Properties

Label 48.13.g.a.31.2
Level $48$
Weight $13$
Character 48.31
Analytic conductor $43.872$
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] = [48,13,Mod(31,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(43.8717032293\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.2
Root \(33.1599 + 57.4347i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.13.g.a.31.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-420.888i q^{3} +10306.8 q^{5} -25721.1i q^{7} -177147. q^{9} +O(q^{10})\) \(q-420.888i q^{3} +10306.8 q^{5} -25721.1i q^{7} -177147. q^{9} +2.20777e6i q^{11} +5.65602e6 q^{13} -4.33799e6i q^{15} -2.37176e7 q^{17} +5.50828e7i q^{19} -1.08257e7 q^{21} -2.18200e7i q^{23} -1.37911e8 q^{25} +7.45591e7i q^{27} +3.88008e8 q^{29} +3.16995e8i q^{31} +9.29225e8 q^{33} -2.65101e8i q^{35} +1.17320e9 q^{37} -2.38055e9i q^{39} +3.32284e9 q^{41} +6.04589e9i q^{43} -1.82581e9 q^{45} +2.63294e9i q^{47} +1.31797e10 q^{49} +9.98246e9i q^{51} +2.86594e10 q^{53} +2.27550e10i q^{55} +2.31837e10 q^{57} +6.66570e10i q^{59} +4.54516e10 q^{61} +4.55641e9i q^{63} +5.82953e10 q^{65} -5.74377e10i q^{67} -9.18379e9 q^{69} +2.96341e9i q^{71} +2.94406e11 q^{73} +5.80453e10i q^{75} +5.67862e10 q^{77} +2.19409e11i q^{79} +3.13811e10 q^{81} +3.91018e10i q^{83} -2.44452e11 q^{85} -1.63308e11i q^{87} +1.53977e11 q^{89} -1.45479e11i q^{91} +1.33419e11 q^{93} +5.67725e11i q^{95} -1.21087e12 q^{97} -3.91100e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 21960 q^{5} - 708588 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 21960 q^{5} - 708588 q^{9} - 5810072 q^{13} - 33882264 q^{17} - 15664752 q^{21} + 142148300 q^{25} + 42583896 q^{29} + 271362960 q^{33} + 9402865736 q^{37} + 17968882536 q^{41} + 3890148120 q^{45} + 53940845764 q^{49} + 60546956760 q^{53} + 46910963376 q^{57} + 168287201672 q^{61} + 481064975280 q^{65} + 163198209024 q^{69} + 789026629000 q^{73} + 140389989696 q^{77} + 125524238436 q^{81} - 777401136720 q^{85} - 638670460536 q^{89} - 607412656080 q^{93} - 563542043000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 420.888i − 0.577350i
\(4\) 0 0
\(5\) 10306.8 0.659633 0.329816 0.944045i \(-0.393013\pi\)
0.329816 + 0.944045i \(0.393013\pi\)
\(6\) 0 0
\(7\) − 25721.1i − 0.218625i −0.994007 0.109313i \(-0.965135\pi\)
0.994007 0.109313i \(-0.0348650\pi\)
\(8\) 0 0
\(9\) −177147. −0.333333
\(10\) 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
\(15\) − 4.33799e6i − 0.380839i
\(16\) 0 0
\(17\) −2.37176e7 −0.982601 −0.491300 0.870990i \(-0.663478\pi\)
−0.491300 + 0.870990i \(0.663478\pi\)
\(18\) 0 0
\(19\) 5.50828e7i 1.17083i 0.810733 + 0.585415i \(0.199069\pi\)
−0.810733 + 0.585415i \(0.800931\pi\)
\(20\) 0 0
\(21\) −1.08257e7 −0.126223
\(22\) 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
\(27\) 7.45591e7i 0.192450i
\(28\) 0 0
\(29\) 3.88008e8 0.652309 0.326154 0.945317i \(-0.394247\pi\)
0.326154 + 0.945317i \(0.394247\pi\)
\(30\) 0 0
\(31\) 3.16995e8i 0.357176i 0.983924 + 0.178588i \(0.0571529\pi\)
−0.983924 + 0.178588i \(0.942847\pi\)
\(32\) 0 0
\(33\) 9.29225e8 0.719511
\(34\) 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
\(39\) − 2.38055e9i − 0.676535i
\(40\) 0 0
\(41\) 3.32284e9 0.699529 0.349765 0.936838i \(-0.386262\pi\)
0.349765 + 0.936838i \(0.386262\pi\)
\(42\) 0 0
\(43\) 6.04589e9i 0.956422i 0.878245 + 0.478211i \(0.158715\pi\)
−0.878245 + 0.478211i \(0.841285\pi\)
\(44\) 0 0
\(45\) −1.82581e9 −0.219878
\(46\) 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
\(51\) 9.98246e9i 0.567305i
