Properties

Label 5700.119
Modulus $5700$
Conductor $5700$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5700, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,45,81,80]))
 
pari: [g,chi] = znchar(Mod(119,5700))
 

Basic properties

Modulus: \(5700\)
Conductor: \(5700\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 5700.ev

\(\chi_{5700}(119,\cdot)\) \(\chi_{5700}(359,\cdot)\) \(\chi_{5700}(479,\cdot)\) \(\chi_{5700}(719,\cdot)\) \(\chi_{5700}(959,\cdot)\) \(\chi_{5700}(1259,\cdot)\) \(\chi_{5700}(1619,\cdot)\) \(\chi_{5700}(1859,\cdot)\) \(\chi_{5700}(2039,\cdot)\) \(\chi_{5700}(2639,\cdot)\) \(\chi_{5700}(2759,\cdot)\) \(\chi_{5700}(3179,\cdot)\) \(\chi_{5700}(3239,\cdot)\) \(\chi_{5700}(3539,\cdot)\) \(\chi_{5700}(3779,\cdot)\) \(\chi_{5700}(4139,\cdot)\) \(\chi_{5700}(4319,\cdot)\) \(\chi_{5700}(4379,\cdot)\) \(\chi_{5700}(4679,\cdot)\) \(\chi_{5700}(4919,\cdot)\) \(\chi_{5700}(5039,\cdot)\) \(\chi_{5700}(5279,\cdot)\) \(\chi_{5700}(5459,\cdot)\) \(\chi_{5700}(5519,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((2851,1901,3877,4201)\) → \((-1,-1,e\left(\frac{9}{10}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5700 }(119, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5700 }(119,a) \;\) at \(\;a = \) e.g. 2