Properties

Label 1872.2.by.d.1297.1
Level $1872$
Weight $2$
Character 1872.1297
Analytic conductor $14.948$
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] = [1872,2,Mod(433,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("1872.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.by (of order \(6\), degree \(2\), not 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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1297.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1872.1297
Dual form 1872.2.by.d.433.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{5} +O(q^{10})\) \(q-1.73205i q^{5} +(-2.50000 + 2.59808i) q^{13} +(-1.50000 + 2.59808i) q^{17} +(3.00000 + 1.73205i) q^{19} +(3.00000 + 5.19615i) q^{23} +2.00000 q^{25} +(1.50000 + 2.59808i) q^{29} +3.46410i q^{31} +(7.50000 - 4.33013i) q^{37} +(4.50000 - 2.59808i) q^{41} +(4.00000 - 6.92820i) q^{43} +3.46410i q^{47} +(-3.50000 - 6.06218i) q^{49} +3.00000 q^{53} +(6.00000 + 3.46410i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(4.50000 + 4.33013i) q^{65} +(-3.00000 + 1.73205i) q^{67} +(3.00000 + 1.73205i) q^{71} -1.73205i q^{73} -4.00000 q^{79} -13.8564i q^{83} +(4.50000 + 2.59808i) q^{85} +(6.00000 - 3.46410i) q^{89} +(3.00000 - 5.19615i) q^{95} +(6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 5 q^{13} - 3 q^{17} + 6 q^{19} + 6 q^{23} + 4 q^{25} + 3 q^{29} + 15 q^{37} + 9 q^{41} + 8 q^{43} - 7 q^{49} + 6 q^{53} + 12 q^{59} - q^{61} + 9 q^{65} - 6 q^{67} + 6 q^{71} - 8 q^{79} + 9 q^{85} + 12 q^{89} + 6 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.73205i 0.774597i −0.921954 0.387298i \(-0.873408\pi\)
0.921954 0.387298i \(-0.126592\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) −2.50000 + 2.59808i −0.693375 + 0.720577i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) 3.00000 + 1.73205i 0.688247 + 0.397360i 0.802955 0.596040i \(-0.203260\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.50000 4.33013i 1.23299 0.711868i 0.265340 0.964155i \(-0.414516\pi\)
0.967653 + 0.252286i \(0.0811825\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 2.59808i 0.702782 0.405751i −0.105601 0.994409i \(-0.533677\pi\)
0.808383 + 0.588657i \(0.200343\pi\)
\(42\) 0 0
\(43\) 4.00000 6.92820i 0.609994 1.05654i −0.381246 0.924473i \(-0.624505\pi\)
0.991241 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 3.46410i 0.781133 + 0.450988i 0.836832 0.547460i \(-0.184405\pi\)
−0.0556984 + 0.998448i \(0.517739\pi\)
\(60\) 0 0
\(61\) −0.500000 + 0.866025i −0.0640184 + 0.110883i −0.896258 0.443533i \(-0.853725\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.50000 + 4.33013i 0.558156 + 0.537086i
\(66\) 0 0
\(67\) −3.00000 + 1.73205i −0.366508 + 0.211604i −0.671932 0.740613i \(-0.734535\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 + 1.73205i 0.356034 + 0.205557i 0.667340 0.744753i \(-0.267433\pi\)
−0.311305 + 0.950310i \(0.600766\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i −0.994850 0.101361i \(-0.967680\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.8564i 1.52094i −0.649374 0.760469i \(-0.724969\pi\)
0.649374 0.760469i \(-0.275031\pi\)
\(84\) 0 0
\(85\) 4.50000 + 2.59808i 0.488094 + 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 3.46410i 0.635999 0.367194i −0.147073 0.989126i \(-0.546985\pi\)
0.783072 + 0.621932i \(0.213652\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) 6.00000 + 3.46410i 0.609208 + 0.351726i 0.772655 0.634826i \(-0.218928\pi\)
