Properties

Label 2500.1.j.d.1751.1
Level $2500$
Weight $1$
Character 2500.1751
Analytic conductor $1.248$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -20
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,1,Mod(251,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2500.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2500.j (of order \(10\), degree \(4\), not 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: \(1.24766253158\)
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: no (minimal twist has level 500)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.250000.1

Embedding invariants

Embedding label 1751.1
Root \(0.951057 + 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 2500.1751
Dual form 2500.1.j.d.751.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.951057 - 0.309017i) q^{2} +(-0.951057 + 1.30902i) q^{3} +(0.809017 + 0.587785i) q^{4} +(1.30902 - 0.951057i) q^{6} -0.618034i q^{7} +(-0.587785 - 0.809017i) q^{8} +(-0.500000 - 1.53884i) q^{9} +O(q^{10})\) \(q+(-0.951057 - 0.309017i) q^{2} +(-0.951057 + 1.30902i) q^{3} +(0.809017 + 0.587785i) q^{4} +(1.30902 - 0.951057i) q^{6} -0.618034i q^{7} +(-0.587785 - 0.809017i) q^{8} +(-0.500000 - 1.53884i) q^{9} +(-1.53884 + 0.500000i) q^{12} +(-0.190983 + 0.587785i) q^{14} +(0.309017 + 0.951057i) q^{16} +1.61803i q^{18} +(0.809017 + 0.587785i) q^{21} +(0.587785 + 0.190983i) q^{23} +1.61803 q^{24} +(0.951057 + 0.309017i) q^{27} +(0.363271 - 0.500000i) q^{28} +(-1.30902 - 0.951057i) q^{29} -1.00000i q^{32} +(0.500000 - 1.53884i) q^{36} +(-0.500000 - 1.53884i) q^{41} +(-0.587785 - 0.809017i) q^{42} -1.61803i q^{43} +(-0.500000 - 0.363271i) q^{46} +(0.951057 - 1.30902i) q^{47} +(-1.53884 - 0.500000i) q^{48} +0.618034 q^{49} +(-0.809017 - 0.587785i) q^{54} +(-0.500000 + 0.363271i) q^{56} +(0.951057 + 1.30902i) q^{58} +(0.190983 - 0.587785i) q^{61} +(-0.951057 + 0.309017i) q^{63} +(-0.309017 + 0.951057i) q^{64} +(1.17557 + 1.61803i) q^{67} +(-0.809017 + 0.587785i) q^{69} +(-0.951057 + 1.30902i) q^{72} +1.61803i q^{82} +(-0.363271 - 0.500000i) q^{83} +(0.309017 + 0.951057i) q^{84} +(-0.500000 + 1.53884i) q^{86} +(2.48990 - 0.809017i) q^{87} +(-0.190983 + 0.587785i) q^{89} +(0.363271 + 0.500000i) q^{92} +(-1.30902 + 0.951057i) q^{94} +(1.30902 + 0.951057i) q^{96} +(-0.587785 - 0.190983i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 6 q^{6} - 4 q^{9} - 6 q^{14} - 2 q^{16} + 2 q^{21} + 4 q^{24} - 6 q^{29} + 4 q^{36} - 4 q^{41} - 4 q^{46} - 4 q^{49} - 2 q^{54} - 4 q^{56} + 6 q^{61} + 2 q^{64} - 2 q^{69} - 2 q^{84} - 4 q^{86} - 6 q^{89} - 6 q^{94} + 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times\).

