Properties

Label 2304.3.g.z.1279.7
Level $2304$
Weight $3$
Character 2304.1279
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(1279,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1279.7
Root \(0.665665 + 1.24775i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1279
Dual form 2304.3.g.z.1279.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.98203 q^{5} -2.13878i q^{7} +O(q^{10})\) \(q+7.98203 q^{5} -2.13878i q^{7} -8.00000i q^{11} -11.6865 q^{13} -11.8564 q^{17} +14.9282i q^{19} -4.27756i q^{23} +38.7128 q^{25} +0.573084 q^{29} -57.4399i q^{31} -17.0718i q^{35} +27.6506 q^{37} -31.5692 q^{41} -28.7846i q^{43} -59.5787i q^{47} +44.4256 q^{49} -31.3550 q^{53} -63.8562i q^{55} -52.7846i q^{59} +59.5787 q^{61} -93.2820 q^{65} -84.7846i q^{67} -42.4685i q^{71} +5.42563 q^{73} -17.1102 q^{77} +44.6072i q^{79} -67.7128i q^{83} -94.6382 q^{85} -133.138 q^{89} +24.9948i q^{91} +119.157i q^{95} +97.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{17} + 88 q^{25} + 80 q^{41} - 88 q^{49} - 192 q^{65} - 400 q^{73} - 400 q^{89} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 7.98203 1.59641 0.798203 0.602388i \(-0.205784\pi\)
0.798203 + 0.602388i \(0.205784\pi\)
\(6\) 0 0
\(7\) − 2.13878i − 0.305540i −0.988262 0.152770i \(-0.951181\pi\)
0.988262 0.152770i \(-0.0488193\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 8.00000i − 0.727273i −0.931541 0.363636i \(-0.881535\pi\)
0.931541 0.363636i \(-0.118465\pi\)
\(12\) 0 0
\(13\) −11.6865 −0.898962 −0.449481 0.893290i \(-0.648391\pi\)
−0.449481 + 0.893290i \(0.648391\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.8564 −0.697436 −0.348718 0.937228i \(-0.613383\pi\)
−0.348718 + 0.937228i \(0.613383\pi\)
\(18\) 0 0
\(19\) 14.9282i 0.785695i 0.919604 + 0.392847i \(0.128510\pi\)
−0.919604 + 0.392847i \(0.871490\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.27756i − 0.185981i −0.995667 0.0929904i \(-0.970357\pi\)
0.995667 0.0929904i \(-0.0296426\pi\)
\(24\) 0 0
\(25\) 38.7128 1.54851
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.573084 0.0197615 0.00988076 0.999951i \(-0.496855\pi\)
0.00988076 + 0.999951i \(0.496855\pi\)
\(30\) 0 0
\(31\) − 57.4399i − 1.85290i −0.376417 0.926450i \(-0.622844\pi\)
0.376417 0.926450i \(-0.377156\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 17.0718i − 0.487766i
\(36\) 0 0
\(37\) 27.6506 0.747313 0.373656 0.927567i \(-0.378104\pi\)
0.373656 + 0.927567i \(0.378104\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −31.5692 −0.769981 −0.384990 0.922921i \(-0.625795\pi\)
−0.384990 + 0.922921i \(0.625795\pi\)
\(42\) 0 0
\(43\) − 28.7846i − 0.669410i −0.942323 0.334705i \(-0.891363\pi\)
0.942323 0.334705i \(-0.108637\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 59.5787i − 1.26763i −0.773484 0.633816i \(-0.781488\pi\)
0.773484 0.633816i \(-0.218512\pi\)
\(48\) 0 0
\(49\) 44.4256 0.906645
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −31.3550 −0.591604 −0.295802 0.955249i \(-0.595587\pi\)
−0.295802 + 0.955249i \(0.595587\pi\)
\(54\) 0 0
\(55\) − 63.8562i − 1.16102i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 52.7846i − 0.894654i −0.894370 0.447327i \(-0.852376\pi\)
0.894370 0.447327i \(-0.147624\pi\)
\(60\) 0 0
\(61\) 59.5787 0.976700 0.488350 0.872648i \(-0.337599\pi\)
0.488350 + 0.872648i \(0.337599\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −93.2820 −1.43511
\(66\) 0 0
\(67\) − 84.7846i − 1.26544i −0.774380 0.632721i \(-0.781938\pi\)
0.774380 0.632721i \(-0.218062\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 42.4685i − 0.598147i −0.954230 0.299074i \(-0.903322\pi\)
0.954230 0.299074i \(-0.0966776\pi\)
\(72\) 0 0
\(73\) 5.42563 0.0743236 0.0371618 0.999309i \(-0.488168\pi\)
0.0371618 + 0.999309i \(0.488168\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −17.1102 −0.222211
\(78\) 0 0
\(79\) 44.6072i 0.564649i 0.959319 + 0.282324i \(0.0911054\pi\)
−0.959319 + 0.282324i \(0.908895\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 67.7128i − 0.815817i −0.913023 0.407909i \(-0.866258\pi\)
0.913023 0.407909i \(-0.133742\pi\)
\(84\) 0 0
\(85\) −94.6382 −1.11339
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −133.138 −1.49594 −0.747969 0.663734i \(-0.768971\pi\)
