Properties

Label 3024.2.b.c.1567.2
Level $3024$
Weight $2$
Character 3024.1567
Analytic conductor $24.147$
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] = [3024,2,Mod(1567,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.c.1567.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+1.73205i q^{5} +(-2.00000 - 1.73205i) q^{7} +3.46410i q^{11} -3.46410i q^{13} -5.19615i q^{17} +2.00000 q^{19} +6.92820i q^{23} +2.00000 q^{25} -6.00000 q^{29} +10.0000 q^{31} +(3.00000 - 3.46410i) q^{35} -11.0000 q^{37} -1.73205i q^{41} -5.19615i q^{43} +9.00000 q^{47} +(1.00000 + 6.92820i) q^{49} +12.0000 q^{53} -6.00000 q^{55} +3.00000 q^{59} -3.46410i q^{61} +6.00000 q^{65} -3.46410i q^{67} +13.8564i q^{71} -6.92820i q^{73} +(6.00000 - 6.92820i) q^{77} +12.1244i q^{79} +15.0000 q^{83} +9.00000 q^{85} -6.92820i q^{89} +(-6.00000 + 6.92820i) q^{91} +3.46410i q^{95} +17.3205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} + 4 q^{19} + 4 q^{25} - 12 q^{29} + 20 q^{31} + 6 q^{35} - 22 q^{37} + 18 q^{47} + 2 q^{49} + 24 q^{53} - 12 q^{55} + 6 q^{59} + 12 q^{65} + 12 q^{77} + 30 q^{83} + 18 q^{85} - 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.19615i 1.26025i −0.776493 0.630126i \(-0.783003\pi\)
0.776493 0.630126i \(-0.216997\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.92820i 1.44463i 0.691564 + 0.722315i \(0.256922\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 3.46410i 0.507093 0.585540i
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.73205i 0.270501i −0.990811 0.135250i \(-0.956816\pi\)
0.990811 0.135250i \(-0.0431839\pi\)
\(42\) 0 0
\(43\) 5.19615i 0.792406i −0.918163 0.396203i \(-0.870328\pi\)
0.918163 0.396203i \(-0.129672\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 3.46410i 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.8564i 1.64445i 0.569160 + 0.822226i \(0.307268\pi\)
−0.569160 + 0.822226i \(0.692732\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 6.92820i 0.683763 0.789542i
\(78\) 0 0
\(79\) 12.1244i 1.36410i 0.731307 + 0.682048i \(0.238911\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) −6.00000 + 6.92820i −0.628971 + 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 17.3205i 1.75863i 0.476240 + 0.879316i \(0.342000\pi\)
−0.476240 + 0.879316i \(0.658000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 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
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.00000 + 10.3923i −0.825029 + 0.952661i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 5.19615i 0.461084i 0.973062 + 0.230542i \(0.0740499\pi\)
−0.973062 + 0.230542i \(0.925950\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.00000 3.46410i −0.346844 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 12.1244i 0.986666i −0.869841 0.493333i \(-0.835778\pi\)
0.869841 0.493333i \(-0.164222\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.3205i 1.39122i
\(156\) 0 0
\(157\) 10.3923i 0.829396i 0.909959 + 0.414698i \(0.136113\pi\)
−0.909959 + 0.414698i \(0.863887\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 13.8564i 0.945732 1.09204i
\(162\) 0 0
\(163\) 1.73205i 0.135665i −0.997697 0.0678323i \(-0.978392\pi\)
0.997697 0.0678323i \(-0.0216083\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.92820i 0.526742i −0.964695 0.263371i \(-0.915166\pi\)
0.964695 0.263371i \(-0.0848343\pi\)
\(174\) 0 0
\(175\) −4.00000 3.46410i −0.302372 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i −0.857209 0.514969i \(-0.827803\pi\)
0.857209 0.514969i \(-0.172197\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.0526i 1.40077i
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.92820i 0.501307i 0.968077 + 0.250654i \(0.0806455\pi\)
−0.968077 + 0.250654i \(0.919354\pi\)
\(192\) 0 0
\(193\) 25.0000 1.79954 0.899770 0.436365i \(-0.143734\pi\)
0.899770 + 0.436365i \(0.143734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 + 10.3923i 0.842235 + 0.729397i
