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Properties

Label 1764.4.a.a.1.1
Level $1764$
Weight $4$
Character 1764.1
Self dual yes
Analytic conductor $104.079$
Analytic rank $1$
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] = [1764,4,Mod(1,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1764.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(104.079369250\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1764.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-20.0000 q^{5} +O(q^{10})\) \(q-20.0000 q^{5} -44.0000 q^{11} +44.0000 q^{13} +72.0000 q^{17} -100.000 q^{19} +120.000 q^{23} +275.000 q^{25} -218.000 q^{29} +280.000 q^{31} -30.0000 q^{37} +120.000 q^{41} +220.000 q^{43} +88.0000 q^{47} -110.000 q^{53} +880.000 q^{55} +580.000 q^{59} -380.000 q^{61} -880.000 q^{65} -980.000 q^{67} +112.000 q^{71} +640.000 q^{73} -488.000 q^{79} +660.000 q^{83} -1440.00 q^{85} +320.000 q^{89} +2000.00 q^{95} -248.000 q^{97} +O(q^{100})\)

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
\(5\) −20.0000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −44.0000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 44.0000 0.938723 0.469362 − 0.883006i \(-0.344484\pi\)
0.469362 + 0.883006i \(0.344484\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 72.0000 1.02721 0.513605 − 0.858027i \(-0.328310\pi\)
0.513605 + 0.858027i \(0.328310\pi\)
\(18\) 0 0
\(19\) −100.000 −1.20745 −0.603726 − 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 120.000 1.08790 0.543951 − 0.839117i \(-0.316928\pi\)
0.543951 + 0.839117i \(0.316928\pi\)
\(24\) 0 0
\(25\) 275.000 2.20000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −218.000 −1.39592 −0.697958 − 0.716138i \(-0.745908\pi\)
−0.697958 + 0.716138i \(0.745908\pi\)
\(30\) 0 0
\(31\) 280.000 1.62224 0.811121 − 0.584879i \(-0.198858\pi\)
0.811121 + 0.584879i \(0.198858\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −30.0000 −0.133296 −0.0666482 − 0.997777i \(-0.521231\pi\)
−0.0666482 + 0.997777i \(0.521231\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 120.000 0.457094 0.228547 − 0.973533i \(-0.426603\pi\)
0.228547 + 0.973533i \(0.426603\pi\)
\(42\) 0 0
\(43\) 220.000 0.780225 0.390113 − 0.920767i \(-0.372436\pi\)
0.390113 + 0.920767i \(0.372436\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 88.0000 0.273109 0.136554 − 0.990633i \(-0.456397\pi\)
0.136554 + 0.990633i \(0.456397\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −110.000 −0.285088 −0.142544 − 0.989788i \(-0.545528\pi\)
−0.142544 + 0.989788i \(0.545528\pi\)
\(54\) 0 0
\(55\) 880.000 2.15744
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 580.000 1.27982 0.639912 − 0.768449i \(-0.278971\pi\)
0.639912 + 0.768449i \(0.278971\pi\)
\(60\) 0 0
\(61\) −380.000 −0.797607 −0.398803 − 0.917036i \(-0.630574\pi\)
−0.398803 + 0.917036i \(0.630574\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −880.000 −1.67924
\(66\) 0 0
\(67\) −980.000 −1.78696 −0.893478 − 0.449107i \(-0.851742\pi\)
−0.893478 + 0.449107i \(0.851742\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 112.000 0.187211 0.0936053 − 0.995609i \(-0.470161\pi\)
0.0936053 + 0.995609i \(0.470161\pi\)
\(72\) 0 0
\(73\) 640.000 1.02611 0.513057 − 0.858354i \(-0.328513\pi\)
0.513057 + 0.858354i \(0.328513\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −488.000 −0.694991 −0.347496 − 0.937682i \(-0.612968\pi\)
−0.347496 + 0.937682i \(0.612968\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 660.000 0.872824 0.436412 − 0.899747i \(-0.356249\pi\)
0.436412 + 0.899747i \(0.356249\pi\)
\(84\) 0 0
\(85\) −1440.00 −1.83753
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 320.000 0.381123 0.190561 − 0.981675i \(-0.438969\pi\)
