Properties

Label 4840.2.a.bg.1.5
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.713352\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.713352 q^{3} +1.00000 q^{5} +0.376172 q^{7} -2.49113 q^{9} +O(q^{10})\) \(q+0.713352 q^{3} +1.00000 q^{5} +0.376172 q^{7} -2.49113 q^{9} -2.82831 q^{13} +0.713352 q^{15} -4.83768 q^{17} +2.92990 q^{19} +0.268343 q^{21} +3.77226 q^{23} +1.00000 q^{25} -3.91711 q^{27} +8.44396 q^{29} +8.04011 q^{31} +0.376172 q^{35} +2.83990 q^{37} -2.01758 q^{39} -3.72034 q^{41} -6.48484 q^{43} -2.49113 q^{45} +2.58897 q^{47} -6.85849 q^{49} -3.45097 q^{51} -0.0487626 q^{53} +2.09005 q^{57} -1.64929 q^{59} +8.69705 q^{61} -0.937093 q^{63} -2.82831 q^{65} +11.4395 q^{67} +2.69095 q^{69} +14.1216 q^{71} +0.513225 q^{73} +0.713352 q^{75} +12.9823 q^{79} +4.67911 q^{81} +2.73687 q^{83} -4.83768 q^{85} +6.02351 q^{87} +12.0195 q^{89} -1.06393 q^{91} +5.73543 q^{93} +2.92990 q^{95} +16.0745 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + 8 q^{5} - 6 q^{7} + 19 q^{9} + 12 q^{13} + q^{15} + 2 q^{17} - 6 q^{19} + 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} + 8 q^{29} + 19 q^{31} - 6 q^{35} + 12 q^{37} - 21 q^{39} - 3 q^{41} - 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} - 7 q^{51} + 28 q^{53} + 25 q^{57} + 25 q^{59} + 10 q^{61} - 64 q^{63} + 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} + 38 q^{73} + q^{75} - 38 q^{79} + 32 q^{81} - 28 q^{83} + 2 q^{85} + 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} - 6 q^{95} + 21 q^{97}+O(q^{100}) \) 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
\(3\) 0.713352 0.411854 0.205927 0.978567i \(-0.433979\pi\)
0.205927 + 0.978567i \(0.433979\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.376172 0.142180 0.0710899 0.997470i \(-0.477352\pi\)
0.0710899 + 0.997470i \(0.477352\pi\)
\(8\) 0 0
\(9\) −2.49113 −0.830376
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.82831 −0.784432 −0.392216 0.919873i \(-0.628291\pi\)
−0.392216 + 0.919873i \(0.628291\pi\)
\(14\) 0 0
\(15\) 0.713352 0.184187
\(16\) 0 0
\(17\) −4.83768 −1.17331 −0.586655 0.809837i \(-0.699556\pi\)
−0.586655 + 0.809837i \(0.699556\pi\)
\(18\) 0 0
\(19\) 2.92990 0.672165 0.336082 0.941833i \(-0.390898\pi\)
0.336082 + 0.941833i \(0.390898\pi\)
\(20\) 0 0
\(21\) 0.268343 0.0585573
\(22\) 0 0
\(23\) 3.77226 0.786571 0.393285 0.919416i \(-0.371338\pi\)
0.393285 + 0.919416i \(0.371338\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.91711 −0.753848
\(28\) 0 0
\(29\) 8.44396 1.56800 0.784002 0.620759i \(-0.213175\pi\)
0.784002 + 0.620759i \(0.213175\pi\)
\(30\) 0 0
\(31\) 8.04011 1.44405 0.722023 0.691869i \(-0.243213\pi\)
0.722023 + 0.691869i \(0.243213\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.376172 0.0635847
\(36\) 0 0
\(37\) 2.83990 0.466877 0.233438 0.972372i \(-0.425002\pi\)
0.233438 + 0.972372i \(0.425002\pi\)
\(38\) 0 0
\(39\) −2.01758 −0.323071
\(40\) 0 0
\(41\) −3.72034 −0.581020 −0.290510 0.956872i \(-0.593825\pi\)
−0.290510 + 0.956872i \(0.593825\pi\)
\(42\) 0 0
\(43\) −6.48484 −0.988929 −0.494465 0.869198i \(-0.664636\pi\)
−0.494465 + 0.869198i \(0.664636\pi\)
\(44\) 0 0
\(45\) −2.49113 −0.371356
\(46\) 0 0
\(47\) 2.58897 0.377639 0.188820 0.982012i \(-0.439534\pi\)
0.188820 + 0.982012i \(0.439534\pi\)
\(48\) 0 0
\(49\) −6.85849 −0.979785
\(50\) 0 0
\(51\) −3.45097 −0.483233
\(52\) 0 0
\(53\) −0.0487626 −0.00669805 −0.00334903 0.999994i \(-0.501066\pi\)
−0.00334903 + 0.999994i \(0.501066\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.09005 0.276834
\(58\) 0 0
\(59\) −1.64929 −0.214720 −0.107360 0.994220i \(-0.534240\pi\)
−0.107360 + 0.994220i \(0.534240\pi\)
\(60\) 0 0
\(61\) 8.69705 1.11354 0.556772 0.830666i \(-0.312040\pi\)
0.556772 + 0.830666i \(0.312040\pi\)
\(62\) 0 0
\(63\) −0.937093 −0.118063
\(64\) 0 0
\(65\) −2.82831 −0.350809
\(66\) 0 0
\(67\) 11.4395 1.39756 0.698779 0.715338i \(-0.253727\pi\)
0.698779 + 0.715338i \(0.253727\pi\)
\(68\) 0 0
\(69\) 2.69095 0.323952
\(70\) 0 0
\(71\) 14.1216 1.67592 0.837962 0.545728i \(-0.183747\pi\)
0.837962 + 0.545728i \(0.183747\pi\)
\(72\) 0 0
\(73\) 0.513225 0.0600684 0.0300342 0.999549i \(-0.490438\pi\)
