Properties

Label 440.2.b.a.89.1
Level $440$
Weight $2$
Character 440.89
Analytic conductor $3.513$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(89,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 89.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 440.89
Dual form 440.2.b.a.89.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +(-4.00000 + 2.00000i) q^{15} +4.00000i q^{17} -4.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} -6.00000i q^{43} +(1.00000 + 2.00000i) q^{45} +2.00000i q^{47} +3.00000 q^{49} +8.00000 q^{51} -12.0000i q^{53} +(1.00000 + 2.00000i) q^{55} +8.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} +2.00000i q^{63} -10.0000i q^{67} -12.0000 q^{69} +8.00000 q^{71} +4.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +2.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} +2.00000i q^{83} +(8.00000 - 4.00000i) q^{85} +4.00000i q^{87} +10.0000 q^{89} -16.0000i q^{93} +(4.00000 + 8.00000i) q^{95} +8.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} - 8 q^{19} - 8 q^{21} - 6 q^{25} - 4 q^{29} + 16 q^{31} - 8 q^{35} - 12 q^{41} + 2 q^{45} + 6 q^{49} + 16 q^{51} + 2 q^{55} - 8 q^{59} + 28 q^{61} - 24 q^{69} + 16 q^{71} + 16 q^{75} + 16 q^{79} - 22 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 2.00000i 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) −1.00000 2.00000i −0.447214 0.894427i
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −4.00000 + 2.00000i −1.03280 + 0.516398i
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) −4.00000 + 2.00000i −0.676123 + 0.338062i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) 1.00000 + 2.00000i 0.134840 + 0.269680i
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000i 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 8.00000 + 6.00000i 0.923760 + 0.692820i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 8.00000 4.00000i 0.867722 0.433861i
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 16.0000i 1.65912i
\(94\) 0 0
\(95\) 4.00000 + 8.00000i 0.410391 + 0.820783i
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 4.00000 + 8.00000i 0.390360 + 0.780720i
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 8.00000i 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 0 0
\(115\) −12.0000 + 6.00000i −1.11901 + 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) 11.0000 + 2.00000i 0.983870 + 0.178885i
\(126\) 0 0
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) −8.00000 + 4.00000i −0.688530 + 0.344265i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 + 4.00000i 0.166091 + 0.332182i
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) −8.00000 16.0000i −0.642575 1.28515i
\(156\) 0 0
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 0 0
\(165\) 4.00000 2.00000i 0.311400 0.155700i
\(166\) 0 0
\(167\) 6.00000i 0.464294i 0.972681 + 0.232147i \(0.0745750\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 8.00000i 0.608229i −0.952636 0.304114i \(-0.901639\pi\)
0.952636 0.304114i \(-0.0983605\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 8.00000i 0.601317i
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 28.0000i 2.06982i
\(184\) 0 0
\(185\) 8.00000 4.00000i 0.588172 0.294086i
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000i 1.70993i −0.518686 0.854965i \(-0.673579\pi\)
0.518686 0.854965i \(-0.326421\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −20.0000 −1.41069
\(202\) 0 0
\(203\) 4.00000i 0.280745i
\(204\) 0 0
\(205\) 6.00000 + 12.0000i 0.419058 + 0.838116i
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) −12.0000 + 6.00000i −0.818393 + 0.409197i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000i 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 3.00000 4.00000i 0.200000 0.266667i
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 4.00000i 0.262049i −0.991379 0.131024i \(-0.958173\pi\)
0.991379 0.131024i \(-0.0418266\pi\)
\(234\) 0 0
\(235\) 4.00000 2.00000i 0.260931 0.130466i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) −3.00000 6.00000i −0.191663 0.383326i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) −8.00000 16.0000i −0.500979 1.00196i
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) −24.0000 + 12.0000i −1.47431 + 0.737154i
\(266\) 0 0
\(267\) 20.0000i 1.22398i
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 4.00000i 0.180907 0.241209i
\(276\) 0 0
\(277\) 32.0000i 1.92269i −0.275340 0.961347i \(-0.588791\pi\)
0.275340 0.961347i \(-0.411209\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 26.0000i 1.54554i 0.634686 + 0.772770i \(0.281129\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(284\) 0 0
\(285\) 16.0000 8.00000i 0.947758 0.473879i