0.190561 + 0.981675i \(0.438969\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2000.00 2.15995
\(96\) 0 0
\(97\) −248.000 −0.259594 −0.129797 − 0.991541i \(-0.541433\pi\)
−0.129797 + 0.991541i \(0.541433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 220.000 0.216741 0.108370 − 0.994111i \(-0.465437\pi\)
0.108370 + 0.994111i \(0.465437\pi\)
\(102\) 0 0
\(103\) −1336.00 −1.27806 −0.639029 − 0.769183i \(-0.720664\pi\)
−0.639029 + 0.769183i \(0.720664\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −100.000 −0.0903492 −0.0451746 − 0.998979i \(-0.514384\pi\)
−0.0451746 + 0.998979i \(0.514384\pi\)
\(108\) 0 0
\(109\) 682.000 0.599300 0.299650 − 0.954049i \(-0.403130\pi\)
0.299650 + 0.954049i \(0.403130\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −370.000 −0.308024 −0.154012 − 0.988069i \(-0.549219\pi\)
−0.154012 + 0.988069i \(0.549219\pi\)
\(114\) 0 0
\(115\) −2400.00 −1.94610
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 605.000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3000.00 −2.14663
\(126\) 0 0
\(127\) −160.000 −0.111793 −0.0558965 − 0.998437i \(-0.517802\pi\)
−0.0558965 + 0.998437i \(0.517802\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −300.000 −0.200085 −0.100042 − 0.994983i \(-0.531898\pi\)
−0.100042 + 0.994983i \(0.531898\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1190.00 −0.742106 −0.371053 − 0.928612i \(-0.621003\pi\)
−0.371053 + 0.928612i \(0.621003\pi\)
\(138\) 0 0
\(139\) −2220.00 −1.35466 −0.677331 − 0.735679i \(-0.736863\pi\)
−0.677331 + 0.735679i \(0.736863\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1936.00 −1.13214
\(144\) 0 0
\(145\) 4360.00 2.49709
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2414.00 −1.32727 −0.663633 − 0.748058i \(-0.730986\pi\)
−0.663633 + 0.748058i \(0.730986\pi\)
\(150\) 0 0
\(151\) 2056.00 1.10805 0.554023 − 0.832501i \(-0.313092\pi\)
0.554023 + 0.832501i \(0.313092\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5600.00 −2.90195
\(156\) 0 0
\(157\) −1060.00 −0.538836 −0.269418 − 0.963023i \(-0.586831\pi\)
−0.269418 + 0.963023i \(0.586831\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1980.00 −0.951445 −0.475723 − 0.879595i \(-0.657813\pi\)
−0.475723 + 0.879595i \(0.657813\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 488.000 0.226123 0.113062 − 0.993588i \(-0.463934\pi\)
0.113062 + 0.993588i \(0.463934\pi\)
\(168\) 0 0
\(169\) −261.000 −0.118798
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3476.00 1.52760 0.763802 − 0.645451i \(-0.223331\pi\)
0.763802 + 0.645451i \(0.223331\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 84.0000 0.0350752 0.0175376 − 0.999846i \(-0.494417\pi\)
0.0175376 + 0.999846i \(0.494417\pi\)
\(180\) 0 0
\(181\) −2180.00 −0.895238 −0.447619 − 0.894224i \(-0.647728\pi\)
−0.447619 + 0.894224i \(0.647728\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 600.000 0.238448
\(186\) 0 0
\(187\) −3168.00 −1.23886
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −696.000 −0.263669 −0.131835 − 0.991272i \(-0.542087\pi\)
−0.131835 + 0.991272i \(0.542087\pi\)
\(192\) 0 0
\(193\) 2690.00 1.00327 0.501633 − 0.865080i \(-0.332733\pi\)
0.501633 + 0.865080i \(0.332733\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5310.00 −1.92042 −0.960208 − 0.279287i \(-0.909902\pi\)
−0.960208 + 0.279287i \(0.909902\pi\)
\(198\) 0 0
\(199\) −3080.00 −1.09716 −0.548581 − 0.836097i \(-0.684832\pi\)
−0.548581 + 0.836097i \(0.684832\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2400.00 −0.817674
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4400.00 1.45624
\(210\) 0 0
\(211\) −4.00000 −0.00130508 −0.000652539 − 1.00000i \(-0.500208\pi\)
−0.000652539 1.00000i \(0.500208\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4400.00 −1.39571
