Properties

Label 1800.1.cr.a.253.1
Level $1800$
Weight $1$
Character 1800.253
Analytic conductor $0.898$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,1,Mod(37,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 10, 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("1800.37");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1800.cr (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: \(0.898317022739\)
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 253.1
Root \(-0.891007 - 0.453990i\) of defining polynomial
Character \(\chi\) \(=\) 1800.253
Dual form 1800.1.cr.a.1117.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.891007 + 0.453990i) q^{2} +(0.587785 - 0.809017i) q^{4} +(0.453990 + 0.891007i) q^{5} +(0.221232 - 0.221232i) q^{7} +(-0.156434 + 0.987688i) q^{8} +O(q^{10})\) \(q+(-0.891007 + 0.453990i) q^{2} +(0.587785 - 0.809017i) q^{4} +(0.453990 + 0.891007i) q^{5} +(0.221232 - 0.221232i) q^{7} +(-0.156434 + 0.987688i) q^{8} +(-0.809017 - 0.587785i) q^{10} +(1.69480 + 0.550672i) q^{11} +(-0.0966818 + 0.297556i) q^{14} +(-0.309017 - 0.951057i) q^{16} +(0.987688 + 0.156434i) q^{20} +(-1.76007 + 0.278768i) q^{22} +(-0.587785 + 0.809017i) q^{25} +(-0.0489435 - 0.309017i) q^{28} +(-1.14412 - 0.831254i) q^{29} +(0.951057 - 0.690983i) q^{31} +(0.707107 + 0.707107i) q^{32} +(0.297556 + 0.0966818i) q^{35} +(-0.951057 + 0.309017i) q^{40} +(1.44168 - 1.04744i) q^{44} +0.902113i q^{49} +(0.156434 - 0.987688i) q^{50} +(-1.87869 + 0.297556i) q^{53} +(0.278768 + 1.76007i) q^{55} +(0.183900 + 0.253116i) q^{56} +(1.39680 + 0.221232i) q^{58} +(0.280582 + 0.863541i) q^{59} +(-0.533698 + 1.04744i) q^{62} +(-0.951057 - 0.309017i) q^{64} +(-0.309017 + 0.0489435i) q^{70} +(1.26007 - 0.642040i) q^{73} +(0.496769 - 0.253116i) q^{77} +(1.11803 - 1.53884i) q^{79} +(0.707107 - 0.707107i) q^{80} +(0.253116 - 1.59811i) q^{83} +(-0.809017 + 1.58779i) q^{88} +(-1.95106 + 0.309017i) q^{97} +(-0.409551 - 0.803789i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{7} - 4 q^{10} + 4 q^{16} - 4 q^{22} - 16 q^{28} + 4 q^{55} + 4 q^{58} + 4 q^{70} - 4 q^{73} - 4 q^{88} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(e\left(\frac{7}{20}\right)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(3\) 0 0
\(4\) 0.587785 0.809017i 0.587785 0.809017i
\(5\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(6\) 0 0
\(7\) 0.221232 0.221232i 0.221232 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(8\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(9\) 0 0
\(10\) −0.809017 0.587785i −0.809017 0.587785i
\(11\) 1.69480 + 0.550672i 1.69480 + 0.550672i 0.987688 0.156434i \(-0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) 0 0
\(13\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(14\) −0.0966818 + 0.297556i −0.0966818 + 0.297556i
\(15\) 0 0
\(16\) −0.309017 0.951057i −0.309017 0.951057i
\(17\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(20\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(21\) 0 0
\(22\) −1.76007 + 0.278768i −1.76007 + 0.278768i
\(23\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(24\) 0 0
\(25\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.0489435 0.309017i −0.0489435 0.309017i
\(29\) −1.14412 0.831254i −1.14412 0.831254i −0.156434 0.987688i \(-0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(30\) 0 0
\(31\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(32\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(36\) 0 0
\(37\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(41\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 1.44168 1.04744i 1.44168 1.04744i
\(45\) 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
\(49\) 0.902113i 0.902113i
\(50\) 0.156434 0.987688i 0.156434 0.987688i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.87869 + 0.297556i −1.87869 + 0.297556i −0.987688 0.156434i \(-0.950000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(54\) 0 0
