Properties

Label 2268.2.t.a.1781.4
Level $2268$
Weight $2$
Character 2268.1781
Analytic conductor $18.110$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(1781,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.t (of order \(6\), degree \(2\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1781.4
Root \(1.68124 - 0.416458i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1781
Dual form 2268.2.t.a.2105.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.349828 - 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +O(q^{10})\) \(q+(-0.349828 - 0.605920i) q^{5} +(-2.02556 - 1.70209i) q^{7} +(0.229685 + 0.132608i) q^{11} +1.31431i q^{13} +(1.86392 - 3.22840i) q^{17} +(-0.382449 + 0.220807i) q^{19} +(-4.29949 + 2.48231i) q^{23} +(2.25524 - 3.90619i) q^{25} -0.315564i q^{29} +(4.85521 + 2.80316i) q^{31} +(-0.322736 + 1.82276i) q^{35} +(-0.351124 - 0.608164i) q^{37} -10.7871 q^{41} -7.46263 q^{43} +(-3.50285 - 6.06712i) q^{47} +(1.20576 + 6.89537i) q^{49} +(-8.51919 - 4.91856i) q^{53} -0.185561i q^{55} +(-6.73182 + 11.6598i) q^{59} +(4.89484 - 2.82604i) q^{61} +(0.796368 - 0.459783i) q^{65} +(2.97060 - 5.14523i) q^{67} +13.4323i q^{71} +(-6.66182 - 3.84620i) q^{73} +(-0.239527 - 0.659550i) q^{77} +(-0.698360 - 1.20959i) q^{79} -7.44798 q^{83} -2.60820 q^{85} +(-5.59261 - 9.68668i) q^{89} +(2.23708 - 2.66221i) q^{91} +(0.267582 + 0.154489i) q^{95} +10.6028i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{7} - 6 q^{11} - 9 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{31} + 15 q^{35} + q^{37} - 12 q^{41} + 4 q^{43} - 18 q^{47} - 8 q^{49} - 15 q^{59} - 3 q^{61} + 39 q^{65} - 7 q^{67} + 48 q^{77} - q^{79} - 12 q^{85} - 21 q^{89} + 9 q^{91} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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
\(5\) −0.349828 0.605920i −0.156448 0.270975i 0.777137 0.629331i \(-0.216671\pi\)
−0.933585 + 0.358355i \(0.883338\pi\)
\(6\) 0 0
\(7\) −2.02556 1.70209i −0.765588 0.643331i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.229685 + 0.132608i 0.0692525 + 0.0399829i 0.534226 0.845341i \(-0.320603\pi\)
−0.464974 + 0.885324i \(0.653936\pi\)
\(12\) 0 0
\(13\) 1.31431i 0.364525i 0.983250 + 0.182262i \(0.0583420\pi\)
−0.983250 + 0.182262i \(0.941658\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.86392 3.22840i 0.452067 0.783003i −0.546447 0.837493i \(-0.684020\pi\)
0.998514 + 0.0544906i \(0.0173535\pi\)
\(18\) 0 0
\(19\) −0.382449 + 0.220807i −0.0877398 + 0.0506566i −0.543228 0.839585i \(-0.682798\pi\)
0.455488 + 0.890242i \(0.349465\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.29949 + 2.48231i −0.896507 + 0.517598i −0.876065 0.482193i \(-0.839840\pi\)
−0.0204414 + 0.999791i \(0.506507\pi\)
\(24\) 0 0
\(25\) 2.25524 3.90619i 0.451048 0.781238i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.315564i 0.0585988i −0.999571 0.0292994i \(-0.990672\pi\)
0.999571 0.0292994i \(-0.00932762\pi\)
\(30\) 0 0
\(31\) 4.85521 + 2.80316i 0.872022 + 0.503462i 0.868020 0.496530i \(-0.165393\pi\)
0.00400255 + 0.999992i \(0.498726\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.322736 + 1.82276i −0.0545523 + 0.308103i
\(36\) 0 0
\(37\) −0.351124 0.608164i −0.0577244 0.0999816i 0.835719 0.549157i \(-0.185051\pi\)
−0.893444 + 0.449175i \(0.851718\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.7871 −1.68466 −0.842330 0.538962i \(-0.818817\pi\)
−0.842330 + 0.538962i \(0.818817\pi\)
\(42\) 0 0
\(43\) −7.46263 −1.13804 −0.569020 0.822324i \(-0.692677\pi\)
−0.569020 + 0.822324i \(0.692677\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.50285 6.06712i −0.510943 0.884980i −0.999920 0.0126827i \(-0.995963\pi\)
0.488976 0.872297i \(-0.337370\pi\)
\(48\) 0 0
\(49\) 1.20576 + 6.89537i 0.172251 + 0.985053i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.51919 4.91856i −1.17020 0.675616i −0.216474 0.976288i \(-0.569456\pi\)
−0.953727 + 0.300672i \(0.902789\pi\)
\(54\) 0 0
\(55\) 0.185561i 0.0250210i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.73182 + 11.6598i −0.876408 + 1.51798i −0.0211522 + 0.999776i \(0.506733\pi\)
−0.855256 + 0.518206i \(0.826600\pi\)
\(60\) 0 0
\(61\) 4.89484 2.82604i 0.626720 0.361837i −0.152761 0.988263i \(-0.548816\pi\)
0.779481 + 0.626426i \(0.215483\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.796368 0.459783i 0.0987773 0.0570291i
\(66\) 0 0
\(67\) 2.97060 5.14523i 0.362916 0.628590i −0.625523 0.780206i \(-0.715114\pi\)
0.988439 + 0.151616i \(0.0484477\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4323i 1.59412i 0.603900 + 0.797060i \(0.293613\pi\)
−0.603900 + 0.797060i \(0.706387\pi\)
\(72\) 0 0
\(73\) −6.66182 3.84620i −0.779707 0.450164i 0.0566194 0.998396i \(-0.481968\pi\)
−0.836326 + 0.548232i \(0.815301\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.239527 0.659550i −0.0272966 0.0751627i
\(78\) 0 0
\(79\) −0.698360 1.20959i −0.0785716 0.136090i 0.824062 0.566499i \(-0.191703\pi\)
