Properties

Label 3920.2.a.bo.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3920,2,Mod(1,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3920.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

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: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 980)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} -1.82843 q^{11} +6.41421 q^{13} +2.41421 q^{15} +3.58579 q^{17} +7.65685 q^{19} -3.41421 q^{23} +1.00000 q^{25} +0.414214 q^{27} -4.65685 q^{29} -7.41421 q^{31} +4.41421 q^{33} -0.585786 q^{37} -15.4853 q^{39} +3.41421 q^{41} -0.343146 q^{43} -2.82843 q^{45} -10.8995 q^{47} -8.65685 q^{51} -12.2426 q^{53} +1.82843 q^{55} -18.4853 q^{57} -0.585786 q^{59} +10.8284 q^{61} -6.41421 q^{65} -3.07107 q^{67} +8.24264 q^{69} +10.4853 q^{71} +10.8284 q^{73} -2.41421 q^{75} +15.1421 q^{79} -9.48528 q^{81} +8.00000 q^{83} -3.58579 q^{85} +11.2426 q^{87} -16.9706 q^{89} +17.8995 q^{93} -7.65685 q^{95} -9.72792 q^{97} -5.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{11} + 10 q^{13} + 2 q^{15} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 12 q^{31} + 6 q^{33} - 4 q^{37} - 14 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{47} - 6 q^{51} - 16 q^{53} - 2 q^{55} - 20 q^{57} - 4 q^{59} + 16 q^{61} - 10 q^{65} + 8 q^{67} + 8 q^{69} + 4 q^{71} + 16 q^{73} - 2 q^{75} + 2 q^{79} - 2 q^{81} + 16 q^{83} - 10 q^{85} + 14 q^{87} + 16 q^{93} - 4 q^{95} + 6 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −1.82843 −0.551292 −0.275646 0.961259i \(-0.588892\pi\)
−0.275646 + 0.961259i \(0.588892\pi\)
\(12\) 0 0
\(13\) 6.41421 1.77898 0.889491 0.456952i \(-0.151059\pi\)
0.889491 + 0.456952i \(0.151059\pi\)
\(14\) 0 0
\(15\) 2.41421 0.623347
\(16\) 0 0
\(17\) 3.58579 0.869681 0.434840 0.900508i \(-0.356805\pi\)
0.434840 + 0.900508i \(0.356805\pi\)
\(18\) 0 0
\(19\) 7.65685 1.75660 0.878301 0.478107i \(-0.158677\pi\)
0.878301 + 0.478107i \(0.158677\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.41421 −0.711913 −0.355956 0.934503i \(-0.615845\pi\)
−0.355956 + 0.934503i \(0.615845\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −4.65685 −0.864756 −0.432378 0.901692i \(-0.642325\pi\)
−0.432378 + 0.901692i \(0.642325\pi\)
\(30\) 0 0
\(31\) −7.41421 −1.33163 −0.665816 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 0 0
\(33\) 4.41421 0.768416
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.585786 −0.0963027 −0.0481513 0.998840i \(-0.515333\pi\)
−0.0481513 + 0.998840i \(0.515333\pi\)
\(38\) 0 0
\(39\) −15.4853 −2.47963
\(40\) 0 0
\(41\) 3.41421 0.533211 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(42\) 0 0
\(43\) −0.343146 −0.0523292 −0.0261646 0.999658i \(-0.508329\pi\)
−0.0261646 + 0.999658i \(0.508329\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) −10.8995 −1.58985 −0.794927 0.606705i \(-0.792491\pi\)
−0.794927 + 0.606705i \(0.792491\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.65685 −1.21220
\(52\) 0 0
\(53\) −12.2426 −1.68166 −0.840828 0.541302i \(-0.817931\pi\)
−0.840828 + 0.541302i \(0.817931\pi\)
\(54\) 0 0
\(55\) 1.82843 0.246545
\(56\) 0 0
\(57\) −18.4853 −2.44844
\(58\) 0 0
\(59\) −0.585786 −0.0762629 −0.0381314 0.999273i \(-0.512141\pi\)
−0.0381314 + 0.999273i \(0.512141\pi\)
\(60\) 0 0
\(61\) 10.8284 1.38644 0.693219 0.720727i \(-0.256192\pi\)
0.693219 + 0.720727i \(0.256192\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.41421 −0.795585
\(66\) 0 0
\(67\) −3.07107 −0.375191 −0.187595 0.982246i \(-0.560069\pi\)
−0.187595 + 0.982246i \(0.560069\pi\)
\(68\) 0 0
\(69\) 8.24264 0.992297
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) 10.8284 1.26737 0.633686 0.773591i \(-0.281541\pi\)
0.633686 + 0.773591i \(0.281541\pi\)
\(74\) 0 0
\(75\) −2.41421 −0.278769
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.1421 1.70362 0.851812 0.523848i \(-0.175504\pi\)
0.851812 + 0.523848i \(0.175504\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −3.58579 −0.388933
