Properties

Label 1200.4.a.l.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, 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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +4.00000 q^{7} +9.00000 q^{9} +28.0000 q^{11} +16.0000 q^{13} -108.000 q^{17} -32.0000 q^{19} -12.0000 q^{21} -28.0000 q^{23} -27.0000 q^{27} -238.000 q^{29} +180.000 q^{31} -84.0000 q^{33} +40.0000 q^{37} -48.0000 q^{39} +422.000 q^{41} +276.000 q^{43} +60.0000 q^{47} -327.000 q^{49} +324.000 q^{51} -220.000 q^{53} +96.0000 q^{57} +804.000 q^{59} -358.000 q^{61} +36.0000 q^{63} -884.000 q^{67} +84.0000 q^{69} +64.0000 q^{71} +152.000 q^{73} +112.000 q^{77} +932.000 q^{79} +81.0000 q^{81} -1292.00 q^{83} +714.000 q^{87} -1146.00 q^{89} +64.0000 q^{91} -540.000 q^{93} -824.000 q^{97} +252.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 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 \(-0.779947\pi\)
−0.770407 + 0.637552i \(0.779947\pi\)
\(18\) 0 0
\(19\) −32.0000 −0.386384 −0.193192 0.981161i \(-0.561884\pi\)
−0.193192 + 0.981161i \(0.561884\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 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
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −238.000 −1.52398 −0.761991 0.647587i \(-0.775778\pi\)
−0.761991 + 0.647587i \(0.775778\pi\)
\(30\) 0 0
\(31\) 180.000 1.04287 0.521435 0.853291i \(-0.325397\pi\)
0.521435 + 0.853291i \(0.325397\pi\)
\(32\) 0 0
\(33\) −84.0000 −0.443107
\(34\) 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
\(39\) −48.0000 −0.197081
\(40\) 0 0
\(41\) 422.000 1.60745 0.803724 0.595003i \(-0.202849\pi\)
0.803724 + 0.595003i \(0.202849\pi\)
\(42\) 0 0
\(43\) 276.000 0.978828 0.489414 0.872052i \(-0.337211\pi\)
0.489414 + 0.872052i \(0.337211\pi\)
\(44\) 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
\(51\) 324.000 0.889590
\(52\) 0 0
\(53\) −220.000 −0.570176 −0.285088 0.958501i \(-0.592023\pi\)
−0.285088 + 0.958501i \(0.592023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 96.0000 0.223079
\(58\) 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
\(63\) 36.0000 0.0719932
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −884.000 −1.61191 −0.805954 0.591979i \(-0.798347\pi\)
−0.805954 + 0.591979i \(0.798347\pi\)
\(68\) 0 0
\(69\) 84.0000 0.146557
\(70\) 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 \(-0.269002\pi\)
0.663659 + 0.748035i \(0.269002\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 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
\(87\) 714.000 0.879872
\(88\) 0 0
\(89\) −1146.00 −1.36490 −0.682448 0.730934i \(-0.739085\pi\)
−0.682448 + 0.730934i \(0.739085\pi\)
\(90\) 0 0
\(91\) 64.0000 0.0737255
\(92\) 0 0
\(93\) −540.000 −0.602101
\(94\) 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
\(99\) 252.000 0.255828
\(100\) 0 0
\(101\) −1290.00 −1.27089 −0.635445 0.772147i \(-0.719183\pi\)
−0.635445 + 0.772147i \(0.719183\pi\)
\(102\) 0 0
\(103\) −1604.00 −1.53444 −0.767218 0.641387i \(-0.778359\pi\)
−0.767218 + 0.641387i \(0.778359\pi\)
\(104\) 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
\(111\) −120.000 −0.102612
\(112\) 0 0
\(113\) −1124.00 −0.935726 −0.467863 0.883801i \(-0.654976\pi\)
−0.467863 + 0.883801i \(0.654976\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 144.000 0.113785
