Properties

Label 4032.2.k.e.3905.1
Level $4032$
Weight $2$
Character 4032.3905
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3905,4032)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3905");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.k (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: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3905.1
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3905
Dual form 4032.2.k.e.3905.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.69552 q^{5} +(-2.41421 - 1.08239i) q^{7} +O(q^{10})\) \(q-3.69552 q^{5} +(-2.41421 - 1.08239i) q^{7} +3.41421i q^{11} +5.22625i q^{13} -6.75699 q^{17} +2.16478i q^{19} -6.24264i q^{23} +8.65685 q^{25} +2.58579i q^{29} +10.4525i q^{31} +(8.92177 + 4.00000i) q^{35} -4.00000 q^{37} +0.634051 q^{41} -6.48528 q^{43} -3.06147 q^{47} +(4.65685 + 5.22625i) q^{49} -2.58579i q^{53} -12.6173i q^{55} -19.3137i q^{65} +1.65685 q^{67} +7.41421i q^{71} +0.896683i q^{73} +(3.69552 - 8.24264i) q^{77} +13.6569 q^{79} -11.7206 q^{83} +24.9706 q^{85} -6.75699 q^{89} +(5.65685 - 12.6173i) q^{91} -8.00000i q^{95} -9.55582i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 24 q^{25} - 32 q^{37} + 16 q^{43} - 8 q^{49} - 32 q^{67} + 64 q^{79} + 64 q^{85}+O(q^{100}) \) 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
\(5\) −3.69552 −1.65269 −0.826343 0.563167i \(-0.809583\pi\)
−0.826343 + 0.563167i \(0.809583\pi\)
\(6\) 0 0
\(7\) −2.41421 1.08239i −0.912487 0.409106i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.41421i 1.02942i 0.857363 + 0.514712i \(0.172101\pi\)
−0.857363 + 0.514712i \(0.827899\pi\)
\(12\) 0 0
\(13\) 5.22625i 1.44950i 0.689011 + 0.724751i \(0.258045\pi\)
−0.689011 + 0.724751i \(0.741955\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.75699 −1.63881 −0.819405 0.573215i \(-0.805696\pi\)
−0.819405 + 0.573215i \(0.805696\pi\)
\(18\) 0 0
\(19\) 2.16478i 0.496636i 0.968679 + 0.248318i \(0.0798777\pi\)
−0.968679 + 0.248318i \(0.920122\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.24264i 1.30168i −0.759215 0.650840i \(-0.774417\pi\)
0.759215 0.650840i \(-0.225583\pi\)
\(24\) 0 0
\(25\) 8.65685 1.73137
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.58579i 0.480168i 0.970752 + 0.240084i \(0.0771751\pi\)
−0.970752 + 0.240084i \(0.922825\pi\)
\(30\) 0 0
\(31\) 10.4525i 1.87733i 0.344837 + 0.938663i \(0.387934\pi\)
−0.344837 + 0.938663i \(0.612066\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.92177 + 4.00000i 1.50805 + 0.676123i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.634051 0.0990221 0.0495110 0.998774i \(-0.484234\pi\)
0.0495110 + 0.998774i \(0.484234\pi\)
\(42\) 0 0
\(43\) −6.48528 −0.988996 −0.494498 0.869179i \(-0.664648\pi\)
−0.494498 + 0.869179i \(0.664648\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.06147 −0.446561 −0.223280 0.974754i \(-0.571677\pi\)
−0.223280 + 0.974754i \(0.571677\pi\)
\(48\) 0 0
\(49\) 4.65685 + 5.22625i 0.665265 + 0.746607i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.58579i 0.355185i −0.984104 0.177593i \(-0.943169\pi\)
0.984104 0.177593i \(-0.0568309\pi\)
\(54\) 0 0
\(55\) 12.6173i 1.70131i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 19.3137i 2.39557i
\(66\) 0 0
\(67\) 1.65685 0.202417 0.101208 0.994865i \(-0.467729\pi\)
0.101208 + 0.994865i \(0.467729\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.41421i 0.879905i 0.898021 + 0.439953i \(0.145005\pi\)
−0.898021 + 0.439953i \(0.854995\pi\)
\(72\) 0 0
\(73\) 0.896683i 0.104949i 0.998622 + 0.0524744i \(0.0167108\pi\)
−0.998622 + 0.0524744i \(0.983289\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.69552 8.24264i 0.421143 0.939336i
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.7206 −1.28650 −0.643252 0.765655i \(-0.722415\pi\)
−0.643252 + 0.765655i \(0.722415\pi\)
\(84\) 0 0
\(85\) 24.9706 2.70844
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.75699 −0.716239 −0.358120 0.933676i \(-0.616582\pi\)
−0.358120 + 0.933676i \(0.616582\pi\)
\(90\) 0 0
\(91\) 5.65685 12.6173i 0.592999 1.32265i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000i 0.820783i
