Properties

Label 1152.3.m.e.991.5
Level $1152$
Weight $3$
Character 1152.991
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(415,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 991.5
Root \(0.136762 + 1.99532i\) of defining polynomial
Character \(\chi\) \(=\) 1152.991
Dual form 1152.3.m.e.415.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +O(q^{10})\) \(q+(0.227650 - 0.227650i) q^{5} +3.90219 q^{7} +(-2.21045 - 2.21045i) q^{11} +(-5.08526 - 5.08526i) q^{13} +18.8341 q^{17} +(-11.7651 + 11.7651i) q^{19} +35.4354 q^{23} +24.8964i q^{25} +(-21.2499 - 21.2499i) q^{29} -35.9691i q^{31} +(0.888334 - 0.888334i) q^{35} +(34.4199 - 34.4199i) q^{37} -44.1055i q^{41} +(28.3018 + 28.3018i) q^{43} +32.8802i q^{47} -33.7729 q^{49} +(42.1450 - 42.1450i) q^{53} -1.00642 q^{55} +(66.9935 + 66.9935i) q^{59} +(-17.3728 - 17.3728i) q^{61} -2.31532 q^{65} +(-39.6756 + 39.6756i) q^{67} +63.0272 q^{71} +75.4549i q^{73} +(-8.62561 - 8.62561i) q^{77} -59.1583i q^{79} +(71.2155 - 71.2155i) q^{83} +(4.28757 - 4.28757i) q^{85} -150.671i q^{89} +(-19.8437 - 19.8437i) q^{91} +5.35665i q^{95} +51.5586 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{19} - 96 q^{37} + 32 q^{43} + 112 q^{49} - 256 q^{55} + 32 q^{61} + 256 q^{67} - 160 q^{85} - 288 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 0.227650 0.227650i 0.0455300 0.0455300i −0.683975 0.729505i \(-0.739750\pi\)
0.729505 + 0.683975i \(0.239750\pi\)
\(6\) 0 0
\(7\) 3.90219 0.557456 0.278728 0.960370i \(-0.410087\pi\)
0.278728 + 0.960370i \(0.410087\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.21045 2.21045i −0.200950 0.200950i 0.599457 0.800407i \(-0.295383\pi\)
−0.800407 + 0.599457i \(0.795383\pi\)
\(12\) 0 0
\(13\) −5.08526 5.08526i −0.391174 0.391174i 0.483932 0.875106i \(-0.339208\pi\)
−0.875106 + 0.483932i \(0.839208\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.8341 1.10789 0.553943 0.832555i \(-0.313123\pi\)
0.553943 + 0.832555i \(0.313123\pi\)
\(18\) 0 0
\(19\) −11.7651 + 11.7651i −0.619216 + 0.619216i −0.945330 0.326114i \(-0.894261\pi\)
0.326114 + 0.945330i \(0.394261\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 35.4354 1.54067 0.770335 0.637640i \(-0.220089\pi\)
0.770335 + 0.637640i \(0.220089\pi\)
\(24\) 0 0
\(25\) 24.8964i 0.995854i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −21.2499 21.2499i −0.732755 0.732755i 0.238410 0.971165i \(-0.423374\pi\)
−0.971165 + 0.238410i \(0.923374\pi\)
\(30\) 0 0
\(31\) 35.9691i 1.16029i −0.814512 0.580146i \(-0.802995\pi\)
0.814512 0.580146i \(-0.197005\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.888334 0.888334i 0.0253810 0.0253810i
\(36\) 0 0
\(37\) 34.4199 34.4199i 0.930267 0.930267i −0.0674555 0.997722i \(-0.521488\pi\)
0.997722 + 0.0674555i \(0.0214881\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 44.1055i 1.07574i −0.843026 0.537872i \(-0.819228\pi\)
0.843026 0.537872i \(-0.180772\pi\)
\(42\) 0 0
\(43\) 28.3018 + 28.3018i 0.658180 + 0.658180i 0.954949 0.296769i \(-0.0959091\pi\)
−0.296769 + 0.954949i \(0.595909\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 32.8802i 0.699579i 0.936828 + 0.349790i \(0.113747\pi\)
−0.936828 + 0.349790i \(0.886253\pi\)
\(48\) 0 0
\(49\) −33.7729 −0.689242
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 42.1450 42.1450i 0.795189 0.795189i −0.187144 0.982333i \(-0.559923\pi\)
0.982333 + 0.187144i \(0.0599230\pi\)
\(54\) 0 0
\(55\) −1.00642 −0.0182985
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 66.9935 + 66.9935i 1.13548 + 1.13548i 0.989250 + 0.146233i \(0.0467149\pi\)
0.146233 + 0.989250i \(0.453285\pi\)
\(60\) 0 0
\(61\) −17.3728 17.3728i −0.284800 0.284800i 0.550220 0.835020i \(-0.314544\pi\)
−0.835020 + 0.550220i \(0.814544\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.31532 −0.0356203
\(66\) 0 0
\(67\) −39.6756 + 39.6756i −0.592174 + 0.592174i −0.938218 0.346044i \(-0.887525\pi\)
0.346044 + 0.938218i \(0.387525\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 63.0272 0.887707 0.443853 0.896099i \(-0.353611\pi\)
0.443853 + 0.896099i \(0.353611\pi\)
\(72\) 0 0
\(73\) 75.4549i 1.03363i 0.856097 + 0.516815i \(0.172882\pi\)
−0.856097 + 0.516815i \(0.827118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.62561 8.62561i −0.112021 0.112021i
\(78\) 0 0
\(79\) 59.1583i 0.748839i −0.927259 0.374419i \(-0.877842\pi\)
0.927259 0.374419i \(-0.122158\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 71.2155 71.2155i 0.858018 0.858018i −0.133086 0.991104i \(-0.542489\pi\)
0.991104 + 0.133086i \(0.0424887\pi\)
