Properties

Label 1456.2.r.g.417.1
Level $1456$
Weight $2$
Character 1456.417
Analytic conductor $11.626$
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] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), 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: \(11.6262185343\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1456.417
Dual form 1456.2.r.g.625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +(-1.50000 + 2.59808i) q^{11} -1.00000 q^{13} +(-3.50000 + 6.06218i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} -5.00000 q^{29} +(-4.00000 - 6.92820i) q^{37} -2.00000 q^{43} +(3.50000 + 6.06218i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.50000 - 2.59808i) q^{53} +(-3.50000 + 6.06218i) q^{59} +(3.50000 + 6.06218i) q^{61} +(-7.50000 + 2.59808i) q^{63} +(-1.50000 + 2.59808i) q^{67} +5.00000 q^{71} +(-7.00000 + 12.1244i) q^{73} +(-6.00000 - 5.19615i) q^{77} +(-3.00000 - 5.19615i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(0.500000 - 2.59808i) q^{91} -14.0000 q^{97} -9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{17} - 7 q^{19} - 6 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{37} - 4 q^{43} + 7 q^{47} - 13 q^{49} + 3 q^{53} - 7 q^{59} + 7 q^{61} - 15 q^{63} - 3 q^{67} + 10 q^{71} - 14 q^{73} - 12 q^{77} - 6 q^{79} - 9 q^{81} + q^{91} - 28 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.50000 + 6.06218i −0.848875 + 1.47029i 0.0333386 + 0.999444i \(0.489386\pi\)
−0.882213 + 0.470850i \(0.843947\pi\)
\(18\) 0 0
\(19\) −3.50000 6.06218i −0.802955 1.39076i −0.917663 0.397360i \(-0.869927\pi\)
0.114708 0.993399i \(-0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 6.92820i −0.657596 1.13899i −0.981236 0.192809i \(-0.938240\pi\)
0.323640 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.50000 + 6.06218i 0.510527 + 0.884260i 0.999926 + 0.0121990i \(0.00388317\pi\)
−0.489398 + 0.872060i \(0.662783\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 2.59808i 0.206041 0.356873i −0.744423 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.50000 + 6.06218i −0.455661 + 0.789228i −0.998726 0.0504625i \(-0.983930\pi\)
0.543065 + 0.839691i \(0.317264\pi\)
\(60\) 0 0
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) 0 0
\(63\) −7.50000 + 2.59808i −0.944911 + 0.327327i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.50000 + 2.59808i −0.183254 + 0.317406i −0.942987 0.332830i \(-0.891996\pi\)
0.759733 + 0.650236i \(0.225330\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 0 0
\(73\) −7.00000 + 12.1244i −0.819288 + 1.41905i 0.0869195 + 0.996215i \(0.472298\pi\)
−0.906208 + 0.422833i \(0.861036\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 5.19615i −0.683763 0.592157i
\(78\) 0 0
\(79\) −3.00000 5.19615i −0.337526 0.584613i 0.646440 0.762964i \(-0.276257\pi\)
−0.983967 + 0.178352i \(0.942924\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0.500000 2.59808i 0.0524142 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) −9.00000 −0.904534
\(100\) 0 0
\(101\) 7.00000 12.1244i 0.696526 1.20642i −0.273138 0.961975i \(-0.588061\pi\)
0.969664 0.244443i \(-0.0786053\pi\)
\(102\) 0 0
\(103\) 7.00000 + 12.1244i 0.689730 + 1.19465i 0.971925 + 0.235291i \(0.0756043\pi\)
−0.282194 + 0.959357i \(0.591062\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 + 6.92820i 0.386695 + 0.669775i 0.992003 0.126217i \(-0.0402834\pi\)
−0.605308 + 0.795991i \(0.706950\pi\)
\(108\) 0 0
\(109\) −2.00000 + 3.46410i −0.191565 + 0.331801i −0.945769 0.324840i \(-0.894690\pi\)
0.754204 + 0.656640i \(0.228023\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.50000 2.59808i −0.138675 0.240192i
\(118\) 0 0
\(119\) −14.0000 12.1244i −1.28338 1.11144i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.00000 + 12.1244i 0.611593 + 1.05931i 0.990972 + 0.134069i \(0.0428042\pi\)
−0.379379 + 0.925241i \(0.623862\pi\)
\(132\) 0 0
