Properties

Label 1280.2.f.i.129.3
Level $1280$
Weight $2$
Character 1280.129
Analytic conductor $10.221$
Analytic rank $0$
Dimension $6$
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] = [1280,2,Mod(129,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.f (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.3
Root \(0.403032 + 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 1280.129
Dual form 1280.2.f.i.129.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.806063 q^{3} +(-1.67513 - 1.48119i) q^{5} +2.15633i q^{7} -2.35026 q^{9} +O(q^{10})\) \(q-0.806063 q^{3} +(-1.67513 - 1.48119i) q^{5} +2.15633i q^{7} -2.35026 q^{9} +0.387873i q^{11} +0.962389 q^{13} +(1.35026 + 1.19394i) q^{15} -1.61213i q^{17} -6.31265i q^{19} -1.73813i q^{21} +6.15633i q^{23} +(0.612127 + 4.96239i) q^{25} +4.31265 q^{27} +2.00000i q^{29} +9.92478 q^{31} -0.312650i q^{33} +(3.19394 - 3.61213i) q^{35} -6.57452 q^{37} -0.775746 q^{39} -4.57452 q^{41} +11.5066 q^{43} +(3.93700 + 3.48119i) q^{45} +4.54420i q^{47} +2.35026 q^{49} +1.29948i q^{51} +8.96239 q^{53} +(0.574515 - 0.649738i) q^{55} +5.08840i q^{57} -6.31265i q^{59} -0.261865i q^{61} -5.06793i q^{63} +(-1.61213 - 1.42548i) q^{65} +9.89446 q^{67} -4.96239i q^{69} +10.7005 q^{71} -13.0884i q^{73} +(-0.493413 - 4.00000i) q^{75} -0.836381 q^{77} +1.92478 q^{79} +3.57452 q^{81} -2.88129 q^{83} +(-2.38787 + 2.70052i) q^{85} -1.61213i q^{87} +10.6253 q^{89} +2.07522i q^{91} -8.00000 q^{93} +(-9.35026 + 10.5745i) q^{95} -9.61213i q^{97} -0.911603i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} + 6 q^{9} - 16 q^{13} - 12 q^{15} + 2 q^{25} - 16 q^{27} + 16 q^{31} + 20 q^{35} - 16 q^{37} - 8 q^{39} - 4 q^{41} + 28 q^{43} + 32 q^{45} - 6 q^{49} + 32 q^{53} - 20 q^{55} - 8 q^{65} + 20 q^{67} + 24 q^{71} - 44 q^{75} - 32 q^{79} - 2 q^{81} - 60 q^{83} - 16 q^{85} - 20 q^{89} - 48 q^{93} - 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.806063 −0.465381 −0.232690 0.972551i \(-0.574753\pi\)
−0.232690 + 0.972551i \(0.574753\pi\)
\(4\) 0 0
\(5\) −1.67513 1.48119i −0.749141 0.662410i
\(6\) 0 0
\(7\) 2.15633i 0.815014i 0.913202 + 0.407507i \(0.133602\pi\)
−0.913202 + 0.407507i \(0.866398\pi\)
\(8\) 0 0
\(9\) −2.35026 −0.783421
\(10\) 0 0
\(11\) 0.387873i 0.116948i 0.998289 + 0.0584741i \(0.0186235\pi\)
−0.998289 + 0.0584741i \(0.981377\pi\)
\(12\) 0 0
\(13\) 0.962389 0.266919 0.133459 0.991054i \(-0.457391\pi\)
0.133459 + 0.991054i \(0.457391\pi\)
\(14\) 0 0
\(15\) 1.35026 + 1.19394i 0.348636 + 0.308273i
\(16\) 0 0
\(17\) 1.61213i 0.390998i −0.980704 0.195499i \(-0.937367\pi\)
0.980704 0.195499i \(-0.0626327\pi\)
\(18\) 0 0
\(19\) 6.31265i 1.44822i −0.689684 0.724111i \(-0.742250\pi\)
0.689684 0.724111i \(-0.257750\pi\)
\(20\) 0 0
\(21\) 1.73813i 0.379292i
\(22\) 0 0
\(23\) 6.15633i 1.28368i 0.766838 + 0.641841i \(0.221829\pi\)
−0.766838 + 0.641841i \(0.778171\pi\)
\(24\) 0 0
\(25\) 0.612127 + 4.96239i 0.122425 + 0.992478i
\(26\) 0 0
\(27\) 4.31265 0.829970
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 9.92478 1.78254 0.891271 0.453470i \(-0.149814\pi\)
0.891271 + 0.453470i \(0.149814\pi\)
\(32\) 0 0
\(33\) 0.312650i 0.0544254i
\(34\) 0 0
\(35\) 3.19394 3.61213i 0.539874 0.610561i
\(36\) 0 0
\(37\) −6.57452 −1.08084 −0.540422 0.841394i \(-0.681735\pi\)
−0.540422 + 0.841394i \(0.681735\pi\)
\(38\) 0 0
\(39\) −0.775746 −0.124219
\(40\) 0 0
\(41\) −4.57452 −0.714419 −0.357210 0.934024i \(-0.616272\pi\)
−0.357210 + 0.934024i \(0.616272\pi\)
\(42\) 0 0
\(43\) 11.5066 1.75474 0.877369 0.479816i \(-0.159297\pi\)
0.877369 + 0.479816i \(0.159297\pi\)
\(44\) 0 0
\(45\) 3.93700 + 3.48119i 0.586893 + 0.518946i
\(46\) 0 0
\(47\) 4.54420i 0.662839i 0.943483 + 0.331420i \(0.107528\pi\)
−0.943483 + 0.331420i \(0.892472\pi\)
\(48\) 0 0
\(49\) 2.35026 0.335752
\(50\) 0 0
\(51\) 1.29948i 0.181963i
\(52\) 0 0
\(53\) 8.96239 1.23108 0.615539 0.788106i \(-0.288938\pi\)
0.615539 + 0.788106i \(0.288938\pi\)
\(54\) 0 0
\(55\) 0.574515 0.649738i 0.0774677 0.0876107i
\(56\) 0 0
\(57\) 5.08840i 0.673975i
\(58\) 0 0
\(59\) 6.31265i 0.821837i −0.911672 0.410919i \(-0.865208\pi\)
0.911672 0.410919i \(-0.134792\pi\)
\(60\) 0 0
\(61\) 0.261865i 0.0335284i −0.999859 0.0167642i \(-0.994664\pi\)
0.999859 0.0167642i \(-0.00533646\pi\)
\(62\) 0 0
\(63\) 5.06793i 0.638499i
\(64\) 0 0
\(65\) −1.61213 1.42548i −0.199960 0.176810i
\(66\) 0 0
\(67\) 9.89446 1.20880 0.604400 0.796681i \(-0.293413\pi\)
0.604400 + 0.796681i \(0.293413\pi\)
\(68\) 0 0
\(69\) 4.96239i 0.597401i
\(70\) 0 0
\(71\) 10.7005 1.26992 0.634959 0.772546i \(-0.281017\pi\)
0.634959 + 0.772546i \(0.281017\pi\)
\(72\) 0 0
