Properties

Label 768.4.d.g.385.1
Level $768$
Weight $4$
Character 768.385
Analytic conductor $45.313$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(385,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.385");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 385.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 768.385
Dual form 768.4.d.g.385.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -18.0000i q^{5} -8.00000 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -18.0000i q^{5} -8.00000 q^{7} -9.00000 q^{9} +36.0000i q^{11} +10.0000i q^{13} -54.0000 q^{15} +18.0000 q^{17} +100.000i q^{19} +24.0000i q^{21} -72.0000 q^{23} -199.000 q^{25} +27.0000i q^{27} +234.000i q^{29} -16.0000 q^{31} +108.000 q^{33} +144.000i q^{35} -226.000i q^{37} +30.0000 q^{39} -90.0000 q^{41} +452.000i q^{43} +162.000i q^{45} +432.000 q^{47} -279.000 q^{49} -54.0000i q^{51} +414.000i q^{53} +648.000 q^{55} +300.000 q^{57} -684.000i q^{59} -422.000i q^{61} +72.0000 q^{63} +180.000 q^{65} -332.000i q^{67} +216.000i q^{69} +360.000 q^{71} -26.0000 q^{73} +597.000i q^{75} -288.000i q^{77} +512.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} -324.000i q^{85} +702.000 q^{87} +630.000 q^{89} -80.0000i q^{91} +48.0000i q^{93} +1800.00 q^{95} -1054.00 q^{97} -324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{7} - 18 q^{9} - 108 q^{15} + 36 q^{17} - 144 q^{23} - 398 q^{25} - 32 q^{31} + 216 q^{33} + 60 q^{39} - 180 q^{41} + 864 q^{47} - 558 q^{49} + 1296 q^{55} + 600 q^{57} + 144 q^{63} + 360 q^{65} + 720 q^{71} - 52 q^{73} + 1024 q^{79} + 162 q^{81} + 1404 q^{87} + 1260 q^{89} + 3600 q^{95} - 2108 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) − 18.0000i − 1.60997i −0.593296 0.804984i \(-0.702174\pi\)
0.593296 0.804984i \(-0.297826\pi\)
\(6\) 0 0
\(7\) −8.00000 −0.431959 −0.215980 0.976398i \(-0.569295\pi\)
−0.215980 + 0.976398i \(0.569295\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 36.0000i 0.986764i 0.869813 + 0.493382i \(0.164240\pi\)
−0.869813 + 0.493382i \(0.835760\pi\)
\(12\) 0 0
\(13\) 10.0000i 0.213346i 0.994294 + 0.106673i \(0.0340198\pi\)
−0.994294 + 0.106673i \(0.965980\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) 0 0
\(17\) 18.0000 0.256802 0.128401 0.991722i \(-0.459015\pi\)
0.128401 + 0.991722i \(0.459015\pi\)
\(18\) 0 0
\(19\) 100.000i 1.20745i 0.797192 + 0.603726i \(0.206318\pi\)
−0.797192 + 0.603726i \(0.793682\pi\)
\(20\) 0 0
\(21\) 24.0000i 0.249392i
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) −199.000 −1.59200
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 234.000i 1.49837i 0.662361 + 0.749185i \(0.269554\pi\)
−0.662361 + 0.749185i \(0.730446\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) 108.000 0.569709
\(34\) 0 0
\(35\) 144.000i 0.695441i
\(36\) 0 0
\(37\) − 226.000i − 1.00417i −0.864819 0.502083i \(-0.832567\pi\)
0.864819 0.502083i \(-0.167433\pi\)
\(38\) 0 0
\(39\) 30.0000 0.123176
\(40\) 0 0
\(41\) −90.0000 −0.342820 −0.171410 0.985200i \(-0.554832\pi\)
−0.171410 + 0.985200i \(0.554832\pi\)
\(42\) 0 0
\(43\) 452.000i 1.60301i 0.597989 + 0.801504i \(0.295967\pi\)
−0.597989 + 0.801504i \(0.704033\pi\)
\(44\) 0 0
\(45\) 162.000i 0.536656i
\(46\) 0 0
\(47\) 432.000 1.34072 0.670358 0.742038i \(-0.266140\pi\)
0.670358 + 0.742038i \(0.266140\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) − 54.0000i − 0.148265i
\(52\) 0 0
\(53\) 414.000i 1.07297i 0.843911 + 0.536484i \(0.180248\pi\)
−0.843911 + 0.536484i \(0.819752\pi\)
\(54\) 0 0
\(55\) 648.000 1.58866
\(56\) 0 0
\(57\) 300.000 0.697122
\(58\) 0 0
\(59\) − 684.000i − 1.50931i −0.656123 0.754654i \(-0.727805\pi\)
0.656123 0.754654i \(-0.272195\pi\)
\(60\) 0 0
\(61\) − 422.000i − 0.885763i −0.896580 0.442882i \(-0.853956\pi\)
0.896580 0.442882i \(-0.146044\pi\)
\(62\) 0 0
\(63\) 72.0000 0.143986
\(64\) 0 0
\(65\) 180.000 0.343481
\(66\) 0 0
\(67\) − 332.000i − 0.605377i −0.953090 0.302688i \(-0.902116\pi\)
0.953090 0.302688i \(-0.0978842\pi\)
\(68\) 0 0
\(69\) 216.000i 0.376860i
\(70\) 0 0
\(71\) 360.000 0.601748 0.300874 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) −26.0000 −0.0416859 −0.0208429 0.999783i \(-0.506635\pi\)
−0.0208429 + 0.999783i \(0.506635\pi\)
\(74\) 0 0
\(75\) 597.000i 0.919142i
\(76\) 0 0
\(77\) − 288.000i − 0.426242i
\(78\) 0 0
\(79\) 512.000 0.729171 0.364585 0.931170i \(-0.381211\pi\)
0.364585 + 0.931170i \(0.381211\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1188.00i 1.57108i 0.618809 + 0.785542i \(0.287616\pi\)
