Properties

Label 9680.2.a.cy.1.3
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $6$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.36422\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.843136 q^{3} -1.00000 q^{5} +4.75693 q^{7} -2.28912 q^{9} -3.78308 q^{13} +0.843136 q^{15} -4.98203 q^{17} +3.12115 q^{19} -4.01074 q^{21} -9.39001 q^{23} +1.00000 q^{25} +4.45945 q^{27} +2.60693 q^{29} +3.68869 q^{31} -4.75693 q^{35} -3.44757 q^{37} +3.18965 q^{39} -9.32837 q^{41} -11.7051 q^{43} +2.28912 q^{45} +1.66156 q^{47} +15.6284 q^{49} +4.20053 q^{51} -1.93994 q^{53} -2.63156 q^{57} -0.103915 q^{59} +10.1513 q^{61} -10.8892 q^{63} +3.78308 q^{65} +4.99277 q^{67} +7.91706 q^{69} +6.22929 q^{71} +7.30650 q^{73} -0.843136 q^{75} +10.5445 q^{79} +3.10744 q^{81} +5.69010 q^{83} +4.98203 q^{85} -2.19800 q^{87} +13.6371 q^{89} -17.9958 q^{91} -3.11007 q^{93} -3.12115 q^{95} +13.6967 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 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 0
\(3\) −0.843136 −0.486785 −0.243392 0.969928i \(-0.578260\pi\)
−0.243392 + 0.969928i \(0.578260\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.75693 1.79795 0.898975 0.438000i \(-0.144313\pi\)
0.898975 + 0.438000i \(0.144313\pi\)
\(8\) 0 0
\(9\) −2.28912 −0.763040
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.78308 −1.04924 −0.524619 0.851337i \(-0.675792\pi\)
−0.524619 + 0.851337i \(0.675792\pi\)
\(14\) 0 0
\(15\) 0.843136 0.217697
\(16\) 0 0
\(17\) −4.98203 −1.20832 −0.604160 0.796863i \(-0.706491\pi\)
−0.604160 + 0.796863i \(0.706491\pi\)
\(18\) 0 0
\(19\) 3.12115 0.716041 0.358021 0.933714i \(-0.383452\pi\)
0.358021 + 0.933714i \(0.383452\pi\)
\(20\) 0 0
\(21\) −4.01074 −0.875215
\(22\) 0 0
\(23\) −9.39001 −1.95795 −0.978976 0.203975i \(-0.934614\pi\)
−0.978976 + 0.203975i \(0.934614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.45945 0.858222
\(28\) 0 0
\(29\) 2.60693 0.484095 0.242047 0.970264i \(-0.422181\pi\)
0.242047 + 0.970264i \(0.422181\pi\)
\(30\) 0 0
\(31\) 3.68869 0.662508 0.331254 0.943542i \(-0.392528\pi\)
0.331254 + 0.943542i \(0.392528\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.75693 −0.804068
\(36\) 0 0
\(37\) −3.44757 −0.566776 −0.283388 0.959005i \(-0.591458\pi\)
−0.283388 + 0.959005i \(0.591458\pi\)
\(38\) 0 0
\(39\) 3.18965 0.510753
\(40\) 0 0
\(41\) −9.32837 −1.45685 −0.728423 0.685127i \(-0.759747\pi\)
−0.728423 + 0.685127i \(0.759747\pi\)
\(42\) 0 0
\(43\) −11.7051 −1.78500 −0.892502 0.451043i \(-0.851052\pi\)
−0.892502 + 0.451043i \(0.851052\pi\)
\(44\) 0 0
\(45\) 2.28912 0.341242
\(46\) 0 0
\(47\) 1.66156 0.242364 0.121182 0.992630i \(-0.461332\pi\)
0.121182 + 0.992630i \(0.461332\pi\)
\(48\) 0 0
\(49\) 15.6284 2.23262
\(50\) 0 0
\(51\) 4.20053 0.588192
\(52\) 0 0
\(53\) −1.93994 −0.266472 −0.133236 0.991084i \(-0.542537\pi\)
−0.133236 + 0.991084i \(0.542537\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.63156 −0.348558
\(58\) 0 0
\(59\) −0.103915 −0.0135285 −0.00676427 0.999977i \(-0.502153\pi\)
−0.00676427 + 0.999977i \(0.502153\pi\)
\(60\) 0 0
\(61\) 10.1513 1.29974 0.649870 0.760045i \(-0.274823\pi\)
0.649870 + 0.760045i \(0.274823\pi\)
\(62\) 0 0
\(63\) −10.8892 −1.37191
\(64\) 0 0
\(65\) 3.78308 0.469233
\(66\) 0 0
\(67\) 4.99277 0.609964 0.304982 0.952358i \(-0.401350\pi\)
0.304982 + 0.952358i \(0.401350\pi\)
\(68\) 0 0
\(69\) 7.91706 0.953102
\(70\) 0 0
\(71\) 6.22929 0.739281 0.369640 0.929175i \(-0.379481\pi\)
0.369640 + 0.929175i \(0.379481\pi\)
\(72\) 0 0
\(73\) 7.30650 0.855161 0.427580 0.903977i \(-0.359366\pi\)
0.427580 + 0.903977i \(0.359366\pi\)
\(74\) 0 0
\(75\) −0.843136 −0.0973570
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.5445 1.18635 0.593175 0.805073i \(-0.297874\pi\)
0.593175 + 0.805073i \(0.297874\pi\)
\(80\) 0 0
\(81\) 3.10744 0.345271
\(82\) 0 0
\(83\) 5.69010 0.624569 0.312285 0.949989i \(-0.398906\pi\)
0.312285 + 0.949989i \(0.398906\pi\)
\(84\) 0 0
\(85\) 4.98203 0.540377
\(86\) 0 0
\(87\) −2.19800 −0.235650
\(88\) 0 0
\(89\) 13.6371 1.44553 0.722767 0.691091i \(-0.242870\pi\)
