Properties

Label 600.1.z.a.389.2
Level $600$
Weight $1$
Character 600.389
Analytic conductor $0.299$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,1,Mod(29,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 5, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.29");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 600.z (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: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.759375000000000000.15

Embedding invariants

Embedding label 389.2
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 600.389
Dual form 600.1.z.a.509.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.587785 + 0.809017i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(-0.587785 - 0.809017i) q^{5} +(-0.309017 - 0.951057i) q^{6} -1.90211i q^{7} +(-0.951057 + 0.309017i) q^{8} +(0.809017 + 0.587785i) q^{9} +O(q^{10})\) \(q+(0.587785 + 0.809017i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(-0.587785 - 0.809017i) q^{5} +(-0.309017 - 0.951057i) q^{6} -1.90211i q^{7} +(-0.951057 + 0.309017i) q^{8} +(0.809017 + 0.587785i) q^{9} +(0.309017 - 0.951057i) q^{10} +(0.951057 - 0.690983i) q^{11} +(0.587785 - 0.809017i) q^{12} +(1.53884 - 1.11803i) q^{14} +(0.309017 + 0.951057i) q^{15} +(-0.809017 - 0.587785i) q^{16} +1.00000i q^{18} +(0.951057 - 0.309017i) q^{20} +(-0.587785 + 1.80902i) q^{21} +(1.11803 + 0.363271i) q^{22} +1.00000 q^{24} +(-0.309017 + 0.951057i) q^{25} +(-0.587785 - 0.809017i) q^{27} +(1.80902 + 0.587785i) q^{28} +(-0.587785 + 0.809017i) q^{30} +(-0.190983 - 0.587785i) q^{31} -1.00000i q^{32} +(-1.11803 + 0.363271i) q^{33} +(-1.53884 + 1.11803i) q^{35} +(-0.809017 + 0.587785i) q^{36} +(0.809017 + 0.587785i) q^{40} +(-1.80902 + 0.587785i) q^{42} +(0.363271 + 1.11803i) q^{44} -1.00000i q^{45} +(0.587785 + 0.809017i) q^{48} -2.61803 q^{49} +(-0.951057 + 0.309017i) q^{50} +(1.53884 + 0.500000i) q^{53} +(0.309017 - 0.951057i) q^{54} +(-1.11803 - 0.363271i) q^{55} +(0.587785 + 1.80902i) q^{56} +(0.951057 + 0.690983i) q^{59} -1.00000 q^{60} +(0.363271 - 0.500000i) q^{62} +(1.11803 - 1.53884i) q^{63} +(0.809017 - 0.587785i) q^{64} +(-0.951057 - 0.690983i) q^{66} +(-1.80902 - 0.587785i) q^{70} +(-0.951057 - 0.309017i) q^{72} +(0.587785 - 0.809017i) q^{75} +(-1.31433 - 1.80902i) q^{77} +(-0.500000 + 1.53884i) q^{79} +1.00000i q^{80} +(0.309017 + 0.951057i) q^{81} +(-0.587785 + 0.190983i) q^{83} +(-1.53884 - 1.11803i) q^{84} +(-0.690983 + 0.951057i) q^{88} +(0.809017 - 0.587785i) q^{90} +0.618034i q^{93} +(-0.309017 + 0.951057i) q^{96} +(1.80902 + 0.587785i) q^{97} +(-1.53884 - 2.11803i) q^{98} +1.17557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 2 q^{6} + 2 q^{9} - 2 q^{10} - 2 q^{15} - 2 q^{16} + 8 q^{24} + 2 q^{25} + 10 q^{28} - 6 q^{31} - 2 q^{36} + 2 q^{40} - 10 q^{42} - 12 q^{49} - 2 q^{54} - 8 q^{60} + 2 q^{64} - 10 q^{70} - 4 q^{79} - 2 q^{81} - 10 q^{88} + 2 q^{90} + 2 q^{96} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(3\) −0.951057 0.309017i −0.951057 0.309017i
\(4\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(5\) −0.587785 0.809017i −0.587785 0.809017i
\(6\) −0.309017 0.951057i −0.309017 0.951057i
\(7\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(8\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(9\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(10\) 0.309017 0.951057i 0.309017 0.951057i
\(11\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(12\) 0.587785 0.809017i 0.587785 0.809017i
\(13\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 1.53884 1.11803i 1.53884 1.11803i
\(15\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(16\) −0.809017 0.587785i −0.809017 0.587785i
\(17\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(18\) 1.00000i 1.00000i
\(19\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) 0.951057 0.309017i 0.951057 0.309017i
\(21\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(22\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(23\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(24\) 1.00000 1.00000
\(25\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(26\) 0 0
\(27\) −0.587785 0.809017i −0.587785 0.809017i
\(28\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(29\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(30\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(31\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(32\) 1.00000i 1.00000i
\(33\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(34\) 0 0
\(35\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(36\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(37\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(41\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(42\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(45\) 1.00000i 1.00000i