\(52\) 0 0
\(53\) 2.86594e10 1.29304 0.646519 0.762898i \(-0.276224\pi\)
0.646519 + 0.762898i \(0.276224\pi\)
\(54\) 0 0
\(55\) 2.27550e10i 0.822054i
\(56\) 0 0
\(57\) 2.31837e10 0.675980
\(58\) 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
\(63\) 4.55641e9i 0.0728751i
\(64\) 0 0
\(65\) 5.82953e10 0.772953
\(66\) 0 0
\(67\) − 5.74377e10i − 0.634962i −0.948265 0.317481i \(-0.897163\pi\)
0.948265 0.317481i \(-0.102837\pi\)
\(68\) 0 0
\(69\) −9.18379e9 −0.0850996
\(70\) 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
\(75\) 5.80453e10i 0.326136i
\(76\) 0 0
\(77\) 5.67862e10 0.272457
\(78\) 0 0
\(79\) 2.19409e11i 0.902594i 0.892374 + 0.451297i \(0.149038\pi\)
−0.892374 + 0.451297i \(0.850962\pi\)
\(80\) 0 0
\(81\) 3.13811e10 0.111111
\(82\) 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
\(87\) − 1.63308e11i − 0.376611i
\(88\) 0 0
\(89\) 1.53977e11 0.309825 0.154912 0.987928i \(-0.450490\pi\)
0.154912 + 0.987928i \(0.450490\pi\)
\(90\) 0 0
\(91\) − 1.45479e11i − 0.256184i
\(92\) 0 0
\(93\) 1.33419e11 0.206216
\(94\) 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
\(99\) − 3.91100e11i − 0.415410i
\(100\) 0 0
\(101\) −1.88925e12 −1.77976 −0.889878 0.456199i \(-0.849211\pi\)
−0.889878 + 0.456199i \(0.849211\pi\)
\(102\) 0 0
\(103\) − 8.42992e11i − 0.705993i −0.935625 0.352996i \(-0.885163\pi\)
0.935625 0.352996i \(-0.114837\pi\)
\(104\) 0 0
\(105\) −1.11578e11 −0.0832611
\(106\) 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
\(111\) − 4.93787e11i − 0.263998i
\(112\) 0 0
\(113\) −6.52840e11 −0.313571 −0.156786 0.987633i \(-0.550113\pi\)
−0.156786 + 0.987633i \(0.550113\pi\)
\(114\) 0 0
\(115\) − 2.24894e11i − 0.0972277i
\(116\) 0 0
\(117\) −1.00195e12 −0.390598
\(118\) 0 0
\(119\) 6.10042e11i 0.214821i
\(120\) 0 0
\(121\) −1.73583e12 −0.553088
\(122\) 0 0
\(123\) − 1.39854e12i − 0.403873i
\(124\) 0 0
\(125\) −3.93772e12 −1.03225
\(126\) 0 0
\(127\) − 6.71362e12i − 1.60005i −0.599965 0.800026i \(-0.704819\pi\)
0.599965 0.800026i \(-0.295181\pi\)
\(128\) 0 0
\(129\) 2.54464e12 0.552190
\(130\) 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
\(135\) 7.68463e11i 0.126946i
\(136\) 0 0
\(137\) −6.50551e12 −0.983916 −0.491958 0.870619i \(-0.663719\pi\)
−0.491958 + 0.870619i \(0.663719\pi\)
\(138\) 0 0
\(139\) − 3.14897e12i − 0.436596i −0.975882 0.218298i \(-0.929949\pi\)
0.975882 0.218298i \(-0.0700505\pi\)
\(140\) 0 0
\(141\) 1.10817e12 0.141024
\(142\) 0 0
\(143\) 1.24872e13i 1.46032i
\(144\) 0 0
\(145\) 3.99911e12 0.430284
\(146\) 0 0
\(147\) − 5.54719e12i − 0.549755i
\(148\) 0 0
\(149\) 1.53701e13 1.40462 0.702309 0.711872i \(-0.252152\pi\)
0.702309 + 0.711872i \(0.252152\pi\)
\(150\) 0 0
\(151\) 1.62889e13i 1.37414i 0.726591 + 0.687071i \(0.241104\pi\)
−0.726591 + 0.687071i \(0.758896\pi\)
\(152\) 0 0
\(153\) 4.20150e12 0.327534
\(154\) 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
\(159\) − 1.20624e13i − 0.746536i
\(160\) 0 0
\(161\) −5.61234e11 −0.0322247
\(162\) 0 0
\(163\) − 1.75586e13i − 0.936193i −0.883677 0.468097i \(-0.844940\pi\)
0.883677 0.468097i \(-0.155060\pi\)
\(164\) 0 0
\(165\) 9.57730e12 0.474613
\(166\) 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
\(171\) − 9.75775e12i − 0.390277i
\(172\) 0 0
\(173\) −2.88795e13 −1.07724 −0.538620 0.842549i \(-0.681054\pi\)
−0.538620 + 0.842549i \(0.681054\pi\)
\(174\) 0 0
\(175\) 3.54722e12i 0.123498i
\(176\) 0 0
\(177\) 2.80552e13 0.912375
\(178\) 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
\(183\) − 1.91300e13i − 0.509342i