−0.618809 + 0.785542i \(0.712384\pi\)
\(84\) 0 0
\(85\) − 324.000i − 0.413444i
\(86\) 0 0
\(87\) 702.000 0.865084
\(88\) 0 0
\(89\) 630.000 0.750336 0.375168 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 0 0
\(91\) − 80.0000i − 0.0921569i
\(92\) 0 0
\(93\) 48.0000i 0.0535201i
\(94\) 0 0
\(95\) 1800.00 1.94396
\(96\) 0 0
\(97\) −1054.00 −1.10327 −0.551637 0.834085i \(-0.685996\pi\)
−0.551637 + 0.834085i \(0.685996\pi\)
\(98\) 0 0
\(99\) − 324.000i − 0.328921i
\(100\) 0 0
\(101\) 558.000i 0.549733i 0.961482 + 0.274867i \(0.0886337\pi\)
−0.961482 + 0.274867i \(0.911366\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.00765304 −0.00382652 0.999993i \(-0.501218\pi\)
−0.00382652 + 0.999993i \(0.501218\pi\)
\(104\) 0 0
\(105\) 432.000 0.401513
\(106\) 0 0
\(107\) 1764.00i 1.59376i 0.604138 + 0.796880i \(0.293518\pi\)
−0.604138 + 0.796880i \(0.706482\pi\)
\(108\) 0 0
\(109\) − 1622.00i − 1.42532i −0.701512 0.712658i \(-0.747491\pi\)
0.701512 0.712658i \(-0.252509\pi\)
\(110\) 0 0
\(111\) −678.000 −0.579756
\(112\) 0 0
\(113\) −1134.00 −0.944051 −0.472025 0.881585i \(-0.656477\pi\)
−0.472025 + 0.881585i \(0.656477\pi\)
\(114\) 0 0
\(115\) 1296.00i 1.05089i
\(116\) 0 0
\(117\) − 90.0000i − 0.0711154i
\(118\) 0 0
\(119\) −144.000 −0.110928
\(120\) 0 0
\(121\) 35.0000 0.0262960
\(122\) 0 0
\(123\) 270.000i 0.197927i
\(124\) 0 0
\(125\) 1332.00i 0.953102i
\(126\) 0 0
\(127\) −592.000 −0.413634 −0.206817 0.978380i \(-0.566310\pi\)
−0.206817 + 0.978380i \(0.566310\pi\)
\(128\) 0 0
\(129\) 1356.00 0.925497
\(130\) 0 0
\(131\) 1908.00i 1.27254i 0.771466 + 0.636270i \(0.219524\pi\)
−0.771466 + 0.636270i \(0.780476\pi\)
\(132\) 0 0
\(133\) − 800.000i − 0.521570i
\(134\) 0 0
\(135\) 486.000 0.309839
\(136\) 0 0
\(137\) −954.000 −0.594932 −0.297466 0.954732i \(-0.596142\pi\)
−0.297466 + 0.954732i \(0.596142\pi\)
\(138\) 0 0
\(139\) 2564.00i 1.56457i 0.622919 + 0.782286i \(0.285947\pi\)
−0.622919 + 0.782286i \(0.714053\pi\)
\(140\) 0 0
\(141\) − 1296.00i − 0.774063i
\(142\) 0 0
\(143\) −360.000 −0.210522
\(144\) 0 0
\(145\) 4212.00 2.41233
\(146\) 0 0
\(147\) 837.000i 0.469623i
\(148\) 0 0
\(149\) − 738.000i − 0.405767i −0.979203 0.202884i \(-0.934969\pi\)
0.979203 0.202884i \(-0.0650313\pi\)
\(150\) 0 0
\(151\) 2440.00 1.31500 0.657498 0.753456i \(-0.271615\pi\)
0.657498 + 0.753456i \(0.271615\pi\)
\(152\) 0 0
\(153\) −162.000 −0.0856008
\(154\) 0 0
\(155\) 288.000i 0.149243i
\(156\) 0 0
\(157\) 2554.00i 1.29829i 0.760665 + 0.649145i \(0.224873\pi\)
−0.760665 + 0.649145i \(0.775127\pi\)
\(158\) 0 0
\(159\) 1242.00 0.619478
\(160\) 0 0
\(161\) 576.000 0.281958
\(162\) 0 0
\(163\) 820.000i 0.394033i 0.980400 + 0.197016i \(0.0631252\pi\)
−0.980400 + 0.197016i \(0.936875\pi\)
\(164\) 0 0
\(165\) − 1944.00i − 0.917213i
\(166\) 0 0
\(167\) −1944.00 −0.900786 −0.450393 0.892830i \(-0.648716\pi\)
−0.450393 + 0.892830i \(0.648716\pi\)
\(168\) 0 0
\(169\) 2097.00 0.954483
\(170\) 0 0
\(171\) − 900.000i − 0.402484i
\(172\) 0 0
\(173\) 1242.00i 0.545824i 0.962039 + 0.272912i \(0.0879867\pi\)
−0.962039 + 0.272912i \(0.912013\pi\)
\(174\) 0 0
\(175\) 1592.00 0.687679
\(176\) 0 0
\(177\) −2052.00 −0.871400
\(178\) 0 0
\(179\) − 1116.00i − 0.465999i −0.972477 0.232999i \(-0.925146\pi\)
0.972477 0.232999i \(-0.0748540\pi\)
\(180\) 0 0
\(181\) 1070.00i 0.439406i 0.975567 + 0.219703i \(0.0705088\pi\)
−0.975567 + 0.219703i \(0.929491\pi\)
\(182\) 0 0
\(183\) −1266.00 −0.511396
\(184\) 0 0
\(185\) −4068.00 −1.61668
\(186\) 0 0
\(187\) 648.000i 0.253403i
\(188\) 0 0
\(189\) − 216.000i − 0.0831306i
\(190\) 0 0
\(191\) −576.000 −0.218209 −0.109104 0.994030i \(-0.534798\pi\)
−0.109104 + 0.994030i \(0.534798\pi\)
\(192\) 0 0
\(193\) −1342.00 −0.500514 −0.250257 0.968179i \(-0.580515\pi\)
−0.250257 + 0.968179i \(0.580515\pi\)
\(194\) 0 0
\(195\) − 540.000i − 0.198309i
\(196\) 0 0
\(197\) 1422.00i 0.514281i 0.966374 + 0.257140i \(0.0827803\pi\)
−0.966374 + 0.257140i \(0.917220\pi\)
\(198\) 0 0
\(199\) −872.000 −0.310625 −0.155313 0.987865i \(-0.549639\pi\)
−0.155313 + 0.987865i \(0.549639\pi\)
\(200\) 0 0
\(201\) −996.000 −0.349515
\(202\) 0 0
\(203\) − 1872.00i − 0.647235i
\(204\) 0 0
\(205\) 1620.00i 0.551930i
\(206\) 0 0
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) − 1340.00i − 0.437201i −0.975814 0.218600i \(-0.929851\pi\)
0.975814 0.218600i \(-0.0701491\pi\)
\(212\) 0 0
\(213\) − 1080.00i − 0.347420i
\(214\) 0 0
\(215\) 8136.00 2.58079
\(216\) 0 0
\(217\) 128.000 0.0400424
\(218\) 0 0
\(219\) 78.0000i 0.0240674i
