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Properties

Label 1944.4.a.a.1.1
Level $1944$
Weight $4$
Character 1944.1
Self dual yes
Analytic conductor $114.700$
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] = [1944,4,Mod(1,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, 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("1944.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1944.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: \(114.699713051\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1944.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-11.0000 q^{5} +30.0000 q^{7} +O(q^{10})\) \(q-11.0000 q^{5} +30.0000 q^{7} +4.00000 q^{11} +16.0000 q^{13} -70.0000 q^{17} +5.00000 q^{19} -53.0000 q^{23} -4.00000 q^{25} +15.0000 q^{29} +74.0000 q^{31} -330.000 q^{35} -156.000 q^{37} -366.000 q^{41} +292.000 q^{43} +201.000 q^{47} +557.000 q^{49} -557.000 q^{53} -44.0000 q^{55} +364.000 q^{59} -518.000 q^{61} -176.000 q^{65} -7.00000 q^{67} +473.000 q^{71} -359.000 q^{73} +120.000 q^{77} +1024.00 q^{79} -1078.00 q^{83} +770.000 q^{85} -1044.00 q^{89} +480.000 q^{91} -55.0000 q^{95} -193.000 q^{97} +O(q^{100})\)

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
\(5\) −11.0000 −0.983870 −0.491935 − 0.870632i \(-0.663710\pi\)
−0.491935 + 0.870632i \(0.663710\pi\)
\(6\) 0 0
\(7\) 30.0000 1.61985 0.809924 − 0.586535i \(-0.199508\pi\)
0.809924 + 0.586535i \(0.199508\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 0.109640 0.0548202 − 0.998496i \(-0.482541\pi\)
0.0548202 + 0.998496i \(0.482541\pi\)
\(12\) 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
\(17\) −70.0000 −0.998676 −0.499338 − 0.866407i \(-0.666423\pi\)
−0.499338 + 0.866407i \(0.666423\pi\)
\(18\) 0 0
\(19\) 5.00000 0.0603726 0.0301863 − 0.999544i \(-0.490390\pi\)
0.0301863 + 0.999544i \(0.490390\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −53.0000 −0.480490 −0.240245 − 0.970712i \(-0.577228\pi\)
−0.240245 + 0.970712i \(0.577228\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.0320000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 15.0000 0.0960493 0.0480247 − 0.998846i \(-0.484707\pi\)
0.0480247 + 0.998846i \(0.484707\pi\)
\(30\) 0 0
\(31\) 74.0000 0.428735 0.214368 − 0.976753i \(-0.431231\pi\)
0.214368 + 0.976753i \(0.431231\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −330.000 −1.59372
\(36\) 0 0
\(37\) −156.000 −0.693142 −0.346571 − 0.938024i \(-0.612654\pi\)
−0.346571 + 0.938024i \(0.612654\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −366.000 −1.39414 −0.697068 − 0.717005i \(-0.745513\pi\)
−0.697068 + 0.717005i \(0.745513\pi\)
\(42\) 0 0
\(43\) 292.000 1.03557 0.517786 − 0.855510i \(-0.326756\pi\)
0.517786 + 0.855510i \(0.326756\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 201.000 0.623806 0.311903 − 0.950114i \(-0.399034\pi\)
0.311903 + 0.950114i \(0.399034\pi\)
\(48\) 0 0
\(49\) 557.000 1.62391
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −557.000 −1.44358 −0.721791 − 0.692111i \(-0.756681\pi\)
−0.721791 + 0.692111i \(0.756681\pi\)
\(54\) 0 0
\(55\) −44.0000 −0.107872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 364.000 0.803199 0.401600 − 0.915815i \(-0.368454\pi\)
0.401600 + 0.915815i \(0.368454\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 − 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −176.000 −0.335848
\(66\) 0 0
\(67\) −7.00000 −0.0127640 −0.00638199 − 0.999980i \(-0.502031\pi\)
−0.00638199 + 0.999980i \(0.502031\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 473.000 0.790631 0.395315 − 0.918545i \(-0.370635\pi\)
0.395315 + 0.918545i \(0.370635\pi\)
\(72\) 0 0
\(73\) −359.000 −0.575586 −0.287793 − 0.957693i \(-0.592921\pi\)
−0.287793 + 0.957693i \(0.592921\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 120.000 0.177601
\(78\) 0 0
\(79\) 1024.00 1.45834 0.729171 − 0.684332i \(-0.239906\pi\)
