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Properties

Label 1296.2.a.b.1.1
Level $1296$
Weight $2$
Character 1296.1
Self dual yes
Analytic conductor $10.349$
Analytic rank $1$
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] = [1296,2,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1296.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(10.3486121020\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1296.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{5} +1.00000 q^{7} +3.00000 q^{11} -1.00000 q^{13} -6.00000 q^{17} +4.00000 q^{19} -3.00000 q^{23} +4.00000 q^{25} -3.00000 q^{29} -5.00000 q^{31} -3.00000 q^{35} +2.00000 q^{37} -3.00000 q^{41} +1.00000 q^{43} -9.00000 q^{47} -6.00000 q^{49} +6.00000 q^{53} -9.00000 q^{55} -3.00000 q^{59} -13.0000 q^{61} +3.00000 q^{65} +7.00000 q^{67} -12.0000 q^{71} -10.0000 q^{73} +3.00000 q^{77} -11.0000 q^{79} -9.00000 q^{83} +18.0000 q^{85} -6.00000 q^{89} -1.00000 q^{91} -12.0000 q^{95} +11.0000 q^{97} +O(q^{100})\)

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 \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 − 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 − 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 − 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 − 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.0000 1.83478 0.917389 − 0.397991i \(-0.130293\pi\)
0.917389 + 0.397991i \(0.130293\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 − 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 − 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.0000 −1.22885 −0.614424 − 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 − 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 15.0000 1.20483
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 0.696441 0.348220 − 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 − 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 − 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 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
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −17.0000 −1.17033 −0.585164 − 0.810915i \(-0.698970\pi\)
−0.585164 + 0.810915i \(0.698970\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) −5.00000 −0.339422
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.0000 1.79205 0.896026 − 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 27.0000 1.76129
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −27.0000 −1.74648 −0.873242 − 0.487286i \(-0.837987\pi\)
−0.873242 + 0.487286i \(0.837987\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 − 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 0.561405 0.280702 − 0.959795i \(-0.409433\pi\)
0.280702 + 0.959795i \(0.409433\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −21.0000 −1.29492 −0.647458 − 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 − 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 − 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 − 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) −5.00000 −0.297219 −0.148610 − 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.0000 1.22683 0.613417 − 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 39.0000 2.23313
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 − 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.0000 1.17948 0.589739 − 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −11.0000 −0.604615 −0.302307 − 0.953211i \(-0.597757\pi\)
−0.302307 + 0.953211i \(0.597757\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −21.0000 −1.14735
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000 0.483145 0.241573 − 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 − 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 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
\(355\) 36.0000 1.91068
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 30.0000 1.57027
