Properties

Label 5292.2.x.b.881.5
Level $5292$
Weight $2$
Character 5292.881
Analytic conductor $42.257$
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] = [5292,2,Mod(881,5292)]
 
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, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5292.881");
 
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.x (of order \(6\), 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: \(16\)
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: \(\Z[a_1, \ldots, a_{19}]\)
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 881.5
Root \(1.68124 - 0.416458i\) of defining polynomial
Character \(\chi\) \(=\) 5292.881
Dual form 5292.2.x.b.4409.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.349828 - 0.605920i) q^{5} +O(q^{10})\) \(q+(0.349828 - 0.605920i) q^{5} +(0.229685 - 0.132608i) q^{11} +(-1.13823 - 0.657156i) q^{13} +3.72784 q^{17} -0.441614i q^{19} +(-4.29949 - 2.48231i) q^{23} +(2.25524 + 3.90619i) q^{25} +(0.273287 - 0.157782i) q^{29} +(4.85521 + 2.80316i) q^{31} +0.702248 q^{37} +(-5.39354 + 9.34189i) q^{41} +(3.73131 + 6.46283i) q^{43} +(3.50285 + 6.06712i) q^{47} -9.83712i q^{53} -0.185561i q^{55} +(6.73182 - 11.6598i) q^{59} +(4.89484 - 2.82604i) q^{61} +(-0.796368 + 0.459783i) q^{65} +(2.97060 - 5.14523i) q^{67} -13.4323i q^{71} +7.69241i q^{73} +(-0.698360 - 1.20959i) q^{79} +(-3.72399 - 6.45014i) q^{83} +(1.30410 - 2.25877i) q^{85} -11.1852 q^{89} +(-0.267582 - 0.154489i) q^{95} +(9.18225 - 5.30138i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{11} + 3 q^{13} - 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} - 6 q^{31} - 2 q^{37} - 6 q^{41} - 2 q^{43} + 18 q^{47} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 42 q^{89} + 6 q^{95} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\) \(2647\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\)

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 \(-0.783329\pi\)
0.933585 + 0.358355i \(0.116662\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.229685 0.132608i 0.0692525 0.0399829i −0.464974 0.885324i \(-0.653936\pi\)
0.534226 + 0.845341i \(0.320603\pi\)
\(12\) 0 0
\(13\) −1.13823 0.657156i −0.315688 0.182262i 0.333781 0.942651i \(-0.391675\pi\)
−0.649469 + 0.760388i \(0.725009\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.72784 0.904134 0.452067 0.891984i \(-0.350687\pi\)
0.452067 + 0.891984i \(0.350687\pi\)
\(18\) 0 0
\(19\) 0.441614i 0.101313i −0.998716 0.0506566i \(-0.983869\pi\)
0.998716 0.0506566i \(-0.0161314\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.29949 2.48231i −0.896507 0.517598i −0.0204414 0.999791i \(-0.506507\pi\)
−0.876065 + 0.482193i \(0.839840\pi\)
\(24\) 0 0
\(25\) 2.25524 + 3.90619i 0.451048 + 0.781238i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.273287 0.157782i 0.0507480 0.0292994i −0.474411 0.880303i \(-0.657339\pi\)
0.525159 + 0.851004i \(0.324006\pi\)
\(30\) 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 0
\(36\) 0 0
\(37\) 0.702248 0.115449 0.0577244 0.998333i \(-0.481616\pi\)
0.0577244 + 0.998333i \(0.481616\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.39354 + 9.34189i −0.842330 + 1.45896i 0.0455900 + 0.998960i \(0.485483\pi\)
−0.887920 + 0.459998i \(0.847850\pi\)
\(42\) 0 0
\(43\) 3.73131 + 6.46283i 0.569020 + 0.985572i 0.996663 + 0.0816240i \(0.0260106\pi\)
−0.427643 + 0.903948i \(0.640656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.50285 + 6.06712i 0.510943 + 0.884980i 0.999920 + 0.0126827i \(0.00403713\pi\)
−0.488976 + 0.872297i \(0.662630\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.83712i 1.35123i −0.737254 0.675616i \(-0.763878\pi\)
0.737254 0.675616i \(-0.236122\pi\)
\(54\) 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 \(-0.493267\pi\)
