Properties

Label 693.1.bp.a.62.1
Level $693$
Weight $1$
Character 693.62
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 62.1
Root \(-0.987688 - 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 693.62
Dual form 693.1.bp.a.503.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.44168 - 1.04744i) q^{2} +(0.672288 + 2.06909i) q^{4} +(-0.951057 + 0.309017i) q^{7} +(0.647354 - 1.99235i) q^{8} +O(q^{10})\) \(q+(-1.44168 - 1.04744i) q^{2} +(0.672288 + 2.06909i) q^{4} +(-0.951057 + 0.309017i) q^{7} +(0.647354 - 1.99235i) q^{8} +(-0.156434 + 0.987688i) q^{11} +(1.69480 + 0.550672i) q^{14} +(-1.26007 + 0.915497i) q^{16} +(1.26007 - 1.26007i) q^{22} +0.907981i q^{23} +(-0.309017 + 0.951057i) q^{25} +(-1.27877 - 1.76007i) q^{28} +(0.610425 + 1.87869i) q^{29} +0.680668 q^{32} +(0.587785 + 1.80902i) q^{37} -1.61803i q^{43} +(-2.14879 + 0.340334i) q^{44} +(0.951057 - 1.30902i) q^{46} +(0.809017 - 0.587785i) q^{49} +(1.44168 - 1.04744i) q^{50} +(1.16110 - 1.59811i) q^{53} +2.09488i q^{56} +(1.08779 - 3.34786i) q^{58} +(0.278768 + 0.202537i) q^{64} -1.61803 q^{67} +(0.183900 + 0.253116i) q^{71} +(1.04744 - 3.22369i) q^{74} +(-0.156434 - 0.987688i) q^{77} +(-0.690983 + 0.951057i) q^{79} +(-1.69480 + 2.33269i) q^{86} +(1.86655 + 0.951057i) q^{88} +(-1.87869 + 0.610425i) q^{92} -1.78201 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{7}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.44168 1.04744i −1.44168 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(3\) 0 0
\(4\) 0.672288 + 2.06909i 0.672288 + 2.06909i
\(5\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(6\) 0 0
\(7\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(8\) 0.647354 1.99235i 0.647354 1.99235i
\(9\) 0 0
\(10\) 0 0
\(11\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(12\) 0 0
\(13\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 1.69480 + 0.550672i 1.69480 + 0.550672i
\(15\) 0 0
\(16\) −1.26007 + 0.915497i −1.26007 + 0.915497i
\(17\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.26007 1.26007i 1.26007 1.26007i
\(23\) 0.907981i 0.907981i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(24\) 0 0
\(25\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.27877 1.76007i −1.27877 1.76007i
\(29\) 0.610425 + 1.87869i 0.610425 + 1.87869i 0.453990 + 0.891007i \(0.350000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 0.680668 0.680668
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(42\) 0 0
\(43\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(44\) −2.14879 + 0.340334i −2.14879 + 0.340334i
\(45\) 0 0
\(46\) 0.951057 1.30902i 0.951057 1.30902i
\(47\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) 0 0
\(49\) 0.809017 0.587785i 0.809017 0.587785i
\(50\) 1.44168 1.04744i 1.44168 1.04744i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.16110 1.59811i 1.16110 1.59811i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.09488i 2.09488i
\(57\) 0 0
\(58\) 1.08779 3.34786i 1.08779 3.34786i
\(59\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.278768 + 0.202537i 0.278768 + 0.202537i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.183900 + 0.253116i 0.183900 + 0.253116i 0.891007 0.453990i \(-0.150000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(72\) 0 0
\(73\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) 1.04744 3.22369i 1.04744 3.22369i
\(75\) 0 0
\(76\) 0 0
\(77\) −0.156434 0.987688i −0.156434 0.987688i
\(78\) 0 0
\(79\) −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i \(0.400000\pi\)
−1.00000 \(1.00000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.69480 + 2.33269i −1.69480 + 2.33269i
\(87\) 0 0
\(88\) 1.86655 + 0.951057i 1.86655 + 0.951057i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.87869 + 0.610425i −1.87869 + 0.610425i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(98\) −1.78201 −1.78201
\(99\) 0 0
\(100\) −2.17557 −2.17557
\(101\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.34786 + 1.08779i −3.34786 + 1.08779i
\(107\) −0.280582 + 0.863541i −0.280582 + 0.863541i 0.707107 + 0.707107i \(0.250000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(108\) 0 0
\(109\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.915497 1.26007i 0.915497 1.26007i
\(113\) −0.297556 0.0966818i −0.297556 0.0966818i 0.156434 0.987688i \(-0.450000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.47681 + 2.52605i −3.47681 + 2.52605i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.951057 0.309017i −0.951057 0.309017i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.11803 + 1.53884i 1.11803 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(128\) −0.400087 1.23134i −0.400087 1.23134i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.33269 + 1.69480i 2.33269 + 1.69480i
