Properties

Label 2028.4.b.c.337.2
Level $2028$
Weight $4$
Character 2028.337
Analytic conductor $119.656$
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] = [2028,4,Mod(337,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2028.b (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: \(119.655873492\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2028.337
Dual form 2028.4.b.c.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +18.0000i q^{5} +8.00000i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +18.0000i q^{5} +8.00000i q^{7} +9.00000 q^{9} +36.0000i q^{11} +54.0000i q^{15} -18.0000 q^{17} +100.000i q^{19} +24.0000i q^{21} -72.0000 q^{23} -199.000 q^{25} +27.0000 q^{27} -234.000 q^{29} +16.0000i q^{31} +108.000i q^{33} -144.000 q^{35} -226.000i q^{37} -90.0000i q^{41} -452.000 q^{43} +162.000i q^{45} +432.000i q^{47} +279.000 q^{49} -54.0000 q^{51} +414.000 q^{53} -648.000 q^{55} +300.000i q^{57} -684.000i q^{59} +422.000 q^{61} +72.0000i q^{63} -332.000i q^{67} -216.000 q^{69} +360.000i q^{71} +26.0000i q^{73} -597.000 q^{75} -288.000 q^{77} +512.000 q^{79} +81.0000 q^{81} +1188.00i q^{83} -324.000i q^{85} -702.000 q^{87} -630.000i q^{89} +48.0000i q^{93} -1800.00 q^{95} +1054.00i q^{97} +324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 18 q^{9} - 36 q^{17} - 144 q^{23} - 398 q^{25} + 54 q^{27} - 468 q^{29} - 288 q^{35} - 904 q^{43} + 558 q^{49} - 108 q^{51} + 828 q^{53} - 1296 q^{55} + 844 q^{61} - 432 q^{69} - 1194 q^{75} - 576 q^{77} + 1024 q^{79} + 162 q^{81} - 1404 q^{87} - 3600 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 18.0000i 1.60997i 0.593296 + 0.804984i \(0.297826\pi\)
−0.593296 + 0.804984i \(0.702174\pi\)
\(6\) 0 0
\(7\) 8.00000i 0.431959i 0.976398 + 0.215980i \(0.0692945\pi\)
−0.976398 + 0.215980i \(0.930705\pi\)
\(8\) 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
\(13\) 0 0
\(14\) 0 0
\(15\) 54.0000i 0.929516i
\(16\) 0 0
\(17\) −18.0000 −0.256802 −0.128401 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) 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
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −234.000 −1.49837 −0.749185 0.662361i \(-0.769554\pi\)
−0.749185 + 0.662361i \(0.769554\pi\)
\(30\) 0 0
\(31\) 16.0000i 0.0926995i 0.998925 + 0.0463498i \(0.0147589\pi\)
−0.998925 + 0.0463498i \(0.985241\pi\)
\(32\) 0 0
\(33\) 108.000i 0.569709i
\(34\) 0 0
\(35\) −144.000 −0.695441
\(36\) 0 0
\(37\) − 226.000i − 1.00417i −0.864819 0.502083i \(-0.832567\pi\)
0.864819 0.502083i \(-0.167433\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 90.0000i − 0.342820i −0.985200 0.171410i \(-0.945168\pi\)
0.985200 0.171410i \(-0.0548323\pi\)
\(42\) 0 0
\(43\) −452.000 −1.60301 −0.801504 0.597989i \(-0.795967\pi\)
−0.801504 + 0.597989i \(0.795967\pi\)
\(44\) 0 0
\(45\) 162.000i 0.536656i
\(46\) 0 0
\(47\) 432.000i 1.34072i 0.742038 + 0.670358i \(0.233860\pi\)
−0.742038 + 0.670358i \(0.766140\pi\)
\(48\) 0 0
\(49\) 279.000 0.813411
\(50\) 0 0
\(51\) −54.0000 −0.148265
\(52\) 0 0
\(53\) 414.000 1.07297 0.536484 0.843911i \(-0.319752\pi\)
0.536484 + 0.843911i \(0.319752\pi\)
\(54\) 0 0
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) 300.000i 0.697122i
\(58\) 0 0
\(59\) − 684.000i − 1.50931i −0.656123 0.754654i \(-0.727805\pi\)
0.656123 0.754654i \(-0.272195\pi\)
\(60\) 0 0
\(61\) 422.000 0.885763 0.442882 0.896580i \(-0.353956\pi\)
0.442882 + 0.896580i \(0.353956\pi\)
\(62\) 0 0
\(63\) 72.0000i 0.143986i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 332.000i − 0.605377i −0.953090 0.302688i \(-0.902116\pi\)
0.953090 0.302688i \(-0.0978842\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 360.000i 0.601748i 0.953664 + 0.300874i \(0.0972784\pi\)
−0.953664 + 0.300874i \(0.902722\pi\)
\(72\) 0 0
\(73\) 26.0000i 0.0416859i 0.999783 + 0.0208429i \(0.00663500\pi\)
−0.999783 + 0.0208429i \(0.993365\pi\)
\(74\) 0 0
\(75\) −597.000 −0.919142
