Properties

Label 4032.2.h.a.575.1
Level $4032$
Weight $2$
Character 4032.575
Analytic conductor $32.196$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(575,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4032.575
Dual form 4032.2.h.a.575.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-2.82843i q^{5} -1.00000i q^{7} +1.41421 q^{11} -6.00000 q^{13} -5.65685i q^{17} +8.00000i q^{19} -1.41421 q^{23} -3.00000 q^{25} +4.24264i q^{29} -4.00000i q^{31} -2.82843 q^{35} -8.00000 q^{37} +8.48528i q^{41} -12.0000i q^{43} -2.82843 q^{47} -1.00000 q^{49} -1.41421i q^{53} -4.00000i q^{55} -14.1421 q^{59} -2.00000 q^{61} +16.9706i q^{65} +14.0000i q^{67} +12.7279 q^{71} +2.00000 q^{73} -1.41421i q^{77} -10.0000i q^{79} -16.0000 q^{85} +14.1421i q^{89} +6.00000i q^{91} +22.6274 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{13} - 12 q^{25} - 32 q^{37} - 4 q^{49} - 8 q^{61} + 8 q^{73} - 64 q^{85} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) − 2.82843i − 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.65685i − 1.37199i −0.727607 0.685994i \(-0.759367\pi\)
0.727607 0.685994i \(-0.240633\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.41421 −0.294884 −0.147442 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.48528i 1.32518i 0.748983 + 0.662589i \(0.230542\pi\)
−0.748983 + 0.662589i \(0.769458\pi\)
\(42\) 0 0
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.41421i − 0.194257i −0.995272 0.0971286i \(-0.969034\pi\)
0.995272 0.0971286i \(-0.0309658\pi\)
\(54\) 0 0
\(55\) − 4.00000i − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.1421 −1.84115 −0.920575 0.390567i \(-0.872279\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.9706i 2.10494i
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7279 1.51053 0.755263 0.655422i \(-0.227509\pi\)
0.755263 + 0.655422i \(0.227509\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.41421i − 0.161165i
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 0.562544i \(-0.809823\pi\)
0.826767 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.1421i 1.49906i 0.661968 + 0.749532i \(0.269721\pi\)
−0.661968 + 0.749532i \(0.730279\pi\)
\(90\) 0 0
\(91\) 6.00000i 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 22.6274 2.32152
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.65685i 0.562878i 0.959579 + 0.281439i \(0.0908117\pi\)
−0.959579 + 0.281439i \(0.909188\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.07107 0.683586 0.341793 0.939775i \(-0.388966\pi\)
0.341793 + 0.939775i \(0.388966\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264i 0.399114i 0.979886 + 0.199557i \(0.0639503\pi\)
−0.979886 + 0.199557i \(0.936050\pi\)
\(114\) 0 0
\(115\) 4.00000i 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.65685 −0.518563
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.41421i − 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.48528 −0.709575
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.3848i 1.50614i 0.657941 + 0.753070i \(0.271428\pi\)
−0.657941 + 0.753070i \(0.728572\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.3137 −0.908739
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.41421i 0.111456i
\(162\) 0 0
\(163\) − 2.00000i − 0.156652i −0.996928 0.0783260i \(-0.975042\pi\)
0.996928 0.0783260i \(-0.0249575\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.1421 −1.09435 −0.547176 0.837018i \(-0.684297\pi\)
−0.547176 + 0.837018i \(0.684297\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 5.65685i − 0.430083i −0.976605 0.215041i \(-0.931011\pi\)
0.976605 0.215041i \(-0.0689886\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.41421 0.105703 0.0528516 0.998602i \(-0.483169\pi\)
0.0528516 + 0.998602i \(0.483169\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.6274i 1.66360i
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.8701 1.94425 0.972125 0.234465i \(-0.0753338\pi\)
0.972125 + 0.234465i \(0.0753338\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.7279i 0.906827i 0.891300 + 0.453413i \(0.149794\pi\)
−0.891300 + 0.453413i \(0.850206\pi\)
\(198\) 0 0
\(199\) − 4.00000i − 0.283552i −0.989899 0.141776i \(-0.954719\pi\)
0.989899 0.141776i \(-0.0452813\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.24264 0.297775
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.3137i 0.782586i
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −33.9411 −2.31477