\(135\) 0 0
\(136\) 0 0
\(137\) −1.04744 1.44168i −1.04744 1.44168i −0.891007 0.453990i \(-0.850000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(138\) 0 0
\(139\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.557537i 0.557537i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −3.34786 + 2.43236i −3.34786 + 2.43236i
\(149\) 1.14412 0.831254i 1.14412 0.831254i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(150\) 0 0
\(151\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.809017 + 1.58779i −0.809017 + 1.58779i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(158\) 1.99235 0.647354i 1.99235 0.647354i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.280582 0.863541i −0.280582 0.863541i
\(162\) 0 0
\(163\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(168\) 0 0
\(169\) −0.309017 0.951057i −0.309017 0.951057i
\(170\) 0 0
\(171\) 0 0
\(172\) 3.34786 1.08779i 3.34786 1.08779i
\(173\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(174\) 0 0
\(175\) 1.00000i 1.00000i
\(176\) −0.707107 1.38778i −0.707107 1.38778i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.69480 0.550672i −1.69480 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(180\) 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
\(184\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.87869 0.610425i 1.87869 0.610425i 0.891007 0.453990i \(-0.150000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(192\) 0 0
\(193\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.76007 + 1.27877i 1.76007 + 1.27877i
\(197\) −0.907981 −0.907981 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.69480 + 1.23134i 1.69480 + 1.23134i
\(201\) 0 0
\(202\) 0 0
\(203\) −1.16110 1.59811i −1.16110 1.59811i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(212\) 4.08723 + 1.32802i 4.08723 + 1.32802i
\(213\) 0 0
\(214\) 1.30902 0.951057i 1.30902 0.951057i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.647354 + 0.891007i −0.647354 + 0.891007i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(224\) −0.647354 + 0.210338i −0.647354 + 0.210338i
\(225\) 0 0
\(226\) 0.327712 + 0.451057i 0.327712 + 0.451057i
\(227\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.13818 4.13818
\(233\) 1.14412 + 0.831254i 1.14412 + 0.831254i 0.987688 0.156434i \(-0.0500000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.0966818 + 0.297556i −0.0966818 + 0.297556i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1.04744 + 1.44168i 1.04744 + 1.44168i
\(243\) 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 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(252\) 0 0
\(253\) −0.896802 0.142040i −0.896802 0.142040i
\(254\) 3.38959i 3.38959i
\(255\) 0 0
\(256\) −0.606480 + 1.86655i −0.606480 + 1.86655i
\(257\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(258\) 0 0
\(259\) −1.11803 1.53884i −1.11803 1.53884i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.97538 1.97538 0.987688 0.156434i \(-0.0500000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.08779 3.34786i −1.08779 3.34786i
\(269\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(270\) 0 0
\(271\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 3.17557i 3.17557i
\(275\) −0.891007 0.453990i −0.891007 0.453990i
\(276\) 0 0
\(277\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.253116 + 0.183900i −0.253116 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(284\) −0.400087 + 0.550672i −0.400087 + 0.550672i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.309017 0.951057i 0.309017 0.951057i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.98470 3.98470
\(297\) 0 0
\(298\) −2.52015 −2.52015
\(299\) 0 0
\(300\) 0 0
\(301\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(302\) 0.647354 + 0.891007i 0.647354 + 0.891007i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 1.93845 0.987688i 1.93845 0.987688i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(312\) 0 0
\(313\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.43236 0.790322i −2.43236 0.790322i