−0.747969 + 0.663734i \(0.768971\pi\)
\(90\) 0 0
\(91\) 24.9948i 0.274669i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 119.157i 1.25429i
\(96\) 0 0
\(97\) 97.1384 1.00143 0.500714 0.865613i \(-0.333071\pi\)
0.500714 + 0.865613i \(0.333071\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 62.1370 0.615218 0.307609 0.951513i \(-0.400471\pi\)
0.307609 + 0.951513i \(0.400471\pi\)
\(102\) 0 0
\(103\) 27.8041i 0.269943i 0.990849 + 0.134971i \(0.0430943\pi\)
−0.990849 + 0.134971i \(0.956906\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 37.7795i 0.353079i 0.984294 + 0.176540i \(0.0564903\pi\)
−0.984294 + 0.176540i \(0.943510\pi\)
\(108\) 0 0
\(109\) −141.691 −1.29992 −0.649960 0.759968i \(-0.725214\pi\)
−0.649960 + 0.759968i \(0.725214\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 58.2872 0.515816 0.257908 0.966170i \(-0.416967\pi\)
0.257908 + 0.966170i \(0.416967\pi\)
\(114\) 0 0
\(115\) − 34.1436i − 0.296901i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 25.3582i 0.213094i
\(120\) 0 0
\(121\) 57.0000 0.471074
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 109.456 0.875649
\(126\) 0 0
\(127\) 185.152i 1.45789i 0.684571 + 0.728946i \(0.259990\pi\)
−0.684571 + 0.728946i \(0.740010\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 125.359i − 0.956939i −0.878104 0.478469i \(-0.841192\pi\)
0.878104 0.478469i \(-0.158808\pi\)
\(132\) 0 0
\(133\) 31.9281 0.240061
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 99.5692 0.726783 0.363391 0.931637i \(-0.381619\pi\)
0.363391 + 0.931637i \(0.381619\pi\)
\(138\) 0 0
\(139\) − 177.492i − 1.27692i −0.769654 0.638461i \(-0.779571\pi\)
0.769654 0.638461i \(-0.220429\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 93.4920i 0.653790i
\(144\) 0 0
\(145\) 4.57437 0.0315474
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −87.8023 −0.589277 −0.294639 0.955609i \(-0.595199\pi\)
−0.294639 + 0.955609i \(0.595199\pi\)
\(150\) 0 0
\(151\) 219.066i 1.45077i 0.688345 + 0.725383i \(0.258337\pi\)
−0.688345 + 0.725383i \(0.741663\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 458.487i − 2.95798i
\(156\) 0 0
\(157\) 253.440 1.61427 0.807133 0.590370i \(-0.201018\pi\)
0.807133 + 0.590370i \(0.201018\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.14875 −0.0568245
\(162\) 0 0
\(163\) − 102.354i − 0.627938i −0.949433 0.313969i \(-0.898341\pi\)
0.949433 0.313969i \(-0.101659\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 281.090i 1.68318i 0.540120 + 0.841588i \(0.318379\pi\)
−0.540120 + 0.841588i \(0.681621\pi\)
\(168\) 0 0
\(169\) −32.4256 −0.191868
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −242.858 −1.40381 −0.701903 0.712273i \(-0.747666\pi\)
−0.701903 + 0.712273i \(0.747666\pi\)
\(174\) 0 0
\(175\) − 82.7981i − 0.473132i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 318.354i − 1.77851i −0.457409 0.889257i \(-0.651222\pi\)
0.457409 0.889257i \(-0.348778\pi\)
\(180\) 0 0
\(181\) 79.5132 0.439299 0.219650 0.975579i \(-0.429509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 220.708 1.19301
\(186\) 0 0
\(187\) 94.8513i 0.507226i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 352.887i − 1.84758i −0.382902 0.923789i \(-0.625075\pi\)
0.382902 0.923789i \(-0.374925\pi\)
\(192\) 0 0
\(193\) −284.277 −1.47294 −0.736469 0.676472i \(-0.763508\pi\)
−0.736469 + 0.676472i \(0.763508\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −75.8087 −0.384816 −0.192408 0.981315i \(-0.561630\pi\)
−0.192408 + 0.981315i \(0.561630\pi\)
\(198\) 0 0
\(199\) 104.186i 0.523547i 0.965129 + 0.261774i \(0.0843074\pi\)
−0.965129 + 0.261774i \(0.915693\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.22570i − 0.00603793i
\(204\) 0 0
\(205\) −251.986 −1.22920
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 119.426 0.571414
\(210\) 0 0
\(211\) 136.918i 0.648900i 0.945903 + 0.324450i \(0.105179\pi\)
−0.945903 + 0.324450i \(0.894821\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 229.760i − 1.06865i
\(216\) 0 0
\(217\) −122.851 −0.566135
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 138.560 0.626968
\(222\) 0 0