\(96\) 0 0
\(97\) 9.55582i 0.970247i −0.874446 0.485123i \(-0.838775\pi\)
0.874446 0.485123i \(-0.161225\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.42742 −0.241537 −0.120768 0.992681i \(-0.538536\pi\)
−0.120768 + 0.992681i \(0.538536\pi\)
\(102\) 0 0
\(103\) 6.12293i 0.603311i −0.953417 0.301655i \(-0.902461\pi\)
0.953417 0.301655i \(-0.0975392\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.07107i 0.103544i 0.998659 + 0.0517720i \(0.0164869\pi\)
−0.998659 + 0.0517720i \(0.983513\pi\)
\(108\) 0 0
\(109\) −1.65685 −0.158698 −0.0793489 0.996847i \(-0.525284\pi\)
−0.0793489 + 0.996847i \(0.525284\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.7574i 1.10604i −0.833168 0.553020i \(-0.813475\pi\)
0.833168 0.553020i \(-0.186525\pi\)
\(114\) 0 0
\(115\) 23.0698i 2.15127i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.3128 + 7.31371i 1.49539 + 0.670447i
\(120\) 0 0
\(121\) −0.656854 −0.0597140
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −13.5140 −1.20873
\(126\) 0 0
\(127\) 16.9706 1.50589 0.752947 0.658081i \(-0.228632\pi\)
0.752947 + 0.658081i \(0.228632\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.06147 −0.267482 −0.133741 0.991016i \(-0.542699\pi\)
−0.133741 + 0.991016i \(0.542699\pi\)
\(132\) 0 0
\(133\) 2.34315 5.22625i 0.203177 0.453174i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.89949i 0.845771i −0.906183 0.422885i \(-0.861017\pi\)
0.906183 0.422885i \(-0.138983\pi\)
\(138\) 0 0
\(139\) 4.32957i 0.367229i −0.982998 0.183615i \(-0.941220\pi\)
0.982998 0.183615i \(-0.0587798\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −17.8435 −1.49215
\(144\) 0 0
\(145\) 9.55582i 0.793568i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.75736i 0.635508i −0.948173 0.317754i \(-0.897071\pi\)
0.948173 0.317754i \(-0.102929\pi\)
\(150\) 0 0
\(151\) −16.1421 −1.31363 −0.656814 0.754052i \(-0.728096\pi\)
−0.656814 + 0.754052i \(0.728096\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 38.6274i 3.10263i
\(156\) 0 0
\(157\) 19.1116i 1.52528i −0.646826 0.762638i \(-0.723904\pi\)
0.646826 0.762638i \(-0.276096\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.75699 + 15.0711i −0.532525 + 1.18777i
\(162\) 0 0
\(163\) −17.6569 −1.38299 −0.691496 0.722380i \(-0.743048\pi\)
−0.691496 + 0.722380i \(0.743048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.7206 −0.906968 −0.453484 0.891265i \(-0.649819\pi\)
−0.453484 + 0.891265i \(0.649819\pi\)
\(168\) 0 0
\(169\) −14.3137 −1.10105
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.3547 0.939307 0.469654 0.882851i \(-0.344379\pi\)
0.469654 + 0.882851i \(0.344379\pi\)
\(174\) 0 0
\(175\) −20.8995 9.37011i −1.57985 0.708314i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.8995i 1.18838i 0.804323 + 0.594192i \(0.202528\pi\)
−0.804323 + 0.594192i \(0.797472\pi\)
\(180\) 0 0
\(181\) 24.3379i 1.80902i −0.426451 0.904511i \(-0.640236\pi\)
0.426451 0.904511i \(-0.359764\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.7821 1.08680
\(186\) 0 0
\(187\) 23.0698i 1.68703i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.72792i 0.197386i −0.995118 0.0986928i \(-0.968534\pi\)
0.995118 0.0986928i \(-0.0314661\pi\)
\(192\) 0 0
\(193\) 5.65685 0.407189 0.203595 0.979055i \(-0.434738\pi\)
0.203595 + 0.979055i \(0.434738\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.3848i 1.59485i −0.603419 0.797425i \(-0.706195\pi\)
0.603419 0.797425i \(-0.293805\pi\)
\(198\) 0 0
\(199\) 23.0698i 1.63537i 0.575663 + 0.817687i \(0.304744\pi\)
−0.575663 + 0.817687i \(0.695256\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.79884 6.24264i 0.196440 0.438147i
\(204\) 0 0
\(205\) −2.34315 −0.163652
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.39104 −0.511249
\(210\) 0 0
\(211\) 19.1716 1.31983 0.659913 0.751342i \(-0.270593\pi\)
0.659913 + 0.751342i \(0.270593\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 23.9665 1.63450
\(216\) 0 0
\(217\) 11.3137 25.2346i 0.768025 1.71303i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 35.3137i 2.37546i
\(222\) 0 0
\(223\) 23.0698i 1.54487i 0.635095 + 0.772434i \(0.280961\pi\)
