Properties

Label 144.13.g.h.127.2
Level $144$
Weight $13$
Character 144.127
Analytic conductor $131.615$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,13,Mod(127,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.127");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 144.g (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: \(131.615109688\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4333x^{2} + 4332x + 18766224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(33.1599 - 57.4347i\) of defining polynomial
Character \(\chi\) \(=\) 144.127
Dual form 144.13.g.h.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10306.8 q^{5} +25721.1i q^{7} +O(q^{10})\) \(q-10306.8 q^{5} +25721.1i q^{7} +2.20777e6i q^{11} +5.65602e6 q^{13} +2.37176e7 q^{17} -5.50828e7i q^{19} -2.18200e7i q^{23} -1.37911e8 q^{25} -3.88008e8 q^{29} -3.16995e8i q^{31} -2.65101e8i q^{35} +1.17320e9 q^{37} -3.32284e9 q^{41} -6.04589e9i q^{43} +2.63294e9i q^{47} +1.31797e10 q^{49} -2.86594e10 q^{53} -2.27550e10i q^{55} +6.66570e10i q^{59} +4.54516e10 q^{61} -5.82953e10 q^{65} +5.74377e10i q^{67} +2.96341e9i q^{71} +2.94406e11 q^{73} -5.67862e10 q^{77} -2.19409e11i q^{79} +3.91018e10i q^{83} -2.44452e11 q^{85} -1.53977e11 q^{89} +1.45479e11i q^{91} +5.67725e11i q^{95} -1.21087e12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 21960 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 21960 q^{5} - 5810072 q^{13} + 33882264 q^{17} + 142148300 q^{25} - 42583896 q^{29} + 9402865736 q^{37} - 17968882536 q^{41} + 53940845764 q^{49} - 60546956760 q^{53} + 168287201672 q^{61} - 481064975280 q^{65} + 789026629000 q^{73} - 140389989696 q^{77} - 777401136720 q^{85} + 638670460536 q^{89} - 563542043000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −10306.8 −0.659633 −0.329816 0.944045i \(-0.606987\pi\)
−0.329816 + 0.944045i \(0.606987\pi\)
\(6\) 0 0
\(7\) 25721.1i 0.218625i 0.994007 + 0.109313i \(0.0348650\pi\)
−0.994007 + 0.109313i \(0.965135\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.20777e6i 1.24623i 0.782130 + 0.623115i \(0.214133\pi\)
−0.782130 + 0.623115i \(0.785867\pi\)
\(12\) 0 0
\(13\) 5.65602e6 1.17179 0.585897 0.810386i \(-0.300742\pi\)
0.585897 + 0.810386i \(0.300742\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.37176e7 0.982601 0.491300 0.870990i \(-0.336522\pi\)
0.491300 + 0.870990i \(0.336522\pi\)
\(18\) 0 0
\(19\) − 5.50828e7i − 1.17083i −0.810733 0.585415i \(-0.800931\pi\)
0.810733 0.585415i \(-0.199069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.18200e7i − 0.147397i −0.997281 0.0736984i \(-0.976520\pi\)
0.997281 0.0736984i \(-0.0234802\pi\)
\(24\) 0 0
\(25\) −1.37911e8 −0.564885
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.88008e8 −0.652309 −0.326154 0.945317i \(-0.605753\pi\)
−0.326154 + 0.945317i \(0.605753\pi\)
\(30\) 0 0
\(31\) − 3.16995e8i − 0.357176i −0.983924 0.178588i \(-0.942847\pi\)
0.983924 0.178588i \(-0.0571529\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.65101e8i − 0.144212i
\(36\) 0 0
\(37\) 1.17320e9 0.457259 0.228629 0.973514i \(-0.426576\pi\)
0.228629 + 0.973514i \(0.426576\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.32284e9 −0.699529 −0.349765 0.936838i \(-0.613738\pi\)
−0.349765 + 0.936838i \(0.613738\pi\)
\(42\) 0 0
\(43\) − 6.04589e9i − 0.956422i −0.878245 0.478211i \(-0.841285\pi\)
0.878245 0.478211i \(-0.158715\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.63294e9i 0.244261i 0.992514 + 0.122130i \(0.0389726\pi\)
−0.992514 + 0.122130i \(0.961027\pi\)
\(48\) 0 0
\(49\) 1.31797e10 0.952203
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.86594e10 −1.29304 −0.646519 0.762898i \(-0.723776\pi\)
−0.646519 + 0.762898i \(0.723776\pi\)
\(54\) 0 0
\(55\) − 2.27550e10i − 0.822054i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.66570e10i 1.58028i 0.612927 + 0.790140i \(0.289992\pi\)
−0.612927 + 0.790140i \(0.710008\pi\)
\(60\) 0 0
\(61\) 4.54516e10 0.882206 0.441103 0.897456i \(-0.354587\pi\)
0.441103 + 0.897456i \(0.354587\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.82953e10 −0.772953
\(66\) 0 0
\(67\) 5.74377e10i 0.634962i 0.948265 + 0.317481i \(0.102837\pi\)
