Properties

Label 1280.2.n.p.767.3
Level $1280$
Weight $2$
Character 1280.767
Analytic conductor $10.221$
Analytic rank $0$
Dimension $8$
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] = [1280,2,Mod(767,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.767");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.n (of order \(4\), degree \(2\), 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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 767.3
Root \(-0.228425 + 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 1280.767
Dual form 1280.2.n.p.1023.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.456850 + 0.456850i) q^{3} +(0.456850 + 2.18890i) q^{5} +(2.79129 - 2.79129i) q^{7} -2.58258i q^{9} +O(q^{10})\) \(q+(0.456850 + 0.456850i) q^{3} +(0.456850 + 2.18890i) q^{5} +(2.79129 - 2.79129i) q^{7} -2.58258i q^{9} +4.37780i q^{11} +(-1.73205 + 1.73205i) q^{13} +(-0.791288 + 1.20871i) q^{15} +(3.00000 + 3.00000i) q^{17} +3.46410 q^{19} +2.55040 q^{21} +(0.791288 + 0.791288i) q^{23} +(-4.58258 + 2.00000i) q^{25} +(2.55040 - 2.55040i) q^{27} -5.29150i q^{29} +1.58258i q^{31} +(-2.00000 + 2.00000i) q^{33} +(7.38505 + 4.83465i) q^{35} +(-5.19615 - 5.19615i) q^{37} -1.58258 q^{39} +7.58258 q^{41} +(8.29875 + 8.29875i) q^{43} +(5.65300 - 1.17985i) q^{45} +(0.791288 - 0.791288i) q^{47} -8.58258i q^{49} +2.74110i q^{51} +(2.64575 - 2.64575i) q^{53} +(-9.58258 + 2.00000i) q^{55} +(1.58258 + 1.58258i) q^{57} +5.29150 q^{59} -9.66930 q^{61} +(-7.20871 - 7.20871i) q^{63} +(-4.58258 - 3.00000i) q^{65} +(-8.29875 + 8.29875i) q^{67} +0.723000i q^{69} +13.5826i q^{71} +(-0.582576 + 0.582576i) q^{73} +(-3.00725 - 1.17985i) q^{75} +(12.2197 + 12.2197i) q^{77} +12.0000 q^{79} -5.41742 q^{81} +(4.83465 + 4.83465i) q^{83} +(-5.19615 + 7.93725i) q^{85} +(2.41742 - 2.41742i) q^{87} +3.16515i q^{89} +9.66930i q^{91} +(-0.723000 + 0.723000i) q^{93} +(1.58258 + 7.58258i) q^{95} +(0.582576 + 0.582576i) q^{97} +11.3060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 12 q^{15} + 24 q^{17} - 12 q^{23} - 16 q^{33} + 24 q^{39} + 24 q^{41} - 12 q^{47} - 40 q^{55} - 24 q^{57} - 76 q^{63} + 32 q^{73} + 96 q^{79} - 80 q^{81} + 56 q^{87} - 24 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(3\) 0.456850 + 0.456850i 0.263763 + 0.263763i 0.826581 0.562818i \(-0.190283\pi\)
−0.562818 + 0.826581i \(0.690283\pi\)
\(4\) 0 0
\(5\) 0.456850 + 2.18890i 0.204310 + 0.978906i
\(6\) 0 0
\(7\) 2.79129 2.79129i 1.05501 1.05501i 0.0566113 0.998396i \(-0.481970\pi\)
0.998396 0.0566113i \(-0.0180296\pi\)
\(8\) 0 0
\(9\) 2.58258i 0.860859i
\(10\) 0 0
\(11\) 4.37780i 1.31996i 0.751284 + 0.659979i \(0.229435\pi\)
−0.751284 + 0.659979i \(0.770565\pi\)
\(12\) 0 0
\(13\) −1.73205 + 1.73205i −0.480384 + 0.480384i −0.905254 0.424870i \(-0.860320\pi\)
0.424870 + 0.905254i \(0.360320\pi\)
\(14\) 0 0
\(15\) −0.791288 + 1.20871i −0.204310 + 0.312088i
\(16\) 0 0
\(17\) 3.00000 + 3.00000i 0.727607 + 0.727607i 0.970143 0.242536i \(-0.0779791\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) 2.55040 0.556543
\(22\) 0 0
\(23\) 0.791288 + 0.791288i 0.164995 + 0.164995i 0.784775 0.619780i \(-0.212778\pi\)
−0.619780 + 0.784775i \(0.712778\pi\)
\(24\) 0 0
\(25\) −4.58258 + 2.00000i −0.916515 + 0.400000i
\(26\) 0 0
\(27\) 2.55040 2.55040i 0.490825 0.490825i
\(28\) 0 0
\(29\) 5.29150i 0.982607i −0.870988 0.491304i \(-0.836521\pi\)
0.870988 0.491304i \(-0.163479\pi\)
\(30\) 0 0
\(31\) 1.58258i 0.284239i 0.989850 + 0.142119i \(0.0453917\pi\)
−0.989850 + 0.142119i \(0.954608\pi\)
\(32\) 0 0
\(33\) −2.00000 + 2.00000i −0.348155 + 0.348155i
\(34\) 0 0
\(35\) 7.38505 + 4.83465i 1.24830 + 0.817205i
\(36\) 0 0
\(37\) −5.19615 5.19615i −0.854242 0.854242i 0.136410 0.990652i \(-0.456443\pi\)
−0.990652 + 0.136410i \(0.956443\pi\)
\(38\) 0 0
\(39\) −1.58258 −0.253415
\(40\) 0 0
\(41\) 7.58258 1.18420 0.592100 0.805865i \(-0.298299\pi\)
0.592100 + 0.805865i \(0.298299\pi\)
\(42\) 0 0
\(43\) 8.29875 + 8.29875i 1.26555 + 1.26555i 0.948364 + 0.317184i \(0.102737\pi\)
0.317184 + 0.948364i \(0.397263\pi\)
\(44\) 0 0
\(45\) 5.65300 1.17985i 0.842700 0.175882i
\(46\) 0 0
\(47\) 0.791288 0.791288i 0.115421 0.115421i −0.647037 0.762458i \(-0.723992\pi\)
0.762458 + 0.647037i \(0.223992\pi\)
\(48\) 0 0
\(49\) 8.58258i 1.22608i
\(50\) 0 0
\(51\) 2.74110i 0.383831i
\(52\) 0 0
\(53\) 2.64575 2.64575i 0.363422 0.363422i −0.501649 0.865071i \(-0.667273\pi\)
0.865071 + 0.501649i \(0.167273\pi\)
\(54\) 0 0
\(55\) −9.58258 + 2.00000i −1.29211 + 0.269680i
\(56\) 0 0
\(57\) 1.58258 + 1.58258i 0.209617 + 0.209617i
\(58\) 0 0
\(59\) 5.29150 0.688895 0.344447 0.938806i \(-0.388066\pi\)
0.344447 + 0.938806i \(0.388066\pi\)
\(60\) 0 0
\(61\) −9.66930 −1.23803 −0.619014 0.785380i \(-0.712468\pi\)
−0.619014 + 0.785380i \(0.712468\pi\)
\(62\) 0 0
\(63\) −7.20871 7.20871i −0.908212 0.908212i
\(64\) 0 0
\(65\) −4.58258 3.00000i −0.568399 0.372104i
\(66\) 0 0
\(67\) −8.29875 + 8.29875i −1.01385 + 1.01385i −0.0139515 + 0.999903i \(0.504441\pi\)
−0.999903 + 0.0139515i \(0.995559\pi\)
\(68\) 0 0
\(69\) 0.723000i 0.0870390i
\(70\) 0 0
\(71\) 13.5826i 1.61196i 0.591946 + 0.805978i \(0.298360\pi\)
−0.591946 + 0.805978i \(0.701640\pi\)