0.855256 0.518206i \(-0.173400\pi\)
\(60\) 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 \(-0.706387\pi\)
0.603900 0.797060i \(-0.293613\pi\)
\(72\) 0 0
\(73\) 7.69241i 0.900328i 0.892946 + 0.450164i \(0.148634\pi\)
−0.892946 + 0.450164i \(0.851366\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 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
\(83\) −3.72399 6.45014i −0.408761 0.707995i 0.585990 0.810318i \(-0.300706\pi\)
−0.994751 + 0.102323i \(0.967372\pi\)
\(84\) 0 0
\(85\) 1.30410 2.25877i 0.141450 0.244998i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.1852 −1.18563 −0.592815 0.805339i \(-0.701984\pi\)
−0.592815 + 0.805339i \(0.701984\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.267582 0.154489i −0.0274534 0.0158502i
\(96\) 0 0
\(97\) 9.18225 5.30138i 0.932316 0.538273i 0.0447729 0.998997i \(-0.485744\pi\)
0.887543 + 0.460724i \(0.152410\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.75357 + 15.1616i 0.871013 + 1.50864i 0.860950 + 0.508690i \(0.169870\pi\)
0.0100634 + 0.999949i \(0.496797\pi\)
\(102\) 0 0
\(103\) 7.39775 + 4.27110i 0.728922 + 0.420844i 0.818028 0.575179i \(-0.195067\pi\)
−0.0891054 + 0.996022i \(0.528401\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.5019i 1.01525i 0.861577 + 0.507627i \(0.169477\pi\)
−0.861577 + 0.507627i \(0.830523\pi\)
\(108\) 0 0
\(109\) 14.2422 1.36416 0.682078 0.731279i \(-0.261076\pi\)
0.682078 + 0.731279i \(0.261076\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.3783 7.72396i −1.25852 0.726609i −0.285737 0.958308i \(-0.592238\pi\)
−0.972788 + 0.231699i \(0.925572\pi\)
\(114\) 0 0
\(115\) −3.00817 + 1.73677i −0.280513 + 0.161954i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 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 \(-0.760257\pi\)
0.957086 + 0.289803i \(0.0935899\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.33589 + 1.34863i −0.199568 + 0.115221i −0.596454 0.802647i \(-0.703424\pi\)
0.396886 + 0.917868i \(0.370091\pi\)
\(138\) 0 0
\(139\) 10.1448 + 5.85710i 0.860470 + 0.496793i 0.864170 0.503200i \(-0.167844\pi\)
−0.00369951 + 0.999993i \(0.501178\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.348578 −0.0291495
\(144\) 0 0
\(145\) 0.220786i 0.0183353i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.3055 + 9.41399i 1.33580 + 0.771224i 0.986182 0.165668i \(-0.0529781\pi\)
0.349618 + 0.936892i \(0.386311\pi\)
\(150\) 0 0
\(151\) −5.00143 8.66273i −0.407010 0.704963i 0.587543 0.809193i \(-0.300095\pi\)
−0.994553 + 0.104230i \(0.966762\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.39698 1.96125i 0.272852 0.157531i
\(156\) 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
\(161\) 0 0
\(162\) 0 0
\(163\) −8.59559 −0.673259 −0.336629 0.941637i \(-0.609287\pi\)
−0.336629 + 0.941637i \(0.609287\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.24437 3.88736i 0.173674 0.300813i −0.766027 0.642808i \(-0.777769\pi\)
0.939702 + 0.341995i \(0.111103\pi\)
\(168\) 0 0
\(169\) −5.63629 9.76234i −0.433561 0.750949i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.56072 + 6.16736i 0.270717 + 0.468895i 0.969046 0.246882i \(-0.0794060\pi\)
−0.698329 + 0.715777i \(0.746073\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.5500i 1.90970i −0.297093 0.954848i \(-0.596017\pi\)
0.297093 0.954848i \(-0.403983\pi\)
\(180\) 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 −0.590229 0.807236i \(-0.700963\pi\)
0.403972 + 0.914771i \(0.367629\pi\)
\(192\) 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 \(-0.831475\pi\)
0.863091 0.505048i \(-0.168525\pi\)
\(198\) 0 0
\(199\) 23.7052i 1.68042i 0.542262 + 0.840209i \(0.317568\pi\)
−0.542262 + 0.840209i \(0.682432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.77362 + 6.53611i 0.263561 + 0.456502i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.0585617 0.101432i −0.00405080 0.00701619i
