Properties

Label 1728.2.s.a.1151.1
Level $1728$
Weight $2$
Character 1728.1151
Analytic conductor $13.798$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(575,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (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: \(13.7981494693\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1151
Dual form 1728.2.s.a.575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 1.73205i) q^{5} +(-3.00000 + 1.73205i) q^{7} +(-1.50000 - 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} -1.73205i q^{17} +1.73205i q^{19} +(3.50000 + 6.06218i) q^{25} +(-3.00000 + 1.73205i) q^{29} +12.0000 q^{35} -2.00000 q^{37} +(4.50000 + 2.59808i) q^{41} +(-4.50000 + 2.59808i) q^{43} +(6.00000 + 10.3923i) q^{47} +(2.50000 - 4.33013i) q^{49} +10.3923i q^{55} +(-7.50000 + 12.9904i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-12.0000 + 6.92820i) q^{65} +(7.50000 + 4.33013i) q^{67} +6.00000 q^{71} -11.0000 q^{73} +(9.00000 + 5.19615i) q^{77} +(3.00000 - 1.73205i) q^{79} +(6.00000 + 10.3923i) q^{83} +(-3.00000 + 5.19615i) q^{85} -13.8564i q^{89} +13.8564i q^{91} +(3.00000 - 5.19615i) q^{95} +(-6.50000 - 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5} - 6 q^{7} - 3 q^{11} + 4 q^{13} + 7 q^{25} - 6 q^{29} + 24 q^{35} - 4 q^{37} + 9 q^{41} - 9 q^{43} + 12 q^{47} + 5 q^{49} - 15 q^{59} + 8 q^{61} - 24 q^{65} + 15 q^{67} + 12 q^{71} - 22 q^{73}+ \cdots - 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.00000 1.73205i −1.34164 0.774597i −0.354593 0.935021i \(-0.615380\pi\)
−0.987048 + 0.160424i \(0.948714\pi\)
\(6\) 0 0
\(7\) −3.00000 + 1.73205i −1.13389 + 0.654654i −0.944911 0.327327i \(-0.893852\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) 2.00000 3.46410i 0.554700 0.960769i −0.443227 0.896410i \(-0.646166\pi\)
0.997927 0.0643593i \(-0.0205004\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205i 0.420084i −0.977692 0.210042i \(-0.932640\pi\)
0.977692 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) 1.73205i 0.397360i 0.980064 + 0.198680i \(0.0636654\pi\)
−0.980064 + 0.198680i \(0.936335\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 3.50000 + 6.06218i 0.700000 + 1.21244i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 + 1.73205i −0.557086 + 0.321634i −0.751975 0.659192i \(-0.770899\pi\)
0.194889 + 0.980825i \(0.437565\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 + 2.59808i 0.702782 + 0.405751i 0.808383 0.588657i \(-0.200343\pi\)
−0.105601 + 0.994409i \(0.533677\pi\)
\(42\) 0 0
\(43\) −4.50000 + 2.59808i −0.686244 + 0.396203i −0.802203 0.597051i \(-0.796339\pi\)
0.115960 + 0.993254i \(0.463006\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i \(0.172597\pi\)
0.0186297 + 0.999826i \(0.494070\pi\)
\(48\) 0 0
\(49\) 2.50000 4.33013i 0.357143 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 10.3923i 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.50000 + 12.9904i −0.976417 + 1.69120i −0.301239 + 0.953549i \(0.597400\pi\)
−0.675178 + 0.737655i \(0.735933\pi\)
\(60\) 0 0
\(61\) 4.00000 + 6.92820i 0.512148 + 0.887066i 0.999901 + 0.0140840i \(0.00448323\pi\)
−0.487753 + 0.872982i \(0.662183\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 + 6.92820i −1.48842 + 0.859338i
\(66\) 0 0
\(67\) 7.50000 + 4.33013i 0.916271 + 0.529009i 0.882443 0.470418i \(-0.155897\pi\)
0.0338274 + 0.999428i \(0.489230\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.00000 + 5.19615i 1.02565 + 0.592157i
\(78\) 0 0
\(79\) 3.00000 1.73205i 0.337526 0.194871i −0.321651 0.946858i \(-0.604238\pi\)
0.659178 + 0.751987i \(0.270905\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −3.00000 + 5.19615i −0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.8564i 1.46878i −0.678730 0.734388i \(-0.737469\pi\)
0.678730 0.734388i \(-0.262531\pi\)