0.729171 + 0.684332i \(0.239906\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1078.00 −1.42561 −0.712806 − 0.701361i \(-0.752576\pi\)
−0.712806 + 0.701361i \(0.752576\pi\)
\(84\) 0 0
\(85\) 770.000 0.982567
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1044.00 −1.24341 −0.621707 − 0.783250i \(-0.713560\pi\)
−0.621707 + 0.783250i \(0.713560\pi\)
\(90\) 0 0
\(91\) 480.000 0.552941
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −55.0000 −0.0593987
\(96\) 0 0
\(97\) −193.000 −0.202022 −0.101011 − 0.994885i \(-0.532208\pi\)
−0.101011 + 0.994885i \(0.532208\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 387.000 0.381267 0.190633 − 0.981661i \(-0.438946\pi\)
0.190633 + 0.981661i \(0.438946\pi\)
\(102\) 0 0
\(103\) 34.0000 0.0325254 0.0162627 − 0.999868i \(-0.494823\pi\)
0.0162627 + 0.999868i \(0.494823\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1668.00 1.50702 0.753512 − 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(108\) 0 0
\(109\) −644.000 −0.565908 −0.282954 − 0.959133i \(-0.591314\pi\)
−0.282954 + 0.959133i \(0.591314\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1896.00 1.57841 0.789207 − 0.614128i \(-0.210492\pi\)
0.789207 + 0.614128i \(0.210492\pi\)
\(114\) 0 0
\(115\) 583.000 0.472739
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2100.00 −1.61770
\(120\) 0 0
\(121\) −1315.00 −0.987979
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1419.00 1.01535
\(126\) 0 0
\(127\) 2248.00 1.57069 0.785345 − 0.619058i \(-0.212485\pi\)
0.785345 + 0.619058i \(0.212485\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1214.00 0.809677 0.404838 − 0.914388i \(-0.367328\pi\)
0.404838 + 0.914388i \(0.367328\pi\)
\(132\) 0 0
\(133\) 150.000 0.0977944
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2964.00 −1.84841 −0.924203 − 0.381902i \(-0.875269\pi\)
−0.924203 + 0.381902i \(0.875269\pi\)
\(138\) 0 0
\(139\) −2568.00 −1.56701 −0.783507 − 0.621383i \(-0.786571\pi\)
−0.783507 + 0.621383i \(0.786571\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 64.0000 0.0374262
\(144\) 0 0
\(145\) −165.000 −0.0945000
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2266.00 −1.24589 −0.622946 − 0.782265i \(-0.714064\pi\)
−0.622946 + 0.782265i \(0.714064\pi\)
\(150\) 0 0
\(151\) −3202.00 −1.72566 −0.862831 − 0.505492i \(-0.831311\pi\)
−0.862831 + 0.505492i \(0.831311\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −814.000 −0.421820
\(156\) 0 0
\(157\) 1540.00 0.782837 0.391418 − 0.920213i \(-0.371985\pi\)
0.391418 + 0.920213i \(0.371985\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1590.00 −0.778320
\(162\) 0 0
\(163\) −3879.00 −1.86397 −0.931984 − 0.362500i \(-0.881923\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2943.00 1.36369 0.681845 − 0.731497i \(-0.261178\pi\)
0.681845 + 0.731497i \(0.261178\pi\)
\(168\) 0 0
\(169\) −1941.00 −0.883477
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1875.00 −0.824009 −0.412005 − 0.911182i \(-0.635171\pi\)
−0.412005 + 0.911182i \(0.635171\pi\)
\(174\) 0 0
\(175\) −120.000 −0.0518351
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1686.00 −0.704009 −0.352004 − 0.935998i \(-0.614500\pi\)
−0.352004 + 0.935998i \(0.614500\pi\)
\(180\) 0 0
\(181\) 696.000 0.285819 0.142910 − 0.989736i \(-0.454354\pi\)
0.142910 + 0.989736i \(0.454354\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1716.00 0.681961
\(186\) 0 0
\(187\) −280.000 −0.109495
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2163.00 −0.819420 −0.409710 − 0.912216i \(-0.634370\pi\)
−0.409710 + 0.912216i \(0.634370\pi\)
\(192\) 0 0
\(193\) −670.000 −0.249884 −0.124942 − 0.992164i \(-0.539875\pi\)
−0.124942 + 0.992164i \(0.539875\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2850.00 −1.03073 −0.515366 − 0.856970i \(-0.672344\pi\)
−0.515366 + 0.856970i \(0.672344\pi\)
\(198\) 0 0
\(199\) 4204.00 1.49756 0.748778 − 0.662821i \(-0.230641\pi\)
0.748778 + 0.662821i \(0.230641\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 450.000 0.155585
