Properties

Label 2160.2.f.k.1729.4
Level $2160$
Weight $2$
Character 2160.1729
Analytic conductor $17.248$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(1729,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.1729");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.f (of order \(2\), degree \(1\), 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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1729.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2160.1729
Dual form 2160.2.f.k.1729.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.12132 + 0.707107i) q^{5} +3.00000i q^{7} +O(q^{10})\) \(q+(2.12132 + 0.707107i) q^{5} +3.00000i q^{7} +4.24264 q^{11} +3.00000i q^{13} -2.82843i q^{17} +1.00000 q^{19} +7.07107i q^{23} +(4.00000 + 3.00000i) q^{25} -4.24264 q^{29} -2.00000 q^{31} +(-2.12132 + 6.36396i) q^{35} -9.00000i q^{37} +4.24264 q^{41} -6.00000i q^{43} +2.82843i q^{47} -2.00000 q^{49} +9.89949i q^{53} +(9.00000 + 3.00000i) q^{55} +8.48528 q^{59} -13.0000 q^{61} +(-2.12132 + 6.36396i) q^{65} -3.00000i q^{67} -12.7279 q^{71} +9.00000i q^{73} +12.7279i q^{77} -5.00000 q^{79} -1.41421i q^{83} +(2.00000 - 6.00000i) q^{85} -9.00000 q^{91} +(2.12132 + 0.707107i) q^{95} -3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{19} + 16 q^{25} - 8 q^{31} - 8 q^{49} + 36 q^{55} - 52 q^{61} - 20 q^{79} + 8 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) 2.12132 + 0.707107i 0.948683 + 0.316228i
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264 1.27920 0.639602 0.768706i \(-0.279099\pi\)
0.639602 + 0.768706i \(0.279099\pi\)
\(12\) 0 0
\(13\) 3.00000i 0.832050i 0.909353 + 0.416025i \(0.136577\pi\)
−0.909353 + 0.416025i \(0.863423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843i 0.685994i −0.939336 0.342997i \(-0.888558\pi\)
0.939336 0.342997i \(-0.111442\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.07107i 1.47442i 0.675664 + 0.737210i \(0.263857\pi\)
−0.675664 + 0.737210i \(0.736143\pi\)
\(24\) 0 0
\(25\) 4.00000 + 3.00000i 0.800000 + 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.24264 −0.787839 −0.393919 0.919145i \(-0.628881\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.12132 + 6.36396i −0.358569 + 1.07571i
\(36\) 0 0
\(37\) 9.00000i 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) 0 0
\(55\) 9.00000 + 3.00000i 1.21356 + 0.404520i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.48528 1.10469 0.552345 0.833616i \(-0.313733\pi\)
0.552345 + 0.833616i \(0.313733\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.12132 + 6.36396i −0.263117 + 0.789352i
\(66\) 0 0
\(67\) 3.00000i 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.7279 −1.51053 −0.755263 0.655422i \(-0.772491\pi\)
−0.755263 + 0.655422i \(0.772491\pi\)
\(72\) 0 0
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7279i 1.45048i
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.41421i 0.155230i −0.996983 0.0776151i \(-0.975269\pi\)
0.996983 0.0776151i \(-0.0247305\pi\)
\(84\) 0 0
\(85\) 2.00000 6.00000i 0.216930 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.943456
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.12132 + 0.707107i 0.217643 + 0.0725476i
\(96\) 0 0
\(97\) 3.00000i 0.304604i −0.988334 0.152302i \(-0.951331\pi\)
0.988334 0.152302i \(-0.0486686\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.48528 0.844317 0.422159 0.906522i \(-0.361273\pi\)
0.422159 + 0.906522i \(0.361273\pi\)
\(102\) 0 0
\(103\) 3.00000i 0.295599i −0.989017 0.147799i \(-0.952781\pi\)
0.989017 0.147799i \(-0.0472190\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421i 0.133038i 0.997785 + 0.0665190i \(0.0211893\pi\)
−0.997785 + 0.0665190i \(0.978811\pi\)
\(114\) 0 0
\(115\) −5.00000 + 15.0000i −0.466252 + 1.39876i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.48528 0.777844
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.36396 + 9.19239i 0.569210 + 0.822192i
\(126\) 0 0
\(127\) 18.0000i 1.59724i −0.601834 0.798621i \(-0.705563\pi\)
0.601834 0.798621i \(-0.294437\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.48528 0.741362 0.370681 0.928760i \(-0.379124\pi\)
0.370681 + 0.928760i \(0.379124\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.41421i 0.120824i 0.998174 + 0.0604122i \(0.0192415\pi\)