0.0300342 + 0.999549i \(0.490438\pi\)
\(74\) 0 0
\(75\) 0.713352 0.0823708
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.9823 1.46063 0.730313 0.683113i \(-0.239374\pi\)
0.730313 + 0.683113i \(0.239374\pi\)
\(80\) 0 0
\(81\) 4.67911 0.519901
\(82\) 0 0
\(83\) 2.73687 0.300411 0.150205 0.988655i \(-0.452007\pi\)
0.150205 + 0.988655i \(0.452007\pi\)
\(84\) 0 0
\(85\) −4.83768 −0.524720
\(86\) 0 0
\(87\) 6.02351 0.645789
\(88\) 0 0
\(89\) 12.0195 1.27407 0.637034 0.770835i \(-0.280161\pi\)
0.637034 + 0.770835i \(0.280161\pi\)
\(90\) 0 0
\(91\) −1.06393 −0.111530
\(92\) 0 0
\(93\) 5.73543 0.594736
\(94\) 0 0
\(95\) 2.92990 0.300601
\(96\) 0 0
\(97\) 16.0745 1.63212 0.816060 0.577967i \(-0.196154\pi\)
0.816060 + 0.577967i \(0.196154\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.5883 1.55109 0.775546 0.631291i \(-0.217475\pi\)
0.775546 + 0.631291i \(0.217475\pi\)
\(102\) 0 0
\(103\) −9.25907 −0.912323 −0.456161 0.889897i \(-0.650776\pi\)
−0.456161 + 0.889897i \(0.650776\pi\)
\(104\) 0 0
\(105\) 0.268343 0.0261876
\(106\) 0 0
\(107\) −2.15917 −0.208735 −0.104368 0.994539i \(-0.533282\pi\)
−0.104368 + 0.994539i \(0.533282\pi\)
\(108\) 0 0
\(109\) −5.79052 −0.554631 −0.277315 0.960779i \(-0.589445\pi\)
−0.277315 + 0.960779i \(0.589445\pi\)
\(110\) 0 0
\(111\) 2.02585 0.192285
\(112\) 0 0
\(113\) 19.0148 1.78876 0.894381 0.447306i \(-0.147616\pi\)
0.894381 + 0.447306i \(0.147616\pi\)
\(114\) 0 0
\(115\) 3.77226 0.351765
\(116\) 0 0
\(117\) 7.04568 0.651373
\(118\) 0 0
\(119\) −1.81980 −0.166821
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −2.65391 −0.239295
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.0973 −0.895993 −0.447996 0.894035i \(-0.647862\pi\)
−0.447996 + 0.894035i \(0.647862\pi\)
\(128\) 0 0
\(129\) −4.62598 −0.407294
\(130\) 0 0
\(131\) −6.29651 −0.550128 −0.275064 0.961426i \(-0.588699\pi\)
−0.275064 + 0.961426i \(0.588699\pi\)
\(132\) 0 0
\(133\) 1.10215 0.0955682
\(134\) 0 0
\(135\) −3.91711 −0.337131
\(136\) 0 0
\(137\) −8.27329 −0.706835 −0.353418 0.935466i \(-0.614981\pi\)
−0.353418 + 0.935466i \(0.614981\pi\)
\(138\) 0 0
\(139\) −10.2671 −0.870843 −0.435422 0.900227i \(-0.643401\pi\)
−0.435422 + 0.900227i \(0.643401\pi\)
\(140\) 0 0
\(141\) 1.84684 0.155532
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 8.44396 0.701232
\(146\) 0 0
\(147\) −4.89252 −0.403528
\(148\) 0 0
\(149\) −4.90160 −0.401554 −0.200777 0.979637i \(-0.564347\pi\)
−0.200777 + 0.979637i \(0.564347\pi\)
\(150\) 0 0
\(151\) −9.35625 −0.761401 −0.380701 0.924698i \(-0.624317\pi\)
−0.380701 + 0.924698i \(0.624317\pi\)
\(152\) 0 0
\(153\) 12.0513 0.974289
\(154\) 0 0
\(155\) 8.04011 0.645797
\(156\) 0 0
\(157\) 9.65817 0.770806 0.385403 0.922748i \(-0.374062\pi\)
0.385403 + 0.922748i \(0.374062\pi\)
\(158\) 0 0
\(159\) −0.0347849 −0.00275862
\(160\) 0 0
\(161\) 1.41902 0.111834
\(162\) 0 0
\(163\) 4.48802 0.351529 0.175764 0.984432i \(-0.443760\pi\)
0.175764 + 0.984432i \(0.443760\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.4716 −1.81628 −0.908142 0.418662i \(-0.862499\pi\)
−0.908142 + 0.418662i \(0.862499\pi\)
\(168\) 0 0
\(169\) −5.00067 −0.384667
\(170\) 0 0
\(171\) −7.29875 −0.558150
\(172\) 0 0
\(173\) −12.9509 −0.984637 −0.492318 0.870415i \(-0.663850\pi\)
−0.492318 + 0.870415i \(0.663850\pi\)
\(174\) 0 0
\(175\) 0.376172 0.0284359
\(176\) 0 0
\(177\) −1.17653 −0.0884332
\(178\) 0 0
\(179\) −3.52933 −0.263795 −0.131897 0.991263i \(-0.542107\pi\)
−0.131897 + 0.991263i \(0.542107\pi\)
\(180\) 0 0
\(181\) 1.79178 0.133182 0.0665910 0.997780i \(-0.478788\pi\)
0.0665910 + 0.997780i \(0.478788\pi\)
\(182\) 0 0
\(183\) 6.20406 0.458617
\(184\) 0 0
\(185\) 2.83990 0.208794
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.47351 −0.107182
\(190\) 0 0
\(191\) 3.40113 0.246097 0.123048 0.992401i \(-0.460733\pi\)
0.123048 + 0.992401i \(0.460733\pi\)
\(192\) 0 0
\(193\) −7.77543 −0.559688 −0.279844 0.960046i \(-0.590283\pi\)
−0.279844 + 0.960046i \(0.590283\pi\)
\(194\) 0 0
\(195\) −2.01758 −0.144482
\(196\) 0 0
\(197\) −7.09410 −0.505434 −0.252717 0.967540i \(-0.581324\pi\)
−0.252717 + 0.967540i \(0.581324\pi\)
