Properties

Label 2304.3.h.k.2177.3
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2177.3
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.k.2177.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.04989 q^{5} -7.79796 q^{7} +O(q^{10})\) \(q-2.04989 q^{5} -7.79796 q^{7} +4.09978 q^{11} -6.69694i q^{13} +19.3704i q^{17} +1.79796i q^{19} -14.1421i q^{23} -20.7980 q^{25} -28.4914 q^{29} +20.2020 q^{31} +15.9849 q^{35} -41.5959i q^{37} +4.94253i q^{41} +75.1918i q^{43} -13.5707i q^{47} +11.8082 q^{49} +20.8633 q^{53} -8.40408 q^{55} -54.8542 q^{59} +89.1918i q^{61} +13.7280i q^{65} -37.7980i q^{67} -117.937i q^{71} -48.4041 q^{73} -31.9699 q^{77} +92.6061 q^{79} +158.806 q^{83} -39.7071i q^{85} -121.036i q^{89} +52.2225i q^{91} -3.68561i q^{95} +167.373 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 88 q^{25} + 240 q^{31} + 408 q^{49} - 224 q^{55} - 544 q^{73} + 976 q^{79} + 320 q^{97}+O(q^{100}) \) 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
\(5\) −2.04989 −0.409978 −0.204989 0.978764i \(-0.565716\pi\)
−0.204989 + 0.978764i \(0.565716\pi\)
\(6\) 0 0
\(7\) −7.79796 −1.11399 −0.556997 0.830514i \(-0.688047\pi\)
−0.556997 + 0.830514i \(0.688047\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.09978 0.372707 0.186353 0.982483i \(-0.440333\pi\)
0.186353 + 0.982483i \(0.440333\pi\)
\(12\) 0 0
\(13\) − 6.69694i − 0.515149i −0.966258 0.257575i \(-0.917077\pi\)
0.966258 0.257575i \(-0.0829233\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.3704i 1.13944i 0.821841 + 0.569718i \(0.192947\pi\)
−0.821841 + 0.569718i \(0.807053\pi\)
\(18\) 0 0
\(19\) 1.79796i 0.0946294i 0.998880 + 0.0473147i \(0.0150664\pi\)
−0.998880 + 0.0473147i \(0.984934\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 14.1421i − 0.614875i −0.951568 0.307438i \(-0.900528\pi\)
0.951568 0.307438i \(-0.0994716\pi\)
\(24\) 0 0
\(25\) −20.7980 −0.831918
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −28.4914 −0.982460 −0.491230 0.871030i \(-0.663453\pi\)
−0.491230 + 0.871030i \(0.663453\pi\)
\(30\) 0 0
\(31\) 20.2020 0.651679 0.325839 0.945425i \(-0.394353\pi\)
0.325839 + 0.945425i \(0.394353\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 15.9849 0.456713
\(36\) 0 0
\(37\) − 41.5959i − 1.12421i −0.827065 0.562107i \(-0.809991\pi\)
0.827065 0.562107i \(-0.190009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.94253i 0.120550i 0.998182 + 0.0602748i \(0.0191977\pi\)
−0.998182 + 0.0602748i \(0.980802\pi\)
\(42\) 0 0
\(43\) 75.1918i 1.74865i 0.485343 + 0.874324i \(0.338695\pi\)
−0.485343 + 0.874324i \(0.661305\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13.5707i − 0.288738i −0.989524 0.144369i \(-0.953885\pi\)
0.989524 0.144369i \(-0.0461152\pi\)
\(48\) 0 0
\(49\) 11.8082 0.240983
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 20.8633 0.393646 0.196823 0.980439i \(-0.436938\pi\)
0.196823 + 0.980439i \(0.436938\pi\)
\(54\) 0 0
\(55\) −8.40408 −0.152801
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −54.8542 −0.929732 −0.464866 0.885381i \(-0.653897\pi\)
−0.464866 + 0.885381i \(0.653897\pi\)
\(60\) 0 0
\(61\) 89.1918i 1.46216i 0.682291 + 0.731081i \(0.260984\pi\)
−0.682291 + 0.731081i \(0.739016\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.7280i 0.211200i
\(66\) 0 0
\(67\) − 37.7980i − 0.564149i −0.959393 0.282074i \(-0.908978\pi\)
0.959393 0.282074i \(-0.0910225\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 117.937i − 1.66108i −0.556958 0.830541i \(-0.688032\pi\)
0.556958 0.830541i \(-0.311968\pi\)
\(72\) 0 0
\(73\) −48.4041 −0.663070 −0.331535 0.943443i \(-0.607566\pi\)
−0.331535 + 0.943443i \(0.607566\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −31.9699 −0.415193
\(78\) 0 0
\(79\) 92.6061 1.17223 0.586115 0.810228i \(-0.300657\pi\)
0.586115 + 0.810228i \(0.300657\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 158.806 1.91333 0.956663 0.291197i \(-0.0940535\pi\)
0.956663 + 0.291197i \(0.0940535\pi\)
\(84\) 0 0
\(85\) − 39.7071i − 0.467143i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 121.036i − 1.35996i −0.733230 0.679980i \(-0.761988\pi\)
