Properties

Label 8568.2.a.bj.1.1
Level $8568$
Weight $2$
Character 8568.1
Self dual yes
Analytic conductor $68.416$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8568,2,Mod(1,8568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8568.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8568.a (trivial)

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: yes
Analytic conductor: \(68.4158244518\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48718\) of defining polynomial
Character \(\chi\) \(=\) 8568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.02435 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-4.02435 q^{5} -1.00000 q^{7} +1.23607 q^{11} -6.47214 q^{13} -1.00000 q^{17} +6.97437 q^{19} -7.07433 q^{23} +11.1954 q^{25} +6.31040 q^{29} +4.88824 q^{31} +4.02435 q^{35} -2.63387 q^{37} +10.1954 q^{41} -2.69762 q^{43} +3.67652 q^{47} +1.00000 q^{49} +1.39047 q^{53} -4.97437 q^{55} -5.07433 q^{59} +3.93369 q^{61} +26.0461 q^{65} -5.63259 q^{67} +6.84431 q^{71} +0.246650 q^{73} -1.23607 q^{77} -11.6765 q^{79} +6.65089 q^{83} +4.02435 q^{85} -2.27872 q^{89} +6.47214 q^{91} -28.0673 q^{95} +10.7069 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61} + 8 q^{65} + 15 q^{67} - 2 q^{71} - 5 q^{73} + 4 q^{77} - 24 q^{79} - 10 q^{83} - q^{85} + 16 q^{89} + 8 q^{91} - 22 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −4.02435 −1.79974 −0.899872 0.436154i \(-0.856340\pi\)
−0.899872 + 0.436154i \(0.856340\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.23607 0.372689 0.186344 0.982485i \(-0.440336\pi\)
0.186344 + 0.982485i \(0.440336\pi\)
\(12\) 0 0
\(13\) −6.47214 −1.79505 −0.897524 0.440966i \(-0.854636\pi\)
−0.897524 + 0.440966i \(0.854636\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 6.97437 1.60003 0.800015 0.599980i \(-0.204825\pi\)
0.800015 + 0.599980i \(0.204825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.07433 −1.47510 −0.737550 0.675293i \(-0.764017\pi\)
−0.737550 + 0.675293i \(0.764017\pi\)
\(24\) 0 0
\(25\) 11.1954 2.23908
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.31040 1.17181 0.585906 0.810379i \(-0.300739\pi\)
0.585906 + 0.810379i \(0.300739\pi\)
\(30\) 0 0
\(31\) 4.88824 0.877954 0.438977 0.898498i \(-0.355341\pi\)
0.438977 + 0.898498i \(0.355341\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.02435 0.680239
\(36\) 0 0
\(37\) −2.63387 −0.433006 −0.216503 0.976282i \(-0.569465\pi\)
−0.216503 + 0.976282i \(0.569465\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.1954 1.59225 0.796126 0.605131i \(-0.206879\pi\)
0.796126 + 0.605131i \(0.206879\pi\)
\(42\) 0 0
\(43\) −2.69762 −0.411384 −0.205692 0.978617i \(-0.565944\pi\)
−0.205692 + 0.978617i \(0.565944\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.67652 0.536276 0.268138 0.963381i \(-0.413592\pi\)
0.268138 + 0.963381i \(0.413592\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.39047 0.190996 0.0954982 0.995430i \(-0.469556\pi\)
0.0954982 + 0.995430i \(0.469556\pi\)
\(54\) 0 0
\(55\) −4.97437 −0.670744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.07433 −0.660621 −0.330311 0.943872i \(-0.607154\pi\)
−0.330311 + 0.943872i \(0.607154\pi\)
\(60\) 0 0
\(61\) 3.93369 0.503657 0.251829 0.967772i \(-0.418968\pi\)
0.251829 + 0.967772i \(0.418968\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 26.0461 3.23063
\(66\) 0 0
\(67\) −5.63259 −0.688131 −0.344065 0.938946i \(-0.611804\pi\)
−0.344065 + 0.938946i \(0.611804\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.84431 0.812270 0.406135 0.913813i \(-0.366876\pi\)
0.406135 + 0.913813i \(0.366876\pi\)
\(72\) 0 0
\(73\) 0.246650 0.0288682 0.0144341 0.999896i \(-0.495405\pi\)
0.0144341 + 0.999896i \(0.495405\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.23607 −0.140863
\(78\) 0 0
\(79\) −11.6765 −1.31371 −0.656856 0.754016i \(-0.728114\pi\)
