Properties

Label 1872.2.by.d.433.1
Level $1872$
Weight $2$
Character 1872.433
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 433.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1872.433
Dual form 1872.2.by.d.1297.1

$q$-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 \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

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

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 \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 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 0.624780 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 0 0
\(19\) 3.00000 1.73205i 0.688247 0.397360i −0.114708 0.993399i \(-0.536593\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 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 −0.692480 0.721437i \(-0.743482\pi\)
0.971023 + 0.238987i \(0.0768152\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.50000 + 4.33013i 1.23299 + 0.711868i 0.967653 0.252286i \(-0.0811825\pi\)
0.265340 + 0.964155i \(0.414516\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 + 2.59808i 0.702782 + 0.405751i 0.808383 0.588657i \(-0.200343\pi\)
−0.105601 + 0.994409i \(0.533677\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410i 0.505291i −0.967559 0.252646i \(-0.918699\pi\)
0.967559 0.252646i \(-0.0813007\pi\)
\(48\) 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 −0.0556984 0.998448i \(-0.517739\pi\)
0.836832 + 0.547460i \(0.184405\pi\)
\(60\) 0 0
\(61\) −0.500000 0.866025i −0.0640184 0.110883i 0.832240 0.554416i \(-0.187058\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 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 0.305424 0.952217i \(-0.401202\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 1.73205i 0.356034 0.205557i −0.311305 0.950310i \(-0.600766\pi\)
0.667340 + 0.744753i \(0.267433\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 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 \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 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 0.783072 0.621932i \(-0.213652\pi\)
−0.147073 + 0.989126i \(0.546985\pi\)
\(90\) 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 −0.163448 0.986552i \(-0.552261\pi\)
0.772655 + 0.634826i \(0.218928\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.50000 + 2.59808i −0.149256 + 0.258518i −0.930953 0.365140i \(-0.881021\pi\)
0.781697 + 0.623658i \(0.214354\pi\)
\(102\) 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 −0.973814 0.227345i \(-0.926996\pi\)
0.683793 + 0.729676i \(0.260329\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i −0.748086 0.663602i \(-0.769027\pi\)
0.748086 0.663602i \(-0.230973\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.50000 12.9904i −0.705541 1.22203i −0.966496 0.256681i \(-0.917371\pi\)
0.260955 0.965351i \(-0.415962\pi\)
\(114\) 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 −0.906977 0.421180i \(-0.861616\pi\)
0.818241 + 0.574875i \(0.194949\pi\)
\(128\) 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 0.203674 0.979039i \(-0.434712\pi\)
0.949709 + 0.313133i \(0.101379\pi\)
\(138\) 0 0
\(139\) −2.00000 3.46410i −0.169638 0.293821i 0.768655 0.639664i \(-0.220926\pi\)
−0.938293 + 0.345843i \(0.887593\pi\)
\(140\) 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 0.363241 0.931695i \(-0.381670\pi\)
0.988492 + 0.151272i \(0.0483370\pi\)
\(150\) 0 0
\(151\) 17.3205i 1.40952i −0.709444 0.704761i \(-0.751054\pi\)
0.709444 0.704761i \(-0.248946\pi\)
\(152\) 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 −0.995375 0.0960641i \(-0.969375\pi\)
−0.414494 + 0.910052i \(0.636041\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 6.92820i −0.928588 0.536120i −0.0422232 0.999108i \(-0.513444\pi\)
−0.886365 + 0.462988i \(0.846777\pi\)