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3168.00 0.964266
\(222\) 0 0
\(223\) 2416.00 0.725504 0.362752 − 0.931886i \(-0.381837\pi\)
0.362752 + 0.931886i \(0.381837\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4972.00 −1.45376 −0.726879 − 0.686765i \(-0.759030\pi\)
−0.726879 + 0.686765i \(0.759030\pi\)
\(228\) 0 0
\(229\) 4460.00 1.28701 0.643505 − 0.765442i \(-0.277480\pi\)
0.643505 + 0.765442i \(0.277480\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4550.00 −1.27932 −0.639658 − 0.768660i \(-0.720924\pi\)
−0.639658 + 0.768660i \(0.720924\pi\)
\(234\) 0 0
\(235\) −1760.00 −0.488552
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2112.00 0.571606 0.285803 − 0.958288i \(-0.407740\pi\)
0.285803 + 0.958288i \(0.407740\pi\)
\(240\) 0 0
\(241\) 4840.00 1.29366 0.646829 − 0.762635i \(-0.276095\pi\)
0.646829 + 0.762635i \(0.276095\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4400.00 −1.13346
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4340.00 −1.09139 −0.545694 − 0.837985i \(-0.683734\pi\)
−0.545694 + 0.837985i \(0.683734\pi\)
\(252\) 0 0
\(253\) −5280.00 −1.31206
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3520.00 −0.854364 −0.427182 − 0.904166i \(-0.640494\pi\)
−0.427182 + 0.904166i \(0.640494\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2560.00 0.600214 0.300107 − 0.953905i \(-0.402978\pi\)
0.300107 + 0.953905i \(0.402978\pi\)
\(264\) 0 0
\(265\) 2200.00 0.509981
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6820.00 1.54581 0.772905 − 0.634522i \(-0.218803\pi\)
0.772905 + 0.634522i \(0.218803\pi\)
\(270\) 0 0
\(271\) −3520.00 −0.789021 −0.394511 − 0.918891i \(-0.629086\pi\)
−0.394511 + 0.918891i \(0.629086\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12100.0 −2.65330
\(276\) 0 0
\(277\) −8690.00 −1.88495 −0.942476 − 0.334275i \(-0.891509\pi\)
−0.942476 + 0.334275i \(0.891509\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3894.00 −0.826678 −0.413339 − 0.910577i \(-0.635638\pi\)
−0.413339 + 0.910577i \(0.635638\pi\)
\(282\) 0 0
\(283\) 6556.00 1.37708 0.688540 − 0.725198i \(-0.258252\pi\)
0.688540 + 0.725198i \(0.258252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2484.00 −0.495279 −0.247640 − 0.968852i \(-0.579655\pi\)
−0.247640 + 0.968852i \(0.579655\pi\)
\(294\) 0 0
\(295\) −11600.0 −2.28942
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5280.00 1.02124
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7600.00 1.42680
\(306\) 0 0
\(307\) −308.000 −0.0572589 −0.0286295 − 0.999590i \(-0.509114\pi\)
−0.0286295 + 0.999590i \(0.509114\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1760.00 0.320902 0.160451 − 0.987044i \(-0.448705\pi\)
0.160451 + 0.987044i \(0.448705\pi\)
\(312\) 0 0
\(313\) −24.0000 −0.00433406 −0.00216703 − 0.999998i \(-0.500690\pi\)
−0.00216703 + 0.999998i \(0.500690\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4890.00 0.866403 0.433202 − 0.901297i \(-0.357384\pi\)
0.433202 + 0.901297i \(0.357384\pi\)
\(318\) 0 0
\(319\) 9592.00 1.68354
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7200.00 −1.24031
\(324\) 0 0
\(325\) 12100.0 2.06519
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8108.00 1.34639 0.673196 − 0.739464i \(-0.264921\pi\)
0.673196 + 0.739464i \(0.264921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 19600.0 3.19660
\(336\) 0 0
\(337\) 2990.00 0.483311 0.241655 − 0.970362i \(-0.422310\pi\)
0.241655 + 0.970362i \(0.422310\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12320.0 −1.95650
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7860.00 1.21599 0.607993 − 0.793943i \(-0.291975\pi\)
0.607993 + 0.793943i \(0.291975\pi\)
\(348\) 0 0
\(349\) 6060.00 0.929468 0.464734 − 0.885450i \(-0.346150\pi\)
