Properties

Label 4140.3.d.c.2161.4
Level $4140$
Weight $3$
Character 4140.2161
Analytic conductor $112.807$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,3,Mod(2161,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.2161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.d (of order \(2\), degree \(1\), 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: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2161.4
Character \(\chi\) \(=\) 4140.2161
Dual form 4140.3.d.c.2161.29

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{5} -4.23458i q^{7} +O(q^{10})\) \(q-2.23607i q^{5} -4.23458i q^{7} -3.42048i q^{11} -13.7692 q^{13} +5.48070i q^{17} +4.66120i q^{19} +(-15.0674 - 17.3774i) q^{23} -5.00000 q^{25} -51.3592 q^{29} +36.0494 q^{31} -9.46881 q^{35} +70.1247i q^{37} +29.1052 q^{41} +24.5556i q^{43} -58.9027 q^{47} +31.0683 q^{49} +26.4800i q^{53} -7.64843 q^{55} +8.79966 q^{59} -30.1183i q^{61} +30.7888i q^{65} -5.12183i q^{67} +45.7745 q^{71} +75.2188 q^{73} -14.4843 q^{77} +62.2814i q^{79} -137.687i q^{83} +12.2552 q^{85} +64.5505i q^{89} +58.3066i q^{91} +10.4228 q^{95} +171.496i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 4.23458i 0.604940i −0.953159 0.302470i \(-0.902189\pi\)
0.953159 0.302470i \(-0.0978112\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.42048i 0.310953i −0.987840 0.155476i \(-0.950309\pi\)
0.987840 0.155476i \(-0.0496913\pi\)
\(12\) 0 0
\(13\) −13.7692 −1.05917 −0.529583 0.848258i \(-0.677652\pi\)
−0.529583 + 0.848258i \(0.677652\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.48070i 0.322394i 0.986922 + 0.161197i \(0.0515355\pi\)
−0.986922 + 0.161197i \(0.948465\pi\)
\(18\) 0 0
\(19\) 4.66120i 0.245326i 0.992448 + 0.122663i \(0.0391435\pi\)
−0.992448 + 0.122663i \(0.960857\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −15.0674 17.3774i −0.655104 0.755539i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −51.3592 −1.77101 −0.885504 0.464633i \(-0.846186\pi\)
−0.885504 + 0.464633i \(0.846186\pi\)
\(30\) 0 0
\(31\) 36.0494 1.16288 0.581442 0.813588i \(-0.302489\pi\)
0.581442 + 0.813588i \(0.302489\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.46881 −0.270537
\(36\) 0 0
\(37\) 70.1247i 1.89526i 0.319369 + 0.947631i \(0.396529\pi\)
−0.319369 + 0.947631i \(0.603471\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 29.1052 0.709883 0.354942 0.934888i \(-0.384501\pi\)
0.354942 + 0.934888i \(0.384501\pi\)
\(42\) 0 0
\(43\) 24.5556i 0.571062i 0.958370 + 0.285531i \(0.0921699\pi\)
−0.958370 + 0.285531i \(0.907830\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −58.9027 −1.25325 −0.626624 0.779322i \(-0.715564\pi\)
−0.626624 + 0.779322i \(0.715564\pi\)
\(48\) 0 0
\(49\) 31.0683 0.634048
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 26.4800i 0.499622i 0.968295 + 0.249811i \(0.0803685\pi\)
−0.968295 + 0.249811i \(0.919632\pi\)
\(54\) 0 0
\(55\) −7.64843 −0.139062
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.79966 0.149147 0.0745734 0.997216i \(-0.476241\pi\)
0.0745734 + 0.997216i \(0.476241\pi\)
\(60\) 0 0
\(61\) 30.1183i 0.493743i −0.969048 0.246872i \(-0.920597\pi\)
0.969048 0.246872i \(-0.0794026\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 30.7888i 0.473674i
\(66\) 0 0
\(67\) 5.12183i 0.0764452i −0.999269 0.0382226i \(-0.987830\pi\)
0.999269 0.0382226i \(-0.0121696\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 45.7745 0.644711 0.322355 0.946619i \(-0.395525\pi\)
0.322355 + 0.946619i \(0.395525\pi\)
\(72\) 0 0
\(73\) 75.2188 1.03039 0.515197 0.857072i \(-0.327719\pi\)
0.515197 + 0.857072i \(0.327719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.4843 −0.188108
\(78\) 0 0
\(79\) 62.2814i 0.788372i 0.919031 + 0.394186i \(0.128973\pi\)
−0.919031 + 0.394186i \(0.871027\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 137.687i 1.65888i −0.558596 0.829440i \(-0.688660\pi\)
0.558596 0.829440i \(-0.311340\pi\)
\(84\) 0 0
