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Properties

Label 4320.2.a.l.1.1
Level $4320$
Weight $2$
Character 4320.1
Self dual yes
Analytic conductor $34.495$
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] = [4320,2,Mod(1,4320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4320, 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("4320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4320.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: \(34.4953736732\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +3.00000 q^{7} +2.00000 q^{11} +5.00000 q^{13} +2.00000 q^{17} +5.00000 q^{19} +2.00000 q^{23} +1.00000 q^{25} +3.00000 q^{35} -1.00000 q^{37} +4.00000 q^{43} -6.00000 q^{47} +2.00000 q^{49} -12.0000 q^{53} +2.00000 q^{55} -6.00000 q^{59} -1.00000 q^{61} +5.00000 q^{65} -3.00000 q^{67} +12.0000 q^{71} -7.00000 q^{73} +6.00000 q^{77} +1.00000 q^{79} +4.00000 q^{83} +2.00000 q^{85} +10.0000 q^{89} +15.0000 q^{91} +5.00000 q^{95} +1.00000 q^{97} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 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
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 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
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 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
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 15.0000 1.57243
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.00000 0.512989
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 − 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 − 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 − 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) 15.0000 1.30066
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 19.0000 1.61156 0.805779 − 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.0000 0.836242
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.0000 0.983078 0.491539 − 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 − 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 − 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.00000 0.608229 0.304114 − 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 − 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 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
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 − 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 − 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 − 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 29.0000 1.86805 0.934027 − 0.357202i \(-0.116269\pi\)
0.934027 + 0.357202i \(0.116269\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 25.0000 1.59071
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0000 −0.748539 −0.374270 − 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.0000 1.72655 0.863277 − 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 − 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 − 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 − 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 − 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 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
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 7.00000 0.395663 0.197832 − 0.980236i \(-0.436610\pi\)
0.197832 + 0.980236i \(0.436610\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 − 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) 5.00000 0.277350
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) 27.0000 1.48405 0.742027 − 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) 17.0000 0.926049 0.463025 − 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) 31.0000 1.65939 0.829696 − 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.0000 1.49029 0.745145 − 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.0000 1.16112 0.580558 − 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) 3.00000 0.156599 0.0782994 − 0.996930i \(-0.475051\pi\)
0.0782994 + 0.996930i \(0.475051\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 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
\(377\) 0 0
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 − 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 − 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 − 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 − 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.0000 −0.885722
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 − 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 − 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −3.00000 −0.145180
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −22.0000 −1.05970 −0.529851 − 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 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
\(437\) 10.0000 0.478365
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 − 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 − 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 − 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.0000 0.703211
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 − 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 − 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 31.0000 1.44069 0.720346 − 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 − 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) −1.00000 −0.0453143 −0.0226572 − 0.999743i \(-0.507213\pi\)
−0.0226572 + 0.999743i \(0.507213\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 − 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 36.0000 1.61482
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 − 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.0000 1.51599 0.757993 − 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −21.0000 −0.928985
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.00000 −0.0440653
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 − 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −18.0000 −0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −15.0000 −0.644900 −0.322450 − 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 7.00000 0.299298 0.149649 − 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.00000 0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.00000 0.0838444 0.0419222 − 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 − 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −13.0000 −0.541197 −0.270599 − 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −42.0000 −1.73353 −0.866763 − 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 − 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 − 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 − 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −31.0000 −1.24600 −0.622998 − 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.0000 1.20192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −35.0000 −1.39333 −0.696664 − 0.717398i \(-0.745333\pi\)
