Properties

Label 208.2.i.a.81.1
Level $208$
Weight $2$
Character 208.81
Analytic conductor $1.661$
Analytic rank $1$
Dimension $2$
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] = [208,2,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.66088836204\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 208.81
Dual form 208.2.i.a.113.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{3} -3.00000 q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{3} -3.00000 q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-3.50000 + 0.866025i) q^{13} +(3.00000 + 5.19615i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(1.00000 - 1.73205i) q^{19} +8.00000 q^{21} +(-3.00000 - 5.19615i) q^{23} +4.00000 q^{25} -4.00000 q^{27} +(-4.50000 - 7.79423i) q^{29} -2.00000 q^{31} +(6.00000 - 10.3923i) q^{35} +(3.50000 + 6.06218i) q^{37} +(5.00000 + 5.19615i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(1.50000 - 2.59808i) q^{45} +6.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +6.00000 q^{51} +9.00000 q^{53} -4.00000 q^{57} +(-2.50000 + 4.33013i) q^{61} +(-2.00000 - 3.46410i) q^{63} +(10.5000 - 2.59808i) q^{65} +(1.00000 + 1.73205i) q^{67} +(-6.00000 + 10.3923i) q^{69} +(-3.00000 + 5.19615i) q^{71} -1.00000 q^{73} +(-4.00000 - 6.92820i) q^{75} +4.00000 q^{79} +(5.50000 + 9.52628i) q^{81} -12.0000 q^{83} +(4.50000 - 7.79423i) q^{85} +(-9.00000 + 15.5885i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(4.00000 - 13.8564i) q^{91} +(2.00000 + 3.46410i) q^{93} +(-3.00000 + 5.19615i) q^{95} +(-7.00000 + 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 6 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 6 q^{5} - 4 q^{7} - q^{9} - 7 q^{13} + 6 q^{15} - 3 q^{17} + 2 q^{19} + 16 q^{21} - 6 q^{23} + 8 q^{25} - 8 q^{27} - 9 q^{29} - 4 q^{31} + 12 q^{35} + 7 q^{37} + 10 q^{39} - 3 q^{41} - 4 q^{43} + 3 q^{45} + 12 q^{47} - 9 q^{49} + 12 q^{51} + 18 q^{53} - 8 q^{57} - 5 q^{61} - 4 q^{63} + 21 q^{65} + 2 q^{67} - 12 q^{69} - 6 q^{71} - 2 q^{73} - 8 q^{75} + 8 q^{79} + 11 q^{81} - 24 q^{83} + 9 q^{85} - 18 q^{87} - 6 q^{89} + 8 q^{91} + 4 q^{93} - 6 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −1.00000 1.73205i −0.577350 1.00000i −0.995782 0.0917517i \(-0.970753\pi\)
0.418432 0.908248i \(-0.362580\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −3.50000 + 0.866025i −0.970725 + 0.240192i
\(14\) 0 0
\(15\) 3.00000 + 5.19615i 0.774597 + 1.34164i
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) 1.00000 1.73205i 0.229416 0.397360i −0.728219 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −4.50000 7.79423i −0.835629 1.44735i −0.893517 0.449029i \(-0.851770\pi\)
0.0578882 0.998323i \(-0.481563\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 10.3923i 1.01419 1.75662i
\(36\) 0 0
\(37\) 3.50000 + 6.06218i 0.575396 + 0.996616i 0.995998 + 0.0893706i \(0.0284856\pi\)
−0.420602 + 0.907245i \(0.638181\pi\)
\(38\) 0 0
\(39\) 5.00000 + 5.19615i 0.800641 + 0.832050i
\(40\) 0 0
\(41\) −1.50000 2.59808i −0.234261 0.405751i 0.724797 0.688963i \(-0.241934\pi\)
−0.959058 + 0.283211i \(0.908600\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 1.50000 2.59808i 0.223607 0.387298i
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −4.50000 7.79423i −0.642857 1.11346i
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −2.50000 + 4.33013i −0.320092 + 0.554416i −0.980507 0.196485i \(-0.937047\pi\)
0.660415 + 0.750901i \(0.270381\pi\)
\(62\) 0 0
\(63\) −2.00000 3.46410i −0.251976 0.436436i
\(64\) 0 0
\(65\) 10.5000 2.59808i 1.30236 0.322252i
\(66\) 0 0
\(67\) 1.00000 + 1.73205i 0.122169 + 0.211604i 0.920623 0.390453i \(-0.127682\pi\)
−0.798454 + 0.602056i \(0.794348\pi\)
\(68\) 0 0
\(69\) −6.00000 + 10.3923i −0.722315 + 1.25109i
\(70\) 0 0
\(71\) −3.00000 + 5.19615i −0.356034 + 0.616670i −0.987294 0.158901i \(-0.949205\pi\)
0.631260 + 0.775571i \(0.282538\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) −4.00000 6.92820i −0.461880 0.800000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.50000 7.79423i 0.488094 0.845403i
\(86\) 0 0
\(87\) −9.00000 + 15.5885i −0.964901 + 1.67126i
\(88\) 0 0
\(89\) −3.00000 5.19615i −0.317999 0.550791i 0.662071 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) 4.00000 13.8564i 0.419314 1.45255i
\(92\) 0 0
\(93\) 2.00000 + 3.46410i 0.207390 + 0.359211i
\(94\) 0 0
\(95\) −3.00000 + 5.19615i −0.307794 + 0.533114i
\(96\) 0 0
