Properties

Label 384.5.b.a.319.3
Level $384$
Weight $5$
Character 384.319
Analytic conductor $39.694$
Analytic rank $0$
Dimension $4$
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] = [384,5,Mod(319,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.b (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: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 5x^{2} - 4x + 61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.3
Root \(2.23205 - 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.5.b.a.319.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.19615 q^{3} -30.1993i q^{5} -52.3068i q^{7} +27.0000 q^{9} +O(q^{10})\) \(q+5.19615 q^{3} -30.1993i q^{5} -52.3068i q^{7} +27.0000 q^{9} -90.0666 q^{11} -60.3987i q^{13} -156.920i q^{15} -338.000 q^{17} -6.92820 q^{19} -271.794i q^{21} -732.295i q^{23} -287.000 q^{25} +140.296 q^{27} +1298.57i q^{29} +1307.67i q^{31} -468.000 q^{33} -1579.63 q^{35} -241.595i q^{37} -313.841i q^{39} +578.000 q^{41} -2029.96 q^{43} -815.382i q^{45} +2196.89i q^{47} -335.000 q^{49} -1756.30 q^{51} +2446.15i q^{53} +2719.95i q^{55} -36.0000 q^{57} -1198.58 q^{59} -6402.26i q^{61} -1412.28i q^{63} -1824.00 q^{65} -8265.35 q^{67} -3805.12i q^{69} -4289.16i q^{71} +8734.00 q^{73} -1491.30 q^{75} +4711.10i q^{77} -11246.0i q^{79} +729.000 q^{81} -13198.2 q^{83} +10207.4i q^{85} +6747.58i q^{87} +910.000 q^{89} -3159.26 q^{91} +6794.85i q^{93} +209.227i q^{95} +5422.00 q^{97} -2431.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 108 q^{9} - 1352 q^{17} - 1148 q^{25} - 1872 q^{33} + 2312 q^{41} - 1340 q^{49} - 144 q^{57} - 7296 q^{65} + 34936 q^{73} + 2916 q^{81} + 3640 q^{89} + 21688 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 5.19615 0.577350
\(4\) 0 0
\(5\) − 30.1993i − 1.20797i −0.796994 0.603987i \(-0.793578\pi\)
0.796994 0.603987i \(-0.206422\pi\)
\(6\) 0 0
\(7\) − 52.3068i − 1.06749i −0.845647 0.533743i \(-0.820785\pi\)
0.845647 0.533743i \(-0.179215\pi\)
\(8\) 0 0
\(9\) 27.0000 0.333333
\(10\) 0 0
\(11\) −90.0666 −0.744352 −0.372176 0.928162i \(-0.621388\pi\)
−0.372176 + 0.928162i \(0.621388\pi\)
\(12\) 0 0
\(13\) − 60.3987i − 0.357389i −0.983905 0.178694i \(-0.942813\pi\)
0.983905 0.178694i \(-0.0571873\pi\)
\(14\) 0 0
\(15\) − 156.920i − 0.697424i
\(16\) 0 0
\(17\) −338.000 −1.16955 −0.584775 0.811195i \(-0.698817\pi\)
−0.584775 + 0.811195i \(0.698817\pi\)
\(18\) 0 0
\(19\) −6.92820 −0.0191917 −0.00959585 0.999954i \(-0.503055\pi\)
−0.00959585 + 0.999954i \(0.503055\pi\)
\(20\) 0 0
\(21\) − 271.794i − 0.616313i
\(22\) 0 0
\(23\) − 732.295i − 1.38430i −0.721753 0.692150i \(-0.756664\pi\)
0.721753 0.692150i \(-0.243336\pi\)
\(24\) 0 0
\(25\) −287.000 −0.459200
\(26\) 0 0
\(27\) 140.296 0.192450
\(28\) 0 0
\(29\) 1298.57i 1.54408i 0.635574 + 0.772040i \(0.280764\pi\)
−0.635574 + 0.772040i \(0.719236\pi\)
\(30\) 0 0
\(31\) 1307.67i 1.36074i 0.732869 + 0.680369i \(0.238181\pi\)
−0.732869 + 0.680369i \(0.761819\pi\)
\(32\) 0 0
\(33\) −468.000 −0.429752
\(34\) 0 0
\(35\) −1579.63 −1.28949
\(36\) 0 0
\(37\) − 241.595i − 0.176475i −0.996099 0.0882377i \(-0.971877\pi\)
0.996099 0.0882377i \(-0.0281235\pi\)
\(38\) 0 0
\(39\) − 313.841i − 0.206338i
\(40\) 0 0
\(41\) 578.000 0.343843 0.171921 0.985111i \(-0.445002\pi\)
0.171921 + 0.985111i \(0.445002\pi\)
\(42\) 0 0
\(43\) −2029.96 −1.09787 −0.548936 0.835865i \(-0.684967\pi\)
−0.548936 + 0.835865i \(0.684967\pi\)
\(44\) 0 0
\(45\) − 815.382i − 0.402658i
\(46\) 0 0
\(47\) 2196.89i 0.994516i 0.867603 + 0.497258i \(0.165660\pi\)
−0.867603 + 0.497258i \(0.834340\pi\)
\(48\) 0 0
\(49\) −335.000 −0.139525
\(50\) 0 0
\(51\) −1756.30 −0.675240
\(52\) 0 0
\(53\) 2446.15i 0.870825i 0.900231 + 0.435412i \(0.143397\pi\)
−0.900231 + 0.435412i \(0.856603\pi\)
\(54\) 0 0
\(55\) 2719.95i 0.899158i
\(56\) 0 0
\(57\) −36.0000 −0.0110803
\(58\) 0 0
\(59\) −1198.58 −0.344320 −0.172160 0.985069i \(-0.555075\pi\)
−0.172160 + 0.985069i \(0.555075\pi\)
\(60\) 0 0
\(61\) − 6402.26i − 1.72058i −0.509809 0.860288i \(-0.670284\pi\)
0.509809 0.860288i \(-0.329716\pi\)
\(62\) 0 0
\(63\) − 1412.28i − 0.355828i
\(64\) 0 0
\(65\) −1824.00 −0.431716
\(66\) 0 0
\(67\) −8265.35 −1.84124 −0.920622 0.390454i \(-0.872318\pi\)
−0.920622 + 0.390454i \(0.872318\pi\)
\(68\) 0 0
\(69\) − 3805.12i − 0.799226i
\(70\) 0 0
\(71\) − 4289.16i − 0.850854i −0.904993 0.425427i \(-0.860124\pi\)