0.722767 + 0.691091i \(0.242870\pi\)
\(90\) 0 0
\(91\) −17.9958 −1.88648
\(92\) 0 0
\(93\) −3.11007 −0.322499
\(94\) 0 0
\(95\) −3.12115 −0.320223
\(96\) 0 0
\(97\) 13.6967 1.39069 0.695345 0.718676i \(-0.255252\pi\)
0.695345 + 0.718676i \(0.255252\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.84287 −0.481884 −0.240942 0.970540i \(-0.577456\pi\)
−0.240942 + 0.970540i \(0.577456\pi\)
\(102\) 0 0
\(103\) −0.904576 −0.0891305 −0.0445653 0.999006i \(-0.514190\pi\)
−0.0445653 + 0.999006i \(0.514190\pi\)
\(104\) 0 0
\(105\) 4.01074 0.391408
\(106\) 0 0
\(107\) −14.7915 −1.42995 −0.714974 0.699151i \(-0.753561\pi\)
−0.714974 + 0.699151i \(0.753561\pi\)
\(108\) 0 0
\(109\) −3.39429 −0.325114 −0.162557 0.986699i \(-0.551974\pi\)
−0.162557 + 0.986699i \(0.551974\pi\)
\(110\) 0 0
\(111\) 2.90677 0.275898
\(112\) 0 0
\(113\) −10.1755 −0.957231 −0.478616 0.878025i \(-0.658861\pi\)
−0.478616 + 0.878025i \(0.658861\pi\)
\(114\) 0 0
\(115\) 9.39001 0.875623
\(116\) 0 0
\(117\) 8.65993 0.800611
\(118\) 0 0
\(119\) −23.6992 −2.17250
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 7.86509 0.709171
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.9587 −0.972430 −0.486215 0.873839i \(-0.661623\pi\)
−0.486215 + 0.873839i \(0.661623\pi\)
\(128\) 0 0
\(129\) 9.86896 0.868913
\(130\) 0 0
\(131\) 12.9021 1.12726 0.563631 0.826027i \(-0.309404\pi\)
0.563631 + 0.826027i \(0.309404\pi\)
\(132\) 0 0
\(133\) 14.8471 1.28741
\(134\) 0 0
\(135\) −4.45945 −0.383808
\(136\) 0 0
\(137\) 18.8824 1.61324 0.806618 0.591073i \(-0.201295\pi\)
0.806618 + 0.591073i \(0.201295\pi\)
\(138\) 0 0
\(139\) 10.8892 0.923611 0.461806 0.886981i \(-0.347202\pi\)
0.461806 + 0.886981i \(0.347202\pi\)
\(140\) 0 0
\(141\) −1.40092 −0.117979
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.60693 −0.216494
\(146\) 0 0
\(147\) −13.1768 −1.08681
\(148\) 0 0
\(149\) −1.69412 −0.138788 −0.0693941 0.997589i \(-0.522107\pi\)
−0.0693941 + 0.997589i \(0.522107\pi\)
\(150\) 0 0
\(151\) −3.48984 −0.283999 −0.142000 0.989867i \(-0.545353\pi\)
−0.142000 + 0.989867i \(0.545353\pi\)
\(152\) 0 0
\(153\) 11.4045 0.921997
\(154\) 0 0
\(155\) −3.68869 −0.296283
\(156\) 0 0
\(157\) 20.7003 1.65207 0.826033 0.563622i \(-0.190592\pi\)
0.826033 + 0.563622i \(0.190592\pi\)
\(158\) 0 0
\(159\) 1.63564 0.129714
\(160\) 0 0
\(161\) −44.6676 −3.52030
\(162\) 0 0
\(163\) −16.3831 −1.28323 −0.641613 0.767028i \(-0.721735\pi\)
−0.641613 + 0.767028i \(0.721735\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.4012 1.11440 0.557199 0.830379i \(-0.311876\pi\)
0.557199 + 0.830379i \(0.311876\pi\)
\(168\) 0 0
\(169\) 1.31169 0.100900
\(170\) 0 0
\(171\) −7.14470 −0.546369
\(172\) 0 0
\(173\) 8.12354 0.617621 0.308811 0.951124i \(-0.400069\pi\)
0.308811 + 0.951124i \(0.400069\pi\)
\(174\) 0 0
\(175\) 4.75693 0.359590
\(176\) 0 0
\(177\) 0.0876143 0.00658549
\(178\) 0 0
\(179\) −2.99340 −0.223737 −0.111869 0.993723i \(-0.535684\pi\)
−0.111869 + 0.993723i \(0.535684\pi\)
\(180\) 0 0
\(181\) −14.5264 −1.07974 −0.539870 0.841749i \(-0.681526\pi\)
−0.539870 + 0.841749i \(0.681526\pi\)
\(182\) 0 0
\(183\) −8.55893 −0.632694
\(184\) 0 0
\(185\) 3.44757 0.253470
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 21.2133 1.54304
\(190\) 0 0
\(191\) 7.96741 0.576502 0.288251 0.957555i \(-0.406926\pi\)
0.288251 + 0.957555i \(0.406926\pi\)
\(192\) 0 0
\(193\) 19.4962 1.40337 0.701684 0.712489i \(-0.252432\pi\)
0.701684 + 0.712489i \(0.252432\pi\)
\(194\) 0 0
\(195\) −3.18965 −0.228416
\(196\) 0 0
\(197\) −17.9751 −1.28067 −0.640336 0.768095i \(-0.721205\pi\)
−0.640336 + 0.768095i \(0.721205\pi\)
\(198\) 0 0
\(199\) 7.17717 0.508776 0.254388 0.967102i \(-0.418126\pi\)
0.254388 + 0.967102i \(0.418126\pi\)
\(200\) 0 0
\(201\) −4.20959 −0.296921
\(202\) 0 0
\(203\) 12.4010 0.870378
\(204\) 0 0
\(205\) 9.32837 0.651522
\(206\) 0 0
\(207\) 21.4949 1.49400
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −4.43285 −0.305170 −0.152585 0.988290i \(-0.548760\pi\)
−0.152585 + 0.988290i \(0.548760\pi\)
\(212\) 0 0