\(86\) 0 0
\(87\) 11.2426 1.20534
\(88\) 0 0
\(89\) −16.9706 −1.79888 −0.899438 0.437048i \(-0.856024\pi\)
−0.899438 + 0.437048i \(0.856024\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 17.8995 1.85609
\(94\) 0 0
\(95\) −7.65685 −0.785577
\(96\) 0 0
\(97\) −9.72792 −0.987721 −0.493860 0.869541i \(-0.664415\pi\)
−0.493860 + 0.869541i \(0.664415\pi\)
\(98\) 0 0
\(99\) −5.17157 −0.519763
\(100\) 0 0
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 0 0
\(103\) −4.41421 −0.434945 −0.217473 0.976066i \(-0.569781\pi\)
−0.217473 + 0.976066i \(0.569781\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.1716 1.08000 0.539998 0.841666i \(-0.318425\pi\)
0.539998 + 0.841666i \(0.318425\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) 1.41421 0.134231
\(112\) 0 0
\(113\) 9.07107 0.853334 0.426667 0.904409i \(-0.359688\pi\)
0.426667 + 0.904409i \(0.359688\pi\)
\(114\) 0 0
\(115\) 3.41421 0.318377
\(116\) 0 0
\(117\) 18.1421 1.67724
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.65685 −0.696078
\(122\) 0 0
\(123\) −8.24264 −0.743214
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.10051 0.186390 0.0931948 0.995648i \(-0.470292\pi\)
0.0931948 + 0.995648i \(0.470292\pi\)
\(128\) 0 0
\(129\) 0.828427 0.0729389
\(130\) 0 0
\(131\) 20.2426 1.76861 0.884304 0.466912i \(-0.154633\pi\)
0.884304 + 0.466912i \(0.154633\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.414214 −0.0356498
\(136\) 0 0
\(137\) −20.9706 −1.79164 −0.895818 0.444421i \(-0.853409\pi\)
−0.895818 + 0.444421i \(0.853409\pi\)
\(138\) 0 0
\(139\) −5.89949 −0.500389 −0.250194 0.968196i \(-0.580495\pi\)
−0.250194 + 0.968196i \(0.580495\pi\)
\(140\) 0 0
\(141\) 26.3137 2.21601
\(142\) 0 0
\(143\) −11.7279 −0.980738
\(144\) 0 0
\(145\) 4.65685 0.386731
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 20.4853 1.67822 0.839110 0.543962i \(-0.183076\pi\)
0.839110 + 0.543962i \(0.183076\pi\)
\(150\) 0 0
\(151\) 6.17157 0.502235 0.251118 0.967957i \(-0.419202\pi\)
0.251118 + 0.967957i \(0.419202\pi\)
\(152\) 0 0
\(153\) 10.1421 0.819943
\(154\) 0 0
\(155\) 7.41421 0.595524
\(156\) 0 0
\(157\) −6.48528 −0.517582 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(158\) 0 0
\(159\) 29.5563 2.34397
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.55635 0.748511 0.374256 0.927326i \(-0.377898\pi\)
0.374256 + 0.927326i \(0.377898\pi\)
\(164\) 0 0
\(165\) −4.41421 −0.343646
\(166\) 0 0
\(167\) 0.414214 0.0320528 0.0160264 0.999872i \(-0.494898\pi\)
0.0160264 + 0.999872i \(0.494898\pi\)
\(168\) 0 0
\(169\) 28.1421 2.16478
\(170\) 0 0
\(171\) 21.6569 1.65614
\(172\) 0 0
\(173\) −12.5563 −0.954642 −0.477321 0.878729i \(-0.658392\pi\)
−0.477321 + 0.878729i \(0.658392\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.41421 0.106299
\(178\) 0 0
\(179\) 10.4853 0.783707 0.391853 0.920028i \(-0.371834\pi\)
0.391853 + 0.920028i \(0.371834\pi\)
\(180\) 0 0
\(181\) 10.2426 0.761329 0.380665 0.924713i \(-0.375695\pi\)
0.380665 + 0.924713i \(0.375695\pi\)
\(182\) 0 0
\(183\) −26.1421 −1.93248
\(184\) 0 0
\(185\) 0.585786 0.0430679
\(186\) 0 0
\(187\) −6.55635 −0.479448
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.34315 −0.386616 −0.193308 0.981138i \(-0.561922\pi\)
−0.193308 + 0.981138i \(0.561922\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) 0 0
\(195\) 15.4853 1.10892
\(196\) 0 0
\(197\) 3.55635 0.253379 0.126690 0.991942i \(-0.459565\pi\)
0.126690 + 0.991942i \(0.459565\pi\)
\(198\) 0 0
\(199\) 1.27208 0.0901752 0.0450876 0.998983i \(-0.485643\pi\)
0.0450876 + 0.998983i \(0.485643\pi\)
\(200\) 0 0
\(201\) 7.41421 0.522958
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.41421 −0.238459
\(206\) 0 0
\(207\) −9.65685 −0.671198
\(208\) 0 0
\(209\) −14.0000 −0.968400
\(210\) 0 0
\(211\) 1.68629 0.116089 0.0580445 0.998314i \(-0.481513\pi\)
0.0580445 + 0.998314i \(0.481513\pi\)
\(212\) 0 0
\(213\) −25.3137 −1.73447
\(214\) 0 0
\(215\) 0.343146 0.0234023
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −26.1421 −1.76652