\(72\) 0 0
\(73\) −0.582576 + 0.582576i −0.0681853 + 0.0681853i −0.740377 0.672192i \(-0.765353\pi\)
0.672192 + 0.740377i \(0.265353\pi\)
\(74\) 0 0
\(75\) −3.00725 1.17985i −0.347247 0.136237i
\(76\) 0 0
\(77\) 12.2197 + 12.2197i 1.39256 + 1.39256i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −5.41742 −0.601936
\(82\) 0 0
\(83\) 4.83465 + 4.83465i 0.530672 + 0.530672i 0.920772 0.390100i \(-0.127560\pi\)
−0.390100 + 0.920772i \(0.627560\pi\)
\(84\) 0 0
\(85\) −5.19615 + 7.93725i −0.563602 + 0.860916i
\(86\) 0 0
\(87\) 2.41742 2.41742i 0.259175 0.259175i
\(88\) 0 0
\(89\) 3.16515i 0.335505i 0.985829 + 0.167753i \(0.0536510\pi\)
−0.985829 + 0.167753i \(0.946349\pi\)
\(90\) 0 0
\(91\) 9.66930i 1.01362i
\(92\) 0 0
\(93\) −0.723000 + 0.723000i −0.0749716 + 0.0749716i
\(94\) 0 0
\(95\) 1.58258 + 7.58258i 0.162369 + 0.777956i
\(96\) 0 0
\(97\) 0.582576 + 0.582576i 0.0591516 + 0.0591516i 0.736064 0.676912i \(-0.236682\pi\)
−0.676912 + 0.736064i \(0.736682\pi\)
\(98\) 0 0
\(99\) 11.3060 1.13630
\(100\) 0 0
\(101\) −3.65480 −0.363666 −0.181833 0.983329i \(-0.558203\pi\)
−0.181833 + 0.983329i \(0.558203\pi\)
\(102\) 0 0
\(103\) −10.3739 10.3739i −1.02217 1.02217i −0.999749 0.0224185i \(-0.992863\pi\)
−0.0224185 0.999749i \(-0.507137\pi\)
\(104\) 0 0
\(105\) 1.16515 + 5.58258i 0.113707 + 0.544804i
\(106\) 0 0
\(107\) 8.29875 8.29875i 0.802271 0.802271i −0.181179 0.983450i \(-0.557991\pi\)
0.983450 + 0.181179i \(0.0579914\pi\)
\(108\) 0 0
\(109\) 6.20520i 0.594351i 0.954823 + 0.297175i \(0.0960446\pi\)
−0.954823 + 0.297175i \(0.903955\pi\)
\(110\) 0 0
\(111\) 4.74773i 0.450634i
\(112\) 0 0
\(113\) −4.58258 + 4.58258i −0.431092 + 0.431092i −0.889000 0.457907i \(-0.848599\pi\)
0.457907 + 0.889000i \(0.348599\pi\)
\(114\) 0 0
\(115\) −1.37055 + 2.09355i −0.127805 + 0.195225i
\(116\) 0 0
\(117\) 4.47315 + 4.47315i 0.413543 + 0.413543i
\(118\) 0 0
\(119\) 16.7477 1.53526
\(120\) 0 0
\(121\) −8.16515 −0.742286
\(122\) 0 0
\(123\) 3.46410 + 3.46410i 0.312348 + 0.312348i
\(124\) 0 0
\(125\) −6.47135 9.11710i −0.578815 0.815459i
\(126\) 0 0
\(127\) 1.20871 1.20871i 0.107256 0.107256i −0.651442 0.758698i \(-0.725836\pi\)
0.758698 + 0.651442i \(0.225836\pi\)
\(128\) 0 0
\(129\) 7.58258i 0.667609i
\(130\) 0 0
\(131\) 11.3060i 0.987810i −0.869516 0.493905i \(-0.835569\pi\)
0.869516 0.493905i \(-0.164431\pi\)
\(132\) 0 0
\(133\) 9.66930 9.66930i 0.838435 0.838435i
\(134\) 0 0
\(135\) 6.74773 + 4.41742i 0.580752 + 0.380191i
\(136\) 0 0
\(137\) −12.1652 12.1652i −1.03934 1.03934i −0.999194 0.0401452i \(-0.987218\pi\)
−0.0401452 0.999194i \(-0.512782\pi\)
\(138\) 0 0
\(139\) −8.94630 −0.758816 −0.379408 0.925230i \(-0.623872\pi\)
−0.379408 + 0.925230i \(0.623872\pi\)
\(140\) 0 0
\(141\) 0.723000 0.0608876
\(142\) 0 0
\(143\) −7.58258 7.58258i −0.634087 0.634087i
\(144\) 0 0
\(145\) 11.5826 2.41742i 0.961881 0.200756i
\(146\) 0 0
\(147\) 3.92095 3.92095i 0.323395 0.323395i
\(148\) 0 0
\(149\) 4.37780i 0.358644i 0.983790 + 0.179322i \(0.0573903\pi\)
−0.983790 + 0.179322i \(0.942610\pi\)
\(150\) 0 0
\(151\) 12.7477i 1.03740i −0.854958 0.518698i \(-0.826417\pi\)
0.854958 0.518698i \(-0.173583\pi\)
\(152\) 0 0
\(153\) 7.74773 7.74773i 0.626367 0.626367i
\(154\) 0 0
\(155\) −3.46410 + 0.723000i −0.278243 + 0.0580728i
\(156\) 0 0
\(157\) −11.4014 11.4014i −0.909927 0.909927i 0.0863386 0.996266i \(-0.472483\pi\)
−0.996266 + 0.0863386i \(0.972483\pi\)
\(158\) 0 0
\(159\) 2.41742 0.191714
\(160\) 0 0
\(161\) 4.41742 0.348142
\(162\) 0 0
\(163\) −8.29875 8.29875i −0.650009 0.650009i 0.302986 0.952995i \(-0.402016\pi\)
−0.952995 + 0.302986i \(0.902016\pi\)
\(164\) 0 0
\(165\) −5.29150 3.46410i −0.411943 0.269680i
\(166\) 0 0
\(167\) 11.2087 11.2087i 0.867356 0.867356i −0.124823 0.992179i \(-0.539836\pi\)
0.992179 + 0.124823i \(0.0398363\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 0 0
\(171\) 8.94630i 0.684141i
\(172\) 0 0
\(173\) 8.66025 8.66025i 0.658427 0.658427i −0.296581 0.955008i \(-0.595846\pi\)
0.955008 + 0.296581i \(0.0958464\pi\)
\(174\) 0 0
\(175\) −7.20871 + 18.3739i −0.544927 + 1.38893i
\(176\) 0 0
\(177\) 2.41742 + 2.41742i 0.181705 + 0.181705i
\(178\) 0 0
\(179\) −8.56490 −0.640171 −0.320085 0.947389i \(-0.603712\pi\)
−0.320085 + 0.947389i \(0.603712\pi\)
\(180\) 0 0
\(181\) 6.92820 0.514969 0.257485 0.966282i \(-0.417106\pi\)
0.257485 + 0.966282i \(0.417106\pi\)
\(182\) 0 0
\(183\) −4.41742 4.41742i −0.326545 0.326545i
\(184\) 0 0
\(185\) 9.00000 13.7477i 0.661693 1.01075i
\(186\) 0 0
\(187\) −13.1334 + 13.1334i −0.960410 + 0.960410i
\(188\) 0 0
\(189\) 14.2378i 1.03565i
\(190\) 0 0
\(191\) 13.5826i 0.982801i 0.870934 + 0.491400i \(0.163515\pi\)
−0.870934 + 0.491400i \(0.836485\pi\)
\(192\) 0 0
\(193\) 14.1652 14.1652i 1.01963 1.01963i 0.0198265 0.999803i \(-0.493689\pi\)
0.999803 0.0198265i \(-0.00631138\pi\)
\(194\) 0 0
\(195\) −0.723000 3.46410i −0.0517751 0.248069i
\(196\) 0 0
\(197\) −2.64575 2.64575i −0.188502 0.188502i 0.606546 0.795048i \(-0.292554\pi\)
−0.795048 + 0.606546i \(0.792554\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −7.58258 −0.534834
\(202\) 0 0
\(203\) −14.7701 14.7701i −1.03666 1.03666i
\(204\) 0 0
