Properties

Label 5733.2.a.ca.1.7
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5733,2,Mod(1,5733)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5733.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.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: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} + 92x^{8} - 228x^{6} + 225x^{4} - 60x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.318530\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} -1.24180 q^{8} +O(q^{10})\) \(q+0.318530 q^{2} -1.89854 q^{4} -0.278702 q^{5} -1.24180 q^{8} -0.0887749 q^{10} +4.26805 q^{11} -1.00000 q^{13} +3.40153 q^{16} -6.27885 q^{17} +0.935311 q^{19} +0.529127 q^{20} +1.35950 q^{22} -0.554198 q^{23} -4.92233 q^{25} -0.318530 q^{26} +4.49614 q^{29} +0.861767 q^{31} +3.56709 q^{32} -2.00000 q^{34} +4.40123 q^{37} +0.297924 q^{38} +0.346092 q^{40} +0.979596 q^{41} -1.71659 q^{43} -8.10306 q^{44} -0.176529 q^{46} -4.42810 q^{47} -1.56791 q^{50} +1.89854 q^{52} -9.30603 q^{53} -1.18951 q^{55} +1.43215 q^{58} +12.5106 q^{59} +1.97463 q^{61} +0.274498 q^{62} -5.66683 q^{64} +0.278702 q^{65} -3.97361 q^{67} +11.9206 q^{68} +2.43794 q^{71} +4.83639 q^{73} +1.40192 q^{74} -1.77572 q^{76} +12.5478 q^{79} -0.948013 q^{80} +0.312030 q^{82} -11.1808 q^{83} +1.74993 q^{85} -0.546784 q^{86} -5.30007 q^{88} -12.1828 q^{89} +1.05217 q^{92} -1.41048 q^{94} -0.260673 q^{95} -12.9099 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 8 q^{10} - 12 q^{13} - 28 q^{19} + 16 q^{25} - 12 q^{31} - 24 q^{34} - 32 q^{40} - 20 q^{43} - 8 q^{46} - 8 q^{52} - 56 q^{55} + 8 q^{58} - 24 q^{61} - 40 q^{64} + 8 q^{67} - 12 q^{73} - 64 q^{76} - 20 q^{79} - 40 q^{82} + 32 q^{85} + 8 q^{88} - 8 q^{94} - 68 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.318530 0.225235 0.112617 0.993638i \(-0.464077\pi\)
0.112617 + 0.993638i \(0.464077\pi\)
\(3\) 0 0
\(4\) −1.89854 −0.949269
\(5\) −0.278702 −0.124639 −0.0623197 0.998056i \(-0.519850\pi\)
−0.0623197 + 0.998056i \(0.519850\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.24180 −0.439043
\(9\) 0 0
\(10\) −0.0887749 −0.0280731
\(11\) 4.26805 1.28687 0.643433 0.765503i \(-0.277510\pi\)
0.643433 + 0.765503i \(0.277510\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 3.40153 0.850382
\(17\) −6.27885 −1.52284 −0.761422 0.648256i \(-0.775499\pi\)
−0.761422 + 0.648256i \(0.775499\pi\)
\(18\) 0 0
\(19\) 0.935311 0.214575 0.107287 0.994228i \(-0.465783\pi\)
0.107287 + 0.994228i \(0.465783\pi\)
\(20\) 0.529127 0.118316
\(21\) 0 0
\(22\) 1.35950 0.289847
\(23\) −0.554198 −0.115558 −0.0577792 0.998329i \(-0.518402\pi\)
−0.0577792 + 0.998329i \(0.518402\pi\)
\(24\) 0 0
\(25\) −4.92233 −0.984465
\(26\) −0.318530 −0.0624688
\(27\) 0 0
\(28\) 0 0
\(29\) 4.49614 0.834913 0.417456 0.908697i \(-0.362922\pi\)
0.417456 + 0.908697i \(0.362922\pi\)
\(30\) 0 0
\(31\) 0.861767 0.154778 0.0773890 0.997001i \(-0.475342\pi\)
0.0773890 + 0.997001i \(0.475342\pi\)
\(32\) 3.56709 0.630578
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 4.40123 0.723557 0.361779 0.932264i \(-0.382170\pi\)
0.361779 + 0.932264i \(0.382170\pi\)
\(38\) 0.297924 0.0483297
\(39\) 0 0
\(40\) 0.346092 0.0547220
\(41\) 0.979596 0.152987 0.0764936 0.997070i \(-0.475628\pi\)
0.0764936 + 0.997070i \(0.475628\pi\)
\(42\) 0 0
\(43\) −1.71659 −0.261777 −0.130889 0.991397i \(-0.541783\pi\)
−0.130889 + 0.991397i \(0.541783\pi\)
\(44\) −8.10306 −1.22158
\(45\) 0 0
\(46\) −0.176529 −0.0260277
\(47\) −4.42810 −0.645905 −0.322953 0.946415i \(-0.604675\pi\)
−0.322953 + 0.946415i \(0.604675\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.56791 −0.221736
\(51\) 0 0
\(52\) 1.89854 0.263280
\(53\) −9.30603 −1.27828 −0.639141 0.769090i \(-0.720710\pi\)
−0.639141 + 0.769090i \(0.720710\pi\)
\(54\) 0 0
\(55\) −1.18951 −0.160394
\(56\) 0 0
\(57\) 0 0
\(58\) 1.43215 0.188051
\(59\) 12.5106 1.62874 0.814372 0.580343i \(-0.197081\pi\)
0.814372 + 0.580343i \(0.197081\pi\)
\(60\) 0 0
\(61\) 1.97463 0.252825 0.126413 0.991978i \(-0.459654\pi\)
0.126413 + 0.991978i \(0.459654\pi\)
\(62\) 0.274498 0.0348613
\(63\) 0 0
\(64\) −5.66683 −0.708354
\(65\) 0.278702 0.0345687
\(66\) 0 0
\(67\) −3.97361 −0.485453 −0.242727 0.970095i \(-0.578042\pi\)
−0.242727 + 0.970095i \(0.578042\pi\)
\(68\) 11.9206 1.44559
\(69\) 0 0
\(70\) 0 0
\(71\) 2.43794 0.289330 0.144665 0.989481i \(-0.453790\pi\)
0.144665 + 0.989481i \(0.453790\pi\)