0.464734 + 0.885450i \(0.346150\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3024.00 −0.455953 −0.227976 − 0.973667i \(-0.573211\pi\)
−0.227976 + 0.973667i \(0.573211\pi\)
\(354\) 0 0
\(355\) −2240.00 −0.334893
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5896.00 −0.866794 −0.433397 − 0.901203i \(-0.642685\pi\)
−0.433397 + 0.901203i \(0.642685\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12800.0 −1.83557
\(366\) 0 0
\(367\) −2192.00 −0.311775 −0.155888 − 0.987775i \(-0.549824\pi\)
−0.155888 + 0.987775i \(0.549824\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3410.00 −0.473360 −0.236680 − 0.971588i \(-0.576059\pi\)
−0.236680 + 0.971588i \(0.576059\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9592.00 −1.31038
\(378\) 0 0
\(379\) 5916.00 0.801806 0.400903 − 0.916120i \(-0.368696\pi\)
0.400903 + 0.916120i \(0.368696\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2040.00 0.272165 0.136082 − 0.990698i \(-0.456549\pi\)
0.136082 + 0.990698i \(0.456549\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −11778.0 −1.53514 −0.767569 − 0.640967i \(-0.778534\pi\)
−0.767569 + 0.640967i \(0.778534\pi\)
\(390\) 0 0
\(391\) 8640.00 1.11750
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9760.00 1.24324
\(396\) 0 0
\(397\) −4708.00 −0.595183 −0.297592 − 0.954693i \(-0.596183\pi\)
−0.297592 + 0.954693i \(0.596183\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5874.00 0.731505 0.365753 − 0.930712i \(-0.380812\pi\)
0.365753 + 0.930712i \(0.380812\pi\)
\(402\) 0 0
\(403\) 12320.0 1.52284
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1320.00 0.160762
\(408\) 0 0
\(409\) −14040.0 −1.69739 −0.848696 − 0.528881i \(-0.822612\pi\)
−0.848696 + 0.528881i \(0.822612\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −13200.0 −1.56136
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7940.00 −0.925762 −0.462881 − 0.886420i \(-0.653184\pi\)
−0.462881 + 0.886420i \(0.653184\pi\)
\(420\) 0 0
\(421\) 5214.00 0.603598 0.301799 − 0.953372i \(-0.402413\pi\)
0.301799 + 0.953372i \(0.402413\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 19800.0 2.25986
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −10384.0 −1.16051 −0.580255 − 0.814435i \(-0.697047\pi\)
−0.580255 + 0.814435i \(0.697047\pi\)
\(432\) 0 0
\(433\) −6520.00 −0.723629 −0.361814 − 0.932250i \(-0.617843\pi\)
−0.361814 + 0.932250i \(0.617843\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12000.0 −1.31359
\(438\) 0 0
\(439\) 11520.0 1.25244 0.626218 − 0.779648i \(-0.284602\pi\)
0.626218 + 0.779648i \(0.284602\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13380.0 −1.43500 −0.717498 − 0.696561i \(-0.754713\pi\)
−0.717498 + 0.696561i \(0.754713\pi\)
\(444\) 0 0
\(445\) −6400.00 −0.681773
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4098.00 −0.430727 −0.215364 − 0.976534i \(-0.569094\pi\)
−0.215364 + 0.976534i \(0.569094\pi\)
\(450\) 0 0
\(451\) −5280.00 −0.551276
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16390.0 −1.67766 −0.838831 − 0.544391i \(-0.816761\pi\)
−0.838831 + 0.544391i \(0.816761\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9540.00 0.963822 0.481911 − 0.876220i \(-0.339943\pi\)
0.481911 + 0.876220i \(0.339943\pi\)
\(462\) 0 0
\(463\) −8920.00 −0.895351 −0.447676 − 0.894196i \(-0.647748\pi\)
−0.447676 + 0.894196i \(0.647748\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8428.00 −0.835121 −0.417560 − 0.908649i \(-0.637115\pi\)
−0.417560 + 0.908649i \(0.637115\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9680.00 −0.940987
\(474\) 0 0
\(475\) −27500.0 −2.65639
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14520.0 −1.38504 −0.692522 − 0.721397i \(-0.743500\pi\)
−0.692522 + 0.721397i \(0.743500\pi\)
