Properties

Label 2700.79
Modulus $2700$
Conductor $2700$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,50,9]))
 
pari: [g,chi] = znchar(Mod(79,2700))
 

Basic properties

Modulus: \(2700\)
Conductor: \(2700\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2700.cj

\(\chi_{2700}(79,\cdot)\) \(\chi_{2700}(139,\cdot)\) \(\chi_{2700}(259,\cdot)\) \(\chi_{2700}(319,\cdot)\) \(\chi_{2700}(439,\cdot)\) \(\chi_{2700}(619,\cdot)\) \(\chi_{2700}(679,\cdot)\) \(\chi_{2700}(859,\cdot)\) \(\chi_{2700}(979,\cdot)\) \(\chi_{2700}(1039,\cdot)\) \(\chi_{2700}(1159,\cdot)\) \(\chi_{2700}(1219,\cdot)\) \(\chi_{2700}(1339,\cdot)\) \(\chi_{2700}(1519,\cdot)\) \(\chi_{2700}(1579,\cdot)\) \(\chi_{2700}(1759,\cdot)\) \(\chi_{2700}(1879,\cdot)\) \(\chi_{2700}(1939,\cdot)\) \(\chi_{2700}(2059,\cdot)\) \(\chi_{2700}(2119,\cdot)\) \(\chi_{2700}(2239,\cdot)\) \(\chi_{2700}(2419,\cdot)\) \(\chi_{2700}(2479,\cdot)\) \(\chi_{2700}(2659,\cdot)\)

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

\((1351,1001,2377)\) → \((-1,e\left(\frac{5}{9}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2700 }(79, a) \) \(-1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{38}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2700 }(79,a) \;\) at \(\;a = \) e.g. 2