0.733230 0.679980i \(-0.238012\pi\)
\(90\) 0 0
\(91\) 52.2225i 0.573873i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 3.68561i − 0.0387959i
\(96\) 0 0
\(97\) 167.373 1.72550 0.862750 0.505631i \(-0.168740\pi\)
0.862750 + 0.505631i \(0.168740\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 91.7024 0.907944 0.453972 0.891016i \(-0.350007\pi\)
0.453972 + 0.891016i \(0.350007\pi\)
\(102\) 0 0
\(103\) −58.9898 −0.572716 −0.286358 0.958123i \(-0.592445\pi\)
−0.286358 + 0.958123i \(0.592445\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 182.676 1.70725 0.853626 0.520886i \(-0.174398\pi\)
0.853626 + 0.520886i \(0.174398\pi\)
\(108\) 0 0
\(109\) 62.6969i 0.575201i 0.957750 + 0.287601i \(0.0928576\pi\)
−0.957750 + 0.287601i \(0.907142\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.7567i 0.0951919i 0.998867 + 0.0475959i \(0.0151560\pi\)
−0.998867 + 0.0475959i \(0.984844\pi\)
\(114\) 0 0
\(115\) 28.9898i 0.252085i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 151.050i − 1.26932i
\(120\) 0 0
\(121\) −104.192 −0.861090
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 93.8807 0.751046
\(126\) 0 0
\(127\) 222.182 1.74946 0.874731 0.484609i \(-0.161038\pi\)
0.874731 + 0.484609i \(0.161038\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 111.166 0.848594 0.424297 0.905523i \(-0.360521\pi\)
0.424297 + 0.905523i \(0.360521\pi\)
\(132\) 0 0
\(133\) − 14.0204i − 0.105417i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 76.5104i − 0.558470i −0.960223 0.279235i \(-0.909919\pi\)
0.960223 0.279235i \(-0.0900809\pi\)
\(138\) 0 0
\(139\) 223.778i 1.60991i 0.593336 + 0.804955i \(0.297811\pi\)
−0.593336 + 0.804955i \(0.702189\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 27.4559i − 0.192000i
\(144\) 0 0
\(145\) 58.4041 0.402787
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 121.672 0.816592 0.408296 0.912850i \(-0.366123\pi\)
0.408296 + 0.912850i \(0.366123\pi\)
\(150\) 0 0
\(151\) 82.9898 0.549601 0.274801 0.961501i \(-0.411388\pi\)
0.274801 + 0.961501i \(0.411388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −41.4119 −0.267174
\(156\) 0 0
\(157\) 32.4245i 0.206525i 0.994654 + 0.103263i \(0.0329282\pi\)
−0.994654 + 0.103263i \(0.967072\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 110.280i 0.684968i
\(162\) 0 0
\(163\) − 13.1714i − 0.0808063i −0.999183 0.0404031i \(-0.987136\pi\)
0.999183 0.0404031i \(-0.0128642\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 50.6548i − 0.303322i −0.988433 0.151661i \(-0.951538\pi\)
0.988433 0.151661i \(-0.0484622\pi\)
\(168\) 0 0
\(169\) 124.151 0.734621
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −294.335 −1.70136 −0.850678 0.525687i \(-0.823808\pi\)
−0.850678 + 0.525687i \(0.823808\pi\)
\(174\) 0 0
\(175\) 162.182 0.926752
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 205.047 1.14551 0.572756 0.819726i \(-0.305874\pi\)
0.572756 + 0.819726i \(0.305874\pi\)
\(180\) 0 0
\(181\) − 107.464i − 0.593725i −0.954920 0.296863i \(-0.904060\pi\)
0.954920 0.296863i \(-0.0959404\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 85.2670i 0.460903i
\(186\) 0 0
\(187\) 79.4143i 0.424675i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 219.731i − 1.15043i −0.818004 0.575213i \(-0.804919\pi\)
0.818004 0.575213i \(-0.195081\pi\)
\(192\) 0 0
\(193\) −177.151 −0.917881 −0.458940 0.888467i \(-0.651771\pi\)
−0.458940 + 0.888467i \(0.651771\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −182.055 −0.924136 −0.462068 0.886845i \(-0.652892\pi\)
−0.462068 + 0.886845i \(0.652892\pi\)
\(198\) 0 0
\(199\) 267.757 1.34551 0.672757 0.739864i \(-0.265110\pi\)
0.672757 + 0.739864i \(0.265110\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 222.174 1.09446
\(204\) 0 0
\(205\) − 10.1316i − 0.0494226i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 7.37123i 0.0352690i
\(210\) 0 0
\(211\) 40.2225i 0.190628i 0.995447 + 0.0953139i \(0.0303855\pi\)
−0.995447 + 0.0953139i \(0.969615\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 154.135i − 0.716906i
\(216\) 0 0
\(217\) −157.535 −0.725966
