Properties

Label 2200.1.fr.b.1107.1
Level $2200$
Weight $1$
Character 2200.1107
Analytic conductor $1.098$
Analytic rank $0$
Dimension $16$
Projective image $D_{10}$
CM discriminant -40
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,1,Mod(107,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 5, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.fr (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.09794302779\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.30181730444800000.2

Embedding invariants

Embedding label 1107.1
Root \(-0.156434 + 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 2200.1107
Dual form 2200.1.fr.b.2043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.156434 - 0.987688i) q^{2} +(-0.951057 + 0.309017i) q^{4} +(0.280582 - 0.550672i) q^{7} +(0.453990 + 0.891007i) q^{8} +(0.587785 - 0.809017i) q^{9} +O(q^{10})\) \(q+(-0.156434 - 0.987688i) q^{2} +(-0.951057 + 0.309017i) q^{4} +(0.280582 - 0.550672i) q^{7} +(0.453990 + 0.891007i) q^{8} +(0.587785 - 0.809017i) q^{9} +(-0.309017 - 0.951057i) q^{11} +(-1.59811 + 0.253116i) q^{13} +(-0.587785 - 0.190983i) q^{14} +(0.809017 - 0.587785i) q^{16} +(-0.891007 - 0.453990i) q^{18} +(0.363271 - 1.11803i) q^{19} +(-0.891007 + 0.453990i) q^{22} +(1.34500 + 1.34500i) q^{23} +(0.500000 + 1.53884i) q^{26} +(-0.0966818 + 0.610425i) q^{28} +(-0.707107 - 0.707107i) q^{32} +(-0.309017 + 0.951057i) q^{36} +(-1.04744 - 0.533698i) q^{37} +(-1.16110 - 0.183900i) q^{38} +(-1.80902 - 0.587785i) q^{41} +(0.587785 + 0.809017i) q^{44} +(1.11803 - 1.53884i) q^{46} +(-0.533698 - 1.04744i) q^{47} +(0.363271 + 0.500000i) q^{49} +(1.44168 - 0.734572i) q^{52} +(-0.297556 - 1.87869i) q^{53} +0.618034 q^{56} +(1.53884 - 0.500000i) q^{59} +(-0.280582 - 0.550672i) q^{63} +(-0.587785 + 0.809017i) q^{64} +(0.987688 + 0.156434i) q^{72} +(-0.363271 + 1.11803i) q^{74} +1.17557i q^{76} +(-0.610425 - 0.0966818i) q^{77} +(-0.309017 - 0.951057i) q^{81} +(-0.297556 + 1.87869i) q^{82} +(0.707107 - 0.707107i) q^{88} -0.618034i q^{89} +(-0.309017 + 0.951057i) q^{91} +(-1.69480 - 0.863541i) q^{92} +(-0.951057 + 0.690983i) q^{94} +(0.437016 - 0.437016i) q^{98} +(-0.951057 - 0.309017i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{11} + 4 q^{16} + 8 q^{26} + 4 q^{36} - 20 q^{41} - 8 q^{56} + 4 q^{81} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(-1\) \(e\left(\frac{7}{10}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.156434 0.987688i −0.156434 0.987688i
\(3\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(4\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.280582 0.550672i 0.280582 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(8\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(9\) 0.587785 0.809017i 0.587785 0.809017i
\(10\) 0 0
\(11\) −0.309017 0.951057i −0.309017 0.951057i
\(12\) 0 0
\(13\) −1.59811 + 0.253116i −1.59811 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) −0.587785 0.190983i −0.587785 0.190983i
\(15\) 0 0
\(16\) 0.809017 0.587785i 0.809017 0.587785i
\(17\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(18\) −0.891007 0.453990i −0.891007 0.453990i
\(19\) 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(23\) 1.34500 + 1.34500i 1.34500 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(27\) 0 0
\(28\) −0.0966818 + 0.610425i −0.0966818 + 0.610425i
\(29\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) −0.707107 0.707107i −0.707107 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(37\) −1.04744 0.533698i −1.04744 0.533698i −0.156434 0.987688i \(-0.550000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(38\) −1.16110 0.183900i −1.16110 0.183900i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(45\) 0 0
\(46\) 1.11803 1.53884i 1.11803 1.53884i
\(47\) −0.533698 1.04744i −0.533698 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(48\) 0 0
\(49\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.44168 0.734572i 1.44168 0.734572i
\(53\) −0.297556 1.87869i −0.297556 1.87869i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.618034 0.618034
\(57\) 0 0
\(58\) 0 0
\(59\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) −0.280582 0.550672i −0.280582 0.550672i
\(64\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(73\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(74\) −0.363271 + 1.11803i −0.363271 + 1.11803i
\(75\) 0 0
