Properties

Label 448.2.b.a.225.1
Level $448$
Weight $2$
Character 448.225
Analytic conductor $3.577$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(225,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, 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("448.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.b (of order \(2\), degree \(1\), 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: \(3.57729801055\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 225.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 448.225
Dual form 448.2.b.a.225.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +4.00000i q^{13} +8.00000 q^{15} +2.00000 q^{17} +6.00000i q^{19} +2.00000i q^{21} -11.0000 q^{25} -4.00000i q^{27} -8.00000i q^{29} +8.00000 q^{31} +4.00000 q^{33} -4.00000i q^{35} +8.00000i q^{37} +8.00000 q^{39} +10.0000 q^{41} +2.00000i q^{43} -4.00000i q^{45} -8.00000 q^{47} +1.00000 q^{49} -4.00000i q^{51} -8.00000 q^{55} +12.0000 q^{57} -10.0000i q^{59} -4.00000i q^{61} +1.00000 q^{63} -16.0000 q^{65} +2.00000i q^{67} -8.00000 q^{71} -6.00000 q^{73} +22.0000i q^{75} -2.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} +6.00000i q^{83} +8.00000i q^{85} -16.0000 q^{87} +10.0000 q^{89} -4.00000i q^{91} -16.0000i q^{93} -24.0000 q^{95} +2.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 2 q^{9} + 16 q^{15} + 4 q^{17} - 22 q^{25} + 16 q^{31} + 8 q^{33} + 16 q^{39} + 20 q^{41} - 16 q^{47} + 2 q^{49} - 16 q^{55} + 24 q^{57} + 2 q^{63} - 32 q^{65} - 16 q^{71} - 12 q^{73} - 16 q^{79} - 22 q^{81} - 32 q^{87} + 20 q^{89} - 48 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 8.00000 2.06559
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) − 8.00000i − 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 4.00000i − 0.560112i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 12.0000 1.58944
\(58\) 0 0
\(59\) − 10.0000i − 1.30189i −0.759125 0.650945i \(-0.774373\pi\)
0.759125 0.650945i \(-0.225627\pi\)
\(60\) 0 0
\(61\) − 4.00000i − 0.512148i −0.966657 0.256074i \(-0.917571\pi\)
0.966657 0.256074i \(-0.0824290\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 22.0000i 2.54034i
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 8.00000i 0.867722i
\(86\) 0 0
\(87\) −16.0000 −1.71538
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) − 4.00000i − 0.419314i
\(92\) 0 0
\(93\) − 16.0000i − 1.65912i
\(94\) 0 0
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) − 2.00000i − 0.201008i
\(100\) 0 0
\(101\) 4.00000i 0.398015i 0.979998 + 0.199007i \(0.0637718\pi\)
−0.979998 + 0.199007i \(0.936228\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) − 8.00000i − 0.766261i −0.923694 0.383131i \(-0.874846\pi\)
0.923694 0.383131i \(-0.125154\pi\)
\(110\) 0 0
\(111\) 16.0000 1.51865
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) − 20.0000i − 1.80334i
\(124\) 0 0
\(125\) − 24.0000i − 2.14663i
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 6.00000i 0.524222i 0.965038 + 0.262111i \(0.0844187\pi\)
−0.965038 + 0.262111i \(0.915581\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) − 2.00000i − 0.169638i −0.996396 0.0848189i \(-0.972969\pi\)
0.996396 0.0848189i \(-0.0270312\pi\)
\(140\) 0 0
\(141\) 16.0000i 1.34744i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 32.0000 2.65746
\(146\) 0 0
\(147\) − 2.00000i − 0.164957i
\(148\) 0 0
\(149\) − 16.0000i − 1.31077i −0.755295 0.655386i \(-0.772506\pi\)
0.755295 0.655386i \(-0.227494\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 32.0000i 2.57030i
\(156\) 0 0
\(157\) 4.00000i 0.319235i 0.987179 + 0.159617i \(0.0510260\pi\)
−0.987179 + 0.159617i \(0.948974\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 22.0000i − 1.72317i −0.507611 0.861586i \(-0.669471\pi\)
0.507611 0.861586i \(-0.330529\pi\)
\(164\) 0 0
\(165\) 16.0000i 1.24560i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) − 6.00000i − 0.458831i
\(172\) 0 0
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) −20.0000 −1.50329
\(178\) 0 0
\(179\) − 14.0000i − 1.04641i −0.852207 0.523205i \(-0.824736\pi\)
0.852207 0.523205i \(-0.175264\pi\)
