 0.933553i \(0.383309\pi\)
\(462\) 0 0
\(463\) −7.22059e27 −0.741263 −0.370632 0.928780i \(-0.620859\pi\)
−0.370632 + 0.928780i \(0.620859\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.14336e27 0.294827 0.147413 0.989075i \(-0.452905\pi\)
0.147413 + 0.989075i \(0.452905\pi\)
\(468\) 0 0
\(469\) −2.90787e27 −0.260771
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.81634e26 0.0148989
\(474\) 0 0
\(475\) −4.07076e26 −0.0319443
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.98918e28 −1.42939 −0.714696 0.699435i \(-0.753435\pi\)
−0.714696 + 0.699435i \(0.753435\pi\)
\(480\) 0 0
\(481\) −1.71419e27 −0.117906
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.75536e27 0.362898
\(486\) 0 0
\(487\) 1.49710e28 0.904058 0.452029 0.892003i \(-0.350700\pi\)
0.452029 + 0.892003i \(0.350700\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.87261e27 0.325444 0.162722 0.986672i \(-0.447973\pi\)
0.162722 + 0.986672i \(0.447973\pi\)
\(492\) 0 0
\(493\) 1.62096e28 0.860756
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.91952e28 0.936375
\(498\) 0 0
\(499\) 3.56403e28 1.66681 0.833404 0.552665i \(-0.186389\pi\)
0.833404 + 0.552665i \(0.186389\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.66833e28 0.717495 0.358748 0.933435i \(-0.383204\pi\)
0.358748 + 0.933435i \(0.383204\pi\)
\(504\) 0 0
\(505\) 2.50053e28 1.03151
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.66847e28 1.39299 0.696495 0.717562i \(-0.254742\pi\)
0.696495 + 0.717562i \(0.254742\pi\)
\(510\) 0 0
\(511\) 1.00901e28 0.367686
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.07413e28 −1.70367
\(516\) 0 0
\(517\) 1.16078e27 0.0374194
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.16512e28 0.941009 0.470505 0.882398i \(-0.344072\pi\)
0.470505 + 0.882398i \(0.344072\pi\)
\(522\) 0 0
\(523\) 4.41639e28 1.26125 0.630623 0.776090i \(-0.282800\pi\)
0.630623 + 0.776090i \(0.282800\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.26362e28 −1.38763
\(528\) 0 0
\(529\) 3.93269e28 0.996334
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.34758e27 −0.101768
\(534\) 0 0
\(535\) −8.48766e27 −0.191018
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.19115e26 −0.00247905
\(540\) 0 0
\(541\) 8.65965e28 1.73352 0.866760 0.498725i \(-0.166198\pi\)
0.866760 + 0.498725i \(0.166198\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.99322e28 1.66630
\(546\) 0 0
\(547\) −6.71643e28 −1.19749 −0.598745 0.800940i \(-0.704334\pi\)
−0.598745 + 0.800940i \(0.704334\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.66328e28 −0.274710
\(552\) 0 0
\(553\) −8.66811e28 −1.37820
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.28272e28 −0.926122 −0.463061 0.886326i \(-0.653249\pi\)
−0.463061 + 0.886326i \(0.653249\pi\)
\(558\) 0 0
\(559\) −2.04538e27 −0.0290369
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.00019e28 −1.18553 −0.592766 0.805375i \(-0.701964\pi\)
−0.592766 + 0.805375i \(0.701964\pi\)
\(564\) 0 0
\(565\) 7.12816e28 0.904625
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.19277e29 1.40565 0.702824 0.711364i \(-0.251922\pi\)
0.702824 + 0.711364i \(0.251922\pi\)
\(570\) 0 0
\(571\) −1.14674e29 −1.30253 −0.651265 0.758851i \(-0.725761\pi\)
−0.651265 + 0.758851i \(0.725761\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.21928e27 −0.0867604
\(576\) 0 0
\(577\) −2.51949e28 −0.256429 −0.128214 0.991746i \(-0.540925\pi\)
−0.128214 + 0.991746i \(0.540925\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.17974e28 0.111670
\(582\) 0 0
\(583\) 3.57590e27 0.0326487
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.22050e28 0.358644 0.179322 0.983790i \(-0.442610\pi\)
0.179322 + 0.983790i \(0.442610\pi\)
\(588\) 0 0
\(589\) 5.40104e28 0.442860
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.61710e29 1.23499 0.617495 0.786575i \(-0.288148\pi\)
0.617495 + 0.786575i \(0.288148\pi\)
\(594\) 0 0
\(595\) −2.10100e29 −1.54881
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.18099e28 0.562113 0.281057 0.959691i \(-0.409315\pi\)
0.281057 + 0.959691i \(0.409315\pi\)
