Properties

Label 1600.2.n.g.1343.1
Level $1600$
Weight $2$
Character 1600.1343
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
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.g.1407.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{9} +(-5.00000 - 5.00000i) q^{13} +(5.00000 - 5.00000i) q^{17} -4.00000i q^{29} +(5.00000 - 5.00000i) q^{37} +8.00000 q^{41} -7.00000i q^{49} +(5.00000 + 5.00000i) q^{53} +12.0000 q^{61} +(-5.00000 - 5.00000i) q^{73} -9.00000 q^{81} -16.0000i q^{89} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{13} + 10 q^{17} + 10 q^{37} + 16 q^{41} + 10 q^{53} + 24 q^{61} - 10 q^{73} - 18 q^{81} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −5.00000 5.00000i −1.38675 1.38675i −0.832050 0.554700i \(-0.812833\pi\)
−0.554700 0.832050i \(-0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 5.00000i 1.21268 1.21268i 0.242536 0.970143i \(-0.422021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i −0.928477 0.371391i \(-0.878881\pi\)
0.928477 0.371391i \(-0.121119\pi\)
\(30\) 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
\(37\) 5.00000 5.00000i 0.821995 0.821995i −0.164399 0.986394i \(-0.552568\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000 + 5.00000i 0.686803 + 0.686803i 0.961524 0.274721i \(-0.0885855\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 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
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −5.00000 5.00000i −0.585206 0.585206i 0.351123 0.936329i \(-0.385800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 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
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000i 1.69600i −0.529999 0.847998i \(-0.677808\pi\)
0.529999 0.847998i \(-0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00000 + 5.00000i −0.507673 + 0.507673i −0.913812 0.406138i \(-0.866875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.0000 15.0000i −1.41108 1.41108i −0.752577 0.658505i \(-0.771189\pi\)
−0.658505 0.752577i \(-0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 15.0000 15.0000i 1.38675 1.38675i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 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
\(137\) 15.0000 15.0000i 1.28154 1.28154i 0.341743 0.939793i \(-0.388983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.0000i 1.14692i 0.819232 + 0.573462i \(0.194400\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 15.0000 + 15.0000i 1.21268 + 1.21268i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.00000 + 5.00000i −0.399043 + 0.399043i −0.877896 0.478852i \(-0.841053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 37.0000i 2.84615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.0000 15.0000i −1.14043 1.14043i −0.988372 0.152057i \(-0.951410\pi\)
−0.152057 0.988372i \(-0.548590\pi\)
\(174\) 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
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 5.00000 + 5.00000i 0.359908 + 0.359908i 0.863779 0.503871i \(-0.168091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 + 15.0000i −1.06871 + 1.06871i −0.0712470 + 0.997459i \(0.522698\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 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
\(221\) −50.0000 −3.36336
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.00000 + 5.00000i 0.327561 + 0.327561i 0.851658 0.524097i \(-0.175597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 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
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(257\) −15.0000 + 15.0000i −0.935674 + 0.935674i −0.998053 0.0623783i \(-0.980131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000i 1.58525i −0.609711 0.792624i \(-0.708714\pi\)
0.609711 0.792624i \(-0.291286\pi\)
\(270\) 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
\(277\) −5.00000 + 5.00000i −0.300421 + 0.300421i −0.841178 0.540758i \(-0.818138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000i 1.94118i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000 + 15.0000i 0.876309 + 0.876309i 0.993151 0.116841i \(-0.0372769\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 25.0000 + 25.0000i 1.41308 + 1.41308i 0.734803 + 0.678280i \(0.237274\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.0000 + 25.0000i −1.40414 + 1.40414i −0.617822 + 0.786318i \(0.711985\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 15.0000 + 15.0000i 0.821995 + 0.821995i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −25.0000 + 25.0000i −1.36184 + 1.36184i −0.490261 + 0.871576i \(0.663099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 36.0000i 1.92704i 0.267644 + 0.963518i \(0.413755\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.0000 25.0000i −1.33062 1.33062i −0.904819 0.425797i \(-0.859994\pi\)
−0.425797 0.904819i \(-0.640006\pi\)