−0.656856 + 0.754016i \(0.728114\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.65089 0.730030 0.365015 0.931002i \(-0.381064\pi\)
0.365015 + 0.931002i \(0.381064\pi\)
\(84\) 0 0
\(85\) 4.02435 0.436502
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.27872 −0.241543 −0.120772 0.992680i \(-0.538537\pi\)
−0.120772 + 0.992680i \(0.538537\pi\)
\(90\) 0 0
\(91\) 6.47214 0.678464
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −28.0673 −2.87964
\(96\) 0 0
\(97\) 10.7069 1.08712 0.543562 0.839369i \(-0.317075\pi\)
0.543562 + 0.839369i \(0.317075\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.7931 1.67097 0.835486 0.549512i \(-0.185187\pi\)
0.835486 + 0.549512i \(0.185187\pi\)
\(102\) 0 0
\(103\) 9.57656 0.943607 0.471803 0.881704i \(-0.343603\pi\)
0.471803 + 0.881704i \(0.343603\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.9613 −1.63971 −0.819855 0.572571i \(-0.805946\pi\)
−0.819855 + 0.572571i \(0.805946\pi\)
\(108\) 0 0
\(109\) −19.6082 −1.87813 −0.939065 0.343741i \(-0.888306\pi\)
−0.939065 + 0.343741i \(0.888306\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4952 1.26952 0.634761 0.772709i \(-0.281099\pi\)
0.634761 + 0.772709i \(0.281099\pi\)
\(114\) 0 0
\(115\) 28.4696 2.65480
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.9324 −2.23002
\(126\) 0 0
\(127\) 4.36484 0.387317 0.193659 0.981069i \(-0.437965\pi\)
0.193659 + 0.981069i \(0.437965\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.6826 −0.933341 −0.466670 0.884431i \(-0.654547\pi\)
−0.466670 + 0.884431i \(0.654547\pi\)
\(132\) 0 0
\(133\) −6.97437 −0.604755
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.1954 −1.55454 −0.777268 0.629169i \(-0.783395\pi\)
−0.777268 + 0.629169i \(0.783395\pi\)
\(138\) 0 0
\(139\) 13.8430 1.17415 0.587075 0.809532i \(-0.300279\pi\)
0.587075 + 0.809532i \(0.300279\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −25.3952 −2.10896
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.42987 0.280986 0.140493 0.990082i \(-0.455131\pi\)
0.140493 + 0.990082i \(0.455131\pi\)
\(150\) 0 0
\(151\) −9.16976 −0.746224 −0.373112 0.927786i \(-0.621709\pi\)
−0.373112 + 0.927786i \(0.621709\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −19.6720 −1.58009
\(156\) 0 0
\(157\) −6.70215 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.07433 0.557535
\(162\) 0 0
\(163\) −14.4104 −1.12871 −0.564353 0.825533i \(-0.690874\pi\)
−0.564353 + 0.825533i \(0.690874\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.25112 0.251579 0.125789 0.992057i \(-0.459854\pi\)
0.125789 + 0.992057i \(0.459854\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.25717 −0.323666 −0.161833 0.986818i \(-0.551741\pi\)
−0.161833 + 0.986818i \(0.551741\pi\)
\(174\) 0 0
\(175\) −11.1954 −0.846292
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.72772 0.203879 0.101940 0.994791i \(-0.467495\pi\)
0.101940 + 0.994791i \(0.467495\pi\)
\(180\) 0 0
\(181\) 3.15471 0.234488 0.117244 0.993103i \(-0.462594\pi\)
0.117244 + 0.993103i \(0.462594\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.5996 0.779300
\(186\) 0 0
\(187\) −1.23607 −0.0903902
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.53263 −0.255612 −0.127806 0.991799i \(-0.540794\pi\)
−0.127806 + 0.991799i \(0.540794\pi\)
\(192\) 0 0
\(193\) 16.2275 1.16808 0.584039 0.811726i \(-0.301472\pi\)
0.584039 + 0.811726i \(0.301472\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.7569 −1.26513 −0.632563 0.774509i \(-0.717997\pi\)
−0.632563 + 0.774509i \(0.717997\pi\)
\(198\) 0 0
\(199\) −15.1486 −1.07386 −0.536928 0.843628i \(-0.680415\pi\)
−0.536928 + 0.843628i \(0.680415\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.31040 −0.442903
\(204\) 0 0
\(205\) −41.0298 −2.86565
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.62079 0.596313
\(210\) 0 0
\(211\) 10.5826 0.728537 0.364269 0.931294i \(-0.381319\pi\)