\(133\) 17.5000 6.06218i 1.51744 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 + 3.46410i −0.170872 + 0.295958i −0.938725 0.344668i \(-0.887992\pi\)
0.767853 + 0.640626i \(0.221325\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.50000 2.59808i 0.125436 0.217262i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) −1.50000 + 2.59808i −0.122068 + 0.211428i −0.920583 0.390547i \(-0.872286\pi\)
0.798515 + 0.601975i \(0.205619\pi\)
\(152\) 0 0
\(153\) −21.0000 −1.69775
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.50000 + 6.06218i −0.279330 + 0.483814i −0.971219 0.238190i \(-0.923446\pi\)
0.691888 + 0.722005i \(0.256779\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0000 5.19615i 1.18217 0.409514i
\(162\) 0 0
\(163\) −6.50000 11.2583i −0.509119 0.881820i −0.999944 0.0105623i \(-0.996638\pi\)
0.490825 0.871258i \(-0.336695\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.00000 0.541676 0.270838 0.962625i \(-0.412699\pi\)
0.270838 + 0.962625i \(0.412699\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 10.5000 18.1865i 0.802955 1.39076i
\(172\) 0 0
\(173\) −3.50000 6.06218i −0.266100 0.460899i 0.701751 0.712422i \(-0.252402\pi\)
−0.967851 + 0.251523i \(0.919068\pi\)
\(174\) 0 0
\(175\) 10.0000 + 8.66025i 0.755929 + 0.654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.00000 + 8.66025i −0.373718 + 0.647298i −0.990134 0.140122i \(-0.955250\pi\)
0.616417 + 0.787420i \(0.288584\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.5000 18.1865i −0.767836 1.32993i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 17.3205i −0.723575 1.25327i −0.959558 0.281511i \(-0.909164\pi\)
0.235983 0.971757i \(-0.424169\pi\)
\(192\) 0 0
\(193\) −2.00000 + 3.46410i −0.143963 + 0.249351i −0.928986 0.370116i \(-0.879318\pi\)
0.785022 + 0.619467i \(0.212651\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −7.00000 + 12.1244i −0.496217 + 0.859473i −0.999990 0.00436292i \(-0.998611\pi\)
0.503774 + 0.863836i \(0.331945\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.50000 12.9904i 0.175466 0.911746i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.00000 15.5885i 0.625543 1.08347i
\(208\) 0 0
\(209\) 21.0000 1.45260
\(210\) 0 0
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.50000 6.06218i 0.235435 0.407786i
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 0 0
\(227\) −14.0000 + 24.2487i −0.929213 + 1.60944i −0.144571 + 0.989494i \(0.546180\pi\)
−0.784642 + 0.619949i \(0.787153\pi\)
\(228\) 0 0
\(229\) −7.00000 12.1244i −0.462573 0.801200i 0.536515 0.843891i \(-0.319740\pi\)
−0.999088 + 0.0426906i \(0.986407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.5000 + 23.3827i 0.884414 + 1.53185i 0.846383 + 0.532574i \(0.178775\pi\)
0.0380310 + 0.999277i \(0.487891\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) 0 0
\(241\) −14.0000 + 24.2487i −0.901819 + 1.56200i −0.0766885 + 0.997055i \(0.524435\pi\)
−0.825131 + 0.564942i \(0.808899\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.50000 + 6.06218i 0.222700 + 0.385727i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.00000 + 12.1244i 0.436648 + 0.756297i 0.997429 0.0716680i \(-0.0228322\pi\)
−0.560781 + 0.827964i \(0.689499\pi\)
\(258\) 0 0
\(259\) 20.0000 6.92820i 1.24274 0.430498i
\(260\) 0 0
\(261\) −7.50000 12.9904i −0.464238 0.804084i
\(262\) 0 0
\(263\) −12.0000 + 20.7846i −0.739952 + 1.28163i 0.212565 + 0.977147i \(0.431818\pi\)
−0.952517 + 0.304487i \(0.901515\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.5000 18.1865i 0.640196 1.10885i −0.345192 0.938532i \(-0.612186\pi\)
0.985389 0.170321i \(-0.0544803\pi\)
\(270\) 0 0
\(271\) −3.50000 6.06218i −0.212610 0.368251i 0.739921 0.672694i \(-0.234863\pi\)
−0.952531 + 0.304443i \(0.901530\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.50000 + 12.9904i 0.452267 + 0.783349i
\(276\) 0 0
\(277\) 8.50000 14.7224i 0.510716 0.884585i −0.489207 0.872167i \(-0.662714\pi\)