\(600\) 0 0
\(601\) 2.09704e28 0.139131 0.0695657 0.997577i \(-0.477839\pi\)
0.0695657 + 0.997577i \(0.477839\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.55894e29 −0.964711
\(606\) 0 0
\(607\) 1.39211e29 0.832131 0.416066 0.909335i \(-0.363409\pi\)
0.416066 + 0.909335i \(0.363409\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.30715e28 −0.0729280
\(612\) 0 0
\(613\) −1.57634e29 −0.849795 −0.424898 0.905241i \(-0.639690\pi\)
−0.424898 + 0.905241i \(0.639690\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.96390e29 −0.988836 −0.494418 0.869224i \(-0.664619\pi\)
−0.494418 + 0.869224i \(0.664619\pi\)
\(618\) 0 0
\(619\) −2.32870e29 −1.13334 −0.566671 0.823944i \(-0.691769\pi\)
−0.566671 + 0.823944i \(0.691769\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.68661e29 0.767168
\(624\) 0 0
\(625\) −2.12554e29 −0.934824
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.68539e29 −1.51583
\(630\) 0 0
\(631\) −2.44143e29 −0.971261 −0.485631 0.874164i \(-0.661410\pi\)
−0.485631 + 0.874164i \(0.661410\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.44123e27 −0.00908835
\(636\) 0 0
\(637\) 1.34136e27 0.00483150
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.44014e28 0.116029 0.0580145 0.998316i \(-0.481523\pi\)
0.0580145 + 0.998316i \(0.481523\pi\)
\(642\) 0 0
\(643\) 4.55690e29 1.48749 0.743745 0.668463i \(-0.233048\pi\)
0.743745 + 0.668463i \(0.233048\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.99700e29 −1.52832 −0.764161 0.645026i \(-0.776847\pi\)
−0.764161 + 0.645026i \(0.776847\pi\)
\(648\) 0 0
\(649\) 1.70470e28 0.0504754
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.87688e29 −0.798607 −0.399303 0.916819i \(-0.630748\pi\)
−0.399303 + 0.916819i \(0.630748\pi\)
\(654\) 0 0
\(655\) −2.39740e29 −0.644475
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.60817e29 −1.16207 −0.581034 0.813879i \(-0.697352\pi\)
−0.581034 + 0.813879i \(0.697352\pi\)
\(660\) 0 0
\(661\) 2.94237e29 0.718756 0.359378 0.933192i \(-0.382989\pi\)
0.359378 + 0.933192i \(0.382989\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.15585e29 0.494300
\(666\) 0 0
\(667\) −3.35833e29 −0.746109
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.15912e28 −0.0659143
\(672\) 0 0
\(673\) −4.64032e29 −0.938403 −0.469202 0.883091i \(-0.655458\pi\)
−0.469202 + 0.883091i \(0.655458\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.90684e29 0.742411 0.371206 0.928551i \(-0.378944\pi\)
0.371206 + 0.928551i \(0.378944\pi\)
\(678\) 0 0
\(679\) 1.99408e29 0.367374
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.80255e29 −0.312227 −0.156113 0.987739i \(-0.549897\pi\)
−0.156113 + 0.987739i \(0.549897\pi\)
\(684\) 0 0
\(685\) −1.52044e29 −0.255399
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.02684e28 −0.0636301
\(690\) 0 0
\(691\) −9.69163e28 −0.148552 −0.0742758 0.997238i \(-0.523665\pi\)
−0.0742758 + 0.997238i \(0.523665\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.96350e29 0.716057
\(696\) 0 0
\(697\) −9.34700e29 −1.30836
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.44452e29 −0.717663 −0.358831 0.933402i \(-0.616825\pi\)
−0.358831 + 0.933402i \(0.616825\pi\)
\(702\) 0 0
\(703\) 3.78160e29 0.483777
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.66363e29 1.04423
\(708\) 0 0
\(709\) 1.34223e30 1.57052 0.785258 0.619168i \(-0.212530\pi\)
0.785258 + 0.619168i \(0.212530\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.09052e30 1.20281
\(714\) 0 0
\(715\) −7.46126e27 −0.00799095
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.27979e29 0.937313 0.468657 0.883380i \(-0.344738\pi\)
0.468657 + 0.883380i \(0.344738\pi\)
\(720\) 0 0
\(721\) −1.75805e30 −1.72469
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.50300e28 0.0324258
\(726\) 0 0
\(727\) 1.97796e29 0.177871 0.0889357 0.996037i \(-0.471653\pi\)
0.0889357 + 0.996037i \(0.471653\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.39744e29 −0.373308
\(732\) 0 0
\(733\) −8.95031e28 −0.0738323 −0.0369161 0.999318i \(-0.511753\pi\)
