Properties

Label 24.4.f.a.11.2
Level $24$
Weight $4$
Character 24.11
Analytic conductor $1.416$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,4,Mod(11,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 24.f (of order \(2\), degree \(1\), 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: \(1.41604584014\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 11.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 24.11
Dual form 24.4.f.a.11.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} +O(q^{10})\) \(q+2.82843i q^{2} +(5.00000 + 1.41421i) q^{3} -8.00000 q^{4} +(-4.00000 + 14.1421i) q^{6} -22.6274i q^{8} +(23.0000 + 14.1421i) q^{9} -70.7107i q^{11} +(-40.0000 - 11.3137i) q^{12} +64.0000 q^{16} +107.480i q^{17} +(-40.0000 + 65.0538i) q^{18} -106.000 q^{19} +200.000 q^{22} +(32.0000 - 113.137i) q^{24} -125.000 q^{25} +(95.0000 + 103.238i) q^{27} +181.019i q^{32} +(100.000 - 353.553i) q^{33} -304.000 q^{34} +(-184.000 - 113.137i) q^{36} -299.813i q^{38} +56.5685i q^{41} +290.000 q^{43} +565.685i q^{44} +(320.000 + 90.5097i) q^{48} +343.000 q^{49} -353.553i q^{50} +(-152.000 + 537.401i) q^{51} +(-292.000 + 268.701i) q^{54} +(-530.000 - 149.907i) q^{57} -325.269i q^{59} -512.000 q^{64} +(1000.00 + 282.843i) q^{66} -70.0000 q^{67} -859.842i q^{68} +(320.000 - 520.431i) q^{72} -430.000 q^{73} +(-625.000 - 176.777i) q^{75} +848.000 q^{76} +(329.000 + 650.538i) q^{81} -160.000 q^{82} -681.651i q^{83} +820.244i q^{86} -1600.00 q^{88} +1329.36i q^{89} +(-256.000 + 905.097i) q^{96} +1910.00 q^{97} +970.151i q^{98} +(1000.00 - 1626.35i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{3} - 16 q^{4} - 8 q^{6} + 46 q^{9} - 80 q^{12} + 128 q^{16} - 80 q^{18} - 212 q^{19} + 400 q^{22} + 64 q^{24} - 250 q^{25} + 190 q^{27} + 200 q^{33} - 608 q^{34} - 368 q^{36} + 580 q^{43} + 640 q^{48} + 686 q^{49} - 304 q^{51} - 584 q^{54} - 1060 q^{57} - 1024 q^{64} + 2000 q^{66} - 140 q^{67} + 640 q^{72} - 860 q^{73} - 1250 q^{75} + 1696 q^{76} + 658 q^{81} - 320 q^{82} - 3200 q^{88} - 512 q^{96} + 3820 q^{97} + 2000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 1.00000i
\(3\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(4\) −8.00000 −1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −4.00000 + 14.1421i −0.272166 + 0.962250i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 22.6274i 1.00000i
\(9\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(10\) 0 0
\(11\) 70.7107i 1.93819i −0.246691 0.969094i \(-0.579343\pi\)
0.246691 0.969094i \(-0.420657\pi\)
\(12\) −40.0000 11.3137i −0.962250 0.272166i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 107.480i 1.53340i 0.642006 + 0.766700i \(0.278102\pi\)
−0.642006 + 0.766700i \(0.721898\pi\)
\(18\) −40.0000 + 65.0538i −0.523783 + 0.851852i
\(19\) −106.000 −1.27990 −0.639949 0.768417i \(-0.721045\pi\)
−0.639949 + 0.768417i \(0.721045\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 200.000 1.93819
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 32.0000 113.137i 0.272166 0.962250i
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 95.0000 + 103.238i 0.677139 + 0.735855i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 181.019i 1.00000i
\(33\) 100.000 353.553i 0.527508 1.86502i
\(34\) −304.000 −1.53340
\(35\) 0 0