0.904993 0.425427i \(-0.139876\pi\)
\(72\) 0 0
\(73\) 8734.00 1.63896 0.819478 0.573110i \(-0.194263\pi\)
0.819478 + 0.573110i \(0.194263\pi\)
\(74\) 0 0
\(75\) −1491.30 −0.265119
\(76\) 0 0
\(77\) 4711.10i 0.794585i
\(78\) 0 0
\(79\) − 11246.0i − 1.80195i −0.433873 0.900974i \(-0.642853\pi\)
0.433873 0.900974i \(-0.357147\pi\)
\(80\) 0 0
\(81\) 729.000 0.111111
\(82\) 0 0
\(83\) −13198.2 −1.91584 −0.957920 0.287034i \(-0.907331\pi\)
−0.957920 + 0.287034i \(0.907331\pi\)
\(84\) 0 0
\(85\) 10207.4i 1.41279i
\(86\) 0 0
\(87\) 6747.58i 0.891475i
\(88\) 0 0
\(89\) 910.000 0.114884 0.0574422 0.998349i \(-0.481705\pi\)
0.0574422 + 0.998349i \(0.481705\pi\)
\(90\) 0 0
\(91\) −3159.26 −0.381507
\(92\) 0 0
\(93\) 6794.85i 0.785623i
\(94\) 0 0
\(95\) 209.227i 0.0231831i
\(96\) 0 0
\(97\) 5422.00 0.576257 0.288128 0.957592i \(-0.406967\pi\)
0.288128 + 0.957592i \(0.406967\pi\)
\(98\) 0 0
\(99\) −2431.80 −0.248117
\(100\) 0 0
\(101\) − 10962.4i − 1.07464i −0.843380 0.537318i \(-0.819438\pi\)
0.843380 0.537318i \(-0.180562\pi\)
\(102\) 0 0
\(103\) − 5387.60i − 0.507833i −0.967226 0.253916i \(-0.918281\pi\)
0.967226 0.253916i \(-0.0817188\pi\)
\(104\) 0 0
\(105\) −8208.00 −0.744490
\(106\) 0 0
\(107\) −6436.30 −0.562171 −0.281086 0.959683i \(-0.590695\pi\)
−0.281086 + 0.959683i \(0.590695\pi\)
\(108\) 0 0
\(109\) 60.3987i 0.00508364i 0.999997 + 0.00254182i \(0.000809087\pi\)
−0.999997 + 0.00254182i \(0.999191\pi\)
\(110\) 0 0
\(111\) − 1255.36i − 0.101888i
\(112\) 0 0
\(113\) −3166.00 −0.247944 −0.123972 0.992286i \(-0.539563\pi\)
−0.123972 + 0.992286i \(0.539563\pi\)
\(114\) 0 0
\(115\) −22114.8 −1.67220
\(116\) 0 0
\(117\) − 1630.76i − 0.119130i
\(118\) 0 0
\(119\) 17679.7i 1.24848i
\(120\) 0 0
\(121\) −6529.00 −0.445939
\(122\) 0 0
\(123\) 3003.38 0.198518
\(124\) 0 0
\(125\) − 10207.4i − 0.653272i
\(126\) 0 0
\(127\) 12919.8i 0.801028i 0.916291 + 0.400514i \(0.131168\pi\)
−0.916291 + 0.400514i \(0.868832\pi\)
\(128\) 0 0
\(129\) −10548.0 −0.633856
\(130\) 0 0
\(131\) −22676.0 −1.32137 −0.660684 0.750664i \(-0.729734\pi\)
−0.660684 + 0.750664i \(0.729734\pi\)
\(132\) 0 0
\(133\) 362.392i 0.0204869i
\(134\) 0 0
\(135\) − 4236.85i − 0.232475i
\(136\) 0 0
\(137\) −15550.0 −0.828494 −0.414247 0.910165i \(-0.635955\pi\)
−0.414247 + 0.910165i \(0.635955\pi\)
\(138\) 0 0
\(139\) 31502.5 1.63048 0.815241 0.579122i \(-0.196604\pi\)
0.815241 + 0.579122i \(0.196604\pi\)
\(140\) 0 0
\(141\) 11415.3i 0.574184i
\(142\) 0 0
\(143\) 5439.91i 0.266023i
\(144\) 0 0
\(145\) 39216.0 1.86521
\(146\) 0 0
\(147\) −1740.71 −0.0805549
\(148\) 0 0
\(149\) − 22619.3i − 1.01884i −0.860518 0.509421i \(-0.829860\pi\)
0.860518 0.509421i \(-0.170140\pi\)
\(150\) 0 0
\(151\) 7689.10i 0.337226i 0.985682 + 0.168613i \(0.0539289\pi\)
−0.985682 + 0.168613i \(0.946071\pi\)
\(152\) 0 0
\(153\) −9126.00 −0.389850
\(154\) 0 0
\(155\) 39490.8 1.64374
\(156\) 0 0
\(157\) − 1207.97i − 0.0490070i −0.999700 0.0245035i \(-0.992200\pi\)
0.999700 0.0245035i \(-0.00780049\pi\)
\(158\) 0 0
\(159\) 12710.5i 0.502771i
\(160\) 0 0
\(161\) −38304.0 −1.47772
\(162\) 0 0
\(163\) −26749.8 −1.00680 −0.503402 0.864052i \(-0.667919\pi\)
−0.503402 + 0.864052i \(0.667919\pi\)
\(164\) 0 0
\(165\) 14133.3i 0.519129i
\(166\) 0 0
\(167\) − 44983.8i − 1.61296i −0.591261 0.806480i \(-0.701370\pi\)
0.591261 0.806480i \(-0.298630\pi\)
\(168\) 0 0
\(169\) 24913.0 0.872273
\(170\) 0 0
\(171\) −187.061 −0.00639723
\(172\) 0 0
\(173\) − 4197.71i − 0.140256i −0.997538 0.0701278i \(-0.977659\pi\)
0.997538 0.0701278i \(-0.0223407\pi\)
\(174\) 0 0
\(175\) 15012.0i 0.490189i
\(176\) 0 0
\(177\) −6228.00 −0.198793
\(178\) 0 0
\(179\) 12491.6 0.389861 0.194931 0.980817i \(-0.437552\pi\)
0.194931 + 0.980817i \(0.437552\pi\)
\(180\) 0 0
\(181\) 37870.0i 1.15595i 0.816056 + 0.577973i \(0.196156\pi\)
−0.816056 + 0.577973i \(0.803844\pi\)
\(182\) 0 0
\(183\) − 33267.1i − 0.993374i
\(184\) 0 0
\(185\) −7296.00 −0.213178
\(186\) 0 0
\(187\) 30442.5 0.870557
\(188\) 0 0
\(189\) − 7338.44i − 0.205438i
\(190\) 0 0
\(191\) − 32848.7i − 0.900432i −0.892920 0.450216i \(-0.851347\pi\)
0.892920 0.450216i \(-0.148653\pi\)
\(192\) 0 0
\(193\) 44830.0 1.20352 0.601761 0.798676i \(-0.294466\pi\)
0.601761 + 0.798676i \(0.294466\pi\)
\(194\) 0 0
\(195\) −9477.78 −0.249251
\(196\) 0 0
\(197\) 42188.5i 1.08708i 0.839383 + 0.543540i \(0.182916\pi\)