−0.998174 + 0.0604122i \(0.980758\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.7279i 1.06436i
\(144\) 0 0
\(145\) −9.00000 3.00000i −0.747409 0.249136i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.9706 1.39028 0.695141 0.718873i \(-0.255342\pi\)
0.695141 + 0.718873i \(0.255342\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788 0.0406894 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.24264 1.41421i −0.340777 0.113592i
\(156\) 0 0
\(157\) 6.00000i 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −21.2132 −1.67183
\(162\) 0 0
\(163\) 9.00000i 0.704934i 0.935824 + 0.352467i \(0.114657\pi\)
−0.935824 + 0.352467i \(0.885343\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.65685i 0.437741i −0.975754 0.218870i \(-0.929763\pi\)
0.975754 0.218870i \(-0.0702371\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.1421i 1.07521i 0.843198 + 0.537603i \(0.180670\pi\)
−0.843198 + 0.537603i \(0.819330\pi\)
\(174\) 0 0
\(175\) −9.00000 + 12.0000i −0.680336 + 0.907115i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.7279 −0.951330 −0.475665 0.879627i \(-0.657792\pi\)
−0.475665 + 0.879627i \(0.657792\pi\)
\(180\) 0 0
\(181\) −1.00000 −0.0743294 −0.0371647 0.999309i \(-0.511833\pi\)
−0.0371647 + 0.999309i \(0.511833\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.36396 19.0919i 0.467888 1.40366i
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) 3.00000i 0.215945i 0.994154 + 0.107972i \(0.0344358\pi\)
−0.994154 + 0.107972i \(0.965564\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.6274i 1.61214i 0.591822 + 0.806068i \(0.298409\pi\)
−0.591822 + 0.806068i \(0.701591\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.7279i 0.893325i
\(204\) 0 0
\(205\) 9.00000 + 3.00000i 0.628587 + 0.209529i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.24264 0.293470
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.24264 12.7279i 0.289346 0.868037i
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.48528 0.570782
\(222\) 0 0
\(223\) 6.00000i 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0416i 1.59570i 0.602857 + 0.797850i \(0.294029\pi\)
−0.602857 + 0.797850i \(0.705971\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.7990i 1.29707i −0.761183 0.648537i \(-0.775381\pi\)
0.761183 0.648537i \(-0.224619\pi\)
\(234\) 0 0
\(235\) −2.00000 + 6.00000i −0.130466 + 0.391397i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.24264 −0.274434 −0.137217 0.990541i \(-0.543816\pi\)
−0.137217 + 0.990541i \(0.543816\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.24264 1.41421i −0.271052 0.0903508i
\(246\) 0 0
\(247\) 3.00000i 0.190885i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.7279 −0.803379 −0.401690 0.915776i \(-0.631577\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(252\) 0 0
\(253\) 30.0000i 1.88608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.7990i 1.23503i −0.786560 0.617514i \(-0.788140\pi\)
0.786560 0.617514i \(-0.211860\pi\)
\(258\) 0 0
\(259\) 27.0000 1.67770
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.82843i 0.174408i 0.996190 + 0.0872041i \(0.0277932\pi\)
−0.996190 + 0.0872041i \(0.972207\pi\)
\(264\) 0 0
\(265\) −7.00000 + 21.0000i −0.430007 + 1.29002i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.4558 −1.55207 −0.776035 0.630690i \(-0.782772\pi\)
−0.776035 + 0.630690i \(0.782772\pi\)
\(270\) 0 0
\(271\) 19.0000 1.15417 0.577084 0.816685i \(-0.304191\pi\)
0.577084 + 0.816685i \(0.304191\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.9706 + 12.7279i 1.02336 + 0.767523i
\(276\) 0 0
\(277\) 12.0000i 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.48528 0.506189 0.253095 0.967442i \(-0.418552\pi\)
0.253095 + 0.967442i \(0.418552\pi\)
\(282\) 0 0
\(283\) 6.00000i 0.356663i 0.983970 + 0.178331i \(0.0570699\pi\)
−0.983970 + 0.178331i \(0.942930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.7279i 0.751305i
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.07107i 0.413096i −0.978436 0.206548i \(-0.933777\pi\)
0.978436 0.206548i \(-0.0662230\pi\)
\(294\) 0 0
\(295\) 18.0000 + 6.00000i 1.04800 + 0.349334i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −21.2132 −1.22679
\(300\) 0 0
