Properties

Label 1368.2.s.j.505.2
Level $1368$
Weight $2$
Character 1368.505
Analytic conductor $10.924$
Analytic rank $0$
Dimension $6$
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] = [1368,2,Mod(505,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.s (of order \(3\), 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: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.2
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 1368.505
Dual form 1368.2.s.j.577.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.347296 - 0.601535i) q^{5} +0.305407 q^{7} +O(q^{10})\) \(q+(0.347296 - 0.601535i) q^{5} +0.305407 q^{7} -4.82295 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-3.75877 + 6.51038i) q^{17} +(3.06418 + 3.10013i) q^{19} +(0.347296 + 0.601535i) q^{23} +(2.25877 + 3.91231i) q^{25} +(5.06418 + 8.77141i) q^{29} -1.82295 q^{31} +(0.106067 - 0.183713i) q^{35} +6.51754 q^{37} +(2.69459 - 4.66717i) q^{41} +(-1.84730 + 3.19961i) q^{43} +(-3.00000 - 5.19615i) q^{47} -6.90673 q^{49} +(2.71688 + 4.70578i) q^{53} +(-1.67499 + 2.90117i) q^{55} +(-2.04189 + 3.53666i) q^{59} +(-0.194593 - 0.337044i) q^{61} -0.694593 q^{65} +(3.91147 + 6.77487i) q^{67} +(5.45336 - 9.44550i) q^{71} +(2.19459 - 3.80115i) q^{73} -1.47296 q^{77} +(-7.21688 + 12.5000i) q^{79} -0.739170 q^{83} +(2.61081 + 4.52206i) q^{85} +(-0.411474 - 0.712694i) q^{89} +(-0.152704 - 0.264490i) q^{91} +(2.92902 - 0.766546i) q^{95} +(-5.45336 + 9.44550i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77} - 27 q^{79} + 24 q^{83} + 24 q^{85} + 18 q^{89} - 3 q^{91} - 48 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.347296 0.601535i 0.155316 0.269015i −0.777858 0.628440i \(-0.783694\pi\)
0.933174 + 0.359425i \(0.117027\pi\)
\(6\) 0 0
\(7\) 0.305407 0.115433 0.0577166 0.998333i \(-0.481618\pi\)
0.0577166 + 0.998333i \(0.481618\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.82295 −1.45417 −0.727087 0.686546i \(-0.759126\pi\)
−0.727087 + 0.686546i \(0.759126\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.138675 0.240192i 0.788320 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.75877 + 6.51038i −0.911636 + 1.57900i −0.0998822 + 0.994999i \(0.531847\pi\)
−0.811754 + 0.584000i \(0.801487\pi\)
\(18\) 0 0
\(19\) 3.06418 + 3.10013i 0.702971 + 0.711219i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.347296 + 0.601535i 0.0724163 + 0.125429i 0.899960 0.435973i \(-0.143596\pi\)
−0.827544 + 0.561402i \(0.810262\pi\)
\(24\) 0 0
\(25\) 2.25877 + 3.91231i 0.451754 + 0.782461i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.06418 + 8.77141i 0.940394 + 1.62881i 0.764721 + 0.644362i \(0.222877\pi\)
0.175674 + 0.984448i \(0.443790\pi\)
\(30\) 0 0
\(31\) −1.82295 −0.327411 −0.163706 0.986509i \(-0.552345\pi\)
−0.163706 + 0.986509i \(0.552345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.106067 0.183713i 0.0179286 0.0310532i
\(36\) 0 0
\(37\) 6.51754 1.07148 0.535739 0.844384i \(-0.320033\pi\)
0.535739 + 0.844384i \(0.320033\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.69459 4.66717i 0.420825 0.728890i −0.575196 0.818016i \(-0.695074\pi\)
0.996020 + 0.0891261i \(0.0284074\pi\)
\(42\) 0 0
\(43\) −1.84730 + 3.19961i −0.281710 + 0.487936i −0.971806 0.235782i \(-0.924235\pi\)
0.690096 + 0.723718i \(0.257568\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −6.90673 −0.986675
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.71688 + 4.70578i 0.373192 + 0.646388i 0.990055 0.140683i \(-0.0449298\pi\)
−0.616862 + 0.787071i \(0.711596\pi\)
\(54\) 0 0
\(55\) −1.67499 + 2.90117i −0.225856 + 0.391194i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.04189 + 3.53666i −0.265831 + 0.460433i −0.967781 0.251793i \(-0.918980\pi\)
0.701950 + 0.712226i \(0.252313\pi\)
\(60\) 0 0
\(61\) −0.194593 0.337044i −0.0249150 0.0431541i 0.853299 0.521422i \(-0.174598\pi\)
−0.878214 + 0.478268i \(0.841265\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.694593 −0.0861536
\(66\) 0 0
\(67\) 3.91147 + 6.77487i 0.477863 + 0.827682i 0.999678 0.0253761i \(-0.00807834\pi\)
−0.521815 + 0.853058i \(0.674745\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.45336 9.44550i 0.647195 1.12097i −0.336595 0.941650i \(-0.609275\pi\)
0.983790 0.179325i \(-0.0573914\pi\)
\(72\) 0 0
\(73\) 2.19459 3.80115i 0.256858 0.444890i −0.708541 0.705670i \(-0.750646\pi\)
0.965398 + 0.260779i \(0.0839795\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.47296 −0.167860
\(78\) 0 0
\(79\) −7.21688 + 12.5000i −0.811963 + 1.40636i 0.0995259 + 0.995035i \(0.468267\pi\)
