Properties

Label 2793.1004
Modulus 27932793
Conductor 27932793
Order 126126
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2793, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,111,28]))
 
pari: [g,chi] = znchar(Mod(1004,2793))
 

Basic properties

Modulus: 27932793
Conductor: 27932793
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 126126
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2793.ef

χ2793(17,)\chi_{2793}(17,\cdot) χ2793(47,)\chi_{2793}(47,\cdot) χ2793(194,)\chi_{2793}(194,\cdot) χ2793(206,)\chi_{2793}(206,\cdot) χ2793(290,)\chi_{2793}(290,\cdot) χ2793(416,)\chi_{2793}(416,\cdot) χ2793(446,)\chi_{2793}(446,\cdot) χ2793(593,)\chi_{2793}(593,\cdot) χ2793(605,)\chi_{2793}(605,\cdot) χ2793(614,)\chi_{2793}(614,\cdot) χ2793(689,)\chi_{2793}(689,\cdot) χ2793(845,)\chi_{2793}(845,\cdot) χ2793(992,)\chi_{2793}(992,\cdot) χ2793(1004,)\chi_{2793}(1004,\cdot) χ2793(1013,)\chi_{2793}(1013,\cdot) χ2793(1088,)\chi_{2793}(1088,\cdot) χ2793(1214,)\chi_{2793}(1214,\cdot) χ2793(1412,)\chi_{2793}(1412,\cdot) χ2793(1487,)\chi_{2793}(1487,\cdot) χ2793(1613,)\chi_{2793}(1613,\cdot) χ2793(1643,)\chi_{2793}(1643,\cdot) χ2793(1790,)\chi_{2793}(1790,\cdot) χ2793(1802,)\chi_{2793}(1802,\cdot) χ2793(1811,)\chi_{2793}(1811,\cdot) χ2793(1886,)\chi_{2793}(1886,\cdot) χ2793(2012,)\chi_{2793}(2012,\cdot) χ2793(2042,)\chi_{2793}(2042,\cdot) χ2793(2189,)\chi_{2793}(2189,\cdot) χ2793(2201,)\chi_{2793}(2201,\cdot) χ2793(2210,)\chi_{2793}(2210,\cdot) ...

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

Related number fields

Field of values: Q(ζ63)\Q(\zeta_{63})
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

(932,2110,2206)(932,2110,2206)(1,e(3742),e(29))(-1,e\left(\frac{37}{42}\right),e\left(\frac{2}{9}\right))

First values

aa 1-11122445588101011111313161617172020
χ2793(1004,a) \chi_{ 2793 }(1004, a) 1111e(79126)e\left(\frac{79}{126}\right)e(1663)e\left(\frac{16}{63}\right)e(3863)e\left(\frac{38}{63}\right)e(3742)e\left(\frac{37}{42}\right)e(29126)e\left(\frac{29}{126}\right)e(1742)e\left(\frac{17}{42}\right)e(23126)e\left(\frac{23}{126}\right)e(3263)e\left(\frac{32}{63}\right)e(4763)e\left(\frac{47}{63}\right)e(67)e\left(\frac{6}{7}\right)
sage: chi.jacobi_sum(n)
 
χ2793(1004,a)   \chi_{ 2793 }(1004,a) \; at   a=\;a = e.g. 2