\(46\) 0 0
\(47\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(49\) −2.61803 −2.61803
\(50\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(51\) 0 0
\(52\) 0 0
\(53\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(54\) 0.309017 0.951057i 0.309017 0.951057i
\(55\) −1.11803 0.363271i −1.11803 0.363271i
\(56\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(57\) 0 0
\(58\) 0 0
\(59\) 0.951057 + 0.690983i 0.951057 + 0.690983i 0.951057 0.309017i \(-0.100000\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −1.00000
\(61\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0.363271 0.500000i 0.363271 0.500000i
\(63\) 1.11803 1.53884i 1.11803 1.53884i
\(64\) 0.809017 0.587785i 0.809017 0.587785i
\(65\) 0 0
\(66\) −0.951057 0.690983i −0.951057 0.690983i
\(67\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.80902 0.587785i −1.80902 0.587785i
\(71\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(72\) −0.951057 0.309017i −0.951057 0.309017i
\(73\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) 0.587785 0.809017i 0.587785 0.809017i
\(76\) 0 0
\(77\) −1.31433 1.80902i −1.31433 1.80902i
\(78\) 0 0
\(79\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 1.00000i 1.00000i
\(81\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(82\) 0 0
\(83\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.53884 1.11803i −1.53884 1.11803i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(89\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(90\) 0.809017 0.587785i 0.809017 0.587785i
\(91\) 0 0
\(92\) 0 0
\(93\) 0.618034i 0.618034i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(97\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(98\) −1.53884 2.11803i −1.53884 2.11803i
\(99\) 1.17557 1.17557
\(100\) −0.809017 0.587785i −0.809017 0.587785i
\(101\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(102\) 0 0
\(103\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 1.80902 0.587785i 1.80902 0.587785i
\(106\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(107\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) 0.951057 0.309017i 0.951057 0.309017i
\(109\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(110\) −0.363271 1.11803i −0.363271 1.11803i
\(111\) 0 0
\(112\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(113\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.17557i 1.17557i
\(119\) 0 0
\(120\) −0.587785 0.809017i −0.587785 0.809017i
\(121\) 0.118034 0.363271i 0.118034 0.363271i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.618034 0.618034
\(125\) 0.951057 0.309017i 0.951057 0.309017i
\(126\) 1.90211 1.90211
\(127\) 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 \(0\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(128\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(132\) 1.17557i 1.17557i
\(133\) 0 0
\(134\) 0 0
\(135\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(136\) 0 0
\(137\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(138\) 0 0
\(139\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) −0.587785 1.80902i −0.587785 1.80902i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.309017 0.951057i −0.309017 0.951057i
\(145\) 0 0
\(146\) 0 0
\(147\) 2.48990 + 0.809017i 2.48990 + 0.809017i
\(148\) 0 0
\(149\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(150\) 1.00000 1.00000
\(151\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.690983 2.12663i 0.690983 2.12663i
\(155\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(159\) −1.30902 0.951057i −1.30902 0.951057i
\(160\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(161\) 0 0
\(162\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(163\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(164\) 0 0
\(165\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(166\) −0.500000 0.363271i −0.500000 0.363271i
\(167\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(168\) 1.90211i 1.90211i
\(169\) −0.309017 0.951057i −0.309017 0.951057i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 0 0
\(175\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(176\) −1.17557 −1.17557
\(177\) −0.690983 0.951057i −0.690983 0.951057i
\(178\) 0 0
\(179\) 0.587785 1.80902i 0.587785 1.80902i 1.00000i \(-0.5\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(180\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(181\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(187\) 0 0
\(188\) 0 0
\(189\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(190\) 0 0
\(191\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(192\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(193\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(194\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(195\) 0 0