−0.839383 + 0.543540i \(0.817084\pi\)
\(198\) 0 0
\(199\) − 18778.1i − 0.474183i −0.971487 0.237092i \(-0.923806\pi\)
0.971487 0.237092i \(-0.0761942\pi\)
\(200\) 0 0
\(201\) −42948.0 −1.06304
\(202\) 0 0
\(203\) 67924.1 1.64828
\(204\) 0 0
\(205\) − 17455.2i − 0.415353i
\(206\) 0 0
\(207\) − 19772.0i − 0.461434i
\(208\) 0 0
\(209\) 624.000 0.0142854
\(210\) 0 0
\(211\) 63704.8 1.43089 0.715447 0.698667i \(-0.246223\pi\)
0.715447 + 0.698667i \(0.246223\pi\)
\(212\) 0 0
\(213\) − 22287.1i − 0.491241i
\(214\) 0 0
\(215\) 61303.6i 1.32620i
\(216\) 0 0
\(217\) 68400.0 1.45257
\(218\) 0 0
\(219\) 45383.2 0.946252
\(220\) 0 0
\(221\) 20414.8i 0.417984i
\(222\) 0 0
\(223\) 19405.8i 0.390231i 0.980780 + 0.195116i \(0.0625082\pi\)
−0.980780 + 0.195116i \(0.937492\pi\)
\(224\) 0 0
\(225\) −7749.00 −0.153067
\(226\) 0 0
\(227\) −12089.7 −0.234620 −0.117310 0.993095i \(-0.537427\pi\)
−0.117310 + 0.993095i \(0.537427\pi\)
\(228\) 0 0
\(229\) − 91020.8i − 1.73568i −0.496844 0.867840i \(-0.665508\pi\)
0.496844 0.867840i \(-0.334492\pi\)
\(230\) 0 0
\(231\) 24479.6i 0.458754i
\(232\) 0 0
\(233\) 45166.0 0.831955 0.415977 0.909375i \(-0.363440\pi\)
0.415977 + 0.909375i \(0.363440\pi\)
\(234\) 0 0
\(235\) 66344.5 1.20135
\(236\) 0 0
\(237\) − 58435.7i − 1.04036i
\(238\) 0 0
\(239\) − 12135.2i − 0.212447i −0.994342 0.106223i \(-0.966124\pi\)
0.994342 0.106223i \(-0.0338759\pi\)
\(240\) 0 0
\(241\) 85822.0 1.47763 0.738813 0.673910i \(-0.235387\pi\)
0.738813 + 0.673910i \(0.235387\pi\)
\(242\) 0 0
\(243\) 3788.00 0.0641500
\(244\) 0 0
\(245\) 10116.8i 0.168543i
\(246\) 0 0
\(247\) 418.454i 0.00685889i
\(248\) 0 0
\(249\) −68580.0 −1.10611
\(250\) 0 0
\(251\) 58190.0 0.923636 0.461818 0.886975i \(-0.347197\pi\)
0.461818 + 0.886975i \(0.347197\pi\)
\(252\) 0 0
\(253\) 65955.4i 1.03041i
\(254\) 0 0
\(255\) 53039.1i 0.815672i
\(256\) 0 0
\(257\) −121726. −1.84297 −0.921483 0.388420i \(-0.873021\pi\)
−0.921483 + 0.388420i \(0.873021\pi\)
\(258\) 0 0
\(259\) −12637.0 −0.188385
\(260\) 0 0
\(261\) 35061.4i 0.514693i
\(262\) 0 0
\(263\) 97918.3i 1.41564i 0.706394 + 0.707819i \(0.250321\pi\)
−0.706394 + 0.707819i \(0.749679\pi\)
\(264\) 0 0
\(265\) 73872.0 1.05193
\(266\) 0 0
\(267\) 4728.50 0.0663286
\(268\) 0 0
\(269\) 14465.5i 0.199907i 0.994992 + 0.0999536i \(0.0318694\pi\)
−0.994992 + 0.0999536i \(0.968131\pi\)
\(270\) 0 0
\(271\) − 55497.5i − 0.755675i −0.925872 0.377837i \(-0.876668\pi\)
0.925872 0.377837i \(-0.123332\pi\)
\(272\) 0 0
\(273\) −16416.0 −0.220263
\(274\) 0 0
\(275\) 25849.1 0.341807
\(276\) 0 0
\(277\) − 102013.i − 1.32953i −0.747053 0.664764i \(-0.768532\pi\)
0.747053 0.664764i \(-0.231468\pi\)
\(278\) 0 0
\(279\) 35307.1i 0.453579i
\(280\) 0 0
\(281\) −40082.0 −0.507618 −0.253809 0.967254i \(-0.581683\pi\)
−0.253809 + 0.967254i \(0.581683\pi\)
\(282\) 0 0
\(283\) 48324.2 0.603381 0.301691 0.953406i \(-0.402449\pi\)
0.301691 + 0.953406i \(0.402449\pi\)
\(284\) 0 0
\(285\) 1087.18i 0.0133847i
\(286\) 0 0
\(287\) − 30233.3i − 0.367047i
\(288\) 0 0
\(289\) 30723.0 0.367848
\(290\) 0 0
\(291\) 28173.5 0.332702
\(292\) 0 0
\(293\) 127592.i 1.48624i 0.669158 + 0.743120i \(0.266655\pi\)
−0.669158 + 0.743120i \(0.733345\pi\)
\(294\) 0 0
\(295\) 36196.3i 0.415930i
\(296\) 0 0
\(297\) −12636.0 −0.143251
\(298\) 0 0
\(299\) −44229.6 −0.494733
\(300\) 0 0
\(301\) 106181.i 1.17196i
\(302\) 0 0
\(303\) − 56962.1i − 0.620441i
\(304\) 0 0
\(305\) −193344. −2.07841
\(306\) 0 0
\(307\) 39788.7 0.422165 0.211083 0.977468i \(-0.432301\pi\)
0.211083 + 0.977468i \(0.432301\pi\)
\(308\) 0 0
\(309\) − 27994.8i − 0.293197i
\(310\) 0 0
\(311\) − 174705.i − 1.80627i −0.429352 0.903137i \(-0.641258\pi\)
0.429352 0.903137i \(-0.358742\pi\)
\(312\) 0 0
\(313\) 26930.0 0.274883 0.137441 0.990510i \(-0.456112\pi\)
0.137441 + 0.990510i \(0.456112\pi\)
\(314\) 0 0
\(315\) −42650.0 −0.429831
\(316\) 0 0
\(317\) 10841.6i 0.107888i 0.998544 + 0.0539440i \(0.0171793\pi\)
−0.998544 + 0.0539440i \(0.982821\pi\)
\(318\) 0 0
\(319\) − 116958.i − 1.14934i
\(320\) 0 0
\(321\) −33444.0 −0.324570
\(322\) 0 0
\(323\) 2341.73 0.0224457
\(324\) 0 0
\(325\) 17334.4i 0.164113i
\(326\) 0 0
\(327\) 313.841i 0.00293504i
\(328\) 0 0
\(329\) 114912. 1.06163
\(330\) 0 0
\(331\) −127597. −1.16462 −0.582309 0.812968i \(-0.697851\pi\)
−0.582309 + 0.812968i \(0.697851\pi\)
\(332\) 0 0
