↑

Properties

Label 1800.4.a.o.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 $1$

Related objects

Downloads

Learn more

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 120)
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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{7} +O(q^{10})\) \(q-4.00000 q^{7} +28.0000 q^{11} +16.0000 q^{13} +108.000 q^{17} +32.0000 q^{19} -28.0000 q^{23} +238.000 q^{29} -180.000 q^{31} +40.0000 q^{37} -422.000 q^{41} -276.000 q^{43} +60.0000 q^{47} -327.000 q^{49} +220.000 q^{53} +804.000 q^{59} -358.000 q^{61} +884.000 q^{67} +64.0000 q^{71} +152.000 q^{73} -112.000 q^{77} -932.000 q^{79} -1292.00 q^{83} +1146.00 q^{89} -64.0000 q^{91} -824.000 q^{97} +O(q^{100})\)

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
\(7\) −4.00000 −0.215980 −0.107990 − 0.994152i \(-0.534441\pi\)
−0.107990 + 0.994152i \(0.534441\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 − 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 16.0000 0.341354 0.170677 − 0.985327i \(-0.445405\pi\)
0.170677 + 0.985327i \(0.445405\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 108.000 1.54081 0.770407 − 0.637552i \(-0.220053\pi\)
0.770407 + 0.637552i \(0.220053\pi\)
\(18\) 0 0
\(19\) 32.0000 0.386384 0.193192 − 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −28.0000 −0.253844 −0.126922 − 0.991913i \(-0.540510\pi\)
−0.126922 + 0.991913i \(0.540510\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 238.000 1.52398 0.761991 − 0.647587i \(-0.224222\pi\)
0.761991 + 0.647587i \(0.224222\pi\)
\(30\) 0 0
\(31\) −180.000 −1.04287 −0.521435 − 0.853291i \(-0.674603\pi\)
−0.521435 + 0.853291i \(0.674603\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 40.0000 0.177729 0.0888643 − 0.996044i \(-0.471676\pi\)
0.0888643 + 0.996044i \(0.471676\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −422.000 −1.60745 −0.803724 − 0.595003i \(-0.797151\pi\)
−0.803724 + 0.595003i \(0.797151\pi\)
\(42\) 0 0
\(43\) −276.000 −0.978828 −0.489414 − 0.872052i \(-0.662789\pi\)
−0.489414 + 0.872052i \(0.662789\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 60.0000 0.186211 0.0931053 − 0.995656i \(-0.470321\pi\)
0.0931053 + 0.995656i \(0.470321\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 220.000 0.570176 0.285088 − 0.958501i \(-0.407977\pi\)
0.285088 + 0.958501i \(0.407977\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 804.000 1.77410 0.887050 − 0.461674i \(-0.152751\pi\)
0.887050 + 0.461674i \(0.152751\pi\)
\(60\) 0 0
\(61\) −358.000 −0.751430 −0.375715 − 0.926735i \(-0.622603\pi\)
−0.375715 + 0.926735i \(0.622603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 884.000 1.61191 0.805954 − 0.591979i \(-0.201653\pi\)
0.805954 + 0.591979i \(0.201653\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 64.0000 0.106978 0.0534888 − 0.998568i \(-0.482966\pi\)
0.0534888 + 0.998568i \(0.482966\pi\)
\(72\) 0 0
\(73\) 152.000 0.243702 0.121851 − 0.992548i \(-0.461117\pi\)
0.121851 + 0.992548i \(0.461117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −112.000 −0.165761
\(78\) 0 0
\(79\) −932.000 −1.32732 −0.663659 − 0.748035i \(-0.730998\pi\)
−0.663659 + 0.748035i \(0.730998\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1292.00 −1.70862 −0.854310 − 0.519764i \(-0.826020\pi\)
−0.854310 + 0.519764i \(0.826020\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1146.00 1.36490 0.682448 − 0.730934i \(-0.260915\pi\)
0.682448 + 0.730934i \(0.260915\pi\)
\(90\) 0 0
\(91\) −64.0000 −0.0737255
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −824.000 −0.862521 −0.431260 − 0.902227i \(-0.641931\pi\)