−0.163448 + 0.986552i \(0.552261\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.50000 2.59808i −0.149256 0.258518i 0.781697 0.623658i \(-0.214354\pi\)
−0.930953 + 0.365140i \(0.881021\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.00000 5.19615i −0.290021 0.502331i 0.683793 0.729676i \(-0.260329\pi\)
−0.973814 + 0.227345i \(0.926996\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.50000 + 12.9904i −0.705541 + 1.22203i 0.260955 + 0.965351i \(0.415962\pi\)
−0.966496 + 0.256681i \(0.917371\pi\)
\(114\) 0 0
\(115\) 9.00000 5.19615i 0.839254 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 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
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −1.00000 1.73205i −0.0887357 0.153695i 0.818241 0.574875i \(-0.194949\pi\)
−0.906977 + 0.421180i \(0.861616\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.5000 + 7.79423i 1.15338 + 0.665906i 0.949709 0.313133i \(-0.101379\pi\)
0.203674 + 0.979039i \(0.434712\pi\)
\(138\) 0 0
\(139\) −2.00000 + 3.46410i −0.169638 + 0.293821i −0.938293 0.345843i \(-0.887593\pi\)
0.768655 + 0.639664i \(0.220926\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.50000 2.59808i 0.373705 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.5000 + 9.52628i 1.35173 + 0.780423i 0.988492 0.151272i \(-0.0483370\pi\)
0.363241 + 0.931695i \(0.381670\pi\)
\(150\) 0 0
\(151\) 17.3205i 1.40952i 0.709444 + 0.704761i \(0.248946\pi\)
−0.709444 + 0.704761i \(0.751054\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −18.0000 10.3923i −1.40987 0.813988i −0.414494 0.910052i \(-0.636041\pi\)
−0.995375 + 0.0960641i \(0.969375\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 + 6.92820i −0.928588 + 0.536120i −0.886365 0.462988i \(-0.846777\pi\)
−0.0422232 + 0.999108i \(0.513444\pi\)
\(168\) 0 0
\(169\) −0.500000 12.9904i −0.0384615 0.999260i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.00000 5.19615i 0.228086 0.395056i −0.729155 0.684349i \(-0.760087\pi\)
0.957241 + 0.289292i \(0.0934200\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.50000 12.9904i −0.551411 0.955072i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 + 15.5885i −0.651217 + 1.12794i 0.331611 + 0.943416i \(0.392408\pi\)
−0.982828 + 0.184525i \(0.940925\pi\)
\(192\) 0 0
\(193\) −4.50000 + 2.59808i −0.323917 + 0.187014i −0.653137 0.757240i \(-0.726548\pi\)
0.329220 + 0.944253i \(0.393214\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 + 6.92820i −0.854965 + 0.493614i −0.862323 0.506359i \(-0.830991\pi\)
0.00735824 + 0.999973i \(0.497658\pi\)
\(198\) 0 0
\(199\) −1.00000 + 1.73205i −0.0708881 + 0.122782i −0.899291 0.437351i \(-0.855917\pi\)
0.828403 + 0.560133i \(0.189250\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.50000 7.79423i −0.314294 0.544373i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000 + 8.66025i 0.344214 + 0.596196i 0.985211 0.171347i \(-0.0548120\pi\)
−0.640996 + 0.767544i \(0.721479\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.0000 6.92820i −0.818393 0.472500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00000 10.3923i −0.201802 0.699062i
\(222\) 0 0
\(223\) 9.00000 5.19615i 0.602685 0.347960i −0.167412 0.985887i \(-0.553541\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.0000 + 12.1244i 1.39382 + 0.804722i 0.993736 0.111757i \(-0.0356478\pi\)
0.400083 + 0.916479i \(0.368981\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.7846i 1.34444i 0.740349 + 0.672222i \(0.234660\pi\)
−0.740349 + 0.672222i \(0.765340\pi\)
\(240\) 0 0
