Properties

Label 1815.323
Modulus $1815$
Conductor $165$
Order $20$
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(1815, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,15,4]))
 
pari: [g,chi] = znchar(Mod(323,1815))
 

Basic properties

Modulus: \(1815\)
Conductor: \(165\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{165}(158,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1815.w

\(\chi_{1815}(323,\cdot)\) \(\chi_{1815}(608,\cdot)\) \(\chi_{1815}(632,\cdot)\) \(\chi_{1815}(977,\cdot)\) \(\chi_{1815}(1358,\cdot)\) \(\chi_{1815}(1412,\cdot)\) \(\chi_{1815}(1697,\cdot)\) \(\chi_{1815}(1703,\cdot)\)

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

\((1211,727,1696)\) → \((-1,-i,e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.20.82802905234194108120391845703125.1