Properties

Label 3800.2.d.j.3649.1
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.59105344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 21x^{4} + 116x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.1
Root \(3.08387i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.j.3649.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.29707i q^{3} +1.78680i q^{7} -7.87067 q^{9} +O(q^{10})\) \(q-3.29707i q^{3} +1.78680i q^{7} -7.87067 q^{9} -5.08387 q^{11} +1.29707i q^{13} +0.213198i q^{17} +1.00000 q^{19} +5.89121 q^{21} +3.72347i q^{23} +16.0590i q^{27} -0.870674 q^{29} +16.7619i q^{33} -2.00000i q^{37} +4.27653 q^{39} +8.59414 q^{41} +3.67801i q^{43} -4.65748i q^{47} +3.80734 q^{49} +0.702929 q^{51} -11.0384i q^{53} -3.29707i q^{57} +4.70293 q^{59} +3.51027 q^{61} -14.0633i q^{63} +1.12933i q^{67} +12.2765 q^{69} +8.76189 q^{71} -6.80734i q^{73} -9.08387i q^{77} -14.5941 q^{79} +29.3355 q^{81} +9.74135i q^{83} +2.87067i q^{87} +6.76189 q^{89} -2.31761 q^{91} +4.16774i q^{97} +40.0135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.29707i − 1.90356i −0.306774 0.951782i \(-0.599250\pi\)
0.306774 0.951782i \(-0.400750\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.78680i 0.675348i 0.941263 + 0.337674i \(0.109640\pi\)
−0.941263 + 0.337674i \(0.890360\pi\)
\(8\) 0 0
\(9\) −7.87067 −2.62356
\(10\) 0 0
\(11\) −5.08387 −1.53285 −0.766423 0.642337i \(-0.777965\pi\)
−0.766423 + 0.642337i \(0.777965\pi\)
\(12\) 0 0
\(13\) 1.29707i 0.359743i 0.983690 + 0.179871i \(0.0575681\pi\)
−0.983690 + 0.179871i \(0.942432\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.213198i 0.0517082i 0.999666 + 0.0258541i \(0.00823053\pi\)
−0.999666 + 0.0258541i \(0.991769\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 5.89121 1.28557
\(22\) 0 0
\(23\) 3.72347i 0.776397i 0.921576 + 0.388198i \(0.126902\pi\)
−0.921576 + 0.388198i \(0.873098\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 16.0590i 3.09055i
\(28\) 0 0
\(29\) −0.870674 −0.161680 −0.0808401 0.996727i \(-0.525760\pi\)
−0.0808401 + 0.996727i \(0.525760\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 16.7619i 2.91787i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 4.27653 0.684793
\(40\) 0 0
\(41\) 8.59414 1.34218 0.671090 0.741376i \(-0.265827\pi\)
0.671090 + 0.741376i \(0.265827\pi\)
\(42\) 0 0
\(43\) 3.67801i 0.560892i 0.959870 + 0.280446i \(0.0904823\pi\)
−0.959870 + 0.280446i \(0.909518\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.65748i − 0.679363i −0.940541 0.339681i \(-0.889681\pi\)
0.940541 0.339681i \(-0.110319\pi\)
\(48\) 0 0
\(49\) 3.80734 0.543906
\(50\) 0 0
\(51\) 0.702929 0.0984298
\(52\) 0 0
\(53\) − 11.0384i − 1.51624i −0.652113 0.758122i \(-0.726117\pi\)
0.652113 0.758122i \(-0.273883\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.29707i − 0.436708i
\(58\) 0 0
\(59\) 4.70293 0.612269 0.306135 0.951988i \(-0.400964\pi\)
0.306135 + 0.951988i \(0.400964\pi\)
\(60\) 0 0
\(61\) 3.51027 0.449444 0.224722 0.974423i \(-0.427853\pi\)
0.224722 + 0.974423i \(0.427853\pi\)
\(62\) 0 0
\(63\) − 14.0633i − 1.77181i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.12933i 0.137969i 0.997618 + 0.0689846i \(0.0219759\pi\)
−0.997618 + 0.0689846i \(0.978024\pi\)
\(68\) 0 0
\(69\) 12.2765 1.47792
\(70\) 0 0
\(71\) 8.76189 1.03984 0.519922 0.854214i \(-0.325961\pi\)
0.519922 + 0.854214i \(0.325961\pi\)
\(72\) 0 0
\(73\) − 6.80734i − 0.796739i −0.917225 0.398369i \(-0.869576\pi\)
0.917225 0.398369i \(-0.130424\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 9.08387i − 1.03520i
\(78\) 0 0
\(79\) −14.5941 −1.64197 −0.820985 0.570950i \(-0.806575\pi\)
−0.820985 + 0.570950i \(0.806575\pi\)
\(80\) 0 0
\(81\) 29.3355 3.25950
\(82\) 0 0
\(83\) 9.74135i 1.06925i 0.845089 + 0.534626i \(0.179547\pi\)
−0.845089 + 0.534626i \(0.820453\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.87067i 0.307769i
\(88\) 0 0
\(89\) 6.76189 0.716758 0.358379 0.933576i \(-0.383329\pi\)
0.358379 + 0.933576i \(0.383329\pi\)