\(73\) 13.0884i 1.53188i −0.642911 0.765940i \(-0.722274\pi\)
0.642911 0.765940i \(-0.277726\pi\)
\(74\) 0 0
\(75\) −0.493413 4.00000i −0.0569744 0.461880i
\(76\) 0 0
\(77\) −0.836381 −0.0953144
\(78\) 0 0
\(79\) 1.92478 0.216554 0.108277 0.994121i \(-0.465467\pi\)
0.108277 + 0.994121i \(0.465467\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 0 0
\(83\) −2.88129 −0.316262 −0.158131 0.987418i \(-0.550547\pi\)
−0.158131 + 0.987418i \(0.550547\pi\)
\(84\) 0 0
\(85\) −2.38787 + 2.70052i −0.259001 + 0.292913i
\(86\) 0 0
\(87\) 1.61213i 0.172838i
\(88\) 0 0
\(89\) 10.6253 1.12628 0.563140 0.826362i \(-0.309593\pi\)
0.563140 + 0.826362i \(0.309593\pi\)
\(90\) 0 0
\(91\) 2.07522i 0.217542i
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −9.35026 + 10.5745i −0.959317 + 1.08492i
\(96\) 0 0
\(97\) 9.61213i 0.975964i −0.872854 0.487982i \(-0.837733\pi\)
0.872854 0.487982i \(-0.162267\pi\)
\(98\) 0 0
\(99\) 0.911603i 0.0916196i
\(100\) 0 0
\(101\) 11.4010i 1.13445i 0.823564 + 0.567223i \(0.191982\pi\)
−0.823564 + 0.567223i \(0.808018\pi\)
\(102\) 0 0
\(103\) 10.9321i 1.07717i 0.842572 + 0.538585i \(0.181041\pi\)
−0.842572 + 0.538585i \(0.818959\pi\)
\(104\) 0 0
\(105\) −2.57452 + 2.91160i −0.251247 + 0.284143i
\(106\) 0 0
\(107\) −11.0435 −1.06761 −0.533807 0.845606i \(-0.679239\pi\)
−0.533807 + 0.845606i \(0.679239\pi\)
\(108\) 0 0
\(109\) 5.03761i 0.482516i −0.970461 0.241258i \(-0.922440\pi\)
0.970461 0.241258i \(-0.0775599\pi\)
\(110\) 0 0
\(111\) 5.29948 0.503004
\(112\) 0 0
\(113\) 19.8496i 1.86729i 0.358201 + 0.933645i \(0.383390\pi\)
−0.358201 + 0.933645i \(0.616610\pi\)
\(114\) 0 0
\(115\) 9.11871 10.3127i 0.850324 0.961660i
\(116\) 0 0
\(117\) −2.26187 −0.209110
\(118\) 0 0
\(119\) 3.47627 0.318669
\(120\) 0 0
\(121\) 10.8496 0.986323
\(122\) 0 0
\(123\) 3.68735 0.332477
\(124\) 0 0
\(125\) 6.32487 9.21933i 0.565713 0.824602i
\(126\) 0 0
\(127\) 11.7685i 1.04428i 0.852859 + 0.522141i \(0.174866\pi\)
−0.852859 + 0.522141i \(0.825134\pi\)
\(128\) 0 0
\(129\) −9.27504 −0.816622
\(130\) 0 0
\(131\) 15.0884i 1.31828i 0.752021 + 0.659140i \(0.229079\pi\)
−0.752021 + 0.659140i \(0.770921\pi\)
\(132\) 0 0
\(133\) 13.6121 1.18032
\(134\) 0 0
\(135\) −7.22425 6.38787i −0.621765 0.549781i
\(136\) 0 0
\(137\) 18.5501i 1.58484i −0.609976 0.792420i \(-0.708821\pi\)
0.609976 0.792420i \(-0.291179\pi\)
\(138\) 0 0
\(139\) 7.61213i 0.645652i −0.946458 0.322826i \(-0.895367\pi\)
0.946458 0.322826i \(-0.104633\pi\)
\(140\) 0 0
\(141\) 3.66291i 0.308473i
\(142\) 0 0
\(143\) 0.373285i 0.0312156i
\(144\) 0 0
\(145\) 2.96239 3.35026i 0.246013 0.278224i
\(146\) 0 0
\(147\) −1.89446 −0.156252
\(148\) 0 0
\(149\) 13.6629i 1.11931i 0.828726 + 0.559655i \(0.189066\pi\)
−0.828726 + 0.559655i \(0.810934\pi\)
\(150\) 0 0
\(151\) 15.2243 1.23893 0.619466 0.785023i \(-0.287349\pi\)
0.619466 + 0.785023i \(0.287349\pi\)
\(152\) 0 0
\(153\) 3.78892i 0.306316i
\(154\) 0 0
\(155\) −16.6253 14.7005i −1.33538 1.18077i
\(156\) 0 0
\(157\) 1.42548 0.113766 0.0568830 0.998381i \(-0.481884\pi\)
0.0568830 + 0.998381i \(0.481884\pi\)
\(158\) 0 0
\(159\) −7.22425 −0.572921
\(160\) 0 0
\(161\) −13.2750 −1.04622
\(162\) 0 0
\(163\) −8.80606 −0.689744 −0.344872 0.938650i \(-0.612078\pi\)
−0.344872 + 0.938650i \(0.612078\pi\)
\(164\) 0 0
\(165\) −0.463096 + 0.523730i −0.0360520 + 0.0407723i
\(166\) 0 0
\(167\) 10.4690i 0.810114i −0.914292 0.405057i \(-0.867252\pi\)
0.914292 0.405057i \(-0.132748\pi\)
\(168\) 0 0
\(169\) −12.0738 −0.928754
\(170\) 0 0
\(171\) 14.8364i 1.13457i
\(172\) 0 0
\(173\) −19.9756 −1.51871 −0.759357 0.650674i \(-0.774486\pi\)
−0.759357 + 0.650674i \(0.774486\pi\)
\(174\) 0 0
\(175\) −10.7005 + 1.31994i −0.808884 + 0.0997784i
\(176\) 0 0
\(177\) 5.08840i 0.382467i
\(178\) 0 0
\(179\) 12.2374i 0.914668i 0.889295 + 0.457334i \(0.151196\pi\)
−0.889295 + 0.457334i \(0.848804\pi\)
\(180\) 0 0
\(181\) 13.8496i 1.02943i −0.857362 0.514715i \(-0.827898\pi\)
0.857362 0.514715i \(-0.172102\pi\)
\(182\) 0 0
\(183\) 0.211080i 0.0156035i
\(184\) 0 0
\(185\) 11.0132 + 9.73813i 0.809705 + 0.715962i
\(186\) 0 0
\(187\) 0.625301 0.0457265
\(188\) 0 0
\(189\) 9.29948i 0.676437i
\(190\) 0 0
\(191\) −3.84955 −0.278544 −0.139272 0.990254i \(-0.544476\pi\)
−0.139272 + 0.990254i \(0.544476\pi\)
\(192\) 0 0
\(193\) 9.61213i 0.691896i 0.938254 + 0.345948i \(0.112443\pi\)
−0.938254 + 0.345948i \(0.887557\pi\)
\(194\) 0 0
\(195\) 1.29948 + 1.14903i 0.0930574 + 0.0822838i
\(196\) 0 0
\(197\) 9.58769 0.683095 0.341547 0.939865i \(-0.389049\pi\)
0.341547 + 0.939865i \(0.389049\pi\)
\(198\) 0 0
\(199\) −5.29948 −0.375670 −0.187835 0.982201i \(-0.560147\pi\)
−0.187835 + 0.982201i \(0.560147\pi\)