\(84\) 0 0
\(85\) 4.28757 4.28757i 0.0504420 0.0504420i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 150.671i 1.69293i −0.532443 0.846466i \(-0.678726\pi\)
0.532443 0.846466i \(-0.321274\pi\)
\(90\) 0 0
\(91\) −19.8437 19.8437i −0.218062 0.218062i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.35665i 0.0563858i
\(96\) 0 0
\(97\) 51.5586 0.531532 0.265766 0.964038i \(-0.414375\pi\)
0.265766 + 0.964038i \(0.414375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 139.414 139.414i 1.38034 1.38034i 0.536327 0.844010i \(-0.319811\pi\)
0.844010 0.536327i \(-0.180189\pi\)
\(102\) 0 0
\(103\) 67.9691 0.659894 0.329947 0.943999i \(-0.392969\pi\)
0.329947 + 0.943999i \(0.392969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 129.801 + 129.801i 1.21309 + 1.21309i 0.970004 + 0.243090i \(0.0781609\pi\)
0.243090 + 0.970004i \(0.421839\pi\)
\(108\) 0 0
\(109\) −30.1323 30.1323i −0.276443 0.276443i 0.555244 0.831687i \(-0.312625\pi\)
−0.831687 + 0.555244i \(0.812625\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.11034 −0.0452243 −0.0226121 0.999744i \(-0.507198\pi\)
−0.0226121 + 0.999744i \(0.507198\pi\)
\(114\) 0 0
\(115\) 8.06687 8.06687i 0.0701467 0.0701467i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 73.4942 0.617598
\(120\) 0 0
\(121\) 111.228i 0.919238i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3589 + 11.3589i 0.0908712 + 0.0908712i
\(126\) 0 0
\(127\) 123.225i 0.970277i 0.874437 + 0.485138i \(0.161231\pi\)
−0.874437 + 0.485138i \(0.838769\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 76.2899 76.2899i 0.582365 0.582365i −0.353187 0.935553i \(-0.614902\pi\)
0.935553 + 0.353187i \(0.114902\pi\)
\(132\) 0 0
\(133\) −45.9098 + 45.9098i −0.345186 + 0.345186i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 73.0194i 0.532988i −0.963837 0.266494i \(-0.914135\pi\)
0.963837 0.266494i \(-0.0858653\pi\)
\(138\) 0 0
\(139\) −16.5250 16.5250i −0.118885 0.118885i 0.645162 0.764046i \(-0.276790\pi\)
−0.764046 + 0.645162i \(0.776790\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 22.4814i 0.157213i
\(144\) 0 0
\(145\) −9.67507 −0.0667246
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.0334 22.0334i 0.147875 0.147875i −0.629293 0.777168i \(-0.716656\pi\)
0.777168 + 0.629293i \(0.216656\pi\)
\(150\) 0 0
\(151\) −161.017 −1.06634 −0.533168 0.846010i \(-0.678999\pi\)
−0.533168 + 0.846010i \(0.678999\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.18836 8.18836i −0.0528281 0.0528281i
\(156\) 0 0
\(157\) 151.888 + 151.888i 0.967441 + 0.967441i 0.999486 0.0320455i \(-0.0102022\pi\)
−0.0320455 + 0.999486i \(0.510202\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 138.276 0.858856
\(162\) 0 0
\(163\) 186.334 186.334i 1.14315 1.14315i 0.155283 0.987870i \(-0.450371\pi\)
0.987870 0.155283i \(-0.0496290\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 199.590 1.19515 0.597576 0.801813i \(-0.296131\pi\)
0.597576 + 0.801813i \(0.296131\pi\)
\(168\) 0 0
\(169\) 117.280i 0.693966i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −138.631 138.631i −0.801333 0.801333i 0.181971 0.983304i \(-0.441752\pi\)
−0.983304 + 0.181971i \(0.941752\pi\)
\(174\) 0 0
\(175\) 97.1504i 0.555145i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −148.572 + 148.572i −0.830010 + 0.830010i −0.987518 0.157508i \(-0.949654\pi\)
0.157508 + 0.987518i \(0.449654\pi\)
\(180\) 0 0
\(181\) −99.6006 + 99.6006i −0.550280 + 0.550280i −0.926522 0.376242i \(-0.877216\pi\)
0.376242 + 0.926522i \(0.377216\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.6714i 0.0847101i
\(186\) 0 0
\(187\) −41.6318 41.6318i −0.222630 0.222630i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 149.517i 0.782812i 0.920218 + 0.391406i \(0.128011\pi\)
−0.920218 + 0.391406i \(0.871989\pi\)
\(192\) 0 0
\(193\) −52.8254 −0.273707 −0.136853 0.990591i \(-0.543699\pi\)
−0.136853 + 0.990591i \(0.543699\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −108.072 + 108.072i −0.548590 + 0.548590i −0.926033 0.377443i \(-0.876803\pi\)
0.377443 + 0.926033i \(0.376803\pi\)
\(198\) 0 0
\(199\) −2.24707 −0.0112918 −0.00564590 0.999984i \(-0.501797\pi\)
−0.00564590 + 0.999984i \(0.501797\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −82.9212 82.9212i −0.408479 0.408479i
\(204\) 0 0
\(205\) −10.0406 10.0406i −0.0489787 0.0489787i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 52.0124 0.248863
\(210\) 0 0
\(211\) −50.4041 + 50.4041i −0.238882 + 0.238882i −0.816387 0.577505i \(-0.804026\pi\)
