Properties

Label 1340.339
Modulus $1340$
Conductor $1340$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1340, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,33,2]))
 
pari: [g,chi] = znchar(Mod(339,1340))
 

Basic properties

Modulus: \(1340\)
Conductor: \(1340\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 1340.bl

\(\chi_{1340}(19,\cdot)\) \(\chi_{1340}(39,\cdot)\) \(\chi_{1340}(199,\cdot)\) \(\chi_{1340}(339,\cdot)\) \(\chi_{1340}(419,\cdot)\) \(\chi_{1340}(479,\cdot)\) \(\chi_{1340}(559,\cdot)\) \(\chi_{1340}(619,\cdot)\) \(\chi_{1340}(639,\cdot)\) \(\chi_{1340}(659,\cdot)\) \(\chi_{1340}(719,\cdot)\) \(\chi_{1340}(839,\cdot)\) \(\chi_{1340}(859,\cdot)\) \(\chi_{1340}(959,\cdot)\) \(\chi_{1340}(1059,\cdot)\) \(\chi_{1340}(1119,\cdot)\) \(\chi_{1340}(1199,\cdot)\) \(\chi_{1340}(1239,\cdot)\) \(\chi_{1340}(1279,\cdot)\) \(\chi_{1340}(1299,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((671,537,1141)\) → \((-1,-1,e\left(\frac{1}{33}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1340 }(339,a) \;\) at \(\;a = \) e.g. 2