−0.911489 + 0.411326i \(0.865066\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.739170 −0.0811345 −0.0405672 0.999177i \(-0.512916\pi\)
−0.0405672 + 0.999177i \(0.512916\pi\)
\(84\) 0 0
\(85\) 2.61081 + 4.52206i 0.283183 + 0.490487i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.411474 0.712694i −0.0436162 0.0755454i 0.843393 0.537297i \(-0.180555\pi\)
−0.887009 + 0.461751i \(0.847221\pi\)
\(90\) 0 0
\(91\) −0.152704 0.264490i −0.0160077 0.0277261i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.92902 0.766546i 0.300511 0.0786459i
\(96\) 0 0
\(97\) −5.45336 + 9.44550i −0.553705 + 0.959045i 0.444298 + 0.895879i \(0.353453\pi\)
−0.998003 + 0.0631663i \(0.979880\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.75877 + 8.24243i 0.473515 + 0.820153i 0.999540 0.0303164i \(-0.00965150\pi\)
−0.526025 + 0.850469i \(0.676318\pi\)
\(102\) 0 0
\(103\) −2.30541 −0.227159 −0.113579 0.993529i \(-0.536232\pi\)
−0.113579 + 0.993529i \(0.536232\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.51754 −0.920095 −0.460048 0.887894i \(-0.652168\pi\)
−0.460048 + 0.887894i \(0.652168\pi\)
\(108\) 0 0
\(109\) −6.75877 + 11.7065i −0.647373 + 1.12128i 0.336375 + 0.941728i \(0.390799\pi\)
−0.983748 + 0.179555i \(0.942534\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 21.0797 1.98301 0.991504 0.130078i \(-0.0415228\pi\)
0.991504 + 0.130078i \(0.0415228\pi\)
\(114\) 0 0
\(115\) 0.482459 0.0449895
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.14796 + 1.98832i −0.105233 + 0.182269i
\(120\) 0 0
\(121\) 12.2608 1.11462
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.61081 0.591289
\(126\) 0 0
\(127\) −4.69459 8.13127i −0.416578 0.721534i 0.579015 0.815317i \(-0.303437\pi\)
−0.995593 + 0.0937831i \(0.970104\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.06418 + 7.03936i −0.355089 + 0.615032i −0.987133 0.159900i \(-0.948883\pi\)
0.632044 + 0.774932i \(0.282216\pi\)
\(132\) 0 0
\(133\) 0.935822 + 0.946803i 0.0811461 + 0.0820982i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.67499 4.63322i −0.228540 0.395843i 0.728836 0.684689i \(-0.240062\pi\)
−0.957376 + 0.288846i \(0.906729\pi\)
\(138\) 0 0
\(139\) −6.97565 12.0822i −0.591667 1.02480i −0.994008 0.109308i \(-0.965137\pi\)
0.402341 0.915490i \(-0.368197\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.41147 + 4.17680i 0.201658 + 0.349281i
\(144\) 0 0
\(145\) 7.03508 0.584232
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.34730 9.26179i 0.438068 0.758755i −0.559473 0.828849i \(-0.688996\pi\)
0.997540 + 0.0700934i \(0.0223297\pi\)
\(150\) 0 0
\(151\) −17.6459 −1.43600 −0.718001 0.696042i \(-0.754943\pi\)
−0.718001 + 0.696042i \(0.754943\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.633103 + 1.09657i −0.0508521 + 0.0880784i
\(156\) 0 0
\(157\) −7.95336 + 13.7756i −0.634747 + 1.09941i 0.351821 + 0.936067i \(0.385563\pi\)
−0.986569 + 0.163348i \(0.947771\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.106067 + 0.183713i 0.00835924 + 0.0144786i
\(162\) 0 0
\(163\) −2.17705 −0.170520 −0.0852599 0.996359i \(-0.527172\pi\)
−0.0852599 + 0.996359i \(0.527172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.47565 9.48411i −0.423719 0.733902i 0.572581 0.819848i \(-0.305942\pi\)
−0.996300 + 0.0859458i \(0.972609\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.06418 + 8.77141i −0.385022 + 0.666878i −0.991772 0.128015i \(-0.959140\pi\)
0.606750 + 0.794893i \(0.292473\pi\)
\(174\) 0 0
\(175\) 0.689845 + 1.19485i 0.0521474 + 0.0903219i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.69459 −0.649864 −0.324932 0.945737i \(-0.605341\pi\)
−0.324932 + 0.945737i \(0.605341\pi\)
\(180\) 0 0
\(181\) −3.93582 6.81704i −0.292547 0.506707i 0.681864 0.731479i \(-0.261170\pi\)
−0.974411 + 0.224772i \(0.927836\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.26352 3.92053i 0.166417 0.288243i
\(186\) 0 0
\(187\) 18.1284 31.3992i 1.32568 2.29614i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.8580 −1.00273 −0.501366 0.865235i \(-0.667169\pi\)
−0.501366 + 0.865235i \(0.667169\pi\)
\(192\) 0 0
\(193\) −1.04664 + 1.81283i −0.0753386 + 0.130490i −0.901233 0.433334i \(-0.857337\pi\)
0.825895 + 0.563824i \(0.190670\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.0446 −1.00063 −0.500317 0.865842i \(-0.666783\pi\)
−0.500317 + 0.865842i \(0.666783\pi\)
\(198\) 0 0
\(199\) 10.9953 + 19.0443i 0.779433 + 1.35002i 0.932269 + 0.361766i \(0.117826\pi\)
−0.152836 + 0.988252i \(0.548841\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.54664 + 2.67885i 0.108553 + 0.188019i
\(204\) 0 0
\(205\) −1.87164 3.24178i −0.130721 0.226416i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −14.7784 14.9518i −1.02224 1.03424i