\(196\) 0.809017 2.48990i 0.809017 2.48990i
\(197\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(198\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(199\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(200\) 1.00000i 1.00000i
\(201\) 0 0
\(202\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −0.587785 1.80902i −0.587785 1.80902i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(211\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(212\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(213\) 0 0
\(214\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(215\) 0 0
\(216\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(217\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(218\) 0 0
\(219\) 0 0
\(220\) 0.690983 0.951057i 0.690983 0.951057i
\(221\) 0 0
\(222\) 0 0
\(223\) 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 \(0\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(224\) −1.90211 −1.90211
\(225\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(226\) 0 0
\(227\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(230\) 0 0
\(231\) 0.690983 + 2.12663i 0.690983 + 2.12663i
\(232\) 0 0
\(233\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(237\) 0.951057 1.30902i 0.951057 1.30902i
\(238\) 0 0
\(239\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) 0.309017 0.951057i 0.309017 0.951057i
\(241\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(242\) 0.363271 0.118034i 0.363271 0.118034i
\(243\) 1.00000i 1.00000i
\(244\) 0 0
\(245\) 1.53884 + 2.11803i 1.53884 + 2.11803i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(249\) 0.618034 0.618034
\(250\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(251\) −1.90211 −1.90211 −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(252\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(253\) 0 0
\(254\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(255\) 0 0
\(256\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(264\) 0.951057 0.690983i 0.951057 0.690983i
\(265\) −0.500000 1.53884i −0.500000 1.53884i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.587785 1.80902i −0.587785 1.80902i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(-0.5\pi\)
\(270\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(271\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(276\) 0 0
\(277\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(278\) 0 0
\(279\) 0.190983 0.587785i 0.190983 0.587785i
\(280\) 1.11803 1.53884i 1.11803 1.53884i
\(281\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(282\) 0 0
\(283\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.587785 0.809017i 0.587785 0.809017i
\(289\) 0.809017 0.587785i 0.809017 0.587785i
\(290\) 0 0
\(291\) −1.53884 1.11803i −1.53884 1.11803i
\(292\) 0 0
\(293\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(294\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(295\) 1.17557i 1.17557i
\(296\) 0 0
\(297\) −1.11803 0.363271i −1.11803 0.363271i
\(298\) −0.690983 0.951057i −0.690983 0.951057i
\(299\) 0 0
\(300\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(301\) 0 0
\(302\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(303\) −1.11803 0.363271i −1.11803 0.363271i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 2.12663 0.690983i 2.12663 0.690983i
\(309\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(310\) −0.618034 −0.618034
\(311\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i \(0.400000\pi\)
−1.00000 \(1.00000\pi\)
\(314\) 0 0
\(315\) −1.90211 −1.90211
\(316\) −1.30902 0.951057i −1.30902 0.951057i
\(317\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(318\) 1.61803i 1.61803i
\(319\) 0 0
\(320\) −0.951057 0.309017i −0.951057 0.309017i
\(321\) 0.500000 1.53884i 0.500000 1.53884i
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 1.17557i 1.17557i
\(331\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(332\) 0.618034i 0.618034i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 1.53884 1.11803i 1.53884 1.11803i
\(337\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(338\) 0.587785 0.809017i 0.587785 0.809017i
\(339\) 0 0
\(340\) 0 0
\(341\) −0.587785 0.427051i −0.587785 0.427051i
\(342\) 0 0
\(343\) 3.07768i 3.07768i
\(344\) 0 0
\(345\) 0 0
\(346\) 0.190983 0.587785i 0.190983 0.587785i
\(347\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(351\) 0 0
\(352\) −0.690983 0.951057i −0.690983 0.951057i
\(353\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(354\) 0.363271 1.11803i 0.363271 1.11803i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.80902 0.587785i 1.80902 0.587785i
\(359\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(360\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(361\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(362\) 0 0