\(72\) 0 0
\(73\) 4.83639 0.566057 0.283029 0.959111i \(-0.408661\pi\)
0.283029 + 0.959111i \(0.408661\pi\)
\(74\) 1.40192 0.162970
\(75\) 0 0
\(76\) −1.77572 −0.203689
\(77\) 0 0
\(78\) 0 0
\(79\) 12.5478 1.41174 0.705871 0.708341i \(-0.250556\pi\)
0.705871 + 0.708341i \(0.250556\pi\)
\(80\) −0.948013 −0.105991
\(81\) 0 0
\(82\) 0.312030 0.0344580
\(83\) −11.1808 −1.22725 −0.613623 0.789599i \(-0.710289\pi\)
−0.613623 + 0.789599i \(0.710289\pi\)
\(84\) 0 0
\(85\) 1.74993 0.189806
\(86\) −0.546784 −0.0589613
\(87\) 0 0
\(88\) −5.30007 −0.564989
\(89\) −12.1828 −1.29137 −0.645686 0.763603i \(-0.723428\pi\)
−0.645686 + 0.763603i \(0.723428\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.05217 0.109696
\(93\) 0 0
\(94\) −1.41048 −0.145480
\(95\) −0.260673 −0.0267445
\(96\) 0 0
\(97\) −12.9099 −1.31081 −0.655403 0.755280i \(-0.727501\pi\)
−0.655403 + 0.755280i \(0.727501\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 9.34523 0.934523
\(101\) −5.10341 −0.507809 −0.253904 0.967229i \(-0.581715\pi\)
−0.253904 + 0.967229i \(0.581715\pi\)
\(102\) 0 0
\(103\) −0.361087 −0.0355789 −0.0177895 0.999842i \(-0.505663\pi\)
−0.0177895 + 0.999842i \(0.505663\pi\)
\(104\) 1.24180 0.121769
\(105\) 0 0
\(106\) −2.96425 −0.287913
\(107\) 13.2663 1.28250 0.641252 0.767330i \(-0.278415\pi\)
0.641252 + 0.767330i \(0.278415\pi\)
\(108\) 0 0
\(109\) −13.7826 −1.32014 −0.660069 0.751205i \(-0.729473\pi\)
−0.660069 + 0.751205i \(0.729473\pi\)
\(110\) −0.378896 −0.0361263
\(111\) 0 0
\(112\) 0 0
\(113\) 2.99313 0.281570 0.140785 0.990040i \(-0.455037\pi\)
0.140785 + 0.990040i \(0.455037\pi\)
\(114\) 0 0
\(115\) 0.154456 0.0144031
\(116\) −8.53610 −0.792557
\(117\) 0 0
\(118\) 3.98501 0.366849
\(119\) 0 0
\(120\) 0 0
\(121\) 7.21625 0.656022
\(122\) 0.628978 0.0569449
\(123\) 0 0
\(124\) −1.63610 −0.146926
\(125\) 2.76537 0.247342
\(126\) 0 0
\(127\) −6.71157 −0.595556 −0.297778 0.954635i \(-0.596245\pi\)
−0.297778 + 0.954635i \(0.596245\pi\)
\(128\) −8.93923 −0.790124
\(129\) 0 0
\(130\) 0.0887749 0.00778607
\(131\) 9.52731 0.832405 0.416203 0.909272i \(-0.363361\pi\)
0.416203 + 0.909272i \(0.363361\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.26571 −0.109341
\(135\) 0 0
\(136\) 7.79708 0.668594
\(137\) −21.1133 −1.80383 −0.901915 0.431913i \(-0.857839\pi\)
−0.901915 + 0.431913i \(0.857839\pi\)
\(138\) 0 0
\(139\) −14.6811 −1.24524 −0.622618 0.782526i \(-0.713931\pi\)
−0.622618 + 0.782526i \(0.713931\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.776556 0.0651671
\(143\) −4.26805 −0.356912
\(144\) 0 0
\(145\) −1.25308 −0.104063
\(146\) 1.54054 0.127496
\(147\) 0 0
\(148\) −8.35590 −0.686851
\(149\) 13.5524 1.11026 0.555130 0.831764i \(-0.312669\pi\)
0.555130 + 0.831764i \(0.312669\pi\)
\(150\) 0 0
\(151\) −8.52294 −0.693587 −0.346794 0.937941i \(-0.612730\pi\)
−0.346794 + 0.937941i \(0.612730\pi\)
\(152\) −1.16147 −0.0942076
\(153\) 0 0
\(154\) 0 0
\(155\) −0.240176 −0.0192914
\(156\) 0 0
\(157\) −13.9303 −1.11176 −0.555879 0.831263i \(-0.687618\pi\)
−0.555879 + 0.831263i \(0.687618\pi\)
\(158\) 3.99686 0.317973
\(159\) 0 0
\(160\) −0.994155 −0.0785948
\(161\) 0 0
\(162\) 0 0
\(163\) 17.6518 1.38260 0.691298 0.722569i \(-0.257039\pi\)
0.691298 + 0.722569i \(0.257039\pi\)
\(164\) −1.85980 −0.145226
\(165\) 0 0
\(166\) −3.56140 −0.276418
\(167\) −7.45254 −0.576695 −0.288348 0.957526i \(-0.593106\pi\)
−0.288348 + 0.957526i \(0.593106\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0.557404 0.0427509
\(171\) 0 0
\(172\) 3.25901 0.248497
\(173\) 19.1846 1.45858 0.729290 0.684205i \(-0.239851\pi\)
0.729290 + 0.684205i \(0.239851\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.5179 1.09433
\(177\) 0 0
\(178\) −3.88058 −0.290861
\(179\) 19.6322 1.46738 0.733692 0.679483i \(-0.237796\pi\)
0.733692 + 0.679483i \(0.237796\pi\)
\(180\) 0 0
\(181\) −13.5654 −1.00831 −0.504153 0.863614i \(-0.668195\pi\)
−0.504153 + 0.863614i \(0.668195\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.688204 0.0507351
\(185\) −1.22663 −0.0901837
\(186\) 0 0
\(187\) −26.7984 −1.95970
\(188\) 8.40693 0.613138
\(189\) 0 0
\(190\) −0.0830321 −0.00602378
\(191\) 2.86026 0.206961 0.103481 0.994631i \(-0.467002\pi\)
0.103481 + 0.994631i \(0.467002\pi\)
\(192\) 0 0
\(193\) 0.318196 0.0229043 0.0114521 0.999934i \(-0.496355\pi\)
0.0114521 + 0.999934i \(0.496355\pi\)
