Properties

Label 693.1.bp.a.314.1
Level $693$
Weight $1$
Character 693.314
Analytic conductor $0.346$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,1,Mod(62,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.62");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 693.bp (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: \(0.345852053755\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 314.1
Root \(-0.453990 - 0.891007i\) of defining polynomial
Character \(\chi\) \(=\) 693.314
Dual form 693.1.bp.a.629.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.610425 - 1.87869i) q^{2} +(-2.34786 + 1.70582i) q^{4} +(0.587785 + 0.809017i) q^{7} +(3.03979 + 2.20854i) q^{8} +O(q^{10})\) \(q+(-0.610425 - 1.87869i) q^{2} +(-2.34786 + 1.70582i) q^{4} +(0.587785 + 0.809017i) q^{7} +(3.03979 + 2.20854i) q^{8} +(0.891007 - 0.453990i) q^{11} +(1.16110 - 1.59811i) q^{14} +(1.39680 - 4.29892i) q^{16} +(-1.39680 - 1.39680i) q^{22} -0.312869i q^{23} +(0.809017 + 0.587785i) q^{25} +(-2.76007 - 0.896802i) q^{28} +(-0.734572 + 0.533698i) q^{29} -5.17160 q^{32} +(0.951057 - 0.690983i) q^{37} -0.618034i q^{43} +(-1.31753 + 2.58580i) q^{44} +(-0.587785 + 0.190983i) q^{46} +(-0.309017 + 0.951057i) q^{49} +(0.610425 - 1.87869i) q^{50} +(0.863541 - 0.280582i) q^{53} +3.75739i q^{56} +(1.45106 + 1.05425i) q^{58} +(1.76007 + 5.41695i) q^{64} +0.618034 q^{67} +(-1.69480 - 0.550672i) q^{71} +(-1.87869 - 1.36495i) q^{74} +(0.891007 + 0.453990i) q^{77} +(-1.80902 + 0.587785i) q^{79} +(-1.16110 + 0.377263i) q^{86} +(3.71113 + 0.587785i) q^{88} +(0.533698 + 0.734572i) q^{92} +1.97538 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 4 q^{16} - 4 q^{22} + 4 q^{25} - 20 q^{28} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 8 q^{67} - 20 q^{79} + 20 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.610425 1.87869i −0.610425 1.87869i −0.453990 0.891007i \(-0.650000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(3\) 0 0
\(4\) −2.34786 + 1.70582i −2.34786 + 1.70582i
\(5\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(6\) 0 0
\(7\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(8\) 3.03979 + 2.20854i 3.03979 + 2.20854i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.891007 0.453990i 0.891007 0.453990i
\(12\) 0 0
\(13\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) 1.16110 1.59811i 1.16110 1.59811i
\(15\) 0 0
\(16\) 1.39680 4.29892i 1.39680 4.29892i
\(17\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(18\) 0 0
\(19\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.39680 1.39680i −1.39680 1.39680i
\(23\) 0.312869i 0.312869i −0.987688 0.156434i \(-0.950000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(24\) 0 0
\(25\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(26\) 0 0
\(27\) 0 0
\(28\) −2.76007 0.896802i −2.76007 0.896802i
\(29\) −0.734572 + 0.533698i −0.734572 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(32\) −5.17160 −5.17160
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(42\) 0 0
\(43\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(44\) −1.31753 + 2.58580i −1.31753 + 2.58580i
\(45\) 0 0
\(46\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(47\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(48\) 0 0
\(49\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(50\) 0.610425 1.87869i 0.610425 1.87869i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.75739i 3.75739i
\(57\) 0 0
\(58\) 1.45106 + 1.05425i 1.45106 + 1.05425i
\(59\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.76007 + 5.41695i 1.76007 + 5.41695i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.69480 0.550672i −1.69480 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(74\) −1.87869 1.36495i −1.87869 1.36495i
\(75\) 0 0
\(76\) 0 0
\(77\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(78\) 0 0
\(79\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.16110 + 0.377263i −1.16110 + 0.377263i
\(87\) 0 0
\(88\) 3.71113 + 0.587785i 3.71113 + 0.587785i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.533698 + 0.734572i 0.533698 + 0.734572i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(98\) 1.97538 1.97538
\(99\) 0 0
\(100\) −2.90211 −2.90211
\(101\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.05425 1.45106i −1.05425 1.45106i
\(107\) 0.253116 + 0.183900i 0.253116 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(108\) 0 0