\(97\) −7.00000 + 12.1244i −0.710742 + 1.23104i 0.253837 + 0.967247i \(0.418307\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50000 7.79423i −0.447767 0.775555i 0.550474 0.834853i \(-0.314447\pi\)
−0.998240 + 0.0592978i \(0.981114\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) −24.0000 −2.34216
\(106\) 0 0
\(107\) 3.00000 + 5.19615i 0.290021 + 0.502331i 0.973814 0.227345i \(-0.0730044\pi\)
−0.683793 + 0.729676i \(0.739671\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 7.00000 12.1244i 0.664411 1.15079i
\(112\) 0 0
\(113\) 4.50000 7.79423i 0.423324 0.733219i −0.572938 0.819599i \(-0.694196\pi\)
0.996262 + 0.0863794i \(0.0275297\pi\)
\(114\) 0 0
\(115\) 9.00000 + 15.5885i 0.839254 + 1.45363i
\(116\) 0 0
\(117\) 1.00000 3.46410i 0.0924500 0.320256i
\(118\) 0 0
\(119\) −6.00000 10.3923i −0.550019 0.952661i
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) −3.00000 + 5.19615i −0.270501 + 0.468521i
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −5.00000 8.66025i −0.443678 0.768473i 0.554281 0.832330i \(-0.312993\pi\)
−0.997959 + 0.0638564i \(0.979660\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 4.00000 + 6.92820i 0.346844 + 0.600751i
\(134\) 0 0
\(135\) 12.0000 1.03280
\(136\) 0 0
\(137\) −7.50000 + 12.9904i −0.640768 + 1.10984i 0.344493 + 0.938789i \(0.388051\pi\)
−0.985262 + 0.171054i \(0.945283\pi\)
\(138\) 0 0
\(139\) −8.00000 + 13.8564i −0.678551 + 1.17529i 0.296866 + 0.954919i \(0.404058\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) −6.00000 10.3923i −0.505291 0.875190i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 13.5000 + 23.3827i 1.12111 + 1.94183i
\(146\) 0 0
\(147\) −9.00000 + 15.5885i −0.742307 + 1.28571i
\(148\) 0 0
\(149\) −4.50000 + 7.79423i −0.368654 + 0.638528i −0.989355 0.145519i \(-0.953515\pi\)
0.620701 + 0.784047i \(0.286848\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −1.50000 2.59808i −0.121268 0.210042i
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 0 0
\(159\) −9.00000 15.5885i −0.713746 1.23625i
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) 4.00000 6.92820i 0.313304 0.542659i −0.665771 0.746156i \(-0.731897\pi\)
0.979076 + 0.203497i \(0.0652307\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 20.7846i −0.928588 1.60836i −0.785687 0.618624i \(-0.787690\pi\)
−0.142901 0.989737i \(-0.545643\pi\)
\(168\) 0 0
\(169\) 11.5000 6.06218i 0.884615 0.466321i
\(170\) 0 0
\(171\) 1.00000 + 1.73205i 0.0764719 + 0.132453i
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) −8.00000 + 13.8564i −0.604743 + 1.04745i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 + 20.7846i 0.896922 + 1.55351i 0.831408 + 0.555663i \(0.187536\pi\)
0.0655145 + 0.997852i \(0.479131\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) −10.5000 18.1865i −0.771975 1.33710i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.00000 13.8564i 0.581914 1.00791i
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) 0 0
\(193\) −5.50000 9.52628i −0.395899 0.685717i 0.597317 0.802005i \(-0.296234\pi\)
−0.993215 + 0.116289i \(0.962900\pi\)
\(194\) 0 0
\(195\) −15.0000 15.5885i −1.07417 1.11631i
\(196\) 0 0
\(197\) −3.00000 5.19615i −0.213741 0.370211i 0.739141 0.673550i \(-0.235232\pi\)
−0.952882 + 0.303340i \(0.901898\pi\)
\(198\) 0 0
\(199\) 13.0000 22.5167i 0.921546 1.59616i 0.124521 0.992217i \(-0.460261\pi\)
0.797025 0.603947i \(-0.206406\pi\)
\(200\) 0 0
\(201\) 2.00000 3.46410i 0.141069 0.244339i
\(202\) 0 0
\(203\) 36.0000 2.52670
\(204\) 0 0
\(205\) 4.50000 + 7.79423i 0.314294 + 0.544373i
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.00000 + 1.73205i 0.0688428 + 0.119239i 0.898392 0.439194i \(-0.144736\pi\)
−0.829549 + 0.558433i \(0.811403\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 6.00000 10.3923i 0.409197 0.708749i
\(216\) 0 0
\(217\) 4.00000 6.92820i 0.271538 0.470317i
\(218\) 0 0
\(219\) 1.00000 + 1.73205i 0.0675737 + 0.117041i
\(220\) 0 0
\(221\) 3.00000 10.3923i 0.201802 0.699062i
\(222\) 0 0
\(223\) 13.0000 + 22.5167i 0.870544 + 1.50783i 0.861435 + 0.507869i \(0.169566\pi\)
0.00910984 + 0.999959i \(0.497100\pi\)
\(224\) 0 0
\(225\) −2.00000 + 3.46410i −0.133333 + 0.230940i
\(226\) 0 0
\(227\) 3.00000 5.19615i 0.199117 0.344881i −0.749125 0.662428i \(-0.769526\pi\)
0.948242 + 0.317547i \(0.102859\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) −4.00000 6.92820i −0.259828 0.450035i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.0322078 0.0557856i −0.849472 0.527633i \(-0.823079\pi\)