\(n\) \(1251\) \(1877\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.951057 0.309017i −0.951057 0.309017i
\(3\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(4\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(5\) 0 0
\(6\) 1.30902 0.951057i 1.30902 0.951057i
\(7\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(8\) −0.587785 0.809017i −0.587785 0.809017i
\(9\) −0.500000 1.53884i −0.500000 1.53884i
\(10\) 0 0
\(11\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(12\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(13\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(15\) 0 0
\(16\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(17\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 1.61803i 1.61803i
\(19\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) 0 0
\(21\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(22\) 0 0
\(23\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.61803 1.61803
\(25\) 0 0
\(26\) 0 0
\(27\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(28\) 0.363271 0.500000i 0.363271 0.500000i
\(29\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.500000 1.53884i 0.500000 1.53884i
\(37\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(42\) −0.587785 0.809017i −0.587785 0.809017i
\(43\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.500000 0.363271i −0.500000 0.363271i
\(47\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(48\) −1.53884 0.500000i −1.53884 0.500000i
\(49\) 0.618034 0.618034
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(54\) −0.809017 0.587785i −0.809017 0.587785i
\(55\) 0 0
\(56\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(57\) 0 0
\(58\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(59\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(64\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.17557 + 1.61803i 1.17557 + 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(68\) 0 0
\(69\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(70\) 0 0
\(71\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(72\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(73\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.61803i 1.61803i
\(83\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(84\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(85\) 0 0
\(86\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(87\) 2.48990 0.809017i 2.48990 0.809017i
\(88\) 0 0
\(89\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(93\) 0 0
\(94\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(95\) 0 0
\(96\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(97\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(98\) −0.587785 0.190983i −0.587785 0.190983i
\(99\) 0 0
\(100\) 0 0
\(101\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) 1.17557 1.61803i 1.17557 1.61803i 0.587785 0.809017i \(-0.300000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(109\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.587785 0.190983i 0.587785 0.190983i
\(113\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.500000 1.53884i −0.500000 1.53884i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.809017 0.587785i −0.809017 0.587785i
\(122\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(123\) 2.48990 + 0.809017i 2.48990 + 0.809017i
\(124\) 0 0
\(125\) 0 0
\(126\) 1.00000 1.00000
\(127\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(128\) 0.587785 0.809017i 0.587785 0.809017i
\(129\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(130\) 0 0
\(131\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.618034 1.90211i −0.618034 1.90211i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(138\) 0.951057 0.309017i 0.951057 0.309017i
\(139\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(140\) 0 0
\(141\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.30902 0.951057i 1.30902 0.951057i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(148\) 0 0
\(149\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.118034 0.363271i 0.118034 0.363271i
\(162\) 0 0
\(163\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(164\) 0.500000 1.53884i 0.500000 1.53884i
\(165\) 0 0
\(166\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(167\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(168\) 1.00000i 1.00000i
\(169\) 0.809017 0.587785i 0.809017 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.951057 1.30902i 0.951057 1.30902i
\(173\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(174\) −2.61803 −2.61803
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.363271 0.500000i 0.363271 0.500000i
\(179\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(180\) 0 0
