Properties

Label 1568.3.g.f.687.1
Level $1568$
Weight $3$
Character 1568.687
Analytic conductor $42.725$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1568,3,Mod(687,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.687");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.g (of order \(2\), degree \(1\), 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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 687.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1568.687
Dual form 1568.3.g.f.687.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -5.19615i q^{5} -8.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -5.19615i q^{5} -8.00000 q^{9} -17.0000 q^{11} -13.8564i q^{13} -5.19615i q^{15} +25.0000 q^{17} -7.00000 q^{19} +5.19615i q^{23} -2.00000 q^{25} -17.0000 q^{27} -13.8564i q^{29} +32.9090i q^{31} -17.0000 q^{33} -8.66025i q^{37} -13.8564i q^{39} -26.0000 q^{41} -14.0000 q^{43} +41.5692i q^{45} +50.2295i q^{47} +25.0000 q^{51} +91.7987i q^{53} +88.3346i q^{55} -7.00000 q^{57} -55.0000 q^{59} +22.5167i q^{61} -72.0000 q^{65} -17.0000 q^{67} +5.19615i q^{69} -119.000 q^{73} -2.00000 q^{75} +74.4782i q^{79} +55.0000 q^{81} +110.000 q^{83} -129.904i q^{85} -13.8564i q^{87} -71.0000 q^{89} +32.9090i q^{93} +36.3731i q^{95} +22.0000 q^{97} +136.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 16 q^{9} - 34 q^{11} + 50 q^{17} - 14 q^{19} - 4 q^{25} - 34 q^{27} - 34 q^{33} - 52 q^{41} - 28 q^{43} + 50 q^{51} - 14 q^{57} - 110 q^{59} - 144 q^{65} - 34 q^{67} - 238 q^{73} - 4 q^{75} + 110 q^{81} + 220 q^{83} - 142 q^{89} + 44 q^{97} + 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.333333 0.166667 0.986013i \(-0.446700\pi\)
0.166667 + 0.986013i \(0.446700\pi\)
\(4\) 0 0
\(5\) − 5.19615i − 1.03923i −0.854400 0.519615i \(-0.826075\pi\)
0.854400 0.519615i \(-0.173925\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −8.00000 −0.888889
\(10\) 0 0
\(11\) −17.0000 −1.54545 −0.772727 0.634738i \(-0.781108\pi\)
−0.772727 + 0.634738i \(0.781108\pi\)
\(12\) 0 0
\(13\) − 13.8564i − 1.06588i −0.846154 0.532939i \(-0.821088\pi\)
0.846154 0.532939i \(-0.178912\pi\)
\(14\) 0 0
\(15\) − 5.19615i − 0.346410i
\(16\) 0 0
\(17\) 25.0000 1.47059 0.735294 0.677748i \(-0.237044\pi\)
0.735294 + 0.677748i \(0.237044\pi\)
\(18\) 0 0
\(19\) −7.00000 −0.368421 −0.184211 0.982887i \(-0.558973\pi\)
−0.184211 + 0.982887i \(0.558973\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615i 0.225920i 0.993600 + 0.112960i \(0.0360331\pi\)
−0.993600 + 0.112960i \(0.963967\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.0800000
\(26\) 0 0
\(27\) −17.0000 −0.629630
\(28\) 0 0
\(29\) − 13.8564i − 0.477807i −0.971043 0.238904i \(-0.923212\pi\)
0.971043 0.238904i \(-0.0767880\pi\)
\(30\) 0 0
\(31\) 32.9090i 1.06158i 0.847504 + 0.530790i \(0.178105\pi\)
−0.847504 + 0.530790i \(0.821895\pi\)
\(32\) 0 0
\(33\) −17.0000 −0.515152
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.66025i − 0.234061i −0.993128 0.117030i \(-0.962662\pi\)
0.993128 0.117030i \(-0.0373375\pi\)
\(38\) 0 0
\(39\) − 13.8564i − 0.355292i
\(40\) 0 0
\(41\) −26.0000 −0.634146 −0.317073 0.948401i \(-0.602700\pi\)
−0.317073 + 0.948401i \(0.602700\pi\)
\(42\) 0 0
\(43\) −14.0000 −0.325581 −0.162791 0.986661i \(-0.552050\pi\)
−0.162791 + 0.986661i \(0.552050\pi\)
\(44\) 0 0
\(45\) 41.5692i 0.923760i
\(46\) 0 0
\(47\) 50.2295i 1.06871i 0.845259 + 0.534356i \(0.179446\pi\)
−0.845259 + 0.534356i \(0.820554\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 25.0000 0.490196
\(52\) 0 0
\(53\) 91.7987i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 88.3346i 1.60608i
\(56\) 0 0
\(57\) −7.00000 −0.122807
\(58\) 0 0
\(59\) −55.0000 −0.932203 −0.466102 0.884731i \(-0.654342\pi\)
−0.466102 + 0.884731i \(0.654342\pi\)
\(60\) 0 0
\(61\) 22.5167i 0.369126i 0.982821 + 0.184563i \(0.0590869\pi\)
−0.982821 + 0.184563i \(0.940913\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −72.0000 −1.10769
\(66\) 0 0
\(67\) −17.0000 −0.253731 −0.126866 0.991920i \(-0.540492\pi\)
−0.126866 + 0.991920i \(0.540492\pi\)
\(68\) 0 0
\(69\) 5.19615i 0.0753066i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −119.000 −1.63014 −0.815068 0.579365i \(-0.803301\pi\)
−0.815068 + 0.579365i \(0.803301\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.0266667
