Properties

Label 2736.2.k.b.2431.2
Level $2736$
Weight $2$
Character 2736.2431
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(2431,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2736.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.b.2431.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{5} +1.73205i q^{7} +O(q^{10})\) \(q-3.00000 q^{5} +1.73205i q^{7} +5.19615i q^{11} +6.92820i q^{13} -3.00000 q^{17} +(4.00000 - 1.73205i) q^{19} -3.46410i q^{23} +4.00000 q^{25} -4.00000 q^{31} -5.19615i q^{35} -6.92820i q^{37} +6.92820i q^{41} +8.66025i q^{43} -8.66025i q^{47} +4.00000 q^{49} -6.92820i q^{53} -15.5885i q^{55} -12.0000 q^{59} +7.00000 q^{61} -20.7846i q^{65} +8.00000 q^{67} -12.0000 q^{71} -5.00000 q^{73} -9.00000 q^{77} -8.00000 q^{79} +3.46410i q^{83} +9.00000 q^{85} +6.92820i q^{89} -12.0000 q^{91} +(-12.0000 + 5.19615i) q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{5} - 6 q^{17} + 8 q^{19} + 8 q^{25} - 8 q^{31} + 8 q^{49} - 24 q^{59} + 14 q^{61} + 16 q^{67} - 24 q^{71} - 10 q^{73} - 18 q^{77} - 16 q^{79} + 18 q^{85} - 24 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.19615i 1.56670i 0.621582 + 0.783349i \(0.286490\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(12\) 0 0
\(13\) 6.92820i 1.92154i 0.277350 + 0.960769i \(0.410544\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 4.00000 1.73205i 0.917663 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410i 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.19615i 0.878310i
\(36\) 0 0
\(37\) 6.92820i 1.13899i −0.821995 0.569495i \(-0.807139\pi\)
0.821995 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i 0.841021 + 0.541002i \(0.181955\pi\)
−0.841021 + 0.541002i \(0.818045\pi\)
\(42\) 0 0
\(43\) 8.66025i 1.32068i 0.750968 + 0.660338i \(0.229587\pi\)
−0.750968 + 0.660338i \(0.770413\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.66025i 1.26323i −0.775283 0.631614i \(-0.782393\pi\)
0.775283 0.631614i \(-0.217607\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.92820i 0.951662i −0.879537 0.475831i \(-0.842147\pi\)
0.879537 0.475831i \(-0.157853\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 20.7846i 2.57801i
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.00000 −1.02565
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.0000 + 5.19615i −1.23117 + 0.533114i
\(96\) 0 0
\(97\) 6.92820i 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.8564i 1.30350i 0.758433 + 0.651751i \(0.225965\pi\)
−0.758433 + 0.651751i \(0.774035\pi\)
\(114\) 0 0
\(115\) 10.3923i 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.19615i 0.476331i
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.5885i 1.36197i −0.732297 0.680985i \(-0.761552\pi\)
0.732297 0.680985i \(-0.238448\pi\)
\(132\) 0 0
\(133\) 3.00000 + 6.92820i 0.260133 + 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.0000 1.79415 0.897076 0.441877i \(-0.145687\pi\)
0.897076 + 0.441877i \(0.145687\pi\)
\(138\) 0 0
\(139\) 19.0526i 1.61602i −0.589171 0.808008i \(-0.700546\pi\)
0.589171 0.808008i \(-0.299454\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −36.0000 −3.01047
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 3.46410i 0.271329i −0.990755 0.135665i \(-0.956683\pi\)
0.990755 0.135665i \(-0.0433170\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −35.0000 −2.69231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 6.92820i 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i −0.857209 0.514969i \(-0.827803\pi\)
0.857209 0.514969i \(-0.172197\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.7846i 1.52811i
\(186\) 0 0
\(187\) 15.5885i 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.1244i 0.877288i 0.898661 + 0.438644i \(0.144541\pi\)
−0.898661 + 0.438644i \(0.855459\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 19.0526i 1.35060i −0.737543 0.675300i \(-0.764014\pi\)
0.737543 0.675300i \(-0.235986\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 20.7846i 1.45166i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.00000 + 20.7846i 0.622543 + 1.43770i
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 25.9808i 1.77187i
\(216\) 0 0
\(217\) 6.92820i 0.470317i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.7846i 1.39812i
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 25.9808i 1.69480i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 22.5167i 1.45648i −0.685321 0.728241i \(-0.740338\pi\)
0.685321 0.728241i \(-0.259662\pi\)
\(240\) 0 0
\(241\) 6.92820i 0.446285i −0.974786 0.223142i \(-0.928369\pi\)
0.974786 0.223142i \(-0.0716315\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 12.0000 + 27.7128i 0.763542 + 1.76332i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.73205i 0.109326i −0.998505 0.0546630i \(-0.982592\pi\)
0.998505 0.0546630i \(-0.0174085\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 27.7128i 1.72868i −0.502910 0.864339i \(-0.667737\pi\)
0.502910 0.864339i \(-0.332263\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.0526i 1.17483i 0.809285 + 0.587416i \(0.199855\pi\)