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 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
\(89\) − 630.000i − 0.750336i −0.926957 0.375168i \(-0.877585\pi\)
0.926957 0.375168i \(-0.122415\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 48.0000i 0.0535201i
\(94\) 0 0
\(95\) −1800.00 −1.94396
\(96\) 0 0
\(97\) 1054.00i 1.10327i 0.834085 + 0.551637i \(0.185996\pi\)
−0.834085 + 0.551637i \(0.814004\pi\)
\(98\) 0 0
\(99\) 324.000i 0.328921i
\(100\) 0 0
\(101\) −558.000 −0.549733 −0.274867 0.961482i \(-0.588634\pi\)
−0.274867 + 0.961482i \(0.588634\pi\)
\(102\) 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
\(107\) 1764.00 1.59376 0.796880 0.604138i \(-0.206482\pi\)
0.796880 + 0.604138i \(0.206482\pi\)
\(108\) 0 0
\(109\) − 1622.00i − 1.42532i −0.701512 0.712658i \(-0.747491\pi\)
0.701512 0.712658i \(-0.252509\pi\)
\(110\) 0 0
\(111\) − 678.000i − 0.579756i
\(112\) 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
\(117\) 0 0
\(118\) 0 0
\(119\) − 144.000i − 0.110928i
\(120\) 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 \(-0.433690\pi\)
0.206817 + 0.978380i \(0.433690\pi\)
\(128\) 0 0
\(129\) −1356.00 −0.925497
\(130\) 0 0
\(131\) −1908.00 −1.27254 −0.636270 0.771466i \(-0.719524\pi\)
−0.636270 + 0.771466i \(0.719524\pi\)
\(132\) 0 0
\(133\) −800.000 −0.521570
\(134\) 0 0
\(135\) 486.000i 0.309839i
\(136\) 0 0
\(137\) 954.000i 0.594932i 0.954732 + 0.297466i \(0.0961415\pi\)
−0.954732 + 0.297466i \(0.903858\pi\)
\(138\) 0 0
\(139\) 2564.00 1.56457 0.782286 0.622919i \(-0.214053\pi\)
0.782286 + 0.622919i \(0.214053\pi\)
\(140\) 0 0
\(141\) 1296.00i 0.774063i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 4212.00i − 2.41233i
\(146\) 0 0
\(147\) 837.000 0.469623
\(148\) 0 0
\(149\) 738.000i 0.405767i 0.979203 + 0.202884i \(0.0650313\pi\)
−0.979203 + 0.202884i \(0.934969\pi\)
\(150\) 0 0
\(151\) − 2440.00i − 1.31500i −0.753456 0.657498i \(-0.771615\pi\)
0.753456 0.657498i \(-0.228385\pi\)
\(152\) 0 0
\(153\) −162.000 −0.0856008
\(154\) 0 0
\(155\) −288.000 −0.149243
\(156\) 0 0
\(157\) −2554.00 −1.29829 −0.649145 0.760665i \(-0.724873\pi\)
−0.649145 + 0.760665i \(0.724873\pi\)
\(158\) 0 0
\(159\) 1242.00 0.619478
\(160\) 0 0
\(161\) − 576.000i − 0.281958i
\(162\) 0 0
\(163\) − 820.000i − 0.394033i −0.980400 0.197016i \(-0.936875\pi\)
0.980400 0.197016i \(-0.0631252\pi\)
\(164\) 0 0
\(165\) −1944.00 −0.917213
\(166\) 0 0
\(167\) 1944.00i 0.900786i 0.892830 + 0.450393i \(0.148716\pi\)
−0.892830 + 0.450393i \(0.851284\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 900.000i 0.402484i
\(172\) 0 0
\(173\) 1242.00 0.545824 0.272912 0.962039i \(-0.412013\pi\)
0.272912 + 0.962039i \(0.412013\pi\)
\(174\) 0 0
\(175\) − 1592.00i − 0.687679i
\(176\) 0 0
\(177\) − 2052.00i − 0.871400i
\(178\) 0 0
\(179\) −1116.00 −0.465999 −0.232999 0.972477i \(-0.574854\pi\)
−0.232999 + 0.972477i \(0.574854\pi\)
\(180\) 0 0
\(181\) −1070.00 −0.439406 −0.219703 0.975567i \(-0.570509\pi\)
−0.219703 + 0.975567i \(0.570509\pi\)
\(182\) 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
\(193\) − 1342.00i − 0.500514i −0.968179 0.250257i \(-0.919485\pi\)
0.968179 0.250257i \(-0.0805152\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1422.00i − 0.514281i −0.966374 0.257140i \(-0.917220\pi\)
0.966374 0.257140i \(-0.0827803\pi\)
\(198\) 0 0
\(199\) −872.000 −0.310625 −0.155313 0.987865i \(-0.549639\pi\)
−0.155313 + 0.987865i \(0.549639\pi\)
\(200\) 0 0
\(201\) − 996.000i − 0.349515i
\(202\) 0 0
\(203\) − 1872.00i − 0.647235i
\(204\) 0 0
\(205\) 1620.00 0.551930
\(206\) 0 0
\(207\) −648.000 −0.217580
\(208\) 0 0
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) 1340.00 0.437201 0.218600 0.975814i \(-0.429851\pi\)
0.218600 + 0.975814i \(0.429851\pi\)
\(212\) 0 0
\(213\) 1080.00i 0.347420i
\(214\) 0 0
\(215\) − 8136.00i − 2.58079i
\(216\) 0 0
\(217\) −128.000 −0.0400424
\(218\) 0 0
\(219\) 78.0000i 0.0240674i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 4880.00i − 1.46542i −0.680540 0.732711i \(-0.738255\pi\)
