Properties

Label 1568.2.i.t.961.1
Level $1568$
Weight $2$
Character 1568.961
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

Related objects

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1568.961
Dual form 1568.2.i.t.1537.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 1.22474i) q^{5} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.707107 + 1.22474i) q^{5} +(1.50000 - 2.59808i) q^{9} -1.41421 q^{13} +(3.53553 + 6.12372i) q^{17} +(1.50000 + 2.59808i) q^{25} -4.00000 q^{29} +(6.00000 - 10.3923i) q^{37} +12.7279 q^{41} +(2.12132 + 3.67423i) q^{45} +(7.00000 + 12.1244i) q^{53} +(-7.77817 + 13.4722i) q^{61} +(1.00000 - 1.73205i) q^{65} +(7.77817 + 13.4722i) q^{73} +(-4.50000 - 7.79423i) q^{81} -10.0000 q^{85} +(-2.12132 + 3.67423i) q^{89} +7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 6 q^{25} - 16 q^{29} + 24 q^{37} + 28 q^{53} + 4 q^{65} - 18 q^{81} - 40 q^{85}+O(q^{100}) \) 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
\(5\) −0.707107 + 1.22474i −0.316228 + 0.547723i −0.979698 0.200480i \(-0.935750\pi\)
0.663470 + 0.748203i \(0.269083\pi\)
\(6\) 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
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.53553 + 6.12372i 0.857493 + 1.48522i 0.874313 + 0.485363i \(0.161312\pi\)
−0.0168199 + 0.999859i \(0.505354\pi\)
\(18\) 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
\(25\) 1.50000 + 2.59808i 0.300000 + 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 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
\(37\) 6.00000 10.3923i 0.986394 1.70848i 0.350823 0.936442i \(-0.385902\pi\)
0.635571 0.772043i \(-0.280765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.7279 1.98777 0.993884 0.110432i \(-0.0352233\pi\)
0.993884 + 0.110432i \(0.0352233\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 2.12132 + 3.67423i 0.316228 + 0.547723i
\(46\) 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 \(0.244748\pi\)
0.242846 + 0.970065i \(0.421919\pi\)
\(54\) 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
\(61\) −7.77817 + 13.4722i −0.995893 + 1.72494i −0.419540 + 0.907737i \(0.637809\pi\)
−0.576353 + 0.817201i \(0.695525\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 1.73205i 0.124035 0.214834i
\(66\) 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
\(73\) 7.77817 + 13.4722i 0.910366 + 1.57680i 0.813547 + 0.581499i \(0.197534\pi\)
0.0968194 + 0.995302i \(0.469133\pi\)
\(74\) 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
\(85\) −10.0000 −1.08465
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.12132 + 3.67423i −0.224860 + 0.389468i −0.956277 0.292462i \(-0.905526\pi\)
0.731418 + 0.681930i \(0.238859\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.07107 0.717958 0.358979 0.933346i \(-0.383125\pi\)
0.358979 + 0.933346i \(0.383125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.36396 + 11.0227i 0.633238 + 1.09680i 0.986886 + 0.161421i \(0.0516078\pi\)
−0.353648 + 0.935379i \(0.615059\pi\)
\(102\) 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
\(109\) 10.0000 + 17.3205i 0.957826 + 1.65900i 0.727764 + 0.685828i \(0.240560\pi\)
0.230063 + 0.973176i \(0.426107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.12132 + 3.67423i −0.196116 + 0.339683i
\(118\) 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
\(125\) −11.3137 −1.01193
\(126\) 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
\(137\) −4.00000 6.92820i −0.341743 0.591916i 0.643013 0.765855i \(-0.277684\pi\)
−0.984757 + 0.173939i \(0.944351\pi\)
\(138\) 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
\(145\) 2.82843 4.89898i 0.234888 0.406838i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.00000 12.1244i 0.573462 0.993266i −0.422744 0.906249i \(-0.638933\pi\)
0.996207 0.0870170i \(-0.0277334\pi\)
\(150\) 0 0
\(151\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 21.2132 1.71499
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0208 20.8207i −0.959366 1.66167i −0.724046 0.689752i \(-0.757720\pi\)
−0.235320 0.971918i \(-0.575614\pi\)
\(158\) 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
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.6066 18.3712i 0.806405 1.39673i −0.108933 0.994049i \(-0.534744\pi\)
0.915338 0.402685i \(-0.131923\pi\)
\(174\) 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
\(181\) −26.8701 −1.99724 −0.998618 0.0525588i \(-0.983262\pi\)
−0.998618 + 0.0525588i \(0.983262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.48528 + 14.6969i 0.623850 + 1.08054i
