Properties

Label 2023.1.l.b.468.2
Level $2023$
Weight $1$
Character 2023.468
Analytic conductor $1.010$
Analytic rank $0$
Dimension $16$
Projective image $D_{5}$
CM discriminant -119
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,1,Mod(468,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2023.468");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2023.l (of order \(8\), 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: \(1.00960852056\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.109951162777600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 47x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 119)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.14161.1
Artin image: $D_5\times C_{16}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\)

Embedding invariants

Embedding label 468.2
Root \(-0.236511 + 0.570989i\) of defining polynomial
Character \(\chi\) \(=\) 2023.468
Dual form 2023.1.l.b.1889.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.437016 + 0.437016i) q^{2} +(0.236511 - 0.570989i) q^{3} +0.618034i q^{4} +(-1.49487 - 0.619195i) q^{5} +(0.146172 + 0.352891i) q^{6} +(0.923880 - 0.382683i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.437016 + 0.437016i) q^{9} +O(q^{10})\) \(q+(-0.437016 + 0.437016i) q^{2} +(0.236511 - 0.570989i) q^{3} +0.618034i q^{4} +(-1.49487 - 0.619195i) q^{5} +(0.146172 + 0.352891i) q^{6} +(0.923880 - 0.382683i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.437016 + 0.437016i) q^{9} +(0.923880 - 0.382683i) q^{10} +(0.352891 + 0.146172i) q^{12} +(-0.236511 + 0.570989i) q^{14} +(-0.707107 + 0.707107i) q^{15} -0.381966 q^{18} +(0.382683 - 0.923880i) q^{20} -0.618034i q^{21} +(-0.570989 + 0.236511i) q^{24} +(1.14412 + 1.14412i) q^{25} +(0.923880 - 0.382683i) q^{27} +(0.236511 + 0.570989i) q^{28} -0.618034i q^{30} +(0.619195 - 1.49487i) q^{31} +(0.707107 - 0.707107i) q^{32} -1.61803 q^{35} +(-0.270091 + 0.270091i) q^{36} +(0.619195 + 1.49487i) q^{40} +(0.570989 - 0.236511i) q^{41} +(0.270091 + 0.270091i) q^{42} +(-0.437016 - 0.437016i) q^{43} +(-0.382683 - 0.923880i) q^{45} +(0.707107 - 0.707107i) q^{49} -1.00000 q^{50} +(1.14412 - 1.14412i) q^{53} +(-0.236511 + 0.570989i) q^{54} +(-0.923880 - 0.382683i) q^{56} +(-0.437016 - 0.437016i) q^{60} +(-0.570989 + 0.236511i) q^{61} +(0.382683 + 0.923880i) q^{62} +(0.570989 + 0.236511i) q^{63} +0.618034i q^{64} +1.61803 q^{67} +(0.707107 - 0.707107i) q^{70} -0.618034i q^{72} +(0.570989 + 0.236511i) q^{73} +(0.923880 - 0.382683i) q^{75} +(-0.146172 + 0.352891i) q^{82} +0.381966 q^{84} +0.381966 q^{86} +(0.570989 + 0.236511i) q^{90} +(-0.707107 - 0.707107i) q^{93} +(-0.236511 - 0.570989i) q^{96} +(1.49487 + 0.619195i) q^{97} +0.618034i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{18} - 8 q^{35} - 16 q^{50} + 8 q^{67} + 24 q^{84} + 24 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(1737\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.437016 + 0.437016i −0.437016 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(3\) 0.236511 0.570989i 0.236511 0.570989i −0.760406 0.649448i \(-0.775000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(4\) 0.618034i 0.618034i
\(5\) −1.49487 0.619195i −1.49487 0.619195i −0.522499 0.852640i \(-0.675000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(6\) 0.146172 + 0.352891i 0.146172 + 0.352891i
\(7\) 0.923880 0.382683i 0.923880 0.382683i
\(8\) −0.707107 0.707107i −0.707107 0.707107i
\(9\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(10\) 0.923880 0.382683i 0.923880 0.382683i
\(11\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(12\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(15\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(16\) 0 0
\(17\) 0 0
\(18\) −0.381966 −0.381966
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0.382683 0.923880i 0.382683 0.923880i
\(21\) 0.618034i 0.618034i
\(22\) 0 0
\(23\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(24\) −0.570989 + 0.236511i −0.570989 + 0.236511i
\(25\) 1.14412 + 1.14412i 1.14412 + 1.14412i
\(26\) 0 0
\(27\) 0.923880 0.382683i 0.923880 0.382683i
\(28\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(29\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0.618034i 0.618034i
\(31\) 0.619195 1.49487i 0.619195 1.49487i −0.233445 0.972370i \(-0.575000\pi\)
0.852640 0.522499i \(-0.175000\pi\)
\(32\) 0.707107 0.707107i 0.707107 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) −1.61803 −1.61803