0.881680 + 0.471848i \(0.156413\pi\)
\(242\) 0 0
\(243\) 5.00000 8.66025i 0.320750 0.555556i
\(244\) 0 0
\(245\) 13.5000 + 23.3827i 0.862483 + 1.49387i
\(246\) 0 0
\(247\) −2.00000 + 6.92820i −0.127257 + 0.440831i
\(248\) 0 0
\(249\) 12.0000 + 20.7846i 0.760469 + 1.31717i
\(250\) 0 0
\(251\) −3.00000 + 5.19615i −0.189358 + 0.327978i −0.945036 0.326965i \(-0.893974\pi\)
0.755678 + 0.654943i \(0.227307\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −18.0000 −1.12720
\(256\) 0 0
\(257\) −1.50000 2.59808i −0.0935674 0.162064i 0.815442 0.578838i \(-0.196494\pi\)
−0.909010 + 0.416775i \(0.863160\pi\)
\(258\) 0 0
\(259\) −28.0000 −1.73984
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) −27.0000 −1.65860
\(266\) 0 0
\(267\) −6.00000 + 10.3923i −0.367194 + 0.635999i
\(268\) 0 0
\(269\) −15.0000 + 25.9808i −0.914566 + 1.58408i −0.107031 + 0.994256i \(0.534134\pi\)
−0.807535 + 0.589819i \(0.799199\pi\)
\(270\) 0 0
\(271\) 4.00000 + 6.92820i 0.242983 + 0.420858i 0.961563 0.274586i \(-0.0885408\pi\)
−0.718580 + 0.695444i \(0.755208\pi\)
\(272\) 0 0
\(273\) −28.0000 + 6.92820i −1.69464 + 0.419314i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.50000 16.4545i 0.570800 0.988654i −0.425684 0.904872i \(-0.639967\pi\)
0.996484 0.0837823i \(-0.0267000\pi\)
\(278\) 0 0
\(279\) 1.00000 1.73205i 0.0598684 0.103695i
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −8.00000 13.8564i −0.475551 0.823678i 0.524057 0.851683i \(-0.324418\pi\)
−0.999608 + 0.0280052i \(0.991084\pi\)
\(284\) 0 0
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) 28.0000 1.64139
\(292\) 0 0
\(293\) −4.50000 + 7.79423i −0.262893 + 0.455344i −0.967009 0.254741i \(-0.918010\pi\)
0.704117 + 0.710084i \(0.251343\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000 + 15.5885i 0.867472 + 0.901504i
\(300\) 0 0
\(301\) −8.00000 13.8564i −0.461112 0.798670i
\(302\) 0 0
\(303\) −9.00000 + 15.5885i −0.517036 + 0.895533i
\(304\) 0 0
\(305\) 7.50000 12.9904i 0.429449 0.743827i
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) 2.00000 + 3.46410i 0.113776 + 0.197066i
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 6.00000 + 10.3923i 0.338062 + 0.585540i
\(316\) 0 0
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.00000 10.3923i 0.334887 0.580042i
\(322\) 0 0
\(323\) 3.00000 + 5.19615i 0.166924 + 0.289122i
\(324\) 0 0
\(325\) −14.0000 + 3.46410i −0.776580 + 0.192154i
\(326\) 0 0
\(327\) 10.0000 + 17.3205i 0.553001 + 0.957826i
\(328\) 0 0
\(329\) −12.0000 + 20.7846i −0.661581 + 1.14589i
\(330\) 0 0
\(331\) 16.0000 27.7128i 0.879440 1.52323i 0.0274825 0.999622i \(-0.491251\pi\)
0.851957 0.523612i \(-0.175416\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) −3.00000 5.19615i −0.163908 0.283896i
\(336\) 0 0
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 18.0000 31.1769i 0.969087 1.67851i
\(346\) 0 0
\(347\) 9.00000 15.5885i 0.483145 0.836832i −0.516667 0.856186i \(-0.672828\pi\)
0.999813 + 0.0193540i \(0.00616095\pi\)
\(348\) 0 0
\(349\) −7.00000 12.1244i −0.374701 0.649002i 0.615581 0.788074i \(-0.288921\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 14.0000 3.46410i 0.747265 0.184900i
\(352\) 0 0
\(353\) −13.5000 23.3827i −0.718532 1.24453i −0.961581 0.274521i \(-0.911481\pi\)
0.243049 0.970014i \(-0.421853\pi\)
\(354\) 0 0
\(355\) 9.00000 15.5885i 0.477670 0.827349i
\(356\) 0 0
\(357\) −12.0000 + 20.7846i −0.635107 + 1.10004i
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) −11.0000 19.0526i −0.574195 0.994535i −0.996129 0.0879086i \(-0.971982\pi\)
0.421933 0.906627i \(-0.361352\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −18.0000 + 31.1769i −0.934513 + 1.61862i
\(372\) 0 0
\(373\) 3.50000 6.06218i 0.181223 0.313888i −0.761074 0.648665i \(-0.775328\pi\)
0.942297 + 0.334777i \(0.108661\pi\)
\(374\) 0 0
\(375\) −3.00000 5.19615i −0.154919 0.268328i
\(376\) 0 0
\(377\) 22.5000 + 23.3827i 1.15881 + 1.20427i
\(378\) 0 0
\(379\) −5.00000 8.66025i −0.256833 0.444847i 0.708559 0.705652i \(-0.249346\pi\)
−0.965392 + 0.260804i \(0.916012\pi\)
\(380\) 0 0
\(381\) −10.0000 + 17.3205i −0.512316 + 0.887357i
\(382\) 0 0
\(383\) −18.0000 + 31.1769i −0.919757 + 1.59307i −0.119974 + 0.992777i \(0.538281\pi\)
−0.799783 + 0.600289i \(0.795052\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 3.46410i −0.101666 0.176090i
\(388\) 0 0