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 33.9411i 2.28313i
\(222\) 0 0
\(223\) − 12.0000i − 0.803579i −0.915732 0.401790i \(-0.868388\pi\)
0.915732 0.401790i \(-0.131612\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −19.7990 −1.31411 −0.657053 0.753845i \(-0.728197\pi\)
−0.657053 + 0.753845i \(0.728197\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.8701i − 1.76032i −0.474681 0.880158i \(-0.657437\pi\)
0.474681 0.880158i \(-0.342563\pi\)
\(234\) 0 0
\(235\) 8.00000i 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.41421 0.0914779 0.0457389 0.998953i \(-0.485436\pi\)
0.0457389 + 0.998953i \(0.485436\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.82843i 0.180702i
\(246\) 0 0
\(247\) − 48.0000i − 3.05417i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.7990i 1.23503i 0.786560 + 0.617514i \(0.211860\pi\)
−0.786560 + 0.617514i \(0.788140\pi\)
\(258\) 0 0
\(259\) 8.00000i 0.497096i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.07107 −0.436021 −0.218010 0.975946i \(-0.569957\pi\)
−0.218010 + 0.975946i \(0.569957\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.7990i 1.20717i 0.797300 + 0.603583i \(0.206261\pi\)
−0.797300 + 0.603583i \(0.793739\pi\)
\(270\) 0 0
\(271\) 32.0000i 1.94386i 0.235267 + 0.971931i \(0.424404\pi\)
−0.235267 + 0.971931i \(0.575596\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7279i 0.759284i 0.925133 + 0.379642i \(0.123953\pi\)
−0.925133 + 0.379642i \(0.876047\pi\)
\(282\) 0 0
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.3137i 0.660954i 0.943814 + 0.330477i \(0.107210\pi\)
−0.943814 + 0.330477i \(0.892790\pi\)
\(294\) 0 0
\(295\) 40.0000i 2.32889i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.48528 0.490716
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.65685i 0.323911i
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −19.7990 −1.12270 −0.561349 0.827579i \(-0.689717\pi\)
−0.561349 + 0.827579i \(0.689717\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.3848i − 1.03259i −0.856410 0.516296i \(-0.827310\pi\)
0.856410 0.516296i \(-0.172690\pi\)
\(318\) 0 0
\(319\) 6.00000i 0.335936i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 45.2548 2.51805
\(324\) 0 0
\(325\) 18.0000 0.998460
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.82843i 0.155936i
\(330\) 0 0
\(331\) − 20.0000i − 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 39.5980 2.16347
\(336\) 0 0
\(337\) −36.0000 −1.96104 −0.980522 0.196407i \(-0.937073\pi\)
−0.980522 + 0.196407i \(0.937073\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 5.65685i − 0.306336i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.8701 −1.44246 −0.721230 0.692696i \(-0.756423\pi\)
−0.721230 + 0.692696i \(0.756423\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 22.6274i − 1.20434i −0.798369 0.602168i \(-0.794304\pi\)
0.798369 0.602168i \(-0.205696\pi\)
\(354\) 0 0
\(355\) − 36.0000i − 1.91068i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.07107 −0.373197 −0.186598 0.982436i \(-0.559746\pi\)
−0.186598 + 0.982436i \(0.559746\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 5.65685i − 0.296093i
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.41421 −0.0734223
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 25.4558i − 1.31104i
\(378\) 0 0
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 15.5563i − 0.788738i −0.918952 0.394369i \(-0.870963\pi\)
0.918952 0.394369i \(-0.129037\pi\)
\(390\) 0 0
\(391\) 8.00000i 0.404577i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −28.2843 −1.42314
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.41421i − 0.0706225i −0.999376 0.0353112i \(-0.988758\pi\)
0.999376 0.0353112i \(-0.0112422\pi\)
\(402\) 0 0
\(403\) 24.0000i 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.1421i 0.695889i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 25.4558 1.24360 0.621800 0.783176i \(-0.286402\pi\)
0.621800 + 0.783176i \(0.286402\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.9706i 0.823193i
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −35.3553 −1.70301 −0.851503 0.524349i \(-0.824309\pi\)
−0.851503 + 0.524349i \(0.824309\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 11.3137i − 0.541208i
\(438\) 0 0
\(439\) − 8.00000i − 0.381819i −0.981608 0.190910i \(-0.938856\pi\)