\(480\) 0 0
\(481\) −1320.00 −0.125129
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4960.00 0.464375
\(486\) 0 0
\(487\) −10520.0 −0.978864 −0.489432 − 0.872042i \(-0.662796\pi\)
−0.489432 + 0.872042i \(0.662796\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5436.00 0.499640 0.249820 − 0.968292i \(-0.419629\pi\)
0.249820 + 0.968292i \(0.419629\pi\)
\(492\) 0 0
\(493\) −15696.0 −1.43390
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 18348.0 1.64603 0.823015 − 0.568019i \(-0.192290\pi\)
0.823015 + 0.568019i \(0.192290\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20944.0 −1.85655 −0.928277 − 0.371889i \(-0.878710\pi\)
−0.928277 + 0.371889i \(0.878710\pi\)
\(504\) 0 0
\(505\) −4400.00 −0.387718
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 260.000 0.0226411 0.0113205 − 0.999936i \(-0.496396\pi\)
0.0113205 + 0.999936i \(0.496396\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26720.0 2.28626
\(516\) 0 0
\(517\) −3872.00 −0.329382
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14600.0 1.22771 0.613856 − 0.789418i \(-0.289618\pi\)
0.613856 + 0.789418i \(0.289618\pi\)
\(522\) 0 0
\(523\) −6820.00 −0.570206 −0.285103 − 0.958497i \(-0.592028\pi\)
−0.285103 + 0.958497i \(0.592028\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20160.0 1.66638
\(528\) 0 0
\(529\) 2233.00 0.183529
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5280.00 0.429085
\(534\) 0 0
\(535\) 2000.00 0.161622
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12166.0 0.966834 0.483417 − 0.875390i \(-0.339395\pi\)
0.483417 + 0.875390i \(0.339395\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13640.0 −1.07206
\(546\) 0 0
\(547\) 22660.0 1.77125 0.885623 − 0.464405i \(-0.153732\pi\)
0.885623 + 0.464405i \(0.153732\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21800.0 1.68550
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8490.00 0.645840 0.322920 − 0.946426i \(-0.395336\pi\)
0.322920 + 0.946426i \(0.395336\pi\)
\(558\) 0 0
\(559\) 9680.00 0.732416
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14300.0 1.07047 0.535234 − 0.844704i \(-0.320224\pi\)
0.535234 + 0.844704i \(0.320224\pi\)
\(564\) 0 0
\(565\) 7400.00 0.551009
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18294.0 −1.34785 −0.673923 − 0.738802i \(-0.735392\pi\)
−0.673923 + 0.738802i \(0.735392\pi\)
\(570\) 0 0
\(571\) −14388.0 −1.05450 −0.527250 − 0.849710i \(-0.676777\pi\)
−0.527250 + 0.849710i \(0.676777\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 33000.0 2.39338
\(576\) 0 0
\(577\) −25232.0 −1.82049 −0.910244 − 0.414072i \(-0.864106\pi\)
−0.910244 + 0.414072i \(0.864106\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4840.00 0.343829
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16060.0 1.12925 0.564623 − 0.825349i \(-0.309022\pi\)
0.564623 + 0.825349i \(0.309022\pi\)
\(588\) 0 0
\(589\) −28000.0 −1.95878
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −23856.0 −1.65202 −0.826011 − 0.563655i \(-0.809395\pi\)
−0.826011 + 0.563655i \(0.809395\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3168.00 −0.216095 −0.108048 − 0.994146i \(-0.534460\pi\)
−0.108048 + 0.994146i \(0.534460\pi\)
\(600\) 0 0
\(601\) 12320.0 0.836179 0.418089 − 0.908406i \(-0.362700\pi\)
0.418089 + 0.908406i \(0.362700\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12100.0 −0.813116
\(606\) 0 0
\(607\) 8928.00 0.596996 0.298498 − 0.954410i \(-0.403514\pi\)
0.298498 + 0.954410i \(0.403514\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3872.00 0.256374
\(612\) 0 0
\(613\) −4510.00 −0.297157 −0.148578 − 0.988901i \(-0.547470\pi\)
−0.148578 + 0.988901i \(0.547470\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5830.00 −0.380400 −0.190200 − 0.981745i \(-0.560914\pi\)