\(90\) 0 0
\(91\) −2.31761 −0.242951
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.16774i 0.423170i 0.977360 + 0.211585i \(0.0678626\pi\)
−0.977360 + 0.211585i \(0.932137\pi\)
\(98\) 0 0
\(99\) 40.0135 4.02151
\(100\) 0 0
\(101\) 19.1883 1.90931 0.954653 0.297722i \(-0.0962267\pi\)
0.954653 + 0.297722i \(0.0962267\pi\)
\(102\) 0 0
\(103\) 15.9091i 1.56757i 0.621033 + 0.783785i \(0.286713\pi\)
−0.621033 + 0.783785i \(0.713287\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.29707i − 0.318740i −0.987219 0.159370i \(-0.949054\pi\)
0.987219 0.159370i \(-0.0509463\pi\)
\(108\) 0 0
\(109\) −4.44428 −0.425685 −0.212842 0.977087i \(-0.568272\pi\)
−0.212842 + 0.977087i \(0.568272\pi\)
\(110\) 0 0
\(111\) −6.59414 −0.625888
\(112\) 0 0
\(113\) 11.1883i 1.05251i 0.850328 + 0.526253i \(0.176403\pi\)
−0.850328 + 0.526253i \(0.823597\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 10.2088i − 0.943806i
\(118\) 0 0
\(119\) −0.380943 −0.0349210
\(120\) 0 0
\(121\) 14.8458 1.34961
\(122\) 0 0
\(123\) − 28.3355i − 2.55493i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.7619i 1.48738i 0.668526 + 0.743688i \(0.266925\pi\)
−0.668526 + 0.743688i \(0.733075\pi\)
\(128\) 0 0
\(129\) 12.1267 1.06769
\(130\) 0 0
\(131\) 3.67801 0.321350 0.160675 0.987007i \(-0.448633\pi\)
0.160675 + 0.987007i \(0.448633\pi\)
\(132\) 0 0
\(133\) 1.78680i 0.154935i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.4015i 1.82845i 0.405205 + 0.914226i \(0.367200\pi\)
−0.405205 + 0.914226i \(0.632800\pi\)
\(138\) 0 0
\(139\) 10.8252 0.918183 0.459092 0.888389i \(-0.348175\pi\)
0.459092 + 0.888389i \(0.348175\pi\)
\(140\) 0 0
\(141\) −15.3560 −1.29321
\(142\) 0 0
\(143\) − 6.59414i − 0.551430i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 12.5531i − 1.03536i
\(148\) 0 0
\(149\) 21.2516 1.74100 0.870500 0.492168i \(-0.163795\pi\)
0.870500 + 0.492168i \(0.163795\pi\)
\(150\) 0 0
\(151\) −15.9091 −1.29466 −0.647332 0.762208i \(-0.724115\pi\)
−0.647332 + 0.762208i \(0.724115\pi\)
\(152\) 0 0
\(153\) − 1.67801i − 0.135659i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) −36.3944 −2.88627
\(160\) 0 0
\(161\) −6.65310 −0.524338
\(162\) 0 0
\(163\) − 9.18828i − 0.719682i −0.933014 0.359841i \(-0.882831\pi\)
0.933014 0.359841i \(-0.117169\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 5.83226i − 0.451313i −0.974207 0.225657i \(-0.927547\pi\)
0.974207 0.225657i \(-0.0724528\pi\)
\(168\) 0 0
\(169\) 11.3176 0.870585
\(170\) 0 0
\(171\) −7.87067 −0.601885
\(172\) 0 0
\(173\) 7.18828i 0.546515i 0.961941 + 0.273257i \(0.0881011\pi\)
−0.961941 + 0.273257i \(0.911899\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 15.5059i − 1.16549i
\(178\) 0 0
\(179\) 7.87333 0.588480 0.294240 0.955732i \(-0.404933\pi\)
0.294240 + 0.955732i \(0.404933\pi\)
\(180\) 0 0
\(181\) −22.3355 −1.66018 −0.830092 0.557627i \(-0.811712\pi\)
−0.830092 + 0.557627i \(0.811712\pi\)
\(182\) 0 0
\(183\) − 11.5736i − 0.855545i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.08387i − 0.0792606i
\(188\) 0 0
\(189\) −28.6942 −2.08719
\(190\) 0 0
\(191\) −23.7370 −1.71755 −0.858773 0.512356i \(-0.828773\pi\)
−0.858773 + 0.512356i \(0.828773\pi\)
\(192\) 0 0
\(193\) 7.31495i 0.526542i 0.964722 + 0.263271i \(0.0848013\pi\)
−0.964722 + 0.263271i \(0.915199\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 8.59414i − 0.612307i −0.951982 0.306154i \(-0.900958\pi\)
0.951982 0.306154i \(-0.0990421\pi\)
\(198\) 0 0
\(199\) 7.06599 0.500895 0.250447 0.968130i \(-0.419422\pi\)
0.250447 + 0.968130i \(0.419422\pi\)
\(200\) 0 0
\(201\) 3.72347 0.262633
\(202\) 0 0
\(203\) − 1.55572i − 0.109190i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 29.3062i − 2.03692i
\(208\) 0 0
\(209\) −5.08387 −0.351659
\(210\) 0 0
\(211\) −5.55572 −0.382472 −0.191236 0.981544i \(-0.561250\pi\)
−0.191236 + 0.981544i \(0.561250\pi\)
\(212\) 0 0
\(213\) − 28.8886i − 1.97941i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −22.4443 −1.51664
\(220\) 0 0