\(200\) 0 0
\(201\) −7.97556 −0.562553
\(202\) 0 0
\(203\) −4.31265 −0.302689
\(204\) 0 0
\(205\) 7.66291 + 6.77575i 0.535201 + 0.473239i
\(206\) 0 0
\(207\) 14.4690i 1.00566i
\(208\) 0 0
\(209\) 2.44851 0.169367
\(210\) 0 0
\(211\) 15.0884i 1.03873i 0.854553 + 0.519364i \(0.173831\pi\)
−0.854553 + 0.519364i \(0.826169\pi\)
\(212\) 0 0
\(213\) −8.62530 −0.590996
\(214\) 0 0
\(215\) −19.2750 17.0435i −1.31455 1.16236i
\(216\) 0 0
\(217\) 21.4010i 1.45280i
\(218\) 0 0
\(219\) 10.5501i 0.712908i
\(220\) 0 0
\(221\) 1.55149i 0.104365i
\(222\) 0 0
\(223\) 22.5296i 1.50869i −0.656476 0.754347i \(-0.727954\pi\)
0.656476 0.754347i \(-0.272046\pi\)
\(224\) 0 0
\(225\) −1.43866 11.6629i −0.0959106 0.777527i
\(226\) 0 0
\(227\) −2.35756 −0.156476 −0.0782382 0.996935i \(-0.524929\pi\)
−0.0782382 + 0.996935i \(0.524929\pi\)
\(228\) 0 0
\(229\) 3.55149i 0.234689i −0.993091 0.117345i \(-0.962562\pi\)
0.993091 0.117345i \(-0.0374382\pi\)
\(230\) 0 0
\(231\) 0.674176 0.0443575
\(232\) 0 0
\(233\) 13.0884i 0.857449i −0.903435 0.428725i \(-0.858963\pi\)
0.903435 0.428725i \(-0.141037\pi\)
\(234\) 0 0
\(235\) 6.73084 7.61213i 0.439072 0.496560i
\(236\) 0 0
\(237\) −1.55149 −0.100780
\(238\) 0 0
\(239\) −15.3258 −0.991345 −0.495673 0.868509i \(-0.665078\pi\)
−0.495673 + 0.868509i \(0.665078\pi\)
\(240\) 0 0
\(241\) 2.12601 0.136948 0.0684741 0.997653i \(-0.478187\pi\)
0.0684741 + 0.997653i \(0.478187\pi\)
\(242\) 0 0
\(243\) −15.8192 −1.01480
\(244\) 0 0
\(245\) −3.93700 3.48119i −0.251525 0.222405i
\(246\) 0 0
\(247\) 6.07522i 0.386557i
\(248\) 0 0
\(249\) 2.32250 0.147182
\(250\) 0 0
\(251\) 26.6859i 1.68440i 0.539164 + 0.842201i \(0.318740\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(252\) 0 0
\(253\) −2.38787 −0.150124
\(254\) 0 0
\(255\) 1.92478 2.17679i 0.120534 0.136316i
\(256\) 0 0
\(257\) 5.40105i 0.336908i −0.985710 0.168454i \(-0.946123\pi\)
0.985710 0.168454i \(-0.0538775\pi\)
\(258\) 0 0
\(259\) 14.1768i 0.880903i
\(260\) 0 0
\(261\) 4.70052i 0.290955i
\(262\) 0 0
\(263\) 14.4690i 0.892195i −0.894984 0.446098i \(-0.852813\pi\)
0.894984 0.446098i \(-0.147187\pi\)
\(264\) 0 0
\(265\) −15.0132 13.2750i −0.922252 0.815479i
\(266\) 0 0
\(267\) −8.56467 −0.524149
\(268\) 0 0
\(269\) 28.1114i 1.71398i −0.515330 0.856992i \(-0.672331\pi\)
0.515330 0.856992i \(-0.327669\pi\)
\(270\) 0 0
\(271\) 24.8773 1.51119 0.755595 0.655039i \(-0.227348\pi\)
0.755595 + 0.655039i \(0.227348\pi\)
\(272\) 0 0
\(273\) 1.67276i 0.101240i
\(274\) 0 0
\(275\) −1.92478 + 0.237428i −0.116068 + 0.0143174i
\(276\) 0 0
\(277\) −19.9756 −1.20022 −0.600108 0.799919i \(-0.704876\pi\)
−0.600108 + 0.799919i \(0.704876\pi\)
\(278\) 0 0
\(279\) −23.3258 −1.39648
\(280\) 0 0
\(281\) 14.4993 0.864955 0.432478 0.901645i \(-0.357639\pi\)
0.432478 + 0.901645i \(0.357639\pi\)
\(282\) 0 0
\(283\) 8.80606 0.523466 0.261733 0.965140i \(-0.415706\pi\)
0.261733 + 0.965140i \(0.415706\pi\)
\(284\) 0 0
\(285\) 7.53690 8.52373i 0.446448 0.504902i
\(286\) 0 0
\(287\) 9.86414i 0.582262i
\(288\) 0 0
\(289\) 14.4010 0.847120
\(290\) 0 0
\(291\) 7.74798i 0.454195i
\(292\) 0 0
\(293\) −3.35026 −0.195724 −0.0978622 0.995200i \(-0.531200\pi\)
−0.0978622 + 0.995200i \(0.531200\pi\)
\(294\) 0 0
\(295\) −9.35026 + 10.5745i −0.544393 + 0.615672i
\(296\) 0 0
\(297\) 1.67276i 0.0970634i
\(298\) 0 0
\(299\) 5.92478i 0.342639i
\(300\) 0 0
\(301\) 24.8119i 1.43014i
\(302\) 0 0
\(303\) 9.18997i 0.527950i
\(304\) 0 0
\(305\) −0.387873 + 0.438658i −0.0222096 + 0.0251175i
\(306\) 0 0
\(307\) −1.95509 −0.111583 −0.0557916 0.998442i \(-0.517768\pi\)
−0.0557916 + 0.998442i \(0.517768\pi\)
\(308\) 0 0
\(309\) 8.81194i 0.501294i
\(310\) 0 0
\(311\) 8.77575 0.497627 0.248813 0.968551i \(-0.419959\pi\)
0.248813 + 0.968551i \(0.419959\pi\)
\(312\) 0 0
\(313\) 18.5501i 1.04851i 0.851561 + 0.524256i \(0.175657\pi\)
−0.851561 + 0.524256i \(0.824343\pi\)
\(314\) 0 0
\(315\) −7.50659 + 8.48944i −0.422948 + 0.478326i
\(316\) 0 0
\(317\) 9.58769 0.538498 0.269249 0.963071i \(-0.413224\pi\)
0.269249 + 0.963071i \(0.413224\pi\)
\(318\) 0 0
\(319\) −0.775746 −0.0434335
\(320\) 0 0
\(321\) 8.90175 0.496847
\(322\) 0 0
\(323\) −10.1768 −0.566252
\(324\) 0 0
\(325\) 0.589104 + 4.77575i 0.0326776 + 0.264911i
\(326\) 0 0
\(327\) 4.06063i 0.224554i
\(328\) 0 0
\(329\) −9.79877 −0.540224
\(330\) 0 0
\(331\) 23.0884i 1.26905i 0.772901 + 0.634527i \(0.218805\pi\)
−0.772901 + 0.634527i \(0.781195\pi\)
\(332\) 0 0
\(333\) 15.4518 0.846755
\(334\) 0 0
\(335\) −16.5745 14.6556i −0.905563 0.800722i
\(336\) 0 0
\(337\) 4.77575i 0.260151i −0.991504 0.130076i \(-0.958478\pi\)
0.991504 0.130076i \(-0.0415220\pi\)
\(338\) 0 0