0.680540 0.732711i \(-0.261745\pi\)
\(224\) 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
\(235\) −7776.00 −2.15851
\(236\) 0 0
\(237\) 1536.00 0.420987
\(238\) 0 0
\(239\) 3888.00i 1.05228i 0.850399 + 0.526138i \(0.176360\pi\)
−0.850399 + 0.526138i \(0.823640\pi\)
\(240\) 0 0
\(241\) 5138.00i 1.37331i 0.726984 + 0.686655i \(0.240922\pi\)
−0.726984 + 0.686655i \(0.759078\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 5022.00i 1.30957i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3564.00i 0.907066i
\(250\) 0 0
\(251\) −4788.00 −1.20405 −0.602024 0.798478i \(-0.705639\pi\)
−0.602024 + 0.798478i \(0.705639\pi\)
\(252\) 0 0
\(253\) − 2592.00i − 0.644101i
\(254\) 0 0
\(255\) − 972.000i − 0.238702i
\(256\) 0 0
\(257\) 5886.00 1.42863 0.714316 0.699823i \(-0.246738\pi\)
0.714316 + 0.699823i \(0.246738\pi\)
\(258\) 0 0
\(259\) 1808.00 0.433759
\(260\) 0 0
\(261\) −2106.00 −0.499456
\(262\) 0 0
\(263\) 2232.00 0.523312 0.261656 0.965161i \(-0.415731\pi\)
0.261656 + 0.965161i \(0.415731\pi\)
\(264\) 0 0
\(265\) 7452.00i 1.72744i
\(266\) 0 0
\(267\) − 1890.00i − 0.433206i
\(268\) 0 0
\(269\) −666.000 −0.150954 −0.0754772 0.997148i \(-0.524048\pi\)
−0.0754772 + 0.997148i \(0.524048\pi\)
\(270\) 0 0
\(271\) − 5536.00i − 1.24092i −0.784240 0.620458i \(-0.786947\pi\)
0.784240 0.620458i \(-0.213053\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 7164.00i − 1.57093i
\(276\) 0 0
\(277\) −2126.00 −0.461151 −0.230576 0.973054i \(-0.574061\pi\)
−0.230576 + 0.973054i \(0.574061\pi\)
\(278\) 0 0
\(279\) 144.000i 0.0308998i
\(280\) 0 0
\(281\) − 2934.00i − 0.622875i −0.950267 0.311437i \(-0.899190\pi\)
0.950267 0.311437i \(-0.100810\pi\)
\(282\) 0 0
\(283\) −2036.00 −0.427659 −0.213830 0.976871i \(-0.568594\pi\)
−0.213830 + 0.976871i \(0.568594\pi\)
\(284\) 0 0
\(285\) −5400.00 −1.12235
\(286\) 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
\(297\) 972.000i 0.189903i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 3616.00i − 0.692434i
\(302\) 0 0
\(303\) −1674.00 −0.317389
\(304\) 0 0
\(305\) 7596.00i 1.42605i
\(306\) 0 0
\(307\) 1244.00i 0.231267i 0.993292 + 0.115633i \(0.0368897\pi\)
−0.993292 + 0.115633i \(0.963110\pi\)
\(308\) 0 0
\(309\) −24.0000 −0.00441849
\(310\) 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 \(-0.445179\pi\)
0.171376 + 0.985206i \(0.445179\pi\)
\(314\) 0 0
\(315\) −1296.00 −0.231814
\(316\) 0 0
\(317\) 9162.00i 1.62331i 0.584137 + 0.811655i \(0.301433\pi\)
−0.584137 + 0.811655i \(0.698567\pi\)
\(318\) 0 0
\(319\) − 8424.00i − 1.47854i
\(320\) 0 0
\(321\) 5292.00 0.920158
\(322\) 0 0
\(323\) − 1800.00i − 0.310076i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4866.00i − 0.822906i
\(328\) 0 0
\(329\) −3456.00 −0.579135
\(330\) 0 0
\(331\) 4348.00i 0.722017i 0.932562 + 0.361009i \(0.117568\pi\)
−0.932562 + 0.361009i \(0.882432\pi\)
\(332\) 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 \(-0.696243\pi\)
−0.578195 + 0.815899i \(0.696243\pi\)
\(338\) 0 0
\(339\) −3402.00 −0.545048
\(340\) 0 0
\(341\) −576.000 −0.0914726
\(342\) 0 0
\(343\) 4976.00i 0.783320i
\(344\) 0 0
\(345\) − 3888.00i − 0.606733i
\(346\) 0 0
\(347\) −1836.00 −0.284039 −0.142020 0.989864i \(-0.545360\pi\)
−0.142020 + 0.989864i \(0.545360\pi\)
\(348\) 0 0
\(349\) 5894.00i 0.904007i 0.892016 + 0.452004i \(0.149291\pi\)
−0.892016 + 0.452004i \(0.850709\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 11106.0i − 1.67454i −0.546789 0.837270i \(-0.684150\pi\)
0.546789 0.837270i \(-0.315850\pi\)
\(354\) 0 0
\(355\) −6480.00 −0.968796
\(356\) 0 0
\(357\) − 432.000i − 0.0640444i
\(358\) 0 0
\(359\) 13176.0i 1.93705i 0.248907 + 0.968527i \(0.419929\pi\)
−0.248907 + 0.968527i \(0.580071\pi\)
\(360\) 0 0
\(361\) −3141.00 −0.457938
\(362\) 0 0
\(363\) 105.000 0.0151820
\(364\) 0 0
\(365\) −468.000 −0.0671130
\(366\) 0 0
\(367\) −6112.00 −0.869329 −0.434665 0.900592i \(-0.643133\pi\)