−0.948265 + 0.317481i \(0.897163\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.96341e9i 0.0231335i 0.999933 + 0.0115667i \(0.00368189\pi\)
−0.999933 + 0.0115667i \(0.996318\pi\)
\(72\) 0 0
\(73\) 2.94406e11 1.94540 0.972702 0.232058i \(-0.0745459\pi\)
0.972702 + 0.232058i \(0.0745459\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.67862e10 −0.272457
\(78\) 0 0
\(79\) − 2.19409e11i − 0.902594i −0.892374 0.451297i \(-0.850962\pi\)
0.892374 0.451297i \(-0.149038\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.91018e10i 0.119599i 0.998210 + 0.0597995i \(0.0190461\pi\)
−0.998210 + 0.0597995i \(0.980954\pi\)
\(84\) 0 0
\(85\) −2.44452e11 −0.648156
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.53977e11 −0.309825 −0.154912 0.987928i \(-0.549510\pi\)
−0.154912 + 0.987928i \(0.549510\pi\)
\(90\) 0 0
\(91\) 1.45479e11i 0.256184i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.67725e11i 0.772318i
\(96\) 0 0
\(97\) −1.21087e12 −1.45368 −0.726840 0.686807i \(-0.759012\pi\)
−0.726840 + 0.686807i \(0.759012\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.88925e12 1.77976 0.889878 0.456199i \(-0.150789\pi\)
0.889878 + 0.456199i \(0.150789\pi\)
\(102\) 0 0
\(103\) 8.42992e11i 0.705993i 0.935625 + 0.352996i \(0.114837\pi\)
−0.935625 + 0.352996i \(0.885163\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.79271e12i − 1.19456i −0.802034 0.597279i \(-0.796249\pi\)
0.802034 0.597279i \(-0.203751\pi\)
\(108\) 0 0
\(109\) 6.23866e11 0.371991 0.185996 0.982551i \(-0.440449\pi\)
0.185996 + 0.982551i \(0.440449\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.52840e11 0.313571 0.156786 0.987633i \(-0.449887\pi\)
0.156786 + 0.987633i \(0.449887\pi\)
\(114\) 0 0
\(115\) 2.24894e11i 0.0972277i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.10042e11i 0.214821i
\(120\) 0 0
\(121\) −1.73583e12 −0.553088
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.93772e12 1.03225
\(126\) 0 0
\(127\) 6.71362e12i 1.60005i 0.599965 + 0.800026i \(0.295181\pi\)
−0.599965 + 0.800026i \(0.704819\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.69909e12i 1.52339i 0.647935 + 0.761696i \(0.275633\pi\)
−0.647935 + 0.761696i \(0.724367\pi\)
\(132\) 0 0
\(133\) 1.41679e12 0.255973
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.50551e12 0.983916 0.491958 0.870619i \(-0.336281\pi\)
0.491958 + 0.870619i \(0.336281\pi\)
\(138\) 0 0
\(139\) 3.14897e12i 0.436596i 0.975882 + 0.218298i \(0.0700505\pi\)
−0.975882 + 0.218298i \(0.929949\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.24872e13i 1.46032i
\(144\) 0 0
\(145\) 3.99911e12 0.430284
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.53701e13 −1.40462 −0.702309 0.711872i \(-0.747848\pi\)
−0.702309 + 0.711872i \(0.747848\pi\)
\(150\) 0 0
\(151\) − 1.62889e13i − 1.37414i −0.726591 0.687071i \(-0.758896\pi\)
0.726591 0.687071i \(-0.241104\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.26719e12i 0.235605i
\(156\) 0 0
\(157\) 2.68898e13 1.79552 0.897758 0.440489i \(-0.145195\pi\)
0.897758 + 0.440489i \(0.145195\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.61234e11 0.0322247
\(162\) 0 0
\(163\) 1.75586e13i 0.936193i 0.883677 + 0.468097i \(0.155060\pi\)
−0.883677 + 0.468097i \(0.844940\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.47614e13i 1.60250i 0.598328 + 0.801251i \(0.295832\pi\)
−0.598328 + 0.801251i \(0.704168\pi\)
\(168\) 0 0
\(169\) 8.69252e12 0.373100
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.88795e13 1.07724 0.538620 0.842549i \(-0.318946\pi\)
0.538620 + 0.842549i \(0.318946\pi\)
\(174\) 0 0
\(175\) − 3.54722e12i − 0.123498i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.60079e13i 1.09466i 0.836917 + 0.547330i \(0.184356\pi\)
−0.836917 + 0.547330i \(0.815644\pi\)
\(180\) 0 0
\(181\) 3.45184e12 0.0981700 0.0490850 0.998795i \(-0.484369\pi\)
0.0490850 + 0.998795i \(0.484369\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.20919e13 −0.301623
\(186\) 0 0
\(187\) 5.23630e13i 1.22455i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.34011e13i 1.51183i 0.654671 + 0.755914i \(0.272807\pi\)
−0.654671 + 0.755914i \(0.727193\pi\)
\(192\) 0 0
\(193\) −1.96054e13 −0.379343 −0.189672 0.981848i \(-0.560742\pi\)