\(180\) 0 0
\(181\) 12.0000i 0.891953i 0.895045 + 0.445976i \(0.147144\pi\)
−0.895045 + 0.445976i \(0.852856\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −32.0000 −2.35269
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 32.0000i 2.29157i
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 8.00000i 0.561490i
\(204\) 0 0
\(205\) 40.0000i 2.79372i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 2.00000i 0.137686i 0.997628 + 0.0688428i \(0.0219307\pi\)
−0.997628 + 0.0688428i \(0.978069\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 12.0000i 0.810885i
\(220\) 0 0
\(221\) 8.00000i 0.538138i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 6.00000i 0.398234i 0.979976 + 0.199117i \(0.0638074\pi\)
−0.979976 + 0.199117i \(0.936193\pi\)
\(228\) 0 0
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) − 32.0000i − 2.08745i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 4.00000i 0.255551i
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) − 18.0000i − 1.13615i −0.822977 0.568075i \(-0.807688\pi\)
0.822977 0.568075i \(-0.192312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 16.0000 1.00196
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) − 8.00000i − 0.497096i
\(260\) 0 0
\(261\) 8.00000i 0.495188i
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 20.0000i − 1.22398i
\(268\) 0 0
\(269\) − 12.0000i − 0.731653i −0.930683 0.365826i \(-0.880786\pi\)
0.930683 0.365826i \(-0.119214\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 0 0
\(275\) − 22.0000i − 1.32665i
\(276\) 0 0
\(277\) − 16.0000i − 0.961347i −0.876900 0.480673i \(-0.840392\pi\)
0.876900 0.480673i \(-0.159608\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 0 0
\(285\) 48.0000i 2.84327i
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 4.00000i − 0.234484i
\(292\) 0 0
\(293\) 4.00000i 0.233682i 0.993151 + 0.116841i \(0.0372769\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 0 0
\(295\) 40.0000 2.32889
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 2.00000i − 0.115278i
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 0 0
\(309\) − 32.0000i − 1.82042i
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 4.00000i 0.225374i
\(316\) 0 0
\(317\) − 32.0000i − 1.79730i −0.438667 0.898650i \(-0.644549\pi\)
0.438667 0.898650i \(-0.355451\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) − 44.0000i − 2.44068i
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 18.0000i 0.989369i 0.869072 + 0.494685i \(0.164716\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) − 8.00000i − 0.438397i
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) − 4.00000i − 0.217250i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 20.0000i 1.07058i 0.844670 + 0.535288i \(0.179797\pi\)
−0.844670 + 0.535288i \(0.820203\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) − 32.0000i − 1.69838i
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) − 14.0000i − 0.734809i
\(364\) 0 0
\(365\) − 24.0000i − 1.25622i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −48.0000 −2.47871
\(376\) 0 0
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) 34.0000i 1.74646i 0.487306 + 0.873231i \(0.337980\pi\)
−0.487306 + 0.873231i \(0.662020\pi\)
\(380\) 0 0
\(381\) 32.0000i 1.63941i
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) − 2.00000i − 0.101666i
\(388\) 0 0
\(389\) − 24.0000i − 1.21685i −0.793612 0.608424i \(-0.791802\pi\)
0.793612 0.608424i \(-0.208198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) − 32.0000i − 1.61009i
\(396\) 0 0
\(397\) − 28.0000i − 1.40528i −0.711546 0.702640i \(-0.752005\pi\)
0.711546 0.702640i \(-0.247995\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 32.0000i 1.59403i
\(404\) 0 0
\(405\) − 44.0000i − 2.18638i
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 12.0000i 0.591916i
\(412\) 0 0
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 14.0000i 0.683945i 0.939710 + 0.341972i \(0.111095\pi\)
−0.939710 + 0.341972i \(0.888905\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −22.0000 −1.06716
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 16.0000i 0.772487i