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 6.00000 + 10.3923i 0.302660 + 0.524222i
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 5.00000 8.66025i 0.250943 0.434646i −0.712843 0.701324i \(-0.752593\pi\)
0.963786 + 0.266678i \(0.0859261\pi\)
\(398\) 0 0
\(399\) 8.00000 13.8564i 0.400501 0.693688i
\(400\) 0 0
\(401\) 16.5000 + 28.5788i 0.823971 + 1.42716i 0.902703 + 0.430263i \(0.141579\pi\)
−0.0787327 + 0.996896i \(0.525087\pi\)
\(402\) 0 0
\(403\) 7.00000 1.73205i 0.348695 0.0862796i
\(404\) 0 0
\(405\) −16.5000 28.5788i −0.819892 1.42009i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.5000 21.6506i 0.618085 1.07056i −0.371750 0.928333i \(-0.621242\pi\)
0.989835 0.142222i \(-0.0454247\pi\)
\(410\) 0 0
\(411\) 30.0000 1.47979
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) 32.0000 1.56705
\(418\) 0 0
\(419\) 15.0000 + 25.9808i 0.732798 + 1.26924i 0.955683 + 0.294398i \(0.0951193\pi\)
−0.222885 + 0.974845i \(0.571547\pi\)
\(420\) 0 0
\(421\) 5.00000 0.243685 0.121843 0.992549i \(-0.461120\pi\)
0.121843 + 0.992549i \(0.461120\pi\)
\(422\) 0 0
\(423\) −3.00000 + 5.19615i −0.145865 + 0.252646i
\(424\) 0 0
\(425\) −6.00000 + 10.3923i −0.291043 + 0.504101i
\(426\) 0 0
\(427\) −10.0000 17.3205i −0.483934 0.838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 10.3923i −0.289010 0.500580i 0.684564 0.728953i \(-0.259993\pi\)
−0.973574 + 0.228373i \(0.926659\pi\)
\(432\) 0 0
\(433\) 0.500000 0.866025i 0.0240285 0.0416185i −0.853761 0.520665i \(-0.825684\pi\)
0.877790 + 0.479046i \(0.159017\pi\)
\(434\) 0 0
\(435\) 27.0000 46.7654i 1.29455 2.24223i
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 16.0000 + 27.7128i 0.763638 + 1.32266i 0.940963 + 0.338508i \(0.109922\pi\)
−0.177325 + 0.984152i \(0.556744\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 9.00000 + 15.5885i 0.426641 + 0.738964i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −15.0000 + 25.9808i −0.707894 + 1.22611i 0.257743 + 0.966213i \(0.417021\pi\)
−0.965637 + 0.259895i \(0.916312\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −10.0000 17.3205i −0.469841 0.813788i
\(454\) 0 0
\(455\) −12.0000 + 41.5692i −0.562569 + 1.94880i
\(456\) 0 0
\(457\) −11.5000 19.9186i −0.537947 0.931752i −0.999014 0.0443868i \(-0.985867\pi\)
0.461067 0.887365i \(-0.347467\pi\)
\(458\) 0 0
\(459\) 6.00000 10.3923i 0.280056 0.485071i
\(460\) 0 0
\(461\) 19.5000 33.7750i 0.908206 1.57306i 0.0916500 0.995791i \(-0.470786\pi\)
0.816556 0.577267i \(-0.195881\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) −6.00000 10.3923i −0.278243 0.481932i
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 7.00000 + 12.1244i 0.322543 + 0.558661i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 6.92820i 0.183533 0.317888i
\(476\) 0 0
\(477\) −4.50000 + 7.79423i −0.206041 + 0.356873i
\(478\) 0 0
\(479\) 9.00000 + 15.5885i 0.411220 + 0.712255i 0.995023 0.0996406i \(-0.0317693\pi\)
−0.583803 + 0.811895i \(0.698436\pi\)
\(480\) 0 0
\(481\) −17.5000 18.1865i −0.797931 0.829235i
\(482\) 0 0
\(483\) −24.0000 41.5692i −1.09204 1.89146i
\(484\) 0 0
\(485\) 21.0000 36.3731i 0.953561 1.65162i
\(486\) 0 0
\(487\) −8.00000 + 13.8564i −0.362515 + 0.627894i −0.988374 0.152042i \(-0.951415\pi\)
0.625859 + 0.779936i \(0.284748\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −12.0000 20.7846i −0.541552 0.937996i −0.998815 0.0486647i \(-0.984503\pi\)
0.457263 0.889332i \(-0.348830\pi\)
\(492\) 0 0
\(493\) 27.0000 1.21602
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 20.7846i −0.538274 0.932317i
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) −24.0000 + 41.5692i −1.07224 + 1.85718i
\(502\) 0 0
\(503\) 6.00000 10.3923i 0.267527 0.463370i −0.700696 0.713460i \(-0.747127\pi\)
0.968223 + 0.250090i \(0.0804603\pi\)
\(504\) 0 0
\(505\) 13.5000 + 23.3827i 0.600742 + 1.04052i
\(506\) 0 0
\(507\) −22.0000 13.8564i −0.977054 0.615385i
\(508\) 0 0
\(509\) 7.50000 + 12.9904i 0.332432 + 0.575789i 0.982988 0.183669i \(-0.0587976\pi\)
−0.650556 + 0.759458i \(0.725464\pi\)
\(510\) 0 0
\(511\) 2.00000 3.46410i 0.0884748 0.153243i
\(512\) 0 0
\(513\) −4.00000 + 6.92820i −0.176604 + 0.305888i
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) 10.0000 + 17.3205i 0.437269 + 0.757373i 0.997478 0.0709788i \(-0.0226123\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(524\) 0 0
\(525\) 32.0000 1.39659
\(526\) 0 0
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.50000 + 7.79423i 0.324861 + 0.337606i