\(181\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(182\) 0 0
\(183\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(184\) −0.190983 0.587785i −0.190983 0.587785i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 1.53884 0.500000i 1.53884 0.500000i
\(189\) 0.190983 0.587785i 0.190983 0.587785i
\(190\) 0 0
\(191\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(192\) −0.951057 1.30902i −0.951057 1.30902i
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(197\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −3.23607 −3.23607
\(202\) −0.587785 0.190983i −0.587785 0.190983i
\(203\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(204\) 0 0
\(205\) 0 0
\(206\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(207\) 1.00000i 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.500000 1.53884i 0.500000 1.53884i
\(215\) 0 0
\(216\) −0.309017 0.951057i −0.309017 0.951057i
\(217\) 0 0
\(218\) 1.61803i 1.61803i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(224\) −0.618034 −0.618034
\(225\) 0 0
\(226\) 0 0
\(227\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) 0 0
\(229\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.61803i 1.61803i
\(233\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(240\) 0 0
\(241\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(243\) 1.00000i 1.00000i
\(244\) 0.500000 0.363271i 0.500000 0.363271i
\(245\) 0 0
\(246\) −2.11803 1.53884i −2.11803 1.53884i
\(247\) 0 0
\(248\) 0 0
\(249\) 1.00000 1.00000
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −0.951057 0.309017i −0.951057 0.309017i
\(253\) 0 0
\(254\) −1.30902 0.951057i −1.30902 0.951057i
\(255\) 0 0
\(256\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) −1.53884 2.11803i −1.53884 2.11803i
\(259\) 0 0
\(260\) 0 0
\(261\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(262\) 0 0
\(263\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.587785 0.809017i −0.587785 0.809017i
\(268\) 2.00000i 2.00000i
\(269\) 1.61803 1.17557i 1.61803 1.17557i 0.809017 0.587785i \(-0.200000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −1.00000 −1.00000
\(277\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(282\) 2.61803i 2.61803i
\(283\) −1.17557 1.61803i −1.17557 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(288\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(289\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0.809017 0.587785i 0.809017 0.587785i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −1.00000
\(302\) 0 0
\(303\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(308\) 0 0
\(309\) 1.00000 + 3.07768i 1.00000 + 3.07768i
\(310\) 0 0
\(311\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(312\) 0 0
\(313\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.11803 1.53884i −2.11803 1.53884i
\(322\) −0.224514 + 0.309017i −0.224514 + 0.309017i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0.618034 0.618034
\(327\) −2.48990 0.809017i −2.48990 0.809017i
\(328\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(329\) −0.809017 0.587785i −0.809017 0.587785i
\(330\) 0 0
\(331\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) 0.618034i 0.618034i
\(333\) 0 0
\(334\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(335\) 0 0
\(336\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(337\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(338\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(348\) 2.48990 + 0.809017i 2.48990 + 0.809017i
\(349\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(360\) 0 0
\(361\) 0.309017 0.951057i 0.309017 0.951057i
\(362\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(363\) 1.53884 0.500000i 1.53884 0.500000i
\(364\) 0 0
\(365\) 0 0
\(366\) −0.309017 0.951057i −0.309017 0.951057i
\(367\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(368\) 0.618034i 0.618034i
\(369\) −2.11803 + 1.53884i −2.11803 + 1.53884i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.61803 −1.61803
\(377\) 0 0
\(378\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(379\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(380\) 0 0
\(381\) −2.11803 + 1.53884i −2.11803 + 1.53884i
\(382\) 0 0
\(383\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(384\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(385\) 0 0
\(386\) 0 0
\(387\) −2.48990 + 0.809017i −2.48990 + 0.809017i
\(388\) 0 0
\(389\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.363271 0.500000i −0.363271 0.500000i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(402\) 3.07768 + 1.00000i 3.07768 + 1.00000i
\(403\) 0 0
\(404\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(405\) 0 0
\(406\) 0.809017 0.587785i 0.809017 0.587785i
\(407\) 0 0
\(408\) 0 0