\(198\) 0 0
\(199\) 12.7734 0.905481 0.452740 0.891642i \(-0.350446\pi\)
0.452740 + 0.891642i \(0.350446\pi\)
\(200\) 0 0
\(201\) 8.16039 0.575590
\(202\) 0 0
\(203\) 3.17638 0.222938
\(204\) 0 0
\(205\) −3.72034 −0.259840
\(206\) 0 0
\(207\) −9.39719 −0.653150
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 7.34047 0.505339 0.252669 0.967553i \(-0.418692\pi\)
0.252669 + 0.967553i \(0.418692\pi\)
\(212\) 0 0
\(213\) 10.0737 0.690236
\(214\) 0 0
\(215\) −6.48484 −0.442263
\(216\) 0 0
\(217\) 3.02446 0.205314
\(218\) 0 0
\(219\) 0.366110 0.0247394
\(220\) 0 0
\(221\) 13.6825 0.920382
\(222\) 0 0
\(223\) 6.53875 0.437867 0.218934 0.975740i \(-0.429742\pi\)
0.218934 + 0.975740i \(0.429742\pi\)
\(224\) 0 0
\(225\) −2.49113 −0.166075
\(226\) 0 0
\(227\) 20.5461 1.36369 0.681847 0.731495i \(-0.261177\pi\)
0.681847 + 0.731495i \(0.261177\pi\)
\(228\) 0 0
\(229\) −4.33094 −0.286197 −0.143098 0.989708i \(-0.545706\pi\)
−0.143098 + 0.989708i \(0.545706\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 28.6595 1.87755 0.938774 0.344533i \(-0.111963\pi\)
0.938774 + 0.344533i \(0.111963\pi\)
\(234\) 0 0
\(235\) 2.58897 0.168886
\(236\) 0 0
\(237\) 9.26097 0.601565
\(238\) 0 0
\(239\) −10.4766 −0.677677 −0.338838 0.940845i \(-0.610034\pi\)
−0.338838 + 0.940845i \(0.610034\pi\)
\(240\) 0 0
\(241\) 15.5202 0.999745 0.499873 0.866099i \(-0.333380\pi\)
0.499873 + 0.866099i \(0.333380\pi\)
\(242\) 0 0
\(243\) 15.0892 0.967971
\(244\) 0 0
\(245\) −6.85849 −0.438173
\(246\) 0 0
\(247\) −8.28666 −0.527267
\(248\) 0 0
\(249\) 1.95235 0.123725
\(250\) 0 0
\(251\) −11.9701 −0.755547 −0.377774 0.925898i \(-0.623310\pi\)
−0.377774 + 0.925898i \(0.623310\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3.45097 −0.216108
\(256\) 0 0
\(257\) −8.47342 −0.528557 −0.264279 0.964446i \(-0.585134\pi\)
−0.264279 + 0.964446i \(0.585134\pi\)
\(258\) 0 0
\(259\) 1.06829 0.0663804
\(260\) 0 0
\(261\) −21.0350 −1.30203
\(262\) 0 0
\(263\) 12.5189 0.771950 0.385975 0.922509i \(-0.373865\pi\)
0.385975 + 0.922509i \(0.373865\pi\)
\(264\) 0 0
\(265\) −0.0487626 −0.00299546
\(266\) 0 0
\(267\) 8.57416 0.524730
\(268\) 0 0
\(269\) −24.0649 −1.46726 −0.733630 0.679549i \(-0.762176\pi\)
−0.733630 + 0.679549i \(0.762176\pi\)
\(270\) 0 0
\(271\) −9.42221 −0.572358 −0.286179 0.958176i \(-0.592385\pi\)
−0.286179 + 0.958176i \(0.592385\pi\)
\(272\) 0 0
\(273\) −0.758957 −0.0459342
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.7811 1.42887 0.714434 0.699703i \(-0.246684\pi\)
0.714434 + 0.699703i \(0.246684\pi\)
\(278\) 0 0
\(279\) −20.0289 −1.19910
\(280\) 0 0
\(281\) 0.496904 0.0296428 0.0148214 0.999890i \(-0.495282\pi\)
0.0148214 + 0.999890i \(0.495282\pi\)
\(282\) 0 0
\(283\) 4.68030 0.278215 0.139108 0.990277i \(-0.455577\pi\)
0.139108 + 0.990277i \(0.455577\pi\)
\(284\) 0 0
\(285\) 2.09005 0.123804
\(286\) 0 0
\(287\) −1.39949 −0.0826092
\(288\) 0 0
\(289\) 6.40318 0.376658
\(290\) 0 0
\(291\) 11.4668 0.672195
\(292\) 0 0
\(293\) 16.9203 0.988493 0.494246 0.869322i \(-0.335444\pi\)
0.494246 + 0.869322i \(0.335444\pi\)
\(294\) 0 0
\(295\) −1.64929 −0.0960256
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.6691 −0.617011
\(300\) 0 0
\(301\) −2.43942 −0.140606
\(302\) 0 0
\(303\) 11.1199 0.638824
\(304\) 0 0
\(305\) 8.69705 0.497992
\(306\) 0 0
\(307\) −34.3667 −1.96141 −0.980707 0.195486i \(-0.937372\pi\)
−0.980707 + 0.195486i \(0.937372\pi\)
\(308\) 0 0
\(309\) −6.60497 −0.375744
\(310\) 0 0
\(311\) 7.36790 0.417796 0.208898 0.977937i \(-0.433012\pi\)
0.208898 + 0.977937i \(0.433012\pi\)
\(312\) 0 0
\(313\) −25.0606 −1.41651 −0.708253 0.705958i \(-0.750517\pi\)
−0.708253 + 0.705958i \(0.750517\pi\)
\(314\) 0 0
\(315\) −0.937093 −0.0527992
\(316\) 0 0
\(317\) 28.0728 1.57673 0.788364 0.615209i \(-0.210929\pi\)
0.788364 + 0.615209i \(0.210929\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.54025 −0.0859685
\(322\) 0 0
\(323\) −14.1739 −0.788658
\(324\) 0 0
\(325\) −2.82831 −0.156886
\(326\) 0 0
\(327\) −4.13068 −0.228427
\(328\) 0 0
\(329\) 0.973897 0.0536927
\(330\) 0 0
\(331\) 21.9644 1.20727 0.603637 0.797259i \(-0.293718\pi\)
