Properties

Label 900.1.bh.a.503.1
Level $900$
Weight $1$
Character 900.503
Analytic conductor $0.449$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,1,Mod(287,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.287");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 900.bh (of order \(20\), degree \(8\), 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: \(0.449158511370\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label 503.1
Root \(0.453990 + 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 900.503
Dual form 900.1.bh.a.467.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.891007 + 0.453990i) q^{2} +(0.587785 - 0.809017i) q^{4} +(-0.987688 - 0.156434i) q^{5} +(-0.156434 + 0.987688i) q^{8} +O(q^{10})\) \(q+(-0.891007 + 0.453990i) q^{2} +(0.587785 - 0.809017i) q^{4} +(-0.987688 - 0.156434i) q^{5} +(-0.156434 + 0.987688i) q^{8} +(0.951057 - 0.309017i) q^{10} +(-0.896802 + 1.76007i) q^{13} +(-0.309017 - 0.951057i) q^{16} +(1.87869 + 0.297556i) q^{17} +(-0.707107 + 0.707107i) q^{20} +(0.951057 + 0.309017i) q^{25} -1.97538i q^{26} +(1.44168 + 1.04744i) q^{29} +(0.707107 + 0.707107i) q^{32} +(-1.80902 + 0.587785i) q^{34} +(0.809017 + 0.412215i) q^{37} +(0.309017 - 0.951057i) q^{40} +(-0.297556 + 0.0966818i) q^{41} -1.00000i q^{49} +(-0.987688 + 0.156434i) q^{50} +(0.896802 + 1.76007i) q^{52} +(-1.16110 + 0.183900i) q^{53} +(-1.76007 - 0.278768i) q^{58} +(-0.363271 + 1.11803i) q^{61} +(-0.951057 - 0.309017i) q^{64} +(1.16110 - 1.59811i) q^{65} +(1.34500 - 1.34500i) q^{68} +(-0.278768 + 0.142040i) q^{73} -0.907981 q^{74} +(0.156434 + 0.987688i) q^{80} +(0.221232 - 0.221232i) q^{82} +(-1.80902 - 0.587785i) q^{85} +(-0.280582 + 0.863541i) q^{89} +(-1.76007 + 0.278768i) q^{97} +(0.453990 + 0.891007i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{13} + 4 q^{16} - 20 q^{34} + 4 q^{37} - 4 q^{40} - 4 q^{52} - 4 q^{58} - 4 q^{73} + 4 q^{82} - 20 q^{85} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{7}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(3\) 0 0
\(4\) 0.587785 0.809017i 0.587785 0.809017i
\(5\) −0.987688 0.156434i −0.987688 0.156434i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(9\) 0 0
\(10\) 0.951057 0.309017i 0.951057 0.309017i
\(11\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(12\) 0 0
\(13\) −0.896802 + 1.76007i −0.896802 + 1.76007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.309017 0.951057i −0.309017 0.951057i
\(17\) 1.87869 + 0.297556i 1.87869 + 0.297556i 0.987688 0.156434i \(-0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(20\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(24\) 0 0
\(25\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(26\) 1.97538i 1.97538i
\(27\) 0 0
\(28\) 0 0
\(29\) 1.44168 + 1.04744i 1.44168 + 1.04744i 0.987688 + 0.156434i \(0.0500000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(30\) 0 0
\(31\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.809017 + 0.412215i 0.809017 + 0.412215i 0.809017 0.587785i \(-0.200000\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.309017 0.951057i 0.309017 0.951057i
\(41\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(51\) 0 0
\(52\) 0.896802 + 1.76007i 0.896802 + 1.76007i
\(53\) −1.16110 + 0.183900i −1.16110 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.76007 0.278768i −1.76007 0.278768i
\(59\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) −0.363271 + 1.11803i −0.363271 + 1.11803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.951057 0.309017i −0.951057 0.309017i
\(65\) 1.16110 1.59811i 1.16110 1.59811i
\(66\) 0 0
\(67\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(68\) 1.34500 1.34500i 1.34500 1.34500i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(72\) 0 0
\(73\) −0.278768 + 0.142040i −0.278768 + 0.142040i −0.587785 0.809017i \(-0.700000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) −0.907981 −0.907981
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(80\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(81\) 0 0
\(82\) 0.221232 0.221232i 0.221232 0.221232i
\(83\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(84\) 0 0
\(85\) −1.80902 0.587785i −1.80902 0.587785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.280582 + 0.863541i −0.280582 + 0.863541i 0.707107 + 0.707107i \(0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.76007 + 0.278768i −1.76007 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(99\) 0 0
\(100\) 0.809017 0.587785i 0.809017 0.587785i
\(101\) 0.907981i 0.907981i −0.891007 0.453990i \(-0.850000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(104\) −1.59811 1.16110i −1.59811 1.16110i
\(105\) 0 0
\(106\) 0.951057 0.690983i 0.951057 0.690983i