−0.635095 + 0.772434i \(0.719039\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.7821 −0.981121 −0.490560 0.871407i \(-0.663208\pi\)
−0.490560 + 0.871407i \(0.663208\pi\)
\(228\) 0 0
\(229\) 3.43289i 0.226851i −0.993546 0.113426i \(-0.963818\pi\)
0.993546 0.113426i \(-0.0361824\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.7279i 1.35793i −0.734170 0.678966i \(-0.762429\pi\)
0.734170 0.678966i \(-0.237571\pi\)
\(234\) 0 0
\(235\) 11.3137 0.738025
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.92893i 0.189457i 0.995503 + 0.0947284i \(0.0301983\pi\)
−0.995503 + 0.0947284i \(0.969802\pi\)
\(240\) 0 0
\(241\) 11.3492i 0.731065i −0.930799 0.365533i \(-0.880887\pi\)
0.930799 0.365533i \(-0.119113\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.2095 19.3137i −1.09947 1.23391i
\(246\) 0 0
\(247\) −11.3137 −0.719874
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.59767 0.353322 0.176661 0.984272i \(-0.443470\pi\)
0.176661 + 0.984272i \(0.443470\pi\)
\(252\) 0 0
\(253\) 21.3137 1.33998
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.8072 1.42267 0.711336 0.702852i \(-0.248091\pi\)
0.711336 + 0.702852i \(0.248091\pi\)
\(258\) 0 0
\(259\) 9.65685 + 4.32957i 0.600048 + 0.269026i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.7279i 1.15481i 0.816457 + 0.577407i \(0.195935\pi\)
−0.816457 + 0.577407i \(0.804065\pi\)
\(264\) 0 0
\(265\) 9.55582i 0.587009i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.0866 −0.675959 −0.337980 0.941153i \(-0.609743\pi\)
−0.337980 + 0.941153i \(0.609743\pi\)
\(270\) 0 0
\(271\) 19.1116i 1.16095i −0.814278 0.580475i \(-0.802867\pi\)
0.814278 0.580475i \(-0.197133\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 29.5563i 1.78231i
\(276\) 0 0
\(277\) −19.6569 −1.18107 −0.590533 0.807014i \(-0.701082\pi\)
−0.590533 + 0.807014i \(0.701082\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.89949i 0.590554i 0.955412 + 0.295277i \(0.0954120\pi\)
−0.955412 + 0.295277i \(0.904588\pi\)
\(282\) 0 0
\(283\) 14.4107i 0.856624i 0.903631 + 0.428312i \(0.140892\pi\)
−0.903631 + 0.428312i \(0.859108\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.53073 0.686292i −0.0903564 0.0405105i
\(288\) 0 0
\(289\) 28.6569 1.68570
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.8686 1.51126 0.755631 0.654998i \(-0.227331\pi\)
0.755631 + 0.654998i \(0.227331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 32.6256 1.88679
\(300\) 0 0
\(301\) 15.6569 + 7.01962i 0.902446 + 0.404604i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.8239i 0.617754i 0.951102 + 0.308877i \(0.0999531\pi\)
−0.951102 + 0.308877i \(0.900047\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.2459 0.694400 0.347200 0.937791i \(-0.387133\pi\)
0.347200 + 0.937791i \(0.387133\pi\)
\(312\) 0 0
\(313\) 20.9050i 1.18162i 0.806810 + 0.590810i \(0.201192\pi\)
−0.806810 + 0.590810i \(0.798808\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.3848i 1.25725i 0.777707 + 0.628627i \(0.216383\pi\)
−0.777707 + 0.628627i \(0.783617\pi\)
\(318\) 0 0
\(319\) −8.82843 −0.494297
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 14.6274i 0.813891i
\(324\) 0 0
\(325\) 45.2429i 2.50962i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.39104 + 3.31371i 0.407481 + 0.182691i
\(330\) 0 0
\(331\) 4.82843 0.265394 0.132697 0.991157i \(-0.457636\pi\)
0.132697 + 0.991157i \(0.457636\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.12293 −0.334532
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −35.6871 −1.93256
\(342\) 0 0
\(343\) −5.58579 17.6578i −0.301604 0.953433i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.5563i 0.727743i 0.931449 + 0.363871i \(0.118545\pi\)
−0.931449 + 0.363871i \(0.881455\pi\)
\(348\) 0 0
\(349\) 19.1116i 1.02302i 0.859277 + 0.511511i \(0.170914\pi\)
−0.859277 + 0.511511i \(0.829086\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.5391 −1.14641 −0.573204 0.819413i \(-0.694300\pi\)
−0.573204 + 0.819413i \(0.694300\pi\)
\(354\) 0 0
\(355\) 27.3994i 1.45421i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.38478i 0.442532i −0.975214 0.221266i \(-0.928981\pi\)