\(184\) 0 0
\(185\) 1.20919e13 0.301623
\(186\) 0 0
\(187\) − 5.23630e13i − 1.22455i
\(188\) 0 0
\(189\) 1.91774e12 0.0420745
\(190\) 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
\(195\) − 2.45358e13i − 0.446265i
\(196\) 0 0
\(197\) −8.33525e13 −1.42601 −0.713003 0.701161i \(-0.752665\pi\)
−0.713003 + 0.701161i \(0.752665\pi\)
\(198\) 0 0
\(199\) 1.92304e13i 0.309650i 0.987942 + 0.154825i \(0.0494813\pi\)
−0.987942 + 0.154825i \(0.950519\pi\)
\(200\) 0 0
\(201\) −2.41748e13 −0.366596
\(202\) 0 0
\(203\) − 9.97998e12i − 0.142611i
\(204\) 0 0
\(205\) 3.42477e13 0.461432
\(206\) 0 0
\(207\) 3.86535e12i 0.0491323i
\(208\) 0 0
\(209\) −1.21610e14 −1.45912
\(210\) 0 0
\(211\) − 1.67562e14i − 1.89880i −0.314062 0.949402i \(-0.601690\pi\)
0.314062 0.949402i \(-0.398310\pi\)
\(212\) 0 0
\(213\) 1.24726e12 0.0133561
\(214\) 0 0
\(215\) 6.23135e13i 0.630887i
\(216\) 0 0
\(217\) 8.15344e12 0.0780877
\(218\) 0 0
\(219\) − 1.23912e14i − 1.12318i
\(220\) 0 0
\(221\) −1.34147e14 −1.15141
\(222\) 0 0
\(223\) − 1.42965e14i − 1.16252i −0.813718 0.581261i \(-0.802560\pi\)
0.813718 0.581261i \(-0.197440\pi\)
\(224\) 0 0
\(225\) 2.44306e13 0.188295
\(226\) 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
\(231\) − 2.39007e13i − 0.157303i
\(232\) 0 0
\(233\) 4.89837e13 0.306137 0.153069 0.988216i \(-0.451084\pi\)
0.153069 + 0.988216i \(0.451084\pi\)
\(234\) 0 0
\(235\) 2.71371e13i 0.161122i
\(236\) 0 0
\(237\) 9.23468e13 0.521113
\(238\) 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
\(243\) − 1.32079e13i − 0.0641500i
\(244\) 0 0
\(245\) 1.35840e14 0.628104
\(246\) 0 0
\(247\) 3.11549e14i 1.37197i
\(248\) 0 0
\(249\) 1.64575e13 0.0690505
\(250\) 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
\(255\) 1.02887e14i 0.374213i
\(256\) 0 0
\(257\) −2.93272e14 −1.01782 −0.508912 0.860819i \(-0.669952\pi\)
−0.508912 + 0.860819i \(0.669952\pi\)
\(258\) 0 0
\(259\) − 3.01760e13i − 0.0999683i
\(260\) 0 0
\(261\) −6.87345e13 −0.217436
\(262\) 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
\(267\) − 6.48072e13i − 0.178878i
\(268\) 0 0
\(269\) 5.21692e14 1.37689 0.688447 0.725287i \(-0.258293\pi\)
0.688447 + 0.725287i \(0.258293\pi\)
\(270\) 0 0
\(271\) 3.63124e14i 0.916726i 0.888765 + 0.458363i \(0.151564\pi\)
−0.888765 + 0.458363i \(0.848436\pi\)
\(272\) 0 0
\(273\) −6.12304e13 −0.147908
\(274\) 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
\(279\) − 5.61547e13i − 0.119059i
\(280\) 0 0
\(281\) 5.70665e14 1.15916 0.579580 0.814916i \(-0.303217\pi\)
0.579580 + 0.814916i \(0.303217\pi\)
\(282\) 0 0
\(283\) − 4.19893e14i − 0.817373i −0.912675 0.408687i \(-0.865987\pi\)
0.912675 0.408687i \(-0.134013\pi\)
\(284\) 0 0
\(285\) 2.38949e14 0.445898
\(286\) 0 0
\(287\) − 8.54669e13i − 0.152935i
\(288\) 0 0
\(289\) −2.00978e13 −0.0344954
\(290\) 0 0
\(291\) 5.09643e14i 0.839282i
\(292\) 0 0
\(293\) 3.50267e14 0.553596 0.276798 0.960928i \(-0.410727\pi\)
0.276798 + 0.960928i \(0.410727\pi\)
\(294\) 0 0
\(295\) 6.87018e14i 1.04240i
\(296\) 0 0
\(297\) −1.64609e14 −0.239837
\(298\) 0 0
\(299\) − 1.23415e14i − 0.172719i
\(300\) 0 0
\(301\) 1.55507e14 0.209098
\(302\) 0 0
\(303\) 7.95162e14i 1.02754i
\(304\) 0 0
\(305\) 4.68459e14 0.581932
\(306\) 0 0
\(307\) − 7.25089e14i − 0.866086i −0.901373 0.433043i \(-0.857440\pi\)
0.901373 0.433043i \(-0.142560\pi\)
\(308\) 0 0
\(309\) −3.54806e14 −0.407605
\(310\) 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
\(315\) 4.69618e13i 0.0480708i
\(316\) 0 0
\(317\) −8.04849e14 −0.793157 −0.396578 0.918001i \(-0.629802\pi\)
