Properties

Label 2736.3.o.j.721.2
Level $2736$
Weight $3$
Character 2736.721
Analytic conductor $74.551$
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] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, 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("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-7}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 15x^{2} + 16x + 134 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.2
Root \(3.66228 + 1.32288i\) of defining polynomial
Character \(\chi\) \(=\) 2736.721
Dual form 2736.3.o.j.721.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.32456 q^{5} +2.00000 q^{7} +O(q^{10})\) \(q-6.32456 q^{5} +2.00000 q^{7} +12.6491 q^{11} +16.7332i q^{13} +(9.00000 + 16.7332i) q^{19} -6.32456 q^{23} +15.0000 q^{25} -21.1660i q^{29} +16.7332i q^{31} -12.6491 q^{35} -50.1996i q^{37} +21.1660i q^{41} -34.0000 q^{43} +82.2192 q^{47} -45.0000 q^{49} -63.4980i q^{53} -80.0000 q^{55} +21.1660i q^{59} +86.0000 q^{61} -105.830i q^{65} +66.9328i q^{67} -126.996i q^{71} -102.000 q^{73} +25.2982 q^{77} +117.132i q^{79} -25.2982 q^{83} +126.996i q^{89} +33.4664i q^{91} +(-56.9210 - 105.830i) q^{95} -33.4664i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 36 q^{19} + 60 q^{25} - 136 q^{43} - 180 q^{49} - 320 q^{55} + 344 q^{61} - 408 q^{73}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) −6.32456 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 2.00000 0.285714 0.142857 0.989743i \(-0.454371\pi\)
0.142857 + 0.989743i \(0.454371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 12.6491 1.14992 0.574960 0.818182i \(-0.305018\pi\)
0.574960 + 0.818182i \(0.305018\pi\)
\(12\) 0 0
\(13\) 16.7332i 1.28717i 0.765375 + 0.643585i \(0.222554\pi\)
−0.765375 + 0.643585i \(0.777446\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 9.00000 + 16.7332i 0.473684 + 0.880695i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.32456 −0.274981 −0.137490 0.990503i \(-0.543904\pi\)
−0.137490 + 0.990503i \(0.543904\pi\)
\(24\) 0 0
\(25\) 15.0000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 21.1660i 0.729862i −0.931034 0.364931i \(-0.881093\pi\)
0.931034 0.364931i \(-0.118907\pi\)
\(30\) 0 0
\(31\) 16.7332i 0.539781i 0.962891 + 0.269890i \(0.0869875\pi\)
−0.962891 + 0.269890i \(0.913013\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −12.6491 −0.361403
\(36\) 0 0
\(37\) 50.1996i 1.35675i −0.734718 0.678373i \(-0.762685\pi\)
0.734718 0.678373i \(-0.237315\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 21.1660i 0.516244i 0.966112 + 0.258122i \(0.0831037\pi\)
−0.966112 + 0.258122i \(0.916896\pi\)
\(42\) 0 0
\(43\) −34.0000 −0.790698 −0.395349 0.918531i \(-0.629376\pi\)
−0.395349 + 0.918531i \(0.629376\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 82.2192 1.74935 0.874673 0.484714i \(-0.161076\pi\)
0.874673 + 0.484714i \(0.161076\pi\)
\(48\) 0 0
\(49\) −45.0000 −0.918367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 63.4980i 1.19808i −0.800721 0.599038i \(-0.795550\pi\)
0.800721 0.599038i \(-0.204450\pi\)
\(54\) 0 0
\(55\) −80.0000 −1.45455
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 21.1660i 0.358746i 0.983781 + 0.179373i \(0.0574069\pi\)
−0.983781 + 0.179373i \(0.942593\pi\)
\(60\) 0 0
\(61\) 86.0000 1.40984 0.704918 0.709289i \(-0.250984\pi\)
0.704918 + 0.709289i \(0.250984\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 105.830i 1.62815i
\(66\) 0 0
\(67\) 66.9328i 0.998997i 0.866315 + 0.499499i \(0.166482\pi\)
−0.866315 + 0.499499i \(0.833518\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 126.996i 1.78868i −0.447391 0.894338i \(-0.647647\pi\)
0.447391 0.894338i \(-0.352353\pi\)
\(72\) 0 0
\(73\) −102.000 −1.39726 −0.698630 0.715483i \(-0.746207\pi\)
−0.698630 + 0.715483i \(0.746207\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 25.2982 0.328548
\(78\) 0 0
\(79\) 117.132i 1.48269i 0.671125 + 0.741344i \(0.265811\pi\)