\(333\) − 6523.06i − 0.0588251i
\(334\) 0 0
\(335\) 249608.i 2.22417i
\(336\) 0 0
\(337\) 186482. 1.64201 0.821007 0.570917i \(-0.193412\pi\)
0.821007 + 0.570917i \(0.193412\pi\)
\(338\) 0 0
\(339\) −16451.0 −0.143151
\(340\) 0 0
\(341\) − 117777.i − 1.01287i
\(342\) 0 0
\(343\) − 108066.i − 0.918544i
\(344\) 0 0
\(345\) −114912. −0.965444
\(346\) 0 0
\(347\) −49224.9 −0.408814 −0.204407 0.978886i \(-0.565527\pi\)
−0.204407 + 0.978886i \(0.565527\pi\)
\(348\) 0 0
\(349\) − 209825.i − 1.72269i −0.508023 0.861343i \(-0.669624\pi\)
0.508023 0.861343i \(-0.330376\pi\)
\(350\) 0 0
\(351\) − 8473.70i − 0.0687795i
\(352\) 0 0
\(353\) −67486.0 −0.541582 −0.270791 0.962638i \(-0.587285\pi\)
−0.270791 + 0.962638i \(0.587285\pi\)
\(354\) 0 0
\(355\) −129530. −1.02781
\(356\) 0 0
\(357\) 91866.4i 0.720809i
\(358\) 0 0
\(359\) − 116016.i − 0.900183i −0.892983 0.450091i \(-0.851391\pi\)
0.892983 0.450091i \(-0.148609\pi\)
\(360\) 0 0
\(361\) −130273. −0.999632
\(362\) 0 0
\(363\) −33925.7 −0.257463
\(364\) 0 0
\(365\) − 263761.i − 1.97982i
\(366\) 0 0
\(367\) 77884.8i 0.578257i 0.957290 + 0.289128i \(0.0933654\pi\)
−0.957290 + 0.289128i \(0.906635\pi\)
\(368\) 0 0
\(369\) 15606.0 0.114614
\(370\) 0 0
\(371\) 127950. 0.929593
\(372\) 0 0
\(373\) 259835.i 1.86758i 0.357816 + 0.933792i \(0.383521\pi\)
−0.357816 + 0.933792i \(0.616479\pi\)
\(374\) 0 0
\(375\) − 53039.1i − 0.377167i
\(376\) 0 0
\(377\) 78432.0 0.551837
\(378\) 0 0
\(379\) −116761. −0.812867 −0.406433 0.913680i \(-0.633228\pi\)
−0.406433 + 0.913680i \(0.633228\pi\)
\(380\) 0 0
\(381\) 67133.1i 0.462474i
\(382\) 0 0
\(383\) − 158803.i − 1.08259i −0.840834 0.541293i \(-0.817935\pi\)
0.840834 0.541293i \(-0.182065\pi\)
\(384\) 0 0
\(385\) 142272. 0.959838
\(386\) 0 0
\(387\) −54809.0 −0.365957
\(388\) 0 0
\(389\) − 121673.i − 0.804073i −0.915624 0.402037i \(-0.868302\pi\)
0.915624 0.402037i \(-0.131698\pi\)
\(390\) 0 0
\(391\) 247516.i 1.61901i
\(392\) 0 0
\(393\) −117828. −0.762893
\(394\) 0 0
\(395\) −339621. −2.17671
\(396\) 0 0
\(397\) 237850.i 1.50911i 0.656234 + 0.754557i \(0.272148\pi\)
−0.656234 + 0.754557i \(0.727852\pi\)
\(398\) 0 0
\(399\) 1883.04i 0.0118281i
\(400\) 0 0
\(401\) −58130.0 −0.361503 −0.180751 0.983529i \(-0.557853\pi\)
−0.180751 + 0.983529i \(0.557853\pi\)
\(402\) 0 0
\(403\) 78981.5 0.486312
\(404\) 0 0
\(405\) − 22015.3i − 0.134219i
\(406\) 0 0
\(407\) 21759.6i 0.131360i
\(408\) 0 0
\(409\) 65186.0 0.389680 0.194840 0.980835i \(-0.437581\pi\)
0.194840 + 0.980835i \(0.437581\pi\)
\(410\) 0 0
\(411\) −80800.2 −0.478331
\(412\) 0 0
\(413\) 62693.8i 0.367557i
\(414\) 0 0
\(415\) 398578.i 2.31428i
\(416\) 0 0
\(417\) 163692. 0.941359
\(418\) 0 0
\(419\) −267519. −1.52379 −0.761897 0.647698i \(-0.775732\pi\)
−0.761897 + 0.647698i \(0.775732\pi\)
\(420\) 0 0
\(421\) − 263399.i − 1.48610i −0.669233 0.743052i \(-0.733377\pi\)
0.669233 0.743052i \(-0.266623\pi\)
\(422\) 0 0
\(423\) 59315.9i 0.331505i
\(424\) 0 0
\(425\) 97006.0 0.537057
\(426\) 0 0
\(427\) −334882. −1.83669
\(428\) 0 0
\(429\) 28266.6i 0.153588i
\(430\) 0 0
\(431\) 105346.i 0.567104i 0.958957 + 0.283552i \(0.0915129\pi\)
−0.958957 + 0.283552i \(0.908487\pi\)
\(432\) 0 0
\(433\) 83758.0 0.446736 0.223368 0.974734i \(-0.428295\pi\)
0.223368 + 0.974734i \(0.428295\pi\)
\(434\) 0 0
\(435\) 203772. 1.07688
\(436\) 0 0
\(437\) 5073.49i 0.0265671i
\(438\) 0 0
\(439\) − 95983.0i − 0.498041i −0.968498 0.249020i \(-0.919891\pi\)
0.968498 0.249020i \(-0.0801087\pi\)
\(440\) 0 0
\(441\) −9045.00 −0.0465084
\(442\) 0 0
\(443\) −222679. −1.13468 −0.567339 0.823484i \(-0.692027\pi\)
−0.567339 + 0.823484i \(0.692027\pi\)
\(444\) 0 0
\(445\) − 27481.4i − 0.138777i
\(446\) 0 0
\(447\) − 117533.i − 0.588229i
\(448\) 0 0
\(449\) −88658.0 −0.439770 −0.219885 0.975526i \(-0.570568\pi\)
−0.219885 + 0.975526i \(0.570568\pi\)
\(450\) 0 0
\(451\) −52058.5 −0.255940
\(452\) 0 0
\(453\) 39953.7i 0.194698i
\(454\) 0 0
\(455\) 95407.6i 0.460851i
\(456\) 0 0
\(457\) −29086.0 −0.139268 −0.0696340 0.997573i \(-0.522183\pi\)
−0.0696340 + 0.997573i \(0.522183\pi\)
\(458\) 0 0
\(459\) −47420.1 −0.225080
\(460\) 0 0
\(461\) 277139.i 1.30406i 0.758195 + 0.652028i \(0.226082\pi\)
−0.758195 + 0.652028i \(0.773918\pi\)
\(462\) 0 0
\(463\) − 279371.i − 1.30322i −0.758553 0.651611i \(-0.774093\pi\)
0.758553 0.651611i \(-0.225907\pi\)
\(464\) 0 0