−0.902634 + 0.430409i \(0.858369\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.44798 −0.817522 −0.408761 0.912641i \(-0.634039\pi\)
−0.408761 + 0.912641i \(0.634039\pi\)
\(84\) 0 0
\(85\) −2.60820 −0.282899
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.59261 9.68668i −0.592815 1.02679i −0.993851 0.110724i \(-0.964683\pi\)
0.401036 0.916062i \(-0.368650\pi\)
\(90\) 0 0
\(91\) 2.23708 2.66221i 0.234510 0.279076i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.267582 + 0.154489i 0.0274534 + 0.0158502i
\(96\) 0 0
\(97\) 10.6028i 1.07655i 0.842770 + 0.538273i \(0.180923\pi\)
−0.842770 + 0.538273i \(0.819077\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.75357 + 15.1616i −0.871013 + 1.50864i −0.0100634 + 0.999949i \(0.503203\pi\)
−0.860950 + 0.508690i \(0.830130\pi\)
\(102\) 0 0
\(103\) −7.39775 + 4.27110i −0.728922 + 0.420844i −0.818028 0.575179i \(-0.804933\pi\)
0.0891054 + 0.996022i \(0.471599\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.09489 + 5.25093i −0.879236 + 0.507627i −0.870406 0.492334i \(-0.836144\pi\)
−0.00882940 + 0.999961i \(0.502811\pi\)
\(108\) 0 0
\(109\) −7.12110 + 12.3341i −0.682078 + 1.18139i 0.292268 + 0.956337i \(0.405590\pi\)
−0.974346 + 0.225057i \(0.927743\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.4479i 1.45322i −0.687051 0.726609i \(-0.741095\pi\)
0.687051 0.726609i \(-0.258905\pi\)
\(114\) 0 0
\(115\) 3.00817 + 1.73677i 0.280513 + 0.161954i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.27052 + 3.36675i −0.849827 + 0.308629i
\(120\) 0 0
\(121\) −5.46483 9.46536i −0.496803 0.860488i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.65406 −0.595157
\(126\) 0 0
\(127\) 21.8304 1.93713 0.968566 0.248758i \(-0.0800225\pi\)
0.968566 + 0.248758i \(0.0800225\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.60461 4.51132i −0.227566 0.394156i 0.729520 0.683959i \(-0.239743\pi\)
−0.957086 + 0.289803i \(0.906410\pi\)
\(132\) 0 0
\(133\) 1.15051 + 0.203707i 0.0997615 + 0.0176636i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.33589 1.34863i −0.199568 0.115221i 0.396886 0.917868i \(-0.370091\pi\)
−0.596454 + 0.802647i \(0.703424\pi\)
\(138\) 0 0
\(139\) 11.7142i 0.993585i −0.867869 0.496793i \(-0.834511\pi\)
0.867869 0.496793i \(-0.165489\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.174289 + 0.301877i −0.0145748 + 0.0252442i
\(144\) 0 0
\(145\) −0.191206 + 0.110393i −0.0158788 + 0.00916765i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.3055 9.41399i 1.33580 0.771224i 0.349618 0.936892i \(-0.386311\pi\)
0.986182 + 0.165668i \(0.0529781\pi\)
\(150\) 0 0
\(151\) −5.00143 + 8.66273i −0.407010 + 0.704963i −0.994553 0.104230i \(-0.966762\pi\)
0.587543 + 0.809193i \(0.300095\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.92249i 0.315062i
\(156\) 0 0
\(157\) 0.218293 + 0.126032i 0.0174217 + 0.0100584i 0.508686 0.860952i \(-0.330132\pi\)
−0.491264 + 0.871011i \(0.663465\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.9340 + 2.29007i 1.01934 + 0.180483i
\(162\) 0 0
\(163\) 4.29780 + 7.44400i 0.336629 + 0.583059i 0.983796 0.179289i \(-0.0573797\pi\)
−0.647167 + 0.762348i \(0.724046\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.48874 0.347349 0.173674 0.984803i \(-0.444436\pi\)
0.173674 + 0.984803i \(0.444436\pi\)
\(168\) 0 0
\(169\) 11.2726 0.867122
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.56072 6.16736i −0.270717 0.468895i 0.698329 0.715777i \(-0.253927\pi\)
−0.969046 + 0.246882i \(0.920594\pi\)
\(174\) 0 0
\(175\) −11.2168 + 4.07358i −0.847912 + 0.307934i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −22.1270 12.7750i −1.65385 0.954848i −0.975470 0.220134i \(-0.929351\pi\)
−0.678376 0.734715i \(-0.737316\pi\)
\(180\) 0 0
\(181\) 0.943175i 0.0701057i −0.999385 0.0350528i \(-0.988840\pi\)
0.999385 0.0350528i \(-0.0111599\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.245666 + 0.425506i −0.0180617 + 0.0312838i
\(186\) 0 0
\(187\) 0.856227 0.494343i 0.0626135 0.0361499i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.57413 1.48617i 0.186258 0.107536i −0.403972 0.914771i \(-0.632371\pi\)
0.590229 + 0.807236i \(0.299037\pi\)
\(192\) 0 0
\(193\) 9.25721 16.0340i 0.666348 1.15415i −0.312570 0.949895i \(-0.601190\pi\)
0.978918 0.204254i \(-0.0654769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1774i 1.01010i 0.863091 + 0.505048i \(0.168525\pi\)
−0.863091 + 0.505048i \(0.831475\pi\)
\(198\) 0 0
\(199\) −20.5293 11.8526i −1.45529 0.840209i −0.456512 0.889717i \(-0.650901\pi\)
−0.998774 + 0.0495081i \(0.984235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.537119 + 0.639193i −0.0376984 + 0.0448625i
\(204\) 0 0
\(205\) 3.77362 + 6.53611i 0.263561 + 0.456502i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.117123 −0.00810160
\(210\) 0 0
\(211\) −6.08007 −0.418569 −0.209285 0.977855i \(-0.567114\pi\)