\(213\) −5.25214 −0.359871
\(214\) 0 0
\(215\) 11.7051 0.798278
\(216\) 0 0
\(217\) 17.5468 1.19116
\(218\) 0 0
\(219\) −6.16037 −0.416280
\(220\) 0 0
\(221\) 18.8474 1.26781
\(222\) 0 0
\(223\) 15.2105 1.01857 0.509286 0.860597i \(-0.329909\pi\)
0.509286 + 0.860597i \(0.329909\pi\)
\(224\) 0 0
\(225\) −2.28912 −0.152608
\(226\) 0 0
\(227\) 4.28398 0.284338 0.142169 0.989842i \(-0.454592\pi\)
0.142169 + 0.989842i \(0.454592\pi\)
\(228\) 0 0
\(229\) −4.09655 −0.270708 −0.135354 0.990797i \(-0.543217\pi\)
−0.135354 + 0.990797i \(0.543217\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.2531 1.13029 0.565143 0.824993i \(-0.308821\pi\)
0.565143 + 0.824993i \(0.308821\pi\)
\(234\) 0 0
\(235\) −1.66156 −0.108388
\(236\) 0 0
\(237\) −8.89046 −0.577498
\(238\) 0 0
\(239\) −3.40626 −0.220332 −0.110166 0.993913i \(-0.535138\pi\)
−0.110166 + 0.993913i \(0.535138\pi\)
\(240\) 0 0
\(241\) 20.1839 1.30016 0.650080 0.759866i \(-0.274735\pi\)
0.650080 + 0.759866i \(0.274735\pi\)
\(242\) 0 0
\(243\) −15.9983 −1.02629
\(244\) 0 0
\(245\) −15.6284 −0.998460
\(246\) 0 0
\(247\) −11.8076 −0.751298
\(248\) 0 0
\(249\) −4.79753 −0.304031
\(250\) 0 0
\(251\) −19.6971 −1.24327 −0.621635 0.783307i \(-0.713531\pi\)
−0.621635 + 0.783307i \(0.713531\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.20053 −0.263047
\(256\) 0 0
\(257\) −10.7134 −0.668285 −0.334143 0.942523i \(-0.608447\pi\)
−0.334143 + 0.942523i \(0.608447\pi\)
\(258\) 0 0
\(259\) −16.3998 −1.01904
\(260\) 0 0
\(261\) −5.96758 −0.369384
\(262\) 0 0
\(263\) 8.86317 0.546527 0.273263 0.961939i \(-0.411897\pi\)
0.273263 + 0.961939i \(0.411897\pi\)
\(264\) 0 0
\(265\) 1.93994 0.119170
\(266\) 0 0
\(267\) −11.4980 −0.703665
\(268\) 0 0
\(269\) −2.15481 −0.131381 −0.0656905 0.997840i \(-0.520925\pi\)
−0.0656905 + 0.997840i \(0.520925\pi\)
\(270\) 0 0
\(271\) 10.2552 0.622961 0.311481 0.950252i \(-0.399175\pi\)
0.311481 + 0.950252i \(0.399175\pi\)
\(272\) 0 0
\(273\) 15.1729 0.918309
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.8253 −0.890766 −0.445383 0.895340i \(-0.646933\pi\)
−0.445383 + 0.895340i \(0.646933\pi\)
\(278\) 0 0
\(279\) −8.44386 −0.505521
\(280\) 0 0
\(281\) 16.8470 1.00501 0.502504 0.864575i \(-0.332412\pi\)
0.502504 + 0.864575i \(0.332412\pi\)
\(282\) 0 0
\(283\) −4.89422 −0.290931 −0.145466 0.989363i \(-0.546468\pi\)
−0.145466 + 0.989363i \(0.546468\pi\)
\(284\) 0 0
\(285\) 2.63156 0.155880
\(286\) 0 0
\(287\) −44.3744 −2.61934
\(288\) 0 0
\(289\) 7.82064 0.460037
\(290\) 0 0
\(291\) −11.5482 −0.676967
\(292\) 0 0
\(293\) −11.9575 −0.698565 −0.349283 0.937017i \(-0.613575\pi\)
−0.349283 + 0.937017i \(0.613575\pi\)
\(294\) 0 0
\(295\) 0.103915 0.00605015
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 35.5232 2.05436
\(300\) 0 0
\(301\) −55.6801 −3.20935
\(302\) 0 0
\(303\) 4.08320 0.234574
\(304\) 0 0
\(305\) −10.1513 −0.581262
\(306\) 0 0
\(307\) 16.8325 0.960680 0.480340 0.877082i \(-0.340513\pi\)
0.480340 + 0.877082i \(0.340513\pi\)
\(308\) 0 0
\(309\) 0.762681 0.0433874
\(310\) 0 0
\(311\) −24.5949 −1.39465 −0.697324 0.716756i \(-0.745626\pi\)
−0.697324 + 0.716756i \(0.745626\pi\)
\(312\) 0 0
\(313\) 7.20995 0.407530 0.203765 0.979020i \(-0.434682\pi\)
0.203765 + 0.979020i \(0.434682\pi\)
\(314\) 0 0
\(315\) 10.8892 0.613536
\(316\) 0 0
\(317\) −8.55901 −0.480722 −0.240361 0.970684i \(-0.577266\pi\)
−0.240361 + 0.970684i \(0.577266\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.4712 0.696077
\(322\) 0 0
\(323\) −15.5497 −0.865207
\(324\) 0 0
\(325\) −3.78308 −0.209848
\(326\) 0 0
\(327\) 2.86185 0.158260
\(328\) 0 0
\(329\) 7.90393 0.435758
\(330\) 0 0
\(331\) 25.4094 1.39662 0.698312 0.715793i \(-0.253935\pi\)
0.698312 + 0.715793i \(0.253935\pi\)
\(332\) 0 0
\(333\) 7.89189 0.432473
\(334\) 0 0
\(335\) −4.99277 −0.272784
\(336\) 0 0
\(337\) 18.9403 1.03175 0.515873 0.856665i \(-0.327468\pi\)
0.515873 + 0.856665i \(0.327468\pi\)
\(338\) 0 0
\(339\) 8.57934 0.465966
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 41.0446 2.21620
\(344\) 0 0
\(345\) −7.91706 −0.426240