\(85\) 12.2552 0.144179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 64.5505i 0.725286i 0.931928 + 0.362643i \(0.118126\pi\)
−0.931928 + 0.362643i \(0.881874\pi\)
\(90\) 0 0
\(91\) 58.3066i 0.640732i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.4228 0.109713
\(96\) 0 0
\(97\) 171.496i 1.76800i 0.467486 + 0.884000i \(0.345160\pi\)
−0.467486 + 0.884000i \(0.654840\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −52.1005 −0.515847 −0.257923 0.966165i \(-0.583038\pi\)
−0.257923 + 0.966165i \(0.583038\pi\)
\(102\) 0 0
\(103\) 32.9011i 0.319429i −0.987163 0.159714i \(-0.948943\pi\)
0.987163 0.159714i \(-0.0510573\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 35.3025i 0.329930i 0.986299 + 0.164965i \(0.0527511\pi\)
−0.986299 + 0.164965i \(0.947249\pi\)
\(108\) 0 0
\(109\) 56.0630i 0.514339i −0.966366 0.257170i \(-0.917210\pi\)
0.966366 0.257170i \(-0.0827899\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 79.6420i 0.704796i 0.935850 + 0.352398i \(0.114634\pi\)
−0.935850 + 0.352398i \(0.885366\pi\)
\(114\) 0 0
\(115\) −38.8570 + 33.6917i −0.337887 + 0.292971i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 23.2085 0.195029
\(120\) 0 0
\(121\) 109.300 0.903308
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −48.5699 −0.382440 −0.191220 0.981547i \(-0.561244\pi\)
−0.191220 + 0.981547i \(0.561244\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.25291 −0.0248314 −0.0124157 0.999923i \(-0.503952\pi\)
−0.0124157 + 0.999923i \(0.503952\pi\)
\(132\) 0 0
\(133\) 19.7382 0.148408
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.210618i 0.00153735i 1.00000 0.000768677i \(0.000244678\pi\)
−1.00000 0.000768677i \(0.999755\pi\)
\(138\) 0 0
\(139\) 34.9353 0.251333 0.125666 0.992073i \(-0.459893\pi\)
0.125666 + 0.992073i \(0.459893\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 47.0972i 0.329351i
\(144\) 0 0
\(145\) 114.843i 0.792018i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 294.015i 1.97326i −0.162983 0.986629i \(-0.552111\pi\)
0.162983 0.986629i \(-0.447889\pi\)
\(150\) 0 0
\(151\) −76.5200 −0.506755 −0.253377 0.967368i \(-0.581541\pi\)
−0.253377 + 0.967368i \(0.581541\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 80.6090i 0.520058i
\(156\) 0 0
\(157\) 118.942i 0.757592i 0.925480 + 0.378796i \(0.123662\pi\)
−0.925480 + 0.378796i \(0.876338\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −73.5859 + 63.8041i −0.457055 + 0.396299i
\(162\) 0 0
\(163\) 285.654 1.75248 0.876240 0.481874i \(-0.160044\pi\)
0.876240 + 0.481874i \(0.160044\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 144.508 0.865316 0.432658 0.901558i \(-0.357576\pi\)
0.432658 + 0.901558i \(0.357576\pi\)
\(168\) 0 0
\(169\) 20.5900 0.121834
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 118.670 0.685954 0.342977 0.939344i \(-0.388565\pi\)
0.342977 + 0.939344i \(0.388565\pi\)
\(174\) 0 0
\(175\) 21.1729i 0.120988i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.2624 0.0685053 0.0342526 0.999413i \(-0.489095\pi\)
0.0342526 + 0.999413i \(0.489095\pi\)
\(180\) 0 0
\(181\) 228.077i 1.26009i −0.776557 0.630047i \(-0.783035\pi\)
0.776557 0.630047i \(-0.216965\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 156.804 0.847587
\(186\) 0 0
\(187\) 18.7466 0.100249
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 266.736i 1.39652i 0.715843 + 0.698261i \(0.246042\pi\)
−0.715843 + 0.698261i \(0.753958\pi\)
\(192\) 0 0
\(193\) 133.378 0.691077 0.345538 0.938405i \(-0.387696\pi\)
0.345538 + 0.938405i \(0.387696\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 119.937 0.608819 0.304409 0.952541i \(-0.401541\pi\)
0.304409 + 0.952541i \(0.401541\pi\)
\(198\) 0 0
\(199\) 37.3932i 0.187905i −0.995577 0.0939526i \(-0.970050\pi\)
0.995577 0.0939526i \(-0.0299502\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 217.485i 1.07135i
\(204\) 0 0
\(205\) 65.0812i 0.317469i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.9435 0.0762849
\(210\) 0 0
\(211\) 145.771 0.690858 0.345429 0.938445i \(-0.387733\pi\)
