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Properties

Label 1800.4.a.z.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,4,Mod(1,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.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: \(106.203438010\)
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 \(\chi\) \(=\) 1800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12.0000 q^{7} +O(q^{10})\) \(q+12.0000 q^{7} -64.0000 q^{11} -58.0000 q^{13} +32.0000 q^{17} -136.000 q^{19} -128.000 q^{23} +144.000 q^{29} +20.0000 q^{31} +18.0000 q^{37} +288.000 q^{41} +200.000 q^{43} +384.000 q^{47} -199.000 q^{49} +496.000 q^{53} +128.000 q^{59} -458.000 q^{61} +496.000 q^{67} -512.000 q^{71} +602.000 q^{73} -768.000 q^{77} +1108.00 q^{79} +704.000 q^{83} +960.000 q^{89} -696.000 q^{91} -206.000 q^{97} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 12.0000 0.647939 0.323970 − 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −64.0000 −1.75425 −0.877124 − 0.480264i \(-0.840541\pi\)
−0.877124 + 0.480264i \(0.840541\pi\)
\(12\) 0 0
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 32.0000 0.456538 0.228269 − 0.973598i \(-0.426693\pi\)
0.228269 + 0.973598i \(0.426693\pi\)
\(18\) 0 0
\(19\) −136.000 −1.64213 −0.821067 − 0.570832i \(-0.806621\pi\)
−0.821067 + 0.570832i \(0.806621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −128.000 −1.16043 −0.580214 − 0.814464i \(-0.697031\pi\)
−0.580214 + 0.814464i \(0.697031\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 144.000 0.922073 0.461037 − 0.887381i \(-0.347478\pi\)
0.461037 + 0.887381i \(0.347478\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 − 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 18.0000 0.0799779 0.0399889 − 0.999200i \(-0.487268\pi\)
0.0399889 + 0.999200i \(0.487268\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 288.000 1.09703 0.548513 − 0.836142i \(-0.315194\pi\)
0.548513 + 0.836142i \(0.315194\pi\)
\(42\) 0 0
\(43\) 200.000 0.709296 0.354648 − 0.935000i \(-0.384601\pi\)
0.354648 + 0.935000i \(0.384601\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 384.000 1.19175 0.595874 − 0.803078i \(-0.296806\pi\)
0.595874 + 0.803078i \(0.296806\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 496.000 1.28549 0.642744 − 0.766081i \(-0.277796\pi\)
0.642744 + 0.766081i \(0.277796\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 128.000 0.282444 0.141222 − 0.989978i \(-0.454897\pi\)
0.141222 + 0.989978i \(0.454897\pi\)
\(60\) 0 0
\(61\) −458.000 −0.961326 −0.480663 − 0.876905i \(-0.659604\pi\)
−0.480663 + 0.876905i \(0.659604\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 496.000 0.904419 0.452209 − 0.891912i \(-0.350636\pi\)
0.452209 + 0.891912i \(0.350636\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −512.000 −0.855820 −0.427910 − 0.903821i \(-0.640750\pi\)
−0.427910 + 0.903821i \(0.640750\pi\)
\(72\) 0 0
\(73\) 602.000 0.965189 0.482594 − 0.875844i \(-0.339695\pi\)
0.482594 + 0.875844i \(0.339695\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −768.000 −1.13665
\(78\) 0 0
\(79\) 1108.00 1.57797 0.788986 − 0.614412i \(-0.210607\pi\)
0.788986 + 0.614412i \(0.210607\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 704.000 0.931013 0.465506 − 0.885045i \(-0.345872\pi\)
0.465506 + 0.885045i \(0.345872\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 960.000 1.14337 0.571684 − 0.820474i \(-0.306290\pi\)
0.571684 + 0.820474i \(0.306290\pi\)
\(90\) 0 0
\(91\) −696.000 −0.801765
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −206.000 −0.215630 −0.107815 − 0.994171i \(-0.534385\pi\)