0.975214 0.221266i \(-0.0710188\pi\)
\(360\) 0 0
\(361\) 14.3137 0.753353
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.31371i 0.173447i
\(366\) 0 0
\(367\) 6.49435i 0.339002i −0.985530 0.169501i \(-0.945784\pi\)
0.985530 0.169501i \(-0.0542157\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.79884 + 6.24264i −0.145308 + 0.324102i
\(372\) 0 0
\(373\) 17.3137 0.896470 0.448235 0.893916i \(-0.352053\pi\)
0.448235 + 0.893916i \(0.352053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.5140 −0.696005
\(378\) 0 0
\(379\) 11.1716 0.573845 0.286923 0.957954i \(-0.407368\pi\)
0.286923 + 0.957954i \(0.407368\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.59767 0.286028 0.143014 0.989721i \(-0.454321\pi\)
0.143014 + 0.989721i \(0.454321\pi\)
\(384\) 0 0
\(385\) −13.6569 + 30.4608i −0.696018 + 1.55243i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.7279i 1.25376i 0.779118 + 0.626878i \(0.215667\pi\)
−0.779118 + 0.626878i \(0.784333\pi\)
\(390\) 0 0
\(391\) 42.1814i 2.13321i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −50.4692 −2.53938
\(396\) 0 0
\(397\) 20.9050i 1.04919i −0.851351 0.524596i \(-0.824216\pi\)
0.851351 0.524596i \(-0.175784\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.443651i 0.0221549i 0.999939 + 0.0110774i \(0.00352613\pi\)
−0.999939 + 0.0110774i \(0.996474\pi\)
\(402\) 0 0
\(403\) −54.6274 −2.72119
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.6569i 0.676945i
\(408\) 0 0
\(409\) 0.896683i 0.0443381i 0.999754 + 0.0221691i \(0.00705721\pi\)
−0.999754 + 0.0221691i \(0.992943\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 43.3137 2.12619
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.9050 1.02128 0.510638 0.859796i \(-0.329409\pi\)
0.510638 + 0.859796i \(0.329409\pi\)
\(420\) 0 0
\(421\) −18.9706 −0.924569 −0.462284 0.886732i \(-0.652970\pi\)
−0.462284 + 0.886732i \(0.652970\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −58.4942 −2.83739
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.75736i 0.0846490i −0.999104 0.0423245i \(-0.986524\pi\)
0.999104 0.0423245i \(-0.0134763\pi\)
\(432\) 0 0
\(433\) 22.6984i 1.09081i 0.838171 + 0.545407i \(0.183625\pi\)
−0.838171 + 0.545407i \(0.816375\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.5140 0.646461
\(438\) 0 0
\(439\) 23.0698i 1.10106i −0.834815 0.550531i \(-0.814425\pi\)
0.834815 0.550531i \(-0.185575\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.4142i 0.542306i −0.962536 0.271153i \(-0.912595\pi\)
0.962536 0.271153i \(-0.0874049\pi\)
\(444\) 0 0
\(445\) 24.9706 1.18372
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.58579i 0.310802i 0.987851 + 0.155401i \(0.0496670\pi\)
−0.987851 + 0.155401i \(0.950333\pi\)
\(450\) 0 0
\(451\) 2.16478i 0.101936i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.9050 + 46.6274i −0.980042 + 2.18593i
\(456\) 0 0
\(457\) 11.3137 0.529233 0.264616 0.964354i \(-0.414755\pi\)
0.264616 + 0.964354i \(0.414755\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.8686 −1.20482 −0.602411 0.798186i \(-0.705793\pi\)
−0.602411 + 0.798186i \(0.705793\pi\)
\(462\) 0 0
\(463\) 18.3431 0.852478 0.426239 0.904611i \(-0.359838\pi\)
0.426239 + 0.904611i \(0.359838\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.0894 1.39237 0.696186 0.717862i \(-0.254879\pi\)
0.696186 + 0.717862i \(0.254879\pi\)
\(468\) 0 0
\(469\) −4.00000 1.79337i −0.184703 0.0828100i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.1421i 1.01810i
\(474\) 0 0
\(475\) 18.7402i 0.859860i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.0279 −1.23494 −0.617469 0.786595i \(-0.711842\pi\)
−0.617469 + 0.786595i \(0.711842\pi\)
\(480\) 0 0
\(481\) 20.9050i 0.953186i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 35.3137i 1.60351i
\(486\) 0 0
\(487\) 11.4558 0.519114 0.259557 0.965728i \(-0.416423\pi\)
0.259557 + 0.965728i \(0.416423\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 38.5269i 1.73870i −0.494201 0.869348i \(-0.664539\pi\)
0.494201 0.869348i \(-0.335461\pi\)
\(492\) 0 0