\(109\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.29892 1.39680i 4.29892 1.39680i
\(113\) −1.04744 + 1.44168i −1.04744 + 1.44168i −0.156434 + 0.987688i \(0.550000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.814279 2.50609i 0.814279 2.50609i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.587785 0.809017i 0.587785 0.809017i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.11803 0.363271i −1.11803 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(128\) 4.91849 3.57349i 4.91849 3.57349i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.377263 1.16110i −0.377263 1.16110i
\(135\) 0 0
\(136\) 0 0
\(137\) 1.87869 + 0.610425i 1.87869 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.52015i 3.52015i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.05425 + 3.24466i −1.05425 + 3.24466i
\(149\) −0.437016 + 1.34500i −0.437016 + 1.34500i 0.453990 + 0.891007i \(0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(150\) 0 0
\(151\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.309017 1.95106i 0.309017 1.95106i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(158\) 2.20854 + 3.03979i 2.20854 + 3.03979i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.253116 0.183900i 0.253116 0.183900i
\(162\) 0 0
\(163\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(168\) 0 0
\(169\) 0.809017 0.587785i 0.809017 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.05425 + 1.45106i 1.05425 + 1.45106i
\(173\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(174\) 0 0
\(175\) 1.00000i 1.00000i
\(176\) −0.707107 4.46450i −0.707107 4.46450i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.16110 + 1.59811i −1.16110 + 1.59811i −0.453990 + 0.891007i \(0.650000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.690983 0.951057i 0.690983 0.951057i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(192\) 0 0
\(193\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.896802 2.76007i −0.896802 2.76007i
\(197\) −0.312869 −0.312869 −0.156434 0.987688i \(-0.550000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.16110 + 3.57349i 1.16110 + 3.57349i
\(201\) 0 0
\(202\) 0 0
\(203\) −0.863541 0.280582i −0.863541 0.280582i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(212\) −1.54885 + 2.13181i −1.54885 + 2.13181i
\(213\) 0 0
\(214\) 0.190983 0.587785i 0.190983 0.587785i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −3.03979 + 0.987688i −3.03979 + 0.987688i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(224\) −3.03979 4.18391i −3.03979 4.18391i
\(225\) 0 0
\(226\) 3.34786 + 1.08779i 3.34786 + 1.08779i
\(227\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.41164 −3.41164
\(233\) −0.437016 1.34500i −0.437016 1.34500i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.44168 1.04744i −1.44168 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −1.87869 0.610425i −1.87869 0.610425i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(252\) 0 0
\(253\) −0.142040 0.278768i −0.142040 0.278768i
\(254\) 2.32219i 2.32219i
\(255\) 0 0
\(256\) −5.10793 3.71113i −5.10793 3.71113i
\(257\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(258\) 0 0
\(259\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.907981 0.907981 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.45106 + 1.05425i −1.45106 + 1.05425i
\(269\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 3.90211i 3.90211i
\(275\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(276\) 0 0
\(277\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.550672 + 1.69480i −0.550672 + 1.69480i 0.156434 + 0.987688i \(0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(284\) 4.91849 1.59811i 4.91849 1.59811i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.809017 0.587785i −0.809017 0.587785i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.41708 4.41708
\(297\) 0 0
\(298\) 2.79360 2.79360
\(299\) 0 0
\(300\) 0 0
\(301\) 0.500000 0.363271i 0.500000 0.363271i
\(302\) 3.03979 + 0.987688i 3.03979 + 0.987688i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −2.86638 + 0.453990i −2.86638 + 0.453990i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(312\) 0 0
\(313\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.24466 4.46589i 3.24466 4.46589i
\(317\) 1.69480 0.550672i 1.69480 0.550672i 0.707107 0.707107i \(-0.250000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(318\) 0 0
