Properties

Label 984.1.cj.b.971.1
Level $984$
Weight $1$
Character 984.971
Analytic conductor $0.491$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [984,1,Mod(11,984)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(984, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 20, 20, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("984.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 984.cj (of order \(40\), degree \(16\), 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: \(0.491079972431\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

Embedding invariants

Embedding label 971.1
Root \(-0.987688 - 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 984.971
Dual form 984.1.cj.b.227.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.453990 - 0.891007i) q^{2} +(0.891007 - 0.453990i) q^{3} +(-0.587785 - 0.809017i) q^{4} -1.00000i q^{6} +(-0.987688 + 0.156434i) q^{8} +(0.587785 - 0.809017i) q^{9} +O(q^{10})\) \(q+(0.453990 - 0.891007i) q^{2} +(0.891007 - 0.453990i) q^{3} +(-0.587785 - 0.809017i) q^{4} -1.00000i q^{6} +(-0.987688 + 0.156434i) q^{8} +(0.587785 - 0.809017i) q^{9} +(-0.243950 - 0.398090i) q^{11} +(-0.891007 - 0.453990i) q^{12} +(-0.309017 + 0.951057i) q^{16} +(0.178671 - 0.744220i) q^{17} +(-0.453990 - 0.891007i) q^{18} +(0.156434 + 1.98769i) q^{19} +(-0.465451 + 0.0366318i) q^{22} +(-0.809017 + 0.587785i) q^{24} +(-0.951057 - 0.309017i) q^{25} +(0.156434 - 0.987688i) q^{27} +(0.707107 + 0.707107i) q^{32} +(-0.398090 - 0.243950i) q^{33} +(-0.581990 - 0.497066i) q^{34} -1.00000 q^{36} +(1.84206 + 0.763007i) q^{38} +(-0.156434 + 0.987688i) q^{41} +(1.69480 + 0.863541i) q^{43} +(-0.178671 + 0.431351i) q^{44} +(0.156434 + 0.987688i) q^{48} +(-0.156434 - 0.987688i) q^{49} +(-0.707107 + 0.707107i) q^{50} +(-0.178671 - 0.744220i) q^{51} +(-0.809017 - 0.587785i) q^{54} +(1.04178 + 1.70002i) q^{57} +(-1.34500 + 0.437016i) q^{59} +(0.951057 - 0.309017i) q^{64} +(-0.398090 + 0.243950i) q^{66} +(-0.546010 + 0.891007i) q^{67} +(-0.707107 + 0.292893i) q^{68} +(-0.453990 + 0.891007i) q^{72} +(0.437016 - 0.437016i) q^{73} +(-0.987688 + 0.156434i) q^{75} +(1.51612 - 1.29489i) q^{76} +(-0.309017 - 0.951057i) q^{81} +(0.809017 + 0.587785i) q^{82} +1.61803i q^{83} +(1.53884 - 1.11803i) q^{86} +(0.303221 + 0.355026i) q^{88} +(1.29671 + 1.10749i) q^{89} +(0.951057 + 0.309017i) q^{96} +(-0.133795 - 0.0819895i) q^{97} +(-0.951057 - 0.309017i) q^{98} +(-0.465451 - 0.0366318i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{16} - 4 q^{17} + 4 q^{22} - 4 q^{24} - 4 q^{33} - 4 q^{34} - 16 q^{36} + 4 q^{44} + 4 q^{51} - 4 q^{54} - 4 q^{66} - 16 q^{67} + 4 q^{76} + 4 q^{81} + 4 q^{82} - 4 q^{89} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/984\mathbb{Z}\right)^\times\).