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.92820i 0.479234i
\(210\) 0 0
\(211\) 3.46410i 0.238479i −0.992866 0.119239i \(-0.961954\pi\)
0.992866 0.119239i \(-0.0380456\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.00000 0.613795
\(216\) 0 0
\(217\) −20.0000 17.3205i −1.35769 1.17579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i 0.726900 + 0.686743i \(0.240960\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 15.5885i 1.01688i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) 3.46410i 0.223142i 0.993756 + 0.111571i \(0.0355883\pi\)
−0.993756 + 0.111571i \(0.964412\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.0000 + 1.73205i −0.766652 + 0.110657i
\(246\) 0 0
\(247\) 6.92820i 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.92820i 0.432169i 0.976375 + 0.216085i \(0.0693287\pi\)
−0.976375 + 0.216085i \(0.930671\pi\)
\(258\) 0 0
\(259\) 22.0000 + 19.0526i 1.36701 + 1.18387i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.46410i 0.213606i −0.994280 0.106803i \(-0.965939\pi\)
0.994280 0.106803i \(-0.0340614\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.5167i 1.37287i −0.727194 0.686433i \(-0.759176\pi\)
0.727194 0.686433i \(-0.240824\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.92820i 0.417786i
\(276\) 0 0
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 + 3.46410i −0.177084 + 0.204479i
\(288\) 0 0
\(289\) −10.0000 −0.588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.9808i 1.51781i 0.651200 + 0.758906i \(0.274266\pi\)
−0.651200 + 0.758906i \(0.725734\pi\)
\(294\) 0 0
\(295\) 5.19615i 0.302532i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −9.00000 + 10.3923i −0.518751 + 0.599002i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −15.0000 −0.850572 −0.425286 0.905059i \(-0.639826\pi\)
−0.425286 + 0.905059i \(0.639826\pi\)
\(312\) 0 0
\(313\) 6.92820i 0.391605i −0.980643 0.195803i \(-0.937269\pi\)
0.980643 0.195803i \(-0.0627312\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.3923i 0.578243i
\(324\) 0 0
\(325\) 6.92820i 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18.0000 15.5885i −0.992372 0.859419i
\(330\) 0 0
\(331\) 19.0526i 1.04722i −0.851957 0.523612i \(-0.824584\pi\)
0.851957 0.523612i \(-0.175416\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 34.6410i 1.87592i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.3205i 0.929814i 0.885360 + 0.464907i \(0.153912\pi\)
−0.885360 + 0.464907i \(0.846088\pi\)
\(348\) 0 0
\(349\) 34.6410i 1.85429i −0.374701 0.927146i \(-0.622255\pi\)
0.374701 0.927146i \(-0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.9090i 1.75157i −0.482704 0.875784i \(-0.660345\pi\)
0.482704 0.875784i \(-0.339655\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846i 1.09697i 0.836160 + 0.548485i \(0.184795\pi\)
−0.836160 + 0.548485i \(0.815205\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 20.7846i −1.24602 1.07908i
\(372\) 0 0
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.7846i 1.07046i
\(378\) 0 0
\(379\) 8.66025i 0.444847i 0.974950 + 0.222424i \(0.0713968\pi\)
−0.974950 + 0.222424i \(0.928603\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −33.0000 −1.68622 −0.843111 0.537740i \(-0.819278\pi\)
−0.843111 + 0.537740i \(0.819278\pi\)
\(384\) 0 0
\(385\) 12.0000 + 10.3923i 0.611577 + 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −21.0000 −1.05662
\(396\) 0 0
\(397\) 20.7846i 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 34.6410i 1.72559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 38.1051i 1.88880i
\(408\) 0 0
\(409\) 6.92820i 0.342578i −0.985221 0.171289i \(-0.945207\pi\)
0.985221 0.171289i \(-0.0547931\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.00000 5.19615i −0.295241 0.255686i
\(414\) 0 0
\(415\) 25.9808i 1.27535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.3923i 0.504101i
\(426\) 0 0
