Properties

Label 3025.1.bj.a.727.1
Level $3025$
Weight $1$
Character 3025.727
Analytic conductor $1.510$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3025,1,Mod(122,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([17, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3025.122");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3025.bj (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: \(1.50967166321\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 727.1
Root \(0.587785 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 3025.727
Dual form 3025.1.bj.a.2663.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.896802 + 1.76007i) q^{3} +(-0.951057 - 0.309017i) q^{4} +1.00000i q^{5} +(-1.70582 + 2.34786i) q^{9} +O(q^{10})\) \(q+(0.896802 + 1.76007i) q^{3} +(-0.951057 - 0.309017i) q^{4} +1.00000i q^{5} +(-1.70582 + 2.34786i) q^{9} +(-0.309017 - 1.95106i) q^{12} +(-1.76007 + 0.896802i) q^{15} +(0.809017 + 0.587785i) q^{16} +(0.309017 - 0.951057i) q^{20} +(-0.142040 + 0.896802i) q^{23} -1.00000 q^{25} +(-3.71113 - 0.587785i) q^{27} +(-0.363271 - 1.11803i) q^{31} +(2.34786 - 1.70582i) q^{36} +(0.221232 + 1.39680i) q^{37} +(-2.34786 - 1.70582i) q^{45} +(1.26007 - 0.642040i) q^{47} +(-0.309017 + 1.95106i) q^{48} -1.00000i q^{49} +(-0.142040 - 0.278768i) q^{53} +(0.690983 - 0.951057i) q^{59} +(1.95106 - 0.309017i) q^{60} +(-0.587785 - 0.809017i) q^{64} +(-0.809017 + 1.58779i) q^{67} +(-1.70582 + 0.554254i) q^{69} +(-0.500000 + 1.53884i) q^{71} +(-0.896802 - 1.76007i) q^{75} +(-0.587785 + 0.809017i) q^{80} +(-1.39680 - 4.29892i) q^{81} +(0.363271 + 0.500000i) q^{89} +(0.412215 - 0.809017i) q^{92} +(1.64204 - 1.64204i) q^{93} +(0.278768 - 0.142040i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{12} - 2 q^{15} + 2 q^{16} - 2 q^{20} + 2 q^{23} - 8 q^{25} - 10 q^{27} + 2 q^{36} + 2 q^{37} - 2 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{53} + 10 q^{59} + 8 q^{60} - 2 q^{67} - 4 q^{71} + 2 q^{75} - 2 q^{81} + 8 q^{92} + 10 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3025\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(2301\)
\(\chi(n)\) \(e\left(\frac{1}{20}\right)\) \(1\)

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 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(3\) 0.896802 + 1.76007i 0.896802 + 1.76007i 0.587785 + 0.809017i \(0.300000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(4\) −0.951057 0.309017i −0.951057 0.309017i
\(5\) 1.00000i 1.00000i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) −1.70582 + 2.34786i −1.70582 + 2.34786i
\(10\) 0 0
\(11\) 0 0
\(12\) −0.309017 1.95106i −0.309017 1.95106i
\(13\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(14\) 0 0
\(15\) −1.76007 + 0.896802i −1.76007 + 0.896802i
\(16\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(17\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) 0.309017 0.951057i 0.309017 0.951057i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.142040 + 0.896802i −0.142040 + 0.896802i 0.809017 + 0.587785i \(0.200000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) −3.71113 0.587785i −3.71113 0.587785i
\(28\) 0 0
\(29\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 2.34786 1.70582i 2.34786 1.70582i
\(37\) 0.221232 + 1.39680i 0.221232 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) −2.34786 1.70582i −2.34786 1.70582i
\(46\) 0 0
\(47\) 1.26007 0.642040i 1.26007 0.642040i 0.309017 0.951057i \(-0.400000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) −0.309017 + 1.95106i −0.309017 + 1.95106i
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.142040 0.278768i −0.142040 0.278768i 0.809017 0.587785i \(-0.200000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.690983 0.951057i 0.690983 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
1.00000 \(0\)
\(60\) 1.95106 0.309017i 1.95106 0.309017i
\(61\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.587785 0.809017i −0.587785 0.809017i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.809017 + 1.58779i −0.809017 + 1.58779i 1.00000i \(0.5\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) 0 0
\(69\) −1.70582 + 0.554254i −1.70582 + 0.554254i
\(70\) 0 0
\(71\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(74\) 0 0
\(75\) −0.896802 1.76007i −0.896802 1.76007i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(80\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(81\) −1.39680 4.29892i −1.39680 4.29892i
\(82\) 0 0
\(83\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.412215 0.809017i 0.412215 0.809017i