0.603637 + 0.797259i \(0.293718\pi\)
\(332\) 0 0
\(333\) −7.07456 −0.387683
\(334\) 0 0
\(335\) 11.4395 0.625007
\(336\) 0 0
\(337\) 22.3799 1.21911 0.609555 0.792744i \(-0.291348\pi\)
0.609555 + 0.792744i \(0.291348\pi\)
\(338\) 0 0
\(339\) 13.5642 0.736709
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5.21318 −0.281485
\(344\) 0 0
\(345\) 2.69095 0.144876
\(346\) 0 0
\(347\) −24.7788 −1.33020 −0.665098 0.746756i \(-0.731610\pi\)
−0.665098 + 0.746756i \(0.731610\pi\)
\(348\) 0 0
\(349\) −22.3282 −1.19520 −0.597600 0.801795i \(-0.703879\pi\)
−0.597600 + 0.801795i \(0.703879\pi\)
\(350\) 0 0
\(351\) 11.0788 0.591342
\(352\) 0 0
\(353\) −5.79232 −0.308294 −0.154147 0.988048i \(-0.549263\pi\)
−0.154147 + 0.988048i \(0.549263\pi\)
\(354\) 0 0
\(355\) 14.1216 0.749496
\(356\) 0 0
\(357\) −1.29816 −0.0687059
\(358\) 0 0
\(359\) 21.0367 1.11027 0.555137 0.831759i \(-0.312666\pi\)
0.555137 + 0.831759i \(0.312666\pi\)
\(360\) 0 0
\(361\) −10.4157 −0.548195
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.513225 0.0268634
\(366\) 0 0
\(367\) −23.6235 −1.23314 −0.616569 0.787301i \(-0.711478\pi\)
−0.616569 + 0.787301i \(0.711478\pi\)
\(368\) 0 0
\(369\) 9.26785 0.482465
\(370\) 0 0
\(371\) −0.0183431 −0.000952327 0
\(372\) 0 0
\(373\) −12.7740 −0.661411 −0.330705 0.943734i \(-0.607287\pi\)
−0.330705 + 0.943734i \(0.607287\pi\)
\(374\) 0 0
\(375\) 0.713352 0.0368373
\(376\) 0 0
\(377\) −23.8821 −1.22999
\(378\) 0 0
\(379\) 24.7773 1.27273 0.636363 0.771390i \(-0.280438\pi\)
0.636363 + 0.771390i \(0.280438\pi\)
\(380\) 0 0
\(381\) −7.20295 −0.369018
\(382\) 0 0
\(383\) 38.1710 1.95045 0.975223 0.221223i \(-0.0710048\pi\)
0.975223 + 0.221223i \(0.0710048\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.1546 0.821183
\(388\) 0 0
\(389\) 2.40973 0.122178 0.0610891 0.998132i \(-0.480543\pi\)
0.0610891 + 0.998132i \(0.480543\pi\)
\(390\) 0 0
\(391\) −18.2490 −0.922892
\(392\) 0 0
\(393\) −4.49163 −0.226573
\(394\) 0 0
\(395\) 12.9823 0.653212
\(396\) 0 0
\(397\) −11.5456 −0.579459 −0.289729 0.957109i \(-0.593565\pi\)
−0.289729 + 0.957109i \(0.593565\pi\)
\(398\) 0 0
\(399\) 0.786218 0.0393601
\(400\) 0 0
\(401\) 25.6006 1.27844 0.639218 0.769026i \(-0.279258\pi\)
0.639218 + 0.769026i \(0.279258\pi\)
\(402\) 0 0
\(403\) −22.7399 −1.13276
\(404\) 0 0
\(405\) 4.67911 0.232507
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 9.36369 0.463005 0.231502 0.972834i \(-0.425636\pi\)
0.231502 + 0.972834i \(0.425636\pi\)
\(410\) 0 0
\(411\) −5.90177 −0.291113
\(412\) 0 0
\(413\) −0.620419 −0.0305288
\(414\) 0 0
\(415\) 2.73687 0.134348
\(416\) 0 0
\(417\) −7.32405 −0.358660
\(418\) 0 0
\(419\) −21.3716 −1.04407 −0.522035 0.852924i \(-0.674827\pi\)
−0.522035 + 0.852924i \(0.674827\pi\)
\(420\) 0 0
\(421\) 27.5205 1.34126 0.670632 0.741790i \(-0.266023\pi\)
0.670632 + 0.741790i \(0.266023\pi\)
\(422\) 0 0
\(423\) −6.44945 −0.313583
\(424\) 0 0
\(425\) −4.83768 −0.234662
\(426\) 0 0
\(427\) 3.27159 0.158323
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.63236 0.126796 0.0633982 0.997988i \(-0.479806\pi\)
0.0633982 + 0.997988i \(0.479806\pi\)
\(432\) 0 0
\(433\) 9.99044 0.480110 0.240055 0.970759i \(-0.422835\pi\)
0.240055 + 0.970759i \(0.422835\pi\)
\(434\) 0 0
\(435\) 6.02351 0.288805
\(436\) 0 0
\(437\) 11.0523 0.528705
\(438\) 0 0
\(439\) 2.85957 0.136480 0.0682400 0.997669i \(-0.478262\pi\)
0.0682400 + 0.997669i \(0.478262\pi\)
\(440\) 0 0
\(441\) 17.0854 0.813590
\(442\) 0 0
\(443\) −40.3056 −1.91498 −0.957489 0.288470i \(-0.906853\pi\)
−0.957489 + 0.288470i \(0.906853\pi\)
\(444\) 0 0
\(445\) 12.0195 0.569781
\(446\) 0 0
\(447\) −3.49656 −0.165382
\(448\) 0 0
\(449\) −1.95041 −0.0920457 −0.0460228 0.998940i \(-0.514655\pi\)
−0.0460228 + 0.998940i \(0.514655\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −6.67430 −0.313586
\(454\) 0 0
\(455\) −1.06393 −0.0498779
\(456\) 0 0
\(457\) 35.7765 1.67355 0.836777 0.547543i \(-0.184437\pi\)
0.836777 + 0.547543i \(0.184437\pi\)
\(458\) 0 0
\(459\) 18.9497 0.884498
\(460\) 0 0
\(461\) 15.1433 0.705295 0.352648 0.935756i \(-0.385281\pi\)
0.352648 + 0.935756i \(0.385281\pi\)
\(462\) 0 0