\(221\) −0.276533 −0.0186016
\(222\) 0 0
\(223\) 16.7619i 1.12246i 0.827660 + 0.561229i \(0.189671\pi\)
−0.827660 + 0.561229i \(0.810329\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.27653i 0.549333i 0.961540 + 0.274666i \(0.0885674\pi\)
−0.961540 + 0.274666i \(0.911433\pi\)
\(228\) 0 0
\(229\) 12.6986 0.839144 0.419572 0.907722i \(-0.362180\pi\)
0.419572 + 0.907722i \(0.362180\pi\)
\(230\) 0 0
\(231\) −29.9502 −1.97058
\(232\) 0 0
\(233\) − 8.36306i − 0.547882i −0.961746 0.273941i \(-0.911673\pi\)
0.961746 0.273941i \(-0.0883274\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 48.1179i 3.12559i
\(238\) 0 0
\(239\) −6.97508 −0.451181 −0.225590 0.974222i \(-0.572431\pi\)
−0.225590 + 0.974222i \(0.572431\pi\)
\(240\) 0 0
\(241\) −19.7413 −1.27165 −0.635826 0.771832i \(-0.719340\pi\)
−0.635826 + 0.771832i \(0.719340\pi\)
\(242\) 0 0
\(243\) − 48.5443i − 3.11412i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.29707i 0.0825306i
\(248\) 0 0
\(249\) 32.1179 2.03539
\(250\) 0 0
\(251\) 10.2722 0.648373 0.324186 0.945993i \(-0.394910\pi\)
0.324186 + 0.945993i \(0.394910\pi\)
\(252\) 0 0
\(253\) − 18.9296i − 1.19010i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.90909i 0.618112i 0.951044 + 0.309056i \(0.100013\pi\)
−0.951044 + 0.309056i \(0.899987\pi\)
\(258\) 0 0
\(259\) 3.57360 0.222053
\(260\) 0 0
\(261\) 6.85279 0.424177
\(262\) 0 0
\(263\) − 23.6780i − 1.46005i −0.683421 0.730024i \(-0.739509\pi\)
0.683421 0.730024i \(-0.260491\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 22.2944i − 1.36440i
\(268\) 0 0
\(269\) 5.02054 0.306108 0.153054 0.988218i \(-0.451089\pi\)
0.153054 + 0.988218i \(0.451089\pi\)
\(270\) 0 0
\(271\) 4.27653 0.259781 0.129890 0.991528i \(-0.458537\pi\)
0.129890 + 0.991528i \(0.458537\pi\)
\(272\) 0 0
\(273\) 7.64132i 0.462474i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.93667i 0.476868i 0.971159 + 0.238434i \(0.0766341\pi\)
−0.971159 + 0.238434i \(0.923366\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.8886 1.36542 0.682708 0.730691i \(-0.260802\pi\)
0.682708 + 0.730691i \(0.260802\pi\)
\(282\) 0 0
\(283\) 5.08387i 0.302205i 0.988518 + 0.151102i \(0.0482823\pi\)
−0.988518 + 0.151102i \(0.951718\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 15.3560i 0.906438i
\(288\) 0 0
\(289\) 16.9545 0.997326
\(290\) 0 0
\(291\) 13.7413 0.805532
\(292\) 0 0
\(293\) 3.55572i 0.207728i 0.994592 + 0.103864i \(0.0331206\pi\)
−0.994592 + 0.103864i \(0.966879\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 81.6417i − 4.73733i
\(298\) 0 0
\(299\) −4.82960 −0.279303
\(300\) 0 0
\(301\) −6.57188 −0.378797
\(302\) 0 0
\(303\) − 63.2651i − 3.63449i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.3355i 0.932316i 0.884702 + 0.466158i \(0.154362\pi\)
−0.884702 + 0.466158i \(0.845638\pi\)
\(308\) 0 0
\(309\) 52.4534 2.98397
\(310\) 0 0
\(311\) −16.3809 −0.928878 −0.464439 0.885605i \(-0.653744\pi\)
−0.464439 + 0.885605i \(0.653744\pi\)
\(312\) 0 0
\(313\) 30.0590i 1.69903i 0.527562 + 0.849516i \(0.323106\pi\)
−0.527562 + 0.849516i \(0.676894\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 12.0179i − 0.674991i −0.941327 0.337496i \(-0.890420\pi\)
0.941327 0.337496i \(-0.109580\pi\)
\(318\) 0 0
\(319\) 4.42640 0.247831
\(320\) 0 0
\(321\) −10.8707 −0.606742
\(322\) 0 0
\(323\) 0.213198i 0.0118627i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.6531i 0.810318i
\(328\) 0 0
\(329\) 8.32199 0.458806
\(330\) 0 0
\(331\) −14.3176 −0.786967 −0.393483 0.919332i \(-0.628730\pi\)
−0.393483 + 0.919332i \(0.628730\pi\)
\(332\) 0 0
\(333\) 15.7413i 0.862621i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11.6147i 0.632692i 0.948644 + 0.316346i \(0.102456\pi\)
−0.948644 + 0.316346i \(0.897544\pi\)
\(338\) 0 0
\(339\) 36.8886 2.00351
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.3106i 1.04267i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.91613i − 0.156546i −0.996932 0.0782730i \(-0.975059\pi\)
0.996932 0.0782730i \(-0.0249406\pi\)