\(339\) 16.0000i 0.869001i
\(340\) 0 0
\(341\) 3.84955i 0.208465i
\(342\) 0 0
\(343\) 20.1622i 1.08866i
\(344\) 0 0
\(345\) −7.35026 + 8.31265i −0.395725 + 0.447538i
\(346\) 0 0
\(347\) −1.89446 −0.101700 −0.0508500 0.998706i \(-0.516193\pi\)
−0.0508500 + 0.998706i \(0.516193\pi\)
\(348\) 0 0
\(349\) 31.4010i 1.68086i 0.541921 + 0.840430i \(0.317697\pi\)
−0.541921 + 0.840430i \(0.682303\pi\)
\(350\) 0 0
\(351\) 4.15045 0.221534
\(352\) 0 0
\(353\) 12.7757i 0.679984i −0.940428 0.339992i \(-0.889576\pi\)
0.940428 0.339992i \(-0.110424\pi\)
\(354\) 0 0
\(355\) −17.9248 15.8496i −0.951348 0.841207i
\(356\) 0 0
\(357\) −2.80209 −0.148303
\(358\) 0 0
\(359\) −16.4749 −0.869510 −0.434755 0.900549i \(-0.643165\pi\)
−0.434755 + 0.900549i \(0.643165\pi\)
\(360\) 0 0
\(361\) −20.8496 −1.09734
\(362\) 0 0
\(363\) −8.74543 −0.459016
\(364\) 0 0
\(365\) −19.3865 + 21.9248i −1.01473 + 1.14760i
\(366\) 0 0
\(367\) 10.0957i 0.526991i 0.964661 + 0.263495i \(0.0848754\pi\)
−0.964661 + 0.263495i \(0.915125\pi\)
\(368\) 0 0
\(369\) 10.7513 0.559691
\(370\) 0 0
\(371\) 19.3258i 1.00335i
\(372\) 0 0
\(373\) 6.82653 0.353464 0.176732 0.984259i \(-0.443447\pi\)
0.176732 + 0.984259i \(0.443447\pi\)
\(374\) 0 0
\(375\) −5.09825 + 7.43136i −0.263272 + 0.383754i
\(376\) 0 0
\(377\) 1.92478i 0.0991311i
\(378\) 0 0
\(379\) 19.4617i 0.999679i −0.866118 0.499840i \(-0.833392\pi\)
0.866118 0.499840i \(-0.166608\pi\)
\(380\) 0 0
\(381\) 9.48612i 0.485989i
\(382\) 0 0
\(383\) 5.90431i 0.301696i −0.988557 0.150848i \(-0.951800\pi\)
0.988557 0.150848i \(-0.0482004\pi\)
\(384\) 0 0
\(385\) 1.40105 + 1.23884i 0.0714040 + 0.0631372i
\(386\) 0 0
\(387\) −27.0435 −1.37470
\(388\) 0 0
\(389\) 14.1866i 0.719291i −0.933089 0.359646i \(-0.882898\pi\)
0.933089 0.359646i \(-0.117102\pi\)
\(390\) 0 0
\(391\) 9.92478 0.501918
\(392\) 0 0
\(393\) 12.1622i 0.613502i
\(394\) 0 0
\(395\) −3.22425 2.85097i −0.162230 0.143448i
\(396\) 0 0
\(397\) 28.1866 1.41465 0.707324 0.706890i \(-0.249902\pi\)
0.707324 + 0.706890i \(0.249902\pi\)
\(398\) 0 0
\(399\) −10.9722 −0.549299
\(400\) 0 0
\(401\) 6.62530 0.330852 0.165426 0.986222i \(-0.447100\pi\)
0.165426 + 0.986222i \(0.447100\pi\)
\(402\) 0 0
\(403\) 9.55149 0.475794
\(404\) 0 0
\(405\) −5.98778 5.29455i −0.297535 0.263088i
\(406\) 0 0
\(407\) 2.55008i 0.126403i
\(408\) 0 0
\(409\) 12.6761 0.626792 0.313396 0.949623i \(-0.398533\pi\)
0.313396 + 0.949623i \(0.398533\pi\)
\(410\) 0 0
\(411\) 14.9525i 0.737554i
\(412\) 0 0
\(413\) 13.6121 0.669809
\(414\) 0 0
\(415\) 4.82653 + 4.26774i 0.236925 + 0.209495i
\(416\) 0 0
\(417\) 6.13586i 0.300474i
\(418\) 0 0
\(419\) 14.8364i 0.724805i 0.932022 + 0.362402i \(0.118043\pi\)
−0.932022 + 0.362402i \(0.881957\pi\)
\(420\) 0 0
\(421\) 0.261865i 0.0127625i 0.999980 + 0.00638126i \(0.00203123\pi\)
−0.999980 + 0.00638126i \(0.997969\pi\)
\(422\) 0 0
\(423\) 10.6801i 0.519282i
\(424\) 0 0
\(425\) 8.00000 0.986826i 0.388057 0.0478681i
\(426\) 0 0
\(427\) 0.564666 0.0273261
\(428\) 0 0
\(429\) 0.300891i 0.0145272i
\(430\) 0 0
\(431\) 4.52373 0.217900 0.108950 0.994047i \(-0.465251\pi\)
0.108950 + 0.994047i \(0.465251\pi\)
\(432\) 0 0
\(433\) 15.6385i 0.751537i −0.926714 0.375769i \(-0.877379\pi\)
0.926714 0.375769i \(-0.122621\pi\)
\(434\) 0 0
\(435\) −2.38787 + 2.70052i −0.114490 + 0.129480i
\(436\) 0 0
\(437\) 38.8627 1.85906
\(438\) 0 0
\(439\) −10.3272 −0.492892 −0.246446 0.969156i \(-0.579263\pi\)
−0.246446 + 0.969156i \(0.579263\pi\)
\(440\) 0 0
\(441\) −5.52373 −0.263035
\(442\) 0 0
\(443\) −15.1939 −0.721886 −0.360943 0.932588i \(-0.617545\pi\)
−0.360943 + 0.932588i \(0.617545\pi\)
\(444\) 0 0
\(445\) −17.7988 15.7381i −0.843743 0.746059i
\(446\) 0 0
\(447\) 11.0132i 0.520905i
\(448\) 0 0
\(449\) −24.6761 −1.16454 −0.582268 0.812997i \(-0.697835\pi\)
−0.582268 + 0.812997i \(0.697835\pi\)
\(450\) 0 0
\(451\) 1.77433i 0.0835500i
\(452\) 0 0
\(453\) −12.2717 −0.576575
\(454\) 0 0
\(455\) 3.07381 3.47627i 0.144102 0.162970i
\(456\) 0 0
\(457\) 8.37328i 0.391686i 0.980635 + 0.195843i \(0.0627443\pi\)
−0.980635 + 0.195843i \(0.937256\pi\)
\(458\) 0 0
\(459\) 6.95254i 0.324517i
\(460\) 0 0
\(461\) 8.29806i 0.386479i −0.981152 0.193240i \(-0.938101\pi\)
0.981152 0.193240i \(-0.0618995\pi\)
\(462\) 0 0
\(463\) 39.4676i 1.83421i 0.398642 + 0.917107i \(0.369482\pi\)
−0.398642 + 0.917107i \(0.630518\pi\)
\(464\) 0 0
\(465\) 13.4010 + 11.8496i 0.621459 + 0.549510i
\(466\) 0 0
\(467\) 2.41819 0.111901 0.0559503 0.998434i \(-0.482181\pi\)
0.0559503 + 0.998434i \(0.482181\pi\)
\(468\) 0 0
\(469\) 21.3357i 0.985190i
\(470\) 0 0
\(471\) −1.14903 −0.0529446
\(472\) 0 0
\(473\) 4.46310i 0.205213i
\(474\) 0 0
