Properties

Label 8450.2.a.w.1.1
Level $8450$
Weight $2$
Character 8450.1
Self dual yes
Analytic conductor $67.474$
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] = [8450,2,Mod(1,8450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.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: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} +2.00000 q^{21} +3.00000 q^{22} +2.00000 q^{24} -4.00000 q^{27} +1.00000 q^{28} -4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -11.0000 q^{37} +5.00000 q^{38} +6.00000 q^{41} +2.00000 q^{42} -2.00000 q^{43} +3.00000 q^{44} +3.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} +12.0000 q^{51} +9.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +10.0000 q^{57} +8.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +16.0000 q^{67} +6.00000 q^{68} +6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -11.0000 q^{74} +5.00000 q^{76} +3.00000 q^{77} -16.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -2.00000 q^{86} +3.00000 q^{88} +9.00000 q^{89} -8.00000 q^{93} +3.00000 q^{94} +2.00000 q^{96} +10.0000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 2.00000 0.308607
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 2.00000 0.288675
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 16.0000 1.95471 0.977356 0.211604i \(-0.0678686\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −11.0000 −1.27872
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −6.00000 −0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 12.0000 1.18818
\(103\) −5.00000 −0.492665 −0.246332 0.969185i \(-0.579225\pi\)
−0.246332 + 0.969185i \(0.579225\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −22.0000 −2.08815
\(112\) 1.00000 0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 10.0000 0.936586
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 8.00000 0.724286
\(123\) 12.0000 1.08200
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 6.00000 0.522233
\(133\) 5.00000 0.433555
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −12.0000 −0.989743
\(148\) −11.0000 −0.904194
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 5.00000 0.405554
\(153\) 6.00000 0.485071
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) −17.0000 −1.35675 −0.678374 0.734717i \(-0.737315\pi\)
−0.678374 + 0.734717i \(0.737315\pi\)
\(158\) −16.0000 −1.27289
\(159\) 18.0000 1.42749
\(160\) 0 0
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 15.0000 1.16073 0.580367 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) −2.00000 −0.152499
\(173\) −15.0000 −1.14043 −0.570214 0.821496i \(-0.693140\pi\)
−0.570214 + 0.821496i \(0.693140\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 9.00000 0.674579
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 16.0000 1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 18.0000 1.31629
\(188\) 3.00000 0.218797
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 2.00000 0.144338
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 3.00000 0.213201
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 32.0000 2.25711
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −5.00000 −0.348367
\(207\) 0 0
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 9.00000 0.618123
\(213\) 12.0000 0.822226
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −4.00000 −0.271538
\(218\) 2.00000 0.135457
\(219\) −28.0000 −1.89206
\(220\) 0 0
\(221\) 0 0
\(222\) −22.0000 −1.47654
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 10.0000 0.662266
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −32.0000 −2.07862
\(238\) 6.00000 0.388922
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) −2.00000 −0.128565
\(243\) −10.0000 −0.641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 1.00000 0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −4.00000 −0.249029
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) 0 0
\(262\) −9.00000 −0.556022
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 5.00000 0.306570
\(267\) 18.0000 1.10158
\(268\) 16.0000 0.977356
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) −19.0000 −1.13954
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 6.00000 0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) −14.0000 −0.819288
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −12.0000 −0.699854
\(295\) 0 0
\(296\) −11.0000 −0.639362
\(297\) −12.0000 −0.696311
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) −16.0000 −0.920697
\(303\) −12.0000 −0.689382
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 3.00000 0.170941
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −17.0000 −0.959366
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 18.0000 1.00939
\(319\) 0 0
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −2.00000 −0.110770
\(327\) 4.00000 0.221201
\(328\) 6.00000 0.331295
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 6.00000 0.329293
\(333\) −11.0000 −0.602796
\(334\) 15.0000 0.820763
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) 24.0000 1.30350
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 5.00000 0.270369
\(343\) −13.0000 −0.701934
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −15.0000 −0.806405
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.00000 0.476999
\(357\) 12.0000 0.635107
\(358\) −24.0000 −1.26844
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 8.00000 0.420471
\(363\) −4.00000 −0.209946
\(364\) 0 0
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) −8.00000 −0.414781
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) −2.00000 −0.101666
\(388\) 10.0000 0.507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) −18.0000 −0.907980
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) −10.0000 −0.501255
\(399\) 10.0000 0.500626
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 32.0000 1.59601
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −33.0000 −1.63575
\(408\) 12.0000 0.594089