\(93\) 1.64204 1.64204i 1.64204 1.64204i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.278768 0.142040i 0.278768 0.142040i −0.309017 0.951057i \(-0.600000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −0.412215 0.809017i −0.412215 0.809017i 0.587785 0.809017i \(-0.300000\pi\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 3.34786 + 1.70582i 3.34786 + 1.70582i
\(109\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(110\) 0 0
\(111\) −2.26007 + 1.64204i −2.26007 + 1.64204i
\(112\) 0 0
\(113\) 1.76007 0.278768i 1.76007 0.278768i 0.809017 0.587785i \(-0.200000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(114\) 0 0
\(115\) −0.896802 0.142040i −0.896802 0.142040i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 1.17557i 1.17557i
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.587785 3.71113i 0.587785 3.71113i
\(136\) 0 0
\(137\) 0.278768 + 1.76007i 0.278768 + 1.76007i 0.587785 + 0.809017i \(0.300000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(138\) 0 0
\(139\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) 2.26007 + 1.64204i 2.26007 + 1.64204i
\(142\) 0 0
\(143\) 0 0
\(144\) −2.76007 + 0.896802i −2.76007 + 0.896802i
\(145\) 0 0
\(146\) 0 0
\(147\) 1.76007 0.896802i 1.76007 0.896802i
\(148\) 0.221232 1.39680i 0.221232 1.39680i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.11803 0.363271i 1.11803 0.363271i
\(156\) 0 0
\(157\) 0.642040 + 0.642040i 0.642040 + 0.642040i 0.951057 0.309017i \(-0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(158\) 0 0
\(159\) 0.363271 0.500000i 0.363271 0.500000i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.309017 0.0489435i 0.309017 0.0489435i 1.00000i \(-0.5\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(168\) 0 0
\(169\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.29360 + 0.363271i 2.29360 + 0.363271i
\(178\) 0 0
\(179\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(180\) 1.70582 + 2.34786i 1.70582 + 2.34786i
\(181\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.39680 + 0.221232i −1.39680 + 0.221232i
\(186\) 0 0
\(187\) 0 0
\(188\) −1.39680 + 0.221232i −1.39680 + 0.221232i
\(189\) 0 0
\(190\) 0 0
\(191\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(192\) 0.896802 1.76007i 0.896802 1.76007i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(197\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(198\) 0 0
\(199\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(200\) 0 0
\(201\) −3.52015 −3.52015
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.86327 1.86327i −1.86327 1.86327i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(212\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i
\(213\) −3.15688 + 0.500000i −3.15688 + 0.500000i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.0489435 0.309017i 0.0489435 0.309017i −0.951057 0.309017i \(-0.900000\pi\)
1.00000 \(0\)
\(224\) 0 0
\(225\) 1.70582 2.34786i 1.70582 2.34786i
\(226\) 0 0
\(227\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(228\) 0 0
\(229\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(234\) 0 0
\(235\) 0.642040 + 1.26007i 0.642040 + 1.26007i
\(236\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) −1.95106 0.309017i −1.95106 0.309017i
\(241\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(242\) 0 0
\(243\) 3.65688 3.65688i 3.65688 3.65688i
\(244\) 0 0
\(245\) 1.00000 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(257\) 1.26007 + 1.26007i 1.26007 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(264\) 0 0
\(265\) 0.278768 0.142040i 0.278768 0.142040i
\(266\) 0 0
\(267\) −0.554254 + 1.08779i −0.554254 + 1.08779i
\(268\) 1.26007 1.26007i 1.26007 1.26007i
\(269\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 1.79360 1.79360
\(277\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(278\) 0 0
\(279\) 3.24466 + 1.05425i 3.24466 + 1.05425i
\(280\) 0 0
\(281\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(282\) 0 0
\(283\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(284\) 0.951057 1.30902i 0.951057 1.30902i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(290\) 0 0
\(291\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.309017 + 1.95106i 0.309017 + 1.95106i
\(301\) 0 0
\(302\) 0 0
\(303\) 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 0
\(309\) 1.05425 1.45106i 1.05425 1.45106i
\(310\) 0 0
\(311\) 0.500000 0.363271i 0.500000 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(312\) 0 0