\(76\) 1.17557i 1.17557i
\(77\) −0.610425 0.0966818i −0.610425 0.0966818i
\(78\) 0 0
\(79\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(80\) 0 0
\(81\) −0.309017 0.951057i −0.309017 0.951057i
\(82\) −0.297556 + 1.87869i −0.297556 + 1.87869i
\(83\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.707107 0.707107i 0.707107 0.707107i
\(89\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(90\) 0 0
\(91\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(92\) −1.69480 0.863541i −1.69480 0.863541i
\(93\) 0 0
\(94\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(98\) 0.437016 0.437016i 0.437016 0.437016i
\(99\) −0.951057 0.309017i −0.951057 0.309017i
\(100\) 0 0
\(101\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(102\) 0 0
\(103\) −0.533698 + 1.04744i −0.533698 + 1.04744i 0.453990 + 0.891007i \(0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(104\) −0.951057 1.30902i −0.951057 1.30902i
\(105\) 0 0
\(106\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(107\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.0966818 0.610425i −0.0966818 0.610425i
\(113\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.734572 + 1.44168i −0.734572 + 1.44168i
\(118\) −0.734572 1.44168i −0.734572 1.44168i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(127\) 1.59811 + 0.253116i 1.59811 + 0.253116i 0.891007 0.453990i \(-0.150000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −0.513743 0.513743i −0.513743 0.513743i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(138\) 0 0
\(139\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.734572 + 1.44168i 0.734572 + 1.44168i
\(144\) 1.00000i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.16110 + 0.183900i 1.16110 + 0.183900i
\(149\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(152\) 1.16110 0.183900i 1.16110 0.183900i
\(153\) 0 0
\(154\) 0.618034i 0.618034i
\(155\) 0 0
\(156\) 0 0
\(157\) 0.863541 + 1.69480i 0.863541 + 1.69480i 0.707107 + 0.707107i \(0.250000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.11803 0.363271i 1.11803 0.363271i
\(162\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(163\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(164\) 1.90211 1.90211
\(165\) 0 0
\(166\) 0 0
\(167\) −0.253116 1.59811i −0.253116 1.59811i −0.707107 0.707107i \(-0.750000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(168\) 0 0
\(169\) 1.53884 0.500000i 1.53884 0.500000i
\(170\) 0 0
\(171\) −0.690983 0.951057i −0.690983 0.951057i
\(172\) 0 0
\(173\) 0.280582 + 0.550672i 0.280582 + 0.550672i 0.987688 0.156434i \(-0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.809017 0.587785i −0.809017 0.587785i
\(177\) 0 0
\(178\) −0.610425 + 0.0966818i −0.610425 + 0.0966818i
\(179\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(182\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(183\) 0 0
\(184\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0.831254 + 0.831254i 0.831254 + 0.831254i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(192\) 0 0
\(193\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.500000 0.363271i −0.500000 0.363271i
\(197\) 0.437016 + 0.437016i 0.437016 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(198\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 1.11803 + 0.363271i 1.11803 + 0.363271i
\(207\) 1.87869 0.297556i 1.87869 0.297556i
\(208\) −1.14412 + 1.14412i −1.14412 + 1.14412i
\(209\) −1.17557 −1.17557
\(210\) 0 0
\(211\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(212\) 0.863541 + 1.69480i 0.863541 + 1.69480i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.69480 0.863541i 1.69480 0.863541i 0.707107 0.707107i \(-0.250000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(224\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(228\) 0 0
\(229\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(234\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(235\) 0 0
\(236\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(237\) 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
\(241\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(242\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.297556 + 1.87869i −0.297556 + 1.87869i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(252\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(253\) 0.863541 1.69480i 0.863541 1.69480i
\(254\) 1.61803i 1.61803i
