Properties

Label 2700.301
Modulus $2700$
Conductor $27$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2,0]))
 
pari: [g,chi] = znchar(Mod(301,2700))
 

Basic properties

Modulus: \(2700\)
Conductor: \(27\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{27}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2700.v

\(\chi_{2700}(301,\cdot)\) \(\chi_{2700}(601,\cdot)\) \(\chi_{2700}(1201,\cdot)\) \(\chi_{2700}(1501,\cdot)\) \(\chi_{2700}(2101,\cdot)\) \(\chi_{2700}(2401,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{27})^+\)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2700 }(301, a) \) \(1\)\(1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2700 }(301,a) \;\) at \(\;a = \) e.g. 2