Properties

Label 416.2.i.e.321.1
Level $416$
Weight $2$
Character 416.321
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(289,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.i (of order \(3\), degree \(2\), 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: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 321.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 416.321
Dual form 416.2.i.e.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.46410 q^{5} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q-2.46410 q^{5} +(1.50000 + 2.59808i) q^{9} +(0.232051 + 3.59808i) q^{13} +(3.96410 + 6.86603i) q^{17} +1.07180 q^{25} +(-4.23205 + 7.33013i) q^{29} +(4.69615 - 8.13397i) q^{37} +(-0.964102 + 1.66987i) q^{41} +(-3.69615 - 6.40192i) q^{45} +(3.50000 - 6.06218i) q^{49} -10.4641 q^{53} +(2.69615 + 4.66987i) q^{61} +(-0.571797 - 8.86603i) q^{65} +16.8564 q^{73} +(-4.50000 + 7.79423i) q^{81} +(-9.76795 - 16.9186i) q^{85} +(-5.00000 + 8.66025i) q^{89} +(-9.00000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 6 q^{9} - 6 q^{13} + 2 q^{17} + 32 q^{25} - 10 q^{29} - 2 q^{37} + 10 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} - 10 q^{61} - 30 q^{65} + 12 q^{73} - 18 q^{81} - 46 q^{85} - 20 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) −2.46410 −1.10198 −0.550990 0.834512i \(-0.685750\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0.232051 + 3.59808i 0.0643593 + 0.997927i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.96410 + 6.86603i 0.961436 + 1.66526i 0.718900 + 0.695113i \(0.244646\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 1.07180 0.214359
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.23205 + 7.33013i −0.785872 + 1.36117i 0.142605 + 0.989780i \(0.454452\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.69615 8.13397i 0.772043 1.33722i −0.164399 0.986394i \(-0.552568\pi\)
0.936442 0.350823i \(-0.114098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.964102 + 1.66987i −0.150567 + 0.260790i −0.931436 0.363905i \(-0.881443\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(44\) 0 0
\(45\) −3.69615 6.40192i −0.550990 0.954342i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 3.50000 6.06218i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.4641 −1.43735 −0.718677 0.695344i \(-0.755252\pi\)
−0.718677 + 0.695344i \(0.755252\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 2.69615 + 4.66987i 0.345207 + 0.597916i 0.985391 0.170305i \(-0.0544754\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.571797 8.86603i −0.0709227 1.09970i
\(66\) 0 0
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0 0
\(73\) 16.8564 1.97289 0.986447 0.164083i \(-0.0524664\pi\)
0.986447 + 0.164083i \(0.0524664\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −9.76795 16.9186i −1.05948 1.83508i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.00000 + 8.66025i −0.529999 + 0.917985i 0.469389 + 0.882992i \(0.344474\pi\)
−0.999388 + 0.0349934i \(0.988859\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.00000 15.5885i −0.913812 1.58277i −0.808632 0.588315i \(-0.799792\pi\)
−0.105180 0.994453i \(-0.533542\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.16025 14.1340i 0.811976 1.40638i −0.0995037 0.995037i \(-0.531726\pi\)
0.911479 0.411346i \(-0.134941\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.4282 18.0622i −0.981003 1.69915i −0.658505 0.752577i \(-0.728811\pi\)
−0.322498 0.946570i \(-0.604523\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −9.00000 + 6.00000i −0.832050 + 0.554700i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.67949 0.865760
\(126\) 0 0
\(127\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.96410 15.5263i −0.765855 1.32650i −0.939793 0.341743i \(-0.888983\pi\)
0.173939 0.984757i \(-0.444351\pi\)
\(138\) 0 0
\(139\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 10.4282 18.0622i 0.866015 1.49998i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.1603 + 21.0622i 0.996207 + 1.72548i 0.573462 + 0.819232i \(0.305600\pi\)
0.422744 + 0.906249i \(0.361067\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −11.8923 + 20.5981i −0.961436 + 1.66526i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.607695 −0.0484994 −0.0242497 0.999706i \(-0.507720\pi\)
−0.0242497 + 0.999706i \(0.507720\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) −12.8923 + 1.66987i −0.991716 + 0.128452i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.0000 + 22.5167i 0.988372 + 1.71191i 0.625871 + 0.779926i \(0.284744\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 26.3205 1.95639 0.978194 0.207693i \(-0.0665956\pi\)
0.978194 + 0.207693i \(0.0665956\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −11.5718 + 20.0429i −0.850775 + 1.47359i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −13.8923 + 24.0622i −0.999990 + 1.73203i −0.496119 + 0.868255i \(0.665242\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00000 1.73205i 0.0712470 0.123404i −0.828201 0.560431i \(-0.810635\pi\)
