Properties

Label 2200.1.dd.b.1499.4
Level $2200$
Weight $1$
Character 2200.1499
Analytic conductor $1.098$
Analytic rank $0$
Dimension $16$
Projective image $D_{15}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,1,Mod(499,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 5, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.dd (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: \(1.09794302779\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

Embedding invariants

Embedding label 1499.4
Root \(-0.406737 - 0.913545i\) of defining polynomial
Character \(\chi\) \(=\) 2200.1499
Dual form 2200.1.dd.b.499.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.587785 + 0.809017i) q^{2} +(0.198825 - 0.0646021i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(0.169131 + 0.122881i) q^{6} +(-0.951057 + 0.309017i) q^{8} +(-0.773659 + 0.562096i) q^{9} +O(q^{10})\) \(q+(0.587785 + 0.809017i) q^{2} +(0.198825 - 0.0646021i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(0.169131 + 0.122881i) q^{6} +(-0.951057 + 0.309017i) q^{8} +(-0.773659 + 0.562096i) q^{9} +(-0.978148 - 0.207912i) q^{11} +0.209057i q^{12} +(-0.809017 - 0.587785i) q^{16} +(-0.786610 + 1.08268i) q^{17} +(-0.909491 - 0.295511i) q^{18} +(0.604528 + 1.86055i) q^{19} +(-0.406737 - 0.913545i) q^{22} +(-0.169131 + 0.122881i) q^{24} +(-0.240391 + 0.330869i) q^{27} -1.00000i q^{32} +(-0.207912 + 0.0218524i) q^{33} -1.33826 q^{34} +(-0.295511 - 0.909491i) q^{36} +(-1.14988 + 1.58268i) q^{38} +(0.564602 + 1.73767i) q^{41} -1.61803i q^{43} +(0.500000 - 0.866025i) q^{44} +(-0.198825 - 0.0646021i) q^{48} +(0.809017 + 0.587785i) q^{49} +(-0.0864545 + 0.266080i) q^{51} -0.408977 q^{54} +(0.240391 + 0.330869i) q^{57} +(-0.190983 + 0.587785i) q^{59} +(0.809017 - 0.587785i) q^{64} +(-0.139886 - 0.155360i) q^{66} -1.33826i q^{67} +(-0.786610 - 1.08268i) q^{68} +(0.562096 - 0.773659i) q^{72} +(1.73767 + 0.564602i) q^{73} -1.95630 q^{76} +(0.269091 - 0.828176i) q^{81} +(-1.07394 + 1.47815i) q^{82} +(1.07394 - 1.47815i) q^{83} +(1.30902 - 0.951057i) q^{86} +(0.994522 - 0.104528i) q^{88} +0.209057 q^{89} +(-0.0646021 - 0.198825i) q^{96} +(-0.951057 - 1.30902i) q^{97} +1.00000i q^{98} +(0.873619 - 0.388960i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{11} - 4 q^{16} + 6 q^{19} + 6 q^{24} - 4 q^{34} - 10 q^{36} + 4 q^{41} + 8 q^{44} + 4 q^{49} - 14 q^{51} + 4 q^{54} - 12 q^{59} + 4 q^{64} - 2 q^{66} + 4 q^{76} - 6 q^{81} + 12 q^{86} - 4 q^{89} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(3\) 0.198825 0.0646021i 0.198825 0.0646021i −0.207912 0.978148i \(-0.566667\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(4\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(5\) 0 0
\(6\) 0.169131 + 0.122881i 0.169131 + 0.122881i
\(7\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(8\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(9\) −0.773659 + 0.562096i −0.773659 + 0.562096i
\(10\) 0 0
\(11\) −0.978148 0.207912i −0.978148 0.207912i
\(12\) 0.209057i 0.209057i
\(13\) 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
\(17\) −0.786610 + 1.08268i −0.786610 + 1.08268i 0.207912 + 0.978148i \(0.433333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(18\) −0.909491 0.295511i −0.909491 0.295511i
\(19\) 0.604528 + 1.86055i 0.604528 + 1.86055i 0.500000 + 0.866025i \(0.333333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.406737 0.913545i −0.406737 0.913545i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.169131 + 0.122881i −0.169131 + 0.122881i
\(25\) 0 0
\(26\) 0 0
\(27\) −0.240391 + 0.330869i −0.240391 + 0.330869i
\(28\) 0 0
\(29\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 1.00000i 1.00000i
\(33\) −0.207912 + 0.0218524i −0.207912 + 0.0218524i
\(34\) −1.33826 −1.33826
\(35\) 0 0
\(36\) −0.295511 0.909491i −0.295511 0.909491i
\(37\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(38\) −1.14988 + 1.58268i −1.14988 + 1.58268i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.564602 + 1.73767i 0.564602 + 1.73767i 0.669131 + 0.743145i \(0.266667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(42\) 0 0