0.577505 + 0.816387i \(0.304026\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.8858 0.0599339
\(216\) 0 0
\(217\) 140.358i 0.646812i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −95.7760 95.7760i −0.433376 0.433376i
\(222\) 0 0
\(223\) 230.340i 1.03291i 0.856313 + 0.516457i \(0.172749\pi\)
−0.856313 + 0.516457i \(0.827251\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.1395 12.1395i 0.0534779 0.0534779i −0.679862 0.733340i \(-0.737960\pi\)
0.733340 + 0.679862i \(0.237960\pi\)
\(228\) 0 0
\(229\) −39.1259 + 39.1259i −0.170855 + 0.170855i −0.787355 0.616500i \(-0.788550\pi\)
0.616500 + 0.787355i \(0.288550\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 36.6784i 0.157418i 0.996898 + 0.0787090i \(0.0250798\pi\)
−0.996898 + 0.0787090i \(0.974920\pi\)
\(234\) 0 0
\(235\) 7.48518 + 7.48518i 0.0318518 + 0.0318518i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 147.098i 0.615474i −0.951471 0.307737i \(-0.900428\pi\)
0.951471 0.307737i \(-0.0995717\pi\)
\(240\) 0 0
\(241\) 205.488 0.852649 0.426324 0.904570i \(-0.359808\pi\)
0.426324 + 0.904570i \(0.359808\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.68839 + 7.68839i −0.0313812 + 0.0313812i
\(246\) 0 0
\(247\) 119.657 0.484442
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −55.4867 55.4867i −0.221063 0.221063i 0.587883 0.808946i \(-0.299961\pi\)
−0.808946 + 0.587883i \(0.799961\pi\)
\(252\) 0 0
\(253\) −78.3282 78.3282i −0.309598 0.309598i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −142.878 −0.555944 −0.277972 0.960589i \(-0.589662\pi\)
−0.277972 + 0.960589i \(0.589662\pi\)
\(258\) 0 0
\(259\) 134.313 134.313i 0.518583 0.518583i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −242.691 −0.922780 −0.461390 0.887197i \(-0.652649\pi\)
−0.461390 + 0.887197i \(0.652649\pi\)
\(264\) 0 0
\(265\) 19.1886i 0.0724099i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 292.665 + 292.665i 1.08797 + 1.08797i 0.995737 + 0.0922353i \(0.0294012\pi\)
0.0922353 + 0.995737i \(0.470599\pi\)
\(270\) 0 0
\(271\) 258.180i 0.952694i −0.879257 0.476347i \(-0.841961\pi\)
0.879257 0.476347i \(-0.158039\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 55.0322 55.0322i 0.200117 0.200117i
\(276\) 0 0
\(277\) −63.2023 + 63.2023i −0.228167 + 0.228167i −0.811927 0.583759i \(-0.801581\pi\)
0.583759 + 0.811927i \(0.301581\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 421.406i 1.49966i 0.661628 + 0.749832i \(0.269866\pi\)
−0.661628 + 0.749832i \(0.730134\pi\)
\(282\) 0 0
\(283\) −275.707 275.707i −0.974229 0.974229i 0.0254475 0.999676i \(-0.491899\pi\)
−0.999676 + 0.0254475i \(0.991899\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 172.108i 0.599681i
\(288\) 0 0
\(289\) 65.7217 0.227411
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −181.330 + 181.330i −0.618873 + 0.618873i −0.945242 0.326369i \(-0.894175\pi\)
0.326369 + 0.945242i \(0.394175\pi\)
\(294\) 0 0
\(295\) 30.5021 0.103397
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −180.198 180.198i −0.602669 0.602669i
\(300\) 0 0
\(301\) 110.439 + 110.439i 0.366907 + 0.366907i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.90985 −0.0259339
\(306\) 0 0
\(307\) −279.027 + 279.027i −0.908883 + 0.908883i −0.996182 0.0872996i \(-0.972176\pi\)
0.0872996 + 0.996182i \(0.472176\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −414.511 −1.33283 −0.666417 0.745579i \(-0.732173\pi\)
−0.666417 + 0.745579i \(0.732173\pi\)
\(312\) 0 0
\(313\) 331.008i 1.05753i 0.848767 + 0.528767i \(0.177346\pi\)
−0.848767 + 0.528767i \(0.822654\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 194.563 + 194.563i 0.613764 + 0.613764i 0.943925 0.330161i \(-0.107103\pi\)
−0.330161 + 0.943925i \(0.607103\pi\)
\(318\) 0 0
\(319\) 93.9437i 0.294494i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −221.585 + 221.585i −0.686021 + 0.686021i
\(324\) 0 0
\(325\) 126.604 126.604i 0.389552 0.389552i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 128.305i 0.389985i
\(330\) 0 0
\(331\) 462.054 + 462.054i 1.39593 + 1.39593i 0.811292 + 0.584641i \(0.198765\pi\)
0.584641 + 0.811292i \(0.301235\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.0643i 0.0539233i
\(336\) 0 0
\(337\) −472.439 −1.40190 −0.700948 0.713212i \(-0.747240\pi\)
−0.700948 + 0.713212i \(0.747240\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −79.5078 + 79.5078i −0.233161 + 0.233161i
\(342\) 0 0
\(343\) −322.996 −0.941679
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −92.8976 92.8976i −0.267716 0.267716i 0.560463 0.828179i \(-0.310623\pi\)