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.550672 0.280582i −0.550672 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.69480 0.550672i 1.69480 0.550672i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(122\) −0.183900 1.16110i −0.183900 1.16110i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.891007 0.453990i −0.891007 0.453990i
\(126\) 0 0
\(127\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(128\) 0.987688 0.156434i 0.987688 0.156434i
\(129\) 0 0
\(130\) −0.309017 + 1.95106i −0.309017 + 1.95106i
\(131\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(137\) 0.734572 1.44168i 0.734572 1.44168i −0.156434 0.987688i \(-0.550000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.26007 1.26007i −1.26007 1.26007i
\(146\) 0.183900 0.253116i 0.183900 0.253116i
\(147\) 0 0
\(148\) 0.809017 0.412215i 0.809017 0.412215i
\(149\) −0.312869 −0.312869 −0.156434 0.987688i \(-0.550000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.26007 1.26007i 1.26007 1.26007i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.587785 0.809017i −0.587785 0.809017i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(164\) −0.0966818 + 0.297556i −0.0966818 + 0.297556i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(168\) 0 0
\(169\) −1.70582 2.34786i −1.70582 2.34786i
\(170\) 1.87869 0.297556i 1.87869 0.297556i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.280582 0.550672i −0.280582 0.550672i 0.707107 0.707107i \(-0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −0.142040 0.896802i −0.142040 0.896802i
\(179\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(180\) 0 0
\(181\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.734572 0.533698i −0.734572 0.533698i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(192\) 0 0
\(193\) −0.221232 0.221232i −0.221232 0.221232i 0.587785 0.809017i \(-0.300000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(194\) 1.44168 1.04744i 1.44168 1.04744i
\(195\) 0 0
\(196\) −0.809017 0.587785i −0.809017 0.587785i
\(197\) −0.253116 1.59811i −0.253116 1.59811i −0.707107 0.707107i \(-0.750000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(201\) 0 0
\(202\) 0.412215 + 0.809017i 0.412215 + 0.809017i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.309017 0.0489435i 0.309017 0.0489435i
\(206\) 0 0
\(207\) 0 0
\(208\) 1.95106 + 0.309017i 1.95106 + 0.309017i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(212\) −0.533698 + 1.04744i −0.533698 + 1.04744i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.831254 + 0.831254i −0.831254 + 0.831254i
\(219\) 0 0
\(220\) 0 0
\(221\) −2.20854 + 3.03979i −2.20854 + 3.03979i
\(222\) 0 0
\(223\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.618034 0.618034
\(227\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(228\) 0 0
\(229\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(233\) −0.183900 + 1.16110i −0.183900 + 1.16110i 0.707107 + 0.707107i \(0.250000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(240\) 0 0
\(241\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(242\) −0.987688 0.156434i −0.987688 0.156434i
\(243\) 0 0
\(244\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(245\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 1.00000
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(257\) 1.14412 + 1.14412i 1.14412 + 1.14412i 0.987688 + 0.156434i \(0.0500000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.610425 1.87869i −0.610425 1.87869i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(264\) 0 0
\(265\) 1.17557 1.17557
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.59811 + 1.16110i −1.59811 + 1.16110i −0.707107 + 0.707107i \(0.750000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(272\) −0.297556 1.87869i −0.297556 1.87869i
\(273\) 0 0
\(274\) 1.61803i 1.61803i
\(275\) 0 0
\(276\) 0 0
\(277\) −0.412215 0.809017i −0.412215 0.809017i 0.587785 0.809017i \(-0.300000\pi\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.16110 + 1.59811i 1.16110 + 1.59811i 0.707107 + 0.707107i \(0.250000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.48990 + 0.809017i 2.48990 + 0.809017i
\(290\) 1.69480 + 0.550672i 1.69480 + 0.550672i
\(291\) 0 0
\(292\) −0.0489435 + 0.309017i −0.0489435 + 0.309017i
\(293\) 0.437016 0.437016i 0.437016 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.533698 + 0.734572i −0.533698 + 0.734572i
\(297\) 0 0
\(298\) 0.278768 0.142040i 0.278768 0.142040i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.533698 1.04744i 0.533698 1.04744i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(312\) 0 0
