Properties

Label 1950.83
Modulus $1950$
Conductor $975$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,3,15]))
 
pari: [g,chi] = znchar(Mod(83,1950))
 

Basic properties

Modulus: \(1950\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{975}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1950.ch

\(\chi_{1950}(47,\cdot)\) \(\chi_{1950}(83,\cdot)\) \(\chi_{1950}(437,\cdot)\) \(\chi_{1950}(473,\cdot)\) \(\chi_{1950}(827,\cdot)\) \(\chi_{1950}(863,\cdot)\) \(\chi_{1950}(1217,\cdot)\) \(\chi_{1950}(1253,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.0.8796562314350871340409503318369388580322265625.2

Values on generators

\((1301,1327,301)\) → \((-1,e\left(\frac{3}{20}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1950 }(83, a) \) \(-1\)\(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1950 }(83,a) \;\) at \(\;a = \) e.g. 2