\(n\) \(247\) \(329\) \(457\) \(493\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{11}{40}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.453990 0.891007i 0.453990 0.891007i
\(3\) 0.891007 0.453990i 0.891007 0.453990i
\(4\) −0.587785 0.809017i −0.587785 0.809017i
\(5\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(6\) 1.00000i 1.00000i
\(7\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(8\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(9\) 0.587785 0.809017i 0.587785 0.809017i
\(10\) 0 0
\(11\) −0.243950 0.398090i −0.243950 0.398090i 0.707107 0.707107i \(-0.250000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(12\) −0.891007 0.453990i −0.891007 0.453990i
\(13\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(17\) 0.178671 0.744220i 0.178671 0.744220i −0.809017 0.587785i \(-0.800000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(18\) −0.453990 0.891007i −0.453990 0.891007i
\(19\) 0.156434 + 1.98769i 0.156434 + 1.98769i 0.156434 + 0.987688i \(0.450000\pi\)
1.00000i \(0.500000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.465451 + 0.0366318i −0.465451 + 0.0366318i
\(23\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(25\) −0.951057 0.309017i −0.951057 0.309017i
\(26\) 0 0
\(27\) 0.156434 0.987688i 0.156434 0.987688i
\(28\) 0 0
\(29\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(30\) 0 0
\(31\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(32\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(33\) −0.398090 0.243950i −0.398090 0.243950i
\(34\) −0.581990 0.497066i −0.581990 0.497066i
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(38\) 1.84206 + 0.763007i 1.84206 + 0.763007i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(42\) 0 0
\(43\) 1.69480 + 0.863541i 1.69480 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) −0.178671 + 0.431351i −0.178671 + 0.431351i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(48\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(49\) −0.156434 0.987688i −0.156434 0.987688i
\(50\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(51\) −0.178671 0.744220i −0.178671 0.744220i
\(52\) 0 0
\(53\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(54\) −0.809017 0.587785i −0.809017 0.587785i
\(55\) 0 0
\(56\) 0 0
\(57\) 1.04178 + 1.70002i 1.04178 + 1.70002i
\(58\) 0 0
\(59\) −1.34500 + 0.437016i −1.34500 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(60\) 0 0
\(61\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.951057 0.309017i 0.951057 0.309017i
\(65\) 0 0
\(66\) −0.398090 + 0.243950i −0.398090 + 0.243950i
\(67\) −0.546010 + 0.891007i −0.546010 + 0.891007i 0.453990 + 0.891007i \(0.350000\pi\)
−1.00000 \(\pi\)
\(68\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(72\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(73\) 0.437016 0.437016i 0.437016 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(74\) 0 0
\(75\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(76\) 1.51612 1.29489i 1.51612 1.29489i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(80\) 0 0
\(81\) −0.309017 0.951057i −0.309017 0.951057i
\(82\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(83\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.53884 1.11803i 1.53884 1.11803i
\(87\) 0 0
\(88\) 0.303221 + 0.355026i 0.303221 + 0.355026i
\(89\) 1.29671 + 1.10749i 1.29671 + 1.10749i 0.987688 + 0.156434i \(0.0500000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(97\) −0.133795 0.0819895i −0.133795 0.0819895i 0.453990 0.891007i \(-0.350000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(98\) −0.951057 0.309017i −0.951057 0.309017i
\(99\) −0.465451 0.0366318i −0.465451 0.0366318i
\(100\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(101\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(102\) −0.744220 0.178671i −0.744220 0.178671i
\(103\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.363271 1.11803i 0.363271 1.11803i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(108\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(109\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.734572 0.533698i −0.734572 0.533698i 0.156434 0.987688i \(-0.450000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(114\) 1.98769 0.156434i 1.98769 0.156434i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −0.221232 + 1.39680i −0.221232 + 1.39680i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.355026 0.696779i 0.355026 0.696779i
\(122\) 0 0
\(123\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(128\) 0.156434 0.987688i 0.156434 0.987688i
