Properties

Label 3332.2.b.a.2549.1
Level $3332$
Weight $2$
Character 3332.2549
Analytic conductor $26.606$
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] = [3332,2,Mod(2549,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3332.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.b (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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 3332.2549
Dual form 3332.2.b.a.2549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421i q^{3} -2.82843i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.41421i q^{3} -2.82843i q^{5} +1.00000 q^{9} -1.41421i q^{11} +4.00000 q^{13} -4.00000 q^{15} +(-3.00000 + 2.82843i) q^{17} +4.00000 q^{19} -1.41421i q^{23} -3.00000 q^{25} -5.65685i q^{27} +2.82843i q^{29} -4.24264i q^{31} -2.00000 q^{33} -8.48528i q^{37} -5.65685i q^{39} -11.3137i q^{41} +8.00000 q^{43} -2.82843i q^{45} +12.0000 q^{47} +(4.00000 + 4.24264i) q^{51} -6.00000 q^{53} -4.00000 q^{55} -5.65685i q^{57} +8.48528i q^{61} -11.3137i q^{65} -4.00000 q^{67} -2.00000 q^{69} +7.07107i q^{71} +4.24264i q^{75} +4.24264i q^{79} -5.00000 q^{81} +(8.00000 + 8.48528i) q^{85} +4.00000 q^{87} -12.0000 q^{89} -6.00000 q^{93} -11.3137i q^{95} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{9} + 8 q^{13} - 8 q^{15} - 6 q^{17} + 8 q^{19} - 6 q^{25} - 4 q^{33} + 16 q^{43} + 24 q^{47} + 8 q^{51} - 12 q^{53} - 8 q^{55} - 8 q^{67} - 4 q^{69} - 10 q^{81} + 16 q^{85} + 8 q^{87} - 24 q^{89} - 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421i 0.816497i −0.912871 0.408248i \(-0.866140\pi\)
0.912871 0.408248i \(-0.133860\pi\)
\(4\) 0 0
\(5\) 2.82843i 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −3.00000 + 2.82843i −0.727607 + 0.685994i
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i −0.989071 0.147442i \(-0.952896\pi\)
0.989071 0.147442i \(-0.0471040\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 4.24264i 0.762001i −0.924575 0.381000i \(-0.875580\pi\)
0.924575 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528i 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 0 0
\(39\) 5.65685i 0.905822i
\(40\) 0 0
\(41\) 11.3137i 1.76690i −0.468521 0.883452i \(-0.655213\pi\)
0.468521 0.883452i \(-0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.82843i 0.421637i
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 + 4.24264i 0.560112 + 0.594089i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 5.65685i 0.749269i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.3137i 1.40329i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 7.07107i 0.839181i 0.907713 + 0.419591i \(0.137826\pi\)
−0.907713 + 0.419591i \(0.862174\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 4.24264i 0.489898i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.24264i 0.477334i 0.971101 + 0.238667i \(0.0767105\pi\)
−0.971101 + 0.238667i \(0.923290\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 8.00000 + 8.48528i 0.867722 + 0.920358i
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 11.3137i 1.16076i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 1.41421i 0.142134i
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 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
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i −0.913708 0.406371i \(-0.866794\pi\)
0.913708 0.406371i \(-0.133206\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 5.65685i 0.532152i −0.963952 0.266076i \(-0.914273\pi\)
0.963952 0.266076i \(-0.0857272\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) −16.0000 −1.44267
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 11.3137i 0.996116i
\(130\) 0 0
\(131\) 9.89949i 0.864923i 0.901652 + 0.432461i \(0.142355\pi\)
−0.901652 + 0.432461i \(0.857645\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −16.0000 −1.37706
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 12.7279i 1.07957i 0.841803 + 0.539784i \(0.181494\pi\)
−0.841803 + 0.539784i \(0.818506\pi\)
\(140\) 0 0
\(141\) 16.9706i 1.42918i
\(142\) 0 0
\(143\) 5.65685i 0.473050i
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −3.00000 + 2.82843i −0.242536 + 0.228665i
\(154\) 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
\(159\) 8.48528i 0.672927i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.24264i 0.332309i 0.986100 + 0.166155i \(0.0531351\pi\)
−0.986100 + 0.166155i \(0.946865\pi\)
\(164\) 0 0
