Properties

Label 256.4.b.b.129.1
Level $256$
Weight $4$
Character 256.129
Analytic conductor $15.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,4,Mod(129,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.4.b.b.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +6.00000i q^{5} -20.0000 q^{7} +23.0000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +6.00000i q^{5} -20.0000 q^{7} +23.0000 q^{9} +14.0000i q^{11} -54.0000i q^{13} +12.0000 q^{15} -66.0000 q^{17} -162.000i q^{19} +40.0000i q^{21} -172.000 q^{23} +89.0000 q^{25} -100.000i q^{27} +2.00000i q^{29} -128.000 q^{31} +28.0000 q^{33} -120.000i q^{35} +158.000i q^{37} -108.000 q^{39} -202.000 q^{41} -298.000i q^{43} +138.000i q^{45} -408.000 q^{47} +57.0000 q^{49} +132.000i q^{51} -690.000i q^{53} -84.0000 q^{55} -324.000 q^{57} -322.000i q^{59} +298.000i q^{61} -460.000 q^{63} +324.000 q^{65} -202.000i q^{67} +344.000i q^{69} +700.000 q^{71} +418.000 q^{73} -178.000i q^{75} -280.000i q^{77} +744.000 q^{79} +421.000 q^{81} +678.000i q^{83} -396.000i q^{85} +4.00000 q^{87} +82.0000 q^{89} +1080.00i q^{91} +256.000i q^{93} +972.000 q^{95} -1122.00 q^{97} +322.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 40 q^{7} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 40 q^{7} + 46 q^{9} + 24 q^{15} - 132 q^{17} - 344 q^{23} + 178 q^{25} - 256 q^{31} + 56 q^{33} - 216 q^{39} - 404 q^{41} - 816 q^{47} + 114 q^{49} - 168 q^{55} - 648 q^{57} - 920 q^{63} + 648 q^{65} + 1400 q^{71} + 836 q^{73} + 1488 q^{79} + 842 q^{81} + 8 q^{87} + 164 q^{89} + 1944 q^{95} - 2244 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 2.00000i − 0.384900i −0.981307 0.192450i \(-0.938357\pi\)
0.981307 0.192450i \(-0.0616434\pi\)
\(4\) 0 0
\(5\) 6.00000i 0.536656i 0.963328 + 0.268328i \(0.0864711\pi\)
−0.963328 + 0.268328i \(0.913529\pi\)
\(6\) 0 0
\(7\) −20.0000 −1.07990 −0.539949 0.841698i \(-0.681557\pi\)
−0.539949 + 0.841698i \(0.681557\pi\)
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) 14.0000i 0.383742i 0.981420 + 0.191871i \(0.0614555\pi\)
−0.981420 + 0.191871i \(0.938545\pi\)
\(12\) 0 0
\(13\) − 54.0000i − 1.15207i −0.817425 0.576035i \(-0.804599\pi\)
0.817425 0.576035i \(-0.195401\pi\)
\(14\) 0 0
\(15\) 12.0000 0.206559
\(16\) 0 0
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) − 162.000i − 1.95607i −0.208438 0.978035i \(-0.566838\pi\)
0.208438 0.978035i \(-0.433162\pi\)
\(20\) 0 0
\(21\) 40.0000i 0.415653i
\(22\) 0 0
\(23\) −172.000 −1.55933 −0.779663 0.626200i \(-0.784609\pi\)
−0.779663 + 0.626200i \(0.784609\pi\)
\(24\) 0 0
\(25\) 89.0000 0.712000
\(26\) 0 0
\(27\) − 100.000i − 0.712778i
\(28\) 0 0
\(29\) 2.00000i 0.0128066i 0.999979 + 0.00640329i \(0.00203824\pi\)
−0.999979 + 0.00640329i \(0.997962\pi\)
\(30\) 0 0
\(31\) −128.000 −0.741596 −0.370798 0.928714i \(-0.620916\pi\)
−0.370798 + 0.928714i \(0.620916\pi\)
\(32\) 0 0
\(33\) 28.0000 0.147702
\(34\) 0 0
\(35\) − 120.000i − 0.579534i
\(36\) 0 0
\(37\) 158.000i 0.702028i 0.936370 + 0.351014i \(0.114163\pi\)
−0.936370 + 0.351014i \(0.885837\pi\)
\(38\) 0 0
\(39\) −108.000 −0.443432
\(40\) 0 0
\(41\) −202.000 −0.769441 −0.384721 0.923033i \(-0.625702\pi\)
−0.384721 + 0.923033i \(0.625702\pi\)
\(42\) 0 0
\(43\) − 298.000i − 1.05685i −0.848980 0.528425i \(-0.822783\pi\)
0.848980 0.528425i \(-0.177217\pi\)
\(44\) 0 0
\(45\) 138.000i 0.457152i
\(46\) 0 0
\(47\) −408.000 −1.26623 −0.633116 0.774057i \(-0.718224\pi\)
−0.633116 + 0.774057i \(0.718224\pi\)
\(48\) 0 0
\(49\) 57.0000 0.166181
\(50\) 0 0
\(51\) 132.000i 0.362425i
\(52\) 0 0
\(53\) − 690.000i − 1.78828i −0.447788 0.894140i \(-0.647788\pi\)
0.447788 0.894140i \(-0.352212\pi\)
\(54\) 0 0
\(55\) −84.0000 −0.205937
\(56\) 0 0
\(57\) −324.000 −0.752892
\(58\) 0 0
\(59\) − 322.000i − 0.710523i −0.934767 0.355261i \(-0.884392\pi\)
0.934767 0.355261i \(-0.115608\pi\)
\(60\) 0 0
\(61\) 298.000i 0.625492i 0.949837 + 0.312746i \(0.101249\pi\)
−0.949837 + 0.312746i \(0.898751\pi\)
\(62\) 0 0
\(63\) −460.000 −0.919914
\(64\) 0 0
\(65\) 324.000 0.618265
\(66\) 0 0
\(67\) − 202.000i − 0.368332i −0.982895 0.184166i \(-0.941042\pi\)
0.982895 0.184166i \(-0.0589584\pi\)
\(68\) 0 0
\(69\) 344.000i 0.600185i
\(70\) 0 0
\(71\) 700.000 1.17007 0.585033 0.811009i \(-0.301081\pi\)
0.585033 + 0.811009i \(0.301081\pi\)
\(72\) 0 0