\(36\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(37\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.619195 + 1.49487i 0.619195 + 1.49487i
\(41\) 0.570989 0.236511i 0.570989 0.236511i −0.0784591 0.996917i \(-0.525000\pi\)
0.649448 + 0.760406i \(0.275000\pi\)
\(42\) 0.270091 + 0.270091i 0.270091 + 0.270091i
\(43\) −0.437016 0.437016i −0.437016 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(44\) 0 0
\(45\) −0.382683 0.923880i −0.382683 0.923880i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.707107 0.707107i 0.707107 0.707107i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 1.14412 1.14412i 1.14412 1.14412i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(54\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(55\) 0 0
\(56\) −0.923880 0.382683i −0.923880 0.382683i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) −0.437016 0.437016i −0.437016 0.437016i
\(61\) −0.570989 + 0.236511i −0.570989 + 0.236511i −0.649448 0.760406i \(-0.725000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(62\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(63\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(64\) 0.618034i 0.618034i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.707107 0.707107i 0.707107 0.707107i
\(71\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(72\) 0.618034i 0.618034i
\(73\) 0.570989 + 0.236511i 0.570989 + 0.236511i 0.649448 0.760406i \(-0.275000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(74\) 0 0
\(75\) 0.923880 0.382683i 0.923880 0.382683i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.146172 + 0.352891i −0.146172 + 0.352891i
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0.381966 0.381966
\(85\) 0 0
\(86\) 0.381966 0.381966
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(91\) 0 0
\(92\) 0 0
\(93\) −0.707107 0.707107i −0.707107 0.707107i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.236511 0.570989i −0.236511 0.570989i
\(97\) 1.49487 + 0.619195i 1.49487 + 0.619195i 0.972370 0.233445i \(-0.0750000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(98\) 0.618034i 0.618034i
\(99\) 0 0
\(100\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(106\) 1.00000i 1.00000i
\(107\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(108\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(109\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 1.00000 1.00000
\(121\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(122\) 0.146172 0.352891i 0.146172 0.352891i
\(123\) 0.381966i 0.381966i
\(124\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(125\) −0.382683 0.923880i −0.382683 0.923880i
\(126\) −0.352891 + 0.146172i −0.352891 + 0.146172i
\(127\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(128\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(129\) −0.352891 + 0.146172i −0.352891 + 0.146172i
\(130\) 0 0
\(131\) −1.84776 0.765367i −1.84776 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 0.382683i \(-0.875000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(135\) −1.61803 −1.61803
\(136\) 0 0
\(137\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) −0.619195 + 1.49487i −0.619195 + 1.49487i 0.233445 + 0.972370i \(0.425000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(140\) 1.00000i 1.00000i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −0.352891 + 0.146172i −0.352891 + 0.146172i
\(147\) −0.236511 0.570989i −0.236511 0.570989i
\(148\) 0 0
\(149\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(150\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(151\) 0.437016 0.437016i 0.437016 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.85123 + 1.85123i −1.85123 + 1.85123i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −0.382683 0.923880i −0.382683 0.923880i
\(160\) −1.49487 + 0.619195i −1.49487 + 0.619195i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(164\) 0.146172 + 0.352891i 0.146172 + 0.352891i
\(165\) 0 0
\(166\) 0 0
\(167\) −0.236511 + 0.570989i −0.236511 + 0.570989i −0.996917 0.0784591i \(-0.975000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(168\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0.270091 0.270091i 0.270091 0.270091i
\(173\) 0.236511 0.570989i 0.236511 0.570989i −0.760406 0.649448i \(-0.775000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(174\) 0 0