\(210\) 0 0
\(211\) 7.97565 13.8142i 0.549067 0.951011i −0.449272 0.893395i \(-0.648317\pi\)
0.998339 0.0576162i \(-0.0183500\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.28312 + 2.22243i 0.0875080 + 0.151568i
\(216\) 0 0
\(217\) −0.556742 −0.0377941
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.51754 0.505685
\(222\) 0 0
\(223\) 13.2520 22.9531i 0.887417 1.53705i 0.0444991 0.999009i \(-0.485831\pi\)
0.842918 0.538042i \(-0.180836\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.65539 −0.176245 −0.0881223 0.996110i \(-0.528087\pi\)
−0.0881223 + 0.996110i \(0.528087\pi\)
\(228\) 0 0
\(229\) 21.2567 1.40468 0.702342 0.711840i \(-0.252138\pi\)
0.702342 + 0.711840i \(0.252138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.75877 11.7065i 0.442782 0.766921i −0.555113 0.831775i \(-0.687325\pi\)
0.997895 + 0.0648544i \(0.0206583\pi\)
\(234\) 0 0
\(235\) −4.16756 −0.271861
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.4688 1.58276 0.791379 0.611326i \(-0.209364\pi\)
0.791379 + 0.611326i \(0.209364\pi\)
\(240\) 0 0
\(241\) −5.93376 10.2776i −0.382227 0.662037i 0.609153 0.793053i \(-0.291510\pi\)
−0.991380 + 0.131016i \(0.958176\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.39868 + 4.15464i −0.153246 + 0.265430i
\(246\) 0 0
\(247\) 1.15270 4.20372i 0.0733448 0.267476i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.69459 + 6.39922i 0.233201 + 0.403915i 0.958748 0.284257i \(-0.0917468\pi\)
−0.725548 + 0.688172i \(0.758413\pi\)
\(252\) 0 0
\(253\) −1.67499 2.90117i −0.105306 0.182395i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.67230 + 4.62857i 0.166694 + 0.288722i 0.937255 0.348643i \(-0.113358\pi\)
−0.770562 + 0.637365i \(0.780024\pi\)
\(258\) 0 0
\(259\) 1.99050 0.123684
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0642 + 20.8958i −0.743909 + 1.28849i 0.206794 + 0.978385i \(0.433697\pi\)
−0.950703 + 0.310104i \(0.899636\pi\)
\(264\) 0 0
\(265\) 3.77425 0.231850
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.80066 + 3.11883i −0.109788 + 0.190159i −0.915684 0.401898i \(-0.868351\pi\)
0.805896 + 0.592057i \(0.201684\pi\)
\(270\) 0 0
\(271\) −9.38919 + 16.2625i −0.570352 + 0.987879i 0.426177 + 0.904640i \(0.359860\pi\)
−0.996530 + 0.0832396i \(0.973473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.8939 18.8688i −0.656929 1.13783i
\(276\) 0 0
\(277\) 20.1284 1.20940 0.604698 0.796455i \(-0.293294\pi\)
0.604698 + 0.796455i \(0.293294\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.17024 12.4192i −0.427741 0.740869i 0.568931 0.822385i \(-0.307357\pi\)
−0.996672 + 0.0815162i \(0.974024\pi\)
\(282\) 0 0
\(283\) −3.38919 + 5.87024i −0.201466 + 0.348950i −0.949001 0.315273i \(-0.897904\pi\)
0.747535 + 0.664223i \(0.231237\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.822948 1.42539i 0.0485771 0.0841380i
\(288\) 0 0
\(289\) −19.7567 34.2196i −1.16216 2.01292i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.7784 −0.980203 −0.490101 0.871665i \(-0.663040\pi\)
−0.490101 + 0.871665i \(0.663040\pi\)
\(294\) 0 0
\(295\) 1.41828 + 2.45654i 0.0825755 + 0.143025i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.347296 0.601535i 0.0200847 0.0347877i
\(300\) 0 0
\(301\) −0.564178 + 0.977185i −0.0325187 + 0.0563240i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.270325 −0.0154788
\(306\) 0 0
\(307\) 8.58172 14.8640i 0.489785 0.848332i −0.510146 0.860088i \(-0.670409\pi\)
0.999931 + 0.0117559i \(0.00374210\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.1729 0.690264 0.345132 0.938554i \(-0.387834\pi\)
0.345132 + 0.938554i \(0.387834\pi\)
\(312\) 0 0
\(313\) −1.85204 3.20783i −0.104684 0.181318i 0.808925 0.587912i \(-0.200050\pi\)
−0.913609 + 0.406594i \(0.866716\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7811 + 20.4054i 0.661690 + 1.14608i 0.980171 + 0.198152i \(0.0634939\pi\)
−0.318481 + 0.947929i \(0.603173\pi\)
\(318\) 0 0
\(319\) −24.4243 42.3041i −1.36750 2.36857i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −31.7006 + 8.29628i −1.76387 + 0.461618i
\(324\) 0 0
\(325\) 2.25877 3.91231i 0.125294 0.217016i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.916222 1.58694i −0.0505129 0.0874910i
\(330\) 0 0
\(331\) −4.30541 −0.236647 −0.118323 0.992975i \(-0.537752\pi\)
−0.118323 + 0.992975i \(0.537752\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.43376 0.296878
\(336\) 0 0
\(337\) −9.77631 + 16.9331i −0.532550 + 0.922403i 0.466728 + 0.884401i \(0.345433\pi\)
−0.999278 + 0.0380021i \(0.987901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.79199 0.476113