\(194\) −4.11220 −0.295239
\(195\) 0 0
\(196\) 0 0
\(197\) −2.15763 −0.153725 −0.0768624 0.997042i \(-0.524490\pi\)
−0.0768624 + 0.997042i \(0.524490\pi\)
\(198\) 0 0
\(199\) −13.7721 −0.976280 −0.488140 0.872765i \(-0.662325\pi\)
−0.488140 + 0.872765i \(0.662325\pi\)
\(200\) 6.11255 0.432222
\(201\) 0 0
\(202\) −1.62559 −0.114376
\(203\) 0 0
\(204\) 0 0
\(205\) −0.273015 −0.0190682
\(206\) −0.115017 −0.00801361
\(207\) 0 0
\(208\) −3.40153 −0.235853
\(209\) 3.99195 0.276129
\(210\) 0 0
\(211\) −5.05624 −0.348086 −0.174043 0.984738i \(-0.555683\pi\)
−0.174043 + 0.984738i \(0.555683\pi\)
\(212\) 17.6679 1.21343
\(213\) 0 0
\(214\) 4.22572 0.288864
\(215\) 0.478417 0.0326277
\(216\) 0 0
\(217\) 0 0
\(218\) −4.39018 −0.297340
\(219\) 0 0
\(220\) 2.25834 0.152257
\(221\) 6.27885 0.422361
\(222\) 0 0
\(223\) −23.9620 −1.60462 −0.802309 0.596910i \(-0.796395\pi\)
−0.802309 + 0.596910i \(0.796395\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.953402 0.0634194
\(227\) −18.8419 −1.25058 −0.625292 0.780391i \(-0.715020\pi\)
−0.625292 + 0.780391i \(0.715020\pi\)
\(228\) 0 0
\(229\) −18.9695 −1.25354 −0.626768 0.779206i \(-0.715623\pi\)
−0.626768 + 0.779206i \(0.715623\pi\)
\(230\) 0.0491989 0.00324408
\(231\) 0 0
\(232\) −5.58331 −0.366562
\(233\) −27.3238 −1.79004 −0.895022 0.446022i \(-0.852840\pi\)
−0.895022 + 0.446022i \(0.852840\pi\)
\(234\) 0 0
\(235\) 1.23412 0.0805052
\(236\) −23.7519 −1.54612
\(237\) 0 0
\(238\) 0 0
\(239\) −23.8749 −1.54434 −0.772170 0.635416i \(-0.780829\pi\)
−0.772170 + 0.635416i \(0.780829\pi\)
\(240\) 0 0
\(241\) −16.3941 −1.05603 −0.528017 0.849234i \(-0.677064\pi\)
−0.528017 + 0.849234i \(0.677064\pi\)
\(242\) 2.29859 0.147759
\(243\) 0 0
\(244\) −3.74891 −0.239999
\(245\) 0 0
\(246\) 0 0
\(247\) −0.935311 −0.0595124
\(248\) −1.07014 −0.0679541
\(249\) 0 0
\(250\) 0.880853 0.0557100
\(251\) 7.88947 0.497979 0.248989 0.968506i \(-0.419902\pi\)
0.248989 + 0.968506i \(0.419902\pi\)
\(252\) 0 0
\(253\) −2.36535 −0.148708
\(254\) −2.13783 −0.134140
\(255\) 0 0
\(256\) 8.48625 0.530391
\(257\) −21.0559 −1.31343 −0.656715 0.754139i \(-0.728054\pi\)
−0.656715 + 0.754139i \(0.728054\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.529127 −0.0328150
\(261\) 0 0
\(262\) 3.03473 0.187486
\(263\) −9.45167 −0.582815 −0.291408 0.956599i \(-0.594124\pi\)
−0.291408 + 0.956599i \(0.594124\pi\)
\(264\) 0 0
\(265\) 2.59361 0.159324
\(266\) 0 0
\(267\) 0 0
\(268\) 7.54405 0.460826
\(269\) −13.4282 −0.818731 −0.409365 0.912371i \(-0.634250\pi\)
−0.409365 + 0.912371i \(0.634250\pi\)
\(270\) 0 0
\(271\) −8.94268 −0.543229 −0.271615 0.962406i \(-0.587558\pi\)
−0.271615 + 0.962406i \(0.587558\pi\)
\(272\) −21.3577 −1.29500
\(273\) 0 0
\(274\) −6.72521 −0.406285
\(275\) −21.0087 −1.26687
\(276\) 0 0
\(277\) −6.08015 −0.365321 −0.182661 0.983176i \(-0.558471\pi\)
−0.182661 + 0.983176i \(0.558471\pi\)
\(278\) −4.67637 −0.280470
\(279\) 0 0
\(280\) 0 0
\(281\) 16.7806 1.00105 0.500523 0.865723i \(-0.333141\pi\)
0.500523 + 0.865723i \(0.333141\pi\)
\(282\) 0 0
\(283\) 4.49011 0.266909 0.133455 0.991055i \(-0.457393\pi\)
0.133455 + 0.991055i \(0.457393\pi\)
\(284\) −4.62852 −0.274652
\(285\) 0 0
\(286\) −1.35950 −0.0803890
\(287\) 0 0
\(288\) 0 0
\(289\) 22.4239 1.31906
\(290\) −0.399144 −0.0234386
\(291\) 0 0
\(292\) −9.18208 −0.537341
\(293\) −7.74333 −0.452370 −0.226185 0.974084i \(-0.572625\pi\)
−0.226185 + 0.974084i \(0.572625\pi\)
\(294\) 0 0
\(295\) −3.48674 −0.203006
\(296\) −5.46545 −0.317673
\(297\) 0 0
\(298\) 4.31686 0.250069
\(299\) 0.554198 0.0320501
\(300\) 0 0
\(301\) 0 0
\(302\) −2.71481 −0.156220
\(303\) 0 0
\(304\) 3.18148 0.182471
\(305\) −0.550333 −0.0315120
\(306\) 0 0
\(307\) −7.02736 −0.401073 −0.200536 0.979686i \(-0.564268\pi\)
−0.200536 + 0.979686i \(0.564268\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.0765033 −0.00434509
\(311\) 33.4422 1.89633 0.948165 0.317777i \(-0.102936\pi\)
0.948165 + 0.317777i \(0.102936\pi\)
\(312\) 0 0
\(313\) −12.8389 −0.725700 −0.362850 0.931848i \(-0.618196\pi\)
−0.362850 + 0.931848i \(0.618196\pi\)
\(314\) −4.43721 −0.250406
\(315\) 0 0
\(316\) −23.8225 −1.34012
\(317\) −3.14685 −0.176745 −0.0883723 0.996088i \(-0.528167\pi\)
−0.0883723 + 0.996088i \(0.528167\pi\)
\(318\) 0 0
\(319\) 19.1898 1.07442
\(320\) 1.57936 0.0882888
\(321\) 0 0
\(322\) 0 0
\(323\) −5.87267 −0.326764
\(324\) 0 0
\(325\) 4.92233 0.273041