\(313\) 0.642040 1.26007i 0.642040 1.26007i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(314\) −0.550672 + 1.69480i −0.550672 + 1.69480i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.39680 + 1.39680i −1.39680 + 1.39680i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.0489435 0.309017i −0.0489435 0.309017i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.26007 0.642040i −1.26007 0.642040i −0.309017 0.951057i \(-0.600000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(338\) 2.58580 + 1.31753i 2.58580 + 1.31753i
\(339\) 0 0
\(340\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(347\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(348\) 0 0
\(349\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.533698 + 0.734572i 0.533698 + 0.734572i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(360\) 0 0
\(361\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(362\) −0.863541 + 1.69480i −0.863541 + 1.69480i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.297556 0.0966818i 0.297556 0.0966818i
\(366\) 0 0
\(367\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0.896802 + 0.142040i 0.896802 + 0.142040i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.26007 0.642040i 1.26007 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.13647 + 1.59811i −3.13647 + 1.59811i
\(378\) 0 0
\(379\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(387\) 0 0
\(388\) −0.809017 + 1.58779i −0.809017 + 1.58779i
\(389\) 0.610425 1.87869i 0.610425 1.87869i 0.156434 0.987688i \(-0.450000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(393\) 0 0
\(394\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.39680 + 0.221232i −1.39680 + 0.221232i −0.809017 0.587785i \(-0.800000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000i 1.00000i
\(401\) 0.312869i 0.312869i −0.987688 0.156434i \(-0.950000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.734572 0.533698i −0.734572 0.533698i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.80902 0.587785i 1.80902 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
1.00000 \(0\)
\(410\) −0.253116 + 0.183900i −0.253116 + 0.183900i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.87869 + 0.610425i −1.87869 + 0.610425i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.17557i 1.17557i
\(425\) 1.69480 + 0.863541i 1.69480 + 0.863541i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(432\) 0 0
\(433\) −1.95106 0.309017i −1.95106 0.309017i −0.951057 0.309017i \(-0.900000\pi\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.363271 1.11803i 0.363271 1.11803i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.587785 3.71113i 0.587785 3.71113i
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) 0 0
\(445\) 0.412215 0.809017i 0.412215 0.809017i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.78201 1.78201 0.891007 0.453990i \(-0.150000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.550672 + 0.280582i −0.550672 + 0.280582i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(458\) 0.253116 1.59811i 0.253116 1.59811i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.69480 0.550672i −1.69480 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(462\) 0 0
\(463\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(464\) 0.550672 1.69480i 0.550672 1.69480i
\(465\) 0 0
\(466\) −0.363271 1.11803i −0.363271 1.11803i
\(467\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(480\) 0 0
\(481\) −1.45106 + 1.05425i −1.45106 + 1.05425i
\(482\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(483\) 0 0
\(484\) 0.951057 0.309017i 0.951057 0.309017i
\(485\) 1.78201 1.78201
\(486\) 0 0
\(487\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(488\) −1.04744 0.533698i −1.04744 0.533698i
\(489\) 0 0
\(490\) −0.309017 0.951057i −0.309017 0.951057i
\(491\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(492\) 0 0
\(493\) 2.39680 + 2.39680i 2.39680 + 2.39680i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(504\) 0 0
\(505\) −0.142040 + 0.896802i −0.142040 + 0.896802i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.280582 0.863541i −0.280582 0.863541i −0.987688 0.156434i \(-0.950000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.453990 0.891007i 0.453990 0.891007i
\(513\) 0 0
\(514\) −1.53884 0.500000i −1.53884 0.500000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(521\) 1.16110 1.59811i 1.16110 1.59811i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(522\) 0 0
\(523\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(530\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.0966818 0.610425i 0.0966818 0.610425i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.896802 1.76007i 0.896802 1.76007i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.587785 1.80902i −0.587785 1.80902i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(-0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(545\) −1.16110 + 0.183900i −1.16110 + 0.183900i