\(241\) −1.50000 0.866025i −0.0966235 0.0557856i 0.450910 0.892570i \(-0.351100\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.5000 + 6.06218i −0.670820 + 0.387298i
\(246\) 0 0
\(247\) −12.0000 + 3.46410i −0.763542 + 0.220416i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.00000 15.5885i 0.568075 0.983935i −0.428681 0.903456i \(-0.641022\pi\)
0.996756 0.0804789i \(-0.0256450\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 + 2.59808i 0.0935674 + 0.162064i 0.909010 0.416775i \(-0.136840\pi\)
−0.815442 + 0.578838i \(0.803506\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 10.3923i −0.369976 0.640817i 0.619586 0.784929i \(-0.287301\pi\)
−0.989561 + 0.144112i \(0.953967\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.00000 + 5.19615i −0.182913 + 0.316815i −0.942871 0.333157i \(-0.891886\pi\)
0.759958 + 0.649972i \(0.225219\pi\)
\(270\) 0 0
\(271\) −18.0000 + 10.3923i −1.09342 + 0.631288i −0.934485 0.356001i \(-0.884140\pi\)
−0.158937 + 0.987289i \(0.550807\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.50000 6.06218i 0.210295 0.364241i −0.741512 0.670940i \(-0.765891\pi\)
0.951807 + 0.306699i \(0.0992243\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.5167i 1.34323i −0.740900 0.671616i \(-0.765601\pi\)
0.740900 0.671616i \(-0.234399\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.50000 2.59808i −0.262893 0.151781i 0.362761 0.931882i \(-0.381834\pi\)
−0.625653 + 0.780101i \(0.715168\pi\)
\(294\) 0 0
\(295\) 6.00000 10.3923i 0.349334 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −21.0000 5.19615i −1.21446 0.300501i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.50000 + 0.866025i 0.0858898 + 0.0495885i
\(306\) 0 0
\(307\) 17.3205i 0.988534i −0.869310 0.494267i \(-0.835437\pi\)
0.869310 0.494267i \(-0.164563\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.19615i 0.291845i −0.989296 0.145922i \(-0.953385\pi\)
0.989296 0.145922i \(-0.0466150\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.00000 + 5.19615i −0.500773 + 0.289122i
\(324\) 0 0
\(325\) −5.00000 + 5.19615i −0.277350 + 0.288231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.0000 + 13.8564i 1.31916 + 0.761617i 0.983593 0.180400i \(-0.0577391\pi\)
0.335566 + 0.942017i \(0.391072\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 + 5.19615i 0.163908 + 0.283896i
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 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
\(347\) 15.0000 25.9808i 0.805242 1.39472i −0.110885 0.993833i \(-0.535369\pi\)
0.916127 0.400887i \(-0.131298\pi\)
\(348\) 0 0
\(349\) 12.0000 6.92820i 0.642345 0.370858i −0.143172 0.989698i \(-0.545730\pi\)
0.785517 + 0.618840i \(0.212397\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.5000 + 16.4545i −1.51690 + 0.875784i −0.517099 + 0.855926i \(0.672988\pi\)
−0.999803 + 0.0198582i \(0.993679\pi\)
\(354\) 0 0
\(355\) 3.00000 5.19615i 0.159223 0.275783i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.92820i 0.365657i 0.983145 + 0.182828i \(0.0585252\pi\)
−0.983145 + 0.182828i \(0.941475\pi\)
\(360\) 0 0
\(361\) −3.50000 6.06218i −0.184211 0.319062i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 −0.157027
\(366\) 0 0
\(367\) −11.0000 19.0526i −0.574195 0.994535i −0.996129 0.0879086i \(-0.971982\pi\)
0.421933 0.906627i \(-0.361352\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.50000 + 16.4545i −0.491891 + 0.851981i −0.999956 0.00933789i \(-0.997028\pi\)
