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Properties

Label 1296.2.a.i.1.1
Level $1296$
Weight $2$
Character 1296.1
Self dual yes
Analytic conductor $10.349$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +3.00000 q^{7} +5.00000 q^{11} -5.00000 q^{13} +2.00000 q^{17} +4.00000 q^{19} -1.00000 q^{23} -4.00000 q^{25} +9.00000 q^{29} +1.00000 q^{31} +3.00000 q^{35} -6.00000 q^{37} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{47} +2.00000 q^{49} -2.00000 q^{53} +5.00000 q^{55} +11.0000 q^{59} +7.00000 q^{61} -5.00000 q^{65} +1.00000 q^{67} +4.00000 q^{71} -2.00000 q^{73} +15.0000 q^{77} -1.00000 q^{79} +1.00000 q^{83} +2.00000 q^{85} +18.0000 q^{89} -15.0000 q^{91} +4.00000 q^{95} -13.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 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
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 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 \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 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
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.0000 1.43208 0.716039 − 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.00000 −0.620174
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 − 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.0000 1.70941
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.00000 0.109764 0.0548821 − 0.998493i \(-0.482522\pi\)
0.0548821 + 0.998493i \(0.482522\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) −15.0000 −1.57243
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 − 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 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
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 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
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.00000 −0.436852 −0.218426 − 0.975854i \(-0.570092\pi\)
−0.218426 + 0.975854i \(0.570092\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 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
\(139\) −3.00000 −0.254457 −0.127228 − 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −25.0000 −2.09061
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.0000 −1.55654 −0.778270 − 0.627929i \(-0.783903\pi\)
−0.778270 + 0.627929i \(0.783903\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 − 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 − 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 − 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 −0.228086 −0.114043 − 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 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
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 10.0000 0.731272
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −27.0000 −1.95365 −0.976826 − 0.214036i \(-0.931339\pi\)
−0.976826 + 0.214036i \(0.931339\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 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
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 27.0000 1.89503
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) −7.00000 −0.481900 −0.240950 − 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) 11.0000 0.736614 0.368307 − 0.929704i \(-0.379937\pi\)
0.368307 + 0.929704i \(0.379937\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.0000 1.39382 0.696909 − 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 − 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 − 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.0000 −1.09964 −0.549819 − 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) 15.0000 0.966235 0.483117 − 0.875556i \(-0.339504\pi\)
0.483117 + 0.875556i \(0.339504\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.00000 −0.436648 −0.218324 − 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) −18.0000 −1.11847
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.00000 −0.431638 −0.215819 − 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 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
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) −29.0000 −1.74244 −0.871221 − 0.490892i \(-0.836671\pi\)
−0.871221 + 0.490892i \(0.836671\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −19.0000 −1.13344 −0.566722 − 0.823909i \(-0.691789\pi\)
−0.566722 + 0.823909i \(0.691789\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 − 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 − 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 − 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 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 \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 0 0
\(319\) 45.0000 2.51952
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 19.0000 1.04433 0.522167 − 0.852843i \(-0.325124\pi\)
0.522167 + 0.852843i \(0.325124\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 0.0546358
\(336\) 0 0
\(337\) −9.00000 −0.490261 −0.245131 − 0.969490i \(-0.578831\pi\)
−0.245131 + 0.969490i \(0.578831\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 −0.483145 −0.241573 − 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) 0 0
\(349\) −21.0000 −1.12410 −0.562052 − 0.827102i \(-0.689988\pi\)
−0.562052 + 0.827102i \(0.689988\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.00000 0.266123 0.133062 − 0.991108i \(-0.457519\pi\)
0.133062 + 0.991108i \(0.457519\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 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
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 23.0000 1.20059 0.600295 − 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 35.0000 1.81223 0.906116 − 0.423030i \(-0.139034\pi\)
0.906116 + 0.423030i \(0.139034\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −45.0000 −2.31762
