Properties

Label 1368.2.f.b.1025.1
Level $1368$
Weight $2$
Character 1368.1025
Analytic conductor $10.924$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1368,2,Mod(1025,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.1025");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.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: \(10.9235349965\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 1368.1025
Dual form 1368.2.f.b.1025.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-1.41421i q^{5} +2.00000 q^{7} +4.24264i q^{11} -2.82843i q^{13} +7.07107i q^{17} +(-1.00000 + 4.24264i) q^{19} +1.41421i q^{23} +3.00000 q^{25} +10.0000 q^{29} -2.82843i q^{31} -2.82843i q^{35} -5.65685i q^{37} -10.0000 q^{41} +12.0000 q^{43} -1.41421i q^{47} -3.00000 q^{49} -10.0000 q^{53} +6.00000 q^{55} +12.0000 q^{59} +8.00000 q^{61} -4.00000 q^{65} +14.1421i q^{67} +8.00000 q^{71} +6.00000 q^{73} +8.48528i q^{77} -11.3137i q^{79} +4.24264i q^{83} +10.0000 q^{85} -6.00000 q^{89} -5.65685i q^{91} +(6.00000 + 1.41421i) q^{95} -8.48528i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} - 2 q^{19} + 6 q^{25} + 20 q^{29} - 20 q^{41} + 24 q^{43} - 6 q^{49} - 20 q^{53} + 12 q^{55} + 24 q^{59} + 16 q^{61} - 8 q^{65} + 16 q^{71} + 12 q^{73} + 20 q^{85} - 12 q^{89} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.07107i 1.71499i 0.514496 + 0.857493i \(0.327979\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) −1.00000 + 4.24264i −0.229416 + 0.973329i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.82843i 0.508001i −0.967204 0.254000i \(-0.918254\pi\)
0.967204 0.254000i \(-0.0817464\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843i 0.478091i
\(36\) 0 0
\(37\) 5.65685i 0.929981i −0.885316 0.464991i \(-0.846058\pi\)
0.885316 0.464991i \(-0.153942\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41421i 0.206284i −0.994667 0.103142i \(-0.967110\pi\)
0.994667 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 14.1421i 1.72774i 0.503718 + 0.863868i \(0.331965\pi\)
−0.503718 + 0.863868i \(0.668035\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) 11.3137i 1.27289i −0.771321 0.636446i \(-0.780404\pi\)
0.771321 0.636446i \(-0.219596\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.24264i 0.465690i 0.972514 + 0.232845i \(0.0748035\pi\)
−0.972514 + 0.232845i \(0.925196\pi\)
\(84\) 0 0
\(85\) 10.0000 1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 5.65685i 0.592999i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00000 + 1.41421i 0.615587 + 0.145095i
\(96\) 0 0
\(97\) 8.48528i 0.861550i −0.902459 0.430775i \(-0.858240\pi\)
0.902459 0.430775i \(-0.141760\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.89949i 0.985037i 0.870302 + 0.492518i \(0.163924\pi\)
−0.870302 + 0.492518i \(0.836076\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i −0.913708 0.406371i \(-0.866794\pi\)
0.913708 0.406371i \(-0.133206\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.1421i 1.29641i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 19.7990i 1.75688i 0.477856 + 0.878438i \(0.341414\pi\)
−0.477856 + 0.878438i \(0.658586\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.89949i 0.864923i −0.901652 0.432461i \(-0.857645\pi\)
0.901652 0.432461i \(-0.142355\pi\)
\(132\) 0 0
\(133\) −2.00000 + 8.48528i −0.173422 + 0.735767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.89949i 0.845771i 0.906183 + 0.422885i \(0.138983\pi\)
−0.906183 + 0.422885i \(0.861017\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 14.1421i 1.17444i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264i 0.347571i 0.984784 + 0.173785i \(0.0555999\pi\)
−0.984784 + 0.173785i \(0.944400\pi\)
\(150\) 0 0
\(151\) 11.3137i 0.920697i 0.887738 + 0.460348i \(0.152275\pi\)
−0.887738 + 0.460348i \(0.847725\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −24.0000 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.82843i 0.222911i
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 6.00000 0.453557
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) 14.1421i 1.05118i 0.850739 + 0.525588i \(0.176155\pi\)
−0.850739 + 0.525588i \(0.823845\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −30.0000 −2.19382
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.5563i 1.12562i −0.826587 0.562809i \(-0.809721\pi\)
0.826587 0.562809i \(-0.190279\pi\)
\(192\) 0 0
\(193\) 16.9706i 1.22157i −0.791797 0.610784i \(-0.790854\pi\)
0.791797 0.610784i \(-0.209146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i −0.654931 0.755689i \(-0.727302\pi\)
0.654931 0.755689i \(-0.272698\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.0000 1.40372
\(204\) 0 0
\(205\) 14.1421i 0.987730i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −18.0000 4.24264i −1.24509 0.293470i
\(210\) 0 0
\(211\) 14.1421i 0.973585i −0.873518 0.486792i \(-0.838167\pi\)