\(493\) 17.4721i 0.786905i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.02509 17.8995i 0.359974 0.802902i
\(498\) 0 0
\(499\) −12.8284 −0.574279 −0.287140 0.957889i \(-0.592704\pi\)
−0.287140 + 0.957889i \(0.592704\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 26.5027 1.18170 0.590848 0.806783i \(-0.298793\pi\)
0.590848 + 0.806783i \(0.298793\pi\)
\(504\) 0 0
\(505\) 8.97056 0.399185
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.2095 0.762797 0.381399 0.924411i \(-0.375443\pi\)
0.381399 + 0.924411i \(0.375443\pi\)
\(510\) 0 0
\(511\) 0.970563 2.16478i 0.0429352 0.0957644i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.6274i 0.997083i
\(516\) 0 0
\(517\) 10.4525i 0.459701i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.75699 0.296029 0.148014 0.988985i \(-0.452712\pi\)
0.148014 + 0.988985i \(0.452712\pi\)
\(522\) 0 0
\(523\) 20.9050i 0.914112i 0.889438 + 0.457056i \(0.151096\pi\)
−0.889438 + 0.457056i \(0.848904\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 70.6274i 3.07658i
\(528\) 0 0
\(529\) −15.9706 −0.694372
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.31371i 0.143533i
\(534\) 0 0
\(535\) 3.95815i 0.171126i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.8435 + 15.8995i −0.768576 + 0.684840i
\(540\) 0 0
\(541\) −27.6569 −1.18906 −0.594531 0.804073i \(-0.702662\pi\)
−0.594531 + 0.804073i \(0.702662\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.12293 0.262278
\(546\) 0 0
\(547\) 17.6569 0.754953 0.377476 0.926019i \(-0.376792\pi\)
0.377476 + 0.926019i \(0.376792\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.59767 −0.238469
\(552\) 0 0
\(553\) −32.9706 14.7821i −1.40205 0.628598i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 45.8995i 1.94482i 0.233270 + 0.972412i \(0.425057\pi\)
−0.233270 + 0.972412i \(0.574943\pi\)
\(558\) 0 0
\(559\) 33.8937i 1.43355i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.2123 1.52617 0.763084 0.646299i \(-0.223684\pi\)
0.763084 + 0.646299i \(0.223684\pi\)
\(564\) 0 0
\(565\) 43.4495i 1.82794i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.24264i 0.177861i 0.996038 + 0.0889304i \(0.0283449\pi\)
−0.996038 + 0.0889304i \(0.971655\pi\)
\(570\) 0 0
\(571\) 12.9706 0.542801 0.271401 0.962466i \(-0.412513\pi\)
0.271401 + 0.962466i \(0.412513\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 54.0416i 2.25369i
\(576\) 0 0
\(577\) 10.4525i 0.435143i 0.976044 + 0.217572i \(0.0698136\pi\)
−0.976044 + 0.217572i \(0.930186\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 28.2960 + 12.6863i 1.17392 + 0.526316i
\(582\) 0 0
\(583\) 8.82843 0.365636
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −32.6256 −1.34660 −0.673302 0.739368i \(-0.735124\pi\)
−0.673302 + 0.739368i \(0.735124\pi\)
\(588\) 0 0
\(589\) −22.6274 −0.932346
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.9301 1.18802 0.594008 0.804459i \(-0.297545\pi\)
0.594008 + 0.804459i \(0.297545\pi\)
\(594\) 0 0
\(595\) −60.2843 27.0279i −2.47141 1.10804i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.2132i 1.27534i 0.770311 + 0.637668i \(0.220101\pi\)
−0.770311 + 0.637668i \(0.779899\pi\)
\(600\) 0 0
\(601\) 22.6984i 0.925886i −0.886388 0.462943i \(-0.846793\pi\)
0.886388 0.462943i \(-0.153207\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.42742 0.0986885
\(606\) 0 0
\(607\) 2.16478i 0.0878659i −0.999034 0.0439329i \(-0.986011\pi\)
0.999034 0.0439329i \(-0.0139888\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000i 0.647291i
\(612\) 0 0
\(613\) 21.3137 0.860853 0.430426 0.902626i \(-0.358363\pi\)
0.430426 + 0.902626i \(0.358363\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.75736i 0.151266i −0.997136 0.0756328i \(-0.975902\pi\)
0.997136 0.0756328i \(-0.0240977\pi\)
\(618\) 0 0
\(619\) 16.5754i 0.666223i −0.942887 0.333112i \(-0.891901\pi\)
0.942887 0.333112i \(-0.108099\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.3128 + 7.31371i 0.653559 + 0.293018i
\(624\) 0 0
\(625\) 6.65685 0.266274
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27.0279 1.07767
\(630\) 0 0
\(631\) −39.5980 −1.57637 −0.788185 0.615438i \(-0.788979\pi\)