\(73\) 418.000 0.670181 0.335090 0.942186i \(-0.391233\pi\)
0.335090 + 0.942186i \(0.391233\pi\)
\(74\) 0 0
\(75\) − 178.000i − 0.274049i
\(76\) 0 0
\(77\) − 280.000i − 0.414402i
\(78\) 0 0
\(79\) 744.000 1.05958 0.529788 0.848130i \(-0.322271\pi\)
0.529788 + 0.848130i \(0.322271\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) 678.000i 0.896629i 0.893876 + 0.448314i \(0.147975\pi\)
−0.893876 + 0.448314i \(0.852025\pi\)
\(84\) 0 0
\(85\) − 396.000i − 0.505320i
\(86\) 0 0
\(87\) 4.00000 0.00492925
\(88\) 0 0
\(89\) 82.0000 0.0976627 0.0488314 0.998807i \(-0.484450\pi\)
0.0488314 + 0.998807i \(0.484450\pi\)
\(90\) 0 0
\(91\) 1080.00i 1.24412i
\(92\) 0 0
\(93\) 256.000i 0.285440i
\(94\) 0 0
\(95\) 972.000 1.04974
\(96\) 0 0
\(97\) −1122.00 −1.17445 −0.587226 0.809423i \(-0.699780\pi\)
−0.587226 + 0.809423i \(0.699780\pi\)
\(98\) 0 0
\(99\) 322.000i 0.326891i
\(100\) 0 0
\(101\) 1390.00i 1.36941i 0.728821 + 0.684704i \(0.240068\pi\)
−0.728821 + 0.684704i \(0.759932\pi\)
\(102\) 0 0
\(103\) −788.000 −0.753825 −0.376912 0.926249i \(-0.623014\pi\)
−0.376912 + 0.926249i \(0.623014\pi\)
\(104\) 0 0
\(105\) −240.000 −0.223063
\(106\) 0 0
\(107\) 1614.00i 1.45824i 0.684388 + 0.729118i \(0.260070\pi\)
−0.684388 + 0.729118i \(0.739930\pi\)
\(108\) 0 0
\(109\) − 2014.00i − 1.76978i −0.465798 0.884891i \(-0.654233\pi\)
0.465798 0.884891i \(-0.345767\pi\)
\(110\) 0 0
\(111\) 316.000 0.270211
\(112\) 0 0
\(113\) −542.000 −0.451213 −0.225607 0.974219i \(-0.572436\pi\)
−0.225607 + 0.974219i \(0.572436\pi\)
\(114\) 0 0
\(115\) − 1032.00i − 0.836822i
\(116\) 0 0
\(117\) − 1242.00i − 0.981393i
\(118\) 0 0
\(119\) 1320.00 1.01684
\(120\) 0 0
\(121\) 1135.00 0.852742
\(122\) 0 0
\(123\) 404.000i 0.296158i
\(124\) 0 0
\(125\) 1284.00i 0.918756i
\(126\) 0 0
\(127\) 1712.00 1.19618 0.598092 0.801427i \(-0.295926\pi\)
0.598092 + 0.801427i \(0.295926\pi\)
\(128\) 0 0
\(129\) −596.000 −0.406782
\(130\) 0 0
\(131\) 2118.00i 1.41260i 0.707913 + 0.706300i \(0.249637\pi\)
−0.707913 + 0.706300i \(0.750363\pi\)
\(132\) 0 0
\(133\) 3240.00i 2.11236i
\(134\) 0 0
\(135\) 600.000 0.382517
\(136\) 0 0
\(137\) 486.000 0.303079 0.151539 0.988451i \(-0.451577\pi\)
0.151539 + 0.988451i \(0.451577\pi\)
\(138\) 0 0
\(139\) 1286.00i 0.784727i 0.919810 + 0.392364i \(0.128343\pi\)
−0.919810 + 0.392364i \(0.871657\pi\)
\(140\) 0 0
\(141\) 816.000i 0.487373i
\(142\) 0 0
\(143\) 756.000 0.442097
\(144\) 0 0
\(145\) −12.0000 −0.00687273
\(146\) 0 0
\(147\) − 114.000i − 0.0639630i
\(148\) 0 0
\(149\) − 2666.00i − 1.46582i −0.680325 0.732910i \(-0.738162\pi\)
0.680325 0.732910i \(-0.261838\pi\)
\(150\) 0 0
\(151\) −172.000 −0.0926964 −0.0463482 0.998925i \(-0.514758\pi\)
−0.0463482 + 0.998925i \(0.514758\pi\)
\(152\) 0 0
\(153\) −1518.00 −0.802111
\(154\) 0 0
\(155\) − 768.000i − 0.397982i
\(156\) 0 0
\(157\) − 838.000i − 0.425985i −0.977054 0.212993i \(-0.931679\pi\)
0.977054 0.212993i \(-0.0683210\pi\)
\(158\) 0 0
\(159\) −1380.00 −0.688309
\(160\) 0 0
\(161\) 3440.00 1.68391
\(162\) 0 0
\(163\) − 1346.00i − 0.646791i −0.946264 0.323395i \(-0.895176\pi\)
0.946264 0.323395i \(-0.104824\pi\)
\(164\) 0 0
\(165\) 168.000i 0.0792653i
\(166\) 0 0
\(167\) 1052.00 0.487462 0.243731 0.969843i \(-0.421629\pi\)
0.243731 + 0.969843i \(0.421629\pi\)
\(168\) 0 0
\(169\) −719.000 −0.327264
\(170\) 0 0
\(171\) − 3726.00i − 1.66628i
\(172\) 0 0
\(173\) − 38.0000i − 0.0166999i −0.999965 0.00834996i \(-0.997342\pi\)
0.999965 0.00834996i \(-0.00265791\pi\)
\(174\) 0 0
\(175\) −1780.00 −0.768888
\(176\) 0 0
\(177\) −644.000 −0.273480
\(178\) 0 0
\(179\) 2790.00i 1.16500i 0.812832 + 0.582498i \(0.197925\pi\)
−0.812832 + 0.582498i \(0.802075\pi\)
\(180\) 0 0
\(181\) − 3418.00i − 1.40364i −0.712357 0.701818i \(-0.752372\pi\)
0.712357 0.701818i \(-0.247628\pi\)
\(182\) 0 0
\(183\) 596.000 0.240752
\(184\) 0 0
\(185\) −948.000 −0.376748
\(186\) 0 0
\(187\) − 924.000i − 0.361335i
\(188\) 0 0
\(189\) 2000.00i 0.769728i
\(190\) 0 0
\(191\) −1968.00 −0.745547 −0.372774 0.927922i \(-0.621593\pi\)
−0.372774 + 0.927922i \(0.621593\pi\)
\(192\) 0 0
\(193\) −1058.00 −0.394593 −0.197297 0.980344i \(-0.563216\pi\)
−0.197297 + 0.980344i \(0.563216\pi\)
\(194\) 0 0
\(195\) − 648.000i − 0.237970i
\(196\) 0 0
\(197\) 726.000i 0.262565i 0.991345 + 0.131283i \(0.0419095\pi\)
−0.991345 + 0.131283i \(0.958090\pi\)
\(198\) 0 0