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 129.722 0.586979
\(222\) 0 0
\(223\) −44.9694 −0.201656 −0.100828 0.994904i \(-0.532149\pi\)
−0.100828 + 0.994904i \(0.532149\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 180.605 0.795618 0.397809 0.917468i \(-0.369771\pi\)
0.397809 + 0.917468i \(0.369771\pi\)
\(228\) 0 0
\(229\) − 129.666i − 0.566228i −0.959086 0.283114i \(-0.908632\pi\)
0.959086 0.283114i \(-0.0913676\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 209.832i 0.900566i 0.892886 + 0.450283i \(0.148677\pi\)
−0.892886 + 0.450283i \(0.851323\pi\)
\(234\) 0 0
\(235\) 27.8184i 0.118376i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 334.583i − 1.39993i −0.714178 0.699964i \(-0.753199\pi\)
0.714178 0.699964i \(-0.246801\pi\)
\(240\) 0 0
\(241\) −51.8184 −0.215014 −0.107507 0.994204i \(-0.534287\pi\)
−0.107507 + 0.994204i \(0.534287\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −24.2054 −0.0987976
\(246\) 0 0
\(247\) 12.0408 0.0487483
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 311.541 1.24120 0.620600 0.784127i \(-0.286889\pi\)
0.620600 + 0.784127i \(0.286889\pi\)
\(252\) 0 0
\(253\) − 57.9796i − 0.229168i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 50.0978i − 0.194933i −0.995239 0.0974665i \(-0.968926\pi\)
0.995239 0.0974665i \(-0.0310739\pi\)
\(258\) 0 0
\(259\) 324.363i 1.25237i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 334.268i − 1.27098i −0.772108 0.635491i \(-0.780798\pi\)
0.772108 0.635491i \(-0.219202\pi\)
\(264\) 0 0
\(265\) −42.7673 −0.161386
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −80.1318 −0.297888 −0.148944 0.988846i \(-0.547587\pi\)
−0.148944 + 0.988846i \(0.547587\pi\)
\(270\) 0 0
\(271\) −61.7775 −0.227961 −0.113981 0.993483i \(-0.536360\pi\)
−0.113981 + 0.993483i \(0.536360\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −85.2670 −0.310062
\(276\) 0 0
\(277\) 423.242i 1.52795i 0.645247 + 0.763974i \(0.276755\pi\)
−0.645247 + 0.763974i \(0.723245\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 63.6396i − 0.226475i −0.993568 0.113238i \(-0.963878\pi\)
0.993568 0.113238i \(-0.0361222\pi\)
\(282\) 0 0
\(283\) − 98.4245i − 0.347790i −0.984764 0.173895i \(-0.944365\pi\)
0.984764 0.173895i \(-0.0556353\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 38.5417i − 0.134291i
\(288\) 0 0
\(289\) −86.2122 −0.298312
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −368.888 −1.25900 −0.629502 0.776999i \(-0.716741\pi\)
−0.629502 + 0.776999i \(0.716741\pi\)
\(294\) 0 0
\(295\) 112.445 0.381169
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −94.7090 −0.316753
\(300\) 0 0
\(301\) − 586.343i − 1.94798i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 182.833i − 0.599453i
\(306\) 0 0
\(307\) 507.737i 1.65387i 0.562301 + 0.826933i \(0.309916\pi\)
−0.562301 + 0.826933i \(0.690084\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 36.7984i 0.118323i 0.998248 + 0.0591614i \(0.0188427\pi\)
−0.998248 + 0.0591614i \(0.981157\pi\)
\(312\) 0 0
\(313\) 287.576 0.918772 0.459386 0.888237i \(-0.348070\pi\)
0.459386 + 0.888237i \(0.348070\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 450.726 1.42185 0.710925 0.703268i \(-0.248277\pi\)
0.710925 + 0.703268i \(0.248277\pi\)
\(318\) 0 0
\(319\) −116.808 −0.366170
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −34.8272 −0.107824
\(324\) 0 0
\(325\) 139.283i 0.428562i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 105.824i 0.321652i
\(330\) 0 0
\(331\) − 133.798i − 0.404223i −0.979362 0.202112i \(-0.935220\pi\)
0.979362 0.202112i \(-0.0647804\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 77.4816i 0.231288i
\(336\) 0 0
\(337\) −124.808 −0.370351 −0.185175 0.982706i \(-0.559285\pi\)
−0.185175 + 0.982706i \(0.559285\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 82.8238 0.242885
\(342\) 0 0
\(343\) 290.020 0.845541
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 571.699 1.64755 0.823773 0.566919i \(-0.191865\pi\)
0.823773 + 0.566919i \(0.191865\pi\)
\(348\) 0 0
\(349\) 499.898i 1.43237i 0.697909 + 0.716186i \(0.254114\pi\)
