Properties

Label 900.1.bb.a.19.1
Level $900$
Weight $1$
Character 900.19
Analytic conductor $0.449$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
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(19,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.19");
 
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.bb (of order \(10\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.47460937500000000.8

Embedding invariants

Embedding label 19.1
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 900.19
Dual form 900.1.bb.a.379.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.951057 + 0.309017i) q^{2} +(0.809017 - 0.587785i) q^{4} +(0.587785 - 0.809017i) q^{5} +(-0.587785 + 0.809017i) q^{8} +O(q^{10})\) \(q+(-0.951057 + 0.309017i) q^{2} +(0.809017 - 0.587785i) q^{4} +(0.587785 - 0.809017i) q^{5} +(-0.587785 + 0.809017i) q^{8} +(-0.309017 + 0.951057i) q^{10} +(1.11803 + 0.363271i) q^{13} +(0.309017 - 0.951057i) q^{16} +(-0.363271 + 0.500000i) q^{17} -1.00000i q^{20} +(-0.309017 - 0.951057i) q^{25} -1.17557 q^{26} +(1.53884 - 1.11803i) q^{29} +1.00000i q^{32} +(0.190983 - 0.587785i) q^{34} +(-1.80902 - 0.587785i) q^{37} +(0.309017 + 0.951057i) q^{40} +(0.363271 - 1.11803i) q^{41} -1.00000 q^{49} +(0.587785 + 0.809017i) q^{50} +(1.11803 - 0.363271i) q^{52} +(0.951057 + 1.30902i) q^{53} +(-1.11803 + 1.53884i) q^{58} +(0.500000 + 1.53884i) q^{61} +(-0.309017 - 0.951057i) q^{64} +(0.951057 - 0.690983i) q^{65} +0.618034i q^{68} +(1.11803 - 0.363271i) q^{73} +1.90211 q^{74} +(-0.587785 - 0.809017i) q^{80} +1.17557i q^{82} +(0.190983 + 0.587785i) q^{85} +(0.587785 + 1.80902i) q^{89} +(-1.11803 - 1.53884i) q^{97} +(0.951057 - 0.309017i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 2 q^{10} - 2 q^{16} + 2 q^{25} + 6 q^{34} - 10 q^{37} - 2 q^{40} - 8 q^{49} + 4 q^{61} + 2 q^{64} + 6 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(3\) 0 0
\(4\) 0.809017 0.587785i 0.809017 0.587785i
\(5\) 0.587785 0.809017i 0.587785 0.809017i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(9\) 0 0
\(10\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(11\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(12\) 0 0
\(13\) 1.11803 + 0.363271i 1.11803 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.309017 0.951057i 0.309017 0.951057i
\(17\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) 1.00000i 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(24\) 0 0
\(25\) −0.309017 0.951057i −0.309017 0.951057i
\(26\) −1.17557 −1.17557
\(27\) 0 0
\(28\) 0 0
\(29\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) 0.190983 0.587785i 0.190983 0.587785i
\(35\) 0 0
\(36\) 0 0
\(37\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(41\) 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(48\) 0 0
\(49\) −1.00000 −1.00000
\(50\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(51\) 0 0
\(52\) 1.11803 0.363271i 1.11803 0.363271i
\(53\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(59\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(60\) 0 0
\(61\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.309017 0.951057i −0.309017 0.951057i
\(65\) 0.951057 0.690983i 0.951057 0.690983i
\(66\) 0 0
\(67\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(68\) 0.618034i 0.618034i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0 0
\(73\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(74\) 1.90211 1.90211
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) −0.587785 0.809017i −0.587785 0.809017i
\(81\) 0 0
\(82\) 1.17557i 1.17557i
\(83\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 0 0
\(85\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(98\) 0.951057 0.309017i 0.951057 0.309017i
\(99\) 0 0
\(100\) −0.809017 0.587785i −0.809017 0.587785i
\(101\) −1.90211 −1.90211 −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(104\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(105\) 0 0
\(106\) −1.30902 0.951057i −1.30902 0.951057i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.587785 1.80902i 0.587785 1.80902i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(122\) −0.951057 1.30902i −0.951057 1.30902i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.951057 0.309017i −0.951057 0.309017i
\(126\) 0 0
\(127\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(128\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(129\) 0 0
\(130\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(131\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.190983 0.587785i −0.190983 0.587785i
\(137\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.90211i 1.90211i
\(146\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(147\) 0 0
\(148\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(149\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(150\) 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
\(157\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(164\) −0.363271 1.11803i −0.363271 1.11803i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(168\) 0 0
\(169\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(170\) −0.363271 0.500000i −0.363271 0.500000i
\(171\) 0 0
\(172\) 0 0
\(173\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.11803 1.53884i −1.11803 1.53884i
\(179\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(180\) 0 0
\(181\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(192\) 0 0
\(193\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(194\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(195\) 0 0
\(196\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(197\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(201\) 0 0
\(202\) 1.80902 0.587785i 1.80902 0.587785i
\(203\) 0 0
\(204\) 0 0
\(205\) −0.690983 0.951057i −0.690983 0.951057i
\(206\) 0 0
\(207\) 0 0
\(208\) 0.690983 0.951057i 0.690983 0.951057i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(212\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.61803i 1.61803i
\(219\) 0 0
\(220\) 0 0