\(326\) 5.62263 0.311409
\(327\) 0 0
\(328\) −1.21646 −0.0671679
\(329\) 0 0
\(330\) 0 0
\(331\) 6.26160 0.344169 0.172084 0.985082i \(-0.444950\pi\)
0.172084 + 0.985082i \(0.444950\pi\)
\(332\) 21.2271 1.16499
\(333\) 0 0
\(334\) −2.37386 −0.129892
\(335\) 1.10745 0.0605066
\(336\) 0 0
\(337\) −15.8596 −0.863929 −0.431964 0.901891i \(-0.642179\pi\)
−0.431964 + 0.901891i \(0.642179\pi\)
\(338\) 0.318530 0.0173257
\(339\) 0 0
\(340\) −3.32231 −0.180177
\(341\) 3.67806 0.199178
\(342\) 0 0
\(343\) 0 0
\(344\) 2.13166 0.114931
\(345\) 0 0
\(346\) 6.11087 0.328522
\(347\) 13.1466 0.705747 0.352873 0.935671i \(-0.385205\pi\)
0.352873 + 0.935671i \(0.385205\pi\)
\(348\) 0 0
\(349\) −29.8740 −1.59912 −0.799559 0.600588i \(-0.794933\pi\)
−0.799559 + 0.600588i \(0.794933\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 15.2245 0.811469
\(353\) −0.661559 −0.0352112 −0.0176056 0.999845i \(-0.505604\pi\)
−0.0176056 + 0.999845i \(0.505604\pi\)
\(354\) 0 0
\(355\) −0.679458 −0.0360619
\(356\) 23.1295 1.22586
\(357\) 0 0
\(358\) 6.25345 0.330505
\(359\) 13.2792 0.700851 0.350425 0.936591i \(-0.386037\pi\)
0.350425 + 0.936591i \(0.386037\pi\)
\(360\) 0 0
\(361\) −18.1252 −0.953958
\(362\) −4.32097 −0.227105
\(363\) 0 0
\(364\) 0 0
\(365\) −1.34791 −0.0705530
\(366\) 0 0
\(367\) −36.0453 −1.88155 −0.940776 0.339029i \(-0.889902\pi\)
−0.940776 + 0.339029i \(0.889902\pi\)
\(368\) −1.88512 −0.0982687
\(369\) 0 0
\(370\) −0.390718 −0.0203125
\(371\) 0 0
\(372\) 0 0
\(373\) −5.95464 −0.308320 −0.154160 0.988046i \(-0.549267\pi\)
−0.154160 + 0.988046i \(0.549267\pi\)
\(374\) −8.53610 −0.441391
\(375\) 0 0
\(376\) 5.49882 0.283580
\(377\) −4.49614 −0.231563
\(378\) 0 0
\(379\) −18.7760 −0.964457 −0.482229 0.876045i \(-0.660173\pi\)
−0.482229 + 0.876045i \(0.660173\pi\)
\(380\) 0.494898 0.0253877
\(381\) 0 0
\(382\) 0.911079 0.0466149
\(383\) 11.2426 0.574473 0.287236 0.957860i \(-0.407264\pi\)
0.287236 + 0.957860i \(0.407264\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.101355 0.00515883
\(387\) 0 0
\(388\) 24.5100 1.24431
\(389\) −10.2488 −0.519632 −0.259816 0.965658i \(-0.583662\pi\)
−0.259816 + 0.965658i \(0.583662\pi\)
\(390\) 0 0
\(391\) 3.47973 0.175977
\(392\) 0 0
\(393\) 0 0
\(394\) −0.687269 −0.0346241
\(395\) −3.49711 −0.175958
\(396\) 0 0
\(397\) 4.24651 0.213126 0.106563 0.994306i \(-0.466015\pi\)
0.106563 + 0.994306i \(0.466015\pi\)
\(398\) −4.38683 −0.219892
\(399\) 0 0
\(400\) −16.7434 −0.837171
\(401\) −16.0419 −0.801092 −0.400546 0.916277i \(-0.631180\pi\)
−0.400546 + 0.916277i \(0.631180\pi\)
\(402\) 0 0
\(403\) −0.861767 −0.0429277
\(404\) 9.68903 0.482047
\(405\) 0 0
\(406\) 0 0
\(407\) 18.7847 0.931121
\(408\) 0 0
\(409\) −10.4352 −0.515986 −0.257993 0.966147i \(-0.583061\pi\)
−0.257993 + 0.966147i \(0.583061\pi\)
\(410\) −0.0869635 −0.00429482
\(411\) 0 0
\(412\) 0.685537 0.0337740
\(413\) 0 0
\(414\) 0 0
\(415\) 3.11610 0.152963
\(416\) −3.56709 −0.174891
\(417\) 0 0
\(418\) 1.27156 0.0621938
\(419\) 26.0460 1.27243 0.636215 0.771512i \(-0.280499\pi\)
0.636215 + 0.771512i \(0.280499\pi\)
\(420\) 0 0
\(421\) 19.4906 0.949912 0.474956 0.880009i \(-0.342464\pi\)
0.474956 + 0.880009i \(0.342464\pi\)
\(422\) −1.61056 −0.0784009
\(423\) 0 0
\(424\) 11.5562 0.561220
\(425\) 30.9065 1.49919
\(426\) 0 0
\(427\) 0 0
\(428\) −25.1866 −1.21744
\(429\) 0 0
\(430\) 0.152390 0.00734889
\(431\) 3.84680 0.185294 0.0926468 0.995699i \(-0.470467\pi\)
0.0926468 + 0.995699i \(0.470467\pi\)
\(432\) 0 0
\(433\) −1.34412 −0.0645943 −0.0322972 0.999478i \(-0.510282\pi\)
−0.0322972 + 0.999478i \(0.510282\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 26.1669 1.25317
\(437\) −0.518348 −0.0247959
\(438\) 0 0
\(439\) −36.1985 −1.72766 −0.863831 0.503781i \(-0.831942\pi\)
−0.863831 + 0.503781i \(0.831942\pi\)
\(440\) 1.47714 0.0704198
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) 11.1890 0.531608 0.265804 0.964027i \(-0.414363\pi\)
0.265804 + 0.964027i \(0.414363\pi\)
\(444\) 0 0
\(445\) 3.39536 0.160956
\(446\) −7.63262 −0.361415
\(447\) 0 0
\(448\) 0 0
\(449\) −10.3847 −0.490082 −0.245041 0.969513i \(-0.578801\pi\)
−0.245041 + 0.969513i \(0.578801\pi\)
\(450\) 0 0
\(451\) 4.18096 0.196874
\(452\) −5.68258 −0.267286
\(453\) 0 0
\(454\) −6.00172 −0.281675
\(455\) 0 0
\(456\) 0 0
\(457\) −27.0132 −1.26363 −0.631813 0.775121i \(-0.717689\pi\)
−0.631813 + 0.775121i \(0.717689\pi\)