−0.809285 + 0.587416i \(0.800145\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.92820i 0.422420i 0.977441 + 0.211210i \(0.0677404\pi\)
−0.977441 + 0.211210i \(0.932260\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.7846i 1.25336i
\(276\) 0 0
\(277\) 23.0000 1.38194 0.690968 0.722885i \(-0.257185\pi\)
0.690968 + 0.722885i \(0.257185\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.8564i 0.826604i −0.910594 0.413302i \(-0.864375\pi\)
0.910594 0.413302i \(-0.135625\pi\)
\(282\) 0 0
\(283\) 8.66025i 0.514799i 0.966305 + 0.257399i \(0.0828656\pi\)
−0.966305 + 0.257399i \(0.917134\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −15.0000 −0.864586
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −21.0000 −1.20246
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.0526i 1.08037i 0.841546 + 0.540186i \(0.181646\pi\)
−0.841546 + 0.540186i \(0.818354\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.6410i 1.94563i 0.231577 + 0.972817i \(0.425612\pi\)
−0.231577 + 0.972817i \(0.574388\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 + 5.19615i −0.667698 + 0.289122i
\(324\) 0 0
\(325\) 27.7128i 1.53723i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.0000 0.826977
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) 6.92820i 0.377403i −0.982034 0.188702i \(-0.939572\pi\)
0.982034 0.188702i \(-0.0604279\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 20.7846i 1.12555i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.0526i 1.02279i 0.859344 + 0.511397i \(0.170872\pi\)
−0.859344 + 0.511397i \(0.829128\pi\)
\(348\) 0 0
\(349\) −13.0000 −0.695874 −0.347937 0.937518i \(-0.613118\pi\)
−0.347937 + 0.937518i \(0.613118\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.5885i 0.822727i −0.911471 0.411364i \(-0.865053\pi\)
0.911471 0.411364i \(-0.134947\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.0000 0.785136
\(366\) 0 0
\(367\) 10.3923i 0.542474i −0.962513 0.271237i \(-0.912567\pi\)
0.962513 0.271237i \(-0.0874327\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 13.8564i 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 27.0000 1.37605
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) 10.3923i 0.525561i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.8564i 0.691956i −0.938243 0.345978i \(-0.887547\pi\)
0.938243 0.345978i \(-0.112453\pi\)
\(402\) 0 0
\(403\) 27.7128i 1.38047i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.0000 1.78445
\(408\) 0 0
\(409\) 6.92820i 0.342578i 0.985221 + 0.171289i \(0.0547931\pi\)
−0.985221 + 0.171289i \(0.945207\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 20.7846i 1.02274i
\(414\) 0 0
\(415\) 10.3923i 0.510138i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.2487i 1.18463i −0.805708 0.592314i \(-0.798215\pi\)
0.805708 0.592314i \(-0.201785\pi\)
\(420\) 0 0
\(421\) 34.6410i 1.68830i 0.536107 + 0.844150i \(0.319894\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 12.1244i 0.586739i
\(428\) 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
\(433\) 13.8564i 0.665896i 0.942945 + 0.332948i \(0.108043\pi\)
−0.942945 + 0.332948i \(0.891957\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 13.8564i −0.287019 0.662842i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.19615i 0.246877i 0.992352 + 0.123438i \(0.0393921\pi\)
−0.992352 + 0.123438i \(0.960608\pi\)
\(444\) 0 0
\(445\) 20.7846i 0.985285i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.8564i 0.653924i 0.945037 + 0.326962i \(0.106025\pi\)
−0.945037 + 0.326962i \(0.893975\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 36.0000 1.68771
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.00000 0.419172 0.209586 0.977790i \(-0.432788\pi\)
0.209586 + 0.977790i \(0.432788\pi\)
\(462\) 0 0
\(463\) 19.0526i 0.885448i −0.896658 0.442724i \(-0.854012\pi\)
0.896658 0.442724i \(-0.145988\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 32.9090i 1.52285i 0.648256 + 0.761423i \(0.275499\pi\)
−0.648256 + 0.761423i \(0.724501\pi\)
\(468\) 0 0
\(469\) 13.8564i 0.639829i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −45.0000 −2.06910
\(474\) 0 0
\(475\) 16.0000 6.92820i 0.734130 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.3205i 0.791394i −0.918381 0.395697i \(-0.870503\pi\)
0.918381 0.395697i \(-0.129497\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.7846i 0.943781i
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.3923i 0.468998i −0.972116 0.234499i \(-0.924655\pi\)
0.972116 0.234499i \(-0.0753450\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.7846i 0.932317i
\(498\) 0 0
\(499\) 36.3731i 1.62828i 0.580667 + 0.814141i \(0.302792\pi\)
−0.580667 + 0.814141i \(0.697208\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.46410i 0.154457i −0.997013 0.0772283i \(-0.975393\pi\)
0.997013 0.0772283i \(-0.0246070\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.92820i 0.307087i 0.988142 + 0.153544i \(0.0490686\pi\)