−0.828179 + 0.560463i \(0.810623\pi\)
\(348\) 0 0
\(349\) −264.708 264.708i −0.758475 0.758475i 0.217570 0.976045i \(-0.430187\pi\)
−0.976045 + 0.217570i \(0.930187\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −413.360 −1.17099 −0.585495 0.810676i \(-0.699100\pi\)
−0.585495 + 0.810676i \(0.699100\pi\)
\(354\) 0 0
\(355\) 14.3481 14.3481i 0.0404173 0.0404173i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −625.198 −1.74150 −0.870749 0.491728i \(-0.836365\pi\)
−0.870749 + 0.491728i \(0.836365\pi\)
\(360\) 0 0
\(361\) 84.1643i 0.233142i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 17.1773 + 17.1773i 0.0470611 + 0.0470611i
\(366\) 0 0
\(367\) 601.904i 1.64007i −0.572316 0.820033i \(-0.693955\pi\)
0.572316 0.820033i \(-0.306045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 164.458 164.458i 0.443283 0.443283i
\(372\) 0 0
\(373\) 493.026 493.026i 1.32179 1.32179i 0.409455 0.912330i \(-0.365719\pi\)
0.912330 0.409455i \(-0.134281\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 216.122i 0.573269i
\(378\) 0 0
\(379\) 144.970 + 144.970i 0.382506 + 0.382506i 0.872004 0.489498i \(-0.162820\pi\)
−0.489498 + 0.872004i \(0.662820\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 417.764i 1.09077i −0.838187 0.545384i \(-0.816384\pi\)
0.838187 0.545384i \(-0.183616\pi\)
\(384\) 0 0
\(385\) −3.92724 −0.0102006
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −144.751 + 144.751i −0.372109 + 0.372109i −0.868245 0.496136i \(-0.834752\pi\)
0.496136 + 0.868245i \(0.334752\pi\)
\(390\) 0 0
\(391\) 667.392 1.70689
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.4674 13.4674i −0.0340946 0.0340946i
\(396\) 0 0
\(397\) 105.531 + 105.531i 0.265822 + 0.265822i 0.827414 0.561592i \(-0.189811\pi\)
−0.561592 + 0.827414i \(0.689811\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −529.383 −1.32016 −0.660078 0.751197i \(-0.729477\pi\)
−0.660078 + 0.751197i \(0.729477\pi\)
\(402\) 0 0
\(403\) −182.912 + 182.912i −0.453876 + 0.453876i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −152.167 −0.373874
\(408\) 0 0
\(409\) 371.204i 0.907590i 0.891106 + 0.453795i \(0.149930\pi\)
−0.891106 + 0.453795i \(0.850070\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 261.422 + 261.422i 0.632982 + 0.632982i
\(414\) 0 0
\(415\) 32.4244i 0.0781311i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −124.486 + 124.486i −0.297103 + 0.297103i −0.839878 0.542775i \(-0.817374\pi\)
0.542775 + 0.839878i \(0.317374\pi\)
\(420\) 0 0
\(421\) −51.7267 + 51.7267i −0.122866 + 0.122866i −0.765866 0.643000i \(-0.777690\pi\)
0.643000 + 0.765866i \(0.277690\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 468.899i 1.10329i
\(426\) 0 0
\(427\) −67.7922 67.7922i −0.158764 0.158764i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 424.978i 0.986029i 0.870021 + 0.493014i \(0.164105\pi\)
−0.870021 + 0.493014i \(0.835895\pi\)
\(432\) 0 0
\(433\) 283.151 0.653927 0.326964 0.945037i \(-0.393975\pi\)
0.326964 + 0.945037i \(0.393975\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −416.901 + 416.901i −0.954008 + 0.954008i
\(438\) 0 0
\(439\) −701.459 −1.59786 −0.798928 0.601427i \(-0.794599\pi\)
−0.798928 + 0.601427i \(0.794599\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −272.651 272.651i −0.615464 0.615464i 0.328900 0.944365i \(-0.393322\pi\)
−0.944365 + 0.328900i \(0.893322\pi\)
\(444\) 0 0
\(445\) −34.3002 34.3002i −0.0770792 0.0770792i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 614.711 1.36907 0.684534 0.728981i \(-0.260006\pi\)
0.684534 + 0.728981i \(0.260006\pi\)
\(450\) 0 0
\(451\) −97.4931 + 97.4931i −0.216171 + 0.216171i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.03482 −0.0198567
\(456\) 0 0
\(457\) 117.859i 0.257898i 0.991651 + 0.128949i \(0.0411603\pi\)
−0.991651 + 0.128949i \(0.958840\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −262.235 262.235i −0.568839 0.568839i 0.362964 0.931803i \(-0.381765\pi\)
−0.931803 + 0.362964i \(0.881765\pi\)
\(462\) 0 0
\(463\) 192.234i 0.415192i −0.978215 0.207596i \(-0.933436\pi\)
0.978215 0.207596i \(-0.0665640\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −529.169 + 529.169i −1.13312 + 1.13312i −0.143470 + 0.989655i \(0.545826\pi\)
−0.989655 + 0.143470i \(0.954174\pi\)
\(468\) 0 0
\(469\) −154.822 + 154.822i −0.330111 + 0.330111i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 125.119i 0.264523i
\(474\) 0 0
\(475\) −292.908 292.908i −0.616649 0.616649i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.43750i 0.00300104i 0.999999 + 0.00150052i \(0.000477630\pi\)