\(199\) −4116.00 −1.46621 −0.733104 0.680116i \(-0.761929\pi\)
−0.733104 + 0.680116i \(0.761929\pi\)
\(200\) 0 0
\(201\) −404.000 −0.141771
\(202\) 0 0
\(203\) − 40.0000i − 0.0138298i
\(204\) 0 0
\(205\) − 1212.00i − 0.412926i
\(206\) 0 0
\(207\) −3956.00 −1.32831
\(208\) 0 0
\(209\) 2268.00 0.750626
\(210\) 0 0
\(211\) − 1482.00i − 0.483531i −0.970335 0.241766i \(-0.922273\pi\)
0.970335 0.241766i \(-0.0777265\pi\)
\(212\) 0 0
\(213\) − 1400.00i − 0.450359i
\(214\) 0 0
\(215\) 1788.00 0.567166
\(216\) 0 0
\(217\) 2560.00 0.800848
\(218\) 0 0
\(219\) − 836.000i − 0.257953i
\(220\) 0 0
\(221\) 3564.00i 1.08480i
\(222\) 0 0
\(223\) −896.000 −0.269061 −0.134530 0.990909i \(-0.542953\pi\)
−0.134530 + 0.990909i \(0.542953\pi\)
\(224\) 0 0
\(225\) 2047.00 0.606519
\(226\) 0 0
\(227\) − 3410.00i − 0.997047i −0.866876 0.498523i \(-0.833876\pi\)
0.866876 0.498523i \(-0.166124\pi\)
\(228\) 0 0
\(229\) 4502.00i 1.29913i 0.760307 + 0.649564i \(0.225049\pi\)
−0.760307 + 0.649564i \(0.774951\pi\)
\(230\) 0 0
\(231\) −560.000 −0.159503
\(232\) 0 0
\(233\) −2302.00 −0.647249 −0.323625 0.946186i \(-0.604901\pi\)
−0.323625 + 0.946186i \(0.604901\pi\)
\(234\) 0 0
\(235\) − 2448.00i − 0.679532i
\(236\) 0 0
\(237\) − 1488.00i − 0.407831i
\(238\) 0 0
\(239\) 4024.00 1.08908 0.544542 0.838734i \(-0.316704\pi\)
0.544542 + 0.838734i \(0.316704\pi\)
\(240\) 0 0
\(241\) −3586.00 −0.958484 −0.479242 0.877683i \(-0.659088\pi\)
−0.479242 + 0.877683i \(0.659088\pi\)
\(242\) 0 0
\(243\) − 3542.00i − 0.935059i
\(244\) 0 0
\(245\) 342.000i 0.0891820i
\(246\) 0 0
\(247\) −8748.00 −2.25353
\(248\) 0 0
\(249\) 1356.00 0.345112
\(250\) 0 0
\(251\) − 1250.00i − 0.314340i −0.987572 0.157170i \(-0.949763\pi\)
0.987572 0.157170i \(-0.0502370\pi\)
\(252\) 0 0
\(253\) − 2408.00i − 0.598378i
\(254\) 0 0
\(255\) −792.000 −0.194498
\(256\) 0 0
\(257\) −6638.00 −1.61116 −0.805578 0.592490i \(-0.798145\pi\)
−0.805578 + 0.592490i \(0.798145\pi\)
\(258\) 0 0
\(259\) − 3160.00i − 0.758119i
\(260\) 0 0
\(261\) 46.0000i 0.0109093i
\(262\) 0 0
\(263\) 1724.00 0.404207 0.202103 0.979364i \(-0.435222\pi\)
0.202103 + 0.979364i \(0.435222\pi\)
\(264\) 0 0
\(265\) 4140.00 0.959691
\(266\) 0 0
\(267\) − 164.000i − 0.0375904i
\(268\) 0 0
\(269\) − 4814.00i − 1.09113i −0.838068 0.545566i \(-0.816315\pi\)
0.838068 0.545566i \(-0.183685\pi\)
\(270\) 0 0
\(271\) −1640.00 −0.367612 −0.183806 0.982963i \(-0.558842\pi\)
−0.183806 + 0.982963i \(0.558842\pi\)
\(272\) 0 0
\(273\) 2160.00 0.478861
\(274\) 0 0
\(275\) 1246.00i 0.273224i
\(276\) 0 0
\(277\) 3982.00i 0.863737i 0.901937 + 0.431869i \(0.142146\pi\)
−0.901937 + 0.431869i \(0.857854\pi\)
\(278\) 0 0
\(279\) −2944.00 −0.631730
\(280\) 0 0
\(281\) −4126.00 −0.875931 −0.437965 0.898992i \(-0.644301\pi\)
−0.437965 + 0.898992i \(0.644301\pi\)
\(282\) 0 0
\(283\) 3446.00i 0.723828i 0.932211 + 0.361914i \(0.117877\pi\)
−0.932211 + 0.361914i \(0.882123\pi\)
\(284\) 0 0
\(285\) − 1944.00i − 0.404044i
\(286\) 0 0
\(287\) 4040.00 0.830919
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) 2244.00i 0.452047i
\(292\) 0 0
\(293\) − 1514.00i − 0.301873i −0.988543 0.150937i \(-0.951771\pi\)
0.988543 0.150937i \(-0.0482289\pi\)
\(294\) 0 0
\(295\) 1932.00 0.381306
\(296\) 0 0
\(297\) 1400.00 0.273523
\(298\) 0 0
\(299\) 9288.00i 1.79645i
\(300\) 0 0
\(301\) 5960.00i 1.14129i
\(302\) 0 0
\(303\) 2780.00 0.527085
\(304\) 0 0
\(305\) −1788.00 −0.335674
\(306\) 0 0
\(307\) − 5490.00i − 1.02062i −0.859990 0.510311i \(-0.829530\pi\)
0.859990 0.510311i \(-0.170470\pi\)
\(308\) 0 0
\(309\) 1576.00i 0.290147i
\(310\) 0 0
\(311\) 5556.00 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(312\) 0 0
\(313\) 2054.00 0.370923 0.185462 0.982652i \(-0.440622\pi\)
0.185462 + 0.982652i \(0.440622\pi\)
\(314\) 0 0
\(315\) − 2760.00i − 0.493677i
\(316\) 0 0
\(317\) − 2494.00i − 0.441883i −0.975287 0.220942i \(-0.929087\pi\)
0.975287 0.220942i \(-0.0709130\pi\)
\(318\) 0 0
\(319\) −28.0000 −0.00491442
\(320\) 0 0
\(321\) 3228.00 0.561275
\(322\) 0 0
\(323\) 10692.0i 1.84185i
\(324\) 0 0
\(325\) − 4806.00i − 0.820274i
\(326\) 0 0
\(327\) −4028.00 −0.681189
\(328\) 0 0
\(329\) 8160.00 1.36740
\(330\) 0 0
\(331\) − 2914.00i − 0.483891i −0.970290 0.241946i \(-0.922214\pi\)
0.970290 0.241946i \(-0.0777855\pi\)
\(332\) 0 0
\(333\) 3634.00i 0.598024i