−0.396578 + 0.918001i \(0.629802\pi\)
\(318\) 0 0
\(319\) 8.56634e14i 0.812926i
\(320\) 0 0
\(321\) −7.54530e14 −0.689678
\(322\) 0 0
\(323\) − 1.30643e15i − 1.15046i
\(324\) 0 0
\(325\) −7.80030e14 −0.661928
\(326\) 0 0
\(327\) − 2.62578e14i − 0.214769i
\(328\) 0 0
\(329\) 6.77220e13 0.0534016
\(330\) 0 0
\(331\) 1.68149e15i 1.27858i 0.768967 + 0.639288i \(0.220771\pi\)
−0.768967 + 0.639288i \(0.779229\pi\)
\(332\) 0 0
\(333\) −2.07829e14 −0.152420
\(334\) 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
\(339\) 2.74773e14i 0.181040i
\(340\) 0 0
\(341\) −6.99852e14 −0.445123
\(342\) 0 0
\(343\) − 6.95009e14i − 0.426801i
\(344\) 0 0
\(345\) −9.46551e13 −0.0561344
\(346\) 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
\(351\) 4.21708e14i 0.225512i
\(352\) 0 0
\(353\) 3.77287e13 0.0194995 0.00974976 0.999952i \(-0.496897\pi\)
0.00974976 + 0.999952i \(0.496897\pi\)
\(354\) 0 0
\(355\) 3.05431e13i 0.0152596i
\(356\) 0 0
\(357\) 2.56759e14 0.124027
\(358\) 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
\(363\) 7.30590e14i 0.319326i
\(364\) 0 0
\(365\) 3.03437e15 1.28325
\(366\) 0 0
\(367\) − 1.63345e15i − 0.668514i −0.942482 0.334257i \(-0.891515\pi\)
0.942482 0.334257i \(-0.108485\pi\)
\(368\) 0 0
\(369\) −5.88631e14 −0.233176
\(370\) 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
\(375\) 1.65734e15i 0.595969i
\(376\) 0 0
\(377\) 2.19458e15 0.764371
\(378\) 0 0
\(379\) 2.47588e15i 0.835401i 0.908585 + 0.417700i \(0.137164\pi\)
−0.908585 + 0.417700i \(0.862836\pi\)
\(380\) 0 0
\(381\) −2.82568e15 −0.923791
\(382\) 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
\(387\) − 1.07101e15i − 0.318807i
\(388\) 0 0
\(389\) −4.47360e15 −1.29110 −0.645549 0.763719i \(-0.723371\pi\)
−0.645549 + 0.763719i \(0.723371\pi\)
\(390\) 0 0
\(391\) 5.17518e14i 0.144832i
\(392\) 0 0
\(393\) 3.24046e15 0.879530
\(394\) 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
\(399\) − 5.96309e14i − 0.147786i
\(400\) 0 0
\(401\) 2.42681e15 0.583673 0.291836 0.956468i \(-0.405734\pi\)
0.291836 + 0.956468i \(0.405734\pi\)
\(402\) 0 0
\(403\) 1.79293e15i 0.418536i
\(404\) 0 0
\(405\) 3.23437e14 0.0732925
\(406\) 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
\(411\) 2.73809e15i 0.568064i
\(412\) 0 0
\(413\) 1.71449e15 0.345489
\(414\) 0 0
\(415\) 4.03012e14i 0.0788914i
\(416\) 0 0
\(417\) −1.32537e15 −0.252069
\(418\) 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
\(423\) − 4.66418e14i − 0.0814203i
\(424\) 0 0
\(425\) 3.27093e15 0.555056
\(426\) 0 0
\(427\) − 1.16906e15i − 0.192873i
\(428\) 0 0
\(429\) 5.25572e15 0.843118
\(430\) 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
\(435\) − 1.68318e15i − 0.248425i
\(436\) 0 0
\(437\) 1.20191e15 0.172577
\(438\) 0 0
\(439\) − 8.34545e15i − 1.16590i −0.812506 0.582952i \(-0.801898\pi\)
0.812506 0.582952i \(-0.198102\pi\)
\(440\) 0 0
\(441\) −2.33475e15 −0.317401
\(442\) 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
\(447\) − 6.46909e15i − 0.810957i
\(448\) 0 0
\(449\) −1.21818e15 −0.148674 −0.0743368 0.997233i \(-0.523684\pi\)
−0.0743368 + 0.997233i \(0.523684\pi\)
\(450\) 0 0
\(451\) 7.33607e15i 0.871774i
\(452\) 0 0
\(453\) 6.85583e15 0.793361
\(454\) 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
\(459\) − 1.76836e15i − 0.189102i
\(460\) 0 0
\(461\) −1.16311e16 −1.21175 −0.605877 0.795558i \(-0.707178\pi\)
−0.605877 + 0.795558i \(0.707178\pi\)
\(462\) 0 0
\(463\) − 1.67988e15i − 0.170527i −0.996358 0.0852636i \(-0.972827\pi\)