\(319\) −0.412215 + 0.809017i −0.412215 + 0.809017i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.500000 0.363271i −0.500000 0.363271i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −2.58580 + 1.87869i −2.58580 + 1.87869i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(338\) −1.59811 1.16110i −1.59811 1.16110i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(344\) 1.36495 1.87869i 1.36495 1.87869i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.550672 1.69480i 0.550672 1.69480i −0.156434 0.987688i \(-0.550000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(350\) 1.87869 0.610425i 1.87869 0.610425i
\(351\) 0 0
\(352\) −4.60793 + 2.34786i −4.60793 + 2.34786i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 3.71113 + 1.20582i 3.71113 + 1.20582i
\(359\) −0.253116 + 0.183900i −0.253116 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(360\) 0 0
\(361\) −0.309017 0.951057i −0.309017 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) −1.34500 0.437016i −1.34500 0.437016i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.734572 + 0.533698i 0.734572 + 0.533698i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.05425 + 1.45106i −1.05425 + 1.45106i
\(383\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.22085i 1.22085i
\(387\) 0 0
\(388\) 0 0
\(389\) −1.16110 1.59811i −1.16110 1.59811i −0.707107 0.707107i \(-0.750000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.03979 + 2.20854i −3.03979 + 2.20854i
\(393\) 0 0
\(394\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.65688 2.65688i 3.65688 2.65688i
\(401\) −0.297556 0.0966818i −0.297556 0.0966818i 0.156434 0.987688i \(-0.450000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 1.79360i 1.79360i
\(407\) 0.533698 1.04744i 0.533698 1.04744i
\(408\) 0 0
\(409\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 1.87869 + 2.58580i 1.87869 + 2.58580i
\(423\) 0 0
\(424\) 3.24466 + 1.05425i 3.24466 + 1.05425i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.907981 −0.907981
\(429\) 0 0
\(430\) 0 0
\(431\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i 0.987688 0.156434i \(-0.0500000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(432\) 0 0
\(433\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.76007 + 3.79892i 2.76007 + 3.79892i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.04744 1.44168i 1.04744 1.44168i 0.156434 0.987688i \(-0.450000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −3.34786 + 4.60793i −3.34786 + 4.60793i
\(449\) 0.297556 0.0966818i 0.297556 0.0966818i −0.156434 0.987688i \(-0.550000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 5.17160i 5.17160i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(464\) 1.26827 + 3.90333i 1.26827 + 3.90333i
\(465\) 0 0
\(466\) −2.26007 + 1.64204i −2.26007 + 1.64204i
\(467\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(468\) 0 0
\(469\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.280582 0.550672i −0.280582 0.550672i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.08779 + 3.34786i −1.08779 + 3.34786i
\(479\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.90211i 2.90211i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.14412 0.831254i 1.14412 0.831254i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.550672 1.69480i −0.550672 1.69480i
\(498\) 0 0
\(499\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(507\) 0 0
\(508\) 3.24466 1.05425i 3.24466 1.05425i
\(509\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.97538 + 6.07958i −1.97538 + 6.07958i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.32219i 2.32219i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.554254 1.70582i −0.554254 1.70582i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.902113 0.902113
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.87869 + 1.36495i 1.87869 + 1.36495i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(540\) 0 0
\(541\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(548\) −5.45218 + 1.77152i −5.45218 + 1.77152i
\(549\) 0 0
\(550\) −0.309017 1.95106i −0.309017 1.95106i
\(551\) 0 0
\(552\) 0 0
\(553\) −1.53884 1.11803i −1.53884 1.11803i
\(554\) 1.36495 + 1.87869i 1.36495 + 1.87869i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.44168 1.04744i 1.44168 1.04744i 0.453990 0.891007i \(-0.350000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 3.52015 3.52015
