Properties

Label 9450.2.a.dd.1.1
Level $9450$
Weight $2$
Character 9450.1
Self dual yes
Analytic conductor $75.459$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9450,2,Mod(1,9450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, 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("9450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9450.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: \(75.4586299101\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -5.00000 q^{11} +5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} -8.00000 q^{19} -5.00000 q^{22} +1.00000 q^{23} +5.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{32} +8.00000 q^{34} -3.00000 q^{37} -8.00000 q^{38} -6.00000 q^{41} +4.00000 q^{43} -5.00000 q^{44} +1.00000 q^{46} +9.00000 q^{47} +1.00000 q^{49} +5.00000 q^{52} +8.00000 q^{53} +1.00000 q^{56} +2.00000 q^{58} -3.00000 q^{59} +5.00000 q^{61} +1.00000 q^{64} +12.0000 q^{67} +8.00000 q^{68} -1.00000 q^{71} +2.00000 q^{73} -3.00000 q^{74} -8.00000 q^{76} -5.00000 q^{77} +10.0000 q^{79} -6.00000 q^{82} -12.0000 q^{83} +4.00000 q^{86} -5.00000 q^{88} -10.0000 q^{89} +5.00000 q^{91} +1.00000 q^{92} +9.00000 q^{94} +13.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −3.00000 −0.348743
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −5.00000 −0.533002
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −3.00000 −0.276172
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 5.00000 0.452679
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.00000 −0.0839181
\(143\) −25.0000 −2.09061
\(144\) 0 0
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) −23.0000 −1.77979 −0.889897 0.456162i \(-0.849224\pi\)
−0.889897 + 0.456162i \(0.849224\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 11.0000 0.822179 0.411089 0.911595i \(-0.365148\pi\)
0.411089 + 0.911595i \(0.365148\pi\)
\(180\) 0 0
\(181\) 3.00000 0.222988 0.111494 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) 5.00000 0.370625
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −40.0000 −2.92509
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 13.0000 0.933346
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 40.0000 2.76686
\(210\) 0 0
\(211\) −18.0000 −1.23917 −0.619586 0.784929i \(-0.712699\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(212\) 8.00000 0.549442
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 0 0
\(221\) 40.0000 2.69069
\(222\) 0 0
\(223\) 28.0000 1.87502 0.937509 0.347960i \(-0.113126\pi\)
0.937509 + 0.347960i \(0.113126\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 19.0000 1.26107 0.630537 0.776159i \(-0.282835\pi\)
0.630537 + 0.776159i \(0.282835\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 0 0
\(247\) −40.0000 −2.54514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 1.00000 0.0617802
\(263\) 29.0000 1.78822 0.894108 0.447851i \(-0.147810\pi\)
0.894108 + 0.447851i \(0.147810\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 14.0000 0.839664
\(279\) 0 0
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 26.0000 1.54554 0.772770 0.634686i \(-0.218871\pi\)
0.772770 + 0.634686i \(0.218871\pi\)
\(284\) −1.00000 −0.0593391
\(285\) 0 0
\(286\) −25.0000 −1.47828
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.00000 −0.174371
\(297\) 0 0
\(298\) 4.00000 0.231714
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) −5.00000 −0.284901
\(309\) 0 0
\(310\) 0 0
\(311\) −31.0000 −1.75785 −0.878924 0.476961i \(-0.841738\pi\)
−0.878924 + 0.476961i \(0.841738\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) −64.0000 −3.56106
\(324\) 0 0
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) −23.0000 −1.25850
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −5.00000 −0.266501
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 11.0000 0.581368
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 3.00000 0.157676
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) 0 0
\(366\) 0 0
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −40.0000 −2.06835
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −35.0000 −1.78842 −0.894208 0.447651i \(-0.852261\pi\)
−0.894208 + 0.447651i \(0.852261\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 13.0000 0.659975
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 0 0
\(397\) 27.0000 1.35509 0.677546 0.735481i \(-0.263044\pi\)
0.677546 + 0.735481i \(0.263044\pi\)
\(398\) 6.00000 0.300753
\(399\) 0 0
