Properties

Label 7920.2.a.bq.1.1
Level $7920$
Weight $2$
Character 7920.1
Self dual yes
Analytic conductor $63.242$
Analytic rank $1$
Dimension $2$
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] = [7920,2,Mod(1,7920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7920, 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("7920.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7920.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: \(63.2415184009\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 7920.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -3.37228 q^{7} -1.00000 q^{11} +2.00000 q^{13} -1.37228 q^{17} -0.627719 q^{19} +2.74456 q^{23} +1.00000 q^{25} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{35} +9.37228 q^{37} +11.4891 q^{41} +4.00000 q^{43} +2.74456 q^{47} +4.37228 q^{49} +4.11684 q^{53} +1.00000 q^{55} -2.74456 q^{59} -5.37228 q^{61} -2.00000 q^{65} -8.00000 q^{67} +10.1168 q^{71} -15.4891 q^{73} +3.37228 q^{77} +1.25544 q^{79} -2.74456 q^{83} +1.37228 q^{85} +1.37228 q^{89} -6.74456 q^{91} +0.627719 q^{95} -12.7446 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - q^{7} - 2 q^{11} + 4 q^{13} + 3 q^{17} - 7 q^{19} - 6 q^{23} + 2 q^{25} + 3 q^{29} - q^{31} + q^{35} + 13 q^{37} + 8 q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{53} + 2 q^{55} + 6 q^{59} - 5 q^{61}+ \cdots - 14 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.37228 −0.332827 −0.166414 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) 0 0
\(19\) −0.627719 −0.144009 −0.0720043 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.74456 0.572281 0.286140 0.958188i \(-0.407628\pi\)
0.286140 + 0.958188i \(0.407628\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.37228 0.570020
\(36\) 0 0
\(37\) 9.37228 1.54079 0.770397 0.637565i \(-0.220058\pi\)
0.770397 + 0.637565i \(0.220058\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.4891 1.79430 0.897150 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.74456 0.400336 0.200168 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.11684 0.565492 0.282746 0.959195i \(-0.408755\pi\)
0.282746 + 0.959195i \(0.408755\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.1168 1.20065 0.600324 0.799757i \(-0.295038\pi\)
0.600324 + 0.799757i \(0.295038\pi\)
\(72\) 0 0
\(73\) −15.4891 −1.81286 −0.906432 0.422351i \(-0.861205\pi\)
−0.906432 + 0.422351i \(0.861205\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.37228 0.384307
\(78\) 0 0
\(79\) 1.25544 0.141248 0.0706239 0.997503i \(-0.477501\pi\)
0.0706239 + 0.997503i \(0.477501\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.74456 −0.301255 −0.150627 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) 0 0
\(85\) 1.37228 0.148845
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.627719 0.0644026
\(96\) 0 0
\(97\) −12.7446 −1.29401 −0.647007 0.762484i \(-0.723980\pi\)
−0.647007 + 0.762484i \(0.723980\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 9.48913 0.934991 0.467496 0.883995i \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −15.4891 −1.48359 −0.741795 0.670627i \(-0.766025\pi\)
−0.741795 + 0.670627i \(0.766025\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.25544 −0.306246 −0.153123 0.988207i \(-0.548933\pi\)
−0.153123 + 0.988207i \(0.548933\pi\)
\(114\) 0 0
\(115\) −2.74456 −0.255932
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.62772 0.424222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 22.1168 1.93236 0.966179 0.257873i \(-0.0830216\pi\)
0.966179 + 0.257873i \(0.0830216\pi\)
\(132\) 0 0
\(133\) 2.11684 0.183554
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.74456 −0.747098 −0.373549 0.927610i \(-0.621859\pi\)