\(205\) 3.46410 + 16.5975i 0.241943 + 1.15922i
\(206\) 0 0
\(207\) 2.04356 2.04356i 0.142037 0.142037i
\(208\) 0 0
\(209\) 15.1652i 1.04900i
\(210\) 0 0
\(211\) 7.65120i 0.526731i 0.964696 + 0.263365i \(0.0848324\pi\)
−0.964696 + 0.263365i \(0.915168\pi\)
\(212\) 0 0
\(213\) −6.20520 + 6.20520i −0.425174 + 0.425174i
\(214\) 0 0
\(215\) −14.3739 + 21.9564i −0.980289 + 1.49742i
\(216\) 0 0
\(217\) 4.41742 + 4.41742i 0.299874 + 0.299874i
\(218\) 0 0
\(219\) −0.532300 −0.0359695
\(220\) 0 0
\(221\) −10.3923 −0.699062
\(222\) 0 0
\(223\) −12.3739 12.3739i −0.828615 0.828615i 0.158710 0.987325i \(-0.449266\pi\)
−0.987325 + 0.158710i \(0.949266\pi\)
\(224\) 0 0
\(225\) 5.16515 + 11.8348i 0.344343 + 0.788990i
\(226\) 0 0
\(227\) 6.66205 6.66205i 0.442176 0.442176i −0.450567 0.892743i \(-0.648778\pi\)
0.892743 + 0.450567i \(0.148778\pi\)
\(228\) 0 0
\(229\) 3.46410i 0.228914i −0.993428 0.114457i \(-0.963487\pi\)
0.993428 0.114457i \(-0.0365129\pi\)
\(230\) 0 0
\(231\) 11.1652i 0.734613i
\(232\) 0 0
\(233\) −6.16515 + 6.16515i −0.403892 + 0.403892i −0.879602 0.475710i \(-0.842191\pi\)
0.475710 + 0.879602i \(0.342191\pi\)
\(234\) 0 0
\(235\) 2.09355 + 1.37055i 0.136568 + 0.0894049i
\(236\) 0 0
\(237\) 5.48220 + 5.48220i 0.356107 + 0.356107i
\(238\) 0 0
\(239\) 3.16515 0.204737 0.102368 0.994747i \(-0.467358\pi\)
0.102368 + 0.994747i \(0.467358\pi\)
\(240\) 0 0
\(241\) −10.7477 −0.692322 −0.346161 0.938175i \(-0.612515\pi\)
−0.346161 + 0.938175i \(0.612515\pi\)
\(242\) 0 0
\(243\) −10.1262 10.1262i −0.649593 0.649593i
\(244\) 0 0
\(245\) 18.7864 3.92095i 1.20022 0.250500i
\(246\) 0 0
\(247\) −6.00000 + 6.00000i −0.381771 + 0.381771i
\(248\) 0 0
\(249\) 4.41742i 0.279943i
\(250\) 0 0
\(251\) 18.2342i 1.15093i −0.817825 0.575467i \(-0.804821\pi\)
0.817825 0.575467i \(-0.195179\pi\)
\(252\) 0 0
\(253\) −3.46410 + 3.46410i −0.217786 + 0.217786i
\(254\) 0 0
\(255\) −6.00000 + 1.25227i −0.375735 + 0.0784204i
\(256\) 0 0
\(257\) −19.7477 19.7477i −1.23183 1.23183i −0.963260 0.268569i \(-0.913449\pi\)
−0.268569 0.963260i \(-0.586551\pi\)
\(258\) 0 0
\(259\) −29.0079 −1.80246
\(260\) 0 0
\(261\) −13.6657 −0.845886
\(262\) 0 0
\(263\) −18.7913 18.7913i −1.15872 1.15872i −0.984751 0.173969i \(-0.944341\pi\)
−0.173969 0.984751i \(-0.555659\pi\)
\(264\) 0 0
\(265\) 7.00000 + 4.58258i 0.430007 + 0.281505i
\(266\) 0 0
\(267\) −1.44600 + 1.44600i −0.0884938 + 0.0884938i
\(268\) 0 0
\(269\) 23.7164i 1.44602i 0.690840 + 0.723008i \(0.257241\pi\)
−0.690840 + 0.723008i \(0.742759\pi\)
\(270\) 0 0
\(271\) 8.74773i 0.531387i 0.964058 + 0.265693i \(0.0856008\pi\)
−0.964058 + 0.265693i \(0.914399\pi\)
\(272\) 0 0
\(273\) −4.41742 + 4.41742i −0.267355 + 0.267355i
\(274\) 0 0
\(275\) −8.75560 20.0616i −0.527983 1.20976i
\(276\) 0 0
\(277\) 5.91915 + 5.91915i 0.355647 + 0.355647i 0.862206 0.506558i \(-0.169083\pi\)
−0.506558 + 0.862206i \(0.669083\pi\)
\(278\) 0 0
\(279\) 4.08712 0.244690
\(280\) 0 0
\(281\) 10.7477 0.641156 0.320578 0.947222i \(-0.396123\pi\)
0.320578 + 0.947222i \(0.396123\pi\)
\(282\) 0 0
\(283\) −17.2451 17.2451i −1.02511 1.02511i −0.999676 0.0254359i \(-0.991903\pi\)
−0.0254359 0.999676i \(-0.508097\pi\)
\(284\) 0 0
\(285\) −2.74110 + 4.18710i −0.162369 + 0.248023i
\(286\) 0 0
\(287\) 21.1652 21.1652i 1.24934 1.24934i
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 0 0
\(291\) 0.532300i 0.0312040i
\(292\) 0 0
\(293\) −23.4304 + 23.4304i −1.36882 + 1.36882i −0.506685 + 0.862131i \(0.669129\pi\)
−0.862131 + 0.506685i \(0.830871\pi\)
\(294\) 0 0
\(295\) 2.41742 + 11.5826i 0.140748 + 0.674364i
\(296\) 0 0
\(297\) 11.1652 + 11.1652i 0.647868 + 0.647868i
\(298\) 0 0
\(299\) −2.74110 −0.158522
\(300\) 0 0
\(301\) 46.3284 2.67033
\(302\) 0 0
\(303\) −1.66970 1.66970i −0.0959216 0.0959216i
\(304\) 0 0
\(305\) −4.41742 21.1652i −0.252941 1.21191i
\(306\) 0 0
\(307\) −1.37055 + 1.37055i −0.0782215 + 0.0782215i −0.745135 0.666914i \(-0.767615\pi\)
0.666914 + 0.745135i \(0.267615\pi\)
\(308\) 0 0
\(309\) 9.47860i 0.539219i
\(310\) 0 0
\(311\) 10.4174i 0.590718i 0.955386 + 0.295359i \(0.0954392\pi\)
−0.955386 + 0.295359i \(0.904561\pi\)
\(312\) 0 0
\(313\) −2.58258 + 2.58258i −0.145976 + 0.145976i −0.776318 0.630342i \(-0.782915\pi\)
0.630342 + 0.776318i \(0.282915\pi\)
\(314\) 0 0
\(315\) 12.4859 19.0725i 0.703498 1.07461i
\(316\) 0 0
\(317\) −17.6066 17.6066i −0.988883 0.988883i 0.0110560 0.999939i \(-0.496481\pi\)
−0.999939 + 0.0110560i \(0.996481\pi\)
\(318\) 0 0
\(319\) 23.1652 1.29700
\(320\) 0 0
\(321\) 7.58258 0.423218
\(322\) 0 0
\(323\) 10.3923 + 10.3923i 0.578243 + 0.578243i
\(324\) 0 0
\(325\) 4.47315 11.4014i 0.248126 0.632433i
\(326\) 0 0
\(327\) −2.83485 + 2.83485i −0.156767 + 0.156767i
\(328\) 0 0
\(329\) 4.41742i 0.243540i
\(330\) 0 0
\(331\) 6.20520i 0.341069i −0.985352 0.170534i \(-0.945451\pi\)
0.985352 0.170534i \(-0.0545494\pi\)
\(332\) 0 0
\(333\) −13.4195 + 13.4195i −0.735382 + 0.735382i
\(334\) 0 0
\(335\) −21.9564 14.3739i −1.19961 0.785328i
\(336\) 0 0
\(337\) 1.00000 + 1.00000i 0.0544735 + 0.0544735i 0.733819 0.679345i \(-0.237736\pi\)
−0.679345 + 0.733819i \(0.737736\pi\)
\(338\) 0 0
\(339\) −4.18710 −0.227412
\(340\) 0 0