−0.107815 + 0.994171i \(0.534385\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −432.000 −0.425600 −0.212800 − 0.977096i \(-0.568258\pi\)
−0.212800 + 0.977096i \(0.568258\pi\)
\(102\) 0 0
\(103\) 68.0000 0.0650509 0.0325254 − 0.999471i \(-0.489645\pi\)
0.0325254 + 0.999471i \(0.489645\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −384.000 −0.346941 −0.173470 − 0.984839i \(-0.555498\pi\)
−0.173470 + 0.984839i \(0.555498\pi\)
\(108\) 0 0
\(109\) −518.000 −0.455187 −0.227594 − 0.973756i \(-0.573086\pi\)
−0.227594 + 0.973756i \(0.573086\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −960.000 −0.799196 −0.399598 − 0.916690i \(-0.630850\pi\)
−0.399598 + 0.916690i \(0.630850\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 384.000 0.295809
\(120\) 0 0
\(121\) 2765.00 2.07739
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −796.000 −0.556170 −0.278085 − 0.960556i \(-0.589700\pi\)
−0.278085 + 0.960556i \(0.589700\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −512.000 −0.341478 −0.170739 − 0.985316i \(-0.554616\pi\)
−0.170739 + 0.985316i \(0.554616\pi\)
\(132\) 0 0
\(133\) −1632.00 −1.06400
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1824.00 1.13748 0.568740 − 0.822517i \(-0.307431\pi\)
0.568740 + 0.822517i \(0.307431\pi\)
\(138\) 0 0
\(139\) −2160.00 −1.31805 −0.659024 − 0.752121i \(-0.729031\pi\)
−0.659024 + 0.752121i \(0.729031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3712.00 2.17072
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 688.000 0.378276 0.189138 − 0.981950i \(-0.439431\pi\)
0.189138 + 0.981950i \(0.439431\pi\)
\(150\) 0 0
\(151\) −844.000 −0.454859 −0.227430 − 0.973795i \(-0.573032\pi\)
−0.227430 + 0.973795i \(0.573032\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −118.000 −0.0599836 −0.0299918 − 0.999550i \(-0.509548\pi\)
−0.0299918 + 0.999550i \(0.509548\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1536.00 −0.751887
\(162\) 0 0
\(163\) −3576.00 −1.71837 −0.859184 − 0.511667i \(-0.829028\pi\)
−0.859184 + 0.511667i \(0.829028\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 384.000 0.177933 0.0889665 − 0.996035i \(-0.471644\pi\)
0.0889665 + 0.996035i \(0.471644\pi\)
\(168\) 0 0
\(169\) 1167.00 0.531179
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2448.00 1.07583 0.537913 − 0.843000i \(-0.319213\pi\)
0.537913 + 0.843000i \(0.319213\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4224.00 1.76378 0.881890 − 0.471455i \(-0.156271\pi\)
0.881890 + 0.471455i \(0.156271\pi\)
\(180\) 0 0
\(181\) −510.000 −0.209436 −0.104718 − 0.994502i \(-0.533394\pi\)
−0.104718 + 0.994502i \(0.533394\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2048.00 −0.800880
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 384.000 0.145473 0.0727363 − 0.997351i \(-0.476827\pi\)
0.0727363 + 0.997351i \(0.476827\pi\)
\(192\) 0 0
\(193\) 3454.00 1.28821 0.644105 − 0.764937i \(-0.277230\pi\)
0.644105 + 0.764937i \(0.277230\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3216.00 −1.16310 −0.581550 − 0.813511i \(-0.697553\pi\)
−0.581550 + 0.813511i \(0.697553\pi\)
\(198\) 0 0
\(199\) 1708.00 0.608427 0.304213 − 0.952604i \(-0.401606\pi\)
0.304213 + 0.952604i \(0.401606\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1728.00 0.597447
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8704.00 2.88071
\(210\) 0 0
\(211\) 2320.00 0.756945 0.378472 − 0.925613i \(-0.376449\pi\)
0.378472 + 0.925613i \(0.376449\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 240.000 0.0750795
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1856.00 −0.564923
\(222\) 0 0
\(223\) 116.000 0.0348338 0.0174169 − 0.999848i \(-0.494456\pi\)