−0.431260 + 0.902227i \(0.641931\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1290.00 1.27089 0.635445 − 0.772147i \(-0.280817\pi\)
0.635445 + 0.772147i \(0.280817\pi\)
\(102\) 0 0
\(103\) 1604.00 1.53444 0.767218 − 0.641387i \(-0.221641\pi\)
0.767218 + 0.641387i \(0.221641\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 892.000 0.805915 0.402957 − 0.915219i \(-0.367982\pi\)
0.402957 + 0.915219i \(0.367982\pi\)
\(108\) 0 0
\(109\) 966.000 0.848863 0.424431 − 0.905460i \(-0.360474\pi\)
0.424431 + 0.905460i \(0.360474\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1124.00 0.935726 0.467863 − 0.883801i \(-0.345024\pi\)
0.467863 + 0.883801i \(0.345024\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −432.000 −0.332785
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1884.00 1.31636 0.658181 − 0.752860i \(-0.271326\pi\)
0.658181 + 0.752860i \(0.271326\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 588.000 0.392166 0.196083 − 0.980587i \(-0.437178\pi\)
0.196083 + 0.980587i \(0.437178\pi\)
\(132\) 0 0
\(133\) −128.000 −0.0834512
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1060.00 0.661036 0.330518 − 0.943800i \(-0.392777\pi\)
0.330518 + 0.943800i \(0.392777\pi\)
\(138\) 0 0
\(139\) −2864.00 −1.74764 −0.873818 − 0.486254i \(-0.838363\pi\)
−0.873818 + 0.486254i \(0.838363\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 448.000 0.261984
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −342.000 −0.188038 −0.0940192 − 0.995570i \(-0.529972\pi\)
−0.0940192 + 0.995570i \(0.529972\pi\)
\(150\) 0 0
\(151\) 1636.00 0.881694 0.440847 − 0.897582i \(-0.354678\pi\)
0.440847 + 0.897582i \(0.354678\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2072.00 1.05327 0.526636 − 0.850091i \(-0.323453\pi\)
0.526636 + 0.850091i \(0.323453\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 112.000 0.0548251
\(162\) 0 0
\(163\) −772.000 −0.370968 −0.185484 − 0.982647i \(-0.559385\pi\)
−0.185484 + 0.982647i \(0.559385\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1044.00 −0.483755 −0.241878 − 0.970307i \(-0.577763\pi\)
−0.241878 + 0.970307i \(0.577763\pi\)
\(168\) 0 0
\(169\) −1941.00 −0.883477
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4404.00 1.93543 0.967717 − 0.252041i \(-0.0811018\pi\)
0.967717 + 0.252041i \(0.0811018\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3452.00 −1.44142 −0.720711 − 0.693235i \(-0.756185\pi\)
−0.720711 + 0.693235i \(0.756185\pi\)
\(180\) 0 0
\(181\) 526.000 0.216007 0.108004 − 0.994151i \(-0.465554\pi\)
0.108004 + 0.994151i \(0.465554\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3024.00 1.18255
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −72.0000 −0.0272761 −0.0136381 − 0.999907i \(-0.504341\pi\)
−0.0136381 + 0.999907i \(0.504341\pi\)
\(192\) 0 0
\(193\) −208.000 −0.0775760 −0.0387880 − 0.999247i \(-0.512350\pi\)
−0.0387880 + 0.999247i \(0.512350\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 372.000 0.134538 0.0672688 − 0.997735i \(-0.478571\pi\)
0.0672688 + 0.997735i \(0.478571\pi\)
\(198\) 0 0
\(199\) 4348.00 1.54885 0.774426 − 0.632665i \(-0.218039\pi\)
0.774426 + 0.632665i \(0.218039\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −952.000 −0.329149
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 896.000 0.296544
\(210\) 0 0
\(211\) −416.000 −0.135728 −0.0678640 − 0.997695i \(-0.521618\pi\)
−0.0678640 + 0.997695i \(0.521618\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 720.000 0.225239
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1728.00 0.525963
\(222\) 0 0
\(223\) 5748.00 1.72607 0.863037 − 0.505141i \(-0.168559\pi\)