\(458\) −6.04233 −0.282340
\(459\) 0 0
\(460\) −0.293241 −0.0136724
\(461\) 17.3981 0.810309 0.405154 0.914248i \(-0.367218\pi\)
0.405154 + 0.914248i \(0.367218\pi\)
\(462\) 0 0
\(463\) 5.57924 0.259289 0.129645 0.991561i \(-0.458616\pi\)
0.129645 + 0.991561i \(0.458616\pi\)
\(464\) 15.2937 0.709994
\(465\) 0 0
\(466\) −8.70345 −0.403180
\(467\) −1.10964 −0.0513479 −0.0256740 0.999670i \(-0.508173\pi\)
−0.0256740 + 0.999670i \(0.508173\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.393104 0.0181326
\(471\) 0 0
\(472\) −15.5357 −0.715088
\(473\) −7.32648 −0.336872
\(474\) 0 0
\(475\) −4.60390 −0.211242
\(476\) 0 0
\(477\) 0 0
\(478\) −7.60487 −0.347839
\(479\) 28.7672 1.31441 0.657204 0.753713i \(-0.271739\pi\)
0.657204 + 0.753713i \(0.271739\pi\)
\(480\) 0 0
\(481\) −4.40123 −0.200679
\(482\) −5.22200 −0.237856
\(483\) 0 0
\(484\) −13.7003 −0.622742
\(485\) 3.59803 0.163378
\(486\) 0 0
\(487\) −8.05808 −0.365147 −0.182573 0.983192i \(-0.558443\pi\)
−0.182573 + 0.983192i \(0.558443\pi\)
\(488\) −2.45209 −0.111001
\(489\) 0 0
\(490\) 0 0
\(491\) 33.6810 1.52000 0.760001 0.649922i \(-0.225198\pi\)
0.760001 + 0.649922i \(0.225198\pi\)
\(492\) 0 0
\(493\) −28.2306 −1.27144
\(494\) −0.297924 −0.0134042
\(495\) 0 0
\(496\) 2.93132 0.131620
\(497\) 0 0
\(498\) 0 0
\(499\) 4.98394 0.223112 0.111556 0.993758i \(-0.464417\pi\)
0.111556 + 0.993758i \(0.464417\pi\)
\(500\) −5.25017 −0.234795
\(501\) 0 0
\(502\) 2.51303 0.112162
\(503\) −10.4861 −0.467554 −0.233777 0.972290i \(-0.575109\pi\)
−0.233777 + 0.972290i \(0.575109\pi\)
\(504\) 0 0
\(505\) 1.42233 0.0632929
\(506\) −0.753433 −0.0334942
\(507\) 0 0
\(508\) 12.7422 0.565343
\(509\) 27.7170 1.22854 0.614268 0.789097i \(-0.289451\pi\)
0.614268 + 0.789097i \(0.289451\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 20.5816 0.909586
\(513\) 0 0
\(514\) −6.70692 −0.295830
\(515\) 0.100636 0.00443454
\(516\) 0 0
\(517\) −18.8994 −0.831193
\(518\) 0 0
\(519\) 0 0
\(520\) −0.346092 −0.0151772
\(521\) −4.84437 −0.212236 −0.106118 0.994354i \(-0.533842\pi\)
−0.106118 + 0.994354i \(0.533842\pi\)
\(522\) 0 0
\(523\) 26.8799 1.17538 0.587688 0.809088i \(-0.300038\pi\)
0.587688 + 0.809088i \(0.300038\pi\)
\(524\) −18.0880 −0.790177
\(525\) 0 0
\(526\) −3.01064 −0.131270
\(527\) −5.41091 −0.235703
\(528\) 0 0
\(529\) −22.6929 −0.986646
\(530\) 0.826142 0.0358853
\(531\) 0 0
\(532\) 0 0
\(533\) −0.979596 −0.0424310
\(534\) 0 0
\(535\) −3.69735 −0.159851
\(536\) 4.93443 0.213135
\(537\) 0 0
\(538\) −4.27727 −0.184406
\(539\) 0 0
\(540\) 0 0
\(541\) 13.9541 0.599933 0.299966 0.953950i \(-0.403025\pi\)
0.299966 + 0.953950i \(0.403025\pi\)
\(542\) −2.84851 −0.122354
\(543\) 0 0
\(544\) −22.3972 −0.960272
\(545\) 3.84125 0.164541
\(546\) 0 0
\(547\) 39.2066 1.67635 0.838177 0.545399i \(-0.183622\pi\)
0.838177 + 0.545399i \(0.183622\pi\)
\(548\) 40.0844 1.71232
\(549\) 0 0
\(550\) −6.69190 −0.285344
\(551\) 4.20529 0.179151
\(552\) 0 0
\(553\) 0 0
\(554\) −1.93671 −0.0822829
\(555\) 0 0
\(556\) 27.8727 1.18206
\(557\) −7.47685 −0.316804 −0.158402 0.987375i \(-0.550634\pi\)
−0.158402 + 0.987375i \(0.550634\pi\)
\(558\) 0 0
\(559\) 1.71659 0.0726039
\(560\) 0 0
\(561\) 0 0
\(562\) 5.34512 0.225470
\(563\) −18.2725 −0.770096 −0.385048 0.922897i \(-0.625815\pi\)
−0.385048 + 0.922897i \(0.625815\pi\)
\(564\) 0 0
\(565\) −0.834192 −0.0350947
\(566\) 1.43023 0.0601172
\(567\) 0 0
\(568\) −3.02743 −0.127028
\(569\) 26.0547 1.09227 0.546136 0.837697i \(-0.316098\pi\)
0.546136 + 0.837697i \(0.316098\pi\)
\(570\) 0 0
\(571\) −2.26065 −0.0946052 −0.0473026 0.998881i \(-0.515063\pi\)
−0.0473026 + 0.998881i \(0.515063\pi\)
\(572\) 8.10306 0.338806
\(573\) 0 0
\(574\) 0 0
\(575\) 2.72794 0.113763
\(576\) 0 0
\(577\) 35.8836 1.49385 0.746927 0.664906i \(-0.231529\pi\)
0.746927 + 0.664906i \(0.231529\pi\)
\(578\) 7.14269 0.297097
\(579\) 0 0
\(580\) 2.37903 0.0987838
\(581\) 0 0
\(582\) 0 0
\(583\) −39.7186 −1.64498
\(584\) −6.00584 −0.248523
\(585\) 0 0
\(586\) −2.46648 −0.101889
\(587\) 21.2452 0.876885 0.438443 0.898759i \(-0.355530\pi\)
0.438443 + 0.898759i \(0.355530\pi\)
\(588\) 0 0
\(589\) 0.806020 0.0332115
\(590\) −1.11063 −0.0457239
\(591\) 0 0
\(592\) 14.9709 0.615300
\(593\) −19.0481 −0.782214 −0.391107 0.920345i \(-0.627908\pi\)
−0.391107 + 0.920345i \(0.627908\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −25.7298 −1.05394