\(90\) 0 0
\(91\) 13.8564i 1.45255i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) −6.50000 11.2583i −0.659975 1.14311i −0.980622 0.195911i \(-0.937234\pi\)
0.320647 0.947199i \(-0.396100\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 5.19615i 0.895533 0.517036i 0.0197851 0.999804i \(-0.493702\pi\)
0.875748 + 0.482768i \(0.160368\pi\)
\(102\) 0 0
\(103\) 12.0000 + 6.92820i 1.18240 + 0.682656i 0.956567 0.291511i \(-0.0941580\pi\)
0.225828 + 0.974167i \(0.427491\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 + 3.46410i 0.564433 + 0.325875i 0.754923 0.655814i \(-0.227674\pi\)
−0.190490 + 0.981689i \(0.561008\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 + 5.19615i 0.275010 + 0.476331i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) −3.00000 5.19615i −0.260133 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.50000 0.866025i 0.128154 0.0739895i −0.434553 0.900646i \(-0.643094\pi\)
0.562706 + 0.826657i \(0.309760\pi\)
\(138\) 0 0
\(139\) −16.5000 9.52628i −1.39951 0.808008i −0.405170 0.914241i \(-0.632788\pi\)
−0.994341 + 0.106233i \(0.966121\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 + 6.92820i 0.983078 + 0.567581i 0.903198 0.429224i \(-0.141213\pi\)
0.0798802 + 0.996804i \(0.474546\pi\)
\(150\) 0 0
\(151\) −6.00000 + 3.46410i −0.488273 + 0.281905i −0.723858 0.689949i \(-0.757633\pi\)
0.235585 + 0.971854i \(0.424299\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000 6.92820i 0.319235 0.552931i −0.661094 0.750303i \(-0.729907\pi\)
0.980329 + 0.197372i \(0.0632408\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.46410i 0.271329i 0.990755 + 0.135665i \(0.0433170\pi\)
−0.990755 + 0.135665i \(0.956683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 5.19615i 0.232147 0.402090i −0.726293 0.687386i \(-0.758758\pi\)
0.958440 + 0.285295i \(0.0920916\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −21.0000 + 12.1244i −1.59660 + 0.921798i −0.604465 + 0.796632i \(0.706613\pi\)
−0.992136 + 0.125166i \(0.960054\pi\)
\(174\) 0 0
\(175\) −21.0000 12.1244i −1.58745 0.916515i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 + 3.46410i 0.441129 + 0.254686i
\(186\) 0 0
\(187\) −4.50000 + 2.59808i −0.329073 + 0.189990i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 5.19615i −0.217072 0.375980i 0.736839 0.676068i \(-0.236317\pi\)
−0.953912 + 0.300088i \(0.902984\pi\)
\(192\) 0 0
\(193\) −11.5000 + 19.9186i −0.827788 + 1.43377i 0.0719816 + 0.997406i \(0.477068\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564i 0.987228i −0.869681 0.493614i \(-0.835676\pi\)
0.869681 0.493614i \(-0.164324\pi\)
\(198\) 0 0
\(199\) 3.46410i 0.245564i −0.992434 0.122782i \(-0.960818\pi\)
0.992434 0.122782i \(-0.0391815\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 10.3923i 0.421117 0.729397i
\(204\) 0 0
\(205\) −9.00000 15.5885i −0.628587 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.50000 2.59808i 0.311272 0.179713i
\(210\) 0 0
\(211\) 15.0000 + 8.66025i 1.03264 + 0.596196i 0.917741 0.397180i \(-0.130011\pi\)
0.114902 + 0.993377i \(0.463345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.0000 1.22759
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 3.46410i −0.403604 0.233021i
\(222\) 0 0
\(223\) −18.0000 + 10.3923i −1.20537 + 0.695920i −0.961744 0.273949i \(-0.911670\pi\)
−0.243625 + 0.969870i \(0.578337\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 2.59808i −0.0995585 0.172440i 0.811943 0.583736i \(-0.198410\pi\)
−0.911502 + 0.411296i \(0.865076\pi\)
\(228\) 0 0
\(229\) −13.0000 + 22.5167i −0.859064 + 1.48794i 0.0137585 + 0.999905i \(0.495620\pi\)
−0.872823 + 0.488037i \(0.837713\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.1244i 0.794293i 0.917755 + 0.397146i \(0.130000\pi\)
−0.917755 + 0.397146i \(0.870000\pi\)
\(234\) 0 0