0.999923 0.0124177i \(-0.00395278\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −7.00000 + 12.1244i −0.416107 + 0.720718i −0.995544 0.0942988i \(-0.969939\pi\)
0.579437 + 0.815017i \(0.303272\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.0000 27.7128i −0.941176 1.63017i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 + 5.19615i 0.173494 + 0.300501i
\(300\) 0 0
\(301\) 1.00000 5.19615i 0.0576390 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.0000 −1.19853 −0.599267 0.800549i \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 7.00000 + 12.1244i 0.395663 + 0.685309i 0.993186 0.116543i \(-0.0371814\pi\)
−0.597522 + 0.801852i \(0.703848\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 + 5.19615i 0.168497 + 0.291845i 0.937892 0.346929i \(-0.112775\pi\)
−0.769395 + 0.638774i \(0.779442\pi\)
\(318\) 0 0
\(319\) 7.50000 12.9904i 0.419919 0.727322i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 49.0000 2.72643
\(324\) 0 0
\(325\) −2.50000 + 4.33013i −0.138675 + 0.240192i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −17.5000 + 6.06218i −0.964806 + 0.334219i
\(330\) 0 0
\(331\) −10.0000 17.3205i −0.549650 0.952021i −0.998298 0.0583130i \(-0.981428\pi\)
0.448649 0.893708i \(-0.351905\pi\)
\(332\) 0 0
\(333\) 12.0000 20.7846i 0.657596 1.13899i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.00000 3.46410i 0.107366 0.185963i −0.807337 0.590091i \(-0.799092\pi\)
0.914702 + 0.404128i \(0.132425\pi\)
\(348\) 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
\(353\) −7.00000 + 12.1244i −0.372572 + 0.645314i −0.989960 0.141344i \(-0.954858\pi\)
0.617388 + 0.786659i \(0.288191\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.00000 + 6.92820i 0.211112 + 0.365657i 0.952063 0.305903i \(-0.0989582\pi\)
−0.740951 + 0.671559i \(0.765625\pi\)
\(360\) 0 0
\(361\) −15.0000 + 25.9808i −0.789474 + 1.36741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.00000 12.1244i 0.365397 0.632886i −0.623443 0.781869i \(-0.714267\pi\)
0.988840 + 0.148983i \(0.0475999\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 + 5.19615i 0.311504 + 0.269771i
\(372\) 0 0
\(373\) −7.50000 12.9904i −0.388335 0.672616i 0.603890 0.797067i \(-0.293616\pi\)
−0.992226 + 0.124451i \(0.960283\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.00000 5.19615i −0.152499 0.264135i
\(388\) 0 0
\(389\) 1.50000 2.59808i 0.0760530 0.131728i −0.825491 0.564416i \(-0.809102\pi\)
0.901544 + 0.432688i \(0.142435\pi\)
\(390\) 0 0
\(391\) 42.0000 2.12403
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 12.1244i −0.351320 0.608504i 0.635161 0.772380i \(-0.280934\pi\)
−0.986481 + 0.163876i \(0.947600\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.0000 19.0526i −0.549314 0.951439i −0.998322 0.0579116i \(-0.981556\pi\)
0.449008 0.893528i \(-0.351777\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 14.0000 24.2487i 0.692255 1.19902i −0.278842 0.960337i \(-0.589950\pi\)
0.971097 0.238685i \(-0.0767162\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.0000 12.1244i −0.688895 0.596601i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) −10.5000 + 18.1865i −0.510527 + 0.884260i
\(424\) 0 0
\(425\) 17.5000 + 30.3109i 0.848875 + 1.47029i
\(426\) 0 0
\(427\) −17.5000 + 6.06218i −0.846884 + 0.293369i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 + 20.7846i −0.578020 + 1.00116i 0.417687 + 0.908591i \(0.362841\pi\)
−0.995706 + 0.0925683i \(0.970492\pi\)
\(432\) 0 0
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −21.0000 + 36.3731i −1.00457 + 1.73996i
\(438\) 0 0
\(439\) 7.00000 + 12.1244i 0.334092 + 0.578664i 0.983310 0.181938i \(-0.0582371\pi\)
−0.649218 + 0.760602i \(0.724904\pi\)
\(440\) 0 0
\(441\) −3.00000 20.7846i −0.142857 0.989743i
\(442\) 0 0
\(443\) −10.0000 17.3205i −0.475114 0.822922i 0.524479 0.851423i \(-0.324260\pi\)
−0.999594 + 0.0285009i \(0.990927\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000 + 5.19615i 0.140334 + 0.243066i 0.927622 0.373519i \(-0.121849\pi\)