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 74.4782i 0.942762i 0.881930 + 0.471381i \(0.156244\pi\)
−0.881930 + 0.471381i \(0.843756\pi\)
\(80\) 0 0
\(81\) 55.0000 0.679012
\(82\) 0 0
\(83\) 110.000 1.32530 0.662651 0.748929i \(-0.269431\pi\)
0.662651 + 0.748929i \(0.269431\pi\)
\(84\) 0 0
\(85\) − 129.904i − 1.52828i
\(86\) 0 0
\(87\) − 13.8564i − 0.159269i
\(88\) 0 0
\(89\) −71.0000 −0.797753 −0.398876 0.917005i \(-0.630600\pi\)
−0.398876 + 0.917005i \(0.630600\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 32.9090i 0.353860i
\(94\) 0 0
\(95\) 36.3731i 0.382874i
\(96\) 0 0
\(97\) 22.0000 0.226804 0.113402 0.993549i \(-0.463825\pi\)
0.113402 + 0.993549i \(0.463825\pi\)
\(98\) 0 0
\(99\) 136.000 1.37374
\(100\) 0 0
\(101\) − 77.9423i − 0.771706i −0.922560 0.385853i \(-0.873907\pi\)
0.922560 0.385853i \(-0.126093\pi\)
\(102\) 0 0
\(103\) 161.081i 1.56389i 0.623347 + 0.781945i \(0.285772\pi\)
−0.623347 + 0.781945i \(0.714228\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −65.0000 −0.607477 −0.303738 0.952755i \(-0.598235\pi\)
−0.303738 + 0.952755i \(0.598235\pi\)
\(108\) 0 0
\(109\) 8.66025i 0.0794519i 0.999211 + 0.0397259i \(0.0126485\pi\)
−0.999211 + 0.0397259i \(0.987352\pi\)
\(110\) 0 0
\(111\) − 8.66025i − 0.0780203i
\(112\) 0 0
\(113\) 122.000 1.07965 0.539823 0.841779i \(-0.318491\pi\)
0.539823 + 0.841779i \(0.318491\pi\)
\(114\) 0 0
\(115\) 27.0000 0.234783
\(116\) 0 0
\(117\) 110.851i 0.947447i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 168.000 1.38843
\(122\) 0 0
\(123\) −26.0000 −0.211382
\(124\) 0 0
\(125\) − 119.512i − 0.956092i
\(126\) 0 0
\(127\) − 166.277i − 1.30927i −0.755947 0.654633i \(-0.772823\pi\)
0.755947 0.654633i \(-0.227177\pi\)
\(128\) 0 0
\(129\) −14.0000 −0.108527
\(130\) 0 0
\(131\) 17.0000 0.129771 0.0648855 0.997893i \(-0.479332\pi\)
0.0648855 + 0.997893i \(0.479332\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 88.3346i 0.654330i
\(136\) 0 0
\(137\) −145.000 −1.05839 −0.529197 0.848499i \(-0.677507\pi\)
−0.529197 + 0.848499i \(0.677507\pi\)
\(138\) 0 0
\(139\) −82.0000 −0.589928 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(140\) 0 0
\(141\) 50.2295i 0.356237i
\(142\) 0 0
\(143\) 235.559i 1.64727i
\(144\) 0 0
\(145\) −72.0000 −0.496552
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.19615i 0.0348735i 0.999848 + 0.0174368i \(0.00555057\pi\)
−0.999848 + 0.0174368i \(0.994449\pi\)
\(150\) 0 0
\(151\) 36.3731i 0.240881i 0.992721 + 0.120441i \(0.0384307\pi\)
−0.992721 + 0.120441i \(0.961569\pi\)
\(152\) 0 0
\(153\) −200.000 −1.30719
\(154\) 0 0
\(155\) 171.000 1.10323
\(156\) 0 0
\(157\) 310.037i 1.97476i 0.158373 + 0.987379i \(0.449375\pi\)
−0.158373 + 0.987379i \(0.550625\pi\)
\(158\) 0 0
\(159\) 91.7987i 0.577350i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.0000 −0.104294 −0.0521472 0.998639i \(-0.516607\pi\)
−0.0521472 + 0.998639i \(0.516607\pi\)
\(164\) 0 0
\(165\) 88.3346i 0.535361i
\(166\) 0 0
\(167\) 13.8564i 0.0829725i 0.999139 + 0.0414862i \(0.0132093\pi\)
−0.999139 + 0.0414862i \(0.986791\pi\)
\(168\) 0 0
\(169\) −23.0000 −0.136095
\(170\) 0 0
\(171\) 56.0000 0.327485
\(172\) 0 0
\(173\) 105.655i 0.610723i 0.952236 + 0.305362i \(0.0987773\pi\)
−0.952236 + 0.305362i \(0.901223\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −55.0000 −0.310734
\(178\) 0 0
\(179\) −89.0000 −0.497207 −0.248603 0.968605i \(-0.579972\pi\)
−0.248603 + 0.968605i \(0.579972\pi\)
\(180\) 0 0
\(181\) 249.415i 1.37799i 0.724768 + 0.688993i \(0.241947\pi\)
−0.724768 + 0.688993i \(0.758053\pi\)
\(182\) 0 0
\(183\) 22.5167i 0.123042i
\(184\) 0 0
\(185\) −45.0000 −0.243243
\(186\) 0 0
\(187\) −425.000 −2.27273
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 216.506i − 1.13354i −0.823876 0.566771i \(-0.808193\pi\)
0.823876 0.566771i \(-0.191807\pi\)
\(192\) 0 0
\(193\) −73.0000 −0.378238 −0.189119 0.981954i \(-0.560563\pi\)
−0.189119 + 0.981954i \(0.560563\pi\)
\(194\) 0 0
\(195\) −72.0000 −0.369231
\(196\) 0 0
\(197\) − 207.846i − 1.05506i −0.849538 0.527528i \(-0.823119\pi\)
0.849538 0.527528i \(-0.176881\pi\)
\(198\) 0 0
\(199\) − 64.0859i − 0.322040i −0.986951 0.161020i \(-0.948522\pi\)
0.986951 0.161020i \(-0.0514783\pi\)
\(200\) 0 0
\(201\) −17.0000 −0.0845771