0.345429 + 0.938445i \(0.387733\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 54.9081 0.255387
\(216\) 0 0
\(217\) 152.654i 0.703476i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 75.4647i 0.341469i
\(222\) 0 0
\(223\) 202.014 0.905894 0.452947 0.891538i \(-0.350373\pi\)
0.452947 + 0.891538i \(0.350373\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 222.735i 0.981213i −0.871381 0.490607i \(-0.836775\pi\)
0.871381 0.490607i \(-0.163225\pi\)
\(228\) 0 0
\(229\) 10.7736i 0.0470461i −0.999723 0.0235231i \(-0.992512\pi\)
0.999723 0.0235231i \(-0.00748832\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −306.711 −1.31636 −0.658179 0.752862i \(-0.728673\pi\)
−0.658179 + 0.752862i \(0.728673\pi\)
\(234\) 0 0
\(235\) 131.710i 0.560470i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 133.517 0.558647 0.279323 0.960197i \(-0.409890\pi\)
0.279323 + 0.960197i \(0.409890\pi\)
\(240\) 0 0
\(241\) 164.609i 0.683023i 0.939878 + 0.341511i \(0.110939\pi\)
−0.939878 + 0.341511i \(0.889061\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 69.4709i 0.283555i
\(246\) 0 0
\(247\) 64.1808i 0.259841i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 214.404i 0.854200i −0.904205 0.427100i \(-0.859535\pi\)
0.904205 0.427100i \(-0.140465\pi\)
\(252\) 0 0
\(253\) −59.4390 + 51.5378i −0.234937 + 0.203707i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −155.697 −0.605827 −0.302913 0.953018i \(-0.597959\pi\)
−0.302913 + 0.953018i \(0.597959\pi\)
\(258\) 0 0
\(259\) 296.948 1.14652
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 508.041i 1.93171i 0.259077 + 0.965857i \(0.416582\pi\)
−0.259077 + 0.965857i \(0.583418\pi\)
\(264\) 0 0
\(265\) 59.2110 0.223438
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 137.722 0.511977 0.255988 0.966680i \(-0.417599\pi\)
0.255988 + 0.966680i \(0.417599\pi\)
\(270\) 0 0
\(271\) −449.193 −1.65754 −0.828769 0.559590i \(-0.810958\pi\)
−0.828769 + 0.559590i \(0.810958\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.1024i 0.0621906i
\(276\) 0 0
\(277\) −265.137 −0.957173 −0.478586 0.878041i \(-0.658851\pi\)
−0.478586 + 0.878041i \(0.658851\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 81.2479i 0.289138i 0.989495 + 0.144569i \(0.0461796\pi\)
−0.989495 + 0.144569i \(0.953820\pi\)
\(282\) 0 0
\(283\) 279.323i 0.987009i 0.869743 + 0.493504i \(0.164284\pi\)
−0.869743 + 0.493504i \(0.835716\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 123.248i 0.429437i
\(288\) 0 0
\(289\) 258.962 0.896062
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 144.258i 0.492348i 0.969226 + 0.246174i \(0.0791734\pi\)
−0.969226 + 0.246174i \(0.920827\pi\)
\(294\) 0 0
\(295\) 19.6766i 0.0667005i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 207.466 + 239.272i 0.693865 + 0.800241i
\(300\) 0 0
\(301\) 103.983 0.345458
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −67.3466 −0.220809
\(306\) 0 0
\(307\) −463.385 −1.50940 −0.754698 0.656072i \(-0.772217\pi\)
−0.754698 + 0.656072i \(0.772217\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 365.413 1.17496 0.587480 0.809238i \(-0.300120\pi\)
0.587480 + 0.809238i \(0.300120\pi\)
\(312\) 0 0
\(313\) 224.561i 0.717449i −0.933444 0.358724i \(-0.883212\pi\)
0.933444 0.358724i \(-0.116788\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −112.230 −0.354037 −0.177018 0.984208i \(-0.556645\pi\)
−0.177018 + 0.984208i \(0.556645\pi\)
\(318\) 0 0
\(319\) 175.673i 0.550700i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −25.5466 −0.0790918
\(324\) 0 0
\(325\) 68.8458 0.211833
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 249.428i 0.758140i
\(330\) 0 0
\(331\) 446.817 1.34990 0.674950 0.737863i \(-0.264165\pi\)
0.674950 + 0.737863i \(0.264165\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.4528 −0.0341873
\(336\) 0 0
\(337\) 615.666i 1.82690i 0.406949 + 0.913451i \(0.366593\pi\)
−0.406949 + 0.913451i \(0.633407\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 123.306i 0.361602i
