Properties

Label 1568.3.h.a.881.3
Level $1568$
Weight $3$
Character 1568.881
Analytic conductor $42.725$
Analytic rank $0$
Dimension $28$
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] = [1568,3,Mod(881,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.881");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.3
Character \(\chi\) \(=\) 1568.881
Dual form 1568.3.h.a.881.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.89635 q^{3} +8.85969 q^{5} +6.18155 q^{9} +O(q^{10})\) \(q-3.89635 q^{3} +8.85969 q^{5} +6.18155 q^{9} -3.64596i q^{11} -7.79378 q^{13} -34.5205 q^{15} +10.4749i q^{17} +10.7853 q^{19} +12.9111 q^{23} +53.4941 q^{25} +10.9817 q^{27} +17.2327i q^{29} -30.2297i q^{31} +14.2059i q^{33} +39.5843i q^{37} +30.3673 q^{39} -73.6801i q^{41} +40.8501i q^{43} +54.7667 q^{45} +41.8030i q^{47} -40.8138i q^{51} +6.41173i q^{53} -32.3020i q^{55} -42.0232 q^{57} -15.9148 q^{59} +12.1570 q^{61} -69.0505 q^{65} +7.79739i q^{67} -50.3060 q^{69} +41.3627 q^{71} -89.6091i q^{73} -208.432 q^{75} -70.7950 q^{79} -98.4224 q^{81} +60.8673 q^{83} +92.8043i q^{85} -67.1446i q^{87} -27.0204i q^{89} +117.785i q^{93} +95.5542 q^{95} +3.26608i q^{97} -22.5377i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −3.89635 −1.29878 −0.649392 0.760454i \(-0.724976\pi\)
−0.649392 + 0.760454i \(0.724976\pi\)
\(4\) 0 0
\(5\) 8.85969 1.77194 0.885969 0.463744i \(-0.153494\pi\)
0.885969 + 0.463744i \(0.153494\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.18155 0.686839
\(10\) 0 0
\(11\) − 3.64596i − 0.331451i −0.986172 0.165725i \(-0.947004\pi\)
0.986172 0.165725i \(-0.0529965\pi\)
\(12\) 0 0
\(13\) −7.79378 −0.599522 −0.299761 0.954014i \(-0.596907\pi\)
−0.299761 + 0.954014i \(0.596907\pi\)
\(14\) 0 0
\(15\) −34.5205 −2.30136
\(16\) 0 0
\(17\) 10.4749i 0.616170i 0.951359 + 0.308085i \(0.0996881\pi\)
−0.951359 + 0.308085i \(0.900312\pi\)
\(18\) 0 0
\(19\) 10.7853 0.567646 0.283823 0.958877i \(-0.408397\pi\)
0.283823 + 0.958877i \(0.408397\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 12.9111 0.561351 0.280675 0.959803i \(-0.409442\pi\)
0.280675 + 0.959803i \(0.409442\pi\)
\(24\) 0 0
\(25\) 53.4941 2.13977
\(26\) 0 0
\(27\) 10.9817 0.406728
\(28\) 0 0
\(29\) 17.2327i 0.594231i 0.954842 + 0.297115i \(0.0960246\pi\)
−0.954842 + 0.297115i \(0.903975\pi\)
\(30\) 0 0
\(31\) − 30.2297i − 0.975151i −0.873081 0.487575i \(-0.837881\pi\)
0.873081 0.487575i \(-0.162119\pi\)
\(32\) 0 0
\(33\) 14.2059i 0.430483i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 39.5843i 1.06985i 0.844900 + 0.534924i \(0.179660\pi\)
−0.844900 + 0.534924i \(0.820340\pi\)
\(38\) 0 0
\(39\) 30.3673 0.778649
\(40\) 0 0
\(41\) − 73.6801i − 1.79707i −0.438897 0.898537i \(-0.644631\pi\)
0.438897 0.898537i \(-0.355369\pi\)
\(42\) 0 0
\(43\) 40.8501i 0.950002i 0.879985 + 0.475001i \(0.157552\pi\)
−0.879985 + 0.475001i \(0.842448\pi\)
\(44\) 0 0
\(45\) 54.7667 1.21704
\(46\) 0 0
\(47\) 41.8030i 0.889426i 0.895673 + 0.444713i \(0.146694\pi\)
−0.895673 + 0.444713i \(0.853306\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 40.8138i − 0.800271i
\(52\) 0 0
\(53\) 6.41173i 0.120976i 0.998169 + 0.0604880i \(0.0192657\pi\)
−0.998169 + 0.0604880i \(0.980734\pi\)
\(54\) 0 0
\(55\) − 32.3020i − 0.587310i
\(56\) 0 0
\(57\) −42.0232 −0.737249
\(58\) 0 0
\(59\) −15.9148 −0.269743 −0.134871 0.990863i \(-0.543062\pi\)
−0.134871 + 0.990863i \(0.543062\pi\)
\(60\) 0 0
\(61\) 12.1570 0.199294 0.0996472 0.995023i \(-0.468229\pi\)
0.0996472 + 0.995023i \(0.468229\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −69.0505 −1.06232
\(66\) 0 0
\(67\) 7.79739i 0.116379i 0.998306 + 0.0581895i \(0.0185328\pi\)
−0.998306 + 0.0581895i \(0.981467\pi\)
\(68\) 0 0
\(69\) −50.3060 −0.729073
\(70\) 0 0
\(71\) 41.3627 0.582574 0.291287 0.956636i \(-0.405917\pi\)
0.291287 + 0.956636i \(0.405917\pi\)
\(72\) 0 0
\(73\) − 89.6091i − 1.22752i −0.789492 0.613761i \(-0.789656\pi\)
0.789492 0.613761i \(-0.210344\pi\)
\(74\) 0 0
\(75\) −208.432 −2.77909
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −70.7950 −0.896140 −0.448070 0.893999i \(-0.647888\pi\)
−0.448070 + 0.893999i \(0.647888\pi\)
\(80\) 0 0
\(81\) −98.4224 −1.21509
\(82\) 0 0
\(83\) 60.8673 0.733341 0.366671 0.930351i \(-0.380498\pi\)