\(235\) 41.5692i 2.71168i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 + 20.7846i −0.776215 + 1.34444i 0.157893 + 0.987456i \(0.449530\pi\)
−0.934109 + 0.356988i \(0.883804\pi\)
\(240\) 0 0
\(241\) 8.50000 + 14.7224i 0.547533 + 0.948355i 0.998443 + 0.0557856i \(0.0177663\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −15.0000 + 8.66025i −0.958315 + 0.553283i
\(246\) 0 0
\(247\) 6.00000 + 3.46410i 0.381771 + 0.220416i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.5000 + 9.52628i 1.02924 + 0.594233i 0.916767 0.399422i \(-0.130789\pi\)
0.112474 + 0.993655i \(0.464122\pi\)
\(258\) 0 0
\(259\) 6.00000 3.46410i 0.372822 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 15.5885i −0.554964 0.961225i −0.997906 0.0646755i \(-0.979399\pi\)
0.442943 0.896550i \(-0.353935\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.92820i 0.422420i −0.977441 0.211210i \(-0.932260\pi\)
0.977441 0.211210i \(-0.0677404\pi\)
\(270\) 0 0
\(271\) 6.92820i 0.420858i −0.977609 0.210429i \(-0.932514\pi\)
0.977609 0.210429i \(-0.0674861\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.5000 18.1865i 0.633174 1.09669i
\(276\) 0 0
\(277\) −4.00000 6.92820i −0.240337 0.416275i 0.720473 0.693482i \(-0.243925\pi\)
−0.960810 + 0.277207i \(0.910591\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 + 3.46410i −0.357930 + 0.206651i −0.668172 0.744007i \(-0.732923\pi\)
0.310242 + 0.950657i \(0.399590\pi\)
\(282\) 0 0
\(283\) 9.00000 + 5.19615i 0.534994 + 0.308879i 0.743048 0.669238i \(-0.233379\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) 14.0000 0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.00000 1.73205i −0.175262 0.101187i 0.409803 0.912174i \(-0.365598\pi\)
−0.585065 + 0.810987i \(0.698931\pi\)
\(294\) 0 0
\(295\) 45.0000 25.9808i 2.62000 1.51266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 9.00000 15.5885i 0.518751 0.898504i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 27.7128i 1.58683i
\(306\) 0 0
\(307\) 25.9808i 1.48280i 0.671063 + 0.741400i \(0.265838\pi\)
−0.671063 + 0.741400i \(0.734162\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.00000 + 5.19615i −0.170114 + 0.294647i −0.938460 0.345389i \(-0.887747\pi\)
0.768345 + 0.640036i \(0.221080\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 + 3.46410i −0.336994 + 0.194563i −0.658942 0.752194i \(-0.728996\pi\)
0.321948 + 0.946757i \(0.395662\pi\)
\(318\) 0 0
\(319\) 9.00000 + 5.19615i 0.503903 + 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 28.0000 1.55316
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −36.0000 20.7846i −1.98474 1.14589i
\(330\) 0 0
\(331\) 21.0000 12.1244i 1.15426 0.666415i 0.204342 0.978900i \(-0.434495\pi\)
0.949923 + 0.312485i \(0.101161\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.0000 25.9808i −0.819538 1.41948i
\(336\) 0 0
\(337\) 5.50000 9.52628i 0.299604 0.518930i −0.676441 0.736497i \(-0.736479\pi\)
0.976045 + 0.217567i \(0.0698121\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.92820i 0.374088i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.5000 + 23.3827i −0.724718 + 1.25525i 0.234372 + 0.972147i \(0.424697\pi\)
−0.959090 + 0.283101i \(0.908637\pi\)
\(348\) 0 0
\(349\) 16.0000 + 27.7128i 0.856460 + 1.48343i 0.875284 + 0.483610i \(0.160675\pi\)
−0.0188232 + 0.999823i \(0.505992\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.5000 + 12.9904i −1.19755 + 0.691408i −0.960009 0.279968i \(-0.909676\pi\)
−0.237545 + 0.971377i \(0.576343\pi\)
\(354\) 0 0
\(355\) −18.0000 10.3923i −0.955341 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 16.0000 0.842105
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 33.0000 + 19.0526i 1.72730 + 0.997257i
\(366\) 0 0
\(367\) −3.00000 + 1.73205i −0.156599 + 0.0904123i −0.576252 0.817272i \(-0.695485\pi\)