\(55\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(56\) 0.183900 + 0.253116i 0.183900 + 0.253116i
\(57\) 0 0
\(58\) 1.39680 + 0.221232i 1.39680 + 0.221232i
\(59\) 0.280582 + 0.863541i 0.280582 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(62\) −0.533698 + 1.04744i −0.533698 + 1.04744i
\(63\) 0 0
\(64\) −0.951057 0.309017i −0.951057 0.309017i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(71\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(72\) 0 0
\(73\) 1.26007 0.642040i 1.26007 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.496769 0.253116i 0.496769 0.253116i
\(78\) 0 0
\(79\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(80\) 0.707107 0.707107i 0.707107 0.707107i
\(81\) 0 0
\(82\) 0 0
\(83\) 0.253116 1.59811i 0.253116 1.59811i −0.453990 0.891007i \(-0.650000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.809017 + 1.58779i −0.809017 + 1.58779i
\(89\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.95106 + 0.309017i −1.95106 + 0.309017i −0.951057 + 0.309017i \(0.900000\pi\)
−1.00000 \(1.00000\pi\)
\(98\) −0.409551 0.803789i −0.409551 0.803789i
\(99\) 0 0
\(100\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(101\) 0.907981i 0.907981i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(102\) 0 0
\(103\) 0.309017 + 1.95106i 0.309017 + 1.95106i 0.309017 + 0.951057i \(0.400000\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.53884 1.11803i 1.53884 1.11803i
\(107\) 0.437016 + 0.437016i 0.437016 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(108\) 0 0
\(109\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(110\) −1.04744 1.44168i −1.04744 1.44168i
\(111\) 0 0
\(112\) −0.278768 0.142040i −0.278768 0.142040i
\(113\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.34500 + 0.437016i −1.34500 + 0.437016i
\(117\) 0 0
\(118\) −0.642040 0.642040i −0.642040 0.642040i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.76007 + 1.27877i 1.76007 + 1.27877i
\(122\) 0 0
\(123\) 0 0
\(124\) 1.17557i 1.17557i
\(125\) −0.987688 0.156434i −0.987688 0.156434i
\(126\) 0 0
\(127\) 0.412215 + 0.809017i 0.412215 + 0.809017i 1.00000 \(0\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(128\) 0.987688 0.156434i 0.987688 0.156434i
\(129\) 0 0
\(130\) 0 0
\(131\) 0.831254 + 1.14412i 0.831254 + 1.14412i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(140\) 0.253116 0.183900i 0.253116 0.183900i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.221232 1.39680i 0.221232 1.39680i
\(146\) −0.831254 + 1.14412i −0.831254 + 1.14412i
\(147\) 0 0
\(148\) 0 0
\(149\) −0.907981 −0.907981 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(150\) 0 0
\(151\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.327712 + 0.451057i −0.327712 + 0.451057i
\(155\) 1.04744 + 0.533698i 1.04744 + 0.533698i
\(156\) 0 0
\(157\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) −0.297556 + 1.87869i −0.297556 + 1.87869i
\(159\) 0 0
\(160\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(167\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(168\) 0 0
\(169\) −0.587785 0.809017i −0.587785 0.809017i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.734572 1.44168i −0.734572 1.44168i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(174\) 0 0
\(175\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i
\(176\) 1.78201i 1.78201i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.59811 1.16110i −1.59811 1.16110i −0.891007 0.453990i \(-0.850000\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(180\) 0 0
\(181\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 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
\(193\) 1.39680 + 1.39680i 1.39680 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(194\) 1.59811 1.16110i 1.59811 1.16110i
\(195\) 0 0
\(196\) 0.729825 + 0.530249i 0.729825 + 0.530249i