\(363\) −0.224514 + 0.309017i −0.224514 + 0.309017i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.951057 2.92705i 0.951057 2.92705i
\(372\) −0.587785 0.190983i −0.587785 0.190983i
\(373\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(374\) 0 0
\(375\) −1.00000 −1.00000
\(376\) 0 0
\(377\) 0 0
\(378\) −1.80902 0.587785i −1.80902 0.587785i
\(379\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(380\) 0 0
\(381\) −0.363271 1.11803i −0.363271 1.11803i
\(382\) 0 0
\(383\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(384\) −0.809017 0.587785i −0.809017 0.587785i
\(385\) −0.690983 + 2.12663i −0.690983 + 2.12663i
\(386\) 1.53884 1.11803i 1.53884 1.11803i
\(387\) 0 0
\(388\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(389\) −0.951057 + 0.690983i −0.951057 + 0.690983i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.48990 0.809017i 2.48990 0.809017i
\(393\) 0 0
\(394\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(395\) 1.53884 0.500000i 1.53884 0.500000i
\(396\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(397\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(398\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(399\) 0 0
\(400\) 0.809017 0.587785i 0.809017 0.587785i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(405\) 0.587785 0.809017i 0.587785 0.809017i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.11803 1.53884i 1.11803 1.53884i
\(413\) 1.31433 1.80902i 1.31433 1.80902i
\(414\) 0 0
\(415\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.587785 1.80902i −0.587785 1.80902i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(-0.5\pi\)
\(420\) 1.90211i 1.90211i
\(421\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.61803 −1.61803
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.53884 0.500000i −1.53884 0.500000i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(432\) 1.00000i 1.00000i
\(433\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(434\) −0.951057 0.690983i −0.951057 0.690983i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(440\) 1.17557 1.17557
\(441\) −2.11803 1.53884i −2.11803 1.53884i
\(442\) 0 0
\(443\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(447\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(448\) −1.11803 1.53884i −1.11803 1.53884i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) −0.951057 0.309017i −0.951057 0.309017i
\(451\) 0 0
\(452\) 0 0
\(453\) −0.587785 0.190983i −0.587785 0.190983i
\(454\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(455\) 0 0
\(456\) 0 0
\(457\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.53884 + 1.11803i −1.53884 + 1.11803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(462\) −1.31433 + 1.80902i −1.31433 + 1.80902i
\(463\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(464\) 0 0
\(465\) 0.500000 0.363271i 0.500000 0.363271i
\(466\) 0 0
\(467\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.11803 0.363271i −1.11803 0.363271i
\(473\) 0 0
\(474\) 1.61803 1.61803
\(475\) 0 0
\(476\) 0 0
\(477\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(478\) 0 0
\(479\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(480\) 0.951057 0.309017i 0.951057 0.309017i
\(481\) 0 0
\(482\) 0.618034i 0.618034i
\(483\) 0 0
\(484\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(485\) −0.587785 1.80902i −0.587785 1.80902i
\(486\) 0.809017 0.587785i 0.809017 0.587785i
\(487\) −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i \(0.400000\pi\)
−1.00000 \(1.00000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(491\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −0.690983 0.951057i −0.690983 0.951057i
\(496\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(497\) 0 0
\(498\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 0 0
\(502\) −1.11803 1.53884i −1.11803 1.53884i
\(503\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(504\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(505\) −0.690983 0.951057i −0.690983 0.951057i
\(506\) 0 0
\(507\) 1.00000i 1.00000i
\(508\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(509\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(513\) 0 0
\(514\) 0 0
\(515\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(520\) 0 0
\(521\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(524\) 0 0
\(525\) −1.53884 1.11803i −1.53884 1.11803i
\(526\) 0 0
\(527\) 0 0
\(528\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(529\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(530\) 0.951057 1.30902i 0.951057 1.30902i
\(531\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.30902 0.951057i 1.30902 0.951057i
\(536\) 0 0
\(537\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(538\) 1.11803 1.53884i 1.11803 1.53884i