0.419653 + 0.907685i \(0.362152\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 17.3205i 0.517780 0.896822i −0.482006 0.876168i \(-0.660092\pi\)
0.999787 0.0206542i \(-0.00657489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.8564i 0.713641i
\(378\) 0 0
\(379\) 19.0526i 0.978664i −0.872098 0.489332i \(-0.837241\pi\)
0.872098 0.489332i \(-0.162759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.00000 + 15.5885i −0.459879 + 0.796533i −0.998954 0.0457244i \(-0.985440\pi\)
0.539076 + 0.842257i \(0.318774\pi\)
\(384\) 0 0
\(385\) −18.0000 31.1769i −0.917365 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.00000 + 5.19615i −0.456318 + 0.263455i −0.710495 0.703702i \(-0.751529\pi\)
0.254177 + 0.967158i \(0.418196\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.50000 + 4.33013i 0.374532 + 0.216236i 0.675437 0.737418i \(-0.263955\pi\)
−0.300904 + 0.953654i \(0.597289\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000 + 5.19615i 0.148704 + 0.257564i
\(408\) 0 0
\(409\) 11.5000 19.9186i 0.568638 0.984911i −0.428063 0.903749i \(-0.640804\pi\)
0.996701 0.0811615i \(-0.0258630\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 51.9615i 2.55686i
\(414\) 0 0
\(415\) 41.5692i 2.04055i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 + 10.3923i −0.293119 + 0.507697i −0.974546 0.224189i \(-0.928027\pi\)
0.681426 + 0.731887i \(0.261360\pi\)
\(420\) 0 0
\(421\) 5.00000 + 8.66025i 0.243685 + 0.422075i 0.961761 0.273890i \(-0.0883103\pi\)
−0.718076 + 0.695965i \(0.754977\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.5000 6.06218i 0.509325 0.294059i
\(426\) 0 0
\(427\) −24.0000 13.8564i −1.16144 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −31.0000 −1.48976 −0.744882 0.667196i \(-0.767494\pi\)
−0.744882 + 0.667196i \(0.767494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −18.0000 + 10.3923i −0.859093 + 0.495998i −0.863708 0.503992i \(-0.831864\pi\)
0.00461537 + 0.999989i \(0.498531\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.50000 7.79423i −0.213801 0.370315i 0.739100 0.673596i \(-0.235251\pi\)
−0.952901 + 0.303281i \(0.901918\pi\)
\(444\) 0 0
\(445\) −24.0000 + 41.5692i −1.13771 + 1.97057i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 25.9808i 1.22611i −0.790041 0.613054i \(-0.789941\pi\)
0.790041 0.613054i \(-0.210059\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24.0000 41.5692i 1.12514 1.94880i
\(456\) 0 0
\(457\) 5.50000 + 9.52628i 0.257279 + 0.445621i 0.965512 0.260358i \(-0.0838407\pi\)
−0.708233 + 0.705979i \(0.750507\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 10.3923i 0.838344 0.484018i −0.0183573 0.999831i \(-0.505844\pi\)
0.856701 + 0.515814i \(0.172510\pi\)
\(462\) 0 0
\(463\) −15.0000 8.66025i −0.697109 0.402476i 0.109161 0.994024i \(-0.465184\pi\)
−0.806270 + 0.591548i \(0.798517\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) −30.0000 −1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13.5000 + 7.79423i 0.620731 + 0.358379i
\(474\) 0 0
\(475\) −10.5000 + 6.06218i −0.481773 + 0.278152i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.0000 25.9808i −0.685367 1.18709i −0.973321 0.229447i \(-0.926308\pi\)
0.287954 0.957644i \(-0.407025\pi\)
\(480\) 0 0
\(481\) −4.00000 + 6.92820i −0.182384 + 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 45.0333i 2.04486i
\(486\) 0 0
\(487\) 17.3205i 0.784867i 0.919780 + 0.392434i \(0.128367\pi\)
−0.919780 + 0.392434i \(0.871633\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.50000 12.9904i 0.338470 0.586248i −0.645675 0.763612i \(-0.723424\pi\)
0.984145 + 0.177365i \(0.0567572\pi\)
\(492\) 0 0
\(493\) 3.00000 + 5.19615i 0.135113 + 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.0000 + 10.3923i −0.807410 + 0.466159i
\(498\) 0 0