0.981608 0.190910i \(-0.0611437\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −26.8701 −1.27663 −0.638317 0.769773i \(-0.720369\pi\)
−0.638317 + 0.769773i \(0.720369\pi\)
\(444\) 0 0
\(445\) 40.0000 1.89618
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 29.6985i − 1.40156i −0.713378 0.700779i \(-0.752836\pi\)
0.713378 0.700779i \(-0.247164\pi\)
\(450\) 0 0
\(451\) 12.0000i 0.565058i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.9706 0.795592
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 16.9706i − 0.790398i −0.918596 0.395199i \(-0.870676\pi\)
0.918596 0.395199i \(-0.129324\pi\)
\(462\) 0 0
\(463\) 30.0000i 1.39422i 0.716965 + 0.697109i \(0.245531\pi\)
−0.716965 + 0.697109i \(0.754469\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.82843 −0.130884 −0.0654420 0.997856i \(-0.520846\pi\)
−0.0654420 + 0.997856i \(0.520846\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 16.9706i − 0.780307i
\(474\) 0 0
\(475\) − 24.0000i − 1.10120i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.7990 −0.904639 −0.452319 0.891856i \(-0.649403\pi\)
−0.452319 + 0.891856i \(0.649403\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.9706i 0.770594i
\(486\) 0 0
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.8701 1.21263 0.606314 0.795225i \(-0.292647\pi\)
0.606314 + 0.795225i \(0.292647\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.7279i − 0.570925i
\(498\) 0 0
\(499\) − 20.0000i − 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.9706 0.756680 0.378340 0.925667i \(-0.376495\pi\)
0.378340 + 0.925667i \(0.376495\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.82843i − 0.125368i −0.998033 0.0626839i \(-0.980034\pi\)
0.998033 0.0626839i \(-0.0199660\pi\)
\(510\) 0 0
\(511\) − 2.00000i − 0.0884748i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.6274 0.997083
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 11.3137i − 0.495663i −0.968803 0.247831i \(-0.920282\pi\)
0.968803 0.247831i \(-0.0797179\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22.6274 −0.985666
\(528\) 0 0
\(529\) −21.0000 −0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 50.9117i − 2.20523i
\(534\) 0 0
\(535\) − 20.0000i − 0.864675i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.41421 −0.0609145
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.3137i 0.484626i
\(546\) 0 0
\(547\) 26.0000i 1.11168i 0.831289 + 0.555840i \(0.187603\pi\)
−0.831289 + 0.555840i \(0.812397\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −33.9411 −1.44594
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.3848i 0.778988i 0.921029 + 0.389494i \(0.127350\pi\)
−0.921029 + 0.389494i \(0.872650\pi\)
\(558\) 0 0
\(559\) 72.0000i 3.04528i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.7990 0.834428 0.417214 0.908808i \(-0.363007\pi\)
0.417214 + 0.908808i \(0.363007\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 32.5269i − 1.36360i −0.731539 0.681800i \(-0.761198\pi\)
0.731539 0.681800i \(-0.238802\pi\)
\(570\) 0 0
\(571\) − 2.00000i − 0.0836974i −0.999124 0.0418487i \(-0.986675\pi\)
0.999124 0.0418487i \(-0.0133247\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.24264 0.176930
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.48528 0.350225 0.175113 0.984548i \(-0.443971\pi\)
0.175113 + 0.984548i \(0.443971\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 39.5980i − 1.62609i −0.582198 0.813047i \(-0.697807\pi\)
0.582198 0.813047i \(-0.302193\pi\)
\(594\) 0 0
\(595\) 16.0000i 0.655936i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 43.8406 1.79128 0.895640 0.444781i \(-0.146718\pi\)
0.895640 + 0.444781i \(0.146718\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25.4558i 1.03493i
\(606\) 0 0
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.9706 0.686555
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 46.6690i 1.87883i 0.342789 + 0.939413i \(0.388629\pi\)
−0.342789 + 0.939413i \(0.611371\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.1421 0.566593
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 45.2548i 1.80443i
\(630\) 0 0
\(631\) − 30.0000i − 1.19428i −0.802137 0.597141i \(-0.796303\pi\)
0.802137 0.597141i \(-0.203697\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.9706 −0.673456
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 15.5563i − 0.614439i −0.951639 0.307219i \(-0.900601\pi\)
0.951639 0.307219i \(-0.0993986\pi\)
\(642\) 0 0