\(43\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(44\) 0.500000 0.866025i 0.500000 0.866025i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(48\) −0.198825 0.0646021i −0.198825 0.0646021i
\(49\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(50\) 0 0
\(51\) −0.0864545 + 0.266080i −0.0864545 + 0.266080i
\(52\) 0 0
\(53\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(54\) −0.408977 −0.408977
\(55\) 0 0
\(56\) 0 0
\(57\) 0.240391 + 0.330869i 0.240391 + 0.330869i
\(58\) 0 0
\(59\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.809017 0.587785i 0.809017 0.587785i
\(65\) 0 0
\(66\) −0.139886 0.155360i −0.139886 0.155360i
\(67\) 1.33826i 1.33826i −0.743145 0.669131i \(-0.766667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(68\) −0.786610 1.08268i −0.786610 1.08268i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(72\) 0.562096 0.773659i 0.562096 0.773659i
\(73\) 1.73767 + 0.564602i 1.73767 + 0.564602i 0.994522 0.104528i \(-0.0333333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.95630 −1.95630
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) 0.269091 0.828176i 0.269091 0.828176i
\(82\) −1.07394 + 1.47815i −1.07394 + 1.47815i
\(83\) 1.07394 1.47815i 1.07394 1.47815i 0.207912 0.978148i \(-0.433333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.30902 0.951057i 1.30902 0.951057i
\(87\) 0 0
\(88\) 0.994522 0.104528i 0.994522 0.104528i
\(89\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.0646021 0.198825i −0.0646021 0.198825i
\(97\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 0.873619 0.388960i 0.873619 0.388960i
\(100\) 0 0
\(101\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(102\) −0.266080 + 0.0864545i −0.266080 + 0.0864545i
\(103\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.198825 + 0.0646021i −0.198825 + 0.0646021i −0.406737 0.913545i \(-0.633333\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(108\) −0.240391 0.330869i −0.240391 0.330869i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.27276 + 0.413545i −1.27276 + 0.413545i −0.866025 0.500000i \(-0.833333\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(114\) −0.126381 + 0.388960i −0.126381 + 0.388960i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(122\) 0 0
\(123\) 0.224514 + 0.309017i 0.224514 + 0.309017i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(128\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(129\) −0.104528 0.321706i −0.104528 0.321706i
\(130\) 0 0
\(131\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(132\) 0.0434654 0.204489i 0.0434654 0.204489i
\(133\) 0 0
\(134\) 1.08268 0.786610i 1.08268 0.786610i
\(135\) 0 0
\(136\) 0.413545 1.27276i 0.413545 1.27276i
\(137\) −1.07394 + 1.47815i −1.07394 + 1.47815i −0.207912 + 0.978148i \(0.566667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0 0
\(139\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.956295 0.956295
\(145\) 0 0
\(146\) 0.564602 + 1.73767i 0.564602 + 1.73767i
\(147\) 0.198825 + 0.0646021i 0.198825 + 0.0646021i
\(148\) 0 0
\(149\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(152\) −1.14988 1.58268i −1.14988 1.58268i
\(153\) 1.27977i 1.27977i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.828176 0.269091i 0.828176 0.269091i
\(163\) 1.14988 + 1.58268i 1.14988 + 1.58268i 0.743145 + 0.669131i \(0.233333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(164\) −1.82709 −1.82709
\(165\) 0 0
\(166\) 1.82709 1.82709
\(167\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(168\) 0 0
\(169\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(170\) 0 0
\(171\) −1.51351 1.09963i −1.51351 1.09963i
\(172\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(173\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(177\) 0.129204i 0.129204i
\(178\) 0.122881 + 0.169131i 0.122881 + 0.169131i
\(179\) −0.413545 1.27276i −0.413545 1.27276i −0.913545 0.406737i \(-0.866667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.994522 0.895472i 0.994522 0.895472i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(192\) 0.122881 0.169131i 0.122881 0.169131i
\(193\) −0.587785 + 0.809017i −0.587785 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(194\) 0.500000 1.53884i 0.500000 1.53884i
