Properties

Label 1568.2.i.g.961.1
Level $1568$
Weight $2$
Character 1568.961
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,2,Mod(961,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (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: \(12.5205430369\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1568.961
Dual form 1568.2.i.g.1537.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{5} +(1.50000 - 2.59808i) q^{9} +6.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(0.500000 + 0.866025i) q^{25} -10.0000 q^{29} +(1.00000 - 1.73205i) q^{37} +10.0000 q^{41} +(-3.00000 - 5.19615i) q^{45} +(-7.00000 - 12.1244i) q^{53} +(5.00000 - 8.66025i) q^{61} +(6.00000 - 10.3923i) q^{65} +(3.00000 + 5.19615i) q^{73} +(-4.50000 - 7.79423i) q^{81} -4.00000 q^{85} +(-5.00000 + 8.66025i) q^{89} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 3 q^{9} + 12 q^{13} - 2 q^{17} + q^{25} - 20 q^{29} + 2 q^{37} + 20 q^{41} - 6 q^{45} - 14 q^{53} + 10 q^{61} + 12 q^{65} + 6 q^{73} - 9 q^{81} - 8 q^{85} - 10 q^{89} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 1.00000 1.73205i 0.447214 0.774597i −0.550990 0.834512i \(-0.685750\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 1.73205i 0.164399 0.284747i −0.772043 0.635571i \(-0.780765\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −3.00000 5.19615i −0.447214 0.774597i
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.00000 12.1244i −0.961524 1.66541i −0.718677 0.695344i \(-0.755252\pi\)
−0.242846 0.970065i \(-0.578081\pi\)
\(54\) 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
\(61\) 5.00000 8.66025i 0.640184 1.10883i −0.345207 0.938527i \(-0.612191\pi\)
0.985391 0.170305i \(-0.0544754\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 10.3923i 0.744208 1.28901i
\(66\) 0 0
\(67\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 3.00000 + 5.19615i 0.351123 + 0.608164i 0.986447 0.164083i \(-0.0524664\pi\)
−0.635323 + 0.772246i \(0.719133\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 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
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.00000 + 8.66025i −0.529999 + 0.917985i 0.469389 + 0.882992i \(0.344474\pi\)
−0.999388 + 0.0349934i \(0.988859\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000 1.82762 0.913812 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.00000 + 1.73205i 0.0995037 + 0.172345i 0.911479 0.411346i \(-0.134941\pi\)
−0.811976 + 0.583691i \(0.801608\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) −3.00000 5.19615i −0.287348 0.497701i 0.685828 0.727764i \(-0.259440\pi\)
−0.973176 + 0.230063i \(0.926107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.00000 15.5885i 0.832050 1.44115i
\(118\) 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
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.0000 + 19.0526i 0.939793 + 1.62777i 0.765855 + 0.643013i \(0.222316\pi\)
0.173939 + 0.984757i \(0.444351\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −10.0000 + 17.3205i −0.830455 + 1.43839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.00000 + 12.1244i −0.573462 + 0.993266i 0.422744 + 0.906249i \(0.361067\pi\)
−0.996207 + 0.0870170i \(0.972267\pi\)
\(150\) 0 0
\(151\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.0000 19.0526i −0.877896 1.52056i −0.853646 0.520854i \(-0.825614\pi\)
−0.0242497 0.999706i \(-0.507720\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.0000 22.5167i 0.988372 1.71191i 0.362500 0.931984i \(-0.381923\pi\)
0.625871 0.779926i \(-0.284744\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 7.00000 + 12.1244i 0.503871 + 0.872730i 0.999990 + 0.00447566i \(0.00142465\pi\)