−0.671125 + 0.741344i \(0.734189\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −25.2982 −0.304798 −0.152399 0.988319i \(-0.548700\pi\)
−0.152399 + 0.988319i \(0.548700\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 126.996i 1.42692i 0.700695 + 0.713461i \(0.252873\pi\)
−0.700695 + 0.713461i \(0.747127\pi\)
\(90\) 0 0
\(91\) 33.4664i 0.367763i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −56.9210 105.830i −0.599168 1.11400i
\(96\) 0 0
\(97\) 33.4664i 0.345014i −0.985008 0.172507i \(-0.944813\pi\)
0.985008 0.172507i \(-0.0551868\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 145.465 1.44025 0.720123 0.693847i \(-0.244086\pi\)
0.720123 + 0.693847i \(0.244086\pi\)
\(102\) 0 0
\(103\) 117.132i 1.13721i −0.822611 0.568604i \(-0.807484\pi\)
0.822611 0.568604i \(-0.192516\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 63.4980i 0.593440i 0.954965 + 0.296720i \(0.0958927\pi\)
−0.954965 + 0.296720i \(0.904107\pi\)
\(108\) 0 0
\(109\) 184.065i 1.68867i 0.535814 + 0.844336i \(0.320005\pi\)
−0.535814 + 0.844336i \(0.679995\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 105.830i 0.936549i 0.883583 + 0.468275i \(0.155124\pi\)
−0.883583 + 0.468275i \(0.844876\pi\)
\(114\) 0 0
\(115\) 40.0000 0.347826
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 39.0000 0.322314
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 63.2456 0.505964
\(126\) 0 0
\(127\) 50.1996i 0.395272i 0.980275 + 0.197636i \(0.0633265\pi\)
−0.980275 + 0.197636i \(0.936674\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −88.5438 −0.675907 −0.337953 0.941163i \(-0.609735\pi\)
−0.337953 + 0.941163i \(0.609735\pi\)
\(132\) 0 0
\(133\) 18.0000 + 33.4664i 0.135338 + 0.251627i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −164.438 −1.20028 −0.600140 0.799895i \(-0.704889\pi\)
−0.600140 + 0.799895i \(0.704889\pi\)
\(138\) 0 0
\(139\) −134.000 −0.964029 −0.482014 0.876163i \(-0.660095\pi\)
−0.482014 + 0.876163i \(0.660095\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 211.660i 1.48014i
\(144\) 0 0
\(145\) 133.866i 0.923211i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −82.2192 −0.551807 −0.275903 0.961185i \(-0.588977\pi\)
−0.275903 + 0.961185i \(0.588977\pi\)
\(150\) 0 0
\(151\) 184.065i 1.21897i 0.792796 + 0.609487i \(0.208625\pi\)
−0.792796 + 0.609487i \(0.791375\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 105.830i 0.682775i
\(156\) 0 0
\(157\) −122.000 −0.777070 −0.388535 0.921434i \(-0.627019\pi\)
−0.388535 + 0.921434i \(0.627019\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.6491 −0.0785659
\(162\) 0 0
\(163\) 138.000 0.846626 0.423313 0.905984i \(-0.360867\pi\)
0.423313 + 0.905984i \(0.360867\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 169.328i 1.01394i 0.861964 + 0.506970i \(0.169235\pi\)
−0.861964 + 0.506970i \(0.830765\pi\)
\(168\) 0 0
\(169\) −111.000 −0.656805
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 148.162i 0.856428i −0.903677 0.428214i \(-0.859143\pi\)
0.903677 0.428214i \(-0.140857\pi\)
\(174\) 0 0
\(175\) 30.0000 0.171429
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 296.324i 1.65544i 0.561140 + 0.827721i \(0.310363\pi\)
−0.561140 + 0.827721i \(0.689637\pi\)
\(180\) 0 0
\(181\) 50.1996i 0.277346i 0.990338 + 0.138673i \(0.0442837\pi\)
−0.990338 + 0.138673i \(0.955716\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 317.490i 1.71616i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −259.307 −1.35763 −0.678814 0.734311i \(-0.737506\pi\)
−0.678814 + 0.734311i \(0.737506\pi\)
\(192\) 0 0
\(193\) 334.664i 1.73401i 0.498299 + 0.867005i \(0.333958\pi\)
−0.498299 + 0.867005i \(0.666042\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −246.658 −1.25207 −0.626035 0.779795i \(-0.715323\pi\)
−0.626035 + 0.779795i \(0.715323\pi\)
\(198\) 0 0
\(199\) −66.0000 −0.331658 −0.165829 0.986154i \(-0.553030\pi\)
−0.165829 + 0.986154i \(0.553030\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 42.3320i 0.208532i