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 0 0
\(293\) 8.00000i 0.467365i 0.972313 + 0.233682i \(0.0750776\pi\)
−0.972313 + 0.233682i \(0.924922\pi\)
\(294\) 0 0
\(295\) 4.00000 + 8.00000i 0.232889 + 0.465778i
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 20.0000i 1.14897i
\(304\) 0 0
\(305\) −14.0000 28.0000i −0.801638 1.60328i
\(306\) 0 0
\(307\) 26.0000i 1.48390i 0.670456 + 0.741949i \(0.266098\pi\)
−0.670456 + 0.741949i \(0.733902\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 24.0000i 1.35656i 0.734803 + 0.678280i \(0.237274\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) 4.00000 2.00000i 0.225374 0.112687i
\(316\) 0 0
\(317\) 20.0000i 1.12331i 0.827371 + 0.561656i \(0.189836\pi\)
−0.827371 + 0.561656i \(0.810164\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 28.0000i 1.54840i
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) −20.0000 + 10.0000i −1.09272 + 0.546358i
\(336\) 0 0
\(337\) 20.0000i 1.08947i 0.838608 + 0.544735i \(0.183370\pi\)
−0.838608 + 0.544735i \(0.816630\pi\)
\(338\) 0 0
\(339\) −16.0000 −0.869001
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 12.0000 + 24.0000i 0.646058 + 1.29212i
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.0000i 1.70319i 0.524202 + 0.851594i \(0.324364\pi\)
−0.524202 + 0.851594i \(0.675636\pi\)
\(354\) 0 0
\(355\) −8.00000 16.0000i −0.424596 0.849192i
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 2.00000i 0.104973i
\(364\) 0 0
\(365\) 8.00000 4.00000i 0.418739 0.209370i
\(366\) 0 0
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 32.0000i 1.65690i 0.560065 + 0.828449i \(0.310776\pi\)
−0.560065 + 0.828449i \(0.689224\pi\)
\(374\) 0 0
\(375\) 4.00000 22.0000i 0.206559 1.13608i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 28.0000 1.43448
\(382\) 0 0
\(383\) 6.00000i 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 0 0
\(385\) 4.00000 2.00000i 0.203859 0.101929i
\(386\) 0 0
\(387\) 6.00000i 0.304997i
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 24.0000i 1.21064i
\(394\) 0 0
\(395\) −8.00000 16.0000i −0.402524 0.805047i
\(396\) 0 0
\(397\) 4.00000i 0.200754i 0.994949 + 0.100377i \(0.0320049\pi\)
−0.994949 + 0.100377i \(0.967995\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 11.0000 + 22.0000i 0.546594 + 1.09319i
\(406\) 0 0
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 0 0
\(415\) 4.00000 2.00000i 0.196352 0.0981761i
\(416\) 0 0
\(417\) 24.0000i 1.17529i
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) −16.0000 12.0000i −0.776114 0.582086i
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 8.00000i 0.384455i −0.981350 0.192228i \(-0.938429\pi\)
0.981350 0.192228i \(-0.0615712\pi\)
\(434\) 0 0
\(435\) 8.00000 4.00000i 0.383571 0.191785i
\(436\) 0 0
\(437\) 24.0000i 1.14808i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 26.0000i 1.23530i −0.786454 0.617649i \(-0.788085\pi\)
0.786454 0.617649i \(-0.211915\pi\)
\(444\) 0 0
\(445\) −10.0000 20.0000i −0.474045 0.948091i
\(446\) 0 0
\(447\) 44.0000i 2.08113i
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000i 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 0 0
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 18.0000i 0.836531i 0.908325 + 0.418265i \(0.137362\pi\)
−0.908325 + 0.418265i \(0.862638\pi\)
\(464\) 0 0
\(465\) −32.0000 + 16.0000i −1.48396 + 0.741982i
\(466\) 0 0
\(467\) 22.0000i 1.01804i 0.860755 + 0.509019i \(0.169992\pi\)
−0.860755 + 0.509019i \(0.830008\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 12.0000 16.0000i 0.550598 0.734130i
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.0000i 1.09204i
\(484\) 0 0
\(485\) 16.0000 8.00000i 0.726523 0.363261i
\(486\) 0 0
\(487\) 6.00000i 0.271886i −0.990717 0.135943i \(-0.956594\pi\)
0.990717 0.135943i \(-0.0434064\pi\)
\(488\) 0 0
\(489\) 28.0000 1.26620
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) −1.00000 2.00000i −0.0449467 0.0898933i
\(496\) 0 0
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 30.0000i 1.33763i 0.743427 + 0.668817i \(0.233199\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(504\) 0 0
\(505\) 10.0000 + 20.0000i 0.444994 + 0.889988i
\(506\) 0 0
\(507\) 26.0000i 1.15470i
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) −28.0000 + 14.0000i −1.23383 + 0.616914i
\(516\) 0 0
\(517\) 2.00000i 0.0879599i
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 14.0000i 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) 0 0
\(525\) 12.0000 16.0000i 0.523723 0.698297i
\(526\) 0 0
\(527\) 32.0000i 1.39394i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.00000 2.00000i 0.172935 0.0864675i
\(536\) 0 0
\(537\) 40.0000i 1.72613i