0.863037 + 0.505141i \(0.168559\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1148.00 0.335663 0.167831 − 0.985816i \(-0.446324\pi\)
0.167831 + 0.985816i \(0.446324\pi\)
\(228\) 0 0
\(229\) 3234.00 0.933226 0.466613 − 0.884462i \(-0.345474\pi\)
0.466613 + 0.884462i \(0.345474\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −228.000 −0.0641063 −0.0320532 − 0.999486i \(-0.510205\pi\)
−0.0320532 + 0.999486i \(0.510205\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4760.00 −1.28828 −0.644140 − 0.764908i \(-0.722784\pi\)
−0.644140 + 0.764908i \(0.722784\pi\)
\(240\) 0 0
\(241\) 3230.00 0.863330 0.431665 − 0.902034i \(-0.357926\pi\)
0.431665 + 0.902034i \(0.357926\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 512.000 0.131894
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1708.00 −0.429514 −0.214757 − 0.976668i \(-0.568896\pi\)
−0.214757 + 0.976668i \(0.568896\pi\)
\(252\) 0 0
\(253\) −784.000 −0.194821
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6372.00 1.54659 0.773297 − 0.634044i \(-0.218606\pi\)
0.773297 + 0.634044i \(0.218606\pi\)
\(258\) 0 0
\(259\) −160.000 −0.0383858
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3036.00 −0.711817 −0.355908 − 0.934521i \(-0.615828\pi\)
−0.355908 + 0.934521i \(0.615828\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 114.000 0.0258390 0.0129195 − 0.999917i \(-0.495887\pi\)
0.0129195 + 0.999917i \(0.495887\pi\)
\(270\) 0 0
\(271\) −5236.00 −1.17367 −0.586835 − 0.809707i \(-0.699626\pi\)
−0.586835 + 0.809707i \(0.699626\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5712.00 1.23899 0.619496 − 0.785000i \(-0.287337\pi\)
0.619496 + 0.785000i \(0.287337\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3222.00 −0.684016 −0.342008 − 0.939697i \(-0.611107\pi\)
−0.342008 + 0.939697i \(0.611107\pi\)
\(282\) 0 0
\(283\) 4620.00 0.970426 0.485213 − 0.874396i \(-0.338742\pi\)
0.485213 + 0.874396i \(0.338742\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1688.00 0.347176
\(288\) 0 0
\(289\) 6751.00 1.37411
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5404.00 −1.07749 −0.538746 − 0.842468i \(-0.681102\pi\)
−0.538746 + 0.842468i \(0.681102\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −448.000 −0.0866505
\(300\) 0 0
\(301\) 1104.00 0.211407
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9700.00 1.80328 0.901642 − 0.432483i \(-0.142362\pi\)
0.901642 + 0.432483i \(0.142362\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9672.00 −1.76350 −0.881750 − 0.471716i \(-0.843635\pi\)
−0.881750 + 0.471716i \(0.843635\pi\)
\(312\) 0 0
\(313\) −4048.00 −0.731011 −0.365506 − 0.930809i \(-0.619104\pi\)
−0.365506 + 0.930809i \(0.619104\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −84.0000 −0.0148830 −0.00744150 − 0.999972i \(-0.502369\pi\)
−0.00744150 + 0.999972i \(0.502369\pi\)
\(318\) 0 0
\(319\) 6664.00 1.16963
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3456.00 0.595347
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −240.000 −0.0402177
\(330\) 0 0
\(331\) 5416.00 0.899366 0.449683 − 0.893188i \(-0.351537\pi\)
0.449683 + 0.893188i \(0.351537\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8216.00 1.32805 0.664027 − 0.747709i \(-0.268846\pi\)
0.664027 + 0.747709i \(0.268846\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5040.00 −0.800385
\(342\) 0 0
\(343\) 2680.00 0.421885
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3836.00 0.593450 0.296725 − 0.954963i \(-0.404105\pi\)
0.296725 + 0.954963i \(0.404105\pi\)
\(348\) 0 0
\(349\) −2038.00 −0.312583 −0.156292 − 0.987711i \(-0.549954\pi\)
−0.156292 + 0.987711i \(0.549954\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5292.00 0.797917 0.398959 − 0.916969i \(-0.369372\pi\)
