Properties

Label 1849.68
Modulus $1849$
Conductor $1849$
Order $903$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(1806))
 
M = H._module
 
chi = DirichletCharacter(H, M([134]))
 
pari: [g,chi] = znchar(Mod(68,1849))
 

Basic properties

Modulus: \(1849\)
Conductor: \(1849\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(903\)
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 1849.o

\(\chi_{1849}(9,\cdot)\) \(\chi_{1849}(10,\cdot)\) \(\chi_{1849}(13,\cdot)\) \(\chi_{1849}(14,\cdot)\) \(\chi_{1849}(15,\cdot)\) \(\chi_{1849}(17,\cdot)\) \(\chi_{1849}(23,\cdot)\) \(\chi_{1849}(24,\cdot)\) \(\chi_{1849}(25,\cdot)\) \(\chi_{1849}(31,\cdot)\) \(\chi_{1849}(38,\cdot)\) \(\chi_{1849}(40,\cdot)\) \(\chi_{1849}(52,\cdot)\) \(\chi_{1849}(53,\cdot)\) \(\chi_{1849}(56,\cdot)\) \(\chi_{1849}(57,\cdot)\) \(\chi_{1849}(58,\cdot)\) \(\chi_{1849}(60,\cdot)\) \(\chi_{1849}(66,\cdot)\) \(\chi_{1849}(67,\cdot)\) \(\chi_{1849}(68,\cdot)\) \(\chi_{1849}(74,\cdot)\) \(\chi_{1849}(81,\cdot)\) \(\chi_{1849}(83,\cdot)\) \(\chi_{1849}(95,\cdot)\) \(\chi_{1849}(96,\cdot)\) \(\chi_{1849}(99,\cdot)\) \(\chi_{1849}(100,\cdot)\) \(\chi_{1849}(101,\cdot)\) \(\chi_{1849}(103,\cdot)\) ...

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: $\Q(\zeta_{903})$
Fixed field: Number field defined by a degree 903 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{67}{903}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1849 }(68, a) \) \(1\)\(1\)\(e\left(\frac{57}{301}\right)\)\(e\left(\frac{67}{903}\right)\)\(e\left(\frac{114}{301}\right)\)\(e\left(\frac{289}{903}\right)\)\(e\left(\frac{34}{129}\right)\)\(e\left(\frac{23}{129}\right)\)\(e\left(\frac{171}{301}\right)\)\(e\left(\frac{134}{903}\right)\)\(e\left(\frac{460}{903}\right)\)\(e\left(\frac{54}{301}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1849 }(68,a) \;\) at \(\;a = \) e.g. 2