−0.189672 + 0.981848i \(0.560742\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.33525e13 1.42601 0.713003 0.701161i \(-0.247335\pi\)
0.713003 + 0.701161i \(0.247335\pi\)
\(198\) 0 0
\(199\) − 1.92304e13i − 0.309650i −0.987942 0.154825i \(-0.950519\pi\)
0.987942 0.154825i \(-0.0494813\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 9.97998e12i − 0.142611i
\(204\) 0 0
\(205\) 3.42477e13 0.461432
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.21610e14 1.45912
\(210\) 0 0
\(211\) 1.67562e14i 1.89880i 0.314062 + 0.949402i \(0.398310\pi\)
−0.314062 + 0.949402i \(0.601690\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.23135e13i 0.630887i
\(216\) 0 0
\(217\) 8.15344e12 0.0780877
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.34147e14 1.15141
\(222\) 0 0
\(223\) 1.42965e14i 1.16252i 0.813718 + 0.581261i \(0.197440\pi\)
−0.813718 + 0.581261i \(0.802560\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.24397e13i 0.237095i 0.992948 + 0.118547i \(0.0378238\pi\)
−0.992948 + 0.118547i \(0.962176\pi\)
\(228\) 0 0
\(229\) −2.11741e14 −1.46822 −0.734110 0.679030i \(-0.762400\pi\)
−0.734110 + 0.679030i \(0.762400\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.89837e13 −0.306137 −0.153069 0.988216i \(-0.548916\pi\)
−0.153069 + 0.988216i \(0.548916\pi\)
\(234\) 0 0
\(235\) − 2.71371e13i − 0.161122i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.66870e14i 0.895347i 0.894197 + 0.447674i \(0.147747\pi\)
−0.894197 + 0.447674i \(0.852253\pi\)
\(240\) 0 0
\(241\) 9.85623e13 0.503047 0.251524 0.967851i \(-0.419068\pi\)
0.251524 + 0.967851i \(0.419068\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.35840e14 −0.628104
\(246\) 0 0
\(247\) − 3.11549e14i − 1.37197i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 3.73220e14i − 1.49253i −0.665649 0.746265i \(-0.731845\pi\)
0.665649 0.746265i \(-0.268155\pi\)
\(252\) 0 0
\(253\) 4.81736e13 0.183690
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.93272e14 1.01782 0.508912 0.860819i \(-0.330048\pi\)
0.508912 + 0.860819i \(0.330048\pi\)
\(258\) 0 0
\(259\) 3.01760e13i 0.0999683i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00708e14i 1.81522i 0.419817 + 0.907609i \(0.362094\pi\)
−0.419817 + 0.907609i \(0.637906\pi\)
\(264\) 0 0
\(265\) 2.95385e14 0.852930
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.21692e14 −1.37689 −0.688447 0.725287i \(-0.741707\pi\)
−0.688447 + 0.725287i \(0.741707\pi\)
\(270\) 0 0
\(271\) − 3.63124e14i − 0.916726i −0.888765 0.458363i \(-0.848436\pi\)
0.888765 0.458363i \(-0.151564\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3.04477e14i − 0.703976i
\(276\) 0 0
\(277\) 2.94181e14 0.651233 0.325616 0.945502i \(-0.394428\pi\)
0.325616 + 0.945502i \(0.394428\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.70665e14 −1.15916 −0.579580 0.814916i \(-0.696783\pi\)
−0.579580 + 0.814916i \(0.696783\pi\)
\(282\) 0 0
\(283\) 4.19893e14i 0.817373i 0.912675 + 0.408687i \(0.134013\pi\)
−0.912675 + 0.408687i \(0.865987\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.54669e13i − 0.152935i
\(288\) 0 0
\(289\) −2.00978e13 −0.0344954
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.50267e14 −0.553596 −0.276798 0.960928i \(-0.589273\pi\)
−0.276798 + 0.960928i \(0.589273\pi\)
\(294\) 0 0
\(295\) − 6.87018e14i − 1.04240i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.23415e14i − 0.172719i
\(300\) 0 0
\(301\) 1.55507e14 0.209098
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.68459e14 −0.581932
\(306\) 0 0
\(307\) 7.25089e14i 0.866086i 0.901373 + 0.433043i \(0.142560\pi\)
−0.901373 + 0.433043i \(0.857440\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 1.30645e15i − 1.44388i −0.691955 0.721941i \(-0.743250\pi\)
0.691955 0.721941i \(-0.256750\pi\)
\(312\) 0 0
\(313\) −9.51812e14 −1.01224 −0.506122 0.862462i \(-0.668921\pi\)
−0.506122 + 0.862462i \(0.668921\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.04849e14 0.793157 0.396578 0.918001i \(-0.370198\pi\)
0.396578 + 0.918001i \(0.370198\pi\)
\(318\) 0 0
\(319\) − 8.56634e14i − 0.812926i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 1.30643e15i − 1.15046i
\(324\) 0 0
\(325\) −7.80030e14 −0.661928
