Properties

Label 567.2.s.a.458.1
Level $567$
Weight $2$
Character 567.458
Analytic conductor $4.528$
Analytic rank $1$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(26,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.s (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: \(4.52751779461\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 458.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 567.458
Dual form 567.2.s.a.26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(-1.50000 - 0.866025i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-4.50000 + 2.59808i) q^{19} -5.00000 q^{25} +(4.00000 + 3.46410i) q^{28} +(-7.50000 + 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{37} +(2.50000 + 4.33013i) q^{43} +(5.50000 - 4.33013i) q^{49} +3.46410i q^{52} +(-6.00000 - 3.46410i) q^{61} +8.00000 q^{64} +(-5.50000 - 9.52628i) q^{67} +(-13.5000 - 7.79423i) q^{73} +(9.00000 + 5.19615i) q^{76} +(6.50000 - 11.2583i) q^{79} +(4.50000 + 0.866025i) q^{91} +(12.0000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 5 q^{7} - 3 q^{13} - 4 q^{16} - 9 q^{19} - 10 q^{25} + 8 q^{28} - 15 q^{31} - q^{37} + 5 q^{43} + 11 q^{49} - 12 q^{61} + 16 q^{64} - 11 q^{67} - 27 q^{73} + 18 q^{76} + 13 q^{79} + 9 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.50000 + 0.866025i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.50000 0.866025i −0.416025 0.240192i 0.277350 0.960769i \(-0.410544\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −4.50000 + 2.59808i −1.03237 + 0.596040i −0.917663 0.397360i \(-0.869927\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 4.00000 + 3.46410i 0.755929 + 0.654654i
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −7.50000 + 4.33013i −1.34704 + 0.777714i −0.987829 0.155543i \(-0.950287\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 0.866025i −0.0821995 0.142374i 0.821995 0.569495i \(-0.192861\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 2.50000 + 4.33013i 0.381246 + 0.660338i 0.991241 0.132068i \(-0.0421616\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.46410i 0.480384i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −6.00000 3.46410i −0.768221 0.443533i 0.0640184 0.997949i \(-0.479608\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 9.52628i −0.671932 1.16382i −0.977356 0.211604i \(-0.932131\pi\)
0.305424 0.952217i \(-0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −13.5000 7.79423i −1.58006 0.912245i −0.994850 0.101361i \(-0.967680\pi\)
−0.585206 0.810885i \(-0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 9.00000 + 5.19615i 1.03237 + 0.596040i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 11.2583i 0.731307 1.26666i −0.225018 0.974355i \(-0.572244\pi\)
0.956325 0.292306i \(-0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 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
\(91\) 4.50000 + 0.866025i 0.471728 + 0.0907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0000 6.92820i 1.21842 0.703452i 0.253837 0.967247i \(-0.418307\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 + 8.66025i 0.500000 + 0.866025i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 19.0526i 1.87730i 0.344865 + 0.938652i \(0.387925\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) 8.50000 14.7224i 0.814152 1.41015i −0.0957826 0.995402i \(-0.530535\pi\)
0.909935 0.414751i \(-0.136131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 10.3923i 0.188982 0.981981i
\(113\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 15.0000 + 8.66025i 1.34704 + 0.777714i
\(125\) 0 0
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 9.00000 10.3923i 0.780399 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 19.5000 + 11.2583i 1.65397 + 0.954919i 0.975417 + 0.220366i \(0.0707252\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 + 1.73205i −0.0821995 + 0.142374i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 + 10.3923i −1.43656 + 0.829396i −0.997609 0.0691164i \(-0.977982\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 6.92820i −0.313304 0.542659i 0.665771 0.746156i \(-0.268103\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(168\) 0 0
