Properties

Label 1200.2.o.c.1199.2
Level $1200$
Weight $2$
Character 1200.1199
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,2,Mod(1199,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.1199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1199.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.1199
Dual form 1200.2.o.c.1199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 1.50000i) q^{3} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 1.50000i) q^{3} +(-1.50000 - 2.59808i) q^{9} -3.00000 q^{11} -2.00000i q^{13} +5.19615 q^{17} -5.19615i q^{19} +6.00000i q^{23} +5.19615 q^{27} -10.3923i q^{29} -3.46410i q^{31} +(2.59808 - 4.50000i) q^{33} -8.00000i q^{37} +(3.00000 + 1.73205i) q^{39} +5.19615i q^{41} +3.46410 q^{43} +6.00000i q^{47} -7.00000 q^{49} +(-4.50000 + 7.79423i) q^{51} +10.3923 q^{53} +(7.79423 + 4.50000i) q^{57} +12.0000 q^{59} +8.00000 q^{61} -12.1244 q^{67} +(-9.00000 - 5.19615i) q^{69} +6.00000 q^{71} +1.00000i q^{73} -6.92820i q^{79} +(-4.50000 + 7.79423i) q^{81} -9.00000i q^{83} +(15.5885 + 9.00000i) q^{87} -5.19615i q^{89} +(5.19615 + 3.00000i) q^{93} -10.0000i q^{97} +(4.50000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} - 12 q^{11} + 12 q^{39} - 28 q^{49} - 18 q^{51} + 48 q^{59} + 32 q^{61} - 36 q^{69} + 24 q^{71} - 18 q^{81} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.866025 + 1.50000i −0.500000 + 0.866025i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.50000 2.59808i −0.500000 0.866025i
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.19615 1.26025 0.630126 0.776493i \(-0.283003\pi\)
0.630126 + 0.776493i \(0.283003\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i −0.802955 0.596040i \(-0.796740\pi\)
0.802955 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 10.3923i 1.92980i −0.262613 0.964901i \(-0.584584\pi\)
0.262613 0.964901i \(-0.415416\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) 2.59808 4.50000i 0.452267 0.783349i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 3.00000 + 1.73205i 0.480384 + 0.277350i
\(40\) 0 0
\(41\) 5.19615i 0.811503i 0.913984 + 0.405751i \(0.132990\pi\)
−0.913984 + 0.405751i \(0.867010\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −4.50000 + 7.79423i −0.630126 + 1.09141i
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.79423 + 4.50000i 1.03237 + 0.596040i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.1244 −1.48123 −0.740613 0.671932i \(-0.765465\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) −9.00000 5.19615i −1.08347 0.625543i
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.92820i 0.779484i −0.920924 0.389742i \(-0.872564\pi\)
0.920924 0.389742i \(-0.127436\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 9.00000i 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 15.5885 + 9.00000i 1.67126 + 0.964901i
\(88\) 0 0
\(89\) 5.19615i 0.550791i −0.961331 0.275396i \(-0.911191\pi\)
0.961331 0.275396i \(-0.0888088\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 5.19615 + 3.00000i 0.538816 + 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 4.50000 + 7.79423i 0.452267 + 0.783349i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 17.3205 1.70664 0.853320 0.521387i \(-0.174585\pi\)
0.853320 + 0.521387i \(0.174585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000i 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 12.0000 + 6.92820i 1.13899 + 0.657596i
\(112\) 0 0
\(113\) 5.19615 0.488813 0.244406 0.969673i \(-0.421407\pi\)
0.244406 + 0.969673i \(0.421407\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.19615 + 3.00000i −0.480384 + 0.277350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −7.79423 4.50000i −0.702782 0.405751i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) −3.00000 + 5.19615i −0.264135 + 0.457496i