\(499\) 13.5000 + 7.79423i 0.604343 + 0.348918i 0.770748 0.637140i \(-0.219883\pi\)
−0.166405 + 0.986057i \(0.553216\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.0000 + 10.3923i 0.797836 + 0.460631i 0.842714 0.538362i \(-0.180957\pi\)
−0.0448779 + 0.998992i \(0.514290\pi\)
\(510\) 0 0
\(511\) 33.0000 19.0526i 1.45983 0.842836i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.0000 41.5692i −1.05757 1.83176i
\(516\) 0 0
\(517\) 18.0000 31.1769i 0.791639 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.5885i 0.682943i 0.939892 + 0.341471i \(0.110925\pi\)
−0.939892 + 0.341471i \(0.889075\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i 0.925525 + 0.378686i \(0.123624\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.0000 10.3923i 0.779667 0.450141i
\(534\) 0 0
\(535\) −9.00000 5.19615i −0.389104 0.224649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −15.0000 −0.646096
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.0000 + 6.92820i 0.514024 + 0.296772i
\(546\) 0 0
\(547\) 7.50000 4.33013i 0.320677 0.185143i −0.331017 0.943625i \(-0.607392\pi\)
0.651694 + 0.758482i \(0.274059\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.00000 5.19615i −0.127804 0.221364i
\(552\) 0 0
\(553\) −6.00000 + 10.3923i −0.255146 + 0.441926i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.1051i 1.61457i 0.590165 + 0.807283i \(0.299063\pi\)
−0.590165 + 0.807283i \(0.700937\pi\)
\(558\) 0 0
\(559\) 20.7846i 0.879095i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.50000 7.79423i 0.189652 0.328488i −0.755482 0.655169i \(-0.772597\pi\)
0.945134 + 0.326682i \(0.105931\pi\)
\(564\) 0 0
\(565\) −12.0000 20.7846i −0.504844 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.5000 9.52628i 0.691716 0.399362i −0.112539 0.993647i \(-0.535898\pi\)
0.804255 + 0.594285i \(0.202565\pi\)
\(570\) 0 0
\(571\) −10.5000 6.06218i −0.439411 0.253694i 0.263937 0.964540i \(-0.414979\pi\)
−0.703348 + 0.710846i \(0.748312\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −36.0000 20.7846i −1.49353 0.862291i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.50000 2.59808i −0.0619116 0.107234i 0.833408 0.552658i \(-0.186386\pi\)
−0.895320 + 0.445424i \(0.853053\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.92820i 0.284507i 0.989830 + 0.142254i \(0.0454349\pi\)
−0.989830 + 0.142254i \(0.954565\pi\)
\(594\) 0 0
\(595\) 20.7846i 0.852086i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.00000 + 15.5885i −0.367730 + 0.636927i −0.989210 0.146503i \(-0.953198\pi\)
0.621480 + 0.783430i \(0.286532\pi\)
\(600\) 0 0
\(601\) 3.50000 + 6.06218i 0.142768 + 0.247281i 0.928538 0.371237i \(-0.121066\pi\)
−0.785770 + 0.618519i \(0.787733\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.00000 + 3.46410i −0.243935 + 0.140836i
\(606\) 0 0
\(607\) −24.0000 13.8564i −0.974130 0.562414i −0.0736371 0.997285i \(-0.523461\pi\)
−0.900493 + 0.434871i \(0.856794\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25.5000 + 14.7224i 1.02659 + 0.592703i 0.916006 0.401164i \(-0.131394\pi\)
0.110585 + 0.993867i \(0.464728\pi\)
\(618\) 0 0
\(619\) −25.5000 + 14.7224i −1.02493 + 0.591744i −0.915529 0.402253i \(-0.868227\pi\)
−0.109403 + 0.993997i \(0.534894\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 + 41.5692i 0.961540 + 1.66544i
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.46410i 0.138123i
\(630\) 0 0
\(631\) 3.46410i 0.137904i 0.997620 + 0.0689519i \(0.0219655\pi\)
−0.997620 + 0.0689519i \(0.978035\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −10.0000 17.3205i −0.396214 0.686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 37.5000 21.6506i 1.48116 0.855149i 0.481389 0.876507i \(-0.340132\pi\)
0.999772 + 0.0213584i \(0.00679909\pi\)
\(642\) 0 0
\(643\) 22.5000 + 12.9904i 0.887313 + 0.512291i 0.873063 0.487608i \(-0.162130\pi\)