−0.496119 + 0.868255i \(0.665242\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 17.3205i 0.698430 1.20972i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 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
\(221\) −6.00000 10.3923i −0.403604 0.699062i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) −15.0000 + 25.9808i −0.991228 + 1.71686i −0.381157 + 0.924510i \(0.624474\pi\)
−0.610071 + 0.792347i \(0.708859\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.0000 + 22.5167i −0.851658 + 1.47512i 0.0280525 + 0.999606i \(0.491069\pi\)
−0.879711 + 0.475509i \(0.842264\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 15.0000 + 25.9808i 0.966235 + 1.67357i 0.706260 + 0.707953i \(0.250381\pi\)
0.259975 + 0.965615i \(0.416286\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 0 0
\(257\) −1.00000 + 1.73205i −0.0623783 + 0.108042i −0.895528 0.445005i \(-0.853202\pi\)
0.833150 + 0.553047i \(0.186535\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −15.0000 + 25.9808i −0.928477 + 1.60817i
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) −28.0000 −1.72003
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.0000 + 22.5167i 0.792624 + 1.37287i 0.924337 + 0.381577i \(0.124619\pi\)
−0.131713 + 0.991288i \(0.542048\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 9.00000 + 15.5885i 0.540758 + 0.936620i 0.998861 + 0.0477206i \(0.0151957\pi\)
−0.458103 + 0.888899i \(0.651471\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −34.0000 −1.98630 −0.993151 0.116841i \(-0.962723\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10.0000 17.3205i −0.572598 0.991769i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 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
\(313\) −13.0000 + 22.5167i −0.734803 + 1.27272i 0.220006 + 0.975499i \(0.429392\pi\)
−0.954810 + 0.297218i \(0.903941\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 + 19.0526i −0.617822 + 1.07010i 0.372061 + 0.928208i \(0.378651\pi\)
−0.989882 + 0.141890i \(0.954682\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.00000 + 5.19615i 0.166410 + 0.288231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(332\) 0 0
\(333\) −3.00000 5.19615i −0.164399 0.284747i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 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
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −17.0000 29.4449i −0.904819 1.56719i −0.821160 0.570697i \(-0.806673\pi\)
−0.0836583 0.996495i \(-0.526660\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(360\) 0 0
\(361\) 9.50000 + 16.4545i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 0 0
\(369\) 15.0000 25.9808i 0.780869 1.35250i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −60.0000 −3.09016
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.0000 + 29.4449i 0.861934 + 1.49291i 0.870059 + 0.492947i \(0.164080\pi\)
−0.00812520 + 0.999967i \(0.502586\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −19.0000 + 32.9090i −0.953583 + 1.65165i −0.216004 + 0.976392i \(0.569302\pi\)
−0.737579 + 0.675261i \(0.764031\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.00000 + 1.73205i −0.0499376 + 0.0864945i −0.889914 0.456129i \(-0.849236\pi\)
0.839976 + 0.542623i \(0.182569\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −18.0000 −0.894427
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 3.00000 + 5.19615i 0.148340 + 0.256933i 0.930614 0.366002i \(-0.119274\pi\)
−0.782274 + 0.622935i \(0.785940\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.00000 1.73205i 0.0485071 0.0840168i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 10.0000 + 17.3205i 0.474045 + 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −21.0000 + 36.3731i −0.982339 + 1.70146i −0.329125 + 0.944286i \(0.606754\pi\)