\(220\) 0 0
\(221\) 23.0000 1.54715
\(222\) 0 0
\(223\) −9.92893 −0.664890 −0.332445 0.943123i \(-0.607874\pi\)
−0.332445 + 0.943123i \(0.607874\pi\)
\(224\) 0 0
\(225\) 2.82843 0.188562
\(226\) 0 0
\(227\) 17.7279 1.17664 0.588322 0.808627i \(-0.299789\pi\)
0.588322 + 0.808627i \(0.299789\pi\)
\(228\) 0 0
\(229\) 4.10051 0.270969 0.135485 0.990779i \(-0.456741\pi\)
0.135485 + 0.990779i \(0.456741\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.8284 −0.971443 −0.485721 0.874114i \(-0.661443\pi\)
−0.485721 + 0.874114i \(0.661443\pi\)
\(234\) 0 0
\(235\) 10.8995 0.711005
\(236\) 0 0
\(237\) −36.5563 −2.37459
\(238\) 0 0
\(239\) −19.8284 −1.28259 −0.641297 0.767293i \(-0.721603\pi\)
−0.641297 + 0.767293i \(0.721603\pi\)
\(240\) 0 0
\(241\) 27.5563 1.77506 0.887530 0.460749i \(-0.152419\pi\)
0.887530 + 0.460749i \(0.152419\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 49.1127 3.12497
\(248\) 0 0
\(249\) −19.3137 −1.22396
\(250\) 0 0
\(251\) 12.9289 0.816067 0.408033 0.912967i \(-0.366215\pi\)
0.408033 + 0.912967i \(0.366215\pi\)
\(252\) 0 0
\(253\) 6.24264 0.392471
\(254\) 0 0
\(255\) 8.65685 0.542113
\(256\) 0 0
\(257\) 13.7990 0.860757 0.430379 0.902648i \(-0.358380\pi\)
0.430379 + 0.902648i \(0.358380\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −13.1716 −0.815300
\(262\) 0 0
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) 0 0
\(265\) 12.2426 0.752059
\(266\) 0 0
\(267\) 40.9706 2.50736
\(268\) 0 0
\(269\) 8.72792 0.532151 0.266075 0.963952i \(-0.414273\pi\)
0.266075 + 0.963952i \(0.414273\pi\)
\(270\) 0 0
\(271\) 4.97056 0.301940 0.150970 0.988538i \(-0.451760\pi\)
0.150970 + 0.988538i \(0.451760\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.82843 −0.110258
\(276\) 0 0
\(277\) −4.58579 −0.275533 −0.137767 0.990465i \(-0.543992\pi\)
−0.137767 + 0.990465i \(0.543992\pi\)
\(278\) 0 0
\(279\) −20.9706 −1.25547
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 27.2426 1.61941 0.809703 0.586839i \(-0.199628\pi\)
0.809703 + 0.586839i \(0.199628\pi\)
\(284\) 0 0
\(285\) 18.4853 1.09497
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.14214 −0.243655
\(290\) 0 0
\(291\) 23.4853 1.37673
\(292\) 0 0
\(293\) −11.5858 −0.676849 −0.338424 0.940994i \(-0.609894\pi\)
−0.338424 + 0.940994i \(0.609894\pi\)
\(294\) 0 0
\(295\) 0.585786 0.0341058
\(296\) 0 0
\(297\) −0.757359 −0.0439464
\(298\) 0 0
\(299\) −21.8995 −1.26648
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) −10.8284 −0.620034
\(306\) 0 0
\(307\) −6.41421 −0.366079 −0.183039 0.983106i \(-0.558594\pi\)
−0.183039 + 0.983106i \(0.558594\pi\)
\(308\) 0 0
\(309\) 10.6569 0.606247
\(310\) 0 0
\(311\) 26.9706 1.52936 0.764680 0.644410i \(-0.222897\pi\)
0.764680 + 0.644410i \(0.222897\pi\)
\(312\) 0 0
\(313\) −14.5563 −0.822774 −0.411387 0.911461i \(-0.634955\pi\)
−0.411387 + 0.911461i \(0.634955\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.9706 1.06549 0.532746 0.846275i \(-0.321160\pi\)
0.532746 + 0.846275i \(0.321160\pi\)
\(318\) 0 0
\(319\) 8.51472 0.476733
\(320\) 0 0
\(321\) −26.9706 −1.50535
\(322\) 0 0
\(323\) 27.4558 1.52768
\(324\) 0 0
\(325\) 6.41421 0.355797
\(326\) 0 0
\(327\) −21.7279 −1.20156
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.828427 0.0455345 0.0227672 0.999741i \(-0.492752\pi\)
0.0227672 + 0.999741i \(0.492752\pi\)
\(332\) 0 0
\(333\) −1.65685 −0.0907951
\(334\) 0 0
\(335\) 3.07107 0.167790
\(336\) 0 0
\(337\) 6.72792 0.366493 0.183247 0.983067i \(-0.441339\pi\)
0.183247 + 0.983067i \(0.441339\pi\)
\(338\) 0 0
\(339\) −21.8995 −1.18942
\(340\) 0 0
\(341\) 13.5563 0.734117
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8.24264 −0.443769
\(346\) 0 0
\(347\) 4.58579 0.246178 0.123089 0.992396i \(-0.460720\pi\)
0.123089 + 0.992396i \(0.460720\pi\)
\(348\) 0 0
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 2.65685 0.141812
\(352\) 0 0
\(353\) 12.0711 0.642478 0.321239 0.946998i \(-0.395901\pi\)
0.321239 + 0.946998i \(0.395901\pi\)