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 20.0000i 0.858282i
\(544\) 0 0
\(545\) −14.0000 28.0000i −0.599694 1.19939i
\(546\) 0 0
\(547\) 30.0000i 1.28271i −0.767245 0.641354i \(-0.778373\pi\)
0.767245 0.641354i \(-0.221627\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −8.00000 16.0000i −0.339581 0.679162i
\(556\) 0 0
\(557\) 32.0000i 1.35588i 0.735116 + 0.677942i \(0.237128\pi\)
−0.735116 + 0.677942i \(0.762872\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 42.0000i 1.77009i 0.465506 + 0.885044i \(0.345872\pi\)
−0.465506 + 0.885044i \(0.654128\pi\)
\(564\) 0 0
\(565\) −16.0000 + 8.00000i −0.673125 + 0.336563i
\(566\) 0 0
\(567\) 22.0000i 0.923913i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) 48.0000i 2.00523i
\(574\) 0 0
\(575\) 24.0000 + 18.0000i 1.00087 + 0.750652i
\(576\) 0 0
\(577\) 8.00000i 0.333044i −0.986038 0.166522i \(-0.946746\pi\)
0.986038 0.166522i \(-0.0532537\pi\)
\(578\) 0 0
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000i 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −48.0000 −1.97446
\(592\) 0 0
\(593\) 36.0000i 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) −8.00000 16.0000i −0.327968 0.655936i
\(596\) 0 0
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 10.0000i 0.407231i
\(604\) 0 0
\(605\) −1.00000 2.00000i −0.0406558 0.0813116i
\(606\) 0 0
\(607\) 18.0000i 0.730597i −0.930890 0.365299i \(-0.880967\pi\)
0.930890 0.365299i \(-0.119033\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 0 0
\(615\) 24.0000 12.0000i 0.967773 0.483887i
\(616\) 0 0
\(617\) 32.0000i 1.28827i −0.764911 0.644136i \(-0.777217\pi\)
0.764911 0.644136i \(-0.222783\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) 20.0000i 0.801283i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 40.0000i 1.58986i
\(634\) 0 0
\(635\) 28.0000 14.0000i 1.11115 0.555573i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 2.00000i 0.0788723i −0.999222 0.0394362i \(-0.987444\pi\)
0.999222 0.0394362i \(-0.0125562\pi\)
\(644\) 0 0
\(645\) 12.0000 + 24.0000i 0.472500 + 0.944999i
\(646\) 0 0
\(647\) 30.0000i 1.17942i −0.807614 0.589711i \(-0.799242\pi\)
0.807614 0.589711i \(-0.200758\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 0 0
\(653\) 36.0000i 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 12.0000 + 24.0000i 0.468879 + 0.937758i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 8.00000i 0.620453 0.310227i
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 44.0000i 1.69608i 0.529936 + 0.848038i \(0.322216\pi\)
−0.529936 + 0.848038i \(0.677784\pi\)
\(674\) 0 0
\(675\) 16.0000 + 12.0000i 0.615840 + 0.461880i
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) 18.0000i 0.688751i −0.938832 0.344375i \(-0.888091\pi\)
0.938832 0.344375i \(-0.111909\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 12.0000i 0.457829i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) −12.0000 24.0000i −0.455186 0.910372i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 16.0000i 0.603451i
\(704\) 0 0
\(705\) −4.00000 8.00000i −0.150649 0.301297i
\(706\) 0 0
\(707\) 20.0000i 0.752177i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 48.0000i 1.79761i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) 60.0000i 2.23142i
\(724\) 0 0
\(725\) 6.00000 8.00000i 0.222834 0.297113i
\(726\) 0 0
\(727\) 30.0000i 1.11264i −0.830969 0.556319i \(-0.812213\pi\)
0.830969 0.556319i \(-0.187787\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 24.0000i 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 0 0
\(735\) −12.0000 + 6.00000i −0.442627 + 0.221313i
\(736\) 0 0
\(737\) 10.0000i 0.368355i
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) −22.0000 44.0000i −0.806018 1.61204i
\(746\) 0 0
\(747\) 2.00000i 0.0731762i
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 24.0000i 0.874609i
\(754\) 0 0
\(755\) 8.00000 + 16.0000i 0.291150 + 0.582300i
\(756\) 0 0
\(757\) 20.0000i 0.726912i −0.931611 0.363456i \(-0.881597\pi\)
0.931611 0.363456i \(-0.118403\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 28.0000i 1.01367i
\(764\) 0 0
\(765\) −8.00000 + 4.00000i −0.289241 + 0.144620i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 0 0
\(775\) −24.0000 + 32.0000i −0.862105 + 1.14947i
\(776\) 0 0
\(777\) 16.0000i 0.573997i
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) 0 0
\(785\) 8.00000 4.00000i 0.285532 0.142766i
\(786\) 0 0
\(787\) 6.00000i 0.213877i −0.994266 0.106938i \(-0.965895\pi\)
0.994266 0.106938i \(-0.0341048\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −16.0000 −0.568895
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 24.0000 + 48.0000i 0.851192 + 1.70238i
\(796\) 0 0
\(797\) 36.0000i