\(175\) 1.49487 + 0.619195i 1.49487 + 0.619195i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −0.437016 0.437016i −0.437016 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(180\) 0.570989 0.236511i 0.570989 0.236511i
\(181\) −0.765367 1.84776i −0.765367 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(182\) 0 0
\(183\) 0.381966i 0.381966i
\(184\) 0 0
\(185\) 0 0
\(186\) 0.618034 0.618034
\(187\) 0 0
\(188\) 0 0
\(189\) 0.707107 0.707107i 0.707107 0.707107i
\(190\) 0 0
\(191\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(192\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(193\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(194\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(195\) 0 0
\(196\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(197\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(198\) 0 0
\(199\) −0.570989 0.236511i −0.570989 0.236511i 0.0784591 0.996917i \(-0.475000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(200\) 1.61803i 1.61803i
\(201\) 0.382683 0.923880i 0.382683 0.923880i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.00000 −1.00000
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) −0.236511 0.570989i −0.236511 0.570989i
\(211\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(212\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(213\) 0 0
\(214\) 0 0
\(215\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(216\) −0.923880 0.382683i −0.923880 0.382683i
\(217\) 1.61803i 1.61803i
\(218\) 0 0
\(219\) 0.270091 0.270091i 0.270091 0.270091i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0.382683 0.923880i 0.382683 0.923880i
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) −0.619195 1.49487i −0.619195 1.49487i −0.852640 0.522499i \(-0.825000\pi\)
0.233445 0.972370i \(-0.425000\pi\)
\(228\) 0 0
\(229\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) 0 0
\(241\) −0.619195 + 1.49487i −0.619195 + 1.49487i 0.233445 + 0.972370i \(0.425000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(242\) 0.618034i 0.618034i
\(243\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(244\) −0.146172 0.352891i −0.146172 0.352891i
\(245\) −1.49487 + 0.619195i −1.49487 + 0.619195i
\(246\) 0.166925 + 0.166925i 0.166925 + 0.166925i
\(247\) 0 0
\(248\) −1.49487 + 0.619195i −1.49487 + 0.619195i
\(249\) 0 0
\(250\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −0.146172 + 0.352891i −0.146172 + 0.352891i
\(253\) 0 0
\(254\) 1.00000 1.00000
\(255\) 0 0
\(256\) −1.00000 −1.00000
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0.0903393 0.218098i 0.0903393 0.218098i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.14198 0.473023i 1.14198 0.473023i
\(263\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) −2.41875 + 1.00188i −2.41875 + 1.00188i
\(266\) 0 0
\(267\) 0 0
\(268\) 1.00000i 1.00000i
\(269\) −0.765367 + 1.84776i −0.765367 + 1.84776i −0.382683 + 0.923880i \(0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(270\) 0.707107 0.707107i 0.707107 0.707107i
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(278\) −0.382683 0.923880i −0.382683 0.923880i
\(279\) 0.923880 0.382683i 0.923880 0.382683i
\(280\) 1.14412 + 1.14412i 1.14412 + 1.14412i
\(281\) 1.14412 + 1.14412i 1.14412 + 1.14412i 0.987688 + 0.156434i \(0.0500000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(282\) 0 0
\(283\) 0.619195 + 1.49487i 0.619195 + 1.49487i 0.852640 + 0.522499i \(0.175000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.437016 0.437016i 0.437016 0.437016i
\(288\) 0.618034 0.618034
\(289\) 0 0
\(290\) 0 0
\(291\) 0.707107 0.707107i 0.707107 0.707107i
\(292\) −0.146172 + 0.352891i −0.146172 + 0.352891i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −0.270091 0.270091i −0.270091 0.270091i
\(299\) 0 0
\(300\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(301\) −0.570989 0.236511i −0.570989 0.236511i
\(302\) 0.381966i 0.381966i
\(303\) 0 0
\(304\) 0 0
\(305\) 1.00000 1.00000
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.61803i 1.61803i
\(311\) −1.49487 0.619195i −1.49487 0.619195i −0.522499 0.852640i \(-0.675000\pi\)
−0.972370 + 0.233445i \(0.925000\pi\)
\(312\) 0 0
\(313\) −1.49487 + 0.619195i −1.49487 + 0.619195i −0.972370 0.233445i \(-0.925000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(314\) 0 0
\(315\) −0.707107 0.707107i −0.707107 0.707107i
\(316\) 0 0
\(317\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(319\) 0 0