\(463\) −12.6301 −0.586972 −0.293486 0.955963i \(-0.594815\pi\)
−0.293486 + 0.955963i \(0.594815\pi\)
\(464\) 0 0
\(465\) 5.73543 0.265974
\(466\) 0 0
\(467\) 31.3546 1.45092 0.725460 0.688264i \(-0.241627\pi\)
0.725460 + 0.688264i \(0.241627\pi\)
\(468\) 0 0
\(469\) 4.30322 0.198704
\(470\) 0 0
\(471\) 6.88968 0.317460
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.92990 0.134433
\(476\) 0 0
\(477\) 0.121474 0.00556190
\(478\) 0 0
\(479\) −12.3355 −0.563625 −0.281812 0.959470i \(-0.590936\pi\)
−0.281812 + 0.959470i \(0.590936\pi\)
\(480\) 0 0
\(481\) −8.03212 −0.366233
\(482\) 0 0
\(483\) 1.01226 0.0460595
\(484\) 0 0
\(485\) 16.0745 0.729906
\(486\) 0 0
\(487\) −21.9102 −0.992847 −0.496424 0.868080i \(-0.665354\pi\)
−0.496424 + 0.868080i \(0.665354\pi\)
\(488\) 0 0
\(489\) 3.20154 0.144779
\(490\) 0 0
\(491\) −27.3534 −1.23444 −0.617220 0.786790i \(-0.711741\pi\)
−0.617220 + 0.786790i \(0.711741\pi\)
\(492\) 0 0
\(493\) −40.8492 −1.83976
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.31215 0.238282
\(498\) 0 0
\(499\) −10.3980 −0.465478 −0.232739 0.972539i \(-0.574769\pi\)
−0.232739 + 0.972539i \(0.574769\pi\)
\(500\) 0 0
\(501\) −16.7435 −0.748044
\(502\) 0 0
\(503\) −10.3868 −0.463123 −0.231561 0.972820i \(-0.574383\pi\)
−0.231561 + 0.972820i \(0.574383\pi\)
\(504\) 0 0
\(505\) 15.5883 0.693670
\(506\) 0 0
\(507\) −3.56724 −0.158427
\(508\) 0 0
\(509\) 23.0777 1.02290 0.511450 0.859313i \(-0.329109\pi\)
0.511450 + 0.859313i \(0.329109\pi\)
\(510\) 0 0
\(511\) 0.193061 0.00854051
\(512\) 0 0
\(513\) −11.4767 −0.506710
\(514\) 0 0
\(515\) −9.25907 −0.408003
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −9.23854 −0.405527
\(520\) 0 0
\(521\) 35.4191 1.55174 0.775868 0.630895i \(-0.217312\pi\)
0.775868 + 0.630895i \(0.217312\pi\)
\(522\) 0 0
\(523\) −18.1523 −0.793745 −0.396872 0.917874i \(-0.629904\pi\)
−0.396872 + 0.917874i \(0.629904\pi\)
\(524\) 0 0
\(525\) 0.268343 0.0117115
\(526\) 0 0
\(527\) −38.8955 −1.69431
\(528\) 0 0
\(529\) −8.77004 −0.381306
\(530\) 0 0
\(531\) 4.10860 0.178298
\(532\) 0 0
\(533\) 10.5223 0.455770
\(534\) 0 0
\(535\) −2.15917 −0.0933493
\(536\) 0 0
\(537\) −2.51766 −0.108645
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 16.2595 0.699052 0.349526 0.936927i \(-0.386343\pi\)
0.349526 + 0.936927i \(0.386343\pi\)
\(542\) 0 0
\(543\) 1.27817 0.0548515
\(544\) 0 0
\(545\) −5.79052 −0.248038
\(546\) 0 0
\(547\) 15.5972 0.666886 0.333443 0.942770i \(-0.391790\pi\)
0.333443 + 0.942770i \(0.391790\pi\)
\(548\) 0 0
\(549\) −21.6655 −0.924660
\(550\) 0 0
\(551\) 24.7399 1.05396
\(552\) 0 0
\(553\) 4.88359 0.207671
\(554\) 0 0
\(555\) 2.02585 0.0859925
\(556\) 0 0
\(557\) 10.5251 0.445962 0.222981 0.974823i \(-0.428421\pi\)
0.222981 + 0.974823i \(0.428421\pi\)
\(558\) 0 0
\(559\) 18.3411 0.775747
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.7231 0.494069 0.247034 0.969007i \(-0.420544\pi\)
0.247034 + 0.969007i \(0.420544\pi\)
\(564\) 0 0
\(565\) 19.0148 0.799959
\(566\) 0 0
\(567\) 1.76015 0.0739194
\(568\) 0 0
\(569\) −9.57510 −0.401409 −0.200705 0.979652i \(-0.564323\pi\)
−0.200705 + 0.979652i \(0.564323\pi\)
\(570\) 0 0
\(571\) 44.0439 1.84318 0.921589 0.388166i \(-0.126891\pi\)
0.921589 + 0.388166i \(0.126891\pi\)
\(572\) 0 0
\(573\) 2.42620 0.101356
\(574\) 0 0
\(575\) 3.77226 0.157314
\(576\) 0 0
\(577\) −28.0466 −1.16760 −0.583798 0.811899i \(-0.698434\pi\)
−0.583798 + 0.811899i \(0.698434\pi\)
\(578\) 0 0
\(579\) −5.54662 −0.230510
\(580\) 0 0
\(581\) 1.02953 0.0427123
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 7.04568 0.291303
\(586\) 0 0
\(587\) −2.70939 −0.111828 −0.0559142 0.998436i \(-0.517807\pi\)
−0.0559142 + 0.998436i \(0.517807\pi\)
\(588\) 0 0
\(589\) 23.5567 0.970637
\(590\) 0 0
\(591\) −5.06059 −0.208165
\(592\) 0 0
\(593\) −35.1817 −1.44474 −0.722369 0.691508i \(-0.756947\pi\)
−0.722369 + 0.691508i \(0.756947\pi\)
\(594\) 0 0
\(595\) −1.81980 −0.0746046
\(596\) 0 0
\(597\) 9.11192 0.372926
\(598\) 0 0
\(599\) −47.2602 −1.93100 −0.965500 0.260404i \(-0.916144\pi\)
−0.965500 + 0.260404i \(0.916144\pi\)
\(600\) 0 0
\(601\) 11.6431 0.474932 0.237466 0.971396i \(-0.423683\pi\)