−0.697909 + 0.716186i \(0.745886\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 5.64242i − 0.0159842i −0.999968 0.00799210i \(-0.997456\pi\)
0.999968 0.00799210i \(-0.00254399\pi\)
\(354\) 0 0
\(355\) 241.757i 0.681006i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 586.769i 1.63445i 0.576316 + 0.817227i \(0.304490\pi\)
−0.576316 + 0.817227i \(0.695510\pi\)
\(360\) 0 0
\(361\) 357.767 0.991045
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 99.2229 0.271844
\(366\) 0 0
\(367\) −26.6265 −0.0725519 −0.0362759 0.999342i \(-0.511550\pi\)
−0.0362759 + 0.999342i \(0.511550\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −162.691 −0.438520
\(372\) 0 0
\(373\) 226.041i 0.606008i 0.952989 + 0.303004i \(0.0979895\pi\)
−0.952989 + 0.303004i \(0.902011\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 190.805i 0.506114i
\(378\) 0 0
\(379\) 343.292i 0.905783i 0.891566 + 0.452892i \(0.149608\pi\)
−0.891566 + 0.452892i \(0.850392\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 654.680i − 1.70935i −0.519166 0.854673i \(-0.673757\pi\)
0.519166 0.854673i \(-0.326243\pi\)
\(384\) 0 0
\(385\) 65.5347 0.170220
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 298.948 0.768504 0.384252 0.923228i \(-0.374459\pi\)
0.384252 + 0.923228i \(0.374459\pi\)
\(390\) 0 0
\(391\) 273.939 0.700611
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −189.832 −0.480588
\(396\) 0 0
\(397\) − 529.596i − 1.33399i −0.745060 0.666997i \(-0.767579\pi\)
0.745060 0.666997i \(-0.232421\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 447.034i − 1.11480i −0.830244 0.557399i \(-0.811799\pi\)
0.830244 0.557399i \(-0.188201\pi\)
\(402\) 0 0
\(403\) − 135.292i − 0.335712i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 170.534i − 0.419002i
\(408\) 0 0
\(409\) 503.292 1.23054 0.615271 0.788316i \(-0.289047\pi\)
0.615271 + 0.788316i \(0.289047\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 427.751 1.03572
\(414\) 0 0
\(415\) −325.535 −0.784421
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −142.722 −0.340624 −0.170312 0.985390i \(-0.554478\pi\)
−0.170312 + 0.985390i \(0.554478\pi\)
\(420\) 0 0
\(421\) 41.3031i 0.0981070i 0.998796 + 0.0490535i \(0.0156205\pi\)
−0.998796 + 0.0490535i \(0.984380\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 402.865i − 0.947917i
\(426\) 0 0
\(427\) − 695.514i − 1.62884i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 441.719i − 1.02487i −0.858726 0.512436i \(-0.828743\pi\)
0.858726 0.512436i \(-0.171257\pi\)
\(432\) 0 0
\(433\) −460.343 −1.06315 −0.531574 0.847012i \(-0.678399\pi\)
−0.531574 + 0.847012i \(0.678399\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.4270 0.0581853
\(438\) 0 0
\(439\) 219.353 0.499665 0.249833 0.968289i \(-0.419624\pi\)
0.249833 + 0.968289i \(0.419624\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.2982 0.0435626 0.0217813 0.999763i \(-0.493066\pi\)
0.0217813 + 0.999763i \(0.493066\pi\)
\(444\) 0 0
\(445\) 248.111i 0.557553i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 573.627i − 1.27756i −0.769387 0.638782i \(-0.779438\pi\)
0.769387 0.638782i \(-0.220562\pi\)
\(450\) 0 0
\(451\) 20.2633i 0.0449296i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 107.050i − 0.235275i
\(456\) 0 0
\(457\) 71.0102 0.155383 0.0776917 0.996977i \(-0.475245\pi\)
0.0776917 + 0.996977i \(0.475245\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −751.869 −1.63095 −0.815476 0.578791i \(-0.803525\pi\)
−0.815476 + 0.578791i \(0.803525\pi\)
\(462\) 0 0
\(463\) −638.182 −1.37836 −0.689181 0.724589i \(-0.742030\pi\)
−0.689181 + 0.724589i \(0.742030\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −225.032 −0.481867 −0.240933 0.970542i \(-0.577453\pi\)
−0.240933 + 0.970542i \(0.577453\pi\)
\(468\) 0 0
\(469\) 294.747i 0.628458i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 308.270i 0.651733i
\(474\) 0 0
\(475\) − 37.3939i − 0.0787240i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 757.675i 1.58178i 0.611955 + 0.790892i \(0.290383\pi\)
−0.611955 + 0.790892i \(0.709617\pi\)
\(480\) 0 0
\(481\) −278.565 −0.579138