0.0174169 + 0.999848i \(0.494456\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1344.00 0.392971 0.196485 − 0.980507i \(-0.437047\pi\)
0.196485 + 0.980507i \(0.437047\pi\)
\(228\) 0 0
\(229\) 4594.00 1.32568 0.662839 − 0.748762i \(-0.269352\pi\)
0.662839 + 0.748762i \(0.269352\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5056.00 1.42159 0.710793 − 0.703401i \(-0.248336\pi\)
0.710793 + 0.703401i \(0.248336\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3712.00 1.00464 0.502321 − 0.864681i \(-0.332480\pi\)
0.502321 + 0.864681i \(0.332480\pi\)
\(240\) 0 0
\(241\) −978.000 −0.261405 −0.130702 − 0.991422i \(-0.541723\pi\)
−0.130702 + 0.991422i \(0.541723\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7888.00 2.03199
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1856.00 −0.466732 −0.233366 − 0.972389i \(-0.574974\pi\)
−0.233366 + 0.972389i \(0.574974\pi\)
\(252\) 0 0
\(253\) 8192.00 2.03568
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7808.00 1.89513 0.947567 − 0.319556i \(-0.103534\pi\)
0.947567 + 0.319556i \(0.103534\pi\)
\(258\) 0 0
\(259\) 216.000 0.0518208
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1024.00 0.240086 0.120043 − 0.992769i \(-0.461697\pi\)
0.120043 + 0.992769i \(0.461697\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1328.00 −0.301002 −0.150501 − 0.988610i \(-0.548089\pi\)
−0.150501 + 0.988610i \(0.548089\pi\)
\(270\) 0 0
\(271\) −5812.00 −1.30278 −0.651391 − 0.758742i \(-0.725814\pi\)
−0.651391 + 0.758742i \(0.725814\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8386.00 −1.81901 −0.909505 − 0.415692i \(-0.863539\pi\)
−0.909505 + 0.415692i \(0.863539\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 640.000 0.135869 0.0679345 − 0.997690i \(-0.478359\pi\)
0.0679345 + 0.997690i \(0.478359\pi\)
\(282\) 0 0
\(283\) −4832.00 −1.01496 −0.507478 − 0.861665i \(-0.669422\pi\)
−0.507478 + 0.861665i \(0.669422\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3456.00 0.710806
\(288\) 0 0
\(289\) −3889.00 −0.791573
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6384.00 1.27289 0.636446 − 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7424.00 1.43592
\(300\) 0 0
\(301\) 2400.00 0.459580
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3312.00 0.615719 0.307860 − 0.951432i \(-0.400387\pi\)
0.307860 + 0.951432i \(0.400387\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9984.00 −1.82039 −0.910194 − 0.414182i \(-0.864068\pi\)
−0.910194 + 0.414182i \(0.864068\pi\)
\(312\) 0 0
\(313\) −2586.00 −0.466995 −0.233497 − 0.972357i \(-0.575017\pi\)
−0.233497 + 0.972357i \(0.575017\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2832.00 −0.501770 −0.250885 − 0.968017i \(-0.580722\pi\)
−0.250885 + 0.968017i \(0.580722\pi\)
\(318\) 0 0
\(319\) −9216.00 −1.61755
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4352.00 −0.749696
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4608.00 0.772180
\(330\) 0 0
\(331\) −5920.00 −0.983059 −0.491530 − 0.870861i \(-0.663562\pi\)
−0.491530 + 0.870861i \(0.663562\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4674.00 0.755516 0.377758 − 0.925904i \(-0.376695\pi\)
0.377758 + 0.925904i \(0.376695\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1280.00 −0.203272
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9024.00 −1.39606 −0.698031 − 0.716067i \(-0.745940\pi\)
−0.698031 + 0.716067i \(0.745940\pi\)
\(348\) 0 0
\(349\) −4362.00 −0.669033 −0.334516 − 0.942390i \(-0.608573\pi\)
−0.334516 + 0.942390i \(0.608573\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8768.00 1.32202 0.661011 − 0.750376i \(-0.270128\pi\)