\(220\) 0 0
\(221\) 180.000i 0.0547878i
\(222\) 0 0
\(223\) 4880.00 1.46542 0.732711 0.680540i \(-0.238255\pi\)
0.732711 + 0.680540i \(0.238255\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) − 2700.00i − 0.789451i −0.918799 0.394725i \(-0.870840\pi\)
0.918799 0.394725i \(-0.129160\pi\)
\(228\) 0 0
\(229\) 254.000i 0.0732960i 0.999328 + 0.0366480i \(0.0116680\pi\)
−0.999328 + 0.0366480i \(0.988332\pi\)
\(230\) 0 0
\(231\) −864.000 −0.246091
\(232\) 0 0
\(233\) −4410.00 −1.23995 −0.619976 0.784621i \(-0.712858\pi\)
−0.619976 + 0.784621i \(0.712858\pi\)
\(234\) 0 0
\(235\) − 7776.00i − 2.15851i
\(236\) 0 0
\(237\) − 1536.00i − 0.420987i
\(238\) 0 0
\(239\) −3888.00 −1.05228 −0.526138 0.850399i \(-0.676360\pi\)
−0.526138 + 0.850399i \(0.676360\pi\)
\(240\) 0 0
\(241\) 5138.00 1.37331 0.686655 0.726984i \(-0.259078\pi\)
0.686655 + 0.726984i \(0.259078\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 5022.00i 1.30957i
\(246\) 0 0
\(247\) −1000.00 −0.257605
\(248\) 0 0
\(249\) 3564.00 0.907066
\(250\) 0 0
\(251\) 4788.00i 1.20405i 0.798478 + 0.602024i \(0.205639\pi\)
−0.798478 + 0.602024i \(0.794361\pi\)
\(252\) 0 0
\(253\) − 2592.00i − 0.644101i
\(254\) 0 0
\(255\) −972.000 −0.238702
\(256\) 0 0
\(257\) −5886.00 −1.42863 −0.714316 0.699823i \(-0.753262\pi\)
−0.714316 + 0.699823i \(0.753262\pi\)
\(258\) 0 0
\(259\) 1808.00i 0.433759i
\(260\) 0 0
\(261\) − 2106.00i − 0.499456i
\(262\) 0 0
\(263\) −2232.00 −0.523312 −0.261656 0.965161i \(-0.584269\pi\)
−0.261656 + 0.965161i \(0.584269\pi\)
\(264\) 0 0
\(265\) 7452.00 1.72744
\(266\) 0 0
\(267\) − 1890.00i − 0.433206i
\(268\) 0 0
\(269\) 666.000i 0.150954i 0.997148 + 0.0754772i \(0.0240480\pi\)
−0.997148 + 0.0754772i \(0.975952\pi\)
\(270\) 0 0
\(271\) −5536.00 −1.24092 −0.620458 0.784240i \(-0.713053\pi\)
−0.620458 + 0.784240i \(0.713053\pi\)
\(272\) 0 0
\(273\) −240.000 −0.0532068
\(274\) 0 0
\(275\) − 7164.00i − 1.57093i
\(276\) 0 0
\(277\) 2126.00i 0.461151i 0.973054 + 0.230576i \(0.0740609\pi\)
−0.973054 + 0.230576i \(0.925939\pi\)
\(278\) 0 0
\(279\) 144.000 0.0308998
\(280\) 0 0
\(281\) 2934.00 0.622875 0.311437 0.950267i \(-0.399190\pi\)
0.311437 + 0.950267i \(0.399190\pi\)
\(282\) 0 0
\(283\) 2036.00i 0.427659i 0.976871 + 0.213830i \(0.0685938\pi\)
−0.976871 + 0.213830i \(0.931406\pi\)
\(284\) 0 0
\(285\) − 5400.00i − 1.12235i
\(286\) 0 0
\(287\) 720.000 0.148085
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 3162.00i 0.636975i
\(292\) 0 0
\(293\) 2286.00i 0.455800i 0.973684 + 0.227900i \(0.0731860\pi\)
−0.973684 + 0.227900i \(0.926814\pi\)
\(294\) 0 0
\(295\) −12312.0 −2.42994
\(296\) 0 0
\(297\) −972.000 −0.189903
\(298\) 0 0
\(299\) − 720.000i − 0.139260i
\(300\) 0 0
\(301\) − 3616.00i − 0.692434i
\(302\) 0 0
\(303\) 1674.00 0.317389
\(304\) 0 0
\(305\) −7596.00 −1.42605
\(306\) 0 0
\(307\) − 1244.00i − 0.231267i −0.993292 0.115633i \(-0.963110\pi\)
0.993292 0.115633i \(-0.0368897\pi\)
\(308\) 0 0
\(309\) 24.0000i 0.00441849i
\(310\) 0 0
\(311\) −1224.00 −0.223173 −0.111586 0.993755i \(-0.535593\pi\)
−0.111586 + 0.993755i \(0.535593\pi\)
\(312\) 0 0
\(313\) −1898.00 −0.342752 −0.171376 0.985206i \(-0.554821\pi\)
−0.171376 + 0.985206i \(0.554821\pi\)
\(314\) 0 0
\(315\) − 1296.00i − 0.231814i
\(316\) 0 0
\(317\) 9162.00i 1.62331i 0.584137 + 0.811655i \(0.301433\pi\)
−0.584137 + 0.811655i \(0.698567\pi\)
\(318\) 0 0
\(319\) −8424.00 −1.47854
\(320\) 0 0
\(321\) 5292.00 0.920158
\(322\) 0 0
\(323\) 1800.00i 0.310076i
\(324\) 0 0
\(325\) − 1990.00i − 0.339647i
\(326\) 0 0
\(327\) −4866.00 −0.822906
\(328\) 0 0
\(329\) −3456.00 −0.579135
\(330\) 0 0
\(331\) − 4348.00i − 0.722017i −0.932562 0.361009i \(-0.882432\pi\)
0.932562 0.361009i \(-0.117568\pi\)
\(332\) 0 0
\(333\) 2034.00i 0.334722i
\(334\) 0 0
\(335\) −5976.00 −0.974638
\(336\) 0 0
\(337\) 7154.00 1.15639 0.578195 0.815899i \(-0.303757\pi\)
0.578195 + 0.815899i \(0.303757\pi\)
\(338\) 0 0
\(339\) 3402.00i 0.545048i
\(340\) 0 0
\(341\) − 576.000i − 0.0914726i
\(342\) 0 0
\(343\) 4976.00 0.783320
\(344\) 0 0
\(345\) 3888.00 0.606733
\(346\) 0 0
\(347\) − 1836.00i − 0.284039i −0.989864 0.142020i \(-0.954640\pi\)
0.989864 0.142020i \(-0.0453596\pi\)
\(348\) 0 0
\(349\) − 5894.00i − 0.904007i −0.892016 0.452004i \(-0.850709\pi\)
0.892016 0.452004i \(-0.149291\pi\)
\(350\) 0 0
\(351\) −270.000 −0.0410585
\(352\) 0 0
\(353\) 11106.0 1.67454 0.837270 0.546789i \(-0.184150\pi\)
0.837270 + 0.546789i \(0.184150\pi\)
\(354\) 0 0
\(355\) − 6480.00i − 0.968796i
\(356\) 0 0