\(317\) −0.183900 + 0.253116i −0.183900 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) −1.95106 + 0.309017i −1.95106 + 0.309017i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0.340334 + 1.04744i 0.340334 + 1.04744i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.90211 1.90211 0.951057 0.309017i \(-0.100000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(338\) −0.550672 + 1.69480i −0.550672 + 1.69480i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(344\) −3.22369 1.04744i −3.22369 1.04744i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.253116 0.183900i 0.253116 0.183900i −0.453990 0.891007i \(-0.650000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(350\) −1.04744 + 1.44168i −1.04744 + 1.44168i
\(351\) 0 0
\(352\) −0.106480 + 0.672288i −0.106480 + 0.672288i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.86655 + 2.56909i 1.86655 + 2.56909i
\(359\) 0.280582 + 0.863541i 0.280582 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0 0
\(361\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) −0.831254 1.14412i −0.831254 1.14412i
\(369\) 0 0
\(370\) 0 0
\(371\) −0.610425 + 1.87869i −0.610425 + 1.87869i
\(372\) 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
\(379\) 1.53884 1.11803i 1.53884 1.11803i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −3.34786 1.08779i −3.34786 1.08779i
\(383\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.88336i 2.88336i
\(387\) 0 0
\(388\) 0 0
\(389\) −1.69480 + 0.550672i −1.69480 + 0.550672i −0.987688 0.156434i \(-0.950000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.647354 1.99235i −0.647354 1.99235i
\(393\) 0 0
\(394\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.481305 1.48131i −0.481305 1.48131i
\(401\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 3.52015i 3.52015i
\(407\) −1.87869 + 0.297556i −1.87869 + 0.297556i
\(408\) 0 0
\(409\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(410\) 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
\(421\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(422\) −1.04744 + 0.340334i −1.04744 + 0.340334i
\(423\) 0 0
\(424\) −2.43236 3.34786i −2.43236 3.34786i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.97538 −1.97538
\(429\) 0 0
\(430\) 0 0
\(431\) −0.734572 0.533698i −0.734572 0.533698i 0.156434 0.987688i \(-0.450000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.27877 0.415497i 1.27877 0.415497i
\(437\) 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
\(443\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i 0.453990 0.891007i \(-0.350000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.327712 0.106480i −0.327712 0.106480i
\(449\) 0.533698 0.734572i 0.533698 0.734572i −0.453990 0.891007i \(-0.650000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.680668i 0.680668i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(464\) −2.48912 1.80845i −2.48912 1.80845i
\(465\) 0 0
\(466\) −0.778768 2.39680i −0.778768 2.39680i
\(467\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(468\) 0 0
\(469\) 1.53884 0.500000i 1.53884 0.500000i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.59811 + 0.253116i 1.59811 + 0.253116i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0.451057 0.327712i 0.451057 0.327712i
\(479\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.17557i 2.17557i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.363271 + 1.11803i −0.363271 + 1.11803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.437016 1.34500i −0.437016 1.34500i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.253116 0.183900i −0.253116 0.183900i
\(498\) 0 0
\(499\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.14412 + 1.14412i 1.14412 + 1.14412i
\(507\) 0 0
\(508\) −2.43236 + 3.34786i −2.43236 + 3.34786i
\(509\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.78201 1.29471i 1.78201 1.29471i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 3.38959i 3.38959i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −2.84786 2.06909i −2.84786 2.06909i
\(527\) 0 0
\(528\) 0 0
\(529\) 0.175571 0.175571
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.04744 + 3.22369i −1.04744 + 3.22369i
\(537\) 0 0
\(538\) 0 0
\(539\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(540\) 0 0
\(541\) −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i \(0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(548\) 2.27878 3.13647i 2.27878 3.13647i
\(549\) 0 0
\(550\) 0.809017 + 1.58779i 0.809017 + 1.58779i
\(551\) 0 0
\(552\) 0 0
\(553\) 0.363271 1.11803i 0.363271 1.11803i
\(554\) −3.22369 + 1.04744i −3.22369 + 1.04744i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i 0.987688 0.156434i \(-0.0500000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.557537 0.557537
\(563\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0.623345 0.202537i 0.623345 0.202537i