0.661011 + 0.750376i \(0.270128\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6144.00 0.903253 0.451627 − 0.892207i \(-0.350844\pi\)
0.451627 + 0.892207i \(0.350844\pi\)
\(360\) 0 0
\(361\) 11637.0 1.69660
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4564.00 −0.649152 −0.324576 − 0.945860i \(-0.605222\pi\)
−0.324576 + 0.945860i \(0.605222\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 5952.00 0.832918
\(372\) 0 0
\(373\) 8770.00 1.21741 0.608704 − 0.793397i \(-0.291690\pi\)
0.608704 + 0.793397i \(0.291690\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8352.00 −1.14098
\(378\) 0 0
\(379\) −1096.00 −0.148543 −0.0742714 − 0.997238i \(-0.523663\pi\)
−0.0742714 + 0.997238i \(0.523663\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10368.0 1.38324 0.691619 − 0.722263i \(-0.256898\pi\)
0.691619 + 0.722263i \(0.256898\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3248.00 0.423342 0.211671 − 0.977341i \(-0.432109\pi\)
0.211671 + 0.977341i \(0.432109\pi\)
\(390\) 0 0
\(391\) −4096.00 −0.529779
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6106.00 0.771918 0.385959 − 0.922516i \(-0.373871\pi\)
0.385959 + 0.922516i \(0.373871\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7008.00 −0.872725 −0.436363 − 0.899771i \(-0.643734\pi\)
−0.436363 + 0.899771i \(0.643734\pi\)
\(402\) 0 0
\(403\) −1160.00 −0.143384
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1152.00 −0.140301
\(408\) 0 0
\(409\) 1590.00 0.192226 0.0961130 − 0.995370i \(-0.469359\pi\)
0.0961130 + 0.995370i \(0.469359\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1536.00 0.183006
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 192.000 0.0223862 0.0111931 − 0.999937i \(-0.496437\pi\)
0.0111931 + 0.999937i \(0.496437\pi\)
\(420\) 0 0
\(421\) 9074.00 1.05045 0.525225 − 0.850963i \(-0.323981\pi\)
0.525225 + 0.850963i \(0.323981\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5496.00 −0.622881
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5248.00 0.586513 0.293257 − 0.956034i \(-0.405261\pi\)
0.293257 + 0.956034i \(0.405261\pi\)
\(432\) 0 0
\(433\) 8222.00 0.912527 0.456263 − 0.889845i \(-0.349187\pi\)
0.456263 + 0.889845i \(0.349187\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17408.0 1.90558
\(438\) 0 0
\(439\) 16236.0 1.76515 0.882576 − 0.470169i \(-0.155807\pi\)
0.882576 + 0.470169i \(0.155807\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14528.0 1.55812 0.779059 − 0.626951i \(-0.215697\pi\)
0.779059 + 0.626951i \(0.215697\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6304.00 −0.662593 −0.331296 − 0.943527i \(-0.607486\pi\)
−0.331296 + 0.943527i \(0.607486\pi\)
\(450\) 0 0
\(451\) −18432.0 −1.92445
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1958.00 −0.200419 −0.100209 − 0.994966i \(-0.531951\pi\)
−0.100209 + 0.994966i \(0.531951\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4048.00 −0.408968 −0.204484 − 0.978870i \(-0.565552\pi\)
−0.204484 + 0.978870i \(0.565552\pi\)
\(462\) 0 0
\(463\) −16988.0 −1.70518 −0.852591 − 0.522579i \(-0.824970\pi\)
−0.852591 + 0.522579i \(0.824970\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6720.00 0.665877 0.332938 − 0.942949i \(-0.391960\pi\)
0.332938 + 0.942949i \(0.391960\pi\)
\(468\) 0 0
\(469\) 5952.00 0.586008
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12800.0 −1.24428
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9728.00 −0.927941 −0.463970 − 0.885851i \(-0.653576\pi\)
−0.463970 + 0.885851i \(0.653576\pi\)
\(480\) 0 0
\(481\) −1044.00 −0.0989653
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8444.00 0.785696 0.392848 − 0.919603i \(-0.371490\pi\)