\(129\) 1.90211 1.90211
\(130\) 0 0
\(131\) −1.76007 + 0.278768i −1.76007 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(132\) 0.0366318 + 0.465451i 0.0366318 + 0.465451i
\(133\) 0 0
\(134\) 0.546010 + 0.891007i 0.546010 + 0.891007i
\(135\) 0 0
\(136\) −0.0600500 + 0.763007i −0.0600500 + 0.763007i
\(137\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(138\) 0 0
\(139\) 0.280582 0.863541i 0.280582 0.863541i −0.707107 0.707107i \(-0.750000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(145\) 0 0
\(146\) −0.190983 0.587785i −0.190983 0.587785i
\(147\) −0.587785 0.809017i −0.587785 0.809017i
\(148\) 0 0
\(149\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(150\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(151\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(152\) −0.465451 1.93874i −0.465451 1.93874i
\(153\) −0.497066 0.581990i −0.497066 0.581990i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.760406 0.649448i \(-0.775000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.987688 0.156434i −0.987688 0.156434i
\(163\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) 0.891007 0.453990i 0.891007 0.453990i
\(165\) 0 0
\(166\) 1.44168 + 0.734572i 1.44168 + 0.734572i
\(167\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(168\) 0 0
\(169\) −0.987688 0.156434i −0.987688 0.156434i
\(170\) 0 0
\(171\) 1.70002 + 1.04178i 1.70002 + 1.04178i
\(172\) −0.297556 1.87869i −0.297556 1.87869i
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.453990 0.108993i 0.453990 0.108993i
\(177\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(178\) 1.57547 0.652583i 1.57547 0.652583i
\(179\) −1.01612 + 1.65816i −1.01612 + 1.65816i −0.309017 + 0.951057i \(0.600000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(180\) 0 0
\(181\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.339853 + 0.110425i −0.339853 + 0.110425i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(192\) 0.707107 0.707107i 0.707107 0.707107i
\(193\) 1.65816 0.398090i 1.65816 0.398090i 0.707107 0.707107i \(-0.250000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(194\) −0.133795 + 0.0819895i −0.133795 + 0.0819895i
\(195\) 0 0
\(196\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(197\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(198\) −0.243950 + 0.398090i −0.243950 + 0.398090i
\(199\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(200\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(201\) −0.0819895 + 1.04178i −0.0819895 + 1.04178i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.497066 + 0.581990i −0.497066 + 0.581990i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.753116 0.547171i 0.753116 0.547171i
\(210\) 0 0
\(211\) −1.20002 1.40505i −1.20002 1.40505i −0.891007 0.453990i \(-0.850000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −0.831254 0.831254i −0.831254 0.831254i
\(215\) 0 0
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 0 0
\(219\) 0.190983 0.587785i 0.190983 0.587785i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(224\) 0 0
\(225\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(226\) −0.809017 + 0.412215i −0.809017 + 0.412215i
\(227\) −0.144974 1.84206i −0.144974 1.84206i −0.453990 0.891007i \(-0.650000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(228\) 0.763007 1.84206i 0.763007 1.84206i
\(229\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.152583 + 1.93874i −0.152583 + 1.93874i 0.156434 + 0.987688i \(0.450000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.14412 + 0.831254i 1.14412 + 0.831254i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(240\) 0 0
\(241\) 0.183900 1.16110i 0.183900 1.16110i −0.707107 0.707107i \(-0.750000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(242\) −0.459656 0.632662i −0.459656 0.632662i
\(243\) −0.707107 0.707107i −0.707107 0.707107i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.734572 + 1.44168i 0.734572 + 1.44168i
\(250\) 0 0
\(251\) −0.0489435 + 0.309017i −0.0489435 + 0.309017i 0.951057 + 0.309017i \(0.100000\pi\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 0.587785i −0.809017 0.587785i
\(257\) 0.0819895 + 0.133795i 0.0819895 + 0.133795i 0.891007 0.453990i \(-0.150000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(258\) 0.863541 1.69480i 0.863541 1.69480i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.550672 + 1.69480i −0.550672 + 1.69480i
\(263\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(264\) 0.431351 + 0.178671i 0.431351 + 0.178671i