\(357\) 432.000i 0.0640444i
\(358\) 0 0
\(359\) −13176.0 −1.93705 −0.968527 0.248907i \(-0.919929\pi\)
−0.968527 + 0.248907i \(0.919929\pi\)
\(360\) 0 0
\(361\) −3141.00 −0.457938
\(362\) 0 0
\(363\) − 105.000i − 0.0151820i
\(364\) 0 0
\(365\) 468.000i 0.0671130i
\(366\) 0 0
\(367\) −6112.00 −0.869329 −0.434665 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(368\) 0 0
\(369\) 810.000 0.114273
\(370\) 0 0
\(371\) − 3312.00i − 0.463478i
\(372\) 0 0
\(373\) − 13618.0i − 1.89038i −0.326515 0.945192i \(-0.605874\pi\)
0.326515 0.945192i \(-0.394126\pi\)
\(374\) 0 0
\(375\) 3996.00 0.550273
\(376\) 0 0
\(377\) −2340.00 −0.319671
\(378\) 0 0
\(379\) 692.000i 0.0937880i 0.998900 + 0.0468940i \(0.0149323\pi\)
−0.998900 + 0.0468940i \(0.985068\pi\)
\(380\) 0 0
\(381\) 1776.00i 0.238812i
\(382\) 0 0
\(383\) −8064.00 −1.07585 −0.537926 0.842992i \(-0.680792\pi\)
−0.537926 + 0.842992i \(0.680792\pi\)
\(384\) 0 0
\(385\) −5184.00 −0.686237
\(386\) 0 0
\(387\) − 4068.00i − 0.534336i
\(388\) 0 0
\(389\) 12654.0i 1.64931i 0.565633 + 0.824657i \(0.308632\pi\)
−0.565633 + 0.824657i \(0.691368\pi\)
\(390\) 0 0
\(391\) −1296.00 −0.167625
\(392\) 0 0
\(393\) 5724.00 0.734701
\(394\) 0 0
\(395\) − 9216.00i − 1.17394i
\(396\) 0 0
\(397\) 106.000i 0.0134005i 0.999978 + 0.00670024i \(0.00213277\pi\)
−0.999978 + 0.00670024i \(0.997867\pi\)
\(398\) 0 0
\(399\) −2400.00 −0.301129
\(400\) 0 0
\(401\) −4014.00 −0.499874 −0.249937 0.968262i \(-0.580410\pi\)
−0.249937 + 0.968262i \(0.580410\pi\)
\(402\) 0 0
\(403\) − 160.000i − 0.0197771i
\(404\) 0 0
\(405\) − 1458.00i − 0.178885i
\(406\) 0 0
\(407\) 8136.00 0.990876
\(408\) 0 0
\(409\) −3914.00 −0.473190 −0.236595 0.971608i \(-0.576032\pi\)
−0.236595 + 0.971608i \(0.576032\pi\)
\(410\) 0 0
\(411\) 2862.00i 0.343484i
\(412\) 0 0
\(413\) 5472.00i 0.651960i
\(414\) 0 0
\(415\) 21384.0 2.52940
\(416\) 0 0
\(417\) 7692.00 0.903307
\(418\) 0 0
\(419\) − 4428.00i − 0.516282i −0.966107 0.258141i \(-0.916890\pi\)
0.966107 0.258141i \(-0.0831098\pi\)
\(420\) 0 0
\(421\) − 15490.0i − 1.79320i −0.442843 0.896599i \(-0.646030\pi\)
0.442843 0.896599i \(-0.353970\pi\)
\(422\) 0 0
\(423\) −3888.00 −0.446906
\(424\) 0 0
\(425\) −3582.00 −0.408829
\(426\) 0 0
\(427\) 3376.00i 0.382614i
\(428\) 0 0
\(429\) 1080.00i 0.121545i
\(430\) 0 0
\(431\) 6768.00 0.756388 0.378194 0.925726i \(-0.376545\pi\)
0.378194 + 0.925726i \(0.376545\pi\)
\(432\) 0 0
\(433\) 1298.00 0.144060 0.0720299 0.997402i \(-0.477052\pi\)
0.0720299 + 0.997402i \(0.477052\pi\)
\(434\) 0 0
\(435\) − 12636.0i − 1.39276i
\(436\) 0 0
\(437\) − 7200.00i − 0.788153i
\(438\) 0 0
\(439\) 2248.00 0.244399 0.122200 0.992506i \(-0.461005\pi\)
0.122200 + 0.992506i \(0.461005\pi\)
\(440\) 0 0
\(441\) 2511.00 0.271137
\(442\) 0 0
\(443\) − 9612.00i − 1.03088i −0.856926 0.515440i \(-0.827628\pi\)
0.856926 0.515440i \(-0.172372\pi\)
\(444\) 0 0
\(445\) − 11340.0i − 1.20802i
\(446\) 0 0
\(447\) −2214.00 −0.234270
\(448\) 0 0
\(449\) 162.000 0.0170273 0.00851364 0.999964i \(-0.497290\pi\)
0.00851364 + 0.999964i \(0.497290\pi\)
\(450\) 0 0
\(451\) − 3240.00i − 0.338283i
\(452\) 0 0
\(453\) − 7320.00i − 0.759213i
\(454\) 0 0
\(455\) −1440.00 −0.148370
\(456\) 0 0
\(457\) −1370.00 −0.140232 −0.0701159 0.997539i \(-0.522337\pi\)
−0.0701159 + 0.997539i \(0.522337\pi\)
\(458\) 0 0
\(459\) 486.000i 0.0494217i
\(460\) 0 0
\(461\) 15354.0i 1.55121i 0.631220 + 0.775604i \(0.282555\pi\)
−0.631220 + 0.775604i \(0.717445\pi\)
\(462\) 0 0
\(463\) −13024.0 −1.30729 −0.653646 0.756800i \(-0.726762\pi\)
−0.653646 + 0.756800i \(0.726762\pi\)
\(464\) 0 0
\(465\) 864.000 0.0861657
\(466\) 0 0
\(467\) 14436.0i 1.43045i 0.698896 + 0.715223i \(0.253675\pi\)
−0.698896 + 0.715223i \(0.746325\pi\)
\(468\) 0 0
\(469\) 2656.00i 0.261498i
\(470\) 0 0
\(471\) 7662.00 0.749568
\(472\) 0 0
\(473\) −16272.0 −1.58179
\(474\) 0 0
\(475\) − 19900.0i − 1.92226i
\(476\) 0 0
\(477\) − 3726.00i − 0.357656i
\(478\) 0 0
\(479\) 12096.0 1.15382 0.576911 0.816807i \(-0.304258\pi\)
0.576911 + 0.816807i \(0.304258\pi\)
\(480\) 0 0
\(481\) 2260.00 0.214235
\(482\) 0 0
\(483\) − 1728.00i − 0.162788i
\(484\) 0 0
\(485\) 18972.0i 1.77624i
\(486\) 0 0
\(487\) −6056.00 −0.563498 −0.281749 0.959488i \(-0.590915\pi\)
−0.281749 + 0.959488i \(0.590915\pi\)
\(488\) 0 0
\(489\) 2460.00 0.227495
\(490\) 0 0
\(491\) 7524.00i 0.691555i 0.938317 + 0.345777i \(0.112385\pi\)
−0.938317 + 0.345777i \(0.887615\pi\)
\(492\) 0 0
\(493\) 4212.00i 0.384785i
\(494\) 0 0
\(495\) −5832.00 −0.529553
\(496\) 0 0