0.392848 + 0.919603i \(0.371490\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15360.0 1.41179 0.705893 − 0.708318i \(-0.250546\pi\)
0.705893 + 0.708318i \(0.250546\pi\)
\(492\) 0 0
\(493\) 4608.00 0.420961
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6144.00 −0.554519
\(498\) 0 0
\(499\) 6624.00 0.594250 0.297125 − 0.954839i \(-0.403972\pi\)
0.297125 + 0.954839i \(0.403972\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6912.00 −0.612705 −0.306353 − 0.951918i \(-0.599109\pi\)
−0.306353 + 0.951918i \(0.599109\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19920.0 1.73465 0.867327 − 0.497739i \(-0.165836\pi\)
0.867327 + 0.497739i \(0.165836\pi\)
\(510\) 0 0
\(511\) 7224.00 0.625383
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −24576.0 −2.09062
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3680.00 −0.309451 −0.154725 − 0.987958i \(-0.549449\pi\)
−0.154725 + 0.987958i \(0.549449\pi\)
\(522\) 0 0
\(523\) 11720.0 0.979885 0.489942 − 0.871755i \(-0.337018\pi\)
0.489942 + 0.871755i \(0.337018\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 640.000 0.0529010
\(528\) 0 0
\(529\) 4217.00 0.346593
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16704.0 −1.35747
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12736.0 1.01777
\(540\) 0 0
\(541\) 11754.0 0.934092 0.467046 − 0.884233i \(-0.345318\pi\)
0.467046 + 0.884233i \(0.345318\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18904.0 1.47765 0.738827 − 0.673895i \(-0.235380\pi\)
0.738827 + 0.673895i \(0.235380\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −19584.0 −1.51417
\(552\) 0 0
\(553\) 13296.0 1.02243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3088.00 0.234906 0.117453 − 0.993078i \(-0.462527\pi\)
0.117453 + 0.993078i \(0.462527\pi\)
\(558\) 0 0
\(559\) −11600.0 −0.877688
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21440.0 −1.60495 −0.802476 − 0.596684i \(-0.796485\pi\)
−0.802476 + 0.596684i \(0.796485\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22624.0 −1.66687 −0.833434 − 0.552620i \(-0.813628\pi\)
−0.833434 + 0.552620i \(0.813628\pi\)
\(570\) 0 0
\(571\) −6000.00 −0.439741 −0.219871 − 0.975529i \(-0.570564\pi\)
−0.219871 + 0.975529i \(0.570564\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19922.0 −1.43737 −0.718686 − 0.695335i \(-0.755256\pi\)
−0.718686 + 0.695335i \(0.755256\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8448.00 0.603239
\(582\) 0 0
\(583\) −31744.0 −2.25506
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3584.00 −0.252006 −0.126003 − 0.992030i \(-0.540215\pi\)
−0.126003 + 0.992030i \(0.540215\pi\)
\(588\) 0 0
\(589\) −2720.00 −0.190281
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1984.00 0.137391 0.0686957 − 0.997638i \(-0.478116\pi\)
0.0686957 + 0.997638i \(0.478116\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14976.0 −1.02154 −0.510770 − 0.859717i \(-0.670640\pi\)
−0.510770 + 0.859717i \(0.670640\pi\)
\(600\) 0 0
\(601\) 25738.0 1.74688 0.873440 − 0.486932i \(-0.161884\pi\)
0.873440 + 0.486932i \(0.161884\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8548.00 0.571586 0.285793 − 0.958291i \(-0.407743\pi\)
0.285793 + 0.958291i \(0.407743\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −22272.0 −1.47468
\(612\) 0 0
\(613\) −8558.00 −0.563873 −0.281937 − 0.959433i \(-0.590977\pi\)
−0.281937 + 0.959433i \(0.590977\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10368.0 −0.676499 −0.338250 − 0.941056i \(-0.609835\pi\)
−0.338250 + 0.941056i \(0.609835\pi\)
\(618\) 0 0
\(619\) −13088.0 −0.849840 −0.424920 − 0.905231i \(-0.639698\pi\)
−0.424920 + 0.905231i \(0.639698\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11520.0 0.740833