0.364269 + 0.931294i \(0.381319\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.8562 0.740385
\(216\) 0 0
\(217\) −4.88824 −0.331835
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 0 0
\(223\) −3.95130 −0.264599 −0.132299 0.991210i \(-0.542236\pi\)
−0.132299 + 0.991210i \(0.542236\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.1031 −0.936059 −0.468029 0.883713i \(-0.655036\pi\)
−0.468029 + 0.883713i \(0.655036\pi\)
\(228\) 0 0
\(229\) −17.5766 −1.16149 −0.580746 0.814085i \(-0.697239\pi\)
−0.580746 + 0.814085i \(0.697239\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.5766 −1.02045 −0.510227 0.860040i \(-0.670439\pi\)
−0.510227 + 0.860040i \(0.670439\pi\)
\(234\) 0 0
\(235\) −14.7956 −0.965159
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.16976 0.334404 0.167202 0.985923i \(-0.446527\pi\)
0.167202 + 0.985923i \(0.446527\pi\)
\(240\) 0 0
\(241\) −2.45830 −0.158353 −0.0791766 0.996861i \(-0.525229\pi\)
−0.0791766 + 0.996861i \(0.525229\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.02435 −0.257106
\(246\) 0 0
\(247\) −45.1391 −2.87213
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3863 1.09741 0.548707 0.836015i \(-0.315120\pi\)
0.548707 + 0.836015i \(0.315120\pi\)
\(252\) 0 0
\(253\) −8.74435 −0.549753
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.2787 0.641169 0.320584 0.947220i \(-0.396121\pi\)
0.320584 + 0.947220i \(0.396121\pi\)
\(258\) 0 0
\(259\) 2.63387 0.163661
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.1487 0.625793 0.312897 0.949787i \(-0.398701\pi\)
0.312897 + 0.949787i \(0.398701\pi\)
\(264\) 0 0
\(265\) −5.59576 −0.343745
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.47055 0.150632 0.0753161 0.997160i \(-0.476003\pi\)
0.0753161 + 0.997160i \(0.476003\pi\)
\(270\) 0 0
\(271\) 21.0864 1.28091 0.640455 0.767996i \(-0.278746\pi\)
0.640455 + 0.767996i \(0.278746\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.8383 0.834479
\(276\) 0 0
\(277\) 25.2335 1.51613 0.758067 0.652176i \(-0.226144\pi\)
0.758067 + 0.652176i \(0.226144\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.5391 −1.64285 −0.821423 0.570319i \(-0.806820\pi\)
−0.821423 + 0.570319i \(0.806820\pi\)
\(282\) 0 0
\(283\) 26.9385 1.60133 0.800665 0.599113i \(-0.204480\pi\)
0.800665 + 0.599113i \(0.204480\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.1954 −0.601815
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.20439 −0.537726 −0.268863 0.963178i \(-0.586648\pi\)
−0.268863 + 0.963178i \(0.586648\pi\)
\(294\) 0 0
\(295\) 20.4209 1.18895
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 45.7860 2.64787
\(300\) 0 0
\(301\) 2.69762 0.155488
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.8305 −0.906454
\(306\) 0 0
\(307\) −5.84878 −0.333807 −0.166904 0.985973i \(-0.553377\pi\)
−0.166904 + 0.985973i \(0.553377\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.5282 −1.73109 −0.865547 0.500828i \(-0.833029\pi\)
−0.865547 + 0.500828i \(0.833029\pi\)
\(312\) 0 0
\(313\) −17.2880 −0.977173 −0.488586 0.872515i \(-0.662487\pi\)
−0.488586 + 0.872515i \(0.662487\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.99395 −0.280488 −0.140244 0.990117i \(-0.544789\pi\)
−0.140244 + 0.990117i \(0.544789\pi\)
\(318\) 0 0
\(319\) 7.80008 0.436721
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.97437 −0.388064
\(324\) 0 0
\(325\) −72.4581 −4.01925
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.67652 −0.202693
\(330\) 0 0
\(331\) 27.5211 1.51270 0.756349 0.654168i \(-0.226981\pi\)
0.756349 + 0.654168i \(0.226981\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 22.6675 1.23846
\(336\) 0 0
\(337\) −19.6439 −1.07007 −0.535035 0.844830i \(-0.679701\pi\)
−0.535035 + 0.844830i \(0.679701\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.04220 0.327203
