Properties

Label 416.2.f.a.129.1
Level $416$
Weight $2$
Character 416.129
Analytic conductor $3.322$
Analytic rank $1$
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] = [416,2,Mod(129,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, 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("416.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.f (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: \(3.32177672409\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 416.129
Dual form 416.2.f.a.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -3.00000i q^{5} +1.00000i q^{7} +6.00000 q^{9} -4.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +9.00000i q^{15} -5.00000 q^{17} +6.00000i q^{19} -3.00000i q^{21} -6.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} +12.0000i q^{33} +3.00000 q^{35} +3.00000i q^{37} +(6.00000 - 9.00000i) q^{39} +12.0000i q^{41} -3.00000 q^{43} -18.0000i q^{45} -7.00000i q^{47} +6.00000 q^{49} +15.0000 q^{51} -2.00000 q^{53} -12.0000 q^{55} -18.0000i q^{57} +2.00000i q^{59} -12.0000 q^{61} +6.00000i q^{63} +(9.00000 + 6.00000i) q^{65} +4.00000i q^{67} +18.0000 q^{69} -11.0000i q^{71} -6.00000i q^{73} +12.0000 q^{75} +4.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} +10.0000i q^{83} +15.0000i q^{85} +12.0000 q^{87} -6.00000i q^{89} +(-3.00000 - 2.00000i) q^{91} +18.0000 q^{95} -24.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 12 q^{9} - 4 q^{13} - 10 q^{17} - 12 q^{23} - 8 q^{25} - 18 q^{27} - 8 q^{29} + 6 q^{35} + 12 q^{39} - 6 q^{43} + 12 q^{49} + 30 q^{51} - 4 q^{53} - 24 q^{55} - 24 q^{61} + 18 q^{65} + 36 q^{69} + 24 q^{75} + 8 q^{77} + 12 q^{79} + 18 q^{81} + 24 q^{87} - 6 q^{91} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 3.00000i 1.34164i −0.741620 0.670820i \(-0.765942\pi\)
0.741620 0.670820i \(-0.234058\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 4.00000i 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.00000i −0.554700 + 0.832050i
\(14\) 0 0
\(15\) 9.00000i 2.32379i
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 3.00000i 0.654654i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 12.0000i 2.08893i
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) 6.00000 9.00000i 0.960769 1.44115i
\(40\) 0 0
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −3.00000 −0.457496 −0.228748 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) 0 0
\(45\) 18.0000i 2.68328i
\(46\) 0 0
\(47\) 7.00000i 1.02105i −0.859861 0.510527i \(-0.829450\pi\)
0.859861 0.510527i \(-0.170550\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 18.0000i 2.38416i
\(58\) 0 0
\(59\) 2.00000i 0.260378i 0.991489 + 0.130189i \(0.0415584\pi\)
−0.991489 + 0.130189i \(0.958442\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 0 0
\(65\) 9.00000 + 6.00000i 1.11631 + 0.744208i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) 11.0000i 1.30546i −0.757591 0.652730i \(-0.773624\pi\)
0.757591 0.652730i \(-0.226376\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 10.0000i 1.09764i 0.835940 + 0.548821i \(0.184923\pi\)
−0.835940 + 0.548821i \(0.815077\pi\)
\(84\) 0 0
\(85\) 15.0000i 1.62698i
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −3.00000 2.00000i −0.314485 0.209657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.0000 1.84676
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 24.0000i 2.41209i
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 15.0000i 1.43674i −0.695662 0.718370i \(-0.744889\pi\)
0.695662 0.718370i \(-0.255111\pi\)
\(110\) 0 0
\(111\) 9.00000i 0.854242i
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 18.0000i 1.67851i
\(116\) 0 0
\(117\) −12.0000 + 18.0000i −1.10940 + 1.66410i
\(118\) 0 0
\(119\) 5.00000i 0.458349i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 36.0000i 3.24601i