\(597\) 0 0
\(598\) 0.176529 0.00721879
\(599\) −21.9481 −0.896776 −0.448388 0.893839i \(-0.648002\pi\)
−0.448388 + 0.893839i \(0.648002\pi\)
\(600\) 0 0
\(601\) 11.1249 0.453794 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.1811 0.658401
\(605\) −2.01118 −0.0817662
\(606\) 0 0
\(607\) −8.29441 −0.336660 −0.168330 0.985731i \(-0.553837\pi\)
−0.168330 + 0.985731i \(0.553837\pi\)
\(608\) 3.33634 0.135306
\(609\) 0 0
\(610\) −0.175297 −0.00709758
\(611\) 4.42810 0.179142
\(612\) 0 0
\(613\) −24.9909 −1.00937 −0.504686 0.863303i \(-0.668392\pi\)
−0.504686 + 0.863303i \(0.668392\pi\)
\(614\) −2.23842 −0.0903354
\(615\) 0 0
\(616\) 0 0
\(617\) −29.5367 −1.18910 −0.594551 0.804058i \(-0.702670\pi\)
−0.594551 + 0.804058i \(0.702670\pi\)
\(618\) 0 0
\(619\) −24.2203 −0.973494 −0.486747 0.873543i \(-0.661817\pi\)
−0.486747 + 0.873543i \(0.661817\pi\)
\(620\) 0.455984 0.0183128
\(621\) 0 0
\(622\) 10.6523 0.427119
\(623\) 0 0
\(624\) 0 0
\(625\) 23.8409 0.953636
\(626\) −4.08958 −0.163453
\(627\) 0 0
\(628\) 26.4472 1.05536
\(629\) −27.6346 −1.10187
\(630\) 0 0
\(631\) 22.9592 0.913992 0.456996 0.889469i \(-0.348925\pi\)
0.456996 + 0.889469i \(0.348925\pi\)
\(632\) −15.5819 −0.619815
\(633\) 0 0
\(634\) −1.00236 −0.0398090
\(635\) 1.87053 0.0742296
\(636\) 0 0
\(637\) 0 0
\(638\) 6.11251 0.241996
\(639\) 0 0
\(640\) 2.49138 0.0984805
\(641\) 40.5768 1.60269 0.801344 0.598204i \(-0.204119\pi\)
0.801344 + 0.598204i \(0.204119\pi\)
\(642\) 0 0
\(643\) 9.87685 0.389505 0.194752 0.980852i \(-0.437610\pi\)
0.194752 + 0.980852i \(0.437610\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.87062 −0.0735986
\(647\) 44.6596 1.75575 0.877876 0.478888i \(-0.158960\pi\)
0.877876 + 0.478888i \(0.158960\pi\)
\(648\) 0 0
\(649\) 53.3960 2.09597
\(650\) 1.56791 0.0614984
\(651\) 0 0
\(652\) −33.5127 −1.31246
\(653\) −0.169205 −0.00662149 −0.00331074 0.999995i \(-0.501054\pi\)
−0.00331074 + 0.999995i \(0.501054\pi\)
\(654\) 0 0
\(655\) −2.65528 −0.103750
\(656\) 3.33212 0.130097
\(657\) 0 0
\(658\) 0 0
\(659\) −33.3418 −1.29881 −0.649405 0.760443i \(-0.724982\pi\)
−0.649405 + 0.760443i \(0.724982\pi\)
\(660\) 0 0
\(661\) 37.0200 1.43991 0.719956 0.694019i \(-0.244162\pi\)
0.719956 + 0.694019i \(0.244162\pi\)
\(662\) 1.99451 0.0775187
\(663\) 0 0
\(664\) 13.8843 0.538814
\(665\) 0 0
\(666\) 0 0
\(667\) −2.49175 −0.0964811
\(668\) 14.1489 0.547439
\(669\) 0 0
\(670\) 0.352756 0.0136282
\(671\) 8.42781 0.325352
\(672\) 0 0
\(673\) 32.6406 1.25820 0.629102 0.777323i \(-0.283423\pi\)
0.629102 + 0.777323i \(0.283423\pi\)
\(674\) −5.05176 −0.194587
\(675\) 0 0
\(676\) −1.89854 −0.0730207
\(677\) −34.7106 −1.33404 −0.667018 0.745042i \(-0.732429\pi\)
−0.667018 + 0.745042i \(0.732429\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.17306 −0.0833331
\(681\) 0 0
\(682\) 1.17157 0.0448618
\(683\) 25.9340 0.992337 0.496169 0.868226i \(-0.334740\pi\)
0.496169 + 0.868226i \(0.334740\pi\)
\(684\) 0 0
\(685\) 5.88432 0.224828
\(686\) 0 0
\(687\) 0 0
\(688\) −5.83902 −0.222611
\(689\) 9.30603 0.354531
\(690\) 0 0
\(691\) 38.3626 1.45938 0.729691 0.683777i \(-0.239664\pi\)
0.729691 + 0.683777i \(0.239664\pi\)
\(692\) −36.4227 −1.38458
\(693\) 0 0
\(694\) 4.18759 0.158959
\(695\) 4.09165 0.155205
\(696\) 0 0
\(697\) −6.15073 −0.232976
\(698\) −9.51575 −0.360176
\(699\) 0 0
\(700\) 0 0
\(701\) 26.2477 0.991363 0.495682 0.868504i \(-0.334918\pi\)
0.495682 + 0.868504i \(0.334918\pi\)
\(702\) 0 0
\(703\) 4.11652 0.155257
\(704\) −24.1863 −0.911556
\(705\) 0 0
\(706\) −0.210726 −0.00793078
\(707\) 0 0
\(708\) 0 0
\(709\) 34.9709 1.31336 0.656679 0.754170i \(-0.271961\pi\)
0.656679 + 0.754170i \(0.271961\pi\)
\(710\) −0.216428 −0.00812238
\(711\) 0 0
\(712\) 15.1286 0.566967
\(713\) −0.477590 −0.0178859
\(714\) 0 0
\(715\) 1.18951 0.0444853
\(716\) −37.2726 −1.39294
\(717\) 0 0
\(718\) 4.22983 0.157856
\(719\) −14.2166 −0.530190 −0.265095 0.964222i \(-0.585403\pi\)
−0.265095 + 0.964222i \(0.585403\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.77341 −0.214864
\(723\) 0 0
\(724\) 25.7544 0.957154
\(725\) −22.1315 −0.821942
\(726\) 0 0
\(727\) −0.566912 −0.0210256 −0.0105128 0.999945i \(-0.503346\pi\)
−0.0105128 + 0.999945i \(0.503346\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.429350 −0.0158910
\(731\) 10.7782 0.398646
\(732\) 0 0
\(733\) −28.2007 −1.04162 −0.520809 0.853673i \(-0.674370\pi\)