\(197\) −0.183900 1.16110i −0.183900 1.16110i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(198\) 0 0
\(199\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(200\) −0.707107 0.707107i −0.707107 0.707107i
\(201\) 0 0
\(202\) −0.412215 0.809017i −0.412215 0.809017i
\(203\) −0.437016 + 0.0692165i −0.437016 + 0.0692165i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.16110 1.59811i −1.16110 1.59811i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(212\) −0.863541 + 1.69480i −0.863541 + 1.69480i
\(213\) 0 0
\(214\) −0.587785 0.190983i −0.587785 0.190983i
\(215\) 0 0
\(216\) 0 0
\(217\) 0.0575365 0.363271i 0.0575365 0.363271i
\(218\) 0 0
\(219\) 0 0
\(220\) 1.58779 + 0.809017i 1.58779 + 0.809017i
\(221\) 0 0
\(222\) 0 0
\(223\) −1.58779 + 0.809017i −1.58779 + 0.809017i −0.587785 + 0.809017i \(0.700000\pi\)
−1.00000 \(\pi\)
\(224\) 0.312869 0.312869
\(225\) 0 0
\(226\) 0 0
\(227\) 1.69480 0.863541i 1.69480 0.863541i 0.707107 0.707107i \(-0.250000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(228\) 0 0
\(229\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 1.00000i 1.00000 1.00000i
\(233\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.863541 + 0.280582i 0.863541 + 0.280582i
\(237\) 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
\(241\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(242\) −2.14879 0.340334i −2.14879 0.340334i
\(243\) 0 0
\(244\) 0 0
\(245\) −0.803789 + 0.409551i −0.803789 + 0.409551i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.533698 + 1.04744i 0.533698 + 1.04744i
\(249\) 0 0
\(250\) 0.951057 0.309017i 0.951057 0.309017i
\(251\) 1.97538i 1.97538i −0.156434 0.987688i \(-0.550000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.734572 0.533698i −0.734572 0.533698i
\(255\) 0 0
\(256\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.26007 0.642040i −1.26007 0.642040i
\(263\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(264\) 0 0
\(265\) −1.11803 1.53884i −1.11803 1.53884i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.253116 + 0.183900i −0.253116 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(270\) 0 0
\(271\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.44168 + 1.04744i −1.44168 + 1.04744i
\(276\) 0 0
\(277\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(281\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(282\) 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
\(289\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(290\) 0.437016 + 1.34500i 0.437016 + 1.34500i
\(291\) 0 0
\(292\) 0.221232 1.39680i 0.221232 1.39680i
\(293\) 1.34500 1.34500i 1.34500 1.34500i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(294\) 0 0
\(295\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(296\) 0 0
\(297\) 0 0
\(298\) 0.809017 0.412215i 0.809017 0.412215i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 1.44168 0.734572i 1.44168 0.734572i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0.0872179 0.550672i 0.0872179 0.550672i
\(309\) 0 0
\(310\) −1.17557 −1.17557
\(311\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(312\) 0 0
\(313\) 0.809017 1.58779i 0.809017 1.58779i 1.00000i \(-0.5\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.587785 1.80902i −0.587785 1.80902i
\(317\) −0.610425 0.0966818i −0.610425 0.0966818i −0.156434 0.987688i \(-0.550000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(318\) 0 0
\(319\) −1.48131 2.03884i −1.48131 2.03884i
\(320\) −0.156434 0.987688i −0.156434 0.987688i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) −1.14412 1.14412i −1.14412 1.14412i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.76007 0.896802i −1.76007 0.896802i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(338\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(339\) 0 0
\(340\) 0 0
\(341\) 1.99235 0.647354i 1.99235 0.647354i