\(165\) 5.65685i 0.440386i
\(166\) 0 0
\(167\) 7.07107i 0.547176i −0.961847 0.273588i \(-0.911790\pi\)
0.961847 0.273588i \(-0.0882104\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 14.1421i 1.07521i 0.843198 + 0.537603i \(0.180670\pi\)
−0.843198 + 0.537603i \(0.819330\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 8.48528i 0.630706i −0.948974 0.315353i \(-0.897877\pi\)
0.948974 0.315353i \(-0.102123\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 4.00000 + 4.24264i 0.292509 + 0.310253i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 16.9706i 1.22157i −0.791797 0.610784i \(-0.790854\pi\)
0.791797 0.610784i \(-0.209146\pi\)
\(194\) 0 0
\(195\) −16.0000 −1.14578
\(196\) 0 0
\(197\) 19.7990i 1.41062i 0.708899 + 0.705310i \(0.249192\pi\)
−0.708899 + 0.705310i \(0.750808\pi\)
\(198\) 0 0
\(199\) 21.2132i 1.50376i −0.659298 0.751882i \(-0.729146\pi\)
0.659298 0.751882i \(-0.270854\pi\)
\(200\) 0 0
\(201\) 5.65685i 0.399004i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −32.0000 −2.23498
\(206\) 0 0
\(207\) 1.41421i 0.0982946i
\(208\) 0 0
\(209\) 5.65685i 0.391293i
\(210\) 0 0
\(211\) 12.7279i 0.876226i 0.898920 + 0.438113i \(0.144353\pi\)
−0.898920 + 0.438113i \(0.855647\pi\)
\(212\) 0 0
\(213\) 10.0000 0.685189
\(214\) 0 0
\(215\) 22.6274i 1.54318i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 + 11.3137i −0.807207 + 0.761042i
\(222\) 0 0
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 18.3848i 1.22024i 0.792309 + 0.610120i \(0.208879\pi\)
−0.792309 + 0.610120i \(0.791121\pi\)
\(228\) 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
\(233\) 5.65685i 0.370593i −0.982683 0.185296i \(-0.940675\pi\)
0.982683 0.185296i \(-0.0593245\pi\)
\(234\) 0 0
\(235\) 33.9411i 2.21407i
\(236\) 0 0
\(237\) 6.00000 0.389742
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 9.89949i 0.635053i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 12.0000 11.3137i 0.751469 0.708492i
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.82843i 0.175075i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 16.9706i 1.04249i
\(266\) 0 0
\(267\) 16.9706i 1.03858i
\(268\) 0 0
\(269\) 2.82843i 0.172452i −0.996276 0.0862261i \(-0.972519\pi\)
0.996276 0.0862261i \(-0.0274808\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.24264i 0.255841i
\(276\) 0 0
\(277\) 8.48528i 0.509831i 0.966963 + 0.254916i \(0.0820477\pi\)
−0.966963 + 0.254916i \(0.917952\pi\)
\(278\) 0 0
\(279\) 4.24264i 0.254000i
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 12.7279i 0.756596i 0.925684 + 0.378298i \(0.123491\pi\)
−0.925684 + 0.378298i \(0.876509\pi\)
\(284\) 0 0
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 16.9706i 0.0588235 0.998268i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) 0 0
\(299\) 5.65685i 0.327144i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 16.9706i 0.974933i
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 11.3137i 0.643614i
\(310\) 0 0
\(311\) 26.8701i 1.52366i 0.647776 + 0.761831i \(0.275699\pi\)
−0.647776 + 0.761831i \(0.724301\pi\)
\(312\) 0 0
\(313\) 16.9706i 0.959233i −0.877478 0.479616i \(-0.840776\pi\)
0.877478 0.479616i \(-0.159224\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843i 0.158860i 0.996840 + 0.0794301i \(0.0253101\pi\)
−0.996840 + 0.0794301i \(0.974690\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) −12.0000 + 11.3137i −0.667698 + 0.629512i
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) −12.0000 −0.663602
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 8.48528i 0.464991i
\(334\) 0 0
\(335\) 11.3137i 0.618134i
\(336\) 0 0
\(337\) 16.9706i 0.924445i 0.886764 + 0.462223i \(0.152948\pi\)
−0.886764 + 0.462223i \(0.847052\pi\)
\(338\) 0 0
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.65685i 0.304555i
\(346\) 0 0
\(347\) 1.41421i 0.0759190i −0.999279 0.0379595i \(-0.987914\pi\)
0.999279 0.0379595i \(-0.0120858\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 22.6274i 1.20776i
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 20.0000 1.06149
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 12.7279i 0.668043i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 12.7279i 0.664392i −0.943210 0.332196i \(-0.892210\pi\)
0.943210 0.332196i \(-0.107790\pi\)
\(368\) 0 0
\(369\) 11.3137i 0.588968i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −8.00000 −0.413118
\(376\) 0 0
