Properties

Label 1444.2.a.b.1.1
Level $1444$
Weight $2$
Character 1444.1
Self dual yes
Analytic conductor $11.530$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,2,Mod(1,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1444.a (trivial)

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: yes
Analytic conductor: \(11.5303980519\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1444.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} +5.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +7.00000 q^{29} +4.00000 q^{31} +4.00000 q^{33} +10.0000 q^{37} +1.00000 q^{39} -5.00000 q^{41} -5.00000 q^{43} +2.00000 q^{45} -7.00000 q^{47} -7.00000 q^{49} -3.00000 q^{51} +11.0000 q^{53} +4.00000 q^{55} +3.00000 q^{59} +11.0000 q^{61} +1.00000 q^{65} -3.00000 q^{67} -5.00000 q^{69} +11.0000 q^{71} +15.0000 q^{73} +4.00000 q^{75} -13.0000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -7.00000 q^{87} +3.00000 q^{89} -4.00000 q^{93} -5.00000 q^{97} +8.00000 q^{99} +O(q^{100})\)

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 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) 0 0
\(73\) 15.0000 1.75562 0.877809 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −7.00000 −0.750479
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 8.00000 0.804030
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −7.00000 −0.581318
\(146\) 0 0
\(147\) 7.00000 0.577350
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −15.0000 −1.16073 −0.580367 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0000 1.14043 0.570214 0.821496i \(-0.306860\pi\)
0.570214 + 0.821496i \(0.306860\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) −11.0000 −0.813143
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 15.0000 1.07972 0.539862 0.841754i \(-0.318476\pi\)
0.539862 + 0.841754i \(0.318476\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) −10.0000 −0.695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 0 0
\(213\) −11.0000 −0.753708
\(214\) 0 0
\(215\) 5.00000 0.340997
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −15.0000 −1.01361
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −25.0000 −1.67412 −0.837062 0.547108i \(-0.815729\pi\)
−0.837062 + 0.547108i \(0.815729\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 0 0
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 7.00000 0.447214
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −31.0000 −1.95670 −0.978351 0.206951i \(-0.933646\pi\)
−0.978351 + 0.206951i \(0.933646\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) 23.0000 1.43470 0.717350 0.696713i \(-0.245355\pi\)
0.717350 + 0.696713i \(0.245355\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −14.0000 −0.866578
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) 0 0
\(269\) 27.0000 1.64622 0.823110 0.567883i \(-0.192237\pi\)
0.823110 + 0.567883i \(0.192237\pi\)
\(270\) 0 0
\(271\) 31.0000 1.88312 0.941558 0.336851i \(-0.109362\pi\)
0.941558 + 0.336851i \(0.109362\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.0000 0.964836
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) 0 0
\(283\) −9.00000 −0.534994 −0.267497 0.963559i \(-0.586197\pi\)
−0.267497 + 0.963559i \(0.586197\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 5.00000 0.293105
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) −5.00000 −0.289157
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.00000 0.0574485
\(304\) 0 0
\(305\) −11.0000 −0.629858
\(306\) 0 0
\(307\) 27.0000 1.54097 0.770486 0.637457i \(-0.220014\pi\)
0.770486 + 0.637457i \(0.220014\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.0000 0.842484 0.421242 0.906948i \(-0.361594\pi\)
0.421242 + 0.906948i \(0.361594\pi\)
\(318\) 0 0
\(319\) −28.0000 −1.56770
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) −3.00000 −0.165900
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) −20.0000 −1.09599
\(334\) 0 0
\(335\) 3.00000 0.163908
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.00000 0.269191