\(334\) 0 0
\(335\) 1212.00 0.197668
\(336\) 0 0
\(337\) 7186.00 1.16156 0.580781 0.814060i \(-0.302747\pi\)
0.580781 + 0.814060i \(0.302747\pi\)
\(338\) 0 0
\(339\) 1084.00i 0.173672i
\(340\) 0 0
\(341\) − 1792.00i − 0.284581i
\(342\) 0 0
\(343\) 5720.00 0.900440
\(344\) 0 0
\(345\) −2064.00 −0.322093
\(346\) 0 0
\(347\) 4446.00i 0.687821i 0.939002 + 0.343910i \(0.111752\pi\)
−0.939002 + 0.343910i \(0.888248\pi\)
\(348\) 0 0
\(349\) − 430.000i − 0.0659524i −0.999456 0.0329762i \(-0.989501\pi\)
0.999456 0.0329762i \(-0.0104985\pi\)
\(350\) 0 0
\(351\) −5400.00 −0.821170
\(352\) 0 0
\(353\) 850.000 0.128161 0.0640806 0.997945i \(-0.479589\pi\)
0.0640806 + 0.997945i \(0.479589\pi\)
\(354\) 0 0
\(355\) 4200.00i 0.627924i
\(356\) 0 0
\(357\) − 2640.00i − 0.391383i
\(358\) 0 0
\(359\) 10988.0 1.61539 0.807694 0.589602i \(-0.200715\pi\)
0.807694 + 0.589602i \(0.200715\pi\)
\(360\) 0 0
\(361\) −19385.0 −2.82621
\(362\) 0 0
\(363\) − 2270.00i − 0.328221i
\(364\) 0 0
\(365\) 2508.00i 0.359657i
\(366\) 0 0
\(367\) 872.000 0.124027 0.0620137 0.998075i \(-0.480248\pi\)
0.0620137 + 0.998075i \(0.480248\pi\)
\(368\) 0 0
\(369\) −4646.00 −0.655450
\(370\) 0 0
\(371\) 13800.0i 1.93116i
\(372\) 0 0
\(373\) 3454.00i 0.479467i 0.970839 + 0.239734i \(0.0770601\pi\)
−0.970839 + 0.239734i \(0.922940\pi\)
\(374\) 0 0
\(375\) 2568.00 0.353629
\(376\) 0 0
\(377\) 108.000 0.0147541
\(378\) 0 0
\(379\) − 1490.00i − 0.201942i −0.994889 0.100971i \(-0.967805\pi\)
0.994889 0.100971i \(-0.0321950\pi\)
\(380\) 0 0
\(381\) − 3424.00i − 0.460412i
\(382\) 0 0
\(383\) 10240.0 1.36616 0.683080 0.730343i \(-0.260640\pi\)
0.683080 + 0.730343i \(0.260640\pi\)
\(384\) 0 0
\(385\) 1680.00 0.222392
\(386\) 0 0
\(387\) − 6854.00i − 0.900280i
\(388\) 0 0
\(389\) − 7458.00i − 0.972071i −0.873939 0.486035i \(-0.838443\pi\)
0.873939 0.486035i \(-0.161557\pi\)
\(390\) 0 0
\(391\) 11352.0 1.46827
\(392\) 0 0
\(393\) 4236.00 0.543710
\(394\) 0 0
\(395\) 4464.00i 0.568628i
\(396\) 0 0
\(397\) 4706.00i 0.594930i 0.954733 + 0.297465i \(0.0961412\pi\)
−0.954733 + 0.297465i \(0.903859\pi\)
\(398\) 0 0
\(399\) 6480.00 0.813047
\(400\) 0 0
\(401\) 5598.00 0.697134 0.348567 0.937284i \(-0.386668\pi\)
0.348567 + 0.937284i \(0.386668\pi\)
\(402\) 0 0
\(403\) 6912.00i 0.854370i
\(404\) 0 0
\(405\) 2526.00i 0.309921i
\(406\) 0 0
\(407\) −2212.00 −0.269397
\(408\) 0 0
\(409\) 838.000 0.101312 0.0506558 0.998716i \(-0.483869\pi\)
0.0506558 + 0.998716i \(0.483869\pi\)
\(410\) 0 0
\(411\) − 972.000i − 0.116655i
\(412\) 0 0
\(413\) 6440.00i 0.767292i
\(414\) 0 0
\(415\) −4068.00 −0.481181
\(416\) 0 0
\(417\) 2572.00 0.302042
\(418\) 0 0
\(419\) − 26.0000i − 0.00303146i −0.999999 0.00151573i \(-0.999518\pi\)
0.999999 0.00151573i \(-0.000482473\pi\)
\(420\) 0 0
\(421\) − 15578.0i − 1.80339i −0.432377 0.901693i \(-0.642325\pi\)
0.432377 0.901693i \(-0.357675\pi\)
\(422\) 0 0
\(423\) −9384.00 −1.07864
\(424\) 0 0
\(425\) −5874.00 −0.670426
\(426\) 0 0
\(427\) − 5960.00i − 0.675467i
\(428\) 0 0
\(429\) − 1512.00i − 0.170163i
\(430\) 0 0
\(431\) −6792.00 −0.759070 −0.379535 0.925177i \(-0.623916\pi\)
−0.379535 + 0.925177i \(0.623916\pi\)
\(432\) 0 0
\(433\) −9314.00 −1.03372 −0.516862 0.856069i \(-0.672900\pi\)
−0.516862 + 0.856069i \(0.672900\pi\)
\(434\) 0 0
\(435\) 24.0000i 0.00264531i
\(436\) 0 0
\(437\) 27864.0i 3.05015i
\(438\) 0 0
\(439\) 3828.00 0.416174 0.208087 0.978110i \(-0.433276\pi\)
0.208087 + 0.978110i \(0.433276\pi\)
\(440\) 0 0
\(441\) 1311.00 0.141561
\(442\) 0 0
\(443\) 15414.0i 1.65314i 0.562834 + 0.826570i \(0.309711\pi\)
−0.562834 + 0.826570i \(0.690289\pi\)
\(444\) 0 0
\(445\) 492.000i 0.0524113i
\(446\) 0 0
\(447\) −5332.00 −0.564195
\(448\) 0 0
\(449\) −3650.00 −0.383640 −0.191820 0.981430i \(-0.561439\pi\)
−0.191820 + 0.981430i \(0.561439\pi\)
\(450\) 0 0
\(451\) − 2828.00i − 0.295267i
\(452\) 0 0
\(453\) 344.000i 0.0356789i
\(454\) 0 0
\(455\) −6480.00 −0.667664
\(456\) 0 0
\(457\) 15862.0 1.62362 0.811809 0.583924i \(-0.198483\pi\)
0.811809 + 0.583924i \(0.198483\pi\)
\(458\) 0 0
\(459\) 6600.00i 0.671158i
\(460\) 0 0
\(461\) − 78.0000i − 0.00788031i −0.999992 0.00394015i \(-0.998746\pi\)
0.999992 0.00394015i \(-0.00125419\pi\)
\(462\) 0 0
\(463\) −4376.00 −0.439244 −0.219622 0.975585i \(-0.570482\pi\)
−0.219622 + 0.975585i \(0.570482\pi\)
\(464\) 0 0