−0.190200 + 0.981745i \(0.560914\pi\)
\(618\) 0 0
\(619\) 24460.0 1.58826 0.794128 − 0.607751i \(-0.207928\pi\)
0.794128 + 0.607751i \(0.207928\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25625.0 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2160.00 −0.136923
\(630\) 0 0
\(631\) 15216.0 0.959967 0.479984 − 0.877277i \(-0.340643\pi\)
0.479984 + 0.877277i \(0.340643\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3200.00 0.199981
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −12478.0 −0.768879 −0.384439 − 0.923150i \(-0.625605\pi\)
−0.384439 + 0.923150i \(0.625605\pi\)
\(642\) 0 0
\(643\) −15996.0 −0.981059 −0.490529 − 0.871425i \(-0.663197\pi\)
−0.490529 + 0.871425i \(0.663197\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5880.00 −0.357290 −0.178645 − 0.983914i \(-0.557171\pi\)
−0.178645 + 0.983914i \(0.557171\pi\)
\(648\) 0 0
\(649\) −25520.0 −1.54352
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7510.00 0.450060 0.225030 − 0.974352i \(-0.427752\pi\)
0.225030 + 0.974352i \(0.427752\pi\)
\(654\) 0 0
\(655\) 6000.00 0.357923
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16508.0 0.975812 0.487906 − 0.872896i \(-0.337761\pi\)
0.487906 + 0.872896i \(0.337761\pi\)
\(660\) 0 0
\(661\) 1220.00 0.0717890 0.0358945 − 0.999356i \(-0.488572\pi\)
0.0358945 + 0.999356i \(0.488572\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −26160.0 −1.51862
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16720.0 0.961950
\(672\) 0 0
\(673\) −13090.0 −0.749751 −0.374875 − 0.927075i \(-0.622315\pi\)
−0.374875 + 0.927075i \(0.622315\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15108.0 0.857677 0.428839 − 0.903381i \(-0.358923\pi\)
0.428839 + 0.903381i \(0.358923\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1540.00 −0.0862759 −0.0431380 − 0.999069i \(-0.513736\pi\)
−0.0431380 + 0.999069i \(0.513736\pi\)
\(684\) 0 0
\(685\) 23800.0 1.32752
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4840.00 −0.267619
\(690\) 0 0
\(691\) −5780.00 −0.318208 −0.159104 − 0.987262i \(-0.550860\pi\)
−0.159104 + 0.987262i \(0.550860\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 44400.0 2.42329
\(696\) 0 0
\(697\) 8640.00 0.469531
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10406.0 0.560669 0.280335 − 0.959902i \(-0.409555\pi\)
0.280335 + 0.959902i \(0.409555\pi\)
\(702\) 0 0
\(703\) 3000.00 0.160949
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 28642.0 1.51717 0.758585 − 0.651575i \(-0.225891\pi\)
0.758585 + 0.651575i \(0.225891\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33600.0 1.76484
\(714\) 0 0
\(715\) 38720.0 2.02524
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7560.00 0.392129 0.196064 − 0.980591i \(-0.437184\pi\)
0.196064 + 0.980591i \(0.437184\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −59950.0 −3.07102
\(726\) 0 0
\(727\) 20360.0 1.03867 0.519333 − 0.854572i \(-0.326180\pi\)
0.519333 + 0.854572i \(0.326180\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15840.0 0.801455
\(732\) 0 0
\(733\) −23276.0 −1.17288 −0.586438 − 0.809994i \(-0.699470\pi\)
−0.586438 + 0.809994i \(0.699470\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 43120.0 2.15515
\(738\) 0 0
\(739\) 3172.00 0.157894 0.0789472 − 0.996879i \(-0.474844\pi\)
0.0789472 + 0.996879i \(0.474844\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6600.00 0.325882 0.162941 − 0.986636i \(-0.447902\pi\)
0.162941 + 0.986636i \(0.447902\pi\)
\(744\) 0 0
\(745\) 48280.0 2.37429
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16416.0 −0.797641 −0.398820 − 0.917029i \(-0.630580\pi\)
−0.398820 + 0.917029i \(0.630580\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −41120.0 −1.98213