\(342\) 0 0
\(343\) −4.24722 −0.229328
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.8452 29.1768i 0.904300 1.56629i 0.0824452 0.996596i \(-0.473727\pi\)
0.821855 0.569697i \(-0.192940\pi\)
\(348\) 0 0
\(349\) 12.5567 0.672147 0.336073 0.941836i \(-0.390901\pi\)
0.336073 + 0.941836i \(0.390901\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.65951 0.301225 0.150613 0.988593i \(-0.451875\pi\)
0.150613 + 0.988593i \(0.451875\pi\)
\(354\) 0 0
\(355\) −3.78787 6.56078i −0.201039 0.348210i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3892 19.7266i 0.601098 1.04113i −0.391557 0.920154i \(-0.628063\pi\)
0.992655 0.120979i \(-0.0386033\pi\)
\(360\) 0 0
\(361\) −0.221629 + 18.9987i −0.0116647 + 0.999932i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.52435 2.64025i −0.0797880 0.138197i
\(366\) 0 0
\(367\) 13.5574 + 23.4821i 0.707689 + 1.22575i 0.965712 + 0.259615i \(0.0835955\pi\)
−0.258023 + 0.966139i \(0.583071\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.829755 + 1.43718i 0.0430788 + 0.0746146i
\(372\) 0 0
\(373\) −13.9418 −0.721879 −0.360940 0.932589i \(-0.617544\pi\)
−0.360940 + 0.932589i \(0.617544\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.06418 8.77141i 0.260818 0.451751i
\(378\) 0 0
\(379\) 1.08378 0.0556699 0.0278350 0.999613i \(-0.491139\pi\)
0.0278350 + 0.999613i \(0.491139\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.5398 28.6478i 0.845146 1.46384i −0.0403487 0.999186i \(-0.512847\pi\)
0.885495 0.464650i \(-0.153820\pi\)
\(384\) 0 0
\(385\) −0.511555 + 0.886039i −0.0260713 + 0.0451567i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.41147 2.44474i −0.0715646 0.123953i 0.828023 0.560695i \(-0.189466\pi\)
−0.899587 + 0.436741i \(0.856133\pi\)
\(390\) 0 0
\(391\) −5.22163 −0.264069
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.01279 + 8.68241i 0.252221 + 0.436860i
\(396\) 0 0
\(397\) 14.7567 25.5594i 0.740618 1.28279i −0.211596 0.977357i \(-0.567866\pi\)
0.952214 0.305431i \(-0.0988005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.1506 19.3135i 0.556837 0.964469i −0.440922 0.897546i \(-0.645348\pi\)
0.997758 0.0669236i \(-0.0213184\pi\)
\(402\) 0 0
\(403\) 0.911474 + 1.57872i 0.0454038 + 0.0786416i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −31.4338 −1.55811
\(408\) 0 0
\(409\) −14.8871 25.7853i −0.736121 1.27500i −0.954230 0.299075i \(-0.903322\pi\)
0.218109 0.975924i \(-0.430011\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.623608 + 1.08012i −0.0306857 + 0.0531493i
\(414\) 0 0
\(415\) −0.256711 + 0.444637i −0.0126015 + 0.0218264i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 40.1147 1.95973 0.979867 0.199653i \(-0.0639815\pi\)
0.979867 + 0.199653i \(0.0639815\pi\)
\(420\) 0 0
\(421\) −17.0155 + 29.4717i −0.829284 + 1.43636i 0.0693170 + 0.997595i \(0.477918\pi\)
−0.898601 + 0.438767i \(0.855415\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −33.9608 −1.64734
\(426\) 0 0
\(427\) −0.0594300 0.102936i −0.00287602 0.00498141i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.12836 + 8.88257i 0.247024 + 0.427858i 0.962699 0.270575i \(-0.0872140\pi\)
−0.715675 + 0.698434i \(0.753881\pi\)
\(432\) 0 0
\(433\) −15.9979 27.7092i −0.768812 1.33162i −0.938208 0.346073i \(-0.887515\pi\)
0.169396 0.985548i \(-0.445818\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.800660 + 2.91987i −0.0383007 + 0.139677i
\(438\) 0 0
\(439\) −6.05943 + 10.4952i −0.289201 + 0.500911i −0.973619 0.228179i \(-0.926723\pi\)
0.684418 + 0.729089i \(0.260056\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.45336 + 4.24935i 0.116563 + 0.201893i 0.918403 0.395645i \(-0.129479\pi\)
−0.801841 + 0.597538i \(0.796146\pi\)
\(444\) 0 0
\(445\) −0.571614 −0.0270971
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.12836 0.289215 0.144607 0.989489i \(-0.453808\pi\)
0.144607 + 0.989489i \(0.453808\pi\)
\(450\) 0 0
\(451\) −12.9959 + 22.5095i −0.611952 + 1.05993i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.212134 −0.00994498
\(456\) 0 0
\(457\) 29.9959 1.40315 0.701574 0.712597i \(-0.252481\pi\)
0.701574 + 0.712597i \(0.252481\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.53983 + 14.7914i −0.397740 + 0.688905i −0.993447 0.114297i \(-0.963539\pi\)
0.595707 + 0.803202i \(0.296872\pi\)
\(462\) 0 0
\(463\) 28.3756 1.31872 0.659362 0.751825i \(-0.270826\pi\)
0.659362 + 0.751825i \(0.270826\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.61081 −0.305912 −0.152956 0.988233i \(-0.548879\pi\)
−0.152956 + 0.988233i \(0.548879\pi\)
\(468\) 0 0
