Properties

Label 768.3.e.h.257.2
Level $768$
Weight $3$
Character 768.257
Analytic conductor $20.926$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,3,Mod(257,768)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.e (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: \(20.9264843029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 257.2
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 768.257
Dual form 768.3.e.h.257.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 2.23607i) q^{3} +4.00000i q^{5} -8.94427 q^{7} +(-1.00000 + 8.94427i) q^{9} +O(q^{10})\) \(q+(-2.00000 - 2.23607i) q^{3} +4.00000i q^{5} -8.94427 q^{7} +(-1.00000 + 8.94427i) q^{9} -4.47214i q^{11} -17.8885 q^{13} +(8.94427 - 8.00000i) q^{15} +17.8885i q^{17} +20.0000 q^{19} +(17.8885 + 20.0000i) q^{21} -16.0000i q^{23} +9.00000 q^{25} +(22.0000 - 15.6525i) q^{27} -52.0000i q^{29} -26.8328 q^{31} +(-10.0000 + 8.94427i) q^{33} -35.7771i q^{35} +53.6656 q^{37} +(35.7771 + 40.0000i) q^{39} +35.7771i q^{41} +36.0000 q^{43} +(-35.7771 - 4.00000i) q^{45} +64.0000i q^{47} +31.0000 q^{49} +(40.0000 - 35.7771i) q^{51} +20.0000i q^{53} +17.8885 q^{55} +(-40.0000 - 44.7214i) q^{57} -102.859i q^{59} -17.8885 q^{61} +(8.94427 - 80.0000i) q^{63} -71.5542i q^{65} +44.0000 q^{67} +(-35.7771 + 32.0000i) q^{69} -80.0000i q^{71} +50.0000 q^{73} +(-18.0000 - 20.1246i) q^{75} +40.0000i q^{77} +80.4984 q^{79} +(-79.0000 - 17.8885i) q^{81} -102.859i q^{83} -71.5542 q^{85} +(-116.276 + 104.000i) q^{87} -160.997i q^{89} +160.000 q^{91} +(53.6656 + 60.0000i) q^{93} +80.0000i q^{95} +50.0000 q^{97} +(40.0000 + 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} - 4 q^{9} + 80 q^{19} + 36 q^{25} + 88 q^{27} - 40 q^{33} + 144 q^{43} + 124 q^{49} + 160 q^{51} - 160 q^{57} + 176 q^{67} + 200 q^{73} - 72 q^{75} - 316 q^{81} + 640 q^{91} + 200 q^{97} + 160 q^{99}+O(q^{100}) \) 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
\(3\) −2.00000 2.23607i −0.666667 0.745356i
\(4\) 0 0
\(5\) 4.00000i 0.800000i 0.916515 + 0.400000i \(0.130990\pi\)
−0.916515 + 0.400000i \(0.869010\pi\)
\(6\) 0 0
\(7\) −8.94427 −1.27775 −0.638877 0.769309i \(-0.720601\pi\)
−0.638877 + 0.769309i \(0.720601\pi\)
\(8\) 0 0
\(9\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(10\) 0 0
\(11\) 4.47214i 0.406558i −0.979121 0.203279i \(-0.934840\pi\)
0.979121 0.203279i \(-0.0651598\pi\)
\(12\) 0 0
\(13\) −17.8885 −1.37604 −0.688021 0.725691i \(-0.741520\pi\)
−0.688021 + 0.725691i \(0.741520\pi\)
\(14\) 0 0
\(15\) 8.94427 8.00000i 0.596285 0.533333i
\(16\) 0 0
\(17\) 17.8885i 1.05227i 0.850402 + 0.526134i \(0.176359\pi\)
−0.850402 + 0.526134i \(0.823641\pi\)
\(18\) 0 0
\(19\) 20.0000 1.05263 0.526316 0.850289i \(-0.323573\pi\)
0.526316 + 0.850289i \(0.323573\pi\)
\(20\) 0 0
\(21\) 17.8885 + 20.0000i 0.851835 + 0.952381i
\(22\) 0 0
\(23\) 16.0000i 0.695652i −0.937559 0.347826i \(-0.886920\pi\)
0.937559 0.347826i \(-0.113080\pi\)
\(24\) 0 0
\(25\) 9.00000 0.360000
\(26\) 0 0
\(27\) 22.0000 15.6525i 0.814815 0.579721i
\(28\) 0 0
\(29\) 52.0000i 1.79310i −0.442939 0.896552i \(-0.646064\pi\)
0.442939 0.896552i \(-0.353936\pi\)
\(30\) 0 0
\(31\) −26.8328 −0.865575 −0.432787 0.901496i \(-0.642470\pi\)
−0.432787 + 0.901496i \(0.642470\pi\)
\(32\) 0 0
\(33\) −10.0000 + 8.94427i −0.303030 + 0.271039i
\(34\) 0 0
\(35\) 35.7771i 1.02220i
\(36\) 0 0
\(37\) 53.6656 1.45042 0.725211 0.688526i \(-0.241742\pi\)
0.725211 + 0.688526i \(0.241742\pi\)
\(38\) 0 0
\(39\) 35.7771 + 40.0000i 0.917361 + 1.02564i
\(40\) 0 0
\(41\) 35.7771i 0.872612i 0.899798 + 0.436306i \(0.143713\pi\)
−0.899798 + 0.436306i \(0.856287\pi\)
\(42\) 0 0
\(43\) 36.0000 0.837209 0.418605 0.908169i \(-0.362519\pi\)
0.418605 + 0.908169i \(0.362519\pi\)
\(44\) 0 0
\(45\) −35.7771 4.00000i −0.795046 0.0888889i
\(46\) 0 0
\(47\) 64.0000i 1.36170i 0.732422 + 0.680851i \(0.238390\pi\)
−0.732422 + 0.680851i \(0.761610\pi\)
\(48\) 0 0
\(49\) 31.0000 0.632653
\(50\) 0 0
\(51\) 40.0000 35.7771i 0.784314 0.701512i
\(52\) 0 0
\(53\) 20.0000i 0.377358i 0.982039 + 0.188679i \(0.0604206\pi\)
−0.982039 + 0.188679i \(0.939579\pi\)
\(54\) 0 0
\(55\) 17.8885 0.325246
\(56\) 0 0
\(57\) −40.0000 44.7214i −0.701754 0.784585i
\(58\) 0 0
\(59\) 102.859i 1.74338i −0.490062 0.871688i \(-0.663026\pi\)
0.490062 0.871688i \(-0.336974\pi\)
\(60\) 0 0
\(61\) −17.8885 −0.293255 −0.146627 0.989192i \(-0.546842\pi\)
−0.146627 + 0.989192i \(0.546842\pi\)
\(62\) 0 0
\(63\) 8.94427 80.0000i 0.141973 1.26984i
\(64\) 0 0
\(65\) 71.5542i 1.10083i
\(66\) 0 0
\(67\) 44.0000 0.656716 0.328358 0.944553i \(-0.393505\pi\)
0.328358 + 0.944553i \(0.393505\pi\)
\(68\) 0 0
\(69\) −35.7771 + 32.0000i −0.518509 + 0.463768i
\(70\) 0 0
\(71\) 80.0000i 1.12676i −0.826198 0.563380i \(-0.809501\pi\)