−0.999999 + 0.00150052i \(0.999522\pi\)
\(480\) 0 0
\(481\) −350.068 −0.727792
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.7373 11.7373i 0.0242006 0.0242006i
\(486\) 0 0
\(487\) 701.238 1.43991 0.719957 0.694019i \(-0.244162\pi\)
0.719957 + 0.694019i \(0.244162\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −519.668 519.668i −1.05839 1.05839i −0.998186 0.0602007i \(-0.980826\pi\)
−0.0602007 0.998186i \(-0.519174\pi\)
\(492\) 0 0
\(493\) −400.222 400.222i −0.811809 0.811809i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 245.944 0.494858
\(498\) 0 0
\(499\) −359.284 + 359.284i −0.720009 + 0.720009i −0.968607 0.248598i \(-0.920030\pi\)
0.248598 + 0.968607i \(0.420030\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 174.376 0.346671 0.173335 0.984863i \(-0.444546\pi\)
0.173335 + 0.984863i \(0.444546\pi\)
\(504\) 0 0
\(505\) 63.4752i 0.125693i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −314.247 314.247i −0.617381 0.617381i 0.327478 0.944859i \(-0.393801\pi\)
−0.944859 + 0.327478i \(0.893801\pi\)
\(510\) 0 0
\(511\) 294.440i 0.576203i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 15.4732 15.4732i 0.0300450 0.0300450i
\(516\) 0 0
\(517\) 72.6801 72.6801i 0.140580 0.140580i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 468.678i 0.899575i 0.893136 + 0.449787i \(0.148500\pi\)
−0.893136 + 0.449787i \(0.851500\pi\)
\(522\) 0 0
\(523\) 417.239 + 417.239i 0.797780 + 0.797780i 0.982745 0.184965i \(-0.0592173\pi\)
−0.184965 + 0.982745i \(0.559217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 677.443i 1.28547i
\(528\) 0 0
\(529\) 726.667 1.37366
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −224.288 + 224.288i −0.420803 + 0.420803i
\(534\) 0 0
\(535\) 59.0984 0.110464
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 74.6533 + 74.6533i 0.138503 + 0.138503i
\(540\) 0 0
\(541\) 163.152 + 163.152i 0.301574 + 0.301574i 0.841629 0.540055i \(-0.181597\pi\)
−0.540055 + 0.841629i \(0.681597\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13.7192 −0.0251729
\(546\) 0 0
\(547\) 595.267 595.267i 1.08824 1.08824i 0.0925300 0.995710i \(-0.470505\pi\)
0.995710 0.0925300i \(-0.0294954\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 500.015 0.907468
\(552\) 0 0
\(553\) 230.847i 0.417445i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 370.177 + 370.177i 0.664590 + 0.664590i 0.956459 0.291868i \(-0.0942769\pi\)
−0.291868 + 0.956459i \(0.594277\pi\)
\(558\) 0 0
\(559\) 287.843i 0.514926i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 426.543 426.543i 0.757625 0.757625i −0.218264 0.975890i \(-0.570040\pi\)
0.975890 + 0.218264i \(0.0700395\pi\)
\(564\) 0 0
\(565\) −1.16337 + 1.16337i −0.00205906 + 0.00205906i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 291.547i 0.512385i −0.966626 0.256192i \(-0.917532\pi\)
0.966626 0.256192i \(-0.0824680\pi\)
\(570\) 0 0
\(571\) −569.328 569.328i −0.997071 0.997071i 0.00292477 0.999996i \(-0.499069\pi\)
−0.999996 + 0.00292477i \(0.999069\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 882.212i 1.53428i
\(576\) 0 0
\(577\) −360.847 −0.625385 −0.312692 0.949854i \(-0.601231\pi\)
−0.312692 + 0.949854i \(0.601231\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 277.897 277.897i 0.478308 0.478308i
\(582\) 0 0
\(583\) −186.319 −0.319586
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 373.439 + 373.439i 0.636183 + 0.636183i 0.949612 0.313429i \(-0.101478\pi\)
−0.313429 + 0.949612i \(0.601478\pi\)
\(588\) 0 0
\(589\) 423.180 + 423.180i 0.718472 + 0.718472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 608.053 1.02538 0.512692 0.858572i \(-0.328648\pi\)
0.512692 + 0.858572i \(0.328648\pi\)
\(594\) 0 0
\(595\) 16.7309 16.7309i 0.0281192 0.0281192i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 211.563 0.353194 0.176597 0.984283i \(-0.443491\pi\)
0.176597 + 0.984283i \(0.443491\pi\)
\(600\) 0 0
\(601\) 343.381i 0.571349i −0.958327 0.285674i \(-0.907782\pi\)
0.958327 0.285674i \(-0.0922176\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.3210 25.3210i −0.0418529 0.0418529i
\(606\) 0 0
\(607\) 1006.23i 1.65771i 0.559461 + 0.828856i \(0.311008\pi\)
−0.559461 + 0.828856i \(0.688992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 167.204 167.204i 0.273657 0.273657i
\(612\) 0 0
\(613\) −380.705 + 380.705i −0.621052 + 0.621052i −0.945801 0.324748i \(-0.894720\pi\)