−0.434665 + 0.900592i \(0.643133\pi\)
\(368\) 0 0
\(369\) − 810.000i − 0.114273i
\(370\) 0 0
\(371\) 3312.00i 0.463478i
\(372\) 0 0
\(373\) −13618.0 −1.89038 −0.945192 0.326515i \(-0.894126\pi\)
−0.945192 + 0.326515i \(0.894126\pi\)
\(374\) 0 0
\(375\) − 3996.00i − 0.550273i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 692.000i − 0.0937880i −0.998900 0.0468940i \(-0.985068\pi\)
0.998900 0.0468940i \(-0.0149323\pi\)
\(380\) 0 0
\(381\) 1776.00 0.238812
\(382\) 0 0
\(383\) 8064.00i 1.07585i 0.842992 + 0.537926i \(0.180792\pi\)
−0.842992 + 0.537926i \(0.819208\pi\)
\(384\) 0 0
\(385\) − 5184.00i − 0.686237i
\(386\) 0 0
\(387\) −4068.00 −0.534336
\(388\) 0 0
\(389\) −12654.0 −1.64931 −0.824657 0.565633i \(-0.808632\pi\)
−0.824657 + 0.565633i \(0.808632\pi\)
\(390\) 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 \(-0.997867\pi\)
0.999978 0.00670024i \(-0.00213277\pi\)
\(398\) 0 0
\(399\) −2400.00 −0.301129
\(400\) 0 0
\(401\) − 4014.00i − 0.499874i −0.968262 0.249937i \(-0.919590\pi\)
0.968262 0.249937i \(-0.0804100\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1458.00i 0.178885i
\(406\) 0 0
\(407\) 8136.00 0.990876
\(408\) 0 0
\(409\) − 3914.00i − 0.473190i −0.971608 0.236595i \(-0.923968\pi\)
0.971608 0.236595i \(-0.0760315\pi\)
\(410\) 0 0
\(411\) 2862.00i 0.343484i
\(412\) 0 0
\(413\) 5472.00 0.651960
\(414\) 0 0
\(415\) −21384.0 −2.52940
\(416\) 0 0
\(417\) 7692.00 0.903307
\(418\) 0 0
\(419\) 4428.00 0.516282 0.258141 0.966107i \(-0.416890\pi\)
0.258141 + 0.966107i \(0.416890\pi\)
\(420\) 0 0
\(421\) 15490.0i 1.79320i 0.442843 + 0.896599i \(0.353970\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(422\) 0 0
\(423\) 3888.00i 0.446906i
\(424\) 0 0
\(425\) 3582.00 0.408829
\(426\) 0 0
\(427\) 3376.00i 0.382614i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 6768.00i − 0.756388i −0.925726 0.378194i \(-0.876545\pi\)
0.925726 0.378194i \(-0.123455\pi\)
\(432\) 0 0
\(433\) −1298.00 −0.144060 −0.0720299 0.997402i \(-0.522948\pi\)
−0.0720299 + 0.997402i \(0.522948\pi\)
\(434\) 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
\(443\) −9612.00 −1.03088 −0.515440 0.856926i \(-0.672372\pi\)
−0.515440 + 0.856926i \(0.672372\pi\)
\(444\) 0 0
\(445\) 11340.0 1.20802
\(446\) 0 0
\(447\) 2214.00i 0.234270i
\(448\) 0 0
\(449\) 162.000i 0.0170273i 0.999964 + 0.00851364i \(0.00271001\pi\)
−0.999964 + 0.00851364i \(0.997290\pi\)
\(450\) 0 0
\(451\) 3240.00 0.338283
\(452\) 0 0
\(453\) − 7320.00i − 0.759213i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 1370.00i − 0.140232i −0.997539 0.0701159i \(-0.977663\pi\)
0.997539 0.0701159i \(-0.0223369\pi\)
\(458\) 0 0
\(459\) −486.000 −0.0494217
\(460\) 0 0
\(461\) 15354.0i 1.55121i 0.631220 + 0.775604i \(0.282555\pi\)
−0.631220 + 0.775604i \(0.717445\pi\)
\(462\) 0 0
\(463\) − 13024.0i − 1.30729i −0.756800 0.653646i \(-0.773238\pi\)
0.756800 0.653646i \(-0.226762\pi\)
\(464\) 0 0
\(465\) −864.000 −0.0861657
\(466\) 0 0
\(467\) 14436.0 1.43045 0.715223 0.698896i \(-0.246325\pi\)
0.715223 + 0.698896i \(0.246325\pi\)
\(468\) 0 0
\(469\) 2656.00 0.261498
\(470\) 0 0
\(471\) −7662.00 −0.749568
\(472\) 0 0
\(473\) − 16272.0i − 1.58179i
\(474\) 0 0
\(475\) − 19900.0i − 1.92226i
\(476\) 0 0
\(477\) 3726.00 0.357656
\(478\) 0 0
\(479\) 12096.0i 1.15382i 0.816807 + 0.576911i \(0.195742\pi\)
−0.816807 + 0.576911i \(0.804258\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 1728.00i − 0.162788i
\(484\) 0 0
\(485\) −18972.0 −1.77624
\(486\) 0 0
\(487\) − 6056.00i − 0.563498i −0.959488 0.281749i \(-0.909085\pi\)
0.959488 0.281749i \(-0.0909146\pi\)
\(488\) 0 0
\(489\) − 2460.00i − 0.227495i
\(490\) 0 0
\(491\) −7524.00 −0.691555 −0.345777 0.938317i \(-0.612385\pi\)
−0.345777 + 0.938317i \(0.612385\pi\)
\(492\) 0 0
\(493\) 4212.00 0.384785
\(494\) 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 \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(504\) 0 0
\(505\) − 10044.0i − 0.885054i
