Properties

Label 5292.2.i.h.2125.6
Level $5292$
Weight $2$
Character 5292.2125
Analytic conductor $42.257$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5292,2,Mod(1549,5292)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5292, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5292.i (of order \(3\), degree \(2\), not 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: \(42.2568327497\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + 16x^{8} - 39x^{6} + 144x^{4} - 162x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 1764)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2125.6
Root \(-0.965975 - 1.43767i\) of defining polynomial
Character \(\chi\) \(=\) 5292.2125
Dual form 5292.2.i.h.1549.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.95014 - 3.37774i) q^{5} +O(q^{10})\) \(q+(1.95014 - 3.37774i) q^{5} +(-2.13378 - 3.69582i) q^{11} +(2.71221 + 4.69768i) q^{13} +(2.49012 - 4.31301i) q^{17} +(0.222091 + 0.384673i) q^{19} +(4.10607 - 7.11191i) q^{23} +(-5.10607 - 8.84396i) q^{25} +(-1.33850 + 2.31835i) q^{29} +0.851993 q^{31} +(4.90135 + 8.48939i) q^{37} +(-2.69402 - 4.66618i) q^{41} +(4.31078 - 7.46649i) q^{43} -3.49246 q^{47} +(-2.83850 + 4.91642i) q^{53} -16.6447 q^{55} +1.89558 q^{59} -11.2566 q^{61} +21.1567 q^{65} +2.59057 q^{67} -0.141862 q^{71} +(5.16595 - 8.94769i) q^{73} -0.409429 q^{79} +(-2.08231 + 3.60666i) q^{83} +(-9.71213 - 16.8219i) q^{85} +(1.20625 + 2.08929i) q^{89} +1.73243 q^{95} +(3.80448 - 6.58955i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{11} - 12 q^{25} - 2 q^{29} + 6 q^{37} + 6 q^{43} - 20 q^{53} + 92 q^{65} + 24 q^{67} - 44 q^{71} - 12 q^{79} - 18 q^{85} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) 1.95014 3.37774i 0.872128 1.51057i 0.0123362 0.999924i \(-0.496073\pi\)
0.859791 0.510645i \(-0.170593\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.13378 3.69582i −0.643360 1.11433i −0.984678 0.174384i \(-0.944207\pi\)
0.341318 0.939948i \(-0.389127\pi\)
\(12\) 0 0
\(13\) 2.71221 + 4.69768i 0.752231 + 1.30290i 0.946739 + 0.322001i \(0.104355\pi\)
−0.194508 + 0.980901i \(0.562311\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.49012 4.31301i 0.603942 1.04606i −0.388276 0.921543i \(-0.626929\pi\)
0.992218 0.124515i \(-0.0397374\pi\)
\(18\) 0 0
\(19\) 0.222091 + 0.384673i 0.0509512 + 0.0882501i 0.890376 0.455226i \(-0.150441\pi\)
−0.839425 + 0.543476i \(0.817108\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.10607 7.11191i 0.856174 1.48294i −0.0193779 0.999812i \(-0.506169\pi\)
0.875552 0.483124i \(-0.160498\pi\)
\(24\) 0 0
\(25\) −5.10607 8.84396i −1.02121 1.76879i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.33850 + 2.31835i −0.248553 + 0.430506i −0.963125 0.269056i \(-0.913288\pi\)
0.714572 + 0.699562i \(0.246622\pi\)
\(30\) 0 0
\(31\) 0.851993 0.153022 0.0765112 0.997069i \(-0.475622\pi\)
0.0765112 + 0.997069i \(0.475622\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.90135 + 8.48939i 0.805777 + 1.39565i 0.915765 + 0.401714i \(0.131585\pi\)
−0.109988 + 0.993933i \(0.535081\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.69402 4.66618i −0.420735 0.728735i 0.575276 0.817959i \(-0.304895\pi\)
−0.996012 + 0.0892242i \(0.971561\pi\)
\(42\) 0 0
\(43\) 4.31078 7.46649i 0.657388 1.13863i −0.323902 0.946091i \(-0.604995\pi\)
0.981289 0.192538i \(-0.0616720\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.49246 −0.509428 −0.254714 0.967016i \(-0.581981\pi\)
−0.254714 + 0.967016i \(0.581981\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.83850 + 4.91642i −0.389898 + 0.675323i −0.992435 0.122768i \(-0.960823\pi\)
0.602538 + 0.798090i \(0.294156\pi\)
\(54\) 0 0
\(55\) −16.6447 −2.24437
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.89558 0.246783 0.123392 0.992358i \(-0.460623\pi\)
0.123392 + 0.992358i \(0.460623\pi\)
\(60\) 0 0
\(61\) −11.2566 −1.44126 −0.720632 0.693317i \(-0.756148\pi\)
−0.720632 + 0.693317i \(0.756148\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.1567 2.62416
\(66\) 0 0
\(67\) 2.59057 0.316489 0.158244 0.987400i \(-0.449417\pi\)
0.158244 + 0.987400i \(0.449417\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.141862 −0.0168359 −0.00841794 0.999965i \(-0.502680\pi\)
−0.00841794 + 0.999965i \(0.502680\pi\)
\(72\) 0 0
\(73\) 5.16595 8.94769i 0.604629 1.04725i −0.387481 0.921878i \(-0.626655\pi\)