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.19615 0.443937 0.221969 0.975054i \(-0.428752\pi\)
0.221969 + 0.975054i \(0.428752\pi\)
\(138\) 0 0
\(139\) 8.66025i 0.734553i −0.930112 0.367277i \(-0.880290\pi\)
0.930112 0.367277i \(-0.119710\pi\)
\(140\) 0 0
\(141\) −9.00000 5.19615i −0.757937 0.437595i
\(142\) 0 0
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.06218 10.5000i 0.500000 0.866025i
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) 20.7846i 1.69143i −0.533637 0.845714i \(-0.679175\pi\)
0.533637 0.845714i \(-0.320825\pi\)
\(152\) 0 0
\(153\) −7.79423 13.5000i −0.630126 1.09141i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) −9.00000 + 15.5885i −0.713746 + 1.23625i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.1244 −0.949653 −0.474826 0.880079i \(-0.657489\pi\)
−0.474826 + 0.880079i \(0.657489\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000i 0.464294i 0.972681 + 0.232147i \(0.0745750\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −13.5000 + 7.79423i −1.03237 + 0.596040i
\(172\) 0 0
\(173\) −20.7846 −1.58022 −0.790112 0.612962i \(-0.789978\pi\)
−0.790112 + 0.612962i \(0.789978\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.3923 + 18.0000i −0.781133 + 1.35296i
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) −6.92820 + 12.0000i −0.512148 + 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −15.5885 −1.13994
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 23.0000i 1.65558i 0.561041 + 0.827788i \(0.310401\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.3923 −0.740421 −0.370211 0.928948i \(-0.620714\pi\)
−0.370211 + 0.928948i \(0.620714\pi\)
\(198\) 0 0
\(199\) 6.92820i 0.491127i −0.969380 0.245564i \(-0.921027\pi\)
0.969380 0.245564i \(-0.0789730\pi\)
\(200\) 0 0
\(201\) 10.5000 18.1865i 0.740613 1.28278i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 15.5885 9.00000i 1.08347 0.625543i
\(208\) 0 0
\(209\) 15.5885i 1.07828i
\(210\) 0 0
\(211\) 1.73205i 0.119239i 0.998221 + 0.0596196i \(0.0189888\pi\)
−0.998221 + 0.0596196i \(0.981011\pi\)
\(212\) 0 0
\(213\) −5.19615 + 9.00000i −0.356034 + 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.50000 0.866025i −0.101361 0.0585206i
\(220\) 0 0
\(221\) 10.3923i 0.699062i
\(222\) 0 0
\(223\) −6.92820 −0.463947 −0.231973 0.972722i \(-0.574518\pi\)
−0.231973 + 0.972722i \(0.574518\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000i 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20.7846 −1.36165 −0.680823 0.732448i \(-0.738378\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.3923 + 6.00000i 0.675053 + 0.389742i
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) −7.79423 13.5000i −0.500000 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.3923 −0.661247
\(248\) 0 0
\(249\) 13.5000 + 7.79423i 0.855528 + 0.493939i
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 18.0000i 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.7846 −1.29651 −0.648254 0.761424i \(-0.724501\pi\)
−0.648254 + 0.761424i \(0.724501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −27.0000 + 15.5885i −1.67126 + 0.964901i
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.79423 + 4.50000i 0.476999 + 0.275396i
\(268\) 0 0
\(269\) 20.7846i 1.26726i 0.773636 + 0.633630i \(0.218436\pi\)
−0.773636 + 0.633630i \(0.781564\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i −0.676435 0.736502i \(-0.736476\pi\)
0.676435 0.736502i \(-0.263524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000i 0.841178i 0.907251 + 0.420589i \(0.138177\pi\)
−0.907251 + 0.420589i \(0.861823\pi\)
\(278\) 0 0