\(756\) 0 0
\(757\) 36850.0 1.76927 0.884634 − 0.466286i \(-0.154408\pi\)
0.884634 + 0.466286i \(0.154408\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −37400.0 −1.78154 −0.890768 − 0.454458i \(-0.849833\pi\)
−0.890768 + 0.454458i \(0.849833\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25520.0 1.20140
\(768\) 0 0
\(769\) 5720.00 0.268229 0.134115 − 0.990966i \(-0.457181\pi\)
0.134115 + 0.990966i \(0.457181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7356.00 0.342273 0.171136 − 0.985247i \(-0.445256\pi\)
0.171136 + 0.985247i \(0.445256\pi\)
\(774\) 0 0
\(775\) 77000.0 3.56893
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12000.0 −0.551919
\(780\) 0 0
\(781\) −4928.00 −0.225785
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 21200.0 0.963899
\(786\) 0 0
\(787\) 38572.0 1.74707 0.873535 − 0.486762i \(-0.161822\pi\)
0.873535 + 0.486762i \(0.161822\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −16720.0 −0.748732
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7812.00 0.347196 0.173598 − 0.984817i \(-0.444461\pi\)
0.173598 + 0.984817i \(0.444461\pi\)
\(798\) 0 0
\(799\) 6336.00 0.280540
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −28160.0 −1.23754
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −45018.0 −1.95643 −0.978213 − 0.207604i \(-0.933433\pi\)
−0.978213 + 0.207604i \(0.933433\pi\)
\(810\) 0 0
\(811\) −2740.00 −0.118637 −0.0593184 − 0.998239i \(-0.518893\pi\)
−0.0593184 + 0.998239i \(0.518893\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39600.0 1.70200
\(816\) 0 0
\(817\) −22000.0 −0.942084
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1454.00 −0.0618087 −0.0309044 − 0.999522i \(-0.509839\pi\)
−0.0309044 + 0.999522i \(0.509839\pi\)
\(822\) 0 0
\(823\) 37120.0 1.57220 0.786101 − 0.618098i \(-0.212097\pi\)
0.786101 + 0.618098i \(0.212097\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7260.00 0.305266 0.152633 − 0.988283i \(-0.451225\pi\)
0.152633 + 0.988283i \(0.451225\pi\)
\(828\) 0 0
\(829\) 11140.0 0.466717 0.233358 − 0.972391i \(-0.425028\pi\)
0.233358 + 0.972391i \(0.425028\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9760.00 −0.404501
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8760.00 0.360463 0.180232 − 0.983624i \(-0.442315\pi\)
0.180232 + 0.983624i \(0.442315\pi\)
\(840\) 0 0
\(841\) 23135.0 0.948583
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5220.00 0.212513
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3600.00 −0.145013
\(852\) 0 0
\(853\) −820.000 −0.0329147 −0.0164574 − 0.999865i \(-0.505239\pi\)
−0.0164574 + 0.999865i \(0.505239\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1320.00 −0.0526142 −0.0263071 − 0.999654i \(-0.508375\pi\)
−0.0263071 + 0.999654i \(0.508375\pi\)
\(858\) 0 0
\(859\) −42780.0 −1.69923 −0.849613 − 0.527407i \(-0.823164\pi\)
−0.849613 + 0.527407i \(0.823164\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29880.0 −1.17859 −0.589297 − 0.807916i \(-0.700595\pi\)
−0.589297 + 0.807916i \(0.700595\pi\)
\(864\) 0 0
\(865\) −69520.0 −2.73266
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 21472.0 0.838191
\(870\) 0 0
\(871\) −43120.0 −1.67746
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28170.0 1.08464 0.542322 − 0.840171i \(-0.317545\pi\)
0.542322 + 0.840171i \(0.317545\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20800.0 0.795425 0.397713 − 0.917510i \(-0.369804\pi\)
0.397713 + 0.917510i \(0.369804\pi\)
\(882\) 0 0
\(883\) −20900.0 −0.796536 −0.398268 − 0.917269i \(-0.630389\pi\)
−0.398268 + 0.917269i \(0.630389\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11640.0 0.440623 0.220312 − 0.975430i \(-0.429293\pi\)
0.220312 + 0.975430i \(0.429293\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8800.00 −0.329766
