Properties

Label 384.2.d.a.193.2
Level $384$
Weight $2$
Character 384.193
Analytic conductor $3.066$
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] = [384,2,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, 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("384.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.d (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: \(3.06625543762\)
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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.2.d.a.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -4.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.00000i q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -8.00000 q^{23} +5.00000 q^{25} -1.00000i q^{27} -8.00000i q^{29} +4.00000 q^{31} -4.00000 q^{33} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +8.00000 q^{47} +9.00000 q^{49} -2.00000i q^{51} +8.00000i q^{53} -4.00000 q^{57} -12.0000i q^{59} +12.0000i q^{61} +4.00000 q^{63} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +6.00000 q^{73} +5.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +8.00000 q^{87} +6.00000 q^{89} -16.0000i q^{91} +4.00000i q^{93} -2.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{7} - 2 q^{9} - 4 q^{17} - 16 q^{23} + 10 q^{25} + 8 q^{31} - 8 q^{33} - 8 q^{39} - 12 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{57} + 8 q^{63} + 16 q^{71} + 12 q^{73} + 8 q^{79} + 2 q^{81} + 16 q^{87} + 12 q^{89} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 8.00000i − 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) − 2.00000i − 0.280056i
\(52\) 0 0
\(53\) 8.00000i 1.09888i 0.835532 + 0.549442i \(0.185160\pi\)
−0.835532 + 0.549442i \(0.814840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 12.0000i 1.53644i 0.640184 + 0.768221i \(0.278858\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) − 8.00000i − 0.963087i
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 5.00000i 0.577350i
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 16.0000i − 1.67726i
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
\(100\) 0 0
\(101\) 8.00000i 0.796030i 0.917379 + 0.398015i \(0.130301\pi\)
−0.917379 + 0.398015i \(0.869699\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 6.00000i − 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i 0.529694 + 0.848189i \(0.322307\pi\)
−0.529694 + 0.848189i \(0.677693\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 32.0000 2.52195
\(162\) 0 0
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −20.0000 −1.51186
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000i 0.298974i 0.988764 + 0.149487i \(0.0477622\pi\)
−0.988764 + 0.149487i \(0.952238\pi\)
\(180\) 0 0
\(181\) − 4.00000i − 0.297318i −0.988889 0.148659i \(-0.952504\pi\)
0.988889 0.148659i \(-0.0474956\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.00000i − 0.585018i
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000i 1.70993i 0.518686 + 0.854965i \(0.326421\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 32.0000i 2.24596i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) 8.00000i 0.548151i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) − 8.00000i − 0.538138i
\(222\) 0 0
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000i 0.259828i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) − 12.0000i − 0.757433i −0.925513 0.378717i \(-0.876365\pi\)
0.925513 0.378717i \(-0.123635\pi\)
\(252\) 0 0
\(253\) − 32.0000i − 2.01182i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) − 16.0000i − 0.994192i
\(260\) 0 0
\(261\) 8.00000i 0.495188i
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 24.0000i 1.46331i 0.681677 + 0.731653i \(0.261251\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 16.0000 0.968364
\(274\) 0 0
\(275\) 20.0000i 1.20605i
\(276\) 0 0
\(277\) − 20.0000i − 1.20168i −0.799368 0.600842i \(-0.794832\pi\)
0.799368 0.600842i \(-0.205168\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 2.00000i − 0.117242i
\(292\) 0 0
\(293\) − 24.0000i − 1.40209i −0.713115 0.701047i \(-0.752716\pi\)
0.713115 0.701047i \(-0.247284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) − 32.0000i − 1.85061i
\(300\) 0 0
\(301\) − 16.0000i − 0.922225i
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 4.00000i 0.227552i
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.00000i 0.449325i 0.974437 + 0.224662i \(0.0721279\pi\)
−0.974437 + 0.224662i \(0.927872\pi\)
\(318\) 0 0
\(319\) 32.0000 1.79166
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 20.0000i 1.10940i
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) − 4.00000i − 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) − 14.0000i − 0.760376i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.00000i 0.423405i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) − 32.0000i − 1.66136i