0.366671 + 0.930351i \(0.380498\pi\)
\(84\) 0 0
\(85\) 92.8043i 1.09182i
\(86\) 0 0
\(87\) − 67.1446i − 0.771777i
\(88\) 0 0
\(89\) − 27.0204i − 0.303600i −0.988411 0.151800i \(-0.951493\pi\)
0.988411 0.151800i \(-0.0485070\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 117.785i 1.26651i
\(94\) 0 0
\(95\) 95.5542 1.00583
\(96\) 0 0
\(97\) 3.26608i 0.0336710i 0.999858 + 0.0168355i \(0.00535916\pi\)
−0.999858 + 0.0168355i \(0.994641\pi\)
\(98\) 0 0
\(99\) − 22.5377i − 0.227653i
\(100\) 0 0
\(101\) 137.714 1.36351 0.681754 0.731582i \(-0.261218\pi\)
0.681754 + 0.731582i \(0.261218\pi\)
\(102\) 0 0
\(103\) − 99.6787i − 0.967754i −0.875136 0.483877i \(-0.839228\pi\)
0.875136 0.483877i \(-0.160772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 94.0107i 0.878605i 0.898339 + 0.439302i \(0.144774\pi\)
−0.898339 + 0.439302i \(0.855226\pi\)
\(108\) 0 0
\(109\) 195.949i 1.79770i 0.438258 + 0.898849i \(0.355595\pi\)
−0.438258 + 0.898849i \(0.644405\pi\)
\(110\) 0 0
\(111\) − 154.234i − 1.38950i
\(112\) 0 0
\(113\) 101.873 0.901527 0.450763 0.892643i \(-0.351152\pi\)
0.450763 + 0.892643i \(0.351152\pi\)
\(114\) 0 0
\(115\) 114.388 0.994679
\(116\) 0 0
\(117\) −48.1777 −0.411775
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 107.707 0.890141
\(122\) 0 0
\(123\) 287.083i 2.33401i
\(124\) 0 0
\(125\) 252.449 2.01960
\(126\) 0 0
\(127\) 139.079 1.09511 0.547554 0.836770i \(-0.315559\pi\)
0.547554 + 0.836770i \(0.315559\pi\)
\(128\) 0 0
\(129\) − 159.166i − 1.23385i
\(130\) 0 0
\(131\) 91.7051 0.700039 0.350020 0.936742i \(-0.386175\pi\)
0.350020 + 0.936742i \(0.386175\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 97.2941 0.720697
\(136\) 0 0
\(137\) 199.581 1.45679 0.728397 0.685155i \(-0.240266\pi\)
0.728397 + 0.685155i \(0.240266\pi\)
\(138\) 0 0
\(139\) 39.4768 0.284006 0.142003 0.989866i \(-0.454646\pi\)
0.142003 + 0.989866i \(0.454646\pi\)
\(140\) 0 0
\(141\) − 162.879i − 1.15517i
\(142\) 0 0
\(143\) 28.4158i 0.198712i
\(144\) 0 0
\(145\) 152.676i 1.05294i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 94.7831i 0.636128i 0.948069 + 0.318064i \(0.103033\pi\)
−0.948069 + 0.318064i \(0.896967\pi\)
\(150\) 0 0
\(151\) −66.5686 −0.440852 −0.220426 0.975404i \(-0.570745\pi\)
−0.220426 + 0.975404i \(0.570745\pi\)
\(152\) 0 0
\(153\) 64.7511i 0.423210i
\(154\) 0 0
\(155\) − 267.826i − 1.72791i
\(156\) 0 0
\(157\) 25.5194 0.162544 0.0812720 0.996692i \(-0.474102\pi\)
0.0812720 + 0.996692i \(0.474102\pi\)
\(158\) 0 0
\(159\) − 24.9823i − 0.157122i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 192.101i − 1.17853i −0.807939 0.589267i \(-0.799417\pi\)
0.807939 0.589267i \(-0.200583\pi\)
\(164\) 0 0
\(165\) 125.860i 0.762789i
\(166\) 0 0
\(167\) 184.150i 1.10269i 0.834276 + 0.551346i \(0.185886\pi\)
−0.834276 + 0.551346i \(0.814114\pi\)
\(168\) 0 0
\(169\) −108.257 −0.640574
\(170\) 0 0
\(171\) 66.6697 0.389882
\(172\) 0 0
\(173\) 69.9037 0.404068 0.202034 0.979379i \(-0.435245\pi\)
0.202034 + 0.979379i \(0.435245\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 62.0098 0.350338
\(178\) 0 0
\(179\) − 239.313i − 1.33695i −0.743736 0.668473i \(-0.766948\pi\)
0.743736 0.668473i \(-0.233052\pi\)
\(180\) 0 0
\(181\) 36.2834 0.200461 0.100230 0.994964i \(-0.468042\pi\)
0.100230 + 0.994964i \(0.468042\pi\)
\(182\) 0 0
\(183\) −47.3678 −0.258840
\(184\) 0 0
\(185\) 350.705i 1.89570i
\(186\) 0 0
\(187\) 38.1910 0.204230
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 325.243 1.70284 0.851422 0.524481i \(-0.175740\pi\)
0.851422 + 0.524481i \(0.175740\pi\)
\(192\) 0 0
\(193\) 199.640 1.03440 0.517201 0.855864i \(-0.326974\pi\)
0.517201 + 0.855864i \(0.326974\pi\)
\(194\) 0 0
\(195\) 269.045 1.37972
\(196\) 0 0
\(197\) 15.5053i 0.0787071i 0.999225 + 0.0393536i \(0.0125299\pi\)
−0.999225 + 0.0393536i \(0.987470\pi\)
\(198\) 0 0
\(199\) − 56.1617i − 0.282220i −0.989994 0.141110i \(-0.954933\pi\)
0.989994 0.141110i \(-0.0450671\pi\)
\(200\) 0 0
\(201\) − 30.3814i − 0.151151i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 652.783i − 3.18431i
\(206\) 0 0
\(207\) 79.8104 0.385558
\(208\) 0 0
\(209\) − 39.3226i − 0.188147i
\(210\) 0 0
\(211\) 370.470i 1.75578i 0.478859 + 0.877892i \(0.341051\pi\)
−0.478859 + 0.877892i \(0.658949\pi\)
\(212\) 0 0
