Properties

Label 1323.109
Modulus $1323$
Conductor $49$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,40]))
 
pari: [g,chi] = znchar(Mod(109,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.bn

\(\chi_{1323}(109,\cdot)\) \(\chi_{1323}(163,\cdot)\) \(\chi_{1323}(298,\cdot)\) \(\chi_{1323}(352,\cdot)\) \(\chi_{1323}(487,\cdot)\) \(\chi_{1323}(541,\cdot)\) \(\chi_{1323}(676,\cdot)\) \(\chi_{1323}(730,\cdot)\) \(\chi_{1323}(865,\cdot)\) \(\chi_{1323}(919,\cdot)\) \(\chi_{1323}(1054,\cdot)\) \(\chi_{1323}(1297,\cdot)\)

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

Values on generators

\((785,1081)\) → \((1,e\left(\frac{20}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)