Properties

Label 1425.29
Modulus $1425$
Conductor $1425$
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(1425, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,9,85]))
 
pari: [g,chi] = znchar(Mod(29,1425))
 

Basic properties

Modulus: \(1425\)
Conductor: \(1425\)
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 1425.cm

\(\chi_{1425}(14,\cdot)\) \(\chi_{1425}(29,\cdot)\) \(\chi_{1425}(59,\cdot)\) \(\chi_{1425}(89,\cdot)\) \(\chi_{1425}(269,\cdot)\) \(\chi_{1425}(314,\cdot)\) \(\chi_{1425}(344,\cdot)\) \(\chi_{1425}(509,\cdot)\) \(\chi_{1425}(554,\cdot)\) \(\chi_{1425}(584,\cdot)\) \(\chi_{1425}(629,\cdot)\) \(\chi_{1425}(659,\cdot)\) \(\chi_{1425}(794,\cdot)\) \(\chi_{1425}(839,\cdot)\) \(\chi_{1425}(869,\cdot)\) \(\chi_{1425}(884,\cdot)\) \(\chi_{1425}(914,\cdot)\) \(\chi_{1425}(944,\cdot)\) \(\chi_{1425}(1079,\cdot)\) \(\chi_{1425}(1154,\cdot)\) \(\chi_{1425}(1169,\cdot)\) \(\chi_{1425}(1229,\cdot)\) \(\chi_{1425}(1364,\cdot)\) \(\chi_{1425}(1409,\cdot)\)

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

\((476,1027,1351)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{17}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 1425 }(29, a) \) \(1\)\(1\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{44}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1425 }(29,a) \;\) at \(\;a = \) e.g. 2