\(313\) 0.896802 0.142040i 0.896802 0.142040i 0.309017 0.951057i \(-0.400000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.142040 0.278768i 0.142040 0.278768i −0.809017 0.587785i \(-0.800000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.809017 0.587785i 0.809017 0.587785i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 4.52015i 4.52015i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.587785 1.80902i −0.587785 1.80902i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(-0.5\pi\)
\(332\) 0 0
\(333\) −3.65688 1.86327i −3.65688 1.86327i
\(334\) 0 0
\(335\) −1.58779 0.809017i −1.58779 0.809017i
\(336\) 0 0
\(337\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(338\) 0 0
\(339\) 2.06909 + 2.84786i 2.06909 + 2.84786i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.554254 1.70582i −0.554254 1.70582i
\(346\) 0 0
\(347\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.642040 + 1.26007i 0.642040 + 1.26007i 0.951057 + 0.309017i \(0.100000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(354\) 0 0
\(355\) −1.53884 0.500000i −1.53884 0.500000i
\(356\) −0.190983 0.587785i −0.190983 0.587785i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(360\) 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
\(367\) −0.809017 + 1.58779i −0.809017 + 1.58779i 1.00000i \(0.5\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.06909 + 1.05425i −2.06909 + 1.05425i
\(373\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(374\) 0 0
\(375\) 1.76007 0.896802i 1.76007 0.896802i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.278768 0.142040i −0.278768 0.142040i 0.309017 0.951057i \(-0.400000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.309017 + 0.0489435i −0.309017 + 0.0489435i
\(389\) −0.690983 0.951057i −0.690983 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.58779 + 0.809017i −1.58779 + 0.809017i −0.587785 + 0.809017i \(0.700000\pi\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.809017 0.587785i −0.809017 0.587785i
\(401\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.29892 1.39680i 4.29892 1.39680i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(410\) 0 0
\(411\) −2.84786 + 2.06909i −2.84786 + 2.06909i
\(412\) 0.142040 + 0.896802i 0.142040 + 0.896802i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(420\) 0 0
\(421\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(422\) 0 0
\(423\) −0.642040 + 4.05368i −0.642040 + 4.05368i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(432\) −2.65688 2.65688i −2.65688 2.65688i
\(433\) 1.26007 + 0.642040i 1.26007 + 0.642040i 0.951057 0.309017i \(-0.100000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(440\) 0 0
\(441\) 2.34786 + 1.70582i 2.34786 + 1.70582i
\(442\) 0 0
\(443\) 0.221232 0.221232i 0.221232 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(444\) 2.65688 0.863271i 2.65688 0.863271i
\(445\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.76007 0.278768i −1.76007 0.278768i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0.809017 + 0.412215i 0.809017 + 0.412215i
\(461\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(462\) 0 0
\(463\) 0.896802 0.142040i 0.896802 0.142040i 0.309017 0.951057i \(-0.400000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(464\) 0 0
\(465\) 1.64204 + 1.64204i 1.64204 + 1.64204i
\(466\) 0 0
\(467\) 0.642040 1.26007i 0.642040 1.26007i −0.309017 0.951057i \(-0.600000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.554254 + 1.70582i −0.554254 + 1.70582i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.896802 + 0.142040i 0.896802 + 0.142040i
\(478\) 0 0
\(479\) 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
\(484\) 0 0
\(485\) 0.142040 + 0.278768i 0.142040 + 0.278768i
\(486\) 0 0
\(487\) −0.221232 1.39680i −0.221232 1.39680i −0.809017 0.587785i \(-0.800000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(488\) 0 0
\(489\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(490\) 0 0
\(491\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.363271 1.11803i 0.363271 1.11803i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(500\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.39680 1.39680i −1.39680 1.39680i
\(508\) 0 0
\(509\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.809017 0.412215i 0.809017 0.412215i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.166977 + 0.0542543i 0.166977 + 0.0542543i
\(530\) 0 0
\(531\) 1.05425 + 3.24466i 1.05425 + 3.24466i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.190983 1.20582i −0.190983 1.20582i
\(538\) 0 0
\(539\) 0 0
\(540\) −1.70582 + 3.34786i −1.70582 + 3.34786i
\(541\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(548\) 0.278768 1.76007i 0.278768 1.76007i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.64204 2.26007i −1.64204 2.26007i
\(556\) 0 0
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(564\) −1.64204 2.26007i −1.64204 2.26007i