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.77220e13 −0.0534016
\(330\) 0 0
\(331\) − 1.68149e15i − 1.27858i −0.768967 0.639288i \(-0.779229\pi\)
0.768967 0.639288i \(-0.220771\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 5.91996e14i − 0.418842i
\(336\) 0 0
\(337\) −8.00353e14 −0.546389 −0.273195 0.961959i \(-0.588080\pi\)
−0.273195 + 0.961959i \(0.588080\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.99852e14 0.445123
\(342\) 0 0
\(343\) 6.95009e14i 0.426801i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.02247e15i − 1.15853i −0.815141 0.579263i \(-0.803340\pi\)
0.815141 0.579263i \(-0.196660\pi\)
\(348\) 0 0
\(349\) −1.47943e15 −0.818732 −0.409366 0.912370i \(-0.634250\pi\)
−0.409366 + 0.912370i \(0.634250\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.77287e13 −0.0194995 −0.00974976 0.999952i \(-0.503103\pi\)
−0.00974976 + 0.999952i \(0.503103\pi\)
\(354\) 0 0
\(355\) − 3.05431e13i − 0.0152596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 2.92222e15i − 1.36504i −0.730865 0.682522i \(-0.760883\pi\)
0.730865 0.682522i \(-0.239117\pi\)
\(360\) 0 0
\(361\) −8.20797e14 −0.370845
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.03437e15 −1.28325
\(366\) 0 0
\(367\) 1.63345e15i 0.668514i 0.942482 + 0.334257i \(0.108485\pi\)
−0.942482 + 0.334257i \(0.891515\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 7.37149e14i − 0.282691i
\(372\) 0 0
\(373\) −1.73734e15 −0.645108 −0.322554 0.946551i \(-0.604541\pi\)
−0.322554 + 0.946551i \(0.604541\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.19458e15 −0.764371
\(378\) 0 0
\(379\) − 2.47588e15i − 0.835401i −0.908585 0.417700i \(-0.862836\pi\)
0.908585 0.417700i \(-0.137164\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.13485e15i 0.359538i 0.983709 + 0.179769i \(0.0575350\pi\)
−0.983709 + 0.179769i \(0.942465\pi\)
\(384\) 0 0
\(385\) 5.85282e14 0.179722
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.47360e15 1.29110 0.645549 0.763719i \(-0.276629\pi\)
0.645549 + 0.763719i \(0.276629\pi\)
\(390\) 0 0
\(391\) − 5.17518e14i − 0.144832i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.26140e15i 0.595380i
\(396\) 0 0
\(397\) 7.57834e15 1.93567 0.967834 0.251591i \(-0.0809536\pi\)
0.967834 + 0.251591i \(0.0809536\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.42681e15 −0.583673 −0.291836 0.956468i \(-0.594266\pi\)
−0.291836 + 0.956468i \(0.594266\pi\)
\(402\) 0 0
\(403\) − 1.79293e15i − 0.418536i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.59016e15i 0.569849i
\(408\) 0 0
\(409\) 3.65591e15 0.781009 0.390505 0.920601i \(-0.372301\pi\)
0.390505 + 0.920601i \(0.372301\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.71449e15 −0.345489
\(414\) 0 0
\(415\) − 4.03012e14i − 0.0788914i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.72429e15i 1.79711i 0.438865 + 0.898553i \(0.355380\pi\)
−0.438865 + 0.898553i \(0.644620\pi\)
\(420\) 0 0
\(421\) 4.09655e15 0.735743 0.367871 0.929877i \(-0.380087\pi\)
0.367871 + 0.929877i \(0.380087\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.27093e15 −0.555056
\(426\) 0 0
\(427\) 1.16906e15i 0.192873i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 7.54740e15i − 1.17743i −0.808342 0.588713i \(-0.799635\pi\)
0.808342 0.588713i \(-0.200365\pi\)
\(432\) 0 0
\(433\) −8.06264e15 −1.22335 −0.611674 0.791110i \(-0.709503\pi\)
−0.611674 + 0.791110i \(0.709503\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.20191e15 −0.172577
\(438\) 0 0
\(439\) 8.34545e15i 1.16590i 0.812506 + 0.582952i \(0.198102\pi\)
−0.812506 + 0.582952i \(0.801898\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.63523e15i 0.745572i 0.927917 + 0.372786i \(0.121597\pi\)
−0.927917 + 0.372786i \(0.878403\pi\)
\(444\) 0 0
\(445\) 1.58701e15 0.204371
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.21818e15 0.148674 0.0743368 0.997233i \(-0.476316\pi\)
0.0743368 + 0.997233i \(0.476316\pi\)
\(450\) 0 0
\(451\) − 7.33607e15i − 0.871774i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.49942e15i − 0.168987i
\(456\) 0 0
\(457\) 1.04213e15 0.114400 0.0571998 0.998363i \(-0.481783\pi\)