\(36\) −184.000 113.137i −0.851852 0.523783i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 299.813i 1.27990i
\(39\) 0 0
\(40\) 0 0
\(41\) 56.5685i 0.215476i 0.994179 + 0.107738i \(0.0343608\pi\)
−0.994179 + 0.107738i \(0.965639\pi\)
\(42\) 0 0
\(43\) 290.000 1.02848 0.514239 0.857647i \(-0.328074\pi\)
0.514239 + 0.857647i \(0.328074\pi\)
\(44\) 565.685i 1.93819i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 320.000 + 90.5097i 0.962250 + 0.272166i
\(49\) 343.000 1.00000
\(50\) 353.553i 1.00000i
\(51\) −152.000 + 537.401i −0.417338 + 1.47551i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −292.000 + 268.701i −0.735855 + 0.677139i
\(55\) 0 0
\(56\) 0 0
\(57\) −530.000 149.907i −1.23158 0.348344i
\(58\) 0 0
\(59\) 325.269i 0.717736i −0.933388 0.358868i \(-0.883163\pi\)
0.933388 0.358868i \(-0.116837\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 1000.00 + 282.843i 1.86502 + 0.527508i
\(67\) −70.0000 −0.127640 −0.0638199 0.997961i \(-0.520328\pi\)
−0.0638199 + 0.997961i \(0.520328\pi\)
\(68\) 859.842i 1.53340i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 320.000 520.431i 0.523783 0.851852i
\(73\) −430.000 −0.689420 −0.344710 0.938709i \(-0.612023\pi\)
−0.344710 + 0.938709i \(0.612023\pi\)
\(74\) 0 0
\(75\) −625.000 176.777i −0.962250 0.272166i
\(76\) 848.000 1.27990
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 329.000 + 650.538i 0.451303 + 0.892371i
\(82\) −160.000 −0.215476
\(83\) 681.651i 0.901457i −0.892661 0.450728i \(-0.851164\pi\)
0.892661 0.450728i \(-0.148836\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 820.244i 1.02848i
\(87\) 0 0
\(88\) −1600.00 −1.93819
\(89\) 1329.36i 1.58328i 0.610988 + 0.791640i \(0.290773\pi\)
−0.610988 + 0.791640i \(0.709227\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −256.000 + 905.097i −0.272166 + 0.962250i
\(97\) 1910.00 1.99929 0.999645 0.0266459i \(-0.00848265\pi\)
0.999645 + 0.0266459i \(0.00848265\pi\)
\(98\) 970.151i 1.00000i
\(99\) 1000.00 1626.35i 1.01519 1.65105i
\(100\) 1000.00 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −1520.00 429.921i −1.47551 0.417338i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1405.73i 1.27006i 0.772486 + 0.635032i \(0.219013\pi\)
−0.772486 + 0.635032i \(0.780987\pi\)
\(108\) −760.000 825.901i −0.677139 0.735855i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2387.19i 1.98733i −0.112387 0.993665i \(-0.535850\pi\)
0.112387 0.993665i \(-0.464150\pi\)
\(114\) 424.000 1499.07i 0.348344 1.23158i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 920.000 0.717736
\(119\) 0 0
\(120\) 0 0
\(121\) −3669.00 −2.75657
\(122\) 0 0
\(123\) −80.0000 + 282.843i −0.0586452 + 0.207342i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1448.15i 1.00000i
\(129\) 1450.00 + 410.122i 0.989654 + 0.279916i
\(130\) 0 0
\(131\) 2729.43i 1.82039i 0.414176 + 0.910197i \(0.364070\pi\)
−0.414176 + 0.910197i \(0.635930\pi\)
\(132\) −800.000 + 2828.43i −0.527508 + 1.86502i
\(133\) 0 0
\(134\) 197.990i 0.127640i
\(135\) 0 0
\(136\) 2432.00 1.53340
\(137\) 2285.37i 1.42520i −0.701571 0.712599i \(-0.747518\pi\)
0.701571 0.712599i \(-0.252482\pi\)
\(138\) 0 0
\(139\) −1474.00 −0.899446 −0.449723 0.893168i \(-0.648477\pi\)
−0.449723 + 0.893168i \(0.648477\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1472.00 + 905.097i 0.851852 + 0.523783i
\(145\) 0 0
\(146\) 1216.22i 0.689420i