0.996358 0.0852636i \(-0.0271732\pi\)
\(464\) 0 0
\(465\) 1.37512e15 0.136027
\(466\) 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
\(471\) − 1.13176e16i − 1.03664i
\(472\) 0 0
\(473\) −1.33479e16 −1.19192
\(474\) 0 0
\(475\) − 7.59654e15i − 0.661385i
\(476\) 0 0
\(477\) −5.07692e15 −0.431013
\(478\) 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
\(483\) 2.36217e14i 0.0186049i
\(484\) 0 0
\(485\) −1.24802e16 −0.958894
\(486\) 0 0
\(487\) − 1.80780e16i − 1.35511i −0.735470 0.677557i \(-0.763039\pi\)
0.735470 0.677557i \(-0.236961\pi\)
\(488\) 0 0
\(489\) −7.39023e15 −0.540511
\(490\) 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
\(495\) − 4.03097e15i − 0.274018i
\(496\) 0 0
\(497\) 7.62219e13 0.00505757
\(498\) 0 0
\(499\) 3.33196e15i 0.215822i 0.994161 + 0.107911i \(0.0344162\pi\)
−0.994161 + 0.107911i \(0.965584\pi\)
\(500\) 0 0
\(501\) 1.46307e16 0.925205
\(502\) 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
\(507\) − 3.65858e15i − 0.215409i
\(508\) 0 0
\(509\) −2.53124e16 −1.45555 −0.727774 0.685817i \(-0.759445\pi\)
−0.727774 + 0.685817i \(0.759445\pi\)
\(510\) 0 0
\(511\) − 7.57244e15i − 0.425315i
\(512\) 0 0
\(513\) −4.10692e15 −0.225327
\(514\) 0 0
\(515\) − 8.68852e15i − 0.465696i
\(516\) 0 0
\(517\) −5.81293e15 −0.304405
\(518\) 0 0
\(519\) 1.21550e16i 0.621945i
\(520\) 0 0
\(521\) −2.07262e15 −0.103632 −0.0518158 0.998657i \(-0.516501\pi\)
−0.0518158 + 0.998657i \(0.516501\pi\)
\(522\) 0 0
\(523\) − 1.96180e16i − 0.958617i −0.877646 0.479309i \(-0.840887\pi\)
0.877646 0.479309i \(-0.159113\pi\)
\(524\) 0 0
\(525\) 1.49299e15 0.0713017
\(526\) 0 0
\(527\) − 7.51836e15i − 0.350961i
\(528\) 0 0
\(529\) 2.14385e16 0.978274
\(530\) 0 0
\(531\) − 1.18081e16i − 0.526760i
\(532\) 0 0
\(533\) 1.87940e16 0.819704
\(534\) 0 0
\(535\) − 1.84770e16i − 0.787969i
\(536\) 0 0
\(537\) 1.51553e16 0.632002
\(538\) 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
\(543\) − 1.45284e15i − 0.0566785i
\(544\) 0 0
\(545\) 6.43004e15 0.245377
\(546\) 0 0
\(547\) 1.90846e16i 0.712458i 0.934399 + 0.356229i \(0.115938\pi\)
−0.934399 + 0.356229i \(0.884062\pi\)
\(548\) 0 0
\(549\) −8.05161e15 −0.294069
\(550\) 0 0
\(551\) 2.13726e16i 0.763743i
\(552\) 0 0
\(553\) 5.64343e15 0.197330
\(554\) 0 0
\(555\) − 5.08934e15i − 0.174142i
\(556\) 0 0
\(557\) −1.48841e16 −0.498415 −0.249208 0.968450i \(-0.580170\pi\)
−0.249208 + 0.968450i \(0.580170\pi\)
\(558\) 0 0
\(559\) 3.41957e16i 1.12073i
\(560\) 0 0
\(561\) −2.20390e16 −0.706992
\(562\) 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
\(567\) − 8.07154e14i − 0.0242917i
\(568\) 0 0
\(569\) 6.41843e16 1.89128 0.945639 0.325219i \(-0.105438\pi\)
0.945639 + 0.325219i \(0.105438\pi\)
\(570\) 0 0
\(571\) 9.82359e15i 0.283435i 0.989907 + 0.141718i \(0.0452625\pi\)
−0.989907 + 0.141718i \(0.954738\pi\)
\(572\) 0 0
\(573\) 3.08937e16 0.872855
\(574\) 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
\(579\) 8.25169e15i 0.219014i
\(580\) 0 0
\(581\) 1.00574e15 0.0261474
\(582\) 0 0
\(583\) 6.32733e16i 1.61142i
\(584\) 0 0
\(585\) −1.03268e16 −0.257651
\(586\) 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
\(591\) 3.50821e16i 0.823305i
\(592\) 0 0
\(593\) 4.15519e16 0.955570 0.477785 0.878477i \(-0.341440\pi\)
0.477785 + 0.878477i \(0.341440\pi\)
\(594\) 0 0
\(595\) 6.28755e15i 0.141703i
\(596\) 0 0
\(597\) 8.09386e15 0.178776
\(598\) 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
\(603\) 1.01749e16i 0.211654i
\(604\) 0 0
\(605\) −1.78908e16 −0.364835
\(606\) 0 0