0.0571998 + 0.998363i \(0.481783\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.16311e16 1.21175 0.605877 0.795558i \(-0.292822\pi\)
0.605877 + 0.795558i \(0.292822\pi\)
\(462\) 0 0
\(463\) 1.67988e15i 0.170527i 0.996358 + 0.0852636i \(0.0271732\pi\)
−0.996358 + 0.0852636i \(0.972827\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.24813e16i 1.20325i 0.798777 + 0.601627i \(0.205481\pi\)
−0.798777 + 0.601627i \(0.794519\pi\)
\(468\) 0 0
\(469\) −1.47736e15 −0.138819
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.33479e16 1.19192
\(474\) 0 0
\(475\) 7.59654e15i 0.661385i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.23267e15i 0.764389i 0.924082 + 0.382195i \(0.124832\pi\)
−0.924082 + 0.382195i \(0.875168\pi\)
\(480\) 0 0
\(481\) 6.63565e15 0.535813
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.24802e16 0.958894
\(486\) 0 0
\(487\) 1.80780e16i 1.35511i 0.735470 + 0.677557i \(0.236961\pi\)
−0.735470 + 0.677557i \(0.763039\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.01221e15i 0.214979i 0.994206 + 0.107490i \(0.0342813\pi\)
−0.994206 + 0.107490i \(0.965719\pi\)
\(492\) 0 0
\(493\) −9.20263e15 −0.640959
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.62219e13 −0.00505757
\(498\) 0 0
\(499\) − 3.33196e15i − 0.215822i −0.994161 0.107911i \(-0.965584\pi\)
0.994161 0.107911i \(-0.0344162\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.62787e15i 0.594459i 0.954806 + 0.297230i \(0.0960627\pi\)
−0.954806 + 0.297230i \(0.903937\pi\)
\(504\) 0 0
\(505\) −1.94720e16 −1.17398
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.53124e16 1.45555 0.727774 0.685817i \(-0.240555\pi\)
0.727774 + 0.685817i \(0.240555\pi\)
\(510\) 0 0
\(511\) 7.57244e15i 0.425315i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 8.68852e15i − 0.465696i
\(516\) 0 0
\(517\) −5.81293e15 −0.304405
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.07262e15 0.103632 0.0518158 0.998657i \(-0.483499\pi\)
0.0518158 + 0.998657i \(0.483499\pi\)
\(522\) 0 0
\(523\) 1.96180e16i 0.958617i 0.877646 + 0.479309i \(0.159113\pi\)
−0.877646 + 0.479309i \(0.840887\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 7.51836e15i − 0.350961i
\(528\) 0 0
\(529\) 2.14385e16 0.978274
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.87940e16 −0.819704
\(534\) 0 0
\(535\) 1.84770e16i 0.787969i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.90978e16i 1.18666i
\(540\) 0 0
\(541\) 3.69504e16 1.47379 0.736895 0.676007i \(-0.236291\pi\)
0.736895 + 0.676007i \(0.236291\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.43004e15 −0.245377
\(546\) 0 0
\(547\) − 1.90846e16i − 0.712458i −0.934399 0.356229i \(-0.884062\pi\)
0.934399 0.356229i \(-0.115938\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.13726e16i 0.763743i
\(552\) 0 0
\(553\) 5.64343e15 0.197330
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.48841e16 0.498415 0.249208 0.968450i \(-0.419830\pi\)
0.249208 + 0.968450i \(0.419830\pi\)
\(558\) 0 0
\(559\) − 3.41957e16i − 1.12073i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1.17234e15i − 0.0368133i −0.999831 0.0184067i \(-0.994141\pi\)
0.999831 0.0184067i \(-0.00585935\pi\)
\(564\) 0 0
\(565\) −6.72866e15 −0.206842
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.41843e16 −1.89128 −0.945639 0.325219i \(-0.894562\pi\)
−0.945639 + 0.325219i \(0.894562\pi\)
\(570\) 0 0
\(571\) − 9.82359e15i − 0.283435i −0.989907 0.141718i \(-0.954738\pi\)
0.989907 0.141718i \(-0.0452625\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00923e15i 0.0832622i
\(576\) 0 0
\(577\) 2.13311e16 0.578040 0.289020 0.957323i \(-0.406671\pi\)
0.289020 + 0.957323i \(0.406671\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.00574e15 −0.0261474
\(582\) 0 0
\(583\) − 6.32733e16i − 1.61142i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.72848e16i − 1.88915i −0.328302 0.944573i \(-0.606476\pi\)
0.328302 0.944573i \(-0.393524\pi\)
\(588\) 0 0
\(589\) −1.74610e16 −0.418193
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.15519e16 −0.955570 −0.477785 0.878477i \(-0.658560\pi\)
−0.477785 + 0.878477i \(0.658560\pi\)
\(594\) 0 0
\(595\) − 6.28755e15i − 0.141703i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 5.90470e15i − 0.127831i −0.997955 0.0639156i \(-0.979641\pi\)