\(147\) 1715.00 + 485.075i 0.962250 + 0.272166i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 500.000 1767.77i 0.272166 0.962250i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 2398.51i 1.27990i
\(153\) −1520.00 + 2472.05i −0.803168 + 1.30623i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −1840.00 + 930.553i −0.892371 + 0.451303i
\(163\) −970.000 −0.466112 −0.233056 0.972463i \(-0.574873\pi\)
−0.233056 + 0.972463i \(0.574873\pi\)
\(164\) 452.548i 0.215476i
\(165\) 0 0
\(166\) 1928.00 0.901457
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) −2438.00 1499.07i −1.09028 0.670389i
\(172\) −2320.00 −1.02848
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4525.48i 1.93819i
\(177\) 460.000 1626.35i 0.195343 0.690642i
\(178\) −3760.00 −1.58328
\(179\) 2870.85i 1.19876i −0.800465 0.599379i \(-0.795414\pi\)
0.800465 0.599379i \(-0.204586\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7600.00 2.97202
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −2560.00 724.077i −0.962250 0.272166i
\(193\) 2090.00 0.779490 0.389745 0.920923i \(-0.372563\pi\)
0.389745 + 0.920923i \(0.372563\pi\)
\(194\) 5402.30i 1.99929i
\(195\) 0 0
\(196\) −2744.00 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 4600.00 + 2828.43i 1.65105 + 1.01519i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 2828.43i 1.00000i
\(201\) −350.000 98.9949i −0.122821 0.0347391i
\(202\) 0 0
\(203\) 0 0
\(204\) 1216.00 4299.21i 0.417338 1.47551i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7495.33i 2.48068i
\(210\) 0 0
\(211\) −6118.00 −1.99612 −0.998058 0.0622910i \(-0.980159\pi\)
−0.998058 + 0.0622910i \(0.980159\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −3976.00 −1.27006
\(215\) 0 0
\(216\) 2336.00 2149.60i 0.735855 0.677139i
\(217\) 0 0
\(218\) 0 0
\(219\) −2150.00 608.112i −0.663395 0.187636i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −2875.00 1767.77i −0.851852 0.523783i
\(226\) 6752.00 1.98733
\(227\) 1903.53i 0.556572i −0.960498 0.278286i \(-0.910234\pi\)
0.960498 0.278286i \(-0.0897663\pi\)
\(228\) 4240.00 + 1199.25i 1.23158 + 0.348344i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3773.12i 1.06088i 0.847722 + 0.530441i \(0.177974\pi\)
−0.847722 + 0.530441i \(0.822026\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2602.15i 0.717736i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1222.00 −0.326622 −0.163311 0.986575i \(-0.552217\pi\)
−0.163311 + 0.986575i \(0.552217\pi\)
\(242\) 10377.5i 2.75657i
\(243\) 725.000 + 3717.97i 0.191394 + 0.981513i
\(244\) 0 0
\(245\) 0 0
\(246\) −800.000 226.274i −0.207342 0.0586452i
\(247\) 0 0
\(248\) 0 0
\(249\) 964.000 3408.25i 0.245345 0.867427i
\(250\) 0 0
\(251\) 6689.23i 1.68215i −0.540916 0.841077i \(-0.681922\pi\)
0.540916 0.841077i \(-0.318078\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 7274.71i 1.76570i −0.469658 0.882849i \(-0.655623\pi\)
0.469658 0.882849i \(-0.344377\pi\)
\(258\) −1160.00 + 4101.22i −0.279916 + 0.989654i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −7720.00 −1.82039
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −8000.00 2262.74i −1.86502 0.527508i
\(265\) 0 0
\(266\) 0 0
\(267\) −1880.00 + 6646.80i −0.430914 + 1.52351i
\(268\) 560.000 0.127640
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 6878.73i 1.53340i