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 135.100i 0.659024i
\(206\) 0 0
\(207\) − 41.5692i − 0.200817i
\(208\) 0 0
\(209\) 119.000 0.569378
\(210\) 0 0
\(211\) −302.000 −1.43128 −0.715640 0.698470i \(-0.753865\pi\)
−0.715640 + 0.698470i \(0.753865\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 72.7461i 0.338354i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −119.000 −0.543379
\(220\) 0 0
\(221\) − 346.410i − 1.56747i
\(222\) 0 0
\(223\) − 138.564i − 0.621364i −0.950514 0.310682i \(-0.899443\pi\)
0.950514 0.310682i \(-0.100557\pi\)
\(224\) 0 0
\(225\) 16.0000 0.0711111
\(226\) 0 0
\(227\) −55.0000 −0.242291 −0.121145 0.992635i \(-0.538657\pi\)
−0.121145 + 0.992635i \(0.538657\pi\)
\(228\) 0 0
\(229\) 327.358i 1.42951i 0.699375 + 0.714755i \(0.253462\pi\)
−0.699375 + 0.714755i \(0.746538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −385.000 −1.65236 −0.826180 0.563406i \(-0.809491\pi\)
−0.826180 + 0.563406i \(0.809491\pi\)
\(234\) 0 0
\(235\) 261.000 1.11064
\(236\) 0 0
\(237\) 74.4782i 0.314254i
\(238\) 0 0
\(239\) − 429.549i − 1.79727i −0.438693 0.898637i \(-0.644558\pi\)
0.438693 0.898637i \(-0.355442\pi\)
\(240\) 0 0
\(241\) 145.000 0.601660 0.300830 0.953678i \(-0.402736\pi\)
0.300830 + 0.953678i \(0.402736\pi\)
\(242\) 0 0
\(243\) 208.000 0.855967
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 96.9948i 0.392692i
\(248\) 0 0
\(249\) 110.000 0.441767
\(250\) 0 0
\(251\) −58.0000 −0.231076 −0.115538 0.993303i \(-0.536859\pi\)
−0.115538 + 0.993303i \(0.536859\pi\)
\(252\) 0 0
\(253\) − 88.3346i − 0.349149i
\(254\) 0 0
\(255\) − 129.904i − 0.509427i
\(256\) 0 0
\(257\) −119.000 −0.463035 −0.231518 0.972831i \(-0.574369\pi\)
−0.231518 + 0.972831i \(0.574369\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 110.851i 0.424717i
\(262\) 0 0
\(263\) 327.358i 1.24471i 0.782737 + 0.622353i \(0.213823\pi\)
−0.782737 + 0.622353i \(0.786177\pi\)
\(264\) 0 0
\(265\) 477.000 1.80000
\(266\) 0 0
\(267\) −71.0000 −0.265918
\(268\) 0 0
\(269\) − 133.368i − 0.495791i −0.968787 0.247896i \(-0.920261\pi\)
0.968787 0.247896i \(-0.0797390\pi\)
\(270\) 0 0
\(271\) − 434.745i − 1.60422i −0.597174 0.802112i \(-0.703710\pi\)
0.597174 0.802112i \(-0.296290\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 34.0000 0.123636
\(276\) 0 0
\(277\) 202.650i 0.731588i 0.930696 + 0.365794i \(0.119203\pi\)
−0.930696 + 0.365794i \(0.880797\pi\)
\(278\) 0 0
\(279\) − 263.272i − 0.943626i
\(280\) 0 0
\(281\) 74.0000 0.263345 0.131673 0.991293i \(-0.457965\pi\)
0.131673 + 0.991293i \(0.457965\pi\)
\(282\) 0 0
\(283\) −463.000 −1.63604 −0.818021 0.575188i \(-0.804929\pi\)
−0.818021 + 0.575188i \(0.804929\pi\)
\(284\) 0 0
\(285\) 36.3731i 0.127625i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 336.000 1.16263
\(290\) 0 0
\(291\) 22.0000 0.0756014
\(292\) 0 0
\(293\) 110.851i 0.378332i 0.981945 + 0.189166i \(0.0605784\pi\)
−0.981945 + 0.189166i \(0.939422\pi\)
\(294\) 0 0
\(295\) 285.788i 0.968774i
\(296\) 0 0
\(297\) 289.000 0.973064
\(298\) 0 0
\(299\) 72.0000 0.240803
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 77.9423i − 0.257235i
\(304\) 0 0
\(305\) 117.000 0.383607
\(306\) 0 0
\(307\) −274.000 −0.892508 −0.446254 0.894906i \(-0.647242\pi\)
−0.446254 + 0.894906i \(0.647242\pi\)
\(308\) 0 0
\(309\) 161.081i 0.521297i
\(310\) 0 0
\(311\) − 50.2295i − 0.161510i −0.996734 0.0807548i \(-0.974267\pi\)
0.996734 0.0807548i \(-0.0257331\pi\)
\(312\) 0 0
\(313\) 409.000 1.30671 0.653355 0.757052i \(-0.273361\pi\)
0.653355 + 0.757052i \(0.273361\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 188.794i − 0.595563i −0.954634 0.297782i \(-0.903753\pi\)
0.954634 0.297782i \(-0.0962467\pi\)
\(318\) 0 0
\(319\) 235.559i 0.738429i
\(320\) 0 0
\(321\) −65.0000 −0.202492
\(322\) 0 0
\(323\) −175.000 −0.541796
\(324\) 0 0
\(325\) 27.7128i 0.0852702i
\(326\) 0 0
\(327\) 8.66025i 0.0264840i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 295.000 0.891239 0.445619 0.895223i \(-0.352983\pi\)
0.445619 + 0.895223i \(0.352983\pi\)
\(332\) 0 0
\(333\) 69.2820i 0.208054i
\(334\) 0 0
\(335\) 88.3346i 0.263685i
\(336\) 0 0
\(337\) 26.0000 0.0771513 0.0385757 0.999256i \(-0.487718\pi\)
0.0385757 + 0.999256i \(0.487718\pi\)
\(338\) 0 0
\(339\) 122.000 0.359882