0.324748 + 0.945801i \(0.394720\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 693.808i 1.12449i −0.826972 0.562243i \(-0.809939\pi\)
0.826972 0.562243i \(-0.190061\pi\)
\(618\) 0 0
\(619\) 619.279 + 619.279i 1.00045 + 1.00045i 1.00000 0.000450619i \(0.000143436\pi\)
0.000450619 1.00000i \(0.499857\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 587.947i 0.943736i
\(624\) 0 0
\(625\) −617.237 −0.987579
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 648.266 648.266i 1.03063 1.03063i
\(630\) 0 0
\(631\) −1007.91 −1.59732 −0.798661 0.601781i \(-0.794458\pi\)
−0.798661 + 0.601781i \(0.794458\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 28.0522 + 28.0522i 0.0441767 + 0.0441767i
\(636\) 0 0
\(637\) 171.744 + 171.744i 0.269613 + 0.269613i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −511.481 −0.797943 −0.398971 0.916963i \(-0.630633\pi\)
−0.398971 + 0.916963i \(0.630633\pi\)
\(642\) 0 0
\(643\) 497.727 497.727i 0.774071 0.774071i −0.204745 0.978815i \(-0.565636\pi\)
0.978815 + 0.204745i \(0.0656365\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 59.8694 0.0925339 0.0462669 0.998929i \(-0.485268\pi\)
0.0462669 + 0.998929i \(0.485268\pi\)
\(648\) 0 0
\(649\) 296.172i 0.456351i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −489.952 489.952i −0.750310 0.750310i 0.224227 0.974537i \(-0.428014\pi\)
−0.974537 + 0.224227i \(0.928014\pi\)
\(654\) 0 0
\(655\) 34.7348i 0.0530302i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −234.984 + 234.984i −0.356577 + 0.356577i −0.862549 0.505973i \(-0.831134\pi\)
0.505973 + 0.862549i \(0.331134\pi\)
\(660\) 0 0
\(661\) −305.060 + 305.060i −0.461512 + 0.461512i −0.899151 0.437639i \(-0.855815\pi\)
0.437639 + 0.899151i \(0.355815\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.9027i 0.0314326i
\(666\) 0 0
\(667\) −752.998 752.998i −1.12893 1.12893i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 76.8036i 0.114461i
\(672\) 0 0
\(673\) 458.738 0.681632 0.340816 0.940130i \(-0.389297\pi\)
0.340816 + 0.940130i \(0.389297\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −171.344 + 171.344i −0.253094 + 0.253094i −0.822238 0.569144i \(-0.807275\pi\)
0.569144 + 0.822238i \(0.307275\pi\)
\(678\) 0 0
\(679\) 201.192 0.296306
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 769.447 + 769.447i 1.12657 + 1.12657i 0.990731 + 0.135838i \(0.0433728\pi\)
0.135838 + 0.990731i \(0.456627\pi\)
\(684\) 0 0
\(685\) −16.6229 16.6229i −0.0242670 0.0242670i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −428.636 −0.622114
\(690\) 0 0
\(691\) −44.7951 + 44.7951i −0.0648264 + 0.0648264i −0.738777 0.673950i \(-0.764596\pi\)
0.673950 + 0.738777i \(0.264596\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.52381 −0.0108256
\(696\) 0 0
\(697\) 830.686i 1.19180i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.7012 31.7012i −0.0452229 0.0452229i 0.684134 0.729357i \(-0.260180\pi\)
−0.729357 + 0.684134i \(0.760180\pi\)
\(702\) 0 0
\(703\) 809.907i 1.15207i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 544.021 544.021i 0.769478 0.769478i
\(708\) 0 0
\(709\) −256.706 + 256.706i −0.362068 + 0.362068i −0.864574 0.502506i \(-0.832411\pi\)
0.502506 + 0.864574i \(0.332411\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1274.58i 1.78763i
\(714\) 0 0
\(715\) 5.11789 + 5.11789i 0.00715789 + 0.00715789i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1368.29i 1.90305i 0.307574 + 0.951524i \(0.400483\pi\)
−0.307574 + 0.951524i \(0.599517\pi\)
\(720\) 0 0
\(721\) 265.229 0.367862
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 529.045 529.045i 0.729717 0.729717i
\(726\) 0 0
\(727\) 37.5336 0.0516281 0.0258141 0.999667i \(-0.491782\pi\)
0.0258141 + 0.999667i \(0.491782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 533.037 + 533.037i 0.729189 + 0.729189i
\(732\) 0 0
\(733\) 385.216 + 385.216i 0.525534 + 0.525534i 0.919237 0.393704i \(-0.128806\pi\)
−0.393704 + 0.919237i \(0.628806\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 175.402 0.237995
\(738\) 0 0
\(739\) 889.039 889.039i 1.20303 1.20303i 0.229790 0.973240i \(-0.426196\pi\)
0.973240 0.229790i \(-0.0738040\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 506.252 0.681362 0.340681 0.940179i \(-0.389342\pi\)
0.340681 + 0.940179i \(0.389342\pi\)
\(744\) 0 0
\(745\) 10.0318i 0.0134655i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 506.509 + 506.509i 0.676247 + 0.676247i
\(750\) 0 0
\(751\) 456.844i 0.608314i −0.952622 0.304157i \(-0.901625\pi\)
0.952622 0.304157i \(-0.0983748\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.6554 + 36.6554i −0.0485502 + 0.0485502i