\(213\) −161.164 −0.756638
\(214\) 0 0
\(215\) 361.919i 1.68335i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 349.149i 1.59429i
\(220\) 0 0
\(221\) − 81.6390i − 0.369407i
\(222\) 0 0
\(223\) 6.78533i 0.0304275i 0.999884 + 0.0152137i \(0.00484287\pi\)
−0.999884 + 0.0152137i \(0.995157\pi\)
\(224\) 0 0
\(225\) 330.677 1.46968
\(226\) 0 0
\(227\) 296.618 1.30669 0.653344 0.757061i \(-0.273365\pi\)
0.653344 + 0.757061i \(0.273365\pi\)
\(228\) 0 0
\(229\) −178.193 −0.778135 −0.389067 0.921209i \(-0.627203\pi\)
−0.389067 + 0.921209i \(0.627203\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 117.802 0.505589 0.252795 0.967520i \(-0.418650\pi\)
0.252795 + 0.967520i \(0.418650\pi\)
\(234\) 0 0
\(235\) 370.362i 1.57601i
\(236\) 0 0
\(237\) 275.842 1.16389
\(238\) 0 0
\(239\) −46.3543 −0.193951 −0.0969755 0.995287i \(-0.530917\pi\)
−0.0969755 + 0.995287i \(0.530917\pi\)
\(240\) 0 0
\(241\) − 366.618i − 1.52124i −0.649199 0.760619i \(-0.724896\pi\)
0.649199 0.760619i \(-0.275104\pi\)
\(242\) 0 0
\(243\) 284.653 1.17141
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −84.0581 −0.340316
\(248\) 0 0
\(249\) −237.160 −0.952452
\(250\) 0 0
\(251\) −129.896 −0.517513 −0.258756 0.965943i \(-0.583313\pi\)
−0.258756 + 0.965943i \(0.583313\pi\)
\(252\) 0 0
\(253\) − 47.0732i − 0.186060i
\(254\) 0 0
\(255\) − 361.598i − 1.41803i
\(256\) 0 0
\(257\) 268.345i 1.04415i 0.852901 + 0.522073i \(0.174841\pi\)
−0.852901 + 0.522073i \(0.825159\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 106.525i 0.408141i
\(262\) 0 0
\(263\) 235.383 0.894991 0.447495 0.894286i \(-0.352316\pi\)
0.447495 + 0.894286i \(0.352316\pi\)
\(264\) 0 0
\(265\) 56.8059i 0.214362i
\(266\) 0 0
\(267\) 105.281i 0.394311i
\(268\) 0 0
\(269\) 354.695 1.31857 0.659285 0.751893i \(-0.270859\pi\)
0.659285 + 0.751893i \(0.270859\pi\)
\(270\) 0 0
\(271\) 421.870i 1.55672i 0.627820 + 0.778358i \(0.283947\pi\)
−0.627820 + 0.778358i \(0.716053\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 195.037i − 0.709226i
\(276\) 0 0
\(277\) − 368.529i − 1.33043i −0.746653 0.665214i \(-0.768340\pi\)
0.746653 0.665214i \(-0.231660\pi\)
\(278\) 0 0
\(279\) − 186.866i − 0.669772i
\(280\) 0 0
\(281\) −35.2868 −0.125576 −0.0627879 0.998027i \(-0.519999\pi\)
−0.0627879 + 0.998027i \(0.519999\pi\)
\(282\) 0 0
\(283\) 196.617 0.694761 0.347380 0.937724i \(-0.387071\pi\)
0.347380 + 0.937724i \(0.387071\pi\)
\(284\) 0 0
\(285\) −372.313 −1.30636
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 179.277 0.620335
\(290\) 0 0
\(291\) − 12.7258i − 0.0437313i
\(292\) 0 0
\(293\) −317.573 −1.08387 −0.541933 0.840421i \(-0.682307\pi\)
−0.541933 + 0.840421i \(0.682307\pi\)
\(294\) 0 0
\(295\) −141.001 −0.477968
\(296\) 0 0
\(297\) − 40.0386i − 0.134810i
\(298\) 0 0
\(299\) −100.626 −0.336542
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −536.583 −1.77090
\(304\) 0 0
\(305\) 107.707 0.353137
\(306\) 0 0
\(307\) −132.193 −0.430596 −0.215298 0.976548i \(-0.569072\pi\)
−0.215298 + 0.976548i \(0.569072\pi\)
\(308\) 0 0
\(309\) 388.383i 1.25690i
\(310\) 0 0
\(311\) 462.404i 1.48683i 0.668831 + 0.743414i \(0.266795\pi\)
−0.668831 + 0.743414i \(0.733205\pi\)
\(312\) 0 0
\(313\) − 566.042i − 1.80844i −0.427067 0.904220i \(-0.640453\pi\)
0.427067 0.904220i \(-0.359547\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 177.033i 0.558463i 0.960224 + 0.279231i \(0.0900796\pi\)
−0.960224 + 0.279231i \(0.909920\pi\)
\(318\) 0 0
\(319\) 62.8296 0.196958
\(320\) 0 0
\(321\) − 366.299i − 1.14112i
\(322\) 0 0
\(323\) 112.975i 0.349766i
\(324\) 0 0
\(325\) −416.922 −1.28284
\(326\) 0 0
\(327\) − 763.486i − 2.33482i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 496.160i 1.49897i 0.662019 + 0.749487i \(0.269700\pi\)
−0.662019 + 0.749487i \(0.730300\pi\)
\(332\) 0 0
\(333\) 244.693i 0.734813i
\(334\) 0 0
\(335\) 69.0825i 0.206216i
\(336\) 0 0
\(337\) −206.191 −0.611843 −0.305922 0.952057i \(-0.598965\pi\)
−0.305922 + 0.952057i \(0.598965\pi\)
\(338\) 0 0
\(339\) −396.931 −1.17089
\(340\) 0 0
\(341\) −110.216 −0.323214
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −445.696 −1.29187
\(346\) 0 0
\(347\) − 606.190i − 1.74694i −0.486874 0.873472i \(-0.661863\pi\)
0.486874 0.873472i \(-0.338137\pi\)
\(348\) 0 0
