Properties

Label 3630.323
Modulus $3630$
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(3630)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,15,4]))
 
pari: [g,chi] = znchar(Mod(323,3630))
 

Basic properties

Modulus: \(3630\)
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 3630.x

\(\chi_{3630}(323,\cdot)\) \(\chi_{3630}(977,\cdot)\) \(\chi_{3630}(1697,\cdot)\) \(\chi_{3630}(1703,\cdot)\) \(\chi_{3630}(2423,\cdot)\) \(\chi_{3630}(2447,\cdot)\) \(\chi_{3630}(3173,\cdot)\) \(\chi_{3630}(3227,\cdot)\)

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

\((1211,727,3511)\) → \((-1,-i,e\left(\frac{1}{5}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)
value at e.g. 2

Related number fields

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