\(320\) 0.382683 0.923880i 0.382683 0.923880i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −0.570989 0.236511i −0.570989 0.236511i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −0.146172 0.352891i −0.146172 0.352891i
\(335\) −2.41875 1.00188i −2.41875 1.00188i
\(336\) 0 0
\(337\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(338\) 0.437016 0.437016i 0.437016 0.437016i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.382683 0.923880i 0.382683 0.923880i
\(344\) 0.618034i 0.618034i
\(345\) 0 0
\(346\) 0.146172 + 0.352891i 0.146172 + 0.352891i
\(347\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.381966 0.381966
\(359\) 1.14412 1.14412i 1.14412 1.14412i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(360\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(361\) 1.00000i 1.00000i
\(362\) 1.14198 + 0.473023i 1.14198 + 0.473023i
\(363\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(364\) 0 0
\(365\) −0.707107 0.707107i −0.707107 0.707107i
\(366\) −0.166925 0.166925i −0.166925 0.166925i
\(367\) −0.570989 + 0.236511i −0.570989 + 0.236511i −0.649448 0.760406i \(-0.725000\pi\)
0.0784591 + 0.996917i \(0.475000\pi\)
\(368\) 0 0
\(369\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(370\) 0 0
\(371\) 0.619195 1.49487i 0.619195 1.49487i
\(372\) 0.437016 0.437016i 0.437016 0.437016i
\(373\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(374\) 0 0
\(375\) −0.618034 −0.618034
\(376\) 0 0
\(377\) 0 0
\(378\) 0.618034i 0.618034i
\(379\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(380\) 0 0
\(381\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(382\) −0.707107 0.707107i −0.707107 0.707107i
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0.352891 0.146172i 0.352891 0.146172i
\(385\) 0 0
\(386\) 0 0
\(387\) 0.381966i 0.381966i
\(388\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(389\) −1.14412 + 1.14412i −1.14412 + 1.14412i −0.156434 + 0.987688i \(0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −1.00000
\(393\) −0.874032 + 0.874032i −0.874032 + 0.874032i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.619195 1.49487i −0.619195 1.49487i −0.852640 0.522499i \(-0.825000\pi\)
0.233445 0.972370i \(-0.425000\pi\)
\(398\) 0.352891 0.146172i 0.352891 0.146172i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(402\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0.437016 0.437016i 0.437016 0.437016i
\(411\) 0.146172 0.352891i 0.146172 0.352891i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(418\) 0 0
\(419\) 0.619195 + 1.49487i 0.619195 + 1.49487i 0.852640 + 0.522499i \(0.175000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(420\) −0.570989 0.236511i −0.570989 0.236511i
\(421\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.61803 −1.61803
\(425\) 0 0
\(426\) 0 0
\(427\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(428\) 0 0
\(429\) 0 0
\(430\) −0.570989 0.236511i −0.570989 0.236511i
\(431\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.236068i 0.236068i
\(439\) −0.236511 + 0.570989i −0.236511 + 0.570989i −0.996917 0.0784591i \(-0.975000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(440\) 0 0
\(441\) 0.618034 0.618034
\(442\) 0 0
\(443\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(448\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(449\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(450\) −0.437016 0.437016i −0.437016 0.437016i
\(451\) 0 0
\(452\) 0 0
\(453\) −0.146172 0.352891i −0.146172 0.352891i
\(454\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(455\) 0 0
\(456\) 0 0
\(457\) −1.14412 + 1.14412i −1.14412 + 1.14412i −0.156434 + 0.987688i \(0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(464\) 0 0
\(465\) 0.619195 + 1.49487i 0.619195 + 1.49487i
\(466\) 0 0
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 1.49487 0.619195i 1.49487 0.619195i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.00000 1.00000
\(478\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(479\) 0.236511 0.570989i 0.236511 0.570989i −0.760406 0.649448i \(-0.775000\pi\)
0.996917 + 0.0784591i \(0.0250000\pi\)
\(480\) 1.00000i 1.00000i
\(481\) 0 0
\(482\) −0.382683 0.923880i −0.382683 0.923880i
\(483\) 0 0
\(484\) −0.437016 0.437016i −0.437016 0.437016i
\(485\) −1.85123 1.85123i −1.85123 1.85123i
\(486\) −0.570989 + 0.236511i −0.570989 + 0.236511i
\(487\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(488\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(489\) 0 0
\(490\) 0.382683 0.923880i 0.382683 0.923880i
