Properties

Label 2816.2.c.h.1409.1
Level $2816$
Weight $2$
Character 2816.1409
Analytic conductor $22.486$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2816,2,Mod(1409,2816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2816, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2816.1409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2816 = 2^{8} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2816.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(22.4858732092\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1409.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2816.1409
Dual form 2816.2.c.h.1409.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +1.00000i q^{5} -6.00000 q^{9} -1.00000i q^{11} +6.00000i q^{13} +3.00000 q^{15} -4.00000 q^{17} -6.00000i q^{19} -3.00000 q^{23} +4.00000 q^{25} +9.00000i q^{27} +4.00000i q^{29} -9.00000 q^{31} -3.00000 q^{33} +7.00000i q^{37} +18.0000 q^{39} +2.00000 q^{41} +6.00000i q^{43} -6.00000i q^{45} +12.0000 q^{47} -7.00000 q^{49} +12.0000i q^{51} +2.00000i q^{53} +1.00000 q^{55} -18.0000 q^{57} +9.00000i q^{59} -8.00000i q^{61} -6.00000 q^{65} +15.0000i q^{67} +9.00000i q^{69} +3.00000 q^{71} +6.00000 q^{73} -12.0000i q^{75} -6.00000 q^{79} +9.00000 q^{81} +6.00000i q^{83} -4.00000i q^{85} +12.0000 q^{87} +5.00000 q^{89} +27.0000i q^{93} +6.00000 q^{95} -3.00000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9} + 6 q^{15} - 8 q^{17} - 6 q^{23} + 8 q^{25} - 18 q^{31} - 6 q^{33} + 36 q^{39} + 4 q^{41} + 24 q^{47} - 14 q^{49} + 2 q^{55} - 36 q^{57} - 12 q^{65} + 6 q^{71} + 12 q^{73} - 12 q^{79} + 18 q^{81}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(1541\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i 0.974679 + 0.223607i \(0.0717831\pi\)
−0.974679 + 0.223607i \(0.928217\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) − 1.00000i − 0.301511i
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) − 6.00000i − 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) − 6.00000i − 0.894427i
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 12.0000i 1.68034i
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) 9.00000i 1.17170i 0.810419 + 0.585850i \(0.199239\pi\)
−0.810419 + 0.585850i \(0.800761\pi\)
\(60\) 0 0
\(61\) − 8.00000i − 1.02430i −0.858898 0.512148i \(-0.828850\pi\)
0.858898 0.512148i \(-0.171150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 15.0000i 1.83254i 0.400559 + 0.916271i \(0.368816\pi\)
−0.400559 + 0.916271i \(0.631184\pi\)
\(68\) 0 0
\(69\) 9.00000i 1.08347i
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) − 12.0000i − 1.38564i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 5.00000 0.529999 0.264999 0.964249i \(-0.414628\pi\)
0.264999 + 0.964249i \(0.414628\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 27.0000i 2.79977i
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) 16.0000i 1.59206i 0.605257 + 0.796030i \(0.293070\pi\)
−0.605257 + 0.796030i \(0.706930\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 21.0000 1.99323
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) − 3.00000i − 0.279751i
\(116\) 0 0
\(117\) − 36.0000i − 3.32820i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 9.00000i 0.804984i
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −9.00000 −0.774597
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) − 12.0000i − 1.01783i −0.860818 0.508913i \(-0.830047\pi\)
0.860818 0.508913i \(-0.169953\pi\)
\(140\) 0 0
\(141\) − 36.0000i − 3.03175i
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 21.0000i 1.73205i
\(148\) 0 0
\(149\) − 16.0000i − 1.31077i −0.755295 0.655386i \(-0.772506\pi\)