\(348\) 0 0
\(349\) −12.8252 −0.686518 −0.343259 0.939241i \(-0.611531\pi\)
−0.343259 + 0.939241i \(0.611531\pi\)
\(350\) 0 0
\(351\) −20.8296 −1.11180
\(352\) 0 0
\(353\) − 25.7592i − 1.37103i −0.728061 0.685513i \(-0.759578\pi\)
0.728061 0.685513i \(-0.240422\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.25600i 0.0664743i
\(358\) 0 0
\(359\) −22.4220 −1.18339 −0.591694 0.806162i \(-0.701541\pi\)
−0.591694 + 0.806162i \(0.701541\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 48.9475i − 2.56908i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.18828i 0.270826i 0.990789 + 0.135413i \(0.0432361\pi\)
−0.990789 + 0.135413i \(0.956764\pi\)
\(368\) 0 0
\(369\) −67.6417 −3.52129
\(370\) 0 0
\(371\) 19.7235 1.02399
\(372\) 0 0
\(373\) − 36.0091i − 1.86448i −0.361838 0.932241i \(-0.617851\pi\)
0.361838 0.932241i \(-0.382149\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.12933i − 0.0581632i
\(378\) 0 0
\(379\) 25.0384 1.28614 0.643069 0.765809i \(-0.277661\pi\)
0.643069 + 0.765809i \(0.277661\pi\)
\(380\) 0 0
\(381\) 55.2651 2.83132
\(382\) 0 0
\(383\) 21.7413i 1.11093i 0.831540 + 0.555466i \(0.187460\pi\)
−0.831540 + 0.555466i \(0.812540\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 28.9484i − 1.47153i
\(388\) 0 0
\(389\) −11.5103 −0.583594 −0.291797 0.956480i \(-0.594253\pi\)
−0.291797 + 0.956480i \(0.594253\pi\)
\(390\) 0 0
\(391\) −0.793836 −0.0401460
\(392\) 0 0
\(393\) − 12.1267i − 0.611710i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 7.21054i − 0.361887i −0.983494 0.180943i \(-0.942085\pi\)
0.983494 0.180943i \(-0.0579151\pi\)
\(398\) 0 0
\(399\) 5.89121 0.294929
\(400\) 0 0
\(401\) 1.23811 0.0618285 0.0309142 0.999522i \(-0.490158\pi\)
0.0309142 + 0.999522i \(0.490158\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.1677i 0.503996i
\(408\) 0 0
\(409\) −32.8030 −1.62200 −0.811001 0.585045i \(-0.801077\pi\)
−0.811001 + 0.585045i \(0.801077\pi\)
\(410\) 0 0
\(411\) 70.5622 3.48058
\(412\) 0 0
\(413\) 8.40320i 0.413495i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 35.6915i − 1.74782i
\(418\) 0 0
\(419\) 17.1883 0.839703 0.419851 0.907593i \(-0.362082\pi\)
0.419851 + 0.907593i \(0.362082\pi\)
\(420\) 0 0
\(421\) 19.8912 0.969438 0.484719 0.874670i \(-0.338922\pi\)
0.484719 + 0.874670i \(0.338922\pi\)
\(422\) 0 0
\(423\) 36.6575i 1.78235i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.27215i 0.303531i
\(428\) 0 0
\(429\) −21.7413 −1.04968
\(430\) 0 0
\(431\) 8.55307 0.411987 0.205993 0.978553i \(-0.433957\pi\)
0.205993 + 0.978553i \(0.433957\pi\)
\(432\) 0 0
\(433\) 31.5238i 1.51494i 0.652872 + 0.757468i \(0.273564\pi\)
−0.652872 + 0.757468i \(0.726436\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.72347i 0.178118i
\(438\) 0 0
\(439\) 4.12667 0.196955 0.0984776 0.995139i \(-0.468603\pi\)
0.0984776 + 0.995139i \(0.468603\pi\)
\(440\) 0 0
\(441\) −29.9663 −1.42697
\(442\) 0 0
\(443\) − 37.7549i − 1.79379i −0.442246 0.896894i \(-0.645818\pi\)
0.442246 0.896894i \(-0.354182\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 70.0681i − 3.31411i
\(448\) 0 0
\(449\) −33.3560 −1.57417 −0.787084 0.616846i \(-0.788410\pi\)
−0.787084 + 0.616846i \(0.788410\pi\)
\(450\) 0 0
\(451\) −43.6915 −2.05735
\(452\) 0 0
\(453\) 52.4534i 2.46448i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.1223i 1.50262i 0.659951 + 0.751309i \(0.270577\pi\)
−0.659951 + 0.751309i \(0.729423\pi\)
\(458\) 0 0
\(459\) −3.42374 −0.159807
\(460\) 0 0
\(461\) −7.81000 −0.363748 −0.181874 0.983322i \(-0.558216\pi\)
−0.181874 + 0.983322i \(0.558216\pi\)
\(462\) 0 0
\(463\) − 18.4897i − 0.859291i −0.902998 0.429645i \(-0.858639\pi\)
0.902998 0.429645i \(-0.141361\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0839i 0.975645i 0.872943 + 0.487823i \(0.162209\pi\)
−0.872943 + 0.487823i \(0.837791\pi\)
\(468\) 0 0
\(469\) −2.01788 −0.0931771
\(470\) 0 0
\(471\) 46.1590 2.12689
\(472\) 0 0
\(473\) − 18.6986i − 0.859760i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 86.8798i 3.97795i