−0.520809 + 0.853673i \(0.674370\pi\)
\(734\) −11.4815 −0.423790
\(735\) 0 0
\(736\) −1.97687 −0.0728686
\(737\) −16.9595 −0.624713
\(738\) 0 0
\(739\) −44.6598 −1.64284 −0.821419 0.570325i \(-0.806817\pi\)
−0.821419 + 0.570325i \(0.806817\pi\)
\(740\) 2.32881 0.0856086
\(741\) 0 0
\(742\) 0 0
\(743\) 3.31876 0.121754 0.0608768 0.998145i \(-0.480610\pi\)
0.0608768 + 0.998145i \(0.480610\pi\)
\(744\) 0 0
\(745\) −3.77709 −0.138382
\(746\) −1.89673 −0.0694443
\(747\) 0 0
\(748\) 50.8779 1.86028
\(749\) 0 0
\(750\) 0 0
\(751\) −33.1595 −1.21001 −0.605004 0.796223i \(-0.706828\pi\)
−0.605004 + 0.796223i \(0.706828\pi\)
\(752\) −15.0623 −0.549266
\(753\) 0 0
\(754\) −1.43215 −0.0521560
\(755\) 2.37536 0.0864482
\(756\) 0 0
\(757\) 18.8780 0.686133 0.343067 0.939311i \(-0.388534\pi\)
0.343067 + 0.939311i \(0.388534\pi\)
\(758\) −5.98071 −0.217229
\(759\) 0 0
\(760\) 0.323704 0.0117420
\(761\) −28.3937 −1.02927 −0.514635 0.857409i \(-0.672073\pi\)
−0.514635 + 0.857409i \(0.672073\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.43032 −0.196462
\(765\) 0 0
\(766\) 3.58112 0.129391
\(767\) −12.5106 −0.451732
\(768\) 0 0
\(769\) −36.8171 −1.32766 −0.663830 0.747884i \(-0.731070\pi\)
−0.663830 + 0.747884i \(0.731070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.604108 −0.0217423
\(773\) 14.3835 0.517337 0.258669 0.965966i \(-0.416716\pi\)
0.258669 + 0.965966i \(0.416716\pi\)
\(774\) 0 0
\(775\) −4.24190 −0.152373
\(776\) 16.0316 0.575500
\(777\) 0 0
\(778\) −3.26453 −0.117039
\(779\) 0.916226 0.0328272
\(780\) 0 0
\(781\) 10.4052 0.372329
\(782\) 1.10840 0.0396362
\(783\) 0 0
\(784\) 0 0
\(785\) 3.88240 0.138569
\(786\) 0 0
\(787\) −37.1507 −1.32428 −0.662140 0.749380i \(-0.730352\pi\)
−0.662140 + 0.749380i \(0.730352\pi\)
\(788\) 4.09634 0.145926
\(789\) 0 0
\(790\) −1.11393 −0.0396319
\(791\) 0 0
\(792\) 0 0
\(793\) −1.97463 −0.0701211
\(794\) 1.35264 0.0480034
\(795\) 0 0
\(796\) 26.1469 0.926753
\(797\) 23.7554 0.841460 0.420730 0.907186i \(-0.361774\pi\)
0.420730 + 0.907186i \(0.361774\pi\)
\(798\) 0 0
\(799\) 27.8034 0.983613
\(800\) −17.5584 −0.620782
\(801\) 0 0
\(802\) −5.10981 −0.180434
\(803\) 20.6420 0.728439
\(804\) 0 0
\(805\) 0 0
\(806\) −0.274498 −0.00966879
\(807\) 0 0
\(808\) 6.33742 0.222950
\(809\) 45.9182 1.61440 0.807200 0.590279i \(-0.200982\pi\)
0.807200 + 0.590279i \(0.200982\pi\)
\(810\) 0 0
\(811\) 9.21250 0.323495 0.161747 0.986832i \(-0.448287\pi\)
0.161747 + 0.986832i \(0.448287\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 5.98347 0.209721
\(815\) −4.91960 −0.172326
\(816\) 0 0
\(817\) −1.60554 −0.0561708
\(818\) −3.32391 −0.116218
\(819\) 0 0
\(820\) 0.518330 0.0181009
\(821\) −9.22432 −0.321931 −0.160965 0.986960i \(-0.551461\pi\)
−0.160965 + 0.986960i \(0.551461\pi\)
\(822\) 0 0
\(823\) 32.3510 1.12768 0.563842 0.825883i \(-0.309323\pi\)
0.563842 + 0.825883i \(0.309323\pi\)
\(824\) 0.448398 0.0156207
\(825\) 0 0
\(826\) 0 0
\(827\) −25.5118 −0.887133 −0.443566 0.896242i \(-0.646287\pi\)
−0.443566 + 0.896242i \(0.646287\pi\)
\(828\) 0 0
\(829\) 37.2508 1.29377 0.646887 0.762586i \(-0.276071\pi\)
0.646887 + 0.762586i \(0.276071\pi\)
\(830\) 0.992570 0.0344526
\(831\) 0 0
\(832\) 5.66683 0.196462
\(833\) 0 0
\(834\) 0 0
\(835\) 2.07704 0.0718789
\(836\) −7.57888 −0.262121
\(837\) 0 0
\(838\) 8.29642 0.286595
\(839\) −24.1576 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(840\) 0 0
\(841\) −8.78471 −0.302921
\(842\) 6.20833 0.213953
\(843\) 0 0
\(844\) 9.59946 0.330427
\(845\) −0.278702 −0.00958764
\(846\) 0 0
\(847\) 0 0
\(848\) −31.6547 −1.08703
\(849\) 0 0
\(850\) 9.84465 0.337669
\(851\) −2.43915 −0.0836131
\(852\) 0 0
\(853\) 39.3757 1.34820 0.674100 0.738640i \(-0.264532\pi\)
0.674100 + 0.738640i \(0.264532\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −16.4741 −0.563074
\(857\) −50.6221 −1.72922 −0.864610 0.502444i \(-0.832434\pi\)
−0.864610 + 0.502444i \(0.832434\pi\)
\(858\) 0 0
\(859\) −16.7350 −0.570991 −0.285496 0.958380i \(-0.592158\pi\)
−0.285496 + 0.958380i \(0.592158\pi\)
\(860\) −0.908292 −0.0309725
\(861\) 0 0
\(862\) 1.22532 0.0417345
\(863\) 19.8012 0.674040 0.337020 0.941497i \(-0.390581\pi\)
0.337020 + 0.941497i \(0.390581\pi\)
\(864\) 0 0
\(865\) −5.34679 −0.181796
\(866\) −0.428142 −0.0145489
\(867\) 0 0
\(868\) 0 0
\(869\) 53.5548 1.81672
\(870\) 0 0
\(871\) 3.97361 0.134641