0.992110 0.125370i \(-0.0400118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.409429 −0.0460644 −0.0230322 0.999735i \(-0.507332\pi\)
−0.0230322 + 0.999735i \(0.507332\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.08231 + 3.60666i −0.228563 + 0.395882i −0.957382 0.288823i \(-0.906736\pi\)
0.728820 + 0.684706i \(0.240069\pi\)
\(84\) 0 0
\(85\) −9.71213 16.8219i −1.05343 1.82459i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.20625 + 2.08929i 0.127863 + 0.221464i 0.922848 0.385164i \(-0.125855\pi\)
−0.794986 + 0.606628i \(0.792522\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.73243 0.177744
\(96\) 0 0
\(97\) 3.80448 6.58955i 0.386286 0.669067i −0.605661 0.795723i \(-0.707091\pi\)
0.991947 + 0.126656i \(0.0404244\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.56185 + 4.43726i 0.254914 + 0.441524i 0.964872 0.262720i \(-0.0846196\pi\)
−0.709958 + 0.704244i \(0.751286\pi\)
\(102\) 0 0
\(103\) 6.48031 11.2242i 0.638524 1.10596i −0.347233 0.937779i \(-0.612879\pi\)
0.985757 0.168177i \(-0.0537880\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.76757 9.98972i −0.557572 0.965743i −0.997698 0.0678070i \(-0.978400\pi\)
0.440127 0.897936i \(-0.354934\pi\)
\(108\) 0 0
\(109\) 0.204714 0.354576i 0.0196081 0.0339622i −0.856055 0.516885i \(-0.827091\pi\)
0.875663 + 0.482923i \(0.160425\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.94456 + 12.0283i 0.653290 + 1.13153i 0.982320 + 0.187211i \(0.0599449\pi\)
−0.329030 + 0.944319i \(0.606722\pi\)
\(114\) 0 0
\(115\) −16.0148 27.7384i −1.49339 2.58662i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.60607 + 6.24589i −0.327824 + 0.567808i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −20.3287 −1.81826
\(126\) 0 0
\(127\) 11.6216 1.03125 0.515623 0.856815i \(-0.327560\pi\)
0.515623 + 0.856815i \(0.327560\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.02775 3.51216i 0.177165 0.306859i −0.763743 0.645520i \(-0.776641\pi\)
0.940908 + 0.338661i \(0.109974\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.57835 + 7.92993i 0.391155 + 0.677500i 0.992602 0.121413i \(-0.0387424\pi\)
−0.601448 + 0.798912i \(0.705409\pi\)
\(138\) 0 0
\(139\) 5.23869 + 9.07369i 0.444340 + 0.769620i 0.998006 0.0631191i \(-0.0201048\pi\)
−0.553666 + 0.832739i \(0.686771\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.5745 20.0477i 0.967910 1.67647i
\(144\) 0 0
\(145\) 5.22051 + 9.04219i 0.433540 + 0.750913i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.13378 + 14.0881i −0.666346 + 1.15414i 0.312573 + 0.949894i \(0.398809\pi\)
−0.978919 + 0.204251i \(0.934524\pi\)
\(150\) 0 0
\(151\) 3.60607 + 6.24589i 0.293457 + 0.508283i 0.974625 0.223844i \(-0.0718608\pi\)
−0.681167 + 0.732128i \(0.738527\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.66150 2.87781i 0.133455 0.231151i
\(156\) 0 0
\(157\) 1.18706 0.0947376 0.0473688 0.998877i \(-0.484916\pi\)
0.0473688 + 0.998877i \(0.484916\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.5074 18.1994i −0.823004 1.42549i −0.903435 0.428725i \(-0.858963\pi\)
0.0804306 0.996760i \(-0.474370\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.0322 17.3763i −0.776315 1.34462i −0.934053 0.357136i \(-0.883753\pi\)
0.157738 0.987481i \(-0.449580\pi\)
\(168\) 0 0
\(169\) −8.21213 + 14.2238i −0.631702 + 1.09414i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.06019 −0.460748 −0.230374 0.973102i \(-0.573995\pi\)
−0.230374 + 0.973102i \(0.573995\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.41685 7.65020i 0.330131 0.571803i −0.652407 0.757869i \(-0.726241\pi\)
0.982537 + 0.186066i \(0.0595739\pi\)
\(180\) 0 0
\(181\) −24.1809 −1.79735 −0.898676 0.438614i \(-0.855470\pi\)
−0.898676 + 0.438614i \(0.855470\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 38.2332 2.81096
\(186\) 0 0
\(187\) −21.2535 −1.55421
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.2270 −1.46357 −0.731786 0.681535i \(-0.761313\pi\)
−0.731786 + 0.681535i \(0.761313\pi\)
\(192\) 0 0
\(193\) −12.3933 −0.892087 −0.446044 0.895011i \(-0.647167\pi\)
−0.446044 + 0.895011i \(0.647167\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.3933 −1.02548 −0.512739 0.858544i \(-0.671369\pi\)
−0.512739 + 0.858544i \(0.671369\pi\)
\(198\) 0 0
\(199\) −5.25688 + 9.10518i −0.372650 + 0.645449i −0.989972 0.141261i \(-0.954884\pi\)
0.617322 + 0.786711i \(0.288218\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −21.0148 −1.46774
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.947789 1.64162i 0.0655599 0.113553i