\(409\) 5.00000 0.247234 0.123617 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) −5.00000 −0.246332
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −38.0000 −1.86087
\(418\) 15.0000 0.733674
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 23.0000 1.11962
\(423\) 3.00000 0.145865
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −4.00000 −0.192450
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −28.0000 −1.33789
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −22.0000 −1.04407
\(445\) 0 0
\(446\) 19.0000 0.899676
\(447\) −36.0000 −1.70274
\(448\) 1.00000 0.0472456
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 12.0000 0.564433
\(453\) −32.0000 −1.50349
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 10.0000 0.468293
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) −4.00000 −0.186908
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 6.00000 0.279145
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −34.0000 −1.56664
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) −32.0000 −1.46981
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 23.0000 1.04762
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 19.0000 0.860972 0.430486 0.902597i \(-0.358342\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 8.00000 0.362143
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 27.0000 1.21849 0.609246 0.792981i \(-0.291472\pi\)
0.609246 + 0.792981i \(0.291472\pi\)
\(492\) 12.0000 0.541002
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 12.0000 0.537733
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 30.0000 1.34030
\(502\) −15.0000 −0.669483
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1.00000 0.0443678
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 1.00000 0.0441942
\(513\) −20.0000 −0.883022
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 9.00000 0.395820
\(518\) −11.0000 −0.483312
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −9.00000 −0.393167
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) −24.0000 −1.04546
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 5.00000 0.216777
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 16.0000 0.691095
\(537\) −48.0000 −2.07135
\(538\) 6.00000 0.258678
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 20.0000 0.859074
\(543\) 16.0000 0.686626
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) −6.00000 −0.256307
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) −21.0000 −0.889799 −0.444899 0.895581i \(-0.646761\pi\)
−0.444899 + 0.895581i \(0.646761\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) −6.00000 −0.253095
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) −11.0000 −0.461957
\(568\) 6.00000 0.251754
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) 17.0000 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 19.0000 0.790296
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 20.0000 0.829027
\(583\) 27.0000 1.11823
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) −12.0000 −0.494872
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 54.0000 2.22126
\(592\) −11.0000 −0.452097
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 16.0000 0.651570
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 13.0000 0.527654 0.263827 0.964570i \(-0.415015\pi\)
0.263827 + 0.964570i \(0.415015\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 13.0000 0.525065 0.262533 0.964923i \(-0.415442\pi\)
0.262533 + 0.964923i \(0.415442\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −10.0000 −0.402259
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.0000 1.20289
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) 0 0
\(626\) −14.0000 −0.559553
\(627\) 30.0000 1.19808
\(628\) −17.0000 −0.678374
\(629\) −66.0000 −2.63159
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −16.0000 −0.636446
\(633\) 46.0000 1.82834
\(634\) −15.0000 −0.595726
\(635\) 0 0
\(636\) 18.0000 0.713746
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 24.0000 0.947204
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 1.18033
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −2.00000 −0.0783260
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −14.0000 −0.546192
\(658\) 3.00000 0.116952
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −11.0000 −0.426241
\(667\) 0 0
\(668\) 15.0000 0.580367
\(669\) 38.0000 1.46916
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 2.00000 0.0771517
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 24.0000 0.921714
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) −12.0000 −0.459504
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 5.00000 0.191180
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −8.00000 −0.305219
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) −13.0000 −0.494543 −0.247272 0.968946i \(-0.579534\pi\)
−0.247272 + 0.968946i \(0.579534\pi\)
\(692\) −15.0000 −0.570214
\(693\) 3.00000 0.113961
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 2.00000 0.0757011
\(699\) 48.0000 1.81553
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −55.0000 −2.07436
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 9.00000 0.337289
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −6.00000 −0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) 6.00000 0.223297
\(723\) 46.0000 1.71076
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) −4.00000 −0.148454
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 16.0000 0.591377
\(733\) 25.0000 0.923396 0.461698 0.887037i \(-0.347240\pi\)
0.461698 + 0.887037i \(0.347240\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 6.00000 0.219529
\(748\) 18.0000 0.658145
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 3.00000 0.109399
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −23.0000 −0.835949 −0.417975 0.908459i \(-0.637260\pi\)
−0.417975 + 0.908459i \(0.637260\pi\)
\(758\) −19.0000 −0.690111
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 2.00000 0.0724524
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 4.00000 0.143963