\(349\) −136.343 −0.390669 −0.195335 0.980737i \(-0.562579\pi\)
−0.195335 + 0.980737i \(0.562579\pi\)
\(350\) 0 0
\(351\) −85.5887 −0.243842
\(352\) 0 0
\(353\) 10.0743i 0.0285390i 0.999898 + 0.0142695i \(0.00454228\pi\)
−0.999898 + 0.0142695i \(0.995458\pi\)
\(354\) 0 0
\(355\) 366.461 1.03229
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −395.616 −1.10199 −0.550997 0.834507i \(-0.685752\pi\)
−0.550997 + 0.834507i \(0.685752\pi\)
\(360\) 0 0
\(361\) −244.678 −0.677778
\(362\) 0 0
\(363\) −419.664 −1.15610
\(364\) 0 0
\(365\) − 793.909i − 2.17509i
\(366\) 0 0
\(367\) 189.932i 0.517526i 0.965941 + 0.258763i \(0.0833149\pi\)
−0.965941 + 0.258763i \(0.916685\pi\)
\(368\) 0 0
\(369\) − 455.457i − 1.23430i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 360.104i 0.965425i 0.875779 + 0.482713i \(0.160348\pi\)
−0.875779 + 0.482713i \(0.839652\pi\)
\(374\) 0 0
\(375\) −983.631 −2.62302
\(376\) 0 0
\(377\) − 134.308i − 0.356254i
\(378\) 0 0
\(379\) − 11.2929i − 0.0297966i −0.999889 0.0148983i \(-0.995258\pi\)
0.999889 0.0148983i \(-0.00474246\pi\)
\(380\) 0 0
\(381\) −541.900 −1.42231
\(382\) 0 0
\(383\) − 434.253i − 1.13382i −0.823779 0.566910i \(-0.808139\pi\)
0.823779 0.566910i \(-0.191861\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 252.517i 0.652499i
\(388\) 0 0
\(389\) 43.1631i 0.110959i 0.998460 + 0.0554796i \(0.0176688\pi\)
−0.998460 + 0.0554796i \(0.982331\pi\)
\(390\) 0 0
\(391\) 135.242i 0.345887i
\(392\) 0 0
\(393\) −357.315 −0.909199
\(394\) 0 0
\(395\) −627.222 −1.58790
\(396\) 0 0
\(397\) 486.790 1.22617 0.613086 0.790016i \(-0.289928\pi\)
0.613086 + 0.790016i \(0.289928\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −546.914 −1.36388 −0.681938 0.731410i \(-0.738862\pi\)
−0.681938 + 0.731410i \(0.738862\pi\)
\(402\) 0 0
\(403\) 235.603i 0.584624i
\(404\) 0 0
\(405\) −871.992 −2.15307
\(406\) 0 0
\(407\) 144.323 0.354601
\(408\) 0 0
\(409\) − 66.7668i − 0.163244i −0.996663 0.0816220i \(-0.973990\pi\)
0.996663 0.0816220i \(-0.0260100\pi\)
\(410\) 0 0
\(411\) −777.637 −1.89206
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 539.266 1.29944
\(416\) 0 0
\(417\) −153.815 −0.368862
\(418\) 0 0
\(419\) −550.169 −1.31305 −0.656527 0.754303i \(-0.727975\pi\)
−0.656527 + 0.754303i \(0.727975\pi\)
\(420\) 0 0
\(421\) 579.599i 1.37672i 0.725369 + 0.688360i \(0.241669\pi\)
−0.725369 + 0.688360i \(0.758331\pi\)
\(422\) 0 0
\(423\) 258.408i 0.610893i
\(424\) 0 0
\(425\) 560.345i 1.31846i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 110.718i − 0.258084i
\(430\) 0 0
\(431\) 431.870 1.00202 0.501009 0.865442i \(-0.332962\pi\)
0.501009 + 0.865442i \(0.332962\pi\)
\(432\) 0 0
\(433\) − 0.143463i 0 0.000331322i −1.00000 0.000165661i \(-0.999947\pi\)
1.00000 0.000165661i \(-5.27316e-5\pi\)
\(434\) 0 0
\(435\) − 594.881i − 1.36754i
\(436\) 0 0
\(437\) 139.249 0.318648
\(438\) 0 0
\(439\) 191.349i 0.435874i 0.975963 + 0.217937i \(0.0699328\pi\)
−0.975963 + 0.217937i \(0.930067\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 393.501i − 0.888265i −0.895961 0.444133i \(-0.853512\pi\)
0.895961 0.444133i \(-0.146488\pi\)
\(444\) 0 0
\(445\) − 239.393i − 0.537961i
\(446\) 0 0
\(447\) − 369.308i − 0.826193i
\(448\) 0 0
\(449\) −725.831 −1.61655 −0.808275 0.588805i \(-0.799598\pi\)
−0.808275 + 0.588805i \(0.799598\pi\)
\(450\) 0 0
\(451\) −268.634 −0.595641
\(452\) 0 0
\(453\) 259.375 0.572571
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −69.3427 −0.151735 −0.0758673 0.997118i \(-0.524173\pi\)
−0.0758673 + 0.997118i \(0.524173\pi\)
\(458\) 0 0
\(459\) 115.032i 0.250614i
\(460\) 0 0
\(461\) 768.006 1.66596 0.832978 0.553306i \(-0.186634\pi\)
0.832978 + 0.553306i \(0.186634\pi\)
\(462\) 0 0
\(463\) −215.717 −0.465911 −0.232956 0.972487i \(-0.574840\pi\)
−0.232956 + 0.972487i \(0.574840\pi\)
\(464\) 0 0
\(465\) 1043.54i 2.24418i
\(466\) 0 0
\(467\) 28.9376 0.0619648 0.0309824 0.999520i \(-0.490136\pi\)
0.0309824 + 0.999520i \(0.490136\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −99.4325 −0.211109
\(472\) 0 0
\(473\) 148.938 0.314879
\(474\) 0 0
\(475\) 576.949 1.21463
\(476\) 0 0
\(477\) 39.6344i 0.0830911i
\(478\) 0 0
\(479\) − 802.952i − 1.67631i −0.545433 0.838154i \(-0.683635\pi\)
0.545433 0.838154i \(-0.316365\pi\)
\(480\) 0 0
