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Properties

Label 1600.2.n.l.1343.1
Level $1600$
Weight $2$
Character 1600.1343
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1343,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 1343.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1343
Dual form 1600.2.n.l.1407.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\) \(q+(1.00000 - 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +6.00000 q^{21} +(-1.00000 + 1.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +6.00000i q^{29} +12.0000 q^{41} +(-9.00000 + 9.00000i) q^{43} +(-7.00000 - 7.00000i) q^{47} +11.0000i q^{49} +8.00000 q^{61} +(-3.00000 + 3.00000i) q^{63} +(-3.00000 - 3.00000i) q^{67} +2.00000i q^{69} +5.00000 q^{81} +(11.0000 - 11.0000i) q^{83} +(6.00000 + 6.00000i) q^{87} -6.00000i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{7} + 12 q^{21} - 2 q^{23} + 8 q^{27} + 24 q^{41} - 18 q^{43} - 14 q^{47} + 16 q^{61} - 6 q^{63} - 6 q^{67} + 10 q^{81} + 22 q^{83} + 12 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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
\(3\) 1.00000 − 1.00000i 0.577350 − 0.577350i −0.356822 − 0.934172i \(-0.616140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000 + 3.00000i 1.13389 + 1.13389i 0.989524 + 0.144370i \(0.0461154\pi\)
0.144370 + 0.989524i \(0.453885\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) −1.00000 + 1.00000i −0.208514 + 0.208514i −0.803636 − 0.595121i \(-0.797104\pi\)
0.595121 + 0.803636i \(0.297104\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) −9.00000 + 9.00000i −1.37249 + 1.37249i −0.515745 + 0.856742i \(0.672485\pi\)
−0.856742 + 0.515745i \(0.827515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.00000 − 7.00000i −1.02105 − 1.02105i −0.999774 − 0.0212814i \(-0.993225\pi\)
−0.0212814 − 0.999774i \(-0.506775\pi\)
\(48\) 0 0
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −3.00000 + 3.00000i −0.377964 + 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 − 3.00000i −0.366508 − 0.366508i 0.499694 − 0.866202i \(-0.333446\pi\)
−0.866202 + 0.499694i \(0.833446\pi\)
\(68\) 0 0
\(69\) 2.00000i 0.240772i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 11.0000 − 11.0000i 1.20741 − 1.20741i 0.235543 − 0.971864i \(-0.424313\pi\)
0.971864 − 0.235543i \(-0.0756868\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 + 6.00000i 0.643268 + 0.643268i
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 9.00000 − 9.00000i 0.886796 − 0.886796i −0.107418 − 0.994214i \(-0.534258\pi\)
0.994214 + 0.107418i \(0.0342582\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.0000 − 13.0000i −1.25676 − 1.25676i −0.952632 − 0.304125i \(-0.901636\pi\)
−0.304125 − 0.952632i \(-0.598364\pi\)
\(108\) 0 0
\(109\) 16.0000i 1.53252i 0.642529 + 0.766261i \(0.277885\pi\)
−0.642529 + 0.766261i \(0.722115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 12.0000 − 12.0000i 1.08200 − 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.00000 + 3.00000i 0.266207 + 0.266207i 0.827570 − 0.561363i \(-0.189723\pi\)
−0.561363 + 0.827570i \(0.689723\pi\)
\(128\) 0 0
\(129\) 18.0000i 1.58481i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −14.0000 −1.17901
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.0000 + 11.0000i 0.907265 + 0.907265i
\(148\) 0 0
\(149\) − 24.0000i − 1.96616i −0.183186 − 0.983078i \(-0.558641\pi\)
0.183186 − 0.983078i \(-0.441359\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −9.00000 + 9.00000i −0.704934 + 0.704934i −0.965465 − 0.260531i \(-0.916102\pi\)
0.260531 + 0.965465i \(0.416102\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.0000 − 17.0000i −1.31550 − 1.31550i −0.917298 − 0.398202i \(-0.869634\pi\)
−0.398202 − 0.917298i \(-0.630366\pi\)
\(168\) 0 0
\(169\) − 13.0000i − 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 8.00000 − 8.00000i 0.591377 − 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 24.0000i 1.74574i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) −18.0000 + 18.0000i −1.26335 + 1.26335i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 − 1.00000i −0.0695048 − 0.0695048i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −21.0000 + 21.0000i −1.40626 + 1.40626i −0.628263 + 0.778001i \(0.716234\pi\)