−0.373549 + 0.927610i \(0.621859\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 1.37228 0.113962
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −21.6060 −1.77003 −0.885015 0.465563i \(-0.845852\pi\)
−0.885015 + 0.465563i \(0.845852\pi\)
\(150\) 0 0
\(151\) 12.2337 0.995563 0.497782 0.867302i \(-0.334148\pi\)
0.497782 + 0.867302i \(0.334148\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.37228 0.270868
\(156\) 0 0
\(157\) 9.37228 0.747989 0.373995 0.927431i \(-0.377988\pi\)
0.373995 + 0.927431i \(0.377988\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.25544 −0.729431
\(162\) 0 0
\(163\) 5.88316 0.460804 0.230402 0.973095i \(-0.425996\pi\)
0.230402 + 0.973095i \(0.425996\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.62772 −0.358104 −0.179052 0.983840i \(-0.557303\pi\)
−0.179052 + 0.983840i \(0.557303\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −3.37228 −0.254921
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.37228 −0.689064
\(186\) 0 0
\(187\) 1.37228 0.100351
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.48913 −0.397179 −0.198590 0.980083i \(-0.563636\pi\)
−0.198590 + 0.980083i \(0.563636\pi\)
\(192\) 0 0
\(193\) 14.8614 1.06975 0.534874 0.844932i \(-0.320359\pi\)
0.534874 + 0.844932i \(0.320359\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7446 1.47799 0.738994 0.673712i \(-0.235301\pi\)
0.738994 + 0.673712i \(0.235301\pi\)
\(198\) 0 0
\(199\) −18.1168 −1.28427 −0.642135 0.766592i \(-0.721951\pi\)
−0.642135 + 0.766592i \(0.721951\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.62772 0.324802
\(204\) 0 0
\(205\) −11.4891 −0.802435
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) −6.11684 −0.421101 −0.210550 0.977583i \(-0.567526\pi\)
−0.210550 + 0.977583i \(0.567526\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 11.3723 0.772001
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.74456 −0.184619
\(222\) 0 0
\(223\) 18.7446 1.25523 0.627614 0.778524i \(-0.284031\pi\)
0.627614 + 0.778524i \(0.284031\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.74456 −0.182163 −0.0910815 0.995843i \(-0.529032\pi\)
−0.0910815 + 0.995843i \(0.529032\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.37228 −0.0899011 −0.0449506 0.998989i \(-0.514313\pi\)
−0.0449506 + 0.998989i \(0.514313\pi\)
\(234\) 0 0
\(235\) −2.74456 −0.179036
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.7446 −0.953746 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.37228 −0.279335
\(246\) 0 0
\(247\) −1.25544 −0.0798816
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.74456 0.173235 0.0866176 0.996242i \(-0.472394\pi\)
0.0866176 + 0.996242i \(0.472394\pi\)
\(252\) 0 0
\(253\) −2.74456 −0.172549
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −31.6060 −1.96390
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.8614 −1.53302 −0.766510 0.642232i \(-0.778008\pi\)
−0.766510 + 0.642232i \(0.778008\pi\)
\(264\) 0 0
\(265\) −4.11684 −0.252896
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −12.7446 −0.765747 −0.382873 0.923801i \(-0.625065\pi\)
−0.382873 + 0.923801i \(0.625065\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −5.25544 −0.312403 −0.156202 0.987725i \(-0.549925\pi\)
−0.156202 + 0.987725i \(0.549925\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −38.7446 −2.28702
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −23.4891 −1.37225 −0.686125 0.727484i \(-0.740690\pi\)
−0.686125 + 0.727484i \(0.740690\pi\)
\(294\) 0 0
\(295\) 2.74456 0.159795
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.48913 0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.37228 0.307616
\(306\) 0 0