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 − 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −1.00000 −0.0517780 −0.0258890 − 0.999665i \(-0.508242\pi\)
−0.0258890 + 0.999665i \(0.508242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.0000 −0.766464 −0.383232 − 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 − 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 33.0000 1.66041
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 23.0000 1.13728 0.568638 − 0.822588i \(-0.307470\pi\)
0.568638 + 0.822588i \(0.307470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 27.0000 1.32538
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.00000 0.439679 0.219839 − 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 35.0000 1.70580 0.852898 − 0.522078i \(-0.174843\pi\)
0.852898 + 0.522078i \(0.174843\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) −13.0000 −0.629114
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 − 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 0.140642
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 − 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 − 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 19.0000 0.883005 0.441502 − 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.0000 1.23366 0.616831 − 0.787096i \(-0.288416\pi\)
0.616831 + 0.787096i \(0.288416\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −33.0000 −1.49845
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 − 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 − 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 − 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −39.0000 −1.72864 −0.864322 − 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −21.0000 −0.925371
\(516\) 0 0
\(517\) −27.0000 −1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.00000 0.129944
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 − 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 − 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 − 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) −27.0000 −1.13590
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 − 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 − 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.0000 −1.59350 −0.796748 − 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 − 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) −41.0000 −1.66414 −0.832069 − 0.554672i \(-0.812844\pi\)
−0.832069 + 0.554672i \(0.812844\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 −0.120775 −0.0603877 − 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 0 0
\(619\) 13.0000 0.522514 0.261257 − 0.965269i \(-0.415863\pi\)
0.261257 + 0.965269i \(0.415863\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 − 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −48.0000 −1.90482
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 − 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) −41.0000 −1.61688 −0.808441 − 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.0000 0.821794 0.410897 − 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) 0 0
\(655\) −63.0000 −2.46161
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 − 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 − 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −39.0000 −1.50558
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 − 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.0000 −0.576497 −0.288248 − 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) 0 0
\(679\) 11.0000 0.422141
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 1.00000 0.0380418 0.0190209 − 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.0000 0.568982
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 15.0000 0.561754
\(714\) 0 0
\(715\) 9.00000 0.336581
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 37.0000 1.37225 0.686127 − 0.727482i \(-0.259309\pi\)
0.686127 + 0.727482i \(0.259309\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.00000 −0.221918
\(732\) 0 0