\(204\) 0 0
\(205\) 4026.00 1.37165
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.0000 0.00661928
\(210\) 0 0
\(211\) −2948.00 −0.961842 −0.480921 − 0.876764i \(-0.659698\pi\)
−0.480921 + 0.876764i \(0.659698\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3212.00 −1.01887
\(216\) 0 0
\(217\) 2220.00 0.694486
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1120.00 −0.340902
\(222\) 0 0
\(223\) −3008.00 −0.903276 −0.451638 − 0.892201i \(-0.649160\pi\)
−0.451638 + 0.892201i \(0.649160\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2070.00 −0.605245 −0.302623 − 0.953110i \(-0.597862\pi\)
−0.302623 + 0.953110i \(0.597862\pi\)
\(228\) 0 0
\(229\) −3794.00 −1.09482 −0.547412 − 0.836863i \(-0.684387\pi\)
−0.547412 + 0.836863i \(0.684387\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6014.00 −1.69095 −0.845473 − 0.534019i \(-0.820681\pi\)
−0.845473 + 0.534019i \(0.820681\pi\)
\(234\) 0 0
\(235\) −2211.00 −0.613744
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2899.00 −0.784606 −0.392303 − 0.919836i \(-0.628321\pi\)
−0.392303 + 0.919836i \(0.628321\pi\)
\(240\) 0 0
\(241\) 5730.00 1.53154 0.765771 − 0.643113i \(-0.222357\pi\)
0.765771 + 0.643113i \(0.222357\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6127.00 −1.59771
\(246\) 0 0
\(247\) 80.0000 0.0206084
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4090.00 −1.02852 −0.514260 − 0.857634i \(-0.671933\pi\)
−0.514260 + 0.857634i \(0.671933\pi\)
\(252\) 0 0
\(253\) −212.000 −0.0526811
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1898.00 0.460677 0.230338 − 0.973111i \(-0.426017\pi\)
0.230338 + 0.973111i \(0.426017\pi\)
\(258\) 0 0
\(259\) −4680.00 −1.12278
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4892.00 −1.14697 −0.573486 − 0.819215i \(-0.694409\pi\)
−0.573486 + 0.819215i \(0.694409\pi\)
\(264\) 0 0
\(265\) 6127.00 1.42030
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2350.00 −0.532647 −0.266323 − 0.963884i \(-0.585809\pi\)
−0.266323 + 0.963884i \(0.585809\pi\)
\(270\) 0 0
\(271\) −256.000 −0.0573834 −0.0286917 − 0.999588i \(-0.509134\pi\)
−0.0286917 + 0.999588i \(0.509134\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.0000 −0.00350850
\(276\) 0 0
\(277\) 586.000 0.127109 0.0635547 − 0.997978i \(-0.479756\pi\)
0.0635547 + 0.997978i \(0.479756\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 530.000 0.112517 0.0562583 − 0.998416i \(-0.482083\pi\)
0.0562583 + 0.998416i \(0.482083\pi\)
\(282\) 0 0
\(283\) −607.000 −0.127500 −0.0637498 − 0.997966i \(-0.520306\pi\)
−0.0637498 + 0.997966i \(0.520306\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10980.0 −2.25829
\(288\) 0 0
\(289\) −13.0000 −0.00264604
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1311.00 0.261397 0.130699 − 0.991422i \(-0.458278\pi\)
0.130699 + 0.991422i \(0.458278\pi\)
\(294\) 0 0
\(295\) −4004.00 −0.790244
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −848.000 −0.164017
\(300\) 0 0
\(301\) 8760.00 1.67747
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5698.00 1.06973
\(306\) 0 0
\(307\) −5883.00 −1.09368 −0.546841 − 0.837236i \(-0.684170\pi\)
−0.546841 + 0.837236i \(0.684170\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2799.00 0.510343 0.255172 − 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(312\) 0 0
\(313\) 8613.00 1.55539 0.777693 − 0.628645i \(-0.216390\pi\)
0.777693 + 0.628645i \(0.216390\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10518.0 −1.86356 −0.931782 − 0.363019i \(-0.881746\pi\)
−0.931782 + 0.363019i \(0.881746\pi\)
\(318\) 0 0
\(319\) 60.0000 0.0105309
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −350.000 −0.0602926
\(324\) 0 0
\(325\) −64.0000 −0.0109233
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6030.00 1.01047
\(330\) 0 0
\(331\) 4331.00 0.719194 0.359597 − 0.933108i \(-0.382914\pi\)
0.359597 + 0.933108i \(0.382914\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 77.0000 0.0125581
