Properties

Label 9036.2279
Modulus $9036$
Conductor $9036$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9036, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,5,12]))
 
pari: [g,chi] = znchar(Mod(2279,9036))
 

Basic properties

Modulus: \(9036\)
Conductor: \(9036\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 9036.be

\(\chi_{9036}(2279,\cdot)\) \(\chi_{9036}(4235,\cdot)\) \(\chi_{9036}(4631,\cdot)\) \(\chi_{9036}(4667,\cdot)\) \(\chi_{9036}(7247,\cdot)\) \(\chi_{9036}(7643,\cdot)\) \(\chi_{9036}(7679,\cdot)\) \(\chi_{9036}(8303,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((4519,2009,1261)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 9036 }(2279, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9036 }(2279,a) \;\) at \(\;a = \) e.g. 2