\(340\) 0 0
\(341\) − 559.452i − 1.64062i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 27.0000 0.0782609
\(346\) 0 0
\(347\) −377.000 −1.08646 −0.543228 0.839585i \(-0.682798\pi\)
−0.543228 + 0.839585i \(0.682798\pi\)
\(348\) 0 0
\(349\) 96.9948i 0.277922i 0.990298 + 0.138961i \(0.0443763\pi\)
−0.990298 + 0.138961i \(0.955624\pi\)
\(350\) 0 0
\(351\) 235.559i 0.671108i
\(352\) 0 0
\(353\) −503.000 −1.42493 −0.712465 0.701708i \(-0.752421\pi\)
−0.712465 + 0.701708i \(0.752421\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 185.329i 0.516238i 0.966113 + 0.258119i \(0.0831027\pi\)
−0.966113 + 0.258119i \(0.916897\pi\)
\(360\) 0 0
\(361\) −312.000 −0.864266
\(362\) 0 0
\(363\) 168.000 0.462810
\(364\) 0 0
\(365\) 618.342i 1.69409i
\(366\) 0 0
\(367\) 296.181i 0.807032i 0.914973 + 0.403516i \(0.132212\pi\)
−0.914973 + 0.403516i \(0.867788\pi\)
\(368\) 0 0
\(369\) 208.000 0.563686
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 119.512i − 0.320406i −0.987084 0.160203i \(-0.948785\pi\)
0.987084 0.160203i \(-0.0512149\pi\)
\(374\) 0 0
\(375\) − 119.512i − 0.318697i
\(376\) 0 0
\(377\) −192.000 −0.509284
\(378\) 0 0
\(379\) 634.000 1.67282 0.836412 0.548102i \(-0.184649\pi\)
0.836412 + 0.548102i \(0.184649\pi\)
\(380\) 0 0
\(381\) − 166.277i − 0.436422i
\(382\) 0 0
\(383\) 244.219i 0.637648i 0.947814 + 0.318824i \(0.103288\pi\)
−0.947814 + 0.318824i \(0.896712\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 112.000 0.289406
\(388\) 0 0
\(389\) − 587.165i − 1.50942i −0.656057 0.754711i \(-0.727777\pi\)
0.656057 0.754711i \(-0.272223\pi\)
\(390\) 0 0
\(391\) 129.904i 0.332235i
\(392\) 0 0
\(393\) 17.0000 0.0432570
\(394\) 0 0
\(395\) 387.000 0.979747
\(396\) 0 0
\(397\) − 240.755i − 0.606436i −0.952921 0.303218i \(-0.901939\pi\)
0.952921 0.303218i \(-0.0980610\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 119.000 0.296758 0.148379 0.988931i \(-0.452594\pi\)
0.148379 + 0.988931i \(0.452594\pi\)
\(402\) 0 0
\(403\) 456.000 1.13151
\(404\) 0 0
\(405\) − 285.788i − 0.705650i
\(406\) 0 0
\(407\) 147.224i 0.361731i
\(408\) 0 0
\(409\) 145.000 0.354523 0.177262 0.984164i \(-0.443276\pi\)
0.177262 + 0.984164i \(0.443276\pi\)
\(410\) 0 0
\(411\) −145.000 −0.352798
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 571.577i − 1.37729i
\(416\) 0 0
\(417\) −82.0000 −0.196643
\(418\) 0 0
\(419\) 302.000 0.720764 0.360382 0.932805i \(-0.382646\pi\)
0.360382 + 0.932805i \(0.382646\pi\)
\(420\) 0 0
\(421\) 401.836i 0.954479i 0.878773 + 0.477240i \(0.158363\pi\)
−0.878773 + 0.477240i \(0.841637\pi\)
\(422\) 0 0
\(423\) − 401.836i − 0.949966i
\(424\) 0 0
\(425\) −50.0000 −0.117647
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 235.559i 0.549088i
\(430\) 0 0
\(431\) − 808.868i − 1.87672i −0.345655 0.938362i \(-0.612343\pi\)
0.345655 0.938362i \(-0.387657\pi\)
\(432\) 0 0
\(433\) −410.000 −0.946882 −0.473441 0.880825i \(-0.656988\pi\)
−0.473441 + 0.880825i \(0.656988\pi\)
\(434\) 0 0
\(435\) −72.0000 −0.165517
\(436\) 0 0
\(437\) − 36.3731i − 0.0832336i
\(438\) 0 0
\(439\) − 490.170i − 1.11656i −0.829652 0.558281i \(-0.811461\pi\)
0.829652 0.558281i \(-0.188539\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −401.000 −0.905192 −0.452596 0.891716i \(-0.649502\pi\)
−0.452596 + 0.891716i \(0.649502\pi\)
\(444\) 0 0
\(445\) 368.927i 0.829049i
\(446\) 0 0
\(447\) 5.19615i 0.0116245i
\(448\) 0 0
\(449\) −310.000 −0.690423 −0.345212 0.938525i \(-0.612193\pi\)
−0.345212 + 0.938525i \(0.612193\pi\)
\(450\) 0 0
\(451\) 442.000 0.980044
\(452\) 0 0
\(453\) 36.3731i 0.0802937i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 167.000 0.365427 0.182713 0.983166i \(-0.441512\pi\)
0.182713 + 0.983166i \(0.441512\pi\)
\(458\) 0 0
\(459\) −425.000 −0.925926
\(460\) 0 0
\(461\) 13.8564i 0.0300573i 0.999887 + 0.0150286i \(0.00478394\pi\)
−0.999887 + 0.0150286i \(0.995216\pi\)
\(462\) 0 0
\(463\) 609.682i 1.31681i 0.752665 + 0.658404i \(0.228768\pi\)
−0.752665 + 0.658404i \(0.771232\pi\)
\(464\) 0 0
\(465\) 171.000 0.367742
\(466\) 0 0
\(467\) 785.000 1.68094 0.840471 0.541856i \(-0.182278\pi\)
0.840471 + 0.541856i \(0.182278\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 310.037i 0.658253i
\(472\) 0 0
\(473\) 238.000 0.503171
\(474\) 0 0
\(475\) 14.0000 0.0294737