\(346\) 0 0
\(347\) −7.89209 −0.423670 −0.211835 0.977305i \(-0.567944\pi\)
−0.211835 + 0.977305i \(0.567944\pi\)
\(348\) 0 0
\(349\) −7.45858 −0.399249 −0.199624 0.979873i \(-0.563972\pi\)
−0.199624 + 0.979873i \(0.563972\pi\)
\(350\) 0 0
\(351\) −16.8705 −0.900478
\(352\) 0 0
\(353\) −2.03405 −0.108262 −0.0541308 0.998534i \(-0.517239\pi\)
−0.0541308 + 0.998534i \(0.517239\pi\)
\(354\) 0 0
\(355\) −6.22929 −0.330616
\(356\) 0 0
\(357\) 19.9816 1.05754
\(358\) 0 0
\(359\) 16.3025 0.860413 0.430206 0.902731i \(-0.358441\pi\)
0.430206 + 0.902731i \(0.358441\pi\)
\(360\) 0 0
\(361\) −9.25841 −0.487285
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −7.30650 −0.382440
\(366\) 0 0
\(367\) 0.770896 0.0402404 0.0201202 0.999798i \(-0.493595\pi\)
0.0201202 + 0.999798i \(0.493595\pi\)
\(368\) 0 0
\(369\) 21.3538 1.11163
\(370\) 0 0
\(371\) −9.22817 −0.479103
\(372\) 0 0
\(373\) 5.01958 0.259904 0.129952 0.991520i \(-0.458518\pi\)
0.129952 + 0.991520i \(0.458518\pi\)
\(374\) 0 0
\(375\) 0.843136 0.0435394
\(376\) 0 0
\(377\) −9.86222 −0.507930
\(378\) 0 0
\(379\) −3.65481 −0.187735 −0.0938676 0.995585i \(-0.529923\pi\)
−0.0938676 + 0.995585i \(0.529923\pi\)
\(380\) 0 0
\(381\) 9.23970 0.473364
\(382\) 0 0
\(383\) −5.48146 −0.280090 −0.140045 0.990145i \(-0.544725\pi\)
−0.140045 + 0.990145i \(0.544725\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 26.7943 1.36203
\(388\) 0 0
\(389\) 0.840557 0.0426179 0.0213090 0.999773i \(-0.493217\pi\)
0.0213090 + 0.999773i \(0.493217\pi\)
\(390\) 0 0
\(391\) 46.7813 2.36583
\(392\) 0 0
\(393\) −10.8782 −0.548734
\(394\) 0 0
\(395\) −10.5445 −0.530552
\(396\) 0 0
\(397\) 3.61935 0.181650 0.0908249 0.995867i \(-0.471050\pi\)
0.0908249 + 0.995867i \(0.471050\pi\)
\(398\) 0 0
\(399\) −12.5181 −0.626690
\(400\) 0 0
\(401\) 10.0879 0.503768 0.251884 0.967757i \(-0.418950\pi\)
0.251884 + 0.967757i \(0.418950\pi\)
\(402\) 0 0
\(403\) −13.9546 −0.695129
\(404\) 0 0
\(405\) −3.10744 −0.154410
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −12.3840 −0.612352 −0.306176 0.951975i \(-0.599050\pi\)
−0.306176 + 0.951975i \(0.599050\pi\)
\(410\) 0 0
\(411\) −15.9205 −0.785299
\(412\) 0 0
\(413\) −0.494315 −0.0243237
\(414\) 0 0
\(415\) −5.69010 −0.279316
\(416\) 0 0
\(417\) −9.18109 −0.449600
\(418\) 0 0
\(419\) 0.612942 0.0299442 0.0149721 0.999888i \(-0.495234\pi\)
0.0149721 + 0.999888i \(0.495234\pi\)
\(420\) 0 0
\(421\) −33.5449 −1.63488 −0.817440 0.576013i \(-0.804608\pi\)
−0.817440 + 0.576013i \(0.804608\pi\)
\(422\) 0 0
\(423\) −3.80352 −0.184933
\(424\) 0 0
\(425\) −4.98203 −0.241664
\(426\) 0 0
\(427\) 48.2890 2.33687
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.72012 0.420034 0.210017 0.977698i \(-0.432648\pi\)
0.210017 + 0.977698i \(0.432648\pi\)
\(432\) 0 0
\(433\) −11.3703 −0.546420 −0.273210 0.961954i \(-0.588085\pi\)
−0.273210 + 0.961954i \(0.588085\pi\)
\(434\) 0 0
\(435\) 2.19800 0.105386
\(436\) 0 0
\(437\) −29.3076 −1.40197
\(438\) 0 0
\(439\) 24.3176 1.16061 0.580307 0.814398i \(-0.302933\pi\)
0.580307 + 0.814398i \(0.302933\pi\)
\(440\) 0 0
\(441\) −35.7752 −1.70358
\(442\) 0 0
\(443\) 0.227915 0.0108285 0.00541427 0.999985i \(-0.498277\pi\)
0.00541427 + 0.999985i \(0.498277\pi\)
\(444\) 0 0
\(445\) −13.6371 −0.646463
\(446\) 0 0
\(447\) 1.42838 0.0675600
\(448\) 0 0
\(449\) 40.4036 1.90676 0.953382 0.301767i \(-0.0975764\pi\)
0.953382 + 0.301767i \(0.0975764\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.94241 0.138247
\(454\) 0 0
\(455\) 17.9958 0.843658
\(456\) 0 0
\(457\) −4.54698 −0.212699 −0.106349 0.994329i \(-0.533916\pi\)
−0.106349 + 0.994329i \(0.533916\pi\)
\(458\) 0 0
\(459\) −22.2171 −1.03701
\(460\) 0 0
\(461\) 36.5792 1.70366 0.851832 0.523815i \(-0.175492\pi\)
0.851832 + 0.523815i \(0.175492\pi\)
\(462\) 0 0
\(463\) 5.18502 0.240968 0.120484 0.992715i \(-0.461555\pi\)
0.120484 + 0.992715i \(0.461555\pi\)
\(464\) 0 0
\(465\) 3.11007 0.144226
\(466\) 0 0
\(467\) 10.0802 0.466454 0.233227 0.972422i \(-0.425071\pi\)
0.233227 + 0.972422i \(0.425071\pi\)
\(468\) 0 0
\(469\) 23.7503 1.09668