\(427\) −6.00000 + 6.92820i −0.290360 + 0.335279i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.6410i 1.66860i −0.551311 0.834300i \(-0.685872\pi\)
0.551311 0.834300i \(-0.314128\pi\)
\(432\) 0 0
\(433\) 31.1769i 1.49827i −0.662419 0.749133i \(-0.730470\pi\)
0.662419 0.749133i \(-0.269530\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.3205i 0.822922i −0.911427 0.411461i \(-0.865019\pi\)
0.911427 0.411461i \(-0.134981\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 10.3923i −0.562569 0.487199i
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 39.8372i 1.85540i 0.373324 + 0.927701i \(0.378218\pi\)
−0.373324 + 0.927701i \(0.621782\pi\)
\(462\) 0 0
\(463\) 22.5167i 1.04644i −0.852198 0.523219i \(-0.824731\pi\)
0.852198 0.523219i \(-0.175269\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −6.00000 + 6.92820i −0.277054 + 0.319915i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.0000 0.827641
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) 38.1051i 1.73744i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) 38.1051i 1.72671i 0.504599 + 0.863354i \(0.331640\pi\)
−0.504599 + 0.863354i \(0.668360\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.46410i 0.156333i −0.996940 0.0781664i \(-0.975093\pi\)
0.996940 0.0781664i \(-0.0249065\pi\)
\(492\) 0 0
\(493\) 31.1769i 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 27.7128i 1.07655 1.24309i
\(498\) 0 0
\(499\) 5.19615i 0.232612i −0.993213 0.116306i \(-0.962895\pi\)
0.993213 0.116306i \(-0.0371053\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 25.9808i 1.15158i 0.817599 + 0.575789i \(0.195305\pi\)
−0.817599 + 0.575789i \(0.804695\pi\)
\(510\) 0 0
\(511\) −12.0000 + 13.8564i −0.530849 + 0.612971i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.92820i 0.305293i
\(516\) 0 0
\(517\) 31.1769i 1.37116i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.1244i 0.531178i 0.964086 + 0.265589i \(0.0855664\pi\)
−0.964086 + 0.265589i \(0.914434\pi\)
\(522\) 0 0
\(523\) −38.0000 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 51.9615i 2.26348i
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −24.0000 + 3.46410i −1.03375 + 0.149209i
\(540\) 0 0
\(541\) −29.0000 −1.24681 −0.623404 0.781900i \(-0.714251\pi\)
−0.623404 + 0.781900i \(0.714251\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.66025i 0.370965i
\(546\) 0 0
\(547\) 1.73205i 0.0740571i −0.999314 0.0370286i \(-0.988211\pi\)
0.999314 0.0370286i \(-0.0117893\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 21.0000 24.2487i 0.893011 1.03116i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 31.1769i 1.31162i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 32.9090i 1.37720i 0.725143 + 0.688599i \(0.241774\pi\)
−0.725143 + 0.688599i \(0.758226\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.8564i 0.577852i
\(576\) 0 0
\(577\) 34.6410i 1.44212i −0.692870 0.721062i \(-0.743654\pi\)
0.692870 0.721062i \(-0.256346\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −30.0000 25.9808i −1.24461 1.07786i
\(582\) 0 0
\(583\) 41.5692i 1.72162i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.4449i 1.20916i −0.796546 0.604578i \(-0.793342\pi\)
0.796546 0.604578i \(-0.206658\pi\)
\(594\) 0 0
\(595\) −18.0000 15.5885i −0.737928 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.46410i 0.141539i −0.997493 0.0707697i \(-0.977454\pi\)
0.997493 0.0707697i \(-0.0225455\pi\)
\(600\) 0 0
\(601\) 10.3923i 0.423911i 0.977279 + 0.211955i \(0.0679832\pi\)
−0.977279 + 0.211955i \(0.932017\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.73205i 0.0704179i
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 31.1769i 1.26128i
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 + 13.8564i −0.480770 + 0.555145i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 57.1577i 2.27903i
\(630\) 0 0
\(631\) 12.1244i 0.482663i 0.970443 + 0.241331i \(0.0775841\pi\)