\(534\) 0 0
\(535\) −9.00000 15.5885i −0.389104 0.673948i
\(536\) 0 0
\(537\) 24.0000 41.5692i 1.03568 1.79384i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19.0000 −0.816874 −0.408437 0.912787i \(-0.633926\pi\)
−0.408437 + 0.912787i \(0.633926\pi\)
\(542\) 0 0
\(543\) 7.00000 + 12.1244i 0.300399 + 0.520306i
\(544\) 0 0
\(545\) 30.0000 1.28506
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) −2.50000 4.33013i −0.106697 0.184805i
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −8.00000 + 13.8564i −0.340195 + 0.589234i
\(554\) 0 0
\(555\) −21.0000 + 36.3731i −0.891400 + 1.54395i
\(556\) 0 0
\(557\) 1.50000 + 2.59808i 0.0635570 + 0.110084i 0.896053 0.443947i \(-0.146422\pi\)
−0.832496 + 0.554031i \(0.813089\pi\)
\(558\) 0 0
\(559\) 4.00000 13.8564i 0.169182 0.586064i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000 31.1769i 0.758610 1.31395i −0.184950 0.982748i \(-0.559212\pi\)
0.943560 0.331202i \(-0.107454\pi\)
\(564\) 0 0
\(565\) −13.5000 + 23.3827i −0.567949 + 0.983717i
\(566\) 0 0
\(567\) −44.0000 −1.84783
\(568\) 0 0
\(569\) 9.00000 + 15.5885i 0.377300 + 0.653502i 0.990668 0.136295i \(-0.0435194\pi\)
−0.613369 + 0.789797i \(0.710186\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −12.0000 20.7846i −0.500435 0.866778i
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) −11.0000 + 19.0526i −0.457144 + 0.791797i
\(580\) 0 0
\(581\) 24.0000 41.5692i 0.995688 1.72458i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.00000 + 10.3923i −0.124035 + 0.429669i
\(586\) 0 0
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) 0 0
\(589\) −2.00000 + 3.46410i −0.0824086 + 0.142736i
\(590\) 0 0
\(591\) −6.00000 + 10.3923i −0.246807 + 0.427482i
\(592\) 0 0
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) 18.0000 + 31.1769i 0.737928 + 1.27813i
\(596\) 0 0
\(597\) −52.0000 −2.12822
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.0203954 + 0.0353259i 0.876043 0.482233i \(-0.160174\pi\)
−0.855648 + 0.517559i \(0.826841\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) −16.5000 + 28.5788i −0.670820 + 1.16190i
\(606\) 0 0
\(607\) −11.0000 + 19.0526i −0.446476 + 0.773320i −0.998154 0.0607380i \(-0.980655\pi\)
0.551678 + 0.834058i \(0.313988\pi\)
\(608\) 0 0
\(609\) −36.0000 62.3538i −1.45879 2.52670i
\(610\) 0 0
\(611\) −21.0000 + 5.19615i −0.849569 + 0.210214i
\(612\) 0 0
\(613\) 9.50000 + 16.4545i 0.383701 + 0.664590i 0.991588 0.129433i \(-0.0413159\pi\)
−0.607887 + 0.794024i \(0.707983\pi\)
\(614\) 0 0
\(615\) 9.00000 15.5885i 0.362915 0.628587i
\(616\) 0 0
\(617\) −13.5000 + 23.3827i −0.543490 + 0.941351i 0.455211 + 0.890384i \(0.349564\pi\)
−0.998700 + 0.0509678i \(0.983769\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 12.0000 + 20.7846i 0.481543 + 0.834058i
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −21.0000 −0.837325
\(630\) 0 0
\(631\) −14.0000 + 24.2487i −0.557331 + 0.965326i 0.440387 + 0.897808i \(0.354841\pi\)
−0.997718 + 0.0675178i \(0.978492\pi\)
\(632\) 0 0
\(633\) 2.00000 3.46410i 0.0794929 0.137686i
\(634\) 0 0
\(635\) 15.0000 + 25.9808i 0.595257 + 1.03102i
\(636\) 0 0
\(637\) 22.5000 + 23.3827i 0.891482 + 0.926456i
\(638\) 0 0
\(639\) −3.00000 5.19615i −0.118678 0.205557i
\(640\) 0 0
\(641\) −7.50000 + 12.9904i −0.296232 + 0.513089i −0.975271 0.221013i \(-0.929064\pi\)
0.679039 + 0.734103i \(0.262397\pi\)
\(642\) 0 0
\(643\) −8.00000 + 13.8564i −0.315489 + 0.546443i −0.979541 0.201243i \(-0.935502\pi\)
0.664052 + 0.747686i \(0.268835\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) 3.00000 + 5.19615i 0.117942 + 0.204282i 0.918952 0.394369i \(-0.129037\pi\)
−0.801010 + 0.598651i \(0.795704\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) 21.0000 + 36.3731i 0.821794 + 1.42339i 0.904345 + 0.426801i \(0.140360\pi\)
−0.0825519 + 0.996587i \(0.526307\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) 0.500000 0.866025i 0.0195069 0.0337869i
\(658\) 0 0
\(659\) −6.00000 + 10.3923i −0.233727 + 0.404827i −0.958902 0.283738i \(-0.908425\pi\)
0.725175 + 0.688565i \(0.241759\pi\)
\(660\) 0 0
\(661\) 3.50000 + 6.06218i 0.136134 + 0.235791i 0.926030 0.377450i \(-0.123199\pi\)
−0.789896 + 0.613241i \(0.789865\pi\)
\(662\) 0 0
\(663\) −21.0000 + 5.19615i −0.815572 + 0.201802i
\(664\) 0 0
\(665\) −12.0000 20.7846i −0.465340 0.805993i
\(666\) 0 0
\(667\) −27.0000 + 46.7654i −1.04544 + 1.81076i
\(668\) 0 0