\(470\) 0 0
\(471\) −17.4532 −0.804201
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.12115 0.143208
\(476\) 0 0
\(477\) 4.44077 0.203329
\(478\) 0 0
\(479\) 2.56337 0.117123 0.0585617 0.998284i \(-0.481349\pi\)
0.0585617 + 0.998284i \(0.481349\pi\)
\(480\) 0 0
\(481\) 13.0424 0.594683
\(482\) 0 0
\(483\) 37.6609 1.71363
\(484\) 0 0
\(485\) −13.6967 −0.621935
\(486\) 0 0
\(487\) −2.41271 −0.109330 −0.0546651 0.998505i \(-0.517409\pi\)
−0.0546651 + 0.998505i \(0.517409\pi\)
\(488\) 0 0
\(489\) 13.8132 0.624656
\(490\) 0 0
\(491\) 15.0326 0.678412 0.339206 0.940712i \(-0.389842\pi\)
0.339206 + 0.940712i \(0.389842\pi\)
\(492\) 0 0
\(493\) −12.9878 −0.584941
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29.6323 1.32919
\(498\) 0 0
\(499\) −7.43047 −0.332634 −0.166317 0.986072i \(-0.553187\pi\)
−0.166317 + 0.986072i \(0.553187\pi\)
\(500\) 0 0
\(501\) −12.1422 −0.542472
\(502\) 0 0
\(503\) 15.1978 0.677637 0.338818 0.940852i \(-0.389973\pi\)
0.338818 + 0.940852i \(0.389973\pi\)
\(504\) 0 0
\(505\) 4.84287 0.215505
\(506\) 0 0
\(507\) −1.10594 −0.0491164
\(508\) 0 0
\(509\) −23.0293 −1.02076 −0.510379 0.859950i \(-0.670495\pi\)
−0.510379 + 0.859950i \(0.670495\pi\)
\(510\) 0 0
\(511\) 34.7565 1.53754
\(512\) 0 0
\(513\) 13.9186 0.614522
\(514\) 0 0
\(515\) 0.904576 0.0398604
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −6.84925 −0.300649
\(520\) 0 0
\(521\) 1.10634 0.0484698 0.0242349 0.999706i \(-0.492285\pi\)
0.0242349 + 0.999706i \(0.492285\pi\)
\(522\) 0 0
\(523\) −1.16929 −0.0511294 −0.0255647 0.999673i \(-0.508138\pi\)
−0.0255647 + 0.999673i \(0.508138\pi\)
\(524\) 0 0
\(525\) −4.01074 −0.175043
\(526\) 0 0
\(527\) −18.3772 −0.800522
\(528\) 0 0
\(529\) 65.1723 2.83358
\(530\) 0 0
\(531\) 0.237873 0.0103228
\(532\) 0 0
\(533\) 35.2900 1.52858
\(534\) 0 0
\(535\) 14.7915 0.639492
\(536\) 0 0
\(537\) 2.52384 0.108912
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3.43127 0.147522 0.0737609 0.997276i \(-0.476500\pi\)
0.0737609 + 0.997276i \(0.476500\pi\)
\(542\) 0 0
\(543\) 12.2477 0.525601
\(544\) 0 0
\(545\) 3.39429 0.145395
\(546\) 0 0
\(547\) −41.9440 −1.79340 −0.896699 0.442642i \(-0.854041\pi\)
−0.896699 + 0.442642i \(0.854041\pi\)
\(548\) 0 0
\(549\) −23.2376 −0.991755
\(550\) 0 0
\(551\) 8.13662 0.346632
\(552\) 0 0
\(553\) 50.1595 2.13300
\(554\) 0 0
\(555\) −2.90677 −0.123385
\(556\) 0 0
\(557\) 32.5066 1.37735 0.688675 0.725070i \(-0.258193\pi\)
0.688675 + 0.725070i \(0.258193\pi\)
\(558\) 0 0
\(559\) 44.2812 1.87289
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −37.2583 −1.57025 −0.785124 0.619338i \(-0.787401\pi\)
−0.785124 + 0.619338i \(0.787401\pi\)
\(564\) 0 0
\(565\) 10.1755 0.428087
\(566\) 0 0
\(567\) 14.7819 0.620780
\(568\) 0 0
\(569\) 37.5360 1.57359 0.786796 0.617213i \(-0.211738\pi\)
0.786796 + 0.617213i \(0.211738\pi\)
\(570\) 0 0
\(571\) −21.6905 −0.907718 −0.453859 0.891073i \(-0.649953\pi\)
−0.453859 + 0.891073i \(0.649953\pi\)
\(572\) 0 0
\(573\) −6.71762 −0.280632
\(574\) 0 0
\(575\) −9.39001 −0.391590
\(576\) 0 0
\(577\) −26.9306 −1.12114 −0.560568 0.828108i \(-0.689417\pi\)
−0.560568 + 0.828108i \(0.689417\pi\)
\(578\) 0 0
\(579\) −16.4380 −0.683138
\(580\) 0 0
\(581\) 27.0674 1.12294
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −8.65993 −0.358044
\(586\) 0 0
\(587\) 15.4757 0.638751 0.319376 0.947628i \(-0.396527\pi\)
0.319376 + 0.947628i \(0.396527\pi\)
\(588\) 0 0
\(589\) 11.5130 0.474383
\(590\) 0 0
\(591\) 15.1554 0.623412
\(592\) 0 0
\(593\) −24.7646 −1.01696 −0.508481 0.861073i \(-0.669793\pi\)
−0.508481 + 0.861073i \(0.669793\pi\)
\(594\) 0 0
\(595\) 23.6992 0.971571
\(596\) 0 0
\(597\) −6.05133 −0.247665
\(598\) 0 0
\(599\) −12.9401 −0.528716 −0.264358 0.964425i \(-0.585160\pi\)
−0.264358 + 0.964425i \(0.585160\pi\)
\(600\) 0 0
\(601\) 18.7917 0.766531 0.383266 0.923638i \(-0.374799\pi\)
0.383266 + 0.923638i \(0.374799\pi\)
\(602\) 0 0
\(603\) −11.4291 −0.465427
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −14.3465 −0.582308 −0.291154 0.956676i \(-0.594039\pi\)