\(569\) −0.437016 + 1.34500i −0.437016 + 1.34500i 0.453990 + 0.891007i \(0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(570\) 0 0
\(571\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.863541 0.280582i −0.863541 0.280582i
\(576\) 0 0
\(577\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) −1.44168 + 1.04744i −1.44168 + 1.04744i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.39680 1.74138i −2.39680 1.74138i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.48912 + 1.80845i 2.48912 + 1.80845i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.04744 + 1.44168i 1.04744 + 1.44168i 0.891007 + 0.453990i \(0.150000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(600\) 0 0
\(601\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(602\) 0.891007 2.74224i 0.891007 2.74224i
\(603\) 0 0
\(604\) 1.34458i 1.34458i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.11803 + 0.363271i 1.11803 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −2.06909 0.327712i −2.06909 0.327712i
\(617\) 1.97538i 1.97538i 0.156434 + 0.987688i \(0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(618\) 0 0
\(619\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(620\) 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
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(632\) 1.44753 + 1.99235i 1.44753 + 1.99235i
\(633\) 0 0
\(634\) 0.530249 0.172288i 0.530249 0.172288i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 3.13647 + 1.59811i 3.13647 + 1.59811i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.87869 + 0.610425i 1.87869 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(642\) 0 0
\(643\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(644\) 1.59811 1.16110i 1.59811 1.16110i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.415497 1.27877i 0.415497 1.27877i
\(653\) −0.863541 + 0.280582i −0.863541 + 0.280582i −0.707107 0.707107i \(-0.750000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(654\) 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 \(\pi\)
\(662\) −2.74224 1.99235i −2.74224 1.99235i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.70582 + 0.554254i −1.70582 + 0.554254i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(674\) −1.04744 0.340334i −1.04744 0.340334i
\(675\) 0 0
\(676\) 1.76007 1.27877i 1.76007 1.27877i
\(677\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.78201i 1.78201i 0.453990 + 0.891007i \(0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.69480 0.550672i 1.69480 0.550672i
\(687\) 0 0
\(688\) 1.48131 + 2.03884i 1.48131 + 2.03884i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −0.557537 −0.557537
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.06909 0.672288i 2.06909 0.672288i
\(701\) 0.280582 0.863541i 0.280582 0.863541i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.243652 + 0.243652i −0.243652 + 0.243652i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 3.87690i 3.87690i
\(717\) 0 0
\(718\) 0.500000 1.53884i 0.500000 1.53884i
\(719\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.550672 1.69480i −0.550672 1.69480i
\(723\) 0 0
\(724\) 0 0
\(725\) −1.97538 −1.97538
\(726\) 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 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0.618034i 0.618034i
\(737\) 0.253116 1.59811i 0.253116 1.59811i
\(738\) 0 0
\(739\) 0.690983 0.951057i 0.690983 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
1.00000 \(0\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.84786 2.06909i 2.84786 2.06909i
\(743\) 1.59811 1.16110i 1.59811 1.16110i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.907981i 0.907981i
\(750\) 0 0
\(751\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(758\) −3.38959 −3.38959
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(762\) 0 0
\(763\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(764\) 2.52605 + 3.47681i 2.52605 + 3.47681i
\(765\) 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
\(772\) −2.06909 + 2.84786i −2.06909 + 2.84786i
\(773\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 3.02015 + 0.981305i 3.02015 + 0.981305i
\(779\) 0 0
\(780\) 0 0
\(781\) −0.278768 + 0.142040i −0.278768 + 0.142040i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.481305 + 1.48131i −0.481305 + 1.48131i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(788\) −0.610425 1.87869i −0.610425 1.87869i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.312869 0.312869
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.210338 + 0.647354i −0.210338 + 0.647354i
\(801\) 0 0