\(476\) 0 0
\(477\) − 734.390i − 1.53960i
\(478\) 0 0
\(479\) − 618.342i − 1.29090i −0.763802 0.645451i \(-0.776669\pi\)
0.763802 0.645451i \(-0.223331\pi\)
\(480\) 0 0
\(481\) −120.000 −0.249480
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 114.315i − 0.235702i
\(486\) 0 0
\(487\) − 393.176i − 0.807342i −0.914904 0.403671i \(-0.867734\pi\)
0.914904 0.403671i \(-0.132266\pi\)
\(488\) 0 0
\(489\) −17.0000 −0.0347648
\(490\) 0 0
\(491\) −422.000 −0.859470 −0.429735 0.902955i \(-0.641393\pi\)
−0.429735 + 0.902955i \(0.641393\pi\)
\(492\) 0 0
\(493\) − 346.410i − 0.702658i
\(494\) 0 0
\(495\) − 706.677i − 1.42763i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −65.0000 −0.130261 −0.0651303 0.997877i \(-0.520746\pi\)
−0.0651303 + 0.997877i \(0.520746\pi\)
\(500\) 0 0
\(501\) 13.8564i 0.0276575i
\(502\) 0 0
\(503\) 249.415i 0.495855i 0.968779 + 0.247928i \(0.0797496\pi\)
−0.968779 + 0.247928i \(0.920250\pi\)
\(504\) 0 0
\(505\) −405.000 −0.801980
\(506\) 0 0
\(507\) −23.0000 −0.0453649
\(508\) 0 0
\(509\) − 545.596i − 1.07190i −0.844250 0.535949i \(-0.819954\pi\)
0.844250 0.535949i \(-0.180046\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 119.000 0.231969
\(514\) 0 0
\(515\) 837.000 1.62524
\(516\) 0 0
\(517\) − 853.901i − 1.65165i
\(518\) 0 0
\(519\) 105.655i 0.203574i
\(520\) 0 0
\(521\) 25.0000 0.0479846 0.0239923 0.999712i \(-0.492362\pi\)
0.0239923 + 0.999712i \(0.492362\pi\)
\(522\) 0 0
\(523\) 593.000 1.13384 0.566922 0.823772i \(-0.308134\pi\)
0.566922 + 0.823772i \(0.308134\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 822.724i 1.56115i
\(528\) 0 0
\(529\) 502.000 0.948960
\(530\) 0 0
\(531\) 440.000 0.828625
\(532\) 0 0
\(533\) 360.267i 0.675922i
\(534\) 0 0
\(535\) 337.750i 0.631308i
\(536\) 0 0
\(537\) −89.0000 −0.165736
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 756.906i − 1.39909i −0.714590 0.699544i \(-0.753387\pi\)
0.714590 0.699544i \(-0.246613\pi\)
\(542\) 0 0
\(543\) 249.415i 0.459328i
\(544\) 0 0
\(545\) 45.0000 0.0825688
\(546\) 0 0
\(547\) −662.000 −1.21024 −0.605119 0.796135i \(-0.706874\pi\)
−0.605119 + 0.796135i \(0.706874\pi\)
\(548\) 0 0
\(549\) − 180.133i − 0.328112i
\(550\) 0 0
\(551\) 96.9948i 0.176034i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −45.0000 −0.0810811
\(556\) 0 0
\(557\) 590.629i 1.06038i 0.847880 + 0.530188i \(0.177879\pi\)
−0.847880 + 0.530188i \(0.822121\pi\)
\(558\) 0 0
\(559\) 193.990i 0.347030i
\(560\) 0 0
\(561\) −425.000 −0.757576
\(562\) 0 0
\(563\) 737.000 1.30906 0.654529 0.756037i \(-0.272867\pi\)
0.654529 + 0.756037i \(0.272867\pi\)
\(564\) 0 0
\(565\) − 633.931i − 1.12200i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −121.000 −0.212654 −0.106327 0.994331i \(-0.533909\pi\)
−0.106327 + 0.994331i \(0.533909\pi\)
\(570\) 0 0
\(571\) −737.000 −1.29072 −0.645359 0.763879i \(-0.723292\pi\)
−0.645359 + 0.763879i \(0.723292\pi\)
\(572\) 0 0
\(573\) − 216.506i − 0.377847i
\(574\) 0 0
\(575\) − 10.3923i − 0.0180736i
\(576\) 0 0
\(577\) −47.0000 −0.0814558 −0.0407279 0.999170i \(-0.512968\pi\)
−0.0407279 + 0.999170i \(0.512968\pi\)
\(578\) 0 0
\(579\) −73.0000 −0.126079
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 1560.58i − 2.67681i
\(584\) 0 0
\(585\) 576.000 0.984615
\(586\) 0 0
\(587\) 446.000 0.759796 0.379898 0.925028i \(-0.375959\pi\)
0.379898 + 0.925028i \(0.375959\pi\)
\(588\) 0 0
\(589\) − 230.363i − 0.391108i
\(590\) 0 0
\(591\) − 207.846i − 0.351685i
\(592\) 0 0
\(593\) −215.000 −0.362563 −0.181282 0.983431i \(-0.558025\pi\)
−0.181282 + 0.983431i \(0.558025\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 64.0859i − 0.107347i
\(598\) 0 0
\(599\) − 282.324i − 0.471326i −0.971835 0.235663i \(-0.924274\pi\)
0.971835 0.235663i \(-0.0757262\pi\)
\(600\) 0 0
\(601\) −266.000 −0.442596 −0.221298 0.975206i \(-0.571029\pi\)
−0.221298 + 0.975206i \(0.571029\pi\)
\(602\) 0 0
\(603\) 136.000 0.225539
\(604\) 0 0
\(605\) − 872.954i − 1.44290i
\(606\) 0 0
\(607\) 659.911i 1.08717i 0.839355 + 0.543584i \(0.182933\pi\)
−0.839355 + 0.543584i \(0.817067\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 696.000 1.13912
\(612\) 0 0
\(613\) − 698.016i − 1.13869i −0.822099 0.569345i \(-0.807197\pi\)
0.822099 0.569345i \(-0.192803\pi\)
\(614\) 0 0
\(615\) 135.100i 0.219675i