0.826198 0.563380i \(-0.190499\pi\)
\(72\) 0 0
\(73\) 50.0000 0.684932 0.342466 0.939530i \(-0.388738\pi\)
0.342466 + 0.939530i \(0.388738\pi\)
\(74\) 0 0
\(75\) −18.0000 20.1246i −0.240000 0.268328i
\(76\) 0 0
\(77\) 40.0000i 0.519481i
\(78\) 0 0
\(79\) 80.4984 1.01897 0.509484 0.860480i \(-0.329836\pi\)
0.509484 + 0.860480i \(0.329836\pi\)
\(80\) 0 0
\(81\) −79.0000 17.8885i −0.975309 0.220846i
\(82\) 0 0
\(83\) 102.859i 1.23927i −0.784891 0.619633i \(-0.787281\pi\)
0.784891 0.619633i \(-0.212719\pi\)
\(84\) 0 0
\(85\) −71.5542 −0.841814
\(86\) 0 0
\(87\) −116.276 + 104.000i −1.33650 + 1.19540i
\(88\) 0 0
\(89\) 160.997i 1.80895i −0.426523 0.904477i \(-0.640262\pi\)
0.426523 0.904477i \(-0.359738\pi\)
\(90\) 0 0
\(91\) 160.000 1.75824
\(92\) 0 0
\(93\) 53.6656 + 60.0000i 0.577050 + 0.645161i
\(94\) 0 0
\(95\) 80.0000i 0.842105i
\(96\) 0 0
\(97\) 50.0000 0.515464 0.257732 0.966216i \(-0.417025\pi\)
0.257732 + 0.966216i \(0.417025\pi\)
\(98\) 0 0
\(99\) 40.0000 + 4.47214i 0.404040 + 0.0451731i
\(100\) 0 0
\(101\) 92.0000i 0.910891i −0.890264 0.455446i \(-0.849480\pi\)
0.890264 0.455446i \(-0.150520\pi\)
\(102\) 0 0
\(103\) −44.7214 −0.434188 −0.217094 0.976151i \(-0.569658\pi\)
−0.217094 + 0.976151i \(0.569658\pi\)
\(104\) 0 0
\(105\) −80.0000 + 71.5542i −0.761905 + 0.681468i
\(106\) 0 0
\(107\) 40.2492i 0.376161i 0.982154 + 0.188080i \(0.0602266\pi\)
−0.982154 + 0.188080i \(0.939773\pi\)
\(108\) 0 0
\(109\) 125.220 1.14881 0.574403 0.818573i \(-0.305234\pi\)
0.574403 + 0.818573i \(0.305234\pi\)
\(110\) 0 0
\(111\) −107.331 120.000i −0.966948 1.08108i
\(112\) 0 0
\(113\) 35.7771i 0.316611i 0.987390 + 0.158306i \(0.0506031\pi\)
−0.987390 + 0.158306i \(0.949397\pi\)
\(114\) 0 0
\(115\) 64.0000 0.556522
\(116\) 0 0
\(117\) 17.8885 160.000i 0.152894 1.36752i
\(118\) 0 0
\(119\) 160.000i 1.34454i
\(120\) 0 0
\(121\) 101.000 0.834711
\(122\) 0 0
\(123\) 80.0000 71.5542i 0.650407 0.581741i
\(124\) 0 0
\(125\) 136.000i 1.08800i
\(126\) 0 0
\(127\) −26.8328 −0.211282 −0.105641 0.994404i \(-0.533689\pi\)
−0.105641 + 0.994404i \(0.533689\pi\)
\(128\) 0 0
\(129\) −72.0000 80.4984i −0.558140 0.624019i
\(130\) 0 0
\(131\) 67.0820i 0.512077i 0.966667 + 0.256038i \(0.0824173\pi\)
−0.966667 + 0.256038i \(0.917583\pi\)
\(132\) 0 0
\(133\) −178.885 −1.34500
\(134\) 0 0
\(135\) 62.6099 + 88.0000i 0.463777 + 0.651852i
\(136\) 0 0
\(137\) 143.108i 1.04459i 0.852766 + 0.522293i \(0.174923\pi\)
−0.852766 + 0.522293i \(0.825077\pi\)
\(138\) 0 0
\(139\) −260.000 −1.87050 −0.935252 0.353983i \(-0.884827\pi\)
−0.935252 + 0.353983i \(0.884827\pi\)
\(140\) 0 0
\(141\) 143.108 128.000i 1.01495 0.907801i
\(142\) 0 0
\(143\) 80.0000i 0.559441i
\(144\) 0 0
\(145\) 208.000 1.43448
\(146\) 0 0
\(147\) −62.0000 69.3181i −0.421769 0.471552i
\(148\) 0 0
\(149\) 28.0000i 0.187919i −0.995576 0.0939597i \(-0.970048\pi\)
0.995576 0.0939597i \(-0.0299525\pi\)
\(150\) 0 0
\(151\) 241.495 1.59931 0.799653 0.600462i \(-0.205017\pi\)
0.799653 + 0.600462i \(0.205017\pi\)
\(152\) 0 0
\(153\) −160.000 17.8885i −1.04575 0.116919i
\(154\) 0 0
\(155\) 107.331i 0.692460i
\(156\) 0 0
\(157\) 53.6656 0.341819 0.170910 0.985287i \(-0.445329\pi\)
0.170910 + 0.985287i \(0.445329\pi\)
\(158\) 0 0
\(159\) 44.7214 40.0000i 0.281266 0.251572i
\(160\) 0 0
\(161\) 143.108i 0.888872i
\(162\) 0 0
\(163\) −124.000 −0.760736 −0.380368 0.924835i \(-0.624203\pi\)
−0.380368 + 0.924835i \(0.624203\pi\)
\(164\) 0 0
\(165\) −35.7771 40.0000i −0.216831 0.242424i
\(166\) 0 0
\(167\) 16.0000i 0.0958084i 0.998852 + 0.0479042i \(0.0152542\pi\)
−0.998852 + 0.0479042i \(0.984746\pi\)
\(168\) 0 0
\(169\) 151.000 0.893491
\(170\) 0 0
\(171\) −20.0000 + 178.885i −0.116959 + 1.04611i
\(172\) 0 0
\(173\) 140.000i 0.809249i 0.914483 + 0.404624i \(0.132598\pi\)
−0.914483 + 0.404624i \(0.867402\pi\)
\(174\) 0 0
\(175\) −80.4984 −0.459991
\(176\) 0 0
\(177\) −230.000 + 205.718i −1.29944 + 1.16225i
\(178\) 0 0
\(179\) 67.0820i 0.374760i 0.982288 + 0.187380i \(0.0599996\pi\)
−0.982288 + 0.187380i \(0.940000\pi\)
\(180\) 0 0
\(181\) 125.220 0.691822 0.345911 0.938267i \(-0.387570\pi\)
0.345911 + 0.938267i \(0.387570\pi\)
\(182\) 0 0
\(183\) 35.7771 + 40.0000i 0.195503 + 0.218579i
\(184\) 0 0
\(185\) 214.663i 1.16034i
\(186\) 0 0
\(187\) 80.0000 0.427807
\(188\) 0 0
\(189\) −196.774 + 140.000i −1.04113 + 0.740741i
\(190\) 0 0
\(191\) 160.000i 0.837696i 0.908056 + 0.418848i \(0.137566\pi\)
−0.908056 + 0.418848i \(0.862434\pi\)
\(192\) 0 0
\(193\) 30.0000 0.155440 0.0777202 0.996975i \(-0.475236\pi\)
0.0777202 + 0.996975i \(0.475236\pi\)
\(194\) 0 0
\(195\) −160.000 + 143.108i −0.820513 + 0.733889i
\(196\) 0 0
\(197\) 180.000i 0.913706i 0.889542 + 0.456853i \(0.151023\pi\)
−0.889542 + 0.456853i \(0.848977\pi\)
\(198\) 0 0