\(607\) 9.38322e15i 0.187594i 0.995591 + 0.0937971i \(0.0299005\pi\)
−0.995591 + 0.0937971i \(0.970099\pi\)
\(608\) 0 0
\(609\) −4.20046e15 −0.0823366
\(610\) 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
\(615\) − 1.44145e16i − 0.266408i
\(616\) 0 0
\(617\) 8.82180e16 1.59899 0.799496 0.600672i \(-0.205100\pi\)
0.799496 + 0.600672i \(0.205100\pi\)
\(618\) 0 0
\(619\) − 4.76892e16i − 0.847766i −0.905717 0.423883i \(-0.860667\pi\)
0.905717 0.423883i \(-0.139333\pi\)
\(620\) 0 0
\(621\) 1.62688e15 0.0283665
\(622\) 0 0
\(623\) − 3.96046e15i − 0.0677356i
\(624\) 0 0
\(625\) −6.91535e15 −0.116020
\(626\) 0 0
\(627\) 5.11843e16i 0.842426i
\(628\) 0 0
\(629\) −2.78255e16 −0.449303
\(630\) 0 0
\(631\) 6.79965e15i 0.107723i 0.998548 + 0.0538617i \(0.0171530\pi\)
−0.998548 + 0.0538617i \(0.982847\pi\)
\(632\) 0 0
\(633\) −7.05248e16 −1.09628
\(634\) 0 0
\(635\) − 6.91956e16i − 1.05545i
\(636\) 0 0
\(637\) 7.45448e16 1.11579
\(638\) 0 0
\(639\) − 5.24959e14i − 0.00771116i
\(640\) 0 0
\(641\) −3.57968e16 −0.516055 −0.258027 0.966138i \(-0.583072\pi\)
−0.258027 + 0.966138i \(0.583072\pi\)
\(642\) 0 0
\(643\) − 4.41166e16i − 0.624218i −0.950046 0.312109i \(-0.898965\pi\)
0.950046 0.312109i \(-0.101035\pi\)
\(644\) 0 0
\(645\) 2.62270e16 0.364243
\(646\) 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
\(651\) − 3.43169e15i − 0.0450840i
\(652\) 0 0
\(653\) −9.19791e16 −1.18634 −0.593171 0.805076i \(-0.702124\pi\)
−0.593171 + 0.805076i \(0.702124\pi\)
\(654\) 0 0
\(655\) 7.93526e16i 1.00488i
\(656\) 0 0
\(657\) −5.21532e16 −0.648468
\(658\) 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
\(663\) 5.64610e16i 0.664764i
\(664\) 0 0
\(665\) 1.46025e16 0.168848
\(666\) 0 0
\(667\) − 8.46635e15i − 0.0961482i
\(668\) 0 0
\(669\) −6.01723e16 −0.671182
\(670\) 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
\(675\) − 1.02825e16i − 0.108712i
\(676\) 0 0
\(677\) −1.78154e16 −0.185039 −0.0925196 0.995711i \(-0.529492\pi\)
−0.0925196 + 0.995711i \(0.529492\pi\)
\(678\) 0 0
\(679\) 3.11450e16i 0.317811i
\(680\) 0 0
\(681\) 1.36535e16 0.136887
\(682\) 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
\(687\) 8.91191e16i 0.847677i
\(688\) 0 0
\(689\) 1.62098e17 1.51517
\(690\) 0 0
\(691\) − 1.18406e17i − 1.08769i −0.839184 0.543847i \(-0.816967\pi\)
0.839184 0.543847i \(-0.183033\pi\)
\(692\) 0 0
\(693\) −1.00595e16 −0.0908191
\(694\) 0 0
\(695\) − 3.24557e16i − 0.287993i
\(696\) 0 0
\(697\) −7.88097e16 −0.687358
\(698\) 0 0
\(699\) − 2.06167e16i − 0.176748i
\(700\) 0 0
\(701\) −3.50131e16 −0.295068 −0.147534 0.989057i \(-0.547134\pi\)
−0.147534 + 0.989057i \(0.547134\pi\)
\(702\) 0 0
\(703\) 6.46231e16i 0.535373i
\(704\) 0 0
\(705\) 1.14217e16 0.0930241
\(706\) 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
\(711\) − 3.88677e16i − 0.300865i
\(712\) 0 0
\(713\) 6.91683e15 0.0526466
\(714\) 0 0
\(715\) 1.28703e17i 0.963277i
\(716\) 0 0
\(717\) 7.02337e16 0.516929
\(718\) 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
\(723\) − 4.14837e16i − 0.290434i
\(724\) 0 0
\(725\) −5.35108e16 −0.368479
\(726\) 0 0
\(727\) − 7.77948e16i − 0.526919i −0.964670 0.263460i \(-0.915136\pi\)
0.964670 0.263460i \(-0.0848636\pi\)
\(728\) 0 0
\(729\) −5.55906e15 −0.0370370
\(730\) 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
\(735\) − 5.71735e16i − 0.362636i
\(736\) 0 0
\(737\) 1.26809e17 0.791309
\(738\) 0 0
\(739\) 2.10708e17i 1.29365i 0.762640 + 0.646823i \(0.223903\pi\)
−0.762640 + 0.646823i \(0.776097\pi\)
\(740\) 0 0
\(741\) 1.31128e17 0.792108