\(341\) −6.92820 −0.375183
\(342\) 0 0
\(343\) −4.41742 4.41742i −0.238518 0.238518i
\(344\) 0 0
\(345\) −1.58258 + 0.330303i −0.0852030 + 0.0177829i
\(346\) 0 0
\(347\) 3.92095 3.92095i 0.210488 0.210488i −0.593987 0.804475i \(-0.702447\pi\)
0.804475 + 0.593987i \(0.202447\pi\)
\(348\) 0 0
\(349\) 10.3923i 0.556287i −0.960539 0.278144i \(-0.910281\pi\)
0.960539 0.278144i \(-0.0897191\pi\)
\(350\) 0 0
\(351\) 8.83485i 0.471569i
\(352\) 0 0
\(353\) 5.83485 5.83485i 0.310558 0.310558i −0.534568 0.845126i \(-0.679526\pi\)
0.845126 + 0.534568i \(0.179526\pi\)
\(354\) 0 0
\(355\) −29.7309 + 6.20520i −1.57795 + 0.329338i
\(356\) 0 0
\(357\) 7.65120 + 7.65120i 0.404945 + 0.404945i
\(358\) 0 0
\(359\) 8.83485 0.466285 0.233143 0.972443i \(-0.425099\pi\)
0.233143 + 0.972443i \(0.425099\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) −3.73025 3.73025i −0.195787 0.195787i
\(364\) 0 0
\(365\) −1.54135 1.00905i −0.0806780 0.0528161i
\(366\) 0 0
\(367\) 15.5390 15.5390i 0.811130 0.811130i −0.173673 0.984803i \(-0.555564\pi\)
0.984803 + 0.173673i \(0.0555637\pi\)
\(368\) 0 0
\(369\) 19.5826i 1.01943i
\(370\) 0 0
\(371\) 14.7701i 0.766826i
\(372\) 0 0
\(373\) 5.19615 5.19615i 0.269047 0.269047i −0.559669 0.828716i \(-0.689072\pi\)
0.828716 + 0.559669i \(0.189072\pi\)
\(374\) 0 0
\(375\) 1.20871 7.12159i 0.0624176 0.367757i
\(376\) 0 0
\(377\) 9.16515 + 9.16515i 0.472029 + 0.472029i
\(378\) 0 0
\(379\) −10.3923 −0.533817 −0.266908 0.963722i \(-0.586002\pi\)
−0.266908 + 0.963722i \(0.586002\pi\)
\(380\) 0 0
\(381\) 1.10440 0.0565802
\(382\) 0 0
\(383\) 3.62614 + 3.62614i 0.185287 + 0.185287i 0.793655 0.608368i \(-0.208176\pi\)
−0.608368 + 0.793655i \(0.708176\pi\)
\(384\) 0 0
\(385\) −21.1652 + 32.3303i −1.07868 + 1.64770i
\(386\) 0 0
\(387\) 21.4322 21.4322i 1.08946 1.08946i
\(388\) 0 0
\(389\) 30.2632i 1.53441i −0.641404 0.767203i \(-0.721648\pi\)
0.641404 0.767203i \(-0.278352\pi\)
\(390\) 0 0
\(391\) 4.74773i 0.240103i
\(392\) 0 0
\(393\) 5.16515 5.16515i 0.260547 0.260547i
\(394\) 0 0
\(395\) 5.48220 + 26.2668i 0.275840 + 1.32163i
\(396\) 0 0
\(397\) −7.93725 7.93725i −0.398359 0.398359i 0.479295 0.877654i \(-0.340893\pi\)
−0.877654 + 0.479295i \(0.840893\pi\)
\(398\) 0 0
\(399\) 8.83485 0.442296
\(400\) 0 0
\(401\) 12.3303 0.615746 0.307873 0.951427i \(-0.400383\pi\)
0.307873 + 0.951427i \(0.400383\pi\)
\(402\) 0 0
\(403\) −2.74110 2.74110i −0.136544 0.136544i
\(404\) 0 0
\(405\) −2.47495 11.8582i −0.122981 0.589239i
\(406\) 0 0
\(407\) 22.7477 22.7477i 1.12756 1.12756i
\(408\) 0 0
\(409\) 25.5826i 1.26498i −0.774570 0.632488i \(-0.782034\pi\)
0.774570 0.632488i \(-0.217966\pi\)
\(410\) 0 0
\(411\) 11.1153i 0.548278i
\(412\) 0 0
\(413\) 14.7701 14.7701i 0.726789 0.726789i
\(414\) 0 0
\(415\) −8.37386 + 12.7913i −0.411057 + 0.627900i
\(416\) 0 0
\(417\) −4.08712 4.08712i −0.200147 0.200147i
\(418\) 0 0
\(419\) 0.190700 0.00931632 0.00465816 0.999989i \(-0.498517\pi\)
0.00465816 + 0.999989i \(0.498517\pi\)
\(420\) 0 0
\(421\) 2.74110 0.133593 0.0667966 0.997767i \(-0.478722\pi\)
0.0667966 + 0.997767i \(0.478722\pi\)
\(422\) 0 0
\(423\) −2.04356 2.04356i −0.0993613 0.0993613i
\(424\) 0 0
\(425\) −19.7477 7.74773i −0.957905 0.375820i
\(426\) 0 0
\(427\) −26.9898 + 26.9898i −1.30613 + 1.30613i
\(428\) 0 0
\(429\) 6.92820i 0.334497i
\(430\) 0 0
\(431\) 16.7477i 0.806710i 0.915044 + 0.403355i \(0.132156\pi\)
−0.915044 + 0.403355i \(0.867844\pi\)
\(432\) 0 0
\(433\) −14.5826 + 14.5826i −0.700794 + 0.700794i −0.964581 0.263787i \(-0.915028\pi\)
0.263787 + 0.964581i \(0.415028\pi\)
\(434\) 0 0
\(435\) 6.39590 + 4.18710i 0.306660 + 0.200756i
\(436\) 0 0
\(437\) 2.74110 + 2.74110i 0.131125 + 0.131125i
\(438\) 0 0
\(439\) 29.4955 1.40774 0.703871 0.710328i \(-0.251453\pi\)
0.703871 + 0.710328i \(0.251453\pi\)
\(440\) 0 0
\(441\) −22.1652 −1.05548
\(442\) 0 0
\(443\) −1.17985 1.17985i −0.0560564 0.0560564i 0.678523 0.734579i \(-0.262620\pi\)
−0.734579 + 0.678523i \(0.762620\pi\)
\(444\) 0 0
\(445\) −6.92820 + 1.44600i −0.328428 + 0.0685470i
\(446\) 0 0
\(447\) −2.00000 + 2.00000i −0.0945968 + 0.0945968i
\(448\) 0 0
\(449\) 10.4174i 0.491629i 0.969317 + 0.245814i \(0.0790553\pi\)
−0.969317 + 0.245814i \(0.920945\pi\)
\(450\) 0 0
\(451\) 33.1950i 1.56309i
\(452\) 0 0
\(453\) 5.82380 5.82380i 0.273626 0.273626i
\(454\) 0 0
\(455\) −21.1652 + 4.41742i −0.992238 + 0.207092i
\(456\) 0 0
\(457\) −3.41742 3.41742i −0.159860 0.159860i 0.622644 0.782505i \(-0.286058\pi\)
−0.782505 + 0.622644i \(0.786058\pi\)
\(458\) 0 0
\(459\) 15.3024 0.714255
\(460\) 0 0
\(461\) −17.5112 −0.815578 −0.407789 0.913076i \(-0.633700\pi\)
−0.407789 + 0.913076i \(0.633700\pi\)
\(462\) 0 0
\(463\) −16.7913 16.7913i −0.780357 0.780357i 0.199534 0.979891i \(-0.436057\pi\)
−0.979891 + 0.199534i \(0.936057\pi\)
\(464\) 0 0
\(465\) −1.91288 1.25227i −0.0887076 0.0580728i
\(466\) 0 0
\(467\) −7.57575 + 7.57575i −0.350564 + 0.350564i −0.860319 0.509755i \(-0.829736\pi\)
0.509755 + 0.860319i \(0.329736\pi\)
\(468\) 0 0
\(469\) 46.3284i 2.13925i
\(470\) 0 0
\(471\) 10.4174i 0.480010i
\(472\) 0 0
\(473\) −36.3303 + 36.3303i −1.67047 + 1.67047i
\(474\) 0 0
\(475\) −15.8745 + 6.92820i −0.728372 + 0.317888i