\(204\) 0 0
\(205\) 133.866i 0.653003i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 113.842 + 211.660i 0.544699 + 1.01273i
\(210\) 0 0
\(211\) 66.9328i 0.317217i −0.987342 0.158609i \(-0.949299\pi\)
0.987342 0.158609i \(-0.0507008\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 215.035 1.00016
\(216\) 0 0
\(217\) 33.4664i 0.154223i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 284.464i 1.27563i 0.770192 + 0.637813i \(0.220161\pi\)
−0.770192 + 0.637813i \(0.779839\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 169.328i 0.745939i −0.927844 0.372969i \(-0.878340\pi\)
0.927844 0.372969i \(-0.121660\pi\)
\(228\) 0 0
\(229\) −298.000 −1.30131 −0.650655 0.759373i \(-0.725506\pi\)
−0.650655 + 0.759373i \(0.725506\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −265.631 −1.14005 −0.570024 0.821628i \(-0.693066\pi\)
−0.570024 + 0.821628i \(0.693066\pi\)
\(234\) 0 0
\(235\) −520.000 −2.21277
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −44.2719 −0.185238 −0.0926190 0.995702i \(-0.529524\pi\)
−0.0926190 + 0.995702i \(0.529524\pi\)
\(240\) 0 0
\(241\) 334.664i 1.38865i 0.719663 + 0.694324i \(0.244296\pi\)
−0.719663 + 0.694324i \(0.755704\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 284.605 1.16165
\(246\) 0 0
\(247\) −280.000 + 150.599i −1.13360 + 0.609712i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 151.789 0.604738 0.302369 0.953191i \(-0.402222\pi\)
0.302369 + 0.953191i \(0.402222\pi\)
\(252\) 0 0
\(253\) −80.0000 −0.316206
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 169.328i 0.658864i −0.944179 0.329432i \(-0.893143\pi\)
0.944179 0.329432i \(-0.106857\pi\)
\(258\) 0 0
\(259\) 100.399i 0.387642i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 309.903 1.17834 0.589170 0.808009i \(-0.299455\pi\)
0.589170 + 0.808009i \(0.299455\pi\)
\(264\) 0 0
\(265\) 401.597i 1.51546i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 317.490i 1.18026i −0.807308 0.590130i \(-0.799076\pi\)
0.807308 0.590130i \(-0.200924\pi\)
\(270\) 0 0
\(271\) 366.000 1.35055 0.675277 0.737564i \(-0.264024\pi\)
0.675277 + 0.737564i \(0.264024\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 189.737 0.689951
\(276\) 0 0
\(277\) −122.000 −0.440433 −0.220217 0.975451i \(-0.570676\pi\)
−0.220217 + 0.975451i \(0.570676\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 105.830i 0.376619i −0.982110 0.188310i \(-0.939699\pi\)
0.982110 0.188310i \(-0.0603009\pi\)
\(282\) 0 0
\(283\) 326.000 1.15194 0.575972 0.817470i \(-0.304624\pi\)
0.575972 + 0.817470i \(0.304624\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 42.3320i 0.147498i
\(288\) 0 0
\(289\) −289.000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 105.830i 0.361195i 0.983557 + 0.180597i \(0.0578031\pi\)
−0.983557 + 0.180597i \(0.942197\pi\)
\(294\) 0 0
\(295\) 133.866i 0.453782i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 105.830i 0.353947i
\(300\) 0 0
\(301\) −68.0000 −0.225914
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −543.912 −1.78332
\(306\) 0 0
\(307\) 200.798i 0.654066i 0.945013 + 0.327033i \(0.106049\pi\)
−0.945013 + 0.327033i \(0.893951\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −309.903 −0.996473 −0.498237 0.867041i \(-0.666019\pi\)
−0.498237 + 0.867041i \(0.666019\pi\)
\(312\) 0 0
\(313\) 86.0000 0.274760 0.137380 0.990518i \(-0.456132\pi\)
0.137380 + 0.990518i \(0.456132\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 486.818i 1.53570i −0.640627 0.767852i \(-0.721326\pi\)
0.640627 0.767852i \(-0.278674\pi\)
\(318\) 0 0
\(319\) 267.731i 0.839283i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 250.998i 0.772302i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 164.438 0.499813
\(330\) 0 0
\(331\) 267.731i 0.808856i 0.914570 + 0.404428i \(0.132529\pi\)
−0.914570 + 0.404428i \(0.867471\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 423.320i 1.26364i
\(336\) 0 0