0.873518 0.486792i \(-0.161833\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.9706i 1.15738i
\(216\) 0 0
\(217\) 5.65685i 0.384012i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.7279i 0.833834i −0.908945 0.416917i \(-0.863111\pi\)
0.908945 0.416917i \(-0.136889\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.24264i 0.274434i −0.990541 0.137217i \(-0.956184\pi\)
0.990541 0.137217i \(-0.0438157\pi\)
\(240\) 0 0
\(241\) 8.48528i 0.546585i 0.961931 + 0.273293i \(0.0881127\pi\)
−0.961931 + 0.273293i \(0.911887\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.24264i 0.271052i
\(246\) 0 0
\(247\) 12.0000 + 2.82843i 0.763542 + 0.179969i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.5563i 0.981908i −0.871185 0.490954i \(-0.836648\pi\)
0.871185 0.490954i \(-0.163352\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 11.3137i 0.703000i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.07107i 0.436021i 0.975946 + 0.218010i \(0.0699567\pi\)
−0.975946 + 0.218010i \(0.930043\pi\)
\(264\) 0 0
\(265\) 14.1421i 0.868744i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.7279i 0.767523i
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) −33.0000 −1.94118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 16.9706i 0.988064i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.3137i 0.647821i
\(306\) 0 0
\(307\) 33.9411i 1.93712i 0.248776 + 0.968561i \(0.419972\pi\)
−0.248776 + 0.968561i \(0.580028\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.5563i 0.882120i 0.897478 + 0.441060i \(0.145397\pi\)
−0.897478 + 0.441060i \(0.854603\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 42.4264i 2.37542i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −30.0000 7.07107i −1.66924 0.393445i
\(324\) 0 0
\(325\) 8.48528i 0.470679i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.82843i 0.155936i
\(330\) 0 0
\(331\) 5.65685i 0.310929i −0.987841 0.155464i \(-0.950313\pi\)
0.987841 0.155464i \(-0.0496874\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 20.0000 1.09272
\(336\) 0 0
\(337\) 25.4558i 1.38667i −0.720616 0.693334i \(-0.756141\pi\)
0.720616 0.693334i \(-0.243859\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.07107i 0.379595i −0.981823 0.189797i \(-0.939217\pi\)
0.981823 0.189797i \(-0.0607831\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.3848i 0.978523i 0.872137 + 0.489261i \(0.162734\pi\)
−0.872137 + 0.489261i \(0.837266\pi\)
\(354\) 0 0
\(355\) 11.3137i 0.600469i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.5269i 1.71670i −0.513061 0.858352i \(-0.671488\pi\)
0.513061 0.858352i \(-0.328512\pi\)
\(360\) 0 0
\(361\) −17.0000 8.48528i −0.894737 0.446594i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.48528i 0.444140i
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.2843i 1.45671i
\(378\) 0 0
\(379\) 11.3137i 0.581146i 0.956853 + 0.290573i \(0.0938459\pi\)
−0.956853 + 0.290573i \(0.906154\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.0416i 1.21896i −0.792802 0.609480i \(-0.791378\pi\)
0.792802 0.609480i \(-0.208622\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 8.48528i 0.419570i 0.977748 + 0.209785i \(0.0672764\pi\)
−0.977748 + 0.209785i \(0.932724\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.5563i 0.759977i 0.924991 + 0.379989i \(0.124072\pi\)
−0.924991 + 0.379989i \(0.875928\pi\)
\(420\) 0 0
\(421\) 8.48528i 0.413547i 0.978389 + 0.206774i \(0.0662964\pi\)
−0.978389 + 0.206774i \(0.933704\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 21.2132i 1.02899i
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 1.41421i −0.287019 0.0676510i
\(438\) 0 0
\(439\) 16.9706i 0.809961i −0.914325 0.404980i \(-0.867278\pi\)
0.914325 0.404980i \(-0.132722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.07107i 0.335957i 0.985791 + 0.167978i \(0.0537239\pi\)
−0.985791 + 0.167978i \(0.946276\pi\)
\(444\) 0 0
\(445\) 8.48528i 0.402241i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 42.4264i 1.99778i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.3848i 0.856264i −0.903716 0.428132i \(-0.859172\pi\)
0.903716 0.428132i \(-0.140828\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.41421i 0.0654420i −0.999465 0.0327210i \(-0.989583\pi\)
0.999465 0.0327210i \(-0.0104173\pi\)
\(468\) 0 0
\(469\) 28.2843i 1.30605i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 50.9117i 2.34092i
\(474\) 0 0
\(475\) −3.00000 + 12.7279i −0.137649 + 0.583997i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.5563i 0.710788i 0.934717 + 0.355394i \(0.115653\pi\)
−0.934717 + 0.355394i \(0.884347\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.0000 −0.544892
\(486\) 0 0
\(487\) 31.1127i 1.40985i −0.709281 0.704925i \(-0.750980\pi\)
0.709281 0.704925i \(-0.249020\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.7279i 0.574403i 0.957870 + 0.287202i \(0.0927249\pi\)
−0.957870 + 0.287202i \(0.907275\pi\)
\(492\) 0 0