\(478\) 0 0
\(479\) 25.5238 1.16621 0.583105 0.812396i \(-0.301837\pi\)
0.583105 + 0.812396i \(0.301837\pi\)
\(480\) 0 0
\(481\) 2.59414 0.118283
\(482\) 0 0
\(483\) 21.9357i 0.998110i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 10.4675i − 0.474326i −0.971470 0.237163i \(-0.923782\pi\)
0.971470 0.237163i \(-0.0762176\pi\)
\(488\) 0 0
\(489\) −30.2944 −1.36996
\(490\) 0 0
\(491\) 12.6710 0.571833 0.285917 0.958254i \(-0.407702\pi\)
0.285917 + 0.958254i \(0.407702\pi\)
\(492\) 0 0
\(493\) − 0.185626i − 0.00836018i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.6558i 0.702257i
\(498\) 0 0
\(499\) 13.8458 0.619821 0.309911 0.950766i \(-0.399701\pi\)
0.309911 + 0.950766i \(0.399701\pi\)
\(500\) 0 0
\(501\) −19.2294 −0.859104
\(502\) 0 0
\(503\) 9.76454i 0.435379i 0.976018 + 0.217690i \(0.0698521\pi\)
−0.976018 + 0.217690i \(0.930148\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 37.3150i − 1.65722i
\(508\) 0 0
\(509\) 28.1677 1.24851 0.624257 0.781219i \(-0.285402\pi\)
0.624257 + 0.781219i \(0.285402\pi\)
\(510\) 0 0
\(511\) 12.1634 0.538076
\(512\) 0 0
\(513\) 16.0590i 0.709020i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 23.6780i 1.04136i
\(518\) 0 0
\(519\) 23.7003 1.04033
\(520\) 0 0
\(521\) −28.2035 −1.23562 −0.617809 0.786328i \(-0.711980\pi\)
−0.617809 + 0.786328i \(0.711980\pi\)
\(522\) 0 0
\(523\) − 1.22023i − 0.0533571i −0.999644 0.0266785i \(-0.991507\pi\)
0.999644 0.0266785i \(-0.00849305\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 9.13579 0.397208
\(530\) 0 0
\(531\) −37.0152 −1.60632
\(532\) 0 0
\(533\) 11.1472i 0.482839i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 25.9589i − 1.12021i
\(538\) 0 0
\(539\) −19.3560 −0.833723
\(540\) 0 0
\(541\) 12.1812 0.523713 0.261856 0.965107i \(-0.415665\pi\)
0.261856 + 0.965107i \(0.415665\pi\)
\(542\) 0 0
\(543\) 73.6417i 3.16027i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) −27.6282 −1.17914
\(550\) 0 0
\(551\) −0.870674 −0.0370920
\(552\) 0 0
\(553\) − 26.0768i − 1.10890i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.60764i 0.364718i 0.983232 + 0.182359i \(0.0583732\pi\)
−0.983232 + 0.182359i \(0.941627\pi\)
\(558\) 0 0
\(559\) −4.77064 −0.201777
\(560\) 0 0
\(561\) −3.57360 −0.150878
\(562\) 0 0
\(563\) 19.3560i 0.815759i 0.913036 + 0.407880i \(0.133732\pi\)
−0.913036 + 0.407880i \(0.866268\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 52.4167i 2.20129i
\(568\) 0 0
\(569\) −9.23811 −0.387282 −0.193641 0.981072i \(-0.562030\pi\)
−0.193641 + 0.981072i \(0.562030\pi\)
\(570\) 0 0
\(571\) 14.3766 0.601640 0.300820 0.953681i \(-0.402740\pi\)
0.300820 + 0.953681i \(0.402740\pi\)
\(572\) 0 0
\(573\) 78.2625i 3.26946i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 5.19266i 0.216173i 0.994141 + 0.108087i \(0.0344724\pi\)
−0.994141 + 0.108087i \(0.965528\pi\)
\(578\) 0 0
\(579\) 24.1179 1.00231
\(580\) 0 0
\(581\) −17.4059 −0.722117
\(582\) 0 0
\(583\) 56.1179i 2.32417i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.3079i 1.25094i 0.780248 + 0.625471i \(0.215093\pi\)
−0.780248 + 0.625471i \(0.784907\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −28.3355 −1.16557
\(592\) 0 0
\(593\) 4.04107i 0.165947i 0.996552 + 0.0829735i \(0.0264417\pi\)
−0.996552 + 0.0829735i \(0.973558\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 23.2971i − 0.953486i
\(598\) 0 0
\(599\) 6.29441 0.257183 0.128591 0.991698i \(-0.458954\pi\)
0.128591 + 0.991698i \(0.458954\pi\)
\(600\) 0 0
\(601\) −10.0358 −0.409367 −0.204684 0.978828i \(-0.565617\pi\)
−0.204684 + 0.978828i \(0.565617\pi\)
\(602\) 0 0
\(603\) − 8.88856i − 0.361970i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 45.3062i − 1.83892i −0.393182 0.919461i \(-0.628626\pi\)
0.393182 0.919461i \(-0.371374\pi\)
\(608\) 0 0
\(609\) −5.12933 −0.207851
\(610\) 0 0
\(611\) 6.04107 0.244396
\(612\) 0 0
\(613\) − 25.1607i − 1.01623i −0.861289 0.508116i \(-0.830342\pi\)