\(169\) −5.00000 8.66025i −0.384615 0.666173i
\(170\) 0 0
\(171\) 0 0
\(172\) 5.00000 8.66025i 0.381246 0.660338i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 12.5000 4.33013i 0.944911 0.327327i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 25.9808i 1.93113i 0.260153 + 0.965567i \(0.416227\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) −12.5000 21.6506i −0.899770 1.55845i −0.827788 0.561041i \(-0.810401\pi\)
−0.0719816 0.997406i \(-0.522932\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −13.0000 5.19615i −0.928571 0.371154i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.00000 1.73205i −0.212664 0.122782i 0.389885 0.920864i \(-0.372515\pi\)
−0.602549 + 0.798082i \(0.705848\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 6.00000 3.46410i 0.416025 0.240192i
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 + 13.8564i −0.550743 + 0.953914i 0.447478 + 0.894295i \(0.352322\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.0000 17.3205i 1.01827 1.17579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −9.00000 + 5.19615i −0.602685 + 0.347960i −0.770097 0.637927i \(-0.779792\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 29.4449i 1.94577i −0.231287 0.972886i \(-0.574293\pi\)
0.231287 0.972886i \(-0.425707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 27.7128i 1.78514i 0.450910 + 0.892570i \(0.351100\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 13.8564i 0.887066i
\(245\) 0 0
\(246\) 0 0
\(247\) 9.00000 0.572656
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 2.00000 + 1.73205i 0.124274 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −11.0000 + 19.0526i −0.671932 + 1.16382i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) −15.0000 + 8.66025i −0.911185 + 0.526073i −0.880812 0.473466i \(-0.843003\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −31.0000 −1.86261 −0.931305 0.364241i \(-0.881328\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(282\) 0 0
\(283\) −28.5000 + 16.4545i −1.69415 + 0.978117i −0.743048 + 0.669238i \(0.766621\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 + 14.7224i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 31.1769i 1.82449i
\(293\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −10.0000 8.66025i −0.576390 0.499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 20.7846i 1.19208i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.73205i 0.0988534i 0.998778 + 0.0494267i \(0.0157394\pi\)
−0.998778 + 0.0494267i \(0.984261\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 4.50000 + 2.59808i 0.254355 + 0.146852i 0.621757 0.783210i \(-0.286419\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −26.0000 −1.46261
\(317\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 7.50000 + 4.33013i 0.416025 + 0.240192i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.500000 + 0.866025i −0.0274825 + 0.0476011i −0.879440 0.476011i \(-0.842082\pi\)
0.851957 + 0.523612i \(0.175416\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.50000 4.33013i 0.136184 0.235877i −0.789865 0.613280i \(-0.789850\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 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
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) −30.0000 + 17.3205i −1.60586 + 0.927146i −0.615581 + 0.788074i \(0.711079\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 4.00000 6.92820i 0.210526 0.364642i
\(362\) 0 0
\(363\) 0 0
\(364\) −3.00000 8.66025i −0.157243 0.453921i
\(365\) 0 0
\(366\) 0 0
\(367\) 15.5885i 0.813711i 0.913493 + 0.406855i \(0.133375\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 37.0000 1.90056 0.950281 0.311393i \(-0.100796\pi\)
0.950281 + 0.311393i \(0.100796\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −24.0000 13.8564i −1.21842 0.703452i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.5000 9.52628i 0.828111 0.478110i −0.0250943 0.999685i \(-0.507989\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 10.0000 17.3205i 0.500000 0.866025i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 15.0000 0.747203