\(255\) 0 0
\(256\) 0.309017 0.951057i 0.309017 0.951057i
\(257\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(258\) 0 0
\(259\) −0.587785 + 0.427051i −0.587785 + 0.427051i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.14412 1.14412i 1.14412 1.14412i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.427051 + 0.587785i −0.427051 + 0.587785i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(270\) 0 0
\(271\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.0966818 0.610425i −0.0966818 0.610425i −0.987688 0.156434i \(-0.950000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(278\) 1.69480 0.863541i 1.69480 0.863541i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.30902 0.951057i 1.30902 0.951057i
\(287\) −0.831254 + 0.831254i −0.831254 + 0.831254i
\(288\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(289\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.550672 0.280582i −0.550672 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.17557i 1.17557i
\(297\) 0 0
\(298\) 0 0
\(299\) −2.48990 1.80902i −2.48990 1.80902i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.363271 1.11803i −0.363271 1.11803i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0.610425 0.0966818i 0.610425 0.0966818i
\(309\) 0 0
\(310\) 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 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(314\) 1.53884 1.11803i 1.53884 1.11803i
\(315\) 0 0
\(316\) 0 0
\(317\) 1.16110 0.183900i 1.16110 0.183900i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −0.533698 1.04744i −0.533698 1.04744i
\(323\) 0 0
\(324\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −0.297556 1.87869i −0.297556 1.87869i
\(329\) −0.726543 −0.726543
\(330\) 0 0
\(331\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(332\) 0 0
\(333\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(334\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(338\) −0.734572 1.44168i −0.734572 1.44168i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −0.831254 + 0.831254i −0.831254 + 0.831254i
\(343\) 0.987688 0.156434i 0.987688 0.156434i
\(344\) 0 0
\(345\) 0 0
\(346\) 0.500000 0.363271i 0.500000 0.363271i
\(347\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(348\) 0 0
\(349\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(357\) 0 0
\(358\) 0.0966818 0.610425i 0.0966818 0.610425i
\(359\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(360\) 0 0
\(361\) −0.309017 0.224514i −0.309017 0.224514i
\(362\) 0 0
\(363\) 0 0
\(364\) 1.00000i 1.00000i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(368\) 1.87869 + 0.297556i 1.87869 + 0.297556i
\(369\) −1.53884 + 1.11803i −1.53884 + 1.11803i
\(370\) 0 0
\(371\) −1.11803 0.363271i −1.11803 0.363271i
\(372\) 0 0
\(373\) −1.14412 + 1.14412i −1.14412 + 1.14412i −0.156434 + 0.987688i \(0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.690983 0.951057i 0.690983 0.951057i
\(377\) 0 0
\(378\) 0 0
\(379\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.297556 + 1.87869i 0.297556 + 1.87869i 0.453990 + 0.891007i \(0.350000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.280582 + 0.550672i −0.280582 + 0.550672i
\(393\) 0 0
\(394\) 0.363271 0.500000i 0.363271 0.500000i
\(395\) 0 0
\(396\) 1.00000 1.00000
\(397\) −1.34500 + 1.34500i −1.34500 + 1.34500i −0.453990 + 0.891007i \(0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.183900 + 1.16110i −0.183900 + 1.16110i
\(408\) 0 0
\(409\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.183900 1.16110i 0.183900 1.16110i
\(413\) 0.156434 0.987688i 0.156434 0.987688i
\(414\) −0.587785 1.80902i −0.587785 1.80902i
\(415\) 0 0
\(416\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(417\) 0 0
\(418\) 0.183900 + 1.16110i 0.183900 + 1.16110i
\(419\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(420\) 0 0
\(421\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(422\) 0 0
\(423\) −1.16110 0.183900i −1.16110 0.183900i
\(424\) 1.53884 1.11803i 1.53884 1.11803i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(432\) 0 0
\(433\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.99235 1.01515i 1.99235 1.01515i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0.618034 0.618034
\(442\) 0 0
\(443\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.11803 1.53884i −1.11803 1.53884i
\(447\) 0 0
\(448\) 0.280582 + 0.550672i 0.280582 + 0.550672i