0.861289 0.508116i \(-0.169658\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.01350i 0.242095i 0.992647 + 0.121047i \(0.0386253\pi\)
−0.992647 + 0.121047i \(0.961375\pi\)
\(618\) 0 0
\(619\) −37.2240 −1.49616 −0.748080 0.663608i \(-0.769024\pi\)
−0.748080 + 0.663608i \(0.769024\pi\)
\(620\) 0 0
\(621\) −59.7950 −2.39949
\(622\) 0 0
\(623\) 12.0821i 0.484061i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 16.7619i 0.669405i
\(628\) 0 0
\(629\) 0.426396 0.0170015
\(630\) 0 0
\(631\) −12.0135 −0.478250 −0.239125 0.970989i \(-0.576861\pi\)
−0.239125 + 0.970989i \(0.576861\pi\)
\(632\) 0 0
\(633\) 18.3176i 0.728060i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.93839i 0.195666i
\(638\) 0 0
\(639\) −68.9619 −2.72809
\(640\) 0 0
\(641\) 6.97946 0.275672 0.137836 0.990455i \(-0.455985\pi\)
0.137836 + 0.990455i \(0.455985\pi\)
\(642\) 0 0
\(643\) − 40.7754i − 1.60802i −0.594613 0.804012i \(-0.702695\pi\)
0.594613 0.804012i \(-0.297305\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.9340i 0.980257i 0.871650 + 0.490129i \(0.163050\pi\)
−0.871650 + 0.490129i \(0.836950\pi\)
\(648\) 0 0
\(649\) −23.9091 −0.938514
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.3988i 1.42440i 0.701979 + 0.712198i \(0.252300\pi\)
−0.701979 + 0.712198i \(0.747700\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 53.5783i 2.09029i
\(658\) 0 0
\(659\) −11.8501 −0.461616 −0.230808 0.972999i \(-0.574137\pi\)
−0.230808 + 0.972999i \(0.574137\pi\)
\(660\) 0 0
\(661\) 34.4854 1.34132 0.670662 0.741763i \(-0.266010\pi\)
0.670662 + 0.741763i \(0.266010\pi\)
\(662\) 0 0
\(663\) 0.911749i 0.0354094i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.24193i − 0.125528i
\(668\) 0 0
\(669\) 55.2651 2.13667
\(670\) 0 0
\(671\) −17.8458 −0.688928
\(672\) 0 0
\(673\) − 17.7003i − 0.682295i −0.940010 0.341148i \(-0.889184\pi\)
0.940010 0.341148i \(-0.110816\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 26.0590i − 1.00153i −0.865584 0.500764i \(-0.833053\pi\)
0.865584 0.500764i \(-0.166947\pi\)
\(678\) 0 0
\(679\) −7.44693 −0.285787
\(680\) 0 0
\(681\) 27.2883 1.04569
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 41.8680i − 1.59736i
\(688\) 0 0
\(689\) 14.3176 0.545457
\(690\) 0 0
\(691\) 13.9367 0.530176 0.265088 0.964224i \(-0.414599\pi\)
0.265088 + 0.964224i \(0.414599\pi\)
\(692\) 0 0
\(693\) 71.4962i 2.71592i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.83226i 0.0694016i
\(698\) 0 0
\(699\) −27.5736 −1.04293
\(700\) 0 0
\(701\) 6.55307 0.247506 0.123753 0.992313i \(-0.460507\pi\)
0.123753 + 0.992313i \(0.460507\pi\)
\(702\) 0 0
\(703\) − 2.00000i − 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 34.2857i 1.28944i
\(708\) 0 0
\(709\) −23.5238 −0.883454 −0.441727 0.897150i \(-0.645634\pi\)
−0.441727 + 0.897150i \(0.645634\pi\)
\(710\) 0 0
\(711\) 114.866 4.30780
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.9973i 0.858852i
\(718\) 0 0
\(719\) 14.8842 0.555086 0.277543 0.960713i \(-0.410480\pi\)
0.277543 + 0.960713i \(0.410480\pi\)
\(720\) 0 0
\(721\) −28.4264 −1.05865
\(722\) 0 0
\(723\) 65.0886i 2.42067i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.32464i 0.0491283i 0.999698 + 0.0245641i \(0.00781979\pi\)
−0.999698 + 0.0245641i \(0.992180\pi\)
\(728\) 0 0
\(729\) −72.0475 −2.66843
\(730\) 0 0
\(731\) −0.784146 −0.0290027
\(732\) 0 0
\(733\) 13.7824i 0.509065i 0.967064 + 0.254533i \(0.0819217\pi\)
−0.967064 + 0.254533i \(0.918078\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.74135i − 0.211485i
\(738\) 0 0
\(739\) −23.5513 −0.866350 −0.433175 0.901310i \(-0.642607\pi\)
−0.433175 + 0.901310i \(0.642607\pi\)
\(740\) 0 0
\(741\) 4.27653 0.157102
\(742\) 0 0
\(743\) 14.0411i 0.515117i 0.966263 + 0.257559i \(0.0829180\pi\)
−0.966263 + 0.257559i \(0.917082\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 76.6710i − 2.80524i
\(748\) 0 0
\(749\) 5.89121 0.215260
\(750\) 0 0
\(751\) −17.7056 −0.646086 −0.323043 0.946384i \(-0.604706\pi\)
−0.323043 + 0.946384i \(0.604706\pi\)