\(346\) 0 0
\(347\) −5.00000 −0.268414 −0.134207 0.990953i \(-0.542849\pi\)
−0.134207 + 0.990953i \(0.542849\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) −11.0000 −0.583819
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) 0 0
\(367\) 25.0000 1.30499 0.652495 0.757793i \(-0.273722\pi\)
0.652495 + 0.757793i \(0.273722\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) −7.00000 −0.360518
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 3.00000 0.153695
\(382\) 0 0
\(383\) −29.0000 −1.48183 −0.740915 0.671598i \(-0.765608\pi\)
−0.740915 + 0.671598i \(0.765608\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 15.0000 0.758583
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) 13.0000 0.654101
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 19.0000 0.948815 0.474407 0.880305i \(-0.342662\pi\)
0.474407 + 0.880305i \(0.342662\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −17.0000 −0.840596 −0.420298 0.907386i \(-0.638074\pi\)
−0.420298 + 0.907386i \(0.638074\pi\)
\(410\) 0 0
\(411\) 5.00000 0.246632
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −9.00000 −0.440732
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 14.0000 0.680703
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 7.00000 0.335624
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −13.0000 −0.620456 −0.310228 0.950662i \(-0.600405\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 14.0000 0.666667
\(442\) 0 0
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) −3.00000 −0.141895
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) 11.0000 0.512321 0.256161 0.966634i \(-0.417542\pi\)
0.256161 + 0.966634i \(0.417542\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −7.00000 −0.322543
\(472\) 0 0
\(473\) 20.0000 0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −22.0000 −1.00731
\(478\) 0 0
\(479\) −23.0000 −1.05090 −0.525448 0.850825i \(-0.676102\pi\)
−0.525448 + 0.850825i \(0.676102\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.00000 0.227038
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −1.00000 −0.0451294 −0.0225647 0.999745i \(-0.507183\pi\)
−0.0225647 + 0.999745i \(0.507183\pi\)
\(492\) 0 0
\(493\) 21.0000 0.945792
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 15.0000 0.670151
\(502\) 0 0
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 1.00000 0.0444994
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 28.0000 1.23144
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) −19.0000 −0.830812 −0.415406 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 5.00000 0.216574
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) 0 0
\(543\) 5.00000 0.214571
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 25.0000 1.06892 0.534461 0.845193i \(-0.320514\pi\)
0.534461 + 0.845193i \(0.320514\pi\)
\(548\) 0 0
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.0000 0.424476
\(556\) 0 0
\(557\) −5.00000 −0.211857 −0.105928 0.994374i \(-0.533781\pi\)
−0.105928 + 0.994374i \(0.533781\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) −15.0000 −0.623379
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −44.0000 −1.82229
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −5.00000 −0.205325 −0.102663 0.994716i \(-0.532736\pi\)
−0.102663 + 0.994716i \(0.532736\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000 0.286491
\(598\) 0 0
\(599\) 45.0000 1.83865 0.919325 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 0 0
\(615\) −5.00000 −0.201619
\(616\) 0 0
\(617\) −45.0000 −1.81163 −0.905816 0.423672i \(-0.860741\pi\)
−0.905816 + 0.423672i \(0.860741\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 25.0000 1.00322
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) 41.0000 1.63218 0.816092 0.577922i \(-0.196136\pi\)
0.816092 + 0.577922i \(0.196136\pi\)
\(632\) 0 0
\(633\) 9.00000 0.357718
\(634\) 0 0
\(635\) 3.00000 0.119051
\(636\) 0 0
\(637\) 7.00000 0.277350
\(638\) 0 0