\(210\) 0 0
\(211\) 3.04004 5.26550i 0.209285 0.362492i −0.742205 0.670173i \(-0.766220\pi\)
0.951489 + 0.307681i \(0.0995531\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.22127 0.356088
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.24313 2.44977i −0.285424 0.164790i
\(222\) 0 0
\(223\) −0.796137 + 0.459650i −0.0533133 + 0.0307804i −0.526420 0.850225i \(-0.676466\pi\)
0.473106 + 0.881005i \(0.343133\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.00297 + 8.66540i 0.332059 + 0.575143i 0.982915 0.184058i \(-0.0589234\pi\)
−0.650857 + 0.759201i \(0.725590\pi\)
\(228\) 0 0
\(229\) −2.38179 1.37513i −0.157393 0.0908710i 0.419235 0.907878i \(-0.362298\pi\)
−0.576628 + 0.817007i \(0.695632\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.41451i 0.420228i −0.977677 0.210114i \(-0.932616\pi\)
0.977677 0.210114i \(-0.0673836\pi\)
\(234\) 0 0
\(235\) 4.90158 0.319744
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.4288 + 6.59844i 0.739270 + 0.426818i 0.821804 0.569770i \(-0.192968\pi\)
−0.0825337 + 0.996588i \(0.526301\pi\)
\(240\) 0 0
\(241\) −2.20722 + 1.27434i −0.142180 + 0.0820874i −0.569402 0.822059i \(-0.692825\pi\)
0.427223 + 0.904146i \(0.359492\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 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 \(-0.297949\pi\)
0.592986 + 0.805213i \(0.297949\pi\)
\(252\) 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 \(-0.685130\pi\)
0.998318 + 0.0579725i \(0.0184636\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.79810 3.92488i 0.419189 0.242019i −0.275542 0.961289i \(-0.588857\pi\)
0.694730 + 0.719271i \(0.255524\pi\)
\(264\) 0 0
\(265\) −5.96050 3.44130i −0.366151 0.211397i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.4453 0.941719 0.470859 0.882208i \(-0.343944\pi\)
0.470859 + 0.882208i \(0.343944\pi\)
\(270\) 0 0
\(271\) 12.6411i 0.767895i 0.923355 + 0.383947i \(0.125436\pi\)
−0.923355 + 0.383947i \(0.874564\pi\)
\(272\) 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 0.987495 0.157649i \(-0.0503915\pi\)
−0.630276 + 0.776371i \(0.717058\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.75411 1.59009i 0.164297 0.0948568i −0.415597 0.909549i \(-0.636427\pi\)
0.579894 + 0.814692i \(0.303094\pi\)
\(282\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) −3.10322 −0.182542
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.42975 + 2.47639i −0.0835266 + 0.144672i −0.904762 0.425917i \(-0.859952\pi\)
0.821236 + 0.570589i \(0.193285\pi\)
\(294\) 0 0
\(295\) −4.70995 8.15788i −0.274224 0.474970i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.26254 + 5.65088i 0.188677 + 0.326799i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.95451i 0.226434i
\(306\) 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 \(-0.598881\pi\)
0.977411 0.211348i \(-0.0677852\pi\)
\(312\) 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 −0.959289 0.282426i \(-0.908861\pi\)
−0.235057 + 0.971982i \(0.575528\pi\)
\(318\) 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
\(325\) 5.92818i 0.328836i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 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
\(335\) −2.07840 3.59989i −0.113555 0.196683i
\(336\) 0 0
\(337\) 13.6580 23.6563i 0.743998 1.28864i −0.206663 0.978412i \(-0.566261\pi\)
0.950661 0.310230i \(-0.100406\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.48689 0.0805196
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.37986 3.10606i −0.288806 0.166742i 0.348597 0.937273i \(-0.386658\pi\)
−0.637403 + 0.770530i \(0.719991\pi\)
\(348\) 0 0
\(349\) −24.6529 + 14.2334i −1.31964 + 0.761896i −0.983671 0.179977i \(-0.942398\pi\)