\(476\) 0 0
\(477\) −6.83285 6.83285i −0.312855 0.312855i
\(478\) 0 0
\(479\) −21.4955 −0.982152 −0.491076 0.871117i \(-0.663396\pi\)
−0.491076 + 0.871117i \(0.663396\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 2.01810 + 2.01810i 0.0918268 + 0.0918268i
\(484\) 0 0
\(485\) −1.00905 + 1.54135i −0.0458186 + 0.0699891i
\(486\) 0 0
\(487\) −0.373864 + 0.373864i −0.0169414 + 0.0169414i −0.715527 0.698585i \(-0.753813\pi\)
0.698585 + 0.715527i \(0.253813\pi\)
\(488\) 0 0
\(489\) 7.58258i 0.342896i
\(490\) 0 0
\(491\) 9.86001i 0.444976i −0.974935 0.222488i \(-0.928582\pi\)
0.974935 0.222488i \(-0.0714178\pi\)
\(492\) 0 0
\(493\) 15.8745 15.8745i 0.714952 0.714952i
\(494\) 0 0
\(495\) 5.16515 + 24.7477i 0.232156 + 1.11233i
\(496\) 0 0
\(497\) 37.9129 + 37.9129i 1.70063 + 1.70063i
\(498\) 0 0
\(499\) 24.2487 1.08552 0.542761 0.839887i \(-0.317379\pi\)
0.542761 + 0.839887i \(0.317379\pi\)
\(500\) 0 0
\(501\) 10.2414 0.457552
\(502\) 0 0
\(503\) −17.5390 17.5390i −0.782026 0.782026i 0.198146 0.980172i \(-0.436508\pi\)
−0.980172 + 0.198146i \(0.936508\pi\)
\(504\) 0 0
\(505\) −1.66970 8.00000i −0.0743006 0.355995i
\(506\) 0 0
\(507\) −3.19795 + 3.19795i −0.142026 + 0.142026i
\(508\) 0 0
\(509\) 38.4865i 1.70588i 0.522005 + 0.852942i \(0.325184\pi\)
−0.522005 + 0.852942i \(0.674816\pi\)
\(510\) 0 0
\(511\) 3.25227i 0.143872i
\(512\) 0 0
\(513\) 8.83485 8.83485i 0.390068 0.390068i
\(514\) 0 0
\(515\) 17.9681 27.4467i 0.791767 1.20944i
\(516\) 0 0
\(517\) 3.46410 + 3.46410i 0.152351 + 0.152351i
\(518\) 0 0
\(519\) 7.91288 0.347337
\(520\) 0 0
\(521\) 33.1652 1.45299 0.726496 0.687171i \(-0.241148\pi\)
0.726496 + 0.687171i \(0.241148\pi\)
\(522\) 0 0
\(523\) 15.9500 + 15.9500i 0.697443 + 0.697443i 0.963858 0.266415i \(-0.0858393\pi\)
−0.266415 + 0.963858i \(0.585839\pi\)
\(524\) 0 0
\(525\) −11.6874 + 5.10080i −0.510080 + 0.222617i
\(526\) 0 0
\(527\) −4.74773 + 4.74773i −0.206814 + 0.206814i
\(528\) 0 0
\(529\) 21.7477i 0.945553i
\(530\) 0 0
\(531\) 13.6657i 0.593041i
\(532\) 0 0
\(533\) −13.1334 + 13.1334i −0.568871 + 0.568871i
\(534\) 0 0
\(535\) 21.9564 + 14.3739i 0.949260 + 0.621436i
\(536\) 0 0
\(537\) −3.91288 3.91288i −0.168853 0.168853i
\(538\) 0 0
\(539\) 37.5728 1.61838
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 3.16515 + 3.16515i 0.135830 + 0.135830i
\(544\) 0 0
\(545\) −13.5826 + 2.83485i −0.581814 + 0.121432i
\(546\) 0 0
\(547\) −0.647551 + 0.647551i −0.0276873 + 0.0276873i −0.720815 0.693128i \(-0.756232\pi\)
0.693128 + 0.720815i \(0.256232\pi\)
\(548\) 0 0
\(549\) 24.9717i 1.06577i
\(550\) 0 0
\(551\) 18.3303i 0.780897i
\(552\) 0 0
\(553\) 33.4955 33.4955i 1.42437 1.42437i
\(554\) 0 0
\(555\) 10.3923 2.16900i 0.441129 0.0920689i
\(556\) 0 0
\(557\) 10.2970 + 10.2970i 0.436296 + 0.436296i 0.890763 0.454467i \(-0.150170\pi\)
−0.454467 + 0.890763i \(0.650170\pi\)
\(558\) 0 0
\(559\) −28.7477 −1.21590
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) 14.3133 + 14.3133i 0.603232 + 0.603232i 0.941169 0.337937i \(-0.109729\pi\)
−0.337937 + 0.941169i \(0.609729\pi\)
\(564\) 0 0
\(565\) −12.1244 7.93725i −0.510075 0.333923i
\(566\) 0 0
\(567\) −15.1216 + 15.1216i −0.635047 + 0.635047i
\(568\) 0 0
\(569\) 4.74773i 0.199035i −0.995036 0.0995175i \(-0.968270\pi\)
0.995036 0.0995175i \(-0.0317299\pi\)
\(570\) 0 0
\(571\) 39.4002i 1.64885i 0.565973 + 0.824424i \(0.308501\pi\)
−0.565973 + 0.824424i \(0.691499\pi\)
\(572\) 0 0
\(573\) −6.20520 + 6.20520i −0.259226 + 0.259226i
\(574\) 0 0
\(575\) −5.20871 2.04356i −0.217218 0.0852224i
\(576\) 0 0
\(577\) −11.7477 11.7477i −0.489064 0.489064i 0.418947 0.908011i \(-0.362399\pi\)
−0.908011 + 0.418947i \(0.862399\pi\)
\(578\) 0 0
\(579\) 12.9427 0.537881
\(580\) 0 0
\(581\) 26.9898 1.11973
\(582\) 0 0
\(583\) 11.5826 + 11.5826i 0.479701 + 0.479701i
\(584\) 0 0
\(585\) −7.74773 + 11.8348i −0.320329 + 0.489311i
\(586\) 0 0
\(587\) 28.3604 28.3604i 1.17056 1.17056i 0.188481 0.982077i \(-0.439644\pi\)
0.982077 0.188481i \(-0.0603562\pi\)
\(588\) 0 0
\(589\) 5.48220i 0.225890i
\(590\) 0 0
\(591\) 2.41742i 0.0994395i
\(592\) 0 0
\(593\) 0.165151 0.165151i 0.00678195 0.00678195i −0.703708 0.710490i \(-0.748474\pi\)
0.710490 + 0.703708i \(0.248474\pi\)
\(594\) 0 0
\(595\) 7.65120 + 36.6591i 0.313669 + 1.50288i
\(596\) 0 0
\(597\) 1.82740 + 1.82740i 0.0747905 + 0.0747905i
\(598\) 0 0
\(599\) −42.3303 −1.72957 −0.864785 0.502143i \(-0.832545\pi\)
−0.864785 + 0.502143i \(0.832545\pi\)
\(600\) 0 0
\(601\) 2.74773 0.112082 0.0560411 0.998428i \(-0.482152\pi\)
0.0560411 + 0.998428i \(0.482152\pi\)
\(602\) 0 0
\(603\) 21.4322 + 21.4322i 0.872785 + 0.872785i
\(604\) 0 0
\(605\) −3.73025 17.8727i −0.151656 0.726629i
\(606\) 0 0
\(607\) −6.37386 + 6.37386i −0.258707 + 0.258707i −0.824528 0.565821i \(-0.808559\pi\)
0.565821 + 0.824528i \(0.308559\pi\)
\(608\) 0 0
\(609\) 13.4955i 0.546863i
\(610\) 0 0
\(611\) 2.74110i 0.110893i
\(612\) 0 0
\(613\) 8.66025 8.66025i 0.349784 0.349784i −0.510245 0.860029i \(-0.670445\pi\)
0.860029 + 0.510245i \(0.170445\pi\)
\(614\) 0 0
\(615\) −6.00000 + 9.16515i −0.241943 + 0.369575i
\(616\) 0 0