\(199\) −259.384 −1.30344 −0.651718 0.758461i \(-0.725952\pi\)
−0.651718 + 0.758461i \(0.725952\pi\)
\(200\) 0 0
\(201\) −88.0000 98.3870i −0.437811 0.489488i
\(202\) 0 0
\(203\) 465.102i 2.29114i
\(204\) 0 0
\(205\) −143.108 −0.698090
\(206\) 0 0
\(207\) 143.108 + 16.0000i 0.691345 + 0.0772947i
\(208\) 0 0
\(209\) 89.4427i 0.427956i
\(210\) 0 0
\(211\) 60.0000 0.284360 0.142180 0.989841i \(-0.454589\pi\)
0.142180 + 0.989841i \(0.454589\pi\)
\(212\) 0 0
\(213\) −178.885 + 160.000i −0.839838 + 0.751174i
\(214\) 0 0
\(215\) 144.000i 0.669767i
\(216\) 0 0
\(217\) 240.000 1.10599
\(218\) 0 0
\(219\) −100.000 111.803i −0.456621 0.510518i
\(220\) 0 0
\(221\) 320.000i 1.44796i
\(222\) 0 0
\(223\) 152.053 0.681850 0.340925 0.940090i \(-0.389260\pi\)
0.340925 + 0.940090i \(0.389260\pi\)
\(224\) 0 0
\(225\) −9.00000 + 80.4984i −0.0400000 + 0.357771i
\(226\) 0 0
\(227\) 254.912i 1.12296i 0.827491 + 0.561480i \(0.189768\pi\)
−0.827491 + 0.561480i \(0.810232\pi\)
\(228\) 0 0
\(229\) 196.774 0.859275 0.429638 0.903001i \(-0.358641\pi\)
0.429638 + 0.903001i \(0.358641\pi\)
\(230\) 0 0
\(231\) 89.4427 80.0000i 0.387198 0.346320i
\(232\) 0 0
\(233\) 160.997i 0.690974i −0.938424 0.345487i \(-0.887714\pi\)
0.938424 0.345487i \(-0.112286\pi\)
\(234\) 0 0
\(235\) −256.000 −1.08936
\(236\) 0 0
\(237\) −160.997 180.000i −0.679312 0.759494i
\(238\) 0 0
\(239\) 320.000i 1.33891i −0.742852 0.669456i \(-0.766527\pi\)
0.742852 0.669456i \(-0.233473\pi\)
\(240\) 0 0
\(241\) 318.000 1.31950 0.659751 0.751484i \(-0.270662\pi\)
0.659751 + 0.751484i \(0.270662\pi\)
\(242\) 0 0
\(243\) 118.000 + 212.426i 0.485597 + 0.874183i
\(244\) 0 0
\(245\) 124.000i 0.506122i
\(246\) 0 0
\(247\) −357.771 −1.44847
\(248\) 0 0
\(249\) −230.000 + 205.718i −0.923695 + 0.826178i
\(250\) 0 0
\(251\) 147.580i 0.587970i −0.955810 0.293985i \(-0.905018\pi\)
0.955810 0.293985i \(-0.0949816\pi\)
\(252\) 0 0
\(253\) −71.5542 −0.282823
\(254\) 0 0
\(255\) 143.108 + 160.000i 0.561209 + 0.627451i
\(256\) 0 0
\(257\) 107.331i 0.417631i −0.977955 0.208816i \(-0.933039\pi\)
0.977955 0.208816i \(-0.0669609\pi\)
\(258\) 0 0
\(259\) −480.000 −1.85328
\(260\) 0 0
\(261\) 465.102 + 52.0000i 1.78200 + 0.199234i
\(262\) 0 0
\(263\) 144.000i 0.547529i −0.961797 0.273764i \(-0.911731\pi\)
0.961797 0.273764i \(-0.0882688\pi\)
\(264\) 0 0
\(265\) −80.0000 −0.301887
\(266\) 0 0
\(267\) −360.000 + 321.994i −1.34831 + 1.20597i
\(268\) 0 0
\(269\) 132.000i 0.490706i −0.969434 0.245353i \(-0.921096\pi\)
0.969434 0.245353i \(-0.0789039\pi\)
\(270\) 0 0
\(271\) −277.272 −1.02315 −0.511573 0.859240i \(-0.670937\pi\)
−0.511573 + 0.859240i \(0.670937\pi\)
\(272\) 0 0
\(273\) −320.000 357.771i −1.17216 1.31052i
\(274\) 0 0
\(275\) 40.2492i 0.146361i
\(276\) 0 0
\(277\) 268.328 0.968694 0.484347 0.874876i \(-0.339057\pi\)
0.484347 + 0.874876i \(0.339057\pi\)
\(278\) 0 0
\(279\) 26.8328 240.000i 0.0961750 0.860215i
\(280\) 0 0
\(281\) 196.774i 0.700263i −0.936701 0.350132i \(-0.886137\pi\)
0.936701 0.350132i \(-0.113863\pi\)
\(282\) 0 0
\(283\) 76.0000 0.268551 0.134276 0.990944i \(-0.457129\pi\)
0.134276 + 0.990944i \(0.457129\pi\)
\(284\) 0 0
\(285\) 178.885 160.000i 0.627668 0.561404i
\(286\) 0 0
\(287\) 320.000i 1.11498i
\(288\) 0 0
\(289\) −31.0000 −0.107266
\(290\) 0 0
\(291\) −100.000 111.803i −0.343643 0.384204i
\(292\) 0 0
\(293\) 140.000i 0.477816i −0.971042 0.238908i \(-0.923211\pi\)
0.971042 0.238908i \(-0.0767894\pi\)
\(294\) 0 0
\(295\) 411.437 1.39470
\(296\) 0 0
\(297\) −70.0000 98.3870i −0.235690 0.331269i
\(298\) 0 0
\(299\) 286.217i 0.957246i
\(300\) 0 0
\(301\) −321.994 −1.06975
\(302\) 0 0
\(303\) −205.718 + 184.000i −0.678938 + 0.607261i
\(304\) 0 0
\(305\) 71.5542i 0.234604i
\(306\) 0 0
\(307\) −244.000 −0.794788 −0.397394 0.917648i \(-0.630085\pi\)
−0.397394 + 0.917648i \(0.630085\pi\)
\(308\) 0 0
\(309\) 89.4427 + 100.000i 0.289459 + 0.323625i
\(310\) 0 0
\(311\) 400.000i 1.28617i 0.765793 + 0.643087i \(0.222347\pi\)
−0.765793 + 0.643087i \(0.777653\pi\)
\(312\) 0 0
\(313\) 290.000 0.926518 0.463259 0.886223i \(-0.346680\pi\)
0.463259 + 0.886223i \(0.346680\pi\)
\(314\) 0 0
\(315\) 320.000 + 35.7771i 1.01587 + 0.113578i
\(316\) 0 0
\(317\) 420.000i 1.32492i −0.749097 0.662461i \(-0.769512\pi\)
0.749097 0.662461i \(-0.230488\pi\)
\(318\) 0 0
\(319\) −232.551 −0.729000
\(320\) 0 0
\(321\) 90.0000 80.4984i 0.280374 0.250774i
\(322\) 0 0
\(323\) 357.771i 1.10765i
\(324\) 0 0
\(325\) −160.997 −0.495375
\(326\) 0 0
\(327\) −250.440 280.000i −0.765870 0.856269i
\(328\) 0 0
\(329\) 572.433i 1.73992i
\(330\) 0 0
\(331\) −340.000 −1.02719 −0.513595 0.858033i \(-0.671687\pi\)
−0.513595 + 0.858033i \(0.671687\pi\)
\(332\) 0 0
\(333\) −53.6656 + 480.000i −0.161158 + 1.44144i
\(334\) 0 0
\(335\) 176.000i 0.525373i