\(301\) 18.0000 1.03750
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −27.5772 9.19239i −1.57906 0.526355i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.48528 0.481156 0.240578 0.970630i \(-0.422663\pi\)
0.240578 + 0.970630i \(0.422663\pi\)
\(312\) 0 0
\(313\) 21.0000i 1.18699i −0.804838 0.593495i \(-0.797748\pi\)
0.804838 0.593495i \(-0.202252\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.3137i 0.635441i −0.948184 0.317721i \(-0.897083\pi\)
0.948184 0.317721i \(-0.102917\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.82843i 0.157378i
\(324\) 0 0
\(325\) −9.00000 + 12.0000i −0.499230 + 0.665640i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.48528 −0.467809
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.12132 6.36396i 0.115900 0.347700i
\(336\) 0 0
\(337\) 33.0000i 1.79762i −0.438334 0.898812i \(-0.644431\pi\)
0.438334 0.898812i \(-0.355569\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.48528 −0.459504
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.8701i 1.44246i −0.692696 0.721230i \(-0.743577\pi\)
0.692696 0.721230i \(-0.256423\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.2843i 1.50542i −0.658352 0.752710i \(-0.728746\pi\)
0.658352 0.752710i \(-0.271254\pi\)
\(354\) 0 0
\(355\) −27.0000 9.00000i −1.43301 0.477670i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.7279 −0.671754 −0.335877 0.941906i \(-0.609033\pi\)
−0.335877 + 0.941906i \(0.609033\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.36396 + 19.0919i −0.333105 + 0.999315i
\(366\) 0 0
\(367\) 15.0000i 0.782994i −0.920179 0.391497i \(-0.871957\pi\)
0.920179 0.391497i \(-0.128043\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.6985 −1.54187
\(372\) 0 0
\(373\) 21.0000i 1.08734i 0.839299 + 0.543669i \(0.182965\pi\)
−0.839299 + 0.543669i \(0.817035\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.7279i 0.655521i
\(378\) 0 0
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.65685i 0.289052i −0.989501 0.144526i \(-0.953834\pi\)
0.989501 0.144526i \(-0.0461657\pi\)
\(384\) 0 0
\(385\) −9.00000 + 27.0000i −0.458682 + 1.37605i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.9411 1.72088 0.860442 0.509549i \(-0.170188\pi\)
0.860442 + 0.509549i \(0.170188\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.6066 3.53553i −0.533676 0.177892i
\(396\) 0 0
\(397\) 36.0000i 1.80679i −0.428811 0.903394i \(-0.641067\pi\)
0.428811 0.903394i \(-0.358933\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.24264 0.211867 0.105934 0.994373i \(-0.466217\pi\)
0.105934 + 0.994373i \(0.466217\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 38.1838i 1.89270i
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 25.4558i 1.25260i
\(414\) 0 0
\(415\) 1.00000 3.00000i 0.0490881 0.147264i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.9411 1.65813 0.829066 0.559150i \(-0.188873\pi\)
0.829066 + 0.559150i \(0.188873\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.48528 11.3137i 0.411597 0.548795i
\(426\) 0 0
\(427\) 39.0000i 1.88734i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.7279 0.613082 0.306541 0.951857i \(-0.400828\pi\)
0.306541 + 0.951857i \(0.400828\pi\)
\(432\) 0 0
\(433\) 36.0000i 1.73005i −0.501729 0.865025i \(-0.667303\pi\)
0.501729 0.865025i \(-0.332697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.07107i 0.338255i
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.1421i 0.671913i −0.941877 0.335957i \(-0.890940\pi\)
0.941877 0.335957i \(-0.109060\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −19.0919 6.36396i −0.895041 0.298347i
\(456\) 0 0
\(457\) 12.0000i 0.561336i −0.959805 0.280668i \(-0.909444\pi\)
0.959805 0.280668i \(-0.0905560\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.9706 −0.790398 −0.395199 0.918596i \(-0.629324\pi\)
−0.395199 + 0.918596i \(0.629324\pi\)
\(462\) 0 0
\(463\) 15.0000i 0.697109i 0.937288 + 0.348555i \(0.113327\pi\)
−0.937288 + 0.348555i \(0.886673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.6274i 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 25.4558i 1.17046i
\(474\) 0 0
\(475\) 4.00000 + 3.00000i 0.183533 + 0.137649i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.24264 0.193851 0.0969256 0.995292i \(-0.469099\pi\)