\(617\) −3.00000 3.00000i −0.120775 0.120775i 0.644136 0.764911i \(-0.277217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) −29.7309 −1.19499 −0.597493 0.801874i \(-0.703836\pi\)
−0.597493 + 0.801874i \(0.703836\pi\)
\(620\) 0 0
\(621\) 4.03620 0.161967
\(622\) 0 0
\(623\) 8.83485 + 8.83485i 0.353961 + 0.353961i
\(624\) 0 0
\(625\) 17.0000 18.3303i 0.680000 0.733212i
\(626\) 0 0
\(627\) −6.92820 + 6.92820i −0.276686 + 0.276686i
\(628\) 0 0
\(629\) 31.1769i 1.24310i
\(630\) 0 0
\(631\) 47.0780i 1.87415i 0.349132 + 0.937073i \(0.386476\pi\)
−0.349132 + 0.937073i \(0.613524\pi\)
\(632\) 0 0
\(633\) −3.49545 + 3.49545i −0.138932 + 0.138932i
\(634\) 0 0
\(635\) 3.19795 + 2.09355i 0.126907 + 0.0830800i
\(636\) 0 0
\(637\) 14.8655 + 14.8655i 0.588991 + 0.588991i
\(638\) 0 0
\(639\) 35.0780 1.38767
\(640\) 0 0
\(641\) 25.9129 1.02350 0.511749 0.859135i \(-0.328998\pi\)
0.511749 + 0.859135i \(0.328998\pi\)
\(642\) 0 0
\(643\) 18.6911 + 18.6911i 0.737103 + 0.737103i 0.972016 0.234913i \(-0.0754805\pi\)
−0.234913 + 0.972016i \(0.575481\pi\)
\(644\) 0 0
\(645\) −16.5975 + 3.46410i −0.653526 + 0.136399i
\(646\) 0 0
\(647\) 0.460985 0.460985i 0.0181232 0.0181232i −0.697987 0.716110i \(-0.745921\pi\)
0.716110 + 0.697987i \(0.245921\pi\)
\(648\) 0 0
\(649\) 23.1652i 0.909312i
\(650\) 0 0
\(651\) 4.03620i 0.158191i
\(652\) 0 0
\(653\) −11.4014 + 11.4014i −0.446170 + 0.446170i −0.894079 0.447909i \(-0.852169\pi\)
0.447909 + 0.894079i \(0.352169\pi\)
\(654\) 0 0
\(655\) 24.7477 5.16515i 0.966974 0.201819i
\(656\) 0 0
\(657\) 1.50455 + 1.50455i 0.0586979 + 0.0586979i
\(658\) 0 0
\(659\) 3.84550 0.149800 0.0748998 0.997191i \(-0.476136\pi\)
0.0748998 + 0.997191i \(0.476136\pi\)
\(660\) 0 0
\(661\) −41.4183 −1.61099 −0.805493 0.592605i \(-0.798099\pi\)
−0.805493 + 0.592605i \(0.798099\pi\)
\(662\) 0 0
\(663\) −4.74773 4.74773i −0.184386 0.184386i
\(664\) 0 0
\(665\) 25.5826 + 16.7477i 0.992050 + 0.649449i
\(666\) 0 0
\(667\) 4.18710 4.18710i 0.162125 0.162125i
\(668\) 0 0
\(669\) 11.3060i 0.437115i
\(670\) 0 0
\(671\) 42.3303i 1.63414i
\(672\) 0 0
\(673\) −18.5826 + 18.5826i −0.716306 + 0.716306i −0.967847 0.251541i \(-0.919063\pi\)
0.251541 + 0.967847i \(0.419063\pi\)
\(674\) 0 0
\(675\) −6.58660 + 16.7882i −0.253519 + 0.646178i
\(676\) 0 0
\(677\) −18.3296 18.3296i −0.704462 0.704462i 0.260903 0.965365i \(-0.415980\pi\)
−0.965365 + 0.260903i \(0.915980\pi\)
\(678\) 0 0
\(679\) 3.25227 0.124811
\(680\) 0 0
\(681\) 6.08712 0.233259
\(682\) 0 0
\(683\) 6.85275 + 6.85275i 0.262213 + 0.262213i 0.825953 0.563739i \(-0.190638\pi\)
−0.563739 + 0.825953i \(0.690638\pi\)
\(684\) 0 0
\(685\) 21.0707 32.1860i 0.805069 1.22976i
\(686\) 0 0
\(687\) 1.58258 1.58258i 0.0603790 0.0603790i
\(688\) 0 0
\(689\) 9.16515i 0.349164i
\(690\) 0 0
\(691\) 32.4720i 1.23529i 0.786456 + 0.617647i \(0.211914\pi\)
−0.786456 + 0.617647i \(0.788086\pi\)
\(692\) 0 0
\(693\) 31.5583 31.5583i 1.19880 1.19880i
\(694\) 0 0
\(695\) −4.08712 19.5826i −0.155033 0.742809i
\(696\) 0 0
\(697\) 22.7477 + 22.7477i 0.861632 + 0.861632i
\(698\) 0 0
\(699\) −5.63310 −0.213063
\(700\) 0 0
\(701\) 19.8709 0.750514 0.375257 0.926921i \(-0.377554\pi\)
0.375257 + 0.926921i \(0.377554\pi\)
\(702\) 0 0
\(703\) −18.0000 18.0000i −0.678883 0.678883i
\(704\) 0 0
\(705\) 0.330303 + 1.58258i 0.0124399 + 0.0596032i
\(706\) 0 0
\(707\) −10.2016 + 10.2016i −0.383671 + 0.383671i
\(708\) 0 0
\(709\) 4.91010i 0.184403i −0.995740 0.0922014i \(-0.970610\pi\)
0.995740 0.0922014i \(-0.0293904\pi\)
\(710\) 0 0
\(711\) 30.9909i 1.16225i
\(712\) 0 0
\(713\) −1.25227 + 1.25227i −0.0468980 + 0.0468980i
\(714\) 0 0
\(715\) 13.1334 20.0616i 0.491162 0.750262i
\(716\) 0 0
\(717\) 1.44600 + 1.44600i 0.0540019 + 0.0540019i
\(718\) 0 0
\(719\) 39.8258 1.48525 0.742625 0.669707i \(-0.233580\pi\)
0.742625 + 0.669707i \(0.233580\pi\)
\(720\) 0 0
\(721\) −57.9129 −2.15679
\(722\) 0 0
\(723\) −4.91010 4.91010i −0.182609 0.182609i
\(724\) 0 0
\(725\) 10.5830 + 24.2487i 0.393043 + 0.900575i
\(726\) 0 0
\(727\) 5.95644 5.95644i 0.220912 0.220912i −0.587970 0.808882i \(-0.700073\pi\)
0.808882 + 0.587970i \(0.200073\pi\)
\(728\) 0 0
\(729\) 7.00000i 0.259259i
\(730\) 0 0
\(731\) 49.7925i 1.84164i
\(732\) 0 0
\(733\) −17.6066 + 17.6066i −0.650313 + 0.650313i −0.953068 0.302755i \(-0.902094\pi\)
0.302755 + 0.953068i \(0.402094\pi\)
\(734\) 0 0
\(735\) 10.3739 + 6.79129i 0.382646 + 0.250500i
\(736\) 0 0
\(737\) −36.3303 36.3303i −1.33824 1.33824i
\(738\) 0 0
\(739\) −2.01810 −0.0742371 −0.0371185 0.999311i \(-0.511818\pi\)
−0.0371185 + 0.999311i \(0.511818\pi\)
\(740\) 0 0
\(741\) −5.48220 −0.201394
\(742\) 0 0
\(743\) 17.2087 + 17.2087i 0.631326 + 0.631326i 0.948401 0.317074i \(-0.102700\pi\)
−0.317074 + 0.948401i \(0.602700\pi\)
\(744\) 0 0
\(745\) −9.58258 + 2.00000i −0.351078 + 0.0732743i
\(746\) 0 0
\(747\) 12.4859 12.4859i 0.456834 0.456834i
\(748\) 0 0
\(749\) 46.3284i 1.69280i
\(750\) 0 0
\(751\) 23.0780i 0.842129i −0.907031 0.421065i \(-0.861657\pi\)
0.907031 0.421065i \(-0.138343\pi\)
\(752\) 0 0
\(753\) 8.33030 8.33030i 0.303573 0.303573i
\(754\) 0 0
\(755\) 27.9035 5.82380i 1.01551 0.211950i
\(756\) 0 0
\(757\) −20.3477 20.3477i −0.739548 0.739548i 0.232942 0.972491i \(-0.425165\pi\)
