Properties

Label 2736.2.dc.c.1889.1
Level $2736$
Weight $2$
Character 2736.1889
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
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] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
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: \( x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.1
Root \(0.733987 - 1.27130i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1889
Dual form 2736.2.dc.c.449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.12320 - 1.80318i) q^{5} +3.25512 q^{7} +O(q^{10})\) \(q+(-3.12320 - 1.80318i) q^{5} +3.25512 q^{7} -2.41128i q^{11} +(-3.29789 + 1.90404i) q^{13} +(5.21143 + 3.00882i) q^{17} +(1.75219 + 3.99122i) q^{19} +(1.58600 - 0.915680i) q^{23} +(4.00293 + 6.93327i) q^{25} +(-1.46797 - 2.54261i) q^{29} +1.11969i q^{31} +(-10.1664 - 5.86957i) q^{35} +10.5658i q^{37} +(0.433003 - 0.749984i) q^{41} +(0.875369 - 1.51618i) q^{43} +(5.51672 - 3.18508i) q^{47} +3.59579 q^{49} +(-2.08823 - 3.61692i) q^{53} +(-4.34798 + 7.53092i) q^{55} +(3.55621 - 6.15953i) q^{59} +(-3.59286 - 6.22302i) q^{61} +13.7333 q^{65} +(13.5936 - 7.84825i) q^{67} +(0.502227 - 0.869883i) q^{71} +(7.05594 - 12.2212i) q^{73} -7.84900i q^{77} +(2.03032 + 1.17221i) q^{79} +16.8931i q^{83} +(-10.8509 - 18.7943i) q^{85} +(2.62097 + 4.53966i) q^{89} +(-10.7350 + 6.19787i) q^{91} +(1.72445 - 15.6249i) q^{95} +(-0.737257 - 0.425655i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −3.12320 1.80318i −1.39674 0.806407i −0.402689 0.915337i \(-0.631924\pi\)
−0.994049 + 0.108930i \(0.965258\pi\)
\(6\) 0 0
\(7\) 3.25512 1.23032 0.615159 0.788403i \(-0.289092\pi\)
0.615159 + 0.788403i \(0.289092\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.41128i 0.727029i −0.931589 0.363514i \(-0.881577\pi\)
0.931589 0.363514i \(-0.118423\pi\)
\(12\) 0 0
\(13\) −3.29789 + 1.90404i −0.914671 + 0.528086i −0.881931 0.471378i \(-0.843757\pi\)
−0.0327401 + 0.999464i \(0.510423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.21143 + 3.00882i 1.26396 + 0.729747i 0.973838 0.227244i \(-0.0729714\pi\)
0.290120 + 0.956990i \(0.406305\pi\)
\(18\) 0 0
\(19\) 1.75219 + 3.99122i 0.401980 + 0.915649i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.58600 0.915680i 0.330705 0.190932i −0.325449 0.945559i \(-0.605516\pi\)
0.656154 + 0.754627i \(0.272182\pi\)
\(24\) 0 0
\(25\) 4.00293 + 6.93327i 0.800586 + 1.38665i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.46797 2.54261i −0.272596 0.472150i 0.696930 0.717139i \(-0.254549\pi\)
−0.969526 + 0.244989i \(0.921216\pi\)
\(30\) 0 0
\(31\) 1.11969i 0.201103i 0.994932 + 0.100551i \(0.0320606\pi\)
−0.994932 + 0.100551i \(0.967939\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10.1664 5.86957i −1.71843 0.992138i
\(36\) 0 0
\(37\) 10.5658i 1.73701i 0.495683 + 0.868503i \(0.334918\pi\)
−0.495683 + 0.868503i \(0.665082\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.433003 0.749984i 0.0676238 0.117128i −0.830231 0.557419i \(-0.811792\pi\)
0.897855 + 0.440292i \(0.145125\pi\)
\(42\) 0 0
\(43\) 0.875369 1.51618i 0.133493 0.231216i −0.791528 0.611133i \(-0.790714\pi\)
0.925021 + 0.379917i \(0.124047\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.51672 3.18508i 0.804696 0.464591i −0.0404148 0.999183i \(-0.512868\pi\)
0.845110 + 0.534592i \(0.179535\pi\)
\(48\) 0 0
\(49\) 3.59579 0.513684
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.08823 3.61692i −0.286841 0.496822i 0.686213 0.727400i \(-0.259272\pi\)
−0.973054 + 0.230578i \(0.925938\pi\)
\(54\) 0 0
\(55\) −4.34798 + 7.53092i −0.586281 + 1.01547i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.55621 6.15953i 0.462979 0.801902i −0.536129 0.844136i \(-0.680114\pi\)
0.999108 + 0.0422336i \(0.0134474\pi\)
\(60\) 0 0
\(61\) −3.59286 6.22302i −0.460019 0.796776i 0.538943 0.842342i \(-0.318824\pi\)
−0.998961 + 0.0455667i \(0.985491\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.7333 1.70341
\(66\) 0 0
\(67\) 13.5936 7.84825i 1.66072 0.958816i 0.688345 0.725383i \(-0.258337\pi\)
0.972373 0.233433i \(-0.0749960\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.502227 0.869883i 0.0596034 0.103236i −0.834684 0.550729i \(-0.814350\pi\)
0.894287 + 0.447493i \(0.147683\pi\)
\(72\) 0 0
\(73\) 7.05594 12.2212i 0.825835 1.43039i −0.0754440 0.997150i \(-0.524037\pi\)
0.901279 0.433239i \(-0.142629\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.84900i 0.894477i
\(78\) 0 0
\(79\) 2.03032 + 1.17221i 0.228429 + 0.131883i 0.609847 0.792519i \(-0.291231\pi\)
−0.381418 + 0.924403i \(0.624564\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.8931i 1.85426i 0.374743 + 0.927129i \(0.377731\pi\)