0.508065 + 0.861319i \(0.330361\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.5000 2.59808i −0.540778 0.133808i
\(378\) 0 0
\(379\) 21.0000 12.1244i 1.07870 0.622786i 0.148153 0.988964i \(-0.452667\pi\)
0.930545 + 0.366178i \(0.119334\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 10.3923i −0.919757 0.531022i −0.0361995 0.999345i \(-0.511525\pi\)
−0.883558 + 0.468323i \(0.844859\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.92820i 0.348596i
\(396\) 0 0
\(397\) −12.0000 6.92820i −0.602263 0.347717i 0.167668 0.985843i \(-0.446376\pi\)
−0.769931 + 0.638127i \(0.779710\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 0.866025i 0.0749064 0.0432472i −0.462079 0.886839i \(-0.652896\pi\)
0.536985 + 0.843592i \(0.319563\pi\)
\(402\) 0 0
\(403\) −9.00000 8.66025i −0.448322 0.431398i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 13.5000 + 7.79423i 0.667532 + 0.385400i 0.795141 0.606425i \(-0.207397\pi\)
−0.127609 + 0.991825i \(0.540730\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.00000 15.5885i −0.439679 0.761546i 0.557986 0.829851i \(-0.311574\pi\)
−0.997665 + 0.0683046i \(0.978241\pi\)
\(420\) 0 0
\(421\) 15.5885i 0.759735i −0.925041 0.379867i \(-0.875970\pi\)
0.925041 0.379867i \(-0.124030\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 + 5.19615i −0.145521 + 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 + 3.46410i −0.289010 + 0.166860i −0.637495 0.770454i \(-0.720029\pi\)
0.348485 + 0.937314i \(0.386696\pi\)
\(432\) 0 0
\(433\) 8.50000 14.7224i 0.408484 0.707515i −0.586236 0.810140i \(-0.699391\pi\)
0.994720 + 0.102625i \(0.0327243\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7846i 0.994263i
\(438\) 0 0
\(439\) −14.0000 24.2487i −0.668184 1.15733i −0.978412 0.206666i \(-0.933739\pi\)
0.310228 0.950662i \(-0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 + 3.46410i 0.283158 + 0.163481i 0.634852 0.772634i \(-0.281061\pi\)
−0.351694 + 0.936115i \(0.614394\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.50000 0.866025i 0.0701670 0.0405110i −0.464506 0.885570i \(-0.653768\pi\)
0.534673 + 0.845059i \(0.320435\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19.5000 11.2583i −0.908206 0.524353i −0.0283522 0.999598i \(-0.509026\pi\)
−0.879853 + 0.475245i \(0.842359\pi\)
\(462\) 0 0
\(463\) 13.8564i 0.643962i 0.946746 + 0.321981i \(0.104349\pi\)
−0.946746 + 0.321981i \(0.895651\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 + 3.46410i 0.275299 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.0000 12.1244i 0.959514 0.553976i 0.0634909 0.997982i \(-0.479777\pi\)
0.896024 + 0.444006i \(0.146443\pi\)
\(480\) 0 0
\(481\) −7.50000 + 30.3109i −0.341971 + 1.38206i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 10.3923i 0.272446 0.471890i
\(486\) 0 0
\(487\) 6.00000 + 3.46410i 0.271886 + 0.156973i 0.629744 0.776802i \(-0.283160\pi\)
−0.357858 + 0.933776i \(0.616493\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 10.3923i −0.270776 0.468998i 0.698285 0.715820i \(-0.253947\pi\)
−0.969061 + 0.246822i \(0.920614\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.0000 + 31.1769i −0.802580 + 1.39011i 0.115332 + 0.993327i \(0.463207\pi\)
−0.917912 + 0.396783i \(0.870127\pi\)
\(504\) 0 0
\(505\) −4.50000 + 2.59808i −0.200247 + 0.115613i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16.5000 + 9.52628i −0.731350 + 0.422245i −0.818916 0.573914i \(-0.805424\pi\)
0.0875661 + 0.996159i \(0.472091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.3205i 0.763233i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) −8.00000 13.8564i −0.349816 0.605898i 0.636401 0.771358i \(-0.280422\pi\)