−0.291154 + 0.956676i \(0.594039\pi\)
\(608\) 0 0
\(609\) −10.4557 −0.423687
\(610\) 0 0
\(611\) −6.28582 −0.254297
\(612\) 0 0
\(613\) 39.4276 1.59247 0.796233 0.604990i \(-0.206823\pi\)
0.796233 + 0.604990i \(0.206823\pi\)
\(614\) 0 0
\(615\) −7.86509 −0.317151
\(616\) 0 0
\(617\) 32.9316 1.32578 0.662888 0.748718i \(-0.269330\pi\)
0.662888 + 0.748718i \(0.269330\pi\)
\(618\) 0 0
\(619\) 41.2089 1.65632 0.828162 0.560489i \(-0.189387\pi\)
0.828162 + 0.560489i \(0.189387\pi\)
\(620\) 0 0
\(621\) −41.8743 −1.68036
\(622\) 0 0
\(623\) 64.8709 2.59900
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.1759 0.684847
\(630\) 0 0
\(631\) −7.79380 −0.310266 −0.155133 0.987894i \(-0.549581\pi\)
−0.155133 + 0.987894i \(0.549581\pi\)
\(632\) 0 0
\(633\) 3.73749 0.148552
\(634\) 0 0
\(635\) 10.9587 0.434884
\(636\) 0 0
\(637\) −59.1234 −2.34255
\(638\) 0 0
\(639\) −14.2596 −0.564101
\(640\) 0 0
\(641\) −16.9699 −0.670269 −0.335135 0.942170i \(-0.608782\pi\)
−0.335135 + 0.942170i \(0.608782\pi\)
\(642\) 0 0
\(643\) −0.346553 −0.0136667 −0.00683336 0.999977i \(-0.502175\pi\)
−0.00683336 + 0.999977i \(0.502175\pi\)
\(644\) 0 0
\(645\) −9.86896 −0.388590
\(646\) 0 0
\(647\) −8.23214 −0.323639 −0.161819 0.986820i \(-0.551736\pi\)
−0.161819 + 0.986820i \(0.551736\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −14.7944 −0.579837
\(652\) 0 0
\(653\) 32.9455 1.28926 0.644629 0.764496i \(-0.277012\pi\)
0.644629 + 0.764496i \(0.277012\pi\)
\(654\) 0 0
\(655\) −12.9021 −0.504127
\(656\) 0 0
\(657\) −16.7255 −0.652522
\(658\) 0 0
\(659\) 18.7687 0.731127 0.365563 0.930786i \(-0.380876\pi\)
0.365563 + 0.930786i \(0.380876\pi\)
\(660\) 0 0
\(661\) 8.66066 0.336860 0.168430 0.985714i \(-0.446130\pi\)
0.168430 + 0.985714i \(0.446130\pi\)
\(662\) 0 0
\(663\) −15.8909 −0.617153
\(664\) 0 0
\(665\) −14.8471 −0.575746
\(666\) 0 0
\(667\) −24.4791 −0.947834
\(668\) 0 0
\(669\) −12.8246 −0.495826
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 11.7088 0.451341 0.225670 0.974204i \(-0.427543\pi\)
0.225670 + 0.974204i \(0.427543\pi\)
\(674\) 0 0
\(675\) 4.45945 0.171644
\(676\) 0 0
\(677\) 31.0721 1.19420 0.597099 0.802167i \(-0.296320\pi\)
0.597099 + 0.802167i \(0.296320\pi\)
\(678\) 0 0
\(679\) 65.1542 2.50039
\(680\) 0 0
\(681\) −3.61198 −0.138411
\(682\) 0 0
\(683\) 31.0210 1.18698 0.593492 0.804840i \(-0.297749\pi\)
0.593492 + 0.804840i \(0.297749\pi\)
\(684\) 0 0
\(685\) −18.8824 −0.721461
\(686\) 0 0
\(687\) 3.45395 0.131776
\(688\) 0 0
\(689\) 7.33896 0.279592
\(690\) 0 0
\(691\) −41.9523 −1.59594 −0.797969 0.602698i \(-0.794092\pi\)
−0.797969 + 0.602698i \(0.794092\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.8892 −0.413051
\(696\) 0 0
\(697\) 46.4742 1.76034
\(698\) 0 0
\(699\) −14.5467 −0.550206
\(700\) 0 0
\(701\) 9.77487 0.369192 0.184596 0.982815i \(-0.440902\pi\)
0.184596 + 0.982815i \(0.440902\pi\)
\(702\) 0 0
\(703\) −10.7604 −0.405835
\(704\) 0 0
\(705\) 1.40092 0.0527618
\(706\) 0 0
\(707\) −23.0372 −0.866404
\(708\) 0 0
\(709\) 17.6328 0.662214 0.331107 0.943593i \(-0.392578\pi\)
0.331107 + 0.943593i \(0.392578\pi\)
\(710\) 0 0
\(711\) −24.1377 −0.905233
\(712\) 0 0
\(713\) −34.6368 −1.29716
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.87194 0.107255
\(718\) 0 0
\(719\) −51.4088 −1.91723 −0.958613 0.284713i \(-0.908102\pi\)
−0.958613 + 0.284713i \(0.908102\pi\)
\(720\) 0 0
\(721\) −4.30300 −0.160252
\(722\) 0 0
\(723\) −17.0178 −0.632898
\(724\) 0 0
\(725\) 2.60693 0.0968189
\(726\) 0 0
\(727\) 17.3527 0.643577 0.321788 0.946812i \(-0.395716\pi\)
0.321788 + 0.946812i \(0.395716\pi\)
\(728\) 0 0
\(729\) 4.16647 0.154314
\(730\) 0 0
\(731\) 58.3150 2.15686
\(732\) 0 0
\(733\) −42.8536 −1.58284 −0.791418 0.611276i \(-0.790657\pi\)
−0.791418 + 0.611276i \(0.790657\pi\)
\(734\) 0 0
\(735\) 13.1768 0.486035
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −24.5873 −0.904458 −0.452229 0.891902i \(-0.649371\pi\)
−0.452229 + 0.891902i \(0.649371\pi\)
\(740\) 0 0
\(741\) 9.95539 0.365720
\(742\) 0 0
\(743\) −4.69027 −0.172069 −0.0860347 0.996292i \(-0.527420\pi\)