\(546\) 0 0
\(547\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(548\) −0.734572 1.44168i −0.734572 1.44168i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.734572 + 0.533698i 0.734572 + 0.533698i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.34500 1.34500i −1.34500 1.34500i −0.891007 0.453990i \(-0.850000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.76007 0.896802i −1.76007 0.896802i
\(563\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(564\) 0 0
\(565\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.734572 + 0.533698i −0.734572 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(578\) −2.58580 + 0.409551i −2.58580 + 0.409551i
\(579\) 0 0
\(580\) −1.76007 + 0.278768i −1.76007 + 0.278768i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.0966818 0.297556i −0.0966818 0.297556i
\(585\) 0 0
\(586\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(587\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.142040 0.896802i 0.142040 0.896802i
\(593\) −0.437016 + 0.437016i −0.437016 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.183900 + 0.253116i −0.183900 + 0.253116i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.707107 0.707107i −0.707107 0.707107i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.17557i 1.17557i
\(611\) 0 0
\(612\) 0 0
\(613\) 0.809017 1.58779i 0.809017 1.58779i 1.00000i \(-0.5\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.16110 + 0.183900i 1.16110 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(626\) 1.41421i 1.41421i
\(627\) 0 0
\(628\) −0.278768 1.76007i −0.278768 1.76007i
\(629\) 1.39724 + 1.01515i 1.39724 + 1.01515i
\(630\) 0 0
\(631\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −1.00000
\(641\) 1.34500 0.437016i 1.34500 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.610425 1.87869i 0.610425 1.87869i
\(651\) 0 0
\(652\) 0 0
\(653\) 1.87869 0.297556i 1.87869 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.183900 + 0.253116i 0.183900 + 0.253116i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.809017 + 0.412215i −0.809017 + 0.412215i −0.809017 0.587785i \(-0.800000\pi\)
1.00000i \(0.5\pi\)
\(674\) 1.41421 1.41421
\(675\) 0 0
\(676\) −2.90211 −2.90211
\(677\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.863541 1.69480i 0.863541 1.69480i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(684\) 0 0
\(685\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.717598 2.20854i 0.717598 2.20854i
\(690\) 0 0
\(691\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) −0.610425 0.0966818i −0.610425 0.0966818i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.587785 + 0.0930960i −0.587785 + 0.0930960i
\(698\) −0.863541 1.69480i −0.863541 1.69480i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.312869i 0.312869i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.809017 0.412215i −0.809017 0.412215i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.156434 0.987688i −0.156434 0.987688i
\(723\) 0 0
\(724\) 1.90211i 1.90211i
\(725\) 1.04744 + 1.44168i 1.04744 + 1.44168i
\(726\) 0 0
\(727\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.221232 + 0.221232i −0.221232 + 0.221232i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.39680 + 0.221232i 1.39680 + 0.221232i 0.809017 0.587785i \(-0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(740\) −0.863541 + 0.280582i −0.863541 + 0.280582i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0.309017 + 0.0489435i 0.309017 + 0.0489435i
\(746\) −0.831254 + 1.14412i −0.831254 + 1.14412i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 2.06909 2.84786i 2.06909 2.84786i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.221232 + 0.221232i −0.221232 + 0.221232i −0.809017 0.587785i \(-0.800000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.87869 + 0.610425i 1.87869 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(773\) −0.533698 1.04744i −0.533698 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.78201i 1.78201i
\(777\) 0 0
\(778\) 0.309017 + 1.95106i 0.309017 + 1.95106i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(785\) −1.44168 + 1.04744i −1.44168 + 1.04744i
\(786\) 0 0
\(787\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(788\) −1.44168 0.734572i −1.44168 0.734572i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.64204 1.64204i −1.64204 1.64204i
\(794\) 1.14412 0.831254i 1.14412 0.831254i
\(795\) 0 0
\(796\) 0 0
\(797\) 0.253116 + 1.59811i 0.253116 + 1.59811i 0.707107 + 0.707107i \(0.250000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(801\) 0 0