\(469\) 1.19459 + 2.06910i 0.0551612 + 0.0955419i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.90941 15.4316i 0.409655 0.709544i
\(474\) 0 0
\(475\) −5.20739 + 18.9905i −0.238931 + 0.871343i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.45336 + 9.44550i 0.249171 + 0.431576i 0.963296 0.268442i \(-0.0865087\pi\)
−0.714125 + 0.700018i \(0.753175\pi\)
\(480\) 0 0
\(481\) −3.25877 5.64436i −0.148587 0.257360i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.78787 + 6.56078i 0.171998 + 0.297910i
\(486\) 0 0
\(487\) 11.8324 0.536179 0.268090 0.963394i \(-0.413608\pi\)
0.268090 + 0.963394i \(0.413608\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.45336 + 12.9096i −0.336366 + 0.582602i −0.983746 0.179565i \(-0.942531\pi\)
0.647381 + 0.762167i \(0.275864\pi\)
\(492\) 0 0
\(493\) −76.1403 −3.42919
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.66550 2.88473i 0.0747077 0.129398i
\(498\) 0 0
\(499\) −19.9115 + 34.4877i −0.891360 + 1.54388i −0.0531141 + 0.998588i \(0.516915\pi\)
−0.838246 + 0.545292i \(0.816419\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.06418 8.77141i −0.225801 0.391098i 0.730759 0.682636i \(-0.239166\pi\)
−0.956559 + 0.291538i \(0.905833\pi\)
\(504\) 0 0
\(505\) 6.61081 0.294177
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.5175 + 21.6810i 0.554830 + 0.960994i 0.997917 + 0.0645153i \(0.0205501\pi\)
−0.443086 + 0.896479i \(0.646117\pi\)
\(510\) 0 0
\(511\) 0.670245 1.16090i 0.0296499 0.0513551i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.800660 + 1.38678i −0.0352813 + 0.0611090i
\(516\) 0 0
\(517\) 14.4688 + 25.0608i 0.636339 + 1.10217i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.9162 −1.04779 −0.523894 0.851783i \(-0.675521\pi\)
−0.523894 + 0.851783i \(0.675521\pi\)
\(522\) 0 0
\(523\) 18.4290 + 31.9200i 0.805845 + 1.39576i 0.915719 + 0.401818i \(0.131622\pi\)
−0.109875 + 0.993945i \(0.535045\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.85204 11.8681i 0.298480 0.516982i
\(528\) 0 0
\(529\) 11.2588 19.5008i 0.489512 0.847859i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.38919 −0.233432
\(534\) 0 0
\(535\) −3.30541 + 5.72513i −0.142905 + 0.247519i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 33.3108 1.43480
\(540\) 0 0
\(541\) −11.2101 19.4164i −0.481959 0.834777i 0.517827 0.855485i \(-0.326741\pi\)
−0.999786 + 0.0207084i \(0.993408\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.69459 + 8.13127i 0.201094 + 0.348305i
\(546\) 0 0
\(547\) 3.18779 + 5.52141i 0.136300 + 0.236078i 0.926093 0.377295i \(-0.123146\pi\)
−0.789793 + 0.613373i \(0.789812\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.6750 + 42.5768i −0.497371 + 1.81383i
\(552\) 0 0
\(553\) −2.20409 + 3.81759i −0.0937274 + 0.162341i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.97090 15.5381i −0.380109 0.658369i 0.610968 0.791655i \(-0.290780\pi\)
−0.991078 + 0.133286i \(0.957447\pi\)
\(558\) 0 0
\(559\) 3.69459 0.156265
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.5175 0.738276 0.369138 0.929375i \(-0.379653\pi\)
0.369138 + 0.929375i \(0.379653\pi\)
\(564\) 0 0
\(565\) 7.32089 12.6802i 0.307992 0.533458i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.6810 −0.531614 −0.265807 0.964026i \(-0.585638\pi\)
−0.265807 + 0.964026i \(0.585638\pi\)
\(570\) 0 0
\(571\) 39.0256 1.63317 0.816585 0.577225i \(-0.195865\pi\)
0.816585 + 0.577225i \(0.195865\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.56893 + 2.71746i −0.0654287 + 0.113326i
\(576\) 0 0
\(577\) 28.2959 1.17797 0.588987 0.808142i \(-0.299527\pi\)
0.588987 + 0.808142i \(0.299527\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.225748 −0.00936560
\(582\) 0 0
\(583\) −13.1034 22.6957i −0.542686 0.939961i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.8648 22.2826i 0.530989 0.919699i −0.468357 0.883539i \(-0.655154\pi\)
0.999346 0.0361602i \(-0.0115127\pi\)
\(588\) 0 0
\(589\) −5.58584 5.65138i −0.230160 0.232861i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.49525 11.2501i −0.266728 0.461987i 0.701287 0.712879i \(-0.252609\pi\)
−0.968015 + 0.250893i \(0.919276\pi\)
\(594\) 0 0
\(595\) 0.797362 + 1.38107i 0.0326886 + 0.0566184i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.18984 8.98908i −0.212051 0.367284i 0.740305 0.672271i \(-0.234681\pi\)
−0.952356 + 0.304987i \(0.901348\pi\)
\(600\) 0 0
\(601\) −7.29591 −0.297606 −0.148803 0.988867i \(-0.547542\pi\)
−0.148803 + 0.988867i \(0.547542\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.25814 7.37532i 0.173118 0.299849i
\(606\) 0 0
\(607\) 39.1147 1.58762 0.793809 0.608167i \(-0.208095\pi\)