0.899448 + 0.437028i \(0.143969\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.37564 4.11474i 0.165922 0.287386i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −23.7846 + 15.8564i −1.59993 + 1.06662i
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0 0
\(225\) 1.60770 + 2.78461i 0.107180 + 0.185641i
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.03590 6.99038i −0.259975 0.450290i 0.706260 0.707953i \(-0.250381\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8.62436 + 14.9378i −0.550990 + 0.954342i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.3564 24.8660i 0.895528 1.55110i 0.0623783 0.998053i \(-0.480131\pi\)
0.833150 0.553047i \(-0.186535\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −25.3923 −1.57174
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 25.7846 1.58394
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.0000 + 22.5167i 0.792624 + 1.37287i 0.924337 + 0.381577i \(0.124619\pi\)
−0.131713 + 0.991288i \(0.542048\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.6244 28.7942i −0.998861 1.73008i −0.540758 0.841178i \(-0.681862\pi\)
−0.458103 0.888899i \(-0.651471\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.7128 −1.95148 −0.975741 0.218926i \(-0.929745\pi\)
−0.975741 + 0.218926i \(0.929745\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −22.9282 + 39.7128i −1.34872 + 2.33605i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.2321 17.7224i −0.597763 1.03536i −0.993151 0.116841i \(-0.962723\pi\)
0.395388 0.918514i \(-0.370610\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.64359 11.5070i −0.380411 0.658891i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.2487 0.744122 0.372061 0.928208i \(-0.378651\pi\)
0.372061 + 0.928208i \(0.378651\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0.248711 + 3.85641i 0.0137960 + 0.213915i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0 0
\(333\) 28.1769 1.54409
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.7128 1.01935 0.509676 0.860366i \(-0.329765\pi\)
0.509676 + 0.860366i \(0.329765\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 5.00000 8.66025i 0.267644 0.463573i −0.700609 0.713545i \(-0.747088\pi\)
0.968253 + 0.249973i \(0.0804216\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.57180 2.72243i 0.0836583 0.144900i −0.821160 0.570697i \(-0.806673\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 9.50000 16.4545i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −41.5359 −2.17409
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0 0
\(369\) −5.78461 −0.301135
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −12.0885 20.9378i −0.625917 1.08412i −0.988363 0.152115i \(-0.951392\pi\)
0.362446 0.932005i \(-0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −27.3564 13.5263i −1.40893 0.696639i
\(378\) 0 0
\(379\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.320508 −0.0162504 −0.00812520 0.999967i \(-0.502586\pi\)
−0.00812520 + 0.999967i \(0.502586\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −19.0000 32.9090i −0.953583 1.65165i −0.737579 0.675261i \(-0.764031\pi\)
−0.216004 0.976392i \(-0.569302\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.8205 30.8660i 0.889914 1.54138i 0.0499376 0.998752i \(-0.484098\pi\)
0.839976 0.542623i \(-0.182569\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 11.0885 19.2058i 0.550990 0.954342i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −18.8205 32.5981i −0.930614 1.61187i −0.782274 0.622935i \(-0.785940\pi\)
−0.148340 0.988936i \(-0.547393\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −39.2487 −1.91287 −0.956433 0.291953i \(-0.905695\pi\)
−0.956433 + 0.291953i \(0.905695\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.24871 + 7.35898i 0.206093 + 0.356963i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(432\) 0 0
\(433\) 18.8923 + 32.7224i 0.907906 + 1.57254i 0.816968 + 0.576683i \(0.195653\pi\)
0.0909384 + 0.995857i \(0.471013\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 21.0000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 12.3205 21.3397i 0.584048 1.01160i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.00000 + 12.1244i 0.330350 + 0.572184i 0.982581 0.185837i \(-0.0594997\pi\)
−0.652230 + 0.758021i \(0.726166\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.9641 24.1865i 0.653213 1.13140i −0.329125 0.944286i \(-0.606754\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.839746 + 1.45448i 0.0391109 + 0.0677420i 0.884918 0.465746i \(-0.154214\pi\)
−0.845807 + 0.533488i \(0.820881\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.6962 27.1865i −0.718677 1.24479i
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 30.3564 + 15.0096i 1.38413 + 0.684380i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 22.1769 + 38.4115i 1.00700 + 1.74418i
\(486\) 0 0
\(487\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) −67.1051 −3.02226
