Properties

Label 3380.2607
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([78,39,95]))
 
pari: [g,chi] = znchar(Mod(2607,3380))
 

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.cs

\(\chi_{3380}(7,\cdot)\) \(\chi_{3380}(123,\cdot)\) \(\chi_{3380}(167,\cdot)\) \(\chi_{3380}(223,\cdot)\) \(\chi_{3380}(267,\cdot)\) \(\chi_{3380}(383,\cdot)\) \(\chi_{3380}(483,\cdot)\) \(\chi_{3380}(527,\cdot)\) \(\chi_{3380}(643,\cdot)\) \(\chi_{3380}(687,\cdot)\) \(\chi_{3380}(743,\cdot)\) \(\chi_{3380}(787,\cdot)\) \(\chi_{3380}(903,\cdot)\) \(\chi_{3380}(947,\cdot)\) \(\chi_{3380}(1003,\cdot)\) \(\chi_{3380}(1047,\cdot)\) \(\chi_{3380}(1163,\cdot)\) \(\chi_{3380}(1207,\cdot)\) \(\chi_{3380}(1307,\cdot)\) \(\chi_{3380}(1423,\cdot)\) \(\chi_{3380}(1467,\cdot)\) \(\chi_{3380}(1523,\cdot)\) \(\chi_{3380}(1567,\cdot)\) \(\chi_{3380}(1683,\cdot)\) \(\chi_{3380}(1727,\cdot)\) \(\chi_{3380}(1783,\cdot)\) \(\chi_{3380}(1827,\cdot)\) \(\chi_{3380}(1943,\cdot)\) \(\chi_{3380}(1987,\cdot)\) \(\chi_{3380}(2043,\cdot)\) ...

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

\((1691,677,1861)\) → \((-1,i,e\left(\frac{95}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3380 }(2607, a) \) \(-1\)\(1\)\(e\left(\frac{119}{156}\right)\)\(e\left(\frac{71}{78}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{35}{156}\right)\)\(e\left(\frac{25}{156}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{15}{52}\right)\)\(e\left(\frac{67}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3380 }(2607,a) \;\) at \(\;a = \) e.g. 2