\(378\) 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
\(383\) 27.0000 1.37964 0.689818 − 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −35.0000 −1.77457 −0.887285 − 0.461221i \(-0.847411\pi\)
−0.887285 + 0.461221i \(0.847411\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.0000 1.04869 0.524345 − 0.851506i \(-0.324310\pi\)
0.524345 + 0.851506i \(0.324310\pi\)
\(402\) 0 0
\(403\) −5.00000 −0.249068
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −30.0000 −1.48704
\(408\) 0 0
\(409\) −33.0000 −1.63174 −0.815872 − 0.578232i \(-0.803743\pi\)
−0.815872 + 0.578232i \(0.803743\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 33.0000 1.62382
\(414\) 0 0
\(415\) 1.00000 0.0490881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 − 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 − 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 21.0000 1.01626
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 − 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −33.0000 −1.57500 −0.787502 − 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.0000 −1.09276 −0.546381 − 0.837536i \(-0.683995\pi\)
−0.546381 + 0.837536i \(0.683995\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 − 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −15.0000 −0.706322
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −15.0000 −0.703211
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 − 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 25.0000 1.16437 0.582183 − 0.813058i \(-0.302199\pi\)
0.582183 + 0.813058i \(0.302199\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 − 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) 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
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.00000 −0.229900
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.00000 0.0456912 0.0228456 − 0.999739i \(-0.492727\pi\)
0.0228456 + 0.999739i \(0.492727\pi\)
\(480\) 0 0
\(481\) 30.0000 1.36788
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.0000 −0.590300
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 − 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −27.0000 −1.20869 −0.604343 − 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −3.00000 −0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 − 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.00000 0.220326
\(516\) 0 0
\(517\) −15.0000 −0.659699
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −40.0000 −1.74908 −0.874539 − 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.0000 0.649722
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 19.0000 0.812381 0.406191 − 0.913788i \(-0.366857\pi\)
0.406191 + 0.913788i \(0.366857\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 − 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.00000 −0.0421450 −0.0210725 − 0.999778i \(-0.506708\pi\)
−0.0210725 + 0.999778i \(0.506708\pi\)
\(564\) 0 0
\(565\) 9.00000 0.378633
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.0000 1.04805 0.524027 − 0.851701i \(-0.324429\pi\)
0.524027 + 0.851701i \(0.324429\pi\)
\(570\) 0 0
\(571\) −15.0000 −0.627730 −0.313865 − 0.949468i \(-0.601624\pi\)
−0.313865 + 0.949468i \(0.601624\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 − 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.0000 0.701665 0.350833 − 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −21.0000 −0.858037 −0.429018 − 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 − 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 13.0000 0.527654 0.263827 − 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 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
\(619\) 11.0000 0.442127 0.221064 − 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 54.0000 2.16346
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 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 \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −10.0000 −0.396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.00000 0.0394976 0.0197488 − 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) 0 0
\(643\) −15.0000 −0.591542 −0.295771 − 0.955259i \(-0.595577\pi\)
−0.295771 + 0.955259i \(0.595577\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 55.0000 2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.0000 1.29139 0.645695 − 0.763596i \(-0.276568\pi\)
0.645695 + 0.763596i \(0.276568\pi\)
\(654\) 0 0
\(655\) −5.00000 −0.195366
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.0000 0.818044 0.409022 − 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) −25.0000 −0.972387 −0.486194 − 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) 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
\(671\) 35.0000 1.35116
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 − 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −35.0000 −1.34516 −0.672580 − 0.740025i \(-0.734814\pi\)
−0.672580 + 0.740025i \(0.734814\pi\)
\(678\) 0 0
\(679\) −39.0000 −1.49668
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 − 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.00000 −0.113796
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.00000 −0.338480
\(708\) 0 0
\(709\) 11.0000 0.413114 0.206557 − 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.00000 −0.0374503
\(714\) 0 0
\(715\) −25.0000 −0.934947
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.0000 1.19340 0.596699 − 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) 15.0000 0.558629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 39.0000 1.44643 0.723215 − 0.690623i \(-0.242664\pi\)
0.723215 + 0.690623i \(0.242664\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.00000 −0.0739727
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 − 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.00000 0.184177