−0.0860347 + 0.996292i \(0.527420\pi\)
\(744\) 0 0
\(745\) 1.69412 0.0620679
\(746\) 0 0
\(747\) −13.0253 −0.476571
\(748\) 0 0
\(749\) −70.3621 −2.57097
\(750\) 0 0
\(751\) 48.2922 1.76221 0.881103 0.472924i \(-0.156801\pi\)
0.881103 + 0.472924i \(0.156801\pi\)
\(752\) 0 0
\(753\) 16.6073 0.605205
\(754\) 0 0
\(755\) 3.48984 0.127008
\(756\) 0 0
\(757\) 38.8242 1.41109 0.705544 0.708666i \(-0.250703\pi\)
0.705544 + 0.708666i \(0.250703\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.8876 1.01093 0.505463 0.862848i \(-0.331322\pi\)
0.505463 + 0.862848i \(0.331322\pi\)
\(762\) 0 0
\(763\) −16.1464 −0.584538
\(764\) 0 0
\(765\) −11.4045 −0.412330
\(766\) 0 0
\(767\) 0.393118 0.0141947
\(768\) 0 0
\(769\) 38.2240 1.37839 0.689197 0.724574i \(-0.257964\pi\)
0.689197 + 0.724574i \(0.257964\pi\)
\(770\) 0 0
\(771\) 9.03288 0.325311
\(772\) 0 0
\(773\) 8.39582 0.301977 0.150988 0.988536i \(-0.451754\pi\)
0.150988 + 0.988536i \(0.451754\pi\)
\(774\) 0 0
\(775\) 3.68869 0.132502
\(776\) 0 0
\(777\) 13.8273 0.496051
\(778\) 0 0
\(779\) −29.1153 −1.04316
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 11.6255 0.415460
\(784\) 0 0
\(785\) −20.7003 −0.738826
\(786\) 0 0
\(787\) 35.4248 1.26276 0.631379 0.775474i \(-0.282489\pi\)
0.631379 + 0.775474i \(0.282489\pi\)
\(788\) 0 0
\(789\) −7.47286 −0.266041
\(790\) 0 0
\(791\) −48.4042 −1.72105
\(792\) 0 0
\(793\) −38.4032 −1.36374
\(794\) 0 0
\(795\) −1.63564 −0.0580101
\(796\) 0 0
\(797\) 53.3896 1.89116 0.945579 0.325392i \(-0.105496\pi\)
0.945579 + 0.325392i \(0.105496\pi\)
\(798\) 0 0
\(799\) −8.27796 −0.292853
\(800\) 0 0
\(801\) −31.2171 −1.10300
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 44.6676 1.57433
\(806\) 0 0
\(807\) 1.81680 0.0639543
\(808\) 0 0
\(809\) 3.84086 0.135038 0.0675188 0.997718i \(-0.478492\pi\)
0.0675188 + 0.997718i \(0.478492\pi\)
\(810\) 0 0
\(811\) 23.7250 0.833097 0.416548 0.909114i \(-0.363240\pi\)
0.416548 + 0.909114i \(0.363240\pi\)
\(812\) 0 0
\(813\) −8.64657 −0.303248
\(814\) 0 0
\(815\) 16.3831 0.573877
\(816\) 0 0
\(817\) −36.5333 −1.27814
\(818\) 0 0
\(819\) 41.1947 1.43946
\(820\) 0 0
\(821\) −0.152300 −0.00531531 −0.00265765 0.999996i \(-0.500846\pi\)
−0.00265765 + 0.999996i \(0.500846\pi\)
\(822\) 0 0
\(823\) 50.1736 1.74894 0.874472 0.485077i \(-0.161208\pi\)
0.874472 + 0.485077i \(0.161208\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.2010 0.772006 0.386003 0.922498i \(-0.373856\pi\)
0.386003 + 0.922498i \(0.373856\pi\)
\(828\) 0 0
\(829\) 8.18588 0.284307 0.142154 0.989845i \(-0.454597\pi\)
0.142154 + 0.989845i \(0.454597\pi\)
\(830\) 0 0
\(831\) 12.4998 0.433612
\(832\) 0 0
\(833\) −77.8610 −2.69773
\(834\) 0 0
\(835\) −14.4012 −0.498374
\(836\) 0 0
\(837\) 16.4495 0.568579
\(838\) 0 0
\(839\) −7.95392 −0.274600 −0.137300 0.990530i \(-0.543842\pi\)
−0.137300 + 0.990530i \(0.543842\pi\)
\(840\) 0 0
\(841\) −22.2039 −0.765653
\(842\) 0 0
\(843\) −14.2043 −0.489223
\(844\) 0 0
\(845\) −1.31169 −0.0451237
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.12650 0.141621
\(850\) 0 0
\(851\) 32.3727 1.10972
\(852\) 0 0
\(853\) 5.87193 0.201051 0.100525 0.994934i \(-0.467948\pi\)
0.100525 + 0.994934i \(0.467948\pi\)
\(854\) 0 0
\(855\) 7.14470 0.244343
\(856\) 0 0
\(857\) −30.9089 −1.05583 −0.527914 0.849298i \(-0.677026\pi\)
−0.527914 + 0.849298i \(0.677026\pi\)
\(858\) 0 0
\(859\) 10.9133 0.372358 0.186179 0.982516i \(-0.440390\pi\)
0.186179 + 0.982516i \(0.440390\pi\)
\(860\) 0 0
\(861\) 37.4137 1.27505
\(862\) 0 0
\(863\) 12.3339 0.419850 0.209925 0.977718i \(-0.432678\pi\)
0.209925 + 0.977718i \(0.432678\pi\)
\(864\) 0 0
\(865\) −8.12354 −0.276209
\(866\) 0 0
\(867\) −6.59386 −0.223939
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −18.8881 −0.639997
\(872\) 0 0
\(873\) −31.3534 −1.06115
\(874\) 0 0
\(875\) −4.75693 −0.160814
\(876\) 0 0
\(877\) 45.3557 1.53155 0.765777 0.643106i \(-0.222355\pi\)
0.765777 + 0.643106i \(0.222355\pi\)
\(878\) 0 0
\(879\) 10.0818 0.340051
\(880\) 0 0
\(881\) −24.4098 −0.822388 −0.411194 0.911548i \(-0.634888\pi\)