−0.374743 + 0.927129i \(0.622269\pi\)
\(84\) 0 0
\(85\) −10.8509 18.7943i −1.17695 2.03853i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.62097 + 4.53966i 0.277823 + 0.481203i 0.970843 0.239715i \(-0.0770539\pi\)
−0.693021 + 0.720918i \(0.743721\pi\)
\(90\) 0 0
\(91\) −10.7350 + 6.19787i −1.12534 + 0.649714i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.72445 15.6249i 0.176925 1.60308i
\(96\) 0 0
\(97\) −0.737257 0.425655i −0.0748571 0.0432188i 0.462104 0.886826i \(-0.347095\pi\)
−0.536961 + 0.843607i \(0.680428\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.87118 + 2.23503i −0.385197 + 0.222393i −0.680077 0.733141i \(-0.738054\pi\)
0.294880 + 0.955534i \(0.404720\pi\)
\(102\) 0 0
\(103\) 7.34778i 0.723998i −0.932178 0.361999i \(-0.882094\pi\)
0.932178 0.361999i \(-0.117906\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.41841 0.910512 0.455256 0.890361i \(-0.349548\pi\)
0.455256 + 0.890361i \(0.349548\pi\)
\(108\) 0 0
\(109\) 14.8524 + 8.57501i 1.42260 + 0.821337i 0.996520 0.0833495i \(-0.0265618\pi\)
0.426077 + 0.904687i \(0.359895\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.69989 −0.253984 −0.126992 0.991904i \(-0.540532\pi\)
−0.126992 + 0.991904i \(0.540532\pi\)
\(114\) 0 0
\(115\) −6.60455 −0.615877
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.9638 + 9.79407i 1.55507 + 0.897821i
\(120\) 0 0
\(121\) 5.18572 0.471429
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.8402i 0.969578i
\(126\) 0 0
\(127\) 7.65025 4.41687i 0.678850 0.391934i −0.120572 0.992705i \(-0.538473\pi\)
0.799422 + 0.600770i \(0.205139\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.3246 8.27029i −1.25154 0.722579i −0.280127 0.959963i \(-0.590377\pi\)
−0.971416 + 0.237384i \(0.923710\pi\)
\(132\) 0 0
\(133\) 5.70358 + 12.9919i 0.494563 + 1.12654i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.66978 + 1.54140i −0.228095 + 0.131691i −0.609693 0.792638i \(-0.708707\pi\)
0.381598 + 0.924328i \(0.375374\pi\)
\(138\) 0 0
\(139\) −4.27050 7.39673i −0.362219 0.627382i 0.626106 0.779738i \(-0.284648\pi\)
−0.988326 + 0.152355i \(0.951314\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.59118 + 7.95215i 0.383933 + 0.664992i
\(144\) 0 0
\(145\) 10.5881i 0.879294i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.747978 0.431845i −0.0612767 0.0353781i 0.469049 0.883172i \(-0.344597\pi\)
−0.530325 + 0.847794i \(0.677930\pi\)
\(150\) 0 0
\(151\) 1.75435i 0.142767i 0.997449 + 0.0713835i \(0.0227414\pi\)
−0.997449 + 0.0713835i \(0.977259\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.01901 3.49702i 0.162171 0.280888i
\(156\) 0 0
\(157\) 7.41445 12.8422i 0.591737 1.02492i −0.402261 0.915525i \(-0.631776\pi\)
0.993998 0.109394i \(-0.0348910\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.16263 2.98064i 0.406872 0.234908i
\(162\) 0 0
\(163\) 6.63532 0.519718 0.259859 0.965647i \(-0.416324\pi\)
0.259859 + 0.965647i \(0.416324\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.59118 7.95215i −0.355276 0.615356i 0.631889 0.775059i \(-0.282280\pi\)
−0.987165 + 0.159703i \(0.948946\pi\)
\(168\) 0 0
\(169\) 0.750738 1.30032i 0.0577491 0.100024i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.18235 15.9043i 0.698121 1.20918i −0.270996 0.962581i \(-0.587353\pi\)
0.969117 0.246601i \(-0.0793138\pi\)
\(174\) 0 0
\(175\) 13.0300 + 22.5686i 0.984975 + 1.70603i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 22.6796 1.69515 0.847577 0.530672i \(-0.178061\pi\)
0.847577 + 0.530672i \(0.178061\pi\)
\(180\) 0 0
\(181\) −4.16348 + 2.40379i −0.309469 + 0.178672i −0.646689 0.762754i \(-0.723847\pi\)
0.337220 + 0.941426i \(0.390513\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.0521 32.9991i 1.40074 2.42614i
\(186\) 0 0
\(187\) 7.25512 12.5662i 0.530547 0.918934i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.26376i 0.308515i −0.988031 0.154257i \(-0.950702\pi\)
0.988031 0.154257i \(-0.0492985\pi\)
\(192\) 0 0
\(193\) 12.4675 + 7.19809i 0.897427 + 0.518130i 0.876365 0.481648i \(-0.159962\pi\)
0.0210626 + 0.999778i \(0.493295\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.14399i 0.508988i 0.967074 + 0.254494i \(0.0819089\pi\)
−0.967074 + 0.254494i \(0.918091\pi\)
\(198\) 0 0
\(199\) −2.92983 5.07462i −0.207690 0.359730i 0.743296 0.668962i \(-0.233261\pi\)
−0.950987 + 0.309232i \(0.899928\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.77843 8.27648i −0.335380 0.580895i
\(204\) 0 0
\(205\) −2.70471 + 1.56157i −0.188905 + 0.109065i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.62395 4.22502i 0.665703 0.292251i