\(409\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.90211 0.618034i 1.90211 0.618034i
\(413\) 0 0
\(414\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) −2.48990 0.809017i −2.48990 0.809017i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.363271 0.118034i −0.363271 0.118034i
\(428\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) 1.00000i 1.00000i
\(433\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(440\) 0 0
\(441\) −0.309017 0.951057i −0.309017 0.951057i
\(442\) 0 0
\(443\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(447\) 0.587785 0.809017i 0.587785 0.809017i
\(448\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(449\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) −0.363271 0.500000i −0.363271 0.500000i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(462\) 0 0
\(463\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(464\) 0.500000 1.53884i 0.500000 1.53884i
\(465\) 0 0
\(466\) 0 0
\(467\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(468\) 0 0
\(469\) 1.00000 0.726543i 1.00000 0.726543i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.618034i 0.618034i
\(483\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(484\) −0.309017 0.951057i −0.309017 0.951057i
\(485\) 0 0
\(486\) 0.309017 0.951057i 0.309017 0.951057i
\(487\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(488\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(489\) 0.309017 0.951057i 0.309017 0.951057i
\(490\) 0 0
\(491\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(492\) 1.53884 + 2.11803i 1.53884 + 2.11803i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.951057 0.309017i −0.951057 0.309017i
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 2.61803 2.61803
\(502\) 0 0
\(503\) 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(504\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(505\) 0 0
\(506\) 0 0
\(507\) 1.61803i 1.61803i
\(508\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(509\) −0.618034 1.90211i −0.618034 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.951057 0.309017i 0.951057 0.309017i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 1.53884 2.11803i 1.53884 2.11803i
\(523\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.61803 −1.61803
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.363271i −0.500000 0.363271i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(535\) 0 0
\(536\) 0.618034 1.90211i 0.618034 1.90211i
\(537\) 0 0
\(538\) −1.90211 + 0.618034i −1.90211 + 0.618034i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(542\) 0 0
\(543\) 2.61803i 2.61803i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(548\) 0 0
\(549\) −1.00000 −1.00000
\(550\) 0 0
\(551\) 0 0
\(552\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.587785 0.190983i 0.587785 0.190983i
\(563\) −1.90211 + 0.618034i −1.90211 + 0.618034i −0.951057 + 0.309017i \(0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(564\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(565\) 0 0
\(566\) 0.618034 + 1.90211i 0.618034 + 1.90211i
\(567\) 0 0
\(568\) 0 0
\(569\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.00000 1.00000
\(575\) 0 0
\(576\) 1.61803 1.61803
\(577\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(578\) 0.587785 0.809017i 0.587785 0.809017i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.90211 0.618034i 1.90211 0.618034i 0.951057 0.309017i \(-0.100000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(588\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.500000 0.363271i −0.500000 0.363271i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(603\) 1.90211 2.61803i 1.90211 2.61803i
\(604\) 0 0
\(605\) 0 0
\(606\) 0.809017 0.587785i 0.809017 0.587785i
\(607\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(608\) 0 0
\(609\) −0.500000 1.53884i −0.500000 1.53884i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(614\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(618\) 3.23607i 3.23607i
\(619\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(622\) 0 0
\(623\) 0.363271 + 0.118034i 0.363271 + 0.118034i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(642\) 1.53884 + 2.11803i 1.53884 + 2.11803i
\(643\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(644\) 0.309017 0.224514i 0.309017 0.224514i
\(645\) 0 0
\(646\) 0 0
\(647\) −1.17557 + 1.61803i −1.17557 + 1.61803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.587785 0.190983i −0.587785 0.190983i
\(653\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(654\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(655\) 0 0
\(656\) 1.30902 0.951057i 1.30902 0.951057i
\(657\) 0 0
\(658\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(659\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 0 0
