Properties

Label 1800.347
Modulus $1800$
Conductor $1800$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,30,50,51]))
 
pari: [g,chi] = znchar(Mod(347,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(1800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.dm

\(\chi_{1800}(83,\cdot)\) \(\chi_{1800}(203,\cdot)\) \(\chi_{1800}(227,\cdot)\) \(\chi_{1800}(347,\cdot)\) \(\chi_{1800}(563,\cdot)\) \(\chi_{1800}(587,\cdot)\) \(\chi_{1800}(803,\cdot)\) \(\chi_{1800}(923,\cdot)\) \(\chi_{1800}(947,\cdot)\) \(\chi_{1800}(1067,\cdot)\) \(\chi_{1800}(1163,\cdot)\) \(\chi_{1800}(1283,\cdot)\) \(\chi_{1800}(1427,\cdot)\) \(\chi_{1800}(1523,\cdot)\) \(\chi_{1800}(1667,\cdot)\) \(\chi_{1800}(1787,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1351,901,1001,577)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(347, a) \) \(-1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(347,a) \;\) at \(\;a = \) e.g. 2