−0.653213 + 0.757174i \(0.726579\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −42.0000 −1.92305
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 6.00000 10.3923i 0.273576 0.473848i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.0000 31.1769i 0.817338 1.41567i
\(486\) 0 0
\(487\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 10.0000 + 17.3205i 0.450377 + 0.780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.00000 8.66025i 0.221621 0.383859i −0.733679 0.679496i \(-0.762199\pi\)
0.955300 + 0.295637i \(0.0955319\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) 11.0000 + 19.0526i 0.481919 + 0.834708i 0.999785 0.0207541i \(-0.00660670\pi\)
−0.517866 + 0.855462i \(0.673273\pi\)
\(522\) 0 0
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 21.0000 36.3731i 0.902861 1.56380i 0.0790969 0.996867i \(-0.474796\pi\)
0.823764 0.566933i \(-0.191870\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −15.0000 25.9808i −0.640184 1.10883i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.0000 32.9090i −0.805056 1.39440i −0.916253 0.400599i \(-0.868802\pi\)
0.111198 0.993798i \(-0.464531\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 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
\(565\) −14.0000 + 24.2487i −0.588984 + 1.02015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.0000 + 22.5167i −0.544988 + 0.943948i 0.453619 + 0.891196i \(0.350133\pi\)
−0.998608 + 0.0527519i \(0.983201\pi\)
\(570\) 0 0
\(571\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.00000 1.73205i −0.0416305 0.0721062i 0.844459 0.535620i \(-0.179922\pi\)
−0.886090 + 0.463513i \(0.846589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −18.0000 31.1769i −0.744208 1.28901i
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.0000 39.8372i 0.944497 1.63592i 0.187741 0.982219i \(-0.439883\pi\)
0.756756 0.653698i \(-0.226783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 19.0526i −0.447214 0.774597i
\(606\) 0 0
\(607\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17.0000 + 29.4449i 0.686624 + 1.18927i 0.972924 + 0.231127i \(0.0742412\pi\)
−0.286300 + 0.958140i \(0.592425\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.50000 16.4545i 0.380000 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −25.0000 43.3013i −0.987441 1.71030i −0.630544 0.776153i \(-0.717168\pi\)
−0.356897 0.934144i \(-0.616165\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.0000 22.5167i 0.508729 0.881145i −0.491220 0.871036i \(-0.663449\pi\)
0.999949 0.0101092i \(-0.00321793\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 25.0000 + 43.3013i 0.972387 + 1.68422i 0.688301 + 0.725426i \(0.258357\pi\)
0.284087 + 0.958799i \(0.408310\pi\)
\(662\) 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
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.00000 1.73205i 0.0384331 0.0665681i −0.846169 0.532915i \(-0.821097\pi\)
0.884602 + 0.466347i \(0.154430\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0 0
\(685\) 44.0000 1.68115
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −42.0000 72.7461i −1.60007 2.77141i
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −10.0000 17.3205i −0.378777 0.656061i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.0000 + 25.9808i −0.563337 + 0.975728i 0.433865 + 0.900978i \(0.357149\pi\)
−0.997202 + 0.0747503i \(0.976184\pi\)
\(710\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.00000 8.66025i −0.185695 0.321634i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −27.0000 + 46.7654i −0.997268 + 1.72732i −0.434659 + 0.900595i \(0.643131\pi\)
−0.562609 + 0.826723i \(0.690202\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 14.0000 + 24.2487i 0.512920 + 0.888404i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.0000 32.9090i 0.688749 1.19295i −0.283493 0.958974i \(-0.591493\pi\)