\(336\) 0 0
\(337\) 110.000 0.326409 0.163205 0.986592i \(-0.447817\pi\)
0.163205 + 0.986592i \(0.447817\pi\)
\(338\) 0 0
\(339\) 80.0000 71.5542i 0.235988 0.211074i
\(340\) 0 0
\(341\) 120.000i 0.351906i
\(342\) 0 0
\(343\) 160.997 0.469379
\(344\) 0 0
\(345\) −128.000 143.108i −0.371014 0.414807i
\(346\) 0 0
\(347\) 147.580i 0.425304i −0.977128 0.212652i \(-0.931790\pi\)
0.977128 0.212652i \(-0.0682101\pi\)
\(348\) 0 0
\(349\) −17.8885 −0.0512566 −0.0256283 0.999672i \(-0.508159\pi\)
−0.0256283 + 0.999672i \(0.508159\pi\)
\(350\) 0 0
\(351\) −393.548 + 280.000i −1.12122 + 0.797721i
\(352\) 0 0
\(353\) 214.663i 0.608109i 0.952655 + 0.304055i \(0.0983405\pi\)
−0.952655 + 0.304055i \(0.901659\pi\)
\(354\) 0 0
\(355\) 320.000 0.901408
\(356\) 0 0
\(357\) −357.771 + 320.000i −1.00216 + 0.896359i
\(358\) 0 0
\(359\) 560.000i 1.55989i 0.625849 + 0.779944i \(0.284753\pi\)
−0.625849 + 0.779944i \(0.715247\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 0 0
\(363\) −202.000 225.843i −0.556474 0.622157i
\(364\) 0 0
\(365\) 200.000i 0.547945i
\(366\) 0 0
\(367\) 617.155 1.68162 0.840810 0.541330i \(-0.182079\pi\)
0.840810 + 0.541330i \(0.182079\pi\)
\(368\) 0 0
\(369\) −320.000 35.7771i −0.867209 0.0969569i
\(370\) 0 0
\(371\) 178.885i 0.482171i
\(372\) 0 0
\(373\) −375.659 −1.00713 −0.503565 0.863957i \(-0.667979\pi\)
−0.503565 + 0.863957i \(0.667979\pi\)
\(374\) 0 0
\(375\) 304.105 272.000i 0.810947 0.725333i
\(376\) 0 0
\(377\) 930.204i 2.46739i
\(378\) 0 0
\(379\) 260.000 0.686016 0.343008 0.939333i \(-0.388554\pi\)
0.343008 + 0.939333i \(0.388554\pi\)
\(380\) 0 0
\(381\) 53.6656 + 60.0000i 0.140855 + 0.157480i
\(382\) 0 0
\(383\) 544.000i 1.42037i −0.704017 0.710183i \(-0.748612\pi\)
0.704017 0.710183i \(-0.251388\pi\)
\(384\) 0 0
\(385\) −160.000 −0.415584
\(386\) 0 0
\(387\) −36.0000 + 321.994i −0.0930233 + 0.832025i
\(388\) 0 0
\(389\) 332.000i 0.853470i −0.904377 0.426735i \(-0.859664\pi\)
0.904377 0.426735i \(-0.140336\pi\)
\(390\) 0 0
\(391\) 286.217 0.732012
\(392\) 0 0
\(393\) 150.000 134.164i 0.381679 0.341384i
\(394\) 0 0
\(395\) 321.994i 0.815174i
\(396\) 0 0
\(397\) 268.328 0.675890 0.337945 0.941166i \(-0.390268\pi\)
0.337945 + 0.941166i \(0.390268\pi\)
\(398\) 0 0
\(399\) 357.771 + 400.000i 0.896669 + 1.00251i
\(400\) 0 0
\(401\) 160.997i 0.401489i −0.979644 0.200744i \(-0.935664\pi\)
0.979644 0.200744i \(-0.0643360\pi\)
\(402\) 0 0
\(403\) 480.000 1.19107
\(404\) 0 0
\(405\) 71.5542 316.000i 0.176677 0.780247i
\(406\) 0 0
\(407\) 240.000i 0.589681i
\(408\) 0 0
\(409\) −178.000 −0.435208 −0.217604 0.976037i \(-0.569824\pi\)
−0.217604 + 0.976037i \(0.569824\pi\)
\(410\) 0 0
\(411\) 320.000 286.217i 0.778589 0.696391i
\(412\) 0 0
\(413\) 920.000i 2.22760i
\(414\) 0 0
\(415\) 411.437 0.991413
\(416\) 0 0
\(417\) 520.000 + 581.378i 1.24700 + 1.39419i
\(418\) 0 0
\(419\) 245.967i 0.587035i −0.955954 0.293517i \(-0.905174\pi\)
0.955954 0.293517i \(-0.0948258\pi\)
\(420\) 0 0
\(421\) 53.6656 0.127472 0.0637359 0.997967i \(-0.479698\pi\)
0.0637359 + 0.997967i \(0.479698\pi\)
\(422\) 0 0
\(423\) −572.433 64.0000i −1.35327 0.151300i
\(424\) 0 0
\(425\) 160.997i 0.378816i
\(426\) 0 0
\(427\) 160.000 0.374707
\(428\) 0 0
\(429\) 178.885 160.000i 0.416982 0.372960i
\(430\) 0 0
\(431\) 320.000i 0.742459i −0.928541 0.371230i \(-0.878936\pi\)
0.928541 0.371230i \(-0.121064\pi\)
\(432\) 0 0
\(433\) 530.000 1.22402 0.612009 0.790851i \(-0.290362\pi\)
0.612009 + 0.790851i \(0.290362\pi\)
\(434\) 0 0
\(435\) −416.000 465.102i −0.956322 1.06920i
\(436\) 0 0
\(437\) 320.000i 0.732265i
\(438\) 0 0
\(439\) −474.046 −1.07983 −0.539916 0.841719i \(-0.681544\pi\)
−0.539916 + 0.841719i \(0.681544\pi\)
\(440\) 0 0
\(441\) −31.0000 + 277.272i −0.0702948 + 0.628736i
\(442\) 0 0
\(443\) 505.351i 1.14075i −0.821385 0.570374i \(-0.806798\pi\)
0.821385 0.570374i \(-0.193202\pi\)
\(444\) 0 0
\(445\) 643.988 1.44716
\(446\) 0 0
\(447\) −62.6099 + 56.0000i −0.140067 + 0.125280i
\(448\) 0 0
\(449\) 268.328i 0.597613i −0.954314 0.298806i \(-0.903412\pi\)
0.954314 0.298806i \(-0.0965885\pi\)
\(450\) 0 0
\(451\) 160.000 0.354767
\(452\) 0 0
\(453\) −482.991 540.000i −1.06620 1.19205i
\(454\) 0 0
\(455\) 640.000i 1.40659i
\(456\) 0 0
\(457\) −210.000 −0.459519 −0.229759 0.973247i \(-0.573794\pi\)
−0.229759 + 0.973247i \(0.573794\pi\)
\(458\) 0 0
\(459\) 280.000 + 393.548i 0.610022 + 0.857403i
\(460\) 0 0
\(461\) 372.000i 0.806941i −0.914993 0.403471i \(-0.867804\pi\)
0.914993 0.403471i \(-0.132196\pi\)
\(462\) 0 0
\(463\) −169.941 −0.367044 −0.183522 0.983016i \(-0.558750\pi\)
−0.183522 + 0.983016i \(0.558750\pi\)
\(464\) 0 0
\(465\) −240.000 + 214.663i −0.516129 + 0.461640i
\(466\) 0 0
\(467\) 460.630i 0.986360i −0.869927 0.493180i \(-0.835834\pi\)