\(742\) 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
\(747\) − 6.92676e15i − 0.0398663i
\(748\) 0 0
\(749\) −4.61103e16 −0.261160
\(750\) 0 0
\(751\) − 8.57383e16i − 0.477898i −0.971032 0.238949i \(-0.923197\pi\)
0.971032 0.238949i \(-0.0768029\pi\)
\(752\) 0 0
\(753\) −1.57084e17 −0.861712
\(754\) 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
\(759\) − 2.02757e16i − 0.106054i
\(760\) 0 0
\(761\) 2.12682e17 1.09502 0.547511 0.836798i \(-0.315575\pi\)
0.547511 + 0.836798i \(0.315575\pi\)
\(762\) 0 0
\(763\) − 1.60465e16i − 0.0813267i
\(764\) 0 0
\(765\) 4.33039e16 0.216052
\(766\) 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
\(771\) 1.23435e17i 0.587641i
\(772\) 0 0
\(773\) −1.73938e17 −0.815299 −0.407649 0.913139i \(-0.633651\pi\)
−0.407649 + 0.913139i \(0.633651\pi\)
\(774\) 0 0
\(775\) − 4.37172e16i − 0.201763i
\(776\) 0 0
\(777\) −1.27007e16 −0.0577167
\(778\) 0 0
\(779\) 1.83031e17i 0.819031i
\(780\) 0 0
\(781\) −6.54253e15 −0.0288296
\(782\) 0 0
\(783\) 2.89296e16i 0.125537i
\(784\) 0 0
\(785\) 2.77146e17 1.18438
\(786\) 0 0
\(787\) − 1.47492e16i − 0.0620754i −0.999518 0.0310377i \(-0.990119\pi\)
0.999518 0.0310377i \(-0.00988119\pi\)
\(788\) 0 0
\(789\) 2.52831e17 1.04802
\(790\) 0 0
\(791\) 1.67917e16i 0.0685546i
\(792\) 0 0
\(793\) 2.57075e17 1.03376
\(794\) 0 0
\(795\) − 1.24324e17i − 0.492439i
\(796\) 0 0
\(797\) −3.41145e17 −1.33103 −0.665517 0.746382i \(-0.731789\pi\)
−0.665517 + 0.746382i \(0.731789\pi\)
\(798\) 0 0
\(799\) − 6.24470e16i − 0.240011i
\(800\) 0 0
\(801\) −2.72766e16 −0.103275
\(802\) 0 0
\(803\) 6.49982e17i 2.42442i
\(804\) 0 0
\(805\) −5.78450e15 −0.0212564
\(806\) 0 0
\(807\) − 2.19574e17i − 0.794950i
\(808\) 0 0
\(809\) −1.84056e16 −0.0656537 −0.0328269 0.999461i \(-0.510451\pi\)
−0.0328269 + 0.999461i \(0.510451\pi\)
\(810\) 0 0
\(811\) − 3.79008e17i − 1.33206i −0.745926 0.666029i \(-0.767993\pi\)
0.745926 0.666029i \(-0.232007\pi\)
\(812\) 0 0
\(813\) 1.52835e17 0.529272
\(814\) 0 0
\(815\) − 1.80973e17i − 0.617544i
\(816\) 0 0
\(817\) −3.33024e17 −1.11981
\(818\) 0 0
\(819\) 2.57711e16i 0.0853946i
\(820\) 0 0
\(821\) −4.57140e17 −1.49276 −0.746381 0.665519i \(-0.768210\pi\)
−0.746381 + 0.665519i \(0.768210\pi\)
\(822\) 0 0
\(823\) − 3.70495e17i − 1.19229i −0.802876 0.596147i \(-0.796698\pi\)
0.802876 0.596147i \(-0.203302\pi\)
\(824\) 0 0
\(825\) −1.28151e17 −0.406441
\(826\) 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
\(831\) − 1.23817e17i − 0.375989i
\(832\) 0 0
\(833\) −3.12591e17 −0.935636
\(834\) 0 0
\(835\) 3.58277e17i 1.05706i
\(836\) 0 0
\(837\) −2.36349e16 −0.0687385
\(838\) 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
\(843\) − 2.40186e17i − 0.669241i
\(844\) 0 0
\(845\) 8.95917e16 0.246109
\(846\) 0 0
\(847\) 4.46473e16i 0.120919i
\(848\) 0 0
\(849\) −1.76728e17 −0.471911
\(850\) 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
\(855\) − 1.00571e17i − 0.257439i
\(856\) 0 0
\(857\) −1.57808e16 −0.0398330 −0.0199165 0.999802i \(-0.506340\pi\)
−0.0199165 + 0.999802i \(0.506340\pi\)
\(858\) 0 0
\(859\) − 6.52158e17i − 1.62328i −0.584157 0.811641i \(-0.698575\pi\)
0.584157 0.811641i \(-0.301425\pi\)
\(860\) 0 0
\(861\) −3.59720e16 −0.0882970
\(862\) 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
\(867\) 8.45892e15i 0.0199159i
\(868\) 0 0
\(869\) −4.84405e17 −1.12484
\(870\) 0 0
\(871\) − 3.24869e17i − 0.744045i
\(872\) 0 0
\(873\) 2.14503e17 0.484560
\(874\) 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
\(879\) − 1.47423e17i − 0.319619i