\(506\) 0 0
\(507\) 0 0
\(508\) 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
\(513\) 2700.00i 0.232374i
\(514\) 0 0
\(515\) − 144.000i − 0.0123212i
\(516\) 0 0
\(517\) −15552.0 −1.32297
\(518\) 0 0
\(519\) 3726.00 0.315131
\(520\) 0 0
\(521\) −8838.00 −0.743186 −0.371593 0.928396i \(-0.621188\pi\)
−0.371593 + 0.928396i \(0.621188\pi\)
\(522\) 0 0
\(523\) 22436.0 1.87583 0.937914 0.346869i \(-0.112755\pi\)
0.937914 + 0.346869i \(0.112755\pi\)
\(524\) 0 0
\(525\) − 4776.00i − 0.397032i
\(526\) 0 0
\(527\) − 288.000i − 0.0238055i
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) − 6156.00i − 0.503103i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 31752.0i 2.56590i
\(536\) 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 \(-0.939405\pi\)
0.981935 0.189218i \(-0.0605954\pi\)
\(542\) 0 0
\(543\) −3210.00 −0.253691
\(544\) 0 0
\(545\) 29196.0 2.29471
\(546\) 0 0
\(547\) −6004.00 −0.469310 −0.234655 0.972079i \(-0.575396\pi\)
−0.234655 + 0.972079i \(0.575396\pi\)
\(548\) 0 0
\(549\) 3798.00 0.295254
\(550\) 0 0
\(551\) − 23400.0i − 1.80921i
\(552\) 0 0
\(553\) 4096.00i 0.314972i
\(554\) 0 0
\(555\) 12204.0 0.933389
\(556\) 0 0
\(557\) − 5274.00i − 0.401197i −0.979674 0.200598i \(-0.935711\pi\)
0.979674 0.200598i \(-0.0642886\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 1944.00i − 0.146303i
\(562\) 0 0
\(563\) 12420.0 0.929735 0.464867 0.885380i \(-0.346102\pi\)
0.464867 + 0.885380i \(0.346102\pi\)
\(564\) 0 0
\(565\) − 20412.0i − 1.51989i
\(566\) 0 0
\(567\) 648.000i 0.0479955i
\(568\) 0 0
\(569\) 21366.0 1.57418 0.787091 0.616837i \(-0.211586\pi\)
0.787091 + 0.616837i \(0.211586\pi\)
\(570\) 0 0
\(571\) −21140.0 −1.54935 −0.774677 0.632357i \(-0.782088\pi\)
−0.774677 + 0.632357i \(0.782088\pi\)
\(572\) 0 0
\(573\) −1728.00 −0.125983
\(574\) 0 0
\(575\) 14328.0 1.03916
\(576\) 0 0
\(577\) − 3266.00i − 0.235642i −0.993035 0.117821i \(-0.962409\pi\)
0.993035 0.117821i \(-0.0375909\pi\)
\(578\) 0 0
\(579\) − 4026.00i − 0.288972i
\(580\) 0 0
\(581\) −9504.00 −0.678644
\(582\) 0 0
\(583\) 14904.0i 1.05877i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 17028.0i − 1.19731i −0.801007 0.598655i \(-0.795702\pi\)
0.801007 0.598655i \(-0.204298\pi\)
\(588\) 0 0
\(589\) −1600.00 −0.111930
\(590\) 0 0
\(591\) − 4266.00i − 0.296920i
\(592\) 0 0
\(593\) 9522.00i 0.659396i 0.944086 + 0.329698i \(0.106947\pi\)
−0.944086 + 0.329698i \(0.893053\pi\)
\(594\) 0 0
\(595\) 2592.00 0.178591
\(596\) 0 0
\(597\) −2616.00 −0.179340
\(598\) 0 0
\(599\) −10296.0 −0.702309 −0.351155 0.936318i \(-0.614211\pi\)
−0.351155 + 0.936318i \(0.614211\pi\)
\(600\) 0 0
\(601\) −3382.00 −0.229542 −0.114771 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(602\) 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
\(609\) − 5616.00i − 0.373681i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 22114.0i 1.45706i 0.685015 + 0.728529i \(0.259795\pi\)
−0.685015 + 0.728529i \(0.740205\pi\)
\(614\) 0 0
\(615\) 4860.00 0.318657
\(616\) 0 0
\(617\) − 19962.0i − 1.30250i −0.758865 0.651248i \(-0.774246\pi\)
0.758865 0.651248i \(-0.225754\pi\)
\(618\) 0 0
\(619\) − 604.000i − 0.0392194i −0.999808 0.0196097i \(-0.993758\pi\)
0.999808 0.0196097i \(-0.00624236\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) 5040.00 0.324115
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) −10800.0 −0.687895
\(628\) 0 0
\(629\) 4068.00i 0.257872i
\(630\) 0 0
\(631\) 152.000i 0.00958958i 0.999989 + 0.00479479i \(0.00152623\pi\)
−0.999989 + 0.00479479i \(0.998474\pi\)
\(632\) 0 0
\(633\) 4020.00 0.252418
\(634\) 0 0
\(635\) 10656.0i 0.665938i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3240.00i 0.200583i
\(640\) 0 0
\(641\) −4194.00 −0.258429 −0.129215 0.991617i \(-0.541246\pi\)
−0.129215 + 0.991617i \(0.541246\pi\)
\(642\) 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
\(651\) −384.000 −0.0231185
\(652\) 0 0
\(653\) 28422.0 1.70328 0.851638 0.524131i \(-0.175610\pi\)