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) − 64.0000i − 3.06857i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 0 0
\(445\) 40.0000i 1.89618i
\(446\) 0 0
\(447\) −32.0000 −1.51355
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 20.0000i 0.941763i
\(452\) 0 0
\(453\) − 32.0000i − 1.50349i
\(454\) 0 0
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) − 8.00000i − 0.373408i
\(460\) 0 0
\(461\) 20.0000i 0.931493i 0.884918 + 0.465746i \(0.154214\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 64.0000 2.96793
\(466\) 0 0
\(467\) − 26.0000i − 1.20314i −0.798821 0.601568i \(-0.794543\pi\)
0.798821 0.601568i \(-0.205457\pi\)
\(468\) 0 0
\(469\) − 2.00000i − 0.0923514i
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) − 66.0000i − 3.02829i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000i 0.363261i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −44.0000 −1.98975
\(490\) 0 0
\(491\) − 6.00000i − 0.270776i −0.990793 0.135388i \(-0.956772\pi\)
0.990793 0.135388i \(-0.0432281\pi\)
\(492\) 0 0
\(493\) − 16.0000i − 0.720604i
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 34.0000i 1.52205i 0.648723 + 0.761025i \(0.275303\pi\)
−0.648723 + 0.761025i \(0.724697\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) 6.00000i 0.266469i
\(508\) 0 0
\(509\) − 12.0000i − 0.531891i −0.963988 0.265945i \(-0.914316\pi\)
0.963988 0.265945i \(-0.0856841\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) 64.0000i 2.82018i
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 22.0000i 0.961993i 0.876723 + 0.480996i \(0.159725\pi\)
−0.876723 + 0.480996i \(0.840275\pi\)
\(524\) 0 0
\(525\) − 22.0000i − 0.960159i
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 10.0000i 0.433963i
\(532\) 0 0
\(533\) 40.0000i 1.73259i
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) −28.0000 −1.20829
\(538\) 0 0
\(539\) 2.00000i 0.0861461i
\(540\) 0 0
\(541\) − 16.0000i − 0.687894i −0.938989 0.343947i \(-0.888236\pi\)
0.938989 0.343947i \(-0.111764\pi\)
\(542\) 0 0
\(543\) 24.0000 1.02994
\(544\) 0 0
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) − 22.0000i − 0.940652i −0.882493 0.470326i \(-0.844136\pi\)
0.882493 0.470326i \(-0.155864\pi\)
\(548\) 0 0
\(549\) 4.00000i 0.170716i
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 64.0000i 2.71665i
\(556\) 0 0
\(557\) − 16.0000i − 0.677942i −0.940797 0.338971i \(-0.889921\pi\)
0.940797 0.338971i \(-0.110079\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) − 18.0000i − 0.758610i −0.925272 0.379305i \(-0.876163\pi\)
0.925272 0.379305i \(-0.123837\pi\)
\(564\) 0 0
\(565\) 8.00000i 0.336563i
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 18.0000i 0.753277i 0.926360 + 0.376638i \(0.122920\pi\)
−0.926360 + 0.376638i \(0.877080\pi\)
\(572\) 0 0
\(573\) − 16.0000i − 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 0 0
\(579\) − 20.0000i − 0.831172i
\(580\) 0 0
\(581\) − 6.00000i − 0.248922i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 16.0000 0.661519
\(586\) 0 0
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) 0 0
\(589\) 48.0000i 1.97781i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) − 8.00000i − 0.327968i
\(596\) 0 0
\(597\) − 32.0000i − 1.30967i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) − 2.00000i − 0.0814463i
\(604\) 0 0
\(605\) 28.0000i 1.13836i
\(606\) 0 0
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 0 0
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) − 32.0000i − 1.29458i
\(612\) 0 0
\(613\) − 8.00000i − 0.323117i −0.986863 0.161558i \(-0.948348\pi\)
0.986863 0.161558i \(-0.0516520\pi\)
\(614\) 0 0
\(615\) 80.0000 3.22591
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) − 10.0000i − 0.401934i −0.979598 0.200967i \(-0.935592\pi\)
0.979598 0.200967i \(-0.0644084\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 24.0000i 0.958468i
\(628\) 0 0
\(629\) 16.0000i 0.637962i
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) − 64.0000i − 2.53976i
\(636\) 0 0
\(637\) 4.00000i 0.158486i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) − 34.0000i − 1.34083i −0.741987 0.670415i \(-0.766116\pi\)