\(756\) 0 0
\(757\) 221.708 221.708i 0.292877 0.292877i −0.545339 0.838216i \(-0.683599\pi\)
0.838216 + 0.545339i \(0.183599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 205.900i 0.270566i −0.990807 0.135283i \(-0.956806\pi\)
0.990807 0.135283i \(-0.0431943\pi\)
\(762\) 0 0
\(763\) −117.582 117.582i −0.154105 0.154105i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 681.359i 0.888342i
\(768\) 0 0
\(769\) 900.133 1.17052 0.585262 0.810844i \(-0.300992\pi\)
0.585262 + 0.810844i \(0.300992\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 870.407 870.407i 1.12601 1.12601i 0.135193 0.990819i \(-0.456835\pi\)
0.990819 0.135193i \(-0.0431654\pi\)
\(774\) 0 0
\(775\) 895.498 1.15548
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 518.907 + 518.907i 0.666119 + 0.666119i
\(780\) 0 0
\(781\) −139.318 139.318i −0.178385 0.178385i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 69.1547 0.0880951
\(786\) 0 0
\(787\) −1005.24 + 1005.24i −1.27730 + 1.27730i −0.335133 + 0.942171i \(0.608781\pi\)
−0.942171 + 0.335133i \(0.891219\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −19.9415 −0.0252106
\(792\) 0 0
\(793\) 176.691i 0.222813i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −434.779 434.779i −0.545520 0.545520i 0.379622 0.925142i \(-0.376054\pi\)
−0.925142 + 0.379622i \(0.876054\pi\)
\(798\) 0 0
\(799\) 619.268i 0.775054i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 166.789 166.789i 0.207708 0.207708i
\(804\) 0 0
\(805\) 31.4785 31.4785i 0.0391037 0.0391037i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.0002i 0.0618049i −0.999522 0.0309024i \(-0.990162\pi\)
0.999522 0.0309024i \(-0.00983812\pi\)
\(810\) 0 0
\(811\) 36.9957 + 36.9957i 0.0456173 + 0.0456173i 0.729547 0.683930i \(-0.239731\pi\)
−0.683930 + 0.729547i \(0.739731\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 84.8378i 0.104096i
\(816\) 0 0
\(817\) −665.947 −0.815112
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 297.373 297.373i 0.362208 0.362208i −0.502417 0.864625i \(-0.667556\pi\)
0.864625 + 0.502417i \(0.167556\pi\)
\(822\) 0 0
\(823\) −345.052 −0.419262 −0.209631 0.977781i \(-0.567226\pi\)
−0.209631 + 0.977781i \(0.567226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 426.416 + 426.416i 0.515618 + 0.515618i 0.916242 0.400624i \(-0.131207\pi\)
−0.400624 + 0.916242i \(0.631207\pi\)
\(828\) 0 0
\(829\) −897.159 897.159i −1.08222 1.08222i −0.996302 0.0859157i \(-0.972618\pi\)
−0.0859157 0.996302i \(-0.527382\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −636.080 −0.763602
\(834\) 0 0
\(835\) 45.4367 45.4367i 0.0544152 0.0544152i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 708.037 0.843906 0.421953 0.906618i \(-0.361345\pi\)
0.421953 + 0.906618i \(0.361345\pi\)
\(840\) 0 0
\(841\) 62.1159i 0.0738595i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.6989 26.6989i −0.0315963 0.0315963i
\(846\) 0 0
\(847\) 434.033i 0.512435i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1219.68 1219.68i 1.43323 1.43323i
\(852\) 0 0
\(853\) −795.484 + 795.484i −0.932572 + 0.932572i −0.997866 0.0652942i \(-0.979201\pi\)
0.0652942 + 0.997866i \(0.479201\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1447.28i 1.68877i −0.535735 0.844386i \(-0.679966\pi\)
0.535735 0.844386i \(-0.320034\pi\)
\(858\) 0 0
\(859\) 99.6061 + 99.6061i 0.115956 + 0.115956i 0.762704 0.646748i \(-0.223871\pi\)
−0.646748 + 0.762704i \(0.723871\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 549.773i 0.637049i 0.947915 + 0.318525i \(0.103187\pi\)
−0.947915 + 0.318525i \(0.896813\pi\)
\(864\) 0 0
\(865\) −63.1185 −0.0729694
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −130.766 + 130.766i −0.150479 + 0.150479i
\(870\) 0 0
\(871\) 403.522 0.463286
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 44.3246 + 44.3246i 0.0506567 + 0.0506567i
\(876\) 0 0
\(877\) 165.469 + 165.469i 0.188677 + 0.188677i 0.795124 0.606447i \(-0.207406\pi\)
−0.606447 + 0.795124i \(0.707406\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1046.60 1.18796 0.593982 0.804479i \(-0.297555\pi\)
0.593982 + 0.804479i \(0.297555\pi\)
\(882\) 0 0
\(883\) −338.581 + 338.581i −0.383444 + 0.383444i −0.872341 0.488897i \(-0.837399\pi\)
0.488897 + 0.872341i \(0.337399\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1486.13 1.67546 0.837728 0.546088i \(-0.183883\pi\)
0.837728 + 0.546088i \(0.183883\pi\)
\(888\) 0 0
\(889\) 480.848i 0.540887i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −386.840 386.840i −0.433191 0.433191i