0.755295 0.655386i \(-0.227494\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 24.0000 1.94029
\(154\) 0 0
\(155\) − 9.00000i − 0.722897i
\(156\) 0 0
\(157\) − 13.0000i − 1.03751i −0.854922 0.518756i \(-0.826395\pi\)
0.854922 0.518756i \(-0.173605\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) 0 0
\(165\) − 3.00000i − 0.233550i
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 36.0000i 2.75299i
\(172\) 0 0
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 27.0000 2.02944
\(178\) 0 0
\(179\) 15.0000i 1.12115i 0.828103 + 0.560576i \(0.189420\pi\)
−0.828103 + 0.560576i \(0.810580\pi\)
\(180\) 0 0
\(181\) 3.00000i 0.222988i 0.993765 + 0.111494i \(0.0355636\pi\)
−0.993765 + 0.111494i \(0.964436\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 18.0000i 1.28901i
\(196\) 0 0
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 45.0000 3.17406
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000i 0.139686i
\(206\) 0 0
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) − 12.0000i − 0.826114i −0.910705 0.413057i \(-0.864461\pi\)
0.910705 0.413057i \(-0.135539\pi\)
\(212\) 0 0
\(213\) − 9.00000i − 0.616670i
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 18.0000i − 1.21633i
\(220\) 0 0
\(221\) − 24.0000i − 1.61441i
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 15.0000i 0.991228i 0.868543 + 0.495614i \(0.165057\pi\)
−0.868543 + 0.495614i \(0.834943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 12.0000i 0.782794i
\(236\) 0 0
\(237\) 18.0000i 1.16923i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 7.00000i − 0.447214i
\(246\) 0 0
\(247\) 36.0000 2.29063
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) 21.0000i 1.32551i 0.748837 + 0.662754i \(0.230613\pi\)
−0.748837 + 0.662754i \(0.769387\pi\)
\(252\) 0 0
\(253\) 3.00000i 0.188608i
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) − 24.0000i − 1.48556i
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) − 15.0000i − 0.917985i
\(268\) 0 0
\(269\) 2.00000i 0.121942i 0.998140 + 0.0609711i \(0.0194197\pi\)
−0.998140 + 0.0609711i \(0.980580\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) 54.0000 3.23290
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) − 18.0000i − 1.06623i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 9.00000i 0.527589i
\(292\) 0 0
\(293\) − 22.0000i − 1.28525i −0.766179 0.642627i \(-0.777845\pi\)
0.766179 0.642627i \(-0.222155\pi\)
\(294\) 0 0
\(295\) −9.00000 −0.524000
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) − 18.0000i − 1.04097i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 48.0000 2.75753
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) − 6.00000i − 0.342438i −0.985233 0.171219i \(-0.945229\pi\)
0.985233 0.171219i \(-0.0547706\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.0000i 1.29181i 0.763418 + 0.645904i \(0.223520\pi\)
−0.763418 + 0.645904i \(0.776480\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 24.0000i 1.33128i
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.0000i 0.824475i 0.911077 + 0.412237i \(0.135253\pi\)
−0.911077 + 0.412237i \(0.864747\pi\)
\(332\) 0 0
\(333\) − 42.0000i − 2.30159i
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) 0 0
\(339\) − 3.00000i − 0.162938i
\(340\) 0 0
\(341\) 9.00000i 0.487377i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −9.00000 −0.484544
\(346\) 0 0
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) 0 0
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 0 0
\(353\) −5.00000 −0.266123 −0.133062 0.991108i \(-0.542481\pi\)
−0.133062 + 0.991108i \(0.542481\pi\)
\(354\) 0 0