\(307\) −5.25544 −0.299944 −0.149972 0.988690i \(-0.547918\pi\)
−0.149972 + 0.988690i \(0.547918\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.3723 1.09850 0.549251 0.835658i \(-0.314913\pi\)
0.549251 + 0.835658i \(0.314913\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.3505 1.36766 0.683831 0.729640i \(-0.260313\pi\)
0.683831 + 0.729640i \(0.260313\pi\)
\(318\) 0 0
\(319\) 1.37228 0.0768330
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.861407 0.0479299
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) −30.9783 −1.70272 −0.851359 0.524583i \(-0.824221\pi\)
−0.851359 + 0.524583i \(0.824221\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 24.1168 1.31373 0.656864 0.754009i \(-0.271883\pi\)
0.656864 + 0.754009i \(0.271883\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.37228 0.182619
\(342\) 0 0
\(343\) 8.86141 0.478471
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.2337 −1.73040 −0.865198 0.501431i \(-0.832807\pi\)
−0.865198 + 0.501431i \(0.832807\pi\)
\(348\) 0 0
\(349\) 19.4891 1.04323 0.521614 0.853181i \(-0.325330\pi\)
0.521614 + 0.853181i \(0.325330\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.510875 0.0271911 0.0135956 0.999908i \(-0.495672\pi\)
0.0135956 + 0.999908i \(0.495672\pi\)
\(354\) 0 0
\(355\) −10.1168 −0.536946
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.4891 0.810738
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.8832 −0.720778
\(372\) 0 0
\(373\) 31.4891 1.63045 0.815223 0.579148i \(-0.196615\pi\)
0.815223 + 0.579148i \(0.196615\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.74456 −0.141352
\(378\) 0 0
\(379\) 0.233688 0.0120037 0.00600187 0.999982i \(-0.498090\pi\)
0.00600187 + 0.999982i \(0.498090\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.2337 1.64706 0.823532 0.567269i \(-0.192000\pi\)
0.823532 + 0.567269i \(0.192000\pi\)
\(384\) 0 0
\(385\) −3.37228 −0.171867
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −3.76631 −0.190471
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.25544 −0.0631679
\(396\) 0 0
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13.3723 −0.667780 −0.333890 0.942612i \(-0.608361\pi\)
−0.333890 + 0.942612i \(0.608361\pi\)
\(402\) 0 0
\(403\) −6.74456 −0.335971
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9.37228 −0.464567
\(408\) 0 0
\(409\) −1.76631 −0.0873385 −0.0436693 0.999046i \(-0.513905\pi\)
−0.0436693 + 0.999046i \(0.513905\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.25544 0.455430
\(414\) 0 0
\(415\) 2.74456 0.134725
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 10.2337 0.498759 0.249380 0.968406i \(-0.419773\pi\)
0.249380 + 0.968406i \(0.419773\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.37228 −0.0665654
\(426\) 0 0
\(427\) 18.1168 0.876736
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −34.9783 −1.68484 −0.842422 0.538819i \(-0.818871\pi\)
−0.842422 + 0.538819i \(0.818871\pi\)
\(432\) 0 0
\(433\) 27.7228 1.33227 0.666137 0.745830i \(-0.267947\pi\)
0.666137 + 0.745830i \(0.267947\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.72281 −0.0824133
\(438\) 0 0
\(439\) −18.9783 −0.905782 −0.452891 0.891566i \(-0.649607\pi\)
−0.452891 + 0.891566i \(0.649607\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.4891 1.40107 0.700535 0.713618i \(-0.252945\pi\)
0.700535 + 0.713618i \(0.252945\pi\)
\(444\) 0 0
\(445\) −1.37228 −0.0650524
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.9783 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(450\) 0 0
\(451\) −11.4891 −0.541002
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.74456 0.316190
\(456\) 0 0
\(457\) −16.3505 −0.764846 −0.382423 0.923987i \(-0.624910\pi\)