−0.778001 + 0.628263i \(0.783766\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.00000 + 7.00000i 0.464606 + 0.464606i 0.900162 − 0.435556i \(-0.143448\pi\)
−0.435556 + 0.900162i \(0.643448\pi\)
\(228\) 0 0
\(229\) − 14.0000i − 0.925146i −0.886581 − 0.462573i \(-0.846926\pi\)
0.886581 − 0.462573i \(-0.153074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −28.0000 −1.80364 −0.901819 − 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) −7.00000 + 7.00000i −0.449050 + 0.449050i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 22.0000i − 1.39419i
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −11.0000 + 11.0000i −0.678289 + 0.678289i −0.959613 − 0.281324i \(-0.909226\pi\)
0.281324 + 0.959613i \(0.409226\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 − 6.00000i −0.367194 − 0.367194i
\(268\) 0 0
\(269\) − 24.0000i − 1.46331i −0.681677 − 0.731653i \(-0.738749\pi\)
0.681677 − 0.731653i \(-0.261251\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 21.0000 − 21.0000i 1.24832 − 1.24832i 0.291859 − 0.956461i \(-0.405726\pi\)
0.956461 − 0.291859i \(-0.0942738\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.0000 + 36.0000i 2.12501 + 2.12501i
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −54.0000 −3.11251
\(302\) 0 0
\(303\) 18.0000 − 18.0000i 1.03407 − 1.03407i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.00000 − 3.00000i −0.171219 − 0.171219i 0.616296 − 0.787515i \(-0.288633\pi\)
−0.787515 + 0.616296i \(0.788633\pi\)
\(308\) 0 0
\(309\) − 18.0000i − 1.02398i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −26.0000 −1.45118
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 16.0000 + 16.0000i 0.884802 + 0.884802i
\(328\) 0 0
\(329\) − 42.0000i − 2.31553i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −12.0000 + 12.0000i −0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.0000 + 17.0000i 0.912608 + 0.912608i 0.996477 − 0.0838690i \(-0.0267277\pi\)
−0.0838690 + 0.996477i \(0.526728\pi\)
\(348\) 0 0
\(349\) 26.0000i 1.39175i 0.718164 + 0.695874i \(0.244983\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 11.0000 − 11.0000i 0.577350 − 0.577350i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −27.0000 − 27.0000i −1.40939 − 1.40939i −0.763055 − 0.646333i \(-0.776302\pi\)
−0.646333 − 0.763055i \(-0.723698\pi\)
\(368\) 0 0
\(369\) 12.0000i 0.624695i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) 19.0000 − 19.0000i 0.970855 − 0.970855i −0.0287325 − 0.999587i \(-0.509147\pi\)
0.999587 + 0.0287325i \(0.00914709\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.00000 − 9.00000i −0.457496 − 0.457496i
\(388\) 0 0
\(389\) − 24.0000i − 1.21685i −0.793612 − 0.608424i \(-0.791802\pi\)
0.793612 − 0.608424i \(-0.208198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000i 0.197787i 0.995098 + 0.0988936i \(0.0315304\pi\)
−0.995098 + 0.0988936i \(0.968470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 − 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 7.00000 − 7.00000i 0.340352 − 0.340352i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24.0000 + 24.0000i 1.16144 + 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) −29.0000 + 29.0000i −1.37783 + 1.37783i −0.529558 + 0.848274i \(0.677642\pi\)
−0.848274 + 0.529558i \(0.822358\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −24.0000 − 24.0000i −1.13516 − 1.13516i
\(448\) 0 0
\(449\) − 36.0000i − 1.69895i −0.527633 − 0.849473i \(-0.676920\pi\)
0.527633 − 0.849473i \(-0.323080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 − 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) 9.00000 − 9.00000i 0.418265 − 0.418265i −0.466340 − 0.884606i \(-0.654428\pi\)
0.884606 + 0.466340i \(0.154428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.0000 − 23.0000i −1.06431 − 1.06431i −0.997785 − 0.0665285i \(-0.978808\pi\)
−0.0665285 − 0.997785i \(-0.521192\pi\)
\(468\) 0 0
\(469\) − 18.0000i − 0.831163i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −6.00000 + 6.00000i −0.273009 + 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −27.0000 − 27.0000i −1.22349 − 1.22349i −0.966384 − 0.257103i \(-0.917232\pi\)
−0.257103 − 0.966384i \(-0.582768\pi\)
\(488\) 0 0
\(489\) 18.0000i 0.813988i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −34.0000 −1.51901
\(502\) 0 0