0.0142506 + 0.999898i \(0.495464\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) 45.0000 1.76640
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.0000 15.5885i −1.05659 0.610023i −0.132104 0.991236i \(-0.542173\pi\)
−0.924487 + 0.381212i \(0.875507\pi\)
\(654\) 0 0
\(655\) −36.0000 + 20.7846i −1.40664 + 0.812122i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 + 20.7846i 0.467454 + 0.809653i 0.999309 0.0371821i \(-0.0118382\pi\)
−0.531855 + 0.846836i \(0.678505\pi\)
\(660\) 0 0
\(661\) 7.00000 12.1244i 0.272268 0.471583i −0.697174 0.716902i \(-0.745559\pi\)
0.969442 + 0.245319i \(0.0788928\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.7846i 0.805993i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 20.7846i 0.463255 0.802381i
\(672\) 0 0
\(673\) 23.0000 + 39.8372i 0.886585 + 1.53561i 0.843886 + 0.536522i \(0.180262\pi\)
0.0426985 + 0.999088i \(0.486405\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 10.3923i 0.691796 0.399409i −0.112488 0.993653i \(-0.535882\pi\)
0.804285 + 0.594244i \(0.202549\pi\)
\(678\) 0 0
\(679\) 39.0000 + 22.5167i 1.49668 + 0.864110i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.0000 −0.573959 −0.286980 0.957937i \(-0.592651\pi\)
−0.286980 + 0.957937i \(0.592651\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 27.0000 15.5885i 1.02713 0.593013i 0.110968 0.993824i \(-0.464605\pi\)
0.916161 + 0.400811i \(0.131272\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.0000 + 57.1577i 1.25176 + 2.16811i
\(696\) 0 0
\(697\) 4.50000 7.79423i 0.170450 0.295227i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.7128i 1.04670i −0.852118 0.523349i \(-0.824682\pi\)
0.852118 0.523349i \(-0.175318\pi\)
\(702\) 0 0
\(703\) 3.46410i 0.130651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.0000 + 31.1769i −0.676960 + 1.17253i
\(708\) 0 0
\(709\) 14.0000 + 24.2487i 0.525781 + 0.910679i 0.999549 + 0.0300298i \(0.00956021\pi\)
−0.473768 + 0.880650i \(0.657106\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 36.0000 + 20.7846i 1.34632 + 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −21.0000 12.1244i −0.779920 0.450287i
\(726\) 0 0
\(727\) −36.0000 + 20.7846i −1.33517 + 0.770859i −0.986086 0.166234i \(-0.946839\pi\)
−0.349080 + 0.937093i \(0.613506\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.50000 + 7.79423i 0.166439 + 0.288280i
\(732\) 0 0
\(733\) 11.0000 19.0526i 0.406294 0.703722i −0.588177 0.808732i \(-0.700154\pi\)
0.994471 + 0.105010i \(0.0334875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.9808i 0.957014i
\(738\) 0 0
\(739\) 25.9808i 0.955718i 0.878437 + 0.477859i \(0.158587\pi\)
−0.878437 + 0.477859i \(0.841413\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 15.5885i 0.330178 0.571885i −0.652369 0.757902i \(-0.726225\pi\)
0.982547 + 0.186017i \(0.0595579\pi\)
\(744\) 0 0
\(745\) −24.0000 41.5692i −0.879292 1.52298i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.00000 + 5.19615i −0.328853 + 0.189863i
\(750\) 0 0
\(751\) 33.0000 + 19.0526i 1.20419 + 0.695238i 0.961483 0.274863i \(-0.0886324\pi\)
0.242704 + 0.970100i \(0.421966\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 24.2487i −1.52250 0.879015i −0.999646 0.0265919i \(-0.991535\pi\)
−0.522852 0.852423i \(-0.675132\pi\)
\(762\) 0 0
\(763\) 12.0000 6.92820i 0.434429 0.250818i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.0000 + 51.9615i 1.08324 + 1.87622i
\(768\) 0 0
\(769\) −7.00000 + 12.1244i −0.252426 + 0.437215i −0.964193 0.265200i \(-0.914562\pi\)
0.711767 + 0.702416i \(0.247895\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.92820i 0.249190i 0.992208 + 0.124595i \(0.0397632\pi\)
−0.992208 + 0.124595i \(0.960237\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.50000 + 7.79423i −0.161229 + 0.279257i
\(780\) 0 0