\(465\) 205200. 0.949011
\(466\) 0 0
\(467\) 254702. 1.16788 0.583939 0.811797i \(-0.301511\pi\)
0.583939 + 0.811797i \(0.301511\pi\)
\(468\) 0 0
\(469\) 432334.i 1.96550i
\(470\) 0 0
\(471\) − 6276.81i − 0.0282942i
\(472\) 0 0
\(473\) 182832. 0.817203
\(474\) 0 0
\(475\) 1988.39 0.00881283
\(476\) 0 0
\(477\) 66046.0i 0.290275i
\(478\) 0 0
\(479\) − 52620.6i − 0.229343i −0.993403 0.114671i \(-0.963418\pi\)
0.993403 0.114671i \(-0.0365815\pi\)
\(480\) 0 0
\(481\) −14592.0 −0.0630703
\(482\) 0 0
\(483\) −199033. −0.853162
\(484\) 0 0
\(485\) − 163741.i − 0.696103i
\(486\) 0 0
\(487\) − 147557.i − 0.622162i −0.950383 0.311081i \(-0.899309\pi\)
0.950383 0.311081i \(-0.100691\pi\)
\(488\) 0 0
\(489\) −138996. −0.581279
\(490\) 0 0
\(491\) −78240.2 −0.324539 −0.162270 0.986746i \(-0.551881\pi\)
−0.162270 + 0.986746i \(0.551881\pi\)
\(492\) 0 0
\(493\) − 438917.i − 1.80588i
\(494\) 0 0
\(495\) 73438.7i 0.299719i
\(496\) 0 0
\(497\) −224352. −0.908275
\(498\) 0 0
\(499\) 298044. 1.19696 0.598480 0.801138i \(-0.295771\pi\)
0.598480 + 0.801138i \(0.295771\pi\)
\(500\) 0 0
\(501\) − 233743.i − 0.931243i
\(502\) 0 0
\(503\) 132964.i 0.525530i 0.964860 + 0.262765i \(0.0846344\pi\)
−0.964860 + 0.262765i \(0.915366\pi\)
\(504\) 0 0
\(505\) −331056. −1.29813
\(506\) 0 0
\(507\) 129452. 0.503607
\(508\) 0 0
\(509\) 12411.9i 0.0479075i 0.999713 + 0.0239538i \(0.00762545\pi\)
−0.999713 + 0.0239538i \(0.992375\pi\)
\(510\) 0 0
\(511\) − 456847.i − 1.74956i
\(512\) 0 0
\(513\) −972.000 −0.00369344
\(514\) 0 0
\(515\) −162702. −0.613449
\(516\) 0 0
\(517\) − 197866.i − 0.740270i
\(518\) 0 0
\(519\) − 21811.9i − 0.0809766i
\(520\) 0 0
\(521\) 170018. 0.626353 0.313177 0.949695i \(-0.398607\pi\)
0.313177 + 0.949695i \(0.398607\pi\)
\(522\) 0 0
\(523\) −401621. −1.46829 −0.734147 0.678991i \(-0.762418\pi\)
−0.734147 + 0.678991i \(0.762418\pi\)
\(524\) 0 0
\(525\) 78004.9i 0.283011i
\(526\) 0 0
\(527\) − 441992.i − 1.59145i
\(528\) 0 0
\(529\) −256415. −0.916288
\(530\) 0 0
\(531\) −32361.6 −0.114773
\(532\) 0 0
\(533\) − 34910.4i − 0.122886i
\(534\) 0 0
\(535\) 194372.i 0.679088i
\(536\) 0 0
\(537\) 64908.0 0.225087
\(538\) 0 0
\(539\) 30172.3 0.103856
\(540\) 0 0
\(541\) − 138857.i − 0.474430i −0.971457 0.237215i \(-0.923765\pi\)
0.971457 0.237215i \(-0.0762345\pi\)
\(542\) 0 0
\(543\) 196778.i 0.667386i
\(544\) 0 0
\(545\) 1824.00 0.00614090
\(546\) 0 0
\(547\) 99094.1 0.331187 0.165593 0.986194i \(-0.447046\pi\)
0.165593 + 0.986194i \(0.447046\pi\)
\(548\) 0 0
\(549\) − 172861.i − 0.573525i
\(550\) 0 0
\(551\) − 8996.77i − 0.0296335i
\(552\) 0 0
\(553\) −588240. −1.92355
\(554\) 0 0
\(555\) −37911.1 −0.123078
\(556\) 0 0
\(557\) 199104.i 0.641756i 0.947120 + 0.320878i \(0.103978\pi\)
−0.947120 + 0.320878i \(0.896022\pi\)
\(558\) 0 0
\(559\) 122607.i 0.392367i
\(560\) 0 0
\(561\) 158184. 0.502617
\(562\) 0 0
\(563\) 572457. 1.80603 0.903017 0.429605i \(-0.141347\pi\)
0.903017 + 0.429605i \(0.141347\pi\)
\(564\) 0 0
\(565\) 95611.1i 0.299510i
\(566\) 0 0
\(567\) − 38131.6i − 0.118609i
\(568\) 0 0
\(569\) 563330. 1.73996 0.869978 0.493090i \(-0.164133\pi\)
0.869978 + 0.493090i \(0.164133\pi\)
\(570\) 0 0
\(571\) 113117. 0.346940 0.173470 0.984839i \(-0.444502\pi\)
0.173470 + 0.984839i \(0.444502\pi\)
\(572\) 0 0
\(573\) − 170687.i − 0.519865i
\(574\) 0 0
\(575\) 210169.i 0.635671i
\(576\) 0 0
\(577\) 155858. 0.468142 0.234071 0.972220i \(-0.424795\pi\)
0.234071 + 0.972220i \(0.424795\pi\)
\(578\) 0 0
\(579\) 232944. 0.694854
\(580\) 0 0
\(581\) 690357.i 2.04513i
\(582\) 0 0
\(583\) − 220316.i − 0.648200i
\(584\) 0 0
\(585\) −49248.0 −0.143905
\(586\) 0 0
\(587\) 270914. 0.786239 0.393119 0.919487i \(-0.371396\pi\)
0.393119 + 0.919487i \(0.371396\pi\)
\(588\) 0 0
\(589\) − 9059.80i − 0.0261149i
\(590\) 0 0
\(591\) 219218.i 0.627626i
\(592\) 0 0
\(593\) −418078. −1.18891 −0.594454 0.804130i \(-0.702632\pi\)
−0.594454 + 0.804130i \(0.702632\pi\)
\(594\) 0 0
\(595\) 533915. 1.50813
\(596\) 0 0
\(597\) − 97574.1i − 0.273770i
\(598\) 0 0
\(599\) 539701.i 1.50418i 0.659060 + 0.752090i \(0.270954\pi\)
−0.659060 + 0.752090i \(0.729046\pi\)
\(600\) 0 0
\(601\) −439490. −1.21675 −0.608373 0.793651i \(-0.708178\pi\)
−0.608373 + 0.793651i \(0.708178\pi\)
\(602\) 0 0
\(603\) −223164. −0.613748
\(604\) 0 0
\(605\) 197171.i 0.538683i
\(606\) 0 0