\(337\) 167.332i 0.496534i 0.968692 + 0.248267i \(0.0798611\pi\)
−0.968692 + 0.248267i \(0.920139\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 211.660i 0.620704i
\(342\) 0 0
\(343\) −188.000 −0.548105
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.6491 0.0364528 0.0182264 0.999834i \(-0.494198\pi\)
0.0182264 + 0.999834i \(0.494198\pi\)
\(348\) 0 0
\(349\) 506.000 1.44986 0.724928 0.688824i \(-0.241873\pi\)
0.724928 + 0.688824i \(0.241873\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 619.806 1.75583 0.877913 0.478821i \(-0.158936\pi\)
0.877913 + 0.478821i \(0.158936\pi\)
\(354\) 0 0
\(355\) 803.194i 2.26252i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −94.8683 −0.264257 −0.132129 0.991233i \(-0.542181\pi\)
−0.132129 + 0.991233i \(0.542181\pi\)
\(360\) 0 0
\(361\) −199.000 + 301.198i −0.551247 + 0.834342i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 645.105 1.76741
\(366\) 0 0
\(367\) −386.000 −1.05177 −0.525886 0.850555i \(-0.676266\pi\)
−0.525886 + 0.850555i \(0.676266\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 126.996i 0.342307i
\(372\) 0 0
\(373\) 317.931i 0.852361i −0.904638 0.426181i \(-0.859859\pi\)
0.904638 0.426181i \(-0.140141\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 354.175 0.939456
\(378\) 0 0
\(379\) 301.198i 0.794717i −0.917664 0.397358i \(-0.869927\pi\)
0.917664 0.397358i \(-0.130073\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 592.648i 1.54738i 0.633562 + 0.773692i \(0.281592\pi\)
−0.633562 + 0.773692i \(0.718408\pi\)
\(384\) 0 0
\(385\) −160.000 −0.415584
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −221.359 −0.569047 −0.284524 0.958669i \(-0.591835\pi\)
−0.284524 + 0.958669i \(0.591835\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 740.810i 1.87547i
\(396\) 0 0
\(397\) 186.000 0.468514 0.234257 0.972175i \(-0.424734\pi\)
0.234257 + 0.972175i \(0.424734\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 105.830i 0.263915i −0.991255 0.131958i \(-0.957874\pi\)
0.991255 0.131958i \(-0.0421263\pi\)
\(402\) 0 0
\(403\) −280.000 −0.694789
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 634.980i 1.56015i
\(408\) 0 0
\(409\) 167.332i 0.409125i −0.978854 0.204562i \(-0.934423\pi\)
0.978854 0.204562i \(-0.0655771\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 42.3320i 0.102499i
\(414\) 0 0
\(415\) 160.000 0.385542
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −771.596 −1.84152 −0.920759 0.390133i \(-0.872429\pi\)
−0.920759 + 0.390133i \(0.872429\pi\)
\(420\) 0 0
\(421\) 284.464i 0.675687i −0.941202 0.337844i \(-0.890302\pi\)
0.941202 0.337844i \(-0.109698\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 172.000 0.402810
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 761.976i 1.76793i 0.467556 + 0.883963i \(0.345135\pi\)
−0.467556 + 0.883963i \(0.654865\pi\)
\(432\) 0 0
\(433\) 100.399i 0.231869i 0.993257 + 0.115934i \(0.0369862\pi\)
−0.993257 + 0.115934i \(0.963014\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −56.9210 105.830i −0.130254 0.242174i
\(438\) 0 0
\(439\) 485.263i 1.10538i 0.833386 + 0.552691i \(0.186399\pi\)
−0.833386 + 0.552691i \(0.813601\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 442.719 0.999365 0.499683 0.866209i \(-0.333450\pi\)
0.499683 + 0.866209i \(0.333450\pi\)
\(444\) 0 0
\(445\) 803.194i 1.80493i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 296.324i 0.659965i 0.943987 + 0.329982i \(0.107043\pi\)
−0.943987 + 0.329982i \(0.892957\pi\)
\(450\) 0 0
\(451\) 267.731i 0.593639i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 211.660i 0.465187i
\(456\) 0 0
\(457\) 82.0000 0.179431 0.0897155 0.995967i \(-0.471404\pi\)
0.0897155 + 0.995967i \(0.471404\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −423.745 −0.919187 −0.459593 0.888129i \(-0.652005\pi\)
−0.459593 + 0.888129i \(0.652005\pi\)
\(462\) 0 0
\(463\) −34.0000 −0.0734341 −0.0367171 0.999326i \(-0.511690\pi\)