−0.209285 + 0.977855i \(0.567114\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.61063 + 4.52175i 0.178044 + 0.308381i
\(216\) 0 0
\(217\) −5.06327 13.9420i −0.343717 0.946444i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.24313 + 2.44977i 0.285424 + 0.164790i
\(222\) 0 0
\(223\) 0.919300i 0.0615609i −0.999526 0.0307804i \(-0.990201\pi\)
0.999526 0.0307804i \(-0.00979926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.00297 + 8.66540i −0.332059 + 0.575143i −0.982915 0.184058i \(-0.941077\pi\)
0.650857 + 0.759201i \(0.274410\pi\)
\(228\) 0 0
\(229\) 2.38179 1.37513i 0.157393 0.0908710i −0.419235 0.907878i \(-0.637702\pi\)
0.576628 + 0.817007i \(0.304368\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.55513 3.20725i 0.363928 0.210114i −0.306874 0.951750i \(-0.599283\pi\)
0.670803 + 0.741636i \(0.265950\pi\)
\(234\) 0 0
\(235\) −2.45079 + 4.24489i −0.159872 + 0.276906i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.1969i 0.853636i 0.904338 + 0.426818i \(0.140365\pi\)
−0.904338 + 0.426818i \(0.859635\pi\)
\(240\) 0 0
\(241\) 2.20722 + 1.27434i 0.142180 + 0.0820874i 0.569402 0.822059i \(-0.307175\pi\)
−0.427223 + 0.904146i \(0.640508\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.75623 3.14279i 0.239977 0.200785i
\(246\) 0 0
\(247\) −0.290209 0.502657i −0.0184656 0.0319833i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −18.7893 −1.18597 −0.592986 0.805213i \(-0.702051\pi\)
−0.592986 + 0.805213i \(0.702051\pi\)
\(252\) 0 0
\(253\) −1.31670 −0.0827804
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.19727 12.4660i −0.448953 0.777610i 0.549365 0.835583i \(-0.314870\pi\)
−0.998318 + 0.0579725i \(0.981536\pi\)
\(258\) 0 0
\(259\) −0.323931 + 1.82952i −0.0201281 + 0.113681i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.79810 + 3.92488i 0.419189 + 0.242019i 0.694730 0.719271i \(-0.255524\pi\)
−0.275542 + 0.961289i \(0.588857\pi\)
\(264\) 0 0
\(265\) 6.88260i 0.422794i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.72267 13.3760i 0.470859 0.815552i −0.528585 0.848880i \(-0.677277\pi\)
0.999444 + 0.0333281i \(0.0106106\pi\)
\(270\) 0 0
\(271\) 10.9476 6.32057i 0.665016 0.383947i −0.129169 0.991623i \(-0.541231\pi\)
0.794186 + 0.607675i \(0.207898\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.03599 0.598128i 0.0624724 0.0360685i
\(276\) 0 0
\(277\) 5.94531 10.2976i 0.357219 0.618722i −0.630276 0.776371i \(-0.717058\pi\)
0.987495 + 0.157649i \(0.0503915\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.18018i 0.189714i −0.995491 0.0948568i \(-0.969761\pi\)
0.995491 0.0948568i \(-0.0302393\pi\)
\(282\) 0 0
\(283\) 16.0195 + 9.24889i 0.952263 + 0.549789i 0.893783 0.448499i \(-0.148041\pi\)
0.0584799 + 0.998289i \(0.481375\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.8499 + 18.3606i 1.28976 + 1.08379i
\(288\) 0 0
\(289\) 1.55161 + 2.68746i 0.0912711 + 0.158086i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.85949 −0.167053 −0.0835266 0.996506i \(-0.526618\pi\)
−0.0835266 + 0.996506i \(0.526618\pi\)
\(294\) 0 0
\(295\) 9.41991 0.548448
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.26254 5.65088i −0.188677 0.326799i
\(300\) 0 0
\(301\) 15.1160 + 12.7021i 0.871270 + 0.732136i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.42470 1.97725i −0.196098 0.113217i
\(306\) 0 0
\(307\) 21.6746i 1.23704i 0.785771 + 0.618518i \(0.212266\pi\)
−0.785771 + 0.618518i \(0.787734\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.8462 + 20.5183i −0.671738 + 1.16348i 0.305673 + 0.952136i \(0.401119\pi\)
−0.977411 + 0.211348i \(0.932215\pi\)
\(312\) 0 0
\(313\) −23.6283 + 13.6418i −1.33555 + 0.771081i −0.986144 0.165890i \(-0.946950\pi\)
−0.349407 + 0.936971i \(0.613617\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.2647 12.2772i 1.19435 0.689556i 0.235057 0.971982i \(-0.424472\pi\)
0.959289 + 0.282426i \(0.0911391\pi\)
\(318\) 0 0
\(319\) 0.0418465 0.0724802i 0.00234295 0.00405811i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.64626i 0.0916006i
\(324\) 0 0
\(325\) 5.13396 + 2.96409i 0.284781 + 0.164418i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.23157 + 18.2515i −0.178162 + 1.00624i
\(330\) 0 0
\(331\) −8.15579 14.1262i −0.448283 0.776449i 0.549991 0.835170i \(-0.314631\pi\)
−0.998274 + 0.0587215i \(0.981298\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.15679 −0.227110
\(336\) 0 0
\(337\) −27.3160 −1.48800 −0.743998 0.668182i \(-0.767073\pi\)
−0.743998 + 0.668182i \(0.767073\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.743445 + 1.28768i 0.0402598 + 0.0697320i
\(342\) 0 0
\(343\) 9.29424 16.0193i 0.501842 0.864960i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.37986 + 3.10606i 0.288806 + 0.166742i 0.637403 0.770530i \(-0.280009\pi\)
−0.348597 + 0.937273i \(0.613342\pi\)
\(348\) 0 0
\(349\) 28.4668i 1.52379i −0.647700 0.761896i \(-0.724269\pi\)