\(669\) 26.0000 45.0333i 1.00522 1.74109i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 6.50000 + 11.2583i 0.250557 + 0.433977i 0.963679 0.267063i \(-0.0860531\pi\)
−0.713123 + 0.701039i \(0.752720\pi\)
\(674\) 0 0
\(675\) −16.0000 −0.615840
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −28.0000 48.4974i −1.07454 1.86116i
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −3.00000 + 5.19615i −0.114792 + 0.198825i −0.917697 0.397282i \(-0.869953\pi\)
0.802905 + 0.596107i \(0.203287\pi\)
\(684\) 0 0
\(685\) 22.5000 38.9711i 0.859681 1.48901i
\(686\) 0 0
\(687\) −14.0000 24.2487i −0.534133 0.925146i
\(688\) 0 0
\(689\) −31.5000 + 7.79423i −1.20005 + 0.296936i
\(690\) 0 0
\(691\) −20.0000 34.6410i −0.760836 1.31781i −0.942420 0.334431i \(-0.891456\pi\)
0.181584 0.983375i \(-0.441877\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 41.5692i 0.910372 1.57681i
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) 18.0000 + 31.1769i 0.680823 + 1.17922i
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) 18.0000 + 31.1769i 0.677919 + 1.17419i
\(706\) 0 0
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) 15.5000 26.8468i 0.582115 1.00825i −0.413114 0.910679i \(-0.635559\pi\)
0.995228 0.0975728i \(-0.0311079\pi\)
\(710\) 0 0
\(711\) −2.00000 + 3.46410i −0.0750059 + 0.129914i
\(712\) 0 0
\(713\) 6.00000 + 10.3923i 0.224702 + 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 + 41.5692i 0.896296 + 1.55243i
\(718\) 0 0
\(719\) 6.00000 10.3923i 0.223762 0.387568i −0.732185 0.681106i \(-0.761499\pi\)
0.955947 + 0.293538i \(0.0948328\pi\)
\(720\) 0 0
\(721\) 4.00000 6.92820i 0.148968 0.258020i
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) −18.0000 31.1769i −0.668503 1.15788i
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) 29.0000 1.07114 0.535570 0.844491i \(-0.320097\pi\)
0.535570 + 0.844491i \(0.320097\pi\)
\(734\) 0 0
\(735\) 27.0000 46.7654i 0.995910 1.72497i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 10.0000 + 17.3205i 0.367856 + 0.637145i 0.989230 0.146369i \(-0.0467586\pi\)
−0.621374 + 0.783514i \(0.713425\pi\)
\(740\) 0 0
\(741\) 14.0000 3.46410i 0.514303 0.127257i
\(742\) 0 0
\(743\) −12.0000 20.7846i −0.440237 0.762513i 0.557470 0.830197i \(-0.311772\pi\)
−0.997707 + 0.0676840i \(0.978439\pi\)
\(744\) 0 0
\(745\) 13.5000 23.3827i 0.494602 0.856675i
\(746\) 0 0
\(747\) 6.00000 10.3923i 0.219529 0.380235i
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −2.00000 3.46410i −0.0729810 0.126407i 0.827225 0.561870i \(-0.189918\pi\)
−0.900207 + 0.435463i \(0.856585\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) −19.0000 32.9090i −0.690567 1.19610i −0.971652 0.236414i \(-0.924028\pi\)
0.281086 0.959683i \(-0.409305\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.00000 15.5885i 0.326250 0.565081i −0.655515 0.755182i \(-0.727548\pi\)
0.981764 + 0.190101i \(0.0608816\pi\)
\(762\) 0 0
\(763\) 20.0000 34.6410i 0.724049 1.25409i
\(764\) 0 0
\(765\) 4.50000 + 7.79423i 0.162698 + 0.281801i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −7.00000 12.1244i −0.252426 0.437215i 0.711767 0.702416i \(-0.247895\pi\)
−0.964193 + 0.265200i \(0.914562\pi\)
\(770\) 0 0
\(771\) −3.00000 + 5.19615i −0.108042 + 0.187135i
\(772\) 0 0
\(773\) 3.00000 5.19615i 0.107903 0.186893i −0.807018 0.590527i \(-0.798920\pi\)
0.914920 + 0.403634i \(0.132253\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 28.0000 + 48.4974i 1.00449 + 1.73984i
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 18.0000 + 31.1769i 0.643268 + 1.11417i
\(784\) 0 0
\(785\) 21.0000 0.749522
\(786\) 0 0
\(787\) 19.0000 32.9090i 0.677277 1.17308i −0.298521 0.954403i \(-0.596493\pi\)
0.975798 0.218675i \(-0.0701734\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 + 31.1769i 0.640006 + 1.10852i
\(792\) 0 0
\(793\) 5.00000 17.3205i 0.177555 0.615069i
\(794\) 0 0
\(795\) 27.0000 + 46.7654i 0.957591 + 1.65860i
\(796\) 0 0
\(797\) 9.00000 15.5885i 0.318796 0.552171i −0.661441 0.749997i \(-0.730055\pi\)
0.980237 + 0.197826i \(0.0633881\pi\)
\(798\) 0 0
\(799\) −9.00000 + 15.5885i −0.318397 + 0.551480i
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −72.0000 −2.53767
\(806\) 0 0
\(807\) 60.0000 2.11210
\(808\) 0 0
\(809\) −7.50000 12.9904i −0.263686 0.456717i 0.703533 0.710663i \(-0.251605\pi\)
−0.967219 + 0.253946i \(0.918272\pi\)
\(810\) 0 0
\(811\) 10.0000 0.351147 0.175574 0.984466i \(-0.443822\pi\)