\(872\) 17.1153 0.579597
\(873\) 0 0
\(874\) −0.165109 −0.00558490
\(875\) 0 0
\(876\) 0 0
\(877\) −20.7398 −0.700333 −0.350167 0.936687i \(-0.613875\pi\)
−0.350167 + 0.936687i \(0.613875\pi\)
\(878\) −11.5303 −0.389129
\(879\) 0 0
\(880\) −4.04616 −0.136396
\(881\) 1.62325 0.0546888 0.0273444 0.999626i \(-0.491295\pi\)
0.0273444 + 0.999626i \(0.491295\pi\)
\(882\) 0 0
\(883\) −11.8833 −0.399904 −0.199952 0.979806i \(-0.564079\pi\)
−0.199952 + 0.979806i \(0.564079\pi\)
\(884\) −11.9206 −0.400934
\(885\) 0 0
\(886\) 3.56404 0.119736
\(887\) −31.7030 −1.06448 −0.532241 0.846593i \(-0.678650\pi\)
−0.532241 + 0.846593i \(0.678650\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.08152 0.0362528
\(891\) 0 0
\(892\) 45.4929 1.52321
\(893\) −4.14165 −0.138595
\(894\) 0 0
\(895\) −5.47155 −0.182894
\(896\) 0 0
\(897\) 0 0
\(898\) −3.30782 −0.110383
\(899\) 3.87463 0.129226
\(900\) 0 0
\(901\) 58.4311 1.94662
\(902\) 1.33176 0.0443428
\(903\) 0 0
\(904\) −3.71688 −0.123621
\(905\) 3.78070 0.125675
\(906\) 0 0
\(907\) −4.45928 −0.148068 −0.0740341 0.997256i \(-0.523587\pi\)
−0.0740341 + 0.997256i \(0.523587\pi\)
\(908\) 35.7722 1.18714
\(909\) 0 0
\(910\) 0 0
\(911\) −46.4395 −1.53861 −0.769305 0.638882i \(-0.779397\pi\)
−0.769305 + 0.638882i \(0.779397\pi\)
\(912\) 0 0
\(913\) −47.7200 −1.57930
\(914\) −8.60452 −0.284612
\(915\) 0 0
\(916\) 36.0142 1.18994
\(917\) 0 0
\(918\) 0 0
\(919\) −19.5628 −0.645317 −0.322658 0.946515i \(-0.604577\pi\)
−0.322658 + 0.946515i \(0.604577\pi\)
\(920\) −0.191804 −0.00632358
\(921\) 0 0
\(922\) 5.54180 0.182509
\(923\) −2.43794 −0.0802457
\(924\) 0 0
\(925\) −21.6643 −0.712317
\(926\) 1.77715 0.0584009
\(927\) 0 0
\(928\) 16.0381 0.526478
\(929\) −18.1377 −0.595078 −0.297539 0.954710i \(-0.596166\pi\)
−0.297539 + 0.954710i \(0.596166\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 51.8754 1.69923
\(933\) 0 0
\(934\) −0.353453 −0.0115653
\(935\) 7.46878 0.244255
\(936\) 0 0
\(937\) 40.3697 1.31882 0.659410 0.751784i \(-0.270806\pi\)
0.659410 + 0.751784i \(0.270806\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.34303 −0.0764211
\(941\) 21.5312 0.701898 0.350949 0.936395i \(-0.385859\pi\)
0.350949 + 0.936395i \(0.385859\pi\)
\(942\) 0 0
\(943\) −0.542890 −0.0176789
\(944\) 42.5552 1.38505
\(945\) 0 0
\(946\) −2.33370 −0.0758752
\(947\) 55.9123 1.81691 0.908454 0.417986i \(-0.137264\pi\)
0.908454 + 0.417986i \(0.137264\pi\)
\(948\) 0 0
\(949\) −4.83639 −0.156996
\(950\) −1.46648 −0.0475789
\(951\) 0 0
\(952\) 0 0
\(953\) −21.2497 −0.688346 −0.344173 0.938906i \(-0.611841\pi\)
−0.344173 + 0.938906i \(0.611841\pi\)
\(954\) 0 0
\(955\) −0.797161 −0.0257955
\(956\) 45.3274 1.46599
\(957\) 0 0
\(958\) 9.16321 0.296050
\(959\) 0 0
\(960\) 0 0
\(961\) −30.2574 −0.976044
\(962\) −1.40192 −0.0451998
\(963\) 0 0
\(964\) 31.1248 1.00246
\(965\) −0.0886819 −0.00285477
\(966\) 0 0
\(967\) 52.4550 1.68684 0.843419 0.537257i \(-0.180539\pi\)
0.843419 + 0.537257i \(0.180539\pi\)
\(968\) −8.96114 −0.288022
\(969\) 0 0
\(970\) 1.14608 0.0367984
\(971\) 16.7283 0.536836 0.268418 0.963302i \(-0.413499\pi\)
0.268418 + 0.963302i \(0.413499\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.56674 −0.0822436
\(975\) 0 0
\(976\) 6.71675 0.214998
\(977\) −10.9472 −0.350232 −0.175116 0.984548i \(-0.556030\pi\)
−0.175116 + 0.984548i \(0.556030\pi\)
\(978\) 0 0
\(979\) −51.9967 −1.66182
\(980\) 0 0
\(981\) 0 0
\(982\) 10.7284 0.342357
\(983\) 2.17536 0.0693832 0.0346916 0.999398i \(-0.488955\pi\)
0.0346916 + 0.999398i \(0.488955\pi\)
\(984\) 0 0
\(985\) 0.601336 0.0191601
\(986\) −8.99228 −0.286373
\(987\) 0 0
\(988\) 1.77572 0.0564933
\(989\) 0.951330 0.0302505
\(990\) 0 0
\(991\) 24.0099 0.762700 0.381350 0.924431i \(-0.375459\pi\)
0.381350 + 0.924431i \(0.375459\pi\)
\(992\) 3.07400 0.0975996
\(993\) 0 0
\(994\) 0 0
\(995\) 3.83832 0.121683
\(996\) 0 0
\(997\) −26.5083 −0.839527 −0.419763 0.907634i \(-0.637887\pi\)
−0.419763 + 0.907634i \(0.637887\pi\)
\(998\) 1.58753 0.0502524
\(999\) 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 5733.2.a.ca.1.7 yes 12
3.2 odd 2 inner 5733.2.a.ca.1.6 12
7.6 odd 2 5733.2.a.cb.1.7 yes 12
21.20 even 2 5733.2.a.cb.1.6 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5733.2.a.ca.1.6 12 3.2 odd 2 inner
5733.2.a.ca.1.7 yes 12 1.1 even 1 trivial
5733.2.a.cb.1.6 yes 12 21.20 even 2
5733.2.a.cb.1.7 yes 12 7.6 odd 2