\(475\) 31.3258 3.86414i 1.43733 0.177299i
\(476\) 0 0
\(477\) −21.0640 −0.964452
\(478\) 0 0
\(479\) 6.44851 0.294640 0.147320 0.989089i \(-0.452935\pi\)
0.147320 + 0.989089i \(0.452935\pi\)
\(480\) 0 0
\(481\) −6.32724 −0.288497
\(482\) 0 0
\(483\) 10.7005 0.486891
\(484\) 0 0
\(485\) −14.2374 + 16.1016i −0.646488 + 0.731135i
\(486\) 0 0
\(487\) 12.3331i 0.558867i 0.960165 + 0.279433i \(0.0901466\pi\)
−0.960165 + 0.279433i \(0.909853\pi\)
\(488\) 0 0
\(489\) 7.09825 0.320994
\(490\) 0 0
\(491\) 14.3127i 0.645921i −0.946412 0.322960i \(-0.895322\pi\)
0.946412 0.322960i \(-0.104678\pi\)
\(492\) 0 0
\(493\) 3.22425 0.145213
\(494\) 0 0
\(495\) −1.35026 + 1.52705i −0.0606898 + 0.0686360i
\(496\) 0 0
\(497\) 23.0738i 1.03500i
\(498\) 0 0
\(499\) 14.0606i 0.629440i −0.949184 0.314720i \(-0.898089\pi\)
0.949184 0.314720i \(-0.101911\pi\)
\(500\) 0 0
\(501\) 8.43866i 0.377011i
\(502\) 0 0
\(503\) 31.8700i 1.42101i −0.703690 0.710507i \(-0.748466\pi\)
0.703690 0.710507i \(-0.251534\pi\)
\(504\) 0 0
\(505\) 16.8872 19.0982i 0.751469 0.849861i
\(506\) 0 0
\(507\) 9.73226 0.432225
\(508\) 0 0
\(509\) 26.0000i 1.15243i 0.817298 + 0.576215i \(0.195471\pi\)
−0.817298 + 0.576215i \(0.804529\pi\)
\(510\) 0 0
\(511\) 28.2228 1.24850
\(512\) 0 0
\(513\) 27.2243i 1.20198i
\(514\) 0 0
\(515\) 16.1925 18.3127i 0.713528 0.806952i
\(516\) 0 0
\(517\) −1.76257 −0.0775178
\(518\) 0 0
\(519\) 16.1016 0.706780
\(520\) 0 0
\(521\) 19.4010 0.849975 0.424988 0.905199i \(-0.360278\pi\)
0.424988 + 0.905199i \(0.360278\pi\)
\(522\) 0 0
\(523\) 7.75860 0.339260 0.169630 0.985508i \(-0.445743\pi\)
0.169630 + 0.985508i \(0.445743\pi\)
\(524\) 0 0
\(525\) 8.62530 1.06396i 0.376439 0.0464350i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) −14.9003 −0.647841
\(530\) 0 0
\(531\) 14.8364i 0.643844i
\(532\) 0 0
\(533\) −4.40246 −0.190692
\(534\) 0 0
\(535\) 18.4993 + 16.3576i 0.799794 + 0.707199i
\(536\) 0 0
\(537\) 9.86414i 0.425669i
\(538\) 0 0
\(539\) 0.911603i 0.0392655i
\(540\) 0 0
\(541\) 25.8496i 1.11136i −0.831397 0.555680i \(-0.812458\pi\)
0.831397 0.555680i \(-0.187542\pi\)
\(542\) 0 0
\(543\) 11.1636i 0.479077i
\(544\) 0 0
\(545\) −7.46168 + 8.43866i −0.319623 + 0.361472i
\(546\) 0 0
\(547\) 38.2677 1.63621 0.818105 0.575068i \(-0.195025\pi\)
0.818105 + 0.575068i \(0.195025\pi\)
\(548\) 0 0
\(549\) 0.615452i 0.0262668i
\(550\) 0 0
\(551\) 12.6253 0.537856
\(552\) 0 0
\(553\) 4.15045i 0.176495i
\(554\) 0 0
\(555\) −8.87732 7.84955i −0.376821 0.333195i
\(556\) 0 0
\(557\) −37.5271 −1.59007 −0.795036 0.606562i \(-0.792548\pi\)
−0.795036 + 0.606562i \(0.792548\pi\)
\(558\) 0 0
\(559\) 11.0738 0.468372
\(560\) 0 0
\(561\) −0.504032 −0.0212802
\(562\) 0 0
\(563\) 43.6688 1.84042 0.920210 0.391425i \(-0.128018\pi\)
0.920210 + 0.391425i \(0.128018\pi\)
\(564\) 0 0
\(565\) 29.4010 33.2506i 1.23691 1.39886i
\(566\) 0 0
\(567\) 7.70782i 0.323698i
\(568\) 0 0
\(569\) −8.42407 −0.353155 −0.176578 0.984287i \(-0.556503\pi\)
−0.176578 + 0.984287i \(0.556503\pi\)
\(570\) 0 0
\(571\) 24.8627i 1.04047i −0.854022 0.520236i \(-0.825844\pi\)
0.854022 0.520236i \(-0.174156\pi\)
\(572\) 0 0
\(573\) 3.10299 0.129629
\(574\) 0 0
\(575\) −30.5501 + 3.76845i −1.27403 + 0.157155i
\(576\) 0 0
\(577\) 14.0263i 0.583924i 0.956430 + 0.291962i \(0.0943082\pi\)
−0.956430 + 0.291962i \(0.905692\pi\)
\(578\) 0 0
\(579\) 7.74798i 0.321995i
\(580\) 0 0
\(581\) 6.21299i 0.257758i
\(582\) 0 0
\(583\) 3.47627i 0.143972i
\(584\) 0 0
\(585\) 3.78892 + 3.35026i 0.156653 + 0.138516i
\(586\) 0 0
\(587\) 34.5804 1.42729 0.713643 0.700510i \(-0.247044\pi\)
0.713643 + 0.700510i \(0.247044\pi\)
\(588\) 0 0
\(589\) 62.6516i 2.58152i
\(590\) 0 0
\(591\) −7.72829 −0.317899
\(592\) 0 0
\(593\) 8.00000i 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) −5.82321 5.14903i −0.238728 0.211090i
\(596\) 0 0
\(597\) 4.27171 0.174830
\(598\) 0 0
\(599\) 42.3996 1.73240 0.866201 0.499696i \(-0.166555\pi\)
0.866201 + 0.499696i \(0.166555\pi\)
\(600\) 0 0
\(601\) −2.75131 −0.112228 −0.0561141 0.998424i \(-0.517871\pi\)
−0.0561141 + 0.998424i \(0.517871\pi\)
\(602\) 0 0
\(603\) −23.2546 −0.946999
\(604\) 0 0
\(605\) −18.1744 16.0703i −0.738895 0.653351i
\(606\) 0 0
\(607\) 37.7948i 1.53404i 0.641621 + 0.767022i \(0.278262\pi\)
−0.641621 + 0.767022i \(0.721738\pi\)
\(608\) 0 0
\(609\) 3.47627 0.140866
\(610\) 0 0
\(611\) 4.37328i 0.176924i
\(612\) 0 0
\(613\) −18.7659 −0.757947 −0.378974 0.925407i \(-0.623723\pi\)
−0.378974 + 0.925407i \(0.623723\pi\)
\(614\) 0 0
\(615\) −6.17679 5.46168i −0.249072 0.220236i
\(616\) 0 0
\(617\) 18.6107i 0.749239i −0.927179 0.374620i \(-0.877773\pi\)