\(355\) 3.00000i 0.159223i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 3.00000i 0.157459i
\(364\) 0 0
\(365\) 6.00000i 0.314054i
\(366\) 0 0
\(367\) 15.0000 0.782994 0.391497 0.920179i \(-0.371957\pi\)
0.391497 + 0.920179i \(0.371957\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 27.0000 1.39427
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) − 21.0000i − 1.07870i −0.842082 0.539349i \(-0.818670\pi\)
0.842082 0.539349i \(-0.181330\pi\)
\(380\) 0 0
\(381\) 18.0000i 0.922168i
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 36.0000i − 1.82998i
\(388\) 0 0
\(389\) 5.00000i 0.253510i 0.991934 + 0.126755i \(0.0404562\pi\)
−0.991934 + 0.126755i \(0.959544\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) − 6.00000i − 0.301893i
\(396\) 0 0
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) − 54.0000i − 2.68993i
\(404\) 0 0
\(405\) 9.00000i 0.447214i
\(406\) 0 0
\(407\) 7.00000 0.346977
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 15.0000i 0.739895i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −36.0000 −1.76293
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 6.00000i 0.292422i 0.989253 + 0.146211i \(0.0467079\pi\)
−0.989253 + 0.146211i \(0.953292\pi\)
\(422\) 0 0
\(423\) −72.0000 −3.50076
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 18.0000i − 0.869048i
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) 12.0000i 0.575356i
\(436\) 0 0
\(437\) 18.0000i 0.861057i
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) 42.0000 2.00000
\(442\) 0 0
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 0 0
\(445\) 5.00000i 0.237023i
\(446\) 0 0
\(447\) −48.0000 −2.27032
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) − 2.00000i − 0.0941763i
\(452\) 0 0
\(453\) − 18.0000i − 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 0 0
\(459\) − 36.0000i − 1.68034i
\(460\) 0 0
\(461\) 40.0000i 1.86299i 0.363760 + 0.931493i \(0.381493\pi\)
−0.363760 + 0.931493i \(0.618507\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) 0 0
\(465\) −27.0000 −1.25210
\(466\) 0 0
\(467\) 15.0000i 0.694117i 0.937843 + 0.347059i \(0.112820\pi\)
−0.937843 + 0.347059i \(0.887180\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −39.0000 −1.79703
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) − 24.0000i − 1.10120i
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −42.0000 −1.91504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 3.00000i − 0.136223i
\(486\) 0 0
\(487\) −3.00000 −0.135943 −0.0679715 0.997687i \(-0.521653\pi\)
−0.0679715 + 0.997687i \(0.521653\pi\)
\(488\) 0 0
\(489\) −36.0000 −1.62798
\(490\) 0 0
\(491\) − 24.0000i − 1.08310i −0.840667 0.541552i \(-0.817837\pi\)
0.840667 0.541552i \(-0.182163\pi\)
\(492\) 0 0
\(493\) − 16.0000i − 0.720604i
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 12.0000i − 0.537194i −0.963253 0.268597i \(-0.913440\pi\)
0.963253 0.268597i \(-0.0865599\pi\)
\(500\) 0 0
\(501\) 54.0000i 2.41254i
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) 69.0000i 3.06440i
\(508\) 0 0
\(509\) 1.00000i 0.0443242i 0.999754 + 0.0221621i \(0.00705500\pi\)
−0.999754 + 0.0221621i \(0.992945\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 12.0000i − 0.527759i
\(518\) 0 0
\(519\) −30.0000 −1.31685
\(520\) 0 0
\(521\) 31.0000 1.35813 0.679067 0.734076i \(-0.262384\pi\)
0.679067 + 0.734076i \(0.262384\pi\)
\(522\) 0 0
\(523\) − 30.0000i − 1.31181i −0.754844 0.655904i \(-0.772288\pi\)
0.754844 0.655904i \(-0.227712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 36.0000 1.56818
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) − 54.0000i − 2.34340i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) 45.0000 1.94189