\(565\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(572\) 0 0
\(573\) −0.587785 + 3.71113i −0.587785 + 3.71113i
\(574\) 0 0
\(575\) 0.142040 0.896802i 0.142040 0.896802i
\(576\) 2.90211 2.90211
\(577\) 1.95106 + 0.309017i 1.95106 + 0.309017i 1.00000 \(0\)
0.951057 + 0.309017i \(0.100000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.309017 1.95106i −0.309017 1.95106i −0.309017 0.951057i \(-0.600000\pi\)
1.00000i \(-0.5\pi\)
\(588\) −1.95106 + 0.309017i −1.95106 + 0.309017i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.642040 + 1.26007i −0.642040 + 1.26007i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.34786 + 1.70582i −3.34786 + 1.70582i
\(598\) 0 0
\(599\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) −2.34786 4.60793i −2.34786 4.60793i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.896802 1.76007i 0.896802 1.76007i 0.309017 0.951057i \(-0.400000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(618\) 0 0
\(619\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(620\) −1.17557 −1.17557
\(621\) 1.05425 3.24466i 1.05425 3.24466i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −0.412215 0.809017i −0.412215 0.809017i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(637\) 0 0
\(638\) 0 0
\(639\) −2.76007 3.79892i −2.76007 3.79892i
\(640\) 0 0
\(641\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(642\) 0 0
\(643\) 1.39680 1.39680i 1.39680 1.39680i 0.587785 0.809017i \(-0.300000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.58779 0.809017i 1.58779 0.809017i 0.587785 0.809017i \(-0.300000\pi\)
1.00000 \(0\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.309017 0.0489435i −0.309017 0.0489435i
\(653\) −0.896802 1.76007i −0.896802 1.76007i −0.587785 0.809017i \(-0.700000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(660\) 0 0
\(661\) 0.500000 0.363271i 0.500000 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.587785 0.190983i 0.587785 0.190983i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(674\) 0 0
\(675\) 3.71113 + 0.587785i 3.71113 + 0.587785i
\(676\) 1.00000 1.00000
\(677\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.809017 0.412215i −0.809017 0.412215i 1.00000i \(-0.5\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(684\) 0 0
\(685\) −1.76007 + 0.278768i −1.76007 + 0.278768i
\(686\) 0 0
\(687\) −0.587785 3.71113i −0.587785 3.71113i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −1.64204 + 2.26007i −1.64204 + 2.26007i
\(706\) 0 0
\(707\) 0 0
\(708\) −2.06909 1.05425i −2.06909 1.05425i
\(709\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.05425 0.166977i 1.05425 0.166977i
\(714\) 0 0
\(715\) 0 0
\(716\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(717\) 0 0
\(718\) 0 0
\(719\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(720\) −0.896802 2.76007i −0.896802 2.76007i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.95106 0.309017i −1.95106 0.309017i −0.951057 0.309017i \(-0.900000\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 5.41695 + 1.76007i 5.41695 + 1.76007i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(734\) 0 0
\(735\) 0.896802 + 1.76007i 0.896802 + 1.76007i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 1.39680 + 0.221232i 1.39680 + 0.221232i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) 1.39680 + 0.221232i 1.39680 + 0.221232i
\(753\) 1.45106 + 2.84786i 1.45106 + 2.84786i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.39680 + 1.39680i 1.39680 + 1.39680i 0.809017 + 0.587785i \(0.200000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.11803 1.53884i −1.11803 1.53884i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.39680 + 1.39680i −1.39680 + 1.39680i
\(769\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(770\) 0 0
\(771\) −1.08779 + 3.34786i −1.08779 + 3.34786i
\(772\) 0 0
\(773\) 0.309017 1.95106i 0.309017 1.95106i 1.00000i \(-0.5\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(774\) 0 0
\(775\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.587785 0.809017i 0.587785 0.809017i
\(785\) −0.642040 + 0.642040i −0.642040 + 0.642040i
\(786\) 0 0
\(787\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(796\) 0.587785 1.80902i 0.587785 1.80902i
\(797\) 0.809017 0.412215i 0.809017 0.412215i 1.00000i \(-0.5\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.79360 −1.79360
\(802\) 0 0
\(803\) 0 0
\(804\) 3.34786 + 1.08779i 3.34786 + 1.08779i
\(805\) 0 0
\(806\) 0 0
\(807\) −2.65688 2.65688i −2.65688 2.65688i
\(808\) 0 0
\(809\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(822\) 0 0
