Properties

Label 24.2.f.a.11.1
Level $24$
Weight $2$
Character 24.11
Analytic conductor $0.192$
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] = [24,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(0.191640964851\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 11.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 24.11
Dual form 24.2.f.a.11.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{2} +(-1.00000 + 1.41421i) q^{3} -2.00000 q^{4} +(2.00000 + 1.41421i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\) \(q-1.41421i q^{2} +(-1.00000 + 1.41421i) q^{3} -2.00000 q^{4} +(2.00000 + 1.41421i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} -2.82843i q^{11} +(2.00000 - 2.82843i) q^{12} +4.00000 q^{16} +5.65685i q^{17} +(-4.00000 + 1.41421i) q^{18} +2.00000 q^{19} -4.00000 q^{22} +(-4.00000 - 2.82843i) q^{24} -5.00000 q^{25} +(5.00000 + 1.41421i) q^{27} -5.65685i q^{32} +(4.00000 + 2.82843i) q^{33} +8.00000 q^{34} +(2.00000 + 5.65685i) q^{36} -2.82843i q^{38} -11.3137i q^{41} -10.0000 q^{43} +5.65685i q^{44} +(-4.00000 + 5.65685i) q^{48} +7.00000 q^{49} +7.07107i q^{50} +(-8.00000 - 5.65685i) q^{51} +(2.00000 - 7.07107i) q^{54} +(-2.00000 + 2.82843i) q^{57} +14.1421i q^{59} -8.00000 q^{64} +(4.00000 - 5.65685i) q^{66} +14.0000 q^{67} -11.3137i q^{68} +(8.00000 - 2.82843i) q^{72} +2.00000 q^{73} +(5.00000 - 7.07107i) q^{75} -4.00000 q^{76} +(-7.00000 + 5.65685i) q^{81} -16.0000 q^{82} -2.82843i q^{83} +14.1421i q^{86} +8.00000 q^{88} +5.65685i q^{89} +(8.00000 + 5.65685i) q^{96} -10.0000 q^{97} -9.89949i q^{98} +(-8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{12} + 8 q^{16} - 8 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} - 10 q^{25} + 10 q^{27} + 8 q^{33} + 16 q^{34} + 4 q^{36} - 20 q^{43} - 8 q^{48} + 14 q^{49} - 16 q^{51} + 4 q^{54} - 4 q^{57} - 16 q^{64} + 8 q^{66} + 28 q^{67} + 16 q^{72} + 4 q^{73} + 10 q^{75} - 8 q^{76} - 14 q^{81} - 32 q^{82} + 16 q^{88} + 16 q^{96} - 20 q^{97} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) −1.00000 + 1.41421i −0.577350 + 0.816497i
\(4\) −2.00000 −1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 + 1.41421i 0.816497 + 0.577350i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −1.00000 2.82843i −0.333333 0.942809i
\(10\) 0 0
\(11\) 2.82843i 0.852803i −0.904534 0.426401i \(-0.859781\pi\)
0.904534 0.426401i \(-0.140219\pi\)
\(12\) 2.00000 2.82843i 0.577350 0.816497i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 5.65685i 1.37199i 0.727607 + 0.685994i \(0.240633\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −4.00000 + 1.41421i −0.942809 + 0.333333i
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −4.00000 2.82843i −0.816497 0.577350i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 5.00000 + 1.41421i 0.962250 + 0.272166i
\(28\) 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 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 4.00000 + 2.82843i 0.696311 + 0.492366i
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 2.00000 + 5.65685i 0.333333 + 0.942809i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 2.82843i 0.458831i
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3137i 1.76690i −0.468521 0.883452i \(-0.655213\pi\)
0.468521 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.00000 + 5.65685i −0.577350 + 0.816497i
\(49\) 7.00000 1.00000
\(50\) 7.07107i 1.00000i
\(51\) −8.00000 5.65685i −1.12022 0.792118i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 2.00000 7.07107i 0.272166 0.962250i
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 + 2.82843i −0.264906 + 0.374634i
\(58\) 0 0
\(59\) 14.1421i 1.84115i 0.390567 + 0.920575i \(0.372279\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 4.00000 5.65685i 0.492366 0.696311i
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 11.3137i 1.37199i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 8.00000 2.82843i 0.942809 0.333333i