0.741987 0.670415i \(-0.233884\pi\)
\(644\) 0 0
\(645\) 16.0000i 0.629999i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 10.0000i 0.389545i 0.980848 + 0.194772i \(0.0623968\pi\)
−0.980848 + 0.194772i \(0.937603\pi\)
\(660\) 0 0
\(661\) 36.0000i 1.40024i 0.714026 + 0.700119i \(0.246870\pi\)
−0.714026 + 0.700119i \(0.753130\pi\)
\(662\) 0 0
\(663\) 16.0000 0.621389
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 0 0
\(675\) 44.0000i 1.69356i
\(676\) 0 0
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 26.0000i 0.994862i 0.867503 + 0.497431i \(0.165723\pi\)
−0.867503 + 0.497431i \(0.834277\pi\)
\(684\) 0 0
\(685\) − 24.0000i − 0.916993i
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 10.0000i − 0.380418i −0.981744 0.190209i \(-0.939083\pi\)
0.981744 0.190209i \(-0.0609166\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) − 36.0000i − 1.36165i
\(700\) 0 0
\(701\) 24.0000i 0.906467i 0.891392 + 0.453234i \(0.149730\pi\)
−0.891392 + 0.453234i \(0.850270\pi\)
\(702\) 0 0
\(703\) −48.0000 −1.81035
\(704\) 0 0
\(705\) −64.0000 −2.41038
\(706\) 0 0
\(707\) − 4.00000i − 0.150435i
\(708\) 0 0
\(709\) − 8.00000i − 0.300446i −0.988652 0.150223i \(-0.952001\pi\)
0.988652 0.150223i \(-0.0479992\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 32.0000i − 1.19673i
\(716\) 0 0
\(717\) − 32.0000i − 1.19506i
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) − 4.00000i − 0.148762i
\(724\) 0 0
\(725\) 88.0000i 3.26824i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 4.00000i 0.147945i
\(732\) 0 0
\(733\) 28.0000i 1.03420i 0.855924 + 0.517102i \(0.172989\pi\)
−0.855924 + 0.517102i \(0.827011\pi\)
\(734\) 0 0
\(735\) 8.00000 0.295084
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) − 22.0000i − 0.809283i −0.914475 0.404642i \(-0.867396\pi\)
0.914475 0.404642i \(-0.132604\pi\)
\(740\) 0 0
\(741\) 48.0000i 1.76332i
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 64.0000 2.34478
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) 6.00000i 0.219235i
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 64.0000i 2.32920i
\(756\) 0 0
\(757\) − 40.0000i − 1.45382i −0.686730 0.726912i \(-0.740955\pi\)
0.686730 0.726912i \(-0.259045\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 8.00000i 0.289619i
\(764\) 0 0
\(765\) − 8.00000i − 0.289241i
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) − 36.0000i − 1.29651i
\(772\) 0 0
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 0 0
\(775\) −88.0000 −3.16105
\(776\) 0 0
\(777\) −16.0000 −0.573997
\(778\) 0 0
\(779\) 60.0000i 2.14972i
\(780\) 0 0
\(781\) − 16.0000i − 0.572525i
\(782\) 0 0
\(783\) −32.0000 −1.14359
\(784\) 0 0
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) 38.0000i 1.35455i 0.735728 + 0.677277i \(0.236840\pi\)
−0.735728 + 0.677277i \(0.763160\pi\)
\(788\) 0 0
\(789\) − 16.0000i − 0.569615i
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 28.0000i − 0.991811i −0.868377 0.495905i \(-0.834836\pi\)
0.868377 0.495905i \(-0.165164\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) − 12.0000i − 0.423471i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 6.00000i 0.210688i 0.994436 + 0.105344i \(0.0335944\pi\)
−0.994436 + 0.105344i \(0.966406\pi\)
\(812\) 0 0
\(813\) − 32.0000i − 1.12229i
\(814\) 0 0
\(815\) 88.0000 3.08251
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 4.00000i 0.139771i
\(820\) 0 0
\(821\) 48.0000i 1.67521i 0.546275 + 0.837606i \(0.316045\pi\)
−0.546275 + 0.837606i \(0.683955\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) −44.0000 −1.53188
\(826\) 0 0
\(827\) 42.0000i 1.46048i 0.683189 + 0.730242i \(0.260592\pi\)
−0.683189 + 0.730242i \(0.739408\pi\)
\(828\) 0 0
\(829\) 12.0000i 0.416777i 0.978046 + 0.208389i \(0.0668219\pi\)
−0.978046 + 0.208389i \(0.933178\pi\)
\(830\) 0 0
\(831\) −32.0000 −1.11007
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 32.0000i − 1.10608i
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) − 4.00000i − 0.137767i
\(844\) 0 0
\(845\) − 12.0000i − 0.412813i