\(168\) 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 0.957241 0.289292i \(-0.0934200\pi\)
−0.729155 + 0.684349i \(0.760087\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 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 −0.982828 0.184525i \(-0.940925\pi\)
0.331611 0.943416i \(-0.392408\pi\)
\(192\) 0 0
\(193\) −4.50000 2.59808i −0.323917 0.187014i 0.329220 0.944253i \(-0.393214\pi\)
−0.653137 + 0.757240i \(0.726548\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 6.92820i −0.854965 0.493614i 0.00735824 0.999973i \(-0.497658\pi\)
−0.862323 + 0.506359i \(0.830991\pi\)
\(198\) 0 0
\(199\) −1.00000 1.73205i −0.0708881 0.122782i 0.828403 0.560133i \(-0.189250\pi\)
−0.899291 + 0.437351i \(0.855917\pi\)
\(200\) 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 −0.640996 0.767544i \(-0.721479\pi\)
0.985211 + 0.171347i \(0.0548120\pi\)
\(212\) 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 0.770097 0.637927i \(-0.220208\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.0000 12.1244i 1.39382 0.804722i 0.400083 0.916479i \(-0.368981\pi\)
0.993736 + 0.111757i \(0.0356478\pi\)
\(228\) 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 \(-0.765340\pi\)
0.740349 0.672222i \(-0.234660\pi\)
\(240\) 0 0
\(241\) −1.50000 + 0.866025i −0.0966235 + 0.0557856i −0.547533 0.836784i \(-0.684433\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 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 0.996756 + 0.0804789i \(0.0256450\pi\)
−0.428681 + 0.903456i \(0.641022\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 2.59808i 0.0935674 0.162064i −0.815442 0.578838i \(-0.803506\pi\)
0.909010 + 0.416775i \(0.136840\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 + 10.3923i −0.369976 + 0.640817i −0.989561 0.144112i \(-0.953967\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(264\) 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 0.759958 0.649972i \(-0.225219\pi\)
−0.942871 + 0.333157i \(0.891886\pi\)
\(270\) 0 0
\(271\) −18.0000 10.3923i −1.09342 0.631288i −0.158937 0.987289i \(-0.550807\pi\)
−0.934485 + 0.356001i \(0.884140\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.50000 + 6.06218i 0.210295 + 0.364241i 0.951807 0.306699i \(-0.0992243\pi\)
−0.741512 + 0.670940i \(0.765891\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.5167i 1.34323i 0.740900 + 0.671616i \(0.234399\pi\)
−0.740900 + 0.671616i \(0.765601\pi\)
\(282\) 0 0
\(283\) 2.00000 3.46410i 0.118888 0.205919i −0.800439 0.599414i \(-0.795400\pi\)
0.919327 + 0.393494i \(0.128734\pi\)
\(284\) 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 −0.625653 0.780101i \(-0.715168\pi\)
0.362761 + 0.931882i \(0.381834\pi\)
\(294\) 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 \(0.164563\pi\)
−0.869310 + 0.494267i \(0.835437\pi\)
\(308\) 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 \(0.0466150\pi\)
−0.989296 + 0.145922i \(0.953385\pi\)
\(318\) 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 0.335566 0.942017i \(-0.391072\pi\)
0.983593 + 0.180400i \(0.0577391\pi\)
\(332\) 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 0.916127 + 0.400887i \(0.131298\pi\)
−0.110885 + 0.993833i \(0.535369\pi\)
\(348\) 0 0
\(349\) 12.0000 + 6.92820i 0.642345 + 0.370858i 0.785517 0.618840i \(-0.212397\pi\)
−0.143172 + 0.989698i \(0.545730\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −28.5000 16.4545i −1.51690 0.875784i −0.999803 0.0198582i \(-0.993679\pi\)
−0.517099 0.855926i \(-0.672988\pi\)
\(354\) 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 \(-0.941475\pi\)
0.983145 0.182828i \(-0.0585252\pi\)
\(360\) 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 0.421933 + 0.906627i \(0.361352\pi\)
−0.996129 + 0.0879086i \(0.971982\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −9.50000 16.4545i −0.491891 0.851981i 0.508065 0.861319i \(-0.330361\pi\)