−0.787288 + 0.616585i \(0.788516\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.00000 + 12.1244i 0.323921 + 0.561048i 0.981293 0.192518i \(-0.0616653\pi\)
−0.657372 + 0.753566i \(0.728332\pi\)
\(468\) 0 0
\(469\) −6.00000 5.19615i −0.277054 0.239936i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 5.19615i 0.137940 0.238919i
\(474\) 0 0
\(475\) −35.0000 −1.60591
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) 3.50000 6.06218i 0.159919 0.276988i −0.774920 0.632059i \(-0.782210\pi\)
0.934839 + 0.355071i \(0.115543\pi\)
\(480\) 0 0
\(481\) 4.00000 + 6.92820i 0.182384 + 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.5000 21.6506i 0.566429 0.981084i −0.430486 0.902597i \(-0.641658\pi\)
0.996915 0.0784867i \(-0.0250088\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 17.5000 30.3109i 0.788160 1.36513i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.50000 + 12.9904i −0.112140 + 0.582698i
\(498\) 0 0
\(499\) 4.00000 + 6.92820i 0.179065 + 0.310149i 0.941560 0.336844i \(-0.109360\pi\)
−0.762496 + 0.646993i \(0.776026\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 + 24.2487i 0.620539 + 1.07481i 0.989385 + 0.145315i \(0.0464195\pi\)
−0.368846 + 0.929490i \(0.620247\pi\)
\(510\) 0 0
\(511\) −28.0000 24.2487i −1.23865 1.07270i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −21.0000 −0.923579
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.00000 12.1244i 0.306676 0.531178i −0.670957 0.741496i \(-0.734117\pi\)
0.977633 + 0.210318i \(0.0674500\pi\)
\(522\) 0 0
\(523\) −7.00000 12.1244i −0.306089 0.530161i 0.671414 0.741082i \(-0.265687\pi\)
−0.977503 + 0.210921i \(0.932354\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) −21.0000 −0.911322
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 16.5000 12.9904i 0.710705 0.559535i
\(540\) 0 0
\(541\) −4.00000 6.92820i −0.171973 0.297867i 0.767136 0.641484i \(-0.221681\pi\)
−0.939110 + 0.343617i \(0.888348\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) −10.5000 + 18.1865i −0.448129 + 0.776182i
\(550\) 0 0
\(551\) 17.5000 + 30.3109i 0.745525 + 1.29129i
\(552\) 0 0
\(553\) 15.0000 5.19615i 0.637865 0.220963i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00000 + 15.5885i −0.381342 + 0.660504i −0.991254 0.131965i \(-0.957871\pi\)
0.609912 + 0.792469i \(0.291205\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.0000 24.2487i 0.590030 1.02196i −0.404198 0.914671i \(-0.632449\pi\)
0.994228 0.107290i \(-0.0342173\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −18.0000 15.5885i −0.755929 0.654654i
\(568\) 0 0
\(569\) −0.500000 0.866025i −0.0209611 0.0363057i 0.855355 0.518043i \(-0.173339\pi\)
−0.876316 + 0.481737i \(0.840006\pi\)
\(570\) 0 0
\(571\) 2.00000 3.46410i 0.0836974 0.144968i −0.821138 0.570730i \(-0.806660\pi\)
0.904835 + 0.425762i \(0.139994\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −30.0000 −1.25109
\(576\) 0 0
\(577\) −7.00000 + 12.1244i −0.291414 + 0.504744i −0.974144 0.225927i \(-0.927459\pi\)
0.682730 + 0.730670i \(0.260792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.50000 + 7.79423i 0.186371 + 0.322804i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.0000 27.7128i 0.653742 1.13231i −0.328465 0.944516i \(-0.606531\pi\)
0.982208 0.187799i \(-0.0601353\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) −9.00000 −0.366508
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.00000 + 12.1244i 0.284121 + 0.492112i 0.972396 0.233338i \(-0.0749648\pi\)
−0.688274 + 0.725450i \(0.741632\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.50000 6.06218i −0.141595 0.245249i
\(612\) 0 0
\(613\) −16.0000 + 27.7128i −0.646234 + 1.11931i 0.337781 + 0.941225i \(0.390324\pi\)
−0.984015 + 0.178085i \(0.943010\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −14.0000 + 24.2487i −0.562708 + 0.974638i 0.434551 + 0.900647i \(0.356907\pi\)