−0.335971 + 0.941872i \(0.609064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.49346 + 2.58674i 0.0794887 + 0.137678i 0.903029 0.429579i \(-0.141338\pi\)
−0.823541 + 0.567257i \(0.808005\pi\)
\(354\) 0 0
\(355\) −8.13889 4.69899i −0.431968 0.249397i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.7123i 1.62094i −0.585784 0.810468i \(-0.699213\pi\)
0.585784 0.810468i \(-0.300787\pi\)
\(360\) 0 0
\(361\) 18.8050 0.989736
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.66098 + 2.69102i 0.243967 + 0.140854i
\(366\) 0 0
\(367\) 16.4877 9.51918i 0.860651 0.496897i −0.00357920 0.999994i \(-0.501139\pi\)
0.864230 + 0.503096i \(0.167806\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.05869 + 3.56576i −0.106595 + 0.184628i −0.914389 0.404837i \(-0.867328\pi\)
0.807794 + 0.589465i \(0.200661\pi\)
\(374\) 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 −0.106956 0.994264i \(-0.534110\pi\)
0.914536 0.404505i \(-0.132556\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.4018 10.6243i 0.933007 0.538672i 0.0452458 0.998976i \(-0.485593\pi\)
0.887761 + 0.460304i \(0.152260\pi\)
\(390\) 0 0
\(391\) −16.0278 9.25367i −0.810562 0.467978i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.977223 −0.0491694
\(396\) 0 0
\(397\) 23.8939i 1.19920i 0.800300 + 0.599599i \(0.204673\pi\)
−0.800300 + 0.599599i \(0.795327\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.0121 + 12.7087i 1.09923 + 0.634642i 0.936019 0.351948i \(-0.114481\pi\)
0.163213 + 0.986591i \(0.447814\pi\)
\(402\) 0 0
\(403\) −3.68423 6.38127i −0.183524 0.317874i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.161295 0.0931240i 0.00799512 0.00461598i
\(408\) 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
\(413\) 0 0
\(414\) 0 0
\(415\) −5.21102 −0.255799
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.04181 12.1968i 0.344015 0.595851i −0.641159 0.767408i \(-0.721546\pi\)
0.985174 + 0.171556i \(0.0548796\pi\)
\(420\) 0 0
\(421\) 8.07639 + 13.9887i 0.393619 + 0.681768i 0.992924 0.118753i \(-0.0378896\pi\)
−0.599305 + 0.800521i \(0.704556\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.40717 + 14.5617i 0.407808 + 0.706344i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.26972i 0.398338i 0.979965 + 0.199169i \(0.0638243\pi\)
−0.979965 + 0.199169i \(0.936176\pi\)
\(432\) 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 −0.975544 0.219803i \(-0.929459\pi\)
−0.297417 + 0.954748i \(0.596125\pi\)
\(444\) 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 \(0.164346\pi\)
−0.869648 + 0.493673i \(0.835654\pi\)
\(450\) 0 0
\(451\) 2.86092i 0.134715i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 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
\(461\) 8.92497 + 15.4585i 0.415677 + 0.719974i 0.995499 0.0947688i \(-0.0302112\pi\)
−0.579822 + 0.814743i \(0.696878\pi\)
\(462\) 0 0
\(463\) −6.24034 + 10.8086i −0.290013 + 0.502318i −0.973813 0.227353i \(-0.926993\pi\)
0.683799 + 0.729670i \(0.260326\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.85599 0.224708 0.112354 0.993668i \(-0.464161\pi\)
0.112354 + 0.993668i \(0.464161\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.71405 + 0.989607i 0.0788121 + 0.0455022i
\(474\) 0 0
\(475\) 1.72503 0.995945i 0.0791497 0.0456971i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.40542 + 7.63041i 0.201289 + 0.348642i 0.948944 0.315445i \(-0.102154\pi\)
−0.747655 + 0.664087i \(0.768820\pi\)
\(480\) 0 0
\(481\) −0.799318 0.461486i −0.0364458 0.0210420i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.41827i 0.336847i
\(486\) 0 0
\(487\) −9.32370 −0.422497 −0.211249 0.977432i \(-0.567753\pi\)
−0.211249 + 0.977432i \(0.567753\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.9192 + 15.5418i 1.21485 + 0.701391i 0.963811 0.266587i \(-0.0858960\pi\)