\(538\) 0 0
\(539\) 7.00000i 0.301511i
\(540\) 0 0
\(541\) 6.00000i 0.257960i 0.991647 + 0.128980i \(0.0411703\pi\)
−0.991647 + 0.128980i \(0.958830\pi\)
\(542\) 0 0
\(543\) 9.00000 0.386227
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 0 0
\(549\) 48.0000i 2.04859i
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 21.0000i 0.891400i
\(556\) 0 0
\(557\) 16.0000i 0.677942i 0.940797 + 0.338971i \(0.110079\pi\)
−0.940797 + 0.338971i \(0.889921\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.00000i 0.0420703i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −44.0000 −1.84458 −0.922288 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) 0 0
\(571\) − 12.0000i − 0.502184i −0.967963 0.251092i \(-0.919210\pi\)
0.967963 0.251092i \(-0.0807897\pi\)
\(572\) 0 0
\(573\) − 63.0000i − 2.63186i
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 29.0000 1.20729 0.603643 0.797255i \(-0.293715\pi\)
0.603643 + 0.797255i \(0.293715\pi\)
\(578\) 0 0
\(579\) − 12.0000i − 0.498703i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) 36.0000 1.48842
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 54.0000i 2.22503i
\(590\) 0 0
\(591\) 66.0000 2.71488
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) − 90.0000i − 3.66508i
\(604\) 0 0
\(605\) − 1.00000i − 0.0406558i
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 72.0000i 2.91281i
\(612\) 0 0
\(613\) − 16.0000i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(614\) 0 0
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) 15.0000i 0.602901i 0.953482 + 0.301450i \(0.0974708\pi\)
−0.953482 + 0.301450i \(0.902529\pi\)
\(620\) 0 0
\(621\) − 27.0000i − 1.08347i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 18.0000i 0.718851i
\(628\) 0 0
\(629\) − 28.0000i − 1.11643i
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) −36.0000 −1.43087
\(634\) 0 0
\(635\) − 6.00000i − 0.238103i
\(636\) 0 0
\(637\) − 42.0000i − 1.66410i
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −37.0000 −1.46141 −0.730706 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(642\) 0 0
\(643\) − 21.0000i − 0.828159i −0.910241 0.414080i \(-0.864104\pi\)
0.910241 0.414080i \(-0.135896\pi\)
\(644\) 0 0
\(645\) 18.0000i 0.708749i
\(646\) 0 0
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0000i 0.665261i 0.943057 + 0.332631i \(0.107936\pi\)
−0.943057 + 0.332631i \(0.892064\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) 18.0000i 0.701180i 0.936529 + 0.350590i \(0.114019\pi\)
−0.936529 + 0.350590i \(0.885981\pi\)
\(660\) 0 0
\(661\) − 23.0000i − 0.894596i −0.894385 0.447298i \(-0.852386\pi\)
0.894385 0.447298i \(-0.147614\pi\)
\(662\) 0 0
\(663\) −72.0000 −2.79625
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) 63.0000i 2.43572i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 36.0000i 1.38564i
\(676\) 0 0
\(677\) − 8.00000i − 0.307465i −0.988113 0.153732i \(-0.950871\pi\)
0.988113 0.153732i \(-0.0491294\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) − 5.00000i − 0.191040i
\(686\) 0 0
\(687\) 45.0000 1.71686
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) − 33.0000i − 1.25538i −0.778464 0.627690i \(-0.784001\pi\)
0.778464 0.627690i \(-0.215999\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) − 12.0000i − 0.453882i
\(700\) 0 0
\(701\) 40.0000i 1.51078i 0.655276 + 0.755390i \(0.272552\pi\)
−0.655276 + 0.755390i \(0.727448\pi\)
\(702\) 0 0
\(703\) 42.0000 1.58406
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 33.0000i − 1.23934i −0.784862 0.619671i \(-0.787266\pi\)
0.784862 0.619671i \(-0.212734\pi\)