\(118\) 0 0
\(119\) −432.000 −0.332785
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) −1266.00 −0.928060
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1884.00 −1.31636 −0.658181 0.752860i \(-0.728674\pi\)
−0.658181 + 0.752860i \(0.728674\pi\)
\(128\) 0 0
\(129\) −828.000 −0.565127
\(130\) 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 \(-0.607223\pi\)
−0.330518 + 0.943800i \(0.607223\pi\)
\(138\) 0 0
\(139\) 2864.00 1.74764 0.873818 0.486254i \(-0.161637\pi\)
0.873818 + 0.486254i \(0.161637\pi\)
\(140\) 0 0
\(141\) −180.000 −0.107509
\(142\) 0 0
\(143\) 448.000 0.261984
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 981.000 0.550418
\(148\) 0 0
\(149\) 342.000 0.188038 0.0940192 0.995570i \(-0.470028\pi\)
0.0940192 + 0.995570i \(0.470028\pi\)
\(150\) 0 0
\(151\) −1636.00 −0.881694 −0.440847 0.897582i \(-0.645322\pi\)
−0.440847 + 0.897582i \(0.645322\pi\)
\(152\) 0 0
\(153\) −972.000 −0.513605
\(154\) 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
\(159\) 660.000 0.329191
\(160\) 0 0
\(161\) −112.000 −0.0548251
\(162\) 0 0
\(163\) 772.000 0.370968 0.185484 0.982647i \(-0.440615\pi\)
0.185484 + 0.982647i \(0.440615\pi\)
\(164\) 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
\(171\) −288.000 −0.128795
\(172\) 0 0
\(173\) −4404.00 −1.93543 −0.967717 0.252041i \(-0.918898\pi\)
−0.967717 + 0.252041i \(0.918898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2412.00 −1.02428
\(178\) 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
\(183\) 1074.00 0.433838
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3024.00 −1.18255
\(188\) 0 0
\(189\) −108.000 −0.0415653
\(190\) 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 \(-0.521429\pi\)
−0.0672688 + 0.997735i \(0.521429\pi\)
\(198\) 0 0
\(199\) −4348.00 −1.54885 −0.774426 0.632665i \(-0.781961\pi\)
−0.774426 + 0.632665i \(0.781961\pi\)
\(200\) 0 0
\(201\) 2652.00 0.930635
\(202\) 0 0
\(203\) −952.000 −0.329149
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −252.000 −0.0846145
\(208\) 0 0
\(209\) −896.000 −0.296544
\(210\) 0 0
\(211\) 416.000 0.135728 0.0678640 0.997695i \(-0.478382\pi\)
0.0678640 + 0.997695i \(0.478382\pi\)
\(212\) 0 0
\(213\) −192.000 −0.0617635
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 720.000 0.225239
\(218\) 0 0
\(219\) −456.000 −0.140701
\(220\) 0 0
\(221\) −1728.00 −0.525963
\(222\) 0 0
\(223\) −5748.00 −1.72607 −0.863037 0.505141i \(-0.831441\pi\)
−0.863037 + 0.505141i \(0.831441\pi\)
\(224\) 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
\(231\) −336.000 −0.0957021
\(232\) 0 0
\(233\) 228.000 0.0641063 0.0320532 0.999486i \(-0.489795\pi\)
0.0320532 + 0.999486i \(0.489795\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2796.00 −0.766328
\(238\) 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
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −512.000 −0.131894
\(248\) 0 0
\(249\) 3876.00 0.986472
\(250\) 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 \(-0.781394\pi\)
−0.773297 + 0.634044i \(0.781394\pi\)
\(258\) 0 0
\(259\) 160.000 0.0383858
\(260\) 0 0
\(261\) −2142.00 −0.507994
\(262\) 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
\(267\) 3438.00 0.788023