\(210\) 0 0
\(211\) 18.9872 + 10.9623i 1.30714 + 0.754675i 0.981617 0.190860i \(-0.0611277\pi\)
0.325519 + 0.945536i \(0.394461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.46791 + 3.15690i −0.372908 + 0.215299i
\(216\) 0 0
\(217\) 3.64473i 0.247420i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −22.9157 −1.54148
\(222\) 0 0
\(223\) 7.87215 + 4.54499i 0.527158 + 0.304355i 0.739858 0.672763i \(-0.234892\pi\)
−0.212700 + 0.977117i \(0.568226\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.58898 0.570071 0.285035 0.958517i \(-0.407995\pi\)
0.285035 + 0.958517i \(0.407995\pi\)
\(228\) 0 0
\(229\) −16.2361 −1.07291 −0.536454 0.843929i \(-0.680237\pi\)
−0.536454 + 0.843929i \(0.680237\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.27065 4.19771i −0.476316 0.275001i 0.242564 0.970135i \(-0.422012\pi\)
−0.718880 + 0.695134i \(0.755345\pi\)
\(234\) 0 0
\(235\) −22.9731 −1.49860
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.75382i 0.307499i −0.988110 0.153750i \(-0.950865\pi\)
0.988110 0.153750i \(-0.0491349\pi\)
\(240\) 0 0
\(241\) 4.80668 2.77514i 0.309625 0.178762i −0.337133 0.941457i \(-0.609457\pi\)
0.646759 + 0.762695i \(0.276124\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.2304 6.48386i −0.717482 0.414239i
\(246\) 0 0
\(247\) −13.3780 9.82638i −0.851220 0.625238i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.48174 2.58754i 0.282885 0.163324i −0.351844 0.936059i \(-0.614445\pi\)
0.634729 + 0.772735i \(0.281112\pi\)
\(252\) 0 0
\(253\) −2.20796 3.82430i −0.138813 0.240432i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.9047 + 18.8875i 0.680215 + 1.17817i 0.974915 + 0.222577i \(0.0714469\pi\)
−0.294700 + 0.955590i \(0.595220\pi\)
\(258\) 0 0
\(259\) 34.3929i 2.13707i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −17.2691 9.97032i −1.06486 0.614797i −0.138087 0.990420i \(-0.544095\pi\)
−0.926772 + 0.375624i \(0.877429\pi\)
\(264\) 0 0
\(265\) 15.0618i 0.925241i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −13.2408 + 22.9337i −0.807305 + 1.39829i 0.107419 + 0.994214i \(0.465741\pi\)
−0.914724 + 0.404079i \(0.867592\pi\)
\(270\) 0 0
\(271\) 6.92030 11.9863i 0.420378 0.728117i −0.575598 0.817733i \(-0.695231\pi\)
0.995976 + 0.0896161i \(0.0285640\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.7181 9.65219i 1.00814 0.582049i
\(276\) 0 0
\(277\) 7.98475 0.479757 0.239879 0.970803i \(-0.422892\pi\)
0.239879 + 0.970803i \(0.422892\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.23116 15.9888i −0.550685 0.953814i −0.998225 0.0595503i \(-0.981033\pi\)
0.447541 0.894264i \(-0.352300\pi\)
\(282\) 0 0
\(283\) 2.30990 4.00086i 0.137309 0.237827i −0.789168 0.614177i \(-0.789488\pi\)
0.926477 + 0.376351i \(0.122821\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.40948 2.44129i 0.0831988 0.144105i
\(288\) 0 0
\(289\) 9.60602 + 16.6381i 0.565060 + 0.978713i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.727559 −0.0425045 −0.0212522 0.999774i \(-0.506765\pi\)
−0.0212522 + 0.999774i \(0.506765\pi\)
\(294\) 0 0
\(295\) −22.2135 + 12.8250i −1.29332 + 0.746699i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.48698 + 6.03963i −0.201657 + 0.349281i
\(300\) 0 0
\(301\) 2.84943 4.93536i 0.164238 0.284469i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 25.9143i 1.48385i
\(306\) 0 0
\(307\) 16.3269 + 9.42632i 0.931824 + 0.537989i 0.887388 0.461024i \(-0.152518\pi\)
0.0444359 + 0.999012i \(0.485851\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.9565i 0.791401i 0.918380 + 0.395700i \(0.129498\pi\)
−0.918380 + 0.395700i \(0.870502\pi\)
\(312\) 0 0
\(313\) −1.03222 1.78786i −0.0583447 0.101056i 0.835378 0.549676i \(-0.185249\pi\)
−0.893722 + 0.448620i \(0.851916\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.96013 + 13.7873i 0.447085 + 0.774375i 0.998195 0.0600583i \(-0.0191287\pi\)
−0.551109 + 0.834433i \(0.685795\pi\)
\(318\) 0 0
\(319\) −6.13094 + 3.53970i −0.343267 + 0.198185i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.87745 + 26.0720i −0.160106 + 1.45068i
\(324\) 0 0
\(325\) −26.4025 15.2435i −1.46455 0.845556i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 17.9576 10.3678i 0.990032 0.571595i
\(330\) 0 0
\(331\) 24.6561i 1.35522i −0.735420 0.677612i \(-0.763015\pi\)
0.735420 0.677612i \(-0.236985\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −56.6073 −3.09279
\(336\) 0 0
\(337\) −15.6046 9.00930i −0.850035 0.490768i 0.0106275 0.999944i \(-0.496617\pi\)
−0.860663 + 0.509175i \(0.829950\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.69989 0.146207