\(710\) 0 0
\(711\) 36.0000 1.35011
\(712\) 0 0
\(713\) 27.0000 1.01116
\(714\) 0 0
\(715\) 6.00000i 0.224387i
\(716\) 0 0
\(717\) 18.0000i 0.672222i
\(718\) 0 0
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 36.0000i 1.33885i
\(724\) 0 0
\(725\) 16.0000i 0.594225i
\(726\) 0 0
\(727\) −3.00000 −0.111264 −0.0556319 0.998451i \(-0.517717\pi\)
−0.0556319 + 0.998451i \(0.517717\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) − 24.0000i − 0.887672i
\(732\) 0 0
\(733\) − 48.0000i − 1.77292i −0.462805 0.886460i \(-0.653157\pi\)
0.462805 0.886460i \(-0.346843\pi\)
\(734\) 0 0
\(735\) −21.0000 −0.774597
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) 6.00000i 0.220714i 0.993892 + 0.110357i \(0.0351994\pi\)
−0.993892 + 0.110357i \(0.964801\pi\)
\(740\) 0 0
\(741\) − 108.000i − 3.96748i
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 16.0000 0.586195
\(746\) 0 0
\(747\) − 36.0000i − 1.31717i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.00000 −0.109472 −0.0547358 0.998501i \(-0.517432\pi\)
−0.0547358 + 0.998501i \(0.517432\pi\)
\(752\) 0 0
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) 6.00000i 0.218362i
\(756\) 0 0
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) 0 0
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 24.0000i 0.867722i
\(766\) 0 0
\(767\) −54.0000 −1.94983
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) − 42.0000i − 1.51259i
\(772\) 0 0
\(773\) 2.00000i 0.0719350i 0.999353 + 0.0359675i \(0.0114513\pi\)
−0.999353 + 0.0359675i \(0.988549\pi\)
\(774\) 0 0
\(775\) −36.0000 −1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 12.0000i − 0.429945i
\(780\) 0 0
\(781\) − 3.00000i − 0.107348i
\(782\) 0 0
\(783\) −36.0000 −1.28654
\(784\) 0 0
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) 0 0
\(789\) 72.0000i 2.56327i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) 6.00000i 0.212798i
\(796\) 0 0
\(797\) − 49.0000i − 1.73567i −0.496853 0.867835i \(-0.665511\pi\)
0.496853 0.867835i \(-0.334489\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) − 6.00000i − 0.211735i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) −46.0000 −1.61727 −0.808637 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 30.0000i 1.05344i 0.850038 + 0.526721i \(0.176579\pi\)
−0.850038 + 0.526721i \(0.823421\pi\)
\(812\) 0 0
\(813\) 18.0000i 0.631288i
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000i 0.767805i 0.923374 + 0.383903i \(0.125420\pi\)
−0.923374 + 0.383903i \(0.874580\pi\)
\(822\) 0 0
\(823\) 33.0000 1.15031 0.575154 0.818045i \(-0.304942\pi\)
0.575154 + 0.818045i \(0.304942\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) 0 0
\(829\) − 21.0000i − 0.729360i −0.931133 0.364680i \(-0.881178\pi\)
0.931133 0.364680i \(-0.118822\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 28.0000 0.970143
\(834\) 0 0
\(835\) − 18.0000i − 0.622916i
\(836\) 0 0
\(837\) − 81.0000i − 2.79977i
\(838\) 0 0
\(839\) 33.0000 1.13929 0.569643 0.821892i \(-0.307081\pi\)
0.569643 + 0.821892i \(0.307081\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) 30.0000i 1.03325i
\(844\) 0 0
\(845\) − 23.0000i − 0.791224i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 21.0000i − 0.719871i
\(852\) 0 0
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) − 27.0000i − 0.921228i −0.887601 0.460614i \(-0.847629\pi\)
0.887601 0.460614i \(-0.152371\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 0 0
\(867\) 3.00000i 0.101885i
\(868\) 0 0
\(869\) 6.00000i 0.203536i
\(870\) 0 0
\(871\) −90.0000 −3.04953
\(872\) 0 0
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) 0 0
