Properties

Label 3800.1299
Modulus $3800$
Conductor $760$
Order $6$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3800, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,3,3,2]))
 
pari: [g,chi] = znchar(Mod(1299,3800))
 

Basic properties

Modulus: \(3800\)
Conductor: \(760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{760}(539,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3800.bn

\(\chi_{3800}(1299,\cdot)\) \(\chi_{3800}(2899,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.8340544000.3

Values on generators

\((951,1901,1977,401)\) → \((-1,-1,-1,e\left(\frac{1}{3}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3800 }(1299, a) \) \(-1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3800 }(1299,a) \;\) at \(\;a = \) e.g. 2