\(563\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −3.93564 5.41695i −3.93564 5.41695i
\(569\) 1.14412 + 0.831254i 1.14412 + 0.831254i 0.987688 0.156434i \(-0.0500000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(570\) 0 0
\(571\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.183900 0.253116i 0.183900 0.253116i
\(576\) 0 0
\(577\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(578\) −0.610425 + 1.87869i −0.610425 + 1.87869i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.642040 0.642040i 0.642040 0.642040i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.64204 5.05368i −1.64204 5.05368i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.26827 3.90333i −1.26827 3.90333i
\(597\) 0 0
\(598\) 0 0
\(599\) −1.87869 0.610425i −1.87869 0.610425i −0.987688 0.156434i \(-0.950000\pi\)
−0.891007 0.453990i \(-0.850000\pi\)
\(600\) 0 0
\(601\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(602\) −0.987688 0.717598i −0.987688 0.717598i
\(603\) 0 0
\(604\) 4.69572i 4.69572i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i \(0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.70582 + 3.34786i 1.70582 + 3.34786i
\(617\) 0.907981i 0.907981i −0.891007 0.453990i \(-0.850000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(618\) 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
\(625\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(632\) −6.79718 2.20854i −6.79718 2.20854i
\(633\) 0 0
\(634\) −2.06909 2.84786i −2.06909 2.84786i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.77152 + 0.280582i 1.77152 + 0.280582i
\(639\) 0 0
\(640\) 0 0
\(641\) −0.533698 + 0.734572i −0.533698 + 0.734572i −0.987688 0.156434i \(-0.950000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(642\) 0 0
\(643\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(644\) −0.280582 + 0.863541i −0.280582 + 0.863541i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 3.79892 + 2.76007i 3.79892 + 2.76007i
\(653\) 0.183900 + 0.253116i 0.183900 + 0.253116i 0.891007 0.453990i \(-0.150000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0.717598 + 2.20854i 0.717598 + 2.20854i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.166977 + 0.229825i 0.166977 + 0.229825i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(674\) 1.87869 2.58580i 1.87869 2.58580i
\(675\) 0 0
\(676\) −0.896802 + 2.76007i −0.896802 + 2.76007i
\(677\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.97538i 1.97538i 0.156434 + 0.987688i \(0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.16110 + 1.59811i 1.16110 + 1.59811i
\(687\) 0 0
\(688\) −2.65688 0.863271i −2.65688 0.863271i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −3.52015 −3.52015
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.70582 2.34786i −1.70582 2.34786i
\(701\) −0.253116 0.183900i −0.253116 0.183900i 0.453990 0.891007i \(-0.350000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.02748 + 4.02748i 4.02748 + 4.02748i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 5.73277i 5.73277i
\(717\) 0 0
\(718\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(719\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.59811 + 1.16110i −1.59811 + 1.16110i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.907981 −0.907981
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.61803i 1.61803i
\(737\) 0.550672 0.280582i 0.550672 0.280582i
\(738\) 0 0
\(739\) 1.80902 0.587785i 1.80902 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
1.00000 \(0\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.554254 1.70582i 0.554254 1.70582i
\(743\) −0.280582 + 0.863541i −0.280582 + 0.863541i 0.707107 + 0.707107i \(0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.312869i 0.312869i
\(750\) 0 0
\(751\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(758\) −2.32219 −2.32219
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(762\) 0 0
\(763\) 1.30902 0.951057i 1.30902 0.951057i
\(764\) 2.50609 + 0.814279i 2.50609 + 0.814279i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.70582 0.554254i 1.70582 0.554254i
\(773\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −2.29360 + 3.15688i −2.29360 + 3.15688i
\(779\) 0 0
\(780\) 0 0
\(781\) −1.76007 + 0.278768i −1.76007 + 0.278768i
\(782\) 0 0
\(783\) 0 0
\(784\) 3.65688 + 2.65688i 3.65688 + 2.65688i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(788\) 0.734572 0.533698i 0.734572 0.533698i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.78201 −1.78201