\(342\) 0 0
\(343\) 0.420808 + 0.420808i 0.420808 + 0.420808i
\(344\) 0 0
\(345\) 0 0
\(346\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(347\) −0.297556 1.87869i −0.297556 1.87869i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) −0.183900 0.253116i −0.183900 0.253116i
\(351\) 0 0
\(352\) 0.809017 + 1.58779i 0.809017 + 1.58779i
\(353\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.95106 + 0.309017i 1.95106 + 0.309017i
\(359\) 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 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.14412 + 0.831254i 1.14412 + 0.831254i
\(366\) 0 0
\(367\) 0.309017 1.95106i 0.309017 1.95106i 1.00000i \(-0.5\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.349798 + 0.481456i −0.349798 + 0.481456i
\(372\) 0 0
\(373\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 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
\(385\) 0.451057 + 0.327712i 0.451057 + 0.327712i
\(386\) −1.87869 0.610425i −1.87869 0.610425i
\(387\) 0 0
\(388\) −0.896802 + 1.76007i −0.896802 + 1.76007i
\(389\) −0.550672 + 1.69480i −0.550672 + 1.69480i 0.156434 + 0.987688i \(0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.891007 0.141122i −0.891007 0.141122i
\(393\) 0 0
\(394\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(395\) 1.87869 + 0.297556i 1.87869 + 0.297556i
\(396\) 0 0
\(397\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(398\) 0.280582 + 0.550672i 0.280582 + 0.550672i
\(399\) 0 0
\(400\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.734572 + 0.533698i 0.734572 + 0.533698i
\(405\) 0 0
\(406\) 0.357960 0.260074i 0.357960 0.260074i
\(407\) 0 0
\(408\) 0 0
\(409\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(413\) 0.253116 + 0.128969i 0.253116 + 0.128969i
\(414\) 0 0
\(415\) 1.53884 0.500000i 1.53884 0.500000i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.253116 0.183900i 0.253116 0.183900i −0.453990 0.891007i \(-0.650000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) 0 0
\(421\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.90211i 1.90211i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.610425 0.0966818i 0.610425 0.0966818i
\(429\) 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
\(433\) −1.76007 0.278768i −1.76007 0.278768i −0.809017 0.587785i \(-0.800000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(434\) 0.113656 + 0.349798i 0.113656 + 0.349798i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(440\) −1.78201 −1.78201
\(441\) 0 0
\(442\) 0 0
\(443\) −0.831254 + 0.831254i −0.831254 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.04744 1.44168i 1.04744 1.44168i
\(447\) 0 0
\(448\) −0.278768 + 0.142040i −0.278768 + 0.142040i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(455\) 0 0
\(456\) 0 0
\(457\) −0.642040 + 0.642040i −0.642040 + 0.642040i −0.951057 0.309017i \(-0.900000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i 0.453990 0.891007i \(-0.350000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(462\) 0 0
\(463\) 0.642040 1.26007i 0.642040 1.26007i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(464\) −0.437016 + 1.34500i −0.437016 + 1.34500i
\(465\) 0 0
\(466\) 0 0
\(467\) 0.610425 + 0.0966818i 0.610425 + 0.0966818i 0.453990 0.891007i \(-0.350000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.896802 + 0.142040i −0.896802 + 0.142040i
\(473\) 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
\(481\) 0 0
\(482\) 0.831254 + 0.831254i 0.831254 + 0.831254i
\(483\) 0 0
\(484\) 2.06909 0.672288i 2.06909 0.672288i
\(485\) −1.16110 1.59811i −1.16110 1.59811i
\(486\) 0 0
\(487\) 0.809017 + 0.412215i 0.809017 + 0.412215i 0.809017 0.587785i \(-0.200000\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.530249 0.729825i 0.530249 0.729825i