\(497\) −2880.00 −0.259931
\(498\) 0 0
\(499\) − 5276.00i − 0.473319i −0.971593 0.236660i \(-0.923947\pi\)
0.971593 0.236660i \(-0.0760526\pi\)
\(500\) 0 0
\(501\) 5832.00i 0.520069i
\(502\) 0 0
\(503\) −4968.00 −0.440382 −0.220191 0.975457i \(-0.570668\pi\)
−0.220191 + 0.975457i \(0.570668\pi\)
\(504\) 0 0
\(505\) 10044.0 0.885054
\(506\) 0 0
\(507\) − 6291.00i − 0.551071i
\(508\) 0 0
\(509\) − 10998.0i − 0.957717i −0.877892 0.478858i \(-0.841051\pi\)
0.877892 0.478858i \(-0.158949\pi\)
\(510\) 0 0
\(511\) 208.000 0.0180066
\(512\) 0 0
\(513\) −2700.00 −0.232374
\(514\) 0 0
\(515\) 144.000i 0.0123212i
\(516\) 0 0
\(517\) 15552.0i 1.32297i
\(518\) 0 0
\(519\) 3726.00 0.315131
\(520\) 0 0
\(521\) 8838.00 0.743186 0.371593 0.928396i \(-0.378812\pi\)
0.371593 + 0.928396i \(0.378812\pi\)
\(522\) 0 0
\(523\) 22436.0i 1.87583i 0.346869 + 0.937914i \(0.387245\pi\)
−0.346869 + 0.937914i \(0.612755\pi\)
\(524\) 0 0
\(525\) − 4776.00i − 0.397032i
\(526\) 0 0
\(527\) −288.000 −0.0238055
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 6156.00i 0.503103i
\(532\) 0 0
\(533\) − 900.000i − 0.0731395i
\(534\) 0 0
\(535\) 31752.0 2.56590
\(536\) 0 0
\(537\) −3348.00 −0.269044
\(538\) 0 0
\(539\) − 10044.0i − 0.802645i
\(540\) 0 0
\(541\) 4762.00i 0.378437i 0.981935 + 0.189218i \(0.0605954\pi\)
−0.981935 + 0.189218i \(0.939405\pi\)
\(542\) 0 0
\(543\) 3210.00 0.253691
\(544\) 0 0
\(545\) −29196.0 −2.29471
\(546\) 0 0
\(547\) 6004.00i 0.469310i 0.972079 + 0.234655i \(0.0753960\pi\)
−0.972079 + 0.234655i \(0.924604\pi\)
\(548\) 0 0
\(549\) 3798.00i 0.295254i
\(550\) 0 0
\(551\) −23400.0 −1.80921
\(552\) 0 0
\(553\) −4096.00 −0.314972
\(554\) 0 0
\(555\) 12204.0i 0.933389i
\(556\) 0 0
\(557\) 5274.00i 0.401197i 0.979674 + 0.200598i \(0.0642886\pi\)
−0.979674 + 0.200598i \(0.935711\pi\)
\(558\) 0 0
\(559\) −4520.00 −0.341996
\(560\) 0 0
\(561\) 1944.00 0.146303
\(562\) 0 0
\(563\) 12420.0i 0.929735i 0.885380 + 0.464867i \(0.153898\pi\)
−0.885380 + 0.464867i \(0.846102\pi\)
\(564\) 0 0
\(565\) 20412.0i 1.51989i
\(566\) 0 0
\(567\) −648.000 −0.0479955
\(568\) 0 0
\(569\) 21366.0 1.57418 0.787091 0.616837i \(-0.211586\pi\)
0.787091 + 0.616837i \(0.211586\pi\)
\(570\) 0 0
\(571\) 21140.0i 1.54935i 0.632357 + 0.774677i \(0.282088\pi\)
−0.632357 + 0.774677i \(0.717912\pi\)
\(572\) 0 0
\(573\) 1728.00i 0.125983i
\(574\) 0 0
\(575\) 14328.0 1.03916
\(576\) 0 0
\(577\) 3266.00 0.235642 0.117821 0.993035i \(-0.462409\pi\)
0.117821 + 0.993035i \(0.462409\pi\)
\(578\) 0 0
\(579\) 4026.00i 0.288972i
\(580\) 0 0
\(581\) − 9504.00i − 0.678644i
\(582\) 0 0
\(583\) −14904.0 −1.05877
\(584\) 0 0
\(585\) −1620.00 −0.114494
\(586\) 0 0
\(587\) 17028.0i 1.19731i 0.801007 + 0.598655i \(0.204298\pi\)
−0.801007 + 0.598655i \(0.795702\pi\)
\(588\) 0 0
\(589\) − 1600.00i − 0.111930i
\(590\) 0 0
\(591\) 4266.00 0.296920
\(592\) 0 0
\(593\) 9522.00 0.659396 0.329698 0.944086i \(-0.393053\pi\)
0.329698 + 0.944086i \(0.393053\pi\)
\(594\) 0 0
\(595\) 2592.00i 0.178591i
\(596\) 0 0
\(597\) 2616.00i 0.179340i
\(598\) 0 0
\(599\) 10296.0 0.702309 0.351155 0.936318i \(-0.385789\pi\)
0.351155 + 0.936318i \(0.385789\pi\)
\(600\) 0 0
\(601\) 3382.00 0.229542 0.114771 0.993392i \(-0.463387\pi\)
0.114771 + 0.993392i \(0.463387\pi\)
\(602\) 0 0
\(603\) 2988.00i 0.201792i
\(604\) 0 0
\(605\) − 630.000i − 0.0423358i
\(606\) 0 0
\(607\) −20656.0 −1.38122 −0.690611 0.723227i \(-0.742658\pi\)
−0.690611 + 0.723227i \(0.742658\pi\)
\(608\) 0 0
\(609\) −5616.00 −0.373681
\(610\) 0 0
\(611\) 4320.00i 0.286037i
\(612\) 0 0
\(613\) − 22114.0i − 1.45706i −0.685015 0.728529i \(-0.740205\pi\)
0.685015 0.728529i \(-0.259795\pi\)
\(614\) 0 0
\(615\) 4860.00 0.318657
\(616\) 0 0
\(617\) −19962.0 −1.30250 −0.651248 0.758865i \(-0.725754\pi\)
−0.651248 + 0.758865i \(0.725754\pi\)
\(618\) 0 0
\(619\) − 604.000i − 0.0392194i −0.999808 0.0196097i \(-0.993758\pi\)
0.999808 0.0196097i \(-0.00624236\pi\)
\(620\) 0 0
\(621\) − 1944.00i − 0.125620i
\(622\) 0 0
\(623\) −5040.00 −0.324115
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) 10800.0i 0.687895i
\(628\) 0 0
\(629\) − 4068.00i − 0.257872i
\(630\) 0 0
\(631\) −152.000 −0.00958958 −0.00479479 0.999989i \(-0.501526\pi\)
−0.00479479 + 0.999989i \(0.501526\pi\)
\(632\) 0 0
\(633\) −4020.00 −0.252418
\(634\) 0 0
\(635\) 10656.0i 0.665938i
\(636\) 0 0
\(637\) − 2790.00i − 0.173538i
\(638\) 0 0
\(639\) −3240.00 −0.200583
\(640\) 0 0
\(641\) 4194.00 0.258429 0.129215 0.991617i \(-0.458754\pi\)
0.129215 + 0.991617i \(0.458754\pi\)