0.175574 + 0.984466i \(0.443822\pi\)
\(812\) 0 0
\(813\) 8.00000 13.8564i 0.280572 0.485965i
\(814\) 0 0
\(815\) −12.0000 + 20.7846i −0.420342 + 0.728053i
\(816\) 0 0
\(817\) 4.00000 + 6.92820i 0.139942 + 0.242387i
\(818\) 0 0
\(819\) 10.0000 + 10.3923i 0.349428 + 0.363137i
\(820\) 0 0
\(821\) 9.00000 + 15.5885i 0.314102 + 0.544041i 0.979246 0.202674i \(-0.0649632\pi\)
−0.665144 + 0.746715i \(0.731630\pi\)
\(822\) 0 0
\(823\) 16.0000 27.7128i 0.557725 0.966008i −0.439961 0.898017i \(-0.645008\pi\)
0.997686 0.0679910i \(-0.0216589\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 9.50000 + 16.4545i 0.329949 + 0.571488i 0.982501 0.186256i \(-0.0596352\pi\)
−0.652553 + 0.757743i \(0.726302\pi\)
\(830\) 0 0
\(831\) −38.0000 −1.31821
\(832\) 0 0
\(833\) 27.0000 0.935495
\(834\) 0 0
\(835\) 36.0000 + 62.3538i 1.24583 + 2.15784i
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 21.0000 36.3731i 0.725001 1.25574i −0.233973 0.972243i \(-0.575173\pi\)
0.958974 0.283495i \(-0.0914938\pi\)
\(840\) 0 0
\(841\) −26.0000 + 45.0333i −0.896552 + 1.55287i
\(842\) 0 0
\(843\) −3.00000 5.19615i −0.103325 0.178965i
\(844\) 0 0
\(845\) −34.5000 + 18.1865i −1.18684 + 0.625636i
\(846\) 0 0
\(847\) 22.0000 + 38.1051i 0.755929 + 1.30931i
\(848\) 0 0
\(849\) −16.0000 + 27.7128i −0.549119 + 0.951101i
\(850\) 0 0
\(851\) 21.0000 36.3731i 0.719871 1.24685i
\(852\) 0 0
\(853\) 5.00000 0.171197 0.0855984 0.996330i \(-0.472720\pi\)
0.0855984 + 0.996330i \(0.472720\pi\)
\(854\) 0 0
\(855\) −3.00000 5.19615i −0.102598 0.177705i
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) 0 0
\(861\) −12.0000 20.7846i −0.408959 0.708338i
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 9.00000 15.5885i 0.306009 0.530023i
\(866\) 0 0
\(867\) 8.00000 13.8564i 0.271694 0.470588i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −5.00000 5.19615i −0.169419 0.176065i
\(872\) 0 0
\(873\) −7.00000 12.1244i −0.236914 0.410347i
\(874\) 0 0
\(875\) −6.00000 + 10.3923i −0.202837 + 0.351324i
\(876\) 0 0
\(877\) −2.50000 + 4.33013i −0.0844190 + 0.146218i −0.905143 0.425106i \(-0.860237\pi\)
0.820724 + 0.571324i \(0.193570\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −25.5000 44.1673i −0.859117 1.48803i −0.872772 0.488127i \(-0.837680\pi\)
0.0136556 0.999907i \(-0.495653\pi\)
\(882\) 0 0
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000 + 10.3923i 0.201460 + 0.348939i 0.948999 0.315279i \(-0.102098\pi\)
−0.747539 + 0.664218i \(0.768765\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 10.3923i 0.200782 0.347765i
\(894\) 0 0
\(895\) −36.0000 62.3538i −1.20335 2.08426i
\(896\) 0 0
\(897\) 12.0000 41.5692i 0.400668 1.38796i
\(898\) 0 0
\(899\) 9.00000 + 15.5885i 0.300167 + 0.519904i
\(900\) 0 0
\(901\) −13.5000 + 23.3827i −0.449750 + 0.778990i
\(902\) 0 0
\(903\) −16.0000 + 27.7128i −0.532447 + 0.922225i
\(904\) 0 0
\(905\) 21.0000 0.698064
\(906\) 0 0
\(907\) 4.00000 + 6.92820i 0.132818 + 0.230047i 0.924762 0.380547i \(-0.124264\pi\)
−0.791944 + 0.610594i \(0.790931\pi\)
\(908\) 0 0
\(909\) 9.00000 0.298511
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −30.0000 −0.991769
\(916\) 0 0
\(917\) 12.0000 20.7846i 0.396275 0.686368i
\(918\) 0 0
\(919\) 19.0000 32.9090i 0.626752 1.08557i −0.361447 0.932393i \(-0.617717\pi\)
0.988199 0.153174i \(-0.0489495\pi\)
\(920\) 0 0
\(921\) 26.0000 + 45.0333i 0.856729 + 1.48390i
\(922\) 0 0
\(923\) 6.00000 20.7846i 0.197492 0.684134i
\(924\) 0 0
\(925\) 14.0000 + 24.2487i 0.460317 + 0.797293i
\(926\) 0 0
\(927\) 1.00000 1.73205i 0.0328443 0.0568880i
\(928\) 0 0
\(929\) 10.5000 18.1865i 0.344494 0.596681i −0.640768 0.767735i \(-0.721384\pi\)
0.985262 + 0.171054i \(0.0547172\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) 6.00000 + 10.3923i 0.196431 + 0.340229i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 0 0
\(939\) −14.0000 24.2487i −0.456873 0.791327i
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −9.00000 + 15.5885i −0.293080 + 0.507630i
\(944\) 0 0
\(945\) −24.0000 + 41.5692i −0.780720 + 1.35225i
\(946\) 0 0
\(947\) −9.00000 15.5885i −0.292461 0.506557i 0.681930 0.731417i \(-0.261141\pi\)
−0.974391 + 0.224860i \(0.927807\pi\)
\(948\) 0 0
\(949\) 3.50000 0.866025i 0.113615 0.0281124i
\(950\) 0 0
\(951\) 15.0000 + 25.9808i 0.486408 + 0.842484i
\(952\) 0 0
\(953\) 3.00000 5.19615i 0.0971795 0.168320i −0.813337 0.581793i \(-0.802351\pi\)
0.910516 + 0.413473i \(0.135685\pi\)
\(954\) 0 0