0.793809 + 0.608167i \(0.208095\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.00000 + 5.19615i −0.121367 + 0.210214i
\(612\) 0 0
\(613\) −20.3209 + 35.1968i −0.820753 + 1.42159i 0.0843693 + 0.996435i \(0.473112\pi\)
−0.905122 + 0.425151i \(0.860221\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.38238 14.5187i −0.337462 0.584501i 0.646493 0.762920i \(-0.276235\pi\)
−0.983955 + 0.178419i \(0.942902\pi\)
\(618\) 0 0
\(619\) −14.0797 −0.565909 −0.282955 0.959133i \(-0.591315\pi\)
−0.282955 + 0.959133i \(0.591315\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.125667 0.217662i −0.00503475 0.00872044i
\(624\) 0 0
\(625\) −8.99794 + 15.5849i −0.359918 + 0.623396i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.4979 + 42.4317i −0.976797 + 1.69186i
\(630\) 0 0
\(631\) −10.4486 18.0975i −0.415953 0.720451i 0.579575 0.814919i \(-0.303219\pi\)
−0.995528 + 0.0944674i \(0.969885\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.52166 −0.258804
\(636\) 0 0
\(637\) 3.45336 + 5.98140i 0.136827 + 0.236992i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.69459 2.93512i 0.0669324 0.115930i −0.830617 0.556844i \(-0.812012\pi\)
0.897550 + 0.440914i \(0.145345\pi\)
\(642\) 0 0
\(643\) 8.82770 15.2900i 0.348130 0.602979i −0.637787 0.770213i \(-0.720150\pi\)
0.985917 + 0.167233i \(0.0534833\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.6500 0.654580 0.327290 0.944924i \(-0.393865\pi\)
0.327290 + 0.944924i \(0.393865\pi\)
\(648\) 0 0
\(649\) 9.84793 17.0571i 0.386565 0.669550i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.5877 1.39265 0.696327 0.717724i \(-0.254816\pi\)
0.696327 + 0.717724i \(0.254816\pi\)
\(654\) 0 0
\(655\) 2.82295 + 4.88949i 0.110302 + 0.191048i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.4561 + 35.4309i 0.796855 + 1.38019i 0.921655 + 0.388011i \(0.126838\pi\)
−0.124800 + 0.992182i \(0.539829\pi\)
\(660\) 0 0
\(661\) 0.645897 + 1.11873i 0.0251225 + 0.0435134i 0.878313 0.478085i \(-0.158669\pi\)
−0.853191 + 0.521599i \(0.825336\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.894543 0.234109i 0.0346889 0.00907834i
\(666\) 0 0
\(667\) −3.51754 + 6.09256i −0.136200 + 0.235905i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.938511 + 1.62555i 0.0362308 + 0.0627536i
\(672\) 0 0
\(673\) 7.99588 0.308219 0.154109 0.988054i \(-0.450749\pi\)
0.154109 + 0.988054i \(0.450749\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −46.7701 −1.79752 −0.898761 0.438439i \(-0.855532\pi\)
−0.898761 + 0.438439i \(0.855532\pi\)
\(678\) 0 0
\(679\) −1.66550 + 2.88473i −0.0639159 + 0.110706i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.69047 0.217740 0.108870 0.994056i \(-0.465277\pi\)
0.108870 + 0.994056i \(0.465277\pi\)
\(684\) 0 0
\(685\) −3.71606 −0.141983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.71688 4.70578i 0.103505 0.179276i
\(690\) 0 0
\(691\) −28.8485 −1.09745 −0.548725 0.836003i \(-0.684887\pi\)
−0.548725 + 0.836003i \(0.684887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.69047 −0.367581
\(696\) 0 0
\(697\) 20.2567 + 35.0857i 0.767278 + 1.32896i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.7716 + 39.4415i −0.860070 + 1.48969i 0.0117902 + 0.999930i \(0.496247\pi\)
−0.871860 + 0.489755i \(0.837086\pi\)
\(702\) 0 0
\(703\) 19.9709 + 20.2052i 0.753217 + 0.762055i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.45336 + 2.51730i 0.0546593 + 0.0946728i
\(708\) 0 0
\(709\) 15.1013 + 26.1563i 0.567142 + 0.982319i 0.996847 + 0.0793493i \(0.0252842\pi\)
−0.429705 + 0.902969i \(0.641382\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −0.633103 1.09657i −0.0237099 0.0410668i
\(714\) 0 0
\(715\) 3.34998 0.125282
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0770 + 31.3102i −0.674157 + 1.16767i 0.302557 + 0.953131i \(0.402160\pi\)
−0.976714 + 0.214543i \(0.931174\pi\)
\(720\) 0 0
\(721\) −0.704088 −0.0262216
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −22.8776 + 39.6252i −0.849654 + 1.47164i
\(726\) 0 0
\(727\) 3.65064 6.32310i 0.135395 0.234511i −0.790353 0.612651i \(-0.790103\pi\)
0.925748 + 0.378140i \(0.123436\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.8871 24.0532i −0.513634 0.889640i
\(732\) 0 0
\(733\) 51.4986 1.90214 0.951071 0.308972i \(-0.0999849\pi\)
0.951071 + 0.308972i \(0.0999849\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.8648 32.6749i −0.694895 1.20359i
\(738\) 0 0
\(739\) −16.3452 + 28.3108i −0.601269 + 1.04143i 0.391360 + 0.920238i \(0.372005\pi\)
−0.992629 + 0.121191i \(0.961329\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −22.7965 + 39.4848i −0.836324 + 1.44856i 0.0566239 + 0.998396i \(0.481966\pi\)