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 34.5000 19.9186i 1.70592 0.984911i 0.766426 0.642333i \(-0.222033\pi\)
0.939490 0.342578i \(-0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 33.0000 19.0526i 1.62579 0.938652i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) 9.50000 + 16.4545i 0.463002 + 0.801942i 0.999109 0.0422075i \(-0.0134391\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.0000 + 3.46410i 0.871081 + 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 22.5167i 1.08208i −0.840996 0.541041i \(-0.818030\pi\)
0.840996 0.541041i \(-0.181970\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −34.0000 −1.62830
\(437\) 0 0
\(438\) 0 0
\(439\) −27.0000 15.5885i −1.28864 0.743996i −0.310228 0.950662i \(-0.600405\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −20.0000 + 6.92820i −0.944911 + 0.327327i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.5000 35.5070i 0.958950 1.66095i 0.233890 0.972263i \(-0.424854\pi\)
0.725059 0.688686i \(-0.241812\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) −11.5000 + 19.9186i −0.534450 + 0.925695i 0.464739 + 0.885448i \(0.346148\pi\)
−0.999190 + 0.0402476i \(0.987185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 22.0000 + 19.0526i 1.01587 + 0.879765i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 22.5000 12.9904i 1.03237 0.596040i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 1.73205i 0.0789747i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 19.0526i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −12.5000 + 21.6506i −0.566429 + 0.981084i 0.430486 + 0.902597i \(0.358342\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 34.6410i 1.55543i
\(497\) 0 0
\(498\) 0 0
\(499\) 43.0000 1.92494 0.962472 0.271380i \(-0.0874801\pi\)
0.962472 + 0.271380i \(0.0874801\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 19.0000 + 32.9090i 0.842989 + 1.46010i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 40.5000 + 7.79423i 1.79161 + 0.344796i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) −25.5000 + 14.7224i −1.11504 + 0.643767i −0.940129 0.340818i \(-0.889296\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −27.0000 5.19615i −1.17060 0.225282i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −14.5000 25.1147i −0.623404 1.07977i −0.988847 0.148933i \(-0.952416\pi\)
0.365444 0.930834i \(-0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000 + 34.6410i 0.855138 + 1.48114i 0.876517 + 0.481371i \(0.159861\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.50000 + 33.7750i −0.276408 + 1.43626i
\(554\) 0 0
\(555\) 0 0
\(556\) 45.0333i 1.90984i
\(557\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 8.66025i 0.366290i
\(560\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 15.5000 + 26.8468i 0.648655 + 1.12350i 0.983444 + 0.181210i \(0.0580014\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.5000 + 16.4545i 1.18647 + 0.685009i 0.957503 0.288425i \(-0.0931316\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 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
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) 0 0
\(589\) 22.5000 38.9711i 0.927096 1.60578i
\(590\) 0 0
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) 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
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 1.50000 0.866025i 0.0611863 0.0353259i −0.469095 0.883148i \(-0.655420\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.00000 6.92820i −0.162758 0.281905i
\(605\) 0 0
\(606\) 0 0
\(607\) 5.19615i 0.210905i −0.994424 0.105453i \(-0.966371\pi\)
0.994424 0.105453i \(-0.0336291\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 5.00000 8.66025i 0.201948 0.349784i −0.747208 0.664590i \(-0.768606\pi\)
0.949156 + 0.314806i \(0.101939\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 8.66025i 0.348085i −0.984738 0.174042i \(-0.944317\pi\)
0.984738 0.174042i \(-0.0556830\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 36.0000 + 20.7846i 1.43656 + 0.829396i
\(629\) 0 0