0.398959 + 0.916969i \(0.369372\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3896.00 0.572766 0.286383 − 0.958115i \(-0.407547\pi\)
0.286383 + 0.958115i \(0.407547\pi\)
\(360\) 0 0
\(361\) −5835.00 −0.850707
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −7652.00 −1.08837 −0.544184 − 0.838966i \(-0.683161\pi\)
−0.544184 + 0.838966i \(0.683161\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −880.000 −0.123146
\(372\) 0 0
\(373\) 1576.00 0.218773 0.109386 − 0.993999i \(-0.465111\pi\)
0.109386 + 0.993999i \(0.465111\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3808.00 0.520217
\(378\) 0 0
\(379\) −5416.00 −0.734040 −0.367020 − 0.930213i \(-0.619622\pi\)
−0.367020 + 0.930213i \(0.619622\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8292.00 1.10627 0.553135 − 0.833092i \(-0.313431\pi\)
0.553135 + 0.833092i \(0.313431\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9642.00 1.25673 0.628366 − 0.777918i \(-0.283724\pi\)
0.628366 + 0.777918i \(0.283724\pi\)
\(390\) 0 0
\(391\) −3024.00 −0.391126
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13032.0 1.64750 0.823750 − 0.566954i \(-0.191878\pi\)
0.823750 + 0.566954i \(0.191878\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13358.0 1.66351 0.831754 − 0.555144i \(-0.187337\pi\)
0.831754 + 0.555144i \(0.187337\pi\)
\(402\) 0 0
\(403\) −2880.00 −0.355988
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1120.00 0.136404
\(408\) 0 0
\(409\) 6410.00 0.774949 0.387474 − 0.921880i \(-0.373348\pi\)
0.387474 + 0.921880i \(0.373348\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3216.00 −0.383170
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7644.00 −0.891250 −0.445625 − 0.895220i \(-0.647019\pi\)
−0.445625 + 0.895220i \(0.647019\pi\)
\(420\) 0 0
\(421\) 14674.0 1.69873 0.849367 − 0.527803i \(-0.176984\pi\)
0.849367 + 0.527803i \(0.176984\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1432.00 0.162294
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9704.00 1.08451 0.542257 − 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(432\) 0 0
\(433\) −1296.00 −0.143838 −0.0719189 − 0.997410i \(-0.522912\pi\)
−0.0719189 + 0.997410i \(0.522912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −896.000 −0.0980812
\(438\) 0 0
\(439\) −15684.0 −1.70514 −0.852570 − 0.522613i \(-0.824957\pi\)
−0.852570 + 0.522613i \(0.824957\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5772.00 −0.619043 −0.309521 − 0.950892i \(-0.600169\pi\)
−0.309521 + 0.950892i \(0.600169\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4782.00 0.502620 0.251310 − 0.967907i \(-0.419139\pi\)
0.251310 + 0.967907i \(0.419139\pi\)
\(450\) 0 0
\(451\) −11816.0 −1.23369
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15000.0 −1.53538 −0.767692 − 0.640819i \(-0.778595\pi\)
−0.767692 + 0.640819i \(0.778595\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3762.00 0.380073 0.190037 − 0.981777i \(-0.439139\pi\)
0.190037 + 0.981777i \(0.439139\pi\)
\(462\) 0 0
\(463\) −5036.00 −0.505492 −0.252746 − 0.967533i \(-0.581334\pi\)
−0.252746 + 0.967533i \(0.581334\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2268.00 0.224733 0.112367 − 0.993667i \(-0.464157\pi\)
0.112367 + 0.993667i \(0.464157\pi\)
\(468\) 0 0
\(469\) −3536.00 −0.348139
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7728.00 −0.751234
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16208.0 −1.54606 −0.773030 − 0.634370i \(-0.781260\pi\)
−0.773030 + 0.634370i \(0.781260\pi\)
\(480\) 0 0
\(481\) 640.000 0.0606684
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −11572.0 −1.07675 −0.538375 − 0.842705i \(-0.680962\pi\)
