Properties

Label 1950.113
Modulus $1950$
Conductor $975$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,57,40]))
 
pari: [g,chi] = znchar(Mod(113,1950))
 

Basic properties

Modulus: \(1950\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{975}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1950.cs

\(\chi_{1950}(113,\cdot)\) \(\chi_{1950}(263,\cdot)\) \(\chi_{1950}(347,\cdot)\) \(\chi_{1950}(497,\cdot)\) \(\chi_{1950}(503,\cdot)\) \(\chi_{1950}(653,\cdot)\) \(\chi_{1950}(737,\cdot)\) \(\chi_{1950}(887,\cdot)\) \(\chi_{1950}(1127,\cdot)\) \(\chi_{1950}(1277,\cdot)\) \(\chi_{1950}(1283,\cdot)\) \(\chi_{1950}(1433,\cdot)\) \(\chi_{1950}(1517,\cdot)\) \(\chi_{1950}(1667,\cdot)\) \(\chi_{1950}(1673,\cdot)\) \(\chi_{1950}(1823,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1301,1327,301)\) → \((-1,e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1950 }(113, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1950 }(113,a) \;\) at \(\;a = \) e.g. 2