\(265\) 0 0
\(266\) 0 0
\(267\) 1.65816 + 0.398090i 1.65816 + 0.398090i
\(268\) 1.04178 0.0819895i 1.04178 0.0819895i
\(269\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 0.652583 + 0.399903i 0.652583 + 0.399903i
\(273\) 0 0
\(274\) −1.84206 0.144974i −1.84206 0.144974i
\(275\) 0.108993 + 0.453990i 0.108993 + 0.453990i
\(276\) 0 0
\(277\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(278\) −0.642040 0.642040i −0.642040 0.642040i
\(279\) 0 0
\(280\) 0 0
\(281\) −0.678671 0.794622i −0.678671 0.794622i 0.309017 0.951057i \(-0.400000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(282\) 0 0
\(283\) −1.14412 + 0.831254i −1.14412 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.987688 0.156434i 0.987688 0.156434i
\(289\) 0.369067 + 0.188049i 0.369067 + 0.188049i
\(290\) 0 0
\(291\) −0.156434 0.0123117i −0.156434 0.0123117i
\(292\) −0.610425 0.0966818i −0.610425 0.0966818i
\(293\) 0 0 0.760406 0.649448i \(-0.225000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(294\) −0.987688 + 0.156434i −0.987688 + 0.156434i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.431351 + 0.178671i −0.431351 + 0.178671i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.93874 0.465451i −1.93874 0.465451i
\(305\) 0 0
\(306\) −0.744220 + 0.178671i −0.744220 + 0.178671i
\(307\) −0.809017 1.58779i −0.809017 1.58779i −0.809017 0.587785i \(-0.800000\pi\)
1.00000i \(-0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(312\) 0 0
\(313\) −1.01612 + 1.65816i −1.01612 + 1.65816i −0.309017 + 0.951057i \(0.600000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.183900 1.16110i −0.183900 1.16110i
\(322\) 0 0
\(323\) 1.50723 + 0.238721i 1.50723 + 0.238721i
\(324\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(325\) 0 0
\(326\) 1.04744 + 0.533698i 1.04744 + 0.533698i
\(327\) 0 0
\(328\) 1.00000i 1.00000i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.144974 0.0600500i −0.144974 0.0600500i 0.309017 0.951057i \(-0.400000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(332\) 1.30902 0.951057i 1.30902 0.951057i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.39680 1.39680i −1.39680 1.39680i −0.809017 0.587785i \(-0.800000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(338\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(339\) −0.896802 0.142040i −0.896802 0.142040i
\(340\) 0 0
\(341\) 0 0
\(342\) 1.70002 1.04178i 1.70002 1.04178i
\(343\) 0 0
\(344\) −1.80902 0.587785i −1.80902 0.587785i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.70002 0.133795i 1.70002 0.133795i 0.809017 0.587785i \(-0.200000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(348\) 0 0
\(349\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.108993 0.453990i 0.108993 0.453990i
\(353\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(354\) 0.437016 + 1.34500i 0.437016 + 1.34500i
\(355\) 0 0
\(356\) 0.133795 1.70002i 0.133795 1.70002i
\(357\) 0 0
\(358\) 1.01612 + 1.65816i 1.01612 + 1.65816i
\(359\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(360\) 0 0
\(361\) −2.93874 + 0.465451i −2.93874 + 0.465451i
\(362\) 0 0
\(363\) 0.782013i 0.782013i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(368\) 0 0
\(369\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(374\) −0.0559007 + 0.352943i −0.0559007 + 0.352943i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.253116 0.183900i −0.253116 0.183900i 0.453990 0.891007i \(-0.350000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(384\) −0.309017 0.951057i −0.309017 0.951057i
\(385\) 0 0
\(386\) 0.398090 1.65816i 0.398090 1.65816i
\(387\) 1.69480 0.863541i 1.69480 0.863541i
\(388\) 0.0123117 + 0.156434i 0.0123117 + 0.156434i
\(389\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(393\) −1.44168 + 1.04744i −1.44168 + 1.04744i
\(394\) 0 0
\(395\) 0 0
\(396\) 0.243950 + 0.398090i 0.243950 + 0.398090i
\(397\) 0 0 −0.996917 0.0784591i \(-0.975000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.587785 0.809017i 0.587785 0.809017i
\(401\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(402\) 0.891007 + 0.546010i 0.891007 + 0.546010i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(409\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(410\) 0 0
\(411\) −1.40505 1.20002i −1.40505 1.20002i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.142040 0.896802i −0.142040 0.896802i
\(418\) −0.145625 0.919442i −0.145625 0.919442i
\(419\) 1.34500 1.34500i 1.34500 1.34500i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(420\) 0 0