\(279\) −9.00000 + 5.19615i −0.538816 + 0.311086i
\(280\) 0 0
\(281\) 20.7846i 1.23991i −0.784639 0.619953i \(-0.787152\pi\)
0.784639 0.619953i \(-0.212848\pi\)
\(282\) 0 0
\(283\) 12.1244 0.720718 0.360359 0.932814i \(-0.382654\pi\)
0.360359 + 0.932814i \(0.382654\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) 0 0
\(291\) 15.0000 + 8.66025i 0.879316 + 0.507673i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −15.5885 −0.904534
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −12.1244 −0.691974 −0.345987 0.938239i \(-0.612456\pi\)
−0.345987 + 0.938239i \(0.612456\pi\)
\(308\) 0 0
\(309\) −15.0000 + 25.9808i −0.853320 + 1.47799i
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 31.1769i 1.74557i
\(320\) 0 0
\(321\) 4.50000 + 2.59808i 0.251166 + 0.145010i
\(322\) 0 0
\(323\) 27.0000i 1.50232i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.92820 + 12.0000i −0.383131 + 0.663602i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.66025i 0.476011i 0.971264 + 0.238005i \(0.0764936\pi\)
−0.971264 + 0.238005i \(0.923506\pi\)
\(332\) 0 0
\(333\) −20.7846 + 12.0000i −1.13899 + 0.657596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 25.0000i 1.36184i −0.732359 0.680918i \(-0.761581\pi\)
0.732359 0.680918i \(-0.238419\pi\)
\(338\) 0 0
\(339\) −4.50000 + 7.79423i −0.244406 + 0.423324i
\(340\) 0 0
\(341\) 10.3923i 0.562775i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.00000i 0.483145i −0.970383 0.241573i \(-0.922337\pi\)
0.970383 0.241573i \(-0.0776632\pi\)
\(348\) 0 0
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) 10.3923i 0.554700i
\(352\) 0 0
\(353\) 20.7846 1.10625 0.553127 0.833097i \(-0.313435\pi\)
0.553127 + 0.833097i \(0.313435\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) 1.73205 3.00000i 0.0909091 0.157459i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.92820 0.361649 0.180825 0.983515i \(-0.442123\pi\)
0.180825 + 0.983515i \(0.442123\pi\)
\(368\) 0 0
\(369\) 13.5000 7.79423i 0.702782 0.405751i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −20.7846 −1.07046
\(378\) 0 0
\(379\) 19.0526i 0.978664i −0.872098 0.489332i \(-0.837241\pi\)
0.872098 0.489332i \(-0.162759\pi\)
\(380\) 0 0
\(381\) 9.00000 15.5885i 0.461084 0.798621i
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.19615 9.00000i −0.264135 0.457496i
\(388\) 0 0
\(389\) 10.3923i 0.526911i 0.964672 + 0.263455i \(0.0848622\pi\)
−0.964672 + 0.263455i \(0.915138\pi\)
\(390\) 0 0
\(391\) 31.1769i 1.57668i
\(392\) 0 0
\(393\) 10.3923 18.0000i 0.524222 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000i 1.40528i 0.711546 + 0.702640i \(0.247995\pi\)
−0.711546 + 0.702640i \(0.752005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.19615i 0.259483i 0.991548 + 0.129742i \(0.0414148\pi\)
−0.991548 + 0.129742i \(0.958585\pi\)
\(402\) 0 0
\(403\) −6.92820 −0.345118
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) −4.50000 + 7.79423i −0.221969 + 0.384461i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.9904 + 7.50000i 0.636142 + 0.367277i
\(418\) 0 0
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 15.5885 9.00000i 0.757937 0.437595i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.00000 5.19615i −0.434524 0.250873i
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 17.0000i 0.816968i 0.912766 + 0.408484i \(0.133942\pi\)
−0.912766 + 0.408484i \(0.866058\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.1769 1.49139
\(438\) 0 0
\(439\) 17.3205i 0.826663i −0.910581 0.413331i \(-0.864365\pi\)
0.910581 0.413331i \(-0.135635\pi\)
\(440\) 0 0
\(441\) 10.5000 + 18.1865i 0.500000 + 0.866025i
\(442\) 0 0
\(443\) 15.0000i 0.712672i 0.934358 + 0.356336i \(0.115974\pi\)