−0.411194 + 0.911548i \(0.634888\pi\)
\(882\) 0 0
\(883\) −34.9932 −1.17761 −0.588807 0.808274i \(-0.700402\pi\)
−0.588807 + 0.808274i \(0.700402\pi\)
\(884\) 0 0
\(885\) −0.0876143 −0.00294512
\(886\) 0 0
\(887\) −44.8014 −1.50428 −0.752142 0.659001i \(-0.770979\pi\)
−0.752142 + 0.659001i \(0.770979\pi\)
\(888\) 0 0
\(889\) −52.1299 −1.74838
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.18599 0.173542
\(894\) 0 0
\(895\) 2.99340 0.100058
\(896\) 0 0
\(897\) −29.9509 −1.00003
\(898\) 0 0
\(899\) 9.61615 0.320717
\(900\) 0 0
\(901\) 9.66486 0.321983
\(902\) 0 0
\(903\) 46.9459 1.56226
\(904\) 0 0
\(905\) 14.5264 0.482874
\(906\) 0 0
\(907\) −6.93726 −0.230348 −0.115174 0.993345i \(-0.536743\pi\)
−0.115174 + 0.993345i \(0.536743\pi\)
\(908\) 0 0
\(909\) 11.0859 0.367697
\(910\) 0 0
\(911\) 26.7451 0.886106 0.443053 0.896496i \(-0.353895\pi\)
0.443053 + 0.896496i \(0.353895\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 8.55893 0.282950
\(916\) 0 0
\(917\) 61.3744 2.02676
\(918\) 0 0
\(919\) 55.1263 1.81845 0.909225 0.416305i \(-0.136675\pi\)
0.909225 + 0.416305i \(0.136675\pi\)
\(920\) 0 0
\(921\) −14.1921 −0.467645
\(922\) 0 0
\(923\) −23.5659 −0.775681
\(924\) 0 0
\(925\) −3.44757 −0.113355
\(926\) 0 0
\(927\) 2.07068 0.0680102
\(928\) 0 0
\(929\) 13.6870 0.449054 0.224527 0.974468i \(-0.427916\pi\)
0.224527 + 0.974468i \(0.427916\pi\)
\(930\) 0 0
\(931\) 48.7785 1.59865
\(932\) 0 0
\(933\) 20.7368 0.678894
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −18.9104 −0.617774 −0.308887 0.951099i \(-0.599957\pi\)
−0.308887 + 0.951099i \(0.599957\pi\)
\(938\) 0 0
\(939\) −6.07897 −0.198380
\(940\) 0 0
\(941\) 10.9993 0.358569 0.179284 0.983797i \(-0.442622\pi\)
0.179284 + 0.983797i \(0.442622\pi\)
\(942\) 0 0
\(943\) 87.5935 2.85244
\(944\) 0 0
\(945\) −21.2133 −0.690068
\(946\) 0 0
\(947\) 11.9776 0.389219 0.194610 0.980881i \(-0.437656\pi\)
0.194610 + 0.980881i \(0.437656\pi\)
\(948\) 0 0
\(949\) −27.6411 −0.897267
\(950\) 0 0
\(951\) 7.21642 0.234008
\(952\) 0 0
\(953\) 26.2762 0.851170 0.425585 0.904918i \(-0.360068\pi\)
0.425585 + 0.904918i \(0.360068\pi\)
\(954\) 0 0
\(955\) −7.96741 −0.257819
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 89.8224 2.90052
\(960\) 0 0
\(961\) −17.3936 −0.561083
\(962\) 0 0
\(963\) 33.8595 1.09111
\(964\) 0 0
\(965\) −19.4962 −0.627605
\(966\) 0 0
\(967\) −11.2656 −0.362277 −0.181138 0.983458i \(-0.557978\pi\)
−0.181138 + 0.983458i \(0.557978\pi\)
\(968\) 0 0
\(969\) 13.1105 0.421170
\(970\) 0 0
\(971\) 57.0037 1.82934 0.914669 0.404204i \(-0.132452\pi\)
0.914669 + 0.404204i \(0.132452\pi\)
\(972\) 0 0
\(973\) 51.7992 1.66061
\(974\) 0 0
\(975\) 3.18965 0.102151
\(976\) 0 0
\(977\) −34.9823 −1.11918 −0.559591 0.828769i \(-0.689042\pi\)
−0.559591 + 0.828769i \(0.689042\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 7.76993 0.248075
\(982\) 0 0
\(983\) −17.9629 −0.572927 −0.286464 0.958091i \(-0.592480\pi\)
−0.286464 + 0.958091i \(0.592480\pi\)
\(984\) 0 0
\(985\) 17.9751 0.572734
\(986\) 0 0
\(987\) −6.66409 −0.212120
\(988\) 0 0
\(989\) 109.911 3.49495
\(990\) 0 0
\(991\) −47.2159 −1.49986 −0.749931 0.661516i \(-0.769913\pi\)
−0.749931 + 0.661516i \(0.769913\pi\)
\(992\) 0 0
\(993\) −21.4235 −0.679856
\(994\) 0 0
\(995\) −7.17717 −0.227532
\(996\) 0 0
\(997\) 20.1669 0.638693 0.319347 0.947638i \(-0.396537\pi\)
0.319347 + 0.947638i \(0.396537\pi\)
\(998\) 0 0
\(999\) −15.3742 −0.486420
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 9680.2.a.cy.1.3 6
4.3 odd 2 4840.2.a.be.1.4 6
11.2 odd 10 880.2.bo.j.81.2 12
11.6 odd 10 880.2.bo.j.641.2 12
11.10 odd 2 9680.2.a.cx.1.3 6
44.35 even 10 440.2.y.b.81.2 12
44.39 even 10 440.2.y.b.201.2 yes 12
44.43 even 2 4840.2.a.bf.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.2 12 44.35 even 10
440.2.y.b.201.2 yes 12 44.39 even 10
880.2.bo.j.81.2 12 11.2 odd 10
880.2.bo.j.641.2 12 11.6 odd 10
4840.2.a.be.1.4 6 4.3 odd 2
4840.2.a.bf.1.4 6 44.43 even 2
9680.2.a.cx.1.3 6 11.10 odd 2
9680.2.a.cy.1.3 6 1.1 even 1 trivial