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 15.0000 0.743522
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 40.0000 1.95646
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) −18.0000 −0.876226
\(423\) 0 0
\(424\) 8.00000 0.388514
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 40.0000 1.90261
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 19.0000 0.891714
\(455\) 0 0
\(456\) 0 0
\(457\) 35.0000 1.63723 0.818615 0.574342i \(-0.194742\pi\)
0.818615 + 0.574342i \(0.194742\pi\)
\(458\) −1.00000 −0.0467269
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) −3.00000 −0.138086
\(473\) −20.0000 −0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) −3.00000 −0.137217
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) −5.00000 −0.227744
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 5.00000 0.226339
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) −40.0000 −1.79969
\(495\) 0 0
\(496\) 0 0
\(497\) −1.00000 −0.0448561
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −15.0000 −0.669483
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.00000 −0.222277
\(507\) 0 0
\(508\) −20.0000 −0.887357
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 28.0000 1.23503
\(515\) 0 0
\(516\) 0 0
\(517\) −45.0000 −1.97910
\(518\) −3.00000 −0.131812
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 1.00000 0.0436852
\(525\) 0 0
\(526\) 29.0000 1.26446
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −30.0000 −1.29944
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) 0 0
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) 17.0000 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 26.0000 1.09286
\(567\) 0 0
\(568\) −1.00000 −0.0419591
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −25.0000 −1.04530
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 47.0000 1.95494
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.00000 −0.123299
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 5.00000 0.204465
\(599\) −48.0000 −1.96123 −0.980613 0.195952i \(-0.937220\pi\)
−0.980613 + 0.195952i \(0.937220\pi\)
\(600\) 0 0
\(601\) 13.0000 0.530281 0.265141 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) 45.0000 1.82051
\(612\) 0 0
\(613\) 33.0000 1.33286 0.666429 0.745569i \(-0.267822\pi\)
0.666429 + 0.745569i \(0.267822\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −5.00000 −0.201456
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −31.0000 −1.24299
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) −10.0000 −0.395904
\(639\) 0 0
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) −64.0000 −2.51805
\(647\) −15.0000 −0.589711 −0.294855 0.955542i \(-0.595271\pi\)
−0.294855 + 0.955542i \(0.595271\pi\)
\(648\) 0 0
\(649\) 15.0000 0.588802
\(650\) 0 0
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 9.00000 0.350857
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −31.0000 −1.20576 −0.602880 0.797832i \(-0.705980\pi\)
−0.602880 + 0.797832i \(0.705980\pi\)
\(662\) 6.00000 0.233197
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) −23.0000 −0.889897
\(669\) 0 0
\(670\) 0 0
\(671\) −25.0000 −0.965114
\(672\) 0 0
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 0 0
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.0000 −1.26271 −0.631355 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 40.0000 1.52388
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 24.0000 0.905177
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 37.0000 1.38956 0.694782 0.719220i \(-0.255501\pi\)
0.694782 + 0.719220i \(0.255501\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 11.0000 0.411089
\(717\) 0 0
\(718\) −3.00000 −0.111959
\(719\) 19.0000 0.708580 0.354290 0.935136i \(-0.384723\pi\)
0.354290 + 0.935136i \(0.384723\pi\)
\(720\) 0 0
\(721\) 2.00000 0.0744839
\(722\) 45.0000 1.67473
\(723\) 0 0
\(724\) 3.00000 0.111494
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 5.00000 0.185312
\(729\) 0 0
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −7.00000 −0.258551 −0.129275 0.991609i \(-0.541265\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) −34.0000 −1.25496
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) 33.0000 1.21065 0.605326 0.795977i \(-0.293043\pi\)
0.605326 + 0.795977i \(0.293043\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) −40.0000 −1.46254
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 9.00000 0.328196
\(753\) 0 0