−0.788185 + 0.615438i \(0.788979\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −62.7150 −2.48877
\(636\) 0 0
\(637\) −27.3137 + 24.3379i −1.08221 + 0.964302i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.8701i 1.37728i 0.725101 + 0.688642i \(0.241793\pi\)
−0.725101 + 0.688642i \(0.758207\pi\)
\(642\) 0 0
\(643\) 18.7402i 0.739042i 0.929222 + 0.369521i \(0.120478\pi\)
−0.929222 + 0.369521i \(0.879522\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −50.4692 −1.98415 −0.992074 0.125658i \(-0.959896\pi\)
−0.992074 + 0.125658i \(0.959896\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.21320i 0.0474763i 0.999718 + 0.0237382i \(0.00755680\pi\)
−0.999718 + 0.0237382i \(0.992443\pi\)
\(654\) 0 0
\(655\) 11.3137 0.442063
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 44.5858i 1.73682i 0.495851 + 0.868408i \(0.334856\pi\)
−0.495851 + 0.868408i \(0.665144\pi\)
\(660\) 0 0
\(661\) 27.7708i 1.08016i −0.841614 0.540079i \(-0.818394\pi\)
0.841614 0.540079i \(-0.181606\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.65914 + 19.3137i −0.335787 + 0.748953i
\(666\) 0 0
\(667\) 16.1421 0.625026
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.34315 −0.0903216 −0.0451608 0.998980i \(-0.514380\pi\)
−0.0451608 + 0.998980i \(0.514380\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.6733 0.563940 0.281970 0.959423i \(-0.409012\pi\)
0.281970 + 0.959423i \(0.409012\pi\)
\(678\) 0 0
\(679\) −10.3431 + 23.0698i −0.396934 + 0.885337i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.27208i 0.354786i −0.984140 0.177393i \(-0.943234\pi\)
0.984140 0.177393i \(-0.0567664\pi\)
\(684\) 0 0
\(685\) 36.5838i 1.39779i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.5140 0.514841
\(690\) 0 0
\(691\) 12.9887i 0.494114i 0.969001 + 0.247057i \(0.0794634\pi\)
−0.969001 + 0.247057i \(0.920537\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000i 0.606915i
\(696\) 0 0
\(697\) −4.28427 −0.162278
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 43.0711i 1.62677i −0.581725 0.813386i \(-0.697622\pi\)
0.581725 0.813386i \(-0.302378\pi\)
\(702\) 0 0
\(703\) 8.65914i 0.326586i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.86030 + 2.62742i 0.220399 + 0.0988142i
\(708\) 0 0
\(709\) −32.2843 −1.21246 −0.606231 0.795289i \(-0.707319\pi\)
−0.606231 + 0.795289i \(0.707319\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 65.2512 2.44368
\(714\) 0 0
\(715\) 65.9411 2.46606
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.65914 0.322931 0.161466 0.986878i \(-0.448378\pi\)
0.161466 + 0.986878i \(0.448378\pi\)
\(720\) 0 0
\(721\) −6.62742 + 14.7821i −0.246818 + 0.550513i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.3848i 0.831350i
\(726\) 0 0
\(727\) 14.7821i 0.548237i −0.961696 0.274118i \(-0.911614\pi\)
0.961696 0.274118i \(-0.0883860\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 43.8210 1.62078
\(732\) 0 0
\(733\) 15.6788i 0.579108i −0.957162 0.289554i \(-0.906493\pi\)
0.957162 0.289554i \(-0.0935070\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) −16.2843 −0.599027 −0.299513 0.954092i \(-0.596824\pi\)
−0.299513 + 0.954092i \(0.596824\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.6985i 0.429176i −0.976705 0.214588i \(-0.931159\pi\)
0.976705 0.214588i \(-0.0688408\pi\)
\(744\) 0 0
\(745\) 28.6675i 1.05029i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.15932 2.58579i 0.0423605 0.0944826i
\(750\) 0 0
\(751\) −30.4853 −1.11242 −0.556212 0.831041i \(-0.687746\pi\)
−0.556212 + 0.831041i \(0.687746\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 59.6536 2.17102
\(756\) 0 0
\(757\) −51.5980 −1.87536 −0.937680 0.347499i \(-0.887031\pi\)
−0.937680 + 0.347499i \(0.887031\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33.7849 1.22470 0.612351 0.790586i \(-0.290224\pi\)
0.612351 + 0.790586i \(0.290224\pi\)
\(762\) 0 0
\(763\) 4.00000 + 1.79337i 0.144810 + 0.0649242i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.6984i 0.818524i −0.912417 0.409262i \(-0.865786\pi\)
0.912417 0.409262i \(-0.134214\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.42742 −0.0873081 −0.0436541 0.999047i \(-0.513900\pi\)