0.251034 + 0.967978i \(0.419229\pi\)
\(492\) 0 0
\(493\) 1.01877 0.588186i 0.0458830 0.0264906i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11.1694 19.3459i 0.500010 0.866043i −0.499990 0.866031i \(-0.666663\pi\)
1.00000 1.16519e-5i \(-3.70891e-6\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.2396 −0.545738 −0.272869 0.962051i \(-0.587973\pi\)
−0.272869 + 0.962051i \(0.587973\pi\)
\(504\) 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 \(-0.934569\pi\)
0.666242 + 0.745736i \(0.267902\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.17588 2.98830i 0.228077 0.131680i
\(516\) 0 0
\(517\) 1.60910 + 0.929015i 0.0707682 + 0.0408580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.63263 −0.246770 −0.123385 0.992359i \(-0.539375\pi\)
−0.123385 + 0.992359i \(0.539375\pi\)
\(522\) 0 0
\(523\) 38.3699i 1.67780i −0.544288 0.838899i \(-0.683200\pi\)
0.544288 0.838899i \(-0.316800\pi\)
\(524\) 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
\(533\) 12.2782 7.08880i 0.531826 0.307050i
\(534\) 0 0
\(535\) 6.36329 + 3.67385i 0.275109 + 0.158834i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6.43346 0.276596 0.138298 0.990391i \(-0.455837\pi\)
0.138298 + 0.990391i \(0.455837\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.98232 8.62963i 0.213419 0.369653i
\(546\) 0 0
\(547\) −6.52889 11.3084i −0.279155 0.483511i 0.692020 0.721878i \(-0.256721\pi\)
−0.971175 + 0.238368i \(0.923388\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.0696787 0.120687i −0.00296841 0.00514144i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.4920i 1.24962i 0.780778 + 0.624809i \(0.214823\pi\)
−0.780778 + 0.624809i \(0.785177\pi\)
\(558\) 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 −0.733656 0.679521i \(-0.762188\pi\)
0.955311 + 0.295604i \(0.0955209\pi\)
\(564\) 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 −0.894956 0.446155i \(-0.852793\pi\)
−0.0610967 + 0.998132i \(0.519460\pi\)
\(570\) 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
\(577\) 14.0337i 0.584229i −0.956383 0.292115i \(-0.905641\pi\)
0.956383 0.292115i \(-0.0943589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.30448 2.25943i −0.0540262 0.0935762i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.52469 + 2.64085i 0.0629308 + 0.108999i 0.895774 0.444509i \(-0.146622\pi\)
−0.832843 + 0.553509i \(0.813289\pi\)
\(588\) 0 0
\(589\) 1.23791 2.14413i 0.0510073 0.0883473i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.6082 −1.09267 −0.546334 0.837568i \(-0.683977\pi\)
−0.546334 + 0.837568i \(0.683977\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.86333 2.23050i −0.157852 0.0911356i 0.418993 0.907989i \(-0.362383\pi\)
−0.576845 + 0.816854i \(0.695716\pi\)
\(600\) 0 0
\(601\) −5.25019 + 3.03120i −0.214160 + 0.123645i −0.603243 0.797557i \(-0.706125\pi\)
0.389083 + 0.921203i \(0.372792\pi\)
\(602\) 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 0.992711 + 0.120516i \(0.0384548\pi\)
0.600725 + 0.799455i \(0.294879\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.20768i 0.372503i
\(612\) 0 0
\(613\) −33.2588 −1.34331 −0.671654 0.740865i \(-0.734416\pi\)
−0.671654 + 0.740865i \(0.734416\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.3001 18.0711i −1.26010 0.727516i −0.287002 0.957930i \(-0.592659\pi\)
−0.973093 + 0.230414i \(0.925992\pi\)
\(618\) 0 0
\(619\) −22.9031 + 13.2231i −0.920554 + 0.531482i −0.883812 0.467843i \(-0.845031\pi\)
−0.0367423 + 0.999325i \(0.511698\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 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
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −21.1444 + 12.2077i −0.835153 + 0.482176i −0.855614 0.517615i \(-0.826820\pi\)
0.0204610 + 0.999791i \(0.493487\pi\)
\(642\) 0 0