−0.0369161 + 0.999318i \(0.511753\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.22017e28 −0.0172973
\(738\) 0 0
\(739\) 1.59180e29 0.120537 0.0602687 0.998182i \(-0.480804\pi\)
0.0602687 + 0.998182i \(0.480804\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.50012e28 −0.0679746 −0.0339873 0.999422i \(-0.510821\pi\)
−0.0339873 + 0.999422i \(0.510821\pi\)
\(744\) 0 0
\(745\) 5.43956e29 0.378376
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.94074e29 −0.193374
\(750\) 0 0
\(751\) 2.97935e30 1.90503 0.952516 0.304488i \(-0.0984855\pi\)
0.952516 + 0.304488i \(0.0984855\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.92664e30 1.76981
\(756\) 0 0
\(757\) 8.10128e29 0.476482 0.238241 0.971206i \(-0.423429\pi\)
0.238241 + 0.971206i \(0.423429\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.84872e29 −0.492426 −0.246213 0.969216i \(-0.579186\pi\)
−0.246213 + 0.969216i \(0.579186\pi\)
\(762\) 0 0
\(763\) 3.11590e30 1.68685
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.91968e29 −0.0983732
\(768\) 0 0
\(769\) 2.59763e29 0.129524 0.0647620 0.997901i \(-0.479371\pi\)
0.0647620 + 0.997901i \(0.479371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.97566e30 −1.87726 −0.938630 0.344925i \(-0.887904\pi\)
−0.938630 + 0.344925i \(0.887904\pi\)
\(774\) 0 0
\(775\) −1.13750e29 −0.0522738
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.59102e29 0.417562
\(780\) 0 0
\(781\) 1.46556e29 0.0621109
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.41300e30 −0.969228
\(786\) 0 0
\(787\) 1.54080e30 0.602577 0.301288 0.953533i \(-0.402583\pi\)
0.301288 + 0.953533i \(0.402583\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.46971e30 0.915784
\(792\) 0 0
\(793\) 3.55750e29 0.128463
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.29587e29 −0.0443862 −0.0221931 0.999754i \(-0.507065\pi\)
−0.0221931 + 0.999754i \(0.507065\pi\)
\(798\) 0 0
\(799\) −2.81029e30 −0.937585
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7.70386e28 0.0243891
\(804\) 0 0
\(805\) 4.35287e30 1.34251
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.25920e30 0.661449 0.330724 0.943727i \(-0.392707\pi\)
0.330724 + 0.943727i \(0.392707\pi\)
\(810\) 0 0
\(811\) 4.17856e30 1.19209 0.596043 0.802953i \(-0.296739\pi\)
0.596043 + 0.802953i \(0.296739\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.39600e30 0.378209
\(816\) 0 0
\(817\) 4.51224e29 0.119141
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.32971e29 0.0333543 0.0166772 0.999861i \(-0.494691\pi\)
0.0166772 + 0.999861i \(0.494691\pi\)
\(822\) 0 0
\(823\) −1.12434e30 −0.274915 −0.137457 0.990508i \(-0.543893\pi\)
−0.137457 + 0.990508i \(0.543893\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.53339e30 0.821077 0.410539 0.911843i \(-0.365341\pi\)
0.410539 + 0.911843i \(0.365341\pi\)
\(828\) 0 0
\(829\) 3.22988e30 0.731751 0.365875 0.930664i \(-0.380770\pi\)
0.365875 + 0.930664i \(0.380770\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.88384e29 0.0621152
\(834\) 0 0
\(835\) −4.43031e29 −0.0930521
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.24571e30 −0.248838 −0.124419 0.992230i \(-0.539707\pi\)
−0.124419 + 0.992230i \(0.539707\pi\)
\(840\) 0 0
\(841\) −3.70155e30 −0.721150
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.14276e30 −0.953237
\(846\) 0 0
\(847\) −5.40129e30 −0.976611
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7.63544e30 1.31393
\(852\) 0 0
\(853\) −6.54885e30 −1.09951 −0.549757 0.835325i \(-0.685280\pi\)
−0.549757 + 0.835325i \(0.685280\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.91673e30 −0.306381 −0.153191 0.988197i \(-0.548955\pi\)
−0.153191 + 0.988197i \(0.548955\pi\)
\(858\) 0 0
\(859\) −2.28266e30 −0.356051 −0.178026 0.984026i \(-0.556971\pi\)
−0.178026 + 0.984026i \(0.556971\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.87662e30 1.17011 0.585055 0.810994i \(-0.301073\pi\)
0.585055 + 0.810994i \(0.301073\pi\)
\(864\) 0 0
\(865\) 3.59092e30 0.520639
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.61814e29 −0.0914173
\(870\) 0 0
\(871\) 2.50014e29 0.0337112