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.53838 −0.189950 −0.0949751 0.995480i \(-0.530277\pi\)
−0.0949751 + 0.995480i \(0.530277\pi\)
\(348\) 0 0
\(349\) −15.9699 −0.854849 −0.427425 0.904051i \(-0.640579\pi\)
−0.427425 + 0.904051i \(0.640579\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.3440 0.923127 0.461564 0.887107i \(-0.347289\pi\)
0.461564 + 0.887107i \(0.347289\pi\)
\(354\) 0 0
\(355\) −27.5439 −1.46188
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.2579 −0.805279 −0.402639 0.915359i \(-0.631907\pi\)
−0.402639 + 0.915359i \(0.631907\pi\)
\(360\) 0 0
\(361\) 29.6418 1.56010
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.992604 −0.0519553
\(366\) 0 0
\(367\) −20.8162 −1.08660 −0.543298 0.839540i \(-0.682825\pi\)
−0.543298 + 0.839540i \(0.682825\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.39047 −0.0721899
\(372\) 0 0
\(373\) −18.0929 −0.936813 −0.468407 0.883513i \(-0.655172\pi\)
−0.468407 + 0.883513i \(0.655172\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −40.8417 −2.10346
\(378\) 0 0
\(379\) −32.2265 −1.65536 −0.827681 0.561198i \(-0.810341\pi\)
−0.827681 + 0.561198i \(0.810341\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.77648 −0.295164 −0.147582 0.989050i \(-0.547149\pi\)
−0.147582 + 0.989050i \(0.547149\pi\)
\(384\) 0 0
\(385\) 4.97437 0.253517
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.6070 0.842006 0.421003 0.907059i \(-0.361678\pi\)
0.421003 + 0.907059i \(0.361678\pi\)
\(390\) 0 0
\(391\) 7.07433 0.357764
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 46.9904 2.36434
\(396\) 0 0
\(397\) −9.80083 −0.491890 −0.245945 0.969284i \(-0.579098\pi\)
−0.245945 + 0.969284i \(0.579098\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.4696 −1.32183 −0.660914 0.750462i \(-0.729831\pi\)
−0.660914 + 0.750462i \(0.729831\pi\)
\(402\) 0 0
\(403\) −31.6374 −1.57597
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.25565 −0.161376
\(408\) 0 0
\(409\) 22.9296 1.13380 0.566898 0.823788i \(-0.308143\pi\)
0.566898 + 0.823788i \(0.308143\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.07433 0.249691
\(414\) 0 0
\(415\) −26.7655 −1.31387
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.11334 −0.249803 −0.124902 0.992169i \(-0.539862\pi\)
−0.124902 + 0.992169i \(0.539862\pi\)
\(420\) 0 0
\(421\) −22.2928 −1.08648 −0.543242 0.839576i \(-0.682803\pi\)
−0.543242 + 0.839576i \(0.682803\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −11.1954 −0.543056
\(426\) 0 0
\(427\) −3.93369 −0.190365
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.4510 −0.792415 −0.396208 0.918161i \(-0.629674\pi\)
−0.396208 + 0.918161i \(0.629674\pi\)
\(432\) 0 0
\(433\) 10.1999 0.490177 0.245088 0.969501i \(-0.421183\pi\)
0.245088 + 0.969501i \(0.421183\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −49.3390 −2.36020
\(438\) 0 0
\(439\) 18.4074 0.878538 0.439269 0.898356i \(-0.355238\pi\)
0.439269 + 0.898356i \(0.355238\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.43047 −0.0679635 −0.0339817 0.999422i \(-0.510819\pi\)
−0.0339817 + 0.999422i \(0.510819\pi\)
\(444\) 0 0
\(445\) 9.17035 0.434716
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.72778 −0.175925 −0.0879625 0.996124i \(-0.528036\pi\)
−0.0879625 + 0.996124i \(0.528036\pi\)
\(450\) 0 0
\(451\) 12.6022 0.593414
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −26.0461 −1.22106
\(456\) 0 0
\(457\) 19.7601 0.924338 0.462169 0.886792i \(-0.347071\pi\)
0.462169 + 0.886792i \(0.347071\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.40227 0.205034 0.102517 0.994731i \(-0.467310\pi\)
0.102517 + 0.994731i \(0.467310\pi\)
\(462\) 0 0
\(463\) 4.51887 0.210010 0.105005 0.994472i \(-0.466514\pi\)
0.105005 + 0.994472i \(0.466514\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.51893 −0.0702877 −0.0351438 0.999382i \(-0.511189\pi\)
−0.0351438 + 0.999382i \(0.511189\pi\)