0.647700 0.761896i \(-0.275731\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.49346 + 2.58674i −0.0794887 + 0.137678i −0.903029 0.429579i \(-0.858662\pi\)
0.823541 + 0.567257i \(0.191995\pi\)
\(354\) 0 0
\(355\) 8.13889 4.69899i 0.431968 0.249397i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.5977 15.3562i 1.40377 0.810468i 0.408994 0.912537i \(-0.365880\pi\)
0.994777 + 0.102070i \(0.0325465\pi\)
\(360\) 0 0
\(361\) −9.40249 + 16.2856i −0.494868 + 0.857136i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.38204i 0.281709i
\(366\) 0 0
\(367\) −16.4877 9.51918i −0.860651 0.496897i 0.00357920 0.999994i \(-0.498861\pi\)
−0.864230 + 0.503096i \(0.832194\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.88426 + 24.4633i 0.461248 + 1.27007i
\(372\) 0 0
\(373\) −2.05869 3.56576i −0.106595 0.184628i 0.807794 0.589465i \(-0.200661\pi\)
−0.914389 + 0.404837i \(0.867328\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.414750 0.0213607
\(378\) 0 0
\(379\) −11.2436 −0.577546 −0.288773 0.957398i \(-0.593247\pi\)
−0.288773 + 0.957398i \(0.593247\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.8046 27.3745i −0.807580 1.39877i −0.914536 0.404505i \(-0.867444\pi\)
0.106956 0.994264i \(-0.465890\pi\)
\(384\) 0 0
\(385\) −0.315841 + 0.375863i −0.0160968 + 0.0191558i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.4018 + 10.6243i 0.933007 + 0.538672i 0.887761 0.460304i \(-0.152260\pi\)
0.0452458 + 0.998976i \(0.485593\pi\)
\(390\) 0 0
\(391\) 18.5073i 0.935956i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.488611 + 0.846300i −0.0245847 + 0.0425820i
\(396\) 0 0
\(397\) 20.6927 11.9469i 1.03854 0.599599i 0.119118 0.992880i \(-0.461993\pi\)
0.919419 + 0.393281i \(0.128660\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.0121 12.7087i 1.09923 0.634642i 0.163213 0.986591i \(-0.447814\pi\)
0.936019 + 0.351948i \(0.114481\pi\)
\(402\) 0 0
\(403\) −3.68423 + 6.38127i −0.183524 + 0.317874i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.186248i 0.00923197i
\(408\) 0 0
\(409\) 19.3831 + 11.1908i 0.958433 + 0.553351i 0.895690 0.444678i \(-0.146682\pi\)
0.0627424 + 0.998030i \(0.480015\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 33.4818 12.1595i 1.64753 0.598330i
\(414\) 0 0
\(415\) 2.60551 + 4.51288i 0.127900 + 0.221528i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0836 0.688030 0.344015 0.938964i \(-0.388213\pi\)
0.344015 + 0.938964i \(0.388213\pi\)
\(420\) 0 0
\(421\) −16.1528 −0.787238 −0.393619 0.919274i \(-0.628777\pi\)
−0.393619 + 0.919274i \(0.628777\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.40717 14.5617i −0.407808 0.706344i
\(426\) 0 0
\(427\) −14.7250 2.60718i −0.712590 0.126170i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.16179 + 4.13486i 0.344971 + 0.199169i 0.662468 0.749090i \(-0.269509\pi\)
−0.317497 + 0.948259i \(0.602842\pi\)
\(432\) 0 0
\(433\) 4.35102i 0.209097i −0.994520 0.104548i \(-0.966660\pi\)
0.994520 0.104548i \(-0.0333397\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.09622 1.89872i 0.0524395 0.0908279i
\(438\) 0 0
\(439\) 18.0200 10.4039i 0.860048 0.496549i −0.00398054 0.999992i \(-0.501267\pi\)
0.864028 + 0.503443i \(0.167934\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.7927 15.4688i 1.27296 0.734945i 0.297417 0.954748i \(-0.403875\pi\)
0.975544 + 0.219803i \(0.0705414\pi\)
\(444\) 0 0
\(445\) −3.91290 + 6.77734i −0.185489 + 0.321277i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.9215i 0.987346i −0.869648 0.493673i \(-0.835654\pi\)
0.869648 0.493673i \(-0.164346\pi\)
\(450\) 0 0
\(451\) −2.47763 1.43046i −0.116667 0.0673577i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.39568 0.424175i −0.112311 0.0198857i
\(456\) 0 0
\(457\) −1.15058 1.99286i −0.0538217 0.0932218i 0.837859 0.545886i \(-0.183807\pi\)
−0.891681 + 0.452664i \(0.850474\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.8499 0.831355 0.415677 0.909512i \(-0.363545\pi\)
0.415677 + 0.909512i \(0.363545\pi\)
\(462\) 0 0
\(463\) 12.4807 0.580026 0.290013 0.957023i \(-0.406340\pi\)
0.290013 + 0.957023i \(0.406340\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.42799 + 4.20541i 0.112354 + 0.194603i 0.916719 0.399533i \(-0.130828\pi\)
−0.804365 + 0.594136i \(0.797494\pi\)
\(468\) 0 0
\(469\) −14.7748 + 5.36571i −0.682236 + 0.247766i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.71405 0.989607i −0.0788121 0.0455022i
\(474\) 0 0
\(475\) 1.99189i 0.0913942i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.40542 + 7.63041i −0.201289 + 0.348642i −0.948944 0.315445i \(-0.897846\pi\)
0.747655 + 0.664087i \(0.231180\pi\)
\(480\) 0 0
\(481\) 0.799318 0.461486i 0.0364458 0.0210420i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.42441 3.70914i 0.291718 0.168423i
\(486\) 0 0
\(487\) 4.66185 8.07456i 0.211249 0.365893i −0.740857 0.671663i \(-0.765580\pi\)
0.952106 + 0.305770i \(0.0989137\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.0836i 1.40278i 0.712777 + 0.701391i \(0.247437\pi\)