\(493\) 70.7107i 3.18465i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.5563i 0.693623i −0.937935 0.346812i \(-0.887264\pi\)
0.937935 0.346812i \(-0.112736\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 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
\(523\) 14.1421i 0.618392i 0.950998 + 0.309196i \(0.100060\pi\)
−0.950998 + 0.309196i \(0.899940\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.2843i 1.22513i
\(534\) 0 0
\(535\) 11.3137i 0.489134i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.7279i 0.548230i
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 16.9706i 0.725609i −0.931865 0.362804i \(-0.881819\pi\)
0.931865 0.362804i \(-0.118181\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.0000 + 42.4264i −0.426014 + 1.80743i
\(552\) 0 0
\(553\) 22.6274i 0.962216i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.3848i 0.778988i −0.921029 0.389494i \(-0.872650\pi\)
0.921029 0.389494i \(-0.127350\pi\)
\(558\) 0 0
\(559\) 33.9411i 1.43556i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 8.48528i 0.356978i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.24264i 0.176930i
\(576\) 0 0
\(577\) 44.0000 1.83174 0.915872 0.401470i \(-0.131501\pi\)
0.915872 + 0.401470i \(0.131501\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.48528i 0.352029i
\(582\) 0 0
\(583\) 42.4264i 1.75712i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.3848i 0.758821i −0.925228 0.379410i \(-0.876127\pi\)
0.925228 0.379410i \(-0.123873\pi\)
\(588\) 0 0
\(589\) 12.0000 + 2.82843i 0.494451 + 0.116543i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 43.8406i 1.80032i −0.435561 0.900159i \(-0.643450\pi\)
0.435561 0.900159i \(-0.356550\pi\)
\(594\) 0 0
\(595\) 20.0000 0.819920
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 33.9411i 1.38449i −0.721664 0.692244i \(-0.756622\pi\)
0.721664 0.692244i \(-0.243378\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.89949i 0.402472i
\(606\) 0 0
\(607\) 31.1127i 1.26283i 0.775447 + 0.631413i \(0.217525\pi\)
−0.775447 + 0.631413i \(0.782475\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.7279i 0.512407i −0.966623 0.256203i \(-0.917528\pi\)
0.966623 0.256203i \(-0.0824717\pi\)
\(618\) 0 0
\(619\) 34.0000 1.36658 0.683288 0.730149i \(-0.260549\pi\)
0.683288 + 0.730149i \(0.260549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 28.0000 1.11115
\(636\) 0 0
\(637\) 8.48528i 0.336199i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.3848i 0.722780i −0.932415 0.361390i \(-0.882302\pi\)
0.932415 0.361390i \(-0.117698\pi\)
\(648\) 0 0
\(649\) 50.9117i 1.99846i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.24264i 0.166027i −0.996548 0.0830137i \(-0.973545\pi\)
0.996548 0.0830137i \(-0.0264545\pi\)
\(654\) 0 0
\(655\) −14.0000 −0.547025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.6274i 0.880105i 0.897972 + 0.440052i \(0.145040\pi\)
−0.897972 + 0.440052i \(0.854960\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 + 2.82843i 0.465340 + 0.109682i
\(666\) 0 0
\(667\) 14.1421i 0.547586i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.9411i 1.31028i
\(672\) 0 0
\(673\) 22.6274i 0.872223i −0.899893 0.436111i \(-0.856355\pi\)
0.899893 0.436111i \(-0.143645\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28.2843i 1.07754i
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.82843i 0.107288i
\(696\) 0 0
\(697\) 70.7107i 2.67836i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7279i 0.480727i 0.970683 + 0.240363i \(0.0772666\pi\)
−0.970683 + 0.240363i \(0.922733\pi\)
\(702\) 0 0
\(703\) 24.0000 + 5.65685i 0.905177 + 0.213352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.7990i 0.744618i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 16.9706i 0.634663i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.0122i 1.52950i 0.644329 + 0.764748i \(0.277137\pi\)
−0.644329 + 0.764748i \(0.722863\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 84.8528i 3.13839i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 45.2548i 1.65137i 0.564130 + 0.825686i \(0.309212\pi\)
−0.564130 + 0.825686i \(0.690788\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.41421i 0.0512652i 0.999671 + 0.0256326i \(0.00816000\pi\)
−0.999671 + 0.0256326i \(0.991840\pi\)
\(762\) 0 0
\(763\) 16.9706i 0.614376i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.9411i 1.22554i
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 8.48528i 0.304800i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.0000 42.4264i 0.358287 1.52008i
\(780\) 0 0
\(781\) 33.9411i 1.21451i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 33.9411i 1.21141i
\(786\) 0 0
\(787\) 8.48528i 0.302468i −0.988498 0.151234i \(-0.951675\pi\)
0.988498 0.151234i \(-0.0483246\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 22.6274i 0.803523i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) 0 0
\(801\) 0 0
\(802\)