\(195\) 0 0
\(196\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0.828176 + 0.478148i 0.828176 + 0.478148i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −0.0864545 0.266080i −0.0864545 0.266080i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.226341 0.164446i −0.226341 0.164446i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.204489 1.94558i −0.204489 1.94558i
\(210\) 0 0
\(211\) −1.47815 + 1.07394i −1.47815 + 1.07394i −0.500000 + 0.866025i \(0.666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −0.169131 0.122881i −0.169131 0.122881i
\(215\) 0 0
\(216\) 0.126381 0.388960i 0.126381 0.388960i
\(217\) 0 0
\(218\) 0 0
\(219\) 0.381966 0.381966
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.08268 0.786610i −1.08268 0.786610i
\(227\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) −0.388960 + 0.126381i −0.388960 + 0.126381i
\(229\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.500000 0.363271i −0.500000 0.363271i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) 0 0
\(241\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(242\) 0.207912 + 0.978148i 0.207912 + 0.978148i
\(243\) 0.591023i 0.591023i
\(244\) 0 0
\(245\) 0 0
\(246\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.118034 0.363271i 0.118034 0.363271i
\(250\) 0 0
\(251\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(257\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(258\) 0.198825 0.273659i 0.198825 0.273659i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0.190983 0.0850311i 0.190983 0.0850311i
\(265\) 0 0
\(266\) 0 0
\(267\) 0.0415657 0.0135055i 0.0415657 0.0135055i
\(268\) 1.27276 + 0.413545i 1.27276 + 0.413545i
\(269\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(270\) 0 0
\(271\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) 1.27276 0.413545i 1.27276 0.413545i
\(273\) 0 0
\(274\) −1.82709 −1.82709
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(278\) 0.951057 0.309017i 0.951057 0.309017i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(282\) 0 0
\(283\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.562096 + 0.773659i 0.562096 + 0.773659i
\(289\) −0.244415 0.752232i −0.244415 0.752232i
\(290\) 0 0
\(291\) −0.273659 0.198825i −0.273659 0.198825i
\(292\) −1.07394 + 1.47815i −1.07394 + 1.47815i
\(293\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(294\) 0.0646021 + 0.198825i 0.0646021 + 0.198825i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.303929 0.273659i 0.303929 0.273659i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0.604528 1.86055i 0.604528 1.86055i
\(305\) 0 0
\(306\) 1.03536 0.752232i 1.03536 0.752232i
\(307\) 0.209057i 0.209057i 0.994522 + 0.104528i \(0.0333333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(314\) 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
\(321\) −0.0353579 + 0.0256890i −0.0353579 + 0.0256890i
\(322\) 0 0
\(323\) −2.48990 0.809017i −2.48990 0.809017i
\(324\) 0.704489 + 0.511841i 0.704489 + 0.511841i
\(325\) 0 0
\(326\) −0.604528 + 1.86055i −0.604528 + 1.86055i
\(327\) 0 0
\(328\) −1.07394 1.47815i −1.07394 1.47815i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(332\) 1.07394 + 1.47815i 1.07394 + 1.47815i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.86055 + 0.604528i 1.86055 + 0.604528i 0.994522 + 0.104528i \(0.0333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(338\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(339\) −0.226341 + 0.164446i −0.226341 + 0.164446i
\(340\) 0 0
\(341\) 0 0
\(342\) 1.87080i 1.87080i
\(343\) 0 0
\(344\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.14988 1.58268i 1.14988 1.58268i 0.406737 0.913545i \(-0.366667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(348\) 0 0
\(349\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.207912 + 0.978148i −0.207912 + 0.978148i
\(353\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(354\) −0.104528 + 0.0759444i −0.104528 + 0.0759444i
\(355\) 0 0
\(356\) −0.0646021 + 0.198825i −0.0646021 + 0.198825i
\(357\) 0 0
\(358\) 0.786610 1.08268i 0.786610 1.08268i
\(359\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0 0
\(361\) −2.28716 + 1.66172i −2.28716 + 1.66172i