\(894\) 0 0
\(895\) −1680.00 −0.0627444
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −61040.0 −2.26451
\(900\) 0 0
\(901\) −7920.00 −0.292845
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 43600.0 1.60145
\(906\) 0 0
\(907\) −37580.0 −1.37577 −0.687885 − 0.725820i \(-0.741461\pi\)
−0.687885 + 0.725820i \(0.741461\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 40832.0 1.48499 0.742494 − 0.669852i \(-0.233643\pi\)
0.742494 + 0.669852i \(0.233643\pi\)
\(912\) 0 0
\(913\) −29040.0 −1.05267
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 42768.0 1.53513 0.767566 − 0.640970i \(-0.221468\pi\)
0.767566 + 0.640970i \(0.221468\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4928.00 0.175739
\(924\) 0 0
\(925\) −8250.00 −0.293252
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −8760.00 −0.309372 −0.154686 − 0.987964i \(-0.549437\pi\)
−0.154686 + 0.987964i \(0.549437\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 63360.0 2.21614
\(936\) 0 0
\(937\) −49632.0 −1.73042 −0.865212 − 0.501407i \(-0.832816\pi\)
−0.865212 + 0.501407i \(0.832816\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −36900.0 −1.27833 −0.639163 − 0.769071i \(-0.720719\pi\)
−0.639163 + 0.769071i \(0.720719\pi\)
\(942\) 0 0
\(943\) 14400.0 0.497273
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33340.0 1.14404 0.572019 − 0.820240i \(-0.306160\pi\)
0.572019 + 0.820240i \(0.306160\pi\)
\(948\) 0 0
\(949\) 28160.0 0.963237
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5610.00 −0.190688 −0.0953440 − 0.995444i \(-0.530395\pi\)
−0.0953440 + 0.995444i \(0.530395\pi\)
\(954\) 0 0
\(955\) 13920.0 0.471666
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 48609.0 1.63167
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −53800.0 −1.79470
\(966\) 0 0
\(967\) −25160.0 −0.836702 −0.418351 − 0.908285i \(-0.637392\pi\)
−0.418351 + 0.908285i \(0.637392\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1060.00 0.0350330 0.0175165 − 0.999847i \(-0.494424\pi\)
0.0175165 + 0.999847i \(0.494424\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38770.0 1.26956 0.634781 − 0.772692i \(-0.281090\pi\)
0.634781 + 0.772692i \(0.281090\pi\)
\(978\) 0 0
\(979\) −14080.0 −0.459651
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −24296.0 −0.788324 −0.394162 − 0.919041i \(-0.628965\pi\)
−0.394162 + 0.919041i \(0.628965\pi\)
\(984\) 0 0
\(985\) 106200. 3.43534
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 26400.0 0.848808
\(990\) 0 0
\(991\) −1672.00 −0.0535952 −0.0267976 − 0.999641i \(-0.508531\pi\)
−0.0267976 + 0.999641i \(0.508531\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 61600.0 1.96266
\(996\) 0 0
\(997\) −45628.0 −1.44940 −0.724701 − 0.689064i \(-0.758022\pi\)
−0.724701 + 0.689064i \(0.758022\pi\)
\(998\) 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 1764.4.a.a.1.1 1
3.2 odd 2 196.4.a.c.1.1 yes 1
7.2 even 3 1764.4.k.p.361.1 2
7.3 odd 6 1764.4.k.a.1549.1 2
7.4 even 3 1764.4.k.p.1549.1 2
7.5 odd 6 1764.4.k.a.361.1 2
7.6 odd 2 1764.4.a.m.1.1 1
12.11 even 2 784.4.a.f.1.1 1
21.2 odd 6 196.4.e.b.165.1 2
21.5 even 6 196.4.e.e.165.1 2
21.11 odd 6 196.4.e.b.177.1 2
21.17 even 6 196.4.e.e.177.1 2
21.20 even 2 196.4.a.a.1.1 ✓ 1
84.83 odd 2 784.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.4.a.a.1.1 ✓ 1 21.20 even 2
196.4.a.c.1.1 yes 1 3.2 odd 2
196.4.e.b.165.1 2 21.2 odd 6
196.4.e.b.177.1 2 21.11 odd 6
196.4.e.e.165.1 2 21.5 even 6
196.4.e.e.177.1 2 21.17 even 6
784.4.a.f.1.1 1 12.11 even 2
784.4.a.m.1.1 1 84.83 odd 2
1764.4.a.a.1.1 1 1.1 even 1 trivial
1764.4.a.m.1.1 1 7.6 odd 2
1764.4.k.a.361.1 2 7.5 odd 6
1764.4.k.a.1549.1 2 7.3 odd 6
1764.4.k.p.361.1 2 7.2 even 3
1764.4.k.p.1549.1 2 7.4 even 3