−0.712777 + 0.701391i \(0.752563\pi\)
\(492\) 0 0
\(493\) −1.01877 0.588186i −0.0458830 0.0264906i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.8630 27.2079i 1.02555 1.22044i
\(498\) 0 0
\(499\) 11.1694 + 19.3459i 0.500010 + 0.866043i 1.00000 1.16519e-5i \(3.70891e-6\pi\)
−0.499990 + 0.866031i \(0.666663\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.2396 0.545738 0.272869 0.962051i \(-0.412027\pi\)
0.272869 + 0.962051i \(0.412027\pi\)
\(504\) 0 0
\(505\) 12.2490 0.545072
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.05496 + 12.2195i 0.312706 + 0.541622i 0.978947 0.204114i \(-0.0654314\pi\)
−0.666242 + 0.745736i \(0.732098\pi\)
\(510\) 0 0
\(511\) 6.94729 + 19.1297i 0.307330 + 0.846250i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.17588 + 2.98830i 0.228077 + 0.131680i
\(516\) 0 0
\(517\) 1.85803i 0.0817161i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.81632 + 4.87800i −0.123385 + 0.213709i −0.921101 0.389325i \(-0.872708\pi\)
0.797715 + 0.603034i \(0.206042\pi\)
\(522\) 0 0
\(523\) −33.2293 + 19.1849i −1.45302 + 0.838899i −0.998651 0.0519176i \(-0.983467\pi\)
−0.454364 + 0.890816i \(0.650133\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0995 10.4497i 0.788425 0.455197i
\(528\) 0 0
\(529\) 0.823769 1.42681i 0.0358161 0.0620352i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.1776i 0.614100i
\(534\) 0 0
\(535\) 6.36329 + 3.67385i 0.275109 + 0.158834i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.637441 + 1.74365i −0.0274565 + 0.0751045i
\(540\) 0 0
\(541\) −3.21673 5.57154i −0.138298 0.239539i 0.788555 0.614965i \(-0.210830\pi\)
−0.926852 + 0.375426i \(0.877496\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.96464 0.426838
\(546\) 0 0
\(547\) 13.0578 0.558310 0.279155 0.960246i \(-0.409946\pi\)
0.279155 + 0.960246i \(0.409946\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.0696787 + 0.120687i 0.00296841 + 0.00514144i
\(552\) 0 0
\(553\) −0.644276 + 3.63878i −0.0273974 + 0.154736i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.5409 + 14.7460i 1.08220 + 0.624809i 0.931489 0.363769i \(-0.118510\pi\)
0.150712 + 0.988578i \(0.451843\pi\)
\(558\) 0 0
\(559\) 9.80822i 0.414844i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.25934 + 9.10944i −0.221655 + 0.383917i −0.955311 0.295604i \(-0.904479\pi\)
0.733656 + 0.679521i \(0.237812\pi\)
\(564\) 0 0
\(565\) −9.36020 + 5.40411i −0.393787 + 0.227353i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.8054 13.1667i 0.956053 0.551977i 0.0610967 0.998132i \(-0.480540\pi\)
0.894956 + 0.446155i \(0.147207\pi\)
\(570\) 0 0
\(571\) 22.0295 38.1562i 0.921906 1.59679i 0.125444 0.992101i \(-0.459965\pi\)
0.796463 0.604688i \(-0.206702\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.3929i 0.933847i
\(576\) 0 0
\(577\) 12.1535 + 7.01684i 0.505957 + 0.292115i 0.731170 0.682195i \(-0.238974\pi\)
−0.225213 + 0.974310i \(0.572308\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.0863 + 12.6772i 0.625885 + 0.525937i
\(582\) 0 0
\(583\) −1.30448 2.25943i −0.0540262 0.0935762i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.04939 0.125862 0.0629308 0.998018i \(-0.479955\pi\)
0.0629308 + 0.998018i \(0.479955\pi\)
\(588\) 0 0
\(589\) −2.47583 −0.102015
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.3041 23.0434i −0.546334 0.946278i −0.998522 0.0543552i \(-0.982690\pi\)
0.452188 0.891923i \(-0.350644\pi\)
\(594\) 0 0
\(595\) 5.28306 + 4.43941i 0.216585 + 0.181998i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.86333 + 2.23050i 0.157852 + 0.0911356i 0.576845 0.816854i \(-0.304284\pi\)
−0.418993 + 0.907989i \(0.637617\pi\)
\(600\) 0 0
\(601\) 6.06239i 0.247290i −0.992327 0.123645i \(-0.960542\pi\)
0.992327 0.123645i \(-0.0394584\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.82350 + 6.62250i −0.155447 + 0.269243i
\(606\) 0 0
\(607\) −39.2581 + 22.6657i −1.59344 + 0.919971i −0.600725 + 0.799455i \(0.705121\pi\)
−0.992711 + 0.120516i \(0.961545\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.97409 4.60384i 0.322597 0.186251i
\(612\) 0 0
\(613\) 16.6294 28.8029i 0.671654 1.16334i −0.305781 0.952102i \(-0.598917\pi\)
0.977435 0.211237i \(-0.0677492\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.1423i 1.45503i −0.686090 0.727516i \(-0.740675\pi\)
0.686090 0.727516i \(-0.259325\pi\)
\(618\) 0 0
\(619\) 22.9031 + 13.2231i 0.920554 + 0.531482i 0.883812 0.467843i \(-0.154969\pi\)
0.0367423 + 0.999325i \(0.488302\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.15949 + 29.1401i −0.206711 + 1.16747i
\(624\) 0 0
\(625\) −8.94843 15.4991i −0.357937 0.619965i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.61787 −0.104381
\(630\) 0 0
\(631\) −32.0484 −1.27583 −0.637914 0.770107i \(-0.720203\pi\)
−0.637914 + 0.770107i \(0.720203\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.63687 13.2274i −0.303060 0.524915i