\(336\) 0 0
\(337\) 4125.00 0.666775 0.333387 − 0.942790i \(-0.391808\pi\)
0.333387 + 0.942790i \(0.391808\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 296.000 0.0470067
\(342\) 0 0
\(343\) 6420.00 1.01063
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5946.00 −0.919879 −0.459939 − 0.887950i \(-0.652129\pi\)
−0.459939 + 0.887950i \(0.652129\pi\)
\(348\) 0 0
\(349\) −1584.00 −0.242950 −0.121475 − 0.992594i \(-0.538762\pi\)
−0.121475 + 0.992594i \(0.538762\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8776.00 −1.32323 −0.661614 − 0.749845i \(-0.730128\pi\)
−0.661614 + 0.749845i \(0.730128\pi\)
\(354\) 0 0
\(355\) −5203.00 −0.777878
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10233.0 1.50439 0.752196 − 0.658939i \(-0.228994\pi\)
0.752196 + 0.658939i \(0.228994\pi\)
\(360\) 0 0
\(361\) −6834.00 −0.996355
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3949.00 0.566302
\(366\) 0 0
\(367\) −4070.00 −0.578889 −0.289445 − 0.957195i \(-0.593471\pi\)
−0.289445 + 0.957195i \(0.593471\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −16710.0 −2.33838
\(372\) 0 0
\(373\) 7172.00 0.995582 0.497791 − 0.867297i \(-0.334145\pi\)
0.497791 + 0.867297i \(0.334145\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 240.000 0.0327868
\(378\) 0 0
\(379\) −2095.00 −0.283939 −0.141970 − 0.989871i \(-0.545344\pi\)
−0.141970 + 0.989871i \(0.545344\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5553.00 0.740849 0.370424 − 0.928863i \(-0.379212\pi\)
0.370424 + 0.928863i \(0.379212\pi\)
\(384\) 0 0
\(385\) −1320.00 −0.174736
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 889.000 0.115872 0.0579358 − 0.998320i \(-0.481548\pi\)
0.0579358 + 0.998320i \(0.481548\pi\)
\(390\) 0 0
\(391\) 3710.00 0.479854
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11264.0 −1.43482
\(396\) 0 0
\(397\) −9604.00 −1.21413 −0.607067 − 0.794651i \(-0.707654\pi\)
−0.607067 + 0.794651i \(0.707654\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10926.0 1.36064 0.680322 − 0.732913i \(-0.261840\pi\)
0.680322 + 0.732913i \(0.261840\pi\)
\(402\) 0 0
\(403\) 1184.00 0.146350
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −624.000 −0.0759964
\(408\) 0 0
\(409\) 3255.00 0.393519 0.196760 − 0.980452i \(-0.436958\pi\)
0.196760 + 0.980452i \(0.436958\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10920.0 1.30106
\(414\) 0 0
\(415\) 11858.0 1.40262
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8070.00 0.940920 0.470460 − 0.882421i \(-0.344088\pi\)
0.470460 + 0.882421i \(0.344088\pi\)
\(420\) 0 0
\(421\) 2000.00 0.231530 0.115765 − 0.993277i \(-0.463068\pi\)
0.115765 + 0.993277i \(0.463068\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 280.000 0.0319576
\(426\) 0 0
\(427\) −15540.0 −1.76120
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1199.00 0.134000 0.0669998 − 0.997753i \(-0.478657\pi\)
0.0669998 + 0.997753i \(0.478657\pi\)
\(432\) 0 0
\(433\) 2269.00 0.251827 0.125914 − 0.992041i \(-0.459814\pi\)
0.125914 + 0.992041i \(0.459814\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −265.000 −0.0290084
\(438\) 0 0
\(439\) 7470.00 0.812127 0.406063 − 0.913845i \(-0.366901\pi\)
0.406063 + 0.913845i \(0.366901\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1196.00 0.128270 0.0641351 − 0.997941i \(-0.479571\pi\)
0.0641351 + 0.997941i \(0.479571\pi\)
\(444\) 0 0
\(445\) 11484.0 1.22336
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −68.0000 −0.00714726 −0.00357363 − 0.999994i \(-0.501138\pi\)
−0.00357363 + 0.999994i \(0.501138\pi\)
\(450\) 0 0
\(451\) −1464.00 −0.152854
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5280.00 −0.544022
\(456\) 0 0
\(457\) 2702.00 0.276574 0.138287 − 0.990392i \(-0.455840\pi\)
0.138287 + 0.990392i \(0.455840\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19094.0 −1.92906 −0.964530 − 0.263975i \(-0.914966\pi\)
−0.964530 + 0.263975i \(0.914966\pi\)
\(462\) 0 0