0.0969256 + 0.995292i \(0.469099\pi\)
\(480\) 0 0
\(481\) 27.0000 1.23109
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.12132 6.36396i 0.0963242 0.288973i
\(486\) 0 0
\(487\) 9.00000i 0.407829i −0.978989 0.203914i \(-0.934634\pi\)
0.978989 0.203914i \(-0.0653664\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.2132 0.957338 0.478669 0.877995i \(-0.341119\pi\)
0.478669 + 0.877995i \(0.341119\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 38.1838i 1.71278i
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 35.3553i 1.57642i −0.615409 0.788208i \(-0.711009\pi\)
0.615409 0.788208i \(-0.288991\pi\)
\(504\) 0 0
\(505\) 18.0000 + 6.00000i 0.800989 + 0.266996i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.6985 1.31636 0.658181 0.752860i \(-0.271326\pi\)
0.658181 + 0.752860i \(0.271326\pi\)
\(510\) 0 0
\(511\) −27.0000 −1.19441
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.12132 6.36396i 0.0934765 0.280430i
\(516\) 0 0
\(517\) 12.0000i 0.527759i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.7279 −0.557620 −0.278810 0.960346i \(-0.589940\pi\)
−0.278810 + 0.960346i \(0.589940\pi\)
\(522\) 0 0
\(523\) 9.00000i 0.393543i 0.980449 + 0.196771i \(0.0630456\pi\)
−0.980449 + 0.196771i \(0.936954\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.65685i 0.246416i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.7279i 0.551308i
\(534\) 0 0
\(535\) −5.00000 + 15.0000i −0.216169 + 0.648507i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.48528 −0.365487
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16.9706 + 5.65685i 0.726939 + 0.242313i
\(546\) 0 0
\(547\) 3.00000i 0.128271i 0.997941 + 0.0641354i \(0.0204289\pi\)
−0.997941 + 0.0641354i \(0.979571\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.24264 −0.180743
\(552\) 0 0
\(553\) 15.0000i 0.637865i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.7990i 0.838910i −0.907776 0.419455i \(-0.862221\pi\)
0.907776 0.419455i \(-0.137779\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11.3137i 0.476816i 0.971165 + 0.238408i \(0.0766255\pi\)
−0.971165 + 0.238408i \(0.923374\pi\)
\(564\) 0 0
\(565\) −1.00000 + 3.00000i −0.0420703 + 0.126211i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −42.4264 −1.77861 −0.889304 0.457317i \(-0.848810\pi\)
−0.889304 + 0.457317i \(0.848810\pi\)
\(570\) 0 0
\(571\) 25.0000 1.04622 0.523109 0.852266i \(-0.324772\pi\)
0.523109 + 0.852266i \(0.324772\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −21.2132 + 28.2843i −0.884652 + 1.17954i
\(576\) 0 0
\(577\) 45.0000i 1.87337i 0.350167 + 0.936687i \(0.386125\pi\)
−0.350167 + 0.936687i \(0.613875\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.24264 0.176014
\(582\) 0 0
\(583\) 42.0000i 1.73946i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.8701i 1.10905i −0.832168 0.554523i \(-0.812901\pi\)
0.832168 0.554523i \(-0.187099\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.89949i 0.406524i 0.979124 + 0.203262i \(0.0651542\pi\)
−0.979124 + 0.203262i \(0.934846\pi\)
\(594\) 0 0
\(595\) 18.0000 + 6.00000i 0.737928 + 0.245976i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −29.6985 −1.21345 −0.606724 0.794913i \(-0.707517\pi\)
−0.606724 + 0.794913i \(0.707517\pi\)
\(600\) 0 0
\(601\) −40.0000 −1.63163 −0.815817 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.8492 + 4.94975i 0.603708 + 0.201236i
\(606\) 0 0
\(607\) 39.0000i 1.58296i −0.611194 0.791481i \(-0.709311\pi\)
0.611194 0.791481i \(-0.290689\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.48528 −0.343278
\(612\) 0 0
\(613\) 9.00000i 0.363507i 0.983344 + 0.181753i \(0.0581772\pi\)
−0.983344 + 0.181753i \(0.941823\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.07107i 0.284670i −0.989819 0.142335i \(-0.954539\pi\)
0.989819 0.142335i \(-0.0454611\pi\)
\(618\) 0 0
\(619\) 19.0000 0.763674 0.381837 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25.4558 −1.01499
\(630\) 0 0
\(631\) −17.0000 −0.676759 −0.338380 0.941010i \(-0.609879\pi\)
−0.338380 + 0.941010i \(0.609879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.7279 38.1838i 0.505092 1.51528i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −16.9706 −0.670297 −0.335148 0.942165i \(-0.608786\pi\)