\(802\) 0.142040 + 0.278768i 0.142040 + 0.278768i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.896802 + 0.142040i 0.896802 + 0.142040i
\(809\) −0.550672 1.69480i −0.550672 1.69480i −0.707107 0.707107i \(-0.750000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(810\) 0 0
\(811\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(819\) 0 0
\(820\) 0.142040 0.278768i 0.142040 0.278768i
\(821\) 0.831254 1.14412i 0.831254 1.14412i −0.156434 0.987688i \(-0.550000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(822\) 0 0
\(823\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(828\) 0 0
\(829\) −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i \(0.400000\pi\)
−1.00000 \(1.00000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.39680 1.39680i 1.39680 1.39680i
\(833\) 0.297556 1.87869i 0.297556 1.87869i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(840\) 0 0
\(841\) 0.672288 + 2.06909i 0.672288 + 2.06909i
\(842\) 0.610425 + 0.0966818i 0.610425 + 0.0966818i
\(843\) 0 0
\(844\) 0 0
\(845\) 1.31753 + 2.58580i 1.31753 + 2.58580i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.533698 + 1.04744i 0.533698 + 1.04744i
\(849\) 0 0
\(850\) −1.90211 −1.90211
\(851\) 0 0
\(852\) 0 0
\(853\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i 1.00000 \(0\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(864\) 0 0
\(865\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(866\) 1.87869 0.610425i 1.87869 0.610425i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.183900 + 1.16110i 0.183900 + 1.16110i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.896802 1.76007i −0.896802 1.76007i −0.587785 0.809017i \(-0.700000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.831254 1.14412i −0.831254 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(882\) 0 0
\(883\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(884\) 1.16110 + 3.57349i 1.16110 + 3.57349i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0.907981i 0.907981i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.58779 + 0.809017i −1.58779 + 0.809017i
\(899\) 0 0
\(900\) 0 0
\(901\) −2.23607 −2.23607
\(902\) 0 0
\(903\) 0 0
\(904\) 0.363271 0.500000i 0.363271 0.500000i
\(905\) −1.69480 + 0.863541i −1.69480 + 0.863541i
\(906\) 0 0
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.437016 + 1.34500i −0.437016 + 1.34500i
\(915\) 0 0
\(916\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.76007 0.278768i 1.76007 0.278768i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.642040 + 0.642040i 0.642040 + 0.642040i
\(926\) 0 0
\(927\) 0 0
\(928\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(929\) −1.44168 1.04744i −1.44168 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.831254 + 0.831254i 0.831254 + 0.831254i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.58779 + 0.809017i 1.58779 + 0.809017i 1.00000 \(0\)
0.587785 + 0.809017i \(0.300000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(948\) 0 0
\(949\) 0.618034i 0.618034i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.610425 + 0.0966818i −0.610425 + 0.0966818i −0.453990 0.891007i \(-0.650000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.309017 0.951057i 0.309017 0.951057i
\(962\) 0.814279 1.59811i 0.814279 1.59811i
\(963\) 0 0
\(964\) −0.587785 0.190983i −0.587785 0.190983i
\(965\) 0.183900 + 0.253116i 0.183900 + 0.253116i
\(966\) 0 0
\(967\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(968\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(969\) 0 0
\(970\) −1.58779 + 0.809017i −1.58779 + 0.809017i
\(971\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.17557 1.17557
\(977\) −1.04744 + 0.533698i −1.04744 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(984\) 0 0
\(985\) 1.61803i 1.61803i
\(986\) −3.22369 1.04744i −3.22369 1.04744i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.39680 0.221232i 1.39680 0.221232i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(998\) 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 900.1.bh.a.503.1 yes 16
3.2 odd 2 inner 900.1.bh.a.503.2 yes 16
4.3 odd 2 CM 900.1.bh.a.503.1 yes 16
12.11 even 2 inner 900.1.bh.a.503.2 yes 16
25.17 odd 20 inner 900.1.bh.a.467.2 yes 16
75.17 even 20 inner 900.1.bh.a.467.1 16
100.67 even 20 inner 900.1.bh.a.467.2 yes 16
300.167 odd 20 inner 900.1.bh.a.467.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.1.bh.a.467.1 16 75.17 even 20 inner
900.1.bh.a.467.1 16 300.167 odd 20 inner
900.1.bh.a.467.2 yes 16 25.17 odd 20 inner
900.1.bh.a.467.2 yes 16 100.67 even 20 inner
900.1.bh.a.503.1 yes 16 1.1 even 1 trivial
900.1.bh.a.503.1 yes 16 4.3 odd 2 CM
900.1.bh.a.503.2 yes 16 3.2 odd 2 inner
900.1.bh.a.503.2 yes 16 12.11 even 2 inner