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 0 0
\(129\) 9.00000 0.792406
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 27.0000i 2.32379i
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −9.00000 −0.763370 −0.381685 0.924292i \(-0.624656\pi\)
−0.381685 + 0.924292i \(0.624656\pi\)
\(140\) 0 0
\(141\) 21.0000i 1.76852i
\(142\) 0 0
\(143\) 12.0000 + 8.00000i 1.00349 + 0.668994i
\(144\) 0 0
\(145\) 12.0000i 0.996546i
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) 6.00000i 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) 3.00000i 0.244137i 0.992522 + 0.122068i \(0.0389527\pi\)
−0.992522 + 0.122068i \(0.961047\pi\)
\(152\) 0 0
\(153\) −30.0000 −2.42536
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 6.00000i 0.472866i
\(162\) 0 0
\(163\) 18.0000i 1.40987i −0.709273 0.704934i \(-0.750976\pi\)
0.709273 0.704934i \(-0.249024\pi\)
\(164\) 0 0
\(165\) 36.0000 2.80260
\(166\) 0 0
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) −5.00000 12.0000i −0.384615 0.923077i
\(170\) 0 0
\(171\) 36.0000i 2.75299i
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 4.00000i 0.302372i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 21.0000 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 36.0000 2.66120
\(184\) 0 0
\(185\) 9.00000 0.661693
\(186\) 0 0
\(187\) 20.0000i 1.46254i
\(188\) 0 0
\(189\) 9.00000i 0.654654i
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 18.0000i 1.29567i −0.761781 0.647834i \(-0.775675\pi\)
0.761781 0.647834i \(-0.224325\pi\)
\(194\) 0 0
\(195\) −27.0000 18.0000i −1.93351 1.28901i
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) 4.00000i 0.280745i
\(204\) 0 0
\(205\) 36.0000 2.51435
\(206\) 0 0
\(207\) −36.0000 −2.50217
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 0 0
\(213\) 33.0000i 2.26112i
\(214\) 0 0
\(215\) 9.00000i 0.613795i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 18.0000i 1.21633i
\(220\) 0 0
\(221\) 10.0000 15.0000i 0.672673 1.00901i
\(222\) 0 0
\(223\) 21.0000i 1.40626i 0.711059 + 0.703132i \(0.248216\pi\)
−0.711059 + 0.703132i \(0.751784\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 0 0
\(229\) 9.00000i 0.594737i 0.954763 + 0.297368i \(0.0961089\pi\)
−0.954763 + 0.297368i \(0.903891\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 19.0000 1.24473 0.622366 0.782727i \(-0.286172\pi\)
0.622366 + 0.782727i \(0.286172\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) 5.00000i 0.323423i −0.986838 0.161712i \(-0.948299\pi\)
0.986838 0.161712i \(-0.0517014\pi\)
\(240\) 0 0
\(241\) 6.00000i 0.386494i −0.981150 0.193247i \(-0.938098\pi\)
0.981150 0.193247i \(-0.0619019\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.0000i 1.14998i
\(246\) 0 0
\(247\) −18.0000 12.0000i −1.14531 0.763542i
\(248\) 0 0
\(249\) 30.0000i 1.90117i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) 0 0
\(255\) 45.0000i 2.81801i
\(256\) 0 0
\(257\) −5.00000 −0.311891 −0.155946 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −24.0000 −1.48556
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) 18.0000i 1.10158i
\(268\) 0 0
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 11.0000i 0.668202i −0.942537 0.334101i \(-0.891567\pi\)
0.942537 0.334101i \(-0.108433\pi\)
\(272\) 0 0
\(273\) 9.00000 + 6.00000i 0.544705 + 0.363137i
\(274\) 0 0
\(275\) 16.0000i 0.964836i
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000i 1.07379i 0.843649 + 0.536895i \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) −54.0000 −3.19868
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.00000i 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 36.0000i 2.08893i
\(298\) 0 0
\(299\) 12.0000 18.0000i 0.693978 1.04097i
\(300\) 0 0
\(301\) 3.00000i 0.172917i