\(273\) 0 0
\(274\) 6464.00 1.42520
\(275\) 8838.83i 1.93819i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 4169.10i 0.899446i
\(279\) 0 0
\(280\) 0 0
\(281\) 1216.22i 0.258199i −0.991632 0.129099i \(-0.958791\pi\)
0.991632 0.129099i \(-0.0412086\pi\)
\(282\) 0 0
\(283\) 8030.00 1.68669 0.843346 0.537371i \(-0.180582\pi\)
0.843346 + 0.537371i \(0.180582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −2560.00 + 4163.44i −0.523783 + 0.851852i
\(289\) −6639.00 −1.35131
\(290\) 0 0
\(291\) 9550.00 + 2701.15i 1.92382 + 0.544138i
\(292\) 3440.00 0.689420
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −1372.00 + 4850.75i −0.272166 + 0.962250i
\(295\) 0 0
\(296\) 0 0
\(297\) 7300.00 6717.51i 1.42623 1.31242i
\(298\) 0 0
\(299\) 0 0
\(300\) 5000.00 + 1414.21i 0.962250 + 0.272166i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −6784.00 −1.27990
\(305\) 0 0
\(306\) −6992.00 4299.21i −1.30623 0.803168i
\(307\) −7990.00 −1.48539 −0.742693 0.669632i \(-0.766452\pi\)
−0.742693 + 0.669632i \(0.766452\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8390.00 1.51511 0.757557 0.652769i \(-0.226393\pi\)
0.757557 + 0.652769i \(0.226393\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1988.00 + 7028.64i −0.345668 + 1.22212i
\(322\) 0 0
\(323\) 11392.9i 1.96259i
\(324\) −2632.00 5204.31i −0.451303 0.892371i
\(325\) 0 0
\(326\) 2743.57i 0.466112i
\(327\) 0 0
\(328\) 1280.00 0.215476
\(329\) 0 0
\(330\) 0 0
\(331\) −8242.00 −1.36864 −0.684322 0.729180i \(-0.739902\pi\)
−0.684322 + 0.729180i \(0.739902\pi\)
\(332\) 5453.21i 0.901457i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11410.0 −1.84434 −0.922170 0.386786i \(-0.873585\pi\)
−0.922170 + 0.386786i \(0.873585\pi\)
\(338\) 6214.05i 1.00000i
\(339\) 3376.00 11936.0i 0.540882 1.91231i
\(340\) 0 0
\(341\) 0 0
\(342\) 4240.00 6895.71i 0.670389 1.09028i
\(343\) 0 0
\(344\) 6561.95i 1.02848i
\(345\) 0 0
\(346\) 0 0
\(347\) 11435.3i 1.76911i 0.466437 + 0.884554i \(0.345537\pi\)
−0.466437 + 0.884554i \(0.654463\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 12800.0 1.93819
\(353\) 12377.2i 1.86621i 0.359605 + 0.933104i \(0.382911\pi\)
−0.359605 + 0.933104i \(0.617089\pi\)
\(354\) 4600.00 + 1301.08i 0.690642 + 0.195343i
\(355\) 0 0
\(356\) 10634.9i 1.58328i
\(357\) 0 0
\(358\) 8120.00 1.19876
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 4377.00 0.638140
\(362\) 0 0
\(363\) −18345.0 5188.75i −2.65251 0.750244i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −800.000 + 1301.08i −0.112863 + 0.183554i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 21496.0i 2.97202i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11666.0 1.58111 0.790557 0.612389i \(-0.209791\pi\)
0.790557 + 0.612389i \(0.209791\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 2048.00 7240.77i 0.272166 0.962250i
\(385\) 0 0
\(386\) 5911.41i 0.779490i
\(387\) 6670.00 + 4101.22i 0.876112 + 0.538699i
\(388\) −15280.0 −1.99929
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 7761.20i 1.00000i
\(393\) −3860.00 + 13647.2i −0.495448 + 1.75167i
\(394\) 0 0
\(395\) 0 0
\(396\) −8000.00 + 13010.8i −1.01519 + 1.65105i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8000.00 −1.00000
\(401\) 14453.3i 1.79990i −0.435989 0.899952i \(-0.643601\pi\)
0.435989 0.899952i \(-0.356399\pi\)