−0.0367171 + 0.999326i \(0.511690\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 404.772 0.866748 0.433374 0.901214i \(-0.357323\pi\)
0.433374 + 0.901214i \(0.357323\pi\)
\(468\) 0 0
\(469\) 133.866i 0.285428i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −430.070 −0.909238
\(474\) 0 0
\(475\) 135.000 + 250.998i 0.284211 + 0.528417i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 31.6228 0.0660183 0.0330092 0.999455i \(-0.489491\pi\)
0.0330092 + 0.999455i \(0.489491\pi\)
\(480\) 0 0
\(481\) 840.000 1.74636
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 211.660i 0.436413i
\(486\) 0 0
\(487\) 552.196i 1.13387i 0.823762 + 0.566936i \(0.191871\pi\)
−0.823762 + 0.566936i \(0.808129\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 290.930 0.592525 0.296262 0.955107i \(-0.404260\pi\)
0.296262 + 0.955107i \(0.404260\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 253.992i 0.511051i
\(498\) 0 0
\(499\) −254.000 −0.509018 −0.254509 0.967070i \(-0.581914\pi\)
−0.254509 + 0.967070i \(0.581914\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −284.605 −0.565815 −0.282908 0.959147i \(-0.591299\pi\)
−0.282908 + 0.959147i \(0.591299\pi\)
\(504\) 0 0
\(505\) −920.000 −1.82178
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 867.806i 1.70492i 0.522789 + 0.852462i \(0.324892\pi\)
−0.522789 + 0.852462i \(0.675108\pi\)
\(510\) 0 0
\(511\) −204.000 −0.399217
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 740.810i 1.43847i
\(516\) 0 0
\(517\) 1040.00 2.01161
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 296.324i 0.568760i 0.958712 + 0.284380i \(0.0917878\pi\)
−0.958712 + 0.284380i \(0.908212\pi\)
\(522\) 0 0
\(523\) 501.996i 0.959839i −0.877312 0.479920i \(-0.840666\pi\)
0.877312 0.479920i \(-0.159334\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −489.000 −0.924386
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −354.175 −0.664494
\(534\) 0 0
\(535\) 401.597i 0.750648i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −569.210 −1.05605
\(540\) 0 0
\(541\) −198.000 −0.365989 −0.182994 0.983114i \(-0.558579\pi\)
−0.182994 + 0.983114i \(0.558579\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1164.13i 2.13602i
\(546\) 0 0
\(547\) 234.265i 0.428272i −0.976804 0.214136i \(-0.931306\pi\)
0.976804 0.214136i \(-0.0686936\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 354.175 190.494i 0.642786 0.345724i
\(552\) 0 0
\(553\) 234.265i 0.423625i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 917.061 1.64643 0.823214 0.567731i \(-0.192179\pi\)
0.823214 + 0.567731i \(0.192179\pi\)
\(558\) 0 0
\(559\) 568.929i 1.01776i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 275.158i 0.488736i −0.969683 0.244368i \(-0.921420\pi\)
0.969683 0.244368i \(-0.0785804\pi\)
\(564\) 0 0
\(565\) 669.328i 1.18465i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 84.6640i 0.148794i 0.997229 + 0.0743972i \(0.0237033\pi\)
−0.997229 + 0.0743972i \(0.976297\pi\)
\(570\) 0 0
\(571\) 46.0000 0.0805604 0.0402802 0.999188i \(-0.487175\pi\)
0.0402802 + 0.999188i \(0.487175\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −94.8683 −0.164988
\(576\) 0 0
\(577\) −66.0000 −0.114385 −0.0571924 0.998363i \(-0.518215\pi\)
−0.0571924 + 0.998363i \(0.518215\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −50.5964 −0.0870851
\(582\) 0 0
\(583\) 803.194i 1.37769i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −177.088 −0.301682 −0.150841 0.988558i \(-0.548198\pi\)
−0.150841 + 0.988558i \(0.548198\pi\)
\(588\) 0 0
\(589\) −280.000 + 150.599i −0.475382 + 0.255686i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −493.315 −0.831898 −0.415949 0.909388i \(-0.636550\pi\)
−0.415949 + 0.909388i \(0.636550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 423.320i 0.706712i −0.935489 0.353356i \(-0.885041\pi\)
0.935489 0.353356i \(-0.114959\pi\)
\(600\) 0 0
\(601\) 167.332i 0.278423i −0.990263 0.139211i \(-0.955543\pi\)