\(616\) 0 0
\(617\) −118.000 −0.191248 −0.0956240 0.995418i \(-0.530485\pi\)
−0.0956240 + 0.995418i \(0.530485\pi\)
\(618\) 0 0
\(619\) −919.000 −1.48465 −0.742326 0.670039i \(-0.766278\pi\)
−0.742326 + 0.670039i \(0.766278\pi\)
\(620\) 0 0
\(621\) − 88.3346i − 0.142246i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −671.000 −1.07360
\(626\) 0 0
\(627\) 119.000 0.189793
\(628\) 0 0
\(629\) − 216.506i − 0.344207i
\(630\) 0 0
\(631\) − 166.277i − 0.263513i −0.991282 0.131757i \(-0.957938\pi\)
0.991282 0.131757i \(-0.0420617\pi\)
\(632\) 0 0
\(633\) −302.000 −0.477093
\(634\) 0 0
\(635\) −864.000 −1.36063
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.00000 −0.00156006 −0.000780031 1.00000i \(-0.500248\pi\)
−0.000780031 1.00000i \(0.500248\pi\)
\(642\) 0 0
\(643\) −514.000 −0.799378 −0.399689 0.916651i \(-0.630882\pi\)
−0.399689 + 0.916651i \(0.630882\pi\)
\(644\) 0 0
\(645\) 72.7461i 0.112785i
\(646\) 0 0
\(647\) 60.6218i 0.0936967i 0.998902 + 0.0468484i \(0.0149178\pi\)
−0.998902 + 0.0468484i \(0.985082\pi\)
\(648\) 0 0
\(649\) 935.000 1.44068
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 327.358i − 0.501313i −0.968076 0.250657i \(-0.919353\pi\)
0.968076 0.250657i \(-0.0806465\pi\)
\(654\) 0 0
\(655\) − 88.3346i − 0.134862i
\(656\) 0 0
\(657\) 952.000 1.44901
\(658\) 0 0
\(659\) −542.000 −0.822458 −0.411229 0.911532i \(-0.634900\pi\)
−0.411229 + 0.911532i \(0.634900\pi\)
\(660\) 0 0
\(661\) 1182.99i 1.78970i 0.446368 + 0.894849i \(0.352717\pi\)
−0.446368 + 0.894849i \(0.647283\pi\)
\(662\) 0 0
\(663\) − 346.410i − 0.522489i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 72.0000 0.107946
\(668\) 0 0
\(669\) − 138.564i − 0.207121i
\(670\) 0 0
\(671\) − 382.783i − 0.570467i
\(672\) 0 0
\(673\) 218.000 0.323923 0.161961 0.986797i \(-0.448218\pi\)
0.161961 + 0.986797i \(0.448218\pi\)
\(674\) 0 0
\(675\) 34.0000 0.0503704
\(676\) 0 0
\(677\) − 642.591i − 0.949174i −0.880209 0.474587i \(-0.842597\pi\)
0.880209 0.474587i \(-0.157403\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −55.0000 −0.0807636
\(682\) 0 0
\(683\) 367.000 0.537335 0.268668 0.963233i \(-0.413417\pi\)
0.268668 + 0.963233i \(0.413417\pi\)
\(684\) 0 0
\(685\) 753.442i 1.09992i
\(686\) 0 0
\(687\) 327.358i 0.476503i
\(688\) 0 0
\(689\) 1272.00 1.84615
\(690\) 0 0
\(691\) 497.000 0.719247 0.359624 0.933097i \(-0.382905\pi\)
0.359624 + 0.933097i \(0.382905\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 426.084i 0.613071i
\(696\) 0 0
\(697\) −650.000 −0.932568
\(698\) 0 0
\(699\) −385.000 −0.550787
\(700\) 0 0
\(701\) 332.554i 0.474399i 0.971461 + 0.237200i \(0.0762295\pi\)
−0.971461 + 0.237200i \(0.923770\pi\)
\(702\) 0 0
\(703\) 60.6218i 0.0862330i
\(704\) 0 0
\(705\) 261.000 0.370213
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 396.640i − 0.559435i −0.960082 0.279718i \(-0.909759\pi\)
0.960082 0.279718i \(-0.0902409\pi\)
\(710\) 0 0
\(711\) − 595.825i − 0.838011i
\(712\) 0 0
\(713\) −171.000 −0.239832
\(714\) 0 0
\(715\) 1224.00 1.71189
\(716\) 0 0
\(717\) − 429.549i − 0.599091i
\(718\) 0 0
\(719\) 64.0859i 0.0891320i 0.999006 + 0.0445660i \(0.0141905\pi\)
−0.999006 + 0.0445660i \(0.985810\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 145.000 0.200553
\(724\) 0 0
\(725\) 27.7128i 0.0382246i
\(726\) 0 0
\(727\) − 55.4256i − 0.0762388i −0.999273 0.0381194i \(-0.987863\pi\)
0.999273 0.0381194i \(-0.0121367\pi\)
\(728\) 0 0
\(729\) −287.000 −0.393690
\(730\) 0 0
\(731\) −350.000 −0.478796
\(732\) 0 0
\(733\) 826.188i 1.12713i 0.826071 + 0.563566i \(0.190571\pi\)
−0.826071 + 0.563566i \(0.809429\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 289.000 0.392130
\(738\) 0 0
\(739\) −713.000 −0.964817 −0.482409 0.875946i \(-0.660238\pi\)
−0.482409 + 0.875946i \(0.660238\pi\)
\(740\) 0 0
\(741\) 96.9948i 0.130897i
\(742\) 0 0
\(743\) − 637.395i − 0.857866i −0.903336 0.428933i \(-0.858890\pi\)
0.903336 0.428933i \(-0.141110\pi\)
\(744\) 0 0
\(745\) 27.0000 0.0362416
\(746\) 0 0
\(747\) −880.000 −1.17805
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1169.13i 1.55677i 0.627787 + 0.778385i \(0.283961\pi\)
−0.627787 + 0.778385i \(0.716039\pi\)
\(752\) 0 0
\(753\) −58.0000 −0.0770252
\(754\) 0 0
\(755\) 189.000 0.250331
\(756\) 0 0