\(468\) 0 0
\(469\) 5.63259 0.260089
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.33444 −0.153318
\(474\) 0 0
\(475\) 78.0808 3.58259
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 39.8418 1.82042 0.910209 0.414148i \(-0.135920\pi\)
0.910209 + 0.414148i \(0.135920\pi\)
\(480\) 0 0
\(481\) 17.0468 0.777267
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −43.0884 −1.95654
\(486\) 0 0
\(487\) 9.19532 0.416680 0.208340 0.978056i \(-0.433194\pi\)
0.208340 + 0.978056i \(0.433194\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −26.5282 −1.19720 −0.598600 0.801048i \(-0.704276\pi\)
−0.598600 + 0.801048i \(0.704276\pi\)
\(492\) 0 0
\(493\) −6.31040 −0.284206
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.84431 −0.307009
\(498\) 0 0
\(499\) 28.9312 1.29514 0.647569 0.762007i \(-0.275786\pi\)
0.647569 + 0.762007i \(0.275786\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.1977 −1.39103 −0.695517 0.718509i \(-0.744825\pi\)
−0.695517 + 0.718509i \(0.744825\pi\)
\(504\) 0 0
\(505\) −67.5811 −3.00732
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.87888 −0.171928 −0.0859641 0.996298i \(-0.527397\pi\)
−0.0859641 + 0.996298i \(0.527397\pi\)
\(510\) 0 0
\(511\) −0.246650 −0.0109111
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −38.5394 −1.69825
\(516\) 0 0
\(517\) 4.54443 0.199864
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −44.1050 −1.93227 −0.966137 0.258030i \(-0.916927\pi\)
−0.966137 + 0.258030i \(0.916927\pi\)
\(522\) 0 0
\(523\) −0.735418 −0.0321576 −0.0160788 0.999871i \(-0.505118\pi\)
−0.0160788 + 0.999871i \(0.505118\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.88824 −0.212935
\(528\) 0 0
\(529\) 27.0461 1.17592
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −65.9859 −2.85817
\(534\) 0 0
\(535\) 68.2582 2.95106
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.23607 0.0532412
\(540\) 0 0
\(541\) 5.95682 0.256104 0.128052 0.991767i \(-0.459128\pi\)
0.128052 + 0.991767i \(0.459128\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 78.9104 3.38015
\(546\) 0 0
\(547\) −39.8844 −1.70533 −0.852667 0.522455i \(-0.825016\pi\)
−0.852667 + 0.522455i \(0.825016\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 44.0110 1.87493
\(552\) 0 0
\(553\) 11.6765 0.496536
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.5740 −1.25309 −0.626545 0.779385i \(-0.715532\pi\)
−0.626545 + 0.779385i \(0.715532\pi\)
\(558\) 0 0
\(559\) 17.4594 0.738453
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −43.7900 −1.84553 −0.922763 0.385367i \(-0.874075\pi\)
−0.922763 + 0.385367i \(0.874075\pi\)
\(564\) 0 0
\(565\) −54.3094 −2.28481
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 44.3088 1.85752 0.928761 0.370678i \(-0.120875\pi\)
0.928761 + 0.370678i \(0.120875\pi\)
\(570\) 0 0
\(571\) −25.9543 −1.08615 −0.543076 0.839684i \(-0.682740\pi\)
−0.543076 + 0.839684i \(0.682740\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −79.1999 −3.30286
\(576\) 0 0
\(577\) −11.1442 −0.463939 −0.231969 0.972723i \(-0.574517\pi\)
−0.231969 + 0.972723i \(0.574517\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.65089 −0.275926
\(582\) 0 0
\(583\) 1.71872 0.0711822
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.6676 −0.811768 −0.405884 0.913925i \(-0.633036\pi\)
−0.405884 + 0.913925i \(0.633036\pi\)
\(588\) 0 0
\(589\) 34.0924 1.40475
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.1396 1.15555 0.577777 0.816194i \(-0.303920\pi\)
0.577777 + 0.816194i \(0.303920\pi\)
\(594\) 0 0
\(595\) −4.02435 −0.164982
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.43434 −0.0994645 −0.0497322 0.998763i \(-0.515837\pi\)
−0.0497322 + 0.998763i \(0.515837\pi\)
\(600\) 0 0
\(601\) −15.8462 −0.646381 −0.323190 0.946334i \(-0.604755\pi\)
−0.323190 + 0.946334i \(0.604755\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 38.1192 1.54977