0.851638 + 0.524131i \(0.175610\pi\)
\(654\) 0 0
\(655\) − 34344.0i − 2.04875i
\(656\) 0 0
\(657\) 234.000i 0.0138953i
\(658\) 0 0
\(659\) −19908.0 −1.17679 −0.588396 0.808573i \(-0.700240\pi\)
−0.588396 + 0.808573i \(0.700240\pi\)
\(660\) 0 0
\(661\) 14318.0i 0.842520i 0.906940 + 0.421260i \(0.138412\pi\)
−0.906940 + 0.421260i \(0.861588\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 14400.0i − 0.839711i
\(666\) 0 0
\(667\) 16848.0 0.978047
\(668\) 0 0
\(669\) − 14640.0i − 0.846061i
\(670\) 0 0
\(671\) 15192.0i 0.874040i
\(672\) 0 0
\(673\) −30050.0 −1.72116 −0.860581 0.509313i \(-0.829899\pi\)
−0.860581 + 0.509313i \(0.829899\pi\)
\(674\) 0 0
\(675\) −5373.00 −0.306381
\(676\) 0 0
\(677\) 22158.0 1.25790 0.628952 0.777444i \(-0.283484\pi\)
0.628952 + 0.777444i \(0.283484\pi\)
\(678\) 0 0
\(679\) −8432.00 −0.476569
\(680\) 0 0
\(681\) − 8100.00i − 0.455790i
\(682\) 0 0
\(683\) − 3132.00i − 0.175465i −0.996144 0.0877325i \(-0.972038\pi\)
0.996144 0.0877325i \(-0.0279621\pi\)
\(684\) 0 0
\(685\) −17172.0 −0.957822
\(686\) 0 0
\(687\) 762.000i 0.0423175i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20932.0i 1.15237i 0.817318 + 0.576187i \(0.195460\pi\)
−0.817318 + 0.576187i \(0.804540\pi\)
\(692\) 0 0
\(693\) −2592.00 −0.142081
\(694\) 0 0
\(695\) 46152.0i 2.51891i
\(696\) 0 0
\(697\) 1620.00i 0.0880371i
\(698\) 0 0
\(699\) −13230.0 −0.715886
\(700\) 0 0
\(701\) 21834.0 1.17640 0.588202 0.808714i \(-0.299836\pi\)
0.588202 + 0.808714i \(0.299836\pi\)
\(702\) 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
\(713\) − 1152.00i − 0.0605088i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11664.0i 0.607531i
\(718\) 0 0
\(719\) 12528.0 0.649813 0.324907 0.945746i \(-0.394667\pi\)
0.324907 + 0.945746i \(0.394667\pi\)
\(720\) 0 0
\(721\) − 64.0000i − 0.00330580i
\(722\) 0 0
\(723\) 15414.0i 0.792881i
\(724\) 0 0
\(725\) 46566.0 2.38540
\(726\) 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
\(731\) 8136.00 0.411656
\(732\) 0 0
\(733\) 29338.0i 1.47834i 0.673519 + 0.739170i \(0.264782\pi\)
−0.673519 + 0.739170i \(0.735218\pi\)
\(734\) 0 0
\(735\) 15066.0i 0.756079i
\(736\) 0 0
\(737\) 11952.0 0.597364
\(738\) 0 0
\(739\) 2540.00i 0.126435i 0.998000 + 0.0632175i \(0.0201362\pi\)
−0.998000 + 0.0632175i \(0.979864\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18792.0i 0.927876i 0.885868 + 0.463938i \(0.153564\pi\)
−0.885868 + 0.463938i \(0.846436\pi\)
\(744\) 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 \(-0.537453\pi\)
−0.117392 + 0.993086i \(0.537453\pi\)
\(752\) 0 0
\(753\) −14364.0 −0.695157
\(754\) 0 0
\(755\) 43920.0 2.11710
\(756\) 0 0
\(757\) −20818.0 −0.999529 −0.499764 0.866161i \(-0.666580\pi\)
−0.499764 + 0.866161i \(0.666580\pi\)
\(758\) 0 0
\(759\) − 7776.00i − 0.371872i
\(760\) 0 0
\(761\) 12042.0i 0.573617i 0.957988 + 0.286808i \(0.0925943\pi\)
−0.957988 + 0.286808i \(0.907406\pi\)
\(762\) 0 0
\(763\) 12976.0 0.615679
\(764\) 0 0
\(765\) − 2916.00i − 0.137815i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 13058.0i − 0.612332i −0.951978 0.306166i \(-0.900954\pi\)
0.951978 0.306166i \(-0.0990463\pi\)
\(770\) 0 0
\(771\) 17658.0 0.824821
\(772\) 0 0
\(773\) 11826.0i 0.550261i 0.961407 + 0.275130i \(0.0887210\pi\)
−0.961407 + 0.275130i \(0.911279\pi\)
\(774\) 0 0
\(775\) − 3184.00i − 0.147578i
\(776\) 0 0
\(777\) 5424.00 0.250431
\(778\) 0 0
\(779\) 9000.00 0.413939
\(780\) 0 0
\(781\) −12960.0 −0.593784
\(782\) 0 0
\(783\) −6318.00 −0.288361
\(784\) 0 0
\(785\) − 45972.0i − 2.09021i
\(786\) 0 0
\(787\) 11996.0i 0.543343i 0.962390 + 0.271672i \(0.0875765\pi\)
−0.962390 + 0.271672i \(0.912424\pi\)
\(788\) 0 0
\(789\) 6696.00 0.302134
\(790\) 0 0
\(791\) − 9072.00i − 0.407792i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 22356.0i 0.997340i
\(796\) 0 0
\(797\) −6966.00 −0.309596 −0.154798 0.987946i \(-0.549473\pi\)
−0.154798 + 0.987946i \(0.549473\pi\)
\(798\) 0 0
\(799\) − 7776.00i − 0.344299i