\(481\) − 308.512i − 0.641396i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.9365i 0.0596629i
\(486\) 0 0
\(487\) −19.9239 −0.0409116 −0.0204558 0.999791i \(-0.506512\pi\)
−0.0204558 + 0.999791i \(0.506512\pi\)
\(488\) 0 0
\(489\) 748.493i 1.53066i
\(490\) 0 0
\(491\) − 76.2017i − 0.155197i −0.996985 0.0775985i \(-0.975275\pi\)
0.996985 0.0775985i \(-0.0247252\pi\)
\(492\) 0 0
\(493\) −180.510 −0.366147
\(494\) 0 0
\(495\) − 199.677i − 0.403387i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 522.871i − 1.04784i −0.851768 0.523919i \(-0.824470\pi\)
0.851768 0.523919i \(-0.175530\pi\)
\(500\) 0 0
\(501\) − 717.512i − 1.43216i
\(502\) 0 0
\(503\) 132.060i 0.262545i 0.991346 + 0.131273i \(0.0419064\pi\)
−0.991346 + 0.131273i \(0.958094\pi\)
\(504\) 0 0
\(505\) 1220.11 2.41605
\(506\) 0 0
\(507\) 421.807 0.831967
\(508\) 0 0
\(509\) 310.157 0.609346 0.304673 0.952457i \(-0.401453\pi\)
0.304673 + 0.952457i \(0.401453\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 118.440 0.230878
\(514\) 0 0
\(515\) − 883.122i − 1.71480i
\(516\) 0 0
\(517\) 152.412 0.294801
\(518\) 0 0
\(519\) −272.369 −0.524797
\(520\) 0 0
\(521\) − 61.0976i − 0.117270i −0.998279 0.0586349i \(-0.981325\pi\)
0.998279 0.0586349i \(-0.0186748\pi\)
\(522\) 0 0
\(523\) 513.846 0.982497 0.491249 0.871019i \(-0.336541\pi\)
0.491249 + 0.871019i \(0.336541\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 316.652 0.600858
\(528\) 0 0
\(529\) −362.304 −0.684886
\(530\) 0 0
\(531\) −98.3784 −0.185270
\(532\) 0 0
\(533\) 574.246i 1.07739i
\(534\) 0 0
\(535\) 832.906i 1.55683i
\(536\) 0 0
\(537\) 932.449i 1.73640i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 107.078i − 0.197926i −0.995091 0.0989630i \(-0.968447\pi\)
0.995091 0.0989630i \(-0.0315525\pi\)
\(542\) 0 0
\(543\) −141.373 −0.260355
\(544\) 0 0
\(545\) 1736.05i 3.18541i
\(546\) 0 0
\(547\) 43.3240i 0.0792030i 0.999216 + 0.0396015i \(0.0126088\pi\)
−0.999216 + 0.0396015i \(0.987391\pi\)
\(548\) 0 0
\(549\) 75.1489 0.136883
\(550\) 0 0
\(551\) 185.859i 0.337313i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 1366.47i − 2.46211i
\(556\) 0 0
\(557\) 72.5011i 0.130164i 0.997880 + 0.0650818i \(0.0207308\pi\)
−0.997880 + 0.0650818i \(0.979269\pi\)
\(558\) 0 0
\(559\) − 318.377i − 0.569547i
\(560\) 0 0
\(561\) −148.805 −0.265250
\(562\) 0 0
\(563\) −584.943 −1.03897 −0.519487 0.854478i \(-0.673877\pi\)
−0.519487 + 0.854478i \(0.673877\pi\)
\(564\) 0 0
\(565\) 902.559 1.59745
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 743.529 1.30673 0.653365 0.757043i \(-0.273357\pi\)
0.653365 + 0.757043i \(0.273357\pi\)
\(570\) 0 0
\(571\) − 1032.06i − 1.80747i −0.428097 0.903733i \(-0.640816\pi\)
0.428097 0.903733i \(-0.359184\pi\)
\(572\) 0 0
\(573\) −1267.26 −2.21163
\(574\) 0 0
\(575\) 690.666 1.20116
\(576\) 0 0
\(577\) 953.094i 1.65181i 0.563810 + 0.825905i \(0.309335\pi\)
−0.563810 + 0.825905i \(0.690665\pi\)
\(578\) 0 0
\(579\) −777.866 −1.34346
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 23.3769 0.0400976
\(584\) 0 0
\(585\) −426.839 −0.729640
\(586\) 0 0
\(587\) −96.2876 −0.164033 −0.0820167 0.996631i \(-0.526136\pi\)
−0.0820167 + 0.996631i \(0.526136\pi\)
\(588\) 0 0
\(589\) − 326.035i − 0.553540i
\(590\) 0 0
\(591\) − 60.4141i − 0.102224i
\(592\) 0 0
\(593\) 51.0193i 0.0860358i 0.999074 + 0.0430179i \(0.0136973\pi\)
−0.999074 + 0.0430179i \(0.986303\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 218.826i 0.366542i
\(598\) 0 0
\(599\) −902.236 −1.50624 −0.753119 0.657885i \(-0.771451\pi\)
−0.753119 + 0.657885i \(0.771451\pi\)
\(600\) 0 0
\(601\) − 903.595i − 1.50349i −0.659456 0.751743i \(-0.729213\pi\)
0.659456 0.751743i \(-0.270787\pi\)
\(602\) 0 0
\(603\) 48.2000i 0.0799337i
\(604\) 0 0
\(605\) 954.251 1.57727
\(606\) 0 0
\(607\) 354.410i 0.583872i 0.956438 + 0.291936i \(0.0942994\pi\)
−0.956438 + 0.291936i \(0.905701\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 325.804i − 0.533230i
\(612\) 0 0
\(613\) − 335.999i − 0.548123i −0.961712 0.274061i \(-0.911633\pi\)
0.961712 0.274061i \(-0.0883671\pi\)
\(614\) 0 0
\(615\) 2543.47i 4.13573i
\(616\) 0 0
\(617\) 223.359 0.362008 0.181004 0.983482i \(-0.442065\pi\)
0.181004 + 0.983482i \(0.442065\pi\)
\(618\) 0 0
\(619\) −726.053 −1.17294 −0.586472 0.809969i \(-0.699484\pi\)