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −343.097 −0.707416
\(486\) 0 0
\(487\) −266.100 −0.546407 −0.273203 0.961956i \(-0.588083\pi\)
−0.273203 + 0.961956i \(0.588083\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14.3701 −0.0292671 −0.0146335 0.999893i \(-0.504658\pi\)
−0.0146335 + 0.999893i \(0.504658\pi\)
\(492\) 0 0
\(493\) − 551.889i − 1.11945i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 919.666i 1.85043i
\(498\) 0 0
\(499\) − 603.151i − 1.20872i −0.796712 0.604360i \(-0.793429\pi\)
0.796712 0.604360i \(-0.206571\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 781.388i 1.55345i 0.629837 + 0.776727i \(0.283122\pi\)
−0.629837 + 0.776727i \(0.716878\pi\)
\(504\) 0 0
\(505\) −187.980 −0.372237
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 930.002 1.82712 0.913558 0.406709i \(-0.133324\pi\)
0.913558 + 0.406709i \(0.133324\pi\)
\(510\) 0 0
\(511\) 377.453 0.738656
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 120.922 0.234801
\(516\) 0 0
\(517\) − 55.6367i − 0.107615i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 463.947i 0.890494i 0.895408 + 0.445247i \(0.146884\pi\)
−0.895408 + 0.445247i \(0.853116\pi\)
\(522\) 0 0
\(523\) − 426.524i − 0.815534i −0.913086 0.407767i \(-0.866307\pi\)
0.913086 0.407767i \(-0.133693\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 391.322i 0.742546i
\(528\) 0 0
\(529\) 329.000 0.621928
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 33.0998 0.0621010
\(534\) 0 0
\(535\) −374.465 −0.699935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 48.4108 0.0898160
\(540\) 0 0
\(541\) 281.930i 0.521127i 0.965457 + 0.260563i \(0.0839083\pi\)
−0.965457 + 0.260563i \(0.916092\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 128.522i − 0.235820i
\(546\) 0 0
\(547\) 204.727i 0.374272i 0.982334 + 0.187136i \(0.0599204\pi\)
−0.982334 + 0.187136i \(0.940080\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 51.2263i − 0.0929697i
\(552\) 0 0
\(553\) −722.139 −1.30586
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −813.465 −1.46044 −0.730220 0.683212i \(-0.760582\pi\)
−0.730220 + 0.683212i \(0.760582\pi\)
\(558\) 0 0
\(559\) 503.555 0.900814
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −666.523 −1.18388 −0.591939 0.805983i \(-0.701637\pi\)
−0.591939 + 0.805983i \(0.701637\pi\)
\(564\) 0 0
\(565\) − 22.0500i − 0.0390265i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 308.756i − 0.542629i −0.962491 0.271315i \(-0.912542\pi\)
0.962491 0.271315i \(-0.0874584\pi\)
\(570\) 0 0
\(571\) − 534.747i − 0.936510i −0.883594 0.468255i \(-0.844883\pi\)
0.883594 0.468255i \(-0.155117\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 294.128i 0.511526i
\(576\) 0 0
\(577\) 158.000 0.273830 0.136915 0.990583i \(-0.456281\pi\)
0.136915 + 0.990583i \(0.456281\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1238.36 −2.13143
\(582\) 0 0
\(583\) 85.5347 0.146715
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −783.503 −1.33476 −0.667379 0.744718i \(-0.732584\pi\)
−0.667379 + 0.744718i \(0.732584\pi\)
\(588\) 0 0
\(589\) 36.3224i 0.0616680i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 26.1283i − 0.0440613i −0.999757 0.0220306i \(-0.992987\pi\)
0.999757 0.0220306i \(-0.00701313\pi\)
\(594\) 0 0
\(595\) 309.635i 0.520394i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 440.063i 0.734663i 0.930090 + 0.367331i \(0.119728\pi\)
−0.930090 + 0.367331i \(0.880272\pi\)
\(600\) 0 0
\(601\) −66.0000 −0.109817 −0.0549085 0.998491i \(-0.517487\pi\)
−0.0549085 + 0.998491i \(0.517487\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 213.582 0.353027
\(606\) 0 0
\(607\) −465.373 −0.766678 −0.383339 0.923608i \(-0.625226\pi\)
−0.383339 + 0.923608i \(0.625226\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −90.8820 −0.148743
\(612\) 0 0
\(613\) − 580.706i − 0.947318i −0.880708 0.473659i \(-0.842933\pi\)
0.880708 0.473659i \(-0.157067\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 43.8118i − 0.0710077i −0.999370 0.0355039i \(-0.988696\pi\)
0.999370 0.0355039i \(-0.0113036\pi\)