0.927179 0.374620i \(-0.122227\pi\)
\(618\) 0 0
\(619\) 19.2097i 0.772102i −0.922478 0.386051i \(-0.873839\pi\)
0.922478 0.386051i \(-0.126161\pi\)
\(620\) 0 0
\(621\) 26.5501i 1.06542i
\(622\) 0 0
\(623\) 22.9116i 0.917934i
\(624\) 0 0
\(625\) −24.2506 + 6.07522i −0.970024 + 0.243009i
\(626\) 0 0
\(627\) −1.97365 −0.0788201
\(628\) 0 0
\(629\) 10.5990i 0.422608i
\(630\) 0 0
\(631\) 26.0263 1.03609 0.518046 0.855353i \(-0.326659\pi\)
0.518046 + 0.855353i \(0.326659\pi\)
\(632\) 0 0
\(633\) 12.1622i 0.483404i
\(634\) 0 0
\(635\) 17.4314 19.7137i 0.691743 0.782314i
\(636\) 0 0
\(637\) 2.26187 0.0896184
\(638\) 0 0
\(639\) −25.1490 −0.994880
\(640\) 0 0
\(641\) 11.1735 0.441325 0.220663 0.975350i \(-0.429178\pi\)
0.220663 + 0.975350i \(0.429178\pi\)
\(642\) 0 0
\(643\) 35.1451 1.38599 0.692993 0.720944i \(-0.256292\pi\)
0.692993 + 0.720944i \(0.256292\pi\)
\(644\) 0 0
\(645\) 15.5369 + 13.7381i 0.611765 + 0.540939i
\(646\) 0 0
\(647\) 6.93207i 0.272528i 0.990673 + 0.136264i \(0.0435095\pi\)
−0.990673 + 0.136264i \(0.956490\pi\)
\(648\) 0 0
\(649\) 2.44851 0.0961123
\(650\) 0 0
\(651\) 17.2506i 0.676104i
\(652\) 0 0
\(653\) 29.7381 1.16374 0.581872 0.813281i \(-0.302321\pi\)
0.581872 + 0.813281i \(0.302321\pi\)
\(654\) 0 0
\(655\) 22.3488 25.2750i 0.873242 0.987577i
\(656\) 0 0
\(657\) 30.7612i 1.20011i
\(658\) 0 0
\(659\) 24.3879i 0.950017i 0.879981 + 0.475008i \(0.157555\pi\)
−0.879981 + 0.475008i \(0.842445\pi\)
\(660\) 0 0
\(661\) 34.1378i 1.32781i 0.747819 + 0.663903i \(0.231101\pi\)
−0.747819 + 0.663903i \(0.768899\pi\)
\(662\) 0 0
\(663\) 1.25060i 0.0485693i
\(664\) 0 0
\(665\) −22.8021 20.1622i −0.884227 0.781857i
\(666\) 0 0
\(667\) −12.3127 −0.476748
\(668\) 0 0
\(669\) 18.1603i 0.702118i
\(670\) 0 0
\(671\) 0.101570 0.00392108
\(672\) 0 0
\(673\) 23.6385i 0.911196i 0.890186 + 0.455598i \(0.150575\pi\)
−0.890186 + 0.455598i \(0.849425\pi\)
\(674\) 0 0
\(675\) 2.63989 + 21.4010i 0.101609 + 0.823727i
\(676\) 0 0
\(677\) −2.88717 −0.110963 −0.0554814 0.998460i \(-0.517669\pi\)
−0.0554814 + 0.998460i \(0.517669\pi\)
\(678\) 0 0
\(679\) 20.7269 0.795424
\(680\) 0 0
\(681\) 1.90034 0.0728212
\(682\) 0 0
\(683\) −13.7440 −0.525900 −0.262950 0.964809i \(-0.584695\pi\)
−0.262950 + 0.964809i \(0.584695\pi\)
\(684\) 0 0
\(685\) −27.4763 + 31.0738i −1.04981 + 1.18727i
\(686\) 0 0
\(687\) 2.86273i 0.109220i
\(688\) 0 0
\(689\) 8.62530 0.328598
\(690\) 0 0
\(691\) 31.5633i 1.20072i −0.799729 0.600361i \(-0.795023\pi\)
0.799729 0.600361i \(-0.204977\pi\)
\(692\) 0 0
\(693\) 1.96571 0.0746713
\(694\) 0 0
\(695\) −11.2750 + 12.7513i −0.427687 + 0.483685i
\(696\) 0 0
\(697\) 7.37470i 0.279337i
\(698\) 0 0
\(699\) 10.5501i 0.399041i
\(700\) 0 0
\(701\) 23.8397i 0.900413i −0.892924 0.450207i \(-0.851350\pi\)
0.892924 0.450207i \(-0.148650\pi\)
\(702\) 0 0
\(703\) 41.5026i 1.56530i
\(704\) 0 0
\(705\) −5.42548 + 6.13586i −0.204336 + 0.231090i
\(706\) 0 0
\(707\) −24.5844 −0.924590
\(708\) 0 0
\(709\) 21.8496i 0.820577i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(710\) 0 0
\(711\) −4.52373 −0.169653
\(712\) 0 0
\(713\) 61.1002i 2.28822i
\(714\) 0 0
\(715\) 0.552907 0.625301i 0.0206776 0.0233849i
\(716\) 0 0
\(717\) 12.3536 0.461353
\(718\) 0 0
\(719\) 33.9248 1.26518 0.632590 0.774487i \(-0.281992\pi\)
0.632590 + 0.774487i \(0.281992\pi\)
\(720\) 0 0
\(721\) −23.5731 −0.877908
\(722\) 0 0
\(723\) −1.71370 −0.0637331
\(724\) 0 0
\(725\) −9.92478 + 1.22425i −0.368597 + 0.0454676i
\(726\) 0 0
\(727\) 2.78163i 0.103165i 0.998669 + 0.0515824i \(0.0164265\pi\)
−0.998669 + 0.0515824i \(0.983574\pi\)
\(728\) 0 0
\(729\) 2.02776 0.0751023
\(730\) 0 0
\(731\) 18.5501i 0.686099i
\(732\) 0 0
\(733\) −4.90175 −0.181050 −0.0905252 0.995894i \(-0.528855\pi\)
−0.0905252 + 0.995894i \(0.528855\pi\)
\(734\) 0 0
\(735\) 3.17347 + 2.80606i 0.117055 + 0.103503i
\(736\) 0 0
\(737\) 3.83780i 0.141367i
\(738\) 0 0
\(739\) 25.9102i 0.953122i −0.879141 0.476561i \(-0.841883\pi\)
0.879141 0.476561i \(-0.158117\pi\)
\(740\) 0 0
\(741\) 4.89701i 0.179896i
\(742\) 0 0
\(743\) 9.06793i 0.332670i −0.986069 0.166335i \(-0.946807\pi\)
0.986069 0.166335i \(-0.0531933\pi\)
\(744\) 0 0
\(745\) 20.2374 22.8872i 0.741442 0.838521i
\(746\) 0 0
\(747\) 6.77178 0.247766
\(748\) 0 0
\(749\) 23.8134i 0.870121i
\(750\) 0 0
\(751\) −11.8496 −0.432396 −0.216198 0.976349i \(-0.569366\pi\)
−0.216198 + 0.976349i \(0.569366\pi\)
\(752\) 0 0
\(753\) 21.5106i 0.783888i
\(754\) 0 0
\(755\) −25.5026 22.5501i −0.928135 0.820681i
\(756\) 0 0
\(757\) 2.05079 0.0745371 0.0372685 0.999305i \(-0.488134\pi\)
0.0372685 + 0.999305i \(0.488134\pi\)
\(758\) 0 0
\(759\) 1.92478 0.0698650
\(760\) 0 0