−0.972491 + 0.232942i \(0.925165\pi\)
\(758\) 0 0
\(759\) −3.16515 −0.114888
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 17.3205 + 17.3205i 0.627044 + 0.627044i
\(764\) 0 0
\(765\) 20.4986 + 13.4195i 0.741127 + 0.485181i
\(766\) 0 0
\(767\) −9.16515 + 9.16515i −0.330934 + 0.330934i
\(768\) 0 0
\(769\) 37.4955i 1.35212i 0.736846 + 0.676060i \(0.236314\pi\)
−0.736846 + 0.676060i \(0.763686\pi\)
\(770\) 0 0
\(771\) 18.0435i 0.649821i
\(772\) 0 0
\(773\) 0.286051 0.286051i 0.0102885 0.0102885i −0.701944 0.712232i \(-0.747684\pi\)
0.712232 + 0.701944i \(0.247684\pi\)
\(774\) 0 0
\(775\) −3.16515 7.25227i −0.113696 0.260509i
\(776\) 0 0
\(777\) −13.2523 13.2523i −0.475423 0.475423i
\(778\) 0 0
\(779\) 26.2668 0.941106
\(780\) 0 0
\(781\) −59.4618 −2.12771
\(782\) 0 0
\(783\) −13.4955 13.4955i −0.482288 0.482288i
\(784\) 0 0
\(785\) 19.7477 30.1652i 0.704827 1.07664i
\(786\) 0 0
\(787\) −20.7092 + 20.7092i −0.738202 + 0.738202i −0.972230 0.234028i \(-0.924809\pi\)
0.234028 + 0.972230i \(0.424809\pi\)
\(788\) 0 0
\(789\) 17.1696i 0.611254i
\(790\) 0 0
\(791\) 25.5826i 0.909612i
\(792\) 0 0
\(793\) 16.7477 16.7477i 0.594729 0.594729i
\(794\) 0 0
\(795\) 1.10440 + 5.29150i 0.0391691 + 0.187670i
\(796\) 0 0
\(797\) 34.9271 + 34.9271i 1.23718 + 1.23718i 0.961149 + 0.276032i \(0.0890195\pi\)
0.276032 + 0.961149i \(0.410981\pi\)
\(798\) 0 0
\(799\) 4.74773 0.167963
\(800\) 0 0
\(801\) 8.17424 0.288823
\(802\) 0 0
\(803\) −2.55040 2.55040i −0.0900017 0.0900017i
\(804\) 0 0
\(805\) 2.01810 + 9.66930i 0.0711287 + 0.340798i
\(806\) 0 0
\(807\) −10.8348 + 10.8348i −0.381405 + 0.381405i
\(808\) 0 0
\(809\) 30.3303i 1.06636i 0.846003 + 0.533178i \(0.179003\pi\)
−0.846003 + 0.533178i \(0.820997\pi\)
\(810\) 0 0
\(811\) 2.16900i 0.0761639i 0.999275 + 0.0380820i \(0.0121248\pi\)
−0.999275 + 0.0380820i \(0.987875\pi\)
\(812\) 0 0
\(813\) −3.99640 + 3.99640i −0.140160 + 0.140160i
\(814\) 0 0
\(815\) 14.3739 21.9564i 0.503494 0.769101i
\(816\) 0 0
\(817\) 28.7477 + 28.7477i 1.00576 + 1.00576i
\(818\) 0 0
\(819\) 24.9717 0.872582
\(820\) 0 0
\(821\) 54.8933 1.91579 0.957895 0.287118i \(-0.0926973\pi\)
0.957895 + 0.287118i \(0.0926973\pi\)
\(822\) 0 0
\(823\) −3.20871 3.20871i −0.111849 0.111849i 0.648967 0.760816i \(-0.275201\pi\)
−0.760816 + 0.648967i \(0.775201\pi\)
\(824\) 0 0
\(825\) 5.16515 13.1652i 0.179827 0.458352i
\(826\) 0 0
\(827\) 14.1226 14.1226i 0.491089 0.491089i −0.417560 0.908649i \(-0.637115\pi\)
0.908649 + 0.417560i \(0.137115\pi\)
\(828\) 0 0
\(829\) 7.65120i 0.265737i −0.991134 0.132869i \(-0.957581\pi\)
0.991134 0.132869i \(-0.0424188\pi\)
\(830\) 0 0
\(831\) 5.40833i 0.187613i
\(832\) 0 0
\(833\) 25.7477 25.7477i 0.892106 0.892106i
\(834\) 0 0
\(835\) 29.6555 + 19.4141i 1.02627 + 0.671851i
\(836\) 0 0
\(837\) 4.03620 + 4.03620i 0.139512 + 0.139512i
\(838\) 0 0
\(839\) 30.3303 1.04712 0.523559 0.851989i \(-0.324604\pi\)
0.523559 + 0.851989i \(0.324604\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 4.91010 + 4.91010i 0.169113 + 0.169113i
\(844\) 0 0
\(845\) −15.3223 + 3.19795i −0.527103 + 0.110013i
\(846\) 0 0
\(847\) −22.7913 + 22.7913i −0.783118 + 0.783118i
\(848\) 0 0
\(849\) 15.7568i 0.540773i
\(850\) 0 0
\(851\) 8.22330i 0.281891i
\(852\) 0 0
\(853\) −36.9452 + 36.9452i −1.26498 + 1.26498i −0.316329 + 0.948650i \(0.602450\pi\)
−0.948650 + 0.316329i \(0.897550\pi\)
\(854\) 0 0
\(855\) 19.5826 4.08712i 0.669710 0.139777i
\(856\) 0 0
\(857\) −39.3303 39.3303i −1.34350 1.34350i −0.892551 0.450947i \(-0.851086\pi\)
−0.450947 0.892551i \(-0.648914\pi\)
\(858\) 0 0
\(859\) −2.01810 −0.0688567 −0.0344284 0.999407i \(-0.510961\pi\)
−0.0344284 + 0.999407i \(0.510961\pi\)
\(860\) 0 0
\(861\) 19.3386 0.659058
\(862\) 0 0
\(863\) 32.7042 + 32.7042i 1.11326 + 1.11326i 0.992706 + 0.120556i \(0.0384678\pi\)
0.120556 + 0.992706i \(0.461532\pi\)
\(864\) 0 0
\(865\) 22.9129 + 15.0000i 0.779061 + 0.510015i
\(866\) 0 0
\(867\) −0.456850 + 0.456850i −0.0155154 + 0.0155154i
\(868\) 0 0
\(869\) 52.5336i 1.78208i
\(870\) 0 0
\(871\) 28.7477i 0.974080i
\(872\) 0 0
\(873\) 1.50455 1.50455i 0.0509212 0.0509212i
\(874\) 0 0
\(875\) −43.5119 7.38505i −1.47097 0.249660i
\(876\) 0 0
\(877\) −19.7756 19.7756i −0.667773 0.667773i 0.289427 0.957200i \(-0.406535\pi\)
−0.957200 + 0.289427i \(0.906535\pi\)
\(878\) 0 0
\(879\) −21.4083 −0.722085
\(880\) 0 0
\(881\) −55.5826 −1.87262 −0.936312 0.351169i \(-0.885784\pi\)
−0.936312 + 0.351169i \(0.885784\pi\)
\(882\) 0 0
\(883\) −22.1552 22.1552i −0.745581 0.745581i 0.228065 0.973646i \(-0.426760\pi\)
−0.973646 + 0.228065i \(0.926760\pi\)
\(884\) 0 0
\(885\) −4.18710 + 6.39590i −0.140748 + 0.214996i
\(886\) 0 0
\(887\) 21.9564 21.9564i 0.737225 0.737225i −0.234815 0.972040i \(-0.575448\pi\)
0.972040 + 0.234815i \(0.0754484\pi\)
\(888\) 0 0
\(889\) 6.74773i 0.226312i
\(890\) 0 0
\(891\) 23.7164i 0.794530i
\(892\) 0 0
\(893\) 2.74110 2.74110i 0.0917275 0.0917275i
\(894\) 0 0
\(895\) −3.91288 18.7477i −0.130793 0.626667i
\(896\) 0 0
\(897\) −1.25227 1.25227i −0.0418122 0.0418122i
\(898\) 0 0
\(899\) 8.37420 0.279295
\(900\) 0 0
\(901\) 15.8745 0.528857
\(902\) 0 0
\(903\) 21.1652 + 21.1652i 0.704332 + 0.704332i
\(904\) 0 0