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 576.000 0.0365129
\(630\) 0 0
\(631\) −4412.00 −0.278350 −0.139175 − 0.990268i \(-0.544445\pi\)
−0.139175 + 0.990268i \(0.544445\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 11542.0 0.717913
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30176.0 1.85941 0.929704 − 0.368308i \(-0.120063\pi\)
0.929704 + 0.368308i \(0.120063\pi\)
\(642\) 0 0
\(643\) −21288.0 −1.30562 −0.652812 − 0.757520i \(-0.726411\pi\)
−0.652812 + 0.757520i \(0.726411\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17024.0 1.03444 0.517220 − 0.855853i \(-0.326967\pi\)
0.517220 + 0.855853i \(0.326967\pi\)
\(648\) 0 0
\(649\) −8192.00 −0.495476
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2256.00 0.135198 0.0675989 − 0.997713i \(-0.478466\pi\)
0.0675989 + 0.997713i \(0.478466\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −23808.0 −1.40733 −0.703663 − 0.710534i \(-0.748454\pi\)
−0.703663 + 0.710534i \(0.748454\pi\)
\(660\) 0 0
\(661\) −26242.0 −1.54417 −0.772084 − 0.635520i \(-0.780786\pi\)
−0.772084 + 0.635520i \(0.780786\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18432.0 −1.07000
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 29312.0 1.68640
\(672\) 0 0
\(673\) 24590.0 1.40843 0.704216 − 0.709986i \(-0.251299\pi\)
0.704216 + 0.709986i \(0.251299\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2864.00 −0.162589 −0.0812943 − 0.996690i \(-0.525905\pi\)
−0.0812943 + 0.996690i \(0.525905\pi\)
\(678\) 0 0
\(679\) −2472.00 −0.139715
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7616.00 −0.426674 −0.213337 − 0.976979i \(-0.568433\pi\)
−0.213337 + 0.976979i \(0.568433\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28768.0 −1.59067
\(690\) 0 0
\(691\) 2168.00 0.119355 0.0596777 − 0.998218i \(-0.480993\pi\)
0.0596777 + 0.998218i \(0.480993\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9216.00 0.500833
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18000.0 0.969830 0.484915 − 0.874561i \(-0.338851\pi\)
0.484915 + 0.874561i \(0.338851\pi\)
\(702\) 0 0
\(703\) −2448.00 −0.131334
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5184.00 −0.275763
\(708\) 0 0
\(709\) 3506.00 0.185713 0.0928566 − 0.995679i \(-0.470400\pi\)
0.0928566 + 0.995679i \(0.470400\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2560.00 −0.134464
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15616.0 −0.809984 −0.404992 − 0.914320i \(-0.632726\pi\)
−0.404992 + 0.914320i \(0.632726\pi\)
\(720\) 0 0
\(721\) 816.000 0.0421490
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15036.0 0.767062 0.383531 − 0.923528i \(-0.374708\pi\)
0.383531 + 0.923528i \(0.374708\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6400.00 0.323820
\(732\) 0 0
\(733\) 19126.0 0.963758 0.481879 − 0.876238i \(-0.339954\pi\)
0.481879 + 0.876238i \(0.339954\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −31744.0 −1.58657
\(738\) 0 0
\(739\) −17392.0 −0.865731 −0.432865 − 0.901459i \(-0.642497\pi\)
−0.432865 + 0.901459i \(0.642497\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32384.0 −1.59900 −0.799498 − 0.600669i \(-0.794901\pi\)
−0.799498 + 0.600669i \(0.794901\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4608.00 −0.224797
\(750\) 0 0
\(751\) −27708.0 −1.34631 −0.673155 − 0.739501i \(-0.735061\pi\)
−0.673155 + 0.739501i \(0.735061\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37246.0 1.78828 0.894141 − 0.447786i \(-0.147787\pi\)
0.894141 + 0.447786i \(0.147787\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4192.00 0.199684 0.0998422 − 0.995003i \(-0.468166\pi\)