\(268\) 0 0
\(269\) −114.000 −0.0258390 −0.0129195 0.999917i \(-0.504113\pi\)
−0.0129195 + 0.999917i \(0.504113\pi\)
\(270\) 0 0
\(271\) 5236.00 1.17367 0.586835 0.809707i \(-0.300374\pi\)
0.586835 + 0.809707i \(0.300374\pi\)
\(272\) 0 0
\(273\) −192.000 −0.0425655
\(274\) 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
\(279\) 1620.00 0.347623
\(280\) 0 0
\(281\) 3222.00 0.684016 0.342008 0.939697i \(-0.388893\pi\)
0.342008 + 0.939697i \(0.388893\pi\)
\(282\) 0 0
\(283\) −4620.00 −0.970426 −0.485213 0.874396i \(-0.661258\pi\)
−0.485213 + 0.874396i \(0.661258\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1688.00 0.347176
\(288\) 0 0
\(289\) 6751.00 1.37411
\(290\) 0 0
\(291\) 2472.00 0.497977
\(292\) 0 0
\(293\) 5404.00 1.07749 0.538746 0.842468i \(-0.318898\pi\)
0.538746 + 0.842468i \(0.318898\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −756.000 −0.147702
\(298\) 0 0
\(299\) −448.000 −0.0866505
\(300\) 0 0
\(301\) 1104.00 0.211407
\(302\) 0 0
\(303\) 3870.00 0.733748
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9700.00 −1.80328 −0.901642 0.432483i \(-0.857638\pi\)
−0.901642 + 0.432483i \(0.857638\pi\)
\(308\) 0 0
\(309\) 4812.00 0.885907
\(310\) 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 \(-0.497631\pi\)
0.00744150 + 0.999972i \(0.497631\pi\)
\(318\) 0 0
\(319\) −6664.00 −1.16963
\(320\) 0 0
\(321\) −2676.00 −0.465295
\(322\) 0 0
\(323\) 3456.00 0.595347
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2898.00 −0.490091
\(328\) 0 0
\(329\) 240.000 0.0402177
\(330\) 0 0
\(331\) −5416.00 −0.899366 −0.449683 0.893188i \(-0.648463\pi\)
−0.449683 + 0.893188i \(0.648463\pi\)
\(332\) 0 0
\(333\) 360.000 0.0592429
\(334\) 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
\(339\) 3372.00 0.540242
\(340\) 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
\(351\) −432.000 −0.0656936
\(352\) 0 0
\(353\) −5292.00 −0.797917 −0.398959 0.916969i \(-0.630628\pi\)
−0.398959 + 0.916969i \(0.630628\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1296.00 0.192133
\(358\) 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
\(363\) 1641.00 0.237273
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7652.00 1.08837 0.544184 0.838966i \(-0.316839\pi\)
0.544184 + 0.838966i \(0.316839\pi\)
\(368\) 0 0
\(369\) 3798.00 0.535816
\(370\) 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 \(-0.380378\pi\)
0.367020 + 0.930213i \(0.380378\pi\)
\(380\) 0 0
\(381\) 5652.00 0.760002
\(382\) 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
\(387\) 2484.00 0.326276
\(388\) 0 0
\(389\) −9642.00 −1.25673 −0.628366 0.777918i \(-0.716276\pi\)
−0.628366 + 0.777918i \(0.716276\pi\)
\(390\) 0 0
\(391\) 3024.00 0.391126
\(392\) 0 0
\(393\) −1764.00 −0.226417
\(394\) 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
\(399\) 384.000 0.0481806
\(400\) 0 0
\(401\) −13358.0 −1.66351 −0.831754 0.555144i \(-0.812663\pi\)
−0.831754 + 0.555144i \(0.812663\pi\)
\(402\) 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
\(411\) 3180.00 0.381649
\(412\) 0 0
\(413\) 3216.00 0.383170
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8592.00 −1.00900
\(418\) 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
\(423\) 540.000 0.0620702