\(421\) 0 0 0.852640 0.522499i \(-0.175000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(422\) −1.79671 + 0.431351i −1.79671 + 0.431351i
\(423\) 0 0
\(424\) 0 0
\(425\) −0.399903 + 0.652583i −0.399903 + 0.652583i
\(426\) 0 0
\(427\) 0 0
\(428\) −1.11803 + 0.363271i −1.11803 + 0.363271i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(432\) 0.891007 + 0.453990i 0.891007 + 0.453990i
\(433\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.437016 0.437016i −0.437016 0.437016i
\(439\) 0 0 0.972370 0.233445i \(-0.0750000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(440\) 0 0
\(441\) −0.891007 0.453990i −0.891007 0.453990i
\(442\) 0 0
\(443\) −0.309017 1.95106i −0.309017 1.95106i −0.309017 0.951057i \(-0.600000\pi\)
1.00000i \(-0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.58779 0.809017i −1.58779 0.809017i −0.587785 0.809017i \(-0.700000\pi\)
−1.00000 \(\pi\)
\(450\) 0.156434 + 0.987688i 0.156434 + 0.987688i
\(451\) 0.431351 0.178671i 0.431351 0.178671i
\(452\) 0.907981i 0.907981i
\(453\) 0 0
\(454\) −1.70711 0.707107i −1.70711 0.707107i
\(455\) 0 0
\(456\) −1.29489 1.51612i −1.29489 1.51612i
\(457\) 0.678671 + 0.794622i 0.678671 + 0.794622i 0.987688 0.156434i \(-0.0500000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(458\) 0 0
\(459\) −0.707107 0.292893i −0.707107 0.292893i
\(460\) 0 0
\(461\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.65816 + 1.01612i 1.65816 + 1.01612i
\(467\) 1.87869 + 0.610425i 1.87869 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.26007 0.642040i 1.26007 0.642040i
\(473\) −0.0696779 0.885341i −0.0696779 0.885341i
\(474\) 0 0
\(475\) 0.465451 1.93874i 0.465451 1.93874i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.951057 0.690983i −0.951057 0.690983i
\(483\) 0 0
\(484\) −0.772385 + 0.122334i −0.772385 + 0.122334i
\(485\) 0 0
\(486\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(487\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(488\) 0 0
\(489\) 0.533698 + 1.04744i 0.533698 + 1.04744i
\(490\) 0 0
\(491\) −0.312869 −0.312869 −0.156434 0.987688i \(-0.550000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(492\) 0.587785 0.809017i 0.587785 0.809017i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 1.61803 1.61803
\(499\) −1.10749 + 1.29671i −1.10749 + 1.29671i −0.156434 + 0.987688i \(0.550000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.253116 + 0.183900i 0.253116 + 0.183900i
\(503\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(508\) 0 0
\(509\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(513\) 1.98769 + 0.156434i 1.98769 + 0.156434i
\(514\) 0.156434 0.0123117i 0.156434 0.0123117i
\(515\) 0 0
\(516\) −1.11803 1.53884i −1.11803 1.53884i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.453990 + 1.89101i 0.453990 + 1.89101i 0.453990 + 0.891007i \(0.350000\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(524\) 1.26007 + 1.26007i 1.26007 + 1.26007i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0.355026 0.303221i 0.355026 0.303221i
\(529\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(530\) 0 0
\(531\) −0.437016 + 1.34500i −0.437016 + 1.34500i
\(532\) 0 0
\(533\) 0 0
\(534\) 1.10749 1.29671i 1.10749 1.29671i
\(535\) 0 0
\(536\) 0.399903 0.965451i 0.399903 0.965451i
\(537\) −0.152583 + 1.93874i −0.152583 + 1.93874i
\(538\) 0 0
\(539\) −0.355026 + 0.303221i −0.355026 + 0.303221i
\(540\) 0 0
\(541\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.652583 0.399903i 0.652583 0.399903i
\(545\) 0 0
\(546\) 0 0
\(547\) −0.144974 + 0.0600500i −0.144974 + 0.0600500i −0.453990 0.891007i \(-0.650000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(548\) −0.965451 + 1.57547i −0.965451 + 1.57547i
\(549\) 0 0
\(550\) 0.453990 + 0.108993i 0.453990 + 0.108993i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.863541 + 0.280582i −0.863541 + 0.280582i
\(557\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −0.252679 + 0.252679i −0.252679 + 0.252679i
\(562\) −1.01612 + 0.243950i −1.01612 + 0.243950i
\(563\) 1.10749 0.678671i 1.10749 0.678671i 0.156434 0.987688i \(-0.450000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.221232 + 1.39680i 0.221232 + 1.39680i
\(567\) 0 0
\(568\) 0 0
\(569\) 1.16110 + 0.183900i 1.16110 + 0.183900i 0.707107 0.707107i \(-0.250000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(570\) 0 0
\(571\) 0.581990 1.40505i 0.581990 1.40505i −0.309017 0.951057i \(-0.600000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.309017 0.951057i 0.309017 0.951057i