\(539\) −2.48990 + 1.80902i −2.48990 + 1.80902i
\(540\) −0.809017 0.587785i −0.809017 0.587785i
\(541\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(542\) 0.587785 0.190983i 0.587785 0.190983i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(551\) 0 0
\(552\) 0 0
\(553\) 2.92705 + 0.951057i 2.92705 + 0.951057i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(558\) 0.587785 0.190983i 0.587785 0.190983i
\(559\) 0 0
\(560\) 1.90211 1.90211
\(561\) 0 0
\(562\) 0 0
\(563\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.80902 0.587785i 1.80902 0.587785i
\(568\) 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 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 1.00000
\(577\) 1.11803 + 1.53884i 1.11803 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(578\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(579\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(580\) 0 0
\(581\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(582\) 1.90211i 1.90211i
\(583\) 1.80902 0.587785i 1.80902 0.587785i
\(584\) 0 0
\(585\) 0 0
\(586\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(587\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(588\) −1.53884 + 2.11803i −1.53884 + 2.11803i
\(589\) 0 0
\(590\) 0.951057 0.690983i 0.951057 0.690983i
\(591\) −0.500000 0.363271i −0.500000 0.363271i
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) −0.363271 1.11803i −0.363271 1.11803i
\(595\) 0 0
\(596\) 0.363271 1.11803i 0.363271 1.11803i
\(597\) −1.53884 0.500000i −1.53884 0.500000i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(601\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(605\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(606\) −0.363271 1.11803i −0.363271 1.11803i
\(607\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 1.80902 + 1.31433i 1.80902 + 1.31433i
\(617\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(618\) 1.90211i 1.90211i
\(619\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(620\) −0.363271 0.500000i −0.363271 0.500000i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 0.587785i −0.809017 0.587785i
\(626\) −1.17557 −1.17557
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −1.11803 1.53884i −1.11803 1.53884i
\(631\) −0.618034 1.90211i −0.618034 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(632\) 1.61803i 1.61803i
\(633\) 0 0
\(634\) −1.30902 0.951057i −1.30902 0.951057i
\(635\) 0.363271 1.11803i 0.363271 1.11803i
\(636\) 1.30902 0.951057i 1.30902 0.951057i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −0.309017 0.951057i −0.309017 0.951057i
\(641\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(642\) 1.53884 0.500000i 1.53884 0.500000i
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(648\) −0.587785 0.809017i −0.587785 0.809017i
\(649\) 1.38197 1.38197
\(650\) 0 0
\(651\) 1.17557 1.17557
\(652\) 0 0
\(653\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(660\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(661\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0.500000 0.363271i 0.500000 0.363271i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.363271 1.11803i −0.363271 1.11803i
\(670\) 0 0
\(671\) 0 0
\(672\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(673\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(674\) 1.90211 1.90211
\(675\) 0.951057 0.309017i 0.951057 0.309017i
\(676\) 1.00000 1.00000
\(677\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(678\) 0 0
\(679\) 1.11803 3.44095i 1.11803 3.44095i
\(680\) 0 0
\(681\) −0.500000 1.53884i −0.500000 1.53884i
\(682\) 0.726543i 0.726543i
\(683\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.48990 + 1.80902i −2.48990 + 1.80902i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(692\) 0.587785 0.190983i 0.587785 0.190983i
\(693\) 2.23607i 2.23607i
\(694\) −0.500000 1.53884i −0.500000 1.53884i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.363271 1.11803i 0.363271 1.11803i
\(705\) 0 0
\(706\) 0 0
\(707\) 2.23607i 2.23607i
\(708\) 1.11803 0.363271i 1.11803 0.363271i
\(709\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(710\) 0 0
\(711\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(720\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(721\) −1.11803 + 3.44095i −1.11803 + 3.44095i
\(722\) −0.951057 0.309017i −0.951057 0.309017i
\(723\) −0.363271 0.500000i −0.363271 0.500000i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.381966 −0.381966
\(727\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(728\) 0 0
\(729\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(734\) −1.53884 1.11803i −1.53884 1.11803i
\(735\) −0.809017 2.48990i −0.809017 2.48990i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.92705 0.951057i 2.92705 0.951057i