−0.986216 + 0.165460i \(0.947089\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.00000 5.19615i −0.392046 0.226348i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.50000 + 18.1865i −0.194917 + 0.787746i
\(534\) 0 0
\(535\) −9.00000 + 5.19615i −0.389104 + 0.224649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.4449i 1.26593i −0.774179 0.632967i \(-0.781837\pi\)
0.774179 0.632967i \(-0.218163\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.3923i 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.5000 7.79423i 0.572013 0.330252i −0.185940 0.982561i \(-0.559533\pi\)
0.757953 + 0.652309i \(0.226200\pi\)
\(558\) 0 0
\(559\) 8.00000 + 27.7128i 0.338364 + 1.17213i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 22.5000 + 12.9904i 0.946582 + 0.546509i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21.0000 36.3731i −0.880366 1.52484i −0.850935 0.525271i \(-0.823964\pi\)
−0.0294311 0.999567i \(-0.509370\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 + 10.3923i 0.250217 + 0.433389i
\(576\) 0 0
\(577\) 19.0526i 0.793168i 0.917998 + 0.396584i \(0.129805\pi\)
−0.917998 + 0.396584i \(0.870195\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.0000 + 10.3923i −0.742940 + 0.428936i −0.823137 0.567843i \(-0.807778\pi\)
0.0801976 + 0.996779i \(0.474445\pi\)
\(588\) 0 0
\(589\) −6.00000 + 10.3923i −0.247226 + 0.428207i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.9808i 1.06690i −0.845831 0.533451i \(-0.820895\pi\)
0.845831 0.533451i \(-0.179105\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −12.5000 21.6506i −0.509886 0.883148i −0.999934 0.0114528i \(-0.996354\pi\)
0.490049 0.871695i \(-0.336979\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.5000 + 9.52628i 0.670820 + 0.387298i
\(606\) 0 0
\(607\) −17.0000 + 29.4449i −0.690009 + 1.19513i 0.281826 + 0.959466i \(0.409060\pi\)
−0.971834 + 0.235665i \(0.924273\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 8.66025i −0.364101 0.350356i
\(612\) 0 0
\(613\) −10.5000 + 6.06218i −0.424091 + 0.244849i −0.696826 0.717240i \(-0.745405\pi\)
0.272735 + 0.962089i \(0.412072\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.5000 + 11.2583i 0.785040 + 0.453243i 0.838214 0.545342i \(-0.183600\pi\)
−0.0531732 + 0.998585i \(0.516934\pi\)
\(618\) 0 0
\(619\) 20.7846i 0.835404i −0.908584 0.417702i \(-0.862836\pi\)
0.908584 0.417702i \(-0.137164\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.9808i 1.03592i
\(630\) 0 0
\(631\) 42.0000 + 24.2487i 1.67199 + 0.965326i 0.966521 + 0.256589i \(0.0825987\pi\)
0.705473 + 0.708737i \(0.250735\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.00000 + 1.73205i −0.119051 + 0.0687343i
\(636\) 0 0
\(637\) 24.5000 + 6.06218i 0.970725 + 0.240192i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5000 28.5788i 0.651711 1.12880i −0.330997 0.943632i \(-0.607385\pi\)
0.982708 0.185164i \(-0.0592817\pi\)
\(642\) 0 0
\(643\) −12.0000 6.92820i −0.473234 0.273222i 0.244359 0.969685i \(-0.421423\pi\)
−0.717592 + 0.696463i \(0.754756\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.00000 15.5885i −0.353827 0.612845i 0.633090 0.774078i \(-0.281786\pi\)
−0.986916 + 0.161233i \(0.948453\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0000 25.9808i −0.586995 1.01671i −0.994623 0.103558i \(-0.966977\pi\)
0.407628 0.913148i \(-0.366356\pi\)
\(654\) 0 0