\(354\) 0 0
\(355\) −10.4853 −0.556501
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) 39.6274 2.08565
\(362\) 0 0
\(363\) 18.4853 0.970226
\(364\) 0 0
\(365\) −10.8284 −0.566786
\(366\) 0 0
\(367\) 5.72792 0.298995 0.149498 0.988762i \(-0.452234\pi\)
0.149498 + 0.988762i \(0.452234\pi\)
\(368\) 0 0
\(369\) 9.65685 0.502716
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.485281 −0.0251269 −0.0125635 0.999921i \(-0.503999\pi\)
−0.0125635 + 0.999921i \(0.503999\pi\)
\(374\) 0 0
\(375\) 2.41421 0.124669
\(376\) 0 0
\(377\) −29.8701 −1.53839
\(378\) 0 0
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) −5.07107 −0.259799
\(382\) 0 0
\(383\) 20.4853 1.04675 0.523374 0.852103i \(-0.324673\pi\)
0.523374 + 0.852103i \(0.324673\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.970563 −0.0493365
\(388\) 0 0
\(389\) −15.1421 −0.767737 −0.383868 0.923388i \(-0.625408\pi\)
−0.383868 + 0.923388i \(0.625408\pi\)
\(390\) 0 0
\(391\) −12.2426 −0.619137
\(392\) 0 0
\(393\) −48.8701 −2.46517
\(394\) 0 0
\(395\) −15.1421 −0.761884
\(396\) 0 0
\(397\) 11.5858 0.581474 0.290737 0.956803i \(-0.406100\pi\)
0.290737 + 0.956803i \(0.406100\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.4853 0.873173 0.436587 0.899662i \(-0.356187\pi\)
0.436587 + 0.899662i \(0.356187\pi\)
\(402\) 0 0
\(403\) −47.5563 −2.36895
\(404\) 0 0
\(405\) 9.48528 0.471327
\(406\) 0 0
\(407\) 1.07107 0.0530909
\(408\) 0 0
\(409\) 27.1716 1.34355 0.671774 0.740756i \(-0.265533\pi\)
0.671774 + 0.740756i \(0.265533\pi\)
\(410\) 0 0
\(411\) 50.6274 2.49727
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 0 0
\(417\) 14.2426 0.697465
\(418\) 0 0
\(419\) −21.5563 −1.05310 −0.526548 0.850145i \(-0.676514\pi\)
−0.526548 + 0.850145i \(0.676514\pi\)
\(420\) 0 0
\(421\) −2.31371 −0.112763 −0.0563816 0.998409i \(-0.517956\pi\)
−0.0563816 + 0.998409i \(0.517956\pi\)
\(422\) 0 0
\(423\) −30.8284 −1.49893
\(424\) 0 0
\(425\) 3.58579 0.173936
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 28.3137 1.36700
\(430\) 0 0
\(431\) 4.17157 0.200938 0.100469 0.994940i \(-0.467966\pi\)
0.100469 + 0.994940i \(0.467966\pi\)
\(432\) 0 0
\(433\) 34.2843 1.64760 0.823798 0.566883i \(-0.191851\pi\)
0.823798 + 0.566883i \(0.191851\pi\)
\(434\) 0 0
\(435\) −11.2426 −0.539043
\(436\) 0 0
\(437\) −26.1421 −1.25055
\(438\) 0 0
\(439\) −15.7574 −0.752058 −0.376029 0.926608i \(-0.622711\pi\)
−0.376029 + 0.926608i \(0.622711\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.4558 −0.734329 −0.367165 0.930156i \(-0.619671\pi\)
−0.367165 + 0.930156i \(0.619671\pi\)
\(444\) 0 0
\(445\) 16.9706 0.804482
\(446\) 0 0
\(447\) −49.4558 −2.33918
\(448\) 0 0
\(449\) 18.1716 0.857570 0.428785 0.903407i \(-0.358942\pi\)
0.428785 + 0.903407i \(0.358942\pi\)
\(450\) 0 0
\(451\) −6.24264 −0.293954
\(452\) 0 0
\(453\) −14.8995 −0.700039
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.928932 −0.0434536 −0.0217268 0.999764i \(-0.506916\pi\)
−0.0217268 + 0.999764i \(0.506916\pi\)
\(458\) 0 0
\(459\) 1.48528 0.0693270
\(460\) 0 0
\(461\) 33.6569 1.56756 0.783778 0.621041i \(-0.213290\pi\)
0.783778 + 0.621041i \(0.213290\pi\)
\(462\) 0 0
\(463\) −37.4558 −1.74072 −0.870360 0.492415i \(-0.836114\pi\)
−0.870360 + 0.492415i \(0.836114\pi\)
\(464\) 0 0
\(465\) −17.8995 −0.830069
\(466\) 0 0
\(467\) 18.8995 0.874564 0.437282 0.899324i \(-0.355941\pi\)
0.437282 + 0.899324i \(0.355941\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.6569 0.721430
\(472\) 0 0
\(473\) 0.627417 0.0288487
\(474\) 0 0
\(475\) 7.65685 0.351321
\(476\) 0 0
\(477\) −34.6274 −1.58548
\(478\) 0 0
\(479\) −6.58579 −0.300912 −0.150456 0.988617i \(-0.548074\pi\)
−0.150456 + 0.988617i \(0.548074\pi\)
\(480\) 0 0
\(481\) −3.75736 −0.171321
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.72792 0.441722
\(486\) 0 0
\(487\) 4.58579 0.207802 0.103901 0.994588i \(-0.466868\pi\)
0.103901 + 0.994588i \(0.466868\pi\)
\(488\) 0 0
\(489\) −23.0711 −1.04331
\(490\) 0 0