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) −20.1077 + 34.8275i −0.894781 + 1.54981i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.5526 + 37.3301i −0.955300 + 1.65463i −0.221621 + 0.975133i \(0.571135\pi\)
−0.733679 + 0.679496i \(0.762199\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 45.6410 1.99957 0.999785 0.0207541i \(-0.00660670\pi\)
0.999785 + 0.0207541i \(0.00660670\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.23205 3.08142i −0.269940 0.133471i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.3205 1.64753 0.823764 0.566933i \(-0.191870\pi\)
0.823764 + 0.566933i \(0.191870\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.7846 −0.633303
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −8.08846 + 14.0096i −0.345207 + 0.597916i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.6244 37.4545i 0.916253 1.58700i 0.111198 0.993798i \(-0.464531\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 25.6962 + 44.5070i 1.08105 + 1.87243i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.0000 + 22.5167i −0.544988 + 0.943948i 0.453619 + 0.891196i \(0.350133\pi\)
−0.998608 + 0.0527519i \(0.983201\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −42.5692 −1.77218 −0.886090 0.463513i \(-0.846589\pi\)
−0.886090 + 0.463513i \(0.846589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 22.1769 14.7846i 0.916903 0.611268i
\(586\) 0 0
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.14359 0.375482 0.187741 0.982219i \(-0.439883\pi\)
0.187741 + 0.982219i \(0.439883\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −18.2846 + 31.6699i −0.745845 + 1.29184i 0.203954 + 0.978980i \(0.434621\pi\)
−0.949799 + 0.312861i \(0.898713\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −13.5526 23.4737i −0.550990 0.954342i
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −24.0885 + 41.7224i −0.972924 + 1.68515i −0.286300 + 0.958140i \(0.592425\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.3564 40.4545i −0.940294 1.62864i −0.764911 0.644136i \(-0.777217\pi\)
−0.175382 0.984500i \(-0.556116\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.2102 −1.16841
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 74.4641 2.96908
\(630\) 0 0
\(631\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.6244 + 11.1865i 0.896410 + 0.443227i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.03590 + 15.6506i 0.356897 + 0.618163i 0.987441 0.157991i \(-0.0505015\pi\)
−0.630544 + 0.776153i \(0.717168\pi\)
\(642\) 0 0
\(643\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.0000 22.5167i 0.508729 0.881145i −0.491220 0.871036i \(-0.663449\pi\)
0.999949 0.0101092i \(-0.00321793\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 25.2846 + 43.7942i 0.986447 + 1.70858i
\(658\) 0 0
\(659\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) −17.6962 + 30.6506i −0.688301 + 1.19217i 0.284087 + 0.958799i \(0.408310\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.10770 + 1.91858i −0.0426985 + 0.0739560i −0.886585 0.462566i \(-0.846929\pi\)
0.843886 + 0.536522i \(0.180262\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 22.0885 + 38.2583i 0.843957 + 1.46178i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.42820 37.6506i −0.0925072 1.43437i
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −15.2872 −0.579044
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 26.5526 + 45.9904i 0.997202 + 1.72721i 0.563337 + 0.826227i \(0.309517\pi\)
0.433865 + 0.900978i \(0.357149\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.53590 + 7.85641i −0.168459 + 0.291780i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −23.5359 −0.869318 −0.434659 0.900595i \(-0.643131\pi\)
−0.434659 + 0.900595i \(0.643131\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) −29.9641 51.8993i −1.09780 1.90144i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.00000 15.5885i 0.327111 0.566572i −0.654827 0.755779i \(-0.727258\pi\)
0.981937 + 0.189207i \(0.0605917\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.0000 + 32.9090i 0.688749 + 1.19295i 0.972243 + 0.233975i \(0.0751733\pi\)
−0.283493 + 0.958974i \(0.591493\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 29.3038 50.7558i 1.05948 1.83508i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −25.0000 + 43.3013i −0.901523 + 1.56148i −0.0760054 + 0.997107i \(0.524217\pi\)
−0.825518 + 0.564376i \(0.809117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.0000 + 29.4449i 0.611448 + 1.05906i 0.990997 + 0.133887i \(0.0427458\pi\)
−0.379549 + 0.925172i \(0.623921\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.49742 0.0534453
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16.1769 + 10.7846i −0.574459 + 0.382973i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.0000 19.0526i −0.389640 0.674876i 0.602761 0.797922i \(-0.294067\pi\)
−0.992401 + 0.123045i \(0.960734\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 0 0
\(804\) 0