−0.970443 + 0.241331i \(0.922416\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.00000 −0.357154
\(636\) 0 0
\(637\) 24.0000 3.46410i 0.950915 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 8.00000 0.315489 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 10.3923i 0.407934i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.92820i 0.269884i −0.990853 0.134942i \(-0.956915\pi\)
0.990853 0.134942i \(-0.0430849\pi\)
\(660\) 0 0
\(661\) 3.46410i 0.134738i 0.997728 + 0.0673690i \(0.0214605\pi\)
−0.997728 + 0.0673690i \(0.978540\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 6.92820i 0.232670 0.268664i
\(666\) 0 0
\(667\) 41.5692i 1.60957i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.7128i 1.06509i 0.846402 + 0.532545i \(0.178764\pi\)
−0.846402 + 0.532545i \(0.821236\pi\)
\(678\) 0 0
\(679\) 30.0000 34.6410i 1.15129 1.32940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.3205i 0.662751i −0.943499 0.331375i \(-0.892487\pi\)
0.943499 0.331375i \(-0.107513\pi\)
\(684\) 0 0
\(685\) 10.3923i 0.397070i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 41.5692i 1.58366i
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.46410i 0.131401i
\(696\) 0 0
\(697\) −9.00000 −0.340899
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −22.0000 −0.829746
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 69.2820i 2.59463i
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) −8.00000 6.92820i −0.297936 0.258020i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −27.0000 −0.998631
\(732\) 0 0
\(733\) 41.5692i 1.53539i −0.640813 0.767697i \(-0.721403\pi\)
0.640813 0.767697i \(-0.278597\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.2487i 0.889599i 0.895630 + 0.444799i \(0.146725\pi\)
−0.895630 + 0.444799i \(0.853275\pi\)
\(744\) 0 0
\(745\) 10.3923i 0.380745i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 17.3205i 0.632034i −0.948753 0.316017i \(-0.897654\pi\)
0.948753 0.316017i \(-0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.0000 0.764268
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 46.7654i 1.69524i 0.530601 + 0.847622i \(0.321966\pi\)
−0.530601 + 0.847622i \(0.678034\pi\)
\(762\) 0 0
\(763\) −10.0000 8.66025i −0.362024 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.3923i 0.375244i
\(768\) 0 0
\(769\) 13.8564i 0.499675i −0.968288 0.249837i \(-0.919623\pi\)
0.968288 0.249837i \(-0.0803772\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.8372i 1.43284i −0.697668 0.716422i \(-0.745779\pi\)
0.697668 0.716422i \(-0.254221\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.46410i 0.124114i
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −36.0000 31.1769i −1.28001 1.10852i
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27.7128i 0.981638i 0.871262 + 0.490819i \(0.163302\pi\)
−0.871262 + 0.490819i \(0.836698\pi\)
\(798\) 0 0
\(799\) 46.7654i 1.65444i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 24.0000 + 20.7846i 0.845889 + 0.732561i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 22.0000 0.772524 0.386262 0.922389i \(-0.373766\pi\)
0.386262 + 0.922389i \(0.373766\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.00000 0.105085
\(816\) 0 0
\(817\) 10.3923i 0.363581i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −48.0000 −1.67521 −0.837606 0.546275i \(-0.816045\pi\)
−0.837606 + 0.546275i \(0.816045\pi\)
\(822\) 0 0
\(823\) 12.1244i 0.422628i 0.977418 + 0.211314i \(0.0677743\pi\)
−0.977418 + 0.211314i \(0.932226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.2487i 0.843210i −0.906780 0.421605i \(-0.861467\pi\)
0.906780 0.421605i \(-0.138533\pi\)
\(828\) 0 0
\(829\) 3.46410i 0.120313i −0.998189 0.0601566i \(-0.980840\pi\)
0.998189 0.0601566i \(-0.0191600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 5.19615i 1.24733 0.180036i
\(834\) 0 0
\(835\) 36.3731i 1.25874i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15.0000 −0.517858 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.73205i 0.0595844i