0.0998422 + 0.995003i \(0.468166\pi\)
\(762\) 0 0
\(763\) −6216.00 −0.294934
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7424.00 −0.349498
\(768\) 0 0
\(769\) 26882.0 1.26058 0.630292 − 0.776358i \(-0.282935\pi\)
0.630292 + 0.776358i \(0.282935\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17232.0 −0.801801 −0.400900 − 0.916122i \(-0.631303\pi\)
−0.400900 + 0.916122i \(0.631303\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −39168.0 −1.80146
\(780\) 0 0
\(781\) 32768.0 1.50132
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −31816.0 −1.44106 −0.720532 − 0.693421i \(-0.756102\pi\)
−0.720532 + 0.693421i \(0.756102\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11520.0 −0.517831
\(792\) 0 0
\(793\) 26564.0 1.18955
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8272.00 0.367640 0.183820 − 0.982960i \(-0.441154\pi\)
0.183820 + 0.982960i \(0.441154\pi\)
\(798\) 0 0
\(799\) 12288.0 0.544078
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −38528.0 −1.69318
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9184.00 0.399125 0.199563 − 0.979885i \(-0.436048\pi\)
0.199563 + 0.979885i \(0.436048\pi\)
\(810\) 0 0
\(811\) −19832.0 −0.858688 −0.429344 − 0.903141i \(-0.641255\pi\)
−0.429344 + 0.903141i \(0.641255\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −27200.0 −1.16476
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15216.0 −0.646823 −0.323412 − 0.946258i \(-0.604830\pi\)
−0.323412 + 0.946258i \(0.604830\pi\)
\(822\) 0 0
\(823\) 39772.0 1.68453 0.842263 − 0.539067i \(-0.181223\pi\)
0.842263 + 0.539067i \(0.181223\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18304.0 −0.769640 −0.384820 − 0.922992i \(-0.625737\pi\)
−0.384820 + 0.922992i \(0.625737\pi\)
\(828\) 0 0
\(829\) 4906.00 0.205540 0.102770 − 0.994705i \(-0.467229\pi\)
0.102770 + 0.994705i \(0.467229\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6368.00 −0.264872
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15360.0 −0.632045 −0.316023 − 0.948752i \(-0.602348\pi\)
−0.316023 + 0.948752i \(0.602348\pi\)
\(840\) 0 0
\(841\) −3653.00 −0.149781
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 33180.0 1.34602
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2304.00 −0.0928086
\(852\) 0 0
\(853\) 24802.0 0.995550 0.497775 − 0.867306i \(-0.334150\pi\)
0.497775 + 0.867306i \(0.334150\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15072.0 0.600758 0.300379 − 0.953820i \(-0.402887\pi\)
0.300379 + 0.953820i \(0.402887\pi\)
\(858\) 0 0
\(859\) 1800.00 0.0714962 0.0357481 − 0.999361i \(-0.488619\pi\)
0.0357481 + 0.999361i \(0.488619\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7552.00 0.297883 0.148942 − 0.988846i \(-0.452413\pi\)
0.148942 + 0.988846i \(0.452413\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −70912.0 −2.76815
\(870\) 0 0
\(871\) −28768.0 −1.11913
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −20838.0 −0.802337 −0.401168 − 0.916004i \(-0.631396\pi\)
−0.401168 + 0.916004i \(0.631396\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −47744.0 −1.82581 −0.912904 − 0.408175i \(-0.866165\pi\)
−0.912904 + 0.408175i \(0.866165\pi\)
\(882\) 0 0
\(883\) −28280.0 −1.07780 −0.538900 − 0.842370i \(-0.681160\pi\)
−0.538900 + 0.842370i \(0.681160\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7424.00 0.281030 0.140515 − 0.990079i \(-0.455124\pi\)
0.140515 + 0.990079i \(0.455124\pi\)
\(888\) 0 0
\(889\) −9552.00 −0.360364
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −52224.0 −1.95701
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2880.00 0.106845
\(900\) 0 0
\(901\) 15872.0 0.586873