−0.538375 + 0.842705i \(0.680962\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5636.00 −0.518023 −0.259011 − 0.965874i \(-0.583397\pi\)
−0.259011 + 0.965874i \(0.583397\pi\)
\(492\) 0 0
\(493\) 25704.0 2.34817
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −256.000 −0.0231050
\(498\) 0 0
\(499\) −5560.00 −0.498797 −0.249399 − 0.968401i \(-0.580233\pi\)
−0.249399 + 0.968401i \(0.580233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15172.0 1.34490 0.672451 − 0.740141i \(-0.265241\pi\)
0.672451 + 0.740141i \(0.265241\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17342.0 1.51016 0.755079 − 0.655634i \(-0.227598\pi\)
0.755079 + 0.655634i \(0.227598\pi\)
\(510\) 0 0
\(511\) −608.000 −0.0526347
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1680.00 0.142914
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4886.00 0.410863 0.205431 − 0.978672i \(-0.434140\pi\)
0.205431 + 0.978672i \(0.434140\pi\)
\(522\) 0 0
\(523\) −18548.0 −1.55076 −0.775380 − 0.631495i \(-0.782442\pi\)
−0.775380 + 0.631495i \(0.782442\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19440.0 −1.60687
\(528\) 0 0
\(529\) −11383.0 −0.935563
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6752.00 −0.548708
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −9156.00 −0.731682
\(540\) 0 0
\(541\) −15770.0 −1.25324 −0.626622 − 0.779323i \(-0.715563\pi\)
−0.626622 + 0.779323i \(0.715563\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7700.00 0.601880 0.300940 − 0.953643i \(-0.402700\pi\)
0.300940 + 0.953643i \(0.402700\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7616.00 0.588843
\(552\) 0 0
\(553\) 3728.00 0.286674
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19236.0 −1.46330 −0.731648 − 0.681683i \(-0.761248\pi\)
−0.731648 + 0.681683i \(0.761248\pi\)
\(558\) 0 0
\(559\) −4416.00 −0.334127
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8388.00 0.627908 0.313954 − 0.949438i \(-0.398346\pi\)
0.313954 + 0.949438i \(0.398346\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16758.0 1.23468 0.617339 − 0.786697i \(-0.288211\pi\)
0.617339 + 0.786697i \(0.288211\pi\)
\(570\) 0 0
\(571\) 8056.00 0.590426 0.295213 − 0.955432i \(-0.404609\pi\)
0.295213 + 0.955432i \(0.404609\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5728.00 −0.413275 −0.206638 − 0.978418i \(-0.566252\pi\)
−0.206638 + 0.978418i \(0.566252\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5168.00 0.369027
\(582\) 0 0
\(583\) 6160.00 0.437601
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12172.0 0.855864 0.427932 − 0.903811i \(-0.359242\pi\)
0.427932 + 0.903811i \(0.359242\pi\)
\(588\) 0 0
\(589\) −5760.00 −0.402948
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10708.0 0.741526 0.370763 − 0.928728i \(-0.379096\pi\)
0.370763 + 0.928728i \(0.379096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9416.00 0.642283 0.321141 − 0.947031i \(-0.395934\pi\)
0.321141 + 0.947031i \(0.395934\pi\)
\(600\) 0 0
\(601\) 9270.00 0.629170 0.314585 − 0.949229i \(-0.398135\pi\)
0.314585 + 0.949229i \(0.398135\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7996.00 0.534675 0.267337 − 0.963603i \(-0.413856\pi\)
0.267337 + 0.963603i \(0.413856\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 960.000 0.0635637
\(612\) 0 0
\(613\) 232.000 0.0152861 0.00764306 − 0.999971i \(-0.497567\pi\)
0.00764306 + 0.999971i \(0.497567\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3740.00 0.244030 0.122015 − 0.992528i \(-0.461064\pi\)
0.122015 + 0.992528i \(0.461064\pi\)
\(618\) 0 0
\(619\) −26000.0 −1.68825 −0.844126 − 0.536145i \(-0.819880\pi\)