\(186\) 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 \(-0.998575\pi\)
0.496119 0.868255i \(-0.334758\pi\)
\(194\) 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
\(205\) −9.00000 + 15.5885i −0.628587 + 1.08875i
\(206\) 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
\(221\) −5.00000 8.66025i −0.336336 0.582552i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 9.00000 0.600000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) −9.19239 + 15.9217i −0.607450 + 1.05213i 0.384209 + 0.923246i \(0.374474\pi\)
−0.991659 + 0.128888i \(0.958859\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.00000 + 13.8564i −0.524097 + 0.907763i 0.475509 + 0.879711i \(0.342264\pi\)
−0.999606 + 0.0280525i \(0.991069\pi\)
\(234\) 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
\(241\) −13.4350 23.2702i −0.865426 1.49896i −0.866623 0.498963i \(-0.833714\pi\)
0.00119700 0.999999i \(-0.499619\pi\)
\(242\) 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
\(257\) 10.6066 18.3712i 0.661622 1.14596i −0.318568 0.947900i \(-0.603202\pi\)
0.980189 0.198062i \(-0.0634648\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 + 10.3923i −0.371391 + 0.643268i
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) −19.7990 −1.21624
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −16.2635 28.1691i −0.991600 1.71750i −0.607811 0.794082i \(-0.707952\pi\)
−0.383789 0.923421i \(-0.625381\pi\)
\(270\) 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
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) 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
\(289\) −16.5000 + 28.5788i −0.970588 + 1.68111i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.2132 1.23929 0.619644 0.784883i \(-0.287277\pi\)
0.619644 + 0.784883i \(0.287277\pi\)
\(294\) 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
\(305\) −11.0000 19.0526i −0.629858 1.09095i
\(306\) 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
\(313\) −0.707107 + 1.22474i −0.0399680 + 0.0692267i −0.885317 0.464987i \(-0.846059\pi\)
0.845349 + 0.534214i \(0.179392\pi\)
\(314\) 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
\(325\) −2.12132 3.67423i −0.117670 0.203810i
\(326\) 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
\(333\) −18.0000 31.1769i −0.986394 1.70848i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 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
\(349\) −32.5269 −1.74113 −0.870563 0.492057i \(-0.836245\pi\)
−0.870563 + 0.492057i \(0.836245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −17.6777 30.6186i −0.940887 1.62966i −0.763783 0.645473i \(-0.776660\pi\)
−0.177104 0.984192i \(-0.556673\pi\)
\(354\) 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
\(365\) −22.0000 −1.15153
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 0 0
\(369\) 19.0919 33.0681i 0.993884 1.72146i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7.00000 12.1244i 0.362446 0.627775i −0.625917 0.779890i \(-0.715275\pi\)
0.988363 + 0.152115i \(0.0486083\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.65685 0.291343
\(378\) 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
\(389\) −10.0000 17.3205i −0.507020 0.878185i −0.999967 0.00812520i \(-0.997414\pi\)
0.492947 0.870059i \(-0.335920\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.6777 30.6186i 0.887217 1.53670i 0.0440652 0.999029i \(-0.485969\pi\)
0.843152 0.537676i \(-0.180698\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.0000 + 34.6410i −0.998752 + 1.72989i −0.456129 + 0.889914i \(0.650764\pi\)
−0.542623 + 0.839976i \(0.682569\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 12.7279 0.632456
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −12.0208 20.8207i −0.594391 1.02952i −0.993632 0.112670i \(-0.964060\pi\)
0.399241 0.916846i \(-0.369274\pi\)
\(410\) 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
\(425\) −10.6066 + 18.3712i −0.514496 + 0.891133i
\(426\) 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
\(433\) −41.0122 −1.97092 −0.985460 0.169907i \(-0.945653\pi\)
−0.985460 + 0.169907i \(0.945653\pi\)
\(434\) 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
\(445\) −3.00000 5.19615i −0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 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 \(-0.393246\pi\)
0.653213 0.757174i \(-0.273421\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7279 0.592798 0.296399 0.955064i \(-0.404214\pi\)
0.296399 + 0.955064i \(0.404214\pi\)
\(462\) 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