\(643\) 31.9014 + 18.4183i 1.25807 + 0.726346i 0.972699 0.232071i \(-0.0745503\pi\)
0.285370 + 0.958418i \(0.407884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26.5694 −1.04455 −0.522276 0.852777i \(-0.674917\pi\)
−0.522276 + 0.852777i \(0.674917\pi\)
\(648\) 0 0
\(649\) 3.57078i 0.140165i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.2767 + 15.1709i 1.02829 + 0.593683i 0.916494 0.400049i \(-0.131007\pi\)
0.111794 + 0.993731i \(0.464340\pi\)
\(654\) 0 0
\(655\) −1.82233 3.15637i −0.0712044 0.123330i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.9873 + 23.6640i −1.59664 + 0.921820i −0.604511 + 0.796597i \(0.706631\pi\)
−0.992129 + 0.125223i \(0.960035\pi\)
\(660\) 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 0
\(666\) 0 0
\(667\) −1.56666 −0.0606613
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.749513 1.29819i 0.0289346 0.0501162i
\(672\) 0 0
\(673\) −2.54758 4.41254i −0.0982020 0.170091i 0.812738 0.582629i \(-0.197976\pi\)
−0.910940 + 0.412538i \(0.864642\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.42072 + 14.5851i 0.323635 + 0.560551i 0.981235 0.192815i \(-0.0617618\pi\)
−0.657601 + 0.753367i \(0.728429\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.1929i 0.696132i −0.937470 0.348066i \(-0.886838\pi\)
0.937470 0.348066i \(-0.113162\pi\)
\(684\) 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 \(-0.918874\pi\)
0.967697 0.252114i \(-0.0811259\pi\)
\(702\) 0 0
\(703\) 0.310122i 0.0116965i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 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
\(713\) −13.9166 24.1043i −0.521182 0.902715i
\(714\) 0 0
\(715\) −0.121942 + 0.211210i −0.00456038 + 0.00789881i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.4069 −1.13399 −0.566994 0.823722i \(-0.691894\pi\)
−0.566994 + 0.823722i \(0.691894\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.23265 + 0.711673i 0.0457796 + 0.0264309i
\(726\) 0 0
\(727\) 11.3671 6.56280i 0.421583 0.243401i −0.274171 0.961681i \(-0.588404\pi\)
0.695754 + 0.718280i \(0.255070\pi\)
\(728\) 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 \(-0.578961\pi\)
−0.962280 + 0.272063i \(0.912294\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.57571i 0.0580419i
\(738\) 0 0
\(739\) 26.2430 0.965366 0.482683 0.875795i \(-0.339662\pi\)
0.482683 + 0.875795i \(0.339662\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.78379 + 5.07132i 0.322246 + 0.186049i 0.652393 0.757881i \(-0.273765\pi\)
−0.330147 + 0.943929i \(0.607098\pi\)
\(744\) 0 0
\(745\) 11.4082 6.58655i 0.417966 0.241313i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.95369 + 6.84798i −0.144272 + 0.249886i −0.929101 0.369826i \(-0.879417\pi\)
0.784829 + 0.619712i \(0.212751\pi\)
\(752\) 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 \(-0.868678\pi\)
0.805286 + 0.592886i \(0.202012\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.3247 + 8.84771i −0.553342 + 0.319472i
\(768\) 0 0
\(769\) 9.79863 + 5.65724i 0.353348 + 0.204005i 0.666159 0.745810i \(-0.267937\pi\)
−0.312811 + 0.949815i \(0.601271\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.4211 −1.38191 −0.690956 0.722896i \(-0.742810\pi\)
−0.690956 + 0.722896i \(0.742810\pi\)
\(774\) 0 0
\(775\) 25.2872i 0.908343i
\(776\) 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
\(785\) 0.152730 0.0881788i 0.00545117 0.00314723i
\(786\) 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
\(791\) 0 0
\(792\) 0 0
\(793\) −7.42859 −0.263797
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15.1359 + 26.2161i −0.536139 + 0.928621i 0.462968 + 0.886375i \(0.346785\pi\)
−0.999107 + 0.0422457i \(0.986549\pi\)
\(798\) 0 0
\(799\) 13.0581 + 22.6172i 0.461961 + 0.800140i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.02008 + 1.76683i 0.0359978 + 0.0623500i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.253310i 0.00890590i −0.999990 0.00445295i \(-0.998583\pi\)
