Properties

Label 1950.7
Modulus $1950$
Conductor $65$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,11]))
 
pari: [g,chi] = znchar(Mod(7,1950))
 

Basic properties

Modulus: \(1950\)
Conductor: \(65\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{65}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1950.bl

\(\chi_{1950}(7,\cdot)\) \(\chi_{1950}(643,\cdot)\) \(\chi_{1950}(1207,\cdot)\) \(\chi_{1950}(1393,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.3500313269603515625.1

Values on generators

\((1301,1327,301)\) → \((1,i,e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1950 }(7, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1950 }(7,a) \;\) at \(\;a = \) e.g. 2