\(223\) − 53.1624i − 0.238396i −0.992870 0.119198i \(-0.961968\pi\)
0.992870 0.119198i \(-0.0380324\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 119.846i 0.527956i 0.964529 + 0.263978i \(0.0850347\pi\)
−0.964529 + 0.263978i \(0.914965\pi\)
\(228\) 0 0
\(229\) 214.103 0.934946 0.467473 0.884007i \(-0.345164\pi\)
0.467473 + 0.884007i \(0.345164\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −127.436 −0.546935 −0.273468 0.961881i \(-0.588171\pi\)
−0.273468 + 0.961881i \(0.588171\pi\)
\(234\) 0 0
\(235\) − 475.559i − 2.02365i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 319.281i 1.33590i 0.744204 + 0.667952i \(0.232829\pi\)
−0.744204 + 0.667952i \(0.767171\pi\)
\(240\) 0 0
\(241\) −247.415 −1.02662 −0.513310 0.858203i \(-0.671581\pi\)
−0.513310 + 0.858203i \(0.671581\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 354.607 1.44737
\(246\) 0 0
\(247\) − 174.459i − 0.706310i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 214.851i − 0.855981i −0.903783 0.427991i \(-0.859222\pi\)
0.903783 0.427991i \(-0.140778\pi\)
\(252\) 0 0
\(253\) −34.2205 −0.135259
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 84.2769 0.327926 0.163963 0.986467i \(-0.447572\pi\)
0.163963 + 0.986467i \(0.447572\pi\)
\(258\) 0 0
\(259\) − 59.1384i − 0.228334i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 277.120i − 1.05369i −0.849962 0.526844i \(-0.823375\pi\)
0.849962 0.526844i \(-0.176625\pi\)
\(264\) 0 0
\(265\) −250.277 −0.944441
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 123.701 0.459855 0.229927 0.973208i \(-0.426151\pi\)
0.229927 + 0.973208i \(0.426151\pi\)
\(270\) 0 0
\(271\) − 197.985i − 0.730572i −0.930895 0.365286i \(-0.880971\pi\)
0.930895 0.365286i \(-0.119029\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 309.703i − 1.12619i
\(276\) 0 0
\(277\) −247.709 −0.894256 −0.447128 0.894470i \(-0.647553\pi\)
−0.447128 + 0.894470i \(0.647553\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 443.128 1.57697 0.788484 0.615055i \(-0.210866\pi\)
0.788484 + 0.615055i \(0.210866\pi\)
\(282\) 0 0
\(283\) 294.620i 1.04106i 0.853843 + 0.520531i \(0.174266\pi\)
−0.853843 + 0.520531i \(0.825734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 67.5196i 0.235260i
\(288\) 0 0
\(289\) −148.426 −0.513583
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 66.7217 0.227719 0.113859 0.993497i \(-0.463679\pi\)
0.113859 + 0.993497i \(0.463679\pi\)
\(294\) 0 0
\(295\) − 421.328i − 1.42823i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.9897i 0.167190i
\(300\) 0 0
\(301\) −61.5639 −0.204531
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 475.559 1.55921
\(306\) 0 0
\(307\) − 524.210i − 1.70753i −0.520662 0.853763i \(-0.674315\pi\)
0.520662 0.853763i \(-0.325685\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 362.057i 1.16417i 0.813128 + 0.582085i \(0.197763\pi\)
−0.813128 + 0.582085i \(0.802237\pi\)
\(312\) 0 0
\(313\) −252.277 −0.805996 −0.402998 0.915201i \(-0.632032\pi\)
−0.402998 + 0.915201i \(0.632032\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −80.3934 −0.253607 −0.126803 0.991928i \(-0.540472\pi\)
−0.126803 + 0.991928i \(0.540472\pi\)
\(318\) 0 0
\(319\) − 4.58467i − 0.0143720i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 176.995i − 0.547972i
\(324\) 0 0
\(325\) −452.417 −1.39205
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −127.426 −0.387312
\(330\) 0 0
\(331\) − 172.056i − 0.519808i −0.965635 0.259904i \(-0.916309\pi\)
0.965635 0.259904i \(-0.0836909\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 676.753i − 2.02016i
\(336\) 0 0
\(337\) 564.277 1.67441 0.837206 0.546888i \(-0.184187\pi\)
0.837206 + 0.546888i \(0.184187\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −459.519 −1.34756
\(342\) 0 0
\(343\) − 199.817i − 0.582556i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 286.123i − 0.824562i −0.911057 0.412281i \(-0.864732\pi\)
0.911057 0.412281i \(-0.135268\pi\)
\(348\) 0 0
\(349\) 421.021 1.20636 0.603182 0.797603i \(-0.293899\pi\)
0.603182 + 0.797603i \(0.293899\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −429.138 −1.21569 −0.607845 0.794056i \(-0.707966\pi\)
−0.607845 + 0.794056i \(0.707966\pi\)