\(342\) 0 0
\(343\) 339.056i 0.988501i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 499.046 1.43817 0.719087 0.694920i \(-0.244560\pi\)
0.719087 + 0.694920i \(0.244560\pi\)
\(348\) 0 0
\(349\) −380.579 −1.09048 −0.545242 0.838279i \(-0.683562\pi\)
−0.545242 + 0.838279i \(0.683562\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 419.035 1.18707 0.593535 0.804809i \(-0.297732\pi\)
0.593535 + 0.804809i \(0.297732\pi\)
\(354\) 0 0
\(355\) 102.355i 0.288323i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 319.353i 0.889561i 0.895640 + 0.444781i \(0.146718\pi\)
−0.895640 + 0.444781i \(0.853282\pi\)
\(360\) 0 0
\(361\) 339.273 0.939815
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 168.194i 0.460807i
\(366\) 0 0
\(367\) 109.262i 0.297716i 0.988859 + 0.148858i \(0.0475597\pi\)
−0.988859 + 0.148858i \(0.952440\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 112.131 0.302241
\(372\) 0 0
\(373\) 597.530i 1.60196i 0.598694 + 0.800978i \(0.295687\pi\)
−0.598694 + 0.800978i \(0.704313\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 707.174 1.87579
\(378\) 0 0
\(379\) 518.720i 1.36865i −0.729175 0.684327i \(-0.760096\pi\)
0.729175 0.684327i \(-0.239904\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 130.657i 0.341142i 0.985345 + 0.170571i \(0.0545612\pi\)
−0.985345 + 0.170571i \(0.945439\pi\)
\(384\) 0 0
\(385\) 32.3879i 0.0841244i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 641.056i 1.64796i 0.566619 + 0.823980i \(0.308251\pi\)
−0.566619 + 0.823980i \(0.691749\pi\)
\(390\) 0 0
\(391\) 95.2403 82.5799i 0.243581 0.211202i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 139.265 0.352571
\(396\) 0 0
\(397\) 66.6553 0.167897 0.0839487 0.996470i \(-0.473247\pi\)
0.0839487 + 0.996470i \(0.473247\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 121.218i 0.302290i −0.988512 0.151145i \(-0.951704\pi\)
0.988512 0.151145i \(-0.0482960\pi\)
\(402\) 0 0
\(403\) −496.371 −1.23169
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 239.860 0.589337
\(408\) 0 0
\(409\) 534.724 1.30739 0.653697 0.756756i \(-0.273217\pi\)
0.653697 + 0.756756i \(0.273217\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 37.2628i 0.0902248i
\(414\) 0 0
\(415\) −307.877 −0.741873
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 431.194i 1.02910i 0.857460 + 0.514551i \(0.172041\pi\)
−0.857460 + 0.514551i \(0.827959\pi\)
\(420\) 0 0
\(421\) 111.757i 0.265456i 0.991153 + 0.132728i \(0.0423737\pi\)
−0.991153 + 0.132728i \(0.957626\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 27.4035i 0.0644788i
\(426\) 0 0
\(427\) −127.538 −0.298685
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 811.648i 1.88317i 0.336769 + 0.941587i \(0.390666\pi\)
−0.336769 + 0.941587i \(0.609334\pi\)
\(432\) 0 0
\(433\) 583.128i 1.34672i −0.739317 0.673358i \(-0.764851\pi\)
0.739317 0.673358i \(-0.235149\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 80.9994 70.2321i 0.185353 0.160714i
\(438\) 0 0
\(439\) −107.124 −0.244017 −0.122009 0.992529i \(-0.538934\pi\)
−0.122009 + 0.992529i \(0.538934\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 704.580 1.59047 0.795237 0.606298i \(-0.207346\pi\)
0.795237 + 0.606298i \(0.207346\pi\)
\(444\) 0 0
\(445\) 144.339 0.324358
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −744.078 −1.65719 −0.828595 0.559849i \(-0.810859\pi\)
−0.828595 + 0.559849i \(0.810859\pi\)
\(450\) 0 0
\(451\) 99.5539i 0.220740i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 130.378 0.286544
\(456\) 0 0
\(457\) 315.363i 0.690073i 0.938589 + 0.345037i \(0.112133\pi\)
−0.938589 + 0.345037i \(0.887867\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 135.783 0.294541 0.147270 0.989096i \(-0.452951\pi\)
0.147270 + 0.989096i \(0.452951\pi\)
\(462\) 0 0
\(463\) 182.888 0.395006 0.197503 0.980302i \(-0.436717\pi\)
0.197503 + 0.980302i \(0.436717\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 590.693i 1.26487i 0.774614 + 0.632434i \(0.217944\pi\)