\(362\) 0 0
\(363\) 0.207912 + 0.0218524i 0.207912 + 0.0218524i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(368\) 0 0
\(369\) −1.41355 1.02700i −1.41355 1.02700i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 1.30902 + 0.278240i 1.30902 + 0.278240i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.47815 + 1.07394i 1.47815 + 1.07394i 0.978148 + 0.207912i \(0.0666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(384\) 0.209057 0.209057
\(385\) 0 0
\(386\) −1.00000 −1.00000
\(387\) 0.909491 + 1.25181i 0.909491 + 1.25181i
\(388\) 1.53884 0.500000i 1.53884 0.500000i
\(389\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.951057 0.309017i −0.951057 0.309017i
\(393\) 0.122881 0.0399263i 0.122881 0.0399263i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.0999601 + 0.951057i 0.0999601 + 0.951057i
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.47815 1.07394i −1.47815 1.07394i −0.978148 0.207912i \(-0.933333\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(402\) 0.164446 0.226341i 0.164446 0.226341i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.279773i 0.279773i
\(409\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(410\) 0 0
\(411\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.209057i 0.209057i
\(418\) 1.45381 1.30902i 1.45381 1.30902i
\(419\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(420\) 0 0
\(421\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) −1.73767 0.564602i −1.73767 0.564602i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.209057i 0.209057i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) 0.388960 0.126381i 0.388960 0.126381i
\(433\) −1.86055 0.604528i −1.86055 0.604528i −0.994522 0.104528i \(-0.966667\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.224514 + 0.309017i 0.224514 + 0.309017i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −0.956295 −0.956295
\(442\) 0 0
\(443\) −1.73767 + 0.564602i −1.73767 + 0.564602i −0.994522 0.104528i \(-0.966667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.08268 0.786610i 1.08268 0.786610i 0.104528 0.994522i \(-0.466667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(450\) 0 0
\(451\) −0.190983 1.81708i −0.190983 1.81708i
\(452\) 1.33826i 1.33826i
\(453\) 0 0
\(454\) −0.190983 0.587785i −0.190983 0.587785i
\(455\) 0 0
\(456\) −0.330869 0.240391i −0.330869 0.240391i
\(457\) 0.122881 0.169131i 0.122881 0.169131i −0.743145 0.669131i \(-0.766667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(458\) 0 0
\(459\) −0.169131 0.520530i −0.169131 0.520530i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(467\) −1.17557 + 1.61803i −1.17557 + 1.61803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.618034i 0.618034i
\(473\) −0.336408 + 1.58268i −0.336408 + 1.58268i
\(474\) 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
\(482\) 0.786610 + 1.08268i 0.786610 + 1.08268i
\(483\) 0 0
\(484\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(485\) 0 0
\(486\) 0.478148 0.347395i 0.478148 0.347395i
\(487\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(488\) 0 0
\(489\) 0.330869 + 0.240391i 0.330869 + 0.240391i
\(490\) 0 0
\(491\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(492\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0.363271 0.118034i 0.363271 0.118034i
\(499\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(503\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.209057i 0.209057i
\(508\) 0 0
\(509\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(513\) −0.760921 0.247238i −0.760921 0.247238i
\(514\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(515\) 0 0
\(516\) 0.338261 0.338261
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.413545 1.27276i 0.413545 1.27276i −0.500000 0.866025i \(-0.666667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(522\) 0 0
\(523\) −1.14988 + 1.58268i −1.14988 + 1.58268i −0.406737 + 0.913545i \(0.633333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(524\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.181049 + 0.104528i 0.181049 + 0.104528i
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) −0.182636 0.562096i −0.182636 0.562096i