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −0.190983 0.587785i −0.190983 0.587785i
\(745\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(746\) 0 0
\(747\) −0.587785 0.190983i −0.587785 0.190983i
\(748\) 0 0
\(749\) 3.07768 3.07768
\(750\) −0.587785 0.809017i −0.587785 0.809017i
\(751\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(752\) 0 0
\(753\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(754\) 0 0
\(755\) −0.363271 0.500000i −0.363271 0.500000i
\(756\) −0.587785 1.80902i −0.587785 1.80902i
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(762\) 0.690983 0.951057i 0.690983 0.951057i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000i 1.00000i
\(769\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(770\) −2.12663 + 0.690983i −2.12663 + 0.690983i
\(771\) 0 0
\(772\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(773\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0.618034 0.618034
\(776\) −1.90211 −1.90211
\(777\) 0 0
\(778\) −1.11803 0.363271i −1.11803 0.363271i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(788\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(789\) 0 0
\(790\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(791\) 0 0
\(792\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(793\) 0 0
\(794\) 0 0
\(795\) 1.61803i 1.61803i
\(796\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(797\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.90211i 1.90211i
\(808\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(809\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(810\) 1.00000 1.00000
\(811\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(812\) 0 0
\(813\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.61803i 1.61803i
\(819\) 0 0
\(820\) 0 0
\(821\) 0.587785 1.80902i 0.587785 1.80902i 1.00000i \(-0.5\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(822\) 0 0
\(823\) −0.690983 0.951057i −0.690983 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
−1.00000 \(\pi\)
\(824\) 1.90211 1.90211
\(825\) 1.17557i 1.17557i
\(826\) 2.23607 2.23607
\(827\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(828\) 0 0
\(829\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(830\) 0.618034i 0.618034i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(838\) 1.11803 1.53884i 1.11803 1.53884i
\(839\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(840\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(841\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(846\) 0 0
\(847\) −0.690983 0.224514i −0.690983 0.224514i
\(848\) −0.951057 1.30902i −0.951057 1.30902i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 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
\(856\) −0.500000 1.53884i −0.500000 1.53884i
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(864\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(865\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(866\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(867\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(868\) 1.17557i 1.17557i
\(869\) 0.587785 + 1.80902i 0.587785 + 1.80902i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(874\) 0 0
\(875\) −0.587785 1.80902i −0.587785 1.80902i
\(876\) 0 0
\(877\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(878\) −1.53884 0.500000i −1.53884 0.500000i
\(879\) 0.500000 1.53884i 0.500000 1.53884i
\(880\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(881\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) 2.61803i 2.61803i
\(883\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(884\) 0 0
\(885\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(886\) 0.500000 0.363271i 0.500000 0.363271i
\(887\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(888\) 0 0
\(889\) 1.80902 1.31433i 1.80902 1.31433i
\(890\) 0 0
\(891\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(892\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(893\) 0 0
\(894\) 0.363271 + 1.11803i 0.363271 + 1.11803i
\(895\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(896\) 0.587785 1.80902i 0.587785 1.80902i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.309017 0.951057i −0.309017 0.951057i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −0.190983 0.587785i −0.190983 0.587785i
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(909\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(910\) 0 0
\(911\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) 0 0
\(913\) −0.427051 + 0.587785i −0.427051 + 0.587785i
\(914\) 0.951057 0.690983i 0.951057 0.690983i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.80902 0.587785i −1.80902 0.587785i
\(923\) 0 0
\(924\) −2.23607 −2.23607
\(925\) 0 0
\(926\) 0 0
\(927\) −1.11803 1.53884i −1.11803 1.53884i
\(928\) 0 0