\(463\) −5374.00 −0.539419 −0.269709 − 0.962942i \(-0.586928\pi\)
−0.269709 + 0.962942i \(0.586928\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6258.00 0.620098 0.310049 − 0.950721i \(-0.399655\pi\)
0.310049 + 0.950721i \(0.399655\pi\)
\(468\) 0 0
\(469\) −210.000 −0.0206757
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1168.00 0.113541
\(474\) 0 0
\(475\) −20.0000 −0.00193192
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3980.00 0.379647 0.189823 − 0.981818i \(-0.439208\pi\)
0.189823 + 0.981818i \(0.439208\pi\)
\(480\) 0 0
\(481\) −2496.00 −0.236607
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2123.00 0.198764
\(486\) 0 0
\(487\) 19078.0 1.77517 0.887584 − 0.460646i \(-0.152382\pi\)
0.887584 + 0.460646i \(0.152382\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10740.0 0.987147 0.493574 − 0.869704i \(-0.335690\pi\)
0.493574 + 0.869704i \(0.335690\pi\)
\(492\) 0 0
\(493\) −1050.00 −0.0959222
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14190.0 1.28070
\(498\) 0 0
\(499\) −18969.0 −1.70174 −0.850871 − 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15681.0 −1.39002 −0.695011 − 0.718999i \(-0.744601\pi\)
−0.695011 + 0.718999i \(0.744601\pi\)
\(504\) 0 0
\(505\) −4257.00 −0.375117
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18945.0 −1.64975 −0.824875 − 0.565316i \(-0.808754\pi\)
−0.824875 + 0.565316i \(0.808754\pi\)
\(510\) 0 0
\(511\) −10770.0 −0.932362
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −374.000 −0.0320008
\(516\) 0 0
\(517\) 804.000 0.0683944
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7204.00 −0.605783 −0.302892 − 0.953025i \(-0.597952\pi\)
−0.302892 + 0.953025i \(0.597952\pi\)
\(522\) 0 0
\(523\) −5051.00 −0.422304 −0.211152 − 0.977453i \(-0.567721\pi\)
−0.211152 + 0.977453i \(0.567721\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5180.00 −0.428168
\(528\) 0 0
\(529\) −9358.00 −0.769130
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5856.00 −0.475894
\(534\) 0 0
\(535\) −18348.0 −1.48272
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2228.00 0.178046
\(540\) 0 0
\(541\) −14400.0 −1.14437 −0.572185 − 0.820124i \(-0.693904\pi\)
−0.572185 + 0.820124i \(0.693904\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7084.00 0.556780
\(546\) 0 0
\(547\) 1949.00 0.152346 0.0761730 − 0.997095i \(-0.475730\pi\)
0.0761730 + 0.997095i \(0.475730\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 75.0000 0.00579874
\(552\) 0 0
\(553\) 30720.0 2.36229
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −22802.0 −1.73456 −0.867282 − 0.497818i \(-0.834135\pi\)
−0.867282 + 0.497818i \(0.834135\pi\)
\(558\) 0 0
\(559\) 4672.00 0.353497
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3410.00 −0.255265 −0.127633 − 0.991822i \(-0.540738\pi\)
−0.127633 + 0.991822i \(0.540738\pi\)
\(564\) 0 0
\(565\) −20856.0 −1.55295
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12550.0 0.924646 0.462323 − 0.886712i \(-0.347016\pi\)
0.462323 + 0.886712i \(0.347016\pi\)
\(570\) 0 0
\(571\) −2451.00 −0.179634 −0.0898171 − 0.995958i \(-0.528628\pi\)
−0.0898171 + 0.995958i \(0.528628\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 212.000 0.0153757
\(576\) 0 0
\(577\) −553.000 −0.0398989 −0.0199495 − 0.999801i \(-0.506351\pi\)
−0.0199495 + 0.999801i \(0.506351\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −32340.0 −2.30928
\(582\) 0 0
\(583\) −2228.00 −0.158275
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17590.0 1.23683 0.618413 − 0.785853i \(-0.287776\pi\)
0.618413 + 0.785853i \(0.287776\pi\)
\(588\) 0 0
\(589\) 370.000 0.0258838
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14918.0 −1.03307 −0.516534 − 0.856267i \(-0.672778\pi\)
−0.516534 + 0.856267i \(0.672778\pi\)
\(594\) 0 0
\(595\) 23100.0 1.59161
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3507.00 0.239219 0.119609 − 0.992821i \(-0.461836\pi\)
0.119609 + 0.992821i \(0.461836\pi\)
\(600\) 0 0