0.990263 0.139211i \(-0.0444567\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −246.658 −0.407699
\(606\) 0 0
\(607\) 719.528i 1.18538i 0.805429 + 0.592692i \(0.201935\pi\)
−0.805429 + 0.592692i \(0.798065\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1375.79i 2.25170i
\(612\) 0 0
\(613\) 342.000 0.557912 0.278956 0.960304i \(-0.410012\pi\)
0.278956 + 0.960304i \(0.410012\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 505.964 0.820040 0.410020 0.912077i \(-0.365522\pi\)
0.410020 + 0.912077i \(0.365522\pi\)
\(618\) 0 0
\(619\) −494.000 −0.798061 −0.399031 0.916938i \(-0.630653\pi\)
−0.399031 + 0.916938i \(0.630653\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 253.992i 0.407692i
\(624\) 0 0
\(625\) −775.000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −78.0000 −0.123613 −0.0618067 0.998088i \(-0.519686\pi\)
−0.0618067 + 0.998088i \(0.519686\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 317.490i 0.499984i
\(636\) 0 0
\(637\) 752.994i 1.18209i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 423.320i 0.660406i −0.943910 0.330203i \(-0.892883\pi\)
0.943910 0.330203i \(-0.107117\pi\)
\(642\) 0 0
\(643\) 502.000 0.780715 0.390358 0.920663i \(-0.372351\pi\)
0.390358 + 0.920663i \(0.372351\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −56.9210 −0.0879768 −0.0439884 0.999032i \(-0.514006\pi\)
−0.0439884 + 0.999032i \(0.514006\pi\)
\(648\) 0 0
\(649\) 267.731i 0.412529i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1283.88 −1.96613 −0.983066 0.183250i \(-0.941338\pi\)
−0.983066 + 0.183250i \(0.941338\pi\)
\(654\) 0 0
\(655\) 560.000 0.854962
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1058.30i 1.60592i 0.596034 + 0.802959i \(0.296742\pi\)
−0.596034 + 0.802959i \(0.703258\pi\)
\(660\) 0 0
\(661\) 384.864i 0.582244i −0.956686 0.291122i \(-0.905971\pi\)
0.956686 0.291122i \(-0.0940286\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −113.842 211.660i −0.171191 0.318286i
\(666\) 0 0
\(667\) 133.866i 0.200698i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1087.82 1.62120
\(672\) 0 0
\(673\) 769.727i 1.14373i 0.820349 + 0.571863i \(0.193779\pi\)
−0.820349 + 0.571863i \(0.806221\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 910.138i 1.34437i 0.740383 + 0.672185i \(0.234644\pi\)
−0.740383 + 0.672185i \(0.765356\pi\)
\(678\) 0 0
\(679\) 66.9328i 0.0985756i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 380.988i 0.557816i −0.960318 0.278908i \(-0.910028\pi\)
0.960318 0.278908i \(-0.0899724\pi\)
\(684\) 0 0
\(685\) 1040.00 1.51825
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1062.53 1.54213
\(690\) 0 0
\(691\) −242.000 −0.350217 −0.175109 0.984549i \(-0.556028\pi\)
−0.175109 + 0.984549i \(0.556028\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 847.490 1.21941
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1081.50 −1.54279 −0.771397 0.636354i \(-0.780442\pi\)
−0.771397 + 0.636354i \(0.780442\pi\)
\(702\) 0 0
\(703\) 840.000 451.796i 1.19488 0.642669i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 290.930 0.411499
\(708\) 0 0
\(709\) 586.000 0.826516 0.413258 0.910614i \(-0.364391\pi\)
0.413258 + 0.910614i \(0.364391\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 105.830i 0.148429i
\(714\) 0 0
\(715\) 1338.66i 1.87225i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −929.710 −1.29306 −0.646530 0.762889i \(-0.723780\pi\)
−0.646530 + 0.762889i \(0.723780\pi\)
\(720\) 0 0
\(721\) 234.265i 0.324917i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 317.490i 0.437917i
\(726\) 0 0
\(727\) 1198.00 1.64787 0.823934 0.566686i \(-0.191775\pi\)
0.823934 + 0.566686i \(0.191775\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 298.000 0.406548 0.203274 0.979122i \(-0.434842\pi\)
0.203274 + 0.979122i \(0.434842\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 846.640i 1.14877i
\(738\) 0 0
\(739\) 106.000 0.143437 0.0717185 0.997425i \(-0.477152\pi\)