−0.586472 + 0.809969i \(0.699484\pi\)
\(620\) 0 0
\(621\) 141.785 0.228317
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 899.270 1.43883
\(626\) 0 0
\(627\) 153.215i 0.244362i
\(628\) 0 0
\(629\) −414.642 −0.659208
\(630\) 0 0
\(631\) 326.157 0.516888 0.258444 0.966026i \(-0.416790\pi\)
0.258444 + 0.966026i \(0.416790\pi\)
\(632\) 0 0
\(633\) − 1443.48i − 2.28038i
\(634\) 0 0
\(635\) 1232.19 1.94046
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 255.686 0.400135
\(640\) 0 0
\(641\) −598.375 −0.933502 −0.466751 0.884389i \(-0.654576\pi\)
−0.466751 + 0.884389i \(0.654576\pi\)
\(642\) 0 0
\(643\) 1008.20 1.56796 0.783979 0.620787i \(-0.213187\pi\)
0.783979 + 0.620787i \(0.213187\pi\)
\(644\) 0 0
\(645\) − 1410.16i − 2.18630i
\(646\) 0 0
\(647\) 663.235i 1.02509i 0.858660 + 0.512546i \(0.171298\pi\)
−0.858660 + 0.512546i \(0.828702\pi\)
\(648\) 0 0
\(649\) 58.0248i 0.0894064i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 990.608i − 1.51701i −0.651666 0.758506i \(-0.725930\pi\)
0.651666 0.758506i \(-0.274070\pi\)
\(654\) 0 0
\(655\) 812.479 1.24043
\(656\) 0 0
\(657\) − 553.924i − 0.843110i
\(658\) 0 0
\(659\) − 82.2318i − 0.124783i −0.998052 0.0623914i \(-0.980127\pi\)
0.998052 0.0623914i \(-0.0198727\pi\)
\(660\) 0 0
\(661\) −626.220 −0.947382 −0.473691 0.880691i \(-0.657079\pi\)
−0.473691 + 0.880691i \(0.657079\pi\)
\(662\) 0 0
\(663\) 318.094i 0.479780i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 222.492i 0.333572i
\(668\) 0 0
\(669\) − 26.4380i − 0.0395187i
\(670\) 0 0
\(671\) − 44.3237i − 0.0660562i
\(672\) 0 0
\(673\) −150.211 −0.223196 −0.111598 0.993753i \(-0.535597\pi\)
−0.111598 + 0.993753i \(0.535597\pi\)
\(674\) 0 0
\(675\) 587.455 0.870303
\(676\) 0 0
\(677\) −556.414 −0.821882 −0.410941 0.911662i \(-0.634800\pi\)
−0.410941 + 0.911662i \(0.634800\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1155.73 −1.69711
\(682\) 0 0
\(683\) 791.356i 1.15865i 0.815098 + 0.579323i \(0.196683\pi\)
−0.815098 + 0.579323i \(0.803317\pi\)
\(684\) 0 0
\(685\) 1768.22 2.58135
\(686\) 0 0
\(687\) 694.302 1.01063
\(688\) 0 0
\(689\) − 49.9716i − 0.0725277i
\(690\) 0 0
\(691\) −976.534 −1.41322 −0.706609 0.707604i \(-0.749776\pi\)
−0.706609 + 0.707604i \(0.749776\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 349.752 0.503241
\(696\) 0 0
\(697\) 771.790 1.10730
\(698\) 0 0
\(699\) −458.999 −0.656651
\(700\) 0 0
\(701\) 855.098i 1.21983i 0.792468 + 0.609913i \(0.208796\pi\)
−0.792468 + 0.609913i \(0.791204\pi\)
\(702\) 0 0
\(703\) 426.928i 0.607294i
\(704\) 0 0
\(705\) − 1443.06i − 2.04689i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 333.471i 0.470340i 0.971954 + 0.235170i \(0.0755647\pi\)
−0.971954 + 0.235170i \(0.924435\pi\)
\(710\) 0 0
\(711\) −437.623 −0.615504
\(712\) 0 0
\(713\) − 390.297i − 0.547401i
\(714\) 0 0
\(715\) 251.755i 0.352105i
\(716\) 0 0
\(717\) 180.613 0.251900
\(718\) 0 0
\(719\) 39.8230i 0.0553866i 0.999616 + 0.0276933i \(0.00881618\pi\)
−0.999616 + 0.0276933i \(0.991184\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1428.47i 1.97576i
\(724\) 0 0
\(725\) 921.848i 1.27151i
\(726\) 0 0
\(727\) − 489.402i − 0.673180i −0.941651 0.336590i \(-0.890726\pi\)
0.941651 0.336590i \(-0.109274\pi\)
\(728\) 0 0
\(729\) −223.307 −0.306320
\(730\) 0 0
\(731\) −427.900 −0.585363
\(732\) 0 0
\(733\) 178.318 0.243272 0.121636 0.992575i \(-0.461186\pi\)
0.121636 + 0.992575i \(0.461186\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.4289 0.0385739
\(738\) 0 0
\(739\) 882.401i 1.19405i 0.802224 + 0.597024i \(0.203650\pi\)
−0.802224 + 0.597024i \(0.796350\pi\)
\(740\) 0 0
\(741\) 327.520 0.441997
\(742\) 0 0
\(743\) −1404.00 −1.88964 −0.944819 0.327591i \(-0.893763\pi\)
−0.944819 + 0.327591i \(0.893763\pi\)
\(744\) 0 0
\(745\) 839.749i 1.12718i
\(746\) 0 0
\(747\) 376.255 0.503687
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −205.680 −0.273875 −0.136938 0.990580i \(-0.543726\pi\)
−0.136938 + 0.990580i \(0.543726\pi\)
\(752\) 0 0
\(753\) 506.119 0.672137
\(754\) 0 0
\(755\) −589.777 −0.781162
\(756\) 0 0
\(757\) 15.0345i 0.0198606i 0.999951 + 0.00993032i \(0.00316097\pi\)
−0.999951 + 0.00993032i \(0.996839\pi\)
\(758\) 0 0
\(759\) 183.414i 0.241652i
\(760\) 0 0