\(823\) −0.0489435 + 0.309017i −0.0489435 + 0.309017i 0.951057 + 0.309017i \(0.100000\pi\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(828\) 1.19629 + 2.34786i 1.19629 + 2.34786i
\(829\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.690983 + 4.36269i 0.690983 + 4.36269i
\(838\) 0 0
\(839\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(840\) 0 0
\(841\) −0.809017 0.587785i −0.809017 0.587785i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.309017 0.951057i −0.309017 0.951057i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.0489435 0.309017i 0.0489435 0.309017i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.28408 −1.28408
\(852\) 3.15688 + 0.500000i 3.15688 + 0.500000i
\(853\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(858\) 0 0
\(859\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.95106 0.309017i 1.95106 0.309017i 0.951057 0.309017i \(-0.100000\pi\)
1.00000 \(0\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.896802 + 1.76007i −0.896802 + 1.76007i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.142040 + 0.896802i −0.142040 + 0.896802i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(882\) 0 0
\(883\) −1.76007 0.896802i −1.76007 0.896802i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(884\) 0 0
\(885\) −0.363271 + 2.29360i −0.363271 + 2.29360i
\(886\) 0 0
\(887\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(893\) 0 0
\(894\) 0 0
\(895\) 0.190983 0.587785i 0.190983 0.587785i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −2.34786 + 1.70582i −2.34786 + 1.70582i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.53884 + 1.11803i −1.53884 + 1.11803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.221232 1.39680i −0.221232 1.39680i
\(926\) 0 0
\(927\) 2.60262 + 0.412215i 2.60262 + 0.412215i
\(928\) 0 0
\(929\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.08779 + 0.554254i 1.08779 + 0.554254i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(938\) 0 0
\(939\) 1.05425 + 1.45106i 1.05425 + 1.45106i
\(940\) −0.221232 1.39680i −0.221232 1.39680i
\(941\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.11803 0.363271i 1.11803 0.363271i
\(945\) 0 0
\(946\) 0 0
\(947\) 1.76007 0.896802i 1.76007 0.896802i 0.809017 0.587785i \(-0.200000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.618034 0.618034
\(952\) 0 0
\(953\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(954\) 0 0
\(955\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(961\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(972\) −4.60793 + 2.34786i −4.60793 + 2.34786i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.39680 0.221232i −1.39680 0.221232i −0.587785 0.809017i \(-0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.951057 0.309017i −0.951057 0.309017i
\(981\) 0 0
\(982\) 0 0
\(983\) 0.809017 + 0.412215i 0.809017 + 0.412215i 0.809017 0.587785i \(-0.200000\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(992\) 0 0
\(993\) 2.65688 2.65688i 2.65688 2.65688i
\(994\) 0 0
\(995\) −1.90211 −1.90211
\(996\) 0 0
\(997\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(998\) 0 0
\(999\) 5.31375i 5.31375i
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 3025.1.bj.a.727.1 8
11.2 odd 10 3025.1.bf.a.202.1 8
11.3 even 5 3025.1.bk.a.977.1 8
11.4 even 5 3025.1.bq.a.27.1 8
11.5 even 5 3025.1.bi.a.2302.1 8
11.6 odd 10 3025.1.bi.a.2302.1 8
11.7 odd 10 3025.1.bq.a.27.1 8
11.8 odd 10 3025.1.bk.a.977.1 8
11.9 even 5 3025.1.bf.a.202.1 8
11.10 odd 2 CM 3025.1.bj.a.727.1 8
25.13 odd 20 inner 3025.1.bj.a.2663.1 yes 8
275.13 even 20 3025.1.bi.a.2138.1 8
275.38 odd 20 3025.1.bf.a.1213.1 8
275.63 even 20 3025.1.bq.a.2913.1 8
275.113 odd 20 3025.1.bq.a.2913.1 8
275.138 even 20 3025.1.bf.a.1213.1 8
275.163 odd 20 3025.1.bi.a.2138.1 8
275.213 odd 20 3025.1.bk.a.1963.1 8
275.238 even 20 3025.1.bk.a.1963.1 8
275.263 even 20 inner 3025.1.bj.a.2663.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3025.1.bf.a.202.1 8 11.2 odd 10
3025.1.bf.a.202.1 8 11.9 even 5
3025.1.bf.a.1213.1 8 275.38 odd 20
3025.1.bf.a.1213.1 8 275.138 even 20
3025.1.bi.a.2138.1 8 275.13 even 20
3025.1.bi.a.2138.1 8 275.163 odd 20
3025.1.bi.a.2302.1 8 11.5 even 5
3025.1.bi.a.2302.1 8 11.6 odd 10
3025.1.bj.a.727.1 8 1.1 even 1 trivial
3025.1.bj.a.727.1 8 11.10 odd 2 CM
3025.1.bj.a.2663.1 yes 8 25.13 odd 20 inner
3025.1.bj.a.2663.1 yes 8 275.263 even 20 inner
3025.1.bk.a.977.1 8 11.3 even 5
3025.1.bk.a.977.1 8 11.8 odd 10
3025.1.bk.a.1963.1 8 275.213 odd 20
3025.1.bk.a.1963.1 8 275.238 even 20
3025.1.bq.a.27.1 8 11.4 even 5
3025.1.bq.a.27.1 8 11.7 odd 10
3025.1.bq.a.2913.1 8 275.63 even 20
3025.1.bq.a.2913.1 8 275.113 odd 20