\(607\) 256879.i 0.697189i 0.937274 + 0.348595i \(0.113341\pi\)
−0.937274 + 0.348595i \(0.886659\pi\)
\(608\) 0 0
\(609\) 352944. 0.951637
\(610\) 0 0
\(611\) 132689. 0.355429
\(612\) 0 0
\(613\) 222871.i 0.593107i 0.955016 + 0.296553i \(0.0958373\pi\)
−0.955016 + 0.296553i \(0.904163\pi\)
\(614\) 0 0
\(615\) − 90700.0i − 0.239804i
\(616\) 0 0
\(617\) 588718. 1.54645 0.773227 0.634129i \(-0.218641\pi\)
0.773227 + 0.634129i \(0.218641\pi\)
\(618\) 0 0
\(619\) 417570. 1.08980 0.544901 0.838500i \(-0.316567\pi\)
0.544901 + 0.838500i \(0.316567\pi\)
\(620\) 0 0
\(621\) − 102738.i − 0.266409i
\(622\) 0 0
\(623\) − 47599.2i − 0.122638i
\(624\) 0 0
\(625\) −487631. −1.24834
\(626\) 0 0
\(627\) 3242.40 0.00824767
\(628\) 0 0
\(629\) 81659.0i 0.206397i
\(630\) 0 0
\(631\) − 204467.i − 0.513529i −0.966474 0.256765i \(-0.917344\pi\)
0.966474 0.256765i \(-0.0826565\pi\)
\(632\) 0 0
\(633\) 331020. 0.826127
\(634\) 0 0
\(635\) 390169. 0.967620
\(636\) 0 0
\(637\) 20233.6i 0.0498647i
\(638\) 0 0
\(639\) − 115807.i − 0.283618i
\(640\) 0 0
\(641\) −270578. −0.658531 −0.329266 0.944237i \(-0.606801\pi\)
−0.329266 + 0.944237i \(0.606801\pi\)
\(642\) 0 0
\(643\) −584969. −1.41485 −0.707426 0.706788i \(-0.750144\pi\)
−0.707426 + 0.706788i \(0.750144\pi\)
\(644\) 0 0
\(645\) 318543.i 0.765681i
\(646\) 0 0
\(647\) − 96976.8i − 0.231664i −0.993269 0.115832i \(-0.963047\pi\)
0.993269 0.115832i \(-0.0369535\pi\)
\(648\) 0 0
\(649\) 107952. 0.256296
\(650\) 0 0
\(651\) 355417. 0.838641
\(652\) 0 0
\(653\) − 532082.i − 1.24782i −0.781496 0.623911i \(-0.785543\pi\)
0.781496 0.623911i \(-0.214457\pi\)
\(654\) 0 0
\(655\) 684800.i 1.59618i
\(656\) 0 0
\(657\) 235818. 0.546319
\(658\) 0 0
\(659\) 253593. 0.583938 0.291969 0.956428i \(-0.405690\pi\)
0.291969 + 0.956428i \(0.405690\pi\)
\(660\) 0 0
\(661\) − 339441.i − 0.776892i −0.921471 0.388446i \(-0.873012\pi\)
0.921471 0.388446i \(-0.126988\pi\)
\(662\) 0 0
\(663\) 106078.i 0.241323i
\(664\) 0 0
\(665\) 10944.0 0.0247476
\(666\) 0 0
\(667\) 950937. 2.13747
\(668\) 0 0
\(669\) 100836.i 0.225300i
\(670\) 0 0
\(671\) 576630.i 1.28071i
\(672\) 0 0
\(673\) −191570. −0.422958 −0.211479 0.977383i \(-0.567828\pi\)
−0.211479 + 0.977383i \(0.567828\pi\)
\(674\) 0 0
\(675\) −40265.0 −0.0883731
\(676\) 0 0
\(677\) − 498923.i − 1.08857i −0.838900 0.544285i \(-0.816801\pi\)
0.838900 0.544285i \(-0.183199\pi\)
\(678\) 0 0
\(679\) − 283607.i − 0.615146i
\(680\) 0 0
\(681\) −62820.0 −0.135458
\(682\) 0 0
\(683\) −96059.5 −0.205920 −0.102960 0.994685i \(-0.532831\pi\)
−0.102960 + 0.994685i \(0.532831\pi\)
\(684\) 0 0
\(685\) 469600.i 1.00080i
\(686\) 0 0
\(687\) − 472958.i − 1.00210i
\(688\) 0 0
\(689\) 147744. 0.311223
\(690\) 0 0
\(691\) 517474. 1.08376 0.541880 0.840456i \(-0.317713\pi\)
0.541880 + 0.840456i \(0.317713\pi\)
\(692\) 0 0
\(693\) 127200.i 0.264862i
\(694\) 0 0
\(695\) − 951356.i − 1.96958i
\(696\) 0 0
\(697\) −195364. −0.402142
\(698\) 0 0
\(699\) 234689. 0.480329
\(700\) 0 0
\(701\) − 214204.i − 0.435904i −0.975959 0.217952i \(-0.930062\pi\)
0.975959 0.217952i \(-0.0699377\pi\)
\(702\) 0 0
\(703\) 1673.82i 0.00338686i
\(704\) 0 0
\(705\) 344736. 0.693599
\(706\) 0 0
\(707\) −573406. −1.14716
\(708\) 0 0
\(709\) − 413550.i − 0.822688i −0.911480 0.411344i \(-0.865059\pi\)
0.911480 0.411344i \(-0.134941\pi\)
\(710\) 0 0
\(711\) − 303641.i − 0.600649i
\(712\) 0 0
\(713\) 957600. 1.88367
\(714\) 0 0
\(715\) 164282. 0.321349
\(716\) 0 0
\(717\) − 63056.2i − 0.122656i
\(718\) 0 0
\(719\) − 734073.i − 1.41998i −0.704213 0.709989i \(-0.748700\pi\)
0.704213 0.709989i \(-0.251300\pi\)
\(720\) 0 0
\(721\) −281808. −0.542104
\(722\) 0 0
\(723\) 445944. 0.853108
\(724\) 0 0
\(725\) − 372690.i − 0.709042i
\(726\) 0 0
\(727\) 1.03479e6i 1.95786i 0.204199 + 0.978929i \(0.434541\pi\)
−0.204199 + 0.978929i \(0.565459\pi\)
\(728\) 0 0
\(729\) 19683.0 0.0370370
\(730\) 0 0
\(731\) 686128. 1.28402
\(732\) 0 0
\(733\) 200222.i 0.372652i 0.982488 + 0.186326i \(0.0596580\pi\)
−0.982488 + 0.186326i \(0.940342\pi\)
\(734\) 0 0
\(735\) 52568.3i 0.0973082i
\(736\) 0 0
\(737\) 744432. 1.37053
\(738\) 0 0
\(739\) −526855. −0.964722 −0.482361 0.875972i \(-0.660221\pi\)
−0.482361 + 0.875972i \(0.660221\pi\)
\(740\) 0 0
\(741\) 2174.35i 0.00395998i
\(742\) 0 0
\(743\) − 380375.i − 0.689024i −0.938782 0.344512i \(-0.888044\pi\)
0.938782 0.344512i \(-0.111956\pi\)