\(342\) 0 0
\(343\) −11.0811 −0.598324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.8051 15.4760i −1.43898 0.830793i −0.441197 0.897410i \(-0.645446\pi\)
−0.997779 + 0.0666174i \(0.978779\pi\)
\(348\) 0 0
\(349\) 23.3181 1.24819 0.624094 0.781349i \(-0.285468\pi\)
0.624094 + 0.781349i \(0.285468\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.5917i 0.563738i 0.959453 + 0.281869i \(0.0909544\pi\)
−0.959453 + 0.281869i \(0.909046\pi\)
\(354\) 0 0
\(355\) −3.13711 + 1.81121i −0.166501 + 0.0961293i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.67800 5.01024i −0.458007 0.264431i 0.253199 0.967414i \(-0.418517\pi\)
−0.711206 + 0.702984i \(0.751851\pi\)
\(360\) 0 0
\(361\) −12.8597 + 13.9867i −0.676824 + 0.736144i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −44.0743 + 25.4463i −2.30695 + 1.33192i
\(366\) 0 0
\(367\) 10.6806 + 18.4993i 0.557521 + 0.965655i 0.997703 + 0.0677462i \(0.0215808\pi\)
−0.440181 + 0.897909i \(0.645086\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.79744 11.7735i −0.352905 0.611250i
\(372\) 0 0
\(373\) 2.27182i 0.117631i −0.998269 0.0588153i \(-0.981268\pi\)
0.998269 0.0588153i \(-0.0187323\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.68245 + 5.59017i 0.498672 + 0.287908i
\(378\) 0 0
\(379\) 16.1937i 0.831814i 0.909407 + 0.415907i \(0.136536\pi\)
−0.909407 + 0.415907i \(0.863464\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.8160 + 32.5903i −0.961453 + 1.66528i −0.242595 + 0.970128i \(0.577999\pi\)
−0.718858 + 0.695157i \(0.755335\pi\)
\(384\) 0 0
\(385\) −14.1532 + 24.5140i −0.721313 + 1.24935i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.15532 + 1.24438i −0.109279 + 0.0630924i −0.553643 0.832754i \(-0.686763\pi\)
0.444364 + 0.895846i \(0.353430\pi\)
\(390\) 0 0
\(391\) 11.0205 0.557329
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.22740 7.32207i −0.212703 0.368413i
\(396\) 0 0
\(397\) 9.14294 15.8360i 0.458871 0.794788i −0.540030 0.841646i \(-0.681587\pi\)
0.998902 + 0.0468572i \(0.0149206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.14886 3.72193i 0.107309 0.185864i −0.807370 0.590045i \(-0.799110\pi\)
0.914679 + 0.404181i \(0.132443\pi\)
\(402\) 0 0
\(403\) −2.13194 3.69262i −0.106199 0.183943i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.4771 1.26285
\(408\) 0 0
\(409\) −16.0826 + 9.28527i −0.795231 + 0.459127i −0.841801 0.539788i \(-0.818504\pi\)
0.0465697 + 0.998915i \(0.485171\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.5759 20.0500i 0.569611 0.986595i
\(414\) 0 0
\(415\) 30.4613 52.7605i 1.49529 2.58991i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.6142i 1.00707i 0.863976 + 0.503534i \(0.167967\pi\)
−0.863976 + 0.503534i \(0.832033\pi\)
\(420\) 0 0
\(421\) 4.45259 + 2.57071i 0.217006 + 0.125289i 0.604563 0.796557i \(-0.293348\pi\)
−0.387557 + 0.921846i \(0.626681\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 48.1764i 2.33690i
\(426\) 0 0
\(427\) −11.6952 20.2567i −0.565970 0.980288i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.83621 4.91246i −0.136615 0.236625i 0.789598 0.613624i \(-0.210289\pi\)
−0.926213 + 0.377000i \(0.876956\pi\)
\(432\) 0 0
\(433\) −26.8444 + 15.4986i −1.29006 + 0.744817i −0.978665 0.205463i \(-0.934130\pi\)
−0.311396 + 0.950280i \(0.600797\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.43366 + 4.72564i 0.307764 + 0.226058i
\(438\) 0 0
\(439\) −27.2376 15.7257i −1.29998 0.750545i −0.319581 0.947559i \(-0.603542\pi\)
−0.980401 + 0.197014i \(0.936876\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.3895 16.9681i 1.39634 0.806177i 0.402332 0.915494i \(-0.368200\pi\)
0.994007 + 0.109317i \(0.0348663\pi\)
\(444\) 0 0
\(445\) 18.9044i 0.896153i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.7937 1.64201 0.821007 0.570918i \(-0.193413\pi\)
0.821007 + 0.570918i \(0.193413\pi\)
\(450\) 0 0
\(451\) −1.80842 1.04409i −0.0851553 0.0491644i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 44.7036 2.09574
\(456\) 0 0
\(457\) −16.0849 −0.752421 −0.376210 0.926534i \(-0.622773\pi\)
−0.376210 + 0.926534i \(0.622773\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0278 10.4084i −0.839639 0.484766i 0.0175025 0.999847i \(-0.494428\pi\)
−0.857141 + 0.515081i \(0.827762\pi\)
\(462\) 0 0
\(463\) −6.27622 −0.291681 −0.145840 0.989308i \(-0.546589\pi\)
−0.145840 + 0.989308i \(0.546589\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.3684i 1.26646i 0.773964 + 0.633229i \(0.218271\pi\)
−0.773964 + 0.633229i \(0.781729\pi\)
\(468\) 0 0
