Properties

Label 475.1.m.a.284.1
Level $475$
Weight $1$
Character 475.284
Analytic conductor $0.237$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,1,Mod(94,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.94");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 475.m (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: \(0.237055881001\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.1889113616943359375.4

Embedding invariants

Embedding label 284.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 475.284
Dual form 475.1.m.a.189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.309017 - 0.951057i) q^{4} +(-0.809017 + 0.587785i) q^{5} -1.17557i q^{7} +(0.809017 - 0.587785i) q^{9} +O(q^{10})\) \(q+(-0.309017 - 0.951057i) q^{4} +(-0.809017 + 0.587785i) q^{5} -1.17557i q^{7} +(0.809017 - 0.587785i) q^{9} +(-1.30902 - 0.951057i) q^{11} +(-0.809017 + 0.587785i) q^{16} +(1.80902 + 0.587785i) q^{17} +(0.309017 - 0.951057i) q^{19} +(0.809017 + 0.587785i) q^{20} +(-0.690983 + 0.951057i) q^{23} +(0.309017 - 0.951057i) q^{25} +(-1.11803 + 0.363271i) q^{28} +(0.690983 + 0.951057i) q^{35} +(-0.809017 - 0.587785i) q^{36} +1.90211i q^{43} +(-0.500000 + 1.53884i) q^{44} +(-0.309017 + 0.951057i) q^{45} -0.381966 q^{49} +1.61803 q^{55} +(0.500000 + 0.363271i) q^{61} +(-0.690983 - 0.951057i) q^{63} +(0.809017 + 0.587785i) q^{64} -1.90211i q^{68} -1.00000 q^{76} +(-1.11803 + 1.53884i) q^{77} +(0.309017 - 0.951057i) q^{80} +(0.309017 - 0.951057i) q^{81} +(1.80902 + 0.587785i) q^{83} +(-1.80902 + 0.587785i) q^{85} +(1.11803 + 0.363271i) q^{92} +(0.309017 + 0.951057i) q^{95} -1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{4} - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{4} - q^{5} + q^{9} - 3 q^{11} - q^{16} + 5 q^{17} - q^{19} + q^{20} - 5 q^{23} - q^{25} + 5 q^{35} - q^{36} - 2 q^{44} + q^{45} - 6 q^{49} + 2 q^{55} + 2 q^{61} - 5 q^{63} + q^{64} - 4 q^{76} - q^{80} - q^{81} + 5 q^{83} - 5 q^{85} - q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(-1\)

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 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(3\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(4\) −0.309017 0.951057i −0.309017 0.951057i
\(5\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(6\) 0 0
\(7\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(8\) 0 0
\(9\) 0.809017 0.587785i 0.809017 0.587785i
\(10\) 0 0
\(11\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(17\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(18\) 0 0
\(19\) 0.309017 0.951057i 0.309017 0.951057i
\(20\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(21\) 0 0
\(22\) 0 0
\(23\) −0.690983 + 0.951057i −0.690983 + 0.951057i 0.309017 + 0.951057i \(0.400000\pi\)
−1.00000 \(1.00000\pi\)
\(24\) 0 0
\(25\) 0.309017 0.951057i 0.309017 0.951057i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(29\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(36\) −0.809017 0.587785i −0.809017 0.587785i
\(37\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(42\) 0 0
\(43\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(44\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(45\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(46\) 0 0
\(47\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) 0 0
\(49\) −0.381966 −0.381966
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(54\) 0 0
\(55\) 1.61803 1.61803
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(62\) 0 0
\(63\) −0.690983 0.951057i −0.690983 0.951057i
\(64\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(68\) 1.90211i 1.90211i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(78\) 0 0
\(79\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(80\) 0.309017 0.951057i 0.309017 0.951057i
\(81\) 0.309017 0.951057i 0.309017 0.951057i
\(82\) 0 0
\(83\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(84\) 0 0
\(85\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(93\) 0 0
\(94\) 0 0
\(95\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(96\) 0 0
\(97\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(98\) 0 0
\(99\) −1.61803 −1.61803
\(100\) −1.00000 −1.00000
\(101\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(102\) 0 0
\(103\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(113\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(114\) 0 0
\(115\) 1.17557i 1.17557i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.690983 2.12663i 0.690983 2.12663i
\(120\) 0 0
\(121\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(126\) 0 0
\(127\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(132\) 0 0
\(133\) −1.11803 0.363271i −1.11803 0.363271i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.690983 0.951057i −0.690983 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(140\) 0.690983 0.951057i 0.690983 0.951057i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.80902 0.587785i 1.80902 0.587785i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.90211i 1.90211i −0.309017 0.951057i \(-0.600000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.11803 + 0.812299i 1.11803 + 0.812299i
\(162\) 0 0
\(163\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(168\) 0 0
\(169\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(170\) 0 0
\(171\) −0.309017 0.951057i −0.309017 0.951057i
\(172\) 1.80902 0.587785i 1.80902 0.587785i
\(173\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(174\) 0 0
\(175\) −1.11803 0.363271i −1.11803 0.363271i
\(176\) 1.61803 1.61803
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(180\) 1.00000 1.00000
\(181\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.80902 2.48990i −1.80902 2.48990i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.118034 + 0.363271i 0.118034 + 0.363271i
