Properties

Label 3744.505
Modulus $3744$
Conductor $208$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3744)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,0,7]))
 
pari: [g,chi] = znchar(Mod(505,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(208\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{208}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3744.er

\(\chi_{3744}(505,\cdot)\) \(\chi_{3744}(2953,\cdot)\) \(\chi_{3744}(3529,\cdot)\) \(\chi_{3744}(3673,\cdot)\)

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

\((703,2341,2081,2017)\) → \((1,i,1,e\left(\frac{7}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{12}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.15394540563150776827904.2