0.237466 + 0.971396i \(0.423683\pi\)
\(602\) 0 0
\(603\) −28.4973 −1.16050
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −45.9362 −1.86449 −0.932246 0.361825i \(-0.882154\pi\)
−0.932246 + 0.361825i \(0.882154\pi\)
\(608\) 0 0
\(609\) 2.26588 0.0918180
\(610\) 0 0
\(611\) −7.32239 −0.296232
\(612\) 0 0
\(613\) 3.35657 0.135571 0.0677854 0.997700i \(-0.478407\pi\)
0.0677854 + 0.997700i \(0.478407\pi\)
\(614\) 0 0
\(615\) −2.65391 −0.107016
\(616\) 0 0
\(617\) −16.4880 −0.663783 −0.331892 0.943318i \(-0.607687\pi\)
−0.331892 + 0.943318i \(0.607687\pi\)
\(618\) 0 0
\(619\) 2.05765 0.0827039 0.0413520 0.999145i \(-0.486834\pi\)
0.0413520 + 0.999145i \(0.486834\pi\)
\(620\) 0 0
\(621\) −14.7764 −0.592955
\(622\) 0 0
\(623\) 4.52142 0.181147
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13.7385 −0.547791
\(630\) 0 0
\(631\) 35.2154 1.40190 0.700951 0.713209i \(-0.252759\pi\)
0.700951 + 0.713209i \(0.252759\pi\)
\(632\) 0 0
\(633\) 5.23634 0.208126
\(634\) 0 0
\(635\) −10.0973 −0.400700
\(636\) 0 0
\(637\) 19.3979 0.768574
\(638\) 0 0
\(639\) −35.1787 −1.39165
\(640\) 0 0
\(641\) −33.6602 −1.32950 −0.664750 0.747066i \(-0.731462\pi\)
−0.664750 + 0.747066i \(0.731462\pi\)
\(642\) 0 0
\(643\) −2.01326 −0.0793951 −0.0396976 0.999212i \(-0.512639\pi\)
−0.0396976 + 0.999212i \(0.512639\pi\)
\(644\) 0 0
\(645\) −4.62598 −0.182148
\(646\) 0 0
\(647\) 34.0410 1.33829 0.669145 0.743131i \(-0.266660\pi\)
0.669145 + 0.743131i \(0.266660\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.15751 0.0845594
\(652\) 0 0
\(653\) −21.1626 −0.828156 −0.414078 0.910241i \(-0.635896\pi\)
−0.414078 + 0.910241i \(0.635896\pi\)
\(654\) 0 0
\(655\) −6.29651 −0.246025
\(656\) 0 0
\(657\) −1.27851 −0.0498794
\(658\) 0 0
\(659\) 21.3464 0.831539 0.415770 0.909470i \(-0.363512\pi\)
0.415770 + 0.909470i \(0.363512\pi\)
\(660\) 0 0
\(661\) 11.3318 0.440757 0.220379 0.975414i \(-0.429271\pi\)
0.220379 + 0.975414i \(0.429271\pi\)
\(662\) 0 0
\(663\) 9.76041 0.379063
\(664\) 0 0
\(665\) 1.10215 0.0427394
\(666\) 0 0
\(667\) 31.8528 1.23335
\(668\) 0 0
\(669\) 4.66443 0.180337
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19.4852 −0.751100 −0.375550 0.926802i \(-0.622546\pi\)
−0.375550 + 0.926802i \(0.622546\pi\)
\(674\) 0 0
\(675\) −3.91711 −0.150770
\(676\) 0 0
\(677\) −25.7596 −0.990021 −0.495010 0.868887i \(-0.664836\pi\)
−0.495010 + 0.868887i \(0.664836\pi\)
\(678\) 0 0
\(679\) 6.04679 0.232054
\(680\) 0 0
\(681\) 14.6566 0.561643
\(682\) 0 0
\(683\) 13.0179 0.498117 0.249059 0.968488i \(-0.419879\pi\)
0.249059 + 0.968488i \(0.419879\pi\)
\(684\) 0 0
\(685\) −8.27329 −0.316106
\(686\) 0 0
\(687\) −3.08948 −0.117871
\(688\) 0 0
\(689\) 0.137916 0.00525417
\(690\) 0 0
\(691\) −48.7463 −1.85440 −0.927198 0.374573i \(-0.877789\pi\)
−0.927198 + 0.374573i \(0.877789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.2671 −0.389453
\(696\) 0 0
\(697\) 17.9978 0.681717
\(698\) 0 0
\(699\) 20.4443 0.773276
\(700\) 0 0
\(701\) 45.1532 1.70541 0.852706 0.522391i \(-0.174960\pi\)
0.852706 + 0.522391i \(0.174960\pi\)
\(702\) 0 0
\(703\) 8.32062 0.313818
\(704\) 0 0
\(705\) 1.84684 0.0695562
\(706\) 0 0
\(707\) 5.86388 0.220534
\(708\) 0 0
\(709\) −7.84526 −0.294635 −0.147318 0.989089i \(-0.547064\pi\)
−0.147318 + 0.989089i \(0.547064\pi\)
\(710\) 0 0
\(711\) −32.3406 −1.21287
\(712\) 0 0
\(713\) 30.3294 1.13584
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.47352 −0.279104
\(718\) 0 0
\(719\) 36.4371 1.35888 0.679438 0.733733i \(-0.262224\pi\)
0.679438 + 0.733733i \(0.262224\pi\)
\(720\) 0 0
\(721\) −3.48300 −0.129714
\(722\) 0 0
\(723\) 11.0714 0.411749
\(724\) 0 0
\(725\) 8.44396 0.313601
\(726\) 0 0
\(727\) −15.4970 −0.574753 −0.287376 0.957818i \(-0.592783\pi\)
−0.287376 + 0.957818i \(0.592783\pi\)
\(728\) 0 0
\(729\) −3.27343 −0.121238
\(730\) 0 0
\(731\) 31.3716 1.16032
\(732\) 0 0
\(733\) 42.3920 1.56579 0.782893 0.622157i \(-0.213743\pi\)
0.782893 + 0.622157i \(0.213743\pi\)
\(734\) 0 0
\(735\) −4.89252 −0.180463
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −30.3196 −1.11532 −0.557661 0.830069i \(-0.688301\pi\)
−0.557661 + 0.830069i \(0.688301\pi\)