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1432.00 −0.162294
\(428\) 0 0
\(429\) −1344.00 −0.151256
\(430\) 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 \(-0.175043\pi\)
0.852570 + 0.522613i \(0.175043\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 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
\(447\) −1026.00 −0.108564
\(448\) 0 0
\(449\) −4782.00 −0.502620 −0.251310 0.967907i \(-0.580861\pi\)
−0.251310 + 0.967907i \(0.580861\pi\)
\(450\) 0 0
\(451\) 11816.0 1.23369
\(452\) 0 0
\(453\) 4908.00 0.509046
\(454\) 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
\(459\) 2916.00 0.296530
\(460\) 0 0
\(461\) −3762.00 −0.380073 −0.190037 0.981777i \(-0.560861\pi\)
−0.190037 + 0.981777i \(0.560861\pi\)
\(462\) 0 0
\(463\) 5036.00 0.505492 0.252746 0.967533i \(-0.418666\pi\)
0.252746 + 0.967533i \(0.418666\pi\)
\(464\) 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
\(471\) −6216.00 −0.608106
\(472\) 0 0
\(473\) 7728.00 0.751234
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1980.00 −0.190059
\(478\) 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
\(483\) 336.000 0.0316533
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 11572.0 1.07675 0.538375 0.842705i \(-0.319038\pi\)
0.538375 + 0.842705i \(0.319038\pi\)
\(488\) 0 0
\(489\) −2316.00 −0.214178
\(490\) 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 \(-0.419767\pi\)
0.249399 + 0.968401i \(0.419767\pi\)
\(500\) 0 0
\(501\) 3132.00 0.279296
\(502\) 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
\(507\) 5823.00 0.510076
\(508\) 0 0
\(509\) −17342.0 −1.51016 −0.755079 0.655634i \(-0.772402\pi\)
−0.755079 + 0.655634i \(0.772402\pi\)
\(510\) 0 0
\(511\) 608.000 0.0526347
\(512\) 0 0
\(513\) 864.000 0.0743597
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1680.00 0.142914
\(518\) 0 0
\(519\) 13212.0 1.11742
\(520\) 0 0
\(521\) −4886.00 −0.410863 −0.205431 0.978672i \(-0.565860\pi\)
−0.205431 + 0.978672i \(0.565860\pi\)
\(522\) 0 0
\(523\) 18548.0 1.55076 0.775380 0.631495i \(-0.217558\pi\)
0.775380 + 0.631495i \(0.217558\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19440.0 −1.60687
\(528\) 0 0
\(529\) −11383.0 −0.935563
\(530\) 0 0
\(531\) 7236.00 0.591367
\(532\) 0 0
\(533\) 6752.00 0.548708
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10356.0 0.832206
\(538\) 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
\(543\) −1578.00 −0.124712
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7700.00 −0.601880 −0.300940 0.953643i \(-0.597300\pi\)
−0.300940 + 0.953643i \(0.597300\pi\)
\(548\) 0 0
\(549\) −3222.00 −0.250477
\(550\) 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 \(-0.238752\pi\)
0.731648 + 0.681683i \(0.238752\pi\)
\(558\) 0 0
\(559\) 4416.00 0.334127
\(560\) 0 0
\(561\) 9072.00 0.682745
\(562\) 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
\(567\) 324.000 0.0239977
\(568\) 0 0
\(569\) −16758.0 −1.23468 −0.617339 0.786697i \(-0.711789\pi\)
−0.617339 + 0.786697i \(0.711789\pi\)
\(570\) 0 0
\(571\) −8056.00 −0.590426 −0.295213 0.955432i \(-0.595391\pi\)
−0.295213 + 0.955432i \(0.595391\pi\)
\(572\) 0 0
\(573\) 216.000 0.0157479
\(574\) 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
\(579\) 624.000 0.0447885