\(733\) 23.0000 0.849524 0.424762 − 0.905305i \(-0.360358\pi\)
0.424762 + 0.905305i \(0.360358\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.0000 0.773545
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 0.330178 0.165089 − 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) 45.0000 1.64867
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 31.0000 1.13121 0.565603 − 0.824678i \(-0.308643\pi\)
0.565603 + 0.824678i \(0.308643\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.0000 −1.41936
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 − 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.00000 0.108324
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 − 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 39.0000 1.39197
\(786\) 0 0
\(787\) 43.0000 1.53278 0.766392 − 0.642373i \(-0.222050\pi\)
0.766392 + 0.642373i \(0.222050\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 13.0000 0.461644
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.00000 0.318796 0.159398 − 0.987214i \(-0.449045\pi\)
0.159398 + 0.987214i \(0.449045\pi\)
\(798\) 0 0
\(799\) 54.0000 1.91038
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30.0000 −1.05868
\(804\) 0 0
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −51.0000 −1.77991 −0.889956 − 0.456046i \(-0.849265\pi\)
−0.889956 + 0.456046i \(0.849265\pi\)
\(822\) 0 0
\(823\) 19.0000 0.662298 0.331149 − 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 − 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 − 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) −27.0000 −0.934374
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.00000 −0.310715 −0.155357 − 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) −13.0000 −0.445112 −0.222556 − 0.974920i \(-0.571440\pi\)
−0.222556 + 0.974920i \(0.571440\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.0000 −0.922302 −0.461151 − 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 0 0
\(859\) −41.0000 −1.39890 −0.699451 − 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −33.0000 −1.11945
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 23.0000 0.776655 0.388327 − 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.0000 0.705111 0.352555 − 0.935791i \(-0.385313\pi\)
0.352555 + 0.935791i \(0.385313\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −36.0000 −1.20469
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −47.0000 −1.56061 −0.780305 − 0.625400i \(-0.784936\pi\)
−0.780305 + 0.625400i \(0.784936\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −45.0000 −1.49092 −0.745458 − 0.666552i \(-0.767769\pi\)
−0.745458 + 0.666552i \(0.767769\pi\)
\(912\) 0 0
\(913\) −27.0000 −0.893570
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.0000 0.693481
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −27.0000 −0.885841 −0.442921 − 0.896561i \(-0.646058\pi\)
−0.442921 + 0.896561i \(0.646058\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 − 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 − 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) 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
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 − 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) −45.0000 −1.45617
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −33.0000 −1.06231
\(966\) 0 0
\(967\) 43.0000 1.38279 0.691393 − 0.722478i \(-0.256997\pi\)
0.691393 + 0.722478i \(0.256997\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −5.00000 −0.160293
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 57.0000 1.82359 0.911796 − 0.410644i \(-0.134696\pi\)
0.911796 + 0.410644i \(0.134696\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 51.0000 1.62665 0.813324 − 0.581811i \(-0.197656\pi\)
0.813324 + 0.581811i \(0.197656\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) −1.00000 −0.0316703 −0.0158352 − 0.999875i \(-0.505041\pi\)
−0.0158352 + 0.999875i \(0.505041\pi\)
\(998\) 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 1296.2.a.b.1.1 1