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 + 1.73205i −0.475457 + 0.0686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −43.5000 25.1147i −1.71547 0.990429i −0.926750 0.375680i \(-0.877409\pi\)
−0.788723 0.614749i \(-0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −8.00000 + 13.8564i −0.313304 + 0.542659i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 13.5000 7.79423i 0.525089 0.303160i −0.213925 0.976850i \(-0.568625\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −18.5000 32.0429i −0.713123 1.23516i −0.963679 0.267063i \(-0.913947\pi\)
0.250557 0.968102i \(-0.419386\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −10.0000 + 17.3205i −0.384615 + 0.666173i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) −24.0000 + 27.7128i −0.921035 + 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 0 0
\(691\) −16.5000 9.52628i −0.627690 0.362397i 0.152167 0.988355i \(-0.451375\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −20.0000 17.3205i −0.755929 0.654654i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 4.50000 + 2.59808i 0.169721 + 0.0979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.0000 + 19.0526i −0.413114 + 0.715534i −0.995228 0.0975728i \(-0.968892\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) −16.5000 47.6314i −0.614492 1.77389i
\(722\) 0 0
\(723\) 0 0
\(724\) 45.0000 25.9808i 1.67241 0.965567i
\(725\) 0 0
\(726\) 0 0
\(727\) −19.5000 + 11.2583i −0.723215 + 0.417548i −0.815935 0.578144i \(-0.803777\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.6936i 1.98322i −0.129275 0.991609i \(-0.541265\pi\)
0.129275 0.991609i \(-0.458735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −26.5000 + 45.8993i −0.974818 + 1.68843i −0.294285 + 0.955718i \(0.595081\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −8.50000 + 44.1673i −0.307721 + 1.59896i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −22.5000 12.9904i −0.811371 0.468445i 0.0360609 0.999350i \(-0.488519\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25.0000 + 43.3013i −0.899770 + 1.55845i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 37.5000 21.6506i 1.34704 0.777714i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.00000 + 27.7128i 0.142857 + 0.989743i
\(785\) 0 0
\(786\) 0 0
\(787\) 3.00000 1.73205i 0.106938 0.0617409i −0.445577 0.895244i \(-0.647001\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 + 10.3923i 0.213066 + 0.369042i
\(794\) 0 0
\(795\) 0 0
\(796\) 6.92820i 0.245564i
\(797\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 10.3923i 0.364923i −0.983213 0.182462i \(-0.941593\pi\)
0.983213 0.182462i \(-0.0584065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −22.5000 12.9904i −0.787175 0.454476i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 26.0000 + 45.0333i 0.906303 + 1.56976i 0.819159 + 0.573567i \(0.194441\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 49.5000 + 28.5788i 1.71921 + 0.992584i 0.920383 + 0.391018i \(0.127877\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −12.0000 6.92820i −0.416025 0.240192i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −14.5000 + 25.1147i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 32.0000 1.10149
\(845\) 0 0
\(846\) 0 0
\(847\) −27.5000 + 9.52628i −0.944911 + 0.327327i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −40.5000 + 23.3827i −1.38669 + 0.800608i −0.992941 0.118609i \(-0.962157\pi\)
−0.393753 + 0.919216i \(0.628823\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 17.3205i 0.590968i −0.955348 0.295484i \(-0.904519\pi\)
0.955348 0.295484i \(-0.0954809\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −45.0000 8.66025i −1.52740 0.293948i
\(869\) 0 0
\(870\) 0 0
\(871\) 19.0526i 0.645571i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −47.0000 −1.58168 −0.790838 0.612026i \(-0.790355\pi\)
−0.790838 + 0.612026i \(0.790355\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 47.5000 16.4545i 1.59310 0.551866i
\(890\) 0 0
\(891\) 0 0
\(892\) 18.0000 + 10.3923i 0.602685 + 0.347960i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −59.0000 −1.95906 −0.979531 0.201291i \(-0.935486\pi\)