−0.999956 + 0.00933789i \(0.997028\pi\)
\(374\) 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 0.930545 0.366178i \(-0.119334\pi\)
0.148153 + 0.988964i \(0.452667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 + 10.3923i −0.919757 + 0.531022i −0.883558 0.468323i \(-0.844859\pi\)
−0.0361995 + 0.999345i \(0.511525\pi\)
\(384\) 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 −0.769931 0.638127i \(-0.779710\pi\)
0.167668 + 0.985843i \(0.446376\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 + 0.866025i 0.0749064 + 0.0432472i 0.536985 0.843592i \(-0.319563\pi\)
−0.462079 + 0.886839i \(0.652896\pi\)
\(402\) 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 −0.127609 0.991825i \(-0.540730\pi\)
0.795141 + 0.606425i \(0.207397\pi\)
\(410\) 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 −0.997665 0.0683046i \(-0.978241\pi\)
0.557986 + 0.829851i \(0.311574\pi\)
\(420\) 0 0
\(421\) 15.5885i 0.759735i 0.925041 + 0.379867i \(0.124030\pi\)
−0.925041 + 0.379867i \(0.875970\pi\)
\(422\) 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 0.348485 0.937314i \(-0.386696\pi\)
−0.637495 + 0.770454i \(0.720029\pi\)
\(432\) 0 0
\(433\) 8.50000 + 14.7224i 0.408484 + 0.707515i 0.994720 0.102625i \(-0.0327243\pi\)
−0.586236 + 0.810140i \(0.699391\pi\)
\(434\) 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 0.310228 + 0.950662i \(0.399595\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 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 −0.351694 0.936115i \(-0.614394\pi\)
0.634852 + 0.772634i \(0.281061\pi\)
\(450\) 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 0.534673 0.845059i \(-0.320435\pi\)
−0.464506 + 0.885570i \(0.653768\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19.5000 + 11.2583i −0.908206 + 0.524353i −0.879853 0.475245i \(-0.842359\pi\)
−0.0283522 + 0.999598i \(0.509026\pi\)
\(462\) 0 0
\(463\) 13.8564i 0.643962i −0.946746 0.321981i \(-0.895651\pi\)
0.946746 0.321981i \(-0.104349\pi\)
\(464\) 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 0.896024 0.444006i \(-0.146443\pi\)
0.0634909 + 0.997982i \(0.479777\pi\)
\(480\) 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 −0.357858 0.933776i \(-0.616493\pi\)
0.629744 + 0.776802i \(0.283160\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 + 10.3923i −0.270776 + 0.468998i −0.969061 0.246822i \(-0.920614\pi\)
0.698285 + 0.715820i \(0.253947\pi\)
\(492\) 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 \(-0.754147\pi\)
0.716258 0.697835i \(-0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.0000 31.1769i −0.802580 1.39011i −0.917912 0.396783i \(-0.870127\pi\)
0.115332 0.993327i \(-0.463207\pi\)
\(504\) 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 0.0875661 0.996159i \(-0.472091\pi\)
−0.818916 + 0.573914i \(0.805424\pi\)
\(510\) 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 −0.986216 0.165460i \(-0.947089\pi\)
0.636401 + 0.771358i \(0.280422\pi\)
\(524\) 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 \(0.218163\pi\)
−0.774179 + 0.632967i \(0.781837\pi\)
\(542\) 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 0.757953 0.652309i \(-0.226200\pi\)
−0.185940 + 0.982561i \(0.559533\pi\)
\(558\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 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 −0.0294311 + 0.999567i \(0.509370\pi\)
−0.850935 + 0.525271i \(0.823964\pi\)
\(570\) 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 \(-0.870195\pi\)
0.917998 0.396584i \(-0.129805\pi\)
\(578\) 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 0.0801976 0.996779i \(-0.474445\pi\)
−0.823137 + 0.567843i \(0.807778\pi\)
\(588\) 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 \(0.179105\pi\)
−0.845831 + 0.533451i \(0.820895\pi\)
\(594\) 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 0.490049 + 0.871695i \(0.336979\pi\)
