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Properties

Label 792.2.a.g.1.1
Level $792$
Weight $2$
Character 792.1
Self dual yes
Analytic conductor $6.324$
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] = [792,2,Mod(1,792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("792.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 792.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: \(6.32415184009\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 792.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{5} -2.00000 q^{7} +1.00000 q^{11} +6.00000 q^{17} +4.00000 q^{19} -1.00000 q^{23} +4.00000 q^{25} +8.00000 q^{29} -7.00000 q^{31} -6.00000 q^{35} -1.00000 q^{37} -4.00000 q^{41} +6.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} -2.00000 q^{53} +3.00000 q^{55} +1.00000 q^{59} +4.00000 q^{61} -5.00000 q^{67} -3.00000 q^{71} +16.0000 q^{73} -2.00000 q^{77} +2.00000 q^{79} +2.00000 q^{83} +18.0000 q^{85} -15.0000 q^{89} +12.0000 q^{95} -7.00000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.00000 0.130189 0.0650945 − 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 − 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 − 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 − 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.00000 0.658505 0.329252 − 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 − 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.0000 1.28154 0.640768 − 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 − 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 − 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.0000 −1.68676
\(156\) 0 0
\(157\) −11.0000 −0.877896 −0.438948 − 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.00000 0.373718 0.186859 − 0.982387i \(-0.440169\pi\)
0.186859 + 0.982387i \(0.440169\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 − 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.0000 −1.12298
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 18.0000 1.22759
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.0000 1.94198 0.970992 − 0.239113i \(-0.0768565\pi\)
0.970992 + 0.239113i \(0.0768565\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −21.0000 −1.38772 −0.693860 − 0.720110i \(-0.744091\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 − 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 − 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 − 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.0000 0.623783 0.311891 − 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 − 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 − 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.0000 −0.701047 −0.350524 − 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 3.00000 0.174667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 − 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 − 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 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
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −35.0000 −1.92377 −0.961887 − 0.273447i \(-0.911836\pi\)
−0.961887 + 0.273447i \(0.911836\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.0000 1.71785 0.858925 − 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 − 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 48.0000 2.51243
\(366\) 0 0
\(367\) 33.0000 1.72259 0.861293 − 0.508109i \(-0.169655\pi\)
0.861293 + 0.508109i \(0.169655\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 − 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 −0.0510976 −0.0255488 − 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.0000 −0.659126 −0.329563 − 0.944134i \(-0.606901\pi\)
−0.329563 + 0.944134i \(0.606901\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 −0.0495682
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 − 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 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
\(425\) 24.0000 1.16417
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 26.0000 1.25238 0.626188 − 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) 13.0000 0.624740 0.312370 − 0.949960i \(-0.398877\pi\)
0.312370 + 0.949960i \(0.398877\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 − 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 − 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 0 0
\(445\) −45.0000 −2.13320
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 − 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 − 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 27.0000 1.25480 0.627398 − 0.778699i \(-0.284120\pi\)
0.627398 + 0.778699i \(0.284120\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0000 1.52706 0.763529 − 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.00000 −0.365529 −0.182765 − 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −21.0000 −0.953561
\(486\) 0 0
\(487\) 9.00000 0.407829 0.203914 − 0.978989i \(-0.434634\pi\)
0.203914 + 0.978989i \(0.434634\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 − 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 − 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 30.0000 1.33498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.0000 0.576215 0.288107 − 0.957598i \(-0.406974\pi\)
0.288107 + 0.957598i \(0.406974\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −48.0000 −2.11513
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.0000 −1.62100 −0.810500 − 0.585739i \(-0.800804\pi\)
−0.810500 + 0.585739i \(0.800804\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −42.0000 −1.82955
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 − 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −42.0000 −1.79908
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 21.0000 0.883477
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 − 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.00000 0.164260 0.0821302 − 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −48.0000 −1.96123 −0.980613 − 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 − 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.00000 0.121967
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −3.00000 −0.120580 −0.0602901 − 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.0000 1.20192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 9.00000 0.358284 0.179142 − 0.983823i \(-0.442668\pi\)
0.179142 + 0.983823i \(0.442668\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 − 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) 7.00000 0.276053 0.138027 − 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.0000 −0.589711 −0.294855 − 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) 0 0