0.999990 0.00445295i \(-0.00141742\pi\)
\(810\) 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 −0.0131576 0.999913i \(-0.504188\pi\)
0.859372 + 0.511352i \(0.170855\pi\)
\(822\) 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 \(0.00355124\pi\)
−0.999938 + 0.0111563i \(0.996449\pi\)
\(828\) 0 0
\(829\) 11.0526i 0.383871i −0.981408 0.191936i \(-0.938523\pi\)
0.981408 0.191936i \(-0.0614765\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.57029 2.71981i −0.0543420 0.0941230i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.62330 8.00780i −0.159614 0.276460i 0.775115 0.631820i \(-0.217692\pi\)
−0.934730 + 0.355360i \(0.884358\pi\)
\(840\) 0 0
\(841\) −14.4502 + 25.0285i −0.498283 + 0.863052i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.88693 −0.271319
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.01931 1.74320i −0.103501 0.0597561i
\(852\) 0 0
\(853\) 34.3256 19.8179i 1.17529 0.678551i 0.220366 0.975417i \(-0.429275\pi\)
0.954919 + 0.296866i \(0.0959414\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.9260 20.6565i −0.407385 0.705612i 0.587211 0.809434i \(-0.300226\pi\)
−0.994596 + 0.103822i \(0.966893\pi\)
\(858\) 0 0
\(859\) −9.62480 5.55688i −0.328394 0.189598i 0.326734 0.945116i \(-0.394052\pi\)
−0.655128 + 0.755518i \(0.727385\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 44.7770i 1.52423i −0.647444 0.762113i \(-0.724162\pi\)
0.647444 0.762113i \(-0.275838\pi\)
\(864\) 0 0
\(865\) 4.98256 0.169412
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.320805 0.185217i −0.0108826 0.00628305i
\(870\) 0 0
\(871\) −6.76244 + 3.90429i −0.229136 + 0.132292i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.84096 + 3.18863i −0.0621647 + 0.107672i −0.895433 0.445197i \(-0.853134\pi\)
0.833268 + 0.552869i \(0.186467\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 17.3992 0.586194 0.293097 0.956083i \(-0.405314\pi\)
0.293097 + 0.956083i \(0.405314\pi\)
\(882\) 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 −0.154132 0.988050i \(-0.549258\pi\)
0.932743 0.360542i \(-0.117408\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.67932 1.54691i 0.0896601 0.0517653i
\(894\) 0 0
\(895\) −15.4812 8.93810i −0.517481 0.298768i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.76915 0.0590046
\(900\) 0 0
\(901\) 36.6712i 1.22169i
\(902\) 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 0.701614 0.712557i \(-0.252463\pi\)
−0.967900 + 0.251337i \(0.919130\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −17.7833 + 10.2672i −0.589187 + 0.340167i −0.764776 0.644296i \(-0.777150\pi\)
0.175589 + 0.984464i \(0.443817\pi\)
\(912\) 0 0
\(913\) −1.71069 0.987665i −0.0566154 0.0326869i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 35.4138 1.16819 0.584097 0.811684i \(-0.301449\pi\)
0.584097 + 0.811684i \(0.301449\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.82712 + 15.2890i −0.290548 + 0.503244i
\(924\) 0 0
\(925\) 1.58374 + 2.74311i 0.0520730 + 0.0901930i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 19.8626 + 34.4030i 0.651670 + 1.12873i 0.982718 + 0.185111i \(0.0592646\pi\)
−0.331048 + 0.943614i \(0.607402\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.691740i 0.0226223i
\(936\) 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 \(-0.634219\pi\)
0.994809 0.101760i \(-0.0324474\pi\)
\(942\) 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 −0.866955 0.498386i \(-0.833926\pi\)
−0.00186277 + 0.999998i \(0.500593\pi\)
\(948\) 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 \(0.164251\pi\)
−0.869795 + 0.493413i \(0.835749\pi\)
\(954\) 0 0