\(738\) 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
\(743\) 11.0000 0.403551 0.201775 − 0.979432i \(-0.435329\pi\)
0.201775 + 0.979432i \(0.435329\pi\)
\(744\) 0 0
\(745\) −19.0000 −0.696106
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 36.0000 1.31541
\(750\) 0 0
\(751\) −27.0000 −0.985244 −0.492622 − 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −17.0000 −0.618693
\(756\) 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
\(761\) 5.00000 0.181250 0.0906249 − 0.995885i \(-0.471114\pi\)
0.0906249 + 0.995885i \(0.471114\pi\)
\(762\) 0 0
\(763\) 30.0000 1.08607
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −55.0000 −1.98593
\(768\) 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
\(773\) −42.0000 −1.51064 −0.755318 − 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) 53.0000 1.88925 0.944623 − 0.328158i \(-0.106428\pi\)
0.944623 + 0.328158i \(0.106428\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.0000 0.960009
\(792\) 0 0
\(793\) −35.0000 −1.24289
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.0000 0.460484 0.230242 − 0.973133i \(-0.426048\pi\)
0.230242 + 0.973133i \(0.426048\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 − 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 − 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) 49.0000 1.70803 0.854016 − 0.520246i \(-0.174160\pi\)
0.854016 + 0.520246i \(0.174160\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −21.0000 −0.726735
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.0000 0.725001 0.362500 − 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 42.0000 1.44314
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −1.00000 −0.0342393 −0.0171197 − 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) 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
\(859\) 9.00000 0.307076 0.153538 − 0.988143i \(-0.450933\pi\)
0.153538 + 0.988143i \(0.450933\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 56.0000 1.90626 0.953131 − 0.302558i \(-0.0978405\pi\)
0.953131 + 0.302558i \(0.0978405\pi\)
\(864\) 0 0
\(865\) −3.00000 −0.102003
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.00000 −0.169613
\(870\) 0 0
\(871\) −5.00000 −0.169419
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −27.0000 −0.912767
\(876\) 0 0
\(877\) 3.00000 0.101303 0.0506514 − 0.998716i \(-0.483870\pi\)
0.0506514 + 0.998716i \(0.483870\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 − 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 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
\(887\) 55.0000 1.84672 0.923360 − 0.383936i \(-0.125432\pi\)
0.923360 + 0.383936i \(0.125432\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −49.0000 −1.62702 −0.813509 − 0.581552i \(-0.802446\pi\)
−0.813509 + 0.581552i \(0.802446\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 41.0000 1.35839 0.679195 − 0.733958i \(-0.262329\pi\)
0.679195 + 0.733958i \(0.262329\pi\)
\(912\) 0 0
\(913\) 5.00000 0.165476
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.0000 −0.658308
\(924\) 0 0
\(925\) 24.0000 0.789115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 29.0000 0.951459 0.475730 − 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.0000 0.327035
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 33.0000 1.07577 0.537885 − 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 0 0
\(943\) 3.00000 0.0976934
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 − 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) −27.0000 −0.873699
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.0000 −0.418485
\(966\) 0 0
\(967\) 25.0000 0.803946 0.401973 − 0.915652i \(-0.368325\pi\)
0.401973 + 0.915652i \(0.368325\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.0000 −1.24772 −0.623860 − 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) 0 0
\(979\) 90.0000 2.87641
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −15.0000 −0.478426 −0.239213 − 0.970967i \(-0.576889\pi\)
−0.239213 + 0.970967i \(0.576889\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.00000 0.0317982
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 19.0000 0.601736 0.300868 − 0.953666i \(-0.402724\pi\)
0.300868 + 0.953666i \(0.402724\pi\)
\(998\) 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.i.1.1 1
3.2 odd 2 1296.2.a.e.1.1 1
4.3 odd 2 648.2.a.c.1.1 1
8.3 odd 2 5184.2.a.i.1.1 1
8.5 even 2 5184.2.a.n.1.1 1
9.2 odd 6 144.2.i.b.49.1 2
9.4 even 3 432.2.i.a.289.1 2
9.5 odd 6 144.2.i.b.97.1 2
9.7 even 3 432.2.i.a.145.1 2
12.11 even 2 648.2.a.a.1.1 1
24.5 odd 2 5184.2.a.x.1.1 1
24.11 even 2 5184.2.a.s.1.1 1
36.7 odd 6 216.2.i.a.145.1 2
36.11 even 6 72.2.i.a.49.1 yes 2
36.23 even 6 72.2.i.a.25.1 ✓ 2
36.31 odd 6 216.2.i.a.73.1 2
72.5 odd 6 576.2.i.c.385.1 2
72.11 even 6 576.2.i.d.193.1 2
72.13 even 6 1728.2.i.g.1153.1 2
72.29 odd 6 576.2.i.c.193.1 2
72.43 odd 6 1728.2.i.h.577.1 2
72.59 even 6 576.2.i.d.385.1 2
72.61 even 6 1728.2.i.g.577.1 2
72.67 odd 6 1728.2.i.h.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.a.25.1 ✓ 2 36.23 even 6
72.2.i.a.49.1 yes 2 36.11 even 6
144.2.i.b.49.1 2 9.2 odd 6
144.2.i.b.97.1 2 9.5 odd 6
216.2.i.a.73.1 2 36.31 odd 6
216.2.i.a.145.1 2 36.7 odd 6
432.2.i.a.145.1 2 9.7 even 3
432.2.i.a.289.1 2 9.4 even 3
576.2.i.c.193.1 2 72.29 odd 6
576.2.i.c.385.1 2 72.5 odd 6
576.2.i.d.193.1 2 72.11 even 6
576.2.i.d.385.1 2 72.59 even 6
648.2.a.a.1.1 1 12.11 even 2
648.2.a.c.1.1 1 4.3 odd 2
1296.2.a.e.1.1 1 3.2 odd 2
1296.2.a.i.1.1 1 1.1 even 1 trivial
1728.2.i.g.577.1 2 72.61 even 6
1728.2.i.g.1153.1 2 72.13 even 6
1728.2.i.h.577.1 2 72.43 odd 6
1728.2.i.h.1153.1 2 72.67 odd 6
5184.2.a.i.1.1 1 8.3 odd 2
5184.2.a.n.1.1 1 8.5 even 2
5184.2.a.s.1.1 1 24.11 even 2
5184.2.a.x.1.1 1 24.5 odd 2