\(491\) 39.2843 1.77287 0.886437 0.462849i \(-0.153173\pi\)
0.886437 + 0.462849i \(0.153173\pi\)
\(492\) 0 0
\(493\) −16.6985 −0.752062
\(494\) 0 0
\(495\) 5.17157 0.232445
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −31.6274 −1.41584 −0.707919 0.706294i \(-0.750366\pi\)
−0.707919 + 0.706294i \(0.750366\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 9.04163 0.403146 0.201573 0.979473i \(-0.435395\pi\)
0.201573 + 0.979473i \(0.435395\pi\)
\(504\) 0 0
\(505\) −0.828427 −0.0368645
\(506\) 0 0
\(507\) −67.9411 −3.01737
\(508\) 0 0
\(509\) −20.9289 −0.927659 −0.463829 0.885925i \(-0.653525\pi\)
−0.463829 + 0.885925i \(0.653525\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.17157 0.140028
\(514\) 0 0
\(515\) 4.41421 0.194513
\(516\) 0 0
\(517\) 19.9289 0.876473
\(518\) 0 0
\(519\) 30.3137 1.33062
\(520\) 0 0
\(521\) 29.3137 1.28426 0.642128 0.766597i \(-0.278052\pi\)
0.642128 + 0.766597i \(0.278052\pi\)
\(522\) 0 0
\(523\) −30.1421 −1.31802 −0.659012 0.752133i \(-0.729025\pi\)
−0.659012 + 0.752133i \(0.729025\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.5858 −1.15810
\(528\) 0 0
\(529\) −11.3431 −0.493180
\(530\) 0 0
\(531\) −1.65685 −0.0719014
\(532\) 0 0
\(533\) 21.8995 0.948572
\(534\) 0 0
\(535\) −11.1716 −0.482989
\(536\) 0 0
\(537\) −25.3137 −1.09237
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −17.9706 −0.772615 −0.386307 0.922370i \(-0.626250\pi\)
−0.386307 + 0.922370i \(0.626250\pi\)
\(542\) 0 0
\(543\) −24.7279 −1.06118
\(544\) 0 0
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) −24.4853 −1.04692 −0.523458 0.852052i \(-0.675358\pi\)
−0.523458 + 0.852052i \(0.675358\pi\)
\(548\) 0 0
\(549\) 30.6274 1.30715
\(550\) 0 0
\(551\) −35.6569 −1.51903
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.41421 −0.0600300
\(556\) 0 0
\(557\) 31.7990 1.34737 0.673683 0.739020i \(-0.264711\pi\)
0.673683 + 0.739020i \(0.264711\pi\)
\(558\) 0 0
\(559\) −2.20101 −0.0930928
\(560\) 0 0
\(561\) 15.8284 0.668277
\(562\) 0 0
\(563\) −24.6274 −1.03792 −0.518961 0.854798i \(-0.673681\pi\)
−0.518961 + 0.854798i \(0.673681\pi\)
\(564\) 0 0
\(565\) −9.07107 −0.381623
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.79899 0.326951 0.163475 0.986547i \(-0.447730\pi\)
0.163475 + 0.986547i \(0.447730\pi\)
\(570\) 0 0
\(571\) 0.828427 0.0346686 0.0173343 0.999850i \(-0.494482\pi\)
0.0173343 + 0.999850i \(0.494482\pi\)
\(572\) 0 0
\(573\) 12.8995 0.538884
\(574\) 0 0
\(575\) −3.41421 −0.142383
\(576\) 0 0
\(577\) 0.0710678 0.00295859 0.00147930 0.999999i \(-0.499529\pi\)
0.00147930 + 0.999999i \(0.499529\pi\)
\(578\) 0 0
\(579\) 13.6569 0.567559
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 22.3848 0.927083
\(584\) 0 0
\(585\) −18.1421 −0.750085
\(586\) 0 0
\(587\) 25.1716 1.03894 0.519471 0.854488i \(-0.326129\pi\)
0.519471 + 0.854488i \(0.326129\pi\)
\(588\) 0 0
\(589\) −56.7696 −2.33915
\(590\) 0 0
\(591\) −8.58579 −0.353172
\(592\) 0 0
\(593\) −7.72792 −0.317348 −0.158674 0.987331i \(-0.550722\pi\)
−0.158674 + 0.987331i \(0.550722\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.07107 −0.125690
\(598\) 0 0
\(599\) −26.7990 −1.09498 −0.547489 0.836813i \(-0.684416\pi\)
−0.547489 + 0.836813i \(0.684416\pi\)
\(600\) 0 0
\(601\) 28.3431 1.15614 0.578071 0.815987i \(-0.303806\pi\)
0.578071 + 0.815987i \(0.303806\pi\)
\(602\) 0 0
\(603\) −8.68629 −0.353733
\(604\) 0 0
\(605\) 7.65685 0.311295
\(606\) 0 0
\(607\) −28.0711 −1.13937 −0.569685 0.821863i \(-0.692935\pi\)
−0.569685 + 0.821863i \(0.692935\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −69.9117 −2.82832
\(612\) 0 0
\(613\) −18.6863 −0.754732 −0.377366 0.926064i \(-0.623170\pi\)
−0.377366 + 0.926064i \(0.623170\pi\)
\(614\) 0 0
\(615\) 8.24264 0.332375
\(616\) 0 0
\(617\) −12.8701 −0.518129 −0.259065 0.965860i \(-0.583414\pi\)
−0.259065 + 0.965860i \(0.583414\pi\)
\(618\) 0 0
\(619\) 42.8701 1.72309 0.861547 0.507679i \(-0.169496\pi\)
0.861547 + 0.507679i \(0.169496\pi\)
\(620\) 0 0