\(532\) 0 0
\(533\) 0 0
\(534\) 0.0353579 + 0.0256890i 0.0353579 + 0.0256890i
\(535\) 0 0
\(536\) 0.413545 + 1.27276i 0.413545 + 1.27276i
\(537\) −0.164446 0.226341i −0.164446 0.226341i
\(538\) 0 0
\(539\) −0.669131 0.743145i −0.669131 0.743145i
\(540\) 0 0
\(541\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.08268 + 0.786610i 1.08268 + 0.786610i
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27276 0.413545i 1.27276 0.413545i 0.406737 0.913545i \(-0.366667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(548\) −1.07394 1.47815i −1.07394 1.47815i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(557\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.139886 0.242290i 0.139886 0.242290i
\(562\) 1.61803i 1.61803i
\(563\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(567\) 0 0
\(568\) 0 0
\(569\) 0.604528 + 1.86055i 0.604528 + 1.86055i 0.500000 + 0.866025i \(0.333333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(570\) 0 0
\(571\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.295511 + 0.909491i −0.295511 + 0.909491i
\(577\) 1.14988 1.58268i 1.14988 1.58268i 0.406737 0.913545i \(-0.366667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(578\) 0.464905 0.639886i 0.464905 0.639886i
\(579\) −0.0646021 + 0.198825i −0.0646021 + 0.198825i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.338261i 0.338261i
\(583\) 0 0
\(584\) −1.82709 −1.82709
\(585\) 0 0
\(586\) 0 0
\(587\) −1.27276 0.413545i −1.27276 0.413545i −0.406737 0.913545i \(-0.633333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) −0.122881 + 0.169131i −0.122881 + 0.169131i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.95630i 1.95630i −0.207912 0.978148i \(-0.566667\pi\)
0.207912 0.978148i \(-0.433333\pi\)
\(594\) 0.400040 + 0.0850311i 0.400040 + 0.0850311i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(600\) 0 0
\(601\) 0.564602 1.73767i 0.564602 1.73767i −0.104528 0.994522i \(-0.533333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(602\) 0 0
\(603\) 0.752232 + 1.03536i 0.752232 + 1.03536i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(608\) 1.86055 0.604528i 1.86055 0.604528i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.21714 + 0.395472i 1.21714 + 0.395472i
\(613\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(614\) −0.169131 + 0.122881i −0.169131 + 0.122881i
\(615\) 0 0
\(616\) 0 0
\(617\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(618\) 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 0
\(626\) 0.618034 0.618034
\(627\) −0.166346 0.373619i −0.166346 0.373619i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(632\) 0 0
\(633\) −0.224514 + 0.309017i −0.224514 + 0.309017i
\(634\) 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 \(-0.400000\pi\)
\(642\) −0.0415657 0.0135055i −0.0415657 0.0135055i
\(643\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −0.809017 2.48990i −0.809017 2.48990i
\(647\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(648\) 0.870796i 0.870796i
\(649\) 0.309017 0.535233i 0.309017 0.535233i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.86055 + 0.604528i −1.86055 + 0.604528i
\(653\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.564602 1.73767i 0.564602 1.73767i
\(657\) −1.66172 + 0.539926i −1.66172 + 0.539926i
\(658\) 0 0
\(659\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −0.122881 0.169131i −0.122881 0.169131i
\(663\) 0 0
\(664\) −0.564602 + 1.73767i −0.564602 + 1.73767i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(674\) 0.604528 + 1.86055i 0.604528 + 1.86055i
\(675\) 0 0
\(676\) −0.809017 0.587785i −0.809017 0.587785i
\(677\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(678\) −0.266080 0.0864545i −0.266080 0.0864545i
\(679\) 0 0
\(680\) 0 0
\(681\) −0.129204 −0.129204
\(682\) 0 0
\(683\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(684\) 1.51351 1.09963i 1.51351 1.09963i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.58268 1.14988i 1.58268 1.14988i 0.669131 0.743145i \(-0.266667\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.95630 1.95630
\(695\) 0 0
\(696\) 0 0
\(697\) −2.32545 0.755585i −2.32545 0.755585i