\(577\) 1.40505 + 0.581990i 1.40505 + 0.581990i 0.951057 0.309017i \(-0.100000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(578\) 0.335106 0.243469i 0.335106 0.243469i
\(579\) 1.29671 1.10749i 1.29671 1.10749i
\(580\) 0 0
\(581\) 0 0
\(582\) −0.0819895 + 0.133795i −0.0819895 + 0.133795i
\(583\) 0 0
\(584\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(585\) 0 0
\(586\) 0 0
\(587\) 1.04178 + 0.0819895i 1.04178 + 0.0819895i 0.587785 0.809017i \(-0.300000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(588\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.29489 0.101910i 1.29489 0.101910i 0.587785 0.809017i \(-0.300000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) −0.0366318 + 0.465451i −0.0366318 + 0.465451i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(600\) 0.951057 0.309017i 0.951057 0.309017i
\(601\) 0.744220 + 1.79671i 0.744220 + 1.79671i 0.587785 + 0.809017i \(0.300000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(602\) 0 0
\(603\) 0.399903 + 0.965451i 0.399903 + 0.965451i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(608\) −1.29489 + 1.51612i −1.29489 + 1.51612i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.178671 + 0.744220i −0.178671 + 0.744220i
\(613\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(614\) −1.78201 −1.78201
\(615\) 0 0
\(616\) 0 0
\(617\) −0.142040 + 0.278768i −0.142040 + 0.278768i −0.951057 0.309017i \(-0.900000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(618\) 0 0
\(619\) −0.183900 0.253116i −0.183900 0.253116i 0.707107 0.707107i \(-0.250000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(626\) 1.01612 + 1.65816i 1.01612 + 1.65816i
\(627\) 0.422621 0.829441i 0.422621 0.829441i
\(628\) 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
\(633\) −1.70711 0.707107i −1.70711 0.707107i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.10749 0.678671i −1.10749 0.678671i −0.156434 0.987688i \(-0.550000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(642\) −1.11803 0.363271i −1.11803 0.363271i
\(643\) 1.84206 + 0.144974i 1.84206 + 0.144974i 0.951057 0.309017i \(-0.100000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.896969 1.23457i 0.896969 1.23457i
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(649\) 0.502083 + 0.428820i 0.502083 + 0.428820i
\(650\) 0 0
\(651\) 0 0
\(652\) 0.951057 0.690983i 0.951057 0.690983i
\(653\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.891007 0.453990i −0.891007 0.453990i
\(657\) −0.0966818 0.610425i −0.0966818 0.610425i
\(658\) 0 0
\(659\) −0.707107 + 1.70711i −0.707107 + 1.70711i 1.00000i \(0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 0 0 −0.987688 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(662\) −0.119322 + 0.101910i −0.119322 + 0.101910i
\(663\) 0 0
\(664\) −0.253116 1.59811i −0.253116 1.59811i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.47879 + 0.355026i 1.47879 + 0.355026i 0.891007 0.453990i \(-0.150000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(674\) −1.87869 + 0.610425i −1.87869 + 0.610425i
\(675\) −0.453990 + 0.891007i −0.453990 + 0.891007i
\(676\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(677\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(678\) −0.533698 + 0.734572i −0.533698 + 0.734572i
\(679\) 0 0
\(680\) 0 0
\(681\) −0.965451 1.57547i −0.965451 1.57547i
\(682\) 0 0
\(683\) 1.79671 0.744220i 1.79671 0.744220i 0.809017 0.587785i \(-0.200000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(684\) −0.156434 1.98769i −0.156434 1.98769i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.34500 + 1.34500i −1.34500 + 1.34500i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.15643 0.987688i 1.15643 0.987688i 0.156434 0.987688i \(-0.450000\pi\)
1.00000 \(0\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.652583 1.57547i 0.652583 1.57547i
\(695\) 0 0
\(696\) 0 0
\(697\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(698\) 0 0
\(699\) 0.744220 + 1.79671i 0.744220 + 1.79671i
\(700\) 0 0
\(701\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.355026 0.303221i −0.355026 0.303221i
\(705\) 0 0
\(706\) 1.34500 + 1.34500i 1.34500 + 1.34500i
\(707\) 0 0
\(708\) 1.39680 + 0.221232i 1.39680 + 0.221232i
\(709\) 0 0 −0.233445 0.972370i \(-0.575000\pi\)
0.233445 + 0.972370i \(0.425000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.45399 0.891007i −1.45399 0.891007i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.93874 0.152583i 1.93874 0.152583i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.919442 + 2.82975i −0.919442 + 2.82975i