−0.988142 + 0.153544i \(0.950931\pi\)
\(510\) 0 0
\(511\) 8.66025i 0.383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.0000 1.05757
\(516\) 0 0
\(517\) 45.0000 1.97910
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.8564i 0.607060i 0.952822 + 0.303530i \(0.0981653\pi\)
−0.952822 + 0.303530i \(0.901835\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.7846i 0.895257i
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 41.5692i 1.78063i
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 13.8564i 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) −60.0000 −2.53773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 41.5692i 1.74883i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.92820i 0.290445i −0.989399 0.145223i \(-0.953610\pi\)
0.989399 0.145223i \(-0.0463899\pi\)
\(570\) 0 0
\(571\) 45.0333i 1.88459i −0.334790 0.942293i \(-0.608665\pi\)
0.334790 0.942293i \(-0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.8564i 0.577852i
\(576\) 0 0
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.9808i 1.07234i 0.844110 + 0.536170i \(0.180130\pi\)
−0.844110 + 0.536170i \(0.819870\pi\)
\(588\) 0 0
\(589\) −16.0000 + 6.92820i −0.659269 + 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 15.5885i 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 34.6410i 1.41304i 0.707695 + 0.706518i \(0.249735\pi\)
−0.707695 + 0.706518i \(0.750265\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 48.0000 1.95148
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 60.0000 2.42734
\(612\) 0 0
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.00000 0.362326 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i 0.977947 + 0.208851i \(0.0669724\pi\)
−0.977947 + 0.208851i \(0.933028\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.7846i 0.828737i
\(630\) 0 0
\(631\) 8.66025i 0.344759i 0.985031 + 0.172380i \(0.0551456\pi\)
−0.985031 + 0.172380i \(0.944854\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 48.0000 1.90482
\(636\) 0 0
\(637\) 27.7128i 1.09802i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.7846i 0.820943i 0.911873 + 0.410471i \(0.134636\pi\)
−0.911873 + 0.410471i \(0.865364\pi\)
\(642\) 0 0
\(643\) 36.3731i 1.43441i 0.696860 + 0.717207i \(0.254580\pi\)
−0.696860 + 0.717207i \(0.745420\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.9808i 1.02141i 0.859756 + 0.510705i \(0.170615\pi\)
−0.859756 + 0.510705i \(0.829385\pi\)
\(648\) 0 0
\(649\) 62.3538i 2.44760i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) 0 0
\(655\) 46.7654i 1.82727i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 41.5692i 1.61686i 0.588596 + 0.808428i \(0.299681\pi\)
−0.588596 + 0.808428i \(0.700319\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −9.00000 20.7846i −0.349005 0.805993i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 36.3731i 1.40417i
\(672\) 0 0
\(673\) 34.6410i 1.33531i 0.744469 + 0.667657i \(0.232703\pi\)
−0.744469 + 0.667657i \(0.767297\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.7128i 1.06509i 0.846402 + 0.532545i \(0.178764\pi\)
−0.846402 + 0.532545i \(0.821236\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −63.0000 −2.40711
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 1.73205i 0.0658903i 0.999457 + 0.0329452i \(0.0104887\pi\)
−0.999457 + 0.0329452i \(0.989511\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 57.1577i 2.16811i
\(696\) 0 0
\(697\) 20.7846i 0.787273i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −12.0000 27.7128i −0.452589 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.3923i 0.390843i
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.8564i 0.518927i
\(714\) 0 0
\(715\) 108.000 4.03897
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.73205i 0.0645946i −0.999478 0.0322973i \(-0.989718\pi\)
0.999478 0.0322973i \(-0.0102823\pi\)
\(720\) 0 0
\(721\) 13.8564i 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25.9808i 0.963573i −0.876289 0.481787i \(-0.839988\pi\)
0.876289 0.481787i \(-0.160012\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 25.9808i 0.960933i
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 41.5692i 1.53122i
\(738\) 0 0
\(739\) 12.1244i 0.446002i −0.974818 0.223001i \(-0.928415\pi\)
0.974818 0.223001i \(-0.0715853\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −27.0000 −0.989203
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 60.0000 2.18362
\(756\) 0 0
\(757\) −13.0000 −0.472493 −0.236247 0.971693i \(-0.575917\pi\)
−0.236247 + 0.971693i \(0.575917\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 83.1384i 3.00196i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.6410i 1.24595i −0.782241 0.622975i \(-0.785924\pi\)
0.782241 0.622975i \(-0.214076\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 + 27.7128i 0.429945 + 0.992915i
\(780\) 0 0
\(781\) 62.3538i 2.23120i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 48.4974i 1.72219i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 25.9808i 0.919133i
\(800\) 0 0
\(801\)