\(491\) −1.14412 + 1.14412i −1.14412 + 1.14412i −0.156434 + 0.987688i \(0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(492\) 0.236068 0.236068
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(500\) 0.570989 0.236511i 0.570989 0.236511i
\(501\) 0.270091 + 0.270091i 0.270091 + 0.270091i
\(502\) 0 0
\(503\) 1.49487 0.619195i 1.49487 0.619195i 0.522499 0.852640i \(-0.325000\pi\)
0.972370 + 0.233445i \(0.0750000\pi\)
\(504\) −0.236511 0.570989i −0.236511 0.570989i
\(505\) 0 0
\(506\) 0 0
\(507\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(508\) 0.707107 0.707107i 0.707107 0.707107i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0.618034 0.618034
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −0.0903393 0.218098i −0.0903393 0.218098i
\(517\) 0 0
\(518\) 0 0
\(519\) −0.270091 0.270091i −0.270091 0.270091i
\(520\) 0 0
\(521\) −0.236511 0.570989i −0.236511 0.570989i 0.760406 0.649448i \(-0.225000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0.473023 1.14198i 0.473023 1.14198i
\(525\) 0.707107 0.707107i 0.707107 0.707107i
\(526\) −1.23607 −1.23607
\(527\) 0 0
\(528\) 0 0
\(529\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(530\) 0.619195 1.49487i 0.619195 1.49487i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −1.14412 1.14412i −1.14412 1.14412i
\(537\) −0.352891 + 0.146172i −0.352891 + 0.146172i
\(538\) −0.473023 1.14198i −0.473023 1.14198i
\(539\) 0 0
\(540\) 1.00000i 1.00000i
\(541\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(542\) 0 0
\(543\) −1.23607 −1.23607
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(548\) 0.381966i 0.381966i
\(549\) −0.352891 0.146172i −0.352891 0.146172i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.923880 0.382683i −0.923880 0.382683i
\(557\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.00000 −1.00000
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.923880 0.382683i −0.923880 0.382683i
\(567\) 0 0
\(568\) 0 0
\(569\) 0.437016 + 0.437016i 0.437016 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(570\) 0 0
\(571\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(572\) 0 0
\(573\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(574\) 0.381966i 0.381966i
\(575\) 0 0
\(576\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0.618034i 0.618034i
\(583\) 0 0
\(584\) −0.236511 0.570989i −0.236511 0.570989i
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0.352891 0.146172i 0.352891 0.146172i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.381966 −0.381966
\(597\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(598\) 0 0
\(599\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(600\) −0.923880 0.382683i −0.923880 0.382683i
\(601\) 0.765367 + 1.84776i 0.765367 + 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(602\) 0.352891 0.146172i 0.352891 0.146172i
\(603\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(604\) 0.270091 + 0.270091i 0.270091 + 0.270091i
\(605\) 1.49487 0.619195i 1.49487 0.619195i
\(606\) 0 0
\(607\) 1.49487 + 0.619195i 1.49487 + 0.619195i 0.972370 0.233445i \(-0.0750000\pi\)
0.522499 + 0.852640i \(0.325000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(611\) 0 0
\(612\) 0 0
\(613\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(616\) 0 0
\(617\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(618\) 0 0
\(619\) 1.84776 0.765367i 1.84776 0.765367i 0.923880 0.382683i \(-0.125000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(620\) −1.14412 1.14412i −1.14412 1.14412i
\(621\) 0 0
\(622\) 0.923880 0.382683i 0.923880 0.382683i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0.382683 0.923880i 0.382683 0.923880i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.618034 0.618034
\(631\) 1.14412 1.14412i 1.14412 1.14412i 0.156434 0.987688i \(-0.450000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00188 + 2.41875i 1.00188 + 2.41875i
\(636\) 0.570989 0.236511i 0.570989 0.236511i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −0.382683 0.923880i −0.382683 0.923880i
\(641\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(642\) 0 0
\(643\) −0.236511 + 0.570989i −0.236511 + 0.570989i −0.996917 0.0784591i \(-0.975000\pi\)
0.760406 + 0.649448i \(0.225000\pi\)
\(644\) 0 0
\(645\) 0.618034 0.618034
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.923880 0.382683i −0.923880 0.382683i
\(652\) 0 0
\(653\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(654\) 0 0
\(655\) 2.28825 + 2.28825i 2.28825 + 2.28825i