\(197\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.17557i 1.17557i
\(208\) 0 0
\(209\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(210\) 0 0
\(211\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.11803 1.53884i −1.11803 1.53884i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −0.500000 1.53884i −0.500000 1.53884i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(224\) 0 0
\(225\) −0.309017 0.951057i −0.309017 0.951057i
\(226\) 0 0
\(227\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) 0 0
\(229\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) 0 0
\(241\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.190983 0.587785i 0.190983 0.587785i
\(245\) 0.309017 0.224514i 0.309017 0.224514i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(252\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(253\) 1.80902 0.587785i 1.80902 0.587785i
\(254\) 0 0
\(255\) 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
\(263\) 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 \(0\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(270\) 0 0
\(271\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(273\) 0 0
\(274\) 0 0
\(275\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(276\) 0 0
\(277\) −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i \(0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(282\) 0 0
\(283\) −1.11803 0.363271i −1.11803 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.23607 2.23607
\(302\) 0 0
\(303\) 0 0
\(304\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(305\) −0.618034 −0.618034
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(309\) 0 0
\(310\) 0 0
\(311\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(312\) 0 0
\(313\) 1.11803 + 1.53884i 1.11803 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(314\) 0 0
\(315\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(316\) 0 0
\(317\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 1.11803 1.53884i 1.11803 1.53884i
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(332\) 1.90211i 1.90211i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(341\) 0 0
\(342\) 0 0
\(343\) 0.726543i 0.726543i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(348\) 0 0
\(349\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.80902 + 0.587785i −1.80902 + 0.587785i −0.809017 + 0.587785i \(0.800000\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −0.809017 0.587785i −0.809017 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(368\) 1.17557i 1.17557i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(380\) 0.809017 0.587785i 0.809017 0.587785i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(384\) 0 0
\(385\) 1.90211i 1.90211i
\(386\) 0 0
\(387\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(388\) 0 0
\(389\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(390\) 0 0
\(391\) −1.80902 + 1.31433i −1.80902 + 1.31433i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(397\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(405\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.11803 1.53884i 1.11803 1.53884i
\(426\) 0 0
\(427\) 0.427051 0.587785i 0.427051 0.587785i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(438\) 0 0
\(439\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(440\) 0 0
\(441\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(442\) 0 0
\(443\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.690983 0.951057i 0.690983 0.951057i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(461\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(462\) 0 0
\(463\) −0.690983 0.951057i −0.690983 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.80902 2.48990i 1.80902 2.48990i
\(474\) 0 0
\(475\) −0.809017 0.587785i −0.809017 0.587785i
\(476\) −2.23607 −2.23607
\(477\) 0 0
\(478\) 0 0
\(479\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.30902 0.951057i 1.30902 0.951057i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.30902 0.951057i 1.30902 0.951057i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(500\) 0.809017 0.587785i 0.809017 0.587785i
\(501\) 0 0
\(502\) 0 0
\(503\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(504\) 0 0
\(505\) 0.500000 0.363271i 0.500000 0.363271i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 0 0
\(523\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(524\) −0.618034 −0.618034
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.118034 0.363271i −0.118034 0.363271i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.17557i 1.17557i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(540\) 0 0
\(541\) 1.61803 1.17557i 1.61803 1.17557i 0.809017 0.587785i \(-0.200000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(548\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(549\) 0.618034 0.618034
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(557\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.11803 0.363271i −1.11803 0.363271i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.11803 0.363271i −1.11803 0.363271i
\(568\) 0 0
\(569\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(570\) 0 0
\(571\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(576\) 1.00000 1.00000
\(577\) 0.690983 0.951057i 0.690983 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
1.00000 \(0\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.690983 2.12663i 0.690983 2.12663i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.17557i 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(594\) 0 0
\(595\) 0.690983 + 2.12663i 0.690983 + 2.12663i
\(596\) −0.500000 1.53884i −0.500000 1.53884i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.30902 0.951057i −1.30902 0.951057i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.11803 1.53884i −1.11803 1.53884i