\(377\) 11.3137i 0.582686i
\(378\) 0 0
\(379\) 38.1838i 1.96137i −0.195598 0.980684i \(-0.562665\pi\)
0.195598 0.980684i \(-0.437335\pi\)
\(380\) 0 0
\(381\) 11.3137i 0.579619i
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 4.00000 + 4.24264i 0.202289 + 0.214560i
\(392\) 0 0
\(393\) 14.0000 0.706207
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 8.48528i 0.425864i −0.977067 0.212932i \(-0.931699\pi\)
0.977067 0.212932i \(-0.0683013\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i −0.825105 0.564980i \(-0.808884\pi\)
0.825105 0.564980i \(-0.191116\pi\)
\(402\) 0 0
\(403\) 16.9706i 0.845364i
\(404\) 0 0
\(405\) 14.1421i 0.702728i
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 16.9706i 0.837096i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 18.0000 0.881464
\(418\) 0 0
\(419\) 24.0416i 1.17451i −0.809402 0.587255i \(-0.800208\pi\)
0.809402 0.587255i \(-0.199792\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 9.00000 8.48528i 0.436564 0.411597i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 18.3848i 0.885564i −0.896629 0.442782i \(-0.853992\pi\)
0.896629 0.442782i \(-0.146008\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 11.3137i 0.542451i
\(436\) 0 0
\(437\) 5.65685i 0.270604i
\(438\) 0 0
\(439\) 38.1838i 1.82241i 0.411951 + 0.911206i \(0.364847\pi\)
−0.411951 + 0.911206i \(0.635153\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 33.9411i 1.60896i
\(446\) 0 0
\(447\) 8.48528i 0.401340i
\(448\) 0 0
\(449\) 5.65685i 0.266963i −0.991051 0.133482i \(-0.957384\pi\)
0.991051 0.133482i \(-0.0426157\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 0 0
\(453\) 11.3137i 0.531564i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 0 0
\(459\) 16.0000 + 16.9706i 0.746816 + 0.792118i
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 16.9706i 0.786991i
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.82843i 0.130327i
\(472\) 0 0
\(473\) 11.3137i 0.520205i
\(474\) 0 0
\(475\) −12.0000 −0.550598
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 18.3848i 0.840022i 0.907519 + 0.420011i \(0.137974\pi\)
−0.907519 + 0.420011i \(0.862026\pi\)
\(480\) 0 0
\(481\) 33.9411i 1.54758i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 29.6985i 1.34577i 0.739749 + 0.672883i \(0.234944\pi\)
−0.739749 + 0.672883i \(0.765056\pi\)
\(488\) 0 0
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −8.00000 8.48528i −0.360302 0.382158i
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 21.2132i 0.949633i −0.880085 0.474817i \(-0.842514\pi\)
0.880085 0.474817i \(-0.157486\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) 0 0
\(503\) 7.07107i 0.315283i −0.987496 0.157642i \(-0.949611\pi\)
0.987496 0.157642i \(-0.0503891\pi\)
\(504\) 0 0
\(505\) 33.9411i 1.51036i
\(506\) 0 0
\(507\) 4.24264i 0.188422i
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 22.6274i 0.999025i
\(514\) 0 0
\(515\) 22.6274i 0.997083i
\(516\) 0 0
\(517\) 16.9706i 0.746364i
\(518\) 0 0
\(519\) 20.0000 0.877903
\(520\) 0 0
\(521\) 22.6274i 0.991325i 0.868515 + 0.495663i \(0.165075\pi\)
−0.868515 + 0.495663i \(0.834925\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 + 12.7279i 0.522728 + 0.554437i
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 45.2548i 1.96020i
\(534\) 0 0
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) 16.9706i 0.732334i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.48528i 0.364811i −0.983223 0.182405i \(-0.941612\pi\)
0.983223 0.182405i \(-0.0583883\pi\)
\(542\) 0 0
\(543\) −12.0000 −0.514969
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 12.7279i 0.544207i 0.962268 + 0.272103i \(0.0877193\pi\)
−0.962268 + 0.272103i \(0.912281\pi\)
\(548\) 0 0
\(549\) 8.48528i 0.362143i
\(550\) 0 0
\(551\) 11.3137i 0.481980i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 33.9411i 1.44072i
\(556\) 0 0
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 6.00000 5.65685i 0.253320 0.238833i
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −16.0000 −0.673125
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 29.6985i 1.24284i 0.783476 + 0.621421i \(0.213445\pi\)
−0.783476 + 0.621421i \(0.786555\pi\)
\(572\) 0 0
\(573\) 16.9706i 0.708955i
\(574\) 0 0
\(575\) 4.24264i 0.176930i
\(576\) 0 0
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.48528i 0.351424i
\(584\) 0 0
\(585\) 11.3137i 0.467764i
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 16.9706i 0.699260i