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 44.0000 1.51008
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 12.0000i 0.410872i 0.978671 + 0.205436i \(0.0658613\pi\)
−0.978671 + 0.205436i \(0.934139\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) − 34.0000i − 1.16007i −0.814593 0.580033i \(-0.803040\pi\)
0.814593 0.580033i \(-0.196960\pi\)
\(860\) 0 0
\(861\) 20.0000i 0.681598i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 48.0000 1.63205
\(866\) 0 0
\(867\) 26.0000i 0.883006i
\(868\) 0 0
\(869\) − 16.0000i − 0.542763i
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 24.0000i 0.811348i
\(876\) 0 0
\(877\) − 8.00000i − 0.270141i −0.990836 0.135070i \(-0.956874\pi\)
0.990836 0.135070i \(-0.0431261\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 30.0000i − 1.00958i −0.863242 0.504790i \(-0.831570\pi\)
0.863242 0.504790i \(-0.168430\pi\)
\(884\) 0 0
\(885\) − 80.0000i − 2.68917i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) − 22.0000i − 0.737028i
\(892\) 0 0
\(893\) − 48.0000i − 1.60626i
\(894\) 0 0
\(895\) 56.0000 1.87187
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 64.0000i − 2.13452i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) −48.0000 −1.59557
\(906\) 0 0
\(907\) − 22.0000i − 0.730498i −0.930910 0.365249i \(-0.880984\pi\)
0.930910 0.365249i \(-0.119016\pi\)
\(908\) 0 0
\(909\) − 4.00000i − 0.132672i
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) − 32.0000i − 1.05789i
\(916\) 0 0
\(917\) − 6.00000i − 0.198137i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 44.0000 1.44985
\(922\) 0 0
\(923\) − 32.0000i − 1.05329i
\(924\) 0 0
\(925\) − 88.0000i − 2.89342i
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 6.00000i 0.196642i
\(932\) 0 0
\(933\) 16.0000i 0.523816i
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 12.0000i 0.391605i
\(940\) 0 0
\(941\) − 52.0000i − 1.69515i −0.530674 0.847576i \(-0.678061\pi\)
0.530674 0.847576i \(-0.321939\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) − 38.0000i − 1.23483i −0.786636 0.617417i \(-0.788179\pi\)
0.786636 0.617417i \(-0.211821\pi\)
\(948\) 0 0
\(949\) − 24.0000i − 0.779073i
\(950\) 0 0
\(951\) −64.0000 −2.07534
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 32.0000i 1.03550i
\(956\) 0 0
\(957\) − 32.0000i − 1.03441i
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 40.0000i 1.28765i
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 6.00000i 0.192549i 0.995355 + 0.0962746i \(0.0306927\pi\)
−0.995355 + 0.0962746i \(0.969307\pi\)
\(972\) 0 0
\(973\) 2.00000i 0.0641171i
\(974\) 0 0
\(975\) −88.0000 −2.81826
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 20.0000i 0.639203i
\(980\) 0 0
\(981\) 8.00000i 0.255420i
\(982\) 0 0
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 36.0000 1.14243
\(994\) 0 0
\(995\) 64.0000i 2.02894i
\(996\) 0 0
\(997\) 20.0000i 0.633406i 0.948525 + 0.316703i \(0.102576\pi\)
−0.948525 + 0.316703i \(0.897424\pi\)
\(998\) 0 0
\(999\) 32.0000 1.01244
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 448.2.b.a.225.1 2
3.2 odd 2 4032.2.c.b.2017.1 2
4.3 odd 2 448.2.b.b.225.2 yes 2
7.6 odd 2 3136.2.b.c.1569.2 2
8.3 odd 2 448.2.b.b.225.1 yes 2
8.5 even 2 inner 448.2.b.a.225.2 yes 2
12.11 even 2 4032.2.c.f.2017.1 2
16.3 odd 4 1792.2.a.d.1.1 1
16.5 even 4 1792.2.a.a.1.1 1
16.11 odd 4 1792.2.a.e.1.1 1
16.13 even 4 1792.2.a.h.1.1 1
24.5 odd 2 4032.2.c.b.2017.2 2
24.11 even 2 4032.2.c.f.2017.2 2
28.27 even 2 3136.2.b.a.1569.1 2
56.13 odd 2 3136.2.b.c.1569.1 2
56.27 even 2 3136.2.b.a.1569.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
448.2.b.a.225.1 2 1.1 even 1 trivial
448.2.b.a.225.2 yes 2 8.5 even 2 inner
448.2.b.b.225.1 yes 2 8.3 odd 2
448.2.b.b.225.2 yes 2 4.3 odd 2
1792.2.a.a.1.1 1 16.5 even 4
1792.2.a.d.1.1 1 16.3 odd 4
1792.2.a.e.1.1 1 16.11 odd 4
1792.2.a.h.1.1 1 16.13 even 4
3136.2.b.a.1569.1 2 28.27 even 2
3136.2.b.a.1569.2 2 56.27 even 2
3136.2.b.c.1569.1 2 56.13 odd 2
3136.2.b.c.1569.2 2 7.6 odd 2
4032.2.c.b.2017.1 2 3.2 odd 2
4032.2.c.b.2017.2 2 24.5 odd 2
4032.2.c.f.2017.1 2 12.11 even 2
4032.2.c.f.2017.2 2 24.11 even 2