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 5.00000 7.07107i 0.577350 0.816497i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) −16.0000 −1.76690
\(83\) 2.82843i 0.310460i −0.987878 0.155230i \(-0.950388\pi\)
0.987878 0.155230i \(-0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.1421i 1.52499i
\(87\) 0 0
\(88\) 8.00000 0.852803
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 8.00000 + 5.65685i 0.816497 + 0.577350i
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.89949i 1.00000i
\(99\) −8.00000 + 2.82843i −0.804030 + 0.284268i
\(100\) 10.0000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −8.00000 + 11.3137i −0.792118 + 1.12022i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.7990i 1.91404i −0.290021 0.957020i \(-0.593662\pi\)
0.290021 0.957020i \(-0.406338\pi\)
\(108\) −10.0000 2.82843i −0.962250 0.272166i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.3137i 1.06430i −0.846649 0.532152i \(-0.821383\pi\)
0.846649 0.532152i \(-0.178617\pi\)
\(114\) 4.00000 + 2.82843i 0.374634 + 0.264906i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 20.0000 1.84115
\(119\) 0 0
\(120\) 0 0
\(121\) 3.00000 0.272727
\(122\) 0 0
\(123\) 16.0000 + 11.3137i 1.44267 + 1.02012i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 10.0000 14.1421i 0.880451 1.24515i
\(130\) 0 0
\(131\) 14.1421i 1.23560i 0.786334 + 0.617802i \(0.211977\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −8.00000 5.65685i −0.696311 0.492366i
\(133\) 0 0
\(134\) 19.7990i 1.71037i
\(135\) 0 0
\(136\) −16.0000 −1.37199
\(137\) 22.6274i 1.93319i 0.256307 + 0.966595i \(0.417494\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −4.00000 11.3137i −0.333333 0.942809i
\(145\) 0 0
\(146\) 2.82843i 0.234082i
\(147\) −7.00000 + 9.89949i −0.577350 + 0.816497i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −10.0000 7.07107i −0.816497 0.577350i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 5.65685i 0.458831i
\(153\) 16.0000 5.65685i 1.29352 0.457330i
\(154\) 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
\(162\) 8.00000 + 9.89949i 0.628539 + 0.777778i
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 22.6274i 1.76690i
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −2.00000 5.65685i −0.152944 0.432590i
\(172\) 20.0000 1.52499
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.3137i 0.852803i
\(177\) −20.0000 14.1421i −1.50329 1.06299i
\(178\) 8.00000 0.599625
\(179\) 19.7990i 1.47985i −0.672692 0.739923i \(-0.734862\pi\)
0.672692 0.739923i \(-0.265138\pi\)
\(180\) 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
\(187\) 16.0000 1.17004
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 8.00000 11.3137i 0.577350 0.816497i
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 14.1421i 1.01535i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 4.00000 + 11.3137i 0.284268 + 0.804030i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 14.1421i 1.00000i
\(201\) −14.0000 + 19.7990i −0.987484 + 1.39651i
\(202\) 0 0
\(203\) 0 0
\(204\) 16.0000 + 11.3137i 1.12022 + 0.792118i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.65685i 0.391293i
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −28.0000 −1.91404
\(215\) 0 0
\(216\) −4.00000 + 14.1421i −0.272166 + 0.962250i
\(217\) 0 0
\(218\) 0 0
\(219\) −2.00000 + 2.82843i −0.135147 + 0.191127i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 5.00000 + 14.1421i 0.333333 + 0.942809i
\(226\) −16.0000 −1.06430
\(227\) 2.82843i 0.187729i −0.995585 0.0938647i \(-0.970078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) 4.00000 5.65685i 0.264906 0.374634i
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.65685i 0.370593i 0.982683 + 0.185296i \(0.0593245\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 28.2843i 1.84115i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 4.24264i 0.272727i
\(243\) −1.00000 15.5563i −0.0641500 0.997940i