\(661\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(665\) 0 0
\(666\) 0 0
\(667\) −0.587785 0.809017i −0.587785 0.809017i
\(668\) 1.61803i 1.61803i
\(669\) 2.11803 1.53884i 2.11803 1.53884i
\(670\) 0 0
\(671\) 0 0
\(672\) 0.587785 0.809017i 0.587785 0.809017i
\(673\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.809017 0.587785i 0.809017 0.587785i
\(682\) 0 0
\(683\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(687\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(688\) 1.53884 0.500000i 1.53884 0.500000i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.500000 0.363271i 0.500000 0.363271i
\(695\) 0 0
\(696\) −2.11803 1.53884i −2.11803 1.53884i
\(697\) 0 0
\(698\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.381966i 0.381966i
\(708\) 0 0
\(709\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.587785 0.190983i 0.587785 0.190983i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(720\) 0 0
\(721\) −1.00000 0.726543i −1.00000 0.726543i
\(722\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(723\) −0.951057 0.309017i −0.951057 0.309017i
\(724\) 1.61803 1.61803
\(725\) 0 0
\(726\) −1.61803 −1.61803
\(727\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(728\) 0 0
\(729\) −1.30902 0.951057i −1.30902 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) 1.00000i 1.00000i
\(733\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(734\) −0.190983 0.587785i −0.190983 0.587785i
\(735\) 0 0
\(736\) 0.190983 0.587785i 0.190983 0.587785i
\(737\) 0 0
\(738\) 2.48990 0.809017i 2.48990 0.809017i
\(739\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(748\) 0 0
\(749\) 1.00000 1.00000
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.500000 0.363271i 0.500000 0.363271i
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(762\) 2.48990 0.809017i 2.48990 0.809017i
\(763\) 0.951057 0.309017i 0.951057 0.309017i
\(764\) 0 0
\(765\) 0 0
\(766\) −0.500000 1.53884i −0.500000 1.53884i
\(767\) 0 0
\(768\) 1.61803i 1.61803i
\(769\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(774\) 2.61803 2.61803
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.951057 1.30902i −0.951057 1.30902i
\(784\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(788\) 0 0
\(789\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.00000 1.00000
\(802\) −0.587785 0.190983i −0.587785 0.190983i
\(803\) 0 0
\(804\) −2.61803 1.90211i −2.61803 1.90211i
\(805\) 0 0
\(806\) 0 0
\(807\) 3.23607i 3.23607i
\(808\) −0.363271 0.500000i −0.363271 0.500000i
\(809\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(812\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.618034i 0.618034i
\(819\) 0 0
\(820\) 0 0
\(821\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(822\) 0 0
\(823\) 1.90211 + 0.618034i 1.90211 + 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(824\) −2.00000 −2.00000
\(825\) 0 0
\(826\) 0 0
\(827\) −1.90211 0.618034i −1.90211 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 0.309017i \(-0.900000\pi\)
\(828\) 0.587785 0.809017i 0.587785 0.809017i
\(829\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(830\) 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 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(840\) 0 0
\(841\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(842\) −0.951057 1.30902i −0.951057 1.30902i
\(843\) 1.00000i 1.00000i
\(844\) 0 0
\(845\) 0 0
\(846\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(847\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(848\) 0 0
\(849\) 3.23607 3.23607
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(854\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(855\) 0 0
\(856\) 1.30902 0.951057i 1.30902 0.951057i
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(860\) 0 0
\(861\) 0.500000 1.53884i 0.500000 1.53884i
\(862\) 0 0
\(863\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(864\) 0.309017 0.951057i 0.309017 0.951057i
\(865\) 0 0
\(866\) 0 0
\(867\) −0.951057 1.30902i −0.951057 1.30902i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.951057 1.30902i 0.951057 1.30902i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) 1.00000i 1.00000i
\(883\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.190983 0.587785i 0.190983 0.587785i
\(887\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(888\) 0 0
\(889\) 0.309017 0.951057i 0.309017 0.951057i
\(890\) 0 0
\(891\) 0 0
\(892\) −0.951057 1.30902i −0.951057 1.30902i
\(893\) 0 0
\(894\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(895\) 0 0
\(896\) −0.500000 0.363271i −0.500000 0.363271i
\(897\) 0 0