\(210\) 0 0
\(211\) 11.7121 + 20.2860i 0.806296 + 1.39655i 0.915412 + 0.402517i \(0.131865\pi\)
−0.109116 + 0.994029i \(0.534802\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −16.8132 29.1214i −1.14665 1.98606i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 27.0148 1.81721
\(222\) 0 0
\(223\) −12.6727 + 21.9497i −0.848625 + 1.46986i 0.0338111 + 0.999428i \(0.489236\pi\)
−0.882436 + 0.470433i \(0.844098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.01267 15.6104i −0.598192 1.03610i −0.993088 0.117373i \(-0.962553\pi\)
0.394896 0.918726i \(-0.370781\pi\)
\(228\) 0 0
\(229\) 2.71221 4.69768i 0.179228 0.310431i −0.762389 0.647120i \(-0.775973\pi\)
0.941616 + 0.336688i \(0.109307\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.42165 2.46237i −0.0931356 0.161316i 0.815693 0.578484i \(-0.196356\pi\)
−0.908829 + 0.417169i \(0.863022\pi\)
\(234\) 0 0
\(235\) −6.81078 + 11.7966i −0.444286 + 0.769526i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.09057 + 10.5492i 0.393966 + 0.682370i 0.992969 0.118378i \(-0.0377693\pi\)
−0.599002 + 0.800747i \(0.704436\pi\)
\(240\) 0 0
\(241\) −9.31237 16.1295i −0.599863 1.03899i −0.992841 0.119445i \(-0.961888\pi\)
0.392978 0.919548i \(-0.371445\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.20471 + 2.08663i −0.0766541 + 0.132769i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.14015 −0.450682 −0.225341 0.974280i \(-0.572350\pi\)
−0.225341 + 0.974280i \(0.572350\pi\)
\(252\) 0 0
\(253\) −35.0458 −2.20331
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.672148 + 1.16419i −0.0419274 + 0.0726204i −0.886228 0.463250i \(-0.846683\pi\)
0.844300 + 0.535871i \(0.180016\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.1493 21.0432i −0.749157 1.29758i −0.948227 0.317592i \(-0.897126\pi\)
0.199071 0.979985i \(-0.436208\pi\)
\(264\) 0 0
\(265\) 11.0709 + 19.1754i 0.680081 + 1.17794i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.69989 4.67636i 0.164615 0.285122i −0.771903 0.635740i \(-0.780695\pi\)
0.936519 + 0.350618i \(0.114028\pi\)
\(270\) 0 0
\(271\) −6.78589 11.7535i −0.412213 0.713974i 0.582918 0.812531i \(-0.301911\pi\)
−0.995131 + 0.0985565i \(0.968577\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −21.7905 + 37.7422i −1.31402 + 2.27594i
\(276\) 0 0
\(277\) 14.5074 + 25.1276i 0.871666 + 1.50977i 0.860272 + 0.509835i \(0.170294\pi\)
0.0113940 + 0.999935i \(0.496373\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.885857 + 1.53435i −0.0528458 + 0.0915316i −0.891238 0.453536i \(-0.850163\pi\)
0.838392 + 0.545067i \(0.183496\pi\)
\(282\) 0 0
\(283\) 24.2536 1.44173 0.720864 0.693076i \(-0.243745\pi\)
0.720864 + 0.693076i \(0.243745\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.90135 6.75734i −0.229491 0.397490i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.52491 + 13.0335i 0.439610 + 0.761426i 0.997659 0.0683813i \(-0.0217835\pi\)
−0.558050 + 0.829808i \(0.688450\pi\)
\(294\) 0 0
\(295\) 3.69664 6.40276i 0.215226 0.372783i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 44.5460 2.57616
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −21.9520 + 38.0219i −1.25697 + 2.17713i
\(306\) 0 0
\(307\) −4.01912 −0.229383 −0.114692 0.993401i \(-0.536588\pi\)
−0.114692 + 0.993401i \(0.536588\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.59688 −0.0905510 −0.0452755 0.998975i \(-0.514417\pi\)
−0.0452755 + 0.998975i \(0.514417\pi\)
\(312\) 0 0
\(313\) −4.87111 −0.275332 −0.137666 0.990479i \(-0.543960\pi\)
−0.137666 + 0.990479i \(0.543960\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.37844 0.470580 0.235290 0.971925i \(-0.424396\pi\)
0.235290 + 0.971925i \(0.424396\pi\)
\(318\) 0 0
\(319\) 11.4243 0.639636
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.21213 0.123086
\(324\) 0 0
\(325\) 27.6974 47.9733i 1.53638 2.66108i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.8337 1.41995 0.709974 0.704228i \(-0.248707\pi\)
0.709974 + 0.704228i \(0.248707\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.05197 8.75026i 0.276018 0.478078i
\(336\) 0 0
\(337\) 9.81820 + 17.0056i 0.534831 + 0.926355i 0.999171 + 0.0406980i \(0.0129582\pi\)
−0.464340 + 0.885657i \(0.653709\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.81797 3.14881i −0.0984485 0.170518i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.4094 1.20300 0.601501 0.798872i \(-0.294570\pi\)