\(636\) 0 0
\(637\) −9.06267 + 1.58474i −0.359076 + 0.0627898i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21.1444 12.2077i −0.835153 0.482176i 0.0204610 0.999791i \(-0.493487\pi\)
−0.855614 + 0.517615i \(0.826820\pi\)
\(642\) 0 0
\(643\) 36.8366i 1.45269i −0.687329 0.726346i \(-0.741217\pi\)
0.687329 0.726346i \(-0.258783\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.2847 + 23.0098i −0.522276 + 0.904608i 0.477389 + 0.878692i \(0.341583\pi\)
−0.999664 + 0.0259155i \(0.991750\pi\)
\(648\) 0 0
\(649\) −3.09239 + 1.78539i −0.121387 + 0.0700827i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.2767 15.1709i 1.02829 0.593683i 0.111794 0.993731i \(-0.464340\pi\)
0.916494 + 0.400049i \(0.131007\pi\)
\(654\) 0 0
\(655\) −1.82233 + 3.15637i −0.0712044 + 0.123330i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.3281i 1.84364i 0.387618 + 0.921820i \(0.373298\pi\)
−0.387618 + 0.921820i \(0.626702\pi\)
\(660\) 0 0
\(661\) −30.4187 17.5623i −1.18315 0.683092i −0.226409 0.974032i \(-0.572699\pi\)
−0.956741 + 0.290940i \(0.906032\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.279049 0.768376i −0.0108211 0.0297963i
\(666\) 0 0
\(667\) 0.783329 + 1.35677i 0.0303306 + 0.0525342i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.49903 0.0578692
\(672\) 0 0
\(673\) 5.09516 0.196404 0.0982020 0.995167i \(-0.468691\pi\)
0.0982020 + 0.995167i \(0.468691\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.42072 14.5851i −0.323635 0.560551i 0.657601 0.753367i \(-0.271571\pi\)
−0.981235 + 0.192815i \(0.938238\pi\)
\(678\) 0 0
\(679\) 18.0469 21.4765i 0.692575 0.824191i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.7555 9.09645i −0.602868 0.348066i 0.167301 0.985906i \(-0.446495\pi\)
−0.770169 + 0.637840i \(0.779828\pi\)
\(684\) 0 0
\(685\) 1.88715i 0.0721042i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.46452 11.1969i 0.246279 0.426567i
\(690\) 0 0
\(691\) 3.05405 1.76326i 0.116182 0.0670775i −0.440783 0.897614i \(-0.645299\pi\)
0.556965 + 0.830536i \(0.311966\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.09786 + 4.09795i −0.269237 + 0.155444i
\(696\) 0 0
\(697\) −20.1063 + 34.8251i −0.761579 + 1.31909i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.3502i 0.504229i 0.967697 + 0.252114i \(0.0811259\pi\)
−0.967697 + 0.252114i \(0.918874\pi\)
\(702\) 0 0
\(703\) 0.268574 + 0.155061i 0.0101295 + 0.00584824i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 43.5374 15.8113i 1.63739 0.594647i
\(708\) 0 0
\(709\) 21.1447 + 36.6237i 0.794107 + 1.37543i 0.923405 + 0.383827i \(0.125394\pi\)
−0.129298 + 0.991606i \(0.541273\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −27.8333 −1.04236
\(714\) 0 0
\(715\) 0.243884 0.00912076
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.2035 26.3332i −0.566994 0.982062i −0.996861 0.0791697i \(-0.974773\pi\)
0.429868 0.902892i \(-0.358560\pi\)
\(720\) 0 0
\(721\) 22.2544 + 3.94032i 0.828796 + 0.146745i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.23265 0.711673i −0.0457796 0.0264309i
\(726\) 0 0
\(727\) 13.1256i 0.486802i 0.969926 + 0.243401i \(0.0782631\pi\)
−0.969926 + 0.243401i \(0.921737\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.9097 + 24.0924i −0.514470 + 0.891089i
\(732\) 0 0
\(733\) 32.7001 18.8794i 1.20781 0.697327i 0.245527 0.969390i \(-0.421039\pi\)
0.962280 + 0.272063i \(0.0877058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.36460 0.787853i 0.0502657 0.0290209i
\(738\) 0 0
\(739\) −13.1215 + 22.7271i −0.482683 + 0.836031i −0.999802 0.0198820i \(-0.993671\pi\)
0.517119 + 0.855913i \(0.327004\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.1426i 0.372098i 0.982541 + 0.186049i \(0.0595682\pi\)
−0.982541 + 0.186049i \(0.940432\pi\)
\(744\) 0 0
\(745\) −11.4082 6.58655i −0.417966 0.241313i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 27.3598 + 4.84428i 0.999705 + 0.177006i
\(750\) 0 0
\(751\) −3.95369 6.84798i −0.144272 0.249886i 0.784829 0.619712i \(-0.212751\pi\)
−0.929101 + 0.369826i \(0.879417\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.99855 0.254703
\(756\) 0 0
\(757\) 29.8903 1.08638 0.543191 0.839609i \(-0.317216\pi\)
0.543191 + 0.839609i \(0.317216\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.05687 + 5.29465i 0.110811 + 0.191931i 0.916098 0.400955i \(-0.131322\pi\)
−0.805286 + 0.592886i \(0.797988\pi\)
\(762\) 0 0
\(763\) 35.4180 12.8627i 1.28222 0.465660i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.3247 8.84771i −0.553342 0.319472i
\(768\) 0 0
\(769\) 11.3145i 0.408011i −0.978970 0.204005i \(-0.934604\pi\)
0.978970 0.204005i \(-0.0653960\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.2106 + 33.2737i −0.690956 + 1.19677i 0.280569 + 0.959834i \(0.409477\pi\)
−0.971525 + 0.236937i \(0.923856\pi\)
\(774\) 0 0
\(775\) 21.8994 12.6436i 0.786648 0.454172i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.12551 2.38186i 0.147812 0.0853391i