\(800\) 0 0
\(801\) − 5670.00i − 0.250112i
\(802\) 0 0
\(803\) −936.000 −0.0411342
\(804\) 0 0
\(805\) 10368.0 0.453943
\(806\) 0 0
\(807\) −1998.00 −0.0871536
\(808\) 0 0
\(809\) −40806.0 −1.77338 −0.886689 0.462367i \(-0.847000\pi\)
−0.886689 + 0.462367i \(0.847000\pi\)
\(810\) 0 0
\(811\) 17980.0i 0.778500i 0.921132 + 0.389250i \(0.127266\pi\)
−0.921132 + 0.389250i \(0.872734\pi\)
\(812\) 0 0
\(813\) − 16608.0i − 0.716443i
\(814\) 0 0
\(815\) 14760.0 0.634381
\(816\) 0 0
\(817\) − 45200.0i − 1.93555i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12834.0i 0.545566i 0.962076 + 0.272783i \(0.0879441\pi\)
−0.962076 + 0.272783i \(0.912056\pi\)
\(822\) 0 0
\(823\) 37864.0 1.60371 0.801857 0.597516i \(-0.203846\pi\)
0.801857 + 0.597516i \(0.203846\pi\)
\(824\) 0 0
\(825\) − 21492.0i − 0.906976i
\(826\) 0 0
\(827\) 42516.0i 1.78770i 0.448368 + 0.893849i \(0.352005\pi\)
−0.448368 + 0.893849i \(0.647995\pi\)
\(828\) 0 0
\(829\) −45638.0 −1.91203 −0.956015 0.293317i \(-0.905241\pi\)
−0.956015 + 0.293317i \(0.905241\pi\)
\(830\) 0 0
\(831\) −6378.00 −0.266246
\(832\) 0 0
\(833\) −5022.00 −0.208886
\(834\) 0 0
\(835\) −34992.0 −1.45024
\(836\) 0 0
\(837\) 432.000i 0.0178400i
\(838\) 0 0
\(839\) 17496.0i 0.719939i 0.932964 + 0.359970i \(0.117213\pi\)
−0.932964 + 0.359970i \(0.882787\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) − 8802.00i − 0.359617i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 280.000i 0.0113588i
\(848\) 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
\(859\) 29780.0 1.18286 0.591432 0.806355i \(-0.298563\pi\)
0.591432 + 0.806355i \(0.298563\pi\)
\(860\) 0 0
\(861\) 2160.00 0.0854966
\(862\) 0 0
\(863\) 48096.0i 1.89711i 0.316611 + 0.948556i \(0.397455\pi\)
−0.316611 + 0.948556i \(0.602545\pi\)
\(864\) 0 0
\(865\) 22356.0i 0.878759i
\(866\) 0 0
\(867\) −13767.0 −0.539275
\(868\) 0 0
\(869\) 18432.0i 0.719520i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9486.00i 0.367758i
\(874\) 0 0
\(875\) 10656.0 0.411701
\(876\) 0 0
\(877\) − 21302.0i − 0.820202i −0.912040 0.410101i \(-0.865493\pi\)
0.912040 0.410101i \(-0.134507\pi\)
\(878\) 0 0
\(879\) 6858.00i 0.263157i
\(880\) 0 0
\(881\) 7470.00 0.285665 0.142832 0.989747i \(-0.454379\pi\)
0.142832 + 0.989747i \(0.454379\pi\)
\(882\) 0 0
\(883\) −764.000 −0.0291174 −0.0145587 0.999894i \(-0.504634\pi\)
−0.0145587 + 0.999894i \(0.504634\pi\)
\(884\) 0 0
\(885\) 36936.0 1.40293
\(886\) 0 0
\(887\) 32328.0 1.22375 0.611876 0.790954i \(-0.290415\pi\)
0.611876 + 0.790954i \(0.290415\pi\)
\(888\) 0 0
\(889\) 4736.00i 0.178673i
\(890\) 0 0
\(891\) 2916.00i 0.109640i
\(892\) 0 0
\(893\) −43200.0 −1.61885
\(894\) 0 0
\(895\) − 20088.0i − 0.750243i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 3744.00i − 0.138898i
\(900\) 0 0
\(901\) −7452.00 −0.275541
\(902\) 0 0
\(903\) − 10848.0i − 0.399777i
\(904\) 0 0
\(905\) − 19260.0i − 0.707430i
\(906\) 0 0
\(907\) 36316.0 1.32950 0.664748 0.747068i \(-0.268539\pi\)
0.664748 + 0.747068i \(0.268539\pi\)
\(908\) 0 0
\(909\) −5022.00 −0.183244
\(910\) 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 \(-0.260550\pi\)
0.683286 + 0.730151i \(0.260550\pi\)
\(920\) 0 0
\(921\) 3732.00i 0.133522i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 44974.0i 1.59863i
\(926\) 0 0
\(927\) −72.0000 −0.00255101
\(928\) 0 0
\(929\) 12798.0i 0.451979i 0.974130 + 0.225990i \(0.0725616\pi\)
−0.974130 + 0.225990i \(0.927438\pi\)
\(930\) 0 0
\(931\) 27900.0i 0.982154i
\(932\) 0 0
\(933\) −3672.00 −0.128849
\(934\) 0 0
\(935\) 11664.0 0.407972
\(936\) 0 0
\(937\) 34874.0 1.21588 0.607942 0.793981i \(-0.291995\pi\)
0.607942 + 0.793981i \(0.291995\pi\)
\(938\) 0 0
\(939\) 5694.00 0.197888
\(940\) 0 0
\(941\) − 17190.0i − 0.595513i −0.954642 0.297757i \(-0.903762\pi\)
0.954642 0.297757i \(-0.0962384\pi\)
\(942\) 0 0
\(943\) 6480.00i 0.223773i
\(944\) 0 0
\(945\) −3888.00 −0.133838
\(946\) 0 0
\(947\) 40284.0i 1.38232i 0.722703 + 0.691158i \(0.242899\pi\)