−0.382423 + 0.923987i \(0.624910\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.1168 −0.750636 −0.375318 0.926896i \(-0.622467\pi\)
−0.375318 + 0.926896i \(0.622467\pi\)
\(462\) 0 0
\(463\) 0.233688 0.0108604 0.00543020 0.999985i \(-0.498272\pi\)
0.00543020 + 0.999985i \(0.498272\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.3723 −0.896442 −0.448221 0.893923i \(-0.647942\pi\)
−0.448221 + 0.893923i \(0.647942\pi\)
\(468\) 0 0
\(469\) 26.9783 1.24574
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −0.627719 −0.0288017
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.48913 0.250805 0.125402 0.992106i \(-0.459978\pi\)
0.125402 + 0.992106i \(0.459978\pi\)
\(480\) 0 0
\(481\) 18.7446 0.854678
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.7446 0.578701
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.37228 −0.332706 −0.166353 0.986066i \(-0.553199\pi\)
−0.166353 + 0.986066i \(0.553199\pi\)
\(492\) 0 0
\(493\) 1.88316 0.0848131
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −34.1168 −1.53035
\(498\) 0 0
\(499\) 33.4891 1.49918 0.749590 0.661903i \(-0.230251\pi\)
0.749590 + 0.661903i \(0.230251\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −34.9783 −1.55960 −0.779802 0.626027i \(-0.784680\pi\)
−0.779802 + 0.626027i \(0.784680\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.76631 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(510\) 0 0
\(511\) 52.2337 2.31068
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −9.48913 −0.418141
\(516\) 0 0
\(517\) −2.74456 −0.120706
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.5109 −0.548111 −0.274056 0.961714i \(-0.588365\pi\)
−0.274056 + 0.961714i \(0.588365\pi\)
\(522\) 0 0
\(523\) −30.9783 −1.35458 −0.677292 0.735714i \(-0.736847\pi\)
−0.677292 + 0.735714i \(0.736847\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.62772 0.201587
\(528\) 0 0
\(529\) −15.4674 −0.672495
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 22.9783 0.995299
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.37228 −0.188327
\(540\) 0 0
\(541\) −20.1168 −0.864891 −0.432445 0.901660i \(-0.642349\pi\)
−0.432445 + 0.901660i \(0.642349\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 15.4891 0.663481
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.861407 0.0366972
\(552\) 0 0
\(553\) −4.23369 −0.180035
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.97825 −0.210935 −0.105468 0.994423i \(-0.533634\pi\)
−0.105468 + 0.994423i \(0.533634\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.23369 −0.347009 −0.173504 0.984833i \(-0.555509\pi\)
−0.173504 + 0.984833i \(0.555509\pi\)
\(564\) 0 0
\(565\) 3.25544 0.136957
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) −15.3723 −0.643310 −0.321655 0.946857i \(-0.604239\pi\)
−0.321655 + 0.946857i \(0.604239\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.74456 0.114456
\(576\) 0 0
\(577\) 36.9783 1.53942 0.769712 0.638391i \(-0.220400\pi\)
0.769712 + 0.638391i \(0.220400\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.25544 0.383980
\(582\) 0 0
\(583\) −4.11684 −0.170502
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.8614 1.02614 0.513070 0.858347i \(-0.328508\pi\)
0.513070 + 0.858347i \(0.328508\pi\)
\(588\) 0 0
\(589\) 2.11684 0.0872230
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.5109 0.513760 0.256880 0.966443i \(-0.417305\pi\)
0.256880 + 0.966443i \(0.417305\pi\)
\(594\) 0 0
\(595\) −4.62772 −0.189718
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.6060 −1.61826 −0.809128 0.587632i \(-0.800060\pi\)
−0.809128 + 0.587632i \(0.800060\pi\)
\(600\) 0 0