0.0717185 + 0.997425i \(0.477152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 42.3320i 0.0569745i 0.999594 + 0.0284872i \(0.00906899\pi\)
−0.999594 + 0.0284872i \(0.990931\pi\)
\(744\) 0 0
\(745\) 520.000 0.697987
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 126.996i 0.169554i
\(750\) 0 0
\(751\) 217.532i 0.289656i −0.989457 0.144828i \(-0.953737\pi\)
0.989457 0.144828i \(-0.0462629\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1164.13i 1.54189i
\(756\) 0 0
\(757\) 918.000 1.21268 0.606341 0.795205i \(-0.292637\pi\)
0.606341 + 0.795205i \(0.292637\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −923.385 −1.21338 −0.606692 0.794937i \(-0.707504\pi\)
−0.606692 + 0.794937i \(0.707504\pi\)
\(762\) 0 0
\(763\) 368.130i 0.482478i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −354.175 −0.461767
\(768\) 0 0
\(769\) −986.000 −1.28218 −0.641092 0.767464i \(-0.721518\pi\)
−0.641092 + 0.767464i \(0.721518\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 275.158i 0.355961i 0.984034 + 0.177981i \(0.0569565\pi\)
−0.984034 + 0.177981i \(0.943044\pi\)
\(774\) 0 0
\(775\) 250.998i 0.323868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −354.175 + 190.494i −0.454654 + 0.244537i
\(780\) 0 0
\(781\) 1606.39i 2.05683i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 771.596 0.982925
\(786\) 0 0
\(787\) 568.929i 0.722908i −0.932390 0.361454i \(-0.882280\pi\)
0.932390 0.361454i \(-0.117720\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 211.660i 0.267585i
\(792\) 0 0
\(793\) 1439.06i 1.81470i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 275.158i 0.345242i 0.984988 + 0.172621i \(0.0552236\pi\)
−0.984988 + 0.172621i \(0.944776\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1290.21 −1.60674
\(804\) 0 0
\(805\) 80.0000 0.0993789
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 632.456 0.781774 0.390887 0.920439i \(-0.372168\pi\)
0.390887 + 0.920439i \(0.372168\pi\)
\(810\) 0 0
\(811\) 301.198i 0.371390i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −872.789 −1.07091
\(816\) 0 0
\(817\) −306.000 568.929i −0.374541 0.696363i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 537.587 0.654796 0.327398 0.944887i \(-0.393828\pi\)
0.327398 + 0.944887i \(0.393828\pi\)
\(822\) 0 0
\(823\) −382.000 −0.464156 −0.232078 0.972697i \(-0.574552\pi\)
−0.232078 + 0.972697i \(0.574552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 634.980i 0.767812i 0.923372 + 0.383906i \(0.125421\pi\)
−0.923372 + 0.383906i \(0.874579\pi\)
\(828\) 0 0
\(829\) 50.1996i 0.0605544i −0.999542 0.0302772i \(-0.990361\pi\)
0.999542 0.0302772i \(-0.00963901\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1070.92i 1.28254i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1354.62i 1.61457i −0.590161 0.807285i \(-0.700936\pi\)
0.590161 0.807285i \(-0.299064\pi\)
\(840\) 0 0
\(841\) 393.000 0.467301
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 702.026 0.830800
\(846\) 0 0
\(847\) 78.0000 0.0920897
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 317.490i 0.373079i
\(852\) 0 0
\(853\) −1142.00 −1.33880 −0.669402 0.742900i \(-0.733450\pi\)
−0.669402 + 0.742900i \(0.733450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1100.63i 1.28429i −0.766585 0.642143i \(-0.778046\pi\)
0.766585 0.642143i \(-0.221954\pi\)
\(858\) 0 0
\(859\) 866.000 1.00815 0.504075 0.863660i \(-0.331834\pi\)
0.504075 + 0.863660i \(0.331834\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1015.97i 1.17725i 0.808405 + 0.588626i \(0.200331\pi\)
−0.808405 + 0.588626i \(0.799669\pi\)
\(864\) 0 0
\(865\) 937.059i 1.08331i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1481.62i 1.70497i
\(870\) 0 0
\(871\) −1120.00 −1.28588
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 126.491 0.144561
\(876\) 0 0
\(877\) 786.460i 0.896762i −0.893842 0.448381i \(-0.852001\pi\)
0.893842 0.448381i \(-0.147999\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 265.631 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(882\) 0 0