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 36.0000i 2.06135i
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) 6.00000i 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 16.0000i 0.895828i
\(320\) 0 0
\(321\) −36.0000 −2.00932
\(322\) 0 0
\(323\) 30.0000i 1.66924i
\(324\) 0 0
\(325\) 8.00000 12.0000i 0.443760 0.665640i
\(326\) 0 0
\(327\) 45.0000i 2.48851i
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) 26.0000i 1.42909i 0.699590 + 0.714545i \(0.253366\pi\)
−0.699590 + 0.714545i \(0.746634\pi\)
\(332\) 0 0
\(333\) 18.0000i 0.986394i
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) 42.0000 2.28113
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 54.0000i 2.90726i
\(346\) 0 0
\(347\) −21.0000 −1.12734 −0.563670 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) 0 0
\(349\) 21.0000i 1.12410i −0.827102 0.562052i \(-0.810012\pi\)
0.827102 0.562052i \(-0.189988\pi\)
\(350\) 0 0
\(351\) 18.0000 27.0000i 0.960769 1.44115i
\(352\) 0 0
\(353\) 18.0000i 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) −33.0000 −1.75146
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) 0 0
\(359\) 32.0000i 1.68890i −0.535638 0.844448i \(-0.679929\pi\)
0.535638 0.844448i \(-0.320071\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 15.0000 0.787296
\(364\) 0 0
\(365\) −18.0000 −0.942163
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 72.0000i 3.74817i
\(370\) 0 0
\(371\) 2.00000i 0.103835i
\(372\) 0 0
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) 9.00000i 0.464758i
\(376\) 0 0
\(377\) 8.00000 12.0000i 0.412021 0.618031i
\(378\) 0 0
\(379\) 30.0000i 1.54100i −0.637442 0.770498i \(-0.720007\pi\)
0.637442 0.770498i \(-0.279993\pi\)
\(380\) 0 0
\(381\) 54.0000 2.76650
\(382\) 0 0
\(383\) 11.0000i 0.562074i −0.959697 0.281037i \(-0.909322\pi\)
0.959697 0.281037i \(-0.0906783\pi\)
\(384\) 0 0
\(385\) 12.0000i 0.611577i
\(386\) 0 0
\(387\) −18.0000 −0.914991
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 30.0000 1.51717
\(392\) 0 0
\(393\) 27.0000 1.36197
\(394\) 0 0
\(395\) 18.0000i 0.905678i
\(396\) 0 0
\(397\) 30.0000i 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 0 0
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) 6.00000i 0.299626i −0.988714 0.149813i \(-0.952133\pi\)
0.988714 0.149813i \(-0.0478671\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 27.0000i 1.34164i
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 6.00000i 0.296681i 0.988936 + 0.148340i \(0.0473931\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) 30.0000 1.47264
\(416\) 0 0
\(417\) 27.0000 1.32220
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 9.00000i 0.438633i 0.975654 + 0.219317i \(0.0703828\pi\)
−0.975654 + 0.219317i \(0.929617\pi\)
\(422\) 0 0
\(423\) 42.0000i 2.04211i
\(424\) 0 0
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) 12.0000i 0.580721i
\(428\) 0 0
\(429\) −36.0000 24.0000i −1.73810 1.15873i
\(430\) 0 0
\(431\) 19.0000i 0.915198i 0.889159 + 0.457599i \(0.151290\pi\)
−0.889159 + 0.457599i \(0.848710\pi\)
\(432\) 0 0
\(433\) −3.00000 −0.144171 −0.0720854 0.997398i \(-0.522965\pi\)
−0.0720854 + 0.997398i \(0.522965\pi\)
\(434\) 0 0
\(435\) 36.0000i 1.72607i
\(436\) 0 0
\(437\) 36.0000i 1.72211i
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 36.0000 1.71429
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 18.0000i 0.851371i
\(448\) 0 0
\(449\) 12.0000i 0.566315i 0.959073 + 0.283158i \(0.0913819\pi\)
−0.959073 + 0.283158i \(0.908618\pi\)
\(450\) 0 0
\(451\) 48.0000 2.26023
\(452\) 0 0
\(453\) 9.00000i 0.422857i
\(454\) 0 0
\(455\) −6.00000 + 9.00000i −0.281284 + 0.421927i
\(456\) 0 0
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 0 0
\(459\) 45.0000 2.10042
\(460\) 0 0
\(461\) 33.0000i 1.53696i 0.639872 + 0.768482i \(0.278987\pi\)