\(580\) 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
\(591\) 1116.00 0.0776753
\(592\) 0 0
\(593\) −10708.0 −0.741526 −0.370763 0.928728i \(-0.620904\pi\)
−0.370763 + 0.928728i \(0.620904\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13044.0 0.894230
\(598\) 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
\(603\) −7956.00 −0.537302
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −7996.00 −0.534675 −0.267337 0.963603i \(-0.586144\pi\)
−0.267337 + 0.963603i \(0.586144\pi\)
\(608\) 0 0
\(609\) 2856.00 0.190034
\(610\) 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 \(-0.538936\pi\)
−0.122015 + 0.992528i \(0.538936\pi\)
\(618\) 0 0
\(619\) 26000.0 1.68825 0.844126 0.536145i \(-0.180120\pi\)
0.844126 + 0.536145i \(0.180120\pi\)
\(620\) 0 0
\(621\) 756.000 0.0488522
\(622\) 0 0
\(623\) −4584.00 −0.294790
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2688.00 0.171210
\(628\) 0 0
\(629\) −4320.00 −0.273847
\(630\) 0 0
\(631\) −11660.0 −0.735622 −0.367811 0.929901i \(-0.619893\pi\)
−0.367811 + 0.929901i \(0.619893\pi\)
\(632\) 0 0
\(633\) −1248.00 −0.0783626
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5232.00 −0.325431
\(638\) 0 0
\(639\) 576.000 0.0356592
\(640\) 0 0
\(641\) −7602.00 −0.468426 −0.234213 0.972185i \(-0.575251\pi\)
−0.234213 + 0.972185i \(0.575251\pi\)
\(642\) 0 0
\(643\) 29268.0 1.79505 0.897525 0.440963i \(-0.145363\pi\)
0.897525 + 0.440963i \(0.145363\pi\)
\(644\) 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
\(651\) −2160.00 −0.130042
\(652\) 0 0
\(653\) 19188.0 1.14990 0.574950 0.818189i \(-0.305022\pi\)
0.574950 + 0.818189i \(0.305022\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1368.00 0.0812340
\(658\) 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
\(663\) 5184.00 0.303665
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6664.00 0.386853
\(668\) 0 0
\(669\) 17244.0 0.996549
\(670\) 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 \(-0.0748691\pi\)
0.972466 + 0.233045i \(0.0748691\pi\)
\(678\) 0 0
\(679\) −3296.00 −0.186287
\(680\) 0 0
\(681\) −3444.00 −0.193795
\(682\) 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
\(687\) −9702.00 −0.538798
\(688\) 0 0
\(689\) −3520.00 −0.194632
\(690\) 0 0
\(691\) −4608.00 −0.253685 −0.126843 0.991923i \(-0.540484\pi\)
−0.126843 + 0.991923i \(0.540484\pi\)
\(692\) 0 0
\(693\) 1008.00 0.0552536
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −45576.0 −2.47678
\(698\) 0 0
\(699\) −684.000 −0.0370118
\(700\) 0 0
\(701\) −2318.00 −0.124893 −0.0624463 0.998048i \(-0.519890\pi\)
−0.0624463 + 0.998048i \(0.519890\pi\)
\(702\) 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
\(711\) 8388.00 0.442440
\(712\) 0 0
\(713\) −5040.00 −0.264726
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 14280.0 0.743789
\(718\) 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
\(723\) −9690.00 −0.498444
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 20340.0 1.03765 0.518823 0.854882i \(-0.326370\pi\)
0.518823 + 0.854882i \(0.326370\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 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 \(-0.724438\pi\)
−0.648103 + 0.761552i \(0.724438\pi\)
\(740\) 0 0
\(741\) 1536.00 0.0761489
\(742\) 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