\(601\) −16.5109 −0.673493 −0.336746 0.941595i \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 5.88316 0.238790 0.119395 0.992847i \(-0.461905\pi\)
0.119395 + 0.992847i \(0.461905\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.48913 0.222066
\(612\) 0 0
\(613\) 20.5109 0.828426 0.414213 0.910180i \(-0.364057\pi\)
0.414213 + 0.910180i \(0.364057\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.23369 0.0899249 0.0449624 0.998989i \(-0.485683\pi\)
0.0449624 + 0.998989i \(0.485683\pi\)
\(618\) 0 0
\(619\) 44.4674 1.78729 0.893647 0.448770i \(-0.148138\pi\)
0.893647 + 0.448770i \(0.148138\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.62772 −0.185406
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) −42.1168 −1.67665 −0.838323 0.545175i \(-0.816463\pi\)
−0.838323 + 0.545175i \(0.816463\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 8.74456 0.346472
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.0951 1.07019 0.535096 0.844791i \(-0.320275\pi\)
0.535096 + 0.844791i \(0.320275\pi\)
\(642\) 0 0
\(643\) 5.88316 0.232009 0.116005 0.993249i \(-0.462991\pi\)
0.116005 + 0.993249i \(0.462991\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −37.7228 −1.48304 −0.741518 0.670933i \(-0.765894\pi\)
−0.741518 + 0.670933i \(0.765894\pi\)
\(648\) 0 0
\(649\) 2.74456 0.107734
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.6277 −0.415895 −0.207947 0.978140i \(-0.566678\pi\)
−0.207947 + 0.978140i \(0.566678\pi\)
\(654\) 0 0
\(655\) −22.1168 −0.864177
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.8614 0.501009 0.250505 0.968115i \(-0.419403\pi\)
0.250505 + 0.968115i \(0.419403\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.11684 −0.0820877
\(666\) 0 0
\(667\) −3.76631 −0.145832
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.37228 0.207395
\(672\) 0 0
\(673\) 14.8614 0.572865 0.286433 0.958100i \(-0.407531\pi\)
0.286433 + 0.958100i \(0.407531\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.25544 −0.125117 −0.0625583 0.998041i \(-0.519926\pi\)
−0.0625583 + 0.998041i \(0.519926\pi\)
\(678\) 0 0
\(679\) 42.9783 1.64935
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −28.6277 −1.09541 −0.547705 0.836672i \(-0.684498\pi\)
−0.547705 + 0.836672i \(0.684498\pi\)
\(684\) 0 0
\(685\) 8.74456 0.334113
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) −40.2337 −1.53056 −0.765281 0.643697i \(-0.777400\pi\)
−0.765281 + 0.643697i \(0.777400\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −15.7663 −0.597192
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 37.3723 1.41153 0.705766 0.708445i \(-0.250603\pi\)
0.705766 + 0.708445i \(0.250603\pi\)
\(702\) 0 0
\(703\) −5.88316 −0.221887
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.2337 0.760966
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.25544 −0.346619
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.8832 0.517754 0.258877 0.965910i \(-0.416648\pi\)
0.258877 + 0.965910i \(0.416648\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.37228 −0.0509652
\(726\) 0 0
\(727\) 24.2337 0.898778 0.449389 0.893336i \(-0.351642\pi\)
0.449389 + 0.893336i \(0.351642\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.48913 −0.203023
\(732\) 0 0
\(733\) 46.2337 1.70768 0.853840 0.520535i \(-0.174268\pi\)
0.853840 + 0.520535i \(0.174268\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 20.4674 0.752905 0.376452 0.926436i \(-0.377144\pi\)
0.376452 + 0.926436i \(0.377144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.62772 −0.169775 −0.0848873 0.996391i \(-0.527053\pi\)
−0.0848873 + 0.996391i \(0.527053\pi\)
\(744\) 0 0
\(745\) 21.6060 0.791581
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 40.4674 1.47865