\(449\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(450\) 0 0
\(451\) 1.90211i 1.90211i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −0.831254 0.831254i −0.831254 0.831254i 0.156434 0.987688i \(-0.450000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(468\) 0.253116 1.59811i 0.253116 1.59811i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.14412 + 1.14412i 1.14412 + 1.14412i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.69480 0.863541i −1.69480 0.863541i
\(478\) 0 0
\(479\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) 1.80902 + 0.587785i 1.80902 + 0.587785i
\(482\) 1.87869 0.297556i 1.87869 0.297556i
\(483\) 0 0
\(484\) 0.587785 0.809017i 0.587785 0.809017i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.11803 0.363271i 1.11803 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 1.90211 1.90211
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.280582 0.550672i 0.280582 0.550672i
\(503\) −0.734572 1.44168i −0.734572 1.44168i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(504\) 0.363271 0.500000i 0.363271 0.500000i
\(505\) 0 0
\(506\) −1.80902 0.587785i −1.80902 0.587785i
\(507\) 0 0
\(508\) −1.59811 + 0.253116i −1.59811 + 0.253116i
\(509\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.987688 0.156434i −0.987688 0.156434i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.831254 + 0.831254i −0.831254 + 0.831254i
\(518\) 0.513743 + 0.513743i 0.513743 + 0.513743i
\(519\) 0 0
\(520\) 0 0
\(521\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.30902 0.951057i −1.30902 0.951057i
\(527\) 0 0
\(528\) 0 0
\(529\) 2.61803i 2.61803i
\(530\) 0 0
\(531\) 0.500000 1.53884i 0.500000 1.53884i
\(532\) 0.647354 + 0.329843i 0.647354 + 0.329843i
\(533\) 3.03979 + 0.481456i 3.03979 + 0.481456i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.363271 0.500000i 0.363271 0.500000i
\(540\) 0 0
\(541\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(555\) 0 0
\(556\) −1.11803 1.53884i −1.11803 1.53884i
\(557\) 0.734572 1.44168i 0.734572 1.44168i −0.156434 0.987688i \(-0.550000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.610425 0.0966818i −0.610425 0.0966818i
\(568\) 0 0
\(569\) 0.587785 1.80902i 0.587785 1.80902i 1.00000i \(-0.5\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(570\) 0 0
\(571\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(572\) −1.14412 1.14412i −1.14412 1.14412i
\(573\) 0 0
\(574\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(575\) 0 0
\(576\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(577\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(578\) 0.156434 0.987688i 0.156434 0.987688i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.69480 + 0.863541i −1.69480 + 0.863541i
\(584\) 0 0
\(585\) 0 0
\(586\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(587\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.16110 + 0.183900i −1.16110 + 0.183900i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −1.39724 + 2.74224i −1.39724 + 2.74224i
\(599\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(600\) 0 0
\(601\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.312869 1.97538i −0.312869 1.97538i −0.156434 0.987688i \(-0.550000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(608\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.11803 + 1.53884i 1.11803 + 1.53884i
\(612\) 0 0
\(613\) 0.907981 + 1.78201i 0.907981 + 1.78201i 0.453990 + 0.891007i \(0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.190983 0.587785i −0.190983 0.587785i
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −0.340334 0.173409i −0.340334 0.173409i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −1.34500 1.34500i −1.34500 1.34500i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.363271 1.11803i −0.363271 1.11803i
\(635\) 0 0
\(636\) 0 0
\(637\) −0.707107 0.707107i −0.707107 0.707107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(644\) −0.951057 + 0.690983i −0.951057 + 0.690983i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(648\) 0.707107 0.707107i 0.707107 0.707107i
\(649\) −0.951057 1.30902i −0.951057 1.30902i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.533698 1.04744i 0.533698 1.04744i −0.453990 0.891007i \(-0.650000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.80902 + 0.587785i −1.80902 + 0.587785i
\(657\) 0 0