\(621\) −1.41421 −0.0567504
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 33.7990 1.34980
\(628\) 0 0
\(629\) −2.10051 −0.0837526
\(630\) 0 0
\(631\) 22.5147 0.896297 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(632\) 0 0
\(633\) −4.07107 −0.161810
\(634\) 0 0
\(635\) −2.10051 −0.0833560
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 29.6569 1.17321
\(640\) 0 0
\(641\) 16.6863 0.659069 0.329534 0.944144i \(-0.393108\pi\)
0.329534 + 0.944144i \(0.393108\pi\)
\(642\) 0 0
\(643\) 38.2132 1.50698 0.753491 0.657458i \(-0.228368\pi\)
0.753491 + 0.657458i \(0.228368\pi\)
\(644\) 0 0
\(645\) −0.828427 −0.0326193
\(646\) 0 0
\(647\) 27.1127 1.06591 0.532955 0.846144i \(-0.321081\pi\)
0.532955 + 0.846144i \(0.321081\pi\)
\(648\) 0 0
\(649\) 1.07107 0.0420431
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −49.3137 −1.92979 −0.964897 0.262628i \(-0.915411\pi\)
−0.964897 + 0.262628i \(0.915411\pi\)
\(654\) 0 0
\(655\) −20.2426 −0.790945
\(656\) 0 0
\(657\) 30.6274 1.19489
\(658\) 0 0
\(659\) 9.34315 0.363957 0.181979 0.983302i \(-0.441750\pi\)
0.181979 + 0.983302i \(0.441750\pi\)
\(660\) 0 0
\(661\) −5.85786 −0.227845 −0.113922 0.993490i \(-0.536341\pi\)
−0.113922 + 0.993490i \(0.536341\pi\)
\(662\) 0 0
\(663\) −55.5269 −2.15649
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15.8995 0.615631
\(668\) 0 0
\(669\) 23.9706 0.926755
\(670\) 0 0
\(671\) −19.7990 −0.764332
\(672\) 0 0
\(673\) −44.4853 −1.71478 −0.857391 0.514666i \(-0.827916\pi\)
−0.857391 + 0.514666i \(0.827916\pi\)
\(674\) 0 0
\(675\) 0.414214 0.0159431
\(676\) 0 0
\(677\) −35.7279 −1.37314 −0.686568 0.727066i \(-0.740884\pi\)
−0.686568 + 0.727066i \(0.740884\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −42.7990 −1.64006
\(682\) 0 0
\(683\) 27.1127 1.03744 0.518719 0.854945i \(-0.326409\pi\)
0.518719 + 0.854945i \(0.326409\pi\)
\(684\) 0 0
\(685\) 20.9706 0.801244
\(686\) 0 0
\(687\) −9.89949 −0.377689
\(688\) 0 0
\(689\) −78.5269 −2.99164
\(690\) 0 0
\(691\) −20.4853 −0.779297 −0.389648 0.920964i \(-0.627403\pi\)
−0.389648 + 0.920964i \(0.627403\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.89949 0.223781
\(696\) 0 0
\(697\) 12.2426 0.463723
\(698\) 0 0
\(699\) 35.7990 1.35404
\(700\) 0 0
\(701\) 17.4853 0.660410 0.330205 0.943909i \(-0.392882\pi\)
0.330205 + 0.943909i \(0.392882\pi\)
\(702\) 0 0
\(703\) −4.48528 −0.169166
\(704\) 0 0
\(705\) −26.3137 −0.991032
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −41.6274 −1.56335 −0.781675 0.623686i \(-0.785635\pi\)
−0.781675 + 0.623686i \(0.785635\pi\)
\(710\) 0 0
\(711\) 42.8284 1.60619
\(712\) 0 0
\(713\) 25.3137 0.948006
\(714\) 0 0
\(715\) 11.7279 0.438599
\(716\) 0 0
\(717\) 47.8701 1.78774
\(718\) 0 0
\(719\) −24.2426 −0.904098 −0.452049 0.891993i \(-0.649307\pi\)
−0.452049 + 0.891993i \(0.649307\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −66.5269 −2.47416
\(724\) 0 0
\(725\) −4.65685 −0.172951
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −1.23045 −0.0455097
\(732\) 0 0
\(733\) 28.6985 1.06000 0.530001 0.847997i \(-0.322191\pi\)
0.530001 + 0.847997i \(0.322191\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.61522 0.206839
\(738\) 0 0
\(739\) −9.34315 −0.343693 −0.171847 0.985124i \(-0.554973\pi\)
−0.171847 + 0.985124i \(0.554973\pi\)
\(740\) 0 0
\(741\) −118.569 −4.35572
\(742\) 0 0
\(743\) 23.0711 0.846395 0.423198 0.906037i \(-0.360908\pi\)
0.423198 + 0.906037i \(0.360908\pi\)
\(744\) 0 0
\(745\) −20.4853 −0.750523
\(746\) 0 0
\(747\) 22.6274 0.827894
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 27.1421 0.990431 0.495215 0.868770i \(-0.335089\pi\)
0.495215 + 0.868770i \(0.335089\pi\)
\(752\) 0 0
\(753\) −31.2132 −1.13747
\(754\) 0 0
\(755\) −6.17157 −0.224607
\(756\) 0 0
\(757\) 42.8284 1.55663 0.778313 0.627877i \(-0.216076\pi\)
0.778313 + 0.627877i \(0.216076\pi\)
\(758\) 0 0
\(759\) −15.0711 −0.547045
\(760\) 0 0
\(761\) 34.8701 1.26404 0.632019 0.774953i \(-0.282226\pi\)