0.997955 0.0639156i \(-0.0203588\pi\)
\(600\) 0 0
\(601\) −6.32387e16 −1.34195 −0.670975 0.741480i \(-0.734124\pi\)
−0.670975 + 0.741480i \(0.734124\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.78908e16 0.364835
\(606\) 0 0
\(607\) − 9.38322e15i − 0.187594i −0.995591 0.0937971i \(-0.970099\pi\)
0.995591 0.0937971i \(-0.0299005\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.48920e16i 0.286223i
\(612\) 0 0
\(613\) 4.29006e16 0.808538 0.404269 0.914640i \(-0.367526\pi\)
0.404269 + 0.914640i \(0.367526\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.82180e16 −1.59899 −0.799496 0.600672i \(-0.794900\pi\)
−0.799496 + 0.600672i \(0.794900\pi\)
\(618\) 0 0
\(619\) 4.76892e16i 0.847766i 0.905717 + 0.423883i \(0.139333\pi\)
−0.905717 + 0.423883i \(0.860667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 3.96046e15i − 0.0677356i
\(624\) 0 0
\(625\) −6.91535e15 −0.116020
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.78255e16 0.449303
\(630\) 0 0
\(631\) − 6.79965e15i − 0.107723i −0.998548 0.0538617i \(-0.982847\pi\)
0.998548 0.0538617i \(-0.0171530\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 6.91956e16i − 1.05545i
\(636\) 0 0
\(637\) 7.45448e16 1.11579
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.57968e16 0.516055 0.258027 0.966138i \(-0.416928\pi\)
0.258027 + 0.966138i \(0.416928\pi\)
\(642\) 0 0
\(643\) 4.41166e16i 0.624218i 0.950046 + 0.312109i \(0.101035\pi\)
−0.950046 + 0.312109i \(0.898965\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.27368e17i − 1.73634i −0.496267 0.868170i \(-0.665296\pi\)
0.496267 0.868170i \(-0.334704\pi\)
\(648\) 0 0
\(649\) −1.47164e17 −1.96939
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.19791e16 1.18634 0.593171 0.805076i \(-0.297876\pi\)
0.593171 + 0.805076i \(0.297876\pi\)
\(654\) 0 0
\(655\) − 7.93526e16i − 1.00488i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.36757e17i − 1.66970i −0.550477 0.834850i \(-0.685554\pi\)
0.550477 0.834850i \(-0.314446\pi\)
\(660\) 0 0
\(661\) −2.59246e16 −0.310816 −0.155408 0.987850i \(-0.549669\pi\)
−0.155408 + 0.987850i \(0.549669\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.46025e16 −0.168848
\(666\) 0 0
\(667\) 8.46635e15i 0.0961482i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.00347e17i 1.09943i
\(672\) 0 0
\(673\) −6.31003e16 −0.679111 −0.339555 0.940586i \(-0.610277\pi\)
−0.339555 + 0.940586i \(0.610277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.78154e16 0.185039 0.0925196 0.995711i \(-0.470508\pi\)
0.0925196 + 0.995711i \(0.470508\pi\)
\(678\) 0 0
\(679\) − 3.11450e16i − 0.317811i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 8.17507e16i − 0.805318i −0.915350 0.402659i \(-0.868086\pi\)
0.915350 0.402659i \(-0.131914\pi\)
\(684\) 0 0
\(685\) −6.70507e16 −0.649023
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.62098e17 −1.51517
\(690\) 0 0
\(691\) 1.18406e17i 1.08769i 0.839184 + 0.543847i \(0.183033\pi\)
−0.839184 + 0.543847i \(0.816967\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 3.24557e16i − 0.287993i
\(696\) 0 0
\(697\) −7.88097e16 −0.687358
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.50131e16 0.295068 0.147534 0.989057i \(-0.452866\pi\)
0.147534 + 0.989057i \(0.452866\pi\)
\(702\) 0 0
\(703\) − 6.46231e16i − 0.535373i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.85934e16i 0.389100i
\(708\) 0 0
\(709\) 6.04581e16 0.475967 0.237984 0.971269i \(-0.423514\pi\)
0.237984 + 0.971269i \(0.423514\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.91683e15 −0.0526466
\(714\) 0 0
\(715\) − 1.28703e17i − 0.963277i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.15269e17i 0.834330i 0.908831 + 0.417165i \(0.136976\pi\)
−0.908831 + 0.417165i \(0.863024\pi\)
\(720\) 0 0
\(721\) −2.16826e16 −0.154348
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.35108e16 0.368479
\(726\) 0 0
\(727\) 7.77948e16i 0.526919i 0.964670 + 0.263460i \(0.0848636\pi\)
−0.964670 + 0.263460i \(0.915136\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 1.43394e17i − 0.939781i
\(732\) 0 0
\(733\) 2.20976e17 1.42469 0.712345 0.701829i \(-0.247633\pi\)