0.601501 + 0.798872i \(0.294570\pi\)
\(348\) 0 0
\(349\) −5.38804 + 9.33236i −0.288415 + 0.499550i −0.973432 0.228978i \(-0.926462\pi\)
0.685016 + 0.728528i \(0.259795\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.70106 9.87453i −0.303437 0.525568i 0.673475 0.739210i \(-0.264801\pi\)
−0.976912 + 0.213642i \(0.931468\pi\)
\(354\) 0 0
\(355\) −0.276650 + 0.479171i −0.0146830 + 0.0254318i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.38171 16.2496i −0.495148 0.857621i 0.504837 0.863215i \(-0.331553\pi\)
−0.999984 + 0.00559386i \(0.998219\pi\)
\(360\) 0 0
\(361\) 9.40135 16.2836i 0.494808 0.857033i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −20.1486 34.8984i −1.05463 1.82667i
\(366\) 0 0
\(367\) −0.833806 1.44420i −0.0435243 0.0753864i 0.843443 0.537219i \(-0.180525\pi\)
−0.886967 + 0.461833i \(0.847192\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.5229 21.6903i 0.648412 1.12308i −0.335090 0.942186i \(-0.608767\pi\)
0.983502 0.180896i \(-0.0578998\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.5211 −0.747876
\(378\) 0 0
\(379\) 5.21213 0.267729 0.133865 0.991000i \(-0.457261\pi\)
0.133865 + 0.991000i \(0.457261\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.2385 19.4656i 0.574258 0.994644i −0.421864 0.906659i \(-0.638624\pi\)
0.996122 0.0879849i \(-0.0280427\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.20471 9.01483i −0.263889 0.457070i 0.703383 0.710811i \(-0.251672\pi\)
−0.967272 + 0.253741i \(0.918339\pi\)
\(390\) 0 0
\(391\) −20.4492 35.4190i −1.03416 1.79121i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.798442 + 1.38294i −0.0401740 + 0.0695834i
\(396\) 0 0
\(397\) −1.31436 2.27654i −0.0659659 0.114256i 0.831156 0.556039i \(-0.187680\pi\)
−0.897122 + 0.441783i \(0.854346\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.59865 + 9.69714i −0.279583 + 0.484252i −0.971281 0.237935i \(-0.923530\pi\)
0.691698 + 0.722187i \(0.256863\pi\)
\(402\) 0 0
\(403\) 2.31078 + 4.00239i 0.115108 + 0.199373i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.9168 36.2290i 1.03681 1.79581i
\(408\) 0 0
\(409\) −31.7309 −1.56899 −0.784495 0.620135i \(-0.787078\pi\)
−0.784495 + 0.620135i \(0.787078\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.12156 + 14.0670i 0.398672 + 0.690520i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −19.0449 32.9867i −0.930403 1.61151i −0.782633 0.622483i \(-0.786124\pi\)
−0.147770 0.989022i \(-0.547209\pi\)
\(420\) 0 0
\(421\) −0.614143 + 1.06373i −0.0299315 + 0.0518429i −0.880603 0.473855i \(-0.842862\pi\)
0.850672 + 0.525697i \(0.176196\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −50.8588 −2.46701
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.66150 + 8.07396i −0.224537 + 0.388909i −0.956180 0.292778i \(-0.905420\pi\)
0.731644 + 0.681687i \(0.238754\pi\)
\(432\) 0 0
\(433\) −15.3849 −0.739350 −0.369675 0.929161i \(-0.620531\pi\)
−0.369675 + 0.929161i \(0.620531\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.64768 0.174492
\(438\) 0 0
\(439\) −5.08935 −0.242901 −0.121451 0.992597i \(-0.538755\pi\)
−0.121451 + 0.992597i \(0.538755\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 34.7865 1.65276 0.826379 0.563114i \(-0.190397\pi\)
0.826379 + 0.563114i \(0.190397\pi\)
\(444\) 0 0
\(445\) 9.40943 0.446050
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 23.8337 1.12478 0.562391 0.826872i \(-0.309882\pi\)
0.562391 + 0.826872i \(0.309882\pi\)
\(450\) 0 0
\(451\) −11.4969 + 19.9132i −0.541369 + 0.937678i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.8337 0.787447 0.393723 0.919229i \(-0.371187\pi\)
0.393723 + 0.919229i \(0.371187\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.13780 + 10.6310i −0.285866 + 0.495134i −0.972819 0.231568i \(-0.925614\pi\)
0.686953 + 0.726702i \(0.258948\pi\)
\(462\) 0 0
\(463\) 6.10607 + 10.5760i 0.283773 + 0.491509i 0.972311 0.233691i \(-0.0750805\pi\)
−0.688538 + 0.725200i \(0.741747\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.8791 + 29.2354i 0.781071 + 1.35285i 0.931318 + 0.364206i \(0.118660\pi\)
−0.150248 + 0.988648i \(0.548007\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −36.7931 −1.69175
\(474\) 0 0
\(475\) 2.26802 3.92833i 0.104064 0.180244i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.9593 + 20.7141i 0.546434 + 0.946451i 0.998515 + 0.0544741i \(0.0173482\pi\)
−0.452082 + 0.891977i \(0.649318\pi\)
\(480\) 0 0