\(955\) 2.07962i 0.0672950i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.215406 + 0.373095i 0.00694859 + 0.0120353i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.47686 11.2182i −0.208497 0.361128i
\(966\) 0 0
\(967\) −6.75865 + 11.7063i −0.217343 + 0.376450i −0.953995 0.299823i \(-0.903072\pi\)
0.736652 + 0.676272i \(0.236406\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33.2855 −1.06818 −0.534092 0.845427i \(-0.679346\pi\)
−0.534092 + 0.845427i \(0.679346\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38.9127 + 22.4662i 1.24493 + 0.718758i 0.970093 0.242734i \(-0.0780441\pi\)
0.274833 + 0.961492i \(0.411377\pi\)
\(978\) 0 0
\(979\) −2.56907 + 1.48325i −0.0821079 + 0.0474050i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.4474 23.2916i −0.428907 0.742888i 0.567870 0.823118i \(-0.307768\pi\)
−0.996776 + 0.0802305i \(0.974434\pi\)
\(984\) 0 0
\(985\) −8.59035 4.95964i −0.273711 0.158027i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 37.0492i 1.17810i
\(990\) 0 0
\(991\) −35.4401 −1.12579 −0.562896 0.826528i \(-0.690313\pi\)
−0.562896 + 0.826528i \(0.690313\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.3635 + 8.29275i 0.455352 + 0.262898i
\(996\) 0 0
\(997\) −28.1418 + 16.2477i −0.891259 + 0.514568i −0.874354 0.485289i \(-0.838714\pi\)
−0.0169046 + 0.999857i \(0.505381\pi\)
\(998\) 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.x.b.881.5 16
3.2 odd 2 1764.2.x.b.293.3 16
7.2 even 3 756.2.bm.a.17.4 16
7.3 odd 6 756.2.w.a.341.4 16
7.4 even 3 5292.2.w.b.1097.5 16
7.5 odd 6 5292.2.bm.a.2285.5 16
7.6 odd 2 5292.2.x.a.881.4 16
9.2 odd 6 5292.2.x.a.4409.4 16
9.7 even 3 1764.2.x.a.1469.6 16
21.2 odd 6 252.2.bm.a.185.8 yes 16
21.5 even 6 1764.2.bm.a.1697.1 16
21.11 odd 6 1764.2.w.b.509.3 16
21.17 even 6 252.2.w.a.5.6 16
21.20 even 2 1764.2.x.a.293.6 16
28.3 even 6 3024.2.ca.d.2609.4 16
28.23 odd 6 3024.2.df.d.17.4 16
63.2 odd 6 756.2.w.a.521.4 16
63.11 odd 6 5292.2.bm.a.4625.5 16
63.16 even 3 252.2.w.a.101.6 yes 16
63.20 even 6 inner 5292.2.x.b.4409.5 16
63.23 odd 6 2268.2.t.a.1781.4 16
63.25 even 3 1764.2.bm.a.1685.1 16
63.31 odd 6 2268.2.t.a.2105.4 16
63.34 odd 6 1764.2.x.b.1469.3 16
63.38 even 6 756.2.bm.a.89.4 16
63.47 even 6 5292.2.w.b.521.5 16
63.52 odd 6 252.2.bm.a.173.8 yes 16
63.58 even 3 2268.2.t.b.1781.5 16
63.59 even 6 2268.2.t.b.2105.5 16
63.61 odd 6 1764.2.w.b.1109.3 16
84.23 even 6 1008.2.df.d.689.1 16
84.59 odd 6 1008.2.ca.d.257.3 16
252.79 odd 6 1008.2.ca.d.353.3 16
252.115 even 6 1008.2.df.d.929.1 16
252.191 even 6 3024.2.ca.d.2033.4 16
252.227 odd 6 3024.2.df.d.1601.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.6 16 21.17 even 6
252.2.w.a.101.6 yes 16 63.16 even 3
252.2.bm.a.173.8 yes 16 63.52 odd 6
252.2.bm.a.185.8 yes 16 21.2 odd 6
756.2.w.a.341.4 16 7.3 odd 6
756.2.w.a.521.4 16 63.2 odd 6
756.2.bm.a.17.4 16 7.2 even 3
756.2.bm.a.89.4 16 63.38 even 6
1008.2.ca.d.257.3 16 84.59 odd 6
1008.2.ca.d.353.3 16 252.79 odd 6
1008.2.df.d.689.1 16 84.23 even 6
1008.2.df.d.929.1 16 252.115 even 6
1764.2.w.b.509.3 16 21.11 odd 6
1764.2.w.b.1109.3 16 63.61 odd 6
1764.2.x.a.293.6 16 21.20 even 2
1764.2.x.a.1469.6 16 9.7 even 3
1764.2.x.b.293.3 16 3.2 odd 2
1764.2.x.b.1469.3 16 63.34 odd 6
1764.2.bm.a.1685.1 16 63.25 even 3
1764.2.bm.a.1697.1 16 21.5 even 6
2268.2.t.a.1781.4 16 63.23 odd 6
2268.2.t.a.2105.4 16 63.31 odd 6
2268.2.t.b.1781.5 16 63.58 even 3
2268.2.t.b.2105.5 16 63.59 even 6
3024.2.ca.d.2033.4 16 252.191 even 6
3024.2.ca.d.2609.4 16 28.3 even 6
3024.2.df.d.17.4 16 28.23 odd 6
3024.2.df.d.1601.4 16 252.227 odd 6
5292.2.w.b.521.5 16 63.47 even 6
5292.2.w.b.1097.5 16 7.4 even 3
5292.2.x.a.881.4 16 7.6 odd 2
5292.2.x.a.4409.4 16 9.2 odd 6
5292.2.x.b.881.5 16 1.1 even 1 trivial
5292.2.x.b.4409.5 16 63.20 even 6 inner
5292.2.bm.a.2285.5 16 7.5 odd 6
5292.2.bm.a.4625.5 16 63.11 odd 6