\(698\) 0 0
\(699\) 0.273659 + 0.198825i 0.273659 + 0.198825i
\(700\) 0 0
\(701\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(705\) 0 0
\(706\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(707\) 0 0
\(708\) −0.122881 0.0399263i −0.122881 0.0399263i
\(709\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.198825 + 0.0646021i −0.198825 + 0.0646021i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.33826 1.33826
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.68872 0.873619i −2.68872 0.873619i
\(723\) 0.266080 0.0864545i 0.266080 0.0864545i
\(724\) 0 0
\(725\) 0 0
\(726\) 0.104528 + 0.181049i 0.104528 + 0.181049i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0.230909 + 0.710666i 0.230909 + 0.710666i
\(730\) 0 0
\(731\) 1.75181 + 1.27276i 1.75181 + 1.27276i
\(732\) 0 0
\(733\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.278240 + 1.30902i −0.278240 + 1.30902i
\(738\) 1.74724i 1.74724i
\(739\) 0.500000 0.363271i 0.500000 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.74724i 1.74724i
\(748\) 0.544320 + 1.22256i 0.544320 + 1.22256i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) 0 0
\(753\) 0.122881 0.169131i 0.122881 0.169131i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(758\) 1.82709i 1.82709i
\(759\) 0 0
\(760\) 0 0
\(761\) −1.08268 + 0.786610i −1.08268 + 0.786610i −0.978148 0.207912i \(-0.933333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.122881 + 0.169131i 0.122881 + 0.169131i
\(769\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(770\) 0 0
\(771\) 0.338261 0.338261
\(772\) −0.587785 0.809017i −0.587785 0.809017i
\(773\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(774\) −0.478148 + 1.47159i −0.478148 + 1.47159i
\(775\) 0 0
\(776\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(777\) 0 0
\(778\) 0 0
\(779\) −2.89169 + 2.10094i −2.89169 + 2.10094i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.309017 0.951057i −0.309017 0.951057i
\(785\) 0 0
\(786\) 0.104528 + 0.0759444i 0.104528 + 0.0759444i
\(787\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.710666 + 0.639886i −0.710666 + 0.639886i
\(793\) 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
\(801\) −0.161739 + 0.117510i −0.161739 + 0.117510i
\(802\) 1.82709i 1.82709i
\(803\) −1.58231 0.913545i −1.58231 0.913545i
\(804\) 0.279773 0.279773
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(810\) 0 0
\(811\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0.226341 0.164446i 0.226341 0.164446i
\(817\) 3.01043 0.978148i 3.01043 0.978148i
\(818\) −0.951057 0.309017i −0.951057 0.309017i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(822\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(823\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.14988 1.58268i −1.14988 1.58268i −0.743145 0.669131i \(-0.766667\pi\)
−0.406737 0.913545i \(-0.633333\pi\)
\(828\) 0 0
\(829\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.27276 + 0.413545i −1.27276 + 0.413545i
\(834\) 0.169131 0.122881i 0.169131 0.122881i
\(835\) 0 0
\(836\) 1.91355 + 0.406737i 1.91355 + 0.406737i
\(837\) 0 0
\(838\) −1.07394 1.47815i −1.07394 1.47815i
\(839\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(840\) 0 0
\(841\) −0.809017 0.587785i −0.809017 0.587785i
\(842\) 0 0
\(843\) 0.321706 + 0.104528i 0.321706 + 0.104528i
\(844\) −0.564602 1.73767i −0.564602 1.73767i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.169131 0.122881i 0.169131 0.122881i
\(850\) 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 0
\(855\) 0 0
\(856\) 0.169131 0.122881i 0.169131 0.122881i
\(857\) 0.209057i 0.209057i 0.994522 + 0.104528i \(0.0333333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(858\) 0 0
\(859\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(864\) 0.330869 + 0.240391i 0.330869 + 0.240391i
\(865\) 0 0
\(866\) −0.604528 1.86055i −0.604528 1.86055i
\(867\) −0.0971915 0.133773i −0.0971915 0.133773i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.47159 + 0.478148i 1.47159 + 0.478148i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(877\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(882\) −0.562096 0.773659i −0.562096 0.773659i