−0.774614 + 0.632434i \(0.782056\pi\)
\(468\) 0 0
\(469\) −21.6888 −0.0462447
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 83.9922 0.177573
\(474\) 0 0
\(475\) 23.3060i 0.0490652i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 690.631i 1.44182i 0.693030 + 0.720909i \(0.256275\pi\)
−0.693030 + 0.720909i \(0.743725\pi\)
\(480\) 0 0
\(481\) 965.558i 2.00740i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 383.477 0.790674
\(486\) 0 0
\(487\) 415.787 0.853772 0.426886 0.904305i \(-0.359611\pi\)
0.426886 + 0.904305i \(0.359611\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 395.472 0.805443 0.402721 0.915323i \(-0.368064\pi\)
0.402721 + 0.915323i \(0.368064\pi\)
\(492\) 0 0
\(493\) 281.485i 0.570963i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 193.836i 0.390011i
\(498\) 0 0
\(499\) −901.606 −1.80683 −0.903413 0.428772i \(-0.858946\pi\)
−0.903413 + 0.428772i \(0.858946\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 76.8603i 0.152804i −0.997077 0.0764019i \(-0.975657\pi\)
0.997077 0.0764019i \(-0.0243432\pi\)
\(504\) 0 0
\(505\) 116.500i 0.230694i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −530.564 −1.04237 −0.521183 0.853445i \(-0.674509\pi\)
−0.521183 + 0.853445i \(0.674509\pi\)
\(510\) 0 0
\(511\) 318.520i 0.623327i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −73.5692 −0.142853
\(516\) 0 0
\(517\) 201.475i 0.389701i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 194.897i 0.374083i −0.982352 0.187042i \(-0.940110\pi\)
0.982352 0.187042i \(-0.0598899\pi\)
\(522\) 0 0
\(523\) 418.691i 0.800557i 0.916394 + 0.400279i \(0.131087\pi\)
−0.916394 + 0.400279i \(0.868913\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 197.576i 0.374907i
\(528\) 0 0
\(529\) −74.9472 + 523.664i −0.141677 + 0.989913i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −400.755 −0.751885
\(534\) 0 0
\(535\) 78.9387 0.147549
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 106.269i 0.197159i
\(540\) 0 0
\(541\) −221.165 −0.408808 −0.204404 0.978887i \(-0.565526\pi\)
−0.204404 + 0.978887i \(0.565526\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −125.361 −0.230020
\(546\) 0 0
\(547\) −606.362 −1.10852 −0.554262 0.832342i \(-0.686999\pi\)
−0.554262 + 0.832342i \(0.686999\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 239.395i 0.434474i
\(552\) 0 0
\(553\) 263.736 0.476918
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 474.639i 0.852135i 0.904691 + 0.426068i \(0.140101\pi\)
−0.904691 + 0.426068i \(0.859899\pi\)
\(558\) 0 0
\(559\) 338.111i 0.604849i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 831.865i 1.47756i 0.673948 + 0.738779i \(0.264597\pi\)
−0.673948 + 0.738779i \(0.735403\pi\)
\(564\) 0 0
\(565\) 178.085 0.315194
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 58.3947i 0.102627i 0.998683 + 0.0513135i \(0.0163408\pi\)
−0.998683 + 0.0513135i \(0.983659\pi\)
\(570\) 0 0
\(571\) 423.510i 0.741698i −0.928693 0.370849i \(-0.879067\pi\)
0.928693 0.370849i \(-0.120933\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 75.3370 + 86.8869i 0.131021 + 0.151108i
\(576\) 0 0
\(577\) −434.749 −0.753465 −0.376733 0.926322i \(-0.622952\pi\)
−0.376733 + 0.926322i \(0.622952\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −583.046 −1.00352
\(582\) 0 0
\(583\) 90.5742 0.155359
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 196.392 0.334568 0.167284 0.985909i \(-0.446500\pi\)
0.167284 + 0.985909i \(0.446500\pi\)
\(588\) 0 0
\(589\) 168.034i 0.285286i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 779.875 1.31514 0.657568 0.753395i \(-0.271585\pi\)
0.657568 + 0.753395i \(0.271585\pi\)
\(594\) 0 0
\(595\) 51.8957i 0.0872197i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 729.208 1.21738 0.608688 0.793410i \(-0.291696\pi\)
0.608688 + 0.793410i \(0.291696\pi\)
\(600\) 0 0
\(601\) −98.3505 −0.163645 −0.0818224 0.996647i \(-0.526074\pi\)
−0.0818224 + 0.996647i \(0.526074\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 244.403i 0.403972i