\(481\) −8.48528 + 14.6969i −0.386896 + 0.670123i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.00000 + 8.66025i −0.227038 + 0.393242i
\(486\) 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
\(493\) −14.1421 24.4949i −0.636930 1.10319i
\(494\) 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
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.0208 20.8207i 0.532813 0.922860i −0.466453 0.884546i \(-0.654468\pi\)
0.999266 0.0383134i \(-0.0121985\pi\)
\(510\) 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
\(521\) 21.9203 + 37.9671i 0.960346 + 1.66337i 0.721630 + 0.692279i \(0.243393\pi\)
0.238716 + 0.971089i \(0.423273\pi\)
\(522\) 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
\(533\) −18.0000 −0.779667
\(534\) 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 \(0.525204\pi\)
−0.823764 + 0.566933i \(0.808130\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −28.2843 −1.21157
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 23.3345 + 40.4166i 0.995893 + 1.72494i
\(550\) 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
\(565\) −9.89949 + 17.1464i −0.416475 + 0.721356i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000 34.6410i 0.838444 1.45223i −0.0527519 0.998608i \(-0.516799\pi\)
0.891196 0.453619i \(-0.149867\pi\)
\(570\) 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
\(577\) −16.2635 28.1691i −0.677057 1.17270i −0.975863 0.218383i \(-0.929922\pi\)
0.298807 0.954314i \(-0.403411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.00000 5.19615i −0.124035 0.214834i
\(586\) 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
\(593\) −21.9203 + 37.9671i −0.900159 + 1.55912i −0.0728728 + 0.997341i \(0.523217\pi\)
−0.827286 + 0.561780i \(0.810117\pi\)
\(594\) 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
\(601\) −26.8701 −1.09605 −0.548026 0.836461i \(-0.684621\pi\)
−0.548026 + 0.836461i \(0.684621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.77817 + 13.4722i 0.316228 + 0.547723i
\(606\) 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
\(613\) 18.0000 + 31.1769i 0.727013 + 1.25922i 0.958140 + 0.286300i \(0.0924254\pi\)
−0.231127 + 0.972924i \(0.574241\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 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
\(625\) 0.500000 0.866025i 0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 84.8528 3.38330
\(630\) 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
\(641\) 4.00000 + 6.92820i 0.157991 + 0.273648i 0.934144 0.356897i \(-0.116165\pi\)
−0.776153 + 0.630544i \(0.782832\pi\)
\(642\) 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
\(653\) 22.0000 38.1051i 0.860927 1.49117i −0.0101092 0.999949i \(-0.503218\pi\)
0.871036 0.491220i \(-0.163449\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 46.6690 1.82073
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 21.9203 + 37.9671i 0.852601 + 1.47675i 0.878853 + 0.477093i \(0.158310\pi\)
−0.0262514 + 0.999655i \(0.508357\pi\)
\(662\) 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
\(673\) −24.0000 −0.925132 −0.462566 0.886585i \(-0.653071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.0919 33.0681i 0.733761 1.27091i −0.221504 0.975159i \(-0.571097\pi\)
0.955265 0.295751i \(-0.0955700\pi\)
\(678\) 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
\(685\) 11.3137 0.432275
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.89949 17.1464i −0.377141 0.653227i
\(690\) 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
\(697\) 45.0000 + 77.9423i 1.70450 + 2.95227i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.0000 1.96401 0.982006 0.188847i \(-0.0604752\pi\)
0.982006 + 0.188847i \(0.0604752\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.0000 38.1051i 0.826227 1.43107i −0.0747503 0.997202i \(-0.523816\pi\)
0.900978 0.433865i \(-0.142851\pi\)
\(710\) 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
\(725\) −6.00000 10.3923i −0.222834 0.385961i
\(726\) 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
\(733\) −20.5061 + 35.5176i −0.757410 + 1.31187i 0.186757 + 0.982406i \(0.440202\pi\)
−0.944167 + 0.329466i \(0.893131\pi\)
\(734\) 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
\(745\) 9.89949 + 17.1464i 0.362689 + 0.628197i
\(746\) 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
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.5772 + 47.7650i −0.999671 + 1.73148i −0.477637 + 0.878557i \(0.658507\pi\)