\(643\) 24.0000i 0.946468i 0.880937 + 0.473234i \(0.156913\pi\)
−0.880937 + 0.473234i \(0.843087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.7990 −0.778379 −0.389189 0.921158i \(-0.627245\pi\)
−0.389189 + 0.921158i \(0.627245\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.5563i 0.608767i 0.952550 + 0.304383i \(0.0984504\pi\)
−0.952550 + 0.304383i \(0.901550\pi\)
\(654\) 0 0
\(655\) − 16.0000i − 0.625172i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 38.1838 1.48743 0.743714 0.668498i \(-0.233062\pi\)
0.743714 + 0.668498i \(0.233062\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 22.6274i − 0.877454i
\(666\) 0 0
\(667\) − 6.00000i − 0.232321i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.82843 −0.109190
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.65685i 0.217411i 0.994074 + 0.108705i \(0.0346705\pi\)
−0.994074 + 0.108705i \(0.965330\pi\)
\(678\) 0 0
\(679\) 6.00000i 0.230259i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.07107 −0.270567 −0.135283 0.990807i \(-0.543195\pi\)
−0.135283 + 0.990807i \(0.543195\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.48528i 0.323263i
\(690\) 0 0
\(691\) − 44.0000i − 1.67384i −0.547326 0.836919i \(-0.684354\pi\)
0.547326 0.836919i \(-0.315646\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 18.3848i − 0.694383i −0.937794 0.347192i \(-0.887135\pi\)
0.937794 0.347192i \(-0.112865\pi\)
\(702\) 0 0
\(703\) − 64.0000i − 2.41381i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.65685 0.212748
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.65685i 0.211851i
\(714\) 0 0
\(715\) 24.0000i 0.897549i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −39.5980 −1.47676 −0.738378 0.674387i \(-0.764408\pi\)
−0.738378 + 0.674387i \(0.764408\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 12.7279i − 0.472703i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −67.8823 −2.51072
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.7990i 0.729305i
\(738\) 0 0
\(739\) 10.0000i 0.367856i 0.982940 + 0.183928i \(0.0588813\pi\)
−0.982940 + 0.183928i \(0.941119\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.7279 −0.466942 −0.233471 0.972364i \(-0.575008\pi\)
−0.233471 + 0.972364i \(0.575008\pi\)
\(744\) 0 0
\(745\) 52.0000 1.90513
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 7.07107i − 0.258371i
\(750\) 0 0
\(751\) − 8.00000i − 0.291924i −0.989290 0.145962i \(-0.953372\pi\)
0.989290 0.145962i \(-0.0466277\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.6274 0.823496
\(756\) 0 0
\(757\) −36.0000 −1.30844 −0.654221 0.756303i \(-0.727003\pi\)
−0.654221 + 0.756303i \(0.727003\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 14.1421i − 0.512652i −0.966590 0.256326i \(-0.917488\pi\)
0.966590 0.256326i \(-0.0825121\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 84.8528 3.06386
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 45.2548i − 1.62770i −0.581073 0.813852i \(-0.697367\pi\)
0.581073 0.813852i \(-0.302633\pi\)
\(774\) 0 0
\(775\) 12.0000i 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −67.8823 −2.43213
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 39.5980i 1.41331i
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.24264 0.150851
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 50.9117i − 1.80338i −0.432378 0.901692i \(-0.642326\pi\)
0.432378 0.901692i \(-0.357674\pi\)
\(798\) 0 0
\(799\) 16.0000i 0.566039i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.82843 0.0998130
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 32.5269i − 1.14359i −0.820398 0.571793i \(-0.806248\pi\)
0.820398 0.571793i \(-0.193752\pi\)
\(810\) 0 0
\(811\) 12.0000i 0.421377i 0.977553 + 0.210688i \(0.0675706\pi\)
−0.977553 + 0.210688i \(0.932429\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.65685 −0.198151
\(816\) 0 0
\(817\) 96.0000 3.35861
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.3848i 0.641633i 0.947141 + 0.320817i \(0.103957\pi\)
−0.947141 + 0.320817i \(0.896043\pi\)
\(822\) 0 0
\(823\) 22.0000i 0.766872i 0.923567 + 0.383436i \(0.125259\pi\)
−0.923567 + 0.383436i \(0.874741\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.2132 0.737655 0.368828 0.929498i \(-0.379759\pi\)
0.368828 + 0.929498i \(0.379759\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.65685i 0.195998i
\(834\) 0 0