\(618\) 0 0
\(619\) 401.980i 0.649402i 0.945817 + 0.324701i \(0.105264\pi\)
−0.945817 + 0.324701i \(0.894736\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 943.837i 1.51499i
\(624\) 0 0
\(625\) 327.504 0.524007
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 805.729 1.28097
\(630\) 0 0
\(631\) −94.5857 −0.149898 −0.0749491 0.997187i \(-0.523879\pi\)
−0.0749491 + 0.997187i \(0.523879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −455.447 −0.717240
\(636\) 0 0
\(637\) − 79.0785i − 0.124142i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 866.698i 1.35210i 0.736854 + 0.676051i \(0.236310\pi\)
−0.736854 + 0.676051i \(0.763690\pi\)
\(642\) 0 0
\(643\) − 802.020i − 1.24731i −0.781700 0.623655i \(-0.785647\pi\)
0.781700 0.623655i \(-0.214353\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 109.223i 0.168815i 0.996431 + 0.0844076i \(0.0268998\pi\)
−0.996431 + 0.0844076i \(0.973100\pi\)
\(648\) 0 0
\(649\) −224.890 −0.346517
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −976.727 −1.49575 −0.747877 0.663837i \(-0.768927\pi\)
−0.747877 + 0.663837i \(0.768927\pi\)
\(654\) 0 0
\(655\) −227.878 −0.347905
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 528.887 0.802560 0.401280 0.915955i \(-0.368565\pi\)
0.401280 + 0.915955i \(0.368565\pi\)
\(660\) 0 0
\(661\) − 287.535i − 0.435000i −0.976060 0.217500i \(-0.930210\pi\)
0.976060 0.217500i \(-0.0697901\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 28.7403i 0.0432185i
\(666\) 0 0
\(667\) 402.929i 0.604091i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 365.667i 0.544958i
\(672\) 0 0
\(673\) −597.029 −0.887115 −0.443558 0.896246i \(-0.646284\pi\)
−0.443558 + 0.896246i \(0.646284\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 652.630 0.964003 0.482001 0.876170i \(-0.339910\pi\)
0.482001 + 0.876170i \(0.339910\pi\)
\(678\) 0 0
\(679\) −1305.17 −1.92220
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −288.971 −0.423091 −0.211546 0.977368i \(-0.567850\pi\)
−0.211546 + 0.977368i \(0.567850\pi\)
\(684\) 0 0
\(685\) 156.838i 0.228960i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 139.720i − 0.202787i
\(690\) 0 0
\(691\) − 752.241i − 1.08863i −0.838882 0.544313i \(-0.816790\pi\)
0.838882 0.544313i \(-0.183210\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 458.719i − 0.660027i
\(696\) 0 0
\(697\) −95.7388 −0.137358
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 808.152 1.15286 0.576428 0.817148i \(-0.304446\pi\)
0.576428 + 0.817148i \(0.304446\pi\)
\(702\) 0 0
\(703\) 74.7878 0.106384
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −715.091 −1.01144
\(708\) 0 0
\(709\) − 737.748i − 1.04055i −0.854000 0.520274i \(-0.825830\pi\)
0.854000 0.520274i \(-0.174170\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 285.700i − 0.400701i
\(714\) 0 0
\(715\) 56.2816i 0.0787156i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 236.275i 0.328616i 0.986409 + 0.164308i \(0.0525390\pi\)
−0.986409 + 0.164308i \(0.947461\pi\)
\(720\) 0 0
\(721\) 460.000 0.638003
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 592.562 0.817327
\(726\) 0 0
\(727\) −444.606 −0.611563 −0.305781 0.952102i \(-0.598918\pi\)
−0.305781 + 0.952102i \(0.598918\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1456.50 −1.99247
\(732\) 0 0
\(733\) 1196.74i 1.63265i 0.577590 + 0.816327i \(0.303993\pi\)
−0.577590 + 0.816327i \(0.696007\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 154.963i − 0.210262i
\(738\) 0 0
\(739\) − 1437.78i − 1.94557i −0.231712 0.972784i \(-0.574433\pi\)
0.231712 0.972784i \(-0.425567\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1265.05i 1.70262i 0.524661 + 0.851311i \(0.324192\pi\)
−0.524661 + 0.851311i \(0.675808\pi\)
\(744\) 0 0
\(745\) −249.414 −0.334784
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1424.50 −1.90187
\(750\) 0 0
\(751\) −578.990 −0.770958 −0.385479 0.922717i \(-0.625964\pi\)
−0.385479 + 0.922717i \(0.625964\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −170.120 −0.225324
\(756\) 0 0
\(757\) 82.6561i 0.109189i 0.998509 + 0.0545945i \(0.0173866\pi\)