−0.522034 + 0.852925i \(0.674827\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −15.0000 + 25.9808i −0.542326 + 0.939336i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −18.3848 −0.662972 −0.331486 0.943460i \(-0.607550\pi\)
−0.331486 + 0.943460i \(0.607550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.5772 + 47.7650i 0.991882 + 1.71799i 0.606068 + 0.795413i \(0.292746\pi\)
0.385813 + 0.922577i \(0.373921\pi\)
\(774\) 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
\(785\) 34.0000 1.21351
\(786\) 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
\(793\) 11.0000 19.0526i 0.390621 0.676576i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.2132 0.751410 0.375705 0.926739i \(-0.377401\pi\)
0.375705 + 0.926739i \(0.377401\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.36396 + 11.0227i 0.224860 + 0.389468i
\(802\) 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
\(829\) 26.1630 + 45.3156i 0.908677 + 1.57387i 0.815904 + 0.578187i \(0.196240\pi\)
0.0927727 + 0.995687i \(0.470427\pi\)
\(830\) 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
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.77817 13.4722i 0.267577 0.463458i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 7.07107 0.242109 0.121054 0.992646i \(-0.461372\pi\)
0.121054 + 0.992646i \(0.461372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.3345 + 40.4166i 0.797092 + 1.38060i 0.921502 + 0.388373i \(0.126963\pi\)
−0.124410 + 0.992231i \(0.539704\pi\)
\(858\) 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
\(865\) 15.0000 + 25.9808i 0.510015 + 0.883372i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.6066 18.3712i 0.358979 0.621770i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.00000 10.3923i 0.202606 0.350923i −0.746762 0.665092i \(-0.768392\pi\)
0.949367 + 0.314169i \(0.101726\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −57.9828 −1.95349 −0.976744 0.214407i \(-0.931218\pi\)
−0.976744 + 0.214407i \(0.931218\pi\)
\(882\) 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
\(901\) −49.4975 + 85.7321i −1.64900 + 2.85615i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.0000 32.9090i 0.631581 1.09393i
\(906\) 0 0
\(907\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(908\) 0 0
\(909\) 38.1838 1.26648
\(910\) 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
\(925\) 36.0000 1.18367
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.12132 + 3.67423i −0.0695983 + 0.120548i −0.898725 0.438514i \(-0.855505\pi\)
0.829126 + 0.559061i \(0.188838\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.07107 0.231002 0.115501 0.993307i \(-0.463153\pi\)
0.115501 + 0.993307i \(0.463153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −13.4350 23.2702i −0.437969 0.758585i 0.559563 0.828788i \(-0.310969\pi\)
−0.997533 + 0.0702023i \(0.977636\pi\)
\(942\) 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
\(949\) −11.0000 19.0526i −0.357075 0.618472i
\(950\) 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
\(965\) 19.7990 0.637352
\(966\) 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
\(977\) 4.00000 + 6.92820i 0.127971 + 0.221653i 0.922890 0.385063i \(-0.125820\pi\)
−0.794919 + 0.606715i \(0.792487\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 1.41421 2.44949i 0.0450606 0.0780472i
\(986\) 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
\(997\) −17.6777 30.6186i −0.559857 0.969701i −0.997508 0.0705563i \(-0.977523\pi\)
0.437650 0.899145i \(-0.355811\pi\)
\(998\) 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.t.961.1 4
4.3 odd 2 CM 1568.2.i.t.961.1 4
7.2 even 3 1568.2.a.o.1.2 yes 2
7.3 odd 6 inner 1568.2.i.t.1537.2 4
7.4 even 3 inner 1568.2.i.t.1537.1 4
7.5 odd 6 1568.2.a.o.1.1 2
7.6 odd 2 inner 1568.2.i.t.961.2 4
28.3 even 6 inner 1568.2.i.t.1537.2 4
28.11 odd 6 inner 1568.2.i.t.1537.1 4
28.19 even 6 1568.2.a.o.1.1 2
28.23 odd 6 1568.2.a.o.1.2 yes 2
28.27 even 2 inner 1568.2.i.t.961.2 4
56.5 odd 6 3136.2.a.bi.1.2 2
56.19 even 6 3136.2.a.bi.1.2 2
56.37 even 6 3136.2.a.bi.1.1 2
56.51 odd 6 3136.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1568.2.a.o.1.1 2 7.5 odd 6
1568.2.a.o.1.1 2 28.19 even 6
1568.2.a.o.1.2 yes 2 7.2 even 3
1568.2.a.o.1.2 yes 2 28.23 odd 6
1568.2.i.t.961.1 4 1.1 even 1 trivial
1568.2.i.t.961.1 4 4.3 odd 2 CM
1568.2.i.t.961.2 4 7.6 odd 2 inner
1568.2.i.t.961.2 4 28.27 even 2 inner
1568.2.i.t.1537.1 4 7.4 even 3 inner
1568.2.i.t.1537.1 4 28.11 odd 6 inner
1568.2.i.t.1537.2 4 7.3 odd 6 inner
1568.2.i.t.1537.2 4 28.3 even 6 inner
3136.2.a.bi.1.1 2 56.37 even 6
3136.2.a.bi.1.1 2 56.51 odd 6
3136.2.a.bi.1.2 2 56.5 odd 6
3136.2.a.bi.1.2 2 56.19 even 6