\(642\) 0 0
\(643\) 7252.00i 0.444776i 0.974958 + 0.222388i \(0.0713852\pi\)
−0.974958 + 0.222388i \(0.928615\pi\)
\(644\) 0 0
\(645\) − 24408.0i − 1.49002i
\(646\) 0 0
\(647\) 6696.00 0.406873 0.203437 0.979088i \(-0.434789\pi\)
0.203437 + 0.979088i \(0.434789\pi\)
\(648\) 0 0
\(649\) 24624.0 1.48933
\(650\) 0 0
\(651\) − 384.000i − 0.0231185i
\(652\) 0 0
\(653\) − 28422.0i − 1.70328i −0.524131 0.851638i \(-0.675610\pi\)
0.524131 0.851638i \(-0.324390\pi\)
\(654\) 0 0
\(655\) 34344.0 2.04875
\(656\) 0 0
\(657\) 234.000 0.0138953
\(658\) 0 0
\(659\) 19908.0i 1.17679i 0.808573 + 0.588396i \(0.200240\pi\)
−0.808573 + 0.588396i \(0.799760\pi\)
\(660\) 0 0
\(661\) 14318.0i 0.842520i 0.906940 + 0.421260i \(0.138412\pi\)
−0.906940 + 0.421260i \(0.861588\pi\)
\(662\) 0 0
\(663\) 540.000 0.0316318
\(664\) 0 0
\(665\) −14400.0 −0.839711
\(666\) 0 0
\(667\) − 16848.0i − 0.978047i
\(668\) 0 0
\(669\) − 14640.0i − 0.846061i
\(670\) 0 0
\(671\) 15192.0 0.874040
\(672\) 0 0
\(673\) 30050.0 1.72116 0.860581 0.509313i \(-0.170101\pi\)
0.860581 + 0.509313i \(0.170101\pi\)
\(674\) 0 0
\(675\) − 5373.00i − 0.306381i
\(676\) 0 0
\(677\) 22158.0i 1.25790i 0.777444 + 0.628952i \(0.216516\pi\)
−0.777444 + 0.628952i \(0.783484\pi\)
\(678\) 0 0
\(679\) 8432.00 0.476569
\(680\) 0 0
\(681\) −8100.00 −0.455790
\(682\) 0 0
\(683\) − 3132.00i − 0.175465i −0.996144 0.0877325i \(-0.972038\pi\)
0.996144 0.0877325i \(-0.0279621\pi\)
\(684\) 0 0
\(685\) 17172.0i 0.957822i
\(686\) 0 0
\(687\) 762.000 0.0423175
\(688\) 0 0
\(689\) −4140.00 −0.228914
\(690\) 0 0
\(691\) 20932.0i 1.15237i 0.817318 + 0.576187i \(0.195460\pi\)
−0.817318 + 0.576187i \(0.804540\pi\)
\(692\) 0 0
\(693\) 2592.00i 0.142081i
\(694\) 0 0
\(695\) 46152.0 2.51891
\(696\) 0 0
\(697\) −1620.00 −0.0880371
\(698\) 0 0
\(699\) 13230.0i 0.715886i
\(700\) 0 0
\(701\) 21834.0i 1.17640i 0.808714 + 0.588202i \(0.200164\pi\)
−0.808714 + 0.588202i \(0.799836\pi\)
\(702\) 0 0
\(703\) 22600.0 1.21248
\(704\) 0 0
\(705\) −23328.0 −1.24622
\(706\) 0 0
\(707\) − 4464.00i − 0.237463i
\(708\) 0 0
\(709\) 12446.0i 0.659266i 0.944109 + 0.329633i \(0.106925\pi\)
−0.944109 + 0.329633i \(0.893075\pi\)
\(710\) 0 0
\(711\) −4608.00 −0.243057
\(712\) 0 0
\(713\) 1152.00 0.0605088
\(714\) 0 0
\(715\) 6480.00i 0.338935i
\(716\) 0 0
\(717\) 11664.0i 0.607531i
\(718\) 0 0
\(719\) −12528.0 −0.649813 −0.324907 0.945746i \(-0.605333\pi\)
−0.324907 + 0.945746i \(0.605333\pi\)
\(720\) 0 0
\(721\) 64.0000 0.00330580
\(722\) 0 0
\(723\) − 15414.0i − 0.792881i
\(724\) 0 0
\(725\) − 46566.0i − 2.38540i
\(726\) 0 0
\(727\) −11576.0 −0.590550 −0.295275 0.955412i \(-0.595411\pi\)
−0.295275 + 0.955412i \(0.595411\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 8136.00i 0.411656i
\(732\) 0 0
\(733\) 29338.0i 1.47834i 0.673519 + 0.739170i \(0.264782\pi\)
−0.673519 + 0.739170i \(0.735218\pi\)
\(734\) 0 0
\(735\) 15066.0 0.756079
\(736\) 0 0
\(737\) 11952.0 0.597364
\(738\) 0 0
\(739\) − 2540.00i − 0.126435i −0.998000 0.0632175i \(-0.979864\pi\)
0.998000 0.0632175i \(-0.0201362\pi\)
\(740\) 0 0
\(741\) 3000.00i 0.148728i
\(742\) 0 0
\(743\) 18792.0 0.927876 0.463938 0.885868i \(-0.346436\pi\)
0.463938 + 0.885868i \(0.346436\pi\)
\(744\) 0 0
\(745\) −13284.0 −0.653273
\(746\) 0 0
\(747\) − 10692.0i − 0.523695i
\(748\) 0 0
\(749\) − 14112.0i − 0.688440i
\(750\) 0 0
\(751\) 4832.00 0.234783 0.117392 0.993086i \(-0.462547\pi\)
0.117392 + 0.993086i \(0.462547\pi\)
\(752\) 0 0
\(753\) 14364.0 0.695157
\(754\) 0 0
\(755\) − 43920.0i − 2.11710i
\(756\) 0 0
\(757\) − 20818.0i − 0.999529i −0.866161 0.499764i \(-0.833420\pi\)
0.866161 0.499764i \(-0.166580\pi\)
\(758\) 0 0
\(759\) −7776.00 −0.371872
\(760\) 0 0
\(761\) −12042.0 −0.573617 −0.286808 0.957988i \(-0.592594\pi\)
−0.286808 + 0.957988i \(0.592594\pi\)
\(762\) 0 0
\(763\) 12976.0i 0.615679i
\(764\) 0 0
\(765\) 2916.00i 0.137815i
\(766\) 0 0
\(767\) 6840.00 0.322005
\(768\) 0 0
\(769\) 13058.0 0.612332 0.306166 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(770\) 0 0
\(771\) 17658.0i 0.824821i
\(772\) 0 0
\(773\) − 11826.0i − 0.550261i −0.961407 0.275130i \(-0.911279\pi\)
0.961407 0.275130i \(-0.0887210\pi\)
\(774\) 0 0
\(775\) 3184.00 0.147578
\(776\) 0 0
\(777\) 5424.00 0.250431
\(778\) 0 0
\(779\) − 9000.00i − 0.413939i
\(780\) 0 0
\(781\) 12960.0i 0.593784i
\(782\) 0 0
\(783\) −6318.00 −0.288361
\(784\) 0 0
\(785\) 45972.0 2.09021
\(786\) 0 0
\(787\) − 11996.0i − 0.543343i −0.962390 0.271672i \(-0.912424\pi\)
0.962390 0.271672i \(-0.0875765\pi\)
