Properties

Label 3380.323
Modulus $3380$
Conductor $3380$
Order $156$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,117,55]))
 
pari: [g,chi] = znchar(Mod(323,3380))
 

Basic properties

Modulus: \(3380\)
Conductor: \(3380\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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 3380.cz

\(\chi_{3380}(63,\cdot)\) \(\chi_{3380}(67,\cdot)\) \(\chi_{3380}(163,\cdot)\) \(\chi_{3380}(227,\cdot)\) \(\chi_{3380}(323,\cdot)\) \(\chi_{3380}(327,\cdot)\) \(\chi_{3380}(423,\cdot)\) \(\chi_{3380}(487,\cdot)\) \(\chi_{3380}(583,\cdot)\) \(\chi_{3380}(683,\cdot)\) \(\chi_{3380}(747,\cdot)\) \(\chi_{3380}(843,\cdot)\) \(\chi_{3380}(847,\cdot)\) \(\chi_{3380}(943,\cdot)\) \(\chi_{3380}(1007,\cdot)\) \(\chi_{3380}(1107,\cdot)\) \(\chi_{3380}(1203,\cdot)\) \(\chi_{3380}(1267,\cdot)\) \(\chi_{3380}(1363,\cdot)\) \(\chi_{3380}(1367,\cdot)\) \(\chi_{3380}(1463,\cdot)\) \(\chi_{3380}(1527,\cdot)\) \(\chi_{3380}(1623,\cdot)\) \(\chi_{3380}(1627,\cdot)\) \(\chi_{3380}(1723,\cdot)\) \(\chi_{3380}(1787,\cdot)\) \(\chi_{3380}(1883,\cdot)\) \(\chi_{3380}(1887,\cdot)\) \(\chi_{3380}(1983,\cdot)\) \(\chi_{3380}(2143,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1691,677,1861)\) → \((-1,-i,e\left(\frac{55}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3380 }(323, a) \) \(-1\)\(1\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{73}{78}\right)\)\(e\left(\frac{127}{156}\right)\)\(e\left(\frac{35}{156}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{21}{52}\right)\)\(e\left(\frac{47}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3380 }(323,a) \;\) at \(\;a = \) e.g. 2