\(465\) −1536.00 −0.153183
\(466\) 0 0
\(467\) − 13714.0i − 1.35890i −0.733720 0.679452i \(-0.762218\pi\)
0.733720 0.679452i \(-0.237782\pi\)
\(468\) 0 0
\(469\) 4040.00i 0.397761i
\(470\) 0 0
\(471\) −1676.00 −0.163962
\(472\) 0 0
\(473\) 4172.00 0.405558
\(474\) 0 0
\(475\) − 14418.0i − 1.39272i
\(476\) 0 0
\(477\) − 15870.0i − 1.52335i
\(478\) 0 0
\(479\) 19072.0 1.81925 0.909626 0.415428i \(-0.136368\pi\)
0.909626 + 0.415428i \(0.136368\pi\)
\(480\) 0 0
\(481\) 8532.00 0.808785
\(482\) 0 0
\(483\) − 6880.00i − 0.648138i
\(484\) 0 0
\(485\) − 6732.00i − 0.630277i
\(486\) 0 0
\(487\) −19428.0 −1.80773 −0.903867 0.427813i \(-0.859284\pi\)
−0.903867 + 0.427813i \(0.859284\pi\)
\(488\) 0 0
\(489\) −2692.00 −0.248950
\(490\) 0 0
\(491\) − 2490.00i − 0.228864i −0.993431 0.114432i \(-0.963495\pi\)
0.993431 0.114432i \(-0.0365048\pi\)
\(492\) 0 0
\(493\) − 132.000i − 0.0120588i
\(494\) 0 0
\(495\) −1932.00 −0.175428
\(496\) 0 0
\(497\) −14000.0 −1.26355
\(498\) 0 0
\(499\) − 2826.00i − 0.253525i −0.991933 0.126763i \(-0.959541\pi\)
0.991933 0.126763i \(-0.0404587\pi\)
\(500\) 0 0
\(501\) − 2104.00i − 0.187624i
\(502\) 0 0
\(503\) −2268.00 −0.201044 −0.100522 0.994935i \(-0.532051\pi\)
−0.100522 + 0.994935i \(0.532051\pi\)
\(504\) 0 0
\(505\) −8340.00 −0.734901
\(506\) 0 0
\(507\) 1438.00i 0.125964i
\(508\) 0 0
\(509\) − 10534.0i − 0.917311i −0.888614 0.458656i \(-0.848331\pi\)
0.888614 0.458656i \(-0.151669\pi\)
\(510\) 0 0
\(511\) −8360.00 −0.723727
\(512\) 0 0
\(513\) −16200.0 −1.39424
\(514\) 0 0
\(515\) − 4728.00i − 0.404545i
\(516\) 0 0
\(517\) − 5712.00i − 0.485906i
\(518\) 0 0
\(519\) −76.0000 −0.00642780
\(520\) 0 0
\(521\) 9478.00 0.797003 0.398502 0.917168i \(-0.369530\pi\)
0.398502 + 0.917168i \(0.369530\pi\)
\(522\) 0 0
\(523\) − 5858.00i − 0.489775i −0.969551 0.244888i \(-0.921249\pi\)
0.969551 0.244888i \(-0.0787511\pi\)
\(524\) 0 0
\(525\) 3560.00i 0.295945i
\(526\) 0 0
\(527\) 8448.00 0.698293
\(528\) 0 0
\(529\) 17417.0 1.43150
\(530\) 0 0
\(531\) − 7406.00i − 0.605260i
\(532\) 0 0
\(533\) 10908.0i 0.886450i
\(534\) 0 0
\(535\) −9684.00 −0.782572
\(536\) 0 0
\(537\) 5580.00 0.448407
\(538\) 0 0
\(539\) 798.000i 0.0637705i
\(540\) 0 0
\(541\) − 1910.00i − 0.151788i −0.997116 0.0758940i \(-0.975819\pi\)
0.997116 0.0758940i \(-0.0241811\pi\)
\(542\) 0 0
\(543\) −6836.00 −0.540259
\(544\) 0 0
\(545\) 12084.0 0.949765
\(546\) 0 0
\(547\) − 1754.00i − 0.137104i −0.997648 0.0685518i \(-0.978162\pi\)
0.997648 0.0685518i \(-0.0218378\pi\)
\(548\) 0 0
\(549\) 6854.00i 0.532826i
\(550\) 0 0
\(551\) 324.000 0.0250506
\(552\) 0 0
\(553\) −14880.0 −1.14424
\(554\) 0 0
\(555\) 1896.00i 0.145010i
\(556\) 0 0
\(557\) − 2294.00i − 0.174506i −0.996186 0.0872531i \(-0.972191\pi\)
0.996186 0.0872531i \(-0.0278089\pi\)
\(558\) 0 0
\(559\) −16092.0 −1.21757
\(560\) 0 0
\(561\) −1848.00 −0.139078
\(562\) 0 0
\(563\) − 16242.0i − 1.21584i −0.793998 0.607921i \(-0.792004\pi\)
0.793998 0.607921i \(-0.207996\pi\)
\(564\) 0 0
\(565\) − 3252.00i − 0.242146i
\(566\) 0 0
\(567\) −8420.00 −0.623645
\(568\) 0 0
\(569\) 15990.0 1.17809 0.589047 0.808099i \(-0.299503\pi\)
0.589047 + 0.808099i \(0.299503\pi\)
\(570\) 0 0
\(571\) − 21674.0i − 1.58849i −0.607597 0.794246i \(-0.707866\pi\)
0.607597 0.794246i \(-0.292134\pi\)
\(572\) 0 0
\(573\) 3936.00i 0.286961i
\(574\) 0 0
\(575\) −15308.0 −1.11024
\(576\) 0 0
\(577\) −12542.0 −0.904905 −0.452453 0.891788i \(-0.649451\pi\)
−0.452453 + 0.891788i \(0.649451\pi\)
\(578\) 0 0
\(579\) 2116.00i 0.151879i
\(580\) 0 0
\(581\) − 13560.0i − 0.968268i
\(582\) 0 0
\(583\) 9660.00 0.686237
\(584\) 0 0
\(585\) 7452.00 0.526671
\(586\) 0 0
\(587\) − 18578.0i − 1.30630i −0.757230 0.653148i \(-0.773448\pi\)
0.757230 0.653148i \(-0.226552\pi\)
\(588\) 0 0
\(589\) 20736.0i 1.45061i
\(590\) 0 0
\(591\) 1452.00 0.101061
\(592\) 0 0
\(593\) 14514.0 1.00509 0.502545 0.864551i \(-0.332397\pi\)
0.502545 + 0.864551i \(0.332397\pi\)
\(594\) 0 0
\(595\) 7920.00i 0.545695i
\(596\) 0 0
\(597\) 8232.00i 0.564344i
\(598\) 0 0
\(599\) 3460.00 0.236013 0.118006 0.993013i \(-0.462350\pi\)
0.118006 + 0.993013i \(0.462350\pi\)
\(600\) 0 0
\(601\) −20686.0 −1.40399 −0.701996 0.712181i \(-0.747708\pi\)
−0.701996 + 0.712181i \(0.747708\pi\)
\(602\) 0 0
\(603\) − 4646.00i − 0.313764i
\(604\) 0 0
\(605\) 6810.00i 0.457630i