−0.0436541 + 0.999047i \(0.513900\pi\)
\(774\) 0 0
\(775\) 90.4858i 3.25035i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.37258i 0.0491779i
\(780\) 0 0
\(781\) −25.3137 −0.905796
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 70.6274i 2.52080i
\(786\) 0 0
\(787\) 33.1509i 1.18170i −0.806781 0.590851i \(-0.798792\pi\)
0.806781 0.590851i \(-0.201208\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.7261 + 28.3848i −0.452487 + 1.00925i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.3547 −0.437624 −0.218812 0.975767i \(-0.570218\pi\)
−0.218812 + 0.975767i \(0.570218\pi\)
\(798\) 0 0
\(799\) 20.6863 0.731828
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.06147 −0.108037
\(804\) 0 0
\(805\) 24.9706 55.6954i 0.880097 1.96301i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.07107i 0.248606i 0.992244 + 0.124303i \(0.0396694\pi\)
−0.992244 + 0.124303i \(0.960331\pi\)
\(810\) 0 0
\(811\) 29.5641i 1.03814i −0.854732 0.519069i \(-0.826279\pi\)
0.854732 0.519069i \(-0.173721\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 65.2512 2.28565
\(816\) 0 0
\(817\) 14.0392i 0.491171i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31.3553i 1.09431i −0.837032 0.547154i \(-0.815711\pi\)
0.837032 0.547154i \(-0.184289\pi\)
\(822\) 0 0
\(823\) −10.3431 −0.360539 −0.180270 0.983617i \(-0.557697\pi\)
−0.180270 + 0.983617i \(0.557697\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.0122i 0.939306i −0.882851 0.469653i \(-0.844379\pi\)
0.882851 0.469653i \(-0.155621\pi\)
\(828\) 0 0
\(829\) 26.1313i 0.907576i −0.891110 0.453788i \(-0.850072\pi\)
0.891110 0.453788i \(-0.149928\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −31.4663 35.3137i −1.09024 1.22355i
\(834\) 0 0
\(835\) 43.3137 1.49893
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.4303 0.739855 0.369928 0.929061i \(-0.379382\pi\)
0.369928 + 0.929061i \(0.379382\pi\)
\(840\) 0 0
\(841\) 22.3137 0.769438
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 52.8966 1.81970
\(846\) 0 0
\(847\) 1.58579 + 0.710974i 0.0544883 + 0.0244294i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.9706i 0.855980i
\(852\) 0 0
\(853\) 12.2459i 0.419291i −0.977778 0.209645i \(-0.932769\pi\)
0.977778 0.209645i \(-0.0672309\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.75699 0.230814 0.115407 0.993318i \(-0.463183\pi\)
0.115407 + 0.993318i \(0.463183\pi\)
\(858\) 0 0
\(859\) 18.7402i 0.639408i 0.947517 + 0.319704i \(0.103583\pi\)
−0.947517 + 0.319704i \(0.896417\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 23.4142i 0.797029i 0.917162 + 0.398515i \(0.130474\pi\)
−0.917162 + 0.398515i \(0.869526\pi\)
\(864\) 0 0
\(865\) −45.6569 −1.55238
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 46.6274i 1.58173i
\(870\) 0 0
\(871\) 8.65914i 0.293404i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 32.6256 + 14.6274i 1.10295 + 0.494497i
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31.4663 −1.06013 −0.530063 0.847958i \(-0.677832\pi\)
−0.530063 + 0.847958i \(0.677832\pi\)
\(882\) 0 0
\(883\) 24.1421 0.812448 0.406224 0.913774i \(-0.366845\pi\)
0.406224 + 0.913774i \(0.366845\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −23.9665 −0.804715 −0.402358 0.915483i \(-0.631809\pi\)
−0.402358 + 0.915483i \(0.631809\pi\)
\(888\) 0 0
\(889\) −40.9706 18.3688i −1.37411 0.616070i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.62742i 0.221778i
\(894\) 0 0
\(895\) 58.7569i 1.96403i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −27.0279 −0.901432
\(900\) 0 0
\(901\) 17.4721i 0.582081i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 89.9411i 2.98974i
\(906\) 0 0
\(907\) −49.7990 −1.65355 −0.826774 0.562534i \(-0.809827\pi\)
−0.826774 + 0.562534i \(0.809827\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 56.1838i 1.86145i −0.365718 0.930726i \(-0.619177\pi\)
0.365718 0.930726i \(-0.380823\pi\)
\(912\) 0 0
\(913\) 40.0166i 1.32436i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.39104 + 3.31371i 0.244074 + 0.109428i
\(918\) 0 0