\(606\) 0 0
\(607\) −949.398 −1.56408 −0.782042 0.623226i \(-0.785822\pi\)
−0.782042 + 0.623226i \(0.785822\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 811.041 1.32740
\(612\) 0 0
\(613\) 354.757i 0.578722i −0.957220 0.289361i \(-0.906557\pi\)
0.957220 0.289361i \(-0.0934428\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1080.07i 1.75052i −0.483654 0.875259i \(-0.660691\pi\)
0.483654 0.875259i \(-0.339309\pi\)
\(618\) 0 0
\(619\) 207.082i 0.334542i 0.985911 + 0.167271i \(0.0534955\pi\)
−0.985911 + 0.167271i \(0.946504\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 273.344 0.438755
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −384.332 −0.611021
\(630\) 0 0
\(631\) 899.817i 1.42602i 0.701155 + 0.713009i \(0.252668\pi\)
−0.701155 + 0.713009i \(0.747332\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 108.606i 0.171032i
\(636\) 0 0
\(637\) −427.785 −0.671562
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 102.776i 0.160337i 0.996781 + 0.0801683i \(0.0255458\pi\)
−0.996781 + 0.0801683i \(0.974454\pi\)
\(642\) 0 0
\(643\) 120.818i 0.187897i −0.995577 0.0939484i \(-0.970051\pi\)
0.995577 0.0939484i \(-0.0299489\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1042.16 1.61076 0.805379 0.592760i \(-0.201962\pi\)
0.805379 + 0.592760i \(0.201962\pi\)
\(648\) 0 0
\(649\) 30.0991i 0.0463776i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −694.904 −1.06417 −0.532085 0.846691i \(-0.678591\pi\)
−0.532085 + 0.846691i \(0.678591\pi\)
\(654\) 0 0
\(655\) 7.27372i 0.0111049i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 541.740i 0.822063i 0.911621 + 0.411032i \(0.134831\pi\)
−0.911621 + 0.411032i \(0.865169\pi\)
\(660\) 0 0
\(661\) 243.406i 0.368239i −0.982904 0.184120i \(-0.941057\pi\)
0.982904 0.184120i \(-0.0589434\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 44.1360i 0.0663699i
\(666\) 0 0
\(667\) 773.850 + 892.489i 1.16019 + 1.33806i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −103.019 −0.153531
\(672\) 0 0
\(673\) 335.090 0.497905 0.248952 0.968516i \(-0.419914\pi\)
0.248952 + 0.968516i \(0.419914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 639.438i 0.944517i −0.881460 0.472258i \(-0.843439\pi\)
0.881460 0.472258i \(-0.156561\pi\)
\(678\) 0 0
\(679\) 726.214 1.06953
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −366.380 −0.536428 −0.268214 0.963359i \(-0.586433\pi\)
−0.268214 + 0.963359i \(0.586433\pi\)
\(684\) 0 0
\(685\) 0.470955 0.000687526
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 364.607i 0.529183i
\(690\) 0 0
\(691\) 243.486 0.352367 0.176184 0.984357i \(-0.443625\pi\)
0.176184 + 0.984357i \(0.443625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 78.1176i 0.112399i
\(696\) 0 0
\(697\) 159.517i 0.228862i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 213.429i 0.304463i 0.988345 + 0.152232i \(0.0486460\pi\)
−0.988345 + 0.152232i \(0.951354\pi\)
\(702\) 0 0
\(703\) −326.865 −0.464957
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 220.624i 0.312056i
\(708\) 0 0
\(709\) 804.957i 1.13534i −0.823256 0.567671i \(-0.807845\pi\)
0.823256 0.567671i \(-0.192155\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −543.171 626.445i −0.761811 0.878604i
\(714\) 0 0
\(715\) 105.313 0.147290
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −550.092 −0.765079 −0.382540 0.923939i \(-0.624950\pi\)
−0.382540 + 0.923939i \(0.624950\pi\)
\(720\) 0 0
\(721\) −139.323 −0.193235
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 256.796 0.354201
\(726\) 0 0
\(727\) 719.138i 0.989185i 0.869125 + 0.494593i \(0.164683\pi\)
−0.869125 + 0.494593i \(0.835317\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −134.582 −0.184107
\(732\) 0 0
\(733\) 1139.86i 1.55506i 0.628844 + 0.777532i \(0.283529\pi\)
−0.628844 + 0.777532i \(0.716471\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −17.5191 −0.0237708
\(738\) 0 0
\(739\) 805.483 1.08996 0.544982 0.838448i \(-0.316537\pi\)