\(780\) 0 0
\(781\) −1.78124 + 3.08519i −0.0637376 + 0.110397i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.176358i 0.00629447i
\(786\) 0 0
\(787\) −41.2747 23.8300i −1.47129 0.849447i −0.471806 0.881703i \(-0.656398\pi\)
−0.999480 + 0.0322557i \(0.989731\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −26.2938 + 31.2906i −0.934900 + 1.11257i
\(792\) 0 0
\(793\) 3.71430 + 6.43335i 0.131898 + 0.228455i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.2717 −1.07228 −0.536139 0.844129i \(-0.680118\pi\)
−0.536139 + 0.844129i \(0.680118\pi\)
\(798\) 0 0
\(799\) −26.1161 −0.923922
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.02008 1.76683i −0.0359978 0.0623500i
\(804\) 0 0
\(805\) −3.13707 8.63810i −0.110567 0.304453i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.219373 0.126655i −0.00771273 0.00445295i 0.496139 0.868243i \(-0.334751\pi\)
−0.503851 + 0.863790i \(0.668084\pi\)
\(810\) 0 0
\(811\) 22.0629i 0.774735i −0.921925 0.387367i \(-0.873385\pi\)
0.921925 0.387367i \(-0.126615\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.00698 5.20824i 0.105330 0.182437i
\(816\) 0 0
\(817\) 2.85407 1.64780i 0.0998514 0.0576492i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −24.2467 + 13.9988i −0.846214 + 0.488562i −0.859372 0.511352i \(-0.829145\pi\)
0.0131576 + 0.999913i \(0.495812\pi\)
\(822\) 0 0
\(823\) −24.4771 + 42.3955i −0.853217 + 1.47782i 0.0250719 + 0.999686i \(0.492019\pi\)
−0.878289 + 0.478130i \(0.841315\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.641658i 0.0223126i −0.999938 0.0111563i \(-0.996449\pi\)
0.999938 0.0111563i \(-0.00355124\pi\)
\(828\) 0 0
\(829\) 9.57180 + 5.52628i 0.332442 + 0.191936i 0.656925 0.753956i \(-0.271857\pi\)
−0.324483 + 0.945892i \(0.605190\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.5085 + 8.95975i 0.849168 + 0.310437i
\(834\) 0 0
\(835\) −1.57029 2.71981i −0.0543420 0.0941230i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.24661 −0.319228 −0.159614 0.987179i \(-0.551025\pi\)
−0.159614 + 0.987179i \(0.551025\pi\)
\(840\) 0 0
\(841\) 28.9004 0.996566
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.94346 6.83028i −0.135659 0.234969i
\(846\) 0 0
\(847\) −5.04161 + 28.4743i −0.173232 + 0.978388i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.01931 + 1.74320i 0.103501 + 0.0597561i
\(852\) 0 0
\(853\) 39.6358i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.9260 20.6565i 0.407385 0.705612i −0.587211 0.809434i \(-0.699774\pi\)
0.994596 + 0.103822i \(0.0331074\pi\)
\(858\) 0 0
\(859\) 9.62480 5.55688i 0.328394 0.189598i −0.326734 0.945116i \(-0.605948\pi\)
0.655128 + 0.755518i \(0.272615\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.7780 22.3885i 1.32002 0.762113i 0.336287 0.941759i \(-0.390829\pi\)
0.983731 + 0.179646i \(0.0574954\pi\)
\(864\) 0 0
\(865\) −2.49128 + 4.31503i −0.0847061 + 0.146715i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.370434i 0.0125661i
\(870\) 0 0
\(871\) 6.76244 + 3.90429i 0.229136 + 0.132292i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 13.4782 + 11.3258i 0.455646 + 0.382883i
\(876\) 0 0
\(877\) −1.84096 3.18863i −0.0621647 0.107672i 0.833268 0.552869i \(-0.186467\pi\)
−0.895433 + 0.445197i \(0.853134\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17.3992 −0.586194 −0.293097 0.956083i \(-0.594686\pi\)
−0.293097 + 0.956083i \(0.594686\pi\)
\(882\) 0 0
\(883\) −2.02834 −0.0682592 −0.0341296 0.999417i \(-0.510866\pi\)
−0.0341296 + 0.999417i \(0.510866\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.1890 40.1645i −0.778610 1.34859i −0.932743 0.360542i \(-0.882592\pi\)
0.154132 0.988050i \(-0.450742\pi\)
\(888\) 0 0
\(889\) −44.2186 37.1573i −1.48305 1.24622i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.67932 + 1.54691i 0.0896601 + 0.0517653i
\(894\) 0 0
\(895\) 17.8762i 0.597536i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.884576 1.53213i 0.0295023 0.0510994i
\(900\) 0 0
\(901\) −31.7582 + 18.3356i −1.05802 + 0.610847i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −0.571488 + 0.329949i −0.0189969 + 0.0109679i
\(906\) 0 0
\(907\) −8.01957 + 13.8903i −0.266285 + 0.461220i −0.967900 0.251337i \(-0.919130\pi\)
0.701614 + 0.712557i \(0.252463\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.5344i 0.680335i 0.940365 + 0.340167i \(0.110484\pi\)
−0.940365 + 0.340167i \(0.889516\pi\)
\(912\) 0 0
\(913\) −1.71069 0.987665i −0.0566154 0.0326869i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.40290 + 13.5712i −0.0793507 + 0.448161i
\(918\) 0 0
\(919\) −17.7069 30.6693i −0.584097 1.01169i −0.994987 0.100001i \(-0.968115\pi\)
0.410890 0.911685i \(-0.365218\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −17.6542 −0.581096
\(924\) 0 0
\(925\) −3.16748 −0.104146
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −19.8626 34.4030i −0.651670 1.12873i −0.982718 0.185111i \(-0.940735\pi\)