3.2 odd 2 1296.2.a.k.1.1 1
4.3 odd 2 324.2.a.a.1.1 1
8.3 odd 2 5184.2.a.ba.1.1 1
8.5 even 2 5184.2.a.bb.1.1 1
9.2 odd 6 144.2.i.a.49.1 2
9.4 even 3 432.2.i.c.289.1 2
9.5 odd 6 144.2.i.a.97.1 2
9.7 even 3 432.2.i.c.145.1 2
12.11 even 2 324.2.a.c.1.1 1
20.3 even 4 8100.2.d.c.649.2 2
20.7 even 4 8100.2.d.c.649.1 2
20.19 odd 2 8100.2.a.g.1.1 1
24.5 odd 2 5184.2.a.f.1.1 1
24.11 even 2 5184.2.a.e.1.1 1
36.7 odd 6 108.2.e.a.37.1 2
36.11 even 6 36.2.e.a.13.1 ✓ 2
36.23 even 6 36.2.e.a.25.1 yes 2
36.31 odd 6 108.2.e.a.73.1 2
60.23 odd 4 8100.2.d.h.649.2 2
60.47 odd 4 8100.2.d.h.649.1 2
60.59 even 2 8100.2.a.j.1.1 1
72.5 odd 6 576.2.i.e.385.1 2
72.11 even 6 576.2.i.f.193.1 2
72.13 even 6 1728.2.i.c.1153.1 2
72.29 odd 6 576.2.i.e.193.1 2
72.43 odd 6 1728.2.i.d.577.1 2
72.59 even 6 576.2.i.f.385.1 2
72.61 even 6 1728.2.i.c.577.1 2
72.67 odd 6 1728.2.i.d.1153.1 2
180.7 even 12 2700.2.s.b.1549.1 4
180.23 odd 12 900.2.s.b.349.1 4
180.43 even 12 2700.2.s.b.1549.2 4
180.47 odd 12 900.2.s.b.49.1 4
180.59 even 6 900.2.i.b.601.1 2
180.67 even 12 2700.2.s.b.2449.2 4
180.79 odd 6 2700.2.i.b.901.1 2
180.83 odd 12 900.2.s.b.49.2 4
180.103 even 12 2700.2.s.b.2449.1 4
180.119 even 6 900.2.i.b.301.1 2
180.139 odd 6 2700.2.i.b.1801.1 2
180.167 odd 12 900.2.s.b.349.2 4
252.11 even 6 1764.2.i.a.373.1 2
252.23 even 6 1764.2.i.a.1537.1 2
252.31 even 6 5292.2.l.c.3313.1 2
252.47 odd 6 1764.2.l.a.949.1 2
252.59 odd 6 1764.2.l.a.961.1 2
252.67 odd 6 5292.2.l.a.3313.1 2
252.79 odd 6 5292.2.l.a.361.1 2
252.83 odd 6 1764.2.j.b.589.1 2
252.95 even 6 1764.2.l.c.961.1 2
252.103 even 6 5292.2.i.a.2125.1 2
252.115 even 6 5292.2.i.a.1549.1 2
252.131 odd 6 1764.2.i.c.1537.1 2
252.139 even 6 5292.2.j.a.3529.1 2
252.151 odd 6 5292.2.i.c.1549.1 2
252.167 odd 6 1764.2.j.b.1177.1 2
252.187 even 6 5292.2.l.c.361.1 2
252.191 even 6 1764.2.l.c.949.1 2
252.223 even 6 5292.2.j.a.1765.1 2
252.227 odd 6 1764.2.i.c.373.1 2
252.247 odd 6 5292.2.i.c.2125.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.e.a.13.1 ✓ 2 36.11 even 6
36.2.e.a.25.1 yes 2 36.23 even 6
108.2.e.a.37.1 2 36.7 odd 6
108.2.e.a.73.1 2 36.31 odd 6
144.2.i.a.49.1 2 9.2 odd 6
144.2.i.a.97.1 2 9.5 odd 6
324.2.a.a.1.1 1 4.3 odd 2
324.2.a.c.1.1 1 12.11 even 2
432.2.i.c.145.1 2 9.7 even 3
432.2.i.c.289.1 2 9.4 even 3
576.2.i.e.193.1 2 72.29 odd 6
576.2.i.e.385.1 2 72.5 odd 6
576.2.i.f.193.1 2 72.11 even 6
576.2.i.f.385.1 2 72.59 even 6
900.2.i.b.301.1 2 180.119 even 6
900.2.i.b.601.1 2 180.59 even 6
900.2.s.b.49.1 4 180.47 odd 12
900.2.s.b.49.2 4 180.83 odd 12
900.2.s.b.349.1 4 180.23 odd 12
900.2.s.b.349.2 4 180.167 odd 12
1296.2.a.b.1.1 1 1.1 even 1 trivial
1296.2.a.k.1.1 1 3.2 odd 2
1728.2.i.c.577.1 2 72.61 even 6
1728.2.i.c.1153.1 2 72.13 even 6
1728.2.i.d.577.1 2 72.43 odd 6
1728.2.i.d.1153.1 2 72.67 odd 6
1764.2.i.a.373.1 2 252.11 even 6
1764.2.i.a.1537.1 2 252.23 even 6
1764.2.i.c.373.1 2 252.227 odd 6
1764.2.i.c.1537.1 2 252.131 odd 6
1764.2.j.b.589.1 2 252.83 odd 6
1764.2.j.b.1177.1 2 252.167 odd 6
1764.2.l.a.949.1 2 252.47 odd 6
1764.2.l.a.961.1 2 252.59 odd 6
1764.2.l.c.949.1 2 252.191 even 6
1764.2.l.c.961.1 2 252.95 even 6
2700.2.i.b.901.1 2 180.79 odd 6
2700.2.i.b.1801.1 2 180.139 odd 6
2700.2.s.b.1549.1 4 180.7 even 12
2700.2.s.b.1549.2 4 180.43 even 12
2700.2.s.b.2449.1 4 180.103 even 12
2700.2.s.b.2449.2 4 180.67 even 12
5184.2.a.e.1.1 1 24.11 even 2
5184.2.a.f.1.1 1 24.5 odd 2
5184.2.a.ba.1.1 1 8.3 odd 2
5184.2.a.bb.1.1 1 8.5 even 2
5292.2.i.a.1549.1 2 252.115 even 6
5292.2.i.a.2125.1 2 252.103 even 6
5292.2.i.c.1549.1 2 252.151 odd 6
5292.2.i.c.2125.1 2 252.247 odd 6
5292.2.j.a.1765.1 2 252.223 even 6
5292.2.j.a.3529.1 2 252.139 even 6
5292.2.l.a.361.1 2 252.79 odd 6
5292.2.l.a.3313.1 2 252.67 odd 6
5292.2.l.c.361.1 2 252.187 even 6
5292.2.l.c.3313.1 2 252.31 even 6
8100.2.a.g.1.1 1 20.19 odd 2
8100.2.a.j.1.1 1 60.59 even 2
8100.2.d.c.649.1 2 20.7 even 4
8100.2.d.c.649.2 2 20.3 even 4
8100.2.d.h.649.1 2 60.47 odd 4
8100.2.d.h.649.2 2 60.23 odd 4