\(658\) 0.113656 + 0.717598i 0.113656 + 0.717598i
\(659\) −1.90211 −1.90211 −0.951057 0.309017i \(-0.900000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −0.253116 1.59811i −0.253116 1.59811i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(667\) 0 0
\(668\) 0.734572 + 1.44168i 0.734572 + 1.44168i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(677\) 1.97538 + 0.312869i 1.97538 + 0.312869i 0.987688 + 0.156434i \(0.0500000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0.951057 + 0.690983i 0.951057 + 0.690983i
\(685\) 0 0
\(686\) −0.309017 0.951057i −0.309017 0.951057i
\(687\) 0 0
\(688\) 0 0
\(689\) 0.951057 + 2.92705i 0.951057 + 2.92705i
\(690\) 0 0
\(691\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(692\) −0.437016 0.437016i −0.437016 0.437016i
\(693\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(702\) 0 0
\(703\) −0.977198 + 0.977198i −0.977198 + 0.977198i
\(704\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.550672 0.280582i 0.550672 0.280582i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −0.618034 −0.618034
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(720\) 0 0
\(721\) 0.427051 + 0.587785i 0.427051 + 0.587785i
\(722\) −0.173409 + 0.340334i −0.173409 + 0.340334i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.34500 1.34500i 1.34500 1.34500i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(728\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(729\) −0.951057 0.309017i −0.951057 0.309017i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.78201 + 0.907981i 1.78201 + 0.907981i 0.891007 + 0.453990i \(0.150000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.90211i 1.90211i
\(737\) 0 0
\(738\) 1.34500 + 1.34500i 1.34500 + 1.34500i
\(739\) 1.53884 + 1.11803i 1.53884 + 1.11803i 0.951057 + 0.309017i \(0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.183900 + 1.16110i −0.183900 + 1.16110i
\(743\) 0.0966818 0.610425i 0.0966818 0.610425i −0.891007 0.453990i \(-0.850000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(752\) −1.04744 0.533698i −1.04744 0.533698i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.16110 + 0.183900i −1.16110 + 0.183900i −0.707107 0.707107i \(-0.750000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(758\) 0.437016 0.437016i 0.437016 0.437016i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 1.80902 0.587785i 1.80902 0.587785i
\(767\) −2.33269 + 1.18856i −2.33269 + 1.18856i
\(768\) 0 0
\(769\) −1.17557 −1.17557 −0.587785 0.809017i \(-0.700000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.69480 + 0.863541i −1.69480 + 0.863541i −0.707107 + 0.707107i \(0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.31433 + 1.80902i −1.31433 + 1.80902i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(788\) −0.550672 0.280582i −0.550672 0.280582i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.156434 0.987688i −0.156434 0.987688i
\(793\) 0 0
\(794\) 1.53884 + 1.11803i 1.53884 + 1.11803i
\(795\) 0 0
\(796\) 0 0
\(797\) 0.183900 1.16110i 0.183900 1.16110i −0.707107 0.707107i \(-0.750000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.500000 0.363271i −0.500000 0.363271i
\(802\) −1.14412 1.14412i −1.14412 1.14412i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(810\) 0 0
\(811\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.17557 1.17557
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.533698 + 1.04744i −0.533698 + 1.04744i
\(819\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(820\) 0 0
\(821\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(822\) 0 0
\(823\) −0.183900 1.16110i −0.183900 1.16110i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(824\) −1.17557 −1.17557
\(825\) 0 0
\(826\) −1.00000 −1.00000
\(827\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(828\) −1.69480 + 0.863541i −1.69480 + 0.863541i
\(829\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.734572 1.44168i 0.734572 1.44168i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 1.11803 0.363271i 1.11803 0.363271i
\(837\) 0 0
\(838\) 1.59811 0.253116i 1.59811 0.253116i
\(839\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(840\) 0 0
\(841\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 1.17557i 1.17557i
\(847\) 0.0966818 + 0.610425i 0.0966818 + 0.610425i
\(848\) −1.34500 1.34500i −1.34500 1.34500i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.690983 2.12663i −0.690983 2.12663i