\(481\) −26.5870 + 46.0500i −1.21226 + 2.09970i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.8385 25.7010i −0.673781 1.16702i
\(486\) 0 0
\(487\) 5.19664 9.00084i 0.235482 0.407867i −0.723931 0.689873i \(-0.757666\pi\)
0.959413 + 0.282006i \(0.0909998\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.68922 + 4.65787i 0.121363 + 0.210207i 0.920305 0.391201i \(-0.127940\pi\)
−0.798943 + 0.601407i \(0.794607\pi\)
\(492\) 0 0
\(493\) 6.66603 + 11.5459i 0.300223 + 0.520001i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 9.62156 16.6650i 0.430720 0.746029i −0.566215 0.824257i \(-0.691593\pi\)
0.996935 + 0.0782282i \(0.0249263\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −22.3334 −0.995798 −0.497899 0.867235i \(-0.665895\pi\)
−0.497899 + 0.867235i \(0.665895\pi\)
\(504\) 0 0
\(505\) 19.9838 0.889269
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.1131 + 34.8369i −0.891497 + 1.54412i −0.0534152 + 0.998572i \(0.517011\pi\)
−0.838081 + 0.545545i \(0.816323\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.2750 43.7776i −1.11375 1.92907i
\(516\) 0 0
\(517\) 7.45216 + 12.9075i 0.327746 + 0.567672i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.3497 31.7826i 0.803914 1.39242i −0.113108 0.993583i \(-0.536081\pi\)
0.917022 0.398837i \(-0.130586\pi\)
\(522\) 0 0
\(523\) −14.1727 24.5479i −0.619731 1.07341i −0.989535 0.144296i \(-0.953908\pi\)
0.369804 0.929110i \(-0.379425\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.12156 3.67465i 0.0924166 0.160070i
\(528\) 0 0
\(529\) −22.2195 38.4854i −0.966067 1.67328i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.6135 25.3113i 0.632980 1.09635i
\(534\) 0 0
\(535\) −44.9902 −1.94509
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.3027 + 21.3089i 0.528934 + 0.916141i 0.999431 + 0.0337394i \(0.0107416\pi\)
−0.470496 + 0.882402i \(0.655925\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.798442 1.38294i −0.0342015 0.0592388i
\(546\) 0 0
\(547\) −0.106065 + 0.183711i −0.00453503 + 0.00785491i −0.868284 0.496067i \(-0.834777\pi\)
0.863749 + 0.503922i \(0.168110\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.18907 −0.0506563
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.40877 16.2965i 0.398662 0.690503i −0.594899 0.803801i \(-0.702808\pi\)
0.993561 + 0.113297i \(0.0361412\pi\)
\(558\) 0 0
\(559\) 46.7669 1.97803
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.8781 0.542749 0.271375 0.962474i \(-0.412522\pi\)
0.271375 + 0.962474i \(0.412522\pi\)
\(564\) 0 0
\(565\) 54.1714 2.27901
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0000 1.13190 0.565949 0.824440i \(-0.308510\pi\)
0.565949 + 0.824440i \(0.308510\pi\)
\(570\) 0 0
\(571\) −10.8498 −0.454052 −0.227026 0.973889i \(-0.572900\pi\)
−0.227026 + 0.973889i \(0.572900\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −83.8634 −3.49734
\(576\) 0 0
\(577\) 10.6085 18.3745i 0.441640 0.764942i −0.556172 0.831067i \(-0.687730\pi\)
0.997811 + 0.0661250i \(0.0210636\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.2270 1.00338
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.92684 + 13.7297i −0.327176 + 0.566685i −0.981950 0.189139i \(-0.939430\pi\)
0.654775 + 0.755824i \(0.272764\pi\)
\(588\) 0 0
\(589\) 0.189220 + 0.327739i 0.00779668 + 0.0135042i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.2580 21.2314i −0.503375 0.871871i −0.999992 0.00390123i \(-0.998758\pi\)
0.496618 0.867969i \(-0.334575\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.42426 −0.180770 −0.0903852 0.995907i \(-0.528810\pi\)
−0.0903852 + 0.995907i \(0.528810\pi\)
\(600\) 0 0
\(601\) −4.47075 + 7.74357i −0.182366 + 0.315867i −0.942686 0.333682i \(-0.891709\pi\)
0.760320 + 0.649549i \(0.225042\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0646 + 24.3607i 0.571809 + 0.990402i
\(606\) 0 0
\(607\) 18.9127 32.7578i 0.767643 1.32960i −0.171195 0.985237i \(-0.554763\pi\)
0.938838 0.344359i \(-0.111904\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.47228 16.4065i −0.383208 0.663735i
\(612\) 0 0
\(613\) 9.33369 16.1664i 0.376984 0.652956i −0.613638 0.789588i \(-0.710294\pi\)
0.990622 + 0.136632i \(0.0436278\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.9365 + 25.8708i 0.601320 + 1.04152i 0.992621 + 0.121255i \(0.0386917\pi\)
−0.391301 + 0.920263i \(0.627975\pi\)
\(618\) 0 0
\(619\) −1.90789 3.30456i −0.0766846 0.132822i 0.825133 0.564938i \(-0.191100\pi\)