−0.639872 + 0.768482i \(0.721013\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −42.0000 −1.93526
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 24.0000i 1.10120i
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 37.0000i 1.69057i 0.534313 + 0.845287i \(0.320570\pi\)
−0.534313 + 0.845287i \(0.679430\pi\)
\(480\) 0 0
\(481\) −9.00000 6.00000i −0.410365 0.273576i
\(482\) 0 0
\(483\) 18.0000i 0.819028i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 54.0000i 2.44196i
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −72.0000 −3.23616
\(496\) 0 0
\(497\) 11.0000 0.493417
\(498\) 0 0
\(499\) 32.0000i 1.43252i 0.697835 + 0.716258i \(0.254147\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 12.0000i 0.533993i
\(506\) 0 0
\(507\) 15.0000 + 36.0000i 0.666173 + 1.59882i
\(508\) 0 0
\(509\) 18.0000i 0.797836i 0.916987 + 0.398918i \(0.130614\pi\)
−0.916987 + 0.398918i \(0.869386\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 54.0000i 2.38416i
\(514\) 0 0
\(515\) 18.0000i 0.793175i
\(516\) 0 0
\(517\) −28.0000 −1.23144
\(518\) 0 0
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 0 0
\(525\) 12.0000i 0.523723i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) −36.0000 24.0000i −1.55933 1.03956i
\(534\) 0 0
\(535\) 36.0000i 1.55642i
\(536\) 0 0
\(537\) −63.0000 −2.71865
\(538\) 0 0
\(539\) 24.0000i 1.03375i
\(540\) 0 0
\(541\) 33.0000i 1.41878i 0.704816 + 0.709390i \(0.251030\pi\)
−0.704816 + 0.709390i \(0.748970\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) −45.0000 −1.92759
\(546\) 0 0
\(547\) 33.0000 1.41098 0.705489 0.708721i \(-0.250727\pi\)
0.705489 + 0.708721i \(0.250727\pi\)
\(548\) 0 0
\(549\) −72.0000 −3.07289
\(550\) 0 0
\(551\) 24.0000i 1.02243i
\(552\) 0 0
\(553\) 6.00000i 0.255146i
\(554\) 0 0
\(555\) −27.0000 −1.14609
\(556\) 0 0
\(557\) 3.00000i 0.127114i −0.997978 0.0635570i \(-0.979756\pi\)
0.997978 0.0635570i \(-0.0202445\pi\)
\(558\) 0 0
\(559\) 6.00000 9.00000i 0.253773 0.380659i
\(560\) 0 0
\(561\) 60.0000i 2.53320i
\(562\) 0 0
\(563\) −3.00000 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) 0 0
\(565\) 42.0000i 1.76695i
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) −21.0000 −0.878823 −0.439411 0.898286i \(-0.644813\pi\)
−0.439411 + 0.898286i \(0.644813\pi\)
\(572\) 0 0
\(573\) 72.0000 3.00784
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) 6.00000i 0.249783i −0.992170 0.124892i \(-0.960142\pi\)
0.992170 0.124892i \(-0.0398583\pi\)
\(578\) 0 0
\(579\) 54.0000i 2.24416i
\(580\) 0 0
\(581\) −10.0000 −0.414870
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 54.0000 + 36.0000i 2.23263 + 1.48842i
\(586\) 0 0
\(587\) 2.00000i 0.0825488i 0.999148 + 0.0412744i \(0.0131418\pi\)
−0.999148 + 0.0412744i \(0.986858\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 9.00000i 0.370211i
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −15.0000 −0.614940
\(596\) 0 0
\(597\) 36.0000 1.47338
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 0 0
\(603\) 24.0000i 0.977356i
\(604\) 0 0
\(605\) 15.0000i 0.609837i
\(606\) 0 0
\(607\) −30.0000 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 0 0
\(609\) 12.0000i 0.486265i
\(610\) 0 0
\(611\) 21.0000 + 14.0000i 0.849569 + 0.566379i
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 0 0
\(615\) −108.000 −4.35498
\(616\) 0 0
\(617\) 36.0000i 1.44931i 0.689114 + 0.724653i \(0.258000\pi\)
−0.689114 + 0.724653i \(0.742000\pi\)
\(618\) 0 0
\(619\) 44.0000i 1.76851i −0.467005 0.884255i \(-0.654667\pi\)
0.467005 0.884255i \(-0.345333\pi\)
\(620\) 0 0
\(621\) 54.0000 2.16695
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −72.0000 −2.87540
\(628\) 0 0
\(629\) 15.0000i 0.598089i
\(630\) 0 0
\(631\) 21.0000i 0.835997i −0.908448 0.417998i \(-0.862732\pi\)