−0.934358 + 0.356336i \(0.884026\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −15.5885 9.00000i −0.737309 0.425685i
\(448\) 0 0
\(449\) 5.19615i 0.245222i −0.992455 0.122611i \(-0.960873\pi\)
0.992455 0.122611i \(-0.0391267\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) 31.1769 + 18.0000i 1.46482 + 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000i 0.0467780i −0.999726 0.0233890i \(-0.992554\pi\)
0.999726 0.0233890i \(-0.00744563\pi\)
\(458\) 0 0
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 20.7846i 0.968036i 0.875058 + 0.484018i \(0.160823\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(462\) 0 0
\(463\) 27.7128 1.28792 0.643962 0.765058i \(-0.277290\pi\)
0.643962 + 0.765058i \(0.277290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.00000 + 1.73205i 0.138233 + 0.0798087i
\(472\) 0 0
\(473\) −10.3923 −0.477839
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.5885 27.0000i −0.713746 1.23625i
\(478\) 0 0
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 38.1051 1.72671 0.863354 0.504599i \(-0.168360\pi\)
0.863354 + 0.504599i \(0.168360\pi\)
\(488\) 0 0
\(489\) 10.5000 18.1865i 0.474826 0.822423i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 54.0000i 2.43204i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 38.1051i 1.70582i 0.522059 + 0.852910i \(0.325164\pi\)
−0.522059 + 0.852910i \(0.674836\pi\)
\(500\) 0 0
\(501\) −9.00000 5.19615i −0.402090 0.232147i
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −7.79423 + 13.5000i −0.346154 + 0.599556i
\(508\) 0 0
\(509\) 31.1769i 1.38189i −0.722906 0.690946i \(-0.757194\pi\)
0.722906 0.690946i \(-0.242806\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 27.0000i 1.19208i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000i 0.791639i
\(518\) 0 0
\(519\) 18.0000 31.1769i 0.790112 1.36851i
\(520\) 0 0
\(521\) 5.19615i 0.227648i −0.993501 0.113824i \(-0.963690\pi\)
0.993501 0.113824i \(-0.0363099\pi\)
\(522\) 0 0
\(523\) −25.9808 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0000i 0.784092i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) −18.0000 31.1769i −0.781133 1.35296i
\(532\) 0 0
\(533\) 10.3923 0.450141
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.9904 22.5000i 0.560576 0.970947i
\(538\) 0 0
\(539\) 21.0000 0.904534
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 0 0
\(543\) −8.66025 + 15.0000i −0.371647 + 0.643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19.0526 0.814629 0.407314 0.913288i \(-0.366465\pi\)
0.407314 + 0.913288i \(0.366465\pi\)
\(548\) 0 0
\(549\) −12.0000 20.7846i −0.512148 0.887066i
\(550\) 0 0
\(551\) −54.0000 −2.30048
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1769 1.32101 0.660504 0.750822i \(-0.270343\pi\)
0.660504 + 0.750822i \(0.270343\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) 0 0
\(561\) 13.5000 23.3827i 0.569970 0.987218i
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.9808i 1.08917i −0.838706 0.544585i \(-0.816687\pi\)
0.838706 0.544585i \(-0.183313\pi\)
\(570\) 0 0
\(571\) 31.1769i 1.30471i 0.757912 + 0.652357i \(0.226220\pi\)
−0.757912 + 0.652357i \(0.773780\pi\)
\(572\) 0 0
\(573\) −15.5885 + 27.0000i −0.651217 + 1.12794i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000i 0.291414i 0.989328 + 0.145707i \(0.0465456\pi\)
−0.989328 + 0.145707i \(0.953454\pi\)
\(578\) 0 0
\(579\) −34.5000 19.9186i −1.43377 0.827788i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −31.1769 −1.29122
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.0000i 1.11441i 0.830375 + 0.557205i \(0.188126\pi\)
−0.830375 + 0.557205i \(0.811874\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) 15.5885 0.640141 0.320071 0.947394i \(-0.396293\pi\)