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) 0 0
\(381\) − 4.00000i − 0.204926i
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) − 32.0000i − 1.62246i −0.584724 0.811232i \(-0.698797\pi\)
0.584724 0.811232i \(-0.301203\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.0000i − 1.00377i −0.864934 0.501886i \(-0.832640\pi\)
0.864934 0.501886i \(-0.167360\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) 16.0000i 0.797017i
\(404\) 0 0
\(405\) 0 0
\(406\) 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
\(411\) − 6.00000i − 0.295958i
\(412\) 0 0
\(413\) 48.0000i 2.36193i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) 20.0000i 0.977064i 0.872546 + 0.488532i \(0.162467\pi\)
−0.872546 + 0.488532i \(0.837533\pi\)
\(420\) 0 0
\(421\) − 20.0000i − 0.974740i −0.873195 0.487370i \(-0.837956\pi\)
0.873195 0.487370i \(-0.162044\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −10.0000 −0.485071
\(426\) 0 0
\(427\) − 48.0000i − 2.32288i
\(428\) 0 0
\(429\) − 16.0000i − 0.772487i
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 32.0000i − 1.53077i
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 12.0000i 0.563809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) 48.0000i 2.21643i
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 20.0000i 0.917663i
\(476\) 0 0
\(477\) − 8.00000i − 0.366295i
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 32.0000i 1.45605i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) 16.0000i 0.720604i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 3.00000i − 0.133235i
\(508\) 0 0
\(509\) − 24.0000i − 1.06378i −0.846813 0.531891i \(-0.821482\pi\)
0.846813 0.531891i \(-0.178518\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) − 20.0000i − 0.872872i
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 36.0000i 1.55063i
\(540\) 0 0
\(541\) − 28.0000i − 1.20381i −0.798566 0.601907i \(-0.794408\pi\)
0.798566 0.601907i \(-0.205592\pi\)
\(542\) 0 0
\(543\) 4.00000 0.171656
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 36.0000i 1.53925i 0.638497 + 0.769624i \(0.279557\pi\)
−0.638497 + 0.769624i \(0.720443\pi\)
\(548\) 0 0
\(549\) − 12.0000i − 0.512148i
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0000i 0.677942i 0.940797 + 0.338971i \(0.110079\pi\)
−0.940797 + 0.338971i \(0.889921\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 4.00000i 0.167395i 0.996491 + 0.0836974i \(0.0266729\pi\)
−0.996491 + 0.0836974i \(0.973327\pi\)
\(572\) 0 0
\(573\) − 16.0000i − 0.668410i
\(574\) 0 0
\(575\) −40.0000 −1.66812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) − 18.0000i − 0.748054i
\(580\) 0 0
\(581\) − 16.0000i − 0.663792i
\(582\) 0 0
\(583\) −32.0000 −1.32530
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) 16.0000i 0.659269i
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) −32.0000 −1.29671
\(610\) 0 0
\(611\) 32.0000i 1.29458i
\(612\) 0 0
\(613\) 36.0000i 1.45403i 0.686624 + 0.727013i \(0.259092\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) − 28.0000i − 1.12542i −0.826656 0.562708i \(-0.809760\pi\)
0.826656 0.562708i \(-0.190240\pi\)
\(620\) 0 0
\(621\) 8.00000i 0.321029i
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) − 16.0000i − 0.638978i
\(628\) 0 0
\(629\) − 8.00000i − 0.318981i
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 36.0000i 1.42637i
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) − 16.0000i − 0.627089i
\(652\) 0 0
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) 4.00000i 0.155582i 0.996970 + 0.0777910i \(0.0247867\pi\)
−0.996970 + 0.0777910i \(0.975213\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 64.0000i 2.47809i
\(668\) 0 0
\(669\) − 20.0000i − 0.773245i
\(670\) 0 0
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) − 5.00000i − 0.192450i
\(676\) 0 0
\(677\) 16.0000i 0.614930i 0.951559 + 0.307465i \(0.0994807\pi\)
−0.951559 + 0.307465i \(0.900519\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) − 44.0000i − 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 0 0
\(689\) −32.0000 −1.21910
\(690\) 0 0
\(691\) − 44.0000i − 1.67384i −0.547326 0.836919i \(-0.684354\pi\)
0.547326 0.836919i \(-0.315646\pi\)
\(692\) 0 0
\(693\) 16.0000i 0.607790i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000 0.454532
\(698\) 0 0
\(699\) 22.0000i 0.832116i
\(700\) 0 0
\(701\) 8.00000i 0.302156i 0.988522 + 0.151078i \(0.0482744\pi\)
−0.988522 + 0.151078i \(0.951726\pi\)
\(702\) 0 0
\(703\) −16.0000 −0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 32.0000i − 1.20348i
\(708\) 0 0
\(709\) − 20.0000i − 0.751116i −0.926799 0.375558i \(-0.877451\pi\)
0.926799 0.375558i \(-0.122549\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) − 2.00000i − 0.0743808i
\(724\) 0 0