−0.696664 + 0.717398i \(0.745333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.0000 0.793676
\(636\) 0 0
\(637\) 10.0000 0.396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 28.0000 1.10593 0.552967 − 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 − 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 44.0000 1.72185 0.860927 − 0.508729i \(-0.169885\pi\)
0.860927 + 0.508729i \(0.169885\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −5.00000 −0.194477 −0.0972387 − 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.0000 0.581675
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 39.0000 1.50334 0.751670 − 0.659540i \(-0.229249\pi\)
0.751670 + 0.659540i \(0.229249\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 3.00000 0.115129
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.0000 0.720711
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −48.0000 −1.80523
\(708\) 0 0
\(709\) 13.0000 0.488225 0.244113 − 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 10.0000 0.373979
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 −0.745874 −0.372937 − 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 − 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 − 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.0000 1.54083 0.770415 − 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 0 0
\(745\) 12.0000 0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −54.0000 −1.97312
\(750\) 0 0
\(751\) −43.0000 −1.56909 −0.784546 − 0.620070i \(-0.787104\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 13.0000 0.472493 0.236247 − 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −30.0000 −1.08607
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −30.0000 −1.08324
\(768\) 0 0
\(769\) −17.0000 −0.613036 −0.306518 − 0.951865i \(-0.599164\pi\)
−0.306518 + 0.951865i \(0.599164\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4.00000 −0.143870 −0.0719350 − 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −43.0000 −1.53278 −0.766392 − 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −5.00000 −0.177555
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 − 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 52.0000 1.82822 0.914111 − 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 19.0000 0.665541
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −23.0000 −0.801730 −0.400865 − 0.916137i \(-0.631290\pi\)
−0.400865 + 0.916137i \(0.631290\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 − 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) −3.00000 −0.104194 −0.0520972 − 0.998642i \(-0.516591\pi\)
−0.0520972 + 0.998642i \(0.516591\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −21.0000 −0.721569
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 3.00000 0.102718 0.0513590 − 0.998680i \(-0.483645\pi\)
0.0513590 + 0.998680i \(0.483645\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.0000 0.956462 0.478231 − 0.878234i \(-0.341278\pi\)
0.478231 + 0.878234i \(0.341278\pi\)
\(858\) 0 0
\(859\) −29.0000 −0.989467 −0.494734 − 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.0000 −0.748889 −0.374444 − 0.927249i \(-0.622167\pi\)
−0.374444 + 0.927249i \(0.622167\pi\)
\(864\) 0 0
\(865\) 8.00000 0.272008
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) −15.0000 −0.508256
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −5.00000 −0.168838 −0.0844190 − 0.996430i \(-0.526903\pi\)
−0.0844190 + 0.996430i \(0.526903\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 23.0000 0.774012 0.387006 − 0.922077i \(-0.373509\pi\)
0.387006 + 0.922077i \(0.373509\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.0000 −0.940148 −0.470074 − 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 0 0
\(889\) 60.0000 2.01234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.0000 −1.00391
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −23.0000 −0.764546
\(906\) 0 0
\(907\) −39.0000 −1.29497 −0.647487 − 0.762077i \(-0.724180\pi\)
−0.647487 + 0.762077i \(0.724180\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 60.0000 1.97492
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 10.0000 0.327737
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −33.0000 −1.07806 −0.539032 − 0.842286i \(-0.681210\pi\)
−0.539032 + 0.842286i \(0.681210\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 − 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 − 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) −35.0000 −1.13615
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 − 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) −10.0000 −0.323592
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.0000 −0.418485
\(966\) 0 0
\(967\) 1.00000 0.0321578 0.0160789 − 0.999871i \(-0.494882\pi\)
0.0160789 + 0.999871i \(0.494882\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −10.0000 −0.320915 −0.160458 − 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) 0 0
\(973\) 57.0000 1.82734
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 − 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 46.0000 1.46717 0.733586 − 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −29.0000 −0.921215 −0.460608 − 0.887604i \(-0.652368\pi\)
−0.460608 + 0.887604i \(0.652368\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 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 4320.2.a.l.1.1 yes 1
3.2 odd 2 4320.2.a.f.1.1 yes 1
4.3 odd 2 4320.2.a.g.1.1 yes 1
8.3 odd 2 8640.2.a.d.1.1 1
8.5 even 2 8640.2.a.ba.1.1 1
12.11 even 2 4320.2.a.a.1.1 ✓ 1
24.5 odd 2 8640.2.a.cf.1.1 1
24.11 even 2 8640.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4320.2.a.a.1.1 ✓ 1 12.11 even 2
4320.2.a.f.1.1 yes 1 3.2 odd 2
4320.2.a.g.1.1 yes 1 4.3 odd 2
4320.2.a.l.1.1 yes 1 1.1 even 1 trivial
8640.2.a.d.1.1 1 8.3 odd 2
8640.2.a.ba.1.1 1 8.5 even 2
8640.2.a.bg.1.1 1 24.11 even 2
8640.2.a.cf.1.1 1 24.5 odd 2