\(835\) 40.0000i 1.38426i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25.4558 −0.878833 −0.439417 0.898283i \(-0.644815\pi\)
−0.439417 + 0.898283i \(0.644815\pi\)
\(840\) 0 0
\(841\) 11.0000 0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 65.0538i − 2.23792i
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137 0.387829
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 19.7990i − 0.676321i −0.941089 0.338160i \(-0.890195\pi\)
0.941089 0.338160i \(-0.109805\pi\)
\(858\) 0 0
\(859\) − 52.0000i − 1.77422i −0.461561 0.887109i \(-0.652710\pi\)
0.461561 0.887109i \(-0.347290\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.41421 −0.0481404 −0.0240702 0.999710i \(-0.507663\pi\)
−0.0240702 + 0.999710i \(0.507663\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 14.1421i − 0.479739i
\(870\) 0 0
\(871\) − 84.0000i − 2.84623i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.65685 −0.191237
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 5.65685i − 0.190584i −0.995449 0.0952921i \(-0.969621\pi\)
0.995449 0.0952921i \(-0.0303785\pi\)
\(882\) 0 0
\(883\) − 36.0000i − 1.21150i −0.795656 0.605748i \(-0.792874\pi\)
0.795656 0.605748i \(-0.207126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.1127 1.04466 0.522331 0.852743i \(-0.325063\pi\)
0.522331 + 0.852743i \(0.325063\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 22.6274i − 0.757198i
\(894\) 0 0
\(895\) − 4.00000i − 0.133705i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.9706 0.566000
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 5.65685i − 0.188040i
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 46.6690 1.54621 0.773107 0.634275i \(-0.218701\pi\)
0.773107 + 0.634275i \(0.218701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5.65685i − 0.186806i
\(918\) 0 0
\(919\) 16.0000i 0.527791i 0.964551 + 0.263896i \(0.0850075\pi\)
−0.964551 + 0.263896i \(0.914993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −76.3675 −2.51367
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 31.1127i 1.02077i 0.859945 + 0.510387i \(0.170498\pi\)
−0.859945 + 0.510387i \(0.829502\pi\)
\(930\) 0 0
\(931\) − 8.00000i − 0.262189i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −22.6274 −0.739996
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 31.1127i − 1.01424i −0.861874 0.507122i \(-0.830709\pi\)
0.861874 0.507122i \(-0.169291\pi\)
\(942\) 0 0
\(943\) − 12.0000i − 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49.4975 −1.60845 −0.804226 0.594324i \(-0.797420\pi\)
−0.804226 + 0.594324i \(0.797420\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.7279i 0.412298i 0.978521 + 0.206149i \(0.0660931\pi\)
−0.978521 + 0.206149i \(0.933907\pi\)
\(954\) 0 0
\(955\) − 76.0000i − 2.45930i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.41421 −0.0456673
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 11.3137i − 0.364201i
\(966\) 0 0
\(967\) − 18.0000i − 0.578841i −0.957202 0.289420i \(-0.906537\pi\)
0.957202 0.289420i \(-0.0934626\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.65685 0.181537 0.0907685 0.995872i \(-0.471068\pi\)
0.0907685 + 0.995872i \(0.471068\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 9.89949i − 0.316713i −0.987382 0.158356i \(-0.949380\pi\)
0.987382 0.158356i \(-0.0506195\pi\)
\(978\) 0 0
\(979\) 20.0000i 0.639203i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.6274 −0.721703 −0.360851 0.932623i \(-0.617514\pi\)
−0.360851 + 0.932623i \(0.617514\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706i 0.539633i
\(990\) 0 0
\(991\) 16.0000i 0.508257i 0.967170 + 0.254128i \(0.0817886\pi\)
−0.967170 + 0.254128i \(0.918211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.3137 −0.358669
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 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 4032.2.h.a.575.1 4
3.2 odd 2 inner 4032.2.h.a.575.3 4
4.3 odd 2 inner 4032.2.h.a.575.2 4
8.3 odd 2 2016.2.h.d.575.4 yes 4
8.5 even 2 2016.2.h.d.575.3 yes 4
12.11 even 2 inner 4032.2.h.a.575.4 4
24.5 odd 2 2016.2.h.d.575.1 4
24.11 even 2 2016.2.h.d.575.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.h.d.575.1 4 24.5 odd 2
2016.2.h.d.575.2 yes 4 24.11 even 2
2016.2.h.d.575.3 yes 4 8.5 even 2
2016.2.h.d.575.4 yes 4 8.3 odd 2
4032.2.h.a.575.1 4 1.1 even 1 trivial
4032.2.h.a.575.2 4 4.3 odd 2 inner
4032.2.h.a.575.3 4 3.2 odd 2 inner
4032.2.h.a.575.4 4 12.11 even 2 inner