\(354\) 0 0
\(355\) − 338.985i − 0.954886i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 263.673i − 0.734465i −0.930129 0.367233i \(-0.880305\pi\)
0.930129 0.367233i \(-0.119695\pi\)
\(360\) 0 0
\(361\) 138.149 0.382684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 43.3075 0.118651
\(366\) 0 0
\(367\) 129.544i 0.352981i 0.984302 + 0.176491i \(0.0564745\pi\)
−0.984302 + 0.176491i \(0.943525\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 67.0615i 0.180759i
\(372\) 0 0
\(373\) 302.478 0.810933 0.405467 0.914110i \(-0.367109\pi\)
0.405467 + 0.914110i \(0.367109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.69735 −0.0177649
\(378\) 0 0
\(379\) 116.210i 0.306623i 0.988178 + 0.153312i \(0.0489938\pi\)
−0.988178 + 0.153312i \(0.951006\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 566.151i 1.47820i 0.673595 + 0.739101i \(0.264749\pi\)
−0.673595 + 0.739101i \(0.735251\pi\)
\(384\) 0 0
\(385\) −136.574 −0.354739
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −350.104 −0.900011 −0.450006 0.893026i \(-0.648578\pi\)
−0.450006 + 0.893026i \(0.648578\pi\)
\(390\) 0 0
\(391\) 50.7165i 0.129710i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 356.056i 0.901408i
\(396\) 0 0
\(397\) 544.149 1.37065 0.685326 0.728236i \(-0.259660\pi\)
0.685326 + 0.728236i \(0.259660\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 296.431 0.739229 0.369614 0.929185i \(-0.379490\pi\)
0.369614 + 0.929185i \(0.379490\pi\)
\(402\) 0 0
\(403\) 671.272i 1.66569i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 221.205i − 0.543500i
\(408\) 0 0
\(409\) 247.415 0.604927 0.302464 0.953161i \(-0.402191\pi\)
0.302464 + 0.953161i \(0.402191\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −112.895 −0.273353
\(414\) 0 0
\(415\) − 540.486i − 1.30238i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 92.1333i 0.219889i 0.993938 + 0.109944i \(0.0350672\pi\)
−0.993938 + 0.109944i \(0.964933\pi\)
\(420\) 0 0
\(421\) 445.540 1.05829 0.529145 0.848531i \(-0.322513\pi\)
0.529145 + 0.848531i \(0.322513\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −458.995 −1.07999
\(426\) 0 0
\(427\) − 127.426i − 0.298421i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 186.677i − 0.433125i −0.976269 0.216563i \(-0.930515\pi\)
0.976269 0.216563i \(-0.0694845\pi\)
\(432\) 0 0
\(433\) 291.128 0.672351 0.336176 0.941799i \(-0.390866\pi\)
0.336176 + 0.941799i \(0.390866\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 63.8562 0.146124
\(438\) 0 0
\(439\) 87.6899i 0.199749i 0.995000 + 0.0998746i \(0.0318442\pi\)
−0.995000 + 0.0998746i \(0.968156\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.7077i 0.0467441i 0.999727 + 0.0233721i \(0.00744024\pi\)
−0.999727 + 0.0233721i \(0.992560\pi\)
\(444\) 0 0
\(445\) −1062.72 −2.38812
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −584.410 −1.30158 −0.650791 0.759257i \(-0.725563\pi\)
−0.650791 + 0.759257i \(0.725563\pi\)
\(450\) 0 0
\(451\) 252.554i 0.559986i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 199.510i 0.438483i
\(456\) 0 0
\(457\) 269.692 0.590136 0.295068 0.955476i \(-0.404658\pi\)
0.295068 + 0.955476i \(0.404658\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 44.4948 0.0965181 0.0482590 0.998835i \(-0.484633\pi\)
0.0482590 + 0.998835i \(0.484633\pi\)
\(462\) 0 0
\(463\) 611.065i 1.31980i 0.751355 + 0.659898i \(0.229400\pi\)
−0.751355 + 0.659898i \(0.770600\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 146.410i − 0.313512i −0.987637 0.156756i \(-0.949896\pi\)
0.987637 0.156756i \(-0.0501036\pi\)
\(468\) 0 0
\(469\) −181.336 −0.386643
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −230.277 −0.486843
\(474\) 0 0
\(475\) 577.913i 1.21666i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 191.876i − 0.400576i −0.979737 0.200288i \(-0.935812\pi\)
0.979737 0.200288i \(-0.0641878\pi\)
\(480\) 0 0
\(481\) −323.138 −0.671805
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 775.362 1.59868
\(486\) 0 0
\(487\) 610.758i 1.25412i 0.778969 + 0.627062i \(0.215743\pi\)
−0.778969 + 0.627062i \(0.784257\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 142.354i − 0.289926i −0.989437 0.144963i \(-0.953694\pi\)