\(503\) −31.0000 + 31.0000i −1.38222 + 1.38222i −0.541559 + 0.840663i \(0.682166\pi\)
−0.840663 + 0.541559i \(0.817834\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.0000 − 13.0000i −0.577350 − 0.577350i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 21.0000 − 21.0000i 0.918266 − 0.918266i −0.0786374 − 0.996903i \(-0.525057\pi\)
0.996903 + 0.0786374i \(0.0250569\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000i 0.913043i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −2.00000 + 2.00000i −0.0858282 + 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −33.0000 − 33.0000i −1.41098 − 1.41098i −0.753293 − 0.657685i \(-0.771536\pi\)
−0.657685 − 0.753293i \(-0.728464\pi\)
\(548\) 0 0
\(549\) 8.00000i 0.341432i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.00000 − 1.00000i 0.0421450 − 0.0421450i −0.685720 − 0.727865i \(-0.740513\pi\)
0.727865 + 0.685720i \(0.240513\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 15.0000 + 15.0000i 0.629941 + 0.629941i
\(568\) 0 0
\(569\) − 36.0000i − 1.50920i −0.656186 − 0.754599i \(-0.727831\pi\)
0.656186 − 0.754599i \(-0.272169\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 66.0000 2.73814
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.00000 + 7.00000i 0.288921 + 0.288921i 0.836653 − 0.547733i \(-0.184509\pi\)
−0.547733 + 0.836653i \(0.684509\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 − 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 3.00000 − 3.00000i 0.122169 − 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 33.0000 + 33.0000i 1.33943 + 1.33943i 0.896612 + 0.442816i \(0.146021\pi\)
0.442816 + 0.896612i \(0.353979\pi\)
\(608\) 0 0
\(609\) 36.0000i 1.45879i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 18.0000 − 18.0000i 0.721155 − 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) 21.0000 − 21.0000i 0.828159 − 0.828159i −0.159103 − 0.987262i \(-0.550860\pi\)
0.987262 + 0.159103i \(0.0508601\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.0000 + 13.0000i 0.511083 + 0.511083i 0.914858 − 0.403775i \(-0.132302\pi\)
−0.403775 + 0.914858i \(0.632302\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 − 6.00000i −0.232321 − 0.232321i
\(668\) 0 0
\(669\) 42.0000i 1.62381i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 31.0000 − 31.0000i 1.18618 − 1.18618i 0.208068 − 0.978114i \(-0.433283\pi\)
0.978114 − 0.208068i \(-0.0667174\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −14.0000 − 14.0000i −0.534133 − 0.534133i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 − 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 54.0000 + 54.0000i 2.03088 + 2.03088i
\(708\) 0 0
\(709\) 46.0000i 1.72757i 0.503864 + 0.863783i \(0.331911\pi\)
−0.503864 + 0.863783i \(0.668089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 54.0000 2.01107
\(722\) 0 0
\(723\) −28.0000 + 28.0000i −1.04133 + 1.04133i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000 + 3.00000i 0.111264 + 0.111264i 0.760547 − 0.649283i \(-0.224931\pi\)
−0.649283 + 0.760547i \(0.724931\pi\)
\(728\) 0 0
\(729\) 29.0000i 1.07407i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.0000 − 19.0000i 0.697042 − 0.697042i −0.266729 − 0.963772i \(-0.585943\pi\)
0.963772 + 0.266729i \(0.0859429\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.0000 + 11.0000i 0.402469 + 0.402469i
\(748\) 0 0
\(749\) − 78.0000i − 2.85006i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −48.0000 + 48.0000i −1.73772 + 1.73772i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000i 0.504853i 0.967616 + 0.252426i \(0.0812286\pi\)
−0.967616 + 0.252426i \(0.918771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.0000 + 24.0000i −0.857690 + 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.0000 + 27.0000i 0.962446 + 0.962446i 0.999320 − 0.0368739i \(-0.0117400\pi\)
−0.0368739 + 0.999320i \(0.511740\pi\)
\(788\) 0 0
\(789\) 22.0000i 0.783221i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 − 24.0000i −0.844840 − 0.844840i
\(808\) 0 0
\(809\) 54.0000i 1.89854i 0.314464 + 0.949269i \(0.398175\pi\)
−0.314464 + 0.949269i \(0.601825\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 48.0000 1.67521 0.837606 − 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(822\) 0 0