0.331048 0.943614i \(-0.392598\pi\)
\(930\) 0 0
\(931\) −1.98369 2.37089i −0.0650127 0.0777027i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.599064 0.345870i −0.0195915 0.0113112i
\(936\) 0 0
\(937\) 23.2142i 0.758376i −0.925320 0.379188i \(-0.876203\pi\)
0.925320 0.379188i \(-0.123797\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −17.9616 + 31.1104i −0.585531 + 1.01417i 0.409278 + 0.912410i \(0.365781\pi\)
−0.994809 + 0.101760i \(0.967553\pi\)
\(942\) 0 0
\(943\) 46.3790 26.7769i 1.51031 0.871977i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.7365 15.4363i 0.868818 0.501612i 0.00186277 0.999998i \(-0.499407\pi\)
0.866955 + 0.498386i \(0.166074\pi\)
\(948\) 0 0
\(949\) 5.05511 8.75571i 0.164096 0.284222i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.4640i 0.986826i −0.869795 0.493413i \(-0.835749\pi\)
0.869795 0.493413i \(-0.164251\pi\)
\(954\) 0 0
\(955\) −1.80100 1.03981i −0.0582791 0.0336475i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.43599 + 6.70762i 0.0786621 + 0.216600i
\(960\) 0 0
\(961\) 0.215406 + 0.373095i 0.00694859 + 0.0120353i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.9537 −0.416995
\(966\) 0 0
\(967\) 13.5173 0.434687 0.217343 0.976095i \(-0.430261\pi\)
0.217343 + 0.976095i \(0.430261\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −16.6428 28.8261i −0.534092 0.925074i −0.999207 0.0398238i \(-0.987320\pi\)
0.465115 0.885250i \(-0.346013\pi\)
\(972\) 0 0
\(973\) −19.9387 + 23.7278i −0.639204 + 0.760677i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38.9127 22.4662i −1.24493 0.718758i −0.274833 0.961492i \(-0.588623\pi\)
−0.970093 + 0.242734i \(0.921956\pi\)
\(978\) 0 0
\(979\) 2.96651i 0.0948100i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.4474 23.2916i 0.428907 0.742888i −0.567870 0.823118i \(-0.692232\pi\)
0.996776 + 0.0802305i \(0.0255656\pi\)
\(984\) 0 0
\(985\) 8.59035 4.95964i 0.273711 0.158027i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0855 18.5246i 1.02026 0.589048i
\(990\) 0 0
\(991\) 17.7201 30.6920i 0.562896 0.974965i −0.434346 0.900746i \(-0.643020\pi\)
0.997242 0.0742186i \(-0.0236463\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.5855i 0.525796i
\(996\) 0 0
\(997\) 28.1418 + 16.2477i 0.891259 + 0.514568i 0.874354 0.485289i \(-0.161286\pi\)
0.0169046 + 0.999857i \(0.494619\pi\)
\(998\) 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 2268.2.t.a.1781.4 16
3.2 odd 2 2268.2.t.b.1781.5 16
7.5 odd 6 2268.2.t.b.2105.5 16
9.2 odd 6 756.2.bm.a.17.4 16
9.4 even 3 756.2.w.a.521.4 16
9.5 odd 6 252.2.w.a.101.6 yes 16
9.7 even 3 252.2.bm.a.185.8 yes 16
21.5 even 6 inner 2268.2.t.a.2105.4 16
36.7 odd 6 1008.2.df.d.689.1 16
36.11 even 6 3024.2.df.d.17.4 16
36.23 even 6 1008.2.ca.d.353.3 16
36.31 odd 6 3024.2.ca.d.2033.4 16
63.2 odd 6 5292.2.w.b.1097.5 16
63.4 even 3 5292.2.x.a.4409.4 16
63.5 even 6 252.2.bm.a.173.8 yes 16
63.11 odd 6 5292.2.x.b.881.5 16
63.13 odd 6 5292.2.w.b.521.5 16
63.16 even 3 1764.2.w.b.509.3 16
63.20 even 6 5292.2.bm.a.2285.5 16
63.23 odd 6 1764.2.bm.a.1685.1 16
63.25 even 3 1764.2.x.b.293.3 16
63.31 odd 6 5292.2.x.b.4409.5 16
63.32 odd 6 1764.2.x.a.1469.6 16
63.34 odd 6 1764.2.bm.a.1697.1 16
63.38 even 6 5292.2.x.a.881.4 16
63.40 odd 6 756.2.bm.a.89.4 16
63.41 even 6 1764.2.w.b.1109.3 16
63.47 even 6 756.2.w.a.341.4 16
63.52 odd 6 1764.2.x.a.293.6 16
63.58 even 3 5292.2.bm.a.4625.5 16
63.59 even 6 1764.2.x.b.1469.3 16
63.61 odd 6 252.2.w.a.5.6 16
252.47 odd 6 3024.2.ca.d.2609.4 16
252.103 even 6 3024.2.df.d.1601.4 16
252.131 odd 6 1008.2.df.d.929.1 16
252.187 even 6 1008.2.ca.d.257.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.6 16 63.61 odd 6
252.2.w.a.101.6 yes 16 9.5 odd 6
252.2.bm.a.173.8 yes 16 63.5 even 6
252.2.bm.a.185.8 yes 16 9.7 even 3
756.2.w.a.341.4 16 63.47 even 6
756.2.w.a.521.4 16 9.4 even 3
756.2.bm.a.17.4 16 9.2 odd 6
756.2.bm.a.89.4 16 63.40 odd 6
1008.2.ca.d.257.3 16 252.187 even 6
1008.2.ca.d.353.3 16 36.23 even 6
1008.2.df.d.689.1 16 36.7 odd 6
1008.2.df.d.929.1 16 252.131 odd 6
1764.2.w.b.509.3 16 63.16 even 3
1764.2.w.b.1109.3 16 63.41 even 6
1764.2.x.a.293.6 16 63.52 odd 6
1764.2.x.a.1469.6 16 63.32 odd 6
1764.2.x.b.293.3 16 63.25 even 3
1764.2.x.b.1469.3 16 63.59 even 6
1764.2.bm.a.1685.1 16 63.23 odd 6
1764.2.bm.a.1697.1 16 63.34 odd 6
2268.2.t.a.1781.4 16 1.1 even 1 trivial
2268.2.t.a.2105.4 16 21.5 even 6 inner
2268.2.t.b.1781.5 16 3.2 odd 2
2268.2.t.b.2105.5 16 7.5 odd 6
3024.2.ca.d.2033.4 16 36.31 odd 6
3024.2.ca.d.2609.4 16 252.47 odd 6
3024.2.df.d.17.4 16 36.11 even 6
3024.2.df.d.1601.4 16 252.103 even 6
5292.2.w.b.521.5 16 63.13 odd 6
5292.2.w.b.1097.5 16 63.2 odd 6
5292.2.x.a.881.4 16 63.38 even 6
5292.2.x.a.4409.4 16 63.4 even 3
5292.2.x.b.881.5 16 63.11 odd 6
5292.2.x.b.4409.5 16 63.31 odd 6
5292.2.bm.a.2285.5 16 63.20 even 6
5292.2.bm.a.4625.5 16 63.58 even 3