\(740\) 0 0
\(741\) −5.91130 −0.217157
\(742\) 0 0
\(743\) −20.8300 −0.764179 −0.382089 0.924125i \(-0.624795\pi\)
−0.382089 + 0.924125i \(0.624795\pi\)
\(744\) 0 0
\(745\) −4.90160 −0.179581
\(746\) 0 0
\(747\) −6.81790 −0.249454
\(748\) 0 0
\(749\) −0.812221 −0.0296779
\(750\) 0 0
\(751\) −30.7746 −1.12298 −0.561491 0.827483i \(-0.689772\pi\)
−0.561491 + 0.827483i \(0.689772\pi\)
\(752\) 0 0
\(753\) −8.53891 −0.311175
\(754\) 0 0
\(755\) −9.35625 −0.340509
\(756\) 0 0
\(757\) −25.3851 −0.922637 −0.461318 0.887235i \(-0.652623\pi\)
−0.461318 + 0.887235i \(0.652623\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.6390 1.21941 0.609707 0.792627i \(-0.291287\pi\)
0.609707 + 0.792627i \(0.291287\pi\)
\(762\) 0 0
\(763\) −2.17823 −0.0788573
\(764\) 0 0
\(765\) 12.0513 0.435715
\(766\) 0 0
\(767\) 4.66471 0.168433
\(768\) 0 0
\(769\) −29.4388 −1.06159 −0.530795 0.847500i \(-0.678107\pi\)
−0.530795 + 0.847500i \(0.678107\pi\)
\(770\) 0 0
\(771\) −6.04453 −0.217688
\(772\) 0 0
\(773\) 24.7585 0.890501 0.445251 0.895406i \(-0.353115\pi\)
0.445251 + 0.895406i \(0.353115\pi\)
\(774\) 0 0
\(775\) 8.04011 0.288809
\(776\) 0 0
\(777\) 0.762068 0.0273390
\(778\) 0 0
\(779\) −10.9002 −0.390541
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −33.0759 −1.18204
\(784\) 0 0
\(785\) 9.65817 0.344715
\(786\) 0 0
\(787\) 37.4598 1.33530 0.667649 0.744476i \(-0.267301\pi\)
0.667649 + 0.744476i \(0.267301\pi\)
\(788\) 0 0
\(789\) 8.93040 0.317931
\(790\) 0 0
\(791\) 7.15284 0.254326
\(792\) 0 0
\(793\) −24.5979 −0.873499
\(794\) 0 0
\(795\) −0.0347849 −0.00123369
\(796\) 0 0
\(797\) −48.1456 −1.70541 −0.852703 0.522397i \(-0.825038\pi\)
−0.852703 + 0.522397i \(0.825038\pi\)
\(798\) 0 0
\(799\) −12.5246 −0.443088
\(800\) 0 0
\(801\) −29.9422 −1.05796
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.41902 0.0500139
\(806\) 0 0
\(807\) −17.1667 −0.604297
\(808\) 0 0
\(809\) 37.2876 1.31096 0.655481 0.755211i \(-0.272466\pi\)
0.655481 + 0.755211i \(0.272466\pi\)
\(810\) 0 0
\(811\) 8.87605 0.311680 0.155840 0.987782i \(-0.450192\pi\)
0.155840 + 0.987782i \(0.450192\pi\)
\(812\) 0 0
\(813\) −6.72135 −0.235728
\(814\) 0 0
\(815\) 4.48802 0.157208
\(816\) 0 0
\(817\) −18.9999 −0.664723
\(818\) 0 0
\(819\) 2.65039 0.0926121
\(820\) 0 0
\(821\) −31.1530 −1.08725 −0.543624 0.839329i \(-0.682948\pi\)
−0.543624 + 0.839329i \(0.682948\pi\)
\(822\) 0 0
\(823\) −32.6349 −1.13758 −0.568790 0.822483i \(-0.692588\pi\)
−0.568790 + 0.822483i \(0.692588\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.7594 −1.00006 −0.500031 0.866008i \(-0.666678\pi\)
−0.500031 + 0.866008i \(0.666678\pi\)
\(828\) 0 0
\(829\) −42.7970 −1.48640 −0.743200 0.669069i \(-0.766693\pi\)
−0.743200 + 0.669069i \(0.766693\pi\)
\(830\) 0 0
\(831\) 16.9643 0.588485
\(832\) 0 0
\(833\) 33.1792 1.14959
\(834\) 0 0
\(835\) −23.4716 −0.812267
\(836\) 0 0
\(837\) −31.4940 −1.08859
\(838\) 0 0
\(839\) −28.5327 −0.985059 −0.492529 0.870296i \(-0.663928\pi\)
−0.492529 + 0.870296i \(0.663928\pi\)
\(840\) 0 0
\(841\) 42.3004 1.45863
\(842\) 0 0
\(843\) 0.354468 0.0122085
\(844\) 0 0
\(845\) −5.00067 −0.172028
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.33870 0.114584
\(850\) 0 0
\(851\) 10.7128 0.367232
\(852\) 0 0
\(853\) 33.4101 1.14394 0.571970 0.820275i \(-0.306179\pi\)
0.571970 + 0.820275i \(0.306179\pi\)
\(854\) 0 0
\(855\) −7.29875 −0.249612
\(856\) 0 0
\(857\) 11.9632 0.408656 0.204328 0.978902i \(-0.434499\pi\)
0.204328 + 0.978902i \(0.434499\pi\)
\(858\) 0 0
\(859\) 23.4984 0.801755 0.400878 0.916132i \(-0.368705\pi\)
0.400878 + 0.916132i \(0.368705\pi\)
\(860\) 0 0
\(861\) −0.998328 −0.0340229
\(862\) 0 0
\(863\) 4.09414 0.139366 0.0696830 0.997569i \(-0.477801\pi\)
0.0696830 + 0.997569i \(0.477801\pi\)
\(864\) 0 0
\(865\) −12.9509 −0.440343
\(866\) 0 0
\(867\) 4.56772 0.155128
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −32.3544 −1.09629
\(872\) 0 0
\(873\) −40.0437 −1.35527
\(874\) 0 0
\(875\) 0.376172 0.0127169
\(876\) 0 0
\(877\) 44.3667 1.49816 0.749079 0.662481i \(-0.230497\pi\)
0.749079 + 0.662481i \(0.230497\pi\)
\(878\) 0 0
\(879\) 12.0701 0.407115