\(601\) 11959.0 0.811677 0.405838 − 0.913945i \(-0.366980\pi\)
0.405838 + 0.913945i \(0.366980\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14465.0 0.972043
\(606\) 0 0
\(607\) −14668.0 −0.980817 −0.490408 − 0.871493i \(-0.663152\pi\)
−0.490408 + 0.871493i \(0.663152\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3216.00 0.212939
\(612\) 0 0
\(613\) −2122.00 −0.139815 −0.0699076 − 0.997553i \(-0.522270\pi\)
−0.0699076 + 0.997553i \(0.522270\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 492.000 0.0321024 0.0160512 − 0.999871i \(-0.494891\pi\)
0.0160512 + 0.999871i \(0.494891\pi\)
\(618\) 0 0
\(619\) −14984.0 −0.972953 −0.486476 − 0.873694i \(-0.661718\pi\)
−0.486476 + 0.873694i \(0.661718\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −31320.0 −2.01414
\(624\) 0 0
\(625\) −15109.0 −0.966976
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10920.0 0.692224
\(630\) 0 0
\(631\) −21080.0 −1.32992 −0.664962 − 0.746878i \(-0.731552\pi\)
−0.664962 + 0.746878i \(0.731552\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24728.0 −1.54536
\(636\) 0 0
\(637\) 8912.00 0.554327
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19552.0 1.20477 0.602385 − 0.798206i \(-0.294217\pi\)
0.602385 + 0.798206i \(0.294217\pi\)
\(642\) 0 0
\(643\) −14211.0 −0.871582 −0.435791 − 0.900048i \(-0.643531\pi\)
−0.435791 + 0.900048i \(0.643531\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18089.0 1.09915 0.549576 − 0.835443i \(-0.314789\pi\)
0.549576 + 0.835443i \(0.314789\pi\)
\(648\) 0 0
\(649\) 1456.00 0.0880632
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23238.0 1.39261 0.696304 − 0.717747i \(-0.254826\pi\)
0.696304 + 0.717747i \(0.254826\pi\)
\(654\) 0 0
\(655\) −13354.0 −0.796617
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32850.0 1.94181 0.970906 − 0.239460i \(-0.0769705\pi\)
0.970906 + 0.239460i \(0.0769705\pi\)
\(660\) 0 0
\(661\) −4030.00 −0.237139 −0.118569 − 0.992946i \(-0.537831\pi\)
−0.118569 + 0.992946i \(0.537831\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1650.00 −0.0962169
\(666\) 0 0
\(667\) −795.000 −0.0461507
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2072.00 −0.119208
\(672\) 0 0
\(673\) 30697.0 1.75822 0.879110 − 0.476618i \(-0.158138\pi\)
0.879110 + 0.476618i \(0.158138\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1223.00 −0.0694294 −0.0347147 − 0.999397i \(-0.511052\pi\)
−0.0347147 + 0.999397i \(0.511052\pi\)
\(678\) 0 0
\(679\) −5790.00 −0.327246
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6398.00 −0.358437 −0.179219 − 0.983809i \(-0.557357\pi\)
−0.179219 + 0.983809i \(0.557357\pi\)
\(684\) 0 0
\(685\) 32604.0 1.81859
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8912.00 −0.492772
\(690\) 0 0
\(691\) 1655.00 0.0911131 0.0455566 − 0.998962i \(-0.485494\pi\)
0.0455566 + 0.998962i \(0.485494\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28248.0 1.54174
\(696\) 0 0
\(697\) 25620.0 1.39229
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3570.00 0.192350 0.0961748 − 0.995364i \(-0.469339\pi\)
0.0961748 + 0.995364i \(0.469339\pi\)
\(702\) 0 0
\(703\) −780.000 −0.0418467
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11610.0 0.617594
\(708\) 0 0
\(709\) 13766.0 0.729186 0.364593 − 0.931167i \(-0.381208\pi\)
0.364593 + 0.931167i \(0.381208\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3922.00 −0.206003
\(714\) 0 0
\(715\) −704.000 −0.0368225
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7792.00 0.404162 0.202081 − 0.979369i \(-0.435230\pi\)
0.202081 + 0.979369i \(0.435230\pi\)
\(720\) 0 0
\(721\) 1020.00 0.0526862
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −60.0000 −0.00307358
\(726\) 0 0
\(727\) 3204.00 0.163452 0.0817261 − 0.996655i \(-0.473957\pi\)
0.0817261 + 0.996655i \(0.473957\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20440.0 −1.03420
\(732\) 0 0
\(733\) 28532.0 1.43773 0.718863 − 0.695152i \(-0.244663\pi\)