\(649\) 1.00000 0.0392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.0000 −0.430463 −0.215232 − 0.976563i \(-0.569051\pi\)
−0.215232 + 0.976563i \(0.569051\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22.0000 −0.856998 −0.428499 − 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 − 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30.0000 1.15299 0.576497 − 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 − 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) 45.0000 1.71936
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −29.0000 −1.10321 −0.551606 − 0.834105i \(-0.685985\pi\)
−0.551606 + 0.834105i \(0.685985\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −66.0000 −2.50352
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 − 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) 19.0000 0.713560 0.356780 − 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.0000 0.857755 0.428878 − 0.903363i \(-0.358909\pi\)
0.428878 + 0.903363i \(0.358909\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 32.0000 1.18845
\(726\) 0 0
\(727\) −35.0000 −1.29808 −0.649039 − 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 − 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 −0.184177
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 − 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −54.0000 −1.97841
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 23.0000 0.839282 0.419641 − 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −54.0000 −1.96526
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 − 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 16.0000 0.576975 0.288487 − 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −33.0000 −1.17782
\(786\) 0 0
\(787\) 24.0000 0.855508 0.427754 − 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.00000 −0.0354218 −0.0177109 − 0.999843i \(-0.505638\pi\)
−0.0177109 + 0.999843i \(0.505638\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.0000 0.564628
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 − 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −26.0000 −0.912983 −0.456492 − 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 − 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 − 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 − 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) −35.0000 −1.21560 −0.607800 − 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.0000 −0.725001 −0.362500 − 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −39.0000 −1.34164
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.00000 0.0342796
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000 0.819824 0.409912 − 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) 11.0000 0.375315 0.187658 − 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 − 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 0 0
\(865\) −54.0000 −1.83606
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.00000 0.202837
\(876\) 0 0
\(877\) 56.0000 1.89099 0.945493 − 0.325643i \(-0.105581\pi\)
0.945493 + 0.325643i \(0.105581\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.0000 0.640126 0.320063 − 0.947396i \(-0.396296\pi\)
0.320063 + 0.947396i \(0.396296\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 34.0000 1.14161 0.570804 − 0.821086i \(-0.306632\pi\)
0.570804 + 0.821086i \(0.306632\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 15.0000 0.501395
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −56.0000 −1.86770
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −15.0000 −0.498617
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 2.00000 0.0661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 − 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.0000 0.721797 0.360898 − 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 − 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.0000 −1.52729 −0.763647 − 0.645634i \(-0.776593\pi\)
−0.763647 + 0.645634i \(0.776593\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 27.0000 0.873699
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 − 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −53.0000 −1.70085 −0.850425 − 0.526096i \(-0.823655\pi\)
−0.850425 + 0.526096i \(0.823655\pi\)
\(972\) 0 0
\(973\) 44.0000 1.41058
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.0000 0.351921 0.175961 − 0.984397i \(-0.443697\pi\)
0.175961 + 0.984397i \(0.443697\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −25.0000 −0.797376 −0.398688 − 0.917087i \(-0.630534\pi\)
−0.398688 + 0.917087i \(0.630534\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 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 792.2.a.g.1.1 1
3.2 odd 2 88.2.a.a.1.1 ✓ 1
4.3 odd 2 1584.2.a.q.1.1 1
8.3 odd 2 6336.2.a.k.1.1 1
8.5 even 2 6336.2.a.h.1.1 1
11.10 odd 2 8712.2.a.x.1.1 1
12.11 even 2 176.2.a.c.1.1 1
15.2 even 4 2200.2.b.a.1849.2 2
15.8 even 4 2200.2.b.a.1849.1 2
15.14 odd 2 2200.2.a.k.1.1 1
21.20 even 2 4312.2.a.l.1.1 1
24.5 odd 2 704.2.a.l.1.1 1
24.11 even 2 704.2.a.b.1.1 1
33.2 even 10 968.2.i.i.81.1 4
33.5 odd 10 968.2.i.j.729.1 4
33.8 even 10 968.2.i.i.9.1 4
33.14 odd 10 968.2.i.j.9.1 4
33.17 even 10 968.2.i.i.729.1 4
33.20 odd 10 968.2.i.j.81.1 4
33.26 odd 10 968.2.i.j.753.1 4
33.29 even 10 968.2.i.i.753.1 4
33.32 even 2 968.2.a.a.1.1 1
48.5 odd 4 2816.2.c.i.1409.2 2
48.11 even 4 2816.2.c.d.1409.1 2
48.29 odd 4 2816.2.c.i.1409.1 2
48.35 even 4 2816.2.c.d.1409.2 2
60.23 odd 4 4400.2.b.b.4049.2 2
60.47 odd 4 4400.2.b.b.4049.1 2
60.59 even 2 4400.2.a.a.1.1 1
84.83 odd 2 8624.2.a.c.1.1 1
132.131 odd 2 1936.2.a.l.1.1 1
264.131 odd 2 7744.2.a.b.1.1 1
264.197 even 2 7744.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.a.1.1 ✓ 1 3.2 odd 2
176.2.a.c.1.1 1 12.11 even 2
704.2.a.b.1.1 1 24.11 even 2
704.2.a.l.1.1 1 24.5 odd 2
792.2.a.g.1.1 1 1.1 even 1 trivial
968.2.a.a.1.1 1 33.32 even 2
968.2.i.i.9.1 4 33.8 even 10
968.2.i.i.81.1 4 33.2 even 10
968.2.i.i.729.1 4 33.17 even 10
968.2.i.i.753.1 4 33.29 even 10
968.2.i.j.9.1 4 33.14 odd 10
968.2.i.j.81.1 4 33.20 odd 10
968.2.i.j.729.1 4 33.5 odd 10
968.2.i.j.753.1 4 33.26 odd 10
1584.2.a.q.1.1 1 4.3 odd 2
1936.2.a.l.1.1 1 132.131 odd 2
2200.2.a.k.1.1 1 15.14 odd 2
2200.2.b.a.1849.1 2 15.8 even 4
2200.2.b.a.1849.2 2 15.2 even 4
2816.2.c.d.1409.1 2 48.11 even 4
2816.2.c.d.1409.2 2 48.35 even 4
2816.2.c.i.1409.1 2 48.29 odd 4
2816.2.c.i.1409.2 2 48.5 odd 4
4312.2.a.l.1.1 1 21.20 even 2
4400.2.a.a.1.1 1 60.59 even 2
4400.2.b.b.4049.1 2 60.47 odd 4
4400.2.b.b.4049.2 2 60.23 odd 4
6336.2.a.h.1.1 1 8.5 even 2
6336.2.a.k.1.1 1 8.3 odd 2
7744.2.a.b.1.1 1 264.131 odd 2
7744.2.a.bk.1.1 1 264.197 even 2
8624.2.a.c.1.1 1 84.83 odd 2
8712.2.a.x.1.1 1 11.10 odd 2