−0.722703 + 0.691158i \(0.757101\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 27486.0i 0.937218i
\(952\) 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
\(963\) 15876.0 0.531253
\(964\) 0 0
\(965\) 24156.0 0.805813
\(966\) 0 0
\(967\) − 37160.0i − 1.23577i −0.786270 0.617883i \(-0.787991\pi\)
0.786270 0.617883i \(-0.212009\pi\)
\(968\) 0 0
\(969\) − 5400.00i − 0.179023i
\(970\) 0 0
\(971\) 18468.0 0.610367 0.305183 0.952294i \(-0.401282\pi\)
0.305183 + 0.952294i \(0.401282\pi\)
\(972\) 0 0
\(973\) 20512.0i 0.675832i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 10386.0i − 0.340100i −0.985435 0.170050i \(-0.945607\pi\)
0.985435 0.170050i \(-0.0543929\pi\)
\(978\) 0 0
\(979\) 22680.0 0.740404
\(980\) 0 0
\(981\) − 14598.0i − 0.475105i
\(982\) 0 0
\(983\) 44136.0i 1.43206i 0.698067 + 0.716032i \(0.254044\pi\)
−0.698067 + 0.716032i \(0.745956\pi\)
\(984\) 0 0
\(985\) 25596.0 0.827976
\(986\) 0 0
\(987\) −10368.0 −0.334364
\(988\) 0 0
\(989\) 32544.0 1.04635
\(990\) 0 0
\(991\) −28432.0 −0.911375 −0.455687 0.890140i \(-0.650606\pi\)
−0.455687 + 0.890140i \(0.650606\pi\)
\(992\) 0 0
\(993\) 13044.0i 0.416857i
\(994\) 0 0
\(995\) − 15696.0i − 0.500097i
\(996\) 0 0
\(997\) −39778.0 −1.26357 −0.631786 0.775143i \(-0.717678\pi\)
−0.631786 + 0.775143i \(0.717678\pi\)
\(998\) 0 0
\(999\) − 6102.00i − 0.193252i
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 2028.4.b.c.337.2 2
13.5 odd 4 2028.4.a.c.1.1 1
13.8 odd 4 12.4.a.a.1.1 1
13.12 even 2 inner 2028.4.b.c.337.1 2
39.8 even 4 36.4.a.a.1.1 1
52.47 even 4 48.4.a.a.1.1 1
65.8 even 4 300.4.d.e.49.2 2
65.34 odd 4 300.4.a.b.1.1 1
65.47 even 4 300.4.d.e.49.1 2
91.34 even 4 588.4.a.c.1.1 1
91.47 even 12 588.4.i.e.361.1 2
91.60 odd 12 588.4.i.d.373.1 2
91.73 even 12 588.4.i.e.373.1 2
91.86 odd 12 588.4.i.d.361.1 2
104.21 odd 4 192.4.a.f.1.1 1
104.99 even 4 192.4.a.l.1.1 1
117.34 odd 12 324.4.e.h.109.1 2
117.47 even 12 324.4.e.a.109.1 2
117.86 even 12 324.4.e.a.217.1 2
117.112 odd 12 324.4.e.h.217.1 2
143.21 even 4 1452.4.a.d.1.1 1
156.47 odd 4 144.4.a.g.1.1 1
195.8 odd 4 900.4.d.c.649.1 2
195.47 odd 4 900.4.d.c.649.2 2
195.164 even 4 900.4.a.g.1.1 1
208.21 odd 4 768.4.d.g.385.1 2
208.99 even 4 768.4.d.j.385.1 2
208.125 odd 4 768.4.d.g.385.2 2
208.203 even 4 768.4.d.j.385.2 2
260.47 odd 4 1200.4.f.d.49.2 2
260.99 even 4 1200.4.a.be.1.1 1
260.203 odd 4 1200.4.f.d.49.1 2
273.47 odd 12 1764.4.k.o.361.1 2
273.86 even 12 1764.4.k.b.361.1 2
273.125 odd 4 1764.4.a.b.1.1 1
273.164 odd 12 1764.4.k.o.1549.1 2
273.242 even 12 1764.4.k.b.1549.1 2
312.125 even 4 576.4.a.b.1.1 1
312.203 odd 4 576.4.a.a.1.1 1
364.307 odd 4 2352.4.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 1 13.8 odd 4
36.4.a.a.1.1 1 39.8 even 4
48.4.a.a.1.1 1 52.47 even 4
144.4.a.g.1.1 1 156.47 odd 4
192.4.a.f.1.1 1 104.21 odd 4
192.4.a.l.1.1 1 104.99 even 4
300.4.a.b.1.1 1 65.34 odd 4
300.4.d.e.49.1 2 65.47 even 4
300.4.d.e.49.2 2 65.8 even 4
324.4.e.a.109.1 2 117.47 even 12
324.4.e.a.217.1 2 117.86 even 12
324.4.e.h.109.1 2 117.34 odd 12
324.4.e.h.217.1 2 117.112 odd 12
576.4.a.a.1.1 1 312.203 odd 4
576.4.a.b.1.1 1 312.125 even 4
588.4.a.c.1.1 1 91.34 even 4
588.4.i.d.361.1 2 91.86 odd 12
588.4.i.d.373.1 2 91.60 odd 12
588.4.i.e.361.1 2 91.47 even 12
588.4.i.e.373.1 2 91.73 even 12
768.4.d.g.385.1 2 208.21 odd 4
768.4.d.g.385.2 2 208.125 odd 4
768.4.d.j.385.1 2 208.99 even 4
768.4.d.j.385.2 2 208.203 even 4
900.4.a.g.1.1 1 195.164 even 4
900.4.d.c.649.1 2 195.8 odd 4
900.4.d.c.649.2 2 195.47 odd 4
1200.4.a.be.1.1 1 260.99 even 4
1200.4.f.d.49.1 2 260.203 odd 4
1200.4.f.d.49.2 2 260.47 odd 4
1452.4.a.d.1.1 1 143.21 even 4
1764.4.a.b.1.1 1 273.125 odd 4
1764.4.k.b.361.1 2 273.86 even 12
1764.4.k.b.1549.1 2 273.242 even 12
1764.4.k.o.361.1 2 273.47 odd 12
1764.4.k.o.1549.1 2 273.164 odd 12
2028.4.a.c.1.1 1 13.5 odd 4
2028.4.b.c.337.1 2 13.12 even 2 inner
2028.4.b.c.337.2 2 1.1 even 1 trivial
2352.4.a.bk.1.1 1 364.307 odd 4