\(402\) 280.000 989.949i 0.0347391 0.122821i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 12160.0 + 3439.37i 1.47551 + 0.417338i
\(409\) 16346.0 1.97618 0.988090 0.153877i \(-0.0491758\pi\)
0.988090 + 0.153877i \(0.0491758\pi\)
\(410\) 0 0
\(411\) 3232.00 11426.8i 0.387890 1.37140i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7370.00 2084.55i −0.865493 0.244798i
\(418\) −21200.0 −2.48068
\(419\) 3493.11i 0.407278i 0.979046 + 0.203639i \(0.0652769\pi\)
−0.979046 + 0.203639i \(0.934723\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 17304.3i 1.99612i
\(423\) 0 0
\(424\) 0 0
\(425\) 13435.0i 1.53340i
\(426\) 0 0
\(427\) 0 0
\(428\) 11245.8i 1.27006i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 6080.00 + 6607.21i 0.677139 + 0.735855i
\(433\) 5510.00 0.611533 0.305766 0.952107i \(-0.401087\pi\)
0.305766 + 0.952107i \(0.401087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1720.00 6081.12i 0.187636 0.663395i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 7889.00 + 4850.75i 0.851852 + 0.523783i
\(442\) 0 0
\(443\) 3736.35i 0.400721i −0.979722 0.200361i \(-0.935789\pi\)
0.979722 0.200361i \(-0.0642114\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7438.76i 0.781864i 0.920420 + 0.390932i \(0.127847\pi\)
−0.920420 + 0.390932i \(0.872153\pi\)
\(450\) 5000.00 8131.73i 0.523783 0.851852i
\(451\) 4000.00 0.417633
\(452\) 19097.5i 1.98733i
\(453\) 0 0
\(454\) 5384.00 0.556572
\(455\) 0 0
\(456\) −3392.00 + 11992.5i −0.348344 + 1.23158i
\(457\) −18070.0 −1.84963 −0.924813 0.380422i \(-0.875779\pi\)
−0.924813 + 0.380422i \(0.875779\pi\)
\(458\) 0 0
\(459\) −11096.0 + 10210.6i −1.12836 + 1.03832i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −10672.0 −1.06088
\(467\) 13471.8i 1.33490i 0.744653 + 0.667452i \(0.232615\pi\)
−0.744653 + 0.667452i \(0.767385\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −7360.00 −0.717736
\(473\) 20506.1i 1.99339i
\(474\) 0 0
\(475\) 13250.0 1.27990
\(476\) 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
\(482\) 3456.34i 0.326622i
\(483\) 0 0
\(484\) 29352.0 2.75657
\(485\) 0 0
\(486\) −10516.0 + 2050.61i −0.981513 + 0.191394i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −4850.00 1371.79i −0.448517 0.126860i
\(490\) 0 0
\(491\) 18002.9i 1.65471i 0.561681 + 0.827354i \(0.310155\pi\)
−0.561681 + 0.827354i \(0.689845\pi\)
\(492\) 640.000 2262.74i 0.0586452 0.207342i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 9640.00 + 2726.60i 0.867427 + 0.245345i
\(499\) −18214.0 −1.63401 −0.817005 0.576631i \(-0.804367\pi\)
−0.817005 + 0.576631i \(0.804367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18920.0 1.68215
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 10985.0 + 3107.03i 0.962250 + 0.272166i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11585.2i 1.00000i
\(513\) −10070.0 10943.2i −0.866669 0.941819i
\(514\) 20576.0 1.76570
\(515\) 0 0
\(516\) −11600.0 3280.98i −0.989654 0.279916i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21948.6i 1.84565i 0.385215 + 0.922827i \(0.374127\pi\)
−0.385215 + 0.922827i \(0.625873\pi\)
\(522\) 0 0
\(523\) −4750.00 −0.397138 −0.198569 0.980087i \(-0.563629\pi\)
−0.198569 + 0.980087i \(0.563629\pi\)
\(524\) 21835.5i 1.82039i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 6400.00 22627.4i 0.527508 1.86502i