\(905\) 3.16515 + 15.1652i 0.105213 + 0.504107i
\(906\) 0 0
\(907\) −11.0399 + 11.0399i −0.366572 + 0.366572i −0.866226 0.499653i \(-0.833461\pi\)
0.499653 + 0.866226i \(0.333461\pi\)
\(908\) 0 0
\(909\) 9.43880i 0.313065i
\(910\) 0 0
\(911\) 49.5826i 1.64274i 0.570393 + 0.821372i \(0.306791\pi\)
−0.570393 + 0.821372i \(0.693209\pi\)
\(912\) 0 0
\(913\) −21.1652 + 21.1652i −0.700464 + 0.700464i
\(914\) 0 0
\(915\) 7.65120 11.6874i 0.252941 0.386374i
\(916\) 0 0
\(917\) −31.5583 31.5583i −1.04215 1.04215i
\(918\) 0 0
\(919\) −42.3303 −1.39635 −0.698174 0.715928i \(-0.746004\pi\)
−0.698174 + 0.715928i \(0.746004\pi\)
\(920\) 0 0
\(921\) −1.25227 −0.0412638
\(922\) 0 0
\(923\) −23.5257 23.5257i −0.774358 0.774358i
\(924\) 0 0
\(925\) 34.2041 + 13.4195i 1.12462 + 0.441229i
\(926\) 0 0
\(927\) −26.7913 + 26.7913i −0.879941 + 0.879941i
\(928\) 0 0
\(929\) 19.9129i 0.653320i 0.945142 + 0.326660i \(0.105923\pi\)
−0.945142 + 0.326660i \(0.894077\pi\)
\(930\) 0 0
\(931\) 29.7309i 0.974391i
\(932\) 0 0
\(933\) −4.75920 + 4.75920i −0.155809 + 0.155809i
\(934\) 0 0
\(935\) −34.7477 22.7477i −1.13637 0.743930i
\(936\) 0 0
\(937\) −34.1652 34.1652i −1.11613 1.11613i −0.992304 0.123822i \(-0.960485\pi\)
−0.123822 0.992304i \(-0.539515\pi\)
\(938\) 0 0
\(939\) −2.35970 −0.0770059
\(940\) 0 0
\(941\) −24.4394 −0.796702 −0.398351 0.917233i \(-0.630417\pi\)
−0.398351 + 0.917233i \(0.630417\pi\)
\(942\) 0 0
\(943\) 6.00000 + 6.00000i 0.195387 + 0.195387i
\(944\) 0 0
\(945\) 31.1652 6.50455i 1.01380 0.211593i
\(946\) 0 0
\(947\) 14.6947 14.6947i 0.477512 0.477512i −0.426823 0.904335i \(-0.640367\pi\)
0.904335 + 0.426823i \(0.140367\pi\)
\(948\) 0 0
\(949\) 2.01810i 0.0655103i
\(950\) 0 0
\(951\) 16.0871i 0.521661i
\(952\) 0 0
\(953\) 10.9129 10.9129i 0.353503 0.353503i −0.507908 0.861411i \(-0.669581\pi\)
0.861411 + 0.507908i \(0.169581\pi\)
\(954\) 0 0
\(955\) −29.7309 + 6.20520i −0.962070 + 0.200796i
\(956\) 0 0
\(957\) 10.5830 + 10.5830i 0.342100 + 0.342100i
\(958\) 0 0
\(959\) −67.9129 −2.19302
\(960\) 0 0
\(961\) 28.4955 0.919208
\(962\) 0 0
\(963\) −21.4322 21.4322i −0.690642 0.690642i
\(964\) 0 0
\(965\) 37.4775 + 24.5348i 1.20644 + 0.789802i
\(966\) 0 0
\(967\) −8.79129 + 8.79129i −0.282709 + 0.282709i −0.834188 0.551480i \(-0.814063\pi\)
0.551480 + 0.834188i \(0.314063\pi\)
\(968\) 0 0
\(969\) 9.49545i 0.305038i
\(970\) 0 0
\(971\) 37.5728i 1.20577i 0.797828 + 0.602885i \(0.205982\pi\)
−0.797828 + 0.602885i \(0.794018\pi\)
\(972\) 0 0
\(973\) −24.9717 + 24.9717i −0.800556 + 0.800556i
\(974\) 0 0
\(975\) 7.25227 3.16515i 0.232259 0.101366i
\(976\) 0 0
\(977\) 28.5826 + 28.5826i 0.914438 + 0.914438i 0.996618 0.0821799i \(-0.0261882\pi\)
−0.0821799 + 0.996618i \(0.526188\pi\)
\(978\) 0 0
\(979\) −13.8564 −0.442853
\(980\) 0 0
\(981\) 16.0254 0.511652
\(982\) 0 0
\(983\) 26.7042 + 26.7042i 0.851731 + 0.851731i 0.990346 0.138616i \(-0.0442653\pi\)
−0.138616 + 0.990346i \(0.544265\pi\)
\(984\) 0 0
\(985\) 4.58258 7.00000i 0.146013 0.223039i
\(986\) 0 0
\(987\) 2.01810 2.01810i 0.0642369 0.0642369i
\(988\) 0 0
\(989\) 13.1334i 0.417618i
\(990\) 0 0
\(991\) 3.25227i 0.103312i −0.998665 0.0516559i \(-0.983550\pi\)
0.998665 0.0516559i \(-0.0164499\pi\)
\(992\) 0 0
\(993\) 2.83485 2.83485i 0.0899612 0.0899612i
\(994\) 0 0
\(995\) 1.82740 + 8.75560i 0.0579325 + 0.277571i
\(996\) 0 0
\(997\) 17.6066 + 17.6066i 0.557605 + 0.557605i 0.928625 0.371020i \(-0.120992\pi\)
−0.371020 + 0.928625i \(0.620992\pi\)
\(998\) 0 0
\(999\) −26.5045 −0.838567
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 1280.2.n.p.767.3 8
4.3 odd 2 1280.2.n.n.767.2 8
5.3 odd 4 1280.2.n.n.1023.2 8
8.3 odd 2 1280.2.n.n.767.3 8
8.5 even 2 inner 1280.2.n.p.767.2 8
16.3 odd 4 320.2.o.f.287.3 yes 8
16.5 even 4 320.2.o.e.287.3 yes 8
16.11 odd 4 320.2.o.f.287.2 yes 8
16.13 even 4 320.2.o.e.287.2 yes 8
20.3 even 4 inner 1280.2.n.p.1023.3 8
40.3 even 4 inner 1280.2.n.p.1023.2 8
40.13 odd 4 1280.2.n.n.1023.3 8
80.3 even 4 320.2.o.e.223.3 yes 8
80.13 odd 4 320.2.o.f.223.2 yes 8
80.19 odd 4 1600.2.o.e.607.2 8
80.27 even 4 1600.2.o.l.543.3 8
80.29 even 4 1600.2.o.l.607.3 8
80.37 odd 4 1600.2.o.e.543.2 8
80.43 even 4 320.2.o.e.223.2 8
80.53 odd 4 320.2.o.f.223.3 yes 8
80.59 odd 4 1600.2.o.e.607.3 8
80.67 even 4 1600.2.o.l.543.2 8
80.69 even 4 1600.2.o.l.607.2 8
80.77 odd 4 1600.2.o.e.543.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.o.e.223.2 8 80.43 even 4
320.2.o.e.223.3 yes 8 80.3 even 4
320.2.o.e.287.2 yes 8 16.13 even 4
320.2.o.e.287.3 yes 8 16.5 even 4
320.2.o.f.223.2 yes 8 80.13 odd 4
320.2.o.f.223.3 yes 8 80.53 odd 4
320.2.o.f.287.2 yes 8 16.11 odd 4
320.2.o.f.287.3 yes 8 16.3 odd 4
1280.2.n.n.767.2 8 4.3 odd 2
1280.2.n.n.767.3 8 8.3 odd 2
1280.2.n.n.1023.2 8 5.3 odd 4
1280.2.n.n.1023.3 8 40.13 odd 4
1280.2.n.p.767.2 8 8.5 even 2 inner
1280.2.n.p.767.3 8 1.1 even 1 trivial
1280.2.n.p.1023.2 8 40.3 even 4 inner
1280.2.n.p.1023.3 8 20.3 even 4 inner
1600.2.o.e.543.2 8 80.37 odd 4
1600.2.o.e.543.3 8 80.77 odd 4
1600.2.o.e.607.2 8 80.19 odd 4
1600.2.o.e.607.3 8 80.59 odd 4
1600.2.o.l.543.2 8 80.67 even 4
1600.2.o.l.543.3 8 80.27 even 4
1600.2.o.l.607.2 8 80.69 even 4
1600.2.o.l.607.3 8 80.29 even 4