\(880\) 0 0
\(881\) −43.0315 −1.44977 −0.724884 0.688871i \(-0.758107\pi\)
−0.724884 + 0.688871i \(0.758107\pi\)
\(882\) 0 0
\(883\) −36.5444 −1.22982 −0.614909 0.788598i \(-0.710807\pi\)
−0.614909 + 0.788598i \(0.710807\pi\)
\(884\) 0 0
\(885\) −1.17653 −0.0395485
\(886\) 0 0
\(887\) −1.79439 −0.0602496 −0.0301248 0.999546i \(-0.509590\pi\)
−0.0301248 + 0.999546i \(0.509590\pi\)
\(888\) 0 0
\(889\) −3.79833 −0.127392
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.58541 0.253836
\(894\) 0 0
\(895\) −3.52933 −0.117973
\(896\) 0 0
\(897\) −7.61084 −0.254119
\(898\) 0 0
\(899\) 67.8903 2.26427
\(900\) 0 0
\(901\) 0.235898 0.00785890
\(902\) 0 0
\(903\) −1.74016 −0.0579090
\(904\) 0 0
\(905\) 1.79178 0.0595608
\(906\) 0 0
\(907\) −51.2671 −1.70230 −0.851149 0.524924i \(-0.824094\pi\)
−0.851149 + 0.524924i \(0.824094\pi\)
\(908\) 0 0
\(909\) −38.8324 −1.28799
\(910\) 0 0
\(911\) 40.0803 1.32792 0.663960 0.747768i \(-0.268875\pi\)
0.663960 + 0.747768i \(0.268875\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 6.20406 0.205100
\(916\) 0 0
\(917\) −2.36857 −0.0782171
\(918\) 0 0
\(919\) 1.08348 0.0357406 0.0178703 0.999840i \(-0.494311\pi\)
0.0178703 + 0.999840i \(0.494311\pi\)
\(920\) 0 0
\(921\) −24.5156 −0.807816
\(922\) 0 0
\(923\) −39.9402 −1.31465
\(924\) 0 0
\(925\) 2.83990 0.0933754
\(926\) 0 0
\(927\) 23.0655 0.757571
\(928\) 0 0
\(929\) 14.2742 0.468320 0.234160 0.972198i \(-0.424766\pi\)
0.234160 + 0.972198i \(0.424766\pi\)
\(930\) 0 0
\(931\) −20.0947 −0.658577
\(932\) 0 0
\(933\) 5.25591 0.172071
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27.8901 −0.911130 −0.455565 0.890203i \(-0.650563\pi\)
−0.455565 + 0.890203i \(0.650563\pi\)
\(938\) 0 0
\(939\) −17.8770 −0.583394
\(940\) 0 0
\(941\) 3.62440 0.118152 0.0590760 0.998253i \(-0.481185\pi\)
0.0590760 + 0.998253i \(0.481185\pi\)
\(942\) 0 0
\(943\) −14.0341 −0.457013
\(944\) 0 0
\(945\) −1.47351 −0.0479332
\(946\) 0 0
\(947\) −54.9559 −1.78583 −0.892913 0.450230i \(-0.851342\pi\)
−0.892913 + 0.450230i \(0.851342\pi\)
\(948\) 0 0
\(949\) −1.45156 −0.0471196
\(950\) 0 0
\(951\) 20.0258 0.649382
\(952\) 0 0
\(953\) −9.03643 −0.292719 −0.146359 0.989231i \(-0.546756\pi\)
−0.146359 + 0.989231i \(0.546756\pi\)
\(954\) 0 0
\(955\) 3.40113 0.110058
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.11218 −0.100498
\(960\) 0 0
\(961\) 33.6433 1.08527
\(962\) 0 0
\(963\) 5.37878 0.173329
\(964\) 0 0
\(965\) −7.77543 −0.250300
\(966\) 0 0
\(967\) −44.4619 −1.42980 −0.714900 0.699227i \(-0.753528\pi\)
−0.714900 + 0.699227i \(0.753528\pi\)
\(968\) 0 0
\(969\) −10.1110 −0.324812
\(970\) 0 0
\(971\) 1.66867 0.0535502 0.0267751 0.999641i \(-0.491476\pi\)
0.0267751 + 0.999641i \(0.491476\pi\)
\(972\) 0 0
\(973\) −3.86219 −0.123816
\(974\) 0 0
\(975\) −2.01758 −0.0646143
\(976\) 0 0
\(977\) 22.6537 0.724755 0.362377 0.932031i \(-0.381965\pi\)
0.362377 + 0.932031i \(0.381965\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.4249 0.460552
\(982\) 0 0
\(983\) 46.9510 1.49750 0.748752 0.662850i \(-0.230653\pi\)
0.748752 + 0.662850i \(0.230653\pi\)
\(984\) 0 0
\(985\) −7.09410 −0.226037
\(986\) 0 0
\(987\) 0.694731 0.0221135
\(988\) 0 0
\(989\) −24.4625 −0.777863
\(990\) 0 0
\(991\) −0.122612 −0.00389489 −0.00194745 0.999998i \(-0.500620\pi\)
−0.00194745 + 0.999998i \(0.500620\pi\)
\(992\) 0 0
\(993\) 15.6684 0.497221
\(994\) 0 0
\(995\) 12.7734 0.404943
\(996\) 0 0
\(997\) −20.1831 −0.639205 −0.319602 0.947552i \(-0.603549\pi\)
−0.319602 + 0.947552i \(0.603549\pi\)
\(998\) 0 0
\(999\) −11.1242 −0.351954
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 4840.2.a.bg.1.5 8
4.3 odd 2 9680.2.a.df.1.4 8
11.5 even 5 440.2.y.d.201.3 yes 16
11.9 even 5 440.2.y.d.81.3 16
11.10 odd 2 4840.2.a.bh.1.5 8
44.27 odd 10 880.2.bo.k.641.2 16
44.31 odd 10 880.2.bo.k.81.2 16
44.43 even 2 9680.2.a.de.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.3 16 11.9 even 5
440.2.y.d.201.3 yes 16 11.5 even 5
880.2.bo.k.81.2 16 44.31 odd 10
880.2.bo.k.641.2 16 44.27 odd 10
4840.2.a.bg.1.5 8 1.1 even 1 trivial
4840.2.a.bh.1.5 8 11.10 odd 2
9680.2.a.de.1.4 8 44.43 even 2
9680.2.a.df.1.4 8 4.3 odd 2