\(613\) 0.690983 + 0.951057i 0.690983 + 0.951057i 1.00000 \(0\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 0.587785i −0.809017 0.587785i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(632\) 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 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(642\) 0 0
\(643\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(644\) 0.427051 1.31433i 0.427051 1.31433i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(654\) 0 0
\(655\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.11803 0.363271i 1.11803 0.363271i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.309017 0.951057i −0.309017 0.951057i
\(672\) 0 0
\(673\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(684\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(685\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(686\) 0 0
\(687\) 0 0
\(688\) −1.11803 1.53884i −1.11803 1.53884i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.500000 0.363271i 0.500000 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(692\) 0 0
\(693\) 1.90211i 1.90211i
\(694\) 0 0
\(695\) 1.61803 1.61803
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.17557i 1.17557i
\(701\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.500000 1.53884i −0.500000 1.53884i
\(705\) 0 0
\(706\) 0 0
\(707\) 0.726543i 0.726543i
\(708\) 0 0
\(709\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(720\) −0.309017 0.951057i −0.309017 0.951057i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(728\) 0 0
\(729\) −0.309017 0.951057i −0.309017 0.951057i
\(730\) 0 0
\(731\) −1.11803 + 3.44095i −1.11803 + 3.44095i
\(732\) 0 0
\(733\) 1.11803 + 0.363271i 1.11803 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(746\) 0 0
\(747\) 1.80902 0.587785i 1.80902 0.587785i
\(748\) −1.80902 + 2.48990i −1.80902 + 2.48990i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(765\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.309017 0.224514i 0.309017 0.224514i
\(785\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(786\) 0 0
\(787\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(788\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.618034 1.90211i −0.618034 1.90211i
\(797\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.38197 −1.38197
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(822\) 0 0
\(823\) −1.11803 + 1.53884i −1.11803 + 1.53884i −0.309017 + 0.951057i \(0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(828\) 1.11803 0.363271i 1.11803 0.363271i
\(829\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.690983 0.224514i −0.690983 0.224514i
\(834\) 0 0
\(835\) 0 0
\(836\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(840\) 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
\(847\) 1.80902 0.587785i 1.80902 0.587785i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(854\) 0 0
\(855\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(860\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.11803 0.363271i 1.11803 0.363271i
\(876\) 0 0
\(877\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(881\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(910\) 0 0
\(911\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(912\) 0 0
\(913\) −1.80902 2.48990i −1.80902 2.48990i
\(914\) 0 0
\(915\) 0 0
\(916\) 1.30902 0.951057i 1.30902 0.951057i
\(917\) −0.690983 0.224514i −0.690983 0.224514i
\(918\) 0 0
\(919\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(930\) 0 0
\(931\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(932\) 1.90211i 1.90211i
\(933\) 0 0
\(934\) 0 0
\(935\) 2.92705 + 0.951057i 2.92705 + 0.951057i
\(936\) 0 0
\(937\) 1.11803 + 1.53884i 1.11803 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(954\) 0 0
\(955\) 0.190983 0.587785i 0.190983 0.587785i
\(956\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(957\) 0 0
\(958\) 0 0
\(959\) −1.11803 + 0.812299i −1.11803 + 0.812299i
\(960\) 0 0
\(961\) −0.809017 0.587785i −0.809017 0.587785i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(972\) 0 0
\(973\) −1.11803 + 1.53884i −1.11803 + 1.53884i
\(974\) 0 0
\(975\) 0 0
\(976\) −0.618034 −0.618034
\(977\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.309017 0.224514i −0.309017 0.224514i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(984\) 0 0
\(985\) 1.11803 1.53884i 1.11803 1.53884i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.80902 1.31433i −1.80902 1.31433i
\(990\) 0 0
\(991\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(996\) 0 0
\(997\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(998\) 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 475.1.m.a.284.1 yes 4
5.2 odd 4 2375.1.o.b.1576.2 8
5.3 odd 4 2375.1.o.b.1576.1 8
5.4 even 2 2375.1.m.a.1424.1 4
19.18 odd 2 CM 475.1.m.a.284.1 yes 4
25.2 odd 20 2375.1.o.b.2051.2 8
25.11 even 5 2375.1.m.a.949.1 4
25.14 even 10 inner 475.1.m.a.189.1 4
25.23 odd 20 2375.1.o.b.2051.1 8
95.18 even 4 2375.1.o.b.1576.1 8
95.37 even 4 2375.1.o.b.1576.2 8
95.94 odd 2 2375.1.m.a.1424.1 4
475.189 odd 10 inner 475.1.m.a.189.1 4
475.227 even 20 2375.1.o.b.2051.2 8
475.398 even 20 2375.1.o.b.2051.1 8
475.436 odd 10 2375.1.m.a.949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.1.m.a.189.1 4 25.14 even 10 inner
475.1.m.a.189.1 4 475.189 odd 10 inner
475.1.m.a.284.1 yes 4 1.1 even 1 trivial
475.1.m.a.284.1 yes 4 19.18 odd 2 CM
2375.1.m.a.949.1 4 25.11 even 5
2375.1.m.a.949.1 4 475.436 odd 10
2375.1.m.a.1424.1 4 5.4 even 2
2375.1.m.a.1424.1 4 95.94 odd 2
2375.1.o.b.1576.1 8 5.3 odd 4
2375.1.o.b.1576.1 8 95.18 even 4
2375.1.o.b.1576.2 8 5.2 odd 4
2375.1.o.b.1576.2 8 95.37 even 4
2375.1.o.b.2051.1 8 25.23 odd 20
2375.1.o.b.2051.1 8 475.398 even 20
2375.1.o.b.2051.2 8 25.2 odd 20
2375.1.o.b.2051.2 8 475.227 even 20