0.869927 0.493180i \(-0.164166\pi\)
\(468\) 0 0
\(469\) −393.548 −0.839121
\(470\) 0 0
\(471\) −107.331 120.000i −0.227880 0.254777i
\(472\) 0 0
\(473\) 160.997i 0.340374i
\(474\) 0 0
\(475\) 180.000 0.378947
\(476\) 0 0
\(477\) −178.885 20.0000i −0.375022 0.0419287i
\(478\) 0 0
\(479\) 320.000i 0.668058i 0.942563 + 0.334029i \(0.108408\pi\)
−0.942563 + 0.334029i \(0.891592\pi\)
\(480\) 0 0
\(481\) −960.000 −1.99584
\(482\) 0 0
\(483\) 320.000 286.217i 0.662526 0.592581i
\(484\) 0 0
\(485\) 200.000i 0.412371i
\(486\) 0 0
\(487\) 62.6099 0.128562 0.0642812 0.997932i \(-0.479525\pi\)
0.0642812 + 0.997932i \(0.479525\pi\)
\(488\) 0 0
\(489\) 248.000 + 277.272i 0.507157 + 0.567019i
\(490\) 0 0
\(491\) 889.955i 1.81254i −0.422704 0.906268i \(-0.638919\pi\)
0.422704 0.906268i \(-0.361081\pi\)
\(492\) 0 0
\(493\) 930.204 1.88682
\(494\) 0 0
\(495\) −17.8885 + 160.000i −0.0361385 + 0.323232i
\(496\) 0 0
\(497\) 715.542i 1.43972i
\(498\) 0 0
\(499\) −100.000 −0.200401 −0.100200 0.994967i \(-0.531948\pi\)
−0.100200 + 0.994967i \(0.531948\pi\)
\(500\) 0 0
\(501\) 35.7771 32.0000i 0.0714114 0.0638723i
\(502\) 0 0
\(503\) 16.0000i 0.0318091i 0.999874 + 0.0159046i \(0.00506280\pi\)
−0.999874 + 0.0159046i \(0.994937\pi\)
\(504\) 0 0
\(505\) 368.000 0.728713
\(506\) 0 0
\(507\) −302.000 337.646i −0.595661 0.665969i
\(508\) 0 0
\(509\) 332.000i 0.652259i 0.945325 + 0.326130i \(0.105745\pi\)
−0.945325 + 0.326130i \(0.894255\pi\)
\(510\) 0 0
\(511\) −447.214 −0.875173
\(512\) 0 0
\(513\) 440.000 313.050i 0.857700 0.610233i
\(514\) 0 0
\(515\) 178.885i 0.347350i
\(516\) 0 0
\(517\) 286.217 0.553611
\(518\) 0 0
\(519\) 313.050 280.000i 0.603178 0.539499i
\(520\) 0 0
\(521\) 214.663i 0.412020i 0.978550 + 0.206010i \(0.0660480\pi\)
−0.978550 + 0.206010i \(0.933952\pi\)
\(522\) 0 0
\(523\) −76.0000 −0.145315 −0.0726577 0.997357i \(-0.523148\pi\)
−0.0726577 + 0.997357i \(0.523148\pi\)
\(524\) 0 0
\(525\) 160.997 + 180.000i 0.306661 + 0.342857i
\(526\) 0 0
\(527\) 480.000i 0.910816i
\(528\) 0 0
\(529\) 273.000 0.516068
\(530\) 0 0
\(531\) 920.000 + 102.859i 1.73258 + 0.193708i
\(532\) 0 0
\(533\) 640.000i 1.20075i
\(534\) 0 0
\(535\) −160.997 −0.300929
\(536\) 0 0
\(537\) 150.000 134.164i 0.279330 0.249840i
\(538\) 0 0
\(539\) 138.636i 0.257210i
\(540\) 0 0
\(541\) −1019.65 −1.88474 −0.942372 0.334566i \(-0.891410\pi\)
−0.942372 + 0.334566i \(0.891410\pi\)
\(542\) 0 0
\(543\) −250.440 280.000i −0.461215 0.515654i
\(544\) 0 0
\(545\) 500.879i 0.919044i
\(546\) 0 0
\(547\) 244.000 0.446069 0.223035 0.974810i \(-0.428404\pi\)
0.223035 + 0.974810i \(0.428404\pi\)
\(548\) 0 0
\(549\) 17.8885 160.000i 0.0325839 0.291439i
\(550\) 0 0
\(551\) 1040.00i 1.88748i
\(552\) 0 0
\(553\) −720.000 −1.30199
\(554\) 0 0
\(555\) 480.000 429.325i 0.864865 0.773559i
\(556\) 0 0
\(557\) 60.0000i 0.107720i 0.998548 + 0.0538600i \(0.0171525\pi\)
−0.998548 + 0.0538600i \(0.982848\pi\)
\(558\) 0 0
\(559\) −643.988 −1.15204
\(560\) 0 0
\(561\) −160.000 178.885i −0.285205 0.318869i
\(562\) 0 0
\(563\) 111.803i 0.198585i 0.995058 + 0.0992925i \(0.0316580\pi\)
−0.995058 + 0.0992925i \(0.968342\pi\)
\(564\) 0 0
\(565\) −143.108 −0.253289
\(566\) 0 0
\(567\) 706.597 + 160.000i 1.24620 + 0.282187i
\(568\) 0 0
\(569\) 858.650i 1.50905i −0.656271 0.754526i \(-0.727867\pi\)
0.656271 0.754526i \(-0.272133\pi\)
\(570\) 0 0
\(571\) 940.000 1.64623 0.823117 0.567871i \(-0.192233\pi\)
0.823117 + 0.567871i \(0.192233\pi\)
\(572\) 0 0
\(573\) 357.771 320.000i 0.624382 0.558464i
\(574\) 0 0
\(575\) 144.000i 0.250435i
\(576\) 0 0
\(577\) −370.000 −0.641248 −0.320624 0.947207i \(-0.603893\pi\)
−0.320624 + 0.947207i \(0.603893\pi\)
\(578\) 0 0
\(579\) −60.0000 67.0820i −0.103627 0.115858i
\(580\) 0 0
\(581\) 920.000i 1.58348i
\(582\) 0 0
\(583\) 89.4427 0.153418
\(584\) 0 0
\(585\) 640.000 + 71.5542i 1.09402 + 0.122315i
\(586\) 0 0
\(587\) 469.574i 0.799956i 0.916525 + 0.399978i \(0.130982\pi\)
−0.916525 + 0.399978i \(0.869018\pi\)
\(588\) 0 0
\(589\) −536.656 −0.911131
\(590\) 0 0
\(591\) 402.492 360.000i 0.681036 0.609137i
\(592\) 0 0
\(593\) 572.433i 0.965318i −0.875808 0.482659i \(-0.839671\pi\)
0.875808 0.482659i \(-0.160329\pi\)
\(594\) 0 0
\(595\) 640.000 1.07563
\(596\) 0 0
\(597\) 518.768 + 580.000i 0.868958 + 0.971524i
\(598\) 0 0
\(599\) 560.000i 0.934891i 0.884022 + 0.467446i \(0.154826\pi\)
−0.884022 + 0.467446i \(0.845174\pi\)
\(600\) 0 0
\(601\) −302.000 −0.502496 −0.251248 0.967923i \(-0.580841\pi\)
−0.251248 + 0.967923i \(0.580841\pi\)
\(602\) 0 0
\(603\) −44.0000 + 393.548i −0.0729685 + 0.652650i
\(604\) 0 0
\(605\) 404.000i 0.667769i
\(606\) 0 0
\(607\) 44.7214 0.0736760 0.0368380 0.999321i \(-0.488271\pi\)
0.0368380 + 0.999321i \(0.488271\pi\)
\(608\) 0 0
\(609\) 1040.00 930.204i 1.70772 1.52743i