0.989437 0.144963i \(-0.0463064\pi\)
\(492\) 0 0
\(493\) −6.79472 −0.0137824
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −90.8306 −0.182758
\(498\) 0 0
\(499\) 91.3693i 0.183105i 0.995800 + 0.0915524i \(0.0291829\pi\)
−0.995800 + 0.0915524i \(0.970817\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 230.067i 0.457389i 0.973498 + 0.228695i \(0.0734457\pi\)
−0.973498 + 0.228695i \(0.926554\pi\)
\(504\) 0 0
\(505\) 495.979 0.982137
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −527.387 −1.03612 −0.518062 0.855343i \(-0.673346\pi\)
−0.518062 + 0.855343i \(0.673346\pi\)
\(510\) 0 0
\(511\) − 11.6042i − 0.0227088i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 221.933i 0.430939i
\(516\) 0 0
\(517\) −476.630 −0.921914
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 191.856 0.368246 0.184123 0.982903i \(-0.441055\pi\)
0.184123 + 0.982903i \(0.441055\pi\)
\(522\) 0 0
\(523\) − 105.492i − 0.201706i −0.994901 0.100853i \(-0.967843\pi\)
0.994901 0.100853i \(-0.0321572\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 681.031i 1.29228i
\(528\) 0 0
\(529\) 510.703 0.965411
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 368.934 0.692184
\(534\) 0 0
\(535\) 301.557i 0.563658i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 355.405i − 0.659378i
\(540\) 0 0
\(541\) −459.744 −0.849804 −0.424902 0.905239i \(-0.639691\pi\)
−0.424902 + 0.905239i \(0.639691\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1130.98 −2.07520
\(546\) 0 0
\(547\) 67.3693i 0.123161i 0.998102 + 0.0615807i \(0.0196142\pi\)
−0.998102 + 0.0615807i \(0.980386\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.55511i 0.0155265i
\(552\) 0 0
\(553\) 95.4050 0.172523
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 476.671 0.855782 0.427891 0.903830i \(-0.359257\pi\)
0.427891 + 0.903830i \(0.359257\pi\)
\(558\) 0 0
\(559\) 336.391i 0.601774i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 910.123i 1.61656i 0.588799 + 0.808280i \(0.299601\pi\)
−0.588799 + 0.808280i \(0.700399\pi\)
\(564\) 0 0
\(565\) 465.250 0.823452
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −124.123 −0.218142 −0.109071 0.994034i \(-0.534788\pi\)
−0.109071 + 0.994034i \(0.534788\pi\)
\(570\) 0 0
\(571\) − 945.031i − 1.65504i −0.561433 0.827522i \(-0.689750\pi\)
0.561433 0.827522i \(-0.310250\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 165.596i − 0.287994i
\(576\) 0 0
\(577\) 215.682 0.373799 0.186899 0.982379i \(-0.440156\pi\)
0.186899 + 0.982379i \(0.440156\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −144.823 −0.249265
\(582\) 0 0
\(583\) 250.840i 0.430258i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 900.785i 1.53456i 0.641314 + 0.767278i \(0.278389\pi\)
−0.641314 + 0.767278i \(0.721611\pi\)
\(588\) 0 0
\(589\) 857.475 1.45581
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 508.585 0.857647 0.428824 0.903388i \(-0.358928\pi\)
0.428824 + 0.903388i \(0.358928\pi\)
\(594\) 0 0
\(595\) 202.410i 0.340185i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 846.934i 1.41391i 0.707257 + 0.706957i \(0.249933\pi\)
−0.707257 + 0.706957i \(0.750067\pi\)
\(600\) 0 0
\(601\) 406.000 0.675541 0.337770 0.941229i \(-0.390327\pi\)
0.337770 + 0.941229i \(0.390327\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 454.976 0.752026
\(606\) 0 0
\(607\) 771.156i 1.27044i 0.772332 + 0.635219i \(0.219090\pi\)
−0.772332 + 0.635219i \(0.780910\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 696.267i 1.13955i
\(612\) 0 0
\(613\) 336.699 0.549264 0.274632 0.961549i \(-0.411444\pi\)
0.274632 + 0.961549i \(0.411444\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −908.831 −1.47298 −0.736492 0.676447i \(-0.763519\pi\)
−0.736492 + 0.676447i \(0.763519\pi\)
\(618\) 0 0
\(619\) 1047.77i 1.69268i 0.532643 + 0.846340i \(0.321199\pi\)
−0.532643 + 0.846340i \(0.678801\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 284.754i 0.457068i
\(624\) 0 0
\(625\) −94.1384 −0.150622
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −327.836 −0.521202
\(630\) 0 0
\(631\) 610.758i 0.967921i 0.875090 + 0.483961i \(0.160802\pi\)
−0.875090 + 0.483961i \(0.839198\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1477.89i 2.32739i