\(752\) 0 0
\(753\) − 33.8680i − 1.23422i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.3220i 1.10207i 0.834482 + 0.551036i \(0.185767\pi\)
−0.834482 + 0.551036i \(0.814233\pi\)
\(758\) 0 0
\(759\) −62.4123 −2.26542
\(760\) 0 0
\(761\) 12.5487 0.454890 0.227445 0.973791i \(-0.426963\pi\)
0.227445 + 0.973791i \(0.426963\pi\)
\(762\) 0 0
\(763\) − 7.94104i − 0.287485i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.10003i 0.220259i
\(768\) 0 0
\(769\) 50.0725 1.80566 0.902830 0.429999i \(-0.141486\pi\)
0.902830 + 0.429999i \(0.141486\pi\)
\(770\) 0 0
\(771\) 32.6710 1.17662
\(772\) 0 0
\(773\) − 11.8003i − 0.424427i −0.977223 0.212214i \(-0.931933\pi\)
0.977223 0.212214i \(-0.0680673\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 11.7824i − 0.422692i
\(778\) 0 0
\(779\) 8.59414 0.307917
\(780\) 0 0
\(781\) −44.5443 −1.59392
\(782\) 0 0
\(783\) − 13.9821i − 0.499680i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 32.9475i 1.17445i 0.809423 + 0.587226i \(0.199780\pi\)
−0.809423 + 0.587226i \(0.800220\pi\)
\(788\) 0 0
\(789\) −78.0681 −2.77930
\(790\) 0 0
\(791\) −19.9912 −0.710807
\(792\) 0 0
\(793\) 4.55307i 0.161684i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 18.8208i − 0.666668i −0.942809 0.333334i \(-0.891826\pi\)
0.942809 0.333334i \(-0.108174\pi\)
\(798\) 0 0
\(799\) 0.992965 0.0351286
\(800\) 0 0
\(801\) −53.2206 −1.88046
\(802\) 0 0
\(803\) 34.6076i 1.22128i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 16.5531i − 0.582696i
\(808\) 0 0
\(809\) 39.2607 1.38033 0.690167 0.723650i \(-0.257537\pi\)
0.690167 + 0.723650i \(0.257537\pi\)
\(810\) 0 0
\(811\) 33.9182 1.19103 0.595515 0.803344i \(-0.296948\pi\)
0.595515 + 0.803344i \(0.296948\pi\)
\(812\) 0 0
\(813\) − 14.1000i − 0.494510i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.67801i 0.128677i
\(818\) 0 0
\(819\) 18.2411 0.637397
\(820\) 0 0
\(821\) 38.6487 1.34885 0.674425 0.738344i \(-0.264392\pi\)
0.674425 + 0.738344i \(0.264392\pi\)
\(822\) 0 0
\(823\) 8.47185i 0.295310i 0.989039 + 0.147655i \(0.0471725\pi\)
−0.989039 + 0.147655i \(0.952827\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 44.7299i − 1.55541i −0.628628 0.777706i \(-0.716383\pi\)
0.628628 0.777706i \(-0.283617\pi\)
\(828\) 0 0
\(829\) 38.8566 1.34955 0.674773 0.738025i \(-0.264242\pi\)
0.674773 + 0.738025i \(0.264242\pi\)
\(830\) 0 0
\(831\) 26.1677 0.907749
\(832\) 0 0
\(833\) 0.811718i 0.0281244i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.53784 0.260235 0.130118 0.991499i \(-0.458465\pi\)
0.130118 + 0.991499i \(0.458465\pi\)
\(840\) 0 0
\(841\) −28.2419 −0.973860
\(842\) 0 0
\(843\) − 75.4652i − 2.59916i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 26.5264i 0.911459i
\(848\) 0 0
\(849\) 16.7619 0.575266
\(850\) 0 0
\(851\) 7.44693 0.255278
\(852\) 0 0
\(853\) 43.5238i 1.49023i 0.666938 + 0.745113i \(0.267604\pi\)
−0.666938 + 0.745113i \(0.732396\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 14.5531i − 0.497123i −0.968616 0.248562i \(-0.920042\pi\)
0.968616 0.248562i \(-0.0799579\pi\)
\(858\) 0 0
\(859\) 24.3132 0.829557 0.414778 0.909922i \(-0.363859\pi\)
0.414778 + 0.909922i \(0.363859\pi\)
\(860\) 0 0
\(861\) 50.6299 1.72546
\(862\) 0 0
\(863\) 12.4264i 0.422999i 0.977378 + 0.211500i \(0.0678347\pi\)
−0.977378 + 0.211500i \(0.932165\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 55.9003i − 1.89847i
\(868\) 0 0
\(869\) 74.1948 2.51688
\(870\) 0 0
\(871\) −1.46482 −0.0496334
\(872\) 0 0
\(873\) − 32.8030i − 1.11021i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 54.6120i 1.84412i 0.387051 + 0.922058i \(0.373494\pi\)
−0.387051 + 0.922058i \(0.626506\pi\)
\(878\) 0 0
\(879\) 11.7235 0.395423
\(880\) 0 0
\(881\) 35.4551 1.19451 0.597257 0.802050i \(-0.296257\pi\)
0.597257 + 0.802050i \(0.296257\pi\)
\(882\) 0 0
\(883\) − 21.5460i − 0.725082i −0.931968 0.362541i \(-0.881909\pi\)
0.931968 0.362541i \(-0.118091\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.6147i 1.66590i 0.553350 + 0.832949i \(0.313349\pi\)