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 4600.00 7481.19i 0.375938 0.611405i
\(532\) 0 0
\(533\) 0 0
\(534\) −18800.0 5317.44i −1.52351 0.430914i
\(535\) 0 0
\(536\) 1583.92i 0.127640i
\(537\) 4060.00 14354.3i 0.326261 1.15351i
\(538\) 0 0
\(539\) 24253.8i 1.93819i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −19456.0 −1.53340
\(545\) 0 0
\(546\) 0 0
\(547\) −21850.0 −1.70793 −0.853966 0.520329i \(-0.825809\pi\)
−0.853966 + 0.520329i \(0.825809\pi\)
\(548\) 18283.0i 1.42520i
\(549\) 0 0
\(550\) −25000.0 −1.93819
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 11792.0 0.899446
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 38000.0 + 10748.0i 2.85982 + 0.808880i
\(562\) 3440.00 0.258199
\(563\) 12391.3i 0.927589i −0.885943 0.463795i \(-0.846488\pi\)
0.885943 0.463795i \(-0.153512\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22712.3i 1.68669i
\(567\) 0 0
\(568\) 0 0
\(569\) 27039.8i 1.99221i 0.0881913 + 0.996104i \(0.471891\pi\)
−0.0881913 + 0.996104i \(0.528109\pi\)
\(570\) 0 0
\(571\) 27038.0 1.98162 0.990810 0.135261i \(-0.0431872\pi\)
0.990810 + 0.135261i \(0.0431872\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −11776.0 7240.77i −0.851852 0.523783i
\(577\) 19550.0 1.41053 0.705266 0.708943i \(-0.250827\pi\)
0.705266 + 0.708943i \(0.250827\pi\)
\(578\) 18777.9i 1.35131i
\(579\) 10450.0 + 2955.71i 0.750064 + 0.212150i
\(580\) 0 0
\(581\) 0 0
\(582\) −7640.00 + 27011.5i −0.544138 + 1.92382i
\(583\) 0 0
\(584\) 9729.79i 0.689420i
\(585\) 0 0
\(586\) 0 0
\(587\) 26493.9i 1.86289i −0.363876 0.931447i \(-0.618547\pi\)
0.363876 0.931447i \(-0.381453\pi\)
\(588\) −13720.0 3880.60i −0.962250 0.272166i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12173.6i 0.843015i 0.906825 + 0.421507i \(0.138499\pi\)
−0.906825 + 0.421507i \(0.861501\pi\)
\(594\) 19000.0 + 20647.5i 1.31242 + 1.42623i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −4000.00 + 14142.1i −0.272166 + 0.962250i
\(601\) −14398.0 −0.977216 −0.488608 0.872503i \(-0.662495\pi\)
−0.488608 + 0.872503i \(0.662495\pi\)
\(602\) 0 0
\(603\) −1610.00 989.949i −0.108730 0.0668555i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 19188.0i 1.27990i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 12160.0 19776.4i 0.803168 1.30623i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 22599.1i 1.48539i
\(615\) 0 0
\(616\) 0 0
\(617\) 11206.2i 0.731192i 0.930774 + 0.365596i \(0.119135\pi\)
−0.930774 + 0.365596i \(0.880865\pi\)
\(618\) 0 0
\(619\) −30706.0 −1.99383 −0.996913 0.0785136i \(-0.974983\pi\)
−0.996913 + 0.0785136i \(0.974983\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 23730.5i 1.51511i
\(627\) −10600.0 + 37476.7i −0.675157 + 2.38704i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −30590.0 8652.16i −1.92076 0.543274i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31763.2i 1.95721i −0.205745 0.978606i \(-0.565962\pi\)
0.205745 0.978606i \(-0.434038\pi\)
\(642\) −19880.0 5622.91i −1.22212 0.345668i
\(643\) 28550.0 1.75101 0.875507 0.483205i \(-0.160528\pi\)
0.875507 + 0.483205i \(0.160528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 32224.0 1.96259
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 14720.0 7444.42i 0.892371 0.451303i
\(649\) −23000.0 −1.39111
\(650\) 0 0
\(651\) 0 0
\(652\) 7760.00 0.466112