\(469\) 44.2486 25.5470i 2.04321 1.17965i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.65595 2.11076i −0.168101 0.0970529i
\(474\) 0 0
\(475\) −20.6583 + 28.1250i −0.947869 + 1.29046i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.44232 + 1.41007i −0.111592 + 0.0644279i −0.554757 0.832012i \(-0.687189\pi\)
0.443165 + 0.896440i \(0.353856\pi\)
\(480\) 0 0
\(481\) −20.1177 34.8449i −0.917289 1.58879i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.53507 + 2.65882i 0.0697039 + 0.120731i
\(486\) 0 0
\(487\) 2.78930i 0.126395i 0.998001 + 0.0631976i \(0.0201298\pi\)
−0.998001 + 0.0631976i \(0.979870\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.2780 + 5.93398i 0.463838 + 0.267797i 0.713656 0.700496i \(-0.247038\pi\)
−0.249819 + 0.968293i \(0.580371\pi\)
\(492\) 0 0
\(493\) 17.6675i 0.795704i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.63481 2.83157i 0.0733312 0.127013i
\(498\) 0 0
\(499\) −0.396132 + 0.686120i −0.0177333 + 0.0307150i −0.874756 0.484564i \(-0.838978\pi\)
0.857023 + 0.515279i \(0.172312\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13.8589 8.00145i 0.617938 0.356767i −0.158128 0.987419i \(-0.550546\pi\)
0.776066 + 0.630652i \(0.217212\pi\)
\(504\) 0 0
\(505\) 16.1206 0.717359
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.75277 9.96410i −0.254987 0.441651i 0.709905 0.704298i \(-0.248738\pi\)
−0.964892 + 0.262647i \(0.915405\pi\)
\(510\) 0 0
\(511\) 22.9679 39.7816i 1.01604 1.75983i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.2494 + 22.9486i −0.583838 + 1.01124i
\(516\) 0 0
\(517\) −7.68012 13.3024i −0.337771 0.585037i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −17.3968 −0.762169 −0.381084 0.924540i \(-0.624449\pi\)
−0.381084 + 0.924540i \(0.624449\pi\)
\(522\) 0 0
\(523\) 18.1876 10.5006i 0.795289 0.459161i −0.0465320 0.998917i \(-0.514817\pi\)
0.841821 + 0.539756i \(0.181484\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.36895 + 5.83520i −0.146754 + 0.254185i
\(528\) 0 0
\(529\) −9.82306 + 17.0140i −0.427090 + 0.739741i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.29782i 0.142845i
\(534\) 0 0
\(535\) −29.4156 16.9831i −1.27175 0.734244i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.67046i 0.373463i
\(540\) 0 0
\(541\) −18.1857 31.4985i −0.781862 1.35423i −0.930856 0.365387i \(-0.880937\pi\)
0.148993 0.988838i \(-0.452397\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −30.9246 53.5630i −1.32466 2.29439i
\(546\) 0 0
\(547\) −32.3308 + 18.6662i −1.38237 + 0.798109i −0.992439 0.122736i \(-0.960833\pi\)
−0.389927 + 0.920846i \(0.627500\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.57593 10.3141i 0.322746 0.439397i
\(552\) 0 0
\(553\) 6.60893 + 3.81567i 0.281040 + 0.162259i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.04479 + 3.48996i −0.256126 + 0.147874i −0.622566 0.782567i \(-0.713910\pi\)
0.366440 + 0.930442i \(0.380576\pi\)
\(558\) 0 0
\(559\) 6.66695i 0.281982i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.3463 0.646768 0.323384 0.946268i \(-0.395179\pi\)
0.323384 + 0.946268i \(0.395179\pi\)
\(564\) 0 0
\(565\) 8.43231 + 4.86839i 0.354750 + 0.204815i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.5078 0.859732 0.429866 0.902893i \(-0.358561\pi\)
0.429866 + 0.902893i \(0.358561\pi\)
\(570\) 0 0
\(571\) 42.8357 1.79262 0.896308 0.443432i \(-0.146239\pi\)
0.896308 + 0.443432i \(0.146239\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.6973 + 7.33080i 0.529515 + 0.305715i
\(576\) 0 0
\(577\) 1.56592 0.0651903 0.0325951 0.999469i \(-0.489623\pi\)
0.0325951 + 0.999469i \(0.489623\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 54.9890i 2.28133i
\(582\) 0 0
\(583\) −8.72142 + 5.03531i −0.361204 + 0.208541i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0867 6.97825i −0.498871 0.288023i 0.229376 0.973338i \(-0.426331\pi\)
−0.728247 + 0.685315i \(0.759665\pi\)
\(588\) 0 0
\(589\) −4.46893 + 1.96191i −0.184139 + 0.0808392i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.7031 14.8397i 1.05550 0.609393i 0.131315 0.991341i \(-0.458080\pi\)
0.924184 + 0.381948i \(0.124747\pi\)
\(594\) 0 0
\(595\) −35.3210 61.1777i −1.44802 2.50804i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.56765 9.64345i −0.227488 0.394021i 0.729575 0.683901i \(-0.239718\pi\)
−0.957063 + 0.289880i \(0.906385\pi\)
\(600\) 0 0
\(601\) 42.1221i 1.71820i 0.511809 + 0.859099i \(0.328975\pi\)
−0.511809 + 0.859099i \(0.671025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −16.1961 9.35080i −0.658463 0.380164i
\(606\) 0 0
\(607\) 27.0336i 1.09726i 0.836066 + 0.548629i \(0.184850\pi\)