\(636\) 0 0
\(637\) −519.180 −0.815040
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.60015 −0.0149768 −0.00748842 0.999972i \(-0.502384\pi\)
−0.00748842 + 0.999972i \(0.502384\pi\)
\(642\) 0 0
\(643\) 86.1999i 0.134059i 0.997751 + 0.0670295i \(0.0213522\pi\)
−0.997751 + 0.0670295i \(0.978648\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 352.580i 0.544946i 0.962163 + 0.272473i \(0.0878416\pi\)
−0.962163 + 0.272473i \(0.912158\pi\)
\(648\) 0 0
\(649\) −422.277 −0.650658
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 319.322 0.489008 0.244504 0.969648i \(-0.421375\pi\)
0.244504 + 0.969648i \(0.421375\pi\)
\(654\) 0 0
\(655\) − 1000.62i − 1.52766i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 275.328i − 0.417797i −0.977937 0.208898i \(-0.933012\pi\)
0.977937 0.208898i \(-0.0669878\pi\)
\(660\) 0 0
\(661\) 133.668 0.202221 0.101111 0.994875i \(-0.467760\pi\)
0.101111 + 0.994875i \(0.467760\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 254.851 0.383235
\(666\) 0 0
\(667\) − 2.45140i − 0.00367526i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 476.630i − 0.710327i
\(672\) 0 0
\(673\) 187.703 0.278904 0.139452 0.990229i \(-0.455466\pi\)
0.139452 + 0.990229i \(0.455466\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1169.84 −1.72797 −0.863986 0.503515i \(-0.832040\pi\)
−0.863986 + 0.503515i \(0.832040\pi\)
\(678\) 0 0
\(679\) − 207.758i − 0.305976i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 89.4566i 0.130976i 0.997853 + 0.0654880i \(0.0208604\pi\)
−0.997853 + 0.0654880i \(0.979140\pi\)
\(684\) 0 0
\(685\) 794.765 1.16024
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 366.431 0.531830
\(690\) 0 0
\(691\) 139.103i 0.201306i 0.994922 + 0.100653i \(0.0320933\pi\)
−0.994922 + 0.100653i \(0.967907\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1416.75i − 2.03849i
\(696\) 0 0
\(697\) 374.297 0.537012
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1218.34 −1.73801 −0.869004 0.494804i \(-0.835240\pi\)
−0.869004 + 0.494804i \(0.835240\pi\)
\(702\) 0 0
\(703\) 412.773i 0.587160i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 132.897i − 0.187974i
\(708\) 0 0
\(709\) −1080.13 −1.52346 −0.761730 0.647895i \(-0.775649\pi\)
−0.761730 + 0.647895i \(0.775649\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −245.703 −0.344604
\(714\) 0 0
\(715\) 746.256i 1.04372i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.7736i 0.0288923i 0.999896 + 0.0144461i \(0.00459851\pi\)
−0.999896 + 0.0144461i \(0.995401\pi\)
\(720\) 0 0
\(721\) 59.4669 0.0824783
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 22.1857 0.0306010
\(726\) 0 0
\(727\) − 1031.17i − 1.41838i −0.705015 0.709192i \(-0.749060\pi\)
0.705015 0.709192i \(-0.250940\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 341.282i 0.466870i
\(732\) 0 0
\(733\) 881.072 1.20201 0.601004 0.799246i \(-0.294767\pi\)
0.601004 + 0.799246i \(0.294767\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −678.277 −0.920321
\(738\) 0 0
\(739\) − 671.195i − 0.908247i −0.890939 0.454124i \(-0.849952\pi\)
0.890939 0.454124i \(-0.150048\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 254.197i − 0.342122i −0.985260 0.171061i \(-0.945281\pi\)
0.985260 0.171061i \(-0.0547195\pi\)
\(744\) 0 0
\(745\) −700.841 −0.940726
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 80.8019 0.107880
\(750\) 0 0
\(751\) − 728.994i − 0.970698i −0.874320 0.485349i \(-0.838693\pi\)
0.874320 0.485349i \(-0.161307\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1748.59i 2.31601i
\(756\) 0 0
\(757\) 372.679 0.492311 0.246155 0.969230i \(-0.420833\pi\)
0.246155 + 0.969230i \(0.420833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1257.80 1.65283 0.826416 0.563060i \(-0.190376\pi\)
0.826416 + 0.563060i \(0.190376\pi\)
\(762\) 0 0
\(763\) 303.046i 0.397177i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 616.868i 0.804260i
\(768\) 0 0
\(769\) 247.703 0.322110 0.161055 0.986945i \(-0.448510\pi\)
0.161055 + 0.986945i \(0.448510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 587.805 0.760420 0.380210 0.924900i \(-0.375852\pi\)