0.718863 + 0.695152i \(0.244663\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −28.0000 −0.00139945
\(738\) 0 0
\(739\) −10684.0 −0.531823 −0.265912 − 0.963997i \(-0.585673\pi\)
−0.265912 + 0.963997i \(0.585673\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12653.0 −0.624756 −0.312378 − 0.949958i \(-0.601126\pi\)
−0.312378 + 0.949958i \(0.601126\pi\)
\(744\) 0 0
\(745\) 24926.0 1.22580
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 50040.0 2.44115
\(750\) 0 0
\(751\) 2496.00 0.121279 0.0606394 − 0.998160i \(-0.480686\pi\)
0.0606394 + 0.998160i \(0.480686\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 35222.0 1.69783
\(756\) 0 0
\(757\) −34564.0 −1.65951 −0.829756 − 0.558127i \(-0.811520\pi\)
−0.829756 + 0.558127i \(0.811520\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23792.0 1.13332 0.566662 − 0.823950i \(-0.308235\pi\)
0.566662 + 0.823950i \(0.308235\pi\)
\(762\) 0 0
\(763\) −19320.0 −0.916685
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5824.00 0.274175
\(768\) 0 0
\(769\) 8693.00 0.407643 0.203822 − 0.979008i \(-0.434664\pi\)
0.203822 + 0.979008i \(0.434664\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8979.00 0.417791 0.208895 − 0.977938i \(-0.433013\pi\)
0.208895 + 0.977938i \(0.433013\pi\)
\(774\) 0 0
\(775\) −296.000 −0.0137195
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1830.00 −0.0841676
\(780\) 0 0
\(781\) 1892.00 0.0866851
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −16940.0 −0.770210
\(786\) 0 0
\(787\) 37795.0 1.71188 0.855938 − 0.517079i \(-0.172981\pi\)
0.855938 + 0.517079i \(0.172981\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 56880.0 2.55679
\(792\) 0 0
\(793\) −8288.00 −0.371142
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29483.0 −1.31034 −0.655170 − 0.755481i \(-0.727403\pi\)
−0.655170 + 0.755481i \(0.727403\pi\)
\(798\) 0 0
\(799\) −14070.0 −0.622980
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1436.00 −0.0631075
\(804\) 0 0
\(805\) 17490.0 0.765766
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 41462.0 1.80189 0.900943 − 0.433937i \(-0.142876\pi\)
0.900943 + 0.433937i \(0.142876\pi\)
\(810\) 0 0
\(811\) 30961.0 1.34055 0.670276 − 0.742112i \(-0.266176\pi\)
0.670276 + 0.742112i \(0.266176\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 42669.0 1.83390
\(816\) 0 0
\(817\) 1460.00 0.0625201
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22263.0 0.946387 0.473194 − 0.880958i \(-0.343101\pi\)
0.473194 + 0.880958i \(0.343101\pi\)
\(822\) 0 0
\(823\) −33112.0 −1.40244 −0.701222 − 0.712943i \(-0.747362\pi\)
−0.701222 + 0.712943i \(0.747362\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22970.0 0.965835 0.482917 − 0.875666i \(-0.339577\pi\)
0.482917 + 0.875666i \(0.339577\pi\)
\(828\) 0 0
\(829\) −24386.0 −1.02167 −0.510833 − 0.859680i \(-0.670663\pi\)
−0.510833 + 0.859680i \(0.670663\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −38990.0 −1.62176
\(834\) 0 0
\(835\) −32373.0 −1.34169
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 35331.0 1.45383 0.726914 − 0.686729i \(-0.240954\pi\)
0.726914 + 0.686729i \(0.240954\pi\)
\(840\) 0 0
\(841\) −24164.0 −0.990775
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21351.0 0.869227
\(846\) 0 0
\(847\) −39450.0 −1.60038
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8268.00 0.333047
\(852\) 0 0
\(853\) −334.000 −0.0134067 −0.00670337 − 0.999978i \(-0.502134\pi\)
−0.00670337 + 0.999978i \(0.502134\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6822.00 −0.271920 −0.135960 − 0.990714i \(-0.543412\pi\)
−0.135960 + 0.990714i \(0.543412\pi\)
\(858\) 0 0
\(859\) 11553.0 0.458886 0.229443 − 0.973322i \(-0.426310\pi\)
0.229443 + 0.973322i \(0.426310\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31367.0 −1.23725 −0.618624 − 0.785687i \(-0.712310\pi\)
−0.618624 + 0.785687i \(0.712310\pi\)
\(864\) 0 0
\(865\) 20625.0 0.810718
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4096.00 0.159893