0.320071 + 0.947394i \(0.396293\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.3923 + 6.00000i 0.425329 + 0.245564i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 18.1865 + 31.5000i 0.740613 + 1.28278i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.92820 −0.281207 −0.140604 0.990066i \(-0.544904\pi\)
−0.140604 + 0.990066i \(0.544904\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 46.0000i 1.85792i −0.370177 0.928961i \(-0.620703\pi\)
0.370177 0.928961i \(-0.379297\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.5692 1.67351 0.836757 0.547575i \(-0.184449\pi\)
0.836757 + 0.547575i \(0.184449\pi\)
\(618\) 0 0
\(619\) 10.3923i 0.417702i −0.977947 0.208851i \(-0.933028\pi\)
0.977947 0.208851i \(-0.0669724\pi\)
\(620\) 0 0
\(621\) 31.1769i 1.25109i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −23.3827 13.5000i −0.933815 0.539138i
\(628\) 0 0
\(629\) 41.5692i 1.65747i
\(630\) 0 0
\(631\) 3.46410i 0.137904i 0.997620 + 0.0689519i \(0.0219655\pi\)
−0.997620 + 0.0689519i \(0.978035\pi\)
\(632\) 0 0
\(633\) −2.59808 1.50000i −0.103264 0.0596196i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.0000i 0.554700i
\(638\) 0 0
\(639\) −9.00000 15.5885i −0.356034 0.616670i
\(640\) 0 0
\(641\) 20.7846i 0.820943i 0.911873 + 0.410471i \(0.134636\pi\)
−0.911873 + 0.410471i \(0.865364\pi\)
\(642\) 0 0
\(643\) 31.1769 1.22950 0.614749 0.788723i \(-0.289257\pi\)
0.614749 + 0.788723i \(0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.7846 0.813365 0.406682 0.913570i \(-0.366686\pi\)
0.406682 + 0.913570i \(0.366686\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.59808 1.50000i 0.101361 0.0585206i
\(658\) 0 0
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) 15.5885 + 9.00000i 0.605406 + 0.349531i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 62.3538 2.41435
\(668\) 0 0
\(669\) 6.00000 10.3923i 0.231973 0.401790i
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −41.5692 −1.59763 −0.798817 0.601574i \(-0.794541\pi\)
−0.798817 + 0.601574i \(0.794541\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 36.0000 + 20.7846i 1.37952 + 0.796468i
\(682\) 0 0
\(683\) 21.0000i 0.803543i 0.915740 + 0.401771i \(0.131605\pi\)
−0.915740 + 0.401771i \(0.868395\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.3205 + 30.0000i −0.660819 + 1.14457i
\(688\) 0 0
\(689\) 20.7846i 0.791831i
\(690\) 0 0
\(691\) 5.19615i 0.197671i 0.995104 + 0.0988355i \(0.0315118\pi\)
−0.995104 + 0.0988355i \(0.968488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 27.0000i 1.02270i
\(698\) 0 0
\(699\) 18.0000 31.1769i 0.680823 1.17922i
\(700\) 0 0
\(701\) 31.1769i 1.17754i −0.808302 0.588768i \(-0.799613\pi\)
0.808302 0.588768i \(-0.200387\pi\)
\(702\) 0 0
\(703\) −41.5692 −1.56781
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −18.0000 + 10.3923i −0.675053 + 0.389742i
\(712\) 0 0
\(713\) 20.7846 0.778390
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 25.9808 45.0000i 0.970269 1.68056i
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 4.33013 7.50000i 0.161039 0.278928i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.3205 0.642382 0.321191 0.947014i \(-0.395917\pi\)
0.321191 + 0.947014i \(0.395917\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 18.0000 0.665754
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.3731 1.33982
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) 9.00000 15.5885i 0.330623 0.572656i
\(742\) 0 0
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −23.3827 + 13.5000i −0.855528 + 0.493939i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 38.1051i 1.39048i 0.718780 + 0.695238i \(0.244701\pi\)
−0.718780 + 0.695238i \(0.755299\pi\)
\(752\) 0 0