0.380210 + 0.924900i \(0.375852\pi\)
\(774\) 0 0
\(775\) − 2223.66i − 2.86924i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 471.272i − 0.604970i
\(780\) 0 0
\(781\) −339.748 −0.435016
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2022.96 2.57702
\(786\) 0 0
\(787\) − 31.0821i − 0.0394944i −0.999805 0.0197472i \(-0.993714\pi\)
0.999805 0.0197472i \(-0.00628614\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 124.663i − 0.157602i
\(792\) 0 0
\(793\) −696.267 −0.878016
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 490.260 0.615132 0.307566 0.951527i \(-0.400486\pi\)
0.307566 + 0.951527i \(0.400486\pi\)
\(798\) 0 0
\(799\) 706.389i 0.884092i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 43.4050i − 0.0540536i
\(804\) 0 0
\(805\) −73.0256 −0.0907150
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 676.102 0.835726 0.417863 0.908510i \(-0.362779\pi\)
0.417863 + 0.908510i \(0.362779\pi\)
\(810\) 0 0
\(811\) 74.1793i 0.0914665i 0.998954 + 0.0457332i \(0.0145624\pi\)
−0.998954 + 0.0457332i \(0.985438\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 816.991i − 1.00244i
\(816\) 0 0
\(817\) 429.703 0.525952
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1130.58 1.37708 0.688540 0.725198i \(-0.258252\pi\)
0.688540 + 0.725198i \(0.258252\pi\)
\(822\) 0 0
\(823\) 82.1839i 0.0998589i 0.998753 + 0.0499295i \(0.0158997\pi\)
−0.998753 + 0.0499295i \(0.984100\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1504.57i 1.81931i 0.415365 + 0.909655i \(0.363654\pi\)
−0.415365 + 0.909655i \(0.636346\pi\)
\(828\) 0 0
\(829\) −1409.65 −1.70042 −0.850209 0.526445i \(-0.823525\pi\)
−0.850209 + 0.526445i \(0.823525\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −526.728 −0.632327
\(834\) 0 0
\(835\) 2243.67i 2.68703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 288.110i − 0.343397i −0.985150 0.171698i \(-0.945075\pi\)
0.985150 0.171698i \(-0.0549254\pi\)
\(840\) 0 0
\(841\) −840.672 −0.999609
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −258.822 −0.306299
\(846\) 0 0
\(847\) − 121.910i − 0.143932i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 118.277i − 0.138986i
\(852\) 0 0
\(853\) −645.132 −0.756310 −0.378155 0.925742i \(-0.623441\pi\)
−0.378155 + 0.925742i \(0.623441\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −995.549 −1.16167 −0.580833 0.814022i \(-0.697273\pi\)
−0.580833 + 0.814022i \(0.697273\pi\)
\(858\) 0 0
\(859\) 774.354i 0.901460i 0.892660 + 0.450730i \(0.148836\pi\)
−0.892660 + 0.450730i \(0.851164\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1007.33i − 1.16724i −0.812025 0.583622i \(-0.801635\pi\)
0.812025 0.583622i \(-0.198365\pi\)
\(864\) 0 0
\(865\) −1938.50 −2.24104
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 356.858 0.410654
\(870\) 0 0
\(871\) 990.836i 1.13758i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 234.102i − 0.267546i
\(876\) 0 0
\(877\) −681.645 −0.777246 −0.388623 0.921397i \(-0.627049\pi\)
−0.388623 + 0.921397i \(0.627049\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 679.108 0.770837 0.385419 0.922742i \(-0.374057\pi\)
0.385419 + 0.922742i \(0.374057\pi\)
\(882\) 0 0
\(883\) − 1059.44i − 1.19982i −0.800068 0.599910i \(-0.795203\pi\)
0.800068 0.599910i \(-0.204797\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 856.411i − 0.965514i −0.875754 0.482757i \(-0.839635\pi\)
0.875754 0.482757i \(-0.160365\pi\)
\(888\) 0 0
\(889\) 396.000 0.445444
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 889.403 0.995972
\(894\) 0 0
\(895\) − 2541.11i − 2.83923i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 32.9179i − 0.0366161i
\(900\) 0 0
\(901\) 371.758 0.412606
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 634.677 0.701300
\(906\) 0 0
\(907\) 761.492i 0.839573i 0.907623 + 0.419786i \(0.137895\pi\)
−0.907623 + 0.419786i \(0.862105\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 897.344i 0.985009i 0.870310 + 0.492505i \(0.163919\pi\)
−0.870310 + 0.492505i \(0.836081\pi\)
\(912\) 0 0
\(913\) −541.703 −0.593321
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −268.115 −0.292383
\(918\) 0 0
\(919\) − 62.6388i − 0.0681598i −0.999419 0.0340799i \(-0.989150\pi\)