\(919\) −1.79899 −0.0593432 −0.0296716 0.999560i \(-0.509446\pi\)
−0.0296716 + 0.999560i \(0.509446\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −38.7485 −1.27542
\(924\) 0 0
\(925\) −34.6274 −1.13854
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 55.9580 1.83592 0.917962 0.396669i \(-0.129834\pi\)
0.917962 + 0.396669i \(0.129834\pi\)
\(930\) 0 0
\(931\) −11.3137 + 10.0811i −0.370792 + 0.330394i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 85.2548i 2.78813i
\(936\) 0 0
\(937\) 50.4692i 1.64876i −0.566040 0.824378i \(-0.691525\pi\)
0.566040 0.824378i \(-0.308475\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.3547 0.402750 0.201375 0.979514i \(-0.435459\pi\)
0.201375 + 0.979514i \(0.435459\pi\)
\(942\) 0 0
\(943\) 3.95815i 0.128895i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.4142i 0.890842i 0.895321 + 0.445421i \(0.146946\pi\)
−0.895321 + 0.445421i \(0.853054\pi\)
\(948\) 0 0
\(949\) −4.68629 −0.152123
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 10.7868i 0.349419i 0.984620 + 0.174709i \(0.0558986\pi\)
−0.984620 + 0.174709i \(0.944101\pi\)
\(954\) 0 0
\(955\) 10.0811i 0.326216i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.7151 + 23.8995i −0.346010 + 0.771755i
\(960\) 0 0
\(961\) −78.2548 −2.52435
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −20.9050 −0.672956
\(966\) 0 0
\(967\) 18.3431 0.589876 0.294938 0.955516i \(-0.404701\pi\)
0.294938 + 0.955516i \(0.404701\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.0279 −0.867368 −0.433684 0.901065i \(-0.642787\pi\)
−0.433684 + 0.901065i \(0.642787\pi\)
\(972\) 0 0
\(973\) −4.68629 + 10.4525i −0.150236 + 0.335092i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.89949i 0.0607702i 0.999538 + 0.0303851i \(0.00967337\pi\)
−0.999538 + 0.0303851i \(0.990327\pi\)
\(978\) 0 0
\(979\) 23.0698i 0.737314i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 59.6536 1.90265 0.951326 0.308185i \(-0.0997216\pi\)
0.951326 + 0.308185i \(0.0997216\pi\)
\(984\) 0 0
\(985\) 82.7233i 2.63578i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 40.4853i 1.28736i
\(990\) 0 0
\(991\) −56.1421 −1.78341 −0.891707 0.452613i \(-0.850492\pi\)
−0.891707 + 0.452613i \(0.850492\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 85.2548i 2.70276i
\(996\) 0 0
\(997\) 46.8824i 1.48478i 0.669967 + 0.742391i \(0.266308\pi\)
−0.669967 + 0.742391i \(0.733692\pi\)
\(998\) 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.k.e.3905.1 8
3.2 odd 2 inner 4032.2.k.e.3905.7 8
4.3 odd 2 4032.2.k.f.3905.2 8
7.6 odd 2 inner 4032.2.k.e.3905.8 8
8.3 odd 2 504.2.k.a.377.8 yes 8
8.5 even 2 1008.2.k.c.881.7 8
12.11 even 2 4032.2.k.f.3905.8 8
21.20 even 2 inner 4032.2.k.e.3905.2 8
24.5 odd 2 1008.2.k.c.881.1 8
24.11 even 2 504.2.k.a.377.2 yes 8
28.27 even 2 4032.2.k.f.3905.7 8
56.3 even 6 3528.2.bl.b.1097.8 16
56.11 odd 6 3528.2.bl.b.1097.2 16
56.13 odd 2 1008.2.k.c.881.2 8
56.19 even 6 3528.2.bl.b.521.7 16
56.27 even 2 504.2.k.a.377.1 8
56.51 odd 6 3528.2.bl.b.521.1 16
84.83 odd 2 4032.2.k.f.3905.1 8
168.11 even 6 3528.2.bl.b.1097.7 16
168.59 odd 6 3528.2.bl.b.1097.1 16
168.83 odd 2 504.2.k.a.377.7 yes 8
168.107 even 6 3528.2.bl.b.521.8 16
168.125 even 2 1008.2.k.c.881.8 8
168.131 odd 6 3528.2.bl.b.521.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.k.a.377.1 8 56.27 even 2
504.2.k.a.377.2 yes 8 24.11 even 2
504.2.k.a.377.7 yes 8 168.83 odd 2
504.2.k.a.377.8 yes 8 8.3 odd 2
1008.2.k.c.881.1 8 24.5 odd 2
1008.2.k.c.881.2 8 56.13 odd 2
1008.2.k.c.881.7 8 8.5 even 2
1008.2.k.c.881.8 8 168.125 even 2
3528.2.bl.b.521.1 16 56.51 odd 6
3528.2.bl.b.521.2 16 168.131 odd 6
3528.2.bl.b.521.7 16 56.19 even 6
3528.2.bl.b.521.8 16 168.107 even 6
3528.2.bl.b.1097.1 16 168.59 odd 6
3528.2.bl.b.1097.2 16 56.11 odd 6
3528.2.bl.b.1097.7 16 168.11 even 6
3528.2.bl.b.1097.8 16 56.3 even 6
4032.2.k.e.3905.1 8 1.1 even 1 trivial
4032.2.k.e.3905.2 8 21.20 even 2 inner
4032.2.k.e.3905.7 8 3.2 odd 2 inner
4032.2.k.e.3905.8 8 7.6 odd 2 inner
4032.2.k.f.3905.1 8 84.83 odd 2
4032.2.k.f.3905.2 8 4.3 odd 2
4032.2.k.f.3905.7 8 28.27 even 2
4032.2.k.f.3905.8 8 12.11 even 2