\(788\) 0 0
\(789\) 6696.00i 0.302134i
\(790\) 0 0
\(791\) 9072.00 0.407792
\(792\) 0 0
\(793\) 4220.00 0.188974
\(794\) 0 0
\(795\) − 22356.0i − 0.997340i
\(796\) 0 0
\(797\) − 6966.00i − 0.309596i −0.987946 0.154798i \(-0.950527\pi\)
0.987946 0.154798i \(-0.0494727\pi\)
\(798\) 0 0
\(799\) 7776.00 0.344299
\(800\) 0 0
\(801\) −5670.00 −0.250112
\(802\) 0 0
\(803\) − 936.000i − 0.0411342i
\(804\) 0 0
\(805\) − 10368.0i − 0.453943i
\(806\) 0 0
\(807\) 1998.00 0.0871536
\(808\) 0 0
\(809\) 40806.0 1.77338 0.886689 0.462367i \(-0.153000\pi\)
0.886689 + 0.462367i \(0.153000\pi\)
\(810\) 0 0
\(811\) − 17980.0i − 0.778500i −0.921132 0.389250i \(-0.872734\pi\)
0.921132 0.389250i \(-0.127266\pi\)
\(812\) 0 0
\(813\) 16608.0i 0.716443i
\(814\) 0 0
\(815\) 14760.0 0.634381
\(816\) 0 0
\(817\) −45200.0 −1.93555
\(818\) 0 0
\(819\) 720.000i 0.0307190i
\(820\) 0 0
\(821\) − 12834.0i − 0.545566i −0.962076 0.272783i \(-0.912056\pi\)
0.962076 0.272783i \(-0.0879441\pi\)
\(822\) 0 0
\(823\) 37864.0 1.60371 0.801857 0.597516i \(-0.203846\pi\)
0.801857 + 0.597516i \(0.203846\pi\)
\(824\) 0 0
\(825\) −21492.0 −0.906976
\(826\) 0 0
\(827\) 42516.0i 1.78770i 0.448368 + 0.893849i \(0.352005\pi\)
−0.448368 + 0.893849i \(0.647995\pi\)
\(828\) 0 0
\(829\) − 45638.0i − 1.91203i −0.293317 0.956015i \(-0.594759\pi\)
0.293317 0.956015i \(-0.405241\pi\)
\(830\) 0 0
\(831\) 6378.00 0.266246
\(832\) 0 0
\(833\) −5022.00 −0.208886
\(834\) 0 0
\(835\) 34992.0i 1.45024i
\(836\) 0 0
\(837\) − 432.000i − 0.0178400i
\(838\) 0 0
\(839\) −17496.0 −0.719939 −0.359970 0.932964i \(-0.617213\pi\)
−0.359970 + 0.932964i \(0.617213\pi\)
\(840\) 0 0
\(841\) −30367.0 −1.24511
\(842\) 0 0
\(843\) − 8802.00i − 0.359617i
\(844\) 0 0
\(845\) − 37746.0i − 1.53669i
\(846\) 0 0
\(847\) −280.000 −0.0113588
\(848\) 0 0
\(849\) 6108.00 0.246909
\(850\) 0 0
\(851\) 16272.0i 0.655461i
\(852\) 0 0
\(853\) 32174.0i 1.29146i 0.763565 + 0.645731i \(0.223447\pi\)
−0.763565 + 0.645731i \(0.776553\pi\)
\(854\) 0 0
\(855\) −16200.0 −0.647986
\(856\) 0 0
\(857\) 38934.0 1.55188 0.775939 0.630807i \(-0.217276\pi\)
0.775939 + 0.630807i \(0.217276\pi\)
\(858\) 0 0
\(859\) 29780.0i 1.18286i 0.806355 + 0.591432i \(0.201437\pi\)
−0.806355 + 0.591432i \(0.798563\pi\)
\(860\) 0 0
\(861\) − 2160.00i − 0.0854966i
\(862\) 0 0
\(863\) −48096.0 −1.89711 −0.948556 0.316611i \(-0.897455\pi\)
−0.948556 + 0.316611i \(0.897455\pi\)
\(864\) 0 0
\(865\) 22356.0 0.878759
\(866\) 0 0
\(867\) 13767.0i 0.539275i
\(868\) 0 0
\(869\) 18432.0i 0.719520i
\(870\) 0 0
\(871\) 3320.00 0.129155
\(872\) 0 0
\(873\) 9486.00 0.367758
\(874\) 0 0
\(875\) − 10656.0i − 0.411701i
\(876\) 0 0
\(877\) − 21302.0i − 0.820202i −0.912040 0.410101i \(-0.865493\pi\)
0.912040 0.410101i \(-0.134507\pi\)
\(878\) 0 0
\(879\) 6858.00 0.263157
\(880\) 0 0
\(881\) −7470.00 −0.285665 −0.142832 0.989747i \(-0.545621\pi\)
−0.142832 + 0.989747i \(0.545621\pi\)
\(882\) 0 0
\(883\) − 764.000i − 0.0291174i −0.999894 0.0145587i \(-0.995366\pi\)
0.999894 0.0145587i \(-0.00463434\pi\)
\(884\) 0 0
\(885\) 36936.0i 1.40293i
\(886\) 0 0
\(887\) −32328.0 −1.22375 −0.611876 0.790954i \(-0.709585\pi\)
−0.611876 + 0.790954i \(0.709585\pi\)
\(888\) 0 0
\(889\) 4736.00 0.178673
\(890\) 0 0
\(891\) 2916.00i 0.109640i
\(892\) 0 0
\(893\) 43200.0i 1.61885i
\(894\) 0 0
\(895\) −20088.0 −0.750243
\(896\) 0 0
\(897\) −2160.00 −0.0804017
\(898\) 0 0
\(899\) − 3744.00i − 0.138898i
\(900\) 0 0
\(901\) 7452.00i 0.275541i
\(902\) 0 0
\(903\) −10848.0 −0.399777
\(904\) 0 0
\(905\) 19260.0 0.707430
\(906\) 0 0
\(907\) − 36316.0i − 1.32950i −0.747068 0.664748i \(-0.768539\pi\)
0.747068 0.664748i \(-0.231461\pi\)
\(908\) 0 0
\(909\) − 5022.00i − 0.183244i
\(910\) 0 0
\(911\) −13392.0 −0.487044 −0.243522 0.969895i \(-0.578303\pi\)
−0.243522 + 0.969895i \(0.578303\pi\)
\(912\) 0 0
\(913\) −42768.0 −1.55029
\(914\) 0 0
\(915\) 22788.0i 0.823331i
\(916\) 0 0
\(917\) − 15264.0i − 0.549686i
\(918\) 0 0
\(919\) −38072.0 −1.36657 −0.683286 0.730151i \(-0.739450\pi\)
−0.683286 + 0.730151i \(0.739450\pi\)
\(920\) 0 0
\(921\) −3732.00 −0.133522
\(922\) 0 0
\(923\) 3600.00i 0.128381i
\(924\) 0 0
\(925\) 44974.0i 1.59863i
\(926\) 0 0
\(927\) 72.0000 0.00255101
\(928\) 0 0
\(929\) −12798.0 −0.451979 −0.225990 0.974130i \(-0.572562\pi\)
−0.225990 + 0.974130i \(0.572562\pi\)
\(930\) 0 0
\(931\) − 27900.0i − 0.982154i
\(932\) 0 0
\(933\) 3672.00i 0.128849i
\(934\) 0 0
\(935\) 11664.0 0.407972
\(936\) 0 0
\(937\) −34874.0 −1.21588 −0.607942 0.793981i \(-0.708005\pi\)