\(781\) −9.00000 15.5885i −0.322045 0.557799i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −24.0000 + 13.8564i −0.856597 + 0.494556i
\(786\) 0 0
\(787\) −15.0000 8.66025i −0.534692 0.308705i 0.208233 0.978079i \(-0.433229\pi\)
−0.742925 + 0.669375i \(0.766562\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.00000 + 1.73205i 0.106265 + 0.0613524i 0.552191 0.833718i \(-0.313792\pi\)
−0.445925 + 0.895070i \(0.647125\pi\)
\(798\) 0 0
\(799\) 18.0000 10.3923i 0.636794 0.367653i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.5000 + 28.5788i 0.582272 + 1.00853i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.0526i 0.669852i −0.942244 0.334926i \(-0.891289\pi\)
0.942244 0.334926i \(-0.108711\pi\)
\(810\) 0 0
\(811\) 36.3731i 1.27723i 0.769526 + 0.638616i \(0.220493\pi\)
−0.769526 + 0.638616i \(0.779507\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.00000 10.3923i 0.210171 0.364027i
\(816\) 0 0
\(817\) −4.50000 7.79423i −0.157435 0.272686i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 1.73205i 0.104701 0.0604490i −0.446735 0.894666i \(-0.647413\pi\)
0.551436 + 0.834217i \(0.314080\pi\)
\(822\) 0 0
\(823\) −21.0000 12.1244i −0.732014 0.422628i 0.0871445 0.996196i \(-0.472226\pi\)
−0.819159 + 0.573567i \(0.805559\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −7.50000 4.33013i −0.259860 0.150030i
\(834\) 0 0
\(835\) −18.0000 + 10.3923i −0.622916 + 0.359641i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18.0000 31.1769i −0.621429 1.07635i −0.989220 0.146438i \(-0.953219\pi\)
0.367791 0.929909i \(-0.380114\pi\)
\(840\) 0 0
\(841\) −8.50000 + 14.7224i −0.293103 + 0.507670i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.3923i 0.357506i
\(846\) 0 0
\(847\) 6.92820i 0.238056i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 16.0000 + 27.7128i 0.547830 + 0.948869i 0.998423 + 0.0561393i \(0.0178791\pi\)
−0.450593 + 0.892729i \(0.648788\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.0000 + 17.3205i −1.02478 + 0.591657i −0.915485 0.402352i \(-0.868193\pi\)
−0.109295 + 0.994009i \(0.534859\pi\)
\(858\) 0 0
\(859\) −25.5000 14.7224i −0.870049 0.502323i −0.00268433 0.999996i \(-0.500854\pi\)
−0.867364 + 0.497674i \(0.834188\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 0 0
\(865\) 84.0000 2.85609
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.00000 5.19615i −0.305304 0.176267i
\(870\) 0 0
\(871\) 30.0000 17.3205i 1.01651 0.586883i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 + 20.7846i 0.405674 + 0.702648i
\(876\) 0 0
\(877\) −20.0000 + 34.6410i −0.675352 + 1.16974i 0.301014 + 0.953620i \(0.402675\pi\)
−0.976366 + 0.216124i \(0.930658\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20.7846i 0.700251i 0.936703 + 0.350126i \(0.113861\pi\)
−0.936703 + 0.350126i \(0.886139\pi\)
\(882\) 0 0
\(883\) 8.66025i 0.291441i 0.989326 + 0.145720i \(0.0465500\pi\)
−0.989326 + 0.145720i \(0.953450\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.0000 + 36.3731i −0.705111 + 1.22129i 0.261540 + 0.965193i \(0.415770\pi\)
−0.966651 + 0.256096i \(0.917564\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −18.0000 + 10.3923i −0.602347 + 0.347765i
\(894\) 0 0
\(895\) 36.0000 + 20.7846i 1.20335 + 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −24.0000 13.8564i −0.797787 0.460603i
\(906\) 0 0
\(907\) −34.5000 + 19.9186i −1.14555 + 0.661386i −0.947800 0.318866i \(-0.896698\pi\)
−0.197754 + 0.980252i \(0.563365\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 20.7846i −0.397578 0.688625i 0.595849 0.803097i \(-0.296816\pi\)
−0.993426 + 0.114472i \(0.963482\pi\)
\(912\) 0 0
\(913\) 18.0000 31.1769i 0.595713 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 41.5692i 1.37274i
\(918\) 0 0
\(919\) 41.5692i 1.37124i 0.727959 + 0.685621i \(0.240469\pi\)