\(723\) −0.363271 1.11803i −0.363271 1.11803i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.696779 0.355026i −0.696779 0.355026i
\(727\) 0 0 −0.522499 0.852640i \(-0.675000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(728\) 0 0
\(729\) −0.951057 0.309017i −0.951057 0.309017i
\(730\) 0 0
\(731\) 0.945476 1.10701i 0.945476 1.10701i
\(732\) 0 0
\(733\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.487899 0.487899
\(738\) 0.951057 0.309017i 0.951057 0.309017i
\(739\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(748\) 0.289096 + 0.210041i 0.289096 + 0.210041i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(752\) 0 0
\(753\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.0784591 0.996917i \(-0.525000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(758\) −0.278768 + 0.142040i −0.278768 + 0.142040i
\(759\) 0 0
\(760\) 0 0
\(761\) 0.437016 + 1.34500i 0.437016 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.987688 0.156434i −0.987688 0.156434i
\(769\) 1.11803 1.53884i 1.11803 1.53884i 0.309017 0.951057i \(-0.400000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(770\) 0 0
\(771\) 0.133795 + 0.0819895i 0.133795 + 0.0819895i
\(772\) −1.29671 1.10749i −1.29671 1.10749i
\(773\) 0 0 −0.649448 0.760406i \(-0.725000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(774\) 1.90211i 1.90211i
\(775\) 0 0
\(776\) 0.144974 + 0.0600500i 0.144974 + 0.0600500i
\(777\) 0 0
\(778\) 0 0
\(779\) −1.98769 0.156434i −1.98769 0.156434i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.987688 + 0.156434i 0.987688 + 0.156434i
\(785\) 0 0
\(786\) 0.278768 + 1.76007i 0.278768 + 1.76007i
\(787\) 0.0489435 + 0.309017i 0.0489435 + 0.309017i 1.00000 \(0\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.465451 0.0366318i 0.465451 0.0366318i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.453990 0.891007i −0.453990 0.891007i
\(801\) 1.65816 0.398090i 1.65816 0.398090i
\(802\) −1.53884 + 0.500000i −1.53884 + 0.500000i
\(803\) −0.280582 0.0673617i −0.280582 0.0673617i
\(804\) 0.891007 0.546010i 0.891007 0.546010i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.70002 1.04178i 1.70002 1.04178i 0.809017 0.587785i \(-0.200000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(810\) 0 0
\(811\) −1.39680 + 1.39680i −1.39680 + 1.39680i −0.587785 + 0.809017i \(0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0.763007 + 0.0600500i 0.763007 + 0.0600500i
\(817\) −1.45133 + 3.50381i −1.45133 + 3.50381i
\(818\) −1.44168 0.734572i −1.44168 0.734572i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(823\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(824\) 0 0
\(825\) 0.303221 + 0.355026i 0.303221 + 0.355026i
\(826\) 0 0
\(827\) 0.987688 + 0.843566i 0.987688 + 0.843566i 0.987688 0.156434i \(-0.0500000\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.763007 0.0600500i −0.763007 0.0600500i
\(834\) −0.863541 0.280582i −0.863541 0.280582i
\(835\) 0 0
\(836\) −0.885341 0.287665i −0.885341 0.287665i
\(837\) 0 0
\(838\) −0.587785 1.80902i −0.587785 1.80902i
\(839\) 0 0 0.996917 0.0784591i \(-0.0250000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(840\) 0 0
\(841\) −0.891007 + 0.453990i −0.891007 + 0.453990i
\(842\) 0 0
\(843\) −0.965451 0.399903i −0.965451 0.399903i
\(844\) −0.431351 + 1.79671i −0.431351 + 1.79671i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.642040 + 1.26007i −0.642040 + 1.26007i
\(850\) 0.399903 + 0.652583i 0.399903 + 0.652583i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.987688 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.183900 + 1.16110i −0.183900 + 1.16110i
\(857\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(858\) 0 0
\(859\) −0.896802 + 1.76007i −0.896802 + 1.76007i −0.309017 + 0.951057i \(0.600000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(864\) 0.809017 0.587785i 0.809017 0.587785i
\(865\) 0 0
\(866\) 0.142040 0.896802i 0.142040 0.896802i
\(867\) 0.414214 0.414214
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.144974 + 0.0600500i −0.144974 + 0.0600500i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(877\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.04744 0.533698i 1.04744 0.533698i 0.156434 0.987688i \(-0.450000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(882\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(883\) −1.51612 + 0.119322i −1.51612 + 0.119322i −0.809017 0.587785i \(-0.800000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.87869 0.610425i −1.87869 0.610425i
\(887\) 0 0 −0.852640 0.522499i \(-0.825000\pi\)