\(491\) −1.87869 + 0.610425i −1.87869 + 0.610425i −0.891007 + 0.453990i \(0.850000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.951057 0.690983i −0.951057 0.690983i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(501\) 0 0
\(502\) 0.896802 + 1.76007i 0.896802 + 1.76007i
\(503\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(504\) 0 0
\(505\) −0.809017 + 0.412215i −0.809017 + 0.412215i
\(506\) 0 0
\(507\) 0 0
\(508\) 0.896802 + 0.142040i 0.896802 + 0.142040i
\(509\) 0.550672 + 1.69480i 0.550672 + 1.69480i 0.707107 + 0.707107i \(0.250000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(510\) 0 0
\(511\) 0.136729 0.420808i 0.136729 0.420808i
\(512\) 0.453990 0.891007i 0.453990 0.891007i
\(513\) 0 0
\(514\) 0 0
\(515\) −1.59811 + 1.16110i −1.59811 + 1.16110i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(522\) 0 0
\(523\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(524\) 1.41421 1.41421
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(530\) 1.69480 + 0.863541i 1.69480 + 0.863541i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(536\) 0 0
\(537\) 0 0
\(538\) 0.142040 0.278768i 0.142040 0.278768i
\(539\) −0.496769 + 1.52890i −0.496769 + 1.52890i
\(540\) 0 0
\(541\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) 1.59811 + 0.253116i 1.59811 + 0.253116i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.809017 1.58779i 0.809017 1.58779i
\(551\) 0 0
\(552\) 0 0
\(553\) −0.0930960 0.587785i −0.0930960 0.587785i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.831254 + 0.831254i 0.831254 + 0.831254i 0.987688 0.156434i \(-0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.312869i 0.312869i
\(561\) 0 0
\(562\) 0 0
\(563\) −1.04744 0.533698i −1.04744 0.533698i −0.156434 0.987688i \(-0.550000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(570\) 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
\(577\) 0.142040 + 0.278768i 0.142040 + 0.278768i 0.951057 0.309017i \(-0.100000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(579\) 0 0
\(580\) −1.00000 1.00000i −1.00000 1.00000i
\(581\) −0.297556 0.409551i −0.297556 0.409551i
\(582\) 0 0
\(583\) −3.34786 0.530249i −3.34786 0.530249i
\(584\) 0.437016 + 1.34500i 0.437016 + 1.34500i
\(585\) 0 0
\(586\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(587\) 0.280582 0.550672i 0.280582 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0.280582 0.863541i 0.280582 0.863541i
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.533698 + 0.734572i −0.533698 + 0.734572i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(605\) −0.340334 + 2.14879i −0.340334 + 2.14879i
\(606\) 0 0
\(607\) 1.26007 1.26007i 1.26007 1.26007i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.172288 + 0.530249i 0.172288 + 0.530249i
\(617\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(620\) 1.04744 0.533698i 1.04744 0.533698i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.309017 0.951057i −0.309017 0.951057i
\(626\) 1.78201i 1.78201i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(632\) 1.34500 + 1.34500i 1.34500 + 1.34500i
\(633\) 0 0
\(634\) 0.587785 0.190983i 0.587785 0.190983i
\(635\) −0.533698 + 0.734572i −0.533698 + 0.734572i
\(636\) 0 0
\(637\) 0 0
\(638\) 2.24547 + 1.14412i 2.24547 + 1.14412i
\(639\) 0 0
\(640\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(641\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(642\) 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
\(649\) 1.61803i 1.61803i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.59811 0.253116i 1.59811 0.253116i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(654\) 0 0
\(655\) −0.642040 + 1.26007i −0.642040 + 1.26007i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.0966818 0.297556i −0.0966818 0.297556i 0.891007 0.453990i \(-0.150000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(660\) 0 0
\(661\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.278768 0.142040i 0.278768 0.142040i −0.309017 0.951057i \(-0.600000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(674\) 1.97538 1.97538