\(883\) 1458.00 1.65119 0.825595 0.564264i \(-0.190840\pi\)
0.825595 + 0.564264i \(0.190840\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 423.320i 0.477249i −0.971112 0.238625i \(-0.923303\pi\)
0.971112 0.238625i \(-0.0766966\pi\)
\(888\) 0 0
\(889\) 100.399i 0.112935i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 739.973 + 1375.79i 0.828637 + 1.54064i
\(894\) 0 0
\(895\) 1874.12i 2.09399i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 354.175 0.393966
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 317.490i 0.350818i
\(906\) 0 0
\(907\) 1773.72i 1.95559i −0.209566 0.977795i \(-0.567205\pi\)
0.209566 0.977795i \(-0.432795\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 550.316i 0.604079i −0.953295 0.302040i \(-0.902332\pi\)
0.953295 0.302040i \(-0.0976675\pi\)
\(912\) 0 0
\(913\) −320.000 −0.350493
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −177.088 −0.193116
\(918\) 0 0
\(919\) −1374.00 −1.49510 −0.747552 0.664204i \(-0.768771\pi\)
−0.747552 + 0.664204i \(0.768771\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2125.05 2.30233
\(924\) 0 0
\(925\) 752.994i 0.814048i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −670.403 −0.721639 −0.360820 0.932636i \(-0.617503\pi\)
−0.360820 + 0.932636i \(0.617503\pi\)
\(930\) 0 0
\(931\) −405.000 752.994i −0.435016 0.808801i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −918.000 −0.979723 −0.489861 0.871800i \(-0.662953\pi\)
−0.489861 + 0.871800i \(0.662953\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1164.13i 1.23712i −0.785737 0.618560i \(-0.787716\pi\)
0.785737 0.618560i \(-0.212284\pi\)
\(942\) 0 0
\(943\) 133.866i 0.141957i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1682.33 −1.77649 −0.888243 0.459374i \(-0.848074\pi\)
−0.888243 + 0.459374i \(0.848074\pi\)
\(948\) 0 0
\(949\) 1706.79i 1.79851i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1206.46i 1.26596i −0.774167 0.632981i \(-0.781831\pi\)
0.774167 0.632981i \(-0.218169\pi\)
\(954\) 0 0
\(955\) 1640.00 1.71728
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −328.877 −0.342937
\(960\) 0 0
\(961\) 681.000 0.708637
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2116.60i 2.19337i
\(966\) 0 0
\(967\) 1682.00 1.73940 0.869700 0.493581i \(-0.164312\pi\)
0.869700 + 0.493581i \(0.164312\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1142.96i 1.17710i −0.808461 0.588550i \(-0.799699\pi\)
0.808461 0.588550i \(-0.200301\pi\)
\(972\) 0 0
\(973\) −268.000 −0.275437
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 740.810i 0.758250i −0.925345 0.379125i \(-0.876225\pi\)
0.925345 0.379125i \(-0.123775\pi\)
\(978\) 0 0
\(979\) 1606.39i 1.64084i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 423.320i 0.430641i 0.976543 + 0.215321i \(0.0690796\pi\)
−0.976543 + 0.215321i \(0.930920\pi\)
\(984\) 0 0
\(985\) 1560.00 1.58376
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 215.035 0.217427
\(990\) 0 0
\(991\) 853.393i 0.861144i −0.902556 0.430572i \(-0.858312\pi\)
0.902556 0.430572i \(-0.141688\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 417.421 0.419518
\(996\) 0 0
\(997\) 454.000 0.455366 0.227683 0.973735i \(-0.426885\pi\)
0.227683 + 0.973735i \(0.426885\pi\)
\(998\) 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 2736.3.o.j.721.2 4
3.2 odd 2 inner 2736.3.o.j.721.4 4
4.3 odd 2 171.3.c.e.37.1 4
12.11 even 2 171.3.c.e.37.4 yes 4
19.18 odd 2 inner 2736.3.o.j.721.1 4
57.56 even 2 inner 2736.3.o.j.721.3 4
76.75 even 2 171.3.c.e.37.3 yes 4
228.227 odd 2 171.3.c.e.37.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.3.c.e.37.1 4 4.3 odd 2
171.3.c.e.37.2 yes 4 228.227 odd 2
171.3.c.e.37.3 yes 4 76.75 even 2
171.3.c.e.37.4 yes 4 12.11 even 2
2736.3.o.j.721.1 4 19.18 odd 2 inner
2736.3.o.j.721.2 4 1.1 even 1 trivial
2736.3.o.j.721.3 4 57.56 even 2 inner
2736.3.o.j.721.4 4 3.2 odd 2 inner