\(750\) 0 0
\(751\) −8.86141 −0.323357 −0.161679 0.986843i \(-0.551691\pi\)
−0.161679 + 0.986843i \(0.551691\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.2337 −0.445229
\(756\) 0 0
\(757\) −20.9783 −0.762467 −0.381234 0.924479i \(-0.624501\pi\)
−0.381234 + 0.924479i \(0.624501\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.97825 −0.180461 −0.0902307 0.995921i \(-0.528760\pi\)
−0.0902307 + 0.995921i \(0.528760\pi\)
\(762\) 0 0
\(763\) 52.2337 1.89099
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.48913 −0.198201
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 33.6060 1.20872 0.604361 0.796710i \(-0.293428\pi\)
0.604361 + 0.796710i \(0.293428\pi\)
\(774\) 0 0
\(775\) −3.37228 −0.121136
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.21194 −0.258395
\(780\) 0 0
\(781\) −10.1168 −0.362009
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.37228 −0.334511
\(786\) 0 0
\(787\) 44.4674 1.58509 0.792545 0.609813i \(-0.208755\pi\)
0.792545 + 0.609813i \(0.208755\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.9783 0.390342
\(792\) 0 0
\(793\) −10.7446 −0.381551
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.4891 −0.406966 −0.203483 0.979079i \(-0.565226\pi\)
−0.203483 + 0.979079i \(0.565226\pi\)
\(798\) 0 0
\(799\) −3.76631 −0.133243
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.4891 0.546599
\(804\) 0 0
\(805\) 9.25544 0.326211
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −44.8614 −1.57530 −0.787649 0.616125i \(-0.788702\pi\)
−0.787649 + 0.616125i \(0.788702\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.88316 −0.206078
\(816\) 0 0
\(817\) −2.51087 −0.0878444
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.4891 −0.400973 −0.200487 0.979696i \(-0.564252\pi\)
−0.200487 + 0.979696i \(0.564252\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.9783 1.63359 0.816797 0.576925i \(-0.195748\pi\)
0.816797 + 0.576925i \(0.195748\pi\)
\(828\) 0 0
\(829\) −24.7446 −0.859414 −0.429707 0.902968i \(-0.641383\pi\)
−0.429707 + 0.902968i \(0.641383\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 4.62772 0.160149
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −10.9783 −0.379011 −0.189506 0.981880i \(-0.560689\pi\)
−0.189506 + 0.981880i \(0.560689\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −3.37228 −0.115873
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 25.7228 0.881767
\(852\) 0 0
\(853\) −38.4674 −1.31710 −0.658549 0.752538i \(-0.728829\pi\)
−0.658549 + 0.752538i \(0.728829\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −36.3505 −1.24171 −0.620855 0.783925i \(-0.713215\pi\)
−0.620855 + 0.783925i \(0.713215\pi\)
\(858\) 0 0
\(859\) 42.7446 1.45843 0.729213 0.684287i \(-0.239886\pi\)
0.729213 + 0.684287i \(0.239886\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.2554 0.723544 0.361772 0.932267i \(-0.382172\pi\)
0.361772 + 0.932267i \(0.382172\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.37228 0.114004
\(876\) 0 0
\(877\) 36.9783 1.24867 0.624333 0.781158i \(-0.285371\pi\)
0.624333 + 0.781158i \(0.285371\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −3.37228 −0.113486 −0.0567432 0.998389i \(-0.518072\pi\)
−0.0567432 + 0.998389i \(0.518072\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.9783 −0.368614 −0.184307 0.982869i \(-0.559004\pi\)
−0.184307 + 0.982869i \(0.559004\pi\)
\(888\) 0 0
\(889\) 26.9783 0.904821
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.72281 −0.0576517
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.62772 0.154343
\(900\) 0 0
\(901\) −5.64947 −0.188211
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) 0.394031 0.0130836 0.00654179 0.999979i \(-0.497918\pi\)