0.972243 0.233975i \(-0.0751733\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 + 10.3923i −0.216930 + 0.375735i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.0000 + 29.4449i 0.611448 + 1.05906i 0.990997 + 0.133887i \(0.0427458\pi\)
−0.379549 + 0.925172i \(0.623921\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −44.0000 −1.57043
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 30.0000 51.9615i 1.06533 1.84521i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 15.0000 + 25.9808i 0.529999 + 0.917985i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 8.66025i −0.175791 0.304478i 0.764644 0.644453i \(-0.222915\pi\)
−0.940435 + 0.339975i \(0.889582\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 43.3013i 0.872506 1.51122i 0.0131101 0.999914i \(-0.495827\pi\)
0.859396 0.511311i \(-0.170840\pi\)
\(822\) 0 0
\(823\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −27.0000 46.7654i −0.937749 1.62423i −0.769657 0.638457i \(-0.779573\pi\)
−0.168091 0.985771i \(-0.553760\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 23.0000 39.8372i 0.791224 1.37044i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.0000 50.2295i −0.990621 1.71581i −0.613642 0.789584i \(-0.710296\pi\)
−0.376979 0.926222i \(-0.623037\pi\)
\(858\) 0 0
\(859\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(864\) 0 0
\(865\) −26.0000 45.0333i −0.884027 1.53118i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 27.0000 46.7654i 0.913812 1.58277i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 29.0000 50.2295i 0.979260 1.69613i 0.314169 0.949367i \(-0.398274\pi\)
0.665092 0.746762i \(-0.268392\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.0000 + 24.2487i −0.466408 + 0.807842i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 + 31.1769i −0.598340 + 1.03636i
\(906\) 0 0
\(907\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23.0000 39.8372i 0.754606 1.30702i −0.190965 0.981597i \(-0.561162\pi\)
0.945570 0.325418i \(-0.105505\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.0000 + 50.2295i 0.945373 + 1.63743i 0.755003 + 0.655722i \(0.227636\pi\)
0.190370 + 0.981712i \(0.439031\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) 0 0
\(949\) 18.0000 + 31.1769i 0.584305 + 1.01205i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.5000 26.8468i 0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31.0000 + 53.6936i 0.991778 + 1.71781i 0.606715 + 0.794919i \(0.292487\pi\)
0.385063 + 0.922890i \(0.374180\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) −2.00000 + 3.46410i −0.0637253 + 0.110375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −31.0000 53.6936i −0.981780 1.70049i −0.655454 0.755235i \(-0.727523\pi\)
−0.326326 0.945257i \(-0.605811\pi\)
\(998\) 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 1568.2.i.g.961.1 2
4.3 odd 2 CM 1568.2.i.g.961.1 2
7.2 even 3 32.2.a.a.1.1 1
7.3 odd 6 1568.2.i.f.1537.1 2
7.4 even 3 inner 1568.2.i.g.1537.1 2
7.5 odd 6 1568.2.a.e.1.1 1
7.6 odd 2 1568.2.i.f.961.1 2
21.2 odd 6 288.2.a.d.1.1 1
28.3 even 6 1568.2.i.f.1537.1 2
28.11 odd 6 inner 1568.2.i.g.1537.1 2
28.19 even 6 1568.2.a.e.1.1 1
28.23 odd 6 32.2.a.a.1.1 1
28.27 even 2 1568.2.i.f.961.1 2
35.2 odd 12 800.2.c.e.449.1 2
35.9 even 6 800.2.a.d.1.1 1
35.23 odd 12 800.2.c.e.449.2 2
56.5 odd 6 3136.2.a.m.1.1 1
56.19 even 6 3136.2.a.m.1.1 1
56.37 even 6 64.2.a.a.1.1 1
56.51 odd 6 64.2.a.a.1.1 1
63.2 odd 6 2592.2.i.e.1729.1 2
63.16 even 3 2592.2.i.t.1729.1 2
63.23 odd 6 2592.2.i.e.865.1 2
63.58 even 3 2592.2.i.t.865.1 2
77.65 odd 6 3872.2.a.f.1.1 1
84.23 even 6 288.2.a.d.1.1 1
91.51 even 6 5408.2.a.g.1.1 1