\(753\) 18.1865 31.5000i 0.662754 1.14792i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 16.0000i 0.581530i −0.956795 0.290765i \(-0.906090\pi\)
0.956795 0.290765i \(-0.0939098\pi\)
\(758\) 0 0
\(759\) 27.0000 + 15.5885i 0.980038 + 0.565825i
\(760\) 0 0
\(761\) 15.5885i 0.565081i 0.959255 + 0.282541i \(0.0911772\pi\)
−0.959255 + 0.282541i \(0.908823\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) 18.0000 31.1769i 0.648254 1.12281i
\(772\) 0 0
\(773\) −10.3923 −0.373785 −0.186893 0.982380i \(-0.559842\pi\)
−0.186893 + 0.982380i \(0.559842\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.0000 0.967375
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 54.0000i 1.92980i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 10.3923 0.370446 0.185223 0.982697i \(-0.440699\pi\)
0.185223 + 0.982697i \(0.440699\pi\)
\(788\) 0 0
\(789\) −9.00000 5.19615i −0.320408 0.184988i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000i 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −51.9615 −1.84057 −0.920286 0.391247i \(-0.872044\pi\)
−0.920286 + 0.391247i \(0.872044\pi\)
\(798\) 0 0
\(799\) 31.1769i 1.10296i
\(800\) 0 0
\(801\) −13.5000 + 7.79423i −0.476999 + 0.275396i
\(802\) 0 0
\(803\) 3.00000i 0.105868i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.1769 18.0000i −1.09748 0.633630i
\(808\) 0 0
\(809\) 20.7846i 0.730748i 0.930861 + 0.365374i \(0.119059\pi\)
−0.930861 + 0.365374i \(0.880941\pi\)
\(810\) 0 0
\(811\) 17.3205i 0.608205i −0.952639 0.304103i \(-0.901643\pi\)
0.952639 0.304103i \(-0.0983566\pi\)
\(812\) 0 0
\(813\) 36.3731 + 21.0000i 1.27566 + 0.736502i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 18.0000i 0.629740i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.3923i 0.362694i 0.983419 + 0.181347i \(0.0580457\pi\)
−0.983419 + 0.181347i \(0.941954\pi\)
\(822\) 0 0
\(823\) 48.4974 1.69051 0.845257 0.534360i \(-0.179447\pi\)
0.845257 + 0.534360i \(0.179447\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.0000i 1.35616i 0.734987 + 0.678081i \(0.237188\pi\)
−0.734987 + 0.678081i \(0.762812\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −21.0000 12.1244i −0.728482 0.420589i
\(832\) 0 0
\(833\) −36.3731 −1.26025
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 18.0000i 0.622171i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −79.0000 −2.72414
\(842\) 0 0
\(843\) 31.1769 + 18.0000i 1.07379 + 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.5000 + 18.1865i −0.360359 + 0.624160i
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 22.0000i 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.5885 0.532492 0.266246 0.963905i \(-0.414217\pi\)
0.266246 + 0.963905i \(0.414217\pi\)
\(858\) 0 0
\(859\) 22.5167i 0.768259i 0.923279 + 0.384129i \(0.125498\pi\)
−0.923279 + 0.384129i \(0.874502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 54.0000i 1.83818i −0.394046 0.919091i \(-0.628925\pi\)
0.394046 0.919091i \(-0.371075\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.66025 + 15.0000i −0.294118 + 0.509427i
\(868\) 0 0
\(869\) 20.7846i 0.705070i
\(870\) 0 0
\(871\) 24.2487i 0.821636i
\(872\) 0 0
\(873\) −25.9808 + 15.0000i −0.879316 + 0.507673i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40.0000i 1.35070i 0.737496 + 0.675352i \(0.236008\pi\)
−0.737496 + 0.675352i \(0.763992\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 20.7846i 0.700251i 0.936703 + 0.350126i \(0.113861\pi\)
−0.936703 + 0.350126i \(0.886139\pi\)
\(882\) 0 0
\(883\) −25.9808 −0.874322 −0.437161 0.899383i \(-0.644016\pi\)
−0.437161 + 0.899383i \(0.644016\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 13.5000 23.3827i 0.452267 0.783349i
\(892\) 0 0
\(893\) 31.1769 1.04330