\(610\) 0 0
\(611\) 1144.87i 1.87376i
\(612\) 0 0
\(613\) −447.214 −0.729549 −0.364775 0.931096i \(-0.618854\pi\)
−0.364775 + 0.931096i \(0.618854\pi\)
\(614\) 0 0
\(615\) 286.217 + 320.000i 0.465393 + 0.520325i
\(616\) 0 0
\(617\) 447.214i 0.724819i 0.932019 + 0.362410i \(0.118046\pi\)
−0.932019 + 0.362410i \(0.881954\pi\)
\(618\) 0 0
\(619\) 780.000 1.26010 0.630048 0.776556i \(-0.283035\pi\)
0.630048 + 0.776556i \(0.283035\pi\)
\(620\) 0 0
\(621\) −250.440 352.000i −0.403284 0.566828i
\(622\) 0 0
\(623\) 1440.00i 2.31140i
\(624\) 0 0
\(625\) −319.000 −0.510400
\(626\) 0 0
\(627\) −200.000 + 178.885i −0.318979 + 0.285304i
\(628\) 0 0
\(629\) 960.000i 1.52623i
\(630\) 0 0
\(631\) −80.4984 −0.127573 −0.0637864 0.997964i \(-0.520318\pi\)
−0.0637864 + 0.997964i \(0.520318\pi\)
\(632\) 0 0
\(633\) −120.000 134.164i −0.189573 0.211950i
\(634\) 0 0
\(635\) 107.331i 0.169026i
\(636\) 0 0
\(637\) −554.545 −0.870557
\(638\) 0 0
\(639\) 715.542 + 80.0000i 1.11978 + 0.125196i
\(640\) 0 0
\(641\) 661.876i 1.03257i −0.856417 0.516284i \(-0.827315\pi\)
0.856417 0.516284i \(-0.172685\pi\)
\(642\) 0 0
\(643\) −844.000 −1.31260 −0.656299 0.754501i \(-0.727879\pi\)
−0.656299 + 0.754501i \(0.727879\pi\)
\(644\) 0 0
\(645\) 321.994 288.000i 0.499215 0.446512i
\(646\) 0 0
\(647\) 16.0000i 0.0247295i 0.999924 + 0.0123648i \(0.00393593\pi\)
−0.999924 + 0.0123648i \(0.996064\pi\)
\(648\) 0 0
\(649\) −460.000 −0.708783
\(650\) 0 0
\(651\) −480.000 536.656i −0.737327 0.824357i
\(652\) 0 0
\(653\) 660.000i 1.01072i −0.862909 0.505360i \(-0.831360\pi\)
0.862909 0.505360i \(-0.168640\pi\)
\(654\) 0 0
\(655\) −268.328 −0.409661
\(656\) 0 0
\(657\) −50.0000 + 447.214i −0.0761035 + 0.680690i
\(658\) 0 0
\(659\) 210.190i 0.318954i 0.987202 + 0.159477i \(0.0509807\pi\)
−0.987202 + 0.159477i \(0.949019\pi\)
\(660\) 0 0
\(661\) 769.207 1.16370 0.581851 0.813295i \(-0.302329\pi\)
0.581851 + 0.813295i \(0.302329\pi\)
\(662\) 0 0
\(663\) −715.542 + 640.000i −1.07925 + 0.965309i
\(664\) 0 0
\(665\) 715.542i 1.07600i
\(666\) 0 0
\(667\) −832.000 −1.24738
\(668\) 0 0
\(669\) −304.105 340.000i −0.454567 0.508221i
\(670\) 0 0
\(671\) 80.0000i 0.119225i
\(672\) 0 0
\(673\) −910.000 −1.35215 −0.676077 0.736831i \(-0.736321\pi\)
−0.676077 + 0.736831i \(0.736321\pi\)
\(674\) 0 0
\(675\) 198.000 140.872i 0.293333 0.208700i
\(676\) 0 0
\(677\) 260.000i 0.384047i 0.981390 + 0.192024i \(0.0615050\pi\)
−0.981390 + 0.192024i \(0.938495\pi\)
\(678\) 0 0
\(679\) −447.214 −0.658636
\(680\) 0 0
\(681\) 570.000 509.823i 0.837004 0.748639i
\(682\) 0 0
\(683\) 934.676i 1.36849i −0.729254 0.684243i \(-0.760133\pi\)
0.729254 0.684243i \(-0.239867\pi\)
\(684\) 0 0
\(685\) −572.433 −0.835669
\(686\) 0 0
\(687\) −393.548 440.000i −0.572850 0.640466i
\(688\) 0 0
\(689\) 357.771i 0.519261i
\(690\) 0 0
\(691\) 100.000 0.144718 0.0723589 0.997379i \(-0.476947\pi\)
0.0723589 + 0.997379i \(0.476947\pi\)
\(692\) 0 0
\(693\) −357.771 40.0000i −0.516264 0.0577201i
\(694\) 0 0
\(695\) 1040.00i 1.49640i
\(696\) 0 0
\(697\) −640.000 −0.918221
\(698\) 0 0
\(699\) −360.000 + 321.994i −0.515021 + 0.460649i
\(700\) 0 0
\(701\) 1028.00i 1.46648i −0.679972 0.733238i \(-0.738008\pi\)
0.679972 0.733238i \(-0.261992\pi\)
\(702\) 0 0
\(703\) 1073.31 1.52676
\(704\) 0 0
\(705\) 512.000 + 572.433i 0.726241 + 0.811962i
\(706\) 0 0
\(707\) 822.873i 1.16389i
\(708\) 0 0
\(709\) 912.316 1.28676 0.643382 0.765545i \(-0.277531\pi\)
0.643382 + 0.765545i \(0.277531\pi\)
\(710\) 0 0
\(711\) −80.4984 + 720.000i −0.113219 + 1.01266i
\(712\) 0 0
\(713\) 429.325i 0.602139i
\(714\) 0 0
\(715\) −320.000 −0.447552
\(716\) 0 0
\(717\) −715.542 + 640.000i −0.997966 + 0.892608i
\(718\) 0 0
\(719\) 480.000i 0.667594i −0.942645 0.333797i \(-0.891670\pi\)
0.942645 0.333797i \(-0.108330\pi\)
\(720\) 0 0
\(721\) 400.000 0.554785
\(722\) 0 0
\(723\) −636.000 711.070i −0.879668 0.983499i
\(724\) 0 0
\(725\) 468.000i 0.645517i
\(726\) 0 0
\(727\) 241.495 0.332181 0.166090 0.986111i \(-0.446886\pi\)
0.166090 + 0.986111i \(0.446886\pi\)
\(728\) 0 0
\(729\) 239.000 688.709i 0.327846 0.944731i
\(730\) 0 0
\(731\) 643.988i 0.880968i
\(732\) 0 0
\(733\) 1126.98 1.53749 0.768744 0.639557i \(-0.220882\pi\)
0.768744 + 0.639557i \(0.220882\pi\)
\(734\) 0 0
\(735\) 277.272 248.000i 0.377241 0.337415i
\(736\) 0 0
\(737\) 196.774i 0.266993i
\(738\) 0 0
\(739\) 1020.00 1.38024 0.690122 0.723693i \(-0.257557\pi\)
0.690122 + 0.723693i \(0.257557\pi\)
\(740\) 0 0
\(741\) 715.542 + 800.000i 0.965643 + 1.07962i
\(742\) 0 0
\(743\) 176.000i 0.236878i −0.992961 0.118439i \(-0.962211\pi\)
0.992961 0.118439i \(-0.0377889\pi\)
\(744\) 0 0
\(745\) 112.000 0.150336
\(746\) 0 0
\(747\) 920.000 + 102.859i 1.23159 + 0.137696i
\(748\) 0 0
\(749\) 360.000i 0.480641i
\(750\) 0 0