−0.901818 + 0.432117i \(0.857767\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14.1135 + 24.4453i −0.564539 + 0.977811i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.8197 1.94657
\(630\) 0 0
\(631\) 20.0458 0.798012 0.399006 0.916948i \(-0.369355\pi\)
0.399006 + 0.916948i \(0.369355\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.6636 39.2546i 0.899379 1.55777i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.75207 8.23083i −0.187696 0.325098i 0.756786 0.653663i \(-0.226769\pi\)
−0.944482 + 0.328564i \(0.893435\pi\)
\(642\) 0 0
\(643\) −15.0611 26.0866i −0.593952 1.02876i −0.993694 0.112128i \(-0.964233\pi\)
0.399741 0.916628i \(-0.369100\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −10.6267 + 18.4060i −0.417780 + 0.723616i −0.995716 0.0924659i \(-0.970525\pi\)
0.577936 + 0.816082i \(0.303858\pi\)
\(648\) 0 0
\(649\) −4.04475 7.00572i −0.158770 0.274999i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.225016 + 0.389739i −0.00880556 + 0.0152517i −0.870395 0.492355i \(-0.836136\pi\)
0.861589 + 0.507606i \(0.169470\pi\)
\(654\) 0 0
\(655\) −7.90877 13.6984i −0.309021 0.535240i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.68441 + 15.0418i −0.338297 + 0.585947i −0.984112 0.177546i \(-0.943184\pi\)
0.645816 + 0.763493i \(0.276517\pi\)
\(660\) 0 0
\(661\) 34.0096 1.32282 0.661411 0.750024i \(-0.269958\pi\)
0.661411 + 0.750024i \(0.269958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.9919 + 19.0386i 0.425609 + 0.737176i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 24.0192 + 41.6025i 0.927252 + 1.60605i
\(672\) 0 0
\(673\) −10.5825 + 18.3294i −0.407925 + 0.706547i −0.994657 0.103234i \(-0.967081\pi\)
0.586732 + 0.809781i \(0.300414\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.7149 0.680839 0.340420 0.940274i \(-0.389431\pi\)
0.340420 + 0.940274i \(0.389431\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.2676 + 28.1763i −0.622461 + 1.07813i 0.366565 + 0.930393i \(0.380534\pi\)
−0.989026 + 0.147742i \(0.952800\pi\)
\(684\) 0 0
\(685\) 35.7136 1.36455
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −30.7944 −1.17317
\(690\) 0 0
\(691\) 31.3820 1.19383 0.596914 0.802305i \(-0.296393\pi\)
0.596914 + 0.802305i \(0.296393\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 40.8647 1.55009
\(696\) 0 0
\(697\) −26.8337 −1.01640
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11.7163 0.442518 0.221259 0.975215i \(-0.428983\pi\)
0.221259 + 0.975215i \(0.428983\pi\)
\(702\) 0 0
\(703\) −2.17709 + 3.77084i −0.0821106 + 0.142220i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.18114 0.269693 0.134847 0.990866i \(-0.456946\pi\)
0.134847 + 0.990866i \(0.456946\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.49834 6.05930i 0.131014 0.226922i
\(714\) 0 0
\(715\) −45.1438 78.1914i −1.68828 2.92419i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.82360 + 13.5509i 0.291771 + 0.505362i 0.974229 0.225563i \(-0.0724222\pi\)
−0.682458 + 0.730925i \(0.739089\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 27.3378 1.01530
\(726\) 0 0
\(727\) −22.5678 + 39.0886i −0.836995 + 1.44972i 0.0554015 + 0.998464i \(0.482356\pi\)
−0.892396 + 0.451253i \(0.850977\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −21.4687 37.1848i −0.794048 1.37533i
\(732\) 0 0
\(733\) −20.6167 + 35.7092i −0.761495 + 1.31895i 0.180585 + 0.983559i \(0.442201\pi\)
−0.942080 + 0.335388i \(0.891133\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.52772 9.57429i −0.203616 0.352673i
\(738\) 0 0
\(739\) −16.6209 + 28.7882i −0.611410 + 1.05899i 0.379593 + 0.925153i \(0.376064\pi\)
−0.991003 + 0.133839i \(0.957269\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.3263 + 28.2779i 0.598953 + 1.03742i 0.992976 + 0.118316i \(0.0377497\pi\)
−0.394023 + 0.919101i \(0.628917\pi\)
\(744\) 0 0
\(745\) 31.7240 + 54.9475i 1.16228 + 2.01312i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0384 + 34.7075i −0.731212 + 1.26650i 0.225154 + 0.974323i \(0.427712\pi\)
−0.956366 + 0.292173i \(0.905622\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 28.1293 1.02373
\(756\) 0 0
\(757\) 21.8337 0.793559 0.396779 0.917914i \(-0.370128\pi\)
0.396779 + 0.917914i \(0.370128\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.56067 7.89932i 0.165324 0.286350i −0.771446 0.636295i \(-0.780466\pi\)
0.936770 + 0.349945i \(0.113800\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.14120 + 8.90482i 0.185638 + 0.321534i