\(354\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 24.0000i 1.24939i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0000 + 25.0000i 1.29445 + 1.29445i 0.932005 + 0.362446i \(0.118058\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.0000 + 20.0000i −1.03005 + 1.03005i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 34.0000i 1.72387i 0.507020 + 0.861934i \(0.330747\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0000 25.0000i 1.25471 1.25471i 0.301131 0.953583i \(-0.402636\pi\)
0.953583 0.301131i \(-0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000i 0.296681i 0.988936 + 0.148340i \(0.0473931\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 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
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −5.00000 5.00000i −0.240285 0.240285i 0.576683 0.816968i \(-0.304347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 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
\(441\) 21.0000 1.00000
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000i 0.660701i −0.943858 0.330350i \(-0.892833\pi\)
0.943858 0.330350i \(-0.107167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0000 25.0000i 1.16945 1.16945i 0.187112 0.982339i \(-0.440087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.0000 + 15.0000i −0.686803 + 0.686803i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −50.0000 −2.27980
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −20.0000 20.0000i −0.900755 0.900755i
\(494\) 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
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 44.0000i 1.95027i −0.221621 0.975133i \(-0.571135\pi\)
0.221621 0.975133i \(-0.428865\pi\)
\(510\) 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
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −40.0000 40.0000i −1.73259 1.73259i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 36.0000i 1.53644i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.00000 5.00000i 0.211857 0.211857i −0.593199 0.805056i \(-0.702135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000i 1.08998i 0.838444 + 0.544988i \(0.183466\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 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
\(577\) 25.0000 25.0000i 1.04076 1.04076i 0.0416305 0.999133i \(-0.486745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.0000 + 15.0000i 0.615976 + 0.615976i 0.944497 0.328521i \(-0.106550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 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
\(601\) 48.0000 1.95796 0.978980 0.203954i \(-0.0653794\pi\)
0.978980 + 0.203954i \(0.0653794\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 35.0000 + 35.0000i 1.41364 + 1.41364i 0.727013 + 0.686624i \(0.240908\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.0000 + 35.0000i −1.40905 + 1.40905i −0.644136 + 0.764911i \(0.722783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.0000i 1.99363i
\(630\) 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
\(637\) −35.0000 + 35.0000i −1.38675 + 1.38675i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −35.0000 35.0000i −1.36966 1.36966i −0.860927 0.508729i \(-0.830115\pi\)
−0.508729 0.860927i \(-0.669885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.0000 15.0000i 0.585206 0.585206i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 35.0000 + 35.0000i 1.34915 + 1.34915i 0.886585 + 0.462566i \(0.153071\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.0000 25.0000i 0.960828 0.960828i −0.0384331 0.999261i \(-0.512237\pi\)
0.999261 + 0.0384331i \(0.0122367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 50.0000i 1.90485i
\(690\) 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
\(697\) 40.0000 40.0000i 1.51511 1.51511i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.0000 1.96401 0.982006 0.188847i \(-0.0604752\pi\)
0.982006 + 0.188847i \(0.0604752\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000i 1.65245i −0.563337 0.826227i \(-0.690483\pi\)
0.563337 0.826227i \(-0.309517\pi\)
\(710\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 25.0000 + 25.0000i 0.923396 + 0.923396i 0.997268 0.0738717i \(-0.0235355\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 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
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 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
\(757\) 35.0000 35.0000i 1.27210 1.27210i 0.327111 0.944986i \(-0.393925\pi\)
0.944986 0.327111i \(-0.106075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 24.0000i 0.865462i 0.901523 + 0.432731i \(0.142450\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.00000 5.00000i −0.179838 0.179838i 0.611448 0.791285i \(-0.290588\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 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
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −60.0000 60.0000i −2.13066 2.13066i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.0000 15.0000i 0.531327 0.531327i −0.389640 0.920967i \(-0.627401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 48.0000 1.69600
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0