\(898\) 1.90211 + 0.618034i 1.90211 + 0.618034i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0.951057 1.30902i 0.951057 1.30902i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(908\) −0.363271 0.500000i −0.363271 0.500000i
\(909\) −0.309017 0.951057i −0.309017 0.951057i
\(910\) 0 0
\(911\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(920\) 0 0
\(921\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(922\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.618034 0.618034
\(927\) −3.07768 1.00000i −3.07768 1.00000i
\(928\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(929\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −0.190983 0.587785i −0.190983 0.587785i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(938\) −1.17557 + 0.381966i −1.17557 + 0.381966i
\(939\) 0 0
\(940\) 0 0
\(941\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(942\) 0 0
\(943\) 1.00000i 1.00000i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.309017 0.951057i 0.309017 0.951057i
\(962\) 0 0
\(963\) 2.48990 0.809017i 2.48990 0.809017i
\(964\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(965\) 0 0
\(966\) −0.190983 0.587785i −0.190983 0.587785i
\(967\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(968\) 1.00000i 1.00000i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(972\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(973\) 0 0
\(974\) 1.61803 1.61803
\(975\) 0 0
\(976\) 0.618034 0.618034
\(977\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(978\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(979\) 0 0
\(980\) 0 0
\(981\) 2.11803 1.53884i 2.11803 1.53884i
\(982\) 0 0
\(983\) −1.17557 1.61803i −1.17557 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(984\) −0.809017 2.48990i −0.809017 2.48990i
\(985\) 0 0
\(986\) 0 0
\(987\) 1.53884 0.500000i 1.53884 0.500000i
\(988\) 0 0
\(989\) 0.309017 0.951057i 0.309017 0.951057i
\(990\) 0 0
\(991\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(997\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(998\) 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 2500.1.j.d.1751.1 8
4.3 odd 2 inner 2500.1.j.d.1751.2 8
5.2 odd 4 2500.1.h.b.1999.1 4
5.3 odd 4 2500.1.h.c.1999.1 4
5.4 even 2 inner 2500.1.j.d.1751.2 8
20.3 even 4 2500.1.h.b.1999.1 4
20.7 even 4 2500.1.h.c.1999.1 4
20.19 odd 2 CM 2500.1.j.d.1751.1 8
25.2 odd 20 2500.1.h.a.1499.1 4
25.3 odd 20 2500.1.h.d.999.1 4
25.4 even 10 2500.1.j.c.251.1 8
25.6 even 5 inner 2500.1.j.d.751.2 8
25.8 odd 20 2500.1.h.c.499.1 4
25.9 even 10 500.1.b.a.251.4 yes 4
25.11 even 5 2500.1.j.c.2251.1 8
25.12 odd 20 500.1.d.b.499.1 2
25.13 odd 20 500.1.d.a.499.2 2
25.14 even 10 2500.1.j.c.2251.2 8
25.16 even 5 500.1.b.a.251.1 4
25.17 odd 20 2500.1.h.b.499.1 4
25.19 even 10 inner 2500.1.j.d.751.1 8
25.21 even 5 2500.1.j.c.251.2 8
25.22 odd 20 2500.1.h.a.999.1 4
25.23 odd 20 2500.1.h.d.1499.1 4
100.3 even 20 2500.1.h.a.999.1 4
100.11 odd 10 2500.1.j.c.2251.2 8
100.19 odd 10 inner 2500.1.j.d.751.2 8
100.23 even 20 2500.1.h.a.1499.1 4
100.27 even 20 2500.1.h.d.1499.1 4
100.31 odd 10 inner 2500.1.j.d.751.1 8
100.39 odd 10 2500.1.j.c.2251.1 8
100.47 even 20 2500.1.h.d.999.1 4
100.59 odd 10 500.1.b.a.251.1 4
100.63 even 20 500.1.d.b.499.1 2
100.67 even 20 2500.1.h.c.499.1 4
100.71 odd 10 2500.1.j.c.251.1 8
100.79 odd 10 2500.1.j.c.251.2 8
100.83 even 20 2500.1.h.b.499.1 4
100.87 even 20 500.1.d.a.499.2 2
100.91 odd 10 500.1.b.a.251.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.1.b.a.251.1 4 25.16 even 5
500.1.b.a.251.1 4 100.59 odd 10
500.1.b.a.251.4 yes 4 25.9 even 10
500.1.b.a.251.4 yes 4 100.91 odd 10
500.1.d.a.499.2 2 25.13 odd 20
500.1.d.a.499.2 2 100.87 even 20
500.1.d.b.499.1 2 25.12 odd 20
500.1.d.b.499.1 2 100.63 even 20
2500.1.h.a.999.1 4 25.22 odd 20
2500.1.h.a.999.1 4 100.3 even 20
2500.1.h.a.1499.1 4 25.2 odd 20
2500.1.h.a.1499.1 4 100.23 even 20
2500.1.h.b.499.1 4 25.17 odd 20
2500.1.h.b.499.1 4 100.83 even 20
2500.1.h.b.1999.1 4 5.2 odd 4
2500.1.h.b.1999.1 4 20.3 even 4
2500.1.h.c.499.1 4 25.8 odd 20
2500.1.h.c.499.1 4 100.67 even 20
2500.1.h.c.1999.1 4 5.3 odd 4
2500.1.h.c.1999.1 4 20.7 even 4
2500.1.h.d.999.1 4 25.3 odd 20
2500.1.h.d.999.1 4 100.47 even 20
2500.1.h.d.1499.1 4 25.23 odd 20
2500.1.h.d.1499.1 4 100.27 even 20
2500.1.j.c.251.1 8 25.4 even 10
2500.1.j.c.251.1 8 100.71 odd 10
2500.1.j.c.251.2 8 25.21 even 5
2500.1.j.c.251.2 8 100.79 odd 10
2500.1.j.c.2251.1 8 25.11 even 5
2500.1.j.c.2251.1 8 100.39 odd 10
2500.1.j.c.2251.2 8 25.14 even 10
2500.1.j.c.2251.2 8 100.11 odd 10
2500.1.j.d.751.1 8 25.19 even 10 inner
2500.1.j.d.751.1 8 100.31 odd 10 inner
2500.1.j.d.751.2 8 25.6 even 5 inner
2500.1.j.d.751.2 8 100.19 odd 10 inner
2500.1.j.d.1751.1 8 1.1 even 1 trivial
2500.1.j.d.1751.1 8 20.19 odd 2 CM
2500.1.j.d.1751.2 8 4.3 odd 2 inner
2500.1.j.d.1751.2 8 5.4 even 2 inner