−0.844126 + 0.536145i \(0.819880\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4584.00 −0.294790
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4320.00 0.273847
\(630\) 0 0
\(631\) 11660.0 0.735622 0.367811 − 0.929901i \(-0.380107\pi\)
0.367811 + 0.929901i \(0.380107\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5232.00 −0.325431
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7602.00 0.468426 0.234213 − 0.972185i \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(642\) 0 0
\(643\) −29268.0 −1.79505 −0.897525 − 0.440963i \(-0.854637\pi\)
−0.897525 + 0.440963i \(0.854637\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17836.0 1.08378 0.541890 − 0.840449i \(-0.317709\pi\)
0.541890 + 0.840449i \(0.317709\pi\)
\(648\) 0 0
\(649\) 22512.0 1.36159
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19188.0 −1.14990 −0.574950 − 0.818189i \(-0.694978\pi\)
−0.574950 + 0.818189i \(0.694978\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13860.0 −0.819285 −0.409643 − 0.912246i \(-0.634347\pi\)
−0.409643 + 0.912246i \(0.634347\pi\)
\(660\) 0 0
\(661\) −16558.0 −0.974329 −0.487165 − 0.873310i \(-0.661969\pi\)
−0.487165 + 0.873310i \(0.661969\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6664.00 −0.386853
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10024.0 −0.576710
\(672\) 0 0
\(673\) 4640.00 0.265764 0.132882 − 0.991132i \(-0.457577\pi\)
0.132882 + 0.991132i \(0.457577\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34260.0 −1.94493 −0.972466 − 0.233045i \(-0.925131\pi\)
−0.972466 + 0.233045i \(0.925131\pi\)
\(678\) 0 0
\(679\) 3296.00 0.186287
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19420.0 −1.08797 −0.543987 − 0.839094i \(-0.683086\pi\)
−0.543987 + 0.839094i \(0.683086\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3520.00 0.194632
\(690\) 0 0
\(691\) 4608.00 0.253685 0.126843 − 0.991923i \(-0.459516\pi\)
0.126843 + 0.991923i \(0.459516\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −45576.0 −2.47678
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2318.00 0.124893 0.0624463 − 0.998048i \(-0.480110\pi\)
0.0624463 + 0.998048i \(0.480110\pi\)
\(702\) 0 0
\(703\) 1280.00 0.0686716
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5160.00 −0.274486
\(708\) 0 0
\(709\) 16834.0 0.891698 0.445849 − 0.895108i \(-0.352902\pi\)
0.445849 + 0.895108i \(0.352902\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5040.00 0.264726
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7400.00 0.383830 0.191915 − 0.981412i \(-0.438530\pi\)
0.191915 + 0.981412i \(0.438530\pi\)
\(720\) 0 0
\(721\) −6416.00 −0.331407
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −20340.0 −1.03765 −0.518823 − 0.854882i \(-0.673630\pi\)
−0.518823 + 0.854882i \(0.673630\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −29808.0 −1.50819
\(732\) 0 0
\(733\) −4896.00 −0.246709 −0.123355 − 0.992363i \(-0.539365\pi\)
−0.123355 + 0.992363i \(0.539365\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24752.0 1.23711
\(738\) 0 0
\(739\) 26040.0 1.29621 0.648103 − 0.761552i \(-0.275562\pi\)
0.648103 + 0.761552i \(0.275562\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −6780.00 −0.334770 −0.167385 − 0.985892i \(-0.553532\pi\)
−0.167385 + 0.985892i \(0.553532\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3568.00 −0.174061
\(750\) 0 0
\(751\) −20692.0 −1.00541 −0.502704 − 0.864458i \(-0.667662\pi\)
−0.502704 + 0.864458i \(0.667662\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10816.0 0.519305 0.259653 − 0.965702i \(-0.416392\pi\)
0.259653 + 0.965702i \(0.416392\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13978.0 −0.665837 −0.332919 − 0.942956i \(-0.608033\pi\)