0.544982 + 0.838448i \(0.316537\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 75.8211i 0.102047i 0.998697 + 0.0510236i \(0.0162484\pi\)
−0.998697 + 0.0510236i \(0.983752\pi\)
\(744\) 0 0
\(745\) −657.438 −0.882468
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 149.491 0.199588
\(750\) 0 0
\(751\) 543.632i 0.723878i −0.932202 0.361939i \(-0.882115\pi\)
0.932202 0.361939i \(-0.117885\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 171.104i 0.226628i
\(756\) 0 0
\(757\) 650.293i 0.859039i 0.903058 + 0.429520i \(0.141317\pi\)
−0.903058 + 0.429520i \(0.858683\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 59.2182 0.0778162 0.0389081 0.999243i \(-0.487612\pi\)
0.0389081 + 0.999243i \(0.487612\pi\)
\(762\) 0 0
\(763\) −237.403 −0.311144
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −121.164 −0.157971
\(768\) 0 0
\(769\) 822.423i 1.06947i −0.845019 0.534736i \(-0.820411\pi\)
0.845019 0.534736i \(-0.179589\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 62.2749i 0.0805626i 0.999188 + 0.0402813i \(0.0128254\pi\)
−0.999188 + 0.0402813i \(0.987175\pi\)
\(774\) 0 0
\(775\) −180.247 −0.232577
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 135.665i 0.174153i
\(780\) 0 0
\(781\) 156.571i 0.200475i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 265.962 0.338805
\(786\) 0 0
\(787\) 404.988i 0.514597i 0.966332 + 0.257299i \(0.0828324\pi\)
−0.966332 + 0.257299i \(0.917168\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 337.250 0.426359
\(792\) 0 0
\(793\) 414.704i 0.522956i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 232.050i 0.291154i −0.989347 0.145577i \(-0.953496\pi\)
0.989347 0.145577i \(-0.0465039\pi\)
\(798\) 0 0
\(799\) 322.828i 0.404040i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 257.285i 0.320404i
\(804\) 0 0
\(805\) 142.670 + 164.543i 0.177230 + 0.204401i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −276.617 −0.341925 −0.170962 0.985278i \(-0.554688\pi\)
−0.170962 + 0.985278i \(0.554688\pi\)
\(810\) 0 0
\(811\) −53.1869 −0.0655819 −0.0327909 0.999462i \(-0.510440\pi\)
−0.0327909 + 0.999462i \(0.510440\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 638.743i 0.783733i
\(816\) 0 0
\(817\) −114.459 −0.140096
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −831.053 −1.01224 −0.506122 0.862462i \(-0.668922\pi\)
−0.506122 + 0.862462i \(0.668922\pi\)
\(822\) 0 0
\(823\) 225.779 0.274336 0.137168 0.990548i \(-0.456200\pi\)
0.137168 + 0.990548i \(0.456200\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 454.191i 0.549203i −0.961558 0.274602i \(-0.911454\pi\)
0.961558 0.274602i \(-0.0885460\pi\)
\(828\) 0 0
\(829\) 1391.70 1.67877 0.839386 0.543536i \(-0.182915\pi\)
0.839386 + 0.543536i \(0.182915\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 170.276i 0.204413i
\(834\) 0 0
\(835\) 323.129i 0.386981i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 132.957i 0.158471i −0.996856 0.0792355i \(-0.974752\pi\)
0.996856 0.0792355i \(-0.0252479\pi\)
\(840\) 0 0
\(841\) 1796.77 2.13647
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 46.0406i 0.0544860i
\(846\) 0 0
\(847\) 462.841i 0.546447i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1218.58 1056.60i 1.43194 1.24159i
\(852\) 0 0
\(853\) 104.622 0.122651 0.0613256 0.998118i \(-0.480467\pi\)
0.0613256 + 0.998118i \(0.480467\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −31.9854 −0.0373226 −0.0186613 0.999826i \(-0.505940\pi\)
−0.0186613 + 0.999826i \(0.505940\pi\)
\(858\) 0 0
\(859\) 984.302 1.14587 0.572935 0.819601i \(-0.305805\pi\)
0.572935 + 0.819601i \(0.305805\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −668.524 −0.774651 −0.387326 0.921943i \(-0.626601\pi\)
−0.387326 + 0.921943i \(0.626601\pi\)
\(864\) 0 0
\(865\) 265.354i 0.306768i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 213.032 0.245147
\(870\) 0 0
\(871\) 70.5233i 0.0809682i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 47.3440 0.0541075