0.852640 + 0.522499i \(0.175000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.303221 + 0.355026i −0.303221 + 0.355026i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.44168 + 1.04744i −1.44168 + 1.04744i
\(899\) 0 0
\(900\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(901\) 0 0
\(902\) 0.0366318 0.465451i 0.0366318 0.465451i
\(903\) 0 0
\(904\) 0.809017 + 0.412215i 0.809017 + 0.412215i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.896802 + 0.142040i 0.896802 + 0.142040i 0.587785 0.809017i \(-0.300000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(908\) −1.40505 + 1.20002i −1.40505 + 1.20002i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) −1.93874 + 0.465451i −1.93874 + 0.465451i
\(913\) 0.644123 0.394719i 0.644123 0.394719i
\(914\) 1.01612 0.243950i 1.01612 0.243950i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.581990 + 0.497066i −0.581990 + 0.497066i
\(919\) 0 0 −0.972370 0.233445i \(-0.925000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(920\) 0 0
\(921\) −1.44168 1.04744i −1.44168 1.04744i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.70711 0.707107i 1.70711 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
1.00000 \(0\)
\(930\) 0 0
\(931\) 1.93874 0.465451i 1.93874 0.465451i
\(932\) 1.65816 1.01612i 1.65816 1.01612i
\(933\) 0 0
\(934\) 1.39680 1.39680i 1.39680 1.39680i
\(935\) 0 0
\(936\) 0 0
\(937\) 0.581990 0.497066i 0.581990 0.497066i −0.309017 0.951057i \(-0.600000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(938\) 0 0
\(939\) −0.152583 + 1.93874i −0.152583 + 1.93874i
\(940\) 0 0
\(941\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.41421i 1.41421i
\(945\) 0 0
\(946\) −0.820478 0.339853i −0.820478 0.339853i
\(947\) −1.44168 + 1.04744i −1.44168 + 1.04744i −0.453990 + 0.891007i \(0.650000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −1.51612 1.29489i −1.51612 1.29489i
\(951\) 0 0
\(952\) 0 0
\(953\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(954\) 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
\(963\) −0.690983 0.951057i −0.690983 0.951057i
\(964\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.233445 0.972370i \(-0.425000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(968\) −0.241655 + 0.743739i −0.241655 + 0.743739i
\(969\) 1.45133 0.471565i 1.45133 0.471565i
\(970\) 0 0
\(971\) 0.0819895 1.04178i 0.0819895 1.04178i −0.809017 0.587785i \(-0.800000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(972\) −0.156434 + 0.987688i −0.156434 + 0.987688i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.843566 + 0.987688i −0.843566 + 0.987688i 0.156434 + 0.987688i \(0.450000\pi\)
−1.00000 \(\pi\)
\(978\) 1.17557 1.17557
\(979\) 0.124550 0.786377i 0.124550 0.786377i
\(980\) 0 0
\(981\) 0 0
\(982\) −0.142040 + 0.278768i −0.142040 + 0.278768i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −0.453990 0.891007i −0.453990 0.891007i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.649448 0.760406i \(-0.275000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(992\) 0 0
\(993\) −0.156434 + 0.0123117i −0.156434 + 0.0123117i
\(994\) 0 0
\(995\) 0 0
\(996\) 0.734572 1.44168i 0.734572 1.44168i
\(997\) 0 0 0.0784591 0.996917i \(-0.475000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(998\) 0.652583 + 1.57547i 0.652583 + 1.57547i
\(999\) 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 984.1.cj.b.971.1 yes 16
3.2 odd 2 984.1.cj.a.971.1 yes 16
4.3 odd 2 3936.1.fx.a.2447.1 16
8.3 odd 2 CM 984.1.cj.b.971.1 yes 16
8.5 even 2 3936.1.fx.a.2447.1 16
12.11 even 2 3936.1.fx.b.2447.1 16
24.5 odd 2 3936.1.fx.b.2447.1 16
24.11 even 2 984.1.cj.a.971.1 yes 16
41.22 odd 40 984.1.cj.a.227.1 16
123.104 even 40 inner 984.1.cj.b.227.1 yes 16
164.63 even 40 3936.1.fx.b.719.1 16
328.227 even 40 984.1.cj.a.227.1 16
328.309 odd 40 3936.1.fx.b.719.1 16
492.227 odd 40 3936.1.fx.a.719.1 16
984.227 odd 40 inner 984.1.cj.b.227.1 yes 16
984.965 even 40 3936.1.fx.a.719.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
984.1.cj.a.227.1 16 41.22 odd 40
984.1.cj.a.227.1 16 328.227 even 40
984.1.cj.a.971.1 yes 16 3.2 odd 2
984.1.cj.a.971.1 yes 16 24.11 even 2
984.1.cj.b.227.1 yes 16 123.104 even 40 inner
984.1.cj.b.227.1 yes 16 984.227 odd 40 inner
984.1.cj.b.971.1 yes 16 1.1 even 1 trivial
984.1.cj.b.971.1 yes 16 8.3 odd 2 CM
3936.1.fx.a.719.1 16 492.227 odd 40
3936.1.fx.a.719.1 16 984.965 even 40
3936.1.fx.a.2447.1 16 4.3 odd 2
3936.1.fx.a.2447.1 16 8.5 even 2
3936.1.fx.b.719.1 16 164.63 even 40
3936.1.fx.b.719.1 16 328.309 odd 40
3936.1.fx.b.2447.1 16 12.11 even 2
3936.1.fx.b.2447.1 16 24.5 odd 2