\(773\) −39.0000 −1.40273 −0.701366 0.712801i \(-0.747426\pi\)
−0.701366 + 0.712801i \(0.747426\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) −22.0000 −0.789246
\(778\) −30.0000 −1.07555
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 27.0000 0.961835
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) 13.0000 0.461353
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 10.0000 0.353996
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 9.00000 0.317999
\(802\) −15.0000 −0.529668
\(803\) −42.0000 −1.48215
\(804\) 32.0000 1.12855
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) −6.00000 −0.211079
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −49.0000 −1.72062 −0.860311 0.509769i \(-0.829731\pi\)
−0.860311 + 0.509769i \(0.829731\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) −33.0000 −1.15665
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) −10.0000 −0.349856
\(818\) 5.00000 0.174821
\(819\) 0 0
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) −12.0000 −0.418548
\(823\) 43.0000 1.49889 0.749443 0.662069i \(-0.230321\pi\)
0.749443 + 0.662069i \(0.230321\pi\)
\(824\) −5.00000 −0.174183
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −36.0000 −1.24733
\(834\) −38.0000 −1.31583
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) 16.0000 0.553041
\(838\) −36.0000 −1.24360
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) −12.0000 −0.413302
\(844\) 23.0000 0.791693
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −2.00000 −0.0687208
\(848\) 9.00000 0.309061
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 12.0000 0.411113
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 30.0000 1.02180
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 38.0000 1.29055
\(868\) −4.00000 −0.135769
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −28.0000 −0.946032
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 20.0000 0.674967
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) −6.00000 −0.202031
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) 27.0000 0.906571 0.453286 0.891365i \(-0.350252\pi\)
0.453286 + 0.891365i \(0.350252\pi\)
\(888\) −22.0000 −0.738272
\(889\) 1.00000 0.0335389
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) 19.0000 0.636167
\(893\) 15.0000 0.501956
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 9.00000 0.300334
\(899\) 0 0
\(900\) 0 0
\(901\) 54.0000 1.79900
\(902\) 18.0000 0.599334
\(903\) −4.00000 −0.133112
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 24.0000 0.796468
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 10.0000 0.331133
\(913\) 18.0000 0.595713
\(914\) 4.00000 0.132308
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) −9.00000 −0.297206
\(918\) −24.0000 −0.792118
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) −42.0000 −1.38320
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) −5.00000 −0.164222
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 24.0000 0.786146
\(933\) 60.0000 1.96431
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −50.0000 −1.63343 −0.816714 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) 16.0000 0.522419
\(939\) −28.0000 −0.913745
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −34.0000 −1.10778
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −32.0000 −1.03931
\(949\) 0 0
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 6.00000 0.194461
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 23.0000 0.740780
\(965\) 0 0
\(966\) 0 0
\(967\) −5.00000 −0.160789 −0.0803946 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 60.0000 1.92748
\(970\) 0 0
\(971\) 33.0000 1.05902 0.529510 0.848304i \(-0.322376\pi\)
0.529510 + 0.848304i \(0.322376\pi\)
\(972\) −10.0000 −0.320750
\(973\) −19.0000 −0.609112
\(974\) 19.0000 0.608799
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) −4.00000 −0.127906
\(979\) 27.0000 0.862924
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 27.0000 0.861605
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −22.0000 −0.698853 −0.349427 0.936964i \(-0.613624\pi\)
−0.349427 + 0.936964i \(0.613624\pi\)
\(992\) −4.00000 −0.127000
\(993\) 40.0000 1.26936
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −41.0000 −1.29848 −0.649242 0.760582i \(-0.724914\pi\)
−0.649242 + 0.760582i \(0.724914\pi\)
\(998\) −4.00000 −0.126618
\(999\) 44.0000 1.39210
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 8450.2.a.w.1.1 1
5.4 even 2 1690.2.a.a.1.1 1
13.3 even 3 650.2.e.a.451.1 2
13.9 even 3 650.2.e.a.601.1 2
13.12 even 2 8450.2.a.k.1.1 1
65.3 odd 12 650.2.o.b.399.2 4
65.4 even 6 1690.2.e.e.991.1 2
65.9 even 6 130.2.e.b.81.1 yes 2
65.19 odd 12 1690.2.l.i.361.2 4
65.22 odd 12 650.2.o.b.549.2 4
65.24 odd 12 1690.2.l.i.1161.2 4
65.29 even 6 130.2.e.b.61.1 2
65.34 odd 4 1690.2.d.a.1351.1 2
65.42 odd 12 650.2.o.b.399.1 4
65.44 odd 4 1690.2.d.a.1351.2 2
65.48 odd 12 650.2.o.b.549.1 4
65.49 even 6 1690.2.e.e.191.1 2
65.54 odd 12 1690.2.l.i.1161.1 4
65.59 odd 12 1690.2.l.i.361.1 4
65.64 even 2 1690.2.a.g.1.1 1
195.29 odd 6 1170.2.i.f.451.1 2
195.74 odd 6 1170.2.i.f.991.1 2
260.139 odd 6 1040.2.q.c.81.1 2
260.159 odd 6 1040.2.q.c.321.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.2.e.b.61.1 2 65.29 even 6
130.2.e.b.81.1 yes 2 65.9 even 6
650.2.e.a.451.1 2 13.3 even 3
650.2.e.a.601.1 2 13.9 even 3
650.2.o.b.399.1 4 65.42 odd 12
650.2.o.b.399.2 4 65.3 odd 12
650.2.o.b.549.1 4 65.48 odd 12
650.2.o.b.549.2 4 65.22 odd 12
1040.2.q.c.81.1 2 260.139 odd 6
1040.2.q.c.321.1 2 260.159 odd 6
1170.2.i.f.451.1 2 195.29 odd 6
1170.2.i.f.991.1 2 195.74 odd 6
1690.2.a.a.1.1 1 5.4 even 2
1690.2.a.g.1.1 1 65.64 even 2
1690.2.d.a.1351.1 2 65.34 odd 4
1690.2.d.a.1351.2 2 65.44 odd 4
1690.2.e.e.191.1 2 65.49 even 6
1690.2.e.e.991.1 2 65.4 even 6
1690.2.l.i.361.1 4 65.59 odd 12
1690.2.l.i.361.2 4 65.19 odd 12
1690.2.l.i.1161.1 4 65.54 odd 12
1690.2.l.i.1161.2 4 65.24 odd 12
8450.2.a.k.1.1 1 13.12 even 2
8450.2.a.w.1.1 1 1.1 even 1 trivial