\(244\) 0 0
\(245\) 0 0
\(246\) 16.0000 22.6274i 1.02012 1.44267i
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 + 2.82843i 0.253490 + 0.179244i
\(250\) 0 0
\(251\) 31.1127i 1.96382i 0.189358 + 0.981908i \(0.439359\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 11.3137i 0.705730i −0.935674 0.352865i \(-0.885208\pi\)
0.935674 0.352865i \(-0.114792\pi\)
\(258\) −20.0000 14.1421i −1.24515 0.880451i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −8.00000 + 11.3137i −0.492366 + 0.696311i
\(265\) 0 0
\(266\) 0 0
\(267\) −8.00000 5.65685i −0.489592 0.346194i
\(268\) −28.0000 −1.71037
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 22.6274i 1.37199i
\(273\) 0 0
\(274\) 32.0000 1.93319
\(275\) 14.1421i 0.852803i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 31.1127i 1.86602i
\(279\) 0 0
\(280\) 0 0
\(281\) 28.2843i 1.68730i −0.536895 0.843649i \(-0.680403\pi\)
0.536895 0.843649i \(-0.319597\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −16.0000 + 5.65685i −0.942809 + 0.333333i
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 10.0000 14.1421i 0.586210 0.829027i
\(292\) −4.00000 −0.234082
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 14.0000 + 9.89949i 0.816497 + 0.577350i
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 14.1421i 0.232104 0.820610i
\(298\) 0 0
\(299\) 0 0
\(300\) −10.0000 + 14.1421i −0.577350 + 0.816497i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −8.00000 22.6274i −0.457330 1.29352i
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 28.0000 + 19.7990i 1.56281 + 1.10507i
\(322\) 0 0
\(323\) 11.3137i 0.629512i
\(324\) 14.0000 11.3137i 0.777778 0.628539i
\(325\) 0 0
\(326\) 2.82843i 0.156652i
\(327\) 0 0
\(328\) 32.0000 1.76690
\(329\) 0 0
\(330\) 0 0
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) 5.65685i 0.310460i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 16.0000 + 11.3137i 0.869001 + 0.614476i
\(340\) 0 0
\(341\) 0 0
\(342\) −8.00000 + 2.82843i −0.432590 + 0.152944i
\(343\) 0 0
\(344\) 28.2843i 1.52499i
\(345\) 0 0
\(346\) 0 0
\(347\) 36.7696i 1.97389i −0.161048 0.986947i \(-0.551488\pi\)
0.161048 0.986947i \(-0.448512\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −16.0000 −0.852803
\(353\) 22.6274i 1.20434i 0.798369 + 0.602168i \(0.205696\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −20.0000 + 28.2843i −1.06299 + 1.50329i
\(355\) 0 0
\(356\) 11.3137i 0.599625i
\(357\) 0 0
\(358\) −28.0000 −1.47985
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −3.00000 + 4.24264i −0.157459 + 0.222681i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −32.0000 + 11.3137i −1.66585 + 0.588968i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 22.6274i 1.17004i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 38.0000 1.95193 0.975964 0.217930i \(-0.0699304\pi\)
0.975964 + 0.217930i \(0.0699304\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −16.0000 11.3137i −0.816497 0.577350i
\(385\) 0 0
\(386\) 31.1127i 1.58359i
\(387\) 10.0000 + 28.2843i 0.508329 + 1.43777i
\(388\) 20.0000 1.01535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 19.7990i 1.00000i
\(393\) −20.0000 14.1421i −1.00887 0.713376i
\(394\) 0 0
\(395\) 0 0
\(396\) 16.0000 5.65685i 0.804030 0.284268i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) 39.5980i 1.97743i 0.149813 + 0.988714i \(0.452133\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 28.0000 + 19.7990i 1.39651 + 0.987484i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 16.0000 22.6274i 0.792118 1.12022i
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −32.0000 22.6274i −1.57844 1.11613i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.0000 31.1127i 1.07734 1.52360i
\(418\) −8.00000 −0.391293
\(419\) 36.7696i 1.79631i −0.439679 0.898155i \(-0.644908\pi\)
0.439679 0.898155i \(-0.355092\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 19.7990i 0.963800i
\(423\) 0 0