\(761\) − 628.491i − 0.825876i −0.910759 0.412938i \(-0.864503\pi\)
0.910759 0.412938i \(-0.135497\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 573.675i 0.749901i
\(766\) 0 0
\(767\) 124.037 0.161717
\(768\) 0 0
\(769\) − 442.918i − 0.575967i −0.957635 0.287983i \(-0.907015\pi\)
0.957635 0.287983i \(-0.0929848\pi\)
\(770\) 0 0
\(771\) − 1045.57i − 1.35612i
\(772\) 0 0
\(773\) −169.198 −0.218885 −0.109442 0.993993i \(-0.534907\pi\)
−0.109442 + 0.993993i \(0.534907\pi\)
\(774\) 0 0
\(775\) − 1617.11i − 2.08659i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 794.660i − 1.02010i
\(780\) 0 0
\(781\) − 150.807i − 0.193094i
\(782\) 0 0
\(783\) 189.244i 0.241690i
\(784\) 0 0
\(785\) 226.094 0.288018
\(786\) 0 0
\(787\) 46.4874 0.0590691 0.0295346 0.999564i \(-0.490597\pi\)
0.0295346 + 0.999564i \(0.490597\pi\)
\(788\) 0 0
\(789\) −917.133 −1.16240
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −94.7487 −0.119481
\(794\) 0 0
\(795\) − 221.336i − 0.278410i
\(796\) 0 0
\(797\) −1351.86 −1.69618 −0.848092 0.529850i \(-0.822248\pi\)
−0.848092 + 0.529850i \(0.822248\pi\)
\(798\) 0 0
\(799\) −437.882 −0.548038
\(800\) 0 0
\(801\) − 167.028i − 0.208525i
\(802\) 0 0
\(803\) −326.711 −0.406863
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1382.02 −1.71254
\(808\) 0 0
\(809\) −1403.13 −1.73440 −0.867198 0.497963i \(-0.834081\pi\)
−0.867198 + 0.497963i \(0.834081\pi\)
\(810\) 0 0
\(811\) 689.037 0.849614 0.424807 0.905284i \(-0.360342\pi\)
0.424807 + 0.905284i \(0.360342\pi\)
\(812\) 0 0
\(813\) − 1643.75i − 2.02184i
\(814\) 0 0
\(815\) − 1701.95i − 2.08829i
\(816\) 0 0
\(817\) 440.579i 0.539265i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.3423i 0.0272135i 0.999907 + 0.0136067i \(0.00433129\pi\)
−0.999907 + 0.0136067i \(0.995669\pi\)
\(822\) 0 0
\(823\) −1024.22 −1.24450 −0.622249 0.782819i \(-0.713781\pi\)
−0.622249 + 0.782819i \(0.713781\pi\)
\(824\) 0 0
\(825\) 759.934i 0.921132i
\(826\) 0 0
\(827\) − 466.377i − 0.563938i −0.959424 0.281969i \(-0.909012\pi\)
0.959424 0.281969i \(-0.0909875\pi\)
\(828\) 0 0
\(829\) −1500.70 −1.81025 −0.905126 0.425142i \(-0.860224\pi\)
−0.905126 + 0.425142i \(0.860224\pi\)
\(830\) 0 0
\(831\) 1435.92i 1.72794i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1631.51i 1.95390i
\(836\) 0 0
\(837\) − 331.972i − 0.396621i
\(838\) 0 0
\(839\) − 1068.18i − 1.27316i −0.771212 0.636579i \(-0.780349\pi\)
0.771212 0.636579i \(-0.219651\pi\)
\(840\) 0 0
\(841\) 544.034 0.646890
\(842\) 0 0
\(843\) 137.490 0.163096
\(844\) 0 0
\(845\) −959.123 −1.13506
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −766.090 −0.902344
\(850\) 0 0
\(851\) 511.076i 0.600559i
\(852\) 0 0
\(853\) −918.640 −1.07695 −0.538476 0.842641i \(-0.681000\pi\)
−0.538476 + 0.842641i \(0.681000\pi\)
\(854\) 0 0
\(855\) 590.673 0.690846
\(856\) 0 0
\(857\) 505.952i 0.590375i 0.955439 + 0.295188i \(0.0953822\pi\)
−0.955439 + 0.295188i \(0.904618\pi\)
\(858\) 0 0
\(859\) 1377.03 1.60306 0.801532 0.597952i \(-0.204019\pi\)
0.801532 + 0.597952i \(0.204019\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −917.635 −1.06331 −0.531654 0.846962i \(-0.678429\pi\)
−0.531654 + 0.846962i \(0.678429\pi\)
\(864\) 0 0
\(865\) 619.325 0.715983
\(866\) 0 0
\(867\) −698.525 −0.805681
\(868\) 0 0
\(869\) 258.116i 0.297026i
\(870\) 0 0
\(871\) − 60.7712i − 0.0697717i
\(872\) 0 0
\(873\) 20.1895i 0.0231265i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 699.948i 0.798116i 0.916926 + 0.399058i \(0.130663\pi\)
−0.916926 + 0.399058i \(0.869337\pi\)
\(878\) 0 0
\(879\) 1237.38 1.40771
\(880\) 0 0
\(881\) 6.37652i 0.00723783i 0.999993 + 0.00361891i \(0.00115194\pi\)
−0.999993 + 0.00361891i \(0.998848\pi\)
\(882\) 0 0
\(883\) 1548.35i 1.75351i 0.480935 + 0.876756i \(0.340297\pi\)
−0.480935 + 0.876756i \(0.659703\pi\)
\(884\) 0 0
\(885\) 549.388 0.620777
\(886\) 0 0
\(887\) − 732.369i − 0.825670i −0.910806 0.412835i \(-0.864539\pi\)
0.910806 0.412835i \(-0.135461\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 358.844i 0.402743i
\(892\) 0 0
\(893\) 450.857i 0.504879i
\(894\) 0 0
\(895\) − 2120.24i − 2.36899i
\(896\) 0 0
\(897\) 392.074 0.437095
\(898\) 0 0
\(899\) 520.938 0.579464
\(900\) 0 0
\(901\) −67.1621 −0.0745418
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 321.459 0.355204