\(768\) 0 0
\(769\) −14.3654 24.8815i −0.518028 0.897250i −0.999781 0.0209433i \(-0.993333\pi\)
0.481753 0.876307i \(-0.340000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.7283 + 34.1705i −0.709578 + 1.22903i 0.255435 + 0.966826i \(0.417781\pi\)
−0.965014 + 0.262200i \(0.915552\pi\)
\(774\) 0 0
\(775\) −4.35033 7.53499i −0.156268 0.270665i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.19664 2.07264i 0.0428740 0.0742599i
\(780\) 0 0
\(781\) 0.302702 + 0.524295i 0.0108315 + 0.0187608i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.31493 4.00957i 0.0826232 0.143108i
\(786\) 0 0
\(787\) 11.0307 0.393202 0.196601 0.980484i \(-0.437010\pi\)
0.196601 + 0.980484i \(0.437010\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −30.5303 52.8801i −1.08416 1.87783i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.15757 + 2.00497i 0.0410032 + 0.0710196i 0.885799 0.464070i \(-0.153611\pi\)
−0.844796 + 0.535089i \(0.820278\pi\)
\(798\) 0 0
\(799\) −8.69664 + 15.0630i −0.307665 + 0.532891i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −44.0921 −1.55598
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.70471 + 6.41675i −0.130251 + 0.225601i −0.923773 0.382940i \(-0.874912\pi\)
0.793522 + 0.608541i \(0.208245\pi\)
\(810\) 0 0
\(811\) 12.8879 0.452555 0.226277 0.974063i \(-0.427344\pi\)
0.226277 + 0.974063i \(0.427344\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −81.9636 −2.87106
\(816\) 0 0
\(817\) 3.82955 0.133979
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.8743 0.903019 0.451510 0.892266i \(-0.350886\pi\)
0.451510 + 0.892266i \(0.350886\pi\)
\(822\) 0 0
\(823\) 44.0620 1.53590 0.767952 0.640507i \(-0.221276\pi\)
0.767952 + 0.640507i \(0.221276\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.81753 0.237069 0.118534 0.992950i \(-0.462180\pi\)
0.118534 + 0.992950i \(0.462180\pi\)
\(828\) 0 0
\(829\) −4.27759 + 7.40900i −0.148567 + 0.257325i −0.930698 0.365789i \(-0.880799\pi\)
0.782131 + 0.623114i \(0.214133\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −78.2566 −2.70818
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.0971 31.3451i 0.624781 1.08215i −0.363803 0.931476i \(-0.618522\pi\)
0.988583 0.150676i \(-0.0481450\pi\)
\(840\) 0 0
\(841\) 10.9168 + 18.9085i 0.376443 + 0.652018i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 32.0296 + 55.4768i 1.10185 + 1.90846i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 80.5011 2.75954
\(852\) 0 0
\(853\) 3.97889 6.89164i 0.136235 0.235965i −0.789834 0.613321i \(-0.789833\pi\)
0.926068 + 0.377356i \(0.123167\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.7899 + 39.4733i 0.778489 + 1.34838i 0.932813 + 0.360362i \(0.117347\pi\)
−0.154324 + 0.988020i \(0.549320\pi\)
\(858\) 0 0
\(859\) 10.4518 18.1030i 0.356609 0.617666i −0.630783 0.775960i \(-0.717266\pi\)
0.987392 + 0.158294i \(0.0505994\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.88586 6.73050i −0.132276 0.229109i 0.792278 0.610161i \(-0.208895\pi\)
−0.924554 + 0.381052i \(0.875562\pi\)
\(864\) 0 0
\(865\) −11.8182 + 20.4697i −0.401831 + 0.695991i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.873633 + 1.51318i 0.0296360 + 0.0513310i
\(870\) 0 0
\(871\) 7.02616 + 12.1697i 0.238072 + 0.412354i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.6886 34.1016i 0.664835 1.15153i −0.314495 0.949259i \(-0.601835\pi\)
0.979330 0.202269i \(-0.0648317\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.7967 0.902805 0.451402 0.892320i \(-0.350924\pi\)
0.451402 + 0.892320i \(0.350924\pi\)
\(882\) 0 0
\(883\) −43.0755 −1.44961 −0.724803 0.688956i \(-0.758069\pi\)
−0.724803 + 0.688956i \(0.758069\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.6253 + 32.2600i −0.625377 + 1.08318i 0.363091 + 0.931754i \(0.381721\pi\)
−0.988468 + 0.151431i \(0.951612\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.775645 1.34346i −0.0259560 0.0449571i
\(894\) 0 0
\(895\) −17.2269 29.8379i −0.575832 0.997370i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.14039 + 1.97521i −0.0380342 + 0.0658771i
\(900\) 0 0
\(901\) 14.1364 + 24.4849i 0.470951 + 0.815711i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −47.1560 + 81.6766i −1.56752 + 2.71502i
\(906\) 0 0
\(907\) 4.89327 + 8.47540i 0.162478 + 0.281421i 0.935757 0.352646i \(-0.114718\pi\)