0.632019 + 0.774953i \(0.282226\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −10.1421 −0.366690
\(766\) 0 0
\(767\) −3.75736 −0.135670
\(768\) 0 0
\(769\) −0.142136 −0.00512554 −0.00256277 0.999997i \(-0.500816\pi\)
−0.00256277 + 0.999997i \(0.500816\pi\)
\(770\) 0 0
\(771\) −33.3137 −1.19976
\(772\) 0 0
\(773\) −18.2721 −0.657201 −0.328600 0.944469i \(-0.606577\pi\)
−0.328600 + 0.944469i \(0.606577\pi\)
\(774\) 0 0
\(775\) −7.41421 −0.266326
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.1421 0.936639
\(780\) 0 0
\(781\) −19.1716 −0.686013
\(782\) 0 0
\(783\) −1.92893 −0.0689344
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) 18.2721 0.651329 0.325665 0.945485i \(-0.394412\pi\)
0.325665 + 0.945485i \(0.394412\pi\)
\(788\) 0 0
\(789\) 24.1421 0.859483
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 69.4558 2.46645
\(794\) 0 0
\(795\) −29.5563 −1.04826
\(796\) 0 0
\(797\) 1.38478 0.0490513 0.0245256 0.999699i \(-0.492192\pi\)
0.0245256 + 0.999699i \(0.492192\pi\)
\(798\) 0 0
\(799\) −39.0833 −1.38267
\(800\) 0 0
\(801\) −48.0000 −1.69600
\(802\) 0 0
\(803\) −19.7990 −0.698691
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −21.0711 −0.741737
\(808\) 0 0
\(809\) 39.9706 1.40529 0.702645 0.711541i \(-0.252002\pi\)
0.702645 + 0.711541i \(0.252002\pi\)
\(810\) 0 0
\(811\) −11.5563 −0.405798 −0.202899 0.979200i \(-0.565036\pi\)
−0.202899 + 0.979200i \(0.565036\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) −9.55635 −0.334744
\(816\) 0 0
\(817\) −2.62742 −0.0919217
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.1716 1.47180 0.735899 0.677091i \(-0.236760\pi\)
0.735899 + 0.677091i \(0.236760\pi\)
\(822\) 0 0
\(823\) 28.5269 0.994386 0.497193 0.867640i \(-0.334364\pi\)
0.497193 + 0.867640i \(0.334364\pi\)
\(824\) 0 0
\(825\) 4.41421 0.153683
\(826\) 0 0
\(827\) −38.0416 −1.32284 −0.661419 0.750017i \(-0.730045\pi\)
−0.661419 + 0.750017i \(0.730045\pi\)
\(828\) 0 0
\(829\) −2.72792 −0.0947446 −0.0473723 0.998877i \(-0.515085\pi\)
−0.0473723 + 0.998877i \(0.515085\pi\)
\(830\) 0 0
\(831\) 11.0711 0.384051
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.414214 −0.0143345
\(836\) 0 0
\(837\) −3.07107 −0.106152
\(838\) 0 0
\(839\) 14.3848 0.496618 0.248309 0.968681i \(-0.420125\pi\)
0.248309 + 0.968681i \(0.420125\pi\)
\(840\) 0 0
\(841\) −7.31371 −0.252197
\(842\) 0 0
\(843\) 2.41421 0.0831499
\(844\) 0 0
\(845\) −28.1421 −0.968119
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −65.7696 −2.25721
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 26.2843 0.899956 0.449978 0.893040i \(-0.351432\pi\)
0.449978 + 0.893040i \(0.351432\pi\)
\(854\) 0 0
\(855\) −21.6569 −0.740649
\(856\) 0 0
\(857\) −15.1716 −0.518251 −0.259126 0.965844i \(-0.583434\pi\)
−0.259126 + 0.965844i \(0.583434\pi\)
\(858\) 0 0
\(859\) −5.45584 −0.186151 −0.0930755 0.995659i \(-0.529670\pi\)
−0.0930755 + 0.995659i \(0.529670\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.2426 0.961391 0.480695 0.876888i \(-0.340384\pi\)
0.480695 + 0.876888i \(0.340384\pi\)
\(864\) 0 0
\(865\) 12.5563 0.426929
\(866\) 0 0
\(867\) 10.0000 0.339618
\(868\) 0 0
\(869\) −27.6863 −0.939193
\(870\) 0 0
\(871\) −19.6985 −0.667458
\(872\) 0 0
\(873\) −27.5147 −0.931232
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 33.3137 1.12492 0.562462 0.826823i \(-0.309854\pi\)
0.562462 + 0.826823i \(0.309854\pi\)
\(878\) 0 0
\(879\) 27.9706 0.943424
\(880\) 0 0
\(881\) 0.284271 0.00957734 0.00478867 0.999989i \(-0.498476\pi\)
0.00478867 + 0.999989i \(0.498476\pi\)
\(882\) 0 0
\(883\) −55.1127 −1.85469 −0.927345 0.374208i \(-0.877915\pi\)
−0.927345 + 0.374208i \(0.877915\pi\)
\(884\) 0 0
\(885\) −1.41421 −0.0475383
\(886\) 0 0
\(887\) 21.5980 0.725189 0.362595 0.931947i \(-0.381891\pi\)
0.362595 + 0.931947i \(0.381891\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 17.3431 0.581017
\(892\) 0 0
\(893\) −83.4558 −2.79274
\(894\) 0 0
\(895\) −10.4853 −0.350484
\(896\) 0 0
\(897\) 52.8701 1.76528
\(898\) 0 0