−0.335148 + 0.942165i \(0.608786\pi\)
\(642\) 0 0
\(643\) 42.0000i 1.65632i 0.560493 + 0.828159i \(0.310612\pi\)
−0.560493 + 0.828159i \(0.689388\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.89949i 0.389189i −0.980884 0.194595i \(-0.937661\pi\)
0.980884 0.194595i \(-0.0623391\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.65685i 0.221370i 0.993856 + 0.110685i \(0.0353044\pi\)
−0.993856 + 0.110685i \(0.964696\pi\)
\(654\) 0 0
\(655\) 18.0000 + 6.00000i 0.703318 + 0.234439i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.9706 0.661079 0.330540 0.943792i \(-0.392769\pi\)
0.330540 + 0.943792i \(0.392769\pi\)
\(660\) 0 0
\(661\) 23.0000 0.894596 0.447298 0.894385i \(-0.352386\pi\)
0.447298 + 0.894385i \(0.352386\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.12132 + 6.36396i −0.0822613 + 0.246784i
\(666\) 0 0
\(667\) 30.0000i 1.16160i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −55.1543 −2.12921
\(672\) 0 0
\(673\) 15.0000i 0.578208i 0.957298 + 0.289104i \(0.0933573\pi\)
−0.957298 + 0.289104i \(0.906643\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.0833i 1.84799i 0.382405 + 0.923995i \(0.375096\pi\)
−0.382405 + 0.923995i \(0.624904\pi\)
\(678\) 0 0
\(679\) 9.00000 0.345388
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.5563i 0.595247i 0.954683 + 0.297624i \(0.0961940\pi\)
−0.954683 + 0.297624i \(0.903806\pi\)
\(684\) 0 0
\(685\) −1.00000 + 3.00000i −0.0382080 + 0.114624i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −29.6985 −1.13142
\(690\) 0 0
\(691\) −50.0000 −1.90209 −0.951045 0.309053i \(-0.899988\pi\)
−0.951045 + 0.309053i \(0.899988\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.5772 + 9.19239i 1.04606 + 0.348687i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.1838 1.44218 0.721090 0.692841i \(-0.243641\pi\)
0.721090 + 0.692841i \(0.243641\pi\)
\(702\) 0 0
\(703\) 9.00000i 0.339441i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25.4558i 0.957366i
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.1421i 0.529627i
\(714\) 0 0
\(715\) −9.00000 + 27.0000i −0.336581 + 1.00974i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.7279 0.474671 0.237336 0.971428i \(-0.423726\pi\)
0.237336 + 0.971428i \(0.423726\pi\)
\(720\) 0 0
\(721\) 9.00000 0.335178
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −16.9706 12.7279i −0.630271 0.472703i
\(726\) 0 0
\(727\) 42.0000i 1.55769i −0.627214 0.778847i \(-0.715805\pi\)
0.627214 0.778847i \(-0.284195\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) 0 0
\(733\) 24.0000i 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.7279i 0.468839i
\(738\) 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
\(743\) 14.1421i 0.518825i −0.965767 0.259412i \(-0.916471\pi\)
0.965767 0.259412i \(-0.0835289\pi\)
\(744\) 0 0
\(745\) 36.0000 + 12.0000i 1.31894 + 0.439646i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −21.2132 −0.775114
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.12132 + 0.707107i 0.0772028 + 0.0257343i
\(756\) 0 0
\(757\) 27.0000i 0.981332i −0.871348 0.490666i \(-0.836754\pi\)
0.871348 0.490666i \(-0.163246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.24264 0.153796 0.0768978 0.997039i \(-0.475498\pi\)
0.0768978 + 0.997039i \(0.475498\pi\)
\(762\) 0 0
\(763\) 24.0000i 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.4558i 0.919157i
\(768\) 0 0
\(769\) 11.0000 0.396670 0.198335 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 41.0122i 1.47511i −0.675289 0.737553i \(-0.735981\pi\)
0.675289 0.737553i \(-0.264019\pi\)
\(774\) 0 0
\(775\) −8.00000 6.00000i −0.287368 0.215526i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.24264 0.152008
\(780\) 0 0
\(781\) −54.0000 −1.93227
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.24264 12.7279i 0.151426 0.454279i
\(786\) 0 0
\(787\) 15.0000i 0.534692i 0.963601 + 0.267346i \(0.0861467\pi\)
−0.963601 + 0.267346i \(0.913853\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.24264 −0.150851
\(792\) 0 0
\(793\) 39.0000i 1.38493i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.7990i 0.701316i −0.936504 0.350658i \(-0.885958\pi\)
0.936504 0.350658i \(-0.114042\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0