\(675\) 0 0
\(676\) −1.00000 −1.00000
\(677\) 1.44168 0.734572i 1.44168 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(678\) 0 0
\(679\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(680\) 0 0
\(681\) 0 0
\(682\) −1.48131 + 1.48131i −1.48131 + 1.48131i
\(683\) −0.0966818 + 0.610425i −0.0966818 + 0.610425i 0.891007 + 0.453990i \(0.150000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.565985 0.183900i −0.565985 0.183900i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) −1.59811 0.253116i −1.59811 0.253116i
\(693\) 0 0
\(694\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.278768 + 0.142040i 0.278768 + 0.142040i
\(701\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.44168 1.04744i −1.44168 1.04744i
\(705\) 0 0
\(706\) 0 0
\(707\) 0.200874 + 0.200874i 0.200874 + 0.200874i
\(708\) 0 0
\(709\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.87869 + 0.610425i −1.87869 + 0.610425i
\(717\) 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
\(721\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(722\) −0.156434 0.987688i −0.156434 0.987688i
\(723\) 0 0
\(724\) 0 0
\(725\) 1.34500 0.437016i 1.34500 0.437016i
\(726\) 0 0
\(727\) 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.39680 0.221232i −1.39680 0.221232i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(734\) 0.610425 + 1.87869i 0.610425 + 1.87869i
\(735\) 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 0
\(741\) 0 0
\(742\) 0.0930960 0.587785i 0.0930960 0.587785i
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) −0.412215 0.809017i −0.412215 0.809017i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.193364 0.193364
\(750\) 0 0
\(751\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.734572 1.44168i −0.734572 1.44168i
\(756\) 0 0
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(770\) −0.550672 0.0872179i −0.550672 0.0872179i
\(771\) 0 0
\(772\) 1.95106 0.309017i 1.95106 0.309017i
\(773\) 0.280582 + 0.550672i 0.280582 + 0.550672i 0.987688 0.156434i \(-0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 1.17557i 1.17557i
\(776\) 1.97538i 1.97538i
\(777\) 0 0
\(778\) −0.278768 1.76007i −0.278768 1.76007i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.857960 0.278768i 0.857960 0.278768i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(788\) −1.04744 0.533698i −1.04744 0.533698i
\(789\) 0 0
\(790\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.500000 0.363271i −0.500000 0.363271i
\(797\) −0.0966818 0.610425i −0.0966818 0.610425i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(801\) 0 0
\(802\) 0 0
\(803\) 2.48912 0.394238i 2.48912 0.394238i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.896802 0.142040i −0.896802 0.142040i
\(809\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0 0
\(811\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(812\) −0.200874 + 0.394238i −0.200874 + 0.394238i
\(813\) 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 0
\(821\) −0.183900 + 0.253116i −0.183900 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(822\) 0 0
\(823\) −0.809017 + 0.412215i −0.809017 + 0.412215i −0.809017 0.587785i \(-0.800000\pi\)
1.00000i \(0.5\pi\)
\(824\) −1.97538 −1.97538
\(825\) 0 0
\(826\) −0.284079 −0.284079
\(827\) −1.04744 + 0.533698i −1.04744 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(828\) 0 0
\(829\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(830\) −1.14412 + 1.14412i −1.14412 + 1.14412i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(839\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(840\) 0 0
\(841\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.453990 0.891007i 0.453990 0.891007i
\(846\) 0 0
\(847\) 0.672288 0.106480i 0.672288 0.106480i