\(870\) 0 0
\(871\) −112.000 −0.00435703
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 42570.0 1.64472
\(876\) 0 0
\(877\) 24162.0 0.930322 0.465161 − 0.885226i \(-0.345996\pi\)
0.465161 + 0.885226i \(0.345996\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −25072.0 −0.958794 −0.479397 − 0.877598i \(-0.659145\pi\)
−0.479397 + 0.877598i \(0.659145\pi\)
\(882\) 0 0
\(883\) 20612.0 0.785559 0.392780 − 0.919633i \(-0.371513\pi\)
0.392780 + 0.919633i \(0.371513\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9776.00 0.370063 0.185032 − 0.982733i \(-0.440761\pi\)
0.185032 + 0.982733i \(0.440761\pi\)
\(888\) 0 0
\(889\) 67440.0 2.54428
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1005.00 0.0376607
\(894\) 0 0
\(895\) 18546.0 0.692653
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1110.00 0.0411797
\(900\) 0 0
\(901\) 38990.0 1.44167
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7656.00 −0.281209
\(906\) 0 0
\(907\) −1385.00 −0.0507036 −0.0253518 − 0.999679i \(-0.508071\pi\)
−0.0253518 + 0.999679i \(0.508071\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17239.0 0.626952 0.313476 − 0.949596i \(-0.398506\pi\)
0.313476 + 0.949596i \(0.398506\pi\)
\(912\) 0 0
\(913\) −4312.00 −0.156305
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36420.0 1.31155
\(918\) 0 0
\(919\) −40430.0 −1.45121 −0.725605 − 0.688111i \(-0.758440\pi\)
−0.725605 + 0.688111i \(0.758440\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7568.00 0.269885
\(924\) 0 0
\(925\) 624.000 0.0221805
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 40442.0 1.42827 0.714133 − 0.700010i \(-0.246821\pi\)
0.714133 + 0.700010i \(0.246821\pi\)
\(930\) 0 0
\(931\) 2785.00 0.0980394
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3080.00 0.107729
\(936\) 0 0
\(937\) 21798.0 0.759989 0.379994 − 0.924989i \(-0.375926\pi\)
0.379994 + 0.924989i \(0.375926\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12075.0 −0.418314 −0.209157 − 0.977882i \(-0.567072\pi\)
−0.209157 + 0.977882i \(0.567072\pi\)
\(942\) 0 0
\(943\) 19398.0 0.669868
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46662.0 −1.60117 −0.800587 − 0.599217i \(-0.795479\pi\)
−0.800587 + 0.599217i \(0.795479\pi\)
\(948\) 0 0
\(949\) −5744.00 −0.196479
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29628.0 1.00708 0.503539 − 0.863973i \(-0.332031\pi\)
0.503539 + 0.863973i \(0.332031\pi\)
\(954\) 0 0
\(955\) 23793.0 0.806203
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −88920.0 −2.99414
\(960\) 0 0
\(961\) −24315.0 −0.816186
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 7370.00 0.245854
\(966\) 0 0
\(967\) −18040.0 −0.599925 −0.299962 − 0.953951i \(-0.596974\pi\)
−0.299962 + 0.953951i \(0.596974\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6276.00 −0.207422 −0.103711 − 0.994607i \(-0.533072\pi\)
−0.103711 + 0.994607i \(0.533072\pi\)
\(972\) 0 0
\(973\) −77040.0 −2.53832
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 20928.0 0.685308 0.342654 − 0.939462i \(-0.388674\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(978\) 0 0
\(979\) −4176.00 −0.136328
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34128.0 1.10734 0.553669 − 0.832737i \(-0.313227\pi\)
0.553669 + 0.832737i \(0.313227\pi\)
\(984\) 0 0
\(985\) 31350.0 1.01411
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15476.0 −0.497582
\(990\) 0 0
\(991\) 52.0000 0.00166684 0.000833418 − 1.00000i \(-0.499735\pi\)
0.000833418 1.00000i \(0.499735\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −46244.0 −1.47340
\(996\) 0 0
\(997\) −49754.0 −1.58047 −0.790233 − 0.612806i \(-0.790041\pi\)
−0.790233 + 0.612806i \(0.790041\pi\)
\(998\) 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 1944.4.a.a.1.1 ✓ 1
3.2 odd 2 1944.4.a.f.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1944.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
1944.4.a.f.1.1 yes 1 3.2 odd 2