\(590\) 0 0
\(591\) 28.0000 1.15177
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.0000 −1.22782
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 16.9706i 0.692244i 0.938190 + 0.346122i \(0.112502\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 25.4558i 1.03493i
\(606\) 0 0
\(607\) 29.6985i 1.20542i 0.797959 + 0.602712i \(0.205913\pi\)
−0.797959 + 0.602712i \(0.794087\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 0 0
\(615\) 45.2548i 1.82485i
\(616\) 0 0
\(617\) 39.5980i 1.59415i −0.603877 0.797077i \(-0.706378\pi\)
0.603877 0.797077i \(-0.293622\pi\)
\(618\) 0 0
\(619\) 12.7279i 0.511578i 0.966733 + 0.255789i \(0.0823353\pi\)
−0.966733 + 0.255789i \(0.917665\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 24.0000 + 25.4558i 0.956943 + 1.01499i
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 18.0000 0.715436
\(634\) 0 0
\(635\) 22.6274i 0.897942i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7.07107i 0.279727i
\(640\) 0 0
\(641\) 45.2548i 1.78746i 0.448607 + 0.893729i \(0.351920\pi\)
−0.448607 + 0.893729i \(0.648080\pi\)
\(642\) 0 0
\(643\) 46.6690i 1.84045i −0.391393 0.920224i \(-0.628007\pi\)
0.391393 0.920224i \(-0.371993\pi\)
\(644\) 0 0
\(645\) −32.0000 −1.26000
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.7990i 0.774794i 0.921913 + 0.387397i \(0.126626\pi\)
−0.921913 + 0.387397i \(0.873374\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 16.0000 + 16.9706i 0.621389 + 0.659082i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 28.2843i 1.09353i
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 16.9706i 0.654167i 0.944995 + 0.327084i \(0.106066\pi\)
−0.944995 + 0.327084i \(0.893934\pi\)
\(674\) 0 0
\(675\) 16.9706i 0.653197i
\(676\) 0 0
\(677\) 2.82843i 0.108705i −0.998522 0.0543526i \(-0.982690\pi\)
0.998522 0.0543526i \(-0.0173095\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) 0 0
\(683\) 41.0122i 1.56929i 0.619947 + 0.784644i \(0.287154\pi\)
−0.619947 + 0.784644i \(0.712846\pi\)
\(684\) 0 0
\(685\) 33.9411i 1.29682i
\(686\) 0 0
\(687\) 5.65685i 0.215822i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 4.24264i 0.161398i −0.996739 0.0806988i \(-0.974285\pi\)
0.996739 0.0806988i \(-0.0257152\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 36.0000 1.36556
\(696\) 0 0
\(697\) 32.0000 + 33.9411i 1.21209 + 1.28561i
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 33.9411i 1.28011i
\(704\) 0 0
\(705\) −48.0000 −1.80778
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.48528i 0.318671i −0.987224 0.159336i \(-0.949065\pi\)
0.987224 0.159336i \(-0.0509352\pi\)
\(710\) 0 0
\(711\) 4.24264i 0.159111i
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 16.9706i 0.633777i
\(718\) 0 0
\(719\) 24.0416i 0.896602i −0.893883 0.448301i \(-0.852029\pi\)
0.893883 0.448301i \(-0.147971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.48528i 0.315135i
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) −24.0000 + 22.6274i −0.887672 + 0.836905i
\(732\) 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
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 0 0
\(741\) 22.6274i 0.831239i
\(742\) 0 0
\(743\) 1.41421i 0.0518825i −0.999663 0.0259412i \(-0.991742\pi\)
0.999663 0.0259412i \(-0.00825828\pi\)
\(744\) 0 0
\(745\) 16.9706i 0.621753i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.7279i 0.464448i 0.972662 + 0.232224i \(0.0746003\pi\)
−0.972662 + 0.232224i \(0.925400\pi\)
\(752\) 0 0
\(753\) 16.9706i 0.618442i
\(754\) 0 0
\(755\) 22.6274i 0.823496i
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) 0 0
\(759\) 2.82843i 0.102665i
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000 + 8.48528i 0.289241 + 0.306786i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) 16.9706i 0.611180i
\(772\) 0 0
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 12.7279i 0.457200i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 45.2548i 1.62142i
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) 16.0000 0.571793
\(784\) 0 0
\(785\) 5.65685i 0.201902i
\(786\) 0 0
\(787\) 29.6985i 1.05864i 0.848423 + 0.529318i \(0.177552\pi\)
−0.848423 + 0.529318i \(0.822448\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 33.9411i 1.20528i
\(794\) 0 0
\(795\) 24.0000 0.851192
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −36.0000 + 33.9411i −1.27359 + 1.20075i