\(761\) −25.0738 −0.908925 −0.454462 0.890766i \(-0.650169\pi\)
−0.454462 + 0.890766i \(0.650169\pi\)
\(762\) 0 0
\(763\) 10.8627 0.393257
\(764\) 0 0
\(765\) 5.61213 6.34694i 0.202907 0.229474i
\(766\) 0 0
\(767\) 6.07522i 0.219364i
\(768\) 0 0
\(769\) 14.7466 0.531775 0.265887 0.964004i \(-0.414335\pi\)
0.265887 + 0.964004i \(0.414335\pi\)
\(770\) 0 0
\(771\) 4.35359i 0.156791i
\(772\) 0 0
\(773\) −10.8872 −0.391584 −0.195792 0.980645i \(-0.562728\pi\)
−0.195792 + 0.980645i \(0.562728\pi\)
\(774\) 0 0
\(775\) 6.07522 + 49.2506i 0.218228 + 1.76913i
\(776\) 0 0
\(777\) 11.4274i 0.409955i
\(778\) 0 0
\(779\) 28.8773i 1.03464i
\(780\) 0 0
\(781\) 4.15045i 0.148515i
\(782\) 0 0
\(783\) 8.62530i 0.308243i
\(784\) 0 0
\(785\) −2.38787 2.11142i −0.0852268 0.0753598i
\(786\) 0 0
\(787\) 36.8178 1.31241 0.656207 0.754581i \(-0.272160\pi\)
0.656207 + 0.754581i \(0.272160\pi\)
\(788\) 0 0
\(789\) 11.6629i 0.415211i
\(790\) 0 0
\(791\) −42.8021 −1.52187
\(792\) 0 0
\(793\) 0.252016i 0.00894935i
\(794\) 0 0
\(795\) 12.1016 + 10.7005i 0.429198 + 0.379508i
\(796\) 0 0
\(797\) 12.8119 0.453822 0.226911 0.973915i \(-0.427137\pi\)
0.226911 + 0.973915i \(0.427137\pi\)
\(798\) 0 0
\(799\) 7.32582 0.259169
\(800\) 0 0
\(801\) −24.9722 −0.882351
\(802\) 0 0
\(803\) 5.07664 0.179151
\(804\) 0 0
\(805\) 22.2374 + 19.6629i 0.783766 + 0.693027i
\(806\) 0 0
\(807\) 22.6596i 0.797655i
\(808\) 0 0
\(809\) 16.2981 0.573009 0.286505 0.958079i \(-0.407507\pi\)
0.286505 + 0.958079i \(0.407507\pi\)
\(810\) 0 0
\(811\) 2.21108i 0.0776415i −0.999246 0.0388208i \(-0.987640\pi\)
0.999246 0.0388208i \(-0.0123601\pi\)
\(812\) 0 0
\(813\) −20.0527 −0.703279
\(814\) 0 0
\(815\) 14.7513 + 13.0435i 0.516716 + 0.456894i
\(816\) 0 0
\(817\) 72.6371i 2.54125i
\(818\) 0 0
\(819\) 4.87732i 0.170427i
\(820\) 0 0
\(821\) 50.6615i 1.76810i −0.467394 0.884049i \(-0.654807\pi\)
0.467394 0.884049i \(-0.345193\pi\)
\(822\) 0 0
\(823\) 0.917483i 0.0319814i −0.999872 0.0159907i \(-0.994910\pi\)
0.999872 0.0159907i \(-0.00509022\pi\)
\(824\) 0 0
\(825\) 1.55149 0.191382i 0.0540160 0.00666305i
\(826\) 0 0
\(827\) −10.5198 −0.365808 −0.182904 0.983131i \(-0.558550\pi\)
−0.182904 + 0.983131i \(0.558550\pi\)
\(828\) 0 0
\(829\) 38.6907i 1.34378i 0.740650 + 0.671891i \(0.234518\pi\)
−0.740650 + 0.671891i \(0.765482\pi\)
\(830\) 0 0
\(831\) 16.1016 0.558557
\(832\) 0 0
\(833\) 3.78892i 0.131278i
\(834\) 0 0
\(835\) −15.5066 + 17.5369i −0.536628 + 0.606890i
\(836\) 0 0
\(837\) 42.8021 1.47946
\(838\) 0 0
\(839\) −10.0263 −0.346148 −0.173074 0.984909i \(-0.555370\pi\)
−0.173074 + 0.984909i \(0.555370\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) −11.6873 −0.402534
\(844\) 0 0
\(845\) 20.2252 + 17.8837i 0.695768 + 0.615216i
\(846\) 0 0
\(847\) 23.3952i 0.803867i
\(848\) 0 0
\(849\) −7.09825 −0.243611
\(850\) 0 0
\(851\) 40.4749i 1.38746i
\(852\) 0 0
\(853\) −49.5388 −1.69618 −0.848088 0.529855i \(-0.822246\pi\)
−0.848088 + 0.529855i \(0.822246\pi\)
\(854\) 0 0
\(855\) 21.9756 24.8529i 0.751548 0.849951i
\(856\) 0 0
\(857\) 43.4763i 1.48512i −0.669779 0.742561i \(-0.733611\pi\)
0.669779 0.742561i \(-0.266389\pi\)
\(858\) 0 0
\(859\) 23.5633i 0.803968i −0.915647 0.401984i \(-0.868321\pi\)
0.915647 0.401984i \(-0.131679\pi\)
\(860\) 0 0
\(861\) 7.95112i 0.270974i
\(862\) 0 0
\(863\) 13.3317i 0.453816i −0.973916 0.226908i \(-0.927138\pi\)
0.973916 0.226908i \(-0.0728617\pi\)
\(864\) 0 0
\(865\) 33.4617 + 29.5877i 1.13773 + 1.00601i
\(866\) 0 0
\(867\) −11.6082 −0.394234
\(868\) 0 0
\(869\) 0.746569i 0.0253256i
\(870\) 0 0
\(871\) 9.52232 0.322651
\(872\) 0 0
\(873\) 22.5910i 0.764590i
\(874\) 0 0
\(875\) 19.8799 + 13.6385i 0.672062 + 0.461065i
\(876\) 0 0
\(877\) −13.6483 −0.460871 −0.230436 0.973088i \(-0.574015\pi\)
−0.230436 + 0.973088i \(0.574015\pi\)
\(878\) 0 0
\(879\) 2.70052 0.0910864
\(880\) 0 0
\(881\) 16.3028 0.549255 0.274628 0.961551i \(-0.411445\pi\)
0.274628 + 0.961551i \(0.411445\pi\)
\(882\) 0 0
\(883\) −13.5818 −0.457064 −0.228532 0.973536i \(-0.573393\pi\)
−0.228532 + 0.973536i \(0.573393\pi\)
\(884\) 0 0
\(885\) 7.53690 8.52373i 0.253350 0.286522i
\(886\) 0 0
\(887\) 37.0797i 1.24501i 0.782614 + 0.622507i \(0.213886\pi\)
−0.782614 + 0.622507i \(0.786114\pi\)
\(888\) 0 0
\(889\) −25.3766 −0.851104
\(890\) 0 0
\(891\) 1.38646i 0.0464481i
\(892\) 0 0
\(893\) 28.6859 0.959938
\(894\) 0 0
\(895\) 18.1260 20.4993i 0.605886 0.685216i
\(896\) 0 0
\(897\) 4.77575i 0.159458i
\(898\) 0 0
\(899\) 19.8496i 0.662020i
\(900\) 0 0
\(901\) 14.4485i 0.481350i
\(902\) 0 0
\(903\) 20.0000i 0.665558i
\(904\) 0 0
\(905\) −20.5139 + 23.1998i −0.681904 + 0.771188i
\(906\) 0 0
