Properties

Modulus 1157
Conductor 1157
Order 66
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 1157.bp

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1157)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,9]))
 
pari: [g,chi] = znchar(Mod(22,1157))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 1157
Conductor = 1157
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 66
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 1157.bp
Orbit index = 42

Galois orbit

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

\(\chi_{1157}(22,\cdot)\) \(\chi_{1157}(81,\cdot)\) \(\chi_{1157}(87,\cdot)\) \(\chi_{1157}(100,\cdot)\) \(\chi_{1157}(133,\cdot)\) \(\chi_{1157}(139,\cdot)\) \(\chi_{1157}(146,\cdot)\) \(\chi_{1157}(263,\cdot)\) \(\chi_{1157}(289,\cdot)\) \(\chi_{1157}(354,\cdot)\) \(\chi_{1157}(367,\cdot)\) \(\chi_{1157}(406,\cdot)\) \(\chi_{1157}(607,\cdot)\) \(\chi_{1157}(737,\cdot)\) \(\chi_{1157}(874,\cdot)\) \(\chi_{1157}(971,\cdot)\) \(\chi_{1157}(1004,\cdot)\) \(\chi_{1157}(1023,\cdot)\) \(\chi_{1157}(1036,\cdot)\) \(\chi_{1157}(1153,\cdot)\)

Values on generators

\((535,92)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{22}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{4}{33}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{33})\)