\(655\) 31.1769i 1.21818i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 10.3923i 0.233727 0.404827i −0.725175 0.688565i \(-0.758241\pi\)
0.958902 + 0.283738i \(0.0915745\pi\)
\(660\) 0 0
\(661\) −40.5000 + 23.3827i −1.57527 + 0.909481i −0.579761 + 0.814787i \(0.696854\pi\)
−0.995506 + 0.0946945i \(0.969813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.00000 + 15.5885i −0.348481 + 0.603587i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9.50000 + 16.4545i 0.366198 + 0.634274i 0.988968 0.148132i \(-0.0473259\pi\)
−0.622770 + 0.782405i \(0.713993\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.0000 + 12.1244i 0.803543 + 0.463926i 0.844708 0.535227i \(-0.179774\pi\)
−0.0411658 + 0.999152i \(0.513107\pi\)
\(684\) 0 0
\(685\) 13.5000 23.3827i 0.515808 0.893407i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.50000 + 7.79423i −0.285727 + 0.296936i
\(690\) 0 0
\(691\) −12.0000 + 6.92820i −0.456502 + 0.263561i −0.710572 0.703624i \(-0.751564\pi\)
0.254071 + 0.967186i \(0.418230\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.00000 + 3.46410i 0.227593 + 0.131401i
\(696\) 0 0
\(697\) 15.5885i 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.50000 2.59808i −0.169001 0.0975728i 0.413114 0.910679i \(-0.364441\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 + 10.3923i −0.674105 + 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 41.5692i 0.895049 1.55027i 0.0613050 0.998119i \(-0.480474\pi\)
0.833744 0.552151i \(-0.186193\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.00000 + 5.19615i 0.111417 + 0.192980i
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000 + 20.7846i 0.443836 + 0.768747i
\(732\) 0 0
\(733\) 12.1244i 0.447823i −0.974609 0.223912i \(-0.928117\pi\)
0.974609 0.223912i \(-0.0718827\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18.0000 10.3923i 0.662141 0.382287i −0.130951 0.991389i \(-0.541803\pi\)
0.793092 + 0.609102i \(0.208470\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30.0000 + 17.3205i −1.10059 + 0.635428i −0.936377 0.350997i \(-0.885843\pi\)
−0.164216 + 0.986424i \(0.552510\pi\)
\(744\) 0 0
\(745\) 16.5000 28.5788i 0.604513 1.04705i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 13.8564i −0.291924 0.505627i 0.682341 0.731034i \(-0.260962\pi\)
−0.974265 + 0.225407i \(0.927629\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) 13.0000 + 22.5167i 0.472493 + 0.818382i 0.999505 0.0314762i \(-0.0100208\pi\)
−0.527011 + 0.849858i \(0.676688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 17.3205i −1.08750 0.627868i −0.154590 0.987979i \(-0.549406\pi\)
−0.932910 + 0.360111i \(0.882739\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 + 6.92820i −0.866590 + 0.250163i
\(768\) 0 0
\(769\) 6.00000 3.46410i 0.216366 0.124919i −0.387901 0.921701i \(-0.626800\pi\)
0.604266 + 0.796782i \(0.293466\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.0000 + 17.3205i 1.07903 + 0.622975i 0.930633 0.365953i \(-0.119257\pi\)
0.148392 + 0.988929i \(0.452590\pi\)
\(774\) 0 0
\(775\) 6.92820i 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.5167i 0.803654i
\(786\) 0 0
\(787\) −33.0000 19.0526i −1.17632 0.679150i −0.221162 0.975237i \(-0.570985\pi\)
−0.955161 + 0.296087i \(0.904318\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.00000 3.46410i −0.0355110 0.123014i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0000 36.3731i 0.743858 1.28840i −0.206868 0.978369i \(-0.566327\pi\)
0.950726 0.310031i \(-0.100340\pi\)
\(798\) 0 0
\(799\) −9.00000 5.19615i −0.318397 &minu