\(606\) 0 0
\(607\) −9776.00 −0.653700 −0.326850 0.945076i \(-0.605987\pi\)
−0.326850 + 0.945076i \(0.605987\pi\)
\(608\) 0 0
\(609\) −80.0000 −0.00532309
\(610\) 0 0
\(611\) 22032.0i 1.45879i
\(612\) 0 0
\(613\) − 8794.00i − 0.579423i −0.957114 0.289712i \(-0.906441\pi\)
0.957114 0.289712i \(-0.0935594\pi\)
\(614\) 0 0
\(615\) −2424.00 −0.158935
\(616\) 0 0
\(617\) −16398.0 −1.06995 −0.534975 0.844868i \(-0.679679\pi\)
−0.534975 + 0.844868i \(0.679679\pi\)
\(618\) 0 0
\(619\) 21374.0i 1.38787i 0.720036 + 0.693937i \(0.244125\pi\)
−0.720036 + 0.693937i \(0.755875\pi\)
\(620\) 0 0
\(621\) 17200.0i 1.11145i
\(622\) 0 0
\(623\) −1640.00 −0.105466
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) − 4536.00i − 0.288916i
\(628\) 0 0
\(629\) − 10428.0i − 0.661036i
\(630\) 0 0
\(631\) 18916.0 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(632\) 0 0
\(633\) −2964.00 −0.186111
\(634\) 0 0
\(635\) 10272.0i 0.641940i
\(636\) 0 0
\(637\) − 3078.00i − 0.191452i
\(638\) 0 0
\(639\) 16100.0 0.996723
\(640\) 0 0
\(641\) −5250.00 −0.323498 −0.161749 0.986832i \(-0.551714\pi\)
−0.161749 + 0.986832i \(0.551714\pi\)
\(642\) 0 0
\(643\) 4502.00i 0.276114i 0.990424 + 0.138057i \(0.0440858\pi\)
−0.990424 + 0.138057i \(0.955914\pi\)
\(644\) 0 0
\(645\) − 3576.00i − 0.218302i
\(646\) 0 0
\(647\) −11076.0 −0.673018 −0.336509 0.941680i \(-0.609246\pi\)
−0.336509 + 0.941680i \(0.609246\pi\)
\(648\) 0 0
\(649\) 4508.00 0.272657
\(650\) 0 0
\(651\) − 5120.00i − 0.308247i
\(652\) 0 0
\(653\) − 30766.0i − 1.84375i −0.387491 0.921873i \(-0.626658\pi\)
0.387491 0.921873i \(-0.373342\pi\)
\(654\) 0 0
\(655\) −12708.0 −0.758080
\(656\) 0 0
\(657\) 9614.00 0.570895
\(658\) 0 0
\(659\) 20518.0i 1.21285i 0.795141 + 0.606425i \(0.207397\pi\)
−0.795141 + 0.606425i \(0.792603\pi\)
\(660\) 0 0
\(661\) 70.0000i 0.00411904i 0.999998 + 0.00205952i \(0.000655566\pi\)
−0.999998 + 0.00205952i \(0.999344\pi\)
\(662\) 0 0
\(663\) 7128.00 0.417539
\(664\) 0 0
\(665\) −19440.0 −1.13361
\(666\) 0 0
\(667\) − 344.000i − 0.0199696i
\(668\) 0 0
\(669\) 1792.00i 0.103562i
\(670\) 0 0
\(671\) −4172.00 −0.240027
\(672\) 0 0
\(673\) 23070.0 1.32137 0.660686 0.750663i \(-0.270266\pi\)
0.660686 + 0.750663i \(0.270266\pi\)
\(674\) 0 0
\(675\) − 8900.00i − 0.507498i
\(676\) 0 0
\(677\) 2622.00i 0.148850i 0.997227 + 0.0744251i \(0.0237122\pi\)
−0.997227 + 0.0744251i \(0.976288\pi\)
\(678\) 0 0
\(679\) 22440.0 1.26829
\(680\) 0 0
\(681\) −6820.00 −0.383764
\(682\) 0 0
\(683\) − 14682.0i − 0.822535i −0.911515 0.411267i \(-0.865086\pi\)
0.911515 0.411267i \(-0.134914\pi\)
\(684\) 0 0
\(685\) 2916.00i 0.162649i
\(686\) 0 0
\(687\) 9004.00 0.500035
\(688\) 0 0
\(689\) −37260.0 −2.06022
\(690\) 0 0
\(691\) 23270.0i 1.28109i 0.767921 + 0.640545i \(0.221291\pi\)
−0.767921 + 0.640545i \(0.778709\pi\)
\(692\) 0 0
\(693\) − 6440.00i − 0.353009i
\(694\) 0 0
\(695\) −7716.00 −0.421129
\(696\) 0 0
\(697\) 13332.0 0.724513
\(698\) 0 0
\(699\) 4604.00i 0.249126i
\(700\) 0 0
\(701\) 31530.0i 1.69882i 0.527735 + 0.849409i \(0.323041\pi\)
−0.527735 + 0.849409i \(0.676959\pi\)
\(702\) 0 0
\(703\) 25596.0 1.37322
\(704\) 0 0
\(705\) −4896.00 −0.261552
\(706\) 0 0
\(707\) − 27800.0i − 1.47882i
\(708\) 0 0
\(709\) 22014.0i 1.16608i 0.812442 + 0.583042i \(0.198138\pi\)
−0.812442 + 0.583042i \(0.801862\pi\)
\(710\) 0 0
\(711\) 17112.0 0.902602
\(712\) 0 0
\(713\) 22016.0 1.15639
\(714\) 0 0
\(715\) 4536.00i 0.237254i
\(716\) 0 0
\(717\) − 8048.00i − 0.419188i
\(718\) 0 0
\(719\) 19016.0 0.986338 0.493169 0.869933i \(-0.335838\pi\)
0.493169 + 0.869933i \(0.335838\pi\)
\(720\) 0 0
\(721\) 15760.0 0.814054
\(722\) 0 0
\(723\) 7172.00i 0.368921i
\(724\) 0 0
\(725\) 178.000i 0.00911828i
\(726\) 0 0
\(727\) −15996.0 −0.816037 −0.408018 0.912974i \(-0.633780\pi\)
−0.408018 + 0.912974i \(0.633780\pi\)
\(728\) 0 0
\(729\) 4283.00 0.217599
\(730\) 0 0
\(731\) 19668.0i 0.995140i
\(732\) 0 0
\(733\) − 10622.0i − 0.535242i −0.963524 0.267621i \(-0.913762\pi\)
0.963524 0.267621i \(-0.0862375\pi\)
\(734\) 0 0
\(735\) 684.000 0.0343261
\(736\) 0 0
\(737\) 2828.00 0.141344
\(738\) 0 0
\(739\) 8878.00i 0.441925i 0.975282 + 0.220962i \(0.0709198\pi\)
−0.975282 + 0.220962i \(0.929080\pi\)
\(740\) 0 0
\(741\) 17496.0i 0.867384i
\(742\) 0 0
\(743\) 21852.0 1.07897 0.539483 0.841996i \(-0.318620\pi\)