\(754\) 10.0000 0.364179
\(755\) 0 0
\(756\) 0 0
\(757\) −1.00000 −0.0363456 −0.0181728 0.999835i \(-0.505785\pi\)
−0.0181728 + 0.999835i \(0.505785\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −35.0000 −1.26460
\(767\) −15.0000 −0.541619
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 13.0000 0.466673
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 25.0000 0.887776
\(794\) 27.0000 0.958194
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) −28.0000 −0.988714
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) 16.0000 0.562530 0.281265 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 15.0000 0.525750
\(815\) 0 0
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 19.0000 0.664319
\(819\) 0 0
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) −3.00000 −0.104383
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 0 0
\(829\) 27.0000 0.937749 0.468874 0.883265i \(-0.344660\pi\)
0.468874 + 0.883265i \(0.344660\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 8.00000 0.277184
\(834\) 0 0
\(835\) 0 0
\(836\) 40.0000 1.38343
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) 7.00000 0.241667 0.120833 0.992673i \(-0.461443\pi\)
0.120833 + 0.992673i \(0.461443\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −35.0000 −1.20618
\(843\) 0 0
\(844\) −18.0000 −0.619586
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 8.00000 0.274721
\(849\) 0 0
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) −37.0000 −1.26686 −0.633428 0.773802i \(-0.718353\pi\)
−0.633428 + 0.773802i \(0.718353\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 44.0000 1.50301 0.751506 0.659727i \(-0.229328\pi\)
0.751506 + 0.659727i \(0.229328\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 21.0000 0.715263
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.00000 −0.0339814
\(867\) 0 0
\(868\) 0 0
\(869\) −50.0000 −1.69613
\(870\) 0 0
\(871\) 60.0000 2.03302
\(872\) 14.0000 0.474100
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 0 0
\(877\) −23.0000 −0.776655 −0.388327 0.921521i \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) 40.0000 1.34993
\(879\) 0 0
\(880\) 0 0
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 40.0000 1.34535
\(885\) 0 0
\(886\) 9.00000 0.302361
\(887\) −49.0000 −1.64526 −0.822629 0.568578i \(-0.807494\pi\)
−0.822629 + 0.568578i \(0.807494\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) −72.0000 −2.40939
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 0 0
\(900\) 0 0
\(901\) 64.0000 2.13215
\(902\) 30.0000 0.998891
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 19.0000 0.630537
\(909\) 0 0
\(910\) 0 0
\(911\) −11.0000 −0.364446 −0.182223 0.983257i \(-0.558329\pi\)
−0.182223 + 0.983257i \(0.558329\pi\)
\(912\) 0 0
\(913\) 60.0000 1.98571
\(914\) 35.0000 1.15770
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) 1.00000 0.0330229
\(918\) 0 0
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 10.0000 0.329332
\(923\) −5.00000 −0.164577
\(924\) 0 0
\(925\) 0 0
\(926\) 18.0000 0.591517
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 20.0000 0.656179 0.328089 0.944647i \(-0.393595\pi\)
0.328089 + 0.944647i \(0.393595\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 8.00000 0.262049
\(933\) 0 0
\(934\) −21.0000 −0.687141
\(935\) 0 0
\(936\) 0 0
\(937\) 11.0000 0.359354 0.179677 0.983726i \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −3.00000 −0.0970269
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −15.0000 −0.483619
\(963\) 0 0
\(964\) −5.00000 −0.161039
\(965\) 0 0
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) 0 0
\(971\) 53.0000 1.70085 0.850425 0.526096i \(-0.176345\pi\)
0.850425 + 0.526096i \(0.176345\pi\)
\(972\) 0 0
\(973\) 14.0000 0.448819
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 50.0000 1.59801
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) 0 0
\(988\) −40.0000 −1.27257
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −1.00000 −0.0317181
\(995\) 0 0
\(996\) 0 0
\(997\) 11.0000 0.348373 0.174187 0.984713i \(-0.444270\pi\)
0.174187 + 0.984713i \(0.444270\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 9450.2.a.dd.1.1 yes 1
3.2 odd 2 9450.2.a.by.1.1 yes 1
5.4 even 2 9450.2.a.c.1.1 1
15.14 odd 2 9450.2.a.cw.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9450.2.a.c.1.1 1 5.4 even 2
9450.2.a.by.1.1 yes 1 3.2 odd 2
9450.2.a.cw.1.1 yes 1 15.14 odd 2
9450.2.a.dd.1.1 yes 1 1.1 even 1 trivial