−0.607942 + 0.793981i \(0.708005\pi\)
\(938\) 0 0
\(939\) 5694.00i 0.197888i
\(940\) 0 0
\(941\) − 17190.0i − 0.595513i −0.954642 0.297757i \(-0.903762\pi\)
0.954642 0.297757i \(-0.0962384\pi\)
\(942\) 0 0
\(943\) 6480.00 0.223773
\(944\) 0 0
\(945\) −3888.00 −0.133838
\(946\) 0 0
\(947\) − 40284.0i − 1.38232i −0.722703 0.691158i \(-0.757101\pi\)
0.722703 0.691158i \(-0.242899\pi\)
\(948\) 0 0
\(949\) − 260.000i − 0.00889353i
\(950\) 0 0
\(951\) 27486.0 0.937218
\(952\) 0 0
\(953\) −15498.0 −0.526789 −0.263394 0.964688i \(-0.584842\pi\)
−0.263394 + 0.964688i \(0.584842\pi\)
\(954\) 0 0
\(955\) 10368.0i 0.351310i
\(956\) 0 0
\(957\) 25272.0i 0.853634i
\(958\) 0 0
\(959\) 7632.00 0.256987
\(960\) 0 0
\(961\) −29535.0 −0.991407
\(962\) 0 0
\(963\) − 15876.0i − 0.531253i
\(964\) 0 0
\(965\) 24156.0i 0.805813i
\(966\) 0 0
\(967\) −37160.0 −1.23577 −0.617883 0.786270i \(-0.712009\pi\)
−0.617883 + 0.786270i \(0.712009\pi\)
\(968\) 0 0
\(969\) 5400.00 0.179023
\(970\) 0 0
\(971\) 18468.0i 0.610367i 0.952294 + 0.305183i \(0.0987178\pi\)
−0.952294 + 0.305183i \(0.901282\pi\)
\(972\) 0 0
\(973\) − 20512.0i − 0.675832i
\(974\) 0 0
\(975\) −5970.00 −0.196095
\(976\) 0 0
\(977\) 10386.0 0.340100 0.170050 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(978\) 0 0
\(979\) 22680.0i 0.740404i
\(980\) 0 0
\(981\) 14598.0i 0.475105i
\(982\) 0 0
\(983\) −44136.0 −1.43206 −0.716032 0.698067i \(-0.754044\pi\)
−0.716032 + 0.698067i \(0.754044\pi\)
\(984\) 0 0
\(985\) 25596.0 0.827976
\(986\) 0 0
\(987\) 10368.0i 0.334364i
\(988\) 0 0
\(989\) − 32544.0i − 1.04635i
\(990\) 0 0
\(991\) −28432.0 −0.911375 −0.455687 0.890140i \(-0.650606\pi\)
−0.455687 + 0.890140i \(0.650606\pi\)
\(992\) 0 0
\(993\) −13044.0 −0.416857
\(994\) 0 0
\(995\) 15696.0i 0.500097i
\(996\) 0 0
\(997\) − 39778.0i − 1.26357i −0.775143 0.631786i \(-0.782322\pi\)
0.775143 0.631786i \(-0.217678\pi\)
\(998\) 0 0
\(999\) 6102.00 0.193252
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 768.4.d.g.385.1 2
4.3 odd 2 768.4.d.j.385.2 2
8.3 odd 2 768.4.d.j.385.1 2
8.5 even 2 inner 768.4.d.g.385.2 2
16.3 odd 4 48.4.a.a.1.1 1
16.5 even 4 192.4.a.f.1.1 1
16.11 odd 4 192.4.a.l.1.1 1
16.13 even 4 12.4.a.a.1.1 1
48.5 odd 4 576.4.a.b.1.1 1
48.11 even 4 576.4.a.a.1.1 1
48.29 odd 4 36.4.a.a.1.1 1
48.35 even 4 144.4.a.g.1.1 1
80.3 even 4 1200.4.f.d.49.1 2
80.13 odd 4 300.4.d.e.49.2 2
80.19 odd 4 1200.4.a.be.1.1 1
80.29 even 4 300.4.a.b.1.1 1
80.67 even 4 1200.4.f.d.49.2 2
80.77 odd 4 300.4.d.e.49.1 2
112.13 odd 4 588.4.a.c.1.1 1
112.45 odd 12 588.4.i.e.373.1 2
112.61 odd 12 588.4.i.e.361.1 2
112.83 even 4 2352.4.a.bk.1.1 1
112.93 even 12 588.4.i.d.361.1 2
112.109 even 12 588.4.i.d.373.1 2
144.13 even 12 324.4.e.h.217.1 2
144.29 odd 12 324.4.e.a.109.1 2
144.61 even 12 324.4.e.h.109.1 2
144.77 odd 12 324.4.e.a.217.1 2
176.109 odd 4 1452.4.a.d.1.1 1
208.77 even 4 2028.4.a.c.1.1 1
208.109 odd 4 2028.4.b.c.337.2 2
208.125 odd 4 2028.4.b.c.337.1 2
240.29 odd 4 900.4.a.g.1.1 1
240.77 even 4 900.4.d.c.649.2 2
240.173 even 4 900.4.d.c.649.1 2
336.125 even 4 1764.4.a.b.1.1 1
336.173 even 12 1764.4.k.o.361.1 2
336.221 odd 12 1764.4.k.b.1549.1 2
336.269 even 12 1764.4.k.o.1549.1 2
336.317 odd 12 1764.4.k.b.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 1 16.13 even 4
36.4.a.a.1.1 1 48.29 odd 4
48.4.a.a.1.1 1 16.3 odd 4
144.4.a.g.1.1 1 48.35 even 4
192.4.a.f.1.1 1 16.5 even 4
192.4.a.l.1.1 1 16.11 odd 4
300.4.a.b.1.1 1 80.29 even 4
300.4.d.e.49.1 2 80.77 odd 4
300.4.d.e.49.2 2 80.13 odd 4
324.4.e.a.109.1 2 144.29 odd 12
324.4.e.a.217.1 2 144.77 odd 12
324.4.e.h.109.1 2 144.61 even 12
324.4.e.h.217.1 2 144.13 even 12
576.4.a.a.1.1 1 48.11 even 4
576.4.a.b.1.1 1 48.5 odd 4
588.4.a.c.1.1 1 112.13 odd 4
588.4.i.d.361.1 2 112.93 even 12
588.4.i.d.373.1 2 112.109 even 12
588.4.i.e.361.1 2 112.61 odd 12
588.4.i.e.373.1 2 112.45 odd 12
768.4.d.g.385.1 2 1.1 even 1 trivial
768.4.d.g.385.2 2 8.5 even 2 inner
768.4.d.j.385.1 2 8.3 odd 2
768.4.d.j.385.2 2 4.3 odd 2
900.4.a.g.1.1 1 240.29 odd 4
900.4.d.c.649.1 2 240.173 even 4
900.4.d.c.649.2 2 240.77 even 4
1200.4.a.be.1.1 1 80.19 odd 4
1200.4.f.d.49.1 2 80.3 even 4
1200.4.f.d.49.2 2 80.67 even 4
1452.4.a.d.1.1 1 176.109 odd 4
1764.4.a.b.1.1 1 336.125 even 4
1764.4.k.b.361.1 2 336.317 odd 12
1764.4.k.b.1549.1 2 336.221 odd 12
1764.4.k.o.361.1 2 336.173 even 12
1764.4.k.o.1549.1 2 336.269 even 12
2028.4.a.c.1.1 1 208.77 even 4
2028.4.b.c.337.1 2 208.125 odd 4
2028.4.b.c.337.2 2 208.109 odd 4
2352.4.a.bk.1.1 1 112.83 even 4