0.539483 + 0.841996i \(0.318620\pi\)
\(744\) 0 0
\(745\) 15996.0 0.786642
\(746\) 0 0
\(747\) 15594.0i 0.763795i
\(748\) 0 0
\(749\) − 32280.0i − 1.57475i
\(750\) 0 0
\(751\) 32024.0 1.55602 0.778011 0.628251i \(-0.216229\pi\)
0.778011 + 0.628251i \(0.216229\pi\)
\(752\) 0 0
\(753\) −2500.00 −0.120989
\(754\) 0 0
\(755\) − 1032.00i − 0.0497461i
\(756\) 0 0
\(757\) − 26602.0i − 1.27723i −0.769525 0.638617i \(-0.779507\pi\)
0.769525 0.638617i \(-0.220493\pi\)
\(758\) 0 0
\(759\) −4816.00 −0.230316
\(760\) 0 0
\(761\) 13958.0 0.664885 0.332442 0.943124i \(-0.392127\pi\)
0.332442 + 0.943124i \(0.392127\pi\)
\(762\) 0 0
\(763\) 40280.0i 1.91118i
\(764\) 0 0
\(765\) − 9108.00i − 0.430458i
\(766\) 0 0
\(767\) −17388.0 −0.818571
\(768\) 0 0
\(769\) −11970.0 −0.561312 −0.280656 0.959808i \(-0.590552\pi\)
−0.280656 + 0.959808i \(0.590552\pi\)
\(770\) 0 0
\(771\) 13276.0i 0.620134i
\(772\) 0 0
\(773\) 12318.0i 0.573154i 0.958057 + 0.286577i \(0.0925173\pi\)
−0.958057 + 0.286577i \(0.907483\pi\)
\(774\) 0 0
\(775\) −11392.0 −0.528016
\(776\) 0 0
\(777\) −6320.00 −0.291800
\(778\) 0 0
\(779\) 32724.0i 1.50508i
\(780\) 0 0
\(781\) 9800.00i 0.449003i
\(782\) 0 0
\(783\) 200.000 0.00912825
\(784\) 0 0
\(785\) 5028.00 0.228608
\(786\) 0 0
\(787\) − 26698.0i − 1.20925i −0.796510 0.604626i \(-0.793323\pi\)
0.796510 0.604626i \(-0.206677\pi\)
\(788\) 0 0
\(789\) − 3448.00i − 0.155579i
\(790\) 0 0
\(791\) 10840.0 0.487264
\(792\) 0 0
\(793\) 16092.0 0.720610
\(794\) 0 0
\(795\) − 8280.00i − 0.369385i
\(796\) 0 0
\(797\) 4794.00i 0.213064i 0.994309 + 0.106532i \(0.0339747\pi\)
−0.994309 + 0.106532i \(0.966025\pi\)
\(798\) 0 0
\(799\) 26928.0 1.19230
\(800\) 0 0
\(801\) 1886.00 0.0831942
\(802\) 0 0
\(803\) 5852.00i 0.257176i
\(804\) 0 0
\(805\) 20640.0i 0.903683i
\(806\) 0 0
\(807\) −9628.00 −0.419977
\(808\) 0 0
\(809\) −39114.0 −1.69985 −0.849923 0.526907i \(-0.823351\pi\)
−0.849923 + 0.526907i \(0.823351\pi\)
\(810\) 0 0
\(811\) − 1090.00i − 0.0471949i −0.999722 0.0235975i \(-0.992488\pi\)
0.999722 0.0235975i \(-0.00751200\pi\)
\(812\) 0 0
\(813\) 3280.00i 0.141494i
\(814\) 0 0
\(815\) 8076.00 0.347104
\(816\) 0 0
\(817\) −48276.0 −2.06727
\(818\) 0 0
\(819\) 24840.0i 1.05980i
\(820\) 0 0
\(821\) − 9730.00i − 0.413617i −0.978381 0.206808i \(-0.933692\pi\)
0.978381 0.206808i \(-0.0663077\pi\)
\(822\) 0 0
\(823\) −1100.00 −0.0465900 −0.0232950 0.999729i \(-0.507416\pi\)
−0.0232950 + 0.999729i \(0.507416\pi\)
\(824\) 0 0
\(825\) 2492.00 0.105164
\(826\) 0 0
\(827\) − 38074.0i − 1.60092i −0.599385 0.800461i \(-0.704588\pi\)
0.599385 0.800461i \(-0.295412\pi\)
\(828\) 0 0
\(829\) − 24230.0i − 1.01513i −0.861613 0.507565i \(-0.830546\pi\)
0.861613 0.507565i \(-0.169454\pi\)
\(830\) 0 0
\(831\) 7964.00 0.332453
\(832\) 0 0
\(833\) −3762.00 −0.156477
\(834\) 0 0
\(835\) 6312.00i 0.261600i
\(836\) 0 0
\(837\) 12800.0i 0.528593i
\(838\) 0 0
\(839\) −16820.0 −0.692123 −0.346061 0.938212i \(-0.612481\pi\)
−0.346061 + 0.938212i \(0.612481\pi\)
\(840\) 0 0
\(841\) 24385.0 0.999836
\(842\) 0 0
\(843\) 8252.00i 0.337146i
\(844\) 0 0
\(845\) − 4314.00i − 0.175629i
\(846\) 0 0
\(847\) −22700.0 −0.920875
\(848\) 0 0
\(849\) 6892.00 0.278602
\(850\) 0 0
\(851\) − 27176.0i − 1.09469i
\(852\) 0 0
\(853\) − 22162.0i − 0.889581i −0.895635 0.444790i \(-0.853278\pi\)
0.895635 0.444790i \(-0.146722\pi\)
\(854\) 0 0
\(855\) 22356.0 0.894221
\(856\) 0 0
\(857\) 8790.00 0.350363 0.175181 0.984536i \(-0.443949\pi\)
0.175181 + 0.984536i \(0.443949\pi\)
\(858\) 0 0
\(859\) 10558.0i 0.419365i 0.977770 + 0.209682i \(0.0672430\pi\)
−0.977770 + 0.209682i \(0.932757\pi\)
\(860\) 0 0
\(861\) − 8080.00i − 0.319821i
\(862\) 0 0
\(863\) −7392.00 −0.291572 −0.145786 0.989316i \(-0.546571\pi\)
−0.145786 + 0.989316i \(0.546571\pi\)
\(864\) 0 0
\(865\) 228.000 0.00896212
\(866\) 0 0
\(867\) 1114.00i 0.0436372i
\(868\) 0 0
\(869\) 10416.0i 0.406604i
\(870\) 0 0
\(871\) −10908.0 −0.424344
\(872\) 0 0
\(873\) −25806.0 −1.00046
\(874\) 0 0
\(875\) − 25680.0i − 0.992163i
\(876\) 0 0
\(877\) − 6574.00i − 0.253122i −0.991959 0.126561i \(-0.959606\pi\)
0.991959 0.126561i \(-0.0403940\pi\)
\(878\) 0 0
\(879\) −3028.00 −0.116191
\(880\) 0 0
\(881\) 47154.0 1.80324 0.901622 0.432524i \(-0.142377\pi\)
0.901622 + 0.432524i \(0.142377\pi\)
\(882\) 0 0
\(883\) − 642.000i − 0.0244677i −0.999925 0.0122339i \(-0.996106\pi\)