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.18391 3.03979i −4.18391 3.03979i
\(801\) 0 0
\(802\) 0.618034i 0.618034i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.0966818 + 0.297556i −0.0966818 + 0.297556i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(810\) 0 0
\(811\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(812\) 2.50609 0.814279i 2.50609 0.814279i
\(813\) 0 0
\(814\) −2.29360 0.363271i −2.29360 0.363271i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.14412 + 0.831254i −1.14412 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(822\) 0 0
\(823\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.437016 + 1.34500i 0.437016 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(828\) 0 0
\(829\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(840\) 0 0
\(841\) −0.0542543 + 0.166977i −0.0542543 + 0.166977i
\(842\) 0.987688 3.03979i 0.987688 3.03979i
\(843\) 0 0
\(844\) 2.76007 3.79892i 2.76007 3.79892i
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000 1.00000
\(848\) 4.10421i 4.10421i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.216187 0.297556i −0.216187 0.297556i
\(852\) 0 0
\(853\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0.500000 0.363271i 0.500000 0.363271i
\(863\) −1.34500 0.437016i −1.34500 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(870\) 0 0
\(871\) 0 0
\(872\) 3.57349 4.91849i 3.57349 4.91849i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.11803 1.53884i 1.11803 1.53884i 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.34786 1.08779i −3.34786 1.08779i
\(887\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(888\) 0 0
\(889\) −0.363271 1.11803i −0.363271 1.11803i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 5.78203 + 1.87869i 5.78203 + 1.87869i
\(897\) 0 0
\(898\) −0.363271 0.500000i −0.363271 0.500000i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −6.36801 + 2.06909i −6.36801 + 2.06909i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.587785 1.80902i 0.587785 1.80902i 1.00000i \(-0.5\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.34500 0.437016i 1.34500 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.19623i 3.19623i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.17557 1.17557
\(926\) 0.377263 + 1.16110i 0.377263 + 1.16110i
\(927\) 0 0
\(928\) 3.79892 2.76007i 3.79892 2.76007i
\(929\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.32037 + 2.41239i 3.32037 + 2.41239i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(938\) 0.717598 0.987688i 0.717598 0.987688i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −0.863271 + 0.863271i −0.863271 + 0.863271i
\(947\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.59811 1.16110i 1.59811 1.16110i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.17160 5.17160
\(957\) 0 0
\(958\) 0 0
\(959\) 0.610425 + 1.87869i 0.610425 + 1.87869i
\(960\) 0 0
\(961\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(968\) 3.57349 1.16110i 3.57349 1.16110i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.16110 3.57349i 1.16110 3.57349i
\(975\) 0 0
\(976\) 0 0
\(977\) −1.87869 + 0.610425i −1.87869 + 0.610425i −0.891007 + 0.453990i \(0.850000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −2.26007 1.64204i −2.26007 1.64204i
\(983\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.193364 −0.193364
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −2.84786 + 2.06909i −2.84786 + 2.06909i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(998\) −3.03979 2.20854i −3.03979 2.20854i
\(999\) 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 693.1.bp.a.314.1 16
3.2 odd 2 inner 693.1.bp.a.314.4 yes 16
7.6 odd 2 CM 693.1.bp.a.314.1 16
11.2 odd 10 inner 693.1.bp.a.629.4 yes 16
21.20 even 2 inner 693.1.bp.a.314.4 yes 16
33.2 even 10 inner 693.1.bp.a.629.1 yes 16
77.13 even 10 inner 693.1.bp.a.629.4 yes 16
231.167 odd 10 inner 693.1.bp.a.629.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.1.bp.a.314.1 16 1.1 even 1 trivial
693.1.bp.a.314.1 16 7.6 odd 2 CM
693.1.bp.a.314.4 yes 16 3.2 odd 2 inner
693.1.bp.a.314.4 yes 16 21.20 even 2 inner
693.1.bp.a.629.1 yes 16 33.2 even 10 inner
693.1.bp.a.629.1 yes 16 231.167 odd 10 inner
693.1.bp.a.629.4 yes 16 11.2 odd 10 inner
693.1.bp.a.629.4 yes 16 77.13 even 10 inner