\(606\) 0 0
\(607\) 18.1413 0.736334 0.368167 0.929760i \(-0.379986\pi\)
0.368167 + 0.929760i \(0.379986\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.7950 −0.962641
\(612\) 0 0
\(613\) 14.9702 0.604641 0.302320 0.953206i \(-0.402239\pi\)
0.302320 + 0.953206i \(0.402239\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.43440 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(618\) 0 0
\(619\) −12.9548 −0.520697 −0.260348 0.965515i \(-0.583837\pi\)
−0.260348 + 0.965515i \(0.583837\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.27872 0.0912948
\(624\) 0 0
\(625\) 44.3598 1.77439
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.63387 0.105019
\(630\) 0 0
\(631\) 0.899738 0.0358180 0.0179090 0.999840i \(-0.494299\pi\)
0.0179090 + 0.999840i \(0.494299\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −17.5657 −0.697072
\(636\) 0 0
\(637\) −6.47214 −0.256435
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.8322 0.585837 0.292919 0.956137i \(-0.405374\pi\)
0.292919 + 0.956137i \(0.405374\pi\)
\(642\) 0 0
\(643\) −17.5168 −0.690793 −0.345397 0.938457i \(-0.612256\pi\)
−0.345397 + 0.938457i \(0.612256\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.05829 −0.0416057 −0.0208028 0.999784i \(-0.506622\pi\)
−0.0208028 + 0.999784i \(0.506622\pi\)
\(648\) 0 0
\(649\) −6.27222 −0.246206
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −35.2848 −1.38080 −0.690400 0.723428i \(-0.742565\pi\)
−0.690400 + 0.723428i \(0.742565\pi\)
\(654\) 0 0
\(655\) 42.9904 1.67977
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.1047 0.783169 0.391585 0.920142i \(-0.371927\pi\)
0.391585 + 0.920142i \(0.371927\pi\)
\(660\) 0 0
\(661\) 12.6541 0.492186 0.246093 0.969246i \(-0.420853\pi\)
0.246093 + 0.969246i \(0.420853\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 28.0673 1.08840
\(666\) 0 0
\(667\) −44.6418 −1.72854
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.86231 0.187707
\(672\) 0 0
\(673\) −21.6342 −0.833937 −0.416968 0.908921i \(-0.636907\pi\)
−0.416968 + 0.908921i \(0.636907\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −27.9965 −1.07599 −0.537996 0.842948i \(-0.680818\pi\)
−0.537996 + 0.842948i \(0.680818\pi\)
\(678\) 0 0
\(679\) −10.7069 −0.410894
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.2225 1.53907 0.769536 0.638603i \(-0.220487\pi\)
0.769536 + 0.638603i \(0.220487\pi\)
\(684\) 0 0
\(685\) 73.2246 2.79777
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.99934 −0.342848
\(690\) 0 0
\(691\) −30.7414 −1.16946 −0.584729 0.811229i \(-0.698799\pi\)
−0.584729 + 0.811229i \(0.698799\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −55.7092 −2.11317
\(696\) 0 0
\(697\) −10.1954 −0.386178
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −24.3696 −0.920428 −0.460214 0.887808i \(-0.652227\pi\)
−0.460214 + 0.887808i \(0.652227\pi\)
\(702\) 0 0
\(703\) −18.3696 −0.692823
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.7931 −0.631568
\(708\) 0 0
\(709\) −22.2957 −0.837334 −0.418667 0.908140i \(-0.637503\pi\)
−0.418667 + 0.908140i \(0.637503\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −34.5810 −1.29507
\(714\) 0 0
\(715\) 32.1948 1.20402
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14.2466 0.531310 0.265655 0.964068i \(-0.414412\pi\)
0.265655 + 0.964068i \(0.414412\pi\)
\(720\) 0 0
\(721\) −9.57656 −0.356650
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 70.6474 2.62378
\(726\) 0 0
\(727\) 26.2762 0.974529 0.487264 0.873255i \(-0.337995\pi\)
0.487264 + 0.873255i \(0.337995\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.69762 0.0997752
\(732\) 0 0
\(733\) −29.7464 −1.09871 −0.549354 0.835590i \(-0.685126\pi\)
−0.549354 + 0.835590i \(0.685126\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.96227 −0.256458
\(738\) 0 0
\(739\) 20.0628 0.738021 0.369010 0.929425i \(-0.379697\pi\)