\(852\) 0 0
\(853\) −0.0966818 + 0.610425i −0.0966818 + 0.610425i 0.891007 + 0.453990i \(0.150000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.16110 0.183900i −1.16110 0.183900i −0.453990 0.891007i \(-0.650000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 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
\(874\) −1.31433 1.80902i −1.31433 1.80902i
\(875\) 0 0
\(876\) 0 0
\(877\) 1.44168 0.734572i 1.44168 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(882\) −0.0966818 0.610425i −0.0966818 0.610425i
\(883\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.280582 + 0.550672i −0.280582 + 0.550672i −0.987688 0.156434i \(-0.950000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) 0.587785 0.809017i 0.587785 0.809017i
\(890\) 0 0
\(891\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(892\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(893\) −1.36495 + 0.216187i −1.36495 + 0.216187i
\(894\) 0 0
\(895\) 0 0
\(896\) 0.500000 0.363271i 0.500000 0.363271i
\(897\) 0 0
\(898\) 1.44168 + 0.734572i 1.44168 + 0.734572i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 1.87869 0.297556i 1.87869 0.297556i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 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 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −0.690983 + 0.951057i −0.690983 + 0.951057i
\(927\) 0.533698 + 1.04744i 0.533698 + 1.04744i
\(928\) 0 0
\(929\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(930\) 0 0
\(931\) 0.690983 0.224514i 0.690983 0.224514i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.61803 −1.61803
\(937\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(942\) 0 0
\(943\) −1.64255 3.22369i −1.64255 3.22369i
\(944\) 0.951057 1.30902i 0.951057 1.30902i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(954\) −0.587785 + 1.80902i −0.587785 + 1.80902i
\(955\) 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
\(962\) 0.297556 1.87869i 0.297556 1.87869i
\(963\) 0 0
\(964\) −0.587785 1.80902i −0.587785 1.80902i
\(965\) 0 0
\(966\) 0 0
\(967\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(968\) −0.891007 0.453990i −0.891007 0.453990i
\(969\) 0 0
\(970\) 0 0
\(971\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1.16110 + 0.183900i 1.16110 + 0.183900i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(978\) 0 0
\(979\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(980\) 0 0
\(981\) 0 0
\(982\) −0.533698 1.04744i −0.533698 1.04744i
\(983\) −0.863541 + 1.69480i −0.863541 + 1.69480i −0.156434 + 0.987688i \(0.550000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −0.297556 1.87869i −0.297556 1.87869i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.907981 1.78201i 0.907981 1.78201i 0.453990 0.891007i \(-0.350000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(998\) −0.280582 0.550672i −0.280582 0.550672i
\(999\) 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.fr.b.1107.1 yes 16
5.2 odd 4 inner 2200.1.fr.b.843.2 yes 16
5.3 odd 4 inner 2200.1.fr.b.843.1 yes 16
5.4 even 2 inner 2200.1.fr.b.1107.2 yes 16
8.3 odd 2 inner 2200.1.fr.b.1107.2 yes 16
11.8 odd 10 inner 2200.1.fr.b.107.2 yes 16
40.3 even 4 inner 2200.1.fr.b.843.2 yes 16
40.19 odd 2 CM 2200.1.fr.b.1107.1 yes 16
40.27 even 4 inner 2200.1.fr.b.843.1 yes 16
55.8 even 20 inner 2200.1.fr.b.2043.2 yes 16
55.19 odd 10 inner 2200.1.fr.b.107.1 16
55.52 even 20 inner 2200.1.fr.b.2043.1 yes 16
88.19 even 10 inner 2200.1.fr.b.107.1 16
440.19 even 10 inner 2200.1.fr.b.107.2 yes 16
440.107 odd 20 inner 2200.1.fr.b.2043.2 yes 16
440.283 odd 20 inner 2200.1.fr.b.2043.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.1.fr.b.107.1 16 55.19 odd 10 inner
2200.1.fr.b.107.1 16 88.19 even 10 inner
2200.1.fr.b.107.2 yes 16 11.8 odd 10 inner
2200.1.fr.b.107.2 yes 16 440.19 even 10 inner
2200.1.fr.b.843.1 yes 16 5.3 odd 4 inner
2200.1.fr.b.843.1 yes 16 40.27 even 4 inner
2200.1.fr.b.843.2 yes 16 5.2 odd 4 inner
2200.1.fr.b.843.2 yes 16 40.3 even 4 inner
2200.1.fr.b.1107.1 yes 16 1.1 even 1 trivial
2200.1.fr.b.1107.1 yes 16 40.19 odd 2 CM
2200.1.fr.b.1107.2 yes 16 5.4 even 2 inner
2200.1.fr.b.1107.2 yes 16 8.3 odd 2 inner
2200.1.fr.b.2043.1 yes 16 55.52 even 20 inner
2200.1.fr.b.2043.1 yes 16 440.283 odd 20 inner
2200.1.fr.b.2043.2 yes 16 55.8 even 20 inner
2200.1.fr.b.2043.2 yes 16 440.107 odd 20 inner