105.2 even 12 7200.2.f.m.6049.1 2
105.23 even 12 7200.2.f.m.6049.2 2
105.44 odd 6 7200.2.a.v.1.1 1
112.37 even 12 256.2.b.b.129.1 2
112.51 odd 12 256.2.b.b.129.2 2
112.93 even 12 256.2.b.b.129.2 2
112.107 odd 12 256.2.b.b.129.1 2
119.16 even 6 9248.2.a.f.1.1 1
140.23 even 12 800.2.c.e.449.2 2
140.79 odd 6 800.2.a.d.1.1 1
140.107 even 12 800.2.c.e.449.1 2
168.107 even 6 576.2.a.c.1.1 1
168.149 odd 6 576.2.a.c.1.1 1
224.37 even 24 1024.2.e.j.769.1 4
224.51 odd 24 1024.2.e.j.257.1 4
224.93 even 24 1024.2.e.j.257.2 4
224.107 odd 24 1024.2.e.j.769.2 4
224.149 even 24 1024.2.e.j.769.2 4
224.163 odd 24 1024.2.e.j.257.2 4
224.205 even 24 1024.2.e.j.257.1 4
224.219 odd 24 1024.2.e.j.769.1 4
252.23 even 6 2592.2.i.e.865.1 2
252.79 odd 6 2592.2.i.t.1729.1 2
252.191 even 6 2592.2.i.e.1729.1 2
252.247 odd 6 2592.2.i.t.865.1 2
280.37 odd 12 1600.2.c.l.449.2 2
280.93 odd 12 1600.2.c.l.449.1 2
280.107 even 12 1600.2.c.l.449.2 2
280.149 even 6 1600.2.a.n.1.1 1
280.163 even 12 1600.2.c.l.449.1 2
280.219 odd 6 1600.2.a.n.1.1 1
308.219 even 6 3872.2.a.f.1.1 1
336.107 even 12 2304.2.d.j.1153.2 2
336.149 odd 12 2304.2.d.j.1153.2 2
336.275 even 12 2304.2.d.j.1153.1 2
336.317 odd 12 2304.2.d.j.1153.1 2
364.51 odd 6 5408.2.a.g.1.1 1
420.23 odd 12 7200.2.f.m.6049.2 2
420.107 odd 12 7200.2.f.m.6049.1 2
420.359 even 6 7200.2.a.v.1.1 1
476.135 odd 6 9248.2.a.f.1.1 1
616.219 even 6 7744.2.a.v.1.1 1
616.373 odd 6 7744.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.a.a.1.1 1 7.2 even 3
32.2.a.a.1.1 1 28.23 odd 6
64.2.a.a.1.1 1 56.37 even 6
64.2.a.a.1.1 1 56.51 odd 6
256.2.b.b.129.1 2 112.37 even 12
256.2.b.b.129.1 2 112.107 odd 12
256.2.b.b.129.2 2 112.51 odd 12
256.2.b.b.129.2 2 112.93 even 12
288.2.a.d.1.1 1 21.2 odd 6
288.2.a.d.1.1 1 84.23 even 6
576.2.a.c.1.1 1 168.107 even 6
576.2.a.c.1.1 1 168.149 odd 6
800.2.a.d.1.1 1 35.9 even 6
800.2.a.d.1.1 1 140.79 odd 6
800.2.c.e.449.1 2 35.2 odd 12
800.2.c.e.449.1 2 140.107 even 12
800.2.c.e.449.2 2 35.23 odd 12
800.2.c.e.449.2 2 140.23 even 12
1024.2.e.j.257.1 4 224.51 odd 24
1024.2.e.j.257.1 4 224.205 even 24
1024.2.e.j.257.2 4 224.93 even 24
1024.2.e.j.257.2 4 224.163 odd 24
1024.2.e.j.769.1 4 224.37 even 24
1024.2.e.j.769.1 4 224.219 odd 24
1024.2.e.j.769.2 4 224.107 odd 24
1024.2.e.j.769.2 4 224.149 even 24
1568.2.a.e.1.1 1 7.5 odd 6
1568.2.a.e.1.1 1 28.19 even 6
1568.2.i.f.961.1 2 7.6 odd 2
1568.2.i.f.961.1 2 28.27 even 2
1568.2.i.f.1537.1 2 7.3 odd 6
1568.2.i.f.1537.1 2 28.3 even 6
1568.2.i.g.961.1 2 1.1 even 1 trivial
1568.2.i.g.961.1 2 4.3 odd 2 CM
1568.2.i.g.1537.1 2 7.4 even 3 inner
1568.2.i.g.1537.1 2 28.11 odd 6 inner
1600.2.a.n.1.1 1 280.149 even 6
1600.2.a.n.1.1 1 280.219 odd 6
1600.2.c.l.449.1 2 280.93 odd 12
1600.2.c.l.449.1 2 280.163 even 12
1600.2.c.l.449.2 2 280.37 odd 12
1600.2.c.l.449.2 2 280.107 even 12
2304.2.d.j.1153.1 2 336.275 even 12
2304.2.d.j.1153.1 2 336.317 odd 12
2304.2.d.j.1153.2 2 336.107 even 12
2304.2.d.j.1153.2 2 336.149 odd 12
2592.2.i.e.865.1 2 63.23 odd 6
2592.2.i.e.865.1 2 252.23 even 6
2592.2.i.e.1729.1 2 63.2 odd 6
2592.2.i.e.1729.1 2 252.191 even 6
2592.2.i.t.865.1 2 63.58 even 3
2592.2.i.t.865.1 2 252.247 odd 6
2592.2.i.t.1729.1 2 63.16 even 3
2592.2.i.t.1729.1 2 252.79 odd 6
3136.2.a.m.1.1 1 56.5 odd 6
3136.2.a.m.1.1 1 56.19 even 6
3872.2.a.f.1.1 1 77.65 odd 6
3872.2.a.f.1.1 1 308.219 even 6
5408.2.a.g.1.1 1 91.51 even 6
5408.2.a.g.1.1 1 364.51 odd 6
7200.2.a.v.1.1 1 105.44 odd 6
7200.2.a.v.1.1 1 420.359 even 6
7200.2.f.m.6049.1 2 105.2 even 12
7200.2.f.m.6049.1 2 420.107 odd 12
7200.2.f.m.6049.2 2 105.23 even 12
7200.2.f.m.6049.2 2 420.23 odd 12
7744.2.a.v.1.1 1 616.219 even 6
7744.2.a.v.1.1 1 616.373 odd 6
9248.2.a.f.1.1 1 119.16 even 6
9248.2.a.f.1.1 1 476.135 odd 6