−0.979531 + 0.201291i \(0.935486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −51.0000 + 29.4449i −1.68509 + 0.972886i
\(917\) 0 0
\(918\) 0 0
\(919\) −0.500000 0.866025i −0.0164935 0.0285675i 0.857661 0.514216i \(-0.171917\pi\)
−0.874154 + 0.485648i \(0.838584\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.50000 + 4.33013i 0.0821995 + 0.142374i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) −13.5000 + 33.7750i −0.442445 + 1.10693i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 50.2295i 1.64093i 0.571700 + 0.820463i \(0.306284\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 13.5000 + 23.3827i 0.438229 + 0.759034i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 22.0000 38.1051i 0.709677 1.22920i
\(962\) 0 0
\(963\) 0 0
\(964\) 48.0000 27.7128i 1.54598 0.892570i
\(965\) 0 0
\(966\) 0 0
\(967\) 30.5000 52.8275i 0.980814 1.69882i 0.321578 0.946883i \(-0.395787\pi\)
0.659236 0.751936i \(-0.270880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(972\) 0 0
\(973\) −58.5000 11.2583i −1.87542 0.360925i
\(974\) 0 0
\(975\) 0 0
\(976\) 24.0000 13.8564i 0.768221 0.443533i
\(977\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −9.00000 15.5885i −0.286328 0.495935i
\(989\) 0 0
\(990\) 0 0
\(991\) 8.50000 14.7224i 0.270011 0.467673i −0.698853 0.715265i \(-0.746306\pi\)
0.968864 + 0.247592i \(0.0796392\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 39.8372i 1.26166i 0.775923 + 0.630828i \(0.217285\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 567.2.s.a.458.1 2
3.2 odd 2 CM 567.2.s.a.458.1 2
7.5 odd 6 567.2.i.b.215.1 2
9.2 odd 6 567.2.i.b.269.1 2
9.4 even 3 21.2.g.a.17.1 yes 2
9.5 odd 6 21.2.g.a.17.1 yes 2
9.7 even 3 567.2.i.b.269.1 2
21.5 even 6 567.2.i.b.215.1 2
36.23 even 6 336.2.bc.c.17.1 2
36.31 odd 6 336.2.bc.c.17.1 2
45.4 even 6 525.2.t.c.101.1 2
45.13 odd 12 525.2.q.d.374.2 4
45.14 odd 6 525.2.t.c.101.1 2
45.22 odd 12 525.2.q.d.374.1 4
45.23 even 12 525.2.q.d.374.2 4
45.32 even 12 525.2.q.d.374.1 4
63.4 even 3 147.2.c.a.146.2 2
63.5 even 6 21.2.g.a.5.1 2
63.13 odd 6 147.2.g.a.80.1 2
63.23 odd 6 147.2.g.a.68.1 2
63.31 odd 6 147.2.c.a.146.1 2
63.32 odd 6 147.2.c.a.146.2 2
63.40 odd 6 21.2.g.a.5.1 2
63.41 even 6 147.2.g.a.80.1 2
63.47 even 6 inner 567.2.s.a.26.1 2
63.58 even 3 147.2.g.a.68.1 2
63.59 even 6 147.2.c.a.146.1 2
63.61 odd 6 inner 567.2.s.a.26.1 2
252.31 even 6 2352.2.k.c.881.2 2
252.59 odd 6 2352.2.k.c.881.2 2
252.67 odd 6 2352.2.k.c.881.1 2
252.95 even 6 2352.2.k.c.881.1 2
252.103 even 6 336.2.bc.c.257.1 2
252.131 odd 6 336.2.bc.c.257.1 2
315.68 odd 12 525.2.q.d.299.1 4
315.103 even 12 525.2.q.d.299.1 4
315.194 even 6 525.2.t.c.26.1 2
315.229 odd 6 525.2.t.c.26.1 2
315.257 odd 12 525.2.q.d.299.2 4
315.292 even 12 525.2.q.d.299.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.g.a.5.1 2 63.5 even 6
21.2.g.a.5.1 2 63.40 odd 6
21.2.g.a.17.1 yes 2 9.4 even 3
21.2.g.a.17.1 yes 2 9.5 odd 6
147.2.c.a.146.1 2 63.31 odd 6
147.2.c.a.146.1 2 63.59 even 6
147.2.c.a.146.2 2 63.4 even 3
147.2.c.a.146.2 2 63.32 odd 6
147.2.g.a.68.1 2 63.23 odd 6
147.2.g.a.68.1 2 63.58 even 3
147.2.g.a.80.1 2 63.13 odd 6
147.2.g.a.80.1 2 63.41 even 6
336.2.bc.c.17.1 2 36.23 even 6
336.2.bc.c.17.1 2 36.31 odd 6
336.2.bc.c.257.1 2 252.103 even 6
336.2.bc.c.257.1 2 252.131 odd 6
525.2.q.d.299.1 4 315.68 odd 12
525.2.q.d.299.1 4 315.103 even 12
525.2.q.d.299.2 4 315.257 odd 12
525.2.q.d.299.2 4 315.292 even 12
525.2.q.d.374.1 4 45.22 odd 12
525.2.q.d.374.1 4 45.32 even 12
525.2.q.d.374.2 4 45.13 odd 12
525.2.q.d.374.2 4 45.23 even 12
525.2.t.c.26.1 2 315.194 even 6
525.2.t.c.26.1 2 315.229 odd 6
525.2.t.c.101.1 2 45.4 even 6
525.2.t.c.101.1 2 45.14 odd 6
567.2.i.b.215.1 2 7.5 odd 6
567.2.i.b.215.1 2 21.5 even 6
567.2.i.b.269.1 2 9.2 odd 6
567.2.i.b.269.1 2 9.7 even 3
567.2.s.a.26.1 2 63.47 even 6 inner
567.2.s.a.26.1 2 63.61 odd 6 inner
567.2.s.a.458.1 2 1.1 even 1 trivial
567.2.s.a.458.1 2 3.2 odd 2 CM
2352.2.k.c.881.1 2 252.67 odd 6
2352.2.k.c.881.1 2 252.95 even 6
2352.2.k.c.881.2 2 252.31 even 6
2352.2.k.c.881.2 2 252.59 odd 6