−0.998509 + 0.0545945i \(0.982613\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 512.088i 0.672915i 0.941699 + 0.336457i \(0.109229\pi\)
−0.941699 + 0.336457i \(0.890771\pi\)
\(762\) 0 0
\(763\) − 488.908i − 0.640771i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 367.355i 0.478950i
\(768\) 0 0
\(769\) 1267.49 1.64824 0.824118 0.566418i \(-0.191671\pi\)
0.824118 + 0.566418i \(0.191671\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −437.342 −0.565772 −0.282886 0.959154i \(-0.591292\pi\)
−0.282886 + 0.959154i \(0.591292\pi\)
\(774\) 0 0
\(775\) −420.161 −0.542144
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.88647 −0.0114075
\(780\) 0 0
\(781\) − 483.514i − 0.619096i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 66.4666i − 0.0846708i
\(786\) 0 0
\(787\) − 965.031i − 1.22621i −0.790000 0.613107i \(-0.789919\pi\)
0.790000 0.613107i \(-0.210081\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 83.8802i − 0.106043i
\(792\) 0 0
\(793\) 597.312 0.753231
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 996.513 1.25033 0.625165 0.780492i \(-0.285032\pi\)
0.625165 + 0.780492i \(0.285032\pi\)
\(798\) 0 0
\(799\) 262.869 0.328998
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −198.446 −0.247131
\(804\) 0 0
\(805\) − 226.061i − 0.280821i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1530.32i − 1.89162i −0.324723 0.945809i \(-0.605271\pi\)
0.324723 0.945809i \(-0.394729\pi\)
\(810\) 0 0
\(811\) 833.716i 1.02801i 0.857787 + 0.514005i \(0.171839\pi\)
−0.857787 + 0.514005i \(0.828161\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.9999i 0.0331288i
\(816\) 0 0
\(817\) −135.192 −0.165473
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1414.40 1.72278 0.861391 0.507942i \(-0.169593\pi\)
0.861391 + 0.507942i \(0.169593\pi\)
\(822\) 0 0
\(823\) −1168.16 −1.41939 −0.709697 0.704507i \(-0.751168\pi\)
−0.709697 + 0.704507i \(0.751168\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −624.110 −0.754667 −0.377334 0.926077i \(-0.623159\pi\)
−0.377334 + 0.926077i \(0.623159\pi\)
\(828\) 0 0
\(829\) 1531.46i 1.84736i 0.383161 + 0.923682i \(0.374836\pi\)
−0.383161 + 0.923682i \(0.625164\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 228.729i 0.274584i
\(834\) 0 0
\(835\) 103.837i 0.124355i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1013.95i − 1.20852i −0.796787 0.604260i \(-0.793469\pi\)
0.796787 0.604260i \(-0.206531\pi\)
\(840\) 0 0
\(841\) −29.2429 −0.0347715
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −254.496 −0.301178
\(846\) 0 0
\(847\) 812.484 0.959249
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −588.255 −0.691252
\(852\) 0 0
\(853\) 258.808i 0.303409i 0.988426 + 0.151705i \(0.0484763\pi\)
−0.988426 + 0.151705i \(0.951524\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1453.90i − 1.69650i −0.529600 0.848248i \(-0.677658\pi\)
0.529600 0.848248i \(-0.322342\pi\)
\(858\) 0 0
\(859\) − 740.645i − 0.862218i −0.902300 0.431109i \(-0.858123\pi\)
0.902300 0.431109i \(-0.141877\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 675.593i 0.782842i 0.920212 + 0.391421i \(0.128016\pi\)
−0.920212 + 0.391421i \(0.871984\pi\)
\(864\) 0 0
\(865\) 603.353 0.697518
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 379.664 0.436898
\(870\) 0 0
\(871\) −253.131 −0.290621
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −732.078 −0.836660
\(876\) 0 0
\(877\) − 321.678i − 0.366793i −0.983039 0.183397i \(-0.941291\pi\)
0.983039 0.183397i \(-0.0587092\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 586.899i 0.666173i 0.942896 + 0.333087i \(0.108090\pi\)
−0.942896 + 0.333087i \(0.891910\pi\)
\(882\) 0 0
\(883\) 1186.95i 1.34422i 0.740451 + 0.672110i \(0.234612\pi\)
−0.740451 + 0.672110i \(0.765388\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 682.707i 0.769681i 0.922983 + 0.384841i \(0.125744\pi\)
−0.922983 + 0.384841i \(0.874256\pi\)
\(888\) 0 0
\(889\) −1732.56 −1.94889
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24.3995 0.0273231
\(894\) 0 0
\(895\) −420.322 −0.469634
