Properties

Label 1013.123
Modulus 10131013
Conductor 10131013
Order 506506
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1013, base_ring=CyclotomicField(506)) M = H._module chi = DirichletCharacter(H, M([113]))
 
Copy content pari:[g,chi] = znchar(Mod(123,1013))
 

Basic properties

Modulus: 10131013
Conductor: 10131013
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 506506
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1013.k

χ1013(9,)\chi_{1013}(9,\cdot) χ1013(13,)\chi_{1013}(13,\cdot) χ1013(15,)\chi_{1013}(15,\cdot) χ1013(21,)\chi_{1013}(21,\cdot) χ1013(24,)\chi_{1013}(24,\cdot) χ1013(25,)\chi_{1013}(25,\cdot) χ1013(35,)\chi_{1013}(35,\cdot) χ1013(40,)\chi_{1013}(40,\cdot) χ1013(43,)\chi_{1013}(43,\cdot) χ1013(49,)\chi_{1013}(49,\cdot) χ1013(51,)\chi_{1013}(51,\cdot) χ1013(53,)\chi_{1013}(53,\cdot) χ1013(54,)\chi_{1013}(54,\cdot) χ1013(56,)\chi_{1013}(56,\cdot) χ1013(66,)\chi_{1013}(66,\cdot) χ1013(71,)\chi_{1013}(71,\cdot) χ1013(73,)\chi_{1013}(73,\cdot) χ1013(74,)\chi_{1013}(74,\cdot) χ1013(76,)\chi_{1013}(76,\cdot) χ1013(78,)\chi_{1013}(78,\cdot) χ1013(79,)\chi_{1013}(79,\cdot) χ1013(85,)\chi_{1013}(85,\cdot) χ1013(87,)\chi_{1013}(87,\cdot) χ1013(93,)\chi_{1013}(93,\cdot) χ1013(110,)\chi_{1013}(110,\cdot) χ1013(119,)\chi_{1013}(119,\cdot) χ1013(123,)\chi_{1013}(123,\cdot) χ1013(126,)\chi_{1013}(126,\cdot) χ1013(130,)\chi_{1013}(130,\cdot) χ1013(136,)\chi_{1013}(136,\cdot) ...

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ253)\Q(\zeta_{253})
Fixed field: Number field defined by a degree 506 polynomial (not computed)

Values on generators

33e(113506)e\left(\frac{113}{506}\right)

First values

aa 1-111223344556677889910101111
χ1013(123,a) \chi_{ 1013 }(123, a) 1111e(346)e\left(\frac{3}{46}\right)e(113506)e\left(\frac{113}{506}\right)e(323)e\left(\frac{3}{23}\right)e(27506)e\left(\frac{27}{506}\right)e(73253)e\left(\frac{73}{253}\right)e(175506)e\left(\frac{175}{506}\right)e(946)e\left(\frac{9}{46}\right)e(113253)e\left(\frac{113}{253}\right)e(30253)e\left(\frac{30}{253}\right)e(1523)e\left(\frac{15}{23}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ1013(123,a)   \chi_{ 1013 }(123,a) \; at   a=\;a = e.g. 2