−0.332919 + 0.942956i \(0.608033\pi\)
\(762\) 0 0
\(763\) −3864.00 −0.183337
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12864.0 0.605596
\(768\) 0 0
\(769\) 2926.00 0.137210 0.0686048 − 0.997644i \(-0.478145\pi\)
0.0686048 + 0.997644i \(0.478145\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13916.0 0.647508 0.323754 − 0.946141i \(-0.395055\pi\)
0.323754 + 0.946141i \(0.395055\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13504.0 −0.621092
\(780\) 0 0
\(781\) 1792.00 0.0821035
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −29996.0 −1.35863 −0.679315 − 0.733847i \(-0.737723\pi\)
−0.679315 + 0.733847i \(0.737723\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4496.00 −0.202098
\(792\) 0 0
\(793\) −5728.00 −0.256503
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8940.00 −0.397329 −0.198664 − 0.980068i \(-0.563660\pi\)
−0.198664 + 0.980068i \(0.563660\pi\)
\(798\) 0 0
\(799\) 6480.00 0.286916
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4256.00 0.187037
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10698.0 0.464922 0.232461 − 0.972606i \(-0.425322\pi\)
0.232461 + 0.972606i \(0.425322\pi\)
\(810\) 0 0
\(811\) −6408.00 −0.277454 −0.138727 − 0.990331i \(-0.544301\pi\)
−0.138727 + 0.990331i \(0.544301\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8832.00 −0.378204
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23130.0 −0.983243 −0.491622 − 0.870809i \(-0.663596\pi\)
−0.491622 + 0.870809i \(0.663596\pi\)
\(822\) 0 0
\(823\) −11852.0 −0.501986 −0.250993 − 0.967989i \(-0.580757\pi\)
−0.250993 + 0.967989i \(0.580757\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32628.0 −1.37193 −0.685965 − 0.727634i \(-0.740620\pi\)
−0.685965 + 0.727634i \(0.740620\pi\)
\(828\) 0 0
\(829\) 36694.0 1.53732 0.768658 − 0.639660i \(-0.220925\pi\)
0.768658 + 0.639660i \(0.220925\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −35316.0 −1.46894
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1704.00 0.0701175 0.0350588 − 0.999385i \(-0.488838\pi\)
0.0350588 + 0.999385i \(0.488838\pi\)
\(840\) 0 0
\(841\) 32255.0 1.32252
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2188.00 0.0887610
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1120.00 −0.0451153
\(852\) 0 0
\(853\) −31880.0 −1.27966 −0.639830 − 0.768516i \(-0.720995\pi\)
−0.639830 + 0.768516i \(0.720995\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7972.00 −0.317758 −0.158879 − 0.987298i \(-0.550788\pi\)
−0.158879 + 0.987298i \(0.550788\pi\)
\(858\) 0 0
\(859\) −6008.00 −0.238638 −0.119319 − 0.992856i \(-0.538071\pi\)
−0.119319 + 0.992856i \(0.538071\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1716.00 0.0676863 0.0338432 − 0.999427i \(-0.489225\pi\)
0.0338432 + 0.999427i \(0.489225\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −26096.0 −1.01870
\(870\) 0 0
\(871\) 14144.0 0.550231
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9032.00 0.347764 0.173882 − 0.984767i \(-0.444369\pi\)
0.173882 + 0.984767i \(0.444369\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27838.0 −1.06457 −0.532285 − 0.846565i \(-0.678666\pi\)
−0.532285 + 0.846565i \(0.678666\pi\)
\(882\) 0 0
\(883\) 4316.00 0.164490 0.0822452 − 0.996612i \(-0.473791\pi\)
0.0822452 + 0.996612i \(0.473791\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43524.0 −1.64757 −0.823784 − 0.566904i \(-0.808141\pi\)
−0.823784 + 0.566904i \(0.808141\pi\)
\(888\) 0 0
\(889\) −7536.00 −0.284307
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1920.00 0.0719489
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −42840.0 −1.58931
\(900\) 0 0
\(901\) 23760.0 0.878535