\(656\) 0 0
\(657\) 0.146172 + 0.352891i 0.146172 + 0.352891i
\(658\) 0 0
\(659\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(660\) 0 0
\(661\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 1.00000 1.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −0.352891 0.146172i −0.352891 0.146172i
\(669\) 0 0
\(670\) 1.49487 0.619195i 1.49487 0.619195i
\(671\) 0 0
\(672\) −0.437016 0.437016i −0.437016 0.437016i
\(673\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(674\) 0 0
\(675\) 1.49487 + 0.619195i 1.49487 + 0.619195i
\(676\) 0.618034i 0.618034i
\(677\) −0.765367 + 1.84776i −0.765367 + 1.84776i −0.382683 + 0.923880i \(0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(678\) 0 0
\(679\) 1.61803 1.61803
\(680\) 0 0
\(681\) −1.00000 −1.00000
\(682\) 0 0
\(683\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(684\) 0 0
\(685\) −0.923880 0.382683i −0.923880 0.382683i
\(686\) 0.236511 + 0.570989i 0.236511 + 0.570989i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.236511 0.570989i −0.236511 0.570989i 0.760406 0.649448i \(-0.225000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(692\) 0.352891 + 0.146172i 0.352891 + 0.146172i
\(693\) 0 0
\(694\) 0 0
\(695\) 1.85123 1.85123i 1.85123 1.85123i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(701\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.270091 0.270091i 0.270091 0.270091i
\(717\) 0.146172 0.352891i 0.146172 0.352891i
\(718\) 1.00000i 1.00000i
\(719\) 0.570989 + 0.236511i 0.570989 + 0.236511i 0.649448 0.760406i \(-0.275000\pi\)
−0.0784591 + 0.996917i \(0.525000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(723\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(724\) 1.14198 0.473023i 1.14198 0.473023i
\(725\) 0 0
\(726\) −0.352891 0.146172i −0.352891 0.146172i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0.437016 0.437016i 0.437016 0.437016i
\(730\) 0.618034 0.618034
\(731\) 0 0
\(732\) −0.236068 −0.236068
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0.146172 0.352891i 0.146172 0.352891i
\(735\) 1.00000i 1.00000i
\(736\) 0 0
\(737\) 0 0
\(738\) −0.218098 + 0.0903393i −0.218098 + 0.0903393i
\(739\) 0.437016 + 0.437016i 0.437016 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(743\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(744\) 1.00000i 1.00000i
\(745\) 0.382683 0.923880i 0.382683 0.923880i
\(746\) 0.270091 0.270091i 0.270091 0.270091i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0.270091 0.270091i 0.270091 0.270091i
\(751\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(756\) 0.437016 + 0.437016i 0.437016 + 0.437016i
\(757\) −0.437016 0.437016i −0.437016 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0.236511 0.570989i 0.236511 0.570989i
\(763\) 0 0
\(764\) −1.00000 −1.00000
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0.166925 + 0.166925i 0.166925 + 0.166925i
\(775\) 2.41875 1.00188i 2.41875 1.00188i
\(776\) −0.619195 1.49487i −0.619195 1.49487i
\(777\) 0 0
\(778\) 1.00000i 1.00000i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0.763932i 0.763932i
\(787\) 1.84776 + 0.765367i 1.84776 + 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(788\) 0 0
\(789\) 1.14198 0.473023i 1.14198 0.473023i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(795\) 1.61803i 1.61803i
\(796\) 0.146172 0.352891i 0.146172 0.352891i
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.61803 1.61803
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(805\) 0 0
\(806\) 0 0
\(807\) 0.874032 + 0.874032i 0.874032 + 0.874032i
\(808\) 0 0
\(809\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(810\) 0 0
\(811\) −0.570989 0.236511i −0.570989 0.236511i 0.0784591 0.996917i \(-0.475000\pi\)
−0.649448 + 0.760406i \(0.725000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0.618034i 0.618034i
\(821\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(822\) 0.0903393 + 0.218098i 0.0903393 + 0.218098i
\(823\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) −0.618034 −0.618034
\(835\) 0.707107 0.707107i 0.707107 0.707107i
\(836\) 0 0
\(837\) 1.61803i 1.61803i
\(838\) −0.923880 0.382683i −0.923880 0.382683i
\(839\) 0.765367 + 1.84776i 0.765367 + 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(840\) 0.923880 0.382683i 0.923880 0.382683i
\(841\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(842\) −0.270091 0.270091i −0.270091 0.270091i
\(843\) 0.923880 0.382683i 0.923880 0.382683i
\(844\) 0 0