0.712345 + 0.701829i \(0.247633\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.26809e17 −0.791309
\(738\) 0 0
\(739\) − 2.10708e17i − 1.29365i −0.762640 0.646823i \(-0.776097\pi\)
0.762640 0.646823i \(-0.223903\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1.48321e17i − 0.881594i −0.897607 0.440797i \(-0.854696\pi\)
0.897607 0.440797i \(-0.145304\pi\)
\(744\) 0 0
\(745\) 1.58416e17 0.926532
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.61103e16 0.261160
\(750\) 0 0
\(751\) 8.57383e16i 0.477898i 0.971032 + 0.238949i \(0.0768029\pi\)
−0.971032 + 0.238949i \(0.923197\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.67886e17i 0.906428i
\(756\) 0 0
\(757\) 1.75448e16 0.0932335 0.0466168 0.998913i \(-0.485156\pi\)
0.0466168 + 0.998913i \(0.485156\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.12682e17 −1.09502 −0.547511 0.836798i \(-0.684425\pi\)
−0.547511 + 0.836798i \(0.684425\pi\)
\(762\) 0 0
\(763\) 1.60465e16i 0.0813267i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.77014e17i 1.85176i
\(768\) 0 0
\(769\) 7.21877e16 0.349064 0.174532 0.984651i \(-0.444159\pi\)
0.174532 + 0.984651i \(0.444159\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.73938e17 0.815299 0.407649 0.913139i \(-0.366349\pi\)
0.407649 + 0.913139i \(0.366349\pi\)
\(774\) 0 0
\(775\) 4.37172e16i 0.201763i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.83031e17i 0.819031i
\(780\) 0 0
\(781\) −6.54253e15 −0.0288296
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.77146e17 −1.18438
\(786\) 0 0
\(787\) 1.47492e16i 0.0620754i 0.999518 + 0.0310377i \(0.00988119\pi\)
−0.999518 + 0.0310377i \(0.990119\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.67917e16i 0.0685546i
\(792\) 0 0
\(793\) 2.57075e17 1.03376
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.41145e17 1.33103 0.665517 0.746382i \(-0.268211\pi\)
0.665517 + 0.746382i \(0.268211\pi\)
\(798\) 0 0
\(799\) 6.24470e16i 0.240011i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.49982e17i 2.42442i
\(804\) 0 0
\(805\) −5.78450e15 −0.0212564
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.84056e16 0.0656537 0.0328269 0.999461i \(-0.489549\pi\)
0.0328269 + 0.999461i \(0.489549\pi\)
\(810\) 0 0
\(811\) 3.79008e17i 1.33206i 0.745926 + 0.666029i \(0.232007\pi\)
−0.745926 + 0.666029i \(0.767993\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.80973e17i − 0.617544i
\(816\) 0 0
\(817\) −3.33024e17 −1.11981
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.57140e17 1.49276 0.746381 0.665519i \(-0.231790\pi\)
0.746381 + 0.665519i \(0.231790\pi\)
\(822\) 0 0
\(823\) 3.70495e17i 1.19229i 0.802876 + 0.596147i \(0.203302\pi\)
−0.802876 + 0.596147i \(0.796698\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.82136e17i 0.569328i 0.958627 + 0.284664i \(0.0918821\pi\)
−0.958627 + 0.284664i \(0.908118\pi\)
\(828\) 0 0
\(829\) 2.11760e17 0.652404 0.326202 0.945300i \(-0.394231\pi\)
0.326202 + 0.945300i \(0.394231\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.12591e17 0.935636
\(834\) 0 0
\(835\) − 3.58277e17i − 1.05706i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1.42876e17i − 0.409626i −0.978801 0.204813i \(-0.934341\pi\)
0.978801 0.204813i \(-0.0656587\pi\)
\(840\) 0 0
\(841\) −2.03264e17 −0.574493
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.95917e16 −0.246109
\(846\) 0 0
\(847\) − 4.46473e16i − 0.120919i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.55993e16i − 0.0673985i
\(852\) 0 0
\(853\) 7.50904e16 0.194935 0.0974675 0.995239i \(-0.468926\pi\)
0.0974675 + 0.995239i \(0.468926\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.57808e16 0.0398330 0.0199165 0.999802i \(-0.493660\pi\)
0.0199165 + 0.999802i \(0.493660\pi\)
\(858\) 0 0
\(859\) 6.52158e17i 1.62328i 0.584157 + 0.811641i \(0.301425\pi\)
−0.584157 + 0.811641i \(0.698575\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 3.77726e17i − 0.914350i −0.889377 0.457175i \(-0.848861\pi\)
0.889377 0.457175i \(-0.151139\pi\)
\(864\) 0 0
\(865\) −2.97654e17 −0.710583
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.84405e17 1.12484
\(870\) 0 0
\(871\) 3.24869e17i 0.744045i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.01282e17i 0.225676i