−0.997259 + 0.0739910i \(0.976426\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 56.0000 2.23287
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.50000 + 2.59808i 0.257539 + 0.102940i
\(638\) 0 0
\(639\) 7.50000 + 12.9904i 0.296695 + 0.513892i
\(640\) 0 0
\(641\) −9.00000 + 15.5885i −0.355479 + 0.615707i −0.987200 0.159489i \(-0.949015\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.0000 + 36.3731i −0.825595 + 1.42997i 0.0758684 + 0.997118i \(0.475827\pi\)
−0.901464 + 0.432855i \(0.857506\pi\)
\(648\) 0 0
\(649\) −10.5000 18.1865i −0.412161 0.713884i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 + 5.19615i 0.117399 + 0.203341i 0.918736 0.394872i \(-0.129211\pi\)
−0.801337 + 0.598213i \(0.795878\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −42.0000 −1.63858
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15.0000 + 25.9808i 0.580802 + 1.00598i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −21.0000 −0.810696
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.5000 + 30.3109i 0.672580 + 1.16494i 0.977170 + 0.212459i \(0.0681471\pi\)
−0.304590 + 0.952483i \(0.598520\pi\)
\(678\) 0 0
\(679\) 7.00000 36.3731i 0.268635 1.39587i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 + 20.7846i −0.459167 + 0.795301i −0.998917 0.0465244i \(-0.985185\pi\)
0.539750 + 0.841825i \(0.318519\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.50000 + 2.59808i −0.0571454 + 0.0989788i
\(690\) 0 0
\(691\) 17.5000 + 30.3109i 0.665731 + 1.15308i 0.979086 + 0.203445i \(0.0652137\pi\)
−0.313355 + 0.949636i \(0.601453\pi\)
\(692\) 0 0
\(693\) 4.50000 23.3827i 0.170941 0.888235i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −28.0000 + 48.4974i −1.05604 + 1.82911i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.0000 + 24.2487i 1.05305 + 0.911967i
\(708\) 0 0
\(709\) −25.0000 43.3013i −0.938895 1.62621i −0.767537 0.641004i \(-0.778518\pi\)
−0.171358 0.985209i \(-0.554815\pi\)
\(710\) 0 0
\(711\) 9.00000 15.5885i 0.337526 0.584613i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −21.0000 36.3731i −0.783168 1.35649i −0.930087 0.367338i \(-0.880269\pi\)
0.146920 0.989148i \(-0.453064\pi\)
\(720\) 0 0
\(721\) −35.0000 + 12.1244i −1.30347 + 0.451535i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.5000 + 21.6506i −0.464238 + 0.804084i
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 7.00000 12.1244i 0.258904 0.448435i
\(732\) 0 0
\(733\) −21.0000 36.3731i −0.775653 1.34347i −0.934427 0.356155i \(-0.884088\pi\)
0.158774 0.987315i \(-0.449246\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.50000 7.79423i −0.165760 0.287104i
\(738\) 0 0
\(739\) 2.00000 3.46410i 0.0735712 0.127429i −0.826893 0.562360i \(-0.809894\pi\)
0.900464 + 0.434930i \(0.143227\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20.0000 + 6.92820i −0.730784 + 0.253151i
\(750\) 0 0
\(751\) −10.0000 17.3205i −0.364905 0.632034i 0.623856 0.781540i \(-0.285565\pi\)
−0.988761 + 0.149505i \(0.952232\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.00000 0.327111 0.163555 0.986534i \(-0.447704\pi\)
0.163555 + 0.986534i \(0.447704\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) −8.00000 6.92820i −0.289619 0.250818i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.50000 6.06218i 0.126378 0.218893i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −21.0000 + 36.3731i −0.755318 + 1.30825i 0.189899 + 0.981804i \(0.439184\pi\)
−0.945216 + 0.326445i \(0.894149\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −7.50000 + 12.9904i −0.268371 + 0.464832i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3.50000 6.06218i 0.124762 0.216093i −0.796878 0.604140i \(-0.793517\pi\)
0.921640 + 0.388047i \(0.126850\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.50000 + 23.3827i −0.160002 + 0.831393i
\(792\) 0 0
\(793\) −3.50000 6.06218i −0.124289 0.215274i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0