\(846\) 0 0
\(847\) 2.00000 + 1.73205i 0.0687208 + 0.0595140i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 76.2102i 2.61245i
\(852\) 0 0
\(853\) 34.6410i 1.18609i 0.805171 + 0.593043i \(0.202074\pi\)
−0.805171 + 0.593043i \(0.797926\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 53.6936i 1.83414i −0.398729 0.917069i \(-0.630549\pi\)
0.398729 0.917069i \(-0.369451\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 21.0000 24.2487i 0.709930 0.819756i
\(876\) 0 0
\(877\) 11.0000 0.371444 0.185722 0.982602i \(-0.440538\pi\)
0.185722 + 0.982602i \(0.440538\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.92820i 0.233417i −0.993166 0.116709i \(-0.962766\pi\)
0.993166 0.116709i \(-0.0372343\pi\)
\(882\) 0 0
\(883\) 43.3013i 1.45720i 0.684937 + 0.728602i \(0.259830\pi\)
−0.684937 + 0.728602i \(0.740170\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.00000 −0.100730 −0.0503651 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) 0 0
\(889\) 9.00000 10.3923i 0.301850 0.348547i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) 62.3538i 2.07731i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 53.6936i 1.78287i −0.453153 0.891433i \(-0.649701\pi\)
0.453153 0.891433i \(-0.350299\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10.3923i 0.344312i 0.985070 + 0.172156i \(0.0550734\pi\)
−0.985070 + 0.172156i \(0.944927\pi\)
\(912\) 0 0
\(913\) 51.9615i 1.71968i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 46.7654i 1.54265i −0.636443 0.771324i \(-0.719595\pi\)
0.636443 0.771324i \(-0.280405\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.5885i 0.511441i 0.966751 + 0.255720i \(0.0823126\pi\)
−0.966751 + 0.255720i \(0.917687\pi\)
\(930\) 0 0
\(931\) 2.00000 + 13.8564i 0.0655474 + 0.454125i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 31.1769i 1.01959i
\(936\) 0 0
\(937\) 17.3205i 0.565836i 0.959144 + 0.282918i \(0.0913025\pi\)
−0.959144 + 0.282918i \(0.908698\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.66025i 0.282316i −0.989987 0.141158i \(-0.954917\pi\)
0.989987 0.141158i \(-0.0450826\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.7128i 0.900545i 0.892891 + 0.450273i \(0.148673\pi\)
−0.892891 + 0.450273i \(0.851327\pi\)
\(948\) 0 0
\(949\) −24.0000 −0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 10.3923i −0.387500 0.335585i
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 43.3013i 1.39392i
\(966\) 0 0
\(967\) 24.2487i 0.779786i −0.920860 0.389893i \(-0.872512\pi\)
0.920860 0.389893i \(-0.127488\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0000 0.866471 0.433236 0.901281i \(-0.357372\pi\)
0.433236 + 0.901281i \(0.357372\pi\)
\(972\) 0 0
\(973\) 4.00000 + 3.46410i 0.128234 + 0.111054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 27.0000 0.861166 0.430583 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) 0 0
\(985\) 31.1769i 0.993379i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 43.3013i 1.37551i 0.725943 + 0.687755i \(0.241404\pi\)
−0.725943 + 0.687755i \(0.758596\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 27.7128i 0.878555i
\(996\) 0 0
\(997\) 51.9615i 1.64564i −0.568304 0.822819i \(-0.692400\pi\)
0.568304 0.822819i \(-0.307600\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3024.2.b.c.1567.2 yes 2
3.2 odd 2 3024.2.b.e.1567.1 yes 2
4.3 odd 2 3024.2.b.l.1567.2 yes 2
7.6 odd 2 3024.2.b.l.1567.1 yes 2
12.11 even 2 3024.2.b.n.1567.1 yes 2
21.20 even 2 3024.2.b.n.1567.2 yes 2
28.27 even 2 inner 3024.2.b.c.1567.1 2
84.83 odd 2 3024.2.b.e.1567.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.c.1567.1 2 28.27 even 2 inner
3024.2.b.c.1567.2 yes 2 1.1 even 1 trivial
3024.2.b.e.1567.1 yes 2 3.2 odd 2
3024.2.b.e.1567.2 yes 2 84.83 odd 2
3024.2.b.l.1567.1 yes 2 7.6 odd 2
3024.2.b.l.1567.2 yes 2 4.3 odd 2
3024.2.b.n.1567.1 yes 2 12.11 even 2
3024.2.b.n.1567.2 yes 2 21.20 even 2