0.999925 0.0122339i \(-0.00389426\pi\)
\(884\) 0 0
\(885\) − 3864.00i − 0.146765i
\(886\) 0 0
\(887\) −23308.0 −0.882307 −0.441153 0.897432i \(-0.645431\pi\)
−0.441153 + 0.897432i \(0.645431\pi\)
\(888\) 0 0
\(889\) −34240.0 −1.29176
\(890\) 0 0
\(891\) 5894.00i 0.221612i
\(892\) 0 0
\(893\) 66096.0i 2.47684i
\(894\) 0 0
\(895\) −16740.0 −0.625203
\(896\) 0 0
\(897\) 18576.0 0.691454
\(898\) 0 0
\(899\) − 256.000i − 0.00949731i
\(900\) 0 0
\(901\) 45540.0i 1.68386i
\(902\) 0 0
\(903\) 11920.0 0.439283
\(904\) 0 0
\(905\) 20508.0 0.753270
\(906\) 0 0
\(907\) − 21450.0i − 0.785265i −0.919695 0.392633i \(-0.871564\pi\)
0.919695 0.392633i \(-0.128436\pi\)
\(908\) 0 0
\(909\) 31970.0i 1.16653i
\(910\) 0 0
\(911\) −40904.0 −1.48761 −0.743804 0.668398i \(-0.766980\pi\)
−0.743804 + 0.668398i \(0.766980\pi\)
\(912\) 0 0
\(913\) −9492.00 −0.344074
\(914\) 0 0
\(915\) 3576.00i 0.129201i
\(916\) 0 0
\(917\) − 42360.0i − 1.52546i
\(918\) 0 0
\(919\) 27380.0 0.982789 0.491394 0.870937i \(-0.336487\pi\)
0.491394 + 0.870937i \(0.336487\pi\)
\(920\) 0 0
\(921\) −10980.0 −0.392837
\(922\) 0 0
\(923\) − 37800.0i − 1.34800i
\(924\) 0 0
\(925\) 14062.0i 0.499844i
\(926\) 0 0
\(927\) −18124.0 −0.642147
\(928\) 0 0
\(929\) 10302.0 0.363830 0.181915 0.983314i \(-0.441770\pi\)
0.181915 + 0.983314i \(0.441770\pi\)
\(930\) 0 0
\(931\) − 9234.00i − 0.325061i
\(932\) 0 0
\(933\) − 11112.0i − 0.389915i
\(934\) 0 0
\(935\) 5544.00 0.193912
\(936\) 0 0
\(937\) −5054.00 −0.176208 −0.0881040 0.996111i \(-0.528081\pi\)
−0.0881040 + 0.996111i \(0.528081\pi\)
\(938\) 0 0
\(939\) − 4108.00i − 0.142768i
\(940\) 0 0
\(941\) − 30462.0i − 1.05530i −0.849463 0.527648i \(-0.823074\pi\)
0.849463 0.527648i \(-0.176926\pi\)
\(942\) 0 0
\(943\) 34744.0 1.19981
\(944\) 0 0
\(945\) −12000.0 −0.413079
\(946\) 0 0
\(947\) − 32082.0i − 1.10087i −0.834878 0.550436i \(-0.814462\pi\)
0.834878 0.550436i \(-0.185538\pi\)
\(948\) 0 0
\(949\) − 22572.0i − 0.772095i
\(950\) 0 0
\(951\) −4988.00 −0.170081
\(952\) 0 0
\(953\) −12970.0 −0.440860 −0.220430 0.975403i \(-0.570746\pi\)
−0.220430 + 0.975403i \(0.570746\pi\)
\(954\) 0 0
\(955\) − 11808.0i − 0.400103i
\(956\) 0 0
\(957\) 56.0000i 0.00189156i
\(958\) 0 0
\(959\) −9720.00 −0.327294
\(960\) 0 0
\(961\) −13407.0 −0.450035
\(962\) 0 0
\(963\) 37122.0i 1.24220i
\(964\) 0 0
\(965\) − 6348.00i − 0.211761i
\(966\) 0 0
\(967\) −1652.00 −0.0549377 −0.0274688 0.999623i \(-0.508745\pi\)
−0.0274688 + 0.999623i \(0.508745\pi\)
\(968\) 0 0
\(969\) 21384.0 0.708930
\(970\) 0 0
\(971\) − 24650.0i − 0.814682i −0.913276 0.407341i \(-0.866456\pi\)
0.913276 0.407341i \(-0.133544\pi\)
\(972\) 0 0
\(973\) − 25720.0i − 0.847426i
\(974\) 0 0
\(975\) −9612.00 −0.315723
\(976\) 0 0
\(977\) 23646.0 0.774312 0.387156 0.922014i \(-0.373458\pi\)
0.387156 + 0.922014i \(0.373458\pi\)
\(978\) 0 0
\(979\) 1148.00i 0.0374773i
\(980\) 0 0
\(981\) − 46322.0i − 1.50759i
\(982\) 0 0
\(983\) −38108.0 −1.23648 −0.618238 0.785991i \(-0.712153\pi\)
−0.618238 + 0.785991i \(0.712153\pi\)
\(984\) 0 0
\(985\) −4356.00 −0.140907
\(986\) 0 0
\(987\) − 16320.0i − 0.526313i
\(988\) 0 0
\(989\) 51256.0i 1.64797i
\(990\) 0 0
\(991\) −18640.0 −0.597497 −0.298748 0.954332i \(-0.596569\pi\)
−0.298748 + 0.954332i \(0.596569\pi\)
\(992\) 0 0
\(993\) −5828.00 −0.186250
\(994\) 0 0
\(995\) − 24696.0i − 0.786850i
\(996\) 0 0
\(997\) 29022.0i 0.921902i 0.887426 + 0.460951i \(0.152492\pi\)
−0.887426 + 0.460951i \(0.847508\pi\)
\(998\) 0 0
\(999\) 15800.0 0.500390
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 256.4.b.b.129.1 2
4.3 odd 2 256.4.b.f.129.2 2
8.3 odd 2 256.4.b.f.129.1 2
8.5 even 2 inner 256.4.b.b.129.2 2
16.3 odd 4 128.4.a.b.1.1 yes 1
16.5 even 4 128.4.a.a.1.1 1
16.11 odd 4 128.4.a.c.1.1 yes 1
16.13 even 4 128.4.a.d.1.1 yes 1
48.5 odd 4 1152.4.a.j.1.1 1
48.11 even 4 1152.4.a.i.1.1 1
48.29 odd 4 1152.4.a.d.1.1 1
48.35 even 4 1152.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.a.a.1.1 1 16.5 even 4
128.4.a.b.1.1 yes 1 16.3 odd 4
128.4.a.c.1.1 yes 1 16.11 odd 4
128.4.a.d.1.1 yes 1 16.13 even 4
256.4.b.b.129.1 2 1.1 even 1 trivial
256.4.b.b.129.2 2 8.5 even 2 inner
256.4.b.f.129.1 2 8.3 odd 2
256.4.b.f.129.2 2 4.3 odd 2
1152.4.a.c.1.1 1 48.35 even 4
1152.4.a.d.1.1 1 48.29 odd 4
1152.4.a.i.1.1 1 48.11 even 4
1152.4.a.j.1.1 1 48.5 odd 4