−0.727959 + 0.685621i \(0.759531\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.0000 20.7846i 0.394985 0.684134i
\(924\) 0 0
\(925\) −7.00000 12.1244i −0.230159 0.398646i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 + 3.46410i −0.196854 + 0.113653i −0.595187 0.803587i \(-0.702922\pi\)
0.398333 + 0.917241i \(0.369589\pi\)
\(930\) 0 0
\(931\) 7.50000 + 4.33013i 0.245803 + 0.141914i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.0000 + 6.92820i 0.391189 + 0.225853i 0.682675 0.730722i \(-0.260816\pi\)
−0.291486 + 0.956575i \(0.594150\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.50000 + 7.79423i 0.146230 + 0.253278i 0.929831 0.367986i \(-0.119953\pi\)
−0.783601 + 0.621264i \(0.786619\pi\)
\(948\) 0 0
\(949\) −22.0000 + 38.1051i −0.714150 + 1.23694i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.1244i 0.392746i 0.980529 + 0.196373i \(0.0629164\pi\)
−0.980529 + 0.196373i \(0.937084\pi\)
\(954\) 0 0
\(955\) 20.7846i 0.672574i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 + 5.19615i −0.0968751 + 0.167793i
\(960\) 0 0
\(961\) −15.5000 26.8468i −0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 69.0000 39.8372i 2.22119 1.28240i
\(966\) 0 0
\(967\) −21.0000 12.1244i −0.675314 0.389893i 0.122773 0.992435i \(-0.460821\pi\)
−0.798087 + 0.602542i \(0.794155\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 66.0000 2.11586
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.5000 + 7.79423i 0.431903 + 0.249359i 0.700157 0.713989i \(-0.253113\pi\)
−0.268254 + 0.963348i \(0.586447\pi\)
\(978\) 0 0
\(979\) −36.0000 + 20.7846i −1.15056 + 0.664279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.00000 + 5.19615i 0.0956851 + 0.165732i 0.909894 0.414840i \(-0.136162\pi\)
−0.814209 + 0.580572i \(0.802829\pi\)
\(984\) 0 0
\(985\) −24.0000 + 41.5692i −0.764704 + 1.32451i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 48.4974i 1.54057i 0.637699 + 0.770286i \(0.279886\pi\)
−0.637699 + 0.770286i \(0.720114\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.00000 + 10.3923i −0.190213 + 0.329458i
\(996\) 0 0
\(997\) 10.0000 + 17.3205i 0.316703 + 0.548546i 0.979798 0.199989i \(-0.0640908\pi\)
−0.663095 + 0.748535i \(0.730757\pi\)
\(998\) 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 1728.2.s.a.1151.1 2
3.2 odd 2 576.2.s.d.383.1 2
4.3 odd 2 1728.2.s.b.1151.1 2
8.3 odd 2 432.2.s.d.287.1 2
8.5 even 2 432.2.s.c.287.1 2
9.2 odd 6 1728.2.s.b.575.1 2
9.4 even 3 5184.2.c.c.5183.2 2
9.5 odd 6 5184.2.c.a.5183.1 2
9.7 even 3 576.2.s.a.191.1 2
12.11 even 2 576.2.s.a.383.1 2
24.5 odd 2 144.2.s.a.95.1 yes 2
24.11 even 2 144.2.s.d.95.1 yes 2
36.7 odd 6 576.2.s.d.191.1 2
36.11 even 6 inner 1728.2.s.a.575.1 2
36.23 even 6 5184.2.c.c.5183.1 2
36.31 odd 6 5184.2.c.a.5183.2 2
72.5 odd 6 1296.2.c.d.1295.2 2
72.11 even 6 432.2.s.c.143.1 2
72.13 even 6 1296.2.c.b.1295.1 2
72.29 odd 6 432.2.s.d.143.1 2
72.43 odd 6 144.2.s.a.47.1 2
72.59 even 6 1296.2.c.b.1295.2 2
72.61 even 6 144.2.s.d.47.1 yes 2
72.67 odd 6 1296.2.c.d.1295.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.a.47.1 2 72.43 odd 6
144.2.s.a.95.1 yes 2 24.5 odd 2
144.2.s.d.47.1 yes 2 72.61 even 6
144.2.s.d.95.1 yes 2 24.11 even 2
432.2.s.c.143.1 2 72.11 even 6
432.2.s.c.287.1 2 8.5 even 2
432.2.s.d.143.1 2 72.29 odd 6
432.2.s.d.287.1 2 8.3 odd 2
576.2.s.a.191.1 2 9.7 even 3
576.2.s.a.383.1 2 12.11 even 2
576.2.s.d.191.1 2 36.7 odd 6
576.2.s.d.383.1 2 3.2 odd 2
1296.2.c.b.1295.1 2 72.13 even 6
1296.2.c.b.1295.2 2 72.59 even 6
1296.2.c.d.1295.1 2 72.67 odd 6
1296.2.c.d.1295.2 2 72.5 odd 6
1728.2.s.a.575.1 2 36.11 even 6 inner
1728.2.s.a.1151.1 2 1.1 even 1 trivial
1728.2.s.b.575.1 2 9.2 odd 6
1728.2.s.b.1151.1 2 4.3 odd 2
5184.2.c.a.5183.1 2 9.5 odd 6
5184.2.c.a.5183.2 2 36.31 odd 6
5184.2.c.c.5183.1 2 36.23 even 6
5184.2.c.c.5183.2 2 9.4 even 3