−0.773279 + 0.634067i \(0.781385\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.2979 + 24.7647i −0.473710 + 0.820490i −0.999547 0.0300951i \(-0.990419\pi\)
0.525837 + 0.850586i \(0.323752\pi\)
\(912\) 0 0
\(913\) 17.7728 0.588192
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.61348 + 14.9190i 0.284133 + 0.492132i 0.972398 0.233326i \(-0.0749611\pi\)
−0.688266 + 0.725459i \(0.741628\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.384758 0.666420i −0.0126645 0.0219355i
\(924\) 0 0
\(925\) 50.0532 86.6948i 1.64574 2.85051i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 26.8597 0.881240 0.440620 0.897694i \(-0.354759\pi\)
0.440620 + 0.897694i \(0.354759\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −41.4472 + 71.7886i −1.35547 + 2.34774i
\(936\) 0 0
\(937\) 51.5653 1.68456 0.842282 0.539037i \(-0.181212\pi\)
0.842282 + 0.539037i \(0.181212\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 41.8004 1.36266 0.681328 0.731979i \(-0.261403\pi\)
0.681328 + 0.731979i \(0.261403\pi\)
\(942\) 0 0
\(943\) −44.2473 −1.44089
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.50415 −0.308843 −0.154422 0.988005i \(-0.549351\pi\)
−0.154422 + 0.988005i \(0.549351\pi\)
\(948\) 0 0
\(949\) 56.0445 1.81928
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 58.4539 1.89351 0.946754 0.321957i \(-0.104341\pi\)
0.946754 + 0.321957i \(0.104341\pi\)
\(954\) 0 0
\(955\) −39.4453 + 68.3213i −1.27642 + 2.21083i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.2741 −0.976584
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −24.1686 + 41.8612i −0.778014 + 1.34756i
\(966\) 0 0
\(967\) −5.09799 8.82997i −0.163940 0.283953i 0.772338 0.635212i \(-0.219087\pi\)
−0.936278 + 0.351259i \(0.885754\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.4238 33.6429i −0.623338 1.07965i −0.988860 0.148850i \(-0.952443\pi\)
0.365522 0.930803i \(-0.380891\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.1322 1.34793 0.673965 0.738763i \(-0.264590\pi\)
0.673965 + 0.738763i \(0.264590\pi\)
\(978\) 0 0
\(979\) 5.14776 8.91619i 0.164523 0.284963i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.0284 29.4941i −0.543123 0.940716i −0.998722 0.0505312i \(-0.983909\pi\)
0.455600 0.890185i \(-0.349425\pi\)
\(984\) 0 0
\(985\) −28.0688 + 48.6167i −0.894348 + 1.54906i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −35.4007 61.3158i −1.12568 1.94973i
\(990\) 0 0
\(991\) −20.5980 + 35.6768i −0.654317 + 1.13331i 0.327748 + 0.944765i \(0.393710\pi\)
−0.982065 + 0.188544i \(0.939623\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.5033 + 35.5127i 0.649997 + 1.12583i
\(996\) 0 0
\(997\) 15.7199 + 27.2277i 0.497856 + 0.862311i 0.999997 0.00247444i \(-0.000787640\pi\)
−0.502141 + 0.864786i \(0.667454\pi\)
\(998\) 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 5292.2.i.h.2125.6 12
3.2 odd 2 1764.2.i.h.1537.2 12
7.2 even 3 5292.2.l.h.3313.1 12
7.3 odd 6 5292.2.j.f.3529.1 12
7.4 even 3 5292.2.j.f.3529.6 12
7.5 odd 6 5292.2.l.h.3313.6 12
7.6 odd 2 inner 5292.2.i.h.2125.1 12
9.4 even 3 5292.2.l.h.361.1 12
9.5 odd 6 1764.2.l.h.949.3 12
21.2 odd 6 1764.2.l.h.961.3 12
21.5 even 6 1764.2.l.h.961.4 12
21.11 odd 6 1764.2.j.f.1177.6 yes 12
21.17 even 6 1764.2.j.f.1177.1 yes 12
21.20 even 2 1764.2.i.h.1537.5 12
63.4 even 3 5292.2.j.f.1765.6 12
63.5 even 6 1764.2.i.h.373.5 12
63.13 odd 6 5292.2.l.h.361.6 12
63.23 odd 6 1764.2.i.h.373.2 12
63.31 odd 6 5292.2.j.f.1765.1 12
63.32 odd 6 1764.2.j.f.589.6 yes 12
63.40 odd 6 inner 5292.2.i.h.1549.1 12
63.41 even 6 1764.2.l.h.949.4 12
63.58 even 3 inner 5292.2.i.h.1549.6 12
63.59 even 6 1764.2.j.f.589.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.i.h.373.2 12 63.23 odd 6
1764.2.i.h.373.5 12 63.5 even 6
1764.2.i.h.1537.2 12 3.2 odd 2
1764.2.i.h.1537.5 12 21.20 even 2
1764.2.j.f.589.1 12 63.59 even 6
1764.2.j.f.589.6 yes 12 63.32 odd 6
1764.2.j.f.1177.1 yes 12 21.17 even 6
1764.2.j.f.1177.6 yes 12 21.11 odd 6
1764.2.l.h.949.3 12 9.5 odd 6
1764.2.l.h.949.4 12 63.41 even 6
1764.2.l.h.961.3 12 21.2 odd 6
1764.2.l.h.961.4 12 21.5 even 6
5292.2.i.h.1549.1 12 63.40 odd 6 inner
5292.2.i.h.1549.6 12 63.58 even 3 inner
5292.2.i.h.2125.1 12 7.6 odd 2 inner
5292.2.i.h.2125.6 12 1.1 even 1 trivial
5292.2.j.f.1765.1 12 63.31 odd 6
5292.2.j.f.1765.6 12 63.4 even 3
5292.2.j.f.3529.1 12 7.3 odd 6
5292.2.j.f.3529.6 12 7.4 even 3
5292.2.l.h.361.1 12 9.4 even 3
5292.2.l.h.361.6 12 63.13 odd 6
5292.2.l.h.3313.1 12 7.2 even 3
5292.2.l.h.3313.6 12 7.5 odd 6