\(747\) −11628.0 −0.569540
\(748\) 0 0
\(749\) 3568.00 0.174061
\(750\) 0 0
\(751\) 20692.0 1.00541 0.502704 0.864458i \(-0.332338\pi\)
0.502704 + 0.864458i \(0.332338\pi\)
\(752\) 0 0
\(753\) 5124.00 0.247980
\(754\) 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
\(759\) 2352.00 0.112480
\(760\) 0 0
\(761\) 13978.0 0.665837 0.332919 0.942956i \(-0.391967\pi\)
0.332919 + 0.942956i \(0.391967\pi\)
\(762\) 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
\(771\) 19116.0 0.892926
\(772\) 0 0
\(773\) −13916.0 −0.647508 −0.323754 0.946141i \(-0.604945\pi\)
−0.323754 + 0.946141i \(0.604945\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −480.000 −0.0221620
\(778\) 0 0
\(779\) −13504.0 −0.621092
\(780\) 0 0
\(781\) 1792.00 0.0821035
\(782\) 0 0
\(783\) 6426.00 0.293291
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29996.0 1.35863 0.679315 0.733847i \(-0.262277\pi\)
0.679315 + 0.733847i \(0.262277\pi\)
\(788\) 0 0
\(789\) 9108.00 0.410968
\(790\) 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 \(-0.436340\pi\)
0.198664 + 0.980068i \(0.436340\pi\)
\(798\) 0 0
\(799\) −6480.00 −0.286916
\(800\) 0 0
\(801\) −10314.0 −0.454965
\(802\) 0 0
\(803\) 4256.00 0.187037
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 342.000 0.0149182
\(808\) 0 0
\(809\) −10698.0 −0.464922 −0.232461 0.972606i \(-0.574678\pi\)
−0.232461 + 0.972606i \(0.574678\pi\)
\(810\) 0 0
\(811\) 6408.00 0.277454 0.138727 0.990331i \(-0.455699\pi\)
0.138727 + 0.990331i \(0.455699\pi\)
\(812\) 0 0
\(813\) −15708.0 −0.677618
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8832.00 −0.378204
\(818\) 0 0
\(819\) 576.000 0.0245752
\(820\) 0 0
\(821\) 23130.0 0.983243 0.491622 0.870809i \(-0.336404\pi\)
0.491622 + 0.870809i \(0.336404\pi\)
\(822\) 0 0
\(823\) 11852.0 0.501986 0.250993 0.967989i \(-0.419243\pi\)
0.250993 + 0.967989i \(0.419243\pi\)
\(824\) 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
\(831\) −17136.0 −0.715332
\(832\) 0 0
\(833\) 35316.0 1.46894
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4860.00 −0.200700
\(838\) 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
\(843\) −9666.00 −0.394917
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2188.00 −0.0887610
\(848\) 0 0
\(849\) 13860.0 0.560276
\(850\) 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 \(-0.449212\pi\)
0.158879 + 0.987298i \(0.449212\pi\)
\(858\) 0 0
\(859\) 6008.00 0.238638 0.119319 0.992856i \(-0.461929\pi\)
0.119319 + 0.992856i \(0.461929\pi\)
\(860\) 0 0
\(861\) −5064.00 −0.200442
\(862\) 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
\(867\) −20253.0 −0.793342
\(868\) 0 0
\(869\) 26096.0 1.01870
\(870\) 0 0
\(871\) −14144.0 −0.550231
\(872\) 0 0
\(873\) −7416.00 −0.287507
\(874\) 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
\(879\) −16212.0 −0.622090
\(880\) 0 0
\(881\) 27838.0 1.06457 0.532285 0.846565i \(-0.321334\pi\)
0.532285 + 0.846565i \(0.321334\pi\)
\(882\) 0 0
\(883\) −4316.00 −0.164490 −0.0822452 0.996612i \(-0.526209\pi\)
−0.0822452 + 0.996612i \(0.526209\pi\)
\(884\) 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
\(891\) 2268.00 0.0852759
\(892\) 0 0
\(893\) −1920.00 −0.0719489
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1344.00 0.0500277