\(744\) 0 0
\(745\) −683088. −1.23073
\(746\) 0 0
\(747\) −356352. −0.638614
\(748\) 0 0
\(749\) 336662.i 0.600110i
\(750\) 0 0
\(751\) − 591851.i − 1.04938i −0.851293 0.524690i \(-0.824181\pi\)
0.851293 0.524690i \(-0.175819\pi\)
\(752\) 0 0
\(753\) 302364. 0.533261
\(754\) 0 0
\(755\) 232206. 0.407360
\(756\) 0 0
\(757\) − 373445.i − 0.651681i −0.945425 0.325840i \(-0.894353\pi\)
0.945425 0.325840i \(-0.105647\pi\)
\(758\) 0 0
\(759\) 342714.i 0.594906i
\(760\) 0 0
\(761\) 37154.0 0.0641558 0.0320779 0.999485i \(-0.489788\pi\)
0.0320779 + 0.999485i \(0.489788\pi\)
\(762\) 0 0
\(763\) 3159.26 0.00542671
\(764\) 0 0
\(765\) 275599.i 0.470929i
\(766\) 0 0
\(767\) 72392.6i 0.123056i
\(768\) 0 0
\(769\) −455330. −0.769970 −0.384985 0.922923i \(-0.625793\pi\)
−0.384985 + 0.922923i \(0.625793\pi\)
\(770\) 0 0
\(771\) −632507. −1.06404
\(772\) 0 0
\(773\) 57046.5i 0.0954708i 0.998860 + 0.0477354i \(0.0152004\pi\)
−0.998860 + 0.0477354i \(0.984800\pi\)
\(774\) 0 0
\(775\) − 375301.i − 0.624851i
\(776\) 0 0
\(777\) −65664.0 −0.108764
\(778\) 0 0
\(779\) −4004.50 −0.00659893
\(780\) 0 0
\(781\) 386310.i 0.633335i
\(782\) 0 0
\(783\) 182185.i 0.297158i
\(784\) 0 0
\(785\) −36480.0 −0.0591992
\(786\) 0 0
\(787\) 99953.2 0.161379 0.0806895 0.996739i \(-0.474288\pi\)
0.0806895 + 0.996739i \(0.474288\pi\)
\(788\) 0 0
\(789\) 508798.i 0.817319i
\(790\) 0 0
\(791\) 165603.i 0.264677i
\(792\) 0 0
\(793\) −386688. −0.614914
\(794\) 0 0
\(795\) 383850. 0.607334
\(796\) 0 0
\(797\) − 517345.i − 0.814448i −0.913328 0.407224i \(-0.866497\pi\)
0.913328 0.407224i \(-0.133503\pi\)
\(798\) 0 0
\(799\) − 742547.i − 1.16314i
\(800\) 0 0
\(801\) 24570.0 0.0382948
\(802\) 0 0
\(803\) −786642. −1.21996
\(804\) 0 0
\(805\) 1.15676e6i 1.78505i
\(806\) 0 0
\(807\) 75164.9i 0.115416i
\(808\) 0 0
\(809\) −570046. −0.870989 −0.435495 0.900191i \(-0.643427\pi\)
−0.435495 + 0.900191i \(0.643427\pi\)
\(810\) 0 0
\(811\) 174418. 0.265185 0.132592 0.991171i \(-0.457670\pi\)
0.132592 + 0.991171i \(0.457670\pi\)
\(812\) 0 0
\(813\) − 288373.i − 0.436289i
\(814\) 0 0
\(815\) 807826.i 1.21619i
\(816\) 0 0
\(817\) 14064.0 0.0210700
\(818\) 0 0
\(819\) −85300.0 −0.127169
\(820\) 0 0
\(821\) − 438827.i − 0.651038i −0.945535 0.325519i \(-0.894461\pi\)
0.945535 0.325519i \(-0.105539\pi\)
\(822\) 0 0
\(823\) 608380.i 0.898205i 0.893480 + 0.449102i \(0.148256\pi\)
−0.893480 + 0.449102i \(0.851744\pi\)
\(824\) 0 0
\(825\) 134316. 0.197342
\(826\) 0 0
\(827\) 727898. 1.06429 0.532144 0.846654i \(-0.321386\pi\)
0.532144 + 0.846654i \(0.321386\pi\)
\(828\) 0 0
\(829\) 619509.i 0.901444i 0.892664 + 0.450722i \(0.148833\pi\)
−0.892664 + 0.450722i \(0.851167\pi\)
\(830\) 0 0
\(831\) − 530077.i − 0.767603i
\(832\) 0 0
\(833\) 113230. 0.163182
\(834\) 0 0
\(835\) −1.35848e6 −1.94841
\(836\) 0 0
\(837\) 183461.i 0.261874i
\(838\) 0 0
\(839\) 1.10064e6i 1.56358i 0.623539 + 0.781792i \(0.285694\pi\)
−0.623539 + 0.781792i \(0.714306\pi\)
\(840\) 0 0
\(841\) −979007. −1.38418
\(842\) 0 0
\(843\) −208272. −0.293073
\(844\) 0 0
\(845\) − 752356.i − 1.05368i
\(846\) 0 0
\(847\) 341511.i 0.476034i
\(848\) 0 0
\(849\) 251100. 0.348362
\(850\) 0 0
\(851\) −176919. −0.244295
\(852\) 0 0
\(853\) − 634911.i − 0.872599i −0.899802 0.436299i \(-0.856289\pi\)
0.899802 0.436299i \(-0.143711\pi\)
\(854\) 0 0
\(855\) 5649.13i 0.00772769i
\(856\) 0 0
\(857\) −405310. −0.551856 −0.275928 0.961178i \(-0.588985\pi\)
−0.275928 + 0.961178i \(0.588985\pi\)
\(858\) 0 0
\(859\) 100230. 0.135835 0.0679177 0.997691i \(-0.478364\pi\)
0.0679177 + 0.997691i \(0.478364\pi\)
\(860\) 0 0
\(861\) − 157097.i − 0.211915i
\(862\) 0 0
\(863\) 317816.i 0.426731i 0.976972 + 0.213366i \(0.0684425\pi\)
−0.976972 + 0.213366i \(0.931557\pi\)
\(864\) 0 0
\(865\) −126768. −0.169425
\(866\) 0 0
\(867\) 159641. 0.212377
\(868\) 0 0
\(869\) 1.01289e6i 1.34128i
\(870\) 0 0
\(871\) 499216.i 0.658040i
\(872\) 0 0
\(873\) 146394. 0.192086
\(874\) 0 0
\(875\) −533915. −0.697358
\(876\) 0 0
\(877\) − 323133.i − 0.420128i −0.977688 0.210064i \(-0.932633\pi\)
0.977688 0.210064i \(-0.0673673\pi\)
\(878\) 0 0
\(879\) 662989.i 0.858081i
\(880\) 0 0
\(881\) −1.10419e6 −1.42263 −0.711315 0.702873i \(-0.751900\pi\)
−0.711315 + 0.702873i \(0.751900\pi\)
\(882\) 0 0
\(883\) −813877. −1.04385 −0.521924 0.852992i \(-0.674786\pi\)
−0.521924 + 0.852992i \(0.674786\pi\)