\(845\) 1.49487 + 0.619195i 1.49487 + 0.619195i
\(846\) 0 0
\(847\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(848\) 0 0
\(849\) 1.00000 1.00000
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.765367 1.84776i 0.765367 1.84776i 0.382683 0.923880i \(-0.375000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(854\) 0.381966i 0.381966i
\(855\) 0 0
\(856\) 0 0
\(857\) −1.49487 + 0.619195i −1.49487 + 0.619195i −0.972370 0.233445i \(-0.925000\pi\)
−0.522499 + 0.852640i \(0.675000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(860\) −0.570989 + 0.236511i −0.570989 + 0.236511i
\(861\) −0.146172 0.352891i −0.146172 0.352891i
\(862\) 0 0
\(863\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(864\) 0.382683 0.923880i 0.382683 0.923880i
\(865\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(866\) 0 0
\(867\) 0 0
\(868\) 1.00000 1.00000
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(874\) 0 0
\(875\) −0.707107 0.707107i −0.707107 0.707107i
\(876\) 0.166925 + 0.166925i 0.166925 + 0.166925i
\(877\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(878\) −0.146172 0.352891i −0.146172 0.352891i
\(879\) 0 0
\(880\) 0 0
\(881\) 0.619195 1.49487i 0.619195 1.49487i −0.233445 0.972370i \(-0.575000\pi\)
0.852640 0.522499i \(-0.175000\pi\)
\(882\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(883\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.874032 + 0.874032i −0.874032 + 0.874032i
\(887\) −0.619195 + 1.49487i −0.619195 + 1.49487i 0.233445 + 0.972370i \(0.425000\pi\)
−0.852640 + 0.522499i \(0.825000\pi\)
\(888\) 0 0
\(889\) −1.49487 0.619195i −1.49487 0.619195i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) −0.218098 + 0.0903393i −0.218098 + 0.0903393i
\(895\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(896\) 0.570989 + 0.236511i 0.570989 + 0.236511i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.618034 −0.618034
\(901\) 0 0
\(902\) 0 0
\(903\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(904\) 0 0
\(905\) 3.23607i 3.23607i
\(906\) 0.218098 + 0.0903393i 0.218098 + 0.0903393i
\(907\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(908\) 0.923880 0.382683i 0.923880 0.382683i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.00000i 1.00000i
\(915\) 0.236511 0.570989i 0.236511 0.570989i
\(916\) 0 0
\(917\) −2.00000 −2.00000
\(918\) 0 0
\(919\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.270091 + 0.270091i 0.270091 + 0.270091i
\(927\) 0 0
\(928\) 0 0
\(929\) 0.619195 + 1.49487i 0.619195 + 1.49487i 0.852640 + 0.522499i \(0.175000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(930\) −0.923880 0.382683i −0.923880 0.382683i
\(931\) 0 0
\(932\) 0 0
\(933\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(939\) 1.00000i 1.00000i
\(940\) 0 0
\(941\) 0.236511 + 0.570989i 0.236511 + 0.570989i 0.996917 0.0784591i \(-0.0250000\pi\)
−0.760406 + 0.649448i \(0.775000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.49487 + 0.619195i −1.49487 + 0.619195i
\(946\) 0 0
\(947\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(954\) −0.437016 + 0.437016i −0.437016 + 0.437016i
\(955\) 1.00188 2.41875i 1.00188 2.41875i
\(956\) 0.381966i 0.381966i
\(957\) 0 0
\(958\) 0.146172 + 0.352891i 0.146172 + 0.352891i
\(959\) 0.570989 0.236511i 0.570989 0.236511i
\(960\) −0.437016 0.437016i −0.437016 0.437016i
\(961\) −1.14412 1.14412i −1.14412 1.14412i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.923880 0.382683i −0.923880 0.382683i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.437016 0.437016i 0.437016 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(968\) 1.00000 1.00000
\(969\) 0 0
\(970\) 1.61803 1.61803
\(971\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) −0.236511 + 0.570989i −0.236511 + 0.570989i
\(973\) 1.61803i 1.61803i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.14412 1.14412i −1.14412 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 0.987688i \(-0.550000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.382683 0.923880i −0.382683 0.923880i
\(981\) 0 0
\(982\) 1.00000i 1.00000i
\(983\) 0.619195 1.49487i 0.619195 1.49487i −0.233445 0.972370i \(-0.575000\pi\)
0.852640 0.522499i \(-0.175000\pi\)
\(984\) −0.270091 + 0.270091i −0.270091 + 0.270091i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(992\) −0.619195 1.49487i −0.619195 1.49487i
\(993\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(994\) 0 0
\(995\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(996\) 0 0