−0.999934 + 0.0114528i \(0.996354\pi\)
\(602\) 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 −0.971834 0.235665i \(-0.924273\pi\)
0.281826 0.959466i \(-0.409060\pi\)
\(608\) 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 0.272735 0.962089i \(-0.412072\pi\)
−0.696826 + 0.717240i \(0.745405\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.5000 11.2583i 0.785040 0.453243i −0.0531732 0.998585i \(-0.516934\pi\)
0.838214 + 0.545342i \(0.183600\pi\)
\(618\) 0 0
\(619\) 20.7846i 0.835404i 0.908584 + 0.417702i \(0.137164\pi\)
−0.908584 + 0.417702i \(0.862836\pi\)
\(620\) 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 0.705473 0.708737i \(-0.250735\pi\)
0.966521 0.256589i \(-0.0825987\pi\)
\(632\) 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 0.982708 + 0.185164i \(0.0592817\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(642\) 0 0
\(643\) −12.0000 + 6.92820i −0.473234 + 0.273222i −0.717592 0.696463i \(-0.754756\pi\)
0.244359 + 0.969685i \(0.421423\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.00000 + 15.5885i −0.353827 + 0.612845i −0.986916 0.161233i \(-0.948453\pi\)
0.633090 + 0.774078i \(0.281786\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0000 + 25.9808i −0.586995 + 1.01671i 0.407628 + 0.913148i \(0.366356\pi\)
−0.994623 + 0.103558i \(0.966977\pi\)
\(654\) 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 0.958902 0.283738i \(-0.0915745\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(660\) 0 0
\(661\) −40.5000 23.3827i −1.57527 0.909481i −0.995506 0.0946945i \(-0.969813\pi\)
−0.579761 0.814787i \(-0.696854\pi\)
\(662\) 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 −0.622770 0.782405i \(-0.713993\pi\)
0.988968 + 0.148132i \(0.0473259\pi\)
\(674\) 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 −0.0411658 0.999152i \(-0.513107\pi\)
0.844708 + 0.535227i \(0.179774\pi\)
\(684\) 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 0.254071 0.967186i \(-0.418230\pi\)
−0.710572 + 0.703624i \(0.751564\pi\)
\(692\) 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 −0.582115 0.813107i \(-0.697775\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 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 0.833744 + 0.552151i \(0.186193\pi\)
0.0613050 + 0.998119i \(0.480474\pi\)
\(720\) 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 \(0.0718827\pi\)
−0.974609 + 0.223912i \(0.928117\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 18.0000 + 10.3923i 0.662141 + 0.382287i 0.793092 0.609102i \(-0.208470\pi\)
−0.130951 + 0.991389i \(0.541803\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30.0000 17.3205i −1.10059 0.635428i −0.164216 0.986424i \(-0.552510\pi\)
−0.936377 + 0.350997i \(0.885843\pi\)
\(744\) 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 −0.974265 0.225407i \(-0.927629\pi\)
0.682341 + 0.731034i \(0.260962\pi\)
\(752\) 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 −0.527011 0.849858i \(-0.676688\pi\)
0.999505 + 0.0314762i \(0.0100208\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 + 17.3205i −1.08750 + 0.627868i −0.932910 0.360111i \(-0.882739\pi\)
−0.154590 + 0.987979i \(0.549406\pi\)
\(762\) 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 0.604266 0.796782i \(-0.293466\pi\)
−0.387901 + 0.921701i \(0.626800\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.0000 17.3205i 1.07903 0.622975i 0.148392 0.988929i \(-0.452590\pi\)
0.930633 + 0.365953i \(0.119257\pi\)
\(774\) 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 −0.955161 0.296087i \(-0.904318\pi\)
−0.221162 + 0.975237i \(0.570985\pi\)
\(788\) 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 0.950726 + 0.310031i \(0.100340\pi\)
−0.206868 + 0.978369i \(0.566327\pi\)
\(798\) 0 0
\(799\) −9.00000 + 5.19615i −0.318397