\(884\) 0 0
\(885\) 188081.i 0.240137i
\(886\) 0 0
\(887\) 221781.i 0.281888i 0.990018 + 0.140944i \(0.0450138\pi\)
−0.990018 + 0.140944i \(0.954986\pi\)
\(888\) 0 0
\(889\) 675792. 0.855085
\(890\) 0 0
\(891\) −65658.6 −0.0827058
\(892\) 0 0
\(893\) − 15220.5i − 0.0190864i
\(894\) 0 0
\(895\) − 377237.i − 0.470942i
\(896\) 0 0
\(897\) −229824. −0.285634
\(898\) 0 0
\(899\) −1.69810e6 −2.10109
\(900\) 0 0
\(901\) − 826797.i − 1.01847i
\(902\) 0 0
\(903\) 551732.i 0.676632i
\(904\) 0 0
\(905\) 1.14365e6 1.39635
\(906\) 0 0
\(907\) −577071. −0.701479 −0.350739 0.936473i \(-0.614070\pi\)
−0.350739 + 0.936473i \(0.614070\pi\)
\(908\) 0 0
\(909\) − 295984.i − 0.358212i
\(910\) 0 0
\(911\) − 922692.i − 1.11178i −0.831255 0.555891i \(-0.812377\pi\)
0.831255 0.555891i \(-0.187623\pi\)
\(912\) 0 0
\(913\) 1.18872e6 1.42606
\(914\) 0 0
\(915\) −1.00464e6 −1.19997
\(916\) 0 0
\(917\) 1.18611e6i 1.41054i
\(918\) 0 0
\(919\) − 1.47845e6i − 1.75056i −0.483620 0.875278i \(-0.660678\pi\)
0.483620 0.875278i \(-0.339322\pi\)
\(920\) 0 0
\(921\) 206748. 0.243737
\(922\) 0 0
\(923\) −259059. −0.304086
\(924\) 0 0
\(925\) 69337.7i 0.0810375i
\(926\) 0 0
\(927\) − 145465.i − 0.169278i
\(928\) 0 0
\(929\) 241006. 0.279252 0.139626 0.990204i \(-0.455410\pi\)
0.139626 + 0.990204i \(0.455410\pi\)
\(930\) 0 0
\(931\) 2320.95 0.00267773
\(932\) 0 0
\(933\) − 907792.i − 1.04285i
\(934\) 0 0
\(935\) − 919344.i − 1.05161i
\(936\) 0 0
\(937\) −931058. −1.06047 −0.530234 0.847851i \(-0.677896\pi\)
−0.530234 + 0.847851i \(0.677896\pi\)
\(938\) 0 0
\(939\) 139932. 0.158704
\(940\) 0 0
\(941\) 245490.i 0.277240i 0.990346 + 0.138620i \(0.0442666\pi\)
−0.990346 + 0.138620i \(0.955733\pi\)
\(942\) 0 0
\(943\) − 423267.i − 0.475982i
\(944\) 0 0
\(945\) −221616. −0.248163
\(946\) 0 0
\(947\) −207853. −0.231770 −0.115885 0.993263i \(-0.536970\pi\)
−0.115885 + 0.993263i \(0.536970\pi\)
\(948\) 0 0
\(949\) − 527522.i − 0.585744i
\(950\) 0 0
\(951\) 56334.4i 0.0622892i
\(952\) 0 0
\(953\) −1.23446e6 −1.35923 −0.679613 0.733570i \(-0.737852\pi\)
−0.679613 + 0.733570i \(0.737852\pi\)
\(954\) 0 0
\(955\) −992008. −1.08770
\(956\) 0 0
\(957\) − 607731.i − 0.663572i
\(958\) 0 0
\(959\) 813371.i 0.884405i
\(960\) 0 0
\(961\) −786479. −0.851609
\(962\) 0 0
\(963\) −173780. −0.187390
\(964\) 0 0
\(965\) − 1.35384e6i − 1.45382i
\(966\) 0 0
\(967\) 181034.i 0.193601i 0.995304 + 0.0968003i \(0.0308608\pi\)
−0.995304 + 0.0968003i \(0.969139\pi\)
\(968\) 0 0
\(969\) 12168.0 0.0129590
\(970\) 0 0
\(971\) 97708.5 0.103632 0.0518160 0.998657i \(-0.483499\pi\)
0.0518160 + 0.998657i \(0.483499\pi\)
\(972\) 0 0
\(973\) − 1.64780e6i − 1.74052i
\(974\) 0 0
\(975\) 90072.3i 0.0947506i
\(976\) 0 0
\(977\) 1.67313e6 1.75284 0.876419 0.481550i \(-0.159926\pi\)
0.876419 + 0.481550i \(0.159926\pi\)
\(978\) 0 0
\(979\) −81960.6 −0.0855145
\(980\) 0 0
\(981\) 1630.76i 0.00169455i
\(982\) 0 0
\(983\) − 1.03358e6i − 1.06964i −0.844966 0.534820i \(-0.820379\pi\)
0.844966 0.534820i \(-0.179621\pi\)
\(984\) 0 0
\(985\) 1.27406e6 1.31316
\(986\) 0 0
\(987\) 597100. 0.612933
\(988\) 0 0
\(989\) 1.48653e6i 1.51978i
\(990\) 0 0
\(991\) 907889.i 0.924454i 0.886762 + 0.462227i \(0.152950\pi\)
−0.886762 + 0.462227i \(0.847050\pi\)
\(992\) 0 0
\(993\) −663012. −0.672393
\(994\) 0 0
\(995\) −567087. −0.572801
\(996\) 0 0
\(997\) 1.47844e6i 1.48735i 0.668542 + 0.743675i \(0.266919\pi\)
−0.668542 + 0.743675i \(0.733081\pi\)
\(998\) 0 0
\(999\) − 33894.8i − 0.0339627i
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 384.5.b.a.319.3 yes 4
3.2 odd 2 1152.5.b.j.703.3 4
4.3 odd 2 inner 384.5.b.a.319.1 4
8.3 odd 2 inner 384.5.b.a.319.4 yes 4
8.5 even 2 inner 384.5.b.a.319.2 yes 4
12.11 even 2 1152.5.b.j.703.4 4
16.3 odd 4 768.5.g.d.511.1 4
16.5 even 4 768.5.g.d.511.2 4
16.11 odd 4 768.5.g.d.511.4 4
16.13 even 4 768.5.g.d.511.3 4
24.5 odd 2 1152.5.b.j.703.1 4
24.11 even 2 1152.5.b.j.703.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.5.b.a.319.1 4 4.3 odd 2 inner
384.5.b.a.319.2 yes 4 8.5 even 2 inner
384.5.b.a.319.3 yes 4 1.1 even 1 trivial
384.5.b.a.319.4 yes 4 8.3 odd 2 inner
768.5.g.d.511.1 4 16.3 odd 4
768.5.g.d.511.2 4 16.5 even 4
768.5.g.d.511.3 4 16.13 even 4
768.5.g.d.511.4 4 16.11 odd 4
1152.5.b.j.703.1 4 24.5 odd 2
1152.5.b.j.703.2 4 24.11 even 2
1152.5.b.j.703.3 4 3.2 odd 2
1152.5.b.j.703.4 4 12.11 even 2