0.369010 + 0.929425i \(0.379697\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.0948 0.370344 0.185172 0.982706i \(-0.440716\pi\)
0.185172 + 0.982706i \(0.440716\pi\)
\(744\) 0 0
\(745\) −13.8030 −0.505703
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 16.9613 0.619752
\(750\) 0 0
\(751\) −2.58103 −0.0941831 −0.0470916 0.998891i \(-0.514995\pi\)
−0.0470916 + 0.998891i \(0.514995\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 36.9023 1.34301
\(756\) 0 0
\(757\) 42.6072 1.54859 0.774293 0.632828i \(-0.218106\pi\)
0.774293 + 0.632828i \(0.218106\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.81361 0.246993 0.123497 0.992345i \(-0.460589\pi\)
0.123497 + 0.992345i \(0.460589\pi\)
\(762\) 0 0
\(763\) 19.6082 0.709866
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 32.8417 1.18585
\(768\) 0 0
\(769\) −9.62782 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32.6842 1.17557 0.587784 0.809018i \(-0.300001\pi\)
0.587784 + 0.809018i \(0.300001\pi\)
\(774\) 0 0
\(775\) 54.7258 1.96581
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 71.1064 2.54765
\(780\) 0 0
\(781\) 8.46003 0.302724
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 26.9718 0.962665
\(786\) 0 0
\(787\) −40.6653 −1.44956 −0.724782 0.688979i \(-0.758059\pi\)
−0.724782 + 0.688979i \(0.758059\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13.4952 −0.479834
\(792\) 0 0
\(793\) −25.4594 −0.904089
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.0019 −0.673082 −0.336541 0.941669i \(-0.609257\pi\)
−0.336541 + 0.941669i \(0.609257\pi\)
\(798\) 0 0
\(799\) −3.67652 −0.130066
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.304876 0.0107588
\(804\) 0 0
\(805\) −28.4696 −1.00342
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.95386 0.279643 0.139821 0.990177i \(-0.455347\pi\)
0.139821 + 0.990177i \(0.455347\pi\)
\(810\) 0 0
\(811\) −40.6097 −1.42600 −0.712999 0.701165i \(-0.752664\pi\)
−0.712999 + 0.701165i \(0.752664\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 57.9923 2.03138
\(816\) 0 0
\(817\) −18.8142 −0.658226
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −50.7051 −1.76962 −0.884810 0.465951i \(-0.845712\pi\)
−0.884810 + 0.465951i \(0.845712\pi\)
\(822\) 0 0
\(823\) 46.3922 1.61713 0.808564 0.588408i \(-0.200245\pi\)
0.808564 + 0.588408i \(0.200245\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.6166 1.20374 0.601869 0.798595i \(-0.294423\pi\)
0.601869 + 0.798595i \(0.294423\pi\)
\(828\) 0 0
\(829\) 2.04614 0.0710653 0.0355326 0.999369i \(-0.488687\pi\)
0.0355326 + 0.999369i \(0.488687\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −13.0836 −0.452778
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.11462 −0.0730049 −0.0365024 0.999334i \(-0.511622\pi\)
−0.0365024 + 0.999334i \(0.511622\pi\)
\(840\) 0 0
\(841\) 10.8211 0.373142
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −116.258 −3.99938
\(846\) 0 0
\(847\) 9.47214 0.325466
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 18.6329 0.638727
\(852\) 0 0
\(853\) 22.7338 0.778392 0.389196 0.921155i \(-0.372753\pi\)
0.389196 + 0.921155i \(0.372753\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.03743 −0.0695973 −0.0347986 0.999394i \(-0.511079\pi\)
−0.0347986 + 0.999394i \(0.511079\pi\)
\(858\) 0 0
\(859\) −15.8648 −0.541301 −0.270650 0.962678i \(-0.587239\pi\)
−0.270650 + 0.962678i \(0.587239\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.8857 0.506714 0.253357 0.967373i \(-0.418465\pi\)
0.253357 + 0.967373i \(0.418465\pi\)
\(864\) 0 0
\(865\) 17.1323 0.582517
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −14.4330 −0.489605
\(870\) 0 0
\(871\) 36.4549 1.23523
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.9324 0.842869
\(876\) 0 0
\(877\) −3.66844 −0.123874 −0.0619372 0.998080i \(-0.519728\pi\)