\(757\) 1039.23i 1.37283i 0.727211 + 0.686414i \(0.240816\pi\)
−0.727211 + 0.686414i \(0.759184\pi\)
\(758\) 0 0
\(759\) − 88.3346i − 0.116383i
\(760\) 0 0
\(761\) −863.000 −1.13403 −0.567017 0.823706i \(-0.691903\pi\)
−0.567017 + 0.823706i \(0.691903\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1039.23i 1.35847i
\(766\) 0 0
\(767\) 762.102i 0.993615i
\(768\) 0 0
\(769\) −410.000 −0.533160 −0.266580 0.963813i \(-0.585894\pi\)
−0.266580 + 0.963813i \(0.585894\pi\)
\(770\) 0 0
\(771\) −119.000 −0.154345
\(772\) 0 0
\(773\) − 798.475i − 1.03296i −0.856300 0.516478i \(-0.827243\pi\)
0.856300 0.516478i \(-0.172757\pi\)
\(774\) 0 0
\(775\) − 65.8179i − 0.0849264i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 182.000 0.233633
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 235.559i 0.300842i
\(784\) 0 0
\(785\) 1611.00 2.05223
\(786\) 0 0
\(787\) −31.0000 −0.0393901 −0.0196950 0.999806i \(-0.506270\pi\)
−0.0196950 + 0.999806i \(0.506270\pi\)
\(788\) 0 0
\(789\) 327.358i 0.414902i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 312.000 0.393443
\(794\) 0 0
\(795\) 477.000 0.600000
\(796\) 0 0
\(797\) − 595.825i − 0.747585i −0.927512 0.373793i \(-0.878057\pi\)
0.927512 0.373793i \(-0.121943\pi\)
\(798\) 0 0
\(799\) 1255.74i 1.57164i
\(800\) 0 0
\(801\) 568.000 0.709114
\(802\) 0 0
\(803\) 2023.00 2.51930
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 133.368i − 0.165264i
\(808\) 0 0
\(809\) −313.000 −0.386897 −0.193449 0.981110i \(-0.561967\pi\)
−0.193449 + 0.981110i \(0.561967\pi\)
\(810\) 0 0
\(811\) −1138.00 −1.40321 −0.701603 0.712568i \(-0.747532\pi\)
−0.701603 + 0.712568i \(0.747532\pi\)
\(812\) 0 0
\(813\) − 434.745i − 0.534741i
\(814\) 0 0
\(815\) 88.3346i 0.108386i
\(816\) 0 0
\(817\) 98.0000 0.119951
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1224.56i 1.49155i 0.666200 + 0.745773i \(0.267920\pi\)
−0.666200 + 0.745773i \(0.732080\pi\)
\(822\) 0 0
\(823\) − 116.047i − 0.141005i −0.997512 0.0705027i \(-0.977540\pi\)
0.997512 0.0705027i \(-0.0224603\pi\)
\(824\) 0 0
\(825\) 34.0000 0.0412121
\(826\) 0 0
\(827\) 754.000 0.911729 0.455865 0.890049i \(-0.349330\pi\)
0.455865 + 0.890049i \(0.349330\pi\)
\(828\) 0 0
\(829\) 905.863i 1.09272i 0.837551 + 0.546359i \(0.183986\pi\)
−0.837551 + 0.546359i \(0.816014\pi\)
\(830\) 0 0
\(831\) 202.650i 0.243863i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 72.0000 0.0862275
\(836\) 0 0
\(837\) − 559.452i − 0.668402i
\(838\) 0 0
\(839\) − 1053.09i − 1.25517i −0.778548 0.627585i \(-0.784044\pi\)
0.778548 0.627585i \(-0.215956\pi\)
\(840\) 0 0
\(841\) 649.000 0.771700
\(842\) 0 0
\(843\) 74.0000 0.0877817
\(844\) 0 0
\(845\) 119.512i 0.141434i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −463.000 −0.545347
\(850\) 0 0
\(851\) 45.0000 0.0528790
\(852\) 0 0
\(853\) 845.241i 0.990904i 0.868635 + 0.495452i \(0.164997\pi\)
−0.868635 + 0.495452i \(0.835003\pi\)
\(854\) 0 0
\(855\) − 290.985i − 0.340333i
\(856\) 0 0
\(857\) −887.000 −1.03501 −0.517503 0.855681i \(-0.673138\pi\)
−0.517503 + 0.855681i \(0.673138\pi\)
\(858\) 0 0
\(859\) −1663.00 −1.93597 −0.967986 0.251004i \(-0.919239\pi\)
−0.967986 + 0.251004i \(0.919239\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 562.917i − 0.652279i −0.945322 0.326139i \(-0.894252\pi\)
0.945322 0.326139i \(-0.105748\pi\)
\(864\) 0 0
\(865\) 549.000 0.634682
\(866\) 0 0
\(867\) 336.000 0.387543
\(868\) 0 0
\(869\) − 1266.13i − 1.45700i
\(870\) 0 0
\(871\) 235.559i 0.270447i
\(872\) 0 0
\(873\) −176.000 −0.201604
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 119.512i − 0.136273i −0.997676 0.0681365i \(-0.978295\pi\)
0.997676 0.0681365i \(-0.0217054\pi\)
\(878\) 0 0
\(879\) 110.851i 0.126111i
\(880\) 0 0
\(881\) 574.000 0.651532 0.325766 0.945450i \(-0.394378\pi\)
0.325766 + 0.945450i \(0.394378\pi\)
\(882\) 0 0
\(883\) −1166.00 −1.32050 −0.660249 0.751047i \(-0.729549\pi\)
−0.660249 + 0.751047i \(0.729549\pi\)
\(884\) 0 0
\(885\) 285.788i 0.322925i
\(886\) 0 0
\(887\) − 545.596i − 0.615103i −0.951532 0.307551i \(-0.900490\pi\)
0.951532 0.307551i \(-0.0995096\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −935.000 −1.04938
\(892\) 0 0
\(893\) − 351.606i − 0.393736i
\(894\) 0 0
\(895\) 462.458i 0.516712i
\(896\) 0 0
\(897\) 72.0000 0.0802676
\(898\) 0 0