0.908448 0.417998i \(-0.137268\pi\)
\(632\) 0 0
\(633\) −9.00000 −0.357718
\(634\) 0 0
\(635\) 54.0000i 2.14292i
\(636\) 0 0
\(637\) −12.0000 + 18.0000i −0.475457 + 0.713186i
\(638\) 0 0
\(639\) 66.0000i 2.61092i
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 27.0000i 1.06312i
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.0000 1.09572 0.547862 0.836569i \(-0.315442\pi\)
0.547862 + 0.836569i \(0.315442\pi\)
\(654\) 0 0
\(655\) 27.0000i 1.05498i
\(656\) 0 0
\(657\) 36.0000i 1.40449i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 42.0000i 1.63361i −0.576913 0.816805i \(-0.695743\pi\)
0.576913 0.816805i \(-0.304257\pi\)
\(662\) 0 0
\(663\) −30.0000 + 45.0000i −1.16510 + 1.74766i
\(664\) 0 0
\(665\) 18.0000i 0.698010i
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 63.0000i 2.43572i
\(670\) 0 0
\(671\) 48.0000i 1.85302i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 36.0000 1.38564
\(676\) 0 0
\(677\) −14.0000 −0.538064 −0.269032 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000i 0.919682i
\(682\) 0 0
\(683\) 14.0000i 0.535695i −0.963461 0.267848i \(-0.913688\pi\)
0.963461 0.267848i \(-0.0863124\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 27.0000i 1.03011i
\(688\) 0 0
\(689\) 4.00000 6.00000i 0.152388 0.228582i
\(690\) 0 0
\(691\) 8.00000i 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 27.0000i 1.02417i
\(696\) 0 0
\(697\) 60.0000i 2.27266i
\(698\) 0 0
\(699\) −57.0000 −2.15594
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) 0 0
\(703\) −18.0000 −0.678883
\(704\) 0 0
\(705\) 63.0000 2.37272
\(706\) 0 0
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) 42.0000i 1.57734i 0.614815 + 0.788672i \(0.289231\pi\)
−0.614815 + 0.788672i \(0.710769\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 36.0000i 0.897549 1.34632i
\(716\) 0 0
\(717\) 15.0000i 0.560185i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 6.00000i 0.223452i
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 15.0000 0.554795
\(732\) 0 0
\(733\) 33.0000i 1.21888i −0.792831 0.609441i \(-0.791394\pi\)
0.792831 0.609441i \(-0.208606\pi\)
\(734\) 0 0
\(735\) 54.0000i 1.99182i
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 16.0000i 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 54.0000 + 36.0000i 1.98374 + 1.32249i
\(742\) 0 0
\(743\) 11.0000i 0.403551i 0.979432 + 0.201775i \(0.0646711\pi\)
−0.979432 + 0.201775i \(0.935329\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) 60.0000i 2.19529i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 36.0000 1.31366 0.656829 0.754039i \(-0.271897\pi\)
0.656829 + 0.754039i \(0.271897\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 9.00000 0.327544
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 72.0000i 2.61343i
\(760\) 0 0
\(761\) 30.0000i 1.08750i 0.839248 + 0.543750i \(0.182996\pi\)
−0.839248 + 0.543750i \(0.817004\pi\)
\(762\) 0 0
\(763\) 15.0000 0.543036
\(764\) 0 0
\(765\) 90.0000i 3.25396i
\(766\) 0 0
\(767\) −6.00000 4.00000i −0.216647 0.144432i
\(768\) 0 0
\(769\) 48.0000i 1.73092i 0.500974 + 0.865462i \(0.332975\pi\)
−0.500974 + 0.865462i \(0.667025\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) 0 0
\(773\) 27.0000i 0.971123i −0.874203 0.485561i \(-0.838615\pi\)
0.874203 0.485561i \(-0.161385\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.00000 0.322873
\(778\) 0 0
\(779\) −72.0000 −2.57967
\(780\) 0 0
\(781\) −44.0000 −1.57444
\(782\) 0 0
\(783\) 36.0000 1.28654
\(784\) 0 0
\(785\) 42.0000i 1.49904i
\(786\) 0 0
\(787\) 32.0000i 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 0 0
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) 14.0000i 0.497783i
\(792\) 0 0
\(793\) 24.0000 36.0000i 0.852265 1.27840i
\(794\) 0 0
\(795\) 18.0000i 0.638394i