0.999419 0.0340799i \(-0.0108501\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 496.308i 0.537712i
\(924\) 0 0
\(925\) 1070.43 1.15722
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −253.313 −0.272673 −0.136336 0.990663i \(-0.543533\pi\)
−0.136336 + 0.990663i \(0.543533\pi\)
\(930\) 0 0
\(931\) 663.195i 0.712347i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 757.106i 0.809739i
\(936\) 0 0
\(937\) 30.5538 0.0326081 0.0163040 0.999867i \(-0.494810\pi\)
0.0163040 + 0.999867i \(0.494810\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −388.745 −0.413119 −0.206559 0.978434i \(-0.566227\pi\)
−0.206559 + 0.978434i \(0.566227\pi\)
\(942\) 0 0
\(943\) 135.039i 0.143202i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 263.615i 0.278369i 0.990266 + 0.139184i \(0.0444481\pi\)
−0.990266 + 0.139184i \(0.955552\pi\)
\(948\) 0 0
\(949\) −63.4066 −0.0668141
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1292.41 1.35615 0.678075 0.734993i \(-0.262815\pi\)
0.678075 + 0.734993i \(0.262815\pi\)
\(954\) 0 0
\(955\) − 2816.76i − 2.94949i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 212.957i − 0.222061i
\(960\) 0 0
\(961\) −2338.34 −2.43324
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2269.11 −2.35141
\(966\) 0 0
\(967\) 1110.00i 1.14788i 0.818896 + 0.573942i \(0.194587\pi\)
−0.818896 + 0.573942i \(0.805413\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1341.66i − 1.38173i −0.722983 0.690866i \(-0.757230\pi\)
0.722983 0.690866i \(-0.242770\pi\)
\(972\) 0 0
\(973\) −379.617 −0.390151
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 920.431 0.942099 0.471050 0.882107i \(-0.343875\pi\)
0.471050 + 0.882107i \(0.343875\pi\)
\(978\) 0 0
\(979\) 1065.11i 1.08795i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1697.84i 1.72720i 0.504176 + 0.863601i \(0.331796\pi\)
−0.504176 + 0.863601i \(0.668204\pi\)
\(984\) 0 0
\(985\) −605.108 −0.614322
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −123.128 −0.124497
\(990\) 0 0
\(991\) 284.765i 0.287351i 0.989625 + 0.143675i \(0.0458921\pi\)
−0.989625 + 0.143675i \(0.954108\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 831.615i 0.835794i
\(996\) 0 0
\(997\) 1093.80 1.09710 0.548548 0.836119i \(-0.315181\pi\)
0.548548 + 0.836119i \(0.315181\pi\)
\(998\) 0 0
\(999\) 0 0
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 2304.3.g.z.1279.7 8
3.2 odd 2 768.3.g.h.511.1 8
4.3 odd 2 inner 2304.3.g.z.1279.8 8
8.3 odd 2 inner 2304.3.g.z.1279.2 8
8.5 even 2 inner 2304.3.g.z.1279.1 8
12.11 even 2 768.3.g.h.511.5 8
16.3 odd 4 72.3.b.b.19.2 4
16.5 even 4 72.3.b.b.19.1 4
16.11 odd 4 288.3.b.b.271.4 4
16.13 even 4 288.3.b.b.271.1 4
24.5 odd 2 768.3.g.h.511.8 8
24.11 even 2 768.3.g.h.511.4 8
48.5 odd 4 24.3.b.a.19.4 yes 4
48.11 even 4 96.3.b.a.79.3 4
48.29 odd 4 96.3.b.a.79.4 4
48.35 even 4 24.3.b.a.19.3 4
240.29 odd 4 2400.3.g.a.751.1 4
240.53 even 4 600.3.p.a.499.5 8
240.59 even 4 2400.3.g.a.751.2 4
240.77 even 4 2400.3.p.a.1999.2 8
240.83 odd 4 600.3.p.a.499.3 8
240.107 odd 4 2400.3.p.a.1999.3 8
240.149 odd 4 600.3.g.a.451.1 4
240.173 even 4 2400.3.p.a.1999.7 8
240.179 even 4 600.3.g.a.451.2 4
240.197 even 4 600.3.p.a.499.4 8
240.203 odd 4 2400.3.p.a.1999.6 8
240.227 odd 4 600.3.p.a.499.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.b.a.19.3 4 48.35 even 4
24.3.b.a.19.4 yes 4 48.5 odd 4
72.3.b.b.19.1 4 16.5 even 4
72.3.b.b.19.2 4 16.3 odd 4
96.3.b.a.79.3 4 48.11 even 4
96.3.b.a.79.4 4 48.29 odd 4
288.3.b.b.271.1 4 16.13 even 4
288.3.b.b.271.4 4 16.11 odd 4
600.3.g.a.451.1 4 240.149 odd 4
600.3.g.a.451.2 4 240.179 even 4
600.3.p.a.499.3 8 240.83 odd 4
600.3.p.a.499.4 8 240.197 even 4
600.3.p.a.499.5 8 240.53 even 4
600.3.p.a.499.6 8 240.227 odd 4
768.3.g.h.511.1 8 3.2 odd 2
768.3.g.h.511.4 8 24.11 even 2
768.3.g.h.511.5 8 12.11 even 2
768.3.g.h.511.8 8 24.5 odd 2
2304.3.g.z.1279.1 8 8.5 even 2 inner
2304.3.g.z.1279.2 8 8.3 odd 2 inner
2304.3.g.z.1279.7 8 1.1 even 1 trivial
2304.3.g.z.1279.8 8 4.3 odd 2 inner
2400.3.g.a.751.1 4 240.29 odd 4
2400.3.g.a.751.2 4 240.59 even 4
2400.3.p.a.1999.2 8 240.77 even 4
2400.3.p.a.1999.3 8 240.107 odd 4
2400.3.p.a.1999.6 8 240.203 odd 4
2400.3.p.a.1999.7 8 240.173 even 4