−0.553350 + 0.832949i \(0.686651\pi\)
\(888\) 0 0
\(889\) −29.9502 −1.00450
\(890\) 0 0
\(891\) −149.138 −4.99631
\(892\) 0 0
\(893\) − 4.65748i − 0.155856i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 15.9235i 0.531671i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 2.35337 0.0784022
\(902\) 0 0
\(903\) 21.6680i 0.721064i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 11.0883i − 0.368179i −0.982909 0.184090i \(-0.941066\pi\)
0.982909 0.184090i \(-0.0589337\pi\)
\(908\) 0 0
\(909\) −151.025 −5.00917
\(910\) 0 0
\(911\) 12.5443 0.415612 0.207806 0.978170i \(-0.433368\pi\)
0.207806 + 0.978170i \(0.433368\pi\)
\(912\) 0 0
\(913\) − 49.5238i − 1.63900i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.57188i 0.217023i
\(918\) 0 0
\(919\) −42.8707 −1.41417 −0.707087 0.707127i \(-0.749991\pi\)
−0.707087 + 0.707127i \(0.749991\pi\)
\(920\) 0 0
\(921\) 53.8593 1.77472
\(922\) 0 0
\(923\) 11.3648i 0.374076i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 125.215i − 4.11261i
\(928\) 0 0
\(929\) 27.9182 0.915967 0.457984 0.888961i \(-0.348572\pi\)
0.457984 + 0.888961i \(0.348572\pi\)
\(930\) 0 0
\(931\) 3.80734 0.124781
\(932\) 0 0
\(933\) 54.0091i 1.76818i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.8982i − 0.356030i −0.984028 0.178015i \(-0.943032\pi\)
0.984028 0.178015i \(-0.0569676\pi\)
\(938\) 0 0
\(939\) 99.1065 3.23422
\(940\) 0 0
\(941\) −30.9475 −1.00886 −0.504430 0.863453i \(-0.668297\pi\)
−0.504430 + 0.863453i \(0.668297\pi\)
\(942\) 0 0
\(943\) 32.0000i 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 18.2944i − 0.594489i −0.954801 0.297244i \(-0.903932\pi\)
0.954801 0.297244i \(-0.0960676\pi\)
\(948\) 0 0
\(949\) 8.82960 0.286621
\(950\) 0 0
\(951\) −39.6238 −1.28489
\(952\) 0 0
\(953\) 23.0616i 0.747039i 0.927622 + 0.373519i \(0.121849\pi\)
−0.927622 + 0.373519i \(0.878151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 14.5941i − 0.471762i
\(958\) 0 0
\(959\) −38.2402 −1.23484
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 25.9502i 0.836232i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 26.3766i 0.848213i 0.905612 + 0.424107i \(0.139412\pi\)
−0.905612 + 0.424107i \(0.860588\pi\)
\(968\) 0 0
\(969\) 0.702929 0.0225813
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 19.3425i 0.620093i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 26.4264i − 0.845455i −0.906257 0.422728i \(-0.861073\pi\)
0.906257 0.422728i \(-0.138927\pi\)
\(978\) 0 0
\(979\) −34.3766 −1.09868
\(980\) 0 0
\(981\) 34.9795 1.11681
\(982\) 0 0
\(983\) − 25.4059i − 0.810321i −0.914246 0.405161i \(-0.867216\pi\)
0.914246 0.405161i \(-0.132784\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 27.4382i − 0.873367i
\(988\) 0 0
\(989\) −13.6950 −0.435474
\(990\) 0 0
\(991\) 29.6504 0.941877 0.470939 0.882166i \(-0.343915\pi\)
0.470939 + 0.882166i \(0.343915\pi\)
\(992\) 0 0
\(993\) 47.2062i 1.49804i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 38.7754i − 1.22803i −0.789295 0.614014i \(-0.789554\pi\)
0.789295 0.614014i \(-0.210446\pi\)
\(998\) 0 0
\(999\) 32.1179 1.01617
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 3800.2.d.j.3649.1 6
5.2 odd 4 3800.2.a.r.1.1 3
5.3 odd 4 152.2.a.c.1.3 3
5.4 even 2 inner 3800.2.d.j.3649.6 6
15.8 even 4 1368.2.a.n.1.3 3
20.3 even 4 304.2.a.g.1.1 3
20.7 even 4 7600.2.a.bv.1.3 3
35.13 even 4 7448.2.a.bf.1.1 3
40.3 even 4 1216.2.a.v.1.3 3
40.13 odd 4 1216.2.a.u.1.1 3
60.23 odd 4 2736.2.a.bd.1.3 3
95.18 even 4 2888.2.a.o.1.1 3
380.303 odd 4 5776.2.a.bp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.3 3 5.3 odd 4
304.2.a.g.1.1 3 20.3 even 4
1216.2.a.u.1.1 3 40.13 odd 4
1216.2.a.v.1.3 3 40.3 even 4
1368.2.a.n.1.3 3 15.8 even 4
2736.2.a.bd.1.3 3 60.23 odd 4
2888.2.a.o.1.1 3 95.18 even 4
3800.2.a.r.1.1 3 5.2 odd 4
3800.2.d.j.3649.1 6 1.1 even 1 trivial
3800.2.d.j.3649.6 6 5.4 even 2 inner
5776.2.a.bp.1.3 3 380.303 odd 4
7448.2.a.bf.1.1 3 35.13 even 4
7600.2.a.bv.1.3 3 20.7 even 4