−0.892948 + 0.450160i \(0.851367\pi\)
\(744\) 0 0
\(745\) −3.71419 6.43317i −0.136078 0.235693i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.90673 −0.106209
\(750\) 0 0
\(751\) 9.15270 + 15.8529i 0.333987 + 0.578482i 0.983290 0.182048i \(-0.0582725\pi\)
−0.649303 + 0.760530i \(0.724939\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.12836 + 10.6146i −0.223034 + 0.386306i
\(756\) 0 0
\(757\) 15.7121 27.2142i 0.571067 0.989117i −0.425390 0.905010i \(-0.639863\pi\)
0.996457 0.0841071i \(-0.0268038\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.5580 −1.39773 −0.698863 0.715255i \(-0.746310\pi\)
−0.698863 + 0.715255i \(0.746310\pi\)
\(762\) 0 0
\(763\) −2.06418 + 3.57526i −0.0747283 + 0.129433i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.08378 0.147457
\(768\) 0 0
\(769\) −7.37164 12.7681i −0.265828 0.460428i 0.701952 0.712224i \(-0.252312\pi\)
−0.967780 + 0.251796i \(0.918979\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.95130 3.37976i −0.0701835 0.121561i 0.828798 0.559548i \(-0.189025\pi\)
−0.898982 + 0.437986i \(0.855692\pi\)
\(774\) 0 0
\(775\) −4.11762 7.13193i −0.147909 0.256186i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.7256 5.94745i 0.814228 0.213090i
\(780\) 0 0
\(781\) −26.3013 + 45.5552i −0.941134 + 1.63009i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.52435 + 9.56845i 0.197172 + 0.341513i
\(786\) 0 0
\(787\) −10.7879 −0.384546 −0.192273 0.981341i \(-0.561586\pi\)
−0.192273 + 0.981341i \(0.561586\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.43788 0.228905
\(792\) 0 0
\(793\) −0.194593 + 0.337044i −0.00691019 + 0.0119688i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.5526 1.71982 0.859911 0.510444i \(-0.170519\pi\)
0.859911 + 0.510444i \(0.170519\pi\)
\(798\) 0 0
\(799\) 45.1052 1.59571
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.5844 + 18.3327i −0.373516 + 0.646948i
\(804\) 0 0
\(805\) 0.147347 0.00519328
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 35.8972 1.26208 0.631040 0.775751i \(-0.282628\pi\)
0.631040 + 0.775751i \(0.282628\pi\)
\(810\) 0 0
\(811\) 3.17705 + 5.50282i 0.111561 + 0.193230i 0.916400 0.400264i \(-0.131082\pi\)
−0.804839 + 0.593494i \(0.797748\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.756082 + 1.30957i −0.0264844 + 0.0458723i
\(816\) 0 0
\(817\) −15.5797 + 4.07732i −0.545063 + 0.142647i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.93582 12.0132i −0.242062 0.419263i 0.719240 0.694762i \(-0.244490\pi\)
−0.961301 + 0.275499i \(0.911157\pi\)
\(822\) 0 0
\(823\) −2.95130 5.11181i −0.102876 0.178186i 0.809992 0.586440i \(-0.199471\pi\)
−0.912868 + 0.408254i \(0.866138\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.9709 + 20.7342i 0.416269 + 0.720999i 0.995561 0.0941211i \(-0.0300041\pi\)
−0.579292 + 0.815120i \(0.696671\pi\)
\(828\) 0 0
\(829\) 20.4593 0.710583 0.355291 0.934756i \(-0.384382\pi\)
0.355291 + 0.934756i \(0.384382\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25.9608 44.9654i 0.899488 1.55796i
\(834\) 0 0
\(835\) −7.60670 −0.263241
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −14.3628 + 24.8771i −0.495858 + 0.858852i −0.999989 0.00477600i \(-0.998480\pi\)
0.504130 + 0.863628i \(0.331813\pi\)
\(840\) 0 0
\(841\) −36.7918 + 63.7253i −1.26868 + 2.19742i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.16756 7.21842i −0.143368 0.248321i
\(846\) 0 0
\(847\) 3.74455 0.128664
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.26352 + 3.92053i 0.0775924 + 0.134394i
\(852\) 0 0
\(853\) 1.22369 2.11949i 0.0418983 0.0725700i −0.844316 0.535846i \(-0.819993\pi\)
0.886214 + 0.463276i \(0.153326\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.0351 + 17.3813i −0.342792 + 0.593733i −0.984950 0.172839i \(-0.944706\pi\)
0.642158 + 0.766572i \(0.278039\pi\)
\(858\) 0 0
\(859\) −22.5128 38.9933i −0.768127 1.33043i −0.938578 0.345068i \(-0.887856\pi\)
0.170451 0.985366i \(-0.445478\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.5918 0.632873 0.316437 0.948614i \(-0.397514\pi\)
0.316437 + 0.948614i \(0.397514\pi\)
\(864\) 0 0
\(865\) 3.51754 + 6.09256i 0.119600 + 0.207153i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.8066 60.2869i 1.18073 2.04509i
\(870\) 0 0
\(871\) 3.91147 6.77487i 0.132535 0.229558i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.01899 0.0682544
\(876\) 0 0
\(877\) −3.86959 + 6.70232i −0.130667 + 0.226321i −0.923934 0.382553i \(-0.875045\pi\)
0.793267 + 0.608874i \(0.208378\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15.0487 −0.507003 −0.253502 0.967335i \(-0.581582\pi\)