0.00654179 + 0.999979i \(0.497918\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.39403 −0.278107 −0.139053 0.990285i \(-0.544406\pi\)
−0.139053 + 0.990285i \(0.544406\pi\)
\(912\) 0 0
\(913\) 2.74456 0.0908318
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −74.5842 −2.46299
\(918\) 0 0
\(919\) −18.9783 −0.626035 −0.313017 0.949747i \(-0.601340\pi\)
−0.313017 + 0.949747i \(0.601340\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 20.2337 0.666000
\(924\) 0 0
\(925\) 9.37228 0.308159
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24.3505 −0.798915 −0.399458 0.916752i \(-0.630801\pi\)
−0.399458 + 0.916752i \(0.630801\pi\)
\(930\) 0 0
\(931\) −2.74456 −0.0899494
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.37228 −0.0448784
\(936\) 0 0
\(937\) −28.5109 −0.931410 −0.465705 0.884940i \(-0.654199\pi\)
−0.465705 + 0.884940i \(0.654199\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.0951 0.492086 0.246043 0.969259i \(-0.420870\pi\)
0.246043 + 0.969259i \(0.420870\pi\)
\(942\) 0 0
\(943\) 31.5326 1.02684
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 48.8614 1.58778 0.793891 0.608060i \(-0.208052\pi\)
0.793891 + 0.608060i \(0.208052\pi\)
\(948\) 0 0
\(949\) −30.9783 −1.00560
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −40.1168 −1.29951 −0.649756 0.760143i \(-0.725129\pi\)
−0.649756 + 0.760143i \(0.725129\pi\)
\(954\) 0 0
\(955\) 5.48913 0.177624
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 29.4891 0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.8614 −0.478406
\(966\) 0 0
\(967\) −47.6060 −1.53090 −0.765452 0.643493i \(-0.777485\pi\)
−0.765452 + 0.643493i \(0.777485\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.02175 −0.0327895 −0.0163947 0.999866i \(-0.505219\pi\)
−0.0163947 + 0.999866i \(0.505219\pi\)
\(972\) 0 0
\(973\) −13.4891 −0.432442
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −14.2337 −0.455376 −0.227688 0.973734i \(-0.573117\pi\)
−0.227688 + 0.973734i \(0.573117\pi\)
\(978\) 0 0
\(979\) −1.37228 −0.0438583
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −13.7228 −0.437690 −0.218845 0.975760i \(-0.570229\pi\)
−0.218845 + 0.975760i \(0.570229\pi\)
\(984\) 0 0
\(985\) −20.7446 −0.660977
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.1168 0.574343
\(996\) 0 0
\(997\) 22.2337 0.704148 0.352074 0.935972i \(-0.385477\pi\)
0.352074 + 0.935972i \(0.385477\pi\)
\(998\) 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 7920.2.a.bq.1.1 2
3.2 odd 2 880.2.a.n.1.2 2
4.3 odd 2 990.2.a.m.1.2 2
12.11 even 2 110.2.a.d.1.1 2
15.2 even 4 4400.2.b.p.4049.1 4
15.8 even 4 4400.2.b.p.4049.4 4
15.14 odd 2 4400.2.a.bl.1.1 2
20.3 even 4 4950.2.c.bc.199.1 4
20.7 even 4 4950.2.c.bc.199.4 4
20.19 odd 2 4950.2.a.bw.1.1 2
24.5 odd 2 3520.2.a.bj.1.1 2
24.11 even 2 3520.2.a.bq.1.2 2
33.32 even 2 9680.2.a.bt.1.2 2
60.23 odd 4 550.2.b.f.199.3 4
60.47 odd 4 550.2.b.f.199.2 4
60.59 even 2 550.2.a.n.1.2 2
84.83 odd 2 5390.2.a.bp.1.2 2
132.131 odd 2 1210.2.a.r.1.1 2
660.659 odd 2 6050.2.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 12.11 even 2
550.2.a.n.1.2 2 60.59 even 2
550.2.b.f.199.2 4 60.47 odd 4
550.2.b.f.199.3 4 60.23 odd 4
880.2.a.n.1.2 2 3.2 odd 2
990.2.a.m.1.2 2 4.3 odd 2
1210.2.a.r.1.1 2 132.131 odd 2
3520.2.a.bj.1.1 2 24.5 odd 2
3520.2.a.bq.1.2 2 24.11 even 2
4400.2.a.bl.1.1 2 15.14 odd 2
4400.2.b.p.4049.1 4 15.2 even 4
4400.2.b.p.4049.4 4 15.8 even 4
4950.2.a.bw.1.1 2 20.19 odd 2
4950.2.c.bc.199.1 4 20.3 even 4
4950.2.c.bc.199.4 4 20.7 even 4
5390.2.a.bp.1.2 2 84.83 odd 2
6050.2.a.cb.1.2 2 660.659 odd 2
7920.2.a.bq.1.1 2 1.1 even 1 trivial
9680.2.a.bt.1.2 2 33.32 even 2