\(906\) 0 0
\(907\) − 723.838i − 0.798057i −0.916939 0.399029i \(-0.869347\pi\)
0.916939 0.399029i \(-0.130653\pi\)
\(908\) 0 0
\(909\) 851.288 0.936511
\(910\) 0 0
\(911\) −1600.04 −1.75636 −0.878179 0.478332i \(-0.841242\pi\)
−0.878179 + 0.478332i \(0.841242\pi\)
\(912\) 0 0
\(913\) − 221.920i − 0.243066i
\(914\) 0 0
\(915\) −419.664 −0.458649
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 524.085 0.570277 0.285139 0.958486i \(-0.407960\pi\)
0.285139 + 0.958486i \(0.407960\pi\)
\(920\) 0 0
\(921\) 515.070 0.559251
\(922\) 0 0
\(923\) −322.372 −0.349266
\(924\) 0 0
\(925\) 2117.53i 2.28922i
\(926\) 0 0
\(927\) − 616.169i − 0.664691i
\(928\) 0 0
\(929\) 637.154i 0.685849i 0.939363 + 0.342924i \(0.111417\pi\)
−0.939363 + 0.342924i \(0.888583\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 1801.69i − 1.93107i
\(934\) 0 0
\(935\) 338.360 0.361883
\(936\) 0 0
\(937\) 383.587i 0.409378i 0.978827 + 0.204689i \(0.0656182\pi\)
−0.978827 + 0.204689i \(0.934382\pi\)
\(938\) 0 0
\(939\) 2205.50i 2.34877i
\(940\) 0 0
\(941\) −260.591 −0.276930 −0.138465 0.990367i \(-0.544217\pi\)
−0.138465 + 0.990367i \(0.544217\pi\)
\(942\) 0 0
\(943\) − 951.288i − 1.00879i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 983.163i 1.03819i 0.854717 + 0.519093i \(0.173730\pi\)
−0.854717 + 0.519093i \(0.826270\pi\)
\(948\) 0 0
\(949\) 698.394i 0.735926i
\(950\) 0 0
\(951\) − 689.781i − 0.725322i
\(952\) 0 0
\(953\) −1137.60 −1.19370 −0.596851 0.802352i \(-0.703582\pi\)
−0.596851 + 0.802352i \(0.703582\pi\)
\(954\) 0 0
\(955\) 2881.56 3.01734
\(956\) 0 0
\(957\) −244.806 −0.255806
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 47.1670 0.0490812
\(962\) 0 0
\(963\) 581.132i 0.603460i
\(964\) 0 0
\(965\) 1768.75 1.83290
\(966\) 0 0
\(967\) 1296.35 1.34059 0.670297 0.742093i \(-0.266167\pi\)
0.670297 + 0.742093i \(0.266167\pi\)
\(968\) 0 0
\(969\) − 440.188i − 0.454271i
\(970\) 0 0
\(971\) −1330.47 −1.37021 −0.685105 0.728445i \(-0.740244\pi\)
−0.685105 + 0.728445i \(0.740244\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1624.47 1.66613
\(976\) 0 0
\(977\) 1386.16 1.41879 0.709397 0.704809i \(-0.248967\pi\)
0.709397 + 0.704809i \(0.248967\pi\)
\(978\) 0 0
\(979\) −98.5153 −0.100629
\(980\) 0 0
\(981\) 1211.27i 1.23473i
\(982\) 0 0
\(983\) − 694.161i − 0.706166i −0.935592 0.353083i \(-0.885133\pi\)
0.935592 0.353083i \(-0.114867\pi\)
\(984\) 0 0
\(985\) 137.372i 0.139464i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 527.418i 0.533284i
\(990\) 0 0
\(991\) 934.514 0.943001 0.471500 0.881866i \(-0.343713\pi\)
0.471500 + 0.881866i \(0.343713\pi\)
\(992\) 0 0
\(993\) − 1933.22i − 1.94684i
\(994\) 0 0
\(995\) − 497.576i − 0.500076i
\(996\) 0 0
\(997\) 1538.21 1.54283 0.771417 0.636330i \(-0.219548\pi\)
0.771417 + 0.636330i \(0.219548\pi\)
\(998\) 0 0
\(999\) 434.702i 0.435137i
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 1568.3.h.a.881.3 28
4.3 odd 2 392.3.h.a.293.18 28
7.2 even 3 224.3.n.a.17.13 28
7.3 odd 6 224.3.n.a.145.2 28
7.6 odd 2 inner 1568.3.h.a.881.25 28
8.3 odd 2 392.3.h.a.293.19 28
8.5 even 2 inner 1568.3.h.a.881.26 28
28.3 even 6 56.3.j.a.5.10 yes 28
28.11 odd 6 392.3.j.e.117.10 28
28.19 even 6 392.3.j.e.325.1 28
28.23 odd 6 56.3.j.a.45.1 yes 28
28.27 even 2 392.3.h.a.293.17 28
56.3 even 6 56.3.j.a.5.1 28
56.11 odd 6 392.3.j.e.117.1 28
56.13 odd 2 inner 1568.3.h.a.881.4 28
56.19 even 6 392.3.j.e.325.10 28
56.27 even 2 392.3.h.a.293.20 28
56.37 even 6 224.3.n.a.17.2 28
56.45 odd 6 224.3.n.a.145.13 28
56.51 odd 6 56.3.j.a.45.10 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.1 28 56.3 even 6
56.3.j.a.5.10 yes 28 28.3 even 6
56.3.j.a.45.1 yes 28 28.23 odd 6
56.3.j.a.45.10 yes 28 56.51 odd 6
224.3.n.a.17.2 28 56.37 even 6
224.3.n.a.17.13 28 7.2 even 3
224.3.n.a.145.2 28 7.3 odd 6
224.3.n.a.145.13 28 56.45 odd 6
392.3.h.a.293.17 28 28.27 even 2
392.3.h.a.293.18 28 4.3 odd 2
392.3.h.a.293.19 28 8.3 odd 2
392.3.h.a.293.20 28 56.27 even 2
392.3.j.e.117.1 28 56.11 odd 6
392.3.j.e.117.10 28 28.11 odd 6
392.3.j.e.325.1 28 28.19 even 6
392.3.j.e.325.10 28 56.19 even 6
1568.3.h.a.881.3 28 1.1 even 1 trivial
1568.3.h.a.881.4 28 56.13 odd 2 inner
1568.3.h.a.881.25 28 7.6 odd 2 inner
1568.3.h.a.881.26 28 8.5 even 2 inner