\(899\) 34.5269 1.15154
\(900\) 0 0
\(901\) −43.8995 −1.46250
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.2426 −0.340477
\(906\) 0 0
\(907\) 42.5269 1.41208 0.706041 0.708170i \(-0.250479\pi\)
0.706041 + 0.708170i \(0.250479\pi\)
\(908\) 0 0
\(909\) 2.34315 0.0777172
\(910\) 0 0
\(911\) −23.5980 −0.781836 −0.390918 0.920426i \(-0.627842\pi\)
−0.390918 + 0.920426i \(0.627842\pi\)
\(912\) 0 0
\(913\) −14.6274 −0.484097
\(914\) 0 0
\(915\) 26.1421 0.864232
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 35.4853 1.17055 0.585276 0.810834i \(-0.300986\pi\)
0.585276 + 0.810834i \(0.300986\pi\)
\(920\) 0 0
\(921\) 15.4853 0.510257
\(922\) 0 0
\(923\) 67.2548 2.21372
\(924\) 0 0
\(925\) −0.585786 −0.0192605
\(926\) 0 0
\(927\) −12.4853 −0.410070
\(928\) 0 0
\(929\) −46.7279 −1.53309 −0.766547 0.642189i \(-0.778027\pi\)
−0.766547 + 0.642189i \(0.778027\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −65.1127 −2.13169
\(934\) 0 0
\(935\) 6.55635 0.214416
\(936\) 0 0
\(937\) 17.7279 0.579146 0.289573 0.957156i \(-0.406487\pi\)
0.289573 + 0.957156i \(0.406487\pi\)
\(938\) 0 0
\(939\) 35.1421 1.14682
\(940\) 0 0
\(941\) −4.00000 −0.130396 −0.0651981 0.997872i \(-0.520768\pi\)
−0.0651981 + 0.997872i \(0.520768\pi\)
\(942\) 0 0
\(943\) −11.6569 −0.379599
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.0416 −0.521283 −0.260641 0.965436i \(-0.583934\pi\)
−0.260641 + 0.965436i \(0.583934\pi\)
\(948\) 0 0
\(949\) 69.4558 2.25463
\(950\) 0 0
\(951\) −45.7990 −1.48513
\(952\) 0 0
\(953\) 14.8701 0.481688 0.240844 0.970564i \(-0.422576\pi\)
0.240844 + 0.970564i \(0.422576\pi\)
\(954\) 0 0
\(955\) 5.34315 0.172900
\(956\) 0 0
\(957\) −20.5563 −0.664492
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 0 0
\(963\) 31.5980 1.01823
\(964\) 0 0
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) −43.4558 −1.39745 −0.698723 0.715392i \(-0.746248\pi\)
−0.698723 + 0.715392i \(0.746248\pi\)
\(968\) 0 0
\(969\) −66.2843 −2.12936
\(970\) 0 0
\(971\) −42.5269 −1.36475 −0.682377 0.731001i \(-0.739054\pi\)
−0.682377 + 0.731001i \(0.739054\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −15.4853 −0.495926
\(976\) 0 0
\(977\) 20.1421 0.644404 0.322202 0.946671i \(-0.395577\pi\)
0.322202 + 0.946671i \(0.395577\pi\)
\(978\) 0 0
\(979\) 31.0294 0.991705
\(980\) 0 0
\(981\) 25.4558 0.812743
\(982\) 0 0
\(983\) −40.0122 −1.27619 −0.638095 0.769957i \(-0.720277\pi\)
−0.638095 + 0.769957i \(0.720277\pi\)
\(984\) 0 0
\(985\) −3.55635 −0.113315
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.17157 0.0372539
\(990\) 0 0
\(991\) −2.68629 −0.0853329 −0.0426664 0.999089i \(-0.513585\pi\)
−0.0426664 + 0.999089i \(0.513585\pi\)
\(992\) 0 0
\(993\) −2.00000 −0.0634681
\(994\) 0 0
\(995\) −1.27208 −0.0403276
\(996\) 0 0
\(997\) 36.8406 1.16675 0.583377 0.812201i \(-0.301731\pi\)
0.583377 + 0.812201i \(0.301731\pi\)
\(998\) 0 0
\(999\) −0.242641 −0.00767681
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 3920.2.a.bo.1.1 2
4.3 odd 2 980.2.a.k.1.2 yes 2
7.6 odd 2 3920.2.a.bx.1.2 2
12.11 even 2 8820.2.a.bl.1.1 2
20.3 even 4 4900.2.e.r.2549.4 4
20.7 even 4 4900.2.e.r.2549.1 4
20.19 odd 2 4900.2.a.x.1.1 2
28.3 even 6 980.2.i.l.961.2 4
28.11 odd 6 980.2.i.k.961.1 4
28.19 even 6 980.2.i.l.361.2 4
28.23 odd 6 980.2.i.k.361.1 4
28.27 even 2 980.2.a.j.1.1 2
84.83 odd 2 8820.2.a.bg.1.1 2
140.27 odd 4 4900.2.e.q.2549.4 4
140.83 odd 4 4900.2.e.q.2549.1 4
140.139 even 2 4900.2.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.j.1.1 2 28.27 even 2
980.2.a.k.1.2 yes 2 4.3 odd 2
980.2.i.k.361.1 4 28.23 odd 6
980.2.i.k.961.1 4 28.11 odd 6
980.2.i.l.361.2 4 28.19 even 6
980.2.i.l.961.2 4 28.3 even 6
3920.2.a.bo.1.1 2 1.1 even 1 trivial
3920.2.a.bx.1.2 2 7.6 odd 2
4900.2.a.x.1.1 2 20.19 odd 2
4900.2.a.z.1.2 2 140.139 even 2
4900.2.e.q.2549.1 4 140.83 odd 4
4900.2.e.q.2549.4 4 140.27 odd 4
4900.2.e.r.2549.1 4 20.7 even 4
4900.2.e.r.2549.4 4 20.3 even 4
8820.2.a.bg.1.1 2 84.83 odd 2
8820.2.a.bl.1.1 2 12.11 even 2