\(221\) −0.587785 + 0.427051i −0.587785 + 0.427051i
\(222\) 0 0
\(223\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.618034 0.618034
\(227\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(228\) 0 0
\(229\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.90211i 1.90211i
\(233\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) 0 0
\(241\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(242\) 0.587785 0.809017i 0.587785 0.809017i
\(243\) 0 0
\(244\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(245\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 0.587785i −0.809017 0.587785i
\(257\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.363271 1.11803i 0.363271 1.11803i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(264\) 0 0
\(265\) 1.61803 1.61803
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(270\) 0 0
\(271\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(273\) 0 0
\(274\) 1.61803 1.61803
\(275\) 0 0
\(276\) 0 0
\(277\) 1.80902 0.587785i 1.80902 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.951057 + 0.690983i 0.951057 + 0.690983i 0.951057 0.309017i \(-0.100000\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(290\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(291\) 0 0
\(292\) 0.690983 0.951057i 0.690983 0.951057i
\(293\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.53884 1.11803i 1.53884 1.11803i
\(297\) 0 0
\(298\) 1.11803 0.363271i 1.11803 0.363271i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(314\) −0.587785 1.80902i −0.587785 1.80902i
\(315\) 0 0
\(316\) 0 0
\(317\) −1.17557 + 1.61803i −1.17557 + 1.61803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.951057 0.309017i −0.951057 0.309017i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.17557i 1.17557i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(338\) −0.363271 0.118034i −0.363271 0.118034i
\(339\) 0 0
\(340\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(341\) 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 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(348\) 0 0
\(349\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.17557 1.61803i −1.17557 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0 0
\(361\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(362\) −0.587785 0.190983i −0.587785 0.190983i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.363271 1.11803i 0.363271 1.11803i
\(366\) 0 0
\(367\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 1.11803 1.53884i 1.11803 1.53884i
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.12663 0.690983i 2.12663 0.690983i
\(378\) 0 0
\(379\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(387\) 0 0
\(388\) −1.80902 0.587785i −1.80902 0.587785i
\(389\) 0.363271 + 1.11803i 0.363271 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.587785 0.809017i 0.587785 0.809017i
\(393\) 0 0
\(394\) −1.30902 0.951057i −1.30902 0.951057i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(410\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(420\) 0 0
\(421\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.61803 −1.61803
\(425\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(432\) 0 0
\(433\) 0.690983 0.951057i 0.690983 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
1.00000 \(0\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.427051 0.587785i 0.427051 0.587785i
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.90211 −1.90211 −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0.951057 1.30902i 0.951057 1.30902i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(464\) −0.587785 1.80902i −0.587785 1.80902i
\(465\) 0 0
\(466\) 0.500000 1.53884i 0.500000 1.53884i
\(467\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(468\) 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 \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) −1.80902 1.31433i −1.80902 1.31433i
\(482\) 0.618034i 0.618034i
\(483\) 0 0
\(484\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(485\) −1.90211 −1.90211
\(486\) 0 0
\(487\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(488\) −1.53884 0.500000i −1.53884 0.500000i
\(489\) 0 0
\(490\) 0.309017 0.951057i 0.309017 0.951057i
\(491\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(492\) 0 0
\(493\) 1.17557i 1.17557i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(504\) 0 0
\(505\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.587785 1.80902i 0.587785 1.80902i 1.00000i \(-0.5\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(513\) 0 0
\(514\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 1.17557i 1.17557i
\(521\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(522\) 0 0
\(523\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.809017 0.587785i 0.809017 0.587785i
\(530\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.812299 1.11803i 0.812299 1.11803i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.500000 0.363271i −0.500000 0.363271i
\(545\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(546\) 0 0
\(547\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(548\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.11803 0.363271i −1.11803 0.363271i
\(563\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(564\) 0 0
\(565\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.53884 1.11803i −1.53884 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(570\) 0 0
\(571\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(578\) −0.363271 0.500000i −0.363271 0.500000i
\(579\) 0 0
\(580\) −1.11803 1.53884i −1.11803 1.53884i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(585\) 0 0
\(586\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(587\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(593\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.00000i 1.00000i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −1.61803 −1.61803
\(611\) 0 0
\(612\) 0 0
\(613\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(626\) 0 0