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.84817e30 −1.00852
\(876\) 0 0
\(877\) −1.25742e31 −1.57756 −0.788778 0.614678i \(-0.789286\pi\)
−0.788778 + 0.614678i \(0.789286\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 5.18000e29 0.0619560 0.0309780 0.999520i \(-0.490138\pi\)
0.0309780 + 0.999520i \(0.490138\pi\)
\(882\) 0 0
\(883\) −4.26502e30 −0.498120 −0.249060 0.968488i \(-0.580122\pi\)
−0.249060 + 0.968488i \(0.580122\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.39563e31 −1.55444 −0.777218 0.629232i \(-0.783370\pi\)
−0.777218 + 0.629232i \(0.783370\pi\)
\(888\) 0 0
\(889\) −8.45819e28 −0.00920045
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.88366e30 0.299229
\(894\) 0 0
\(895\) −1.18160e31 −1.19765
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.64773e30 −0.449536
\(900\) 0 0
\(901\) −8.65743e30 −0.818048
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.29006e30 −0.477145
\(906\) 0 0
\(907\) −7.99799e30 −0.704862 −0.352431 0.935838i \(-0.614645\pi\)
−0.352431 + 0.935838i \(0.614645\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.04758e30 −0.508907 −0.254454 0.967085i \(-0.581896\pi\)
−0.254454 + 0.967085i \(0.581896\pi\)
\(912\) 0 0
\(913\) 9.00733e28 0.00740718
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.30632e30 −0.652425
\(918\) 0 0
\(919\) 2.36741e31 1.81744 0.908719 0.417408i \(-0.137061\pi\)
0.908719 + 0.417408i \(0.137061\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.65037e30 −0.121050
\(924\) 0 0
\(925\) −7.96435e29 −0.0571034
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.41968e31 1.65803 0.829016 0.559226i \(-0.188901\pi\)
0.829016 + 0.559226i \(0.188901\pi\)
\(930\) 0 0
\(931\) −2.95912e29 −0.0198240
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.60412e30 −0.102734
\(936\) 0 0
\(937\) −8.58736e30 −0.537767 −0.268883 0.963173i \(-0.586655\pi\)
−0.268883 + 0.963173i \(0.586655\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.47302e30 0.327745 0.163872 0.986482i \(-0.447601\pi\)
0.163872 + 0.986482i \(0.447601\pi\)
\(942\) 0 0
\(943\) 1.93652e31 1.13410
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.44243e31 −1.36819 −0.684096 0.729392i \(-0.739803\pi\)
−0.684096 + 0.729392i \(0.739803\pi\)
\(948\) 0 0
\(949\) −8.67536e29 −0.0475327
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.04469e31 −0.547661 −0.273831 0.961778i \(-0.588291\pi\)
−0.273831 + 0.961778i \(0.588291\pi\)
\(954\) 0 0
\(955\) 3.52889e30 0.180968
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.26791e30 −0.258549
\(960\) 0 0
\(961\) −5.73329e30 −0.275301
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.81884e31 1.29579
\(966\) 0 0
\(967\) −3.41008e31 −1.53386 −0.766931 0.641729i \(-0.778217\pi\)
−0.766931 + 0.641729i \(0.778217\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.17245e31 −1.36645 −0.683224 0.730209i \(-0.739423\pi\)
−0.683224 + 0.730209i \(0.739423\pi\)
\(972\) 0 0
\(973\) 1.71972e31 0.724890
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.33987e31 −0.944710 −0.472355 0.881408i \(-0.656596\pi\)
−0.472355 + 0.881408i \(0.656596\pi\)
\(978\) 0 0
\(979\) 1.28774e30 0.0508872
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.15166e28 −0.000436024 0 −0.000218012 1.00000i \(-0.500069\pi\)
−0.000218012 1.00000i \(0.500069\pi\)
\(984\) 0 0
\(985\) 1.35156e31 0.500906
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.11067e30 0.323586
\(990\) 0 0
\(991\) −1.52057e31 −0.528730 −0.264365 0.964423i \(-0.585162\pi\)
−0.264365 + 0.964423i \(0.585162\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.90579e31 1.30187
\(996\) 0 0
\(997\) −4.15395e31 −1.35569 −0.677847 0.735203i \(-0.737087\pi\)
−0.677847 + 0.735203i \(0.737087\pi\)
\(998\) 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.f.1.2 3
3.2 odd 2 8.22.a.b.1.3 3
12.11 even 2 16.22.a.f.1.1 3
24.5 odd 2 64.22.a.l.1.1 3
24.11 even 2 64.22.a.m.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.22.a.b.1.3 3 3.2 odd 2
16.22.a.f.1.1 3 12.11 even 2
64.22.a.l.1.1 3 24.5 odd 2
64.22.a.m.1.3 3 24.11 even 2
72.22.a.f.1.2 3 1.1 even 1 trivial