Properties

Label 1323.37
Modulus $1323$
Conductor $441$
Order $21$
Real no
Primitive no
Minimal no
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([14,32]))
 
pari: [g,chi] = znchar(Mod(37,1323))
 

Basic properties

Modulus: \(1323\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(184,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1323.bk

\(\chi_{1323}(37,\cdot)\) \(\chi_{1323}(46,\cdot)\) \(\chi_{1323}(235,\cdot)\) \(\chi_{1323}(415,\cdot)\) \(\chi_{1323}(424,\cdot)\) \(\chi_{1323}(604,\cdot)\) \(\chi_{1323}(613,\cdot)\) \(\chi_{1323}(793,\cdot)\) \(\chi_{1323}(982,\cdot)\) \(\chi_{1323}(991,\cdot)\) \(\chi_{1323}(1171,\cdot)\) \(\chi_{1323}(1180,\cdot)\)

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

\((785,1081)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{16}{21}\right))\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 21.21.2972491714150324080426899160865869074720055489.1