\(627\) 0 0
\(628\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(629\) 0.951057 0.690983i 0.951057 0.690983i
\(630\) 0 0
\(631\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.618034 1.90211i 0.618034 1.90211i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.11803 0.363271i −1.11803 0.363271i
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.951057 0.690983i −0.951057 0.690983i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(660\) 0 0
\(661\) −0.618034 1.90211i −0.618034 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(662\) 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
\(673\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.381966 0.381966
\(677\) 1.90211 0.618034i 1.90211 0.618034i 0.951057 0.309017i \(-0.100000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.587785 0.190983i −0.587785 0.190983i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(684\) 0 0
\(685\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(690\) 0 0
\(691\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(692\) 0.363271 0.500000i 0.363271 0.500000i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.427051 + 0.587785i 0.427051 + 0.587785i
\(698\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.61803 + 1.17557i 1.61803 + 1.17557i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.80902 0.587785i −1.80902 0.587785i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.587785 0.809017i −0.587785 0.809017i
\(723\) 0 0
\(724\) 0.618034 0.618034
\(725\) −1.53884 1.11803i −1.53884 1.11803i
\(726\) 0 0
\(727\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.17557i 1.17557i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(746\) 0 0
\(747\) 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
\(754\) −1.80902 + 1.31433i −1.80902 + 1.31433i
\(755\) 0 0
\(756\) 0 0
\(757\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.690983 0.951057i −0.690983 0.951057i
\(773\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.90211 1.90211
\(777\) 0 0
\(778\) −0.690983 0.951057i −0.690983 0.951057i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(785\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(786\) 0 0
\(787\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(788\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.90211i 1.90211i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.951057 0.309017i 0.951057 0.309017i
\(801\) 0 0
\(802\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.11803 1.53884i 1.11803 1.53884i
\(809\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.618034i 0.618034i
\(819\) 0 0
\(820\) −1.11803 0.363271i −1.11803 0.363271i
\(821\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(822\) 0 0
\(823\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(828\) 0 0
\(829\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.17557i 1.17557i
\(833\) 0.363271 0.500000i 0.363271 0.500000i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(840\) 0 0
\(841\) 0.809017 2.48990i 0.809017 2.48990i
\(842\) 0.363271 0.500000i 0.363271 0.500000i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.363271 0.118034i 0.363271 0.118034i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.53884 0.500000i 1.53884 0.500000i
\(849\) 0 0
\(850\) −0.618034 −0.618034
\(851\) 0 0
\(852\) 0 0
\(853\) 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 \(0\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(858\) 0 0
\(859\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(864\) 0 0
\(865\) 0.190983 0.587785i 0.190983 0.587785i
\(866\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.951057 1.30902i −0.951057 1.30902i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(882\) 0 0
\(883\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(884\) −0.224514 + 0.690983i −0.224514 + 0.690983i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.90211 −1.90211
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.80902 0.587785i 1.80902 0.587785i
\(899\) 0 0
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) 0.500000 0.363271i 0.500000 0.363271i
\(905\) 0.587785 0.190983i 0.587785 0.190983i
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.11803 1.53884i −1.11803 1.53884i
\(923\) 0 0
\(924\) 0 0
\(925\) 1.90211i 1.90211i
\(926\) 0 0
\(927\) 0 0
\(928\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(929\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.61803i 1.61803i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(948\) 0 0
\(949\) 1.38197 1.38197
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(954\) 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
\(962\) 2.12663 + 0.690983i 2.12663 + 0.690983i
\(963\) 0 0
\(964\) −0.190983 0.587785i −0.190983 0.587785i
\(965\) −0.951057 0.690983i −0.951057 0.690983i
\(966\) 0 0
\(967\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(968\) 1.00000i 1.00000i
\(969\) 0 0
\(970\) 1.80902 0.587785i 1.80902 0.587785i
\(971\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.61803 1.61803
\(977\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000i 1.00000i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(984\) 0 0
\(985\) 1.61803 1.61803
\(986\) −0.363271 1.11803i −0.363271 1.11803i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(998\) 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.bb.a.19.1 8
3.2 odd 2 inner 900.1.bb.a.19.2 yes 8
4.3 odd 2 CM 900.1.bb.a.19.1 8
12.11 even 2 inner 900.1.bb.a.19.2 yes 8
25.4 even 10 inner 900.1.bb.a.379.1 yes 8
75.29 odd 10 inner 900.1.bb.a.379.2 yes 8
100.79 odd 10 inner 900.1.bb.a.379.1 yes 8
300.179 even 10 inner 900.1.bb.a.379.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.1.bb.a.19.1 8 1.1 even 1 trivial
900.1.bb.a.19.1 8 4.3 odd 2 CM
900.1.bb.a.19.2 yes 8 3.2 odd 2 inner
900.1.bb.a.19.2 yes 8 12.11 even 2 inner
900.1.bb.a.379.1 yes 8 25.4 even 10 inner
900.1.bb.a.379.1 yes 8 100.79 odd 10 inner
900.1.bb.a.379.2 yes 8 75.29 odd 10 inner
900.1.bb.a.379.2 yes 8 300.179 even 10 inner