\(876\) 0 0
\(877\) 2.37602e17 0.522220 0.261110 0.965309i \(-0.415912\pi\)
0.261110 + 0.965309i \(0.415912\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.87143e17 0.400238 0.200119 0.979772i \(-0.435867\pi\)
0.200119 + 0.979772i \(0.435867\pi\)
\(882\) 0 0
\(883\) − 5.84655e17i − 1.23349i −0.787163 0.616745i \(-0.788451\pi\)
0.787163 0.616745i \(-0.211549\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.95742e17i 0.607254i 0.952791 + 0.303627i \(0.0981977\pi\)
−0.952791 + 0.303627i \(0.901802\pi\)
\(888\) 0 0
\(889\) −1.72681e17 −0.349812
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.45030e17 0.285988
\(894\) 0 0
\(895\) − 3.71124e17i − 0.722074i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.22997e17i 0.232989i
\(900\) 0 0
\(901\) −6.79731e17 −1.27054
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.55772e16 −0.0647561
\(906\) 0 0
\(907\) − 8.53775e14i − 0.00153356i −1.00000 0.000766778i \(-0.999756\pi\)
1.00000 0.000766778i \(-0.000244073\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.24702e17i 0.568035i 0.958819 + 0.284018i \(0.0916674\pi\)
−0.958819 + 0.284018i \(0.908333\pi\)
\(912\) 0 0
\(913\) −8.63277e16 −0.149048
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.98029e17 −0.333052
\(918\) 0 0
\(919\) 1.05696e18i 1.75454i 0.479996 + 0.877271i \(0.340638\pi\)
−0.479996 + 0.877271i \(0.659362\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.67611e16i 0.0271077i
\(924\) 0 0
\(925\) −1.61798e17 −0.258299
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.48690e17 −0.386869 −0.193435 0.981113i \(-0.561963\pi\)
−0.193435 + 0.981113i \(0.561963\pi\)
\(930\) 0 0
\(931\) − 7.25975e17i − 1.11487i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 5.39693e17i − 0.807751i
\(936\) 0 0
\(937\) 4.48490e17 0.662697 0.331348 0.943508i \(-0.392496\pi\)
0.331348 + 0.943508i \(0.392496\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.19998e17 −0.316869 −0.158435 0.987369i \(-0.550645\pi\)
−0.158435 + 0.987369i \(0.550645\pi\)
\(942\) 0 0
\(943\) 7.25043e16i 0.103108i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.50630e17i 0.763414i 0.924283 + 0.381707i \(0.124664\pi\)
−0.924283 + 0.381707i \(0.875336\pi\)
\(948\) 0 0
\(949\) 1.66517e18 2.27961
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −9.59450e17 −1.28075 −0.640376 0.768062i \(-0.721221\pi\)
−0.640376 + 0.768062i \(0.721221\pi\)
\(954\) 0 0
\(955\) − 7.56528e17i − 0.997251i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.67329e17i 0.215109i
\(960\) 0 0
\(961\) 6.87177e17 0.872425
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.02068e17 0.250227
\(966\) 0 0
\(967\) 1.36871e17i 0.167399i 0.996491 + 0.0836993i \(0.0266735\pi\)
−0.996491 + 0.0836993i \(0.973326\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1.45694e18i − 1.73831i −0.494538 0.869156i \(-0.664663\pi\)
0.494538 0.869156i \(-0.335337\pi\)
\(972\) 0 0
\(973\) −8.09949e16 −0.0954510
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.57226e18 −1.80783 −0.903915 0.427713i \(-0.859319\pi\)
−0.903915 + 0.427713i \(0.859319\pi\)
\(978\) 0 0
\(979\) − 3.39947e17i − 0.386113i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1.13899e18i − 1.26240i −0.775618 0.631202i \(-0.782562\pi\)
0.775618 0.631202i \(-0.217438\pi\)
\(984\) 0 0
\(985\) −8.59094e17 −0.940640
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.31921e17 −0.140973
\(990\) 0 0
\(991\) − 1.72529e18i − 1.82146i −0.413001 0.910731i \(-0.635519\pi\)
0.413001 0.910731i \(-0.364481\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.98203e17i 0.204255i
\(996\) 0 0
\(997\) 9.17404e17 0.934092 0.467046 0.884233i \(-0.345318\pi\)
0.467046 + 0.884233i \(0.345318\pi\)
\(998\) 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 144.13.g.h.127.2 4
3.2 odd 2 48.13.g.a.31.4 yes 4
4.3 odd 2 inner 144.13.g.h.127.1 4
12.11 even 2 48.13.g.a.31.2 4
24.5 odd 2 192.13.g.c.127.1 4
24.11 even 2 192.13.g.c.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.13.g.a.31.2 4 12.11 even 2
48.13.g.a.31.4 yes 4 3.2 odd 2
144.13.g.h.127.1 4 4.3 odd 2 inner
144.13.g.h.127.2 4 1.1 even 1 trivial
192.13.g.c.127.1 4 24.5 odd 2
192.13.g.c.127.3 4 24.11 even 2