Properties

Label 3744.37
Modulus $3744$
Conductor $416$
Order $24$
Real no
Primitive no
Minimal yes
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([0,3,0,14]))
 
pari: [g,chi] = znchar(Mod(37,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(416\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{416}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3744.io

\(\chi_{3744}(37,\cdot)\) \(\chi_{3744}(253,\cdot)\) \(\chi_{3744}(1333,\cdot)\) \(\chi_{3744}(1549,\cdot)\) \(\chi_{3744}(1909,\cdot)\) \(\chi_{3744}(2125,\cdot)\) \(\chi_{3744}(3205,\cdot)\) \(\chi_{3744}(3421,\cdot)\)

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

\((703,2341,2081,2017)\) → \((1,e\left(\frac{1}{8}\right),1,e\left(\frac{7}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{7}{12}\right)\)\(-i\)\(e\left(\frac{17}{24}\right)\)\(i\)\(e\left(\frac{1}{24}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{24})\)
Fixed field: 24.0.31808511574029960248322509834333516654369310400053248.2