\(929\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(930\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −1.30902 0.951057i −1.30902 0.951057i
\(935\) 0 0
\(936\) 0 0
\(937\) −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i \(0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(938\) 0 0
\(939\) 0.951057 0.690983i 0.951057 0.690983i
\(940\) 0 0
\(941\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −0.363271 1.11803i −0.363271 1.11803i
\(945\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(946\) 0 0
\(947\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(948\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(949\) 0 0
\(950\) 0 0
\(951\) 1.61803 1.61803
\(952\) 0 0
\(953\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(954\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(961\) 0.500000 0.363271i 0.500000 0.363271i
\(962\) 0 0
\(963\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(964\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(965\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(966\) 0 0
\(967\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(968\) 0.381966i 0.381966i
\(969\) 0 0
\(970\) 1.11803 1.53884i 1.11803 1.53884i
\(971\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(972\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(973\) 0 0
\(974\) −1.17557 −1.17557
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −2.48990 + 0.809017i −2.48990 + 0.809017i
\(981\) 0 0
\(982\) 1.90211i 1.90211i
\(983\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(984\) 0 0
\(985\) −0.190983 0.587785i −0.190983 0.587785i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.363271 1.11803i 0.363271 1.11803i
\(991\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(992\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(993\) 0 0
\(994\) 0 0
\(995\) −0.951057 1.30902i −0.951057 1.30902i
\(996\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(997\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 600.1.z.a.389.2 yes 8
3.2 odd 2 inner 600.1.z.a.389.1 8
4.3 odd 2 2400.1.cf.a.689.2 8
5.2 odd 4 3000.1.bj.c.1301.2 8
5.3 odd 4 3000.1.bj.d.1301.1 8
5.4 even 2 3000.1.z.c.1949.1 8
8.3 odd 2 2400.1.cf.a.689.1 8
8.5 even 2 inner 600.1.z.a.389.1 8
12.11 even 2 2400.1.cf.a.689.1 8
15.2 even 4 3000.1.bj.d.1301.2 8
15.8 even 4 3000.1.bj.c.1301.1 8
15.14 odd 2 3000.1.z.c.1949.2 8
24.5 odd 2 CM 600.1.z.a.389.2 yes 8
24.11 even 2 2400.1.cf.a.689.2 8
25.9 even 10 inner 600.1.z.a.509.2 yes 8
25.12 odd 20 3000.1.bj.c.701.2 8
25.13 odd 20 3000.1.bj.d.701.1 8
25.16 even 5 3000.1.z.c.2549.1 8
40.13 odd 4 3000.1.bj.c.1301.1 8
40.29 even 2 3000.1.z.c.1949.2 8
40.37 odd 4 3000.1.bj.d.1301.2 8
75.38 even 20 3000.1.bj.c.701.1 8
75.41 odd 10 3000.1.z.c.2549.2 8
75.59 odd 10 inner 600.1.z.a.509.1 yes 8
75.62 even 20 3000.1.bj.d.701.2 8
100.59 odd 10 2400.1.cf.a.209.2 8
120.29 odd 2 3000.1.z.c.1949.1 8
120.53 even 4 3000.1.bj.d.1301.1 8
120.77 even 4 3000.1.bj.c.1301.2 8
200.13 odd 20 3000.1.bj.c.701.1 8
200.37 odd 20 3000.1.bj.d.701.2 8
200.59 odd 10 2400.1.cf.a.209.1 8
200.109 even 10 inner 600.1.z.a.509.1 yes 8
200.141 even 10 3000.1.z.c.2549.2 8
300.59 even 10 2400.1.cf.a.209.1 8
600.59 even 10 2400.1.cf.a.209.2 8
600.341 odd 10 3000.1.z.c.2549.1 8
600.413 even 20 3000.1.bj.d.701.1 8
600.437 even 20 3000.1.bj.c.701.2 8
600.509 odd 10 inner 600.1.z.a.509.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.1.z.a.389.1 8 3.2 odd 2 inner
600.1.z.a.389.1 8 8.5 even 2 inner
600.1.z.a.389.2 yes 8 1.1 even 1 trivial
600.1.z.a.389.2 yes 8 24.5 odd 2 CM
600.1.z.a.509.1 yes 8 75.59 odd 10 inner
600.1.z.a.509.1 yes 8 200.109 even 10 inner
600.1.z.a.509.2 yes 8 25.9 even 10 inner
600.1.z.a.509.2 yes 8 600.509 odd 10 inner
2400.1.cf.a.209.1 8 200.59 odd 10
2400.1.cf.a.209.1 8 300.59 even 10
2400.1.cf.a.209.2 8 100.59 odd 10
2400.1.cf.a.209.2 8 600.59 even 10
2400.1.cf.a.689.1 8 8.3 odd 2
2400.1.cf.a.689.1 8 12.11 even 2
2400.1.cf.a.689.2 8 4.3 odd 2
2400.1.cf.a.689.2 8 24.11 even 2
3000.1.z.c.1949.1 8 5.4 even 2
3000.1.z.c.1949.1 8 120.29 odd 2
3000.1.z.c.1949.2 8 15.14 odd 2
3000.1.z.c.1949.2 8 40.29 even 2
3000.1.z.c.2549.1 8 25.16 even 5
3000.1.z.c.2549.1 8 600.341 odd 10
3000.1.z.c.2549.2 8 75.41 odd 10
3000.1.z.c.2549.2 8 200.141 even 10
3000.1.bj.c.701.1 8 75.38 even 20
3000.1.bj.c.701.1 8 200.13 odd 20
3000.1.bj.c.701.2 8 25.12 odd 20
3000.1.bj.c.701.2 8 600.437 even 20
3000.1.bj.c.1301.1 8 15.8 even 4
3000.1.bj.c.1301.1 8 40.13 odd 4
3000.1.bj.c.1301.2 8 5.2 odd 4
3000.1.bj.c.1301.2 8 120.77 even 4
3000.1.bj.d.701.1 8 25.13 odd 20
3000.1.bj.d.701.1 8 600.413 even 20
3000.1.bj.d.701.2 8 75.62 even 20
3000.1.bj.d.701.2 8 200.37 odd 20
3000.1.bj.d.1301.1 8 5.3 odd 4
3000.1.bj.d.1301.1 8 120.53 even 4
3000.1.bj.d.1301.2 8 15.2 even 4
3000.1.bj.d.1301.2 8 40.37 odd 4