\(751\) 1296.92 1.72692 0.863462 0.504414i \(-0.168292\pi\)
0.863462 + 0.504414i \(0.168292\pi\)
\(752\) 0 0
\(753\) −330.000 + 295.161i −0.438247 + 0.391980i
\(754\) 0 0
\(755\) 965.981i 1.27945i
\(756\) 0 0
\(757\) 983.870 1.29970 0.649848 0.760064i \(-0.274833\pi\)
0.649848 + 0.760064i \(0.274833\pi\)
\(758\) 0 0
\(759\) 143.108 + 160.000i 0.188549 + 0.210804i
\(760\) 0 0
\(761\) 107.331i 0.141040i −0.997510 0.0705199i \(-0.977534\pi\)
0.997510 0.0705199i \(-0.0224658\pi\)
\(762\) 0 0
\(763\) −1120.00 −1.46789
\(764\) 0 0
\(765\) 71.5542 640.000i 0.0935349 0.836601i
\(766\) 0 0
\(767\) 1840.00i 2.39896i
\(768\) 0 0
\(769\) −1378.00 −1.79194 −0.895969 0.444117i \(-0.853517\pi\)
−0.895969 + 0.444117i \(0.853517\pi\)
\(770\) 0 0
\(771\) −240.000 + 214.663i −0.311284 + 0.278421i
\(772\) 0 0
\(773\) 1180.00i 1.52652i −0.646091 0.763260i \(-0.723598\pi\)
0.646091 0.763260i \(-0.276402\pi\)
\(774\) 0 0
\(775\) −241.495 −0.311607
\(776\) 0 0
\(777\) 960.000 + 1073.31i 1.23552 + 1.38135i
\(778\) 0 0
\(779\) 715.542i 0.918539i
\(780\) 0 0
\(781\) −357.771 −0.458093
\(782\) 0 0
\(783\) −813.929 1144.00i −1.03950 1.46105i
\(784\) 0 0
\(785\) 214.663i 0.273455i
\(786\) 0 0
\(787\) −1244.00 −1.58069 −0.790343 0.612665i \(-0.790098\pi\)
−0.790343 + 0.612665i \(0.790098\pi\)
\(788\) 0 0
\(789\) −321.994 + 288.000i −0.408104 + 0.365019i
\(790\) 0 0
\(791\) 320.000i 0.404551i
\(792\) 0 0
\(793\) 320.000 0.403531
\(794\) 0 0
\(795\) 160.000 + 178.885i 0.201258 + 0.225013i
\(796\) 0 0
\(797\) 580.000i 0.727729i −0.931452 0.363864i \(-0.881457\pi\)
0.931452 0.363864i \(-0.118543\pi\)
\(798\) 0 0
\(799\) −1144.87 −1.43287
\(800\) 0 0
\(801\) 1440.00 + 160.997i 1.79775 + 0.200995i
\(802\) 0 0
\(803\) 223.607i 0.278464i
\(804\) 0 0
\(805\) −572.433 −0.711097
\(806\) 0 0
\(807\) −295.161 + 264.000i −0.365751 + 0.327138i
\(808\) 0 0
\(809\) 250.440i 0.309567i 0.987948 + 0.154783i \(0.0494680\pi\)
−0.987948 + 0.154783i \(0.950532\pi\)
\(810\) 0 0
\(811\) 980.000 1.20838 0.604192 0.796839i \(-0.293496\pi\)
0.604192 + 0.796839i \(0.293496\pi\)
\(812\) 0 0
\(813\) 554.545 + 620.000i 0.682097 + 0.762608i
\(814\) 0 0
\(815\) 496.000i 0.608589i
\(816\) 0 0
\(817\) 720.000 0.881273
\(818\) 0 0
\(819\) −160.000 + 1431.08i −0.195360 + 1.74735i
\(820\) 0 0
\(821\) 932.000i 1.13520i 0.823304 + 0.567600i \(0.192128\pi\)
−0.823304 + 0.567600i \(0.807872\pi\)
\(822\) 0 0
\(823\) 1565.25 1.90188 0.950940 0.309375i \(-0.100120\pi\)
0.950940 + 0.309375i \(0.100120\pi\)
\(824\) 0 0
\(825\) −90.0000 + 80.4984i −0.109091 + 0.0975739i
\(826\) 0 0
\(827\) 1542.89i 1.86564i 0.360339 + 0.932822i \(0.382661\pi\)
−0.360339 + 0.932822i \(0.617339\pi\)
\(828\) 0 0
\(829\) 482.991 0.582618 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(830\) 0 0
\(831\) −536.656 600.000i −0.645796 0.722022i
\(832\) 0 0
\(833\) 554.545i 0.665720i
\(834\) 0 0
\(835\) −64.0000 −0.0766467
\(836\) 0 0
\(837\) −590.322 + 420.000i −0.705283 + 0.501792i
\(838\) 0 0
\(839\) 1360.00i 1.62098i −0.585754 0.810489i \(-0.699202\pi\)
0.585754 0.810489i \(-0.300798\pi\)
\(840\) 0 0
\(841\) −1863.00 −2.21522
\(842\) 0 0
\(843\) −440.000 + 393.548i −0.521945 + 0.466842i
\(844\) 0 0
\(845\) 604.000i 0.714793i
\(846\) 0 0
\(847\) −903.371 −1.06655
\(848\) 0 0
\(849\) −152.000 169.941i −0.179034 0.200166i
\(850\) 0 0
\(851\) 858.650i 1.00899i
\(852\) 0 0
\(853\) −948.093 −1.11148 −0.555740 0.831356i \(-0.687565\pi\)
−0.555740 + 0.831356i \(0.687565\pi\)
\(854\) 0 0
\(855\) −715.542 80.0000i −0.836891 0.0935673i
\(856\) 0 0
\(857\) 858.650i 1.00193i 0.865469 + 0.500963i \(0.167021\pi\)
−0.865469 + 0.500963i \(0.832979\pi\)
\(858\) 0 0
\(859\) −1340.00 −1.55995 −0.779977 0.625809i \(-0.784769\pi\)
−0.779977 + 0.625809i \(0.784769\pi\)
\(860\) 0 0
\(861\) −715.542 + 640.000i −0.831059 + 0.743322i
\(862\) 0 0
\(863\) 224.000i 0.259560i 0.991543 + 0.129780i \(0.0414271\pi\)
−0.991543 + 0.129780i \(0.958573\pi\)
\(864\) 0 0
\(865\) −560.000 −0.647399
\(866\) 0 0
\(867\) 62.0000 + 69.3181i 0.0715110 + 0.0799517i
\(868\) 0 0
\(869\) 360.000i 0.414269i
\(870\) 0 0
\(871\) −787.096 −0.903669
\(872\) 0 0
\(873\) −50.0000 + 447.214i −0.0572738 + 0.512272i
\(874\) 0 0
\(875\) 1216.42i 1.39020i
\(876\) 0 0
\(877\) −948.093 −1.08106 −0.540532 0.841324i \(-0.681777\pi\)
−0.540532 + 0.841324i \(0.681777\pi\)
\(878\) 0 0
\(879\) −313.050 + 280.000i −0.356143 + 0.318544i
\(880\) 0 0
\(881\) 1538.41i 1.74621i 0.487528 + 0.873107i \(0.337899\pi\)
−0.487528 + 0.873107i \(0.662101\pi\)
\(882\) 0 0
\(883\) −1164.00 −1.31823 −0.659117 0.752041i \(-0.729070\pi\)
−0.659117 + 0.752041i \(0.729070\pi\)
\(884\) 0 0
\(885\) −822.873 920.000i −0.929800 1.03955i
\(886\) 0 0
\(887\) 336.000i 0.378805i 0.981900 + 0.189402i \(0.0606551\pi\)