−0.253502 + 0.967335i \(0.581582\pi\)
\(882\) 0 0
\(883\) 0.281059 + 0.486809i 0.00945839 + 0.0163824i 0.870716 0.491786i \(-0.163656\pi\)
−0.861257 + 0.508169i \(0.830323\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.43107 + 2.47869i 0.0480508 + 0.0832264i 0.889050 0.457809i \(-0.151366\pi\)
−0.841000 + 0.541036i \(0.818032\pi\)
\(888\) 0 0
\(889\) −1.43376 2.48335i −0.0480869 0.0832889i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.91622 25.2223i 0.231443 0.844033i
\(894\) 0 0
\(895\) −3.01960 + 5.23010i −0.100934 + 0.174823i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9.23173 15.9898i −0.307896 0.533291i
\(900\) 0 0
\(901\) −40.8485 −1.36086
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.46759 −0.181749
\(906\) 0 0
\(907\) 17.0351 29.5056i 0.565641 0.979718i −0.431349 0.902185i \(-0.641962\pi\)
0.996990 0.0775332i \(-0.0247044\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.90261 0.195562 0.0977811 0.995208i \(-0.468826\pi\)
0.0977811 + 0.995208i \(0.468826\pi\)
\(912\) 0 0
\(913\) 3.56498 0.117984
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.24123 + 2.14987i −0.0409890 + 0.0709950i
\(918\) 0 0
\(919\) −0.394562 −0.0130154 −0.00650770 0.999979i \(-0.502071\pi\)
−0.00650770 + 0.999979i \(0.502071\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.9067 −0.358999
\(924\) 0 0
\(925\) 14.7216 + 25.4986i 0.484044 + 0.838389i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −16.8999 + 29.2715i −0.554468 + 0.960367i 0.443476 + 0.896286i \(0.353745\pi\)
−0.997945 + 0.0640813i \(0.979588\pi\)
\(930\) 0 0
\(931\) −21.1634 21.4118i −0.693604 0.701742i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.5918 21.8097i −0.411797 0.713253i
\(936\) 0 0
\(937\) −20.8601 36.1307i −0.681469 1.18034i −0.974532 0.224247i \(-0.928008\pi\)
0.293063 0.956093i \(-0.405325\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.73917 + 8.20848i 0.154493 + 0.267589i 0.932874 0.360203i \(-0.117292\pi\)
−0.778382 + 0.627791i \(0.783959\pi\)
\(942\) 0 0
\(943\) 3.74329 0.121898
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.71007 6.42604i 0.120561 0.208818i −0.799428 0.600762i \(-0.794864\pi\)
0.919989 + 0.391944i \(0.128197\pi\)
\(948\) 0 0
\(949\) −4.38919 −0.142479
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.909415 1.57515i 0.0294588 0.0510242i −0.850920 0.525295i \(-0.823955\pi\)
0.880379 + 0.474271i \(0.157288\pi\)
\(954\) 0 0
\(955\) −4.81284 + 8.33609i −0.155740 + 0.269750i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −0.816962 1.41502i −0.0263811 0.0456934i
\(960\) 0 0
\(961\) −27.6769 −0.892802
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.726986 + 1.25918i 0.0234025 + 0.0405343i
\(966\) 0 0
\(967\) −2.95067 + 5.11072i −0.0948873 + 0.164350i −0.909562 0.415569i \(-0.863582\pi\)
0.814674 + 0.579919i \(0.196916\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.6851 39.2917i 0.727999 1.26093i −0.229728 0.973255i \(-0.573784\pi\)
0.957727 0.287677i \(-0.0928830\pi\)
\(972\) 0 0
\(973\) −2.13041 3.68999i −0.0682980 0.118296i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 36.2377 1.15935 0.579674 0.814849i \(-0.303180\pi\)
0.579674 + 0.814849i \(0.303180\pi\)
\(978\) 0 0
\(979\) 1.98452 + 3.43729i 0.0634255 + 0.109856i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.32770 + 7.49579i −0.138032 + 0.239079i −0.926752 0.375675i \(-0.877411\pi\)
0.788720 + 0.614753i \(0.210744\pi\)
\(984\) 0 0
\(985\) −4.87763 + 8.44830i −0.155414 + 0.269185i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.56624 −0.0816016
\(990\) 0 0
\(991\) −29.9020 + 51.7917i −0.949868 + 1.64522i −0.204169 + 0.978936i \(0.565449\pi\)
−0.745699 + 0.666283i \(0.767884\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 15.2744 0.484232
\(996\) 0 0
\(997\) 7.52498 + 13.0336i 0.238318 + 0.412780i 0.960232 0.279204i \(-0.0900705\pi\)
−0.721913 + 0.691983i \(0.756737\pi\)
\(998\) 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 1368.2.s.j.505.2 6
3.2 odd 2 456.2.q.f.49.2 6
4.3 odd 2 2736.2.s.x.1873.2 6
12.11 even 2 912.2.q.k.49.2 6
19.7 even 3 inner 1368.2.s.j.577.2 6
57.8 even 6 8664.2.a.z.1.2 3
57.11 odd 6 8664.2.a.x.1.2 3
57.26 odd 6 456.2.q.f.121.2 yes 6
76.7 odd 6 2736.2.s.x.577.2 6
228.83 even 6 912.2.q.k.577.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.q.f.49.2 6 3.2 odd 2
456.2.q.f.121.2 yes 6 57.26 odd 6
912.2.q.k.49.2 6 12.11 even 2
912.2.q.k.577.2 6 228.83 even 6
1368.2.s.j.505.2 6 1.1 even 1 trivial
1368.2.s.j.577.2 6 19.7 even 3 inner
2736.2.s.x.577.2 6 76.7 odd 6
2736.2.s.x.1873.2 6 4.3 odd 2
8664.2.a.x.1.2 3 57.11 odd 6
8664.2.a.z.1.2 3 57.8 even 6