\(725\) − 40.0000i − 1.48556i
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) − 28.0000i − 1.03420i −0.855924 0.517102i \(-0.827011\pi\)
0.855924 0.517102i \(-0.172989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) − 28.0000i − 1.03000i −0.857191 0.514998i \(-0.827793\pi\)
0.857191 0.514998i \(-0.172207\pi\)
\(740\) 0 0
\(741\) − 16.0000i − 0.587775i
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) − 16.0000i − 0.584627i
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.0000i 0.436147i 0.975932 + 0.218074i \(0.0699773\pi\)
−0.975932 + 0.218074i \(0.930023\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) − 16.0000i − 0.579239i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) − 8.00000i − 0.287740i −0.989597 0.143870i \(-0.954045\pi\)
0.989597 0.143870i \(-0.0459547\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 12.0000i − 0.427754i −0.976861 0.213877i \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 0 0
\(789\) − 16.0000i − 0.569615i
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 16.0000i − 0.566749i −0.959009 0.283375i \(-0.908546\pi\)
0.959009 0.283375i \(-0.0914540\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 24.0000i 0.846942i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) − 44.0000i − 1.54505i −0.634985 0.772524i \(-0.718994\pi\)
0.634985 0.772524i \(-0.281006\pi\)
\(812\) 0 0
\(813\) 12.0000i 0.420858i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 16.0000i 0.559085i
\(820\) 0 0
\(821\) 24.0000i 0.837606i 0.908077 + 0.418803i \(0.137550\pi\)
−0.908077 + 0.418803i \(0.862450\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) 0 0
\(829\) 4.00000i 0.138926i 0.997585 + 0.0694629i \(0.0221285\pi\)
−0.997585 + 0.0694629i \(0.977871\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) 6.00000i 0.206651i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) − 32.0000i − 1.09695i
\(852\) 0 0
\(853\) − 44.0000i − 1.50653i −0.657716 0.753266i \(-0.728477\pi\)
0.657716 0.753266i \(-0.271523\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) − 28.0000i − 0.955348i −0.878537 0.477674i \(-0.841480\pi\)
0.878537 0.477674i \(-0.158520\pi\)
\(860\) 0 0
\(861\) 24.0000i 0.817918i
\(862\) 0 0
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 16.0000i 0.542763i
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) − 32.0000i − 1.06726i
\(900\) 0 0
\(901\) − 16.0000i − 0.533037i
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 0 0
\(909\) − 8.00000i − 0.265343i
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 48.0000i 1.58510i
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 32.0000i 1.05329i
\(924\) 0 0
\(925\) 20.0000i 0.657596i
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 36.0000i 1.17985i
\(932\) 0 0
\(933\) 32.0000i 1.04763i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) − 22.0000i − 0.717943i
\(940\) 0 0
\(941\) − 32.0000i − 1.04317i −0.853199 0.521585i \(-0.825341\pi\)
0.853199 0.521585i \(-0.174659\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) 24.0000i 0.779073i
\(950\) 0 0
\(951\) −8.00000 −0.259418
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.0000i 1.03441i
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 20.0000i 0.641831i 0.947108 + 0.320915i \(0.103990\pi\)
−0.947108 + 0.320915i \(0.896010\pi\)
\(972\) 0 0
\(973\) − 80.0000i − 2.56468i
\(974\) 0 0
\(975\) −20.0000 −0.640513
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) − 4.00000i − 0.127710i
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 32.0000i − 1.01857i
\(988\) 0 0
\(989\) − 32.0000i − 1.01754i
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 44.0000i − 1.39349i −0.717317 0.696747i \(-0.754630\pi\)
0.717317 0.696747i \(-0.245370\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 384.2.d.a.193.2 yes 2
3.2 odd 2 1152.2.d.a.577.1 2
4.3 odd 2 384.2.d.b.193.1 yes 2
8.3 odd 2 384.2.d.b.193.2 yes 2
8.5 even 2 inner 384.2.d.a.193.1 2
12.11 even 2 1152.2.d.f.577.2 2
16.3 odd 4 768.2.a.f.1.1 1
16.5 even 4 768.2.a.g.1.1 1
16.11 odd 4 768.2.a.b.1.1 1
16.13 even 4 768.2.a.c.1.1 1
24.5 odd 2 1152.2.d.a.577.2 2
24.11 even 2 1152.2.d.f.577.1 2
48.5 odd 4 2304.2.a.k.1.1 1
48.11 even 4 2304.2.a.f.1.1 1
48.29 odd 4 2304.2.a.j.1.1 1
48.35 even 4 2304.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.d.a.193.1 2 8.5 even 2 inner
384.2.d.a.193.2 yes 2 1.1 even 1 trivial
384.2.d.b.193.1 yes 2 4.3 odd 2
384.2.d.b.193.2 yes 2 8.3 odd 2
768.2.a.b.1.1 1 16.11 odd 4
768.2.a.c.1.1 1 16.13 even 4
768.2.a.f.1.1 1 16.3 odd 4
768.2.a.g.1.1 1 16.5 even 4
1152.2.d.a.577.1 2 3.2 odd 2
1152.2.d.a.577.2 2 24.5 odd 2
1152.2.d.f.577.1 2 24.11 even 2
1152.2.d.f.577.2 2 12.11 even 2
2304.2.a.f.1.1 1 48.11 even 4
2304.2.a.g.1.1 1 48.35 even 4
2304.2.a.j.1.1 1 48.29 odd 4
2304.2.a.k.1.1 1 48.5 odd 4