\(802\) 1.61803i 1.61803i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.734572 0.533698i 0.734572 0.533698i −0.156434 0.987688i \(-0.550000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(812\) 2.52605 3.47681i 2.52605 3.47681i
\(813\) 0 0
\(814\) 3.02015 + 1.53884i 3.02015 + 1.53884i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.437016 + 1.34500i 0.437016 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(822\) 0 0
\(823\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.14412 0.831254i −1.14412 0.831254i −0.156434 0.987688i \(-0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(828\) 0 0
\(829\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 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 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(840\) 0 0
\(841\) −2.34786 + 1.70582i −2.34786 + 1.70582i
\(842\) −0.891007 + 0.647354i −0.891007 + 0.647354i
\(843\) 0 0
\(844\) 1.27877 + 0.415497i 1.27877 + 0.415497i
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000 1.00000
\(848\) 3.07672i 3.07672i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.64255 + 0.533698i −1.64255 + 0.533698i
\(852\) 0 0
\(853\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.53884 + 1.11803i 1.53884 + 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 + 1.53884i 0.500000 + 1.53884i
\(863\) −0.831254 1.14412i −0.831254 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.831254 0.831254i −0.831254 0.831254i
\(870\) 0 0
\(871\) 0 0
\(872\) −1.23134 0.400087i −1.23134 0.400087i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.11803 0.363271i −1.11803 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(878\) 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 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.327712 0.451057i −0.327712 0.451057i
\(887\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(888\) 0 0
\(889\) −1.53884 1.11803i −1.53884 1.11803i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0.761010 + 1.04744i 0.761010 + 1.04744i
\(897\) 0 0
\(898\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −0.385248 + 0.530249i −0.385248 + 0.530249i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.951057 + 0.690983i −0.951057 + 0.690983i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.831254 1.14412i 0.831254 1.14412i −0.156434 0.987688i \(-0.550000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.10134i 1.10134i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.90211 −1.90211
\(926\) −2.33269 1.69480i −2.33269 1.69480i
\(927\) 0 0
\(928\) 0.415497 + 1.27877i 0.415497 + 1.27877i
\(929\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.950759 + 2.92614i −0.950759 + 2.92614i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(938\) −2.74224 0.891007i −2.74224 0.891007i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −2.03884 2.03884i −2.03884 2.03884i
\(947\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.550672 + 1.69480i 0.550672 + 1.69480i 0.707107 + 0.707107i \(0.250000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −0.680668 −0.680668
\(957\) 0 0
\(958\) 0 0
\(959\) 1.44168 + 1.04744i 1.44168 + 1.04744i
\(960\) 0 0
\(961\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(968\) −1.23134 + 1.69480i −1.23134 + 1.69480i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.69480 1.23134i 1.69480 1.23134i
\(975\) 0 0
\(976\) 0 0
\(977\) 1.04744 1.44168i 1.04744 1.44168i 0.156434 0.987688i \(-0.450000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −0.778768 + 2.39680i −0.778768 + 2.39680i
\(983\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.46914 1.46914
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0.172288 + 0.530249i 0.172288 + 0.530249i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(998\) −0.647354 + 1.99235i −0.647354 + 1.99235i
\(999\) 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.62.1 16
3.2 odd 2 inner 693.1.bp.a.62.4 yes 16
7.6 odd 2 CM 693.1.bp.a.62.1 16
11.8 odd 10 inner 693.1.bp.a.503.4 yes 16
21.20 even 2 inner 693.1.bp.a.62.4 yes 16
33.8 even 10 inner 693.1.bp.a.503.1 yes 16
77.41 even 10 inner 693.1.bp.a.503.4 yes 16
231.41 odd 10 inner 693.1.bp.a.503.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.1.bp.a.62.1 16 1.1 even 1 trivial
693.1.bp.a.62.1 16 7.6 odd 2 CM
693.1.bp.a.62.4 yes 16 3.2 odd 2 inner
693.1.bp.a.62.4 yes 16 21.20 even 2 inner
693.1.bp.a.503.1 yes 16 33.8 even 10 inner
693.1.bp.a.503.1 yes 16 231.41 odd 10 inner
693.1.bp.a.503.4 yes 16 11.8 odd 10 inner
693.1.bp.a.503.4 yes 16 77.41 even 10 inner