−0.836066 + 0.548629i \(0.815150\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.1290 + 21.0081i −0.490688 + 0.849897i
\(612\) 0 0
\(613\) 10.4631 18.1226i 0.422600 0.731964i −0.573593 0.819140i \(-0.694451\pi\)
0.996193 + 0.0871761i \(0.0277843\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.7102 10.8023i 0.753244 0.434886i −0.0736207 0.997286i \(-0.523455\pi\)
0.826865 + 0.562401i \(0.190122\pi\)
\(618\) 0 0
\(619\) −8.24636 −0.331449 −0.165725 0.986172i \(-0.552996\pi\)
−0.165725 + 0.986172i \(0.552996\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.53158 + 14.7771i 0.341811 + 0.592033i
\(624\) 0 0
\(625\) 0.467776 0.810213i 0.0187111 0.0324085i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −31.7906 + 55.0630i −1.26757 + 2.19550i
\(630\) 0 0
\(631\) −8.72187 15.1067i −0.347212 0.601389i 0.638541 0.769588i \(-0.279538\pi\)
−0.985753 + 0.168199i \(0.946205\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −31.8577 −1.26423
\(636\) 0 0
\(637\) −11.8585 + 6.84653i −0.469852 + 0.271269i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12.5620 + 21.7581i −0.496170 + 0.859392i −0.999990 0.00441635i \(-0.998594\pi\)
0.503820 + 0.863809i \(0.331928\pi\)
\(642\) 0 0
\(643\) −1.48464 + 2.57147i −0.0585485 + 0.101409i −0.893814 0.448438i \(-0.851981\pi\)
0.835266 + 0.549847i \(0.185314\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.0704751i 0.00277066i −0.999999 0.00138533i \(-0.999559\pi\)
0.999999 0.00138533i \(-0.000440965\pi\)
\(648\) 0 0
\(649\) −14.8524 8.57501i −0.583006 0.336599i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.9759i 0.781717i 0.920451 + 0.390858i \(0.127822\pi\)
−0.920451 + 0.390858i \(0.872178\pi\)
\(654\) 0 0
\(655\) 29.8257 + 51.6596i 1.16539 + 2.01851i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 18.3916 + 31.8552i 0.716435 + 1.24090i 0.962404 + 0.271624i \(0.0875606\pi\)
−0.245969 + 0.969278i \(0.579106\pi\)
\(660\) 0 0
\(661\) 0.578222 0.333836i 0.0224902 0.0129847i −0.488713 0.872445i \(-0.662533\pi\)
0.511203 + 0.859460i \(0.329200\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.61329 50.8609i 0.217674 1.97230i
\(666\) 0 0
\(667\) −4.65643 2.68839i −0.180298 0.104095i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0054 + 8.66340i −0.579279 + 0.334447i
\(672\) 0 0
\(673\) 17.3752i 0.669767i 0.942260 + 0.334883i \(0.108697\pi\)
−0.942260 + 0.334883i \(0.891303\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.63427 −0.370275 −0.185138 0.982713i \(-0.559273\pi\)
−0.185138 + 0.982713i \(0.559273\pi\)
\(678\) 0 0
\(679\) −2.39986 1.38556i −0.0920981 0.0531729i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −32.6304 −1.24857 −0.624283 0.781198i \(-0.714609\pi\)
−0.624283 + 0.781198i \(0.714609\pi\)
\(684\) 0 0
\(685\) 11.1177 0.424785
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.7735 + 7.95215i 0.524730 + 0.302953i
\(690\) 0 0
\(691\) −1.14755 −0.0436548 −0.0218274 0.999762i \(-0.506948\pi\)
−0.0218274 + 0.999762i \(0.506948\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 30.8020i 1.16839i
\(696\) 0 0
\(697\) 4.51314 2.60566i 0.170947 0.0986964i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8.40173 4.85074i −0.317329 0.183210i 0.332872 0.942972i \(-0.391982\pi\)
−0.650201 + 0.759762i \(0.725315\pi\)
\(702\) 0 0
\(703\) −42.1704 + 18.5133i −1.59049 + 0.698242i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.6011 + 7.27527i −0.473915 + 0.273615i
\(708\) 0 0
\(709\) −1.37142 2.37537i −0.0515047 0.0892087i 0.839124 0.543941i \(-0.183068\pi\)
−0.890628 + 0.454732i \(0.849735\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.02528 + 1.77583i 0.0383970 + 0.0665055i
\(714\) 0 0
\(715\) 33.1149i 1.23843i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.5543 11.2897i −0.729252 0.421034i 0.0888966 0.996041i \(-0.471666\pi\)
−0.818148 + 0.575007i \(0.804999\pi\)
\(720\) 0 0
\(721\) 23.9179i 0.890749i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 11.7524 20.3557i 0.436473 0.755993i
\(726\) 0 0
\(727\) 0.909044 1.57451i 0.0337146 0.0583953i −0.848676 0.528913i \(-0.822600\pi\)
0.882390 + 0.470518i \(0.155933\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.12386 5.26766i 0.337458 0.194831i
\(732\) 0 0
\(733\) −4.71411 −0.174120 −0.0870598 0.996203i \(-0.527747\pi\)
−0.0870598 + 0.996203i \(0.527747\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.9243 32.7779i −0.697087 1.20739i
\(738\) 0 0
\(739\) −20.3144 + 35.1856i −0.747279 + 1.29432i 0.201844 + 0.979418i \(0.435307\pi\)
−0.949123 + 0.314907i \(0.898027\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.86525 + 15.3551i −0.325234 + 0.563323i −0.981560 0.191155i \(-0.938777\pi\)