\(955\) −9.00000 + 15.5885i −0.291233 + 0.504431i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −30.0000 51.9615i −0.968751 1.67793i
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 16.5000 + 28.5788i 0.531154 + 0.919985i
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) 0 0
\(969\) 6.00000 10.3923i 0.192748 0.333849i
\(970\) 0 0
\(971\) 15.0000 25.9808i 0.481373 0.833762i −0.518399 0.855139i \(-0.673472\pi\)
0.999771 + 0.0213768i \(0.00680496\pi\)
\(972\) 0 0
\(973\) −32.0000 55.4256i −1.02587 1.77686i
\(974\) 0 0
\(975\) 20.0000 + 20.7846i 0.640513 + 0.665640i
\(976\) 0 0
\(977\) −19.5000 33.7750i −0.623860 1.08056i −0.988760 0.149511i \(-0.952230\pi\)
0.364900 0.931047i \(-0.381103\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.00000 8.66025i 0.159638 0.276501i
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 9.00000 + 15.5885i 0.286764 + 0.496690i
\(986\) 0 0
\(987\) 48.0000 1.52786
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 1.00000 + 1.73205i 0.0317660 + 0.0550204i 0.881471 0.472237i \(-0.156554\pi\)
−0.849705 + 0.527258i \(0.823220\pi\)
\(992\) 0 0
\(993\) −64.0000 −2.03098
\(994\) 0 0
\(995\) −39.0000 + 67.5500i −1.23638 + 2.14148i
\(996\) 0 0
\(997\) −2.50000 + 4.33013i −0.0791758 + 0.137136i −0.902895 0.429862i \(-0.858562\pi\)
0.823719 + 0.566999i \(0.191896\pi\)
\(998\) 0 0
\(999\) −14.0000 24.2487i −0.442940 0.767195i
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 208.2.i.a.81.1 2
3.2 odd 2 1872.2.t.m.289.1 2
4.3 odd 2 52.2.e.b.29.1 yes 2
8.3 odd 2 832.2.i.c.705.1 2
8.5 even 2 832.2.i.i.705.1 2
12.11 even 2 468.2.l.d.289.1 2
13.2 odd 12 2704.2.f.i.337.2 2
13.3 even 3 2704.2.a.l.1.1 1
13.9 even 3 inner 208.2.i.a.113.1 2
13.10 even 6 2704.2.a.m.1.1 1
13.11 odd 12 2704.2.f.i.337.1 2
20.3 even 4 1300.2.bb.d.549.1 4
20.7 even 4 1300.2.bb.d.549.2 4
20.19 odd 2 1300.2.i.b.601.1 2
28.3 even 6 2548.2.i.b.1745.1 2
28.11 odd 6 2548.2.i.g.1745.1 2
28.19 even 6 2548.2.l.g.1537.1 2
28.23 odd 6 2548.2.l.b.1537.1 2
28.27 even 2 2548.2.k.a.393.1 2
39.35 odd 6 1872.2.t.m.1153.1 2
52.3 odd 6 676.2.a.a.1.1 1
52.7 even 12 676.2.h.d.485.1 4
52.11 even 12 676.2.d.a.337.1 2
52.15 even 12 676.2.d.a.337.2 2
52.19 even 12 676.2.h.d.485.2 4
52.23 odd 6 676.2.a.b.1.1 1
52.31 even 4 676.2.h.d.361.2 4
52.35 odd 6 52.2.e.b.9.1 2
52.43 odd 6 676.2.e.d.529.1 2
52.47 even 4 676.2.h.d.361.1 4
52.51 odd 2 676.2.e.d.653.1 2
104.35 odd 6 832.2.i.c.321.1 2
104.61 even 6 832.2.i.i.321.1 2
156.11 odd 12 6084.2.b.k.4393.2 2
156.23 even 6 6084.2.a.c.1.1 1
156.35 even 6 468.2.l.d.217.1 2
156.107 even 6 6084.2.a.o.1.1 1
156.119 odd 12 6084.2.b.k.4393.1 2
260.87 even 12 1300.2.bb.d.1049.1 4
260.139 odd 6 1300.2.i.b.1101.1 2
260.243 even 12 1300.2.bb.d.1049.2 4
364.87 even 6 2548.2.l.g.373.1 2
364.139 even 6 2548.2.k.a.1569.1 2
364.191 odd 6 2548.2.i.g.165.1 2
364.243 even 6 2548.2.i.b.165.1 2
364.347 odd 6 2548.2.l.b.373.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
52.2.e.b.9.1 2 52.35 odd 6
52.2.e.b.29.1 yes 2 4.3 odd 2
208.2.i.a.81.1 2 1.1 even 1 trivial
208.2.i.a.113.1 2 13.9 even 3 inner
468.2.l.d.217.1 2 156.35 even 6
468.2.l.d.289.1 2 12.11 even 2
676.2.a.a.1.1 1 52.3 odd 6
676.2.a.b.1.1 1 52.23 odd 6
676.2.d.a.337.1 2 52.11 even 12
676.2.d.a.337.2 2 52.15 even 12
676.2.e.d.529.1 2 52.43 odd 6
676.2.e.d.653.1 2 52.51 odd 2
676.2.h.d.361.1 4 52.47 even 4
676.2.h.d.361.2 4 52.31 even 4
676.2.h.d.485.1 4 52.7 even 12
676.2.h.d.485.2 4 52.19 even 12
832.2.i.c.321.1 2 104.35 odd 6
832.2.i.c.705.1 2 8.3 odd 2
832.2.i.i.321.1 2 104.61 even 6
832.2.i.i.705.1 2 8.5 even 2
1300.2.i.b.601.1 2 20.19 odd 2
1300.2.i.b.1101.1 2 260.139 odd 6
1300.2.bb.d.549.1 4 20.3 even 4
1300.2.bb.d.549.2 4 20.7 even 4
1300.2.bb.d.1049.1 4 260.87 even 12
1300.2.bb.d.1049.2 4 260.243 even 12
1872.2.t.m.289.1 2 3.2 odd 2
1872.2.t.m.1153.1 2 39.35 odd 6
2548.2.i.b.165.1 2 364.243 even 6
2548.2.i.b.1745.1 2 28.3 even 6
2548.2.i.g.165.1 2 364.191 odd 6
2548.2.i.g.1745.1 2 28.11 odd 6
2548.2.k.a.393.1 2 28.27 even 2
2548.2.k.a.1569.1 2 364.139 even 6
2548.2.l.b.373.1 2 364.347 odd 6
2548.2.l.b.1537.1 2 28.23 odd 6
2548.2.l.g.373.1 2 364.87 even 6
2548.2.l.g.1537.1 2 28.19 even 6
2704.2.a.l.1.1 1 13.3 even 3
2704.2.a.m.1.1 1 13.10 even 6
2704.2.f.i.337.1 2 13.11 odd 12
2704.2.f.i.337.2 2 13.2 odd 12
6084.2.a.c.1.1 1 156.23 even 6
6084.2.a.o.1.1 1 156.107 even 6
6084.2.b.k.4393.1 2 156.119 odd 12
6084.2.b.k.4393.2 2 156.11 odd 12