\(823\) 39.0000 − 39.0000i 1.35945 − 1.35945i 0.484866 − 0.874588i \(-0.338868\pi\)
0.874588 − 0.484866i \(-0.161132\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23.0000 − 23.0000i −0.799788 − 0.799788i 0.183274 − 0.983062i \(-0.441331\pi\)
−0.983062 + 0.183274i \(0.941331\pi\)
\(828\) 0 0
\(829\) 56.0000i 1.94496i 0.232986 + 0.972480i \(0.425151\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 12.0000 − 12.0000i 0.413302 − 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 33.0000 + 33.0000i 1.13389 + 1.13389i
\(848\) 0 0
\(849\) − 42.0000i − 1.44144i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 72.0000 2.45375
\(862\) 0 0
\(863\) −41.0000 + 41.0000i −1.39566 + 1.39566i −0.583653 + 0.812003i \(0.698377\pi\)
−0.812003 + 0.583653i \(0.801623\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.0000 + 17.0000i 0.577350 + 0.577350i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 − 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) −39.0000 + 39.0000i −1.31245 + 1.31245i −0.392853 + 0.919601i \(0.628512\pi\)
−0.919601 + 0.392853i \(0.871488\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37.0000 − 37.0000i −1.24234 − 1.24234i −0.959028 − 0.283310i \(-0.908567\pi\)
−0.283310 − 0.959028i \(-0.591433\pi\)
\(888\) 0 0
\(889\) 18.0000i 0.603701i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −54.0000 + 54.0000i −1.79701 + 1.79701i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.00000 − 3.00000i −0.0996134 − 0.0996134i 0.655544 − 0.755157i \(-0.272439\pi\)
−0.755157 + 0.655544i \(0.772439\pi\)
\(908\) 0 0
\(909\) 18.0000i 0.597022i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.00000 + 9.00000i 0.295599 + 0.295599i
\(928\) 0 0
\(929\) − 36.0000i − 1.18112i −0.806993 − 0.590561i \(-0.798907\pi\)
0.806993 − 0.590561i \(-0.201093\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) −12.0000 + 12.0000i −0.390774 + 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −43.0000 − 43.0000i −1.39731 − 1.39731i −0.807650 − 0.589662i \(-0.799261\pi\)
−0.589662 − 0.807650i \(-0.700739\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 13.0000 − 13.0000i 0.418919 − 0.418919i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.0000 + 33.0000i 1.06121 + 1.06121i 0.998000 + 0.0632081i \(0.0201332\pi\)
0.0632081 + 0.998000i \(0.479867\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) 29.0000 − 29.0000i 0.924956 − 0.924956i −0.0724180 − 0.997374i \(-0.523072\pi\)
0.997374 + 0.0724180i \(0.0230716\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −42.0000 − 42.0000i −1.33687 − 1.33687i
\(988\) 0 0
\(989\) − 18.0000i − 0.572367i
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 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 1600.2.n.l.1343.1 2
4.3 odd 2 1600.2.n.b.1343.1 2
5.2 odd 4 1600.2.n.b.1407.1 2
5.3 odd 4 inner 1600.2.n.l.1407.1 2
5.4 even 2 1600.2.n.b.1343.1 2
8.3 odd 2 100.2.e.c.43.1 yes 2
8.5 even 2 100.2.e.a.43.1 yes 2
20.3 even 4 1600.2.n.b.1407.1 2
20.7 even 4 inner 1600.2.n.l.1407.1 2
20.19 odd 2 CM 1600.2.n.l.1343.1 2
24.5 odd 2 900.2.k.e.343.1 2
24.11 even 2 900.2.k.a.343.1 2
40.3 even 4 100.2.e.c.7.1 yes 2
40.13 odd 4 100.2.e.a.7.1 ✓ 2
40.19 odd 2 100.2.e.a.43.1 yes 2
40.27 even 4 100.2.e.a.7.1 ✓ 2
40.29 even 2 100.2.e.c.43.1 yes 2
40.37 odd 4 100.2.e.c.7.1 yes 2
120.29 odd 2 900.2.k.a.343.1 2
120.53 even 4 900.2.k.e.307.1 2
120.59 even 2 900.2.k.e.343.1 2
120.77 even 4 900.2.k.a.307.1 2
120.83 odd 4 900.2.k.a.307.1 2
120.107 odd 4 900.2.k.e.307.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.2.e.a.7.1 ✓ 2 40.13 odd 4
100.2.e.a.7.1 ✓ 2 40.27 even 4
100.2.e.a.43.1 yes 2 8.5 even 2
100.2.e.a.43.1 yes 2 40.19 odd 2
100.2.e.c.7.1 yes 2 40.3 even 4
100.2.e.c.7.1 yes 2 40.37 odd 4
100.2.e.c.43.1 yes 2 8.3 odd 2
100.2.e.c.43.1 yes 2 40.29 even 2
900.2.k.a.307.1 2 120.77 even 4
900.2.k.a.307.1 2 120.83 odd 4
900.2.k.a.343.1 2 24.11 even 2
900.2.k.a.343.1 2 120.29 odd 2
900.2.k.e.307.1 2 120.53 even 4
900.2.k.e.307.1 2 120.107 odd 4
900.2.k.e.343.1 2 24.5 odd 2
900.2.k.e.343.1 2 120.59 even 2
1600.2.n.b.1343.1 2 4.3 odd 2
1600.2.n.b.1343.1 2 5.4 even 2
1600.2.n.b.1407.1 2 5.2 odd 4
1600.2.n.b.1407.1 2 20.3 even 4
1600.2.n.l.1343.1 2 1.1 even 1 trivial
1600.2.n.l.1343.1 2 20.19 odd 2 CM
1600.2.n.l.1407.1 2 5.3 odd 4 inner
1600.2.n.l.1407.1 2 20.7 even 4 inner