\(899\) 456.000 0.507230
\(900\) 0 0
\(901\) 2294.97i 2.54713i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1296.00 1.43204
\(906\) 0 0
\(907\) −521.000 −0.574421 −0.287211 0.957867i \(-0.592728\pi\)
−0.287211 + 0.957867i \(0.592728\pi\)
\(908\) 0 0
\(909\) 623.538i 0.685961i
\(910\) 0 0
\(911\) − 1191.65i − 1.30807i −0.756465 0.654035i \(-0.773075\pi\)
0.756465 0.654035i \(-0.226925\pi\)
\(912\) 0 0
\(913\) −1870.00 −2.04819
\(914\) 0 0
\(915\) 117.000 0.127869
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 1394.30i − 1.51719i −0.651560 0.758597i \(-0.725885\pi\)
0.651560 0.758597i \(-0.274115\pi\)
\(920\) 0 0
\(921\) −274.000 −0.297503
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 17.3205i 0.0187249i
\(926\) 0 0
\(927\) − 1288.65i − 1.39012i
\(928\) 0 0
\(929\) 961.000 1.03445 0.517223 0.855851i \(-0.326966\pi\)
0.517223 + 0.855851i \(0.326966\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 50.2295i − 0.0538365i
\(934\) 0 0
\(935\) 2208.36i 2.36189i
\(936\) 0 0
\(937\) 142.000 0.151547 0.0757737 0.997125i \(-0.475857\pi\)
0.0757737 + 0.997125i \(0.475857\pi\)
\(938\) 0 0
\(939\) 409.000 0.435570
\(940\) 0 0
\(941\) 1224.56i 1.30134i 0.759361 + 0.650669i \(0.225512\pi\)
−0.759361 + 0.650669i \(0.774488\pi\)
\(942\) 0 0
\(943\) − 135.100i − 0.143266i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 175.000 0.184794 0.0923970 0.995722i \(-0.470547\pi\)
0.0923970 + 0.995722i \(0.470547\pi\)
\(948\) 0 0
\(949\) 1648.91i 1.73753i
\(950\) 0 0
\(951\) − 188.794i − 0.198521i
\(952\) 0 0
\(953\) −454.000 −0.476390 −0.238195 0.971217i \(-0.576556\pi\)
−0.238195 + 0.971217i \(0.576556\pi\)
\(954\) 0 0
\(955\) −1125.00 −1.17801
\(956\) 0 0
\(957\) 235.559i 0.246143i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −122.000 −0.126951
\(962\) 0 0
\(963\) 520.000 0.539979
\(964\) 0 0
\(965\) 379.319i 0.393077i
\(966\) 0 0
\(967\) − 720.533i − 0.745122i −0.928008 0.372561i \(-0.878480\pi\)
0.928008 0.372561i \(-0.121520\pi\)
\(968\) 0 0
\(969\) −175.000 −0.180599
\(970\) 0 0
\(971\) −1639.00 −1.68795 −0.843975 0.536382i \(-0.819791\pi\)
−0.843975 + 0.536382i \(0.819791\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 27.7128i 0.0284234i
\(976\) 0 0
\(977\) −793.000 −0.811668 −0.405834 0.913947i \(-0.633019\pi\)
−0.405834 + 0.913947i \(0.633019\pi\)
\(978\) 0 0
\(979\) 1207.00 1.23289
\(980\) 0 0
\(981\) − 69.2820i − 0.0706239i
\(982\) 0 0
\(983\) 1543.26i 1.56995i 0.619530 + 0.784973i \(0.287323\pi\)
−0.619530 + 0.784973i \(0.712677\pi\)
\(984\) 0 0
\(985\) −1080.00 −1.09645
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 72.7461i − 0.0735552i
\(990\) 0 0
\(991\) 895.470i 0.903603i 0.892119 + 0.451801i \(0.149218\pi\)
−0.892119 + 0.451801i \(0.850782\pi\)
\(992\) 0 0
\(993\) 295.000 0.297080
\(994\) 0 0
\(995\) −333.000 −0.334673
\(996\) 0 0
\(997\) 795.011i 0.797404i 0.917081 + 0.398702i \(0.130539\pi\)
−0.917081 + 0.398702i \(0.869461\pi\)
\(998\) 0 0
\(999\) 147.224i 0.147372i
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 1568.3.g.f.687.1 2
4.3 odd 2 392.3.g.d.99.2 2
7.3 odd 6 224.3.o.a.79.1 2
7.5 odd 6 224.3.o.b.207.1 2
7.6 odd 2 1568.3.g.c.687.2 2
8.3 odd 2 inner 1568.3.g.f.687.2 2
8.5 even 2 392.3.g.d.99.1 2
28.3 even 6 56.3.k.a.51.1 yes 2
28.11 odd 6 392.3.k.a.275.1 2
28.19 even 6 56.3.k.b.11.1 yes 2
28.23 odd 6 392.3.k.c.67.1 2
28.27 even 2 392.3.g.e.99.2 2
56.3 even 6 224.3.o.b.79.1 2
56.5 odd 6 56.3.k.a.11.1 2
56.13 odd 2 392.3.g.e.99.1 2
56.19 even 6 224.3.o.a.207.1 2
56.27 even 2 1568.3.g.c.687.1 2
56.37 even 6 392.3.k.a.67.1 2
56.45 odd 6 56.3.k.b.51.1 yes 2
56.53 even 6 392.3.k.c.275.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.a.11.1 2 56.5 odd 6
56.3.k.a.51.1 yes 2 28.3 even 6
56.3.k.b.11.1 yes 2 28.19 even 6
56.3.k.b.51.1 yes 2 56.45 odd 6
224.3.o.a.79.1 2 7.3 odd 6
224.3.o.a.207.1 2 56.19 even 6
224.3.o.b.79.1 2 56.3 even 6
224.3.o.b.207.1 2 7.5 odd 6
392.3.g.d.99.1 2 8.5 even 2
392.3.g.d.99.2 2 4.3 odd 2
392.3.g.e.99.1 2 56.13 odd 2
392.3.g.e.99.2 2 28.27 even 2
392.3.k.a.67.1 2 56.37 even 6
392.3.k.a.275.1 2 28.11 odd 6
392.3.k.c.67.1 2 28.23 odd 6
392.3.k.c.275.1 2 56.53 even 6
1568.3.g.c.687.1 2 56.27 even 2
1568.3.g.c.687.2 2 7.6 odd 2
1568.3.g.f.687.1 2 1.1 even 1 trivial
1568.3.g.f.687.2 2 8.3 odd 2 inner