\(876\) 0 0
\(877\) 1103.98 1.25881 0.629405 0.777077i \(-0.283299\pi\)
0.629405 + 0.777077i \(0.283299\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 607.647i 0.689724i 0.938653 + 0.344862i \(0.112074\pi\)
−0.938653 + 0.344862i \(0.887926\pi\)
\(882\) 0 0
\(883\) 85.7974 0.0971659 0.0485829 0.998819i \(-0.484529\pi\)
0.0485829 + 0.998819i \(0.484529\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −689.893 −0.777782 −0.388891 0.921284i \(-0.627142\pi\)
−0.388891 + 0.921284i \(0.627142\pi\)
\(888\) 0 0
\(889\) 205.673i 0.231353i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 274.557i 0.307455i
\(894\) 0 0
\(895\) 27.4197i 0.0306365i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1851.47 −2.05948
\(900\) 0 0
\(901\) −145.129 −0.161075
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −509.996 −0.563531
\(906\) 0 0
\(907\) 1182.25i 1.30348i 0.758444 + 0.651738i \(0.225960\pi\)
−0.758444 + 0.651738i \(0.774040\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 611.652i 0.671407i −0.941968 0.335704i \(-0.891026\pi\)
0.941968 0.335704i \(-0.108974\pi\)
\(912\) 0 0
\(913\) −470.956 −0.515833
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.7747i 0.0150215i
\(918\) 0 0
\(919\) 1037.12i 1.12854i −0.825592 0.564268i \(-0.809159\pi\)
0.825592 0.564268i \(-0.190841\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −630.276 −0.682856
\(924\) 0 0
\(925\) 350.623i 0.379052i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −944.065 −1.01622 −0.508108 0.861293i \(-0.669655\pi\)
−0.508108 + 0.861293i \(0.669655\pi\)
\(930\) 0 0
\(931\) 144.816i 0.155549i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 41.9188i 0.0448329i
\(936\) 0 0
\(937\) 1307.12i 1.39500i 0.716583 + 0.697502i \(0.245705\pi\)
−0.716583 + 0.697502i \(0.754295\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 710.928i 0.755502i −0.925907 0.377751i \(-0.876697\pi\)
0.925907 0.377751i \(-0.123303\pi\)
\(942\) 0 0
\(943\) −438.540 505.773i −0.465048 0.536344i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −674.787 −0.712552 −0.356276 0.934381i \(-0.615954\pi\)
−0.356276 + 0.934381i \(0.615954\pi\)
\(948\) 0 0
\(949\) −1035.70 −1.09136
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 671.206i 0.704309i −0.935942 0.352154i \(-0.885449\pi\)
0.935942 0.352154i \(-0.114551\pi\)
\(954\) 0 0
\(955\) 596.439 0.624543
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.891877 0.000930007
\(960\) 0 0
\(961\) 338.562 0.352301
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 298.242i 0.309059i
\(966\) 0 0
\(967\) −1886.45 −1.95083 −0.975413 0.220385i \(-0.929269\pi\)
−0.975413 + 0.220385i \(0.929269\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1134.32i 1.16819i −0.811684 0.584096i \(-0.801449\pi\)
0.811684 0.584096i \(-0.198551\pi\)
\(972\) 0 0
\(973\) 147.936i 0.152041i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.6770i 0.0385640i −0.999814 0.0192820i \(-0.993862\pi\)
0.999814 0.0192820i \(-0.00613803\pi\)
\(978\) 0 0
\(979\) 220.794 0.225530
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1653.42i 1.68201i 0.541027 + 0.841005i \(0.318036\pi\)
−0.541027 + 0.841005i \(0.681964\pi\)
\(984\) 0 0
\(985\) 268.188i 0.272272i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 426.713 369.990i 0.431459 0.374105i
\(990\) 0 0
\(991\) 1681.88 1.69715 0.848576 0.529073i \(-0.177460\pi\)
0.848576 + 0.529073i \(0.177460\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −83.6136 −0.0840338
\(996\) 0 0
\(997\) −1474.49 −1.47893 −0.739463 0.673198i \(-0.764920\pi\)
−0.739463 + 0.673198i \(0.764920\pi\)
\(998\) 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 4140.3.d.c.2161.4 32
3.2 odd 2 1380.3.d.a.781.26 yes 32
23.22 odd 2 inner 4140.3.d.c.2161.29 32
69.68 even 2 1380.3.d.a.781.23 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.3.d.a.781.23 32 69.68 even 2
1380.3.d.a.781.26 yes 32 3.2 odd 2
4140.3.d.c.2161.4 32 1.1 even 1 trivial
4140.3.d.c.2161.29 32 23.22 odd 2 inner