Properties

Label 3630.403
Modulus $3630$
Conductor $55$
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([0,15,14]))
 
pari: [g,chi] = znchar(Mod(403,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(55\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{55}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.v

\(\chi_{3630}(403,\cdot)\) \(\chi_{3630}(457,\cdot)\) \(\chi_{3630}(1183,\cdot)\) \(\chi_{3630}(1207,\cdot)\) \(\chi_{3630}(1927,\cdot)\) \(\chi_{3630}(1933,\cdot)\) \(\chi_{3630}(2653,\cdot)\) \(\chi_{3630}(3307,\cdot)\)

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

\((1211,727,3511)\) → \((1,-i,e\left(\frac{7}{10}\right))\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{55})^+\)