\(894\) 0 0
\(895\) 67.6447i 0.0755807i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −764.339 + 764.339i −0.850210 + 0.850210i
\(900\) 0 0
\(901\) 793.761 793.761i 0.880978 0.880978i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 45.3482i 0.0501085i
\(906\) 0 0
\(907\) 66.8336 + 66.8336i 0.0736864 + 0.0736864i 0.742989 0.669303i \(-0.233407\pi\)
−0.669303 + 0.742989i \(0.733407\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 227.208i 0.249406i −0.992194 0.124703i \(-0.960202\pi\)
0.992194 0.124703i \(-0.0397977\pi\)
\(912\) 0 0
\(913\) −314.837 −0.344838
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 297.698 297.698i 0.324643 0.324643i
\(918\) 0 0
\(919\) −1375.00 −1.49619 −0.748097 0.663589i \(-0.769032\pi\)
−0.748097 + 0.663589i \(0.769032\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −320.509 320.509i −0.347247 0.347247i
\(924\) 0 0
\(925\) 856.929 + 856.929i 0.926410 + 0.926410i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1788.04 1.92470 0.962348 0.271822i \(-0.0876262\pi\)
0.962348 + 0.271822i \(0.0876262\pi\)
\(930\) 0 0
\(931\) 397.342 397.342i 0.426790 0.426790i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −18.9549 −0.0202727
\(936\) 0 0
\(937\) 768.834i 0.820527i 0.911967 + 0.410263i \(0.134563\pi\)
−0.911967 + 0.410263i \(0.865437\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1016.85 1016.85i −1.08061 1.08061i −0.996453 0.0841571i \(-0.973180\pi\)
−0.0841571 0.996453i \(-0.526820\pi\)
\(942\) 0 0
\(943\) 1562.90i 1.65737i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −476.431 + 476.431i −0.503095 + 0.503095i −0.912398 0.409304i \(-0.865772\pi\)
0.409304 + 0.912398i \(0.365772\pi\)
\(948\) 0 0
\(949\) 383.708 383.708i 0.404329 0.404329i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 577.529i 0.606011i 0.952989 + 0.303006i \(0.0979901\pi\)
−0.952989 + 0.303006i \(0.902010\pi\)
\(954\) 0 0
\(955\) 34.0375 + 34.0375i 0.0356414 + 0.0356414i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 284.936i 0.297118i
\(960\) 0 0
\(961\) −332.773 −0.346278
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.0257 + 12.0257i −0.0124619 + 0.0124619i
\(966\) 0 0
\(967\) 864.047 0.893534 0.446767 0.894650i \(-0.352575\pi\)
0.446767 + 0.894650i \(0.352575\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −898.515 898.515i −0.925350 0.925350i 0.0720505 0.997401i \(-0.477046\pi\)
−0.997401 + 0.0720505i \(0.977046\pi\)
\(972\) 0 0
\(973\) −64.4836 64.4836i −0.0662730 0.0662730i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −538.131 −0.550799 −0.275400 0.961330i \(-0.588810\pi\)
−0.275400 + 0.961330i \(0.588810\pi\)
\(978\) 0 0
\(979\) −333.051 + 333.051i −0.340195 + 0.340195i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 759.452 0.772586 0.386293 0.922376i \(-0.373755\pi\)
0.386293 + 0.922376i \(0.373755\pi\)
\(984\) 0 0
\(985\) 49.2052i 0.0499546i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1002.88 + 1002.88i 1.01404 + 1.01404i
\(990\) 0 0
\(991\) 514.565i 0.519238i 0.965711 + 0.259619i \(0.0835969\pi\)
−0.965711 + 0.259619i \(0.916403\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.511545 + 0.511545i −0.000514115 + 0.000514115i
\(996\) 0 0
\(997\) 1249.59 1249.59i 1.25335 1.25335i 0.299145 0.954208i \(-0.403299\pi\)
0.954208 0.299145i \(-0.0967013\pi\)
\(998\) 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 1152.3.m.e.991.5 16
3.2 odd 2 inner 1152.3.m.e.991.4 16
4.3 odd 2 1152.3.m.d.991.5 16
8.3 odd 2 576.3.m.b.559.4 16
8.5 even 2 144.3.m.b.19.1 16
12.11 even 2 1152.3.m.d.991.4 16
16.3 odd 4 144.3.m.b.91.1 yes 16
16.5 even 4 1152.3.m.d.415.5 16
16.11 odd 4 inner 1152.3.m.e.415.5 16
16.13 even 4 576.3.m.b.271.4 16
24.5 odd 2 144.3.m.b.19.8 yes 16
24.11 even 2 576.3.m.b.559.5 16
48.5 odd 4 1152.3.m.d.415.4 16
48.11 even 4 inner 1152.3.m.e.415.4 16
48.29 odd 4 576.3.m.b.271.5 16
48.35 even 4 144.3.m.b.91.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.m.b.19.1 16 8.5 even 2
144.3.m.b.19.8 yes 16 24.5 odd 2
144.3.m.b.91.1 yes 16 16.3 odd 4
144.3.m.b.91.8 yes 16 48.35 even 4
576.3.m.b.271.4 16 16.13 even 4
576.3.m.b.271.5 16 48.29 odd 4
576.3.m.b.559.4 16 8.3 odd 2
576.3.m.b.559.5 16 24.11 even 2
1152.3.m.d.415.4 16 48.5 odd 4
1152.3.m.d.415.5 16 16.5 even 4
1152.3.m.d.991.4 16 12.11 even 2
1152.3.m.d.991.5 16 4.3 odd 2
1152.3.m.e.415.4 16 48.11 even 4 inner
1152.3.m.e.415.5 16 16.11 odd 4 inner
1152.3.m.e.991.4 16 3.2 odd 2 inner
1152.3.m.e.991.5 16 1.1 even 1 trivial