\(907\) 35.7294 1.18638 0.593188 0.805064i \(-0.297869\pi\)
0.593188 + 0.805064i \(0.297869\pi\)
\(908\) 0 0
\(909\) 26.7954i 0.888749i
\(910\) 0 0
\(911\) −20.5237 −0.679982 −0.339991 0.940429i \(-0.610424\pi\)
−0.339991 + 0.940429i \(0.610424\pi\)
\(912\) 0 0
\(913\) 1.11757i 0.0369863i
\(914\) 0 0
\(915\) 0.312650 0.353586i 0.0103359 0.0116892i
\(916\) 0 0
\(917\) −32.5355 −1.07442
\(918\) 0 0
\(919\) −20.9986 −0.692679 −0.346340 0.938109i \(-0.612576\pi\)
−0.346340 + 0.938109i \(0.612576\pi\)
\(920\) 0 0
\(921\) 1.57593 0.0519287
\(922\) 0 0
\(923\) 10.2981 0.338965
\(924\) 0 0
\(925\) −4.02444 32.6253i −0.132323 1.07271i
\(926\) 0 0
\(927\) 25.6932i 0.843876i
\(928\) 0 0
\(929\) 31.5271 1.03437 0.517185 0.855874i \(-0.326980\pi\)
0.517185 + 0.855874i \(0.326980\pi\)
\(930\) 0 0
\(931\) 14.8364i 0.486243i
\(932\) 0 0
\(933\) −7.07381 −0.231586
\(934\) 0 0
\(935\) −1.04746 0.926192i −0.0342556 0.0302897i
\(936\) 0 0
\(937\) 17.8641i 0.583596i 0.956480 + 0.291798i \(0.0942535\pi\)
−0.956480 + 0.291798i \(0.905746\pi\)
\(938\) 0 0
\(939\) 14.9525i 0.487958i
\(940\) 0 0
\(941\) 45.5487i 1.48484i 0.669933 + 0.742422i \(0.266323\pi\)
−0.669933 + 0.742422i \(0.733677\pi\)
\(942\) 0 0
\(943\) 28.1622i 0.917088i
\(944\) 0 0
\(945\) 13.7743 15.5778i 0.448079 0.506747i
\(946\) 0 0
\(947\) −43.2057 −1.40400 −0.701998 0.712179i \(-0.747709\pi\)
−0.701998 + 0.712179i \(0.747709\pi\)
\(948\) 0 0
\(949\) 12.5961i 0.408887i
\(950\) 0 0
\(951\) −7.72829 −0.250607
\(952\) 0 0
\(953\) 10.9722i 0.355426i 0.984082 + 0.177713i \(0.0568698\pi\)
−0.984082 + 0.177713i \(0.943130\pi\)
\(954\) 0 0
\(955\) 6.44851 + 5.70194i 0.208669 + 0.184510i
\(956\) 0 0
\(957\) 0.625301 0.0202131
\(958\) 0 0
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) 67.5012 2.17746
\(962\) 0 0
\(963\) 25.9551 0.836391
\(964\) 0 0
\(965\) 14.2374 16.1016i 0.458319 0.518328i
\(966\) 0 0
\(967\) 44.1417i 1.41950i −0.704452 0.709751i \(-0.748807\pi\)
0.704452 0.709751i \(-0.251193\pi\)
\(968\) 0 0
\(969\) 8.20314 0.263523
\(970\) 0 0
\(971\) 20.1886i 0.647881i 0.946077 + 0.323941i \(0.105008\pi\)
−0.946077 + 0.323941i \(0.894992\pi\)
\(972\) 0 0
\(973\) 16.4142 0.526216
\(974\) 0 0
\(975\) −0.474855 3.84955i −0.0152075 0.123284i
\(976\) 0 0
\(977\) 1.73340i 0.0554562i −0.999616 0.0277281i \(-0.991173\pi\)
0.999616 0.0277281i \(-0.00882727\pi\)
\(978\) 0 0
\(979\) 4.12127i 0.131716i
\(980\) 0 0
\(981\) 11.8397i 0.378013i
\(982\) 0 0
\(983\) 14.0059i 0.446718i 0.974736 + 0.223359i \(0.0717023\pi\)
−0.974736 + 0.223359i \(0.928298\pi\)
\(984\) 0 0
\(985\) −16.0606 14.2012i −0.511734 0.452489i
\(986\) 0 0
\(987\) 7.89843 0.251410
\(988\) 0 0
\(989\) 70.8383i 2.25253i
\(990\) 0 0
\(991\) −42.8021 −1.35965 −0.679827 0.733373i \(-0.737945\pi\)
−0.679827 + 0.733373i \(0.737945\pi\)
\(992\) 0 0
\(993\) 18.6107i 0.590593i
\(994\) 0 0
\(995\) 8.87732 + 7.84955i 0.281430 + 0.248848i
\(996\) 0 0
\(997\) −42.8872 −1.35825 −0.679125 0.734023i \(-0.737641\pi\)
−0.679125 + 0.734023i \(0.737641\pi\)
\(998\) 0 0
\(999\) −28.3536 −0.897068
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.f.i.129.3 6
4.3 odd 2 1280.2.f.k.129.3 6
5.4 even 2 1280.2.f.l.129.3 6
8.3 odd 2 1280.2.f.j.129.4 6
8.5 even 2 1280.2.f.l.129.4 6
16.3 odd 4 640.2.c.c.129.4 yes 6
16.5 even 4 640.2.c.a.129.4 yes 6
16.11 odd 4 640.2.c.b.129.3 yes 6
16.13 even 4 640.2.c.d.129.3 yes 6
20.19 odd 2 1280.2.f.j.129.3 6
40.19 odd 2 1280.2.f.k.129.4 6
40.29 even 2 inner 1280.2.f.i.129.4 6
80.3 even 4 3200.2.a.bo.1.2 3
80.13 odd 4 3200.2.a.bv.1.2 3
80.19 odd 4 640.2.c.c.129.3 yes 6
80.27 even 4 3200.2.a.br.1.2 3
80.29 even 4 640.2.c.d.129.4 yes 6
80.37 odd 4 3200.2.a.bs.1.2 3
80.43 even 4 3200.2.a.bt.1.2 3
80.53 odd 4 3200.2.a.bq.1.2 3
80.59 odd 4 640.2.c.b.129.4 yes 6
80.67 even 4 3200.2.a.bu.1.2 3
80.69 even 4 640.2.c.a.129.3 6
80.77 odd 4 3200.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.3 6 80.69 even 4
640.2.c.a.129.4 yes 6 16.5 even 4
640.2.c.b.129.3 yes 6 16.11 odd 4
640.2.c.b.129.4 yes 6 80.59 odd 4
640.2.c.c.129.3 yes 6 80.19 odd 4
640.2.c.c.129.4 yes 6 16.3 odd 4
640.2.c.d.129.3 yes 6 16.13 even 4
640.2.c.d.129.4 yes 6 80.29 even 4
1280.2.f.i.129.3 6 1.1 even 1 trivial
1280.2.f.i.129.4 6 40.29 even 2 inner
1280.2.f.j.129.3 6 20.19 odd 2
1280.2.f.j.129.4 6 8.3 odd 2
1280.2.f.k.129.3 6 4.3 odd 2
1280.2.f.k.129.4 6 40.19 odd 2
1280.2.f.l.129.3 6 5.4 even 2
1280.2.f.l.129.4 6 8.5 even 2
3200.2.a.bo.1.2 3 80.3 even 4
3200.2.a.bp.1.2 3 80.77 odd 4
3200.2.a.bq.1.2 3 80.53 odd 4
3200.2.a.br.1.2 3 80.27 even 4
3200.2.a.bs.1.2 3 80.37 odd 4
3200.2.a.bt.1.2 3 80.43 even 4
3200.2.a.bu.1.2 3 80.67 even 4
3200.2.a.bv.1.2 3 80.13 odd 4