\(883\) −1.73767 + 0.564602i −1.73767 + 0.564602i −0.994522 0.104528i \(-0.966667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.47815 1.07394i −1.47815 1.07394i
\(887\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.435398 + 0.754131i −0.435398 + 0.754131i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.27276 + 0.413545i 1.27276 + 0.413545i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 1.35779 1.22256i 1.35779 1.22256i
\(903\) 0 0
\(904\) 1.08268 0.786610i 1.08268 0.786610i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(908\) 0.363271 0.500000i 0.363271 0.500000i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) 0.408977i 0.408977i
\(913\) −1.35779 + 1.22256i −1.35779 + 1.22256i
\(914\) 0.209057 0.209057
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0.321706 0.442790i 0.321706 0.442790i
\(919\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(920\) 0 0
\(921\) 0.0135055 + 0.0415657i 0.0135055 + 0.0415657i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 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
\(931\) −0.604528 + 1.86055i −0.604528 + 1.86055i
\(932\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(933\) 0 0
\(934\) −2.00000 −2.00000
\(935\) 0 0
\(936\) 0 0
\(937\) −1.14988 1.58268i −1.14988 1.58268i −0.743145 0.669131i \(-0.766667\pi\)
−0.406737 0.913545i \(-0.633333\pi\)
\(938\) 0 0
\(939\) 0.0399263 0.122881i 0.0399263 0.122881i
\(940\) 0 0
\(941\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.500000 0.363271i 0.500000 0.363271i
\(945\) 0 0
\(946\) −1.47815 + 0.658114i −1.47815 + 0.658114i
\(947\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.73767 + 0.564602i 1.73767 + 0.564602i 0.994522 0.104528i \(-0.0333333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(954\) 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
\(963\) 0.117510 0.161739i 0.117510 0.161739i
\(964\) −0.413545 + 1.27276i −0.413545 + 1.27276i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.994522 0.104528i −0.994522 0.104528i
\(969\) −0.547318 −0.547318
\(970\) 0 0
\(971\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(972\) 0.562096 + 0.182636i 0.562096 + 0.182636i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.587785 0.809017i −0.587785 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(978\) 0.408977i 0.408977i
\(979\) −0.204489 0.0434654i −0.204489 0.0434654i
\(980\) 0 0
\(981\) 0 0
\(982\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(983\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(984\) −0.309017 0.224514i −0.309017 0.224514i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −0.0415657 + 0.0135055i −0.0415657 + 0.0135055i
\(994\) 0 0
\(995\) 0 0
\(996\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(997\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(998\) 1.53884 0.500000i 1.53884 0.500000i
\(999\) 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 2200.1.dd.b.1499.4 16
5.2 odd 4 2200.1.cl.b.1851.1 yes 8
5.3 odd 4 2200.1.cl.d.1851.2 yes 8
5.4 even 2 inner 2200.1.dd.b.1499.1 16
8.3 odd 2 CM 2200.1.dd.b.1499.4 16
11.4 even 5 inner 2200.1.dd.b.499.1 16
40.3 even 4 2200.1.cl.d.1851.2 yes 8
40.19 odd 2 inner 2200.1.dd.b.1499.1 16
40.27 even 4 2200.1.cl.b.1851.1 yes 8
55.4 even 10 inner 2200.1.dd.b.499.4 16
55.37 odd 20 2200.1.cl.b.851.1 8
55.48 odd 20 2200.1.cl.d.851.2 yes 8
88.59 odd 10 inner 2200.1.dd.b.499.1 16
440.59 odd 10 inner 2200.1.dd.b.499.4 16
440.147 even 20 2200.1.cl.b.851.1 8
440.323 even 20 2200.1.cl.d.851.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.1.cl.b.851.1 8 55.37 odd 20
2200.1.cl.b.851.1 8 440.147 even 20
2200.1.cl.b.1851.1 yes 8 5.2 odd 4
2200.1.cl.b.1851.1 yes 8 40.27 even 4
2200.1.cl.d.851.2 yes 8 55.48 odd 20
2200.1.cl.d.851.2 yes 8 440.323 even 20
2200.1.cl.d.1851.2 yes 8 5.3 odd 4
2200.1.cl.d.1851.2 yes 8 40.3 even 4
2200.1.dd.b.499.1 16 11.4 even 5 inner
2200.1.dd.b.499.1 16 88.59 odd 10 inner
2200.1.dd.b.499.4 16 55.4 even 10 inner
2200.1.dd.b.499.4 16 440.59 odd 10 inner
2200.1.dd.b.1499.1 16 5.4 even 2 inner
2200.1.dd.b.1499.1 16 40.19 odd 2 inner
2200.1.dd.b.1499.4 16 1.1 even 1 trivial
2200.1.dd.b.1499.4 16 8.3 odd 2 CM