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −575.583 −0.640249
\(900\) 0 0
\(901\) 404.130i 0.448534i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 220.290i 0.243414i
\(906\) 0 0
\(907\) 803.837i 0.886259i 0.896458 + 0.443129i \(0.146132\pi\)
−0.896458 + 0.443129i \(0.853868\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1090.77i 1.19733i 0.800998 + 0.598667i \(0.204303\pi\)
−0.800998 + 0.598667i \(0.795697\pi\)
\(912\) 0 0
\(913\) 651.069 0.713110
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −866.867 −0.945329
\(918\) 0 0
\(919\) −362.141 −0.394060 −0.197030 0.980397i \(-0.563130\pi\)
−0.197030 + 0.980397i \(0.563130\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −789.815 −0.855704
\(924\) 0 0
\(925\) 865.110i 0.935254i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 421.437i 0.453646i 0.973936 + 0.226823i \(0.0728339\pi\)
−0.973936 + 0.226823i \(0.927166\pi\)
\(930\) 0 0
\(931\) 21.2306i 0.0228041i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 162.790i − 0.174107i
\(936\) 0 0
\(937\) −9.23266 −0.00985342 −0.00492671 0.999988i \(-0.501568\pi\)
−0.00492671 + 0.999988i \(0.501568\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −175.499 −0.186503 −0.0932513 0.995643i \(-0.529726\pi\)
−0.0932513 + 0.995643i \(0.529726\pi\)
\(942\) 0 0
\(943\) 69.8979 0.0741230
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1185.99 1.25237 0.626185 0.779675i \(-0.284616\pi\)
0.626185 + 0.779675i \(0.284616\pi\)
\(948\) 0 0
\(949\) 324.159i 0.341580i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 929.581i 0.975426i 0.873004 + 0.487713i \(0.162169\pi\)
−0.873004 + 0.487713i \(0.837831\pi\)
\(954\) 0 0
\(955\) 450.424i 0.471649i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 596.625i 0.622132i
\(960\) 0 0
\(961\) −552.878 −0.575315
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 363.140 0.376311
\(966\) 0 0
\(967\) −272.284 −0.281576 −0.140788 0.990040i \(-0.544964\pi\)
−0.140788 + 0.990040i \(0.544964\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1439.00 1.48197 0.740986 0.671520i \(-0.234358\pi\)
0.740986 + 0.671520i \(0.234358\pi\)
\(972\) 0 0
\(973\) − 1745.01i − 1.79343i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1448.98i − 1.48309i −0.670902 0.741546i \(-0.734093\pi\)
0.670902 0.741546i \(-0.265907\pi\)
\(978\) 0 0
\(979\) − 496.222i − 0.506867i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 289.013i 0.294011i 0.989136 + 0.147006i \(0.0469636\pi\)
−0.989136 + 0.147006i \(0.953036\pi\)
\(984\) 0 0
\(985\) 373.192 0.378875
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1063.37 1.07520
\(990\) 0 0
\(991\) 340.202 0.343292 0.171646 0.985159i \(-0.445092\pi\)
0.171646 + 0.985159i \(0.445092\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −548.872 −0.551630
\(996\) 0 0
\(997\) − 1664.06i − 1.66906i −0.550959 0.834532i \(-0.685738\pi\)
0.550959 0.834532i \(-0.314262\pi\)
\(998\) 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.h.k.2177.3 8
3.2 odd 2 inner 2304.3.h.k.2177.5 8
4.3 odd 2 2304.3.h.i.2177.3 8
8.3 odd 2 2304.3.h.i.2177.6 8
8.5 even 2 inner 2304.3.h.k.2177.6 8
12.11 even 2 2304.3.h.i.2177.5 8
16.3 odd 4 1152.3.e.f.1025.3 yes 4
16.5 even 4 1152.3.e.d.1025.2 yes 4
16.11 odd 4 1152.3.e.h.1025.2 yes 4
16.13 even 4 1152.3.e.b.1025.3 yes 4
24.5 odd 2 inner 2304.3.h.k.2177.4 8
24.11 even 2 2304.3.h.i.2177.4 8
48.5 odd 4 1152.3.e.d.1025.3 yes 4
48.11 even 4 1152.3.e.h.1025.3 yes 4
48.29 odd 4 1152.3.e.b.1025.2 4
48.35 even 4 1152.3.e.f.1025.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.e.b.1025.2 4 48.29 odd 4
1152.3.e.b.1025.3 yes 4 16.13 even 4
1152.3.e.d.1025.2 yes 4 16.5 even 4
1152.3.e.d.1025.3 yes 4 48.5 odd 4
1152.3.e.f.1025.2 yes 4 48.35 even 4
1152.3.e.f.1025.3 yes 4 16.3 odd 4
1152.3.e.h.1025.2 yes 4 16.11 odd 4
1152.3.e.h.1025.3 yes 4 48.11 even 4
2304.3.h.i.2177.3 8 4.3 odd 2
2304.3.h.i.2177.4 8 24.11 even 2
2304.3.h.i.2177.5 8 12.11 even 2
2304.3.h.i.2177.6 8 8.3 odd 2
2304.3.h.k.2177.3 8 1.1 even 1 trivial
2304.3.h.k.2177.4 8 24.5 odd 2 inner
2304.3.h.k.2177.5 8 3.2 odd 2 inner
2304.3.h.k.2177.6 8 8.5 even 2 inner