\(424\) 0 0
\(425\) 28.2843i 1.37199i
\(426\) 0 0
\(427\) 0 0
\(428\) 39.5980i 1.91404i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 20.0000 + 5.65685i 0.962250 + 0.272166i
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 4.00000 + 2.82843i 0.191127 + 0.135147i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −7.00000 19.7990i −0.333333 0.942809i
\(442\) 0 0
\(443\) 2.82843i 0.134383i −0.997740 0.0671913i \(-0.978596\pi\)
0.997740 0.0671913i \(-0.0214038\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.65685i 0.266963i 0.991051 + 0.133482i \(0.0426157\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) 20.0000 7.07107i 0.942809 0.333333i
\(451\) −32.0000 −1.50682
\(452\) 22.6274i 1.06430i
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −8.00000 5.65685i −0.374634 0.264906i
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 0 0
\(459\) −8.00000 + 28.2843i −0.373408 + 1.32020i
\(460\) 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
\(466\) 8.00000 0.370593
\(467\) 31.1127i 1.43972i 0.694117 + 0.719862i \(0.255795\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −40.0000 −1.84115
\(473\) 28.2843i 1.30051i
\(474\) 0 0
\(475\) −10.0000 −0.458831
\(476\) 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
\(482\) 36.7696i 1.67481i
\(483\) 0 0
\(484\) −6.00000 −0.272727
\(485\) 0 0
\(486\) −22.0000 + 1.41421i −0.997940 + 0.0641500i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −2.00000 + 2.82843i −0.0904431 + 0.127906i
\(490\) 0 0
\(491\) 14.1421i 0.638226i 0.947717 + 0.319113i \(0.103385\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) −32.0000 22.6274i −1.44267 1.02012i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 5.65685i 0.179244 0.253490i
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 44.0000 1.96382
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.0000 + 18.3848i −0.577350 + 0.816497i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 10.0000 + 2.82843i 0.441511 + 0.124878i
\(514\) −16.0000 −0.705730
\(515\) 0 0
\(516\) −20.0000 + 28.2843i −0.880451 + 1.24515i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 45.2548i 1.98265i −0.131432 0.991325i \(-0.541958\pi\)
0.131432 0.991325i \(-0.458042\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 28.2843i 1.23560i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 16.0000 + 11.3137i 0.696311 + 0.492366i
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 40.0000 14.1421i 1.73585 0.613716i
\(532\) 0 0
\(533\) 0 0
\(534\) −8.00000 + 11.3137i −0.346194 + 0.489592i
\(535\) 0 0
\(536\) 39.5980i 1.71037i
\(537\) 28.0000 + 19.7990i 1.20829 + 0.854389i
\(538\) 0 0
\(539\) 19.7990i 0.852803i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 32.0000 1.37199
\(545\) 0 0
\(546\) 0 0
\(547\) −46.0000 −1.96682 −0.983409 0.181402i \(-0.941936\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 45.2548i 1.93319i
\(549\) 0 0
\(550\) 20.0000 0.852803
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 44.0000 1.86602
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −16.0000 + 22.6274i −0.675521 + 0.955330i
\(562\) −40.0000 −1.68730
\(563\) 36.7696i 1.54965i −0.632175 0.774826i \(-0.717837\pi\)
0.632175 0.774826i \(-0.282163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 31.1127i 1.30776i
\(567\) 0 0
\(568\) 0 0
\(569\) 22.6274i 0.948591i 0.880366 + 0.474295i \(0.157297\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 8.00000 + 22.6274i 0.333333 + 0.942809i
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 21.2132i 0.882353i
\(579\) 22.0000 31.1127i 0.914289 1.29300i
\(580\) 0 0
\(581\) 0 0
\(582\) −20.0000 14.1421i −0.829027 0.586210i
\(583\) 0 0
\(584\) 5.65685i 0.234082i
\(585\) 0 0
\(586\) 0 0
\(587\) 48.0833i 1.98461i 0.123823 + 0.992304i \(0.460484\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(588\) 14.0000 19.7990i 0.577350 0.816497i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 45.2548i 1.85839i −0.369586 0.929197i \(-0.620500\pi\)