−0.981900 + 0.189402i \(0.939345\pi\)
\(888\) 0 0
\(889\) 240.000 0.269966
\(890\) 0 0
\(891\) −80.0000 + 353.299i −0.0897868 + 0.396519i
\(892\) 0 0
\(893\) 1280.00i 1.43337i
\(894\) 0 0
\(895\) −268.328 −0.299808
\(896\) 0 0
\(897\) 640.000 572.433i 0.713489 0.638164i
\(898\) 0 0
\(899\) 1395.31i 1.55206i
\(900\) 0 0
\(901\) −357.771 −0.397082
\(902\) 0 0
\(903\) 643.988 + 720.000i 0.713165 + 0.797342i
\(904\) 0 0
\(905\) 500.879i 0.553458i
\(906\) 0 0
\(907\) −764.000 −0.842337 −0.421169 0.906982i \(-0.638380\pi\)
−0.421169 + 0.906982i \(0.638380\pi\)
\(908\) 0 0
\(909\) 822.873 + 92.0000i 0.905251 + 0.101210i
\(910\) 0 0
\(911\) 1280.00i 1.40505i 0.711659 + 0.702525i \(0.247944\pi\)
−0.711659 + 0.702525i \(0.752056\pi\)
\(912\) 0 0
\(913\) −460.000 −0.503834
\(914\) 0 0
\(915\) −160.000 + 143.108i −0.174863 + 0.156403i
\(916\) 0 0
\(917\) 600.000i 0.654308i
\(918\) 0 0
\(919\) 849.706 0.924598 0.462299 0.886724i \(-0.347025\pi\)
0.462299 + 0.886724i \(0.347025\pi\)
\(920\) 0 0
\(921\) 488.000 + 545.601i 0.529859 + 0.592400i
\(922\) 0 0
\(923\) 1431.08i 1.55047i
\(924\) 0 0
\(925\) 482.991 0.522152
\(926\) 0 0
\(927\) 44.7214 400.000i 0.0482431 0.431499i
\(928\) 0 0
\(929\) 375.659i 0.404370i −0.979347 0.202185i \(-0.935196\pi\)
0.979347 0.202185i \(-0.0648042\pi\)
\(930\) 0 0
\(931\) 620.000 0.665951
\(932\) 0 0
\(933\) 894.427 800.000i 0.958657 0.857449i
\(934\) 0 0
\(935\) 320.000i 0.342246i
\(936\) 0 0
\(937\) 510.000 0.544290 0.272145 0.962256i \(-0.412267\pi\)
0.272145 + 0.962256i \(0.412267\pi\)
\(938\) 0 0
\(939\) −580.000 648.460i −0.617678 0.690585i
\(940\) 0 0
\(941\) 812.000i 0.862912i 0.902134 + 0.431456i \(0.142000\pi\)
−0.902134 + 0.431456i \(0.858000\pi\)
\(942\) 0 0
\(943\) 572.433 0.607034
\(944\) 0 0
\(945\) −560.000 787.096i −0.592593 0.832906i
\(946\) 0 0
\(947\) 1569.72i 1.65757i 0.559566 + 0.828785i \(0.310968\pi\)
−0.559566 + 0.828785i \(0.689032\pi\)
\(948\) 0 0
\(949\) −894.427 −0.942494
\(950\) 0 0
\(951\) −939.149 + 840.000i −0.987538 + 0.883281i
\(952\) 0 0
\(953\) 1073.31i 1.12625i −0.826373 0.563123i \(-0.809600\pi\)
0.826373 0.563123i \(-0.190400\pi\)
\(954\) 0 0
\(955\) −640.000 −0.670157
\(956\) 0 0
\(957\) 465.102 + 520.000i 0.486000 + 0.543365i
\(958\) 0 0
\(959\) 1280.00i 1.33472i
\(960\) 0 0
\(961\) −241.000 −0.250780
\(962\) 0 0
\(963\) −360.000 40.2492i −0.373832 0.0417957i
\(964\) 0 0
\(965\) 120.000i 0.124352i
\(966\) 0 0
\(967\) 420.381 0.434727 0.217363 0.976091i \(-0.430254\pi\)
0.217363 + 0.976091i \(0.430254\pi\)
\(968\) 0 0
\(969\) 800.000 715.542i 0.825593 0.738433i
\(970\) 0 0
\(971\) 791.568i 0.815209i −0.913159 0.407605i \(-0.866364\pi\)
0.913159 0.407605i \(-0.133636\pi\)
\(972\) 0 0
\(973\) 2325.51 2.39004
\(974\) 0 0
\(975\) 321.994 + 360.000i 0.330250 + 0.369231i
\(976\) 0 0
\(977\) 661.876i 0.677458i 0.940884 + 0.338729i \(0.109997\pi\)
−0.940884 + 0.338729i \(0.890003\pi\)
\(978\) 0 0
\(979\) −720.000 −0.735444
\(980\) 0 0
\(981\) −125.220 + 1120.00i −0.127645 + 1.14169i
\(982\) 0 0
\(983\) 1776.00i 1.80671i −0.428889 0.903357i \(-0.641095\pi\)
0.428889 0.903357i \(-0.358905\pi\)
\(984\) 0 0
\(985\) −720.000 −0.730964
\(986\) 0 0
\(987\) −1280.00 + 1144.87i −1.29686 + 1.15995i
\(988\) 0 0
\(989\) 576.000i 0.582406i
\(990\) 0 0
\(991\) 1368.47 1.38090 0.690451 0.723379i \(-0.257412\pi\)
0.690451 + 0.723379i \(0.257412\pi\)
\(992\) 0 0
\(993\) 680.000 + 760.263i 0.684794 + 0.765622i
\(994\) 0 0
\(995\) 1037.54i 1.04275i
\(996\) 0 0
\(997\) −1234.31 −1.23802 −0.619012 0.785382i \(-0.712467\pi\)
−0.619012 + 0.785382i \(0.712467\pi\)
\(998\) 0 0
\(999\) 1180.64 840.000i 1.18183 0.840841i
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 768.3.e.h.257.2 4
3.2 odd 2 inner 768.3.e.h.257.3 4
4.3 odd 2 768.3.e.m.257.4 4
8.3 odd 2 inner 768.3.e.h.257.1 4
8.5 even 2 768.3.e.m.257.3 4
12.11 even 2 768.3.e.m.257.1 4
16.3 odd 4 384.3.h.f.65.2 yes 4
16.5 even 4 384.3.h.e.65.2 yes 4
16.11 odd 4 384.3.h.e.65.3 yes 4
16.13 even 4 384.3.h.f.65.3 yes 4
24.5 odd 2 768.3.e.m.257.2 4
24.11 even 2 inner 768.3.e.h.257.4 4
48.5 odd 4 384.3.h.f.65.4 yes 4
48.11 even 4 384.3.h.f.65.1 yes 4
48.29 odd 4 384.3.h.e.65.1 4
48.35 even 4 384.3.h.e.65.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.h.e.65.1 4 48.29 odd 4
384.3.h.e.65.2 yes 4 16.5 even 4
384.3.h.e.65.3 yes 4 16.11 odd 4
384.3.h.e.65.4 yes 4 48.35 even 4
384.3.h.f.65.1 yes 4 48.11 even 4
384.3.h.f.65.2 yes 4 16.3 odd 4
384.3.h.f.65.3 yes 4 16.13 even 4
384.3.h.f.65.4 yes 4 48.5 odd 4
768.3.e.h.257.1 4 8.3 odd 2 inner
768.3.e.h.257.2 4 1.1 even 1 trivial
768.3.e.h.257.3 4 3.2 odd 2 inner
768.3.e.h.257.4 4 24.11 even 2 inner
768.3.e.m.257.1 4 12.11 even 2
768.3.e.m.257.2 4 24.5 odd 2
768.3.e.m.257.3 4 8.5 even 2
768.3.e.m.257.4 4 4.3 odd 2