\(639\) −22.0000 −0.870307
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) −5.00000 −0.196875
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) −15.0000 −0.586098
\(656\) 0 0
\(657\) −30.0000 −1.17041
\(658\) 0 0
\(659\) 5.00000 0.194772 0.0973862 0.995247i \(-0.468952\pi\)
0.0973862 + 0.995247i \(0.468952\pi\)
\(660\) 0 0
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 0 0
\(663\) 3.00000 0.116510
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.0000 1.35521
\(668\) 0 0
\(669\) 25.0000 0.966556
\(670\) 0 0
\(671\) −44.0000 −1.69860
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) 10.0000 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) 0 0
\(685\) 5.00000 0.191040
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.00000 −0.341389
\(696\) 0 0
\(697\) −15.0000 −0.568166
\(698\) 0 0
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −7.00000 −0.263635
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.00000 0.112667 0.0563337 0.998412i \(-0.482059\pi\)
0.0563337 + 0.998412i \(0.482059\pi\)
\(710\) 0 0
\(711\) 26.0000 0.975076
\(712\) 0 0
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 7.00000 0.261056 0.130528 0.991445i \(-0.458333\pi\)
0.130528 + 0.991445i \(0.458333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19.0000 −0.706618
\(724\) 0 0
\(725\) −28.0000 −1.03989
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −15.0000 −0.554795
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 0 0
\(735\) −7.00000 −0.258199
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −13.0000 −0.478213 −0.239106 0.970993i \(-0.576854\pi\)
−0.239106 + 0.970993i \(0.576854\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.0000 −0.917161 −0.458581 0.888653i \(-0.651642\pi\)
−0.458581 + 0.888653i \(0.651642\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29.0000 1.05823 0.529113 0.848552i \(-0.322525\pi\)
0.529113 + 0.848552i \(0.322525\pi\)
\(752\) 0 0
\(753\) 31.0000 1.12970
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) 0 0
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) −3.00000 −0.108324
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) −23.0000 −0.828325
\(772\) 0 0
\(773\) −21.0000 −0.755318 −0.377659 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −44.0000 −1.57444
\(782\) 0 0
\(783\) 35.0000 1.25080
\(784\) 0 0
\(785\) −7.00000 −0.249841
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 0 0
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −11.0000 −0.390621
\(794\) 0 0
\(795\) 11.0000 0.390130
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −21.0000 −0.742927
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −60.0000 −2.11735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −27.0000 −0.950445
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 21.0000 0.737410 0.368705 0.929547i \(-0.379801\pi\)
0.368705 + 0.929547i \(0.379801\pi\)
\(812\) 0 0
\(813\) −31.0000 −1.08722
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.00000 −0.314102 −0.157051 0.987590i \(-0.550199\pi\)
−0.157051 + 0.987590i \(0.550199\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) −16.0000 −0.557048
\(826\) 0 0
\(827\) −7.00000 −0.243414 −0.121707 0.992566i \(-0.538837\pi\)
−0.121707 + 0.992566i \(0.538837\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 0 0
\(837\) 20.0000 0.691301
\(838\) 0 0
\(839\) −25.0000 −0.863096 −0.431548 0.902090i \(-0.642032\pi\)
−0.431548 + 0.902090i \(0.642032\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) −7.00000 −0.241093
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9.00000 0.308879
\(850\) 0 0
\(851\) 50.0000 1.71398
\(852\) 0 0
\(853\) 39.0000 1.33533 0.667667 0.744460i \(-0.267293\pi\)
0.667667 + 0.744460i \(0.267293\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −15.0000 −0.510015
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 52.0000 1.76398
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.00000 −0.168838 −0.0844190 0.996430i \(-0.526903\pi\)