−0.0619372 + 0.998080i \(0.519728\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.9695 1.34661 0.673304 0.739366i \(-0.264874\pi\)
0.673304 + 0.739366i \(0.264874\pi\)
\(882\) 0 0
\(883\) −17.1028 −0.575556 −0.287778 0.957697i \(-0.592917\pi\)
−0.287778 + 0.957697i \(0.592917\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.5622 0.891871 0.445936 0.895065i \(-0.352871\pi\)
0.445936 + 0.895065i \(0.352871\pi\)
\(888\) 0 0
\(889\) −4.36484 −0.146392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.6414 0.858058
\(894\) 0 0
\(895\) −10.9773 −0.366931
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 30.8467 1.02880
\(900\) 0 0
\(901\) −1.39047 −0.0463234
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.6957 −0.422018
\(906\) 0 0
\(907\) 42.8523 1.42289 0.711443 0.702744i \(-0.248042\pi\)
0.711443 + 0.702744i \(0.248042\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.1141 1.22964 0.614822 0.788666i \(-0.289228\pi\)
0.614822 + 0.788666i \(0.289228\pi\)
\(912\) 0 0
\(913\) 8.22096 0.272074
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.6826 0.352770
\(918\) 0 0
\(919\) 23.7655 0.783950 0.391975 0.919976i \(-0.371792\pi\)
0.391975 + 0.919976i \(0.371792\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −44.2973 −1.45806
\(924\) 0 0
\(925\) −29.4872 −0.969535
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24.7467 −0.811912 −0.405956 0.913893i \(-0.633061\pi\)
−0.405956 + 0.913893i \(0.633061\pi\)
\(930\) 0 0
\(931\) 6.97437 0.228576
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.97437 0.162679
\(936\) 0 0
\(937\) 26.9577 0.880671 0.440336 0.897833i \(-0.354859\pi\)
0.440336 + 0.897833i \(0.354859\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.6992 1.00077 0.500383 0.865804i \(-0.333193\pi\)
0.500383 + 0.865804i \(0.333193\pi\)
\(942\) 0 0
\(943\) −72.1255 −2.34873
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −28.3892 −0.922525 −0.461262 0.887264i \(-0.652603\pi\)
−0.461262 + 0.887264i \(0.652603\pi\)
\(948\) 0 0
\(949\) −1.59635 −0.0518197
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39.1278 −1.26747 −0.633737 0.773549i \(-0.718480\pi\)
−0.633737 + 0.773549i \(0.718480\pi\)
\(954\) 0 0
\(955\) 14.2166 0.460037
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.1954 0.587560
\(960\) 0 0
\(961\) −7.10510 −0.229197
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −65.3050 −2.10224
\(966\) 0 0
\(967\) 39.8719 1.28219 0.641097 0.767460i \(-0.278480\pi\)
0.641097 + 0.767460i \(0.278480\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −16.8891 −0.541996 −0.270998 0.962580i \(-0.587354\pi\)
−0.270998 + 0.962580i \(0.587354\pi\)
\(972\) 0 0
\(973\) −13.8430 −0.443787
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21.4487 −0.686205 −0.343103 0.939298i \(-0.611478\pi\)
−0.343103 + 0.939298i \(0.611478\pi\)
\(978\) 0 0
\(979\) −2.81665 −0.0900205
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 48.8004 1.55649 0.778245 0.627960i \(-0.216110\pi\)
0.778245 + 0.627960i \(0.216110\pi\)
\(984\) 0 0
\(985\) 71.4600 2.27690
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.0839 0.606832
\(990\) 0 0
\(991\) 5.00203 0.158895 0.0794474 0.996839i \(-0.474684\pi\)
0.0794474 + 0.996839i \(0.474684\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 60.9632 1.93266
\(996\) 0 0
\(997\) −54.7966 −1.73543 −0.867713 0.497066i \(-0.834411\pi\)
−0.867713 + 0.497066i \(0.834411\pi\)
\(998\) 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 8568.2.a.bj.1.1 4
3.2 odd 2 952.2.a.g.1.3 4
12.11 even 2 1904.2.a.q.1.2 4
21.20 even 2 6664.2.a.o.1.2 4
24.5 odd 2 7616.2.a.bj.1.2 4
24.11 even 2 7616.2.a.bp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.3 4 3.2 odd 2
1904.2.a.q.1.2 4 12.11 even 2
6664.2.a.o.1.2 4 21.20 even 2
7616.2.a.bj.1.2 4 24.5 odd 2
7616.2.a.bp.1.3 4 24.11 even 2
8568.2.a.bj.1.1 4 1.1 even 1 trivial