\(879\) −66.0000 −2.22612
\(880\) 0 0
\(881\) 29.0000 0.977035 0.488517 0.872554i \(-0.337538\pi\)
0.488517 + 0.872554i \(0.337538\pi\)
\(882\) 0 0
\(883\) 12.0000i 0.403832i 0.979403 + 0.201916i \(0.0647168\pi\)
−0.979403 + 0.201916i \(0.935283\pi\)
\(884\) 0 0
\(885\) 27.0000i 0.907595i
\(886\) 0 0
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 9.00000i − 0.301511i
\(892\) 0 0
\(893\) − 72.0000i − 2.40939i
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) −54.0000 −1.80301
\(898\) 0 0
\(899\) − 36.0000i − 1.20067i
\(900\) 0 0
\(901\) − 8.00000i − 0.266519i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.00000 −0.0997234
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 0 0
\(909\) − 96.0000i − 3.18412i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 0 0
\(915\) − 24.0000i − 0.793416i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 18.0000i 0.592477i
\(924\) 0 0
\(925\) 28.0000i 0.920634i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 0 0
\(931\) 42.0000i 1.37649i
\(932\) 0 0
\(933\) − 72.0000i − 2.35717i
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) − 3.00000i − 0.0979013i
\(940\) 0 0
\(941\) 10.0000i 0.325991i 0.986627 + 0.162995i \(0.0521156\pi\)
−0.986627 + 0.162995i \(0.947884\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.0000i 0.487435i 0.969846 + 0.243717i \(0.0783669\pi\)
−0.969846 + 0.243717i \(0.921633\pi\)
\(948\) 0 0
\(949\) 36.0000i 1.16861i
\(950\) 0 0
\(951\) 69.0000 2.23748
\(952\) 0 0
\(953\) 40.0000 1.29573 0.647864 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 0 0
\(955\) 21.0000i 0.679544i
\(956\) 0 0
\(957\) − 12.0000i − 0.387905i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) − 72.0000i − 2.32017i
\(964\) 0 0
\(965\) 4.00000i 0.128765i
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 72.0000 2.31297
\(970\) 0 0
\(971\) 15.0000i 0.481373i 0.970603 + 0.240686i \(0.0773725\pi\)
−0.970603 + 0.240686i \(0.922627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 72.0000 2.30585
\(976\) 0 0
\(977\) −23.0000 −0.735835 −0.367918 0.929858i \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(978\) 0 0
\(979\) − 5.00000i − 0.159801i
\(980\) 0 0
\(981\) − 36.0000i − 1.14939i
\(982\) 0 0
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 18.0000i − 0.572367i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 45.0000 1.42803
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) 0 0
\(999\) −63.0000 −1.99323
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 2816.2.c.h.1409.1 2
4.3 odd 2 2816.2.c.g.1409.2 2
8.3 odd 2 2816.2.c.g.1409.1 2
8.5 even 2 inner 2816.2.c.h.1409.2 2
16.3 odd 4 352.2.a.a.1.1 1
16.5 even 4 704.2.a.a.1.1 1
16.11 odd 4 704.2.a.k.1.1 1
16.13 even 4 352.2.a.f.1.1 yes 1
48.5 odd 4 6336.2.a.bs.1.1 1
48.11 even 4 6336.2.a.bt.1.1 1
48.29 odd 4 3168.2.a.i.1.1 1
48.35 even 4 3168.2.a.h.1.1 1
80.19 odd 4 8800.2.a.bb.1.1 1
80.29 even 4 8800.2.a.a.1.1 1
176.21 odd 4 7744.2.a.a.1.1 1
176.43 even 4 7744.2.a.bj.1.1 1
176.109 odd 4 3872.2.a.m.1.1 1
176.131 even 4 3872.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.a.1.1 1 16.3 odd 4
352.2.a.f.1.1 yes 1 16.13 even 4
704.2.a.a.1.1 1 16.5 even 4
704.2.a.k.1.1 1 16.11 odd 4
2816.2.c.g.1409.1 2 8.3 odd 2
2816.2.c.g.1409.2 2 4.3 odd 2
2816.2.c.h.1409.1 2 1.1 even 1 trivial
2816.2.c.h.1409.2 2 8.5 even 2 inner
3168.2.a.h.1.1 1 48.35 even 4
3168.2.a.i.1.1 1 48.29 odd 4
3872.2.a.a.1.1 1 176.131 even 4
3872.2.a.m.1.1 1 176.109 odd 4
6336.2.a.bs.1.1 1 48.5 odd 4
6336.2.a.bt.1.1 1 48.11 even 4
7744.2.a.a.1.1 1 176.21 odd 4
7744.2.a.bj.1.1 1 176.43 even 4
8800.2.a.a.1.1 1 80.29 even 4
8800.2.a.bb.1.1 1 80.19 odd 4