0.656325 + 0.754478i \(0.272110\pi\)
\(744\) 0 0
\(745\) 1.55739 + 2.69748i 0.0570584 + 0.0988280i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.6580 1.12022
\(750\) 0 0
\(751\) −21.6639 + 12.5076i −0.790526 + 0.456410i −0.840148 0.542358i \(-0.817532\pi\)
0.0496218 + 0.998768i \(0.484198\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.16341 5.47919i 0.115128 0.199408i
\(756\) 0 0
\(757\) 17.6740 30.6122i 0.642371 1.11262i −0.342531 0.939507i \(-0.611284\pi\)
0.984902 0.173113i \(-0.0553826\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.2068i 0.949997i 0.879987 + 0.474998i \(0.157551\pi\)
−0.879987 + 0.474998i \(0.842449\pi\)
\(762\) 0 0
\(763\) 48.3462 + 27.9127i 1.75025 + 1.01051i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.0846i 0.977969i
\(768\) 0 0
\(769\) −1.58555 2.74626i −0.0571765 0.0990327i 0.836020 0.548698i \(-0.184876\pi\)
−0.893197 + 0.449666i \(0.851543\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.9192 20.6447i −0.428705 0.742539i 0.568054 0.822992i \(-0.307697\pi\)
−0.996758 + 0.0804531i \(0.974363\pi\)
\(774\) 0 0
\(775\) −7.76313 + 4.48204i −0.278860 + 0.161000i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.75205 + 0.414098i 0.134431 + 0.0148366i
\(780\) 0 0
\(781\) −2.09753 1.21101i −0.0750556 0.0433334i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −46.3136 + 26.7392i −1.65300 + 0.954362i
\(786\) 0 0
\(787\) 18.1185i 0.645855i −0.946424 0.322928i \(-0.895333\pi\)
0.946424 0.322928i \(-0.104667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.78846 −0.312482
\(792\) 0 0
\(793\) 23.6978 + 13.6819i 0.841532 + 0.485859i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.1831 −1.21083 −0.605414 0.795911i \(-0.706992\pi\)
−0.605414 + 0.795911i \(0.706992\pi\)
\(798\) 0 0
\(799\) 38.3333 1.35614
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −29.4689 17.0139i −1.03993 0.600406i
\(804\) 0 0
\(805\) −21.4986 −0.757725
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.3496i 1.13735i 0.822562 + 0.568675i \(0.192544\pi\)
−0.822562 + 0.568675i \(0.807456\pi\)
\(810\) 0 0
\(811\) −9.70299 + 5.60202i −0.340718 + 0.196714i −0.660589 0.750747i \(-0.729694\pi\)
0.319872 + 0.947461i \(0.396360\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.7234 11.9647i −0.725911 0.419105i
\(816\) 0 0
\(817\) 7.58524 + 0.837149i 0.265374 + 0.0292881i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.549531 + 0.317272i −0.0191788 + 0.0110729i −0.509559 0.860436i \(-0.670191\pi\)
0.490380 + 0.871509i \(0.336858\pi\)
\(822\) 0 0
\(823\) 3.07938 + 5.33364i 0.107340 + 0.185919i 0.914692 0.404152i \(-0.132433\pi\)
−0.807352 + 0.590071i \(0.799100\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.740416 + 1.28244i 0.0257468 + 0.0445947i 0.878612 0.477537i \(-0.158470\pi\)
−0.852865 + 0.522132i \(0.825137\pi\)
\(828\) 0 0
\(829\) 8.23512i 0.286017i 0.989721 + 0.143009i \(0.0456777\pi\)
−0.989721 + 0.143009i \(0.954322\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.7392 + 10.8191i 0.649275 + 0.374859i
\(834\) 0 0
\(835\) 33.1149i 1.14599i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.6827 37.5556i 0.748571 1.29656i −0.199937 0.979809i \(-0.564074\pi\)
0.948508 0.316754i \(-0.102593\pi\)
\(840\) 0 0
\(841\) 10.1901 17.6498i 0.351383 0.608613i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.68942 + 2.70744i −0.161321 + 0.0931386i
\(846\) 0 0
\(847\) 16.8801 0.580008
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.67489 + 16.7574i 0.331651 + 0.574436i
\(852\) 0 0
\(853\) −25.4171 + 44.0237i −0.870264 + 1.50734i −0.00854088 + 0.999964i \(0.502719\pi\)
−0.861723 + 0.507378i \(0.830615\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.3175 + 47.3152i −0.933147 + 1.61626i −0.155242 + 0.987877i \(0.549616\pi\)
−0.777905 + 0.628381i \(0.783718\pi\)
\(858\) 0 0
\(859\) −16.9342 29.3310i −0.577789 1.00076i −0.995732 0.0922865i \(-0.970582\pi\)
0.417944 0.908473i \(-0.362751\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.1330 1.33210 0.666052 0.745905i \(-0.267983\pi\)
0.666052 + 0.745905i \(0.267983\pi\)
\(864\) 0 0
\(865\) −57.3567 + 33.1149i −1.95019 + 1.12594i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.82652 4.89567i 0.0958830 0.166074i
\(870\) 0 0
\(871\) −29.8868 + 51.7654i −1.01267 + 1.75400i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 35.2861i 1.19289i
\(876\) 0 0
\(877\) 15.3022 + 8.83476i 0.516720 + 0.298329i 0.735592 0.677425i \(-0.236904\pi\)
−0.218872 + 0.975754i \(0.570238\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29.6545i 0.999086i 0.866289 + 0.499543i \(0.166499\pi\)