\(848\) 0.863541 + 1.69480i 0.863541 + 1.69480i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 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
\(865\) 0.951057 1.30902i 0.951057 1.30902i
\(866\) 1.69480 0.550672i 1.69480 0.550672i
\(867\) 0 0
\(868\) −0.260074 0.260074i −0.260074 0.260074i
\(869\) 2.74224 1.99235i 2.74224 1.99235i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.253116 + 0.183900i −0.253116 + 0.183900i
\(876\) 0 0
\(877\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(878\) −0.610425 + 0.0966818i −0.610425 + 0.0966818i
\(879\) 0 0
\(880\) 1.58779 0.809017i 1.58779 0.809017i
\(881\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(882\) 0 0
\(883\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.363271 1.11803i 0.363271 1.11803i
\(887\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(888\) 0 0
\(889\) 0.270175 + 0.0877853i 0.270175 + 0.0877853i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.278768 + 1.76007i −0.278768 + 1.76007i
\(893\) 0 0
\(894\) 0 0
\(895\) 0.309017 1.95106i 0.309017 1.95106i
\(896\) 0.183900 0.253116i 0.183900 0.253116i
\(897\) 0 0
\(898\) 0 0
\(899\) −1.66251 −1.66251
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) 0.297556 1.87869i 0.297556 1.87869i
\(909\) 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
\(913\) 1.30902 2.56909i 1.30902 2.56909i
\(914\) 0.280582 0.863541i 0.280582 0.863541i
\(915\) 0 0
\(916\) 0 0
\(917\) 0.437016 + 0.0692165i 0.437016 + 0.0692165i
\(918\) 0 0
\(919\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1.41421i 1.41421i
\(927\) 0 0
\(928\) −0.221232 1.39680i −0.221232 1.39680i
\(929\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(935\) 0 0
\(936\) 0 0
\(937\) 0.278768 + 0.142040i 0.278768 + 0.142040i 0.587785 0.809017i \(-0.300000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.87869 0.610425i 1.87869 0.610425i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.734572 0.533698i 0.734572 0.533698i
\(945\) 0 0
\(946\) 0 0
\(947\) 0.183900 + 1.16110i 0.183900 + 1.16110i 0.891007 + 0.453990i \(0.150000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.118034 0.363271i 0.118034 0.363271i
\(962\) 0 0
\(963\) 0 0
\(964\) −1.11803 0.363271i −1.11803 0.363271i
\(965\) −0.610425 + 1.87869i −0.610425 + 1.87869i
\(966\) 0 0
\(967\) −0.309017 + 1.95106i −0.309017 + 1.95106i 1.00000i \(0.5\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(968\) −1.53836 + 1.53836i −1.53836 + 1.53836i
\(969\) 0 0
\(970\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(971\) 1.16110 1.59811i 1.16110 1.59811i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −0.907981 −0.907981
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.141122 + 0.891007i −0.141122 + 0.891007i
\(981\) 0 0
\(982\) 1.39680 1.39680i 1.39680 1.39680i
\(983\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(984\) 0 0
\(985\) 0.951057 0.690983i 0.951057 0.690983i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(992\) 1.16110 + 0.183900i 1.16110 + 0.183900i
\(993\) 0 0
\(994\) 0 0
\(995\) 0.550672 0.280582i 0.550672 0.280582i
\(996\) 0 0
\(997\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(998\) 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 1800.1.cr.a.253.1 16
3.2 odd 2 inner 1800.1.cr.a.253.2 yes 16
8.5 even 2 inner 1800.1.cr.a.253.2 yes 16
24.5 odd 2 CM 1800.1.cr.a.253.1 16
25.17 odd 20 inner 1800.1.cr.a.1117.2 yes 16
75.17 even 20 inner 1800.1.cr.a.1117.1 yes 16
200.117 odd 20 inner 1800.1.cr.a.1117.1 yes 16
600.317 even 20 inner 1800.1.cr.a.1117.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1800.1.cr.a.253.1 16 1.1 even 1 trivial
1800.1.cr.a.253.1 16 24.5 odd 2 CM
1800.1.cr.a.253.2 yes 16 3.2 odd 2 inner
1800.1.cr.a.253.2 yes 16 8.5 even 2 inner
1800.1.cr.a.1117.1 yes 16 75.17 even 20 inner
1800.1.cr.a.1117.1 yes 16 200.117 odd 20 inner
1800.1.cr.a.1117.2 yes 16 25.17 odd 20 inner
1800.1.cr.a.1117.2 yes 16 600.317 even 20 inner