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −10556.0 −0.386446 −0.193223 − 0.981155i \(-0.561894\pi\)
−0.193223 + 0.981155i \(0.561894\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 47472.0 1.72647 0.863237 − 0.504799i \(-0.168433\pi\)
0.863237 + 0.504799i \(0.168433\pi\)
\(912\) 0 0
\(913\) −36176.0 −1.31134
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2352.00 −0.0847000
\(918\) 0 0
\(919\) 11964.0 0.429441 0.214720 − 0.976676i \(-0.431116\pi\)
0.214720 + 0.976676i \(0.431116\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1024.00 0.0365172
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15214.0 0.537304 0.268652 − 0.963237i \(-0.413422\pi\)
0.268652 + 0.963237i \(0.413422\pi\)
\(930\) 0 0
\(931\) −10464.0 −0.368361
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −39712.0 −1.38456 −0.692281 − 0.721628i \(-0.743394\pi\)
−0.692281 + 0.721628i \(0.743394\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36034.0 1.24833 0.624163 − 0.781294i \(-0.285440\pi\)
0.624163 + 0.781294i \(0.285440\pi\)
\(942\) 0 0
\(943\) 11816.0 0.408040
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2532.00 −0.0868838 −0.0434419 − 0.999056i \(-0.513832\pi\)
−0.0434419 + 0.999056i \(0.513832\pi\)
\(948\) 0 0
\(949\) 2432.00 0.0831887
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 55284.0 1.87914 0.939572 − 0.342351i \(-0.111223\pi\)
0.939572 + 0.342351i \(0.111223\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4240.00 −0.142770
\(960\) 0 0
\(961\) 2609.00 0.0875768
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −4372.00 −0.145392 −0.0726960 − 0.997354i \(-0.523160\pi\)
−0.0726960 + 0.997354i \(0.523160\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24300.0 −0.803114 −0.401557 − 0.915834i \(-0.631531\pi\)
−0.401557 + 0.915834i \(0.631531\pi\)
\(972\) 0 0
\(973\) 11456.0 0.377454
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46204.0 1.51300 0.756498 − 0.653996i \(-0.226909\pi\)
0.756498 + 0.653996i \(0.226909\pi\)
\(978\) 0 0
\(979\) 32088.0 1.04754
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −25468.0 −0.826351 −0.413176 − 0.910651i \(-0.635580\pi\)
−0.413176 + 0.910651i \(0.635580\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7728.00 0.248469
\(990\) 0 0
\(991\) 11668.0 0.374012 0.187006 − 0.982359i \(-0.440122\pi\)
0.187006 + 0.982359i \(0.440122\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 7224.00 0.229475 0.114737 − 0.993396i \(-0.463397\pi\)
0.114737 + 0.993396i \(0.463397\pi\)
\(998\) 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.o.1.1 1
3.2 odd 2 600.4.a.k.1.1 1
5.2 odd 4 360.4.f.a.289.2 2
5.3 odd 4 360.4.f.a.289.1 2
5.4 even 2 1800.4.a.u.1.1 1
12.11 even 2 1200.4.a.l.1.1 1
15.2 even 4 120.4.f.c.49.1 ✓ 2
15.8 even 4 120.4.f.c.49.2 yes 2
15.14 odd 2 600.4.a.f.1.1 1
20.3 even 4 720.4.f.b.289.1 2
20.7 even 4 720.4.f.b.289.2 2
60.23 odd 4 240.4.f.e.49.1 2
60.47 odd 4 240.4.f.e.49.2 2
60.59 even 2 1200.4.a.z.1.1 1
120.53 even 4 960.4.f.b.769.1 2
120.77 even 4 960.4.f.b.769.2 2
120.83 odd 4 960.4.f.a.769.2 2
120.107 odd 4 960.4.f.a.769.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.c.49.1 ✓ 2 15.2 even 4
120.4.f.c.49.2 yes 2 15.8 even 4
240.4.f.e.49.1 2 60.23 odd 4
240.4.f.e.49.2 2 60.47 odd 4
360.4.f.a.289.1 2 5.3 odd 4
360.4.f.a.289.2 2 5.2 odd 4
600.4.a.f.1.1 1 15.14 odd 2
600.4.a.k.1.1 1 3.2 odd 2
720.4.f.b.289.1 2 20.3 even 4
720.4.f.b.289.2 2 20.7 even 4
960.4.f.a.769.1 2 120.107 odd 4
960.4.f.a.769.2 2 120.83 odd 4
960.4.f.b.769.1 2 120.53 even 4
960.4.f.b.769.2 2 120.77 even 4
1200.4.a.l.1.1 1 12.11 even 2
1200.4.a.z.1.1 1 60.59 even 2
1800.4.a.o.1.1 1 1.1 even 1 trivial
1800.4.a.u.1.1 1 5.4 even 2