−0.866289 + 0.499543i \(0.833501\pi\)
\(882\) 0 0
\(883\) −20.7052 35.8624i −0.696784 1.20687i −0.969575 0.244793i \(-0.921280\pi\)
0.272791 0.962073i \(-0.412053\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.1366 + 21.0212i 0.407507 + 0.705823i 0.994610 0.103690i \(-0.0330649\pi\)
−0.587103 + 0.809512i \(0.699732\pi\)
\(888\) 0 0
\(889\) 24.9025 14.3774i 0.835202 0.482204i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 22.3787 + 16.4376i 0.748874 + 0.550062i
\(894\) 0 0
\(895\) −70.8330 40.8955i −2.36769 1.36698i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.84694 1.64368i 0.0949506 0.0548198i
\(900\) 0 0
\(901\) 25.1325i 0.837284i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.3379 0.576330
\(906\) 0 0
\(907\) 1.80842 + 1.04409i 0.0600477 + 0.0346685i 0.529723 0.848171i \(-0.322296\pi\)
−0.469676 + 0.882839i \(0.655629\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 35.8978 1.18935 0.594673 0.803967i \(-0.297281\pi\)
0.594673 + 0.803967i \(0.297281\pi\)
\(912\) 0 0
\(913\) 40.7340 1.34810
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −46.6281 26.9208i −1.53980 0.889002i
\(918\) 0 0
\(919\) 45.9710 1.51644 0.758221 0.651998i \(-0.226069\pi\)
0.758221 + 0.651998i \(0.226069\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.82504i 0.125903i
\(924\) 0 0
\(925\) −73.2556 + 42.2941i −2.40863 + 1.39062i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6.46164 + 3.73063i 0.211999 + 0.122398i 0.602240 0.798315i \(-0.294275\pi\)
−0.390241 + 0.920713i \(0.627608\pi\)
\(930\) 0 0
\(931\) 6.30050 + 14.3516i 0.206491 + 0.470354i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −45.3184 + 26.1646i −1.48207 + 0.855674i
\(936\) 0 0
\(937\) −4.68304 8.11127i −0.152988 0.264984i 0.779336 0.626606i \(-0.215556\pi\)
−0.932325 + 0.361622i \(0.882223\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 14.6883 + 25.4410i 0.478826 + 0.829351i 0.999705 0.0242791i \(-0.00772904\pi\)
−0.520879 + 0.853631i \(0.674396\pi\)
\(942\) 0 0
\(943\) 1.58597i 0.0516463i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.7430 18.9042i −1.06401 0.614304i −0.137468 0.990506i \(-0.543896\pi\)
−0.926538 + 0.376202i \(0.877230\pi\)
\(948\) 0 0
\(949\) 53.7392i 1.74445i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 23.7307 41.1029i 0.768714 1.33145i −0.169546 0.985522i \(-0.554230\pi\)
0.938260 0.345930i \(-0.112437\pi\)
\(954\) 0 0
\(955\) −7.68833 + 13.3166i −0.248789 + 0.430914i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.69045 + 5.01743i −0.280629 + 0.162021i
\(960\) 0 0
\(961\) 29.7463 0.959558
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −25.9589 44.9622i −0.835647 1.44738i
\(966\) 0 0
\(967\) 16.3557 28.3290i 0.525965 0.910998i −0.473577 0.880752i \(-0.657038\pi\)
0.999542 0.0302461i \(-0.00962909\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.502227 0.869883i 0.0161172 0.0279159i −0.857854 0.513893i \(-0.828203\pi\)
0.873972 + 0.485977i \(0.161536\pi\)
\(972\) 0 0
\(973\) −13.9010 24.0772i −0.445645 0.771880i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.57637 0.0504327 0.0252163 0.999682i \(-0.491973\pi\)
0.0252163 + 0.999682i \(0.491973\pi\)
\(978\) 0 0
\(979\) 10.9464 6.31991i 0.349849 0.201985i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.1511 + 29.7065i −0.547034 + 0.947491i 0.451441 + 0.892301i \(0.350910\pi\)
−0.998476 + 0.0551907i \(0.982423\pi\)
\(984\) 0 0
\(985\) 12.8819 22.3121i 0.410452 0.710923i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.20623i 0.101952i
\(990\) 0 0
\(991\) 19.2588 + 11.1191i 0.611776 + 0.353209i 0.773660 0.633601i \(-0.218424\pi\)
−0.161884 + 0.986810i \(0.551757\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 21.1321i 0.669932i
\(996\) 0 0
\(997\) 11.6060 + 20.1022i 0.367566 + 0.636644i 0.989184 0.146677i \(-0.0468577\pi\)
−0.621618 + 0.783321i \(0.713524\pi\)
\(998\) 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 2736.2.dc.c.1889.1 16
3.2 odd 2 inner 2736.2.dc.c.1889.8 16
4.3 odd 2 171.2.m.a.8.3 16
12.11 even 2 171.2.m.a.8.6 yes 16
19.12 odd 6 inner 2736.2.dc.c.449.8 16
57.50 even 6 inner 2736.2.dc.c.449.1 16
76.31 even 6 171.2.m.a.107.6 yes 16
228.107 odd 6 171.2.m.a.107.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.m.a.8.3 16 4.3 odd 2
171.2.m.a.8.6 yes 16 12.11 even 2
171.2.m.a.107.3 yes 16 228.107 odd 6
171.2.m.a.107.6 yes 16 76.31 even 6
2736.2.dc.c.449.1 16 57.50 even 6 inner
2736.2.dc.c.449.8 16 19.12 odd 6 inner
2736.2.dc.c.1889.1 16 1.1 even 1 trivial
2736.2.dc.c.1889.8 16 3.2 odd 2 inner