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3620.39i 0.215476i
\(657\) −9890.00 6081.12i −0.587284 0.361107i
\(658\) 0 0
\(659\) 16107.9i 0.952161i −0.879402 0.476081i \(-0.842057\pi\)
0.879402 0.476081i \(-0.157943\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 23311.9i 1.36864i
\(663\) 0 0
\(664\) −15424.0 −0.901457
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 19190.0 1.09914 0.549569 0.835448i \(-0.314792\pi\)
0.549569 + 0.835448i \(0.314792\pi\)
\(674\) 32272.4i 1.84434i
\(675\) −11875.0 12904.7i −0.677139 0.735855i
\(676\) −17576.0 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 33760.0 + 9548.77i 1.91231 + 0.540882i
\(679\) 0 0
\(680\) 0 0
\(681\) 2692.00 9517.66i 0.151480 0.535562i
\(682\) 0 0
\(683\) 33632.8i 1.88422i 0.335300 + 0.942112i \(0.391162\pi\)
−0.335300 + 0.942112i \(0.608838\pi\)
\(684\) 19504.0 + 11992.5i 1.09028 + 0.670389i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 18560.0 1.02848
\(689\) 0 0
\(690\) 0 0
\(691\) −1978.00 −0.108895 −0.0544477 0.998517i \(-0.517340\pi\)
−0.0544477 + 0.998517i \(0.517340\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −32344.0 −1.76911
\(695\) 0 0
\(696\) 0 0
\(697\) −6080.00 −0.330411
\(698\) 0 0
\(699\) −5336.00 + 18865.6i −0.288735 + 1.02083i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 36203.9i 1.93819i
\(705\) 0 0
\(706\) −35008.0 −1.86621
\(707\) 0 0
\(708\) −3680.00 + 13010.8i −0.195343 + 0.690642i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 30080.0 1.58328
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 22966.8i 1.19876i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12380.0i 0.638140i
\(723\) −6110.00 1728.17i −0.314292 0.0888953i
\(724\) 0 0
\(725\) 0 0
\(726\) 14676.0 51887.5i 0.750244 2.65251i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1633.00 + 19615.1i −0.0829650 + 0.996552i
\(730\) 0 0
\(731\) 31169.3i 1.57707i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4949.75i 0.247390i
\(738\) −3680.00 2262.74i −0.183554 0.112863i
\(739\) 36074.0 1.79567 0.897837 0.440327i \(-0.145138\pi\)
0.897837 + 0.440327i \(0.145138\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9640.00 15678.0i 0.472168 0.767908i
\(748\) −60800.0 −2.97202
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 9460.00 33446.2i 0.457824 1.61865i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 32996.4i 1.58111i
\(759\) 0 0
\(760\) 0 0
\(761\) 24381.0i 1.16138i −0.814124 0.580691i \(-0.802782\pi\)
0.814124 0.580691i \(-0.197218\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 20480.0 + 5792.62i 0.962250 + 0.272166i
\(769\) 40106.0 1.88070 0.940351 0.340207i \(-0.110497\pi\)
0.940351 + 0.340207i \(0.110497\pi\)
\(770\) 0 0
\(771\) 10288.0 36373.6i 0.480562 1.69904i
\(772\) −16720.0 −0.779490
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −11600.0 + 18865.6i −0.538699 + 0.876112i
\(775\) 0 0
\(776\) 43218.4i 1.99929i
\(777\) 0 0
\(778\) 0 0
\(779\) 5996.27i 0.275788i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 21952.0 1.00000
\(785\) 0 0
\(786\) −38600.0 10917.7i −1.75167 0.495448i
\(787\) 6950.00 0.314791 0.157396 0.987536i \(-0.449690\pi\)
0.157396 + 0.987536i \(0.449690\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −36800.0 22627.4i −1.65105 1.01519i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0