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5912.00 0.216433 0.108217 − 0.994127i \(-0.465486\pi\)
0.108217 + 0.994127i \(0.465486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26240.0 0.954303 0.477151 − 0.878821i \(-0.341669\pi\)
0.477151 + 0.878821i \(0.341669\pi\)
\(912\) 0 0
\(913\) −45056.0 −1.63323
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6144.00 −0.221257
\(918\) 0 0
\(919\) 35620.0 1.27856 0.639279 − 0.768975i \(-0.279233\pi\)
0.639279 + 0.768975i \(0.279233\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29696.0 1.05900
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3232.00 0.114143 0.0570713 − 0.998370i \(-0.481824\pi\)
0.0570713 + 0.998370i \(0.481824\pi\)
\(930\) 0 0
\(931\) 27064.0 0.952725
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11478.0 0.400181 0.200091 − 0.979777i \(-0.435876\pi\)
0.200091 + 0.979777i \(0.435876\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 39984.0 1.38517 0.692583 − 0.721338i \(-0.256473\pi\)
0.692583 + 0.721338i \(0.256473\pi\)
\(942\) 0 0
\(943\) −36864.0 −1.27302
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24192.0 0.830131 0.415066 − 0.909791i \(-0.363759\pi\)
0.415066 + 0.909791i \(0.363759\pi\)
\(948\) 0 0
\(949\) −34916.0 −1.19433
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39456.0 −1.34114 −0.670569 − 0.741847i \(-0.733950\pi\)
−0.670569 + 0.741847i \(0.733950\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 21888.0 0.737018
\(960\) 0 0
\(961\) −29391.0 −0.986573
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 41668.0 1.38568 0.692840 − 0.721091i \(-0.256359\pi\)
0.692840 + 0.721091i \(0.256359\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 51648.0 1.70697 0.853483 − 0.521121i \(-0.174486\pi\)
0.853483 + 0.521121i \(0.174486\pi\)
\(972\) 0 0
\(973\) −25920.0 −0.854015
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55776.0 1.82644 0.913220 − 0.407466i \(-0.133588\pi\)
0.913220 + 0.407466i \(0.133588\pi\)
\(978\) 0 0
\(979\) −61440.0 −2.00575
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36096.0 −1.17119 −0.585597 − 0.810602i \(-0.699140\pi\)
−0.585597 + 0.810602i \(0.699140\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −25600.0 −0.823087
\(990\) 0 0
\(991\) 42532.0 1.36334 0.681672 − 0.731658i \(-0.261253\pi\)
0.681672 + 0.731658i \(0.261253\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −29806.0 −0.946806 −0.473403 − 0.880846i \(-0.656975\pi\)
−0.473403 + 0.880846i \(0.656975\pi\)
\(998\) 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 1800.4.a.z.1.1 1
3.2 odd 2 1800.4.a.ba.1.1 1
5.2 odd 4 1800.4.f.b.649.2 2
5.3 odd 4 1800.4.f.b.649.1 2
5.4 even 2 72.4.a.a.1.1 ✓ 1
15.2 even 4 1800.4.f.x.649.2 2
15.8 even 4 1800.4.f.x.649.1 2
15.14 odd 2 72.4.a.d.1.1 yes 1
20.19 odd 2 144.4.a.a.1.1 1
40.19 odd 2 576.4.a.x.1.1 1
40.29 even 2 576.4.a.w.1.1 1
45.4 even 6 648.4.i.l.217.1 2
45.14 odd 6 648.4.i.a.217.1 2
45.29 odd 6 648.4.i.a.433.1 2
45.34 even 6 648.4.i.l.433.1 2
60.59 even 2 144.4.a.f.1.1 1
120.29 odd 2 576.4.a.c.1.1 1
120.59 even 2 576.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.a.a.1.1 ✓ 1 5.4 even 2
72.4.a.d.1.1 yes 1 15.14 odd 2
144.4.a.a.1.1 1 20.19 odd 2
144.4.a.f.1.1 1 60.59 even 2
576.4.a.c.1.1 1 120.29 odd 2
576.4.a.d.1.1 1 120.59 even 2
576.4.a.w.1.1 1 40.29 even 2
576.4.a.x.1.1 1 40.19 odd 2
648.4.i.a.217.1 2 45.14 odd 6
648.4.i.a.433.1 2 45.29 odd 6
648.4.i.l.217.1 2 45.4 even 6
648.4.i.l.433.1 2 45.34 even 6
1800.4.a.z.1.1 1 1.1 even 1 trivial
1800.4.a.ba.1.1 1 3.2 odd 2
1800.4.f.b.649.1 2 5.3 odd 4
1800.4.f.b.649.2 2 5.2 odd 4
1800.4.f.x.649.1 2 15.8 even 4
1800.4.f.x.649.2 2 15.2 even 4