\(880\) 0 0
\(881\) −1.87143e17 −0.400238 −0.200119 0.979772i \(-0.564133\pi\)
−0.200119 + 0.979772i \(0.564133\pi\)
\(882\) 0 0
\(883\) 5.84655e17i 1.23349i 0.787163 + 0.616745i \(0.211549\pi\)
−0.787163 + 0.616745i \(0.788451\pi\)
\(884\) 0 0
\(885\) 2.89158e17 0.601832
\(886\) 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
\(891\) 6.92822e16i 0.138470i
\(892\) 0 0
\(893\) −1.45030e17 −0.285988
\(894\) 0 0
\(895\) 3.71124e17i 0.722074i
\(896\) 0 0
\(897\) −5.19437e16 −0.0997191
\(898\) 0 0
\(899\) 1.22997e17i 0.232989i
\(900\) 0 0
\(901\) −6.79731e17 −1.27054
\(902\) 0 0
\(903\) − 6.54509e16i − 0.120723i
\(904\) 0 0
\(905\) 3.55772e16 0.0647561
\(906\) 0 0
\(907\) 8.53775e14i 0.00153356i 1.00000 0.000766778i \(0.000244073\pi\)
−1.00000 0.000766778i \(0.999756\pi\)
\(908\) 0 0
\(909\) 3.34674e17 0.593252
\(910\) 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
\(915\) − 1.97169e17i − 0.335979i
\(916\) 0 0
\(917\) 1.98029e17 0.333052
\(918\) 0 0
\(919\) − 1.05696e18i − 1.75454i −0.479996 0.877271i \(-0.659362\pi\)
0.479996 0.877271i \(-0.340638\pi\)
\(920\) 0 0
\(921\) −3.05182e17 −0.500035
\(922\) 0 0
\(923\) 1.67611e16i 0.0271077i
\(924\) 0 0
\(925\) −1.61798e17 −0.258299
\(926\) 0 0
\(927\) 1.49334e17i 0.235331i
\(928\) 0 0
\(929\) 2.48690e17 0.386869 0.193435 0.981113i \(-0.438037\pi\)
0.193435 + 0.981113i \(0.438037\pi\)
\(930\) 0 0
\(931\) 7.25975e17i 1.11487i
\(932\) 0 0
\(933\) −5.49871e17 −0.833625
\(934\) 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
\(939\) 4.00607e17i 0.584419i
\(940\) 0 0
\(941\) 2.19998e17 0.316869 0.158435 0.987369i \(-0.449355\pi\)
0.158435 + 0.987369i \(0.449355\pi\)
\(942\) 0 0
\(943\) − 7.25043e16i − 0.103108i
\(944\) 0 0
\(945\) 1.97657e16 0.0277537
\(946\) 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
\(951\) 3.38752e17i 0.457929i
\(952\) 0 0
\(953\) 9.59450e17 1.28075 0.640376 0.768062i \(-0.278779\pi\)
0.640376 + 0.768062i \(0.278779\pi\)
\(954\) 0 0
\(955\) 7.56528e17i 0.997251i
\(956\) 0 0
\(957\) 3.60547e17 0.469343
\(958\) 0 0
\(959\) 1.67329e17i 0.215109i
\(960\) 0 0
\(961\) 6.87177e17 0.872425
\(962\) 0 0
\(963\) 3.17573e17i 0.398186i
\(964\) 0 0
\(965\) −2.02068e17 −0.250227
\(966\) 0 0
\(967\) − 1.36871e17i − 0.167399i −0.996491 0.0836993i \(-0.973326\pi\)
0.996491 0.0836993i \(-0.0266735\pi\)
\(968\) 0 0
\(969\) −5.49862e17 −0.664218
\(970\) 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
\(975\) 3.28305e17i 0.382165i
\(976\) 0 0
\(977\) 1.57226e18 1.80783 0.903915 0.427713i \(-0.140681\pi\)
0.903915 + 0.427713i \(0.140681\pi\)
\(978\) 0 0
\(979\) 3.39947e17i 0.386113i
\(980\) 0 0
\(981\) −1.10516e17 −0.123997
\(982\) 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
\(987\) − 2.85034e16i − 0.0308314i
\(988\) 0 0
\(989\) 1.31921e17 0.140973
\(990\) 0 0
\(991\) 1.72529e18i 1.82146i 0.413001 + 0.910731i \(0.364481\pi\)
−0.413001 + 0.910731i \(0.635519\pi\)
\(992\) 0 0
\(993\) 7.07720e17 0.738187
\(994\) 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
\(999\) 8.74728e16i 0.0879995i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 48.13.g.a.31.2 4
3.2 odd 2 144.13.g.h.127.1 4
4.3 odd 2 inner 48.13.g.a.31.4 yes 4
8.3 odd 2 192.13.g.c.127.1 4
8.5 even 2 192.13.g.c.127.3 4
12.11 even 2 144.13.g.h.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.13.g.a.31.2 4 1.1 even 1 trivial
48.13.g.a.31.4 yes 4 4.3 odd 2 inner
144.13.g.h.127.1 4 3.2 odd 2
144.13.g.h.127.2 4 12.11 even 2
192.13.g.c.127.1 4 8.3 odd 2
192.13.g.c.127.3 4 8.5 even 2