0.369586 0.929197i \(-0.379500\pi\)
\(594\) −20.0000 5.65685i −0.820610 0.232104i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 20.0000 + 14.1421i 0.816497 + 0.577350i
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) −14.0000 39.5980i −0.570124 1.61255i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 11.3137i 0.458831i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −32.0000 + 11.3137i −1.29352 + 0.457330i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 48.0833i 1.94048i
\(615\) 0 0
\(616\) 0 0
\(617\) 39.5980i 1.59415i 0.603877 + 0.797077i \(0.293622\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 14.1421i 0.565233i
\(627\) 8.00000 + 5.65685i 0.319489 + 0.225913i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −14.0000 + 19.7990i −0.556450 + 0.786939i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.2843i 1.11716i −0.829450 0.558581i \(-0.811346\pi\)
0.829450 0.558581i \(-0.188654\pi\)
\(642\) 28.0000 39.5980i 1.10507 1.56281i
\(643\) 50.0000 1.97181 0.985904 0.167313i \(-0.0535092\pi\)
0.985904 + 0.167313i \(0.0535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −16.0000 19.7990i −0.628539 0.777778i
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 45.2548i 1.76690i
\(657\) −2.00000 5.65685i −0.0780274 0.220695i
\(658\) 0 0
\(659\) 48.0833i 1.87306i 0.350590 + 0.936529i \(0.385981\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 36.7696i 1.42909i
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 19.7990i 0.762629i
\(675\) −25.0000 7.07107i −0.962250 0.272166i
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 16.0000 22.6274i 0.614476 0.869001i
\(679\) 0 0
\(680\) 0 0
\(681\) 4.00000 + 2.82843i 0.153280 + 0.108386i
\(682\) 0 0
\(683\) 31.1127i 1.19049i 0.803543 + 0.595247i \(0.202946\pi\)
−0.803543 + 0.595247i \(0.797054\pi\)
\(684\) 4.00000 + 11.3137i 0.152944 + 0.432590i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −52.0000 −1.97389
\(695\) 0 0
\(696\) 0 0
\(697\) 64.0000 2.42417
\(698\) 0 0
\(699\) −8.00000 5.65685i −0.302588 0.213962i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 22.6274i 0.852803i
\(705\) 0 0
\(706\) 32.0000 1.20434
\(707\) 0 0
\(708\) 40.0000 + 28.2843i 1.50329 + 1.06299i
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −16.0000 −0.599625
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 39.5980i 1.47985i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 21.2132i 0.789474i
\(723\) −26.0000 + 36.7696i −0.966950 + 1.36747i
\(724\) 0 0
\(725\) 0 0
\(726\) 6.00000 + 4.24264i 0.222681 + 0.157459i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(730\) 0 0
\(731\) 56.5685i 2.09226i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.5980i 1.45861i
\(738\) 16.0000 + 45.2548i 0.588968 + 1.66585i
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 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
\(747\) −8.00000 + 2.82843i −0.292705 + 0.103487i
\(748\) −32.0000 −1.17004
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −44.0000 31.1127i −1.60345 1.13381i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 53.7401i 1.95193i
\(759\) 0 0
\(760\) 0 0
\(761\) 11.3137i 0.410122i −0.978749 0.205061i \(-0.934261\pi\)
0.978749 0.205061i \(-0.0657392\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 + 22.6274i −0.577350 + 0.816497i
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 16.0000 + 11.3137i 0.576226 + 0.407453i
\(772\) 44.0000 1.58359
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 40.0000 14.1421i 1.43777 0.508329i
\(775\) 0 0
\(776\) 28.2843i 1.01535i
\(777\) 0 0
\(778\) 0 0
\(779\) 22.6274i 0.810711i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) −20.0000 + 28.2843i −0.713376 + 1.00887i
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −8.00000 22.6274i −0.284268 0.804030i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0