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −10.3923 + 18.0000i −0.346989 + 0.601003i
\(898\) 0 0
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 54.0000 1.79900
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 38.1051 1.26526 0.632630 0.774454i \(-0.281975\pi\)
0.632630 + 0.774454i \(0.281975\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 27.0000i 0.893570i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.46410i 0.114270i −0.998366 0.0571351i \(-0.981803\pi\)
0.998366 0.0571351i \(-0.0181966\pi\)
\(920\) 0 0
\(921\) 10.5000 18.1865i 0.345987 0.599267i
\(922\) 0 0
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −25.9808 45.0000i −0.853320 1.47799i
\(928\) 0 0
\(929\) 20.7846i 0.681921i 0.940078 + 0.340960i \(0.110752\pi\)
−0.940078 + 0.340960i \(0.889248\pi\)
\(930\) 0 0
\(931\) 36.3731i 1.19208i
\(932\) 0 0
\(933\) −20.7846 + 36.0000i −0.680458 + 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 25.0000i 0.816714i 0.912822 + 0.408357i \(0.133898\pi\)
−0.912822 + 0.408357i \(0.866102\pi\)
\(938\) 0 0
\(939\) −21.0000 12.1244i −0.685309 0.395663i
\(940\) 0 0
\(941\) 10.3923i 0.338779i −0.985549 0.169390i \(-0.945820\pi\)
0.985549 0.169390i \(-0.0541797\pi\)
\(942\) 0 0
\(943\) −31.1769 −1.01526
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.3731 1.17824 0.589120 0.808046i \(-0.299475\pi\)
0.589120 + 0.808046i \(0.299475\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −46.7654 27.0000i −1.51171 0.872786i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) −7.79423 + 4.50000i −0.251166 + 0.145010i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −13.8564 −0.445592 −0.222796 0.974865i \(-0.571518\pi\)
−0.222796 + 0.974865i \(0.571518\pi\)
\(968\) 0 0
\(969\) 40.5000 + 23.3827i 1.30105 + 0.751160i
\(970\) 0 0
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.9808 −0.831198 −0.415599 0.909548i \(-0.636428\pi\)
−0.415599 + 0.909548i \(0.636428\pi\)
\(978\) 0 0
\(979\) 15.5885i 0.498209i
\(980\) 0 0
\(981\) −12.0000 20.7846i −0.383131 0.663602i
\(982\) 0 0
\(983\) 24.0000i 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.7846i 0.660912i
\(990\) 0 0
\(991\) 20.7846i 0.660245i 0.943938 + 0.330122i \(0.107090\pi\)
−0.943938 + 0.330122i \(0.892910\pi\)
\(992\) 0 0
\(993\) −12.9904 7.50000i −0.412237 0.238005i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0000i 0.886769i 0.896332 + 0.443384i \(0.146222\pi\)
−0.896332 + 0.443384i \(0.853778\pi\)
\(998\) 0 0
\(999\) 41.5692i 1.31519i
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 1200.2.o.c.1199.2 4
3.2 odd 2 1200.2.o.d.1199.1 4
4.3 odd 2 1200.2.o.d.1199.3 4
5.2 odd 4 1200.2.h.h.1151.2 yes 2
5.3 odd 4 1200.2.h.a.1151.1 2
5.4 even 2 inner 1200.2.o.c.1199.3 4
12.11 even 2 inner 1200.2.o.c.1199.4 4
15.2 even 4 1200.2.h.b.1151.2 yes 2
15.8 even 4 1200.2.h.i.1151.1 yes 2
15.14 odd 2 1200.2.o.d.1199.4 4
20.3 even 4 1200.2.h.i.1151.2 yes 2
20.7 even 4 1200.2.h.b.1151.1 yes 2
20.19 odd 2 1200.2.o.d.1199.2 4
60.23 odd 4 1200.2.h.a.1151.2 yes 2
60.47 odd 4 1200.2.h.h.1151.1 yes 2
60.59 even 2 inner 1200.2.o.c.1199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1200.2.h.a.1151.1 2 5.3 odd 4
1200.2.h.a.1151.2 yes 2 60.23 odd 4
1200.2.h.b.1151.1 yes 2 20.7 even 4
1200.2.h.b.1151.2 yes 2 15.2 even 4
1200.2.h.h.1151.1 yes 2 60.47 odd 4
1200.2.h.h.1151.2 yes 2 5.2 odd 4
1200.2.h.i.1151.1 yes 2 15.8 even 4
1200.2.h.i.1151.2 yes 2 20.3 even 4
1200.2.o.c.1199.1 4 60.59 even 2 inner
1200.2.o.c.1199.2 4 1.1 even 1 trivial
1200.2.o.c.1199.3 4 5.4 even 2 inner
1200.2.o.c.1199.4 4 12.11 even 2 inner
1200.2.o.d.1199.1 4 3.2 odd 2
1200.2.o.d.1199.2 4 20.19 odd 2
1200.2.o.d.1199.3 4 4.3 odd 2
1200.2.o.d.1199.4 4 15.14 odd 2