\(898\) 0 0
\(899\) −42840.0 −1.58931
\(900\) 0 0
\(901\) 23760.0 0.878535
\(902\) 0 0
\(903\) −3312.00 −0.122056
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 10556.0 0.386446 0.193223 0.981155i \(-0.438106\pi\)
0.193223 + 0.981155i \(0.438106\pi\)
\(908\) 0 0
\(909\) −11610.0 −0.423630
\(910\) 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 \(-0.568884\pi\)
−0.214720 + 0.976676i \(0.568884\pi\)
\(920\) 0 0
\(921\) 29100.0 1.04113
\(922\) 0 0
\(923\) 1024.00 0.0365172
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −14436.0 −0.511478
\(928\) 0 0
\(929\) −15214.0 −0.537304 −0.268652 0.963237i \(-0.586578\pi\)
−0.268652 + 0.963237i \(0.586578\pi\)
\(930\) 0 0
\(931\) 10464.0 0.368361
\(932\) 0 0
\(933\) 29016.0 1.01816
\(934\) 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
\(939\) 12144.0 0.422049
\(940\) 0 0
\(941\) −36034.0 −1.24833 −0.624163 0.781294i \(-0.714560\pi\)
−0.624163 + 0.781294i \(0.714560\pi\)
\(942\) 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
\(951\) −252.000 −0.00859270
\(952\) 0 0
\(953\) −55284.0 −1.87914 −0.939572 0.342351i \(-0.888777\pi\)
−0.939572 + 0.342351i \(0.888777\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 19992.0 0.675287
\(958\) 0 0
\(959\) −4240.00 −0.142770
\(960\) 0 0
\(961\) 2609.00 0.0875768
\(962\) 0 0
\(963\) 8028.00 0.268638
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4372.00 0.145392 0.0726960 0.997354i \(-0.476840\pi\)
0.0726960 + 0.997354i \(0.476840\pi\)
\(968\) 0 0
\(969\) −10368.0 −0.343724
\(970\) 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 \(-0.773091\pi\)
−0.756498 + 0.653996i \(0.773091\pi\)
\(978\) 0 0
\(979\) −32088.0 −1.04754
\(980\) 0 0
\(981\) 8694.00 0.282954
\(982\) 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
\(987\) −720.000 −0.0232197
\(988\) 0 0
\(989\) −7728.00 −0.248469
\(990\) 0 0
\(991\) −11668.0 −0.374012 −0.187006 0.982359i \(-0.559878\pi\)
−0.187006 + 0.982359i \(0.559878\pi\)
\(992\) 0 0
\(993\) 16248.0 0.519249
\(994\) 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
\(999\) −1080.00 −0.0342039
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 1200.4.a.l.1.1 1
4.3 odd 2 600.4.a.k.1.1 1
5.2 odd 4 240.4.f.e.49.2 2
5.3 odd 4 240.4.f.e.49.1 2
5.4 even 2 1200.4.a.z.1.1 1
12.11 even 2 1800.4.a.o.1.1 1
15.2 even 4 720.4.f.b.289.2 2
15.8 even 4 720.4.f.b.289.1 2
20.3 even 4 120.4.f.c.49.2 yes 2
20.7 even 4 120.4.f.c.49.1 2
20.19 odd 2 600.4.a.f.1.1 1
40.3 even 4 960.4.f.b.769.1 2
40.13 odd 4 960.4.f.a.769.2 2
40.27 even 4 960.4.f.b.769.2 2
40.37 odd 4 960.4.f.a.769.1 2
60.23 odd 4 360.4.f.a.289.1 2
60.47 odd 4 360.4.f.a.289.2 2
60.59 even 2 1800.4.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.c.49.1 2 20.7 even 4
120.4.f.c.49.2 yes 2 20.3 even 4
240.4.f.e.49.1 2 5.3 odd 4
240.4.f.e.49.2 2 5.2 odd 4
360.4.f.a.289.1 2 60.23 odd 4
360.4.f.a.289.2 2 60.47 odd 4
600.4.a.f.1.1 1 20.19 odd 2
600.4.a.k.1.1 1 4.3 odd 2
720.4.f.b.289.1 2 15.8 even 4
720.4.f.b.289.2 2 15.2 even 4
960.4.f.a.769.1 2 40.37 odd 4
960.4.f.a.769.2 2 40.13 odd 4
960.4.f.b.769.1 2 40.3 even 4
960.4.f.b.769.2 2 40.27 even 4
1200.4.a.l.1.1 1 1.1 even 1 trivial
1200.4.a.z.1.1 1 5.4 even 2
1800.4.a.o.1.1 1 12.11 even 2
1800.4.a.u.1.1 1 60.59 even 2