\(997\) −0.236511 0.570989i −0.236511 0.570989i 0.760406 0.649448i \(-0.225000\pi\)
−0.996917 + 0.0784591i \(0.975000\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2023.1.l.b.468.2 16
7.6 odd 2 inner 2023.1.l.b.468.1 16
17.2 even 8 inner 2023.1.l.b.1889.1 16
17.3 odd 16 2023.1.f.b.251.4 8
17.4 even 4 inner 2023.1.l.b.1266.3 16
17.5 odd 16 2023.1.f.b.1483.2 8
17.6 odd 16 2023.1.c.e.1735.1 4
17.7 odd 16 119.1.d.b.118.2 yes 2
17.8 even 8 inner 2023.1.l.b.1868.3 16
17.9 even 8 inner 2023.1.l.b.1868.4 16
17.10 odd 16 119.1.d.a.118.2 2
17.11 odd 16 2023.1.c.e.1735.2 4
17.12 odd 16 2023.1.f.b.1483.1 8
17.13 even 4 inner 2023.1.l.b.1266.4 16
17.14 odd 16 2023.1.f.b.251.3 8
17.15 even 8 inner 2023.1.l.b.1889.2 16
17.16 even 2 inner 2023.1.l.b.468.1 16
51.41 even 16 1071.1.h.a.118.1 2
51.44 even 16 1071.1.h.b.118.1 2
68.7 even 16 1904.1.n.a.1665.2 2
68.27 even 16 1904.1.n.b.1665.1 2
85.7 even 16 2975.1.b.a.2974.3 4
85.24 odd 16 2975.1.h.c.951.1 2
85.27 even 16 2975.1.b.b.2974.3 4
85.44 odd 16 2975.1.h.d.951.1 2
85.58 even 16 2975.1.b.a.2974.2 4
85.78 even 16 2975.1.b.b.2974.2 4
119.6 even 16 2023.1.c.e.1735.2 4
119.10 even 48 833.1.h.a.509.1 4
119.13 odd 4 inner 2023.1.l.b.1266.3 16
119.20 even 16 2023.1.f.b.251.3 8
119.24 even 48 833.1.h.b.509.1 4
119.27 even 16 119.1.d.b.118.2 yes 2
119.41 even 16 119.1.d.a.118.2 2
119.44 odd 48 833.1.h.b.815.1 4
119.48 even 16 2023.1.f.b.251.4 8
119.55 odd 4 inner 2023.1.l.b.1266.4 16
119.58 odd 48 833.1.h.a.815.1 4
119.61 even 48 833.1.h.a.815.1 4
119.62 even 16 2023.1.c.e.1735.1 4
119.75 even 48 833.1.h.b.815.1 4
119.76 odd 8 inner 2023.1.l.b.1868.4 16
119.83 odd 8 inner 2023.1.l.b.1889.1 16
119.90 even 16 2023.1.f.b.1483.1 8
119.95 odd 48 833.1.h.b.509.1 4
119.97 even 16 2023.1.f.b.1483.2 8
119.104 odd 8 inner 2023.1.l.b.1889.2 16
119.109 odd 48 833.1.h.a.509.1 4
119.111 odd 8 inner 2023.1.l.b.1868.3 16
119.118 odd 2 CM 2023.1.l.b.468.2 16
357.41 odd 16 1071.1.h.b.118.1 2
357.146 odd 16 1071.1.h.a.118.1 2
476.27 odd 16 1904.1.n.a.1665.2 2
476.279 odd 16 1904.1.n.b.1665.1 2
595.27 odd 16 2975.1.b.a.2974.3 4
595.279 even 16 2975.1.h.d.951.1 2
595.384 even 16 2975.1.h.c.951.1 2
595.398 odd 16 2975.1.b.b.2974.2 4
595.503 odd 16 2975.1.b.a.2974.2 4
595.517 odd 16 2975.1.b.b.2974.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
119.1.d.a.118.2 2 17.10 odd 16
119.1.d.a.118.2 2 119.41 even 16
119.1.d.b.118.2 yes 2 17.7 odd 16
119.1.d.b.118.2 yes 2 119.27 even 16
833.1.h.a.509.1 4 119.10 even 48
833.1.h.a.509.1 4 119.109 odd 48
833.1.h.a.815.1 4 119.58 odd 48
833.1.h.a.815.1 4 119.61 even 48
833.1.h.b.509.1 4 119.24 even 48
833.1.h.b.509.1 4 119.95 odd 48
833.1.h.b.815.1 4 119.44 odd 48
833.1.h.b.815.1 4 119.75 even 48
1071.1.h.a.118.1 2 51.41 even 16
1071.1.h.a.118.1 2 357.146 odd 16
1071.1.h.b.118.1 2 51.44 even 16
1071.1.h.b.118.1 2 357.41 odd 16
1904.1.n.a.1665.2 2 68.7 even 16
1904.1.n.a.1665.2 2 476.27 odd 16
1904.1.n.b.1665.1 2 68.27 even 16
1904.1.n.b.1665.1 2 476.279 odd 16
2023.1.c.e.1735.1 4 17.6 odd 16
2023.1.c.e.1735.1 4 119.62 even 16
2023.1.c.e.1735.2 4 17.11 odd 16
2023.1.c.e.1735.2 4 119.6 even 16
2023.1.f.b.251.3 8 17.14 odd 16
2023.1.f.b.251.3 8 119.20 even 16
2023.1.f.b.251.4 8 17.3 odd 16
2023.1.f.b.251.4 8 119.48 even 16
2023.1.f.b.1483.1 8 17.12 odd 16
2023.1.f.b.1483.1 8 119.90 even 16
2023.1.f.b.1483.2 8 17.5 odd 16
2023.1.f.b.1483.2 8 119.97 even 16
2023.1.l.b.468.1 16 7.6 odd 2 inner
2023.1.l.b.468.1 16 17.16 even 2 inner
2023.1.l.b.468.2 16 1.1 even 1 trivial
2023.1.l.b.468.2 16 119.118 odd 2 CM
2023.1.l.b.1266.3 16 17.4 even 4 inner
2023.1.l.b.1266.3 16 119.13 odd 4 inner
2023.1.l.b.1266.4 16 17.13 even 4 inner
2023.1.l.b.1266.4 16 119.55 odd 4 inner
2023.1.l.b.1868.3 16 17.8 even 8 inner
2023.1.l.b.1868.3 16 119.111 odd 8 inner
2023.1.l.b.1868.4 16 17.9 even 8 inner
2023.1.l.b.1868.4 16 119.76 odd 8 inner
2023.1.l.b.1889.1 16 17.2 even 8 inner
2023.1.l.b.1889.1 16 119.83 odd 8 inner
2023.1.l.b.1889.2 16 17.15 even 8 inner
2023.1.l.b.1889.2 16 119.104 odd 8 inner
2975.1.b.a.2974.2 4 85.58 even 16
2975.1.b.a.2974.2 4 595.503 odd 16
2975.1.b.a.2974.3 4 85.7 even 16
2975.1.b.a.2974.3 4 595.27 odd 16
2975.1.b.b.2974.2 4 85.78 even 16
2975.1.b.b.2974.2 4 595.398 odd 16
2975.1.b.b.2974.3 4 85.27 even 16
2975.1.b.b.2974.3 4 595.517 odd 16
2975.1.h.c.951.1 2 85.24 odd 16
2975.1.h.c.951.1 2 595.384 even 16
2975.1.h.d.951.1 2 85.44 odd 16
2975.1.h.d.951.1 2 595.279 even 16