−0.0844190 + 0.996430i \(0.526903\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) 25.0000 0.841317 0.420658 0.907219i \(-0.361799\pi\)
0.420658 + 0.907219i \(0.361799\pi\)
\(884\) 0 0
\(885\) 3.00000 0.100844
\(886\) 0 0
\(887\) 25.0000 0.839418 0.419709 0.907659i \(-0.362132\pi\)
0.419709 + 0.907659i \(0.362132\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 5.00000 0.166945
\(898\) 0 0
\(899\) 28.0000 0.933852
\(900\) 0 0
\(901\) 33.0000 1.09939
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) 35.0000 1.16216 0.581078 0.813848i \(-0.302631\pi\)
0.581078 + 0.813848i \(0.302631\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 11.0000 0.363649
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −27.0000 −0.889680
\(922\) 0 0
\(923\) −11.0000 −0.362069
\(924\) 0 0
\(925\) −40.0000 −1.31519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 23.0000 0.754606 0.377303 0.926090i \(-0.376852\pi\)
0.377303 + 0.926090i \(0.376852\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 20.0000 0.654771
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) 23.0000 0.751377 0.375689 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) 0 0
\(939\) −11.0000 −0.358971
\(940\) 0 0
\(941\) 19.0000 0.619382 0.309691 0.950837i \(-0.399774\pi\)
0.309691 + 0.950837i \(0.399774\pi\)
\(942\) 0 0
\(943\) −25.0000 −0.814112
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 0 0
\(949\) −15.0000 −0.486921
\(950\) 0 0
\(951\) −15.0000 −0.486408
\(952\) 0 0
\(953\) −25.0000 −0.809829 −0.404915 0.914354i \(-0.632699\pi\)
−0.404915 + 0.914354i \(0.632699\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 28.0000 0.905111
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −40.0000 −1.28898
\(964\) 0 0
\(965\) −15.0000 −0.482867
\(966\) 0 0
\(967\) −45.0000 −1.44710 −0.723551 0.690271i \(-0.757491\pi\)
−0.723551 + 0.690271i \(0.757491\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 35.0000 1.12320 0.561602 0.827408i \(-0.310185\pi\)
0.561602 + 0.827408i \(0.310185\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −25.0000 −0.794954
\(990\) 0 0
\(991\) −5.00000 −0.158830 −0.0794151 0.996842i \(-0.525305\pi\)
−0.0794151 + 0.996842i \(0.525305\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −53.0000 −1.67853 −0.839263 0.543725i \(-0.817013\pi\)
−0.839263 + 0.543725i \(0.817013\pi\)
\(998\) 0 0
\(999\) 50.0000 1.58193
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 1444.2.a.b.1.1 1
4.3 odd 2 5776.2.a.k.1.1 1
19.7 even 3 76.2.e.a.49.1 yes 2
19.8 odd 6 1444.2.e.b.653.1 2
19.11 even 3 76.2.e.a.45.1 2
19.12 odd 6 1444.2.e.b.429.1 2
19.18 odd 2 1444.2.a.c.1.1 1
57.11 odd 6 684.2.k.b.577.1 2
57.26 odd 6 684.2.k.b.505.1 2
76.7 odd 6 304.2.i.a.49.1 2
76.11 odd 6 304.2.i.a.273.1 2
76.75 even 2 5776.2.a.f.1.1 1
95.7 odd 12 1900.2.s.a.49.1 4
95.49 even 6 1900.2.i.a.501.1 2
95.64 even 6 1900.2.i.a.201.1 2
95.68 odd 12 1900.2.s.a.349.1 4
95.83 odd 12 1900.2.s.a.49.2 4
95.87 odd 12 1900.2.s.a.349.2 4
152.11 odd 6 1216.2.i.g.577.1 2
152.45 even 6 1216.2.i.c.961.1 2
152.83 odd 6 1216.2.i.g.961.1 2
152.125 even 6 1216.2.i.c.577.1 2
228.11 even 6 2736.2.s.g.577.1 2
228.83 even 6 2736.2.s.g.1873.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.e.a.45.1 2 19.11 even 3
76.2.e.a.49.1 yes 2 19.7 even 3
304.2.i.a.49.1 2 76.7 odd 6
304.2.i.a.273.1 2 76.11 odd 6
684.2.k.b.505.1 2 57.26 odd 6
684.2.k.b.577.1 2 57.11 odd 6
1216.2.i.c.577.1 2 152.125 even 6
1216.2.i.c.961.1 2 152.45 even 6
1216.2.i.g.577.1 2 152.11 odd 6
1216.2.i.g.961.1 2 152.83 odd 6
1444.2.a.b.1.1 1 1.1 even 1 trivial
1444.2.a.c.1.1 1 19.18 odd 2
1444.2.e.b.429.1 2 19.12 odd 6
1444.2.e.b.653.1 2 19.8 odd 6
1900.2.i.a.201.1 2 95.64 even 6
1900.2.i.a.501.1 2 95.49 even 6
1900.2.s.a.49.1 4 95.7 odd 12
1900.2.s.a.49.2 4 95.83 odd 12
1900.2.s.a.349.1 4 95.68 odd 12
1900.2.s.a.349.2 4 95.87 odd 12
2736.2.s.g.577.1 2 228.11 even 6
2736.2.s.g.1873.1 2 228.83 even 6
5776.2.a.f.1.1 1 76.75 even 2
5776.2.a.k.1.1 1 4.3 odd 2